TPTP Problem File: ITP222^4.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : ITP222^4 : TPTP v8.2.0. Released v8.0.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer problem VEBT_Definitions 00680_031305
% 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 : 0063_VEBT_Definitions_00680_031305 [Des22]
% Status : Theorem
% Rating : 0.67 v8.2.0, 0.33 v8.1.0
% Syntax : Number of formulae : 9526 (3041 unt; 543 typ; 0 def)
% Number of atoms : 27921 (9127 equ; 5 cnn)
% Maximal formula atoms : 71 ( 3 avg)
% Number of connectives : 171446 (1965 ~; 327 |;2309 &;154093 @)
% ( 0 <=>;12752 =>; 0 <=; 0 <~>)
% Maximal formula depth : 40 ( 7 avg)
% Number of types : 12 ( 11 usr)
% Number of type conns : 4271 (4271 >; 0 *; 0 +; 0 <<)
% Number of symbols : 536 ( 532 usr; 11 con; 0-8 aty)
% Number of variables : 30246 (3300 ^;25665 !; 795 ?;30246 :)
% ( 486 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TH1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% from the van Emde Boas Trees session in the Archive of Formal
% proofs -
% www.isa-afp.org/browser_info/current/AFP/Van_Emde_Boas_Trees
% 2022-02-17 17:57:02.960
%------------------------------------------------------------------------------
% Could-be-implicit typings (16)
thf(ty_t_VEBT__Definitions_OVEBT,type,
vEBT_VEBT: $tType ).
thf(ty_t_Code__Numeral_Ointeger,type,
code_integer: $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_Extended__Nat_Oenat,type,
extended_enat: $tType ).
thf(ty_t_Complex_Ocomplex,type,
complex: $tType ).
thf(ty_t_Option_Ooption,type,
option: $tType > $tType ).
thf(ty_t_Filter_Ofilter,type,
filter: $tType > $tType ).
thf(ty_t_String_Ochar,type,
char: $tType ).
thf(ty_t_Real_Oreal,type,
real: $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 ).
% Explicit typings (527)
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_Rings_Odvd,type,
dvd:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oone,type,
one:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oidom,type,
idom:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oring,type,
ring:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oplus,type,
plus:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ozero,type,
zero:
!>[A: $tType] : $o ).
thf(sy_cl_Num_Onumeral,type,
numeral:
!>[A: $tType] : $o ).
thf(sy_cl_Power_Opower,type,
power:
!>[A: $tType] : $o ).
thf(sy_cl_Fields_Ofield,type,
field:
!>[A: $tType] : $o ).
thf(sy_cl_GCD_Oring__gcd,type,
ring_gcd:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oabs__if,type,
abs_if:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oring__1,type,
ring_1:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ouminus,type,
uminus:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oord,type,
ord:
!>[A: $tType] : $o ).
thf(sy_cl_Fields_Oinverse,type,
inverse:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemiring,type,
semiring:
!>[A: $tType] : $o ).
thf(sy_cl_Nat_Oring__char__0,type,
ring_char_0:
!>[A: $tType] : $o ).
thf(sy_cl_Num_Oneg__numeral,type,
neg_numeral:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oorder,type,
order:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Ocomm__ring,type,
comm_ring:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Omult__zero,type,
mult_zero:
!>[A: $tType] : $o ).
thf(sy_cl_GCD_Osemiring__gcd,type,
semiring_gcd:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ogroup__add,type,
group_add:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Olattice,type,
lattice:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Ono__bot,type,
no_bot:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Ono__top,type,
no_top:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemiring__0,type,
semiring_0:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemiring__1,type,
semiring_1:
!>[A: $tType] : $o ).
thf(sy_cl_Finite__Set_Ofinite,type,
finite_finite:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Omonoid__add,type,
monoid_add:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Ocomm__ring__1,type,
comm_ring_1:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oidom__divide,type,
idom_divide:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oidom__modulo,type,
idom_modulo:
!>[A: $tType] : $o ).
thf(sy_cl_Transcendental_Oln,type,
ln:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Omonoid__mult,type,
monoid_mult:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Olinorder,type,
linorder:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Opreorder,type,
preorder:
!>[A: $tType] : $o ).
thf(sy_cl_Parity_Oring__parity,type,
ring_parity:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oidom__abs__sgn,type,
idom_abs_sgn:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oordered__ring,type,
ordered_ring:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Ozero__neq__one,type,
zero_neq_one:
!>[A: $tType] : $o ).
thf(sy_cl_Fields_Ofield__char__0,type,
field_char_0:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oab__group__add,type,
ab_group_add:
!>[A: $tType] : $o ).
thf(sy_cl_Nat_Osemiring__char__0,type,
semiring_char_0:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oorder__bot,type,
order_bot:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oorder__top,type,
order_top:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Owellorder,type,
wellorder:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Ocomm__semiring,type,
comm_semiring:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Ozero__less__one,type,
zero_less_one:
!>[A: $tType] : $o ).
thf(sy_cl_Fields_Odivision__ring,type,
division_ring:
!>[A: $tType] : $o ).
thf(sy_cl_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_Complete__Lattices_OInf,type,
complete_Inf:
!>[A: $tType] : $o ).
thf(sy_cl_Complete__Lattices_OSup,type,
complete_Sup:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Osemigroup__mult,type,
semigroup_mult:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Odense__order,type,
dense_order:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Ocomm__semiring__0,type,
comm_semiring_0:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Ocomm__semiring__1,type,
comm_semiring_1:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__idom,type,
linordered_idom:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__ring,type,
linordered_ring:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemiring__modulo,type,
semiring_modulo:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocomm__monoid__add,type,
comm_monoid_add:
!>[A: $tType] : $o ).
thf(sy_cl_Parity_Osemiring__parity,type,
semiring_parity:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oordered__ring__abs,type,
ordered_ring_abs:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oordered__semiring,type,
ordered_semiring:
!>[A: $tType] : $o ).
thf(sy_cl_Fields_Olinordered__field,type,
linordered_field:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oab__semigroup__add,type,
ab_semigroup_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocomm__monoid__diff,type,
comm_monoid_diff:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocomm__monoid__mult,type,
comm_monoid_mult:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oalgebraic__semidom,type,
algebraic_semidom:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemiring__1__cancel,type,
semiring_1_cancel:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oab__semigroup__mult,type,
ab_semigroup_mult:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_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_Real__Vector__Spaces_Obanach,type,
real_Vector_banach:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__semiring,type,
linordered_semiring:
!>[A: $tType] : $o ).
thf(sy_cl_Enum_Ofinite__distrib__lattice,type,
finite8700451911770168679attice:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocancel__semigroup__add,type,
cancel_semigroup_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__ab__group__add,type,
ordered_ab_group_add:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__semiring__1,type,
linord6961819062388156250ring_1:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oordered__comm__semiring,type,
ordere2520102378445227354miring:
!>[A: $tType] : $o ).
thf(sy_cl_Topological__Spaces_Ot1__space,type,
topological_t1_space:
!>[A: $tType] : $o ).
thf(sy_cl_Topological__Spaces_Ot2__space,type,
topological_t2_space:
!>[A: $tType] : $o ).
thf(sy_cl_Bit__Operations_Osemiring__bits,type,
bit_semiring_bits:
!>[A: $tType] : $o ).
thf(sy_cl_Limits_Otopological__group__add,type,
topolo1633459387980952147up_add:
!>[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_Limits_Otopological__monoid__add,type,
topolo6943815403480290642id_add:
!>[A: $tType] : $o ).
thf(sy_cl_Real__Vector__Spaces_Oreal__field,type,
real_V7773925162809079976_field:
!>[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_Limits_Otopological__monoid__mult,type,
topolo1898628316856586783d_mult:
!>[A: $tType] : $o ).
thf(sy_cl_Real__Vector__Spaces_Oreal__vector,type,
real_V4867850818363320053vector:
!>[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_Limits_Otopological__ab__group__add,type,
topolo1287966508704411220up_add:
!>[A: $tType] : $o ).
thf(sy_cl_Real__Vector__Spaces_Ometric__space,type,
real_V7819770556892013058_space:
!>[A: $tType] : $o ).
thf(sy_cl_Real__Vector__Spaces_Oreal__algebra,type,
real_V6157519004096292374lgebra:
!>[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_Topological__Spaces_Operfect__space,type,
topolo8386298272705272623_space:
!>[A: $tType] : $o ).
thf(sy_cl_Topological__Spaces_Ouniform__space,type,
topolo7287701948861334536_space:
!>[A: $tType] : $o ).
thf(sy_cl_Euclidean__Division_Oeuclidean__ring,type,
euclid5891614535332579305n_ring:
!>[A: $tType] : $o ).
thf(sy_cl_Limits_Otopological__semigroup__mult,type,
topolo4211221413907600880p_mult:
!>[A: $tType] : $o ).
thf(sy_cl_Real__Vector__Spaces_Ocomplete__space,type,
real_V8037385150606011577_space:
!>[A: $tType] : $o ).
thf(sy_cl_Real__Vector__Spaces_Oreal__algebra__1,type,
real_V2191834092415804123ebra_1:
!>[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_Topological__Spaces_Oorder__topology,type,
topolo2564578578187576103pology:
!>[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_Limits_Otopological__comm__monoid__add,type,
topolo5987344860129210374id_add:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Ounbounded__dense__linorder,type,
unboun7993243217541854897norder:
!>[A: $tType] : $o ).
thf(sy_cl_Real__Vector__Spaces_Ouniformity__dist,type,
real_V768167426530841204y_dist:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__semiring__1__strict,type,
linord715952674999750819strict:
!>[A: $tType] : $o ).
thf(sy_cl_Topological__Spaces_Oopen__uniformity,type,
topolo569519726778239578ormity:
!>[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_Limits_Otopological__comm__monoid__mult,type,
topolo4987421752381908075d_mult:
!>[A: $tType] : $o ).
thf(sy_cl_Real__Vector__Spaces_Oreal__div__algebra,type,
real_V5047593784448816457lgebra:
!>[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_Real__Vector__Spaces_Oreal__normed__field,type,
real_V3459762299906320749_field:
!>[A: $tType] : $o ).
thf(sy_cl_Topological__Spaces_Odiscrete__topology,type,
topolo8865339358273720382pology:
!>[A: $tType] : $o ).
thf(sy_cl_Topological__Spaces_Olinorder__topology,type,
topolo1944317154257567458pology:
!>[A: $tType] : $o ).
thf(sy_cl_Topological__Spaces_Otopological__space,type,
topolo4958980785337419405_space:
!>[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_Real__Vector__Spaces_Oreal__normed__vector,type,
real_V822414075346904944vector:
!>[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_Real__Vector__Spaces_Oordered__real__vector,type,
real_V5355595471888546746vector:
!>[A: $tType] : $o ).
thf(sy_cl_Real__Vector__Spaces_Oreal__normed__algebra,type,
real_V4412858255891104859lgebra:
!>[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_Real__Vector__Spaces_Oreal__normed__algebra__1,type,
real_V2822296259951069270ebra_1:
!>[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_Real__Vector__Spaces_Oreal__normed__div__algebra,type,
real_V8999393235501362500lgebra:
!>[A: $tType] : $o ).
thf(sy_cl_Topological__Spaces_Ofirst__countable__topology,type,
topolo3112930676232923870pology:
!>[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_Topological__Spaces_Olinear__continuum__topology,type,
topolo8458572112393995274pology:
!>[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_Countable__Complete__Lattices_Ocountable__complete__lattice,type,
counta3822494911875563373attice:
!>[A: $tType] : $o ).
thf(sy_cl_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct,type,
semiri1453513574482234551roduct:
!>[A: $tType] : $o ).
thf(sy_cl_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations,type,
bit_un5681908812861735899ations:
!>[A: $tType] : $o ).
thf(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__lattice,type,
condit1219197933456340205attice:
!>[A: $tType] : $o ).
thf(sy_cl_Conditionally__Complete__Lattices_Oconditionally__complete__linorder,type,
condit6923001295902523014norder:
!>[A: $tType] : $o ).
thf(sy_c_Archimedean__Field_Oceiling,type,
archimedean_ceiling:
!>[A: $tType] : ( A > int ) ).
thf(sy_c_Archimedean__Field_Ofloor__ceiling__class_Ofloor,type,
archim6421214686448440834_floor:
!>[A: $tType] : ( A > int ) ).
thf(sy_c_Archimedean__Field_Ofrac,type,
archimedean_frac:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Archimedean__Field_Oround,type,
archimedean_round:
!>[A: $tType] : ( A > int ) ).
thf(sy_c_BNF__Greatest__Fixpoint_OSucc,type,
bNF_Greatest_Succ:
!>[A: $tType] : ( ( set @ ( list @ A ) ) > ( list @ A ) > ( set @ A ) ) ).
thf(sy_c_BNF__Greatest__Fixpoint_Oimage2,type,
bNF_Greatest_image2:
!>[C: $tType,A: $tType,B: $tType] : ( ( set @ C ) > ( C > A ) > ( C > B ) > ( set @ ( product_prod @ A @ B ) ) ) ).
thf(sy_c_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_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_Oor__not__num__neg__rel,type,
bit_or3848514188828904588eg_rel: ( product_prod @ num @ num ) > ( product_prod @ num @ num ) > $o ).
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_Otake__bit__num,type,
bit_take_bit_num: nat > num > ( option @ num ) ).
thf(sy_c_Boolean__Algebras_Oabstract__boolean__algebra,type,
boolea2506097494486148201lgebra:
!>[A: $tType] : ( ( A > A > A ) > ( A > A > A ) > ( A > A ) > A > A > $o ) ).
thf(sy_c_Boolean__Algebras_Oabstract__boolean__algebra__sym__diff,type,
boolea3799213064322606851m_diff:
!>[A: $tType] : ( ( A > A > A ) > ( A > A > A ) > ( A > A ) > A > A > ( A > A > A ) > $o ) ).
thf(sy_c_Code__Numeral_Obit__cut__integer,type,
code_bit_cut_integer: code_integer > ( product_prod @ code_integer @ $o ) ).
thf(sy_c_Code__Numeral_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_Ointeger_Oint__of__integer,type,
code_int_of_integer: code_integer > int ).
thf(sy_c_Code__Numeral_Ointeger_Ointeger__of__int,type,
code_integer_of_int: int > code_integer ).
thf(sy_c_Code__Numeral_Ointeger__of__num,type,
code_integer_of_num: num > code_integer ).
thf(sy_c_Code__Numeral_Onat__of__integer,type,
code_nat_of_integer: code_integer > nat ).
thf(sy_c_Code__Numeral_Onum__of__integer,type,
code_num_of_integer: code_integer > num ).
thf(sy_c_Complete__Lattices_OInf__class_OInf,type,
complete_Inf_Inf:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_Complete__Lattices_OSup__class_OSup,type,
complete_Sup_Sup:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_Complex_OArg,type,
arg: complex > real ).
thf(sy_c_Complex_Ocis,type,
cis: real > complex ).
thf(sy_c_Complex_Ocnj,type,
cnj: complex > complex ).
thf(sy_c_Complex_Ocomplex_OComplex,type,
complex2: real > real > complex ).
thf(sy_c_Complex_Ocomplex_OIm,type,
im: complex > real ).
thf(sy_c_Complex_Ocomplex_ORe,type,
re: complex > real ).
thf(sy_c_Complex_Ocsqrt,type,
csqrt: complex > complex ).
thf(sy_c_Complex_Oimaginary__unit,type,
imaginary_unit: complex ).
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_Deriv_Odifferentiable,type,
differentiable:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( filter @ A ) > $o ) ).
thf(sy_c_Deriv_Ohas__derivative,type,
has_derivative:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( A > B ) > ( filter @ A ) > $o ) ).
thf(sy_c_Deriv_Ohas__field__derivative,type,
has_field_derivative:
!>[A: $tType] : ( ( A > A ) > A > ( filter @ A ) > $o ) ).
thf(sy_c_Divides_Oadjust__div,type,
adjust_div: ( product_prod @ int @ int ) > int ).
thf(sy_c_Divides_Oadjust__mod,type,
adjust_mod: int > int > int ).
thf(sy_c_Divides_Odivmod__nat,type,
divmod_nat: nat > nat > ( product_prod @ nat @ nat ) ).
thf(sy_c_Divides_Oeucl__rel__int,type,
eucl_rel_int: int > int > ( product_prod @ int @ int ) > $o ).
thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivides__aux,type,
unique5940410009612947441es_aux:
!>[A: $tType] : ( ( product_prod @ A @ A ) > $o ) ).
thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod,type,
unique8689654367752047608divmod:
!>[A: $tType] : ( num > num > ( product_prod @ A @ A ) ) ).
thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod__step,type,
unique1321980374590559556d_step:
!>[A: $tType] : ( num > ( product_prod @ A @ A ) > ( product_prod @ A @ A ) ) ).
thf(sy_c_Extended__Nat_Oenat,type,
extended_enat2: nat > extended_enat ).
thf(sy_c_Extended__Nat_Oenat_Ocase__enat,type,
extended_case_enat:
!>[T: $tType] : ( ( nat > T ) > T > extended_enat > T ) ).
thf(sy_c_Extended__Nat_Oenat_Orec__enat,type,
extended_rec_enat:
!>[T: $tType] : ( ( nat > T ) > T > extended_enat > T ) ).
thf(sy_c_Extended__Nat_Oenat_Orec__set__enat,type,
extend4933016492236175606t_enat:
!>[T: $tType] : ( ( nat > T ) > T > extended_enat > T > $o ) ).
thf(sy_c_Extended__Nat_Oinfinity__class_Oinfinity,type,
extend4730790105801354508finity:
!>[A: $tType] : A ).
thf(sy_c_Extended__Nat_Othe__enat,type,
extended_the_enat: extended_enat > nat ).
thf(sy_c_Factorial_Ocomm__semiring__1__class_Opochhammer,type,
comm_s3205402744901411588hammer:
!>[A: $tType] : ( A > nat > A ) ).
thf(sy_c_Factorial_Osemiring__char__0__class_Ofact,type,
semiring_char_0_fact:
!>[A: $tType] : ( nat > A ) ).
thf(sy_c_Fields_Oinverse__class_Oinverse,type,
inverse_inverse:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Filter_Oat__bot,type,
at_bot:
!>[A: $tType] : ( filter @ A ) ).
thf(sy_c_Filter_Oat__top,type,
at_top:
!>[A: $tType] : ( filter @ A ) ).
thf(sy_c_Filter_Oeventually,type,
eventually:
!>[A: $tType] : ( ( A > $o ) > ( filter @ A ) > $o ) ).
thf(sy_c_Filter_Ofilterlim,type,
filterlim:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( filter @ B ) > ( filter @ A ) > $o ) ).
thf(sy_c_Filter_Omap__filter__on,type,
map_filter_on:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B ) > ( filter @ A ) > ( filter @ B ) ) ).
thf(sy_c_Filter_Oprincipal,type,
principal:
!>[A: $tType] : ( ( set @ A ) > ( filter @ A ) ) ).
thf(sy_c_Filter_Oprod__filter,type,
prod_filter:
!>[A: $tType,B: $tType] : ( ( filter @ A ) > ( filter @ B ) > ( filter @ ( product_prod @ A @ B ) ) ) ).
thf(sy_c_Finite__Set_OFpow,type,
finite_Fpow:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( set @ A ) ) ) ).
thf(sy_c_Finite__Set_Ocard,type,
finite_card:
!>[B: $tType] : ( ( set @ B ) > nat ) ).
thf(sy_c_Finite__Set_Ocomp__fun__commute,type,
finite6289374366891150609ommute:
!>[A: $tType,B: $tType] : ( ( A > B > B ) > $o ) ).
thf(sy_c_Finite__Set_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_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_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_Ostrict__mono__on,type,
strict_mono_on:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > $o ) ).
thf(sy_c_Fun_Othe__inv__into,type,
the_inv_into:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B ) > B > A ) ).
thf(sy_c_GCD_Obezw,type,
bezw: nat > nat > ( product_prod @ int @ int ) ).
thf(sy_c_GCD_Obezw__rel,type,
bezw_rel: ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) > $o ).
thf(sy_c_GCD_Ogcd__class_Ogcd,type,
gcd_gcd:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_GCD_Ogcd__nat__rel,type,
gcd_nat_rel: ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) > $o ).
thf(sy_c_Groups_Oabs__class_Oabs,type,
abs_abs:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Groups_Ominus__class_Ominus,type,
minus_minus:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Groups_Oone__class_Oone,type,
one_one:
!>[A: $tType] : A ).
thf(sy_c_Groups_Oplus__class_Oplus,type,
plus_plus:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Groups_Osgn__class_Osgn,type,
sgn_sgn:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Groups_Otimes__class_Otimes,type,
times_times:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Groups_Ouminus__class_Ouminus,type,
uminus_uminus:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Groups_Ozero__class_Ozero,type,
zero_zero:
!>[A: $tType] : A ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__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__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_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_If,type,
if:
!>[A: $tType] : ( $o > A > A > A ) ).
thf(sy_c_Int_OAbs__Integ,type,
abs_Integ: ( product_prod @ nat @ nat ) > int ).
thf(sy_c_Int_ORep__Integ,type,
rep_Integ: int > ( product_prod @ nat @ nat ) ).
thf(sy_c_Int_Oint__ge__less__than,type,
int_ge_less_than: int > ( set @ ( product_prod @ int @ int ) ) ).
thf(sy_c_Int_Oint__ge__less__than2,type,
int_ge_less_than2: int > ( set @ ( product_prod @ int @ int ) ) ).
thf(sy_c_Int_Onat,type,
nat2: int > nat ).
thf(sy_c_Int_Oring__1__class_OInts,type,
ring_1_Ints:
!>[A: $tType] : ( set @ A ) ).
thf(sy_c_Int_Oring__1__class_Oof__int,type,
ring_1_of_int:
!>[A: $tType] : ( int > A ) ).
thf(sy_c_Lattices_Oinf__class_Oinf,type,
inf_inf:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Lattices_Osemilattice__neutr__order,type,
semila1105856199041335345_order:
!>[A: $tType] : ( ( A > A > A ) > A > ( A > A > $o ) > ( A > A > $o ) > $o ) ).
thf(sy_c_Lattices_Osup__class_Osup,type,
sup_sup:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on,type,
lattic7623131987881927897min_on:
!>[B: $tType,A: $tType] : ( ( B > A ) > ( set @ B ) > B ) ).
thf(sy_c_Limits_OBfun,type,
bfun:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( filter @ A ) > $o ) ).
thf(sy_c_Limits_Oat__infinity,type,
at_infinity:
!>[A: $tType] : ( filter @ A ) ).
thf(sy_c_List_Oappend,type,
append:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Oconcat,type,
concat:
!>[A: $tType] : ( ( list @ ( list @ A ) ) > ( list @ A ) ) ).
thf(sy_c_List_Ocount__list,type,
count_list:
!>[A: $tType] : ( ( list @ A ) > A > nat ) ).
thf(sy_c_List_Odistinct,type,
distinct:
!>[A: $tType] : ( ( list @ A ) > $o ) ).
thf(sy_c_List_Odrop,type,
drop:
!>[A: $tType] : ( nat > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_OdropWhile,type,
dropWhile:
!>[A: $tType] : ( ( A > $o ) > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Oenumerate,type,
enumerate:
!>[A: $tType] : ( nat > ( list @ A ) > ( list @ ( product_prod @ nat @ A ) ) ) ).
thf(sy_c_List_Oextract,type,
extract:
!>[A: $tType] : ( ( A > $o ) > ( list @ A ) > ( option @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) ) ) ).
thf(sy_c_List_Ofilter,type,
filter2:
!>[A: $tType] : ( ( A > $o ) > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Ofind,type,
find:
!>[A: $tType] : ( ( A > $o ) > ( list @ A ) > ( option @ A ) ) ).
thf(sy_c_List_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_Olenlex,type,
lenlex:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ) ).
thf(sy_c_List_Olex,type,
lex:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ) ).
thf(sy_c_List_Olexn,type,
lexn:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > nat > ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ) ).
thf(sy_c_List_Olexord,type,
lexord:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ) ).
thf(sy_c_List_Olexordp,type,
lexordp:
!>[A: $tType] : ( ( A > A > $o ) > ( list @ A ) > ( list @ A ) > $o ) ).
thf(sy_c_List_Olinorder_Oinsort__key,type,
insort_key:
!>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( B > A ) > B > ( list @ B ) > ( list @ B ) ) ).
thf(sy_c_List_Olinorder_Osorted__key__list__of__set,type,
sorted8670434370408473282of_set:
!>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( B > A ) > ( set @ B ) > ( list @ B ) ) ).
thf(sy_c_List_Olinorder__class_Oinsort__key,type,
linorder_insort_key:
!>[B: $tType,A: $tType] : ( ( B > A ) > B > ( 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_Ocase__list,type,
case_list:
!>[B: $tType,A: $tType] : ( B > ( A > ( list @ A ) > B ) > ( list @ A ) > B ) ).
thf(sy_c_List_Olist_Ohd,type,
hd:
!>[A: $tType] : ( ( list @ A ) > A ) ).
thf(sy_c_List_Olist_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__update,type,
list_update:
!>[A: $tType] : ( ( list @ A ) > nat > A > ( list @ A ) ) ).
thf(sy_c_List_Olistrel1,type,
listrel1:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ) ).
thf(sy_c_List_Olistrel1p,type,
listrel1p:
!>[A: $tType] : ( ( A > A > $o ) > ( list @ A ) > ( list @ A ) > $o ) ).
thf(sy_c_List_Olistset,type,
listset:
!>[A: $tType] : ( ( list @ ( set @ A ) ) > ( set @ ( list @ A ) ) ) ).
thf(sy_c_List_Omap__filter,type,
map_filter:
!>[A: $tType,B: $tType] : ( ( A > ( option @ B ) ) > ( list @ A ) > ( list @ B ) ) ).
thf(sy_c_List_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_Opartition,type,
partition:
!>[A: $tType] : ( ( A > $o ) > ( list @ A ) > ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ).
thf(sy_c_List_Oproduct,type,
product:
!>[A: $tType,B: $tType] : ( ( list @ A ) > ( list @ B ) > ( list @ ( product_prod @ A @ B ) ) ) ).
thf(sy_c_List_Oproduct__lists,type,
product_lists:
!>[A: $tType] : ( ( list @ ( list @ A ) ) > ( list @ ( list @ A ) ) ) ).
thf(sy_c_List_Oremdups,type,
remdups:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Oremdups__adj,type,
remdups_adj:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Oremdups__adj__rel,type,
remdups_adj_rel:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) > $o ) ).
thf(sy_c_List_Oremove1,type,
remove1:
!>[A: $tType] : ( A > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_OremoveAll,type,
removeAll:
!>[A: $tType] : ( A > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Oreplicate,type,
replicate:
!>[A: $tType] : ( nat > A > ( list @ A ) ) ).
thf(sy_c_List_Orev,type,
rev:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Orotate1,type,
rotate1:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Oset__Cons,type,
set_Cons:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( list @ A ) ) > ( set @ ( list @ A ) ) ) ).
thf(sy_c_List_Oshuffles,type,
shuffles:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) > ( set @ ( list @ A ) ) ) ).
thf(sy_c_List_Oshuffles__rel,type,
shuffles_rel:
!>[A: $tType] : ( ( product_prod @ ( list @ A ) @ ( list @ A ) ) > ( product_prod @ ( list @ A ) @ ( list @ A ) ) > $o ) ).
thf(sy_c_List_Osorted__wrt,type,
sorted_wrt:
!>[A: $tType] : ( ( A > A > $o ) > ( list @ A ) > $o ) ).
thf(sy_c_List_Osorted__wrt__rel,type,
sorted_wrt_rel:
!>[A: $tType] : ( ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) > ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) > $o ) ).
thf(sy_c_List_Osplice,type,
splice:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Osplice__rel,type,
splice_rel:
!>[A: $tType] : ( ( product_prod @ ( list @ A ) @ ( list @ A ) ) > ( product_prod @ ( list @ A ) @ ( list @ A ) ) > $o ) ).
thf(sy_c_List_Osubseqs,type,
subseqs:
!>[A: $tType] : ( ( list @ A ) > ( list @ ( list @ A ) ) ) ).
thf(sy_c_List_Otake,type,
take:
!>[A: $tType] : ( nat > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_OtakeWhile,type,
takeWhile:
!>[A: $tType] : ( ( A > $o ) > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Otranspose,type,
transpose:
!>[A: $tType] : ( ( list @ ( list @ A ) ) > ( list @ ( list @ A ) ) ) ).
thf(sy_c_List_Otranspose__rel,type,
transpose_rel:
!>[A: $tType] : ( ( list @ ( list @ A ) ) > ( list @ ( list @ A ) ) > $o ) ).
thf(sy_c_List_Ounion,type,
union:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Oupt,type,
upt: nat > nat > ( list @ nat ) ).
thf(sy_c_List_Oupto,type,
upto: int > int > ( list @ int ) ).
thf(sy_c_List_Oupto__aux,type,
upto_aux: int > int > ( list @ int ) > ( list @ int ) ).
thf(sy_c_List_Oupto__rel,type,
upto_rel: ( product_prod @ int @ int ) > ( product_prod @ int @ int ) > $o ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Ocompow,type,
compow:
!>[A: $tType] : ( nat > A > A ) ).
thf(sy_c_Nat_Ofunpow,type,
funpow:
!>[A: $tType] : ( nat > ( A > A ) > A > A ) ).
thf(sy_c_Nat_Onat_Ocase__nat,type,
case_nat:
!>[A: $tType] : ( A > ( nat > A ) > nat > A ) ).
thf(sy_c_Nat_Onat_Opred,type,
pred: nat > nat ).
thf(sy_c_Nat_Oold_Onat_Orec__nat,type,
rec_nat:
!>[T: $tType] : ( T > ( nat > T > T ) > nat > T ) ).
thf(sy_c_Nat_Oold_Onat_Orec__set__nat,type,
rec_set_nat:
!>[T: $tType] : ( T > ( nat > T > T ) > nat > T > $o ) ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat,type,
semiring_1_of_nat:
!>[A: $tType] : ( nat > A ) ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat__aux,type,
semiri8178284476397505188at_aux:
!>[A: $tType] : ( ( A > A ) > nat > A > A ) ).
thf(sy_c_Nat_Osize__class_Osize,type,
size_size:
!>[A: $tType] : ( A > nat ) ).
thf(sy_c_Nat__Bijection_Oprod__decode__aux,type,
nat_prod_decode_aux: nat > nat > ( product_prod @ nat @ nat ) ).
thf(sy_c_Nat__Bijection_Oprod__decode__aux__rel,type,
nat_pr5047031295181774490ux_rel: ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) > $o ).
thf(sy_c_Nat__Bijection_Oset__decode,type,
nat_set_decode: nat > ( set @ nat ) ).
thf(sy_c_Nat__Bijection_Oset__encode,type,
nat_set_encode: ( set @ nat ) > nat ).
thf(sy_c_Nat__Bijection_Otriangle,type,
nat_triangle: nat > nat ).
thf(sy_c_NthRoot_Oroot,type,
root: nat > real > real ).
thf(sy_c_NthRoot_Osqrt,type,
sqrt: real > real ).
thf(sy_c_Num_OBitM,type,
bitM: num > num ).
thf(sy_c_Num_Oinc,type,
inc: num > num ).
thf(sy_c_Num_Oneg__numeral__class_Odbl,type,
neg_numeral_dbl:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__dec,type,
neg_numeral_dbl_dec:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__inc,type,
neg_numeral_dbl_inc:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Num_Oneg__numeral__class_Osub,type,
neg_numeral_sub:
!>[A: $tType] : ( num > num > A ) ).
thf(sy_c_Num_Onum_OBit0,type,
bit0: num > num ).
thf(sy_c_Num_Onum_OBit1,type,
bit1: num > num ).
thf(sy_c_Num_Onum_OOne,type,
one2: num ).
thf(sy_c_Num_Onum_Ocase__num,type,
case_num:
!>[A: $tType] : ( A > ( num > A ) > ( num > A ) > num > A ) ).
thf(sy_c_Num_Onum_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_Option_Ooption_ONone,type,
none:
!>[A: $tType] : ( option @ A ) ).
thf(sy_c_Option_Ooption_OSome,type,
some:
!>[A: $tType] : ( A > ( option @ A ) ) ).
thf(sy_c_Option_Ooption_Ocase__option,type,
case_option:
!>[B: $tType,A: $tType] : ( B > ( A > B ) > ( option @ A ) > B ) ).
thf(sy_c_Option_Ooption_Omap__option,type,
map_option:
!>[A: $tType,Aa: $tType] : ( ( A > Aa ) > ( option @ A ) > ( option @ Aa ) ) ).
thf(sy_c_Option_Ooption_Osize__option,type,
size_option:
!>[A: $tType] : ( ( A > nat ) > ( option @ A ) > nat ) ).
thf(sy_c_Order__Continuity_Ocountable__complete__lattice__class_Occlfp,type,
order_532582986084564980_cclfp:
!>[A: $tType] : ( ( A > A ) > A ) ).
thf(sy_c_Orderings_Obot__class_Obot,type,
bot_bot:
!>[A: $tType] : A ).
thf(sy_c_Orderings_Oord_OLeast,type,
least:
!>[A: $tType] : ( ( A > A > $o ) > ( A > $o ) > A ) ).
thf(sy_c_Orderings_Oord__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_Oorder__class_OGreatest,type,
order_Greatest:
!>[A: $tType] : ( ( A > $o ) > A ) ).
thf(sy_c_Orderings_Oorder__class_Oantimono,type,
order_antimono:
!>[A: $tType,B: $tType] : ( ( A > B ) > $o ) ).
thf(sy_c_Orderings_Oorder__class_Omono,type,
order_mono:
!>[A: $tType,B: $tType] : ( ( A > B ) > $o ) ).
thf(sy_c_Orderings_Oorder__class_Ostrict__mono,type,
order_strict_mono:
!>[A: $tType,B: $tType] : ( ( A > B ) > $o ) ).
thf(sy_c_Orderings_Otop__class_Otop,type,
top_top:
!>[A: $tType] : 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_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_Product__Type_Oproduct,type,
product_product:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( set @ B ) > ( set @ ( product_prod @ A @ B ) ) ) ).
thf(sy_c_Rat_OFrct,type,
frct: ( product_prod @ int @ int ) > rat ).
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_Oquotient__of,type,
quotient_of: rat > ( product_prod @ int @ int ) ).
thf(sy_c_Real__Vector__Spaces_OReals,type,
real_Vector_Reals:
!>[A: $tType] : ( set @ A ) ).
thf(sy_c_Real__Vector__Spaces_Obounded__linear,type,
real_V3181309239436604168linear:
!>[A: $tType,B: $tType] : ( ( A > B ) > $o ) ).
thf(sy_c_Real__Vector__Spaces_Obounded__linear__axioms,type,
real_V4916620083959148203axioms:
!>[A: $tType,B: $tType] : ( ( A > B ) > $o ) ).
thf(sy_c_Real__Vector__Spaces_Odist__class_Odist,type,
real_V557655796197034286t_dist:
!>[A: $tType] : ( A > A > real ) ).
thf(sy_c_Real__Vector__Spaces_Onorm__class_Onorm,type,
real_V7770717601297561774m_norm:
!>[A: $tType] : ( A > real ) ).
thf(sy_c_Real__Vector__Spaces_Oof__real,type,
real_Vector_of_real:
!>[A: $tType] : ( real > A ) ).
thf(sy_c_Real__Vector__Spaces_OscaleR__class_OscaleR,type,
real_V8093663219630862766scaleR:
!>[A: $tType] : ( real > A > A ) ).
thf(sy_c_Relation_OId__on,type,
id_on:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) ) ).
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_Rings_Odivide__class_Odivide,type,
divide_divide:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Rings_Odvd__class_Odvd,type,
dvd_dvd:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Rings_Omodulo__class_Omodulo,type,
modulo_modulo:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool,type,
zero_neq_one_of_bool:
!>[A: $tType] : ( $o > A ) ).
thf(sy_c_Series_Osuminf,type,
suminf:
!>[A: $tType] : ( ( nat > A ) > A ) ).
thf(sy_c_Series_Osummable,type,
summable:
!>[A: $tType] : ( ( nat > A ) > $o ) ).
thf(sy_c_Series_Osums,type,
sums:
!>[A: $tType] : ( ( nat > A ) > A > $o ) ).
thf(sy_c_Set_OBall,type,
ball:
!>[A: $tType] : ( ( set @ A ) > ( A > $o ) > $o ) ).
thf(sy_c_Set_OCollect,type,
collect:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
thf(sy_c_Set_OPow,type,
pow2:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( set @ A ) ) ) ).
thf(sy_c_Set_Ofilter,type,
filter3:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) > ( set @ A ) ) ).
thf(sy_c_Set_Oimage,type,
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_Othe__elem,type,
the_elem:
!>[A: $tType] : ( ( set @ A ) > 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_Oascii__of,type,
ascii_of: char > char ).
thf(sy_c_String_Ochar_OChar,type,
char2: $o > $o > $o > $o > $o > $o > $o > $o > char ).
thf(sy_c_String_Ochar__of__integer,type,
char_of_integer: code_integer > char ).
thf(sy_c_String_Ocomm__semiring__1__class_Oof__char,type,
comm_s6883823935334413003f_char:
!>[A: $tType] : ( char > A ) ).
thf(sy_c_String_Ointeger__of__char,type,
integer_of_char: char > code_integer ).
thf(sy_c_String_Ounique__euclidean__semiring__with__bit__operations__class_Ochar__of,type,
unique5772411509450598832har_of:
!>[A: $tType] : ( A > char ) ).
thf(sy_c_Topological__Spaces_Ocontinuous,type,
topolo3448309680560233919inuous:
!>[A: $tType,B: $tType] : ( ( filter @ A ) > ( A > B ) > $o ) ).
thf(sy_c_Topological__Spaces_Ocontinuous__on,type,
topolo81223032696312382ous_on:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B ) > $o ) ).
thf(sy_c_Topological__Spaces_Ogenerate__topology,type,
topolo8378437560675496660pology:
!>[A: $tType] : ( ( set @ ( set @ A ) ) > ( set @ A ) > $o ) ).
thf(sy_c_Topological__Spaces_Omonoseq,type,
topological_monoseq:
!>[A: $tType] : ( ( nat > A ) > $o ) ).
thf(sy_c_Topological__Spaces_Oopen__class_Oopen,type,
topolo1002775350975398744n_open:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Topological__Spaces_Ot2__space__class_OLim,type,
topolo3827282254853284352ce_Lim:
!>[F: $tType,A: $tType] : ( ( filter @ F ) > ( F > A ) > A ) ).
thf(sy_c_Topological__Spaces_Otopological__space__class_Oat__within,type,
topolo174197925503356063within:
!>[A: $tType] : ( A > ( set @ A ) > ( filter @ A ) ) ).
thf(sy_c_Topological__Spaces_Otopological__space__class_Ocompact,type,
topolo2193935891317330818ompact:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Topological__Spaces_Otopological__space__class_Onhds,type,
topolo7230453075368039082e_nhds:
!>[A: $tType] : ( A > ( filter @ A ) ) ).
thf(sy_c_Topological__Spaces_Ouniform__space__class_OCauchy,type,
topolo3814608138187158403Cauchy:
!>[A: $tType] : ( ( nat > A ) > $o ) ).
thf(sy_c_Topological__Spaces_Ouniform__space__class_Ocauchy__filter,type,
topolo6773858410816713723filter:
!>[A: $tType] : ( ( filter @ A ) > $o ) ).
thf(sy_c_Topological__Spaces_Ouniform__space__class_Ocomplete,type,
topolo2479028161051973599mplete:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Topological__Spaces_Ouniform__space__class_Ototally__bounded,type,
topolo6688025880775521714ounded:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Topological__Spaces_Ouniformity__class_Ouniformity,type,
topolo7806501430040627800ormity:
!>[A: $tType] : ( filter @ ( product_prod @ A @ A ) ) ).
thf(sy_c_Transcendental_Oarccos,type,
arccos: real > real ).
thf(sy_c_Transcendental_Oarcosh,type,
arcosh:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Transcendental_Oarcsin,type,
arcsin: real > real ).
thf(sy_c_Transcendental_Oarctan,type,
arctan: real > real ).
thf(sy_c_Transcendental_Oarsinh,type,
arsinh:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Transcendental_Oartanh,type,
artanh:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Transcendental_Ocos,type,
cos:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Transcendental_Ocos__coeff,type,
cos_coeff: nat > real ).
thf(sy_c_Transcendental_Ocosh,type,
cosh:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Transcendental_Ocot,type,
cot:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Transcendental_Odiffs,type,
diffs:
!>[A: $tType] : ( ( nat > A ) > nat > A ) ).
thf(sy_c_Transcendental_Oexp,type,
exp:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Transcendental_Oln__class_Oln,type,
ln_ln:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Transcendental_Olog,type,
log: real > real > real ).
thf(sy_c_Transcendental_Opi,type,
pi: real ).
thf(sy_c_Transcendental_Opowr,type,
powr:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Transcendental_Osin,type,
sin:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Transcendental_Osin__coeff,type,
sin_coeff: nat > real ).
thf(sy_c_Transcendental_Osinh,type,
sinh:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Transcendental_Otan,type,
tan:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Transcendental_Otanh,type,
tanh:
!>[A: $tType] : ( A > A ) ).
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_Otrancl,type,
transitive_trancl:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ A @ A ) ) ) ).
thf(sy_c_VEBT__Definitions_OVEBT_OLeaf,type,
vEBT_Leaf: $o > $o > vEBT_VEBT ).
thf(sy_c_VEBT__Definitions_OVEBT_ONode,type,
vEBT_Node: ( option @ ( product_prod @ nat @ nat ) ) > nat > ( list @ vEBT_VEBT ) > vEBT_VEBT > vEBT_VEBT ).
thf(sy_c_VEBT__Definitions_OVEBT_Osize__VEBT,type,
vEBT_size_VEBT: vEBT_VEBT > nat ).
thf(sy_c_VEBT__Definitions_OVEBT__internal_Oboth__member__options,type,
vEBT_V8194947554948674370ptions: vEBT_VEBT > nat > $o ).
thf(sy_c_VEBT__Definitions_OVEBT__internal_Oelim__dead,type,
vEBT_VEBT_elim_dead: vEBT_VEBT > extended_enat > vEBT_VEBT ).
thf(sy_c_VEBT__Definitions_OVEBT__internal_Oelim__dead__rel,type,
vEBT_V312737461966249ad_rel: ( product_prod @ vEBT_VEBT @ extended_enat ) > ( product_prod @ vEBT_VEBT @ extended_enat ) > $o ).
thf(sy_c_VEBT__Definitions_OVEBT__internal_Ohigh,type,
vEBT_VEBT_high: nat > nat > nat ).
thf(sy_c_VEBT__Definitions_OVEBT__internal_Oin__children,type,
vEBT_V5917875025757280293ildren: nat > ( list @ vEBT_VEBT ) > nat > $o ).
thf(sy_c_VEBT__Definitions_OVEBT__internal_Olow,type,
vEBT_VEBT_low: nat > nat > nat ).
thf(sy_c_VEBT__Definitions_OVEBT__internal_Omembermima,type,
vEBT_VEBT_membermima: vEBT_VEBT > nat > $o ).
thf(sy_c_VEBT__Definitions_OVEBT__internal_Omembermima__rel,type,
vEBT_V4351362008482014158ma_rel: ( product_prod @ vEBT_VEBT @ nat ) > ( product_prod @ vEBT_VEBT @ nat ) > $o ).
thf(sy_c_VEBT__Definitions_OVEBT__internal_Onaive__member,type,
vEBT_V5719532721284313246member: vEBT_VEBT > nat > $o ).
thf(sy_c_VEBT__Definitions_OVEBT__internal_Onaive__member__rel,type,
vEBT_V5765760719290551771er_rel: ( product_prod @ vEBT_VEBT @ nat ) > ( product_prod @ vEBT_VEBT @ nat ) > $o ).
thf(sy_c_VEBT__Definitions_OVEBT__internal_Ovalid_H,type,
vEBT_VEBT_valid: vEBT_VEBT > nat > $o ).
thf(sy_c_VEBT__Definitions_OVEBT__internal_Ovalid_H__rel,type,
vEBT_VEBT_valid_rel: ( product_prod @ vEBT_VEBT @ nat ) > ( product_prod @ vEBT_VEBT @ nat ) > $o ).
thf(sy_c_VEBT__Definitions_Oinvar__vebt,type,
vEBT_invar_vebt: vEBT_VEBT > nat > $o ).
thf(sy_c_VEBT__Definitions_Oset__vebt,type,
vEBT_set_vebt: vEBT_VEBT > ( set @ nat ) ).
thf(sy_c_VEBT__Definitions_Ovebt__buildup,type,
vEBT_vebt_buildup: nat > vEBT_VEBT ).
thf(sy_c_VEBT__Definitions_Ovebt__buildup__rel,type,
vEBT_v4011308405150292612up_rel: nat > nat > $o ).
thf(sy_c_Wellfounded_Oaccp,type,
accp:
!>[A: $tType] : ( ( A > A > $o ) > A > $o ) ).
thf(sy_c_Wellfounded_Ofinite__psubset,type,
finite_psubset:
!>[A: $tType] : ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ).
thf(sy_c_Wellfounded_Olex__prod,type,
lex_prod:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ B @ B ) ) > ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) ) ).
thf(sy_c_Wellfounded_Omax__ext,type,
max_ext:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ) ).
thf(sy_c_Wellfounded_Omax__extp,type,
max_extp:
!>[A: $tType] : ( ( A > A > $o ) > ( set @ A ) > ( set @ A ) > $o ) ).
thf(sy_c_Wellfounded_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_Wfrec_Osame__fst,type,
same_fst:
!>[A: $tType,B: $tType] : ( ( A > $o ) > ( A > ( set @ ( product_prod @ B @ B ) ) ) > ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) ) ).
thf(sy_c_fChoice,type,
fChoice:
!>[A: $tType] : ( ( A > $o ) > A ) ).
thf(sy_c_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_dega____,type,
dega: nat ).
thf(sy_v_m____,type,
m: nat ).
thf(sy_v_na____,type,
na: nat ).
thf(sy_v_summarya____,type,
summarya: vEBT_VEBT ).
thf(sy_v_treeLista____,type,
treeLista: list @ vEBT_VEBT ).
% Relevant facts (8180)
thf(fact_0__C2_Ohyps_C_I3_J,axiom,
m = na ).
% "2.hyps"(3)
thf(fact_1__C2_OIH_C_I2_J,axiom,
( ( vEBT_VEBT_elim_dead @ summarya @ ( extend4730790105801354508finity @ extended_enat ) )
= summarya ) ).
% "2.IH"(2)
thf(fact_2__C2_Ohyps_C_I5_J,axiom,
~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ summarya @ X_1 ) ).
% "2.hyps"(5)
thf(fact_3__C2_Ohyps_C_I1_J,axiom,
vEBT_invar_vebt @ summarya @ m ).
% "2.hyps"(1)
thf(fact_4_not__enat__eq,axiom,
! [X: extended_enat] :
( ( ! [Y: nat] :
( X
!= ( extended_enat2 @ Y ) ) )
= ( X
= ( extend4730790105801354508finity @ extended_enat ) ) ) ).
% not_enat_eq
thf(fact_5_not__infinity__eq,axiom,
! [X: extended_enat] :
( ( X
!= ( extend4730790105801354508finity @ extended_enat ) )
= ( ? [I: nat] :
( X
= ( extended_enat2 @ I ) ) ) ) ).
% not_infinity_eq
thf(fact_6_enat_Osimps_I5_J,axiom,
! [T: $tType,F1: nat > T,F2: T] :
( ( extended_case_enat @ T @ F1 @ F2 @ ( extend4730790105801354508finity @ extended_enat ) )
= F2 ) ).
% enat.simps(5)
thf(fact_7__C2_OIH_C_I1_J,axiom,
! [X2: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ treeLista ) )
=> ( ( vEBT_invar_vebt @ X2 @ na )
& ( ( vEBT_VEBT_elim_dead @ X2 @ ( extend4730790105801354508finity @ extended_enat ) )
= X2 ) ) ) ).
% "2.IH"(1)
thf(fact_8_enat_Osimps_I7_J,axiom,
! [T: $tType,F1: nat > T,F2: T] :
( ( extended_rec_enat @ T @ F1 @ F2 @ ( extend4730790105801354508finity @ extended_enat ) )
= F2 ) ).
% enat.simps(7)
thf(fact_9_enat_Odistinct_I2_J,axiom,
! [Nat: nat] :
( ( extend4730790105801354508finity @ extended_enat )
!= ( extended_enat2 @ Nat ) ) ).
% enat.distinct(2)
thf(fact_10_enat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( ( extended_enat2 @ Nat2 )
!= ( extend4730790105801354508finity @ extended_enat ) ) ).
% enat.distinct(1)
thf(fact_11_enat_Oexhaust,axiom,
! [Y2: extended_enat] :
( ! [Nat3: nat] :
( Y2
!= ( extended_enat2 @ Nat3 ) )
=> ( Y2
= ( extend4730790105801354508finity @ extended_enat ) ) ) ).
% enat.exhaust
thf(fact_12_enat2__cases,axiom,
! [Y2: extended_enat,Ya: extended_enat] :
( ( ? [Nat3: nat] :
( Y2
= ( extended_enat2 @ Nat3 ) )
=> ! [Nata: nat] :
( Ya
!= ( extended_enat2 @ Nata ) ) )
=> ( ( ? [Nat3: nat] :
( Y2
= ( extended_enat2 @ Nat3 ) )
=> ( Ya
!= ( extend4730790105801354508finity @ extended_enat ) ) )
=> ( ( ( Y2
= ( extend4730790105801354508finity @ extended_enat ) )
=> ! [Nat3: nat] :
( Ya
!= ( extended_enat2 @ Nat3 ) ) )
=> ~ ( ( Y2
= ( extend4730790105801354508finity @ extended_enat ) )
=> ( Ya
!= ( extend4730790105801354508finity @ extended_enat ) ) ) ) ) ) ).
% enat2_cases
thf(fact_13_enat3__cases,axiom,
! [Y2: extended_enat,Ya: extended_enat,Yb: extended_enat] :
( ( ? [Nat3: nat] :
( Y2
= ( extended_enat2 @ Nat3 ) )
=> ( ? [Nata: nat] :
( Ya
= ( extended_enat2 @ Nata ) )
=> ! [Natb: nat] :
( Yb
!= ( extended_enat2 @ Natb ) ) ) )
=> ( ( ? [Nat3: nat] :
( Y2
= ( extended_enat2 @ Nat3 ) )
=> ( ? [Nata: nat] :
( Ya
= ( extended_enat2 @ Nata ) )
=> ( Yb
!= ( extend4730790105801354508finity @ extended_enat ) ) ) )
=> ( ( ? [Nat3: nat] :
( Y2
= ( extended_enat2 @ Nat3 ) )
=> ( ( Ya
= ( extend4730790105801354508finity @ extended_enat ) )
=> ! [Nata: nat] :
( Yb
!= ( extended_enat2 @ Nata ) ) ) )
=> ( ( ? [Nat3: nat] :
( Y2
= ( extended_enat2 @ Nat3 ) )
=> ( ( Ya
= ( extend4730790105801354508finity @ extended_enat ) )
=> ( Yb
!= ( extend4730790105801354508finity @ extended_enat ) ) ) )
=> ( ( ( Y2
= ( extend4730790105801354508finity @ extended_enat ) )
=> ( ? [Nat3: nat] :
( Ya
= ( extended_enat2 @ Nat3 ) )
=> ! [Nata: nat] :
( Yb
!= ( extended_enat2 @ Nata ) ) ) )
=> ( ( ( Y2
= ( extend4730790105801354508finity @ extended_enat ) )
=> ( ? [Nat3: nat] :
( Ya
= ( extended_enat2 @ Nat3 ) )
=> ( Yb
!= ( extend4730790105801354508finity @ extended_enat ) ) ) )
=> ( ( ( Y2
= ( extend4730790105801354508finity @ extended_enat ) )
=> ( ( Ya
= ( extend4730790105801354508finity @ extended_enat ) )
=> ! [Nat3: nat] :
( Yb
!= ( extended_enat2 @ Nat3 ) ) ) )
=> ~ ( ( Y2
= ( extend4730790105801354508finity @ extended_enat ) )
=> ( ( Ya
= ( extend4730790105801354508finity @ extended_enat ) )
=> ( Yb
!= ( extend4730790105801354508finity @ extended_enat ) ) ) ) ) ) ) ) ) ) ) ).
% enat3_cases
thf(fact_14__C2_Ohyps_C_I6_J,axiom,
! [X2: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ treeLista ) )
=> ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X_1 ) ) ).
% "2.hyps"(6)
thf(fact_15_enat_Oinject,axiom,
! [Nat2: nat,Nat4: nat] :
( ( ( extended_enat2 @ Nat2 )
= ( extended_enat2 @ Nat4 ) )
= ( Nat2 = Nat4 ) ) ).
% enat.inject
thf(fact_16_enat_Osimps_I4_J,axiom,
! [T: $tType,F1: nat > T,F2: T,Nat2: nat] :
( ( extended_case_enat @ T @ F1 @ F2 @ ( extended_enat2 @ Nat2 ) )
= ( F1 @ Nat2 ) ) ).
% enat.simps(4)
thf(fact_17_enat_Osimps_I6_J,axiom,
! [T: $tType,F1: nat > T,F2: T,Nat2: nat] :
( ( extended_rec_enat @ T @ F1 @ F2 @ ( extended_enat2 @ Nat2 ) )
= ( F1 @ Nat2 ) ) ).
% enat.simps(6)
thf(fact_18__C2_Ohyps_C_I4_J,axiom,
( dega
= ( plus_plus @ nat @ na @ m ) ) ).
% "2.hyps"(4)
thf(fact_19_enat__ex__split,axiom,
( ( ^ [P: extended_enat > $o] :
? [X3: extended_enat] : ( P @ X3 ) )
= ( ^ [P2: extended_enat > $o] :
( ( P2 @ ( extend4730790105801354508finity @ extended_enat ) )
| ? [X4: nat] : ( P2 @ ( extended_enat2 @ X4 ) ) ) ) ) ).
% enat_ex_split
thf(fact_20_the__enat_Osimps,axiom,
! [N: nat] :
( ( extended_the_enat @ ( extended_enat2 @ N ) )
= N ) ).
% the_enat.simps
thf(fact_21_valid__eq,axiom,
vEBT_VEBT_valid = vEBT_invar_vebt ).
% valid_eq
thf(fact_22_valid__eq1,axiom,
! [T2: vEBT_VEBT,D2: nat] :
( ( vEBT_invar_vebt @ T2 @ D2 )
=> ( vEBT_VEBT_valid @ T2 @ D2 ) ) ).
% valid_eq1
thf(fact_23_valid__eq2,axiom,
! [T2: vEBT_VEBT,D2: nat] :
( ( vEBT_VEBT_valid @ T2 @ D2 )
=> ( vEBT_invar_vebt @ T2 @ D2 ) ) ).
% valid_eq2
thf(fact_24__C2_Ohyps_C_I2_J,axiom,
( ( size_size @ ( list @ vEBT_VEBT ) @ treeLista )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ m ) ) ).
% "2.hyps"(2)
thf(fact_25__092_060open_062deg_Adiv_A2_A_061_An_092_060close_062,axiom,
( ( divide_divide @ nat @ dega @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= na ) ).
% \<open>deg div 2 = n\<close>
thf(fact_26_a,axiom,
! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ m ) )
=> ( ( vEBT_VEBT_elim_dead @ ( nth @ vEBT_VEBT @ treeLista @ I2 ) @ ( extended_enat2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ na ) ) )
= ( nth @ vEBT_VEBT @ treeLista @ I2 ) ) ) ).
% a
thf(fact_27__092_060open_0622_A_094_Am_A_061_A2_A_094_Adeg_Adiv_A2_A_094_A_Ideg_Adiv_A2_J_092_060close_062,axiom,
( ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ m )
= ( divide_divide @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ dega ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ dega @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% \<open>2 ^ m = 2 ^ deg div 2 ^ (deg div 2)\<close>
thf(fact_28_calculation,axiom,
( ( take @ vEBT_VEBT @ ( divide_divide @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ dega ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ dega @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
@ ( map @ vEBT_VEBT @ vEBT_VEBT
@ ^ [T3: vEBT_VEBT] : ( vEBT_VEBT_elim_dead @ T3 @ ( extended_enat2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ na ) ) )
@ treeLista ) )
= treeLista ) ).
% calculation
thf(fact_29_elimnum,axiom,
! [Info: option @ ( product_prod @ nat @ nat ),Deg: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) @ N )
=> ( ( vEBT_VEBT_elim_dead @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) @ ( extended_enat2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) )
= ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) ) ) ).
% elimnum
thf(fact_30_infinity__ileE,axiom,
! [M: nat] :
~ ( ord_less_eq @ extended_enat @ ( extend4730790105801354508finity @ extended_enat ) @ ( extended_enat2 @ M ) ) ).
% infinity_ileE
thf(fact_31_enat__ord__code_I5_J,axiom,
! [N: nat] :
~ ( ord_less_eq @ extended_enat @ ( extend4730790105801354508finity @ extended_enat ) @ ( extended_enat2 @ N ) ) ).
% enat_ord_code(5)
thf(fact_32_inthall,axiom,
! [A: $tType,Xs: list @ A,P3: A > $o,N: nat] :
( ! [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Xs ) )
=> ( P3 @ X5 ) )
=> ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( P3 @ ( nth @ A @ Xs @ N ) ) ) ) ).
% inthall
thf(fact_33_VEBT_Oinject_I1_J,axiom,
! [X11: option @ ( product_prod @ nat @ nat ),X12: nat,X13: list @ vEBT_VEBT,X14: vEBT_VEBT,Y11: option @ ( product_prod @ nat @ nat ),Y12: nat,Y13: list @ vEBT_VEBT,Y14: vEBT_VEBT] :
( ( ( vEBT_Node @ X11 @ X12 @ X13 @ X14 )
= ( vEBT_Node @ Y11 @ Y12 @ Y13 @ Y14 ) )
= ( ( X11 = Y11 )
& ( X12 = Y12 )
& ( X13 = Y13 )
& ( X14 = Y14 ) ) ) ).
% VEBT.inject(1)
thf(fact_34_pow__sum,axiom,
! [A2: nat,B2: nat] :
( ( divide_divide @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( plus_plus @ nat @ A2 @ B2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ A2 ) )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ B2 ) ) ).
% pow_sum
thf(fact_35_enat__ord__code_I3_J,axiom,
! [Q: extended_enat] : ( ord_less_eq @ extended_enat @ Q @ ( extend4730790105801354508finity @ extended_enat ) ) ).
% enat_ord_code(3)
thf(fact_36_enat__ord__simps_I5_J,axiom,
! [Q: extended_enat] :
( ( ord_less_eq @ extended_enat @ ( extend4730790105801354508finity @ extended_enat ) @ Q )
= ( Q
= ( extend4730790105801354508finity @ extended_enat ) ) ) ).
% enat_ord_simps(5)
thf(fact_37_b,axiom,
( ( map @ vEBT_VEBT @ vEBT_VEBT
@ ^ [T3: vEBT_VEBT] : ( vEBT_VEBT_elim_dead @ T3 @ ( extended_enat2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ na ) ) )
@ treeLista )
= treeLista ) ).
% b
thf(fact_38_VEBT__internal_Oelim__dead_Osimps_I3_J,axiom,
! [Info: option @ ( product_prod @ nat @ nat ),Deg: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,L: nat] :
( ( vEBT_VEBT_elim_dead @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) @ ( extended_enat2 @ L ) )
= ( vEBT_Node @ Info @ Deg
@ ( take @ vEBT_VEBT @ ( divide_divide @ nat @ L @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
@ ( map @ vEBT_VEBT @ vEBT_VEBT
@ ^ [T3: vEBT_VEBT] : ( vEBT_VEBT_elim_dead @ T3 @ ( extended_enat2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ TreeList ) )
@ ( vEBT_VEBT_elim_dead @ Summary @ ( extended_enat2 @ ( divide_divide @ nat @ L @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.elim_dead.simps(3)
thf(fact_39_VEBT__internal_Oelim__dead_Osimps_I2_J,axiom,
! [Info: option @ ( product_prod @ nat @ nat ),Deg: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( vEBT_VEBT_elim_dead @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) @ ( extend4730790105801354508finity @ extended_enat ) )
= ( vEBT_Node @ Info @ Deg
@ ( map @ vEBT_VEBT @ vEBT_VEBT
@ ^ [T3: vEBT_VEBT] : ( vEBT_VEBT_elim_dead @ T3 @ ( extended_enat2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ TreeList )
@ ( vEBT_VEBT_elim_dead @ Summary @ ( extend4730790105801354508finity @ extended_enat ) ) ) ) ).
% VEBT_internal.elim_dead.simps(2)
thf(fact_40_enat__ile,axiom,
! [N: extended_enat,M: nat] :
( ( ord_less_eq @ extended_enat @ N @ ( extended_enat2 @ M ) )
=> ? [K: nat] :
( N
= ( extended_enat2 @ K ) ) ) ).
% enat_ile
thf(fact_41_enat__ord__simps_I3_J,axiom,
! [Q: extended_enat] : ( ord_less_eq @ extended_enat @ Q @ ( extend4730790105801354508finity @ extended_enat ) ) ).
% enat_ord_simps(3)
thf(fact_42_case__enat__def,axiom,
! [T: $tType] :
( ( extended_case_enat @ T )
= ( extended_rec_enat @ T ) ) ).
% case_enat_def
thf(fact_43_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_44_mem__Collect__eq,axiom,
! [A: $tType,A2: A,P3: A > $o] :
( ( member @ A @ A2 @ ( collect @ A @ P3 ) )
= ( P3 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_45_Collect__mem__eq,axiom,
! [A: $tType,A3: set @ A] :
( ( collect @ A
@ ^ [X4: A] : ( member @ A @ X4 @ A3 ) )
= A3 ) ).
% Collect_mem_eq
thf(fact_46_Collect__cong,axiom,
! [A: $tType,P3: A > $o,Q2: A > $o] :
( ! [X5: A] :
( ( P3 @ X5 )
= ( Q2 @ X5 ) )
=> ( ( collect @ A @ P3 )
= ( collect @ A @ Q2 ) ) ) ).
% Collect_cong
thf(fact_47_ext,axiom,
! [B: $tType,A: $tType,F3: A > B,G: A > B] :
( ! [X5: A] :
( ( F3 @ X5 )
= ( G @ X5 ) )
=> ( F3 = G ) ) ).
% ext
thf(fact_48_nth__map,axiom,
! [B: $tType,A: $tType,N: nat,Xs: list @ A,F3: A > B] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ B @ ( map @ A @ B @ F3 @ Xs ) @ N )
= ( F3 @ ( nth @ A @ Xs @ N ) ) ) ) ).
% nth_map
thf(fact_49_nth__take,axiom,
! [A: $tType,I2: nat,N: nat,Xs: list @ A] :
( ( ord_less @ nat @ I2 @ N )
=> ( ( nth @ A @ ( take @ A @ N @ Xs ) @ I2 )
= ( nth @ A @ Xs @ I2 ) ) ) ).
% nth_take
thf(fact_50_div__exp__eq,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A2: A,M: nat,N: nat] :
( ( divide_divide @ A @ ( divide_divide @ A @ A2 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
= ( divide_divide @ A @ A2 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( plus_plus @ nat @ M @ N ) ) ) ) ) ).
% div_exp_eq
thf(fact_51_field__less__half__sum,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y2: A] :
( ( ord_less @ A @ X @ Y2 )
=> ( ord_less @ A @ X @ ( divide_divide @ A @ ( plus_plus @ A @ X @ Y2 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% field_less_half_sum
thf(fact_52_high__bound__aux,axiom,
! [Ma: nat,N: nat,M: nat] :
( ( ord_less @ nat @ Ma @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( plus_plus @ nat @ N @ M ) ) )
=> ( ord_less @ nat @ ( vEBT_VEBT_high @ Ma @ N ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) ) ) ).
% high_bound_aux
thf(fact_53_length__map,axiom,
! [A: $tType,B: $tType,F3: B > A,Xs: list @ B] :
( ( size_size @ ( list @ A ) @ ( map @ B @ A @ F3 @ Xs ) )
= ( size_size @ ( list @ B ) @ Xs ) ) ).
% length_map
thf(fact_54_map__eq__conv,axiom,
! [A: $tType,B: $tType,F3: B > A,Xs: list @ B,G: B > A] :
( ( ( map @ B @ A @ F3 @ Xs )
= ( map @ B @ A @ G @ Xs ) )
= ( ! [X4: B] :
( ( member @ B @ X4 @ ( set2 @ B @ Xs ) )
=> ( ( F3 @ X4 )
= ( G @ X4 ) ) ) ) ) ).
% map_eq_conv
thf(fact_55_nat__add__left__cancel__less,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less @ nat @ ( plus_plus @ nat @ K2 @ M ) @ ( plus_plus @ nat @ K2 @ N ) )
= ( ord_less @ nat @ M @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_56_less__exp,axiom,
! [N: nat] : ( ord_less @ nat @ N @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ).
% less_exp
thf(fact_57_semiring__norm_I83_J,axiom,
! [N: num] :
( one2
!= ( bit0 @ N ) ) ).
% semiring_norm(83)
thf(fact_58_semiring__norm_I85_J,axiom,
! [M: num] :
( ( bit0 @ M )
!= one2 ) ).
% semiring_norm(85)
thf(fact_59_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_60_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_61_semiring__norm_I87_J,axiom,
! [M: num,N: num] :
( ( ( bit0 @ M )
= ( bit0 @ N ) )
= ( M = N ) ) ).
% semiring_norm(87)
thf(fact_62_semiring__norm_I75_J,axiom,
! [M: num] :
~ ( ord_less @ num @ M @ one2 ) ).
% semiring_norm(75)
thf(fact_63_plus__enat__simps_I2_J,axiom,
! [Q: extended_enat] :
( ( plus_plus @ extended_enat @ ( extend4730790105801354508finity @ extended_enat ) @ Q )
= ( extend4730790105801354508finity @ extended_enat ) ) ).
% plus_enat_simps(2)
thf(fact_64_plus__enat__simps_I3_J,axiom,
! [Q: extended_enat] :
( ( plus_plus @ extended_enat @ Q @ ( extend4730790105801354508finity @ extended_enat ) )
= ( extend4730790105801354508finity @ extended_enat ) ) ).
% plus_enat_simps(3)
thf(fact_65_enat__ord__simps_I4_J,axiom,
! [Q: extended_enat] :
( ( ord_less @ extended_enat @ Q @ ( extend4730790105801354508finity @ extended_enat ) )
= ( Q
!= ( extend4730790105801354508finity @ extended_enat ) ) ) ).
% enat_ord_simps(4)
thf(fact_66_enat__ord__simps_I6_J,axiom,
! [Q: extended_enat] :
~ ( ord_less @ extended_enat @ ( extend4730790105801354508finity @ extended_enat ) @ Q ) ).
% enat_ord_simps(6)
thf(fact_67_high__def,axiom,
( vEBT_VEBT_high
= ( ^ [X4: nat,N2: nat] : ( divide_divide @ nat @ X4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ).
% high_def
thf(fact_68_map__ident,axiom,
! [A: $tType] :
( ( map @ A @ A
@ ^ [X4: A] : X4 )
= ( ^ [Xs2: list @ A] : Xs2 ) ) ).
% map_ident
thf(fact_69_semiring__norm_I2_J,axiom,
( ( plus_plus @ num @ one2 @ one2 )
= ( bit0 @ one2 ) ) ).
% semiring_norm(2)
thf(fact_70_semiring__norm_I76_J,axiom,
! [N: num] : ( ord_less @ num @ one2 @ ( bit0 @ N ) ) ).
% semiring_norm(76)
thf(fact_71_enat__ord__number_I2_J,axiom,
! [M: num,N: num] :
( ( ord_less @ extended_enat @ ( numeral_numeral @ extended_enat @ M ) @ ( numeral_numeral @ extended_enat @ N ) )
= ( ord_less @ nat @ ( numeral_numeral @ nat @ M ) @ ( numeral_numeral @ nat @ N ) ) ) ).
% enat_ord_number(2)
thf(fact_72_enat__ord__simps_I2_J,axiom,
! [M: nat,N: nat] :
( ( ord_less @ extended_enat @ ( extended_enat2 @ M ) @ ( extended_enat2 @ N ) )
= ( ord_less @ nat @ M @ N ) ) ).
% enat_ord_simps(2)
thf(fact_73_plus__enat__simps_I1_J,axiom,
! [M: nat,N: nat] :
( ( plus_plus @ extended_enat @ ( extended_enat2 @ M ) @ ( extended_enat2 @ N ) )
= ( extended_enat2 @ ( plus_plus @ nat @ M @ N ) ) ) ).
% plus_enat_simps(1)
thf(fact_74_numeral__less__enat__iff,axiom,
! [M: num,N: nat] :
( ( ord_less @ extended_enat @ ( numeral_numeral @ extended_enat @ M ) @ ( extended_enat2 @ N ) )
= ( ord_less @ nat @ ( numeral_numeral @ nat @ M ) @ N ) ) ).
% numeral_less_enat_iff
thf(fact_75_enat__add__left__cancel__less,axiom,
! [A2: extended_enat,B2: extended_enat,C2: extended_enat] :
( ( ord_less @ extended_enat @ ( plus_plus @ extended_enat @ A2 @ B2 ) @ ( plus_plus @ extended_enat @ A2 @ C2 ) )
= ( ( A2
!= ( extend4730790105801354508finity @ extended_enat ) )
& ( ord_less @ extended_enat @ B2 @ C2 ) ) ) ).
% enat_add_left_cancel_less
thf(fact_76_plus__eq__infty__iff__enat,axiom,
! [M: extended_enat,N: extended_enat] :
( ( ( plus_plus @ extended_enat @ M @ N )
= ( extend4730790105801354508finity @ extended_enat ) )
= ( ( M
= ( extend4730790105801354508finity @ extended_enat ) )
| ( N
= ( extend4730790105801354508finity @ extended_enat ) ) ) ) ).
% plus_eq_infty_iff_enat
thf(fact_77_enat__add__left__cancel,axiom,
! [A2: extended_enat,B2: extended_enat,C2: extended_enat] :
( ( ( plus_plus @ extended_enat @ A2 @ B2 )
= ( plus_plus @ extended_enat @ A2 @ C2 ) )
= ( ( A2
= ( extend4730790105801354508finity @ extended_enat ) )
| ( B2 = C2 ) ) ) ).
% enat_add_left_cancel
thf(fact_78_numeral__ne__infinity,axiom,
! [K2: num] :
( ( numeral_numeral @ extended_enat @ K2 )
!= ( extend4730790105801354508finity @ extended_enat ) ) ).
% numeral_ne_infinity
thf(fact_79_chain__incr,axiom,
! [A: $tType,Y3: A > extended_enat,K2: nat] :
( ! [I3: A] :
? [J: A] : ( ord_less @ extended_enat @ ( Y3 @ I3 ) @ ( Y3 @ J ) )
=> ? [J2: A] : ( ord_less @ extended_enat @ ( extended_enat2 @ K2 ) @ ( Y3 @ J2 ) ) ) ).
% chain_incr
thf(fact_80_enat__iless,axiom,
! [N: extended_enat,M: nat] :
( ( ord_less @ extended_enat @ N @ ( extended_enat2 @ M ) )
=> ? [K: nat] :
( N
= ( extended_enat2 @ K ) ) ) ).
% enat_iless
thf(fact_81_less__enat__def,axiom,
( ( ord_less @ extended_enat )
= ( ^ [M2: extended_enat,N2: extended_enat] :
( extended_case_enat @ $o
@ ^ [M1: nat] : ( extended_case_enat @ $o @ ( ord_less @ nat @ M1 ) @ $true @ N2 )
@ $false
@ M2 ) ) ) ).
% less_enat_def
thf(fact_82_numeral__eq__enat,axiom,
( ( numeral_numeral @ extended_enat )
= ( ^ [K3: num] : ( extended_enat2 @ ( numeral_numeral @ nat @ K3 ) ) ) ) ).
% numeral_eq_enat
thf(fact_83_less__enatE,axiom,
! [N: extended_enat,M: nat] :
( ( ord_less @ extended_enat @ N @ ( extended_enat2 @ M ) )
=> ~ ! [K: nat] :
( ( N
= ( extended_enat2 @ K ) )
=> ~ ( ord_less @ nat @ K @ M ) ) ) ).
% less_enatE
thf(fact_84_infinity__ilessE,axiom,
! [M: nat] :
~ ( ord_less @ extended_enat @ ( extend4730790105801354508finity @ extended_enat ) @ ( extended_enat2 @ M ) ) ).
% infinity_ilessE
thf(fact_85_less__infinityE,axiom,
! [N: extended_enat] :
( ( ord_less @ extended_enat @ N @ ( extend4730790105801354508finity @ extended_enat ) )
=> ~ ! [K: nat] :
( N
!= ( extended_enat2 @ K ) ) ) ).
% less_infinityE
thf(fact_86_enat__ord__code_I4_J,axiom,
! [M: nat] : ( ord_less @ extended_enat @ ( extended_enat2 @ M ) @ ( extend4730790105801354508finity @ extended_enat ) ) ).
% enat_ord_code(4)
thf(fact_87_enat__add__left__cancel__le,axiom,
! [A2: extended_enat,B2: extended_enat,C2: extended_enat] :
( ( ord_less_eq @ extended_enat @ ( plus_plus @ extended_enat @ A2 @ B2 ) @ ( plus_plus @ extended_enat @ A2 @ C2 ) )
= ( ( A2
= ( extend4730790105801354508finity @ extended_enat ) )
| ( ord_less_eq @ extended_enat @ B2 @ C2 ) ) ) ).
% enat_add_left_cancel_le
thf(fact_88_measure__induct,axiom,
! [B: $tType,A: $tType] :
( ( wellorder @ B )
=> ! [F3: A > B,P3: A > $o,A2: A] :
( ! [X5: A] :
( ! [Y4: A] :
( ( ord_less @ B @ ( F3 @ Y4 ) @ ( F3 @ X5 ) )
=> ( P3 @ Y4 ) )
=> ( P3 @ X5 ) )
=> ( P3 @ A2 ) ) ) ).
% measure_induct
thf(fact_89_measure__induct__rule,axiom,
! [B: $tType,A: $tType] :
( ( wellorder @ B )
=> ! [F3: A > B,P3: A > $o,A2: A] :
( ! [X5: A] :
( ! [Y4: A] :
( ( ord_less @ B @ ( F3 @ Y4 ) @ ( F3 @ X5 ) )
=> ( P3 @ Y4 ) )
=> ( P3 @ X5 ) )
=> ( P3 @ A2 ) ) ) ).
% measure_induct_rule
thf(fact_90_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_91_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ N ) ).
% less_not_refl
thf(fact_92_less__not__refl2,axiom,
! [N: nat,M: nat] :
( ( ord_less @ nat @ N @ M )
=> ( M != N ) ) ).
% less_not_refl2
thf(fact_93_less__not__refl3,axiom,
! [S: nat,T2: nat] :
( ( ord_less @ nat @ S @ T2 )
=> ( S != T2 ) ) ).
% less_not_refl3
thf(fact_94_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_95_nat__less__induct,axiom,
! [P3: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M3: nat] :
( ( ord_less @ nat @ M3 @ N3 )
=> ( P3 @ M3 ) )
=> ( P3 @ N3 ) )
=> ( P3 @ N ) ) ).
% nat_less_induct
thf(fact_96_infinite__descent,axiom,
! [P3: nat > $o,N: nat] :
( ! [N3: nat] :
( ~ ( P3 @ N3 )
=> ? [M3: nat] :
( ( ord_less @ nat @ M3 @ N3 )
& ~ ( P3 @ M3 ) ) )
=> ( P3 @ N ) ) ).
% infinite_descent
thf(fact_97_linorder__neqE__nat,axiom,
! [X: nat,Y2: nat] :
( ( X != Y2 )
=> ( ~ ( ord_less @ nat @ X @ Y2 )
=> ( ord_less @ nat @ Y2 @ X ) ) ) ).
% linorder_neqE_nat
thf(fact_98_infinite__descent__measure,axiom,
! [A: $tType,P3: A > $o,V: A > nat,X: A] :
( ! [X5: A] :
( ~ ( P3 @ X5 )
=> ? [Y4: A] :
( ( ord_less @ nat @ ( V @ Y4 ) @ ( V @ X5 ) )
& ~ ( P3 @ Y4 ) ) )
=> ( P3 @ X ) ) ).
% infinite_descent_measure
thf(fact_99_size__neq__size__imp__neq,axiom,
! [A: $tType] :
( ( size @ A )
=> ! [X: A,Y2: A] :
( ( ( size_size @ A @ X )
!= ( size_size @ A @ Y2 ) )
=> ( X != Y2 ) ) ) ).
% size_neq_size_imp_neq
thf(fact_100_Ex__list__of__length,axiom,
! [A: $tType,N: nat] :
? [Xs3: list @ A] :
( ( size_size @ ( list @ A ) @ Xs3 )
= N ) ).
% Ex_list_of_length
thf(fact_101_neq__if__length__neq,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs )
!= ( size_size @ ( list @ A ) @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_102_take__equalityI,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ! [I3: nat] :
( ( take @ A @ I3 @ Xs )
= ( take @ A @ I3 @ Ys ) )
=> ( Xs = Ys ) ) ).
% take_equalityI
thf(fact_103_plus__enat__def,axiom,
( ( plus_plus @ extended_enat )
= ( ^ [M2: extended_enat,N2: extended_enat] :
( extended_case_enat @ extended_enat
@ ^ [O: nat] :
( extended_case_enat @ extended_enat
@ ^ [P4: nat] : ( extended_enat2 @ ( plus_plus @ nat @ O @ P4 ) )
@ ( extend4730790105801354508finity @ extended_enat )
@ N2 )
@ ( extend4730790105801354508finity @ extended_enat )
@ M2 ) ) ) ).
% plus_enat_def
thf(fact_104_list_Omap__ident,axiom,
! [A: $tType,T2: list @ A] :
( ( map @ A @ A
@ ^ [X4: A] : X4
@ T2 )
= T2 ) ).
% list.map_ident
thf(fact_105_power__divide,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A2: A,B2: A,N: nat] :
( ( power_power @ A @ ( divide_divide @ A @ A2 @ B2 ) @ N )
= ( divide_divide @ A @ ( power_power @ A @ A2 @ N ) @ ( power_power @ A @ B2 @ N ) ) ) ) ).
% power_divide
thf(fact_106_add__lessD1,axiom,
! [I2: nat,J3: nat,K2: nat] :
( ( ord_less @ nat @ ( plus_plus @ nat @ I2 @ J3 ) @ K2 )
=> ( ord_less @ nat @ I2 @ K2 ) ) ).
% add_lessD1
thf(fact_107_add__less__mono,axiom,
! [I2: nat,J3: nat,K2: nat,L: nat] :
( ( ord_less @ nat @ I2 @ J3 )
=> ( ( ord_less @ nat @ K2 @ L )
=> ( ord_less @ nat @ ( plus_plus @ nat @ I2 @ K2 ) @ ( plus_plus @ nat @ J3 @ L ) ) ) ) ).
% add_less_mono
thf(fact_108_not__add__less1,axiom,
! [I2: nat,J3: nat] :
~ ( ord_less @ nat @ ( plus_plus @ nat @ I2 @ J3 ) @ I2 ) ).
% not_add_less1
thf(fact_109_not__add__less2,axiom,
! [J3: nat,I2: nat] :
~ ( ord_less @ nat @ ( plus_plus @ nat @ J3 @ I2 ) @ I2 ) ).
% not_add_less2
thf(fact_110_add__less__mono1,axiom,
! [I2: nat,J3: nat,K2: nat] :
( ( ord_less @ nat @ I2 @ J3 )
=> ( ord_less @ nat @ ( plus_plus @ nat @ I2 @ K2 ) @ ( plus_plus @ nat @ J3 @ K2 ) ) ) ).
% add_less_mono1
thf(fact_111_trans__less__add1,axiom,
! [I2: nat,J3: nat,M: nat] :
( ( ord_less @ nat @ I2 @ J3 )
=> ( ord_less @ nat @ I2 @ ( plus_plus @ nat @ J3 @ M ) ) ) ).
% trans_less_add1
thf(fact_112_trans__less__add2,axiom,
! [I2: nat,J3: nat,M: nat] :
( ( ord_less @ nat @ I2 @ J3 )
=> ( ord_less @ nat @ I2 @ ( plus_plus @ nat @ M @ J3 ) ) ) ).
% trans_less_add2
thf(fact_113_less__add__eq__less,axiom,
! [K2: nat,L: nat,M: nat,N: nat] :
( ( ord_less @ nat @ K2 @ L )
=> ( ( ( plus_plus @ nat @ M @ L )
= ( plus_plus @ nat @ K2 @ N ) )
=> ( ord_less @ nat @ M @ N ) ) ) ).
% less_add_eq_less
thf(fact_114_length__induct,axiom,
! [A: $tType,P3: ( list @ A ) > $o,Xs: list @ A] :
( ! [Xs3: list @ A] :
( ! [Ys2: list @ A] :
( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Ys2 ) @ ( size_size @ ( list @ A ) @ Xs3 ) )
=> ( P3 @ Ys2 ) )
=> ( P3 @ Xs3 ) )
=> ( P3 @ Xs ) ) ).
% length_induct
thf(fact_115_list_Omap__cong,axiom,
! [B: $tType,A: $tType,X: list @ A,Ya: list @ A,F3: A > B,G: A > B] :
( ( X = Ya )
=> ( ! [Z: A] :
( ( member @ A @ Z @ ( set2 @ A @ Ya ) )
=> ( ( F3 @ Z )
= ( G @ Z ) ) )
=> ( ( map @ A @ B @ F3 @ X )
= ( map @ A @ B @ G @ Ya ) ) ) ) ).
% list.map_cong
thf(fact_116_list_Omap__cong0,axiom,
! [B: $tType,A: $tType,X: list @ A,F3: A > B,G: A > B] :
( ! [Z: A] :
( ( member @ A @ Z @ ( set2 @ A @ X ) )
=> ( ( F3 @ Z )
= ( G @ Z ) ) )
=> ( ( map @ A @ B @ F3 @ X )
= ( map @ A @ B @ G @ X ) ) ) ).
% list.map_cong0
thf(fact_117_list_Oinj__map__strong,axiom,
! [B: $tType,A: $tType,X: list @ A,Xa: list @ A,F3: A > B,Fa: A > B] :
( ! [Z: A,Za: A] :
( ( member @ A @ Z @ ( set2 @ A @ X ) )
=> ( ( member @ A @ Za @ ( set2 @ A @ Xa ) )
=> ( ( ( F3 @ Z )
= ( Fa @ Za ) )
=> ( Z = Za ) ) ) )
=> ( ( ( map @ A @ B @ F3 @ X )
= ( map @ A @ B @ Fa @ Xa ) )
=> ( X = Xa ) ) ) ).
% list.inj_map_strong
thf(fact_118_map__ext,axiom,
! [B: $tType,A: $tType,Xs: list @ A,F3: A > B,G: A > B] :
( ! [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Xs ) )
=> ( ( F3 @ X5 )
= ( G @ X5 ) ) )
=> ( ( map @ A @ B @ F3 @ Xs )
= ( map @ A @ B @ G @ Xs ) ) ) ).
% map_ext
thf(fact_119_map__idI,axiom,
! [A: $tType,Xs: list @ A,F3: A > A] :
( ! [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Xs ) )
=> ( ( F3 @ X5 )
= X5 ) )
=> ( ( map @ A @ A @ F3 @ Xs )
= Xs ) ) ).
% map_idI
thf(fact_120_map__cong,axiom,
! [B: $tType,A: $tType,Xs: list @ A,Ys: list @ A,F3: A > B,G: A > B] :
( ( Xs = Ys )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Ys ) )
=> ( ( F3 @ X5 )
= ( G @ X5 ) ) )
=> ( ( map @ A @ B @ F3 @ Xs )
= ( map @ A @ B @ G @ Ys ) ) ) ) ).
% map_cong
thf(fact_121_ex__map__conv,axiom,
! [A: $tType,B: $tType,Ys: list @ B,F3: A > B] :
( ( ? [Xs2: list @ A] :
( Ys
= ( map @ A @ B @ F3 @ Xs2 ) ) )
= ( ! [X4: B] :
( ( member @ B @ X4 @ ( set2 @ B @ Ys ) )
=> ? [Y: A] :
( X4
= ( F3 @ Y ) ) ) ) ) ).
% ex_map_conv
thf(fact_122_map__eq__imp__length__eq,axiom,
! [A: $tType,B: $tType,C: $tType,F3: B > A,Xs: list @ B,G: C > A,Ys: list @ C] :
( ( ( map @ B @ A @ F3 @ Xs )
= ( map @ C @ A @ G @ Ys ) )
=> ( ( size_size @ ( list @ B ) @ Xs )
= ( size_size @ ( list @ C ) @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_123_in__set__takeD,axiom,
! [A: $tType,X: A,N: nat,Xs: list @ A] :
( ( member @ A @ X @ ( set2 @ A @ ( take @ A @ N @ Xs ) ) )
=> ( member @ A @ X @ ( set2 @ A @ Xs ) ) ) ).
% in_set_takeD
thf(fact_124_take__map,axiom,
! [A: $tType,B: $tType,N: nat,F3: B > A,Xs: list @ B] :
( ( take @ A @ N @ ( map @ B @ A @ F3 @ Xs ) )
= ( map @ B @ A @ F3 @ ( take @ B @ N @ Xs ) ) ) ).
% take_map
thf(fact_125_nth__equalityI,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ A ) @ Ys ) )
=> ( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ A @ Xs @ I3 )
= ( nth @ A @ Ys @ I3 ) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
thf(fact_126_Skolem__list__nth,axiom,
! [A: $tType,K2: nat,P3: nat > A > $o] :
( ( ! [I: nat] :
( ( ord_less @ nat @ I @ K2 )
=> ? [X6: A] : ( P3 @ I @ X6 ) ) )
= ( ? [Xs2: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= K2 )
& ! [I: nat] :
( ( ord_less @ nat @ I @ K2 )
=> ( P3 @ I @ ( nth @ A @ Xs2 @ I ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_127_list__eq__iff__nth__eq,axiom,
! [A: $tType] :
( ( ^ [Y5: list @ A,Z2: list @ A] : Y5 = Z2 )
= ( ^ [Xs2: list @ A,Ys3: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ A ) @ Ys3 ) )
& ! [I: nat] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( nth @ A @ Xs2 @ I )
= ( nth @ A @ Ys3 @ I ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_128_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_129_all__set__conv__all__nth,axiom,
! [A: $tType,Xs: list @ A,P3: A > $o] :
( ( ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
=> ( P3 @ X4 ) ) )
= ( ! [I: nat] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( P3 @ ( nth @ A @ Xs @ I ) ) ) ) ) ).
% all_set_conv_all_nth
thf(fact_130_all__nth__imp__all__set,axiom,
! [A: $tType,Xs: list @ A,P3: A > $o,X: A] :
( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( P3 @ ( nth @ A @ Xs @ I3 ) ) )
=> ( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( P3 @ X ) ) ) ).
% all_nth_imp_all_set
thf(fact_131_in__set__conv__nth,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
= ( ? [I: nat] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
& ( ( nth @ A @ Xs @ I )
= X ) ) ) ) ).
% in_set_conv_nth
thf(fact_132_list__ball__nth,axiom,
! [A: $tType,N: nat,Xs: list @ A,P3: A > $o] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Xs ) )
=> ( P3 @ X5 ) )
=> ( P3 @ ( nth @ A @ Xs @ N ) ) ) ) ).
% list_ball_nth
thf(fact_133_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_134_field__sum__of__halves,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A] :
( ( plus_plus @ A @ ( divide_divide @ A @ X @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ A @ X @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
= X ) ) ).
% field_sum_of_halves
thf(fact_135_both__member__options__ding,axiom,
! [Info: option @ ( product_prod @ nat @ nat ),Deg: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) @ N )
=> ( ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg ) )
=> ( ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
=> ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) @ X ) ) ) ) ).
% both_member_options_ding
thf(fact_136_high__inv,axiom,
! [X: nat,N: nat,Y2: nat] :
( ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( vEBT_VEBT_high @ ( plus_plus @ nat @ ( times_times @ nat @ Y2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) @ X ) @ N )
= Y2 ) ) ).
% high_inv
thf(fact_137_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_138_add__numeral__left,axiom,
! [A: $tType] :
( ( numeral @ A )
=> ! [V2: num,W: num,Z3: A] :
( ( plus_plus @ A @ ( numeral_numeral @ A @ V2 ) @ ( plus_plus @ A @ ( numeral_numeral @ A @ W ) @ Z3 ) )
= ( plus_plus @ A @ ( numeral_numeral @ A @ ( plus_plus @ num @ V2 @ W ) ) @ Z3 ) ) ) ).
% add_numeral_left
thf(fact_139_add__less__cancel__left,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ C2 @ A2 ) @ ( plus_plus @ A @ C2 @ B2 ) )
= ( ord_less @ A @ A2 @ B2 ) ) ) ).
% add_less_cancel_left
thf(fact_140_add__less__cancel__right,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) )
= ( ord_less @ A @ A2 @ B2 ) ) ) ).
% add_less_cancel_right
thf(fact_141_add__le__cancel__left,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ C2 @ A2 ) @ ( plus_plus @ A @ C2 @ B2 ) )
= ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% add_le_cancel_left
thf(fact_142_add__le__cancel__right,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) )
= ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% add_le_cancel_right
thf(fact_143_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_144_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_145_VEBT__internal_Oelim__dead_Oelims,axiom,
! [X: vEBT_VEBT,Xa: extended_enat,Y2: vEBT_VEBT] :
( ( ( vEBT_VEBT_elim_dead @ X @ Xa )
= Y2 )
=> ( ! [A4: $o,B3: $o] :
( ( X
= ( vEBT_Leaf @ A4 @ B3 ) )
=> ( Y2
!= ( vEBT_Leaf @ A4 @ B3 ) ) )
=> ( ! [Info2: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( ( Xa
= ( extend4730790105801354508finity @ extended_enat ) )
=> ( Y2
!= ( vEBT_Node @ Info2 @ Deg2
@ ( map @ vEBT_VEBT @ vEBT_VEBT
@ ^ [T3: vEBT_VEBT] : ( vEBT_VEBT_elim_dead @ T3 @ ( extended_enat2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ TreeList2 )
@ ( vEBT_VEBT_elim_dead @ Summary2 @ ( extend4730790105801354508finity @ extended_enat ) ) ) ) ) )
=> ~ ! [Info2: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) )
=> ! [L2: nat] :
( ( Xa
= ( extended_enat2 @ L2 ) )
=> ( Y2
!= ( vEBT_Node @ Info2 @ Deg2
@ ( take @ vEBT_VEBT @ ( divide_divide @ nat @ L2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
@ ( map @ vEBT_VEBT @ vEBT_VEBT
@ ^ [T3: vEBT_VEBT] : ( vEBT_VEBT_elim_dead @ T3 @ ( extended_enat2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ TreeList2 ) )
@ ( vEBT_VEBT_elim_dead @ Summary2 @ ( extended_enat2 @ ( divide_divide @ nat @ L2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.elim_dead.elims
thf(fact_146_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_147_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_148_add__right__cancel,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( ( plus_plus @ A @ B2 @ A2 )
= ( plus_plus @ A @ C2 @ A2 ) )
= ( B2 = C2 ) ) ) ).
% add_right_cancel
thf(fact_149_add__left__cancel,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ( plus_plus @ A @ A2 @ B2 )
= ( plus_plus @ A @ A2 @ C2 ) )
= ( B2 = C2 ) ) ) ).
% add_left_cancel
thf(fact_150_nat__add__left__cancel__le,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ ( plus_plus @ nat @ K2 @ M ) @ ( plus_plus @ nat @ K2 @ N ) )
= ( ord_less_eq @ nat @ M @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_151_VEBT_Oinject_I2_J,axiom,
! [X21: $o,X22: $o,Y21: $o,Y22: $o] :
( ( ( vEBT_Leaf @ X21 @ X22 )
= ( vEBT_Leaf @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% VEBT.inject(2)
thf(fact_152_mult__numeral__left__semiring__numeral,axiom,
! [A: $tType] :
( ( semiring_numeral @ A )
=> ! [V2: num,W: num,Z3: A] :
( ( times_times @ A @ ( numeral_numeral @ A @ V2 ) @ ( times_times @ A @ ( numeral_numeral @ A @ W ) @ Z3 ) )
= ( times_times @ A @ ( numeral_numeral @ A @ ( times_times @ num @ V2 @ W ) ) @ Z3 ) ) ) ).
% mult_numeral_left_semiring_numeral
thf(fact_153_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_154_low__inv,axiom,
! [X: nat,N: nat,Y2: nat] :
( ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( vEBT_VEBT_low @ ( plus_plus @ nat @ ( times_times @ nat @ Y2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) @ X ) @ N )
= X ) ) ).
% low_inv
thf(fact_155_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_156_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_157_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_158_semiring__norm_I68_J,axiom,
! [N: num] : ( ord_less_eq @ num @ one2 @ N ) ).
% semiring_norm(68)
thf(fact_159_enat__ord__simps_I1_J,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ extended_enat @ ( extended_enat2 @ M ) @ ( extended_enat2 @ N ) )
= ( ord_less_eq @ nat @ M @ N ) ) ).
% enat_ord_simps(1)
thf(fact_160_distrib__right__numeral,axiom,
! [A: $tType] :
( ( ( numeral @ A )
& ( semiring @ A ) )
=> ! [A2: A,B2: A,V2: num] :
( ( times_times @ A @ ( plus_plus @ A @ A2 @ B2 ) @ ( numeral_numeral @ A @ V2 ) )
= ( plus_plus @ A @ ( times_times @ A @ A2 @ ( numeral_numeral @ A @ V2 ) ) @ ( times_times @ A @ B2 @ ( numeral_numeral @ A @ V2 ) ) ) ) ) ).
% distrib_right_numeral
thf(fact_161_distrib__left__numeral,axiom,
! [A: $tType] :
( ( ( numeral @ A )
& ( semiring @ A ) )
=> ! [V2: num,B2: A,C2: A] :
( ( times_times @ A @ ( numeral_numeral @ A @ V2 ) @ ( plus_plus @ A @ B2 @ C2 ) )
= ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ V2 ) @ B2 ) @ ( times_times @ A @ ( numeral_numeral @ A @ V2 ) @ C2 ) ) ) ) ).
% distrib_left_numeral
thf(fact_162_semiring__norm_I69_J,axiom,
! [M: num] :
~ ( ord_less_eq @ num @ ( bit0 @ M ) @ one2 ) ).
% semiring_norm(69)
thf(fact_163_enat__ord__number_I1_J,axiom,
! [M: num,N: num] :
( ( ord_less_eq @ extended_enat @ ( numeral_numeral @ extended_enat @ M ) @ ( numeral_numeral @ extended_enat @ N ) )
= ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ M ) @ ( numeral_numeral @ nat @ N ) ) ) ).
% enat_ord_number(1)
thf(fact_164_le__divide__eq__numeral1_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A,W: num] :
( ( ord_less_eq @ A @ A2 @ ( divide_divide @ A @ B2 @ ( numeral_numeral @ A @ W ) ) )
= ( ord_less_eq @ A @ ( times_times @ A @ A2 @ ( numeral_numeral @ A @ W ) ) @ B2 ) ) ) ).
% le_divide_eq_numeral1(1)
thf(fact_165_divide__le__eq__numeral1_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,W: num,A2: A] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ ( numeral_numeral @ A @ W ) ) @ A2 )
= ( ord_less_eq @ A @ B2 @ ( times_times @ A @ A2 @ ( numeral_numeral @ A @ W ) ) ) ) ) ).
% divide_le_eq_numeral1(1)
thf(fact_166_less__divide__eq__numeral1_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A,W: num] :
( ( ord_less @ A @ A2 @ ( divide_divide @ A @ B2 @ ( numeral_numeral @ A @ W ) ) )
= ( ord_less @ A @ ( times_times @ A @ A2 @ ( numeral_numeral @ A @ W ) ) @ B2 ) ) ) ).
% less_divide_eq_numeral1(1)
thf(fact_167_divide__less__eq__numeral1_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,W: num,A2: A] :
( ( ord_less @ A @ ( divide_divide @ A @ B2 @ ( numeral_numeral @ A @ W ) ) @ A2 )
= ( ord_less @ A @ B2 @ ( times_times @ A @ A2 @ ( numeral_numeral @ A @ W ) ) ) ) ) ).
% divide_less_eq_numeral1(1)
thf(fact_168_power__add__numeral2,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A2: A,M: num,N: num,B2: A] :
( ( times_times @ A @ ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ M ) ) @ ( times_times @ A @ ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ N ) ) @ B2 ) )
= ( times_times @ A @ ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ ( plus_plus @ num @ M @ N ) ) ) @ B2 ) ) ) ).
% power_add_numeral2
thf(fact_169_power__add__numeral,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A2: A,M: num,N: num] :
( ( times_times @ A @ ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ M ) ) @ ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ N ) ) )
= ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ ( plus_plus @ num @ M @ N ) ) ) ) ) ).
% power_add_numeral
thf(fact_170_numeral__le__enat__iff,axiom,
! [M: num,N: nat] :
( ( ord_less_eq @ extended_enat @ ( numeral_numeral @ extended_enat @ M ) @ ( extended_enat2 @ N ) )
= ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ M ) @ N ) ) ).
% numeral_le_enat_iff
thf(fact_171_enat__less__induct,axiom,
! [P3: extended_enat > $o,N: extended_enat] :
( ! [N3: extended_enat] :
( ! [M3: extended_enat] :
( ( ord_less @ extended_enat @ M3 @ N3 )
=> ( P3 @ M3 ) )
=> ( P3 @ N3 ) )
=> ( P3 @ N ) ) ).
% enat_less_induct
thf(fact_172_mult__le__mono2,axiom,
! [I2: nat,J3: nat,K2: nat] :
( ( ord_less_eq @ nat @ I2 @ J3 )
=> ( ord_less_eq @ nat @ ( times_times @ nat @ K2 @ I2 ) @ ( times_times @ nat @ K2 @ J3 ) ) ) ).
% mult_le_mono2
thf(fact_173_mult__le__mono1,axiom,
! [I2: nat,J3: nat,K2: nat] :
( ( ord_less_eq @ nat @ I2 @ J3 )
=> ( ord_less_eq @ nat @ ( times_times @ nat @ I2 @ K2 ) @ ( times_times @ nat @ J3 @ K2 ) ) ) ).
% mult_le_mono1
thf(fact_174_mult__le__mono,axiom,
! [I2: nat,J3: nat,K2: nat,L: nat] :
( ( ord_less_eq @ nat @ I2 @ J3 )
=> ( ( ord_less_eq @ nat @ K2 @ L )
=> ( ord_less_eq @ nat @ ( times_times @ nat @ I2 @ K2 ) @ ( times_times @ nat @ J3 @ L ) ) ) ) ).
% mult_le_mono
thf(fact_175_le__square,axiom,
! [M: nat] : ( ord_less_eq @ nat @ M @ ( times_times @ nat @ M @ M ) ) ).
% le_square
thf(fact_176_le__cube,axiom,
! [M: nat] : ( ord_less_eq @ nat @ M @ ( times_times @ nat @ M @ ( times_times @ nat @ M @ M ) ) ) ).
% le_cube
thf(fact_177_mult_Oleft__commute,axiom,
! [A: $tType] :
( ( ab_semigroup_mult @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( times_times @ A @ B2 @ ( times_times @ A @ A2 @ C2 ) )
= ( times_times @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% mult.left_commute
thf(fact_178_mult_Ocommute,axiom,
! [A: $tType] :
( ( ab_semigroup_mult @ A )
=> ( ( times_times @ A )
= ( ^ [A5: A,B4: A] : ( times_times @ A @ B4 @ A5 ) ) ) ) ).
% mult.commute
thf(fact_179_mult_Oassoc,axiom,
! [A: $tType] :
( ( semigroup_mult @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( times_times @ A @ ( times_times @ A @ A2 @ B2 ) @ C2 )
= ( times_times @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% mult.assoc
thf(fact_180_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: $tType] :
( ( ab_semigroup_mult @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( times_times @ A @ ( times_times @ A @ A2 @ B2 ) @ C2 )
= ( times_times @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_181_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_182_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_183_le__num__One__iff,axiom,
! [X: num] :
( ( ord_less_eq @ num @ X @ one2 )
= ( X = one2 ) ) ).
% le_num_One_iff
thf(fact_184_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_185_power__commuting__commutes,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [X: A,Y2: A,N: nat] :
( ( ( times_times @ A @ X @ Y2 )
= ( times_times @ A @ Y2 @ X ) )
=> ( ( times_times @ A @ ( power_power @ A @ X @ N ) @ Y2 )
= ( times_times @ A @ Y2 @ ( power_power @ A @ X @ N ) ) ) ) ) ).
% power_commuting_commutes
thf(fact_186_power__mult__distrib,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A2: A,B2: A,N: nat] :
( ( power_power @ A @ ( times_times @ A @ A2 @ B2 ) @ N )
= ( times_times @ A @ ( power_power @ A @ A2 @ N ) @ ( power_power @ A @ B2 @ N ) ) ) ) ).
% power_mult_distrib
thf(fact_187_power__commutes,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A2: A,N: nat] :
( ( times_times @ A @ ( power_power @ A @ A2 @ N ) @ A2 )
= ( times_times @ A @ A2 @ ( power_power @ A @ A2 @ N ) ) ) ) ).
% power_commutes
thf(fact_188_power__mult,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A2: A,M: nat,N: nat] :
( ( power_power @ A @ A2 @ ( times_times @ nat @ M @ N ) )
= ( power_power @ A @ ( power_power @ A @ A2 @ M ) @ N ) ) ) ).
% power_mult
thf(fact_189_VEBT_Oexhaust,axiom,
! [Y2: vEBT_VEBT] :
( ! [X112: option @ ( product_prod @ nat @ nat ),X122: nat,X132: list @ vEBT_VEBT,X142: vEBT_VEBT] :
( Y2
!= ( vEBT_Node @ X112 @ X122 @ X132 @ X142 ) )
=> ~ ! [X212: $o,X222: $o] :
( Y2
!= ( vEBT_Leaf @ X212 @ X222 ) ) ) ).
% VEBT.exhaust
thf(fact_190_VEBT_Odistinct_I1_J,axiom,
! [X11: option @ ( product_prod @ nat @ nat ),X12: nat,X13: list @ vEBT_VEBT,X14: vEBT_VEBT,X21: $o,X22: $o] :
( ( vEBT_Node @ X11 @ X12 @ X13 @ X14 )
!= ( vEBT_Leaf @ X21 @ X22 ) ) ).
% VEBT.distinct(1)
thf(fact_191_less__mono__imp__le__mono,axiom,
! [F3: nat > nat,I2: nat,J3: nat] :
( ! [I3: nat,J2: nat] :
( ( ord_less @ nat @ I3 @ J2 )
=> ( ord_less @ nat @ ( F3 @ I3 ) @ ( F3 @ J2 ) ) )
=> ( ( ord_less_eq @ nat @ I2 @ J3 )
=> ( ord_less_eq @ nat @ ( F3 @ I2 ) @ ( F3 @ J3 ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_192_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_193_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_194_le__eq__less__or__eq,axiom,
( ( ord_less_eq @ nat )
= ( ^ [M2: nat,N2: nat] :
( ( ord_less @ nat @ M2 @ N2 )
| ( M2 = N2 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_195_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_196_nat__less__le,axiom,
( ( ord_less @ nat )
= ( ^ [M2: nat,N2: nat] :
( ( ord_less_eq @ nat @ M2 @ N2 )
& ( M2 != N2 ) ) ) ) ).
% nat_less_le
thf(fact_197_left__add__mult__distrib,axiom,
! [I2: nat,U: nat,J3: nat,K2: nat] :
( ( plus_plus @ nat @ ( times_times @ nat @ I2 @ U ) @ ( plus_plus @ nat @ ( times_times @ nat @ J3 @ U ) @ K2 ) )
= ( plus_plus @ nat @ ( times_times @ nat @ ( plus_plus @ nat @ I2 @ J3 ) @ U ) @ K2 ) ) ).
% left_add_mult_distrib
thf(fact_198_add__mult__distrib2,axiom,
! [K2: nat,M: nat,N: nat] :
( ( times_times @ nat @ K2 @ ( plus_plus @ nat @ M @ N ) )
= ( plus_plus @ nat @ ( times_times @ nat @ K2 @ M ) @ ( times_times @ nat @ K2 @ N ) ) ) ).
% add_mult_distrib2
thf(fact_199_add__mult__distrib,axiom,
! [M: nat,N: nat,K2: nat] :
( ( times_times @ nat @ ( plus_plus @ nat @ M @ N ) @ K2 )
= ( plus_plus @ nat @ ( times_times @ nat @ M @ K2 ) @ ( times_times @ nat @ N @ K2 ) ) ) ).
% add_mult_distrib
thf(fact_200_subset__code_I1_J,axiom,
! [A: $tType,Xs: list @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ B5 )
= ( ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
=> ( member @ A @ X4 @ B5 ) ) ) ) ).
% subset_code(1)
thf(fact_201_nat__le__iff__add,axiom,
( ( ord_less_eq @ nat )
= ( ^ [M2: nat,N2: nat] :
? [K3: nat] :
( N2
= ( plus_plus @ nat @ M2 @ K3 ) ) ) ) ).
% nat_le_iff_add
thf(fact_202_trans__le__add2,axiom,
! [I2: nat,J3: nat,M: nat] :
( ( ord_less_eq @ nat @ I2 @ J3 )
=> ( ord_less_eq @ nat @ I2 @ ( plus_plus @ nat @ M @ J3 ) ) ) ).
% trans_le_add2
thf(fact_203_trans__le__add1,axiom,
! [I2: nat,J3: nat,M: nat] :
( ( ord_less_eq @ nat @ I2 @ J3 )
=> ( ord_less_eq @ nat @ I2 @ ( plus_plus @ nat @ J3 @ M ) ) ) ).
% trans_le_add1
thf(fact_204_add__le__mono1,axiom,
! [I2: nat,J3: nat,K2: nat] :
( ( ord_less_eq @ nat @ I2 @ J3 )
=> ( ord_less_eq @ nat @ ( plus_plus @ nat @ I2 @ K2 ) @ ( plus_plus @ nat @ J3 @ K2 ) ) ) ).
% add_le_mono1
thf(fact_205_add__le__mono,axiom,
! [I2: nat,J3: nat,K2: nat,L: nat] :
( ( ord_less_eq @ nat @ I2 @ J3 )
=> ( ( ord_less_eq @ nat @ K2 @ L )
=> ( ord_less_eq @ nat @ ( plus_plus @ nat @ I2 @ K2 ) @ ( plus_plus @ nat @ J3 @ L ) ) ) ) ).
% add_le_mono
thf(fact_206_le__Suc__ex,axiom,
! [K2: nat,L: nat] :
( ( ord_less_eq @ nat @ K2 @ L )
=> ? [N3: nat] :
( L
= ( plus_plus @ nat @ K2 @ N3 ) ) ) ).
% le_Suc_ex
thf(fact_207_add__leD2,axiom,
! [M: nat,K2: nat,N: nat] :
( ( ord_less_eq @ nat @ ( plus_plus @ nat @ M @ K2 ) @ N )
=> ( ord_less_eq @ nat @ K2 @ N ) ) ).
% add_leD2
thf(fact_208_add__leD1,axiom,
! [M: nat,K2: nat,N: nat] :
( ( ord_less_eq @ nat @ ( plus_plus @ nat @ M @ K2 ) @ N )
=> ( ord_less_eq @ nat @ M @ N ) ) ).
% add_leD1
thf(fact_209_le__add2,axiom,
! [N: nat,M: nat] : ( ord_less_eq @ nat @ N @ ( plus_plus @ nat @ M @ N ) ) ).
% le_add2
thf(fact_210_le__add1,axiom,
! [N: nat,M: nat] : ( ord_less_eq @ nat @ N @ ( plus_plus @ nat @ N @ M ) ) ).
% le_add1
thf(fact_211_add__leE,axiom,
! [M: nat,K2: nat,N: nat] :
( ( ord_less_eq @ nat @ ( plus_plus @ nat @ M @ K2 ) @ N )
=> ~ ( ( ord_less_eq @ nat @ M @ N )
=> ~ ( ord_less_eq @ nat @ K2 @ N ) ) ) ).
% add_leE
thf(fact_212_div__mult2__eq,axiom,
! [M: nat,N: nat,Q: nat] :
( ( divide_divide @ nat @ M @ ( times_times @ nat @ N @ Q ) )
= ( divide_divide @ nat @ ( divide_divide @ nat @ M @ N ) @ Q ) ) ).
% div_mult2_eq
thf(fact_213_div__le__dividend,axiom,
! [M: nat,N: nat] : ( ord_less_eq @ nat @ ( divide_divide @ nat @ M @ N ) @ M ) ).
% div_le_dividend
thf(fact_214_div__le__mono,axiom,
! [M: nat,N: nat,K2: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ord_less_eq @ nat @ ( divide_divide @ nat @ M @ K2 ) @ ( divide_divide @ nat @ N @ K2 ) ) ) ).
% div_le_mono
thf(fact_215_mult__numeral__1__right,axiom,
! [A: $tType] :
( ( semiring_numeral @ A )
=> ! [A2: A] :
( ( times_times @ A @ A2 @ ( numeral_numeral @ A @ one2 ) )
= A2 ) ) ).
% mult_numeral_1_right
thf(fact_216_mult__numeral__1,axiom,
! [A: $tType] :
( ( semiring_numeral @ A )
=> ! [A2: A] :
( ( times_times @ A @ ( numeral_numeral @ A @ one2 ) @ A2 )
= A2 ) ) ).
% mult_numeral_1
thf(fact_217_VEBT__internal_Oelim__dead_Osimps_I1_J,axiom,
! [A2: $o,B2: $o,Uu: extended_enat] :
( ( vEBT_VEBT_elim_dead @ ( vEBT_Leaf @ A2 @ B2 ) @ Uu )
= ( vEBT_Leaf @ A2 @ B2 ) ) ).
% VEBT_internal.elim_dead.simps(1)
thf(fact_218_less__eq__enat__def,axiom,
( ( ord_less_eq @ extended_enat )
= ( ^ [M2: extended_enat] :
( extended_case_enat @ $o
@ ^ [N1: nat] :
( extended_case_enat @ $o
@ ^ [M1: nat] : ( ord_less_eq @ nat @ M1 @ N1 )
@ $false
@ M2 )
@ $true ) ) ) ).
% less_eq_enat_def
thf(fact_219_power__add,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A2: A,M: nat,N: nat] :
( ( power_power @ A @ A2 @ ( plus_plus @ nat @ M @ N ) )
= ( times_times @ A @ ( power_power @ A @ A2 @ M ) @ ( power_power @ A @ A2 @ N ) ) ) ) ).
% power_add
thf(fact_220_mono__nat__linear__lb,axiom,
! [F3: nat > nat,M: nat,K2: nat] :
( ! [M4: nat,N3: nat] :
( ( ord_less @ nat @ M4 @ N3 )
=> ( ord_less @ nat @ ( F3 @ M4 ) @ ( F3 @ N3 ) ) )
=> ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( F3 @ M ) @ K2 ) @ ( F3 @ ( plus_plus @ nat @ M @ K2 ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_221_less__mult__imp__div__less,axiom,
! [M: nat,I2: nat,N: nat] :
( ( ord_less @ nat @ M @ ( times_times @ nat @ I2 @ N ) )
=> ( ord_less @ nat @ ( divide_divide @ nat @ M @ N ) @ I2 ) ) ).
% less_mult_imp_div_less
thf(fact_222_set__take__subset,axiom,
! [A: $tType,N: nat,Xs: list @ A] : ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ ( take @ A @ N @ Xs ) ) @ ( set2 @ A @ Xs ) ) ).
% set_take_subset
thf(fact_223_left__add__twice,axiom,
! [A: $tType] :
( ( semiring_numeral @ A )
=> ! [A2: A,B2: A] :
( ( plus_plus @ A @ A2 @ ( plus_plus @ A @ A2 @ B2 ) )
= ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) @ B2 ) ) ) ).
% left_add_twice
thf(fact_224_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_225_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_226_add__right__imp__eq,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( ( plus_plus @ A @ B2 @ A2 )
= ( plus_plus @ A @ C2 @ A2 ) )
=> ( B2 = C2 ) ) ) ).
% add_right_imp_eq
thf(fact_227_add__left__imp__eq,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ( plus_plus @ A @ A2 @ B2 )
= ( plus_plus @ A @ A2 @ C2 ) )
=> ( B2 = C2 ) ) ) ).
% add_left_imp_eq
thf(fact_228_add_Oleft__commute,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( plus_plus @ A @ B2 @ ( plus_plus @ A @ A2 @ C2 ) )
= ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% add.left_commute
thf(fact_229_add_Ocommute,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A )
=> ( ( plus_plus @ A )
= ( ^ [A5: A,B4: A] : ( plus_plus @ A @ B4 @ A5 ) ) ) ) ).
% add.commute
thf(fact_230_add_Oright__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( ( plus_plus @ A @ B2 @ A2 )
= ( plus_plus @ A @ C2 @ A2 ) )
= ( B2 = C2 ) ) ) ).
% add.right_cancel
thf(fact_231_add_Oleft__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ( plus_plus @ A @ A2 @ B2 )
= ( plus_plus @ A @ A2 @ C2 ) )
= ( B2 = C2 ) ) ) ).
% add.left_cancel
thf(fact_232_add_Oassoc,axiom,
! [A: $tType] :
( ( semigroup_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 )
= ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% add.assoc
thf(fact_233_group__cancel_Oadd2,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [B5: A,K2: A,B2: A,A2: A] :
( ( B5
= ( plus_plus @ A @ K2 @ B2 ) )
=> ( ( plus_plus @ A @ A2 @ B5 )
= ( plus_plus @ A @ K2 @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ) ).
% group_cancel.add2
thf(fact_234_group__cancel_Oadd1,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A3: A,K2: A,A2: A,B2: A] :
( ( A3
= ( plus_plus @ A @ K2 @ A2 ) )
=> ( ( plus_plus @ A @ A3 @ B2 )
= ( plus_plus @ A @ K2 @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ) ).
% group_cancel.add1
thf(fact_235_add__mono__thms__linordered__semiring_I4_J,axiom,
! [A: $tType] :
( ( ordere6658533253407199908up_add @ A )
=> ! [I2: A,J3: A,K2: A,L: A] :
( ( ( I2 = J3 )
& ( K2 = L ) )
=> ( ( plus_plus @ A @ I2 @ K2 )
= ( plus_plus @ A @ J3 @ L ) ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_236_is__num__normalize_I1_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 )
= ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% is_num_normalize(1)
thf(fact_237_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 )
= ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_238_add__One__commute,axiom,
! [N: num] :
( ( plus_plus @ num @ one2 @ N )
= ( plus_plus @ num @ N @ one2 ) ) ).
% add_One_commute
thf(fact_239_power__numeral__even,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [Z3: A,W: num] :
( ( power_power @ A @ Z3 @ ( numeral_numeral @ nat @ ( bit0 @ W ) ) )
= ( times_times @ A @ ( power_power @ A @ Z3 @ ( numeral_numeral @ nat @ W ) ) @ ( power_power @ A @ Z3 @ ( numeral_numeral @ nat @ W ) ) ) ) ) ).
% power_numeral_even
thf(fact_240_iadd__le__enat__iff,axiom,
! [X: extended_enat,Y2: extended_enat,N: nat] :
( ( ord_less_eq @ extended_enat @ ( plus_plus @ extended_enat @ X @ Y2 ) @ ( extended_enat2 @ N ) )
= ( ? [Y6: nat,X7: nat] :
( ( X
= ( extended_enat2 @ X7 ) )
& ( Y2
= ( extended_enat2 @ Y6 ) )
& ( ord_less_eq @ nat @ ( plus_plus @ nat @ X7 @ Y6 ) @ N ) ) ) ) ).
% iadd_le_enat_iff
thf(fact_241_power2__eq__square,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A2: A] :
( ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( times_times @ A @ A2 @ A2 ) ) ) ).
% power2_eq_square
thf(fact_242_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_243_power__even__eq,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A2: A,N: nat] :
( ( power_power @ A @ A2 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
= ( power_power @ A @ ( power_power @ A @ A2 @ N ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% power_even_eq
thf(fact_244_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_245_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_246_self__le__ge2__pow,axiom,
! [K2: nat,M: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K2 )
=> ( ord_less_eq @ nat @ M @ ( power_power @ nat @ K2 @ M ) ) ) ).
% self_le_ge2_pow
thf(fact_247_nth__take__lemma,axiom,
! [A: $tType,K2: nat,Xs: list @ A,Ys: list @ A] :
( ( ord_less_eq @ nat @ K2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ord_less_eq @ nat @ K2 @ ( size_size @ ( list @ A ) @ Ys ) )
=> ( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ K2 )
=> ( ( nth @ A @ Xs @ I3 )
= ( nth @ A @ Ys @ I3 ) ) )
=> ( ( take @ A @ K2 @ Xs )
= ( take @ A @ K2 @ Ys ) ) ) ) ) ).
% nth_take_lemma
thf(fact_248_power2__sum,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [X: A,Y2: A] :
( ( power_power @ A @ ( plus_plus @ A @ X @ Y2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( plus_plus @ A @ ( plus_plus @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( times_times @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X ) @ Y2 ) ) ) ) ).
% power2_sum
thf(fact_249_add__le__imp__le__right,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) )
=> ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% add_le_imp_le_right
thf(fact_250_add__le__imp__le__left,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ C2 @ A2 ) @ ( plus_plus @ A @ C2 @ B2 ) )
=> ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% add_le_imp_le_left
thf(fact_251_le__iff__add,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A5: A,B4: A] :
? [C3: A] :
( B4
= ( plus_plus @ A @ A5 @ C3 ) ) ) ) ) ).
% le_iff_add
thf(fact_252_add__right__mono,axiom,
! [A: $tType] :
( ( ordere6658533253407199908up_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% add_right_mono
thf(fact_253_less__eqE,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ~ ! [C4: A] :
( B2
!= ( plus_plus @ A @ A2 @ C4 ) ) ) ) ).
% less_eqE
thf(fact_254_add__left__mono,axiom,
! [A: $tType] :
( ( ordere6658533253407199908up_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ C2 @ A2 ) @ ( plus_plus @ A @ C2 @ B2 ) ) ) ) ).
% add_left_mono
thf(fact_255_add__mono,axiom,
! [A: $tType] :
( ( ordere6658533253407199908up_add @ A )
=> ! [A2: A,B2: A,C2: A,D2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ D2 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ D2 ) ) ) ) ) ).
% add_mono
thf(fact_256_add__mono__thms__linordered__semiring_I1_J,axiom,
! [A: $tType] :
( ( ordere6658533253407199908up_add @ A )
=> ! [I2: A,J3: A,K2: A,L: A] :
( ( ( ord_less_eq @ A @ I2 @ J3 )
& ( ord_less_eq @ A @ K2 @ L ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ I2 @ K2 ) @ ( plus_plus @ A @ J3 @ L ) ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_257_add__mono__thms__linordered__semiring_I2_J,axiom,
! [A: $tType] :
( ( ordere6658533253407199908up_add @ A )
=> ! [I2: A,J3: A,K2: A,L: A] :
( ( ( I2 = J3 )
& ( ord_less_eq @ A @ K2 @ L ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ I2 @ K2 ) @ ( plus_plus @ A @ J3 @ L ) ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_258_add__mono__thms__linordered__semiring_I3_J,axiom,
! [A: $tType] :
( ( ordere6658533253407199908up_add @ A )
=> ! [I2: A,J3: A,K2: A,L: A] :
( ( ( ord_less_eq @ A @ I2 @ J3 )
& ( K2 = L ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ I2 @ K2 ) @ ( plus_plus @ A @ J3 @ L ) ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_259_add__less__imp__less__right,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) )
=> ( ord_less @ A @ A2 @ B2 ) ) ) ).
% add_less_imp_less_right
thf(fact_260_add__less__imp__less__left,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ C2 @ A2 ) @ ( plus_plus @ A @ C2 @ B2 ) )
=> ( ord_less @ A @ A2 @ B2 ) ) ) ).
% add_less_imp_less_left
thf(fact_261_add__strict__right__mono,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ord_less @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% add_strict_right_mono
thf(fact_262_add__strict__left__mono,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ord_less @ A @ ( plus_plus @ A @ C2 @ A2 ) @ ( plus_plus @ A @ C2 @ B2 ) ) ) ) ).
% add_strict_left_mono
thf(fact_263_add__strict__mono,axiom,
! [A: $tType] :
( ( strict9044650504122735259up_add @ A )
=> ! [A2: A,B2: A,C2: A,D2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ C2 @ D2 )
=> ( ord_less @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ D2 ) ) ) ) ) ).
% add_strict_mono
thf(fact_264_add__mono__thms__linordered__field_I1_J,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [I2: A,J3: A,K2: A,L: A] :
( ( ( ord_less @ A @ I2 @ J3 )
& ( K2 = L ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I2 @ K2 ) @ ( plus_plus @ A @ J3 @ L ) ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_265_add__mono__thms__linordered__field_I2_J,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [I2: A,J3: A,K2: A,L: A] :
( ( ( I2 = J3 )
& ( ord_less @ A @ K2 @ L ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I2 @ K2 ) @ ( plus_plus @ A @ J3 @ L ) ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_266_add__mono__thms__linordered__field_I5_J,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [I2: A,J3: A,K2: A,L: A] :
( ( ( ord_less @ A @ I2 @ J3 )
& ( ord_less @ A @ K2 @ L ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I2 @ K2 ) @ ( plus_plus @ A @ J3 @ L ) ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_267_add__less__le__mono,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [A2: A,B2: A,C2: A,D2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ D2 )
=> ( ord_less @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ D2 ) ) ) ) ) ).
% add_less_le_mono
thf(fact_268_add__le__less__mono,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [A2: A,B2: A,C2: A,D2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ C2 @ D2 )
=> ( ord_less @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ D2 ) ) ) ) ) ).
% add_le_less_mono
thf(fact_269_add__mono__thms__linordered__field_I3_J,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [I2: A,J3: A,K2: A,L: A] :
( ( ( ord_less @ A @ I2 @ J3 )
& ( ord_less_eq @ A @ K2 @ L ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I2 @ K2 ) @ ( plus_plus @ A @ J3 @ L ) ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_270_add__mono__thms__linordered__field_I4_J,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [I2: A,J3: A,K2: A,L: A] :
( ( ( ord_less_eq @ A @ I2 @ J3 )
& ( ord_less @ A @ K2 @ L ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I2 @ K2 ) @ ( plus_plus @ A @ J3 @ L ) ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_271_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_272_divide__numeral__1,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A2: A] :
( ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ one2 ) )
= A2 ) ) ).
% divide_numeral_1
thf(fact_273_in__children__def,axiom,
( vEBT_V5917875025757280293ildren
= ( ^ [N2: nat,TreeList3: list @ vEBT_VEBT,X4: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ X4 @ N2 ) ) @ ( vEBT_VEBT_low @ X4 @ N2 ) ) ) ) ).
% in_children_def
thf(fact_274_sum__squares__bound,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y2: A] : ( ord_less_eq @ A @ ( times_times @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X ) @ Y2 ) @ ( plus_plus @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% sum_squares_bound
thf(fact_275_set__n__deg__not__0,axiom,
! [TreeList: list @ vEBT_VEBT,N: nat,M: nat] :
( ! [X5: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X5 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ( vEBT_invar_vebt @ X5 @ N ) )
=> ( ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) )
=> ( ord_less_eq @ nat @ ( one_one @ nat ) @ N ) ) ) ).
% set_n_deg_not_0
thf(fact_276_times__divide__eq__right,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( times_times @ A @ A2 @ ( divide_divide @ A @ B2 @ C2 ) )
= ( divide_divide @ A @ ( times_times @ A @ A2 @ B2 ) @ C2 ) ) ) ).
% times_divide_eq_right
thf(fact_277_divide__divide__eq__right,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( divide_divide @ A @ A2 @ ( divide_divide @ A @ B2 @ C2 ) )
= ( divide_divide @ A @ ( times_times @ A @ A2 @ C2 ) @ B2 ) ) ) ).
% divide_divide_eq_right
thf(fact_278_divide__divide__eq__left,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( divide_divide @ A @ ( divide_divide @ A @ A2 @ B2 ) @ C2 )
= ( divide_divide @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% divide_divide_eq_left
thf(fact_279_times__divide__eq__left,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [B2: A,C2: A,A2: A] :
( ( times_times @ A @ ( divide_divide @ A @ B2 @ C2 ) @ A2 )
= ( divide_divide @ A @ ( times_times @ A @ B2 @ A2 ) @ C2 ) ) ) ).
% times_divide_eq_left
thf(fact_280_low__def,axiom,
( vEBT_VEBT_low
= ( ^ [X4: nat,N2: nat] : ( modulo_modulo @ nat @ X4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ).
% low_def
thf(fact_281_set__vebt__def,axiom,
( vEBT_set_vebt
= ( ^ [T3: vEBT_VEBT] : ( collect @ nat @ ( vEBT_V8194947554948674370ptions @ T3 ) ) ) ) ).
% set_vebt_def
thf(fact_282_invar__vebt_Ointros_I2_J,axiom,
! [TreeList: list @ vEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
( ! [X5: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X5 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ( vEBT_invar_vebt @ X5 @ N ) )
=> ( ( vEBT_invar_vebt @ Summary @ M )
=> ( ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) )
=> ( ( M = N )
=> ( ( Deg
= ( plus_plus @ nat @ N @ M ) )
=> ( ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_12 )
=> ( ! [X5: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X5 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_12 ) )
=> ( vEBT_invar_vebt @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg @ TreeList @ Summary ) @ Deg ) ) ) ) ) ) ) ) ).
% invar_vebt.intros(2)
thf(fact_283_power__numeral,axiom,
! [A: $tType] :
( ( semiring_numeral @ A )
=> ! [K2: num,L: num] :
( ( power_power @ A @ ( numeral_numeral @ A @ K2 ) @ ( numeral_numeral @ nat @ L ) )
= ( numeral_numeral @ A @ ( pow @ K2 @ L ) ) ) ) ).
% power_numeral
thf(fact_284_verit__eq__simplify_I8_J,axiom,
! [X23: num,Y23: num] :
( ( ( bit0 @ X23 )
= ( bit0 @ Y23 ) )
= ( X23 = Y23 ) ) ).
% verit_eq_simplify(8)
thf(fact_285_mod__mod__trivial,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A2: A,B2: A] :
( ( modulo_modulo @ A @ ( modulo_modulo @ A @ A2 @ B2 ) @ B2 )
= ( modulo_modulo @ A @ A2 @ B2 ) ) ) ).
% mod_mod_trivial
thf(fact_286_times__enat__simps_I2_J,axiom,
( ( times_times @ extended_enat @ ( extend4730790105801354508finity @ extended_enat ) @ ( extend4730790105801354508finity @ extended_enat ) )
= ( extend4730790105801354508finity @ extended_enat ) ) ).
% times_enat_simps(2)
thf(fact_287_mult_Oright__neutral,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A2: A] :
( ( times_times @ A @ A2 @ ( one_one @ A ) )
= A2 ) ) ).
% mult.right_neutral
thf(fact_288_mult__1,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A2: A] :
( ( times_times @ A @ ( one_one @ A ) @ A2 )
= A2 ) ) ).
% mult_1
thf(fact_289_bits__div__by__1,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A2: A] :
( ( divide_divide @ A @ A2 @ ( one_one @ A ) )
= A2 ) ) ).
% bits_div_by_1
thf(fact_290_power__one,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [N: nat] :
( ( power_power @ A @ ( one_one @ A ) @ N )
= ( one_one @ A ) ) ) ).
% power_one
thf(fact_291_mod__add__self1,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [B2: A,A2: A] :
( ( modulo_modulo @ A @ ( plus_plus @ A @ B2 @ A2 ) @ B2 )
= ( modulo_modulo @ A @ A2 @ B2 ) ) ) ).
% mod_add_self1
thf(fact_292_mod__add__self2,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A2: A,B2: A] :
( ( modulo_modulo @ A @ ( plus_plus @ A @ A2 @ B2 ) @ B2 )
= ( modulo_modulo @ A @ A2 @ B2 ) ) ) ).
% mod_add_self2
thf(fact_293_power__one__right,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A2: A] :
( ( power_power @ A @ A2 @ ( one_one @ nat ) )
= A2 ) ) ).
% power_one_right
thf(fact_294_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_295_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_296_mod__less,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ N )
=> ( ( modulo_modulo @ nat @ M @ N )
= M ) ) ).
% mod_less
thf(fact_297_times__enat__simps_I1_J,axiom,
! [M: nat,N: nat] :
( ( times_times @ extended_enat @ ( extended_enat2 @ M ) @ ( extended_enat2 @ N ) )
= ( extended_enat2 @ ( times_times @ nat @ M @ N ) ) ) ).
% times_enat_simps(1)
thf(fact_298_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_299_semiring__norm_I12_J,axiom,
! [N: num] :
( ( times_times @ num @ one2 @ N )
= N ) ).
% semiring_norm(12)
thf(fact_300_semiring__norm_I11_J,axiom,
! [M: num] :
( ( times_times @ num @ M @ one2 )
= M ) ).
% semiring_norm(11)
thf(fact_301_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_302_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_303_power__inject__exp,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,M: nat,N: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ A2 )
=> ( ( ( power_power @ A @ A2 @ M )
= ( power_power @ A @ A2 @ N ) )
= ( M = N ) ) ) ) ).
% power_inject_exp
thf(fact_304_mod__mult__self4,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [B2: A,C2: A,A2: A] :
( ( modulo_modulo @ A @ ( plus_plus @ A @ ( times_times @ A @ B2 @ C2 ) @ A2 ) @ B2 )
= ( modulo_modulo @ A @ A2 @ B2 ) ) ) ).
% mod_mult_self4
thf(fact_305_mod__mult__self3,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [C2: A,B2: A,A2: A] :
( ( modulo_modulo @ A @ ( plus_plus @ A @ ( times_times @ A @ C2 @ B2 ) @ A2 ) @ B2 )
= ( modulo_modulo @ A @ A2 @ B2 ) ) ) ).
% mod_mult_self3
thf(fact_306_mod__mult__self2,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( modulo_modulo @ A @ ( plus_plus @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) ) @ B2 )
= ( modulo_modulo @ A @ A2 @ B2 ) ) ) ).
% mod_mult_self2
thf(fact_307_mod__mult__self1,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( modulo_modulo @ A @ ( plus_plus @ A @ A2 @ ( times_times @ A @ C2 @ B2 ) ) @ B2 )
= ( modulo_modulo @ A @ A2 @ B2 ) ) ) ).
% mod_mult_self1
thf(fact_308_num__double,axiom,
! [N: num] :
( ( times_times @ num @ ( bit0 @ one2 ) @ N )
= ( bit0 @ N ) ) ).
% num_double
thf(fact_309_power__mult__numeral,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A2: A,M: num,N: num] :
( ( power_power @ A @ ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ M ) ) @ ( numeral_numeral @ nat @ N ) )
= ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ ( times_times @ num @ M @ N ) ) ) ) ) ).
% power_mult_numeral
thf(fact_310_power__strict__increasing__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [B2: A,X: nat,Y2: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ B2 )
=> ( ( ord_less @ A @ ( power_power @ A @ B2 @ X ) @ ( power_power @ A @ B2 @ Y2 ) )
= ( ord_less @ nat @ X @ Y2 ) ) ) ) ).
% power_strict_increasing_iff
thf(fact_311_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_312_power__increasing__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [B2: A,X: nat,Y2: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ B2 )
=> ( ( ord_less_eq @ A @ ( power_power @ A @ B2 @ X ) @ ( power_power @ A @ B2 @ Y2 ) )
= ( ord_less_eq @ nat @ X @ Y2 ) ) ) ) ).
% power_increasing_iff
thf(fact_313_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_314_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_315_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_316_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_317_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_318_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_319_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_320_Nat_Oex__has__greatest__nat,axiom,
! [P3: nat > $o,K2: nat,B2: nat] :
( ( P3 @ K2 )
=> ( ! [Y7: nat] :
( ( P3 @ Y7 )
=> ( ord_less_eq @ nat @ Y7 @ B2 ) )
=> ? [X5: nat] :
( ( P3 @ X5 )
& ! [Y4: nat] :
( ( P3 @ Y4 )
=> ( ord_less_eq @ nat @ Y4 @ X5 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_321_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_322_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_323_eq__imp__le,axiom,
! [M: nat,N: nat] :
( ( M = N )
=> ( ord_less_eq @ nat @ M @ N ) ) ).
% eq_imp_le
thf(fact_324_le__trans,axiom,
! [I2: nat,J3: nat,K2: nat] :
( ( ord_less_eq @ nat @ I2 @ J3 )
=> ( ( ord_less_eq @ nat @ J3 @ K2 )
=> ( ord_less_eq @ nat @ I2 @ K2 ) ) ) ).
% le_trans
thf(fact_325_le__refl,axiom,
! [N: nat] : ( ord_less_eq @ nat @ N @ N ) ).
% le_refl
thf(fact_326_one__reorient,axiom,
! [A: $tType] :
( ( one @ A )
=> ! [X: A] :
( ( ( one_one @ A )
= X )
= ( X
= ( one_one @ A ) ) ) ) ).
% one_reorient
thf(fact_327_mod__mult__right__eq,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( modulo_modulo @ A @ ( times_times @ A @ A2 @ ( modulo_modulo @ A @ B2 @ C2 ) ) @ C2 )
= ( modulo_modulo @ A @ ( times_times @ A @ A2 @ B2 ) @ C2 ) ) ) ).
% mod_mult_right_eq
thf(fact_328_mod__mult__left__eq,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( modulo_modulo @ A @ ( times_times @ A @ ( modulo_modulo @ A @ A2 @ C2 ) @ B2 ) @ C2 )
= ( modulo_modulo @ A @ ( times_times @ A @ A2 @ B2 ) @ C2 ) ) ) ).
% mod_mult_left_eq
thf(fact_329_mult__mod__right,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( times_times @ A @ C2 @ ( modulo_modulo @ A @ A2 @ B2 ) )
= ( modulo_modulo @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) ) ) ) ).
% mult_mod_right
thf(fact_330_mod__mult__mult2,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( modulo_modulo @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) )
= ( times_times @ A @ ( modulo_modulo @ A @ A2 @ B2 ) @ C2 ) ) ) ).
% mod_mult_mult2
thf(fact_331_mod__mult__cong,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A2: A,C2: A,A6: A,B2: A,B6: A] :
( ( ( modulo_modulo @ A @ A2 @ C2 )
= ( modulo_modulo @ A @ A6 @ C2 ) )
=> ( ( ( modulo_modulo @ A @ B2 @ C2 )
= ( modulo_modulo @ A @ B6 @ C2 ) )
=> ( ( modulo_modulo @ A @ ( times_times @ A @ A2 @ B2 ) @ C2 )
= ( modulo_modulo @ A @ ( times_times @ A @ A6 @ B6 ) @ C2 ) ) ) ) ) ).
% mod_mult_cong
thf(fact_332_mod__mult__eq,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( modulo_modulo @ A @ ( times_times @ A @ ( modulo_modulo @ A @ A2 @ C2 ) @ ( modulo_modulo @ A @ B2 @ C2 ) ) @ C2 )
= ( modulo_modulo @ A @ ( times_times @ A @ A2 @ B2 ) @ C2 ) ) ) ).
% mod_mult_eq
thf(fact_333_mod__add__eq,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( modulo_modulo @ A @ ( plus_plus @ A @ ( modulo_modulo @ A @ A2 @ C2 ) @ ( modulo_modulo @ A @ B2 @ C2 ) ) @ C2 )
= ( modulo_modulo @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 ) ) ) ).
% mod_add_eq
thf(fact_334_mod__add__cong,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A2: A,C2: A,A6: A,B2: A,B6: A] :
( ( ( modulo_modulo @ A @ A2 @ C2 )
= ( modulo_modulo @ A @ A6 @ C2 ) )
=> ( ( ( modulo_modulo @ A @ B2 @ C2 )
= ( modulo_modulo @ A @ B6 @ C2 ) )
=> ( ( modulo_modulo @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 )
= ( modulo_modulo @ A @ ( plus_plus @ A @ A6 @ B6 ) @ C2 ) ) ) ) ) ).
% mod_add_cong
thf(fact_335_mod__add__left__eq,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( modulo_modulo @ A @ ( plus_plus @ A @ ( modulo_modulo @ A @ A2 @ C2 ) @ B2 ) @ C2 )
= ( modulo_modulo @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 ) ) ) ).
% mod_add_left_eq
thf(fact_336_mod__add__right__eq,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( modulo_modulo @ A @ ( plus_plus @ A @ A2 @ ( modulo_modulo @ A @ B2 @ C2 ) ) @ C2 )
= ( modulo_modulo @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 ) ) ) ).
% mod_add_right_eq
thf(fact_337_power__mod,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A2: A,B2: A,N: nat] :
( ( modulo_modulo @ A @ ( power_power @ A @ ( modulo_modulo @ A @ A2 @ B2 ) @ N ) @ B2 )
= ( modulo_modulo @ A @ ( power_power @ A @ A2 @ N ) @ B2 ) ) ) ).
% power_mod
thf(fact_338_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_339_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_340_comm__monoid__mult__class_Omult__1,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A2: A] :
( ( times_times @ A @ ( one_one @ A ) @ A2 )
= A2 ) ) ).
% comm_monoid_mult_class.mult_1
thf(fact_341_mult_Ocomm__neutral,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A2: A] :
( ( times_times @ A @ A2 @ ( one_one @ A ) )
= A2 ) ) ).
% mult.comm_neutral
thf(fact_342_nat__mult__1__right,axiom,
! [N: nat] :
( ( times_times @ nat @ N @ ( one_one @ nat ) )
= N ) ).
% nat_mult_1_right
thf(fact_343_nat__mult__1,axiom,
! [N: nat] :
( ( times_times @ nat @ ( one_one @ nat ) @ N )
= N ) ).
% nat_mult_1
thf(fact_344_mod__eqE,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( ( modulo_modulo @ A @ A2 @ C2 )
= ( modulo_modulo @ A @ B2 @ C2 ) )
=> ~ ! [D3: A] :
( B2
!= ( plus_plus @ A @ A2 @ ( times_times @ A @ C2 @ D3 ) ) ) ) ) ).
% mod_eqE
thf(fact_345_div__add1__eq,axiom,
! [A: $tType] :
( ( euclid3128863361964157862miring @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( divide_divide @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 )
= ( plus_plus @ A @ ( plus_plus @ A @ ( divide_divide @ A @ A2 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) ) @ ( divide_divide @ A @ ( plus_plus @ A @ ( modulo_modulo @ A @ A2 @ C2 ) @ ( modulo_modulo @ A @ B2 @ C2 ) ) @ C2 ) ) ) ) ).
% div_add1_eq
thf(fact_346_four__x__squared,axiom,
! [X: real] :
( ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( power_power @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% four_x_squared
thf(fact_347_L2__set__mult__ineq__lemma,axiom,
! [A2: real,C2: real,B2: real,D2: real] : ( ord_less_eq @ real @ ( times_times @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( times_times @ real @ A2 @ C2 ) ) @ ( times_times @ real @ B2 @ D2 ) ) @ ( plus_plus @ real @ ( times_times @ real @ ( power_power @ real @ A2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ D2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( times_times @ real @ ( power_power @ real @ B2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ C2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% L2_set_mult_ineq_lemma
thf(fact_348_div__mod__decomp,axiom,
! [A3: nat,N: nat] :
( A3
= ( plus_plus @ nat @ ( times_times @ nat @ ( divide_divide @ nat @ A3 @ N ) @ N ) @ ( modulo_modulo @ nat @ A3 @ N ) ) ) ).
% div_mod_decomp
thf(fact_349_div__mult2__numeral__eq,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [A2: A,K2: num,L: num] :
( ( divide_divide @ A @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ K2 ) ) @ ( numeral_numeral @ A @ L ) )
= ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( times_times @ num @ K2 @ L ) ) ) ) ) ).
% div_mult2_numeral_eq
thf(fact_350_gt__half__sum,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ord_less @ A @ ( divide_divide @ A @ ( plus_plus @ A @ A2 @ B2 ) @ ( plus_plus @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) @ B2 ) ) ) ).
% gt_half_sum
thf(fact_351_less__half__sum,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ord_less @ A @ A2 @ ( divide_divide @ A @ ( plus_plus @ A @ A2 @ B2 ) @ ( plus_plus @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) ) ) ) ).
% less_half_sum
thf(fact_352_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_353_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_354_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_355_numeral__One,axiom,
! [A: $tType] :
( ( numeral @ A )
=> ( ( numeral_numeral @ A @ one2 )
= ( one_one @ A ) ) ) ).
% numeral_One
thf(fact_356_one__le__power,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,N: nat] :
( ( ord_less_eq @ A @ ( one_one @ A ) @ A2 )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ ( power_power @ A @ A2 @ N ) ) ) ) ).
% one_le_power
thf(fact_357_power__increasing,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N: nat,N4: nat,A2: A] :
( ( ord_less_eq @ nat @ N @ N4 )
=> ( ( ord_less_eq @ A @ ( one_one @ A ) @ A2 )
=> ( ord_less_eq @ A @ ( power_power @ A @ A2 @ N ) @ ( power_power @ A @ A2 @ N4 ) ) ) ) ) ).
% power_increasing
thf(fact_358_left__right__inverse__power,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [X: A,Y2: A,N: nat] :
( ( ( times_times @ A @ X @ Y2 )
= ( one_one @ A ) )
=> ( ( times_times @ A @ ( power_power @ A @ X @ N ) @ ( power_power @ A @ Y2 @ N ) )
= ( one_one @ A ) ) ) ) ).
% left_right_inverse_power
thf(fact_359_power__one__over,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A,N: nat] :
( ( power_power @ A @ ( divide_divide @ A @ ( one_one @ A ) @ A2 ) @ N )
= ( divide_divide @ A @ ( one_one @ A ) @ ( power_power @ A @ A2 @ N ) ) ) ) ).
% power_one_over
thf(fact_360_numerals_I1_J,axiom,
( ( numeral_numeral @ nat @ one2 )
= ( one_one @ nat ) ) ).
% numerals(1)
thf(fact_361_pow_Osimps_I1_J,axiom,
! [X: num] :
( ( pow @ X @ one2 )
= X ) ).
% pow.simps(1)
thf(fact_362_VEBT__internal_Ovalid_H_Osimps_I1_J,axiom,
! [Uu: $o,Uv: $o,D2: nat] :
( ( vEBT_VEBT_valid @ ( vEBT_Leaf @ Uu @ Uv ) @ D2 )
= ( D2
= ( one_one @ nat ) ) ) ).
% VEBT_internal.valid'.simps(1)
thf(fact_363_div__mult1__eq,axiom,
! [A: $tType] :
( ( euclid3128863361964157862miring @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( divide_divide @ A @ ( times_times @ A @ A2 @ B2 ) @ C2 )
= ( plus_plus @ A @ ( times_times @ A @ A2 @ ( divide_divide @ A @ B2 @ C2 ) ) @ ( divide_divide @ A @ ( times_times @ A @ A2 @ ( modulo_modulo @ A @ B2 @ C2 ) ) @ C2 ) ) ) ) ).
% div_mult1_eq
thf(fact_364_mod__mult2__eq,axiom,
! [M: nat,N: nat,Q: nat] :
( ( modulo_modulo @ nat @ M @ ( times_times @ nat @ N @ Q ) )
= ( plus_plus @ nat @ ( times_times @ nat @ N @ ( modulo_modulo @ nat @ ( divide_divide @ nat @ M @ N ) @ Q ) ) @ ( modulo_modulo @ nat @ M @ N ) ) ) ).
% mod_mult2_eq
thf(fact_365_verit__comp__simplify1_I2_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% verit_comp_simplify1(2)
thf(fact_366_verit__la__disequality,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A] :
( ( A2 = B2 )
| ~ ( ord_less_eq @ A @ A2 @ B2 )
| ~ ( ord_less_eq @ A @ B2 @ A2 ) ) ) ).
% verit_la_disequality
thf(fact_367_verit__comp__simplify1_I1_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A] :
~ ( ord_less @ A @ A2 @ A2 ) ) ).
% verit_comp_simplify1(1)
thf(fact_368_linordered__field__no__lb,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X2: A] :
? [Y7: A] : ( ord_less @ A @ Y7 @ X2 ) ) ).
% linordered_field_no_lb
thf(fact_369_linordered__field__no__ub,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X2: A] :
? [X_12: A] : ( ord_less @ A @ X2 @ X_12 ) ) ).
% linordered_field_no_ub
thf(fact_370_power__le__imp__le__exp,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,M: nat,N: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ ( power_power @ A @ A2 @ M ) @ ( power_power @ A @ A2 @ N ) )
=> ( ord_less_eq @ nat @ M @ N ) ) ) ) ).
% power_le_imp_le_exp
thf(fact_371_power__gt1__lemma,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,N: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ A2 )
=> ( ord_less @ A @ ( one_one @ A ) @ ( times_times @ A @ A2 @ ( power_power @ A @ A2 @ N ) ) ) ) ) ).
% power_gt1_lemma
thf(fact_372_power__less__power__Suc,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,N: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ A2 )
=> ( ord_less @ A @ ( power_power @ A @ A2 @ N ) @ ( times_times @ A @ A2 @ ( power_power @ A @ A2 @ N ) ) ) ) ) ).
% power_less_power_Suc
thf(fact_373_power__less__imp__less__exp,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,M: nat,N: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ A2 )
=> ( ( ord_less @ A @ ( power_power @ A @ A2 @ M ) @ ( power_power @ A @ A2 @ N ) )
=> ( ord_less @ nat @ M @ N ) ) ) ) ).
% power_less_imp_less_exp
thf(fact_374_power__strict__increasing,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N: nat,N4: nat,A2: A] :
( ( ord_less @ nat @ N @ N4 )
=> ( ( ord_less @ A @ ( one_one @ A ) @ A2 )
=> ( ord_less @ A @ ( power_power @ A @ A2 @ N ) @ ( power_power @ A @ A2 @ N4 ) ) ) ) ) ).
% power_strict_increasing
thf(fact_375_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_376_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_377_div__exp__mod__exp__eq,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A2: A,N: nat,M: nat] :
( ( modulo_modulo @ A @ ( divide_divide @ A @ A2 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) )
= ( divide_divide @ A @ ( modulo_modulo @ A @ A2 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( plus_plus @ nat @ N @ M ) ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% div_exp_mod_exp_eq
thf(fact_378_ex__power__ivl1,axiom,
! [B2: nat,K2: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ B2 )
=> ( ( ord_less_eq @ nat @ ( one_one @ nat ) @ K2 )
=> ? [N3: nat] :
( ( ord_less_eq @ nat @ ( power_power @ nat @ B2 @ N3 ) @ K2 )
& ( ord_less @ nat @ K2 @ ( power_power @ nat @ B2 @ ( plus_plus @ nat @ N3 @ ( one_one @ nat ) ) ) ) ) ) ) ).
% ex_power_ivl1
thf(fact_379_ex__power__ivl2,axiom,
! [B2: nat,K2: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ B2 )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K2 )
=> ? [N3: nat] :
( ( ord_less @ nat @ ( power_power @ nat @ B2 @ N3 ) @ K2 )
& ( ord_less_eq @ nat @ K2 @ ( power_power @ nat @ B2 @ ( plus_plus @ nat @ N3 @ ( one_one @ nat ) ) ) ) ) ) ) ).
% ex_power_ivl2
thf(fact_380_verit__comp__simplify1_I3_J,axiom,
! [B: $tType] :
( ( linorder @ B )
=> ! [B6: B,A6: B] :
( ( ~ ( ord_less_eq @ B @ B6 @ A6 ) )
= ( ord_less @ B @ A6 @ B6 ) ) ) ).
% verit_comp_simplify1(3)
thf(fact_381_verit__eq__simplify_I10_J,axiom,
! [X23: num] :
( one2
!= ( bit0 @ X23 ) ) ).
% verit_eq_simplify(10)
thf(fact_382_times__divide__times__eq,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [X: A,Y2: A,Z3: A,W: A] :
( ( times_times @ A @ ( divide_divide @ A @ X @ Y2 ) @ ( divide_divide @ A @ Z3 @ W ) )
= ( divide_divide @ A @ ( times_times @ A @ X @ Z3 ) @ ( times_times @ A @ Y2 @ W ) ) ) ) ).
% times_divide_times_eq
thf(fact_383_divide__divide__times__eq,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [X: A,Y2: A,Z3: A,W: A] :
( ( divide_divide @ A @ ( divide_divide @ A @ X @ Y2 ) @ ( divide_divide @ A @ Z3 @ W ) )
= ( divide_divide @ A @ ( times_times @ A @ X @ W ) @ ( times_times @ A @ Y2 @ Z3 ) ) ) ) ).
% divide_divide_times_eq
thf(fact_384_divide__divide__eq__left_H,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( divide_divide @ A @ ( divide_divide @ A @ A2 @ B2 ) @ C2 )
= ( divide_divide @ A @ A2 @ ( times_times @ A @ C2 @ B2 ) ) ) ) ).
% divide_divide_eq_left'
thf(fact_385_add__divide__distrib,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( divide_divide @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 )
= ( plus_plus @ A @ ( divide_divide @ A @ A2 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) ) ) ) ).
% add_divide_distrib
thf(fact_386_div__by__1,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [A2: A] :
( ( divide_divide @ A @ A2 @ ( one_one @ A ) )
= A2 ) ) ).
% div_by_1
thf(fact_387_mod__eq__nat1E,axiom,
! [M: nat,Q: nat,N: nat] :
( ( ( modulo_modulo @ nat @ M @ Q )
= ( modulo_modulo @ nat @ N @ Q ) )
=> ( ( ord_less_eq @ nat @ N @ M )
=> ~ ! [S2: nat] :
( M
!= ( plus_plus @ nat @ N @ ( times_times @ nat @ Q @ S2 ) ) ) ) ) ).
% mod_eq_nat1E
thf(fact_388_mod__eq__nat2E,axiom,
! [M: nat,Q: nat,N: nat] :
( ( ( modulo_modulo @ nat @ M @ Q )
= ( modulo_modulo @ nat @ N @ Q ) )
=> ( ( ord_less_eq @ nat @ M @ N )
=> ~ ! [S2: nat] :
( N
!= ( plus_plus @ nat @ M @ ( times_times @ nat @ Q @ S2 ) ) ) ) ) ).
% mod_eq_nat2E
thf(fact_389_nat__mod__eq__lemma,axiom,
! [X: nat,N: nat,Y2: nat] :
( ( ( modulo_modulo @ nat @ X @ N )
= ( modulo_modulo @ nat @ Y2 @ N ) )
=> ( ( ord_less_eq @ nat @ Y2 @ X )
=> ? [Q3: nat] :
( X
= ( plus_plus @ nat @ Y2 @ ( times_times @ nat @ N @ Q3 ) ) ) ) ) ).
% nat_mod_eq_lemma
thf(fact_390_cancel__div__mod__rules_I2_J,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ ( times_times @ A @ B2 @ ( divide_divide @ A @ A2 @ B2 ) ) @ ( modulo_modulo @ A @ A2 @ B2 ) ) @ C2 )
= ( plus_plus @ A @ A2 @ C2 ) ) ) ).
% cancel_div_mod_rules(2)
thf(fact_391_cancel__div__mod__rules_I1_J,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ ( times_times @ A @ ( divide_divide @ A @ A2 @ B2 ) @ B2 ) @ ( modulo_modulo @ A @ A2 @ B2 ) ) @ C2 )
= ( plus_plus @ A @ A2 @ C2 ) ) ) ).
% cancel_div_mod_rules(1)
thf(fact_392_mod__div__decomp,axiom,
! [A: $tType] :
( ( semiring_modulo @ A )
=> ! [A2: A,B2: A] :
( A2
= ( plus_plus @ A @ ( times_times @ A @ ( divide_divide @ A @ A2 @ B2 ) @ B2 ) @ ( modulo_modulo @ A @ A2 @ B2 ) ) ) ) ).
% mod_div_decomp
thf(fact_393_div__mult__mod__eq,axiom,
! [A: $tType] :
( ( semiring_modulo @ A )
=> ! [A2: A,B2: A] :
( ( plus_plus @ A @ ( times_times @ A @ ( divide_divide @ A @ A2 @ B2 ) @ B2 ) @ ( modulo_modulo @ A @ A2 @ B2 ) )
= A2 ) ) ).
% div_mult_mod_eq
thf(fact_394_mod__div__mult__eq,axiom,
! [A: $tType] :
( ( semiring_modulo @ A )
=> ! [A2: A,B2: A] :
( ( plus_plus @ A @ ( modulo_modulo @ A @ A2 @ B2 ) @ ( times_times @ A @ ( divide_divide @ A @ A2 @ B2 ) @ B2 ) )
= A2 ) ) ).
% mod_div_mult_eq
thf(fact_395_mod__mult__div__eq,axiom,
! [A: $tType] :
( ( semiring_modulo @ A )
=> ! [A2: A,B2: A] :
( ( plus_plus @ A @ ( modulo_modulo @ A @ A2 @ B2 ) @ ( times_times @ A @ B2 @ ( divide_divide @ A @ A2 @ B2 ) ) )
= A2 ) ) ).
% mod_mult_div_eq
thf(fact_396_mult__div__mod__eq,axiom,
! [A: $tType] :
( ( semiring_modulo @ A )
=> ! [B2: A,A2: A] :
( ( plus_plus @ A @ ( times_times @ A @ B2 @ ( divide_divide @ A @ A2 @ B2 ) ) @ ( modulo_modulo @ A @ A2 @ B2 ) )
= A2 ) ) ).
% mult_div_mod_eq
thf(fact_397_zdiv__numeral__Bit0,axiom,
! [V2: num,W: num] :
( ( divide_divide @ int @ ( numeral_numeral @ int @ ( bit0 @ V2 ) ) @ ( numeral_numeral @ int @ ( bit0 @ W ) ) )
= ( divide_divide @ int @ ( numeral_numeral @ int @ V2 ) @ ( numeral_numeral @ int @ W ) ) ) ).
% zdiv_numeral_Bit0
thf(fact_398_real__divide__square__eq,axiom,
! [R: real,A2: real] :
( ( divide_divide @ real @ ( times_times @ real @ R @ A2 ) @ ( times_times @ real @ R @ R ) )
= ( divide_divide @ real @ A2 @ R ) ) ).
% real_divide_square_eq
thf(fact_399_zmod__numeral__Bit0,axiom,
! [V2: num,W: num] :
( ( modulo_modulo @ int @ ( numeral_numeral @ int @ ( bit0 @ V2 ) ) @ ( numeral_numeral @ int @ ( bit0 @ W ) ) )
= ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( modulo_modulo @ int @ ( numeral_numeral @ int @ V2 ) @ ( numeral_numeral @ int @ W ) ) ) ) ).
% zmod_numeral_Bit0
thf(fact_400_less__eq__real__def,axiom,
( ( ord_less_eq @ real )
= ( ^ [X4: real,Y: real] :
( ( ord_less @ real @ X4 @ Y )
| ( X4 = Y ) ) ) ) ).
% less_eq_real_def
thf(fact_401_real__arch__pow,axiom,
! [X: real,Y2: real] :
( ( ord_less @ real @ ( one_one @ real ) @ X )
=> ? [N3: nat] : ( ord_less @ real @ Y2 @ ( power_power @ real @ X @ N3 ) ) ) ).
% real_arch_pow
thf(fact_402_complete__real,axiom,
! [S3: set @ real] :
( ? [X2: real] : ( member @ real @ X2 @ S3 )
=> ( ? [Z4: real] :
! [X5: real] :
( ( member @ real @ X5 @ S3 )
=> ( ord_less_eq @ real @ X5 @ Z4 ) )
=> ? [Y7: real] :
( ! [X2: real] :
( ( member @ real @ X2 @ S3 )
=> ( ord_less_eq @ real @ X2 @ Y7 ) )
& ! [Z4: real] :
( ! [X5: real] :
( ( member @ real @ X5 @ S3 )
=> ( ord_less_eq @ real @ X5 @ Z4 ) )
=> ( ord_less_eq @ real @ Y7 @ Z4 ) ) ) ) ) ).
% complete_real
thf(fact_403_infinity__ne__i1,axiom,
( ( extend4730790105801354508finity @ extended_enat )
!= ( one_one @ extended_enat ) ) ).
% infinity_ne_i1
thf(fact_404_two__realpow__ge__one,axiom,
! [N: nat] : ( ord_less_eq @ real @ ( one_one @ real ) @ ( power_power @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ N ) ) ).
% two_realpow_ge_one
thf(fact_405_one__enat__def,axiom,
( ( one_one @ extended_enat )
= ( extended_enat2 @ ( one_one @ nat ) ) ) ).
% one_enat_def
thf(fact_406_enat__1__iff_I1_J,axiom,
! [X: nat] :
( ( ( extended_enat2 @ X )
= ( one_one @ extended_enat ) )
= ( X
= ( one_one @ nat ) ) ) ).
% enat_1_iff(1)
thf(fact_407_enat__1__iff_I2_J,axiom,
! [X: nat] :
( ( ( one_one @ extended_enat )
= ( extended_enat2 @ X ) )
= ( X
= ( one_one @ nat ) ) ) ).
% enat_1_iff(2)
thf(fact_408_linorder__neqE__linordered__idom,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,Y2: A] :
( ( X != Y2 )
=> ( ~ ( ord_less @ A @ X @ Y2 )
=> ( ord_less @ A @ Y2 @ X ) ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_409_combine__common__factor,axiom,
! [A: $tType] :
( ( semiring @ A )
=> ! [A2: A,E: A,B2: A,C2: A] :
( ( plus_plus @ A @ ( times_times @ A @ A2 @ E ) @ ( plus_plus @ A @ ( times_times @ A @ B2 @ E ) @ C2 ) )
= ( plus_plus @ A @ ( times_times @ A @ ( plus_plus @ A @ A2 @ B2 ) @ E ) @ C2 ) ) ) ).
% combine_common_factor
thf(fact_410_distrib__right,axiom,
! [A: $tType] :
( ( semiring @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( times_times @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 )
= ( plus_plus @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% distrib_right
thf(fact_411_distrib__left,axiom,
! [A: $tType] :
( ( semiring @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( times_times @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) )
= ( plus_plus @ A @ ( times_times @ A @ A2 @ B2 ) @ ( times_times @ A @ A2 @ C2 ) ) ) ) ).
% distrib_left
thf(fact_412_comm__semiring__class_Odistrib,axiom,
! [A: $tType] :
( ( comm_semiring @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( times_times @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 )
= ( plus_plus @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% comm_semiring_class.distrib
thf(fact_413_ring__class_Oring__distribs_I1_J,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( times_times @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) )
= ( plus_plus @ A @ ( times_times @ A @ A2 @ B2 ) @ ( times_times @ A @ A2 @ C2 ) ) ) ) ).
% ring_class.ring_distribs(1)
thf(fact_414_ring__class_Oring__distribs_I2_J,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( times_times @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 )
= ( plus_plus @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% ring_class.ring_distribs(2)
thf(fact_415_lambda__one,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ( ( ^ [X4: A] : X4 )
= ( times_times @ A @ ( one_one @ A ) ) ) ) ).
% lambda_one
thf(fact_416_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_417_add__mono1,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ord_less @ A @ ( plus_plus @ A @ A2 @ ( one_one @ A ) ) @ ( plus_plus @ A @ B2 @ ( one_one @ A ) ) ) ) ) ).
% add_mono1
thf(fact_418_less__add__one,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A] : ( ord_less @ A @ A2 @ ( plus_plus @ A @ A2 @ ( one_one @ A ) ) ) ) ).
% less_add_one
thf(fact_419_cong__exp__iff__simps_I9_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M: num,Q: num,N: num] :
( ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit0 @ M ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q ) ) )
= ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit0 @ N ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q ) ) ) )
= ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ Q ) )
= ( modulo_modulo @ A @ ( numeral_numeral @ A @ N ) @ ( numeral_numeral @ A @ Q ) ) ) ) ) ).
% cong_exp_iff_simps(9)
thf(fact_420_cong__exp__iff__simps_I4_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M: num,N: num] :
( ( modulo_modulo @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ one2 ) )
= ( modulo_modulo @ A @ ( numeral_numeral @ A @ N ) @ ( numeral_numeral @ A @ one2 ) ) ) ) ).
% cong_exp_iff_simps(4)
thf(fact_421_nat__mod__eq__iff,axiom,
! [X: nat,N: nat,Y2: nat] :
( ( ( modulo_modulo @ nat @ X @ N )
= ( modulo_modulo @ nat @ Y2 @ N ) )
= ( ? [Q1: nat,Q22: nat] :
( ( plus_plus @ nat @ X @ ( times_times @ nat @ N @ Q1 ) )
= ( plus_plus @ nat @ Y2 @ ( times_times @ nat @ N @ Q22 ) ) ) ) ) ).
% nat_mod_eq_iff
thf(fact_422_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_423_cong__exp__iff__simps_I6_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [Q: num,N: num] :
( ( modulo_modulo @ A @ ( numeral_numeral @ A @ one2 ) @ ( numeral_numeral @ A @ ( bit0 @ Q ) ) )
!= ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit0 @ N ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q ) ) ) ) ) ).
% cong_exp_iff_simps(6)
thf(fact_424_cong__exp__iff__simps_I8_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M: num,Q: num] :
( ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit0 @ M ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q ) ) )
!= ( modulo_modulo @ A @ ( numeral_numeral @ A @ one2 ) @ ( numeral_numeral @ A @ ( bit0 @ Q ) ) ) ) ) ).
% cong_exp_iff_simps(8)
thf(fact_425_invar__vebt_Ointros_I3_J,axiom,
! [TreeList: list @ vEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
( ! [X5: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X5 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ( vEBT_invar_vebt @ X5 @ N ) )
=> ( ( vEBT_invar_vebt @ Summary @ M )
=> ( ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) )
=> ( ( M
= ( suc @ N ) )
=> ( ( Deg
= ( plus_plus @ nat @ N @ M ) )
=> ( ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_12 )
=> ( ! [X5: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X5 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_12 ) )
=> ( vEBT_invar_vebt @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg @ TreeList @ Summary ) @ Deg ) ) ) ) ) ) ) ) ).
% invar_vebt.intros(3)
thf(fact_426_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_427_ex__has__greatest__nat__lemma,axiom,
! [A: $tType,P3: A > $o,K2: A,F3: A > nat,N: nat] :
( ( P3 @ K2 )
=> ( ! [X5: A] :
( ( P3 @ X5 )
=> ? [Y4: A] :
( ( P3 @ Y4 )
& ~ ( ord_less_eq @ nat @ ( F3 @ Y4 ) @ ( F3 @ X5 ) ) ) )
=> ? [Y7: A] :
( ( P3 @ Y7 )
& ~ ( ord_less @ nat @ ( F3 @ Y7 ) @ ( plus_plus @ nat @ ( F3 @ K2 ) @ N ) ) ) ) ) ).
% ex_has_greatest_nat_lemma
thf(fact_428_divmod__digit__1_I1_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ ( modulo_modulo @ A @ A2 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) )
=> ( ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A2 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) ) @ ( one_one @ A ) )
= ( divide_divide @ A @ A2 @ B2 ) ) ) ) ) ) ).
% divmod_digit_1(1)
thf(fact_429_arith__geo__mean,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [U: A,X: A,Y2: A] :
( ( ( power_power @ A @ U @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( times_times @ A @ X @ Y2 ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y2 )
=> ( ord_less_eq @ A @ U @ ( divide_divide @ A @ ( plus_plus @ A @ X @ Y2 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% arith_geo_mean
thf(fact_430_both__member__options__from__chilf__to__complete__tree,axiom,
! [X: nat,Deg: nat,TreeList: list @ vEBT_VEBT,Mi: nat,Ma: nat,Summary: vEBT_VEBT] :
( ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
=> ( ( ord_less_eq @ nat @ ( one_one @ nat ) @ Deg )
=> ( ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
=> ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X ) ) ) ) ).
% both_member_options_from_chilf_to_complete_tree
thf(fact_431_subset__antisym,axiom,
! [A: $tType,A3: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B5 )
=> ( ( ord_less_eq @ ( set @ A ) @ B5 @ A3 )
=> ( A3 = B5 ) ) ) ).
% subset_antisym
thf(fact_432_subsetI,axiom,
! [A: $tType,A3: set @ A,B5: set @ A] :
( ! [X5: A] :
( ( member @ A @ X5 @ A3 )
=> ( member @ A @ X5 @ B5 ) )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ B5 ) ) ).
% subsetI
thf(fact_433_psubsetI,axiom,
! [A: $tType,A3: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B5 )
=> ( ( A3 != B5 )
=> ( ord_less @ ( set @ A ) @ A3 @ B5 ) ) ) ).
% psubsetI
thf(fact_434_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_435_even__odd__cases,axiom,
! [X: nat] :
( ! [N3: nat] :
( X
!= ( plus_plus @ nat @ N3 @ N3 ) )
=> ~ ! [N3: nat] :
( X
!= ( plus_plus @ nat @ N3 @ ( suc @ N3 ) ) ) ) ).
% even_odd_cases
thf(fact_436_deg__not__0,axiom,
! [T2: vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ).
% deg_not_0
thf(fact_437_old_Onat_Oinject,axiom,
! [Nat2: nat,Nat4: nat] :
( ( ( suc @ Nat2 )
= ( suc @ Nat4 ) )
= ( Nat2 = Nat4 ) ) ).
% old.nat.inject
thf(fact_438_nat_Oinject,axiom,
! [X23: nat,Y23: nat] :
( ( ( suc @ X23 )
= ( suc @ Y23 ) )
= ( X23 = Y23 ) ) ).
% nat.inject
thf(fact_439_bot__nat__0_Onot__eq__extremum,axiom,
! [A2: nat] :
( ( A2
!= ( zero_zero @ nat ) )
= ( ord_less @ nat @ ( zero_zero @ nat ) @ A2 ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_440_neq0__conv,axiom,
! [N: nat] :
( ( N
!= ( zero_zero @ nat ) )
= ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ).
% neq0_conv
thf(fact_441_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% less_nat_zero_code
thf(fact_442_le0,axiom,
! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% le0
thf(fact_443_bot__nat__0_Oextremum,axiom,
! [A2: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ A2 ) ).
% bot_nat_0.extremum
thf(fact_444_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_445_Nat_Oadd__0__right,axiom,
! [M: nat] :
( ( plus_plus @ nat @ M @ ( zero_zero @ nat ) )
= M ) ).
% Nat.add_0_right
thf(fact_446_mult__cancel2,axiom,
! [M: nat,K2: nat,N: nat] :
( ( ( times_times @ nat @ M @ K2 )
= ( times_times @ nat @ N @ K2 ) )
= ( ( M = N )
| ( K2
= ( zero_zero @ nat ) ) ) ) ).
% mult_cancel2
thf(fact_447_mult__cancel1,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ( times_times @ nat @ K2 @ M )
= ( times_times @ nat @ K2 @ N ) )
= ( ( M = N )
| ( K2
= ( zero_zero @ nat ) ) ) ) ).
% mult_cancel1
thf(fact_448_mult__0__right,axiom,
! [M: nat] :
( ( times_times @ nat @ M @ ( zero_zero @ nat ) )
= ( zero_zero @ nat ) ) ).
% mult_0_right
thf(fact_449_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_450_div__pos__pos__trivial,axiom,
! [K2: int,L: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K2 )
=> ( ( ord_less @ int @ K2 @ L )
=> ( ( divide_divide @ int @ K2 @ L )
= ( zero_zero @ int ) ) ) ) ).
% div_pos_pos_trivial
thf(fact_451_div__neg__neg__trivial,axiom,
! [K2: int,L: int] :
( ( ord_less_eq @ int @ K2 @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ L @ K2 )
=> ( ( divide_divide @ int @ K2 @ L )
= ( zero_zero @ int ) ) ) ) ).
% div_neg_neg_trivial
thf(fact_452_mod__pos__pos__trivial,axiom,
! [K2: int,L: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K2 )
=> ( ( ord_less @ int @ K2 @ L )
=> ( ( modulo_modulo @ int @ K2 @ L )
= K2 ) ) ) ).
% mod_pos_pos_trivial
thf(fact_453_mod__neg__neg__trivial,axiom,
! [K2: int,L: int] :
( ( ord_less_eq @ int @ K2 @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ L @ K2 )
=> ( ( modulo_modulo @ int @ K2 @ L )
= K2 ) ) ) ).
% mod_neg_neg_trivial
thf(fact_454_i0__less,axiom,
! [N: extended_enat] :
( ( ord_less @ extended_enat @ ( zero_zero @ extended_enat ) @ N )
= ( N
!= ( zero_zero @ extended_enat ) ) ) ).
% i0_less
thf(fact_455_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_456_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_457_mult__zero__left,axiom,
! [A: $tType] :
( ( mult_zero @ A )
=> ! [A2: A] :
( ( times_times @ A @ ( zero_zero @ A ) @ A2 )
= ( zero_zero @ A ) ) ) ).
% mult_zero_left
thf(fact_458_mult__zero__right,axiom,
! [A: $tType] :
( ( mult_zero @ A )
=> ! [A2: A] :
( ( times_times @ A @ A2 @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% mult_zero_right
thf(fact_459_mult__eq__0__iff,axiom,
! [A: $tType] :
( ( semiri3467727345109120633visors @ A )
=> ! [A2: A,B2: A] :
( ( ( times_times @ A @ A2 @ B2 )
= ( zero_zero @ A ) )
= ( ( A2
= ( zero_zero @ A ) )
| ( B2
= ( zero_zero @ A ) ) ) ) ) ).
% mult_eq_0_iff
thf(fact_460_mult__cancel__left,axiom,
! [A: $tType] :
( ( semiri6575147826004484403cancel @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ( times_times @ A @ C2 @ A2 )
= ( times_times @ A @ C2 @ B2 ) )
= ( ( C2
= ( zero_zero @ A ) )
| ( A2 = B2 ) ) ) ) ).
% mult_cancel_left
thf(fact_461_mult__cancel__right,axiom,
! [A: $tType] :
( ( semiri6575147826004484403cancel @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( ( times_times @ A @ A2 @ C2 )
= ( times_times @ A @ B2 @ C2 ) )
= ( ( C2
= ( zero_zero @ A ) )
| ( A2 = B2 ) ) ) ) ).
% mult_cancel_right
thf(fact_462_add_Oright__neutral,axiom,
! [A: $tType] :
( ( monoid_add @ A )
=> ! [A2: A] :
( ( plus_plus @ A @ A2 @ ( zero_zero @ A ) )
= A2 ) ) ).
% add.right_neutral
thf(fact_463_double__zero__sym,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A2: A] :
( ( ( zero_zero @ A )
= ( plus_plus @ A @ A2 @ A2 ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% double_zero_sym
thf(fact_464_add__cancel__left__left,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add @ A )
=> ! [B2: A,A2: A] :
( ( ( plus_plus @ A @ B2 @ A2 )
= A2 )
= ( B2
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_left_left
thf(fact_465_add__cancel__left__right,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add @ A )
=> ! [A2: A,B2: A] :
( ( ( plus_plus @ A @ A2 @ B2 )
= A2 )
= ( B2
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_left_right
thf(fact_466_add__cancel__right__left,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add @ A )
=> ! [A2: A,B2: A] :
( ( A2
= ( plus_plus @ A @ B2 @ A2 ) )
= ( B2
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_right_left
thf(fact_467_add__cancel__right__right,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add @ A )
=> ! [A2: A,B2: A] :
( ( A2
= ( plus_plus @ A @ A2 @ B2 ) )
= ( B2
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_right_right
thf(fact_468_add__eq__0__iff__both__eq__0,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [X: A,Y2: A] :
( ( ( plus_plus @ A @ X @ Y2 )
= ( zero_zero @ A ) )
= ( ( X
= ( zero_zero @ A ) )
& ( Y2
= ( zero_zero @ A ) ) ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_469_zero__eq__add__iff__both__eq__0,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [X: A,Y2: A] :
( ( ( zero_zero @ A )
= ( plus_plus @ A @ X @ Y2 ) )
= ( ( X
= ( zero_zero @ A ) )
& ( Y2
= ( zero_zero @ A ) ) ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_470_add__0,axiom,
! [A: $tType] :
( ( monoid_add @ A )
=> ! [A2: A] :
( ( plus_plus @ A @ ( zero_zero @ A ) @ A2 )
= A2 ) ) ).
% add_0
thf(fact_471_divide__eq__0__iff,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A2: A,B2: A] :
( ( ( divide_divide @ A @ A2 @ B2 )
= ( zero_zero @ A ) )
= ( ( A2
= ( zero_zero @ A ) )
| ( B2
= ( zero_zero @ A ) ) ) ) ) ).
% divide_eq_0_iff
thf(fact_472_divide__cancel__left,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ( divide_divide @ A @ C2 @ A2 )
= ( divide_divide @ A @ C2 @ B2 ) )
= ( ( C2
= ( zero_zero @ A ) )
| ( A2 = B2 ) ) ) ) ).
% divide_cancel_left
thf(fact_473_divide__cancel__right,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( ( divide_divide @ A @ A2 @ C2 )
= ( divide_divide @ A @ B2 @ C2 ) )
= ( ( C2
= ( zero_zero @ A ) )
| ( A2 = B2 ) ) ) ) ).
% divide_cancel_right
thf(fact_474_division__ring__divide__zero,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A] :
( ( divide_divide @ A @ A2 @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% division_ring_divide_zero
thf(fact_475_bits__div__0,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A2: A] :
( ( divide_divide @ A @ ( zero_zero @ A ) @ A2 )
= ( zero_zero @ A ) ) ) ).
% bits_div_0
thf(fact_476_bits__div__by__0,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A2: A] :
( ( divide_divide @ A @ A2 @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% bits_div_by_0
thf(fact_477_div__0,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [A2: A] :
( ( divide_divide @ A @ ( zero_zero @ A ) @ A2 )
= ( zero_zero @ A ) ) ) ).
% div_0
thf(fact_478_div__by__0,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [A2: A] :
( ( divide_divide @ A @ A2 @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% div_by_0
thf(fact_479_bits__mod__0,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A2: A] :
( ( modulo_modulo @ A @ ( zero_zero @ A ) @ A2 )
= ( zero_zero @ A ) ) ) ).
% bits_mod_0
thf(fact_480_mod__0,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [A2: A] :
( ( modulo_modulo @ A @ ( zero_zero @ A ) @ A2 )
= ( zero_zero @ A ) ) ) ).
% mod_0
thf(fact_481_mod__by__0,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [A2: A] :
( ( modulo_modulo @ A @ A2 @ ( zero_zero @ A ) )
= A2 ) ) ).
% mod_by_0
thf(fact_482_mod__self,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [A2: A] :
( ( modulo_modulo @ A @ A2 @ A2 )
= ( zero_zero @ A ) ) ) ).
% mod_self
thf(fact_483_power__Suc0__right,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A2: A] :
( ( power_power @ A @ A2 @ ( suc @ ( zero_zero @ nat ) ) )
= A2 ) ) ).
% power_Suc0_right
thf(fact_484_zero__less__Suc,axiom,
! [N: nat] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_485_less__Suc0,axiom,
! [N: nat] :
( ( ord_less @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% less_Suc0
thf(fact_486_lessI,axiom,
! [N: nat] : ( ord_less @ nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_487_Suc__mono,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ N )
=> ( ord_less @ nat @ ( suc @ M ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_488_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_489_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_490_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_491_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_492_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_493_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_494_nat__mult__less__cancel__disj,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less @ nat @ ( times_times @ nat @ K2 @ M ) @ ( times_times @ nat @ K2 @ N ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K2 )
& ( ord_less @ nat @ M @ N ) ) ) ).
% nat_mult_less_cancel_disj
thf(fact_495_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_496_mult__less__cancel2,axiom,
! [M: nat,K2: nat,N: nat] :
( ( ord_less @ nat @ ( times_times @ nat @ M @ K2 ) @ ( times_times @ nat @ N @ K2 ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K2 )
& ( ord_less @ nat @ M @ N ) ) ) ).
% mult_less_cancel2
thf(fact_497_not__real__square__gt__zero,axiom,
! [X: real] :
( ( ~ ( ord_less @ real @ ( zero_zero @ real ) @ ( times_times @ real @ X @ X ) ) )
= ( X
= ( zero_zero @ real ) ) ) ).
% not_real_square_gt_zero
thf(fact_498_div__by__Suc__0,axiom,
! [M: nat] :
( ( divide_divide @ nat @ M @ ( suc @ ( zero_zero @ nat ) ) )
= M ) ).
% div_by_Suc_0
thf(fact_499_less__one,axiom,
! [N: nat] :
( ( ord_less @ nat @ N @ ( one_one @ nat ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% less_one
thf(fact_500_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_501_power__Suc__0,axiom,
! [N: nat] :
( ( power_power @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% power_Suc_0
thf(fact_502_div__less,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ N )
=> ( ( divide_divide @ nat @ M @ N )
= ( zero_zero @ nat ) ) ) ).
% div_less
thf(fact_503_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_504_mod__by__Suc__0,axiom,
! [M: nat] :
( ( modulo_modulo @ nat @ M @ ( suc @ ( zero_zero @ nat ) ) )
= ( zero_zero @ nat ) ) ).
% mod_by_Suc_0
thf(fact_505_nat__mult__div__cancel__disj,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ( K2
= ( zero_zero @ nat ) )
=> ( ( divide_divide @ nat @ ( times_times @ nat @ K2 @ M ) @ ( times_times @ nat @ K2 @ N ) )
= ( zero_zero @ nat ) ) )
& ( ( K2
!= ( zero_zero @ nat ) )
=> ( ( divide_divide @ nat @ ( times_times @ nat @ K2 @ M ) @ ( times_times @ nat @ K2 @ N ) )
= ( divide_divide @ nat @ M @ N ) ) ) ) ).
% nat_mult_div_cancel_disj
thf(fact_506_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_507_mi__ma__2__deg,axiom,
! [Mi: nat,Ma: nat,Deg: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ N )
=> ( ( ord_less_eq @ nat @ Mi @ Ma )
& ( ord_less @ nat @ Ma @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg ) ) ) ) ).
% mi_ma_2_deg
thf(fact_508_add__le__same__cancel1,axiom,
! [A: $tType] :
( ( ordere1937475149494474687imp_le @ A )
=> ! [B2: A,A2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ B2 @ A2 ) @ B2 )
= ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% add_le_same_cancel1
thf(fact_509_add__le__same__cancel2,axiom,
! [A: $tType] :
( ( ordere1937475149494474687imp_le @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ B2 ) @ B2 )
= ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% add_le_same_cancel2
thf(fact_510_le__add__same__cancel1,axiom,
! [A: $tType] :
( ( ordere1937475149494474687imp_le @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ ( plus_plus @ A @ A2 @ B2 ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) ) ) ).
% le_add_same_cancel1
thf(fact_511_le__add__same__cancel2,axiom,
! [A: $tType] :
( ( ordere1937475149494474687imp_le @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ ( plus_plus @ A @ B2 @ A2 ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) ) ) ).
% le_add_same_cancel2
thf(fact_512_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ A2 ) @ ( zero_zero @ A ) )
= ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_513_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ A2 ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_514_add__less__same__cancel1,axiom,
! [A: $tType] :
( ( ordere1937475149494474687imp_le @ A )
=> ! [B2: A,A2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ B2 @ A2 ) @ B2 )
= ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% add_less_same_cancel1
thf(fact_515_add__less__same__cancel2,axiom,
! [A: $tType] :
( ( ordere1937475149494474687imp_le @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ A2 @ B2 ) @ B2 )
= ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% add_less_same_cancel2
thf(fact_516_less__add__same__cancel1,axiom,
! [A: $tType] :
( ( ordere1937475149494474687imp_le @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ ( plus_plus @ A @ A2 @ B2 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) ) ) ).
% less_add_same_cancel1
thf(fact_517_less__add__same__cancel2,axiom,
! [A: $tType] :
( ( ordere1937475149494474687imp_le @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ ( plus_plus @ A @ B2 @ A2 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) ) ) ).
% less_add_same_cancel2
thf(fact_518_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ A2 @ A2 ) @ ( zero_zero @ A ) )
= ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_519_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ A2 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_520_mult__cancel__right2,axiom,
! [A: $tType] :
( ( ring_15535105094025558882visors @ A )
=> ! [A2: A,C2: A] :
( ( ( times_times @ A @ A2 @ C2 )
= C2 )
= ( ( C2
= ( zero_zero @ A ) )
| ( A2
= ( one_one @ A ) ) ) ) ) ).
% mult_cancel_right2
thf(fact_521_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_522_mult__cancel__left2,axiom,
! [A: $tType] :
( ( ring_15535105094025558882visors @ A )
=> ! [C2: A,A2: A] :
( ( ( times_times @ A @ C2 @ A2 )
= C2 )
= ( ( C2
= ( zero_zero @ A ) )
| ( A2
= ( one_one @ A ) ) ) ) ) ).
% mult_cancel_left2
thf(fact_523_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_524_sum__squares__eq__zero__iff,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [X: A,Y2: A] :
( ( ( plus_plus @ A @ ( times_times @ A @ X @ X ) @ ( times_times @ A @ Y2 @ Y2 ) )
= ( zero_zero @ A ) )
= ( ( X
= ( zero_zero @ A ) )
& ( Y2
= ( zero_zero @ A ) ) ) ) ) ).
% sum_squares_eq_zero_iff
thf(fact_525_mult__divide__mult__cancel__left__if,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ( C2
= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) )
= ( zero_zero @ A ) ) )
& ( ( C2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) )
= ( divide_divide @ A @ A2 @ B2 ) ) ) ) ) ).
% mult_divide_mult_cancel_left_if
thf(fact_526_nonzero__mult__divide__mult__cancel__left,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) )
= ( divide_divide @ A @ A2 @ B2 ) ) ) ) ).
% nonzero_mult_divide_mult_cancel_left
thf(fact_527_nonzero__mult__divide__mult__cancel__left2,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ B2 @ C2 ) )
= ( divide_divide @ A @ A2 @ B2 ) ) ) ) ).
% nonzero_mult_divide_mult_cancel_left2
thf(fact_528_nonzero__mult__divide__mult__cancel__right,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) )
= ( divide_divide @ A @ A2 @ B2 ) ) ) ) ).
% nonzero_mult_divide_mult_cancel_right
thf(fact_529_nonzero__mult__divide__mult__cancel__right2,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ C2 @ B2 ) )
= ( divide_divide @ A @ A2 @ B2 ) ) ) ) ).
% nonzero_mult_divide_mult_cancel_right2
thf(fact_530_div__mult__mult1__if,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ( C2
= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) )
= ( zero_zero @ A ) ) )
& ( ( C2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) )
= ( divide_divide @ A @ A2 @ B2 ) ) ) ) ) ).
% div_mult_mult1_if
thf(fact_531_div__mult__mult2,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) )
= ( divide_divide @ A @ A2 @ B2 ) ) ) ) ).
% div_mult_mult2
thf(fact_532_div__mult__mult1,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) )
= ( divide_divide @ A @ A2 @ B2 ) ) ) ) ).
% div_mult_mult1
thf(fact_533_nonzero__mult__div__cancel__left,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [A2: A,B2: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ A2 @ B2 ) @ A2 )
= B2 ) ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_534_nonzero__mult__div__cancel__right,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [B2: A,A2: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ A2 @ B2 ) @ B2 )
= A2 ) ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_535_divide__eq__1__iff,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A2: A,B2: A] :
( ( ( divide_divide @ A @ A2 @ B2 )
= ( one_one @ A ) )
= ( ( B2
!= ( zero_zero @ A ) )
& ( A2 = B2 ) ) ) ) ).
% divide_eq_1_iff
thf(fact_536_one__eq__divide__iff,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A2: A,B2: A] :
( ( ( one_one @ A )
= ( divide_divide @ A @ A2 @ B2 ) )
= ( ( B2
!= ( zero_zero @ A ) )
& ( A2 = B2 ) ) ) ) ).
% one_eq_divide_iff
thf(fact_537_divide__self,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ A2 @ A2 )
= ( one_one @ A ) ) ) ) ).
% divide_self
thf(fact_538_divide__self__if,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A] :
( ( ( A2
= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ A2 @ A2 )
= ( zero_zero @ A ) ) )
& ( ( A2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ A2 @ A2 )
= ( one_one @ A ) ) ) ) ) ).
% divide_self_if
thf(fact_539_divide__eq__eq__1,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,A2: A] :
( ( ( divide_divide @ A @ B2 @ A2 )
= ( one_one @ A ) )
= ( ( A2
!= ( zero_zero @ A ) )
& ( A2 = B2 ) ) ) ) ).
% divide_eq_eq_1
thf(fact_540_eq__divide__eq__1,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,A2: A] :
( ( ( one_one @ A )
= ( divide_divide @ A @ B2 @ A2 ) )
= ( ( A2
!= ( zero_zero @ A ) )
& ( A2 = B2 ) ) ) ) ).
% eq_divide_eq_1
thf(fact_541_one__divide__eq__0__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A] :
( ( ( divide_divide @ A @ ( one_one @ A ) @ A2 )
= ( zero_zero @ A ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% one_divide_eq_0_iff
thf(fact_542_zero__eq__1__divide__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A] :
( ( ( zero_zero @ A )
= ( divide_divide @ A @ ( one_one @ A ) @ A2 ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% zero_eq_1_divide_iff
thf(fact_543_div__self,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [A2: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ A2 @ A2 )
= ( one_one @ A ) ) ) ) ).
% div_self
thf(fact_544_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_545_power__zero__numeral,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [K2: num] :
( ( power_power @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ nat @ K2 ) )
= ( zero_zero @ A ) ) ) ).
% power_zero_numeral
thf(fact_546_power__eq__0__iff,axiom,
! [A: $tType] :
( ( semiri2026040879449505780visors @ A )
=> ! [A2: A,N: nat] :
( ( ( power_power @ A @ A2 @ N )
= ( zero_zero @ A ) )
= ( ( A2
= ( zero_zero @ A ) )
& ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% power_eq_0_iff
thf(fact_547_mod__mult__self2__is__0,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A2: A,B2: A] :
( ( modulo_modulo @ A @ ( times_times @ A @ A2 @ B2 ) @ B2 )
= ( zero_zero @ A ) ) ) ).
% mod_mult_self2_is_0
thf(fact_548_mod__mult__self1__is__0,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [B2: A,A2: A] :
( ( modulo_modulo @ A @ ( times_times @ A @ B2 @ A2 ) @ B2 )
= ( zero_zero @ A ) ) ) ).
% mod_mult_self1_is_0
thf(fact_549_bits__mod__by__1,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A2: A] :
( ( modulo_modulo @ A @ A2 @ ( one_one @ A ) )
= ( zero_zero @ A ) ) ) ).
% bits_mod_by_1
thf(fact_550_mod__by__1,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [A2: A] :
( ( modulo_modulo @ A @ A2 @ ( one_one @ A ) )
= ( zero_zero @ A ) ) ) ).
% mod_by_1
thf(fact_551_mod__div__trivial,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A2: A,B2: A] :
( ( divide_divide @ A @ ( modulo_modulo @ A @ A2 @ B2 ) @ B2 )
= ( zero_zero @ A ) ) ) ).
% mod_div_trivial
thf(fact_552_bits__mod__div__trivial,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A2: A,B2: A] :
( ( divide_divide @ A @ ( modulo_modulo @ A @ A2 @ B2 ) @ B2 )
= ( zero_zero @ A ) ) ) ).
% bits_mod_div_trivial
thf(fact_553_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_554_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_555_mult__le__cancel2,axiom,
! [M: nat,K2: nat,N: nat] :
( ( ord_less_eq @ nat @ ( times_times @ nat @ M @ K2 ) @ ( times_times @ nat @ N @ K2 ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K2 )
=> ( ord_less_eq @ nat @ M @ N ) ) ) ).
% mult_le_cancel2
thf(fact_556_nat__mult__le__cancel__disj,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ ( times_times @ nat @ K2 @ M ) @ ( times_times @ nat @ K2 @ N ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K2 )
=> ( ord_less_eq @ nat @ M @ N ) ) ) ).
% nat_mult_le_cancel_disj
thf(fact_557_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_558_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_559_dbl__simps_I5_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [K2: num] :
( ( neg_numeral_dbl @ A @ ( numeral_numeral @ A @ K2 ) )
= ( numeral_numeral @ A @ ( bit0 @ K2 ) ) ) ) ).
% dbl_simps(5)
thf(fact_560_both__member__options__from__complete__tree__to__child,axiom,
! [Deg: nat,Mi: nat,Ma: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ Deg )
=> ( ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
=> ( ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
| ( X = Mi )
| ( X = Ma ) ) ) ) ).
% both_member_options_from_complete_tree_to_child
thf(fact_561_times__enat__simps_I4_J,axiom,
! [M: nat] :
( ( ( M
= ( zero_zero @ nat ) )
=> ( ( times_times @ extended_enat @ ( extended_enat2 @ M ) @ ( extend4730790105801354508finity @ extended_enat ) )
= ( zero_zero @ extended_enat ) ) )
& ( ( M
!= ( zero_zero @ nat ) )
=> ( ( times_times @ extended_enat @ ( extended_enat2 @ M ) @ ( extend4730790105801354508finity @ extended_enat ) )
= ( extend4730790105801354508finity @ extended_enat ) ) ) ) ).
% times_enat_simps(4)
thf(fact_562_times__enat__simps_I3_J,axiom,
! [N: nat] :
( ( ( N
= ( zero_zero @ nat ) )
=> ( ( times_times @ extended_enat @ ( extend4730790105801354508finity @ extended_enat ) @ ( extended_enat2 @ N ) )
= ( zero_zero @ extended_enat ) ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ( ( times_times @ extended_enat @ ( extend4730790105801354508finity @ extended_enat ) @ ( extended_enat2 @ N ) )
= ( extend4730790105801354508finity @ extended_enat ) ) ) ) ).
% times_enat_simps(3)
thf(fact_563_divide__le__0__1__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ ( one_one @ A ) @ A2 ) @ ( zero_zero @ A ) )
= ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% divide_le_0_1_iff
thf(fact_564_zero__le__divide__1__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ ( one_one @ A ) @ A2 ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% zero_le_divide_1_iff
thf(fact_565_divide__less__0__1__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A] :
( ( ord_less @ A @ ( divide_divide @ A @ ( one_one @ A ) @ A2 ) @ ( zero_zero @ A ) )
= ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% divide_less_0_1_iff
thf(fact_566_divide__less__eq__1__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( divide_divide @ A @ B2 @ A2 ) @ ( one_one @ A ) )
= ( ord_less @ A @ A2 @ B2 ) ) ) ) ).
% divide_less_eq_1_neg
thf(fact_567_divide__less__eq__1__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ ( divide_divide @ A @ B2 @ A2 ) @ ( one_one @ A ) )
= ( ord_less @ A @ B2 @ A2 ) ) ) ) ).
% divide_less_eq_1_pos
thf(fact_568_less__divide__eq__1__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B2 @ A2 ) )
= ( ord_less @ A @ B2 @ A2 ) ) ) ) ).
% less_divide_eq_1_neg
thf(fact_569_less__divide__eq__1__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B2 @ A2 ) )
= ( ord_less @ A @ A2 @ B2 ) ) ) ) ).
% less_divide_eq_1_pos
thf(fact_570_zero__less__divide__1__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ ( one_one @ A ) @ A2 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% zero_less_divide_1_iff
thf(fact_571_divide__eq__eq__numeral1_I1_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B2: A,W: num,A2: A] :
( ( ( divide_divide @ A @ B2 @ ( numeral_numeral @ A @ W ) )
= A2 )
= ( ( ( ( numeral_numeral @ A @ W )
!= ( zero_zero @ A ) )
=> ( B2
= ( times_times @ A @ A2 @ ( numeral_numeral @ A @ W ) ) ) )
& ( ( ( numeral_numeral @ A @ W )
= ( zero_zero @ A ) )
=> ( A2
= ( zero_zero @ A ) ) ) ) ) ) ).
% divide_eq_eq_numeral1(1)
thf(fact_572_eq__divide__eq__numeral1_I1_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A,B2: A,W: num] :
( ( A2
= ( divide_divide @ A @ B2 @ ( numeral_numeral @ A @ W ) ) )
= ( ( ( ( numeral_numeral @ A @ W )
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ A2 @ ( numeral_numeral @ A @ W ) )
= B2 ) )
& ( ( ( numeral_numeral @ A @ W )
= ( zero_zero @ A ) )
=> ( A2
= ( zero_zero @ A ) ) ) ) ) ) ).
% eq_divide_eq_numeral1(1)
thf(fact_573_nonzero__divide__mult__cancel__left,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A2: A,B2: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ A2 @ ( times_times @ A @ A2 @ B2 ) )
= ( divide_divide @ A @ ( one_one @ A ) @ B2 ) ) ) ) ).
% nonzero_divide_mult_cancel_left
thf(fact_574_nonzero__divide__mult__cancel__right,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [B2: A,A2: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ B2 @ ( times_times @ A @ A2 @ B2 ) )
= ( divide_divide @ A @ ( one_one @ A ) @ A2 ) ) ) ) ).
% nonzero_divide_mult_cancel_right
thf(fact_575_div__mult__self1,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ A2 @ ( times_times @ A @ C2 @ B2 ) ) @ B2 )
= ( plus_plus @ A @ C2 @ ( divide_divide @ A @ A2 @ B2 ) ) ) ) ) ).
% div_mult_self1
thf(fact_576_div__mult__self2,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) ) @ B2 )
= ( plus_plus @ A @ C2 @ ( divide_divide @ A @ A2 @ B2 ) ) ) ) ) ).
% div_mult_self2
thf(fact_577_div__mult__self3,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [B2: A,C2: A,A2: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ C2 @ B2 ) @ A2 ) @ B2 )
= ( plus_plus @ A @ C2 @ ( divide_divide @ A @ A2 @ B2 ) ) ) ) ) ).
% div_mult_self3
thf(fact_578_div__mult__self4,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [B2: A,C2: A,A2: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ B2 @ C2 ) @ A2 ) @ B2 )
= ( plus_plus @ A @ C2 @ ( divide_divide @ A @ A2 @ B2 ) ) ) ) ) ).
% div_mult_self4
thf(fact_579_power__mono__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,B2: A,N: nat] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ A @ ( power_power @ A @ A2 @ N ) @ ( power_power @ A @ B2 @ N ) )
= ( ord_less_eq @ A @ A2 @ B2 ) ) ) ) ) ) ).
% power_mono_iff
thf(fact_580_half__negative__int__iff,axiom,
! [K2: int] :
( ( ord_less @ int @ ( divide_divide @ int @ K2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( zero_zero @ int ) )
= ( ord_less @ int @ K2 @ ( zero_zero @ int ) ) ) ).
% half_negative_int_iff
thf(fact_581_half__nonnegative__int__iff,axiom,
! [K2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( divide_divide @ int @ K2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) )
= ( ord_less_eq @ int @ ( zero_zero @ int ) @ K2 ) ) ).
% half_nonnegative_int_iff
thf(fact_582_Suc__numeral,axiom,
! [N: num] :
( ( suc @ ( numeral_numeral @ nat @ N ) )
= ( numeral_numeral @ nat @ ( plus_plus @ num @ N @ one2 ) ) ) ).
% Suc_numeral
thf(fact_583_Suc__mod__mult__self1,axiom,
! [M: nat,K2: nat,N: nat] :
( ( modulo_modulo @ nat @ ( suc @ ( plus_plus @ nat @ M @ ( times_times @ nat @ K2 @ N ) ) ) @ N )
= ( modulo_modulo @ nat @ ( suc @ M ) @ N ) ) ).
% Suc_mod_mult_self1
thf(fact_584_Suc__mod__mult__self2,axiom,
! [M: nat,N: nat,K2: nat] :
( ( modulo_modulo @ nat @ ( suc @ ( plus_plus @ nat @ M @ ( times_times @ nat @ N @ K2 ) ) ) @ N )
= ( modulo_modulo @ nat @ ( suc @ M ) @ N ) ) ).
% Suc_mod_mult_self2
thf(fact_585_Suc__mod__mult__self3,axiom,
! [K2: nat,N: nat,M: nat] :
( ( modulo_modulo @ nat @ ( suc @ ( plus_plus @ nat @ ( times_times @ nat @ K2 @ N ) @ M ) ) @ N )
= ( modulo_modulo @ nat @ ( suc @ M ) @ N ) ) ).
% Suc_mod_mult_self3
thf(fact_586_Suc__mod__mult__self4,axiom,
! [N: nat,K2: nat,M: nat] :
( ( modulo_modulo @ nat @ ( suc @ ( plus_plus @ nat @ ( times_times @ nat @ N @ K2 ) @ M ) ) @ N )
= ( modulo_modulo @ nat @ ( suc @ M ) @ N ) ) ).
% Suc_mod_mult_self4
thf(fact_587_divide__le__eq__1__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ A2 ) @ ( one_one @ A ) )
= ( ord_less_eq @ A @ A2 @ B2 ) ) ) ) ).
% divide_le_eq_1_neg
thf(fact_588_divide__le__eq__1__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ A2 ) @ ( one_one @ A ) )
= ( ord_less_eq @ A @ B2 @ A2 ) ) ) ) ).
% divide_le_eq_1_pos
thf(fact_589_le__divide__eq__1__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B2 @ A2 ) )
= ( ord_less_eq @ A @ B2 @ A2 ) ) ) ) ).
% le_divide_eq_1_neg
thf(fact_590_le__divide__eq__1__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B2 @ A2 ) )
= ( ord_less_eq @ A @ A2 @ B2 ) ) ) ) ).
% le_divide_eq_1_pos
thf(fact_591_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_592_zero__eq__power2,axiom,
! [A: $tType] :
( ( semiri2026040879449505780visors @ A )
=> ! [A2: A] :
( ( ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( zero_zero @ A ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% zero_eq_power2
thf(fact_593_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_594_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_595_Suc__1,axiom,
( ( suc @ ( one_one @ nat ) )
= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ).
% Suc_1
thf(fact_596_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_597_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_598_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_599_Suc__times__numeral__mod__eq,axiom,
! [K2: num,N: nat] :
( ( ( numeral_numeral @ nat @ K2 )
!= ( one_one @ nat ) )
=> ( ( modulo_modulo @ nat @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ K2 ) @ N ) ) @ ( numeral_numeral @ nat @ K2 ) )
= ( one_one @ nat ) ) ) ).
% Suc_times_numeral_mod_eq
thf(fact_600_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_601_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_602_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_603_power2__less__eq__zero__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( zero_zero @ A ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% power2_less_eq_zero_iff
thf(fact_604_power2__eq__iff__nonneg,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X: A,Y2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y2 )
=> ( ( ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( power_power @ A @ Y2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( X = Y2 ) ) ) ) ) ).
% power2_eq_iff_nonneg
thf(fact_605_zero__less__power2,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( A2
!= ( zero_zero @ A ) ) ) ) ).
% zero_less_power2
thf(fact_606_sum__power2__eq__zero__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,Y2: A] :
( ( ( plus_plus @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( zero_zero @ A ) )
= ( ( X
= ( zero_zero @ A ) )
& ( Y2
= ( zero_zero @ A ) ) ) ) ) ).
% sum_power2_eq_zero_iff
thf(fact_607_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_608_mod__pos__neg__trivial,axiom,
! [K2: int,L: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ K2 )
=> ( ( ord_less_eq @ int @ ( plus_plus @ int @ K2 @ L ) @ ( zero_zero @ int ) )
=> ( ( modulo_modulo @ int @ K2 @ L )
= ( plus_plus @ int @ K2 @ L ) ) ) ) ).
% mod_pos_neg_trivial
thf(fact_609_zmod__zmult2__eq,axiom,
! [C2: int,A2: int,B2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ C2 )
=> ( ( modulo_modulo @ int @ A2 @ ( times_times @ int @ B2 @ C2 ) )
= ( plus_plus @ int @ ( times_times @ int @ B2 @ ( modulo_modulo @ int @ ( divide_divide @ int @ A2 @ B2 ) @ C2 ) ) @ ( modulo_modulo @ int @ A2 @ B2 ) ) ) ) ).
% zmod_zmult2_eq
thf(fact_610_zdiv__zmult2__eq,axiom,
! [C2: int,A2: int,B2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ C2 )
=> ( ( divide_divide @ int @ A2 @ ( times_times @ int @ B2 @ C2 ) )
= ( divide_divide @ int @ ( divide_divide @ int @ A2 @ B2 ) @ C2 ) ) ) ).
% zdiv_zmult2_eq
thf(fact_611_unique__quotient__lemma__neg,axiom,
! [B2: int,Q4: int,R2: int,Q: int,R: int] :
( ( ord_less_eq @ int @ ( plus_plus @ int @ ( times_times @ int @ B2 @ Q4 ) @ R2 ) @ ( plus_plus @ int @ ( times_times @ int @ B2 @ Q ) @ R ) )
=> ( ( ord_less_eq @ int @ R @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ B2 @ R )
=> ( ( ord_less @ int @ B2 @ R2 )
=> ( ord_less_eq @ int @ Q @ Q4 ) ) ) ) ) ).
% unique_quotient_lemma_neg
thf(fact_612_Euclidean__Division_Opos__mod__bound,axiom,
! [L: int,K2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ L )
=> ( ord_less @ int @ ( modulo_modulo @ int @ K2 @ L ) @ L ) ) ).
% Euclidean_Division.pos_mod_bound
thf(fact_613_neg__mod__bound,axiom,
! [L: int,K2: int] :
( ( ord_less @ int @ L @ ( zero_zero @ int ) )
=> ( ord_less @ int @ L @ ( modulo_modulo @ int @ K2 @ L ) ) ) ).
% neg_mod_bound
thf(fact_614_nonneg1__imp__zdiv__pos__iff,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A2 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ ( divide_divide @ int @ A2 @ B2 ) )
= ( ( ord_less_eq @ int @ B2 @ A2 )
& ( ord_less @ int @ ( zero_zero @ int ) @ B2 ) ) ) ) ).
% nonneg1_imp_zdiv_pos_iff
thf(fact_615_Euclidean__Division_Opos__mod__sign,axiom,
! [L: int,K2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ L )
=> ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( modulo_modulo @ int @ K2 @ L ) ) ) ).
% Euclidean_Division.pos_mod_sign
thf(fact_616_neg__mod__sign,axiom,
! [L: int,K2: int] :
( ( ord_less @ int @ L @ ( zero_zero @ int ) )
=> ( ord_less_eq @ int @ ( modulo_modulo @ int @ K2 @ L ) @ ( zero_zero @ int ) ) ) ).
% neg_mod_sign
thf(fact_617_pos__imp__zdiv__nonneg__iff,axiom,
! [B2: int,A2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( divide_divide @ int @ A2 @ B2 ) )
= ( ord_less_eq @ int @ ( zero_zero @ int ) @ A2 ) ) ) ).
% pos_imp_zdiv_nonneg_iff
thf(fact_618_neg__imp__zdiv__nonneg__iff,axiom,
! [B2: int,A2: int] :
( ( ord_less @ int @ B2 @ ( zero_zero @ int ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( divide_divide @ int @ A2 @ B2 ) )
= ( ord_less_eq @ int @ A2 @ ( zero_zero @ int ) ) ) ) ).
% neg_imp_zdiv_nonneg_iff
thf(fact_619_unique__quotient__lemma,axiom,
! [B2: int,Q4: int,R2: int,Q: int,R: int] :
( ( ord_less_eq @ int @ ( plus_plus @ int @ ( times_times @ int @ B2 @ Q4 ) @ R2 ) @ ( plus_plus @ int @ ( times_times @ int @ B2 @ Q ) @ R ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ R2 )
=> ( ( ord_less @ int @ R2 @ B2 )
=> ( ( ord_less @ int @ R @ B2 )
=> ( ord_less_eq @ int @ Q4 @ Q ) ) ) ) ) ).
% unique_quotient_lemma
thf(fact_620_zdiv__mono2__neg__lemma,axiom,
! [B2: int,Q: int,R: int,B6: int,Q4: int,R2: int] :
( ( ( plus_plus @ int @ ( times_times @ int @ B2 @ Q ) @ R )
= ( plus_plus @ int @ ( times_times @ int @ B6 @ Q4 ) @ R2 ) )
=> ( ( ord_less @ int @ ( plus_plus @ int @ ( times_times @ int @ B6 @ Q4 ) @ R2 ) @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ R @ B2 )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ R2 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B6 )
=> ( ( ord_less_eq @ int @ B6 @ B2 )
=> ( ord_less_eq @ int @ Q4 @ Q ) ) ) ) ) ) ) ).
% zdiv_mono2_neg_lemma
thf(fact_621_pos__imp__zdiv__pos__iff,axiom,
! [K2: int,I2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ K2 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ ( divide_divide @ int @ I2 @ K2 ) )
= ( ord_less_eq @ int @ K2 @ I2 ) ) ) ).
% pos_imp_zdiv_pos_iff
thf(fact_622_div__nonpos__pos__le0,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq @ int @ A2 @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
=> ( ord_less_eq @ int @ ( divide_divide @ int @ A2 @ B2 ) @ ( zero_zero @ int ) ) ) ) ).
% div_nonpos_pos_le0
thf(fact_623_div__nonneg__neg__le0,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A2 )
=> ( ( ord_less @ int @ B2 @ ( zero_zero @ int ) )
=> ( ord_less_eq @ int @ ( divide_divide @ int @ A2 @ B2 ) @ ( zero_zero @ int ) ) ) ) ).
% div_nonneg_neg_le0
thf(fact_624_verit__le__mono__div__int,axiom,
! [A3: int,B5: int,N: int] :
( ( ord_less @ int @ A3 @ B5 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ N )
=> ( ord_less_eq @ int
@ ( plus_plus @ int @ ( divide_divide @ int @ A3 @ N )
@ ( if @ int
@ ( ( modulo_modulo @ int @ B5 @ N )
= ( zero_zero @ int ) )
@ ( one_one @ int )
@ ( zero_zero @ int ) ) )
@ ( divide_divide @ int @ B5 @ N ) ) ) ) ).
% verit_le_mono_div_int
thf(fact_625_int__div__less__self,axiom,
! [X: int,K2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ X )
=> ( ( ord_less @ int @ ( one_one @ int ) @ K2 )
=> ( ord_less @ int @ ( divide_divide @ int @ X @ K2 ) @ X ) ) ) ).
% int_div_less_self
thf(fact_626_zmod__trivial__iff,axiom,
! [I2: int,K2: int] :
( ( ( modulo_modulo @ int @ I2 @ K2 )
= I2 )
= ( ( K2
= ( zero_zero @ int ) )
| ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ I2 )
& ( ord_less @ int @ I2 @ K2 ) )
| ( ( ord_less_eq @ int @ I2 @ ( zero_zero @ int ) )
& ( ord_less @ int @ K2 @ I2 ) ) ) ) ).
% zmod_trivial_iff
thf(fact_627_zdiv__mono2__lemma,axiom,
! [B2: int,Q: int,R: int,B6: int,Q4: int,R2: int] :
( ( ( plus_plus @ int @ ( times_times @ int @ B2 @ Q ) @ R )
= ( plus_plus @ int @ ( times_times @ int @ B6 @ Q4 ) @ R2 ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( plus_plus @ int @ ( times_times @ int @ B6 @ Q4 ) @ R2 ) )
=> ( ( ord_less @ int @ R2 @ B6 )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ R )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B6 )
=> ( ( ord_less_eq @ int @ B6 @ B2 )
=> ( ord_less_eq @ int @ Q @ Q4 ) ) ) ) ) ) ) ).
% zdiv_mono2_lemma
thf(fact_628_div__positive__int,axiom,
! [L: int,K2: int] :
( ( ord_less_eq @ int @ L @ K2 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ L )
=> ( ord_less @ int @ ( zero_zero @ int ) @ ( divide_divide @ int @ K2 @ L ) ) ) ) ).
% div_positive_int
thf(fact_629_split__pos__lemma,axiom,
! [K2: int,P3: int > int > $o,N: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ K2 )
=> ( ( P3 @ ( divide_divide @ int @ N @ K2 ) @ ( modulo_modulo @ int @ N @ K2 ) )
= ( ! [I: int,J4: int] :
( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ J4 )
& ( ord_less @ int @ J4 @ K2 )
& ( N
= ( plus_plus @ int @ ( times_times @ int @ K2 @ I ) @ J4 ) ) )
=> ( P3 @ I @ J4 ) ) ) ) ) ).
% split_pos_lemma
thf(fact_630_split__neg__lemma,axiom,
! [K2: int,P3: int > int > $o,N: int] :
( ( ord_less @ int @ K2 @ ( zero_zero @ int ) )
=> ( ( P3 @ ( divide_divide @ int @ N @ K2 ) @ ( modulo_modulo @ int @ N @ K2 ) )
= ( ! [I: int,J4: int] :
( ( ( ord_less @ int @ K2 @ J4 )
& ( ord_less_eq @ int @ J4 @ ( zero_zero @ int ) )
& ( N
= ( plus_plus @ int @ ( times_times @ int @ K2 @ I ) @ J4 ) ) )
=> ( P3 @ I @ J4 ) ) ) ) ) ).
% split_neg_lemma
thf(fact_631_div__int__pos__iff,axiom,
! [K2: int,L: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( divide_divide @ int @ K2 @ L ) )
= ( ( K2
= ( zero_zero @ int ) )
| ( L
= ( zero_zero @ int ) )
| ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K2 )
& ( ord_less_eq @ int @ ( zero_zero @ int ) @ L ) )
| ( ( ord_less @ int @ K2 @ ( zero_zero @ int ) )
& ( ord_less @ int @ L @ ( zero_zero @ int ) ) ) ) ) ).
% div_int_pos_iff
thf(fact_632_div__mod__decomp__int,axiom,
! [A3: int,N: int] :
( A3
= ( plus_plus @ int @ ( times_times @ int @ ( divide_divide @ int @ A3 @ N ) @ N ) @ ( modulo_modulo @ int @ A3 @ N ) ) ) ).
% div_mod_decomp_int
thf(fact_633_zdiv__mono2__neg,axiom,
! [A2: int,B6: int,B2: int] :
( ( ord_less @ int @ A2 @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B6 )
=> ( ( ord_less_eq @ int @ B6 @ B2 )
=> ( ord_less_eq @ int @ ( divide_divide @ int @ A2 @ B6 ) @ ( divide_divide @ int @ A2 @ B2 ) ) ) ) ) ).
% zdiv_mono2_neg
thf(fact_634_zdiv__mono1__neg,axiom,
! [A2: int,A6: int,B2: int] :
( ( ord_less_eq @ int @ A2 @ A6 )
=> ( ( ord_less @ int @ B2 @ ( zero_zero @ int ) )
=> ( ord_less_eq @ int @ ( divide_divide @ int @ A6 @ B2 ) @ ( divide_divide @ int @ A2 @ B2 ) ) ) ) ).
% zdiv_mono1_neg
thf(fact_635_int__mod__pos__eq,axiom,
! [A2: int,B2: int,Q: int,R: int] :
( ( A2
= ( plus_plus @ int @ ( times_times @ int @ B2 @ Q ) @ R ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ R )
=> ( ( ord_less @ int @ R @ B2 )
=> ( ( modulo_modulo @ int @ A2 @ B2 )
= R ) ) ) ) ).
% int_mod_pos_eq
thf(fact_636_int__mod__neg__eq,axiom,
! [A2: int,B2: int,Q: int,R: int] :
( ( A2
= ( plus_plus @ int @ ( times_times @ int @ B2 @ Q ) @ R ) )
=> ( ( ord_less_eq @ int @ R @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ B2 @ R )
=> ( ( modulo_modulo @ int @ A2 @ B2 )
= R ) ) ) ) ).
% int_mod_neg_eq
thf(fact_637_int__div__pos__eq,axiom,
! [A2: int,B2: int,Q: int,R: int] :
( ( A2
= ( plus_plus @ int @ ( times_times @ int @ B2 @ Q ) @ R ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ R )
=> ( ( ord_less @ int @ R @ B2 )
=> ( ( divide_divide @ int @ A2 @ B2 )
= Q ) ) ) ) ).
% int_div_pos_eq
thf(fact_638_int__div__neg__eq,axiom,
! [A2: int,B2: int,Q: int,R: int] :
( ( A2
= ( plus_plus @ int @ ( times_times @ int @ B2 @ Q ) @ R ) )
=> ( ( ord_less_eq @ int @ R @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ B2 @ R )
=> ( ( divide_divide @ int @ A2 @ B2 )
= Q ) ) ) ) ).
% int_div_neg_eq
thf(fact_639_zdiv__eq__0__iff,axiom,
! [I2: int,K2: int] :
( ( ( divide_divide @ int @ I2 @ K2 )
= ( zero_zero @ int ) )
= ( ( K2
= ( zero_zero @ int ) )
| ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ I2 )
& ( ord_less @ int @ I2 @ K2 ) )
| ( ( ord_less_eq @ int @ I2 @ ( zero_zero @ int ) )
& ( ord_less @ int @ K2 @ I2 ) ) ) ) ).
% zdiv_eq_0_iff
thf(fact_640_zdiv__mono__strict,axiom,
! [A3: int,B5: int,N: int] :
( ( ord_less @ int @ A3 @ B5 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ N )
=> ( ( ( modulo_modulo @ int @ A3 @ N )
= ( zero_zero @ int ) )
=> ( ( ( modulo_modulo @ int @ B5 @ N )
= ( zero_zero @ int ) )
=> ( ord_less @ int @ ( divide_divide @ int @ A3 @ N ) @ ( divide_divide @ int @ B5 @ N ) ) ) ) ) ) ).
% zdiv_mono_strict
thf(fact_641_pos__mod__conj,axiom,
! [B2: int,A2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( modulo_modulo @ int @ A2 @ B2 ) )
& ( ord_less @ int @ ( modulo_modulo @ int @ A2 @ B2 ) @ B2 ) ) ) ).
% pos_mod_conj
thf(fact_642_neg__mod__conj,axiom,
! [B2: int,A2: int] :
( ( ord_less @ int @ B2 @ ( zero_zero @ int ) )
=> ( ( ord_less_eq @ int @ ( modulo_modulo @ int @ A2 @ B2 ) @ ( zero_zero @ int ) )
& ( ord_less @ int @ B2 @ ( modulo_modulo @ int @ A2 @ B2 ) ) ) ) ).
% neg_mod_conj
thf(fact_643_q__pos__lemma,axiom,
! [B6: int,Q4: int,R2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( plus_plus @ int @ ( times_times @ int @ B6 @ Q4 ) @ R2 ) )
=> ( ( ord_less @ int @ R2 @ B6 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B6 )
=> ( ord_less_eq @ int @ ( zero_zero @ int ) @ Q4 ) ) ) ) ).
% q_pos_lemma
thf(fact_644_zdiv__mono2,axiom,
! [A2: int,B6: int,B2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A2 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B6 )
=> ( ( ord_less_eq @ int @ B6 @ B2 )
=> ( ord_less_eq @ int @ ( divide_divide @ int @ A2 @ B2 ) @ ( divide_divide @ int @ A2 @ B6 ) ) ) ) ) ).
% zdiv_mono2
thf(fact_645_zdiv__mono1,axiom,
! [A2: int,A6: int,B2: int] :
( ( ord_less_eq @ int @ A2 @ A6 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
=> ( ord_less_eq @ int @ ( divide_divide @ int @ A2 @ B2 ) @ ( divide_divide @ int @ A6 @ B2 ) ) ) ) ).
% zdiv_mono1
thf(fact_646_split__zmod,axiom,
! [P3: int > $o,N: int,K2: int] :
( ( P3 @ ( modulo_modulo @ int @ N @ K2 ) )
= ( ( ( K2
= ( zero_zero @ int ) )
=> ( P3 @ N ) )
& ( ( ord_less @ int @ ( zero_zero @ int ) @ K2 )
=> ! [I: int,J4: int] :
( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ J4 )
& ( ord_less @ int @ J4 @ K2 )
& ( N
= ( plus_plus @ int @ ( times_times @ int @ K2 @ I ) @ J4 ) ) )
=> ( P3 @ J4 ) ) )
& ( ( ord_less @ int @ K2 @ ( zero_zero @ int ) )
=> ! [I: int,J4: int] :
( ( ( ord_less @ int @ K2 @ J4 )
& ( ord_less_eq @ int @ J4 @ ( zero_zero @ int ) )
& ( N
= ( plus_plus @ int @ ( times_times @ int @ K2 @ I ) @ J4 ) ) )
=> ( P3 @ J4 ) ) ) ) ) ).
% split_zmod
thf(fact_647_split__zdiv,axiom,
! [P3: int > $o,N: int,K2: int] :
( ( P3 @ ( divide_divide @ int @ N @ K2 ) )
= ( ( ( K2
= ( zero_zero @ int ) )
=> ( P3 @ ( zero_zero @ int ) ) )
& ( ( ord_less @ int @ ( zero_zero @ int ) @ K2 )
=> ! [I: int,J4: int] :
( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ J4 )
& ( ord_less @ int @ J4 @ K2 )
& ( N
= ( plus_plus @ int @ ( times_times @ int @ K2 @ I ) @ J4 ) ) )
=> ( P3 @ I ) ) )
& ( ( ord_less @ int @ K2 @ ( zero_zero @ int ) )
=> ! [I: int,J4: int] :
( ( ( ord_less @ int @ K2 @ J4 )
& ( ord_less_eq @ int @ J4 @ ( zero_zero @ int ) )
& ( N
= ( plus_plus @ int @ ( times_times @ int @ K2 @ I ) @ J4 ) ) )
=> ( P3 @ I ) ) ) ) ) ).
% split_zdiv
thf(fact_648_zmod__eq__0__iff,axiom,
! [M: int,D2: int] :
( ( ( modulo_modulo @ int @ M @ D2 )
= ( zero_zero @ int ) )
= ( ? [Q5: int] :
( M
= ( times_times @ int @ D2 @ Q5 ) ) ) ) ).
% zmod_eq_0_iff
thf(fact_649_zmod__eq__0D,axiom,
! [M: int,D2: int] :
( ( ( modulo_modulo @ int @ M @ D2 )
= ( zero_zero @ int ) )
=> ? [Q3: int] :
( M
= ( times_times @ int @ D2 @ Q3 ) ) ) ).
% zmod_eq_0D
thf(fact_650_zmod__le__nonneg__dividend,axiom,
! [M: int,K2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ M )
=> ( ord_less_eq @ int @ ( modulo_modulo @ int @ M @ K2 ) @ M ) ) ).
% zmod_le_nonneg_dividend
thf(fact_651_zero__one__enat__neq_I1_J,axiom,
( ( zero_zero @ extended_enat )
!= ( one_one @ extended_enat ) ) ).
% zero_one_enat_neq(1)
thf(fact_652_zero__reorient,axiom,
! [A: $tType] :
( ( zero @ A )
=> ! [X: A] :
( ( ( zero_zero @ A )
= X )
= ( X
= ( zero_zero @ A ) ) ) ) ).
% zero_reorient
thf(fact_653_vebt__buildup_Ocases,axiom,
! [X: nat] :
( ( X
!= ( zero_zero @ nat ) )
=> ( ( X
!= ( suc @ ( zero_zero @ nat ) ) )
=> ~ ! [Va: nat] :
( X
!= ( suc @ ( suc @ Va ) ) ) ) ) ).
% vebt_buildup.cases
thf(fact_654_not0__implies__Suc,axiom,
! [N: nat] :
( ( N
!= ( zero_zero @ nat ) )
=> ? [M4: nat] :
( N
= ( suc @ M4 ) ) ) ).
% not0_implies_Suc
thf(fact_655_Zero__not__Suc,axiom,
! [M: nat] :
( ( zero_zero @ nat )
!= ( suc @ M ) ) ).
% Zero_not_Suc
thf(fact_656_Zero__neq__Suc,axiom,
! [M: nat] :
( ( zero_zero @ nat )
!= ( suc @ M ) ) ).
% Zero_neq_Suc
thf(fact_657_Suc__neq__Zero,axiom,
! [M: nat] :
( ( suc @ M )
!= ( zero_zero @ nat ) ) ).
% Suc_neq_Zero
thf(fact_658_zero__induct,axiom,
! [P3: nat > $o,K2: nat] :
( ( P3 @ K2 )
=> ( ! [N3: nat] :
( ( P3 @ ( suc @ N3 ) )
=> ( P3 @ N3 ) )
=> ( P3 @ ( zero_zero @ nat ) ) ) ) ).
% zero_induct
thf(fact_659_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_660_diff__induct,axiom,
! [P3: nat > nat > $o,M: nat,N: nat] :
( ! [X5: nat] : ( P3 @ X5 @ ( zero_zero @ nat ) )
=> ( ! [Y7: nat] : ( P3 @ ( zero_zero @ nat ) @ ( suc @ Y7 ) )
=> ( ! [X5: nat,Y7: nat] :
( ( P3 @ X5 @ Y7 )
=> ( P3 @ ( suc @ X5 ) @ ( suc @ Y7 ) ) )
=> ( P3 @ M @ N ) ) ) ) ).
% diff_induct
thf(fact_661_nat__induct,axiom,
! [P3: nat > $o,N: nat] :
( ( P3 @ ( zero_zero @ nat ) )
=> ( ! [N3: nat] :
( ( P3 @ N3 )
=> ( P3 @ ( suc @ N3 ) ) )
=> ( P3 @ N ) ) ) ).
% nat_induct
thf(fact_662_Suc__inject,axiom,
! [X: nat,Y2: nat] :
( ( ( suc @ X )
= ( suc @ Y2 ) )
=> ( X = Y2 ) ) ).
% Suc_inject
thf(fact_663_old_Onat_Oexhaust,axiom,
! [Y2: nat] :
( ( Y2
!= ( zero_zero @ nat ) )
=> ~ ! [Nat3: nat] :
( Y2
!= ( suc @ Nat3 ) ) ) ).
% old.nat.exhaust
thf(fact_664_nat_OdiscI,axiom,
! [Nat2: nat,X23: nat] :
( ( Nat2
= ( suc @ X23 ) )
=> ( Nat2
!= ( zero_zero @ nat ) ) ) ).
% nat.discI
thf(fact_665_old_Onat_Odistinct_I1_J,axiom,
! [Nat4: nat] :
( ( zero_zero @ nat )
!= ( suc @ Nat4 ) ) ).
% old.nat.distinct(1)
thf(fact_666_old_Onat_Odistinct_I2_J,axiom,
! [Nat5: nat] :
( ( suc @ Nat5 )
!= ( zero_zero @ nat ) ) ).
% old.nat.distinct(2)
thf(fact_667_nat_Odistinct_I1_J,axiom,
! [X23: nat] :
( ( zero_zero @ nat )
!= ( suc @ X23 ) ) ).
% nat.distinct(1)
thf(fact_668_less__Suc__eq__0__disj,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ ( suc @ N ) )
= ( ( M
= ( zero_zero @ nat ) )
| ? [J4: nat] :
( ( M
= ( suc @ J4 ) )
& ( ord_less @ nat @ J4 @ N ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_669_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ? [M4: nat] :
( N
= ( suc @ M4 ) ) ) ).
% gr0_implies_Suc
thf(fact_670_All__less__Suc2,axiom,
! [N: nat,P3: nat > $o] :
( ( ! [I: nat] :
( ( ord_less @ nat @ I @ ( suc @ N ) )
=> ( P3 @ I ) ) )
= ( ( P3 @ ( zero_zero @ nat ) )
& ! [I: nat] :
( ( ord_less @ nat @ I @ N )
=> ( P3 @ ( suc @ I ) ) ) ) ) ).
% All_less_Suc2
thf(fact_671_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
= ( ? [M2: nat] :
( N
= ( suc @ M2 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_672_Ex__less__Suc2,axiom,
! [N: nat,P3: nat > $o] :
( ( ? [I: nat] :
( ( ord_less @ nat @ I @ ( suc @ N ) )
& ( P3 @ I ) ) )
= ( ( P3 @ ( zero_zero @ nat ) )
| ? [I: nat] :
( ( ord_less @ nat @ I @ N )
& ( P3 @ ( suc @ I ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_673_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_674_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_675_realpow__pos__nth2,axiom,
! [A2: real,N: nat] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
=> ? [R3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R3 )
& ( ( power_power @ real @ R3 @ ( suc @ N ) )
= A2 ) ) ) ).
% realpow_pos_nth2
thf(fact_676_One__nat__def,axiom,
( ( one_one @ nat )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% One_nat_def
thf(fact_677_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_678_enat__0__iff_I2_J,axiom,
! [X: nat] :
( ( ( zero_zero @ extended_enat )
= ( extended_enat2 @ X ) )
= ( X
= ( zero_zero @ nat ) ) ) ).
% enat_0_iff(2)
thf(fact_679_enat__0__iff_I1_J,axiom,
! [X: nat] :
( ( ( extended_enat2 @ X )
= ( zero_zero @ extended_enat ) )
= ( X
= ( zero_zero @ nat ) ) ) ).
% enat_0_iff(1)
thf(fact_680_zero__enat__def,axiom,
( ( zero_zero @ extended_enat )
= ( extended_enat2 @ ( zero_zero @ nat ) ) ) ).
% zero_enat_def
thf(fact_681_power__inject__base,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,N: nat,B2: A] :
( ( ( power_power @ A @ A2 @ ( suc @ N ) )
= ( power_power @ A @ B2 @ ( suc @ N ) ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( A2 = B2 ) ) ) ) ) ).
% power_inject_base
thf(fact_682_power__le__imp__le__base,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,N: nat,B2: A] :
( ( ord_less_eq @ A @ ( power_power @ A @ A2 @ ( suc @ N ) ) @ ( power_power @ A @ B2 @ ( suc @ N ) ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less_eq @ A @ A2 @ B2 ) ) ) ) ).
% power_le_imp_le_base
thf(fact_683_numeral__1__eq__Suc__0,axiom,
( ( numeral_numeral @ nat @ one2 )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% numeral_1_eq_Suc_0
thf(fact_684_num_Osize_I5_J,axiom,
! [X23: num] :
( ( size_size @ num @ ( bit0 @ X23 ) )
= ( plus_plus @ nat @ ( size_size @ num @ X23 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% num.size(5)
thf(fact_685_ex__least__nat__less,axiom,
! [P3: nat > $o,N: nat] :
( ( P3 @ N )
=> ( ~ ( P3 @ ( zero_zero @ nat ) )
=> ? [K: nat] :
( ( ord_less @ nat @ K @ N )
& ! [I4: nat] :
( ( ord_less_eq @ nat @ I4 @ K )
=> ~ ( P3 @ I4 ) )
& ( P3 @ ( suc @ K ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_686_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_687_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_688_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_689_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_690_nat__induct__non__zero,axiom,
! [N: nat,P3: nat > $o] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( P3 @ ( one_one @ nat ) )
=> ( ! [N3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( ( P3 @ N3 )
=> ( P3 @ ( suc @ N3 ) ) ) )
=> ( P3 @ N ) ) ) ) ).
% nat_induct_non_zero
thf(fact_691_power__gt__expt,axiom,
! [N: nat,K2: nat] :
( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N )
=> ( ord_less @ nat @ K2 @ ( power_power @ nat @ N @ K2 ) ) ) ).
% power_gt_expt
thf(fact_692_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_693_nat__one__le__power,axiom,
! [I2: nat,N: nat] :
( ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ I2 )
=> ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( power_power @ nat @ I2 @ N ) ) ) ).
% nat_one_le_power
thf(fact_694_realpow__pos__nth__unique,axiom,
! [N: nat,A2: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
=> ? [X5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X5 )
& ( ( power_power @ real @ X5 @ N )
= A2 )
& ! [Y4: real] :
( ( ( ord_less @ real @ ( zero_zero @ real ) @ Y4 )
& ( ( power_power @ real @ Y4 @ N )
= A2 ) )
=> ( Y4 = X5 ) ) ) ) ) ).
% realpow_pos_nth_unique
thf(fact_695_realpow__pos__nth,axiom,
! [N: nat,A2: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
=> ? [R3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R3 )
& ( ( power_power @ real @ R3 @ N )
= A2 ) ) ) ) ).
% realpow_pos_nth
thf(fact_696_Nat_OlessE,axiom,
! [I2: nat,K2: nat] :
( ( ord_less @ nat @ I2 @ K2 )
=> ( ( K2
!= ( suc @ I2 ) )
=> ~ ! [J2: nat] :
( ( ord_less @ nat @ I2 @ J2 )
=> ( K2
!= ( suc @ J2 ) ) ) ) ) ).
% Nat.lessE
thf(fact_697_Suc__lessD,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ ( suc @ M ) @ N )
=> ( ord_less @ nat @ M @ N ) ) ).
% Suc_lessD
thf(fact_698_Suc__lessE,axiom,
! [I2: nat,K2: nat] :
( ( ord_less @ nat @ ( suc @ I2 ) @ K2 )
=> ~ ! [J2: nat] :
( ( ord_less @ nat @ I2 @ J2 )
=> ( K2
!= ( suc @ J2 ) ) ) ) ).
% Suc_lessE
thf(fact_699_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_700_less__SucE,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ ( suc @ N ) )
=> ( ~ ( ord_less @ nat @ M @ N )
=> ( M = N ) ) ) ).
% less_SucE
thf(fact_701_less__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ N )
=> ( ord_less @ nat @ M @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_702_Ex__less__Suc,axiom,
! [N: nat,P3: nat > $o] :
( ( ? [I: nat] :
( ( ord_less @ nat @ I @ ( suc @ N ) )
& ( P3 @ I ) ) )
= ( ( P3 @ N )
| ? [I: nat] :
( ( ord_less @ nat @ I @ N )
& ( P3 @ I ) ) ) ) ).
% Ex_less_Suc
thf(fact_703_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_704_not__less__eq,axiom,
! [M: nat,N: nat] :
( ( ~ ( ord_less @ nat @ M @ N ) )
= ( ord_less @ nat @ N @ ( suc @ M ) ) ) ).
% not_less_eq
thf(fact_705_All__less__Suc,axiom,
! [N: nat,P3: nat > $o] :
( ( ! [I: nat] :
( ( ord_less @ nat @ I @ ( suc @ N ) )
=> ( P3 @ I ) ) )
= ( ( P3 @ N )
& ! [I: nat] :
( ( ord_less @ nat @ I @ N )
=> ( P3 @ I ) ) ) ) ).
% All_less_Suc
thf(fact_706_Suc__less__eq2,axiom,
! [N: nat,M: nat] :
( ( ord_less @ nat @ ( suc @ N ) @ M )
= ( ? [M5: nat] :
( ( M
= ( suc @ M5 ) )
& ( ord_less @ nat @ N @ M5 ) ) ) ) ).
% Suc_less_eq2
thf(fact_707_less__antisym,axiom,
! [N: nat,M: nat] :
( ~ ( ord_less @ nat @ N @ M )
=> ( ( ord_less @ nat @ N @ ( suc @ M ) )
=> ( M = N ) ) ) ).
% less_antisym
thf(fact_708_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_709_less__trans__Suc,axiom,
! [I2: nat,J3: nat,K2: nat] :
( ( ord_less @ nat @ I2 @ J3 )
=> ( ( ord_less @ nat @ J3 @ K2 )
=> ( ord_less @ nat @ ( suc @ I2 ) @ K2 ) ) ) ).
% less_trans_Suc
thf(fact_710_less__Suc__induct,axiom,
! [I2: nat,J3: nat,P3: nat > nat > $o] :
( ( ord_less @ nat @ I2 @ J3 )
=> ( ! [I3: nat] : ( P3 @ I3 @ ( suc @ I3 ) )
=> ( ! [I3: nat,J2: nat,K: nat] :
( ( ord_less @ nat @ I3 @ J2 )
=> ( ( ord_less @ nat @ J2 @ K )
=> ( ( P3 @ I3 @ J2 )
=> ( ( P3 @ J2 @ K )
=> ( P3 @ I3 @ K ) ) ) ) )
=> ( P3 @ I2 @ J3 ) ) ) ) ).
% less_Suc_induct
thf(fact_711_strict__inc__induct,axiom,
! [I2: nat,J3: nat,P3: nat > $o] :
( ( ord_less @ nat @ I2 @ J3 )
=> ( ! [I3: nat] :
( ( J3
= ( suc @ I3 ) )
=> ( P3 @ I3 ) )
=> ( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ J3 )
=> ( ( P3 @ ( suc @ I3 ) )
=> ( P3 @ I3 ) ) )
=> ( P3 @ I2 ) ) ) ) ).
% strict_inc_induct
thf(fact_712_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_713_invar__vebt_Ointros_I1_J,axiom,
! [A2: $o,B2: $o] : ( vEBT_invar_vebt @ ( vEBT_Leaf @ A2 @ B2 ) @ ( suc @ ( zero_zero @ nat ) ) ) ).
% invar_vebt.intros(1)
thf(fact_714_transitive__stepwise__le,axiom,
! [M: nat,N: nat,R4: nat > nat > $o] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ! [X5: nat] : ( R4 @ X5 @ X5 )
=> ( ! [X5: nat,Y7: nat,Z: nat] :
( ( R4 @ X5 @ Y7 )
=> ( ( R4 @ Y7 @ Z )
=> ( R4 @ X5 @ Z ) ) )
=> ( ! [N3: nat] : ( R4 @ N3 @ ( suc @ N3 ) )
=> ( R4 @ M @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_715_nat__induct__at__least,axiom,
! [M: nat,N: nat,P3: nat > $o] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( P3 @ M )
=> ( ! [N3: nat] :
( ( ord_less_eq @ nat @ M @ N3 )
=> ( ( P3 @ N3 )
=> ( P3 @ ( suc @ N3 ) ) ) )
=> ( P3 @ N ) ) ) ) ).
% nat_induct_at_least
thf(fact_716_full__nat__induct,axiom,
! [P3: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M3: nat] :
( ( ord_less_eq @ nat @ ( suc @ M3 ) @ N3 )
=> ( P3 @ M3 ) )
=> ( P3 @ N3 ) )
=> ( P3 @ N ) ) ).
% full_nat_induct
thf(fact_717_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_718_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq @ nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_719_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_720_Suc__le__D,axiom,
! [N: nat,M6: nat] :
( ( ord_less_eq @ nat @ ( suc @ N ) @ M6 )
=> ? [M4: nat] :
( M6
= ( suc @ M4 ) ) ) ).
% Suc_le_D
thf(fact_721_le__SucI,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ord_less_eq @ nat @ M @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_722_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_723_Suc__leD,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ ( suc @ M ) @ N )
=> ( ord_less_eq @ nat @ M @ N ) ) ).
% Suc_leD
thf(fact_724_nat__arith_Osuc1,axiom,
! [A3: nat,K2: nat,A2: nat] :
( ( A3
= ( plus_plus @ nat @ K2 @ A2 ) )
=> ( ( suc @ A3 )
= ( plus_plus @ nat @ K2 @ ( suc @ A2 ) ) ) ) ).
% nat_arith.suc1
thf(fact_725_add__Suc,axiom,
! [M: nat,N: nat] :
( ( plus_plus @ nat @ ( suc @ M ) @ N )
= ( suc @ ( plus_plus @ nat @ M @ N ) ) ) ).
% add_Suc
thf(fact_726_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_727_Suc__mult__cancel1,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ( times_times @ nat @ ( suc @ K2 ) @ M )
= ( times_times @ nat @ ( suc @ K2 ) @ N ) )
= ( M = N ) ) ).
% Suc_mult_cancel1
thf(fact_728_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_729_zero__le,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [X: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ X ) ) ).
% zero_le
thf(fact_730_field__lbound__gt__zero,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [D1: A,D22: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ D1 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ D22 )
=> ? [E2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ E2 )
& ( ord_less @ A @ E2 @ D1 )
& ( ord_less @ A @ E2 @ D22 ) ) ) ) ) ).
% field_lbound_gt_zero
thf(fact_731_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_732_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_733_not__less__zero,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [N: A] :
~ ( ord_less @ A @ N @ ( zero_zero @ A ) ) ) ).
% not_less_zero
thf(fact_734_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_735_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_736_zero__neq__numeral,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [N: num] :
( ( zero_zero @ A )
!= ( numeral_numeral @ A @ N ) ) ) ).
% zero_neq_numeral
thf(fact_737_mult__not__zero,axiom,
! [A: $tType] :
( ( mult_zero @ A )
=> ! [A2: A,B2: A] :
( ( ( times_times @ A @ A2 @ B2 )
!= ( zero_zero @ A ) )
=> ( ( A2
!= ( zero_zero @ A ) )
& ( B2
!= ( zero_zero @ A ) ) ) ) ) ).
% mult_not_zero
thf(fact_738_divisors__zero,axiom,
! [A: $tType] :
( ( semiri3467727345109120633visors @ A )
=> ! [A2: A,B2: A] :
( ( ( times_times @ A @ A2 @ B2 )
= ( zero_zero @ A ) )
=> ( ( A2
= ( zero_zero @ A ) )
| ( B2
= ( zero_zero @ A ) ) ) ) ) ).
% divisors_zero
thf(fact_739_no__zero__divisors,axiom,
! [A: $tType] :
( ( semiri3467727345109120633visors @ A )
=> ! [A2: A,B2: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( B2
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ A2 @ B2 )
!= ( zero_zero @ A ) ) ) ) ) ).
% no_zero_divisors
thf(fact_740_mult__left__cancel,axiom,
! [A: $tType] :
( ( semiri6575147826004484403cancel @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( ( times_times @ A @ C2 @ A2 )
= ( times_times @ A @ C2 @ B2 ) )
= ( A2 = B2 ) ) ) ) ).
% mult_left_cancel
thf(fact_741_mult__right__cancel,axiom,
! [A: $tType] :
( ( semiri6575147826004484403cancel @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( ( times_times @ A @ A2 @ C2 )
= ( times_times @ A @ B2 @ C2 ) )
= ( A2 = B2 ) ) ) ) ).
% mult_right_cancel
thf(fact_742_zero__neq__one,axiom,
! [A: $tType] :
( ( zero_neq_one @ A )
=> ( ( zero_zero @ A )
!= ( one_one @ A ) ) ) ).
% zero_neq_one
thf(fact_743_verit__sum__simplify,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add @ A )
=> ! [A2: A] :
( ( plus_plus @ A @ A2 @ ( zero_zero @ A ) )
= A2 ) ) ).
% verit_sum_simplify
thf(fact_744_comm__monoid__add__class_Oadd__0,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A2: A] :
( ( plus_plus @ A @ ( zero_zero @ A ) @ A2 )
= A2 ) ) ).
% comm_monoid_add_class.add_0
thf(fact_745_add_Ocomm__neutral,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A2: A] :
( ( plus_plus @ A @ A2 @ ( zero_zero @ A ) )
= A2 ) ) ).
% add.comm_neutral
thf(fact_746_add_Ogroup__left__neutral,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A] :
( ( plus_plus @ A @ ( zero_zero @ A ) @ A2 )
= A2 ) ) ).
% add.group_left_neutral
thf(fact_747_power__not__zero,axiom,
! [A: $tType] :
( ( semiri2026040879449505780visors @ A )
=> ! [A2: A,N: nat] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( power_power @ A @ A2 @ N )
!= ( zero_zero @ A ) ) ) ) ).
% power_not_zero
thf(fact_748_num_Osize_I4_J,axiom,
( ( size_size @ num @ one2 )
= ( zero_zero @ nat ) ) ).
% num.size(4)
thf(fact_749_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_750_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_751_bot__nat__0_Oextremum__strict,axiom,
! [A2: nat] :
~ ( ord_less @ nat @ A2 @ ( zero_zero @ nat ) ) ).
% bot_nat_0.extremum_strict
thf(fact_752_gr0I,axiom,
! [N: nat] :
( ( N
!= ( zero_zero @ nat ) )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ).
% gr0I
thf(fact_753_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% not_gr0
thf(fact_754_not__less0,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% not_less0
thf(fact_755_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% less_zeroE
thf(fact_756_gr__implies__not0,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ N )
=> ( N
!= ( zero_zero @ nat ) ) ) ).
% gr_implies_not0
thf(fact_757_infinite__descent0,axiom,
! [P3: nat > $o,N: nat] :
( ( P3 @ ( zero_zero @ nat ) )
=> ( ! [N3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( ~ ( P3 @ N3 )
=> ? [M3: nat] :
( ( ord_less @ nat @ M3 @ N3 )
& ~ ( P3 @ M3 ) ) ) )
=> ( P3 @ N ) ) ) ).
% infinite_descent0
thf(fact_758_infinite__descent0__measure,axiom,
! [A: $tType,V: A > nat,P3: A > $o,X: A] :
( ! [X5: A] :
( ( ( V @ X5 )
= ( zero_zero @ nat ) )
=> ( P3 @ X5 ) )
=> ( ! [X5: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( V @ X5 ) )
=> ( ~ ( P3 @ X5 )
=> ? [Y4: A] :
( ( ord_less @ nat @ ( V @ Y4 ) @ ( V @ X5 ) )
& ~ ( P3 @ Y4 ) ) ) )
=> ( P3 @ X ) ) ) ).
% infinite_descent0_measure
thf(fact_759_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq @ nat @ N @ ( zero_zero @ nat ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% le_0_eq
thf(fact_760_bot__nat__0_Oextremum__uniqueI,axiom,
! [A2: nat] :
( ( ord_less_eq @ nat @ A2 @ ( zero_zero @ nat ) )
=> ( A2
= ( zero_zero @ nat ) ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_761_bot__nat__0_Oextremum__unique,axiom,
! [A2: nat] :
( ( ord_less_eq @ nat @ A2 @ ( zero_zero @ nat ) )
= ( A2
= ( zero_zero @ nat ) ) ) ).
% bot_nat_0.extremum_unique
thf(fact_762_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% less_eq_nat.simps(1)
thf(fact_763_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus @ nat @ ( zero_zero @ nat ) @ N )
= N ) ).
% plus_nat.add_0
thf(fact_764_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_765_nat__mult__eq__cancel__disj,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ( times_times @ nat @ K2 @ M )
= ( times_times @ nat @ K2 @ N ) )
= ( ( K2
= ( zero_zero @ nat ) )
| ( M = N ) ) ) ).
% nat_mult_eq_cancel_disj
thf(fact_766_mult__0,axiom,
! [N: nat] :
( ( times_times @ nat @ ( zero_zero @ nat ) @ N )
= ( zero_zero @ nat ) ) ).
% mult_0
thf(fact_767_real__arch__pow__inv,axiom,
! [Y2: real,X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ Y2 )
=> ( ( ord_less @ real @ X @ ( one_one @ real ) )
=> ? [N3: nat] : ( ord_less @ real @ ( power_power @ real @ X @ N3 ) @ Y2 ) ) ) ).
% real_arch_pow_inv
thf(fact_768_infinity__ne__i0,axiom,
( ( extend4730790105801354508finity @ extended_enat )
!= ( zero_zero @ extended_enat ) ) ).
% infinity_ne_i0
thf(fact_769_VEBT_Osize_I4_J,axiom,
! [X21: $o,X22: $o] :
( ( size_size @ vEBT_VEBT @ ( vEBT_Leaf @ X21 @ X22 ) )
= ( zero_zero @ nat ) ) ).
% VEBT.size(4)
thf(fact_770_power__Suc__le__self,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,N: nat] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ A2 @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( power_power @ A @ A2 @ ( suc @ N ) ) @ A2 ) ) ) ) ).
% power_Suc_le_self
thf(fact_771_power__Suc__less__one,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,N: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ A2 @ ( one_one @ A ) )
=> ( ord_less @ A @ ( power_power @ A @ A2 @ ( suc @ N ) ) @ ( one_one @ A ) ) ) ) ) ).
% power_Suc_less_one
thf(fact_772_not__iless0,axiom,
! [N: extended_enat] :
~ ( ord_less @ extended_enat @ N @ ( zero_zero @ extended_enat ) ) ).
% not_iless0
thf(fact_773_numeral__2__eq__2,axiom,
( ( numeral_numeral @ nat @ ( bit0 @ one2 ) )
= ( suc @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% numeral_2_eq_2
thf(fact_774_iadd__is__0,axiom,
! [M: extended_enat,N: extended_enat] :
( ( ( plus_plus @ extended_enat @ M @ N )
= ( zero_zero @ extended_enat ) )
= ( ( M
= ( zero_zero @ extended_enat ) )
& ( N
= ( zero_zero @ extended_enat ) ) ) ) ).
% iadd_is_0
thf(fact_775_ile0__eq,axiom,
! [N: extended_enat] :
( ( ord_less_eq @ extended_enat @ N @ ( zero_zero @ extended_enat ) )
= ( N
= ( zero_zero @ extended_enat ) ) ) ).
% ile0_eq
thf(fact_776_i0__lb,axiom,
! [N: extended_enat] : ( ord_less_eq @ extended_enat @ ( zero_zero @ extended_enat ) @ N ) ).
% i0_lb
thf(fact_777_power__eq__iff__eq__base,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N: nat,A2: A,B2: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ( power_power @ A @ A2 @ N )
= ( power_power @ A @ B2 @ N ) )
= ( A2 = B2 ) ) ) ) ) ) ).
% power_eq_iff_eq_base
thf(fact_778_power__eq__imp__eq__base,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,N: nat,B2: A] :
( ( ( power_power @ A @ A2 @ N )
= ( power_power @ A @ B2 @ N ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( A2 = B2 ) ) ) ) ) ) ).
% power_eq_imp_eq_base
thf(fact_779_imult__is__0,axiom,
! [M: extended_enat,N: extended_enat] :
( ( ( times_times @ extended_enat @ M @ N )
= ( zero_zero @ extended_enat ) )
= ( ( M
= ( zero_zero @ extended_enat ) )
| ( N
= ( zero_zero @ extended_enat ) ) ) ) ).
% imult_is_0
thf(fact_780_lambda__zero,axiom,
! [A: $tType] :
( ( mult_zero @ A )
=> ( ( ^ [H: A] : ( zero_zero @ A ) )
= ( times_times @ A @ ( zero_zero @ A ) ) ) ) ).
% lambda_zero
thf(fact_781_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_782_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_783_power__strict__mono,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,B2: A,N: nat] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ord_less @ A @ ( power_power @ A @ A2 @ N ) @ ( power_power @ A @ B2 @ N ) ) ) ) ) ) ).
% power_strict_mono
thf(fact_784_split__div_H,axiom,
! [P3: nat > $o,M: nat,N: nat] :
( ( P3 @ ( divide_divide @ nat @ M @ N ) )
= ( ( ( N
= ( zero_zero @ nat ) )
& ( P3 @ ( zero_zero @ nat ) ) )
| ? [Q5: nat] :
( ( ord_less_eq @ nat @ ( times_times @ nat @ N @ Q5 ) @ M )
& ( ord_less @ nat @ M @ ( times_times @ nat @ N @ ( suc @ Q5 ) ) )
& ( P3 @ Q5 ) ) ) ) ).
% split_div'
thf(fact_785_power__Suc2,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A2: A,N: nat] :
( ( power_power @ A @ A2 @ ( suc @ N ) )
= ( times_times @ A @ ( power_power @ A @ A2 @ N ) @ A2 ) ) ) ).
% power_Suc2
thf(fact_786_power__Suc,axiom,
! [A: $tType] :
( ( power @ A )
=> ! [A2: A,N: nat] :
( ( power_power @ A @ A2 @ ( suc @ N ) )
= ( times_times @ A @ A2 @ ( power_power @ A @ A2 @ N ) ) ) ) ).
% power_Suc
thf(fact_787_lift__Suc__mono__less,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [F3: nat > A,N: nat,N5: nat] :
( ! [N3: nat] : ( ord_less @ A @ ( F3 @ N3 ) @ ( F3 @ ( suc @ N3 ) ) )
=> ( ( ord_less @ nat @ N @ N5 )
=> ( ord_less @ A @ ( F3 @ N ) @ ( F3 @ N5 ) ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_788_lift__Suc__mono__less__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [F3: nat > A,N: nat,M: nat] :
( ! [N3: nat] : ( ord_less @ A @ ( F3 @ N3 ) @ ( F3 @ ( suc @ N3 ) ) )
=> ( ( ord_less @ A @ ( F3 @ N ) @ ( F3 @ M ) )
= ( ord_less @ nat @ N @ M ) ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_789_lift__Suc__mono__le,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [F3: nat > A,N: nat,N5: nat] :
( ! [N3: nat] : ( ord_less_eq @ A @ ( F3 @ N3 ) @ ( F3 @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq @ nat @ N @ N5 )
=> ( ord_less_eq @ A @ ( F3 @ N ) @ ( F3 @ N5 ) ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_790_lift__Suc__antimono__le,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [F3: nat > A,N: nat,N5: nat] :
( ! [N3: nat] : ( ord_less_eq @ A @ ( F3 @ ( suc @ N3 ) ) @ ( F3 @ N3 ) )
=> ( ( ord_less_eq @ nat @ N @ N5 )
=> ( ord_less_eq @ A @ ( F3 @ N5 ) @ ( F3 @ N ) ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_791_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_792_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_793_less__eq__Suc__le,axiom,
( ( ord_less @ nat )
= ( ^ [N2: nat] : ( ord_less_eq @ nat @ ( suc @ N2 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_794_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_795_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_796_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_797_inc__induct,axiom,
! [I2: nat,J3: nat,P3: nat > $o] :
( ( ord_less_eq @ nat @ I2 @ J3 )
=> ( ( P3 @ J3 )
=> ( ! [N3: nat] :
( ( ord_less_eq @ nat @ I2 @ N3 )
=> ( ( ord_less @ nat @ N3 @ J3 )
=> ( ( P3 @ ( suc @ N3 ) )
=> ( P3 @ N3 ) ) ) )
=> ( P3 @ I2 ) ) ) ) ).
% inc_induct
thf(fact_798_dec__induct,axiom,
! [I2: nat,J3: nat,P3: nat > $o] :
( ( ord_less_eq @ nat @ I2 @ J3 )
=> ( ( P3 @ I2 )
=> ( ! [N3: nat] :
( ( ord_less_eq @ nat @ I2 @ N3 )
=> ( ( ord_less @ nat @ N3 @ J3 )
=> ( ( P3 @ N3 )
=> ( P3 @ ( suc @ N3 ) ) ) ) )
=> ( P3 @ J3 ) ) ) ) ).
% dec_induct
thf(fact_799_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_800_Suc__leI,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ N )
=> ( ord_less_eq @ nat @ ( suc @ M ) @ N ) ) ).
% Suc_leI
thf(fact_801_less__natE,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ N )
=> ~ ! [Q3: nat] :
( N
!= ( suc @ ( plus_plus @ nat @ M @ Q3 ) ) ) ) ).
% less_natE
thf(fact_802_less__add__Suc1,axiom,
! [I2: nat,M: nat] : ( ord_less @ nat @ I2 @ ( suc @ ( plus_plus @ nat @ I2 @ M ) ) ) ).
% less_add_Suc1
thf(fact_803_less__add__Suc2,axiom,
! [I2: nat,M: nat] : ( ord_less @ nat @ I2 @ ( suc @ ( plus_plus @ nat @ M @ I2 ) ) ) ).
% less_add_Suc2
thf(fact_804_less__iff__Suc__add,axiom,
( ( ord_less @ nat )
= ( ^ [M2: nat,N2: nat] :
? [K3: nat] :
( N2
= ( suc @ ( plus_plus @ nat @ M2 @ K3 ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_805_less__imp__Suc__add,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ N )
=> ? [K: nat] :
( N
= ( suc @ ( plus_plus @ nat @ M @ K ) ) ) ) ).
% less_imp_Suc_add
thf(fact_806_Suc__mult__less__cancel1,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less @ nat @ ( times_times @ nat @ ( suc @ K2 ) @ M ) @ ( times_times @ nat @ ( suc @ K2 ) @ N ) )
= ( ord_less @ nat @ M @ N ) ) ).
% Suc_mult_less_cancel1
thf(fact_807_Suc__mult__le__cancel1,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ ( times_times @ nat @ ( suc @ K2 ) @ M ) @ ( times_times @ nat @ ( suc @ K2 ) @ N ) )
= ( ord_less_eq @ nat @ M @ N ) ) ).
% Suc_mult_le_cancel1
thf(fact_808_dbl__def,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( neg_numeral_dbl @ A )
= ( ^ [X4: A] : ( plus_plus @ A @ X4 @ X4 ) ) ) ) ).
% dbl_def
thf(fact_809_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_810_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_811_Suc__eq__plus1__left,axiom,
( suc
= ( plus_plus @ nat @ ( one_one @ nat ) ) ) ).
% Suc_eq_plus1_left
thf(fact_812_plus__1__eq__Suc,axiom,
( ( plus_plus @ nat @ ( one_one @ nat ) )
= suc ) ).
% plus_1_eq_Suc
thf(fact_813_Suc__eq__plus1,axiom,
( suc
= ( ^ [N2: nat] : ( plus_plus @ nat @ N2 @ ( one_one @ nat ) ) ) ) ).
% Suc_eq_plus1
thf(fact_814_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_815_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_816_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: $tType] :
( ( ordere2520102378445227354miring @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_817_zero__le__mult__iff,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B2 ) )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) )
| ( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
& ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) ) ) ) ) ).
% zero_le_mult_iff
thf(fact_818_mult__nonneg__nonpos2,axiom,
! [A: $tType] :
( ( ordered_semiring_0 @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ B2 @ A2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_819_mult__nonpos__nonneg,axiom,
! [A: $tType] :
( ( ordered_semiring_0 @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ A2 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% mult_nonpos_nonneg
thf(fact_820_mult__nonneg__nonpos,axiom,
! [A: $tType] :
( ( ordered_semiring_0 @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ A2 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% mult_nonneg_nonpos
thf(fact_821_mult__nonneg__nonneg,axiom,
! [A: $tType] :
( ( ordered_semiring_0 @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B2 ) ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_822_split__mult__neg__le,axiom,
! [A: $tType] :
( ( ordered_semiring_0 @ A )
=> ! [A2: A,B2: A] :
( ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
& ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) )
| ( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ A2 @ B2 ) @ ( zero_zero @ A ) ) ) ) ).
% split_mult_neg_le
thf(fact_823_mult__le__0__iff,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( times_times @ A @ A2 @ B2 ) @ ( zero_zero @ A ) )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
& ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) )
| ( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) ) ) ) ) ).
% mult_le_0_iff
thf(fact_824_mult__right__mono,axiom,
! [A: $tType] :
( ( ordered_semiring @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ) ).
% mult_right_mono
thf(fact_825_mult__right__mono__neg,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ) ).
% mult_right_mono_neg
thf(fact_826_mult__left__mono,axiom,
! [A: $tType] :
( ( ordered_semiring @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) ) ) ) ) ).
% mult_left_mono
thf(fact_827_mult__nonpos__nonpos,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B2 ) ) ) ) ) ).
% mult_nonpos_nonpos
thf(fact_828_mult__left__mono__neg,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) ) ) ) ) ).
% mult_left_mono_neg
thf(fact_829_split__mult__pos__le,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [A2: A,B2: A] :
( ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) )
| ( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
& ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B2 ) ) ) ) ).
% split_mult_pos_le
thf(fact_830_zero__le__square,axiom,
! [A: $tType] :
( ( linordered_ring @ A )
=> ! [A2: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ A2 ) ) ) ).
% zero_le_square
thf(fact_831_mult__mono_H,axiom,
! [A: $tType] :
( ( ordered_semiring @ A )
=> ! [A2: A,B2: A,C2: A,D2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ D2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ D2 ) ) ) ) ) ) ) ).
% mult_mono'
thf(fact_832_mult__mono,axiom,
! [A: $tType] :
( ( ordered_semiring @ A )
=> ! [A2: A,B2: A,C2: A,D2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ D2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ D2 ) ) ) ) ) ) ) ).
% mult_mono
thf(fact_833_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_834_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_835_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_836_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_837_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_838_add__decreasing,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ C2 @ B2 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C2 ) @ B2 ) ) ) ) ).
% add_decreasing
thf(fact_839_add__increasing,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ B2 @ C2 )
=> ( ord_less_eq @ A @ B2 @ ( plus_plus @ A @ A2 @ C2 ) ) ) ) ) ).
% add_increasing
thf(fact_840_add__decreasing2,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C2 ) @ B2 ) ) ) ) ).
% add_decreasing2
thf(fact_841_add__increasing2,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [C2: A,B2: A,A2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ord_less_eq @ A @ B2 @ ( plus_plus @ A @ A2 @ C2 ) ) ) ) ) ).
% add_increasing2
thf(fact_842_add__nonneg__nonneg,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_843_add__nonpos__nonpos,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% add_nonpos_nonpos
thf(fact_844_add__nonneg__eq__0__iff,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [X: A,Y2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y2 )
=> ( ( ( plus_plus @ A @ X @ Y2 )
= ( zero_zero @ A ) )
= ( ( X
= ( zero_zero @ A ) )
& ( Y2
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_845_add__nonpos__eq__0__iff,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [X: A,Y2: A] :
( ( ord_less_eq @ A @ X @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ Y2 @ ( zero_zero @ A ) )
=> ( ( ( plus_plus @ A @ X @ Y2 )
= ( zero_zero @ A ) )
= ( ( X
= ( zero_zero @ A ) )
& ( Y2
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_846_mult__neg__neg,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B2 ) ) ) ) ) ).
% mult_neg_neg
thf(fact_847_not__square__less__zero,axiom,
! [A: $tType] :
( ( linordered_ring @ A )
=> ! [A2: A] :
~ ( ord_less @ A @ ( times_times @ A @ A2 @ A2 ) @ ( zero_zero @ A ) ) ) ).
% not_square_less_zero
thf(fact_848_mult__less__0__iff,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( times_times @ A @ A2 @ B2 ) @ ( zero_zero @ A ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
& ( ord_less @ A @ B2 @ ( zero_zero @ A ) ) )
| ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
& ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) ) ) ) ) ).
% mult_less_0_iff
thf(fact_849_mult__neg__pos,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less @ A @ ( times_times @ A @ A2 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% mult_neg_pos
thf(fact_850_mult__pos__neg,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ A2 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% mult_pos_neg
thf(fact_851_mult__pos__pos,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B2 ) ) ) ) ) ).
% mult_pos_pos
thf(fact_852_mult__pos__neg2,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ B2 @ A2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% mult_pos_neg2
thf(fact_853_zero__less__mult__iff,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
& ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) )
| ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
& ( ord_less @ A @ B2 @ ( zero_zero @ A ) ) ) ) ) ) ).
% zero_less_mult_iff
thf(fact_854_zero__less__mult__pos,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B2 ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) ) ) ) ).
% zero_less_mult_pos
thf(fact_855_zero__less__mult__pos2,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [B2: A,A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ B2 @ A2 ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) ) ) ) ).
% zero_less_mult_pos2
thf(fact_856_mult__less__cancel__left__neg,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) )
= ( ord_less @ A @ B2 @ A2 ) ) ) ) ).
% mult_less_cancel_left_neg
thf(fact_857_mult__less__cancel__left__pos,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) )
= ( ord_less @ A @ A2 @ B2 ) ) ) ) ).
% mult_less_cancel_left_pos
thf(fact_858_mult__strict__left__mono__neg,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( ord_less @ A @ B2 @ A2 )
=> ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) ) ) ) ) ).
% mult_strict_left_mono_neg
thf(fact_859_mult__strict__left__mono,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) ) ) ) ) ).
% mult_strict_left_mono
thf(fact_860_mult__less__cancel__left__disj,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
& ( ord_less @ A @ A2 @ B2 ) )
| ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
& ( ord_less @ A @ B2 @ A2 ) ) ) ) ) ).
% mult_less_cancel_left_disj
thf(fact_861_mult__strict__right__mono__neg,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( ord_less @ A @ B2 @ A2 )
=> ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ) ).
% mult_strict_right_mono_neg
thf(fact_862_mult__strict__right__mono,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ) ).
% mult_strict_right_mono
thf(fact_863_mult__less__cancel__right__disj,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( ord_less @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
& ( ord_less @ A @ A2 @ B2 ) )
| ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
& ( ord_less @ A @ B2 @ A2 ) ) ) ) ) ).
% mult_less_cancel_right_disj
thf(fact_864_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: $tType] :
( ( linord2810124833399127020strict @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_865_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_866_not__one__less__zero,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ~ ( ord_less @ A @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ).
% not_one_less_zero
thf(fact_867_zero__less__one,axiom,
! [A: $tType] :
( ( zero_less_one @ A )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ).
% zero_less_one
thf(fact_868_add__neg__neg,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( plus_plus @ A @ A2 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% add_neg_neg
thf(fact_869_add__pos__pos,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ) ).
% add_pos_pos
thf(fact_870_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ~ ! [C4: A] :
( ( B2
= ( plus_plus @ A @ A2 @ C4 ) )
=> ( C4
= ( zero_zero @ A ) ) ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
thf(fact_871_pos__add__strict,axiom,
! [A: $tType] :
( ( strict7427464778891057005id_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ B2 @ C2 )
=> ( ord_less @ A @ B2 @ ( plus_plus @ A @ A2 @ C2 ) ) ) ) ) ).
% pos_add_strict
thf(fact_872_add__less__zeroD,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,Y2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ X @ Y2 ) @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ X @ ( zero_zero @ A ) )
| ( ord_less @ A @ Y2 @ ( zero_zero @ A ) ) ) ) ) ).
% add_less_zeroD
thf(fact_873_divide__le__0__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ A2 @ B2 ) @ ( zero_zero @ A ) )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
& ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) )
| ( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) ) ) ) ) ).
% divide_le_0_iff
thf(fact_874_divide__right__mono,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ A2 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) ) ) ) ) ).
% divide_right_mono
thf(fact_875_zero__le__divide__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ A2 @ B2 ) )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) )
| ( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
& ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) ) ) ) ) ).
% zero_le_divide_iff
thf(fact_876_divide__nonneg__nonneg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y2 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X @ Y2 ) ) ) ) ) ).
% divide_nonneg_nonneg
thf(fact_877_divide__nonneg__nonpos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less_eq @ A @ Y2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ X @ Y2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% divide_nonneg_nonpos
thf(fact_878_divide__nonpos__nonneg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y2: A] :
( ( ord_less_eq @ A @ X @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y2 )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ X @ Y2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% divide_nonpos_nonneg
thf(fact_879_divide__nonpos__nonpos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y2: A] :
( ( ord_less_eq @ A @ X @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ Y2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X @ Y2 ) ) ) ) ) ).
% divide_nonpos_nonpos
thf(fact_880_divide__right__mono__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ C2 ) @ ( divide_divide @ A @ A2 @ C2 ) ) ) ) ) ).
% divide_right_mono_neg
thf(fact_881_mod__induct,axiom,
! [P3: nat > $o,N: nat,P5: nat,M: nat] :
( ( P3 @ N )
=> ( ( ord_less @ nat @ N @ P5 )
=> ( ( ord_less @ nat @ M @ P5 )
=> ( ! [N3: nat] :
( ( ord_less @ nat @ N3 @ P5 )
=> ( ( P3 @ N3 )
=> ( P3 @ ( modulo_modulo @ nat @ ( suc @ N3 ) @ P5 ) ) ) )
=> ( P3 @ M ) ) ) ) ) ).
% mod_induct
thf(fact_882_power__mono,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,B2: A,N: nat] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ord_less_eq @ A @ ( power_power @ A @ A2 @ N ) @ ( power_power @ A @ B2 @ N ) ) ) ) ) ).
% power_mono
thf(fact_883_zero__le__power,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,N: nat] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A2 @ N ) ) ) ) ).
% zero_le_power
thf(fact_884_divide__neg__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y2: A] :
( ( ord_less @ A @ X @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ Y2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X @ Y2 ) ) ) ) ) ).
% divide_neg_neg
thf(fact_885_divide__neg__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y2: A] :
( ( ord_less @ A @ X @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ Y2 )
=> ( ord_less @ A @ ( divide_divide @ A @ X @ Y2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% divide_neg_pos
thf(fact_886_divide__pos__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less @ A @ Y2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( divide_divide @ A @ X @ Y2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% divide_pos_neg
thf(fact_887_divide__pos__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ Y2 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X @ Y2 ) ) ) ) ) ).
% divide_pos_pos
thf(fact_888_divide__less__0__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( divide_divide @ A @ A2 @ B2 ) @ ( zero_zero @ A ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
& ( ord_less @ A @ B2 @ ( zero_zero @ A ) ) )
| ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
& ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) ) ) ) ) ).
% divide_less_0_iff
thf(fact_889_divide__less__cancel,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( ord_less @ A @ ( divide_divide @ A @ A2 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ A2 @ B2 ) )
& ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B2 @ A2 ) )
& ( C2
!= ( zero_zero @ A ) ) ) ) ) ).
% divide_less_cancel
thf(fact_890_zero__less__divide__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ A2 @ B2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
& ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) )
| ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
& ( ord_less @ A @ B2 @ ( zero_zero @ A ) ) ) ) ) ) ).
% zero_less_divide_iff
thf(fact_891_divide__strict__right__mono,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( divide_divide @ A @ A2 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) ) ) ) ) ).
% divide_strict_right_mono
thf(fact_892_divide__strict__right__mono__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( ord_less @ A @ B2 @ A2 )
=> ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( divide_divide @ A @ A2 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) ) ) ) ) ).
% divide_strict_right_mono_neg
thf(fact_893_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_894_zero__less__power,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,N: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A2 @ N ) ) ) ) ).
% zero_less_power
thf(fact_895_frac__eq__eq,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [Y2: A,Z3: A,X: A,W: A] :
( ( Y2
!= ( zero_zero @ A ) )
=> ( ( Z3
!= ( zero_zero @ A ) )
=> ( ( ( divide_divide @ A @ X @ Y2 )
= ( divide_divide @ A @ W @ Z3 ) )
= ( ( times_times @ A @ X @ Z3 )
= ( times_times @ A @ W @ Y2 ) ) ) ) ) ) ).
% frac_eq_eq
thf(fact_896_divide__eq__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B2: A,C2: A,A2: A] :
( ( ( divide_divide @ A @ B2 @ C2 )
= A2 )
= ( ( ( C2
!= ( zero_zero @ A ) )
=> ( B2
= ( times_times @ A @ A2 @ C2 ) ) )
& ( ( C2
= ( zero_zero @ A ) )
=> ( A2
= ( zero_zero @ A ) ) ) ) ) ) ).
% divide_eq_eq
thf(fact_897_eq__divide__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( A2
= ( divide_divide @ A @ B2 @ C2 ) )
= ( ( ( C2
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ A2 @ C2 )
= B2 ) )
& ( ( C2
= ( zero_zero @ A ) )
=> ( A2
= ( zero_zero @ A ) ) ) ) ) ) ).
% eq_divide_eq
thf(fact_898_divide__eq__imp,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [C2: A,B2: A,A2: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( B2
= ( times_times @ A @ A2 @ C2 ) )
=> ( ( divide_divide @ A @ B2 @ C2 )
= A2 ) ) ) ) ).
% divide_eq_imp
thf(fact_899_eq__divide__imp,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( ( times_times @ A @ A2 @ C2 )
= B2 )
=> ( A2
= ( divide_divide @ A @ B2 @ C2 ) ) ) ) ) ).
% eq_divide_imp
thf(fact_900_nonzero__divide__eq__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [C2: A,B2: A,A2: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( ( divide_divide @ A @ B2 @ C2 )
= A2 )
= ( B2
= ( times_times @ A @ A2 @ C2 ) ) ) ) ) ).
% nonzero_divide_eq_eq
thf(fact_901_nonzero__eq__divide__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( A2
= ( divide_divide @ A @ B2 @ C2 ) )
= ( ( times_times @ A @ A2 @ C2 )
= B2 ) ) ) ) ).
% nonzero_eq_divide_eq
thf(fact_902_right__inverse__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B2: A,A2: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( ( divide_divide @ A @ A2 @ B2 )
= ( one_one @ A ) )
= ( A2 = B2 ) ) ) ) ).
% right_inverse_eq
thf(fact_903_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_904_neg__zdiv__mult__2,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq @ int @ A2 @ ( zero_zero @ int ) )
=> ( ( divide_divide @ int @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ B2 ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ A2 ) )
= ( divide_divide @ int @ ( plus_plus @ int @ B2 @ ( one_one @ int ) ) @ A2 ) ) ) ).
% neg_zdiv_mult_2
thf(fact_905_pos__zdiv__mult__2,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A2 )
=> ( ( divide_divide @ int @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ B2 ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ A2 ) )
= ( divide_divide @ int @ B2 @ A2 ) ) ) ).
% pos_zdiv_mult_2
thf(fact_906_pos__zmod__mult__2,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A2 )
=> ( ( modulo_modulo @ int @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ B2 ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ A2 ) )
= ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( modulo_modulo @ int @ B2 @ A2 ) ) ) ) ) ).
% pos_zmod_mult_2
thf(fact_907_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ord_less_eq @ A @ ( modulo_modulo @ A @ A2 @ B2 ) @ A2 ) ) ) ).
% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
thf(fact_908_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [B2: A,A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less @ A @ ( modulo_modulo @ A @ A2 @ B2 ) @ B2 ) ) ) ).
% unique_euclidean_semiring_numeral_class.pos_mod_bound
thf(fact_909_power__0,axiom,
! [A: $tType] :
( ( power @ A )
=> ! [A2: A] :
( ( power_power @ A @ A2 @ ( zero_zero @ nat ) )
= ( one_one @ A ) ) ) ).
% power_0
thf(fact_910_mod__eq__self__iff__div__eq__0,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ! [A2: A,B2: A] :
( ( ( modulo_modulo @ A @ A2 @ B2 )
= A2 )
= ( ( divide_divide @ A @ A2 @ B2 )
= ( zero_zero @ A ) ) ) ) ).
% mod_eq_self_iff_div_eq_0
thf(fact_911_ex__least__nat__le,axiom,
! [P3: nat > $o,N: nat] :
( ( P3 @ N )
=> ( ~ ( P3 @ ( zero_zero @ nat ) )
=> ? [K: nat] :
( ( ord_less_eq @ nat @ K @ N )
& ! [I4: nat] :
( ( ord_less @ nat @ I4 @ K )
=> ~ ( P3 @ I4 ) )
& ( P3 @ K ) ) ) ) ).
% ex_least_nat_le
thf(fact_912_less__imp__add__positive,axiom,
! [I2: nat,J3: nat] :
( ( ord_less @ nat @ I2 @ J3 )
=> ? [K: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
& ( ( plus_plus @ nat @ I2 @ K )
= J3 ) ) ) ).
% less_imp_add_positive
thf(fact_913_nat__mult__less__cancel1,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K2 )
=> ( ( ord_less @ nat @ ( times_times @ nat @ K2 @ M ) @ ( times_times @ nat @ K2 @ N ) )
= ( ord_less @ nat @ M @ N ) ) ) ).
% nat_mult_less_cancel1
thf(fact_914_nat__mult__eq__cancel1,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K2 )
=> ( ( ( times_times @ nat @ K2 @ M )
= ( times_times @ nat @ K2 @ N ) )
= ( M = N ) ) ) ).
% nat_mult_eq_cancel1
thf(fact_915_mult__less__mono2,axiom,
! [I2: nat,J3: nat,K2: nat] :
( ( ord_less @ nat @ I2 @ J3 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K2 )
=> ( ord_less @ nat @ ( times_times @ nat @ K2 @ I2 ) @ ( times_times @ nat @ K2 @ J3 ) ) ) ) ).
% mult_less_mono2
thf(fact_916_mult__less__mono1,axiom,
! [I2: nat,J3: nat,K2: nat] :
( ( ord_less @ nat @ I2 @ J3 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K2 )
=> ( ord_less @ nat @ ( times_times @ nat @ I2 @ K2 ) @ ( times_times @ nat @ J3 @ K2 ) ) ) ) ).
% mult_less_mono1
thf(fact_917_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_918_nat__power__less__imp__less,axiom,
! [I2: nat,M: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ I2 )
=> ( ( ord_less @ nat @ ( power_power @ nat @ I2 @ M ) @ ( power_power @ nat @ I2 @ N ) )
=> ( ord_less @ nat @ M @ N ) ) ) ).
% nat_power_less_imp_less
thf(fact_919_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_920_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_921_mod__eq__0D,axiom,
! [M: nat,D2: nat] :
( ( ( modulo_modulo @ nat @ M @ D2 )
= ( zero_zero @ nat ) )
=> ? [Q3: nat] :
( M
= ( times_times @ nat @ D2 @ Q3 ) ) ) ).
% mod_eq_0D
thf(fact_922_nat__bit__induct,axiom,
! [P3: nat > $o,N: nat] :
( ( P3 @ ( zero_zero @ nat ) )
=> ( ! [N3: nat] :
( ( P3 @ N3 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( P3 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) ) ) )
=> ( ! [N3: nat] :
( ( P3 @ N3 )
=> ( P3 @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) ) ) )
=> ( P3 @ N ) ) ) ) ).
% nat_bit_induct
thf(fact_923_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_924_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_925_imult__is__infinity,axiom,
! [A2: extended_enat,B2: extended_enat] :
( ( ( times_times @ extended_enat @ A2 @ B2 )
= ( extend4730790105801354508finity @ extended_enat ) )
= ( ( ( A2
= ( extend4730790105801354508finity @ extended_enat ) )
& ( B2
!= ( zero_zero @ extended_enat ) ) )
| ( ( B2
= ( extend4730790105801354508finity @ extended_enat ) )
& ( A2
!= ( zero_zero @ extended_enat ) ) ) ) ) ).
% imult_is_infinity
thf(fact_926_enat__0__less__mult__iff,axiom,
! [M: extended_enat,N: extended_enat] :
( ( ord_less @ extended_enat @ ( zero_zero @ extended_enat ) @ ( times_times @ extended_enat @ M @ N ) )
= ( ( ord_less @ extended_enat @ ( zero_zero @ extended_enat ) @ M )
& ( ord_less @ extended_enat @ ( zero_zero @ extended_enat ) @ N ) ) ) ).
% enat_0_less_mult_iff
thf(fact_927_odd__0__le__power__imp__0__le,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A,N: nat] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A2 @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% odd_0_le_power_imp_0_le
thf(fact_928_odd__power__less__zero,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A,N: nat] :
( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( power_power @ A @ A2 @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) @ ( zero_zero @ A ) ) ) ) ).
% odd_power_less_zero
thf(fact_929_power__gt1,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,N: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ A2 )
=> ( ord_less @ A @ ( one_one @ A ) @ ( power_power @ A @ A2 @ ( suc @ N ) ) ) ) ) ).
% power_gt1
thf(fact_930_mult__le__cancel__left,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ A2 @ B2 ) )
& ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B2 @ A2 ) ) ) ) ) ).
% mult_le_cancel_left
thf(fact_931_mult__le__cancel__right,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ A2 @ B2 ) )
& ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B2 @ A2 ) ) ) ) ) ).
% mult_le_cancel_right
thf(fact_932_mult__left__less__imp__less,axiom,
! [A: $tType] :
( ( linordered_semiring @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ A2 @ B2 ) ) ) ) ).
% mult_left_less_imp_less
thf(fact_933_mult__strict__mono,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A2: A,B2: A,C2: A,D2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ C2 @ D2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ D2 ) ) ) ) ) ) ) ).
% mult_strict_mono
thf(fact_934_mult__less__cancel__left,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ A2 @ B2 ) )
& ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B2 @ A2 ) ) ) ) ) ).
% mult_less_cancel_left
thf(fact_935_mult__right__less__imp__less,axiom,
! [A: $tType] :
( ( linordered_semiring @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( ord_less @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ A2 @ B2 ) ) ) ) ).
% mult_right_less_imp_less
thf(fact_936_mult__strict__mono_H,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A2: A,B2: A,C2: A,D2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ C2 @ D2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ D2 ) ) ) ) ) ) ) ).
% mult_strict_mono'
thf(fact_937_mult__less__cancel__right,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( ord_less @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ A2 @ B2 ) )
& ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B2 @ A2 ) ) ) ) ) ).
% mult_less_cancel_right
thf(fact_938_mult__le__cancel__left__neg,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) )
= ( ord_less_eq @ A @ B2 @ A2 ) ) ) ) ).
% mult_le_cancel_left_neg
thf(fact_939_mult__le__cancel__left__pos,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) )
= ( ord_less_eq @ A @ A2 @ B2 ) ) ) ) ).
% mult_le_cancel_left_pos
thf(fact_940_mult__left__le__imp__le,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ A2 @ B2 ) ) ) ) ).
% mult_left_le_imp_le
thf(fact_941_mult__right__le__imp__le,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ A2 @ B2 ) ) ) ) ).
% mult_right_le_imp_le
thf(fact_942_mult__le__less__imp__less,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A2: A,B2: A,C2: A,D2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ C2 @ D2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ D2 ) ) ) ) ) ) ) ).
% mult_le_less_imp_less
thf(fact_943_mult__less__le__imp__less,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A2: A,B2: A,C2: A,D2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ D2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ D2 ) ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_944_field__le__epsilon,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y2: A] :
( ! [E2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ E2 )
=> ( ord_less_eq @ A @ X @ ( plus_plus @ A @ Y2 @ E2 ) ) )
=> ( ord_less_eq @ A @ X @ Y2 ) ) ) ).
% field_le_epsilon
thf(fact_945_add__neg__nonpos,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( plus_plus @ A @ A2 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% add_neg_nonpos
thf(fact_946_add__nonneg__pos,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ) ).
% add_nonneg_pos
thf(fact_947_add__nonpos__neg,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( plus_plus @ A @ A2 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% add_nonpos_neg
thf(fact_948_add__pos__nonneg,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ) ).
% add_pos_nonneg
thf(fact_949_add__strict__increasing,axiom,
! [A: $tType] :
( ( ordere8940638589300402666id_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ B2 @ C2 )
=> ( ord_less @ A @ B2 @ ( plus_plus @ A @ A2 @ C2 ) ) ) ) ) ).
% add_strict_increasing
thf(fact_950_add__strict__increasing2,axiom,
! [A: $tType] :
( ( ordere8940638589300402666id_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ B2 @ C2 )
=> ( ord_less @ A @ B2 @ ( plus_plus @ A @ A2 @ C2 ) ) ) ) ) ).
% add_strict_increasing2
thf(fact_951_mult__left__le,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [C2: A,A2: A] :
( ( ord_less_eq @ A @ C2 @ ( one_one @ A ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C2 ) @ A2 ) ) ) ) ).
% mult_left_le
thf(fact_952_mult__le__one,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ ( one_one @ A ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ A2 @ B2 ) @ ( one_one @ A ) ) ) ) ) ) ).
% mult_le_one
thf(fact_953_mult__right__le__one__le,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,Y2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y2 )
=> ( ( ord_less_eq @ A @ Y2 @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ X @ Y2 ) @ X ) ) ) ) ) ).
% mult_right_le_one_le
thf(fact_954_mult__left__le__one__le,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,Y2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y2 )
=> ( ( ord_less_eq @ A @ Y2 @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ Y2 @ X ) @ X ) ) ) ) ) ).
% mult_left_le_one_le
thf(fact_955_sum__squares__le__zero__iff,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [X: A,Y2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ ( times_times @ A @ X @ X ) @ ( times_times @ A @ Y2 @ Y2 ) ) @ ( zero_zero @ A ) )
= ( ( X
= ( zero_zero @ A ) )
& ( Y2
= ( zero_zero @ A ) ) ) ) ) ).
% sum_squares_le_zero_iff
thf(fact_956_sum__squares__ge__zero,axiom,
! [A: $tType] :
( ( linordered_ring @ A )
=> ! [X: A,Y2: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ ( times_times @ A @ X @ X ) @ ( times_times @ A @ Y2 @ Y2 ) ) ) ) ).
% sum_squares_ge_zero
thf(fact_957_frac__le,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y2: A,X: A,W: A,Z3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y2 )
=> ( ( ord_less_eq @ A @ X @ Y2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ W )
=> ( ( ord_less_eq @ A @ W @ Z3 )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ X @ Z3 ) @ ( divide_divide @ A @ Y2 @ W ) ) ) ) ) ) ) ).
% frac_le
thf(fact_958_frac__less,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y2: A,W: A,Z3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less @ A @ X @ Y2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ W )
=> ( ( ord_less_eq @ A @ W @ Z3 )
=> ( ord_less @ A @ ( divide_divide @ A @ X @ Z3 ) @ ( divide_divide @ A @ Y2 @ W ) ) ) ) ) ) ) ).
% frac_less
thf(fact_959_frac__less2,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y2: A,W: A,Z3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less_eq @ A @ X @ Y2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ W )
=> ( ( ord_less @ A @ W @ Z3 )
=> ( ord_less @ A @ ( divide_divide @ A @ X @ Z3 ) @ ( divide_divide @ A @ Y2 @ W ) ) ) ) ) ) ) ).
% frac_less2
thf(fact_960_divide__le__cancel,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ A2 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ A2 @ B2 ) )
& ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B2 @ A2 ) ) ) ) ) ).
% divide_le_cancel
thf(fact_961_divide__nonneg__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less @ A @ Y2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ X @ Y2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% divide_nonneg_neg
thf(fact_962_divide__nonneg__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ Y2 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X @ Y2 ) ) ) ) ) ).
% divide_nonneg_pos
thf(fact_963_divide__nonpos__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y2: A] :
( ( ord_less_eq @ A @ X @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ Y2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X @ Y2 ) ) ) ) ) ).
% divide_nonpos_neg
thf(fact_964_divide__nonpos__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y2: A] :
( ( ord_less_eq @ A @ X @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ Y2 )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ X @ Y2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% divide_nonpos_pos
thf(fact_965_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ A2 @ B2 )
=> ( ( divide_divide @ A @ A2 @ B2 )
= ( zero_zero @ A ) ) ) ) ) ).
% unique_euclidean_semiring_numeral_class.div_less
thf(fact_966_div__positive,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [B2: A,A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ A2 @ B2 ) ) ) ) ) ).
% div_positive
thf(fact_967_sum__squares__gt__zero__iff,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [X: A,Y2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ ( times_times @ A @ X @ X ) @ ( times_times @ A @ Y2 @ Y2 ) ) )
= ( ( X
!= ( zero_zero @ A ) )
| ( Y2
!= ( zero_zero @ A ) ) ) ) ) ).
% sum_squares_gt_zero_iff
thf(fact_968_not__sum__squares__lt__zero,axiom,
! [A: $tType] :
( ( linordered_ring @ A )
=> ! [X: A,Y2: A] :
~ ( ord_less @ A @ ( plus_plus @ A @ ( times_times @ A @ X @ X ) @ ( times_times @ A @ Y2 @ Y2 ) ) @ ( zero_zero @ A ) ) ) ).
% not_sum_squares_lt_zero
thf(fact_969_power__less__imp__less__base,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,N: nat,B2: A] :
( ( ord_less @ A @ ( power_power @ A @ A2 @ N ) @ ( power_power @ A @ B2 @ N ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less @ A @ A2 @ B2 ) ) ) ) ).
% power_less_imp_less_base
thf(fact_970_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_971_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( divide_divide @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) )
= ( divide_divide @ A @ ( divide_divide @ A @ A2 @ B2 ) @ C2 ) ) ) ) ).
% unique_euclidean_semiring_numeral_class.div_mult2_eq
thf(fact_972_divide__less__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,C2: A,A2: A] :
( ( ord_less @ A @ ( divide_divide @ A @ B2 @ C2 ) @ A2 )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ B2 @ ( times_times @ A @ A2 @ C2 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ A2 @ C2 ) @ B2 ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ) ) ) ) ).
% divide_less_eq
thf(fact_973_less__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less @ A @ A2 @ ( divide_divide @ A @ B2 @ C2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( times_times @ A @ A2 @ C2 ) @ B2 ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B2 @ ( times_times @ A @ A2 @ C2 ) ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% less_divide_eq
thf(fact_974_neg__divide__less__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,B2: A,A2: A] :
( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( divide_divide @ A @ B2 @ C2 ) @ A2 )
= ( ord_less @ A @ ( times_times @ A @ A2 @ C2 ) @ B2 ) ) ) ) ).
% neg_divide_less_eq
thf(fact_975_neg__less__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ A2 @ ( divide_divide @ A @ B2 @ C2 ) )
= ( ord_less @ A @ B2 @ ( times_times @ A @ A2 @ C2 ) ) ) ) ) ).
% neg_less_divide_eq
thf(fact_976_pos__divide__less__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,B2: A,A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less @ A @ ( divide_divide @ A @ B2 @ C2 ) @ A2 )
= ( ord_less @ A @ B2 @ ( times_times @ A @ A2 @ C2 ) ) ) ) ) ).
% pos_divide_less_eq
thf(fact_977_pos__less__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less @ A @ A2 @ ( divide_divide @ A @ B2 @ C2 ) )
= ( ord_less @ A @ ( times_times @ A @ A2 @ C2 ) @ B2 ) ) ) ) ).
% pos_less_divide_eq
thf(fact_978_mult__imp__div__pos__less,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y2: A,X: A,Z3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Y2 )
=> ( ( ord_less @ A @ X @ ( times_times @ A @ Z3 @ Y2 ) )
=> ( ord_less @ A @ ( divide_divide @ A @ X @ Y2 ) @ Z3 ) ) ) ) ).
% mult_imp_div_pos_less
thf(fact_979_mult__imp__less__div__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y2: A,Z3: A,X: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Y2 )
=> ( ( ord_less @ A @ ( times_times @ A @ Z3 @ Y2 ) @ X )
=> ( ord_less @ A @ Z3 @ ( divide_divide @ A @ X @ Y2 ) ) ) ) ) ).
% mult_imp_less_div_pos
thf(fact_980_divide__strict__left__mono,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( ord_less @ A @ B2 @ A2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B2 ) )
=> ( ord_less @ A @ ( divide_divide @ A @ C2 @ A2 ) @ ( divide_divide @ A @ C2 @ B2 ) ) ) ) ) ) ).
% divide_strict_left_mono
thf(fact_981_divide__strict__left__mono__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B2 ) )
=> ( ord_less @ A @ ( divide_divide @ A @ C2 @ A2 ) @ ( divide_divide @ A @ C2 @ B2 ) ) ) ) ) ) ).
% divide_strict_left_mono_neg
thf(fact_982_power__le__one,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,N: nat] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ A2 @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( power_power @ A @ A2 @ N ) @ ( one_one @ A ) ) ) ) ) ).
% power_le_one
thf(fact_983_divide__less__eq__1,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,A2: A] :
( ( ord_less @ A @ ( divide_divide @ A @ B2 @ A2 ) @ ( one_one @ A ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
& ( ord_less @ A @ B2 @ A2 ) )
| ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
& ( ord_less @ A @ A2 @ B2 ) )
| ( A2
= ( zero_zero @ A ) ) ) ) ) ).
% divide_less_eq_1
thf(fact_984_less__divide__eq__1,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,A2: A] :
( ( ord_less @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B2 @ A2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
& ( ord_less @ A @ A2 @ B2 ) )
| ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
& ( ord_less @ A @ B2 @ A2 ) ) ) ) ) ).
% less_divide_eq_1
thf(fact_985_divide__eq__eq__numeral_I1_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B2: A,C2: A,W: num] :
( ( ( divide_divide @ A @ B2 @ C2 )
= ( numeral_numeral @ A @ W ) )
= ( ( ( C2
!= ( zero_zero @ A ) )
=> ( B2
= ( times_times @ A @ ( numeral_numeral @ A @ W ) @ C2 ) ) )
& ( ( C2
= ( zero_zero @ A ) )
=> ( ( numeral_numeral @ A @ W )
= ( zero_zero @ A ) ) ) ) ) ) ).
% divide_eq_eq_numeral(1)
thf(fact_986_eq__divide__eq__numeral_I1_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [W: num,B2: A,C2: A] :
( ( ( numeral_numeral @ A @ W )
= ( divide_divide @ A @ B2 @ C2 ) )
= ( ( ( C2
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ ( numeral_numeral @ A @ W ) @ C2 )
= B2 ) )
& ( ( C2
= ( zero_zero @ A ) )
=> ( ( numeral_numeral @ A @ W )
= ( zero_zero @ A ) ) ) ) ) ) ).
% eq_divide_eq_numeral(1)
thf(fact_987_add__divide__eq__if__simps_I2_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z3: A,A2: A,B2: A] :
( ( ( Z3
= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( divide_divide @ A @ A2 @ Z3 ) @ B2 )
= B2 ) )
& ( ( Z3
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( divide_divide @ A @ A2 @ Z3 ) @ B2 )
= ( divide_divide @ A @ ( plus_plus @ A @ A2 @ ( times_times @ A @ B2 @ Z3 ) ) @ Z3 ) ) ) ) ) ).
% add_divide_eq_if_simps(2)
thf(fact_988_add__divide__eq__if__simps_I1_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z3: A,A2: A,B2: A] :
( ( ( Z3
= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ A2 @ ( divide_divide @ A @ B2 @ Z3 ) )
= A2 ) )
& ( ( Z3
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ A2 @ ( divide_divide @ A @ B2 @ Z3 ) )
= ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ A2 @ Z3 ) @ B2 ) @ Z3 ) ) ) ) ) ).
% add_divide_eq_if_simps(1)
thf(fact_989_add__frac__eq,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [Y2: A,Z3: A,X: A,W: A] :
( ( Y2
!= ( zero_zero @ A ) )
=> ( ( Z3
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( divide_divide @ A @ X @ Y2 ) @ ( divide_divide @ A @ W @ Z3 ) )
= ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ X @ Z3 ) @ ( times_times @ A @ W @ Y2 ) ) @ ( times_times @ A @ Y2 @ Z3 ) ) ) ) ) ) ).
% add_frac_eq
thf(fact_990_add__frac__num,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [Y2: A,X: A,Z3: A] :
( ( Y2
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( divide_divide @ A @ X @ Y2 ) @ Z3 )
= ( divide_divide @ A @ ( plus_plus @ A @ X @ ( times_times @ A @ Z3 @ Y2 ) ) @ Y2 ) ) ) ) ).
% add_frac_num
thf(fact_991_add__num__frac,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [Y2: A,Z3: A,X: A] :
( ( Y2
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ Z3 @ ( divide_divide @ A @ X @ Y2 ) )
= ( divide_divide @ A @ ( plus_plus @ A @ X @ ( times_times @ A @ Z3 @ Y2 ) ) @ Y2 ) ) ) ) ).
% add_num_frac
thf(fact_992_add__divide__eq__iff,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z3: A,X: A,Y2: A] :
( ( Z3
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ X @ ( divide_divide @ A @ Y2 @ Z3 ) )
= ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ X @ Z3 ) @ Y2 ) @ Z3 ) ) ) ) ).
% add_divide_eq_iff
thf(fact_993_divide__add__eq__iff,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z3: A,X: A,Y2: A] :
( ( Z3
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( divide_divide @ A @ X @ Z3 ) @ Y2 )
= ( divide_divide @ A @ ( plus_plus @ A @ X @ ( times_times @ A @ Y2 @ Z3 ) ) @ Z3 ) ) ) ) ).
% divide_add_eq_iff
thf(fact_994_div__add__self1,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [B2: A,A2: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ B2 @ A2 ) @ B2 )
= ( plus_plus @ A @ ( divide_divide @ A @ A2 @ B2 ) @ ( one_one @ A ) ) ) ) ) ).
% div_add_self1
thf(fact_995_div__add__self2,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [B2: A,A2: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ A2 @ B2 ) @ B2 )
= ( plus_plus @ A @ ( divide_divide @ A @ A2 @ B2 ) @ ( one_one @ A ) ) ) ) ) ).
% div_add_self2
thf(fact_996_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ A2 @ B2 )
=> ( ( modulo_modulo @ A @ A2 @ B2 )
= A2 ) ) ) ) ).
% unique_euclidean_semiring_numeral_class.mod_less
thf(fact_997_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [B2: A,A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( modulo_modulo @ A @ A2 @ B2 ) ) ) ) ).
% unique_euclidean_semiring_numeral_class.pos_mod_sign
thf(fact_998_subset__iff__psubset__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A7: set @ A,B7: set @ A] :
( ( ord_less @ ( set @ A ) @ A7 @ B7 )
| ( A7 = B7 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_999_subset__psubset__trans,axiom,
! [A: $tType,A3: set @ A,B5: set @ A,C5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B5 )
=> ( ( ord_less @ ( set @ A ) @ B5 @ C5 )
=> ( ord_less @ ( set @ A ) @ A3 @ C5 ) ) ) ).
% subset_psubset_trans
thf(fact_1000_subset__not__subset__eq,axiom,
! [A: $tType] :
( ( ord_less @ ( set @ A ) )
= ( ^ [A7: set @ A,B7: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A7 @ B7 )
& ~ ( ord_less_eq @ ( set @ A ) @ B7 @ A7 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_1001_psubset__subset__trans,axiom,
! [A: $tType,A3: set @ A,B5: set @ A,C5: set @ A] :
( ( ord_less @ ( set @ A ) @ A3 @ B5 )
=> ( ( ord_less_eq @ ( set @ A ) @ B5 @ C5 )
=> ( ord_less @ ( set @ A ) @ A3 @ C5 ) ) ) ).
% psubset_subset_trans
thf(fact_1002_psubset__imp__subset,axiom,
! [A: $tType,A3: set @ A,B5: set @ A] :
( ( ord_less @ ( set @ A ) @ A3 @ B5 )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ B5 ) ) ).
% psubset_imp_subset
thf(fact_1003_psubset__eq,axiom,
! [A: $tType] :
( ( ord_less @ ( set @ A ) )
= ( ^ [A7: set @ A,B7: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A7 @ B7 )
& ( A7 != B7 ) ) ) ) ).
% psubset_eq
thf(fact_1004_psubsetE,axiom,
! [A: $tType,A3: set @ A,B5: set @ A] :
( ( ord_less @ ( set @ A ) @ A3 @ B5 )
=> ~ ( ( ord_less_eq @ ( set @ A ) @ A3 @ B5 )
=> ( ord_less_eq @ ( set @ A ) @ B5 @ A3 ) ) ) ).
% psubsetE
thf(fact_1005_in__mono,axiom,
! [A: $tType,A3: set @ A,B5: set @ A,X: A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B5 )
=> ( ( member @ A @ X @ A3 )
=> ( member @ A @ X @ B5 ) ) ) ).
% in_mono
thf(fact_1006_subsetD,axiom,
! [A: $tType,A3: set @ A,B5: set @ A,C2: A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B5 )
=> ( ( member @ A @ C2 @ A3 )
=> ( member @ A @ C2 @ B5 ) ) ) ).
% subsetD
thf(fact_1007_equalityE,axiom,
! [A: $tType,A3: set @ A,B5: set @ A] :
( ( A3 = B5 )
=> ~ ( ( ord_less_eq @ ( set @ A ) @ A3 @ B5 )
=> ~ ( ord_less_eq @ ( set @ A ) @ B5 @ A3 ) ) ) ).
% equalityE
thf(fact_1008_subset__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A7: set @ A,B7: set @ A] :
! [X4: A] :
( ( member @ A @ X4 @ A7 )
=> ( member @ A @ X4 @ B7 ) ) ) ) ).
% subset_eq
thf(fact_1009_equalityD1,axiom,
! [A: $tType,A3: set @ A,B5: set @ A] :
( ( A3 = B5 )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ B5 ) ) ).
% equalityD1
thf(fact_1010_equalityD2,axiom,
! [A: $tType,A3: set @ A,B5: set @ A] :
( ( A3 = B5 )
=> ( ord_less_eq @ ( set @ A ) @ B5 @ A3 ) ) ).
% equalityD2
thf(fact_1011_subset__iff,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A7: set @ A,B7: set @ A] :
! [T3: A] :
( ( member @ A @ T3 @ A7 )
=> ( member @ A @ T3 @ B7 ) ) ) ) ).
% subset_iff
thf(fact_1012_subset__refl,axiom,
! [A: $tType,A3: set @ A] : ( ord_less_eq @ ( set @ A ) @ A3 @ A3 ) ).
% subset_refl
thf(fact_1013_Collect__mono,axiom,
! [A: $tType,P3: A > $o,Q2: A > $o] :
( ! [X5: A] :
( ( P3 @ X5 )
=> ( Q2 @ X5 ) )
=> ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P3 ) @ ( collect @ A @ Q2 ) ) ) ).
% Collect_mono
thf(fact_1014_subset__trans,axiom,
! [A: $tType,A3: set @ A,B5: set @ A,C5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B5 )
=> ( ( ord_less_eq @ ( set @ A ) @ B5 @ C5 )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ C5 ) ) ) ).
% subset_trans
thf(fact_1015_set__eq__subset,axiom,
! [A: $tType] :
( ( ^ [Y5: set @ A,Z2: set @ A] : Y5 = Z2 )
= ( ^ [A7: set @ A,B7: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A7 @ B7 )
& ( ord_less_eq @ ( set @ A ) @ B7 @ A7 ) ) ) ) ).
% set_eq_subset
thf(fact_1016_Collect__mono__iff,axiom,
! [A: $tType,P3: A > $o,Q2: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ ( collect @ A @ P3 ) @ ( collect @ A @ Q2 ) )
= ( ! [X4: A] :
( ( P3 @ X4 )
=> ( Q2 @ X4 ) ) ) ) ).
% Collect_mono_iff
thf(fact_1017_cong__exp__iff__simps_I2_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [N: num,Q: num] :
( ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit0 @ N ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q ) ) )
= ( zero_zero @ A ) )
= ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ N ) @ ( numeral_numeral @ A @ Q ) )
= ( zero_zero @ A ) ) ) ) ).
% cong_exp_iff_simps(2)
thf(fact_1018_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_1019_bounded__Max__nat,axiom,
! [P3: nat > $o,X: nat,M7: nat] :
( ( P3 @ X )
=> ( ! [X5: nat] :
( ( P3 @ X5 )
=> ( ord_less_eq @ nat @ X5 @ M7 ) )
=> ~ ! [M4: nat] :
( ( P3 @ M4 )
=> ~ ! [X2: nat] :
( ( P3 @ X2 )
=> ( ord_less_eq @ nat @ X2 @ M4 ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_1020_ex__has__least__nat,axiom,
! [A: $tType,P3: A > $o,K2: A,M: A > nat] :
( ( P3 @ K2 )
=> ? [X5: A] :
( ( P3 @ X5 )
& ! [Y4: A] :
( ( P3 @ Y4 )
=> ( ord_less_eq @ nat @ ( M @ X5 ) @ ( M @ Y4 ) ) ) ) ) ).
% ex_has_least_nat
thf(fact_1021_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_1022_times__enat__def,axiom,
( ( times_times @ extended_enat )
= ( ^ [M2: extended_enat,N2: extended_enat] :
( extended_case_enat @ extended_enat
@ ^ [O: nat] :
( extended_case_enat @ extended_enat
@ ^ [P4: nat] : ( extended_enat2 @ ( times_times @ nat @ O @ P4 ) )
@ ( if @ extended_enat
@ ( O
= ( zero_zero @ nat ) )
@ ( zero_zero @ extended_enat )
@ ( extend4730790105801354508finity @ extended_enat ) )
@ N2 )
@ ( if @ extended_enat
@ ( N2
= ( zero_zero @ extended_enat ) )
@ ( zero_zero @ extended_enat )
@ ( extend4730790105801354508finity @ extended_enat ) )
@ M2 ) ) ) ).
% times_enat_def
thf(fact_1023_nat__mult__le__cancel1,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K2 )
=> ( ( ord_less_eq @ nat @ ( times_times @ nat @ K2 @ M ) @ ( times_times @ nat @ K2 @ N ) )
= ( ord_less_eq @ nat @ M @ N ) ) ) ).
% nat_mult_le_cancel1
thf(fact_1024_div__le__mono2,axiom,
! [M: nat,N: nat,K2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
=> ( ( ord_less_eq @ nat @ M @ N )
=> ( ord_less_eq @ nat @ ( divide_divide @ nat @ K2 @ N ) @ ( divide_divide @ nat @ K2 @ M ) ) ) ) ).
% div_le_mono2
thf(fact_1025_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_1026_nat__mult__div__cancel1,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K2 )
=> ( ( divide_divide @ nat @ ( times_times @ nat @ K2 @ M ) @ ( times_times @ nat @ K2 @ N ) )
= ( divide_divide @ nat @ M @ N ) ) ) ).
% nat_mult_div_cancel1
thf(fact_1027_div__less__iff__less__mult,axiom,
! [Q: nat,M: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ Q )
=> ( ( ord_less @ nat @ ( divide_divide @ nat @ M @ Q ) @ N )
= ( ord_less @ nat @ M @ ( times_times @ nat @ N @ Q ) ) ) ) ).
% div_less_iff_less_mult
thf(fact_1028_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_1029_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_1030_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_1031_div__less__mono,axiom,
! [A3: nat,B5: nat,N: nat] :
( ( ord_less @ nat @ A3 @ B5 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ( modulo_modulo @ nat @ A3 @ N )
= ( zero_zero @ nat ) )
=> ( ( ( modulo_modulo @ nat @ B5 @ N )
= ( zero_zero @ nat ) )
=> ( ord_less @ nat @ ( divide_divide @ nat @ A3 @ N ) @ ( divide_divide @ nat @ B5 @ N ) ) ) ) ) ) ).
% div_less_mono
thf(fact_1032_Suc__ile__eq,axiom,
! [M: nat,N: extended_enat] :
( ( ord_less_eq @ extended_enat @ ( extended_enat2 @ ( suc @ M ) ) @ N )
= ( ord_less @ extended_enat @ ( extended_enat2 @ M ) @ N ) ) ).
% Suc_ile_eq
thf(fact_1033_imult__infinity,axiom,
! [N: extended_enat] :
( ( ord_less @ extended_enat @ ( zero_zero @ extended_enat ) @ N )
=> ( ( times_times @ extended_enat @ ( extend4730790105801354508finity @ extended_enat ) @ N )
= ( extend4730790105801354508finity @ extended_enat ) ) ) ).
% imult_infinity
thf(fact_1034_imult__infinity__right,axiom,
! [N: extended_enat] :
( ( ord_less @ extended_enat @ ( zero_zero @ extended_enat ) @ N )
=> ( ( times_times @ extended_enat @ N @ ( extend4730790105801354508finity @ extended_enat ) )
= ( extend4730790105801354508finity @ extended_enat ) ) ) ).
% imult_infinity_right
thf(fact_1035_div__nat__eqI,axiom,
! [N: nat,Q: nat,M: nat] :
( ( ord_less_eq @ nat @ ( times_times @ nat @ N @ Q ) @ M )
=> ( ( ord_less @ nat @ M @ ( times_times @ nat @ N @ ( suc @ Q ) ) )
=> ( ( divide_divide @ nat @ M @ N )
= Q ) ) ) ).
% div_nat_eqI
thf(fact_1036_field__le__mult__one__interval,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y2: A] :
( ! [Z: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Z )
=> ( ( ord_less @ A @ Z @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ Z @ X ) @ Y2 ) ) )
=> ( ord_less_eq @ A @ X @ Y2 ) ) ) ).
% field_le_mult_one_interval
thf(fact_1037_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_1038_mult__le__cancel__left2,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [C2: A,A2: A] :
( ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A2 ) @ C2 )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ A2 @ ( one_one @ A ) ) )
& ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ A2 ) ) ) ) ) ).
% mult_le_cancel_left2
thf(fact_1039_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_1040_mult__le__cancel__right2,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A,C2: A] :
( ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C2 ) @ C2 )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ A2 @ ( one_one @ A ) ) )
& ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ A2 ) ) ) ) ) ).
% mult_le_cancel_right2
thf(fact_1041_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_1042_mult__less__cancel__left2,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [C2: A,A2: A] :
( ( ord_less @ A @ ( times_times @ A @ C2 @ A2 ) @ C2 )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ A2 @ ( one_one @ A ) ) )
& ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( one_one @ A ) @ A2 ) ) ) ) ) ).
% mult_less_cancel_left2
thf(fact_1043_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_1044_mult__less__cancel__right2,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A,C2: A] :
( ( ord_less @ A @ ( times_times @ A @ A2 @ C2 ) @ C2 )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ A2 @ ( one_one @ A ) ) )
& ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( one_one @ A ) @ A2 ) ) ) ) ) ).
% mult_less_cancel_right2
thf(fact_1045_divide__le__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,C2: A,A2: A] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ C2 ) @ A2 )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ B2 @ ( times_times @ A @ A2 @ C2 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C2 ) @ B2 ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ) ) ) ) ).
% divide_le_eq
thf(fact_1046_le__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ ( divide_divide @ A @ B2 @ C2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C2 ) @ B2 ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B2 @ ( times_times @ A @ A2 @ C2 ) ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% le_divide_eq
thf(fact_1047_divide__left__mono,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B2 ) )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ C2 @ A2 ) @ ( divide_divide @ A @ C2 @ B2 ) ) ) ) ) ) ).
% divide_left_mono
thf(fact_1048_neg__divide__le__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,B2: A,A2: A] :
( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ C2 ) @ A2 )
= ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C2 ) @ B2 ) ) ) ) ).
% neg_divide_le_eq
thf(fact_1049_neg__le__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ A2 @ ( divide_divide @ A @ B2 @ C2 ) )
= ( ord_less_eq @ A @ B2 @ ( times_times @ A @ A2 @ C2 ) ) ) ) ) ).
% neg_le_divide_eq
thf(fact_1050_pos__divide__le__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,B2: A,A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ C2 ) @ A2 )
= ( ord_less_eq @ A @ B2 @ ( times_times @ A @ A2 @ C2 ) ) ) ) ) ).
% pos_divide_le_eq
thf(fact_1051_pos__le__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less_eq @ A @ A2 @ ( divide_divide @ A @ B2 @ C2 ) )
= ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C2 ) @ B2 ) ) ) ) ).
% pos_le_divide_eq
thf(fact_1052_mult__imp__div__pos__le,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y2: A,X: A,Z3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Y2 )
=> ( ( ord_less_eq @ A @ X @ ( times_times @ A @ Z3 @ Y2 ) )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ X @ Y2 ) @ Z3 ) ) ) ) ).
% mult_imp_div_pos_le
thf(fact_1053_mult__imp__le__div__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y2: A,Z3: A,X: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Y2 )
=> ( ( ord_less_eq @ A @ ( times_times @ A @ Z3 @ Y2 ) @ X )
=> ( ord_less_eq @ A @ Z3 @ ( divide_divide @ A @ X @ Y2 ) ) ) ) ) ).
% mult_imp_le_div_pos
thf(fact_1054_divide__left__mono__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B2 ) )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ C2 @ A2 ) @ ( divide_divide @ A @ C2 @ B2 ) ) ) ) ) ) ).
% divide_left_mono_neg
thf(fact_1055_convex__bound__le,axiom,
! [A: $tType] :
( ( linord6961819062388156250ring_1 @ A )
=> ! [X: A,A2: A,Y2: A,U: A,V2: A] :
( ( ord_less_eq @ A @ X @ A2 )
=> ( ( ord_less_eq @ A @ Y2 @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ U )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ V2 )
=> ( ( ( plus_plus @ A @ U @ V2 )
= ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ ( times_times @ A @ U @ X ) @ ( times_times @ A @ V2 @ Y2 ) ) @ A2 ) ) ) ) ) ) ) ).
% convex_bound_le
thf(fact_1056_divide__le__eq__1,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,A2: A] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ A2 ) @ ( one_one @ A ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
& ( ord_less_eq @ A @ B2 @ A2 ) )
| ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
& ( ord_less_eq @ A @ A2 @ B2 ) )
| ( A2
= ( zero_zero @ A ) ) ) ) ) ).
% divide_le_eq_1
thf(fact_1057_le__divide__eq__1,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,A2: A] :
( ( ord_less_eq @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B2 @ A2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
& ( ord_less_eq @ A @ A2 @ B2 ) )
| ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
& ( ord_less_eq @ A @ B2 @ A2 ) ) ) ) ) ).
% le_divide_eq_1
thf(fact_1058_Suc__nat__number__of__add,axiom,
! [V2: num,N: nat] :
( ( suc @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ V2 ) @ N ) )
= ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( plus_plus @ num @ V2 @ one2 ) ) @ N ) ) ).
% Suc_nat_number_of_add
thf(fact_1059_divide__less__eq__numeral_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,C2: A,W: num] :
( ( ord_less @ A @ ( divide_divide @ A @ B2 @ C2 ) @ ( numeral_numeral @ A @ W ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ B2 @ ( times_times @ A @ ( numeral_numeral @ A @ W ) @ C2 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ ( numeral_numeral @ A @ W ) @ C2 ) @ B2 ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ W ) ) ) ) ) ) ) ) ).
% divide_less_eq_numeral(1)
thf(fact_1060_less__divide__eq__numeral_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [W: num,B2: A,C2: A] :
( ( ord_less @ A @ ( numeral_numeral @ A @ W ) @ ( divide_divide @ A @ B2 @ C2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( times_times @ A @ ( numeral_numeral @ A @ W ) @ C2 ) @ B2 ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B2 @ ( times_times @ A @ ( numeral_numeral @ A @ W ) @ C2 ) ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( numeral_numeral @ A @ W ) @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% less_divide_eq_numeral(1)
thf(fact_1061_power__Suc__less,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,N: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ A2 @ ( one_one @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ A2 @ ( power_power @ A @ A2 @ N ) ) @ ( power_power @ A @ A2 @ N ) ) ) ) ) ).
% power_Suc_less
thf(fact_1062_power__strict__decreasing,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N: nat,N4: nat,A2: A] :
( ( ord_less @ nat @ N @ N4 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ A2 @ ( one_one @ A ) )
=> ( ord_less @ A @ ( power_power @ A @ A2 @ N4 ) @ ( power_power @ A @ A2 @ N ) ) ) ) ) ) ).
% power_strict_decreasing
thf(fact_1063_power__decreasing,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N: nat,N4: nat,A2: A] :
( ( ord_less_eq @ nat @ N @ N4 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ A2 @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( power_power @ A @ A2 @ N4 ) @ ( power_power @ A @ A2 @ N ) ) ) ) ) ) ).
% power_decreasing
thf(fact_1064_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_1065_self__le__power,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,N: nat] :
( ( ord_less_eq @ A @ ( one_one @ A ) @ A2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ord_less_eq @ A @ A2 @ ( power_power @ A @ A2 @ N ) ) ) ) ) ).
% self_le_power
thf(fact_1066_one__less__power,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,N: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ A2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ord_less @ A @ ( one_one @ A ) @ ( power_power @ A @ A2 @ N ) ) ) ) ) ).
% one_less_power
thf(fact_1067_pos2,axiom,
ord_less @ nat @ ( zero_zero @ nat ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ).
% pos2
thf(fact_1068_invar__vebt_Osimps,axiom,
( vEBT_invar_vebt
= ( ^ [A1: vEBT_VEBT,A22: nat] :
( ( ? [A5: $o,B4: $o] :
( A1
= ( vEBT_Leaf @ A5 @ B4 ) )
& ( A22
= ( suc @ ( zero_zero @ nat ) ) ) )
| ? [TreeList3: list @ vEBT_VEBT,N2: nat,Summary3: vEBT_VEBT] :
( ( A1
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ A22 @ TreeList3 @ Summary3 ) )
& ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
=> ( vEBT_invar_vebt @ X4 @ N2 ) )
& ( vEBT_invar_vebt @ Summary3 @ N2 )
& ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) )
& ( A22
= ( plus_plus @ nat @ N2 @ N2 ) )
& ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X6 )
& ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
=> ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X6 ) ) )
| ? [TreeList3: list @ vEBT_VEBT,N2: nat,Summary3: vEBT_VEBT] :
( ( A1
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ A22 @ TreeList3 @ Summary3 ) )
& ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
=> ( vEBT_invar_vebt @ X4 @ N2 ) )
& ( vEBT_invar_vebt @ Summary3 @ ( suc @ N2 ) )
& ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ N2 ) ) )
& ( A22
= ( plus_plus @ nat @ N2 @ ( suc @ N2 ) ) )
& ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X6 )
& ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
=> ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X6 ) ) )
| ? [TreeList3: list @ vEBT_VEBT,N2: nat,Summary3: vEBT_VEBT,Mi2: nat,Ma2: nat] :
( ( A1
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ A22 @ TreeList3 @ Summary3 ) )
& ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
=> ( vEBT_invar_vebt @ X4 @ N2 ) )
& ( vEBT_invar_vebt @ Summary3 @ N2 )
& ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) )
& ( A22
= ( plus_plus @ nat @ N2 @ N2 ) )
& ! [I: nat] :
( ( ord_less @ nat @ I @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) )
=> ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList3 @ I ) @ X6 ) )
= ( vEBT_V8194947554948674370ptions @ Summary3 @ I ) ) )
& ( ( Mi2 = Ma2 )
=> ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
=> ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X6 ) ) )
& ( ord_less_eq @ nat @ Mi2 @ Ma2 )
& ( ord_less @ nat @ Ma2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ A22 ) )
& ( ( Mi2 != Ma2 )
=> ! [I: nat] :
( ( ord_less @ nat @ I @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) )
=> ( ( ( ( vEBT_VEBT_high @ Ma2 @ N2 )
= I )
=> ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList3 @ I ) @ ( vEBT_VEBT_low @ Ma2 @ N2 ) ) )
& ! [X4: nat] :
( ( ( ( vEBT_VEBT_high @ X4 @ N2 )
= I )
& ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList3 @ I ) @ ( vEBT_VEBT_low @ X4 @ N2 ) ) )
=> ( ( ord_less @ nat @ Mi2 @ X4 )
& ( ord_less_eq @ nat @ X4 @ Ma2 ) ) ) ) ) ) )
| ? [TreeList3: list @ vEBT_VEBT,N2: nat,Summary3: vEBT_VEBT,Mi2: nat,Ma2: nat] :
( ( A1
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ A22 @ TreeList3 @ Summary3 ) )
& ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
=> ( vEBT_invar_vebt @ X4 @ N2 ) )
& ( vEBT_invar_vebt @ Summary3 @ ( suc @ N2 ) )
& ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ N2 ) ) )
& ( A22
= ( plus_plus @ nat @ N2 @ ( suc @ N2 ) ) )
& ! [I: nat] :
( ( ord_less @ nat @ I @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ N2 ) ) )
=> ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList3 @ I ) @ X6 ) )
= ( vEBT_V8194947554948674370ptions @ Summary3 @ I ) ) )
& ( ( Mi2 = Ma2 )
=> ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
=> ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X6 ) ) )
& ( ord_less_eq @ nat @ Mi2 @ Ma2 )
& ( ord_less @ nat @ Ma2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ A22 ) )
& ( ( Mi2 != Ma2 )
=> ! [I: nat] :
( ( ord_less @ nat @ I @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ N2 ) ) )
=> ( ( ( ( vEBT_VEBT_high @ Ma2 @ N2 )
= I )
=> ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList3 @ I ) @ ( vEBT_VEBT_low @ Ma2 @ N2 ) ) )
& ! [X4: nat] :
( ( ( ( vEBT_VEBT_high @ X4 @ N2 )
= I )
& ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList3 @ I ) @ ( vEBT_VEBT_low @ X4 @ N2 ) ) )
=> ( ( ord_less @ nat @ Mi2 @ X4 )
& ( ord_less_eq @ nat @ X4 @ Ma2 ) ) ) ) ) ) ) ) ) ) ).
% invar_vebt.simps
thf(fact_1069_invar__vebt_Ocases,axiom,
! [A12: vEBT_VEBT,A23: nat] :
( ( vEBT_invar_vebt @ A12 @ A23 )
=> ( ( ? [A4: $o,B3: $o] :
( A12
= ( vEBT_Leaf @ A4 @ B3 ) )
=> ( A23
!= ( suc @ ( zero_zero @ nat ) ) ) )
=> ( ! [TreeList2: list @ vEBT_VEBT,N3: nat,Summary2: vEBT_VEBT,M4: nat,Deg2: nat] :
( ( A12
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( ( A23 = Deg2 )
=> ( ! [X2: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ( vEBT_invar_vebt @ X2 @ N3 ) )
=> ( ( vEBT_invar_vebt @ Summary2 @ M4 )
=> ( ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M4 ) )
=> ( ( M4 = N3 )
=> ( ( Deg2
= ( plus_plus @ nat @ N3 @ M4 ) )
=> ( ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X_1 )
=> ~ ! [X2: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X_1 ) ) ) ) ) ) ) ) ) )
=> ( ! [TreeList2: list @ vEBT_VEBT,N3: nat,Summary2: vEBT_VEBT,M4: nat,Deg2: nat] :
( ( A12
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( ( A23 = Deg2 )
=> ( ! [X2: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ( vEBT_invar_vebt @ X2 @ N3 ) )
=> ( ( vEBT_invar_vebt @ Summary2 @ M4 )
=> ( ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M4 ) )
=> ( ( M4
= ( suc @ N3 ) )
=> ( ( Deg2
= ( plus_plus @ nat @ N3 @ M4 ) )
=> ( ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X_1 )
=> ~ ! [X2: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X_1 ) ) ) ) ) ) ) ) ) )
=> ( ! [TreeList2: list @ vEBT_VEBT,N3: nat,Summary2: vEBT_VEBT,M4: nat,Deg2: nat,Mi3: nat,Ma3: nat] :
( ( A12
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ Ma3 ) ) @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( ( A23 = Deg2 )
=> ( ! [X2: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ( vEBT_invar_vebt @ X2 @ N3 ) )
=> ( ( vEBT_invar_vebt @ Summary2 @ M4 )
=> ( ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M4 ) )
=> ( ( M4 = N3 )
=> ( ( Deg2
= ( plus_plus @ nat @ N3 @ M4 ) )
=> ( ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M4 ) )
=> ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I4 ) @ X6 ) )
= ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
=> ( ( ( Mi3 = Ma3 )
=> ! [X2: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X_1 ) ) )
=> ( ( ord_less_eq @ nat @ Mi3 @ Ma3 )
=> ( ( ord_less @ nat @ Ma3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
=> ~ ( ( Mi3 != Ma3 )
=> ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M4 ) )
=> ( ( ( ( vEBT_VEBT_high @ Ma3 @ N3 )
= I4 )
=> ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I4 ) @ ( vEBT_VEBT_low @ Ma3 @ N3 ) ) )
& ! [X2: nat] :
( ( ( ( vEBT_VEBT_high @ X2 @ N3 )
= I4 )
& ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I4 ) @ ( vEBT_VEBT_low @ X2 @ N3 ) ) )
=> ( ( ord_less @ nat @ Mi3 @ X2 )
& ( ord_less_eq @ nat @ X2 @ Ma3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
=> ~ ! [TreeList2: list @ vEBT_VEBT,N3: nat,Summary2: vEBT_VEBT,M4: nat,Deg2: nat,Mi3: nat,Ma3: nat] :
( ( A12
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ Ma3 ) ) @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( ( A23 = Deg2 )
=> ( ! [X2: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ( vEBT_invar_vebt @ X2 @ N3 ) )
=> ( ( vEBT_invar_vebt @ Summary2 @ M4 )
=> ( ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M4 ) )
=> ( ( M4
= ( suc @ N3 ) )
=> ( ( Deg2
= ( plus_plus @ nat @ N3 @ M4 ) )
=> ( ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M4 ) )
=> ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I4 ) @ X6 ) )
= ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
=> ( ( ( Mi3 = Ma3 )
=> ! [X2: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X_1 ) ) )
=> ( ( ord_less_eq @ nat @ Mi3 @ Ma3 )
=> ( ( ord_less @ nat @ Ma3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
=> ~ ( ( Mi3 != Ma3 )
=> ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M4 ) )
=> ( ( ( ( vEBT_VEBT_high @ Ma3 @ N3 )
= I4 )
=> ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I4 ) @ ( vEBT_VEBT_low @ Ma3 @ N3 ) ) )
& ! [X2: nat] :
( ( ( ( vEBT_VEBT_high @ X2 @ N3 )
= I4 )
& ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I4 ) @ ( vEBT_VEBT_low @ X2 @ N3 ) ) )
=> ( ( ord_less @ nat @ Mi3 @ X2 )
& ( ord_less_eq @ nat @ X2 @ Ma3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% invar_vebt.cases
thf(fact_1070_less__eq__div__iff__mult__less__eq,axiom,
! [Q: nat,M: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ Q )
=> ( ( ord_less_eq @ nat @ M @ ( divide_divide @ nat @ N @ Q ) )
= ( ord_less_eq @ nat @ ( times_times @ nat @ M @ Q ) @ N ) ) ) ).
% less_eq_div_iff_mult_less_eq
thf(fact_1071_split__div,axiom,
! [P3: nat > $o,M: nat,N: nat] :
( ( P3 @ ( divide_divide @ nat @ M @ N ) )
= ( ( ( N
= ( zero_zero @ nat ) )
=> ( P3 @ ( zero_zero @ nat ) ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ! [I: nat,J4: nat] :
( ( ord_less @ nat @ J4 @ N )
=> ( ( M
= ( plus_plus @ nat @ ( times_times @ nat @ N @ I ) @ J4 ) )
=> ( P3 @ I ) ) ) ) ) ) ).
% split_div
thf(fact_1072_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_1073_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_1074_split__mod,axiom,
! [P3: nat > $o,M: nat,N: nat] :
( ( P3 @ ( modulo_modulo @ nat @ M @ N ) )
= ( ( ( N
= ( zero_zero @ nat ) )
=> ( P3 @ M ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ! [I: nat,J4: nat] :
( ( ord_less @ nat @ J4 @ N )
=> ( ( M
= ( plus_plus @ nat @ ( times_times @ nat @ N @ I ) @ J4 ) )
=> ( P3 @ J4 ) ) ) ) ) ) ).
% split_mod
thf(fact_1075_Collect__subset,axiom,
! [A: $tType,A3: set @ A,P3: A > $o] :
( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ A3 )
& ( P3 @ X4 ) ) )
@ A3 ) ).
% Collect_subset
thf(fact_1076_less__eq__set__def,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A7: set @ A,B7: set @ A] :
( ord_less_eq @ ( A > $o )
@ ^ [X4: A] : ( member @ A @ X4 @ A7 )
@ ^ [X4: A] : ( member @ A @ X4 @ B7 ) ) ) ) ).
% less_eq_set_def
thf(fact_1077_invar__vebt_Ointros_I5_J,axiom,
! [TreeList: list @ vEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi: nat,Ma: nat] :
( ! [X5: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X5 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ( vEBT_invar_vebt @ X5 @ N ) )
=> ( ( vEBT_invar_vebt @ Summary @ M )
=> ( ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) )
=> ( ( M
= ( suc @ N ) )
=> ( ( Deg
= ( plus_plus @ nat @ N @ M ) )
=> ( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) )
=> ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ I3 ) @ X6 ) )
= ( vEBT_V8194947554948674370ptions @ Summary @ I3 ) ) )
=> ( ( ( Mi = Ma )
=> ! [X5: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X5 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_12 ) ) )
=> ( ( ord_less_eq @ nat @ Mi @ Ma )
=> ( ( ord_less @ nat @ Ma @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg ) )
=> ( ( ( Mi != Ma )
=> ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) )
=> ( ( ( ( vEBT_VEBT_high @ Ma @ N )
= I3 )
=> ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ I3 ) @ ( vEBT_VEBT_low @ Ma @ N ) ) )
& ! [X5: nat] :
( ( ( ( vEBT_VEBT_high @ X5 @ N )
= I3 )
& ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ I3 ) @ ( vEBT_VEBT_low @ X5 @ N ) ) )
=> ( ( ord_less @ nat @ Mi @ X5 )
& ( ord_less_eq @ nat @ X5 @ Ma ) ) ) ) ) )
=> ( vEBT_invar_vebt @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ Deg ) ) ) ) ) ) ) ) ) ) ) ).
% invar_vebt.intros(5)
thf(fact_1078_convex__bound__lt,axiom,
! [A: $tType] :
( ( linord715952674999750819strict @ A )
=> ! [X: A,A2: A,Y2: A,U: A,V2: A] :
( ( ord_less @ A @ X @ A2 )
=> ( ( ord_less @ A @ Y2 @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ U )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ V2 )
=> ( ( ( plus_plus @ A @ U @ V2 )
= ( one_one @ A ) )
=> ( ord_less @ A @ ( plus_plus @ A @ ( times_times @ A @ U @ X ) @ ( times_times @ A @ V2 @ Y2 ) ) @ A2 ) ) ) ) ) ) ) ).
% convex_bound_lt
thf(fact_1079_divide__le__eq__numeral_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,C2: A,W: num] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ C2 ) @ ( numeral_numeral @ A @ W ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ B2 @ ( times_times @ A @ ( numeral_numeral @ A @ W ) @ C2 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ ( numeral_numeral @ A @ W ) @ C2 ) @ B2 ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ W ) ) ) ) ) ) ) ) ).
% divide_le_eq_numeral(1)
thf(fact_1080_le__divide__eq__numeral_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [W: num,B2: A,C2: A] :
( ( ord_less_eq @ A @ ( numeral_numeral @ A @ W ) @ ( divide_divide @ A @ B2 @ C2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ ( numeral_numeral @ A @ W ) @ C2 ) @ B2 ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B2 @ ( times_times @ A @ ( numeral_numeral @ A @ W ) @ C2 ) ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( numeral_numeral @ A @ W ) @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% le_divide_eq_numeral(1)
thf(fact_1081_half__gt__zero,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% half_gt_zero
thf(fact_1082_half__gt__zero__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% half_gt_zero_iff
thf(fact_1083_zero__le__power2,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% zero_le_power2
thf(fact_1084_power2__eq__imp__eq,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X: A,Y2: A] :
( ( ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( power_power @ A @ Y2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y2 )
=> ( X = Y2 ) ) ) ) ) ).
% power2_eq_imp_eq
thf(fact_1085_power2__le__imp__le,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X: A,Y2: A] :
( ( ord_less_eq @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y2 )
=> ( ord_less_eq @ A @ X @ Y2 ) ) ) ) ).
% power2_le_imp_le
thf(fact_1086_power2__less__0,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A] :
~ ( ord_less @ A @ ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( zero_zero @ A ) ) ) ).
% power2_less_0
thf(fact_1087_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( modulo_modulo @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) )
= ( plus_plus @ A @ ( times_times @ A @ B2 @ ( modulo_modulo @ A @ ( divide_divide @ A @ A2 @ B2 ) @ C2 ) ) @ ( modulo_modulo @ A @ A2 @ B2 ) ) ) ) ) ).
% unique_euclidean_semiring_numeral_class.mod_mult2_eq
thf(fact_1088_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_1089_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_1090_power__odd__eq,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A2: A,N: nat] :
( ( power_power @ A @ A2 @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) )
= ( times_times @ A @ A2 @ ( power_power @ A @ ( power_power @ A @ A2 @ N ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% power_odd_eq
thf(fact_1091_power2__less__imp__less,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X: A,Y2: A] :
( ( ord_less @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y2 )
=> ( ord_less @ A @ X @ Y2 ) ) ) ) ).
% power2_less_imp_less
thf(fact_1092_sum__power2__ge__zero,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,Y2: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% sum_power2_ge_zero
thf(fact_1093_sum__power2__le__zero__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,Y2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( zero_zero @ A ) )
= ( ( X
= ( zero_zero @ A ) )
& ( Y2
= ( zero_zero @ A ) ) ) ) ) ).
% sum_power2_le_zero_iff
thf(fact_1094_divmod__digit__0_I2_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [B2: A,A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less @ A @ ( modulo_modulo @ A @ A2 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) @ B2 )
=> ( ( modulo_modulo @ A @ A2 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) )
= ( modulo_modulo @ A @ A2 @ B2 ) ) ) ) ) ).
% divmod_digit_0(2)
thf(fact_1095_not__sum__power2__lt__zero,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,Y2: A] :
~ ( ord_less @ A @ ( plus_plus @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( zero_zero @ A ) ) ) ).
% not_sum_power2_lt_zero
thf(fact_1096_sum__power2__gt__zero__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,Y2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
= ( ( X
!= ( zero_zero @ A ) )
| ( Y2
!= ( zero_zero @ A ) ) ) ) ) ).
% sum_power2_gt_zero_iff
thf(fact_1097_bits__stable__imp__add__self,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A2: A] :
( ( ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= A2 )
=> ( ( plus_plus @ A @ A2 @ ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
= ( zero_zero @ A ) ) ) ) ).
% bits_stable_imp_add_self
thf(fact_1098_zero__le__even__power_H,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A,N: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A2 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% zero_le_even_power'
thf(fact_1099_verit__le__mono__div,axiom,
! [A3: nat,B5: nat,N: nat] :
( ( ord_less @ nat @ A3 @ B5 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ord_less_eq @ nat
@ ( plus_plus @ nat @ ( divide_divide @ nat @ A3 @ N )
@ ( if @ nat
@ ( ( modulo_modulo @ nat @ B5 @ N )
= ( zero_zero @ nat ) )
@ ( one_one @ nat )
@ ( zero_zero @ nat ) ) )
@ ( divide_divide @ nat @ B5 @ N ) ) ) ) ).
% verit_le_mono_div
thf(fact_1100_Lattices__Big_Oex__has__greatest__nat,axiom,
! [A: $tType,P3: A > $o,K2: A,F3: A > nat,B2: nat] :
( ( P3 @ K2 )
=> ( ! [Y7: A] :
( ( P3 @ Y7 )
=> ( ord_less @ nat @ ( F3 @ Y7 ) @ B2 ) )
=> ? [X5: A] :
( ( P3 @ X5 )
& ! [Y4: A] :
( ( P3 @ Y4 )
=> ( ord_less_eq @ nat @ ( F3 @ Y4 ) @ ( F3 @ X5 ) ) ) ) ) ) ).
% Lattices_Big.ex_has_greatest_nat
thf(fact_1101_divmod__digit__0_I1_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [B2: A,A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less @ A @ ( modulo_modulo @ A @ A2 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) @ B2 )
=> ( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A2 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) )
= ( divide_divide @ A @ A2 @ B2 ) ) ) ) ) ).
% divmod_digit_0(1)
thf(fact_1102_invar__vebt_Ointros_I4_J,axiom,
! [TreeList: list @ vEBT_VEBT,N: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi: nat,Ma: nat] :
( ! [X5: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X5 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ( vEBT_invar_vebt @ X5 @ N ) )
=> ( ( vEBT_invar_vebt @ Summary @ M )
=> ( ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) )
=> ( ( M = N )
=> ( ( Deg
= ( plus_plus @ nat @ N @ M ) )
=> ( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) )
=> ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ I3 ) @ X6 ) )
= ( vEBT_V8194947554948674370ptions @ Summary @ I3 ) ) )
=> ( ( ( Mi = Ma )
=> ! [X5: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X5 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_12 ) ) )
=> ( ( ord_less_eq @ nat @ Mi @ Ma )
=> ( ( ord_less @ nat @ Ma @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg ) )
=> ( ( ( Mi != Ma )
=> ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) )
=> ( ( ( ( vEBT_VEBT_high @ Ma @ N )
= I3 )
=> ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ I3 ) @ ( vEBT_VEBT_low @ Ma @ N ) ) )
& ! [X5: nat] :
( ( ( ( vEBT_VEBT_high @ X5 @ N )
= I3 )
& ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ I3 ) @ ( vEBT_VEBT_low @ X5 @ N ) ) )
=> ( ( ord_less @ nat @ Mi @ X5 )
& ( ord_less_eq @ nat @ X5 @ Ma ) ) ) ) ) )
=> ( vEBT_invar_vebt @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ Deg ) ) ) ) ) ) ) ) ) ) ) ).
% invar_vebt.intros(4)
thf(fact_1103_VEBT__internal_Oexp__split__high__low_I1_J,axiom,
! [X: nat,N: nat,M: nat] :
( ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( plus_plus @ nat @ N @ M ) ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
=> ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ N ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) ) ) ) ) ).
% VEBT_internal.exp_split_high_low(1)
thf(fact_1104_VEBT__internal_Oexp__split__high__low_I2_J,axiom,
! [X: nat,N: nat,M: nat] :
( ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( plus_plus @ nat @ N @ M ) ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
=> ( ord_less @ nat @ ( vEBT_VEBT_low @ X @ N ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ) ) ).
% VEBT_internal.exp_split_high_low(2)
thf(fact_1105_not__mod__2__eq__0__eq__1,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A2: A] :
( ( ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
!= ( zero_zero @ A ) )
= ( ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( one_one @ A ) ) ) ) ).
% not_mod_2_eq_0_eq_1
thf(fact_1106_not__mod__2__eq__1__eq__0,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A2: A] :
( ( ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
!= ( one_one @ A ) )
= ( ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( zero_zero @ A ) ) ) ) ).
% not_mod_2_eq_1_eq_0
thf(fact_1107_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_1108_VEBT__internal_Omembermima_Oelims_I3_J,axiom,
! [X: vEBT_VEBT,Xa: nat] :
( ~ ( vEBT_VEBT_membermima @ X @ Xa )
=> ( ! [Uu2: $o,Uv2: $o] :
( X
!= ( vEBT_Leaf @ Uu2 @ Uv2 ) )
=> ( ! [Ux: list @ vEBT_VEBT,Uy: vEBT_VEBT] :
( X
!= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( zero_zero @ nat ) @ Ux @ Uy ) )
=> ( ! [Mi3: nat,Ma3: nat] :
( ? [Va2: list @ vEBT_VEBT,Vb: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ Ma3 ) ) @ ( zero_zero @ nat ) @ Va2 @ Vb ) )
=> ( ( Xa = Mi3 )
| ( Xa = Ma3 ) ) )
=> ( ! [Mi3: nat,Ma3: nat,V3: nat,TreeList2: list @ vEBT_VEBT] :
( ? [Vc: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ Ma3 ) ) @ ( suc @ V3 ) @ TreeList2 @ Vc ) )
=> ( ( Xa = Mi3 )
| ( Xa = Ma3 )
| ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) )
=> ~ ! [V3: nat,TreeList2: list @ vEBT_VEBT] :
( ? [Vd: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V3 ) @ TreeList2 @ Vd ) )
=> ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.membermima.elims(3)
thf(fact_1109_VEBT__internal_Omembermima_Oelims_I1_J,axiom,
! [X: vEBT_VEBT,Xa: nat,Y2: $o] :
( ( ( vEBT_VEBT_membermima @ X @ Xa )
= Y2 )
=> ( ( ? [Uu2: $o,Uv2: $o] :
( X
= ( vEBT_Leaf @ Uu2 @ Uv2 ) )
=> Y2 )
=> ( ( ? [Ux: list @ vEBT_VEBT,Uy: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( zero_zero @ nat ) @ Ux @ Uy ) )
=> Y2 )
=> ( ! [Mi3: nat,Ma3: nat] :
( ? [Va2: list @ vEBT_VEBT,Vb: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ Ma3 ) ) @ ( zero_zero @ nat ) @ Va2 @ Vb ) )
=> ( Y2
= ( ~ ( ( Xa = Mi3 )
| ( Xa = Ma3 ) ) ) ) )
=> ( ! [Mi3: nat,Ma3: nat,V3: nat,TreeList2: list @ vEBT_VEBT] :
( ? [Vc: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ Ma3 ) ) @ ( suc @ V3 ) @ TreeList2 @ Vc ) )
=> ( Y2
= ( ~ ( ( Xa = Mi3 )
| ( Xa = Ma3 )
| ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) )
=> ~ ! [V3: nat,TreeList2: list @ vEBT_VEBT] :
( ? [Vd: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V3 ) @ TreeList2 @ Vd ) )
=> ( Y2
= ( ~ ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.membermima.elims(1)
thf(fact_1110_VEBT__internal_Onaive__member_Oelims_I3_J,axiom,
! [X: vEBT_VEBT,Xa: nat] :
( ~ ( vEBT_V5719532721284313246member @ X @ Xa )
=> ( ! [A4: $o,B3: $o] :
( ( X
= ( vEBT_Leaf @ A4 @ B3 ) )
=> ( ( ( Xa
= ( zero_zero @ nat ) )
=> A4 )
& ( ( Xa
!= ( zero_zero @ nat ) )
=> ( ( ( Xa
= ( one_one @ nat ) )
=> B3 )
& ( Xa
= ( one_one @ nat ) ) ) ) ) )
=> ( ! [Uu2: option @ ( product_prod @ nat @ nat ),Uv2: list @ vEBT_VEBT,Uw: vEBT_VEBT] :
( X
!= ( vEBT_Node @ Uu2 @ ( zero_zero @ nat ) @ Uv2 @ Uw ) )
=> ~ ! [Uy: option @ ( product_prod @ nat @ nat ),V3: nat,TreeList2: list @ vEBT_VEBT] :
( ? [S2: vEBT_VEBT] :
( X
= ( vEBT_Node @ Uy @ ( suc @ V3 ) @ TreeList2 @ S2 ) )
=> ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_V5719532721284313246member @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ).
% VEBT_internal.naive_member.elims(3)
thf(fact_1111_VEBT__internal_Onaive__member_Oelims_I2_J,axiom,
! [X: vEBT_VEBT,Xa: nat] :
( ( vEBT_V5719532721284313246member @ X @ Xa )
=> ( ! [A4: $o,B3: $o] :
( ( X
= ( vEBT_Leaf @ A4 @ B3 ) )
=> ~ ( ( ( Xa
= ( zero_zero @ nat ) )
=> A4 )
& ( ( Xa
!= ( zero_zero @ nat ) )
=> ( ( ( Xa
= ( one_one @ nat ) )
=> B3 )
& ( Xa
= ( one_one @ nat ) ) ) ) ) )
=> ~ ! [Uy: option @ ( product_prod @ nat @ nat ),V3: nat,TreeList2: list @ vEBT_VEBT] :
( ? [S2: vEBT_VEBT] :
( X
= ( vEBT_Node @ Uy @ ( suc @ V3 ) @ TreeList2 @ S2 ) )
=> ~ ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_V5719532721284313246member @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ).
% VEBT_internal.naive_member.elims(2)
thf(fact_1112_VEBT__internal_Onaive__member_Oelims_I1_J,axiom,
! [X: vEBT_VEBT,Xa: nat,Y2: $o] :
( ( ( vEBT_V5719532721284313246member @ X @ Xa )
= Y2 )
=> ( ! [A4: $o,B3: $o] :
( ( X
= ( vEBT_Leaf @ A4 @ B3 ) )
=> ( Y2
= ( ~ ( ( ( Xa
= ( zero_zero @ nat ) )
=> A4 )
& ( ( Xa
!= ( zero_zero @ nat ) )
=> ( ( ( Xa
= ( one_one @ nat ) )
=> B3 )
& ( Xa
= ( one_one @ nat ) ) ) ) ) ) ) )
=> ( ( ? [Uu2: option @ ( product_prod @ nat @ nat ),Uv2: list @ vEBT_VEBT,Uw: vEBT_VEBT] :
( X
= ( vEBT_Node @ Uu2 @ ( zero_zero @ nat ) @ Uv2 @ Uw ) )
=> Y2 )
=> ~ ! [Uy: option @ ( product_prod @ nat @ nat ),V3: nat,TreeList2: list @ vEBT_VEBT] :
( ? [S2: vEBT_VEBT] :
( X
= ( vEBT_Node @ Uy @ ( suc @ V3 ) @ TreeList2 @ S2 ) )
=> ( Y2
= ( ~ ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_V5719532721284313246member @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.naive_member.elims(1)
thf(fact_1113_buildup__gives__valid,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( vEBT_invar_vebt @ ( vEBT_vebt_buildup @ N ) @ N ) ) ).
% buildup_gives_valid
thf(fact_1114_VEBT__internal_Omembermima_Oelims_I2_J,axiom,
! [X: vEBT_VEBT,Xa: nat] :
( ( vEBT_VEBT_membermima @ X @ Xa )
=> ( ! [Mi3: nat,Ma3: nat] :
( ? [Va2: list @ vEBT_VEBT,Vb: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ Ma3 ) ) @ ( zero_zero @ nat ) @ Va2 @ Vb ) )
=> ~ ( ( Xa = Mi3 )
| ( Xa = Ma3 ) ) )
=> ( ! [Mi3: nat,Ma3: nat,V3: nat,TreeList2: list @ vEBT_VEBT] :
( ? [Vc: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ Ma3 ) ) @ ( suc @ V3 ) @ TreeList2 @ Vc ) )
=> ~ ( ( Xa = Mi3 )
| ( Xa = Ma3 )
| ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) )
=> ~ ! [V3: nat,TreeList2: list @ vEBT_VEBT] :
( ? [Vd: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V3 ) @ TreeList2 @ Vd ) )
=> ~ ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ).
% VEBT_internal.membermima.elims(2)
thf(fact_1115_not__None__eq,axiom,
! [A: $tType,X: option @ A] :
( ( X
!= ( none @ A ) )
= ( ? [Y: A] :
( X
= ( some @ A @ Y ) ) ) ) ).
% not_None_eq
thf(fact_1116_buildup__nothing__in__min__max,axiom,
! [N: nat,X: nat] :
~ ( vEBT_VEBT_membermima @ ( vEBT_vebt_buildup @ N ) @ X ) ).
% buildup_nothing_in_min_max
thf(fact_1117_buildup__nothing__in__leaf,axiom,
! [N: nat,X: nat] :
~ ( vEBT_V5719532721284313246member @ ( vEBT_vebt_buildup @ N ) @ X ) ).
% buildup_nothing_in_leaf
thf(fact_1118_both__member__options__def,axiom,
( vEBT_V8194947554948674370ptions
= ( ^ [T3: vEBT_VEBT,X4: nat] :
( ( vEBT_V5719532721284313246member @ T3 @ X4 )
| ( vEBT_VEBT_membermima @ T3 @ X4 ) ) ) ) ).
% both_member_options_def
thf(fact_1119_not__Some__eq,axiom,
! [A: $tType,X: option @ A] :
( ( ! [Y: A] :
( X
!= ( some @ A @ Y ) ) )
= ( X
= ( none @ A ) ) ) ).
% not_Some_eq
thf(fact_1120_verit__la__generic,axiom,
! [A2: int,X: int] :
( ( ord_less_eq @ int @ A2 @ X )
| ( A2 = X )
| ( ord_less_eq @ int @ X @ A2 ) ) ).
% verit_la_generic
thf(fact_1121_VEBT__internal_Omembermima_Osimps_I1_J,axiom,
! [Uu: $o,Uv: $o,Uw2: nat] :
~ ( vEBT_VEBT_membermima @ ( vEBT_Leaf @ Uu @ Uv ) @ Uw2 ) ).
% VEBT_internal.membermima.simps(1)
thf(fact_1122_VEBT__internal_Onaive__member_Osimps_I2_J,axiom,
! [Uu: option @ ( product_prod @ nat @ nat ),Uv: list @ vEBT_VEBT,Uw2: vEBT_VEBT,Ux2: nat] :
~ ( vEBT_V5719532721284313246member @ ( vEBT_Node @ Uu @ ( zero_zero @ nat ) @ Uv @ Uw2 ) @ Ux2 ) ).
% VEBT_internal.naive_member.simps(2)
thf(fact_1123_vebt__buildup_Osimps_I1_J,axiom,
( ( vEBT_vebt_buildup @ ( zero_zero @ nat ) )
= ( vEBT_Leaf @ $false @ $false ) ) ).
% vebt_buildup.simps(1)
thf(fact_1124_VEBT__internal_Omembermima_Osimps_I2_J,axiom,
! [Ux2: list @ vEBT_VEBT,Uy2: vEBT_VEBT,Uz: nat] :
~ ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( zero_zero @ nat ) @ Ux2 @ Uy2 ) @ Uz ) ).
% VEBT_internal.membermima.simps(2)
thf(fact_1125_vebt__buildup_Osimps_I2_J,axiom,
( ( vEBT_vebt_buildup @ ( suc @ ( zero_zero @ nat ) ) )
= ( vEBT_Leaf @ $false @ $false ) ) ).
% vebt_buildup.simps(2)
thf(fact_1126_VEBT__internal_Onaive__member_Osimps_I1_J,axiom,
! [A2: $o,B2: $o,X: nat] :
( ( vEBT_V5719532721284313246member @ ( vEBT_Leaf @ A2 @ B2 ) @ X )
= ( ( ( X
= ( zero_zero @ nat ) )
=> A2 )
& ( ( X
!= ( zero_zero @ nat ) )
=> ( ( ( X
= ( one_one @ nat ) )
=> B2 )
& ( X
= ( one_one @ nat ) ) ) ) ) ) ).
% VEBT_internal.naive_member.simps(1)
thf(fact_1127_VEBT__internal_Omembermima_Osimps_I3_J,axiom,
! [Mi: nat,Ma: nat,Va3: list @ vEBT_VEBT,Vb2: vEBT_VEBT,X: nat] :
( ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( zero_zero @ nat ) @ Va3 @ Vb2 ) @ X )
= ( ( X = Mi )
| ( X = Ma ) ) ) ).
% VEBT_internal.membermima.simps(3)
thf(fact_1128_option_Osize_I3_J,axiom,
! [A: $tType] :
( ( size_size @ ( option @ A ) @ ( none @ A ) )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% option.size(3)
thf(fact_1129_combine__options__cases,axiom,
! [A: $tType,B: $tType,X: option @ A,P3: ( option @ A ) > ( option @ B ) > $o,Y2: option @ B] :
( ( ( X
= ( none @ A ) )
=> ( P3 @ X @ Y2 ) )
=> ( ( ( Y2
= ( none @ B ) )
=> ( P3 @ X @ Y2 ) )
=> ( ! [A4: A,B3: B] :
( ( X
= ( some @ A @ A4 ) )
=> ( ( Y2
= ( some @ B @ B3 ) )
=> ( P3 @ X @ Y2 ) ) )
=> ( P3 @ X @ Y2 ) ) ) ) ).
% combine_options_cases
thf(fact_1130_split__option__all,axiom,
! [A: $tType] :
( ( ^ [P: ( option @ A ) > $o] :
! [X3: option @ A] : ( P @ X3 ) )
= ( ^ [P2: ( option @ A ) > $o] :
( ( P2 @ ( none @ A ) )
& ! [X4: A] : ( P2 @ ( some @ A @ X4 ) ) ) ) ) ).
% split_option_all
thf(fact_1131_split__option__ex,axiom,
! [A: $tType] :
( ( ^ [P: ( option @ A ) > $o] :
? [X3: option @ A] : ( P @ X3 ) )
= ( ^ [P2: ( option @ A ) > $o] :
( ( P2 @ ( none @ A ) )
| ? [X4: A] : ( P2 @ ( some @ A @ X4 ) ) ) ) ) ).
% split_option_ex
thf(fact_1132_option_Oexhaust,axiom,
! [A: $tType,Y2: option @ A] :
( ( Y2
!= ( none @ A ) )
=> ~ ! [X24: A] :
( Y2
!= ( some @ A @ X24 ) ) ) ).
% option.exhaust
thf(fact_1133_option_OdiscI,axiom,
! [A: $tType,Option: option @ A,X23: A] :
( ( Option
= ( some @ A @ X23 ) )
=> ( Option
!= ( none @ A ) ) ) ).
% option.discI
thf(fact_1134_option_Odistinct_I1_J,axiom,
! [A: $tType,X23: A] :
( ( none @ A )
!= ( some @ A @ X23 ) ) ).
% option.distinct(1)
thf(fact_1135_VEBT__internal_Omembermima_Osimps_I5_J,axiom,
! [V2: nat,TreeList: list @ vEBT_VEBT,Vd2: vEBT_VEBT,X: nat] :
( ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V2 ) @ TreeList @ Vd2 ) @ X )
= ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) ) ) ) ).
% VEBT_internal.membermima.simps(5)
thf(fact_1136_VEBT__internal_Onaive__member_Osimps_I3_J,axiom,
! [Uy2: option @ ( product_prod @ nat @ nat ),V2: nat,TreeList: list @ vEBT_VEBT,S: vEBT_VEBT,X: nat] :
( ( vEBT_V5719532721284313246member @ ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList @ S ) @ X )
= ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
=> ( vEBT_V5719532721284313246member @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) ) ) ) ).
% VEBT_internal.naive_member.simps(3)
thf(fact_1137_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_1138_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_1139_nat__induct2,axiom,
! [P3: nat > $o,N: nat] :
( ( P3 @ ( zero_zero @ nat ) )
=> ( ( P3 @ ( one_one @ nat ) )
=> ( ! [N3: nat] :
( ( P3 @ N3 )
=> ( P3 @ ( plus_plus @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
=> ( P3 @ N ) ) ) ) ).
% nat_induct2
thf(fact_1140_VEBT__internal_Omembermima_Osimps_I4_J,axiom,
! [Mi: nat,Ma: nat,V2: nat,TreeList: list @ vEBT_VEBT,Vc2: vEBT_VEBT,X: nat] :
( ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ V2 ) @ TreeList @ Vc2 ) @ X )
= ( ( X = Mi )
| ( X = Ma )
| ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ V2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) ) ) ) ) ).
% VEBT_internal.membermima.simps(4)
thf(fact_1141_zle__add1__eq__le,axiom,
! [W: int,Z3: int] :
( ( ord_less @ int @ W @ ( plus_plus @ int @ Z3 @ ( one_one @ int ) ) )
= ( ord_less_eq @ int @ W @ Z3 ) ) ).
% zle_add1_eq_le
thf(fact_1142_set__bit__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A] :
( ( bit_se5668285175392031749et_bit @ A @ ( zero_zero @ nat ) @ A2 )
= ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% set_bit_0
thf(fact_1143_incr__mult__lemma,axiom,
! [D2: int,P3: int > $o,K2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D2 )
=> ( ! [X5: int] :
( ( P3 @ X5 )
=> ( P3 @ ( plus_plus @ int @ X5 @ D2 ) ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K2 )
=> ! [X2: int] :
( ( P3 @ X2 )
=> ( P3 @ ( plus_plus @ int @ X2 @ ( times_times @ int @ K2 @ D2 ) ) ) ) ) ) ) ).
% incr_mult_lemma
thf(fact_1144_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_1145_double__eq__0__iff,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A2: A] :
( ( ( plus_plus @ A @ A2 @ A2 )
= ( zero_zero @ A ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% double_eq_0_iff
thf(fact_1146_unset__bit__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A] :
( ( bit_se2638667681897837118et_bit @ A @ ( zero_zero @ nat ) @ A2 )
= ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% unset_bit_0
thf(fact_1147_zless__imp__add1__zle,axiom,
! [W: int,Z3: int] :
( ( ord_less @ int @ W @ Z3 )
=> ( ord_less_eq @ int @ ( plus_plus @ int @ W @ ( one_one @ int ) ) @ Z3 ) ) ).
% zless_imp_add1_zle
thf(fact_1148_add1__zle__eq,axiom,
! [W: int,Z3: int] :
( ( ord_less_eq @ int @ ( plus_plus @ int @ W @ ( one_one @ int ) ) @ Z3 )
= ( ord_less @ int @ W @ Z3 ) ) ).
% add1_zle_eq
thf(fact_1149_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_1150_unset__bit__nonnegative__int__iff,axiom,
! [N: nat,K2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se2638667681897837118et_bit @ int @ N @ K2 ) )
= ( ord_less_eq @ int @ ( zero_zero @ int ) @ K2 ) ) ).
% unset_bit_nonnegative_int_iff
thf(fact_1151_set__bit__nonnegative__int__iff,axiom,
! [N: nat,K2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se5668285175392031749et_bit @ int @ N @ K2 ) )
= ( ord_less_eq @ int @ ( zero_zero @ int ) @ K2 ) ) ).
% set_bit_nonnegative_int_iff
thf(fact_1152_unset__bit__negative__int__iff,axiom,
! [N: nat,K2: int] :
( ( ord_less @ int @ ( bit_se2638667681897837118et_bit @ int @ N @ K2 ) @ ( zero_zero @ int ) )
= ( ord_less @ int @ K2 @ ( zero_zero @ int ) ) ) ).
% unset_bit_negative_int_iff
thf(fact_1153_set__bit__negative__int__iff,axiom,
! [N: nat,K2: int] :
( ( ord_less @ int @ ( bit_se5668285175392031749et_bit @ int @ N @ K2 ) @ ( zero_zero @ int ) )
= ( ord_less @ int @ K2 @ ( zero_zero @ int ) ) ) ).
% set_bit_negative_int_iff
thf(fact_1154_unset__bit__less__eq,axiom,
! [N: nat,K2: int] : ( ord_less_eq @ int @ ( bit_se2638667681897837118et_bit @ int @ N @ K2 ) @ K2 ) ).
% unset_bit_less_eq
thf(fact_1155_set__bit__greater__eq,axiom,
! [K2: int,N: nat] : ( ord_less_eq @ int @ K2 @ ( bit_se5668285175392031749et_bit @ int @ N @ K2 ) ) ).
% set_bit_greater_eq
thf(fact_1156_minf_I11_J,axiom,
! [C: $tType,D: $tType] :
( ( ord @ C )
=> ! [F4: D] :
? [Z: C] :
! [X2: C] :
( ( ord_less @ C @ X2 @ Z )
=> ( F4 = F4 ) ) ) ).
% minf(11)
thf(fact_1157_minf_I7_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z: A] :
! [X2: A] :
( ( ord_less @ A @ X2 @ Z )
=> ~ ( ord_less @ A @ T2 @ X2 ) ) ) ).
% minf(7)
thf(fact_1158_minf_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z: A] :
! [X2: A] :
( ( ord_less @ A @ X2 @ Z )
=> ( ord_less @ A @ X2 @ T2 ) ) ) ).
% minf(5)
thf(fact_1159_minf_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z: A] :
! [X2: A] :
( ( ord_less @ A @ X2 @ Z )
=> ( X2 != T2 ) ) ) ).
% minf(4)
thf(fact_1160_minf_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z: A] :
! [X2: A] :
( ( ord_less @ A @ X2 @ Z )
=> ( X2 != T2 ) ) ) ).
% minf(3)
thf(fact_1161_minf_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P3: A > $o,P6: A > $o,Q2: A > $o,Q6: A > $o] :
( ? [Z4: A] :
! [X5: A] :
( ( ord_less @ A @ X5 @ Z4 )
=> ( ( P3 @ X5 )
= ( P6 @ X5 ) ) )
=> ( ? [Z4: A] :
! [X5: A] :
( ( ord_less @ A @ X5 @ Z4 )
=> ( ( Q2 @ X5 )
= ( Q6 @ X5 ) ) )
=> ? [Z: A] :
! [X2: A] :
( ( ord_less @ A @ X2 @ Z )
=> ( ( ( P3 @ X2 )
| ( Q2 @ X2 ) )
= ( ( P6 @ X2 )
| ( Q6 @ X2 ) ) ) ) ) ) ) ).
% minf(2)
thf(fact_1162_minf_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P3: A > $o,P6: A > $o,Q2: A > $o,Q6: A > $o] :
( ? [Z4: A] :
! [X5: A] :
( ( ord_less @ A @ X5 @ Z4 )
=> ( ( P3 @ X5 )
= ( P6 @ X5 ) ) )
=> ( ? [Z4: A] :
! [X5: A] :
( ( ord_less @ A @ X5 @ Z4 )
=> ( ( Q2 @ X5 )
= ( Q6 @ X5 ) ) )
=> ? [Z: A] :
! [X2: A] :
( ( ord_less @ A @ X2 @ Z )
=> ( ( ( P3 @ X2 )
& ( Q2 @ X2 ) )
= ( ( P6 @ X2 )
& ( Q6 @ X2 ) ) ) ) ) ) ) ).
% minf(1)
thf(fact_1163_pinf_I11_J,axiom,
! [C: $tType,D: $tType] :
( ( ord @ C )
=> ! [F4: D] :
? [Z: C] :
! [X2: C] :
( ( ord_less @ C @ Z @ X2 )
=> ( F4 = F4 ) ) ) ).
% pinf(11)
thf(fact_1164_pinf_I7_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z: A] :
! [X2: A] :
( ( ord_less @ A @ Z @ X2 )
=> ( ord_less @ A @ T2 @ X2 ) ) ) ).
% pinf(7)
thf(fact_1165_pinf_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z: A] :
! [X2: A] :
( ( ord_less @ A @ Z @ X2 )
=> ~ ( ord_less @ A @ X2 @ T2 ) ) ) ).
% pinf(5)
thf(fact_1166_pinf_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z: A] :
! [X2: A] :
( ( ord_less @ A @ Z @ X2 )
=> ( X2 != T2 ) ) ) ).
% pinf(4)
thf(fact_1167_pinf_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z: A] :
! [X2: A] :
( ( ord_less @ A @ Z @ X2 )
=> ( X2 != T2 ) ) ) ).
% pinf(3)
thf(fact_1168_pinf_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P3: A > $o,P6: A > $o,Q2: A > $o,Q6: A > $o] :
( ? [Z4: A] :
! [X5: A] :
( ( ord_less @ A @ Z4 @ X5 )
=> ( ( P3 @ X5 )
= ( P6 @ X5 ) ) )
=> ( ? [Z4: A] :
! [X5: A] :
( ( ord_less @ A @ Z4 @ X5 )
=> ( ( Q2 @ X5 )
= ( Q6 @ X5 ) ) )
=> ? [Z: A] :
! [X2: A] :
( ( ord_less @ A @ Z @ X2 )
=> ( ( ( P3 @ X2 )
| ( Q2 @ X2 ) )
= ( ( P6 @ X2 )
| ( Q6 @ X2 ) ) ) ) ) ) ) ).
% pinf(2)
thf(fact_1169_pinf_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P3: A > $o,P6: A > $o,Q2: A > $o,Q6: A > $o] :
( ? [Z4: A] :
! [X5: A] :
( ( ord_less @ A @ Z4 @ X5 )
=> ( ( P3 @ X5 )
= ( P6 @ X5 ) ) )
=> ( ? [Z4: A] :
! [X5: A] :
( ( ord_less @ A @ Z4 @ X5 )
=> ( ( Q2 @ X5 )
= ( Q6 @ X5 ) ) )
=> ? [Z: A] :
! [X2: A] :
( ( ord_less @ A @ Z @ X2 )
=> ( ( ( P3 @ X2 )
& ( Q2 @ X2 ) )
= ( ( P6 @ X2 )
& ( Q6 @ X2 ) ) ) ) ) ) ) ).
% pinf(1)
thf(fact_1170_minf_I8_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z: A] :
! [X2: A] :
( ( ord_less @ A @ X2 @ Z )
=> ~ ( ord_less_eq @ A @ T2 @ X2 ) ) ) ).
% minf(8)
thf(fact_1171_minf_I6_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z: A] :
! [X2: A] :
( ( ord_less @ A @ X2 @ Z )
=> ( ord_less_eq @ A @ X2 @ T2 ) ) ) ).
% minf(6)
thf(fact_1172_pinf_I8_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z: A] :
! [X2: A] :
( ( ord_less @ A @ Z @ X2 )
=> ( ord_less_eq @ A @ T2 @ X2 ) ) ) ).
% pinf(8)
thf(fact_1173_pinf_I6_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z: A] :
! [X2: A] :
( ( ord_less @ A @ Z @ X2 )
=> ~ ( ord_less_eq @ A @ X2 @ T2 ) ) ) ).
% pinf(6)
thf(fact_1174_conj__le__cong,axiom,
! [X: int,X8: int,P3: $o,P6: $o] :
( ( X = X8 )
=> ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X8 )
=> ( P3 = P6 ) )
=> ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
& P3 )
= ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X8 )
& P6 ) ) ) ) ).
% conj_le_cong
thf(fact_1175_imp__le__cong,axiom,
! [X: int,X8: int,P3: $o,P6: $o] :
( ( X = X8 )
=> ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X8 )
=> ( P3 = P6 ) )
=> ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
=> P3 )
= ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X8 )
=> P6 ) ) ) ) ).
% imp_le_cong
thf(fact_1176_less__eq__int__code_I1_J,axiom,
ord_less_eq @ int @ ( zero_zero @ int ) @ ( zero_zero @ int ) ).
% less_eq_int_code(1)
thf(fact_1177_times__int__code_I1_J,axiom,
! [K2: int] :
( ( times_times @ int @ K2 @ ( zero_zero @ int ) )
= ( zero_zero @ int ) ) ).
% times_int_code(1)
thf(fact_1178_times__int__code_I2_J,axiom,
! [L: int] :
( ( times_times @ int @ ( zero_zero @ int ) @ L )
= ( zero_zero @ int ) ) ).
% times_int_code(2)
thf(fact_1179_int__distrib_I1_J,axiom,
! [Z1: int,Z22: int,W: int] :
( ( times_times @ int @ ( plus_plus @ int @ Z1 @ Z22 ) @ W )
= ( plus_plus @ int @ ( times_times @ int @ Z1 @ W ) @ ( times_times @ int @ Z22 @ W ) ) ) ).
% int_distrib(1)
thf(fact_1180_int__distrib_I2_J,axiom,
! [W: int,Z1: int,Z22: int] :
( ( times_times @ int @ W @ ( plus_plus @ int @ Z1 @ Z22 ) )
= ( plus_plus @ int @ ( times_times @ int @ W @ Z1 ) @ ( times_times @ int @ W @ Z22 ) ) ) ).
% int_distrib(2)
thf(fact_1181_zmult__zless__mono2,axiom,
! [I2: int,J3: int,K2: int] :
( ( ord_less @ int @ I2 @ J3 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ K2 )
=> ( ord_less @ int @ ( times_times @ int @ K2 @ I2 ) @ ( times_times @ int @ K2 @ J3 ) ) ) ) ).
% zmult_zless_mono2
thf(fact_1182_odd__nonzero,axiom,
! [Z3: int] :
( ( plus_plus @ int @ ( plus_plus @ int @ ( one_one @ int ) @ Z3 ) @ Z3 )
!= ( zero_zero @ int ) ) ).
% odd_nonzero
thf(fact_1183_int__ge__induct,axiom,
! [K2: int,I2: int,P3: int > $o] :
( ( ord_less_eq @ int @ K2 @ I2 )
=> ( ( P3 @ K2 )
=> ( ! [I3: int] :
( ( ord_less_eq @ int @ K2 @ I3 )
=> ( ( P3 @ I3 )
=> ( P3 @ ( plus_plus @ int @ I3 @ ( one_one @ int ) ) ) ) )
=> ( P3 @ I2 ) ) ) ) ).
% int_ge_induct
thf(fact_1184_int__gr__induct,axiom,
! [K2: int,I2: int,P3: int > $o] :
( ( ord_less @ int @ K2 @ I2 )
=> ( ( P3 @ ( plus_plus @ int @ K2 @ ( one_one @ int ) ) )
=> ( ! [I3: int] :
( ( ord_less @ int @ K2 @ I3 )
=> ( ( P3 @ I3 )
=> ( P3 @ ( plus_plus @ int @ I3 @ ( one_one @ int ) ) ) ) )
=> ( P3 @ I2 ) ) ) ) ).
% int_gr_induct
thf(fact_1185_zless__add1__eq,axiom,
! [W: int,Z3: int] :
( ( ord_less @ int @ W @ ( plus_plus @ int @ Z3 @ ( one_one @ int ) ) )
= ( ( ord_less @ int @ W @ Z3 )
| ( W = Z3 ) ) ) ).
% zless_add1_eq
thf(fact_1186_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_1187_set__bit__Suc,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A2: A] :
( ( bit_se5668285175392031749et_bit @ A @ ( suc @ N ) @ A2 )
= ( plus_plus @ A @ ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5668285175392031749et_bit @ A @ N @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% set_bit_Suc
thf(fact_1188_unset__bit__Suc,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A2: A] :
( ( bit_se2638667681897837118et_bit @ A @ ( suc @ N ) @ A2 )
= ( plus_plus @ A @ ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se2638667681897837118et_bit @ A @ N @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% unset_bit_Suc
thf(fact_1189_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_1190_flip__bit__Suc,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A2: A] :
( ( bit_se8732182000553998342ip_bit @ A @ ( suc @ N ) @ A2 )
= ( plus_plus @ A @ ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se8732182000553998342ip_bit @ A @ N @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% flip_bit_Suc
thf(fact_1191_mult__le__cancel__iff1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z3: A,X: A,Y2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Z3 )
=> ( ( ord_less_eq @ A @ ( times_times @ A @ X @ Z3 ) @ ( times_times @ A @ Y2 @ Z3 ) )
= ( ord_less_eq @ A @ X @ Y2 ) ) ) ) ).
% mult_le_cancel_iff1
thf(fact_1192_mult__le__cancel__iff2,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z3: A,X: A,Y2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Z3 )
=> ( ( ord_less_eq @ A @ ( times_times @ A @ Z3 @ X ) @ ( times_times @ A @ Z3 @ Y2 ) )
= ( ord_less_eq @ A @ X @ Y2 ) ) ) ) ).
% mult_le_cancel_iff2
thf(fact_1193_product__nth,axiom,
! [A: $tType,B: $tType,N: nat,Xs: list @ A,Ys: list @ B] :
( ( ord_less @ nat @ N @ ( times_times @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ B ) @ Ys ) ) )
=> ( ( nth @ ( product_prod @ A @ B ) @ ( product @ A @ B @ Xs @ Ys ) @ N )
= ( product_Pair @ A @ B @ ( nth @ A @ Xs @ ( divide_divide @ nat @ N @ ( size_size @ ( list @ B ) @ Ys ) ) ) @ ( nth @ B @ Ys @ ( modulo_modulo @ nat @ N @ ( size_size @ ( list @ B ) @ Ys ) ) ) ) ) ) ).
% product_nth
thf(fact_1194_predicate1I,axiom,
! [A: $tType,P3: A > $o,Q2: A > $o] :
( ! [X5: A] :
( ( P3 @ X5 )
=> ( Q2 @ X5 ) )
=> ( ord_less_eq @ ( A > $o ) @ P3 @ Q2 ) ) ).
% predicate1I
thf(fact_1195_gcd__nat__induct,axiom,
! [P3: nat > nat > $o,M: nat,N: nat] :
( ! [M4: nat] : ( P3 @ M4 @ ( zero_zero @ nat ) )
=> ( ! [M4: nat,N3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( ( P3 @ N3 @ ( modulo_modulo @ nat @ M4 @ N3 ) )
=> ( P3 @ M4 @ N3 ) ) )
=> ( P3 @ M @ N ) ) ) ).
% gcd_nat_induct
thf(fact_1196_concat__bit__Suc,axiom,
! [N: nat,K2: int,L: int] :
( ( bit_concat_bit @ ( suc @ N ) @ K2 @ L )
= ( plus_plus @ int @ ( modulo_modulo @ int @ K2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_concat_bit @ N @ ( divide_divide @ int @ K2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ L ) ) ) ) ).
% concat_bit_Suc
thf(fact_1197_dual__order_Orefl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% dual_order.refl
thf(fact_1198_order__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% order_refl
thf(fact_1199_flip__bit__nonnegative__int__iff,axiom,
! [N: nat,K2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se8732182000553998342ip_bit @ int @ N @ K2 ) )
= ( ord_less_eq @ int @ ( zero_zero @ int ) @ K2 ) ) ).
% flip_bit_nonnegative_int_iff
thf(fact_1200_flip__bit__negative__int__iff,axiom,
! [N: nat,K2: int] :
( ( ord_less @ int @ ( bit_se8732182000553998342ip_bit @ int @ N @ K2 ) @ ( zero_zero @ int ) )
= ( ord_less @ int @ K2 @ ( zero_zero @ int ) ) ) ).
% flip_bit_negative_int_iff
thf(fact_1201_concat__bit__0,axiom,
! [K2: int,L: int] :
( ( bit_concat_bit @ ( zero_zero @ nat ) @ K2 @ L )
= L ) ).
% concat_bit_0
thf(fact_1202_concat__bit__nonnegative__iff,axiom,
! [N: nat,K2: int,L: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_concat_bit @ N @ K2 @ L ) )
= ( ord_less_eq @ int @ ( zero_zero @ int ) @ L ) ) ).
% concat_bit_nonnegative_iff
thf(fact_1203_concat__bit__negative__iff,axiom,
! [N: nat,K2: int,L: int] :
( ( ord_less @ int @ ( bit_concat_bit @ N @ K2 @ L ) @ ( zero_zero @ int ) )
= ( ord_less @ int @ L @ ( zero_zero @ int ) ) ) ).
% concat_bit_negative_iff
thf(fact_1204_length__product,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B] :
( ( size_size @ ( list @ ( product_prod @ A @ B ) ) @ ( product @ A @ B @ Xs @ Ys ) )
= ( times_times @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ B ) @ Ys ) ) ) ).
% length_product
thf(fact_1205_concat__bit__assoc,axiom,
! [N: nat,K2: int,M: nat,L: int,R: int] :
( ( bit_concat_bit @ N @ K2 @ ( bit_concat_bit @ M @ L @ R ) )
= ( bit_concat_bit @ ( plus_plus @ nat @ M @ N ) @ ( bit_concat_bit @ N @ K2 @ L ) @ R ) ) ).
% concat_bit_assoc
thf(fact_1206_VEBT__internal_Ovalid_H_Ocases,axiom,
! [X: product_prod @ vEBT_VEBT @ nat] :
( ! [Uu2: $o,Uv2: $o,D3: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ D3 ) )
=> ~ ! [Mima: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT,Deg3: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary2 ) @ Deg3 ) ) ) ).
% VEBT_internal.valid'.cases
thf(fact_1207_order__antisym__conv,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Y2: A,X: A] :
( ( ord_less_eq @ A @ Y2 @ X )
=> ( ( ord_less_eq @ A @ X @ Y2 )
= ( X = Y2 ) ) ) ) ).
% order_antisym_conv
thf(fact_1208_linorder__le__cases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y2: A] :
( ~ ( ord_less_eq @ A @ X @ Y2 )
=> ( ord_less_eq @ A @ Y2 @ X ) ) ) ).
% linorder_le_cases
thf(fact_1209_ord__le__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A2: A,B2: A,F3: A > B,C2: B] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ( F3 @ B2 )
= C2 )
=> ( ! [X5: A,Y7: A] :
( ( ord_less_eq @ A @ X5 @ Y7 )
=> ( ord_less_eq @ B @ ( F3 @ X5 ) @ ( F3 @ Y7 ) ) )
=> ( ord_less_eq @ B @ ( F3 @ A2 ) @ C2 ) ) ) ) ) ).
% ord_le_eq_subst
thf(fact_1210_ord__eq__le__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A2: A,F3: B > A,B2: B,C2: B] :
( ( A2
= ( F3 @ B2 ) )
=> ( ( ord_less_eq @ B @ B2 @ C2 )
=> ( ! [X5: B,Y7: B] :
( ( ord_less_eq @ B @ X5 @ Y7 )
=> ( ord_less_eq @ A @ ( F3 @ X5 ) @ ( F3 @ Y7 ) ) )
=> ( ord_less_eq @ A @ A2 @ ( F3 @ C2 ) ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_1211_linorder__linear,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y2: A] :
( ( ord_less_eq @ A @ X @ Y2 )
| ( ord_less_eq @ A @ Y2 @ X ) ) ) ).
% linorder_linear
thf(fact_1212_order__eq__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y2: A] :
( ( X = Y2 )
=> ( ord_less_eq @ A @ X @ Y2 ) ) ) ).
% order_eq_refl
thf(fact_1213_order__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C )
& ( order @ A ) )
=> ! [A2: A,B2: A,F3: A > C,C2: C] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ C @ ( F3 @ B2 ) @ C2 )
=> ( ! [X5: A,Y7: A] :
( ( ord_less_eq @ A @ X5 @ Y7 )
=> ( ord_less_eq @ C @ ( F3 @ X5 ) @ ( F3 @ Y7 ) ) )
=> ( ord_less_eq @ C @ ( F3 @ A2 ) @ C2 ) ) ) ) ) ).
% order_subst2
thf(fact_1214_order__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A2: A,F3: B > A,B2: B,C2: B] :
( ( ord_less_eq @ A @ A2 @ ( F3 @ B2 ) )
=> ( ( ord_less_eq @ B @ B2 @ C2 )
=> ( ! [X5: B,Y7: B] :
( ( ord_less_eq @ B @ X5 @ Y7 )
=> ( ord_less_eq @ A @ ( F3 @ X5 ) @ ( F3 @ Y7 ) ) )
=> ( ord_less_eq @ A @ A2 @ ( F3 @ C2 ) ) ) ) ) ) ).
% order_subst1
thf(fact_1215_Orderings_Oorder__eq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y5: A,Z2: A] : Y5 = Z2 )
= ( ^ [A5: A,B4: A] :
( ( ord_less_eq @ A @ A5 @ B4 )
& ( ord_less_eq @ A @ B4 @ A5 ) ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_1216_antisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ) ).
% antisym
thf(fact_1217_dual__order_Otrans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_less_eq @ A @ C2 @ B2 )
=> ( ord_less_eq @ A @ C2 @ A2 ) ) ) ) ).
% dual_order.trans
thf(fact_1218_dual__order_Oantisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B2: A,A2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_less_eq @ A @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ) ).
% dual_order.antisym
thf(fact_1219_dual__order_Oeq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y5: A,Z2: A] : Y5 = Z2 )
= ( ^ [A5: A,B4: A] :
( ( ord_less_eq @ A @ B4 @ A5 )
& ( ord_less_eq @ A @ A5 @ B4 ) ) ) ) ) ).
% dual_order.eq_iff
thf(fact_1220_linorder__wlog,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P3: A > A > $o,A2: A,B2: A] :
( ! [A4: A,B3: A] :
( ( ord_less_eq @ A @ A4 @ B3 )
=> ( P3 @ A4 @ B3 ) )
=> ( ! [A4: A,B3: A] :
( ( P3 @ B3 @ A4 )
=> ( P3 @ A4 @ B3 ) )
=> ( P3 @ A2 @ B2 ) ) ) ) ).
% linorder_wlog
thf(fact_1221_order__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y2: A,Z3: A] :
( ( ord_less_eq @ A @ X @ Y2 )
=> ( ( ord_less_eq @ A @ Y2 @ Z3 )
=> ( ord_less_eq @ A @ X @ Z3 ) ) ) ) ).
% order_trans
thf(fact_1222_order_Otrans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ C2 )
=> ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% order.trans
thf(fact_1223_order__antisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,Y2: A] :
( ( ord_less_eq @ A @ X @ Y2 )
=> ( ( ord_less_eq @ A @ Y2 @ X )
=> ( X = Y2 ) ) ) ) ).
% order_antisym
thf(fact_1224_ord__le__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( B2 = C2 )
=> ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% ord_le_eq_trans
thf(fact_1225_ord__eq__le__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( A2 = B2 )
=> ( ( ord_less_eq @ A @ B2 @ C2 )
=> ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% ord_eq_le_trans
thf(fact_1226_order__class_Oorder__eq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y5: A,Z2: A] : Y5 = Z2 )
= ( ^ [X4: A,Y: A] :
( ( ord_less_eq @ A @ X4 @ Y )
& ( ord_less_eq @ A @ Y @ X4 ) ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_1227_le__cases3,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y2: A,Z3: A] :
( ( ( ord_less_eq @ A @ X @ Y2 )
=> ~ ( ord_less_eq @ A @ Y2 @ Z3 ) )
=> ( ( ( ord_less_eq @ A @ Y2 @ X )
=> ~ ( ord_less_eq @ A @ X @ Z3 ) )
=> ( ( ( ord_less_eq @ A @ X @ Z3 )
=> ~ ( ord_less_eq @ A @ Z3 @ Y2 ) )
=> ( ( ( ord_less_eq @ A @ Z3 @ Y2 )
=> ~ ( ord_less_eq @ A @ Y2 @ X ) )
=> ( ( ( ord_less_eq @ A @ Y2 @ Z3 )
=> ~ ( ord_less_eq @ A @ Z3 @ X ) )
=> ~ ( ( ord_less_eq @ A @ Z3 @ X )
=> ~ ( ord_less_eq @ A @ X @ Y2 ) ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_1228_nle__le,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A] :
( ( ~ ( ord_less_eq @ A @ A2 @ B2 ) )
= ( ( ord_less_eq @ A @ B2 @ A2 )
& ( B2 != A2 ) ) ) ) ).
% nle_le
thf(fact_1229_order__less__imp__not__less,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y2: A] :
( ( ord_less @ A @ X @ Y2 )
=> ~ ( ord_less @ A @ Y2 @ X ) ) ) ).
% order_less_imp_not_less
thf(fact_1230_order__less__imp__not__eq2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,Y2: A] :
( ( ord_less @ A @ X @ Y2 )
=> ( Y2 != X ) ) ) ).
% order_less_imp_not_eq2
thf(fact_1231_order__less__imp__not__eq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,Y2: A] :
( ( ord_less @ A @ X @ Y2 )
=> ( X != Y2 ) ) ) ).
% order_less_imp_not_eq
thf(fact_1232_linorder__less__linear,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y2: A] :
( ( ord_less @ A @ X @ Y2 )
| ( X = Y2 )
| ( ord_less @ A @ Y2 @ X ) ) ) ).
% linorder_less_linear
thf(fact_1233_order__less__imp__triv,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y2: A,P3: $o] :
( ( ord_less @ A @ X @ Y2 )
=> ( ( ord_less @ A @ Y2 @ X )
=> P3 ) ) ) ).
% order_less_imp_triv
thf(fact_1234_order__less__not__sym,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y2: A] :
( ( ord_less @ A @ X @ Y2 )
=> ~ ( ord_less @ A @ Y2 @ X ) ) ) ).
% order_less_not_sym
thf(fact_1235_order__less__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C )
& ( order @ A ) )
=> ! [A2: A,B2: A,F3: A > C,C2: C] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ C @ ( F3 @ B2 ) @ C2 )
=> ( ! [X5: A,Y7: A] :
( ( ord_less @ A @ X5 @ Y7 )
=> ( ord_less @ C @ ( F3 @ X5 ) @ ( F3 @ Y7 ) ) )
=> ( ord_less @ C @ ( F3 @ A2 ) @ C2 ) ) ) ) ) ).
% order_less_subst2
thf(fact_1236_order__less__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A2: A,F3: B > A,B2: B,C2: B] :
( ( ord_less @ A @ A2 @ ( F3 @ B2 ) )
=> ( ( ord_less @ B @ B2 @ C2 )
=> ( ! [X5: B,Y7: B] :
( ( ord_less @ B @ X5 @ Y7 )
=> ( ord_less @ A @ ( F3 @ X5 ) @ ( F3 @ Y7 ) ) )
=> ( ord_less @ A @ A2 @ ( F3 @ C2 ) ) ) ) ) ) ).
% order_less_subst1
thf(fact_1237_order__less__irrefl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A] :
~ ( ord_less @ A @ X @ X ) ) ).
% order_less_irrefl
thf(fact_1238_ord__less__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A2: A,B2: A,F3: A > B,C2: B] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ( F3 @ B2 )
= C2 )
=> ( ! [X5: A,Y7: A] :
( ( ord_less @ A @ X5 @ Y7 )
=> ( ord_less @ B @ ( F3 @ X5 ) @ ( F3 @ Y7 ) ) )
=> ( ord_less @ B @ ( F3 @ A2 ) @ C2 ) ) ) ) ) ).
% ord_less_eq_subst
thf(fact_1239_ord__eq__less__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A2: A,F3: B > A,B2: B,C2: B] :
( ( A2
= ( F3 @ B2 ) )
=> ( ( ord_less @ B @ B2 @ C2 )
=> ( ! [X5: B,Y7: B] :
( ( ord_less @ B @ X5 @ Y7 )
=> ( ord_less @ A @ ( F3 @ X5 ) @ ( F3 @ Y7 ) ) )
=> ( ord_less @ A @ A2 @ ( F3 @ C2 ) ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_1240_order__less__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y2: A,Z3: A] :
( ( ord_less @ A @ X @ Y2 )
=> ( ( ord_less @ A @ Y2 @ Z3 )
=> ( ord_less @ A @ X @ Z3 ) ) ) ) ).
% order_less_trans
thf(fact_1241_order__less__asym_H,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ~ ( ord_less @ A @ B2 @ A2 ) ) ) ).
% order_less_asym'
thf(fact_1242_linorder__neq__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y2: A] :
( ( X != Y2 )
= ( ( ord_less @ A @ X @ Y2 )
| ( ord_less @ A @ Y2 @ X ) ) ) ) ).
% linorder_neq_iff
thf(fact_1243_order__less__asym,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y2: A] :
( ( ord_less @ A @ X @ Y2 )
=> ~ ( ord_less @ A @ Y2 @ X ) ) ) ).
% order_less_asym
thf(fact_1244_linorder__neqE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y2: A] :
( ( X != Y2 )
=> ( ~ ( ord_less @ A @ X @ Y2 )
=> ( ord_less @ A @ Y2 @ X ) ) ) ) ).
% linorder_neqE
thf(fact_1245_dual__order_Ostrict__implies__not__eq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B2: A,A2: A] :
( ( ord_less @ A @ B2 @ A2 )
=> ( A2 != B2 ) ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_1246_order_Ostrict__implies__not__eq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( A2 != B2 ) ) ) ).
% order.strict_implies_not_eq
thf(fact_1247_dual__order_Ostrict__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( ord_less @ A @ B2 @ A2 )
=> ( ( ord_less @ A @ C2 @ B2 )
=> ( ord_less @ A @ C2 @ A2 ) ) ) ) ).
% dual_order.strict_trans
thf(fact_1248_not__less__iff__gr__or__eq,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y2: A] :
( ( ~ ( ord_less @ A @ X @ Y2 ) )
= ( ( ord_less @ A @ Y2 @ X )
| ( X = Y2 ) ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_1249_order_Ostrict__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ B2 @ C2 )
=> ( ord_less @ A @ A2 @ C2 ) ) ) ) ).
% order.strict_trans
thf(fact_1250_linorder__less__wlog,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P3: A > A > $o,A2: A,B2: A] :
( ! [A4: A,B3: A] :
( ( ord_less @ A @ A4 @ B3 )
=> ( P3 @ A4 @ B3 ) )
=> ( ! [A4: A] : ( P3 @ A4 @ A4 )
=> ( ! [A4: A,B3: A] :
( ( P3 @ B3 @ A4 )
=> ( P3 @ A4 @ B3 ) )
=> ( P3 @ A2 @ B2 ) ) ) ) ) ).
% linorder_less_wlog
thf(fact_1251_exists__least__iff,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ( ( ^ [P: A > $o] :
? [X3: A] : ( P @ X3 ) )
= ( ^ [P2: A > $o] :
? [N2: A] :
( ( P2 @ N2 )
& ! [M2: A] :
( ( ord_less @ A @ M2 @ N2 )
=> ~ ( P2 @ M2 ) ) ) ) ) ) ).
% exists_least_iff
thf(fact_1252_dual__order_Oirrefl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A2: A] :
~ ( ord_less @ A @ A2 @ A2 ) ) ).
% dual_order.irrefl
thf(fact_1253_dual__order_Oasym,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [B2: A,A2: A] :
( ( ord_less @ A @ B2 @ A2 )
=> ~ ( ord_less @ A @ A2 @ B2 ) ) ) ).
% dual_order.asym
thf(fact_1254_linorder__cases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y2: A] :
( ~ ( ord_less @ A @ X @ Y2 )
=> ( ( X != Y2 )
=> ( ord_less @ A @ Y2 @ X ) ) ) ) ).
% linorder_cases
thf(fact_1255_antisym__conv3,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Y2: A,X: A] :
( ~ ( ord_less @ A @ Y2 @ X )
=> ( ( ~ ( ord_less @ A @ X @ Y2 ) )
= ( X = Y2 ) ) ) ) ).
% antisym_conv3
thf(fact_1256_less__induct,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [P3: A > $o,A2: A] :
( ! [X5: A] :
( ! [Y4: A] :
( ( ord_less @ A @ Y4 @ X5 )
=> ( P3 @ Y4 ) )
=> ( P3 @ X5 ) )
=> ( P3 @ A2 ) ) ) ).
% less_induct
thf(fact_1257_ord__less__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( B2 = C2 )
=> ( ord_less @ A @ A2 @ C2 ) ) ) ) ).
% ord_less_eq_trans
thf(fact_1258_ord__eq__less__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( A2 = B2 )
=> ( ( ord_less @ A @ B2 @ C2 )
=> ( ord_less @ A @ A2 @ C2 ) ) ) ) ).
% ord_eq_less_trans
thf(fact_1259_order_Oasym,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ~ ( ord_less @ A @ B2 @ A2 ) ) ) ).
% order.asym
thf(fact_1260_less__imp__neq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,Y2: A] :
( ( ord_less @ A @ X @ Y2 )
=> ( X != Y2 ) ) ) ).
% less_imp_neq
thf(fact_1261_dense,axiom,
! [A: $tType] :
( ( dense_order @ A )
=> ! [X: A,Y2: A] :
( ( ord_less @ A @ X @ Y2 )
=> ? [Z: A] :
( ( ord_less @ A @ X @ Z )
& ( ord_less @ A @ Z @ Y2 ) ) ) ) ).
% dense
thf(fact_1262_gt__ex,axiom,
! [A: $tType] :
( ( no_top @ A )
=> ! [X: A] :
? [X_12: A] : ( ord_less @ A @ X @ X_12 ) ) ).
% gt_ex
thf(fact_1263_lt__ex,axiom,
! [A: $tType] :
( ( no_bot @ A )
=> ! [X: A] :
? [Y7: A] : ( ord_less @ A @ Y7 @ X ) ) ).
% lt_ex
thf(fact_1264_less__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ( ( ord_less @ ( A > B ) )
= ( ^ [F5: A > B,G2: A > B] :
( ( ord_less_eq @ ( A > B ) @ F5 @ G2 )
& ~ ( ord_less_eq @ ( A > B ) @ G2 @ F5 ) ) ) ) ) ).
% less_fun_def
thf(fact_1265_rev__predicate1D,axiom,
! [A: $tType,P3: A > $o,X: A,Q2: A > $o] :
( ( P3 @ X )
=> ( ( ord_less_eq @ ( A > $o ) @ P3 @ Q2 )
=> ( Q2 @ X ) ) ) ).
% rev_predicate1D
thf(fact_1266_predicate1D,axiom,
! [A: $tType,P3: A > $o,Q2: A > $o,X: A] :
( ( ord_less_eq @ ( A > $o ) @ P3 @ Q2 )
=> ( ( P3 @ X )
=> ( Q2 @ X ) ) ) ).
% predicate1D
thf(fact_1267_VEBT__internal_Onaive__member_Ocases,axiom,
! [X: product_prod @ vEBT_VEBT @ nat] :
( ! [A4: $o,B3: $o,X5: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A4 @ B3 ) @ X5 ) )
=> ( ! [Uu2: option @ ( product_prod @ nat @ nat ),Uv2: list @ vEBT_VEBT,Uw: vEBT_VEBT,Ux: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Uu2 @ ( zero_zero @ nat ) @ Uv2 @ Uw ) @ Ux ) )
=> ~ ! [Uy: option @ ( product_prod @ nat @ nat ),V3: nat,TreeList2: list @ vEBT_VEBT,S2: vEBT_VEBT,X5: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Uy @ ( suc @ V3 ) @ TreeList2 @ S2 ) @ X5 ) ) ) ) ).
% VEBT_internal.naive_member.cases
thf(fact_1268_VEBT__internal_Oelim__dead_Ocases,axiom,
! [X: product_prod @ vEBT_VEBT @ extended_enat] :
( ! [A4: $o,B3: $o,Uu2: extended_enat] :
( X
!= ( product_Pair @ vEBT_VEBT @ extended_enat @ ( vEBT_Leaf @ A4 @ B3 ) @ Uu2 ) )
=> ( ! [Info2: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( X
!= ( product_Pair @ vEBT_VEBT @ extended_enat @ ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) @ ( extend4730790105801354508finity @ extended_enat ) ) )
=> ~ ! [Info2: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT,L2: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ extended_enat @ ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) @ ( extended_enat2 @ L2 ) ) ) ) ) ).
% VEBT_internal.elim_dead.cases
thf(fact_1269_order__le__imp__less__or__eq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,Y2: A] :
( ( ord_less_eq @ A @ X @ Y2 )
=> ( ( ord_less @ A @ X @ Y2 )
| ( X = Y2 ) ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_1270_linorder__le__less__linear,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y2: A] :
( ( ord_less_eq @ A @ X @ Y2 )
| ( ord_less @ A @ Y2 @ X ) ) ) ).
% linorder_le_less_linear
thf(fact_1271_order__less__le__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C )
& ( order @ A ) )
=> ! [A2: A,B2: A,F3: A > C,C2: C] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ C @ ( F3 @ B2 ) @ C2 )
=> ( ! [X5: A,Y7: A] :
( ( ord_less @ A @ X5 @ Y7 )
=> ( ord_less @ C @ ( F3 @ X5 ) @ ( F3 @ Y7 ) ) )
=> ( ord_less @ C @ ( F3 @ A2 ) @ C2 ) ) ) ) ) ).
% order_less_le_subst2
thf(fact_1272_order__less__le__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A2: A,F3: B > A,B2: B,C2: B] :
( ( ord_less @ A @ A2 @ ( F3 @ B2 ) )
=> ( ( ord_less_eq @ B @ B2 @ C2 )
=> ( ! [X5: B,Y7: B] :
( ( ord_less_eq @ B @ X5 @ Y7 )
=> ( ord_less_eq @ A @ ( F3 @ X5 ) @ ( F3 @ Y7 ) ) )
=> ( ord_less @ A @ A2 @ ( F3 @ C2 ) ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_1273_order__le__less__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C )
& ( order @ A ) )
=> ! [A2: A,B2: A,F3: A > C,C2: C] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less @ C @ ( F3 @ B2 ) @ C2 )
=> ( ! [X5: A,Y7: A] :
( ( ord_less_eq @ A @ X5 @ Y7 )
=> ( ord_less_eq @ C @ ( F3 @ X5 ) @ ( F3 @ Y7 ) ) )
=> ( ord_less @ C @ ( F3 @ A2 ) @ C2 ) ) ) ) ) ).
% order_le_less_subst2
thf(fact_1274_order__le__less__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A2: A,F3: B > A,B2: B,C2: B] :
( ( ord_less_eq @ A @ A2 @ ( F3 @ B2 ) )
=> ( ( ord_less @ B @ B2 @ C2 )
=> ( ! [X5: B,Y7: B] :
( ( ord_less @ B @ X5 @ Y7 )
=> ( ord_less @ A @ ( F3 @ X5 ) @ ( F3 @ Y7 ) ) )
=> ( ord_less @ A @ A2 @ ( F3 @ C2 ) ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_1275_order__less__le__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y2: A,Z3: A] :
( ( ord_less @ A @ X @ Y2 )
=> ( ( ord_less_eq @ A @ Y2 @ Z3 )
=> ( ord_less @ A @ X @ Z3 ) ) ) ) ).
% order_less_le_trans
thf(fact_1276_order__le__less__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y2: A,Z3: A] :
( ( ord_less_eq @ A @ X @ Y2 )
=> ( ( ord_less @ A @ Y2 @ Z3 )
=> ( ord_less @ A @ X @ Z3 ) ) ) ) ).
% order_le_less_trans
thf(fact_1277_order__neq__le__trans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A,B2: A] :
( ( A2 != B2 )
=> ( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ord_less @ A @ A2 @ B2 ) ) ) ) ).
% order_neq_le_trans
thf(fact_1278_order__le__neq__trans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_less @ A @ A2 @ B2 ) ) ) ) ).
% order_le_neq_trans
thf(fact_1279_order__less__imp__le,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y2: A] :
( ( ord_less @ A @ X @ Y2 )
=> ( ord_less_eq @ A @ X @ Y2 ) ) ) ).
% order_less_imp_le
thf(fact_1280_linorder__not__less,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y2: A] :
( ( ~ ( ord_less @ A @ X @ Y2 ) )
= ( ord_less_eq @ A @ Y2 @ X ) ) ) ).
% linorder_not_less
thf(fact_1281_linorder__not__le,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y2: A] :
( ( ~ ( ord_less_eq @ A @ X @ Y2 ) )
= ( ord_less @ A @ Y2 @ X ) ) ) ).
% linorder_not_le
thf(fact_1282_order__less__le,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less @ A )
= ( ^ [X4: A,Y: A] :
( ( ord_less_eq @ A @ X4 @ Y )
& ( X4 != Y ) ) ) ) ) ).
% order_less_le
thf(fact_1283_order__le__less,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [X4: A,Y: A] :
( ( ord_less @ A @ X4 @ Y )
| ( X4 = Y ) ) ) ) ) ).
% order_le_less
thf(fact_1284_dual__order_Ostrict__implies__order,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [B2: A,A2: A] :
( ( ord_less @ A @ B2 @ A2 )
=> ( ord_less_eq @ A @ B2 @ A2 ) ) ) ).
% dual_order.strict_implies_order
thf(fact_1285_order_Ostrict__implies__order,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% order.strict_implies_order
thf(fact_1286_dual__order_Ostrict__iff__not,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( ( ord_less @ A )
= ( ^ [B4: A,A5: A] :
( ( ord_less_eq @ A @ B4 @ A5 )
& ~ ( ord_less_eq @ A @ A5 @ B4 ) ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_1287_dual__order_Ostrict__trans2,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( ord_less @ A @ B2 @ A2 )
=> ( ( ord_less_eq @ A @ C2 @ B2 )
=> ( ord_less @ A @ C2 @ A2 ) ) ) ) ).
% dual_order.strict_trans2
thf(fact_1288_dual__order_Ostrict__trans1,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_less @ A @ C2 @ B2 )
=> ( ord_less @ A @ C2 @ A2 ) ) ) ) ).
% dual_order.strict_trans1
thf(fact_1289_dual__order_Ostrict__iff__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less @ A )
= ( ^ [B4: A,A5: A] :
( ( ord_less_eq @ A @ B4 @ A5 )
& ( A5 != B4 ) ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_1290_dual__order_Oorder__iff__strict,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B4: A,A5: A] :
( ( ord_less @ A @ B4 @ A5 )
| ( A5 = B4 ) ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_1291_dense__le__bounded,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [X: A,Y2: A,Z3: A] :
( ( ord_less @ A @ X @ Y2 )
=> ( ! [W2: A] :
( ( ord_less @ A @ X @ W2 )
=> ( ( ord_less @ A @ W2 @ Y2 )
=> ( ord_less_eq @ A @ W2 @ Z3 ) ) )
=> ( ord_less_eq @ A @ Y2 @ Z3 ) ) ) ) ).
% dense_le_bounded
thf(fact_1292_dense__ge__bounded,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [Z3: A,X: A,Y2: A] :
( ( ord_less @ A @ Z3 @ X )
=> ( ! [W2: A] :
( ( ord_less @ A @ Z3 @ W2 )
=> ( ( ord_less @ A @ W2 @ X )
=> ( ord_less_eq @ A @ Y2 @ W2 ) ) )
=> ( ord_less_eq @ A @ Y2 @ Z3 ) ) ) ) ).
% dense_ge_bounded
thf(fact_1293_order_Ostrict__iff__not,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( ( ord_less @ A )
= ( ^ [A5: A,B4: A] :
( ( ord_less_eq @ A @ A5 @ B4 )
& ~ ( ord_less_eq @ A @ B4 @ A5 ) ) ) ) ) ).
% order.strict_iff_not
thf(fact_1294_order_Ostrict__trans2,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ C2 )
=> ( ord_less @ A @ A2 @ C2 ) ) ) ) ).
% order.strict_trans2
thf(fact_1295_order_Ostrict__trans1,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ B2 @ C2 )
=> ( ord_less @ A @ A2 @ C2 ) ) ) ) ).
% order.strict_trans1
thf(fact_1296_order_Ostrict__iff__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less @ A )
= ( ^ [A5: A,B4: A] :
( ( ord_less_eq @ A @ A5 @ B4 )
& ( A5 != B4 ) ) ) ) ) ).
% order.strict_iff_order
thf(fact_1297_order_Oorder__iff__strict,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A5: A,B4: A] :
( ( ord_less @ A @ A5 @ B4 )
| ( A5 = B4 ) ) ) ) ) ).
% order.order_iff_strict
thf(fact_1298_not__le__imp__less,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Y2: A,X: A] :
( ~ ( ord_less_eq @ A @ Y2 @ X )
=> ( ord_less @ A @ X @ Y2 ) ) ) ).
% not_le_imp_less
thf(fact_1299_less__le__not__le,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( ( ord_less @ A )
= ( ^ [X4: A,Y: A] :
( ( ord_less_eq @ A @ X4 @ Y )
& ~ ( ord_less_eq @ A @ Y @ X4 ) ) ) ) ) ).
% less_le_not_le
thf(fact_1300_dense__le,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [Y2: A,Z3: A] :
( ! [X5: A] :
( ( ord_less @ A @ X5 @ Y2 )
=> ( ord_less_eq @ A @ X5 @ Z3 ) )
=> ( ord_less_eq @ A @ Y2 @ Z3 ) ) ) ).
% dense_le
thf(fact_1301_dense__ge,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [Z3: A,Y2: A] :
( ! [X5: A] :
( ( ord_less @ A @ Z3 @ X5 )
=> ( ord_less_eq @ A @ Y2 @ X5 ) )
=> ( ord_less_eq @ A @ Y2 @ Z3 ) ) ) ).
% dense_ge
thf(fact_1302_antisym__conv2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,Y2: A] :
( ( ord_less_eq @ A @ X @ Y2 )
=> ( ( ~ ( ord_less @ A @ X @ Y2 ) )
= ( X = Y2 ) ) ) ) ).
% antisym_conv2
thf(fact_1303_antisym__conv1,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,Y2: A] :
( ~ ( ord_less @ A @ X @ Y2 )
=> ( ( ord_less_eq @ A @ X @ Y2 )
= ( X = Y2 ) ) ) ) ).
% antisym_conv1
thf(fact_1304_nless__le,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A,B2: A] :
( ( ~ ( ord_less @ A @ A2 @ B2 ) )
= ( ~ ( ord_less_eq @ A @ A2 @ B2 )
| ( A2 = B2 ) ) ) ) ).
% nless_le
thf(fact_1305_leI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y2: A] :
( ~ ( ord_less @ A @ X @ Y2 )
=> ( ord_less_eq @ A @ Y2 @ X ) ) ) ).
% leI
thf(fact_1306_leD,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Y2: A,X: A] :
( ( ord_less_eq @ A @ Y2 @ X )
=> ~ ( ord_less @ A @ X @ Y2 ) ) ) ).
% leD
thf(fact_1307_Euclid__induct,axiom,
! [P3: nat > nat > $o,A2: nat,B2: nat] :
( ! [A4: nat,B3: nat] :
( ( P3 @ A4 @ B3 )
= ( P3 @ B3 @ A4 ) )
=> ( ! [A4: nat] : ( P3 @ A4 @ ( zero_zero @ nat ) )
=> ( ! [A4: nat,B3: nat] :
( ( P3 @ A4 @ B3 )
=> ( P3 @ A4 @ ( plus_plus @ nat @ A4 @ B3 ) ) )
=> ( P3 @ A2 @ B2 ) ) ) ) ).
% Euclid_induct
thf(fact_1308_le__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ( ( ord_less_eq @ ( A > B ) )
= ( ^ [F5: A > B,G2: A > B] :
! [X4: A] : ( ord_less_eq @ B @ ( F5 @ X4 ) @ ( G2 @ X4 ) ) ) ) ) ).
% le_fun_def
thf(fact_1309_le__funI,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F3: A > B,G: A > B] :
( ! [X5: A] : ( ord_less_eq @ B @ ( F3 @ X5 ) @ ( G @ X5 ) )
=> ( ord_less_eq @ ( A > B ) @ F3 @ G ) ) ) ).
% le_funI
thf(fact_1310_le__funE,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F3: A > B,G: A > B,X: A] :
( ( ord_less_eq @ ( A > B ) @ F3 @ G )
=> ( ord_less_eq @ B @ ( F3 @ X ) @ ( G @ X ) ) ) ) ).
% le_funE
thf(fact_1311_le__funD,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F3: A > B,G: A > B,X: A] :
( ( ord_less_eq @ ( A > B ) @ F3 @ G )
=> ( ord_less_eq @ B @ ( F3 @ X ) @ ( G @ X ) ) ) ) ).
% le_funD
thf(fact_1312_VEBT__internal_Omembermima_Ocases,axiom,
! [X: product_prod @ vEBT_VEBT @ nat] :
( ! [Uu2: $o,Uv2: $o,Uw: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Uw ) )
=> ( ! [Ux: list @ vEBT_VEBT,Uy: vEBT_VEBT,Uz2: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( zero_zero @ nat ) @ Ux @ Uy ) @ Uz2 ) )
=> ( ! [Mi3: nat,Ma3: nat,Va2: list @ vEBT_VEBT,Vb: vEBT_VEBT,X5: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ Ma3 ) ) @ ( zero_zero @ nat ) @ Va2 @ Vb ) @ X5 ) )
=> ( ! [Mi3: nat,Ma3: nat,V3: nat,TreeList2: list @ vEBT_VEBT,Vc: vEBT_VEBT,X5: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ Ma3 ) ) @ ( suc @ V3 ) @ TreeList2 @ Vc ) @ X5 ) )
=> ~ ! [V3: nat,TreeList2: list @ vEBT_VEBT,Vd: vEBT_VEBT,X5: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V3 ) @ TreeList2 @ Vd ) @ X5 ) ) ) ) ) ) ).
% VEBT_internal.membermima.cases
thf(fact_1313_mult__less__iff1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z3: A,X: A,Y2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Z3 )
=> ( ( ord_less @ A @ ( times_times @ A @ X @ Z3 ) @ ( times_times @ A @ Y2 @ Z3 ) )
= ( ord_less @ A @ X @ Y2 ) ) ) ) ).
% mult_less_iff1
thf(fact_1314_pos__eucl__rel__int__mult__2,axiom,
! [B2: int,A2: int,Q: int,R: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ B2 )
=> ( ( eucl_rel_int @ A2 @ B2 @ ( product_Pair @ int @ int @ Q @ R ) )
=> ( eucl_rel_int @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ A2 ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ B2 ) @ ( product_Pair @ int @ int @ Q @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ R ) ) ) ) ) ) ).
% pos_eucl_rel_int_mult_2
thf(fact_1315_VEBT__internal_Oelim__dead_Opelims,axiom,
! [X: vEBT_VEBT,Xa: extended_enat,Y2: vEBT_VEBT] :
( ( ( vEBT_VEBT_elim_dead @ X @ Xa )
= Y2 )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ extended_enat ) @ vEBT_V312737461966249ad_rel @ ( product_Pair @ vEBT_VEBT @ extended_enat @ X @ Xa ) )
=> ( ! [A4: $o,B3: $o] :
( ( X
= ( vEBT_Leaf @ A4 @ B3 ) )
=> ( ( Y2
= ( vEBT_Leaf @ A4 @ B3 ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ extended_enat ) @ vEBT_V312737461966249ad_rel @ ( product_Pair @ vEBT_VEBT @ extended_enat @ ( vEBT_Leaf @ A4 @ B3 ) @ Xa ) ) ) )
=> ( ! [Info2: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( ( Xa
= ( extend4730790105801354508finity @ extended_enat ) )
=> ( ( Y2
= ( vEBT_Node @ Info2 @ Deg2
@ ( map @ vEBT_VEBT @ vEBT_VEBT
@ ^ [T3: vEBT_VEBT] : ( vEBT_VEBT_elim_dead @ T3 @ ( extended_enat2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ TreeList2 )
@ ( vEBT_VEBT_elim_dead @ Summary2 @ ( extend4730790105801354508finity @ extended_enat ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ extended_enat ) @ vEBT_V312737461966249ad_rel @ ( product_Pair @ vEBT_VEBT @ extended_enat @ ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) @ ( extend4730790105801354508finity @ extended_enat ) ) ) ) ) )
=> ~ ! [Info2: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) )
=> ! [L2: nat] :
( ( Xa
= ( extended_enat2 @ L2 ) )
=> ( ( Y2
= ( vEBT_Node @ Info2 @ Deg2
@ ( take @ vEBT_VEBT @ ( divide_divide @ nat @ L2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
@ ( map @ vEBT_VEBT @ vEBT_VEBT
@ ^ [T3: vEBT_VEBT] : ( vEBT_VEBT_elim_dead @ T3 @ ( extended_enat2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ TreeList2 ) )
@ ( vEBT_VEBT_elim_dead @ Summary2 @ ( extended_enat2 @ ( divide_divide @ nat @ L2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ extended_enat ) @ vEBT_V312737461966249ad_rel @ ( product_Pair @ vEBT_VEBT @ extended_enat @ ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) @ ( extended_enat2 @ L2 ) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.elim_dead.pelims
thf(fact_1316_VEBT__internal_Omembermima_Opelims_I1_J,axiom,
! [X: vEBT_VEBT,Xa: nat,Y2: $o] :
( ( ( vEBT_VEBT_membermima @ X @ Xa )
= Y2 )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa ) )
=> ( ! [Uu2: $o,Uv2: $o] :
( ( X
= ( vEBT_Leaf @ Uu2 @ Uv2 ) )
=> ( ~ Y2
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa ) ) ) )
=> ( ! [Ux: list @ vEBT_VEBT,Uy: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( zero_zero @ nat ) @ Ux @ Uy ) )
=> ( ~ Y2
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( zero_zero @ nat ) @ Ux @ Uy ) @ Xa ) ) ) )
=> ( ! [Mi3: nat,Ma3: nat,Va2: list @ vEBT_VEBT,Vb: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ Ma3 ) ) @ ( zero_zero @ nat ) @ Va2 @ Vb ) )
=> ( ( Y2
= ( ( Xa = Mi3 )
| ( Xa = Ma3 ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ Ma3 ) ) @ ( zero_zero @ nat ) @ Va2 @ Vb ) @ Xa ) ) ) )
=> ( ! [Mi3: nat,Ma3: nat,V3: nat,TreeList2: list @ vEBT_VEBT,Vc: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ Ma3 ) ) @ ( suc @ V3 ) @ TreeList2 @ Vc ) )
=> ( ( Y2
= ( ( Xa = Mi3 )
| ( Xa = Ma3 )
| ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ Ma3 ) ) @ ( suc @ V3 ) @ TreeList2 @ Vc ) @ Xa ) ) ) )
=> ~ ! [V3: nat,TreeList2: list @ vEBT_VEBT,Vd: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V3 ) @ TreeList2 @ Vd ) )
=> ( ( Y2
= ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V3 ) @ TreeList2 @ Vd ) @ Xa ) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.membermima.pelims(1)
thf(fact_1317_VEBT__internal_Omembermima_Opelims_I3_J,axiom,
! [X: vEBT_VEBT,Xa: nat] :
( ~ ( vEBT_VEBT_membermima @ X @ Xa )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa ) )
=> ( ! [Uu2: $o,Uv2: $o] :
( ( X
= ( vEBT_Leaf @ Uu2 @ Uv2 ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa ) ) )
=> ( ! [Ux: list @ vEBT_VEBT,Uy: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( zero_zero @ nat ) @ Ux @ Uy ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( zero_zero @ nat ) @ Ux @ Uy ) @ Xa ) ) )
=> ( ! [Mi3: nat,Ma3: nat,Va2: list @ vEBT_VEBT,Vb: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ Ma3 ) ) @ ( zero_zero @ nat ) @ Va2 @ Vb ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ Ma3 ) ) @ ( zero_zero @ nat ) @ Va2 @ Vb ) @ Xa ) )
=> ( ( Xa = Mi3 )
| ( Xa = Ma3 ) ) ) )
=> ( ! [Mi3: nat,Ma3: nat,V3: nat,TreeList2: list @ vEBT_VEBT,Vc: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ Ma3 ) ) @ ( suc @ V3 ) @ TreeList2 @ Vc ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ Ma3 ) ) @ ( suc @ V3 ) @ TreeList2 @ Vc ) @ Xa ) )
=> ( ( Xa = Mi3 )
| ( Xa = Ma3 )
| ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) )
=> ~ ! [V3: nat,TreeList2: list @ vEBT_VEBT,Vd: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V3 ) @ TreeList2 @ Vd ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V3 ) @ TreeList2 @ Vd ) @ Xa ) )
=> ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.membermima.pelims(3)
thf(fact_1318_option_Osize__gen_I2_J,axiom,
! [A: $tType,X: A > nat,X23: A] :
( ( size_option @ A @ X @ ( some @ A @ X23 ) )
= ( plus_plus @ nat @ ( X @ X23 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% option.size_gen(2)
thf(fact_1319_even__succ__mod__exp,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A2: A,N: nat] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( modulo_modulo @ A @ ( plus_plus @ A @ ( one_one @ A ) @ A2 ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
= ( plus_plus @ A @ ( one_one @ A ) @ ( modulo_modulo @ A @ A2 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) ) ) ) ) ) ).
% even_succ_mod_exp
thf(fact_1320_even__succ__div__exp,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A2: A,N: nat] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ ( one_one @ A ) @ A2 ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
= ( divide_divide @ A @ A2 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) ) ) ) ) ).
% even_succ_div_exp
thf(fact_1321_signed__take__bit__Suc,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: nat,A2: A] :
( ( bit_ri4674362597316999326ke_bit @ A @ ( suc @ N ) @ A2 )
= ( plus_plus @ A @ ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_ri4674362597316999326ke_bit @ A @ N @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% signed_take_bit_Suc
thf(fact_1322_VEBT__internal_Omembermima_Opelims_I2_J,axiom,
! [X: vEBT_VEBT,Xa: nat] :
( ( vEBT_VEBT_membermima @ X @ Xa )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa ) )
=> ( ! [Mi3: nat,Ma3: nat,Va2: list @ vEBT_VEBT,Vb: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ Ma3 ) ) @ ( zero_zero @ nat ) @ Va2 @ Vb ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ Ma3 ) ) @ ( zero_zero @ nat ) @ Va2 @ Vb ) @ Xa ) )
=> ~ ( ( Xa = Mi3 )
| ( Xa = Ma3 ) ) ) )
=> ( ! [Mi3: nat,Ma3: nat,V3: nat,TreeList2: list @ vEBT_VEBT,Vc: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ Ma3 ) ) @ ( suc @ V3 ) @ TreeList2 @ Vc ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ Ma3 ) ) @ ( suc @ V3 ) @ TreeList2 @ Vc ) @ Xa ) )
=> ~ ( ( Xa = Mi3 )
| ( Xa = Ma3 )
| ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) )
=> ~ ! [V3: nat,TreeList2: list @ vEBT_VEBT,Vd: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V3 ) @ TreeList2 @ Vd ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V3 ) @ TreeList2 @ Vd ) @ Xa ) )
=> ~ ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.membermima.pelims(2)
thf(fact_1323_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_1324_dvd__add__triv__left__iff,axiom,
! [A: $tType] :
( ( comm_s4317794764714335236cancel @ A )
=> ! [A2: A,B2: A] :
( ( dvd_dvd @ A @ A2 @ ( plus_plus @ A @ A2 @ B2 ) )
= ( dvd_dvd @ A @ A2 @ B2 ) ) ) ).
% dvd_add_triv_left_iff
thf(fact_1325_dvd__add__triv__right__iff,axiom,
! [A: $tType] :
( ( comm_s4317794764714335236cancel @ A )
=> ! [A2: A,B2: A] :
( ( dvd_dvd @ A @ A2 @ ( plus_plus @ A @ B2 @ A2 ) )
= ( dvd_dvd @ A @ A2 @ B2 ) ) ) ).
% dvd_add_triv_right_iff
thf(fact_1326_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_1327_dvd__1__left,axiom,
! [K2: nat] : ( dvd_dvd @ nat @ ( suc @ ( zero_zero @ nat ) ) @ K2 ) ).
% dvd_1_left
thf(fact_1328_div__dvd__div,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A2 @ B2 )
=> ( ( dvd_dvd @ A @ A2 @ C2 )
=> ( ( dvd_dvd @ A @ ( divide_divide @ A @ B2 @ A2 ) @ ( divide_divide @ A @ C2 @ A2 ) )
= ( dvd_dvd @ A @ B2 @ C2 ) ) ) ) ) ).
% div_dvd_div
thf(fact_1329_nat__mult__dvd__cancel__disj,axiom,
! [K2: nat,M: nat,N: nat] :
( ( dvd_dvd @ nat @ ( times_times @ nat @ K2 @ M ) @ ( times_times @ nat @ K2 @ N ) )
= ( ( K2
= ( zero_zero @ nat ) )
| ( dvd_dvd @ nat @ M @ N ) ) ) ).
% nat_mult_dvd_cancel_disj
thf(fact_1330_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_1331_dvd__mult__cancel__left,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( dvd_dvd @ A @ ( times_times @ A @ C2 @ A2 ) @ ( times_times @ A @ C2 @ B2 ) )
= ( ( C2
= ( zero_zero @ A ) )
| ( dvd_dvd @ A @ A2 @ B2 ) ) ) ) ).
% dvd_mult_cancel_left
thf(fact_1332_dvd__mult__cancel__right,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( dvd_dvd @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) )
= ( ( C2
= ( zero_zero @ A ) )
| ( dvd_dvd @ A @ A2 @ B2 ) ) ) ) ).
% dvd_mult_cancel_right
thf(fact_1333_dvd__times__left__cancel__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ ( times_times @ A @ A2 @ B2 ) @ ( times_times @ A @ A2 @ C2 ) )
= ( dvd_dvd @ A @ B2 @ C2 ) ) ) ) ).
% dvd_times_left_cancel_iff
thf(fact_1334_dvd__times__right__cancel__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ ( times_times @ A @ B2 @ A2 ) @ ( times_times @ A @ C2 @ A2 ) )
= ( dvd_dvd @ A @ B2 @ C2 ) ) ) ) ).
% dvd_times_right_cancel_iff
thf(fact_1335_unit__prod,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A,B2: A] :
( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( dvd_dvd @ A @ ( times_times @ A @ A2 @ B2 ) @ ( one_one @ A ) ) ) ) ) ).
% unit_prod
thf(fact_1336_dvd__add__times__triv__left__iff,axiom,
! [A: $tType] :
( ( comm_s4317794764714335236cancel @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( dvd_dvd @ A @ A2 @ ( plus_plus @ A @ ( times_times @ A @ C2 @ A2 ) @ B2 ) )
= ( dvd_dvd @ A @ A2 @ B2 ) ) ) ).
% dvd_add_times_triv_left_iff
thf(fact_1337_dvd__add__times__triv__right__iff,axiom,
! [A: $tType] :
( ( comm_s4317794764714335236cancel @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A2 @ ( plus_plus @ A @ B2 @ ( times_times @ A @ C2 @ A2 ) ) )
= ( dvd_dvd @ A @ A2 @ B2 ) ) ) ).
% dvd_add_times_triv_right_iff
thf(fact_1338_dvd__div__mult__self,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A,B2: A] :
( ( dvd_dvd @ A @ A2 @ B2 )
=> ( ( times_times @ A @ ( divide_divide @ A @ B2 @ A2 ) @ A2 )
= B2 ) ) ) ).
% dvd_div_mult_self
thf(fact_1339_dvd__mult__div__cancel,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A,B2: A] :
( ( dvd_dvd @ A @ A2 @ B2 )
=> ( ( times_times @ A @ A2 @ ( divide_divide @ A @ B2 @ A2 ) )
= B2 ) ) ) ).
% dvd_mult_div_cancel
thf(fact_1340_unit__div,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A,B2: A] :
( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( dvd_dvd @ A @ ( divide_divide @ A @ A2 @ B2 ) @ ( one_one @ A ) ) ) ) ) ).
% unit_div
thf(fact_1341_unit__div__1__unit,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A] :
( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) )
=> ( dvd_dvd @ A @ ( divide_divide @ A @ ( one_one @ A ) @ A2 ) @ ( one_one @ A ) ) ) ) ).
% unit_div_1_unit
thf(fact_1342_unit__div__1__div__1,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A] :
( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) )
=> ( ( divide_divide @ A @ ( one_one @ A ) @ ( divide_divide @ A @ ( one_one @ A ) @ A2 ) )
= A2 ) ) ) ).
% unit_div_1_div_1
thf(fact_1343_div__add,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( dvd_dvd @ A @ C2 @ A2 )
=> ( ( dvd_dvd @ A @ C2 @ B2 )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 )
= ( plus_plus @ A @ ( divide_divide @ A @ A2 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) ) ) ) ) ) ).
% div_add
thf(fact_1344_dvd__imp__mod__0,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [A2: A,B2: A] :
( ( dvd_dvd @ A @ A2 @ B2 )
=> ( ( modulo_modulo @ A @ B2 @ A2 )
= ( zero_zero @ A ) ) ) ) ).
% dvd_imp_mod_0
thf(fact_1345_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_1346_signed__take__bit__numeral__of__1,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [K2: num] :
( ( bit_ri4674362597316999326ke_bit @ A @ ( numeral_numeral @ nat @ K2 ) @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% signed_take_bit_numeral_of_1
thf(fact_1347_unit__mult__div__div,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A,B2: A] :
( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) )
=> ( ( times_times @ A @ B2 @ ( divide_divide @ A @ ( one_one @ A ) @ A2 ) )
= ( divide_divide @ A @ B2 @ A2 ) ) ) ) ).
% unit_mult_div_div
thf(fact_1348_unit__div__mult__self,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A,B2: A] :
( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) )
=> ( ( times_times @ A @ ( divide_divide @ A @ B2 @ A2 ) @ A2 )
= B2 ) ) ) ).
% unit_div_mult_self
thf(fact_1349_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_1350_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_1351_pow__divides__pow__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [N: nat,A2: A,B2: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( dvd_dvd @ A @ ( power_power @ A @ A2 @ N ) @ ( power_power @ A @ B2 @ N ) )
= ( dvd_dvd @ A @ A2 @ B2 ) ) ) ) ).
% pow_divides_pow_iff
thf(fact_1352_even__mult__iff,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A2: A,B2: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( times_times @ A @ A2 @ B2 ) )
= ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
| ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) ) ) ).
% even_mult_iff
thf(fact_1353_odd__add,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A2: A,B2: A] :
( ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( plus_plus @ A @ A2 @ B2 ) ) )
= ( ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) )
!= ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) ) ) ) ).
% odd_add
thf(fact_1354_even__add,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A2: A,B2: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( plus_plus @ A @ A2 @ B2 ) )
= ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
= ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) ) ) ).
% even_add
thf(fact_1355_even__mod__2__iff,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A2: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
= ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) ) ) ).
% even_mod_2_iff
thf(fact_1356_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_1357_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_1358_signed__take__bit__Suc__bit0,axiom,
! [N: nat,K2: num] :
( ( bit_ri4674362597316999326ke_bit @ int @ ( suc @ N ) @ ( numeral_numeral @ int @ ( bit0 @ K2 ) ) )
= ( times_times @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N @ ( numeral_numeral @ int @ K2 ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ).
% signed_take_bit_Suc_bit0
thf(fact_1359_zero__le__power__eq__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A,W: num] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ W ) ) )
= ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W ) )
| ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W ) )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ) ) ).
% zero_le_power_eq_numeral
thf(fact_1360_power__less__zero__eq,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A,N: nat] :
( ( ord_less @ A @ ( power_power @ A @ A2 @ N ) @ ( zero_zero @ A ) )
= ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
& ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ) ).
% power_less_zero_eq
thf(fact_1361_power__less__zero__eq__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A,W: num] :
( ( ord_less @ A @ ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ W ) ) @ ( zero_zero @ A ) )
= ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W ) )
& ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ) ).
% power_less_zero_eq_numeral
thf(fact_1362_even__plus__one__iff,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A2: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( plus_plus @ A @ A2 @ ( one_one @ A ) ) )
= ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) ) ) ) ).
% even_plus_one_iff
thf(fact_1363_zero__less__power__eq__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A,W: num] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ W ) ) )
= ( ( ( numeral_numeral @ nat @ W )
= ( zero_zero @ nat ) )
| ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W ) )
& ( A2
!= ( zero_zero @ A ) ) )
| ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W ) )
& ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ) ) ).
% zero_less_power_eq_numeral
thf(fact_1364_even__succ__div__2,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A2: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ ( one_one @ A ) @ A2 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% even_succ_div_2
thf(fact_1365_odd__succ__div__two,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [A2: A] :
( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ A2 @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( plus_plus @ A @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) ) ) ) ) ).
% odd_succ_div_two
thf(fact_1366_even__succ__div__two,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [A2: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ A2 @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% even_succ_div_two
thf(fact_1367_even__power,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A2: A,N: nat] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( power_power @ A @ A2 @ N ) )
= ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
& ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% even_power
thf(fact_1368_odd__two__times__div__two__succ,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [A2: A] :
( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
=> ( ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( one_one @ A ) )
= A2 ) ) ) ).
% odd_two_times_div_two_succ
thf(fact_1369_power__le__zero__eq__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A,W: num] :
( ( ord_less_eq @ A @ ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ W ) ) @ ( zero_zero @ A ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( numeral_numeral @ nat @ W ) )
& ( ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W ) )
& ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) )
| ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W ) )
& ( A2
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% power_le_zero_eq_numeral
thf(fact_1370_dvd__field__iff,axiom,
! [A: $tType] :
( ( field @ A )
=> ( ( dvd_dvd @ A )
= ( ^ [A5: A,B4: A] :
( ( A5
= ( zero_zero @ A ) )
=> ( B4
= ( zero_zero @ A ) ) ) ) ) ) ).
% dvd_field_iff
thf(fact_1371_dvdE,axiom,
! [A: $tType] :
( ( dvd @ A )
=> ! [B2: A,A2: A] :
( ( dvd_dvd @ A @ B2 @ A2 )
=> ~ ! [K: A] :
( A2
!= ( times_times @ A @ B2 @ K ) ) ) ) ).
% dvdE
thf(fact_1372_dvdI,axiom,
! [A: $tType] :
( ( dvd @ A )
=> ! [A2: A,B2: A,K2: A] :
( ( A2
= ( times_times @ A @ B2 @ K2 ) )
=> ( dvd_dvd @ A @ B2 @ A2 ) ) ) ).
% dvdI
thf(fact_1373_dvd__def,axiom,
! [A: $tType] :
( ( dvd @ A )
=> ( ( dvd_dvd @ A )
= ( ^ [B4: A,A5: A] :
? [K3: A] :
( A5
= ( times_times @ A @ B4 @ K3 ) ) ) ) ) ).
% dvd_def
thf(fact_1374_dvd__mult,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( dvd_dvd @ A @ A2 @ C2 )
=> ( dvd_dvd @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% dvd_mult
thf(fact_1375_dvd__mult2,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A2 @ B2 )
=> ( dvd_dvd @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% dvd_mult2
thf(fact_1376_dvd__mult__left,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ ( times_times @ A @ A2 @ B2 ) @ C2 )
=> ( dvd_dvd @ A @ A2 @ C2 ) ) ) ).
% dvd_mult_left
thf(fact_1377_dvd__triv__left,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A2: A,B2: A] : ( dvd_dvd @ A @ A2 @ ( times_times @ A @ A2 @ B2 ) ) ) ).
% dvd_triv_left
thf(fact_1378_mult__dvd__mono,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A2: A,B2: A,C2: A,D2: A] :
( ( dvd_dvd @ A @ A2 @ B2 )
=> ( ( dvd_dvd @ A @ C2 @ D2 )
=> ( dvd_dvd @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ D2 ) ) ) ) ) ).
% mult_dvd_mono
thf(fact_1379_dvd__mult__right,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ ( times_times @ A @ A2 @ B2 ) @ C2 )
=> ( dvd_dvd @ A @ B2 @ C2 ) ) ) ).
% dvd_mult_right
thf(fact_1380_dvd__triv__right,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A2: A,B2: A] : ( dvd_dvd @ A @ A2 @ ( times_times @ A @ B2 @ A2 ) ) ) ).
% dvd_triv_right
thf(fact_1381_division__decomp,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) )
=> ? [B8: A,C6: A] :
( ( A2
= ( times_times @ A @ B8 @ C6 ) )
& ( dvd_dvd @ A @ B8 @ B2 )
& ( dvd_dvd @ A @ C6 @ C2 ) ) ) ) ).
% division_decomp
thf(fact_1382_dvd__productE,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [P5: A,A2: A,B2: A] :
( ( dvd_dvd @ A @ P5 @ ( times_times @ A @ A2 @ B2 ) )
=> ~ ! [X5: A,Y7: A] :
( ( P5
= ( times_times @ A @ X5 @ Y7 ) )
=> ( ( dvd_dvd @ A @ X5 @ A2 )
=> ~ ( dvd_dvd @ A @ Y7 @ B2 ) ) ) ) ) ).
% dvd_productE
thf(fact_1383_one__dvd,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A2: A] : ( dvd_dvd @ A @ ( one_one @ A ) @ A2 ) ) ).
% one_dvd
thf(fact_1384_unit__imp__dvd,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,A2: A] :
( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( dvd_dvd @ A @ B2 @ A2 ) ) ) ).
% unit_imp_dvd
thf(fact_1385_dvd__unit__imp__unit,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A,B2: A] :
( ( dvd_dvd @ A @ A2 @ B2 )
=> ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( dvd_dvd @ A @ A2 @ ( one_one @ A ) ) ) ) ) ).
% dvd_unit_imp_unit
thf(fact_1386_dvd__add,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A2 @ B2 )
=> ( ( dvd_dvd @ A @ A2 @ C2 )
=> ( dvd_dvd @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ) ).
% dvd_add
thf(fact_1387_dvd__add__left__iff,axiom,
! [A: $tType] :
( ( comm_s4317794764714335236cancel @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( dvd_dvd @ A @ A2 @ C2 )
=> ( ( dvd_dvd @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) )
= ( dvd_dvd @ A @ A2 @ B2 ) ) ) ) ).
% dvd_add_left_iff
thf(fact_1388_dvd__add__right__iff,axiom,
! [A: $tType] :
( ( comm_s4317794764714335236cancel @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A2 @ B2 )
=> ( ( dvd_dvd @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) )
= ( dvd_dvd @ A @ A2 @ C2 ) ) ) ) ).
% dvd_add_right_iff
thf(fact_1389_dvd__div__eq__iff,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( dvd_dvd @ A @ C2 @ A2 )
=> ( ( dvd_dvd @ A @ C2 @ B2 )
=> ( ( ( divide_divide @ A @ A2 @ C2 )
= ( divide_divide @ A @ B2 @ C2 ) )
= ( A2 = B2 ) ) ) ) ) ).
% dvd_div_eq_iff
thf(fact_1390_dvd__div__eq__cancel,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( ( divide_divide @ A @ A2 @ C2 )
= ( divide_divide @ A @ B2 @ C2 ) )
=> ( ( dvd_dvd @ A @ C2 @ A2 )
=> ( ( dvd_dvd @ A @ C2 @ B2 )
=> ( A2 = B2 ) ) ) ) ) ).
% dvd_div_eq_cancel
thf(fact_1391_div__div__div__same,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [D2: A,B2: A,A2: A] :
( ( dvd_dvd @ A @ D2 @ B2 )
=> ( ( dvd_dvd @ A @ B2 @ A2 )
=> ( ( divide_divide @ A @ ( divide_divide @ A @ A2 @ D2 ) @ ( divide_divide @ A @ B2 @ D2 ) )
= ( divide_divide @ A @ A2 @ B2 ) ) ) ) ) ).
% div_div_div_same
thf(fact_1392_dvd__power__same,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [X: A,Y2: A,N: nat] :
( ( dvd_dvd @ A @ X @ Y2 )
=> ( dvd_dvd @ A @ ( power_power @ A @ X @ N ) @ ( power_power @ A @ Y2 @ N ) ) ) ) ).
% dvd_power_same
thf(fact_1393_mod__mod__cancel,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [C2: A,B2: A,A2: A] :
( ( dvd_dvd @ A @ C2 @ B2 )
=> ( ( modulo_modulo @ A @ ( modulo_modulo @ A @ A2 @ B2 ) @ C2 )
= ( modulo_modulo @ A @ A2 @ C2 ) ) ) ) ).
% mod_mod_cancel
thf(fact_1394_dvd__mod,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [K2: A,M: A,N: A] :
( ( dvd_dvd @ A @ K2 @ M )
=> ( ( dvd_dvd @ A @ K2 @ N )
=> ( dvd_dvd @ A @ K2 @ ( modulo_modulo @ A @ M @ N ) ) ) ) ) ).
% dvd_mod
thf(fact_1395_dvd__mod__iff,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [C2: A,B2: A,A2: A] :
( ( dvd_dvd @ A @ C2 @ B2 )
=> ( ( dvd_dvd @ A @ C2 @ ( modulo_modulo @ A @ A2 @ B2 ) )
= ( dvd_dvd @ A @ C2 @ A2 ) ) ) ) ).
% dvd_mod_iff
thf(fact_1396_dvd__mod__imp__dvd,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( dvd_dvd @ A @ C2 @ ( modulo_modulo @ A @ A2 @ B2 ) )
=> ( ( dvd_dvd @ A @ C2 @ B2 )
=> ( dvd_dvd @ A @ C2 @ A2 ) ) ) ) ).
% dvd_mod_imp_dvd
thf(fact_1397_signed__take__bit__mult,axiom,
! [N: nat,K2: int,L: int] :
( ( bit_ri4674362597316999326ke_bit @ int @ N @ ( times_times @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N @ K2 ) @ ( bit_ri4674362597316999326ke_bit @ int @ N @ L ) ) )
= ( bit_ri4674362597316999326ke_bit @ int @ N @ ( times_times @ int @ K2 @ L ) ) ) ).
% signed_take_bit_mult
thf(fact_1398_signed__take__bit__add,axiom,
! [N: nat,K2: int,L: int] :
( ( bit_ri4674362597316999326ke_bit @ int @ N @ ( plus_plus @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N @ K2 ) @ ( bit_ri4674362597316999326ke_bit @ int @ N @ L ) ) )
= ( bit_ri4674362597316999326ke_bit @ int @ N @ ( plus_plus @ int @ K2 @ L ) ) ) ).
% signed_take_bit_add
thf(fact_1399_subset__divisors__dvd,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [C3: A] : ( dvd_dvd @ A @ C3 @ A2 ) )
@ ( collect @ A
@ ^ [C3: A] : ( dvd_dvd @ A @ C3 @ B2 ) ) )
= ( dvd_dvd @ A @ A2 @ B2 ) ) ) ).
% subset_divisors_dvd
thf(fact_1400_strict__subset__divisors__dvd,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ ( set @ A )
@ ( collect @ A
@ ^ [C3: A] : ( dvd_dvd @ A @ C3 @ A2 ) )
@ ( collect @ A
@ ^ [C3: A] : ( dvd_dvd @ A @ C3 @ B2 ) ) )
= ( ( dvd_dvd @ A @ A2 @ B2 )
& ~ ( dvd_dvd @ A @ B2 @ A2 ) ) ) ) ).
% strict_subset_divisors_dvd
thf(fact_1401_even__signed__take__bit__iff,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [M: nat,A2: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_ri4674362597316999326ke_bit @ A @ M @ A2 ) )
= ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) ) ) ).
% even_signed_take_bit_iff
thf(fact_1402_mod__int__unique,axiom,
! [K2: int,L: int,Q: int,R: int] :
( ( eucl_rel_int @ K2 @ L @ ( product_Pair @ int @ int @ Q @ R ) )
=> ( ( modulo_modulo @ int @ K2 @ L )
= R ) ) ).
% mod_int_unique
thf(fact_1403_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_1404_minf_I10_J,axiom,
! [B: $tType] :
( ( ( plus @ B )
& ( linorder @ B )
& ( dvd @ B ) )
=> ! [D2: B,S: B] :
? [Z: B] :
! [X2: B] :
( ( ord_less @ B @ X2 @ Z )
=> ( ( ~ ( dvd_dvd @ B @ D2 @ ( plus_plus @ B @ X2 @ S ) ) )
= ( ~ ( dvd_dvd @ B @ D2 @ ( plus_plus @ B @ X2 @ S ) ) ) ) ) ) ).
% minf(10)
thf(fact_1405_minf_I9_J,axiom,
! [B: $tType] :
( ( ( plus @ B )
& ( linorder @ B )
& ( dvd @ B ) )
=> ! [D2: B,S: B] :
? [Z: B] :
! [X2: B] :
( ( ord_less @ B @ X2 @ Z )
=> ( ( dvd_dvd @ B @ D2 @ ( plus_plus @ B @ X2 @ S ) )
= ( dvd_dvd @ B @ D2 @ ( plus_plus @ B @ X2 @ S ) ) ) ) ) ).
% minf(9)
thf(fact_1406_pinf_I10_J,axiom,
! [B: $tType] :
( ( ( plus @ B )
& ( linorder @ B )
& ( dvd @ B ) )
=> ! [D2: B,S: B] :
? [Z: B] :
! [X2: B] :
( ( ord_less @ B @ Z @ X2 )
=> ( ( ~ ( dvd_dvd @ B @ D2 @ ( plus_plus @ B @ X2 @ S ) ) )
= ( ~ ( dvd_dvd @ B @ D2 @ ( plus_plus @ B @ X2 @ S ) ) ) ) ) ) ).
% pinf(10)
thf(fact_1407_pinf_I9_J,axiom,
! [B: $tType] :
( ( ( plus @ B )
& ( linorder @ B )
& ( dvd @ B ) )
=> ! [D2: B,S: B] :
? [Z: B] :
! [X2: B] :
( ( ord_less @ B @ Z @ X2 )
=> ( ( dvd_dvd @ B @ D2 @ ( plus_plus @ B @ X2 @ S ) )
= ( dvd_dvd @ B @ D2 @ ( plus_plus @ B @ X2 @ S ) ) ) ) ) ).
% pinf(9)
thf(fact_1408_dvd__div__eq__0__iff,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [B2: A,A2: A] :
( ( dvd_dvd @ A @ B2 @ A2 )
=> ( ( ( divide_divide @ A @ A2 @ B2 )
= ( zero_zero @ A ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ) ).
% dvd_div_eq_0_iff
thf(fact_1409_unit__mult__right__cancel,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) )
=> ( ( ( times_times @ A @ B2 @ A2 )
= ( times_times @ A @ C2 @ A2 ) )
= ( B2 = C2 ) ) ) ) ).
% unit_mult_right_cancel
thf(fact_1410_unit__mult__left__cancel,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) )
=> ( ( ( times_times @ A @ A2 @ B2 )
= ( times_times @ A @ A2 @ C2 ) )
= ( B2 = C2 ) ) ) ) ).
% unit_mult_left_cancel
thf(fact_1411_mult__unit__dvd__iff_H,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ ( times_times @ A @ A2 @ B2 ) @ C2 )
= ( dvd_dvd @ A @ B2 @ C2 ) ) ) ) ).
% mult_unit_dvd_iff'
thf(fact_1412_dvd__mult__unit__iff_H,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) )
= ( dvd_dvd @ A @ A2 @ C2 ) ) ) ) ).
% dvd_mult_unit_iff'
thf(fact_1413_mult__unit__dvd__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ ( times_times @ A @ A2 @ B2 ) @ C2 )
= ( dvd_dvd @ A @ A2 @ C2 ) ) ) ) ).
% mult_unit_dvd_iff
thf(fact_1414_dvd__mult__unit__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ A2 @ ( times_times @ A @ C2 @ B2 ) )
= ( dvd_dvd @ A @ A2 @ C2 ) ) ) ) ).
% dvd_mult_unit_iff
thf(fact_1415_is__unit__mult__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A,B2: A] :
( ( dvd_dvd @ A @ ( times_times @ A @ A2 @ B2 ) @ ( one_one @ A ) )
= ( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) )
& ( dvd_dvd @ A @ B2 @ ( one_one @ A ) ) ) ) ) ).
% is_unit_mult_iff
thf(fact_1416_dvd__div__mult,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [C2: A,B2: A,A2: A] :
( ( dvd_dvd @ A @ C2 @ B2 )
=> ( ( times_times @ A @ ( divide_divide @ A @ B2 @ C2 ) @ A2 )
= ( divide_divide @ A @ ( times_times @ A @ B2 @ A2 ) @ C2 ) ) ) ) ).
% dvd_div_mult
thf(fact_1417_div__mult__swap,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [C2: A,B2: A,A2: A] :
( ( dvd_dvd @ A @ C2 @ B2 )
=> ( ( times_times @ A @ A2 @ ( divide_divide @ A @ B2 @ C2 ) )
= ( divide_divide @ A @ ( times_times @ A @ A2 @ B2 ) @ C2 ) ) ) ) ).
% div_mult_swap
thf(fact_1418_div__div__eq__right,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [C2: A,B2: A,A2: A] :
( ( dvd_dvd @ A @ C2 @ B2 )
=> ( ( dvd_dvd @ A @ B2 @ A2 )
=> ( ( divide_divide @ A @ A2 @ ( divide_divide @ A @ B2 @ C2 ) )
= ( times_times @ A @ ( divide_divide @ A @ A2 @ B2 ) @ C2 ) ) ) ) ) ).
% div_div_eq_right
thf(fact_1419_dvd__div__mult2__eq,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,C2: A,A2: A] :
( ( dvd_dvd @ A @ ( times_times @ A @ B2 @ C2 ) @ A2 )
=> ( ( divide_divide @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) )
= ( divide_divide @ A @ ( divide_divide @ A @ A2 @ B2 ) @ C2 ) ) ) ) ).
% dvd_div_mult2_eq
thf(fact_1420_dvd__mult__imp__div,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( dvd_dvd @ A @ ( times_times @ A @ A2 @ C2 ) @ B2 )
=> ( dvd_dvd @ A @ A2 @ ( divide_divide @ A @ B2 @ C2 ) ) ) ) ).
% dvd_mult_imp_div
thf(fact_1421_div__mult__div__if__dvd,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,A2: A,D2: A,C2: A] :
( ( dvd_dvd @ A @ B2 @ A2 )
=> ( ( dvd_dvd @ A @ D2 @ C2 )
=> ( ( times_times @ A @ ( divide_divide @ A @ A2 @ B2 ) @ ( divide_divide @ A @ C2 @ D2 ) )
= ( divide_divide @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ D2 ) ) ) ) ) ) ).
% div_mult_div_if_dvd
thf(fact_1422_unit__div__cancel,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) )
=> ( ( ( divide_divide @ A @ B2 @ A2 )
= ( divide_divide @ A @ C2 @ A2 ) )
= ( B2 = C2 ) ) ) ) ).
% unit_div_cancel
thf(fact_1423_div__unit__dvd__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ ( divide_divide @ A @ A2 @ B2 ) @ C2 )
= ( dvd_dvd @ A @ A2 @ C2 ) ) ) ) ).
% div_unit_dvd_iff
thf(fact_1424_dvd__div__unit__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ A2 @ ( divide_divide @ A @ C2 @ B2 ) )
= ( dvd_dvd @ A @ A2 @ C2 ) ) ) ) ).
% dvd_div_unit_iff
thf(fact_1425_div__plus__div__distrib__dvd__left,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( dvd_dvd @ A @ C2 @ A2 )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 )
= ( plus_plus @ A @ ( divide_divide @ A @ A2 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) ) ) ) ) ).
% div_plus_div_distrib_dvd_left
thf(fact_1426_div__plus__div__distrib__dvd__right,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [C2: A,B2: A,A2: A] :
( ( dvd_dvd @ A @ C2 @ B2 )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 )
= ( plus_plus @ A @ ( divide_divide @ A @ A2 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) ) ) ) ) ).
% div_plus_div_distrib_dvd_right
thf(fact_1427_div__power,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,A2: A,N: nat] :
( ( dvd_dvd @ A @ B2 @ A2 )
=> ( ( power_power @ A @ ( divide_divide @ A @ A2 @ B2 ) @ N )
= ( divide_divide @ A @ ( power_power @ A @ A2 @ N ) @ ( power_power @ A @ B2 @ N ) ) ) ) ) ).
% div_power
thf(fact_1428_mod__eq__0__iff__dvd,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [A2: A,B2: A] :
( ( ( modulo_modulo @ A @ A2 @ B2 )
= ( zero_zero @ A ) )
= ( dvd_dvd @ A @ B2 @ A2 ) ) ) ).
% mod_eq_0_iff_dvd
thf(fact_1429_dvd__eq__mod__eq__0,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ( ( dvd_dvd @ A )
= ( ^ [A5: A,B4: A] :
( ( modulo_modulo @ A @ B4 @ A5 )
= ( zero_zero @ A ) ) ) ) ) ).
% dvd_eq_mod_eq_0
thf(fact_1430_mod__0__imp__dvd,axiom,
! [A: $tType] :
( ( semiring_modulo @ A )
=> ! [A2: A,B2: A] :
( ( ( modulo_modulo @ A @ A2 @ B2 )
= ( zero_zero @ A ) )
=> ( dvd_dvd @ A @ B2 @ A2 ) ) ) ).
% mod_0_imp_dvd
thf(fact_1431_le__imp__power__dvd,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [M: nat,N: nat,A2: A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( dvd_dvd @ A @ ( power_power @ A @ A2 @ M ) @ ( power_power @ A @ A2 @ N ) ) ) ) ).
% le_imp_power_dvd
thf(fact_1432_power__le__dvd,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A2: A,N: nat,B2: A,M: nat] :
( ( dvd_dvd @ A @ ( power_power @ A @ A2 @ N ) @ B2 )
=> ( ( ord_less_eq @ nat @ M @ N )
=> ( dvd_dvd @ A @ ( power_power @ A @ A2 @ M ) @ B2 ) ) ) ) ).
% power_le_dvd
thf(fact_1433_dvd__power__le,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [X: A,Y2: A,N: nat,M: nat] :
( ( dvd_dvd @ A @ X @ Y2 )
=> ( ( ord_less_eq @ nat @ N @ M )
=> ( dvd_dvd @ A @ ( power_power @ A @ X @ N ) @ ( power_power @ A @ Y2 @ M ) ) ) ) ) ).
% dvd_power_le
thf(fact_1434_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_1435_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_1436_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_1437_zdvd__mono,axiom,
! [K2: int,M: int,T2: int] :
( ( K2
!= ( zero_zero @ int ) )
=> ( ( dvd_dvd @ int @ M @ T2 )
= ( dvd_dvd @ int @ ( times_times @ int @ K2 @ M ) @ ( times_times @ int @ K2 @ T2 ) ) ) ) ).
% zdvd_mono
thf(fact_1438_zdvd__mult__cancel,axiom,
! [K2: int,M: int,N: int] :
( ( dvd_dvd @ int @ ( times_times @ int @ K2 @ M ) @ ( times_times @ int @ K2 @ N ) )
=> ( ( K2
!= ( zero_zero @ int ) )
=> ( dvd_dvd @ int @ M @ N ) ) ) ).
% zdvd_mult_cancel
thf(fact_1439_bezout__add__nat,axiom,
! [A2: nat,B2: nat] :
? [D3: nat,X5: nat,Y7: nat] :
( ( dvd_dvd @ nat @ D3 @ A2 )
& ( dvd_dvd @ nat @ D3 @ B2 )
& ( ( ( times_times @ nat @ A2 @ X5 )
= ( plus_plus @ nat @ ( times_times @ nat @ B2 @ Y7 ) @ D3 ) )
| ( ( times_times @ nat @ B2 @ X5 )
= ( plus_plus @ nat @ ( times_times @ nat @ A2 @ Y7 ) @ D3 ) ) ) ) ).
% bezout_add_nat
thf(fact_1440_bezout__lemma__nat,axiom,
! [D2: nat,A2: nat,B2: nat,X: nat,Y2: nat] :
( ( dvd_dvd @ nat @ D2 @ A2 )
=> ( ( dvd_dvd @ nat @ D2 @ B2 )
=> ( ( ( ( times_times @ nat @ A2 @ X )
= ( plus_plus @ nat @ ( times_times @ nat @ B2 @ Y2 ) @ D2 ) )
| ( ( times_times @ nat @ B2 @ X )
= ( plus_plus @ nat @ ( times_times @ nat @ A2 @ Y2 ) @ D2 ) ) )
=> ? [X5: nat,Y7: nat] :
( ( dvd_dvd @ nat @ D2 @ A2 )
& ( dvd_dvd @ nat @ D2 @ ( plus_plus @ nat @ A2 @ B2 ) )
& ( ( ( times_times @ nat @ A2 @ X5 )
= ( plus_plus @ nat @ ( times_times @ nat @ ( plus_plus @ nat @ A2 @ B2 ) @ Y7 ) @ D2 ) )
| ( ( times_times @ nat @ ( plus_plus @ nat @ A2 @ B2 ) @ X5 )
= ( plus_plus @ nat @ ( times_times @ nat @ A2 @ Y7 ) @ D2 ) ) ) ) ) ) ) ).
% bezout_lemma_nat
thf(fact_1441_zdvd__reduce,axiom,
! [K2: int,N: int,M: int] :
( ( dvd_dvd @ int @ K2 @ ( plus_plus @ int @ N @ ( times_times @ int @ K2 @ M ) ) )
= ( dvd_dvd @ int @ K2 @ N ) ) ).
% zdvd_reduce
thf(fact_1442_zdvd__period,axiom,
! [A2: int,D2: int,X: int,T2: int,C2: int] :
( ( dvd_dvd @ int @ A2 @ D2 )
=> ( ( dvd_dvd @ int @ A2 @ ( plus_plus @ int @ X @ T2 ) )
= ( dvd_dvd @ int @ A2 @ ( plus_plus @ int @ ( plus_plus @ int @ X @ ( times_times @ int @ C2 @ D2 ) ) @ T2 ) ) ) ) ).
% zdvd_period
thf(fact_1443_unit__dvdE,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A,B2: A] :
( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) )
=> ~ ( ( A2
!= ( zero_zero @ A ) )
=> ! [C4: A] :
( B2
!= ( times_times @ A @ A2 @ C4 ) ) ) ) ) ).
% unit_dvdE
thf(fact_1444_unity__coeff__ex,axiom,
! [A: $tType] :
( ( ( dvd @ A )
& ( semiring_0 @ A ) )
=> ! [P3: A > $o,L: A] :
( ( ? [X4: A] : ( P3 @ ( times_times @ A @ L @ X4 ) ) )
= ( ? [X4: A] :
( ( dvd_dvd @ A @ L @ ( plus_plus @ A @ X4 @ ( zero_zero @ A ) ) )
& ( P3 @ X4 ) ) ) ) ) ).
% unity_coeff_ex
thf(fact_1445_dvd__div__eq__mult,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ A2 @ B2 )
=> ( ( ( divide_divide @ A @ B2 @ A2 )
= C2 )
= ( B2
= ( times_times @ A @ C2 @ A2 ) ) ) ) ) ) ).
% dvd_div_eq_mult
thf(fact_1446_div__dvd__iff__mult,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ B2 @ A2 )
=> ( ( dvd_dvd @ A @ ( divide_divide @ A @ A2 @ B2 ) @ C2 )
= ( dvd_dvd @ A @ A2 @ ( times_times @ A @ C2 @ B2 ) ) ) ) ) ) ).
% div_dvd_iff_mult
thf(fact_1447_dvd__div__iff__mult,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [C2: A,B2: A,A2: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ C2 @ B2 )
=> ( ( dvd_dvd @ A @ A2 @ ( divide_divide @ A @ B2 @ C2 ) )
= ( dvd_dvd @ A @ ( times_times @ A @ A2 @ C2 ) @ B2 ) ) ) ) ) ).
% dvd_div_iff_mult
thf(fact_1448_dvd__div__div__eq__mult,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A,C2: A,B2: A,D2: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( C2
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ A2 @ B2 )
=> ( ( dvd_dvd @ A @ C2 @ D2 )
=> ( ( ( divide_divide @ A @ B2 @ A2 )
= ( divide_divide @ A @ D2 @ C2 ) )
= ( ( times_times @ A @ B2 @ C2 )
= ( times_times @ A @ A2 @ D2 ) ) ) ) ) ) ) ) ).
% dvd_div_div_eq_mult
thf(fact_1449_unit__div__eq__0__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,A2: A] :
( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( ( divide_divide @ A @ A2 @ B2 )
= ( zero_zero @ A ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ) ).
% unit_div_eq_0_iff
thf(fact_1450_even__numeral,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [N: num] : ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ A @ ( bit0 @ N ) ) ) ) ).
% even_numeral
thf(fact_1451_unit__eq__div1,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( ( divide_divide @ A @ A2 @ B2 )
= C2 )
= ( A2
= ( times_times @ A @ C2 @ B2 ) ) ) ) ) ).
% unit_eq_div1
thf(fact_1452_unit__eq__div2,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( A2
= ( divide_divide @ A @ C2 @ B2 ) )
= ( ( times_times @ A @ A2 @ B2 )
= C2 ) ) ) ) ).
% unit_eq_div2
thf(fact_1453_div__mult__unit2,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [C2: A,B2: A,A2: A] :
( ( dvd_dvd @ A @ C2 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ B2 @ A2 )
=> ( ( divide_divide @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) )
= ( divide_divide @ A @ ( divide_divide @ A @ A2 @ B2 ) @ C2 ) ) ) ) ) ).
% div_mult_unit2
thf(fact_1454_unit__div__commute,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( times_times @ A @ ( divide_divide @ A @ A2 @ B2 ) @ C2 )
= ( divide_divide @ A @ ( times_times @ A @ A2 @ C2 ) @ B2 ) ) ) ) ).
% unit_div_commute
thf(fact_1455_unit__div__mult__swap,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( dvd_dvd @ A @ C2 @ ( one_one @ A ) )
=> ( ( times_times @ A @ A2 @ ( divide_divide @ A @ B2 @ C2 ) )
= ( divide_divide @ A @ ( times_times @ A @ A2 @ B2 ) @ C2 ) ) ) ) ).
% unit_div_mult_swap
thf(fact_1456_is__unit__div__mult2__eq,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,C2: A,A2: A] :
( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ C2 @ ( one_one @ A ) )
=> ( ( divide_divide @ A @ A2 @ ( times_times @ A @ B2 @ C2 ) )
= ( divide_divide @ A @ ( divide_divide @ A @ A2 @ B2 ) @ C2 ) ) ) ) ) ).
% is_unit_div_mult2_eq
thf(fact_1457_unit__imp__mod__eq__0,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ! [B2: A,A2: A] :
( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( modulo_modulo @ A @ A2 @ B2 )
= ( zero_zero @ A ) ) ) ) ).
% unit_imp_mod_eq_0
thf(fact_1458_is__unit__power__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A,N: nat] :
( ( dvd_dvd @ A @ ( power_power @ A @ A2 @ N ) @ ( one_one @ A ) )
= ( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) )
| ( N
= ( zero_zero @ nat ) ) ) ) ) ).
% is_unit_power_iff
thf(fact_1459_eucl__rel__int__dividesI,axiom,
! [L: int,K2: int,Q: int] :
( ( L
!= ( zero_zero @ int ) )
=> ( ( K2
= ( times_times @ int @ Q @ L ) )
=> ( eucl_rel_int @ K2 @ L @ ( product_Pair @ int @ int @ Q @ ( zero_zero @ int ) ) ) ) ) ).
% eucl_rel_int_dividesI
thf(fact_1460_dvd__imp__le,axiom,
! [K2: nat,N: nat] :
( ( dvd_dvd @ nat @ K2 @ N )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ord_less_eq @ nat @ K2 @ N ) ) ) ).
% dvd_imp_le
thf(fact_1461_dvd__mult__cancel,axiom,
! [K2: nat,M: nat,N: nat] :
( ( dvd_dvd @ nat @ ( times_times @ nat @ K2 @ M ) @ ( times_times @ nat @ K2 @ N ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K2 )
=> ( dvd_dvd @ nat @ M @ N ) ) ) ).
% dvd_mult_cancel
thf(fact_1462_nat__mult__dvd__cancel1,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K2 )
=> ( ( dvd_dvd @ nat @ ( times_times @ nat @ K2 @ M ) @ ( times_times @ nat @ K2 @ N ) )
= ( dvd_dvd @ nat @ M @ N ) ) ) ).
% nat_mult_dvd_cancel1
thf(fact_1463_bezout__add__strong__nat,axiom,
! [A2: nat,B2: nat] :
( ( A2
!= ( zero_zero @ nat ) )
=> ? [D3: nat,X5: nat,Y7: nat] :
( ( dvd_dvd @ nat @ D3 @ A2 )
& ( dvd_dvd @ nat @ D3 @ B2 )
& ( ( times_times @ nat @ A2 @ X5 )
= ( plus_plus @ nat @ ( times_times @ nat @ B2 @ Y7 ) @ D3 ) ) ) ) ).
% bezout_add_strong_nat
thf(fact_1464_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_1465_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_1466_eucl__rel__int,axiom,
! [K2: int,L: int] : ( eucl_rel_int @ K2 @ L @ ( product_Pair @ int @ int @ ( divide_divide @ int @ K2 @ L ) @ ( modulo_modulo @ int @ K2 @ L ) ) ) ).
% eucl_rel_int
thf(fact_1467_even__zero,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( zero_zero @ A ) ) ) ).
% even_zero
thf(fact_1468_is__unitE,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A,C2: A] :
( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) )
=> ~ ( ( A2
!= ( zero_zero @ A ) )
=> ! [B3: A] :
( ( B3
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ B3 @ ( one_one @ A ) )
=> ( ( ( divide_divide @ A @ ( one_one @ A ) @ A2 )
= B3 )
=> ( ( ( divide_divide @ A @ ( one_one @ A ) @ B3 )
= A2 )
=> ( ( ( times_times @ A @ A2 @ B3 )
= ( one_one @ A ) )
=> ( ( divide_divide @ A @ C2 @ A2 )
!= ( times_times @ A @ C2 @ B3 ) ) ) ) ) ) ) ) ) ) ).
% is_unitE
thf(fact_1469_is__unit__div__mult__cancel__left,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A,B2: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( divide_divide @ A @ A2 @ ( times_times @ A @ A2 @ B2 ) )
= ( divide_divide @ A @ ( one_one @ A ) @ B2 ) ) ) ) ) ).
% is_unit_div_mult_cancel_left
thf(fact_1470_is__unit__div__mult__cancel__right,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A,B2: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( divide_divide @ A @ A2 @ ( times_times @ A @ B2 @ A2 ) )
= ( divide_divide @ A @ ( one_one @ A ) @ B2 ) ) ) ) ) ).
% is_unit_div_mult_cancel_right
thf(fact_1471_evenE,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A2: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
=> ~ ! [B3: A] :
( A2
!= ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B3 ) ) ) ) ).
% evenE
thf(fact_1472_odd__one,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( one_one @ A ) ) ) ).
% odd_one
thf(fact_1473_odd__even__add,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A2: A,B2: A] :
( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
=> ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 )
=> ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ) ).
% odd_even_add
thf(fact_1474_bit__eq__rec,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ( ( ^ [Y5: A,Z2: A] : Y5 = Z2 )
= ( ^ [A5: A,B4: A] :
( ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A5 )
= ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B4 ) )
& ( ( divide_divide @ A @ A5 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( divide_divide @ A @ B4 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% bit_eq_rec
thf(fact_1475_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_1476_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_1477_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_1478_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_1479_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_1480_power__dvd__imp__le,axiom,
! [I2: nat,M: nat,N: nat] :
( ( dvd_dvd @ nat @ ( power_power @ nat @ I2 @ M ) @ ( power_power @ nat @ I2 @ N ) )
=> ( ( ord_less @ nat @ ( one_one @ nat ) @ I2 )
=> ( ord_less_eq @ nat @ M @ N ) ) ) ).
% power_dvd_imp_le
thf(fact_1481_mod__int__pos__iff,axiom,
! [K2: int,L: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( modulo_modulo @ int @ K2 @ L ) )
= ( ( dvd_dvd @ int @ L @ K2 )
| ( ( L
= ( zero_zero @ int ) )
& ( ord_less_eq @ int @ ( zero_zero @ int ) @ K2 ) )
| ( ord_less @ int @ ( zero_zero @ int ) @ L ) ) ) ).
% mod_int_pos_iff
thf(fact_1482_even__two__times__div__two,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [A2: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
=> ( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
= A2 ) ) ) ).
% even_two_times_div_two
thf(fact_1483_even__iff__mod__2__eq__zero,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A2: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
= ( ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( zero_zero @ A ) ) ) ) ).
% even_iff_mod_2_eq_zero
thf(fact_1484_odd__iff__mod__2__eq__one,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A2: A] :
( ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) )
= ( ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( one_one @ A ) ) ) ) ).
% odd_iff_mod_2_eq_one
thf(fact_1485_power__mono__odd,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: nat,A2: A,B2: A] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ord_less_eq @ A @ ( power_power @ A @ A2 @ N ) @ ( power_power @ A @ B2 @ N ) ) ) ) ) ).
% power_mono_odd
thf(fact_1486_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_1487_dvd__power__iff__le,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K2 )
=> ( ( dvd_dvd @ nat @ ( power_power @ nat @ K2 @ M ) @ ( power_power @ nat @ K2 @ N ) )
= ( ord_less_eq @ nat @ M @ N ) ) ) ).
% dvd_power_iff_le
thf(fact_1488_signed__take__bit__int__less__exp,axiom,
! [N: nat,K2: int] : ( ord_less @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N @ K2 ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ).
% signed_take_bit_int_less_exp
thf(fact_1489_even__unset__bit__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: nat,A2: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se2638667681897837118et_bit @ A @ M @ A2 ) )
= ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
| ( M
= ( zero_zero @ nat ) ) ) ) ) ).
% even_unset_bit_iff
thf(fact_1490_even__set__bit__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: nat,A2: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5668285175392031749et_bit @ A @ M @ A2 ) )
= ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
& ( M
!= ( zero_zero @ nat ) ) ) ) ) ).
% even_set_bit_iff
thf(fact_1491_even__flip__bit__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: nat,A2: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se8732182000553998342ip_bit @ A @ M @ A2 ) )
= ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
!= ( M
= ( zero_zero @ nat ) ) ) ) ) ).
% even_flip_bit_iff
thf(fact_1492_oddE,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A2: A] :
( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
=> ~ ! [B3: A] :
( A2
!= ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B3 ) @ ( one_one @ A ) ) ) ) ) ).
% oddE
thf(fact_1493_mod2__eq__if,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A2: A] :
( ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
=> ( ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( zero_zero @ A ) ) )
& ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
=> ( ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( one_one @ A ) ) ) ) ) ).
% mod2_eq_if
thf(fact_1494_parity__cases,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A2: A] :
( ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
=> ( ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
!= ( zero_zero @ A ) ) )
=> ~ ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
=> ( ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
!= ( one_one @ A ) ) ) ) ) ).
% parity_cases
thf(fact_1495_zero__le__power__eq,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A,N: nat] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A2 @ N ) )
= ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
| ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ) ) ).
% zero_le_power_eq
thf(fact_1496_zero__le__odd__power,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: nat,A2: A] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A2 @ N ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ) ).
% zero_le_odd_power
thf(fact_1497_zero__le__even__power,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: nat,A2: A] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A2 @ N ) ) ) ) ).
% zero_le_even_power
thf(fact_1498_signed__take__bit__int__greater__eq__self__iff,axiom,
! [K2: int,N: nat] :
( ( ord_less_eq @ int @ K2 @ ( bit_ri4674362597316999326ke_bit @ int @ N @ K2 ) )
= ( ord_less @ int @ K2 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ) ).
% signed_take_bit_int_greater_eq_self_iff
thf(fact_1499_signed__take__bit__int__less__self__iff,axiom,
! [N: nat,K2: int] :
( ( ord_less @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N @ K2 ) @ K2 )
= ( ord_less_eq @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) @ K2 ) ) ).
% signed_take_bit_int_less_self_iff
thf(fact_1500_zero__less__power__eq,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A,N: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A2 @ N ) )
= ( ( N
= ( zero_zero @ nat ) )
| ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
& ( A2
!= ( zero_zero @ A ) ) )
| ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
& ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ) ) ).
% zero_less_power_eq
thf(fact_1501_eucl__rel__int__iff,axiom,
! [K2: int,L: int,Q: int,R: int] :
( ( eucl_rel_int @ K2 @ L @ ( product_Pair @ int @ int @ Q @ R ) )
= ( ( K2
= ( plus_plus @ int @ ( times_times @ int @ L @ Q ) @ 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 ) )
=> ( Q
= ( zero_zero @ int ) ) ) ) ) ) ) ).
% eucl_rel_int_iff
thf(fact_1502_power__le__zero__eq,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A,N: nat] :
( ( ord_less_eq @ A @ ( power_power @ A @ A2 @ N ) @ ( zero_zero @ A ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
& ( ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
& ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) )
| ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
& ( A2
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% power_le_zero_eq
thf(fact_1503_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_1504_VEBT__internal_Onaive__member_Opelims_I3_J,axiom,
! [X: vEBT_VEBT,Xa: nat] :
( ~ ( vEBT_V5719532721284313246member @ X @ Xa )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa ) )
=> ( ! [A4: $o,B3: $o] :
( ( X
= ( vEBT_Leaf @ A4 @ B3 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A4 @ B3 ) @ Xa ) )
=> ( ( ( Xa
= ( zero_zero @ nat ) )
=> A4 )
& ( ( Xa
!= ( zero_zero @ nat ) )
=> ( ( ( Xa
= ( one_one @ nat ) )
=> B3 )
& ( Xa
= ( one_one @ nat ) ) ) ) ) ) )
=> ( ! [Uu2: option @ ( product_prod @ nat @ nat ),Uv2: list @ vEBT_VEBT,Uw: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Uu2 @ ( zero_zero @ nat ) @ Uv2 @ Uw ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Uu2 @ ( zero_zero @ nat ) @ Uv2 @ Uw ) @ Xa ) ) )
=> ~ ! [Uy: option @ ( product_prod @ nat @ nat ),V3: nat,TreeList2: list @ vEBT_VEBT,S2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Uy @ ( suc @ V3 ) @ TreeList2 @ S2 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Uy @ ( suc @ V3 ) @ TreeList2 @ S2 ) @ Xa ) )
=> ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_V5719532721284313246member @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.naive_member.pelims(3)
thf(fact_1505_VEBT__internal_Onaive__member_Opelims_I2_J,axiom,
! [X: vEBT_VEBT,Xa: nat] :
( ( vEBT_V5719532721284313246member @ X @ Xa )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa ) )
=> ( ! [A4: $o,B3: $o] :
( ( X
= ( vEBT_Leaf @ A4 @ B3 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A4 @ B3 ) @ Xa ) )
=> ~ ( ( ( Xa
= ( zero_zero @ nat ) )
=> A4 )
& ( ( Xa
!= ( zero_zero @ nat ) )
=> ( ( ( Xa
= ( one_one @ nat ) )
=> B3 )
& ( Xa
= ( one_one @ nat ) ) ) ) ) ) )
=> ~ ! [Uy: option @ ( product_prod @ nat @ nat ),V3: nat,TreeList2: list @ vEBT_VEBT,S2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Uy @ ( suc @ V3 ) @ TreeList2 @ S2 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Uy @ ( suc @ V3 ) @ TreeList2 @ S2 ) @ Xa ) )
=> ~ ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_V5719532721284313246member @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ) ).
% VEBT_internal.naive_member.pelims(2)
thf(fact_1506_VEBT__internal_Onaive__member_Opelims_I1_J,axiom,
! [X: vEBT_VEBT,Xa: nat,Y2: $o] :
( ( ( vEBT_V5719532721284313246member @ X @ Xa )
= Y2 )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa ) )
=> ( ! [A4: $o,B3: $o] :
( ( X
= ( vEBT_Leaf @ A4 @ B3 ) )
=> ( ( Y2
= ( ( ( Xa
= ( zero_zero @ nat ) )
=> A4 )
& ( ( Xa
!= ( zero_zero @ nat ) )
=> ( ( ( Xa
= ( one_one @ nat ) )
=> B3 )
& ( Xa
= ( one_one @ nat ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A4 @ B3 ) @ Xa ) ) ) )
=> ( ! [Uu2: option @ ( product_prod @ nat @ nat ),Uv2: list @ vEBT_VEBT,Uw: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Uu2 @ ( zero_zero @ nat ) @ Uv2 @ Uw ) )
=> ( ~ Y2
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Uu2 @ ( zero_zero @ nat ) @ Uv2 @ Uw ) @ Xa ) ) ) )
=> ~ ! [Uy: option @ ( product_prod @ nat @ nat ),V3: nat,TreeList2: list @ vEBT_VEBT,S2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Uy @ ( suc @ V3 ) @ TreeList2 @ S2 ) )
=> ( ( Y2
= ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_V5719532721284313246member @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Uy @ ( suc @ V3 ) @ TreeList2 @ S2 ) @ Xa ) ) ) ) ) ) ) ) ).
% VEBT_internal.naive_member.pelims(1)
thf(fact_1507_div2__even__ext__nat,axiom,
! [X: nat,Y2: nat] :
( ( ( divide_divide @ nat @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( divide_divide @ nat @ Y2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
=> ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ X )
= ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Y2 ) )
=> ( X = Y2 ) ) ) ).
% div2_even_ext_nat
thf(fact_1508_vebt__buildup_Oelims,axiom,
! [X: nat,Y2: vEBT_VEBT] :
( ( ( vEBT_vebt_buildup @ X )
= Y2 )
=> ( ( ( X
= ( zero_zero @ nat ) )
=> ( Y2
!= ( vEBT_Leaf @ $false @ $false ) ) )
=> ( ( ( X
= ( suc @ ( zero_zero @ nat ) ) )
=> ( Y2
!= ( vEBT_Leaf @ $false @ $false ) ) )
=> ~ ! [Va: nat] :
( ( X
= ( suc @ ( suc @ Va ) ) )
=> ~ ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va ) ) )
=> ( Y2
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ Va ) ) @ ( replicate @ vEBT_VEBT @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) )
& ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va ) ) )
=> ( Y2
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ Va ) ) @ ( replicate @ vEBT_VEBT @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_buildup.elims
thf(fact_1509_neg__eucl__rel__int__mult__2,axiom,
! [B2: int,A2: int,Q: int,R: int] :
( ( ord_less_eq @ int @ B2 @ ( zero_zero @ int ) )
=> ( ( eucl_rel_int @ ( plus_plus @ int @ A2 @ ( one_one @ int ) ) @ B2 @ ( product_Pair @ int @ int @ Q @ R ) )
=> ( eucl_rel_int @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ A2 ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ B2 ) @ ( product_Pair @ int @ int @ Q @ ( minus_minus @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ R ) @ ( one_one @ int ) ) ) ) ) ) ).
% neg_eucl_rel_int_mult_2
thf(fact_1510_flip__bit__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A] :
( ( bit_se8732182000553998342ip_bit @ A @ ( zero_zero @ nat ) @ A2 )
= ( plus_plus @ A @ ( zero_neq_one_of_bool @ A @ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% flip_bit_0
thf(fact_1511_add__scale__eq__noteq,axiom,
! [A: $tType] :
( ( semiri1453513574482234551roduct @ A )
=> ! [R: A,A2: A,B2: A,C2: A,D2: A] :
( ( R
!= ( zero_zero @ A ) )
=> ( ( ( A2 = B2 )
& ( C2 != D2 ) )
=> ( ( plus_plus @ A @ A2 @ ( times_times @ A @ R @ C2 ) )
!= ( plus_plus @ A @ B2 @ ( times_times @ A @ R @ D2 ) ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_1512_intind,axiom,
! [A: $tType,I2: nat,N: nat,P3: A > $o,X: A] :
( ( ord_less @ nat @ I2 @ N )
=> ( ( P3 @ X )
=> ( P3 @ ( nth @ A @ ( replicate @ A @ N @ X ) @ I2 ) ) ) ) ).
% intind
thf(fact_1513_diff__self,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A] :
( ( minus_minus @ A @ A2 @ A2 )
= ( zero_zero @ A ) ) ) ).
% diff_self
thf(fact_1514_diff__0__right,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A] :
( ( minus_minus @ A @ A2 @ ( zero_zero @ A ) )
= A2 ) ) ).
% diff_0_right
thf(fact_1515_zero__diff,axiom,
! [A: $tType] :
( ( comm_monoid_diff @ A )
=> ! [A2: A] :
( ( minus_minus @ A @ ( zero_zero @ A ) @ A2 )
= ( zero_zero @ A ) ) ) ).
% zero_diff
thf(fact_1516_diff__zero,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add @ A )
=> ! [A2: A] :
( ( minus_minus @ A @ A2 @ ( zero_zero @ A ) )
= A2 ) ) ).
% diff_zero
thf(fact_1517_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add @ A )
=> ! [A2: A] :
( ( minus_minus @ A @ A2 @ A2 )
= ( zero_zero @ A ) ) ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_1518_add__diff__cancel__right_H,axiom,
! [A: $tType] :
( ( cancel2418104881723323429up_add @ A )
=> ! [A2: A,B2: A] :
( ( minus_minus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ B2 )
= A2 ) ) ).
% add_diff_cancel_right'
thf(fact_1519_add__diff__cancel__right,axiom,
! [A: $tType] :
( ( cancel2418104881723323429up_add @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( minus_minus @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) )
= ( minus_minus @ A @ A2 @ B2 ) ) ) ).
% add_diff_cancel_right
thf(fact_1520_add__diff__cancel__left_H,axiom,
! [A: $tType] :
( ( cancel2418104881723323429up_add @ A )
=> ! [A2: A,B2: A] :
( ( minus_minus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ A2 )
= B2 ) ) ).
% add_diff_cancel_left'
thf(fact_1521_add__diff__cancel__left,axiom,
! [A: $tType] :
( ( cancel2418104881723323429up_add @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( minus_minus @ A @ ( plus_plus @ A @ C2 @ A2 ) @ ( plus_plus @ A @ C2 @ B2 ) )
= ( minus_minus @ A @ A2 @ B2 ) ) ) ).
% add_diff_cancel_left
thf(fact_1522_diff__add__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A] :
( ( plus_plus @ A @ ( minus_minus @ A @ A2 @ B2 ) @ B2 )
= A2 ) ) ).
% diff_add_cancel
thf(fact_1523_add__diff__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A] :
( ( minus_minus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ B2 )
= A2 ) ) ).
% add_diff_cancel
thf(fact_1524_minus__mod__self2,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel @ A )
=> ! [A2: A,B2: A] :
( ( modulo_modulo @ A @ ( minus_minus @ A @ A2 @ B2 ) @ B2 )
= ( modulo_modulo @ A @ A2 @ B2 ) ) ) ).
% minus_mod_self2
thf(fact_1525_of__bool__less__eq__iff,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [P3: $o,Q2: $o] :
( ( ord_less_eq @ A @ ( zero_neq_one_of_bool @ A @ P3 ) @ ( zero_neq_one_of_bool @ A @ Q2 ) )
= ( P3
=> Q2 ) ) ) ).
% of_bool_less_eq_iff
thf(fact_1526_of__bool__less__iff,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [P3: $o,Q2: $o] :
( ( ord_less @ A @ ( zero_neq_one_of_bool @ A @ P3 ) @ ( zero_neq_one_of_bool @ A @ Q2 ) )
= ( ~ P3
& Q2 ) ) ) ).
% of_bool_less_iff
thf(fact_1527_of__bool__eq__1__iff,axiom,
! [A: $tType] :
( ( zero_neq_one @ A )
=> ! [P3: $o] :
( ( ( zero_neq_one_of_bool @ A @ P3 )
= ( one_one @ A ) )
= P3 ) ) ).
% of_bool_eq_1_iff
thf(fact_1528_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_1529_replicate__eq__replicate,axiom,
! [A: $tType,M: nat,X: A,N: nat,Y2: A] :
( ( ( replicate @ A @ M @ X )
= ( replicate @ A @ N @ Y2 ) )
= ( ( M = N )
& ( ( M
!= ( zero_zero @ nat ) )
=> ( X = Y2 ) ) ) ) ).
% replicate_eq_replicate
thf(fact_1530_length__replicate,axiom,
! [A: $tType,N: nat,X: A] :
( ( size_size @ ( list @ A ) @ ( replicate @ A @ N @ X ) )
= N ) ).
% length_replicate
thf(fact_1531_map__replicate,axiom,
! [A: $tType,B: $tType,F3: B > A,N: nat,X: B] :
( ( map @ B @ A @ F3 @ ( replicate @ B @ N @ X ) )
= ( replicate @ A @ N @ ( F3 @ X ) ) ) ).
% map_replicate
thf(fact_1532_diff__ge__0__iff__ge,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( minus_minus @ A @ A2 @ B2 ) )
= ( ord_less_eq @ A @ B2 @ A2 ) ) ) ).
% diff_ge_0_iff_ge
thf(fact_1533_diff__gt__0__iff__gt,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( minus_minus @ A @ A2 @ B2 ) )
= ( ord_less @ A @ B2 @ A2 ) ) ) ).
% diff_gt_0_iff_gt
thf(fact_1534_le__add__diff__inverse2,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [B2: A,A2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( plus_plus @ A @ ( minus_minus @ A @ A2 @ B2 ) @ B2 )
= A2 ) ) ) ).
% le_add_diff_inverse2
thf(fact_1535_le__add__diff__inverse,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [B2: A,A2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( plus_plus @ A @ B2 @ ( minus_minus @ A @ A2 @ B2 ) )
= A2 ) ) ) ).
% le_add_diff_inverse
thf(fact_1536_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_1537_diff__add__zero,axiom,
! [A: $tType] :
( ( comm_monoid_diff @ A )
=> ! [A2: A,B2: A] :
( ( minus_minus @ A @ A2 @ ( plus_plus @ A @ A2 @ B2 ) )
= ( zero_zero @ A ) ) ) ).
% diff_add_zero
thf(fact_1538_right__diff__distrib__numeral,axiom,
! [A: $tType] :
( ( ( numeral @ A )
& ( ring @ A ) )
=> ! [V2: num,B2: A,C2: A] :
( ( times_times @ A @ ( numeral_numeral @ A @ V2 ) @ ( minus_minus @ A @ B2 @ C2 ) )
= ( minus_minus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ V2 ) @ B2 ) @ ( times_times @ A @ ( numeral_numeral @ A @ V2 ) @ C2 ) ) ) ) ).
% right_diff_distrib_numeral
thf(fact_1539_left__diff__distrib__numeral,axiom,
! [A: $tType] :
( ( ( numeral @ A )
& ( ring @ A ) )
=> ! [A2: A,B2: A,V2: num] :
( ( times_times @ A @ ( minus_minus @ A @ A2 @ B2 ) @ ( numeral_numeral @ A @ V2 ) )
= ( minus_minus @ A @ ( times_times @ A @ A2 @ ( numeral_numeral @ A @ V2 ) ) @ ( times_times @ A @ B2 @ ( numeral_numeral @ A @ V2 ) ) ) ) ) ).
% left_diff_distrib_numeral
thf(fact_1540_div__diff,axiom,
! [A: $tType] :
( ( idom_modulo @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( dvd_dvd @ A @ C2 @ A2 )
=> ( ( dvd_dvd @ A @ C2 @ B2 )
=> ( ( divide_divide @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C2 )
= ( minus_minus @ A @ ( divide_divide @ A @ A2 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) ) ) ) ) ) ).
% div_diff
thf(fact_1541_zero__less__of__bool__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [P3: $o] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( zero_neq_one_of_bool @ A @ P3 ) )
= P3 ) ) ).
% zero_less_of_bool_iff
thf(fact_1542_of__bool__less__one__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [P3: $o] :
( ( ord_less @ A @ ( zero_neq_one_of_bool @ A @ P3 ) @ ( one_one @ A ) )
= ~ P3 ) ) ).
% of_bool_less_one_iff
thf(fact_1543_of__bool__not__iff,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [P3: $o] :
( ( zero_neq_one_of_bool @ A @ ~ P3 )
= ( minus_minus @ A @ ( one_one @ A ) @ ( zero_neq_one_of_bool @ A @ P3 ) ) ) ) ).
% of_bool_not_iff
thf(fact_1544_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_1545_in__set__replicate,axiom,
! [A: $tType,X: A,N: nat,Y2: A] :
( ( member @ A @ X @ ( set2 @ A @ ( replicate @ A @ N @ Y2 ) ) )
= ( ( X = Y2 )
& ( N
!= ( zero_zero @ nat ) ) ) ) ).
% in_set_replicate
thf(fact_1546_Bex__set__replicate,axiom,
! [A: $tType,N: nat,A2: A,P3: A > $o] :
( ( ? [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ ( replicate @ A @ N @ A2 ) ) )
& ( P3 @ X4 ) ) )
= ( ( P3 @ A2 )
& ( N
!= ( zero_zero @ nat ) ) ) ) ).
% Bex_set_replicate
thf(fact_1547_Ball__set__replicate,axiom,
! [A: $tType,N: nat,A2: A,P3: A > $o] :
( ( ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ ( replicate @ A @ N @ A2 ) ) )
=> ( P3 @ X4 ) ) )
= ( ( P3 @ A2 )
| ( N
= ( zero_zero @ nat ) ) ) ) ).
% Ball_set_replicate
thf(fact_1548_nth__replicate,axiom,
! [A: $tType,I2: nat,N: nat,X: A] :
( ( ord_less @ nat @ I2 @ N )
=> ( ( nth @ A @ ( replicate @ A @ N @ X ) @ I2 )
= X ) ) ).
% nth_replicate
thf(fact_1549_zle__diff1__eq,axiom,
! [W: int,Z3: int] :
( ( ord_less_eq @ int @ W @ ( minus_minus @ int @ Z3 @ ( one_one @ int ) ) )
= ( ord_less @ int @ W @ Z3 ) ) ).
% zle_diff1_eq
thf(fact_1550_odd__of__bool__self,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [P5: $o] :
( ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( zero_neq_one_of_bool @ A @ P5 ) ) )
= P5 ) ) ).
% odd_of_bool_self
thf(fact_1551_even__diff,axiom,
! [A: $tType] :
( ( ring_parity @ A )
=> ! [A2: A,B2: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( minus_minus @ A @ A2 @ B2 ) )
= ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ).
% even_diff
thf(fact_1552_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_1553_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_1554_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_1555_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_1556_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_1557_signed__take__bit__diff,axiom,
! [N: nat,K2: int,L: int] :
( ( bit_ri4674362597316999326ke_bit @ int @ N @ ( minus_minus @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N @ K2 ) @ ( bit_ri4674362597316999326ke_bit @ int @ N @ L ) ) )
= ( bit_ri4674362597316999326ke_bit @ int @ N @ ( minus_minus @ int @ K2 @ L ) ) ) ).
% signed_take_bit_diff
thf(fact_1558_dvd__antisym,axiom,
! [M: nat,N: nat] :
( ( dvd_dvd @ nat @ M @ N )
=> ( ( dvd_dvd @ nat @ N @ M )
=> ( M = N ) ) ) ).
% dvd_antisym
thf(fact_1559_diff__eq__diff__eq,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A,C2: A,D2: A] :
( ( ( minus_minus @ A @ A2 @ B2 )
= ( minus_minus @ A @ C2 @ D2 ) )
=> ( ( A2 = B2 )
= ( C2 = D2 ) ) ) ) ).
% diff_eq_diff_eq
thf(fact_1560_diff__right__commute,axiom,
! [A: $tType] :
( ( cancel2418104881723323429up_add @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( minus_minus @ A @ ( minus_minus @ A @ A2 @ C2 ) @ B2 )
= ( minus_minus @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C2 ) ) ) ).
% diff_right_commute
thf(fact_1561_of__bool__conj,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [P3: $o,Q2: $o] :
( ( zero_neq_one_of_bool @ A
@ ( P3
& Q2 ) )
= ( times_times @ A @ ( zero_neq_one_of_bool @ A @ P3 ) @ ( zero_neq_one_of_bool @ A @ Q2 ) ) ) ) ).
% of_bool_conj
thf(fact_1562_diff__eq__diff__less__eq,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B2: A,C2: A,D2: A] :
( ( ( minus_minus @ A @ A2 @ B2 )
= ( minus_minus @ A @ C2 @ D2 ) )
=> ( ( ord_less_eq @ A @ A2 @ B2 )
= ( ord_less_eq @ A @ C2 @ D2 ) ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_1563_diff__right__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ A2 @ C2 ) @ ( minus_minus @ A @ B2 @ C2 ) ) ) ) ).
% diff_right_mono
thf(fact_1564_diff__left__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ C2 @ A2 ) @ ( minus_minus @ A @ C2 @ B2 ) ) ) ) ).
% diff_left_mono
thf(fact_1565_diff__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B2: A,D2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ D2 @ C2 )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ A2 @ C2 ) @ ( minus_minus @ A @ B2 @ D2 ) ) ) ) ) ).
% diff_mono
thf(fact_1566_eq__iff__diff__eq__0,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ( ( ^ [Y5: A,Z2: A] : Y5 = Z2 )
= ( ^ [A5: A,B4: A] :
( ( minus_minus @ A @ A5 @ B4 )
= ( zero_zero @ A ) ) ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_1567_diff__strict__right__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ord_less @ A @ ( minus_minus @ A @ A2 @ C2 ) @ ( minus_minus @ A @ B2 @ C2 ) ) ) ) ).
% diff_strict_right_mono
thf(fact_1568_diff__strict__left__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( ord_less @ A @ B2 @ A2 )
=> ( ord_less @ A @ ( minus_minus @ A @ C2 @ A2 ) @ ( minus_minus @ A @ C2 @ B2 ) ) ) ) ).
% diff_strict_left_mono
thf(fact_1569_diff__eq__diff__less,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B2: A,C2: A,D2: A] :
( ( ( minus_minus @ A @ A2 @ B2 )
= ( minus_minus @ A @ C2 @ D2 ) )
=> ( ( ord_less @ A @ A2 @ B2 )
= ( ord_less @ A @ C2 @ D2 ) ) ) ) ).
% diff_eq_diff_less
thf(fact_1570_diff__strict__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B2: A,D2: A,C2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ D2 @ C2 )
=> ( ord_less @ A @ ( minus_minus @ A @ A2 @ C2 ) @ ( minus_minus @ A @ B2 @ D2 ) ) ) ) ) ).
% diff_strict_mono
thf(fact_1571_right__diff__distrib_H,axiom,
! [A: $tType] :
( ( comm_s4317794764714335236cancel @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( times_times @ A @ A2 @ ( minus_minus @ A @ B2 @ C2 ) )
= ( minus_minus @ A @ ( times_times @ A @ A2 @ B2 ) @ ( times_times @ A @ A2 @ C2 ) ) ) ) ).
% right_diff_distrib'
thf(fact_1572_left__diff__distrib_H,axiom,
! [A: $tType] :
( ( comm_s4317794764714335236cancel @ A )
=> ! [B2: A,C2: A,A2: A] :
( ( times_times @ A @ ( minus_minus @ A @ B2 @ C2 ) @ A2 )
= ( minus_minus @ A @ ( times_times @ A @ B2 @ A2 ) @ ( times_times @ A @ C2 @ A2 ) ) ) ) ).
% left_diff_distrib'
thf(fact_1573_right__diff__distrib,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( times_times @ A @ A2 @ ( minus_minus @ A @ B2 @ C2 ) )
= ( minus_minus @ A @ ( times_times @ A @ A2 @ B2 ) @ ( times_times @ A @ A2 @ C2 ) ) ) ) ).
% right_diff_distrib
thf(fact_1574_left__diff__distrib,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( times_times @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C2 )
= ( minus_minus @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% left_diff_distrib
thf(fact_1575_inf__period_I2_J,axiom,
! [A: $tType] :
( ( ( comm_ring @ A )
& ( dvd @ A ) )
=> ! [P3: A > $o,D4: A,Q2: A > $o] :
( ! [X5: A,K: A] :
( ( P3 @ X5 )
= ( P3 @ ( minus_minus @ A @ X5 @ ( times_times @ A @ K @ D4 ) ) ) )
=> ( ! [X5: A,K: A] :
( ( Q2 @ X5 )
= ( Q2 @ ( minus_minus @ A @ X5 @ ( times_times @ A @ K @ D4 ) ) ) )
=> ! [X2: A,K4: A] :
( ( ( P3 @ X2 )
| ( Q2 @ X2 ) )
= ( ( P3 @ ( minus_minus @ A @ X2 @ ( times_times @ A @ K4 @ D4 ) ) )
| ( Q2 @ ( minus_minus @ A @ X2 @ ( times_times @ A @ K4 @ D4 ) ) ) ) ) ) ) ) ).
% inf_period(2)
thf(fact_1576_inf__period_I1_J,axiom,
! [A: $tType] :
( ( ( comm_ring @ A )
& ( dvd @ A ) )
=> ! [P3: A > $o,D4: A,Q2: A > $o] :
( ! [X5: A,K: A] :
( ( P3 @ X5 )
= ( P3 @ ( minus_minus @ A @ X5 @ ( times_times @ A @ K @ D4 ) ) ) )
=> ( ! [X5: A,K: A] :
( ( Q2 @ X5 )
= ( Q2 @ ( minus_minus @ A @ X5 @ ( times_times @ A @ K @ D4 ) ) ) )
=> ! [X2: A,K4: A] :
( ( ( P3 @ X2 )
& ( Q2 @ X2 ) )
= ( ( P3 @ ( minus_minus @ A @ X2 @ ( times_times @ A @ K4 @ D4 ) ) )
& ( Q2 @ ( minus_minus @ A @ X2 @ ( times_times @ A @ K4 @ D4 ) ) ) ) ) ) ) ) ).
% inf_period(1)
thf(fact_1577_diff__diff__eq,axiom,
! [A: $tType] :
( ( cancel2418104881723323429up_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( minus_minus @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C2 )
= ( minus_minus @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% diff_diff_eq
thf(fact_1578_add__implies__diff,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add @ A )
=> ! [C2: A,B2: A,A2: A] :
( ( ( plus_plus @ A @ C2 @ B2 )
= A2 )
=> ( C2
= ( minus_minus @ A @ A2 @ B2 ) ) ) ) ).
% add_implies_diff
thf(fact_1579_diff__add__eq__diff__diff__swap,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( minus_minus @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) )
= ( minus_minus @ A @ ( minus_minus @ A @ A2 @ C2 ) @ B2 ) ) ) ).
% diff_add_eq_diff_diff_swap
thf(fact_1580_diff__add__eq,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( plus_plus @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C2 )
= ( minus_minus @ A @ ( plus_plus @ A @ A2 @ C2 ) @ B2 ) ) ) ).
% diff_add_eq
thf(fact_1581_diff__diff__eq2,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( minus_minus @ A @ A2 @ ( minus_minus @ A @ B2 @ C2 ) )
= ( minus_minus @ A @ ( plus_plus @ A @ A2 @ C2 ) @ B2 ) ) ) ).
% diff_diff_eq2
thf(fact_1582_add__diff__eq,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( plus_plus @ A @ A2 @ ( minus_minus @ A @ B2 @ C2 ) )
= ( minus_minus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 ) ) ) ).
% add_diff_eq
thf(fact_1583_eq__diff__eq,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( A2
= ( minus_minus @ A @ C2 @ B2 ) )
= ( ( plus_plus @ A @ A2 @ B2 )
= C2 ) ) ) ).
% eq_diff_eq
thf(fact_1584_diff__eq__eq,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ( minus_minus @ A @ A2 @ B2 )
= C2 )
= ( A2
= ( plus_plus @ A @ C2 @ B2 ) ) ) ) ).
% diff_eq_eq
thf(fact_1585_group__cancel_Osub1,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A3: A,K2: A,A2: A,B2: A] :
( ( A3
= ( plus_plus @ A @ K2 @ A2 ) )
=> ( ( minus_minus @ A @ A3 @ B2 )
= ( plus_plus @ A @ K2 @ ( minus_minus @ A @ A2 @ B2 ) ) ) ) ) ).
% group_cancel.sub1
thf(fact_1586_add__diff__add,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A2: A,C2: A,B2: A,D2: A] :
( ( minus_minus @ A @ ( plus_plus @ A @ A2 @ C2 ) @ ( plus_plus @ A @ B2 @ D2 ) )
= ( plus_plus @ A @ ( minus_minus @ A @ A2 @ B2 ) @ ( minus_minus @ A @ C2 @ D2 ) ) ) ) ).
% add_diff_add
thf(fact_1587_diff__divide__distrib,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( divide_divide @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C2 )
= ( minus_minus @ A @ ( divide_divide @ A @ A2 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) ) ) ) ).
% diff_divide_distrib
thf(fact_1588_dvd__diff__commute,axiom,
! [A: $tType] :
( ( euclid5891614535332579305n_ring @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( dvd_dvd @ A @ A2 @ ( minus_minus @ A @ C2 @ B2 ) )
= ( dvd_dvd @ A @ A2 @ ( minus_minus @ A @ B2 @ C2 ) ) ) ) ).
% dvd_diff_commute
thf(fact_1589_mod__diff__eq,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( modulo_modulo @ A @ ( minus_minus @ A @ ( modulo_modulo @ A @ A2 @ C2 ) @ ( modulo_modulo @ A @ B2 @ C2 ) ) @ C2 )
= ( modulo_modulo @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C2 ) ) ) ).
% mod_diff_eq
thf(fact_1590_mod__diff__cong,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel @ A )
=> ! [A2: A,C2: A,A6: A,B2: A,B6: A] :
( ( ( modulo_modulo @ A @ A2 @ C2 )
= ( modulo_modulo @ A @ A6 @ C2 ) )
=> ( ( ( modulo_modulo @ A @ B2 @ C2 )
= ( modulo_modulo @ A @ B6 @ C2 ) )
=> ( ( modulo_modulo @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C2 )
= ( modulo_modulo @ A @ ( minus_minus @ A @ A6 @ B6 ) @ C2 ) ) ) ) ) ).
% mod_diff_cong
thf(fact_1591_mod__diff__left__eq,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( modulo_modulo @ A @ ( minus_minus @ A @ ( modulo_modulo @ A @ A2 @ C2 ) @ B2 ) @ C2 )
= ( modulo_modulo @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C2 ) ) ) ).
% mod_diff_left_eq
thf(fact_1592_mod__diff__right__eq,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( modulo_modulo @ A @ ( minus_minus @ A @ A2 @ ( modulo_modulo @ A @ B2 @ C2 ) ) @ C2 )
= ( modulo_modulo @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C2 ) ) ) ).
% mod_diff_right_eq
thf(fact_1593_int__distrib_I4_J,axiom,
! [W: int,Z1: int,Z22: int] :
( ( times_times @ int @ W @ ( minus_minus @ int @ Z1 @ Z22 ) )
= ( minus_minus @ int @ ( times_times @ int @ W @ Z1 ) @ ( times_times @ int @ W @ Z22 ) ) ) ).
% int_distrib(4)
thf(fact_1594_int__distrib_I3_J,axiom,
! [Z1: int,Z22: int,W: int] :
( ( times_times @ int @ ( minus_minus @ int @ Z1 @ Z22 ) @ W )
= ( minus_minus @ int @ ( times_times @ int @ Z1 @ W ) @ ( times_times @ int @ Z22 @ W ) ) ) ).
% int_distrib(3)
thf(fact_1595_zero__less__eq__of__bool,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [P3: $o] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( zero_neq_one_of_bool @ A @ P3 ) ) ) ).
% zero_less_eq_of_bool
thf(fact_1596_of__bool__less__eq__one,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [P3: $o] : ( ord_less_eq @ A @ ( zero_neq_one_of_bool @ A @ P3 ) @ ( one_one @ A ) ) ) ).
% of_bool_less_eq_one
thf(fact_1597_of__bool__def,axiom,
! [A: $tType] :
( ( zero_neq_one @ A )
=> ( ( zero_neq_one_of_bool @ A )
= ( ^ [P4: $o] : ( if @ A @ P4 @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ) ) ).
% of_bool_def
thf(fact_1598_split__of__bool,axiom,
! [A: $tType] :
( ( zero_neq_one @ A )
=> ! [P3: A > $o,P5: $o] :
( ( P3 @ ( zero_neq_one_of_bool @ A @ P5 ) )
= ( ( P5
=> ( P3 @ ( one_one @ A ) ) )
& ( ~ P5
=> ( P3 @ ( zero_zero @ A ) ) ) ) ) ) ).
% split_of_bool
thf(fact_1599_split__of__bool__asm,axiom,
! [A: $tType] :
( ( zero_neq_one @ A )
=> ! [P3: A > $o,P5: $o] :
( ( P3 @ ( zero_neq_one_of_bool @ A @ P5 ) )
= ( ~ ( ( P5
& ~ ( P3 @ ( one_one @ A ) ) )
| ( ~ P5
& ~ ( P3 @ ( zero_zero @ A ) ) ) ) ) ) ) ).
% split_of_bool_asm
thf(fact_1600_le__iff__diff__le__0,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A5: A,B4: A] : ( ord_less_eq @ A @ ( minus_minus @ A @ A5 @ B4 ) @ ( zero_zero @ A ) ) ) ) ) ).
% le_iff_diff_le_0
thf(fact_1601_less__iff__diff__less__0,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ( ( ord_less @ A )
= ( ^ [A5: A,B4: A] : ( ord_less @ A @ ( minus_minus @ A @ A5 @ B4 ) @ ( zero_zero @ A ) ) ) ) ) ).
% less_iff_diff_less_0
thf(fact_1602_add__le__add__imp__diff__le,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [I2: A,K2: A,N: A,J3: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ I2 @ K2 ) @ N )
=> ( ( ord_less_eq @ A @ N @ ( plus_plus @ A @ J3 @ K2 ) )
=> ( ( ord_less_eq @ A @ ( plus_plus @ A @ I2 @ K2 ) @ N )
=> ( ( ord_less_eq @ A @ N @ ( plus_plus @ A @ J3 @ K2 ) )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ N @ K2 ) @ J3 ) ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_1603_add__le__imp__le__diff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [I2: A,K2: A,N: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ I2 @ K2 ) @ N )
=> ( ord_less_eq @ A @ I2 @ ( minus_minus @ A @ N @ K2 ) ) ) ) ).
% add_le_imp_le_diff
thf(fact_1604_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ( minus_minus @ A @ B2 @ A2 )
= C2 )
= ( B2
= ( plus_plus @ A @ C2 @ A2 ) ) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_1605_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( plus_plus @ A @ A2 @ ( minus_minus @ A @ B2 @ A2 ) )
= B2 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
thf(fact_1606_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( minus_minus @ A @ C2 @ ( minus_minus @ A @ B2 @ A2 ) )
= ( minus_minus @ A @ ( plus_plus @ A @ C2 @ A2 ) @ B2 ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_1607_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( minus_minus @ A @ ( plus_plus @ A @ B2 @ C2 ) @ A2 )
= ( plus_plus @ A @ ( minus_minus @ A @ B2 @ A2 ) @ C2 ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_1608_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( plus_plus @ A @ ( minus_minus @ A @ B2 @ A2 ) @ C2 )
= ( minus_minus @ A @ ( plus_plus @ A @ B2 @ C2 ) @ A2 ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_1609_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( minus_minus @ A @ ( plus_plus @ A @ C2 @ B2 ) @ A2 )
= ( plus_plus @ A @ C2 @ ( minus_minus @ A @ B2 @ A2 ) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_1610_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( plus_plus @ A @ C2 @ ( minus_minus @ A @ B2 @ A2 ) )
= ( minus_minus @ A @ ( plus_plus @ A @ C2 @ B2 ) @ A2 ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_1611_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ ( minus_minus @ A @ B2 @ A2 ) )
= ( ord_less_eq @ A @ ( plus_plus @ A @ C2 @ A2 ) @ B2 ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_1612_le__add__diff,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ord_less_eq @ A @ C2 @ ( minus_minus @ A @ ( plus_plus @ A @ B2 @ C2 ) @ A2 ) ) ) ) ).
% le_add_diff
thf(fact_1613_diff__add,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( plus_plus @ A @ ( minus_minus @ A @ B2 @ A2 ) @ A2 )
= B2 ) ) ) ).
% diff_add
thf(fact_1614_le__diff__eq,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ ( minus_minus @ A @ C2 @ B2 ) )
= ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 ) ) ) ).
% le_diff_eq
thf(fact_1615_diff__le__eq,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C2 )
= ( ord_less_eq @ A @ A2 @ ( plus_plus @ A @ C2 @ B2 ) ) ) ) ).
% diff_le_eq
thf(fact_1616_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,B2: A] :
( ~ ( ord_less @ A @ A2 @ B2 )
=> ( ( plus_plus @ A @ B2 @ ( minus_minus @ A @ A2 @ B2 ) )
= A2 ) ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_1617_less__diff__eq,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( ord_less @ A @ A2 @ ( minus_minus @ A @ C2 @ B2 ) )
= ( ord_less @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 ) ) ) ).
% less_diff_eq
thf(fact_1618_diff__less__eq,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C2 )
= ( ord_less @ A @ A2 @ ( plus_plus @ A @ C2 @ B2 ) ) ) ) ).
% diff_less_eq
thf(fact_1619_square__diff__square__factored,axiom,
! [A: $tType] :
( ( comm_ring @ A )
=> ! [X: A,Y2: A] :
( ( minus_minus @ A @ ( times_times @ A @ X @ X ) @ ( times_times @ A @ Y2 @ Y2 ) )
= ( times_times @ A @ ( plus_plus @ A @ X @ Y2 ) @ ( minus_minus @ A @ X @ Y2 ) ) ) ) ).
% square_diff_square_factored
thf(fact_1620_eq__add__iff2,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A2: A,E: A,C2: A,B2: A,D2: A] :
( ( ( plus_plus @ A @ ( times_times @ A @ A2 @ E ) @ C2 )
= ( plus_plus @ A @ ( times_times @ A @ B2 @ E ) @ D2 ) )
= ( C2
= ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ B2 @ A2 ) @ E ) @ D2 ) ) ) ) ).
% eq_add_iff2
thf(fact_1621_eq__add__iff1,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A2: A,E: A,C2: A,B2: A,D2: A] :
( ( ( plus_plus @ A @ ( times_times @ A @ A2 @ E ) @ C2 )
= ( plus_plus @ A @ ( times_times @ A @ B2 @ E ) @ D2 ) )
= ( ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ A2 @ B2 ) @ E ) @ C2 )
= D2 ) ) ) ).
% eq_add_iff1
thf(fact_1622_mult__diff__mult,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [X: A,Y2: A,A2: A,B2: A] :
( ( minus_minus @ A @ ( times_times @ A @ X @ Y2 ) @ ( times_times @ A @ A2 @ B2 ) )
= ( plus_plus @ A @ ( times_times @ A @ X @ ( minus_minus @ A @ Y2 @ B2 ) ) @ ( times_times @ A @ ( minus_minus @ A @ X @ A2 ) @ B2 ) ) ) ) ).
% mult_diff_mult
thf(fact_1623_replicate__eqI,axiom,
! [A: $tType,Xs: list @ A,N: nat,X: A] :
( ( ( size_size @ ( list @ A ) @ Xs )
= N )
=> ( ! [Y7: A] :
( ( member @ A @ Y7 @ ( set2 @ A @ Xs ) )
=> ( Y7 = X ) )
=> ( Xs
= ( replicate @ A @ N @ X ) ) ) ) ).
% replicate_eqI
thf(fact_1624_replicate__length__same,axiom,
! [A: $tType,Xs: list @ A,X: A] :
( ! [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Xs ) )
=> ( X5 = X ) )
=> ( ( replicate @ A @ ( size_size @ ( list @ A ) @ Xs ) @ X )
= Xs ) ) ).
% replicate_length_same
thf(fact_1625_mod__eq__dvd__iff,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( ( modulo_modulo @ A @ A2 @ C2 )
= ( modulo_modulo @ A @ B2 @ C2 ) )
= ( dvd_dvd @ A @ C2 @ ( minus_minus @ A @ A2 @ B2 ) ) ) ) ).
% mod_eq_dvd_iff
thf(fact_1626_dvd__minus__mod,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [B2: A,A2: A] : ( dvd_dvd @ A @ B2 @ ( minus_minus @ A @ A2 @ ( modulo_modulo @ A @ A2 @ B2 ) ) ) ) ).
% dvd_minus_mod
thf(fact_1627_int__le__induct,axiom,
! [I2: int,K2: int,P3: int > $o] :
( ( ord_less_eq @ int @ I2 @ K2 )
=> ( ( P3 @ K2 )
=> ( ! [I3: int] :
( ( ord_less_eq @ int @ I3 @ K2 )
=> ( ( P3 @ I3 )
=> ( P3 @ ( minus_minus @ int @ I3 @ ( one_one @ int ) ) ) ) )
=> ( P3 @ I2 ) ) ) ) ).
% int_le_induct
thf(fact_1628_int__less__induct,axiom,
! [I2: int,K2: int,P3: int > $o] :
( ( ord_less @ int @ I2 @ K2 )
=> ( ( P3 @ ( minus_minus @ int @ K2 @ ( one_one @ int ) ) )
=> ( ! [I3: int] :
( ( ord_less @ int @ I3 @ K2 )
=> ( ( P3 @ I3 )
=> ( P3 @ ( minus_minus @ int @ I3 @ ( one_one @ int ) ) ) ) )
=> ( P3 @ I2 ) ) ) ) ).
% int_less_induct
thf(fact_1629_map__replicate__const,axiom,
! [B: $tType,A: $tType,K2: A,Lst: list @ B] :
( ( map @ B @ A
@ ^ [X4: B] : K2
@ Lst )
= ( replicate @ A @ ( size_size @ ( list @ B ) @ Lst ) @ K2 ) ) ).
% map_replicate_const
thf(fact_1630_ordered__ring__class_Ole__add__iff2,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [A2: A,E: A,C2: A,B2: A,D2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ ( times_times @ A @ A2 @ 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 @ A2 ) @ E ) @ D2 ) ) ) ) ).
% ordered_ring_class.le_add_iff2
thf(fact_1631_ordered__ring__class_Ole__add__iff1,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [A2: A,E: A,C2: A,B2: A,D2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ ( times_times @ A @ A2 @ E ) @ C2 ) @ ( plus_plus @ A @ ( times_times @ A @ B2 @ E ) @ D2 ) )
= ( ord_less_eq @ A @ ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ A2 @ B2 ) @ E ) @ C2 ) @ D2 ) ) ) ).
% ordered_ring_class.le_add_iff1
thf(fact_1632_less__add__iff2,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [A2: A,E: A,C2: A,B2: A,D2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ ( times_times @ A @ A2 @ 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 @ A2 ) @ E ) @ D2 ) ) ) ) ).
% less_add_iff2
thf(fact_1633_less__add__iff1,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [A2: A,E: A,C2: A,B2: A,D2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ ( times_times @ A @ A2 @ E ) @ C2 ) @ ( plus_plus @ A @ ( times_times @ A @ B2 @ E ) @ D2 ) )
= ( ord_less @ A @ ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ A2 @ B2 ) @ E ) @ C2 ) @ D2 ) ) ) ).
% less_add_iff1
thf(fact_1634_divide__diff__eq__iff,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z3: A,X: A,Y2: A] :
( ( Z3
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( divide_divide @ A @ X @ Z3 ) @ Y2 )
= ( divide_divide @ A @ ( minus_minus @ A @ X @ ( times_times @ A @ Y2 @ Z3 ) ) @ Z3 ) ) ) ) ).
% divide_diff_eq_iff
thf(fact_1635_diff__divide__eq__iff,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z3: A,X: A,Y2: A] :
( ( Z3
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ X @ ( divide_divide @ A @ Y2 @ Z3 ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ X @ Z3 ) @ Y2 ) @ Z3 ) ) ) ) ).
% diff_divide_eq_iff
thf(fact_1636_diff__frac__eq,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [Y2: A,Z3: A,X: A,W: A] :
( ( Y2
!= ( zero_zero @ A ) )
=> ( ( Z3
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( divide_divide @ A @ X @ Y2 ) @ ( divide_divide @ A @ W @ Z3 ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ X @ Z3 ) @ ( times_times @ A @ W @ Y2 ) ) @ ( times_times @ A @ Y2 @ Z3 ) ) ) ) ) ) ).
% diff_frac_eq
thf(fact_1637_add__divide__eq__if__simps_I4_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z3: A,A2: A,B2: A] :
( ( ( Z3
= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ A2 @ ( divide_divide @ A @ B2 @ Z3 ) )
= A2 ) )
& ( ( Z3
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ A2 @ ( divide_divide @ A @ B2 @ Z3 ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ A2 @ Z3 ) @ B2 ) @ Z3 ) ) ) ) ) ).
% add_divide_eq_if_simps(4)
thf(fact_1638_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_1639_inf__period_I3_J,axiom,
! [A: $tType] :
( ( ( comm_ring @ A )
& ( dvd @ A ) )
=> ! [D2: A,D4: A,T2: A] :
( ( dvd_dvd @ A @ D2 @ D4 )
=> ! [X2: A,K4: A] :
( ( dvd_dvd @ A @ D2 @ ( plus_plus @ A @ X2 @ T2 ) )
= ( dvd_dvd @ A @ D2 @ ( plus_plus @ A @ ( minus_minus @ A @ X2 @ ( times_times @ A @ K4 @ D4 ) ) @ T2 ) ) ) ) ) ).
% inf_period(3)
thf(fact_1640_inf__period_I4_J,axiom,
! [A: $tType] :
( ( ( comm_ring @ A )
& ( dvd @ A ) )
=> ! [D2: A,D4: A,T2: A] :
( ( dvd_dvd @ A @ D2 @ D4 )
=> ! [X2: A,K4: A] :
( ( ~ ( dvd_dvd @ A @ D2 @ ( plus_plus @ A @ X2 @ T2 ) ) )
= ( ~ ( dvd_dvd @ A @ D2 @ ( plus_plus @ A @ ( minus_minus @ A @ X2 @ ( times_times @ A @ K4 @ D4 ) ) @ T2 ) ) ) ) ) ) ).
% inf_period(4)
thf(fact_1641_minus__mult__div__eq__mod,axiom,
! [A: $tType] :
( ( semiring_modulo @ A )
=> ! [A2: A,B2: A] :
( ( minus_minus @ A @ A2 @ ( times_times @ A @ B2 @ ( divide_divide @ A @ A2 @ B2 ) ) )
= ( modulo_modulo @ A @ A2 @ B2 ) ) ) ).
% minus_mult_div_eq_mod
thf(fact_1642_minus__mod__eq__mult__div,axiom,
! [A: $tType] :
( ( semiring_modulo @ A )
=> ! [A2: A,B2: A] :
( ( minus_minus @ A @ A2 @ ( modulo_modulo @ A @ A2 @ B2 ) )
= ( times_times @ A @ B2 @ ( divide_divide @ A @ A2 @ B2 ) ) ) ) ).
% minus_mod_eq_mult_div
thf(fact_1643_minus__mod__eq__div__mult,axiom,
! [A: $tType] :
( ( semiring_modulo @ A )
=> ! [A2: A,B2: A] :
( ( minus_minus @ A @ A2 @ ( modulo_modulo @ A @ A2 @ B2 ) )
= ( times_times @ A @ ( divide_divide @ A @ A2 @ B2 ) @ B2 ) ) ) ).
% minus_mod_eq_div_mult
thf(fact_1644_minus__div__mult__eq__mod,axiom,
! [A: $tType] :
( ( semiring_modulo @ A )
=> ! [A2: A,B2: A] :
( ( minus_minus @ A @ A2 @ ( times_times @ A @ ( divide_divide @ A @ A2 @ B2 ) @ B2 ) )
= ( modulo_modulo @ A @ A2 @ B2 ) ) ) ).
% minus_div_mult_eq_mod
thf(fact_1645_plusinfinity,axiom,
! [D2: int,P6: int > $o,P3: int > $o] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D2 )
=> ( ! [X5: int,K: int] :
( ( P6 @ X5 )
= ( P6 @ ( minus_minus @ int @ X5 @ ( times_times @ int @ K @ D2 ) ) ) )
=> ( ? [Z4: int] :
! [X5: int] :
( ( ord_less @ int @ Z4 @ X5 )
=> ( ( P3 @ X5 )
= ( P6 @ X5 ) ) )
=> ( ? [X_1: int] : ( P6 @ X_1 )
=> ? [X_12: int] : ( P3 @ X_12 ) ) ) ) ) ).
% plusinfinity
thf(fact_1646_minusinfinity,axiom,
! [D2: int,P1: int > $o,P3: int > $o] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D2 )
=> ( ! [X5: int,K: int] :
( ( P1 @ X5 )
= ( P1 @ ( minus_minus @ int @ X5 @ ( times_times @ int @ K @ D2 ) ) ) )
=> ( ? [Z4: int] :
! [X5: int] :
( ( ord_less @ int @ X5 @ Z4 )
=> ( ( P3 @ X5 )
= ( P1 @ X5 ) ) )
=> ( ? [X_1: int] : ( P1 @ X_1 )
=> ? [X_12: int] : ( P3 @ X_12 ) ) ) ) ) ).
% minusinfinity
thf(fact_1647_int__induct,axiom,
! [P3: int > $o,K2: int,I2: int] :
( ( P3 @ K2 )
=> ( ! [I3: int] :
( ( ord_less_eq @ int @ K2 @ I3 )
=> ( ( P3 @ I3 )
=> ( P3 @ ( plus_plus @ int @ I3 @ ( one_one @ int ) ) ) ) )
=> ( ! [I3: int] :
( ( ord_less_eq @ int @ I3 @ K2 )
=> ( ( P3 @ I3 )
=> ( P3 @ ( minus_minus @ int @ I3 @ ( one_one @ int ) ) ) ) )
=> ( P3 @ I2 ) ) ) ) ).
% int_induct
thf(fact_1648_frac__le__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y2: A,Z3: A,X: A,W: A] :
( ( Y2
!= ( zero_zero @ A ) )
=> ( ( Z3
!= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( divide_divide @ A @ X @ Y2 ) @ ( divide_divide @ A @ W @ Z3 ) )
= ( ord_less_eq @ A @ ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ X @ Z3 ) @ ( times_times @ A @ W @ Y2 ) ) @ ( times_times @ A @ Y2 @ Z3 ) ) @ ( zero_zero @ A ) ) ) ) ) ) ).
% frac_le_eq
thf(fact_1649_frac__less__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y2: A,Z3: A,X: A,W: A] :
( ( Y2
!= ( zero_zero @ A ) )
=> ( ( Z3
!= ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( divide_divide @ A @ X @ Y2 ) @ ( divide_divide @ A @ W @ Z3 ) )
= ( ord_less @ A @ ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ X @ Z3 ) @ ( times_times @ A @ W @ Y2 ) ) @ ( times_times @ A @ Y2 @ Z3 ) ) @ ( zero_zero @ A ) ) ) ) ) ) ).
% frac_less_eq
thf(fact_1650_power2__commute,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [X: A,Y2: A] :
( ( power_power @ A @ ( minus_minus @ A @ X @ Y2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( power_power @ A @ ( minus_minus @ A @ Y2 @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% power2_commute
thf(fact_1651_even__diff__iff,axiom,
! [K2: int,L: int] :
( ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( minus_minus @ int @ K2 @ L ) )
= ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( plus_plus @ int @ K2 @ L ) ) ) ).
% even_diff_iff
thf(fact_1652_decr__mult__lemma,axiom,
! [D2: int,P3: int > $o,K2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D2 )
=> ( ! [X5: int] :
( ( P3 @ X5 )
=> ( P3 @ ( minus_minus @ int @ X5 @ D2 ) ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K2 )
=> ! [X2: int] :
( ( P3 @ X2 )
=> ( P3 @ ( minus_minus @ int @ X2 @ ( times_times @ int @ K2 @ D2 ) ) ) ) ) ) ) ).
% decr_mult_lemma
thf(fact_1653_mod__pos__geq,axiom,
! [L: int,K2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ L )
=> ( ( ord_less_eq @ int @ L @ K2 )
=> ( ( modulo_modulo @ int @ K2 @ L )
= ( modulo_modulo @ int @ ( minus_minus @ int @ K2 @ L ) @ L ) ) ) ) ).
% mod_pos_geq
thf(fact_1654_of__bool__odd__eq__mod__2,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A2: A] :
( ( zero_neq_one_of_bool @ A
@ ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) )
= ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% of_bool_odd_eq_mod_2
thf(fact_1655_scaling__mono,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [U: A,V2: A,R: A,S: A] :
( ( ord_less_eq @ A @ U @ V2 )
=> ( ( 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 @ V2 @ U ) ) @ S ) ) @ V2 ) ) ) ) ) ).
% scaling_mono
thf(fact_1656_bits__induct,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [P3: A > $o,A2: A] :
( ! [A4: A] :
( ( ( divide_divide @ A @ A4 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= A4 )
=> ( P3 @ A4 ) )
=> ( ! [A4: A,B3: $o] :
( ( P3 @ A4 )
=> ( ( ( divide_divide @ A @ ( plus_plus @ A @ ( zero_neq_one_of_bool @ A @ B3 ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A4 ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= A4 )
=> ( P3 @ ( plus_plus @ A @ ( zero_neq_one_of_bool @ A @ B3 ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A4 ) ) ) ) )
=> ( P3 @ A2 ) ) ) ) ).
% bits_induct
thf(fact_1657_signed__take__bit__int__less__eq,axiom,
! [N: nat,K2: int] :
( ( ord_less_eq @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) @ K2 )
=> ( ord_less_eq @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N @ K2 ) @ ( minus_minus @ int @ K2 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( suc @ N ) ) ) ) ) ).
% signed_take_bit_int_less_eq
thf(fact_1658_div__pos__geq,axiom,
! [L: int,K2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ L )
=> ( ( ord_less_eq @ int @ L @ K2 )
=> ( ( divide_divide @ int @ K2 @ L )
= ( plus_plus @ int @ ( divide_divide @ int @ ( minus_minus @ int @ K2 @ L ) @ L ) @ ( one_one @ int ) ) ) ) ) ).
% div_pos_geq
thf(fact_1659_add__0__iff,axiom,
! [A: $tType] :
( ( semiri1453513574482234551roduct @ A )
=> ! [B2: A,A2: A] :
( ( B2
= ( plus_plus @ A @ B2 @ A2 ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% add_0_iff
thf(fact_1660_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_1661_crossproduct__noteq,axiom,
! [A: $tType] :
( ( semiri1453513574482234551roduct @ A )
=> ! [A2: A,B2: A,C2: A,D2: A] :
( ( ( A2 != B2 )
& ( C2 != D2 ) )
= ( ( plus_plus @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ D2 ) )
!= ( plus_plus @ A @ ( times_times @ A @ A2 @ D2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ) ).
% crossproduct_noteq
thf(fact_1662_crossproduct__eq,axiom,
! [A: $tType] :
( ( semiri1453513574482234551roduct @ A )
=> ! [W: A,Y2: A,X: A,Z3: A] :
( ( ( plus_plus @ A @ ( times_times @ A @ W @ Y2 ) @ ( times_times @ A @ X @ Z3 ) )
= ( plus_plus @ A @ ( times_times @ A @ W @ Z3 ) @ ( times_times @ A @ X @ Y2 ) ) )
= ( ( W = X )
| ( Y2 = Z3 ) ) ) ) ).
% crossproduct_eq
thf(fact_1663_power2__diff,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [X: A,Y2: A] :
( ( power_power @ A @ ( minus_minus @ A @ X @ Y2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( minus_minus @ A @ ( plus_plus @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( times_times @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X ) @ Y2 ) ) ) ) ).
% power2_diff
thf(fact_1664_divmod__digit__1_I2_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ ( modulo_modulo @ A @ A2 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) )
=> ( ( minus_minus @ A @ ( modulo_modulo @ A @ A2 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) @ B2 )
= ( modulo_modulo @ A @ A2 @ B2 ) ) ) ) ) ) ).
% divmod_digit_1(2)
thf(fact_1665_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_1666_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_1667_neg__zmod__mult__2,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq @ int @ A2 @ ( zero_zero @ int ) )
=> ( ( modulo_modulo @ int @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ B2 ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ A2 ) )
= ( minus_minus @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( modulo_modulo @ int @ ( plus_plus @ int @ B2 @ ( one_one @ int ) ) @ A2 ) ) @ ( one_one @ int ) ) ) ) ).
% neg_zmod_mult_2
thf(fact_1668_vebt__buildup_Osimps_I3_J,axiom,
! [Va3: nat] :
( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va3 ) ) )
=> ( ( vEBT_vebt_buildup @ ( suc @ ( suc @ Va3 ) ) )
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ Va3 ) ) @ ( replicate @ vEBT_VEBT @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) )
& ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va3 ) ) )
=> ( ( vEBT_vebt_buildup @ ( suc @ ( suc @ Va3 ) ) )
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ Va3 ) ) @ ( replicate @ vEBT_VEBT @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ Va3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ).
% vebt_buildup.simps(3)
thf(fact_1669_divmod__step__eq,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [L: num,R: A,Q: A] :
( ( ( ord_less_eq @ A @ ( numeral_numeral @ A @ L ) @ R )
=> ( ( unique1321980374590559556d_step @ A @ L @ ( product_Pair @ A @ A @ Q @ R ) )
= ( product_Pair @ A @ A @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Q ) @ ( 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 @ Q @ R ) )
= ( product_Pair @ A @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Q ) @ R ) ) ) ) ) ).
% divmod_step_eq
thf(fact_1670_triangle__def,axiom,
( nat_triangle
= ( ^ [N2: nat] : ( divide_divide @ nat @ ( times_times @ nat @ N2 @ ( suc @ N2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% triangle_def
thf(fact_1671_vebt__buildup_Opelims,axiom,
! [X: nat,Y2: vEBT_VEBT] :
( ( ( vEBT_vebt_buildup @ X )
= Y2 )
=> ( ( accp @ nat @ vEBT_v4011308405150292612up_rel @ X )
=> ( ( ( X
= ( zero_zero @ nat ) )
=> ( ( Y2
= ( vEBT_Leaf @ $false @ $false ) )
=> ~ ( accp @ nat @ vEBT_v4011308405150292612up_rel @ ( zero_zero @ nat ) ) ) )
=> ( ( ( X
= ( suc @ ( zero_zero @ nat ) ) )
=> ( ( Y2
= ( vEBT_Leaf @ $false @ $false ) )
=> ~ ( accp @ nat @ vEBT_v4011308405150292612up_rel @ ( suc @ ( zero_zero @ nat ) ) ) ) )
=> ~ ! [Va: nat] :
( ( X
= ( suc @ ( suc @ Va ) ) )
=> ( ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va ) ) )
=> ( Y2
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ Va ) ) @ ( replicate @ vEBT_VEBT @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) )
& ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va ) ) )
=> ( Y2
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ Va ) ) @ ( replicate @ vEBT_VEBT @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) )
=> ~ ( accp @ nat @ vEBT_v4011308405150292612up_rel @ ( suc @ ( suc @ Va ) ) ) ) ) ) ) ) ) ).
% vebt_buildup.pelims
thf(fact_1672_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_1673_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_1674_set__decode__def,axiom,
( nat_set_decode
= ( ^ [X4: nat] :
( collect @ nat
@ ^ [N2: nat] :
~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ X4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ) ) ).
% set_decode_def
thf(fact_1675_even__mult__exp__div__exp__iff,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A2: A,M: nat,N: nat] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ ( times_times @ A @ A2 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) )
= ( ( ord_less @ nat @ N @ M )
| ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N )
= ( zero_zero @ A ) )
| ( ( ord_less_eq @ nat @ M @ N )
& ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A2 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ N @ M ) ) ) ) ) ) ) ) ).
% even_mult_exp_div_exp_iff
thf(fact_1676_idiff__infinity,axiom,
! [N: extended_enat] :
( ( minus_minus @ extended_enat @ ( extend4730790105801354508finity @ extended_enat ) @ N )
= ( extend4730790105801354508finity @ extended_enat ) ) ).
% idiff_infinity
thf(fact_1677_idiff__0,axiom,
! [N: extended_enat] :
( ( minus_minus @ extended_enat @ ( zero_zero @ extended_enat ) @ N )
= ( zero_zero @ extended_enat ) ) ).
% idiff_0
thf(fact_1678_idiff__0__right,axiom,
! [N: extended_enat] :
( ( minus_minus @ extended_enat @ N @ ( zero_zero @ extended_enat ) )
= N ) ).
% idiff_0_right
thf(fact_1679_Suc__diff__diff,axiom,
! [M: nat,N: nat,K2: nat] :
( ( minus_minus @ nat @ ( minus_minus @ nat @ ( suc @ M ) @ N ) @ ( suc @ K2 ) )
= ( minus_minus @ nat @ ( minus_minus @ nat @ M @ N ) @ K2 ) ) ).
% Suc_diff_diff
thf(fact_1680_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_1681_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus @ nat @ ( zero_zero @ nat ) @ N )
= ( zero_zero @ nat ) ) ).
% diff_0_eq_0
thf(fact_1682_diff__self__eq__0,axiom,
! [M: nat] :
( ( minus_minus @ nat @ M @ M )
= ( zero_zero @ nat ) ) ).
% diff_self_eq_0
thf(fact_1683_diff__diff__cancel,axiom,
! [I2: nat,N: nat] :
( ( ord_less_eq @ nat @ I2 @ N )
=> ( ( minus_minus @ nat @ N @ ( minus_minus @ nat @ N @ I2 ) )
= I2 ) ) ).
% diff_diff_cancel
thf(fact_1684_diff__diff__left,axiom,
! [I2: nat,J3: nat,K2: nat] :
( ( minus_minus @ nat @ ( minus_minus @ nat @ I2 @ J3 ) @ K2 )
= ( minus_minus @ nat @ I2 @ ( plus_plus @ nat @ J3 @ K2 ) ) ) ).
% diff_diff_left
thf(fact_1685_idiff__enat__0,axiom,
! [N: extended_enat] :
( ( minus_minus @ extended_enat @ ( extended_enat2 @ ( zero_zero @ nat ) ) @ N )
= ( extended_enat2 @ ( zero_zero @ nat ) ) ) ).
% idiff_enat_0
thf(fact_1686_idiff__enat__0__right,axiom,
! [N: extended_enat] :
( ( minus_minus @ extended_enat @ N @ ( extended_enat2 @ ( zero_zero @ nat ) ) )
= N ) ).
% idiff_enat_0_right
thf(fact_1687_idiff__enat__enat,axiom,
! [A2: nat,B2: nat] :
( ( minus_minus @ extended_enat @ ( extended_enat2 @ A2 ) @ ( extended_enat2 @ B2 ) )
= ( extended_enat2 @ ( minus_minus @ nat @ A2 @ B2 ) ) ) ).
% idiff_enat_enat
thf(fact_1688_idiff__self,axiom,
! [N: extended_enat] :
( ( N
!= ( extend4730790105801354508finity @ extended_enat ) )
=> ( ( minus_minus @ extended_enat @ N @ N )
= ( zero_zero @ extended_enat ) ) ) ).
% idiff_self
thf(fact_1689_add__diff__cancel__enat,axiom,
! [X: extended_enat,Y2: extended_enat] :
( ( X
!= ( extend4730790105801354508finity @ extended_enat ) )
=> ( ( minus_minus @ extended_enat @ ( plus_plus @ extended_enat @ X @ Y2 ) @ X )
= Y2 ) ) ).
% add_diff_cancel_enat
thf(fact_1690_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_1691_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_1692_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_1693_Nat_Odiff__diff__right,axiom,
! [K2: nat,J3: nat,I2: nat] :
( ( ord_less_eq @ nat @ K2 @ J3 )
=> ( ( minus_minus @ nat @ I2 @ ( minus_minus @ nat @ J3 @ K2 ) )
= ( minus_minus @ nat @ ( plus_plus @ nat @ I2 @ K2 ) @ J3 ) ) ) ).
% Nat.diff_diff_right
thf(fact_1694_Nat_Oadd__diff__assoc2,axiom,
! [K2: nat,J3: nat,I2: nat] :
( ( ord_less_eq @ nat @ K2 @ J3 )
=> ( ( plus_plus @ nat @ ( minus_minus @ nat @ J3 @ K2 ) @ I2 )
= ( minus_minus @ nat @ ( plus_plus @ nat @ J3 @ I2 ) @ K2 ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_1695_Nat_Oadd__diff__assoc,axiom,
! [K2: nat,J3: nat,I2: nat] :
( ( ord_less_eq @ nat @ K2 @ J3 )
=> ( ( plus_plus @ nat @ I2 @ ( minus_minus @ nat @ J3 @ K2 ) )
= ( minus_minus @ nat @ ( plus_plus @ nat @ I2 @ J3 ) @ K2 ) ) ) ).
% Nat.add_diff_assoc
thf(fact_1696_diff__Suc__1,axiom,
! [N: nat] :
( ( minus_minus @ nat @ ( suc @ N ) @ ( one_one @ nat ) )
= N ) ).
% diff_Suc_1
thf(fact_1697_idiff__infinity__right,axiom,
! [A2: nat] :
( ( minus_minus @ extended_enat @ ( extended_enat2 @ A2 ) @ ( extend4730790105801354508finity @ extended_enat ) )
= ( zero_zero @ extended_enat ) ) ).
% idiff_infinity_right
thf(fact_1698_triangle__Suc,axiom,
! [N: nat] :
( ( nat_triangle @ ( suc @ N ) )
= ( plus_plus @ nat @ ( nat_triangle @ N ) @ ( suc @ N ) ) ) ).
% triangle_Suc
thf(fact_1699_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_1700_diff__Suc__diff__eq1,axiom,
! [K2: nat,J3: nat,I2: nat] :
( ( ord_less_eq @ nat @ K2 @ J3 )
=> ( ( minus_minus @ nat @ I2 @ ( suc @ ( minus_minus @ nat @ J3 @ K2 ) ) )
= ( minus_minus @ nat @ ( plus_plus @ nat @ I2 @ K2 ) @ ( suc @ J3 ) ) ) ) ).
% diff_Suc_diff_eq1
thf(fact_1701_diff__Suc__diff__eq2,axiom,
! [K2: nat,J3: nat,I2: nat] :
( ( ord_less_eq @ nat @ K2 @ J3 )
=> ( ( minus_minus @ nat @ ( suc @ ( minus_minus @ nat @ J3 @ K2 ) ) @ I2 )
= ( minus_minus @ nat @ ( suc @ J3 ) @ ( plus_plus @ nat @ K2 @ I2 ) ) ) ) ).
% diff_Suc_diff_eq2
thf(fact_1702_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_1703_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_1704_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_1705_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_1706_diff__commute,axiom,
! [I2: nat,J3: nat,K2: nat] :
( ( minus_minus @ nat @ ( minus_minus @ nat @ I2 @ J3 ) @ K2 )
= ( minus_minus @ nat @ ( minus_minus @ nat @ I2 @ K2 ) @ J3 ) ) ).
% diff_commute
thf(fact_1707_zero__induct__lemma,axiom,
! [P3: nat > $o,K2: nat,I2: nat] :
( ( P3 @ K2 )
=> ( ! [N3: nat] :
( ( P3 @ ( suc @ N3 ) )
=> ( P3 @ N3 ) )
=> ( P3 @ ( minus_minus @ nat @ K2 @ I2 ) ) ) ) ).
% zero_induct_lemma
thf(fact_1708_Diff__mono,axiom,
! [A: $tType,A3: set @ A,C5: set @ A,D4: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ C5 )
=> ( ( ord_less_eq @ ( set @ A ) @ D4 @ B5 )
=> ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ B5 ) @ ( minus_minus @ ( set @ A ) @ C5 @ D4 ) ) ) ) ).
% Diff_mono
thf(fact_1709_Diff__subset,axiom,
! [A: $tType,A3: set @ A,B5: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ B5 ) @ A3 ) ).
% Diff_subset
thf(fact_1710_double__diff,axiom,
! [A: $tType,A3: set @ A,B5: set @ A,C5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B5 )
=> ( ( ord_less_eq @ ( set @ A ) @ B5 @ C5 )
=> ( ( minus_minus @ ( set @ A ) @ B5 @ ( minus_minus @ ( set @ A ) @ C5 @ A3 ) )
= A3 ) ) ) ).
% double_diff
thf(fact_1711_minus__nat_Odiff__0,axiom,
! [M: nat] :
( ( minus_minus @ nat @ M @ ( zero_zero @ nat ) )
= M ) ).
% minus_nat.diff_0
thf(fact_1712_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_1713_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_1714_less__imp__diff__less,axiom,
! [J3: nat,K2: nat,N: nat] :
( ( ord_less @ nat @ J3 @ K2 )
=> ( ord_less @ nat @ ( minus_minus @ nat @ J3 @ N ) @ K2 ) ) ).
% less_imp_diff_less
thf(fact_1715_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_1716_le__diff__iff_H,axiom,
! [A2: nat,C2: nat,B2: nat] :
( ( ord_less_eq @ nat @ A2 @ C2 )
=> ( ( ord_less_eq @ nat @ B2 @ C2 )
=> ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ C2 @ A2 ) @ ( minus_minus @ nat @ C2 @ B2 ) )
= ( ord_less_eq @ nat @ B2 @ A2 ) ) ) ) ).
% le_diff_iff'
thf(fact_1717_diff__le__self,axiom,
! [M: nat,N: nat] : ( ord_less_eq @ nat @ ( minus_minus @ nat @ M @ N ) @ M ) ).
% diff_le_self
thf(fact_1718_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_1719_Nat_Odiff__diff__eq,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ K2 @ M )
=> ( ( ord_less_eq @ nat @ K2 @ N )
=> ( ( minus_minus @ nat @ ( minus_minus @ nat @ M @ K2 ) @ ( minus_minus @ nat @ N @ K2 ) )
= ( minus_minus @ nat @ M @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_1720_le__diff__iff,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ K2 @ M )
=> ( ( ord_less_eq @ nat @ K2 @ N )
=> ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ M @ K2 ) @ ( minus_minus @ nat @ N @ K2 ) )
= ( ord_less_eq @ nat @ M @ N ) ) ) ) ).
% le_diff_iff
thf(fact_1721_eq__diff__iff,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ K2 @ M )
=> ( ( ord_less_eq @ nat @ K2 @ N )
=> ( ( ( minus_minus @ nat @ M @ K2 )
= ( minus_minus @ nat @ N @ K2 ) )
= ( M = N ) ) ) ) ).
% eq_diff_iff
thf(fact_1722_Nat_Odiff__cancel,axiom,
! [K2: nat,M: nat,N: nat] :
( ( minus_minus @ nat @ ( plus_plus @ nat @ K2 @ M ) @ ( plus_plus @ nat @ K2 @ N ) )
= ( minus_minus @ nat @ M @ N ) ) ).
% Nat.diff_cancel
thf(fact_1723_diff__cancel2,axiom,
! [M: nat,K2: nat,N: nat] :
( ( minus_minus @ nat @ ( plus_plus @ nat @ M @ K2 ) @ ( plus_plus @ nat @ N @ K2 ) )
= ( minus_minus @ nat @ M @ N ) ) ).
% diff_cancel2
thf(fact_1724_diff__add__inverse,axiom,
! [N: nat,M: nat] :
( ( minus_minus @ nat @ ( plus_plus @ nat @ N @ M ) @ N )
= M ) ).
% diff_add_inverse
thf(fact_1725_diff__add__inverse2,axiom,
! [M: nat,N: nat] :
( ( minus_minus @ nat @ ( plus_plus @ nat @ M @ N ) @ N )
= M ) ).
% diff_add_inverse2
thf(fact_1726_diff__mult__distrib2,axiom,
! [K2: nat,M: nat,N: nat] :
( ( times_times @ nat @ K2 @ ( minus_minus @ nat @ M @ N ) )
= ( minus_minus @ nat @ ( times_times @ nat @ K2 @ M ) @ ( times_times @ nat @ K2 @ N ) ) ) ).
% diff_mult_distrib2
thf(fact_1727_diff__mult__distrib,axiom,
! [M: nat,N: nat,K2: nat] :
( ( times_times @ nat @ ( minus_minus @ nat @ M @ N ) @ K2 )
= ( minus_minus @ nat @ ( times_times @ nat @ M @ K2 ) @ ( times_times @ nat @ N @ K2 ) ) ) ).
% diff_mult_distrib
thf(fact_1728_dvd__diff__nat,axiom,
! [K2: nat,M: nat,N: nat] :
( ( dvd_dvd @ nat @ K2 @ M )
=> ( ( dvd_dvd @ nat @ K2 @ N )
=> ( dvd_dvd @ nat @ K2 @ ( minus_minus @ nat @ M @ N ) ) ) ) ).
% dvd_diff_nat
thf(fact_1729_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_1730_diff__less__Suc,axiom,
! [M: nat,N: nat] : ( ord_less @ nat @ ( minus_minus @ nat @ M @ N ) @ ( suc @ M ) ) ).
% diff_less_Suc
thf(fact_1731_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_1732_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_1733_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_1734_diff__less__mono,axiom,
! [A2: nat,B2: nat,C2: nat] :
( ( ord_less @ nat @ A2 @ B2 )
=> ( ( ord_less_eq @ nat @ C2 @ A2 )
=> ( ord_less @ nat @ ( minus_minus @ nat @ A2 @ C2 ) @ ( minus_minus @ nat @ B2 @ C2 ) ) ) ) ).
% diff_less_mono
thf(fact_1735_less__diff__iff,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ K2 @ M )
=> ( ( ord_less_eq @ nat @ K2 @ N )
=> ( ( ord_less @ nat @ ( minus_minus @ nat @ M @ K2 ) @ ( minus_minus @ nat @ N @ K2 ) )
= ( ord_less @ nat @ M @ N ) ) ) ) ).
% less_diff_iff
thf(fact_1736_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_1737_less__diff__conv,axiom,
! [I2: nat,J3: nat,K2: nat] :
( ( ord_less @ nat @ I2 @ ( minus_minus @ nat @ J3 @ K2 ) )
= ( ord_less @ nat @ ( plus_plus @ nat @ I2 @ K2 ) @ J3 ) ) ).
% less_diff_conv
thf(fact_1738_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_1739_Nat_Ole__imp__diff__is__add,axiom,
! [I2: nat,J3: nat,K2: nat] :
( ( ord_less_eq @ nat @ I2 @ J3 )
=> ( ( ( minus_minus @ nat @ J3 @ I2 )
= K2 )
= ( J3
= ( plus_plus @ nat @ K2 @ I2 ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_1740_Nat_Odiff__add__assoc2,axiom,
! [K2: nat,J3: nat,I2: nat] :
( ( ord_less_eq @ nat @ K2 @ J3 )
=> ( ( minus_minus @ nat @ ( plus_plus @ nat @ J3 @ I2 ) @ K2 )
= ( plus_plus @ nat @ ( minus_minus @ nat @ J3 @ K2 ) @ I2 ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_1741_Nat_Odiff__add__assoc,axiom,
! [K2: nat,J3: nat,I2: nat] :
( ( ord_less_eq @ nat @ K2 @ J3 )
=> ( ( minus_minus @ nat @ ( plus_plus @ nat @ I2 @ J3 ) @ K2 )
= ( plus_plus @ nat @ I2 @ ( minus_minus @ nat @ J3 @ K2 ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_1742_Nat_Ole__diff__conv2,axiom,
! [K2: nat,J3: nat,I2: nat] :
( ( ord_less_eq @ nat @ K2 @ J3 )
=> ( ( ord_less_eq @ nat @ I2 @ ( minus_minus @ nat @ J3 @ K2 ) )
= ( ord_less_eq @ nat @ ( plus_plus @ nat @ I2 @ K2 ) @ J3 ) ) ) ).
% Nat.le_diff_conv2
thf(fact_1743_le__diff__conv,axiom,
! [J3: nat,K2: nat,I2: nat] :
( ( ord_less_eq @ nat @ ( minus_minus @ nat @ J3 @ K2 ) @ I2 )
= ( ord_less_eq @ nat @ J3 @ ( plus_plus @ nat @ I2 @ K2 ) ) ) ).
% le_diff_conv
thf(fact_1744_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_1745_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_1746_dvd__diffD,axiom,
! [K2: nat,M: nat,N: nat] :
( ( dvd_dvd @ nat @ K2 @ ( minus_minus @ nat @ M @ N ) )
=> ( ( dvd_dvd @ nat @ K2 @ N )
=> ( ( ord_less_eq @ nat @ N @ M )
=> ( dvd_dvd @ nat @ K2 @ M ) ) ) ) ).
% dvd_diffD
thf(fact_1747_dvd__diffD1,axiom,
! [K2: nat,M: nat,N: nat] :
( ( dvd_dvd @ nat @ K2 @ ( minus_minus @ nat @ M @ N ) )
=> ( ( dvd_dvd @ nat @ K2 @ M )
=> ( ( ord_less_eq @ nat @ N @ M )
=> ( dvd_dvd @ nat @ K2 @ N ) ) ) ) ).
% dvd_diffD1
thf(fact_1748_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_1749_bezout1__nat,axiom,
! [A2: nat,B2: nat] :
? [D3: nat,X5: nat,Y7: nat] :
( ( dvd_dvd @ nat @ D3 @ A2 )
& ( dvd_dvd @ nat @ D3 @ B2 )
& ( ( ( minus_minus @ nat @ ( times_times @ nat @ A2 @ X5 ) @ ( times_times @ nat @ B2 @ Y7 ) )
= D3 )
| ( ( minus_minus @ nat @ ( times_times @ nat @ B2 @ X5 ) @ ( times_times @ nat @ A2 @ Y7 ) )
= D3 ) ) ) ).
% bezout1_nat
thf(fact_1750_mod__if,axiom,
( ( modulo_modulo @ nat )
= ( ^ [M2: nat,N2: nat] : ( if @ nat @ ( ord_less @ nat @ M2 @ N2 ) @ M2 @ ( modulo_modulo @ nat @ ( minus_minus @ nat @ M2 @ N2 ) @ N2 ) ) ) ) ).
% mod_if
thf(fact_1751_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_1752_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_1753_diff__enat__def,axiom,
( ( minus_minus @ extended_enat )
= ( ^ [A5: extended_enat,B4: extended_enat] :
( extended_case_enat @ extended_enat
@ ^ [X4: nat] :
( extended_case_enat @ extended_enat
@ ^ [Y: nat] : ( extended_enat2 @ ( minus_minus @ nat @ X4 @ Y ) )
@ ( zero_zero @ extended_enat )
@ B4 )
@ ( extend4730790105801354508finity @ extended_enat )
@ A5 ) ) ) ).
% diff_enat_def
thf(fact_1754_add__diff__assoc__enat,axiom,
! [Z3: extended_enat,Y2: extended_enat,X: extended_enat] :
( ( ord_less_eq @ extended_enat @ Z3 @ Y2 )
=> ( ( plus_plus @ extended_enat @ X @ ( minus_minus @ extended_enat @ Y2 @ Z3 ) )
= ( minus_minus @ extended_enat @ ( plus_plus @ extended_enat @ X @ Y2 ) @ Z3 ) ) ) ).
% add_diff_assoc_enat
thf(fact_1755_diff__Suc__less,axiom,
! [N: nat,I2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ord_less @ nat @ ( minus_minus @ nat @ N @ ( suc @ I2 ) ) @ N ) ) ).
% diff_Suc_less
thf(fact_1756_nat__diff__split,axiom,
! [P3: nat > $o,A2: nat,B2: nat] :
( ( P3 @ ( minus_minus @ nat @ A2 @ B2 ) )
= ( ( ( ord_less @ nat @ A2 @ B2 )
=> ( P3 @ ( zero_zero @ nat ) ) )
& ! [D5: nat] :
( ( A2
= ( plus_plus @ nat @ B2 @ D5 ) )
=> ( P3 @ D5 ) ) ) ) ).
% nat_diff_split
thf(fact_1757_nat__diff__split__asm,axiom,
! [P3: nat > $o,A2: nat,B2: nat] :
( ( P3 @ ( minus_minus @ nat @ A2 @ B2 ) )
= ( ~ ( ( ( ord_less @ nat @ A2 @ B2 )
& ~ ( P3 @ ( zero_zero @ nat ) ) )
| ? [D5: nat] :
( ( A2
= ( plus_plus @ nat @ B2 @ D5 ) )
& ~ ( P3 @ D5 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_1758_less__diff__conv2,axiom,
! [K2: nat,J3: nat,I2: nat] :
( ( ord_less_eq @ nat @ K2 @ J3 )
=> ( ( ord_less @ nat @ ( minus_minus @ nat @ J3 @ K2 ) @ I2 )
= ( ord_less @ nat @ J3 @ ( plus_plus @ nat @ I2 @ K2 ) ) ) ) ).
% less_diff_conv2
thf(fact_1759_nat__eq__add__iff1,axiom,
! [J3: nat,I2: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ J3 @ I2 )
=> ( ( ( plus_plus @ nat @ ( times_times @ nat @ I2 @ U ) @ M )
= ( plus_plus @ nat @ ( times_times @ nat @ J3 @ U ) @ N ) )
= ( ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ I2 @ J3 ) @ U ) @ M )
= N ) ) ) ).
% nat_eq_add_iff1
thf(fact_1760_nat__eq__add__iff2,axiom,
! [I2: nat,J3: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ I2 @ J3 )
=> ( ( ( plus_plus @ nat @ ( times_times @ nat @ I2 @ U ) @ M )
= ( plus_plus @ nat @ ( times_times @ nat @ J3 @ U ) @ N ) )
= ( M
= ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ J3 @ I2 ) @ U ) @ N ) ) ) ) ).
% nat_eq_add_iff2
thf(fact_1761_nat__le__add__iff1,axiom,
! [J3: nat,I2: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ J3 @ I2 )
=> ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I2 @ U ) @ M ) @ ( plus_plus @ nat @ ( times_times @ nat @ J3 @ U ) @ N ) )
= ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ I2 @ J3 ) @ U ) @ M ) @ N ) ) ) ).
% nat_le_add_iff1
thf(fact_1762_nat__le__add__iff2,axiom,
! [I2: nat,J3: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ I2 @ J3 )
=> ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I2 @ U ) @ M ) @ ( plus_plus @ nat @ ( times_times @ nat @ J3 @ U ) @ N ) )
= ( ord_less_eq @ nat @ M @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ J3 @ I2 ) @ U ) @ N ) ) ) ) ).
% nat_le_add_iff2
thf(fact_1763_nat__diff__add__eq1,axiom,
! [J3: nat,I2: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ J3 @ I2 )
=> ( ( minus_minus @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I2 @ U ) @ M ) @ ( plus_plus @ nat @ ( times_times @ nat @ J3 @ U ) @ N ) )
= ( minus_minus @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ I2 @ J3 ) @ U ) @ M ) @ N ) ) ) ).
% nat_diff_add_eq1
thf(fact_1764_nat__diff__add__eq2,axiom,
! [I2: nat,J3: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ I2 @ J3 )
=> ( ( minus_minus @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I2 @ U ) @ M ) @ ( plus_plus @ nat @ ( times_times @ nat @ J3 @ U ) @ N ) )
= ( minus_minus @ nat @ M @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ J3 @ I2 ) @ U ) @ N ) ) ) ) ).
% nat_diff_add_eq2
thf(fact_1765_mod__eq__dvd__iff__nat,axiom,
! [N: nat,M: nat,Q: nat] :
( ( ord_less_eq @ nat @ N @ M )
=> ( ( ( modulo_modulo @ nat @ M @ Q )
= ( modulo_modulo @ nat @ N @ Q ) )
= ( dvd_dvd @ nat @ Q @ ( minus_minus @ nat @ M @ N ) ) ) ) ).
% mod_eq_dvd_iff_nat
thf(fact_1766_modulo__nat__def,axiom,
( ( modulo_modulo @ nat )
= ( ^ [M2: nat,N2: nat] : ( minus_minus @ nat @ M2 @ ( times_times @ nat @ ( divide_divide @ nat @ M2 @ N2 ) @ N2 ) ) ) ) ).
% modulo_nat_def
thf(fact_1767_power__diff,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [A2: A,N: nat,M: nat] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ nat @ N @ M )
=> ( ( power_power @ A @ A2 @ ( minus_minus @ nat @ M @ N ) )
= ( divide_divide @ A @ ( power_power @ A @ A2 @ M ) @ ( power_power @ A @ A2 @ N ) ) ) ) ) ) ).
% power_diff
thf(fact_1768_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_1769_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_1770_div__if,axiom,
( ( divide_divide @ nat )
= ( ^ [M2: nat,N2: nat] :
( if @ nat
@ ( ( ord_less @ nat @ M2 @ N2 )
| ( N2
= ( zero_zero @ nat ) ) )
@ ( zero_zero @ nat )
@ ( suc @ ( divide_divide @ nat @ ( minus_minus @ nat @ M2 @ N2 ) @ N2 ) ) ) ) ) ).
% div_if
thf(fact_1771_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_1772_add__eq__if,axiom,
( ( plus_plus @ nat )
= ( ^ [M2: nat,N2: nat] :
( if @ nat
@ ( M2
= ( zero_zero @ nat ) )
@ N2
@ ( suc @ ( plus_plus @ nat @ ( minus_minus @ nat @ M2 @ ( one_one @ nat ) ) @ N2 ) ) ) ) ) ).
% add_eq_if
thf(fact_1773_nat__less__add__iff1,axiom,
! [J3: nat,I2: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ J3 @ I2 )
=> ( ( ord_less @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I2 @ U ) @ M ) @ ( plus_plus @ nat @ ( times_times @ nat @ J3 @ U ) @ N ) )
= ( ord_less @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ I2 @ J3 ) @ U ) @ M ) @ N ) ) ) ).
% nat_less_add_iff1
thf(fact_1774_nat__less__add__iff2,axiom,
! [I2: nat,J3: nat,U: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ I2 @ J3 )
=> ( ( ord_less @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I2 @ U ) @ M ) @ ( plus_plus @ nat @ ( times_times @ nat @ J3 @ U ) @ N ) )
= ( ord_less @ nat @ M @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ J3 @ I2 ) @ U ) @ N ) ) ) ) ).
% nat_less_add_iff2
thf(fact_1775_mult__eq__if,axiom,
( ( times_times @ nat )
= ( ^ [M2: nat,N2: nat] :
( if @ nat
@ ( M2
= ( zero_zero @ nat ) )
@ ( zero_zero @ nat )
@ ( plus_plus @ nat @ N2 @ ( times_times @ nat @ ( minus_minus @ nat @ M2 @ ( one_one @ nat ) ) @ N2 ) ) ) ) ) ).
% mult_eq_if
thf(fact_1776_dvd__minus__add,axiom,
! [Q: nat,N: nat,R: nat,M: nat] :
( ( ord_less_eq @ nat @ Q @ N )
=> ( ( ord_less_eq @ nat @ Q @ ( times_times @ nat @ R @ M ) )
=> ( ( dvd_dvd @ nat @ M @ ( minus_minus @ nat @ N @ Q ) )
= ( dvd_dvd @ nat @ M @ ( plus_plus @ nat @ N @ ( minus_minus @ nat @ ( times_times @ nat @ R @ M ) @ Q ) ) ) ) ) ) ).
% dvd_minus_add
thf(fact_1777_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_1778_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_1779_power__diff__power__eq,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A2: A,N: nat,M: nat] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( ( ord_less_eq @ nat @ N @ M )
=> ( ( divide_divide @ A @ ( power_power @ A @ A2 @ M ) @ ( power_power @ A @ A2 @ N ) )
= ( power_power @ A @ A2 @ ( minus_minus @ nat @ M @ N ) ) ) )
& ( ~ ( ord_less_eq @ nat @ N @ M )
=> ( ( divide_divide @ A @ ( power_power @ A @ A2 @ M ) @ ( power_power @ A @ A2 @ N ) )
= ( divide_divide @ A @ ( one_one @ A ) @ ( power_power @ A @ A2 @ ( minus_minus @ nat @ N @ M ) ) ) ) ) ) ) ) ).
% power_diff_power_eq
thf(fact_1780_power__eq__if,axiom,
! [A: $tType] :
( ( power @ A )
=> ( ( power_power @ A )
= ( ^ [P4: A,M2: nat] :
( if @ A
@ ( M2
= ( zero_zero @ nat ) )
@ ( one_one @ A )
@ ( times_times @ A @ P4 @ ( power_power @ A @ P4 @ ( minus_minus @ nat @ M2 @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% power_eq_if
thf(fact_1781_power__minus__mult,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [N: nat,A2: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( times_times @ A @ ( power_power @ A @ A2 @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) @ A2 )
= ( power_power @ A @ A2 @ N ) ) ) ) ).
% power_minus_mult
thf(fact_1782_diff__le__diff__pow,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K2 )
=> ( ord_less_eq @ nat @ ( minus_minus @ nat @ M @ N ) @ ( minus_minus @ nat @ ( power_power @ nat @ K2 @ M ) @ ( power_power @ nat @ K2 @ N ) ) ) ) ).
% diff_le_diff_pow
thf(fact_1783_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_1784_mult__exp__mod__exp__eq,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [M: nat,N: nat,A2: A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( modulo_modulo @ A @ ( times_times @ A @ A2 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
= ( times_times @ A @ ( modulo_modulo @ A @ A2 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ N @ M ) ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M ) ) ) ) ) ).
% mult_exp_mod_exp_eq
thf(fact_1785_int__power__div__base,axiom,
! [M: nat,K2: int] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ K2 )
=> ( ( divide_divide @ int @ ( power_power @ int @ K2 @ M ) @ K2 )
= ( power_power @ int @ K2 @ ( minus_minus @ nat @ M @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ).
% int_power_div_base
thf(fact_1786_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_1787_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_1788_real__average__minus__second,axiom,
! [B2: real,A2: real] :
( ( minus_minus @ real @ ( divide_divide @ real @ ( plus_plus @ real @ B2 @ A2 ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ A2 )
= ( divide_divide @ real @ ( minus_minus @ real @ B2 @ A2 ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ).
% real_average_minus_second
thf(fact_1789_real__average__minus__first,axiom,
! [A2: real,B2: real] :
( ( minus_minus @ real @ ( divide_divide @ real @ ( plus_plus @ real @ A2 @ B2 ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ A2 )
= ( divide_divide @ real @ ( minus_minus @ real @ B2 @ A2 ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ).
% real_average_minus_first
thf(fact_1790_Suc__if__eq,axiom,
! [A: $tType,F3: nat > A,H2: nat > A,G: A,N: nat] :
( ! [N3: nat] :
( ( F3 @ ( suc @ N3 ) )
= ( H2 @ N3 ) )
=> ( ( ( F3 @ ( zero_zero @ nat ) )
= G )
=> ( ( ( N
= ( zero_zero @ nat ) )
=> ( ( F3 @ N )
= G ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ( ( F3 @ N )
= ( H2 @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% Suc_if_eq
thf(fact_1791_signed__take__bit__rec,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( ( bit_ri4674362597316999326ke_bit @ A )
= ( ^ [N2: nat,A5: A] :
( if @ A
@ ( N2
= ( zero_zero @ nat ) )
@ ( uminus_uminus @ A @ ( modulo_modulo @ A @ A5 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
@ ( plus_plus @ A @ ( modulo_modulo @ A @ A5 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_ri4674362597316999326ke_bit @ A @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) @ ( divide_divide @ A @ A5 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% signed_take_bit_rec
thf(fact_1792_Bolzano,axiom,
! [A2: real,B2: real,P3: real > real > $o] :
( ( ord_less_eq @ real @ A2 @ B2 )
=> ( ! [A4: real,B3: real,C4: real] :
( ( P3 @ A4 @ B3 )
=> ( ( P3 @ B3 @ C4 )
=> ( ( ord_less_eq @ real @ A4 @ B3 )
=> ( ( ord_less_eq @ real @ B3 @ C4 )
=> ( P3 @ A4 @ C4 ) ) ) ) )
=> ( ! [X5: real] :
( ( ord_less_eq @ real @ A2 @ X5 )
=> ( ( ord_less_eq @ real @ X5 @ B2 )
=> ? [D6: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D6 )
& ! [A4: real,B3: real] :
( ( ( ord_less_eq @ real @ A4 @ X5 )
& ( ord_less_eq @ real @ X5 @ B3 )
& ( ord_less @ real @ ( minus_minus @ real @ B3 @ A4 ) @ D6 ) )
=> ( P3 @ A4 @ B3 ) ) ) ) )
=> ( P3 @ A2 @ B2 ) ) ) ) ).
% Bolzano
thf(fact_1793_divmod__step__def,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ( ( unique1321980374590559556d_step @ A )
= ( ^ [L3: num] :
( product_case_prod @ A @ A @ ( product_prod @ A @ A )
@ ^ [Q5: 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 ) ) @ Q5 ) @ ( one_one @ A ) ) @ ( minus_minus @ A @ R5 @ ( numeral_numeral @ A @ L3 ) ) ) @ ( product_Pair @ A @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Q5 ) @ R5 ) ) ) ) ) ) ).
% divmod_step_def
thf(fact_1794_signed__take__bit__Suc__bit1,axiom,
! [N: nat,K2: num] :
( ( bit_ri4674362597316999326ke_bit @ int @ ( suc @ N ) @ ( numeral_numeral @ int @ ( bit1 @ K2 ) ) )
= ( plus_plus @ int @ ( times_times @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N @ ( numeral_numeral @ int @ K2 ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( one_one @ int ) ) ) ).
% signed_take_bit_Suc_bit1
thf(fact_1795_take__bit__rec,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se2584673776208193580ke_bit @ A )
= ( ^ [N2: nat,A5: A] :
( if @ A
@ ( N2
= ( zero_zero @ nat ) )
@ ( zero_zero @ A )
@ ( plus_plus @ A @ ( times_times @ A @ ( bit_se2584673776208193580ke_bit @ A @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) @ ( divide_divide @ A @ A5 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ A @ A5 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% take_bit_rec
thf(fact_1796_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_1797_add_Oinverse__inverse,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A] :
( ( uminus_uminus @ A @ ( uminus_uminus @ A @ A2 ) )
= A2 ) ) ).
% add.inverse_inverse
thf(fact_1798_neg__equal__iff__equal,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A] :
( ( ( uminus_uminus @ A @ A2 )
= ( uminus_uminus @ A @ B2 ) )
= ( A2 = B2 ) ) ) ).
% neg_equal_iff_equal
thf(fact_1799_Compl__subset__Compl__iff,axiom,
! [A: $tType,A3: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A3 ) @ ( uminus_uminus @ ( set @ A ) @ B5 ) )
= ( ord_less_eq @ ( set @ A ) @ B5 @ A3 ) ) ).
% Compl_subset_Compl_iff
thf(fact_1800_Compl__anti__mono,axiom,
! [A: $tType,A3: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B5 )
=> ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ B5 ) @ ( uminus_uminus @ ( set @ A ) @ A3 ) ) ) ).
% Compl_anti_mono
thf(fact_1801_verit__eq__simplify_I9_J,axiom,
! [X32: num,Y32: num] :
( ( ( bit1 @ X32 )
= ( bit1 @ Y32 ) )
= ( X32 = Y32 ) ) ).
% verit_eq_simplify(9)
thf(fact_1802_semiring__norm_I90_J,axiom,
! [M: num,N: num] :
( ( ( bit1 @ M )
= ( bit1 @ N ) )
= ( M = N ) ) ).
% semiring_norm(90)
thf(fact_1803_neg__le__iff__le,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [B2: A,A2: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A2 ) )
= ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% neg_le_iff_le
thf(fact_1804_add_Oinverse__neutral,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ( ( uminus_uminus @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% add.inverse_neutral
thf(fact_1805_neg__0__equal__iff__equal,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A] :
( ( ( zero_zero @ A )
= ( uminus_uminus @ A @ A2 ) )
= ( ( zero_zero @ A )
= A2 ) ) ) ).
% neg_0_equal_iff_equal
thf(fact_1806_neg__equal__0__iff__equal,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A] :
( ( ( uminus_uminus @ A @ A2 )
= ( zero_zero @ A ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% neg_equal_0_iff_equal
thf(fact_1807_equal__neg__zero,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A2: A] :
( ( A2
= ( uminus_uminus @ A @ A2 ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% equal_neg_zero
thf(fact_1808_neg__equal__zero,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A2: A] :
( ( ( uminus_uminus @ A @ A2 )
= A2 )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% neg_equal_zero
thf(fact_1809_neg__less__iff__less,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [B2: A,A2: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A2 ) )
= ( ord_less @ A @ A2 @ B2 ) ) ) ).
% neg_less_iff_less
thf(fact_1810_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_1811_mult__minus__right,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A2: A,B2: A] :
( ( times_times @ A @ A2 @ ( uminus_uminus @ A @ B2 ) )
= ( uminus_uminus @ A @ ( times_times @ A @ A2 @ B2 ) ) ) ) ).
% mult_minus_right
thf(fact_1812_minus__mult__minus,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A2: A,B2: A] :
( ( times_times @ A @ ( uminus_uminus @ A @ A2 ) @ ( uminus_uminus @ A @ B2 ) )
= ( times_times @ A @ A2 @ B2 ) ) ) ).
% minus_mult_minus
thf(fact_1813_mult__minus__left,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A2: A,B2: A] :
( ( times_times @ A @ ( uminus_uminus @ A @ A2 ) @ B2 )
= ( uminus_uminus @ A @ ( times_times @ A @ A2 @ B2 ) ) ) ) ).
% mult_minus_left
thf(fact_1814_minus__add__distrib,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A2: A,B2: A] :
( ( uminus_uminus @ A @ ( plus_plus @ A @ A2 @ B2 ) )
= ( plus_plus @ A @ ( uminus_uminus @ A @ A2 ) @ ( uminus_uminus @ A @ B2 ) ) ) ) ).
% minus_add_distrib
thf(fact_1815_minus__add__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ A2 ) @ ( plus_plus @ A @ A2 @ B2 ) )
= B2 ) ) ).
% minus_add_cancel
thf(fact_1816_add__minus__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A] :
( ( plus_plus @ A @ A2 @ ( plus_plus @ A @ ( uminus_uminus @ A @ A2 ) @ B2 ) )
= B2 ) ) ).
% add_minus_cancel
thf(fact_1817_minus__diff__eq,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A] :
( ( uminus_uminus @ A @ ( minus_minus @ A @ A2 @ B2 ) )
= ( minus_minus @ A @ B2 @ A2 ) ) ) ).
% minus_diff_eq
thf(fact_1818_div__minus__minus,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel @ A )
=> ! [A2: A,B2: A] :
( ( divide_divide @ A @ ( uminus_uminus @ A @ A2 ) @ ( uminus_uminus @ A @ B2 ) )
= ( divide_divide @ A @ A2 @ B2 ) ) ) ).
% div_minus_minus
thf(fact_1819_semiring__norm_I88_J,axiom,
! [M: num,N: num] :
( ( bit0 @ M )
!= ( bit1 @ N ) ) ).
% semiring_norm(88)
thf(fact_1820_semiring__norm_I89_J,axiom,
! [M: num,N: num] :
( ( bit1 @ M )
!= ( bit0 @ N ) ) ).
% semiring_norm(89)
thf(fact_1821_semiring__norm_I84_J,axiom,
! [N: num] :
( one2
!= ( bit1 @ N ) ) ).
% semiring_norm(84)
thf(fact_1822_semiring__norm_I86_J,axiom,
! [M: num] :
( ( bit1 @ M )
!= one2 ) ).
% semiring_norm(86)
thf(fact_1823_mod__minus__minus,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel @ A )
=> ! [A2: A,B2: A] :
( ( modulo_modulo @ A @ ( uminus_uminus @ A @ A2 ) @ ( uminus_uminus @ A @ B2 ) )
= ( uminus_uminus @ A @ ( modulo_modulo @ A @ A2 @ B2 ) ) ) ) ).
% mod_minus_minus
thf(fact_1824_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_1825_real__add__minus__iff,axiom,
! [X: real,A2: real] :
( ( ( plus_plus @ real @ X @ ( uminus_uminus @ real @ A2 ) )
= ( zero_zero @ real ) )
= ( X = A2 ) ) ).
% real_add_minus_iff
thf(fact_1826_concat__bit__of__zero__2,axiom,
! [N: nat,K2: int] :
( ( bit_concat_bit @ N @ K2 @ ( zero_zero @ int ) )
= ( bit_se2584673776208193580ke_bit @ int @ N @ K2 ) ) ).
% concat_bit_of_zero_2
thf(fact_1827_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_1828_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_1829_neg__0__le__iff__le,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ A2 ) )
= ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% neg_0_le_iff_le
thf(fact_1830_neg__le__0__iff__le,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A2 ) @ ( zero_zero @ A ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% neg_le_0_iff_le
thf(fact_1831_less__eq__neg__nonpos,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ A2 @ ( uminus_uminus @ A @ A2 ) )
= ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% less_eq_neg_nonpos
thf(fact_1832_neg__less__eq__nonneg,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A2 ) @ A2 )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% neg_less_eq_nonneg
thf(fact_1833_neg__less__0__iff__less,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ A2 ) @ ( zero_zero @ A ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% neg_less_0_iff_less
thf(fact_1834_neg__0__less__iff__less,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ A2 ) )
= ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% neg_0_less_iff_less
thf(fact_1835_neg__less__pos,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A2: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ A2 ) @ A2 )
= ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% neg_less_pos
thf(fact_1836_less__neg__neg,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A2: A] :
( ( ord_less @ A @ A2 @ ( uminus_uminus @ A @ A2 ) )
= ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% less_neg_neg
thf(fact_1837_ab__left__minus,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ A2 ) @ A2 )
= ( zero_zero @ A ) ) ) ).
% ab_left_minus
thf(fact_1838_add_Oright__inverse,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A] :
( ( plus_plus @ A @ A2 @ ( uminus_uminus @ A @ A2 ) )
= ( zero_zero @ A ) ) ) ).
% add.right_inverse
thf(fact_1839_diff__0,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A] :
( ( minus_minus @ A @ ( zero_zero @ A ) @ A2 )
= ( uminus_uminus @ A @ A2 ) ) ) ).
% diff_0
thf(fact_1840_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_1841_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_1842_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_1843_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_1844_diff__minus__eq__add,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A] :
( ( minus_minus @ A @ A2 @ ( uminus_uminus @ A @ B2 ) )
= ( plus_plus @ A @ A2 @ B2 ) ) ) ).
% diff_minus_eq_add
thf(fact_1845_uminus__add__conv__diff,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A2: A,B2: A] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ A2 ) @ B2 )
= ( minus_minus @ A @ B2 @ A2 ) ) ) ).
% uminus_add_conv_diff
thf(fact_1846_div__minus1__right,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel @ A )
=> ! [A2: A] :
( ( divide_divide @ A @ A2 @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( uminus_uminus @ A @ A2 ) ) ) ).
% div_minus1_right
thf(fact_1847_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_1848_minus__mod__self1,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel @ A )
=> ! [B2: A,A2: A] :
( ( modulo_modulo @ A @ ( minus_minus @ A @ B2 @ A2 ) @ B2 )
= ( modulo_modulo @ A @ ( uminus_uminus @ A @ A2 ) @ B2 ) ) ) ).
% minus_mod_self1
thf(fact_1849_take__bit__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A] :
( ( bit_se2584673776208193580ke_bit @ A @ ( zero_zero @ nat ) @ A2 )
= ( zero_zero @ A ) ) ) ).
% take_bit_0
thf(fact_1850_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_1851_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_1852_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_1853_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_1854_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_1855_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_1856_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_1857_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_1858_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_1859_semiring__norm_I70_J,axiom,
! [M: num] :
~ ( ord_less_eq @ num @ ( bit1 @ M ) @ one2 ) ).
% semiring_norm(70)
thf(fact_1860_semiring__norm_I77_J,axiom,
! [N: num] : ( ord_less @ num @ one2 @ ( bit1 @ N ) ) ).
% semiring_norm(77)
thf(fact_1861_dbl__simps_I1_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [K2: num] :
( ( neg_numeral_dbl @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ K2 ) ) )
= ( uminus_uminus @ A @ ( neg_numeral_dbl @ A @ ( numeral_numeral @ A @ K2 ) ) ) ) ) ).
% dbl_simps(1)
thf(fact_1862_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_1863_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_1864_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_1865_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_1866_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_1867_left__minus__one__mult__self,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [N: nat,A2: A] :
( ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N ) @ ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N ) @ A2 ) )
= A2 ) ) ).
% left_minus_one_mult_self
thf(fact_1868_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_1869_mod__minus1__right,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel @ A )
=> ! [A2: A] :
( ( modulo_modulo @ A @ A2 @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( zero_zero @ A ) ) ) ).
% mod_minus1_right
thf(fact_1870_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_1871_semiring__norm_I168_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [V2: num,W: num,Y2: A] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) @ ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ Y2 ) )
= ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( plus_plus @ num @ V2 @ W ) ) ) @ Y2 ) ) ) ).
% semiring_norm(168)
thf(fact_1872_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_1873_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_1874_zdiv__numeral__Bit1,axiom,
! [V2: num,W: num] :
( ( divide_divide @ int @ ( numeral_numeral @ int @ ( bit1 @ V2 ) ) @ ( numeral_numeral @ int @ ( bit0 @ W ) ) )
= ( divide_divide @ int @ ( numeral_numeral @ int @ V2 ) @ ( numeral_numeral @ int @ W ) ) ) ).
% zdiv_numeral_Bit1
thf(fact_1875_semiring__norm_I3_J,axiom,
! [N: num] :
( ( plus_plus @ num @ one2 @ ( bit0 @ N ) )
= ( bit1 @ N ) ) ).
% semiring_norm(3)
thf(fact_1876_semiring__norm_I4_J,axiom,
! [N: num] :
( ( plus_plus @ num @ one2 @ ( bit1 @ N ) )
= ( bit0 @ ( plus_plus @ num @ N @ one2 ) ) ) ).
% semiring_norm(4)
thf(fact_1877_semiring__norm_I5_J,axiom,
! [M: num] :
( ( plus_plus @ num @ ( bit0 @ M ) @ one2 )
= ( bit1 @ M ) ) ).
% semiring_norm(5)
thf(fact_1878_semiring__norm_I8_J,axiom,
! [M: num] :
( ( plus_plus @ num @ ( bit1 @ M ) @ one2 )
= ( bit0 @ ( plus_plus @ num @ M @ one2 ) ) ) ).
% semiring_norm(8)
thf(fact_1879_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_1880_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_1881_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_1882_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_1883_semiring__norm_I170_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [V2: num,W: num,Y2: A] :
( ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ W ) @ Y2 ) )
= ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( times_times @ num @ V2 @ W ) ) ) @ Y2 ) ) ) ).
% semiring_norm(170)
thf(fact_1884_semiring__norm_I171_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [V2: num,W: num,Y2: A] :
( ( times_times @ A @ ( numeral_numeral @ A @ V2 ) @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ Y2 ) )
= ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( times_times @ num @ V2 @ W ) ) ) @ Y2 ) ) ) ).
% semiring_norm(171)
thf(fact_1885_semiring__norm_I172_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [V2: num,W: num,Y2: A] :
( ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ Y2 ) )
= ( times_times @ A @ ( numeral_numeral @ A @ ( times_times @ num @ V2 @ W ) ) @ Y2 ) ) ) ).
% semiring_norm(172)
thf(fact_1886_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_1887_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_1888_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_1889_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_1890_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_1891_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_1892_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_1893_le__divide__eq__numeral1_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A,W: num] :
( ( ord_less_eq @ A @ A2 @ ( divide_divide @ A @ B2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) ) )
= ( ord_less_eq @ A @ B2 @ ( times_times @ A @ A2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) ) ) ) ) ).
% le_divide_eq_numeral1(2)
thf(fact_1894_divide__le__eq__numeral1_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,W: num,A2: A] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) ) @ A2 )
= ( ord_less_eq @ A @ ( times_times @ A @ A2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) ) @ B2 ) ) ) ).
% divide_le_eq_numeral1(2)
thf(fact_1895_eq__divide__eq__numeral1_I2_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A,B2: A,W: num] :
( ( A2
= ( divide_divide @ A @ B2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) ) )
= ( ( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) )
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ A2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) )
= B2 ) )
& ( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) )
= ( zero_zero @ A ) )
=> ( A2
= ( zero_zero @ A ) ) ) ) ) ) ).
% eq_divide_eq_numeral1(2)
thf(fact_1896_divide__eq__eq__numeral1_I2_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B2: A,W: num,A2: A] :
( ( ( divide_divide @ A @ B2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) )
= A2 )
= ( ( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) )
!= ( zero_zero @ A ) )
=> ( B2
= ( times_times @ A @ A2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) ) ) )
& ( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) )
= ( zero_zero @ A ) )
=> ( A2
= ( zero_zero @ A ) ) ) ) ) ) ).
% divide_eq_eq_numeral1(2)
thf(fact_1897_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_1898_less__divide__eq__numeral1_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A,W: num] :
( ( ord_less @ A @ A2 @ ( divide_divide @ A @ B2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) ) )
= ( ord_less @ A @ B2 @ ( times_times @ A @ A2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) ) ) ) ) ).
% less_divide_eq_numeral1(2)
thf(fact_1899_divide__less__eq__numeral1_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,W: num,A2: A] :
( ( ord_less @ A @ ( divide_divide @ A @ B2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) ) @ A2 )
= ( ord_less @ A @ ( times_times @ A @ A2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) ) @ B2 ) ) ) ).
% divide_less_eq_numeral1(2)
thf(fact_1900_power2__minus,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [A2: A] :
( ( power_power @ A @ ( uminus_uminus @ A @ A2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% power2_minus
thf(fact_1901_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_1902_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_1903_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_1904_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_1905_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_1906_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_1907_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_1908_Power_Oring__1__class_Opower__minus__even,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [A2: A,N: nat] :
( ( power_power @ A @ ( uminus_uminus @ A @ A2 ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
= ( power_power @ A @ A2 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% Power.ring_1_class.power_minus_even
thf(fact_1909_Parity_Oring__1__class_Opower__minus__even,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [N: nat,A2: A] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( power_power @ A @ ( uminus_uminus @ A @ A2 ) @ N )
= ( power_power @ A @ A2 @ N ) ) ) ) ).
% Parity.ring_1_class.power_minus_even
thf(fact_1910_power__minus__odd,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [N: nat,A2: A] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( power_power @ A @ ( uminus_uminus @ A @ A2 ) @ N )
= ( uminus_uminus @ A @ ( power_power @ A @ A2 @ N ) ) ) ) ) ).
% power_minus_odd
thf(fact_1911_even__take__bit__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A2: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se2584673776208193580ke_bit @ A @ N @ A2 ) )
= ( ( N
= ( zero_zero @ nat ) )
| ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) ) ) ) ).
% even_take_bit_eq
thf(fact_1912_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_1913_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_1914_Suc__div__eq__add3__div__numeral,axiom,
! [M: nat,V2: num] :
( ( divide_divide @ nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ ( numeral_numeral @ nat @ V2 ) )
= ( divide_divide @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) @ M ) @ ( numeral_numeral @ nat @ V2 ) ) ) ).
% Suc_div_eq_add3_div_numeral
thf(fact_1915_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_1916_Suc__mod__eq__add3__mod__numeral,axiom,
! [M: nat,V2: num] :
( ( modulo_modulo @ nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ ( numeral_numeral @ nat @ V2 ) )
= ( modulo_modulo @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) @ M ) @ ( numeral_numeral @ nat @ V2 ) ) ) ).
% Suc_mod_eq_add3_mod_numeral
thf(fact_1917_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_1918_signed__take__bit__Suc__minus__bit0,axiom,
! [N: nat,K2: num] :
( ( bit_ri4674362597316999326ke_bit @ int @ ( suc @ N ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ K2 ) ) ) )
= ( times_times @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K2 ) ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ).
% signed_take_bit_Suc_minus_bit0
thf(fact_1919_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_1920_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_1921_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_1922_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_1923_take__bit__Suc__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A] :
( ( bit_se2584673776208193580ke_bit @ A @ ( suc @ ( zero_zero @ nat ) ) @ A2 )
= ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% take_bit_Suc_0
thf(fact_1924_signed__take__bit__0,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [A2: A] :
( ( bit_ri4674362597316999326ke_bit @ A @ ( zero_zero @ nat ) @ A2 )
= ( uminus_uminus @ A @ ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% signed_take_bit_0
thf(fact_1925_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_1926_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_1927_zmod__numeral__Bit1,axiom,
! [V2: num,W: num] :
( ( modulo_modulo @ int @ ( numeral_numeral @ int @ ( bit1 @ V2 ) ) @ ( numeral_numeral @ int @ ( bit0 @ W ) ) )
= ( plus_plus @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( modulo_modulo @ int @ ( numeral_numeral @ int @ V2 ) @ ( numeral_numeral @ int @ W ) ) ) @ ( one_one @ int ) ) ) ).
% zmod_numeral_Bit1
thf(fact_1928_signed__take__bit__Suc__minus__bit1,axiom,
! [N: nat,K2: num] :
( ( bit_ri4674362597316999326ke_bit @ int @ ( suc @ N ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ K2 ) ) ) )
= ( plus_plus @ int @ ( times_times @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N @ ( minus_minus @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K2 ) ) @ ( one_one @ int ) ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( one_one @ int ) ) ) ).
% signed_take_bit_Suc_minus_bit1
thf(fact_1929_minus__set__def,axiom,
! [A: $tType] :
( ( minus_minus @ ( set @ A ) )
= ( ^ [A7: set @ A,B7: set @ A] :
( collect @ A
@ ( minus_minus @ ( A > $o )
@ ^ [X4: A] : ( member @ A @ X4 @ A7 )
@ ^ [X4: A] : ( member @ A @ X4 @ B7 ) ) ) ) ) ).
% minus_set_def
thf(fact_1930_set__diff__eq,axiom,
! [A: $tType] :
( ( minus_minus @ ( set @ A ) )
= ( ^ [A7: set @ A,B7: set @ A] :
( collect @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ A7 )
& ~ ( member @ A @ X4 @ B7 ) ) ) ) ) ).
% set_diff_eq
thf(fact_1931_verit__negate__coefficient_I3_J,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B2: A] :
( ( A2 = B2 )
=> ( ( uminus_uminus @ A @ A2 )
= ( uminus_uminus @ A @ B2 ) ) ) ) ).
% verit_negate_coefficient(3)
thf(fact_1932_equation__minus__iff,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A] :
( ( A2
= ( uminus_uminus @ A @ B2 ) )
= ( B2
= ( uminus_uminus @ A @ A2 ) ) ) ) ).
% equation_minus_iff
thf(fact_1933_minus__equation__iff,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A] :
( ( ( uminus_uminus @ A @ A2 )
= B2 )
= ( ( uminus_uminus @ A @ B2 )
= A2 ) ) ) ).
% minus_equation_iff
thf(fact_1934_power__minus__Bit1,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X: A,K2: num] :
( ( power_power @ A @ ( uminus_uminus @ A @ X ) @ ( numeral_numeral @ nat @ ( bit1 @ K2 ) ) )
= ( uminus_uminus @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit1 @ K2 ) ) ) ) ) ) ).
% power_minus_Bit1
thf(fact_1935_take__bit__add,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [N: nat,A2: A,B2: A] :
( ( bit_se2584673776208193580ke_bit @ A @ N @ ( plus_plus @ A @ ( bit_se2584673776208193580ke_bit @ A @ N @ A2 ) @ ( bit_se2584673776208193580ke_bit @ A @ N @ B2 ) ) )
= ( bit_se2584673776208193580ke_bit @ A @ N @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ).
% take_bit_add
thf(fact_1936_take__bit__tightened,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A2: A,B2: A,M: nat] :
( ( ( bit_se2584673776208193580ke_bit @ A @ N @ A2 )
= ( bit_se2584673776208193580ke_bit @ A @ N @ B2 ) )
=> ( ( ord_less_eq @ nat @ M @ N )
=> ( ( bit_se2584673776208193580ke_bit @ A @ M @ A2 )
= ( bit_se2584673776208193580ke_bit @ A @ M @ B2 ) ) ) ) ) ).
% take_bit_tightened
thf(fact_1937_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_1938_take__bit__tightened__less__eq__nat,axiom,
! [M: nat,N: nat,Q: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ord_less_eq @ nat @ ( bit_se2584673776208193580ke_bit @ nat @ M @ Q ) @ ( bit_se2584673776208193580ke_bit @ nat @ N @ Q ) ) ) ).
% take_bit_tightened_less_eq_nat
thf(fact_1939_take__bit__mult,axiom,
! [N: nat,K2: int,L: int] :
( ( bit_se2584673776208193580ke_bit @ int @ N @ ( times_times @ int @ ( bit_se2584673776208193580ke_bit @ int @ N @ K2 ) @ ( bit_se2584673776208193580ke_bit @ int @ N @ L ) ) )
= ( bit_se2584673776208193580ke_bit @ int @ N @ ( times_times @ int @ K2 @ L ) ) ) ).
% take_bit_mult
thf(fact_1940_le__imp__neg__le,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A2 ) ) ) ) ).
% le_imp_neg_le
thf(fact_1941_minus__le__iff,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A2 ) @ B2 )
= ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ A2 ) ) ) ).
% minus_le_iff
thf(fact_1942_le__minus__iff,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ ( uminus_uminus @ A @ B2 ) )
= ( ord_less_eq @ A @ B2 @ ( uminus_uminus @ A @ A2 ) ) ) ) ).
% le_minus_iff
thf(fact_1943_minus__less__iff,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ A2 ) @ B2 )
= ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ A2 ) ) ) ).
% minus_less_iff
thf(fact_1944_less__minus__iff,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ ( uminus_uminus @ A @ B2 ) )
= ( ord_less @ A @ B2 @ ( uminus_uminus @ A @ A2 ) ) ) ) ).
% less_minus_iff
thf(fact_1945_verit__negate__coefficient_I2_J,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A2 ) ) ) ) ).
% verit_negate_coefficient(2)
thf(fact_1946_verit__eq__simplify_I14_J,axiom,
! [X23: num,X32: num] :
( ( bit0 @ X23 )
!= ( bit1 @ X32 ) ) ).
% verit_eq_simplify(14)
thf(fact_1947_verit__eq__simplify_I12_J,axiom,
! [X32: num] :
( one2
!= ( bit1 @ X32 ) ) ).
% verit_eq_simplify(12)
thf(fact_1948_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_1949_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_1950_minus__mult__commute,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A2: A,B2: A] :
( ( times_times @ A @ ( uminus_uminus @ A @ A2 ) @ B2 )
= ( times_times @ A @ A2 @ ( uminus_uminus @ A @ B2 ) ) ) ) ).
% minus_mult_commute
thf(fact_1951_square__eq__iff,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [A2: A,B2: A] :
( ( ( times_times @ A @ A2 @ A2 )
= ( times_times @ A @ B2 @ B2 ) )
= ( ( A2 = B2 )
| ( A2
= ( uminus_uminus @ A @ B2 ) ) ) ) ) ).
% square_eq_iff
thf(fact_1952_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_1953_add_Oinverse__distrib__swap,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A] :
( ( uminus_uminus @ A @ ( plus_plus @ A @ A2 @ B2 ) )
= ( plus_plus @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A2 ) ) ) ) ).
% add.inverse_distrib_swap
thf(fact_1954_group__cancel_Oneg1,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A3: A,K2: A,A2: A] :
( ( A3
= ( plus_plus @ A @ K2 @ A2 ) )
=> ( ( uminus_uminus @ A @ A3 )
= ( plus_plus @ A @ ( uminus_uminus @ A @ K2 ) @ ( uminus_uminus @ A @ A2 ) ) ) ) ) ).
% group_cancel.neg1
thf(fact_1955_is__num__normalize_I8_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [A2: A,B2: A] :
( ( uminus_uminus @ A @ ( plus_plus @ A @ A2 @ B2 ) )
= ( plus_plus @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A2 ) ) ) ) ).
% is_num_normalize(8)
thf(fact_1956_take__bit__diff,axiom,
! [N: nat,K2: int,L: int] :
( ( bit_se2584673776208193580ke_bit @ int @ N @ ( minus_minus @ int @ ( bit_se2584673776208193580ke_bit @ int @ N @ K2 ) @ ( bit_se2584673776208193580ke_bit @ int @ N @ L ) ) )
= ( bit_se2584673776208193580ke_bit @ int @ N @ ( minus_minus @ int @ K2 @ L ) ) ) ).
% take_bit_diff
thf(fact_1957_minus__diff__minus,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A2: A,B2: A] :
( ( minus_minus @ A @ ( uminus_uminus @ A @ A2 ) @ ( uminus_uminus @ A @ B2 ) )
= ( uminus_uminus @ A @ ( minus_minus @ A @ A2 @ B2 ) ) ) ) ).
% minus_diff_minus
thf(fact_1958_minus__diff__commute,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [B2: A,A2: A] :
( ( minus_minus @ A @ ( uminus_uminus @ A @ B2 ) @ A2 )
= ( minus_minus @ A @ ( uminus_uminus @ A @ A2 ) @ B2 ) ) ) ).
% minus_diff_commute
thf(fact_1959_div__minus__right,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel @ A )
=> ! [A2: A,B2: A] :
( ( divide_divide @ A @ A2 @ ( uminus_uminus @ A @ B2 ) )
= ( divide_divide @ A @ ( uminus_uminus @ A @ A2 ) @ B2 ) ) ) ).
% div_minus_right
thf(fact_1960_minus__divide__left,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A,B2: A] :
( ( uminus_uminus @ A @ ( divide_divide @ A @ A2 @ B2 ) )
= ( divide_divide @ A @ ( uminus_uminus @ A @ A2 ) @ B2 ) ) ) ).
% minus_divide_left
thf(fact_1961_minus__divide__divide,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A2: A,B2: A] :
( ( divide_divide @ A @ ( uminus_uminus @ A @ A2 ) @ ( uminus_uminus @ A @ B2 ) )
= ( divide_divide @ A @ A2 @ B2 ) ) ) ).
% minus_divide_divide
thf(fact_1962_minus__divide__right,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A2: A,B2: A] :
( ( uminus_uminus @ A @ ( divide_divide @ A @ A2 @ B2 ) )
= ( divide_divide @ A @ A2 @ ( uminus_uminus @ A @ B2 ) ) ) ) ).
% minus_divide_right
thf(fact_1963_mod__minus__eq,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel @ A )
=> ! [A2: A,B2: A] :
( ( modulo_modulo @ A @ ( uminus_uminus @ A @ ( modulo_modulo @ A @ A2 @ B2 ) ) @ B2 )
= ( modulo_modulo @ A @ ( uminus_uminus @ A @ A2 ) @ B2 ) ) ) ).
% mod_minus_eq
thf(fact_1964_mod__minus__cong,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel @ A )
=> ! [A2: A,B2: A,A6: A] :
( ( ( modulo_modulo @ A @ A2 @ B2 )
= ( modulo_modulo @ A @ A6 @ B2 ) )
=> ( ( modulo_modulo @ A @ ( uminus_uminus @ A @ A2 ) @ B2 )
= ( modulo_modulo @ A @ ( uminus_uminus @ A @ A6 ) @ B2 ) ) ) ) ).
% mod_minus_cong
thf(fact_1965_mod__minus__right,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel @ A )
=> ! [A2: A,B2: A] :
( ( modulo_modulo @ A @ A2 @ ( uminus_uminus @ A @ B2 ) )
= ( uminus_uminus @ A @ ( modulo_modulo @ A @ ( uminus_uminus @ A @ A2 ) @ B2 ) ) ) ) ).
% mod_minus_right
thf(fact_1966_minus__real__def,axiom,
( ( minus_minus @ real )
= ( ^ [X4: real,Y: real] : ( plus_plus @ real @ X4 @ ( uminus_uminus @ real @ Y ) ) ) ) ).
% minus_real_def
thf(fact_1967_concat__bit__eq__iff,axiom,
! [N: nat,K2: int,L: int,R: int,S: int] :
( ( ( bit_concat_bit @ N @ K2 @ L )
= ( bit_concat_bit @ N @ R @ S ) )
= ( ( ( bit_se2584673776208193580ke_bit @ int @ N @ K2 )
= ( bit_se2584673776208193580ke_bit @ int @ N @ R ) )
& ( L = S ) ) ) ).
% concat_bit_eq_iff
thf(fact_1968_concat__bit__take__bit__eq,axiom,
! [N: nat,B2: int] :
( ( bit_concat_bit @ N @ ( bit_se2584673776208193580ke_bit @ int @ N @ B2 ) )
= ( bit_concat_bit @ N @ B2 ) ) ).
% concat_bit_take_bit_eq
thf(fact_1969_signed__take__bit__minus,axiom,
! [N: nat,K2: int] :
( ( bit_ri4674362597316999326ke_bit @ int @ N @ ( uminus_uminus @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N @ K2 ) ) )
= ( bit_ri4674362597316999326ke_bit @ int @ N @ ( uminus_uminus @ int @ K2 ) ) ) ).
% signed_take_bit_minus
thf(fact_1970_take__bit__tightened__less__eq__int,axiom,
! [M: nat,N: nat,K2: int] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ord_less_eq @ int @ ( bit_se2584673776208193580ke_bit @ int @ M @ K2 ) @ ( bit_se2584673776208193580ke_bit @ int @ N @ K2 ) ) ) ).
% take_bit_tightened_less_eq_int
thf(fact_1971_take__bit__int__less__eq__self__iff,axiom,
! [N: nat,K2: int] :
( ( ord_less_eq @ int @ ( bit_se2584673776208193580ke_bit @ int @ N @ K2 ) @ K2 )
= ( ord_less_eq @ int @ ( zero_zero @ int ) @ K2 ) ) ).
% take_bit_int_less_eq_self_iff
thf(fact_1972_take__bit__nonnegative,axiom,
! [N: nat,K2: int] : ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se2584673776208193580ke_bit @ int @ N @ K2 ) ) ).
% take_bit_nonnegative
thf(fact_1973_take__bit__int__greater__self__iff,axiom,
! [K2: int,N: nat] :
( ( ord_less @ int @ K2 @ ( bit_se2584673776208193580ke_bit @ int @ N @ K2 ) )
= ( ord_less @ int @ K2 @ ( zero_zero @ int ) ) ) ).
% take_bit_int_greater_self_iff
thf(fact_1974_not__take__bit__negative,axiom,
! [N: nat,K2: int] :
~ ( ord_less @ int @ ( bit_se2584673776208193580ke_bit @ int @ N @ K2 ) @ ( zero_zero @ int ) ) ).
% not_take_bit_negative
thf(fact_1975_signed__take__bit__eq__iff__take__bit__eq,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: nat,A2: A,B2: A] :
( ( ( bit_ri4674362597316999326ke_bit @ A @ N @ A2 )
= ( bit_ri4674362597316999326ke_bit @ A @ N @ B2 ) )
= ( ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N ) @ A2 )
= ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N ) @ B2 ) ) ) ) ).
% signed_take_bit_eq_iff_take_bit_eq
thf(fact_1976_signed__take__bit__take__bit,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [M: nat,N: nat,A2: A] :
( ( bit_ri4674362597316999326ke_bit @ A @ M @ ( bit_se2584673776208193580ke_bit @ A @ N @ A2 ) )
= ( if @ ( A > A ) @ ( ord_less_eq @ nat @ N @ M ) @ ( bit_se2584673776208193580ke_bit @ A @ N ) @ ( bit_ri4674362597316999326ke_bit @ A @ M ) @ A2 ) ) ) ).
% signed_take_bit_take_bit
thf(fact_1977_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_1978_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_1979_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_1980_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_1981_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_1982_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_1983_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_1984_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_1985_neg__eq__iff__add__eq__0,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A] :
( ( ( uminus_uminus @ A @ A2 )
= B2 )
= ( ( plus_plus @ A @ A2 @ B2 )
= ( zero_zero @ A ) ) ) ) ).
% neg_eq_iff_add_eq_0
thf(fact_1986_eq__neg__iff__add__eq__0,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A] :
( ( A2
= ( uminus_uminus @ A @ B2 ) )
= ( ( plus_plus @ A @ A2 @ B2 )
= ( zero_zero @ A ) ) ) ) ).
% eq_neg_iff_add_eq_0
thf(fact_1987_add_Oinverse__unique,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A] :
( ( ( plus_plus @ A @ A2 @ B2 )
= ( zero_zero @ A ) )
=> ( ( uminus_uminus @ A @ A2 )
= B2 ) ) ) ).
% add.inverse_unique
thf(fact_1988_ab__group__add__class_Oab__left__minus,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A2: A] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ A2 ) @ A2 )
= ( zero_zero @ A ) ) ) ).
% ab_group_add_class.ab_left_minus
thf(fact_1989_add__eq__0__iff,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A] :
( ( ( plus_plus @ A @ A2 @ B2 )
= ( zero_zero @ A ) )
= ( B2
= ( uminus_uminus @ A @ A2 ) ) ) ) ).
% add_eq_0_iff
thf(fact_1990_xor__num_Ocases,axiom,
! [X: product_prod @ num @ num] :
( ( X
!= ( product_Pair @ num @ num @ one2 @ one2 ) )
=> ( ! [N3: num] :
( X
!= ( product_Pair @ num @ num @ one2 @ ( bit0 @ N3 ) ) )
=> ( ! [N3: num] :
( X
!= ( product_Pair @ num @ num @ one2 @ ( bit1 @ N3 ) ) )
=> ( ! [M4: num] :
( X
!= ( product_Pair @ num @ num @ ( bit0 @ M4 ) @ one2 ) )
=> ( ! [M4: num,N3: num] :
( X
!= ( product_Pair @ num @ num @ ( bit0 @ M4 ) @ ( bit0 @ N3 ) ) )
=> ( ! [M4: num,N3: num] :
( X
!= ( product_Pair @ num @ num @ ( bit0 @ M4 ) @ ( bit1 @ N3 ) ) )
=> ( ! [M4: num] :
( X
!= ( product_Pair @ num @ num @ ( bit1 @ M4 ) @ one2 ) )
=> ( ! [M4: num,N3: num] :
( X
!= ( product_Pair @ num @ num @ ( bit1 @ M4 ) @ ( bit0 @ N3 ) ) )
=> ~ ! [M4: num,N3: num] :
( X
!= ( product_Pair @ num @ num @ ( bit1 @ M4 ) @ ( bit1 @ N3 ) ) ) ) ) ) ) ) ) ) ) ).
% xor_num.cases
thf(fact_1991_num_Oexhaust,axiom,
! [Y2: num] :
( ( Y2 != one2 )
=> ( ! [X24: num] :
( Y2
!= ( bit0 @ X24 ) )
=> ~ ! [X33: num] :
( Y2
!= ( bit1 @ X33 ) ) ) ) ).
% num.exhaust
thf(fact_1992_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_1993_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_1994_numeral__times__minus__swap,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [W: num,X: A] :
( ( times_times @ A @ ( numeral_numeral @ A @ W ) @ ( uminus_uminus @ A @ X ) )
= ( times_times @ A @ X @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) ) ) ) ).
% numeral_times_minus_swap
thf(fact_1995_nonzero__minus__divide__divide,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B2: A,A2: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( uminus_uminus @ A @ A2 ) @ ( uminus_uminus @ A @ B2 ) )
= ( divide_divide @ A @ A2 @ B2 ) ) ) ) ).
% nonzero_minus_divide_divide
thf(fact_1996_nonzero__minus__divide__right,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B2: A,A2: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( uminus_uminus @ A @ ( divide_divide @ A @ A2 @ B2 ) )
= ( divide_divide @ A @ A2 @ ( uminus_uminus @ A @ B2 ) ) ) ) ) ).
% nonzero_minus_divide_right
thf(fact_1997_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_1998_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_1999_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_2000_group__cancel_Osub2,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [B5: A,K2: A,B2: A,A2: A] :
( ( B5
= ( plus_plus @ A @ K2 @ B2 ) )
=> ( ( minus_minus @ A @ A2 @ B5 )
= ( plus_plus @ A @ ( uminus_uminus @ A @ K2 ) @ ( minus_minus @ A @ A2 @ B2 ) ) ) ) ) ).
% group_cancel.sub2
thf(fact_2001_diff__conv__add__uminus,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ( ( minus_minus @ A )
= ( ^ [A5: A,B4: A] : ( plus_plus @ A @ A5 @ ( uminus_uminus @ A @ B4 ) ) ) ) ) ).
% diff_conv_add_uminus
thf(fact_2002_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ( ( minus_minus @ A )
= ( ^ [A5: A,B4: A] : ( plus_plus @ A @ A5 @ ( uminus_uminus @ A @ B4 ) ) ) ) ) ).
% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_2003_take__bit__unset__bit__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,M: nat,A2: A] :
( ( ( ord_less_eq @ nat @ N @ M )
=> ( ( bit_se2584673776208193580ke_bit @ A @ N @ ( bit_se2638667681897837118et_bit @ A @ M @ A2 ) )
= ( bit_se2584673776208193580ke_bit @ A @ N @ A2 ) ) )
& ( ~ ( ord_less_eq @ nat @ N @ M )
=> ( ( bit_se2584673776208193580ke_bit @ A @ N @ ( bit_se2638667681897837118et_bit @ A @ M @ A2 ) )
= ( bit_se2638667681897837118et_bit @ A @ M @ ( bit_se2584673776208193580ke_bit @ A @ N @ A2 ) ) ) ) ) ) ).
% take_bit_unset_bit_eq
thf(fact_2004_take__bit__set__bit__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,M: nat,A2: A] :
( ( ( ord_less_eq @ nat @ N @ M )
=> ( ( bit_se2584673776208193580ke_bit @ A @ N @ ( bit_se5668285175392031749et_bit @ A @ M @ A2 ) )
= ( bit_se2584673776208193580ke_bit @ A @ N @ A2 ) ) )
& ( ~ ( ord_less_eq @ nat @ N @ M )
=> ( ( bit_se2584673776208193580ke_bit @ A @ N @ ( bit_se5668285175392031749et_bit @ A @ M @ A2 ) )
= ( bit_se5668285175392031749et_bit @ A @ M @ ( bit_se2584673776208193580ke_bit @ A @ N @ A2 ) ) ) ) ) ) ).
% take_bit_set_bit_eq
thf(fact_2005_take__bit__flip__bit__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,M: nat,A2: A] :
( ( ( ord_less_eq @ nat @ N @ M )
=> ( ( bit_se2584673776208193580ke_bit @ A @ N @ ( bit_se8732182000553998342ip_bit @ A @ M @ A2 ) )
= ( bit_se2584673776208193580ke_bit @ A @ N @ A2 ) ) )
& ( ~ ( ord_less_eq @ nat @ N @ M )
=> ( ( bit_se2584673776208193580ke_bit @ A @ N @ ( bit_se8732182000553998342ip_bit @ A @ M @ A2 ) )
= ( bit_se8732182000553998342ip_bit @ A @ M @ ( bit_se2584673776208193580ke_bit @ A @ N @ A2 ) ) ) ) ) ) ).
% take_bit_flip_bit_eq
thf(fact_2006_dvd__div__neg,axiom,
! [A: $tType] :
( ( idom_divide @ A )
=> ! [B2: A,A2: A] :
( ( dvd_dvd @ A @ B2 @ A2 )
=> ( ( divide_divide @ A @ A2 @ ( uminus_uminus @ A @ B2 ) )
= ( uminus_uminus @ A @ ( divide_divide @ A @ A2 @ B2 ) ) ) ) ) ).
% dvd_div_neg
thf(fact_2007_dvd__neg__div,axiom,
! [A: $tType] :
( ( idom_divide @ A )
=> ! [B2: A,A2: A] :
( ( dvd_dvd @ A @ B2 @ A2 )
=> ( ( divide_divide @ A @ ( uminus_uminus @ A @ A2 ) @ B2 )
= ( uminus_uminus @ A @ ( divide_divide @ A @ A2 @ B2 ) ) ) ) ) ).
% dvd_neg_div
thf(fact_2008_real__minus__mult__self__le,axiom,
! [U: real,X: real] : ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( times_times @ real @ U @ U ) ) @ ( times_times @ real @ X @ X ) ) ).
% real_minus_mult_self_le
thf(fact_2009_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_2010_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_2011_zmod__zminus2__not__zero,axiom,
! [K2: int,L: int] :
( ( ( modulo_modulo @ int @ K2 @ ( uminus_uminus @ int @ L ) )
!= ( zero_zero @ int ) )
=> ( ( modulo_modulo @ int @ K2 @ L )
!= ( zero_zero @ int ) ) ) ).
% zmod_zminus2_not_zero
thf(fact_2012_zmod__zminus1__not__zero,axiom,
! [K2: int,L: int] :
( ( ( modulo_modulo @ int @ ( uminus_uminus @ int @ K2 ) @ L )
!= ( zero_zero @ int ) )
=> ( ( modulo_modulo @ int @ K2 @ L )
!= ( zero_zero @ int ) ) ) ).
% zmod_zminus1_not_zero
thf(fact_2013_take__bit__Suc__minus__bit0,axiom,
! [N: nat,K2: num] :
( ( bit_se2584673776208193580ke_bit @ int @ ( suc @ N ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ K2 ) ) ) )
= ( times_times @ int @ ( bit_se2584673776208193580ke_bit @ int @ N @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K2 ) ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ).
% take_bit_Suc_minus_bit0
thf(fact_2014_take__bit__signed__take__bit,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [M: nat,N: nat,A2: A] :
( ( ord_less_eq @ nat @ M @ ( suc @ N ) )
=> ( ( bit_se2584673776208193580ke_bit @ A @ M @ ( bit_ri4674362597316999326ke_bit @ A @ N @ A2 ) )
= ( bit_se2584673776208193580ke_bit @ A @ M @ A2 ) ) ) ) ).
% take_bit_signed_take_bit
thf(fact_2015_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_2016_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_2017_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_2018_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_2019_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_2020_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_2021_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_2022_take__bit__decr__eq,axiom,
! [N: nat,K2: int] :
( ( ( bit_se2584673776208193580ke_bit @ int @ N @ K2 )
!= ( zero_zero @ int ) )
=> ( ( bit_se2584673776208193580ke_bit @ int @ N @ ( minus_minus @ int @ K2 @ ( one_one @ int ) ) )
= ( minus_minus @ int @ ( bit_se2584673776208193580ke_bit @ int @ N @ K2 ) @ ( one_one @ int ) ) ) ) ).
% take_bit_decr_eq
thf(fact_2023_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_2024_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_2025_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_2026_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_2027_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_2028_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_2029_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_2030_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_2031_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_2032_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_2033_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_2034_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_2035_nonzero__neg__divide__eq__eq2,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B2: A,C2: A,A2: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( C2
= ( uminus_uminus @ A @ ( divide_divide @ A @ A2 @ B2 ) ) )
= ( ( times_times @ A @ C2 @ B2 )
= ( uminus_uminus @ A @ A2 ) ) ) ) ) ).
% nonzero_neg_divide_eq_eq2
thf(fact_2036_nonzero__neg__divide__eq__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( ( uminus_uminus @ A @ ( divide_divide @ A @ A2 @ B2 ) )
= C2 )
= ( ( uminus_uminus @ A @ A2 )
= ( times_times @ A @ C2 @ B2 ) ) ) ) ) ).
% nonzero_neg_divide_eq_eq
thf(fact_2037_minus__divide__eq__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B2: A,C2: A,A2: A] :
( ( ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) )
= A2 )
= ( ( ( C2
!= ( zero_zero @ A ) )
=> ( ( uminus_uminus @ A @ B2 )
= ( times_times @ A @ A2 @ C2 ) ) )
& ( ( C2
= ( zero_zero @ A ) )
=> ( A2
= ( zero_zero @ A ) ) ) ) ) ) ).
% minus_divide_eq_eq
thf(fact_2038_eq__minus__divide__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( A2
= ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) )
= ( ( ( C2
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ A2 @ C2 )
= ( uminus_uminus @ A @ B2 ) ) )
& ( ( C2
= ( zero_zero @ A ) )
=> ( A2
= ( zero_zero @ A ) ) ) ) ) ) ).
% eq_minus_divide_eq
thf(fact_2039_divide__eq__minus__1__iff,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A2: A,B2: A] :
( ( ( divide_divide @ A @ A2 @ B2 )
= ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( ( B2
!= ( zero_zero @ A ) )
& ( A2
= ( uminus_uminus @ A @ B2 ) ) ) ) ) ).
% divide_eq_minus_1_iff
thf(fact_2040_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_2041_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_2042_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_2043_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_2044_power__minus,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [A2: A,N: nat] :
( ( power_power @ A @ ( uminus_uminus @ A @ A2 ) @ N )
= ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N ) @ ( power_power @ A @ A2 @ N ) ) ) ) ).
% power_minus
thf(fact_2045_cong__exp__iff__simps_I10_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M: num,Q: num,N: num] :
( ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit0 @ M ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q ) ) )
!= ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit1 @ N ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q ) ) ) ) ) ).
% cong_exp_iff_simps(10)
thf(fact_2046_cong__exp__iff__simps_I12_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M: num,Q: num,N: num] :
( ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit1 @ M ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q ) ) )
!= ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit0 @ N ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q ) ) ) ) ) ).
% cong_exp_iff_simps(12)
thf(fact_2047_cong__exp__iff__simps_I13_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M: num,Q: num,N: num] :
( ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit1 @ M ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q ) ) )
= ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit1 @ N ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q ) ) ) )
= ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ Q ) )
= ( modulo_modulo @ A @ ( numeral_numeral @ A @ N ) @ ( numeral_numeral @ A @ Q ) ) ) ) ) ).
% cong_exp_iff_simps(13)
thf(fact_2048_power__minus__Bit0,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X: A,K2: num] :
( ( power_power @ A @ ( uminus_uminus @ A @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ K2 ) ) )
= ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ K2 ) ) ) ) ) ).
% power_minus_Bit0
thf(fact_2049_take__bit__Suc__bit1,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [N: nat,K2: num] :
( ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N ) @ ( numeral_numeral @ A @ ( bit1 @ K2 ) ) )
= ( plus_plus @ A @ ( times_times @ A @ ( bit_se2584673776208193580ke_bit @ A @ N @ ( numeral_numeral @ A @ K2 ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) ) ) ) ).
% take_bit_Suc_bit1
thf(fact_2050_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_2051_take__bit__numeral__minus__1__eq,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [K2: num] :
( ( bit_se2584673776208193580ke_bit @ A @ ( numeral_numeral @ nat @ K2 ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ K2 ) ) @ ( one_one @ A ) ) ) ) ).
% take_bit_numeral_minus_1_eq
thf(fact_2052_real__0__less__add__iff,axiom,
! [X: real,Y2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( plus_plus @ real @ X @ Y2 ) )
= ( ord_less @ real @ ( uminus_uminus @ real @ X ) @ Y2 ) ) ).
% real_0_less_add_iff
thf(fact_2053_real__add__less__0__iff,axiom,
! [X: real,Y2: real] :
( ( ord_less @ real @ ( plus_plus @ real @ X @ Y2 ) @ ( zero_zero @ real ) )
= ( ord_less @ real @ Y2 @ ( uminus_uminus @ real @ X ) ) ) ).
% real_add_less_0_iff
thf(fact_2054_real__add__le__0__iff,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq @ real @ ( plus_plus @ real @ X @ Y2 ) @ ( zero_zero @ real ) )
= ( ord_less_eq @ real @ Y2 @ ( uminus_uminus @ real @ X ) ) ) ).
% real_add_le_0_iff
thf(fact_2055_real__0__le__add__iff,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( plus_plus @ real @ X @ Y2 ) )
= ( ord_less_eq @ real @ ( uminus_uminus @ real @ X ) @ Y2 ) ) ).
% real_0_le_add_iff
thf(fact_2056_zmod__zminus2__eq__if,axiom,
! [A2: int,B2: int] :
( ( ( ( modulo_modulo @ int @ A2 @ B2 )
= ( zero_zero @ int ) )
=> ( ( modulo_modulo @ int @ A2 @ ( uminus_uminus @ int @ B2 ) )
= ( zero_zero @ int ) ) )
& ( ( ( modulo_modulo @ int @ A2 @ B2 )
!= ( zero_zero @ int ) )
=> ( ( modulo_modulo @ int @ A2 @ ( uminus_uminus @ int @ B2 ) )
= ( minus_minus @ int @ ( modulo_modulo @ int @ A2 @ B2 ) @ B2 ) ) ) ) ).
% zmod_zminus2_eq_if
thf(fact_2057_zmod__zminus1__eq__if,axiom,
! [A2: int,B2: int] :
( ( ( ( modulo_modulo @ int @ A2 @ B2 )
= ( zero_zero @ int ) )
=> ( ( modulo_modulo @ int @ ( uminus_uminus @ int @ A2 ) @ B2 )
= ( zero_zero @ int ) ) )
& ( ( ( modulo_modulo @ int @ A2 @ B2 )
!= ( zero_zero @ int ) )
=> ( ( modulo_modulo @ int @ ( uminus_uminus @ int @ A2 ) @ B2 )
= ( minus_minus @ int @ B2 @ ( modulo_modulo @ int @ A2 @ B2 ) ) ) ) ) ).
% zmod_zminus1_eq_if
thf(fact_2058_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_2059_power__numeral__odd,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [Z3: A,W: num] :
( ( power_power @ A @ Z3 @ ( numeral_numeral @ nat @ ( bit1 @ W ) ) )
= ( times_times @ A @ ( times_times @ A @ Z3 @ ( power_power @ A @ Z3 @ ( numeral_numeral @ nat @ W ) ) ) @ ( power_power @ A @ Z3 @ ( numeral_numeral @ nat @ W ) ) ) ) ) ).
% power_numeral_odd
thf(fact_2060_take__bit__minus__small__eq,axiom,
! [K2: int,N: nat] :
( ( ord_less @ int @ ( zero_zero @ int ) @ K2 )
=> ( ( ord_less_eq @ int @ K2 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( bit_se2584673776208193580ke_bit @ int @ N @ ( uminus_uminus @ int @ K2 ) )
= ( minus_minus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) @ K2 ) ) ) ) ).
% take_bit_minus_small_eq
thf(fact_2061_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_2062_pos__minus__divide__less__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,B2: A,A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) @ A2 )
= ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ ( times_times @ A @ A2 @ C2 ) ) ) ) ) ).
% pos_minus_divide_less_eq
thf(fact_2063_pos__less__minus__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less @ A @ A2 @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) )
= ( ord_less @ A @ ( times_times @ A @ A2 @ C2 ) @ ( uminus_uminus @ A @ B2 ) ) ) ) ) ).
% pos_less_minus_divide_eq
thf(fact_2064_neg__minus__divide__less__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,B2: A,A2: A] :
( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) @ A2 )
= ( ord_less @ A @ ( times_times @ A @ A2 @ C2 ) @ ( uminus_uminus @ A @ B2 ) ) ) ) ) ).
% neg_minus_divide_less_eq
thf(fact_2065_neg__less__minus__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ A2 @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) )
= ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ ( times_times @ A @ A2 @ C2 ) ) ) ) ) ).
% neg_less_minus_divide_eq
thf(fact_2066_minus__divide__less__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,C2: A,A2: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) @ A2 )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ ( times_times @ A @ A2 @ C2 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ A2 @ C2 ) @ ( uminus_uminus @ A @ B2 ) ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ) ) ) ) ).
% minus_divide_less_eq
thf(fact_2067_less__minus__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less @ A @ A2 @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( times_times @ A @ A2 @ C2 ) @ ( uminus_uminus @ A @ B2 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ ( times_times @ A @ A2 @ C2 ) ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% less_minus_divide_eq
thf(fact_2068_eq__divide__eq__numeral_I2_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [W: num,B2: A,C2: A] :
( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) )
= ( divide_divide @ A @ B2 @ C2 ) )
= ( ( ( C2
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ C2 )
= B2 ) )
& ( ( C2
= ( zero_zero @ A ) )
=> ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) )
= ( zero_zero @ A ) ) ) ) ) ) ).
% eq_divide_eq_numeral(2)
thf(fact_2069_divide__eq__eq__numeral_I2_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B2: A,C2: A,W: num] :
( ( ( divide_divide @ A @ B2 @ C2 )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) )
= ( ( ( C2
!= ( zero_zero @ A ) )
=> ( B2
= ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ C2 ) ) )
& ( ( C2
= ( zero_zero @ A ) )
=> ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) )
= ( zero_zero @ A ) ) ) ) ) ) ).
% divide_eq_eq_numeral(2)
thf(fact_2070_odd__numeral,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [N: num] :
~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ A @ ( bit1 @ N ) ) ) ) ).
% odd_numeral
thf(fact_2071_minus__divide__add__eq__iff,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z3: A,X: A,Y2: A] :
( ( Z3
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ X @ Z3 ) ) @ Y2 )
= ( divide_divide @ A @ ( plus_plus @ A @ ( uminus_uminus @ A @ X ) @ ( times_times @ A @ Y2 @ Z3 ) ) @ Z3 ) ) ) ) ).
% minus_divide_add_eq_iff
thf(fact_2072_add__divide__eq__if__simps_I3_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z3: A,A2: A,B2: A] :
( ( ( Z3
= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ A2 @ Z3 ) ) @ B2 )
= B2 ) )
& ( ( Z3
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ A2 @ Z3 ) ) @ B2 )
= ( divide_divide @ A @ ( plus_plus @ A @ ( uminus_uminus @ A @ A2 ) @ ( times_times @ A @ B2 @ Z3 ) ) @ Z3 ) ) ) ) ) ).
% add_divide_eq_if_simps(3)
thf(fact_2073_cong__exp__iff__simps_I3_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [N: num,Q: num] :
( ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit1 @ N ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q ) ) )
!= ( zero_zero @ A ) ) ) ).
% cong_exp_iff_simps(3)
thf(fact_2074_minus__divide__diff__eq__iff,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z3: A,X: A,Y2: A] :
( ( Z3
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ X @ Z3 ) ) @ Y2 )
= ( divide_divide @ A @ ( minus_minus @ A @ ( uminus_uminus @ A @ X ) @ ( times_times @ A @ Y2 @ Z3 ) ) @ Z3 ) ) ) ) ).
% minus_divide_diff_eq_iff
thf(fact_2075_add__divide__eq__if__simps_I5_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z3: A,A2: A,B2: A] :
( ( ( Z3
= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( divide_divide @ A @ A2 @ Z3 ) @ B2 )
= ( uminus_uminus @ A @ B2 ) ) )
& ( ( Z3
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( divide_divide @ A @ A2 @ Z3 ) @ B2 )
= ( divide_divide @ A @ ( minus_minus @ A @ A2 @ ( times_times @ A @ B2 @ Z3 ) ) @ Z3 ) ) ) ) ) ).
% add_divide_eq_if_simps(5)
thf(fact_2076_add__divide__eq__if__simps_I6_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z3: A,A2: A,B2: A] :
( ( ( Z3
= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ A2 @ Z3 ) ) @ B2 )
= ( uminus_uminus @ A @ B2 ) ) )
& ( ( Z3
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ A2 @ Z3 ) ) @ B2 )
= ( divide_divide @ A @ ( minus_minus @ A @ ( uminus_uminus @ A @ A2 ) @ ( times_times @ A @ B2 @ Z3 ) ) @ Z3 ) ) ) ) ) ).
% add_divide_eq_if_simps(6)
thf(fact_2077_power3__eq__cube,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A2: A] :
( ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) )
= ( times_times @ A @ ( times_times @ A @ A2 @ A2 ) @ A2 ) ) ) ).
% power3_eq_cube
thf(fact_2078_even__minus,axiom,
! [A: $tType] :
( ( ring_parity @ A )
=> ! [A2: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( uminus_uminus @ A @ A2 ) )
= ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) ) ) ).
% even_minus
thf(fact_2079_numeral__3__eq__3,axiom,
( ( numeral_numeral @ nat @ ( bit1 @ one2 ) )
= ( suc @ ( suc @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ).
% numeral_3_eq_3
thf(fact_2080_power2__eq__iff,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [X: A,Y2: A] :
( ( ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( power_power @ A @ Y2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( ( X = Y2 )
| ( X
= ( uminus_uminus @ A @ Y2 ) ) ) ) ) ).
% power2_eq_iff
thf(fact_2081_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_2082_verit__less__mono__div__int2,axiom,
! [A3: int,B5: int,N: int] :
( ( ord_less_eq @ int @ A3 @ B5 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ ( uminus_uminus @ int @ N ) )
=> ( ord_less_eq @ int @ ( divide_divide @ int @ B5 @ N ) @ ( divide_divide @ int @ A3 @ N ) ) ) ) ).
% verit_less_mono_div_int2
thf(fact_2083_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_2084_take__bit__Suc__bit0,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [N: nat,K2: num] :
( ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N ) @ ( numeral_numeral @ A @ ( bit0 @ K2 ) ) )
= ( times_times @ A @ ( bit_se2584673776208193580ke_bit @ A @ N @ ( numeral_numeral @ A @ K2 ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% take_bit_Suc_bit0
thf(fact_2085_take__bit__eq__mod,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se2584673776208193580ke_bit @ A )
= ( ^ [N2: nat,A5: A] : ( modulo_modulo @ A @ A5 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ) ).
% take_bit_eq_mod
thf(fact_2086_mult__commute__abs,axiom,
! [A: $tType] :
( ( ab_semigroup_mult @ A )
=> ! [C2: A] :
( ( ^ [X4: A] : ( times_times @ A @ X4 @ C2 ) )
= ( times_times @ A @ C2 ) ) ) ).
% mult_commute_abs
thf(fact_2087_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_2088_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_2089_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_2090_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_2091_take__bit__nat__def,axiom,
( ( bit_se2584673776208193580ke_bit @ nat )
= ( ^ [N2: nat,M2: nat] : ( modulo_modulo @ nat @ M2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ).
% take_bit_nat_def
thf(fact_2092_take__bit__int__less__exp,axiom,
! [N: nat,K2: int] : ( ord_less @ int @ ( bit_se2584673776208193580ke_bit @ int @ N @ K2 ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ).
% take_bit_int_less_exp
thf(fact_2093_pos__minus__divide__le__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,B2: A,A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) @ A2 )
= ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( times_times @ A @ A2 @ C2 ) ) ) ) ) ).
% pos_minus_divide_le_eq
thf(fact_2094_pos__le__minus__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less_eq @ A @ A2 @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) )
= ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C2 ) @ ( uminus_uminus @ A @ B2 ) ) ) ) ) ).
% pos_le_minus_divide_eq
thf(fact_2095_neg__minus__divide__le__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,B2: A,A2: A] :
( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) @ A2 )
= ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C2 ) @ ( uminus_uminus @ A @ B2 ) ) ) ) ) ).
% neg_minus_divide_le_eq
thf(fact_2096_neg__le__minus__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ A2 @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) )
= ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( times_times @ A @ A2 @ C2 ) ) ) ) ) ).
% neg_le_minus_divide_eq
thf(fact_2097_minus__divide__le__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,C2: A,A2: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) @ A2 )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( times_times @ A @ A2 @ C2 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C2 ) @ ( uminus_uminus @ A @ B2 ) ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ) ) ) ) ).
% minus_divide_le_eq
thf(fact_2098_le__minus__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C2 ) @ ( uminus_uminus @ A @ B2 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( times_times @ A @ A2 @ C2 ) ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% le_minus_divide_eq
thf(fact_2099_less__divide__eq__numeral_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [W: num,B2: A,C2: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ ( divide_divide @ A @ B2 @ C2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ C2 ) @ B2 ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B2 @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ C2 ) ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% less_divide_eq_numeral(2)
thf(fact_2100_divide__less__eq__numeral_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,C2: A,W: num] :
( ( ord_less @ A @ ( divide_divide @ A @ B2 @ C2 ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ B2 @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ C2 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ C2 ) @ B2 ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) ) ) ) ) ) ) ) ).
% divide_less_eq_numeral(2)
thf(fact_2101_take__bit__int__def,axiom,
( ( bit_se2584673776208193580ke_bit @ int )
= ( ^ [N2: nat,K3: int] : ( modulo_modulo @ int @ K3 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ).
% take_bit_int_def
thf(fact_2102_cong__exp__iff__simps_I7_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [Q: num,N: num] :
( ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ one2 ) @ ( numeral_numeral @ A @ ( bit0 @ Q ) ) )
= ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit1 @ N ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q ) ) ) )
= ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ N ) @ ( numeral_numeral @ A @ Q ) )
= ( zero_zero @ A ) ) ) ) ).
% cong_exp_iff_simps(7)
thf(fact_2103_cong__exp__iff__simps_I11_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M: num,Q: num] :
( ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit1 @ M ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q ) ) )
= ( modulo_modulo @ A @ ( numeral_numeral @ A @ one2 ) @ ( numeral_numeral @ A @ ( bit0 @ Q ) ) ) )
= ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ Q ) )
= ( zero_zero @ A ) ) ) ) ).
% cong_exp_iff_simps(11)
thf(fact_2104_power2__eq__1__iff,axiom,
! [A: $tType] :
( ( ring_15535105094025558882visors @ A )
=> ! [A2: A] :
( ( ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( one_one @ A ) )
= ( ( A2
= ( one_one @ A ) )
| ( A2
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ) ).
% power2_eq_1_iff
thf(fact_2105_uminus__power__if,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [N: nat,A2: A] :
( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( power_power @ A @ ( uminus_uminus @ A @ A2 ) @ N )
= ( power_power @ A @ A2 @ N ) ) )
& ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( power_power @ A @ ( uminus_uminus @ A @ A2 ) @ N )
= ( uminus_uminus @ A @ ( power_power @ A @ A2 @ N ) ) ) ) ) ) ).
% uminus_power_if
thf(fact_2106_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_2107_neg__one__power__add__eq__neg__one__power__diff,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [K2: nat,N: nat] :
( ( ord_less_eq @ nat @ K2 @ N )
=> ( ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( plus_plus @ nat @ N @ K2 ) )
= ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( minus_minus @ nat @ N @ K2 ) ) ) ) ) ).
% neg_one_power_add_eq_neg_one_power_diff
thf(fact_2108_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_2109_realpow__square__minus__le,axiom,
! [U: real,X: real] : ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( power_power @ real @ U @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% realpow_square_minus_le
thf(fact_2110_take__bit__eq__0__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A2: A] :
( ( ( bit_se2584673776208193580ke_bit @ A @ N @ A2 )
= ( zero_zero @ A ) )
= ( dvd_dvd @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) @ A2 ) ) ) ).
% take_bit_eq_0_iff
thf(fact_2111_minus__mod__int__eq,axiom,
! [L: int,K2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ L )
=> ( ( modulo_modulo @ int @ ( uminus_uminus @ int @ K2 ) @ L )
= ( minus_minus @ int @ ( minus_minus @ int @ L @ ( one_one @ int ) ) @ ( modulo_modulo @ int @ ( minus_minus @ int @ K2 @ ( one_one @ int ) ) @ L ) ) ) ) ).
% minus_mod_int_eq
thf(fact_2112_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_2113_zdiv__zminus2__eq__if,axiom,
! [B2: int,A2: int] :
( ( B2
!= ( zero_zero @ int ) )
=> ( ( ( ( modulo_modulo @ int @ A2 @ B2 )
= ( zero_zero @ int ) )
=> ( ( divide_divide @ int @ A2 @ ( uminus_uminus @ int @ B2 ) )
= ( uminus_uminus @ int @ ( divide_divide @ int @ A2 @ B2 ) ) ) )
& ( ( ( modulo_modulo @ int @ A2 @ B2 )
!= ( zero_zero @ int ) )
=> ( ( divide_divide @ int @ A2 @ ( uminus_uminus @ int @ B2 ) )
= ( minus_minus @ int @ ( uminus_uminus @ int @ ( divide_divide @ int @ A2 @ B2 ) ) @ ( one_one @ int ) ) ) ) ) ) ).
% zdiv_zminus2_eq_if
thf(fact_2114_zdiv__zminus1__eq__if,axiom,
! [B2: int,A2: int] :
( ( B2
!= ( zero_zero @ int ) )
=> ( ( ( ( modulo_modulo @ int @ A2 @ B2 )
= ( zero_zero @ int ) )
=> ( ( divide_divide @ int @ ( uminus_uminus @ int @ A2 ) @ B2 )
= ( uminus_uminus @ int @ ( divide_divide @ int @ A2 @ B2 ) ) ) )
& ( ( ( modulo_modulo @ int @ A2 @ B2 )
!= ( zero_zero @ int ) )
=> ( ( divide_divide @ int @ ( uminus_uminus @ int @ A2 ) @ B2 )
= ( minus_minus @ int @ ( uminus_uminus @ int @ ( divide_divide @ int @ A2 @ B2 ) ) @ ( one_one @ int ) ) ) ) ) ) ).
% zdiv_zminus1_eq_if
thf(fact_2115_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_2116_zminus1__lemma,axiom,
! [A2: int,B2: int,Q: int,R: int] :
( ( eucl_rel_int @ A2 @ B2 @ ( product_Pair @ int @ int @ Q @ R ) )
=> ( ( B2
!= ( zero_zero @ int ) )
=> ( eucl_rel_int @ ( uminus_uminus @ int @ A2 ) @ B2
@ ( product_Pair @ int @ int
@ ( if @ int
@ ( R
= ( zero_zero @ int ) )
@ ( uminus_uminus @ int @ Q )
@ ( minus_minus @ int @ ( uminus_uminus @ int @ Q ) @ ( one_one @ int ) ) )
@ ( if @ int
@ ( R
= ( zero_zero @ int ) )
@ ( zero_zero @ int )
@ ( minus_minus @ int @ B2 @ R ) ) ) ) ) ) ).
% zminus1_lemma
thf(fact_2117_take__bit__int__less__self__iff,axiom,
! [N: nat,K2: int] :
( ( ord_less @ int @ ( bit_se2584673776208193580ke_bit @ int @ N @ K2 ) @ K2 )
= ( ord_less_eq @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) @ K2 ) ) ).
% take_bit_int_less_self_iff
thf(fact_2118_take__bit__int__greater__eq__self__iff,axiom,
! [K2: int,N: nat] :
( ( ord_less_eq @ int @ K2 @ ( bit_se2584673776208193580ke_bit @ int @ N @ K2 ) )
= ( ord_less @ int @ K2 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ) ).
% take_bit_int_greater_eq_self_iff
thf(fact_2119_le__divide__eq__numeral_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [W: num,B2: A,C2: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ ( divide_divide @ A @ B2 @ C2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ C2 ) @ B2 ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B2 @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ C2 ) ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% le_divide_eq_numeral(2)
thf(fact_2120_divide__le__eq__numeral_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,C2: A,W: num] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ C2 ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ B2 @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ C2 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ C2 ) @ B2 ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) ) ) ) ) ) ) ) ).
% divide_le_eq_numeral(2)
thf(fact_2121_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_2122_minus__power__mult__self,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [A2: A,N: nat] :
( ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ A2 ) @ N ) @ ( power_power @ A @ ( uminus_uminus @ A @ A2 ) @ N ) )
= ( power_power @ A @ A2 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% minus_power_mult_self
thf(fact_2123_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_2124_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_2125_div__pos__neg__trivial,axiom,
! [K2: int,L: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ K2 )
=> ( ( ord_less_eq @ int @ ( plus_plus @ int @ K2 @ L ) @ ( zero_zero @ int ) )
=> ( ( divide_divide @ int @ K2 @ L )
= ( uminus_uminus @ int @ ( one_one @ int ) ) ) ) ) ).
% div_pos_neg_trivial
thf(fact_2126_signed__take__bit__int__less__eq__self__iff,axiom,
! [N: nat,K2: int] :
( ( ord_less_eq @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N @ K2 ) @ K2 )
= ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) @ K2 ) ) ).
% signed_take_bit_int_less_eq_self_iff
thf(fact_2127_signed__take__bit__int__greater__eq__minus__exp,axiom,
! [N: nat,K2: int] : ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) @ ( bit_ri4674362597316999326ke_bit @ int @ N @ K2 ) ) ).
% signed_take_bit_int_greater_eq_minus_exp
thf(fact_2128_signed__take__bit__int__greater__self__iff,axiom,
! [K2: int,N: nat] :
( ( ord_less @ int @ K2 @ ( bit_ri4674362597316999326ke_bit @ int @ N @ K2 ) )
= ( ord_less @ int @ K2 @ ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% signed_take_bit_int_greater_self_iff
thf(fact_2129_take__bit__int__eq__self,axiom,
! [K2: int,N: nat] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K2 )
=> ( ( ord_less @ int @ K2 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( bit_se2584673776208193580ke_bit @ int @ N @ K2 )
= K2 ) ) ) ).
% take_bit_int_eq_self
thf(fact_2130_take__bit__int__eq__self__iff,axiom,
! [N: nat,K2: int] :
( ( ( bit_se2584673776208193580ke_bit @ int @ N @ K2 )
= K2 )
= ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K2 )
& ( ord_less @ int @ K2 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% take_bit_int_eq_self_iff
thf(fact_2131_take__bit__incr__eq,axiom,
! [N: nat,K2: int] :
( ( ( bit_se2584673776208193580ke_bit @ int @ N @ K2 )
!= ( minus_minus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ int ) ) )
=> ( ( bit_se2584673776208193580ke_bit @ int @ N @ ( plus_plus @ int @ K2 @ ( one_one @ int ) ) )
= ( plus_plus @ int @ ( one_one @ int ) @ ( bit_se2584673776208193580ke_bit @ int @ N @ K2 ) ) ) ) ).
% take_bit_incr_eq
thf(fact_2132_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_2133_take__bit__Suc,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A2: A] :
( ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N ) @ A2 )
= ( plus_plus @ A @ ( times_times @ A @ ( bit_se2584673776208193580ke_bit @ A @ N @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% take_bit_Suc
thf(fact_2134_int__bit__induct,axiom,
! [P3: int > $o,K2: int] :
( ( P3 @ ( zero_zero @ int ) )
=> ( ( P3 @ ( uminus_uminus @ int @ ( one_one @ int ) ) )
=> ( ! [K: int] :
( ( P3 @ K )
=> ( ( K
!= ( zero_zero @ int ) )
=> ( P3 @ ( times_times @ int @ K @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) )
=> ( ! [K: int] :
( ( P3 @ K )
=> ( ( K
!= ( uminus_uminus @ int @ ( one_one @ int ) ) )
=> ( P3 @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ K @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) )
=> ( P3 @ K2 ) ) ) ) ) ).
% int_bit_induct
thf(fact_2135_signed__take__bit__int__eq__self,axiom,
! [N: nat,K2: int] :
( ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) @ K2 )
=> ( ( ord_less @ int @ K2 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( bit_ri4674362597316999326ke_bit @ int @ N @ K2 )
= K2 ) ) ) ).
% signed_take_bit_int_eq_self
thf(fact_2136_signed__take__bit__int__eq__self__iff,axiom,
! [N: nat,K2: int] :
( ( ( bit_ri4674362597316999326ke_bit @ int @ N @ K2 )
= K2 )
= ( ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) @ K2 )
& ( ord_less @ int @ K2 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% signed_take_bit_int_eq_self_iff
thf(fact_2137_take__bit__int__less__eq,axiom,
! [N: nat,K2: int] :
( ( ord_less_eq @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) @ K2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ord_less_eq @ int @ ( bit_se2584673776208193580ke_bit @ int @ N @ K2 ) @ ( minus_minus @ int @ K2 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ) ) ) ).
% take_bit_int_less_eq
thf(fact_2138_take__bit__int__greater__eq,axiom,
! [K2: int,N: nat] :
( ( ord_less @ int @ K2 @ ( zero_zero @ int ) )
=> ( ord_less_eq @ int @ ( plus_plus @ int @ K2 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) @ ( bit_se2584673776208193580ke_bit @ int @ N @ K2 ) ) ) ).
% take_bit_int_greater_eq
thf(fact_2139_signed__take__bit__eq__take__bit__shift,axiom,
( ( bit_ri4674362597316999326ke_bit @ int )
= ( ^ [N2: nat,K3: int] : ( minus_minus @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( suc @ N2 ) @ ( plus_plus @ int @ K3 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ).
% signed_take_bit_eq_take_bit_shift
thf(fact_2140_stable__imp__take__bit__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A,N: nat] :
( ( ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= A2 )
=> ( ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
=> ( ( bit_se2584673776208193580ke_bit @ A @ N @ A2 )
= ( zero_zero @ A ) ) )
& ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
=> ( ( bit_se2584673776208193580ke_bit @ A @ N @ A2 )
= ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ A ) ) ) ) ) ) ) ).
% stable_imp_take_bit_eq
thf(fact_2141_divmod__step__nat__def,axiom,
( ( unique1321980374590559556d_step @ nat )
= ( ^ [L3: num] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [Q5: 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 ) ) @ Q5 ) @ ( one_one @ nat ) ) @ ( minus_minus @ nat @ R5 @ ( numeral_numeral @ nat @ L3 ) ) ) @ ( product_Pair @ nat @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Q5 ) @ R5 ) ) ) ) ) ).
% divmod_step_nat_def
thf(fact_2142_divmod__step__int__def,axiom,
( ( unique1321980374590559556d_step @ int )
= ( ^ [L3: num] :
( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [Q5: 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 ) ) @ Q5 ) @ ( one_one @ int ) ) @ ( minus_minus @ int @ R5 @ ( numeral_numeral @ int @ L3 ) ) ) @ ( product_Pair @ int @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Q5 ) @ R5 ) ) ) ) ) ).
% divmod_step_int_def
thf(fact_2143_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_2144_signed__take__bit__int__greater__eq,axiom,
! [K2: int,N: nat] :
( ( ord_less @ int @ K2 @ ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) )
=> ( ord_less_eq @ int @ ( plus_plus @ int @ K2 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( suc @ N ) ) ) @ ( bit_ri4674362597316999326ke_bit @ int @ N @ K2 ) ) ) ).
% signed_take_bit_int_greater_eq
thf(fact_2145_mod__exhaust__less__4,axiom,
! [M: nat] :
( ( ( modulo_modulo @ nat @ M @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) )
= ( zero_zero @ nat ) )
| ( ( modulo_modulo @ nat @ M @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) )
= ( one_one @ nat ) )
| ( ( modulo_modulo @ nat @ M @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) )
= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
| ( ( modulo_modulo @ nat @ M @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) )
= ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) ) ) ).
% mod_exhaust_less_4
thf(fact_2146_case__prod__conv,axiom,
! [B: $tType,A: $tType,C: $tType,F3: B > C > A,A2: B,B2: C] :
( ( product_case_prod @ B @ C @ A @ F3 @ ( product_Pair @ B @ C @ A2 @ B2 ) )
= ( F3 @ A2 @ B2 ) ) ).
% case_prod_conv
thf(fact_2147_compl__less__compl__iff,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y2: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ X ) @ ( uminus_uminus @ A @ Y2 ) )
= ( ord_less @ A @ Y2 @ X ) ) ) ).
% compl_less_compl_iff
thf(fact_2148_compl__le__compl__iff,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y2: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ X ) @ ( uminus_uminus @ A @ Y2 ) )
= ( ord_less_eq @ A @ Y2 @ X ) ) ) ).
% compl_le_compl_iff
thf(fact_2149_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 )
@ ^ [Q5: A,R5: A] : ( product_Pair @ A @ A @ Q5 @ ( 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_2150_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_2151_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_2152_case__prodI2,axiom,
! [B: $tType,A: $tType,P5: product_prod @ A @ B,C2: A > B > $o] :
( ! [A4: A,B3: B] :
( ( P5
= ( product_Pair @ A @ B @ A4 @ B3 ) )
=> ( C2 @ A4 @ B3 ) )
=> ( product_case_prod @ A @ B @ $o @ C2 @ P5 ) ) ).
% case_prodI2
thf(fact_2153_case__prodI,axiom,
! [A: $tType,B: $tType,F3: A > B > $o,A2: A,B2: B] :
( ( F3 @ A2 @ B2 )
=> ( product_case_prod @ A @ B @ $o @ F3 @ ( product_Pair @ A @ B @ A2 @ B2 ) ) ) ).
% case_prodI
thf(fact_2154_mem__case__prodI2,axiom,
! [C: $tType,B: $tType,A: $tType,P5: product_prod @ A @ B,Z3: C,C2: A > B > ( set @ C )] :
( ! [A4: A,B3: B] :
( ( P5
= ( product_Pair @ A @ B @ A4 @ B3 ) )
=> ( member @ C @ Z3 @ ( C2 @ A4 @ B3 ) ) )
=> ( member @ C @ Z3 @ ( product_case_prod @ A @ B @ ( set @ C ) @ C2 @ P5 ) ) ) ).
% mem_case_prodI2
thf(fact_2155_mem__case__prodI,axiom,
! [A: $tType,B: $tType,C: $tType,Z3: A,C2: B > C > ( set @ A ),A2: B,B2: C] :
( ( member @ A @ Z3 @ ( C2 @ A2 @ B2 ) )
=> ( member @ A @ Z3 @ ( product_case_prod @ B @ C @ ( set @ A ) @ C2 @ ( product_Pair @ B @ C @ A2 @ B2 ) ) ) ) ).
% mem_case_prodI
thf(fact_2156_case__prodI2_H,axiom,
! [A: $tType,B: $tType,C: $tType,P5: product_prod @ A @ B,C2: A > B > C > $o,X: C] :
( ! [A4: A,B3: B] :
( ( ( product_Pair @ A @ B @ A4 @ B3 )
= P5 )
=> ( C2 @ A4 @ B3 @ X ) )
=> ( product_case_prod @ A @ B @ ( C > $o ) @ C2 @ P5 @ X ) ) ).
% case_prodI2'
thf(fact_2157_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_2158_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_2159_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_2160_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_2161_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_2162_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 )
@ ^ [Q5: A,R5: A] : ( product_Pair @ A @ A @ Q5 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ R5 ) )
@ ( unique8689654367752047608divmod @ A @ M @ N ) ) ) ) ).
% divmod_algorithm_code(5)
thf(fact_2163_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_2164_uminus__set__def,axiom,
! [A: $tType] :
( ( uminus_uminus @ ( set @ A ) )
= ( ^ [A7: set @ A] :
( collect @ A
@ ( uminus_uminus @ ( A > $o )
@ ^ [X4: A] : ( member @ A @ X4 @ A7 ) ) ) ) ) ).
% uminus_set_def
thf(fact_2165_take__bit__minus,axiom,
! [N: nat,K2: int] :
( ( bit_se2584673776208193580ke_bit @ int @ N @ ( uminus_uminus @ int @ ( bit_se2584673776208193580ke_bit @ int @ N @ K2 ) ) )
= ( bit_se2584673776208193580ke_bit @ int @ N @ ( uminus_uminus @ int @ K2 ) ) ) ).
% take_bit_minus
thf(fact_2166_Collect__neg__eq,axiom,
! [A: $tType,P3: A > $o] :
( ( collect @ A
@ ^ [X4: A] :
~ ( P3 @ X4 ) )
= ( uminus_uminus @ ( set @ A ) @ ( collect @ A @ P3 ) ) ) ).
% Collect_neg_eq
thf(fact_2167_Compl__eq,axiom,
! [A: $tType] :
( ( uminus_uminus @ ( set @ A ) )
= ( ^ [A7: set @ A] :
( collect @ A
@ ^ [X4: A] :
~ ( member @ A @ X4 @ A7 ) ) ) ) ).
% Compl_eq
thf(fact_2168_case__prodD,axiom,
! [A: $tType,B: $tType,F3: A > B > $o,A2: A,B2: B] :
( ( product_case_prod @ A @ B @ $o @ F3 @ ( product_Pair @ A @ B @ A2 @ B2 ) )
=> ( F3 @ A2 @ B2 ) ) ).
% case_prodD
thf(fact_2169_case__prodE,axiom,
! [A: $tType,B: $tType,C2: A > B > $o,P5: product_prod @ A @ B] :
( ( product_case_prod @ A @ B @ $o @ C2 @ P5 )
=> ~ ! [X5: A,Y7: B] :
( ( P5
= ( product_Pair @ A @ B @ X5 @ Y7 ) )
=> ~ ( C2 @ X5 @ Y7 ) ) ) ).
% case_prodE
thf(fact_2170_case__prodE_H,axiom,
! [B: $tType,A: $tType,C: $tType,C2: A > B > C > $o,P5: product_prod @ A @ B,Z3: C] :
( ( product_case_prod @ A @ B @ ( C > $o ) @ C2 @ P5 @ Z3 )
=> ~ ! [X5: A,Y7: B] :
( ( P5
= ( product_Pair @ A @ B @ X5 @ Y7 ) )
=> ~ ( C2 @ X5 @ Y7 @ Z3 ) ) ) ).
% case_prodE'
thf(fact_2171_case__prodD_H,axiom,
! [B: $tType,A: $tType,C: $tType,R4: A > B > C > $o,A2: A,B2: B,C2: C] :
( ( product_case_prod @ A @ B @ ( C > $o ) @ R4 @ ( product_Pair @ A @ B @ A2 @ B2 ) @ C2 )
=> ( R4 @ A2 @ B2 @ C2 ) ) ).
% case_prodD'
thf(fact_2172_prod_Ocase__distrib,axiom,
! [C: $tType,D: $tType,B: $tType,A: $tType,H2: C > D,F3: A > B > C,Prod: product_prod @ A @ B] :
( ( H2 @ ( product_case_prod @ A @ B @ C @ F3 @ Prod ) )
= ( product_case_prod @ A @ B @ D
@ ^ [X1: A,X25: B] : ( H2 @ ( F3 @ X1 @ X25 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_2173_divmod__int__def,axiom,
( ( unique8689654367752047608divmod @ int )
= ( ^ [M2: num,N2: num] : ( product_Pair @ int @ int @ ( divide_divide @ int @ ( numeral_numeral @ int @ M2 ) @ ( numeral_numeral @ int @ N2 ) ) @ ( modulo_modulo @ int @ ( numeral_numeral @ int @ M2 ) @ ( numeral_numeral @ int @ N2 ) ) ) ) ) ).
% divmod_int_def
thf(fact_2174_divmod__def,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ( ( unique8689654367752047608divmod @ A )
= ( ^ [M2: num,N2: num] : ( product_Pair @ A @ A @ ( divide_divide @ A @ ( numeral_numeral @ A @ M2 ) @ ( numeral_numeral @ A @ N2 ) ) @ ( modulo_modulo @ A @ ( numeral_numeral @ A @ M2 ) @ ( numeral_numeral @ A @ N2 ) ) ) ) ) ) ).
% divmod_def
thf(fact_2175_divmod_H__nat__def,axiom,
( ( unique8689654367752047608divmod @ nat )
= ( ^ [M2: num,N2: num] : ( product_Pair @ nat @ nat @ ( divide_divide @ nat @ ( numeral_numeral @ nat @ M2 ) @ ( numeral_numeral @ nat @ N2 ) ) @ ( modulo_modulo @ nat @ ( numeral_numeral @ nat @ M2 ) @ ( numeral_numeral @ nat @ N2 ) ) ) ) ) ).
% divmod'_nat_def
thf(fact_2176_dbl__dec__def,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( neg_numeral_dbl_dec @ A )
= ( ^ [X4: A] : ( minus_minus @ A @ ( plus_plus @ A @ X4 @ X4 ) @ ( one_one @ A ) ) ) ) ) ).
% dbl_dec_def
thf(fact_2177_compl__mono,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y2: A] :
( ( ord_less_eq @ A @ X @ Y2 )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ Y2 ) @ ( uminus_uminus @ A @ X ) ) ) ) ).
% compl_mono
thf(fact_2178_compl__le__swap1,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [Y2: A,X: A] :
( ( ord_less_eq @ A @ Y2 @ ( uminus_uminus @ A @ X ) )
=> ( ord_less_eq @ A @ X @ ( uminus_uminus @ A @ Y2 ) ) ) ) ).
% compl_le_swap1
thf(fact_2179_compl__le__swap2,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [Y2: A,X: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ Y2 ) @ X )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ X ) @ Y2 ) ) ) ).
% compl_le_swap2
thf(fact_2180_compl__less__swap1,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [Y2: A,X: A] :
( ( ord_less @ A @ Y2 @ ( uminus_uminus @ A @ X ) )
=> ( ord_less @ A @ X @ ( uminus_uminus @ A @ Y2 ) ) ) ) ).
% compl_less_swap1
thf(fact_2181_compl__less__swap2,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [Y2: A,X: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ Y2 ) @ X )
=> ( ord_less @ A @ ( uminus_uminus @ A @ X ) @ Y2 ) ) ) ).
% compl_less_swap2
thf(fact_2182_cond__case__prod__eta,axiom,
! [C: $tType,B: $tType,A: $tType,F3: A > B > C,G: ( product_prod @ A @ B ) > C] :
( ! [X5: A,Y7: B] :
( ( F3 @ X5 @ Y7 )
= ( G @ ( product_Pair @ A @ B @ X5 @ Y7 ) ) )
=> ( ( product_case_prod @ A @ B @ C @ F3 )
= G ) ) ).
% cond_case_prod_eta
thf(fact_2183_case__prod__eta,axiom,
! [C: $tType,B: $tType,A: $tType,F3: ( product_prod @ A @ B ) > C] :
( ( product_case_prod @ A @ B @ C
@ ^ [X4: A,Y: B] : ( F3 @ ( product_Pair @ A @ B @ X4 @ Y ) ) )
= F3 ) ).
% case_prod_eta
thf(fact_2184_case__prodE2,axiom,
! [B: $tType,A: $tType,C: $tType,Q2: A > $o,P3: B > C > A,Z3: product_prod @ B @ C] :
( ( Q2 @ ( product_case_prod @ B @ C @ A @ P3 @ Z3 ) )
=> ~ ! [X5: B,Y7: C] :
( ( Z3
= ( product_Pair @ B @ C @ X5 @ Y7 ) )
=> ~ ( Q2 @ ( P3 @ X5 @ Y7 ) ) ) ) ).
% case_prodE2
thf(fact_2185_divmod__divmod__step,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ( ( unique8689654367752047608divmod @ A )
= ( ^ [M2: num,N2: num] : ( if @ ( product_prod @ A @ A ) @ ( ord_less @ num @ M2 @ N2 ) @ ( product_Pair @ A @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ M2 ) ) @ ( unique1321980374590559556d_step @ A @ N2 @ ( unique8689654367752047608divmod @ A @ M2 @ ( bit0 @ N2 ) ) ) ) ) ) ) ).
% divmod_divmod_step
thf(fact_2186_dvd__numeral__simp,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M: num,N: num] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ N ) )
= ( unique5940410009612947441es_aux @ A @ ( unique8689654367752047608divmod @ A @ N @ M ) ) ) ) ).
% dvd_numeral_simp
thf(fact_2187_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_2188_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_2189_signed__take__bit__numeral__minus__bit1,axiom,
! [L: num,K2: num] :
( ( bit_ri4674362597316999326ke_bit @ int @ ( numeral_numeral @ nat @ L ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ K2 ) ) ) )
= ( plus_plus @ int @ ( times_times @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( pred_numeral @ L ) @ ( minus_minus @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K2 ) ) @ ( one_one @ int ) ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( one_one @ int ) ) ) ).
% signed_take_bit_numeral_minus_bit1
thf(fact_2190_divmod__nat__if,axiom,
( divmod_nat
= ( ^ [M2: nat,N2: nat] :
( if @ ( product_prod @ nat @ nat )
@ ( ( N2
= ( zero_zero @ nat ) )
| ( ord_less @ nat @ M2 @ N2 ) )
@ ( product_Pair @ nat @ nat @ ( zero_zero @ nat ) @ M2 )
@ ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [Q5: nat] : ( product_Pair @ nat @ nat @ ( suc @ Q5 ) )
@ ( divmod_nat @ ( minus_minus @ nat @ M2 @ N2 ) @ N2 ) ) ) ) ) ).
% divmod_nat_if
thf(fact_2191_take__bit__Suc__minus__bit1,axiom,
! [N: nat,K2: num] :
( ( bit_se2584673776208193580ke_bit @ int @ ( suc @ N ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ K2 ) ) ) )
= ( plus_plus @ int @ ( times_times @ int @ ( bit_se2584673776208193580ke_bit @ int @ N @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( inc @ K2 ) ) ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( one_one @ int ) ) ) ).
% take_bit_Suc_minus_bit1
thf(fact_2192_signed__take__bit__numeral__bit1,axiom,
! [L: num,K2: num] :
( ( bit_ri4674362597316999326ke_bit @ int @ ( numeral_numeral @ nat @ L ) @ ( numeral_numeral @ int @ ( bit1 @ K2 ) ) )
= ( plus_plus @ int @ ( times_times @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( pred_numeral @ L ) @ ( numeral_numeral @ int @ K2 ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( one_one @ int ) ) ) ).
% signed_take_bit_numeral_bit1
thf(fact_2193_split__part,axiom,
! [B: $tType,A: $tType,P3: $o,Q2: A > B > $o] :
( ( product_case_prod @ A @ B @ $o
@ ^ [A5: A,B4: B] :
( P3
& ( Q2 @ A5 @ B4 ) ) )
= ( ^ [Ab: product_prod @ A @ B] :
( P3
& ( product_case_prod @ A @ B @ $o @ Q2 @ Ab ) ) ) ) ).
% split_part
thf(fact_2194_pred__numeral__simps_I1_J,axiom,
( ( pred_numeral @ one2 )
= ( zero_zero @ nat ) ) ).
% pred_numeral_simps(1)
thf(fact_2195_eq__numeral__Suc,axiom,
! [K2: num,N: nat] :
( ( ( numeral_numeral @ nat @ K2 )
= ( suc @ N ) )
= ( ( pred_numeral @ K2 )
= N ) ) ).
% eq_numeral_Suc
thf(fact_2196_Suc__eq__numeral,axiom,
! [N: nat,K2: num] :
( ( ( suc @ N )
= ( numeral_numeral @ nat @ K2 ) )
= ( N
= ( pred_numeral @ K2 ) ) ) ).
% Suc_eq_numeral
thf(fact_2197_pred__numeral__inc,axiom,
! [K2: num] :
( ( pred_numeral @ ( inc @ K2 ) )
= ( numeral_numeral @ nat @ K2 ) ) ).
% pred_numeral_inc
thf(fact_2198_pred__numeral__simps_I3_J,axiom,
! [K2: num] :
( ( pred_numeral @ ( bit1 @ K2 ) )
= ( numeral_numeral @ nat @ ( bit0 @ K2 ) ) ) ).
% pred_numeral_simps(3)
thf(fact_2199_less__numeral__Suc,axiom,
! [K2: num,N: nat] :
( ( ord_less @ nat @ ( numeral_numeral @ nat @ K2 ) @ ( suc @ N ) )
= ( ord_less @ nat @ ( pred_numeral @ K2 ) @ N ) ) ).
% less_numeral_Suc
thf(fact_2200_less__Suc__numeral,axiom,
! [N: nat,K2: num] :
( ( ord_less @ nat @ ( suc @ N ) @ ( numeral_numeral @ nat @ K2 ) )
= ( ord_less @ nat @ N @ ( pred_numeral @ K2 ) ) ) ).
% less_Suc_numeral
thf(fact_2201_le__numeral__Suc,axiom,
! [K2: num,N: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ K2 ) @ ( suc @ N ) )
= ( ord_less_eq @ nat @ ( pred_numeral @ K2 ) @ N ) ) ).
% le_numeral_Suc
thf(fact_2202_le__Suc__numeral,axiom,
! [N: nat,K2: num] :
( ( ord_less_eq @ nat @ ( suc @ N ) @ ( numeral_numeral @ nat @ K2 ) )
= ( ord_less_eq @ nat @ N @ ( pred_numeral @ K2 ) ) ) ).
% le_Suc_numeral
thf(fact_2203_diff__numeral__Suc,axiom,
! [K2: num,N: nat] :
( ( minus_minus @ nat @ ( numeral_numeral @ nat @ K2 ) @ ( suc @ N ) )
= ( minus_minus @ nat @ ( pred_numeral @ K2 ) @ N ) ) ).
% diff_numeral_Suc
thf(fact_2204_diff__Suc__numeral,axiom,
! [N: nat,K2: num] :
( ( minus_minus @ nat @ ( suc @ N ) @ ( numeral_numeral @ nat @ K2 ) )
= ( minus_minus @ nat @ N @ ( pred_numeral @ K2 ) ) ) ).
% diff_Suc_numeral
thf(fact_2205_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_2206_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_2207_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_2208_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_2209_signed__take__bit__numeral__bit0,axiom,
! [L: num,K2: num] :
( ( bit_ri4674362597316999326ke_bit @ int @ ( numeral_numeral @ nat @ L ) @ ( numeral_numeral @ int @ ( bit0 @ K2 ) ) )
= ( times_times @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( pred_numeral @ L ) @ ( numeral_numeral @ int @ K2 ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ).
% signed_take_bit_numeral_bit0
thf(fact_2210_signed__take__bit__numeral__minus__bit0,axiom,
! [L: num,K2: num] :
( ( bit_ri4674362597316999326ke_bit @ int @ ( numeral_numeral @ nat @ L ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ K2 ) ) ) )
= ( times_times @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( pred_numeral @ L ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K2 ) ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ).
% signed_take_bit_numeral_minus_bit0
thf(fact_2211_Collect__case__prod__mono,axiom,
! [B: $tType,A: $tType,A3: A > B > $o,B5: A > B > $o] :
( ( ord_less_eq @ ( A > B > $o ) @ A3 @ B5 )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ A3 ) ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ B5 ) ) ) ) ).
% Collect_case_prod_mono
thf(fact_2212_prod_Odisc__eq__case,axiom,
! [B: $tType,A: $tType,Prod: product_prod @ A @ B] :
( product_case_prod @ A @ B @ $o
@ ^ [Uu3: A,Uv3: B] : $true
@ Prod ) ).
% prod.disc_eq_case
thf(fact_2213_num__induct,axiom,
! [P3: num > $o,X: num] :
( ( P3 @ one2 )
=> ( ! [X5: num] :
( ( P3 @ X5 )
=> ( P3 @ ( inc @ X5 ) ) )
=> ( P3 @ X ) ) ) ).
% num_induct
thf(fact_2214_add__inc,axiom,
! [X: num,Y2: num] :
( ( plus_plus @ num @ X @ ( inc @ Y2 ) )
= ( inc @ ( plus_plus @ num @ X @ Y2 ) ) ) ).
% add_inc
thf(fact_2215_numeral__eq__Suc,axiom,
( ( numeral_numeral @ nat )
= ( ^ [K3: num] : ( suc @ ( pred_numeral @ K3 ) ) ) ) ).
% numeral_eq_Suc
thf(fact_2216_inc_Osimps_I1_J,axiom,
( ( inc @ one2 )
= ( bit0 @ one2 ) ) ).
% inc.simps(1)
thf(fact_2217_inc_Osimps_I2_J,axiom,
! [X: num] :
( ( inc @ ( bit0 @ X ) )
= ( bit1 @ X ) ) ).
% inc.simps(2)
thf(fact_2218_inc_Osimps_I3_J,axiom,
! [X: num] :
( ( inc @ ( bit1 @ X ) )
= ( bit0 @ ( inc @ X ) ) ) ).
% inc.simps(3)
thf(fact_2219_add__One,axiom,
! [X: num] :
( ( plus_plus @ num @ X @ one2 )
= ( inc @ X ) ) ).
% add_One
thf(fact_2220_mult__inc,axiom,
! [X: num,Y2: num] :
( ( times_times @ num @ X @ ( inc @ Y2 ) )
= ( plus_plus @ num @ ( times_times @ num @ X @ Y2 ) @ X ) ) ).
% mult_inc
thf(fact_2221_pred__numeral__def,axiom,
( pred_numeral
= ( ^ [K3: num] : ( minus_minus @ nat @ ( numeral_numeral @ nat @ K3 ) @ ( one_one @ nat ) ) ) ) ).
% pred_numeral_def
thf(fact_2222_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_2223_Divides_Oadjust__div__def,axiom,
( adjust_div
= ( product_case_prod @ int @ int @ int
@ ^ [Q5: int,R5: int] :
( plus_plus @ int @ Q5
@ ( zero_neq_one_of_bool @ int
@ ( R5
!= ( zero_zero @ int ) ) ) ) ) ) ).
% Divides.adjust_div_def
thf(fact_2224_take__bit__numeral__minus__bit1,axiom,
! [L: num,K2: num] :
( ( bit_se2584673776208193580ke_bit @ int @ ( numeral_numeral @ nat @ L ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ K2 ) ) ) )
= ( plus_plus @ int @ ( times_times @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( pred_numeral @ L ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( inc @ K2 ) ) ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( one_one @ int ) ) ) ).
% take_bit_numeral_minus_bit1
thf(fact_2225_take__bit__numeral__bit0,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [L: num,K2: num] :
( ( bit_se2584673776208193580ke_bit @ A @ ( numeral_numeral @ nat @ L ) @ ( numeral_numeral @ A @ ( bit0 @ K2 ) ) )
= ( times_times @ A @ ( bit_se2584673776208193580ke_bit @ A @ ( pred_numeral @ L ) @ ( numeral_numeral @ A @ K2 ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% take_bit_numeral_bit0
thf(fact_2226_divmod__nat__def,axiom,
( divmod_nat
= ( ^ [M2: nat,N2: nat] : ( product_Pair @ nat @ nat @ ( divide_divide @ nat @ M2 @ N2 ) @ ( modulo_modulo @ nat @ M2 @ N2 ) ) ) ) ).
% divmod_nat_def
thf(fact_2227_take__bit__numeral__minus__bit0,axiom,
! [L: num,K2: num] :
( ( bit_se2584673776208193580ke_bit @ int @ ( numeral_numeral @ nat @ L ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ K2 ) ) ) )
= ( times_times @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( pred_numeral @ L ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K2 ) ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ).
% take_bit_numeral_minus_bit0
thf(fact_2228_take__bit__numeral__bit1,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [L: num,K2: num] :
( ( bit_se2584673776208193580ke_bit @ A @ ( numeral_numeral @ nat @ L ) @ ( numeral_numeral @ A @ ( bit1 @ K2 ) ) )
= ( plus_plus @ A @ ( times_times @ A @ ( bit_se2584673776208193580ke_bit @ A @ ( pred_numeral @ L ) @ ( numeral_numeral @ A @ K2 ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) ) ) ) ).
% take_bit_numeral_bit1
thf(fact_2229_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_2230_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_2231_of__int__code__if,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( ( ring_1_of_int @ A )
= ( ^ [K3: int] :
( if @ A
@ ( K3
= ( zero_zero @ int ) )
@ ( zero_zero @ A )
@ ( if @ A @ ( ord_less @ int @ K3 @ ( zero_zero @ int ) ) @ ( uminus_uminus @ A @ ( ring_1_of_int @ A @ ( uminus_uminus @ int @ K3 ) ) )
@ ( if @ A
@ ( ( modulo_modulo @ int @ K3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) )
= ( zero_zero @ int ) )
@ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( ring_1_of_int @ A @ ( divide_divide @ int @ K3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) )
@ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( ring_1_of_int @ A @ ( divide_divide @ int @ K3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) @ ( one_one @ A ) ) ) ) ) ) ) ) ).
% of_int_code_if
thf(fact_2232_int__ge__less__than2__def,axiom,
( int_ge_less_than2
= ( ^ [D5: int] :
( collect @ ( product_prod @ int @ int )
@ ( product_case_prod @ int @ int @ $o
@ ^ [Z5: int,Z6: int] :
( ( ord_less_eq @ int @ D5 @ Z6 )
& ( ord_less @ int @ Z5 @ Z6 ) ) ) ) ) ) ).
% int_ge_less_than2_def
thf(fact_2233_int__ge__less__than__def,axiom,
( int_ge_less_than
= ( ^ [D5: int] :
( collect @ ( product_prod @ int @ int )
@ ( product_case_prod @ int @ int @ $o
@ ^ [Z5: int,Z6: int] :
( ( ord_less_eq @ int @ D5 @ Z5 )
& ( ord_less @ int @ Z5 @ Z6 ) ) ) ) ) ) ).
% int_ge_less_than_def
thf(fact_2234_dbl__dec__simps_I1_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [K2: num] :
( ( neg_numeral_dbl_dec @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ K2 ) ) )
= ( uminus_uminus @ A @ ( neg_numeral_dbl_inc @ A @ ( numeral_numeral @ A @ K2 ) ) ) ) ) ).
% dbl_dec_simps(1)
thf(fact_2235_dbl__inc__simps_I1_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [K2: num] :
( ( neg_numeral_dbl_inc @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ K2 ) ) )
= ( uminus_uminus @ A @ ( neg_numeral_dbl_dec @ A @ ( numeral_numeral @ A @ K2 ) ) ) ) ) ).
% dbl_inc_simps(1)
thf(fact_2236_predicate2I,axiom,
! [B: $tType,A: $tType,P3: A > B > $o,Q2: A > B > $o] :
( ! [X5: A,Y7: B] :
( ( P3 @ X5 @ Y7 )
=> ( Q2 @ X5 @ Y7 ) )
=> ( ord_less_eq @ ( A > B > $o ) @ P3 @ Q2 ) ) ).
% predicate2I
thf(fact_2237_of__int__le__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [W: int,Z3: int] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ W ) @ ( ring_1_of_int @ A @ Z3 ) )
= ( ord_less_eq @ int @ W @ Z3 ) ) ) ).
% of_int_le_iff
thf(fact_2238_of__int__numeral,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [K2: num] :
( ( ring_1_of_int @ A @ ( numeral_numeral @ int @ K2 ) )
= ( numeral_numeral @ A @ K2 ) ) ) ).
% of_int_numeral
thf(fact_2239_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_2240_of__int__less__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [W: int,Z3: int] :
( ( ord_less @ A @ ( ring_1_of_int @ A @ W ) @ ( ring_1_of_int @ A @ Z3 ) )
= ( ord_less @ int @ W @ Z3 ) ) ) ).
% of_int_less_iff
thf(fact_2241_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_2242_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_2243_of__int__mult,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [W: int,Z3: int] :
( ( ring_1_of_int @ A @ ( times_times @ int @ W @ Z3 ) )
= ( times_times @ A @ ( ring_1_of_int @ A @ W ) @ ( ring_1_of_int @ A @ Z3 ) ) ) ) ).
% of_int_mult
thf(fact_2244_of__int__add,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [W: int,Z3: int] :
( ( ring_1_of_int @ A @ ( plus_plus @ int @ W @ Z3 ) )
= ( plus_plus @ A @ ( ring_1_of_int @ A @ W ) @ ( ring_1_of_int @ A @ Z3 ) ) ) ) ).
% of_int_add
thf(fact_2245_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_2246_of__int__eq__of__int__power__cancel__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [B2: int,W: nat,X: int] :
( ( ( power_power @ A @ ( ring_1_of_int @ A @ B2 ) @ W )
= ( ring_1_of_int @ A @ X ) )
= ( ( power_power @ int @ B2 @ W )
= X ) ) ) ).
% of_int_eq_of_int_power_cancel_iff
thf(fact_2247_of__int__power__eq__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [X: int,B2: int,W: nat] :
( ( ( ring_1_of_int @ A @ X )
= ( power_power @ A @ ( ring_1_of_int @ A @ B2 ) @ W ) )
= ( X
= ( power_power @ int @ B2 @ W ) ) ) ) ).
% of_int_power_eq_of_int_cancel_iff
thf(fact_2248_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_2249_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_2250_dbl__inc__simps_I5_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [K2: num] :
( ( neg_numeral_dbl_inc @ A @ ( numeral_numeral @ A @ K2 ) )
= ( numeral_numeral @ A @ ( bit1 @ K2 ) ) ) ) ).
% dbl_inc_simps(5)
thf(fact_2251_dbl__dec__simps_I5_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [K2: num] :
( ( neg_numeral_dbl_dec @ A @ ( numeral_numeral @ A @ K2 ) )
= ( numeral_numeral @ A @ ( bitM @ K2 ) ) ) ) ).
% dbl_dec_simps(5)
thf(fact_2252_pred__numeral__simps_I2_J,axiom,
! [K2: num] :
( ( pred_numeral @ ( bit0 @ K2 ) )
= ( numeral_numeral @ nat @ ( bitM @ K2 ) ) ) ).
% pred_numeral_simps(2)
thf(fact_2253_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_2254_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_2255_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_2256_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_2257_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_2258_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_2259_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_2260_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_2261_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_2262_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_2263_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_2264_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_2265_of__int__le__of__int__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [B2: int,W: nat,X: int] :
( ( ord_less_eq @ A @ ( power_power @ A @ ( ring_1_of_int @ A @ B2 ) @ W ) @ ( ring_1_of_int @ A @ X ) )
= ( ord_less_eq @ int @ ( power_power @ int @ B2 @ W ) @ X ) ) ) ).
% of_int_le_of_int_power_cancel_iff
thf(fact_2266_of__int__power__le__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: int,B2: int,W: nat] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ X ) @ ( power_power @ A @ ( ring_1_of_int @ A @ B2 ) @ W ) )
= ( ord_less_eq @ int @ X @ ( power_power @ int @ B2 @ W ) ) ) ) ).
% of_int_power_le_of_int_cancel_iff
thf(fact_2267_numeral__power__eq__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [X: num,N: nat,Y2: int] :
( ( ( power_power @ A @ ( numeral_numeral @ A @ X ) @ N )
= ( ring_1_of_int @ A @ Y2 ) )
= ( ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N )
= Y2 ) ) ) ).
% numeral_power_eq_of_int_cancel_iff
thf(fact_2268_of__int__eq__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [Y2: int,X: num,N: nat] :
( ( ( ring_1_of_int @ A @ Y2 )
= ( power_power @ A @ ( numeral_numeral @ A @ X ) @ N ) )
= ( Y2
= ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N ) ) ) ) ).
% of_int_eq_numeral_power_cancel_iff
thf(fact_2269_of__int__less__of__int__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [B2: int,W: nat,X: int] :
( ( ord_less @ A @ ( power_power @ A @ ( ring_1_of_int @ A @ B2 ) @ W ) @ ( ring_1_of_int @ A @ X ) )
= ( ord_less @ int @ ( power_power @ int @ B2 @ W ) @ X ) ) ) ).
% of_int_less_of_int_power_cancel_iff
thf(fact_2270_of__int__power__less__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: int,B2: int,W: nat] :
( ( ord_less @ A @ ( ring_1_of_int @ A @ X ) @ ( power_power @ A @ ( ring_1_of_int @ A @ B2 ) @ W ) )
= ( ord_less @ int @ X @ ( power_power @ int @ B2 @ W ) ) ) ) ).
% of_int_power_less_of_int_cancel_iff
thf(fact_2271_numeral__power__le__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: num,N: nat,A2: int] :
( ( ord_less_eq @ A @ ( power_power @ A @ ( numeral_numeral @ A @ X ) @ N ) @ ( ring_1_of_int @ A @ A2 ) )
= ( ord_less_eq @ int @ ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N ) @ A2 ) ) ) ).
% numeral_power_le_of_int_cancel_iff
thf(fact_2272_of__int__le__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: int,X: num,N: nat] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ A2 ) @ ( power_power @ A @ ( numeral_numeral @ A @ X ) @ N ) )
= ( ord_less_eq @ int @ A2 @ ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N ) ) ) ) ).
% of_int_le_numeral_power_cancel_iff
thf(fact_2273_numeral__power__less__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: num,N: nat,A2: int] :
( ( ord_less @ A @ ( power_power @ A @ ( numeral_numeral @ A @ X ) @ N ) @ ( ring_1_of_int @ A @ A2 ) )
= ( ord_less @ int @ ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N ) @ A2 ) ) ) ).
% numeral_power_less_of_int_cancel_iff
thf(fact_2274_of__int__less__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: int,X: num,N: nat] :
( ( ord_less @ A @ ( ring_1_of_int @ A @ A2 ) @ ( power_power @ A @ ( numeral_numeral @ A @ X ) @ N ) )
= ( ord_less @ int @ A2 @ ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N ) ) ) ) ).
% of_int_less_numeral_power_cancel_iff
thf(fact_2275_neg__numeral__power__eq__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [X: num,N: nat,Y2: int] :
( ( ( power_power @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ X ) ) @ N )
= ( ring_1_of_int @ A @ Y2 ) )
= ( ( power_power @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ X ) ) @ N )
= Y2 ) ) ) ).
% neg_numeral_power_eq_of_int_cancel_iff
thf(fact_2276_of__int__eq__neg__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [Y2: int,X: num,N: nat] :
( ( ( ring_1_of_int @ A @ Y2 )
= ( power_power @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ X ) ) @ N ) )
= ( Y2
= ( power_power @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ X ) ) @ N ) ) ) ) ).
% of_int_eq_neg_numeral_power_cancel_iff
thf(fact_2277_neg__numeral__power__le__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: num,N: nat,A2: int] :
( ( ord_less_eq @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ X ) ) @ N ) @ ( ring_1_of_int @ A @ A2 ) )
= ( ord_less_eq @ int @ ( power_power @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ X ) ) @ N ) @ A2 ) ) ) ).
% neg_numeral_power_le_of_int_cancel_iff
thf(fact_2278_of__int__le__neg__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: int,X: num,N: nat] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ A2 ) @ ( power_power @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ X ) ) @ N ) )
= ( ord_less_eq @ int @ A2 @ ( power_power @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ X ) ) @ N ) ) ) ) ).
% of_int_le_neg_numeral_power_cancel_iff
thf(fact_2279_neg__numeral__power__less__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: num,N: nat,A2: int] :
( ( ord_less @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ X ) ) @ N ) @ ( ring_1_of_int @ A @ A2 ) )
= ( ord_less @ int @ ( power_power @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ X ) ) @ N ) @ A2 ) ) ) ).
% neg_numeral_power_less_of_int_cancel_iff
thf(fact_2280_of__int__less__neg__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: int,X: num,N: nat] :
( ( ord_less @ A @ ( ring_1_of_int @ A @ A2 ) @ ( power_power @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ X ) ) @ N ) )
= ( ord_less @ int @ A2 @ ( power_power @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ X ) ) @ N ) ) ) ) ).
% of_int_less_neg_numeral_power_cancel_iff
thf(fact_2281_predicate2D,axiom,
! [A: $tType,B: $tType,P3: A > B > $o,Q2: A > B > $o,X: A,Y2: B] :
( ( ord_less_eq @ ( A > B > $o ) @ P3 @ Q2 )
=> ( ( P3 @ X @ Y2 )
=> ( Q2 @ X @ Y2 ) ) ) ).
% predicate2D
thf(fact_2282_rev__predicate2D,axiom,
! [A: $tType,B: $tType,P3: A > B > $o,X: A,Y2: B,Q2: A > B > $o] :
( ( P3 @ X @ Y2 )
=> ( ( ord_less_eq @ ( A > B > $o ) @ P3 @ Q2 )
=> ( Q2 @ X @ Y2 ) ) ) ).
% rev_predicate2D
thf(fact_2283_mult__of__int__commute,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X: int,Y2: A] :
( ( times_times @ A @ ( ring_1_of_int @ A @ X ) @ Y2 )
= ( times_times @ A @ Y2 @ ( ring_1_of_int @ A @ X ) ) ) ) ).
% mult_of_int_commute
thf(fact_2284_semiring__norm_I26_J,axiom,
( ( bitM @ one2 )
= one2 ) ).
% semiring_norm(26)
thf(fact_2285_take__bit__of__int,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: nat,K2: int] :
( ( bit_se2584673776208193580ke_bit @ A @ N @ ( ring_1_of_int @ A @ K2 ) )
= ( ring_1_of_int @ A @ ( bit_se2584673776208193580ke_bit @ int @ N @ K2 ) ) ) ) ).
% take_bit_of_int
thf(fact_2286_semiring__norm_I28_J,axiom,
! [N: num] :
( ( bitM @ ( bit1 @ N ) )
= ( bit1 @ ( bit0 @ N ) ) ) ).
% semiring_norm(28)
thf(fact_2287_semiring__norm_I27_J,axiom,
! [N: num] :
( ( bitM @ ( bit0 @ N ) )
= ( bit1 @ ( bitM @ N ) ) ) ).
% semiring_norm(27)
thf(fact_2288_inc__BitM__eq,axiom,
! [N: num] :
( ( inc @ ( bitM @ N ) )
= ( bit0 @ N ) ) ).
% inc_BitM_eq
thf(fact_2289_BitM__inc__eq,axiom,
! [N: num] :
( ( bitM @ ( inc @ N ) )
= ( bit1 @ N ) ) ).
% BitM_inc_eq
thf(fact_2290_real__of__int__div4,axiom,
! [N: int,X: int] : ( ord_less_eq @ real @ ( ring_1_of_int @ real @ ( divide_divide @ int @ N @ X ) ) @ ( divide_divide @ real @ ( ring_1_of_int @ real @ N ) @ ( ring_1_of_int @ real @ X ) ) ) ).
% real_of_int_div4
thf(fact_2291_real__of__int__div,axiom,
! [D2: int,N: int] :
( ( dvd_dvd @ int @ D2 @ N )
=> ( ( ring_1_of_int @ real @ ( divide_divide @ int @ N @ D2 ) )
= ( divide_divide @ real @ ( ring_1_of_int @ real @ N ) @ ( ring_1_of_int @ real @ D2 ) ) ) ) ).
% real_of_int_div
thf(fact_2292_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_2293_BitM__plus__one,axiom,
! [N: num] :
( ( plus_plus @ num @ ( bitM @ N ) @ one2 )
= ( bit0 @ N ) ) ).
% BitM_plus_one
thf(fact_2294_one__plus__BitM,axiom,
! [N: num] :
( ( plus_plus @ num @ one2 @ ( bitM @ N ) )
= ( bit0 @ N ) ) ).
% one_plus_BitM
thf(fact_2295_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_2296_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_2297_of__int__neg__numeral,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [K2: num] :
( ( ring_1_of_int @ A @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K2 ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ K2 ) ) ) ) ).
% of_int_neg_numeral
thf(fact_2298_int__le__real__less,axiom,
( ( ord_less_eq @ int )
= ( ^ [N2: int,M2: int] : ( ord_less @ real @ ( ring_1_of_int @ real @ N2 ) @ ( plus_plus @ real @ ( ring_1_of_int @ real @ M2 ) @ ( one_one @ real ) ) ) ) ) ).
% int_le_real_less
thf(fact_2299_int__less__real__le,axiom,
( ( ord_less @ int )
= ( ^ [N2: int,M2: int] : ( ord_less_eq @ real @ ( plus_plus @ real @ ( ring_1_of_int @ real @ N2 ) @ ( one_one @ real ) ) @ ( ring_1_of_int @ real @ M2 ) ) ) ) ).
% int_less_real_le
thf(fact_2300_dbl__inc__def,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( neg_numeral_dbl_inc @ A )
= ( ^ [X4: A] : ( plus_plus @ A @ ( plus_plus @ A @ X4 @ X4 ) @ ( one_one @ A ) ) ) ) ) ).
% dbl_inc_def
thf(fact_2301_real__of__int__div__aux,axiom,
! [X: int,D2: int] :
( ( divide_divide @ real @ ( ring_1_of_int @ real @ X ) @ ( ring_1_of_int @ real @ D2 ) )
= ( plus_plus @ real @ ( ring_1_of_int @ real @ ( divide_divide @ int @ X @ D2 ) ) @ ( divide_divide @ real @ ( ring_1_of_int @ real @ ( modulo_modulo @ int @ X @ D2 ) ) @ ( ring_1_of_int @ real @ D2 ) ) ) ) ).
% real_of_int_div_aux
thf(fact_2302_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_2303_odd__numeral__BitM,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [W: num] :
~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ A @ ( bitM @ W ) ) ) ) ).
% odd_numeral_BitM
thf(fact_2304_real__of__int__div2,axiom,
! [N: int,X: int] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( minus_minus @ real @ ( divide_divide @ real @ ( ring_1_of_int @ real @ N ) @ ( ring_1_of_int @ real @ X ) ) @ ( ring_1_of_int @ real @ ( divide_divide @ int @ N @ X ) ) ) ) ).
% real_of_int_div2
thf(fact_2305_real__of__int__div3,axiom,
! [N: int,X: int] : ( ord_less_eq @ real @ ( minus_minus @ real @ ( divide_divide @ real @ ( ring_1_of_int @ real @ N ) @ ( ring_1_of_int @ real @ X ) ) @ ( ring_1_of_int @ real @ ( divide_divide @ int @ N @ X ) ) ) @ ( one_one @ real ) ) ).
% real_of_int_div3
thf(fact_2306_even__of__int__iff,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [K2: int] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( ring_1_of_int @ A @ K2 ) )
= ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K2 ) ) ) ).
% even_of_int_iff
thf(fact_2307_floor__exists,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X: A] :
? [Z: int] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Z ) @ X )
& ( ord_less @ A @ X @ ( ring_1_of_int @ A @ ( plus_plus @ int @ Z @ ( one_one @ int ) ) ) ) ) ) ).
% floor_exists
thf(fact_2308_floor__exists1,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X: A] :
? [X5: int] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ X5 ) @ X )
& ( ord_less @ A @ X @ ( ring_1_of_int @ A @ ( plus_plus @ int @ X5 @ ( 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 = X5 ) ) ) ) ).
% floor_exists1
thf(fact_2309_pred__subset__eq2,axiom,
! [B: $tType,A: $tType,R4: set @ ( product_prod @ A @ B ),S3: set @ ( product_prod @ A @ B )] :
( ( ord_less_eq @ ( A > B > $o )
@ ^ [X4: A,Y: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y ) @ R4 )
@ ^ [X4: A,Y: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y ) @ S3 ) )
= ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R4 @ S3 ) ) ).
% pred_subset_eq2
thf(fact_2310_accp__subset,axiom,
! [A: $tType,R1: A > A > $o,R22: A > A > $o] :
( ( ord_less_eq @ ( A > A > $o ) @ R1 @ R22 )
=> ( ord_less_eq @ ( A > $o ) @ ( accp @ A @ R22 ) @ ( accp @ A @ R1 ) ) ) ).
% accp_subset
thf(fact_2311_signed__take__bit__eq__take__bit__minus,axiom,
( ( bit_ri4674362597316999326ke_bit @ int )
= ( ^ [N2: nat,K3: int] : ( minus_minus @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( suc @ N2 ) @ K3 ) @ ( times_times @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( suc @ N2 ) ) @ ( zero_neq_one_of_bool @ int @ ( bit_se5641148757651400278ts_bit @ int @ K3 @ N2 ) ) ) ) ) ) ).
% signed_take_bit_eq_take_bit_minus
thf(fact_2312_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_2313_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_2314_mask__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se2239418461657761734s_mask @ A @ ( zero_zero @ nat ) )
= ( zero_zero @ A ) ) ) ).
% mask_0
thf(fact_2315_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_2316_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_2317_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_2318_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_2319_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_2320_signed__take__bit__nonnegative__iff,axiom,
! [N: nat,K2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_ri4674362597316999326ke_bit @ int @ N @ K2 ) )
= ( ~ ( bit_se5641148757651400278ts_bit @ int @ K2 @ N ) ) ) ).
% signed_take_bit_nonnegative_iff
thf(fact_2321_signed__take__bit__negative__iff,axiom,
! [N: nat,K2: int] :
( ( ord_less @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N @ K2 ) @ ( zero_zero @ int ) )
= ( bit_se5641148757651400278ts_bit @ int @ K2 @ N ) ) ).
% signed_take_bit_negative_iff
thf(fact_2322_bit__numeral__simps_I2_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [W: num,N: num] :
( ( bit_se5641148757651400278ts_bit @ A @ ( numeral_numeral @ A @ ( bit0 @ W ) ) @ ( numeral_numeral @ nat @ N ) )
= ( bit_se5641148757651400278ts_bit @ A @ ( numeral_numeral @ A @ W ) @ ( pred_numeral @ N ) ) ) ) ).
% bit_numeral_simps(2)
thf(fact_2323_bit__minus__numeral__Bit0__Suc__iff,axiom,
! [W: num,N: nat] :
( ( bit_se5641148757651400278ts_bit @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ W ) ) ) @ ( suc @ N ) )
= ( bit_se5641148757651400278ts_bit @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ W ) ) @ N ) ) ).
% bit_minus_numeral_Bit0_Suc_iff
thf(fact_2324_bit__numeral__simps_I3_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [W: num,N: num] :
( ( bit_se5641148757651400278ts_bit @ A @ ( numeral_numeral @ A @ ( bit1 @ W ) ) @ ( numeral_numeral @ nat @ N ) )
= ( bit_se5641148757651400278ts_bit @ A @ ( numeral_numeral @ A @ W ) @ ( pred_numeral @ N ) ) ) ) ).
% bit_numeral_simps(3)
thf(fact_2325_bit__minus__numeral__Bit1__Suc__iff,axiom,
! [W: num,N: nat] :
( ( bit_se5641148757651400278ts_bit @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ W ) ) ) @ ( suc @ N ) )
= ( ~ ( bit_se5641148757651400278ts_bit @ int @ ( numeral_numeral @ int @ W ) @ N ) ) ) ).
% bit_minus_numeral_Bit1_Suc_iff
thf(fact_2326_bit__0,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A2: A] :
( ( bit_se5641148757651400278ts_bit @ A @ A2 @ ( zero_zero @ nat ) )
= ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) ) ) ) ).
% bit_0
thf(fact_2327_bit__minus__numeral__int_I1_J,axiom,
! [W: num,N: num] :
( ( bit_se5641148757651400278ts_bit @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ W ) ) ) @ ( numeral_numeral @ nat @ N ) )
= ( bit_se5641148757651400278ts_bit @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ W ) ) @ ( pred_numeral @ N ) ) ) ).
% bit_minus_numeral_int(1)
thf(fact_2328_bit__minus__numeral__int_I2_J,axiom,
! [W: num,N: num] :
( ( bit_se5641148757651400278ts_bit @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ W ) ) ) @ ( numeral_numeral @ nat @ N ) )
= ( ~ ( bit_se5641148757651400278ts_bit @ int @ ( numeral_numeral @ int @ W ) @ ( pred_numeral @ N ) ) ) ) ).
% bit_minus_numeral_int(2)
thf(fact_2329_bit__mod__2__iff,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A2: A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ N )
= ( ( N
= ( zero_zero @ nat ) )
& ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) ) ) ) ).
% bit_mod_2_iff
thf(fact_2330_of__int__mask__eq,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: nat] :
( ( ring_1_of_int @ A @ ( bit_se2239418461657761734s_mask @ int @ N ) )
= ( bit_se2239418461657761734s_mask @ A @ N ) ) ) ).
% of_int_mask_eq
thf(fact_2331_bit__numeral__iff,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [M: num,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( numeral_numeral @ A @ M ) @ N )
= ( bit_se5641148757651400278ts_bit @ nat @ ( numeral_numeral @ nat @ M ) @ N ) ) ) ).
% bit_numeral_iff
thf(fact_2332_bit__disjunctive__add__iff,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A2: A,B2: A,N: nat] :
( ! [N3: nat] :
( ~ ( bit_se5641148757651400278ts_bit @ A @ A2 @ N3 )
| ~ ( bit_se5641148757651400278ts_bit @ A @ B2 @ N3 ) )
=> ( ( bit_se5641148757651400278ts_bit @ A @ ( plus_plus @ A @ A2 @ B2 ) @ N )
= ( ( bit_se5641148757651400278ts_bit @ A @ A2 @ N )
| ( bit_se5641148757651400278ts_bit @ A @ B2 @ N ) ) ) ) ) ).
% bit_disjunctive_add_iff
thf(fact_2333_bit__unset__bit__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: nat,A2: A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( bit_se2638667681897837118et_bit @ A @ M @ A2 ) @ N )
= ( ( bit_se5641148757651400278ts_bit @ A @ A2 @ N )
& ( M != N ) ) ) ) ).
% bit_unset_bit_iff
thf(fact_2334_less__eq__mask,axiom,
! [N: nat] : ( ord_less_eq @ nat @ N @ ( bit_se2239418461657761734s_mask @ nat @ N ) ) ).
% less_eq_mask
thf(fact_2335_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_2336_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_2337_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_2338_bit__take__bit__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: nat,A2: A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( bit_se2584673776208193580ke_bit @ A @ M @ A2 ) @ N )
= ( ( ord_less @ nat @ N @ M )
& ( bit_se5641148757651400278ts_bit @ A @ A2 @ N ) ) ) ) ).
% bit_take_bit_iff
thf(fact_2339_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_2340_signed__take__bit__eq__if__positive,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [A2: A,N: nat] :
( ~ ( bit_se5641148757651400278ts_bit @ A @ A2 @ N )
=> ( ( bit_ri4674362597316999326ke_bit @ A @ N @ A2 )
= ( bit_se2584673776208193580ke_bit @ A @ N @ A2 ) ) ) ) ).
% signed_take_bit_eq_if_positive
thf(fact_2341_mask__nonnegative__int,axiom,
! [N: nat] : ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se2239418461657761734s_mask @ int @ N ) ) ).
% mask_nonnegative_int
thf(fact_2342_not__mask__negative__int,axiom,
! [N: nat] :
~ ( ord_less @ int @ ( bit_se2239418461657761734s_mask @ int @ N ) @ ( zero_zero @ int ) ) ).
% not_mask_negative_int
thf(fact_2343_bit__not__int__iff_H,axiom,
! [K2: int,N: nat] :
( ( bit_se5641148757651400278ts_bit @ int @ ( minus_minus @ int @ ( uminus_uminus @ int @ K2 ) @ ( one_one @ int ) ) @ N )
= ( ~ ( bit_se5641148757651400278ts_bit @ int @ K2 @ N ) ) ) ).
% bit_not_int_iff'
thf(fact_2344_flip__bit__eq__if,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se8732182000553998342ip_bit @ A )
= ( ^ [N2: nat,A5: A] : ( if @ ( nat > A > A ) @ ( bit_se5641148757651400278ts_bit @ A @ A5 @ N2 ) @ ( bit_se2638667681897837118et_bit @ A ) @ ( bit_se5668285175392031749et_bit @ A ) @ N2 @ A5 ) ) ) ) ).
% flip_bit_eq_if
thf(fact_2345_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_2346_bit__imp__take__bit__positive,axiom,
! [N: nat,M: nat,K2: int] :
( ( ord_less @ nat @ N @ M )
=> ( ( bit_se5641148757651400278ts_bit @ int @ K2 @ N )
=> ( ord_less @ int @ ( zero_zero @ int ) @ ( bit_se2584673776208193580ke_bit @ int @ M @ K2 ) ) ) ) ).
% bit_imp_take_bit_positive
thf(fact_2347_bit__concat__bit__iff,axiom,
! [M: nat,K2: int,L: int,N: nat] :
( ( bit_se5641148757651400278ts_bit @ int @ ( bit_concat_bit @ M @ K2 @ L ) @ N )
= ( ( ( ord_less @ nat @ N @ M )
& ( bit_se5641148757651400278ts_bit @ int @ K2 @ N ) )
| ( ( ord_less_eq @ nat @ M @ N )
& ( bit_se5641148757651400278ts_bit @ int @ L @ ( minus_minus @ nat @ N @ M ) ) ) ) ) ).
% bit_concat_bit_iff
thf(fact_2348_signed__take__bit__eq__concat__bit,axiom,
( ( bit_ri4674362597316999326ke_bit @ int )
= ( ^ [N2: nat,K3: int] : ( bit_concat_bit @ N2 @ K3 @ ( uminus_uminus @ int @ ( zero_neq_one_of_bool @ int @ ( bit_se5641148757651400278ts_bit @ int @ K3 @ N2 ) ) ) ) ) ) ).
% signed_take_bit_eq_concat_bit
thf(fact_2349_exp__eq__0__imp__not__bit,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [N: nat,A2: A] :
( ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N )
= ( zero_zero @ A ) )
=> ~ ( bit_se5641148757651400278ts_bit @ A @ A2 @ N ) ) ) ).
% exp_eq_0_imp_not_bit
thf(fact_2350_bit__Suc,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A2: A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ A2 @ ( suc @ N ) )
= ( bit_se5641148757651400278ts_bit @ A @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ N ) ) ) ).
% bit_Suc
thf(fact_2351_stable__imp__bit__iff__odd,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A2: A,N: nat] :
( ( ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= A2 )
=> ( ( bit_se5641148757651400278ts_bit @ A @ A2 @ N )
= ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) ) ) ) ) ).
% stable_imp_bit_iff_odd
thf(fact_2352_bit__iff__idd__imp__stable,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A2: A] :
( ! [N3: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ A2 @ N3 )
= ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) ) )
=> ( ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= A2 ) ) ) ).
% bit_iff_idd_imp_stable
thf(fact_2353_pred__equals__eq2,axiom,
! [B: $tType,A: $tType,R4: set @ ( product_prod @ A @ B ),S3: set @ ( product_prod @ A @ B )] :
( ( ( ^ [X4: A,Y: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y ) @ R4 ) )
= ( ^ [X4: A,Y: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y ) @ S3 ) ) )
= ( R4 = S3 ) ) ).
% pred_equals_eq2
thf(fact_2354_take__bit__eq__mask__iff,axiom,
! [N: nat,K2: int] :
( ( ( bit_se2584673776208193580ke_bit @ int @ N @ K2 )
= ( bit_se2239418461657761734s_mask @ int @ N ) )
= ( ( bit_se2584673776208193580ke_bit @ int @ N @ ( plus_plus @ int @ K2 @ ( one_one @ int ) ) )
= ( zero_zero @ int ) ) ) ).
% take_bit_eq_mask_iff
thf(fact_2355_int__bit__bound,axiom,
! [K2: int] :
~ ! [N3: nat] :
( ! [M3: nat] :
( ( ord_less_eq @ nat @ N3 @ M3 )
=> ( ( bit_se5641148757651400278ts_bit @ int @ K2 @ M3 )
= ( bit_se5641148757651400278ts_bit @ int @ K2 @ N3 ) ) )
=> ~ ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( ( bit_se5641148757651400278ts_bit @ int @ K2 @ ( minus_minus @ nat @ N3 @ ( one_one @ nat ) ) )
= ( ~ ( bit_se5641148757651400278ts_bit @ int @ K2 @ N3 ) ) ) ) ) ).
% int_bit_bound
thf(fact_2356_bit__iff__odd,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ( ( bit_se5641148757651400278ts_bit @ A )
= ( ^ [A5: A,N2: nat] :
~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A5 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ) ) ).
% bit_iff_odd
thf(fact_2357_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_2358_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_2359_bit__int__def,axiom,
( ( bit_se5641148757651400278ts_bit @ int )
= ( ^ [K3: int,N2: nat] :
~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( divide_divide @ int @ K3 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ) ).
% bit_int_def
thf(fact_2360_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_2361_even__bit__succ__iff,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A2: A,N: nat] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
=> ( ( bit_se5641148757651400278ts_bit @ A @ ( plus_plus @ A @ ( one_one @ A ) @ A2 ) @ N )
= ( ( bit_se5641148757651400278ts_bit @ A @ A2 @ N )
| ( N
= ( zero_zero @ nat ) ) ) ) ) ) ).
% even_bit_succ_iff
thf(fact_2362_odd__bit__iff__bit__pred,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A2: A,N: nat] :
( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
=> ( ( bit_se5641148757651400278ts_bit @ A @ A2 @ N )
= ( ( bit_se5641148757651400278ts_bit @ A @ ( minus_minus @ A @ A2 @ ( one_one @ A ) ) @ N )
| ( N
= ( zero_zero @ nat ) ) ) ) ) ) ).
% odd_bit_iff_bit_pred
thf(fact_2363_mask__nat__def,axiom,
( ( bit_se2239418461657761734s_mask @ nat )
= ( ^ [N2: nat] : ( minus_minus @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) @ ( one_one @ nat ) ) ) ) ).
% mask_nat_def
thf(fact_2364_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_2365_mask__int__def,axiom,
( ( bit_se2239418461657761734s_mask @ int )
= ( ^ [N2: nat] : ( minus_minus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) @ ( one_one @ int ) ) ) ) ).
% mask_int_def
thf(fact_2366_ex__le__of__int,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X: A] :
? [Z: int] : ( ord_less_eq @ A @ X @ ( ring_1_of_int @ A @ Z ) ) ) ).
% ex_le_of_int
thf(fact_2367_ex__of__int__less,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X: A] :
? [Z: int] : ( ord_less @ A @ ( ring_1_of_int @ A @ Z ) @ X ) ) ).
% ex_of_int_less
thf(fact_2368_ex__less__of__int,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X: A] :
? [Z: int] : ( ord_less @ A @ X @ ( ring_1_of_int @ A @ Z ) ) ) ).
% ex_less_of_int
thf(fact_2369_subrelI,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ A @ B )] :
( ! [X5: A,Y7: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X5 @ Y7 ) @ R )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X5 @ Y7 ) @ S ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R @ S ) ) ).
% subrelI
thf(fact_2370_bit__sum__mult__2__cases,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A,B2: A,N: nat] :
( ! [J2: nat] :
~ ( bit_se5641148757651400278ts_bit @ A @ A2 @ ( suc @ J2 ) )
=> ( ( bit_se5641148757651400278ts_bit @ A @ ( plus_plus @ A @ A2 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) @ N )
= ( ( ( N
= ( zero_zero @ nat ) )
=> ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ( bit_se5641148757651400278ts_bit @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) @ N ) ) ) ) ) ) ).
% bit_sum_mult_2_cases
thf(fact_2371_bit__rec,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ( ( bit_se5641148757651400278ts_bit @ A )
= ( ^ [A5: A,N2: nat] :
( ( ( N2
= ( zero_zero @ nat ) )
=> ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A5 ) )
& ( ( N2
!= ( zero_zero @ nat ) )
=> ( bit_se5641148757651400278ts_bit @ A @ ( divide_divide @ A @ A5 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% bit_rec
thf(fact_2372_mask__eq__exp__minus__1,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se2239418461657761734s_mask @ A )
= ( ^ [N2: nat] : ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) @ ( one_one @ A ) ) ) ) ) ).
% mask_eq_exp_minus_1
thf(fact_2373_accp__subset__induct,axiom,
! [A: $tType,D4: A > $o,R4: A > A > $o,X: A,P3: A > $o] :
( ( ord_less_eq @ ( A > $o ) @ D4 @ ( accp @ A @ R4 ) )
=> ( ! [X5: A,Z: A] :
( ( D4 @ X5 )
=> ( ( R4 @ Z @ X5 )
=> ( D4 @ Z ) ) )
=> ( ( D4 @ X )
=> ( ! [X5: A] :
( ( D4 @ X5 )
=> ( ! [Z4: A] :
( ( R4 @ Z4 @ X5 )
=> ( P3 @ Z4 ) )
=> ( P3 @ X5 ) ) )
=> ( P3 @ X ) ) ) ) ) ).
% accp_subset_induct
thf(fact_2374_set__bit__eq,axiom,
( ( bit_se5668285175392031749et_bit @ int )
= ( ^ [N2: nat,K3: int] :
( plus_plus @ int @ K3
@ ( times_times @ int
@ ( zero_neq_one_of_bool @ int
@ ~ ( bit_se5641148757651400278ts_bit @ int @ K3 @ N2 ) )
@ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ) ).
% set_bit_eq
thf(fact_2375_unset__bit__eq,axiom,
( ( bit_se2638667681897837118et_bit @ int )
= ( ^ [N2: nat,K3: int] : ( minus_minus @ int @ K3 @ ( times_times @ int @ ( zero_neq_one_of_bool @ int @ ( bit_se5641148757651400278ts_bit @ int @ K3 @ N2 ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ) ).
% unset_bit_eq
thf(fact_2376_take__bit__Suc__from__most,axiom,
! [N: nat,K2: int] :
( ( bit_se2584673776208193580ke_bit @ int @ ( suc @ N ) @ K2 )
= ( 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 @ K2 @ N ) ) ) @ ( bit_se2584673776208193580ke_bit @ int @ N @ K2 ) ) ) ).
% take_bit_Suc_from_most
thf(fact_2377_pred__subset__eq,axiom,
! [A: $tType,R4: set @ A,S3: set @ A] :
( ( ord_less_eq @ ( A > $o )
@ ^ [X4: A] : ( member @ A @ X4 @ R4 )
@ ^ [X4: A] : ( member @ A @ X4 @ S3 ) )
= ( ord_less_eq @ ( set @ A ) @ R4 @ S3 ) ) ).
% pred_subset_eq
thf(fact_2378_take__bit__eq__mask__iff__exp__dvd,axiom,
! [N: nat,K2: int] :
( ( ( bit_se2584673776208193580ke_bit @ int @ N @ K2 )
= ( bit_se2239418461657761734s_mask @ int @ N ) )
= ( dvd_dvd @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) @ ( plus_plus @ int @ K2 @ ( one_one @ int ) ) ) ) ).
% take_bit_eq_mask_iff_exp_dvd
thf(fact_2379_round__unique,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Y2: int] :
( ( ord_less @ A @ ( minus_minus @ A @ X @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( ring_1_of_int @ A @ Y2 ) )
=> ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Y2 ) @ ( plus_plus @ A @ X @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) )
=> ( ( archimedean_round @ A @ X )
= Y2 ) ) ) ) ).
% round_unique
thf(fact_2380_in__measure,axiom,
! [A: $tType,X: A,Y2: A,F3: A > nat] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y2 ) @ ( measure @ A @ F3 ) )
= ( ord_less @ nat @ ( F3 @ X ) @ ( F3 @ Y2 ) ) ) ).
% in_measure
thf(fact_2381_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_2382_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_2383_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_2384_upto_Opinduct,axiom,
! [A0: int,A12: int,P3: int > int > $o] :
( ( accp @ ( product_prod @ int @ int ) @ upto_rel @ ( product_Pair @ int @ int @ A0 @ A12 ) )
=> ( ! [I3: int,J2: int] :
( ( accp @ ( product_prod @ int @ int ) @ upto_rel @ ( product_Pair @ int @ int @ I3 @ J2 ) )
=> ( ( ( ord_less_eq @ int @ I3 @ J2 )
=> ( P3 @ ( plus_plus @ int @ I3 @ ( one_one @ int ) ) @ J2 ) )
=> ( P3 @ I3 @ J2 ) ) )
=> ( P3 @ A0 @ A12 ) ) ) ).
% upto.pinduct
thf(fact_2385_round__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [N: num] :
( ( archimedean_round @ A @ ( numeral_numeral @ A @ N ) )
= ( numeral_numeral @ int @ N ) ) ) ).
% round_numeral
thf(fact_2386_round__1,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archimedean_round @ A @ ( one_one @ A ) )
= ( one_one @ int ) ) ) ).
% round_1
thf(fact_2387_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_2388_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_2389_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_2390_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_2391_round__mono,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Y2: A] :
( ( ord_less_eq @ A @ X @ Y2 )
=> ( ord_less_eq @ int @ ( archimedean_round @ A @ X ) @ ( archimedean_round @ A @ Y2 ) ) ) ) ).
% round_mono
thf(fact_2392_bit__nat__def,axiom,
( ( bit_se5641148757651400278ts_bit @ nat )
= ( ^ [M2: nat,N2: nat] :
~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ M2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ) ).
% bit_nat_def
thf(fact_2393_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_2394_num_Osize__gen_I2_J,axiom,
! [X23: num] :
( ( size_num @ ( bit0 @ X23 ) )
= ( plus_plus @ nat @ ( size_num @ X23 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% num.size_gen(2)
thf(fact_2395_fold__atLeastAtMost__nat_Opinduct,axiom,
! [A: $tType,A0: nat > A > A,A12: nat,A23: nat,A32: A,P3: ( 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 ) @ A12 @ ( product_Pair @ nat @ A @ A23 @ A32 ) ) ) )
=> ( ! [F6: nat > A > A,A4: nat,B3: nat,Acc: A] :
( ( accp @ ( product_prod @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) ) @ ( set_fo1817059534552279752at_rel @ A ) @ ( product_Pair @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) @ F6 @ ( product_Pair @ nat @ ( product_prod @ nat @ A ) @ A4 @ ( product_Pair @ nat @ A @ B3 @ Acc ) ) ) )
=> ( ( ~ ( ord_less @ nat @ B3 @ A4 )
=> ( P3 @ F6 @ ( plus_plus @ nat @ A4 @ ( one_one @ nat ) ) @ B3 @ ( F6 @ A4 @ Acc ) ) )
=> ( P3 @ F6 @ A4 @ B3 @ Acc ) ) )
=> ( P3 @ A0 @ A12 @ A23 @ A32 ) ) ) ).
% fold_atLeastAtMost_nat.pinduct
thf(fact_2396_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_2397_exp__lower__Taylor__quadratic,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ord_less_eq @ real @ ( plus_plus @ real @ ( plus_plus @ real @ ( one_one @ real ) @ X ) @ ( divide_divide @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ ( exp @ real @ X ) ) ) ).
% exp_lower_Taylor_quadratic
thf(fact_2398_sqrt__sum__squares__half__less,axiom,
! [X: real,U: real,Y2: real] :
( ( ord_less @ real @ X @ ( divide_divide @ real @ U @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less @ real @ Y2 @ ( divide_divide @ real @ U @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y2 )
=> ( ord_less @ real @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ U ) ) ) ) ) ).
% sqrt_sum_squares_half_less
thf(fact_2399_and_Oidem,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A] :
( ( bit_se5824344872417868541ns_and @ A @ A2 @ A2 )
= A2 ) ) ).
% and.idem
thf(fact_2400_and_Oleft__idem,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A,B2: A] :
( ( bit_se5824344872417868541ns_and @ A @ A2 @ ( bit_se5824344872417868541ns_and @ A @ A2 @ B2 ) )
= ( bit_se5824344872417868541ns_and @ A @ A2 @ B2 ) ) ) ).
% and.left_idem
thf(fact_2401_and_Oright__idem,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A,B2: A] :
( ( bit_se5824344872417868541ns_and @ A @ ( bit_se5824344872417868541ns_and @ A @ A2 @ B2 ) @ B2 )
= ( bit_se5824344872417868541ns_and @ A @ A2 @ B2 ) ) ) ).
% and.right_idem
thf(fact_2402_real__sqrt__eq__iff,axiom,
! [X: real,Y2: real] :
( ( ( sqrt @ X )
= ( sqrt @ Y2 ) )
= ( X = Y2 ) ) ).
% real_sqrt_eq_iff
thf(fact_2403_and__zero__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A] :
( ( bit_se5824344872417868541ns_and @ A @ A2 @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% and_zero_eq
thf(fact_2404_zero__and__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A] :
( ( bit_se5824344872417868541ns_and @ A @ ( zero_zero @ A ) @ A2 )
= ( zero_zero @ A ) ) ) ).
% zero_and_eq
thf(fact_2405_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_2406_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_2407_real__sqrt__eq__zero__cancel__iff,axiom,
! [X: real] :
( ( ( sqrt @ X )
= ( zero_zero @ real ) )
= ( X
= ( zero_zero @ real ) ) ) ).
% real_sqrt_eq_zero_cancel_iff
thf(fact_2408_real__sqrt__zero,axiom,
( ( sqrt @ ( zero_zero @ real ) )
= ( zero_zero @ real ) ) ).
% real_sqrt_zero
thf(fact_2409_real__sqrt__less__iff,axiom,
! [X: real,Y2: real] :
( ( ord_less @ real @ ( sqrt @ X ) @ ( sqrt @ Y2 ) )
= ( ord_less @ real @ X @ Y2 ) ) ).
% real_sqrt_less_iff
thf(fact_2410_real__sqrt__le__iff,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq @ real @ ( sqrt @ X ) @ ( sqrt @ Y2 ) )
= ( ord_less_eq @ real @ X @ Y2 ) ) ).
% real_sqrt_le_iff
thf(fact_2411_real__sqrt__eq__1__iff,axiom,
! [X: real] :
( ( ( sqrt @ X )
= ( one_one @ real ) )
= ( X
= ( one_one @ real ) ) ) ).
% real_sqrt_eq_1_iff
thf(fact_2412_real__sqrt__one,axiom,
( ( sqrt @ ( one_one @ real ) )
= ( one_one @ real ) ) ).
% real_sqrt_one
thf(fact_2413_take__bit__and,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A2: A,B2: A] :
( ( bit_se2584673776208193580ke_bit @ A @ N @ ( bit_se5824344872417868541ns_and @ A @ A2 @ B2 ) )
= ( bit_se5824344872417868541ns_and @ A @ ( bit_se2584673776208193580ke_bit @ A @ N @ A2 ) @ ( bit_se2584673776208193580ke_bit @ A @ N @ B2 ) ) ) ) ).
% take_bit_and
thf(fact_2414_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_2415_and_Oright__neutral,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [A2: A] :
( ( bit_se5824344872417868541ns_and @ A @ A2 @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= A2 ) ) ).
% and.right_neutral
thf(fact_2416_and_Oleft__neutral,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [A2: A] :
( ( bit_se5824344872417868541ns_and @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ A2 )
= A2 ) ) ).
% and.left_neutral
thf(fact_2417_real__sqrt__lt__0__iff,axiom,
! [X: real] :
( ( ord_less @ real @ ( sqrt @ X ) @ ( zero_zero @ real ) )
= ( ord_less @ real @ X @ ( zero_zero @ real ) ) ) ).
% real_sqrt_lt_0_iff
thf(fact_2418_real__sqrt__gt__0__iff,axiom,
! [Y2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( sqrt @ Y2 ) )
= ( ord_less @ real @ ( zero_zero @ real ) @ Y2 ) ) ).
% real_sqrt_gt_0_iff
thf(fact_2419_real__sqrt__ge__0__iff,axiom,
! [Y2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( sqrt @ Y2 ) )
= ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y2 ) ) ).
% real_sqrt_ge_0_iff
thf(fact_2420_real__sqrt__le__0__iff,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( sqrt @ X ) @ ( zero_zero @ real ) )
= ( ord_less_eq @ real @ X @ ( zero_zero @ real ) ) ) ).
% real_sqrt_le_0_iff
thf(fact_2421_real__sqrt__gt__1__iff,axiom,
! [Y2: real] :
( ( ord_less @ real @ ( one_one @ real ) @ ( sqrt @ Y2 ) )
= ( ord_less @ real @ ( one_one @ real ) @ Y2 ) ) ).
% real_sqrt_gt_1_iff
thf(fact_2422_real__sqrt__lt__1__iff,axiom,
! [X: real] :
( ( ord_less @ real @ ( sqrt @ X ) @ ( one_one @ real ) )
= ( ord_less @ real @ X @ ( one_one @ real ) ) ) ).
% real_sqrt_lt_1_iff
thf(fact_2423_real__sqrt__ge__1__iff,axiom,
! [Y2: real] :
( ( ord_less_eq @ real @ ( one_one @ real ) @ ( sqrt @ Y2 ) )
= ( ord_less_eq @ real @ ( one_one @ real ) @ Y2 ) ) ).
% real_sqrt_ge_1_iff
thf(fact_2424_real__sqrt__le__1__iff,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( sqrt @ X ) @ ( one_one @ real ) )
= ( ord_less_eq @ real @ X @ ( one_one @ real ) ) ) ).
% real_sqrt_le_1_iff
thf(fact_2425_and__nonnegative__int__iff,axiom,
! [K2: int,L: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se5824344872417868541ns_and @ int @ K2 @ L ) )
= ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K2 )
| ( ord_less_eq @ int @ ( zero_zero @ int ) @ L ) ) ) ).
% and_nonnegative_int_iff
thf(fact_2426_and__negative__int__iff,axiom,
! [K2: int,L: int] :
( ( ord_less @ int @ ( bit_se5824344872417868541ns_and @ int @ K2 @ L ) @ ( zero_zero @ int ) )
= ( ( ord_less @ int @ K2 @ ( zero_zero @ int ) )
& ( ord_less @ int @ L @ ( zero_zero @ int ) ) ) ) ).
% and_negative_int_iff
thf(fact_2427_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_2428_and__numerals_I2_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [Y2: num] :
( ( bit_se5824344872417868541ns_and @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit1 @ Y2 ) ) )
= ( one_one @ A ) ) ) ).
% and_numerals(2)
thf(fact_2429_real__sqrt__four,axiom,
( ( sqrt @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ one2 ) ) ) )
= ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ).
% real_sqrt_four
thf(fact_2430_and__numerals_I1_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [Y2: num] :
( ( bit_se5824344872417868541ns_and @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ Y2 ) ) )
= ( zero_zero @ A ) ) ) ).
% and_numerals(1)
thf(fact_2431_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_2432_and__numerals_I3_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X: num,Y2: num] :
( ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ ( bit0 @ X ) ) @ ( numeral_numeral @ A @ ( bit0 @ Y2 ) ) )
= ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ X ) @ ( numeral_numeral @ A @ Y2 ) ) ) ) ) ).
% and_numerals(3)
thf(fact_2433_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_2434_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_2435_and__numerals_I4_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X: num,Y2: num] :
( ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ ( bit0 @ X ) ) @ ( numeral_numeral @ A @ ( bit1 @ Y2 ) ) )
= ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ X ) @ ( numeral_numeral @ A @ Y2 ) ) ) ) ) ).
% and_numerals(4)
thf(fact_2436_and__numerals_I6_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X: num,Y2: num] :
( ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ ( bit1 @ X ) ) @ ( numeral_numeral @ A @ ( bit0 @ Y2 ) ) )
= ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ X ) @ ( numeral_numeral @ A @ Y2 ) ) ) ) ) ).
% and_numerals(6)
thf(fact_2437_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_2438_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_2439_real__sqrt__pow2__iff,axiom,
! [X: real] :
( ( ( power_power @ real @ ( sqrt @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= X )
= ( ord_less_eq @ real @ ( zero_zero @ real ) @ X ) ) ).
% real_sqrt_pow2_iff
thf(fact_2440_real__sqrt__pow2,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( power_power @ real @ ( sqrt @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= X ) ) ).
% real_sqrt_pow2
thf(fact_2441_real__sqrt__sum__squares__mult__squared__eq,axiom,
! [X: real,Y2: real,Xa: real,Ya: real] :
( ( power_power @ real @ ( sqrt @ ( times_times @ real @ ( plus_plus @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( plus_plus @ real @ ( power_power @ real @ Xa @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Ya @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( times_times @ real @ ( plus_plus @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( plus_plus @ real @ ( power_power @ real @ Xa @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Ya @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% real_sqrt_sum_squares_mult_squared_eq
thf(fact_2442_and__numerals_I7_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X: num,Y2: num] :
( ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ ( bit1 @ X ) ) @ ( numeral_numeral @ A @ ( bit1 @ Y2 ) ) )
= ( 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 @ Y2 ) ) ) ) ) ) ).
% and_numerals(7)
thf(fact_2443_and_Oassoc,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( bit_se5824344872417868541ns_and @ A @ ( bit_se5824344872417868541ns_and @ A @ A2 @ B2 ) @ C2 )
= ( bit_se5824344872417868541ns_and @ A @ A2 @ ( bit_se5824344872417868541ns_and @ A @ B2 @ C2 ) ) ) ) ).
% and.assoc
thf(fact_2444_and_Ocommute,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se5824344872417868541ns_and @ A )
= ( ^ [A5: A,B4: A] : ( bit_se5824344872417868541ns_and @ A @ B4 @ A5 ) ) ) ) ).
% and.commute
thf(fact_2445_and_Oleft__commute,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( bit_se5824344872417868541ns_and @ A @ B2 @ ( bit_se5824344872417868541ns_and @ A @ A2 @ C2 ) )
= ( bit_se5824344872417868541ns_and @ A @ A2 @ ( bit_se5824344872417868541ns_and @ A @ B2 @ C2 ) ) ) ) ).
% and.left_commute
thf(fact_2446_of__int__and__eq,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [K2: int,L: int] :
( ( ring_1_of_int @ A @ ( bit_se5824344872417868541ns_and @ int @ K2 @ L ) )
= ( bit_se5824344872417868541ns_and @ A @ ( ring_1_of_int @ A @ K2 ) @ ( ring_1_of_int @ A @ L ) ) ) ) ).
% of_int_and_eq
thf(fact_2447_real__sqrt__less__mono,axiom,
! [X: real,Y2: real] :
( ( ord_less @ real @ X @ Y2 )
=> ( ord_less @ real @ ( sqrt @ X ) @ ( sqrt @ Y2 ) ) ) ).
% real_sqrt_less_mono
thf(fact_2448_real__sqrt__le__mono,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq @ real @ X @ Y2 )
=> ( ord_less_eq @ real @ ( sqrt @ X ) @ ( sqrt @ Y2 ) ) ) ).
% real_sqrt_le_mono
thf(fact_2449_real__sqrt__divide,axiom,
! [X: real,Y2: real] :
( ( sqrt @ ( divide_divide @ real @ X @ Y2 ) )
= ( divide_divide @ real @ ( sqrt @ X ) @ ( sqrt @ Y2 ) ) ) ).
% real_sqrt_divide
thf(fact_2450_real__sqrt__mult,axiom,
! [X: real,Y2: real] :
( ( sqrt @ ( times_times @ real @ X @ Y2 ) )
= ( times_times @ real @ ( sqrt @ X ) @ ( sqrt @ Y2 ) ) ) ).
% real_sqrt_mult
thf(fact_2451_real__sqrt__power,axiom,
! [X: real,K2: nat] :
( ( sqrt @ ( power_power @ real @ X @ K2 ) )
= ( power_power @ real @ ( sqrt @ X ) @ K2 ) ) ).
% real_sqrt_power
thf(fact_2452_real__sqrt__minus,axiom,
! [X: real] :
( ( sqrt @ ( uminus_uminus @ real @ X ) )
= ( uminus_uminus @ real @ ( sqrt @ X ) ) ) ).
% real_sqrt_minus
thf(fact_2453_bit__and__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A,B2: A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( bit_se5824344872417868541ns_and @ A @ A2 @ B2 ) @ N )
= ( ( bit_se5641148757651400278ts_bit @ A @ A2 @ N )
& ( bit_se5641148757651400278ts_bit @ A @ B2 @ N ) ) ) ) ).
% bit_and_iff
thf(fact_2454_bit__and__int__iff,axiom,
! [K2: int,L: int,N: nat] :
( ( bit_se5641148757651400278ts_bit @ int @ ( bit_se5824344872417868541ns_and @ int @ K2 @ L ) @ N )
= ( ( bit_se5641148757651400278ts_bit @ int @ K2 @ N )
& ( bit_se5641148757651400278ts_bit @ int @ L @ N ) ) ) ).
% bit_and_int_iff
thf(fact_2455_and__eq__minus__1__iff,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [A2: A,B2: A] :
( ( ( bit_se5824344872417868541ns_and @ A @ A2 @ B2 )
= ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( ( A2
= ( uminus_uminus @ A @ ( one_one @ A ) ) )
& ( B2
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ) ).
% and_eq_minus_1_iff
thf(fact_2456_real__sqrt__gt__zero,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( sqrt @ X ) ) ) ).
% real_sqrt_gt_zero
thf(fact_2457_real__sqrt__ge__zero,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( sqrt @ X ) ) ) ).
% real_sqrt_ge_zero
thf(fact_2458_real__sqrt__eq__zero__cancel,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ( sqrt @ X )
= ( zero_zero @ real ) )
=> ( X
= ( zero_zero @ real ) ) ) ) ).
% real_sqrt_eq_zero_cancel
thf(fact_2459_real__sqrt__ge__one,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( one_one @ real ) @ X )
=> ( ord_less_eq @ real @ ( one_one @ real ) @ ( sqrt @ X ) ) ) ).
% real_sqrt_ge_one
thf(fact_2460_AND__upper2_H,axiom,
! [Y2: int,Z3: int,X: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y2 )
=> ( ( ord_less_eq @ int @ Y2 @ Z3 )
=> ( ord_less_eq @ int @ ( bit_se5824344872417868541ns_and @ int @ X @ Y2 ) @ Z3 ) ) ) ).
% AND_upper2'
thf(fact_2461_AND__upper1_H,axiom,
! [Y2: int,Z3: int,Ya: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y2 )
=> ( ( ord_less_eq @ int @ Y2 @ Z3 )
=> ( ord_less_eq @ int @ ( bit_se5824344872417868541ns_and @ int @ Y2 @ Ya ) @ Z3 ) ) ) ).
% AND_upper1'
thf(fact_2462_AND__upper2,axiom,
! [Y2: int,X: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y2 )
=> ( ord_less_eq @ int @ ( bit_se5824344872417868541ns_and @ int @ X @ Y2 ) @ Y2 ) ) ).
% AND_upper2
thf(fact_2463_AND__upper1,axiom,
! [X: int,Y2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
=> ( ord_less_eq @ int @ ( bit_se5824344872417868541ns_and @ int @ X @ Y2 ) @ X ) ) ).
% AND_upper1
thf(fact_2464_AND__lower,axiom,
! [X: int,Y2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
=> ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se5824344872417868541ns_and @ int @ X @ Y2 ) ) ) ).
% AND_lower
thf(fact_2465_take__bit__eq__mask,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se2584673776208193580ke_bit @ A )
= ( ^ [N2: nat,A5: A] : ( bit_se5824344872417868541ns_and @ A @ A5 @ ( bit_se2239418461657761734s_mask @ A @ N2 ) ) ) ) ) ).
% take_bit_eq_mask
thf(fact_2466_real__div__sqrt,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( divide_divide @ real @ X @ ( sqrt @ X ) )
= ( sqrt @ X ) ) ) ).
% real_div_sqrt
thf(fact_2467_sqrt__add__le__add__sqrt,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y2 )
=> ( ord_less_eq @ real @ ( sqrt @ ( plus_plus @ real @ X @ Y2 ) ) @ ( plus_plus @ real @ ( sqrt @ X ) @ ( sqrt @ Y2 ) ) ) ) ) ).
% sqrt_add_le_add_sqrt
thf(fact_2468_le__real__sqrt__sumsq,axiom,
! [X: real,Y2: real] : ( ord_less_eq @ real @ X @ ( sqrt @ ( plus_plus @ real @ ( times_times @ real @ X @ X ) @ ( times_times @ real @ Y2 @ Y2 ) ) ) ) ).
% le_real_sqrt_sumsq
thf(fact_2469_AND__upper2_H_H,axiom,
! [Y2: int,Z3: int,X: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y2 )
=> ( ( ord_less @ int @ Y2 @ Z3 )
=> ( ord_less @ int @ ( bit_se5824344872417868541ns_and @ int @ X @ Y2 ) @ Z3 ) ) ) ).
% AND_upper2''
thf(fact_2470_AND__upper1_H_H,axiom,
! [Y2: int,Z3: int,Ya: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y2 )
=> ( ( ord_less @ int @ Y2 @ Z3 )
=> ( ord_less @ int @ ( bit_se5824344872417868541ns_and @ int @ Y2 @ Ya ) @ Z3 ) ) ) ).
% AND_upper1''
thf(fact_2471_and__less__eq,axiom,
! [L: int,K2: int] :
( ( ord_less @ int @ L @ ( zero_zero @ int ) )
=> ( ord_less_eq @ int @ ( bit_se5824344872417868541ns_and @ int @ K2 @ L ) @ K2 ) ) ).
% and_less_eq
thf(fact_2472_even__and__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A,B2: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ A @ A2 @ B2 ) )
= ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
| ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) ) ) ).
% even_and_iff
thf(fact_2473_sqrt2__less__2,axiom,
ord_less @ real @ ( sqrt @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ).
% sqrt2_less_2
thf(fact_2474_even__and__iff__int,axiom,
! [K2: int,L: int] :
( ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ K2 @ L ) )
= ( ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K2 )
| ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ L ) ) ) ).
% even_and_iff_int
thf(fact_2475_one__and__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A] :
( ( bit_se5824344872417868541ns_and @ A @ ( one_one @ A ) @ A2 )
= ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% one_and_eq
thf(fact_2476_and__one__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A] :
( ( bit_se5824344872417868541ns_and @ A @ A2 @ ( one_one @ A ) )
= ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% and_one_eq
thf(fact_2477_real__less__rsqrt,axiom,
! [X: real,Y2: real] :
( ( ord_less @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ Y2 )
=> ( ord_less @ real @ X @ ( sqrt @ Y2 ) ) ) ).
% real_less_rsqrt
thf(fact_2478_real__le__rsqrt,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ Y2 )
=> ( ord_less_eq @ real @ X @ ( sqrt @ Y2 ) ) ) ).
% real_le_rsqrt
thf(fact_2479_sqrt__le__D,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq @ real @ ( sqrt @ X ) @ Y2 )
=> ( ord_less_eq @ real @ X @ ( power_power @ real @ Y2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% sqrt_le_D
thf(fact_2480_num_Osize__gen_I1_J,axiom,
( ( size_num @ one2 )
= ( zero_zero @ nat ) ) ).
% num.size_gen(1)
thf(fact_2481_and__exp__eq__0__iff__not__bit,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A,N: nat] :
( ( ( bit_se5824344872417868541ns_and @ A @ A2 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
= ( zero_zero @ A ) )
= ( ~ ( bit_se5641148757651400278ts_bit @ A @ A2 @ N ) ) ) ) ).
% and_exp_eq_0_iff_not_bit
thf(fact_2482_real__sqrt__unique,axiom,
! [Y2: real,X: real] :
( ( ( power_power @ real @ Y2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= X )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y2 )
=> ( ( sqrt @ X )
= Y2 ) ) ) ).
% real_sqrt_unique
thf(fact_2483_real__le__lsqrt,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y2 )
=> ( ( ord_less_eq @ real @ X @ ( power_power @ real @ Y2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
=> ( ord_less_eq @ real @ ( sqrt @ X ) @ Y2 ) ) ) ) ).
% real_le_lsqrt
thf(fact_2484_lemma__real__divide__sqrt__less,axiom,
! [U: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ U )
=> ( ord_less @ real @ ( divide_divide @ real @ U @ ( sqrt @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ U ) ) ).
% lemma_real_divide_sqrt_less
thf(fact_2485_real__sqrt__sum__squares__eq__cancel2,axiom,
! [X: real,Y2: real] :
( ( ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
= Y2 )
=> ( X
= ( zero_zero @ real ) ) ) ).
% real_sqrt_sum_squares_eq_cancel2
thf(fact_2486_real__sqrt__sum__squares__eq__cancel,axiom,
! [X: real,Y2: real] :
( ( ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
= X )
=> ( Y2
= ( zero_zero @ real ) ) ) ).
% real_sqrt_sum_squares_eq_cancel
thf(fact_2487_real__sqrt__sum__squares__triangle__ineq,axiom,
! [A2: real,C2: real,B2: real,D2: real] : ( ord_less_eq @ real @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ ( plus_plus @ real @ A2 @ C2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ ( plus_plus @ real @ B2 @ D2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( plus_plus @ real @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ A2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ B2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ C2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ D2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% real_sqrt_sum_squares_triangle_ineq
thf(fact_2488_real__sqrt__sum__squares__ge2,axiom,
! [Y2: real,X: real] : ( ord_less_eq @ real @ Y2 @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% real_sqrt_sum_squares_ge2
thf(fact_2489_real__sqrt__sum__squares__ge1,axiom,
! [X: real,Y2: real] : ( ord_less_eq @ real @ X @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% real_sqrt_sum_squares_ge1
thf(fact_2490_real__less__lsqrt,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y2 )
=> ( ( ord_less @ real @ X @ ( power_power @ real @ Y2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
=> ( ord_less @ real @ ( sqrt @ X ) @ Y2 ) ) ) ) ).
% real_less_lsqrt
thf(fact_2491_sqrt__sum__squares__le__sum,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y2 )
=> ( ord_less_eq @ real @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( plus_plus @ real @ X @ Y2 ) ) ) ) ).
% sqrt_sum_squares_le_sum
thf(fact_2492_sqrt__even__pow2,axiom,
! [N: nat] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( sqrt @ ( power_power @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ N ) )
= ( power_power @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% sqrt_even_pow2
thf(fact_2493_real__sqrt__sum__squares__mult__ge__zero,axiom,
! [X: real,Y2: real,Xa: real,Ya: real] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( sqrt @ ( times_times @ real @ ( plus_plus @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( plus_plus @ real @ ( power_power @ real @ Xa @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Ya @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% real_sqrt_sum_squares_mult_ge_zero
thf(fact_2494_real__sqrt__power__even,axiom,
! [N: nat,X: real] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( power_power @ real @ ( sqrt @ X ) @ N )
= ( power_power @ real @ X @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% real_sqrt_power_even
thf(fact_2495_arith__geo__mean__sqrt,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y2 )
=> ( ord_less_eq @ real @ ( sqrt @ ( times_times @ real @ X @ Y2 ) ) @ ( divide_divide @ real @ ( plus_plus @ real @ X @ Y2 ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ).
% arith_geo_mean_sqrt
thf(fact_2496_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_2497_one__le__exp__iff,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( one_one @ real ) @ ( exp @ real @ X ) )
= ( ord_less_eq @ real @ ( zero_zero @ real ) @ X ) ) ).
% one_le_exp_iff
thf(fact_2498_exp__le__one__iff,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( exp @ real @ X ) @ ( one_one @ real ) )
= ( ord_less_eq @ real @ X @ ( zero_zero @ real ) ) ) ).
% exp_le_one_iff
thf(fact_2499_exp__less__one__iff,axiom,
! [X: real] :
( ( ord_less @ real @ ( exp @ real @ X ) @ ( one_one @ real ) )
= ( ord_less @ real @ X @ ( zero_zero @ real ) ) ) ).
% exp_less_one_iff
thf(fact_2500_one__less__exp__iff,axiom,
! [X: real] :
( ( ord_less @ real @ ( one_one @ real ) @ ( exp @ real @ X ) )
= ( ord_less @ real @ ( zero_zero @ real ) @ X ) ) ).
% one_less_exp_iff
thf(fact_2501_real__exp__bound__lemma,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ X @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less_eq @ real @ ( exp @ real @ X ) @ ( plus_plus @ real @ ( one_one @ real ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ X ) ) ) ) ) ).
% real_exp_bound_lemma
thf(fact_2502_exp__bound,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ X @ ( one_one @ real ) )
=> ( ord_less_eq @ real @ ( exp @ real @ X ) @ ( plus_plus @ real @ ( plus_plus @ real @ ( one_one @ real ) @ X ) @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% exp_bound
thf(fact_2503_exp__le__cancel__iff,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq @ real @ ( exp @ real @ X ) @ ( exp @ real @ Y2 ) )
= ( ord_less_eq @ real @ X @ Y2 ) ) ).
% exp_le_cancel_iff
thf(fact_2504_exp__zero,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ( ( exp @ A @ ( zero_zero @ A ) )
= ( one_one @ A ) ) ) ).
% exp_zero
thf(fact_2505_exp__eq__one__iff,axiom,
! [X: real] :
( ( ( exp @ real @ X )
= ( one_one @ real ) )
= ( X
= ( zero_zero @ real ) ) ) ).
% exp_eq_one_iff
thf(fact_2506_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_2507_and__nat__numerals_I1_J,axiom,
! [Y2: num] :
( ( bit_se5824344872417868541ns_and @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( numeral_numeral @ nat @ ( bit0 @ Y2 ) ) )
= ( zero_zero @ nat ) ) ).
% and_nat_numerals(1)
thf(fact_2508_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_2509_and__nat__numerals_I2_J,axiom,
! [Y2: num] :
( ( bit_se5824344872417868541ns_and @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( numeral_numeral @ nat @ ( bit1 @ Y2 ) ) )
= ( one_one @ nat ) ) ).
% and_nat_numerals(2)
thf(fact_2510_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_2511_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_2512_exp__times__arg__commute,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [A3: A] :
( ( times_times @ A @ ( exp @ A @ A3 ) @ A3 )
= ( times_times @ A @ A3 @ ( exp @ A @ A3 ) ) ) ) ).
% exp_times_arg_commute
thf(fact_2513_and__nat__unfold,axiom,
( ( bit_se5824344872417868541ns_and @ nat )
= ( ^ [M2: nat,N2: nat] :
( if @ nat
@ ( ( M2
= ( zero_zero @ nat ) )
| ( N2
= ( zero_zero @ nat ) ) )
@ ( zero_zero @ nat )
@ ( plus_plus @ nat @ ( times_times @ nat @ ( modulo_modulo @ nat @ M2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ nat @ N2 @ ( 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 @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ).
% and_nat_unfold
thf(fact_2514_and__nat__rec,axiom,
( ( bit_se5824344872417868541ns_and @ nat )
= ( ^ [M2: nat,N2: 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 ) ) @ N2 ) ) )
@ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ nat @ ( divide_divide @ nat @ M2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% and_nat_rec
thf(fact_2515_mult__exp__exp,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A,Y2: A] :
( ( times_times @ A @ ( exp @ A @ X ) @ ( exp @ A @ Y2 ) )
= ( exp @ A @ ( plus_plus @ A @ X @ Y2 ) ) ) ) ).
% mult_exp_exp
thf(fact_2516_exp__add__commuting,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X: A,Y2: A] :
( ( ( times_times @ A @ X @ Y2 )
= ( times_times @ A @ Y2 @ X ) )
=> ( ( exp @ A @ ( plus_plus @ A @ X @ Y2 ) )
= ( times_times @ A @ ( exp @ A @ X ) @ ( exp @ A @ Y2 ) ) ) ) ) ).
% exp_add_commuting
thf(fact_2517_exp__diff,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A,Y2: A] :
( ( exp @ A @ ( minus_minus @ A @ X @ Y2 ) )
= ( divide_divide @ A @ ( exp @ A @ X ) @ ( exp @ A @ Y2 ) ) ) ) ).
% exp_diff
thf(fact_2518_exp__ge__zero,axiom,
! [X: real] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( exp @ real @ X ) ) ).
% exp_ge_zero
thf(fact_2519_not__exp__le__zero,axiom,
! [X: real] :
~ ( ord_less_eq @ real @ ( exp @ real @ X ) @ ( zero_zero @ real ) ) ).
% not_exp_le_zero
thf(fact_2520_exp__minus__inverse,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X: A] :
( ( times_times @ A @ ( exp @ A @ X ) @ ( exp @ A @ ( uminus_uminus @ A @ X ) ) )
= ( one_one @ A ) ) ) ).
% exp_minus_inverse
thf(fact_2521_exp__gt__one,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ord_less @ real @ ( one_one @ real ) @ ( exp @ real @ X ) ) ) ).
% exp_gt_one
thf(fact_2522_exp__ge__add__one__self,axiom,
! [X: real] : ( ord_less_eq @ real @ ( plus_plus @ real @ ( one_one @ real ) @ X ) @ ( exp @ real @ X ) ) ).
% exp_ge_add_one_self
thf(fact_2523_exp__ge__add__one__self__aux,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ord_less_eq @ real @ ( plus_plus @ real @ ( one_one @ real ) @ X ) @ ( exp @ real @ X ) ) ) ).
% exp_ge_add_one_self_aux
thf(fact_2524_lemma__exp__total,axiom,
! [Y2: real] :
( ( ord_less_eq @ real @ ( one_one @ real ) @ Y2 )
=> ? [X5: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X5 )
& ( ord_less_eq @ real @ X5 @ ( minus_minus @ real @ Y2 @ ( one_one @ real ) ) )
& ( ( exp @ real @ X5 )
= Y2 ) ) ) ).
% lemma_exp_total
thf(fact_2525_exp__le,axiom,
ord_less_eq @ real @ ( exp @ real @ ( one_one @ real ) ) @ ( numeral_numeral @ real @ ( bit1 @ one2 ) ) ).
% exp_le
thf(fact_2526_exp__double,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [Z3: A] :
( ( exp @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Z3 ) )
= ( power_power @ A @ ( exp @ A @ Z3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% exp_double
thf(fact_2527_exp__half__le2,axiom,
ord_less_eq @ real @ ( exp @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ).
% exp_half_le2
thf(fact_2528_arsinh__real__aux,axiom,
! [X: real] : ( ord_less @ real @ ( zero_zero @ real ) @ ( plus_plus @ real @ X @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ real ) ) ) ) ) ).
% arsinh_real_aux
thf(fact_2529_arcosh__1,axiom,
! [A: $tType] :
( ( ln @ A )
=> ( ( arcosh @ A @ ( one_one @ A ) )
= ( zero_zero @ A ) ) ) ).
% arcosh_1
thf(fact_2530_fold__atLeastAtMost__nat_Opsimps,axiom,
! [A: $tType,F3: nat > A > A,A2: 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 ) ) @ F3 @ ( product_Pair @ nat @ ( product_prod @ nat @ A ) @ A2 @ ( product_Pair @ nat @ A @ B2 @ Acc2 ) ) ) )
=> ( ( ( ord_less @ nat @ B2 @ A2 )
=> ( ( set_fo6178422350223883121st_nat @ A @ F3 @ A2 @ B2 @ Acc2 )
= Acc2 ) )
& ( ~ ( ord_less @ nat @ B2 @ A2 )
=> ( ( set_fo6178422350223883121st_nat @ A @ F3 @ A2 @ B2 @ Acc2 )
= ( set_fo6178422350223883121st_nat @ A @ F3 @ ( plus_plus @ nat @ A2 @ ( one_one @ nat ) ) @ B2 @ ( F3 @ A2 @ Acc2 ) ) ) ) ) ) ).
% fold_atLeastAtMost_nat.psimps
thf(fact_2531_fold__atLeastAtMost__nat_Opelims,axiom,
! [A: $tType,X: nat > A > A,Xa: nat,Xb: nat,Xc: A,Y2: A] :
( ( ( set_fo6178422350223883121st_nat @ A @ X @ Xa @ Xb @ Xc )
= Y2 )
=> ( ( accp @ ( product_prod @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) ) @ ( set_fo1817059534552279752at_rel @ A ) @ ( product_Pair @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) @ X @ ( product_Pair @ nat @ ( product_prod @ nat @ A ) @ Xa @ ( product_Pair @ nat @ A @ Xb @ Xc ) ) ) )
=> ~ ( ( ( ( ord_less @ nat @ Xb @ Xa )
=> ( Y2 = Xc ) )
& ( ~ ( ord_less @ nat @ Xb @ Xa )
=> ( Y2
= ( set_fo6178422350223883121st_nat @ A @ X @ ( plus_plus @ nat @ Xa @ ( one_one @ nat ) ) @ Xb @ ( X @ Xa @ Xc ) ) ) ) )
=> ~ ( accp @ ( product_prod @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) ) @ ( set_fo1817059534552279752at_rel @ A ) @ ( product_Pair @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) @ X @ ( product_Pair @ nat @ ( product_prod @ nat @ A ) @ Xa @ ( product_Pair @ nat @ A @ Xb @ Xc ) ) ) ) ) ) ) ).
% fold_atLeastAtMost_nat.pelims
thf(fact_2532_tanh__real__altdef,axiom,
( ( tanh @ real )
= ( ^ [X4: real] : ( divide_divide @ real @ ( minus_minus @ real @ ( one_one @ real ) @ ( exp @ real @ ( times_times @ real @ ( uminus_uminus @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ X4 ) ) ) @ ( plus_plus @ real @ ( one_one @ real ) @ ( exp @ real @ ( times_times @ real @ ( uminus_uminus @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ X4 ) ) ) ) ) ) ).
% tanh_real_altdef
thf(fact_2533_ln__one__minus__pos__lower__bound,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ X @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less_eq @ real @ ( minus_minus @ real @ ( uminus_uminus @ real @ X ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( ln_ln @ real @ ( minus_minus @ real @ ( one_one @ real ) @ X ) ) ) ) ) ).
% ln_one_minus_pos_lower_bound
thf(fact_2534_arctan__half,axiom,
( arctan
= ( ^ [X4: real] : ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( arctan @ ( divide_divide @ real @ X4 @ ( plus_plus @ real @ ( one_one @ real ) @ ( sqrt @ ( plus_plus @ real @ ( one_one @ real ) @ ( power_power @ real @ X4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ).
% arctan_half
thf(fact_2535_tanh__real__le__iff,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq @ real @ ( tanh @ real @ X ) @ ( tanh @ real @ Y2 ) )
= ( ord_less_eq @ real @ X @ Y2 ) ) ).
% tanh_real_le_iff
thf(fact_2536_ln__one,axiom,
! [A: $tType] :
( ( ln @ A )
=> ( ( ln_ln @ A @ ( one_one @ A ) )
= ( zero_zero @ A ) ) ) ).
% ln_one
thf(fact_2537_zero__le__arctan__iff,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( arctan @ X ) )
= ( ord_less_eq @ real @ ( zero_zero @ real ) @ X ) ) ).
% zero_le_arctan_iff
thf(fact_2538_arctan__le__zero__iff,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( arctan @ X ) @ ( zero_zero @ real ) )
= ( ord_less_eq @ real @ X @ ( zero_zero @ real ) ) ) ).
% arctan_le_zero_iff
thf(fact_2539_tanh__real__nonpos__iff,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( tanh @ real @ X ) @ ( zero_zero @ real ) )
= ( ord_less_eq @ real @ X @ ( zero_zero @ real ) ) ) ).
% tanh_real_nonpos_iff
thf(fact_2540_tanh__real__nonneg__iff,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( tanh @ real @ X ) )
= ( ord_less_eq @ real @ ( zero_zero @ real ) @ X ) ) ).
% tanh_real_nonneg_iff
thf(fact_2541_ln__le__cancel__iff,axiom,
! [X: real,Y2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ Y2 )
=> ( ( ord_less_eq @ real @ ( ln_ln @ real @ X ) @ ( ln_ln @ real @ Y2 ) )
= ( ord_less_eq @ real @ X @ Y2 ) ) ) ) ).
% ln_le_cancel_iff
thf(fact_2542_ln__eq__zero__iff,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ( ln_ln @ real @ X )
= ( zero_zero @ real ) )
= ( X
= ( one_one @ real ) ) ) ) ).
% ln_eq_zero_iff
thf(fact_2543_ln__gt__zero__iff,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( ln_ln @ real @ X ) )
= ( ord_less @ real @ ( one_one @ real ) @ X ) ) ) ).
% ln_gt_zero_iff
thf(fact_2544_ln__less__zero__iff,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( ln_ln @ real @ X ) @ ( zero_zero @ real ) )
= ( ord_less @ real @ X @ ( one_one @ real ) ) ) ) ).
% ln_less_zero_iff
thf(fact_2545_ln__ge__zero__iff,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( ln_ln @ real @ X ) )
= ( ord_less_eq @ real @ ( one_one @ real ) @ X ) ) ) ).
% ln_ge_zero_iff
thf(fact_2546_ln__le__zero__iff,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( ln_ln @ real @ X ) @ ( zero_zero @ real ) )
= ( ord_less_eq @ real @ X @ ( one_one @ real ) ) ) ) ).
% ln_le_zero_iff
thf(fact_2547_arctan__le__iff,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq @ real @ ( arctan @ X ) @ ( arctan @ Y2 ) )
= ( ord_less_eq @ real @ X @ Y2 ) ) ).
% arctan_le_iff
thf(fact_2548_arctan__monotone_H,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq @ real @ X @ Y2 )
=> ( ord_less_eq @ real @ ( arctan @ X ) @ ( arctan @ Y2 ) ) ) ).
% arctan_monotone'
thf(fact_2549_tanh__real__lt__1,axiom,
! [X: real] : ( ord_less @ real @ ( tanh @ real @ X ) @ ( one_one @ real ) ) ).
% tanh_real_lt_1
thf(fact_2550_ln__bound,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ord_less_eq @ real @ ( ln_ln @ real @ X ) @ X ) ) ).
% ln_bound
thf(fact_2551_ln__gt__zero,axiom,
! [X: real] :
( ( ord_less @ real @ ( one_one @ real ) @ X )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( ln_ln @ real @ X ) ) ) ).
% ln_gt_zero
thf(fact_2552_ln__less__zero,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ X @ ( one_one @ real ) )
=> ( ord_less @ real @ ( ln_ln @ real @ X ) @ ( zero_zero @ real ) ) ) ) ).
% ln_less_zero
thf(fact_2553_ln__gt__zero__imp__gt__one,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( ln_ln @ real @ X ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ord_less @ real @ ( one_one @ real ) @ X ) ) ) ).
% ln_gt_zero_imp_gt_one
thf(fact_2554_ln__ge__zero,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( one_one @ real ) @ X )
=> ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( ln_ln @ real @ X ) ) ) ).
% ln_ge_zero
thf(fact_2555_tanh__real__gt__neg1,axiom,
! [X: real] : ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ ( tanh @ real @ X ) ) ).
% tanh_real_gt_neg1
thf(fact_2556_ln__ge__zero__imp__ge__one,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( ln_ln @ real @ X ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ord_less_eq @ real @ ( one_one @ real ) @ X ) ) ) ).
% ln_ge_zero_imp_ge_one
thf(fact_2557_ln__add__one__self__le__self,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ord_less_eq @ real @ ( ln_ln @ real @ ( plus_plus @ real @ ( one_one @ real ) @ X ) ) @ X ) ) ).
% ln_add_one_self_le_self
thf(fact_2558_ln__mult,axiom,
! [X: real,Y2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ Y2 )
=> ( ( ln_ln @ real @ ( times_times @ real @ X @ Y2 ) )
= ( plus_plus @ real @ ( ln_ln @ real @ X ) @ ( ln_ln @ real @ Y2 ) ) ) ) ) ).
% ln_mult
thf(fact_2559_ln__eq__minus__one,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ( ln_ln @ real @ X )
= ( minus_minus @ real @ X @ ( one_one @ real ) ) )
=> ( X
= ( one_one @ real ) ) ) ) ).
% ln_eq_minus_one
thf(fact_2560_ln__ge__iff,axiom,
! [X: real,Y2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ Y2 @ ( ln_ln @ real @ X ) )
= ( ord_less_eq @ real @ ( exp @ real @ Y2 ) @ X ) ) ) ).
% ln_ge_iff
thf(fact_2561_ln__x__over__x__mono,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq @ real @ ( exp @ real @ ( one_one @ real ) ) @ X )
=> ( ( ord_less_eq @ real @ X @ Y2 )
=> ( ord_less_eq @ real @ ( divide_divide @ real @ ( ln_ln @ real @ Y2 ) @ Y2 ) @ ( divide_divide @ real @ ( ln_ln @ real @ X ) @ X ) ) ) ) ).
% ln_x_over_x_mono
thf(fact_2562_ln__2__less__1,axiom,
ord_less @ real @ ( ln_ln @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( one_one @ real ) ).
% ln_2_less_1
thf(fact_2563_ln__le__minus__one,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ord_less_eq @ real @ ( ln_ln @ real @ X ) @ ( minus_minus @ real @ X @ ( one_one @ real ) ) ) ) ).
% ln_le_minus_one
thf(fact_2564_ln__diff__le,axiom,
! [X: real,Y2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ Y2 )
=> ( ord_less_eq @ real @ ( minus_minus @ real @ ( ln_ln @ real @ X ) @ ( ln_ln @ real @ Y2 ) ) @ ( divide_divide @ real @ ( minus_minus @ real @ X @ Y2 ) @ Y2 ) ) ) ) ).
% ln_diff_le
thf(fact_2565_ln__add__one__self__le__self2,axiom,
! [X: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X )
=> ( ord_less_eq @ real @ ( ln_ln @ real @ ( plus_plus @ real @ ( one_one @ real ) @ X ) ) @ X ) ) ).
% ln_add_one_self_le_self2
thf(fact_2566_fold__atLeastAtMost__nat_Osimps,axiom,
! [A: $tType] :
( ( set_fo6178422350223883121st_nat @ A )
= ( ^ [F5: nat > A > A,A5: nat,B4: nat,Acc3: A] : ( if @ A @ ( ord_less @ nat @ B4 @ A5 ) @ Acc3 @ ( set_fo6178422350223883121st_nat @ A @ F5 @ ( plus_plus @ nat @ A5 @ ( one_one @ nat ) ) @ B4 @ ( F5 @ A5 @ Acc3 ) ) ) ) ) ).
% fold_atLeastAtMost_nat.simps
thf(fact_2567_fold__atLeastAtMost__nat_Oelims,axiom,
! [A: $tType,X: nat > A > A,Xa: nat,Xb: nat,Xc: A,Y2: A] :
( ( ( set_fo6178422350223883121st_nat @ A @ X @ Xa @ Xb @ Xc )
= Y2 )
=> ( ( ( ord_less @ nat @ Xb @ Xa )
=> ( Y2 = Xc ) )
& ( ~ ( ord_less @ nat @ Xb @ Xa )
=> ( Y2
= ( set_fo6178422350223883121st_nat @ A @ X @ ( plus_plus @ nat @ Xa @ ( one_one @ nat ) ) @ Xb @ ( X @ Xa @ Xc ) ) ) ) ) ) ).
% fold_atLeastAtMost_nat.elims
thf(fact_2568_tanh__ln__real,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( tanh @ real @ ( ln_ln @ real @ X ) )
= ( divide_divide @ real @ ( minus_minus @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ real ) ) @ ( plus_plus @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ real ) ) ) ) ) ).
% tanh_ln_real
thf(fact_2569_arcosh__real__def,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( one_one @ real ) @ X )
=> ( ( arcosh @ real @ X )
= ( ln_ln @ real @ ( plus_plus @ real @ X @ ( sqrt @ ( minus_minus @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ real ) ) ) ) ) ) ) ).
% arcosh_real_def
thf(fact_2570_ln__one__minus__pos__upper__bound,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ X @ ( one_one @ real ) )
=> ( ord_less_eq @ real @ ( ln_ln @ real @ ( minus_minus @ real @ ( one_one @ real ) @ X ) ) @ ( uminus_uminus @ real @ X ) ) ) ) ).
% ln_one_minus_pos_upper_bound
thf(fact_2571_ln__sqrt,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ln_ln @ real @ ( sqrt @ X ) )
= ( divide_divide @ real @ ( ln_ln @ real @ X ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ).
% ln_sqrt
thf(fact_2572_tanh__altdef,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ( ( tanh @ A )
= ( ^ [X4: A] : ( divide_divide @ A @ ( minus_minus @ A @ ( exp @ A @ X4 ) @ ( exp @ A @ ( uminus_uminus @ A @ X4 ) ) ) @ ( plus_plus @ A @ ( exp @ A @ X4 ) @ ( exp @ A @ ( uminus_uminus @ A @ X4 ) ) ) ) ) ) ) ).
% tanh_altdef
thf(fact_2573_ln__one__plus__pos__lower__bound,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ X @ ( one_one @ real ) )
=> ( ord_less_eq @ real @ ( minus_minus @ real @ X @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( ln_ln @ real @ ( plus_plus @ real @ ( one_one @ real ) @ X ) ) ) ) ) ).
% ln_one_plus_pos_lower_bound
thf(fact_2574_artanh__def,axiom,
! [A: $tType] :
( ( ( real_V3459762299906320749_field @ A )
& ( ln @ A ) )
=> ( ( artanh @ A )
= ( ^ [X4: A] : ( divide_divide @ A @ ( ln_ln @ A @ ( divide_divide @ A @ ( plus_plus @ A @ ( one_one @ A ) @ X4 ) @ ( minus_minus @ A @ ( one_one @ A ) @ X4 ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% artanh_def
thf(fact_2575_abs__ln__one__plus__x__minus__x__bound__nonpos,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X )
=> ( ( ord_less_eq @ real @ X @ ( zero_zero @ real ) )
=> ( ord_less_eq @ real @ ( abs_abs @ real @ ( minus_minus @ real @ ( ln_ln @ real @ ( plus_plus @ real @ ( one_one @ real ) @ X ) ) @ X ) ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% abs_ln_one_plus_x_minus_x_bound_nonpos
thf(fact_2576_arsinh__real__def,axiom,
( ( arsinh @ real )
= ( ^ [X4: real] : ( ln_ln @ real @ ( plus_plus @ real @ X4 @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ real ) ) ) ) ) ) ) ).
% arsinh_real_def
thf(fact_2577_abs__ln__one__plus__x__minus__x__bound,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ X ) @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less_eq @ real @ ( abs_abs @ real @ ( minus_minus @ real @ ( ln_ln @ real @ ( plus_plus @ real @ ( one_one @ real ) @ X ) ) @ X ) ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% abs_ln_one_plus_x_minus_x_bound
thf(fact_2578_arctan__double,axiom,
! [X: real] :
( ( ord_less @ real @ ( abs_abs @ real @ X ) @ ( one_one @ real ) )
=> ( ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( arctan @ X ) )
= ( arctan @ ( divide_divide @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ X ) @ ( minus_minus @ real @ ( one_one @ real ) @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% arctan_double
thf(fact_2579_log__base__10__eq1,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( log @ ( numeral_numeral @ real @ ( bit0 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) @ X )
= ( times_times @ real @ ( divide_divide @ real @ ( ln_ln @ real @ ( exp @ real @ ( one_one @ real ) ) ) @ ( ln_ln @ real @ ( numeral_numeral @ real @ ( bit0 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) ) ) @ ( ln_ln @ real @ X ) ) ) ) ).
% log_base_10_eq1
thf(fact_2580_abs__idempotent,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A2: A] :
( ( abs_abs @ A @ ( abs_abs @ A @ A2 ) )
= ( abs_abs @ A @ A2 ) ) ) ).
% abs_idempotent
thf(fact_2581_abs__zero,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ( ( abs_abs @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% abs_zero
thf(fact_2582_abs__eq__0,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A2: A] :
( ( ( abs_abs @ A @ A2 )
= ( zero_zero @ A ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% abs_eq_0
thf(fact_2583_abs__0__eq,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A2: A] :
( ( ( zero_zero @ A )
= ( abs_abs @ A @ A2 ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% abs_0_eq
thf(fact_2584_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_2585_abs__mult__self__eq,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A] :
( ( times_times @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ A2 ) )
= ( times_times @ A @ A2 @ A2 ) ) ) ).
% abs_mult_self_eq
thf(fact_2586_abs__1,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ( ( abs_abs @ A @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% abs_1
thf(fact_2587_abs__add__abs,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A2: A,B2: A] :
( ( abs_abs @ A @ ( plus_plus @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B2 ) ) )
= ( plus_plus @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B2 ) ) ) ) ).
% abs_add_abs
thf(fact_2588_abs__divide,axiom,
! [A: $tType] :
( ( field_abs_sgn @ A )
=> ! [A2: A,B2: A] :
( ( abs_abs @ A @ ( divide_divide @ A @ A2 @ B2 ) )
= ( divide_divide @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B2 ) ) ) ) ).
% abs_divide
thf(fact_2589_abs__minus__cancel,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A2: A] :
( ( abs_abs @ A @ ( uminus_uminus @ A @ A2 ) )
= ( abs_abs @ A @ A2 ) ) ) ).
% abs_minus_cancel
thf(fact_2590_abs__le__zero__iff,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ ( abs_abs @ A @ A2 ) @ ( zero_zero @ A ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% abs_le_zero_iff
thf(fact_2591_abs__le__self__iff,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ ( abs_abs @ A @ A2 ) @ A2 )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% abs_le_self_iff
thf(fact_2592_abs__of__nonneg,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( abs_abs @ A @ A2 )
= A2 ) ) ) ).
% abs_of_nonneg
thf(fact_2593_zero__less__abs__iff,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( abs_abs @ A @ A2 ) )
= ( A2
!= ( zero_zero @ A ) ) ) ) ).
% zero_less_abs_iff
thf(fact_2594_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_2595_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_2596_abs__power__minus,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A,N: nat] :
( ( abs_abs @ A @ ( power_power @ A @ ( uminus_uminus @ A @ A2 ) @ N ) )
= ( abs_abs @ A @ ( power_power @ A @ A2 @ N ) ) ) ) ).
% abs_power_minus
thf(fact_2597_log__one,axiom,
! [A2: real] :
( ( log @ A2 @ ( one_one @ real ) )
= ( zero_zero @ real ) ) ).
% log_one
thf(fact_2598_real__sqrt__mult__self,axiom,
! [A2: real] :
( ( times_times @ real @ ( sqrt @ A2 ) @ ( sqrt @ A2 ) )
= ( abs_abs @ real @ A2 ) ) ).
% real_sqrt_mult_self
thf(fact_2599_real__sqrt__abs2,axiom,
! [X: real] :
( ( sqrt @ ( times_times @ real @ X @ X ) )
= ( abs_abs @ real @ X ) ) ).
% real_sqrt_abs2
thf(fact_2600_divide__le__0__abs__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ A2 @ ( abs_abs @ A @ B2 ) ) @ ( zero_zero @ A ) )
= ( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
| ( B2
= ( zero_zero @ A ) ) ) ) ) ).
% divide_le_0_abs_iff
thf(fact_2601_zero__le__divide__abs__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ A2 @ ( abs_abs @ A @ B2 ) ) )
= ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
| ( B2
= ( zero_zero @ A ) ) ) ) ) ).
% zero_le_divide_abs_iff
thf(fact_2602_abs__of__nonpos,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( abs_abs @ A @ A2 )
= ( uminus_uminus @ A @ A2 ) ) ) ) ).
% abs_of_nonpos
thf(fact_2603_zero__less__log__cancel__iff,axiom,
! [A2: real,X: real] :
( ( ord_less @ real @ ( one_one @ real ) @ A2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( log @ A2 @ X ) )
= ( ord_less @ real @ ( one_one @ real ) @ X ) ) ) ) ).
% zero_less_log_cancel_iff
thf(fact_2604_log__less__zero__cancel__iff,axiom,
! [A2: real,X: real] :
( ( ord_less @ real @ ( one_one @ real ) @ A2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( log @ A2 @ X ) @ ( zero_zero @ real ) )
= ( ord_less @ real @ X @ ( one_one @ real ) ) ) ) ) ).
% log_less_zero_cancel_iff
thf(fact_2605_one__less__log__cancel__iff,axiom,
! [A2: real,X: real] :
( ( ord_less @ real @ ( one_one @ real ) @ A2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( one_one @ real ) @ ( log @ A2 @ X ) )
= ( ord_less @ real @ A2 @ X ) ) ) ) ).
% one_less_log_cancel_iff
thf(fact_2606_log__less__one__cancel__iff,axiom,
! [A2: real,X: real] :
( ( ord_less @ real @ ( one_one @ real ) @ A2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( log @ A2 @ X ) @ ( one_one @ real ) )
= ( ord_less @ real @ X @ A2 ) ) ) ) ).
% log_less_one_cancel_iff
thf(fact_2607_log__less__cancel__iff,axiom,
! [A2: real,X: real,Y2: real] :
( ( ord_less @ real @ ( one_one @ real ) @ A2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ Y2 )
=> ( ( ord_less @ real @ ( log @ A2 @ X ) @ ( log @ A2 @ Y2 ) )
= ( ord_less @ real @ X @ Y2 ) ) ) ) ) ).
% log_less_cancel_iff
thf(fact_2608_log__eq__one,axiom,
! [A2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
=> ( ( A2
!= ( one_one @ real ) )
=> ( ( log @ A2 @ A2 )
= ( one_one @ real ) ) ) ) ).
% log_eq_one
thf(fact_2609_artanh__minus__real,axiom,
! [X: real] :
( ( ord_less @ real @ ( abs_abs @ real @ X ) @ ( one_one @ real ) )
=> ( ( artanh @ real @ ( uminus_uminus @ real @ X ) )
= ( uminus_uminus @ real @ ( artanh @ real @ X ) ) ) ) ).
% artanh_minus_real
thf(fact_2610_zero__less__power__abs__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A,N: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( power_power @ A @ ( abs_abs @ A @ A2 ) @ N ) )
= ( ( A2
!= ( zero_zero @ A ) )
| ( N
= ( zero_zero @ nat ) ) ) ) ) ).
% zero_less_power_abs_iff
thf(fact_2611_abs__power2,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A] :
( ( abs_abs @ A @ ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% abs_power2
thf(fact_2612_power2__abs,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A] :
( ( power_power @ A @ ( abs_abs @ A @ A2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% power2_abs
thf(fact_2613_zero__le__log__cancel__iff,axiom,
! [A2: real,X: real] :
( ( ord_less @ real @ ( one_one @ real ) @ A2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( log @ A2 @ X ) )
= ( ord_less_eq @ real @ ( one_one @ real ) @ X ) ) ) ) ).
% zero_le_log_cancel_iff
thf(fact_2614_log__le__zero__cancel__iff,axiom,
! [A2: real,X: real] :
( ( ord_less @ real @ ( one_one @ real ) @ A2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( log @ A2 @ X ) @ ( zero_zero @ real ) )
= ( ord_less_eq @ real @ X @ ( one_one @ real ) ) ) ) ) ).
% log_le_zero_cancel_iff
thf(fact_2615_one__le__log__cancel__iff,axiom,
! [A2: real,X: real] :
( ( ord_less @ real @ ( one_one @ real ) @ A2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( one_one @ real ) @ ( log @ A2 @ X ) )
= ( ord_less_eq @ real @ A2 @ X ) ) ) ) ).
% one_le_log_cancel_iff
thf(fact_2616_log__le__one__cancel__iff,axiom,
! [A2: real,X: real] :
( ( ord_less @ real @ ( one_one @ real ) @ A2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( log @ A2 @ X ) @ ( one_one @ real ) )
= ( ord_less_eq @ real @ X @ A2 ) ) ) ) ).
% log_le_one_cancel_iff
thf(fact_2617_log__le__cancel__iff,axiom,
! [A2: real,X: real,Y2: real] :
( ( ord_less @ real @ ( one_one @ real ) @ A2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ Y2 )
=> ( ( ord_less_eq @ real @ ( log @ A2 @ X ) @ ( log @ A2 @ Y2 ) )
= ( ord_less_eq @ real @ X @ Y2 ) ) ) ) ) ).
% log_le_cancel_iff
thf(fact_2618_power__even__abs__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [W: num,A2: A] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W ) )
=> ( ( power_power @ A @ ( abs_abs @ A @ A2 ) @ ( numeral_numeral @ nat @ W ) )
= ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ W ) ) ) ) ) ).
% power_even_abs_numeral
thf(fact_2619_real__sqrt__abs,axiom,
! [X: real] :
( ( sqrt @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( abs_abs @ real @ X ) ) ).
% real_sqrt_abs
thf(fact_2620_abs__le__D1,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( abs_abs @ A @ A2 ) @ B2 )
=> ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% abs_le_D1
thf(fact_2621_abs__ge__self,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A2: A] : ( ord_less_eq @ A @ A2 @ ( abs_abs @ A @ A2 ) ) ) ).
% abs_ge_self
thf(fact_2622_abs__mult,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ! [A2: A,B2: A] :
( ( abs_abs @ A @ ( times_times @ A @ A2 @ B2 ) )
= ( times_times @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B2 ) ) ) ) ).
% abs_mult
thf(fact_2623_abs__one,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( ( abs_abs @ A @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% abs_one
thf(fact_2624_abs__minus__commute,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A2: A,B2: A] :
( ( abs_abs @ A @ ( minus_minus @ A @ A2 @ B2 ) )
= ( abs_abs @ A @ ( minus_minus @ A @ B2 @ A2 ) ) ) ) ).
% abs_minus_commute
thf(fact_2625_power__abs,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A,N: nat] :
( ( abs_abs @ A @ ( power_power @ A @ A2 @ N ) )
= ( power_power @ A @ ( abs_abs @ A @ A2 ) @ N ) ) ) ).
% power_abs
thf(fact_2626_abs__ge__zero,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A2: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( abs_abs @ A @ A2 ) ) ) ).
% abs_ge_zero
thf(fact_2627_abs__of__pos,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( abs_abs @ A @ A2 )
= A2 ) ) ) ).
% abs_of_pos
thf(fact_2628_abs__not__less__zero,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A2: A] :
~ ( ord_less @ A @ ( abs_abs @ A @ A2 ) @ ( zero_zero @ A ) ) ) ).
% abs_not_less_zero
thf(fact_2629_abs__triangle__ineq,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A2: A,B2: A] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( plus_plus @ A @ A2 @ B2 ) ) @ ( plus_plus @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B2 ) ) ) ) ).
% abs_triangle_ineq
thf(fact_2630_abs__mult__less,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A,C2: A,B2: A,D2: A] :
( ( ord_less @ A @ ( abs_abs @ A @ A2 ) @ C2 )
=> ( ( ord_less @ A @ ( abs_abs @ A @ B2 ) @ D2 )
=> ( ord_less @ A @ ( times_times @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B2 ) ) @ ( times_times @ A @ C2 @ D2 ) ) ) ) ) ).
% abs_mult_less
thf(fact_2631_abs__triangle__ineq2,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A2: A,B2: A] : ( ord_less_eq @ A @ ( minus_minus @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B2 ) ) @ ( abs_abs @ A @ ( minus_minus @ A @ A2 @ B2 ) ) ) ) ).
% abs_triangle_ineq2
thf(fact_2632_abs__triangle__ineq3,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A2: A,B2: A] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B2 ) ) ) @ ( abs_abs @ A @ ( minus_minus @ A @ A2 @ B2 ) ) ) ) ).
% abs_triangle_ineq3
thf(fact_2633_abs__triangle__ineq2__sym,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A2: A,B2: A] : ( ord_less_eq @ A @ ( minus_minus @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B2 ) ) @ ( abs_abs @ A @ ( minus_minus @ A @ B2 @ A2 ) ) ) ) ).
% abs_triangle_ineq2_sym
thf(fact_2634_abs__leI,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A2 ) @ B2 )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ A2 ) @ B2 ) ) ) ) ).
% abs_leI
thf(fact_2635_abs__le__D2,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( abs_abs @ A @ A2 ) @ B2 )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ A2 ) @ B2 ) ) ) ).
% abs_le_D2
thf(fact_2636_abs__le__iff,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( abs_abs @ A @ A2 ) @ B2 )
= ( ( ord_less_eq @ A @ A2 @ B2 )
& ( ord_less_eq @ A @ ( uminus_uminus @ A @ A2 ) @ B2 ) ) ) ) ).
% abs_le_iff
thf(fact_2637_abs__ge__minus__self,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A2: A] : ( ord_less_eq @ A @ ( uminus_uminus @ A @ A2 ) @ ( abs_abs @ A @ A2 ) ) ) ).
% abs_ge_minus_self
thf(fact_2638_nonzero__abs__divide,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,A2: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( abs_abs @ A @ ( divide_divide @ A @ A2 @ B2 ) )
= ( divide_divide @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B2 ) ) ) ) ) ).
% nonzero_abs_divide
thf(fact_2639_abs__less__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( abs_abs @ A @ A2 ) @ B2 )
= ( ( ord_less @ A @ A2 @ B2 )
& ( ord_less @ A @ ( uminus_uminus @ A @ A2 ) @ B2 ) ) ) ) ).
% abs_less_iff
thf(fact_2640_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_2641_abs__eq__mult,axiom,
! [A: $tType] :
( ( ordered_ring_abs @ A )
=> ! [A2: A,B2: A] :
( ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
| ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) )
& ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
| ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) ) )
=> ( ( abs_abs @ A @ ( times_times @ A @ A2 @ B2 ) )
= ( times_times @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B2 ) ) ) ) ) ).
% abs_eq_mult
thf(fact_2642_abs__mult__pos,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,Y2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( times_times @ A @ ( abs_abs @ A @ Y2 ) @ X )
= ( abs_abs @ A @ ( times_times @ A @ Y2 @ X ) ) ) ) ) ).
% abs_mult_pos
thf(fact_2643_abs__eq__iff_H,axiom,
! [A: $tType] :
( ( linordered_ring @ A )
=> ! [A2: A,B2: A] :
( ( ( abs_abs @ A @ A2 )
= B2 )
= ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
& ( ( A2 = B2 )
| ( A2
= ( uminus_uminus @ A @ B2 ) ) ) ) ) ) ).
% abs_eq_iff'
thf(fact_2644_eq__abs__iff_H,axiom,
! [A: $tType] :
( ( linordered_ring @ A )
=> ! [A2: A,B2: A] :
( ( A2
= ( abs_abs @ A @ B2 ) )
= ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
& ( ( B2 = A2 )
| ( B2
= ( uminus_uminus @ A @ A2 ) ) ) ) ) ) ).
% eq_abs_iff'
thf(fact_2645_abs__minus__le__zero,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A2: A] : ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( abs_abs @ A @ A2 ) ) @ ( zero_zero @ A ) ) ) ).
% abs_minus_le_zero
thf(fact_2646_zero__le__power__abs,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A,N: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ ( abs_abs @ A @ A2 ) @ N ) ) ) ).
% zero_le_power_abs
thf(fact_2647_abs__div__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y2: A,X: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Y2 )
=> ( ( divide_divide @ A @ ( abs_abs @ A @ X ) @ Y2 )
= ( abs_abs @ A @ ( divide_divide @ A @ X @ Y2 ) ) ) ) ) ).
% abs_div_pos
thf(fact_2648_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_2649_abs__of__neg,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A2: A] :
( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( abs_abs @ A @ A2 )
= ( uminus_uminus @ A @ A2 ) ) ) ) ).
% abs_of_neg
thf(fact_2650_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_2651_abs__triangle__ineq4,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A2: A,B2: A] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ A2 @ B2 ) ) @ ( plus_plus @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B2 ) ) ) ) ).
% abs_triangle_ineq4
thf(fact_2652_abs__diff__triangle__ineq,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A2: A,B2: A,C2: A,D2: A] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ ( plus_plus @ A @ C2 @ D2 ) ) ) @ ( plus_plus @ A @ ( abs_abs @ A @ ( minus_minus @ A @ A2 @ C2 ) ) @ ( abs_abs @ A @ ( minus_minus @ A @ B2 @ D2 ) ) ) ) ) ).
% abs_diff_triangle_ineq
thf(fact_2653_abs__diff__le__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,A2: A,R: A] :
( ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X @ A2 ) ) @ R )
= ( ( ord_less_eq @ A @ ( minus_minus @ A @ A2 @ R ) @ X )
& ( ord_less_eq @ A @ X @ ( plus_plus @ A @ A2 @ R ) ) ) ) ) ).
% abs_diff_le_iff
thf(fact_2654_abs__diff__less__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,A2: A,R: A] :
( ( ord_less @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X @ A2 ) ) @ R )
= ( ( ord_less @ A @ ( minus_minus @ A @ A2 @ R ) @ X )
& ( ord_less @ A @ X @ ( plus_plus @ A @ A2 @ R ) ) ) ) ) ).
% abs_diff_less_iff
thf(fact_2655_abs__real__def,axiom,
( ( abs_abs @ real )
= ( ^ [A5: real] : ( if @ real @ ( ord_less @ real @ A5 @ ( zero_zero @ real ) ) @ ( uminus_uminus @ real @ A5 ) @ A5 ) ) ) ).
% abs_real_def
thf(fact_2656_log__ln,axiom,
( ( ln_ln @ real )
= ( log @ ( exp @ real @ ( one_one @ real ) ) ) ) ).
% log_ln
thf(fact_2657_sin__bound__lemma,axiom,
! [X: real,Y2: real,U: real,V2: real] :
( ( X = Y2 )
=> ( ( ord_less_eq @ real @ ( abs_abs @ real @ U ) @ V2 )
=> ( ord_less_eq @ real @ ( abs_abs @ real @ ( minus_minus @ real @ ( plus_plus @ real @ X @ U ) @ Y2 ) ) @ V2 ) ) ) ).
% sin_bound_lemma
thf(fact_2658_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_2659_log__base__change,axiom,
! [A2: real,B2: real,X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
=> ( ( A2
!= ( one_one @ real ) )
=> ( ( log @ B2 @ X )
= ( divide_divide @ real @ ( log @ A2 @ X ) @ ( log @ A2 @ B2 ) ) ) ) ) ).
% log_base_change
thf(fact_2660_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_2661_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_2662_lemma__interval,axiom,
! [A2: real,X: real,B2: real] :
( ( ord_less @ real @ A2 @ X )
=> ( ( ord_less @ real @ X @ B2 )
=> ? [D3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D3 )
& ! [Y4: real] :
( ( ord_less @ real @ ( abs_abs @ real @ ( minus_minus @ real @ X @ Y4 ) ) @ D3 )
=> ( ( ord_less_eq @ real @ A2 @ Y4 )
& ( ord_less_eq @ real @ Y4 @ B2 ) ) ) ) ) ) ).
% lemma_interval
thf(fact_2663_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_2664_abs__le__square__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,Y2: A] :
( ( ord_less_eq @ A @ ( abs_abs @ A @ X ) @ ( abs_abs @ A @ Y2 ) )
= ( ord_less_eq @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% abs_le_square_iff
thf(fact_2665_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_2666_power__even__abs,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: nat,A2: A] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( power_power @ A @ ( abs_abs @ A @ A2 ) @ N )
= ( power_power @ A @ A2 @ N ) ) ) ) ).
% power_even_abs
thf(fact_2667_log__mult,axiom,
! [A2: real,X: real,Y2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
=> ( ( A2
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ Y2 )
=> ( ( log @ A2 @ ( times_times @ real @ X @ Y2 ) )
= ( plus_plus @ real @ ( log @ A2 @ X ) @ ( log @ A2 @ Y2 ) ) ) ) ) ) ) ).
% log_mult
thf(fact_2668_log__divide,axiom,
! [A2: real,X: real,Y2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
=> ( ( A2
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ Y2 )
=> ( ( log @ A2 @ ( divide_divide @ real @ X @ Y2 ) )
= ( minus_minus @ real @ ( log @ A2 @ X ) @ ( log @ A2 @ Y2 ) ) ) ) ) ) ) ).
% log_divide
thf(fact_2669_power2__le__iff__abs__le,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Y2: A,X: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y2 )
=> ( ( ord_less_eq @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( ord_less_eq @ A @ ( abs_abs @ A @ X ) @ Y2 ) ) ) ) ).
% power2_le_iff_abs_le
thf(fact_2670_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_2671_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_2672_power__mono__even,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: nat,A2: A,B2: A] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( ord_less_eq @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B2 ) )
=> ( ord_less_eq @ A @ ( power_power @ A @ A2 @ N ) @ ( power_power @ A @ B2 @ N ) ) ) ) ) ).
% power_mono_even
thf(fact_2673_log__eq__div__ln__mult__log,axiom,
! [A2: real,B2: real,X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
=> ( ( A2
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ B2 )
=> ( ( B2
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( log @ A2 @ X )
= ( times_times @ real @ ( divide_divide @ real @ ( ln_ln @ real @ B2 ) @ ( ln_ln @ real @ A2 ) ) @ ( log @ B2 @ X ) ) ) ) ) ) ) ) ).
% log_eq_div_ln_mult_log
thf(fact_2674_sqrt__ge__absD,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ X ) @ ( sqrt @ Y2 ) )
=> ( ord_less_eq @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ Y2 ) ) ).
% sqrt_ge_absD
thf(fact_2675_sqrt__sum__squares__le__sum__abs,axiom,
! [X: real,Y2: real] : ( ord_less_eq @ real @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( plus_plus @ real @ ( abs_abs @ real @ X ) @ ( abs_abs @ real @ Y2 ) ) ) ).
% sqrt_sum_squares_le_sum_abs
thf(fact_2676_real__sqrt__ge__abs2,axiom,
! [Y2: real,X: real] : ( ord_less_eq @ real @ ( abs_abs @ real @ Y2 ) @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% real_sqrt_ge_abs2
thf(fact_2677_real__sqrt__ge__abs1,axiom,
! [X: real,Y2: real] : ( ord_less_eq @ real @ ( abs_abs @ real @ X ) @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% real_sqrt_ge_abs1
thf(fact_2678_arctan__add,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ X ) @ ( one_one @ real ) )
=> ( ( ord_less @ real @ ( abs_abs @ real @ Y2 ) @ ( one_one @ real ) )
=> ( ( plus_plus @ real @ ( arctan @ X ) @ ( arctan @ Y2 ) )
= ( arctan @ ( divide_divide @ real @ ( plus_plus @ real @ X @ Y2 ) @ ( minus_minus @ real @ ( one_one @ real ) @ ( times_times @ real @ X @ Y2 ) ) ) ) ) ) ) ).
% arctan_add
thf(fact_2679_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_2680_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_2681_cos__x__y__le__one,axiom,
! [X: real,Y2: real] : ( ord_less_eq @ real @ ( abs_abs @ real @ ( divide_divide @ real @ X @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) @ ( one_one @ real ) ) ).
% cos_x_y_le_one
thf(fact_2682_real__sqrt__sum__squares__less,axiom,
! [X: real,U: real,Y2: real] :
( ( ord_less @ real @ ( abs_abs @ real @ X ) @ ( divide_divide @ real @ U @ ( sqrt @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) )
=> ( ( ord_less @ real @ ( abs_abs @ real @ Y2 ) @ ( divide_divide @ real @ U @ ( sqrt @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) )
=> ( ord_less @ real @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ U ) ) ) ).
% real_sqrt_sum_squares_less
thf(fact_2683_log__base__10__eq2,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( log @ ( numeral_numeral @ real @ ( bit0 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) @ X )
= ( times_times @ real @ ( log @ ( numeral_numeral @ real @ ( bit0 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) @ ( exp @ real @ ( one_one @ real ) ) ) @ ( ln_ln @ real @ X ) ) ) ) ).
% log_base_10_eq2
thf(fact_2684_abs__ln__one__plus__x__minus__x__bound__nonneg,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ X @ ( one_one @ real ) )
=> ( ord_less_eq @ real @ ( abs_abs @ real @ ( minus_minus @ real @ ( ln_ln @ real @ ( plus_plus @ real @ ( one_one @ real ) @ X ) ) @ X ) ) @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% abs_ln_one_plus_x_minus_x_bound_nonneg
thf(fact_2685_abs__sqrt__wlog,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [P3: A > A > $o,X: A] :
( ! [X5: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X5 )
=> ( P3 @ X5 @ ( power_power @ A @ X5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
=> ( P3 @ ( abs_abs @ A @ X ) @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% abs_sqrt_wlog
thf(fact_2686_log2__of__power__le,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
=> ( ord_less_eq @ real @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ M ) ) @ ( semiring_1_of_nat @ real @ N ) ) ) ) ).
% log2_of_power_le
thf(fact_2687_machin,axiom,
( ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ one2 ) ) ) )
= ( minus_minus @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( arctan @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit1 @ ( bit0 @ one2 ) ) ) ) ) ) @ ( arctan @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ).
% machin
thf(fact_2688_machin__Euler,axiom,
( ( plus_plus @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit1 @ ( bit0 @ one2 ) ) ) @ ( arctan @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( arctan @ ( divide_divide @ real @ ( numeral_numeral @ real @ ( bit1 @ one2 ) ) @ ( numeral_numeral @ real @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) )
= ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ).
% machin_Euler
thf(fact_2689_exp__ge__one__minus__x__over__n__power__n,axiom,
! [X: real,N: nat] :
( ( ord_less_eq @ real @ X @ ( semiring_1_of_nat @ real @ N ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ord_less_eq @ real @ ( power_power @ real @ ( minus_minus @ real @ ( one_one @ real ) @ ( divide_divide @ real @ X @ ( semiring_1_of_nat @ real @ N ) ) ) @ N ) @ ( exp @ real @ ( uminus_uminus @ real @ X ) ) ) ) ) ).
% exp_ge_one_minus_x_over_n_power_n
thf(fact_2690_exp__ge__one__plus__x__over__n__power__n,axiom,
! [N: nat,X: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( semiring_1_of_nat @ real @ N ) ) @ X )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ord_less_eq @ real @ ( power_power @ real @ ( plus_plus @ real @ ( one_one @ real ) @ ( divide_divide @ real @ X @ ( semiring_1_of_nat @ real @ N ) ) ) @ N ) @ ( exp @ real @ X ) ) ) ) ).
% exp_ge_one_plus_x_over_n_power_n
thf(fact_2691_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_2692_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_2693_zdvd1__eq,axiom,
! [X: int] :
( ( dvd_dvd @ int @ X @ ( one_one @ int ) )
= ( ( abs_abs @ int @ X )
= ( one_one @ int ) ) ) ).
% zdvd1_eq
thf(fact_2694_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_2695_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_2696_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_2697_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_2698_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_2699_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_2700_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_2701_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_2702_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_2703_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_2704_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_2705_of__nat__power__eq__of__nat__cancel__iff,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [X: nat,B2: nat,W: nat] :
( ( ( semiring_1_of_nat @ A @ X )
= ( power_power @ A @ ( semiring_1_of_nat @ A @ B2 ) @ W ) )
= ( X
= ( power_power @ nat @ B2 @ W ) ) ) ) ).
% of_nat_power_eq_of_nat_cancel_iff
thf(fact_2706_of__nat__eq__of__nat__power__cancel__iff,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [B2: nat,W: nat,X: nat] :
( ( ( power_power @ A @ ( semiring_1_of_nat @ A @ B2 ) @ W )
= ( semiring_1_of_nat @ A @ X ) )
= ( ( power_power @ nat @ B2 @ W )
= X ) ) ) ).
% of_nat_eq_of_nat_power_cancel_iff
thf(fact_2707_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_2708_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_2709_of__nat__of__bool,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [P3: $o] :
( ( semiring_1_of_nat @ A @ ( zero_neq_one_of_bool @ nat @ P3 ) )
= ( zero_neq_one_of_bool @ A @ P3 ) ) ) ).
% of_nat_of_bool
thf(fact_2710_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_2711_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_2712_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_2713_of__nat__power__less__of__nat__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X: nat,B2: nat,W: nat] :
( ( ord_less @ A @ ( semiring_1_of_nat @ A @ X ) @ ( power_power @ A @ ( semiring_1_of_nat @ A @ B2 ) @ W ) )
= ( ord_less @ nat @ X @ ( power_power @ nat @ B2 @ W ) ) ) ) ).
% of_nat_power_less_of_nat_cancel_iff
thf(fact_2714_of__nat__less__of__nat__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [B2: nat,W: nat,X: nat] :
( ( ord_less @ A @ ( power_power @ A @ ( semiring_1_of_nat @ A @ B2 ) @ W ) @ ( semiring_1_of_nat @ A @ X ) )
= ( ord_less @ nat @ ( power_power @ nat @ B2 @ W ) @ X ) ) ) ).
% of_nat_less_of_nat_power_cancel_iff
thf(fact_2715_of__nat__power__le__of__nat__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X: nat,B2: nat,W: nat] :
( ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ X ) @ ( power_power @ A @ ( semiring_1_of_nat @ A @ B2 ) @ W ) )
= ( ord_less_eq @ nat @ X @ ( power_power @ nat @ B2 @ W ) ) ) ) ).
% of_nat_power_le_of_nat_cancel_iff
thf(fact_2716_of__nat__le__of__nat__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [B2: nat,W: nat,X: nat] :
( ( ord_less_eq @ A @ ( power_power @ A @ ( semiring_1_of_nat @ A @ B2 ) @ W ) @ ( semiring_1_of_nat @ A @ X ) )
= ( ord_less_eq @ nat @ ( power_power @ nat @ B2 @ W ) @ X ) ) ) ).
% of_nat_le_of_nat_power_cancel_iff
thf(fact_2717_real__of__nat__eq__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [Y2: nat,X: num,N: nat] :
( ( ( semiring_1_of_nat @ A @ Y2 )
= ( power_power @ A @ ( numeral_numeral @ A @ X ) @ N ) )
= ( Y2
= ( power_power @ nat @ ( numeral_numeral @ nat @ X ) @ N ) ) ) ) ).
% real_of_nat_eq_numeral_power_cancel_iff
thf(fact_2718_numeral__power__eq__of__nat__cancel__iff,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [X: num,N: nat,Y2: nat] :
( ( ( power_power @ A @ ( numeral_numeral @ A @ X ) @ N )
= ( semiring_1_of_nat @ A @ Y2 ) )
= ( ( power_power @ nat @ ( numeral_numeral @ nat @ X ) @ N )
= Y2 ) ) ) ).
% numeral_power_eq_of_nat_cancel_iff
thf(fact_2719_real__of__nat__less__numeral__iff,axiom,
! [N: nat,W: num] :
( ( ord_less @ real @ ( semiring_1_of_nat @ real @ N ) @ ( numeral_numeral @ real @ W ) )
= ( ord_less @ nat @ N @ ( numeral_numeral @ nat @ W ) ) ) ).
% real_of_nat_less_numeral_iff
thf(fact_2720_numeral__less__real__of__nat__iff,axiom,
! [W: num,N: nat] :
( ( ord_less @ real @ ( numeral_numeral @ real @ W ) @ ( semiring_1_of_nat @ real @ N ) )
= ( ord_less @ nat @ ( numeral_numeral @ nat @ W ) @ N ) ) ).
% numeral_less_real_of_nat_iff
thf(fact_2721_numeral__le__real__of__nat__iff,axiom,
! [N: num,M: nat] :
( ( ord_less_eq @ real @ ( numeral_numeral @ real @ N ) @ ( semiring_1_of_nat @ real @ M ) )
= ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ N ) @ M ) ) ).
% numeral_le_real_of_nat_iff
thf(fact_2722_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_2723_log__pow__cancel,axiom,
! [A2: real,B2: nat] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
=> ( ( A2
!= ( one_one @ real ) )
=> ( ( log @ A2 @ ( power_power @ real @ A2 @ B2 ) )
= ( semiring_1_of_nat @ real @ B2 ) ) ) ) ).
% log_pow_cancel
thf(fact_2724_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_2725_of__nat__less__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X: nat,I2: num,N: nat] :
( ( ord_less @ A @ ( semiring_1_of_nat @ A @ X ) @ ( power_power @ A @ ( numeral_numeral @ A @ I2 ) @ N ) )
= ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ I2 ) @ N ) ) ) ) ).
% of_nat_less_numeral_power_cancel_iff
thf(fact_2726_numeral__power__less__of__nat__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [I2: num,N: nat,X: nat] :
( ( ord_less @ A @ ( power_power @ A @ ( numeral_numeral @ A @ I2 ) @ N ) @ ( semiring_1_of_nat @ A @ X ) )
= ( ord_less @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ I2 ) @ N ) @ X ) ) ) ).
% numeral_power_less_of_nat_cancel_iff
thf(fact_2727_of__nat__le__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X: nat,I2: num,N: nat] :
( ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ X ) @ ( power_power @ A @ ( numeral_numeral @ A @ I2 ) @ N ) )
= ( ord_less_eq @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ I2 ) @ N ) ) ) ) ).
% of_nat_le_numeral_power_cancel_iff
thf(fact_2728_numeral__power__le__of__nat__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [I2: num,N: nat,X: nat] :
( ( ord_less_eq @ A @ ( power_power @ A @ ( numeral_numeral @ A @ I2 ) @ N ) @ ( semiring_1_of_nat @ A @ X ) )
= ( ord_less_eq @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ I2 ) @ N ) @ X ) ) ) ).
% numeral_power_le_of_nat_cancel_iff
thf(fact_2729_real__arch__simple,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X: A] :
? [N3: nat] : ( ord_less_eq @ A @ X @ ( semiring_1_of_nat @ A @ N3 ) ) ) ).
% real_arch_simple
thf(fact_2730_reals__Archimedean2,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X: A] :
? [N3: nat] : ( ord_less @ A @ X @ ( semiring_1_of_nat @ A @ N3 ) ) ) ).
% reals_Archimedean2
thf(fact_2731_mult__of__nat__commute,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [X: nat,Y2: A] :
( ( times_times @ A @ ( semiring_1_of_nat @ A @ X ) @ Y2 )
= ( times_times @ A @ Y2 @ ( semiring_1_of_nat @ A @ X ) ) ) ) ).
% mult_of_nat_commute
thf(fact_2732_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_2733_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_2734_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_2735_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_2736_div__mult2__eq_H,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [A2: A,M: nat,N: nat] :
( ( divide_divide @ A @ A2 @ ( times_times @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) ) )
= ( divide_divide @ A @ ( divide_divide @ A @ A2 @ ( semiring_1_of_nat @ A @ M ) ) @ ( semiring_1_of_nat @ A @ N ) ) ) ) ).
% div_mult2_eq'
thf(fact_2737_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_2738_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_2739_of__nat__mono,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [I2: nat,J3: nat] :
( ( ord_less_eq @ nat @ I2 @ J3 )
=> ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ I2 ) @ ( semiring_1_of_nat @ A @ J3 ) ) ) ) ).
% of_nat_mono
thf(fact_2740_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_2741_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_2742_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_2743_pi__ge__zero,axiom,
ord_less_eq @ real @ ( zero_zero @ real ) @ pi ).
% pi_ge_zero
thf(fact_2744_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_2745_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_2746_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_2747_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_2748_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_2749_ex__less__of__nat__mult,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X: A,Y2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X )
=> ? [N3: nat] : ( ord_less @ A @ Y2 @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N3 ) @ X ) ) ) ) ).
% ex_less_of_nat_mult
thf(fact_2750_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_2751_exp__of__nat__mult,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [N: nat,X: A] :
( ( exp @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ X ) )
= ( power_power @ A @ ( exp @ A @ X ) @ N ) ) ) ).
% exp_of_nat_mult
thf(fact_2752_exp__of__nat2__mult,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A,N: nat] :
( ( exp @ A @ ( times_times @ A @ X @ ( semiring_1_of_nat @ A @ N ) ) )
= ( power_power @ A @ ( exp @ A @ X ) @ N ) ) ) ).
% exp_of_nat2_mult
thf(fact_2753_reals__Archimedean3,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ! [Y4: real] :
? [N3: nat] : ( ord_less @ real @ Y4 @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N3 ) @ X ) ) ) ).
% reals_Archimedean3
thf(fact_2754_dvd__imp__le__int,axiom,
! [I2: int,D2: int] :
( ( I2
!= ( zero_zero @ int ) )
=> ( ( dvd_dvd @ int @ D2 @ I2 )
=> ( ord_less_eq @ int @ ( abs_abs @ int @ D2 ) @ ( abs_abs @ int @ I2 ) ) ) ) ).
% dvd_imp_le_int
thf(fact_2755_real__of__nat__div4,axiom,
! [N: nat,X: nat] : ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ ( divide_divide @ nat @ N @ X ) ) @ ( divide_divide @ real @ ( semiring_1_of_nat @ real @ N ) @ ( semiring_1_of_nat @ real @ X ) ) ) ).
% real_of_nat_div4
thf(fact_2756_abs__mod__less,axiom,
! [L: int,K2: int] :
( ( L
!= ( zero_zero @ int ) )
=> ( ord_less @ int @ ( abs_abs @ int @ ( modulo_modulo @ int @ K2 @ L ) ) @ ( abs_abs @ int @ L ) ) ) ).
% abs_mod_less
thf(fact_2757_real__of__nat__div,axiom,
! [D2: nat,N: nat] :
( ( dvd_dvd @ nat @ D2 @ N )
=> ( ( semiring_1_of_nat @ real @ ( divide_divide @ nat @ N @ D2 ) )
= ( divide_divide @ real @ ( semiring_1_of_nat @ real @ N ) @ ( semiring_1_of_nat @ real @ D2 ) ) ) ) ).
% real_of_nat_div
thf(fact_2758_mod__mult2__eq_H,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [A2: A,M: nat,N: nat] :
( ( modulo_modulo @ A @ A2 @ ( times_times @ A @ ( semiring_1_of_nat @ A @ M ) @ ( semiring_1_of_nat @ A @ N ) ) )
= ( plus_plus @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ M ) @ ( modulo_modulo @ A @ ( divide_divide @ A @ A2 @ ( semiring_1_of_nat @ A @ M ) ) @ ( semiring_1_of_nat @ A @ N ) ) ) @ ( modulo_modulo @ A @ A2 @ ( semiring_1_of_nat @ A @ M ) ) ) ) ) ).
% mod_mult2_eq'
thf(fact_2759_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_2760_pi__less__4,axiom,
ord_less @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ one2 ) ) ) ).
% pi_less_4
thf(fact_2761_nat__less__real__le,axiom,
( ( ord_less @ nat )
= ( ^ [N2: nat,M2: nat] : ( ord_less_eq @ real @ ( plus_plus @ real @ ( semiring_1_of_nat @ real @ N2 ) @ ( one_one @ real ) ) @ ( semiring_1_of_nat @ real @ M2 ) ) ) ) ).
% nat_less_real_le
thf(fact_2762_nat__le__real__less,axiom,
( ( ord_less_eq @ nat )
= ( ^ [N2: nat,M2: nat] : ( ord_less @ real @ ( semiring_1_of_nat @ real @ N2 ) @ ( plus_plus @ real @ ( semiring_1_of_nat @ real @ M2 ) @ ( one_one @ real ) ) ) ) ) ).
% nat_le_real_less
thf(fact_2763_pi__ge__two,axiom,
ord_less_eq @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ).
% pi_ge_two
thf(fact_2764_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_2765_pi__half__neq__two,axiom,
( ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) )
!= ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ).
% pi_half_neq_two
thf(fact_2766_log__of__power__eq,axiom,
! [M: nat,B2: real,N: nat] :
( ( ( semiring_1_of_nat @ real @ M )
= ( power_power @ real @ B2 @ N ) )
=> ( ( ord_less @ real @ ( one_one @ real ) @ B2 )
=> ( ( semiring_1_of_nat @ real @ N )
= ( log @ B2 @ ( semiring_1_of_nat @ real @ M ) ) ) ) ) ).
% log_of_power_eq
thf(fact_2767_less__log__of__power,axiom,
! [B2: real,N: nat,M: real] :
( ( ord_less @ real @ ( power_power @ real @ B2 @ N ) @ M )
=> ( ( ord_less @ real @ ( one_one @ real ) @ B2 )
=> ( ord_less @ real @ ( semiring_1_of_nat @ real @ N ) @ ( log @ B2 @ M ) ) ) ) ).
% less_log_of_power
thf(fact_2768_real__of__nat__div__aux,axiom,
! [X: nat,D2: nat] :
( ( divide_divide @ real @ ( semiring_1_of_nat @ real @ X ) @ ( semiring_1_of_nat @ real @ D2 ) )
= ( plus_plus @ real @ ( semiring_1_of_nat @ real @ ( divide_divide @ nat @ X @ D2 ) ) @ ( divide_divide @ real @ ( semiring_1_of_nat @ real @ ( modulo_modulo @ nat @ X @ D2 ) ) @ ( semiring_1_of_nat @ real @ D2 ) ) ) ) ).
% real_of_nat_div_aux
thf(fact_2769_nat__approx__posE,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [E: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ E )
=> ~ ! [N3: nat] :
~ ( ord_less @ A @ ( divide_divide @ A @ ( one_one @ A ) @ ( semiring_1_of_nat @ A @ ( suc @ N3 ) ) ) @ E ) ) ) ).
% nat_approx_posE
thf(fact_2770_of__nat__less__two__power,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: nat] : ( ord_less @ A @ ( semiring_1_of_nat @ A @ N ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) ) ).
% of_nat_less_two_power
thf(fact_2771_inverse__of__nat__le,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [N: nat,M: nat] :
( ( ord_less_eq @ nat @ N @ M )
=> ( ( N
!= ( zero_zero @ nat ) )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ ( one_one @ A ) @ ( semiring_1_of_nat @ A @ M ) ) @ ( divide_divide @ A @ ( one_one @ A ) @ ( semiring_1_of_nat @ A @ N ) ) ) ) ) ) ).
% inverse_of_nat_le
thf(fact_2772_exp__divide__power__eq,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [N: nat,X: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( power_power @ A @ ( exp @ A @ ( divide_divide @ A @ X @ ( semiring_1_of_nat @ A @ N ) ) ) @ N )
= ( exp @ A @ X ) ) ) ) ).
% exp_divide_power_eq
thf(fact_2773_even__abs__add__iff,axiom,
! [K2: int,L: int] :
( ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( plus_plus @ int @ ( abs_abs @ int @ K2 ) @ L ) )
= ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( plus_plus @ int @ K2 @ L ) ) ) ).
% even_abs_add_iff
thf(fact_2774_even__add__abs__iff,axiom,
! [K2: int,L: int] :
( ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( plus_plus @ int @ K2 @ ( abs_abs @ int @ L ) ) )
= ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( plus_plus @ int @ K2 @ L ) ) ) ).
% even_add_abs_iff
thf(fact_2775_real__archimedian__rdiv__eq__0,axiom,
! [X: real,C2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ C2 )
=> ( ! [M4: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M4 )
=> ( ord_less_eq @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ M4 ) @ X ) @ C2 ) )
=> ( X
= ( zero_zero @ real ) ) ) ) ) ).
% real_archimedian_rdiv_eq_0
thf(fact_2776_pi__half__neq__zero,axiom,
( ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) )
!= ( zero_zero @ real ) ) ).
% pi_half_neq_zero
thf(fact_2777_pi__half__less__two,axiom,
ord_less @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ).
% pi_half_less_two
thf(fact_2778_pi__half__le__two,axiom,
ord_less_eq @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ).
% pi_half_le_two
thf(fact_2779_real__of__nat__div2,axiom,
! [N: nat,X: nat] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( minus_minus @ real @ ( divide_divide @ real @ ( semiring_1_of_nat @ real @ N ) @ ( semiring_1_of_nat @ real @ X ) ) @ ( semiring_1_of_nat @ real @ ( divide_divide @ nat @ N @ X ) ) ) ) ).
% real_of_nat_div2
thf(fact_2780_le__log__of__power,axiom,
! [B2: real,N: nat,M: real] :
( ( ord_less_eq @ real @ ( power_power @ real @ B2 @ N ) @ M )
=> ( ( ord_less @ real @ ( one_one @ real ) @ B2 )
=> ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ N ) @ ( log @ B2 @ M ) ) ) ) ).
% le_log_of_power
thf(fact_2781_ln__realpow,axiom,
! [X: real,N: nat] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ln_ln @ real @ ( power_power @ real @ X @ N ) )
= ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ ( ln_ln @ real @ X ) ) ) ) ).
% ln_realpow
thf(fact_2782_log__nat__power,axiom,
! [X: real,B2: real,N: nat] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( log @ B2 @ ( power_power @ real @ X @ N ) )
= ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ ( log @ B2 @ X ) ) ) ) ).
% log_nat_power
thf(fact_2783_real__of__nat__div3,axiom,
! [N: nat,X: nat] : ( ord_less_eq @ real @ ( minus_minus @ real @ ( divide_divide @ real @ ( semiring_1_of_nat @ real @ N ) @ ( semiring_1_of_nat @ real @ X ) ) @ ( semiring_1_of_nat @ real @ ( divide_divide @ nat @ N @ X ) ) ) @ ( one_one @ real ) ) ).
% real_of_nat_div3
thf(fact_2784_nat__intermed__int__val,axiom,
! [M: nat,N: nat,F3: nat > int,K2: int] :
( ! [I3: nat] :
( ( ( ord_less_eq @ nat @ M @ I3 )
& ( ord_less @ nat @ I3 @ N ) )
=> ( ord_less_eq @ int @ ( abs_abs @ int @ ( minus_minus @ int @ ( F3 @ ( suc @ I3 ) ) @ ( F3 @ I3 ) ) ) @ ( one_one @ int ) ) )
=> ( ( ord_less_eq @ nat @ M @ N )
=> ( ( ord_less_eq @ int @ ( F3 @ M ) @ K2 )
=> ( ( ord_less_eq @ int @ K2 @ ( F3 @ N ) )
=> ? [I3: nat] :
( ( ord_less_eq @ nat @ M @ I3 )
& ( ord_less_eq @ nat @ I3 @ N )
& ( ( F3 @ I3 )
= K2 ) ) ) ) ) ) ).
% nat_intermed_int_val
thf(fact_2785_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_2786_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_2787_pi__half__gt__zero,axiom,
ord_less @ real @ ( zero_zero @ real ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ).
% pi_half_gt_zero
thf(fact_2788_linear__plus__1__le__power,axiom,
! [X: real,N: nat] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ord_less_eq @ real @ ( plus_plus @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ X ) @ ( one_one @ real ) ) @ ( power_power @ real @ ( plus_plus @ real @ X @ ( one_one @ real ) ) @ N ) ) ) ).
% linear_plus_1_le_power
thf(fact_2789_pi__half__ge__zero,axiom,
ord_less_eq @ real @ ( zero_zero @ real ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ).
% pi_half_ge_zero
thf(fact_2790_log2__of__power__eq,axiom,
! [M: nat,N: nat] :
( ( M
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( semiring_1_of_nat @ real @ N )
= ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ M ) ) ) ) ).
% log2_of_power_eq
thf(fact_2791_log__of__power__less,axiom,
! [M: nat,B2: real,N: nat] :
( ( ord_less @ real @ ( semiring_1_of_nat @ real @ M ) @ ( power_power @ real @ B2 @ N ) )
=> ( ( ord_less @ real @ ( one_one @ real ) @ B2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
=> ( ord_less @ real @ ( log @ B2 @ ( semiring_1_of_nat @ real @ M ) ) @ ( semiring_1_of_nat @ real @ N ) ) ) ) ) ).
% log_of_power_less
thf(fact_2792_Bernoulli__inequality,axiom,
! [X: real,N: nat] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X )
=> ( ord_less_eq @ real @ ( plus_plus @ real @ ( one_one @ real ) @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ X ) ) @ ( power_power @ real @ ( plus_plus @ real @ ( one_one @ real ) @ X ) @ N ) ) ) ).
% Bernoulli_inequality
thf(fact_2793_m2pi__less__pi,axiom,
ord_less @ real @ ( uminus_uminus @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) ) @ pi ).
% m2pi_less_pi
thf(fact_2794_arctan__ubound,axiom,
! [Y2: real] : ( ord_less @ real @ ( arctan @ Y2 ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ).
% arctan_ubound
thf(fact_2795_arctan__one,axiom,
( ( arctan @ ( one_one @ real ) )
= ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ).
% arctan_one
thf(fact_2796_nat__ivt__aux,axiom,
! [N: nat,F3: nat > int,K2: int] :
( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ N )
=> ( ord_less_eq @ int @ ( abs_abs @ int @ ( minus_minus @ int @ ( F3 @ ( suc @ I3 ) ) @ ( F3 @ I3 ) ) ) @ ( one_one @ int ) ) )
=> ( ( ord_less_eq @ int @ ( F3 @ ( zero_zero @ nat ) ) @ K2 )
=> ( ( ord_less_eq @ int @ K2 @ ( F3 @ N ) )
=> ? [I3: nat] :
( ( ord_less_eq @ nat @ I3 @ N )
& ( ( F3 @ I3 )
= K2 ) ) ) ) ) ).
% nat_ivt_aux
thf(fact_2797_log__of__power__le,axiom,
! [M: nat,B2: real,N: nat] :
( ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ M ) @ ( power_power @ real @ B2 @ N ) )
=> ( ( ord_less @ real @ ( one_one @ real ) @ B2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
=> ( ord_less_eq @ real @ ( log @ B2 @ ( semiring_1_of_nat @ real @ M ) ) @ ( semiring_1_of_nat @ real @ N ) ) ) ) ) ).
% log_of_power_le
thf(fact_2798_minus__pi__half__less__zero,axiom,
ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ ( zero_zero @ real ) ).
% minus_pi_half_less_zero
thf(fact_2799_arctan__bounded,axiom,
! [Y2: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ ( arctan @ Y2 ) )
& ( ord_less @ real @ ( arctan @ Y2 ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ).
% arctan_bounded
thf(fact_2800_arctan__lbound,axiom,
! [Y2: real] : ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ ( arctan @ Y2 ) ) ).
% arctan_lbound
thf(fact_2801_nat0__intermed__int__val,axiom,
! [N: nat,F3: nat > int,K2: int] :
( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ N )
=> ( ord_less_eq @ int @ ( abs_abs @ int @ ( minus_minus @ int @ ( F3 @ ( plus_plus @ nat @ I3 @ ( one_one @ nat ) ) ) @ ( F3 @ I3 ) ) ) @ ( one_one @ int ) ) )
=> ( ( ord_less_eq @ int @ ( F3 @ ( zero_zero @ nat ) ) @ K2 )
=> ( ( ord_less_eq @ int @ K2 @ ( F3 @ N ) )
=> ? [I3: nat] :
( ( ord_less_eq @ nat @ I3 @ N )
& ( ( F3 @ I3 )
= K2 ) ) ) ) ) ).
% nat0_intermed_int_val
thf(fact_2802_less__log2__of__power,axiom,
! [N: nat,M: nat] :
( ( ord_less @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ M )
=> ( ord_less @ real @ ( semiring_1_of_nat @ real @ N ) @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ M ) ) ) ) ).
% less_log2_of_power
thf(fact_2803_le__log2__of__power,axiom,
! [N: nat,M: nat] :
( ( ord_less_eq @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ M )
=> ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ N ) @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ M ) ) ) ) ).
% le_log2_of_power
thf(fact_2804_Bernoulli__inequality__even,axiom,
! [N: nat,X: real] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ord_less_eq @ real @ ( plus_plus @ real @ ( one_one @ real ) @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ X ) ) @ ( power_power @ real @ ( plus_plus @ real @ ( one_one @ real ) @ X ) @ N ) ) ) ).
% Bernoulli_inequality_even
thf(fact_2805_log2__of__power__less,axiom,
! [M: nat,N: nat] :
( ( ord_less @ nat @ M @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
=> ( ord_less @ real @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ M ) ) @ ( semiring_1_of_nat @ real @ N ) ) ) ) ).
% log2_of_power_less
thf(fact_2806_of__nat__code__if,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( ( semiring_1_of_nat @ A )
= ( ^ [N2: nat] :
( if @ A
@ ( N2
= ( zero_zero @ nat ) )
@ ( zero_zero @ A )
@ ( product_case_prod @ nat @ nat @ A
@ ^ [M2: nat,Q5: nat] :
( if @ A
@ ( Q5
= ( zero_zero @ nat ) )
@ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ A @ M2 ) )
@ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ A @ M2 ) ) @ ( one_one @ A ) ) )
@ ( divmod_nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% of_nat_code_if
thf(fact_2807_monoseq__arctan__series,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ X ) @ ( one_one @ real ) )
=> ( topological_monoseq @ real
@ ^ [N2: nat] : ( times_times @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( semiring_1_of_nat @ real @ ( plus_plus @ nat @ ( times_times @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) @ ( power_power @ real @ X @ ( plus_plus @ nat @ ( times_times @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) ) ) ).
% monoseq_arctan_series
thf(fact_2808_lemma__termdiff3,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [H2: A,Z3: A,K5: real,N: nat] :
( ( H2
!= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ Z3 ) @ K5 )
=> ( ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ Z3 @ H2 ) ) @ K5 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( divide_divide @ A @ ( minus_minus @ A @ ( power_power @ A @ ( plus_plus @ A @ Z3 @ H2 ) @ N ) @ ( power_power @ A @ Z3 @ N ) ) @ H2 ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ ( power_power @ A @ Z3 @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) @ ( times_times @ real @ ( times_times @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ ( semiring_1_of_nat @ real @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) ) ) @ ( power_power @ real @ K5 @ ( minus_minus @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( real_V7770717601297561774m_norm @ A @ H2 ) ) ) ) ) ) ) ).
% lemma_termdiff3
thf(fact_2809_sin__cos__npi,axiom,
! [N: nat] :
( ( sin @ real @ ( divide_divide @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) @ pi ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
= ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ N ) ) ).
% sin_cos_npi
thf(fact_2810_ln__series,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ X @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) )
=> ( ( ln_ln @ real @ X )
= ( suminf @ real
@ ^ [N2: nat] : ( times_times @ real @ ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ N2 ) @ ( divide_divide @ real @ ( one_one @ real ) @ ( semiring_1_of_nat @ real @ ( plus_plus @ nat @ N2 @ ( one_one @ nat ) ) ) ) ) @ ( power_power @ real @ ( minus_minus @ real @ X @ ( one_one @ real ) ) @ ( suc @ N2 ) ) ) ) ) ) ) ).
% ln_series
thf(fact_2811_arctan__series,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ X ) @ ( one_one @ real ) )
=> ( ( arctan @ X )
= ( suminf @ real
@ ^ [K3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ K3 ) @ ( times_times @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( semiring_1_of_nat @ real @ ( plus_plus @ nat @ ( times_times @ nat @ K3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) @ ( power_power @ real @ X @ ( plus_plus @ nat @ ( times_times @ nat @ K3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% arctan_series
thf(fact_2812_int__eq__iff__numeral,axiom,
! [M: nat,V2: num] :
( ( ( semiring_1_of_nat @ int @ M )
= ( numeral_numeral @ int @ V2 ) )
= ( M
= ( numeral_numeral @ nat @ V2 ) ) ) ).
% int_eq_iff_numeral
thf(fact_2813_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_2814_sin__npi,axiom,
! [N: nat] :
( ( sin @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ pi ) )
= ( zero_zero @ real ) ) ).
% sin_npi
thf(fact_2815_sin__npi2,axiom,
! [N: nat] :
( ( sin @ real @ ( times_times @ real @ pi @ ( semiring_1_of_nat @ real @ N ) ) )
= ( zero_zero @ real ) ) ).
% sin_npi2
thf(fact_2816_sin__npi__int,axiom,
! [N: int] :
( ( sin @ real @ ( times_times @ real @ pi @ ( ring_1_of_int @ real @ N ) ) )
= ( zero_zero @ real ) ) ).
% sin_npi_int
thf(fact_2817_powser__zero,axiom,
! [A: $tType] :
( ( real_V2822296259951069270ebra_1 @ A )
=> ! [F3: nat > A] :
( ( suminf @ A
@ ^ [N2: nat] : ( times_times @ A @ ( F3 @ N2 ) @ ( power_power @ A @ ( zero_zero @ A ) @ N2 ) ) )
= ( F3 @ ( zero_zero @ nat ) ) ) ) ).
% powser_zero
thf(fact_2818_sin__two__pi,axiom,
( ( sin @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) )
= ( zero_zero @ real ) ) ).
% sin_two_pi
thf(fact_2819_sin__pi__half,axiom,
( ( sin @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
= ( one_one @ real ) ) ).
% sin_pi_half
thf(fact_2820_sin__periodic,axiom,
! [X: real] :
( ( sin @ real @ ( plus_plus @ real @ X @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) ) )
= ( sin @ real @ X ) ) ).
% sin_periodic
thf(fact_2821_sin__2npi,axiom,
! [N: nat] :
( ( sin @ real @ ( times_times @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ N ) ) @ pi ) )
= ( zero_zero @ real ) ) ).
% sin_2npi
thf(fact_2822_sin__2pi__minus,axiom,
! [X: real] :
( ( sin @ real @ ( minus_minus @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) @ X ) )
= ( uminus_uminus @ real @ ( sin @ real @ X ) ) ) ).
% sin_2pi_minus
thf(fact_2823_sin__int__2pin,axiom,
! [N: int] :
( ( sin @ real @ ( times_times @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) @ ( ring_1_of_int @ real @ N ) ) )
= ( zero_zero @ real ) ) ).
% sin_int_2pin
thf(fact_2824_sin__3over2__pi,axiom,
( ( sin @ real @ ( times_times @ real @ ( divide_divide @ real @ ( numeral_numeral @ real @ ( bit1 @ one2 ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ pi ) )
= ( uminus_uminus @ real @ ( one_one @ real ) ) ) ).
% sin_3over2_pi
thf(fact_2825_nat__int__comparison_I1_J,axiom,
( ( ^ [Y5: nat,Z2: nat] : Y5 = Z2 )
= ( ^ [A5: nat,B4: nat] :
( ( semiring_1_of_nat @ int @ A5 )
= ( semiring_1_of_nat @ int @ B4 ) ) ) ) ).
% nat_int_comparison(1)
thf(fact_2826_int__if,axiom,
! [P3: $o,A2: nat,B2: nat] :
( ( P3
=> ( ( semiring_1_of_nat @ int @ ( if @ nat @ P3 @ A2 @ B2 ) )
= ( semiring_1_of_nat @ int @ A2 ) ) )
& ( ~ P3
=> ( ( semiring_1_of_nat @ int @ ( if @ nat @ P3 @ A2 @ B2 ) )
= ( semiring_1_of_nat @ int @ B2 ) ) ) ) ).
% int_if
thf(fact_2827_of__nat__eq__enat,axiom,
( ( semiring_1_of_nat @ extended_enat )
= extended_enat2 ) ).
% of_nat_eq_enat
thf(fact_2828_sin__x__le__x,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ord_less_eq @ real @ ( sin @ real @ X ) @ X ) ) ).
% sin_x_le_x
thf(fact_2829_sin__le__one,axiom,
! [X: real] : ( ord_less_eq @ real @ ( sin @ real @ X ) @ ( one_one @ real ) ) ).
% sin_le_one
thf(fact_2830_abs__sin__x__le__abs__x,axiom,
! [X: real] : ( ord_less_eq @ real @ ( abs_abs @ real @ ( sin @ real @ X ) ) @ ( abs_abs @ real @ X ) ) ).
% abs_sin_x_le_abs_x
thf(fact_2831_int__ops_I1_J,axiom,
( ( semiring_1_of_nat @ int @ ( zero_zero @ nat ) )
= ( zero_zero @ int ) ) ).
% int_ops(1)
thf(fact_2832_complex__mod__minus__le__complex__mod,axiom,
! [X: complex] : ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( real_V7770717601297561774m_norm @ complex @ X ) ) @ ( real_V7770717601297561774m_norm @ complex @ X ) ) ).
% complex_mod_minus_le_complex_mod
thf(fact_2833_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_2834_nat__int__comparison_I2_J,axiom,
( ( ord_less @ nat )
= ( ^ [A5: nat,B4: nat] : ( ord_less @ int @ ( semiring_1_of_nat @ int @ A5 ) @ ( semiring_1_of_nat @ int @ B4 ) ) ) ) ).
% nat_int_comparison(2)
thf(fact_2835_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_2836_nat__int__comparison_I3_J,axiom,
( ( ord_less_eq @ nat )
= ( ^ [A5: nat,B4: nat] : ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ A5 ) @ ( semiring_1_of_nat @ int @ B4 ) ) ) ) ).
% nat_int_comparison(3)
thf(fact_2837_complex__mod__triangle__ineq2,axiom,
! [B2: complex,A2: complex] : ( ord_less_eq @ real @ ( minus_minus @ real @ ( real_V7770717601297561774m_norm @ complex @ ( plus_plus @ complex @ B2 @ A2 ) ) @ ( real_V7770717601297561774m_norm @ complex @ B2 ) ) @ ( real_V7770717601297561774m_norm @ complex @ A2 ) ) ).
% complex_mod_triangle_ineq2
thf(fact_2838_zero__le__imp__eq__int,axiom,
! [K2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K2 )
=> ? [N3: nat] :
( K2
= ( semiring_1_of_nat @ int @ N3 ) ) ) ).
% zero_le_imp_eq_int
thf(fact_2839_nonneg__int__cases,axiom,
! [K2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K2 )
=> ~ ! [N3: nat] :
( K2
!= ( semiring_1_of_nat @ int @ N3 ) ) ) ).
% nonneg_int_cases
thf(fact_2840_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_2841_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_2842_int__ops_I5_J,axiom,
! [A2: nat,B2: nat] :
( ( semiring_1_of_nat @ int @ ( plus_plus @ nat @ A2 @ B2 ) )
= ( plus_plus @ int @ ( semiring_1_of_nat @ int @ A2 ) @ ( semiring_1_of_nat @ int @ B2 ) ) ) ).
% int_ops(5)
thf(fact_2843_int__ops_I7_J,axiom,
! [A2: nat,B2: nat] :
( ( semiring_1_of_nat @ int @ ( times_times @ nat @ A2 @ B2 ) )
= ( times_times @ int @ ( semiring_1_of_nat @ int @ A2 ) @ ( semiring_1_of_nat @ int @ B2 ) ) ) ).
% int_ops(7)
thf(fact_2844_int__ops_I2_J,axiom,
( ( semiring_1_of_nat @ int @ ( one_one @ nat ) )
= ( one_one @ int ) ) ).
% int_ops(2)
thf(fact_2845_zle__iff__zadd,axiom,
( ( ord_less_eq @ int )
= ( ^ [W3: int,Z6: int] :
? [N2: nat] :
( Z6
= ( plus_plus @ int @ W3 @ ( semiring_1_of_nat @ int @ N2 ) ) ) ) ) ).
% zle_iff_zadd
thf(fact_2846_zdiv__int,axiom,
! [A2: nat,B2: nat] :
( ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ A2 @ B2 ) )
= ( divide_divide @ int @ ( semiring_1_of_nat @ int @ A2 ) @ ( semiring_1_of_nat @ int @ B2 ) ) ) ).
% zdiv_int
thf(fact_2847_zmod__int,axiom,
! [A2: nat,B2: nat] :
( ( semiring_1_of_nat @ int @ ( modulo_modulo @ nat @ A2 @ B2 ) )
= ( modulo_modulo @ int @ ( semiring_1_of_nat @ int @ A2 ) @ ( semiring_1_of_nat @ int @ B2 ) ) ) ).
% zmod_int
thf(fact_2848_nat__less__as__int,axiom,
( ( ord_less @ nat )
= ( ^ [A5: nat,B4: nat] : ( ord_less @ int @ ( semiring_1_of_nat @ int @ A5 ) @ ( semiring_1_of_nat @ int @ B4 ) ) ) ) ).
% nat_less_as_int
thf(fact_2849_nat__leq__as__int,axiom,
( ( ord_less_eq @ nat )
= ( ^ [A5: nat,B4: nat] : ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ A5 ) @ ( semiring_1_of_nat @ int @ B4 ) ) ) ) ).
% nat_leq_as_int
thf(fact_2850_sin__x__ge__neg__x,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ord_less_eq @ real @ ( uminus_uminus @ real @ X ) @ ( sin @ real @ X ) ) ) ).
% sin_x_ge_neg_x
thf(fact_2851_sin__ge__zero,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ X @ pi )
=> ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( sin @ real @ X ) ) ) ) ).
% sin_ge_zero
thf(fact_2852_sin__ge__minus__one,axiom,
! [X: real] : ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ ( sin @ real @ X ) ) ).
% sin_ge_minus_one
thf(fact_2853_abs__sin__le__one,axiom,
! [X: real] : ( ord_less_eq @ real @ ( abs_abs @ real @ ( sin @ real @ X ) ) @ ( one_one @ real ) ) ).
% abs_sin_le_one
thf(fact_2854_int__cases4,axiom,
! [M: int] :
( ! [N3: nat] :
( M
!= ( semiring_1_of_nat @ int @ N3 ) )
=> ~ ! [N3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( M
!= ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N3 ) ) ) ) ) ).
% int_cases4
thf(fact_2855_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_2856_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_2857_int__ops_I4_J,axiom,
! [A2: nat] :
( ( semiring_1_of_nat @ int @ ( suc @ A2 ) )
= ( plus_plus @ int @ ( semiring_1_of_nat @ int @ A2 ) @ ( one_one @ int ) ) ) ).
% int_ops(4)
thf(fact_2858_nonpos__int__cases,axiom,
! [K2: int] :
( ( ord_less_eq @ int @ K2 @ ( zero_zero @ int ) )
=> ~ ! [N3: nat] :
( K2
!= ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N3 ) ) ) ) ).
% nonpos_int_cases
thf(fact_2859_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_2860_norm__exp,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( exp @ A @ X ) ) @ ( exp @ real @ ( real_V7770717601297561774m_norm @ A @ X ) ) ) ) ).
% norm_exp
thf(fact_2861_sin__zero__iff__int2,axiom,
! [X: real] :
( ( ( sin @ real @ X )
= ( zero_zero @ real ) )
= ( ? [I: int] :
( X
= ( times_times @ real @ ( ring_1_of_int @ real @ I ) @ pi ) ) ) ) ).
% sin_zero_iff_int2
thf(fact_2862_zero__less__imp__eq__int,axiom,
! [K2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ K2 )
=> ? [N3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
& ( K2
= ( semiring_1_of_nat @ int @ N3 ) ) ) ) ).
% zero_less_imp_eq_int
thf(fact_2863_pos__int__cases,axiom,
! [K2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ K2 )
=> ~ ! [N3: nat] :
( ( K2
= ( semiring_1_of_nat @ int @ N3 ) )
=> ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 ) ) ) ).
% pos_int_cases
thf(fact_2864_int__cases3,axiom,
! [K2: int] :
( ( K2
!= ( zero_zero @ int ) )
=> ( ! [N3: nat] :
( ( K2
= ( semiring_1_of_nat @ int @ N3 ) )
=> ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 ) )
=> ~ ! [N3: nat] :
( ( K2
= ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N3 ) ) )
=> ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 ) ) ) ) ).
% int_cases3
thf(fact_2865_zmult__zless__mono2__lemma,axiom,
! [I2: int,J3: int,K2: nat] :
( ( ord_less @ int @ I2 @ J3 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K2 )
=> ( ord_less @ int @ ( times_times @ int @ ( semiring_1_of_nat @ int @ K2 ) @ I2 ) @ ( times_times @ int @ ( semiring_1_of_nat @ int @ K2 ) @ J3 ) ) ) ) ).
% zmult_zless_mono2_lemma
thf(fact_2866_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_2867_int__ops_I6_J,axiom,
! [A2: nat,B2: nat] :
( ( ( ord_less @ int @ ( semiring_1_of_nat @ int @ A2 ) @ ( semiring_1_of_nat @ int @ B2 ) )
=> ( ( semiring_1_of_nat @ int @ ( minus_minus @ nat @ A2 @ B2 ) )
= ( zero_zero @ int ) ) )
& ( ~ ( ord_less @ int @ ( semiring_1_of_nat @ int @ A2 ) @ ( semiring_1_of_nat @ int @ B2 ) )
=> ( ( semiring_1_of_nat @ int @ ( minus_minus @ nat @ A2 @ B2 ) )
= ( minus_minus @ int @ ( semiring_1_of_nat @ int @ A2 ) @ ( semiring_1_of_nat @ int @ B2 ) ) ) ) ) ).
% int_ops(6)
thf(fact_2868_sin__gt__zero__02,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ X @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( sin @ real @ X ) ) ) ) ).
% sin_gt_zero_02
thf(fact_2869_lemma__NBseq__def2,axiom,
! [A: $tType,B: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [X9: A > B] :
( ( ? [K6: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K6 )
& ! [N2: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( X9 @ N2 ) ) @ K6 ) ) )
= ( ? [N6: nat] :
! [N2: A] : ( ord_less @ real @ ( real_V7770717601297561774m_norm @ B @ ( X9 @ N2 ) ) @ ( semiring_1_of_nat @ real @ ( suc @ N6 ) ) ) ) ) ) ).
% lemma_NBseq_def2
thf(fact_2870_lemma__NBseq__def,axiom,
! [A: $tType,B: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [X9: A > B] :
( ( ? [K6: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K6 )
& ! [N2: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( X9 @ N2 ) ) @ K6 ) ) )
= ( ? [N6: nat] :
! [N2: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( X9 @ N2 ) ) @ ( semiring_1_of_nat @ real @ ( suc @ N6 ) ) ) ) ) ) ).
% lemma_NBseq_def
thf(fact_2871_neg__int__cases,axiom,
! [K2: int] :
( ( ord_less @ int @ K2 @ ( zero_zero @ int ) )
=> ~ ! [N3: nat] :
( ( K2
= ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N3 ) ) )
=> ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 ) ) ) ).
% neg_int_cases
thf(fact_2872_zdiff__int__split,axiom,
! [P3: int > $o,X: nat,Y2: nat] :
( ( P3 @ ( semiring_1_of_nat @ int @ ( minus_minus @ nat @ X @ Y2 ) ) )
= ( ( ( ord_less_eq @ nat @ Y2 @ X )
=> ( P3 @ ( minus_minus @ int @ ( semiring_1_of_nat @ int @ X ) @ ( semiring_1_of_nat @ int @ Y2 ) ) ) )
& ( ( ord_less @ nat @ X @ Y2 )
=> ( P3 @ ( zero_zero @ int ) ) ) ) ) ).
% zdiff_int_split
thf(fact_2873_monoseq__realpow,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ X @ ( one_one @ real ) )
=> ( topological_monoseq @ real @ ( power_power @ real @ X ) ) ) ) ).
% monoseq_realpow
thf(fact_2874_sin__pi__divide__n__ge__0,axiom,
! [N: nat] :
( ( N
!= ( zero_zero @ nat ) )
=> ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( sin @ real @ ( divide_divide @ real @ pi @ ( semiring_1_of_nat @ real @ N ) ) ) ) ) ).
% sin_pi_divide_n_ge_0
thf(fact_2875_sin__45,axiom,
( ( sin @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ one2 ) ) ) ) )
= ( divide_divide @ real @ ( sqrt @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ).
% sin_45
thf(fact_2876_sin__gt__zero2,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ X @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( sin @ real @ X ) ) ) ) ).
% sin_gt_zero2
thf(fact_2877_sin__lt__zero,axiom,
! [X: real] :
( ( ord_less @ real @ pi @ X )
=> ( ( ord_less @ real @ X @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) )
=> ( ord_less @ real @ ( sin @ real @ X ) @ ( zero_zero @ real ) ) ) ) ).
% sin_lt_zero
thf(fact_2878_sin__30,axiom,
( ( sin @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ ( bit1 @ one2 ) ) ) ) )
= ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ).
% sin_30
thf(fact_2879_sin__inj__pi,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X )
=> ( ( ord_less_eq @ real @ X @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ Y2 )
=> ( ( ord_less_eq @ real @ Y2 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ( sin @ real @ X )
= ( sin @ real @ Y2 ) )
=> ( X = Y2 ) ) ) ) ) ) ).
% sin_inj_pi
thf(fact_2880_sin__mono__le__eq,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X )
=> ( ( ord_less_eq @ real @ X @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ Y2 )
=> ( ( ord_less_eq @ real @ Y2 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less_eq @ real @ ( sin @ real @ X ) @ ( sin @ real @ Y2 ) )
= ( ord_less_eq @ real @ X @ Y2 ) ) ) ) ) ) ).
% sin_mono_le_eq
thf(fact_2881_sin__monotone__2pi__le,axiom,
! [Y2: real,X: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ Y2 )
=> ( ( ord_less_eq @ real @ Y2 @ X )
=> ( ( ord_less_eq @ real @ X @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less_eq @ real @ ( sin @ real @ Y2 ) @ ( sin @ real @ X ) ) ) ) ) ).
% sin_monotone_2pi_le
thf(fact_2882_sin__60,axiom,
( ( sin @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit1 @ one2 ) ) ) )
= ( divide_divide @ real @ ( sqrt @ ( numeral_numeral @ real @ ( bit1 @ one2 ) ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ).
% sin_60
thf(fact_2883_exp__bound__half,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [Z3: A] :
( ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ Z3 ) @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( exp @ A @ Z3 ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ).
% exp_bound_half
thf(fact_2884_sin__le__zero,axiom,
! [X: real] :
( ( ord_less_eq @ real @ pi @ X )
=> ( ( ord_less @ real @ X @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) )
=> ( ord_less_eq @ real @ ( sin @ real @ X ) @ ( zero_zero @ real ) ) ) ) ).
% sin_le_zero
thf(fact_2885_sin__less__zero,axiom,
! [X: real] :
( ( ord_less @ real @ ( divide_divide @ real @ ( uminus_uminus @ real @ pi ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ X )
=> ( ( ord_less @ real @ X @ ( zero_zero @ real ) )
=> ( ord_less @ real @ ( sin @ real @ X ) @ ( zero_zero @ real ) ) ) ) ).
% sin_less_zero
thf(fact_2886_sin__mono__less__eq,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X )
=> ( ( ord_less_eq @ real @ X @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ Y2 )
=> ( ( ord_less_eq @ real @ Y2 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less @ real @ ( sin @ real @ X ) @ ( sin @ real @ Y2 ) )
= ( ord_less @ real @ X @ Y2 ) ) ) ) ) ) ).
% sin_mono_less_eq
thf(fact_2887_sin__monotone__2pi,axiom,
! [Y2: real,X: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ Y2 )
=> ( ( ord_less @ real @ Y2 @ X )
=> ( ( ord_less_eq @ real @ X @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less @ real @ ( sin @ real @ Y2 ) @ ( sin @ real @ X ) ) ) ) ) ).
% sin_monotone_2pi
thf(fact_2888_sin__total,axiom,
! [Y2: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y2 )
=> ( ( ord_less_eq @ real @ Y2 @ ( one_one @ real ) )
=> ? [X5: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X5 )
& ( ord_less_eq @ real @ X5 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
& ( ( sin @ real @ X5 )
= Y2 )
& ! [Y4: real] :
( ( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ Y4 )
& ( ord_less_eq @ real @ Y4 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
& ( ( sin @ real @ Y4 )
= Y2 ) )
=> ( Y4 = X5 ) ) ) ) ) ).
% sin_total
thf(fact_2889_sin__pi__divide__n__gt__0,axiom,
! [N: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( sin @ real @ ( divide_divide @ real @ pi @ ( semiring_1_of_nat @ real @ N ) ) ) ) ) ).
% sin_pi_divide_n_gt_0
thf(fact_2890_sin__arctan,axiom,
! [X: real] :
( ( sin @ real @ ( arctan @ X ) )
= ( divide_divide @ real @ X @ ( sqrt @ ( plus_plus @ real @ ( one_one @ real ) @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% sin_arctan
thf(fact_2891_exp__bound__lemma,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [Z3: A] :
( ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ Z3 ) @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( exp @ A @ Z3 ) ) @ ( plus_plus @ real @ ( one_one @ real ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( real_V7770717601297561774m_norm @ A @ Z3 ) ) ) ) ) ) ).
% exp_bound_lemma
thf(fact_2892_sin__zero__iff__int,axiom,
! [X: real] :
( ( ( sin @ real @ X )
= ( zero_zero @ real ) )
= ( ? [I: int] :
( ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ I )
& ( X
= ( times_times @ real @ ( ring_1_of_int @ real @ I ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% sin_zero_iff_int
thf(fact_2893_sin__zero__lemma,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ( sin @ real @ X )
= ( zero_zero @ real ) )
=> ? [N3: nat] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 )
& ( X
= ( times_times @ real @ ( semiring_1_of_nat @ real @ N3 ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% sin_zero_lemma
thf(fact_2894_sin__zero__iff,axiom,
! [X: real] :
( ( ( sin @ real @ X )
= ( zero_zero @ real ) )
= ( ? [N2: nat] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
& ( X
= ( times_times @ real @ ( semiring_1_of_nat @ real @ N2 ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) )
| ? [N2: nat] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
& ( X
= ( uminus_uminus @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N2 ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ).
% sin_zero_iff
thf(fact_2895_pi__series,axiom,
( ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ one2 ) ) ) )
= ( suminf @ real
@ ^ [K3: nat] : ( divide_divide @ real @ ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ K3 ) @ ( one_one @ real ) ) @ ( semiring_1_of_nat @ real @ ( plus_plus @ nat @ ( times_times @ nat @ K3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) ) ) ).
% pi_series
thf(fact_2896_norm__divide__numeral,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [A2: A,W: num] :
( ( real_V7770717601297561774m_norm @ A @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ W ) ) )
= ( divide_divide @ real @ ( real_V7770717601297561774m_norm @ A @ A2 ) @ ( numeral_numeral @ real @ W ) ) ) ) ).
% norm_divide_numeral
thf(fact_2897_norm__mult__numeral1,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [W: num,A2: A] :
( ( real_V7770717601297561774m_norm @ A @ ( times_times @ A @ ( numeral_numeral @ A @ W ) @ A2 ) )
= ( times_times @ real @ ( numeral_numeral @ real @ W ) @ ( real_V7770717601297561774m_norm @ A @ A2 ) ) ) ) ).
% norm_mult_numeral1
thf(fact_2898_norm__mult__numeral2,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [A2: A,W: num] :
( ( real_V7770717601297561774m_norm @ A @ ( times_times @ A @ A2 @ ( numeral_numeral @ A @ W ) ) )
= ( times_times @ real @ ( real_V7770717601297561774m_norm @ A @ A2 ) @ ( numeral_numeral @ real @ W ) ) ) ) ).
% norm_mult_numeral2
thf(fact_2899_norm__neg__numeral,axiom,
! [A: $tType] :
( ( real_V2822296259951069270ebra_1 @ A )
=> ! [W: num] :
( ( real_V7770717601297561774m_norm @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) )
= ( numeral_numeral @ real @ W ) ) ) ).
% norm_neg_numeral
thf(fact_2900_norm__le__zero__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X: A] :
( ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ ( zero_zero @ real ) )
= ( X
= ( zero_zero @ A ) ) ) ) ).
% norm_le_zero_iff
thf(fact_2901_suminf__geometric,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [C2: A] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ C2 ) @ ( one_one @ real ) )
=> ( ( suminf @ A @ ( power_power @ A @ C2 ) )
= ( divide_divide @ A @ ( one_one @ A ) @ ( minus_minus @ A @ ( one_one @ A ) @ C2 ) ) ) ) ) ).
% suminf_geometric
thf(fact_2902_suminf__zero,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topological_t2_space @ A ) )
=> ( ( suminf @ A
@ ^ [N2: nat] : ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% suminf_zero
thf(fact_2903_norm__one,axiom,
! [A: $tType] :
( ( real_V2822296259951069270ebra_1 @ A )
=> ( ( real_V7770717601297561774m_norm @ A @ ( one_one @ A ) )
= ( one_one @ real ) ) ) ).
% norm_one
thf(fact_2904_norm__numeral,axiom,
! [A: $tType] :
( ( real_V2822296259951069270ebra_1 @ A )
=> ! [W: num] :
( ( real_V7770717601297561774m_norm @ A @ ( numeral_numeral @ A @ W ) )
= ( numeral_numeral @ real @ W ) ) ) ).
% norm_numeral
thf(fact_2905_norm__ge__zero,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X: A] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( real_V7770717601297561774m_norm @ A @ X ) ) ) ).
% norm_ge_zero
thf(fact_2906_norm__mult,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [X: A,Y2: A] :
( ( real_V7770717601297561774m_norm @ A @ ( times_times @ A @ X @ Y2 ) )
= ( times_times @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ ( real_V7770717601297561774m_norm @ A @ Y2 ) ) ) ) ).
% norm_mult
thf(fact_2907_norm__divide,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [A2: A,B2: A] :
( ( real_V7770717601297561774m_norm @ A @ ( divide_divide @ A @ A2 @ B2 ) )
= ( divide_divide @ real @ ( real_V7770717601297561774m_norm @ A @ A2 ) @ ( real_V7770717601297561774m_norm @ A @ B2 ) ) ) ) ).
% norm_divide
thf(fact_2908_norm__power,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [X: A,N: nat] :
( ( real_V7770717601297561774m_norm @ A @ ( power_power @ A @ X @ N ) )
= ( power_power @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ N ) ) ) ).
% norm_power
thf(fact_2909_norm__uminus__minus,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X: A,Y2: A] :
( ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( uminus_uminus @ A @ X ) @ Y2 ) )
= ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ X @ Y2 ) ) ) ) ).
% norm_uminus_minus
thf(fact_2910_nonzero__norm__divide,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [B2: A,A2: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( real_V7770717601297561774m_norm @ A @ ( divide_divide @ A @ A2 @ B2 ) )
= ( divide_divide @ real @ ( real_V7770717601297561774m_norm @ A @ A2 ) @ ( real_V7770717601297561774m_norm @ A @ B2 ) ) ) ) ) ).
% nonzero_norm_divide
thf(fact_2911_power__eq__imp__eq__norm,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [W: A,N: nat,Z3: A] :
( ( ( power_power @ A @ W @ N )
= ( power_power @ A @ Z3 @ N ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( real_V7770717601297561774m_norm @ A @ W )
= ( real_V7770717601297561774m_norm @ A @ Z3 ) ) ) ) ) ).
% power_eq_imp_eq_norm
thf(fact_2912_norm__mult__less,axiom,
! [A: $tType] :
( ( real_V4412858255891104859lgebra @ A )
=> ! [X: A,R: real,Y2: A,S: real] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ R )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ Y2 ) @ S )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( times_times @ A @ X @ Y2 ) ) @ ( times_times @ real @ R @ S ) ) ) ) ) ).
% norm_mult_less
thf(fact_2913_norm__mult__ineq,axiom,
! [A: $tType] :
( ( real_V4412858255891104859lgebra @ A )
=> ! [X: A,Y2: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( times_times @ A @ X @ Y2 ) ) @ ( times_times @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ ( real_V7770717601297561774m_norm @ A @ Y2 ) ) ) ) ).
% norm_mult_ineq
thf(fact_2914_norm__triangle__lt,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X: A,Y2: A,E: real] :
( ( ord_less @ real @ ( plus_plus @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ ( real_V7770717601297561774m_norm @ A @ Y2 ) ) @ E )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ X @ Y2 ) ) @ E ) ) ) ).
% norm_triangle_lt
thf(fact_2915_norm__add__less,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X: A,R: real,Y2: A,S: real] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ R )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ Y2 ) @ S )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ X @ Y2 ) ) @ ( plus_plus @ real @ R @ S ) ) ) ) ) ).
% norm_add_less
thf(fact_2916_norm__triangle__mono,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [A2: A,R: real,B2: A,S: real] :
( ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ A2 ) @ R )
=> ( ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ B2 ) @ S )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ A2 @ B2 ) ) @ ( plus_plus @ real @ R @ S ) ) ) ) ) ).
% norm_triangle_mono
thf(fact_2917_norm__triangle__ineq,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X: A,Y2: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ X @ Y2 ) ) @ ( plus_plus @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ ( real_V7770717601297561774m_norm @ A @ Y2 ) ) ) ) ).
% norm_triangle_ineq
thf(fact_2918_norm__triangle__le,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X: A,Y2: A,E: real] :
( ( ord_less_eq @ real @ ( plus_plus @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ ( real_V7770717601297561774m_norm @ A @ Y2 ) ) @ E )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ X @ Y2 ) ) @ E ) ) ) ).
% norm_triangle_le
thf(fact_2919_norm__add__leD,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [A2: A,B2: A,C2: real] :
( ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ A2 @ B2 ) ) @ C2 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ B2 ) @ ( plus_plus @ real @ ( real_V7770717601297561774m_norm @ A @ A2 ) @ C2 ) ) ) ) ).
% norm_add_leD
thf(fact_2920_norm__power__ineq,axiom,
! [A: $tType] :
( ( real_V2822296259951069270ebra_1 @ A )
=> ! [X: A,N: nat] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( power_power @ A @ X @ N ) ) @ ( power_power @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ N ) ) ) ).
% norm_power_ineq
thf(fact_2921_norm__triangle__le__diff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X: A,Y2: A,E: real] :
( ( ord_less_eq @ real @ ( plus_plus @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ ( real_V7770717601297561774m_norm @ A @ Y2 ) ) @ E )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ X @ Y2 ) ) @ E ) ) ) ).
% norm_triangle_le_diff
thf(fact_2922_norm__diff__triangle__le,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X: A,Y2: A,E1: real,Z3: A,E22: real] :
( ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ X @ Y2 ) ) @ E1 )
=> ( ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ Y2 @ Z3 ) ) @ E22 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ X @ Z3 ) ) @ ( plus_plus @ real @ E1 @ E22 ) ) ) ) ) ).
% norm_diff_triangle_le
thf(fact_2923_norm__triangle__ineq4,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [A2: A,B2: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ A2 @ B2 ) ) @ ( plus_plus @ real @ ( real_V7770717601297561774m_norm @ A @ A2 ) @ ( real_V7770717601297561774m_norm @ A @ B2 ) ) ) ) ).
% norm_triangle_ineq4
thf(fact_2924_norm__triangle__sub,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X: A,Y2: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ ( plus_plus @ real @ ( real_V7770717601297561774m_norm @ A @ Y2 ) @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ X @ Y2 ) ) ) ) ) ).
% norm_triangle_sub
thf(fact_2925_norm__diff__ineq,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [A2: A,B2: A] : ( ord_less_eq @ real @ ( minus_minus @ real @ ( real_V7770717601297561774m_norm @ A @ A2 ) @ ( real_V7770717601297561774m_norm @ A @ B2 ) ) @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ).
% norm_diff_ineq
thf(fact_2926_norm__triangle__ineq2,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [A2: A,B2: A] : ( ord_less_eq @ real @ ( minus_minus @ real @ ( real_V7770717601297561774m_norm @ A @ A2 ) @ ( real_V7770717601297561774m_norm @ A @ B2 ) ) @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ A2 @ B2 ) ) ) ) ).
% norm_triangle_ineq2
thf(fact_2927_power__eq__1__iff,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [W: A,N: nat] :
( ( ( power_power @ A @ W @ N )
= ( one_one @ A ) )
=> ( ( ( real_V7770717601297561774m_norm @ A @ W )
= ( one_one @ real ) )
| ( N
= ( zero_zero @ nat ) ) ) ) ) ).
% power_eq_1_iff
thf(fact_2928_norm__diff__triangle__ineq,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [A2: A,B2: A,C2: A,D2: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ ( plus_plus @ A @ C2 @ D2 ) ) ) @ ( plus_plus @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ A2 @ C2 ) ) @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ B2 @ D2 ) ) ) ) ) ).
% norm_diff_triangle_ineq
thf(fact_2929_norm__triangle__ineq3,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [A2: A,B2: A] : ( ord_less_eq @ real @ ( abs_abs @ real @ ( minus_minus @ real @ ( real_V7770717601297561774m_norm @ A @ A2 ) @ ( real_V7770717601297561774m_norm @ A @ B2 ) ) ) @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ A2 @ B2 ) ) ) ) ).
% norm_triangle_ineq3
thf(fact_2930_square__norm__one,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [X: A] :
( ( ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( one_one @ A ) )
=> ( ( real_V7770717601297561774m_norm @ A @ X )
= ( one_one @ real ) ) ) ) ).
% square_norm_one
thf(fact_2931_norm__power__diff,axiom,
! [A: $tType] :
( ( ( comm_monoid_mult @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [Z3: A,W: A,M: nat] :
( ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ Z3 ) @ ( one_one @ real ) )
=> ( ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ W ) @ ( one_one @ real ) )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( power_power @ A @ Z3 @ M ) @ ( power_power @ A @ W @ M ) ) ) @ ( times_times @ real @ ( semiring_1_of_nat @ real @ M ) @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ Z3 @ W ) ) ) ) ) ) ) ).
% norm_power_diff
thf(fact_2932_ceiling__log__nat__eq__powr__iff,axiom,
! [B2: nat,K2: nat,N: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ B2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K2 )
=> ( ( ( archimedean_ceiling @ real @ ( log @ ( semiring_1_of_nat @ real @ B2 ) @ ( semiring_1_of_nat @ real @ K2 ) ) )
= ( plus_plus @ int @ ( semiring_1_of_nat @ int @ N ) @ ( one_one @ int ) ) )
= ( ( ord_less @ nat @ ( power_power @ nat @ B2 @ N ) @ K2 )
& ( ord_less_eq @ nat @ K2 @ ( power_power @ nat @ B2 @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% ceiling_log_nat_eq_powr_iff
thf(fact_2933_summable__arctan__series,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ X ) @ ( one_one @ real ) )
=> ( summable @ real
@ ^ [K3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ K3 ) @ ( times_times @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( semiring_1_of_nat @ real @ ( plus_plus @ nat @ ( times_times @ nat @ K3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) @ ( power_power @ real @ X @ ( plus_plus @ nat @ ( times_times @ nat @ K3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) ) ) ) ).
% summable_arctan_series
thf(fact_2934_cos__pi__eq__zero,axiom,
! [M: nat] :
( ( cos @ real @ ( divide_divide @ real @ ( times_times @ real @ pi @ ( semiring_1_of_nat @ real @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) ) ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
= ( zero_zero @ real ) ) ).
% cos_pi_eq_zero
thf(fact_2935_sincos__total__2pi,axiom,
! [X: real,Y2: real] :
( ( ( plus_plus @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( one_one @ real ) )
=> ~ ! [T4: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ T4 )
=> ( ( ord_less @ real @ T4 @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) )
=> ( ( X
= ( cos @ real @ T4 ) )
=> ( Y2
!= ( sin @ real @ T4 ) ) ) ) ) ) ).
% sincos_total_2pi
thf(fact_2936_sin__tan,axiom,
! [X: real] :
( ( ord_less @ real @ ( abs_abs @ real @ X ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( sin @ real @ X )
= ( divide_divide @ real @ ( tan @ real @ X ) @ ( sqrt @ ( plus_plus @ real @ ( one_one @ real ) @ ( power_power @ real @ ( tan @ real @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% sin_tan
thf(fact_2937_of__int__ceiling__cancel,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ X ) )
= X )
= ( ? [N2: int] :
( X
= ( ring_1_of_int @ A @ N2 ) ) ) ) ) ).
% of_int_ceiling_cancel
thf(fact_2938_summable__zero,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ( summable @ A
@ ^ [N2: nat] : ( zero_zero @ A ) ) ) ).
% summable_zero
thf(fact_2939_summable__single,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [I2: nat,F3: nat > A] :
( summable @ A
@ ^ [R5: nat] : ( if @ A @ ( R5 = I2 ) @ ( F3 @ R5 ) @ ( zero_zero @ A ) ) ) ) ).
% summable_single
thf(fact_2940_summable__iff__shift,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: nat > A,K2: nat] :
( ( summable @ A
@ ^ [N2: nat] : ( F3 @ ( plus_plus @ nat @ N2 @ K2 ) ) )
= ( summable @ A @ F3 ) ) ) ).
% summable_iff_shift
thf(fact_2941_cos__zero,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ( ( cos @ A @ ( zero_zero @ A ) )
= ( one_one @ A ) ) ) ).
% cos_zero
thf(fact_2942_ceiling__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V2: num] :
( ( archimedean_ceiling @ A @ ( numeral_numeral @ A @ V2 ) )
= ( numeral_numeral @ int @ V2 ) ) ) ).
% ceiling_numeral
thf(fact_2943_ceiling__one,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archimedean_ceiling @ A @ ( one_one @ A ) )
= ( one_one @ int ) ) ) ).
% ceiling_one
thf(fact_2944_summable__cmult__iff,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [C2: A,F3: nat > A] :
( ( summable @ A
@ ^ [N2: nat] : ( times_times @ A @ C2 @ ( F3 @ N2 ) ) )
= ( ( C2
= ( zero_zero @ A ) )
| ( summable @ A @ F3 ) ) ) ) ).
% summable_cmult_iff
thf(fact_2945_summable__divide__iff,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: nat > A,C2: A] :
( ( summable @ A
@ ^ [N2: nat] : ( divide_divide @ A @ ( F3 @ N2 ) @ C2 ) )
= ( ( C2
= ( zero_zero @ A ) )
| ( summable @ A @ F3 ) ) ) ) ).
% summable_divide_iff
thf(fact_2946_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_2947_cos__pi,axiom,
( ( cos @ real @ pi )
= ( uminus_uminus @ real @ ( one_one @ real ) ) ) ).
% cos_pi
thf(fact_2948_sin__cos__squared__add3,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A] :
( ( plus_plus @ A @ ( times_times @ A @ ( cos @ A @ X ) @ ( cos @ A @ X ) ) @ ( times_times @ A @ ( sin @ A @ X ) @ ( sin @ A @ X ) ) )
= ( one_one @ A ) ) ) ).
% sin_cos_squared_add3
thf(fact_2949_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_2950_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_2951_ceiling__le__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,V2: num] :
( ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ X ) @ ( numeral_numeral @ int @ V2 ) )
= ( ord_less_eq @ A @ X @ ( numeral_numeral @ A @ V2 ) ) ) ) ).
% ceiling_le_numeral
thf(fact_2952_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_2953_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_2954_numeral__less__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V2: num,X: A] :
( ( ord_less @ int @ ( numeral_numeral @ int @ V2 ) @ ( archimedean_ceiling @ A @ X ) )
= ( ord_less @ A @ ( numeral_numeral @ A @ V2 ) @ X ) ) ) ).
% numeral_less_ceiling
thf(fact_2955_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_2956_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_2957_ceiling__add__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,V2: num] :
( ( archimedean_ceiling @ A @ ( plus_plus @ A @ X @ ( numeral_numeral @ A @ V2 ) ) )
= ( plus_plus @ int @ ( archimedean_ceiling @ A @ X ) @ ( numeral_numeral @ int @ V2 ) ) ) ) ).
% ceiling_add_numeral
thf(fact_2958_ceiling__neg__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V2: num] :
( ( archimedean_ceiling @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) )
= ( uminus_uminus @ int @ ( numeral_numeral @ int @ V2 ) ) ) ) ).
% ceiling_neg_numeral
thf(fact_2959_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_2960_ceiling__diff__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,V2: num] :
( ( archimedean_ceiling @ A @ ( minus_minus @ A @ X @ ( numeral_numeral @ A @ V2 ) ) )
= ( minus_minus @ int @ ( archimedean_ceiling @ A @ X ) @ ( numeral_numeral @ int @ V2 ) ) ) ) ).
% ceiling_diff_numeral
thf(fact_2961_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_2962_ceiling__numeral__power,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: num,N: nat] :
( ( archimedean_ceiling @ A @ ( power_power @ A @ ( numeral_numeral @ A @ X ) @ N ) )
= ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N ) ) ) ).
% ceiling_numeral_power
thf(fact_2963_tan__npi,axiom,
! [N: nat] :
( ( tan @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ pi ) )
= ( zero_zero @ real ) ) ).
% tan_npi
thf(fact_2964_tan__periodic__n,axiom,
! [X: real,N: num] :
( ( tan @ real @ ( plus_plus @ real @ X @ ( times_times @ real @ ( numeral_numeral @ real @ N ) @ pi ) ) )
= ( tan @ real @ X ) ) ).
% tan_periodic_n
thf(fact_2965_tan__periodic__nat,axiom,
! [X: real,N: nat] :
( ( tan @ real @ ( plus_plus @ real @ X @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ pi ) ) )
= ( tan @ real @ X ) ) ).
% tan_periodic_nat
thf(fact_2966_tan__periodic__int,axiom,
! [X: real,I2: int] :
( ( tan @ real @ ( plus_plus @ real @ X @ ( times_times @ real @ ( ring_1_of_int @ real @ I2 ) @ pi ) ) )
= ( tan @ real @ X ) ) ).
% tan_periodic_int
thf(fact_2967_summable__geometric__iff,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [C2: A] :
( ( summable @ A @ ( power_power @ A @ C2 ) )
= ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ C2 ) @ ( one_one @ real ) ) ) ) ).
% summable_geometric_iff
thf(fact_2968_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_2969_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_2970_ceiling__divide__eq__div__numeral,axiom,
! [A2: num,B2: num] :
( ( archimedean_ceiling @ real @ ( divide_divide @ real @ ( numeral_numeral @ real @ A2 ) @ ( numeral_numeral @ real @ B2 ) ) )
= ( uminus_uminus @ int @ ( divide_divide @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ A2 ) ) @ ( numeral_numeral @ int @ B2 ) ) ) ) ).
% ceiling_divide_eq_div_numeral
thf(fact_2971_ceiling__less__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,V2: num] :
( ( ord_less @ int @ ( archimedean_ceiling @ A @ X ) @ ( numeral_numeral @ int @ V2 ) )
= ( ord_less_eq @ A @ X @ ( minus_minus @ A @ ( numeral_numeral @ A @ V2 ) @ ( one_one @ A ) ) ) ) ) ).
% ceiling_less_numeral
thf(fact_2972_numeral__le__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V2: num,X: A] :
( ( ord_less_eq @ int @ ( numeral_numeral @ int @ V2 ) @ ( archimedean_ceiling @ A @ X ) )
= ( ord_less @ A @ ( minus_minus @ A @ ( numeral_numeral @ A @ V2 ) @ ( one_one @ A ) ) @ X ) ) ) ).
% numeral_le_ceiling
thf(fact_2973_ceiling__le__neg__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,V2: num] :
( ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ X ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V2 ) ) )
= ( ord_less_eq @ A @ X @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) ) ) ) ).
% ceiling_le_neg_numeral
thf(fact_2974_neg__numeral__less__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V2: num,X: A] :
( ( ord_less @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V2 ) ) @ ( archimedean_ceiling @ A @ X ) )
= ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) @ X ) ) ) ).
% neg_numeral_less_ceiling
thf(fact_2975_cos__pi__half,axiom,
( ( cos @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
= ( zero_zero @ real ) ) ).
% cos_pi_half
thf(fact_2976_cos__two__pi,axiom,
( ( cos @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) )
= ( one_one @ real ) ) ).
% cos_two_pi
thf(fact_2977_cos__periodic,axiom,
! [X: real] :
( ( cos @ real @ ( plus_plus @ real @ X @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) ) )
= ( cos @ real @ X ) ) ).
% cos_periodic
thf(fact_2978_cos__2pi__minus,axiom,
! [X: real] :
( ( cos @ real @ ( minus_minus @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) @ X ) )
= ( cos @ real @ X ) ) ).
% cos_2pi_minus
thf(fact_2979_tan__periodic,axiom,
! [X: real] :
( ( tan @ real @ ( plus_plus @ real @ X @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) ) )
= ( tan @ real @ X ) ) ).
% tan_periodic
thf(fact_2980_cos__npi2,axiom,
! [N: nat] :
( ( cos @ real @ ( times_times @ real @ pi @ ( semiring_1_of_nat @ real @ N ) ) )
= ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ N ) ) ).
% cos_npi2
thf(fact_2981_cos__npi,axiom,
! [N: nat] :
( ( cos @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ pi ) )
= ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ N ) ) ).
% cos_npi
thf(fact_2982_ceiling__minus__divide__eq__div__numeral,axiom,
! [A2: num,B2: num] :
( ( archimedean_ceiling @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ ( numeral_numeral @ real @ A2 ) @ ( numeral_numeral @ real @ B2 ) ) ) )
= ( uminus_uminus @ int @ ( divide_divide @ int @ ( numeral_numeral @ int @ A2 ) @ ( numeral_numeral @ int @ B2 ) ) ) ) ).
% ceiling_minus_divide_eq_div_numeral
thf(fact_2983_sin__cos__squared__add,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A] :
( ( plus_plus @ A @ ( power_power @ A @ ( sin @ A @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ ( cos @ A @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( one_one @ A ) ) ) ).
% sin_cos_squared_add
thf(fact_2984_sin__cos__squared__add2,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A] :
( ( plus_plus @ A @ ( power_power @ A @ ( cos @ A @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ ( sin @ A @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( one_one @ A ) ) ) ).
% sin_cos_squared_add2
thf(fact_2985_cos__2npi,axiom,
! [N: nat] :
( ( cos @ real @ ( times_times @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ N ) ) @ pi ) )
= ( one_one @ real ) ) ).
% cos_2npi
thf(fact_2986_cos__int__2pin,axiom,
! [N: int] :
( ( cos @ real @ ( times_times @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) @ ( ring_1_of_int @ real @ N ) ) )
= ( one_one @ real ) ) ).
% cos_int_2pin
thf(fact_2987_ceiling__less__neg__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,V2: num] :
( ( ord_less @ int @ ( archimedean_ceiling @ A @ X ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V2 ) ) )
= ( ord_less_eq @ A @ X @ ( minus_minus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) @ ( one_one @ A ) ) ) ) ) ).
% ceiling_less_neg_numeral
thf(fact_2988_neg__numeral__le__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V2: num,X: A] :
( ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V2 ) ) @ ( archimedean_ceiling @ A @ X ) )
= ( ord_less @ A @ ( minus_minus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) @ ( one_one @ A ) ) @ X ) ) ) ).
% neg_numeral_le_ceiling
thf(fact_2989_cos__3over2__pi,axiom,
( ( cos @ real @ ( times_times @ real @ ( divide_divide @ real @ ( numeral_numeral @ real @ ( bit1 @ one2 ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ pi ) )
= ( zero_zero @ real ) ) ).
% cos_3over2_pi
thf(fact_2990_cos__npi__int,axiom,
! [N: int] :
( ( ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N )
=> ( ( cos @ real @ ( times_times @ real @ pi @ ( ring_1_of_int @ real @ N ) ) )
= ( one_one @ real ) ) )
& ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N )
=> ( ( cos @ real @ ( times_times @ real @ pi @ ( ring_1_of_int @ real @ N ) ) )
= ( uminus_uminus @ real @ ( one_one @ real ) ) ) ) ) ).
% cos_npi_int
thf(fact_2991_summable__norm__cancel,axiom,
! [A: $tType] :
( ( real_Vector_banach @ A )
=> ! [F3: nat > A] :
( ( summable @ real
@ ^ [N2: nat] : ( real_V7770717601297561774m_norm @ A @ ( F3 @ N2 ) ) )
=> ( summable @ A @ F3 ) ) ) ).
% summable_norm_cancel
thf(fact_2992_tan__def,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ( ( tan @ A )
= ( ^ [X4: A] : ( divide_divide @ A @ ( sin @ A @ X4 ) @ ( cos @ A @ X4 ) ) ) ) ) ).
% tan_def
thf(fact_2993_summable__const__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [C2: A] :
( ( summable @ A
@ ^ [Uu3: nat] : C2 )
= ( C2
= ( zero_zero @ A ) ) ) ) ).
% summable_const_iff
thf(fact_2994_summable__comparison__test_H,axiom,
! [A: $tType] :
( ( real_Vector_banach @ A )
=> ! [G: nat > real,N4: nat,F3: nat > A] :
( ( summable @ real @ G )
=> ( ! [N3: nat] :
( ( ord_less_eq @ nat @ N4 @ N3 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( F3 @ N3 ) ) @ ( G @ N3 ) ) )
=> ( summable @ A @ F3 ) ) ) ) ).
% summable_comparison_test'
thf(fact_2995_summable__comparison__test,axiom,
! [A: $tType] :
( ( real_Vector_banach @ A )
=> ! [F3: nat > A,G: nat > real] :
( ? [N7: nat] :
! [N3: nat] :
( ( ord_less_eq @ nat @ N7 @ N3 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( F3 @ N3 ) ) @ ( G @ N3 ) ) )
=> ( ( summable @ real @ G )
=> ( summable @ A @ F3 ) ) ) ) ).
% summable_comparison_test
thf(fact_2996_summable__mult,axiom,
! [A: $tType] :
( ( real_V4412858255891104859lgebra @ A )
=> ! [F3: nat > A,C2: A] :
( ( summable @ A @ F3 )
=> ( summable @ A
@ ^ [N2: nat] : ( times_times @ A @ C2 @ ( F3 @ N2 ) ) ) ) ) ).
% summable_mult
thf(fact_2997_summable__mult2,axiom,
! [A: $tType] :
( ( real_V4412858255891104859lgebra @ A )
=> ! [F3: nat > A,C2: A] :
( ( summable @ A @ F3 )
=> ( summable @ A
@ ^ [N2: nat] : ( times_times @ A @ ( F3 @ N2 ) @ C2 ) ) ) ) ).
% summable_mult2
thf(fact_2998_summable__add,axiom,
! [A: $tType] :
( ( ( topolo5987344860129210374id_add @ A )
& ( topological_t2_space @ A ) )
=> ! [F3: nat > A,G: nat > A] :
( ( summable @ A @ F3 )
=> ( ( summable @ A @ G )
=> ( summable @ A
@ ^ [N2: nat] : ( plus_plus @ A @ ( F3 @ N2 ) @ ( G @ N2 ) ) ) ) ) ) ).
% summable_add
thf(fact_2999_summable__diff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: nat > A,G: nat > A] :
( ( summable @ A @ F3 )
=> ( ( summable @ A @ G )
=> ( summable @ A
@ ^ [N2: nat] : ( minus_minus @ A @ ( F3 @ N2 ) @ ( G @ N2 ) ) ) ) ) ) ).
% summable_diff
thf(fact_3000_summable__divide,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: nat > A,C2: A] :
( ( summable @ A @ F3 )
=> ( summable @ A
@ ^ [N2: nat] : ( divide_divide @ A @ ( F3 @ N2 ) @ C2 ) ) ) ) ).
% summable_divide
thf(fact_3001_summable__minus,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: nat > A] :
( ( summable @ A @ F3 )
=> ( summable @ A
@ ^ [N2: nat] : ( uminus_uminus @ A @ ( F3 @ N2 ) ) ) ) ) ).
% summable_minus
thf(fact_3002_summable__minus__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: nat > A] :
( ( summable @ A
@ ^ [N2: nat] : ( uminus_uminus @ A @ ( F3 @ N2 ) ) )
= ( summable @ A @ F3 ) ) ) ).
% summable_minus_iff
thf(fact_3003_summable__Suc__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: nat > A] :
( ( summable @ A
@ ^ [N2: nat] : ( F3 @ ( suc @ N2 ) ) )
= ( summable @ A @ F3 ) ) ) ).
% summable_Suc_iff
thf(fact_3004_summable__ignore__initial__segment,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: nat > A,K2: nat] :
( ( summable @ A @ F3 )
=> ( summable @ A
@ ^ [N2: nat] : ( F3 @ ( plus_plus @ nat @ N2 @ K2 ) ) ) ) ) ).
% summable_ignore_initial_segment
thf(fact_3005_summable__rabs__cancel,axiom,
! [F3: nat > real] :
( ( summable @ real
@ ^ [N2: nat] : ( abs_abs @ real @ ( F3 @ N2 ) ) )
=> ( summable @ real @ F3 ) ) ).
% summable_rabs_cancel
thf(fact_3006_powser__insidea,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [F3: nat > A,X: A,Z3: A] :
( ( summable @ A
@ ^ [N2: nat] : ( times_times @ A @ ( F3 @ N2 ) @ ( power_power @ A @ X @ N2 ) ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ Z3 ) @ ( real_V7770717601297561774m_norm @ A @ X ) )
=> ( summable @ real
@ ^ [N2: nat] : ( real_V7770717601297561774m_norm @ A @ ( times_times @ A @ ( F3 @ N2 ) @ ( power_power @ A @ Z3 @ N2 ) ) ) ) ) ) ) ).
% powser_insidea
thf(fact_3007_suminf__le,axiom,
! [A: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F3: nat > A,G: nat > A] :
( ! [N3: nat] : ( ord_less_eq @ A @ ( F3 @ N3 ) @ ( G @ N3 ) )
=> ( ( summable @ A @ F3 )
=> ( ( summable @ A @ G )
=> ( ord_less_eq @ A @ ( suminf @ A @ F3 ) @ ( suminf @ A @ G ) ) ) ) ) ) ).
% suminf_le
thf(fact_3008_ceiling__mono,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Y2: A,X: A] :
( ( ord_less_eq @ A @ Y2 @ X )
=> ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ Y2 ) @ ( archimedean_ceiling @ A @ X ) ) ) ) ).
% ceiling_mono
thf(fact_3009_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_3010_ceiling__less__cancel,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Y2: A] :
( ( ord_less @ int @ ( archimedean_ceiling @ A @ X ) @ ( archimedean_ceiling @ A @ Y2 ) )
=> ( ord_less @ A @ X @ Y2 ) ) ) ).
% ceiling_less_cancel
thf(fact_3011_cos__le__one,axiom,
! [X: real] : ( ord_less_eq @ real @ ( cos @ real @ X ) @ ( one_one @ real ) ) ).
% cos_le_one
thf(fact_3012_summable__mult__D,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [C2: A,F3: nat > A] :
( ( summable @ A
@ ^ [N2: nat] : ( times_times @ A @ C2 @ ( F3 @ N2 ) ) )
=> ( ( C2
!= ( zero_zero @ A ) )
=> ( summable @ A @ F3 ) ) ) ) ).
% summable_mult_D
thf(fact_3013_polar__Ex,axiom,
! [X: real,Y2: real] :
? [R3: real,A4: real] :
( ( X
= ( times_times @ real @ R3 @ ( cos @ real @ A4 ) ) )
& ( Y2
= ( times_times @ real @ R3 @ ( sin @ real @ A4 ) ) ) ) ).
% polar_Ex
thf(fact_3014_summable__zero__power,axiom,
! [A: $tType] :
( ( ( comm_ring_1 @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ( summable @ A @ ( power_power @ A @ ( zero_zero @ A ) ) ) ) ).
% summable_zero_power
thf(fact_3015_suminf__mult2,axiom,
! [A: $tType] :
( ( real_V4412858255891104859lgebra @ A )
=> ! [F3: nat > A,C2: A] :
( ( summable @ A @ F3 )
=> ( ( times_times @ A @ ( suminf @ A @ F3 ) @ C2 )
= ( suminf @ A
@ ^ [N2: nat] : ( times_times @ A @ ( F3 @ N2 ) @ C2 ) ) ) ) ) ).
% suminf_mult2
thf(fact_3016_suminf__mult,axiom,
! [A: $tType] :
( ( real_V4412858255891104859lgebra @ A )
=> ! [F3: nat > A,C2: A] :
( ( summable @ A @ F3 )
=> ( ( suminf @ A
@ ^ [N2: nat] : ( times_times @ A @ C2 @ ( F3 @ N2 ) ) )
= ( times_times @ A @ C2 @ ( suminf @ A @ F3 ) ) ) ) ) ).
% suminf_mult
thf(fact_3017_suminf__add,axiom,
! [A: $tType] :
( ( ( topolo5987344860129210374id_add @ A )
& ( topological_t2_space @ A ) )
=> ! [F3: nat > A,G: nat > A] :
( ( summable @ A @ F3 )
=> ( ( summable @ A @ G )
=> ( ( plus_plus @ A @ ( suminf @ A @ F3 ) @ ( suminf @ A @ G ) )
= ( suminf @ A
@ ^ [N2: nat] : ( plus_plus @ A @ ( F3 @ N2 ) @ ( G @ N2 ) ) ) ) ) ) ) ).
% suminf_add
thf(fact_3018_suminf__diff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: nat > A,G: nat > A] :
( ( summable @ A @ F3 )
=> ( ( summable @ A @ G )
=> ( ( minus_minus @ A @ ( suminf @ A @ F3 ) @ ( suminf @ A @ G ) )
= ( suminf @ A
@ ^ [N2: nat] : ( minus_minus @ A @ ( F3 @ N2 ) @ ( G @ N2 ) ) ) ) ) ) ) ).
% suminf_diff
thf(fact_3019_suminf__divide,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: nat > A,C2: A] :
( ( summable @ A @ F3 )
=> ( ( suminf @ A
@ ^ [N2: nat] : ( divide_divide @ A @ ( F3 @ N2 ) @ C2 ) )
= ( divide_divide @ A @ ( suminf @ A @ F3 ) @ C2 ) ) ) ) ).
% suminf_divide
thf(fact_3020_suminf__minus,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: nat > A] :
( ( summable @ A @ F3 )
=> ( ( suminf @ A
@ ^ [N2: nat] : ( uminus_uminus @ A @ ( F3 @ N2 ) ) )
= ( uminus_uminus @ A @ ( suminf @ A @ F3 ) ) ) ) ) ).
% suminf_minus
thf(fact_3021_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_3022_add__tan__eq,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A,Y2: A] :
( ( ( cos @ A @ X )
!= ( zero_zero @ A ) )
=> ( ( ( cos @ A @ Y2 )
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( tan @ A @ X ) @ ( tan @ A @ Y2 ) )
= ( divide_divide @ A @ ( sin @ A @ ( plus_plus @ A @ X @ Y2 ) ) @ ( times_times @ A @ ( cos @ A @ X ) @ ( cos @ A @ Y2 ) ) ) ) ) ) ) ).
% add_tan_eq
thf(fact_3023_suminf__nonneg,axiom,
! [A: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F3: nat > A] :
( ( summable @ A @ F3 )
=> ( ! [N3: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F3 @ N3 ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( suminf @ A @ F3 ) ) ) ) ) ).
% suminf_nonneg
thf(fact_3024_suminf__eq__zero__iff,axiom,
! [A: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F3: nat > A] :
( ( summable @ A @ F3 )
=> ( ! [N3: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F3 @ N3 ) )
=> ( ( ( suminf @ A @ F3 )
= ( zero_zero @ A ) )
= ( ! [N2: nat] :
( ( F3 @ N2 )
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% suminf_eq_zero_iff
thf(fact_3025_suminf__pos,axiom,
! [A: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F3: nat > A] :
( ( summable @ A @ F3 )
=> ( ! [N3: nat] : ( ord_less @ A @ ( zero_zero @ A ) @ ( F3 @ N3 ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( suminf @ A @ F3 ) ) ) ) ) ).
% suminf_pos
thf(fact_3026_cos__one__sin__zero,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A] :
( ( ( cos @ A @ X )
= ( one_one @ A ) )
=> ( ( sin @ A @ X )
= ( zero_zero @ A ) ) ) ) ).
% cos_one_sin_zero
thf(fact_3027_sin__add,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A,Y2: A] :
( ( sin @ A @ ( plus_plus @ A @ X @ Y2 ) )
= ( plus_plus @ A @ ( times_times @ A @ ( sin @ A @ X ) @ ( cos @ A @ Y2 ) ) @ ( times_times @ A @ ( cos @ A @ X ) @ ( sin @ A @ Y2 ) ) ) ) ) ).
% sin_add
thf(fact_3028_sin__diff,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A,Y2: A] :
( ( sin @ A @ ( minus_minus @ A @ X @ Y2 ) )
= ( minus_minus @ A @ ( times_times @ A @ ( sin @ A @ X ) @ ( cos @ A @ Y2 ) ) @ ( times_times @ A @ ( cos @ A @ X ) @ ( sin @ A @ Y2 ) ) ) ) ) ).
% sin_diff
thf(fact_3029_lemma__tan__add1,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A,Y2: A] :
( ( ( cos @ A @ X )
!= ( zero_zero @ A ) )
=> ( ( ( cos @ A @ Y2 )
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( tan @ A @ X ) @ ( tan @ A @ Y2 ) ) )
= ( divide_divide @ A @ ( cos @ A @ ( plus_plus @ A @ X @ Y2 ) ) @ ( times_times @ A @ ( cos @ A @ X ) @ ( cos @ A @ Y2 ) ) ) ) ) ) ) ).
% lemma_tan_add1
thf(fact_3030_tan__diff,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A,Y2: A] :
( ( ( cos @ A @ X )
!= ( zero_zero @ A ) )
=> ( ( ( cos @ A @ Y2 )
!= ( zero_zero @ A ) )
=> ( ( ( cos @ A @ ( minus_minus @ A @ X @ Y2 ) )
!= ( zero_zero @ A ) )
=> ( ( tan @ A @ ( minus_minus @ A @ X @ Y2 ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( tan @ A @ X ) @ ( tan @ A @ Y2 ) ) @ ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( tan @ A @ X ) @ ( tan @ A @ Y2 ) ) ) ) ) ) ) ) ) ).
% tan_diff
thf(fact_3031_tan__add,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A,Y2: A] :
( ( ( cos @ A @ X )
!= ( zero_zero @ A ) )
=> ( ( ( cos @ A @ Y2 )
!= ( zero_zero @ A ) )
=> ( ( ( cos @ A @ ( plus_plus @ A @ X @ Y2 ) )
!= ( zero_zero @ A ) )
=> ( ( tan @ A @ ( plus_plus @ A @ X @ Y2 ) )
= ( divide_divide @ A @ ( plus_plus @ A @ ( tan @ A @ X ) @ ( tan @ A @ Y2 ) ) @ ( minus_minus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( tan @ A @ X ) @ ( tan @ A @ Y2 ) ) ) ) ) ) ) ) ) ).
% tan_add
thf(fact_3032_ceiling__le,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,A2: int] :
( ( ord_less_eq @ A @ X @ ( ring_1_of_int @ A @ A2 ) )
=> ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ X ) @ A2 ) ) ) ).
% ceiling_le
thf(fact_3033_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_3034_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_3035_summable__0__powser,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [F3: nat > A] :
( summable @ A
@ ^ [N2: nat] : ( times_times @ A @ ( F3 @ N2 ) @ ( power_power @ A @ ( zero_zero @ A ) @ N2 ) ) ) ) ).
% summable_0_powser
thf(fact_3036_summable__zero__power_H,axiom,
! [A: $tType] :
( ( ( ring_1 @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [F3: nat > A] :
( summable @ A
@ ^ [N2: nat] : ( times_times @ A @ ( F3 @ N2 ) @ ( power_power @ A @ ( zero_zero @ A ) @ N2 ) ) ) ) ).
% summable_zero_power'
thf(fact_3037_ceiling__add__le,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Y2: A] : ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ ( plus_plus @ A @ X @ Y2 ) ) @ ( plus_plus @ int @ ( archimedean_ceiling @ A @ X ) @ ( archimedean_ceiling @ A @ Y2 ) ) ) ) ).
% ceiling_add_le
thf(fact_3038_cos__monotone__0__pi__le,axiom,
! [Y2: real,X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y2 )
=> ( ( ord_less_eq @ real @ Y2 @ X )
=> ( ( ord_less_eq @ real @ X @ pi )
=> ( ord_less_eq @ real @ ( cos @ real @ X ) @ ( cos @ real @ Y2 ) ) ) ) ) ).
% cos_monotone_0_pi_le
thf(fact_3039_cos__mono__le__eq,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ X @ pi )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y2 )
=> ( ( ord_less_eq @ real @ Y2 @ pi )
=> ( ( ord_less_eq @ real @ ( cos @ real @ X ) @ ( cos @ real @ Y2 ) )
= ( ord_less_eq @ real @ Y2 @ X ) ) ) ) ) ) ).
% cos_mono_le_eq
thf(fact_3040_cos__inj__pi,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ X @ pi )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y2 )
=> ( ( ord_less_eq @ real @ Y2 @ pi )
=> ( ( ( cos @ real @ X )
= ( cos @ real @ Y2 ) )
=> ( X = Y2 ) ) ) ) ) ) ).
% cos_inj_pi
thf(fact_3041_cos__ge__minus__one,axiom,
! [X: real] : ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ ( cos @ real @ X ) ) ).
% cos_ge_minus_one
thf(fact_3042_powser__split__head_I3_J,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V8999393235501362500lgebra @ A ) )
=> ! [F3: nat > A,Z3: A] :
( ( summable @ A
@ ^ [N2: nat] : ( times_times @ A @ ( F3 @ N2 ) @ ( power_power @ A @ Z3 @ N2 ) ) )
=> ( summable @ A
@ ^ [N2: nat] : ( times_times @ A @ ( F3 @ ( suc @ N2 ) ) @ ( power_power @ A @ Z3 @ N2 ) ) ) ) ) ).
% powser_split_head(3)
thf(fact_3043_summable__powser__split__head,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [F3: nat > A,Z3: A] :
( ( summable @ A
@ ^ [N2: nat] : ( times_times @ A @ ( F3 @ ( suc @ N2 ) ) @ ( power_power @ A @ Z3 @ N2 ) ) )
= ( summable @ A
@ ^ [N2: nat] : ( times_times @ A @ ( F3 @ N2 ) @ ( power_power @ A @ Z3 @ N2 ) ) ) ) ) ).
% summable_powser_split_head
thf(fact_3044_summable__powser__ignore__initial__segment,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [F3: nat > A,M: nat,Z3: A] :
( ( summable @ A
@ ^ [N2: nat] : ( times_times @ A @ ( F3 @ ( plus_plus @ nat @ N2 @ M ) ) @ ( power_power @ A @ Z3 @ N2 ) ) )
= ( summable @ A
@ ^ [N2: nat] : ( times_times @ A @ ( F3 @ N2 ) @ ( power_power @ A @ Z3 @ N2 ) ) ) ) ) ).
% summable_powser_ignore_initial_segment
thf(fact_3045_abs__cos__le__one,axiom,
! [X: real] : ( ord_less_eq @ real @ ( abs_abs @ real @ ( cos @ real @ X ) ) @ ( one_one @ real ) ) ).
% abs_cos_le_one
thf(fact_3046_summable__norm__comparison__test,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: nat > A,G: nat > real] :
( ? [N7: nat] :
! [N3: nat] :
( ( ord_less_eq @ nat @ N7 @ N3 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( F3 @ N3 ) ) @ ( G @ N3 ) ) )
=> ( ( summable @ real @ G )
=> ( summable @ real
@ ^ [N2: nat] : ( real_V7770717601297561774m_norm @ A @ ( F3 @ N2 ) ) ) ) ) ) ).
% summable_norm_comparison_test
thf(fact_3047_summable__rabs__comparison__test,axiom,
! [F3: nat > real,G: nat > real] :
( ? [N7: nat] :
! [N3: nat] :
( ( ord_less_eq @ nat @ N7 @ N3 )
=> ( ord_less_eq @ real @ ( abs_abs @ real @ ( F3 @ N3 ) ) @ ( G @ N3 ) ) )
=> ( ( summable @ real @ G )
=> ( summable @ real
@ ^ [N2: nat] : ( abs_abs @ real @ ( F3 @ N2 ) ) ) ) ) ).
% summable_rabs_comparison_test
thf(fact_3048_summable__rabs,axiom,
! [F3: nat > real] :
( ( summable @ real
@ ^ [N2: nat] : ( abs_abs @ real @ ( F3 @ N2 ) ) )
=> ( ord_less_eq @ real @ ( abs_abs @ real @ ( suminf @ real @ F3 ) )
@ ( suminf @ real
@ ^ [N2: nat] : ( abs_abs @ real @ ( F3 @ N2 ) ) ) ) ) ).
% summable_rabs
thf(fact_3049_suminf__pos2,axiom,
! [A: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F3: nat > A,I2: nat] :
( ( summable @ A @ F3 )
=> ( ! [N3: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F3 @ N3 ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ ( F3 @ I2 ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( suminf @ A @ F3 ) ) ) ) ) ) ).
% suminf_pos2
thf(fact_3050_suminf__pos__iff,axiom,
! [A: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F3: nat > A] :
( ( summable @ A @ F3 )
=> ( ! [N3: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F3 @ N3 ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ ( suminf @ A @ F3 ) )
= ( ? [I: nat] : ( ord_less @ A @ ( zero_zero @ A ) @ ( F3 @ I ) ) ) ) ) ) ) ).
% suminf_pos_iff
thf(fact_3051_cos__diff,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A,Y2: A] :
( ( cos @ A @ ( minus_minus @ A @ X @ Y2 ) )
= ( plus_plus @ A @ ( times_times @ A @ ( cos @ A @ X ) @ ( cos @ A @ Y2 ) ) @ ( times_times @ A @ ( sin @ A @ X ) @ ( sin @ A @ Y2 ) ) ) ) ) ).
% cos_diff
thf(fact_3052_cos__add,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A,Y2: A] :
( ( cos @ A @ ( plus_plus @ A @ X @ Y2 ) )
= ( minus_minus @ A @ ( times_times @ A @ ( cos @ A @ X ) @ ( cos @ A @ Y2 ) ) @ ( times_times @ A @ ( sin @ A @ X ) @ ( sin @ A @ Y2 ) ) ) ) ) ).
% cos_add
thf(fact_3053_sin__zero__norm__cos__one,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A] :
( ( ( sin @ A @ X )
= ( zero_zero @ A ) )
=> ( ( real_V7770717601297561774m_norm @ A @ ( cos @ A @ X ) )
= ( one_one @ real ) ) ) ) ).
% sin_zero_norm_cos_one
thf(fact_3054_of__int__ceiling__le__add__one,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [R: A] : ( ord_less_eq @ A @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ R ) ) @ ( plus_plus @ A @ R @ ( one_one @ A ) ) ) ) ).
% of_int_ceiling_le_add_one
thf(fact_3055_of__int__ceiling__diff__one__le,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [R: A] : ( ord_less_eq @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ R ) ) @ ( one_one @ A ) ) @ R ) ) ).
% of_int_ceiling_diff_one_le
thf(fact_3056_cos__two__neq__zero,axiom,
( ( cos @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) )
!= ( zero_zero @ real ) ) ).
% cos_two_neq_zero
thf(fact_3057_powser__inside,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V8999393235501362500lgebra @ A ) )
=> ! [F3: nat > A,X: A,Z3: A] :
( ( summable @ A
@ ^ [N2: nat] : ( times_times @ A @ ( F3 @ N2 ) @ ( power_power @ A @ X @ N2 ) ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ Z3 ) @ ( real_V7770717601297561774m_norm @ A @ X ) )
=> ( summable @ A
@ ^ [N2: nat] : ( times_times @ A @ ( F3 @ N2 ) @ ( power_power @ A @ Z3 @ N2 ) ) ) ) ) ) ).
% powser_inside
thf(fact_3058_cos__monotone__0__pi,axiom,
! [Y2: real,X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y2 )
=> ( ( ord_less @ real @ Y2 @ X )
=> ( ( ord_less_eq @ real @ X @ pi )
=> ( ord_less @ real @ ( cos @ real @ X ) @ ( cos @ real @ Y2 ) ) ) ) ) ).
% cos_monotone_0_pi
thf(fact_3059_cos__mono__less__eq,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ X @ pi )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y2 )
=> ( ( ord_less_eq @ real @ Y2 @ pi )
=> ( ( ord_less @ real @ ( cos @ real @ X ) @ ( cos @ real @ Y2 ) )
= ( ord_less @ real @ Y2 @ X ) ) ) ) ) ) ).
% cos_mono_less_eq
thf(fact_3060_tan__half,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ( ( tan @ A )
= ( ^ [X4: A] : ( divide_divide @ A @ ( sin @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X4 ) ) @ ( plus_plus @ A @ ( cos @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X4 ) ) @ ( one_one @ A ) ) ) ) ) ) ).
% tan_half
thf(fact_3061_complete__algebra__summable__geometric,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X: A] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ ( one_one @ real ) )
=> ( summable @ A @ ( power_power @ A @ X ) ) ) ) ).
% complete_algebra_summable_geometric
thf(fact_3062_summable__geometric,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [C2: A] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ C2 ) @ ( one_one @ real ) )
=> ( summable @ A @ ( power_power @ A @ C2 ) ) ) ) ).
% summable_geometric
thf(fact_3063_cos__monotone__minus__pi__0_H,axiom,
! [Y2: real,X: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ pi ) @ Y2 )
=> ( ( ord_less_eq @ real @ Y2 @ X )
=> ( ( ord_less_eq @ real @ X @ ( zero_zero @ real ) )
=> ( ord_less_eq @ real @ ( cos @ real @ Y2 ) @ ( cos @ real @ X ) ) ) ) ) ).
% cos_monotone_minus_pi_0'
thf(fact_3064_suminf__split__head,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: nat > A] :
( ( summable @ A @ F3 )
=> ( ( suminf @ A
@ ^ [N2: nat] : ( F3 @ ( suc @ N2 ) ) )
= ( minus_minus @ A @ ( suminf @ A @ F3 ) @ ( F3 @ ( zero_zero @ nat ) ) ) ) ) ) ).
% suminf_split_head
thf(fact_3065_ceiling__divide__eq__div,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [A2: int,B2: int] :
( ( archimedean_ceiling @ A @ ( divide_divide @ A @ ( ring_1_of_int @ A @ A2 ) @ ( ring_1_of_int @ A @ B2 ) ) )
= ( uminus_uminus @ int @ ( divide_divide @ int @ ( uminus_uminus @ int @ A2 ) @ B2 ) ) ) ) ).
% ceiling_divide_eq_div
thf(fact_3066_sin__zero__abs__cos__one,axiom,
! [X: real] :
( ( ( sin @ real @ X )
= ( zero_zero @ real ) )
=> ( ( abs_abs @ real @ ( cos @ real @ X ) )
= ( one_one @ real ) ) ) ).
% sin_zero_abs_cos_one
thf(fact_3067_summable__norm,axiom,
! [A: $tType] :
( ( real_Vector_banach @ A )
=> ! [F3: nat > A] :
( ( summable @ real
@ ^ [N2: nat] : ( real_V7770717601297561774m_norm @ A @ ( F3 @ N2 ) ) )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( suminf @ A @ F3 ) )
@ ( suminf @ real
@ ^ [N2: nat] : ( real_V7770717601297561774m_norm @ A @ ( F3 @ N2 ) ) ) ) ) ) ).
% summable_norm
thf(fact_3068_sin__double,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A] :
( ( sin @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X ) )
= ( times_times @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( sin @ A @ X ) ) @ ( cos @ A @ X ) ) ) ) ).
% sin_double
thf(fact_3069_ceiling__split,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [P3: int > $o,T2: A] :
( ( P3 @ ( archimedean_ceiling @ A @ T2 ) )
= ( ! [I: int] :
( ( ( ord_less @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ I ) @ ( one_one @ A ) ) @ T2 )
& ( ord_less_eq @ A @ T2 @ ( ring_1_of_int @ A @ I ) ) )
=> ( P3 @ I ) ) ) ) ) ).
% ceiling_split
thf(fact_3070_ceiling__eq__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,A2: int] :
( ( ( archimedean_ceiling @ A @ X )
= A2 )
= ( ( ord_less @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ A2 ) @ ( one_one @ A ) ) @ X )
& ( ord_less_eq @ A @ X @ ( ring_1_of_int @ A @ A2 ) ) ) ) ) ).
% ceiling_eq_iff
thf(fact_3071_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_3072_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_3073_mult__ceiling__le,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ ( times_times @ A @ A2 @ B2 ) ) @ ( times_times @ int @ ( archimedean_ceiling @ A @ A2 ) @ ( archimedean_ceiling @ A @ B2 ) ) ) ) ) ) ).
% mult_ceiling_le
thf(fact_3074_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_3075_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_3076_cos__two__less__zero,axiom,
ord_less @ real @ ( cos @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( zero_zero @ real ) ).
% cos_two_less_zero
thf(fact_3077_cos__two__le__zero,axiom,
ord_less_eq @ real @ ( cos @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( zero_zero @ real ) ).
% cos_two_le_zero
thf(fact_3078_cos__is__zero,axiom,
? [X5: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X5 )
& ( ord_less_eq @ real @ X5 @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) )
& ( ( cos @ real @ X5 )
= ( zero_zero @ real ) )
& ! [Y4: real] :
( ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y4 )
& ( ord_less_eq @ real @ Y4 @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) )
& ( ( cos @ real @ Y4 )
= ( zero_zero @ real ) ) )
=> ( Y4 = X5 ) ) ) ).
% cos_is_zero
thf(fact_3079_tan__double,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A] :
( ( ( cos @ A @ X )
!= ( zero_zero @ A ) )
=> ( ( ( cos @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X ) )
!= ( zero_zero @ A ) )
=> ( ( tan @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X ) )
= ( divide_divide @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( tan @ A @ X ) ) @ ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ ( tan @ A @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ).
% tan_double
thf(fact_3080_cos__monotone__minus__pi__0,axiom,
! [Y2: real,X: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ pi ) @ Y2 )
=> ( ( ord_less @ real @ Y2 @ X )
=> ( ( ord_less_eq @ real @ X @ ( zero_zero @ real ) )
=> ( ord_less @ real @ ( cos @ real @ Y2 ) @ ( cos @ real @ X ) ) ) ) ) ).
% cos_monotone_minus_pi_0
thf(fact_3081_cos__total,axiom,
! [Y2: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y2 )
=> ( ( ord_less_eq @ real @ Y2 @ ( one_one @ real ) )
=> ? [X5: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X5 )
& ( ord_less_eq @ real @ X5 @ pi )
& ( ( cos @ real @ X5 )
= Y2 )
& ! [Y4: real] :
( ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y4 )
& ( ord_less_eq @ real @ Y4 @ pi )
& ( ( cos @ real @ Y4 )
= Y2 ) )
=> ( Y4 = X5 ) ) ) ) ) ).
% cos_total
thf(fact_3082_sincos__principal__value,axiom,
! [X: real] :
? [Y7: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ pi ) @ Y7 )
& ( ord_less_eq @ real @ Y7 @ pi )
& ( ( sin @ real @ Y7 )
= ( sin @ real @ X ) )
& ( ( cos @ real @ Y7 )
= ( cos @ real @ X ) ) ) ).
% sincos_principal_value
thf(fact_3083_ceiling__divide__upper,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Q: A,P5: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Q )
=> ( ord_less_eq @ A @ P5 @ ( times_times @ A @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ ( divide_divide @ A @ P5 @ Q ) ) ) @ Q ) ) ) ) ).
% ceiling_divide_upper
thf(fact_3084_powser__split__head_I1_J,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V8999393235501362500lgebra @ A ) )
=> ! [F3: nat > A,Z3: A] :
( ( summable @ A
@ ^ [N2: nat] : ( times_times @ A @ ( F3 @ N2 ) @ ( power_power @ A @ Z3 @ N2 ) ) )
=> ( ( suminf @ A
@ ^ [N2: nat] : ( times_times @ A @ ( F3 @ N2 ) @ ( power_power @ A @ Z3 @ N2 ) ) )
= ( plus_plus @ A @ ( F3 @ ( zero_zero @ nat ) )
@ ( times_times @ A
@ ( suminf @ A
@ ^ [N2: nat] : ( times_times @ A @ ( F3 @ ( suc @ N2 ) ) @ ( power_power @ A @ Z3 @ N2 ) ) )
@ Z3 ) ) ) ) ) ).
% powser_split_head(1)
thf(fact_3085_powser__split__head_I2_J,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V8999393235501362500lgebra @ A ) )
=> ! [F3: nat > A,Z3: A] :
( ( summable @ A
@ ^ [N2: nat] : ( times_times @ A @ ( F3 @ N2 ) @ ( power_power @ A @ Z3 @ N2 ) ) )
=> ( ( times_times @ A
@ ( suminf @ A
@ ^ [N2: nat] : ( times_times @ A @ ( F3 @ ( suc @ N2 ) ) @ ( power_power @ A @ Z3 @ N2 ) ) )
@ Z3 )
= ( minus_minus @ A
@ ( suminf @ A
@ ^ [N2: nat] : ( times_times @ A @ ( F3 @ N2 ) @ ( power_power @ A @ Z3 @ N2 ) ) )
@ ( F3 @ ( zero_zero @ nat ) ) ) ) ) ) ).
% powser_split_head(2)
thf(fact_3086_cos__tan,axiom,
! [X: real] :
( ( ord_less @ real @ ( abs_abs @ real @ X ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( cos @ real @ X )
= ( divide_divide @ real @ ( one_one @ real ) @ ( sqrt @ ( plus_plus @ real @ ( one_one @ real ) @ ( power_power @ real @ ( tan @ real @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% cos_tan
thf(fact_3087_tan__45,axiom,
( ( tan @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ one2 ) ) ) ) )
= ( one_one @ real ) ) ).
% tan_45
thf(fact_3088_suminf__exist__split,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [R: real,F3: nat > A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R )
=> ( ( summable @ A @ F3 )
=> ? [N8: nat] :
! [N9: nat] :
( ( ord_less_eq @ nat @ N8 @ N9 )
=> ( ord_less @ real
@ ( real_V7770717601297561774m_norm @ A
@ ( suminf @ A
@ ^ [I: nat] : ( F3 @ ( plus_plus @ nat @ I @ N9 ) ) ) )
@ R ) ) ) ) ) ).
% suminf_exist_split
thf(fact_3089_tan__60,axiom,
( ( tan @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit1 @ one2 ) ) ) )
= ( sqrt @ ( numeral_numeral @ real @ ( bit1 @ one2 ) ) ) ) ).
% tan_60
thf(fact_3090_summable__power__series,axiom,
! [F3: nat > real,Z3: real] :
( ! [I3: nat] : ( ord_less_eq @ real @ ( F3 @ I3 ) @ ( one_one @ real ) )
=> ( ! [I3: nat] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( F3 @ I3 ) )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Z3 )
=> ( ( ord_less @ real @ Z3 @ ( one_one @ real ) )
=> ( summable @ real
@ ^ [I: nat] : ( times_times @ real @ ( F3 @ I ) @ ( power_power @ real @ Z3 @ I ) ) ) ) ) ) ) ).
% summable_power_series
thf(fact_3091_Abel__lemma,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [R: real,R0: real,A2: nat > A,M7: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ R )
=> ( ( ord_less @ real @ R @ R0 )
=> ( ! [N3: nat] : ( ord_less_eq @ real @ ( times_times @ real @ ( real_V7770717601297561774m_norm @ A @ ( A2 @ N3 ) ) @ ( power_power @ real @ R0 @ N3 ) ) @ M7 )
=> ( summable @ real
@ ^ [N2: nat] : ( times_times @ real @ ( real_V7770717601297561774m_norm @ A @ ( A2 @ N2 ) ) @ ( power_power @ real @ R @ N2 ) ) ) ) ) ) ) ).
% Abel_lemma
thf(fact_3092_cos__45,axiom,
( ( cos @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ one2 ) ) ) ) )
= ( divide_divide @ real @ ( sqrt @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ).
% cos_45
thf(fact_3093_sin__cos__le1,axiom,
! [X: real,Y2: real] : ( ord_less_eq @ real @ ( abs_abs @ real @ ( plus_plus @ real @ ( times_times @ real @ ( sin @ real @ X ) @ ( sin @ real @ Y2 ) ) @ ( times_times @ real @ ( cos @ real @ X ) @ ( cos @ real @ Y2 ) ) ) ) @ ( one_one @ real ) ) ).
% sin_cos_le1
thf(fact_3094_cos__plus__cos,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [W: A,Z3: A] :
( ( plus_plus @ A @ ( cos @ A @ W ) @ ( cos @ A @ Z3 ) )
= ( times_times @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( cos @ A @ ( divide_divide @ A @ ( plus_plus @ A @ W @ Z3 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) @ ( cos @ A @ ( divide_divide @ A @ ( minus_minus @ A @ W @ Z3 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% cos_plus_cos
thf(fact_3095_cos__times__cos,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [W: A,Z3: A] :
( ( times_times @ A @ ( cos @ A @ W ) @ ( cos @ A @ Z3 ) )
= ( divide_divide @ A @ ( plus_plus @ A @ ( cos @ A @ ( minus_minus @ A @ W @ Z3 ) ) @ ( cos @ A @ ( plus_plus @ A @ W @ Z3 ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% cos_times_cos
thf(fact_3096_summable__ratio__test,axiom,
! [A: $tType] :
( ( real_Vector_banach @ A )
=> ! [C2: real,N4: nat,F3: nat > A] :
( ( ord_less @ real @ C2 @ ( one_one @ real ) )
=> ( ! [N3: nat] :
( ( ord_less_eq @ nat @ N4 @ N3 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( F3 @ ( suc @ N3 ) ) ) @ ( times_times @ real @ C2 @ ( real_V7770717601297561774m_norm @ A @ ( F3 @ N3 ) ) ) ) )
=> ( summable @ A @ F3 ) ) ) ) ).
% summable_ratio_test
thf(fact_3097_cos__squared__eq,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A] :
( ( power_power @ A @ ( cos @ A @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ ( sin @ A @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% cos_squared_eq
thf(fact_3098_sin__squared__eq,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A] :
( ( power_power @ A @ ( sin @ A @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ ( cos @ A @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% sin_squared_eq
thf(fact_3099_ceiling__divide__lower,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Q: A,P5: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Q )
=> ( ord_less @ A @ ( times_times @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ ( divide_divide @ A @ P5 @ Q ) ) ) @ ( one_one @ A ) ) @ Q ) @ P5 ) ) ) ).
% ceiling_divide_lower
thf(fact_3100_tan__gt__zero,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ X @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( tan @ real @ X ) ) ) ) ).
% tan_gt_zero
thf(fact_3101_lemma__tan__total,axiom,
! [Y2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ Y2 )
=> ? [X5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X5 )
& ( ord_less @ real @ X5 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
& ( ord_less @ real @ Y2 @ ( tan @ real @ X5 ) ) ) ) ).
% lemma_tan_total
thf(fact_3102_ceiling__eq,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [N: int,X: A] :
( ( ord_less @ A @ ( ring_1_of_int @ A @ N ) @ X )
=> ( ( ord_less_eq @ A @ X @ ( plus_plus @ A @ ( ring_1_of_int @ A @ N ) @ ( one_one @ A ) ) )
=> ( ( archimedean_ceiling @ A @ X )
= ( plus_plus @ int @ N @ ( one_one @ int ) ) ) ) ) ) ).
% ceiling_eq
thf(fact_3103_lemma__tan__total1,axiom,
! [Y2: real] :
? [X5: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X5 )
& ( ord_less @ real @ X5 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
& ( ( tan @ real @ X5 )
= Y2 ) ) ).
% lemma_tan_total1
thf(fact_3104_tan__mono__lt__eq,axiom,
! [X: real,Y2: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X )
=> ( ( ord_less @ real @ X @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ Y2 )
=> ( ( ord_less @ real @ Y2 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less @ real @ ( tan @ real @ X ) @ ( tan @ real @ Y2 ) )
= ( ord_less @ real @ X @ Y2 ) ) ) ) ) ) ).
% tan_mono_lt_eq
thf(fact_3105_tan__monotone_H,axiom,
! [Y2: real,X: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ Y2 )
=> ( ( ord_less @ real @ Y2 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X )
=> ( ( ord_less @ real @ X @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less @ real @ Y2 @ X )
= ( ord_less @ real @ ( tan @ real @ Y2 ) @ ( tan @ real @ X ) ) ) ) ) ) ) ).
% tan_monotone'
thf(fact_3106_tan__monotone,axiom,
! [Y2: real,X: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ Y2 )
=> ( ( ord_less @ real @ Y2 @ X )
=> ( ( ord_less @ real @ X @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less @ real @ ( tan @ real @ Y2 ) @ ( tan @ real @ X ) ) ) ) ) ).
% tan_monotone
thf(fact_3107_tan__total,axiom,
! [Y2: real] :
? [X5: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X5 )
& ( ord_less @ real @ X5 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
& ( ( tan @ real @ X5 )
= Y2 )
& ! [Y4: real] :
( ( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ Y4 )
& ( ord_less @ real @ Y4 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
& ( ( tan @ real @ Y4 )
= Y2 ) )
=> ( Y4 = X5 ) ) ) ).
% tan_total
thf(fact_3108_tan__minus__45,axiom,
( ( tan @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) )
= ( uminus_uminus @ real @ ( one_one @ real ) ) ) ).
% tan_minus_45
thf(fact_3109_cos__double__less__one,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ X @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) )
=> ( ord_less @ real @ ( cos @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ X ) ) @ ( one_one @ real ) ) ) ) ).
% cos_double_less_one
thf(fact_3110_tan__inverse,axiom,
! [Y2: real] :
( ( divide_divide @ real @ ( one_one @ real ) @ ( tan @ real @ Y2 ) )
= ( tan @ real @ ( minus_minus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ Y2 ) ) ) ).
% tan_inverse
thf(fact_3111_cos__gt__zero,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ X @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( cos @ real @ X ) ) ) ) ).
% cos_gt_zero
thf(fact_3112_cos__60,axiom,
( ( cos @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit1 @ one2 ) ) ) )
= ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ).
% cos_60
thf(fact_3113_cos__30,axiom,
( ( cos @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ ( bit1 @ one2 ) ) ) ) )
= ( divide_divide @ real @ ( sqrt @ ( numeral_numeral @ real @ ( bit1 @ one2 ) ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ).
% cos_30
thf(fact_3114_cos__one__2pi__int,axiom,
! [X: real] :
( ( ( cos @ real @ X )
= ( one_one @ real ) )
= ( ? [X4: int] :
( X
= ( times_times @ real @ ( times_times @ real @ ( ring_1_of_int @ real @ X4 ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ pi ) ) ) ) ).
% cos_one_2pi_int
thf(fact_3115_cos__double__cos,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [W: A] :
( ( cos @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ W ) )
= ( minus_minus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( power_power @ A @ ( cos @ A @ W ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( one_one @ A ) ) ) ) ).
% cos_double_cos
thf(fact_3116_cos__treble__cos,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A] :
( ( cos @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit1 @ one2 ) ) @ X ) )
= ( minus_minus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ ( cos @ A @ X ) @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit1 @ one2 ) ) @ ( cos @ A @ X ) ) ) ) ) ).
% cos_treble_cos
thf(fact_3117_cos__diff__cos,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [W: A,Z3: A] :
( ( minus_minus @ A @ ( cos @ A @ W ) @ ( cos @ A @ Z3 ) )
= ( times_times @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( sin @ A @ ( divide_divide @ A @ ( plus_plus @ A @ W @ Z3 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) @ ( sin @ A @ ( divide_divide @ A @ ( minus_minus @ A @ Z3 @ W ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% cos_diff_cos
thf(fact_3118_sin__diff__sin,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [W: A,Z3: A] :
( ( minus_minus @ A @ ( sin @ A @ W ) @ ( sin @ A @ Z3 ) )
= ( times_times @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( sin @ A @ ( divide_divide @ A @ ( minus_minus @ A @ W @ Z3 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) @ ( cos @ A @ ( divide_divide @ A @ ( plus_plus @ A @ W @ Z3 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% sin_diff_sin
thf(fact_3119_sin__plus__sin,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [W: A,Z3: A] :
( ( plus_plus @ A @ ( sin @ A @ W ) @ ( sin @ A @ Z3 ) )
= ( times_times @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( sin @ A @ ( divide_divide @ A @ ( plus_plus @ A @ W @ Z3 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) @ ( cos @ A @ ( divide_divide @ A @ ( minus_minus @ A @ W @ Z3 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% sin_plus_sin
thf(fact_3120_cos__times__sin,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [W: A,Z3: A] :
( ( times_times @ A @ ( cos @ A @ W ) @ ( sin @ A @ Z3 ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( sin @ A @ ( plus_plus @ A @ W @ Z3 ) ) @ ( sin @ A @ ( minus_minus @ A @ W @ Z3 ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% cos_times_sin
thf(fact_3121_sin__times__cos,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [W: A,Z3: A] :
( ( times_times @ A @ ( sin @ A @ W ) @ ( cos @ A @ Z3 ) )
= ( divide_divide @ A @ ( plus_plus @ A @ ( sin @ A @ ( plus_plus @ A @ W @ Z3 ) ) @ ( sin @ A @ ( minus_minus @ A @ W @ Z3 ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% sin_times_cos
thf(fact_3122_sin__times__sin,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [W: A,Z3: A] :
( ( times_times @ A @ ( sin @ A @ W ) @ ( sin @ A @ Z3 ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( cos @ A @ ( minus_minus @ A @ W @ Z3 ) ) @ ( cos @ A @ ( plus_plus @ A @ W @ Z3 ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% sin_times_sin
thf(fact_3123_cos__double,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A] :
( ( cos @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X ) )
= ( minus_minus @ A @ ( power_power @ A @ ( cos @ A @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ ( sin @ A @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% cos_double
thf(fact_3124_tan__total__pos,axiom,
! [Y2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y2 )
=> ? [X5: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X5 )
& ( ord_less @ real @ X5 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
& ( ( tan @ real @ X5 )
= Y2 ) ) ) ).
% tan_total_pos
thf(fact_3125_tan__pos__pi2__le,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ X @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( tan @ real @ X ) ) ) ) ).
% tan_pos_pi2_le
thf(fact_3126_tan__less__zero,axiom,
! [X: real] :
( ( ord_less @ real @ ( divide_divide @ real @ ( uminus_uminus @ real @ pi ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ X )
=> ( ( ord_less @ real @ X @ ( zero_zero @ real ) )
=> ( ord_less @ real @ ( tan @ real @ X ) @ ( zero_zero @ real ) ) ) ) ).
% tan_less_zero
thf(fact_3127_tan__mono__le,axiom,
! [X: real,Y2: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X )
=> ( ( ord_less_eq @ real @ X @ Y2 )
=> ( ( ord_less @ real @ Y2 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less_eq @ real @ ( tan @ real @ X ) @ ( tan @ real @ Y2 ) ) ) ) ) ).
% tan_mono_le
thf(fact_3128_tan__mono__le__eq,axiom,
! [X: real,Y2: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X )
=> ( ( ord_less @ real @ X @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ Y2 )
=> ( ( ord_less @ real @ Y2 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less_eq @ real @ ( tan @ real @ X ) @ ( tan @ real @ Y2 ) )
= ( ord_less_eq @ real @ X @ Y2 ) ) ) ) ) ) ).
% tan_mono_le_eq
thf(fact_3129_tan__bound__pi2,axiom,
! [X: real] :
( ( ord_less @ real @ ( abs_abs @ real @ X ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ one2 ) ) ) ) )
=> ( ord_less @ real @ ( abs_abs @ real @ ( tan @ real @ X ) ) @ ( one_one @ real ) ) ) ).
% tan_bound_pi2
thf(fact_3130_tan__30,axiom,
( ( tan @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ ( bit1 @ one2 ) ) ) ) )
= ( divide_divide @ real @ ( one_one @ real ) @ ( sqrt @ ( numeral_numeral @ real @ ( bit1 @ one2 ) ) ) ) ) ).
% tan_30
thf(fact_3131_cos__gt__zero__pi,axiom,
! [X: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X )
=> ( ( ord_less @ real @ X @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( cos @ real @ X ) ) ) ) ).
% cos_gt_zero_pi
thf(fact_3132_cos__ge__zero,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X )
=> ( ( ord_less_eq @ real @ X @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( cos @ real @ X ) ) ) ) ).
% cos_ge_zero
thf(fact_3133_arctan__unique,axiom,
! [X: real,Y2: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X )
=> ( ( ord_less @ real @ X @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ( tan @ real @ X )
= Y2 )
=> ( ( arctan @ Y2 )
= X ) ) ) ) ).
% arctan_unique
thf(fact_3134_arctan__tan,axiom,
! [X: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X )
=> ( ( ord_less @ real @ X @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( arctan @ ( tan @ real @ X ) )
= X ) ) ) ).
% arctan_tan
thf(fact_3135_arctan,axiom,
! [Y2: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ ( arctan @ Y2 ) )
& ( ord_less @ real @ ( arctan @ Y2 ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
& ( ( tan @ real @ ( arctan @ Y2 ) )
= Y2 ) ) ).
% arctan
thf(fact_3136_cos__one__2pi,axiom,
! [X: real] :
( ( ( cos @ real @ X )
= ( one_one @ real ) )
= ( ? [X4: nat] :
( X
= ( times_times @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ X4 ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ pi ) )
| ? [X4: nat] :
( X
= ( uminus_uminus @ real @ ( times_times @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ X4 ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ pi ) ) ) ) ) ).
% cos_one_2pi
thf(fact_3137_cos__double__sin,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [W: A] :
( ( cos @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ W ) )
= ( minus_minus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( power_power @ A @ ( sin @ A @ W ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% cos_double_sin
thf(fact_3138_tan__total__pi4,axiom,
! [X: real] :
( ( ord_less @ real @ ( abs_abs @ real @ X ) @ ( one_one @ real ) )
=> ? [Z: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) @ Z )
& ( ord_less @ real @ Z @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ one2 ) ) ) ) )
& ( ( tan @ real @ Z )
= X ) ) ) ).
% tan_total_pi4
thf(fact_3139_cos__arctan,axiom,
! [X: real] :
( ( cos @ real @ ( arctan @ X ) )
= ( divide_divide @ real @ ( one_one @ real ) @ ( sqrt @ ( plus_plus @ real @ ( one_one @ real ) @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% cos_arctan
thf(fact_3140_sincos__total__pi,axiom,
! [Y2: real,X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y2 )
=> ( ( ( plus_plus @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( one_one @ real ) )
=> ? [T4: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ T4 )
& ( ord_less_eq @ real @ T4 @ pi )
& ( X
= ( cos @ real @ T4 ) )
& ( Y2
= ( sin @ real @ T4 ) ) ) ) ) ).
% sincos_total_pi
thf(fact_3141_sin__cos__sqrt,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( sin @ real @ X ) )
=> ( ( sin @ real @ X )
= ( sqrt @ ( minus_minus @ real @ ( one_one @ real ) @ ( power_power @ real @ ( cos @ real @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% sin_cos_sqrt
thf(fact_3142_sin__expansion__lemma,axiom,
! [X: real,M: nat] :
( ( sin @ real @ ( plus_plus @ real @ X @ ( divide_divide @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ ( suc @ M ) ) @ pi ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) )
= ( cos @ real @ ( plus_plus @ real @ X @ ( divide_divide @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ M ) @ pi ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ).
% sin_expansion_lemma
thf(fact_3143_cos__zero__iff__int,axiom,
! [X: real] :
( ( ( cos @ real @ X )
= ( zero_zero @ real ) )
= ( ? [I: int] :
( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ I )
& ( X
= ( times_times @ real @ ( ring_1_of_int @ real @ I ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% cos_zero_iff_int
thf(fact_3144_cos__zero__lemma,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ( cos @ real @ X )
= ( zero_zero @ real ) )
=> ? [N3: nat] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 )
& ( X
= ( times_times @ real @ ( semiring_1_of_nat @ real @ N3 ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% cos_zero_lemma
thf(fact_3145_cos__zero__iff,axiom,
! [X: real] :
( ( ( cos @ real @ X )
= ( zero_zero @ real ) )
= ( ? [N2: nat] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
& ( X
= ( times_times @ real @ ( semiring_1_of_nat @ real @ N2 ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) )
| ? [N2: nat] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 )
& ( X
= ( uminus_uminus @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N2 ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ).
% cos_zero_iff
thf(fact_3146_cos__expansion__lemma,axiom,
! [X: real,M: nat] :
( ( cos @ real @ ( plus_plus @ real @ X @ ( divide_divide @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ ( suc @ M ) ) @ pi ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) )
= ( uminus_uminus @ real @ ( sin @ real @ ( plus_plus @ real @ X @ ( divide_divide @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ M ) @ pi ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% cos_expansion_lemma
thf(fact_3147_sincos__total__pi__half,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y2 )
=> ( ( ( plus_plus @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( one_one @ real ) )
=> ? [T4: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ T4 )
& ( ord_less_eq @ real @ T4 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
& ( X
= ( cos @ real @ T4 ) )
& ( Y2
= ( sin @ real @ T4 ) ) ) ) ) ) ).
% sincos_total_pi_half
thf(fact_3148_sincos__total__2pi__le,axiom,
! [X: real,Y2: real] :
( ( ( plus_plus @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( one_one @ real ) )
=> ? [T4: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ T4 )
& ( ord_less_eq @ real @ T4 @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) )
& ( X
= ( cos @ real @ T4 ) )
& ( Y2
= ( sin @ real @ T4 ) ) ) ) ).
% sincos_total_2pi_le
thf(fact_3149_ceiling__log__nat__eq__if,axiom,
! [B2: nat,N: nat,K2: nat] :
( ( ord_less @ nat @ ( power_power @ nat @ B2 @ N ) @ K2 )
=> ( ( ord_less_eq @ nat @ K2 @ ( power_power @ nat @ B2 @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ B2 )
=> ( ( archimedean_ceiling @ real @ ( log @ ( semiring_1_of_nat @ real @ B2 ) @ ( semiring_1_of_nat @ real @ K2 ) ) )
= ( plus_plus @ int @ ( semiring_1_of_nat @ int @ N ) @ ( one_one @ int ) ) ) ) ) ) ).
% ceiling_log_nat_eq_if
thf(fact_3150_ceiling__log2__div2,axiom,
! [N: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( archimedean_ceiling @ real @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ N ) ) )
= ( plus_plus @ int @ ( archimedean_ceiling @ real @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ ( plus_plus @ nat @ ( divide_divide @ nat @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) ) @ ( one_one @ int ) ) ) ) ).
% ceiling_log2_div2
thf(fact_3151_complex__unimodular__polar,axiom,
! [Z3: complex] :
( ( ( real_V7770717601297561774m_norm @ complex @ Z3 )
= ( one_one @ real ) )
=> ~ ! [T4: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ T4 )
=> ( ( ord_less @ real @ T4 @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) )
=> ( Z3
!= ( complex2 @ ( cos @ real @ T4 ) @ ( sin @ real @ T4 ) ) ) ) ) ) ).
% complex_unimodular_polar
thf(fact_3152_ceiling__log__eq__powr__iff,axiom,
! [X: real,B2: real,K2: nat] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( one_one @ real ) @ B2 )
=> ( ( ( archimedean_ceiling @ real @ ( log @ B2 @ X ) )
= ( plus_plus @ int @ ( semiring_1_of_nat @ int @ K2 ) @ ( one_one @ int ) ) )
= ( ( ord_less @ real @ ( powr @ real @ B2 @ ( semiring_1_of_nat @ real @ K2 ) ) @ X )
& ( ord_less_eq @ real @ X @ ( powr @ real @ B2 @ ( semiring_1_of_nat @ real @ ( plus_plus @ nat @ K2 @ ( one_one @ nat ) ) ) ) ) ) ) ) ) ).
% ceiling_log_eq_powr_iff
thf(fact_3153_cos__arcsin,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X )
=> ( ( ord_less_eq @ real @ X @ ( one_one @ real ) )
=> ( ( cos @ real @ ( arcsin @ X ) )
= ( sqrt @ ( minus_minus @ real @ ( one_one @ real ) @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% cos_arcsin
thf(fact_3154_sin__arccos__abs,axiom,
! [Y2: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ Y2 ) @ ( one_one @ real ) )
=> ( ( sin @ real @ ( arccos @ Y2 ) )
= ( sqrt @ ( minus_minus @ real @ ( one_one @ real ) @ ( power_power @ real @ Y2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% sin_arccos_abs
thf(fact_3155_sin__arccos,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X )
=> ( ( ord_less_eq @ real @ X @ ( one_one @ real ) )
=> ( ( sin @ real @ ( arccos @ X ) )
= ( sqrt @ ( minus_minus @ real @ ( one_one @ real ) @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% sin_arccos
thf(fact_3156_powr__one__eq__one,axiom,
! [A: $tType] :
( ( ln @ A )
=> ! [A2: A] :
( ( powr @ A @ ( one_one @ A ) @ A2 )
= ( one_one @ A ) ) ) ).
% powr_one_eq_one
thf(fact_3157_powr__zero__eq__one,axiom,
! [A: $tType] :
( ( ln @ A )
=> ! [X: A] :
( ( ( X
= ( zero_zero @ A ) )
=> ( ( powr @ A @ X @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) )
& ( ( X
!= ( zero_zero @ A ) )
=> ( ( powr @ A @ X @ ( zero_zero @ A ) )
= ( one_one @ A ) ) ) ) ) ).
% powr_zero_eq_one
thf(fact_3158_powr__nonneg__iff,axiom,
! [A2: real,X: real] :
( ( ord_less_eq @ real @ ( powr @ real @ A2 @ X ) @ ( zero_zero @ real ) )
= ( A2
= ( zero_zero @ real ) ) ) ).
% powr_nonneg_iff
thf(fact_3159_powr__less__cancel__iff,axiom,
! [X: real,A2: real,B2: real] :
( ( ord_less @ real @ ( one_one @ real ) @ X )
=> ( ( ord_less @ real @ ( powr @ real @ X @ A2 ) @ ( powr @ real @ X @ B2 ) )
= ( ord_less @ real @ A2 @ B2 ) ) ) ).
% powr_less_cancel_iff
thf(fact_3160_arccos__1,axiom,
( ( arccos @ ( one_one @ real ) )
= ( zero_zero @ real ) ) ).
% arccos_1
thf(fact_3161_powr__eq__one__iff,axiom,
! [A2: real,X: real] :
( ( ord_less @ real @ ( one_one @ real ) @ A2 )
=> ( ( ( powr @ real @ A2 @ X )
= ( one_one @ real ) )
= ( X
= ( zero_zero @ real ) ) ) ) ).
% powr_eq_one_iff
thf(fact_3162_powr__one__gt__zero__iff,axiom,
! [X: real] :
( ( ( powr @ real @ X @ ( one_one @ real ) )
= X )
= ( ord_less_eq @ real @ ( zero_zero @ real ) @ X ) ) ).
% powr_one_gt_zero_iff
thf(fact_3163_powr__one,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( powr @ real @ X @ ( one_one @ real ) )
= X ) ) ).
% powr_one
thf(fact_3164_powr__le__cancel__iff,axiom,
! [X: real,A2: real,B2: real] :
( ( ord_less @ real @ ( one_one @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( powr @ real @ X @ A2 ) @ ( powr @ real @ X @ B2 ) )
= ( ord_less_eq @ real @ A2 @ B2 ) ) ) ).
% powr_le_cancel_iff
thf(fact_3165_numeral__powr__numeral__real,axiom,
! [M: num,N: num] :
( ( powr @ real @ ( numeral_numeral @ real @ M ) @ ( numeral_numeral @ real @ N ) )
= ( power_power @ real @ ( numeral_numeral @ real @ M ) @ ( numeral_numeral @ nat @ N ) ) ) ).
% numeral_powr_numeral_real
thf(fact_3166_arccos__minus__1,axiom,
( ( arccos @ ( uminus_uminus @ real @ ( one_one @ real ) ) )
= pi ) ).
% arccos_minus_1
thf(fact_3167_log__powr__cancel,axiom,
! [A2: real,Y2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
=> ( ( A2
!= ( one_one @ real ) )
=> ( ( log @ A2 @ ( powr @ real @ A2 @ Y2 ) )
= Y2 ) ) ) ).
% log_powr_cancel
thf(fact_3168_powr__log__cancel,axiom,
! [A2: real,X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
=> ( ( A2
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( powr @ real @ A2 @ ( log @ A2 @ X ) )
= X ) ) ) ) ).
% powr_log_cancel
thf(fact_3169_cos__arccos,axiom,
! [Y2: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y2 )
=> ( ( ord_less_eq @ real @ Y2 @ ( one_one @ real ) )
=> ( ( cos @ real @ ( arccos @ Y2 ) )
= Y2 ) ) ) ).
% cos_arccos
thf(fact_3170_sin__arcsin,axiom,
! [Y2: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y2 )
=> ( ( ord_less_eq @ real @ Y2 @ ( one_one @ real ) )
=> ( ( sin @ real @ ( arcsin @ Y2 ) )
= Y2 ) ) ) ).
% sin_arcsin
thf(fact_3171_norm__cos__sin,axiom,
! [T2: real] :
( ( real_V7770717601297561774m_norm @ complex @ ( complex2 @ ( cos @ real @ T2 ) @ ( sin @ real @ T2 ) ) )
= ( one_one @ real ) ) ).
% norm_cos_sin
thf(fact_3172_powr__numeral,axiom,
! [X: real,N: num] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( powr @ real @ X @ ( numeral_numeral @ real @ N ) )
= ( power_power @ real @ X @ ( numeral_numeral @ nat @ N ) ) ) ) ).
% powr_numeral
thf(fact_3173_arccos__0,axiom,
( ( arccos @ ( zero_zero @ real ) )
= ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ).
% arccos_0
thf(fact_3174_arcsin__1,axiom,
( ( arcsin @ ( one_one @ real ) )
= ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ).
% arcsin_1
thf(fact_3175_arcsin__minus__1,axiom,
( ( arcsin @ ( uminus_uminus @ real @ ( one_one @ real ) ) )
= ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ).
% arcsin_minus_1
thf(fact_3176_square__powr__half,axiom,
! [X: real] :
( ( powr @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
= ( abs_abs @ real @ X ) ) ).
% square_powr_half
thf(fact_3177_powr__powr,axiom,
! [X: real,A2: real,B2: real] :
( ( powr @ real @ ( powr @ real @ X @ A2 ) @ B2 )
= ( powr @ real @ X @ ( times_times @ real @ A2 @ B2 ) ) ) ).
% powr_powr
thf(fact_3178_powr__ge__pzero,axiom,
! [X: real,Y2: real] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( powr @ real @ X @ Y2 ) ) ).
% powr_ge_pzero
thf(fact_3179_powr__mono2,axiom,
! [A2: real,X: real,Y2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ A2 )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ X @ Y2 )
=> ( ord_less_eq @ real @ ( powr @ real @ X @ A2 ) @ ( powr @ real @ Y2 @ A2 ) ) ) ) ) ).
% powr_mono2
thf(fact_3180_powr__less__mono,axiom,
! [A2: real,B2: real,X: real] :
( ( ord_less @ real @ A2 @ B2 )
=> ( ( ord_less @ real @ ( one_one @ real ) @ X )
=> ( ord_less @ real @ ( powr @ real @ X @ A2 ) @ ( powr @ real @ X @ B2 ) ) ) ) ).
% powr_less_mono
thf(fact_3181_powr__less__cancel,axiom,
! [X: real,A2: real,B2: real] :
( ( ord_less @ real @ ( powr @ real @ X @ A2 ) @ ( powr @ real @ X @ B2 ) )
=> ( ( ord_less @ real @ ( one_one @ real ) @ X )
=> ( ord_less @ real @ A2 @ B2 ) ) ) ).
% powr_less_cancel
thf(fact_3182_powr__mono,axiom,
! [A2: real,B2: real,X: real] :
( ( ord_less_eq @ real @ A2 @ B2 )
=> ( ( ord_less_eq @ real @ ( one_one @ real ) @ X )
=> ( ord_less_eq @ real @ ( powr @ real @ X @ A2 ) @ ( powr @ real @ X @ B2 ) ) ) ) ).
% powr_mono
thf(fact_3183_Complex__eq__1,axiom,
! [A2: real,B2: real] :
( ( ( complex2 @ A2 @ B2 )
= ( one_one @ complex ) )
= ( ( A2
= ( one_one @ real ) )
& ( B2
= ( zero_zero @ real ) ) ) ) ).
% Complex_eq_1
thf(fact_3184_one__complex_Ocode,axiom,
( ( one_one @ complex )
= ( complex2 @ ( one_one @ real ) @ ( zero_zero @ real ) ) ) ).
% one_complex.code
thf(fact_3185_powr__mono2_H,axiom,
! [A2: real,X: real,Y2: real] :
( ( ord_less_eq @ real @ A2 @ ( zero_zero @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ X @ Y2 )
=> ( ord_less_eq @ real @ ( powr @ real @ Y2 @ A2 ) @ ( powr @ real @ X @ A2 ) ) ) ) ) ).
% powr_mono2'
thf(fact_3186_powr__less__mono2,axiom,
! [A2: real,X: real,Y2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ X @ Y2 )
=> ( ord_less @ real @ ( powr @ real @ X @ A2 ) @ ( powr @ real @ Y2 @ A2 ) ) ) ) ) ).
% powr_less_mono2
thf(fact_3187_powr__inj,axiom,
! [A2: real,X: real,Y2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
=> ( ( A2
!= ( one_one @ real ) )
=> ( ( ( powr @ real @ A2 @ X )
= ( powr @ real @ A2 @ Y2 ) )
= ( X = Y2 ) ) ) ) ).
% powr_inj
thf(fact_3188_gr__one__powr,axiom,
! [X: real,Y2: real] :
( ( ord_less @ real @ ( one_one @ real ) @ X )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ Y2 )
=> ( ord_less @ real @ ( one_one @ real ) @ ( powr @ real @ X @ Y2 ) ) ) ) ).
% gr_one_powr
thf(fact_3189_ge__one__powr__ge__zero,axiom,
! [X: real,A2: real] :
( ( ord_less_eq @ real @ ( one_one @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ A2 )
=> ( ord_less_eq @ real @ ( one_one @ real ) @ ( powr @ real @ X @ A2 ) ) ) ) ).
% ge_one_powr_ge_zero
thf(fact_3190_powr__mono__both,axiom,
! [A2: real,B2: real,X: real,Y2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ A2 )
=> ( ( ord_less_eq @ real @ A2 @ B2 )
=> ( ( ord_less_eq @ real @ ( one_one @ real ) @ X )
=> ( ( ord_less_eq @ real @ X @ Y2 )
=> ( ord_less_eq @ real @ ( powr @ real @ X @ A2 ) @ ( powr @ real @ Y2 @ B2 ) ) ) ) ) ) ).
% powr_mono_both
thf(fact_3191_powr__le1,axiom,
! [A2: real,X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ A2 )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ X @ ( one_one @ real ) )
=> ( ord_less_eq @ real @ ( powr @ real @ X @ A2 ) @ ( one_one @ real ) ) ) ) ) ).
% powr_le1
thf(fact_3192_powr__divide,axiom,
! [X: real,Y2: real,A2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y2 )
=> ( ( powr @ real @ ( divide_divide @ real @ X @ Y2 ) @ A2 )
= ( divide_divide @ real @ ( powr @ real @ X @ A2 ) @ ( powr @ real @ Y2 @ A2 ) ) ) ) ) ).
% powr_divide
thf(fact_3193_powr__mult,axiom,
! [X: real,Y2: real,A2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y2 )
=> ( ( powr @ real @ ( times_times @ real @ X @ Y2 ) @ A2 )
= ( times_times @ real @ ( powr @ real @ X @ A2 ) @ ( powr @ real @ Y2 @ A2 ) ) ) ) ) ).
% powr_mult
thf(fact_3194_divide__powr__uminus,axiom,
! [A2: real,B2: real,C2: real] :
( ( divide_divide @ real @ A2 @ ( powr @ real @ B2 @ C2 ) )
= ( times_times @ real @ A2 @ ( powr @ real @ B2 @ ( uminus_uminus @ real @ C2 ) ) ) ) ).
% divide_powr_uminus
thf(fact_3195_ln__powr,axiom,
! [X: real,Y2: real] :
( ( X
!= ( zero_zero @ real ) )
=> ( ( ln_ln @ real @ ( powr @ real @ X @ Y2 ) )
= ( times_times @ real @ Y2 @ ( ln_ln @ real @ X ) ) ) ) ).
% ln_powr
thf(fact_3196_log__powr,axiom,
! [X: real,B2: real,Y2: real] :
( ( X
!= ( zero_zero @ real ) )
=> ( ( log @ B2 @ ( powr @ real @ X @ Y2 ) )
= ( times_times @ real @ Y2 @ ( log @ B2 @ X ) ) ) ) ).
% log_powr
thf(fact_3197_Complex__eq__neg__1,axiom,
! [A2: real,B2: real] :
( ( ( complex2 @ A2 @ B2 )
= ( uminus_uminus @ complex @ ( one_one @ complex ) ) )
= ( ( A2
= ( uminus_uminus @ real @ ( one_one @ real ) ) )
& ( B2
= ( zero_zero @ real ) ) ) ) ).
% Complex_eq_neg_1
thf(fact_3198_powr__add,axiom,
! [A: $tType] :
( ( ( real_V3459762299906320749_field @ A )
& ( ln @ A ) )
=> ! [X: A,A2: A,B2: A] :
( ( powr @ A @ X @ ( plus_plus @ A @ A2 @ B2 ) )
= ( times_times @ A @ ( powr @ A @ X @ A2 ) @ ( powr @ A @ X @ B2 ) ) ) ) ).
% powr_add
thf(fact_3199_powr__diff,axiom,
! [A: $tType] :
( ( ( real_V3459762299906320749_field @ A )
& ( ln @ A ) )
=> ! [W: A,Z1: A,Z22: A] :
( ( powr @ A @ W @ ( minus_minus @ A @ Z1 @ Z22 ) )
= ( divide_divide @ A @ ( powr @ A @ W @ Z1 ) @ ( powr @ A @ W @ Z22 ) ) ) ) ).
% powr_diff
thf(fact_3200_complex__mult,axiom,
! [A2: real,B2: real,C2: real,D2: real] :
( ( times_times @ complex @ ( complex2 @ A2 @ B2 ) @ ( complex2 @ C2 @ D2 ) )
= ( complex2 @ ( minus_minus @ real @ ( times_times @ real @ A2 @ C2 ) @ ( times_times @ real @ B2 @ D2 ) ) @ ( plus_plus @ real @ ( times_times @ real @ A2 @ D2 ) @ ( times_times @ real @ B2 @ C2 ) ) ) ) ).
% complex_mult
thf(fact_3201_arccos__le__arccos,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X )
=> ( ( ord_less_eq @ real @ X @ Y2 )
=> ( ( ord_less_eq @ real @ Y2 @ ( one_one @ real ) )
=> ( ord_less_eq @ real @ ( arccos @ Y2 ) @ ( arccos @ X ) ) ) ) ) ).
% arccos_le_arccos
thf(fact_3202_arccos__eq__iff,axiom,
! [X: real,Y2: real] :
( ( ( ord_less_eq @ real @ ( abs_abs @ real @ X ) @ ( one_one @ real ) )
& ( ord_less_eq @ real @ ( abs_abs @ real @ Y2 ) @ ( one_one @ real ) ) )
=> ( ( ( arccos @ X )
= ( arccos @ Y2 ) )
= ( X = Y2 ) ) ) ).
% arccos_eq_iff
thf(fact_3203_arccos__le__mono,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ X ) @ ( one_one @ real ) )
=> ( ( ord_less_eq @ real @ ( abs_abs @ real @ Y2 ) @ ( one_one @ real ) )
=> ( ( ord_less_eq @ real @ ( arccos @ X ) @ ( arccos @ Y2 ) )
= ( ord_less_eq @ real @ Y2 @ X ) ) ) ) ).
% arccos_le_mono
thf(fact_3204_arcsin__minus,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X )
=> ( ( ord_less_eq @ real @ X @ ( one_one @ real ) )
=> ( ( arcsin @ ( uminus_uminus @ real @ X ) )
= ( uminus_uminus @ real @ ( arcsin @ X ) ) ) ) ) ).
% arcsin_minus
thf(fact_3205_arcsin__le__arcsin,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X )
=> ( ( ord_less_eq @ real @ X @ Y2 )
=> ( ( ord_less_eq @ real @ Y2 @ ( one_one @ real ) )
=> ( ord_less_eq @ real @ ( arcsin @ X ) @ ( arcsin @ Y2 ) ) ) ) ) ).
% arcsin_le_arcsin
thf(fact_3206_arcsin__eq__iff,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ X ) @ ( one_one @ real ) )
=> ( ( ord_less_eq @ real @ ( abs_abs @ real @ Y2 ) @ ( one_one @ real ) )
=> ( ( ( arcsin @ X )
= ( arcsin @ Y2 ) )
= ( X = Y2 ) ) ) ) ).
% arcsin_eq_iff
thf(fact_3207_arcsin__le__mono,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ X ) @ ( one_one @ real ) )
=> ( ( ord_less_eq @ real @ ( abs_abs @ real @ Y2 ) @ ( one_one @ real ) )
=> ( ( ord_less_eq @ real @ ( arcsin @ X ) @ ( arcsin @ Y2 ) )
= ( ord_less_eq @ real @ X @ Y2 ) ) ) ) ).
% arcsin_le_mono
thf(fact_3208_less__log__iff,axiom,
! [B2: real,X: real,Y2: real] :
( ( ord_less @ real @ ( one_one @ real ) @ B2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ Y2 @ ( log @ B2 @ X ) )
= ( ord_less @ real @ ( powr @ real @ B2 @ Y2 ) @ X ) ) ) ) ).
% less_log_iff
thf(fact_3209_log__less__iff,axiom,
! [B2: real,X: real,Y2: real] :
( ( ord_less @ real @ ( one_one @ real ) @ B2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( log @ B2 @ X ) @ Y2 )
= ( ord_less @ real @ X @ ( powr @ real @ B2 @ Y2 ) ) ) ) ) ).
% log_less_iff
thf(fact_3210_less__powr__iff,axiom,
! [B2: real,X: real,Y2: real] :
( ( ord_less @ real @ ( one_one @ real ) @ B2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ X @ ( powr @ real @ B2 @ Y2 ) )
= ( ord_less @ real @ ( log @ B2 @ X ) @ Y2 ) ) ) ) ).
% less_powr_iff
thf(fact_3211_powr__less__iff,axiom,
! [B2: real,X: real,Y2: real] :
( ( ord_less @ real @ ( one_one @ real ) @ B2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( powr @ real @ B2 @ Y2 ) @ X )
= ( ord_less @ real @ Y2 @ ( log @ B2 @ X ) ) ) ) ) ).
% powr_less_iff
thf(fact_3212_powr__minus__divide,axiom,
! [A: $tType] :
( ( ( real_V3459762299906320749_field @ A )
& ( ln @ A ) )
=> ! [X: A,A2: A] :
( ( powr @ A @ X @ ( uminus_uminus @ A @ A2 ) )
= ( divide_divide @ A @ ( one_one @ A ) @ ( powr @ A @ X @ A2 ) ) ) ) ).
% powr_minus_divide
thf(fact_3213_arccos__lbound,axiom,
! [Y2: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y2 )
=> ( ( ord_less_eq @ real @ Y2 @ ( one_one @ real ) )
=> ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( arccos @ Y2 ) ) ) ) ).
% arccos_lbound
thf(fact_3214_arccos__less__arccos,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X )
=> ( ( ord_less @ real @ X @ Y2 )
=> ( ( ord_less_eq @ real @ Y2 @ ( one_one @ real ) )
=> ( ord_less @ real @ ( arccos @ Y2 ) @ ( arccos @ X ) ) ) ) ) ).
% arccos_less_arccos
thf(fact_3215_arccos__less__mono,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ X ) @ ( one_one @ real ) )
=> ( ( ord_less_eq @ real @ ( abs_abs @ real @ Y2 ) @ ( one_one @ real ) )
=> ( ( ord_less @ real @ ( arccos @ X ) @ ( arccos @ Y2 ) )
= ( ord_less @ real @ Y2 @ X ) ) ) ) ).
% arccos_less_mono
thf(fact_3216_arccos__ubound,axiom,
! [Y2: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y2 )
=> ( ( ord_less_eq @ real @ Y2 @ ( one_one @ real ) )
=> ( ord_less_eq @ real @ ( arccos @ Y2 ) @ pi ) ) ) ).
% arccos_ubound
thf(fact_3217_arccos__cos,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ X @ pi )
=> ( ( arccos @ ( cos @ real @ X ) )
= X ) ) ) ).
% arccos_cos
thf(fact_3218_powr__neg__one,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( powr @ real @ X @ ( uminus_uminus @ real @ ( one_one @ real ) ) )
= ( divide_divide @ real @ ( one_one @ real ) @ X ) ) ) ).
% powr_neg_one
thf(fact_3219_arcsin__less__arcsin,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X )
=> ( ( ord_less @ real @ X @ Y2 )
=> ( ( ord_less_eq @ real @ Y2 @ ( one_one @ real ) )
=> ( ord_less @ real @ ( arcsin @ X ) @ ( arcsin @ Y2 ) ) ) ) ) ).
% arcsin_less_arcsin
thf(fact_3220_powr__mult__base,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( times_times @ real @ X @ ( powr @ real @ X @ Y2 ) )
= ( powr @ real @ X @ ( plus_plus @ real @ ( one_one @ real ) @ Y2 ) ) ) ) ).
% powr_mult_base
thf(fact_3221_arcsin__less__mono,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ X ) @ ( one_one @ real ) )
=> ( ( ord_less_eq @ real @ ( abs_abs @ real @ Y2 ) @ ( one_one @ real ) )
=> ( ( ord_less @ real @ ( arcsin @ X ) @ ( arcsin @ Y2 ) )
= ( ord_less @ real @ X @ Y2 ) ) ) ) ).
% arcsin_less_mono
thf(fact_3222_le__log__iff,axiom,
! [B2: real,X: real,Y2: real] :
( ( ord_less @ real @ ( one_one @ real ) @ B2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ Y2 @ ( log @ B2 @ X ) )
= ( ord_less_eq @ real @ ( powr @ real @ B2 @ Y2 ) @ X ) ) ) ) ).
% le_log_iff
thf(fact_3223_log__le__iff,axiom,
! [B2: real,X: real,Y2: real] :
( ( ord_less @ real @ ( one_one @ real ) @ B2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( log @ B2 @ X ) @ Y2 )
= ( ord_less_eq @ real @ X @ ( powr @ real @ B2 @ Y2 ) ) ) ) ) ).
% log_le_iff
thf(fact_3224_le__powr__iff,axiom,
! [B2: real,X: real,Y2: real] :
( ( ord_less @ real @ ( one_one @ real ) @ B2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ X @ ( powr @ real @ B2 @ Y2 ) )
= ( ord_less_eq @ real @ ( log @ B2 @ X ) @ Y2 ) ) ) ) ).
% le_powr_iff
thf(fact_3225_powr__le__iff,axiom,
! [B2: real,X: real,Y2: real] :
( ( ord_less @ real @ ( one_one @ real ) @ B2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( powr @ real @ B2 @ Y2 ) @ X )
= ( ord_less_eq @ real @ Y2 @ ( log @ B2 @ X ) ) ) ) ) ).
% powr_le_iff
thf(fact_3226_cos__arccos__abs,axiom,
! [Y2: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ Y2 ) @ ( one_one @ real ) )
=> ( ( cos @ real @ ( arccos @ Y2 ) )
= Y2 ) ) ).
% cos_arccos_abs
thf(fact_3227_arccos__cos__eq__abs,axiom,
! [Theta: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ Theta ) @ pi )
=> ( ( arccos @ ( cos @ real @ Theta ) )
= ( abs_abs @ real @ Theta ) ) ) ).
% arccos_cos_eq_abs
thf(fact_3228_ln__powr__bound,axiom,
! [X: real,A2: real] :
( ( ord_less_eq @ real @ ( one_one @ real ) @ X )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
=> ( ord_less_eq @ real @ ( ln_ln @ real @ X ) @ ( divide_divide @ real @ ( powr @ real @ X @ A2 ) @ A2 ) ) ) ) ).
% ln_powr_bound
thf(fact_3229_ln__powr__bound2,axiom,
! [X: real,A2: real] :
( ( ord_less @ real @ ( one_one @ real ) @ X )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
=> ( ord_less_eq @ real @ ( powr @ real @ ( ln_ln @ real @ X ) @ A2 ) @ ( times_times @ real @ ( powr @ real @ A2 @ A2 ) @ X ) ) ) ) ).
% ln_powr_bound2
thf(fact_3230_add__log__eq__powr,axiom,
! [B2: real,X: real,Y2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ B2 )
=> ( ( B2
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( plus_plus @ real @ Y2 @ ( log @ B2 @ X ) )
= ( log @ B2 @ ( times_times @ real @ ( powr @ real @ B2 @ Y2 ) @ X ) ) ) ) ) ) ).
% add_log_eq_powr
thf(fact_3231_log__add__eq__powr,axiom,
! [B2: real,X: real,Y2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ B2 )
=> ( ( B2
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( plus_plus @ real @ ( log @ B2 @ X ) @ Y2 )
= ( log @ B2 @ ( times_times @ real @ X @ ( powr @ real @ B2 @ Y2 ) ) ) ) ) ) ) ).
% log_add_eq_powr
thf(fact_3232_minus__log__eq__powr,axiom,
! [B2: real,X: real,Y2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ B2 )
=> ( ( B2
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( minus_minus @ real @ Y2 @ ( log @ B2 @ X ) )
= ( log @ B2 @ ( divide_divide @ real @ ( powr @ real @ B2 @ Y2 ) @ X ) ) ) ) ) ) ).
% minus_log_eq_powr
thf(fact_3233_arccos__lt__bounded,axiom,
! [Y2: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y2 )
=> ( ( ord_less @ real @ Y2 @ ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( arccos @ Y2 ) )
& ( ord_less @ real @ ( arccos @ Y2 ) @ pi ) ) ) ) ).
% arccos_lt_bounded
thf(fact_3234_arccos__bounded,axiom,
! [Y2: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y2 )
=> ( ( ord_less_eq @ real @ Y2 @ ( one_one @ real ) )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( arccos @ Y2 ) )
& ( ord_less_eq @ real @ ( arccos @ Y2 ) @ pi ) ) ) ) ).
% arccos_bounded
thf(fact_3235_sin__arccos__nonzero,axiom,
! [X: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X )
=> ( ( ord_less @ real @ X @ ( one_one @ real ) )
=> ( ( sin @ real @ ( arccos @ X ) )
!= ( zero_zero @ real ) ) ) ) ).
% sin_arccos_nonzero
thf(fact_3236_arccos__cos2,axiom,
! [X: real] :
( ( ord_less_eq @ real @ X @ ( zero_zero @ real ) )
=> ( ( ord_less_eq @ real @ ( uminus_uminus @ real @ pi ) @ X )
=> ( ( arccos @ ( cos @ real @ X ) )
= ( uminus_uminus @ real @ X ) ) ) ) ).
% arccos_cos2
thf(fact_3237_arccos__minus,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X )
=> ( ( ord_less_eq @ real @ X @ ( one_one @ real ) )
=> ( ( arccos @ ( uminus_uminus @ real @ X ) )
= ( minus_minus @ real @ pi @ ( arccos @ X ) ) ) ) ) ).
% arccos_minus
thf(fact_3238_cos__arcsin__nonzero,axiom,
! [X: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X )
=> ( ( ord_less @ real @ X @ ( one_one @ real ) )
=> ( ( cos @ real @ ( arcsin @ X ) )
!= ( zero_zero @ real ) ) ) ) ).
% cos_arcsin_nonzero
thf(fact_3239_powr__def,axiom,
! [A: $tType] :
( ( ln @ A )
=> ( ( powr @ A )
= ( ^ [X4: A,A5: A] :
( if @ A
@ ( X4
= ( zero_zero @ A ) )
@ ( zero_zero @ A )
@ ( exp @ A @ ( times_times @ A @ A5 @ ( ln_ln @ A @ X4 ) ) ) ) ) ) ) ).
% powr_def
thf(fact_3240_log__minus__eq__powr,axiom,
! [B2: real,X: real,Y2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ B2 )
=> ( ( B2
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( minus_minus @ real @ ( log @ B2 @ X ) @ Y2 )
= ( log @ B2 @ ( times_times @ real @ X @ ( powr @ real @ B2 @ ( uminus_uminus @ real @ Y2 ) ) ) ) ) ) ) ) ).
% log_minus_eq_powr
thf(fact_3241_arccos,axiom,
! [Y2: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y2 )
=> ( ( ord_less_eq @ real @ Y2 @ ( one_one @ real ) )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( arccos @ Y2 ) )
& ( ord_less_eq @ real @ ( arccos @ Y2 ) @ pi )
& ( ( cos @ real @ ( arccos @ Y2 ) )
= Y2 ) ) ) ) ).
% arccos
thf(fact_3242_arccos__minus__abs,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ X ) @ ( one_one @ real ) )
=> ( ( arccos @ ( uminus_uminus @ real @ X ) )
= ( minus_minus @ real @ pi @ ( arccos @ X ) ) ) ) ).
% arccos_minus_abs
thf(fact_3243_complex__norm,axiom,
! [X: real,Y2: real] :
( ( real_V7770717601297561774m_norm @ complex @ ( complex2 @ X @ Y2 ) )
= ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% complex_norm
thf(fact_3244_powr__half__sqrt,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( powr @ real @ X @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
= ( sqrt @ X ) ) ) ).
% powr_half_sqrt
thf(fact_3245_powr__neg__numeral,axiom,
! [X: real,N: num] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( powr @ real @ X @ ( uminus_uminus @ real @ ( numeral_numeral @ real @ N ) ) )
= ( divide_divide @ real @ ( one_one @ real ) @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ N ) ) ) ) ) ).
% powr_neg_numeral
thf(fact_3246_arccos__le__pi2,axiom,
! [Y2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y2 )
=> ( ( ord_less_eq @ real @ Y2 @ ( one_one @ real ) )
=> ( ord_less_eq @ real @ ( arccos @ Y2 ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ).
% arccos_le_pi2
thf(fact_3247_arcsin__lt__bounded,axiom,
! [Y2: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y2 )
=> ( ( ord_less @ real @ Y2 @ ( one_one @ real ) )
=> ( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ ( arcsin @ Y2 ) )
& ( ord_less @ real @ ( arcsin @ Y2 ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% arcsin_lt_bounded
thf(fact_3248_arcsin__bounded,axiom,
! [Y2: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y2 )
=> ( ( ord_less_eq @ real @ Y2 @ ( one_one @ real ) )
=> ( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ ( arcsin @ Y2 ) )
& ( ord_less_eq @ real @ ( arcsin @ Y2 ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% arcsin_bounded
thf(fact_3249_arcsin__ubound,axiom,
! [Y2: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y2 )
=> ( ( ord_less_eq @ real @ Y2 @ ( one_one @ real ) )
=> ( ord_less_eq @ real @ ( arcsin @ Y2 ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ).
% arcsin_ubound
thf(fact_3250_arcsin__lbound,axiom,
! [Y2: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y2 )
=> ( ( ord_less_eq @ real @ Y2 @ ( one_one @ real ) )
=> ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ ( arcsin @ Y2 ) ) ) ) ).
% arcsin_lbound
thf(fact_3251_arcsin__sin,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X )
=> ( ( ord_less_eq @ real @ X @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( arcsin @ ( sin @ real @ X ) )
= X ) ) ) ).
% arcsin_sin
thf(fact_3252_le__arcsin__iff,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X )
=> ( ( ord_less_eq @ real @ X @ ( one_one @ real ) )
=> ( ( ord_less_eq @ real @ ( divide_divide @ real @ ( uminus_uminus @ real @ pi ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ Y2 )
=> ( ( ord_less_eq @ real @ Y2 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less_eq @ real @ Y2 @ ( arcsin @ X ) )
= ( ord_less_eq @ real @ ( sin @ real @ Y2 ) @ X ) ) ) ) ) ) ).
% le_arcsin_iff
thf(fact_3253_arcsin__le__iff,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X )
=> ( ( ord_less_eq @ real @ X @ ( one_one @ real ) )
=> ( ( ord_less_eq @ real @ ( divide_divide @ real @ ( uminus_uminus @ real @ pi ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ Y2 )
=> ( ( ord_less_eq @ real @ Y2 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less_eq @ real @ ( arcsin @ X ) @ Y2 )
= ( ord_less_eq @ real @ X @ ( sin @ real @ Y2 ) ) ) ) ) ) ) ).
% arcsin_le_iff
thf(fact_3254_arcsin__pi,axiom,
! [Y2: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y2 )
=> ( ( ord_less_eq @ real @ Y2 @ ( one_one @ real ) )
=> ( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ ( arcsin @ Y2 ) )
& ( ord_less_eq @ real @ ( arcsin @ Y2 ) @ pi )
& ( ( sin @ real @ ( arcsin @ Y2 ) )
= Y2 ) ) ) ) ).
% arcsin_pi
thf(fact_3255_arcsin,axiom,
! [Y2: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y2 )
=> ( ( ord_less_eq @ real @ Y2 @ ( one_one @ real ) )
=> ( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ ( arcsin @ Y2 ) )
& ( ord_less_eq @ real @ ( arcsin @ Y2 ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
& ( ( sin @ real @ ( arcsin @ Y2 ) )
= Y2 ) ) ) ) ).
% arcsin
thf(fact_3256_arccos__cos__eq__abs__2pi,axiom,
! [Theta: real] :
~ ! [K: int] :
( ( arccos @ ( cos @ real @ Theta ) )
!= ( abs_abs @ real @ ( minus_minus @ real @ Theta @ ( times_times @ real @ ( ring_1_of_int @ real @ K ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) ) ) ) ) ).
% arccos_cos_eq_abs_2pi
thf(fact_3257_arcosh__def,axiom,
! [A: $tType] :
( ( ln @ A )
=> ( ( arcosh @ A )
= ( ^ [X4: A] : ( ln_ln @ A @ ( plus_plus @ A @ X4 @ ( powr @ A @ ( minus_minus @ A @ ( power_power @ A @ X4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) ) @ ( real_Vector_of_real @ A @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% arcosh_def
thf(fact_3258_geometric__deriv__sums,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [Z3: A] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ Z3 ) @ ( one_one @ real ) )
=> ( sums @ A
@ ^ [N2: nat] : ( times_times @ A @ ( semiring_1_of_nat @ A @ ( suc @ N2 ) ) @ ( power_power @ A @ Z3 @ N2 ) )
@ ( divide_divide @ A @ ( one_one @ A ) @ ( power_power @ A @ ( minus_minus @ A @ ( one_one @ A ) @ Z3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% geometric_deriv_sums
thf(fact_3259_arsinh__def,axiom,
! [A: $tType] :
( ( ln @ A )
=> ( ( arsinh @ A )
= ( ^ [X4: A] : ( ln_ln @ A @ ( plus_plus @ A @ X4 @ ( powr @ A @ ( plus_plus @ A @ ( power_power @ A @ X4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) ) @ ( real_Vector_of_real @ A @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% arsinh_def
thf(fact_3260_floor__log__nat__eq__powr__iff,axiom,
! [B2: nat,K2: nat,N: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ B2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K2 )
=> ( ( ( archim6421214686448440834_floor @ real @ ( log @ ( semiring_1_of_nat @ real @ B2 ) @ ( semiring_1_of_nat @ real @ K2 ) ) )
= ( semiring_1_of_nat @ int @ N ) )
= ( ( ord_less_eq @ nat @ ( power_power @ nat @ B2 @ N ) @ K2 )
& ( ord_less @ nat @ K2 @ ( power_power @ nat @ B2 @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% floor_log_nat_eq_powr_iff
thf(fact_3261_monoseq__def,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( topological_monoseq @ A )
= ( ^ [X6: nat > A] :
( ! [M2: nat,N2: nat] :
( ( ord_less_eq @ nat @ M2 @ N2 )
=> ( ord_less_eq @ A @ ( X6 @ M2 ) @ ( X6 @ N2 ) ) )
| ! [M2: nat,N2: nat] :
( ( ord_less_eq @ nat @ M2 @ N2 )
=> ( ord_less_eq @ A @ ( X6 @ N2 ) @ ( X6 @ M2 ) ) ) ) ) ) ) ).
% monoseq_def
thf(fact_3262_summable__complex__of__real,axiom,
! [F3: nat > real] :
( ( summable @ complex
@ ^ [N2: nat] : ( real_Vector_of_real @ complex @ ( F3 @ N2 ) ) )
= ( summable @ real @ F3 ) ) ).
% summable_complex_of_real
thf(fact_3263_of__int__floor__cancel,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ A @ X ) )
= X )
= ( ? [N2: int] :
( X
= ( ring_1_of_int @ A @ N2 ) ) ) ) ) ).
% of_int_floor_cancel
thf(fact_3264_sums__zero,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ( sums @ A
@ ^ [N2: nat] : ( zero_zero @ A )
@ ( zero_zero @ A ) ) ) ).
% sums_zero
thf(fact_3265_of__real__1,axiom,
! [A: $tType] :
( ( real_V2191834092415804123ebra_1 @ A )
=> ( ( real_Vector_of_real @ A @ ( one_one @ real ) )
= ( one_one @ A ) ) ) ).
% of_real_1
thf(fact_3266_of__real__eq__1__iff,axiom,
! [A: $tType] :
( ( real_V2191834092415804123ebra_1 @ A )
=> ! [X: real] :
( ( ( real_Vector_of_real @ A @ X )
= ( one_one @ A ) )
= ( X
= ( one_one @ real ) ) ) ) ).
% of_real_eq_1_iff
thf(fact_3267_of__real__mult,axiom,
! [A: $tType] :
( ( real_V2191834092415804123ebra_1 @ A )
=> ! [X: real,Y2: real] :
( ( real_Vector_of_real @ A @ ( times_times @ real @ X @ Y2 ) )
= ( times_times @ A @ ( real_Vector_of_real @ A @ X ) @ ( real_Vector_of_real @ A @ Y2 ) ) ) ) ).
% of_real_mult
thf(fact_3268_of__real__numeral,axiom,
! [A: $tType] :
( ( real_V2191834092415804123ebra_1 @ A )
=> ! [W: num] :
( ( real_Vector_of_real @ A @ ( numeral_numeral @ real @ W ) )
= ( numeral_numeral @ A @ W ) ) ) ).
% of_real_numeral
thf(fact_3269_floor__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V2: num] :
( ( archim6421214686448440834_floor @ A @ ( numeral_numeral @ A @ V2 ) )
= ( numeral_numeral @ int @ V2 ) ) ) ).
% floor_numeral
thf(fact_3270_of__real__divide,axiom,
! [A: $tType] :
( ( real_V5047593784448816457lgebra @ A )
=> ! [X: real,Y2: real] :
( ( real_Vector_of_real @ A @ ( divide_divide @ real @ X @ Y2 ) )
= ( divide_divide @ A @ ( real_Vector_of_real @ A @ X ) @ ( real_Vector_of_real @ A @ Y2 ) ) ) ) ).
% of_real_divide
thf(fact_3271_floor__one,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archim6421214686448440834_floor @ A @ ( one_one @ A ) )
= ( one_one @ int ) ) ) ).
% floor_one
thf(fact_3272_of__real__add,axiom,
! [A: $tType] :
( ( real_V2191834092415804123ebra_1 @ A )
=> ! [X: real,Y2: real] :
( ( real_Vector_of_real @ A @ ( plus_plus @ real @ X @ Y2 ) )
= ( plus_plus @ A @ ( real_Vector_of_real @ A @ X ) @ ( real_Vector_of_real @ A @ Y2 ) ) ) ) ).
% of_real_add
thf(fact_3273_of__real__power,axiom,
! [A: $tType] :
( ( real_V2191834092415804123ebra_1 @ A )
=> ! [X: real,N: nat] :
( ( real_Vector_of_real @ A @ ( power_power @ real @ X @ N ) )
= ( power_power @ A @ ( real_Vector_of_real @ A @ X ) @ N ) ) ) ).
% of_real_power
thf(fact_3274_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_3275_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_3276_numeral__le__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V2: num,X: A] :
( ( ord_less_eq @ int @ ( numeral_numeral @ int @ V2 ) @ ( archim6421214686448440834_floor @ A @ X ) )
= ( ord_less_eq @ A @ ( numeral_numeral @ A @ V2 ) @ X ) ) ) ).
% numeral_le_floor
thf(fact_3277_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_3278_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_3279_floor__less__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,V2: num] :
( ( ord_less @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( numeral_numeral @ int @ V2 ) )
= ( ord_less @ A @ X @ ( numeral_numeral @ A @ V2 ) ) ) ) ).
% floor_less_numeral
thf(fact_3280_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_3281_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_3282_floor__neg__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V2: num] :
( ( archim6421214686448440834_floor @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) )
= ( uminus_uminus @ int @ ( numeral_numeral @ int @ V2 ) ) ) ) ).
% floor_neg_numeral
thf(fact_3283_of__real__neg__numeral,axiom,
! [A: $tType] :
( ( real_V2191834092415804123ebra_1 @ A )
=> ! [W: num] :
( ( real_Vector_of_real @ A @ ( uminus_uminus @ real @ ( numeral_numeral @ real @ W ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) ) ) ).
% of_real_neg_numeral
thf(fact_3284_floor__diff__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,V2: num] :
( ( archim6421214686448440834_floor @ A @ ( minus_minus @ A @ X @ ( numeral_numeral @ A @ V2 ) ) )
= ( minus_minus @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( numeral_numeral @ int @ V2 ) ) ) ) ).
% floor_diff_numeral
thf(fact_3285_cos__of__real__pi,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ( ( cos @ A @ ( real_Vector_of_real @ A @ pi ) )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% cos_of_real_pi
thf(fact_3286_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_3287_floor__numeral__power,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: num,N: nat] :
( ( archim6421214686448440834_floor @ A @ ( power_power @ A @ ( numeral_numeral @ A @ X ) @ N ) )
= ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N ) ) ) ).
% floor_numeral_power
thf(fact_3288_powser__sums__zero__iff,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [A2: nat > A,X: A] :
( ( sums @ A
@ ^ [N2: nat] : ( times_times @ A @ ( A2 @ N2 ) @ ( power_power @ A @ ( zero_zero @ A ) @ N2 ) )
@ X )
= ( ( A2 @ ( zero_zero @ nat ) )
= X ) ) ) ).
% powser_sums_zero_iff
thf(fact_3289_floor__divide__eq__div__numeral,axiom,
! [A2: num,B2: num] :
( ( archim6421214686448440834_floor @ real @ ( divide_divide @ real @ ( numeral_numeral @ real @ A2 ) @ ( numeral_numeral @ real @ B2 ) ) )
= ( divide_divide @ int @ ( numeral_numeral @ int @ A2 ) @ ( numeral_numeral @ int @ B2 ) ) ) ).
% floor_divide_eq_div_numeral
thf(fact_3290_numeral__less__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V2: num,X: A] :
( ( ord_less @ int @ ( numeral_numeral @ int @ V2 ) @ ( archim6421214686448440834_floor @ A @ X ) )
= ( ord_less_eq @ A @ ( plus_plus @ A @ ( numeral_numeral @ A @ V2 ) @ ( one_one @ A ) ) @ X ) ) ) ).
% numeral_less_floor
thf(fact_3291_floor__le__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,V2: num] :
( ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( numeral_numeral @ int @ V2 ) )
= ( ord_less @ A @ X @ ( plus_plus @ A @ ( numeral_numeral @ A @ V2 ) @ ( one_one @ A ) ) ) ) ) ).
% floor_le_numeral
thf(fact_3292_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_3293_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_3294_neg__numeral__le__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V2: num,X: A] :
( ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V2 ) ) @ ( archim6421214686448440834_floor @ A @ X ) )
= ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) @ X ) ) ) ).
% neg_numeral_le_floor
thf(fact_3295_floor__less__neg__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,V2: num] :
( ( ord_less @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V2 ) ) )
= ( ord_less @ A @ X @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) ) ) ) ).
% floor_less_neg_numeral
thf(fact_3296_norm__of__real__add1,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [X: real] :
( ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ ( real_Vector_of_real @ A @ X ) @ ( one_one @ A ) ) )
= ( abs_abs @ real @ ( plus_plus @ real @ X @ ( one_one @ real ) ) ) ) ) ).
% norm_of_real_add1
thf(fact_3297_norm__of__real__addn,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [X: real,B2: num] :
( ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ ( real_Vector_of_real @ A @ X ) @ ( numeral_numeral @ A @ B2 ) ) )
= ( abs_abs @ real @ ( plus_plus @ real @ X @ ( numeral_numeral @ real @ B2 ) ) ) ) ) ).
% norm_of_real_addn
thf(fact_3298_floor__one__divide__eq__div__numeral,axiom,
! [B2: num] :
( ( archim6421214686448440834_floor @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ B2 ) ) )
= ( divide_divide @ int @ ( one_one @ int ) @ ( numeral_numeral @ int @ B2 ) ) ) ).
% floor_one_divide_eq_div_numeral
thf(fact_3299_floor__minus__divide__eq__div__numeral,axiom,
! [A2: num,B2: num] :
( ( archim6421214686448440834_floor @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ ( numeral_numeral @ real @ A2 ) @ ( numeral_numeral @ real @ B2 ) ) ) )
= ( divide_divide @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ A2 ) ) @ ( numeral_numeral @ int @ B2 ) ) ) ).
% floor_minus_divide_eq_div_numeral
thf(fact_3300_cos__of__real__pi__half,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V7773925162809079976_field @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ( ( cos @ A @ ( divide_divide @ A @ ( real_Vector_of_real @ A @ pi ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
= ( zero_zero @ A ) ) ) ).
% cos_of_real_pi_half
thf(fact_3301_sin__of__real__pi__half,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V7773925162809079976_field @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ( ( sin @ A @ ( divide_divide @ A @ ( real_Vector_of_real @ A @ pi ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
= ( one_one @ A ) ) ) ).
% sin_of_real_pi_half
thf(fact_3302_neg__numeral__less__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V2: num,X: A] :
( ( ord_less @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V2 ) ) @ ( archim6421214686448440834_floor @ A @ X ) )
= ( ord_less_eq @ A @ ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) @ ( one_one @ A ) ) @ X ) ) ) ).
% neg_numeral_less_floor
thf(fact_3303_floor__le__neg__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,V2: num] :
( ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V2 ) ) )
= ( ord_less @ A @ X @ ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) @ ( one_one @ A ) ) ) ) ) ).
% floor_le_neg_numeral
thf(fact_3304_floor__minus__one__divide__eq__div__numeral,axiom,
! [B2: num] :
( ( archim6421214686448440834_floor @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ B2 ) ) ) )
= ( divide_divide @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( numeral_numeral @ int @ B2 ) ) ) ).
% floor_minus_one_divide_eq_div_numeral
thf(fact_3305_sums__le,axiom,
! [A: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F3: nat > A,G: nat > A,S: A,T2: A] :
( ! [N3: nat] : ( ord_less_eq @ A @ ( F3 @ N3 ) @ ( G @ N3 ) )
=> ( ( sums @ A @ F3 @ S )
=> ( ( sums @ A @ G @ T2 )
=> ( ord_less_eq @ A @ S @ T2 ) ) ) ) ) ).
% sums_le
thf(fact_3306_sums__of__real,axiom,
! [A: $tType] :
( ( ( real_V2191834092415804123ebra_1 @ A )
& ( real_V822414075346904944vector @ A ) )
=> ! [X9: nat > real,A2: real] :
( ( sums @ real @ X9 @ A2 )
=> ( sums @ A
@ ^ [N2: nat] : ( real_Vector_of_real @ A @ ( X9 @ N2 ) )
@ ( real_Vector_of_real @ A @ A2 ) ) ) ) ).
% sums_of_real
thf(fact_3307_sums__of__real__iff,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [F3: nat > real,C2: real] :
( ( sums @ A
@ ^ [N2: nat] : ( real_Vector_of_real @ A @ ( F3 @ N2 ) )
@ ( real_Vector_of_real @ A @ C2 ) )
= ( sums @ real @ F3 @ C2 ) ) ) ).
% sums_of_real_iff
thf(fact_3308_complex__exp__exists,axiom,
! [Z3: complex] :
? [A4: complex,R3: real] :
( Z3
= ( times_times @ complex @ ( real_Vector_of_real @ complex @ R3 ) @ ( exp @ complex @ A4 ) ) ) ).
% complex_exp_exists
thf(fact_3309_sums__single,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [I2: nat,F3: nat > A] :
( sums @ A
@ ^ [R5: nat] : ( if @ A @ ( R5 = I2 ) @ ( F3 @ R5 ) @ ( zero_zero @ A ) )
@ ( F3 @ I2 ) ) ) ).
% sums_single
thf(fact_3310_sums__mult,axiom,
! [A: $tType] :
( ( real_V4412858255891104859lgebra @ A )
=> ! [F3: nat > A,A2: A,C2: A] :
( ( sums @ A @ F3 @ A2 )
=> ( sums @ A
@ ^ [N2: nat] : ( times_times @ A @ C2 @ ( F3 @ N2 ) )
@ ( times_times @ A @ C2 @ A2 ) ) ) ) ).
% sums_mult
thf(fact_3311_sums__mult2,axiom,
! [A: $tType] :
( ( real_V4412858255891104859lgebra @ A )
=> ! [F3: nat > A,A2: A,C2: A] :
( ( sums @ A @ F3 @ A2 )
=> ( sums @ A
@ ^ [N2: nat] : ( times_times @ A @ ( F3 @ N2 ) @ C2 )
@ ( times_times @ A @ A2 @ C2 ) ) ) ) ).
% sums_mult2
thf(fact_3312_sums__add,axiom,
! [A: $tType] :
( ( ( topolo5987344860129210374id_add @ A )
& ( topological_t2_space @ A ) )
=> ! [F3: nat > A,A2: A,G: nat > A,B2: A] :
( ( sums @ A @ F3 @ A2 )
=> ( ( sums @ A @ G @ B2 )
=> ( sums @ A
@ ^ [N2: nat] : ( plus_plus @ A @ ( F3 @ N2 ) @ ( G @ N2 ) )
@ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ) ).
% sums_add
thf(fact_3313_sums__diff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: nat > A,A2: A,G: nat > A,B2: A] :
( ( sums @ A @ F3 @ A2 )
=> ( ( sums @ A @ G @ B2 )
=> ( sums @ A
@ ^ [N2: nat] : ( minus_minus @ A @ ( F3 @ N2 ) @ ( G @ N2 ) )
@ ( minus_minus @ A @ A2 @ B2 ) ) ) ) ) ).
% sums_diff
thf(fact_3314_sums__divide,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: nat > A,A2: A,C2: A] :
( ( sums @ A @ F3 @ A2 )
=> ( sums @ A
@ ^ [N2: nat] : ( divide_divide @ A @ ( F3 @ N2 ) @ C2 )
@ ( divide_divide @ A @ A2 @ C2 ) ) ) ) ).
% sums_divide
thf(fact_3315_sums__minus,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: nat > A,A2: A] :
( ( sums @ A @ F3 @ A2 )
=> ( sums @ A
@ ^ [N2: nat] : ( uminus_uminus @ A @ ( F3 @ N2 ) )
@ ( uminus_uminus @ A @ A2 ) ) ) ) ).
% sums_minus
thf(fact_3316_floor__mono,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Y2: A] :
( ( ord_less_eq @ A @ X @ Y2 )
=> ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( archim6421214686448440834_floor @ A @ Y2 ) ) ) ) ).
% floor_mono
thf(fact_3317_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_3318_floor__less__cancel,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Y2: A] :
( ( ord_less @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( archim6421214686448440834_floor @ A @ Y2 ) )
=> ( ord_less @ A @ X @ Y2 ) ) ) ).
% floor_less_cancel
thf(fact_3319_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_3320_sums__mult__iff,axiom,
! [A: $tType] :
( ( ( field @ A )
& ( real_V4412858255891104859lgebra @ A ) )
=> ! [C2: A,F3: nat > A,D2: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( sums @ A
@ ^ [N2: nat] : ( times_times @ A @ C2 @ ( F3 @ N2 ) )
@ ( times_times @ A @ C2 @ D2 ) )
= ( sums @ A @ F3 @ D2 ) ) ) ) ).
% sums_mult_iff
thf(fact_3321_sums__mult2__iff,axiom,
! [A: $tType] :
( ( ( field @ A )
& ( real_V4412858255891104859lgebra @ A ) )
=> ! [C2: A,F3: nat > A,D2: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( sums @ A
@ ^ [N2: nat] : ( times_times @ A @ ( F3 @ N2 ) @ C2 )
@ ( times_times @ A @ D2 @ C2 ) )
= ( sums @ A @ F3 @ D2 ) ) ) ) ).
% sums_mult2_iff
thf(fact_3322_complex__of__real__mult__Complex,axiom,
! [R: real,X: real,Y2: real] :
( ( times_times @ complex @ ( real_Vector_of_real @ complex @ R ) @ ( complex2 @ X @ Y2 ) )
= ( complex2 @ ( times_times @ real @ R @ X ) @ ( times_times @ real @ R @ Y2 ) ) ) ).
% complex_of_real_mult_Complex
thf(fact_3323_Complex__mult__complex__of__real,axiom,
! [X: real,Y2: real,R: real] :
( ( times_times @ complex @ ( complex2 @ X @ Y2 ) @ ( real_Vector_of_real @ complex @ R ) )
= ( complex2 @ ( times_times @ real @ X @ R ) @ ( times_times @ real @ Y2 @ R ) ) ) ).
% Complex_mult_complex_of_real
thf(fact_3324_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_3325_summable__of__real,axiom,
! [A: $tType] :
( ( ( real_V2191834092415804123ebra_1 @ A )
& ( real_V822414075346904944vector @ A ) )
=> ! [X9: nat > real] :
( ( summable @ real @ X9 )
=> ( summable @ A
@ ^ [N2: nat] : ( real_Vector_of_real @ A @ ( X9 @ N2 ) ) ) ) ) ).
% summable_of_real
thf(fact_3326_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_3327_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_3328_nonzero__of__real__divide,axiom,
! [A: $tType] :
( ( real_V7773925162809079976_field @ A )
=> ! [Y2: real,X: real] :
( ( Y2
!= ( zero_zero @ real ) )
=> ( ( real_Vector_of_real @ A @ ( divide_divide @ real @ X @ Y2 ) )
= ( divide_divide @ A @ ( real_Vector_of_real @ A @ X ) @ ( real_Vector_of_real @ A @ Y2 ) ) ) ) ) ).
% nonzero_of_real_divide
thf(fact_3329_le__floor__add,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Y2: A] : ( ord_less_eq @ int @ ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( archim6421214686448440834_floor @ A @ Y2 ) ) @ ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X @ Y2 ) ) ) ) ).
% le_floor_add
thf(fact_3330_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_3331_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_3332_floor__divide__of__int__eq,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [K2: int,L: int] :
( ( archim6421214686448440834_floor @ A @ ( divide_divide @ A @ ( ring_1_of_int @ A @ K2 ) @ ( ring_1_of_int @ A @ L ) ) )
= ( divide_divide @ int @ K2 @ L ) ) ) ).
% floor_divide_of_int_eq
thf(fact_3333_sums__mult__D,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [C2: A,F3: nat > A,A2: A] :
( ( sums @ A
@ ^ [N2: nat] : ( times_times @ A @ C2 @ ( F3 @ N2 ) )
@ A2 )
=> ( ( C2
!= ( zero_zero @ A ) )
=> ( sums @ A @ F3 @ ( divide_divide @ A @ A2 @ C2 ) ) ) ) ) ).
% sums_mult_D
thf(fact_3334_sums__Suc__imp,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: nat > A,S: A] :
( ( ( F3 @ ( zero_zero @ nat ) )
= ( zero_zero @ A ) )
=> ( ( sums @ A
@ ^ [N2: nat] : ( F3 @ ( suc @ N2 ) )
@ S )
=> ( sums @ A @ F3 @ S ) ) ) ) ).
% sums_Suc_imp
thf(fact_3335_floor__power,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,N: nat] :
( ( X
= ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ A @ X ) ) )
=> ( ( archim6421214686448440834_floor @ A @ ( power_power @ A @ X @ N ) )
= ( power_power @ int @ ( archim6421214686448440834_floor @ A @ X ) @ N ) ) ) ) ).
% floor_power
thf(fact_3336_sums__Suc,axiom,
! [A: $tType] :
( ( ( topolo5987344860129210374id_add @ A )
& ( topological_t2_space @ A ) )
=> ! [F3: nat > A,L: A] :
( ( sums @ A
@ ^ [N2: nat] : ( F3 @ ( suc @ N2 ) )
@ L )
=> ( sums @ A @ F3 @ ( plus_plus @ A @ L @ ( F3 @ ( zero_zero @ nat ) ) ) ) ) ) ).
% sums_Suc
thf(fact_3337_sums__Suc__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: nat > A,S: A] :
( ( sums @ A
@ ^ [N2: nat] : ( F3 @ ( suc @ N2 ) )
@ S )
= ( sums @ A @ F3 @ ( plus_plus @ A @ S @ ( F3 @ ( zero_zero @ nat ) ) ) ) ) ) ).
% sums_Suc_iff
thf(fact_3338_sums__zero__iff__shift,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [N: nat,F3: nat > A,S: A] :
( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ N )
=> ( ( F3 @ I3 )
= ( zero_zero @ A ) ) )
=> ( ( sums @ A
@ ^ [I: nat] : ( F3 @ ( plus_plus @ nat @ I @ N ) )
@ S )
= ( sums @ A @ F3 @ S ) ) ) ) ).
% sums_zero_iff_shift
thf(fact_3339_suminf__of__real,axiom,
! [A: $tType] :
( ( ( real_V2191834092415804123ebra_1 @ A )
& ( real_V822414075346904944vector @ A ) )
=> ! [X9: nat > real] :
( ( summable @ real @ X9 )
=> ( ( real_Vector_of_real @ A @ ( suminf @ real @ X9 ) )
= ( suminf @ A
@ ^ [N2: nat] : ( real_Vector_of_real @ A @ ( X9 @ N2 ) ) ) ) ) ) ).
% suminf_of_real
thf(fact_3340_norm__less__p1,axiom,
! [A: $tType] :
( ( real_V2822296259951069270ebra_1 @ A )
=> ! [X: A] : ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ ( real_Vector_of_real @ A @ ( real_V7770717601297561774m_norm @ A @ X ) ) @ ( one_one @ A ) ) ) ) ) ).
% norm_less_p1
thf(fact_3341_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_3342_powser__sums__if,axiom,
! [A: $tType] :
( ( ( ring_1 @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [M: nat,Z3: A] :
( sums @ A
@ ^ [N2: nat] : ( times_times @ A @ ( if @ A @ ( N2 = M ) @ ( one_one @ A ) @ ( zero_zero @ A ) ) @ ( power_power @ A @ Z3 @ N2 ) )
@ ( power_power @ A @ Z3 @ M ) ) ) ).
% powser_sums_if
thf(fact_3343_powser__sums__zero,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [A2: nat > A] :
( sums @ A
@ ^ [N2: nat] : ( times_times @ A @ ( A2 @ N2 ) @ ( power_power @ A @ ( zero_zero @ A ) @ N2 ) )
@ ( A2 @ ( zero_zero @ nat ) ) ) ) ).
% powser_sums_zero
thf(fact_3344_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_3345_ceiling__altdef,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archimedean_ceiling @ A )
= ( ^ [X4: A] :
( if @ int
@ ( X4
= ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ A @ X4 ) ) )
@ ( archim6421214686448440834_floor @ A @ X4 )
@ ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X4 ) @ ( one_one @ int ) ) ) ) ) ) ).
% ceiling_altdef
thf(fact_3346_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_3347_floor__eq,axiom,
! [N: int,X: real] :
( ( ord_less @ real @ ( ring_1_of_int @ real @ N ) @ X )
=> ( ( ord_less @ real @ X @ ( plus_plus @ real @ ( ring_1_of_int @ real @ N ) @ ( one_one @ real ) ) )
=> ( ( archim6421214686448440834_floor @ real @ X )
= N ) ) ) ).
% floor_eq
thf(fact_3348_real__of__int__floor__add__one__gt,axiom,
! [R: real] : ( ord_less @ real @ R @ ( plus_plus @ real @ ( ring_1_of_int @ real @ ( archim6421214686448440834_floor @ real @ R ) ) @ ( one_one @ real ) ) ) ).
% real_of_int_floor_add_one_gt
thf(fact_3349_real__of__int__floor__add__one__ge,axiom,
! [R: real] : ( ord_less_eq @ real @ R @ ( plus_plus @ real @ ( ring_1_of_int @ real @ ( archim6421214686448440834_floor @ real @ R ) ) @ ( one_one @ real ) ) ) ).
% real_of_int_floor_add_one_ge
thf(fact_3350_real__of__int__floor__gt__diff__one,axiom,
! [R: real] : ( ord_less @ real @ ( minus_minus @ real @ R @ ( one_one @ real ) ) @ ( ring_1_of_int @ real @ ( archim6421214686448440834_floor @ real @ R ) ) ) ).
% real_of_int_floor_gt_diff_one
thf(fact_3351_real__of__int__floor__ge__diff__one,axiom,
! [R: real] : ( ord_less_eq @ real @ ( minus_minus @ real @ R @ ( one_one @ real ) ) @ ( ring_1_of_int @ real @ ( archim6421214686448440834_floor @ real @ R ) ) ) ).
% real_of_int_floor_ge_diff_one
thf(fact_3352_floor__split,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [P3: int > $o,T2: A] :
( ( P3 @ ( archim6421214686448440834_floor @ A @ T2 ) )
= ( ! [I: int] :
( ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ I ) @ T2 )
& ( ord_less @ A @ T2 @ ( plus_plus @ A @ ( ring_1_of_int @ A @ I ) @ ( one_one @ A ) ) ) )
=> ( P3 @ I ) ) ) ) ) ).
% floor_split
thf(fact_3353_floor__eq__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,A2: int] :
( ( ( archim6421214686448440834_floor @ A @ X )
= A2 )
= ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ A2 ) @ X )
& ( ord_less @ A @ X @ ( plus_plus @ A @ ( ring_1_of_int @ A @ A2 ) @ ( one_one @ A ) ) ) ) ) ) ).
% floor_eq_iff
thf(fact_3354_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_3355_le__mult__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less_eq @ int @ ( times_times @ int @ ( archim6421214686448440834_floor @ A @ A2 ) @ ( archim6421214686448440834_floor @ A @ B2 ) ) @ ( archim6421214686448440834_floor @ A @ ( times_times @ A @ A2 @ B2 ) ) ) ) ) ) ).
% le_mult_floor
thf(fact_3356_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_3357_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_3358_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_3359_norm__of__real__diff,axiom,
! [A: $tType] :
( ( real_V2822296259951069270ebra_1 @ A )
=> ! [B2: real,A2: real] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( real_Vector_of_real @ A @ B2 ) @ ( real_Vector_of_real @ A @ A2 ) ) ) @ ( abs_abs @ real @ ( minus_minus @ real @ B2 @ A2 ) ) ) ) ).
% norm_of_real_diff
thf(fact_3360_floor__eq2,axiom,
! [N: int,X: real] :
( ( ord_less_eq @ real @ ( ring_1_of_int @ real @ N ) @ X )
=> ( ( ord_less @ real @ X @ ( plus_plus @ real @ ( ring_1_of_int @ real @ N ) @ ( one_one @ real ) ) )
=> ( ( archim6421214686448440834_floor @ real @ X )
= N ) ) ) ).
% floor_eq2
thf(fact_3361_floor__divide__real__eq__div,axiom,
! [B2: int,A2: real] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ B2 )
=> ( ( archim6421214686448440834_floor @ real @ ( divide_divide @ real @ A2 @ ( ring_1_of_int @ real @ B2 ) ) )
= ( divide_divide @ int @ ( archim6421214686448440834_floor @ real @ A2 ) @ B2 ) ) ) ).
% floor_divide_real_eq_div
thf(fact_3362_floor__divide__lower,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Q: A,P5: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Q )
=> ( ord_less_eq @ A @ ( times_times @ A @ ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ A @ ( divide_divide @ A @ P5 @ Q ) ) ) @ Q ) @ P5 ) ) ) ).
% floor_divide_lower
thf(fact_3363_cos__int__times__real,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [M: int,X: real] :
( ( cos @ A @ ( times_times @ A @ ( ring_1_of_int @ A @ M ) @ ( real_Vector_of_real @ A @ X ) ) )
= ( real_Vector_of_real @ A @ ( cos @ real @ ( times_times @ real @ ( ring_1_of_int @ real @ M ) @ X ) ) ) ) ) ).
% cos_int_times_real
thf(fact_3364_sin__int__times__real,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [M: int,X: real] :
( ( sin @ A @ ( times_times @ A @ ( ring_1_of_int @ A @ M ) @ ( real_Vector_of_real @ A @ X ) ) )
= ( real_Vector_of_real @ A @ ( sin @ real @ ( times_times @ real @ ( ring_1_of_int @ real @ M ) @ X ) ) ) ) ) ).
% sin_int_times_real
thf(fact_3365_floor__divide__upper,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Q: A,P5: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Q )
=> ( ord_less @ A @ P5 @ ( times_times @ A @ ( plus_plus @ A @ ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ A @ ( divide_divide @ A @ P5 @ Q ) ) ) @ ( one_one @ A ) ) @ Q ) ) ) ) ).
% floor_divide_upper
thf(fact_3366_geometric__sums,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [C2: A] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ C2 ) @ ( one_one @ real ) )
=> ( sums @ A @ ( power_power @ A @ C2 ) @ ( divide_divide @ A @ ( one_one @ A ) @ ( minus_minus @ A @ ( one_one @ A ) @ C2 ) ) ) ) ) ).
% geometric_sums
thf(fact_3367_power__half__series,axiom,
( sums @ real
@ ^ [N2: nat] : ( power_power @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( suc @ N2 ) )
@ ( one_one @ real ) ) ).
% power_half_series
thf(fact_3368_round__def,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archimedean_round @ A )
= ( ^ [X4: A] : ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X4 @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% round_def
thf(fact_3369_sums__if_H,axiom,
! [G: nat > real,X: real] :
( ( sums @ real @ G @ X )
=> ( sums @ real
@ ^ [N2: nat] : ( if @ real @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) @ ( zero_zero @ real ) @ ( G @ ( divide_divide @ nat @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
@ X ) ) ).
% sums_if'
thf(fact_3370_cos__sin__eq,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ( ( cos @ A )
= ( ^ [X4: A] : ( sin @ A @ ( minus_minus @ A @ ( divide_divide @ A @ ( real_Vector_of_real @ A @ pi ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ X4 ) ) ) ) ) ).
% cos_sin_eq
thf(fact_3371_sin__cos__eq,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ( ( sin @ A )
= ( ^ [X4: A] : ( cos @ A @ ( minus_minus @ A @ ( divide_divide @ A @ ( real_Vector_of_real @ A @ pi ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ X4 ) ) ) ) ) ).
% sin_cos_eq
thf(fact_3372_sums__if,axiom,
! [G: nat > real,X: real,F3: nat > real,Y2: real] :
( ( sums @ real @ G @ X )
=> ( ( sums @ real @ F3 @ Y2 )
=> ( sums @ real
@ ^ [N2: nat] : ( if @ real @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) @ ( F3 @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( G @ ( divide_divide @ nat @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
@ ( plus_plus @ real @ X @ Y2 ) ) ) ) ).
% sums_if
thf(fact_3373_minus__sin__cos__eq,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A] :
( ( uminus_uminus @ A @ ( sin @ A @ X ) )
= ( cos @ A @ ( plus_plus @ A @ X @ ( divide_divide @ A @ ( real_Vector_of_real @ A @ pi ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% minus_sin_cos_eq
thf(fact_3374_floor__log__eq__powr__iff,axiom,
! [X: real,B2: real,K2: int] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( one_one @ real ) @ B2 )
=> ( ( ( archim6421214686448440834_floor @ real @ ( log @ B2 @ X ) )
= K2 )
= ( ( ord_less_eq @ real @ ( powr @ real @ B2 @ ( ring_1_of_int @ real @ K2 ) ) @ X )
& ( ord_less @ real @ X @ ( powr @ real @ B2 @ ( ring_1_of_int @ real @ ( plus_plus @ int @ K2 @ ( one_one @ int ) ) ) ) ) ) ) ) ) ).
% floor_log_eq_powr_iff
thf(fact_3375_monoseq__minus,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: nat > A] :
( ( topological_monoseq @ A @ A2 )
=> ( topological_monoseq @ A
@ ^ [N2: nat] : ( uminus_uminus @ A @ ( A2 @ N2 ) ) ) ) ) ).
% monoseq_minus
thf(fact_3376_floor__log2__div2,axiom,
! [N: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( archim6421214686448440834_floor @ real @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ N ) ) )
= ( plus_plus @ int @ ( archim6421214686448440834_floor @ real @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( one_one @ int ) ) ) ) ).
% floor_log2_div2
thf(fact_3377_floor__log__nat__eq__if,axiom,
! [B2: nat,N: nat,K2: nat] :
( ( ord_less_eq @ nat @ ( power_power @ nat @ B2 @ N ) @ K2 )
=> ( ( ord_less @ nat @ K2 @ ( power_power @ nat @ B2 @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ B2 )
=> ( ( archim6421214686448440834_floor @ real @ ( log @ ( semiring_1_of_nat @ real @ B2 ) @ ( semiring_1_of_nat @ real @ K2 ) ) )
= ( semiring_1_of_nat @ int @ N ) ) ) ) ) ).
% floor_log_nat_eq_if
thf(fact_3378_mono__SucI1,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X9: nat > A] :
( ! [N3: nat] : ( ord_less_eq @ A @ ( X9 @ N3 ) @ ( X9 @ ( suc @ N3 ) ) )
=> ( topological_monoseq @ A @ X9 ) ) ) ).
% mono_SucI1
thf(fact_3379_mono__SucI2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X9: nat > A] :
( ! [N3: nat] : ( ord_less_eq @ A @ ( X9 @ ( suc @ N3 ) ) @ ( X9 @ N3 ) )
=> ( topological_monoseq @ A @ X9 ) ) ) ).
% mono_SucI2
thf(fact_3380_monoseq__Suc,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( topological_monoseq @ A )
= ( ^ [X6: nat > A] :
( ! [N2: nat] : ( ord_less_eq @ A @ ( X6 @ N2 ) @ ( X6 @ ( suc @ N2 ) ) )
| ! [N2: nat] : ( ord_less_eq @ A @ ( X6 @ ( suc @ N2 ) ) @ ( X6 @ N2 ) ) ) ) ) ) ).
% monoseq_Suc
thf(fact_3381_monoI1,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X9: nat > A] :
( ! [M4: nat,N3: nat] :
( ( ord_less_eq @ nat @ M4 @ N3 )
=> ( ord_less_eq @ A @ ( X9 @ M4 ) @ ( X9 @ N3 ) ) )
=> ( topological_monoseq @ A @ X9 ) ) ) ).
% monoI1
thf(fact_3382_monoI2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X9: nat > A] :
( ! [M4: nat,N3: nat] :
( ( ord_less_eq @ nat @ M4 @ N3 )
=> ( ord_less_eq @ A @ ( X9 @ N3 ) @ ( X9 @ M4 ) ) )
=> ( topological_monoseq @ A @ X9 ) ) ) ).
% monoI2
thf(fact_3383_diffs__equiv,axiom,
! [A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( ring_1 @ A ) )
=> ! [C2: nat > A,X: A] :
( ( summable @ A
@ ^ [N2: nat] : ( times_times @ A @ ( diffs @ A @ C2 @ N2 ) @ ( power_power @ A @ X @ N2 ) ) )
=> ( sums @ A
@ ^ [N2: nat] : ( times_times @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( C2 @ N2 ) ) @ ( power_power @ A @ X @ ( minus_minus @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) ) ) )
@ ( suminf @ A
@ ^ [N2: nat] : ( times_times @ A @ ( diffs @ A @ C2 @ N2 ) @ ( power_power @ A @ X @ N2 ) ) ) ) ) ) ).
% diffs_equiv
thf(fact_3384_round__altdef,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archimedean_round @ A )
= ( ^ [X4: A] : ( if @ int @ ( ord_less_eq @ A @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( archimedean_frac @ A @ X4 ) ) @ ( archimedean_ceiling @ A @ X4 ) @ ( archim6421214686448440834_floor @ A @ X4 ) ) ) ) ) ).
% round_altdef
thf(fact_3385_sin__paired,axiom,
! [X: real] :
( sums @ real
@ ^ [N2: nat] : ( times_times @ real @ ( divide_divide @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ N2 ) @ ( semiring_char_0_fact @ real @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) @ ( one_one @ nat ) ) ) ) @ ( power_power @ real @ X @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) @ ( one_one @ nat ) ) ) )
@ ( sin @ real @ X ) ) ).
% sin_paired
thf(fact_3386_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_3387_of__nat__code,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( ( semiring_1_of_nat @ A )
= ( ^ [N2: nat] :
( semiri8178284476397505188at_aux @ A
@ ^ [I: A] : ( plus_plus @ A @ I @ ( one_one @ A ) )
@ N2
@ ( zero_zero @ A ) ) ) ) ) ).
% of_nat_code
thf(fact_3388_pochhammer__1,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A2: A] :
( ( comm_s3205402744901411588hammer @ A @ A2 @ ( one_one @ nat ) )
= A2 ) ) ).
% pochhammer_1
thf(fact_3389_fact__0,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ( ( semiring_char_0_fact @ A @ ( zero_zero @ nat ) )
= ( one_one @ A ) ) ) ).
% fact_0
thf(fact_3390_fact__1,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ( ( semiring_char_0_fact @ A @ ( one_one @ nat ) )
= ( one_one @ A ) ) ) ).
% fact_1
thf(fact_3391_pochhammer__0,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A2: A] :
( ( comm_s3205402744901411588hammer @ A @ A2 @ ( zero_zero @ nat ) )
= ( one_one @ A ) ) ) ).
% pochhammer_0
thf(fact_3392_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_3393_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_3394_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_3395_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_3396_diffs__of__real,axiom,
! [A: $tType] :
( ( real_V2191834092415804123ebra_1 @ A )
=> ! [F3: nat > real] :
( ( diffs @ A
@ ^ [N2: nat] : ( real_Vector_of_real @ A @ ( F3 @ N2 ) ) )
= ( ^ [N2: nat] : ( real_Vector_of_real @ A @ ( diffs @ real @ F3 @ N2 ) ) ) ) ) ).
% diffs_of_real
thf(fact_3397_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_3398_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_3399_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_3400_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_3401_diffs__minus,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [C2: nat > A] :
( ( diffs @ A
@ ^ [N2: nat] : ( uminus_uminus @ A @ ( C2 @ N2 ) ) )
= ( ^ [N2: nat] : ( uminus_uminus @ A @ ( diffs @ A @ C2 @ N2 ) ) ) ) ) ).
% diffs_minus
thf(fact_3402_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_3403_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_3404_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_3405_pochhammer__neq__0__mono,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A2: A,M: nat,N: nat] :
( ( ( comm_s3205402744901411588hammer @ A @ A2 @ M )
!= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ nat @ N @ M )
=> ( ( comm_s3205402744901411588hammer @ A @ A2 @ N )
!= ( zero_zero @ A ) ) ) ) ) ).
% pochhammer_neq_0_mono
thf(fact_3406_pochhammer__eq__0__mono,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A2: A,N: nat,M: nat] :
( ( ( comm_s3205402744901411588hammer @ A @ A2 @ N )
= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ nat @ N @ M )
=> ( ( comm_s3205402744901411588hammer @ A @ A2 @ M )
= ( zero_zero @ A ) ) ) ) ) ).
% pochhammer_eq_0_mono
thf(fact_3407_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_3408_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_3409_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_3410_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_3411_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_3412_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_3413_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_3414_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_3415_of__nat__aux_Osimps_I2_J,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [Inc: A > A,N: nat,I2: A] :
( ( semiri8178284476397505188at_aux @ A @ Inc @ ( suc @ N ) @ I2 )
= ( semiri8178284476397505188at_aux @ A @ Inc @ N @ ( Inc @ I2 ) ) ) ) ).
% of_nat_aux.simps(2)
thf(fact_3416_of__nat__aux_Osimps_I1_J,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [Inc: A > A,I2: A] :
( ( semiri8178284476397505188at_aux @ A @ Inc @ ( zero_zero @ nat ) @ I2 )
= I2 ) ) ).
% of_nat_aux.simps(1)
thf(fact_3417_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_3418_pochhammer__rec,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A2: A,N: nat] :
( ( comm_s3205402744901411588hammer @ A @ A2 @ ( suc @ N ) )
= ( times_times @ A @ A2 @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ A2 @ ( one_one @ A ) ) @ N ) ) ) ) ).
% pochhammer_rec
thf(fact_3419_diffs__def,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( ( diffs @ A )
= ( ^ [C3: nat > A,N2: nat] : ( times_times @ A @ ( semiring_1_of_nat @ A @ ( suc @ N2 ) ) @ ( C3 @ ( suc @ N2 ) ) ) ) ) ) ).
% diffs_def
thf(fact_3420_pochhammer__Suc,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A2: A,N: nat] :
( ( comm_s3205402744901411588hammer @ A @ A2 @ ( suc @ N ) )
= ( times_times @ A @ ( comm_s3205402744901411588hammer @ A @ A2 @ N ) @ ( plus_plus @ A @ A2 @ ( semiring_1_of_nat @ A @ N ) ) ) ) ) ).
% pochhammer_Suc
thf(fact_3421_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_3422_pochhammer__eq__0__iff,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A2: A,N: nat] :
( ( ( comm_s3205402744901411588hammer @ A @ A2 @ N )
= ( zero_zero @ A ) )
= ( ? [K3: nat] :
( ( ord_less @ nat @ K3 @ N )
& ( A2
= ( uminus_uminus @ A @ ( semiring_1_of_nat @ A @ K3 ) ) ) ) ) ) ) ).
% pochhammer_eq_0_iff
thf(fact_3423_pochhammer__of__nat__eq__0__iff,axiom,
! [A: $tType] :
( ( ( ring_char_0 @ A )
& ( idom @ A ) )
=> ! [N: nat,K2: nat] :
( ( ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ ( semiring_1_of_nat @ A @ N ) ) @ K2 )
= ( zero_zero @ A ) )
= ( ord_less @ nat @ N @ K2 ) ) ) ).
% pochhammer_of_nat_eq_0_iff
thf(fact_3424_pochhammer__of__nat__eq__0__lemma,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [N: nat,K2: nat] :
( ( ord_less @ nat @ N @ K2 )
=> ( ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ ( semiring_1_of_nat @ A @ N ) ) @ K2 )
= ( zero_zero @ A ) ) ) ) ).
% pochhammer_of_nat_eq_0_lemma
thf(fact_3425_fact__numeral,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [K2: num] :
( ( semiring_char_0_fact @ A @ ( numeral_numeral @ nat @ K2 ) )
= ( times_times @ A @ ( numeral_numeral @ A @ K2 ) @ ( semiring_char_0_fact @ A @ ( pred_numeral @ K2 ) ) ) ) ) ).
% fact_numeral
thf(fact_3426_pochhammer__of__nat__eq__0__lemma_H,axiom,
! [A: $tType] :
( ( ( ring_char_0 @ A )
& ( idom @ A ) )
=> ! [K2: nat,N: nat] :
( ( ord_less_eq @ nat @ K2 @ N )
=> ( ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ ( semiring_1_of_nat @ A @ N ) ) @ K2 )
!= ( zero_zero @ A ) ) ) ) ).
% pochhammer_of_nat_eq_0_lemma'
thf(fact_3427_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_3428_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_3429_termdiff__converges__all,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [C2: nat > A,X: A] :
( ! [X5: A] :
( summable @ A
@ ^ [N2: nat] : ( times_times @ A @ ( C2 @ N2 ) @ ( power_power @ A @ X5 @ N2 ) ) )
=> ( summable @ A
@ ^ [N2: nat] : ( times_times @ A @ ( diffs @ A @ C2 @ N2 ) @ ( power_power @ A @ X @ N2 ) ) ) ) ) ).
% termdiff_converges_all
thf(fact_3430_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_3431_frac__add,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Y2: A] :
( ( ( ord_less @ A @ ( plus_plus @ A @ ( archimedean_frac @ A @ X ) @ ( archimedean_frac @ A @ Y2 ) ) @ ( one_one @ A ) )
=> ( ( archimedean_frac @ A @ ( plus_plus @ A @ X @ Y2 ) )
= ( plus_plus @ A @ ( archimedean_frac @ A @ X ) @ ( archimedean_frac @ A @ Y2 ) ) ) )
& ( ~ ( ord_less @ A @ ( plus_plus @ A @ ( archimedean_frac @ A @ X ) @ ( archimedean_frac @ A @ Y2 ) ) @ ( one_one @ A ) )
=> ( ( archimedean_frac @ A @ ( plus_plus @ A @ X @ Y2 ) )
= ( minus_minus @ A @ ( plus_plus @ A @ ( archimedean_frac @ A @ X ) @ ( archimedean_frac @ A @ Y2 ) ) @ ( one_one @ A ) ) ) ) ) ) ).
% frac_add
thf(fact_3432_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_3433_square__fact__le__2__fact,axiom,
! [N: nat] : ( ord_less_eq @ real @ ( times_times @ real @ ( semiring_char_0_fact @ real @ N ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( semiring_char_0_fact @ real @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ).
% square_fact_le_2_fact
thf(fact_3434_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_3435_fact__code,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ( ( semiring_char_0_fact @ A )
= ( ^ [N2: nat] : ( semiring_1_of_nat @ A @ ( set_fo6178422350223883121st_nat @ nat @ ( times_times @ nat ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 @ ( one_one @ nat ) ) ) ) ) ) ).
% fact_code
thf(fact_3436_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_3437_pochhammer__absorb__comp,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [R: A,K2: nat] :
( ( times_times @ A @ ( minus_minus @ A @ R @ ( semiring_1_of_nat @ A @ K2 ) ) @ ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ R ) @ K2 ) )
= ( times_times @ A @ R @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ ( uminus_uminus @ A @ R ) @ ( one_one @ A ) ) @ K2 ) ) ) ) ).
% pochhammer_absorb_comp
thf(fact_3438_pochhammer__minus,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [B2: A,K2: nat] :
( ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ B2 ) @ K2 )
= ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ K2 ) @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ ( minus_minus @ A @ B2 @ ( semiring_1_of_nat @ A @ K2 ) ) @ ( one_one @ A ) ) @ K2 ) ) ) ) ).
% pochhammer_minus
thf(fact_3439_pochhammer__minus_H,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [B2: A,K2: nat] :
( ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ ( minus_minus @ A @ B2 @ ( semiring_1_of_nat @ A @ K2 ) ) @ ( one_one @ A ) ) @ K2 )
= ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ K2 ) @ ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ B2 ) @ K2 ) ) ) ) ).
% pochhammer_minus'
thf(fact_3440_termdiff__converges,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A,K5: real,C2: nat > A] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ K5 )
=> ( ! [X5: A] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X5 ) @ K5 )
=> ( summable @ A
@ ^ [N2: nat] : ( times_times @ A @ ( C2 @ N2 ) @ ( power_power @ A @ X5 @ N2 ) ) ) )
=> ( summable @ A
@ ^ [N2: nat] : ( times_times @ A @ ( diffs @ A @ C2 @ N2 ) @ ( power_power @ A @ X @ N2 ) ) ) ) ) ) ).
% termdiff_converges
thf(fact_3441_floor__add,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Y2: A] :
( ( ( ord_less @ A @ ( plus_plus @ A @ ( archimedean_frac @ A @ X ) @ ( archimedean_frac @ A @ Y2 ) ) @ ( one_one @ A ) )
=> ( ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X @ Y2 ) )
= ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( archim6421214686448440834_floor @ A @ Y2 ) ) ) )
& ( ~ ( ord_less @ A @ ( plus_plus @ A @ ( archimedean_frac @ A @ X ) @ ( archimedean_frac @ A @ Y2 ) ) @ ( one_one @ A ) )
=> ( ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X @ Y2 ) )
= ( plus_plus @ int @ ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( archim6421214686448440834_floor @ A @ Y2 ) ) @ ( one_one @ int ) ) ) ) ) ) ).
% floor_add
thf(fact_3442_pochhammer__code,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ( ( comm_s3205402744901411588hammer @ A )
= ( ^ [A5: A,N2: nat] :
( if @ A
@ ( N2
= ( zero_zero @ nat ) )
@ ( one_one @ A )
@ ( set_fo6178422350223883121st_nat @ A
@ ^ [O: nat] : ( times_times @ A @ ( plus_plus @ A @ A5 @ ( semiring_1_of_nat @ A @ O ) ) )
@ ( zero_zero @ nat )
@ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) )
@ ( one_one @ A ) ) ) ) ) ) ).
% pochhammer_code
thf(fact_3443_cos__paired,axiom,
! [X: real] :
( sums @ real
@ ^ [N2: nat] : ( times_times @ real @ ( divide_divide @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ N2 ) @ ( semiring_char_0_fact @ real @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) @ ( power_power @ real @ X @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) )
@ ( cos @ real @ X ) ) ).
% cos_paired
thf(fact_3444_sin__coeff__def,axiom,
( sin_coeff
= ( ^ [N2: nat] : ( if @ real @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) @ ( zero_zero @ real ) @ ( divide_divide @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ ( divide_divide @ nat @ ( minus_minus @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( semiring_char_0_fact @ real @ N2 ) ) ) ) ) ).
% sin_coeff_def
thf(fact_3445_choose__dvd,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [K2: nat,N: nat] :
( ( ord_less_eq @ nat @ K2 @ N )
=> ( dvd_dvd @ A @ ( times_times @ A @ ( semiring_char_0_fact @ A @ K2 ) @ ( semiring_char_0_fact @ A @ ( minus_minus @ nat @ N @ K2 ) ) ) @ ( semiring_char_0_fact @ A @ N ) ) ) ) ).
% choose_dvd
thf(fact_3446_cos__coeff__def,axiom,
( cos_coeff
= ( ^ [N2: nat] : ( if @ real @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) @ ( divide_divide @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( semiring_char_0_fact @ real @ N2 ) ) @ ( zero_zero @ real ) ) ) ) ).
% cos_coeff_def
thf(fact_3447_fact__fact__dvd__fact,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [K2: nat,N: nat] : ( dvd_dvd @ A @ ( times_times @ A @ ( semiring_char_0_fact @ A @ K2 ) @ ( semiring_char_0_fact @ A @ N ) ) @ ( semiring_char_0_fact @ A @ ( plus_plus @ nat @ K2 @ N ) ) ) ) ).
% fact_fact_dvd_fact
thf(fact_3448_central__binomial__lower__bound,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ord_less_eq @ real @ ( divide_divide @ real @ ( power_power @ real @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ one2 ) ) ) @ N ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ N ) ) ) @ ( semiring_1_of_nat @ real @ ( binomial @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ N ) ) ) ) ).
% central_binomial_lower_bound
thf(fact_3449_binomial__n__n,axiom,
! [N: nat] :
( ( binomial @ N @ N )
= ( one_one @ nat ) ) ).
% binomial_n_n
thf(fact_3450_binomial__eq__0__iff,axiom,
! [N: nat,K2: nat] :
( ( ( binomial @ N @ K2 )
= ( zero_zero @ nat ) )
= ( ord_less @ nat @ N @ K2 ) ) ).
% binomial_eq_0_iff
thf(fact_3451_binomial__Suc__Suc,axiom,
! [N: nat,K2: nat] :
( ( binomial @ ( suc @ N ) @ ( suc @ K2 ) )
= ( plus_plus @ nat @ ( binomial @ N @ K2 ) @ ( binomial @ N @ ( suc @ K2 ) ) ) ) ).
% binomial_Suc_Suc
thf(fact_3452_binomial__n__0,axiom,
! [N: nat] :
( ( binomial @ N @ ( zero_zero @ nat ) )
= ( one_one @ nat ) ) ).
% binomial_n_0
thf(fact_3453_cos__coeff__0,axiom,
( ( cos_coeff @ ( zero_zero @ nat ) )
= ( one_one @ real ) ) ).
% cos_coeff_0
thf(fact_3454_zero__less__binomial__iff,axiom,
! [N: nat,K2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( binomial @ N @ K2 ) )
= ( ord_less_eq @ nat @ K2 @ N ) ) ).
% zero_less_binomial_iff
thf(fact_3455_binomial__fact__lemma,axiom,
! [K2: nat,N: nat] :
( ( ord_less_eq @ nat @ K2 @ N )
=> ( ( times_times @ nat @ ( times_times @ nat @ ( semiring_char_0_fact @ nat @ K2 ) @ ( semiring_char_0_fact @ nat @ ( minus_minus @ nat @ N @ K2 ) ) ) @ ( binomial @ N @ K2 ) )
= ( semiring_char_0_fact @ nat @ N ) ) ) ).
% binomial_fact_lemma
thf(fact_3456_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_3457_fact__ge__self,axiom,
! [N: nat] : ( ord_less_eq @ nat @ N @ ( semiring_char_0_fact @ nat @ N ) ) ).
% fact_ge_self
thf(fact_3458_binomial__eq__0,axiom,
! [N: nat,K2: nat] :
( ( ord_less @ nat @ N @ K2 )
=> ( ( binomial @ N @ K2 )
= ( zero_zero @ nat ) ) ) ).
% binomial_eq_0
thf(fact_3459_choose__one,axiom,
! [N: nat] :
( ( binomial @ N @ ( one_one @ nat ) )
= N ) ).
% choose_one
thf(fact_3460_Suc__times__binomial__eq,axiom,
! [N: nat,K2: nat] :
( ( times_times @ nat @ ( suc @ N ) @ ( binomial @ N @ K2 ) )
= ( times_times @ nat @ ( binomial @ ( suc @ N ) @ ( suc @ K2 ) ) @ ( suc @ K2 ) ) ) ).
% Suc_times_binomial_eq
thf(fact_3461_Suc__times__binomial,axiom,
! [K2: nat,N: nat] :
( ( times_times @ nat @ ( suc @ K2 ) @ ( binomial @ ( suc @ N ) @ ( suc @ K2 ) ) )
= ( times_times @ nat @ ( suc @ N ) @ ( binomial @ N @ K2 ) ) ) ).
% Suc_times_binomial
thf(fact_3462_binomial__symmetric,axiom,
! [K2: nat,N: nat] :
( ( ord_less_eq @ nat @ K2 @ N )
=> ( ( binomial @ N @ K2 )
= ( binomial @ N @ ( minus_minus @ nat @ N @ K2 ) ) ) ) ).
% binomial_symmetric
thf(fact_3463_choose__mult__lemma,axiom,
! [M: nat,R: nat,K2: nat] :
( ( times_times @ nat @ ( binomial @ ( plus_plus @ nat @ ( plus_plus @ nat @ M @ R ) @ K2 ) @ ( plus_plus @ nat @ M @ K2 ) ) @ ( binomial @ ( plus_plus @ nat @ M @ K2 ) @ K2 ) )
= ( times_times @ nat @ ( binomial @ ( plus_plus @ nat @ ( plus_plus @ nat @ M @ R ) @ K2 ) @ K2 ) @ ( binomial @ ( plus_plus @ nat @ M @ R ) @ M ) ) ) ).
% choose_mult_lemma
thf(fact_3464_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_3465_binomial__altdef__nat,axiom,
! [K2: nat,N: nat] :
( ( ord_less_eq @ nat @ K2 @ N )
=> ( ( binomial @ N @ K2 )
= ( divide_divide @ nat @ ( semiring_char_0_fact @ nat @ N ) @ ( times_times @ nat @ ( semiring_char_0_fact @ nat @ K2 ) @ ( semiring_char_0_fact @ nat @ ( minus_minus @ nat @ N @ K2 ) ) ) ) ) ) ).
% binomial_altdef_nat
thf(fact_3466_zero__less__binomial,axiom,
! [K2: nat,N: nat] :
( ( ord_less_eq @ nat @ K2 @ N )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ ( binomial @ N @ K2 ) ) ) ).
% zero_less_binomial
thf(fact_3467_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_3468_Suc__times__binomial__add,axiom,
! [A2: nat,B2: nat] :
( ( times_times @ nat @ ( suc @ A2 ) @ ( binomial @ ( suc @ ( plus_plus @ nat @ A2 @ B2 ) ) @ ( suc @ A2 ) ) )
= ( times_times @ nat @ ( suc @ B2 ) @ ( binomial @ ( suc @ ( plus_plus @ nat @ A2 @ B2 ) ) @ A2 ) ) ) ).
% Suc_times_binomial_add
thf(fact_3469_choose__mult,axiom,
! [K2: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ K2 @ M )
=> ( ( ord_less_eq @ nat @ M @ N )
=> ( ( times_times @ nat @ ( binomial @ N @ M ) @ ( binomial @ M @ K2 ) )
= ( times_times @ nat @ ( binomial @ N @ K2 ) @ ( binomial @ ( minus_minus @ nat @ N @ K2 ) @ ( minus_minus @ nat @ M @ K2 ) ) ) ) ) ) ).
% choose_mult
thf(fact_3470_binomial__Suc__Suc__eq__times,axiom,
! [N: nat,K2: nat] :
( ( binomial @ ( suc @ N ) @ ( suc @ K2 ) )
= ( divide_divide @ nat @ ( times_times @ nat @ ( suc @ N ) @ ( binomial @ N @ K2 ) ) @ ( suc @ K2 ) ) ) ).
% binomial_Suc_Suc_eq_times
thf(fact_3471_binomial__absorb__comp,axiom,
! [N: nat,K2: nat] :
( ( times_times @ nat @ ( minus_minus @ nat @ N @ K2 ) @ ( binomial @ N @ K2 ) )
= ( times_times @ nat @ N @ ( binomial @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ K2 ) ) ) ).
% binomial_absorb_comp
thf(fact_3472_binomial__absorption,axiom,
! [K2: nat,N: nat] :
( ( times_times @ nat @ ( suc @ K2 ) @ ( binomial @ N @ ( suc @ K2 ) ) )
= ( times_times @ nat @ N @ ( binomial @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ K2 ) ) ) ).
% binomial_absorption
thf(fact_3473_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_3474_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_3475_binomial__ge__n__over__k__pow__k,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [K2: nat,N: nat] :
( ( ord_less_eq @ nat @ K2 @ N )
=> ( ord_less_eq @ A @ ( power_power @ A @ ( divide_divide @ A @ ( semiring_1_of_nat @ A @ N ) @ ( semiring_1_of_nat @ A @ K2 ) ) @ K2 ) @ ( semiring_1_of_nat @ A @ ( binomial @ N @ K2 ) ) ) ) ) ).
% binomial_ge_n_over_k_pow_k
thf(fact_3476_binomial__le__pow2,axiom,
! [N: nat,K2: nat] : ( ord_less_eq @ nat @ ( binomial @ N @ K2 ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ).
% binomial_le_pow2
thf(fact_3477_choose__reduce__nat,axiom,
! [N: nat,K2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K2 )
=> ( ( binomial @ N @ K2 )
= ( plus_plus @ nat @ ( binomial @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ ( minus_minus @ nat @ K2 @ ( one_one @ nat ) ) ) @ ( binomial @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ K2 ) ) ) ) ) ).
% choose_reduce_nat
thf(fact_3478_times__binomial__minus1__eq,axiom,
! [K2: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K2 )
=> ( ( times_times @ nat @ K2 @ ( binomial @ N @ K2 ) )
= ( times_times @ nat @ N @ ( binomial @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ ( minus_minus @ nat @ K2 @ ( one_one @ nat ) ) ) ) ) ) ).
% times_binomial_minus1_eq
thf(fact_3479_fact__diff__Suc,axiom,
! [N: nat,M: nat] :
( ( ord_less @ nat @ N @ ( suc @ M ) )
=> ( ( semiring_char_0_fact @ nat @ ( minus_minus @ nat @ ( suc @ M ) @ N ) )
= ( times_times @ nat @ ( minus_minus @ nat @ ( suc @ M ) @ N ) @ ( semiring_char_0_fact @ nat @ ( minus_minus @ nat @ M @ N ) ) ) ) ) ).
% fact_diff_Suc
thf(fact_3480_binomial__code,axiom,
( binomial
= ( ^ [N2: nat,K3: nat] : ( if @ nat @ ( ord_less @ nat @ N2 @ K3 ) @ ( zero_zero @ nat ) @ ( if @ nat @ ( ord_less @ nat @ N2 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K3 ) ) @ ( binomial @ N2 @ ( minus_minus @ nat @ N2 @ K3 ) ) @ ( divide_divide @ nat @ ( set_fo6178422350223883121st_nat @ nat @ ( times_times @ nat ) @ ( plus_plus @ nat @ ( minus_minus @ nat @ N2 @ K3 ) @ ( one_one @ nat ) ) @ N2 @ ( one_one @ nat ) ) @ ( semiring_char_0_fact @ nat @ K3 ) ) ) ) ) ) ).
% binomial_code
thf(fact_3481_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_3482_binomial__addition__formula,axiom,
! [N: nat,K2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( binomial @ N @ ( suc @ K2 ) )
= ( plus_plus @ nat @ ( binomial @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ ( suc @ K2 ) ) @ ( binomial @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ K2 ) ) ) ) ).
% binomial_addition_formula
thf(fact_3483_fact__binomial,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K2: nat,N: nat] :
( ( ord_less_eq @ nat @ K2 @ N )
=> ( ( times_times @ A @ ( semiring_char_0_fact @ A @ K2 ) @ ( semiring_1_of_nat @ A @ ( binomial @ N @ K2 ) ) )
= ( divide_divide @ A @ ( semiring_char_0_fact @ A @ N ) @ ( semiring_char_0_fact @ A @ ( minus_minus @ nat @ N @ K2 ) ) ) ) ) ) ).
% fact_binomial
thf(fact_3484_binomial__fact,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K2: nat,N: nat] :
( ( ord_less_eq @ nat @ K2 @ N )
=> ( ( semiring_1_of_nat @ A @ ( binomial @ N @ K2 ) )
= ( divide_divide @ A @ ( semiring_char_0_fact @ A @ N ) @ ( times_times @ A @ ( semiring_char_0_fact @ A @ K2 ) @ ( semiring_char_0_fact @ A @ ( minus_minus @ nat @ N @ K2 ) ) ) ) ) ) ) ).
% binomial_fact
thf(fact_3485_binomial__mono,axiom,
! [K2: nat,K7: nat,N: nat] :
( ( ord_less_eq @ nat @ K2 @ K7 )
=> ( ( ord_less_eq @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K7 ) @ N )
=> ( ord_less_eq @ nat @ ( binomial @ N @ K2 ) @ ( binomial @ N @ K7 ) ) ) ) ).
% binomial_mono
thf(fact_3486_binomial__maximum_H,axiom,
! [N: nat,K2: nat] : ( ord_less_eq @ nat @ ( binomial @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ K2 ) @ ( binomial @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ N ) ) ).
% binomial_maximum'
thf(fact_3487_binomial__maximum,axiom,
! [N: nat,K2: nat] : ( ord_less_eq @ nat @ ( binomial @ N @ K2 ) @ ( binomial @ N @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% binomial_maximum
thf(fact_3488_binomial__antimono,axiom,
! [K2: nat,K7: nat,N: nat] :
( ( ord_less_eq @ nat @ K2 @ K7 )
=> ( ( ord_less_eq @ nat @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ K2 )
=> ( ( ord_less_eq @ nat @ K7 @ N )
=> ( ord_less_eq @ nat @ ( binomial @ N @ K7 ) @ ( binomial @ N @ K2 ) ) ) ) ) ).
% binomial_antimono
thf(fact_3489_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_3490_binomial__strict__mono,axiom,
! [K2: nat,K7: nat,N: nat] :
( ( ord_less @ nat @ K2 @ K7 )
=> ( ( ord_less_eq @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K7 ) @ N )
=> ( ord_less @ nat @ ( binomial @ N @ K2 ) @ ( binomial @ N @ K7 ) ) ) ) ).
% binomial_strict_mono
thf(fact_3491_binomial__strict__antimono,axiom,
! [K2: nat,K7: nat,N: nat] :
( ( ord_less @ nat @ K2 @ K7 )
=> ( ( ord_less_eq @ nat @ N @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K2 ) )
=> ( ( ord_less_eq @ nat @ K7 @ N )
=> ( ord_less @ nat @ ( binomial @ N @ K7 ) @ ( binomial @ N @ K2 ) ) ) ) ) ).
% binomial_strict_antimono
thf(fact_3492_binomial__less__binomial__Suc,axiom,
! [K2: nat,N: nat] :
( ( ord_less @ nat @ K2 @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
=> ( ord_less @ nat @ ( binomial @ N @ K2 ) @ ( binomial @ N @ ( suc @ K2 ) ) ) ) ).
% binomial_less_binomial_Suc
thf(fact_3493_central__binomial__odd,axiom,
! [N: nat] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( binomial @ N @ ( suc @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
= ( binomial @ N @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% central_binomial_odd
thf(fact_3494_exp__two__pi__i_H,axiom,
( ( exp @ complex @ ( times_times @ complex @ imaginary_unit @ ( times_times @ complex @ ( real_Vector_of_real @ complex @ pi ) @ ( numeral_numeral @ complex @ ( bit0 @ one2 ) ) ) ) )
= ( one_one @ complex ) ) ).
% exp_two_pi_i'
thf(fact_3495_exp__two__pi__i,axiom,
( ( exp @ complex @ ( times_times @ complex @ ( times_times @ complex @ ( numeral_numeral @ complex @ ( bit0 @ one2 ) ) @ ( real_Vector_of_real @ complex @ pi ) ) @ imaginary_unit ) )
= ( one_one @ complex ) ) ).
% exp_two_pi_i
thf(fact_3496_cot__less__zero,axiom,
! [X: real] :
( ( ord_less @ real @ ( divide_divide @ real @ ( uminus_uminus @ real @ pi ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ X )
=> ( ( ord_less @ real @ X @ ( zero_zero @ real ) )
=> ( ord_less @ real @ ( cot @ real @ X ) @ ( zero_zero @ real ) ) ) ) ).
% cot_less_zero
thf(fact_3497_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_3498_cot__periodic,axiom,
! [X: real] :
( ( cot @ real @ ( plus_plus @ real @ X @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) ) )
= ( cot @ real @ X ) ) ).
% cot_periodic
thf(fact_3499_gbinomial__1,axiom,
! [A: $tType] :
( ( ( semiring_char_0 @ A )
& ( semidom_divide @ A ) )
=> ! [A2: A] :
( ( gbinomial @ A @ A2 @ ( one_one @ nat ) )
= A2 ) ) ).
% gbinomial_1
thf(fact_3500_gbinomial__0_I1_J,axiom,
! [A: $tType] :
( ( ( semiring_char_0 @ A )
& ( semidom_divide @ A ) )
=> ! [A2: A] :
( ( gbinomial @ A @ A2 @ ( zero_zero @ nat ) )
= ( one_one @ A ) ) ) ).
% gbinomial_0(1)
thf(fact_3501_norm__ii,axiom,
( ( real_V7770717601297561774m_norm @ complex @ imaginary_unit )
= ( one_one @ real ) ) ).
% norm_ii
thf(fact_3502_divide__i,axiom,
! [X: complex] :
( ( divide_divide @ complex @ X @ imaginary_unit )
= ( times_times @ complex @ ( uminus_uminus @ complex @ imaginary_unit ) @ X ) ) ).
% divide_i
thf(fact_3503_complex__i__mult__minus,axiom,
! [X: complex] :
( ( times_times @ complex @ imaginary_unit @ ( times_times @ complex @ imaginary_unit @ X ) )
= ( uminus_uminus @ complex @ X ) ) ).
% complex_i_mult_minus
thf(fact_3504_i__squared,axiom,
( ( times_times @ complex @ imaginary_unit @ imaginary_unit )
= ( uminus_uminus @ complex @ ( one_one @ complex ) ) ) ).
% i_squared
thf(fact_3505_divide__numeral__i,axiom,
! [Z3: complex,N: num] :
( ( divide_divide @ complex @ Z3 @ ( times_times @ complex @ ( numeral_numeral @ complex @ N ) @ imaginary_unit ) )
= ( divide_divide @ complex @ ( uminus_uminus @ complex @ ( times_times @ complex @ imaginary_unit @ Z3 ) ) @ ( numeral_numeral @ complex @ N ) ) ) ).
% divide_numeral_i
thf(fact_3506_cot__npi,axiom,
! [N: nat] :
( ( cot @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ pi ) )
= ( zero_zero @ real ) ) ).
% cot_npi
thf(fact_3507_power2__i,axiom,
( ( power_power @ complex @ imaginary_unit @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( uminus_uminus @ complex @ ( one_one @ complex ) ) ) ).
% power2_i
thf(fact_3508_i__even__power,axiom,
! [N: nat] :
( ( power_power @ complex @ imaginary_unit @ ( times_times @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( power_power @ complex @ ( uminus_uminus @ complex @ ( one_one @ complex ) ) @ N ) ) ).
% i_even_power
thf(fact_3509_exp__pi__i,axiom,
( ( exp @ complex @ ( times_times @ complex @ ( real_Vector_of_real @ complex @ pi ) @ imaginary_unit ) )
= ( uminus_uminus @ complex @ ( one_one @ complex ) ) ) ).
% exp_pi_i
thf(fact_3510_exp__pi__i_H,axiom,
( ( exp @ complex @ ( times_times @ complex @ imaginary_unit @ ( real_Vector_of_real @ complex @ pi ) ) )
= ( uminus_uminus @ complex @ ( one_one @ complex ) ) ) ).
% exp_pi_i'
thf(fact_3511_complex__i__not__one,axiom,
( imaginary_unit
!= ( one_one @ complex ) ) ).
% complex_i_not_one
thf(fact_3512_i__times__eq__iff,axiom,
! [W: complex,Z3: complex] :
( ( ( times_times @ complex @ imaginary_unit @ W )
= Z3 )
= ( W
= ( uminus_uminus @ complex @ ( times_times @ complex @ imaginary_unit @ Z3 ) ) ) ) ).
% i_times_eq_iff
thf(fact_3513_gbinomial__Suc__Suc,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A2: A,K2: nat] :
( ( gbinomial @ A @ ( plus_plus @ A @ A2 @ ( one_one @ A ) ) @ ( suc @ K2 ) )
= ( plus_plus @ A @ ( gbinomial @ A @ A2 @ K2 ) @ ( gbinomial @ A @ A2 @ ( suc @ K2 ) ) ) ) ) ).
% gbinomial_Suc_Suc
thf(fact_3514_gbinomial__of__nat__symmetric,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K2: nat,N: nat] :
( ( ord_less_eq @ nat @ K2 @ N )
=> ( ( gbinomial @ A @ ( semiring_1_of_nat @ A @ N ) @ K2 )
= ( gbinomial @ A @ ( semiring_1_of_nat @ A @ N ) @ ( minus_minus @ nat @ N @ K2 ) ) ) ) ) ).
% gbinomial_of_nat_symmetric
thf(fact_3515_imaginary__unit_Ocode,axiom,
( imaginary_unit
= ( complex2 @ ( zero_zero @ real ) @ ( one_one @ real ) ) ) ).
% imaginary_unit.code
thf(fact_3516_Complex__eq__i,axiom,
! [X: real,Y2: real] :
( ( ( complex2 @ X @ Y2 )
= imaginary_unit )
= ( ( X
= ( zero_zero @ real ) )
& ( Y2
= ( one_one @ real ) ) ) ) ).
% Complex_eq_i
thf(fact_3517_i__mult__Complex,axiom,
! [A2: real,B2: real] :
( ( times_times @ complex @ imaginary_unit @ ( complex2 @ A2 @ B2 ) )
= ( complex2 @ ( uminus_uminus @ real @ B2 ) @ A2 ) ) ).
% i_mult_Complex
thf(fact_3518_Complex__mult__i,axiom,
! [A2: real,B2: real] :
( ( times_times @ complex @ ( complex2 @ A2 @ B2 ) @ imaginary_unit )
= ( complex2 @ ( uminus_uminus @ real @ B2 ) @ A2 ) ) ).
% Complex_mult_i
thf(fact_3519_gbinomial__addition__formula,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A2: A,K2: nat] :
( ( gbinomial @ A @ A2 @ ( suc @ K2 ) )
= ( plus_plus @ A @ ( gbinomial @ A @ ( minus_minus @ A @ A2 @ ( one_one @ A ) ) @ ( suc @ K2 ) ) @ ( gbinomial @ A @ ( minus_minus @ A @ A2 @ ( one_one @ A ) ) @ K2 ) ) ) ) ).
% gbinomial_addition_formula
thf(fact_3520_gbinomial__absorb__comp,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A2: A,K2: nat] :
( ( times_times @ A @ ( minus_minus @ A @ A2 @ ( semiring_1_of_nat @ A @ K2 ) ) @ ( gbinomial @ A @ A2 @ K2 ) )
= ( times_times @ A @ A2 @ ( gbinomial @ A @ ( minus_minus @ A @ A2 @ ( one_one @ A ) ) @ K2 ) ) ) ) ).
% gbinomial_absorb_comp
thf(fact_3521_gbinomial__ge__n__over__k__pow__k,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [K2: nat,A2: A] :
( ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ K2 ) @ A2 )
=> ( ord_less_eq @ A @ ( power_power @ A @ ( divide_divide @ A @ A2 @ ( semiring_1_of_nat @ A @ K2 ) ) @ K2 ) @ ( gbinomial @ A @ A2 @ K2 ) ) ) ) ).
% gbinomial_ge_n_over_k_pow_k
thf(fact_3522_gbinomial__mult__1,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A2: A,K2: nat] :
( ( times_times @ A @ A2 @ ( gbinomial @ A @ A2 @ K2 ) )
= ( plus_plus @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ K2 ) @ ( gbinomial @ A @ A2 @ K2 ) ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( suc @ K2 ) ) @ ( gbinomial @ A @ A2 @ ( suc @ K2 ) ) ) ) ) ) ).
% gbinomial_mult_1
thf(fact_3523_gbinomial__mult__1_H,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A2: A,K2: nat] :
( ( times_times @ A @ ( gbinomial @ A @ A2 @ K2 ) @ A2 )
= ( plus_plus @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ K2 ) @ ( gbinomial @ A @ A2 @ K2 ) ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( suc @ K2 ) ) @ ( gbinomial @ A @ A2 @ ( suc @ K2 ) ) ) ) ) ) ).
% gbinomial_mult_1'
thf(fact_3524_cot__def,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ( ( cot @ A )
= ( ^ [X4: A] : ( divide_divide @ A @ ( cos @ A @ X4 ) @ ( sin @ A @ X4 ) ) ) ) ) ).
% cot_def
thf(fact_3525_Suc__times__gbinomial,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K2: nat,A2: A] :
( ( times_times @ A @ ( semiring_1_of_nat @ A @ ( suc @ K2 ) ) @ ( gbinomial @ A @ ( plus_plus @ A @ A2 @ ( one_one @ A ) ) @ ( suc @ K2 ) ) )
= ( times_times @ A @ ( plus_plus @ A @ A2 @ ( one_one @ A ) ) @ ( gbinomial @ A @ A2 @ K2 ) ) ) ) ).
% Suc_times_gbinomial
thf(fact_3526_gbinomial__absorption,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K2: nat,A2: A] :
( ( times_times @ A @ ( semiring_1_of_nat @ A @ ( suc @ K2 ) ) @ ( gbinomial @ A @ A2 @ ( suc @ K2 ) ) )
= ( times_times @ A @ A2 @ ( gbinomial @ A @ ( minus_minus @ A @ A2 @ ( one_one @ A ) ) @ K2 ) ) ) ) ).
% gbinomial_absorption
thf(fact_3527_gbinomial__trinomial__revision,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K2: nat,M: nat,A2: A] :
( ( ord_less_eq @ nat @ K2 @ M )
=> ( ( times_times @ A @ ( gbinomial @ A @ A2 @ M ) @ ( gbinomial @ A @ ( semiring_1_of_nat @ A @ M ) @ K2 ) )
= ( times_times @ A @ ( gbinomial @ A @ A2 @ K2 ) @ ( gbinomial @ A @ ( minus_minus @ A @ A2 @ ( semiring_1_of_nat @ A @ K2 ) ) @ ( minus_minus @ nat @ M @ K2 ) ) ) ) ) ) ).
% gbinomial_trinomial_revision
thf(fact_3528_complex__of__real__i,axiom,
! [R: real] :
( ( times_times @ complex @ ( real_Vector_of_real @ complex @ R ) @ imaginary_unit )
= ( complex2 @ ( zero_zero @ real ) @ R ) ) ).
% complex_of_real_i
thf(fact_3529_i__complex__of__real,axiom,
! [R: real] :
( ( times_times @ complex @ imaginary_unit @ ( real_Vector_of_real @ complex @ R ) )
= ( complex2 @ ( zero_zero @ real ) @ R ) ) ).
% i_complex_of_real
thf(fact_3530_Complex__eq,axiom,
( complex2
= ( ^ [A5: real,B4: real] : ( plus_plus @ complex @ ( real_Vector_of_real @ complex @ A5 ) @ ( times_times @ complex @ imaginary_unit @ ( real_Vector_of_real @ complex @ B4 ) ) ) ) ) ).
% Complex_eq
thf(fact_3531_gbinomial__factors,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A2: A,K2: nat] :
( ( gbinomial @ A @ ( plus_plus @ A @ A2 @ ( one_one @ A ) ) @ ( suc @ K2 ) )
= ( times_times @ A @ ( divide_divide @ A @ ( plus_plus @ A @ A2 @ ( one_one @ A ) ) @ ( semiring_1_of_nat @ A @ ( suc @ K2 ) ) ) @ ( gbinomial @ A @ A2 @ K2 ) ) ) ) ).
% gbinomial_factors
thf(fact_3532_gbinomial__rec,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A2: A,K2: nat] :
( ( gbinomial @ A @ ( plus_plus @ A @ A2 @ ( one_one @ A ) ) @ ( suc @ K2 ) )
= ( times_times @ A @ ( gbinomial @ A @ A2 @ K2 ) @ ( divide_divide @ A @ ( plus_plus @ A @ A2 @ ( one_one @ A ) ) @ ( semiring_1_of_nat @ A @ ( suc @ K2 ) ) ) ) ) ) ).
% gbinomial_rec
thf(fact_3533_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_3534_gbinomial__index__swap,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K2: nat,N: nat] :
( ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ K2 ) @ ( gbinomial @ A @ ( minus_minus @ A @ ( uminus_uminus @ A @ ( semiring_1_of_nat @ A @ N ) ) @ ( one_one @ A ) ) @ K2 ) )
= ( 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 @ K2 ) ) @ ( one_one @ A ) ) @ N ) ) ) ) ).
% gbinomial_index_swap
thf(fact_3535_complex__split__polar,axiom,
! [Z3: complex] :
? [R3: real,A4: real] :
( Z3
= ( times_times @ complex @ ( real_Vector_of_real @ complex @ R3 ) @ ( plus_plus @ complex @ ( real_Vector_of_real @ complex @ ( cos @ real @ A4 ) ) @ ( times_times @ complex @ imaginary_unit @ ( real_Vector_of_real @ complex @ ( sin @ real @ A4 ) ) ) ) ) ) ).
% complex_split_polar
thf(fact_3536_gbinomial__minus,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A2: A,K2: nat] :
( ( gbinomial @ A @ ( uminus_uminus @ A @ A2 ) @ K2 )
= ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ K2 ) @ ( gbinomial @ A @ ( minus_minus @ A @ ( plus_plus @ A @ A2 @ ( semiring_1_of_nat @ A @ K2 ) ) @ ( one_one @ A ) ) @ K2 ) ) ) ) ).
% gbinomial_minus
thf(fact_3537_gbinomial__reduce__nat,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K2: nat,A2: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K2 )
=> ( ( gbinomial @ A @ A2 @ K2 )
= ( plus_plus @ A @ ( gbinomial @ A @ ( minus_minus @ A @ A2 @ ( one_one @ A ) ) @ ( minus_minus @ nat @ K2 @ ( one_one @ nat ) ) ) @ ( gbinomial @ A @ ( minus_minus @ A @ A2 @ ( one_one @ A ) ) @ K2 ) ) ) ) ) ).
% gbinomial_reduce_nat
thf(fact_3538_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_3539_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_3540_gbinomial__absorption_H,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K2: nat,A2: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K2 )
=> ( ( gbinomial @ A @ A2 @ K2 )
= ( times_times @ A @ ( divide_divide @ A @ A2 @ ( semiring_1_of_nat @ A @ K2 ) ) @ ( gbinomial @ A @ ( minus_minus @ A @ A2 @ ( one_one @ A ) ) @ ( minus_minus @ nat @ K2 @ ( one_one @ nat ) ) ) ) ) ) ) ).
% gbinomial_absorption'
thf(fact_3541_cmod__unit__one,axiom,
! [A2: real] :
( ( real_V7770717601297561774m_norm @ complex @ ( plus_plus @ complex @ ( real_Vector_of_real @ complex @ ( cos @ real @ A2 ) ) @ ( times_times @ complex @ imaginary_unit @ ( real_Vector_of_real @ complex @ ( sin @ real @ A2 ) ) ) ) )
= ( one_one @ real ) ) ).
% cmod_unit_one
thf(fact_3542_cmod__complex__polar,axiom,
! [R: real,A2: real] :
( ( real_V7770717601297561774m_norm @ complex @ ( times_times @ complex @ ( real_Vector_of_real @ complex @ R ) @ ( plus_plus @ complex @ ( real_Vector_of_real @ complex @ ( cos @ real @ A2 ) ) @ ( times_times @ complex @ imaginary_unit @ ( real_Vector_of_real @ complex @ ( sin @ real @ A2 ) ) ) ) ) )
= ( abs_abs @ real @ R ) ) ).
% cmod_complex_polar
thf(fact_3543_cot__gt__zero,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ X @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( cot @ real @ X ) ) ) ) ).
% cot_gt_zero
thf(fact_3544_tan__cot_H,axiom,
! [X: real] :
( ( tan @ real @ ( minus_minus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ X ) )
= ( cot @ real @ X ) ) ).
% tan_cot'
thf(fact_3545_Arg__minus__ii,axiom,
( ( arg @ ( uminus_uminus @ complex @ imaginary_unit ) )
= ( divide_divide @ real @ ( uminus_uminus @ real @ pi ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ).
% Arg_minus_ii
thf(fact_3546_csqrt__ii,axiom,
( ( csqrt @ imaginary_unit )
= ( divide_divide @ complex @ ( plus_plus @ complex @ ( one_one @ complex ) @ imaginary_unit ) @ ( real_Vector_of_real @ complex @ ( sqrt @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ).
% csqrt_ii
thf(fact_3547_Arg__ii,axiom,
( ( arg @ imaginary_unit )
= ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ).
% Arg_ii
thf(fact_3548_cis__minus__pi__half,axiom,
( ( cis @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) )
= ( uminus_uminus @ complex @ imaginary_unit ) ) ).
% cis_minus_pi_half
thf(fact_3549_arctan__def,axiom,
( arctan
= ( ^ [Y: real] :
( the @ real
@ ^ [X4: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X4 )
& ( ord_less @ real @ X4 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
& ( ( tan @ real @ X4 )
= Y ) ) ) ) ) ).
% arctan_def
thf(fact_3550_csqrt__eq__1,axiom,
! [Z3: complex] :
( ( ( csqrt @ Z3 )
= ( one_one @ complex ) )
= ( Z3
= ( one_one @ complex ) ) ) ).
% csqrt_eq_1
thf(fact_3551_csqrt__1,axiom,
( ( csqrt @ ( one_one @ complex ) )
= ( one_one @ complex ) ) ).
% csqrt_1
thf(fact_3552_norm__cis,axiom,
! [A2: real] :
( ( real_V7770717601297561774m_norm @ complex @ ( cis @ A2 ) )
= ( one_one @ real ) ) ).
% norm_cis
thf(fact_3553_cis__zero,axiom,
( ( cis @ ( zero_zero @ real ) )
= ( one_one @ complex ) ) ).
% cis_zero
thf(fact_3554_cis__pi,axiom,
( ( cis @ pi )
= ( uminus_uminus @ complex @ ( one_one @ complex ) ) ) ).
% cis_pi
thf(fact_3555_power2__csqrt,axiom,
! [Z3: complex] :
( ( power_power @ complex @ ( csqrt @ Z3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= Z3 ) ).
% power2_csqrt
thf(fact_3556_cis__pi__half,axiom,
( ( cis @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
= imaginary_unit ) ).
% cis_pi_half
thf(fact_3557_cis__2pi,axiom,
( ( cis @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) )
= ( one_one @ complex ) ) ).
% cis_2pi
thf(fact_3558_DeMoivre,axiom,
! [A2: real,N: nat] :
( ( power_power @ complex @ ( cis @ A2 ) @ N )
= ( cis @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ A2 ) ) ) ).
% DeMoivre
thf(fact_3559_cis__mult,axiom,
! [A2: real,B2: real] :
( ( times_times @ complex @ ( cis @ A2 ) @ ( cis @ B2 ) )
= ( cis @ ( plus_plus @ real @ A2 @ B2 ) ) ) ).
% cis_mult
thf(fact_3560_ln__real__def,axiom,
( ( ln_ln @ real )
= ( ^ [X4: real] :
( the @ real
@ ^ [U2: real] :
( ( exp @ real @ U2 )
= X4 ) ) ) ) ).
% ln_real_def
thf(fact_3561_suminf__def,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ( ( suminf @ A )
= ( ^ [F5: nat > A] : ( the @ A @ ( sums @ A @ F5 ) ) ) ) ) ).
% suminf_def
thf(fact_3562_ln__neg__is__const,axiom,
! [X: real] :
( ( ord_less_eq @ real @ X @ ( zero_zero @ real ) )
=> ( ( ln_ln @ real @ X )
= ( the @ real
@ ^ [X4: real] : $false ) ) ) ).
% ln_neg_is_const
thf(fact_3563_cis__conv__exp,axiom,
( cis
= ( ^ [B4: real] : ( exp @ complex @ ( times_times @ complex @ imaginary_unit @ ( real_Vector_of_real @ complex @ B4 ) ) ) ) ) ).
% cis_conv_exp
thf(fact_3564_arccos__def,axiom,
( arccos
= ( ^ [Y: real] :
( the @ real
@ ^ [X4: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X4 )
& ( ord_less_eq @ real @ X4 @ pi )
& ( ( cos @ real @ X4 )
= Y ) ) ) ) ) ).
% arccos_def
thf(fact_3565_of__real__sqrt,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( real_Vector_of_real @ complex @ ( sqrt @ X ) )
= ( csqrt @ ( real_Vector_of_real @ complex @ X ) ) ) ) ).
% of_real_sqrt
thf(fact_3566_Arg__bounded,axiom,
! [Z3: complex] :
( ( ord_less @ real @ ( uminus_uminus @ real @ pi ) @ ( arg @ Z3 ) )
& ( ord_less_eq @ real @ ( arg @ Z3 ) @ pi ) ) ).
% Arg_bounded
thf(fact_3567_pi__half,axiom,
( ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) )
= ( the @ real
@ ^ [X4: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X4 )
& ( ord_less_eq @ real @ X4 @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) )
& ( ( cos @ real @ X4 )
= ( zero_zero @ real ) ) ) ) ) ).
% pi_half
thf(fact_3568_pi__def,axiom,
( pi
= ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) )
@ ( the @ real
@ ^ [X4: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X4 )
& ( ord_less_eq @ real @ X4 @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) )
& ( ( cos @ real @ X4 )
= ( zero_zero @ real ) ) ) ) ) ) ).
% pi_def
thf(fact_3569_arcsin__def,axiom,
( arcsin
= ( ^ [Y: real] :
( the @ real
@ ^ [X4: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X4 )
& ( ord_less_eq @ real @ X4 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
& ( ( sin @ real @ X4 )
= Y ) ) ) ) ) ).
% arcsin_def
thf(fact_3570_rec__enat__def,axiom,
! [T: $tType] :
( ( extended_rec_enat @ T )
= ( ^ [F12: nat > T,F22: T,X4: extended_enat] : ( the @ T @ ( extend4933016492236175606t_enat @ T @ F12 @ F22 @ X4 ) ) ) ) ).
% rec_enat_def
thf(fact_3571_the__sym__eq__trivial,axiom,
! [A: $tType,X: A] :
( ( the @ A
@ ( ^ [Y5: A,Z2: A] : Y5 = Z2
@ X ) )
= X ) ).
% the_sym_eq_trivial
thf(fact_3572_the__eq__trivial,axiom,
! [A: $tType,A2: A] :
( ( the @ A
@ ^ [X4: A] : X4 = A2 )
= A2 ) ).
% the_eq_trivial
thf(fact_3573_the__equality,axiom,
! [A: $tType,P3: A > $o,A2: A] :
( ( P3 @ A2 )
=> ( ! [X5: A] :
( ( P3 @ X5 )
=> ( X5 = A2 ) )
=> ( ( the @ A @ P3 )
= A2 ) ) ) ).
% the_equality
thf(fact_3574_modulo__int__unfold,axiom,
! [L: int,K2: int,N: nat,M: nat] :
( ( ( ( ( sgn_sgn @ int @ L )
= ( zero_zero @ int ) )
| ( ( sgn_sgn @ int @ K2 )
= ( zero_zero @ int ) )
| ( N
= ( zero_zero @ nat ) ) )
=> ( ( modulo_modulo @ int @ ( times_times @ int @ ( sgn_sgn @ int @ K2 ) @ ( 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 @ K2 ) @ ( semiring_1_of_nat @ int @ M ) ) ) )
& ( ~ ( ( ( sgn_sgn @ int @ L )
= ( zero_zero @ int ) )
| ( ( sgn_sgn @ int @ K2 )
= ( zero_zero @ int ) )
| ( N
= ( zero_zero @ nat ) ) )
=> ( ( ( ( sgn_sgn @ int @ K2 )
= ( sgn_sgn @ int @ L ) )
=> ( ( modulo_modulo @ int @ ( times_times @ int @ ( sgn_sgn @ int @ K2 ) @ ( 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 @ K2 )
!= ( sgn_sgn @ int @ L ) )
=> ( ( modulo_modulo @ int @ ( times_times @ int @ ( sgn_sgn @ int @ K2 ) @ ( 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_3575_sgn__one,axiom,
! [A: $tType] :
( ( real_V2822296259951069270ebra_1 @ A )
=> ( ( sgn_sgn @ A @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% sgn_one
thf(fact_3576_sgn__1,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ( ( sgn_sgn @ A @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% sgn_1
thf(fact_3577_sgn__divide,axiom,
! [A: $tType] :
( ( field_abs_sgn @ A )
=> ! [A2: A,B2: A] :
( ( sgn_sgn @ A @ ( divide_divide @ A @ A2 @ B2 ) )
= ( divide_divide @ A @ ( sgn_sgn @ A @ A2 ) @ ( sgn_sgn @ A @ B2 ) ) ) ) ).
% sgn_divide
thf(fact_3578_power__sgn,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A,N: nat] :
( ( sgn_sgn @ A @ ( power_power @ A @ A2 @ N ) )
= ( power_power @ A @ ( sgn_sgn @ A @ A2 ) @ N ) ) ) ).
% power_sgn
thf(fact_3579_The__split__eq,axiom,
! [A: $tType,B: $tType,X: A,Y2: B] :
( ( the @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [X7: A,Y6: B] :
( ( X = X7 )
& ( Y2 = Y6 ) ) ) )
= ( product_Pair @ A @ B @ X @ Y2 ) ) ).
% The_split_eq
thf(fact_3580_sgn__less,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A] :
( ( ord_less @ A @ ( sgn_sgn @ A @ A2 ) @ ( zero_zero @ A ) )
= ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% sgn_less
thf(fact_3581_sgn__greater,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( sgn_sgn @ A @ A2 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% sgn_greater
thf(fact_3582_divide__sgn,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( divide_divide @ A @ A2 @ ( sgn_sgn @ A @ B2 ) )
= ( times_times @ A @ A2 @ ( sgn_sgn @ A @ B2 ) ) ) ) ).
% divide_sgn
thf(fact_3583_sgn__pos,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( sgn_sgn @ A @ A2 )
= ( one_one @ A ) ) ) ) ).
% sgn_pos
thf(fact_3584_abs__sgn__eq__1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( abs_abs @ A @ ( sgn_sgn @ A @ A2 ) )
= ( one_one @ A ) ) ) ) ).
% abs_sgn_eq_1
thf(fact_3585_sgn__mult__self__eq,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A] :
( ( times_times @ A @ ( sgn_sgn @ A @ A2 ) @ ( sgn_sgn @ A @ A2 ) )
= ( zero_neq_one_of_bool @ A
@ ( A2
!= ( zero_zero @ A ) ) ) ) ) ).
% sgn_mult_self_eq
thf(fact_3586_sgn__mult__dvd__iff,axiom,
! [R: int,L: int,K2: int] :
( ( dvd_dvd @ int @ ( times_times @ int @ ( sgn_sgn @ int @ R ) @ L ) @ K2 )
= ( ( dvd_dvd @ int @ L @ K2 )
& ( ( R
= ( zero_zero @ int ) )
=> ( K2
= ( zero_zero @ int ) ) ) ) ) ).
% sgn_mult_dvd_iff
thf(fact_3587_mult__sgn__dvd__iff,axiom,
! [L: int,R: int,K2: int] :
( ( dvd_dvd @ int @ ( times_times @ int @ L @ ( sgn_sgn @ int @ R ) ) @ K2 )
= ( ( dvd_dvd @ int @ L @ K2 )
& ( ( R
= ( zero_zero @ int ) )
=> ( K2
= ( zero_zero @ int ) ) ) ) ) ).
% mult_sgn_dvd_iff
thf(fact_3588_dvd__sgn__mult__iff,axiom,
! [L: int,R: int,K2: int] :
( ( dvd_dvd @ int @ L @ ( times_times @ int @ ( sgn_sgn @ int @ R ) @ K2 ) )
= ( ( dvd_dvd @ int @ L @ K2 )
| ( R
= ( zero_zero @ int ) ) ) ) ).
% dvd_sgn_mult_iff
thf(fact_3589_dvd__mult__sgn__iff,axiom,
! [L: int,K2: int,R: int] :
( ( dvd_dvd @ int @ L @ ( times_times @ int @ K2 @ ( sgn_sgn @ int @ R ) ) )
= ( ( dvd_dvd @ int @ L @ K2 )
| ( R
= ( zero_zero @ int ) ) ) ) ).
% dvd_mult_sgn_iff
thf(fact_3590_sgn__neg,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A] :
( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( sgn_sgn @ A @ A2 )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ).
% sgn_neg
thf(fact_3591_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_3592_Real__Vector__Spaces_Osgn__mult,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [X: A,Y2: A] :
( ( sgn_sgn @ A @ ( times_times @ A @ X @ Y2 ) )
= ( times_times @ A @ ( sgn_sgn @ A @ X ) @ ( sgn_sgn @ A @ Y2 ) ) ) ) ).
% Real_Vector_Spaces.sgn_mult
thf(fact_3593_sgn__mult,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ! [A2: A,B2: A] :
( ( sgn_sgn @ A @ ( times_times @ A @ A2 @ B2 ) )
= ( times_times @ A @ ( sgn_sgn @ A @ A2 ) @ ( sgn_sgn @ A @ B2 ) ) ) ) ).
% sgn_mult
thf(fact_3594_same__sgn__sgn__add,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [B2: A,A2: A] :
( ( ( sgn_sgn @ A @ B2 )
= ( sgn_sgn @ A @ A2 ) )
=> ( ( sgn_sgn @ A @ ( plus_plus @ A @ A2 @ B2 ) )
= ( sgn_sgn @ A @ A2 ) ) ) ) ).
% same_sgn_sgn_add
thf(fact_3595_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_3596_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_3597_sgn__mult__abs,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ! [A2: A] :
( ( times_times @ A @ ( sgn_sgn @ A @ A2 ) @ ( abs_abs @ A @ A2 ) )
= A2 ) ) ).
% sgn_mult_abs
thf(fact_3598_abs__mult__sgn,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ! [A2: A] :
( ( times_times @ A @ ( abs_abs @ A @ A2 ) @ ( sgn_sgn @ A @ A2 ) )
= A2 ) ) ).
% abs_mult_sgn
thf(fact_3599_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_3600_int__sgnE,axiom,
! [K2: int] :
~ ! [N3: nat,L2: int] :
( K2
!= ( times_times @ int @ ( sgn_sgn @ int @ L2 ) @ ( semiring_1_of_nat @ int @ N3 ) ) ) ).
% int_sgnE
thf(fact_3601_same__sgn__abs__add,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [B2: A,A2: A] :
( ( ( sgn_sgn @ A @ B2 )
= ( sgn_sgn @ A @ A2 ) )
=> ( ( abs_abs @ A @ ( plus_plus @ A @ A2 @ B2 ) )
= ( plus_plus @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B2 ) ) ) ) ) ).
% same_sgn_abs_add
thf(fact_3602_sgn__1__pos,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A] :
( ( ( sgn_sgn @ A @ A2 )
= ( one_one @ A ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% sgn_1_pos
thf(fact_3603_abs__sgn__eq,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A] :
( ( ( A2
= ( zero_zero @ A ) )
=> ( ( abs_abs @ A @ ( sgn_sgn @ A @ A2 ) )
= ( zero_zero @ A ) ) )
& ( ( A2
!= ( zero_zero @ A ) )
=> ( ( abs_abs @ A @ ( sgn_sgn @ A @ A2 ) )
= ( one_one @ A ) ) ) ) ) ).
% abs_sgn_eq
thf(fact_3604_sgn__mod,axiom,
! [L: int,K2: int] :
( ( L
!= ( zero_zero @ int ) )
=> ( ~ ( dvd_dvd @ int @ L @ K2 )
=> ( ( sgn_sgn @ int @ ( modulo_modulo @ int @ K2 @ L ) )
= ( sgn_sgn @ int @ L ) ) ) ) ).
% sgn_mod
thf(fact_3605_sgn__1__neg,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A] :
( ( ( sgn_sgn @ A @ A2 )
= ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% sgn_1_neg
thf(fact_3606_sgn__if,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( ( sgn_sgn @ A )
= ( ^ [X4: A] :
( if @ A
@ ( X4
= ( zero_zero @ A ) )
@ ( zero_zero @ A )
@ ( if @ A @ ( ord_less @ A @ ( zero_zero @ A ) @ X4 ) @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ) ) ).
% sgn_if
thf(fact_3607_zsgn__def,axiom,
( ( sgn_sgn @ int )
= ( ^ [I: int] :
( if @ int
@ ( I
= ( zero_zero @ int ) )
@ ( zero_zero @ int )
@ ( if @ int @ ( ord_less @ int @ ( zero_zero @ int ) @ I ) @ ( one_one @ int ) @ ( uminus_uminus @ int @ ( one_one @ int ) ) ) ) ) ) ).
% zsgn_def
thf(fact_3608_norm__sgn,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X: A] :
( ( ( X
= ( zero_zero @ A ) )
=> ( ( real_V7770717601297561774m_norm @ A @ ( sgn_sgn @ A @ X ) )
= ( zero_zero @ real ) ) )
& ( ( X
!= ( zero_zero @ A ) )
=> ( ( real_V7770717601297561774m_norm @ A @ ( sgn_sgn @ A @ X ) )
= ( one_one @ real ) ) ) ) ) ).
% norm_sgn
thf(fact_3609_div__sgn__abs__cancel,axiom,
! [V2: int,K2: int,L: int] :
( ( V2
!= ( zero_zero @ int ) )
=> ( ( divide_divide @ int @ ( times_times @ int @ ( sgn_sgn @ int @ V2 ) @ ( abs_abs @ int @ K2 ) ) @ ( times_times @ int @ ( sgn_sgn @ int @ V2 ) @ ( abs_abs @ int @ L ) ) )
= ( divide_divide @ int @ ( abs_abs @ int @ K2 ) @ ( abs_abs @ int @ L ) ) ) ) ).
% div_sgn_abs_cancel
thf(fact_3610_div__dvd__sgn__abs,axiom,
! [L: int,K2: int] :
( ( dvd_dvd @ int @ L @ K2 )
=> ( ( divide_divide @ int @ K2 @ L )
= ( times_times @ int @ ( times_times @ int @ ( sgn_sgn @ int @ K2 ) @ ( sgn_sgn @ int @ L ) ) @ ( divide_divide @ int @ ( abs_abs @ int @ K2 ) @ ( abs_abs @ int @ L ) ) ) ) ) ).
% div_dvd_sgn_abs
thf(fact_3611_theI,axiom,
! [A: $tType,P3: A > $o,A2: A] :
( ( P3 @ A2 )
=> ( ! [X5: A] :
( ( P3 @ X5 )
=> ( X5 = A2 ) )
=> ( P3 @ ( the @ A @ P3 ) ) ) ) ).
% theI
thf(fact_3612_theI_H,axiom,
! [A: $tType,P3: A > $o] :
( ? [X2: A] :
( ( P3 @ X2 )
& ! [Y7: A] :
( ( P3 @ Y7 )
=> ( Y7 = X2 ) ) )
=> ( P3 @ ( the @ A @ P3 ) ) ) ).
% theI'
thf(fact_3613_theI2,axiom,
! [A: $tType,P3: A > $o,A2: A,Q2: A > $o] :
( ( P3 @ A2 )
=> ( ! [X5: A] :
( ( P3 @ X5 )
=> ( X5 = A2 ) )
=> ( ! [X5: A] :
( ( P3 @ X5 )
=> ( Q2 @ X5 ) )
=> ( Q2 @ ( the @ A @ P3 ) ) ) ) ) ).
% theI2
thf(fact_3614_If__def,axiom,
! [A: $tType] :
( ( if @ A )
= ( ^ [P2: $o,X4: A,Y: A] :
( the @ A
@ ^ [Z6: A] :
( ( P2
=> ( Z6 = X4 ) )
& ( ~ P2
=> ( Z6 = Y ) ) ) ) ) ) ).
% If_def
thf(fact_3615_the1I2,axiom,
! [A: $tType,P3: A > $o,Q2: A > $o] :
( ? [X2: A] :
( ( P3 @ X2 )
& ! [Y7: A] :
( ( P3 @ Y7 )
=> ( Y7 = X2 ) ) )
=> ( ! [X5: A] :
( ( P3 @ X5 )
=> ( Q2 @ X5 ) )
=> ( Q2 @ ( the @ A @ P3 ) ) ) ) ).
% the1I2
thf(fact_3616_the1__equality,axiom,
! [A: $tType,P3: A > $o,A2: A] :
( ? [X2: A] :
( ( P3 @ X2 )
& ! [Y7: A] :
( ( P3 @ Y7 )
=> ( Y7 = X2 ) ) )
=> ( ( P3 @ A2 )
=> ( ( the @ A @ P3 )
= A2 ) ) ) ).
% the1_equality
thf(fact_3617_eucl__rel__int__remainderI,axiom,
! [R: int,L: int,K2: int,Q: int] :
( ( ( sgn_sgn @ int @ R )
= ( sgn_sgn @ int @ L ) )
=> ( ( ord_less @ int @ ( abs_abs @ int @ R ) @ ( abs_abs @ int @ L ) )
=> ( ( K2
= ( plus_plus @ int @ ( times_times @ int @ Q @ L ) @ R ) )
=> ( eucl_rel_int @ K2 @ L @ ( product_Pair @ int @ int @ Q @ R ) ) ) ) ) ).
% eucl_rel_int_remainderI
thf(fact_3618_eucl__rel__int_Osimps,axiom,
( eucl_rel_int
= ( ^ [A1: int,A22: int,A33: product_prod @ int @ int] :
( ? [K3: int] :
( ( A1 = K3 )
& ( A22
= ( zero_zero @ int ) )
& ( A33
= ( product_Pair @ int @ int @ ( zero_zero @ int ) @ K3 ) ) )
| ? [L3: int,K3: int,Q5: int] :
( ( A1 = K3 )
& ( A22 = L3 )
& ( A33
= ( product_Pair @ int @ int @ Q5 @ ( zero_zero @ int ) ) )
& ( L3
!= ( zero_zero @ int ) )
& ( K3
= ( times_times @ int @ Q5 @ L3 ) ) )
| ? [R5: int,L3: int,K3: int,Q5: int] :
( ( A1 = K3 )
& ( A22 = L3 )
& ( A33
= ( product_Pair @ int @ int @ Q5 @ 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 @ Q5 @ L3 ) @ R5 ) ) ) ) ) ) ).
% eucl_rel_int.simps
thf(fact_3619_eucl__rel__int_Ocases,axiom,
! [A12: int,A23: int,A32: product_prod @ int @ int] :
( ( eucl_rel_int @ A12 @ A23 @ A32 )
=> ( ( ( A23
= ( zero_zero @ int ) )
=> ( A32
!= ( product_Pair @ int @ int @ ( zero_zero @ int ) @ A12 ) ) )
=> ( ! [Q3: int] :
( ( A32
= ( product_Pair @ int @ int @ Q3 @ ( zero_zero @ int ) ) )
=> ( ( A23
!= ( zero_zero @ int ) )
=> ( A12
!= ( times_times @ int @ Q3 @ A23 ) ) ) )
=> ~ ! [R3: int,Q3: int] :
( ( A32
= ( product_Pair @ int @ int @ Q3 @ R3 ) )
=> ( ( ( sgn_sgn @ int @ R3 )
= ( sgn_sgn @ int @ A23 ) )
=> ( ( ord_less @ int @ ( abs_abs @ int @ R3 ) @ ( abs_abs @ int @ A23 ) )
=> ( A12
!= ( plus_plus @ int @ ( times_times @ int @ Q3 @ A23 ) @ R3 ) ) ) ) ) ) ) ) ).
% eucl_rel_int.cases
thf(fact_3620_floor__real__def,axiom,
( ( archim6421214686448440834_floor @ real )
= ( ^ [X4: real] :
( the @ int
@ ^ [Z6: int] :
( ( ord_less_eq @ real @ ( ring_1_of_int @ real @ Z6 ) @ X4 )
& ( ord_less @ real @ X4 @ ( ring_1_of_int @ real @ ( plus_plus @ int @ Z6 @ ( one_one @ int ) ) ) ) ) ) ) ) ).
% floor_real_def
thf(fact_3621_divide__int__unfold,axiom,
! [L: int,K2: int,N: nat,M: nat] :
( ( ( ( ( sgn_sgn @ int @ L )
= ( zero_zero @ int ) )
| ( ( sgn_sgn @ int @ K2 )
= ( zero_zero @ int ) )
| ( N
= ( zero_zero @ nat ) ) )
=> ( ( divide_divide @ int @ ( times_times @ int @ ( sgn_sgn @ int @ K2 ) @ ( 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 @ K2 )
= ( zero_zero @ int ) )
| ( N
= ( zero_zero @ nat ) ) )
=> ( ( ( ( sgn_sgn @ int @ K2 )
= ( sgn_sgn @ int @ L ) )
=> ( ( divide_divide @ int @ ( times_times @ int @ ( sgn_sgn @ int @ K2 ) @ ( 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 @ K2 )
!= ( sgn_sgn @ int @ L ) )
=> ( ( divide_divide @ int @ ( times_times @ int @ ( sgn_sgn @ int @ K2 ) @ ( 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_3622_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_3623_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_3624_powr__int,axiom,
! [X: real,I2: int] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ I2 )
=> ( ( powr @ real @ X @ ( ring_1_of_int @ real @ I2 ) )
= ( power_power @ real @ X @ ( nat2 @ I2 ) ) ) )
& ( ~ ( ord_less_eq @ int @ ( zero_zero @ int ) @ I2 )
=> ( ( powr @ real @ X @ ( ring_1_of_int @ real @ I2 ) )
= ( divide_divide @ real @ ( one_one @ real ) @ ( power_power @ real @ X @ ( nat2 @ ( uminus_uminus @ int @ I2 ) ) ) ) ) ) ) ) ).
% powr_int
thf(fact_3625_arctan__inverse,axiom,
! [X: real] :
( ( X
!= ( zero_zero @ real ) )
=> ( ( arctan @ ( divide_divide @ real @ ( one_one @ real ) @ X ) )
= ( minus_minus @ real @ ( divide_divide @ real @ ( times_times @ real @ ( sgn_sgn @ real @ X ) @ pi ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( arctan @ X ) ) ) ) ).
% arctan_inverse
thf(fact_3626_cis__multiple__2pi,axiom,
! [N: real] :
( ( member @ real @ N @ ( ring_1_Ints @ real ) )
=> ( ( cis @ ( times_times @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) @ N ) )
= ( one_one @ complex ) ) ) ).
% cis_multiple_2pi
thf(fact_3627_nat__numeral,axiom,
! [K2: num] :
( ( nat2 @ ( numeral_numeral @ int @ K2 ) )
= ( numeral_numeral @ nat @ K2 ) ) ).
% nat_numeral
thf(fact_3628_sgn__le__0__iff,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( sgn_sgn @ real @ X ) @ ( zero_zero @ real ) )
= ( ord_less_eq @ real @ X @ ( zero_zero @ real ) ) ) ).
% sgn_le_0_iff
thf(fact_3629_zero__le__sgn__iff,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( sgn_sgn @ real @ X ) )
= ( ord_less_eq @ real @ ( zero_zero @ real ) @ X ) ) ).
% zero_le_sgn_iff
thf(fact_3630_nat__1,axiom,
( ( nat2 @ ( one_one @ int ) )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% nat_1
thf(fact_3631_nat__0__iff,axiom,
! [I2: int] :
( ( ( nat2 @ I2 )
= ( zero_zero @ nat ) )
= ( ord_less_eq @ int @ I2 @ ( zero_zero @ int ) ) ) ).
% nat_0_iff
thf(fact_3632_nat__le__0,axiom,
! [Z3: int] :
( ( ord_less_eq @ int @ Z3 @ ( zero_zero @ int ) )
=> ( ( nat2 @ Z3 )
= ( zero_zero @ nat ) ) ) ).
% nat_le_0
thf(fact_3633_zless__nat__conj,axiom,
! [W: int,Z3: int] :
( ( ord_less @ nat @ ( nat2 @ W ) @ ( nat2 @ Z3 ) )
= ( ( ord_less @ int @ ( zero_zero @ int ) @ Z3 )
& ( ord_less @ int @ W @ Z3 ) ) ) ).
% zless_nat_conj
thf(fact_3634_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_3635_floor__add2,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Y2: A] :
( ( ( member @ A @ X @ ( ring_1_Ints @ A ) )
| ( member @ A @ Y2 @ ( ring_1_Ints @ A ) ) )
=> ( ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X @ Y2 ) )
= ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( archim6421214686448440834_floor @ A @ Y2 ) ) ) ) ) ).
% floor_add2
thf(fact_3636_of__nat__nat__take__bit__eq,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [N: nat,K2: int] :
( ( semiring_1_of_nat @ A @ ( nat2 @ ( bit_se2584673776208193580ke_bit @ int @ N @ K2 ) ) )
= ( ring_1_of_int @ A @ ( bit_se2584673776208193580ke_bit @ int @ N @ K2 ) ) ) ) ).
% of_nat_nat_take_bit_eq
thf(fact_3637_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_3638_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_3639_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_3640_diff__nat__numeral,axiom,
! [V2: num,V4: num] :
( ( minus_minus @ nat @ ( numeral_numeral @ nat @ V2 ) @ ( numeral_numeral @ nat @ V4 ) )
= ( nat2 @ ( minus_minus @ int @ ( numeral_numeral @ int @ V2 ) @ ( numeral_numeral @ int @ V4 ) ) ) ) ).
% diff_nat_numeral
thf(fact_3641_nat__eq__numeral__power__cancel__iff,axiom,
! [Y2: int,X: num,N: nat] :
( ( ( nat2 @ Y2 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ X ) @ N ) )
= ( Y2
= ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N ) ) ) ).
% nat_eq_numeral_power_cancel_iff
thf(fact_3642_numeral__power__eq__nat__cancel__iff,axiom,
! [X: num,N: nat,Y2: int] :
( ( ( power_power @ nat @ ( numeral_numeral @ nat @ X ) @ N )
= ( nat2 @ Y2 ) )
= ( ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N )
= Y2 ) ) ).
% numeral_power_eq_nat_cancel_iff
thf(fact_3643_nat__ceiling__le__eq,axiom,
! [X: real,A2: nat] :
( ( ord_less_eq @ nat @ ( nat2 @ ( archimedean_ceiling @ real @ X ) ) @ A2 )
= ( ord_less_eq @ real @ X @ ( semiring_1_of_nat @ real @ A2 ) ) ) ).
% nat_ceiling_le_eq
thf(fact_3644_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_3645_nat__numeral__diff__1,axiom,
! [V2: num] :
( ( minus_minus @ nat @ ( numeral_numeral @ nat @ V2 ) @ ( one_one @ nat ) )
= ( nat2 @ ( minus_minus @ int @ ( numeral_numeral @ int @ V2 ) @ ( one_one @ int ) ) ) ) ).
% nat_numeral_diff_1
thf(fact_3646_nat__less__numeral__power__cancel__iff,axiom,
! [A2: int,X: num,N: nat] :
( ( ord_less @ nat @ ( nat2 @ A2 ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ X ) @ N ) )
= ( ord_less @ int @ A2 @ ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N ) ) ) ).
% nat_less_numeral_power_cancel_iff
thf(fact_3647_numeral__power__less__nat__cancel__iff,axiom,
! [X: num,N: nat,A2: int] :
( ( ord_less @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ X ) @ N ) @ ( nat2 @ A2 ) )
= ( ord_less @ int @ ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N ) @ A2 ) ) ).
% numeral_power_less_nat_cancel_iff
thf(fact_3648_numeral__power__le__nat__cancel__iff,axiom,
! [X: num,N: nat,A2: int] :
( ( ord_less_eq @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ X ) @ N ) @ ( nat2 @ A2 ) )
= ( ord_less_eq @ int @ ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N ) @ A2 ) ) ).
% numeral_power_le_nat_cancel_iff
thf(fact_3649_nat__le__numeral__power__cancel__iff,axiom,
! [A2: int,X: num,N: nat] :
( ( ord_less_eq @ nat @ ( nat2 @ A2 ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ X ) @ N ) )
= ( ord_less_eq @ int @ A2 @ ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N ) ) ) ).
% nat_le_numeral_power_cancel_iff
thf(fact_3650_Ints__numeral,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [N: num] : ( member @ A @ ( numeral_numeral @ A @ N ) @ ( ring_1_Ints @ A ) ) ) ).
% Ints_numeral
thf(fact_3651_Ints__add,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [A2: A,B2: A] :
( ( member @ A @ A2 @ ( ring_1_Ints @ A ) )
=> ( ( member @ A @ B2 @ ( ring_1_Ints @ A ) )
=> ( member @ A @ ( plus_plus @ A @ A2 @ B2 ) @ ( ring_1_Ints @ A ) ) ) ) ) ).
% Ints_add
thf(fact_3652_Ints__1,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( member @ A @ ( one_one @ A ) @ ( ring_1_Ints @ A ) ) ) ).
% Ints_1
thf(fact_3653_Ints__mult,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [A2: A,B2: A] :
( ( member @ A @ A2 @ ( ring_1_Ints @ A ) )
=> ( ( member @ A @ B2 @ ( ring_1_Ints @ A ) )
=> ( member @ A @ ( times_times @ A @ A2 @ B2 ) @ ( ring_1_Ints @ A ) ) ) ) ) ).
% Ints_mult
thf(fact_3654_Ints__power,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [A2: A,N: nat] :
( ( member @ A @ A2 @ ( ring_1_Ints @ A ) )
=> ( member @ A @ ( power_power @ A @ A2 @ N ) @ ( ring_1_Ints @ A ) ) ) ) ).
% Ints_power
thf(fact_3655_nat__zero__as__int,axiom,
( ( zero_zero @ nat )
= ( nat2 @ ( zero_zero @ int ) ) ) ).
% nat_zero_as_int
thf(fact_3656_nat__numeral__as__int,axiom,
( ( numeral_numeral @ nat )
= ( ^ [I: num] : ( nat2 @ ( numeral_numeral @ int @ I ) ) ) ) ).
% nat_numeral_as_int
thf(fact_3657_nat__mono,axiom,
! [X: int,Y2: int] :
( ( ord_less_eq @ int @ X @ Y2 )
=> ( ord_less_eq @ nat @ ( nat2 @ X ) @ ( nat2 @ Y2 ) ) ) ).
% nat_mono
thf(fact_3658_eq__nat__nat__iff,axiom,
! [Z3: int,Z7: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z3 )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z7 )
=> ( ( ( nat2 @ Z3 )
= ( nat2 @ Z7 ) )
= ( Z3 = Z7 ) ) ) ) ).
% eq_nat_nat_iff
thf(fact_3659_all__nat,axiom,
( ( ^ [P: nat > $o] :
! [X3: nat] : ( P @ X3 ) )
= ( ^ [P2: nat > $o] :
! [X4: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X4 )
=> ( P2 @ ( nat2 @ X4 ) ) ) ) ) ).
% all_nat
thf(fact_3660_ex__nat,axiom,
( ( ^ [P: nat > $o] :
? [X3: nat] : ( P @ X3 ) )
= ( ^ [P2: nat > $o] :
? [X4: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X4 )
& ( P2 @ ( nat2 @ X4 ) ) ) ) ) ).
% ex_nat
thf(fact_3661_nat__one__as__int,axiom,
( ( one_one @ nat )
= ( nat2 @ ( one_one @ int ) ) ) ).
% nat_one_as_int
thf(fact_3662_Ints__double__eq__0__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [A2: A] :
( ( member @ A @ A2 @ ( ring_1_Ints @ A ) )
=> ( ( ( plus_plus @ A @ A2 @ A2 )
= ( zero_zero @ A ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ) ).
% Ints_double_eq_0_iff
thf(fact_3663_unset__bit__nat__def,axiom,
( ( bit_se2638667681897837118et_bit @ nat )
= ( ^ [M2: nat,N2: nat] : ( nat2 @ ( bit_se2638667681897837118et_bit @ int @ M2 @ ( semiring_1_of_nat @ int @ N2 ) ) ) ) ) ).
% unset_bit_nat_def
thf(fact_3664_nat__mask__eq,axiom,
! [N: nat] :
( ( nat2 @ ( bit_se2239418461657761734s_mask @ int @ N ) )
= ( bit_se2239418461657761734s_mask @ nat @ N ) ) ).
% nat_mask_eq
thf(fact_3665_nat__mono__iff,axiom,
! [Z3: int,W: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ Z3 )
=> ( ( ord_less @ nat @ ( nat2 @ W ) @ ( nat2 @ Z3 ) )
= ( ord_less @ int @ W @ Z3 ) ) ) ).
% nat_mono_iff
thf(fact_3666_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_3667_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_3668_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_3669_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_3670_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_3671_nat__int__add,axiom,
! [A2: nat,B2: nat] :
( ( nat2 @ ( plus_plus @ int @ ( semiring_1_of_nat @ int @ A2 ) @ ( semiring_1_of_nat @ int @ B2 ) ) )
= ( plus_plus @ nat @ A2 @ B2 ) ) ).
% nat_int_add
thf(fact_3672_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_3673_nat__abs__mult__distrib,axiom,
! [W: int,Z3: int] :
( ( nat2 @ ( abs_abs @ int @ ( times_times @ int @ W @ Z3 ) ) )
= ( times_times @ nat @ ( nat2 @ ( abs_abs @ int @ W ) ) @ ( nat2 @ ( abs_abs @ int @ Z3 ) ) ) ) ).
% nat_abs_mult_distrib
thf(fact_3674_Ints__odd__nonzero,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [A2: A] :
( ( member @ A @ A2 @ ( ring_1_Ints @ A ) )
=> ( ( plus_plus @ A @ ( plus_plus @ A @ ( one_one @ A ) @ A2 ) @ A2 )
!= ( zero_zero @ A ) ) ) ) ).
% Ints_odd_nonzero
thf(fact_3675_of__int__divide__in__Ints,axiom,
! [A: $tType] :
( ( idom_divide @ A )
=> ! [B2: int,A2: int] :
( ( dvd_dvd @ int @ B2 @ A2 )
=> ( member @ A @ ( divide_divide @ A @ ( ring_1_of_int @ A @ A2 ) @ ( ring_1_of_int @ A @ B2 ) ) @ ( ring_1_Ints @ A ) ) ) ) ).
% of_int_divide_in_Ints
thf(fact_3676_and__nat__def,axiom,
( ( bit_se5824344872417868541ns_and @ nat )
= ( ^ [M2: nat,N2: nat] : ( nat2 @ ( bit_se5824344872417868541ns_and @ int @ ( semiring_1_of_nat @ int @ M2 ) @ ( semiring_1_of_nat @ int @ N2 ) ) ) ) ) ).
% and_nat_def
thf(fact_3677_nat__plus__as__int,axiom,
( ( plus_plus @ nat )
= ( ^ [A5: nat,B4: nat] : ( nat2 @ ( plus_plus @ int @ ( semiring_1_of_nat @ int @ A5 ) @ ( semiring_1_of_nat @ int @ B4 ) ) ) ) ) ).
% nat_plus_as_int
thf(fact_3678_nat__times__as__int,axiom,
( ( times_times @ nat )
= ( ^ [A5: nat,B4: nat] : ( nat2 @ ( times_times @ int @ ( semiring_1_of_nat @ int @ A5 ) @ ( semiring_1_of_nat @ int @ B4 ) ) ) ) ) ).
% nat_times_as_int
thf(fact_3679_real__nat__ceiling__ge,axiom,
! [X: real] : ( ord_less_eq @ real @ X @ ( semiring_1_of_nat @ real @ ( nat2 @ ( archimedean_ceiling @ real @ X ) ) ) ) ).
% real_nat_ceiling_ge
thf(fact_3680_nat__minus__as__int,axiom,
( ( minus_minus @ nat )
= ( ^ [A5: nat,B4: nat] : ( nat2 @ ( minus_minus @ int @ ( semiring_1_of_nat @ int @ A5 ) @ ( semiring_1_of_nat @ int @ B4 ) ) ) ) ) ).
% nat_minus_as_int
thf(fact_3681_nat__div__as__int,axiom,
( ( divide_divide @ nat )
= ( ^ [A5: nat,B4: nat] : ( nat2 @ ( divide_divide @ int @ ( semiring_1_of_nat @ int @ A5 ) @ ( semiring_1_of_nat @ int @ B4 ) ) ) ) ) ).
% nat_div_as_int
thf(fact_3682_nat__mod__as__int,axiom,
( ( modulo_modulo @ nat )
= ( ^ [A5: nat,B4: nat] : ( nat2 @ ( modulo_modulo @ int @ ( semiring_1_of_nat @ int @ A5 ) @ ( semiring_1_of_nat @ int @ B4 ) ) ) ) ) ).
% nat_mod_as_int
thf(fact_3683_sgn__real__def,axiom,
( ( sgn_sgn @ real )
= ( ^ [A5: real] :
( if @ real
@ ( A5
= ( zero_zero @ real ) )
@ ( zero_zero @ real )
@ ( if @ real @ ( ord_less @ real @ ( zero_zero @ real ) @ A5 ) @ ( one_one @ real ) @ ( uminus_uminus @ real @ ( one_one @ real ) ) ) ) ) ) ).
% sgn_real_def
thf(fact_3684_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_3685_nat__less__eq__zless,axiom,
! [W: int,Z3: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ W )
=> ( ( ord_less @ nat @ ( nat2 @ W ) @ ( nat2 @ Z3 ) )
= ( ord_less @ int @ W @ Z3 ) ) ) ).
% nat_less_eq_zless
thf(fact_3686_nat__le__eq__zle,axiom,
! [W: int,Z3: int] :
( ( ( ord_less @ int @ ( zero_zero @ int ) @ W )
| ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z3 ) )
=> ( ( ord_less_eq @ nat @ ( nat2 @ W ) @ ( nat2 @ Z3 ) )
= ( ord_less_eq @ int @ W @ Z3 ) ) ) ).
% nat_le_eq_zle
thf(fact_3687_nat__eq__iff2,axiom,
! [M: nat,W: int] :
( ( M
= ( nat2 @ W ) )
= ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ W )
=> ( W
= ( semiring_1_of_nat @ int @ M ) ) )
& ( ~ ( ord_less_eq @ int @ ( zero_zero @ int ) @ W )
=> ( M
= ( zero_zero @ nat ) ) ) ) ) ).
% nat_eq_iff2
thf(fact_3688_nat__eq__iff,axiom,
! [W: int,M: nat] :
( ( ( nat2 @ W )
= M )
= ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ W )
=> ( W
= ( semiring_1_of_nat @ int @ M ) ) )
& ( ~ ( ord_less_eq @ int @ ( zero_zero @ int ) @ W )
=> ( M
= ( zero_zero @ nat ) ) ) ) ) ).
% nat_eq_iff
thf(fact_3689_le__mult__nat__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [A2: A,B2: A] : ( ord_less_eq @ nat @ ( times_times @ nat @ ( nat2 @ ( archim6421214686448440834_floor @ A @ A2 ) ) @ ( nat2 @ ( archim6421214686448440834_floor @ A @ B2 ) ) ) @ ( nat2 @ ( archim6421214686448440834_floor @ A @ ( times_times @ A @ A2 @ B2 ) ) ) ) ) ).
% le_mult_nat_floor
thf(fact_3690_le__nat__iff,axiom,
! [K2: int,N: nat] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K2 )
=> ( ( ord_less_eq @ nat @ N @ ( nat2 @ K2 ) )
= ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ N ) @ K2 ) ) ) ).
% le_nat_iff
thf(fact_3691_nat__add__distrib,axiom,
! [Z3: int,Z7: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z3 )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z7 )
=> ( ( nat2 @ ( plus_plus @ int @ Z3 @ Z7 ) )
= ( plus_plus @ nat @ ( nat2 @ Z3 ) @ ( nat2 @ Z7 ) ) ) ) ) ).
% nat_add_distrib
thf(fact_3692_nat__mult__distrib,axiom,
! [Z3: int,Z7: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z3 )
=> ( ( nat2 @ ( times_times @ int @ Z3 @ Z7 ) )
= ( times_times @ nat @ ( nat2 @ Z3 ) @ ( nat2 @ Z7 ) ) ) ) ).
% nat_mult_distrib
thf(fact_3693_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_3694_nat__diff__distrib,axiom,
! [Z7: int,Z3: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z7 )
=> ( ( ord_less_eq @ int @ Z7 @ Z3 )
=> ( ( nat2 @ ( minus_minus @ int @ Z3 @ Z7 ) )
= ( minus_minus @ nat @ ( nat2 @ Z3 ) @ ( nat2 @ Z7 ) ) ) ) ) ).
% nat_diff_distrib
thf(fact_3695_nat__diff__distrib_H,axiom,
! [X: int,Y2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y2 )
=> ( ( nat2 @ ( minus_minus @ int @ X @ Y2 ) )
= ( minus_minus @ nat @ ( nat2 @ X ) @ ( nat2 @ Y2 ) ) ) ) ) ).
% nat_diff_distrib'
thf(fact_3696_nat__abs__triangle__ineq,axiom,
! [K2: int,L: int] : ( ord_less_eq @ nat @ ( nat2 @ ( abs_abs @ int @ ( plus_plus @ int @ K2 @ L ) ) ) @ ( plus_plus @ nat @ ( nat2 @ ( abs_abs @ int @ K2 ) ) @ ( nat2 @ ( abs_abs @ int @ L ) ) ) ) ).
% nat_abs_triangle_ineq
thf(fact_3697_nat__div__distrib,axiom,
! [X: int,Y2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
=> ( ( nat2 @ ( divide_divide @ int @ X @ Y2 ) )
= ( divide_divide @ nat @ ( nat2 @ X ) @ ( nat2 @ Y2 ) ) ) ) ).
% nat_div_distrib
thf(fact_3698_nat__div__distrib_H,axiom,
! [Y2: int,X: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y2 )
=> ( ( nat2 @ ( divide_divide @ int @ X @ Y2 ) )
= ( divide_divide @ nat @ ( nat2 @ X ) @ ( nat2 @ Y2 ) ) ) ) ).
% nat_div_distrib'
thf(fact_3699_nat__floor__neg,axiom,
! [X: real] :
( ( ord_less_eq @ real @ X @ ( zero_zero @ real ) )
=> ( ( nat2 @ ( archim6421214686448440834_floor @ real @ X ) )
= ( zero_zero @ nat ) ) ) ).
% nat_floor_neg
thf(fact_3700_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_3701_nat__mod__distrib,axiom,
! [X: int,Y2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y2 )
=> ( ( nat2 @ ( modulo_modulo @ int @ X @ Y2 ) )
= ( modulo_modulo @ nat @ ( nat2 @ X ) @ ( nat2 @ Y2 ) ) ) ) ) ).
% nat_mod_distrib
thf(fact_3702_div__abs__eq__div__nat,axiom,
! [K2: int,L: int] :
( ( divide_divide @ int @ ( abs_abs @ int @ K2 ) @ ( abs_abs @ int @ L ) )
= ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ ( nat2 @ ( abs_abs @ int @ K2 ) ) @ ( nat2 @ ( abs_abs @ int @ L ) ) ) ) ) ).
% div_abs_eq_div_nat
thf(fact_3703_Ints__odd__less__0,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A] :
( ( member @ A @ A2 @ ( ring_1_Ints @ A ) )
=> ( ( ord_less @ A @ ( plus_plus @ A @ ( plus_plus @ A @ ( one_one @ A ) @ A2 ) @ A2 ) @ ( zero_zero @ A ) )
= ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ) ).
% Ints_odd_less_0
thf(fact_3704_floor__eq3,axiom,
! [N: nat,X: real] :
( ( ord_less @ real @ ( semiring_1_of_nat @ real @ N ) @ X )
=> ( ( ord_less @ real @ X @ ( semiring_1_of_nat @ real @ ( suc @ N ) ) )
=> ( ( nat2 @ ( archim6421214686448440834_floor @ real @ X ) )
= N ) ) ) ).
% floor_eq3
thf(fact_3705_le__nat__floor,axiom,
! [X: nat,A2: real] :
( ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ X ) @ A2 )
=> ( ord_less_eq @ nat @ X @ ( nat2 @ ( archim6421214686448440834_floor @ real @ A2 ) ) ) ) ).
% le_nat_floor
thf(fact_3706_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_3707_mod__abs__eq__div__nat,axiom,
! [K2: int,L: int] :
( ( modulo_modulo @ int @ ( abs_abs @ int @ K2 ) @ ( abs_abs @ int @ L ) )
= ( semiring_1_of_nat @ int @ ( modulo_modulo @ nat @ ( nat2 @ ( abs_abs @ int @ K2 ) ) @ ( nat2 @ ( abs_abs @ int @ L ) ) ) ) ) ).
% mod_abs_eq_div_nat
thf(fact_3708_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_3709_Ints__eq__abs__less1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,Y2: A] :
( ( member @ A @ X @ ( ring_1_Ints @ A ) )
=> ( ( member @ A @ Y2 @ ( ring_1_Ints @ A ) )
=> ( ( X = Y2 )
= ( ord_less @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X @ Y2 ) ) @ ( one_one @ A ) ) ) ) ) ) ).
% Ints_eq_abs_less1
thf(fact_3710_nat__take__bit__eq,axiom,
! [K2: int,N: nat] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K2 )
=> ( ( nat2 @ ( bit_se2584673776208193580ke_bit @ int @ N @ K2 ) )
= ( bit_se2584673776208193580ke_bit @ nat @ N @ ( nat2 @ K2 ) ) ) ) ).
% nat_take_bit_eq
thf(fact_3711_take__bit__nat__eq,axiom,
! [K2: int,N: nat] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K2 )
=> ( ( bit_se2584673776208193580ke_bit @ nat @ N @ ( nat2 @ K2 ) )
= ( nat2 @ ( bit_se2584673776208193580ke_bit @ int @ N @ K2 ) ) ) ) ).
% take_bit_nat_eq
thf(fact_3712_sin__times__pi__eq__0,axiom,
! [X: real] :
( ( ( sin @ real @ ( times_times @ real @ X @ pi ) )
= ( zero_zero @ real ) )
= ( member @ real @ X @ ( ring_1_Ints @ real ) ) ) ).
% sin_times_pi_eq_0
thf(fact_3713_bit__nat__iff,axiom,
! [K2: int,N: nat] :
( ( bit_se5641148757651400278ts_bit @ nat @ ( nat2 @ K2 ) @ N )
= ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K2 )
& ( bit_se5641148757651400278ts_bit @ int @ K2 @ N ) ) ) ).
% bit_nat_iff
thf(fact_3714_nat__2,axiom,
( ( nat2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) )
= ( suc @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% nat_2
thf(fact_3715_sgn__power__injE,axiom,
! [A2: real,N: nat,X: real,B2: real] :
( ( ( times_times @ real @ ( sgn_sgn @ real @ A2 ) @ ( power_power @ real @ ( abs_abs @ real @ A2 ) @ N ) )
= X )
=> ( ( X
= ( times_times @ real @ ( sgn_sgn @ real @ B2 ) @ ( power_power @ real @ ( abs_abs @ real @ B2 ) @ N ) ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( A2 = B2 ) ) ) ) ).
% sgn_power_injE
thf(fact_3716_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_3717_nat__less__iff,axiom,
! [W: int,M: nat] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ W )
=> ( ( ord_less @ nat @ ( nat2 @ W ) @ M )
= ( ord_less @ int @ W @ ( semiring_1_of_nat @ int @ M ) ) ) ) ).
% nat_less_iff
thf(fact_3718_nat__mult__distrib__neg,axiom,
! [Z3: int,Z7: int] :
( ( ord_less_eq @ int @ Z3 @ ( zero_zero @ int ) )
=> ( ( nat2 @ ( times_times @ int @ Z3 @ Z7 ) )
= ( times_times @ nat @ ( nat2 @ ( uminus_uminus @ int @ Z3 ) ) @ ( nat2 @ ( uminus_uminus @ int @ Z7 ) ) ) ) ) ).
% nat_mult_distrib_neg
thf(fact_3719_nat__abs__int__diff,axiom,
! [A2: nat,B2: nat] :
( ( ( ord_less_eq @ nat @ A2 @ B2 )
=> ( ( nat2 @ ( abs_abs @ int @ ( minus_minus @ int @ ( semiring_1_of_nat @ int @ A2 ) @ ( semiring_1_of_nat @ int @ B2 ) ) ) )
= ( minus_minus @ nat @ B2 @ A2 ) ) )
& ( ~ ( ord_less_eq @ nat @ A2 @ B2 )
=> ( ( nat2 @ ( abs_abs @ int @ ( minus_minus @ int @ ( semiring_1_of_nat @ int @ A2 ) @ ( semiring_1_of_nat @ int @ B2 ) ) ) )
= ( minus_minus @ nat @ A2 @ B2 ) ) ) ) ).
% nat_abs_int_diff
thf(fact_3720_floor__eq4,axiom,
! [N: nat,X: real] :
( ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ N ) @ X )
=> ( ( ord_less @ real @ X @ ( semiring_1_of_nat @ real @ ( suc @ N ) ) )
=> ( ( nat2 @ ( archim6421214686448440834_floor @ real @ X ) )
= N ) ) ) ).
% floor_eq4
thf(fact_3721_diff__nat__eq__if,axiom,
! [Z7: int,Z3: int] :
( ( ( ord_less @ int @ Z7 @ ( zero_zero @ int ) )
=> ( ( minus_minus @ nat @ ( nat2 @ Z3 ) @ ( nat2 @ Z7 ) )
= ( nat2 @ Z3 ) ) )
& ( ~ ( ord_less @ int @ Z7 @ ( zero_zero @ int ) )
=> ( ( minus_minus @ nat @ ( nat2 @ Z3 ) @ ( nat2 @ Z7 ) )
= ( if @ nat @ ( ord_less @ int @ ( minus_minus @ int @ Z3 @ Z7 ) @ ( zero_zero @ int ) ) @ ( zero_zero @ nat ) @ ( nat2 @ ( minus_minus @ int @ Z3 @ Z7 ) ) ) ) ) ) ).
% diff_nat_eq_if
thf(fact_3722_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_3723_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_3724_Arg__correct,axiom,
! [Z3: complex] :
( ( Z3
!= ( zero_zero @ complex ) )
=> ( ( ( sgn_sgn @ complex @ Z3 )
= ( cis @ ( arg @ Z3 ) ) )
& ( ord_less @ real @ ( uminus_uminus @ real @ pi ) @ ( arg @ Z3 ) )
& ( ord_less_eq @ real @ ( arg @ Z3 ) @ pi ) ) ) ).
% Arg_correct
thf(fact_3725_cis__Arg__unique,axiom,
! [Z3: complex,X: real] :
( ( ( sgn_sgn @ complex @ Z3 )
= ( cis @ X ) )
=> ( ( ord_less @ real @ ( uminus_uminus @ real @ pi ) @ X )
=> ( ( ord_less_eq @ real @ X @ pi )
=> ( ( arg @ Z3 )
= X ) ) ) ) ).
% cis_Arg_unique
thf(fact_3726_le__mult__floor__Ints,axiom,
! [A: $tType,B: $tType] :
( ( ( archim2362893244070406136eiling @ B )
& ( linordered_idom @ A ) )
=> ! [A2: B,B2: B] :
( ( ord_less_eq @ B @ ( zero_zero @ B ) @ A2 )
=> ( ( member @ B @ A2 @ ( ring_1_Ints @ B ) )
=> ( ord_less_eq @ A @ ( ring_1_of_int @ A @ ( times_times @ int @ ( archim6421214686448440834_floor @ B @ A2 ) @ ( archim6421214686448440834_floor @ B @ B2 ) ) ) @ ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ B @ ( times_times @ B @ A2 @ B2 ) ) ) ) ) ) ) ).
% le_mult_floor_Ints
thf(fact_3727_frac__unique__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,A2: A] :
( ( ( archimedean_frac @ A @ X )
= A2 )
= ( ( member @ A @ ( minus_minus @ A @ X @ A2 ) @ ( ring_1_Ints @ A ) )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
& ( ord_less @ A @ A2 @ ( one_one @ A ) ) ) ) ) ).
% frac_unique_iff
thf(fact_3728_mult__ceiling__le__Ints,axiom,
! [A: $tType,B: $tType] :
( ( ( archim2362893244070406136eiling @ B )
& ( linordered_idom @ A ) )
=> ! [A2: B,B2: B] :
( ( ord_less_eq @ B @ ( zero_zero @ B ) @ A2 )
=> ( ( member @ B @ A2 @ ( ring_1_Ints @ B ) )
=> ( ord_less_eq @ A @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ B @ ( times_times @ B @ A2 @ B2 ) ) ) @ ( ring_1_of_int @ A @ ( times_times @ int @ ( archimedean_ceiling @ B @ A2 ) @ ( archimedean_ceiling @ B @ B2 ) ) ) ) ) ) ) ).
% mult_ceiling_le_Ints
thf(fact_3729_sin__integer__2pi,axiom,
! [N: real] :
( ( member @ real @ N @ ( ring_1_Ints @ real ) )
=> ( ( sin @ real @ ( times_times @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) @ N ) )
= ( zero_zero @ real ) ) ) ).
% sin_integer_2pi
thf(fact_3730_cos__integer__2pi,axiom,
! [N: real] :
( ( member @ real @ N @ ( ring_1_Ints @ real ) )
=> ( ( cos @ real @ ( times_times @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) @ N ) )
= ( one_one @ real ) ) ) ).
% cos_integer_2pi
thf(fact_3731_even__nat__iff,axiom,
! [K2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K2 )
=> ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( nat2 @ K2 ) )
= ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K2 ) ) ) ).
% even_nat_iff
thf(fact_3732_floor__rat__def,axiom,
( ( archim6421214686448440834_floor @ rat )
= ( ^ [X4: rat] :
( the @ int
@ ^ [Z6: int] :
( ( ord_less_eq @ rat @ ( ring_1_of_int @ rat @ Z6 ) @ X4 )
& ( ord_less @ rat @ X4 @ ( ring_1_of_int @ rat @ ( plus_plus @ int @ Z6 @ ( one_one @ int ) ) ) ) ) ) ) ) ).
% floor_rat_def
thf(fact_3733_powr__real__of__int,axiom,
! [X: real,N: int] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ N )
=> ( ( powr @ real @ X @ ( ring_1_of_int @ real @ N ) )
= ( power_power @ real @ X @ ( nat2 @ N ) ) ) )
& ( ~ ( ord_less_eq @ int @ ( zero_zero @ int ) @ N )
=> ( ( powr @ real @ X @ ( ring_1_of_int @ real @ N ) )
= ( inverse_inverse @ real @ ( power_power @ real @ X @ ( nat2 @ ( uminus_uminus @ int @ N ) ) ) ) ) ) ) ) ).
% powr_real_of_int
thf(fact_3734_case__prod__app,axiom,
! [A: $tType,D: $tType,C: $tType,B: $tType] :
( ( product_case_prod @ B @ C @ ( D > A ) )
= ( ^ [F5: B > C > D > A,X4: product_prod @ B @ C,Y: D] :
( product_case_prod @ B @ C @ A
@ ^ [L3: B,R5: C] : ( F5 @ L3 @ R5 @ Y )
@ X4 ) ) ) ).
% case_prod_app
thf(fact_3735_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_3736_inverse__eq__iff__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A,B2: A] :
( ( ( inverse_inverse @ A @ A2 )
= ( inverse_inverse @ A @ B2 ) )
= ( A2 = B2 ) ) ) ).
% inverse_eq_iff_eq
thf(fact_3737_inverse__inverse__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A] :
( ( inverse_inverse @ A @ ( inverse_inverse @ A @ A2 ) )
= A2 ) ) ).
% inverse_inverse_eq
thf(fact_3738_bit_Oxor__left__self,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X: A,Y2: A] :
( ( bit_se5824344971392196577ns_xor @ A @ X @ ( bit_se5824344971392196577ns_xor @ A @ X @ Y2 ) )
= Y2 ) ) ).
% bit.xor_left_self
thf(fact_3739_inverse__nonzero__iff__nonzero,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A] :
( ( ( inverse_inverse @ A @ A2 )
= ( zero_zero @ A ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% inverse_nonzero_iff_nonzero
thf(fact_3740_inverse__zero,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ( ( inverse_inverse @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% inverse_zero
thf(fact_3741_inverse__mult__distrib,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A2: A,B2: A] :
( ( inverse_inverse @ A @ ( times_times @ A @ A2 @ B2 ) )
= ( times_times @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) ) ) ) ).
% inverse_mult_distrib
thf(fact_3742_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_3743_inverse__1,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ( ( inverse_inverse @ A @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% inverse_1
thf(fact_3744_inverse__divide,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A2: A,B2: A] :
( ( inverse_inverse @ A @ ( divide_divide @ A @ A2 @ B2 ) )
= ( divide_divide @ A @ B2 @ A2 ) ) ) ).
% inverse_divide
thf(fact_3745_inverse__minus__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A] :
( ( inverse_inverse @ A @ ( uminus_uminus @ A @ A2 ) )
= ( uminus_uminus @ A @ ( inverse_inverse @ A @ A2 ) ) ) ) ).
% inverse_minus_eq
thf(fact_3746_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_3747_xor__self__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A] :
( ( bit_se5824344971392196577ns_xor @ A @ A2 @ A2 )
= ( zero_zero @ A ) ) ) ).
% xor_self_eq
thf(fact_3748_xor_Oleft__neutral,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A] :
( ( bit_se5824344971392196577ns_xor @ A @ ( zero_zero @ A ) @ A2 )
= A2 ) ) ).
% xor.left_neutral
thf(fact_3749_xor_Oright__neutral,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A] :
( ( bit_se5824344971392196577ns_xor @ A @ A2 @ ( zero_zero @ A ) )
= A2 ) ) ).
% xor.right_neutral
thf(fact_3750_abs__inverse,axiom,
! [A: $tType] :
( ( field_abs_sgn @ A )
=> ! [A2: A] :
( ( abs_abs @ A @ ( inverse_inverse @ A @ A2 ) )
= ( inverse_inverse @ A @ ( abs_abs @ A @ A2 ) ) ) ) ).
% abs_inverse
thf(fact_3751_sgn__inverse,axiom,
! [A: $tType] :
( ( field_abs_sgn @ A )
=> ! [A2: A] :
( ( sgn_sgn @ A @ ( inverse_inverse @ A @ A2 ) )
= ( inverse_inverse @ A @ ( sgn_sgn @ A @ A2 ) ) ) ) ).
% sgn_inverse
thf(fact_3752_inverse__sgn,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A] :
( ( inverse_inverse @ A @ ( sgn_sgn @ A @ A2 ) )
= ( sgn_sgn @ A @ A2 ) ) ) ).
% inverse_sgn
thf(fact_3753_take__bit__xor,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A2: A,B2: A] :
( ( bit_se2584673776208193580ke_bit @ A @ N @ ( bit_se5824344971392196577ns_xor @ A @ A2 @ B2 ) )
= ( bit_se5824344971392196577ns_xor @ A @ ( bit_se2584673776208193580ke_bit @ A @ N @ A2 ) @ ( bit_se2584673776208193580ke_bit @ A @ N @ B2 ) ) ) ) ).
% take_bit_xor
thf(fact_3754_inverse__nonpositive__iff__nonpositive,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ ( inverse_inverse @ A @ A2 ) @ ( zero_zero @ A ) )
= ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% inverse_nonpositive_iff_nonpositive
thf(fact_3755_inverse__nonnegative__iff__nonnegative,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( inverse_inverse @ A @ A2 ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% inverse_nonnegative_iff_nonnegative
thf(fact_3756_inverse__less__iff__less,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) )
= ( ord_less @ A @ B2 @ A2 ) ) ) ) ) ).
% inverse_less_iff_less
thf(fact_3757_inverse__less__iff__less__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) )
= ( ord_less @ A @ B2 @ A2 ) ) ) ) ) ).
% inverse_less_iff_less_neg
thf(fact_3758_inverse__negative__iff__negative,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A] :
( ( ord_less @ A @ ( inverse_inverse @ A @ A2 ) @ ( zero_zero @ A ) )
= ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% inverse_negative_iff_negative
thf(fact_3759_inverse__positive__iff__positive,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( inverse_inverse @ A @ A2 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% inverse_positive_iff_positive
thf(fact_3760_inverse__le__iff__le__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) )
= ( ord_less_eq @ A @ B2 @ A2 ) ) ) ) ) ).
% inverse_le_iff_le_neg
thf(fact_3761_inverse__le__iff__le,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less_eq @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) )
= ( ord_less_eq @ A @ B2 @ A2 ) ) ) ) ) ).
% inverse_le_iff_le
thf(fact_3762_left__inverse,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ ( inverse_inverse @ A @ A2 ) @ A2 )
= ( one_one @ A ) ) ) ) ).
% left_inverse
thf(fact_3763_right__inverse,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ A2 @ ( inverse_inverse @ A @ A2 ) )
= ( one_one @ A ) ) ) ) ).
% right_inverse
thf(fact_3764_inverse__eq__divide__numeral,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [W: num] :
( ( inverse_inverse @ A @ ( numeral_numeral @ A @ W ) )
= ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ W ) ) ) ) ).
% inverse_eq_divide_numeral
thf(fact_3765_inverse__eq__divide__neg__numeral,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [W: num] :
( ( inverse_inverse @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) )
= ( divide_divide @ A @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) ) ) ) ).
% inverse_eq_divide_neg_numeral
thf(fact_3766_xor__numerals_I3_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X: num,Y2: num] :
( ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ ( bit0 @ X ) ) @ ( numeral_numeral @ A @ ( bit0 @ Y2 ) ) )
= ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ X ) @ ( numeral_numeral @ A @ Y2 ) ) ) ) ) ).
% xor_numerals(3)
thf(fact_3767_xor__numerals_I1_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [Y2: num] :
( ( bit_se5824344971392196577ns_xor @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ Y2 ) ) )
= ( numeral_numeral @ A @ ( bit1 @ Y2 ) ) ) ) ).
% xor_numerals(1)
thf(fact_3768_xor__numerals_I2_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [Y2: num] :
( ( bit_se5824344971392196577ns_xor @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit1 @ Y2 ) ) )
= ( numeral_numeral @ A @ ( bit0 @ Y2 ) ) ) ) ).
% xor_numerals(2)
thf(fact_3769_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_3770_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_3771_xor__numerals_I7_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X: num,Y2: num] :
( ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ ( bit1 @ X ) ) @ ( numeral_numeral @ A @ ( bit1 @ Y2 ) ) )
= ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ X ) @ ( numeral_numeral @ A @ Y2 ) ) ) ) ) ).
% xor_numerals(7)
thf(fact_3772_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_3773_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_3774_xor__nat__numerals_I2_J,axiom,
! [Y2: num] :
( ( bit_se5824344971392196577ns_xor @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( numeral_numeral @ nat @ ( bit1 @ Y2 ) ) )
= ( numeral_numeral @ nat @ ( bit0 @ Y2 ) ) ) ).
% xor_nat_numerals(2)
thf(fact_3775_xor__nat__numerals_I1_J,axiom,
! [Y2: num] :
( ( bit_se5824344971392196577ns_xor @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( numeral_numeral @ nat @ ( bit0 @ Y2 ) ) )
= ( numeral_numeral @ nat @ ( bit1 @ Y2 ) ) ) ).
% xor_nat_numerals(1)
thf(fact_3776_xor__numerals_I4_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X: num,Y2: num] :
( ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ ( bit0 @ X ) ) @ ( numeral_numeral @ A @ ( bit1 @ Y2 ) ) )
= ( 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 @ Y2 ) ) ) ) ) ) ).
% xor_numerals(4)
thf(fact_3777_xor__numerals_I6_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X: num,Y2: num] :
( ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ ( bit1 @ X ) ) @ ( numeral_numeral @ A @ ( bit0 @ Y2 ) ) )
= ( 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 @ Y2 ) ) ) ) ) ) ).
% xor_numerals(6)
thf(fact_3778_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_3779_power__inverse,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A,N: nat] :
( ( power_power @ A @ ( inverse_inverse @ A @ A2 ) @ N )
= ( inverse_inverse @ A @ ( power_power @ A @ A2 @ N ) ) ) ) ).
% power_inverse
thf(fact_3780_mult__commute__imp__mult__inverse__commute,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Y2: A,X: A] :
( ( ( times_times @ A @ Y2 @ X )
= ( times_times @ A @ X @ Y2 ) )
=> ( ( times_times @ A @ ( inverse_inverse @ A @ Y2 ) @ X )
= ( times_times @ A @ X @ ( inverse_inverse @ A @ Y2 ) ) ) ) ) ).
% mult_commute_imp_mult_inverse_commute
thf(fact_3781_real__sqrt__inverse,axiom,
! [X: real] :
( ( sqrt @ ( inverse_inverse @ real @ X ) )
= ( inverse_inverse @ real @ ( sqrt @ X ) ) ) ).
% real_sqrt_inverse
thf(fact_3782_nonzero__imp__inverse__nonzero,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( inverse_inverse @ A @ A2 )
!= ( zero_zero @ A ) ) ) ) ).
% nonzero_imp_inverse_nonzero
thf(fact_3783_nonzero__inverse__inverse__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( inverse_inverse @ A @ ( inverse_inverse @ A @ A2 ) )
= A2 ) ) ) ).
% nonzero_inverse_inverse_eq
thf(fact_3784_nonzero__inverse__eq__imp__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A,B2: A] :
( ( ( inverse_inverse @ A @ A2 )
= ( inverse_inverse @ A @ B2 ) )
=> ( ( A2
!= ( zero_zero @ A ) )
=> ( ( B2
!= ( zero_zero @ A ) )
=> ( A2 = B2 ) ) ) ) ) ).
% nonzero_inverse_eq_imp_eq
thf(fact_3785_inverse__zero__imp__zero,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A] :
( ( ( inverse_inverse @ A @ A2 )
= ( zero_zero @ A ) )
=> ( A2
= ( zero_zero @ A ) ) ) ) ).
% inverse_zero_imp_zero
thf(fact_3786_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_3787_less__eq__rat__def,axiom,
( ( ord_less_eq @ rat )
= ( ^ [X4: rat,Y: rat] :
( ( ord_less @ rat @ X4 @ Y )
| ( X4 = Y ) ) ) ) ).
% less_eq_rat_def
thf(fact_3788_inverse__eq__imp__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A,B2: A] :
( ( ( inverse_inverse @ A @ A2 )
= ( inverse_inverse @ A @ B2 ) )
=> ( A2 = B2 ) ) ) ).
% inverse_eq_imp_eq
thf(fact_3789_xor_Oassoc,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( bit_se5824344971392196577ns_xor @ A @ ( bit_se5824344971392196577ns_xor @ A @ A2 @ B2 ) @ C2 )
= ( bit_se5824344971392196577ns_xor @ A @ A2 @ ( bit_se5824344971392196577ns_xor @ A @ B2 @ C2 ) ) ) ) ).
% xor.assoc
thf(fact_3790_xor_Ocommute,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se5824344971392196577ns_xor @ A )
= ( ^ [A5: A,B4: A] : ( bit_se5824344971392196577ns_xor @ A @ B4 @ A5 ) ) ) ) ).
% xor.commute
thf(fact_3791_xor_Oleft__commute,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( bit_se5824344971392196577ns_xor @ A @ B2 @ ( bit_se5824344971392196577ns_xor @ A @ A2 @ C2 ) )
= ( bit_se5824344971392196577ns_xor @ A @ A2 @ ( bit_se5824344971392196577ns_xor @ A @ B2 @ C2 ) ) ) ) ).
% xor.left_commute
thf(fact_3792_of__int__xor__eq,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [K2: int,L: int] :
( ( ring_1_of_int @ A @ ( bit_se5824344971392196577ns_xor @ int @ K2 @ L ) )
= ( bit_se5824344971392196577ns_xor @ A @ ( ring_1_of_int @ A @ K2 ) @ ( ring_1_of_int @ A @ L ) ) ) ) ).
% of_int_xor_eq
thf(fact_3793_obtain__pos__sum,axiom,
! [R: rat] :
( ( ord_less @ rat @ ( zero_zero @ rat ) @ R )
=> ~ ! [S2: rat] :
( ( ord_less @ rat @ ( zero_zero @ rat ) @ S2 )
=> ! [T4: rat] :
( ( ord_less @ rat @ ( zero_zero @ rat ) @ T4 )
=> ( R
!= ( plus_plus @ rat @ S2 @ T4 ) ) ) ) ) ).
% obtain_pos_sum
thf(fact_3794_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_3795_bit__xor__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A,B2: A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( bit_se5824344971392196577ns_xor @ A @ A2 @ B2 ) @ N )
= ( ( bit_se5641148757651400278ts_bit @ A @ A2 @ N )
!= ( bit_se5641148757651400278ts_bit @ A @ B2 @ N ) ) ) ) ).
% bit_xor_iff
thf(fact_3796_bit_Oconj__xor__distrib2,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [Y2: A,Z3: A,X: A] :
( ( bit_se5824344872417868541ns_and @ A @ ( bit_se5824344971392196577ns_xor @ A @ Y2 @ Z3 ) @ X )
= ( bit_se5824344971392196577ns_xor @ A @ ( bit_se5824344872417868541ns_and @ A @ Y2 @ X ) @ ( bit_se5824344872417868541ns_and @ A @ Z3 @ X ) ) ) ) ).
% bit.conj_xor_distrib2
thf(fact_3797_bit_Oconj__xor__distrib,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X: A,Y2: A,Z3: A] :
( ( bit_se5824344872417868541ns_and @ A @ X @ ( bit_se5824344971392196577ns_xor @ A @ Y2 @ Z3 ) )
= ( bit_se5824344971392196577ns_xor @ A @ ( bit_se5824344872417868541ns_and @ A @ X @ Y2 ) @ ( bit_se5824344872417868541ns_and @ A @ X @ Z3 ) ) ) ) ).
% bit.conj_xor_distrib
thf(fact_3798_norm__inverse__le__norm,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [R: real,X: A] :
( ( ord_less_eq @ real @ R @ ( real_V7770717601297561774m_norm @ A @ X ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ R )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( inverse_inverse @ A @ X ) ) @ ( inverse_inverse @ real @ R ) ) ) ) ) ).
% norm_inverse_le_norm
thf(fact_3799_inverse__less__imp__less,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ord_less @ A @ B2 @ A2 ) ) ) ) ).
% inverse_less_imp_less
thf(fact_3800_less__imp__inverse__less,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ord_less @ A @ ( inverse_inverse @ A @ B2 ) @ ( inverse_inverse @ A @ A2 ) ) ) ) ) ).
% less_imp_inverse_less
thf(fact_3801_inverse__less__imp__less__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) )
=> ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B2 @ A2 ) ) ) ) ).
% inverse_less_imp_less_neg
thf(fact_3802_less__imp__inverse__less__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( inverse_inverse @ A @ B2 ) @ ( inverse_inverse @ A @ A2 ) ) ) ) ) ).
% less_imp_inverse_less_neg
thf(fact_3803_inverse__negative__imp__negative,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A] :
( ( ord_less @ A @ ( inverse_inverse @ A @ A2 ) @ ( zero_zero @ A ) )
=> ( ( A2
!= ( zero_zero @ A ) )
=> ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ) ).
% inverse_negative_imp_negative
thf(fact_3804_inverse__positive__imp__positive,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( inverse_inverse @ A @ A2 ) )
=> ( ( A2
!= ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ) ).
% inverse_positive_imp_positive
thf(fact_3805_negative__imp__inverse__negative,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A] :
( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( inverse_inverse @ A @ A2 ) @ ( zero_zero @ A ) ) ) ) ).
% negative_imp_inverse_negative
thf(fact_3806_positive__imp__inverse__positive,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( inverse_inverse @ A @ A2 ) ) ) ) ).
% positive_imp_inverse_positive
thf(fact_3807_nonzero__inverse__mult__distrib,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A,B2: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( B2
!= ( zero_zero @ A ) )
=> ( ( inverse_inverse @ A @ ( times_times @ A @ A2 @ B2 ) )
= ( times_times @ A @ ( inverse_inverse @ A @ B2 ) @ ( inverse_inverse @ A @ A2 ) ) ) ) ) ) ).
% nonzero_inverse_mult_distrib
thf(fact_3808_nonzero__inverse__minus__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( inverse_inverse @ A @ ( uminus_uminus @ A @ A2 ) )
= ( uminus_uminus @ A @ ( inverse_inverse @ A @ A2 ) ) ) ) ) ).
% nonzero_inverse_minus_eq
thf(fact_3809_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_3810_inverse__unique,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A,B2: A] :
( ( ( times_times @ A @ A2 @ B2 )
= ( one_one @ A ) )
=> ( ( inverse_inverse @ A @ A2 )
= B2 ) ) ) ).
% inverse_unique
thf(fact_3811_divide__inverse__commute,axiom,
! [A: $tType] :
( ( field @ A )
=> ( ( divide_divide @ A )
= ( ^ [A5: A,B4: A] : ( times_times @ A @ ( inverse_inverse @ A @ B4 ) @ A5 ) ) ) ) ).
% divide_inverse_commute
thf(fact_3812_divide__inverse,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ( ( divide_divide @ A )
= ( ^ [A5: A,B4: A] : ( times_times @ A @ A5 @ ( inverse_inverse @ A @ B4 ) ) ) ) ) ).
% divide_inverse
thf(fact_3813_field__class_Ofield__divide__inverse,axiom,
! [A: $tType] :
( ( field @ A )
=> ( ( divide_divide @ A )
= ( ^ [A5: A,B4: A] : ( times_times @ A @ A5 @ ( inverse_inverse @ A @ B4 ) ) ) ) ) ).
% field_class.field_divide_inverse
thf(fact_3814_inverse__eq__divide,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ( ( inverse_inverse @ A )
= ( divide_divide @ A @ ( one_one @ A ) ) ) ) ).
% inverse_eq_divide
thf(fact_3815_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_3816_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_3817_mult__inverse__of__nat__commute,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Xa: nat,X: A] :
( ( times_times @ A @ ( inverse_inverse @ A @ ( semiring_1_of_nat @ A @ Xa ) ) @ X )
= ( times_times @ A @ X @ ( inverse_inverse @ A @ ( semiring_1_of_nat @ A @ Xa ) ) ) ) ) ).
% mult_inverse_of_nat_commute
thf(fact_3818_nonzero__abs__inverse,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( abs_abs @ A @ ( inverse_inverse @ A @ A2 ) )
= ( inverse_inverse @ A @ ( abs_abs @ A @ A2 ) ) ) ) ) ).
% nonzero_abs_inverse
thf(fact_3819_mult__inverse__of__int__commute,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Xa: int,X: A] :
( ( times_times @ A @ ( inverse_inverse @ A @ ( ring_1_of_int @ A @ Xa ) ) @ X )
= ( times_times @ A @ X @ ( inverse_inverse @ A @ ( ring_1_of_int @ A @ Xa ) ) ) ) ) ).
% mult_inverse_of_int_commute
thf(fact_3820_divide__real__def,axiom,
( ( divide_divide @ real )
= ( ^ [X4: real,Y: real] : ( times_times @ real @ X4 @ ( inverse_inverse @ real @ Y ) ) ) ) ).
% divide_real_def
thf(fact_3821_exp__fdiffs,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ( ( diffs @ A
@ ^ [N2: nat] : ( inverse_inverse @ A @ ( semiring_char_0_fact @ A @ N2 ) ) )
= ( ^ [N2: nat] : ( inverse_inverse @ A @ ( semiring_char_0_fact @ A @ N2 ) ) ) ) ) ).
% exp_fdiffs
thf(fact_3822_le__imp__inverse__le__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( inverse_inverse @ A @ B2 ) @ ( inverse_inverse @ A @ A2 ) ) ) ) ) ).
% le_imp_inverse_le_neg
thf(fact_3823_inverse__le__imp__le__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) )
=> ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B2 @ A2 ) ) ) ) ).
% inverse_le_imp_le_neg
thf(fact_3824_le__imp__inverse__le,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ord_less_eq @ A @ ( inverse_inverse @ A @ B2 ) @ ( inverse_inverse @ A @ A2 ) ) ) ) ) ).
% le_imp_inverse_le
thf(fact_3825_inverse__le__imp__le,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ord_less_eq @ A @ B2 @ A2 ) ) ) ) ).
% inverse_le_imp_le
thf(fact_3826_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_3827_one__less__inverse,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ A2 @ ( one_one @ A ) )
=> ( ord_less @ A @ ( one_one @ A ) @ ( inverse_inverse @ A @ A2 ) ) ) ) ) ).
% one_less_inverse
thf(fact_3828_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_3829_field__class_Ofield__inverse,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A2: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ ( inverse_inverse @ A @ A2 ) @ A2 )
= ( one_one @ A ) ) ) ) ).
% field_class.field_inverse
thf(fact_3830_division__ring__inverse__add,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A,B2: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( B2
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) )
= ( times_times @ A @ ( times_times @ A @ ( inverse_inverse @ A @ A2 ) @ ( plus_plus @ A @ A2 @ B2 ) ) @ ( inverse_inverse @ A @ B2 ) ) ) ) ) ) ).
% division_ring_inverse_add
thf(fact_3831_inverse__add,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A2: A,B2: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( B2
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) )
= ( times_times @ A @ ( times_times @ A @ ( plus_plus @ A @ A2 @ B2 ) @ ( inverse_inverse @ A @ A2 ) ) @ ( inverse_inverse @ A @ B2 ) ) ) ) ) ) ).
% inverse_add
thf(fact_3832_division__ring__inverse__diff,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A,B2: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( B2
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) )
= ( times_times @ A @ ( times_times @ A @ ( inverse_inverse @ A @ A2 ) @ ( minus_minus @ A @ B2 @ A2 ) ) @ ( inverse_inverse @ A @ B2 ) ) ) ) ) ) ).
% division_ring_inverse_diff
thf(fact_3833_nonzero__inverse__eq__divide,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( inverse_inverse @ A @ A2 )
= ( divide_divide @ A @ ( one_one @ A ) @ A2 ) ) ) ) ).
% nonzero_inverse_eq_divide
thf(fact_3834_inverse__powr,axiom,
! [Y2: real,A2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y2 )
=> ( ( powr @ real @ ( inverse_inverse @ real @ Y2 ) @ A2 )
= ( inverse_inverse @ real @ ( powr @ real @ Y2 @ A2 ) ) ) ) ).
% inverse_powr
thf(fact_3835_inverse__less__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B2 ) )
=> ( ord_less @ A @ B2 @ A2 ) )
& ( ( ord_less_eq @ A @ ( times_times @ A @ A2 @ B2 ) @ ( zero_zero @ A ) )
=> ( ord_less @ A @ A2 @ B2 ) ) ) ) ) ).
% inverse_less_iff
thf(fact_3836_inverse__le__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B2 ) )
=> ( ord_less_eq @ A @ B2 @ A2 ) )
& ( ( ord_less_eq @ A @ ( times_times @ A @ A2 @ B2 ) @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ A2 @ B2 ) ) ) ) ) ).
% inverse_le_iff
thf(fact_3837_one__le__inverse,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ A2 @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ ( inverse_inverse @ A @ A2 ) ) ) ) ) ).
% one_le_inverse
thf(fact_3838_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_3839_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_3840_inverse__diff__inverse,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A,B2: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( B2
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) )
= ( uminus_uminus @ A @ ( times_times @ A @ ( times_times @ A @ ( inverse_inverse @ A @ A2 ) @ ( minus_minus @ A @ A2 @ B2 ) ) @ ( inverse_inverse @ A @ B2 ) ) ) ) ) ) ) ).
% inverse_diff_inverse
thf(fact_3841_reals__Archimedean,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X )
=> ? [N3: nat] : ( ord_less @ A @ ( inverse_inverse @ A @ ( semiring_1_of_nat @ A @ ( suc @ N3 ) ) ) @ X ) ) ) ).
% reals_Archimedean
thf(fact_3842_predicate2D__conj,axiom,
! [A: $tType,B: $tType,P3: A > B > $o,Q2: A > B > $o,R4: $o,X: A,Y2: B] :
( ( ( ord_less_eq @ ( A > B > $o ) @ P3 @ Q2 )
& R4 )
=> ( R4
& ( ( P3 @ X @ Y2 )
=> ( Q2 @ X @ Y2 ) ) ) ) ).
% predicate2D_conj
thf(fact_3843_forall__pos__mono__1,axiom,
! [P3: real > $o,E: real] :
( ! [D3: real,E2: real] :
( ( ord_less @ real @ D3 @ E2 )
=> ( ( P3 @ D3 )
=> ( P3 @ E2 ) ) )
=> ( ! [N3: nat] : ( P3 @ ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ ( suc @ N3 ) ) ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ E )
=> ( P3 @ E ) ) ) ) ).
% forall_pos_mono_1
thf(fact_3844_forall__pos__mono,axiom,
! [P3: real > $o,E: real] :
( ! [D3: real,E2: real] :
( ( ord_less @ real @ D3 @ E2 )
=> ( ( P3 @ D3 )
=> ( P3 @ E2 ) ) )
=> ( ! [N3: nat] :
( ( N3
!= ( zero_zero @ nat ) )
=> ( P3 @ ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ N3 ) ) ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ E )
=> ( P3 @ E ) ) ) ) ).
% forall_pos_mono
thf(fact_3845_real__arch__inverse,axiom,
! [E: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E )
= ( ? [N2: nat] :
( ( N2
!= ( zero_zero @ nat ) )
& ( ord_less @ real @ ( zero_zero @ real ) @ ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ N2 ) ) )
& ( ord_less @ real @ ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ N2 ) ) @ E ) ) ) ) ).
% real_arch_inverse
thf(fact_3846_even__xor__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A,B2: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5824344971392196577ns_xor @ A @ A2 @ B2 ) )
= ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
= ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) ) ) ).
% even_xor_iff
thf(fact_3847_sqrt__divide__self__eq,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( divide_divide @ real @ ( sqrt @ X ) @ X )
= ( inverse_inverse @ real @ ( sqrt @ X ) ) ) ) ).
% sqrt_divide_self_eq
thf(fact_3848_summable__exp,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A] :
( summable @ A
@ ^ [N2: nat] : ( times_times @ A @ ( inverse_inverse @ A @ ( semiring_char_0_fact @ A @ N2 ) ) @ ( power_power @ A @ X @ N2 ) ) ) ) ).
% summable_exp
thf(fact_3849_fun__cong__unused__0,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( zero @ B )
=> ! [F3: ( A > B ) > C,G: C] :
( ( F3
= ( ^ [X4: A > B] : G ) )
=> ( ( F3
@ ^ [X4: A] : ( zero_zero @ B ) )
= G ) ) ) ).
% fun_cong_unused_0
thf(fact_3850_ex__inverse__of__nat__less,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X )
=> ? [N3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
& ( ord_less @ A @ ( inverse_inverse @ A @ ( semiring_1_of_nat @ A @ N3 ) ) @ X ) ) ) ) ).
% ex_inverse_of_nat_less
thf(fact_3851_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_3852_log__inverse,axiom,
! [A2: real,X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
=> ( ( A2
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( log @ A2 @ ( inverse_inverse @ real @ X ) )
= ( uminus_uminus @ real @ ( log @ A2 @ X ) ) ) ) ) ) ).
% log_inverse
thf(fact_3853_eq__subset,axiom,
! [A: $tType,P3: A > A > $o] :
( ord_less_eq @ ( A > A > $o )
@ ^ [Y5: A,Z2: A] : Y5 = Z2
@ ^ [A5: A,B4: A] :
( ( P3 @ A5 @ B4 )
| ( A5 = B4 ) ) ) ).
% eq_subset
thf(fact_3854_exp__plus__inverse__exp,axiom,
! [X: real] : ( ord_less_eq @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( plus_plus @ real @ ( exp @ real @ X ) @ ( inverse_inverse @ real @ ( exp @ real @ X ) ) ) ) ).
% exp_plus_inverse_exp
thf(fact_3855_plus__inverse__ge__2,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ord_less_eq @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( plus_plus @ real @ X @ ( inverse_inverse @ real @ X ) ) ) ) ).
% plus_inverse_ge_2
thf(fact_3856_real__inv__sqrt__pow2,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( power_power @ real @ ( inverse_inverse @ real @ ( sqrt @ X ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( inverse_inverse @ real @ X ) ) ) ).
% real_inv_sqrt_pow2
thf(fact_3857_tan__cot,axiom,
! [X: real] :
( ( tan @ real @ ( minus_minus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ X ) )
= ( inverse_inverse @ real @ ( tan @ real @ X ) ) ) ).
% tan_cot
thf(fact_3858_xor__nat__unfold,axiom,
( ( bit_se5824344971392196577ns_xor @ nat )
= ( ^ [M2: nat,N2: nat] :
( if @ nat
@ ( M2
= ( zero_zero @ nat ) )
@ N2
@ ( if @ nat
@ ( N2
= ( 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 @ N2 @ ( 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 @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% xor_nat_unfold
thf(fact_3859_real__le__x__sinh,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ord_less_eq @ real @ X @ ( divide_divide @ real @ ( minus_minus @ real @ ( exp @ real @ X ) @ ( inverse_inverse @ real @ ( exp @ real @ X ) ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ).
% real_le_x_sinh
thf(fact_3860_xor__nat__rec,axiom,
( ( bit_se5824344971392196577ns_xor @ nat )
= ( ^ [M2: nat,N2: 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 ) ) @ N2 ) ) ) )
@ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( bit_se5824344971392196577ns_xor @ nat @ ( divide_divide @ nat @ M2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% xor_nat_rec
thf(fact_3861_xor__one__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A] :
( ( bit_se5824344971392196577ns_xor @ A @ A2 @ ( one_one @ A ) )
= ( minus_minus @ A @ ( plus_plus @ A @ A2 @ ( zero_neq_one_of_bool @ A @ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) ) )
@ ( zero_neq_one_of_bool @ A
@ ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) ) ) ) ) ).
% xor_one_eq
thf(fact_3862_one__xor__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A] :
( ( bit_se5824344971392196577ns_xor @ A @ ( one_one @ A ) @ A2 )
= ( minus_minus @ A @ ( plus_plus @ A @ A2 @ ( zero_neq_one_of_bool @ A @ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) ) )
@ ( zero_neq_one_of_bool @ A
@ ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) ) ) ) ) ).
% one_xor_eq
thf(fact_3863_real__le__abs__sinh,axiom,
! [X: real] : ( ord_less_eq @ real @ ( abs_abs @ real @ X ) @ ( abs_abs @ real @ ( divide_divide @ real @ ( minus_minus @ real @ ( exp @ real @ X ) @ ( inverse_inverse @ real @ ( exp @ real @ X ) ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ).
% real_le_abs_sinh
thf(fact_3864_tan__sec,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A] :
( ( ( cos @ A @ X )
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( one_one @ A ) @ ( power_power @ A @ ( tan @ A @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( power_power @ A @ ( inverse_inverse @ A @ ( cos @ A @ X ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% tan_sec
thf(fact_3865_case__prod__Pair__iden,axiom,
! [B: $tType,A: $tType,P5: product_prod @ A @ B] :
( ( product_case_prod @ A @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B ) @ P5 )
= P5 ) ).
% case_prod_Pair_iden
thf(fact_3866_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_3867_exp__first__two__terms,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ( ( exp @ A )
= ( ^ [X4: A] :
( plus_plus @ A @ ( plus_plus @ A @ ( one_one @ A ) @ X4 )
@ ( suminf @ A
@ ^ [N2: nat] : ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ ( plus_plus @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( power_power @ A @ X4 @ ( plus_plus @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% exp_first_two_terms
thf(fact_3868_Arg__def,axiom,
( arg
= ( ^ [Z6: complex] :
( if @ real
@ ( Z6
= ( zero_zero @ complex ) )
@ ( zero_zero @ real )
@ ( fChoice @ real
@ ^ [A5: real] :
( ( ( sgn_sgn @ complex @ Z6 )
= ( cis @ A5 ) )
& ( ord_less @ real @ ( uminus_uminus @ real @ pi ) @ A5 )
& ( ord_less_eq @ real @ A5 @ pi ) ) ) ) ) ) ).
% Arg_def
thf(fact_3869_sinh__ln__real,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( sinh @ real @ ( ln_ln @ real @ X ) )
= ( divide_divide @ real @ ( minus_minus @ real @ X @ ( inverse_inverse @ real @ X ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ).
% sinh_ln_real
thf(fact_3870_cosh__ln__real,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( cosh @ real @ ( ln_ln @ real @ X ) )
= ( divide_divide @ real @ ( plus_plus @ real @ X @ ( inverse_inverse @ real @ X ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ).
% cosh_ln_real
thf(fact_3871_mult__scaleR__left,axiom,
! [A: $tType] :
( ( real_V6157519004096292374lgebra @ A )
=> ! [A2: real,X: A,Y2: A] :
( ( times_times @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ X ) @ Y2 )
= ( real_V8093663219630862766scaleR @ A @ A2 @ ( times_times @ A @ X @ Y2 ) ) ) ) ).
% mult_scaleR_left
thf(fact_3872_mult__scaleR__right,axiom,
! [A: $tType] :
( ( real_V6157519004096292374lgebra @ A )
=> ! [X: A,A2: real,Y2: A] :
( ( times_times @ A @ X @ ( real_V8093663219630862766scaleR @ A @ A2 @ Y2 ) )
= ( real_V8093663219630862766scaleR @ A @ A2 @ ( times_times @ A @ X @ Y2 ) ) ) ) ).
% mult_scaleR_right
thf(fact_3873_scaleR__one,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [X: A] :
( ( real_V8093663219630862766scaleR @ A @ ( one_one @ real ) @ X )
= X ) ) ).
% scaleR_one
thf(fact_3874_scaleR__scaleR,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [A2: real,B2: real,X: A] :
( ( real_V8093663219630862766scaleR @ A @ A2 @ ( real_V8093663219630862766scaleR @ A @ B2 @ X ) )
= ( real_V8093663219630862766scaleR @ A @ ( times_times @ real @ A2 @ B2 ) @ X ) ) ) ).
% scaleR_scaleR
thf(fact_3875_sinh__real__le__iff,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq @ real @ ( sinh @ real @ X ) @ ( sinh @ real @ Y2 ) )
= ( ord_less_eq @ real @ X @ Y2 ) ) ).
% sinh_real_le_iff
thf(fact_3876_scaleR__eq__iff,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [B2: A,U: real,A2: A] :
( ( ( plus_plus @ A @ B2 @ ( real_V8093663219630862766scaleR @ A @ U @ A2 ) )
= ( plus_plus @ A @ A2 @ ( real_V8093663219630862766scaleR @ A @ U @ B2 ) ) )
= ( ( A2 = B2 )
| ( U
= ( one_one @ real ) ) ) ) ) ).
% scaleR_eq_iff
thf(fact_3877_cosh__0,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ( ( cosh @ A @ ( zero_zero @ A ) )
= ( one_one @ A ) ) ) ).
% cosh_0
thf(fact_3878_scaleR__power,axiom,
! [A: $tType] :
( ( real_V2822296259951069270ebra_1 @ A )
=> ! [X: real,Y2: A,N: nat] :
( ( power_power @ A @ ( real_V8093663219630862766scaleR @ A @ X @ Y2 ) @ N )
= ( real_V8093663219630862766scaleR @ A @ ( power_power @ real @ X @ N ) @ ( power_power @ A @ Y2 @ N ) ) ) ) ).
% scaleR_power
thf(fact_3879_sinh__real__nonneg__iff,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( sinh @ real @ X ) )
= ( ord_less_eq @ real @ ( zero_zero @ real ) @ X ) ) ).
% sinh_real_nonneg_iff
thf(fact_3880_sinh__real__nonpos__iff,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( sinh @ real @ X ) @ ( zero_zero @ real ) )
= ( ord_less_eq @ real @ X @ ( zero_zero @ real ) ) ) ).
% sinh_real_nonpos_iff
thf(fact_3881_xor__nonnegative__int__iff,axiom,
! [K2: int,L: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se5824344971392196577ns_xor @ int @ K2 @ L ) )
= ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K2 )
= ( ord_less_eq @ int @ ( zero_zero @ int ) @ L ) ) ) ).
% xor_nonnegative_int_iff
thf(fact_3882_xor__negative__int__iff,axiom,
! [K2: int,L: int] :
( ( ord_less @ int @ ( bit_se5824344971392196577ns_xor @ int @ K2 @ L ) @ ( zero_zero @ int ) )
= ( ( ord_less @ int @ K2 @ ( zero_zero @ int ) )
!= ( ord_less @ int @ L @ ( zero_zero @ int ) ) ) ) ).
% xor_negative_int_iff
thf(fact_3883_scaleR__minus1__left,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [X: A] :
( ( real_V8093663219630862766scaleR @ A @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X )
= ( uminus_uminus @ A @ X ) ) ) ).
% scaleR_minus1_left
thf(fact_3884_scaleR__collapse,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [U: real,A2: A] :
( ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ ( minus_minus @ real @ ( one_one @ real ) @ U ) @ A2 ) @ ( real_V8093663219630862766scaleR @ A @ U @ A2 ) )
= A2 ) ) ).
% scaleR_collapse
thf(fact_3885_norm__scaleR,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [A2: real,X: A] :
( ( real_V7770717601297561774m_norm @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ X ) )
= ( times_times @ real @ ( abs_abs @ real @ A2 ) @ ( real_V7770717601297561774m_norm @ A @ X ) ) ) ) ).
% norm_scaleR
thf(fact_3886_scaleR__times,axiom,
! [A: $tType] :
( ( real_V2191834092415804123ebra_1 @ A )
=> ! [U: num,W: num,A2: A] :
( ( real_V8093663219630862766scaleR @ A @ ( numeral_numeral @ real @ U ) @ ( times_times @ A @ ( numeral_numeral @ A @ W ) @ A2 ) )
= ( real_V8093663219630862766scaleR @ A @ ( times_times @ real @ ( numeral_numeral @ real @ U ) @ ( numeral_numeral @ real @ W ) ) @ A2 ) ) ) ).
% scaleR_times
thf(fact_3887_inverse__scaleR__times,axiom,
! [A: $tType] :
( ( real_V2191834092415804123ebra_1 @ A )
=> ! [V2: num,W: num,A2: A] :
( ( real_V8093663219630862766scaleR @ A @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ V2 ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ W ) @ A2 ) )
= ( real_V8093663219630862766scaleR @ A @ ( divide_divide @ real @ ( numeral_numeral @ real @ W ) @ ( numeral_numeral @ real @ V2 ) ) @ A2 ) ) ) ).
% inverse_scaleR_times
thf(fact_3888_fraction__scaleR__times,axiom,
! [A: $tType] :
( ( real_V2191834092415804123ebra_1 @ A )
=> ! [U: num,V2: num,W: num,A2: A] :
( ( real_V8093663219630862766scaleR @ A @ ( divide_divide @ real @ ( numeral_numeral @ real @ U ) @ ( numeral_numeral @ real @ V2 ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ W ) @ A2 ) )
= ( real_V8093663219630862766scaleR @ A @ ( divide_divide @ real @ ( times_times @ real @ ( numeral_numeral @ real @ U ) @ ( numeral_numeral @ real @ W ) ) @ ( numeral_numeral @ real @ V2 ) ) @ A2 ) ) ) ).
% fraction_scaleR_times
thf(fact_3889_scaleR__half__double,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [A2: A] :
( ( real_V8093663219630862766scaleR @ A @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( plus_plus @ A @ A2 @ A2 ) )
= A2 ) ) ).
% scaleR_half_double
thf(fact_3890_sinh__le__cosh__real,axiom,
! [X: real] : ( ord_less_eq @ real @ ( sinh @ real @ X ) @ ( cosh @ real @ X ) ) ).
% sinh_le_cosh_real
thf(fact_3891_bit__xor__int__iff,axiom,
! [K2: int,L: int,N: nat] :
( ( bit_se5641148757651400278ts_bit @ int @ ( bit_se5824344971392196577ns_xor @ int @ K2 @ L ) @ N )
= ( ( bit_se5641148757651400278ts_bit @ int @ K2 @ N )
!= ( bit_se5641148757651400278ts_bit @ int @ L @ N ) ) ) ).
% bit_xor_int_iff
thf(fact_3892_cosh__plus__sinh,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X: A] :
( ( plus_plus @ A @ ( cosh @ A @ X ) @ ( sinh @ A @ X ) )
= ( exp @ A @ X ) ) ) ).
% cosh_plus_sinh
thf(fact_3893_sinh__plus__cosh,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X: A] :
( ( plus_plus @ A @ ( sinh @ A @ X ) @ ( cosh @ A @ X ) )
= ( exp @ A @ X ) ) ) ).
% sinh_plus_cosh
thf(fact_3894_verit__sko__ex_H,axiom,
! [A: $tType,P3: A > $o,A3: $o] :
( ( ( P3 @ ( fChoice @ A @ P3 ) )
= A3 )
=> ( ( ? [X6: A] : ( P3 @ X6 ) )
= A3 ) ) ).
% verit_sko_ex'
thf(fact_3895_verit__sko__forall,axiom,
! [A: $tType] :
( ( ^ [P: A > $o] :
! [X3: A] : ( P @ X3 ) )
= ( ^ [P2: A > $o] :
( P2
@ ( fChoice @ A
@ ^ [X4: A] :
~ ( P2 @ X4 ) ) ) ) ) ).
% verit_sko_forall
thf(fact_3896_verit__sko__forall_H,axiom,
! [A: $tType,P3: A > $o,A3: $o] :
( ( ( P3
@ ( fChoice @ A
@ ^ [X4: A] :
~ ( P3 @ X4 ) ) )
= A3 )
=> ( ( ! [X6: A] : ( P3 @ X6 ) )
= A3 ) ) ).
% verit_sko_forall'
thf(fact_3897_verit__sko__forall_H_H,axiom,
! [A: $tType,B5: A,A3: A,P3: A > $o] :
( ( B5 = A3 )
=> ( ( ( fChoice @ A @ P3 )
= A3 )
= ( ( fChoice @ A @ P3 )
= B5 ) ) ) ).
% verit_sko_forall''
thf(fact_3898_verit__sko__ex__indirect,axiom,
! [A: $tType,X: A,P3: A > $o] :
( ( X
= ( fChoice @ A @ P3 ) )
=> ( ( ? [X6: A] : ( P3 @ X6 ) )
= ( P3 @ X ) ) ) ).
% verit_sko_ex_indirect
thf(fact_3899_verit__sko__ex__indirect2,axiom,
! [A: $tType,X: A,P3: A > $o,P6: A > $o] :
( ( X
= ( fChoice @ A @ P3 ) )
=> ( ! [X5: A] :
( ( P3 @ X5 )
= ( P6 @ X5 ) )
=> ( ( ? [X6: A] : ( P6 @ X6 ) )
= ( P3 @ X ) ) ) ) ).
% verit_sko_ex_indirect2
thf(fact_3900_verit__sko__forall__indirect,axiom,
! [A: $tType,X: A,P3: A > $o] :
( ( X
= ( fChoice @ A
@ ^ [X4: A] :
~ ( P3 @ X4 ) ) )
=> ( ( ! [X6: A] : ( P3 @ X6 ) )
= ( P3 @ X ) ) ) ).
% verit_sko_forall_indirect
thf(fact_3901_verit__sko__forall__indirect2,axiom,
! [A: $tType,X: A,P3: A > $o,P6: A > $o] :
( ( X
= ( fChoice @ A
@ ^ [X4: A] :
~ ( P3 @ X4 ) ) )
=> ( ! [X5: A] :
( ( P3 @ X5 )
= ( P6 @ X5 ) )
=> ( ( ! [X6: A] : ( P6 @ X6 ) )
= ( P3 @ X ) ) ) ) ).
% verit_sko_forall_indirect2
thf(fact_3902_scaleR__right__distrib,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [A2: real,X: A,Y2: A] :
( ( real_V8093663219630862766scaleR @ A @ A2 @ ( plus_plus @ A @ X @ Y2 ) )
= ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ X ) @ ( real_V8093663219630862766scaleR @ A @ A2 @ Y2 ) ) ) ) ).
% scaleR_right_distrib
thf(fact_3903_sinh__diff,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A,Y2: A] :
( ( sinh @ A @ ( minus_minus @ A @ X @ Y2 ) )
= ( minus_minus @ A @ ( times_times @ A @ ( sinh @ A @ X ) @ ( cosh @ A @ Y2 ) ) @ ( times_times @ A @ ( cosh @ A @ X ) @ ( sinh @ A @ Y2 ) ) ) ) ) ).
% sinh_diff
thf(fact_3904_cosh__diff,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A,Y2: A] :
( ( cosh @ A @ ( minus_minus @ A @ X @ Y2 ) )
= ( minus_minus @ A @ ( times_times @ A @ ( cosh @ A @ X ) @ ( cosh @ A @ Y2 ) ) @ ( times_times @ A @ ( sinh @ A @ X ) @ ( sinh @ A @ Y2 ) ) ) ) ) ).
% cosh_diff
thf(fact_3905_sinh__add,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A,Y2: A] :
( ( sinh @ A @ ( plus_plus @ A @ X @ Y2 ) )
= ( plus_plus @ A @ ( times_times @ A @ ( sinh @ A @ X ) @ ( cosh @ A @ Y2 ) ) @ ( times_times @ A @ ( cosh @ A @ X ) @ ( sinh @ A @ Y2 ) ) ) ) ) ).
% sinh_add
thf(fact_3906_cosh__add,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A,Y2: A] :
( ( cosh @ A @ ( plus_plus @ A @ X @ Y2 ) )
= ( plus_plus @ A @ ( times_times @ A @ ( cosh @ A @ X ) @ ( cosh @ A @ Y2 ) ) @ ( times_times @ A @ ( sinh @ A @ X ) @ ( sinh @ A @ Y2 ) ) ) ) ) ).
% cosh_add
thf(fact_3907_real__scaleR__def,axiom,
( ( real_V8093663219630862766scaleR @ real )
= ( times_times @ real ) ) ).
% real_scaleR_def
thf(fact_3908_tanh__def,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ( ( tanh @ A )
= ( ^ [X4: A] : ( divide_divide @ A @ ( sinh @ A @ X4 ) @ ( cosh @ A @ X4 ) ) ) ) ) ).
% tanh_def
thf(fact_3909_summable__scaleR__right,axiom,
! [B: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [X9: nat > B,R: real] :
( ( summable @ B @ X9 )
=> ( summable @ B
@ ^ [N2: nat] : ( real_V8093663219630862766scaleR @ B @ R @ ( X9 @ N2 ) ) ) ) ) ).
% summable_scaleR_right
thf(fact_3910_sums__scaleR__right,axiom,
! [B: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [X9: nat > B,A2: B,R: real] :
( ( sums @ B @ X9 @ A2 )
=> ( sums @ B
@ ^ [N2: nat] : ( real_V8093663219630862766scaleR @ B @ R @ ( X9 @ N2 ) )
@ ( real_V8093663219630862766scaleR @ B @ R @ A2 ) ) ) ) ).
% sums_scaleR_right
thf(fact_3911_XOR__lower,axiom,
! [X: int,Y2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y2 )
=> ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se5824344971392196577ns_xor @ int @ X @ Y2 ) ) ) ) ).
% XOR_lower
thf(fact_3912_scaleR__left_Oadd,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [X: real,Y2: real,Xa: A] :
( ( real_V8093663219630862766scaleR @ A @ ( plus_plus @ real @ X @ Y2 ) @ Xa )
= ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ X @ Xa ) @ ( real_V8093663219630862766scaleR @ A @ Y2 @ Xa ) ) ) ) ).
% scaleR_left.add
thf(fact_3913_scaleR__left__distrib,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [A2: real,B2: real,X: A] :
( ( real_V8093663219630862766scaleR @ A @ ( plus_plus @ real @ A2 @ B2 ) @ X )
= ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ X ) @ ( real_V8093663219630862766scaleR @ A @ B2 @ X ) ) ) ) ).
% scaleR_left_distrib
thf(fact_3914_cosh__real__nonneg,axiom,
! [X: real] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( cosh @ real @ X ) ) ).
% cosh_real_nonneg
thf(fact_3915_cosh__real__nonneg__le__iff,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y2 )
=> ( ( ord_less_eq @ real @ ( cosh @ real @ X ) @ ( cosh @ real @ Y2 ) )
= ( ord_less_eq @ real @ X @ Y2 ) ) ) ) ).
% cosh_real_nonneg_le_iff
thf(fact_3916_cosh__real__nonpos__le__iff,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq @ real @ X @ ( zero_zero @ real ) )
=> ( ( ord_less_eq @ real @ Y2 @ ( zero_zero @ real ) )
=> ( ( ord_less_eq @ real @ ( cosh @ real @ X ) @ ( cosh @ real @ Y2 ) )
= ( ord_less_eq @ real @ Y2 @ X ) ) ) ) ).
% cosh_real_nonpos_le_iff
thf(fact_3917_scaleR__conv__of__real,axiom,
! [A: $tType] :
( ( real_V2191834092415804123ebra_1 @ A )
=> ( ( real_V8093663219630862766scaleR @ A )
= ( ^ [R5: real] : ( times_times @ A @ ( real_Vector_of_real @ A @ R5 ) ) ) ) ) ).
% scaleR_conv_of_real
thf(fact_3918_of__real__def,axiom,
! [A: $tType] :
( ( real_V2191834092415804123ebra_1 @ A )
=> ( ( real_Vector_of_real @ A )
= ( ^ [R5: real] : ( real_V8093663219630862766scaleR @ A @ R5 @ ( one_one @ A ) ) ) ) ) ).
% of_real_def
thf(fact_3919_cosh__real__ge__1,axiom,
! [X: real] : ( ord_less_eq @ real @ ( one_one @ real ) @ ( cosh @ real @ X ) ) ).
% cosh_real_ge_1
thf(fact_3920_sinh__double,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A] :
( ( sinh @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X ) )
= ( times_times @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( sinh @ A @ X ) ) @ ( cosh @ A @ X ) ) ) ) ).
% sinh_double
thf(fact_3921_complex__scaleR,axiom,
! [R: real,A2: real,B2: real] :
( ( real_V8093663219630862766scaleR @ complex @ R @ ( complex2 @ A2 @ B2 ) )
= ( complex2 @ ( times_times @ real @ R @ A2 ) @ ( times_times @ real @ R @ B2 ) ) ) ).
% complex_scaleR
thf(fact_3922_divide__complex__def,axiom,
( ( divide_divide @ complex )
= ( ^ [X4: complex,Y: complex] : ( times_times @ complex @ X4 @ ( inverse_inverse @ complex @ Y ) ) ) ) ).
% divide_complex_def
thf(fact_3923_suminf__scaleR__right,axiom,
! [B: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [X9: nat > B,R: real] :
( ( summable @ B @ X9 )
=> ( ( real_V8093663219630862766scaleR @ B @ R @ ( suminf @ B @ X9 ) )
= ( suminf @ B
@ ^ [N2: nat] : ( real_V8093663219630862766scaleR @ B @ R @ ( X9 @ N2 ) ) ) ) ) ) ).
% suminf_scaleR_right
thf(fact_3924_summable__scaleR__left,axiom,
! [B: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [X9: nat > real,X: B] :
( ( summable @ real @ X9 )
=> ( summable @ B
@ ^ [N2: nat] : ( real_V8093663219630862766scaleR @ B @ ( X9 @ N2 ) @ X ) ) ) ) ).
% summable_scaleR_left
thf(fact_3925_sums__scaleR__left,axiom,
! [B: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [X9: nat > real,A2: real,X: B] :
( ( sums @ real @ X9 @ A2 )
=> ( sums @ B
@ ^ [N2: nat] : ( real_V8093663219630862766scaleR @ B @ ( X9 @ N2 ) @ X )
@ ( real_V8093663219630862766scaleR @ B @ A2 @ X ) ) ) ) ).
% sums_scaleR_left
thf(fact_3926_scaleR__right__mono__neg,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [B2: real,A2: real,C2: A] :
( ( ord_less_eq @ real @ B2 @ A2 )
=> ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ C2 ) @ ( real_V8093663219630862766scaleR @ A @ B2 @ C2 ) ) ) ) ) ).
% scaleR_right_mono_neg
thf(fact_3927_scaleR__right__mono,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A2: real,B2: real,X: A] :
( ( ord_less_eq @ real @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ X ) @ ( real_V8093663219630862766scaleR @ A @ B2 @ X ) ) ) ) ) ).
% scaleR_right_mono
thf(fact_3928_scaleR__le__cancel__left__pos,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C2: real,A2: A,B2: A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ C2 )
=> ( ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ C2 @ A2 ) @ ( real_V8093663219630862766scaleR @ A @ C2 @ B2 ) )
= ( ord_less_eq @ A @ A2 @ B2 ) ) ) ) ).
% scaleR_le_cancel_left_pos
thf(fact_3929_scaleR__le__cancel__left__neg,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C2: real,A2: A,B2: A] :
( ( ord_less @ real @ C2 @ ( zero_zero @ real ) )
=> ( ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ C2 @ A2 ) @ ( real_V8093663219630862766scaleR @ A @ C2 @ B2 ) )
= ( ord_less_eq @ A @ B2 @ A2 ) ) ) ) ).
% scaleR_le_cancel_left_neg
thf(fact_3930_scaleR__le__cancel__left,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C2: real,A2: A,B2: A] :
( ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ C2 @ A2 ) @ ( real_V8093663219630862766scaleR @ A @ C2 @ B2 ) )
= ( ( ( ord_less @ real @ ( zero_zero @ real ) @ C2 )
=> ( ord_less_eq @ A @ A2 @ B2 ) )
& ( ( ord_less @ real @ C2 @ ( zero_zero @ real ) )
=> ( ord_less_eq @ A @ B2 @ A2 ) ) ) ) ) ).
% scaleR_le_cancel_left
thf(fact_3931_scaleR__left__mono__neg,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [B2: A,A2: A,C2: real] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_less_eq @ real @ C2 @ ( zero_zero @ real ) )
=> ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ C2 @ A2 ) @ ( real_V8093663219630862766scaleR @ A @ C2 @ B2 ) ) ) ) ) ).
% scaleR_left_mono_neg
thf(fact_3932_scaleR__left__mono,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [X: A,Y2: A,A2: real] :
( ( ord_less_eq @ A @ X @ Y2 )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ A2 )
=> ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ X ) @ ( real_V8093663219630862766scaleR @ A @ A2 @ Y2 ) ) ) ) ) ).
% scaleR_left_mono
thf(fact_3933_Real__Vector__Spaces_Ole__add__iff2,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A2: real,E: A,C2: A,B2: real,D2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ E ) @ C2 ) @ ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ B2 @ E ) @ D2 ) )
= ( ord_less_eq @ A @ C2 @ ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ ( minus_minus @ real @ B2 @ A2 ) @ E ) @ D2 ) ) ) ) ).
% Real_Vector_Spaces.le_add_iff2
thf(fact_3934_Real__Vector__Spaces_Ole__add__iff1,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A2: real,E: A,C2: A,B2: real,D2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ E ) @ C2 ) @ ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ B2 @ E ) @ D2 ) )
= ( ord_less_eq @ A @ ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ ( minus_minus @ real @ A2 @ B2 ) @ E ) @ C2 ) @ D2 ) ) ) ).
% Real_Vector_Spaces.le_add_iff1
thf(fact_3935_cosh__real__nonpos__less__iff,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq @ real @ X @ ( zero_zero @ real ) )
=> ( ( ord_less_eq @ real @ Y2 @ ( zero_zero @ real ) )
=> ( ( ord_less @ real @ ( cosh @ real @ X ) @ ( cosh @ real @ Y2 ) )
= ( ord_less @ real @ Y2 @ X ) ) ) ) ).
% cosh_real_nonpos_less_iff
thf(fact_3936_cosh__real__nonneg__less__iff,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y2 )
=> ( ( ord_less @ real @ ( cosh @ real @ X ) @ ( cosh @ real @ Y2 ) )
= ( ord_less @ real @ X @ Y2 ) ) ) ) ).
% cosh_real_nonneg_less_iff
thf(fact_3937_cosh__real__strict__mono,axiom,
! [X: real,Y2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ X @ Y2 )
=> ( ord_less @ real @ ( cosh @ real @ X ) @ ( cosh @ real @ Y2 ) ) ) ) ).
% cosh_real_strict_mono
thf(fact_3938_cosh__square__eq,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A] :
( ( power_power @ A @ ( cosh @ A @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( plus_plus @ A @ ( power_power @ A @ ( sinh @ A @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) ) ) ) ).
% cosh_square_eq
thf(fact_3939_sinh__square__eq,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A] :
( ( power_power @ A @ ( sinh @ A @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( minus_minus @ A @ ( power_power @ A @ ( cosh @ A @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) ) ) ) ).
% sinh_square_eq
thf(fact_3940_hyperbolic__pythagoras,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A] :
( ( minus_minus @ A @ ( power_power @ A @ ( cosh @ A @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ ( sinh @ A @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( one_one @ A ) ) ) ).
% hyperbolic_pythagoras
thf(fact_3941_xor__nat__def,axiom,
( ( bit_se5824344971392196577ns_xor @ nat )
= ( ^ [M2: nat,N2: nat] : ( nat2 @ ( bit_se5824344971392196577ns_xor @ int @ ( semiring_1_of_nat @ int @ M2 ) @ ( semiring_1_of_nat @ int @ N2 ) ) ) ) ) ).
% xor_nat_def
thf(fact_3942_arcosh__cosh__real,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( arcosh @ real @ ( cosh @ real @ X ) )
= X ) ) ).
% arcosh_cosh_real
thf(fact_3943_cosh__def,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ( ( cosh @ A )
= ( ^ [X4: A] : ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( plus_plus @ A @ ( exp @ A @ X4 ) @ ( exp @ A @ ( uminus_uminus @ A @ X4 ) ) ) ) ) ) ) ).
% cosh_def
thf(fact_3944_sinh__def,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ( ( sinh @ A )
= ( ^ [X4: A] : ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( minus_minus @ A @ ( exp @ A @ X4 ) @ ( exp @ A @ ( uminus_uminus @ A @ X4 ) ) ) ) ) ) ) ).
% sinh_def
thf(fact_3945_cosh__double,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A] :
( ( cosh @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X ) )
= ( plus_plus @ A @ ( power_power @ A @ ( cosh @ A @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ ( sinh @ A @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% cosh_double
thf(fact_3946_suminf__scaleR__left,axiom,
! [B: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [X9: nat > real,X: B] :
( ( summable @ real @ X9 )
=> ( ( real_V8093663219630862766scaleR @ B @ ( suminf @ real @ X9 ) @ X )
= ( suminf @ B
@ ^ [N2: nat] : ( real_V8093663219630862766scaleR @ B @ ( X9 @ N2 ) @ X ) ) ) ) ) ).
% suminf_scaleR_left
thf(fact_3947_zero__le__scaleR__iff,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A2: real,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( real_V8093663219630862766scaleR @ A @ A2 @ B2 ) )
= ( ( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) )
| ( ( ord_less @ real @ A2 @ ( zero_zero @ real ) )
& ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) )
| ( A2
= ( zero_zero @ real ) ) ) ) ) ).
% zero_le_scaleR_iff
thf(fact_3948_scaleR__le__0__iff,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A2: real,B2: A] :
( ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ B2 ) @ ( zero_zero @ A ) )
= ( ( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
& ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) )
| ( ( ord_less @ real @ A2 @ ( zero_zero @ real ) )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) )
| ( A2
= ( zero_zero @ real ) ) ) ) ) ).
% scaleR_le_0_iff
thf(fact_3949_scaleR__mono,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A2: real,B2: real,X: A,Y2: A] :
( ( ord_less_eq @ real @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ X @ Y2 )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ B2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ X ) @ ( real_V8093663219630862766scaleR @ A @ B2 @ Y2 ) ) ) ) ) ) ) ).
% scaleR_mono
thf(fact_3950_scaleR__mono_H,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A2: real,B2: real,C2: A,D2: A] :
( ( ord_less_eq @ real @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ D2 )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ C2 ) @ ( real_V8093663219630862766scaleR @ A @ B2 @ D2 ) ) ) ) ) ) ) ).
% scaleR_mono'
thf(fact_3951_split__scaleR__neg__le,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A2: real,X: A] :
( ( ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ A2 )
& ( ord_less_eq @ A @ X @ ( zero_zero @ A ) ) )
| ( ( ord_less_eq @ real @ A2 @ ( zero_zero @ real ) )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ X ) ) )
=> ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ X ) @ ( zero_zero @ A ) ) ) ) ).
% split_scaleR_neg_le
thf(fact_3952_split__scaleR__pos__le,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A2: real,B2: A] :
( ( ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ A2 )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) )
| ( ( ord_less_eq @ real @ A2 @ ( zero_zero @ real ) )
& ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( real_V8093663219630862766scaleR @ A @ A2 @ B2 ) ) ) ) ).
% split_scaleR_pos_le
thf(fact_3953_scaleR__nonneg__nonneg,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A2: real,X: A] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( real_V8093663219630862766scaleR @ A @ A2 @ X ) ) ) ) ) ).
% scaleR_nonneg_nonneg
thf(fact_3954_scaleR__nonneg__nonpos,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A2: real,X: A] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ A2 )
=> ( ( ord_less_eq @ A @ X @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ X ) @ ( zero_zero @ A ) ) ) ) ) ).
% scaleR_nonneg_nonpos
thf(fact_3955_scaleR__nonpos__nonneg,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A2: real,X: A] :
( ( ord_less_eq @ real @ A2 @ ( zero_zero @ real ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ X ) @ ( zero_zero @ A ) ) ) ) ) ).
% scaleR_nonpos_nonneg
thf(fact_3956_scaleR__nonpos__nonpos,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A2: real,B2: A] :
( ( ord_less_eq @ real @ A2 @ ( zero_zero @ real ) )
=> ( ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( real_V8093663219630862766scaleR @ A @ A2 @ B2 ) ) ) ) ) ).
% scaleR_nonpos_nonpos
thf(fact_3957_scaleR__left__le__one__le,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [X: A,A2: real] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less_eq @ real @ A2 @ ( one_one @ real ) )
=> ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ X ) @ X ) ) ) ) ).
% scaleR_left_le_one_le
thf(fact_3958_scaleR__2,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [X: A] :
( ( real_V8093663219630862766scaleR @ A @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ X )
= ( plus_plus @ A @ X @ X ) ) ) ).
% scaleR_2
thf(fact_3959_real__vector__eq__affinity,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [M: real,Y2: A,X: A,C2: A] :
( ( M
!= ( zero_zero @ real ) )
=> ( ( Y2
= ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ M @ X ) @ C2 ) )
= ( ( minus_minus @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ M ) @ Y2 ) @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ M ) @ C2 ) )
= X ) ) ) ) ).
% real_vector_eq_affinity
thf(fact_3960_real__vector__affinity__eq,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [M: real,X: A,C2: A,Y2: A] :
( ( M
!= ( zero_zero @ real ) )
=> ( ( ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ M @ X ) @ C2 )
= Y2 )
= ( X
= ( minus_minus @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ M ) @ Y2 ) @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ M ) @ C2 ) ) ) ) ) ) ).
% real_vector_affinity_eq
thf(fact_3961_pos__divideR__le__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C2: real,B2: A,A2: A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ C2 )
=> ( ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C2 ) @ B2 ) @ A2 )
= ( ord_less_eq @ A @ B2 @ ( real_V8093663219630862766scaleR @ A @ C2 @ A2 ) ) ) ) ) ).
% pos_divideR_le_eq
thf(fact_3962_pos__le__divideR__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C2: real,A2: A,B2: A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ C2 )
=> ( ( ord_less_eq @ A @ A2 @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C2 ) @ B2 ) )
= ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ C2 @ A2 ) @ B2 ) ) ) ) ).
% pos_le_divideR_eq
thf(fact_3963_neg__divideR__le__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C2: real,B2: A,A2: A] :
( ( ord_less @ real @ C2 @ ( zero_zero @ real ) )
=> ( ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C2 ) @ B2 ) @ A2 )
= ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ C2 @ A2 ) @ B2 ) ) ) ) ).
% neg_divideR_le_eq
thf(fact_3964_neg__le__divideR__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C2: real,A2: A,B2: A] :
( ( ord_less @ real @ C2 @ ( zero_zero @ real ) )
=> ( ( ord_less_eq @ A @ A2 @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C2 ) @ B2 ) )
= ( ord_less_eq @ A @ B2 @ ( real_V8093663219630862766scaleR @ A @ C2 @ A2 ) ) ) ) ) ).
% neg_le_divideR_eq
thf(fact_3965_pos__divideR__less__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C2: real,B2: A,A2: A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ C2 )
=> ( ( ord_less @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C2 ) @ B2 ) @ A2 )
= ( ord_less @ A @ B2 @ ( real_V8093663219630862766scaleR @ A @ C2 @ A2 ) ) ) ) ) ).
% pos_divideR_less_eq
thf(fact_3966_pos__less__divideR__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C2: real,A2: A,B2: A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ C2 )
=> ( ( ord_less @ A @ A2 @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C2 ) @ B2 ) )
= ( ord_less @ A @ ( real_V8093663219630862766scaleR @ A @ C2 @ A2 ) @ B2 ) ) ) ) ).
% pos_less_divideR_eq
thf(fact_3967_neg__divideR__less__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C2: real,B2: A,A2: A] :
( ( ord_less @ real @ C2 @ ( zero_zero @ real ) )
=> ( ( ord_less @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C2 ) @ B2 ) @ A2 )
= ( ord_less @ A @ ( real_V8093663219630862766scaleR @ A @ C2 @ A2 ) @ B2 ) ) ) ) ).
% neg_divideR_less_eq
thf(fact_3968_neg__less__divideR__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C2: real,A2: A,B2: A] :
( ( ord_less @ real @ C2 @ ( zero_zero @ real ) )
=> ( ( ord_less @ A @ A2 @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C2 ) @ B2 ) )
= ( ord_less @ A @ B2 @ ( real_V8093663219630862766scaleR @ A @ C2 @ A2 ) ) ) ) ) ).
% neg_less_divideR_eq
thf(fact_3969_summable__exp__generic,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X: A] :
( summable @ A
@ ^ [N2: nat] : ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ N2 ) ) @ ( power_power @ A @ X @ N2 ) ) ) ) ).
% summable_exp_generic
thf(fact_3970_sin__converges,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X: A] :
( sums @ A
@ ^ [N2: nat] : ( real_V8093663219630862766scaleR @ A @ ( sin_coeff @ N2 ) @ ( power_power @ A @ X @ N2 ) )
@ ( sin @ A @ X ) ) ) ).
% sin_converges
thf(fact_3971_sin__def,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ( ( sin @ A )
= ( ^ [X4: A] :
( suminf @ A
@ ^ [N2: nat] : ( real_V8093663219630862766scaleR @ A @ ( sin_coeff @ N2 ) @ ( power_power @ A @ X4 @ N2 ) ) ) ) ) ) ).
% sin_def
thf(fact_3972_cos__converges,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X: A] :
( sums @ A
@ ^ [N2: nat] : ( real_V8093663219630862766scaleR @ A @ ( cos_coeff @ N2 ) @ ( power_power @ A @ X @ N2 ) )
@ ( cos @ A @ X ) ) ) ).
% cos_converges
thf(fact_3973_cos__def,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ( ( cos @ A )
= ( ^ [X4: A] :
( suminf @ A
@ ^ [N2: nat] : ( real_V8093663219630862766scaleR @ A @ ( cos_coeff @ N2 ) @ ( power_power @ A @ X4 @ N2 ) ) ) ) ) ) ).
% cos_def
thf(fact_3974_summable__norm__sin,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X: A] :
( summable @ real
@ ^ [N2: nat] : ( real_V7770717601297561774m_norm @ A @ ( real_V8093663219630862766scaleR @ A @ ( sin_coeff @ N2 ) @ ( power_power @ A @ X @ N2 ) ) ) ) ) ).
% summable_norm_sin
thf(fact_3975_summable__norm__cos,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X: A] :
( summable @ real
@ ^ [N2: nat] : ( real_V7770717601297561774m_norm @ A @ ( real_V8093663219630862766scaleR @ A @ ( cos_coeff @ N2 ) @ ( power_power @ A @ X @ N2 ) ) ) ) ) ).
% summable_norm_cos
thf(fact_3976_cosh__converges,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X: A] :
( sums @ A
@ ^ [N2: nat] : ( if @ A @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ N2 ) ) @ ( power_power @ A @ X @ N2 ) ) @ ( zero_zero @ A ) )
@ ( cosh @ A @ X ) ) ) ).
% cosh_converges
thf(fact_3977_neg__minus__divideR__le__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C2: real,B2: A,A2: A] :
( ( ord_less @ real @ C2 @ ( zero_zero @ real ) )
=> ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C2 ) @ B2 ) ) @ A2 )
= ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ C2 @ A2 ) @ ( uminus_uminus @ A @ B2 ) ) ) ) ) ).
% neg_minus_divideR_le_eq
thf(fact_3978_neg__le__minus__divideR__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C2: real,A2: A,B2: A] :
( ( ord_less @ real @ C2 @ ( zero_zero @ real ) )
=> ( ( ord_less_eq @ A @ A2 @ ( uminus_uminus @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C2 ) @ B2 ) ) )
= ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( real_V8093663219630862766scaleR @ A @ C2 @ A2 ) ) ) ) ) ).
% neg_le_minus_divideR_eq
thf(fact_3979_pos__minus__divideR__le__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C2: real,B2: A,A2: A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ C2 )
=> ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C2 ) @ B2 ) ) @ A2 )
= ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( real_V8093663219630862766scaleR @ A @ C2 @ A2 ) ) ) ) ) ).
% pos_minus_divideR_le_eq
thf(fact_3980_pos__le__minus__divideR__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C2: real,A2: A,B2: A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ C2 )
=> ( ( ord_less_eq @ A @ A2 @ ( uminus_uminus @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C2 ) @ B2 ) ) )
= ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ C2 @ A2 ) @ ( uminus_uminus @ A @ B2 ) ) ) ) ) ).
% pos_le_minus_divideR_eq
thf(fact_3981_neg__minus__divideR__less__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C2: real,B2: A,A2: A] :
( ( ord_less @ real @ C2 @ ( zero_zero @ real ) )
=> ( ( ord_less @ A @ ( uminus_uminus @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C2 ) @ B2 ) ) @ A2 )
= ( ord_less @ A @ ( real_V8093663219630862766scaleR @ A @ C2 @ A2 ) @ ( uminus_uminus @ A @ B2 ) ) ) ) ) ).
% neg_minus_divideR_less_eq
thf(fact_3982_neg__less__minus__divideR__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C2: real,A2: A,B2: A] :
( ( ord_less @ real @ C2 @ ( zero_zero @ real ) )
=> ( ( ord_less @ A @ A2 @ ( uminus_uminus @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C2 ) @ B2 ) ) )
= ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ ( real_V8093663219630862766scaleR @ A @ C2 @ A2 ) ) ) ) ) ).
% neg_less_minus_divideR_eq
thf(fact_3983_pos__minus__divideR__less__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C2: real,B2: A,A2: A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ C2 )
=> ( ( ord_less @ A @ ( uminus_uminus @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C2 ) @ B2 ) ) @ A2 )
= ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ ( real_V8093663219630862766scaleR @ A @ C2 @ A2 ) ) ) ) ) ).
% pos_minus_divideR_less_eq
thf(fact_3984_pos__less__minus__divideR__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C2: real,A2: A,B2: A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ C2 )
=> ( ( ord_less @ A @ A2 @ ( uminus_uminus @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C2 ) @ B2 ) ) )
= ( ord_less @ A @ ( real_V8093663219630862766scaleR @ A @ C2 @ A2 ) @ ( uminus_uminus @ A @ B2 ) ) ) ) ) ).
% pos_less_minus_divideR_eq
thf(fact_3985_sinh__converges,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X: A] :
( sums @ A
@ ^ [N2: nat] : ( if @ A @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) @ ( zero_zero @ A ) @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ N2 ) ) @ ( power_power @ A @ X @ N2 ) ) )
@ ( sinh @ A @ X ) ) ) ).
% sinh_converges
thf(fact_3986_exp__converges,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X: A] :
( sums @ A
@ ^ [N2: nat] : ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ N2 ) ) @ ( power_power @ A @ X @ N2 ) )
@ ( exp @ A @ X ) ) ) ).
% exp_converges
thf(fact_3987_exp__def,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ( ( exp @ A )
= ( ^ [X4: A] :
( suminf @ A
@ ^ [N2: nat] : ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ N2 ) ) @ ( power_power @ A @ X4 @ N2 ) ) ) ) ) ) ).
% exp_def
thf(fact_3988_summable__norm__exp,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X: A] :
( summable @ real
@ ^ [N2: nat] : ( real_V7770717601297561774m_norm @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ N2 ) ) @ ( power_power @ A @ X @ N2 ) ) ) ) ) ).
% summable_norm_exp
thf(fact_3989_sin__minus__converges,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X: A] :
( sums @ A
@ ^ [N2: nat] : ( uminus_uminus @ A @ ( real_V8093663219630862766scaleR @ A @ ( sin_coeff @ N2 ) @ ( power_power @ A @ ( uminus_uminus @ A @ X ) @ N2 ) ) )
@ ( sin @ A @ X ) ) ) ).
% sin_minus_converges
thf(fact_3990_cos__minus__converges,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X: A] :
( sums @ A
@ ^ [N2: nat] : ( real_V8093663219630862766scaleR @ A @ ( cos_coeff @ N2 ) @ ( power_power @ A @ ( uminus_uminus @ A @ X ) @ N2 ) )
@ ( cos @ A @ X ) ) ) ).
% cos_minus_converges
thf(fact_3991_tanh__add,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A,Y2: A] :
( ( ( cosh @ A @ X )
!= ( zero_zero @ A ) )
=> ( ( ( cosh @ A @ Y2 )
!= ( zero_zero @ A ) )
=> ( ( tanh @ A @ ( plus_plus @ A @ X @ Y2 ) )
= ( divide_divide @ A @ ( plus_plus @ A @ ( tanh @ A @ X ) @ ( tanh @ A @ Y2 ) ) @ ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( tanh @ A @ X ) @ ( tanh @ A @ Y2 ) ) ) ) ) ) ) ) ).
% tanh_add
thf(fact_3992_XOR__upper,axiom,
! [X: int,N: nat,Y2: 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 @ Y2 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) )
=> ( ord_less @ int @ ( bit_se5824344971392196577ns_xor @ int @ X @ Y2 ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ) ) ) ).
% XOR_upper
thf(fact_3993_cosh__field__def,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ( ( cosh @ A )
= ( ^ [Z6: A] : ( divide_divide @ A @ ( plus_plus @ A @ ( exp @ A @ Z6 ) @ ( exp @ A @ ( uminus_uminus @ A @ Z6 ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% cosh_field_def
thf(fact_3994_complex__inverse,axiom,
! [A2: real,B2: real] :
( ( inverse_inverse @ complex @ ( complex2 @ A2 @ B2 ) )
= ( complex2 @ ( divide_divide @ real @ A2 @ ( plus_plus @ real @ ( power_power @ real @ A2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ B2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( divide_divide @ real @ ( uminus_uminus @ real @ B2 ) @ ( plus_plus @ real @ ( power_power @ real @ A2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ B2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% complex_inverse
thf(fact_3995_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_3996_sinh__field__def,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ( ( sinh @ A )
= ( ^ [Z6: A] : ( divide_divide @ A @ ( minus_minus @ A @ ( exp @ A @ Z6 ) @ ( exp @ A @ ( uminus_uminus @ A @ Z6 ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% sinh_field_def
thf(fact_3997_exp__first__term,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ( ( exp @ A )
= ( ^ [X4: A] :
( plus_plus @ A @ ( one_one @ A )
@ ( suminf @ A
@ ^ [N2: nat] : ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ ( suc @ N2 ) ) ) @ ( power_power @ A @ X4 @ ( suc @ N2 ) ) ) ) ) ) ) ) ).
% exp_first_term
thf(fact_3998_cosh__zero__iff,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A] :
( ( ( cosh @ A @ X )
= ( zero_zero @ A ) )
= ( ( power_power @ A @ ( exp @ A @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ).
% cosh_zero_iff
thf(fact_3999_some__sym__eq__trivial,axiom,
! [A: $tType,X: A] :
( ( fChoice @ A
@ ( ^ [Y5: A,Z2: A] : Y5 = Z2
@ X ) )
= X ) ).
% some_sym_eq_trivial
thf(fact_4000_some__eq__trivial,axiom,
! [A: $tType,X: A] :
( ( fChoice @ A
@ ^ [Y: A] : Y = X )
= X ) ).
% some_eq_trivial
thf(fact_4001_some__equality,axiom,
! [A: $tType,P3: A > $o,A2: A] :
( ( P3 @ A2 )
=> ( ! [X5: A] :
( ( P3 @ X5 )
=> ( X5 = A2 ) )
=> ( ( fChoice @ A @ P3 )
= A2 ) ) ) ).
% some_equality
thf(fact_4002_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_4003_bit_Odouble__compl,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X: A] :
( ( bit_ri4277139882892585799ns_not @ A @ ( bit_ri4277139882892585799ns_not @ A @ X ) )
= X ) ) ).
% bit.double_compl
thf(fact_4004_bit_Ocompl__eq__compl__iff,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X: A,Y2: A] :
( ( ( bit_ri4277139882892585799ns_not @ A @ X )
= ( bit_ri4277139882892585799ns_not @ A @ Y2 ) )
= ( X = Y2 ) ) ) ).
% bit.compl_eq_compl_iff
thf(fact_4005_bit_Oxor__compl__left,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X: A,Y2: A] :
( ( bit_se5824344971392196577ns_xor @ A @ ( bit_ri4277139882892585799ns_not @ A @ X ) @ Y2 )
= ( bit_ri4277139882892585799ns_not @ A @ ( bit_se5824344971392196577ns_xor @ A @ X @ Y2 ) ) ) ) ).
% bit.xor_compl_left
thf(fact_4006_bit_Oxor__compl__right,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X: A,Y2: A] :
( ( bit_se5824344971392196577ns_xor @ A @ X @ ( bit_ri4277139882892585799ns_not @ A @ Y2 ) )
= ( bit_ri4277139882892585799ns_not @ A @ ( bit_se5824344971392196577ns_xor @ A @ X @ Y2 ) ) ) ) ).
% bit.xor_compl_right
thf(fact_4007_Eps__case__prod__eq,axiom,
! [A: $tType,B: $tType,X: A,Y2: B] :
( ( fChoice @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [X7: A,Y6: B] :
( ( X = X7 )
& ( Y2 = Y6 ) ) ) )
= ( product_Pair @ A @ B @ X @ Y2 ) ) ).
% Eps_case_prod_eq
thf(fact_4008_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_4009_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_4010_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_4011_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_4012_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_4013_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_4014_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_4015_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_4016_not__nonnegative__int__iff,axiom,
! [K2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_ri4277139882892585799ns_not @ int @ K2 ) )
= ( ord_less @ int @ K2 @ ( zero_zero @ int ) ) ) ).
% not_nonnegative_int_iff
thf(fact_4017_not__negative__int__iff,axiom,
! [K2: int] :
( ( ord_less @ int @ ( bit_ri4277139882892585799ns_not @ int @ K2 ) @ ( zero_zero @ int ) )
= ( ord_less_eq @ int @ ( zero_zero @ int ) @ K2 ) ) ).
% not_negative_int_iff
thf(fact_4018_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_4019_even__not__iff,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [A2: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_ri4277139882892585799ns_not @ A @ A2 ) )
= ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) ) ) ) ).
% even_not_iff
thf(fact_4020_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_4021_of__int__not__numeral,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [K2: num] :
( ( ring_1_of_int @ A @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ K2 ) ) )
= ( bit_ri4277139882892585799ns_not @ A @ ( numeral_numeral @ A @ K2 ) ) ) ) ).
% of_int_not_numeral
thf(fact_4022_take__bit__not__take__bit,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: nat,A2: A] :
( ( bit_se2584673776208193580ke_bit @ A @ N @ ( bit_ri4277139882892585799ns_not @ A @ ( bit_se2584673776208193580ke_bit @ A @ N @ A2 ) ) )
= ( bit_se2584673776208193580ke_bit @ A @ N @ ( bit_ri4277139882892585799ns_not @ A @ A2 ) ) ) ) ).
% take_bit_not_take_bit
thf(fact_4023_take__bit__not__iff,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: nat,A2: A,B2: A] :
( ( ( bit_se2584673776208193580ke_bit @ A @ N @ ( bit_ri4277139882892585799ns_not @ A @ A2 ) )
= ( bit_se2584673776208193580ke_bit @ A @ N @ ( bit_ri4277139882892585799ns_not @ A @ B2 ) ) )
= ( ( bit_se2584673776208193580ke_bit @ A @ N @ A2 )
= ( bit_se2584673776208193580ke_bit @ A @ N @ B2 ) ) ) ) ).
% take_bit_not_iff
thf(fact_4024_bit__not__int__iff,axiom,
! [K2: int,N: nat] :
( ( bit_se5641148757651400278ts_bit @ int @ ( bit_ri4277139882892585799ns_not @ int @ K2 ) @ N )
= ( ~ ( bit_se5641148757651400278ts_bit @ int @ K2 @ N ) ) ) ).
% bit_not_int_iff
thf(fact_4025_of__int__not__eq,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [K2: int] :
( ( ring_1_of_int @ A @ ( bit_ri4277139882892585799ns_not @ int @ K2 ) )
= ( bit_ri4277139882892585799ns_not @ A @ ( ring_1_of_int @ A @ K2 ) ) ) ) ).
% of_int_not_eq
thf(fact_4026_split__paired__Eps,axiom,
! [B: $tType,A: $tType] :
( ( fChoice @ ( product_prod @ A @ B ) )
= ( ^ [P2: ( product_prod @ A @ B ) > $o] :
( fChoice @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [A5: A,B4: B] : ( P2 @ ( product_Pair @ A @ B @ A5 @ B4 ) ) ) ) ) ) ).
% split_paired_Eps
thf(fact_4027_not__diff__distrib,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [A2: A,B2: A] :
( ( bit_ri4277139882892585799ns_not @ A @ ( minus_minus @ A @ A2 @ B2 ) )
= ( plus_plus @ A @ ( bit_ri4277139882892585799ns_not @ A @ A2 ) @ B2 ) ) ) ).
% not_diff_distrib
thf(fact_4028_not__add__distrib,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [A2: A,B2: A] :
( ( bit_ri4277139882892585799ns_not @ A @ ( plus_plus @ A @ A2 @ B2 ) )
= ( minus_minus @ A @ ( bit_ri4277139882892585799ns_not @ A @ A2 ) @ B2 ) ) ) ).
% not_add_distrib
thf(fact_4029_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_4030_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_4031_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_4032_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_4033_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_4034_disjunctive__diff,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [B2: A,A2: A] :
( ! [N3: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ B2 @ N3 )
=> ( bit_se5641148757651400278ts_bit @ A @ A2 @ N3 ) )
=> ( ( minus_minus @ A @ A2 @ B2 )
= ( bit_se5824344872417868541ns_and @ A @ A2 @ ( bit_ri4277139882892585799ns_not @ A @ B2 ) ) ) ) ) ).
% disjunctive_diff
thf(fact_4035_take__bit__not__eq__mask__diff,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: nat,A2: A] :
( ( bit_se2584673776208193580ke_bit @ A @ N @ ( bit_ri4277139882892585799ns_not @ A @ A2 ) )
= ( minus_minus @ A @ ( bit_se2239418461657761734s_mask @ A @ N ) @ ( bit_se2584673776208193580ke_bit @ A @ N @ A2 ) ) ) ) ).
% take_bit_not_eq_mask_diff
thf(fact_4036_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_4037_not__int__div__2,axiom,
! [K2: int] :
( ( divide_divide @ int @ ( bit_ri4277139882892585799ns_not @ int @ K2 ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) )
= ( bit_ri4277139882892585799ns_not @ int @ ( divide_divide @ int @ K2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ).
% not_int_div_2
thf(fact_4038_even__not__iff__int,axiom,
! [K2: int] :
( ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_ri4277139882892585799ns_not @ int @ K2 ) )
= ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K2 ) ) ) ).
% even_not_iff_int
thf(fact_4039_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_4040_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_4041_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_4042_bit__minus__int__iff,axiom,
! [K2: int,N: nat] :
( ( bit_se5641148757651400278ts_bit @ int @ ( uminus_uminus @ int @ K2 ) @ N )
= ( bit_se5641148757651400278ts_bit @ int @ ( bit_ri4277139882892585799ns_not @ int @ ( minus_minus @ int @ K2 @ ( one_one @ int ) ) ) @ N ) ) ).
% bit_minus_int_iff
thf(fact_4043_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_4044_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_4045_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_4046_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_4047_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_4048_someI2,axiom,
! [A: $tType,P3: A > $o,A2: A,Q2: A > $o] :
( ( P3 @ A2 )
=> ( ! [X5: A] :
( ( P3 @ X5 )
=> ( Q2 @ X5 ) )
=> ( Q2 @ ( fChoice @ A @ P3 ) ) ) ) ).
% someI2
thf(fact_4049_someI__ex,axiom,
! [A: $tType,P3: A > $o] :
( ? [X_1: A] : ( P3 @ X_1 )
=> ( P3 @ ( fChoice @ A @ P3 ) ) ) ).
% someI_ex
thf(fact_4050_someI2__ex,axiom,
! [A: $tType,P3: A > $o,Q2: A > $o] :
( ? [X_1: A] : ( P3 @ X_1 )
=> ( ! [X5: A] :
( ( P3 @ X5 )
=> ( Q2 @ X5 ) )
=> ( Q2 @ ( fChoice @ A @ P3 ) ) ) ) ).
% someI2_ex
thf(fact_4051_someI2__bex,axiom,
! [A: $tType,A3: set @ A,P3: A > $o,Q2: A > $o] :
( ? [X2: A] :
( ( member @ A @ X2 @ A3 )
& ( P3 @ X2 ) )
=> ( ! [X5: A] :
( ( ( member @ A @ X5 @ A3 )
& ( P3 @ X5 ) )
=> ( Q2 @ X5 ) )
=> ( Q2
@ ( fChoice @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ A3 )
& ( P3 @ X4 ) ) ) ) ) ) ).
% someI2_bex
thf(fact_4052_some__eq__ex,axiom,
! [A: $tType,P3: A > $o] :
( ( P3 @ ( fChoice @ A @ P3 ) )
= ( ? [X6: A] : ( P3 @ X6 ) ) ) ).
% some_eq_ex
thf(fact_4053_some1__equality,axiom,
! [A: $tType,P3: A > $o,A2: A] :
( ? [X2: A] :
( ( P3 @ X2 )
& ! [Y7: A] :
( ( P3 @ Y7 )
=> ( Y7 = X2 ) ) )
=> ( ( P3 @ A2 )
=> ( ( fChoice @ A @ P3 )
= A2 ) ) ) ).
% some1_equality
thf(fact_4054_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_4055_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_4056_bit__not__iff__eq,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [A2: A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( bit_ri4277139882892585799ns_not @ A @ A2 ) @ N )
= ( ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N )
!= ( zero_zero @ A ) )
& ~ ( bit_se5641148757651400278ts_bit @ A @ A2 @ N ) ) ) ) ).
% bit_not_iff_eq
thf(fact_4057_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_4058_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_4059_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_4060_sin__x__sin__y,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A,Y2: A] :
( sums @ A
@ ^ [P4: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [N2: nat] :
( if @ A
@ ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ P4 )
& ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) )
@ ( times_times @ A @ ( real_V8093663219630862766scaleR @ A @ ( uminus_uminus @ real @ ( divide_divide @ real @ ( ring_1_of_int @ real @ ( times_times @ int @ ( power_power @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( divide_divide @ nat @ P4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( semiring_1_of_nat @ int @ ( binomial @ P4 @ N2 ) ) ) ) @ ( semiring_char_0_fact @ real @ P4 ) ) ) @ ( power_power @ A @ X @ N2 ) ) @ ( power_power @ A @ Y2 @ ( minus_minus @ nat @ P4 @ N2 ) ) )
@ ( zero_zero @ A ) )
@ ( set_ord_atMost @ nat @ P4 ) )
@ ( times_times @ A @ ( sin @ A @ X ) @ ( sin @ A @ Y2 ) ) ) ) ).
% sin_x_sin_y
thf(fact_4061_sums__cos__x__plus__y,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A,Y2: A] :
( sums @ A
@ ^ [P4: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [N2: nat] : ( if @ A @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ P4 ) @ ( times_times @ A @ ( real_V8093663219630862766scaleR @ A @ ( divide_divide @ real @ ( ring_1_of_int @ real @ ( times_times @ int @ ( power_power @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( divide_divide @ nat @ P4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( semiring_1_of_nat @ int @ ( binomial @ P4 @ N2 ) ) ) ) @ ( semiring_char_0_fact @ real @ P4 ) ) @ ( power_power @ A @ X @ N2 ) ) @ ( power_power @ A @ Y2 @ ( minus_minus @ nat @ P4 @ N2 ) ) ) @ ( zero_zero @ A ) )
@ ( set_ord_atMost @ nat @ P4 ) )
@ ( cos @ A @ ( plus_plus @ A @ X @ Y2 ) ) ) ) ).
% sums_cos_x_plus_y
thf(fact_4062_cos__x__cos__y,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A,Y2: A] :
( sums @ A
@ ^ [P4: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [N2: nat] :
( if @ A
@ ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ P4 )
& ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) )
@ ( times_times @ A @ ( real_V8093663219630862766scaleR @ A @ ( divide_divide @ real @ ( ring_1_of_int @ real @ ( times_times @ int @ ( power_power @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( divide_divide @ nat @ P4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( semiring_1_of_nat @ int @ ( binomial @ P4 @ N2 ) ) ) ) @ ( semiring_char_0_fact @ real @ P4 ) ) @ ( power_power @ A @ X @ N2 ) ) @ ( power_power @ A @ Y2 @ ( minus_minus @ nat @ P4 @ N2 ) ) )
@ ( zero_zero @ A ) )
@ ( set_ord_atMost @ nat @ P4 ) )
@ ( times_times @ A @ ( cos @ A @ X ) @ ( cos @ A @ Y2 ) ) ) ) ).
% cos_x_cos_y
thf(fact_4063_rat__inverse__code,axiom,
! [P5: rat] :
( ( quotient_of @ ( inverse_inverse @ rat @ P5 ) )
= ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [A5: int,B4: int] :
( if @ ( product_prod @ int @ int )
@ ( A5
= ( zero_zero @ int ) )
@ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
@ ( product_Pair @ int @ int @ ( times_times @ int @ ( sgn_sgn @ int @ A5 ) @ B4 ) @ ( abs_abs @ int @ A5 ) ) )
@ ( quotient_of @ P5 ) ) ) ).
% rat_inverse_code
thf(fact_4064_atMost__iff,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [I2: A,K2: A] :
( ( member @ A @ I2 @ ( set_ord_atMost @ A @ K2 ) )
= ( ord_less_eq @ A @ I2 @ K2 ) ) ) ).
% atMost_iff
thf(fact_4065_of__nat__sum,axiom,
! [A: $tType,B: $tType] :
( ( semiring_1 @ A )
=> ! [F3: B > nat,A3: set @ B] :
( ( semiring_1_of_nat @ A @ ( groups7311177749621191930dd_sum @ B @ nat @ F3 @ A3 ) )
= ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X4: B] : ( semiring_1_of_nat @ A @ ( F3 @ X4 ) )
@ A3 ) ) ) ).
% of_nat_sum
thf(fact_4066_of__int__sum,axiom,
! [A: $tType,B: $tType] :
( ( ring_1 @ A )
=> ! [F3: B > int,A3: set @ B] :
( ( ring_1_of_int @ A @ ( groups7311177749621191930dd_sum @ B @ int @ F3 @ A3 ) )
= ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X4: B] : ( ring_1_of_int @ A @ ( F3 @ X4 ) )
@ A3 ) ) ) ).
% of_int_sum
thf(fact_4067_of__real__sum,axiom,
! [A: $tType,B: $tType] :
( ( real_V2191834092415804123ebra_1 @ A )
=> ! [F3: B > real,S: set @ B] :
( ( real_Vector_of_real @ A @ ( groups7311177749621191930dd_sum @ B @ real @ F3 @ S ) )
= ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X4: B] : ( real_Vector_of_real @ A @ ( F3 @ X4 ) )
@ S ) ) ) ).
% of_real_sum
thf(fact_4068_atMost__subset__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y2: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_ord_atMost @ A @ X ) @ ( set_ord_atMost @ A @ Y2 ) )
= ( ord_less_eq @ A @ X @ Y2 ) ) ) ).
% atMost_subset_iff
thf(fact_4069_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_4070_quotient__of__number_I3_J,axiom,
! [K2: num] :
( ( quotient_of @ ( numeral_numeral @ rat @ K2 ) )
= ( product_Pair @ int @ int @ ( numeral_numeral @ int @ K2 ) @ ( one_one @ int ) ) ) ).
% quotient_of_number(3)
thf(fact_4071_rat__one__code,axiom,
( ( quotient_of @ ( one_one @ rat ) )
= ( product_Pair @ int @ int @ ( one_one @ int ) @ ( one_one @ int ) ) ) ).
% rat_one_code
thf(fact_4072_rat__zero__code,axiom,
( ( quotient_of @ ( zero_zero @ rat ) )
= ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) ) ) ).
% rat_zero_code
thf(fact_4073_quotient__of__number_I5_J,axiom,
! [K2: num] :
( ( quotient_of @ ( uminus_uminus @ rat @ ( numeral_numeral @ rat @ K2 ) ) )
= ( product_Pair @ int @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K2 ) ) @ ( one_one @ int ) ) ) ).
% quotient_of_number(5)
thf(fact_4074_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_4075_divide__rat__def,axiom,
( ( divide_divide @ rat )
= ( ^ [Q5: rat,R5: rat] : ( times_times @ rat @ Q5 @ ( inverse_inverse @ rat @ R5 ) ) ) ) ).
% divide_rat_def
thf(fact_4076_scaleR__left_Osum,axiom,
! [A: $tType,C: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [G: C > real,A3: set @ C,X: A] :
( ( real_V8093663219630862766scaleR @ A @ ( groups7311177749621191930dd_sum @ C @ real @ G @ A3 ) @ X )
= ( groups7311177749621191930dd_sum @ C @ A
@ ^ [X4: C] : ( real_V8093663219630862766scaleR @ A @ ( G @ X4 ) @ X )
@ A3 ) ) ) ).
% scaleR_left.sum
thf(fact_4077_scaleR__sum__left,axiom,
! [A: $tType,C: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [F3: C > real,A3: set @ C,X: A] :
( ( real_V8093663219630862766scaleR @ A @ ( groups7311177749621191930dd_sum @ C @ real @ F3 @ A3 ) @ X )
= ( groups7311177749621191930dd_sum @ C @ A
@ ^ [A5: C] : ( real_V8093663219630862766scaleR @ A @ ( F3 @ A5 ) @ X )
@ A3 ) ) ) ).
% scaleR_sum_left
thf(fact_4078_mod__sum__eq,axiom,
! [B: $tType,A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [F3: B > A,A2: A,A3: set @ B] :
( ( modulo_modulo @ A
@ ( groups7311177749621191930dd_sum @ B @ A
@ ^ [I: B] : ( modulo_modulo @ A @ ( F3 @ I ) @ A2 )
@ A3 )
@ A2 )
= ( modulo_modulo @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ A3 ) @ A2 ) ) ) ).
% mod_sum_eq
thf(fact_4079_scaleR__sum__right,axiom,
! [A: $tType,C: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [A2: real,F3: C > A,A3: set @ C] :
( ( real_V8093663219630862766scaleR @ A @ A2 @ ( groups7311177749621191930dd_sum @ C @ A @ F3 @ A3 ) )
= ( groups7311177749621191930dd_sum @ C @ A
@ ^ [X4: C] : ( real_V8093663219630862766scaleR @ A @ A2 @ ( F3 @ X4 ) )
@ A3 ) ) ) ).
% scaleR_sum_right
thf(fact_4080_scaleR__right_Osum,axiom,
! [A: $tType,C: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [A2: real,G: C > A,A3: set @ C] :
( ( real_V8093663219630862766scaleR @ A @ A2 @ ( groups7311177749621191930dd_sum @ C @ A @ G @ A3 ) )
= ( groups7311177749621191930dd_sum @ C @ A
@ ^ [X4: C] : ( real_V8093663219630862766scaleR @ A @ A2 @ ( G @ X4 ) )
@ A3 ) ) ) ).
% scaleR_right.sum
thf(fact_4081_sum__norm__le,axiom,
! [A: $tType,B: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [S3: set @ B,F3: B > A,G: B > real] :
( ! [X5: B] :
( ( member @ B @ X5 @ S3 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( F3 @ X5 ) ) @ ( G @ X5 ) ) )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ S3 ) ) @ ( groups7311177749621191930dd_sum @ B @ real @ G @ S3 ) ) ) ) ).
% sum_norm_le
thf(fact_4082_norm__sum,axiom,
! [A: $tType,B: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: B > A,A3: set @ B] :
( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ A3 ) )
@ ( groups7311177749621191930dd_sum @ B @ real
@ ^ [I: B] : ( real_V7770717601297561774m_norm @ A @ ( F3 @ I ) )
@ A3 ) ) ) ).
% norm_sum
thf(fact_4083_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_4084_summable__sum,axiom,
! [I5: $tType,A: $tType] :
( ( ( topolo5987344860129210374id_add @ A )
& ( topological_t2_space @ A ) )
=> ! [I6: set @ I5,F3: I5 > nat > A] :
( ! [I3: I5] :
( ( member @ I5 @ I3 @ I6 )
=> ( summable @ A @ ( F3 @ I3 ) ) )
=> ( summable @ A
@ ^ [N2: nat] :
( groups7311177749621191930dd_sum @ I5 @ A
@ ^ [I: I5] : ( F3 @ I @ N2 )
@ I6 ) ) ) ) ).
% summable_sum
thf(fact_4085_sums__sum,axiom,
! [A: $tType,I5: $tType] :
( ( ( topolo5987344860129210374id_add @ A )
& ( topological_t2_space @ A ) )
=> ! [I6: set @ I5,F3: I5 > nat > A,X: I5 > A] :
( ! [I3: I5] :
( ( member @ I5 @ I3 @ I6 )
=> ( sums @ A @ ( F3 @ I3 ) @ ( X @ I3 ) ) )
=> ( sums @ A
@ ^ [N2: nat] :
( groups7311177749621191930dd_sum @ I5 @ A
@ ^ [I: I5] : ( F3 @ I @ N2 )
@ I6 )
@ ( groups7311177749621191930dd_sum @ I5 @ A @ X @ I6 ) ) ) ) ).
% sums_sum
thf(fact_4086_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
@ ^ [I: nat] : ( G @ ( suc @ I ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ) ).
% sum.atMost_Suc_shift
thf(fact_4087_sum__telescope,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [F3: nat > A,I2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I: nat] : ( minus_minus @ A @ ( F3 @ I ) @ ( F3 @ ( suc @ I ) ) )
@ ( set_ord_atMost @ nat @ I2 ) )
= ( minus_minus @ A @ ( F3 @ ( zero_zero @ nat ) ) @ ( F3 @ ( suc @ I2 ) ) ) ) ) ).
% sum_telescope
thf(fact_4088_polyfun__eq__coeffs,axiom,
! [A: $tType] :
( ( ( real_V8999393235501362500lgebra @ A )
& ( idom @ A ) )
=> ! [C2: nat > A,N: nat,D2: nat > A] :
( ( ! [X4: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I: nat] : ( times_times @ A @ ( C2 @ I ) @ ( power_power @ A @ X4 @ I ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I: nat] : ( times_times @ A @ ( D2 @ I ) @ ( power_power @ A @ X4 @ I ) )
@ ( set_ord_atMost @ nat @ N ) ) ) )
= ( ! [I: nat] :
( ( ord_less_eq @ nat @ I @ N )
=> ( ( C2 @ I )
= ( D2 @ I ) ) ) ) ) ) ).
% polyfun_eq_coeffs
thf(fact_4089_bounded__imp__summable,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( linord2810124833399127020strict @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [A2: nat > A,B5: A] :
( ! [N3: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( A2 @ N3 ) )
=> ( ! [N3: nat] : ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ A2 @ ( set_ord_atMost @ nat @ N3 ) ) @ B5 )
=> ( summable @ A @ A2 ) ) ) ) ).
% bounded_imp_summable
thf(fact_4090_atMost__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( set_ord_atMost @ A )
= ( ^ [U2: A] :
( collect @ A
@ ^ [X4: A] : ( ord_less_eq @ A @ X4 @ U2 ) ) ) ) ) ).
% atMost_def
thf(fact_4091_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_4092_choose__rising__sum_I2_J,axiom,
! [N: nat,M: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [J4: nat] : ( binomial @ ( plus_plus @ nat @ N @ J4 ) @ 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_4093_choose__rising__sum_I1_J,axiom,
! [N: nat,M: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [J4: nat] : ( binomial @ ( plus_plus @ nat @ N @ J4 ) @ 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_4094_polyfun__eq__0,axiom,
! [A: $tType] :
( ( ( real_V8999393235501362500lgebra @ A )
& ( idom @ A ) )
=> ! [C2: nat > A,N: nat] :
( ( ! [X4: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I: nat] : ( times_times @ A @ ( C2 @ I ) @ ( power_power @ A @ X4 @ I ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( zero_zero @ A ) ) )
= ( ! [I: nat] :
( ( ord_less_eq @ nat @ I @ N )
=> ( ( C2 @ I )
= ( zero_zero @ A ) ) ) ) ) ) ).
% polyfun_eq_0
thf(fact_4095_zero__polynom__imp__zero__coeffs,axiom,
! [A: $tType] :
( ( ( ab_semigroup_mult @ A )
& ( real_V8999393235501362500lgebra @ A ) )
=> ! [C2: nat > A,N: nat,K2: nat] :
( ! [W2: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I: nat] : ( times_times @ A @ ( C2 @ I ) @ ( power_power @ A @ W2 @ I ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ nat @ K2 @ N )
=> ( ( C2 @ K2 )
= ( zero_zero @ A ) ) ) ) ) ).
% zero_polynom_imp_zero_coeffs
thf(fact_4096_gbinomial__parallel__sum,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A2: A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( gbinomial @ A @ ( plus_plus @ A @ A2 @ ( semiring_1_of_nat @ A @ K3 ) ) @ K3 )
@ ( set_ord_atMost @ nat @ N ) )
= ( gbinomial @ A @ ( plus_plus @ A @ ( plus_plus @ A @ A2 @ ( semiring_1_of_nat @ A @ N ) ) @ ( one_one @ A ) ) @ N ) ) ) ).
% gbinomial_parallel_sum
thf(fact_4097_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_4098_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_4099_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_4100_suminf__sum,axiom,
! [A: $tType,I5: $tType] :
( ( ( topolo5987344860129210374id_add @ A )
& ( topological_t2_space @ A ) )
=> ! [I6: set @ I5,F3: I5 > nat > A] :
( ! [I3: I5] :
( ( member @ I5 @ I3 @ I6 )
=> ( summable @ A @ ( F3 @ I3 ) ) )
=> ( ( suminf @ A
@ ^ [N2: nat] :
( groups7311177749621191930dd_sum @ I5 @ A
@ ^ [I: I5] : ( F3 @ I @ N2 )
@ I6 ) )
= ( groups7311177749621191930dd_sum @ I5 @ A
@ ^ [I: I5] : ( suminf @ A @ ( F3 @ I ) )
@ I6 ) ) ) ) ).
% suminf_sum
thf(fact_4101_choose__row__sum,axiom,
! [N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat @ ( binomial @ N ) @ ( set_ord_atMost @ nat @ N ) )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ).
% choose_row_sum
thf(fact_4102_binomial,axiom,
! [A2: nat,B2: nat,N: nat] :
( ( power_power @ nat @ ( plus_plus @ nat @ A2 @ 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 @ A2 @ K3 ) ) @ ( power_power @ nat @ B2 @ ( minus_minus @ nat @ N @ K3 ) ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ).
% binomial
thf(fact_4103_summable__Cauchy__product,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V4412858255891104859lgebra @ A ) )
=> ! [A2: nat > A,B2: nat > A] :
( ( summable @ real
@ ^ [K3: nat] : ( real_V7770717601297561774m_norm @ A @ ( A2 @ K3 ) ) )
=> ( ( summable @ real
@ ^ [K3: nat] : ( real_V7770717601297561774m_norm @ A @ ( B2 @ K3 ) ) )
=> ( summable @ A
@ ^ [K3: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I: nat] : ( times_times @ A @ ( A2 @ I ) @ ( B2 @ ( minus_minus @ nat @ K3 @ I ) ) )
@ ( set_ord_atMost @ nat @ K3 ) ) ) ) ) ) ).
% summable_Cauchy_product
thf(fact_4104_Cauchy__product,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V4412858255891104859lgebra @ A ) )
=> ! [A2: nat > A,B2: nat > A] :
( ( summable @ real
@ ^ [K3: nat] : ( real_V7770717601297561774m_norm @ A @ ( A2 @ K3 ) ) )
=> ( ( summable @ real
@ ^ [K3: nat] : ( real_V7770717601297561774m_norm @ A @ ( B2 @ K3 ) ) )
=> ( ( times_times @ A @ ( suminf @ A @ A2 ) @ ( suminf @ A @ B2 ) )
= ( suminf @ A
@ ^ [K3: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I: nat] : ( times_times @ A @ ( A2 @ I ) @ ( B2 @ ( minus_minus @ nat @ K3 @ I ) ) )
@ ( set_ord_atMost @ nat @ K3 ) ) ) ) ) ) ) ).
% Cauchy_product
thf(fact_4105_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
@ ^ [I: nat] : ( plus_plus @ A @ ( G @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I ) ) @ ( G @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I ) ) ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% sum.in_pairs_0
thf(fact_4106_polynomial__product,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [M: nat,A2: nat > A,N: nat,B2: nat > A,X: A] :
( ! [I3: nat] :
( ( ord_less @ nat @ M @ I3 )
=> ( ( A2 @ I3 )
= ( zero_zero @ A ) ) )
=> ( ! [J2: nat] :
( ( ord_less @ nat @ N @ J2 )
=> ( ( B2 @ J2 )
= ( zero_zero @ A ) ) )
=> ( ( times_times @ A
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I: nat] : ( times_times @ A @ ( A2 @ I ) @ ( power_power @ A @ X @ I ) )
@ ( set_ord_atMost @ nat @ M ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [J4: nat] : ( times_times @ A @ ( B2 @ J4 ) @ ( power_power @ A @ X @ J4 ) )
@ ( set_ord_atMost @ nat @ N ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [R5: nat] :
( times_times @ A
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( times_times @ A @ ( A2 @ K3 ) @ ( B2 @ ( minus_minus @ nat @ R5 @ K3 ) ) )
@ ( set_ord_atMost @ nat @ R5 ) )
@ ( power_power @ A @ X @ R5 ) )
@ ( set_ord_atMost @ nat @ ( plus_plus @ nat @ M @ N ) ) ) ) ) ) ) ).
% polynomial_product
thf(fact_4107_gbinomial__sum__lower__neg,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A2: A,M: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( times_times @ A @ ( gbinomial @ A @ A2 @ 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 @ A2 @ ( one_one @ A ) ) @ M ) ) ) ) ).
% gbinomial_sum_lower_neg
thf(fact_4108_binomial__ring,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A2: A,B2: A,N: nat] :
( ( power_power @ A @ ( plus_plus @ A @ A2 @ B2 ) @ N )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( times_times @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( binomial @ N @ K3 ) ) @ ( power_power @ A @ A2 @ K3 ) ) @ ( power_power @ A @ B2 @ ( minus_minus @ nat @ N @ K3 ) ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% binomial_ring
thf(fact_4109_polynomial__product__nat,axiom,
! [M: nat,A2: nat > nat,N: nat,B2: nat > nat,X: nat] :
( ! [I3: nat] :
( ( ord_less @ nat @ M @ I3 )
=> ( ( A2 @ I3 )
= ( zero_zero @ nat ) ) )
=> ( ! [J2: nat] :
( ( ord_less @ nat @ N @ J2 )
=> ( ( B2 @ J2 )
= ( zero_zero @ nat ) ) )
=> ( ( times_times @ nat
@ ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [I: nat] : ( times_times @ nat @ ( A2 @ I ) @ ( power_power @ nat @ X @ I ) )
@ ( set_ord_atMost @ nat @ M ) )
@ ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [J4: nat] : ( times_times @ nat @ ( B2 @ J4 ) @ ( power_power @ nat @ X @ J4 ) )
@ ( set_ord_atMost @ nat @ N ) ) )
= ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [R5: nat] :
( times_times @ nat
@ ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [K3: nat] : ( times_times @ nat @ ( A2 @ K3 ) @ ( B2 @ ( minus_minus @ nat @ R5 @ K3 ) ) )
@ ( set_ord_atMost @ nat @ R5 ) )
@ ( power_power @ nat @ X @ R5 ) )
@ ( set_ord_atMost @ nat @ ( plus_plus @ nat @ M @ N ) ) ) ) ) ) ).
% polynomial_product_nat
thf(fact_4110_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_4111_pochhammer__binomial__sum,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [A2: A,B2: A,N: nat] :
( ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ A2 @ B2 ) @ N )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( times_times @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( binomial @ N @ K3 ) ) @ ( comm_s3205402744901411588hammer @ A @ A2 @ K3 ) ) @ ( comm_s3205402744901411588hammer @ A @ B2 @ ( minus_minus @ nat @ N @ K3 ) ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% pochhammer_binomial_sum
thf(fact_4112_Cauchy__product__sums,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V4412858255891104859lgebra @ A ) )
=> ! [A2: nat > A,B2: nat > A] :
( ( summable @ real
@ ^ [K3: nat] : ( real_V7770717601297561774m_norm @ A @ ( A2 @ K3 ) ) )
=> ( ( summable @ real
@ ^ [K3: nat] : ( real_V7770717601297561774m_norm @ A @ ( B2 @ K3 ) ) )
=> ( sums @ A
@ ^ [K3: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I: nat] : ( times_times @ A @ ( A2 @ I ) @ ( B2 @ ( minus_minus @ nat @ K3 @ I ) ) )
@ ( set_ord_atMost @ nat @ K3 ) )
@ ( times_times @ A @ ( suminf @ A @ A2 ) @ ( suminf @ A @ B2 ) ) ) ) ) ) ).
% Cauchy_product_sums
thf(fact_4113_sum__power__add,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [X: A,M: nat,I6: set @ nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I: nat] : ( power_power @ A @ X @ ( plus_plus @ nat @ M @ I ) )
@ I6 )
= ( times_times @ A @ ( power_power @ A @ X @ M ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ I6 ) ) ) ) ).
% sum_power_add
thf(fact_4114_sum_Ozero__middle,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [P5: nat,K2: nat,G: nat > A,H2: nat > A] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ P5 )
=> ( ( ord_less_eq @ nat @ K2 @ P5 )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [J4: nat] : ( if @ A @ ( ord_less @ nat @ J4 @ K2 ) @ ( G @ J4 ) @ ( if @ A @ ( J4 = K2 ) @ ( zero_zero @ A ) @ ( H2 @ ( minus_minus @ nat @ J4 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) )
@ ( set_ord_atMost @ nat @ P5 ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [J4: nat] : ( if @ A @ ( ord_less @ nat @ J4 @ K2 ) @ ( G @ J4 ) @ ( H2 @ J4 ) )
@ ( set_ord_atMost @ nat @ ( minus_minus @ nat @ P5 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ) ) ).
% sum.zero_middle
thf(fact_4115_gbinomial__partial__sum__poly,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [M: nat,A2: A,X: A,Y2: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( times_times @ A @ ( times_times @ A @ ( gbinomial @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ M ) @ A2 ) @ K3 ) @ ( power_power @ A @ X @ K3 ) ) @ ( power_power @ A @ Y2 @ ( 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 @ A2 ) @ K3 ) @ ( power_power @ A @ ( uminus_uminus @ A @ X ) @ K3 ) ) @ ( power_power @ A @ ( plus_plus @ A @ X @ Y2 ) @ ( minus_minus @ nat @ M @ K3 ) ) )
@ ( set_ord_atMost @ nat @ M ) ) ) ) ).
% gbinomial_partial_sum_poly
thf(fact_4116_exp__series__add__commuting,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X: A,Y2: A,N: nat] :
( ( ( times_times @ A @ X @ Y2 )
= ( times_times @ A @ Y2 @ X ) )
=> ( ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ A @ ( plus_plus @ A @ X @ Y2 ) @ N ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I: nat] : ( times_times @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ I ) ) @ ( power_power @ A @ X @ I ) ) @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ ( minus_minus @ nat @ N @ I ) ) ) @ ( power_power @ A @ Y2 @ ( minus_minus @ nat @ N @ I ) ) ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ) ).
% exp_series_add_commuting
thf(fact_4117_root__polyfun,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [N: nat,Z3: A,A2: A] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ N )
=> ( ( ( power_power @ A @ Z3 @ N )
= A2 )
= ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I: nat] :
( times_times @ A
@ ( if @ A
@ ( I
= ( zero_zero @ nat ) )
@ ( uminus_uminus @ A @ A2 )
@ ( if @ A @ ( I = N ) @ ( one_one @ A ) @ ( zero_zero @ A ) ) )
@ ( power_power @ A @ Z3 @ I ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( zero_zero @ A ) ) ) ) ) ).
% root_polyfun
thf(fact_4118_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_4119_choose__alternating__linear__sum,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [N: nat] :
( ( N
!= ( one_one @ nat ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I: nat] : ( times_times @ A @ ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ I ) @ ( semiring_1_of_nat @ A @ I ) ) @ ( semiring_1_of_nat @ A @ ( binomial @ N @ I ) ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( zero_zero @ A ) ) ) ) ).
% choose_alternating_linear_sum
thf(fact_4120_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_4121_gbinomial__partial__sum__poly__xpos,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [M: nat,A2: A,X: A,Y2: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( times_times @ A @ ( times_times @ A @ ( gbinomial @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ M ) @ A2 ) @ K3 ) @ ( power_power @ A @ X @ K3 ) ) @ ( power_power @ A @ Y2 @ ( 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 ) @ A2 ) @ ( one_one @ A ) ) @ K3 ) @ ( power_power @ A @ X @ K3 ) ) @ ( power_power @ A @ ( plus_plus @ A @ X @ Y2 ) @ ( minus_minus @ nat @ M @ K3 ) ) )
@ ( set_ord_atMost @ nat @ M ) ) ) ) ).
% gbinomial_partial_sum_poly_xpos
thf(fact_4122_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_4123_choose__linear__sum,axiom,
! [N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [I: nat] : ( times_times @ nat @ I @ ( binomial @ N @ I ) )
@ ( 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_4124_choose__alternating__sum,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I: nat] : ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ I ) @ ( semiring_1_of_nat @ A @ ( binomial @ N @ I ) ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( zero_zero @ A ) ) ) ) ).
% choose_alternating_sum
thf(fact_4125_polyfun__extremal__lemma,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [E: real,C2: nat > A,N: nat] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E )
=> ? [M8: real] :
! [Z4: A] :
( ( ord_less_eq @ real @ M8 @ ( real_V7770717601297561774m_norm @ A @ Z4 ) )
=> ( ord_less_eq @ real
@ ( real_V7770717601297561774m_norm @ A
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I: nat] : ( times_times @ A @ ( C2 @ I ) @ ( power_power @ A @ Z4 @ I ) )
@ ( set_ord_atMost @ nat @ N ) ) )
@ ( times_times @ real @ E @ ( power_power @ real @ ( real_V7770717601297561774m_norm @ A @ Z4 ) @ ( suc @ N ) ) ) ) ) ) ) ).
% polyfun_extremal_lemma
thf(fact_4126_rat__abs__code,axiom,
! [P5: rat] :
( ( quotient_of @ ( abs_abs @ rat @ P5 ) )
= ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [A5: int] : ( product_Pair @ int @ int @ ( abs_abs @ int @ A5 ) )
@ ( quotient_of @ P5 ) ) ) ).
% rat_abs_code
thf(fact_4127_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_4128_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
@ ^ [I: nat] :
( if @ A
@ ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I )
@ ( semiring_1_of_nat @ A @ ( binomial @ N @ I ) )
@ ( zero_zero @ A ) )
@ ( set_ord_atMost @ nat @ N ) ) )
= ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% choose_odd_sum
thf(fact_4129_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
@ ^ [I: nat] : ( if @ A @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I ) @ ( semiring_1_of_nat @ A @ ( binomial @ N @ I ) ) @ ( zero_zero @ A ) )
@ ( set_ord_atMost @ nat @ N ) ) )
= ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% choose_even_sum
thf(fact_4130_gbinomial__partial__row__sum,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A2: A,M: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( times_times @ A @ ( gbinomial @ A @ A2 @ K3 ) @ ( minus_minus @ A @ ( divide_divide @ A @ A2 @ ( 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 @ A2 @ ( plus_plus @ nat @ M @ ( one_one @ nat ) ) ) ) ) ) ).
% gbinomial_partial_row_sum
thf(fact_4131_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_4132_rat__less__code,axiom,
( ( ord_less @ rat )
= ( ^ [P4: rat,Q5: rat] :
( product_case_prod @ int @ int @ $o
@ ^ [A5: int,C3: int] :
( product_case_prod @ int @ int @ $o
@ ^ [B4: int,D5: int] : ( ord_less @ int @ ( times_times @ int @ A5 @ D5 ) @ ( times_times @ int @ C3 @ B4 ) )
@ ( quotient_of @ Q5 ) )
@ ( quotient_of @ P4 ) ) ) ) ).
% rat_less_code
thf(fact_4133_rat__floor__code,axiom,
( ( archim6421214686448440834_floor @ rat )
= ( ^ [P4: rat] : ( product_case_prod @ int @ int @ int @ ( divide_divide @ int ) @ ( quotient_of @ P4 ) ) ) ) ).
% rat_floor_code
thf(fact_4134_rat__less__eq__code,axiom,
( ( ord_less_eq @ rat )
= ( ^ [P4: rat,Q5: rat] :
( product_case_prod @ int @ int @ $o
@ ^ [A5: int,C3: int] :
( product_case_prod @ int @ int @ $o
@ ^ [B4: int,D5: int] : ( ord_less_eq @ int @ ( times_times @ int @ A5 @ D5 ) @ ( times_times @ int @ C3 @ B4 ) )
@ ( quotient_of @ Q5 ) )
@ ( quotient_of @ P4 ) ) ) ) ).
% rat_less_eq_code
thf(fact_4135_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
@ ^ [Q5: nat] : ( ord_less @ nat @ Q5 @ N ) ) ) ) ) ).
% mask_eq_sum_exp
thf(fact_4136_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
@ ^ [Q5: nat] : ( ord_less @ nat @ Q5 @ N ) ) ) ) ).
% mask_eq_sum_exp_nat
thf(fact_4137_sum__abs__ge__zero,axiom,
! [B: $tType,A: $tType] :
( ( ordere166539214618696060dd_abs @ B )
=> ! [F3: A > B,A3: set @ A] :
( ord_less_eq @ B @ ( zero_zero @ B )
@ ( groups7311177749621191930dd_sum @ A @ B
@ ^ [I: A] : ( abs_abs @ B @ ( F3 @ I ) )
@ A3 ) ) ) ).
% sum_abs_ge_zero
thf(fact_4138_sum__abs,axiom,
! [B: $tType,A: $tType] :
( ( ordere166539214618696060dd_abs @ B )
=> ! [F3: A > B,A3: set @ A] :
( ord_less_eq @ B @ ( abs_abs @ B @ ( groups7311177749621191930dd_sum @ A @ B @ F3 @ A3 ) )
@ ( groups7311177749621191930dd_sum @ A @ B
@ ^ [I: A] : ( abs_abs @ B @ ( F3 @ I ) )
@ A3 ) ) ) ).
% sum_abs
thf(fact_4139_convex__sum__bound__le,axiom,
! [A: $tType,B: $tType] :
( ( linordered_idom @ B )
=> ! [I6: set @ A,X: A > B,A2: A > B,B2: B,Delta: B] :
( ! [I3: A] :
( ( member @ A @ I3 @ I6 )
=> ( ord_less_eq @ B @ ( zero_zero @ B ) @ ( X @ I3 ) ) )
=> ( ( ( groups7311177749621191930dd_sum @ A @ B @ X @ I6 )
= ( one_one @ B ) )
=> ( ! [I3: A] :
( ( member @ A @ I3 @ I6 )
=> ( ord_less_eq @ B @ ( abs_abs @ B @ ( minus_minus @ B @ ( A2 @ I3 ) @ B2 ) ) @ Delta ) )
=> ( ord_less_eq @ B
@ ( abs_abs @ B
@ ( minus_minus @ B
@ ( groups7311177749621191930dd_sum @ A @ B
@ ^ [I: A] : ( times_times @ B @ ( A2 @ I ) @ ( X @ I ) )
@ I6 )
@ B2 ) )
@ Delta ) ) ) ) ) ).
% convex_sum_bound_le
thf(fact_4140_abs__sum__abs,axiom,
! [B: $tType,A: $tType] :
( ( ordere166539214618696060dd_abs @ B )
=> ! [F3: A > B,A3: set @ A] :
( ( abs_abs @ B
@ ( groups7311177749621191930dd_sum @ A @ B
@ ^ [A5: A] : ( abs_abs @ B @ ( F3 @ A5 ) )
@ A3 ) )
= ( groups7311177749621191930dd_sum @ A @ B
@ ^ [A5: A] : ( abs_abs @ B @ ( F3 @ A5 ) )
@ A3 ) ) ) ).
% abs_sum_abs
thf(fact_4141_of__nat__id,axiom,
( ( semiring_1_of_nat @ nat )
= ( ^ [N2: nat] : N2 ) ) ).
% of_nat_id
thf(fact_4142_sum_Oneutral__const,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A3: set @ B] :
( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [Uu3: B] : ( zero_zero @ A )
@ A3 )
= ( zero_zero @ A ) ) ) ).
% sum.neutral_const
thf(fact_4143_int__sum,axiom,
! [B: $tType,F3: B > nat,A3: set @ B] :
( ( semiring_1_of_nat @ int @ ( groups7311177749621191930dd_sum @ B @ nat @ F3 @ A3 ) )
= ( groups7311177749621191930dd_sum @ B @ int
@ ^ [X4: B] : ( semiring_1_of_nat @ int @ ( F3 @ X4 ) )
@ A3 ) ) ).
% int_sum
thf(fact_4144_Complex__sum_H,axiom,
! [A: $tType,F3: A > real,S: set @ A] :
( ( groups7311177749621191930dd_sum @ A @ complex
@ ^ [X4: A] : ( complex2 @ ( F3 @ X4 ) @ ( zero_zero @ real ) )
@ S )
= ( complex2 @ ( groups7311177749621191930dd_sum @ A @ real @ F3 @ S ) @ ( zero_zero @ real ) ) ) ).
% Complex_sum'
thf(fact_4145_sum__subtractf__nat,axiom,
! [A: $tType,A3: set @ A,G: A > nat,F3: A > nat] :
( ! [X5: A] :
( ( member @ A @ X5 @ A3 )
=> ( ord_less_eq @ nat @ ( G @ X5 ) @ ( F3 @ X5 ) ) )
=> ( ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [X4: A] : ( minus_minus @ nat @ ( F3 @ X4 ) @ ( G @ X4 ) )
@ A3 )
= ( minus_minus @ nat @ ( groups7311177749621191930dd_sum @ A @ nat @ F3 @ A3 ) @ ( groups7311177749621191930dd_sum @ A @ nat @ G @ A3 ) ) ) ) ).
% sum_subtractf_nat
thf(fact_4146_sum__SucD,axiom,
! [A: $tType,F3: A > nat,A3: set @ A,N: nat] :
( ( ( groups7311177749621191930dd_sum @ A @ nat @ F3 @ A3 )
= ( suc @ N ) )
=> ? [X5: A] :
( ( member @ A @ X5 @ A3 )
& ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F3 @ X5 ) ) ) ) ).
% sum_SucD
thf(fact_4147_sum__nth__roots,axiom,
! [N: nat,C2: complex] :
( ( ord_less @ nat @ ( one_one @ nat ) @ N )
=> ( ( groups7311177749621191930dd_sum @ complex @ complex
@ ^ [X4: complex] : X4
@ ( collect @ complex
@ ^ [Z6: complex] :
( ( power_power @ complex @ Z6 @ N )
= C2 ) ) )
= ( zero_zero @ complex ) ) ) ).
% sum_nth_roots
thf(fact_4148_sum_Oswap,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: B > C > A,B5: set @ C,A3: set @ B] :
( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [I: B] : ( groups7311177749621191930dd_sum @ C @ A @ ( G @ I ) @ B5 )
@ A3 )
= ( groups7311177749621191930dd_sum @ C @ A
@ ^ [J4: C] :
( groups7311177749621191930dd_sum @ B @ A
@ ^ [I: B] : ( G @ I @ J4 )
@ A3 )
@ B5 ) ) ) ).
% sum.swap
thf(fact_4149_sum__roots__unity,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( one_one @ nat ) @ N )
=> ( ( groups7311177749621191930dd_sum @ complex @ complex
@ ^ [X4: complex] : X4
@ ( collect @ complex
@ ^ [Z6: complex] :
( ( power_power @ complex @ Z6 @ N )
= ( one_one @ complex ) ) ) )
= ( zero_zero @ complex ) ) ) ).
% sum_roots_unity
thf(fact_4150_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
@ ^ [I: nat,J4: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ I @ J4 ) @ N ) ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I: nat] : ( G @ I @ ( minus_minus @ nat @ K3 @ I ) )
@ ( set_ord_atMost @ nat @ K3 ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% sum.triangle_reindex_eq
thf(fact_4151_sum__mono,axiom,
! [A: $tType,B: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [K5: set @ B,F3: B > A,G: B > A] :
( ! [I3: B] :
( ( member @ B @ I3 @ K5 )
=> ( ord_less_eq @ A @ ( F3 @ I3 ) @ ( G @ I3 ) ) )
=> ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ K5 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ K5 ) ) ) ) ).
% sum_mono
thf(fact_4152_sum__distrib__left,axiom,
! [A: $tType,B: $tType] :
( ( semiring_0 @ A )
=> ! [R: A,F3: B > A,A3: set @ B] :
( ( times_times @ A @ R @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ A3 ) )
= ( groups7311177749621191930dd_sum @ B @ A
@ ^ [N2: B] : ( times_times @ A @ R @ ( F3 @ N2 ) )
@ A3 ) ) ) ).
% sum_distrib_left
thf(fact_4153_sum__distrib__right,axiom,
! [A: $tType,B: $tType] :
( ( semiring_0 @ A )
=> ! [F3: B > A,A3: set @ B,R: A] :
( ( times_times @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ A3 ) @ R )
= ( groups7311177749621191930dd_sum @ B @ A
@ ^ [N2: B] : ( times_times @ A @ ( F3 @ N2 ) @ R )
@ A3 ) ) ) ).
% sum_distrib_right
thf(fact_4154_sum__product,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( semiring_0 @ B )
=> ! [F3: A > B,A3: set @ A,G: C > B,B5: set @ C] :
( ( times_times @ B @ ( groups7311177749621191930dd_sum @ A @ B @ F3 @ A3 ) @ ( groups7311177749621191930dd_sum @ C @ B @ G @ B5 ) )
= ( groups7311177749621191930dd_sum @ A @ B
@ ^ [I: A] :
( groups7311177749621191930dd_sum @ C @ B
@ ^ [J4: C] : ( times_times @ B @ ( F3 @ I ) @ ( G @ J4 ) )
@ B5 )
@ A3 ) ) ) ).
% sum_product
thf(fact_4155_sum_Odistrib,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: B > A,H2: B > A,A3: set @ B] :
( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X4: B] : ( plus_plus @ A @ ( G @ X4 ) @ ( H2 @ X4 ) )
@ A3 )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G @ A3 ) @ ( groups7311177749621191930dd_sum @ B @ A @ H2 @ A3 ) ) ) ) ).
% sum.distrib
thf(fact_4156_sum__subtractf,axiom,
! [A: $tType,B: $tType] :
( ( ab_group_add @ A )
=> ! [F3: B > A,G: B > A,A3: set @ B] :
( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X4: B] : ( minus_minus @ A @ ( F3 @ X4 ) @ ( G @ X4 ) )
@ A3 )
= ( minus_minus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ A3 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ A3 ) ) ) ) ).
% sum_subtractf
thf(fact_4157_sum__divide__distrib,axiom,
! [A: $tType,B: $tType] :
( ( field @ A )
=> ! [F3: B > A,A3: set @ B,R: A] :
( ( divide_divide @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ A3 ) @ R )
= ( groups7311177749621191930dd_sum @ B @ A
@ ^ [N2: B] : ( divide_divide @ A @ ( F3 @ N2 ) @ R )
@ A3 ) ) ) ).
% sum_divide_distrib
thf(fact_4158_sum__negf,axiom,
! [A: $tType,B: $tType] :
( ( ab_group_add @ A )
=> ! [F3: B > A,A3: set @ B] :
( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X4: B] : ( uminus_uminus @ A @ ( F3 @ X4 ) )
@ A3 )
= ( uminus_uminus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ A3 ) ) ) ) ).
% sum_negf
thf(fact_4159_sum__nonneg,axiom,
! [A: $tType,B: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A3: set @ B,F3: B > A] :
( ! [X5: B] :
( ( member @ B @ X5 @ A3 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F3 @ X5 ) ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ A3 ) ) ) ) ).
% sum_nonneg
thf(fact_4160_sum__nonpos,axiom,
! [B: $tType,A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A3: set @ B,F3: B > A] :
( ! [X5: B] :
( ( member @ B @ X5 @ A3 )
=> ( ord_less_eq @ A @ ( F3 @ X5 ) @ ( zero_zero @ A ) ) )
=> ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ A3 ) @ ( zero_zero @ A ) ) ) ) ).
% sum_nonpos
thf(fact_4161_Maclaurin__minus__cos__expansion,axiom,
! [N: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ X @ ( zero_zero @ real ) )
=> ? [T4: real] :
( ( ord_less @ real @ X @ T4 )
& ( ord_less @ real @ T4 @ ( zero_zero @ real ) )
& ( ( cos @ real @ X )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M2: nat] : ( times_times @ real @ ( cos_coeff @ M2 ) @ ( power_power @ real @ X @ M2 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( cos @ real @ ( plus_plus @ real @ T4 @ ( times_times @ real @ ( times_times @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( semiring_1_of_nat @ real @ N ) ) @ pi ) ) ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ X @ N ) ) ) ) ) ) ) ).
% Maclaurin_minus_cos_expansion
thf(fact_4162_Maclaurin__cos__expansion2,axiom,
! [X: real,N: nat] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ? [T4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ T4 )
& ( ord_less @ real @ T4 @ X )
& ( ( cos @ real @ X )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M2: nat] : ( times_times @ real @ ( cos_coeff @ M2 ) @ ( power_power @ real @ X @ M2 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( cos @ real @ ( plus_plus @ real @ T4 @ ( times_times @ real @ ( times_times @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( semiring_1_of_nat @ real @ N ) ) @ pi ) ) ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ X @ N ) ) ) ) ) ) ) ).
% Maclaurin_cos_expansion2
thf(fact_4163_Maclaurin__sin__expansion3,axiom,
! [N: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ? [T4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ T4 )
& ( ord_less @ real @ T4 @ X )
& ( ( sin @ real @ X )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M2: nat] : ( times_times @ real @ ( sin_coeff @ M2 ) @ ( power_power @ real @ X @ M2 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( sin @ real @ ( plus_plus @ real @ T4 @ ( times_times @ real @ ( times_times @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( semiring_1_of_nat @ real @ N ) ) @ pi ) ) ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ X @ N ) ) ) ) ) ) ) ).
% Maclaurin_sin_expansion3
thf(fact_4164_Maclaurin__sin__expansion4,axiom,
! [X: real,N: nat] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ? [T4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ T4 )
& ( ord_less_eq @ real @ T4 @ X )
& ( ( sin @ real @ X )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M2: nat] : ( times_times @ real @ ( sin_coeff @ M2 ) @ ( power_power @ real @ X @ M2 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( sin @ real @ ( plus_plus @ real @ T4 @ ( times_times @ real @ ( times_times @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( semiring_1_of_nat @ real @ N ) ) @ pi ) ) ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ X @ N ) ) ) ) ) ) ).
% Maclaurin_sin_expansion4
thf(fact_4165_lessThan__iff,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [I2: A,K2: A] :
( ( member @ A @ I2 @ ( set_ord_lessThan @ A @ K2 ) )
= ( ord_less @ A @ I2 @ K2 ) ) ) ).
% lessThan_iff
thf(fact_4166_lessThan__subset__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y2: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_ord_lessThan @ A @ X ) @ ( set_ord_lessThan @ A @ Y2 ) )
= ( ord_less_eq @ A @ X @ Y2 ) ) ) ).
% lessThan_subset_iff
thf(fact_4167_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_4168_sumr__cos__zero__one,axiom,
! [N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M2: nat] : ( times_times @ real @ ( cos_coeff @ M2 ) @ ( power_power @ real @ ( zero_zero @ real ) @ M2 ) )
@ ( set_ord_lessThan @ nat @ ( suc @ N ) ) )
= ( one_one @ real ) ) ).
% sumr_cos_zero_one
thf(fact_4169_lessThan__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( set_ord_lessThan @ A )
= ( ^ [U2: A] :
( collect @ A
@ ^ [X4: A] : ( ord_less @ A @ X4 @ U2 ) ) ) ) ) ).
% lessThan_def
thf(fact_4170_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_4171_Iic__subset__Iio__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_ord_atMost @ A @ A2 ) @ ( set_ord_lessThan @ A @ B2 ) )
= ( ord_less @ A @ A2 @ B2 ) ) ) ).
% Iic_subset_Iio_iff
thf(fact_4172_sum_Onat__diff__reindex,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I: nat] : ( G @ ( minus_minus @ nat @ N @ ( suc @ I ) ) )
@ ( set_ord_lessThan @ nat @ N ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% sum.nat_diff_reindex
thf(fact_4173_sum__diff__distrib,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [Q2: A > nat,P3: A > nat,N: A] :
( ! [X5: A] : ( ord_less_eq @ nat @ ( Q2 @ X5 ) @ ( P3 @ X5 ) )
=> ( ( minus_minus @ nat @ ( groups7311177749621191930dd_sum @ A @ nat @ P3 @ ( set_ord_lessThan @ A @ N ) ) @ ( groups7311177749621191930dd_sum @ A @ nat @ Q2 @ ( set_ord_lessThan @ A @ N ) ) )
= ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [X4: A] : ( minus_minus @ nat @ ( P3 @ X4 ) @ ( Q2 @ X4 ) )
@ ( set_ord_lessThan @ A @ N ) ) ) ) ) ).
% sum_diff_distrib
thf(fact_4174_suminf__le__const,axiom,
! [A: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F3: nat > A,X: A] :
( ( summable @ A @ F3 )
=> ( ! [N3: nat] : ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_ord_lessThan @ nat @ N3 ) ) @ X )
=> ( ord_less_eq @ A @ ( suminf @ A @ F3 ) @ X ) ) ) ) ).
% suminf_le_const
thf(fact_4175_sumr__diff__mult__const2,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [F3: nat > A,N: nat,R: A] :
( ( minus_minus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_ord_lessThan @ nat @ N ) ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ R ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I: nat] : ( minus_minus @ A @ ( F3 @ I ) @ R )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% sumr_diff_mult_const2
thf(fact_4176_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
@ ^ [I: nat] : ( G @ ( suc @ I ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% sum.lessThan_Suc_shift
thf(fact_4177_sum__lessThan__telescope,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [F3: nat > A,M: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [N2: nat] : ( minus_minus @ A @ ( F3 @ ( suc @ N2 ) ) @ ( F3 @ N2 ) )
@ ( set_ord_lessThan @ nat @ M ) )
= ( minus_minus @ A @ ( F3 @ M ) @ ( F3 @ ( zero_zero @ nat ) ) ) ) ) ).
% sum_lessThan_telescope
thf(fact_4178_sum__lessThan__telescope_H,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [F3: nat > A,M: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [N2: nat] : ( minus_minus @ A @ ( F3 @ N2 ) @ ( F3 @ ( suc @ N2 ) ) )
@ ( set_ord_lessThan @ nat @ M ) )
= ( minus_minus @ A @ ( F3 @ ( zero_zero @ nat ) ) @ ( F3 @ M ) ) ) ) ).
% sum_lessThan_telescope'
thf(fact_4179_summableI__nonneg__bounded,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( ordere6911136660526730532id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F3: nat > A,X: A] :
( ! [N3: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F3 @ N3 ) )
=> ( ! [N3: nat] : ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_ord_lessThan @ nat @ N3 ) ) @ X )
=> ( summable @ A @ F3 ) ) ) ) ).
% summableI_nonneg_bounded
thf(fact_4180_sums__iff__shift,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: nat > A,N: nat,S: A] :
( ( sums @ A
@ ^ [I: nat] : ( F3 @ ( plus_plus @ nat @ I @ N ) )
@ S )
= ( sums @ A @ F3 @ ( plus_plus @ A @ S @ ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ) ).
% sums_iff_shift
thf(fact_4181_sums__iff__shift_H,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: nat > A,N: nat,S: A] :
( ( sums @ A
@ ^ [I: nat] : ( F3 @ ( plus_plus @ nat @ I @ N ) )
@ ( minus_minus @ A @ S @ ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_ord_lessThan @ nat @ N ) ) ) )
= ( sums @ A @ F3 @ S ) ) ) ).
% sums_iff_shift'
thf(fact_4182_sums__split__initial__segment,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: nat > A,S: A,N: nat] :
( ( sums @ A @ F3 @ S )
=> ( sums @ A
@ ^ [I: nat] : ( F3 @ ( plus_plus @ nat @ I @ N ) )
@ ( minus_minus @ A @ S @ ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ) ).
% sums_split_initial_segment
thf(fact_4183_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_4184_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_4185_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_4186_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
@ ^ [I: nat] : ( G @ ( suc @ I ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% sum.atMost_shift
thf(fact_4187_suminf__split__initial__segment,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: nat > A,K2: nat] :
( ( summable @ A @ F3 )
=> ( ( suminf @ A @ F3 )
= ( plus_plus @ A
@ ( suminf @ A
@ ^ [N2: nat] : ( F3 @ ( plus_plus @ nat @ N2 @ K2 ) ) )
@ ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_ord_lessThan @ nat @ K2 ) ) ) ) ) ) ).
% suminf_split_initial_segment
thf(fact_4188_suminf__minus__initial__segment,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: nat > A,K2: nat] :
( ( summable @ A @ F3 )
=> ( ( suminf @ A
@ ^ [N2: nat] : ( F3 @ ( plus_plus @ nat @ N2 @ K2 ) ) )
= ( minus_minus @ A @ ( suminf @ A @ F3 ) @ ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_ord_lessThan @ nat @ K2 ) ) ) ) ) ) ).
% suminf_minus_initial_segment
thf(fact_4189_sum__less__suminf,axiom,
! [A: $tType] :
( ( ( ordere8940638589300402666id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F3: nat > A,N: nat] :
( ( summable @ A @ F3 )
=> ( ! [M4: nat] :
( ( ord_less_eq @ nat @ N @ M4 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( F3 @ M4 ) ) )
=> ( ord_less @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_ord_lessThan @ nat @ N ) ) @ ( suminf @ A @ F3 ) ) ) ) ) ).
% sum_less_suminf
thf(fact_4190_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_4191_lemma__termdiff1,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [Z3: A,H2: A,M: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [P4: nat] : ( minus_minus @ A @ ( times_times @ A @ ( power_power @ A @ ( plus_plus @ A @ Z3 @ H2 ) @ ( minus_minus @ nat @ M @ P4 ) ) @ ( power_power @ A @ Z3 @ P4 ) ) @ ( power_power @ A @ Z3 @ M ) )
@ ( set_ord_lessThan @ nat @ M ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [P4: nat] : ( times_times @ A @ ( power_power @ A @ Z3 @ P4 ) @ ( minus_minus @ A @ ( power_power @ A @ ( plus_plus @ A @ Z3 @ H2 ) @ ( minus_minus @ nat @ M @ P4 ) ) @ ( power_power @ A @ Z3 @ ( minus_minus @ nat @ M @ P4 ) ) ) )
@ ( set_ord_lessThan @ nat @ M ) ) ) ) ).
% lemma_termdiff1
thf(fact_4192_diff__power__eq__sum,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [X: A,N: nat,Y2: A] :
( ( minus_minus @ A @ ( power_power @ A @ X @ ( suc @ N ) ) @ ( power_power @ A @ Y2 @ ( suc @ N ) ) )
= ( times_times @ A @ ( minus_minus @ A @ X @ Y2 )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [P4: nat] : ( times_times @ A @ ( power_power @ A @ X @ P4 ) @ ( power_power @ A @ Y2 @ ( minus_minus @ nat @ N @ P4 ) ) )
@ ( set_ord_lessThan @ nat @ ( suc @ N ) ) ) ) ) ) ).
% diff_power_eq_sum
thf(fact_4193_power__diff__sumr2,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [X: A,N: nat,Y2: A] :
( ( minus_minus @ A @ ( power_power @ A @ X @ N ) @ ( power_power @ A @ Y2 @ N ) )
= ( times_times @ A @ ( minus_minus @ A @ X @ Y2 )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I: nat] : ( times_times @ A @ ( power_power @ A @ Y2 @ ( minus_minus @ nat @ N @ ( suc @ I ) ) ) @ ( power_power @ A @ X @ I ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% power_diff_sumr2
thf(fact_4194_polyfun__linear__factor__root,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [C2: nat > A,A2: A,N: nat] :
( ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I: nat] : ( times_times @ A @ ( C2 @ I ) @ ( power_power @ A @ A2 @ I ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( zero_zero @ A ) )
=> ~ ! [B3: nat > A] :
~ ! [Z4: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I: nat] : ( times_times @ A @ ( C2 @ I ) @ ( power_power @ A @ Z4 @ I ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( times_times @ A @ ( minus_minus @ A @ Z4 @ A2 )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I: nat] : ( times_times @ A @ ( B3 @ I ) @ ( power_power @ A @ Z4 @ I ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ) ).
% polyfun_linear_factor_root
thf(fact_4195_polyfun__linear__factor,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [C2: nat > A,N: nat,A2: A] :
? [B3: nat > A] :
! [Z4: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I: nat] : ( times_times @ A @ ( C2 @ I ) @ ( power_power @ A @ Z4 @ I ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( plus_plus @ A
@ ( times_times @ A @ ( minus_minus @ A @ Z4 @ A2 )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I: nat] : ( times_times @ A @ ( B3 @ I ) @ ( power_power @ A @ Z4 @ I ) )
@ ( set_ord_lessThan @ nat @ N ) ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I: nat] : ( times_times @ A @ ( C2 @ I ) @ ( power_power @ A @ A2 @ I ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ) ).
% polyfun_linear_factor
thf(fact_4196_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
@ ^ [I: nat,J4: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ I @ J4 ) @ N ) ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I: nat] : ( G @ I @ ( minus_minus @ nat @ K3 @ I ) )
@ ( set_ord_atMost @ nat @ K3 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% sum.triangle_reindex
thf(fact_4197_real__sum__nat__ivl__bounded2,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N: nat,F3: nat > A,K5: A,K2: nat] :
( ! [P7: nat] :
( ( ord_less @ nat @ P7 @ N )
=> ( ord_less_eq @ A @ ( F3 @ P7 ) @ K5 ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ K5 )
=> ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_ord_lessThan @ nat @ ( minus_minus @ nat @ N @ K2 ) ) ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ K5 ) ) ) ) ) ).
% real_sum_nat_ivl_bounded2
thf(fact_4198_sum__less__suminf2,axiom,
! [A: $tType] :
( ( ( ordere8940638589300402666id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F3: nat > A,N: nat,I2: nat] :
( ( summable @ A @ F3 )
=> ( ! [M4: nat] :
( ( ord_less_eq @ nat @ N @ M4 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F3 @ M4 ) ) )
=> ( ( ord_less_eq @ nat @ N @ I2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ ( F3 @ I2 ) )
=> ( ord_less @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_ord_lessThan @ nat @ N ) ) @ ( suminf @ A @ F3 ) ) ) ) ) ) ) ).
% sum_less_suminf2
thf(fact_4199_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
@ ^ [I: nat] : ( power_power @ A @ X @ ( minus_minus @ nat @ N @ ( suc @ I ) ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% one_diff_power_eq'
thf(fact_4200_Maclaurin__zero,axiom,
! [A: $tType] :
( ( zero @ A )
=> ! [X: real,N: nat,Diff: nat > A > real] :
( ( X
= ( zero_zero @ real ) )
=> ( ( N
!= ( zero_zero @ nat ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M2: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M2 @ ( zero_zero @ A ) ) @ ( semiring_char_0_fact @ real @ M2 ) ) @ ( power_power @ real @ X @ M2 ) )
@ ( set_ord_lessThan @ nat @ N ) )
= ( Diff @ ( zero_zero @ nat ) @ ( zero_zero @ A ) ) ) ) ) ) ).
% Maclaurin_zero
thf(fact_4201_Maclaurin__lemma,axiom,
! [H2: real,F3: real > real,J3: nat > real,N: nat] :
( ( ord_less @ real @ ( zero_zero @ real ) @ H2 )
=> ? [B9: real] :
( ( F3 @ H2 )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M2: nat] : ( times_times @ real @ ( divide_divide @ real @ ( J3 @ M2 ) @ ( semiring_char_0_fact @ real @ M2 ) ) @ ( power_power @ real @ H2 @ M2 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ B9 @ ( divide_divide @ real @ ( power_power @ real @ H2 @ N ) @ ( semiring_char_0_fact @ real @ N ) ) ) ) ) ) ).
% Maclaurin_lemma
thf(fact_4202_sum__split__even__odd,axiom,
! [F3: nat > real,G: nat > real,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I: nat] : ( if @ real @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I ) @ ( F3 @ I ) @ ( G @ I ) )
@ ( set_ord_lessThan @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I: nat] : ( F3 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I: nat] : ( G @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I ) @ ( one_one @ nat ) ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% sum_split_even_odd
thf(fact_4203_Maclaurin__exp__le,axiom,
! [X: real,N: nat] :
? [T4: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ T4 ) @ ( abs_abs @ real @ X ) )
& ( ( exp @ real @ X )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M2: nat] : ( divide_divide @ real @ ( power_power @ real @ X @ M2 ) @ ( semiring_char_0_fact @ real @ M2 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( exp @ real @ T4 ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ X @ N ) ) ) ) ) ).
% Maclaurin_exp_le
thf(fact_4204_polyfun__diff__alt,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [N: nat,A2: nat > A,X: A,Y2: A] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ N )
=> ( ( minus_minus @ A
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I: nat] : ( times_times @ A @ ( A2 @ I ) @ ( power_power @ A @ X @ I ) )
@ ( set_ord_atMost @ nat @ N ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I: nat] : ( times_times @ A @ ( A2 @ I ) @ ( power_power @ A @ Y2 @ I ) )
@ ( set_ord_atMost @ nat @ N ) ) )
= ( times_times @ A @ ( minus_minus @ A @ X @ Y2 )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [J4: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( times_times @ A @ ( times_times @ A @ ( A2 @ ( plus_plus @ nat @ ( plus_plus @ nat @ J4 @ K3 ) @ ( one_one @ nat ) ) ) @ ( power_power @ A @ Y2 @ K3 ) ) @ ( power_power @ A @ X @ J4 ) )
@ ( set_ord_lessThan @ nat @ ( minus_minus @ nat @ N @ J4 ) ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ) ).
% polyfun_diff_alt
thf(fact_4205_exp__first__terms,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [K2: nat] :
( ( exp @ A )
= ( ^ [X4: A] :
( plus_plus @ A
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [N2: nat] : ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ N2 ) ) @ ( power_power @ A @ X4 @ N2 ) )
@ ( set_ord_lessThan @ nat @ K2 ) )
@ ( suminf @ A
@ ^ [N2: nat] : ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ ( plus_plus @ nat @ N2 @ K2 ) ) ) @ ( power_power @ A @ X4 @ ( plus_plus @ nat @ N2 @ K2 ) ) ) ) ) ) ) ) ).
% exp_first_terms
thf(fact_4206_Maclaurin__sin__bound,axiom,
! [X: real,N: nat] :
( ord_less_eq @ real
@ ( abs_abs @ real
@ ( minus_minus @ real @ ( sin @ real @ X )
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M2: nat] : ( times_times @ real @ ( sin_coeff @ M2 ) @ ( power_power @ real @ X @ M2 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) )
@ ( times_times @ real @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ ( abs_abs @ real @ X ) @ N ) ) ) ).
% Maclaurin_sin_bound
thf(fact_4207_sum__pos__lt__pair,axiom,
! [F3: nat > real,K2: nat] :
( ( summable @ real @ F3 )
=> ( ! [D3: nat] : ( ord_less @ real @ ( zero_zero @ real ) @ ( plus_plus @ real @ ( F3 @ ( plus_plus @ nat @ K2 @ ( times_times @ nat @ ( suc @ ( suc @ ( zero_zero @ nat ) ) ) @ D3 ) ) ) @ ( F3 @ ( plus_plus @ nat @ K2 @ ( plus_plus @ nat @ ( times_times @ nat @ ( suc @ ( suc @ ( zero_zero @ nat ) ) ) @ D3 ) @ ( one_one @ nat ) ) ) ) ) )
=> ( ord_less @ real @ ( groups7311177749621191930dd_sum @ nat @ real @ F3 @ ( set_ord_lessThan @ nat @ K2 ) ) @ ( suminf @ real @ F3 ) ) ) ) ).
% sum_pos_lt_pair
thf(fact_4208_Maclaurin__exp__lt,axiom,
! [X: real,N: nat] :
( ( X
!= ( zero_zero @ real ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ? [T4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( abs_abs @ real @ T4 ) )
& ( ord_less @ real @ ( abs_abs @ real @ T4 ) @ ( abs_abs @ real @ X ) )
& ( ( exp @ real @ X )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M2: nat] : ( divide_divide @ real @ ( power_power @ real @ X @ M2 ) @ ( semiring_char_0_fact @ real @ M2 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( exp @ real @ T4 ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ X @ N ) ) ) ) ) ) ) ).
% Maclaurin_exp_lt
thf(fact_4209_lemma__termdiff2,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [H2: A,Z3: A,N: nat] :
( ( H2
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( divide_divide @ A @ ( minus_minus @ A @ ( power_power @ A @ ( plus_plus @ A @ Z3 @ H2 ) @ N ) @ ( power_power @ A @ Z3 @ N ) ) @ H2 ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ ( power_power @ A @ Z3 @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) ) ) )
= ( times_times @ A @ H2
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [P4: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [Q5: nat] : ( times_times @ A @ ( power_power @ A @ ( plus_plus @ A @ Z3 @ H2 ) @ Q5 ) @ ( power_power @ A @ Z3 @ ( minus_minus @ nat @ ( minus_minus @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ Q5 ) ) )
@ ( set_ord_lessThan @ nat @ ( minus_minus @ nat @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) @ P4 ) ) )
@ ( set_ord_lessThan @ nat @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ) ) ).
% lemma_termdiff2
thf(fact_4210_Maclaurin__sin__expansion,axiom,
! [X: real,N: nat] :
? [T4: real] :
( ( sin @ real @ X )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M2: nat] : ( times_times @ real @ ( sin_coeff @ M2 ) @ ( power_power @ real @ X @ M2 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( sin @ real @ ( plus_plus @ real @ T4 @ ( times_times @ real @ ( times_times @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( semiring_1_of_nat @ real @ N ) ) @ pi ) ) ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ X @ N ) ) ) ) ).
% Maclaurin_sin_expansion
thf(fact_4211_Maclaurin__sin__expansion2,axiom,
! [X: real,N: nat] :
? [T4: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ T4 ) @ ( abs_abs @ real @ X ) )
& ( ( sin @ real @ X )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M2: nat] : ( times_times @ real @ ( sin_coeff @ M2 ) @ ( power_power @ real @ X @ M2 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( sin @ real @ ( plus_plus @ real @ T4 @ ( times_times @ real @ ( times_times @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( semiring_1_of_nat @ real @ N ) ) @ pi ) ) ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ X @ N ) ) ) ) ) ).
% Maclaurin_sin_expansion2
thf(fact_4212_Maclaurin__cos__expansion,axiom,
! [X: real,N: nat] :
? [T4: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ T4 ) @ ( abs_abs @ real @ X ) )
& ( ( cos @ real @ X )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M2: nat] : ( times_times @ real @ ( cos_coeff @ M2 ) @ ( power_power @ real @ X @ M2 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( cos @ real @ ( plus_plus @ real @ T4 @ ( times_times @ real @ ( times_times @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( semiring_1_of_nat @ real @ N ) ) @ pi ) ) ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ X @ N ) ) ) ) ) ).
% Maclaurin_cos_expansion
thf(fact_4213_quotient__of__int,axiom,
! [A2: int] :
( ( quotient_of @ ( of_int @ A2 ) )
= ( product_Pair @ int @ int @ A2 @ ( one_one @ int ) ) ) ).
% quotient_of_int
thf(fact_4214_bij__betw__roots__unity,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( bij_betw @ nat @ complex
@ ^ [K3: nat] : ( cis @ ( divide_divide @ real @ ( times_times @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) @ ( semiring_1_of_nat @ real @ K3 ) ) @ ( semiring_1_of_nat @ real @ N ) ) )
@ ( set_ord_lessThan @ nat @ N )
@ ( collect @ complex
@ ^ [Z6: complex] :
( ( power_power @ complex @ Z6 @ N )
= ( one_one @ complex ) ) ) ) ) ).
% bij_betw_roots_unity
thf(fact_4215_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_4216_Frct__code__post_I5_J,axiom,
! [K2: num] :
( ( frct @ ( product_Pair @ int @ int @ ( one_one @ int ) @ ( numeral_numeral @ int @ K2 ) ) )
= ( divide_divide @ rat @ ( one_one @ rat ) @ ( numeral_numeral @ rat @ K2 ) ) ) ).
% Frct_code_post(5)
thf(fact_4217_atLeastAtMost__iff,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [I2: A,L: A,U: A] :
( ( member @ A @ I2 @ ( set_or1337092689740270186AtMost @ A @ L @ U ) )
= ( ( ord_less_eq @ A @ L @ I2 )
& ( ord_less_eq @ A @ I2 @ U ) ) ) ) ).
% atLeastAtMost_iff
thf(fact_4218_Icc__eq__Icc,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,H2: A,L4: A,H3: A] :
( ( ( set_or1337092689740270186AtMost @ A @ L @ H2 )
= ( set_or1337092689740270186AtMost @ A @ L4 @ H3 ) )
= ( ( ( L = L4 )
& ( H2 = H3 ) )
| ( ~ ( ord_less_eq @ A @ L @ H2 )
& ~ ( ord_less_eq @ A @ L4 @ H3 ) ) ) ) ) ).
% Icc_eq_Icc
thf(fact_4219_atLeastatMost__subset__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A2: A,B2: A,C2: A,D2: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) @ ( set_or1337092689740270186AtMost @ A @ C2 @ D2 ) )
= ( ~ ( ord_less_eq @ A @ A2 @ B2 )
| ( ( ord_less_eq @ A @ C2 @ A2 )
& ( ord_less_eq @ A @ B2 @ D2 ) ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_4220_Icc__subset__Iic__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [L: A,H2: A,H3: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ L @ H2 ) @ ( set_ord_atMost @ A @ H3 ) )
= ( ~ ( ord_less_eq @ A @ L @ H2 )
| ( ord_less_eq @ A @ H2 @ H3 ) ) ) ) ).
% Icc_subset_Iic_iff
thf(fact_4221_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_4222_sum_Oreindex__bij__betw,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comm_monoid_add @ A )
=> ! [H2: B > C,S3: set @ B,T5: set @ C,G: C > A] :
( ( bij_betw @ B @ C @ H2 @ S3 @ T5 )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X4: B] : ( G @ ( H2 @ X4 ) )
@ S3 )
= ( groups7311177749621191930dd_sum @ C @ A @ G @ T5 ) ) ) ) ).
% sum.reindex_bij_betw
thf(fact_4223_not__Iic__le__Icc,axiom,
! [A: $tType] :
( ( no_bot @ A )
=> ! [H2: A,L4: A,H3: A] :
~ ( ord_less_eq @ ( set @ A ) @ ( set_ord_atMost @ A @ H2 ) @ ( set_or1337092689740270186AtMost @ A @ L4 @ H3 ) ) ) ).
% not_Iic_le_Icc
thf(fact_4224_ex__nat__less,axiom,
! [N: nat,P3: nat > $o] :
( ( ? [M2: nat] :
( ( ord_less_eq @ nat @ M2 @ N )
& ( P3 @ M2 ) ) )
= ( ? [X4: nat] :
( ( member @ nat @ X4 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
& ( P3 @ X4 ) ) ) ) ).
% ex_nat_less
thf(fact_4225_all__nat__less,axiom,
! [N: nat,P3: nat > $o] :
( ( ! [M2: nat] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( P3 @ M2 ) ) )
= ( ! [X4: nat] :
( ( member @ nat @ X4 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
=> ( P3 @ X4 ) ) ) ) ).
% all_nat_less
thf(fact_4226_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
@ ^ [I: nat] : ( G @ ( suc @ I ) )
@ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ).
% sum.shift_bounds_cl_Suc_ivl
thf(fact_4227_sum_Oshift__bounds__cl__nat__ivl,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,M: nat,K2: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ M @ K2 ) @ ( plus_plus @ nat @ N @ K2 ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I: nat] : ( G @ ( plus_plus @ nat @ I @ K2 ) )
@ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ).
% sum.shift_bounds_cl_nat_ivl
thf(fact_4228_atLeastatMost__psubset__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A2: A,B2: A,C2: A,D2: A] :
( ( ord_less @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) @ ( set_or1337092689740270186AtMost @ A @ C2 @ D2 ) )
= ( ( ~ ( ord_less_eq @ A @ A2 @ B2 )
| ( ( ord_less_eq @ A @ C2 @ A2 )
& ( ord_less_eq @ A @ B2 @ D2 )
& ( ( ord_less @ A @ C2 @ A2 )
| ( ord_less @ A @ B2 @ D2 ) ) ) )
& ( ord_less_eq @ A @ C2 @ D2 ) ) ) ) ).
% atLeastatMost_psubset_iff
thf(fact_4229_sum_OatLeastAtMost__rev,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,N: nat,M: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ N @ M ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I: nat] : ( G @ ( minus_minus @ nat @ ( plus_plus @ nat @ M @ N ) @ I ) )
@ ( set_or1337092689740270186AtMost @ nat @ N @ M ) ) ) ) ).
% sum.atLeastAtMost_rev
thf(fact_4230_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_4231_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_4232_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_4233_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
@ ^ [I: nat] : ( G @ ( suc @ I ) )
@ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ) ) ).
% sum.Suc_reindex_ivl
thf(fact_4234_sum__Suc__diff,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [M: nat,N: nat,F3: nat > A] :
( ( ord_less_eq @ nat @ M @ ( suc @ N ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I: nat] : ( minus_minus @ A @ ( F3 @ ( suc @ I ) ) @ ( F3 @ I ) )
@ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( minus_minus @ A @ ( F3 @ ( suc @ N ) ) @ ( F3 @ M ) ) ) ) ) ).
% sum_Suc_diff
thf(fact_4235_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_4236_sum__bounds__lt__plus1,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [F3: nat > A,Mm: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( F3 @ ( suc @ K3 ) )
@ ( set_ord_lessThan @ nat @ Mm ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ Mm ) ) ) ) ).
% sum_bounds_lt_plus1
thf(fact_4237_sum_Onested__swap_H,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A2: nat > nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I: nat] : ( groups7311177749621191930dd_sum @ nat @ A @ ( A2 @ I ) @ ( set_ord_lessThan @ nat @ I ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [J4: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I: nat] : ( A2 @ I @ J4 )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ J4 ) @ N ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% sum.nested_swap'
thf(fact_4238_sum__atLeastAtMost__code,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [F3: nat > A,A2: nat,B2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_or1337092689740270186AtMost @ nat @ A2 @ B2 ) )
= ( set_fo6178422350223883121st_nat @ A
@ ^ [A5: nat] : ( plus_plus @ A @ ( F3 @ A5 ) )
@ A2
@ B2
@ ( zero_zero @ A ) ) ) ) ).
% sum_atLeastAtMost_code
thf(fact_4239_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_4240_sum__up__index__split,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [F3: nat > A,M: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_ord_atMost @ nat @ ( plus_plus @ nat @ M @ N ) ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_ord_atMost @ nat @ M ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ ( plus_plus @ nat @ M @ N ) ) ) ) ) ) ).
% sum_up_index_split
thf(fact_4241_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_4242_sum__natinterval__diff,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [M: nat,N: nat,F3: nat > A] :
( ( ( ord_less_eq @ nat @ M @ N )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( minus_minus @ A @ ( F3 @ K3 ) @ ( F3 @ ( plus_plus @ nat @ K3 @ ( one_one @ nat ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( minus_minus @ A @ ( F3 @ M ) @ ( F3 @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) ) ) )
& ( ~ ( ord_less_eq @ nat @ M @ N )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( minus_minus @ A @ ( F3 @ K3 ) @ ( F3 @ ( plus_plus @ nat @ K3 @ ( one_one @ nat ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ M @ N ) )
= ( zero_zero @ A ) ) ) ) ) ).
% sum_natinterval_diff
thf(fact_4243_sum__telescope_H_H,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [M: nat,N: nat,F3: nat > A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( minus_minus @ A @ ( F3 @ K3 ) @ ( F3 @ ( minus_minus @ nat @ K3 @ ( one_one @ nat ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M ) @ N ) )
= ( minus_minus @ A @ ( F3 @ N ) @ ( F3 @ M ) ) ) ) ) ).
% sum_telescope''
thf(fact_4244_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_4245_summable__partial__sum__bound,axiom,
! [A: $tType] :
( ( real_Vector_banach @ A )
=> ! [F3: nat > A,E: real] :
( ( summable @ A @ F3 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ E )
=> ~ ! [N8: nat] :
~ ! [M3: nat] :
( ( ord_less_eq @ nat @ N8 @ M3 )
=> ! [N9: nat] : ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_or1337092689740270186AtMost @ nat @ M3 @ N9 ) ) ) @ E ) ) ) ) ) ).
% summable_partial_sum_bound
thf(fact_4246_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_4247_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
@ ^ [I: nat] : ( plus_plus @ A @ ( G @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I ) ) @ ( G @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ).
% sum.in_pairs
thf(fact_4248_polyfun__eq__const,axiom,
! [A: $tType] :
( ( ( real_V8999393235501362500lgebra @ A )
& ( idom @ A ) )
=> ! [C2: nat > A,N: nat,K2: A] :
( ( ! [X4: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I: nat] : ( times_times @ A @ ( C2 @ I ) @ ( power_power @ A @ X4 @ I ) )
@ ( set_ord_atMost @ nat @ N ) )
= K2 ) )
= ( ( ( C2 @ ( zero_zero @ nat ) )
= K2 )
& ! [X4: nat] :
( ( member @ nat @ X4 @ ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ N ) )
=> ( ( C2 @ X4 )
= ( zero_zero @ A ) ) ) ) ) ) ).
% polyfun_eq_const
thf(fact_4249_Frct__code__post_I4_J,axiom,
! [K2: num] :
( ( frct @ ( product_Pair @ int @ int @ ( numeral_numeral @ int @ K2 ) @ ( one_one @ int ) ) )
= ( numeral_numeral @ rat @ K2 ) ) ).
% Frct_code_post(4)
thf(fact_4250_gbinomial__sum__up__index,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K2: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [J4: nat] : ( gbinomial @ A @ ( semiring_1_of_nat @ A @ J4 ) @ K2 )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
= ( gbinomial @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N ) @ ( one_one @ A ) ) @ ( plus_plus @ nat @ K2 @ ( one_one @ nat ) ) ) ) ) ).
% gbinomial_sum_up_index
thf(fact_4251_gauss__sum__nat,axiom,
! [N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [X4: nat] : X4
@ ( 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_4252_double__arith__series,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A2: A,D2: A,N: nat] :
( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I: nat] : ( plus_plus @ A @ A2 @ ( times_times @ A @ ( semiring_1_of_nat @ A @ I ) @ D2 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) )
= ( times_times @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N ) @ ( one_one @ A ) ) @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ D2 ) ) ) ) ) ).
% double_arith_series
thf(fact_4253_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_4254_arith__series__nat,axiom,
! [A2: nat,D2: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [I: nat] : ( plus_plus @ nat @ A2 @ ( times_times @ nat @ I @ D2 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
= ( divide_divide @ nat @ ( times_times @ nat @ ( suc @ N ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ A2 ) @ ( times_times @ nat @ N @ D2 ) ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% arith_series_nat
thf(fact_4255_Sum__Icc__nat,axiom,
! [M: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [X4: nat] : X4
@ ( 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_4256_arith__series,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [A2: A,D2: A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I: nat] : ( plus_plus @ A @ A2 @ ( times_times @ A @ ( semiring_1_of_nat @ A @ I ) @ D2 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
= ( divide_divide @ A @ ( times_times @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N ) @ ( one_one @ A ) ) @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ D2 ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% arith_series
thf(fact_4257_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_4258_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_4259_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_4260_polyfun__diff,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [N: nat,A2: nat > A,X: A,Y2: A] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ N )
=> ( ( minus_minus @ A
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I: nat] : ( times_times @ A @ ( A2 @ I ) @ ( power_power @ A @ X @ I ) )
@ ( set_ord_atMost @ nat @ N ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I: nat] : ( times_times @ A @ ( A2 @ I ) @ ( power_power @ A @ Y2 @ I ) )
@ ( set_ord_atMost @ nat @ N ) ) )
= ( times_times @ A @ ( minus_minus @ A @ X @ Y2 )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [J4: nat] :
( times_times @ A
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I: nat] : ( times_times @ A @ ( A2 @ I ) @ ( power_power @ A @ Y2 @ ( minus_minus @ nat @ ( minus_minus @ nat @ I @ J4 ) @ ( one_one @ nat ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ J4 ) @ N ) )
@ ( power_power @ A @ X @ J4 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ) ).
% polyfun_diff
thf(fact_4261_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_4262_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_4263_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_4264_rat__plus__code,axiom,
! [P5: rat,Q: rat] :
( ( quotient_of @ ( plus_plus @ rat @ P5 @ Q ) )
= ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [A5: int,C3: int] :
( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [B4: int,D5: int] : ( normalize @ ( product_Pair @ int @ int @ ( plus_plus @ int @ ( times_times @ int @ A5 @ D5 ) @ ( times_times @ int @ B4 @ C3 ) ) @ ( times_times @ int @ C3 @ D5 ) ) )
@ ( quotient_of @ Q ) )
@ ( quotient_of @ P5 ) ) ) ).
% rat_plus_code
thf(fact_4265_rat__divide__code,axiom,
! [P5: rat,Q: rat] :
( ( quotient_of @ ( divide_divide @ rat @ P5 @ Q ) )
= ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [A5: int,C3: int] :
( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [B4: int,D5: int] : ( normalize @ ( product_Pair @ int @ int @ ( times_times @ int @ A5 @ D5 ) @ ( times_times @ int @ C3 @ B4 ) ) )
@ ( quotient_of @ Q ) )
@ ( quotient_of @ P5 ) ) ) ).
% rat_divide_code
thf(fact_4266_rat__times__code,axiom,
! [P5: rat,Q: rat] :
( ( quotient_of @ ( times_times @ rat @ P5 @ Q ) )
= ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [A5: int,C3: int] :
( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [B4: int,D5: int] : ( normalize @ ( product_Pair @ int @ int @ ( times_times @ int @ A5 @ B4 ) @ ( times_times @ int @ C3 @ D5 ) ) )
@ ( quotient_of @ Q ) )
@ ( quotient_of @ P5 ) ) ) ).
% rat_times_code
thf(fact_4267_prod_Oneutral__const,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: set @ B] :
( ( groups7121269368397514597t_prod @ B @ A
@ ^ [Uu3: B] : ( one_one @ A )
@ A3 )
= ( one_one @ A ) ) ) ).
% prod.neutral_const
thf(fact_4268_of__nat__prod,axiom,
! [A: $tType,B: $tType] :
( ( comm_semiring_1 @ A )
=> ! [F3: B > nat,A3: set @ B] :
( ( semiring_1_of_nat @ A @ ( groups7121269368397514597t_prod @ B @ nat @ F3 @ A3 ) )
= ( groups7121269368397514597t_prod @ B @ A
@ ^ [X4: B] : ( semiring_1_of_nat @ A @ ( F3 @ X4 ) )
@ A3 ) ) ) ).
% of_nat_prod
thf(fact_4269_of__int__prod,axiom,
! [A: $tType,B: $tType] :
( ( comm_ring_1 @ A )
=> ! [F3: B > int,A3: set @ B] :
( ( ring_1_of_int @ A @ ( groups7121269368397514597t_prod @ B @ int @ F3 @ A3 ) )
= ( groups7121269368397514597t_prod @ B @ A
@ ^ [X4: B] : ( ring_1_of_int @ A @ ( F3 @ X4 ) )
@ A3 ) ) ) ).
% of_int_prod
thf(fact_4270_of__real__prod,axiom,
! [A: $tType,B: $tType] :
( ( ( comm_monoid_mult @ A )
& ( real_V2191834092415804123ebra_1 @ A ) )
=> ! [F3: B > real,S: set @ B] :
( ( real_Vector_of_real @ A @ ( groups7121269368397514597t_prod @ B @ real @ F3 @ S ) )
= ( groups7121269368397514597t_prod @ B @ A
@ ^ [X4: B] : ( real_Vector_of_real @ A @ ( F3 @ X4 ) )
@ S ) ) ) ).
% of_real_prod
thf(fact_4271_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_4272_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_4273_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_4274_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_4275_prod_Oreindex__bij__betw,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comm_monoid_mult @ A )
=> ! [H2: B > C,S3: set @ B,T5: set @ C,G: C > A] :
( ( bij_betw @ B @ C @ H2 @ S3 @ T5 )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [X4: B] : ( G @ ( H2 @ X4 ) )
@ S3 )
= ( groups7121269368397514597t_prod @ C @ A @ G @ T5 ) ) ) ) ).
% prod.reindex_bij_betw
thf(fact_4276_prod_Oneutral,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: set @ B,G: B > A] :
( ! [X5: B] :
( ( member @ B @ X5 @ A3 )
=> ( ( G @ X5 )
= ( one_one @ A ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ A3 )
= ( one_one @ A ) ) ) ) ).
% prod.neutral
thf(fact_4277_prod_Onot__neutral__contains__not__neutral,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: B > A,A3: set @ B] :
( ( ( groups7121269368397514597t_prod @ B @ A @ G @ A3 )
!= ( one_one @ A ) )
=> ~ ! [A4: B] :
( ( member @ B @ A4 @ A3 )
=> ( ( G @ A4 )
= ( one_one @ A ) ) ) ) ) ).
% prod.not_neutral_contains_not_neutral
thf(fact_4278_prod__dividef,axiom,
! [A: $tType,B: $tType] :
( ( field @ A )
=> ! [F3: B > A,G: B > A,A3: set @ B] :
( ( groups7121269368397514597t_prod @ B @ A
@ ^ [X4: B] : ( divide_divide @ A @ ( F3 @ X4 ) @ ( G @ X4 ) )
@ A3 )
= ( divide_divide @ A @ ( groups7121269368397514597t_prod @ B @ A @ F3 @ A3 ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ A3 ) ) ) ) ).
% prod_dividef
thf(fact_4279_prod__power__distrib,axiom,
! [B: $tType,A: $tType] :
( ( comm_semiring_1 @ B )
=> ! [F3: A > B,A3: set @ A,N: nat] :
( ( power_power @ B @ ( groups7121269368397514597t_prod @ A @ B @ F3 @ A3 ) @ N )
= ( groups7121269368397514597t_prod @ A @ B
@ ^ [X4: A] : ( power_power @ B @ ( F3 @ X4 ) @ N )
@ A3 ) ) ) ).
% prod_power_distrib
thf(fact_4280_prod_Odistrib,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: B > A,H2: B > A,A3: set @ B] :
( ( groups7121269368397514597t_prod @ B @ A
@ ^ [X4: B] : ( times_times @ A @ ( G @ X4 ) @ ( H2 @ X4 ) )
@ A3 )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G @ A3 ) @ ( groups7121269368397514597t_prod @ B @ A @ H2 @ A3 ) ) ) ) ).
% prod.distrib
thf(fact_4281_abs__prod,axiom,
! [A: $tType,B: $tType] :
( ( linordered_field @ A )
=> ! [F3: B > A,A3: set @ B] :
( ( abs_abs @ A @ ( groups7121269368397514597t_prod @ B @ A @ F3 @ A3 ) )
= ( groups7121269368397514597t_prod @ B @ A
@ ^ [X4: B] : ( abs_abs @ A @ ( F3 @ X4 ) )
@ A3 ) ) ) ).
% abs_prod
thf(fact_4282_prod_Oswap,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: B > C > A,B5: set @ C,A3: set @ B] :
( ( groups7121269368397514597t_prod @ B @ A
@ ^ [I: B] : ( groups7121269368397514597t_prod @ C @ A @ ( G @ I ) @ B5 )
@ A3 )
= ( groups7121269368397514597t_prod @ C @ A
@ ^ [J4: C] :
( groups7121269368397514597t_prod @ B @ A
@ ^ [I: B] : ( G @ I @ J4 )
@ A3 )
@ B5 ) ) ) ).
% prod.swap
thf(fact_4283_norm__prod__le,axiom,
! [A: $tType,B: $tType] :
( ( ( comm_monoid_mult @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [F3: B > A,A3: set @ B] :
( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( groups7121269368397514597t_prod @ B @ A @ F3 @ A3 ) )
@ ( groups7121269368397514597t_prod @ B @ real
@ ^ [A5: B] : ( real_V7770717601297561774m_norm @ A @ ( F3 @ A5 ) )
@ A3 ) ) ) ).
% norm_prod_le
thf(fact_4284_prod__norm,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V8999393235501362500lgebra @ B )
& ( comm_semiring_1 @ B ) )
=> ! [F3: A > B,A3: set @ A] :
( ( groups7121269368397514597t_prod @ A @ real
@ ^ [X4: A] : ( real_V7770717601297561774m_norm @ B @ ( F3 @ X4 ) )
@ A3 )
= ( real_V7770717601297561774m_norm @ B @ ( groups7121269368397514597t_prod @ A @ B @ F3 @ A3 ) ) ) ) ).
% prod_norm
thf(fact_4285_mod__prod__eq,axiom,
! [B: $tType,A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [F3: B > A,A2: A,A3: set @ B] :
( ( modulo_modulo @ A
@ ( groups7121269368397514597t_prod @ B @ A
@ ^ [I: B] : ( modulo_modulo @ A @ ( F3 @ I ) @ A2 )
@ A3 )
@ A2 )
= ( modulo_modulo @ A @ ( groups7121269368397514597t_prod @ B @ A @ F3 @ A3 ) @ A2 ) ) ) ).
% mod_prod_eq
thf(fact_4286_prod__nonneg,axiom,
! [A: $tType,B: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: set @ B,F3: B > A] :
( ! [X5: B] :
( ( member @ B @ X5 @ A3 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F3 @ X5 ) ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( groups7121269368397514597t_prod @ B @ A @ F3 @ A3 ) ) ) ) ).
% prod_nonneg
thf(fact_4287_prod__mono,axiom,
! [A: $tType,B: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: set @ B,F3: B > A,G: B > A] :
( ! [I3: B] :
( ( member @ B @ I3 @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F3 @ I3 ) )
& ( ord_less_eq @ A @ ( F3 @ I3 ) @ ( G @ I3 ) ) ) )
=> ( ord_less_eq @ A @ ( groups7121269368397514597t_prod @ B @ A @ F3 @ A3 ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ A3 ) ) ) ) ).
% prod_mono
thf(fact_4288_prod__pos,axiom,
! [A: $tType,B: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: set @ B,F3: B > A] :
( ! [X5: B] :
( ( member @ B @ X5 @ A3 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( F3 @ X5 ) ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( groups7121269368397514597t_prod @ B @ A @ F3 @ A3 ) ) ) ) ).
% prod_pos
thf(fact_4289_prod__ge__1,axiom,
! [A: $tType,B: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [A3: set @ B,F3: B > A] :
( ! [X5: B] :
( ( member @ B @ X5 @ A3 )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ ( F3 @ X5 ) ) )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ ( groups7121269368397514597t_prod @ B @ A @ F3 @ A3 ) ) ) ) ).
% prod_ge_1
thf(fact_4290_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
@ ^ [I: nat] : ( G @ ( suc @ I ) )
@ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ).
% prod.shift_bounds_cl_Suc_ivl
thf(fact_4291_power__sum,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [C2: A,F3: B > nat,A3: set @ B] :
( ( power_power @ A @ C2 @ ( groups7311177749621191930dd_sum @ B @ nat @ F3 @ A3 ) )
= ( groups7121269368397514597t_prod @ B @ A
@ ^ [A5: B] : ( power_power @ A @ C2 @ ( F3 @ A5 ) )
@ A3 ) ) ) ).
% power_sum
thf(fact_4292_prod_Oshift__bounds__cl__nat__ivl,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,M: nat,K2: nat,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ M @ K2 ) @ ( plus_plus @ nat @ N @ K2 ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I: nat] : ( G @ ( plus_plus @ nat @ I @ K2 ) )
@ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ).
% prod.shift_bounds_cl_nat_ivl
thf(fact_4293_prod__le__1,axiom,
! [B: $tType,A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [A3: set @ B,F3: B > A] :
( ! [X5: B] :
( ( member @ B @ X5 @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F3 @ X5 ) )
& ( ord_less_eq @ A @ ( F3 @ X5 ) @ ( one_one @ A ) ) ) )
=> ( ord_less_eq @ A @ ( groups7121269368397514597t_prod @ B @ A @ F3 @ A3 ) @ ( one_one @ A ) ) ) ) ).
% prod_le_1
thf(fact_4294_aset_I2_J,axiom,
! [D4: int,A3: set @ int,P3: int > $o,Q2: int > $o] :
( ! [X5: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ A3 )
=> ( X5
!= ( minus_minus @ int @ Xb2 @ Xa2 ) ) ) )
=> ( ( P3 @ X5 )
=> ( P3 @ ( plus_plus @ int @ X5 @ D4 ) ) ) )
=> ( ! [X5: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ A3 )
=> ( X5
!= ( minus_minus @ int @ Xb2 @ Xa2 ) ) ) )
=> ( ( Q2 @ X5 )
=> ( Q2 @ ( plus_plus @ int @ X5 @ D4 ) ) ) )
=> ! [X2: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ A3 )
=> ( X2
!= ( minus_minus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ( P3 @ X2 )
| ( Q2 @ X2 ) )
=> ( ( P3 @ ( plus_plus @ int @ X2 @ D4 ) )
| ( Q2 @ ( plus_plus @ int @ X2 @ D4 ) ) ) ) ) ) ) ).
% aset(2)
thf(fact_4295_aset_I1_J,axiom,
! [D4: int,A3: set @ int,P3: int > $o,Q2: int > $o] :
( ! [X5: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ A3 )
=> ( X5
!= ( minus_minus @ int @ Xb2 @ Xa2 ) ) ) )
=> ( ( P3 @ X5 )
=> ( P3 @ ( plus_plus @ int @ X5 @ D4 ) ) ) )
=> ( ! [X5: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ A3 )
=> ( X5
!= ( minus_minus @ int @ Xb2 @ Xa2 ) ) ) )
=> ( ( Q2 @ X5 )
=> ( Q2 @ ( plus_plus @ int @ X5 @ D4 ) ) ) )
=> ! [X2: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ A3 )
=> ( X2
!= ( minus_minus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ( P3 @ X2 )
& ( Q2 @ X2 ) )
=> ( ( P3 @ ( plus_plus @ int @ X2 @ D4 ) )
& ( Q2 @ ( plus_plus @ int @ X2 @ D4 ) ) ) ) ) ) ) ).
% aset(1)
thf(fact_4296_bset_I2_J,axiom,
! [D4: int,B5: set @ int,P3: int > $o,Q2: int > $o] :
( ! [X5: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ B5 )
=> ( X5
!= ( plus_plus @ int @ Xb2 @ Xa2 ) ) ) )
=> ( ( P3 @ X5 )
=> ( P3 @ ( minus_minus @ int @ X5 @ D4 ) ) ) )
=> ( ! [X5: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ B5 )
=> ( X5
!= ( plus_plus @ int @ Xb2 @ Xa2 ) ) ) )
=> ( ( Q2 @ X5 )
=> ( Q2 @ ( minus_minus @ int @ X5 @ D4 ) ) ) )
=> ! [X2: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ B5 )
=> ( X2
!= ( plus_plus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ( P3 @ X2 )
| ( Q2 @ X2 ) )
=> ( ( P3 @ ( minus_minus @ int @ X2 @ D4 ) )
| ( Q2 @ ( minus_minus @ int @ X2 @ D4 ) ) ) ) ) ) ) ).
% bset(2)
thf(fact_4297_bset_I1_J,axiom,
! [D4: int,B5: set @ int,P3: int > $o,Q2: int > $o] :
( ! [X5: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ B5 )
=> ( X5
!= ( plus_plus @ int @ Xb2 @ Xa2 ) ) ) )
=> ( ( P3 @ X5 )
=> ( P3 @ ( minus_minus @ int @ X5 @ D4 ) ) ) )
=> ( ! [X5: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ B5 )
=> ( X5
!= ( plus_plus @ int @ Xb2 @ Xa2 ) ) ) )
=> ( ( Q2 @ X5 )
=> ( Q2 @ ( minus_minus @ int @ X5 @ D4 ) ) ) )
=> ! [X2: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ B5 )
=> ( X2
!= ( plus_plus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ( P3 @ X2 )
& ( Q2 @ X2 ) )
=> ( ( P3 @ ( minus_minus @ int @ X2 @ D4 ) )
& ( Q2 @ ( minus_minus @ int @ X2 @ D4 ) ) ) ) ) ) ) ).
% bset(1)
thf(fact_4298_prod_Onat__diff__reindex,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I: nat] : ( G @ ( minus_minus @ nat @ N @ ( suc @ I ) ) )
@ ( set_ord_lessThan @ nat @ N ) )
= ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% prod.nat_diff_reindex
thf(fact_4299_prod_OatLeastAtMost__rev,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,N: nat,M: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ N @ M ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I: nat] : ( G @ ( minus_minus @ nat @ ( plus_plus @ nat @ M @ N ) @ I ) )
@ ( set_or1337092689740270186AtMost @ nat @ N @ M ) ) ) ) ).
% prod.atLeastAtMost_rev
thf(fact_4300_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_4301_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_4302_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_4303_aset_I10_J,axiom,
! [D2: int,D4: int,A3: set @ int,T2: int] :
( ( dvd_dvd @ int @ D2 @ D4 )
=> ! [X2: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ A3 )
=> ( X2
!= ( minus_minus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ~ ( dvd_dvd @ int @ D2 @ ( plus_plus @ int @ X2 @ T2 ) )
=> ~ ( dvd_dvd @ int @ D2 @ ( plus_plus @ int @ ( plus_plus @ int @ X2 @ D4 ) @ T2 ) ) ) ) ) ).
% aset(10)
thf(fact_4304_aset_I9_J,axiom,
! [D2: int,D4: int,A3: set @ int,T2: int] :
( ( dvd_dvd @ int @ D2 @ D4 )
=> ! [X2: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ A3 )
=> ( X2
!= ( minus_minus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( dvd_dvd @ int @ D2 @ ( plus_plus @ int @ X2 @ T2 ) )
=> ( dvd_dvd @ int @ D2 @ ( plus_plus @ int @ ( plus_plus @ int @ X2 @ D4 ) @ T2 ) ) ) ) ) ).
% aset(9)
thf(fact_4305_bset_I10_J,axiom,
! [D2: int,D4: int,B5: set @ int,T2: int] :
( ( dvd_dvd @ int @ D2 @ D4 )
=> ! [X2: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ B5 )
=> ( X2
!= ( plus_plus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ~ ( dvd_dvd @ int @ D2 @ ( plus_plus @ int @ X2 @ T2 ) )
=> ~ ( dvd_dvd @ int @ D2 @ ( plus_plus @ int @ ( minus_minus @ int @ X2 @ D4 ) @ T2 ) ) ) ) ) ).
% bset(10)
thf(fact_4306_bset_I9_J,axiom,
! [D2: int,D4: int,B5: set @ int,T2: int] :
( ( dvd_dvd @ int @ D2 @ D4 )
=> ! [X2: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ B5 )
=> ( X2
!= ( plus_plus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( dvd_dvd @ int @ D2 @ ( plus_plus @ int @ X2 @ T2 ) )
=> ( dvd_dvd @ int @ D2 @ ( plus_plus @ int @ ( minus_minus @ int @ X2 @ D4 ) @ T2 ) ) ) ) ) ).
% bset(9)
thf(fact_4307_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
@ ^ [I: nat] : ( G @ ( suc @ I ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% prod.lessThan_Suc_shift
thf(fact_4308_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
@ ^ [I: nat] : ( G @ ( suc @ I ) )
@ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ) ) ).
% prod.Suc_reindex_ivl
thf(fact_4309_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
@ ^ [I: nat] : ( G @ ( suc @ I ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ) ).
% prod.atMost_Suc_shift
thf(fact_4310_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_4311_fact__prod,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ( ( semiring_char_0_fact @ A )
= ( ^ [N2: nat] :
( semiring_1_of_nat @ A
@ ( groups7121269368397514597t_prod @ nat @ nat
@ ^ [X4: nat] : X4
@ ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ N2 ) ) ) ) ) ) ).
% fact_prod
thf(fact_4312_prod_Onested__swap_H,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A2: nat > nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I: nat] : ( groups7121269368397514597t_prod @ nat @ A @ ( A2 @ I ) @ ( set_ord_lessThan @ nat @ I ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [J4: nat] :
( groups7121269368397514597t_prod @ nat @ A
@ ^ [I: nat] : ( A2 @ I @ J4 )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ J4 ) @ N ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% prod.nested_swap'
thf(fact_4313_prod__atLeastAtMost__code,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [F3: nat > A,A2: nat,B2: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ F3 @ ( set_or1337092689740270186AtMost @ nat @ A2 @ B2 ) )
= ( set_fo6178422350223883121st_nat @ A
@ ^ [A5: nat] : ( times_times @ A @ ( F3 @ A5 ) )
@ A2
@ B2
@ ( one_one @ A ) ) ) ) ).
% prod_atLeastAtMost_code
thf(fact_4314_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_4315_periodic__finite__ex,axiom,
! [D2: int,P3: int > $o] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D2 )
=> ( ! [X5: int,K: int] :
( ( P3 @ X5 )
= ( P3 @ ( minus_minus @ int @ X5 @ ( times_times @ int @ K @ D2 ) ) ) )
=> ( ( ? [X6: int] : ( P3 @ X6 ) )
= ( ? [X4: int] :
( ( member @ int @ X4 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D2 ) )
& ( P3 @ X4 ) ) ) ) ) ) ).
% periodic_finite_ex
thf(fact_4316_aset_I7_J,axiom,
! [D4: int,A3: set @ int,T2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
=> ! [X2: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ A3 )
=> ( X2
!= ( minus_minus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ord_less @ int @ T2 @ X2 )
=> ( ord_less @ int @ T2 @ ( plus_plus @ int @ X2 @ D4 ) ) ) ) ) ).
% aset(7)
thf(fact_4317_aset_I5_J,axiom,
! [D4: int,T2: int,A3: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
=> ( ( member @ int @ T2 @ A3 )
=> ! [X2: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ A3 )
=> ( X2
!= ( minus_minus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ord_less @ int @ X2 @ T2 )
=> ( ord_less @ int @ ( plus_plus @ int @ X2 @ D4 ) @ T2 ) ) ) ) ) ).
% aset(5)
thf(fact_4318_aset_I4_J,axiom,
! [D4: int,T2: int,A3: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
=> ( ( member @ int @ T2 @ A3 )
=> ! [X2: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ A3 )
=> ( X2
!= ( minus_minus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( X2 != T2 )
=> ( ( plus_plus @ int @ X2 @ D4 )
!= T2 ) ) ) ) ) ).
% aset(4)
thf(fact_4319_aset_I3_J,axiom,
! [D4: int,T2: int,A3: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
=> ( ( member @ int @ ( plus_plus @ int @ T2 @ ( one_one @ int ) ) @ A3 )
=> ! [X2: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ A3 )
=> ( X2
!= ( minus_minus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( X2 = T2 )
=> ( ( plus_plus @ int @ X2 @ D4 )
= T2 ) ) ) ) ) ).
% aset(3)
thf(fact_4320_bset_I7_J,axiom,
! [D4: int,T2: int,B5: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
=> ( ( member @ int @ T2 @ B5 )
=> ! [X2: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ B5 )
=> ( X2
!= ( plus_plus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ord_less @ int @ T2 @ X2 )
=> ( ord_less @ int @ T2 @ ( minus_minus @ int @ X2 @ D4 ) ) ) ) ) ) ).
% bset(7)
thf(fact_4321_bset_I5_J,axiom,
! [D4: int,B5: set @ int,T2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
=> ! [X2: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ B5 )
=> ( X2
!= ( plus_plus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ord_less @ int @ X2 @ T2 )
=> ( ord_less @ int @ ( minus_minus @ int @ X2 @ D4 ) @ T2 ) ) ) ) ).
% bset(5)
thf(fact_4322_bset_I4_J,axiom,
! [D4: int,T2: int,B5: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
=> ( ( member @ int @ T2 @ B5 )
=> ! [X2: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ B5 )
=> ( X2
!= ( plus_plus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( X2 != T2 )
=> ( ( minus_minus @ int @ X2 @ D4 )
!= T2 ) ) ) ) ) ).
% bset(4)
thf(fact_4323_bset_I3_J,axiom,
! [D4: int,T2: int,B5: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
=> ( ( member @ int @ ( minus_minus @ int @ T2 @ ( one_one @ int ) ) @ B5 )
=> ! [X2: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ B5 )
=> ( X2
!= ( plus_plus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( X2 = T2 )
=> ( ( minus_minus @ int @ X2 @ D4 )
= T2 ) ) ) ) ) ).
% bset(3)
thf(fact_4324_normalize__crossproduct,axiom,
! [Q: int,S: int,P5: int,R: int] :
( ( Q
!= ( zero_zero @ int ) )
=> ( ( S
!= ( zero_zero @ int ) )
=> ( ( ( normalize @ ( product_Pair @ int @ int @ P5 @ Q ) )
= ( normalize @ ( product_Pair @ int @ int @ R @ S ) ) )
=> ( ( times_times @ int @ P5 @ S )
= ( times_times @ int @ R @ Q ) ) ) ) ) ).
% normalize_crossproduct
thf(fact_4325_norm__prod__diff,axiom,
! [A: $tType,I5: $tType] :
( ( ( comm_monoid_mult @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [I6: set @ I5,Z3: I5 > A,W: I5 > A] :
( ! [I3: I5] :
( ( member @ I5 @ I3 @ I6 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( Z3 @ I3 ) ) @ ( one_one @ real ) ) )
=> ( ! [I3: I5] :
( ( member @ I5 @ I3 @ I6 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( W @ I3 ) ) @ ( one_one @ real ) ) )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( groups7121269368397514597t_prod @ I5 @ A @ Z3 @ I6 ) @ ( groups7121269368397514597t_prod @ I5 @ A @ W @ I6 ) ) )
@ ( groups7311177749621191930dd_sum @ I5 @ real
@ ^ [I: I5] : ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( Z3 @ I ) @ ( W @ I ) ) )
@ I6 ) ) ) ) ) ).
% norm_prod_diff
thf(fact_4326_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
@ ^ [I: nat] : ( G @ ( suc @ I ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% prod.atMost_shift
thf(fact_4327_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
@ ^ [X4: nat] : X4
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ N ) @ M ) ) ) ) ) ).
% fact_eq_fact_times
thf(fact_4328_aset_I8_J,axiom,
! [D4: int,A3: set @ int,T2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
=> ! [X2: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ A3 )
=> ( X2
!= ( minus_minus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ord_less_eq @ int @ T2 @ X2 )
=> ( ord_less_eq @ int @ T2 @ ( plus_plus @ int @ X2 @ D4 ) ) ) ) ) ).
% aset(8)
thf(fact_4329_aset_I6_J,axiom,
! [D4: int,T2: int,A3: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
=> ( ( member @ int @ ( plus_plus @ int @ T2 @ ( one_one @ int ) ) @ A3 )
=> ! [X2: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ A3 )
=> ( X2
!= ( minus_minus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ord_less_eq @ int @ X2 @ T2 )
=> ( ord_less_eq @ int @ ( plus_plus @ int @ X2 @ D4 ) @ T2 ) ) ) ) ) ).
% aset(6)
thf(fact_4330_bset_I8_J,axiom,
! [D4: int,T2: int,B5: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
=> ( ( member @ int @ ( minus_minus @ int @ T2 @ ( one_one @ int ) ) @ B5 )
=> ! [X2: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ B5 )
=> ( X2
!= ( plus_plus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ord_less_eq @ int @ T2 @ X2 )
=> ( ord_less_eq @ int @ T2 @ ( minus_minus @ int @ X2 @ D4 ) ) ) ) ) ) ).
% bset(8)
thf(fact_4331_bset_I6_J,axiom,
! [D4: int,B5: set @ int,T2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
=> ! [X2: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb3: int] :
( ( member @ int @ Xb3 @ B5 )
=> ( X2
!= ( plus_plus @ int @ Xb3 @ Xa3 ) ) ) )
=> ( ( ord_less_eq @ int @ X2 @ T2 )
=> ( ord_less_eq @ int @ ( minus_minus @ int @ X2 @ D4 ) @ T2 ) ) ) ) ).
% bset(6)
thf(fact_4332_cpmi,axiom,
! [D4: int,P3: int > $o,P6: int > $o,B5: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
=> ( ? [Z4: int] :
! [X5: int] :
( ( ord_less @ int @ X5 @ Z4 )
=> ( ( P3 @ X5 )
= ( P6 @ X5 ) ) )
=> ( ! [X5: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ B5 )
=> ( X5
!= ( plus_plus @ int @ Xb2 @ Xa2 ) ) ) )
=> ( ( P3 @ X5 )
=> ( P3 @ ( minus_minus @ int @ X5 @ D4 ) ) ) )
=> ( ! [X5: int,K: int] :
( ( P6 @ X5 )
= ( P6 @ ( minus_minus @ int @ X5 @ ( times_times @ int @ K @ D4 ) ) ) )
=> ( ( ? [X6: int] : ( P3 @ X6 ) )
= ( ? [X4: int] :
( ( member @ int @ X4 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
& ( P6 @ X4 ) )
| ? [X4: int] :
( ( member @ int @ X4 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
& ? [Y: int] :
( ( member @ int @ Y @ B5 )
& ( P3 @ ( plus_plus @ int @ Y @ X4 ) ) ) ) ) ) ) ) ) ) ).
% cpmi
thf(fact_4333_cppi,axiom,
! [D4: int,P3: int > $o,P6: int > $o,A3: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D4 )
=> ( ? [Z4: int] :
! [X5: int] :
( ( ord_less @ int @ Z4 @ X5 )
=> ( ( P3 @ X5 )
= ( P6 @ X5 ) ) )
=> ( ! [X5: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ A3 )
=> ( X5
!= ( minus_minus @ int @ Xb2 @ Xa2 ) ) ) )
=> ( ( P3 @ X5 )
=> ( P3 @ ( plus_plus @ int @ X5 @ D4 ) ) ) )
=> ( ! [X5: int,K: int] :
( ( P6 @ X5 )
= ( P6 @ ( minus_minus @ int @ X5 @ ( times_times @ int @ K @ D4 ) ) ) )
=> ( ( ? [X6: int] : ( P3 @ X6 ) )
= ( ? [X4: int] :
( ( member @ int @ X4 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
& ( P6 @ X4 ) )
| ? [X4: int] :
( ( member @ int @ X4 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D4 ) )
& ? [Y: int] :
( ( member @ int @ Y @ A3 )
& ( P3 @ ( minus_minus @ int @ Y @ X4 ) ) ) ) ) ) ) ) ) ) ).
% cppi
thf(fact_4334_pochhammer__Suc__prod,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A2: A,N: nat] :
( ( comm_s3205402744901411588hammer @ A @ A2 @ ( suc @ N ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I: nat] : ( plus_plus @ A @ A2 @ ( semiring_1_of_nat @ A @ I ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% pochhammer_Suc_prod
thf(fact_4335_pochhammer__prod__rev,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ( ( comm_s3205402744901411588hammer @ A )
= ( ^ [A5: A,N2: nat] :
( groups7121269368397514597t_prod @ nat @ A
@ ^ [I: nat] : ( plus_plus @ A @ A5 @ ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ N2 @ I ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ N2 ) ) ) ) ) ).
% pochhammer_prod_rev
thf(fact_4336_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
@ ^ [X4: nat] : X4
@ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) @ M ) ) ) ) ).
% fact_div_fact
thf(fact_4337_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
@ ^ [I: nat] : ( times_times @ A @ ( G @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I ) ) @ ( G @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ).
% prod.in_pairs
thf(fact_4338_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
@ ^ [I: nat] : ( times_times @ A @ ( G @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I ) ) @ ( G @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I ) ) ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% prod.in_pairs_0
thf(fact_4339_pochhammer__Suc__prod__rev,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A2: A,N: nat] :
( ( comm_s3205402744901411588hammer @ A @ A2 @ ( suc @ N ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I: nat] : ( plus_plus @ A @ A2 @ ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ N @ I ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% pochhammer_Suc_prod_rev
thf(fact_4340_prod_Ozero__middle,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [P5: nat,K2: nat,G: nat > A,H2: nat > A] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ P5 )
=> ( ( ord_less_eq @ nat @ K2 @ P5 )
=> ( ( groups7121269368397514597t_prod @ nat @ A
@ ^ [J4: nat] : ( if @ A @ ( ord_less @ nat @ J4 @ K2 ) @ ( G @ J4 ) @ ( if @ A @ ( J4 = K2 ) @ ( one_one @ A ) @ ( H2 @ ( minus_minus @ nat @ J4 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) )
@ ( set_ord_atMost @ nat @ P5 ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [J4: nat] : ( if @ A @ ( ord_less @ nat @ J4 @ K2 ) @ ( G @ J4 ) @ ( H2 @ J4 ) )
@ ( set_ord_atMost @ nat @ ( minus_minus @ nat @ P5 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ) ) ).
% prod.zero_middle
thf(fact_4341_gbinomial__Suc,axiom,
! [A: $tType] :
( ( ( semiring_char_0 @ A )
& ( semidom_divide @ A ) )
=> ! [A2: A,K2: nat] :
( ( gbinomial @ A @ A2 @ ( suc @ K2 ) )
= ( divide_divide @ A
@ ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I: nat] : ( minus_minus @ A @ A2 @ ( semiring_1_of_nat @ A @ I ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ K2 ) )
@ ( semiring_char_0_fact @ A @ ( suc @ K2 ) ) ) ) ) ).
% gbinomial_Suc
thf(fact_4342_Sum__Icc__int,axiom,
! [M: int,N: int] :
( ( ord_less_eq @ int @ M @ N )
=> ( ( groups7311177749621191930dd_sum @ int @ int
@ ^ [X4: int] : X4
@ ( 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_4343_rat__minus__code,axiom,
! [P5: rat,Q: rat] :
( ( quotient_of @ ( minus_minus @ rat @ P5 @ Q ) )
= ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [A5: int,C3: int] :
( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [B4: int,D5: int] : ( normalize @ ( product_Pair @ int @ int @ ( minus_minus @ int @ ( times_times @ int @ A5 @ D5 ) @ ( times_times @ int @ B4 @ C3 ) ) @ ( times_times @ int @ C3 @ D5 ) ) )
@ ( quotient_of @ Q ) )
@ ( quotient_of @ P5 ) ) ) ).
% rat_minus_code
thf(fact_4344_bij__betw__nth__root__unity,axiom,
! [C2: complex,N: nat] :
( ( C2
!= ( zero_zero @ complex ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( bij_betw @ complex @ complex @ ( times_times @ complex @ ( times_times @ complex @ ( real_Vector_of_real @ complex @ ( root @ N @ ( real_V7770717601297561774m_norm @ complex @ C2 ) ) ) @ ( cis @ ( divide_divide @ real @ ( arg @ C2 ) @ ( semiring_1_of_nat @ real @ N ) ) ) ) )
@ ( collect @ complex
@ ^ [Z6: complex] :
( ( power_power @ complex @ Z6 @ N )
= ( one_one @ complex ) ) )
@ ( collect @ complex
@ ^ [Z6: complex] :
( ( power_power @ complex @ Z6 @ N )
= C2 ) ) ) ) ) ).
% bij_betw_nth_root_unity
thf(fact_4345_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_4346_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_4347_set__encode__def,axiom,
( nat_set_encode
= ( groups7311177749621191930dd_sum @ nat @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% set_encode_def
thf(fact_4348_push__bit__nonnegative__int__iff,axiom,
! [N: nat,K2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se4730199178511100633sh_bit @ int @ N @ K2 ) )
= ( ord_less_eq @ int @ ( zero_zero @ int ) @ K2 ) ) ).
% push_bit_nonnegative_int_iff
thf(fact_4349_push__bit__negative__int__iff,axiom,
! [N: nat,K2: int] :
( ( ord_less @ int @ ( bit_se4730199178511100633sh_bit @ int @ N @ K2 ) @ ( zero_zero @ int ) )
= ( ord_less @ int @ K2 @ ( zero_zero @ int ) ) ) ).
% push_bit_negative_int_iff
thf(fact_4350_push__bit__eq__0__iff,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [N: nat,A2: A] :
( ( ( bit_se4730199178511100633sh_bit @ A @ N @ A2 )
= ( zero_zero @ A ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% push_bit_eq_0_iff
thf(fact_4351_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_4352_real__root__zero,axiom,
! [N: nat] :
( ( root @ N @ ( zero_zero @ real ) )
= ( zero_zero @ real ) ) ).
% real_root_zero
thf(fact_4353_push__bit__push__bit,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: nat,N: nat,A2: A] :
( ( bit_se4730199178511100633sh_bit @ A @ M @ ( bit_se4730199178511100633sh_bit @ A @ N @ A2 ) )
= ( bit_se4730199178511100633sh_bit @ A @ ( plus_plus @ nat @ M @ N ) @ A2 ) ) ) ).
% push_bit_push_bit
thf(fact_4354_push__bit__and,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A2: A,B2: A] :
( ( bit_se4730199178511100633sh_bit @ A @ N @ ( bit_se5824344872417868541ns_and @ A @ A2 @ B2 ) )
= ( bit_se5824344872417868541ns_and @ A @ ( bit_se4730199178511100633sh_bit @ A @ N @ A2 ) @ ( bit_se4730199178511100633sh_bit @ A @ N @ B2 ) ) ) ) ).
% push_bit_and
thf(fact_4355_push__bit__xor,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A2: A,B2: A] :
( ( bit_se4730199178511100633sh_bit @ A @ N @ ( bit_se5824344971392196577ns_xor @ A @ A2 @ B2 ) )
= ( bit_se5824344971392196577ns_xor @ A @ ( bit_se4730199178511100633sh_bit @ A @ N @ A2 ) @ ( bit_se4730199178511100633sh_bit @ A @ N @ B2 ) ) ) ) ).
% push_bit_xor
thf(fact_4356_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_4357_real__root__Suc__0,axiom,
! [X: real] :
( ( root @ ( suc @ ( zero_zero @ nat ) ) @ X )
= X ) ).
% real_root_Suc_0
thf(fact_4358_root__0,axiom,
! [X: real] :
( ( root @ ( zero_zero @ nat ) @ X )
= ( zero_zero @ real ) ) ).
% root_0
thf(fact_4359_real__root__eq__iff,axiom,
! [N: nat,X: real,Y2: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ( root @ N @ X )
= ( root @ N @ Y2 ) )
= ( X = Y2 ) ) ) ).
% real_root_eq_iff
thf(fact_4360_push__bit__Suc__numeral,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,K2: num] :
( ( bit_se4730199178511100633sh_bit @ A @ ( suc @ N ) @ ( numeral_numeral @ A @ K2 ) )
= ( bit_se4730199178511100633sh_bit @ A @ N @ ( numeral_numeral @ A @ ( bit0 @ K2 ) ) ) ) ) ).
% push_bit_Suc_numeral
thf(fact_4361_real__root__eq__0__iff,axiom,
! [N: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ( root @ N @ X )
= ( zero_zero @ real ) )
= ( X
= ( zero_zero @ real ) ) ) ) ).
% real_root_eq_0_iff
thf(fact_4362_real__root__less__iff,axiom,
! [N: nat,X: real,Y2: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( root @ N @ X ) @ ( root @ N @ Y2 ) )
= ( ord_less @ real @ X @ Y2 ) ) ) ).
% real_root_less_iff
thf(fact_4363_real__root__le__iff,axiom,
! [N: nat,X: real,Y2: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ real @ ( root @ N @ X ) @ ( root @ N @ Y2 ) )
= ( ord_less_eq @ real @ X @ Y2 ) ) ) ).
% real_root_le_iff
thf(fact_4364_real__root__eq__1__iff,axiom,
! [N: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ( root @ N @ X )
= ( one_one @ real ) )
= ( X
= ( one_one @ real ) ) ) ) ).
% real_root_eq_1_iff
thf(fact_4365_real__root__one,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( root @ N @ ( one_one @ real ) )
= ( one_one @ real ) ) ) ).
% real_root_one
thf(fact_4366_push__bit__Suc__minus__numeral,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: nat,K2: num] :
( ( bit_se4730199178511100633sh_bit @ A @ ( suc @ N ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ K2 ) ) )
= ( bit_se4730199178511100633sh_bit @ A @ N @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( bit0 @ K2 ) ) ) ) ) ) ).
% push_bit_Suc_minus_numeral
thf(fact_4367_real__root__lt__0__iff,axiom,
! [N: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( root @ N @ X ) @ ( zero_zero @ real ) )
= ( ord_less @ real @ X @ ( zero_zero @ real ) ) ) ) ).
% real_root_lt_0_iff
thf(fact_4368_real__root__gt__0__iff,axiom,
! [N: nat,Y2: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( root @ N @ Y2 ) )
= ( ord_less @ real @ ( zero_zero @ real ) @ Y2 ) ) ) ).
% real_root_gt_0_iff
thf(fact_4369_real__root__le__0__iff,axiom,
! [N: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ real @ ( root @ N @ X ) @ ( zero_zero @ real ) )
= ( ord_less_eq @ real @ X @ ( zero_zero @ real ) ) ) ) ).
% real_root_le_0_iff
thf(fact_4370_real__root__ge__0__iff,axiom,
! [N: nat,Y2: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( root @ N @ Y2 ) )
= ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y2 ) ) ) ).
% real_root_ge_0_iff
thf(fact_4371_real__root__lt__1__iff,axiom,
! [N: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( root @ N @ X ) @ ( one_one @ real ) )
= ( ord_less @ real @ X @ ( one_one @ real ) ) ) ) ).
% real_root_lt_1_iff
thf(fact_4372_real__root__gt__1__iff,axiom,
! [N: nat,Y2: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( one_one @ real ) @ ( root @ N @ Y2 ) )
= ( ord_less @ real @ ( one_one @ real ) @ Y2 ) ) ) ).
% real_root_gt_1_iff
thf(fact_4373_real__root__le__1__iff,axiom,
! [N: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ real @ ( root @ N @ X ) @ ( one_one @ real ) )
= ( ord_less_eq @ real @ X @ ( one_one @ real ) ) ) ) ).
% real_root_le_1_iff
thf(fact_4374_real__root__ge__1__iff,axiom,
! [N: nat,Y2: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ real @ ( one_one @ real ) @ ( root @ N @ Y2 ) )
= ( ord_less_eq @ real @ ( one_one @ real ) @ Y2 ) ) ) ).
% real_root_ge_1_iff
thf(fact_4375_push__bit__numeral,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [L: num,K2: num] :
( ( bit_se4730199178511100633sh_bit @ A @ ( numeral_numeral @ nat @ L ) @ ( numeral_numeral @ A @ K2 ) )
= ( bit_se4730199178511100633sh_bit @ A @ ( pred_numeral @ L ) @ ( numeral_numeral @ A @ ( bit0 @ K2 ) ) ) ) ) ).
% push_bit_numeral
thf(fact_4376_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_4377_push__bit__Suc,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A2: A] :
( ( bit_se4730199178511100633sh_bit @ A @ ( suc @ N ) @ A2 )
= ( bit_se4730199178511100633sh_bit @ A @ N @ ( times_times @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% push_bit_Suc
thf(fact_4378_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_4379_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_4380_even__push__bit__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A2: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se4730199178511100633sh_bit @ A @ N @ A2 ) )
= ( ( N
!= ( zero_zero @ nat ) )
| ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) ) ) ) ).
% even_push_bit_iff
thf(fact_4381_real__root__pow__pos2,axiom,
! [N: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( power_power @ real @ ( root @ N @ X ) @ N )
= X ) ) ) ).
% real_root_pow_pos2
thf(fact_4382_push__bit__minus__numeral,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [L: num,K2: num] :
( ( bit_se4730199178511100633sh_bit @ A @ ( numeral_numeral @ nat @ L ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ K2 ) ) )
= ( bit_se4730199178511100633sh_bit @ A @ ( pred_numeral @ L ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( bit0 @ K2 ) ) ) ) ) ) ).
% push_bit_minus_numeral
thf(fact_4383_int__prod,axiom,
! [B: $tType,F3: B > nat,A3: set @ B] :
( ( semiring_1_of_nat @ int @ ( groups7121269368397514597t_prod @ B @ nat @ F3 @ A3 ) )
= ( groups7121269368397514597t_prod @ B @ int
@ ^ [X4: B] : ( semiring_1_of_nat @ int @ ( F3 @ X4 ) )
@ A3 ) ) ).
% int_prod
thf(fact_4384_real__root__inverse,axiom,
! [N: nat,X: real] :
( ( root @ N @ ( inverse_inverse @ real @ X ) )
= ( inverse_inverse @ real @ ( root @ N @ X ) ) ) ).
% real_root_inverse
thf(fact_4385_push__bit__minus,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: nat,A2: A] :
( ( bit_se4730199178511100633sh_bit @ A @ N @ ( uminus_uminus @ A @ A2 ) )
= ( uminus_uminus @ A @ ( bit_se4730199178511100633sh_bit @ A @ N @ A2 ) ) ) ) ).
% push_bit_minus
thf(fact_4386_push__bit__add,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A2: A,B2: A] :
( ( bit_se4730199178511100633sh_bit @ A @ N @ ( plus_plus @ A @ A2 @ B2 ) )
= ( plus_plus @ A @ ( bit_se4730199178511100633sh_bit @ A @ N @ A2 ) @ ( bit_se4730199178511100633sh_bit @ A @ N @ B2 ) ) ) ) ).
% push_bit_add
thf(fact_4387_real__root__minus,axiom,
! [N: nat,X: real] :
( ( root @ N @ ( uminus_uminus @ real @ X ) )
= ( uminus_uminus @ real @ ( root @ N @ X ) ) ) ).
% real_root_minus
thf(fact_4388_push__bit__nat__eq,axiom,
! [N: nat,K2: int] :
( ( bit_se4730199178511100633sh_bit @ nat @ N @ ( nat2 @ K2 ) )
= ( nat2 @ ( bit_se4730199178511100633sh_bit @ int @ N @ K2 ) ) ) ).
% push_bit_nat_eq
thf(fact_4389_real__root__commute,axiom,
! [M: nat,N: nat,X: real] :
( ( root @ M @ ( root @ N @ X ) )
= ( root @ N @ ( root @ M @ X ) ) ) ).
% real_root_commute
thf(fact_4390_push__bit__of__int,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: nat,K2: int] :
( ( bit_se4730199178511100633sh_bit @ A @ N @ ( ring_1_of_int @ A @ K2 ) )
= ( ring_1_of_int @ A @ ( bit_se4730199178511100633sh_bit @ int @ N @ K2 ) ) ) ) ).
% push_bit_of_int
thf(fact_4391_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_4392_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_4393_real__root__mult,axiom,
! [N: nat,X: real,Y2: real] :
( ( root @ N @ ( times_times @ real @ X @ Y2 ) )
= ( times_times @ real @ ( root @ N @ X ) @ ( root @ N @ Y2 ) ) ) ).
% real_root_mult
thf(fact_4394_real__root__mult__exp,axiom,
! [M: nat,N: nat,X: real] :
( ( root @ ( times_times @ nat @ M @ N ) @ X )
= ( root @ M @ ( root @ N @ X ) ) ) ).
% real_root_mult_exp
thf(fact_4395_real__root__divide,axiom,
! [N: nat,X: real,Y2: real] :
( ( root @ N @ ( divide_divide @ real @ X @ Y2 ) )
= ( divide_divide @ real @ ( root @ N @ X ) @ ( root @ N @ Y2 ) ) ) ).
% real_root_divide
thf(fact_4396_real__root__pos__pos__le,axiom,
! [X: real,N: nat] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( root @ N @ X ) ) ) ).
% real_root_pos_pos_le
thf(fact_4397_push__bit__take__bit,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: nat,N: nat,A2: A] :
( ( bit_se4730199178511100633sh_bit @ A @ M @ ( bit_se2584673776208193580ke_bit @ A @ N @ A2 ) )
= ( bit_se2584673776208193580ke_bit @ A @ ( plus_plus @ nat @ M @ N ) @ ( bit_se4730199178511100633sh_bit @ A @ M @ A2 ) ) ) ) ).
% push_bit_take_bit
thf(fact_4398_take__bit__push__bit,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: nat,N: nat,A2: A] :
( ( bit_se2584673776208193580ke_bit @ A @ M @ ( bit_se4730199178511100633sh_bit @ A @ N @ A2 ) )
= ( bit_se4730199178511100633sh_bit @ A @ N @ ( bit_se2584673776208193580ke_bit @ A @ ( minus_minus @ nat @ M @ N ) @ A2 ) ) ) ) ).
% take_bit_push_bit
thf(fact_4399_prod__int__eq,axiom,
! [I2: nat,J3: nat] :
( ( groups7121269368397514597t_prod @ nat @ int @ ( semiring_1_of_nat @ int ) @ ( set_or1337092689740270186AtMost @ nat @ I2 @ J3 ) )
= ( groups7121269368397514597t_prod @ int @ int
@ ^ [X4: int] : X4
@ ( set_or1337092689740270186AtMost @ int @ ( semiring_1_of_nat @ int @ I2 ) @ ( semiring_1_of_nat @ int @ J3 ) ) ) ) ).
% prod_int_eq
thf(fact_4400_flip__bit__nat__def,axiom,
( ( bit_se8732182000553998342ip_bit @ nat )
= ( ^ [M2: nat,N2: nat] : ( bit_se5824344971392196577ns_xor @ nat @ N2 @ ( bit_se4730199178511100633sh_bit @ nat @ M2 @ ( one_one @ nat ) ) ) ) ) ).
% flip_bit_nat_def
thf(fact_4401_real__root__less__mono,axiom,
! [N: nat,X: real,Y2: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ X @ Y2 )
=> ( ord_less @ real @ ( root @ N @ X ) @ ( root @ N @ Y2 ) ) ) ) ).
% real_root_less_mono
thf(fact_4402_real__root__le__mono,axiom,
! [N: nat,X: real,Y2: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ real @ X @ Y2 )
=> ( ord_less_eq @ real @ ( root @ N @ X ) @ ( root @ N @ Y2 ) ) ) ) ).
% real_root_le_mono
thf(fact_4403_real__root__power,axiom,
! [N: nat,X: real,K2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( root @ N @ ( power_power @ real @ X @ K2 ) )
= ( power_power @ real @ ( root @ N @ X ) @ K2 ) ) ) ).
% real_root_power
thf(fact_4404_real__root__abs,axiom,
! [N: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( root @ N @ ( abs_abs @ real @ X ) )
= ( abs_abs @ real @ ( root @ N @ X ) ) ) ) ).
% real_root_abs
thf(fact_4405_sgn__root,axiom,
! [N: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( sgn_sgn @ real @ ( root @ N @ X ) )
= ( sgn_sgn @ real @ X ) ) ) ).
% sgn_root
thf(fact_4406_bit__push__bit__iff__int,axiom,
! [M: nat,K2: int,N: nat] :
( ( bit_se5641148757651400278ts_bit @ int @ ( bit_se4730199178511100633sh_bit @ int @ M @ K2 ) @ N )
= ( ( ord_less_eq @ nat @ M @ N )
& ( bit_se5641148757651400278ts_bit @ int @ K2 @ ( minus_minus @ nat @ N @ M ) ) ) ) ).
% bit_push_bit_iff_int
thf(fact_4407_prod__int__plus__eq,axiom,
! [I2: nat,J3: nat] :
( ( groups7121269368397514597t_prod @ nat @ int @ ( semiring_1_of_nat @ int ) @ ( set_or1337092689740270186AtMost @ nat @ I2 @ ( plus_plus @ nat @ I2 @ J3 ) ) )
= ( groups7121269368397514597t_prod @ int @ int
@ ^ [X4: int] : X4
@ ( set_or1337092689740270186AtMost @ int @ ( semiring_1_of_nat @ int @ I2 ) @ ( semiring_1_of_nat @ int @ ( plus_plus @ nat @ I2 @ J3 ) ) ) ) ) ).
% prod_int_plus_eq
thf(fact_4408_bit__push__bit__iff__nat,axiom,
! [M: nat,Q: nat,N: nat] :
( ( bit_se5641148757651400278ts_bit @ nat @ ( bit_se4730199178511100633sh_bit @ nat @ M @ Q ) @ N )
= ( ( ord_less_eq @ nat @ M @ N )
& ( bit_se5641148757651400278ts_bit @ nat @ Q @ ( minus_minus @ nat @ N @ M ) ) ) ) ).
% bit_push_bit_iff_nat
thf(fact_4409_concat__bit__eq,axiom,
( bit_concat_bit
= ( ^ [N2: nat,K3: int,L3: int] : ( plus_plus @ int @ ( bit_se2584673776208193580ke_bit @ int @ N2 @ K3 ) @ ( bit_se4730199178511100633sh_bit @ int @ N2 @ L3 ) ) ) ) ).
% concat_bit_eq
thf(fact_4410_flip__bit__eq__xor,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se8732182000553998342ip_bit @ A )
= ( ^ [N2: nat,A5: A] : ( bit_se5824344971392196577ns_xor @ A @ A5 @ ( bit_se4730199178511100633sh_bit @ A @ N2 @ ( one_one @ A ) ) ) ) ) ) ).
% flip_bit_eq_xor
thf(fact_4411_flip__bit__int__def,axiom,
( ( bit_se8732182000553998342ip_bit @ int )
= ( ^ [N2: nat,K3: int] : ( bit_se5824344971392196577ns_xor @ int @ K3 @ ( bit_se4730199178511100633sh_bit @ int @ N2 @ ( one_one @ int ) ) ) ) ) ).
% flip_bit_int_def
thf(fact_4412_real__root__gt__zero,axiom,
! [N: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( root @ N @ X ) ) ) ) ).
% real_root_gt_zero
thf(fact_4413_real__root__strict__decreasing,axiom,
! [N: nat,N4: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ nat @ N @ N4 )
=> ( ( ord_less @ real @ ( one_one @ real ) @ X )
=> ( ord_less @ real @ ( root @ N4 @ X ) @ ( root @ N @ X ) ) ) ) ) ).
% real_root_strict_decreasing
thf(fact_4414_sqrt__def,axiom,
( sqrt
= ( root @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% sqrt_def
thf(fact_4415_root__abs__power,axiom,
! [N: nat,Y2: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( abs_abs @ real @ ( root @ N @ ( power_power @ real @ Y2 @ N ) ) )
= ( abs_abs @ real @ Y2 ) ) ) ).
% root_abs_power
thf(fact_4416_push__bit__double,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A2: A] :
( ( bit_se4730199178511100633sh_bit @ A @ N @ ( times_times @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
= ( times_times @ A @ ( bit_se4730199178511100633sh_bit @ A @ N @ A2 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% push_bit_double
thf(fact_4417_bit__iff__and__push__bit__not__eq__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se5641148757651400278ts_bit @ A )
= ( ^ [A5: A,N2: nat] :
( ( bit_se5824344872417868541ns_and @ A @ A5 @ ( bit_se4730199178511100633sh_bit @ A @ N2 @ ( one_one @ A ) ) )
!= ( zero_zero @ A ) ) ) ) ) ).
% bit_iff_and_push_bit_not_eq_0
thf(fact_4418_push__bit__mask__eq,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [M: nat,N: nat] :
( ( bit_se4730199178511100633sh_bit @ A @ M @ ( bit_se2239418461657761734s_mask @ A @ N ) )
= ( bit_se5824344872417868541ns_and @ A @ ( bit_se2239418461657761734s_mask @ A @ ( plus_plus @ nat @ N @ M ) ) @ ( bit_ri4277139882892585799ns_not @ A @ ( bit_se2239418461657761734s_mask @ A @ M ) ) ) ) ) ).
% push_bit_mask_eq
thf(fact_4419_unset__bit__eq__and__not,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( ( bit_se2638667681897837118et_bit @ A )
= ( ^ [N2: nat,A5: A] : ( bit_se5824344872417868541ns_and @ A @ A5 @ ( bit_ri4277139882892585799ns_not @ A @ ( bit_se4730199178511100633sh_bit @ A @ N2 @ ( one_one @ A ) ) ) ) ) ) ) ).
% unset_bit_eq_and_not
thf(fact_4420_unset__bit__int__def,axiom,
( ( bit_se2638667681897837118et_bit @ int )
= ( ^ [N2: nat,K3: int] : ( bit_se5824344872417868541ns_and @ int @ K3 @ ( bit_ri4277139882892585799ns_not @ int @ ( bit_se4730199178511100633sh_bit @ int @ N2 @ ( one_one @ int ) ) ) ) ) ) ).
% unset_bit_int_def
thf(fact_4421_real__root__pos__pos,axiom,
! [N: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( root @ N @ X ) ) ) ) ).
% real_root_pos_pos
thf(fact_4422_real__root__strict__increasing,axiom,
! [N: nat,N4: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ nat @ N @ N4 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ X @ ( one_one @ real ) )
=> ( ord_less @ real @ ( root @ N @ X ) @ ( root @ N4 @ X ) ) ) ) ) ) ).
% real_root_strict_increasing
thf(fact_4423_real__root__decreasing,axiom,
! [N: nat,N4: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ nat @ N @ N4 )
=> ( ( ord_less_eq @ real @ ( one_one @ real ) @ X )
=> ( ord_less_eq @ real @ ( root @ N4 @ X ) @ ( root @ N @ X ) ) ) ) ) ).
% real_root_decreasing
thf(fact_4424_real__root__pow__pos,axiom,
! [N: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( power_power @ real @ ( root @ N @ X ) @ N )
= X ) ) ) ).
% real_root_pow_pos
thf(fact_4425_real__root__power__cancel,axiom,
! [N: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( root @ N @ ( power_power @ real @ X @ N ) )
= X ) ) ) ).
% real_root_power_cancel
thf(fact_4426_real__root__pos__unique,axiom,
! [N: nat,Y2: real,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y2 )
=> ( ( ( power_power @ real @ Y2 @ N )
= X )
=> ( ( root @ N @ X )
= Y2 ) ) ) ) ).
% real_root_pos_unique
thf(fact_4427_odd__real__root__power__cancel,axiom,
! [N: nat,X: real] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( root @ N @ ( power_power @ real @ X @ N ) )
= X ) ) ).
% odd_real_root_power_cancel
thf(fact_4428_odd__real__root__unique,axiom,
! [N: nat,Y2: real,X: real] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( ( power_power @ real @ Y2 @ N )
= X )
=> ( ( root @ N @ X )
= Y2 ) ) ) ).
% odd_real_root_unique
thf(fact_4429_odd__real__root__pow,axiom,
! [N: nat,X: real] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( power_power @ real @ ( root @ N @ X ) @ N )
= X ) ) ).
% odd_real_root_pow
thf(fact_4430_push__bit__int__def,axiom,
( ( bit_se4730199178511100633sh_bit @ int )
= ( ^ [N2: nat,K3: int] : ( times_times @ int @ K3 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ).
% push_bit_int_def
thf(fact_4431_push__bit__nat__def,axiom,
( ( bit_se4730199178511100633sh_bit @ nat )
= ( ^ [N2: nat,M2: nat] : ( times_times @ nat @ M2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ).
% push_bit_nat_def
thf(fact_4432_push__bit__eq__mult,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se4730199178511100633sh_bit @ A )
= ( ^ [N2: nat,A5: A] : ( times_times @ A @ A5 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ) ).
% push_bit_eq_mult
thf(fact_4433_exp__dvdE,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A2: A] :
( ( dvd_dvd @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) @ A2 )
=> ~ ! [B3: A] :
( A2
!= ( bit_se4730199178511100633sh_bit @ A @ N @ B3 ) ) ) ) ).
% exp_dvdE
thf(fact_4434_real__root__increasing,axiom,
! [N: nat,N4: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ nat @ N @ N4 )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ X @ ( one_one @ real ) )
=> ( ord_less_eq @ real @ ( root @ N @ X ) @ ( root @ N4 @ X ) ) ) ) ) ) ).
% real_root_increasing
thf(fact_4435_sgn__power__root,axiom,
! [N: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( times_times @ real @ ( sgn_sgn @ real @ ( root @ N @ X ) ) @ ( power_power @ real @ ( abs_abs @ real @ ( root @ N @ X ) ) @ N ) )
= X ) ) ).
% sgn_power_root
thf(fact_4436_root__sgn__power,axiom,
! [N: nat,Y2: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( root @ N @ ( times_times @ real @ ( sgn_sgn @ real @ Y2 ) @ ( power_power @ real @ ( abs_abs @ real @ Y2 ) @ N ) ) )
= Y2 ) ) ).
% root_sgn_power
thf(fact_4437_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_4438_ln__root,axiom,
! [N: nat,B2: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ B2 )
=> ( ( ln_ln @ real @ ( root @ N @ B2 ) )
= ( divide_divide @ real @ ( ln_ln @ real @ B2 ) @ ( semiring_1_of_nat @ real @ N ) ) ) ) ) ).
% ln_root
thf(fact_4439_log__root,axiom,
! [N: nat,A2: real,B2: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
=> ( ( log @ B2 @ ( root @ N @ A2 ) )
= ( divide_divide @ real @ ( log @ B2 @ A2 ) @ ( semiring_1_of_nat @ real @ N ) ) ) ) ) ).
% log_root
thf(fact_4440_log__base__root,axiom,
! [N: nat,B2: real,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ B2 )
=> ( ( log @ ( root @ N @ B2 ) @ X )
= ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ ( log @ B2 @ X ) ) ) ) ) ).
% log_base_root
thf(fact_4441_split__root,axiom,
! [P3: real > $o,N: nat,X: real] :
( ( P3 @ ( root @ N @ X ) )
= ( ( ( N
= ( zero_zero @ nat ) )
=> ( P3 @ ( zero_zero @ real ) ) )
& ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ! [Y: real] :
( ( ( times_times @ real @ ( sgn_sgn @ real @ Y ) @ ( power_power @ real @ ( abs_abs @ real @ Y ) @ N ) )
= X )
=> ( P3 @ Y ) ) ) ) ) ).
% split_root
thf(fact_4442_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
@ ^ [I: nat,J4: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ I @ J4 ) @ N ) ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [K3: nat] :
( groups7121269368397514597t_prod @ nat @ A
@ ^ [I: nat] : ( G @ I @ ( minus_minus @ nat @ K3 @ I ) )
@ ( set_ord_atMost @ nat @ K3 ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% prod.triangle_reindex_eq
thf(fact_4443_take__bit__horner__sum__bit__eq,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [N: nat,Bs: list @ $o] :
( ( bit_se2584673776208193580ke_bit @ A @ N @ ( groups4207007520872428315er_sum @ $o @ A @ ( zero_neq_one_of_bool @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Bs ) )
= ( groups4207007520872428315er_sum @ $o @ A @ ( zero_neq_one_of_bool @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( take @ $o @ N @ Bs ) ) ) ) ).
% take_bit_horner_sum_bit_eq
thf(fact_4444_root__powr__inverse,axiom,
! [N: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( root @ N @ X )
= ( powr @ real @ X @ ( divide_divide @ real @ ( one_one @ real ) @ ( semiring_1_of_nat @ real @ N ) ) ) ) ) ) ).
% root_powr_inverse
thf(fact_4445_signed__take__bit__code,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( ( bit_ri4674362597316999326ke_bit @ A )
= ( ^ [N2: nat,A5: A] : ( if @ A @ ( bit_se5641148757651400278ts_bit @ A @ ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N2 ) @ A5 ) @ N2 ) @ ( plus_plus @ A @ ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N2 ) @ A5 ) @ ( bit_se4730199178511100633sh_bit @ A @ ( suc @ N2 ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) @ ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N2 ) @ A5 ) ) ) ) ) ).
% signed_take_bit_code
thf(fact_4446_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
@ ^ [I: nat,J4: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ I @ J4 ) @ N ) ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [K3: nat] :
( groups7121269368397514597t_prod @ nat @ A
@ ^ [I: nat] : ( G @ I @ ( minus_minus @ nat @ K3 @ I ) )
@ ( set_ord_atMost @ nat @ K3 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% prod.triangle_reindex
thf(fact_4447_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_4448_VEBT_Osize_I3_J,axiom,
! [X11: option @ ( product_prod @ nat @ nat ),X12: nat,X13: list @ vEBT_VEBT,X14: vEBT_VEBT] :
( ( size_size @ vEBT_VEBT @ ( vEBT_Node @ X11 @ X12 @ X13 @ X14 ) )
= ( plus_plus @ nat @ ( plus_plus @ nat @ ( size_list @ vEBT_VEBT @ ( size_size @ vEBT_VEBT ) @ X13 ) @ ( size_size @ vEBT_VEBT @ X14 ) ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% VEBT.size(3)
thf(fact_4449_horner__sum__bit__eq__take__bit,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A,N: nat] :
( ( groups4207007520872428315er_sum @ $o @ A @ ( zero_neq_one_of_bool @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( map @ nat @ $o @ ( bit_se5641148757651400278ts_bit @ A @ A2 ) @ ( upt @ ( zero_zero @ nat ) @ N ) ) )
= ( bit_se2584673776208193580ke_bit @ A @ N @ A2 ) ) ) ).
% horner_sum_bit_eq_take_bit
thf(fact_4450_Sum__Ico__nat,axiom,
! [M: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [X4: nat] : X4
@ ( 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_4451_sum__power2,axiom,
! [K2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K2 ) )
= ( minus_minus @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K2 ) @ ( one_one @ nat ) ) ) ).
% sum_power2
thf(fact_4452_length__upt,axiom,
! [I2: nat,J3: nat] :
( ( size_size @ ( list @ nat ) @ ( upt @ I2 @ J3 ) )
= ( minus_minus @ nat @ J3 @ I2 ) ) ).
% length_upt
thf(fact_4453_atLeastLessThan__iff,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [I2: A,L: A,U: A] :
( ( member @ A @ I2 @ ( set_or7035219750837199246ssThan @ A @ L @ U ) )
= ( ( ord_less_eq @ A @ L @ I2 )
& ( ord_less @ A @ I2 @ U ) ) ) ) ).
% atLeastLessThan_iff
thf(fact_4454_ivl__subset,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [I2: A,J3: A,M: A,N: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ I2 @ J3 ) @ ( set_or7035219750837199246ssThan @ A @ M @ N ) )
= ( ( ord_less_eq @ A @ J3 @ I2 )
| ( ( ord_less_eq @ A @ M @ I2 )
& ( ord_less_eq @ A @ J3 @ N ) ) ) ) ) ).
% ivl_subset
thf(fact_4455_ivl__diff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [I2: A,N: A,M: A] :
( ( ord_less_eq @ A @ I2 @ N )
=> ( ( minus_minus @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ I2 @ M ) @ ( set_or7035219750837199246ssThan @ A @ I2 @ N ) )
= ( set_or7035219750837199246ssThan @ A @ N @ M ) ) ) ) ).
% ivl_diff
thf(fact_4456_nth__upt,axiom,
! [I2: nat,K2: nat,J3: nat] :
( ( ord_less @ nat @ ( plus_plus @ nat @ I2 @ K2 ) @ J3 )
=> ( ( nth @ nat @ ( upt @ I2 @ J3 ) @ K2 )
= ( plus_plus @ nat @ I2 @ K2 ) ) ) ).
% nth_upt
thf(fact_4457_take__upt,axiom,
! [I2: nat,M: nat,N: nat] :
( ( ord_less_eq @ nat @ ( plus_plus @ nat @ I2 @ M ) @ N )
=> ( ( take @ nat @ M @ ( upt @ I2 @ N ) )
= ( upt @ I2 @ ( plus_plus @ nat @ I2 @ M ) ) ) ) ).
% take_upt
thf(fact_4458_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_4459_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_4460_atLeastLessThan__inj_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A,C2: A,D2: A] :
( ( ( set_or7035219750837199246ssThan @ A @ A2 @ B2 )
= ( set_or7035219750837199246ssThan @ A @ C2 @ D2 ) )
=> ( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ C2 @ D2 )
=> ( B2 = D2 ) ) ) ) ) ).
% atLeastLessThan_inj(2)
thf(fact_4461_atLeastLessThan__inj_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A,C2: A,D2: A] :
( ( ( set_or7035219750837199246ssThan @ A @ A2 @ B2 )
= ( set_or7035219750837199246ssThan @ A @ C2 @ D2 ) )
=> ( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ C2 @ D2 )
=> ( A2 = C2 ) ) ) ) ) ).
% atLeastLessThan_inj(1)
thf(fact_4462_atLeastLessThan__eq__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A,C2: A,D2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ C2 @ D2 )
=> ( ( ( set_or7035219750837199246ssThan @ A @ A2 @ B2 )
= ( set_or7035219750837199246ssThan @ A @ C2 @ D2 ) )
= ( ( A2 = C2 )
& ( B2 = D2 ) ) ) ) ) ) ).
% atLeastLessThan_eq_iff
thf(fact_4463_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_4464_atLeastLessThan__upt,axiom,
( ( set_or7035219750837199246ssThan @ nat )
= ( ^ [I: nat,J4: nat] : ( set2 @ nat @ ( upt @ I @ J4 ) ) ) ) ).
% atLeastLessThan_upt
thf(fact_4465_atLeastLessThan__subset__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A,C2: A,D2: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ A2 @ B2 ) @ ( set_or7035219750837199246ssThan @ A @ C2 @ D2 ) )
=> ( ( ord_less_eq @ A @ B2 @ A2 )
| ( ( ord_less_eq @ A @ C2 @ A2 )
& ( ord_less_eq @ A @ B2 @ D2 ) ) ) ) ) ).
% atLeastLessThan_subset_iff
thf(fact_4466_all__nat__less__eq,axiom,
! [N: nat,P3: nat > $o] :
( ( ! [M2: nat] :
( ( ord_less @ nat @ M2 @ N )
=> ( P3 @ M2 ) ) )
= ( ! [X4: nat] :
( ( member @ nat @ X4 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) )
=> ( P3 @ X4 ) ) ) ) ).
% all_nat_less_eq
thf(fact_4467_ex__nat__less__eq,axiom,
! [N: nat,P3: nat > $o] :
( ( ? [M2: nat] :
( ( ord_less @ nat @ M2 @ N )
& ( P3 @ M2 ) ) )
= ( ? [X4: nat] :
( ( member @ nat @ X4 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) )
& ( P3 @ X4 ) ) ) ) ).
% ex_nat_less_eq
thf(fact_4468_atLeastAtMost__upt,axiom,
( ( set_or1337092689740270186AtMost @ nat )
= ( ^ [N2: nat,M2: nat] : ( set2 @ nat @ ( upt @ N2 @ ( suc @ M2 ) ) ) ) ) ).
% atLeastAtMost_upt
thf(fact_4469_atLeast__upt,axiom,
( ( set_ord_lessThan @ nat )
= ( ^ [N2: nat] : ( set2 @ nat @ ( upt @ ( zero_zero @ nat ) @ N2 ) ) ) ) ).
% atLeast_upt
thf(fact_4470_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
@ ^ [I: nat] : ( G @ ( suc @ I ) )
@ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) ) ) ).
% sum.shift_bounds_Suc_ivl
thf(fact_4471_map__add__upt,axiom,
! [N: nat,M: nat] :
( ( map @ nat @ nat
@ ^ [I: nat] : ( plus_plus @ nat @ I @ N )
@ ( upt @ ( zero_zero @ nat ) @ M ) )
= ( upt @ N @ ( plus_plus @ nat @ M @ N ) ) ) ).
% map_add_upt
thf(fact_4472_sum_Oshift__bounds__nat__ivl,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,M: nat,K2: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( plus_plus @ nat @ M @ K2 ) @ ( plus_plus @ nat @ N @ K2 ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I: nat] : ( G @ ( plus_plus @ nat @ I @ K2 ) )
@ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) ) ) ).
% sum.shift_bounds_nat_ivl
thf(fact_4473_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
@ ^ [I: nat] : ( G @ ( suc @ I ) )
@ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) ) ) ).
% prod.shift_bounds_Suc_ivl
thf(fact_4474_prod_Oshift__bounds__nat__ivl,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,M: nat,K2: nat,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( plus_plus @ nat @ M @ K2 ) @ ( plus_plus @ nat @ N @ K2 ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I: nat] : ( G @ ( plus_plus @ nat @ I @ K2 ) )
@ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) ) ) ).
% prod.shift_bounds_nat_ivl
thf(fact_4475_map__replicate__trivial,axiom,
! [A: $tType,X: A,I2: nat] :
( ( map @ nat @ A
@ ^ [I: nat] : X
@ ( upt @ ( zero_zero @ nat ) @ I2 ) )
= ( replicate @ A @ I2 @ X ) ) ).
% map_replicate_trivial
thf(fact_4476_sum_Oivl__cong,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( comm_monoid_add @ A ) )
=> ! [A2: B,C2: B,B2: B,D2: B,G: B > A,H2: B > A] :
( ( A2 = C2 )
=> ( ( B2 = D2 )
=> ( ! [X5: B] :
( ( ord_less_eq @ B @ C2 @ X5 )
=> ( ( ord_less @ B @ X5 @ D2 )
=> ( ( G @ X5 )
= ( H2 @ X5 ) ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( set_or7035219750837199246ssThan @ B @ A2 @ B2 ) )
= ( groups7311177749621191930dd_sum @ B @ A @ H2 @ ( set_or7035219750837199246ssThan @ B @ C2 @ D2 ) ) ) ) ) ) ) ).
% sum.ivl_cong
thf(fact_4477_prod_Oivl__cong,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( comm_monoid_mult @ A ) )
=> ! [A2: B,C2: B,B2: B,D2: B,G: B > A,H2: B > A] :
( ( A2 = C2 )
=> ( ( B2 = D2 )
=> ( ! [X5: B] :
( ( ord_less_eq @ B @ C2 @ X5 )
=> ( ( ord_less @ B @ X5 @ D2 )
=> ( ( G @ X5 )
= ( H2 @ X5 ) ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( set_or7035219750837199246ssThan @ B @ A2 @ B2 ) )
= ( groups7121269368397514597t_prod @ B @ A @ H2 @ ( set_or7035219750837199246ssThan @ B @ C2 @ D2 ) ) ) ) ) ) ) ).
% prod.ivl_cong
thf(fact_4478_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_4479_sum__diff__nat__ivl,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [M: nat,N: nat,P5: nat,F3: nat > A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( ord_less_eq @ nat @ N @ P5 )
=> ( ( minus_minus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_or7035219750837199246ssThan @ nat @ M @ P5 ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_or7035219750837199246ssThan @ nat @ N @ P5 ) ) ) ) ) ) ).
% sum_diff_nat_ivl
thf(fact_4480_size__list__estimation,axiom,
! [A: $tType,X: A,Xs: list @ A,Y2: nat,F3: A > nat] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ( ord_less @ nat @ Y2 @ ( F3 @ X ) )
=> ( ord_less @ nat @ Y2 @ ( size_list @ A @ F3 @ Xs ) ) ) ) ).
% size_list_estimation
thf(fact_4481_size__list__pointwise,axiom,
! [A: $tType,Xs: list @ A,F3: A > nat,G: A > nat] :
( ! [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Xs ) )
=> ( ord_less_eq @ nat @ ( F3 @ X5 ) @ ( G @ X5 ) ) )
=> ( ord_less_eq @ nat @ ( size_list @ A @ F3 @ Xs ) @ ( size_list @ A @ G @ Xs ) ) ) ).
% size_list_pointwise
thf(fact_4482_size__list__estimation_H,axiom,
! [A: $tType,X: A,Xs: list @ A,Y2: nat,F3: A > nat] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ( ord_less_eq @ nat @ Y2 @ ( F3 @ X ) )
=> ( ord_less_eq @ nat @ Y2 @ ( size_list @ A @ F3 @ Xs ) ) ) ) ).
% size_list_estimation'
thf(fact_4483_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_4484_atMost__upto,axiom,
( ( set_ord_atMost @ nat )
= ( ^ [N2: nat] : ( set2 @ nat @ ( upt @ ( zero_zero @ nat ) @ ( suc @ N2 ) ) ) ) ) ).
% atMost_upto
thf(fact_4485_map__decr__upt,axiom,
! [M: nat,N: nat] :
( ( map @ nat @ nat
@ ^ [N2: nat] : ( minus_minus @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) )
@ ( upt @ ( suc @ M ) @ ( suc @ N ) ) )
= ( upt @ M @ N ) ) ).
% map_decr_upt
thf(fact_4486_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_4487_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A2: A,B2: A,C2: A,D2: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) @ ( set_or7035219750837199246ssThan @ A @ C2 @ D2 ) )
= ( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ A2 )
& ( ord_less @ A @ B2 @ D2 ) ) ) ) ) ).
% atLeastAtMost_subseteq_atLeastLessThan_iff
thf(fact_4488_atLeastLessThan__subseteq__atLeastAtMost__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A2: A,B2: A,C2: A,D2: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ A2 @ B2 ) @ ( set_or1337092689740270186AtMost @ A @ C2 @ D2 ) )
= ( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ A2 )
& ( ord_less_eq @ A @ B2 @ D2 ) ) ) ) ) ).
% atLeastLessThan_subseteq_atLeastAtMost_iff
thf(fact_4489_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_4490_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_4491_sum_OatLeastLessThan__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A2: nat,B2: nat,G: nat > A] :
( ( ord_less_eq @ nat @ A2 @ B2 )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ A2 @ ( suc @ B2 ) ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ A2 @ B2 ) ) @ ( G @ B2 ) ) ) ) ) ).
% sum.atLeastLessThan_Suc
thf(fact_4492_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_4493_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_4494_prod_OatLeastLessThan__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A2: nat,B2: nat,G: nat > A] :
( ( ord_less_eq @ nat @ A2 @ B2 )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ A2 @ ( suc @ B2 ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ A2 @ B2 ) ) @ ( G @ B2 ) ) ) ) ) ).
% prod.atLeastLessThan_Suc
thf(fact_4495_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_4496_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_4497_nth__map__upt,axiom,
! [A: $tType,I2: nat,N: nat,M: nat,F3: nat > A] :
( ( ord_less @ nat @ I2 @ ( minus_minus @ nat @ N @ M ) )
=> ( ( nth @ A @ ( map @ nat @ A @ F3 @ ( upt @ M @ N ) ) @ I2 )
= ( F3 @ ( plus_plus @ nat @ M @ I2 ) ) ) ) ).
% nth_map_upt
thf(fact_4498_sum__Suc__diff_H,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [M: nat,N: nat,F3: nat > A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I: nat] : ( minus_minus @ A @ ( F3 @ ( suc @ I ) ) @ ( F3 @ I ) )
@ ( set_or7035219750837199246ssThan @ nat @ M @ N ) )
= ( minus_minus @ A @ ( F3 @ N ) @ ( F3 @ M ) ) ) ) ) ).
% sum_Suc_diff'
thf(fact_4499_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
@ ^ [I: nat] : ( G @ ( minus_minus @ nat @ ( plus_plus @ nat @ M @ N ) @ ( suc @ I ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ N @ M ) ) ) ) ).
% sum.atLeastLessThan_rev
thf(fact_4500_sum_Onested__swap,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A2: nat > nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I: nat] : ( groups7311177749621191930dd_sum @ nat @ A @ ( A2 @ I ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ I ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [J4: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I: nat] : ( A2 @ I @ J4 )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ J4 ) @ N ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% sum.nested_swap
thf(fact_4501_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
@ ^ [I: nat] : ( G @ ( minus_minus @ nat @ ( plus_plus @ nat @ M @ N ) @ ( suc @ I ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ N @ M ) ) ) ) ).
% prod.atLeastLessThan_rev
thf(fact_4502_prod_Onested__swap,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A2: nat > nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I: nat] : ( groups7121269368397514597t_prod @ nat @ A @ ( A2 @ I ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ I ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [J4: nat] :
( groups7121269368397514597t_prod @ nat @ A
@ ^ [I: nat] : ( A2 @ I @ J4 )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ J4 ) @ N ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% prod.nested_swap
thf(fact_4503_sum_Onat__group,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,K2: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [M2: nat] : ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( times_times @ nat @ M2 @ K2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ M2 @ K2 ) @ K2 ) ) )
@ ( set_ord_lessThan @ nat @ N ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_lessThan @ nat @ ( times_times @ nat @ N @ K2 ) ) ) ) ) ).
% sum.nat_group
thf(fact_4504_prod_Onat__group,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,K2: nat,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A
@ ^ [M2: nat] : ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( times_times @ nat @ M2 @ K2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ M2 @ K2 ) @ K2 ) ) )
@ ( set_ord_lessThan @ nat @ N ) )
= ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_lessThan @ nat @ ( times_times @ nat @ N @ K2 ) ) ) ) ) ).
% prod.nat_group
thf(fact_4505_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_4506_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_4507_map__upt__eqI,axiom,
! [A: $tType,Xs: list @ A,N: nat,M: nat,F3: nat > A] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( minus_minus @ nat @ N @ M ) )
=> ( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ A @ Xs @ I3 )
= ( F3 @ ( plus_plus @ nat @ M @ I3 ) ) ) )
=> ( ( map @ nat @ A @ F3 @ ( upt @ M @ N ) )
= Xs ) ) ) ).
% map_upt_eqI
thf(fact_4508_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
@ ^ [I: nat] : ( G @ ( minus_minus @ nat @ ( plus_plus @ nat @ M @ N ) @ I ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ N ) @ M ) ) ) ) ).
% sum.atLeastLessThan_rev_at_least_Suc_atMost
thf(fact_4509_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
@ ^ [I: nat] : ( G @ ( minus_minus @ nat @ ( plus_plus @ nat @ M @ N ) @ I ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ N ) @ M ) ) ) ) ).
% prod.atLeastLessThan_rev_at_least_Suc_atMost
thf(fact_4510_pochhammer__prod,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ( ( comm_s3205402744901411588hammer @ A )
= ( ^ [A5: A,N2: nat] :
( groups7121269368397514597t_prod @ nat @ A
@ ^ [I: nat] : ( plus_plus @ A @ A5 @ ( semiring_1_of_nat @ A @ I ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ) ) ).
% pochhammer_prod
thf(fact_4511_fact__prod__rev,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ( ( semiring_char_0_fact @ A )
= ( ^ [N2: nat] : ( semiring_1_of_nat @ A @ ( groups7121269368397514597t_prod @ nat @ nat @ ( minus_minus @ nat @ N2 ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ) ) ) ).
% fact_prod_rev
thf(fact_4512_summable__Cauchy,axiom,
! [A: $tType] :
( ( real_Vector_banach @ A )
=> ( ( summable @ A )
= ( ^ [F5: nat > A] :
! [E3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E3 )
=> ? [N6: nat] :
! [M2: nat] :
( ( ord_less_eq @ nat @ N6 @ M2 )
=> ! [N2: nat] : ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F5 @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N2 ) ) ) @ E3 ) ) ) ) ) ) ).
% summable_Cauchy
thf(fact_4513_sums__group,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [F3: nat > A,S: A,K2: nat] :
( ( sums @ A @ F3 @ S )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K2 )
=> ( sums @ A
@ ^ [N2: nat] : ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_or7035219750837199246ssThan @ nat @ ( times_times @ nat @ N2 @ K2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ N2 @ K2 ) @ K2 ) ) )
@ S ) ) ) ) ).
% sums_group
thf(fact_4514_take__bit__sum,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ( ( bit_se2584673776208193580ke_bit @ A )
= ( ^ [N2: 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 ) @ N2 ) ) ) ) ) ).
% take_bit_sum
thf(fact_4515_fact__split,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [K2: nat,N: nat] :
( ( ord_less_eq @ nat @ K2 @ 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 @ K2 ) @ N ) ) ) @ ( semiring_char_0_fact @ A @ ( minus_minus @ nat @ N @ K2 ) ) ) ) ) ) ).
% fact_split
thf(fact_4516_binomial__altdef__of__nat,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K2: nat,N: nat] :
( ( ord_less_eq @ nat @ K2 @ N )
=> ( ( semiring_1_of_nat @ A @ ( binomial @ N @ K2 ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I: nat] : ( divide_divide @ A @ ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ N @ I ) ) @ ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ K2 @ I ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K2 ) ) ) ) ) ).
% binomial_altdef_of_nat
thf(fact_4517_gbinomial__altdef__of__nat,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( ( gbinomial @ A )
= ( ^ [A5: A,K3: nat] :
( groups7121269368397514597t_prod @ nat @ A
@ ^ [I: nat] : ( divide_divide @ A @ ( minus_minus @ A @ A5 @ ( semiring_1_of_nat @ A @ I ) ) @ ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ K3 @ I ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K3 ) ) ) ) ) ).
% gbinomial_altdef_of_nat
thf(fact_4518_gbinomial__mult__fact,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K2: nat,A2: A] :
( ( times_times @ A @ ( semiring_char_0_fact @ A @ K2 ) @ ( gbinomial @ A @ A2 @ K2 ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I: nat] : ( minus_minus @ A @ A2 @ ( semiring_1_of_nat @ A @ I ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K2 ) ) ) ) ).
% gbinomial_mult_fact
thf(fact_4519_gbinomial__mult__fact_H,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A2: A,K2: nat] :
( ( times_times @ A @ ( gbinomial @ A @ A2 @ K2 ) @ ( semiring_char_0_fact @ A @ K2 ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I: nat] : ( minus_minus @ A @ A2 @ ( semiring_1_of_nat @ A @ I ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K2 ) ) ) ) ).
% gbinomial_mult_fact'
thf(fact_4520_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
@ ^ [I: nat] : ( minus_minus @ A @ A5 @ ( semiring_1_of_nat @ A @ I ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K3 ) )
@ ( semiring_char_0_fact @ A @ K3 ) ) ) ) ) ).
% gbinomial_prod_rev
thf(fact_4521_horner__sum__eq__sum,axiom,
! [A: $tType,B: $tType] :
( ( comm_semiring_1 @ A )
=> ( ( groups4207007520872428315er_sum @ B @ A )
= ( ^ [F5: B > A,A5: A,Xs2: list @ B] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [N2: nat] : ( times_times @ A @ ( F5 @ ( nth @ B @ Xs2 @ N2 ) ) @ ( power_power @ A @ A5 @ N2 ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ B ) @ Xs2 ) ) ) ) ) ) ).
% horner_sum_eq_sum
thf(fact_4522_Chebyshev__sum__upper,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: nat,A2: nat > A,B2: nat > A] :
( ! [I3: nat,J2: nat] :
( ( ord_less_eq @ nat @ I3 @ J2 )
=> ( ( ord_less @ nat @ J2 @ N )
=> ( ord_less_eq @ A @ ( A2 @ I3 ) @ ( A2 @ J2 ) ) ) )
=> ( ! [I3: nat,J2: nat] :
( ( ord_less_eq @ nat @ I3 @ J2 )
=> ( ( ord_less @ nat @ J2 @ N )
=> ( ord_less_eq @ A @ ( B2 @ J2 ) @ ( B2 @ I3 ) ) ) )
=> ( ord_less_eq @ A
@ ( times_times @ A @ ( semiring_1_of_nat @ A @ N )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( times_times @ A @ ( A2 @ K3 ) @ ( B2 @ K3 ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) )
@ ( times_times @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ A2 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ B2 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ) ) ) ).
% Chebyshev_sum_upper
thf(fact_4523_Chebyshev__sum__upper__nat,axiom,
! [N: nat,A2: nat > nat,B2: nat > nat] :
( ! [I3: nat,J2: nat] :
( ( ord_less_eq @ nat @ I3 @ J2 )
=> ( ( ord_less @ nat @ J2 @ N )
=> ( ord_less_eq @ nat @ ( A2 @ I3 ) @ ( A2 @ J2 ) ) ) )
=> ( ! [I3: nat,J2: nat] :
( ( ord_less_eq @ nat @ I3 @ J2 )
=> ( ( ord_less @ nat @ J2 @ N )
=> ( ord_less_eq @ nat @ ( B2 @ J2 ) @ ( B2 @ I3 ) ) ) )
=> ( ord_less_eq @ nat
@ ( times_times @ nat @ N
@ ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [I: nat] : ( times_times @ nat @ ( A2 @ I ) @ ( B2 @ I ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) )
@ ( times_times @ nat @ ( groups7311177749621191930dd_sum @ nat @ nat @ A2 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) @ ( groups7311177749621191930dd_sum @ nat @ nat @ B2 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ) ) ).
% Chebyshev_sum_upper_nat
thf(fact_4524_VEBT_Osize__gen_I1_J,axiom,
! [X11: option @ ( product_prod @ nat @ nat ),X12: nat,X13: list @ vEBT_VEBT,X14: vEBT_VEBT] :
( ( vEBT_size_VEBT @ ( vEBT_Node @ X11 @ X12 @ X13 @ X14 ) )
= ( plus_plus @ nat @ ( plus_plus @ nat @ ( size_list @ vEBT_VEBT @ vEBT_size_VEBT @ X13 ) @ ( vEBT_size_VEBT @ X14 ) ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% VEBT.size_gen(1)
thf(fact_4525_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_4526_VEBT_Osize__gen_I2_J,axiom,
! [X21: $o,X22: $o] :
( ( vEBT_size_VEBT @ ( vEBT_Leaf @ X21 @ X22 ) )
= ( zero_zero @ nat ) ) ).
% VEBT.size_gen(2)
thf(fact_4527_Cauchy__iff2,axiom,
( ( topolo3814608138187158403Cauchy @ real )
= ( ^ [X6: nat > real] :
! [J4: nat] :
? [M9: nat] :
! [M2: nat] :
( ( ord_less_eq @ nat @ M9 @ M2 )
=> ! [N2: nat] :
( ( ord_less_eq @ nat @ M9 @ N2 )
=> ( ord_less @ real @ ( abs_abs @ real @ ( minus_minus @ real @ ( X6 @ M2 ) @ ( X6 @ N2 ) ) ) @ ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ ( suc @ J4 ) ) ) ) ) ) ) ) ).
% Cauchy_iff2
thf(fact_4528_length__subseqs,axiom,
! [A: $tType,Xs: list @ A] :
( ( size_size @ ( list @ ( list @ A ) ) @ ( subseqs @ A @ Xs ) )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ).
% length_subseqs
thf(fact_4529_divmod__step__integer__def,axiom,
( ( unique1321980374590559556d_step @ code_integer )
= ( ^ [L3: num] :
( product_case_prod @ code_integer @ code_integer @ ( product_prod @ code_integer @ code_integer )
@ ^ [Q5: 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 ) ) @ Q5 ) @ ( 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 ) ) @ Q5 ) @ R5 ) ) ) ) ) ).
% divmod_step_integer_def
thf(fact_4530_csqrt_Osimps_I1_J,axiom,
! [Z3: complex] :
( ( re @ ( csqrt @ Z3 ) )
= ( sqrt @ ( divide_divide @ real @ ( plus_plus @ real @ ( real_V7770717601297561774m_norm @ complex @ Z3 ) @ ( re @ Z3 ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ).
% csqrt.simps(1)
thf(fact_4531_Re__sum,axiom,
! [A: $tType,F3: A > complex,S: set @ A] :
( ( re @ ( groups7311177749621191930dd_sum @ A @ complex @ F3 @ S ) )
= ( groups7311177749621191930dd_sum @ A @ real
@ ^ [X4: A] : ( re @ ( F3 @ X4 ) )
@ S ) ) ).
% Re_sum
thf(fact_4532_subseqs__refl,axiom,
! [A: $tType,Xs: list @ A] : ( member @ ( list @ A ) @ Xs @ ( set2 @ ( list @ A ) @ ( subseqs @ A @ Xs ) ) ) ).
% subseqs_refl
thf(fact_4533_times__integer__code_I1_J,axiom,
! [K2: code_integer] :
( ( times_times @ code_integer @ K2 @ ( zero_zero @ code_integer ) )
= ( zero_zero @ code_integer ) ) ).
% times_integer_code(1)
thf(fact_4534_times__integer__code_I2_J,axiom,
! [L: code_integer] :
( ( times_times @ code_integer @ ( zero_zero @ code_integer ) @ L )
= ( zero_zero @ code_integer ) ) ).
% times_integer_code(2)
thf(fact_4535_divmod__integer_H__def,axiom,
( ( unique8689654367752047608divmod @ code_integer )
= ( ^ [M2: num,N2: num] : ( product_Pair @ code_integer @ code_integer @ ( divide_divide @ code_integer @ ( numeral_numeral @ code_integer @ M2 ) @ ( numeral_numeral @ code_integer @ N2 ) ) @ ( modulo_modulo @ code_integer @ ( numeral_numeral @ code_integer @ M2 ) @ ( numeral_numeral @ code_integer @ N2 ) ) ) ) ) ).
% divmod_integer'_def
thf(fact_4536_less__eq__integer__code_I1_J,axiom,
ord_less_eq @ code_integer @ ( zero_zero @ code_integer ) @ ( zero_zero @ code_integer ) ).
% less_eq_integer_code(1)
thf(fact_4537_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_4538_sums__Re,axiom,
! [X9: nat > complex,A2: complex] :
( ( sums @ complex @ X9 @ A2 )
=> ( sums @ real
@ ^ [N2: nat] : ( re @ ( X9 @ N2 ) )
@ ( re @ A2 ) ) ) ).
% sums_Re
thf(fact_4539_complex__Re__le__cmod,axiom,
! [X: complex] : ( ord_less_eq @ real @ ( re @ X ) @ ( real_V7770717601297561774m_norm @ complex @ X ) ) ).
% complex_Re_le_cmod
thf(fact_4540_one__complex_Osimps_I1_J,axiom,
( ( re @ ( one_one @ complex ) )
= ( one_one @ real ) ) ).
% one_complex.simps(1)
thf(fact_4541_scaleR__complex_Osimps_I1_J,axiom,
! [R: real,X: complex] :
( ( re @ ( real_V8093663219630862766scaleR @ complex @ R @ X ) )
= ( times_times @ real @ R @ ( re @ X ) ) ) ).
% scaleR_complex.simps(1)
thf(fact_4542_summable__Re,axiom,
! [F3: nat > complex] :
( ( summable @ complex @ F3 )
=> ( summable @ real
@ ^ [X4: nat] : ( re @ ( F3 @ X4 ) ) ) ) ).
% summable_Re
thf(fact_4543_abs__Re__le__cmod,axiom,
! [X: complex] : ( ord_less_eq @ real @ ( abs_abs @ real @ ( re @ X ) ) @ ( real_V7770717601297561774m_norm @ complex @ X ) ) ).
% abs_Re_le_cmod
thf(fact_4544_Re__csqrt,axiom,
! [Z3: complex] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( re @ ( csqrt @ Z3 ) ) ) ).
% Re_csqrt
thf(fact_4545_one__integer_Orsp,axiom,
( ( one_one @ int )
= ( one_one @ int ) ) ).
% one_integer.rsp
thf(fact_4546_one__natural_Orsp,axiom,
( ( one_one @ nat )
= ( one_one @ nat ) ) ).
% one_natural.rsp
thf(fact_4547_cmod__plus__Re__le__0__iff,axiom,
! [Z3: complex] :
( ( ord_less_eq @ real @ ( plus_plus @ real @ ( real_V7770717601297561774m_norm @ complex @ Z3 ) @ ( re @ Z3 ) ) @ ( zero_zero @ real ) )
= ( ( re @ Z3 )
= ( uminus_uminus @ real @ ( real_V7770717601297561774m_norm @ complex @ Z3 ) ) ) ) ).
% cmod_plus_Re_le_0_iff
thf(fact_4548_cos__n__Re__cis__pow__n,axiom,
! [N: nat,A2: real] :
( ( cos @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ A2 ) )
= ( re @ ( power_power @ complex @ ( cis @ A2 ) @ N ) ) ) ).
% cos_n_Re_cis_pow_n
thf(fact_4549_CauchyD,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X9: nat > A,E: real] :
( ( topolo3814608138187158403Cauchy @ A @ X9 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ E )
=> ? [M8: nat] :
! [M3: nat] :
( ( ord_less_eq @ nat @ M8 @ M3 )
=> ! [N9: nat] :
( ( ord_less_eq @ nat @ M8 @ N9 )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( X9 @ M3 ) @ ( X9 @ N9 ) ) ) @ E ) ) ) ) ) ) ).
% CauchyD
thf(fact_4550_CauchyI,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X9: nat > A] :
( ! [E2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E2 )
=> ? [M10: nat] :
! [M4: nat] :
( ( ord_less_eq @ nat @ M10 @ M4 )
=> ! [N3: nat] :
( ( ord_less_eq @ nat @ M10 @ N3 )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( X9 @ M4 ) @ ( X9 @ N3 ) ) ) @ E2 ) ) ) )
=> ( topolo3814608138187158403Cauchy @ A @ X9 ) ) ) ).
% CauchyI
thf(fact_4551_Cauchy__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ( ( topolo3814608138187158403Cauchy @ A )
= ( ^ [X6: nat > A] :
! [E3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E3 )
=> ? [M9: nat] :
! [M2: nat] :
( ( ord_less_eq @ nat @ M9 @ M2 )
=> ! [N2: nat] :
( ( ord_less_eq @ nat @ M9 @ N2 )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( X6 @ M2 ) @ ( X6 @ N2 ) ) ) @ E3 ) ) ) ) ) ) ) ).
% Cauchy_iff
thf(fact_4552_csqrt_Ocode,axiom,
( csqrt
= ( ^ [Z6: complex] :
( complex2 @ ( sqrt @ ( divide_divide @ real @ ( plus_plus @ real @ ( real_V7770717601297561774m_norm @ complex @ Z6 ) @ ( re @ Z6 ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
@ ( times_times @ real
@ ( if @ real
@ ( ( im @ Z6 )
= ( zero_zero @ real ) )
@ ( one_one @ real )
@ ( sgn_sgn @ real @ ( im @ Z6 ) ) )
@ ( sqrt @ ( divide_divide @ real @ ( minus_minus @ real @ ( real_V7770717601297561774m_norm @ complex @ Z6 ) @ ( re @ Z6 ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% csqrt.code
thf(fact_4553_csqrt_Osimps_I2_J,axiom,
! [Z3: complex] :
( ( im @ ( csqrt @ Z3 ) )
= ( times_times @ real
@ ( if @ real
@ ( ( im @ Z3 )
= ( zero_zero @ real ) )
@ ( one_one @ real )
@ ( sgn_sgn @ real @ ( im @ Z3 ) ) )
@ ( sqrt @ ( divide_divide @ real @ ( minus_minus @ real @ ( real_V7770717601297561774m_norm @ complex @ Z3 ) @ ( re @ Z3 ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ).
% csqrt.simps(2)
thf(fact_4554_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_4555_csqrt__of__real__nonpos,axiom,
! [X: complex] :
( ( ( im @ X )
= ( zero_zero @ real ) )
=> ( ( ord_less_eq @ real @ ( re @ X ) @ ( zero_zero @ real ) )
=> ( ( csqrt @ X )
= ( times_times @ complex @ imaginary_unit @ ( real_Vector_of_real @ complex @ ( sqrt @ ( abs_abs @ real @ ( re @ X ) ) ) ) ) ) ) ) ).
% csqrt_of_real_nonpos
thf(fact_4556_Im__sum,axiom,
! [A: $tType,F3: A > complex,S: set @ A] :
( ( im @ ( groups7311177749621191930dd_sum @ A @ complex @ F3 @ S ) )
= ( groups7311177749621191930dd_sum @ A @ real
@ ^ [X4: A] : ( im @ ( F3 @ X4 ) )
@ S ) ) ).
% Im_sum
thf(fact_4557_Im__i__times,axiom,
! [Z3: complex] :
( ( im @ ( times_times @ complex @ imaginary_unit @ Z3 ) )
= ( re @ Z3 ) ) ).
% Im_i_times
thf(fact_4558_Re__i__times,axiom,
! [Z3: complex] :
( ( re @ ( times_times @ complex @ imaginary_unit @ Z3 ) )
= ( uminus_uminus @ real @ ( im @ Z3 ) ) ) ).
% Re_i_times
thf(fact_4559_csqrt__of__real__nonneg,axiom,
! [X: complex] :
( ( ( im @ X )
= ( zero_zero @ real ) )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( re @ X ) )
=> ( ( csqrt @ X )
= ( real_Vector_of_real @ complex @ ( sqrt @ ( re @ X ) ) ) ) ) ) ).
% csqrt_of_real_nonneg
thf(fact_4560_csqrt__minus,axiom,
! [X: complex] :
( ( ( ord_less @ real @ ( im @ X ) @ ( zero_zero @ real ) )
| ( ( ( im @ X )
= ( zero_zero @ real ) )
& ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( re @ X ) ) ) )
=> ( ( csqrt @ ( uminus_uminus @ complex @ X ) )
= ( times_times @ complex @ imaginary_unit @ ( csqrt @ X ) ) ) ) ).
% csqrt_minus
thf(fact_4561_Cauchy__Im,axiom,
! [X9: nat > complex] :
( ( topolo3814608138187158403Cauchy @ complex @ X9 )
=> ( topolo3814608138187158403Cauchy @ real
@ ^ [N2: nat] : ( im @ ( X9 @ N2 ) ) ) ) ).
% Cauchy_Im
thf(fact_4562_modulo__integer_Oabs__eq,axiom,
! [Xa: int,X: int] :
( ( modulo_modulo @ code_integer @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X ) )
= ( code_integer_of_int @ ( modulo_modulo @ int @ Xa @ X ) ) ) ).
% modulo_integer.abs_eq
thf(fact_4563_sums__Im,axiom,
! [X9: nat > complex,A2: complex] :
( ( sums @ complex @ X9 @ A2 )
=> ( sums @ real
@ ^ [N2: nat] : ( im @ ( X9 @ N2 ) )
@ ( im @ A2 ) ) ) ).
% sums_Im
thf(fact_4564_imaginary__unit_Osimps_I2_J,axiom,
( ( im @ imaginary_unit )
= ( one_one @ real ) ) ).
% imaginary_unit.simps(2)
thf(fact_4565_one__complex_Osimps_I2_J,axiom,
( ( im @ ( one_one @ complex ) )
= ( zero_zero @ real ) ) ).
% one_complex.simps(2)
thf(fact_4566_times__integer_Oabs__eq,axiom,
! [Xa: int,X: int] :
( ( times_times @ code_integer @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X ) )
= ( code_integer_of_int @ ( times_times @ int @ Xa @ X ) ) ) ).
% times_integer.abs_eq
thf(fact_4567_one__integer__def,axiom,
( ( one_one @ code_integer )
= ( code_integer_of_int @ ( one_one @ int ) ) ) ).
% one_integer_def
thf(fact_4568_less__eq__integer_Oabs__eq,axiom,
! [Xa: int,X: int] :
( ( ord_less_eq @ code_integer @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X ) )
= ( ord_less_eq @ int @ Xa @ X ) ) ).
% less_eq_integer.abs_eq
thf(fact_4569_scaleR__complex_Osimps_I2_J,axiom,
! [R: real,X: complex] :
( ( im @ ( real_V8093663219630862766scaleR @ complex @ R @ X ) )
= ( times_times @ real @ R @ ( im @ X ) ) ) ).
% scaleR_complex.simps(2)
thf(fact_4570_sums__complex__iff,axiom,
( ( sums @ complex )
= ( ^ [F5: nat > complex,X4: complex] :
( ( sums @ real
@ ^ [Y: nat] : ( re @ ( F5 @ Y ) )
@ ( re @ X4 ) )
& ( sums @ real
@ ^ [Y: nat] : ( im @ ( F5 @ Y ) )
@ ( im @ X4 ) ) ) ) ) ).
% sums_complex_iff
thf(fact_4571_summable__Im,axiom,
! [F3: nat > complex] :
( ( summable @ complex @ F3 )
=> ( summable @ real
@ ^ [X4: nat] : ( im @ ( F3 @ X4 ) ) ) ) ).
% summable_Im
thf(fact_4572_abs__Im__le__cmod,axiom,
! [X: complex] : ( ord_less_eq @ real @ ( abs_abs @ real @ ( im @ X ) ) @ ( real_V7770717601297561774m_norm @ complex @ X ) ) ).
% abs_Im_le_cmod
thf(fact_4573_Cauchy__Re,axiom,
! [X9: nat > complex] :
( ( topolo3814608138187158403Cauchy @ complex @ X9 )
=> ( topolo3814608138187158403Cauchy @ real
@ ^ [N2: nat] : ( re @ ( X9 @ N2 ) ) ) ) ).
% Cauchy_Re
thf(fact_4574_summable__complex__iff,axiom,
( ( summable @ complex )
= ( ^ [F5: nat > complex] :
( ( summable @ real
@ ^ [X4: nat] : ( re @ ( F5 @ X4 ) ) )
& ( summable @ real
@ ^ [X4: nat] : ( im @ ( F5 @ X4 ) ) ) ) ) ) ).
% summable_complex_iff
thf(fact_4575_times__complex_Osimps_I2_J,axiom,
! [X: complex,Y2: complex] :
( ( im @ ( times_times @ complex @ X @ Y2 ) )
= ( plus_plus @ real @ ( times_times @ real @ ( re @ X ) @ ( im @ Y2 ) ) @ ( times_times @ real @ ( im @ X ) @ ( re @ Y2 ) ) ) ) ).
% times_complex.simps(2)
thf(fact_4576_cmod__Im__le__iff,axiom,
! [X: complex,Y2: complex] :
( ( ( re @ X )
= ( re @ Y2 ) )
=> ( ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ complex @ X ) @ ( real_V7770717601297561774m_norm @ complex @ Y2 ) )
= ( ord_less_eq @ real @ ( abs_abs @ real @ ( im @ X ) ) @ ( abs_abs @ real @ ( im @ Y2 ) ) ) ) ) ).
% cmod_Im_le_iff
thf(fact_4577_cmod__Re__le__iff,axiom,
! [X: complex,Y2: complex] :
( ( ( im @ X )
= ( im @ Y2 ) )
=> ( ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ complex @ X ) @ ( real_V7770717601297561774m_norm @ complex @ Y2 ) )
= ( ord_less_eq @ real @ ( abs_abs @ real @ ( re @ X ) ) @ ( abs_abs @ real @ ( re @ Y2 ) ) ) ) ) ).
% cmod_Re_le_iff
thf(fact_4578_times__complex_Osimps_I1_J,axiom,
! [X: complex,Y2: complex] :
( ( re @ ( times_times @ complex @ X @ Y2 ) )
= ( minus_minus @ real @ ( times_times @ real @ ( re @ X ) @ ( re @ Y2 ) ) @ ( times_times @ real @ ( im @ X ) @ ( im @ Y2 ) ) ) ) ).
% times_complex.simps(1)
thf(fact_4579_scaleR__complex_Ocode,axiom,
( ( real_V8093663219630862766scaleR @ complex )
= ( ^ [R5: real,X4: complex] : ( complex2 @ ( times_times @ real @ R5 @ ( re @ X4 ) ) @ ( times_times @ real @ R5 @ ( im @ X4 ) ) ) ) ) ).
% scaleR_complex.code
thf(fact_4580_csqrt__principal,axiom,
! [Z3: complex] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( re @ ( csqrt @ Z3 ) ) )
| ( ( ( re @ ( csqrt @ Z3 ) )
= ( zero_zero @ real ) )
& ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( im @ ( csqrt @ Z3 ) ) ) ) ) ).
% csqrt_principal
thf(fact_4581_cmod__le,axiom,
! [Z3: complex] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ complex @ Z3 ) @ ( plus_plus @ real @ ( abs_abs @ real @ ( re @ Z3 ) ) @ ( abs_abs @ real @ ( im @ Z3 ) ) ) ) ).
% cmod_le
thf(fact_4582_sin__n__Im__cis__pow__n,axiom,
! [N: nat,A2: real] :
( ( sin @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ A2 ) )
= ( im @ ( power_power @ complex @ ( cis @ A2 ) @ N ) ) ) ).
% sin_n_Im_cis_pow_n
thf(fact_4583_Re__exp,axiom,
! [Z3: complex] :
( ( re @ ( exp @ complex @ Z3 ) )
= ( times_times @ real @ ( exp @ real @ ( re @ Z3 ) ) @ ( cos @ real @ ( im @ Z3 ) ) ) ) ).
% Re_exp
thf(fact_4584_Im__exp,axiom,
! [Z3: complex] :
( ( im @ ( exp @ complex @ Z3 ) )
= ( times_times @ real @ ( exp @ real @ ( re @ Z3 ) ) @ ( sin @ real @ ( im @ Z3 ) ) ) ) ).
% Im_exp
thf(fact_4585_complex__eq,axiom,
! [A2: complex] :
( A2
= ( plus_plus @ complex @ ( real_Vector_of_real @ complex @ ( re @ A2 ) ) @ ( times_times @ complex @ imaginary_unit @ ( real_Vector_of_real @ complex @ ( im @ A2 ) ) ) ) ) ).
% complex_eq
thf(fact_4586_fun__complex__eq,axiom,
! [A: $tType,F3: A > complex] :
( F3
= ( ^ [X4: A] : ( plus_plus @ complex @ ( real_Vector_of_real @ complex @ ( re @ ( F3 @ X4 ) ) ) @ ( times_times @ complex @ imaginary_unit @ ( real_Vector_of_real @ complex @ ( im @ ( F3 @ X4 ) ) ) ) ) ) ) ).
% fun_complex_eq
thf(fact_4587_times__complex_Ocode,axiom,
( ( times_times @ complex )
= ( ^ [X4: complex,Y: complex] : ( complex2 @ ( minus_minus @ real @ ( times_times @ real @ ( re @ X4 ) @ ( re @ Y ) ) @ ( times_times @ real @ ( im @ X4 ) @ ( im @ Y ) ) ) @ ( plus_plus @ real @ ( times_times @ real @ ( re @ X4 ) @ ( im @ Y ) ) @ ( times_times @ real @ ( im @ X4 ) @ ( re @ Y ) ) ) ) ) ) ).
% times_complex.code
thf(fact_4588_exp__eq__polar,axiom,
( ( exp @ complex )
= ( ^ [Z6: complex] : ( times_times @ complex @ ( real_Vector_of_real @ complex @ ( exp @ real @ ( re @ Z6 ) ) ) @ ( cis @ ( im @ Z6 ) ) ) ) ) ).
% exp_eq_polar
thf(fact_4589_cmod__power2,axiom,
! [Z3: complex] :
( ( power_power @ real @ ( real_V7770717601297561774m_norm @ complex @ Z3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( plus_plus @ real @ ( power_power @ real @ ( re @ Z3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ ( im @ Z3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% cmod_power2
thf(fact_4590_Im__power2,axiom,
! [X: complex] :
( ( im @ ( power_power @ complex @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( times_times @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( re @ X ) ) @ ( im @ X ) ) ) ).
% Im_power2
thf(fact_4591_Re__power2,axiom,
! [X: complex] :
( ( re @ ( power_power @ complex @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( minus_minus @ real @ ( power_power @ real @ ( re @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ ( im @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% Re_power2
thf(fact_4592_complex__eq__0,axiom,
! [Z3: complex] :
( ( Z3
= ( zero_zero @ complex ) )
= ( ( plus_plus @ real @ ( power_power @ real @ ( re @ Z3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ ( im @ Z3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( zero_zero @ real ) ) ) ).
% complex_eq_0
thf(fact_4593_norm__complex__def,axiom,
( ( real_V7770717601297561774m_norm @ complex )
= ( ^ [Z6: complex] : ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ ( re @ Z6 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ ( im @ Z6 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% norm_complex_def
thf(fact_4594_inverse__complex_Osimps_I1_J,axiom,
! [X: complex] :
( ( re @ ( inverse_inverse @ complex @ X ) )
= ( divide_divide @ real @ ( re @ X ) @ ( plus_plus @ real @ ( power_power @ real @ ( re @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ ( im @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% inverse_complex.simps(1)
thf(fact_4595_complex__neq__0,axiom,
! [Z3: complex] :
( ( Z3
!= ( zero_zero @ complex ) )
= ( ord_less @ real @ ( zero_zero @ real ) @ ( plus_plus @ real @ ( power_power @ real @ ( re @ Z3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ ( im @ Z3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% complex_neq_0
thf(fact_4596_Re__divide,axiom,
! [X: complex,Y2: complex] :
( ( re @ ( divide_divide @ complex @ X @ Y2 ) )
= ( divide_divide @ real @ ( plus_plus @ real @ ( times_times @ real @ ( re @ X ) @ ( re @ Y2 ) ) @ ( times_times @ real @ ( im @ X ) @ ( im @ Y2 ) ) ) @ ( plus_plus @ real @ ( power_power @ real @ ( re @ Y2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ ( im @ Y2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% Re_divide
thf(fact_4597_csqrt__square,axiom,
! [B2: complex] :
( ( ( ord_less @ real @ ( zero_zero @ real ) @ ( re @ B2 ) )
| ( ( ( re @ B2 )
= ( zero_zero @ real ) )
& ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( im @ B2 ) ) ) )
=> ( ( csqrt @ ( power_power @ complex @ B2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= B2 ) ) ).
% csqrt_square
thf(fact_4598_csqrt__unique,axiom,
! [W: complex,Z3: complex] :
( ( ( power_power @ complex @ W @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= Z3 )
=> ( ( ( ord_less @ real @ ( zero_zero @ real ) @ ( re @ W ) )
| ( ( ( re @ W )
= ( zero_zero @ real ) )
& ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( im @ W ) ) ) )
=> ( ( csqrt @ Z3 )
= W ) ) ) ).
% csqrt_unique
thf(fact_4599_inverse__complex_Osimps_I2_J,axiom,
! [X: complex] :
( ( im @ ( inverse_inverse @ complex @ X ) )
= ( divide_divide @ real @ ( uminus_uminus @ real @ ( im @ X ) ) @ ( plus_plus @ real @ ( power_power @ real @ ( re @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ ( im @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% inverse_complex.simps(2)
thf(fact_4600_Im__divide,axiom,
! [X: complex,Y2: complex] :
( ( im @ ( divide_divide @ complex @ X @ Y2 ) )
= ( divide_divide @ real @ ( minus_minus @ real @ ( times_times @ real @ ( im @ X ) @ ( re @ Y2 ) ) @ ( times_times @ real @ ( re @ X ) @ ( im @ Y2 ) ) ) @ ( plus_plus @ real @ ( power_power @ real @ ( re @ Y2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ ( im @ Y2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% Im_divide
thf(fact_4601_complex__abs__le__norm,axiom,
! [Z3: complex] : ( ord_less_eq @ real @ ( plus_plus @ real @ ( abs_abs @ real @ ( re @ Z3 ) ) @ ( abs_abs @ real @ ( im @ Z3 ) ) ) @ ( times_times @ real @ ( sqrt @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( real_V7770717601297561774m_norm @ complex @ Z3 ) ) ) ).
% complex_abs_le_norm
thf(fact_4602_complex__unit__circle,axiom,
! [Z3: complex] :
( ( Z3
!= ( zero_zero @ complex ) )
=> ( ( plus_plus @ real @ ( power_power @ real @ ( divide_divide @ real @ ( re @ Z3 ) @ ( real_V7770717601297561774m_norm @ complex @ Z3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ ( divide_divide @ real @ ( im @ Z3 ) @ ( real_V7770717601297561774m_norm @ complex @ Z3 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( one_one @ real ) ) ) ).
% complex_unit_circle
thf(fact_4603_inverse__complex_Ocode,axiom,
( ( inverse_inverse @ complex )
= ( ^ [X4: complex] : ( complex2 @ ( divide_divide @ real @ ( re @ X4 ) @ ( plus_plus @ real @ ( power_power @ real @ ( re @ X4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ ( im @ X4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( divide_divide @ real @ ( uminus_uminus @ real @ ( im @ X4 ) ) @ ( plus_plus @ real @ ( power_power @ real @ ( re @ X4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ ( im @ X4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% inverse_complex.code
thf(fact_4604_Complex__divide,axiom,
( ( divide_divide @ complex )
= ( ^ [X4: complex,Y: complex] : ( complex2 @ ( divide_divide @ real @ ( plus_plus @ real @ ( times_times @ real @ ( re @ X4 ) @ ( re @ Y ) ) @ ( times_times @ real @ ( im @ X4 ) @ ( im @ Y ) ) ) @ ( plus_plus @ real @ ( power_power @ real @ ( re @ Y ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ ( im @ Y ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( divide_divide @ real @ ( minus_minus @ real @ ( times_times @ real @ ( im @ X4 ) @ ( re @ Y ) ) @ ( times_times @ real @ ( re @ X4 ) @ ( im @ Y ) ) ) @ ( plus_plus @ real @ ( power_power @ real @ ( re @ Y ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ ( im @ Y ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% Complex_divide
thf(fact_4605_length__mul__elem,axiom,
! [A: $tType,Xs: list @ ( list @ A ),N: nat] :
( ! [X5: list @ A] :
( ( member @ ( list @ A ) @ X5 @ ( set2 @ ( list @ A ) @ Xs ) )
=> ( ( size_size @ ( list @ A ) @ X5 )
= N ) )
=> ( ( size_size @ ( list @ A ) @ ( concat @ A @ Xs ) )
= ( times_times @ nat @ ( size_size @ ( list @ ( list @ A ) ) @ Xs ) @ N ) ) ) ).
% length_mul_elem
thf(fact_4606_Im__Reals__divide,axiom,
! [R: complex,Z3: complex] :
( ( member @ complex @ R @ ( real_Vector_Reals @ complex ) )
=> ( ( im @ ( divide_divide @ complex @ R @ Z3 ) )
= ( divide_divide @ real @ ( times_times @ real @ ( uminus_uminus @ real @ ( re @ R ) ) @ ( im @ Z3 ) ) @ ( power_power @ real @ ( real_V7770717601297561774m_norm @ complex @ Z3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% Im_Reals_divide
thf(fact_4607_Re__Reals__divide,axiom,
! [R: complex,Z3: complex] :
( ( member @ complex @ R @ ( real_Vector_Reals @ complex ) )
=> ( ( re @ ( divide_divide @ complex @ R @ Z3 ) )
= ( divide_divide @ real @ ( times_times @ real @ ( re @ R ) @ ( re @ Z3 ) ) @ ( power_power @ real @ ( real_V7770717601297561774m_norm @ complex @ Z3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% Re_Reals_divide
thf(fact_4608_series__comparison__complex,axiom,
! [A: $tType] :
( ( real_Vector_banach @ A )
=> ! [G: nat > complex,N4: nat,F3: nat > A] :
( ( summable @ complex @ G )
=> ( ! [N3: nat] : ( member @ complex @ ( G @ N3 ) @ ( real_Vector_Reals @ complex ) )
=> ( ! [N3: nat] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( re @ ( G @ N3 ) ) )
=> ( ! [N3: nat] :
( ( ord_less_eq @ nat @ N4 @ N3 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( F3 @ N3 ) ) @ ( real_V7770717601297561774m_norm @ complex @ ( G @ N3 ) ) ) )
=> ( summable @ A @ F3 ) ) ) ) ) ) ).
% series_comparison_complex
thf(fact_4609_real__eq__imaginary__iff,axiom,
! [Y2: complex,X: complex] :
( ( member @ complex @ Y2 @ ( real_Vector_Reals @ complex ) )
=> ( ( member @ complex @ X @ ( real_Vector_Reals @ complex ) )
=> ( ( X
= ( times_times @ complex @ imaginary_unit @ Y2 ) )
= ( ( X
= ( zero_zero @ complex ) )
& ( Y2
= ( zero_zero @ complex ) ) ) ) ) ) ).
% real_eq_imaginary_iff
thf(fact_4610_imaginary__eq__real__iff,axiom,
! [Y2: complex,X: complex] :
( ( member @ complex @ Y2 @ ( real_Vector_Reals @ complex ) )
=> ( ( member @ complex @ X @ ( real_Vector_Reals @ complex ) )
=> ( ( ( times_times @ complex @ imaginary_unit @ Y2 )
= X )
= ( ( X
= ( zero_zero @ complex ) )
& ( Y2
= ( zero_zero @ complex ) ) ) ) ) ) ).
% imaginary_eq_real_iff
thf(fact_4611_Reals__divide,axiom,
! [A: $tType] :
( ( real_V7773925162809079976_field @ A )
=> ! [A2: A,B2: A] :
( ( member @ A @ A2 @ ( real_Vector_Reals @ A ) )
=> ( ( member @ A @ B2 @ ( real_Vector_Reals @ A ) )
=> ( member @ A @ ( divide_divide @ A @ A2 @ B2 ) @ ( real_Vector_Reals @ A ) ) ) ) ) ).
% Reals_divide
thf(fact_4612_map__concat,axiom,
! [A: $tType,B: $tType,F3: B > A,Xs: list @ ( list @ B )] :
( ( map @ B @ A @ F3 @ ( concat @ B @ Xs ) )
= ( concat @ A @ ( map @ ( list @ B ) @ ( list @ A ) @ ( map @ B @ A @ F3 ) @ Xs ) ) ) ).
% map_concat
thf(fact_4613_Reals__power,axiom,
! [A: $tType] :
( ( real_V2191834092415804123ebra_1 @ A )
=> ! [A2: A,N: nat] :
( ( member @ A @ A2 @ ( real_Vector_Reals @ A ) )
=> ( member @ A @ ( power_power @ A @ A2 @ N ) @ ( real_Vector_Reals @ A ) ) ) ) ).
% Reals_power
thf(fact_4614_Reals__add,axiom,
! [A: $tType] :
( ( real_V2191834092415804123ebra_1 @ A )
=> ! [A2: A,B2: A] :
( ( member @ A @ A2 @ ( real_Vector_Reals @ A ) )
=> ( ( member @ A @ B2 @ ( real_Vector_Reals @ A ) )
=> ( member @ A @ ( plus_plus @ A @ A2 @ B2 ) @ ( real_Vector_Reals @ A ) ) ) ) ) ).
% Reals_add
thf(fact_4615_Reals__1,axiom,
! [B: $tType] :
( ( real_V2191834092415804123ebra_1 @ B )
=> ( member @ B @ ( one_one @ B ) @ ( real_Vector_Reals @ B ) ) ) ).
% Reals_1
thf(fact_4616_Reals__numeral,axiom,
! [A: $tType] :
( ( real_V2191834092415804123ebra_1 @ A )
=> ! [W: num] : ( member @ A @ ( numeral_numeral @ A @ W ) @ ( real_Vector_Reals @ A ) ) ) ).
% Reals_numeral
thf(fact_4617_Reals__mult,axiom,
! [A: $tType] :
( ( real_V2191834092415804123ebra_1 @ A )
=> ! [A2: A,B2: A] :
( ( member @ A @ A2 @ ( real_Vector_Reals @ A ) )
=> ( ( member @ A @ B2 @ ( real_Vector_Reals @ A ) )
=> ( member @ A @ ( times_times @ A @ A2 @ B2 ) @ ( real_Vector_Reals @ A ) ) ) ) ) ).
% Reals_mult
thf(fact_4618_nonzero__Reals__divide,axiom,
! [A: $tType] :
( ( real_V7773925162809079976_field @ A )
=> ! [A2: A,B2: A] :
( ( member @ A @ A2 @ ( real_Vector_Reals @ A ) )
=> ( ( member @ A @ B2 @ ( real_Vector_Reals @ A ) )
=> ( ( B2
!= ( zero_zero @ A ) )
=> ( member @ A @ ( divide_divide @ A @ A2 @ B2 ) @ ( real_Vector_Reals @ A ) ) ) ) ) ) ).
% nonzero_Reals_divide
thf(fact_4619_product__concat__map,axiom,
! [B: $tType,A: $tType] :
( ( product @ A @ B )
= ( ^ [Xs2: list @ A,Ys3: list @ B] :
( concat @ ( product_prod @ A @ B )
@ ( map @ A @ ( list @ ( product_prod @ A @ B ) )
@ ^ [X4: A] : ( map @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 ) @ Ys3 )
@ Xs2 ) ) ) ) ).
% product_concat_map
thf(fact_4620_Re__prod__Reals,axiom,
! [A: $tType,A3: set @ A,F3: A > complex] :
( ! [X5: A] :
( ( member @ A @ X5 @ A3 )
=> ( member @ complex @ ( F3 @ X5 ) @ ( real_Vector_Reals @ complex ) ) )
=> ( ( re @ ( groups7121269368397514597t_prod @ A @ complex @ F3 @ A3 ) )
= ( groups7121269368397514597t_prod @ A @ real
@ ^ [X4: A] : ( re @ ( F3 @ X4 ) )
@ A3 ) ) ) ).
% Re_prod_Reals
thf(fact_4621_set__n__lists,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( set2 @ ( list @ A ) @ ( n_lists @ A @ N @ Xs ) )
= ( collect @ ( list @ A )
@ ^ [Ys3: list @ A] :
( ( ( size_size @ ( list @ A ) @ Ys3 )
= N )
& ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Ys3 ) @ ( set2 @ A @ Xs ) ) ) ) ) ).
% set_n_lists
thf(fact_4622_complex__mult__cnj,axiom,
! [Z3: complex] :
( ( times_times @ complex @ Z3 @ ( cnj @ Z3 ) )
= ( real_Vector_of_real @ complex @ ( plus_plus @ real @ ( power_power @ real @ ( re @ Z3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ ( im @ Z3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% complex_mult_cnj
thf(fact_4623_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_4624_complex__cnj__mult,axiom,
! [X: complex,Y2: complex] :
( ( cnj @ ( times_times @ complex @ X @ Y2 ) )
= ( times_times @ complex @ ( cnj @ X ) @ ( cnj @ Y2 ) ) ) ).
% complex_cnj_mult
thf(fact_4625_complex__cnj__one,axiom,
( ( cnj @ ( one_one @ complex ) )
= ( one_one @ complex ) ) ).
% complex_cnj_one
thf(fact_4626_complex__cnj__one__iff,axiom,
! [Z3: complex] :
( ( ( cnj @ Z3 )
= ( one_one @ complex ) )
= ( Z3
= ( one_one @ complex ) ) ) ).
% complex_cnj_one_iff
thf(fact_4627_cnj__sum,axiom,
! [A: $tType,F3: A > complex,S: set @ A] :
( ( cnj @ ( groups7311177749621191930dd_sum @ A @ complex @ F3 @ S ) )
= ( groups7311177749621191930dd_sum @ A @ complex
@ ^ [X4: A] : ( cnj @ ( F3 @ X4 ) )
@ S ) ) ).
% cnj_sum
thf(fact_4628_cnj__prod,axiom,
! [A: $tType,F3: A > complex,S: set @ A] :
( ( cnj @ ( groups7121269368397514597t_prod @ A @ complex @ F3 @ S ) )
= ( groups7121269368397514597t_prod @ A @ complex
@ ^ [X4: A] : ( cnj @ ( F3 @ X4 ) )
@ S ) ) ).
% cnj_prod
thf(fact_4629_complex__In__mult__cnj__zero,axiom,
! [Z3: complex] :
( ( im @ ( times_times @ complex @ Z3 @ ( cnj @ Z3 ) ) )
= ( zero_zero @ real ) ) ).
% complex_In_mult_cnj_zero
thf(fact_4630_sums__cnj,axiom,
! [F3: nat > complex,L: complex] :
( ( sums @ complex
@ ^ [X4: nat] : ( cnj @ ( F3 @ X4 ) )
@ ( cnj @ L ) )
= ( sums @ complex @ F3 @ L ) ) ).
% sums_cnj
thf(fact_4631_Re__complex__div__eq__0,axiom,
! [A2: complex,B2: complex] :
( ( ( re @ ( divide_divide @ complex @ A2 @ B2 ) )
= ( zero_zero @ real ) )
= ( ( re @ ( times_times @ complex @ A2 @ ( cnj @ B2 ) ) )
= ( zero_zero @ real ) ) ) ).
% Re_complex_div_eq_0
thf(fact_4632_Im__complex__div__eq__0,axiom,
! [A2: complex,B2: complex] :
( ( ( im @ ( divide_divide @ complex @ A2 @ B2 ) )
= ( zero_zero @ real ) )
= ( ( im @ ( times_times @ complex @ A2 @ ( cnj @ B2 ) ) )
= ( zero_zero @ real ) ) ) ).
% Im_complex_div_eq_0
thf(fact_4633_complex__mod__sqrt__Re__mult__cnj,axiom,
( ( real_V7770717601297561774m_norm @ complex )
= ( ^ [Z6: complex] : ( sqrt @ ( re @ ( times_times @ complex @ Z6 @ ( cnj @ Z6 ) ) ) ) ) ) ).
% complex_mod_sqrt_Re_mult_cnj
thf(fact_4634_length__n__lists__elem,axiom,
! [A: $tType,Ys: list @ A,N: nat,Xs: list @ A] :
( ( member @ ( list @ A ) @ Ys @ ( set2 @ ( list @ A ) @ ( n_lists @ A @ N @ Xs ) ) )
=> ( ( size_size @ ( list @ A ) @ Ys )
= N ) ) ).
% length_n_lists_elem
thf(fact_4635_integer__of__num__triv_I1_J,axiom,
( ( code_integer_of_num @ one2 )
= ( one_one @ code_integer ) ) ).
% integer_of_num_triv(1)
thf(fact_4636_Re__complex__div__gt__0,axiom,
! [A2: complex,B2: complex] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( re @ ( divide_divide @ complex @ A2 @ B2 ) ) )
= ( ord_less @ real @ ( zero_zero @ real ) @ ( re @ ( times_times @ complex @ A2 @ ( cnj @ B2 ) ) ) ) ) ).
% Re_complex_div_gt_0
thf(fact_4637_Re__complex__div__lt__0,axiom,
! [A2: complex,B2: complex] :
( ( ord_less @ real @ ( re @ ( divide_divide @ complex @ A2 @ B2 ) ) @ ( zero_zero @ real ) )
= ( ord_less @ real @ ( re @ ( times_times @ complex @ A2 @ ( cnj @ B2 ) ) ) @ ( zero_zero @ real ) ) ) ).
% Re_complex_div_lt_0
thf(fact_4638_Re__complex__div__ge__0,axiom,
! [A2: complex,B2: complex] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( re @ ( divide_divide @ complex @ A2 @ B2 ) ) )
= ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( re @ ( times_times @ complex @ A2 @ ( cnj @ B2 ) ) ) ) ) ).
% Re_complex_div_ge_0
thf(fact_4639_Re__complex__div__le__0,axiom,
! [A2: complex,B2: complex] :
( ( ord_less_eq @ real @ ( re @ ( divide_divide @ complex @ A2 @ B2 ) ) @ ( zero_zero @ real ) )
= ( ord_less_eq @ real @ ( re @ ( times_times @ complex @ A2 @ ( cnj @ B2 ) ) ) @ ( zero_zero @ real ) ) ) ).
% Re_complex_div_le_0
thf(fact_4640_Im__complex__div__gt__0,axiom,
! [A2: complex,B2: complex] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( im @ ( divide_divide @ complex @ A2 @ B2 ) ) )
= ( ord_less @ real @ ( zero_zero @ real ) @ ( im @ ( times_times @ complex @ A2 @ ( cnj @ B2 ) ) ) ) ) ).
% Im_complex_div_gt_0
thf(fact_4641_Im__complex__div__lt__0,axiom,
! [A2: complex,B2: complex] :
( ( ord_less @ real @ ( im @ ( divide_divide @ complex @ A2 @ B2 ) ) @ ( zero_zero @ real ) )
= ( ord_less @ real @ ( im @ ( times_times @ complex @ A2 @ ( cnj @ B2 ) ) ) @ ( zero_zero @ real ) ) ) ).
% Im_complex_div_lt_0
thf(fact_4642_Im__complex__div__ge__0,axiom,
! [A2: complex,B2: complex] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( im @ ( divide_divide @ complex @ A2 @ B2 ) ) )
= ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( im @ ( times_times @ complex @ A2 @ ( cnj @ B2 ) ) ) ) ) ).
% Im_complex_div_ge_0
thf(fact_4643_Im__complex__div__le__0,axiom,
! [A2: complex,B2: complex] :
( ( ord_less_eq @ real @ ( im @ ( divide_divide @ complex @ A2 @ B2 ) ) @ ( zero_zero @ real ) )
= ( ord_less_eq @ real @ ( im @ ( times_times @ complex @ A2 @ ( cnj @ B2 ) ) ) @ ( zero_zero @ real ) ) ) ).
% Im_complex_div_le_0
thf(fact_4644_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_4645_complex__mod__mult__cnj,axiom,
! [Z3: complex] :
( ( real_V7770717601297561774m_norm @ complex @ ( times_times @ complex @ Z3 @ ( cnj @ Z3 ) ) )
= ( power_power @ real @ ( real_V7770717601297561774m_norm @ complex @ Z3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% complex_mod_mult_cnj
thf(fact_4646_complex__div__gt__0,axiom,
! [A2: complex,B2: complex] :
( ( ( ord_less @ real @ ( zero_zero @ real ) @ ( re @ ( divide_divide @ complex @ A2 @ B2 ) ) )
= ( ord_less @ real @ ( zero_zero @ real ) @ ( re @ ( times_times @ complex @ A2 @ ( cnj @ B2 ) ) ) ) )
& ( ( ord_less @ real @ ( zero_zero @ real ) @ ( im @ ( divide_divide @ complex @ A2 @ B2 ) ) )
= ( ord_less @ real @ ( zero_zero @ real ) @ ( im @ ( times_times @ complex @ A2 @ ( cnj @ B2 ) ) ) ) ) ) ).
% complex_div_gt_0
thf(fact_4647_length__n__lists,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( size_size @ ( list @ ( list @ A ) ) @ ( n_lists @ A @ N @ Xs ) )
= ( power_power @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N ) ) ).
% length_n_lists
thf(fact_4648_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_4649_complex__norm__square,axiom,
! [Z3: complex] :
( ( real_Vector_of_real @ complex @ ( power_power @ real @ ( real_V7770717601297561774m_norm @ complex @ Z3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( times_times @ complex @ Z3 @ ( cnj @ Z3 ) ) ) ).
% complex_norm_square
thf(fact_4650_complex__add__cnj,axiom,
! [Z3: complex] :
( ( plus_plus @ complex @ Z3 @ ( cnj @ Z3 ) )
= ( real_Vector_of_real @ complex @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( re @ Z3 ) ) ) ) ).
% complex_add_cnj
thf(fact_4651_complex__diff__cnj,axiom,
! [Z3: complex] :
( ( minus_minus @ complex @ Z3 @ ( cnj @ Z3 ) )
= ( times_times @ complex @ ( real_Vector_of_real @ complex @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( im @ Z3 ) ) ) @ imaginary_unit ) ) ).
% complex_diff_cnj
thf(fact_4652_complex__div__cnj,axiom,
( ( divide_divide @ complex )
= ( ^ [A5: complex,B4: complex] : ( divide_divide @ complex @ ( times_times @ complex @ A5 @ ( cnj @ B4 ) ) @ ( real_Vector_of_real @ complex @ ( power_power @ real @ ( real_V7770717601297561774m_norm @ complex @ B4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% complex_div_cnj
thf(fact_4653_cnj__add__mult__eq__Re,axiom,
! [Z3: complex,W: complex] :
( ( plus_plus @ complex @ ( times_times @ complex @ Z3 @ ( cnj @ W ) ) @ ( times_times @ complex @ ( cnj @ Z3 ) @ W ) )
= ( real_Vector_of_real @ complex @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( re @ ( times_times @ complex @ Z3 @ ( cnj @ W ) ) ) ) ) ) ).
% cnj_add_mult_eq_Re
thf(fact_4654_product__code,axiom,
! [B: $tType,A: $tType,Xs: list @ A,Ys: list @ B] :
( ( product_product @ A @ B @ ( set2 @ A @ Xs ) @ ( set2 @ B @ Ys ) )
= ( set2 @ ( product_prod @ A @ B )
@ ( concat @ ( product_prod @ A @ B )
@ ( map @ A @ ( list @ ( product_prod @ A @ B ) )
@ ^ [X4: A] : ( map @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 ) @ Ys )
@ Xs ) ) ) ) ).
% product_code
thf(fact_4655_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,J4: code_integer] :
( if @ int
@ ( J4
= ( 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_4656_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_4657_times__integer_Orep__eq,axiom,
! [X: code_integer,Xa: code_integer] :
( ( code_int_of_integer @ ( times_times @ code_integer @ X @ Xa ) )
= ( times_times @ int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa ) ) ) ).
% times_integer.rep_eq
thf(fact_4658_one__integer_Orep__eq,axiom,
( ( code_int_of_integer @ ( one_one @ code_integer ) )
= ( one_one @ int ) ) ).
% one_integer.rep_eq
thf(fact_4659_modulo__integer_Orep__eq,axiom,
! [X: code_integer,Xa: code_integer] :
( ( code_int_of_integer @ ( modulo_modulo @ code_integer @ X @ Xa ) )
= ( modulo_modulo @ int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa ) ) ) ).
% modulo_integer.rep_eq
thf(fact_4660_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_4661_less__eq__integer_Orep__eq,axiom,
( ( ord_less_eq @ code_integer )
= ( ^ [X4: code_integer,Xa4: code_integer] : ( ord_less_eq @ int @ ( code_int_of_integer @ X4 ) @ ( code_int_of_integer @ Xa4 ) ) ) ) ).
% less_eq_integer.rep_eq
thf(fact_4662_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_4663_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,J4: code_integer] :
( if @ num
@ ( J4
= ( 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_4664_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,S4: 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 ) @ S4 ) )
@ ( S4
= ( one_one @ code_integer ) ) )
@ ( code_divmod_abs @ K3 @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% bit_cut_integer_code
thf(fact_4665_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,J4: code_integer] :
( if @ nat
@ ( J4
= ( 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_4666_nat__of__integer__non__positive,axiom,
! [K2: code_integer] :
( ( ord_less_eq @ code_integer @ K2 @ ( zero_zero @ code_integer ) )
=> ( ( code_nat_of_integer @ K2 )
= ( zero_zero @ nat ) ) ) ).
% nat_of_integer_non_positive
thf(fact_4667_nat__of__integer__code__post_I3_J,axiom,
! [K2: num] :
( ( code_nat_of_integer @ ( numeral_numeral @ code_integer @ K2 ) )
= ( numeral_numeral @ nat @ K2 ) ) ).
% nat_of_integer_code_post(3)
thf(fact_4668_nat__of__integer__code__post_I2_J,axiom,
( ( code_nat_of_integer @ ( one_one @ code_integer ) )
= ( one_one @ nat ) ) ).
% nat_of_integer_code_post(2)
thf(fact_4669_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_4670_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,S4: code_integer] :
( if @ ( product_prod @ code_integer @ code_integer )
@ ( S4
= ( 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 @ S4 ) ) )
@ ( 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,S4: code_integer] :
( if @ ( product_prod @ code_integer @ code_integer )
@ ( S4
= ( 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 ) @ S4 ) ) )
@ ( code_divmod_abs @ K3 @ L3 ) ) ) ) ) ) ) ) ) ).
% divmod_integer_code
thf(fact_4671_case__nat__add__eq__if,axiom,
! [A: $tType,A2: A,F3: nat > A,V2: num,N: nat] :
( ( case_nat @ A @ A2 @ F3 @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ V2 ) @ N ) )
= ( F3 @ ( plus_plus @ nat @ ( pred_numeral @ V2 ) @ N ) ) ) ).
% case_nat_add_eq_if
thf(fact_4672_rec__nat__add__eq__if,axiom,
! [A: $tType,A2: A,F3: nat > A > A,V2: num,N: nat] :
( ( rec_nat @ A @ A2 @ F3 @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ V2 ) @ N ) )
= ( F3 @ ( plus_plus @ nat @ ( pred_numeral @ V2 ) @ N ) @ ( rec_nat @ A @ A2 @ F3 @ ( plus_plus @ nat @ ( pred_numeral @ V2 ) @ N ) ) ) ) ).
% rec_nat_add_eq_if
thf(fact_4673_old_Onat_Osimps_I7_J,axiom,
! [T: $tType,F1: T,F2: nat > T > T,Nat2: nat] :
( ( rec_nat @ T @ F1 @ F2 @ ( suc @ Nat2 ) )
= ( F2 @ Nat2 @ ( rec_nat @ T @ F1 @ F2 @ Nat2 ) ) ) ).
% old.nat.simps(7)
thf(fact_4674_old_Onat_Osimps_I6_J,axiom,
! [T: $tType,F1: T,F2: nat > T > T] :
( ( rec_nat @ T @ F1 @ F2 @ ( zero_zero @ nat ) )
= F1 ) ).
% old.nat.simps(6)
thf(fact_4675_case__nat__numeral,axiom,
! [A: $tType,A2: A,F3: nat > A,V2: num] :
( ( case_nat @ A @ A2 @ F3 @ ( numeral_numeral @ nat @ V2 ) )
= ( F3 @ ( pred_numeral @ V2 ) ) ) ).
% case_nat_numeral
thf(fact_4676_rec__nat__numeral,axiom,
! [A: $tType,A2: A,F3: nat > A > A,V2: num] :
( ( rec_nat @ A @ A2 @ F3 @ ( numeral_numeral @ nat @ V2 ) )
= ( F3 @ ( pred_numeral @ V2 ) @ ( rec_nat @ A @ A2 @ F3 @ ( pred_numeral @ V2 ) ) ) ) ).
% rec_nat_numeral
thf(fact_4677_nat_Ocase__distrib,axiom,
! [B: $tType,A: $tType,H2: A > B,F1: A,F2: nat > A,Nat2: nat] :
( ( H2 @ ( case_nat @ A @ F1 @ F2 @ Nat2 ) )
= ( case_nat @ B @ ( H2 @ F1 )
@ ^ [X4: nat] : ( H2 @ ( F2 @ X4 ) )
@ Nat2 ) ) ).
% nat.case_distrib
thf(fact_4678_old_Onat_Osimps_I5_J,axiom,
! [A: $tType,F1: A,F2: nat > A,X23: nat] :
( ( case_nat @ A @ F1 @ F2 @ ( suc @ X23 ) )
= ( F2 @ X23 ) ) ).
% old.nat.simps(5)
thf(fact_4679_old_Onat_Osimps_I4_J,axiom,
! [A: $tType,F1: A,F2: nat > A] :
( ( case_nat @ A @ F1 @ F2 @ ( zero_zero @ nat ) )
= F1 ) ).
% old.nat.simps(4)
thf(fact_4680_nat_Odisc__eq__case_I2_J,axiom,
! [Nat2: nat] :
( ( Nat2
!= ( zero_zero @ nat ) )
= ( case_nat @ $o @ $false
@ ^ [Uu3: nat] : $true
@ Nat2 ) ) ).
% nat.disc_eq_case(2)
thf(fact_4681_nat_Odisc__eq__case_I1_J,axiom,
! [Nat2: nat] :
( ( Nat2
= ( zero_zero @ nat ) )
= ( case_nat @ $o @ $true
@ ^ [Uu3: nat] : $false
@ Nat2 ) ) ).
% nat.disc_eq_case(1)
thf(fact_4682_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_4683_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_4684_bit__numeral__rec_I1_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [W: num,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( numeral_numeral @ A @ ( bit0 @ W ) ) @ N )
= ( case_nat @ $o @ $false @ ( bit_se5641148757651400278ts_bit @ A @ ( numeral_numeral @ A @ W ) ) @ N ) ) ) ).
% bit_numeral_rec(1)
thf(fact_4685_bit__numeral__rec_I2_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [W: num,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( numeral_numeral @ A @ ( bit1 @ W ) ) @ N )
= ( case_nat @ $o @ $true @ ( bit_se5641148757651400278ts_bit @ A @ ( numeral_numeral @ A @ W ) ) @ N ) ) ) ).
% bit_numeral_rec(2)
thf(fact_4686_old_Orec__nat__def,axiom,
! [T: $tType] :
( ( rec_nat @ T )
= ( ^ [F12: T,F22: nat > T > T,X4: nat] : ( the @ T @ ( rec_set_nat @ T @ F12 @ F22 @ X4 ) ) ) ) ).
% old.rec_nat_def
thf(fact_4687_Nitpick_Ocase__nat__unfold,axiom,
! [A: $tType] :
( ( case_nat @ A )
= ( ^ [X4: A,F5: nat > A,N2: nat] :
( if @ A
@ ( N2
= ( zero_zero @ nat ) )
@ X4
@ ( F5 @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) ) ) ) ) ).
% Nitpick.case_nat_unfold
thf(fact_4688_rec__nat__0__imp,axiom,
! [A: $tType,F3: nat > A,F1: A,F2: nat > A > A] :
( ( F3
= ( rec_nat @ A @ F1 @ F2 ) )
=> ( ( F3 @ ( zero_zero @ nat ) )
= F1 ) ) ).
% rec_nat_0_imp
thf(fact_4689_subset__Collect__iff,axiom,
! [A: $tType,B5: set @ A,A3: set @ A,P3: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ B5 @ A3 )
=> ( ( ord_less_eq @ ( set @ A ) @ B5
@ ( collect @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ A3 )
& ( P3 @ X4 ) ) ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ B5 )
=> ( P3 @ X4 ) ) ) ) ) ).
% subset_Collect_iff
thf(fact_4690_subset__CollectI,axiom,
! [A: $tType,B5: set @ A,A3: set @ A,Q2: A > $o,P3: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ B5 @ A3 )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ B5 )
=> ( ( Q2 @ X5 )
=> ( P3 @ X5 ) ) )
=> ( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ B5 )
& ( Q2 @ X4 ) ) )
@ ( collect @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ A3 )
& ( P3 @ X4 ) ) ) ) ) ) ).
% subset_CollectI
thf(fact_4691_rec__nat__Suc__imp,axiom,
! [A: $tType,F3: nat > A,F1: A,F2: nat > A > A,N: nat] :
( ( F3
= ( rec_nat @ A @ F1 @ F2 ) )
=> ( ( F3 @ ( suc @ N ) )
= ( F2 @ N @ ( F3 @ N ) ) ) ) ).
% rec_nat_Suc_imp
thf(fact_4692_nat_Osplit__sels_I2_J,axiom,
! [A: $tType,P3: A > $o,F1: A,F2: nat > A,Nat2: nat] :
( ( P3 @ ( case_nat @ A @ F1 @ F2 @ Nat2 ) )
= ( ~ ( ( ( Nat2
= ( zero_zero @ nat ) )
& ~ ( P3 @ F1 ) )
| ( ( Nat2
= ( suc @ ( pred @ Nat2 ) ) )
& ~ ( P3 @ ( F2 @ ( pred @ Nat2 ) ) ) ) ) ) ) ).
% nat.split_sels(2)
thf(fact_4693_nat_Osplit__sels_I1_J,axiom,
! [A: $tType,P3: A > $o,F1: A,F2: nat > A,Nat2: nat] :
( ( P3 @ ( case_nat @ A @ F1 @ F2 @ Nat2 ) )
= ( ( ( Nat2
= ( zero_zero @ nat ) )
=> ( P3 @ F1 ) )
& ( ( Nat2
= ( suc @ ( pred @ Nat2 ) ) )
=> ( P3 @ ( F2 @ ( pred @ Nat2 ) ) ) ) ) ) ).
% nat.split_sels(1)
thf(fact_4694_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_4695_or_Oright__idem,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A,B2: A] :
( ( bit_se1065995026697491101ons_or @ A @ ( bit_se1065995026697491101ons_or @ A @ A2 @ B2 ) @ B2 )
= ( bit_se1065995026697491101ons_or @ A @ A2 @ B2 ) ) ) ).
% or.right_idem
thf(fact_4696_or_Oleft__idem,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A,B2: A] :
( ( bit_se1065995026697491101ons_or @ A @ A2 @ ( bit_se1065995026697491101ons_or @ A @ A2 @ B2 ) )
= ( bit_se1065995026697491101ons_or @ A @ A2 @ B2 ) ) ) ).
% or.left_idem
thf(fact_4697_or_Oidem,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A] :
( ( bit_se1065995026697491101ons_or @ A @ A2 @ A2 )
= A2 ) ) ).
% or.idem
thf(fact_4698_or_Oright__neutral,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A] :
( ( bit_se1065995026697491101ons_or @ A @ A2 @ ( zero_zero @ A ) )
= A2 ) ) ).
% or.right_neutral
thf(fact_4699_or_Oleft__neutral,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A] :
( ( bit_se1065995026697491101ons_or @ A @ ( zero_zero @ A ) @ A2 )
= A2 ) ) ).
% or.left_neutral
thf(fact_4700_take__bit__or,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A2: A,B2: A] :
( ( bit_se2584673776208193580ke_bit @ A @ N @ ( bit_se1065995026697491101ons_or @ A @ A2 @ B2 ) )
= ( bit_se1065995026697491101ons_or @ A @ ( bit_se2584673776208193580ke_bit @ A @ N @ A2 ) @ ( bit_se2584673776208193580ke_bit @ A @ N @ B2 ) ) ) ) ).
% take_bit_or
thf(fact_4701_push__bit__or,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A2: A,B2: A] :
( ( bit_se4730199178511100633sh_bit @ A @ N @ ( bit_se1065995026697491101ons_or @ A @ A2 @ B2 ) )
= ( bit_se1065995026697491101ons_or @ A @ ( bit_se4730199178511100633sh_bit @ A @ N @ A2 ) @ ( bit_se4730199178511100633sh_bit @ A @ N @ B2 ) ) ) ) ).
% push_bit_or
thf(fact_4702_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_4703_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_4704_or__nonnegative__int__iff,axiom,
! [K2: int,L: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se1065995026697491101ons_or @ int @ K2 @ L ) )
= ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K2 )
& ( ord_less_eq @ int @ ( zero_zero @ int ) @ L ) ) ) ).
% or_nonnegative_int_iff
thf(fact_4705_or__negative__int__iff,axiom,
! [K2: int,L: int] :
( ( ord_less @ int @ ( bit_se1065995026697491101ons_or @ int @ K2 @ L ) @ ( zero_zero @ int ) )
= ( ( ord_less @ int @ K2 @ ( zero_zero @ int ) )
| ( ord_less @ int @ L @ ( zero_zero @ int ) ) ) ) ).
% or_negative_int_iff
thf(fact_4706_bit_Ode__Morgan__conj,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X: A,Y2: A] :
( ( bit_ri4277139882892585799ns_not @ A @ ( bit_se5824344872417868541ns_and @ A @ X @ Y2 ) )
= ( bit_se1065995026697491101ons_or @ A @ ( bit_ri4277139882892585799ns_not @ A @ X ) @ ( bit_ri4277139882892585799ns_not @ A @ Y2 ) ) ) ) ).
% bit.de_Morgan_conj
thf(fact_4707_bit_Ode__Morgan__disj,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X: A,Y2: A] :
( ( bit_ri4277139882892585799ns_not @ A @ ( bit_se1065995026697491101ons_or @ A @ X @ Y2 ) )
= ( bit_se5824344872417868541ns_and @ A @ ( bit_ri4277139882892585799ns_not @ A @ X ) @ ( bit_ri4277139882892585799ns_not @ A @ Y2 ) ) ) ) ).
% bit.de_Morgan_disj
thf(fact_4708_or__numerals_I2_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [Y2: num] :
( ( bit_se1065995026697491101ons_or @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit1 @ Y2 ) ) )
= ( numeral_numeral @ A @ ( bit1 @ Y2 ) ) ) ) ).
% or_numerals(2)
thf(fact_4709_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_4710_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_4711_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_4712_or__numerals_I3_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X: num,Y2: num] :
( ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ ( bit0 @ X ) ) @ ( numeral_numeral @ A @ ( bit0 @ Y2 ) ) )
= ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ X ) @ ( numeral_numeral @ A @ Y2 ) ) ) ) ) ).
% or_numerals(3)
thf(fact_4713_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_4714_or__numerals_I1_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [Y2: num] :
( ( bit_se1065995026697491101ons_or @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ Y2 ) ) )
= ( numeral_numeral @ A @ ( bit1 @ Y2 ) ) ) ) ).
% or_numerals(1)
thf(fact_4715_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_4716_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_4717_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_4718_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_4719_or__numerals_I7_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X: num,Y2: num] :
( ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ ( bit1 @ X ) ) @ ( numeral_numeral @ A @ ( bit1 @ Y2 ) ) )
= ( 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 @ Y2 ) ) ) ) ) ) ).
% or_numerals(7)
thf(fact_4720_or__numerals_I6_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X: num,Y2: num] :
( ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ ( bit1 @ X ) ) @ ( numeral_numeral @ A @ ( bit0 @ Y2 ) ) )
= ( 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 @ Y2 ) ) ) ) ) ) ).
% or_numerals(6)
thf(fact_4721_or__numerals_I4_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X: num,Y2: num] :
( ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ ( bit0 @ X ) ) @ ( numeral_numeral @ A @ ( bit1 @ Y2 ) ) )
= ( 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 @ Y2 ) ) ) ) ) ) ).
% or_numerals(4)
thf(fact_4722_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_4723_or__eq__0__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A,B2: A] :
( ( ( bit_se1065995026697491101ons_or @ A @ A2 @ B2 )
= ( zero_zero @ A ) )
= ( ( A2
= ( zero_zero @ A ) )
& ( B2
= ( zero_zero @ A ) ) ) ) ) ).
% or_eq_0_iff
thf(fact_4724_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_4725_of__int__or__eq,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [K2: int,L: int] :
( ( ring_1_of_int @ A @ ( bit_se1065995026697491101ons_or @ int @ K2 @ L ) )
= ( bit_se1065995026697491101ons_or @ A @ ( ring_1_of_int @ A @ K2 ) @ ( ring_1_of_int @ A @ L ) ) ) ) ).
% of_int_or_eq
thf(fact_4726_bit__or__int__iff,axiom,
! [K2: int,L: int,N: nat] :
( ( bit_se5641148757651400278ts_bit @ int @ ( bit_se1065995026697491101ons_or @ int @ K2 @ L ) @ N )
= ( ( bit_se5641148757651400278ts_bit @ int @ K2 @ N )
| ( bit_se5641148757651400278ts_bit @ int @ L @ N ) ) ) ).
% bit_or_int_iff
thf(fact_4727_bit_Oconj__disj__distrib,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X: A,Y2: A,Z3: A] :
( ( bit_se5824344872417868541ns_and @ A @ X @ ( bit_se1065995026697491101ons_or @ A @ Y2 @ Z3 ) )
= ( bit_se1065995026697491101ons_or @ A @ ( bit_se5824344872417868541ns_and @ A @ X @ Y2 ) @ ( bit_se5824344872417868541ns_and @ A @ X @ Z3 ) ) ) ) ).
% bit.conj_disj_distrib
thf(fact_4728_bit_Odisj__conj__distrib,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X: A,Y2: A,Z3: A] :
( ( bit_se1065995026697491101ons_or @ A @ X @ ( bit_se5824344872417868541ns_and @ A @ Y2 @ Z3 ) )
= ( bit_se5824344872417868541ns_and @ A @ ( bit_se1065995026697491101ons_or @ A @ X @ Y2 ) @ ( bit_se1065995026697491101ons_or @ A @ X @ Z3 ) ) ) ) ).
% bit.disj_conj_distrib
thf(fact_4729_bit_Oconj__disj__distrib2,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [Y2: A,Z3: A,X: A] :
( ( bit_se5824344872417868541ns_and @ A @ ( bit_se1065995026697491101ons_or @ A @ Y2 @ Z3 ) @ X )
= ( bit_se1065995026697491101ons_or @ A @ ( bit_se5824344872417868541ns_and @ A @ Y2 @ X ) @ ( bit_se5824344872417868541ns_and @ A @ Z3 @ X ) ) ) ) ).
% bit.conj_disj_distrib2
thf(fact_4730_bit_Odisj__conj__distrib2,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [Y2: A,Z3: A,X: A] :
( ( bit_se1065995026697491101ons_or @ A @ ( bit_se5824344872417868541ns_and @ A @ Y2 @ Z3 ) @ X )
= ( bit_se5824344872417868541ns_and @ A @ ( bit_se1065995026697491101ons_or @ A @ Y2 @ X ) @ ( bit_se1065995026697491101ons_or @ A @ Z3 @ X ) ) ) ) ).
% bit.disj_conj_distrib2
thf(fact_4731_bit__or__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A,B2: A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( bit_se1065995026697491101ons_or @ A @ A2 @ B2 ) @ N )
= ( ( bit_se5641148757651400278ts_bit @ A @ A2 @ N )
| ( bit_se5641148757651400278ts_bit @ A @ B2 @ N ) ) ) ) ).
% bit_or_iff
thf(fact_4732_or_Oleft__commute,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( bit_se1065995026697491101ons_or @ A @ B2 @ ( bit_se1065995026697491101ons_or @ A @ A2 @ C2 ) )
= ( bit_se1065995026697491101ons_or @ A @ A2 @ ( bit_se1065995026697491101ons_or @ A @ B2 @ C2 ) ) ) ) ).
% or.left_commute
thf(fact_4733_or_Ocommute,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se1065995026697491101ons_or @ A )
= ( ^ [A5: A,B4: A] : ( bit_se1065995026697491101ons_or @ A @ B4 @ A5 ) ) ) ) ).
% or.commute
thf(fact_4734_or_Oassoc,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( bit_se1065995026697491101ons_or @ A @ ( bit_se1065995026697491101ons_or @ A @ A2 @ B2 ) @ C2 )
= ( bit_se1065995026697491101ons_or @ A @ A2 @ ( bit_se1065995026697491101ons_or @ A @ B2 @ C2 ) ) ) ) ).
% or.assoc
thf(fact_4735_OR__lower,axiom,
! [X: int,Y2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y2 )
=> ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se1065995026697491101ons_or @ int @ X @ Y2 ) ) ) ) ).
% OR_lower
thf(fact_4736_or__greater__eq,axiom,
! [L: int,K2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ L )
=> ( ord_less_eq @ int @ K2 @ ( bit_se1065995026697491101ons_or @ int @ K2 @ L ) ) ) ).
% or_greater_eq
thf(fact_4737_disjunctive__add,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A,B2: A] :
( ! [N3: nat] :
( ~ ( bit_se5641148757651400278ts_bit @ A @ A2 @ N3 )
| ~ ( bit_se5641148757651400278ts_bit @ A @ B2 @ N3 ) )
=> ( ( plus_plus @ A @ A2 @ B2 )
= ( bit_se1065995026697491101ons_or @ A @ A2 @ B2 ) ) ) ) ).
% disjunctive_add
thf(fact_4738_plus__and__or,axiom,
! [X: int,Y2: int] :
( ( plus_plus @ int @ ( bit_se5824344872417868541ns_and @ int @ X @ Y2 ) @ ( bit_se1065995026697491101ons_or @ int @ X @ Y2 ) )
= ( plus_plus @ int @ X @ Y2 ) ) ).
% plus_and_or
thf(fact_4739_and__eq__not__not__or,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( ( bit_se5824344872417868541ns_and @ A )
= ( ^ [A5: A,B4: A] : ( bit_ri4277139882892585799ns_not @ A @ ( bit_se1065995026697491101ons_or @ A @ ( bit_ri4277139882892585799ns_not @ A @ A5 ) @ ( bit_ri4277139882892585799ns_not @ A @ B4 ) ) ) ) ) ) ).
% and_eq_not_not_or
thf(fact_4740_or__eq__not__not__and,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( ( bit_se1065995026697491101ons_or @ A )
= ( ^ [A5: A,B4: A] : ( bit_ri4277139882892585799ns_not @ A @ ( bit_se5824344872417868541ns_and @ A @ ( bit_ri4277139882892585799ns_not @ A @ A5 ) @ ( bit_ri4277139882892585799ns_not @ A @ B4 ) ) ) ) ) ) ).
% or_eq_not_not_and
thf(fact_4741_or__int__def,axiom,
( ( bit_se1065995026697491101ons_or @ int )
= ( ^ [K3: int,L3: int] : ( bit_ri4277139882892585799ns_not @ int @ ( bit_se5824344872417868541ns_and @ int @ ( bit_ri4277139882892585799ns_not @ int @ K3 ) @ ( bit_ri4277139882892585799ns_not @ int @ L3 ) ) ) ) ) ).
% or_int_def
thf(fact_4742_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_4743_bit_Oxor__def,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( ( bit_se5824344971392196577ns_xor @ A )
= ( ^ [X4: A,Y: A] : ( bit_se1065995026697491101ons_or @ A @ ( bit_se5824344872417868541ns_and @ A @ X4 @ ( bit_ri4277139882892585799ns_not @ A @ Y ) ) @ ( bit_se5824344872417868541ns_and @ A @ ( bit_ri4277139882892585799ns_not @ A @ X4 ) @ Y ) ) ) ) ) ).
% bit.xor_def
thf(fact_4744_bit_Oxor__def2,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( ( bit_se5824344971392196577ns_xor @ A )
= ( ^ [X4: A,Y: A] : ( bit_se5824344872417868541ns_and @ A @ ( bit_se1065995026697491101ons_or @ A @ X4 @ Y ) @ ( bit_se1065995026697491101ons_or @ A @ ( bit_ri4277139882892585799ns_not @ A @ X4 ) @ ( bit_ri4277139882892585799ns_not @ A @ Y ) ) ) ) ) ) ).
% bit.xor_def2
thf(fact_4745_set__bit__eq__or,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se5668285175392031749et_bit @ A )
= ( ^ [N2: nat,A5: A] : ( bit_se1065995026697491101ons_or @ A @ A5 @ ( bit_se4730199178511100633sh_bit @ A @ N2 @ ( one_one @ A ) ) ) ) ) ) ).
% set_bit_eq_or
thf(fact_4746_xor__int__def,axiom,
( ( bit_se5824344971392196577ns_xor @ int )
= ( ^ [K3: int,L3: int] : ( bit_se1065995026697491101ons_or @ int @ ( bit_se5824344872417868541ns_and @ int @ K3 @ ( bit_ri4277139882892585799ns_not @ int @ L3 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( bit_ri4277139882892585799ns_not @ int @ K3 ) @ L3 ) ) ) ) ).
% xor_int_def
thf(fact_4747_concat__bit__def,axiom,
( bit_concat_bit
= ( ^ [N2: nat,K3: int,L3: int] : ( bit_se1065995026697491101ons_or @ int @ ( bit_se2584673776208193580ke_bit @ int @ N2 @ K3 ) @ ( bit_se4730199178511100633sh_bit @ int @ N2 @ L3 ) ) ) ) ).
% concat_bit_def
thf(fact_4748_set__bit__int__def,axiom,
( ( bit_se5668285175392031749et_bit @ int )
= ( ^ [N2: nat,K3: int] : ( bit_se1065995026697491101ons_or @ int @ K3 @ ( bit_se4730199178511100633sh_bit @ int @ N2 @ ( one_one @ int ) ) ) ) ) ).
% set_bit_int_def
thf(fact_4749_even__or__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A,B2: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ A @ A2 @ B2 ) )
= ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
& ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) ) ) ).
% even_or_iff
thf(fact_4750_bit_Ocomplement__unique,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [A2: A,X: A,Y2: A] :
( ( ( bit_se5824344872417868541ns_and @ A @ A2 @ X )
= ( zero_zero @ A ) )
=> ( ( ( bit_se1065995026697491101ons_or @ A @ A2 @ X )
= ( uminus_uminus @ A @ ( one_one @ A ) ) )
=> ( ( ( bit_se5824344872417868541ns_and @ A @ A2 @ Y2 )
= ( zero_zero @ A ) )
=> ( ( ( bit_se1065995026697491101ons_or @ A @ A2 @ Y2 )
= ( uminus_uminus @ A @ ( one_one @ A ) ) )
=> ( X = Y2 ) ) ) ) ) ) ).
% bit.complement_unique
thf(fact_4751_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_4752_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_4753_bit_Ocompl__unique,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X: A,Y2: A] :
( ( ( bit_se5824344872417868541ns_and @ A @ X @ Y2 )
= ( zero_zero @ A ) )
=> ( ( ( bit_se1065995026697491101ons_or @ A @ X @ Y2 )
= ( uminus_uminus @ A @ ( one_one @ A ) ) )
=> ( ( bit_ri4277139882892585799ns_not @ A @ X )
= Y2 ) ) ) ) ).
% bit.compl_unique
thf(fact_4754_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_4755_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_4756_signed__take__bit__eq__if__negative,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [A2: A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ A2 @ N )
=> ( ( bit_ri4674362597316999326ke_bit @ A @ N @ A2 )
= ( bit_se1065995026697491101ons_or @ A @ ( bit_se2584673776208193580ke_bit @ A @ N @ A2 ) @ ( bit_ri4277139882892585799ns_not @ A @ ( bit_se2239418461657761734s_mask @ A @ N ) ) ) ) ) ) ).
% signed_take_bit_eq_if_negative
thf(fact_4757_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_4758_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_4759_one__or__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A] :
( ( bit_se1065995026697491101ons_or @ A @ ( one_one @ A ) @ A2 )
= ( plus_plus @ A @ A2 @ ( zero_neq_one_of_bool @ A @ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) ) ) ) ) ).
% one_or_eq
thf(fact_4760_or__one__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A] :
( ( bit_se1065995026697491101ons_or @ A @ A2 @ ( one_one @ A ) )
= ( plus_plus @ A @ A2 @ ( zero_neq_one_of_bool @ A @ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) ) ) ) ) ).
% or_one_eq
thf(fact_4761_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_4762_OR__upper,axiom,
! [X: int,N: nat,Y2: 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 @ Y2 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) )
=> ( ord_less @ int @ ( bit_se1065995026697491101ons_or @ int @ X @ Y2 ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ) ) ) ).
% OR_upper
thf(fact_4763_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_4764_signed__take__bit__def,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( ( bit_ri4674362597316999326ke_bit @ A )
= ( ^ [N2: nat,A5: A] : ( bit_se1065995026697491101ons_or @ A @ ( bit_se2584673776208193580ke_bit @ A @ N2 @ A5 ) @ ( times_times @ A @ ( zero_neq_one_of_bool @ A @ ( bit_se5641148757651400278ts_bit @ A @ A5 @ N2 ) ) @ ( bit_ri4277139882892585799ns_not @ A @ ( bit_se2239418461657761734s_mask @ A @ N2 ) ) ) ) ) ) ) ).
% signed_take_bit_def
thf(fact_4765_pred__def,axiom,
( pred
= ( case_nat @ nat @ ( zero_zero @ nat )
@ ^ [X25: nat] : X25 ) ) ).
% pred_def
thf(fact_4766_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_4767_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_4768_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_4769_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_4770_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_4771_max__number__of_I1_J,axiom,
! [A: $tType] :
( ( ( numeral @ A )
& ( ord @ A ) )
=> ! [U: num,V2: num] :
( ( ( ord_less_eq @ A @ ( numeral_numeral @ A @ U ) @ ( numeral_numeral @ A @ V2 ) )
=> ( ( ord_max @ A @ ( numeral_numeral @ A @ U ) @ ( numeral_numeral @ A @ V2 ) )
= ( numeral_numeral @ A @ V2 ) ) )
& ( ~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ U ) @ ( numeral_numeral @ A @ V2 ) )
=> ( ( ord_max @ A @ ( numeral_numeral @ A @ U ) @ ( numeral_numeral @ A @ V2 ) )
= ( numeral_numeral @ A @ U ) ) ) ) ) ).
% max_number_of(1)
thf(fact_4772_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_4773_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_4774_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_4775_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_4776_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_4777_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_4778_max__number__of_I2_J,axiom,
! [A: $tType] :
( ( ( uminus @ A )
& ( numeral @ A )
& ( ord @ A ) )
=> ! [U: num,V2: num] :
( ( ( ord_less_eq @ A @ ( numeral_numeral @ A @ U ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) )
=> ( ( ord_max @ A @ ( numeral_numeral @ A @ U ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ U ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) )
=> ( ( ord_max @ A @ ( numeral_numeral @ A @ U ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) )
= ( numeral_numeral @ A @ U ) ) ) ) ) ).
% max_number_of(2)
thf(fact_4779_max__number__of_I3_J,axiom,
! [A: $tType] :
( ( ( uminus @ A )
& ( numeral @ A )
& ( ord @ A ) )
=> ! [U: num,V2: num] :
( ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( numeral_numeral @ A @ V2 ) )
=> ( ( ord_max @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( numeral_numeral @ A @ V2 ) )
= ( numeral_numeral @ A @ V2 ) ) )
& ( ~ ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( numeral_numeral @ A @ V2 ) )
=> ( ( ord_max @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( numeral_numeral @ A @ V2 ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) ) ) ) ) ).
% max_number_of(3)
thf(fact_4780_max__number__of_I4_J,axiom,
! [A: $tType] :
( ( ( uminus @ A )
& ( numeral @ A )
& ( ord @ A ) )
=> ! [U: num,V2: num] :
( ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) )
=> ( ( ord_max @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) )
=> ( ( ord_max @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) ) ) ) ) ).
% max_number_of(4)
thf(fact_4781_or__nat__numerals_I2_J,axiom,
! [Y2: num] :
( ( bit_se1065995026697491101ons_or @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( numeral_numeral @ nat @ ( bit1 @ Y2 ) ) )
= ( numeral_numeral @ nat @ ( bit1 @ Y2 ) ) ) ).
% or_nat_numerals(2)
thf(fact_4782_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_4783_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_4784_or__nat__numerals_I1_J,axiom,
! [Y2: num] :
( ( bit_se1065995026697491101ons_or @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( numeral_numeral @ nat @ ( bit0 @ Y2 ) ) )
= ( numeral_numeral @ nat @ ( bit1 @ Y2 ) ) ) ).
% or_nat_numerals(1)
thf(fact_4785_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_4786_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_4787_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_4788_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_4789_max__diff__distrib__left,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [X: A,Y2: A,Z3: A] :
( ( minus_minus @ A @ ( ord_max @ A @ X @ Y2 ) @ Z3 )
= ( ord_max @ A @ ( minus_minus @ A @ X @ Z3 ) @ ( minus_minus @ A @ Y2 @ Z3 ) ) ) ) ).
% max_diff_distrib_left
thf(fact_4790_max__absorb2,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X: A,Y2: A] :
( ( ord_less_eq @ A @ X @ Y2 )
=> ( ( ord_max @ A @ X @ Y2 )
= Y2 ) ) ) ).
% max_absorb2
thf(fact_4791_max__absorb1,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Y2: A,X: A] :
( ( ord_less_eq @ A @ Y2 @ X )
=> ( ( ord_max @ A @ X @ Y2 )
= X ) ) ) ).
% max_absorb1
thf(fact_4792_max__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( ord_max @ A )
= ( ^ [A5: A,B4: A] : ( if @ A @ ( ord_less_eq @ A @ A5 @ B4 ) @ B4 @ A5 ) ) ) ) ).
% max_def
thf(fact_4793_max__add__distrib__right,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [X: A,Y2: A,Z3: A] :
( ( plus_plus @ A @ X @ ( ord_max @ A @ Y2 @ Z3 ) )
= ( ord_max @ A @ ( plus_plus @ A @ X @ Y2 ) @ ( plus_plus @ A @ X @ Z3 ) ) ) ) ).
% max_add_distrib_right
thf(fact_4794_max__add__distrib__left,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [X: A,Y2: A,Z3: A] :
( ( plus_plus @ A @ ( ord_max @ A @ X @ Y2 ) @ Z3 )
= ( ord_max @ A @ ( plus_plus @ A @ X @ Z3 ) @ ( plus_plus @ A @ Y2 @ Z3 ) ) ) ) ).
% max_add_distrib_left
thf(fact_4795_or__not__num__neg_Osimps_I1_J,axiom,
( ( bit_or_not_num_neg @ one2 @ one2 )
= one2 ) ).
% or_not_num_neg.simps(1)
thf(fact_4796_of__nat__max,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [X: nat,Y2: nat] :
( ( semiring_1_of_nat @ A @ ( ord_max @ nat @ X @ Y2 ) )
= ( ord_max @ A @ ( semiring_1_of_nat @ A @ X ) @ ( semiring_1_of_nat @ A @ Y2 ) ) ) ) ).
% of_nat_max
thf(fact_4797_max__def__raw,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( ord_max @ A )
= ( ^ [A5: A,B4: A] : ( if @ A @ ( ord_less_eq @ A @ A5 @ B4 ) @ B4 @ A5 ) ) ) ) ).
% max_def_raw
thf(fact_4798_set__bit__nat__def,axiom,
( ( bit_se5668285175392031749et_bit @ nat )
= ( ^ [M2: nat,N2: nat] : ( bit_se1065995026697491101ons_or @ nat @ N2 @ ( bit_se4730199178511100633sh_bit @ nat @ M2 @ ( one_one @ nat ) ) ) ) ) ).
% set_bit_nat_def
thf(fact_4799_or__not__num__neg_Osimps_I4_J,axiom,
! [N: num] :
( ( bit_or_not_num_neg @ ( bit0 @ N ) @ one2 )
= ( bit0 @ one2 ) ) ).
% or_not_num_neg.simps(4)
thf(fact_4800_or__not__num__neg_Osimps_I6_J,axiom,
! [N: num,M: num] :
( ( bit_or_not_num_neg @ ( bit0 @ N ) @ ( bit1 @ M ) )
= ( bit0 @ ( bit_or_not_num_neg @ N @ M ) ) ) ).
% or_not_num_neg.simps(6)
thf(fact_4801_or__not__num__neg_Osimps_I3_J,axiom,
! [M: num] :
( ( bit_or_not_num_neg @ one2 @ ( bit1 @ M ) )
= ( bit1 @ M ) ) ).
% or_not_num_neg.simps(3)
thf(fact_4802_or__not__num__neg_Osimps_I7_J,axiom,
! [N: num] :
( ( bit_or_not_num_neg @ ( bit1 @ N ) @ one2 )
= one2 ) ).
% or_not_num_neg.simps(7)
thf(fact_4803_or__not__num__neg_Osimps_I5_J,axiom,
! [N: num,M: num] :
( ( bit_or_not_num_neg @ ( bit0 @ N ) @ ( bit0 @ M ) )
= ( bitM @ ( bit_or_not_num_neg @ N @ M ) ) ) ).
% or_not_num_neg.simps(5)
thf(fact_4804_or__not__num__neg_Osimps_I9_J,axiom,
! [N: num,M: num] :
( ( bit_or_not_num_neg @ ( bit1 @ N ) @ ( bit1 @ M ) )
= ( bitM @ ( bit_or_not_num_neg @ N @ M ) ) ) ).
% or_not_num_neg.simps(9)
thf(fact_4805_or__nat__def,axiom,
( ( bit_se1065995026697491101ons_or @ nat )
= ( ^ [M2: nat,N2: nat] : ( nat2 @ ( bit_se1065995026697491101ons_or @ int @ ( semiring_1_of_nat @ int @ M2 ) @ ( semiring_1_of_nat @ int @ N2 ) ) ) ) ) ).
% or_nat_def
thf(fact_4806_or__not__num__neg_Osimps_I2_J,axiom,
! [M: num] :
( ( bit_or_not_num_neg @ one2 @ ( bit0 @ M ) )
= ( bit1 @ M ) ) ).
% or_not_num_neg.simps(2)
thf(fact_4807_or__not__num__neg_Osimps_I8_J,axiom,
! [N: num,M: num] :
( ( bit_or_not_num_neg @ ( bit1 @ N ) @ ( bit0 @ M ) )
= ( bitM @ ( bit_or_not_num_neg @ N @ M ) ) ) ).
% or_not_num_neg.simps(8)
thf(fact_4808_or__not__num__neg_Oelims,axiom,
! [X: num,Xa: num,Y2: num] :
( ( ( bit_or_not_num_neg @ X @ Xa )
= Y2 )
=> ( ( ( X = one2 )
=> ( ( Xa = one2 )
=> ( Y2 != one2 ) ) )
=> ( ( ( X = one2 )
=> ! [M4: num] :
( ( Xa
= ( bit0 @ M4 ) )
=> ( Y2
!= ( bit1 @ M4 ) ) ) )
=> ( ( ( X = one2 )
=> ! [M4: num] :
( ( Xa
= ( bit1 @ M4 ) )
=> ( Y2
!= ( bit1 @ M4 ) ) ) )
=> ( ( ? [N3: num] :
( X
= ( bit0 @ N3 ) )
=> ( ( Xa = one2 )
=> ( Y2
!= ( bit0 @ one2 ) ) ) )
=> ( ! [N3: num] :
( ( X
= ( bit0 @ N3 ) )
=> ! [M4: num] :
( ( Xa
= ( bit0 @ M4 ) )
=> ( Y2
!= ( bitM @ ( bit_or_not_num_neg @ N3 @ M4 ) ) ) ) )
=> ( ! [N3: num] :
( ( X
= ( bit0 @ N3 ) )
=> ! [M4: num] :
( ( Xa
= ( bit1 @ M4 ) )
=> ( Y2
!= ( bit0 @ ( bit_or_not_num_neg @ N3 @ M4 ) ) ) ) )
=> ( ( ? [N3: num] :
( X
= ( bit1 @ N3 ) )
=> ( ( Xa = one2 )
=> ( Y2 != one2 ) ) )
=> ( ! [N3: num] :
( ( X
= ( bit1 @ N3 ) )
=> ! [M4: num] :
( ( Xa
= ( bit0 @ M4 ) )
=> ( Y2
!= ( bitM @ ( bit_or_not_num_neg @ N3 @ M4 ) ) ) ) )
=> ~ ! [N3: num] :
( ( X
= ( bit1 @ N3 ) )
=> ! [M4: num] :
( ( Xa
= ( bit1 @ M4 ) )
=> ( Y2
!= ( bitM @ ( bit_or_not_num_neg @ N3 @ M4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% or_not_num_neg.elims
thf(fact_4809_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_4810_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_4811_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_4812_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_4813_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_4814_or__nat__rec,axiom,
( ( bit_se1065995026697491101ons_or @ nat )
= ( ^ [M2: nat,N2: 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 ) ) @ N2 ) ) )
@ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ nat @ ( divide_divide @ nat @ M2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% or_nat_rec
thf(fact_4815_max__less__iff__conj,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y2: A,Z3: A] :
( ( ord_less @ A @ ( ord_max @ A @ X @ Y2 ) @ Z3 )
= ( ( ord_less @ A @ X @ Z3 )
& ( ord_less @ A @ Y2 @ Z3 ) ) ) ) ).
% max_less_iff_conj
thf(fact_4816_max_Oabsorb4,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_max @ A @ A2 @ B2 )
= B2 ) ) ) ).
% max.absorb4
thf(fact_4817_max_Oabsorb3,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,A2: A] :
( ( ord_less @ A @ B2 @ A2 )
=> ( ( ord_max @ A @ A2 @ B2 )
= A2 ) ) ) ).
% max.absorb3
thf(fact_4818_max__enat__simps_I4_J,axiom,
! [Q: extended_enat] :
( ( ord_max @ extended_enat @ Q @ ( extend4730790105801354508finity @ extended_enat ) )
= ( extend4730790105801354508finity @ extended_enat ) ) ).
% max_enat_simps(4)
thf(fact_4819_max__enat__simps_I5_J,axiom,
! [Q: extended_enat] :
( ( ord_max @ extended_enat @ ( extend4730790105801354508finity @ extended_enat ) @ Q )
= ( extend4730790105801354508finity @ extended_enat ) ) ).
% max_enat_simps(5)
thf(fact_4820_max__enat__simps_I2_J,axiom,
! [Q: extended_enat] :
( ( ord_max @ extended_enat @ Q @ ( zero_zero @ extended_enat ) )
= Q ) ).
% max_enat_simps(2)
thf(fact_4821_max__enat__simps_I3_J,axiom,
! [Q: extended_enat] :
( ( ord_max @ extended_enat @ ( zero_zero @ extended_enat ) @ Q )
= Q ) ).
% max_enat_simps(3)
thf(fact_4822_max_Oabsorb1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,A2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_max @ A @ A2 @ B2 )
= A2 ) ) ) ).
% max.absorb1
thf(fact_4823_max_Oabsorb2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_max @ A @ A2 @ B2 )
= B2 ) ) ) ).
% max.absorb2
thf(fact_4824_max_Obounded__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,C2: A,A2: A] :
( ( ord_less_eq @ A @ ( ord_max @ A @ B2 @ C2 ) @ A2 )
= ( ( ord_less_eq @ A @ B2 @ A2 )
& ( ord_less_eq @ A @ C2 @ A2 ) ) ) ) ).
% max.bounded_iff
thf(fact_4825_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_4826_max__0R,axiom,
! [N: nat] :
( ( ord_max @ nat @ N @ ( zero_zero @ nat ) )
= N ) ).
% max_0R
thf(fact_4827_max__0L,axiom,
! [N: nat] :
( ( ord_max @ nat @ ( zero_zero @ nat ) @ N )
= N ) ).
% max_0L
thf(fact_4828_max__nat_Oright__neutral,axiom,
! [A2: nat] :
( ( ord_max @ nat @ A2 @ ( zero_zero @ nat ) )
= A2 ) ).
% max_nat.right_neutral
thf(fact_4829_max__nat_Oneutr__eq__iff,axiom,
! [A2: nat,B2: nat] :
( ( ( zero_zero @ nat )
= ( ord_max @ nat @ A2 @ B2 ) )
= ( ( A2
= ( zero_zero @ nat ) )
& ( B2
= ( zero_zero @ nat ) ) ) ) ).
% max_nat.neutr_eq_iff
thf(fact_4830_max__nat_Oleft__neutral,axiom,
! [A2: nat] :
( ( ord_max @ nat @ ( zero_zero @ nat ) @ A2 )
= A2 ) ).
% max_nat.left_neutral
thf(fact_4831_max__nat_Oeq__neutr__iff,axiom,
! [A2: nat,B2: nat] :
( ( ( ord_max @ nat @ A2 @ B2 )
= ( zero_zero @ nat ) )
= ( ( A2
= ( zero_zero @ nat ) )
& ( B2
= ( zero_zero @ nat ) ) ) ) ).
% max_nat.eq_neutr_iff
thf(fact_4832_max__enat__simps_I1_J,axiom,
! [M: nat,N: nat] :
( ( ord_max @ extended_enat @ ( extended_enat2 @ M ) @ ( extended_enat2 @ N ) )
= ( extended_enat2 @ ( ord_max @ nat @ M @ N ) ) ) ).
% max_enat_simps(1)
thf(fact_4833_max__Suc__numeral,axiom,
! [N: nat,K2: num] :
( ( ord_max @ nat @ ( suc @ N ) @ ( numeral_numeral @ nat @ K2 ) )
= ( suc @ ( ord_max @ nat @ N @ ( pred_numeral @ K2 ) ) ) ) ).
% max_Suc_numeral
thf(fact_4834_max__numeral__Suc,axiom,
! [K2: num,N: nat] :
( ( ord_max @ nat @ ( numeral_numeral @ nat @ K2 ) @ ( suc @ N ) )
= ( suc @ ( ord_max @ nat @ ( pred_numeral @ K2 ) @ N ) ) ) ).
% max_numeral_Suc
thf(fact_4835_nat__mult__max__left,axiom,
! [M: nat,N: nat,Q: nat] :
( ( times_times @ nat @ ( ord_max @ nat @ M @ N ) @ Q )
= ( ord_max @ nat @ ( times_times @ nat @ M @ Q ) @ ( times_times @ nat @ N @ Q ) ) ) ).
% nat_mult_max_left
thf(fact_4836_nat__mult__max__right,axiom,
! [M: nat,N: nat,Q: nat] :
( ( times_times @ nat @ M @ ( ord_max @ nat @ N @ Q ) )
= ( ord_max @ nat @ ( times_times @ nat @ M @ N ) @ ( times_times @ nat @ M @ Q ) ) ) ).
% nat_mult_max_right
thf(fact_4837_nat__add__max__left,axiom,
! [M: nat,N: nat,Q: nat] :
( ( plus_plus @ nat @ ( ord_max @ nat @ M @ N ) @ Q )
= ( ord_max @ nat @ ( plus_plus @ nat @ M @ Q ) @ ( plus_plus @ nat @ N @ Q ) ) ) ).
% nat_add_max_left
thf(fact_4838_nat__add__max__right,axiom,
! [M: nat,N: nat,Q: nat] :
( ( plus_plus @ nat @ M @ ( ord_max @ nat @ N @ Q ) )
= ( ord_max @ nat @ ( plus_plus @ nat @ M @ N ) @ ( plus_plus @ nat @ M @ Q ) ) ) ).
% nat_add_max_right
thf(fact_4839_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_4840_max__Suc1,axiom,
! [N: nat,M: nat] :
( ( ord_max @ nat @ ( suc @ N ) @ M )
= ( case_nat @ nat @ ( suc @ N )
@ ^ [M5: nat] : ( suc @ ( ord_max @ nat @ N @ M5 ) )
@ M ) ) ).
% max_Suc1
thf(fact_4841_max__Suc2,axiom,
! [M: nat,N: nat] :
( ( ord_max @ nat @ M @ ( suc @ N ) )
= ( case_nat @ nat @ ( suc @ N )
@ ^ [M5: nat] : ( suc @ ( ord_max @ nat @ M5 @ N ) )
@ M ) ) ).
% max_Suc2
thf(fact_4842_max_Omono,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C2: A,A2: A,D2: A,B2: A] :
( ( ord_less_eq @ A @ C2 @ A2 )
=> ( ( ord_less_eq @ A @ D2 @ B2 )
=> ( ord_less_eq @ A @ ( ord_max @ A @ C2 @ D2 ) @ ( ord_max @ A @ A2 @ B2 ) ) ) ) ) ).
% max.mono
thf(fact_4843_max_OorderE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,A2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( A2
= ( ord_max @ A @ A2 @ B2 ) ) ) ) ).
% max.orderE
thf(fact_4844_max_OorderI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A] :
( ( A2
= ( ord_max @ A @ A2 @ B2 ) )
=> ( ord_less_eq @ A @ B2 @ A2 ) ) ) ).
% max.orderI
thf(fact_4845_max_OboundedE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,C2: A,A2: A] :
( ( ord_less_eq @ A @ ( ord_max @ A @ B2 @ C2 ) @ A2 )
=> ~ ( ( ord_less_eq @ A @ B2 @ A2 )
=> ~ ( ord_less_eq @ A @ C2 @ A2 ) ) ) ) ).
% max.boundedE
thf(fact_4846_max_OboundedI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_less_eq @ A @ C2 @ A2 )
=> ( ord_less_eq @ A @ ( ord_max @ A @ B2 @ C2 ) @ A2 ) ) ) ) ).
% max.boundedI
thf(fact_4847_max_Oorder__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B4: A,A5: A] :
( A5
= ( ord_max @ A @ A5 @ B4 ) ) ) ) ) ).
% max.order_iff
thf(fact_4848_max_Ocobounded1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A] : ( ord_less_eq @ A @ A2 @ ( ord_max @ A @ A2 @ B2 ) ) ) ).
% max.cobounded1
thf(fact_4849_max_Ocobounded2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,A2: A] : ( ord_less_eq @ A @ B2 @ ( ord_max @ A @ A2 @ B2 ) ) ) ).
% max.cobounded2
thf(fact_4850_le__max__iff__disj,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Z3: A,X: A,Y2: A] :
( ( ord_less_eq @ A @ Z3 @ ( ord_max @ A @ X @ Y2 ) )
= ( ( ord_less_eq @ A @ Z3 @ X )
| ( ord_less_eq @ A @ Z3 @ Y2 ) ) ) ) ).
% le_max_iff_disj
thf(fact_4851_max_Oabsorb__iff1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B4: A,A5: A] :
( ( ord_max @ A @ A5 @ B4 )
= A5 ) ) ) ) ).
% max.absorb_iff1
thf(fact_4852_max_Oabsorb__iff2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A5: A,B4: A] :
( ( ord_max @ A @ A5 @ B4 )
= B4 ) ) ) ) ).
% max.absorb_iff2
thf(fact_4853_max_OcoboundedI1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less_eq @ A @ C2 @ A2 )
=> ( ord_less_eq @ A @ C2 @ ( ord_max @ A @ A2 @ B2 ) ) ) ) ).
% max.coboundedI1
thf(fact_4854_max_OcoboundedI2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C2: A,B2: A,A2: A] :
( ( ord_less_eq @ A @ C2 @ B2 )
=> ( ord_less_eq @ A @ C2 @ ( ord_max @ A @ A2 @ B2 ) ) ) ) ).
% max.coboundedI2
thf(fact_4855_less__max__iff__disj,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Z3: A,X: A,Y2: A] :
( ( ord_less @ A @ Z3 @ ( ord_max @ A @ X @ Y2 ) )
= ( ( ord_less @ A @ Z3 @ X )
| ( ord_less @ A @ Z3 @ Y2 ) ) ) ) ).
% less_max_iff_disj
thf(fact_4856_max_Ostrict__boundedE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,C2: A,A2: A] :
( ( ord_less @ A @ ( ord_max @ A @ B2 @ C2 ) @ A2 )
=> ~ ( ( ord_less @ A @ B2 @ A2 )
=> ~ ( ord_less @ A @ C2 @ A2 ) ) ) ) ).
% max.strict_boundedE
thf(fact_4857_max_Ostrict__order__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less @ A )
= ( ^ [B4: A,A5: A] :
( ( A5
= ( ord_max @ A @ A5 @ B4 ) )
& ( A5 != B4 ) ) ) ) ) ).
% max.strict_order_iff
thf(fact_4858_max_Ostrict__coboundedI1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less @ A @ C2 @ A2 )
=> ( ord_less @ A @ C2 @ ( ord_max @ A @ A2 @ B2 ) ) ) ) ).
% max.strict_coboundedI1
thf(fact_4859_max_Ostrict__coboundedI2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C2: A,B2: A,A2: A] :
( ( ord_less @ A @ C2 @ B2 )
=> ( ord_less @ A @ C2 @ ( ord_max @ A @ A2 @ B2 ) ) ) ) ).
% max.strict_coboundedI2
thf(fact_4860_or__nat__unfold,axiom,
( ( bit_se1065995026697491101ons_or @ nat )
= ( ^ [M2: nat,N2: nat] :
( if @ nat
@ ( M2
= ( zero_zero @ nat ) )
@ N2
@ ( if @ nat
@ ( N2
= ( zero_zero @ nat ) )
@ M2
@ ( plus_plus @ nat @ ( ord_max @ nat @ ( modulo_modulo @ nat @ M2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ nat @ N2 @ ( 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 @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% or_nat_unfold
thf(fact_4861_or__not__num__neg_Opelims,axiom,
! [X: num,Xa: num,Y2: num] :
( ( ( bit_or_not_num_neg @ X @ Xa )
= Y2 )
=> ( ( accp @ ( product_prod @ num @ num ) @ bit_or3848514188828904588eg_rel @ ( product_Pair @ num @ num @ X @ Xa ) )
=> ( ( ( X = one2 )
=> ( ( Xa = one2 )
=> ( ( Y2 = one2 )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_or3848514188828904588eg_rel @ ( product_Pair @ num @ num @ one2 @ one2 ) ) ) ) )
=> ( ( ( X = one2 )
=> ! [M4: num] :
( ( Xa
= ( bit0 @ M4 ) )
=> ( ( Y2
= ( bit1 @ M4 ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_or3848514188828904588eg_rel @ ( product_Pair @ num @ num @ one2 @ ( bit0 @ M4 ) ) ) ) ) )
=> ( ( ( X = one2 )
=> ! [M4: num] :
( ( Xa
= ( bit1 @ M4 ) )
=> ( ( Y2
= ( bit1 @ M4 ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_or3848514188828904588eg_rel @ ( product_Pair @ num @ num @ one2 @ ( bit1 @ M4 ) ) ) ) ) )
=> ( ! [N3: num] :
( ( X
= ( bit0 @ N3 ) )
=> ( ( Xa = one2 )
=> ( ( Y2
= ( bit0 @ one2 ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_or3848514188828904588eg_rel @ ( product_Pair @ num @ num @ ( bit0 @ N3 ) @ one2 ) ) ) ) )
=> ( ! [N3: num] :
( ( X
= ( bit0 @ N3 ) )
=> ! [M4: num] :
( ( Xa
= ( bit0 @ M4 ) )
=> ( ( Y2
= ( bitM @ ( bit_or_not_num_neg @ N3 @ M4 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_or3848514188828904588eg_rel @ ( product_Pair @ num @ num @ ( bit0 @ N3 ) @ ( bit0 @ M4 ) ) ) ) ) )
=> ( ! [N3: num] :
( ( X
= ( bit0 @ N3 ) )
=> ! [M4: num] :
( ( Xa
= ( bit1 @ M4 ) )
=> ( ( Y2
= ( bit0 @ ( bit_or_not_num_neg @ N3 @ M4 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_or3848514188828904588eg_rel @ ( product_Pair @ num @ num @ ( bit0 @ N3 ) @ ( bit1 @ M4 ) ) ) ) ) )
=> ( ! [N3: num] :
( ( X
= ( bit1 @ N3 ) )
=> ( ( Xa = one2 )
=> ( ( Y2 = one2 )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_or3848514188828904588eg_rel @ ( product_Pair @ num @ num @ ( bit1 @ N3 ) @ one2 ) ) ) ) )
=> ( ! [N3: num] :
( ( X
= ( bit1 @ N3 ) )
=> ! [M4: num] :
( ( Xa
= ( bit0 @ M4 ) )
=> ( ( Y2
= ( bitM @ ( bit_or_not_num_neg @ N3 @ M4 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_or3848514188828904588eg_rel @ ( product_Pair @ num @ num @ ( bit1 @ N3 ) @ ( bit0 @ M4 ) ) ) ) ) )
=> ~ ! [N3: num] :
( ( X
= ( bit1 @ N3 ) )
=> ! [M4: num] :
( ( Xa
= ( bit1 @ M4 ) )
=> ( ( Y2
= ( bitM @ ( bit_or_not_num_neg @ N3 @ M4 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_or3848514188828904588eg_rel @ ( product_Pair @ num @ num @ ( bit1 @ N3 ) @ ( bit1 @ M4 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% or_not_num_neg.pelims
thf(fact_4862_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_4863_prod__decode__aux_Oelims,axiom,
! [X: nat,Xa: nat,Y2: product_prod @ nat @ nat] :
( ( ( nat_prod_decode_aux @ X @ Xa )
= Y2 )
=> ( ( ( ord_less_eq @ nat @ Xa @ X )
=> ( Y2
= ( product_Pair @ nat @ nat @ Xa @ ( minus_minus @ nat @ X @ Xa ) ) ) )
& ( ~ ( ord_less_eq @ nat @ Xa @ X )
=> ( Y2
= ( nat_prod_decode_aux @ ( suc @ X ) @ ( minus_minus @ nat @ Xa @ ( suc @ X ) ) ) ) ) ) ) ).
% prod_decode_aux.elims
thf(fact_4864_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_4865_prod__decode__aux_Opelims,axiom,
! [X: nat,Xa: nat,Y2: product_prod @ nat @ nat] :
( ( ( nat_prod_decode_aux @ X @ Xa )
= Y2 )
=> ( ( accp @ ( product_prod @ nat @ nat ) @ nat_pr5047031295181774490ux_rel @ ( product_Pair @ nat @ nat @ X @ Xa ) )
=> ~ ( ( ( ( ord_less_eq @ nat @ Xa @ X )
=> ( Y2
= ( product_Pair @ nat @ nat @ Xa @ ( minus_minus @ nat @ X @ Xa ) ) ) )
& ( ~ ( ord_less_eq @ nat @ Xa @ X )
=> ( Y2
= ( nat_prod_decode_aux @ ( suc @ X ) @ ( minus_minus @ nat @ Xa @ ( suc @ X ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ nat @ nat ) @ nat_pr5047031295181774490ux_rel @ ( product_Pair @ nat @ nat @ X @ Xa ) ) ) ) ) ).
% prod_decode_aux.pelims
thf(fact_4866_bezw__0,axiom,
! [X: nat] :
( ( bezw @ X @ ( zero_zero @ nat ) )
= ( product_Pair @ int @ int @ ( one_one @ int ) @ ( zero_zero @ int ) ) ) ).
% bezw_0
thf(fact_4867_sum__zero__power_H,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A3: set @ nat,C2: nat > A,D2: nat > A] :
( ( ( ( finite_finite2 @ nat @ A3 )
& ( member @ nat @ ( zero_zero @ nat ) @ A3 ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I: nat] : ( divide_divide @ A @ ( times_times @ A @ ( C2 @ I ) @ ( power_power @ A @ ( zero_zero @ A ) @ I ) ) @ ( D2 @ I ) )
@ A3 )
= ( divide_divide @ A @ ( C2 @ ( zero_zero @ nat ) ) @ ( D2 @ ( zero_zero @ nat ) ) ) ) )
& ( ~ ( ( finite_finite2 @ nat @ A3 )
& ( member @ nat @ ( zero_zero @ nat ) @ A3 ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I: nat] : ( divide_divide @ A @ ( times_times @ A @ ( C2 @ I ) @ ( power_power @ A @ ( zero_zero @ A ) @ I ) ) @ ( D2 @ I ) )
@ A3 )
= ( zero_zero @ A ) ) ) ) ) ).
% sum_zero_power'
thf(fact_4868_List_Ofinite__set,axiom,
! [A: $tType,Xs: list @ A] : ( finite_finite2 @ A @ ( set2 @ A @ Xs ) ) ).
% List.finite_set
thf(fact_4869_infinite__Icc__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A2: A,B2: A] :
( ( ~ ( finite_finite2 @ A @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) ) )
= ( ord_less @ A @ A2 @ B2 ) ) ) ).
% infinite_Icc_iff
thf(fact_4870_infinite__Ico__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A2: A,B2: A] :
( ( ~ ( finite_finite2 @ A @ ( set_or7035219750837199246ssThan @ A @ A2 @ B2 ) ) )
= ( ord_less @ A @ A2 @ B2 ) ) ) ).
% infinite_Ico_iff
thf(fact_4871_prod_Oinfinite,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: set @ B,G: B > A] :
( ~ ( finite_finite2 @ B @ A3 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ A3 )
= ( one_one @ A ) ) ) ) ).
% prod.infinite
thf(fact_4872_sum_Odelta_H,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ A )
=> ! [S3: set @ B,A2: B,B2: B > A] :
( ( finite_finite2 @ B @ S3 )
=> ( ( ( member @ B @ A2 @ S3 )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [K3: B] : ( if @ A @ ( A2 = K3 ) @ ( B2 @ K3 ) @ ( zero_zero @ A ) )
@ S3 )
= ( B2 @ A2 ) ) )
& ( ~ ( member @ B @ A2 @ S3 )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [K3: B] : ( if @ A @ ( A2 = K3 ) @ ( B2 @ K3 ) @ ( zero_zero @ A ) )
@ S3 )
= ( zero_zero @ A ) ) ) ) ) ) ).
% sum.delta'
thf(fact_4873_sum_Odelta,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ A )
=> ! [S3: set @ B,A2: B,B2: B > A] :
( ( finite_finite2 @ B @ S3 )
=> ( ( ( member @ B @ A2 @ S3 )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [K3: B] : ( if @ A @ ( K3 = A2 ) @ ( B2 @ K3 ) @ ( zero_zero @ A ) )
@ S3 )
= ( B2 @ A2 ) ) )
& ( ~ ( member @ B @ A2 @ S3 )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [K3: B] : ( if @ A @ ( K3 = A2 ) @ ( B2 @ K3 ) @ ( zero_zero @ A ) )
@ S3 )
= ( zero_zero @ A ) ) ) ) ) ) ).
% sum.delta
thf(fact_4874_prod__eq__1__iff,axiom,
! [A: $tType,A3: set @ A,F3: A > nat] :
( ( finite_finite2 @ A @ A3 )
=> ( ( ( groups7121269368397514597t_prod @ A @ nat @ F3 @ A3 )
= ( one_one @ nat ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ A3 )
=> ( ( F3 @ X4 )
= ( one_one @ nat ) ) ) ) ) ) ).
% prod_eq_1_iff
thf(fact_4875_prod_Odelta,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S3: set @ B,A2: B,B2: B > A] :
( ( finite_finite2 @ B @ S3 )
=> ( ( ( member @ B @ A2 @ S3 )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K3: B] : ( if @ A @ ( K3 = A2 ) @ ( B2 @ K3 ) @ ( one_one @ A ) )
@ S3 )
= ( B2 @ A2 ) ) )
& ( ~ ( member @ B @ A2 @ S3 )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K3: B] : ( if @ A @ ( K3 = A2 ) @ ( B2 @ K3 ) @ ( one_one @ A ) )
@ S3 )
= ( one_one @ A ) ) ) ) ) ) ).
% prod.delta
thf(fact_4876_prod_Odelta_H,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S3: set @ B,A2: B,B2: B > A] :
( ( finite_finite2 @ B @ S3 )
=> ( ( ( member @ B @ A2 @ S3 )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K3: B] : ( if @ A @ ( A2 = K3 ) @ ( B2 @ K3 ) @ ( one_one @ A ) )
@ S3 )
= ( B2 @ A2 ) ) )
& ( ~ ( member @ B @ A2 @ S3 )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K3: B] : ( if @ A @ ( A2 = K3 ) @ ( B2 @ K3 ) @ ( one_one @ A ) )
@ S3 )
= ( one_one @ A ) ) ) ) ) ) ).
% prod.delta'
thf(fact_4877_summable__If__finite,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [P3: nat > $o,F3: nat > A] :
( ( finite_finite2 @ nat @ ( collect @ nat @ P3 ) )
=> ( summable @ A
@ ^ [R5: nat] : ( if @ A @ ( P3 @ R5 ) @ ( F3 @ R5 ) @ ( zero_zero @ A ) ) ) ) ) ).
% summable_If_finite
thf(fact_4878_summable__If__finite__set,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [A3: set @ nat,F3: nat > A] :
( ( finite_finite2 @ nat @ A3 )
=> ( summable @ A
@ ^ [R5: nat] : ( if @ A @ ( member @ nat @ R5 @ A3 ) @ ( F3 @ R5 ) @ ( zero_zero @ A ) ) ) ) ) ).
% summable_If_finite_set
thf(fact_4879_prod__pos__nat__iff,axiom,
! [A: $tType,A3: set @ A,F3: A > nat] :
( ( finite_finite2 @ A @ A3 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( groups7121269368397514597t_prod @ A @ nat @ F3 @ A3 ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ A3 )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F3 @ X4 ) ) ) ) ) ) ).
% prod_pos_nat_iff
thf(fact_4880_sum__zero__power,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: set @ nat,C2: nat > A] :
( ( ( ( finite_finite2 @ nat @ A3 )
& ( member @ nat @ ( zero_zero @ nat ) @ A3 ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I: nat] : ( times_times @ A @ ( C2 @ I ) @ ( power_power @ A @ ( zero_zero @ A ) @ I ) )
@ A3 )
= ( C2 @ ( zero_zero @ nat ) ) ) )
& ( ~ ( ( finite_finite2 @ nat @ A3 )
& ( member @ nat @ ( zero_zero @ nat ) @ A3 ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I: nat] : ( times_times @ A @ ( C2 @ I ) @ ( power_power @ A @ ( zero_zero @ A ) @ I ) )
@ A3 )
= ( zero_zero @ A ) ) ) ) ) ).
% sum_zero_power
thf(fact_4881_finite__lists__length__eq,axiom,
! [A: $tType,A3: set @ A,N: nat] :
( ( finite_finite2 @ A @ A3 )
=> ( finite_finite2 @ ( list @ A )
@ ( collect @ ( list @ A )
@ ^ [Xs2: list @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs2 ) @ A3 )
& ( ( size_size @ ( list @ A ) @ Xs2 )
= N ) ) ) ) ) ).
% finite_lists_length_eq
thf(fact_4882_finite__less__ub,axiom,
! [F3: nat > nat,U: nat] :
( ! [N3: nat] : ( ord_less_eq @ nat @ N3 @ ( F3 @ N3 ) )
=> ( finite_finite2 @ nat
@ ( collect @ nat
@ ^ [N2: nat] : ( ord_less_eq @ nat @ ( F3 @ N2 ) @ U ) ) ) ) ).
% finite_less_ub
thf(fact_4883_sum_Oswap__restrict,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( comm_monoid_add @ A )
=> ! [A3: set @ B,B5: set @ C,G: B > C > A,R4: B > C > $o] :
( ( finite_finite2 @ B @ A3 )
=> ( ( finite_finite2 @ C @ B5 )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X4: B] :
( groups7311177749621191930dd_sum @ C @ A @ ( G @ X4 )
@ ( collect @ C
@ ^ [Y: C] :
( ( member @ C @ Y @ B5 )
& ( R4 @ X4 @ Y ) ) ) )
@ A3 )
= ( groups7311177749621191930dd_sum @ C @ A
@ ^ [Y: C] :
( groups7311177749621191930dd_sum @ B @ A
@ ^ [X4: B] : ( G @ X4 @ Y )
@ ( collect @ B
@ ^ [X4: B] :
( ( member @ B @ X4 @ A3 )
& ( R4 @ X4 @ Y ) ) ) )
@ B5 ) ) ) ) ) ).
% sum.swap_restrict
thf(fact_4884_finite__M__bounded__by__nat,axiom,
! [P3: nat > $o,I2: nat] :
( finite_finite2 @ nat
@ ( collect @ nat
@ ^ [K3: nat] :
( ( P3 @ K3 )
& ( ord_less @ nat @ K3 @ I2 ) ) ) ) ).
% finite_M_bounded_by_nat
thf(fact_4885_prod_Oswap__restrict,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: set @ B,B5: set @ C,G: B > C > A,R4: B > C > $o] :
( ( finite_finite2 @ B @ A3 )
=> ( ( finite_finite2 @ C @ B5 )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [X4: B] :
( groups7121269368397514597t_prod @ C @ A @ ( G @ X4 )
@ ( collect @ C
@ ^ [Y: C] :
( ( member @ C @ Y @ B5 )
& ( R4 @ X4 @ Y ) ) ) )
@ A3 )
= ( groups7121269368397514597t_prod @ C @ A
@ ^ [Y: C] :
( groups7121269368397514597t_prod @ B @ A
@ ^ [X4: B] : ( G @ X4 @ Y )
@ ( collect @ B
@ ^ [X4: B] :
( ( member @ B @ X4 @ A3 )
& ( R4 @ X4 @ Y ) ) ) )
@ B5 ) ) ) ) ) ).
% prod.swap_restrict
thf(fact_4886_finite__nat__set__iff__bounded,axiom,
( ( finite_finite2 @ nat )
= ( ^ [N6: set @ nat] :
? [M2: nat] :
! [X4: nat] :
( ( member @ nat @ X4 @ N6 )
=> ( ord_less @ nat @ X4 @ M2 ) ) ) ) ).
% finite_nat_set_iff_bounded
thf(fact_4887_bounded__nat__set__is__finite,axiom,
! [N4: set @ nat,N: nat] :
( ! [X5: nat] :
( ( member @ nat @ X5 @ N4 )
=> ( ord_less @ nat @ X5 @ N ) )
=> ( finite_finite2 @ nat @ N4 ) ) ).
% bounded_nat_set_is_finite
thf(fact_4888_finite__nat__set__iff__bounded__le,axiom,
( ( finite_finite2 @ nat )
= ( ^ [N6: set @ nat] :
? [M2: nat] :
! [X4: nat] :
( ( member @ nat @ X4 @ N6 )
=> ( ord_less_eq @ nat @ X4 @ M2 ) ) ) ) ).
% finite_nat_set_iff_bounded_le
thf(fact_4889_finite__list,axiom,
! [A: $tType,A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ? [Xs3: list @ A] :
( ( set2 @ A @ Xs3 )
= A3 ) ) ).
% finite_list
thf(fact_4890_sum__mono__inv,axiom,
! [A: $tType,I5: $tType] :
( ( ordere8940638589300402666id_add @ A )
=> ! [F3: I5 > A,I6: set @ I5,G: I5 > A,I2: I5] :
( ( ( groups7311177749621191930dd_sum @ I5 @ A @ F3 @ I6 )
= ( groups7311177749621191930dd_sum @ I5 @ A @ G @ I6 ) )
=> ( ! [I3: I5] :
( ( member @ I5 @ I3 @ I6 )
=> ( ord_less_eq @ A @ ( F3 @ I3 ) @ ( G @ I3 ) ) )
=> ( ( member @ I5 @ I2 @ I6 )
=> ( ( finite_finite2 @ I5 @ I6 )
=> ( ( F3 @ I2 )
= ( G @ I2 ) ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_4891_infinite__Icc,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ~ ( finite_finite2 @ A @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) ) ) ) ).
% infinite_Icc
thf(fact_4892_infinite__Ico,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ~ ( finite_finite2 @ A @ ( set_or7035219750837199246ssThan @ A @ A2 @ B2 ) ) ) ) ).
% infinite_Ico
thf(fact_4893_finite__lists__length__le,axiom,
! [A: $tType,A3: set @ A,N: nat] :
( ( finite_finite2 @ A @ A3 )
=> ( finite_finite2 @ ( list @ A )
@ ( collect @ ( list @ A )
@ ^ [Xs2: list @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs2 ) @ A3 )
& ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ N ) ) ) ) ) ).
% finite_lists_length_le
thf(fact_4894_sum_Ofinite__Collect__op,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [I6: set @ B,X: B > A,Y2: B > A] :
( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [I: B] :
( ( member @ B @ I @ I6 )
& ( ( X @ I )
!= ( zero_zero @ A ) ) ) ) )
=> ( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [I: B] :
( ( member @ B @ I @ I6 )
& ( ( Y2 @ I )
!= ( zero_zero @ A ) ) ) ) )
=> ( finite_finite2 @ B
@ ( collect @ B
@ ^ [I: B] :
( ( member @ B @ I @ I6 )
& ( ( plus_plus @ A @ ( X @ I ) @ ( Y2 @ I ) )
!= ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_4895_prod_Ofinite__Collect__op,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [I6: set @ B,X: B > A,Y2: B > A] :
( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [I: B] :
( ( member @ B @ I @ I6 )
& ( ( X @ I )
!= ( one_one @ A ) ) ) ) )
=> ( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [I: B] :
( ( member @ B @ I @ I6 )
& ( ( Y2 @ I )
!= ( one_one @ A ) ) ) ) )
=> ( finite_finite2 @ B
@ ( collect @ B
@ ^ [I: B] :
( ( member @ B @ I @ I6 )
& ( ( times_times @ A @ ( X @ I ) @ ( Y2 @ I ) )
!= ( one_one @ A ) ) ) ) ) ) ) ) ).
% prod.finite_Collect_op
thf(fact_4896_sum_Ointer__filter,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A3: set @ B,G: B > A,P3: B > $o] :
( ( finite_finite2 @ B @ A3 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G
@ ( collect @ B
@ ^ [X4: B] :
( ( member @ B @ X4 @ A3 )
& ( P3 @ X4 ) ) ) )
= ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X4: B] : ( if @ A @ ( P3 @ X4 ) @ ( G @ X4 ) @ ( zero_zero @ A ) )
@ A3 ) ) ) ) ).
% sum.inter_filter
thf(fact_4897_prod_Ointer__filter,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: set @ B,G: B > A,P3: B > $o] :
( ( finite_finite2 @ B @ A3 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G
@ ( collect @ B
@ ^ [X4: B] :
( ( member @ B @ X4 @ A3 )
& ( P3 @ X4 ) ) ) )
= ( groups7121269368397514597t_prod @ B @ A
@ ^ [X4: B] : ( if @ A @ ( P3 @ X4 ) @ ( G @ X4 ) @ ( one_one @ A ) )
@ A3 ) ) ) ) ).
% prod.inter_filter
thf(fact_4898_finite__int__segment,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [A2: A,B2: A] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ ( ring_1_Ints @ A ) )
& ( ord_less_eq @ A @ A2 @ X4 )
& ( ord_less_eq @ A @ X4 @ B2 ) ) ) ) ) ).
% finite_int_segment
thf(fact_4899_sum__le__included,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [S: set @ B,T2: set @ C,G: C > A,I2: C > B,F3: B > A] :
( ( finite_finite2 @ B @ S )
=> ( ( finite_finite2 @ C @ T2 )
=> ( ! [X5: C] :
( ( member @ C @ X5 @ T2 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( G @ X5 ) ) )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ S )
=> ? [Xa2: C] :
( ( member @ C @ Xa2 @ T2 )
& ( ( I2 @ Xa2 )
= X5 )
& ( ord_less_eq @ A @ ( F3 @ X5 ) @ ( G @ Xa2 ) ) ) )
=> ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ S ) @ ( groups7311177749621191930dd_sum @ C @ A @ G @ T2 ) ) ) ) ) ) ) ).
% sum_le_included
thf(fact_4900_sum__nonneg__eq__0__iff,axiom,
! [A: $tType,B: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A3: set @ B,F3: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ A3 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F3 @ X5 ) ) )
=> ( ( ( groups7311177749621191930dd_sum @ B @ A @ F3 @ A3 )
= ( zero_zero @ A ) )
= ( ! [X4: B] :
( ( member @ B @ X4 @ A3 )
=> ( ( F3 @ X4 )
= ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
thf(fact_4901_sum__strict__mono__ex1,axiom,
! [A: $tType,I5: $tType] :
( ( ordere8940638589300402666id_add @ A )
=> ! [A3: set @ I5,F3: I5 > A,G: I5 > A] :
( ( finite_finite2 @ I5 @ A3 )
=> ( ! [X5: I5] :
( ( member @ I5 @ X5 @ A3 )
=> ( ord_less_eq @ A @ ( F3 @ X5 ) @ ( G @ X5 ) ) )
=> ( ? [X2: I5] :
( ( member @ I5 @ X2 @ A3 )
& ( ord_less @ A @ ( F3 @ X2 ) @ ( G @ X2 ) ) )
=> ( ord_less @ A @ ( groups7311177749621191930dd_sum @ I5 @ A @ F3 @ A3 ) @ ( groups7311177749621191930dd_sum @ I5 @ A @ G @ A3 ) ) ) ) ) ) ).
% sum_strict_mono_ex1
thf(fact_4902_sum_Orelated,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [R4: A > A > $o,S3: set @ B,H2: B > A,G: B > A] :
( ( R4 @ ( zero_zero @ A ) @ ( zero_zero @ A ) )
=> ( ! [X15: A,Y1: A,X24: A,Y24: A] :
( ( ( R4 @ X15 @ X24 )
& ( R4 @ Y1 @ Y24 ) )
=> ( R4 @ ( plus_plus @ A @ X15 @ Y1 ) @ ( plus_plus @ A @ X24 @ Y24 ) ) )
=> ( ( finite_finite2 @ B @ S3 )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ S3 )
=> ( R4 @ ( H2 @ X5 ) @ ( G @ X5 ) ) )
=> ( R4 @ ( groups7311177749621191930dd_sum @ B @ A @ H2 @ S3 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ S3 ) ) ) ) ) ) ) ).
% sum.related
thf(fact_4903_prod_Orelated,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [R4: A > A > $o,S3: set @ B,H2: B > A,G: B > A] :
( ( R4 @ ( one_one @ A ) @ ( one_one @ A ) )
=> ( ! [X15: A,Y1: A,X24: A,Y24: A] :
( ( ( R4 @ X15 @ X24 )
& ( R4 @ Y1 @ Y24 ) )
=> ( R4 @ ( times_times @ A @ X15 @ Y1 ) @ ( times_times @ A @ X24 @ Y24 ) ) )
=> ( ( finite_finite2 @ B @ S3 )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ S3 )
=> ( R4 @ ( H2 @ X5 ) @ ( G @ X5 ) ) )
=> ( R4 @ ( groups7121269368397514597t_prod @ B @ A @ H2 @ S3 ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ S3 ) ) ) ) ) ) ) ).
% prod.related
thf(fact_4904_prod__dvd__prod__subset2,axiom,
! [A: $tType,B: $tType] :
( ( comm_semiring_1 @ A )
=> ! [B5: set @ B,A3: set @ B,F3: B > A,G: B > A] :
( ( finite_finite2 @ B @ B5 )
=> ( ( ord_less_eq @ ( set @ B ) @ A3 @ B5 )
=> ( ! [A4: B] :
( ( member @ B @ A4 @ A3 )
=> ( dvd_dvd @ A @ ( F3 @ A4 ) @ ( G @ A4 ) ) )
=> ( dvd_dvd @ A @ ( groups7121269368397514597t_prod @ B @ A @ F3 @ A3 ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ B5 ) ) ) ) ) ) ).
% prod_dvd_prod_subset2
thf(fact_4905_prod__dvd__prod__subset,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [B5: set @ B,A3: set @ B,F3: B > A] :
( ( finite_finite2 @ B @ B5 )
=> ( ( ord_less_eq @ ( set @ B ) @ A3 @ B5 )
=> ( dvd_dvd @ A @ ( groups7121269368397514597t_prod @ B @ A @ F3 @ A3 ) @ ( groups7121269368397514597t_prod @ B @ A @ F3 @ B5 ) ) ) ) ) ).
% prod_dvd_prod_subset
thf(fact_4906_prod_Oreindex__bij__witness__not__neutral,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S5: set @ B,T6: set @ C,S3: set @ B,I2: C > B,J3: B > C,T5: set @ C,G: B > A,H2: C > A] :
( ( finite_finite2 @ B @ S5 )
=> ( ( finite_finite2 @ C @ T6 )
=> ( ! [A4: B] :
( ( member @ B @ A4 @ ( minus_minus @ ( set @ B ) @ S3 @ S5 ) )
=> ( ( I2 @ ( J3 @ A4 ) )
= A4 ) )
=> ( ! [A4: B] :
( ( member @ B @ A4 @ ( minus_minus @ ( set @ B ) @ S3 @ S5 ) )
=> ( member @ C @ ( J3 @ A4 ) @ ( minus_minus @ ( set @ C ) @ T5 @ T6 ) ) )
=> ( ! [B3: C] :
( ( member @ C @ B3 @ ( minus_minus @ ( set @ C ) @ T5 @ T6 ) )
=> ( ( J3 @ ( I2 @ B3 ) )
= B3 ) )
=> ( ! [B3: C] :
( ( member @ C @ B3 @ ( minus_minus @ ( set @ C ) @ T5 @ T6 ) )
=> ( member @ B @ ( I2 @ B3 ) @ ( minus_minus @ ( set @ B ) @ S3 @ S5 ) ) )
=> ( ! [A4: B] :
( ( member @ B @ A4 @ S5 )
=> ( ( G @ A4 )
= ( one_one @ A ) ) )
=> ( ! [B3: C] :
( ( member @ C @ B3 @ T6 )
=> ( ( H2 @ B3 )
= ( one_one @ A ) ) )
=> ( ! [A4: B] :
( ( member @ B @ A4 @ S3 )
=> ( ( H2 @ ( J3 @ A4 ) )
= ( G @ A4 ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ S3 )
= ( groups7121269368397514597t_prod @ C @ A @ H2 @ T5 ) ) ) ) ) ) ) ) ) ) ) ) ).
% prod.reindex_bij_witness_not_neutral
thf(fact_4907_sum__eq__1__iff,axiom,
! [A: $tType,A3: set @ A,F3: A > nat] :
( ( finite_finite2 @ A @ A3 )
=> ( ( ( groups7311177749621191930dd_sum @ A @ nat @ F3 @ A3 )
= ( one_one @ nat ) )
= ( ? [X4: A] :
( ( member @ A @ X4 @ A3 )
& ( ( F3 @ X4 )
= ( one_one @ nat ) )
& ! [Y: A] :
( ( member @ A @ Y @ A3 )
=> ( ( X4 != Y )
=> ( ( F3 @ Y )
= ( zero_zero @ nat ) ) ) ) ) ) ) ) ).
% sum_eq_1_iff
thf(fact_4908_sum__nonneg__0,axiom,
! [B: $tType,A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [S: set @ B,F3: B > A,I2: B] :
( ( finite_finite2 @ B @ S )
=> ( ! [I3: B] :
( ( member @ B @ I3 @ S )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F3 @ I3 ) ) )
=> ( ( ( groups7311177749621191930dd_sum @ B @ A @ F3 @ S )
= ( zero_zero @ A ) )
=> ( ( member @ B @ I2 @ S )
=> ( ( F3 @ I2 )
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_4909_sum__nonneg__leq__bound,axiom,
! [B: $tType,A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [S: set @ B,F3: B > A,B5: A,I2: B] :
( ( finite_finite2 @ B @ S )
=> ( ! [I3: B] :
( ( member @ B @ I3 @ S )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F3 @ I3 ) ) )
=> ( ( ( groups7311177749621191930dd_sum @ B @ A @ F3 @ S )
= B5 )
=> ( ( member @ B @ I2 @ S )
=> ( ord_less_eq @ A @ ( F3 @ I2 ) @ B5 ) ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_4910_sum_Osetdiff__irrelevant,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A3: set @ B,G: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G
@ ( minus_minus @ ( set @ B ) @ A3
@ ( collect @ B
@ ^ [X4: B] :
( ( G @ X4 )
= ( zero_zero @ A ) ) ) ) )
= ( groups7311177749621191930dd_sum @ B @ A @ G @ A3 ) ) ) ) ).
% sum.setdiff_irrelevant
thf(fact_4911_prod_Osetdiff__irrelevant,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: set @ B,G: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G
@ ( minus_minus @ ( set @ B ) @ A3
@ ( collect @ B
@ ^ [X4: B] :
( ( G @ X4 )
= ( one_one @ A ) ) ) ) )
= ( groups7121269368397514597t_prod @ B @ A @ G @ A3 ) ) ) ) ).
% prod.setdiff_irrelevant
thf(fact_4912_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_4913_sums__finite,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [N4: set @ nat,F3: nat > A] :
( ( finite_finite2 @ nat @ N4 )
=> ( ! [N3: nat] :
( ~ ( member @ nat @ N3 @ N4 )
=> ( ( F3 @ N3 )
= ( zero_zero @ A ) ) )
=> ( sums @ A @ F3 @ ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ N4 ) ) ) ) ) ).
% sums_finite
thf(fact_4914_sums__If__finite,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [P3: nat > $o,F3: nat > A] :
( ( finite_finite2 @ nat @ ( collect @ nat @ P3 ) )
=> ( sums @ A
@ ^ [R5: nat] : ( if @ A @ ( P3 @ R5 ) @ ( F3 @ R5 ) @ ( zero_zero @ A ) )
@ ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( collect @ nat @ P3 ) ) ) ) ) ).
% sums_If_finite
thf(fact_4915_sums__If__finite__set,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [A3: set @ nat,F3: nat > A] :
( ( finite_finite2 @ nat @ A3 )
=> ( sums @ A
@ ^ [R5: nat] : ( if @ A @ ( member @ nat @ R5 @ A3 ) @ ( F3 @ R5 ) @ ( zero_zero @ A ) )
@ ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ A3 ) ) ) ) ).
% sums_If_finite_set
thf(fact_4916_suminf__finite,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topological_t2_space @ A ) )
=> ! [N4: set @ nat,F3: nat > A] :
( ( finite_finite2 @ nat @ N4 )
=> ( ! [N3: nat] :
( ~ ( member @ nat @ N3 @ N4 )
=> ( ( F3 @ N3 )
= ( zero_zero @ A ) ) )
=> ( ( suminf @ A @ F3 )
= ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ N4 ) ) ) ) ) ).
% suminf_finite
thf(fact_4917_finite__abs__int__segment,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [A2: A] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [K3: A] :
( ( member @ A @ K3 @ ( ring_1_Ints @ A ) )
& ( ord_less_eq @ A @ ( abs_abs @ A @ K3 ) @ A2 ) ) ) ) ) ).
% finite_abs_int_segment
thf(fact_4918_exp__sum,axiom,
! [B: $tType,A: $tType] :
( ( ( comm_monoid_mult @ B )
& ( real_Vector_banach @ B )
& ( real_V2822296259951069270ebra_1 @ B ) )
=> ! [I6: set @ A,F3: A > B] :
( ( finite_finite2 @ A @ I6 )
=> ( ( exp @ B @ ( groups7311177749621191930dd_sum @ A @ B @ F3 @ I6 ) )
= ( groups7121269368397514597t_prod @ A @ B
@ ^ [X4: A] : ( exp @ B @ ( F3 @ X4 ) )
@ I6 ) ) ) ) ).
% exp_sum
thf(fact_4919_subset__eq__atLeast0__atMost__finite,axiom,
! [N4: set @ nat,N: nat] :
( ( ord_less_eq @ ( set @ nat ) @ N4 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
=> ( finite_finite2 @ nat @ N4 ) ) ).
% subset_eq_atLeast0_atMost_finite
thf(fact_4920_subset__eq__atLeast0__lessThan__finite,axiom,
! [N4: set @ nat,N: nat] :
( ( ord_less_eq @ ( set @ nat ) @ N4 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) )
=> ( finite_finite2 @ nat @ N4 ) ) ).
% subset_eq_atLeast0_lessThan_finite
thf(fact_4921_sum__pos2,axiom,
! [A: $tType,B: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [I6: set @ B,I2: B,F3: B > A] :
( ( finite_finite2 @ B @ I6 )
=> ( ( member @ B @ I2 @ I6 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ ( F3 @ I2 ) )
=> ( ! [I3: B] :
( ( member @ B @ I3 @ I6 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F3 @ I3 ) ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ I6 ) ) ) ) ) ) ) ).
% sum_pos2
thf(fact_4922_less__1__prod2,axiom,
! [B: $tType,A: $tType] :
( ( linordered_idom @ B )
=> ! [I6: set @ A,I2: A,F3: A > B] :
( ( finite_finite2 @ A @ I6 )
=> ( ( member @ A @ I2 @ I6 )
=> ( ( ord_less @ B @ ( one_one @ B ) @ ( F3 @ I2 ) )
=> ( ! [I3: A] :
( ( member @ A @ I3 @ I6 )
=> ( ord_less_eq @ B @ ( one_one @ B ) @ ( F3 @ I3 ) ) )
=> ( ord_less @ B @ ( one_one @ B ) @ ( groups7121269368397514597t_prod @ A @ B @ F3 @ I6 ) ) ) ) ) ) ) ).
% less_1_prod2
thf(fact_4923_sum_Osame__carrier,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [C5: set @ B,A3: set @ B,B5: set @ B,G: B > A,H2: B > A] :
( ( finite_finite2 @ B @ C5 )
=> ( ( ord_less_eq @ ( set @ B ) @ A3 @ C5 )
=> ( ( ord_less_eq @ ( set @ B ) @ B5 @ C5 )
=> ( ! [A4: B] :
( ( member @ B @ A4 @ ( minus_minus @ ( set @ B ) @ C5 @ A3 ) )
=> ( ( G @ A4 )
= ( zero_zero @ A ) ) )
=> ( ! [B3: B] :
( ( member @ B @ B3 @ ( minus_minus @ ( set @ B ) @ C5 @ B5 ) )
=> ( ( H2 @ B3 )
= ( zero_zero @ A ) ) )
=> ( ( ( groups7311177749621191930dd_sum @ B @ A @ G @ A3 )
= ( groups7311177749621191930dd_sum @ B @ A @ H2 @ B5 ) )
= ( ( groups7311177749621191930dd_sum @ B @ A @ G @ C5 )
= ( groups7311177749621191930dd_sum @ B @ A @ H2 @ C5 ) ) ) ) ) ) ) ) ) ).
% sum.same_carrier
thf(fact_4924_sum_Osame__carrierI,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [C5: set @ B,A3: set @ B,B5: set @ B,G: B > A,H2: B > A] :
( ( finite_finite2 @ B @ C5 )
=> ( ( ord_less_eq @ ( set @ B ) @ A3 @ C5 )
=> ( ( ord_less_eq @ ( set @ B ) @ B5 @ C5 )
=> ( ! [A4: B] :
( ( member @ B @ A4 @ ( minus_minus @ ( set @ B ) @ C5 @ A3 ) )
=> ( ( G @ A4 )
= ( zero_zero @ A ) ) )
=> ( ! [B3: B] :
( ( member @ B @ B3 @ ( minus_minus @ ( set @ B ) @ C5 @ B5 ) )
=> ( ( H2 @ B3 )
= ( zero_zero @ A ) ) )
=> ( ( ( groups7311177749621191930dd_sum @ B @ A @ G @ C5 )
= ( groups7311177749621191930dd_sum @ B @ A @ H2 @ C5 ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ A3 )
= ( groups7311177749621191930dd_sum @ B @ A @ H2 @ B5 ) ) ) ) ) ) ) ) ) ).
% sum.same_carrierI
thf(fact_4925_sum_Omono__neutral__left,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [T5: set @ B,S3: set @ B,G: B > A] :
( ( finite_finite2 @ B @ T5 )
=> ( ( ord_less_eq @ ( set @ B ) @ S3 @ T5 )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ ( minus_minus @ ( set @ B ) @ T5 @ S3 ) )
=> ( ( G @ X5 )
= ( zero_zero @ A ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ S3 )
= ( groups7311177749621191930dd_sum @ B @ A @ G @ T5 ) ) ) ) ) ) ).
% sum.mono_neutral_left
thf(fact_4926_sum_Omono__neutral__right,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [T5: set @ B,S3: set @ B,G: B > A] :
( ( finite_finite2 @ B @ T5 )
=> ( ( ord_less_eq @ ( set @ B ) @ S3 @ T5 )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ ( minus_minus @ ( set @ B ) @ T5 @ S3 ) )
=> ( ( G @ X5 )
= ( zero_zero @ A ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ T5 )
= ( groups7311177749621191930dd_sum @ B @ A @ G @ S3 ) ) ) ) ) ) ).
% sum.mono_neutral_right
thf(fact_4927_sum_Omono__neutral__cong__left,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [T5: set @ B,S3: set @ B,H2: B > A,G: B > A] :
( ( finite_finite2 @ B @ T5 )
=> ( ( ord_less_eq @ ( set @ B ) @ S3 @ T5 )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ ( minus_minus @ ( set @ B ) @ T5 @ S3 ) )
=> ( ( H2 @ X5 )
= ( zero_zero @ A ) ) )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ S3 )
=> ( ( G @ X5 )
= ( H2 @ X5 ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ S3 )
= ( groups7311177749621191930dd_sum @ B @ A @ H2 @ T5 ) ) ) ) ) ) ) ).
% sum.mono_neutral_cong_left
thf(fact_4928_sum_Omono__neutral__cong__right,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [T5: set @ B,S3: set @ B,G: B > A,H2: B > A] :
( ( finite_finite2 @ B @ T5 )
=> ( ( ord_less_eq @ ( set @ B ) @ S3 @ T5 )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ ( minus_minus @ ( set @ B ) @ T5 @ S3 ) )
=> ( ( G @ X5 )
= ( zero_zero @ A ) ) )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ S3 )
=> ( ( G @ X5 )
= ( H2 @ X5 ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ T5 )
= ( groups7311177749621191930dd_sum @ B @ A @ H2 @ S3 ) ) ) ) ) ) ) ).
% sum.mono_neutral_cong_right
thf(fact_4929_sum_Osubset__diff,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [B5: set @ B,A3: set @ B,G: B > A] :
( ( ord_less_eq @ ( set @ B ) @ B5 @ A3 )
=> ( ( finite_finite2 @ B @ A3 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ A3 )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A3 @ B5 ) ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ B5 ) ) ) ) ) ) ).
% sum.subset_diff
thf(fact_4930_sum__diff,axiom,
! [A: $tType,B: $tType] :
( ( ab_group_add @ A )
=> ! [A3: set @ B,B5: set @ B,F3: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ( ord_less_eq @ ( set @ B ) @ B5 @ A3 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ F3 @ ( minus_minus @ ( set @ B ) @ A3 @ B5 ) )
= ( minus_minus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ A3 ) @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ B5 ) ) ) ) ) ) ).
% sum_diff
thf(fact_4931_prod_Osubset__diff,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [B5: set @ B,A3: set @ B,G: B > A] :
( ( ord_less_eq @ ( set @ B ) @ B5 @ A3 )
=> ( ( finite_finite2 @ B @ A3 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ A3 )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A3 @ B5 ) ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ B5 ) ) ) ) ) ) ).
% prod.subset_diff
thf(fact_4932_prod_Osame__carrier,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [C5: set @ B,A3: set @ B,B5: set @ B,G: B > A,H2: B > A] :
( ( finite_finite2 @ B @ C5 )
=> ( ( ord_less_eq @ ( set @ B ) @ A3 @ C5 )
=> ( ( ord_less_eq @ ( set @ B ) @ B5 @ C5 )
=> ( ! [A4: B] :
( ( member @ B @ A4 @ ( minus_minus @ ( set @ B ) @ C5 @ A3 ) )
=> ( ( G @ A4 )
= ( one_one @ A ) ) )
=> ( ! [B3: B] :
( ( member @ B @ B3 @ ( minus_minus @ ( set @ B ) @ C5 @ B5 ) )
=> ( ( H2 @ B3 )
= ( one_one @ A ) ) )
=> ( ( ( groups7121269368397514597t_prod @ B @ A @ G @ A3 )
= ( groups7121269368397514597t_prod @ B @ A @ H2 @ B5 ) )
= ( ( groups7121269368397514597t_prod @ B @ A @ G @ C5 )
= ( groups7121269368397514597t_prod @ B @ A @ H2 @ C5 ) ) ) ) ) ) ) ) ) ).
% prod.same_carrier
thf(fact_4933_prod_Osame__carrierI,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [C5: set @ B,A3: set @ B,B5: set @ B,G: B > A,H2: B > A] :
( ( finite_finite2 @ B @ C5 )
=> ( ( ord_less_eq @ ( set @ B ) @ A3 @ C5 )
=> ( ( ord_less_eq @ ( set @ B ) @ B5 @ C5 )
=> ( ! [A4: B] :
( ( member @ B @ A4 @ ( minus_minus @ ( set @ B ) @ C5 @ A3 ) )
=> ( ( G @ A4 )
= ( one_one @ A ) ) )
=> ( ! [B3: B] :
( ( member @ B @ B3 @ ( minus_minus @ ( set @ B ) @ C5 @ B5 ) )
=> ( ( H2 @ B3 )
= ( one_one @ A ) ) )
=> ( ( ( groups7121269368397514597t_prod @ B @ A @ G @ C5 )
= ( groups7121269368397514597t_prod @ B @ A @ H2 @ C5 ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ A3 )
= ( groups7121269368397514597t_prod @ B @ A @ H2 @ B5 ) ) ) ) ) ) ) ) ) ).
% prod.same_carrierI
thf(fact_4934_prod_Omono__neutral__left,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [T5: set @ B,S3: set @ B,G: B > A] :
( ( finite_finite2 @ B @ T5 )
=> ( ( ord_less_eq @ ( set @ B ) @ S3 @ T5 )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ ( minus_minus @ ( set @ B ) @ T5 @ S3 ) )
=> ( ( G @ X5 )
= ( one_one @ A ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ S3 )
= ( groups7121269368397514597t_prod @ B @ A @ G @ T5 ) ) ) ) ) ) ).
% prod.mono_neutral_left
thf(fact_4935_prod_Omono__neutral__right,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [T5: set @ B,S3: set @ B,G: B > A] :
( ( finite_finite2 @ B @ T5 )
=> ( ( ord_less_eq @ ( set @ B ) @ S3 @ T5 )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ ( minus_minus @ ( set @ B ) @ T5 @ S3 ) )
=> ( ( G @ X5 )
= ( one_one @ A ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ T5 )
= ( groups7121269368397514597t_prod @ B @ A @ G @ S3 ) ) ) ) ) ) ).
% prod.mono_neutral_right
thf(fact_4936_prod_Omono__neutral__cong__left,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [T5: set @ B,S3: set @ B,H2: B > A,G: B > A] :
( ( finite_finite2 @ B @ T5 )
=> ( ( ord_less_eq @ ( set @ B ) @ S3 @ T5 )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ ( minus_minus @ ( set @ B ) @ T5 @ S3 ) )
=> ( ( H2 @ X5 )
= ( one_one @ A ) ) )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ S3 )
=> ( ( G @ X5 )
= ( H2 @ X5 ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ S3 )
= ( groups7121269368397514597t_prod @ B @ A @ H2 @ T5 ) ) ) ) ) ) ) ).
% prod.mono_neutral_cong_left
thf(fact_4937_prod_Omono__neutral__cong__right,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [T5: set @ B,S3: set @ B,G: B > A,H2: B > A] :
( ( finite_finite2 @ B @ T5 )
=> ( ( ord_less_eq @ ( set @ B ) @ S3 @ T5 )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ ( minus_minus @ ( set @ B ) @ T5 @ S3 ) )
=> ( ( G @ X5 )
= ( one_one @ A ) ) )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ S3 )
=> ( ( G @ X5 )
= ( H2 @ X5 ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ T5 )
= ( groups7121269368397514597t_prod @ B @ A @ H2 @ S3 ) ) ) ) ) ) ) ).
% prod.mono_neutral_cong_right
thf(fact_4938_sum__diff__nat,axiom,
! [A: $tType,B5: set @ A,A3: set @ A,F3: A > nat] :
( ( finite_finite2 @ A @ B5 )
=> ( ( ord_less_eq @ ( set @ A ) @ B5 @ A3 )
=> ( ( groups7311177749621191930dd_sum @ A @ nat @ F3 @ ( minus_minus @ ( set @ A ) @ A3 @ B5 ) )
= ( minus_minus @ nat @ ( groups7311177749621191930dd_sum @ A @ nat @ F3 @ A3 ) @ ( groups7311177749621191930dd_sum @ A @ nat @ F3 @ B5 ) ) ) ) ) ).
% sum_diff_nat
thf(fact_4939_finite__roots__unity,axiom,
! [A: $tType] :
( ( ( real_V8999393235501362500lgebra @ A )
& ( idom @ A ) )
=> ! [N: nat] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ N )
=> ( finite_finite2 @ A
@ ( collect @ A
@ ^ [Z6: A] :
( ( power_power @ A @ Z6 @ N )
= ( one_one @ A ) ) ) ) ) ) ).
% finite_roots_unity
thf(fact_4940_sum_Oreindex__bij__betw__not__neutral,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comm_monoid_add @ A )
=> ! [S5: set @ B,T6: set @ C,H2: B > C,S3: set @ B,T5: set @ C,G: C > A] :
( ( finite_finite2 @ B @ S5 )
=> ( ( finite_finite2 @ C @ T6 )
=> ( ( bij_betw @ B @ C @ H2 @ ( minus_minus @ ( set @ B ) @ S3 @ S5 ) @ ( minus_minus @ ( set @ C ) @ T5 @ T6 ) )
=> ( ! [A4: B] :
( ( member @ B @ A4 @ S5 )
=> ( ( G @ ( H2 @ A4 ) )
= ( zero_zero @ A ) ) )
=> ( ! [B3: C] :
( ( member @ C @ B3 @ T6 )
=> ( ( G @ B3 )
= ( zero_zero @ A ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X4: B] : ( G @ ( H2 @ X4 ) )
@ S3 )
= ( groups7311177749621191930dd_sum @ C @ A @ G @ T5 ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_betw_not_neutral
thf(fact_4941_sums__If__finite__set_H,axiom,
! [A: $tType] :
( ( ( topolo1287966508704411220up_add @ A )
& ( topological_t2_space @ A ) )
=> ! [G: nat > A,S3: A,A3: set @ nat,S5: A,F3: nat > A] :
( ( sums @ A @ G @ S3 )
=> ( ( finite_finite2 @ nat @ A3 )
=> ( ( S5
= ( plus_plus @ A @ S3
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [N2: nat] : ( minus_minus @ A @ ( F3 @ N2 ) @ ( G @ N2 ) )
@ A3 ) ) )
=> ( sums @ A
@ ^ [N2: nat] : ( if @ A @ ( member @ nat @ N2 @ A3 ) @ ( F3 @ N2 ) @ ( G @ N2 ) )
@ S5 ) ) ) ) ) ).
% sums_If_finite_set'
thf(fact_4942_prod_Oreindex__bij__betw__not__neutral,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S5: set @ B,T6: set @ C,H2: B > C,S3: set @ B,T5: set @ C,G: C > A] :
( ( finite_finite2 @ B @ S5 )
=> ( ( finite_finite2 @ C @ T6 )
=> ( ( bij_betw @ B @ C @ H2 @ ( minus_minus @ ( set @ B ) @ S3 @ S5 ) @ ( minus_minus @ ( set @ C ) @ T5 @ T6 ) )
=> ( ! [A4: B] :
( ( member @ B @ A4 @ S5 )
=> ( ( G @ ( H2 @ A4 ) )
= ( one_one @ A ) ) )
=> ( ! [B3: C] :
( ( member @ C @ B3 @ T6 )
=> ( ( G @ B3 )
= ( one_one @ A ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [X4: B] : ( G @ ( H2 @ X4 ) )
@ S3 )
= ( groups7121269368397514597t_prod @ C @ A @ G @ T5 ) ) ) ) ) ) ) ) ).
% prod.reindex_bij_betw_not_neutral
thf(fact_4943_sum__mono2,axiom,
! [A: $tType,B: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [B5: set @ B,A3: set @ B,F3: B > A] :
( ( finite_finite2 @ B @ B5 )
=> ( ( ord_less_eq @ ( set @ B ) @ A3 @ B5 )
=> ( ! [B3: B] :
( ( member @ B @ B3 @ ( minus_minus @ ( set @ B ) @ B5 @ A3 ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F3 @ B3 ) ) )
=> ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ A3 ) @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ B5 ) ) ) ) ) ) ).
% sum_mono2
thf(fact_4944_even__prod__iff,axiom,
! [A: $tType,B: $tType] :
( ( semiring_parity @ A )
=> ! [A3: set @ B,F3: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( groups7121269368397514597t_prod @ B @ A @ F3 @ A3 ) )
= ( ? [X4: B] :
( ( member @ B @ X4 @ A3 )
& ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( F3 @ X4 ) ) ) ) ) ) ) ).
% even_prod_iff
thf(fact_4945_sum__le__suminf,axiom,
! [A: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F3: nat > A,I6: set @ nat] :
( ( summable @ A @ F3 )
=> ( ( finite_finite2 @ nat @ I6 )
=> ( ! [N3: nat] :
( ( member @ nat @ N3 @ ( uminus_uminus @ ( set @ nat ) @ I6 ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F3 @ N3 ) ) )
=> ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ I6 ) @ ( suminf @ A @ F3 ) ) ) ) ) ) ).
% sum_le_suminf
thf(fact_4946_sum__strict__mono2,axiom,
! [B: $tType,A: $tType] :
( ( ordere8940638589300402666id_add @ B )
=> ! [B5: set @ A,A3: set @ A,B2: A,F3: A > B] :
( ( finite_finite2 @ A @ B5 )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ B5 )
=> ( ( member @ A @ B2 @ ( minus_minus @ ( set @ A ) @ B5 @ A3 ) )
=> ( ( ord_less @ B @ ( zero_zero @ B ) @ ( F3 @ B2 ) )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ B5 )
=> ( ord_less_eq @ B @ ( zero_zero @ B ) @ ( F3 @ X5 ) ) )
=> ( ord_less @ B @ ( groups7311177749621191930dd_sum @ A @ B @ F3 @ A3 ) @ ( groups7311177749621191930dd_sum @ A @ B @ F3 @ B5 ) ) ) ) ) ) ) ) ).
% sum_strict_mono2
thf(fact_4947_prod__mono2,axiom,
! [B: $tType,A: $tType] :
( ( linordered_idom @ B )
=> ! [B5: set @ A,A3: set @ A,F3: A > B] :
( ( finite_finite2 @ A @ B5 )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ B5 )
=> ( ! [B3: A] :
( ( member @ A @ B3 @ ( minus_minus @ ( set @ A ) @ B5 @ A3 ) )
=> ( ord_less_eq @ B @ ( one_one @ B ) @ ( F3 @ B3 ) ) )
=> ( ! [A4: A] :
( ( member @ A @ A4 @ A3 )
=> ( ord_less_eq @ B @ ( zero_zero @ B ) @ ( F3 @ A4 ) ) )
=> ( ord_less_eq @ B @ ( groups7121269368397514597t_prod @ A @ B @ F3 @ A3 ) @ ( groups7121269368397514597t_prod @ A @ B @ F3 @ B5 ) ) ) ) ) ) ) ).
% prod_mono2
thf(fact_4948_ln__prod,axiom,
! [A: $tType,I6: set @ A,F3: A > real] :
( ( finite_finite2 @ A @ I6 )
=> ( ! [I3: A] :
( ( member @ A @ I3 @ I6 )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( F3 @ I3 ) ) )
=> ( ( ln_ln @ real @ ( groups7121269368397514597t_prod @ A @ real @ F3 @ I6 ) )
= ( groups7311177749621191930dd_sum @ A @ real
@ ^ [X4: A] : ( ln_ln @ real @ ( F3 @ X4 ) )
@ I6 ) ) ) ) ).
% ln_prod
thf(fact_4949_even__set__encode__iff,axiom,
! [A3: set @ nat] :
( ( finite_finite2 @ nat @ A3 )
=> ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( nat_set_encode @ A3 ) )
= ( ~ ( member @ nat @ ( zero_zero @ nat ) @ A3 ) ) ) ) ).
% even_set_encode_iff
thf(fact_4950_polyfun__finite__roots,axiom,
! [A: $tType] :
( ( ( real_V8999393235501362500lgebra @ A )
& ( idom @ A ) )
=> ! [C2: nat > A,N: nat] :
( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X4: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I: nat] : ( times_times @ A @ ( C2 @ I ) @ ( power_power @ A @ X4 @ I ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( zero_zero @ A ) ) ) )
= ( ? [I: nat] :
( ( ord_less_eq @ nat @ I @ N )
& ( ( C2 @ I )
!= ( zero_zero @ A ) ) ) ) ) ) ).
% polyfun_finite_roots
thf(fact_4951_polyfun__roots__finite,axiom,
! [A: $tType] :
( ( ( real_V8999393235501362500lgebra @ A )
& ( idom @ A ) )
=> ! [C2: nat > A,K2: nat,N: nat] :
( ( ( C2 @ K2 )
!= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ nat @ K2 @ N )
=> ( finite_finite2 @ A
@ ( collect @ A
@ ^ [Z6: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I: nat] : ( times_times @ A @ ( C2 @ I ) @ ( power_power @ A @ Z6 @ I ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% polyfun_roots_finite
thf(fact_4952_finite__Collect__le__nat,axiom,
! [K2: nat] :
( finite_finite2 @ nat
@ ( collect @ nat
@ ^ [N2: nat] : ( ord_less_eq @ nat @ N2 @ K2 ) ) ) ).
% finite_Collect_le_nat
thf(fact_4953_finite__Collect__less__nat,axiom,
! [K2: nat] :
( finite_finite2 @ nat
@ ( collect @ nat
@ ^ [N2: nat] : ( ord_less @ nat @ N2 @ K2 ) ) ) ).
% finite_Collect_less_nat
thf(fact_4954_finite__Collect__subsets,axiom,
! [A: $tType,A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( finite_finite2 @ ( set @ A )
@ ( collect @ ( set @ A )
@ ^ [B7: set @ A] : ( ord_less_eq @ ( set @ A ) @ B7 @ A3 ) ) ) ) ).
% finite_Collect_subsets
thf(fact_4955_finite__Collect__disjI,axiom,
! [A: $tType,P3: A > $o,Q2: A > $o] :
( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X4: A] :
( ( P3 @ X4 )
| ( Q2 @ X4 ) ) ) )
= ( ( finite_finite2 @ A @ ( collect @ A @ P3 ) )
& ( finite_finite2 @ A @ ( collect @ A @ Q2 ) ) ) ) ).
% finite_Collect_disjI
thf(fact_4956_finite__Collect__conjI,axiom,
! [A: $tType,P3: A > $o,Q2: A > $o] :
( ( ( finite_finite2 @ A @ ( collect @ A @ P3 ) )
| ( finite_finite2 @ A @ ( collect @ A @ Q2 ) ) )
=> ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X4: A] :
( ( P3 @ X4 )
& ( Q2 @ X4 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_4957_finite__interval__int1,axiom,
! [A2: int,B2: int] :
( finite_finite2 @ int
@ ( collect @ int
@ ^ [I: int] :
( ( ord_less_eq @ int @ A2 @ I )
& ( ord_less_eq @ int @ I @ B2 ) ) ) ) ).
% finite_interval_int1
thf(fact_4958_finite__interval__int4,axiom,
! [A2: int,B2: int] :
( finite_finite2 @ int
@ ( collect @ int
@ ^ [I: int] :
( ( ord_less @ int @ A2 @ I )
& ( ord_less @ int @ I @ B2 ) ) ) ) ).
% finite_interval_int4
thf(fact_4959_finite__interval__int3,axiom,
! [A2: int,B2: int] :
( finite_finite2 @ int
@ ( collect @ int
@ ^ [I: int] :
( ( ord_less @ int @ A2 @ I )
& ( ord_less_eq @ int @ I @ B2 ) ) ) ) ).
% finite_interval_int3
thf(fact_4960_finite__interval__int2,axiom,
! [A2: int,B2: int] :
( finite_finite2 @ int
@ ( collect @ int
@ ^ [I: int] :
( ( ord_less_eq @ int @ A2 @ I )
& ( ord_less @ int @ I @ B2 ) ) ) ) ).
% finite_interval_int2
thf(fact_4961_finite__nth__roots,axiom,
! [N: nat,C2: complex] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( finite_finite2 @ complex
@ ( collect @ complex
@ ^ [Z6: complex] :
( ( power_power @ complex @ Z6 @ N )
= C2 ) ) ) ) ).
% finite_nth_roots
thf(fact_4962_finite__maxlen,axiom,
! [A: $tType,M7: set @ ( list @ A )] :
( ( finite_finite2 @ ( list @ A ) @ M7 )
=> ? [N3: nat] :
! [X2: list @ A] :
( ( member @ ( list @ A ) @ X2 @ M7 )
=> ( ord_less @ nat @ ( size_size @ ( list @ A ) @ X2 ) @ N3 ) ) ) ).
% finite_maxlen
thf(fact_4963_finite__enat__bounded,axiom,
! [A3: set @ extended_enat,N: nat] :
( ! [Y7: extended_enat] :
( ( member @ extended_enat @ Y7 @ A3 )
=> ( ord_less_eq @ extended_enat @ Y7 @ ( extended_enat2 @ N ) ) )
=> ( finite_finite2 @ extended_enat @ A3 ) ) ).
% finite_enat_bounded
thf(fact_4964_finite__divisors__int,axiom,
! [I2: int] :
( ( I2
!= ( zero_zero @ int ) )
=> ( finite_finite2 @ int
@ ( collect @ int
@ ^ [D5: int] : ( dvd_dvd @ int @ D5 @ I2 ) ) ) ) ).
% finite_divisors_int
thf(fact_4965_pigeonhole__infinite__rel,axiom,
! [B: $tType,A: $tType,A3: set @ A,B5: set @ B,R4: A > B > $o] :
( ~ ( finite_finite2 @ A @ A3 )
=> ( ( finite_finite2 @ B @ B5 )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ A3 )
=> ? [Xa2: B] :
( ( member @ B @ Xa2 @ B5 )
& ( R4 @ X5 @ Xa2 ) ) )
=> ? [X5: B] :
( ( member @ B @ X5 @ B5 )
& ~ ( finite_finite2 @ A
@ ( collect @ A
@ ^ [A5: A] :
( ( member @ A @ A5 @ A3 )
& ( R4 @ A5 @ X5 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_4966_not__finite__existsD,axiom,
! [A: $tType,P3: A > $o] :
( ~ ( finite_finite2 @ A @ ( collect @ A @ P3 ) )
=> ? [X_12: A] : ( P3 @ X_12 ) ) ).
% not_finite_existsD
thf(fact_4967_finite__has__minimal2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: set @ A,A2: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( member @ A @ A2 @ A3 )
=> ? [X5: A] :
( ( member @ A @ X5 @ A3 )
& ( ord_less_eq @ A @ X5 @ A2 )
& ! [Xa2: A] :
( ( member @ A @ Xa2 @ A3 )
=> ( ( ord_less_eq @ A @ Xa2 @ X5 )
=> ( X5 = Xa2 ) ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_4968_finite__has__maximal2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: set @ A,A2: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( member @ A @ A2 @ A3 )
=> ? [X5: A] :
( ( member @ A @ X5 @ A3 )
& ( ord_less_eq @ A @ A2 @ X5 )
& ! [Xa2: A] :
( ( member @ A @ Xa2 @ A3 )
=> ( ( ord_less_eq @ A @ X5 @ Xa2 )
=> ( X5 = Xa2 ) ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_4969_rev__finite__subset,axiom,
! [A: $tType,B5: set @ A,A3: set @ A] :
( ( finite_finite2 @ A @ B5 )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ B5 )
=> ( finite_finite2 @ A @ A3 ) ) ) ).
% rev_finite_subset
thf(fact_4970_infinite__super,axiom,
! [A: $tType,S3: set @ A,T5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ S3 @ T5 )
=> ( ~ ( finite_finite2 @ A @ S3 )
=> ~ ( finite_finite2 @ A @ T5 ) ) ) ).
% infinite_super
thf(fact_4971_finite__subset,axiom,
! [A: $tType,A3: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B5 )
=> ( ( finite_finite2 @ A @ B5 )
=> ( finite_finite2 @ A @ A3 ) ) ) ).
% finite_subset
thf(fact_4972_finite__nat__iff__bounded__le,axiom,
( ( finite_finite2 @ nat )
= ( ^ [S6: set @ nat] :
? [K3: nat] : ( ord_less_eq @ ( set @ nat ) @ S6 @ ( set_ord_atMost @ nat @ K3 ) ) ) ) ).
% finite_nat_iff_bounded_le
thf(fact_4973_finite__nat__iff__bounded,axiom,
( ( finite_finite2 @ nat )
= ( ^ [S6: set @ nat] :
? [K3: nat] : ( ord_less_eq @ ( set @ nat ) @ S6 @ ( set_ord_lessThan @ nat @ K3 ) ) ) ) ).
% finite_nat_iff_bounded
thf(fact_4974_finite__nat__bounded,axiom,
! [S3: set @ nat] :
( ( finite_finite2 @ nat @ S3 )
=> ? [K: nat] : ( ord_less_eq @ ( set @ nat ) @ S3 @ ( set_ord_lessThan @ nat @ K ) ) ) ).
% finite_nat_bounded
thf(fact_4975_infinite__int__iff__unbounded__le,axiom,
! [S3: set @ int] :
( ( ~ ( finite_finite2 @ int @ S3 ) )
= ( ! [M2: int] :
? [N2: int] :
( ( ord_less_eq @ int @ M2 @ ( abs_abs @ int @ N2 ) )
& ( member @ int @ N2 @ S3 ) ) ) ) ).
% infinite_int_iff_unbounded_le
thf(fact_4976_unbounded__k__infinite,axiom,
! [K2: nat,S3: set @ nat] :
( ! [M4: nat] :
( ( ord_less @ nat @ K2 @ M4 )
=> ? [N9: nat] :
( ( ord_less @ nat @ M4 @ N9 )
& ( member @ nat @ N9 @ S3 ) ) )
=> ~ ( finite_finite2 @ nat @ S3 ) ) ).
% unbounded_k_infinite
thf(fact_4977_infinite__nat__iff__unbounded,axiom,
! [S3: set @ nat] :
( ( ~ ( finite_finite2 @ nat @ S3 ) )
= ( ! [M2: nat] :
? [N2: nat] :
( ( ord_less @ nat @ M2 @ N2 )
& ( member @ nat @ N2 @ S3 ) ) ) ) ).
% infinite_nat_iff_unbounded
thf(fact_4978_infinite__nat__iff__unbounded__le,axiom,
! [S3: set @ nat] :
( ( ~ ( finite_finite2 @ nat @ S3 ) )
= ( ! [M2: nat] :
? [N2: nat] :
( ( ord_less_eq @ nat @ M2 @ N2 )
& ( member @ nat @ N2 @ S3 ) ) ) ) ).
% infinite_nat_iff_unbounded_le
thf(fact_4979_sum__count__set,axiom,
! [A: $tType,Xs: list @ A,X9: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ X9 )
=> ( ( finite_finite2 @ A @ X9 )
=> ( ( groups7311177749621191930dd_sum @ A @ nat @ ( count_list @ A @ Xs ) @ X9 )
= ( size_size @ ( list @ A ) @ Xs ) ) ) ) ).
% sum_count_set
thf(fact_4980_set__encode__insert,axiom,
! [A3: set @ nat,N: nat] :
( ( finite_finite2 @ nat @ A3 )
=> ( ~ ( member @ nat @ N @ A3 )
=> ( ( nat_set_encode @ ( insert @ nat @ N @ A3 ) )
= ( plus_plus @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ ( nat_set_encode @ A3 ) ) ) ) ) ).
% set_encode_insert
thf(fact_4981_even__sum__iff,axiom,
! [A: $tType,B: $tType] :
( ( semiring_parity @ A )
=> ! [A3: set @ B,F3: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ A3 ) )
= ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) )
@ ( finite_card @ B
@ ( collect @ B
@ ^ [A5: B] :
( ( member @ B @ A5 @ A3 )
& ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( F3 @ A5 ) ) ) ) ) ) ) ) ) ).
% even_sum_iff
thf(fact_4982_card__Collect__less__nat,axiom,
! [N: nat] :
( ( finite_card @ nat
@ ( collect @ nat
@ ^ [I: nat] : ( ord_less @ nat @ I @ N ) ) )
= N ) ).
% card_Collect_less_nat
thf(fact_4983_insert__subset,axiom,
! [A: $tType,X: A,A3: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X @ A3 ) @ B5 )
= ( ( member @ A @ X @ B5 )
& ( ord_less_eq @ ( set @ A ) @ A3 @ B5 ) ) ) ).
% insert_subset
thf(fact_4984_card__Collect__le__nat,axiom,
! [N: nat] :
( ( finite_card @ nat
@ ( collect @ nat
@ ^ [I: nat] : ( ord_less_eq @ nat @ I @ N ) ) )
= ( suc @ N ) ) ).
% card_Collect_le_nat
thf(fact_4985_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_4986_prod__constant,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [Y2: A,A3: set @ B] :
( ( groups7121269368397514597t_prod @ B @ A
@ ^ [X4: B] : Y2
@ A3 )
= ( power_power @ A @ Y2 @ ( finite_card @ B @ A3 ) ) ) ) ).
% prod_constant
thf(fact_4987_sum_Oinsert,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A3: set @ B,X: B,G: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ~ ( member @ B @ X @ A3 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( insert @ B @ X @ A3 ) )
= ( plus_plus @ A @ ( G @ X ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ A3 ) ) ) ) ) ) ).
% sum.insert
thf(fact_4988_prod_Oinsert,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: set @ B,X: B,G: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ~ ( member @ B @ X @ A3 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( insert @ B @ X @ A3 ) )
= ( times_times @ A @ ( G @ X ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ A3 ) ) ) ) ) ) ).
% prod.insert
thf(fact_4989_sum__constant,axiom,
! [B: $tType,A: $tType] :
( ( semiring_1 @ A )
=> ! [Y2: A,A3: set @ B] :
( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X4: B] : Y2
@ A3 )
= ( times_times @ A @ ( semiring_1_of_nat @ A @ ( finite_card @ B @ A3 ) ) @ Y2 ) ) ) ).
% sum_constant
thf(fact_4990_card__Diff__insert,axiom,
! [A: $tType,A2: A,A3: set @ A,B5: set @ A] :
( ( member @ A @ A2 @ A3 )
=> ( ~ ( member @ A @ A2 @ B5 )
=> ( ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ A2 @ B5 ) ) )
= ( minus_minus @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A3 @ B5 ) ) @ ( one_one @ nat ) ) ) ) ) ).
% card_Diff_insert
thf(fact_4991_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_4992_n__subsets,axiom,
! [A: $tType,A3: set @ A,K2: nat] :
( ( finite_finite2 @ A @ A3 )
=> ( ( finite_card @ ( set @ A )
@ ( collect @ ( set @ A )
@ ^ [B7: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ B7 @ A3 )
& ( ( finite_card @ A @ B7 )
= K2 ) ) ) )
= ( binomial @ ( finite_card @ A @ A3 ) @ K2 ) ) ) ).
% n_subsets
thf(fact_4993_card__le__Suc__iff,axiom,
! [A: $tType,N: nat,A3: set @ A] :
( ( ord_less_eq @ nat @ ( suc @ N ) @ ( finite_card @ A @ A3 ) )
= ( ? [A5: A,B7: set @ A] :
( ( A3
= ( insert @ A @ A5 @ B7 ) )
& ~ ( member @ A @ A5 @ B7 )
& ( ord_less_eq @ nat @ N @ ( finite_card @ A @ B7 ) )
& ( finite_finite2 @ A @ B7 ) ) ) ) ).
% card_le_Suc_iff
thf(fact_4994_insert__mono,axiom,
! [A: $tType,C5: set @ A,D4: set @ A,A2: A] :
( ( ord_less_eq @ ( set @ A ) @ C5 @ D4 )
=> ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ A2 @ C5 ) @ ( insert @ A @ A2 @ D4 ) ) ) ).
% insert_mono
thf(fact_4995_subset__insert,axiom,
! [A: $tType,X: A,A3: set @ A,B5: set @ A] :
( ~ ( member @ A @ X @ A3 )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ ( insert @ A @ X @ B5 ) )
= ( ord_less_eq @ ( set @ A ) @ A3 @ B5 ) ) ) ).
% subset_insert
thf(fact_4996_subset__insertI,axiom,
! [A: $tType,B5: set @ A,A2: A] : ( ord_less_eq @ ( set @ A ) @ B5 @ ( insert @ A @ A2 @ B5 ) ) ).
% subset_insertI
thf(fact_4997_subset__insertI2,axiom,
! [A: $tType,A3: set @ A,B5: set @ A,B2: A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B5 )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ ( insert @ A @ B2 @ B5 ) ) ) ).
% subset_insertI2
thf(fact_4998_insert__compr,axiom,
! [A: $tType] :
( ( insert @ A )
= ( ^ [A5: A,B7: set @ A] :
( collect @ A
@ ^ [X4: A] :
( ( X4 = A5 )
| ( member @ A @ X4 @ B7 ) ) ) ) ) ).
% insert_compr
thf(fact_4999_insert__Collect,axiom,
! [A: $tType,A2: A,P3: A > $o] :
( ( insert @ A @ A2 @ ( collect @ A @ P3 ) )
= ( collect @ A
@ ^ [U2: A] :
( ( U2 != A2 )
=> ( P3 @ U2 ) ) ) ) ).
% insert_Collect
thf(fact_5000_card__insert__le,axiom,
! [A: $tType,A3: set @ A,X: A] : ( ord_less_eq @ nat @ ( finite_card @ A @ A3 ) @ ( finite_card @ A @ ( insert @ A @ X @ A3 ) ) ) ).
% card_insert_le
thf(fact_5001_card__subset__eq,axiom,
! [A: $tType,B5: set @ A,A3: set @ A] :
( ( finite_finite2 @ A @ B5 )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ B5 )
=> ( ( ( finite_card @ A @ A3 )
= ( finite_card @ A @ B5 ) )
=> ( A3 = B5 ) ) ) ) ).
% card_subset_eq
thf(fact_5002_infinite__arbitrarily__large,axiom,
! [A: $tType,A3: set @ A,N: nat] :
( ~ ( finite_finite2 @ A @ A3 )
=> ? [B9: set @ A] :
( ( finite_finite2 @ A @ B9 )
& ( ( finite_card @ A @ B9 )
= N )
& ( ord_less_eq @ ( set @ A ) @ B9 @ A3 ) ) ) ).
% infinite_arbitrarily_large
thf(fact_5003_card__le__if__inj__on__rel,axiom,
! [B: $tType,A: $tType,B5: set @ A,A3: set @ B,R: B > A > $o] :
( ( finite_finite2 @ A @ B5 )
=> ( ! [A4: B] :
( ( member @ B @ A4 @ A3 )
=> ? [B10: A] :
( ( member @ A @ B10 @ B5 )
& ( R @ A4 @ B10 ) ) )
=> ( ! [A13: B,A24: B,B3: A] :
( ( member @ B @ A13 @ A3 )
=> ( ( member @ B @ A24 @ A3 )
=> ( ( member @ A @ B3 @ B5 )
=> ( ( R @ A13 @ B3 )
=> ( ( R @ A24 @ B3 )
=> ( A13 = A24 ) ) ) ) ) )
=> ( ord_less_eq @ nat @ ( finite_card @ B @ A3 ) @ ( finite_card @ A @ B5 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_5004_subset__Diff__insert,axiom,
! [A: $tType,A3: set @ A,B5: set @ A,X: A,C5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ ( minus_minus @ ( set @ A ) @ B5 @ ( insert @ A @ X @ C5 ) ) )
= ( ( ord_less_eq @ ( set @ A ) @ A3 @ ( minus_minus @ ( set @ A ) @ B5 @ C5 ) )
& ~ ( member @ A @ X @ A3 ) ) ) ).
% subset_Diff_insert
thf(fact_5005_card__insert__le__m1,axiom,
! [A: $tType,N: nat,Y2: set @ A,X: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ nat @ ( finite_card @ A @ Y2 ) @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) )
=> ( ord_less_eq @ nat @ ( finite_card @ A @ ( insert @ A @ X @ Y2 ) ) @ N ) ) ) ).
% card_insert_le_m1
thf(fact_5006_sum__multicount__gen,axiom,
! [A: $tType,B: $tType,S: set @ A,T2: set @ B,R4: A > B > $o,K2: B > nat] :
( ( finite_finite2 @ A @ S )
=> ( ( finite_finite2 @ B @ T2 )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ T2 )
=> ( ( finite_card @ A
@ ( collect @ A
@ ^ [I: A] :
( ( member @ A @ I @ S )
& ( R4 @ I @ X5 ) ) ) )
= ( K2 @ X5 ) ) )
=> ( ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [I: A] :
( finite_card @ B
@ ( collect @ B
@ ^ [J4: B] :
( ( member @ B @ J4 @ T2 )
& ( R4 @ I @ J4 ) ) ) )
@ S )
= ( groups7311177749621191930dd_sum @ B @ nat @ K2 @ T2 ) ) ) ) ) ).
% sum_multicount_gen
thf(fact_5007_card__lists__length__eq,axiom,
! [A: $tType,A3: set @ A,N: nat] :
( ( finite_finite2 @ A @ A3 )
=> ( ( finite_card @ ( list @ A )
@ ( collect @ ( list @ A )
@ ^ [Xs2: list @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs2 ) @ A3 )
& ( ( size_size @ ( list @ A ) @ Xs2 )
= N ) ) ) )
= ( power_power @ nat @ ( finite_card @ A @ A3 ) @ N ) ) ) ).
% card_lists_length_eq
thf(fact_5008_card__eq__sum,axiom,
! [A: $tType] :
( ( finite_card @ A )
= ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [X4: A] : ( one_one @ nat ) ) ) ).
% card_eq_sum
thf(fact_5009_card__2__iff_H,axiom,
! [A: $tType,S3: set @ A] :
( ( ( finite_card @ A @ S3 )
= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( ? [X4: A] :
( ( member @ A @ X4 @ S3 )
& ? [Y: A] :
( ( member @ A @ Y @ S3 )
& ( X4 != Y )
& ! [Z6: A] :
( ( member @ A @ Z6 @ S3 )
=> ( ( Z6 = X4 )
| ( Z6 = Y ) ) ) ) ) ) ) ).
% card_2_iff'
thf(fact_5010_card__ge__0__finite,axiom,
! [A: $tType,A3: set @ A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( finite_card @ A @ A3 ) )
=> ( finite_finite2 @ A @ A3 ) ) ).
% card_ge_0_finite
thf(fact_5011_card__mono,axiom,
! [A: $tType,B5: set @ A,A3: set @ A] :
( ( finite_finite2 @ A @ B5 )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ B5 )
=> ( ord_less_eq @ nat @ ( finite_card @ A @ A3 ) @ ( finite_card @ A @ B5 ) ) ) ) ).
% card_mono
thf(fact_5012_card__seteq,axiom,
! [A: $tType,B5: set @ A,A3: set @ A] :
( ( finite_finite2 @ A @ B5 )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ B5 )
=> ( ( ord_less_eq @ nat @ ( finite_card @ A @ B5 ) @ ( finite_card @ A @ A3 ) )
=> ( A3 = B5 ) ) ) ) ).
% card_seteq
thf(fact_5013_finite__if__finite__subsets__card__bdd,axiom,
! [A: $tType,F4: set @ A,C5: nat] :
( ! [G3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ G3 @ F4 )
=> ( ( finite_finite2 @ A @ G3 )
=> ( ord_less_eq @ nat @ ( finite_card @ A @ G3 ) @ C5 ) ) )
=> ( ( finite_finite2 @ A @ F4 )
& ( ord_less_eq @ nat @ ( finite_card @ A @ F4 ) @ C5 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_5014_obtain__subset__with__card__n,axiom,
! [A: $tType,N: nat,S3: set @ A] :
( ( ord_less_eq @ nat @ N @ ( finite_card @ A @ S3 ) )
=> ~ ! [T7: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ T7 @ S3 )
=> ( ( ( finite_card @ A @ T7 )
= N )
=> ~ ( finite_finite2 @ A @ T7 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_5015_card__less__sym__Diff,axiom,
! [A: $tType,A3: set @ A,B5: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( finite_finite2 @ A @ B5 )
=> ( ( ord_less @ nat @ ( finite_card @ A @ A3 ) @ ( finite_card @ A @ B5 ) )
=> ( ord_less @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A3 @ B5 ) ) @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ B5 @ A3 ) ) ) ) ) ) ).
% card_less_sym_Diff
thf(fact_5016_card__le__sym__Diff,axiom,
! [A: $tType,A3: set @ A,B5: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( finite_finite2 @ A @ B5 )
=> ( ( ord_less_eq @ nat @ ( finite_card @ A @ A3 ) @ ( finite_card @ A @ B5 ) )
=> ( ord_less_eq @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A3 @ B5 ) ) @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ B5 @ A3 ) ) ) ) ) ) ).
% card_le_sym_Diff
thf(fact_5017_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_5018_psubset__card__mono,axiom,
! [A: $tType,B5: set @ A,A3: set @ A] :
( ( finite_finite2 @ A @ B5 )
=> ( ( ord_less @ ( set @ A ) @ A3 @ B5 )
=> ( ord_less @ nat @ ( finite_card @ A @ A3 ) @ ( finite_card @ A @ B5 ) ) ) ) ).
% psubset_card_mono
thf(fact_5019_sum_Oinsert__if,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A3: set @ B,X: B,G: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ( ( member @ B @ X @ A3 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( insert @ B @ X @ A3 ) )
= ( groups7311177749621191930dd_sum @ B @ A @ G @ A3 ) ) )
& ( ~ ( member @ B @ X @ A3 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( insert @ B @ X @ A3 ) )
= ( plus_plus @ A @ ( G @ X ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ A3 ) ) ) ) ) ) ) ).
% sum.insert_if
thf(fact_5020_prod_Oinsert__if,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: set @ B,X: B,G: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ( ( member @ B @ X @ A3 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( insert @ B @ X @ A3 ) )
= ( groups7121269368397514597t_prod @ B @ A @ G @ A3 ) ) )
& ( ~ ( member @ B @ X @ A3 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( insert @ B @ X @ A3 ) )
= ( times_times @ A @ ( G @ X ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ A3 ) ) ) ) ) ) ) ).
% prod.insert_if
thf(fact_5021_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_5022_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_5023_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_5024_lessThan__nat__numeral,axiom,
! [K2: num] :
( ( set_ord_lessThan @ nat @ ( numeral_numeral @ nat @ K2 ) )
= ( insert @ nat @ ( pred_numeral @ K2 ) @ ( set_ord_lessThan @ nat @ ( pred_numeral @ K2 ) ) ) ) ).
% lessThan_nat_numeral
thf(fact_5025_card__less,axiom,
! [M7: set @ nat,I2: nat] :
( ( member @ nat @ ( zero_zero @ nat ) @ M7 )
=> ( ( finite_card @ nat
@ ( collect @ nat
@ ^ [K3: nat] :
( ( member @ nat @ K3 @ M7 )
& ( ord_less @ nat @ K3 @ ( suc @ I2 ) ) ) ) )
!= ( zero_zero @ nat ) ) ) ).
% card_less
thf(fact_5026_card__less__Suc,axiom,
! [M7: set @ nat,I2: nat] :
( ( member @ nat @ ( zero_zero @ nat ) @ M7 )
=> ( ( suc
@ ( finite_card @ nat
@ ( collect @ nat
@ ^ [K3: nat] :
( ( member @ nat @ ( suc @ K3 ) @ M7 )
& ( ord_less @ nat @ K3 @ I2 ) ) ) ) )
= ( finite_card @ nat
@ ( collect @ nat
@ ^ [K3: nat] :
( ( member @ nat @ K3 @ M7 )
& ( ord_less @ nat @ K3 @ ( suc @ I2 ) ) ) ) ) ) ) ).
% card_less_Suc
thf(fact_5027_card__less__Suc2,axiom,
! [M7: set @ nat,I2: nat] :
( ~ ( member @ nat @ ( zero_zero @ nat ) @ M7 )
=> ( ( finite_card @ nat
@ ( collect @ nat
@ ^ [K3: nat] :
( ( member @ nat @ ( suc @ K3 ) @ M7 )
& ( ord_less @ nat @ K3 @ I2 ) ) ) )
= ( finite_card @ nat
@ ( collect @ nat
@ ^ [K3: nat] :
( ( member @ nat @ K3 @ M7 )
& ( ord_less @ nat @ K3 @ ( suc @ I2 ) ) ) ) ) ) ) ).
% card_less_Suc2
thf(fact_5028_atMost__nat__numeral,axiom,
! [K2: num] :
( ( set_ord_atMost @ nat @ ( numeral_numeral @ nat @ K2 ) )
= ( insert @ nat @ ( numeral_numeral @ nat @ K2 ) @ ( set_ord_atMost @ nat @ ( pred_numeral @ K2 ) ) ) ) ).
% atMost_nat_numeral
thf(fact_5029_sum__constant__scaleR,axiom,
! [C: $tType,A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [Y2: A,A3: set @ C] :
( ( groups7311177749621191930dd_sum @ C @ A
@ ^ [X4: C] : Y2
@ A3 )
= ( real_V8093663219630862766scaleR @ A @ ( semiring_1_of_nat @ real @ ( finite_card @ C @ A3 ) ) @ Y2 ) ) ) ).
% sum_constant_scaleR
thf(fact_5030_sum__Suc,axiom,
! [A: $tType,F3: A > nat,A3: set @ A] :
( ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [X4: A] : ( suc @ ( F3 @ X4 ) )
@ A3 )
= ( plus_plus @ nat @ ( groups7311177749621191930dd_sum @ A @ nat @ F3 @ A3 ) @ ( finite_card @ A @ A3 ) ) ) ).
% sum_Suc
thf(fact_5031_sum__multicount,axiom,
! [A: $tType,B: $tType,S3: set @ A,T5: set @ B,R4: A > B > $o,K2: nat] :
( ( finite_finite2 @ A @ S3 )
=> ( ( finite_finite2 @ B @ T5 )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ T5 )
=> ( ( finite_card @ A
@ ( collect @ A
@ ^ [I: A] :
( ( member @ A @ I @ S3 )
& ( R4 @ I @ X5 ) ) ) )
= K2 ) )
=> ( ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [I: A] :
( finite_card @ B
@ ( collect @ B
@ ^ [J4: B] :
( ( member @ B @ J4 @ T5 )
& ( R4 @ I @ J4 ) ) ) )
@ S3 )
= ( times_times @ nat @ K2 @ ( finite_card @ B @ T5 ) ) ) ) ) ) ).
% sum_multicount
thf(fact_5032_subset__card__intvl__is__intvl,axiom,
! [A3: set @ nat,K2: nat] :
( ( ord_less_eq @ ( set @ nat ) @ A3 @ ( set_or7035219750837199246ssThan @ nat @ K2 @ ( plus_plus @ nat @ K2 @ ( finite_card @ nat @ A3 ) ) ) )
=> ( A3
= ( set_or7035219750837199246ssThan @ nat @ K2 @ ( plus_plus @ nat @ K2 @ ( finite_card @ nat @ A3 ) ) ) ) ) ).
% subset_card_intvl_is_intvl
thf(fact_5033_real__of__card,axiom,
! [A: $tType,A3: set @ A] :
( ( semiring_1_of_nat @ real @ ( finite_card @ A @ A3 ) )
= ( groups7311177749621191930dd_sum @ A @ real
@ ^ [X4: A] : ( one_one @ real )
@ A3 ) ) ).
% real_of_card
thf(fact_5034_sum__bounded__below,axiom,
! [A: $tType,B: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( semiring_1 @ A ) )
=> ! [A3: set @ B,K5: A,F3: B > A] :
( ! [I3: B] :
( ( member @ B @ I3 @ A3 )
=> ( ord_less_eq @ A @ K5 @ ( F3 @ I3 ) ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( finite_card @ B @ A3 ) ) @ K5 ) @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ A3 ) ) ) ) ).
% sum_bounded_below
thf(fact_5035_sum__bounded__above,axiom,
! [B: $tType,A: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( semiring_1 @ A ) )
=> ! [A3: set @ B,F3: B > A,K5: A] :
( ! [I3: B] :
( ( member @ B @ I3 @ A3 )
=> ( ord_less_eq @ A @ ( F3 @ I3 ) @ K5 ) )
=> ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ A3 ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( finite_card @ B @ A3 ) ) @ K5 ) ) ) ) ).
% sum_bounded_above
thf(fact_5036_card__le__Suc0__iff__eq,axiom,
! [A: $tType,A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( ord_less_eq @ nat @ ( finite_card @ A @ A3 ) @ ( suc @ ( zero_zero @ nat ) ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ A3 )
=> ! [Y: A] :
( ( member @ A @ Y @ A3 )
=> ( X4 = Y ) ) ) ) ) ) ).
% card_le_Suc0_iff_eq
thf(fact_5037_card__Diff__subset,axiom,
! [A: $tType,B5: set @ A,A3: set @ A] :
( ( finite_finite2 @ A @ B5 )
=> ( ( ord_less_eq @ ( set @ A ) @ B5 @ A3 )
=> ( ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A3 @ B5 ) )
= ( minus_minus @ nat @ ( finite_card @ A @ A3 ) @ ( finite_card @ A @ B5 ) ) ) ) ) ).
% card_Diff_subset
thf(fact_5038_card__psubset,axiom,
! [A: $tType,B5: set @ A,A3: set @ A] :
( ( finite_finite2 @ A @ B5 )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ B5 )
=> ( ( ord_less @ nat @ ( finite_card @ A @ A3 ) @ ( finite_card @ A @ B5 ) )
=> ( ord_less @ ( set @ A ) @ A3 @ B5 ) ) ) ) ).
% card_psubset
thf(fact_5039_diff__card__le__card__Diff,axiom,
! [A: $tType,B5: set @ A,A3: set @ A] :
( ( finite_finite2 @ A @ B5 )
=> ( ord_less_eq @ nat @ ( minus_minus @ nat @ ( finite_card @ A @ A3 ) @ ( finite_card @ A @ B5 ) ) @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A3 @ B5 ) ) ) ) ).
% diff_card_le_card_Diff
thf(fact_5040_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_5041_card__lists__length__le,axiom,
! [A: $tType,A3: set @ A,N: nat] :
( ( finite_finite2 @ A @ A3 )
=> ( ( finite_card @ ( list @ A )
@ ( collect @ ( list @ A )
@ ^ [Xs2: list @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs2 ) @ A3 )
& ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ N ) ) ) )
= ( groups7311177749621191930dd_sum @ nat @ nat @ ( power_power @ nat @ ( finite_card @ A @ A3 ) ) @ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% card_lists_length_le
thf(fact_5042_ex__bij__betw__nat__finite__1,axiom,
! [A: $tType,M7: set @ A] :
( ( finite_finite2 @ A @ M7 )
=> ? [H4: nat > A] : ( bij_betw @ nat @ A @ H4 @ ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ ( finite_card @ A @ M7 ) ) @ M7 ) ) ).
% ex_bij_betw_nat_finite_1
thf(fact_5043_card__roots__unity,axiom,
! [A: $tType] :
( ( ( real_V8999393235501362500lgebra @ A )
& ( idom @ A ) )
=> ! [N: nat] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ N )
=> ( ord_less_eq @ nat
@ ( finite_card @ A
@ ( collect @ A
@ ^ [Z6: A] :
( ( power_power @ A @ Z6 @ N )
= ( one_one @ A ) ) ) )
@ N ) ) ) ).
% card_roots_unity
thf(fact_5044_subset__eq__atLeast0__lessThan__card,axiom,
! [N4: set @ nat,N: nat] :
( ( ord_less_eq @ ( set @ nat ) @ N4 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) )
=> ( ord_less_eq @ nat @ ( finite_card @ nat @ N4 ) @ N ) ) ).
% subset_eq_atLeast0_lessThan_card
thf(fact_5045_card__sum__le__nat__sum,axiom,
! [S3: set @ nat] :
( ord_less_eq @ nat
@ ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [X4: nat] : X4
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( finite_card @ nat @ S3 ) ) )
@ ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [X4: nat] : X4
@ S3 ) ) ).
% card_sum_le_nat_sum
thf(fact_5046_card__nth__roots,axiom,
! [C2: complex,N: nat] :
( ( C2
!= ( zero_zero @ complex ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( finite_card @ complex
@ ( collect @ complex
@ ^ [Z6: complex] :
( ( power_power @ complex @ Z6 @ N )
= C2 ) ) )
= N ) ) ) ).
% card_nth_roots
thf(fact_5047_card__roots__unity__eq,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( finite_card @ complex
@ ( collect @ complex
@ ^ [Z6: complex] :
( ( power_power @ complex @ Z6 @ N )
= ( one_one @ complex ) ) ) )
= N ) ) ).
% card_roots_unity_eq
thf(fact_5048_sum__norm__bound,axiom,
! [A: $tType,B: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [S3: set @ B,F3: B > A,K5: real] :
( ! [X5: B] :
( ( member @ B @ X5 @ S3 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( F3 @ X5 ) ) @ K5 ) )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ S3 ) ) @ ( times_times @ real @ ( semiring_1_of_nat @ real @ ( finite_card @ B @ S3 ) ) @ K5 ) ) ) ) ).
% sum_norm_bound
thf(fact_5049_prod__le__power,axiom,
! [B: $tType,A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: set @ B,F3: B > A,N: A,K2: nat] :
( ! [I3: B] :
( ( member @ B @ I3 @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F3 @ I3 ) )
& ( ord_less_eq @ A @ ( F3 @ I3 ) @ N ) ) )
=> ( ( ord_less_eq @ nat @ ( finite_card @ B @ A3 ) @ K2 )
=> ( ( ord_less_eq @ A @ ( one_one @ A ) @ N )
=> ( ord_less_eq @ A @ ( groups7121269368397514597t_prod @ B @ A @ F3 @ A3 ) @ ( power_power @ A @ N @ K2 ) ) ) ) ) ) ).
% prod_le_power
thf(fact_5050_sum__bounded__above__strict,axiom,
! [B: $tType,A: $tType] :
( ( ( ordere8940638589300402666id_add @ A )
& ( semiring_1 @ A ) )
=> ! [A3: set @ B,F3: B > A,K5: A] :
( ! [I3: B] :
( ( member @ B @ I3 @ A3 )
=> ( ord_less @ A @ ( F3 @ I3 ) @ K5 ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( finite_card @ B @ A3 ) )
=> ( ord_less @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ A3 ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( finite_card @ B @ A3 ) ) @ K5 ) ) ) ) ) ).
% sum_bounded_above_strict
thf(fact_5051_polyfun__roots__card,axiom,
! [A: $tType] :
( ( ( real_V8999393235501362500lgebra @ A )
& ( idom @ A ) )
=> ! [C2: nat > A,K2: nat,N: nat] :
( ( ( C2 @ K2 )
!= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ nat @ K2 @ N )
=> ( ord_less_eq @ nat
@ ( finite_card @ A
@ ( collect @ A
@ ^ [Z6: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I: nat] : ( times_times @ A @ ( C2 @ I ) @ ( power_power @ A @ Z6 @ I ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( zero_zero @ A ) ) ) )
@ N ) ) ) ) ).
% polyfun_roots_card
thf(fact_5052_prod__gen__delta,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S3: set @ B,A2: B,B2: B > A,C2: A] :
( ( finite_finite2 @ B @ S3 )
=> ( ( ( member @ B @ A2 @ S3 )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K3: B] : ( if @ A @ ( K3 = A2 ) @ ( B2 @ K3 ) @ C2 )
@ S3 )
= ( times_times @ A @ ( B2 @ A2 ) @ ( power_power @ A @ C2 @ ( minus_minus @ nat @ ( finite_card @ B @ S3 ) @ ( one_one @ nat ) ) ) ) ) )
& ( ~ ( member @ B @ A2 @ S3 )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K3: B] : ( if @ A @ ( K3 = A2 ) @ ( B2 @ K3 ) @ C2 )
@ S3 )
= ( power_power @ A @ C2 @ ( finite_card @ B @ S3 ) ) ) ) ) ) ) ).
% prod_gen_delta
thf(fact_5053_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_5054_polyfun__rootbound,axiom,
! [A: $tType] :
( ( ( real_V8999393235501362500lgebra @ A )
& ( idom @ A ) )
=> ! [C2: nat > A,K2: nat,N: nat] :
( ( ( C2 @ K2 )
!= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ nat @ K2 @ N )
=> ( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [Z6: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I: nat] : ( times_times @ A @ ( C2 @ I ) @ ( power_power @ A @ Z6 @ I ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( zero_zero @ A ) ) ) )
& ( ord_less_eq @ nat
@ ( finite_card @ A
@ ( collect @ A
@ ^ [Z6: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I: nat] : ( times_times @ A @ ( C2 @ I ) @ ( power_power @ A @ Z6 @ I ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( zero_zero @ A ) ) ) )
@ N ) ) ) ) ) ).
% polyfun_rootbound
thf(fact_5055_card__lists__distinct__length__eq,axiom,
! [A: $tType,A3: set @ A,K2: nat] :
( ( finite_finite2 @ A @ A3 )
=> ( ( ord_less_eq @ nat @ K2 @ ( finite_card @ A @ A3 ) )
=> ( ( finite_card @ ( list @ A )
@ ( collect @ ( list @ A )
@ ^ [Xs2: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= K2 )
& ( distinct @ A @ Xs2 )
& ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs2 ) @ A3 ) ) ) )
= ( groups7121269368397514597t_prod @ nat @ nat
@ ^ [X4: nat] : X4
@ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ ( minus_minus @ nat @ ( finite_card @ A @ A3 ) @ K2 ) @ ( one_one @ nat ) ) @ ( finite_card @ A @ A3 ) ) ) ) ) ) ).
% card_lists_distinct_length_eq
thf(fact_5056_card__lists__distinct__length__eq_H,axiom,
! [A: $tType,K2: nat,A3: set @ A] :
( ( ord_less @ nat @ K2 @ ( finite_card @ A @ A3 ) )
=> ( ( finite_card @ ( list @ A )
@ ( collect @ ( list @ A )
@ ^ [Xs2: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= K2 )
& ( distinct @ A @ Xs2 )
& ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs2 ) @ A3 ) ) ) )
= ( groups7121269368397514597t_prod @ nat @ nat
@ ^ [X4: nat] : X4
@ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ ( minus_minus @ nat @ ( finite_card @ A @ A3 ) @ K2 ) @ ( one_one @ nat ) ) @ ( finite_card @ A @ A3 ) ) ) ) ) ).
% card_lists_distinct_length_eq'
thf(fact_5057_sum__list__map__eq__sum__count2,axiom,
! [A: $tType,Xs: list @ A,X9: set @ A,F3: A > nat] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ X9 )
=> ( ( finite_finite2 @ A @ X9 )
=> ( ( groups8242544230860333062m_list @ nat @ ( map @ A @ nat @ F3 @ Xs ) )
= ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [X4: A] : ( times_times @ nat @ ( count_list @ A @ Xs @ X4 ) @ ( F3 @ X4 ) )
@ X9 ) ) ) ) ).
% sum_list_map_eq_sum_count2
thf(fact_5058_sum__list__eq__0__iff,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [Ns: list @ A] :
( ( ( groups8242544230860333062m_list @ A @ Ns )
= ( zero_zero @ A ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Ns ) )
=> ( X4
= ( zero_zero @ A ) ) ) ) ) ) ).
% sum_list_eq_0_iff
thf(fact_5059_sum__list__0,axiom,
! [B: $tType,A: $tType] :
( ( monoid_add @ A )
=> ! [Xs: list @ B] :
( ( groups8242544230860333062m_list @ A
@ ( map @ B @ A
@ ^ [X4: B] : ( zero_zero @ A )
@ Xs ) )
= ( zero_zero @ A ) ) ) ).
% sum_list_0
thf(fact_5060_sum__list__upt,axiom,
! [M: nat,N: nat] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( groups8242544230860333062m_list @ nat @ ( upt @ M @ N ) )
= ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [X4: nat] : X4
@ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) ) ) ).
% sum_list_upt
thf(fact_5061_finite__lists__distinct__length__eq,axiom,
! [A: $tType,A3: set @ A,N: nat] :
( ( finite_finite2 @ A @ A3 )
=> ( finite_finite2 @ ( list @ A )
@ ( collect @ ( list @ A )
@ ^ [Xs2: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= N )
& ( distinct @ A @ Xs2 )
& ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs2 ) @ A3 ) ) ) ) ) ).
% finite_lists_distinct_length_eq
thf(fact_5062_distinct__sum__list__conv__Sum,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [Xs: list @ A] :
( ( distinct @ A @ Xs )
=> ( ( groups8242544230860333062m_list @ A @ Xs )
= ( groups7311177749621191930dd_sum @ A @ A
@ ^ [X4: A] : X4
@ ( set2 @ A @ Xs ) ) ) ) ) ).
% distinct_sum_list_conv_Sum
thf(fact_5063_distinct__take,axiom,
! [A: $tType,Xs: list @ A,I2: nat] :
( ( distinct @ A @ Xs )
=> ( distinct @ A @ ( take @ A @ I2 @ Xs ) ) ) ).
% distinct_take
thf(fact_5064_distinct__product,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B] :
( ( distinct @ A @ Xs )
=> ( ( distinct @ B @ Ys )
=> ( distinct @ ( product_prod @ A @ B ) @ ( product @ A @ B @ Xs @ Ys ) ) ) ) ).
% distinct_product
thf(fact_5065_sorted__list__of__set_Odistinct__if__distinct__map,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] :
( ( distinct @ A @ Xs )
=> ( distinct @ A @ Xs ) ) ) ).
% sorted_list_of_set.distinct_if_distinct_map
thf(fact_5066_distinct__upt,axiom,
! [I2: nat,J3: nat] : ( distinct @ nat @ ( upt @ I2 @ J3 ) ) ).
% distinct_upt
thf(fact_5067_distinct__set__subseqs,axiom,
! [A: $tType,Xs: list @ A] :
( ( distinct @ A @ Xs )
=> ( distinct @ ( set @ A ) @ ( map @ ( list @ A ) @ ( set @ A ) @ ( set2 @ A ) @ ( subseqs @ A @ Xs ) ) ) ) ).
% distinct_set_subseqs
thf(fact_5068_sum_Odistinct__set__conv__list,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [Xs: list @ B,G: B > A] :
( ( distinct @ B @ Xs )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( set2 @ B @ Xs ) )
= ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ G @ Xs ) ) ) ) ) ).
% sum.distinct_set_conv_list
thf(fact_5069_sum__list__distinct__conv__sum__set,axiom,
! [C: $tType,B: $tType] :
( ( comm_monoid_add @ C )
=> ! [Xs: list @ B,F3: B > C] :
( ( distinct @ B @ Xs )
=> ( ( groups8242544230860333062m_list @ C @ ( map @ B @ C @ F3 @ Xs ) )
= ( groups7311177749621191930dd_sum @ B @ C @ F3 @ ( set2 @ B @ Xs ) ) ) ) ) ).
% sum_list_distinct_conv_sum_set
thf(fact_5070_finite__distinct__list,axiom,
! [A: $tType,A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ? [Xs3: list @ A] :
( ( ( set2 @ A @ Xs3 )
= A3 )
& ( distinct @ A @ Xs3 ) ) ) ).
% finite_distinct_list
thf(fact_5071_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_5072_length__concat,axiom,
! [B: $tType,Xss: list @ ( list @ B )] :
( ( size_size @ ( list @ B ) @ ( concat @ B @ Xss ) )
= ( groups8242544230860333062m_list @ nat @ ( map @ ( list @ B ) @ nat @ ( size_size @ ( list @ B ) ) @ Xss ) ) ) ).
% length_concat
thf(fact_5073_subseqs__distinctD,axiom,
! [A: $tType,Ys: list @ A,Xs: list @ A] :
( ( member @ ( list @ A ) @ Ys @ ( set2 @ ( list @ A ) @ ( subseqs @ A @ Xs ) ) )
=> ( ( distinct @ A @ Xs )
=> ( distinct @ A @ Ys ) ) ) ).
% subseqs_distinctD
thf(fact_5074_sum__list__const__mult,axiom,
! [A: $tType,B: $tType] :
( ( semiring_0 @ A )
=> ! [C2: A,F3: B > A,Xs: list @ B] :
( ( groups8242544230860333062m_list @ A
@ ( map @ B @ A
@ ^ [X4: B] : ( times_times @ A @ C2 @ ( F3 @ X4 ) )
@ Xs ) )
= ( times_times @ A @ C2 @ ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ F3 @ Xs ) ) ) ) ) ).
% sum_list_const_mult
thf(fact_5075_sum__list__mult__const,axiom,
! [B: $tType,A: $tType] :
( ( semiring_0 @ A )
=> ! [F3: B > A,C2: A,Xs: list @ B] :
( ( groups8242544230860333062m_list @ A
@ ( map @ B @ A
@ ^ [X4: B] : ( times_times @ A @ ( F3 @ X4 ) @ C2 )
@ Xs ) )
= ( times_times @ A @ ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ F3 @ Xs ) ) @ C2 ) ) ) ).
% sum_list_mult_const
thf(fact_5076_sum__list__addf,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [F3: B > A,G: B > A,Xs: list @ B] :
( ( groups8242544230860333062m_list @ A
@ ( map @ B @ A
@ ^ [X4: B] : ( plus_plus @ A @ ( F3 @ X4 ) @ ( G @ X4 ) )
@ Xs ) )
= ( plus_plus @ A @ ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ F3 @ Xs ) ) @ ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ G @ Xs ) ) ) ) ) ).
% sum_list_addf
thf(fact_5077_sum__list__subtractf,axiom,
! [A: $tType,B: $tType] :
( ( ab_group_add @ A )
=> ! [F3: B > A,G: B > A,Xs: list @ B] :
( ( groups8242544230860333062m_list @ A
@ ( map @ B @ A
@ ^ [X4: B] : ( minus_minus @ A @ ( F3 @ X4 ) @ ( G @ X4 ) )
@ Xs ) )
= ( minus_minus @ A @ ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ F3 @ Xs ) ) @ ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ G @ Xs ) ) ) ) ) ).
% sum_list_subtractf
thf(fact_5078_distinct__card,axiom,
! [A: $tType,Xs: list @ A] :
( ( distinct @ A @ Xs )
=> ( ( finite_card @ A @ ( set2 @ A @ Xs ) )
= ( size_size @ ( list @ A ) @ Xs ) ) ) ).
% distinct_card
thf(fact_5079_card__distinct,axiom,
! [A: $tType,Xs: list @ A] :
( ( ( finite_card @ A @ ( set2 @ A @ Xs ) )
= ( size_size @ ( list @ A ) @ Xs ) )
=> ( distinct @ A @ Xs ) ) ).
% card_distinct
thf(fact_5080_distinct__conv__nth,axiom,
! [A: $tType] :
( ( distinct @ A )
= ( ^ [Xs2: list @ A] :
! [I: nat] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ! [J4: nat] :
( ( ord_less @ nat @ J4 @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( ( I != J4 )
=> ( ( nth @ A @ Xs2 @ I )
!= ( nth @ A @ Xs2 @ J4 ) ) ) ) ) ) ) ).
% distinct_conv_nth
thf(fact_5081_nth__eq__iff__index__eq,axiom,
! [A: $tType,Xs: list @ A,I2: nat,J3: nat] :
( ( distinct @ A @ Xs )
=> ( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ord_less @ nat @ J3 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ( nth @ A @ Xs @ I2 )
= ( nth @ A @ Xs @ J3 ) )
= ( I2 = J3 ) ) ) ) ) ).
% nth_eq_iff_index_eq
thf(fact_5082_Groups__List_Osum__list__nonneg,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [Xs: list @ A] :
( ! [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Xs ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ X5 ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( groups8242544230860333062m_list @ A @ Xs ) ) ) ) ).
% Groups_List.sum_list_nonneg
thf(fact_5083_sum__list__nonneg__eq__0__iff,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [Xs: list @ A] :
( ! [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Xs ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ X5 ) )
=> ( ( ( groups8242544230860333062m_list @ A @ Xs )
= ( zero_zero @ A ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
=> ( X4
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% sum_list_nonneg_eq_0_iff
thf(fact_5084_sum__list__nonpos,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [Xs: list @ A] :
( ! [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Xs ) )
=> ( ord_less_eq @ A @ X5 @ ( zero_zero @ A ) ) )
=> ( ord_less_eq @ A @ ( groups8242544230860333062m_list @ A @ Xs ) @ ( zero_zero @ A ) ) ) ) ).
% sum_list_nonpos
thf(fact_5085_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_5086_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_5087_sum__list__mono,axiom,
! [B: $tType,A: $tType] :
( ( ( monoid_add @ B )
& ( ordere6658533253407199908up_add @ B ) )
=> ! [Xs: list @ A,F3: A > B,G: A > B] :
( ! [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Xs ) )
=> ( ord_less_eq @ B @ ( F3 @ X5 ) @ ( G @ X5 ) ) )
=> ( ord_less_eq @ B @ ( groups8242544230860333062m_list @ B @ ( map @ A @ B @ F3 @ Xs ) ) @ ( groups8242544230860333062m_list @ B @ ( map @ A @ B @ G @ Xs ) ) ) ) ) ).
% sum_list_mono
thf(fact_5088_distinct__Ex1,axiom,
! [A: $tType,Xs: list @ A,X: A] :
( ( distinct @ A @ Xs )
=> ( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ? [X5: nat] :
( ( ord_less @ nat @ X5 @ ( size_size @ ( list @ A ) @ Xs ) )
& ( ( nth @ A @ Xs @ X5 )
= X )
& ! [Y4: nat] :
( ( ( ord_less @ nat @ Y4 @ ( size_size @ ( list @ A ) @ Xs ) )
& ( ( nth @ A @ Xs @ Y4 )
= X ) )
=> ( Y4 = X5 ) ) ) ) ) ).
% distinct_Ex1
thf(fact_5089_elem__le__sum__list,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [K2: nat,Ns: list @ A] :
( ( ord_less @ nat @ K2 @ ( size_size @ ( list @ A ) @ Ns ) )
=> ( ord_less_eq @ A @ ( nth @ A @ Ns @ K2 ) @ ( groups8242544230860333062m_list @ A @ Ns ) ) ) ) ).
% elem_le_sum_list
thf(fact_5090_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_5091_size__list__conv__sum__list,axiom,
! [B: $tType] :
( ( size_list @ B )
= ( ^ [F5: B > nat,Xs2: list @ B] : ( plus_plus @ nat @ ( groups8242544230860333062m_list @ nat @ ( map @ B @ nat @ F5 @ Xs2 ) ) @ ( size_size @ ( list @ B ) @ Xs2 ) ) ) ) ).
% size_list_conv_sum_list
thf(fact_5092_sum__list__triv,axiom,
! [C: $tType,B: $tType] :
( ( semiring_1 @ B )
=> ! [R: B,Xs: list @ C] :
( ( groups8242544230860333062m_list @ B
@ ( map @ C @ B
@ ^ [X4: C] : R
@ Xs ) )
= ( times_times @ B @ ( semiring_1_of_nat @ B @ ( size_size @ ( list @ C ) @ Xs ) ) @ R ) ) ) ).
% sum_list_triv
thf(fact_5093_sum__list__Suc,axiom,
! [A: $tType,F3: A > nat,Xs: list @ A] :
( ( groups8242544230860333062m_list @ nat
@ ( map @ A @ nat
@ ^ [X4: A] : ( suc @ ( F3 @ X4 ) )
@ Xs ) )
= ( plus_plus @ nat @ ( groups8242544230860333062m_list @ nat @ ( map @ A @ nat @ F3 @ Xs ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ).
% sum_list_Suc
thf(fact_5094_bij__betw__nth,axiom,
! [A: $tType,Xs: list @ A,A3: set @ nat,B5: set @ A] :
( ( distinct @ A @ Xs )
=> ( ( A3
= ( set_ord_lessThan @ nat @ ( size_size @ ( list @ A ) @ Xs ) ) )
=> ( ( B5
= ( set2 @ A @ Xs ) )
=> ( bij_betw @ nat @ A @ ( nth @ A @ Xs ) @ A3 @ B5 ) ) ) ) ).
% bij_betw_nth
thf(fact_5095_sum__list__sum__nth,axiom,
! [B: $tType] :
( ( comm_monoid_add @ B )
=> ( ( groups8242544230860333062m_list @ B )
= ( ^ [Xs2: list @ B] : ( groups7311177749621191930dd_sum @ nat @ B @ ( nth @ B @ Xs2 ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ B ) @ Xs2 ) ) ) ) ) ) ).
% sum_list_sum_nth
thf(fact_5096_card__length__sum__list__rec,axiom,
! [M: nat,N4: 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 )
= N4 ) ) ) )
= ( 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 )
= N4 ) ) ) )
@ ( 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 ) )
= N4 ) ) ) ) ) ) ) ).
% card_length_sum_list_rec
thf(fact_5097_card__length__sum__list,axiom,
! [M: nat,N4: nat] :
( ( finite_card @ ( list @ nat )
@ ( collect @ ( list @ nat )
@ ^ [L3: list @ nat] :
( ( ( size_size @ ( list @ nat ) @ L3 )
= M )
& ( ( groups8242544230860333062m_list @ nat @ L3 )
= N4 ) ) ) )
= ( binomial @ ( minus_minus @ nat @ ( plus_plus @ nat @ N4 @ M ) @ ( one_one @ nat ) ) @ N4 ) ) ).
% card_length_sum_list
thf(fact_5098_sum__list__map__eq__sum__count,axiom,
! [A: $tType,F3: A > nat,Xs: list @ A] :
( ( groups8242544230860333062m_list @ nat @ ( map @ A @ nat @ F3 @ Xs ) )
= ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [X4: A] : ( times_times @ nat @ ( count_list @ A @ Xs @ X4 ) @ ( F3 @ X4 ) )
@ ( set2 @ A @ Xs ) ) ) ).
% sum_list_map_eq_sum_count
thf(fact_5099_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_5100_and__int_Oelims,axiom,
! [X: int,Xa: int,Y2: int] :
( ( ( bit_se5824344872417868541ns_and @ int @ X @ Xa )
= Y2 )
=> ( ( ( ( member @ int @ X @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
& ( member @ int @ Xa @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) )
=> ( Y2
= ( uminus_uminus @ int
@ ( zero_neq_one_of_bool @ int
@ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ X )
& ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Xa ) ) ) ) ) )
& ( ~ ( ( member @ int @ X @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
& ( member @ int @ Xa @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) )
=> ( Y2
= ( plus_plus @ int
@ ( zero_neq_one_of_bool @ int
@ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ X )
& ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Xa ) ) )
@ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( divide_divide @ int @ X @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ Xa @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% and_int.elims
thf(fact_5101_and__int_Opelims,axiom,
! [X: int,Xa: int,Y2: int] :
( ( ( bit_se5824344872417868541ns_and @ int @ X @ Xa )
= Y2 )
=> ( ( accp @ ( product_prod @ int @ int ) @ bit_and_int_rel @ ( product_Pair @ int @ int @ X @ Xa ) )
=> ~ ( ( ( ( ( member @ int @ X @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
& ( member @ int @ Xa @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) )
=> ( Y2
= ( uminus_uminus @ int
@ ( zero_neq_one_of_bool @ int
@ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ X )
& ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Xa ) ) ) ) ) )
& ( ~ ( ( member @ int @ X @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
& ( member @ int @ Xa @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) )
=> ( Y2
= ( plus_plus @ int
@ ( zero_neq_one_of_bool @ int
@ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ X )
& ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Xa ) ) )
@ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( divide_divide @ int @ X @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ Xa @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ int @ int ) @ bit_and_int_rel @ ( product_Pair @ int @ int @ X @ Xa ) ) ) ) ) ).
% and_int.pelims
thf(fact_5102_empty__subsetI,axiom,
! [A: $tType,A3: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A3 ) ).
% empty_subsetI
thf(fact_5103_subset__empty,axiom,
! [A: $tType,A3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ ( bot_bot @ ( set @ A ) ) )
= ( A3
= ( bot_bot @ ( set @ A ) ) ) ) ).
% subset_empty
thf(fact_5104_singleton__conv,axiom,
! [A: $tType,A2: A] :
( ( collect @ A
@ ^ [X4: A] : X4 = A2 )
= ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% singleton_conv
thf(fact_5105_singleton__conv2,axiom,
! [A: $tType,A2: A] :
( ( collect @ A
@ ( ^ [Y5: A,Z2: A] : Y5 = Z2
@ A2 ) )
= ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% singleton_conv2
thf(fact_5106_atLeastatMost__empty__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A2: A,B2: A] :
( ( ( set_or1337092689740270186AtMost @ A @ A2 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
= ( ~ ( ord_less_eq @ A @ A2 @ B2 ) ) ) ) ).
% atLeastatMost_empty_iff
thf(fact_5107_atLeastatMost__empty__iff2,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A2: A,B2: A] :
( ( ( bot_bot @ ( set @ A ) )
= ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
= ( ~ ( ord_less_eq @ A @ A2 @ B2 ) ) ) ) ).
% atLeastatMost_empty_iff2
thf(fact_5108_atLeastatMost__empty,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B2: A,A2: A] :
( ( ord_less @ A @ B2 @ A2 )
=> ( ( set_or1337092689740270186AtMost @ A @ A2 @ B2 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% atLeastatMost_empty
thf(fact_5109_singleton__insert__inj__eq_H,axiom,
! [A: $tType,A2: A,A3: set @ A,B2: A] :
( ( ( insert @ A @ A2 @ A3 )
= ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) )
= ( ( A2 = B2 )
& ( ord_less_eq @ ( set @ A ) @ A3 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_5110_singleton__insert__inj__eq,axiom,
! [A: $tType,B2: A,A2: A,A3: set @ A] :
( ( ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) )
= ( insert @ A @ A2 @ A3 ) )
= ( ( A2 = B2 )
& ( ord_less_eq @ ( set @ A ) @ A3 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_5111_atLeastLessThan__empty,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B2: A,A2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( set_or7035219750837199246ssThan @ A @ A2 @ B2 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% atLeastLessThan_empty
thf(fact_5112_atLeastLessThan__empty__iff2,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A2: A,B2: A] :
( ( ( bot_bot @ ( set @ A ) )
= ( set_or7035219750837199246ssThan @ A @ A2 @ B2 ) )
= ( ~ ( ord_less @ A @ A2 @ B2 ) ) ) ) ).
% atLeastLessThan_empty_iff2
thf(fact_5113_atLeastLessThan__empty__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A2: A,B2: A] :
( ( ( set_or7035219750837199246ssThan @ A @ A2 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
= ( ~ ( ord_less @ A @ A2 @ B2 ) ) ) ) ).
% atLeastLessThan_empty_iff
thf(fact_5114_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_5115_Diff__eq__empty__iff,axiom,
! [A: $tType,A3: set @ A,B5: set @ A] :
( ( ( minus_minus @ ( set @ A ) @ A3 @ B5 )
= ( bot_bot @ ( set @ A ) ) )
= ( ord_less_eq @ ( set @ A ) @ A3 @ B5 ) ) ).
% Diff_eq_empty_iff
thf(fact_5116_subset__Compl__singleton,axiom,
! [A: $tType,A3: set @ A,B2: A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ ( uminus_uminus @ ( set @ A ) @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( ~ ( member @ A @ B2 @ A3 ) ) ) ).
% subset_Compl_singleton
thf(fact_5117_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_5118_Collect__conv__if,axiom,
! [A: $tType,P3: A > $o,A2: A] :
( ( ( P3 @ A2 )
=> ( ( collect @ A
@ ^ [X4: A] :
( ( X4 = A2 )
& ( P3 @ X4 ) ) )
= ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) )
& ( ~ ( P3 @ A2 )
=> ( ( collect @ A
@ ^ [X4: A] :
( ( X4 = A2 )
& ( P3 @ X4 ) ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Collect_conv_if
thf(fact_5119_Collect__conv__if2,axiom,
! [A: $tType,P3: A > $o,A2: A] :
( ( ( P3 @ A2 )
=> ( ( collect @ A
@ ^ [X4: A] :
( ( A2 = X4 )
& ( P3 @ X4 ) ) )
= ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) )
& ( ~ ( P3 @ A2 )
=> ( ( collect @ A
@ ^ [X4: A] :
( ( A2 = X4 )
& ( P3 @ X4 ) ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Collect_conv_if2
thf(fact_5120_bot_Oextremum__uniqueI,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ A2 @ ( bot_bot @ A ) )
=> ( A2
= ( bot_bot @ A ) ) ) ) ).
% bot.extremum_uniqueI
thf(fact_5121_bot_Oextremum__unique,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ A2 @ ( bot_bot @ A ) )
= ( A2
= ( bot_bot @ A ) ) ) ) ).
% bot.extremum_unique
thf(fact_5122_bot_Oextremum,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [A2: A] : ( ord_less_eq @ A @ ( bot_bot @ A ) @ A2 ) ) ).
% bot.extremum
thf(fact_5123_bot_Onot__eq__extremum,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [A2: A] :
( ( A2
!= ( bot_bot @ A ) )
= ( ord_less @ A @ ( bot_bot @ A ) @ A2 ) ) ) ).
% bot.not_eq_extremum
thf(fact_5124_bot_Oextremum__strict,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [A2: A] :
~ ( ord_less @ A @ A2 @ ( bot_bot @ A ) ) ) ).
% bot.extremum_strict
thf(fact_5125_empty__def,axiom,
! [A: $tType] :
( ( bot_bot @ ( set @ A ) )
= ( collect @ A
@ ^ [X4: A] : $false ) ) ).
% empty_def
thf(fact_5126_some__in__eq,axiom,
! [A: $tType,A3: set @ A] :
( ( member @ A
@ ( fChoice @ A
@ ^ [X4: A] : ( member @ A @ X4 @ A3 ) )
@ A3 )
= ( A3
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% some_in_eq
thf(fact_5127_diff__shunt__var,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y2: A] :
( ( ( minus_minus @ A @ X @ Y2 )
= ( bot_bot @ A ) )
= ( ord_less_eq @ A @ X @ Y2 ) ) ) ).
% diff_shunt_var
thf(fact_5128_finite__has__maximal,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ? [X5: A] :
( ( member @ A @ X5 @ A3 )
& ! [Xa2: A] :
( ( member @ A @ Xa2 @ A3 )
=> ( ( ord_less_eq @ A @ X5 @ Xa2 )
=> ( X5 = Xa2 ) ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_5129_finite__has__minimal,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ? [X5: A] :
( ( member @ A @ X5 @ A3 )
& ! [Xa2: A] :
( ( member @ A @ Xa2 @ A3 )
=> ( ( ord_less_eq @ A @ Xa2 @ X5 )
=> ( X5 = Xa2 ) ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_5130_infinite__growing,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X9: set @ A] :
( ( X9
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ X9 )
=> ? [Xa2: A] :
( ( member @ A @ Xa2 @ X9 )
& ( ord_less @ A @ X5 @ Xa2 ) ) )
=> ~ ( finite_finite2 @ A @ X9 ) ) ) ) ).
% infinite_growing
thf(fact_5131_ex__min__if__finite,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [S3: set @ A] :
( ( finite_finite2 @ A @ S3 )
=> ( ( S3
!= ( bot_bot @ ( set @ A ) ) )
=> ? [X5: A] :
( ( member @ A @ X5 @ S3 )
& ~ ? [Xa2: A] :
( ( member @ A @ Xa2 @ S3 )
& ( ord_less @ A @ Xa2 @ X5 ) ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_5132_subset__singleton__iff,axiom,
! [A: $tType,X9: set @ A,A2: A] :
( ( ord_less_eq @ ( set @ A ) @ X9 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
= ( ( X9
= ( bot_bot @ ( set @ A ) ) )
| ( X9
= ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% subset_singleton_iff
thf(fact_5133_subset__singletonD,axiom,
! [A: $tType,A3: set @ A,X: A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
=> ( ( A3
= ( bot_bot @ ( set @ A ) ) )
| ( A3
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% subset_singletonD
thf(fact_5134_subset__Compl__self__eq,axiom,
! [A: $tType,A3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ ( uminus_uminus @ ( set @ A ) @ A3 ) )
= ( A3
= ( bot_bot @ ( set @ A ) ) ) ) ).
% subset_Compl_self_eq
thf(fact_5135_finite__ranking__induct,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [S3: set @ B,P3: ( set @ B ) > $o,F3: B > A] :
( ( finite_finite2 @ B @ S3 )
=> ( ( P3 @ ( bot_bot @ ( set @ B ) ) )
=> ( ! [X5: B,S7: set @ B] :
( ( finite_finite2 @ B @ S7 )
=> ( ! [Y4: B] :
( ( member @ B @ Y4 @ S7 )
=> ( ord_less_eq @ A @ ( F3 @ Y4 ) @ ( F3 @ X5 ) ) )
=> ( ( P3 @ S7 )
=> ( P3 @ ( insert @ B @ X5 @ S7 ) ) ) ) )
=> ( P3 @ S3 ) ) ) ) ) ).
% finite_ranking_induct
thf(fact_5136_finite__linorder__max__induct,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,P3: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ A3 )
=> ( ( P3 @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [B3: A,A8: set @ A] :
( ( finite_finite2 @ A @ A8 )
=> ( ! [X2: A] :
( ( member @ A @ X2 @ A8 )
=> ( ord_less @ A @ X2 @ B3 ) )
=> ( ( P3 @ A8 )
=> ( P3 @ ( insert @ A @ B3 @ A8 ) ) ) ) )
=> ( P3 @ A3 ) ) ) ) ) ).
% finite_linorder_max_induct
thf(fact_5137_finite__linorder__min__induct,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,P3: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ A3 )
=> ( ( P3 @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [B3: A,A8: set @ A] :
( ( finite_finite2 @ A @ A8 )
=> ( ! [X2: A] :
( ( member @ A @ X2 @ A8 )
=> ( ord_less @ A @ B3 @ X2 ) )
=> ( ( P3 @ A8 )
=> ( P3 @ ( insert @ A @ B3 @ A8 ) ) ) ) )
=> ( P3 @ A3 ) ) ) ) ) ).
% finite_linorder_min_induct
thf(fact_5138_sum__strict__mono,axiom,
! [A: $tType,B: $tType] :
( ( strict7427464778891057005id_add @ A )
=> ! [A3: set @ B,F3: B > A,G: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ A3 )
=> ( ord_less @ A @ ( F3 @ X5 ) @ ( G @ X5 ) ) )
=> ( ord_less @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ A3 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ A3 ) ) ) ) ) ) ).
% sum_strict_mono
thf(fact_5139_finite__subset__induct,axiom,
! [A: $tType,F4: set @ A,A3: set @ A,P3: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ F4 )
=> ( ( ord_less_eq @ ( set @ A ) @ F4 @ A3 )
=> ( ( P3 @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [A4: A,F7: set @ A] :
( ( finite_finite2 @ A @ F7 )
=> ( ( member @ A @ A4 @ A3 )
=> ( ~ ( member @ A @ A4 @ F7 )
=> ( ( P3 @ F7 )
=> ( P3 @ ( insert @ A @ A4 @ F7 ) ) ) ) ) )
=> ( P3 @ F4 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_5140_finite__subset__induct_H,axiom,
! [A: $tType,F4: set @ A,A3: set @ A,P3: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ F4 )
=> ( ( ord_less_eq @ ( set @ A ) @ F4 @ A3 )
=> ( ( P3 @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [A4: A,F7: set @ A] :
( ( finite_finite2 @ A @ F7 )
=> ( ( member @ A @ A4 @ A3 )
=> ( ( ord_less_eq @ ( set @ A ) @ F7 @ A3 )
=> ( ~ ( member @ A @ A4 @ F7 )
=> ( ( P3 @ F7 )
=> ( P3 @ ( insert @ A @ A4 @ F7 ) ) ) ) ) ) )
=> ( P3 @ F4 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_5141_Diff__single__insert,axiom,
! [A: $tType,A3: set @ A,X: A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ B5 )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ ( insert @ A @ X @ B5 ) ) ) ).
% Diff_single_insert
thf(fact_5142_subset__insert__iff,axiom,
! [A: $tType,A3: set @ A,X: A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ ( insert @ A @ X @ B5 ) )
= ( ( ( member @ A @ X @ A3 )
=> ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ B5 ) )
& ( ~ ( member @ A @ X @ A3 )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ B5 ) ) ) ) ).
% subset_insert_iff
thf(fact_5143_card__1__singletonE,axiom,
! [A: $tType,A3: set @ A] :
( ( ( finite_card @ A @ A3 )
= ( one_one @ nat ) )
=> ~ ! [X5: A] :
( A3
!= ( insert @ A @ X5 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% card_1_singletonE
thf(fact_5144_sum__pos,axiom,
! [A: $tType,B: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [I6: set @ B,F3: B > A] :
( ( finite_finite2 @ B @ I6 )
=> ( ( I6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [I3: B] :
( ( member @ B @ I3 @ I6 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( F3 @ I3 ) ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ I6 ) ) ) ) ) ) ).
% sum_pos
thf(fact_5145_less__1__prod,axiom,
! [B: $tType,A: $tType] :
( ( linordered_idom @ B )
=> ! [I6: set @ A,F3: A > B] :
( ( finite_finite2 @ A @ I6 )
=> ( ( I6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [I3: A] :
( ( member @ A @ I3 @ I6 )
=> ( ord_less @ B @ ( one_one @ B ) @ ( F3 @ I3 ) ) )
=> ( ord_less @ B @ ( one_one @ B ) @ ( groups7121269368397514597t_prod @ A @ B @ F3 @ I6 ) ) ) ) ) ) ).
% less_1_prod
thf(fact_5146_card__gt__0__iff,axiom,
! [A: $tType,A3: set @ A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( finite_card @ A @ A3 ) )
= ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
& ( finite_finite2 @ A @ A3 ) ) ) ).
% card_gt_0_iff
thf(fact_5147_finite__remove__induct,axiom,
! [A: $tType,B5: set @ A,P3: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ B5 )
=> ( ( P3 @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [A8: set @ A] :
( ( finite_finite2 @ A @ A8 )
=> ( ( A8
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ A8 @ B5 )
=> ( ! [X2: A] :
( ( member @ A @ X2 @ A8 )
=> ( P3 @ ( minus_minus @ ( set @ A ) @ A8 @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) )
=> ( P3 @ A8 ) ) ) ) )
=> ( P3 @ B5 ) ) ) ) ).
% finite_remove_induct
thf(fact_5148_remove__induct,axiom,
! [A: $tType,P3: ( set @ A ) > $o,B5: set @ A] :
( ( P3 @ ( bot_bot @ ( set @ A ) ) )
=> ( ( ~ ( finite_finite2 @ A @ B5 )
=> ( P3 @ B5 ) )
=> ( ! [A8: set @ A] :
( ( finite_finite2 @ A @ A8 )
=> ( ( A8
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ A8 @ B5 )
=> ( ! [X2: A] :
( ( member @ A @ X2 @ A8 )
=> ( P3 @ ( minus_minus @ ( set @ A ) @ A8 @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) )
=> ( P3 @ A8 ) ) ) ) )
=> ( P3 @ B5 ) ) ) ) ).
% remove_induct
thf(fact_5149_card__Diff1__le,axiom,
! [A: $tType,A3: set @ A,X: A] : ( ord_less_eq @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) @ ( finite_card @ A @ A3 ) ) ).
% card_Diff1_le
thf(fact_5150_psubset__insert__iff,axiom,
! [A: $tType,A3: set @ A,X: A,B5: set @ A] :
( ( ord_less @ ( set @ A ) @ A3 @ ( insert @ A @ X @ B5 ) )
= ( ( ( member @ A @ X @ B5 )
=> ( ord_less @ ( set @ A ) @ A3 @ B5 ) )
& ( ~ ( member @ A @ X @ B5 )
=> ( ( ( member @ A @ X @ A3 )
=> ( ord_less @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ B5 ) )
& ( ~ ( member @ A @ X @ A3 )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ B5 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_5151_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_5152_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_5153_simp__from__to,axiom,
( ( set_or1337092689740270186AtMost @ int )
= ( ^ [I: int,J4: int] : ( if @ ( set @ int ) @ ( ord_less @ int @ J4 @ I ) @ ( bot_bot @ ( set @ int ) ) @ ( insert @ int @ I @ ( set_or1337092689740270186AtMost @ int @ ( plus_plus @ int @ I @ ( one_one @ int ) ) @ J4 ) ) ) ) ) ).
% simp_from_to
thf(fact_5154_prod__mono__strict,axiom,
! [A: $tType,B: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: set @ B,F3: B > A,G: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ! [I3: B] :
( ( member @ B @ I3 @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F3 @ I3 ) )
& ( ord_less @ A @ ( F3 @ I3 ) @ ( G @ I3 ) ) ) )
=> ( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ord_less @ A @ ( groups7121269368397514597t_prod @ B @ A @ F3 @ A3 ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ A3 ) ) ) ) ) ) ).
% prod_mono_strict
thf(fact_5155_card__2__iff,axiom,
! [A: $tType,S3: set @ A] :
( ( ( finite_card @ A @ S3 )
= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( ? [X4: A,Y: A] :
( ( S3
= ( insert @ A @ X4 @ ( insert @ A @ Y @ ( bot_bot @ ( set @ A ) ) ) ) )
& ( X4 != Y ) ) ) ) ).
% card_2_iff
thf(fact_5156_sum_Oremove,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A3: set @ B,X: B,G: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ( member @ B @ X @ A3 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ A3 )
= ( plus_plus @ A @ ( G @ X ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A3 @ ( insert @ B @ X @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ) ).
% sum.remove
thf(fact_5157_sum_Oinsert__remove,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A3: set @ B,G: B > A,X: B] :
( ( finite_finite2 @ B @ A3 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( insert @ B @ X @ A3 ) )
= ( plus_plus @ A @ ( G @ X ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A3 @ ( insert @ B @ X @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ).
% sum.insert_remove
thf(fact_5158_card__3__iff,axiom,
! [A: $tType,S3: set @ A] :
( ( ( finite_card @ A @ S3 )
= ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) )
= ( ? [X4: A,Y: A,Z6: A] :
( ( S3
= ( insert @ A @ X4 @ ( insert @ A @ Y @ ( insert @ A @ Z6 @ ( bot_bot @ ( set @ A ) ) ) ) ) )
& ( X4 != Y )
& ( Y != Z6 )
& ( X4 != Z6 ) ) ) ) ).
% card_3_iff
thf(fact_5159_odd__card__imp__not__empty,axiom,
! [A: $tType,A3: set @ A] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( finite_card @ A @ A3 ) )
=> ( A3
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% odd_card_imp_not_empty
thf(fact_5160_prod_Oinsert__remove,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: set @ B,G: B > A,X: B] :
( ( finite_finite2 @ B @ A3 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( insert @ B @ X @ A3 ) )
= ( times_times @ A @ ( G @ X ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A3 @ ( insert @ B @ X @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ).
% prod.insert_remove
thf(fact_5161_prod_Oremove,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: set @ B,X: B,G: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ( member @ B @ X @ A3 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ A3 )
= ( times_times @ A @ ( G @ X ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A3 @ ( insert @ B @ X @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ) ).
% prod.remove
thf(fact_5162_card__Diff1__less__iff,axiom,
! [A: $tType,A3: set @ A,X: A] :
( ( ord_less @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) @ ( finite_card @ A @ A3 ) )
= ( ( finite_finite2 @ A @ A3 )
& ( member @ A @ X @ A3 ) ) ) ).
% card_Diff1_less_iff
thf(fact_5163_card__Diff2__less,axiom,
! [A: $tType,A3: set @ A,X: A,Y2: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( member @ A @ X @ A3 )
=> ( ( member @ A @ Y2 @ A3 )
=> ( ord_less @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ ( insert @ A @ Y2 @ ( bot_bot @ ( set @ A ) ) ) ) ) @ ( finite_card @ A @ A3 ) ) ) ) ) ).
% card_Diff2_less
thf(fact_5164_card__Diff1__less,axiom,
! [A: $tType,A3: set @ A,X: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( member @ A @ X @ A3 )
=> ( ord_less @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) @ ( finite_card @ A @ A3 ) ) ) ) ).
% card_Diff1_less
thf(fact_5165_card__Diff__singleton__if,axiom,
! [A: $tType,X: A,A3: set @ A] :
( ( ( member @ A @ X @ A3 )
=> ( ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( minus_minus @ nat @ ( finite_card @ A @ A3 ) @ ( one_one @ nat ) ) ) )
& ( ~ ( member @ A @ X @ A3 )
=> ( ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( finite_card @ A @ A3 ) ) ) ) ).
% card_Diff_singleton_if
thf(fact_5166_card__Diff__singleton,axiom,
! [A: $tType,X: A,A3: set @ A] :
( ( member @ A @ X @ A3 )
=> ( ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( minus_minus @ nat @ ( finite_card @ A @ A3 ) @ ( one_one @ nat ) ) ) ) ).
% card_Diff_singleton
thf(fact_5167_sum_Odelta__remove,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [S3: set @ B,A2: B,B2: B > A,C2: B > A] :
( ( finite_finite2 @ B @ S3 )
=> ( ( ( member @ B @ A2 @ S3 )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [K3: B] : ( if @ A @ ( K3 = A2 ) @ ( B2 @ K3 ) @ ( C2 @ K3 ) )
@ S3 )
= ( plus_plus @ A @ ( B2 @ A2 ) @ ( groups7311177749621191930dd_sum @ B @ A @ C2 @ ( minus_minus @ ( set @ B ) @ S3 @ ( insert @ B @ A2 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) )
& ( ~ ( member @ B @ A2 @ S3 )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [K3: B] : ( if @ A @ ( K3 = A2 ) @ ( B2 @ K3 ) @ ( C2 @ K3 ) )
@ S3 )
= ( groups7311177749621191930dd_sum @ B @ A @ C2 @ ( minus_minus @ ( set @ B ) @ S3 @ ( insert @ B @ A2 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ) ).
% sum.delta_remove
thf(fact_5168_prod_Odelta__remove,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S3: set @ B,A2: B,B2: B > A,C2: B > A] :
( ( finite_finite2 @ B @ S3 )
=> ( ( ( member @ B @ A2 @ S3 )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K3: B] : ( if @ A @ ( K3 = A2 ) @ ( B2 @ K3 ) @ ( C2 @ K3 ) )
@ S3 )
= ( times_times @ A @ ( B2 @ A2 ) @ ( groups7121269368397514597t_prod @ B @ A @ C2 @ ( minus_minus @ ( set @ B ) @ S3 @ ( insert @ B @ A2 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) )
& ( ~ ( member @ B @ A2 @ S3 )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K3: B] : ( if @ A @ ( K3 = A2 ) @ ( B2 @ K3 ) @ ( C2 @ K3 ) )
@ S3 )
= ( groups7121269368397514597t_prod @ B @ A @ C2 @ ( minus_minus @ ( set @ B ) @ S3 @ ( insert @ B @ A2 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ) ).
% prod.delta_remove
thf(fact_5169_member__le__sum,axiom,
! [B: $tType,C: $tType] :
( ( ( ordere6911136660526730532id_add @ B )
& ( semiring_1 @ B ) )
=> ! [I2: C,A3: set @ C,F3: C > B] :
( ( member @ C @ I2 @ A3 )
=> ( ! [X5: C] :
( ( member @ C @ X5 @ ( minus_minus @ ( set @ C ) @ A3 @ ( insert @ C @ I2 @ ( bot_bot @ ( set @ C ) ) ) ) )
=> ( ord_less_eq @ B @ ( zero_zero @ B ) @ ( F3 @ X5 ) ) )
=> ( ( finite_finite2 @ C @ A3 )
=> ( ord_less_eq @ B @ ( F3 @ I2 ) @ ( groups7311177749621191930dd_sum @ C @ B @ F3 @ A3 ) ) ) ) ) ) ).
% member_le_sum
thf(fact_5170_sum__bounded__above__divide,axiom,
! [B: $tType,A: $tType] :
( ( linordered_field @ A )
=> ! [A3: set @ B,F3: B > A,K5: A] :
( ! [I3: B] :
( ( member @ B @ I3 @ A3 )
=> ( ord_less_eq @ A @ ( F3 @ I3 ) @ ( divide_divide @ A @ K5 @ ( semiring_1_of_nat @ A @ ( finite_card @ B @ A3 ) ) ) ) )
=> ( ( finite_finite2 @ B @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ A3 ) @ K5 ) ) ) ) ) ).
% sum_bounded_above_divide
thf(fact_5171_prod__diff1,axiom,
! [A: $tType,B: $tType] :
( ( semidom_divide @ A )
=> ! [A3: set @ B,F3: B > A,A2: B] :
( ( finite_finite2 @ B @ A3 )
=> ( ( ( F3 @ A2 )
!= ( zero_zero @ A ) )
=> ( ( ( member @ B @ A2 @ A3 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ F3 @ ( minus_minus @ ( set @ B ) @ A3 @ ( insert @ B @ A2 @ ( bot_bot @ ( set @ B ) ) ) ) )
= ( divide_divide @ A @ ( groups7121269368397514597t_prod @ B @ A @ F3 @ A3 ) @ ( F3 @ A2 ) ) ) )
& ( ~ ( member @ B @ A2 @ A3 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ F3 @ ( minus_minus @ ( set @ B ) @ A3 @ ( insert @ B @ A2 @ ( bot_bot @ ( set @ B ) ) ) ) )
= ( groups7121269368397514597t_prod @ B @ A @ F3 @ A3 ) ) ) ) ) ) ) ).
% prod_diff1
thf(fact_5172_sinh__zero__iff,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A] :
( ( ( sinh @ A @ X )
= ( zero_zero @ A ) )
= ( member @ A @ ( exp @ A @ X ) @ ( insert @ A @ ( one_one @ A ) @ ( insert @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% sinh_zero_iff
thf(fact_5173_and__int_Opinduct,axiom,
! [A0: int,A12: int,P3: int > int > $o] :
( ( accp @ ( product_prod @ int @ int ) @ bit_and_int_rel @ ( product_Pair @ int @ int @ A0 @ A12 ) )
=> ( ! [K: int,L2: int] :
( ( accp @ ( product_prod @ int @ int ) @ bit_and_int_rel @ ( product_Pair @ int @ int @ K @ L2 ) )
=> ( ( ~ ( ( member @ int @ K @ ( 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 ) ) ) ) ) )
=> ( P3 @ ( divide_divide @ int @ K @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) )
=> ( P3 @ K @ L2 ) ) )
=> ( P3 @ A0 @ A12 ) ) ) ).
% and_int.pinduct
thf(fact_5174_and__int_Opsimps,axiom,
! [K2: int,L: int] :
( ( accp @ ( product_prod @ int @ int ) @ bit_and_int_rel @ ( product_Pair @ int @ int @ K2 @ L ) )
=> ( ( ( ( member @ int @ K2 @ ( 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 @ K2 @ L )
= ( uminus_uminus @ int
@ ( zero_neq_one_of_bool @ int
@ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K2 )
& ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ L ) ) ) ) ) )
& ( ~ ( ( member @ int @ K2 @ ( 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 @ K2 @ L )
= ( plus_plus @ int
@ ( zero_neq_one_of_bool @ int
@ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K2 )
& ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ L ) ) )
@ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( divide_divide @ int @ K2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% and_int.psimps
thf(fact_5175_sorted__wrt__less__sum__mono__lowerbound,axiom,
! [B: $tType] :
( ( ordere6911136660526730532id_add @ B )
=> ! [F3: nat > B,Ns: list @ nat] :
( ! [X5: nat,Y7: nat] :
( ( ord_less_eq @ nat @ X5 @ Y7 )
=> ( ord_less_eq @ B @ ( F3 @ X5 ) @ ( F3 @ Y7 ) ) )
=> ( ( sorted_wrt @ nat @ ( ord_less @ nat ) @ Ns )
=> ( ord_less_eq @ B @ ( groups7311177749621191930dd_sum @ nat @ B @ F3 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ nat ) @ Ns ) ) ) @ ( groups8242544230860333062m_list @ B @ ( map @ nat @ B @ F3 @ Ns ) ) ) ) ) ) ).
% sorted_wrt_less_sum_mono_lowerbound
thf(fact_5176_the__elem__def,axiom,
! [A: $tType] :
( ( the_elem @ A )
= ( ^ [X6: set @ A] :
( the @ A
@ ^ [X4: A] :
( X6
= ( insert @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% the_elem_def
thf(fact_5177_sum__diff1_H__aux,axiom,
! [B: $tType,A: $tType] :
( ( ab_group_add @ B )
=> ! [F4: set @ A,I6: set @ A,F3: A > B,I2: A] :
( ( finite_finite2 @ A @ F4 )
=> ( ( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [I: A] :
( ( member @ A @ I @ I6 )
& ( ( F3 @ I )
!= ( zero_zero @ B ) ) ) )
@ F4 )
=> ( ( ( member @ A @ I2 @ I6 )
=> ( ( groups1027152243600224163dd_sum @ A @ B @ F3 @ ( minus_minus @ ( set @ A ) @ I6 @ ( insert @ A @ I2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( minus_minus @ B @ ( groups1027152243600224163dd_sum @ A @ B @ F3 @ I6 ) @ ( F3 @ I2 ) ) ) )
& ( ~ ( member @ A @ I2 @ I6 )
=> ( ( groups1027152243600224163dd_sum @ A @ B @ F3 @ ( minus_minus @ ( set @ A ) @ I6 @ ( insert @ A @ I2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( groups1027152243600224163dd_sum @ A @ B @ F3 @ I6 ) ) ) ) ) ) ) ).
% sum_diff1'_aux
thf(fact_5178_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_5179_sum_Oinsert_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [I6: set @ B,P5: B > A,I2: B] :
( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X4: B] :
( ( member @ B @ X4 @ I6 )
& ( ( P5 @ X4 )
!= ( zero_zero @ A ) ) ) ) )
=> ( ( ( member @ B @ I2 @ I6 )
=> ( ( groups1027152243600224163dd_sum @ B @ A @ P5 @ ( insert @ B @ I2 @ I6 ) )
= ( groups1027152243600224163dd_sum @ B @ A @ P5 @ I6 ) ) )
& ( ~ ( member @ B @ I2 @ I6 )
=> ( ( groups1027152243600224163dd_sum @ B @ A @ P5 @ ( insert @ B @ I2 @ I6 ) )
= ( plus_plus @ A @ ( P5 @ I2 ) @ ( groups1027152243600224163dd_sum @ B @ A @ P5 @ I6 ) ) ) ) ) ) ) ).
% sum.insert'
thf(fact_5180_sorted__wrt__map,axiom,
! [A: $tType,B: $tType,R4: A > A > $o,F3: B > A,Xs: list @ B] :
( ( sorted_wrt @ A @ R4 @ ( map @ B @ A @ F3 @ Xs ) )
= ( sorted_wrt @ B
@ ^ [X4: B,Y: B] : ( R4 @ ( F3 @ X4 ) @ ( F3 @ Y ) )
@ Xs ) ) ).
% sorted_wrt_map
thf(fact_5181_bot__enat__def,axiom,
( ( bot_bot @ extended_enat )
= ( zero_zero @ extended_enat ) ) ).
% bot_enat_def
thf(fact_5182_bot__nat__def,axiom,
( ( bot_bot @ nat )
= ( zero_zero @ nat ) ) ).
% bot_nat_def
thf(fact_5183_bot__empty__eq2,axiom,
! [B: $tType,A: $tType] :
( ( bot_bot @ ( A > B > $o ) )
= ( ^ [X4: A,Y: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y ) @ ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ) ).
% bot_empty_eq2
thf(fact_5184_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_5185_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
@ ^ [X4: B] :
( ( member @ B @ X4 @ I6 )
& ( ( G @ X4 )
!= ( zero_zero @ A ) ) ) ) )
= ( groups1027152243600224163dd_sum @ B @ A @ G @ I6 ) ) ) ).
% sum.non_neutral'
thf(fact_5186_sorted__wrt__true,axiom,
! [A: $tType,Xs: list @ A] :
( sorted_wrt @ A
@ ^ [Uu3: A,Uv3: A] : $true
@ Xs ) ).
% sorted_wrt_true
thf(fact_5187_sorted__wrt__take,axiom,
! [A: $tType,F3: A > A > $o,Xs: list @ A,N: nat] :
( ( sorted_wrt @ A @ F3 @ Xs )
=> ( sorted_wrt @ A @ F3 @ ( take @ A @ N @ Xs ) ) ) ).
% sorted_wrt_take
thf(fact_5188_sorted__wrt__mono__rel,axiom,
! [A: $tType,Xs: list @ A,P3: A > A > $o,Q2: A > A > $o] :
( ! [X5: A,Y7: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Xs ) )
=> ( ( member @ A @ Y7 @ ( set2 @ A @ Xs ) )
=> ( ( P3 @ X5 @ Y7 )
=> ( Q2 @ X5 @ Y7 ) ) ) )
=> ( ( sorted_wrt @ A @ P3 @ Xs )
=> ( sorted_wrt @ A @ Q2 @ Xs ) ) ) ).
% sorted_wrt_mono_rel
thf(fact_5189_strict__sorted__equal,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,Ys: list @ A] :
( ( sorted_wrt @ A @ ( ord_less @ A ) @ Xs )
=> ( ( sorted_wrt @ A @ ( ord_less @ A ) @ Ys )
=> ( ( ( set2 @ A @ Ys )
= ( set2 @ A @ Xs ) )
=> ( Ys = Xs ) ) ) ) ) ).
% strict_sorted_equal
thf(fact_5190_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_5191_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_5192_sorted__wrt__upt,axiom,
! [M: nat,N: nat] : ( sorted_wrt @ nat @ ( ord_less @ nat ) @ ( upt @ M @ N ) ) ).
% sorted_wrt_upt
thf(fact_5193_sorted__upt,axiom,
! [M: nat,N: nat] : ( sorted_wrt @ nat @ ( ord_less_eq @ nat ) @ ( upt @ M @ N ) ) ).
% sorted_upt
thf(fact_5194_sorted__map,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F3: B > A,Xs: list @ B] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F3 @ Xs ) )
= ( sorted_wrt @ B
@ ^ [X4: B,Y: B] : ( ord_less_eq @ A @ ( F3 @ X4 ) @ ( F3 @ Y ) )
@ Xs ) ) ) ).
% sorted_map
thf(fact_5195_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_5196_sorted__distinct__set__unique,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,Ys: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( ( distinct @ A @ Xs )
=> ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Ys )
=> ( ( distinct @ A @ Ys )
=> ( ( ( set2 @ A @ Xs )
= ( set2 @ A @ Ys ) )
=> ( Xs = Ys ) ) ) ) ) ) ) ).
% sorted_distinct_set_unique
thf(fact_5197_sorted__wrt01,axiom,
! [A: $tType,Xs: list @ A,P3: A > A > $o] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) )
=> ( sorted_wrt @ A @ P3 @ Xs ) ) ).
% sorted_wrt01
thf(fact_5198_sorted__wrt__nth__less,axiom,
! [A: $tType,P3: A > A > $o,Xs: list @ A,I2: nat,J3: nat] :
( ( sorted_wrt @ A @ P3 @ Xs )
=> ( ( ord_less @ nat @ I2 @ J3 )
=> ( ( ord_less @ nat @ J3 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( P3 @ ( nth @ A @ Xs @ I2 ) @ ( nth @ A @ Xs @ J3 ) ) ) ) ) ).
% sorted_wrt_nth_less
thf(fact_5199_sorted__wrt__iff__nth__less,axiom,
! [A: $tType] :
( ( sorted_wrt @ A )
= ( ^ [P2: A > A > $o,Xs2: list @ A] :
! [I: nat,J4: nat] :
( ( ord_less @ nat @ I @ J4 )
=> ( ( ord_less @ nat @ J4 @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( P2 @ ( nth @ A @ Xs2 @ I ) @ ( nth @ A @ Xs2 @ J4 ) ) ) ) ) ) ).
% sorted_wrt_iff_nth_less
thf(fact_5200_sum_Odistrib__triv_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [I6: set @ B,G: B > A,H2: B > A] :
( ( finite_finite2 @ B @ I6 )
=> ( ( groups1027152243600224163dd_sum @ B @ A
@ ^ [I: B] : ( plus_plus @ A @ ( G @ I ) @ ( H2 @ I ) )
@ I6 )
= ( plus_plus @ A @ ( groups1027152243600224163dd_sum @ B @ A @ G @ I6 ) @ ( groups1027152243600224163dd_sum @ B @ A @ H2 @ I6 ) ) ) ) ) ).
% sum.distrib_triv'
thf(fact_5201_sum_Omono__neutral__cong__right_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [S3: set @ B,T5: set @ B,G: B > A,H2: B > A] :
( ( ord_less_eq @ ( set @ B ) @ S3 @ T5 )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ ( minus_minus @ ( set @ B ) @ T5 @ S3 ) )
=> ( ( G @ X5 )
= ( zero_zero @ A ) ) )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ S3 )
=> ( ( G @ X5 )
= ( H2 @ X5 ) ) )
=> ( ( groups1027152243600224163dd_sum @ B @ A @ G @ T5 )
= ( groups1027152243600224163dd_sum @ B @ A @ H2 @ S3 ) ) ) ) ) ) ).
% sum.mono_neutral_cong_right'
thf(fact_5202_sum_Omono__neutral__cong__left_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [S3: set @ B,T5: set @ B,H2: B > A,G: B > A] :
( ( ord_less_eq @ ( set @ B ) @ S3 @ T5 )
=> ( ! [I3: B] :
( ( member @ B @ I3 @ ( minus_minus @ ( set @ B ) @ T5 @ S3 ) )
=> ( ( H2 @ I3 )
= ( zero_zero @ A ) ) )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ S3 )
=> ( ( G @ X5 )
= ( H2 @ X5 ) ) )
=> ( ( groups1027152243600224163dd_sum @ B @ A @ G @ S3 )
= ( groups1027152243600224163dd_sum @ B @ A @ H2 @ T5 ) ) ) ) ) ) ).
% sum.mono_neutral_cong_left'
thf(fact_5203_sum_Omono__neutral__right_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [S3: set @ B,T5: set @ B,G: B > A] :
( ( ord_less_eq @ ( set @ B ) @ S3 @ T5 )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ ( minus_minus @ ( set @ B ) @ T5 @ S3 ) )
=> ( ( G @ X5 )
= ( zero_zero @ A ) ) )
=> ( ( groups1027152243600224163dd_sum @ B @ A @ G @ T5 )
= ( groups1027152243600224163dd_sum @ B @ A @ G @ S3 ) ) ) ) ) ).
% sum.mono_neutral_right'
thf(fact_5204_sum_Omono__neutral__left_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [S3: set @ B,T5: set @ B,G: B > A] :
( ( ord_less_eq @ ( set @ B ) @ S3 @ T5 )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ ( minus_minus @ ( set @ B ) @ T5 @ S3 ) )
=> ( ( G @ X5 )
= ( zero_zero @ A ) ) )
=> ( ( groups1027152243600224163dd_sum @ B @ A @ G @ S3 )
= ( groups1027152243600224163dd_sum @ B @ A @ G @ T5 ) ) ) ) ) ).
% sum.mono_neutral_left'
thf(fact_5205_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_5206_sorted__iff__nth__mono__less,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
= ( ! [I: nat,J4: nat] :
( ( ord_less @ nat @ I @ J4 )
=> ( ( ord_less @ nat @ J4 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ord_less_eq @ A @ ( nth @ A @ Xs @ I ) @ ( nth @ A @ Xs @ J4 ) ) ) ) ) ) ) ).
% sorted_iff_nth_mono_less
thf(fact_5207_finite__sorted__distinct__unique,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ? [X5: list @ A] :
( ( ( set2 @ A @ X5 )
= A3 )
& ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ X5 )
& ( distinct @ A @ X5 )
& ! [Y4: list @ A] :
( ( ( ( set2 @ A @ Y4 )
= A3 )
& ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Y4 )
& ( distinct @ A @ Y4 ) )
=> ( Y4 = X5 ) ) ) ) ) ).
% finite_sorted_distinct_unique
thf(fact_5208_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
@ ^ [X4: B] :
( ( member @ B @ X4 @ I6 )
& ( ( G @ X4 )
!= ( zero_zero @ A ) ) ) ) )
=> ( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X4: B] :
( ( member @ B @ X4 @ I6 )
& ( ( H2 @ X4 )
!= ( zero_zero @ A ) ) ) ) )
=> ( ( groups1027152243600224163dd_sum @ B @ A
@ ^ [I: B] : ( plus_plus @ A @ ( G @ I ) @ ( H2 @ I ) )
@ I6 )
= ( plus_plus @ A @ ( groups1027152243600224163dd_sum @ B @ A @ G @ I6 ) @ ( groups1027152243600224163dd_sum @ B @ A @ H2 @ I6 ) ) ) ) ) ) ).
% sum.distrib'
thf(fact_5209_sum_OG__def,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ( ( groups1027152243600224163dd_sum @ B @ A )
= ( ^ [P4: B > A,I7: set @ B] :
( if @ A
@ ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X4: B] :
( ( member @ B @ X4 @ I7 )
& ( ( P4 @ X4 )
!= ( zero_zero @ A ) ) ) ) )
@ ( groups7311177749621191930dd_sum @ B @ A @ P4
@ ( collect @ B
@ ^ [X4: B] :
( ( member @ B @ X4 @ I7 )
& ( ( P4 @ X4 )
!= ( zero_zero @ A ) ) ) ) )
@ ( zero_zero @ A ) ) ) ) ) ).
% sum.G_def
thf(fact_5210_sorted__iff__nth__Suc,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
= ( ! [I: nat] :
( ( ord_less @ nat @ ( suc @ I ) @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ord_less_eq @ A @ ( nth @ A @ Xs @ I ) @ ( nth @ A @ Xs @ ( suc @ I ) ) ) ) ) ) ) ).
% sorted_iff_nth_Suc
thf(fact_5211_sorted__list__of__set_Ofinite__set__strict__sorted,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ~ ! [L2: list @ A] :
( ( sorted_wrt @ A @ ( ord_less @ A ) @ L2 )
=> ( ( ( set2 @ A @ L2 )
= A3 )
=> ( ( size_size @ ( list @ A ) @ L2 )
!= ( finite_card @ A @ A3 ) ) ) ) ) ) ).
% sorted_list_of_set.finite_set_strict_sorted
thf(fact_5212_sorted__nth__mono,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,I2: nat,J3: nat] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( ( ord_less_eq @ nat @ I2 @ J3 )
=> ( ( ord_less @ nat @ J3 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ord_less_eq @ A @ ( nth @ A @ Xs @ I2 ) @ ( nth @ A @ Xs @ J3 ) ) ) ) ) ) ).
% sorted_nth_mono
thf(fact_5213_sorted__iff__nth__mono,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
= ( ! [I: nat,J4: nat] :
( ( ord_less_eq @ nat @ I @ J4 )
=> ( ( ord_less @ nat @ J4 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ord_less_eq @ A @ ( nth @ A @ Xs @ I ) @ ( nth @ A @ Xs @ J4 ) ) ) ) ) ) ) ).
% sorted_iff_nth_mono
thf(fact_5214_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_5215_sorted__wrt__less__idx,axiom,
! [Ns: list @ nat,I2: nat] :
( ( sorted_wrt @ nat @ ( ord_less @ nat ) @ Ns )
=> ( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ nat ) @ Ns ) )
=> ( ord_less_eq @ nat @ I2 @ ( nth @ nat @ Ns @ I2 ) ) ) ) ).
% sorted_wrt_less_idx
thf(fact_5216_atLeastLessThan__nat__numeral,axiom,
! [M: nat,K2: num] :
( ( ( ord_less_eq @ nat @ M @ ( pred_numeral @ K2 ) )
=> ( ( set_or7035219750837199246ssThan @ nat @ M @ ( numeral_numeral @ nat @ K2 ) )
= ( insert @ nat @ ( pred_numeral @ K2 ) @ ( set_or7035219750837199246ssThan @ nat @ M @ ( pred_numeral @ K2 ) ) ) ) )
& ( ~ ( ord_less_eq @ nat @ M @ ( pred_numeral @ K2 ) )
=> ( ( set_or7035219750837199246ssThan @ nat @ M @ ( numeral_numeral @ nat @ K2 ) )
= ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% atLeastLessThan_nat_numeral
thf(fact_5217_sum__diff1_H,axiom,
! [B: $tType,A: $tType] :
( ( ab_group_add @ B )
=> ! [I6: set @ A,F3: A > B,I2: A] :
( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [I: A] :
( ( member @ A @ I @ I6 )
& ( ( F3 @ I )
!= ( zero_zero @ B ) ) ) ) )
=> ( ( ( member @ A @ I2 @ I6 )
=> ( ( groups1027152243600224163dd_sum @ A @ B @ F3 @ ( minus_minus @ ( set @ A ) @ I6 @ ( insert @ A @ I2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( minus_minus @ B @ ( groups1027152243600224163dd_sum @ A @ B @ F3 @ I6 ) @ ( F3 @ I2 ) ) ) )
& ( ~ ( member @ A @ I2 @ I6 )
=> ( ( groups1027152243600224163dd_sum @ A @ B @ F3 @ ( minus_minus @ ( set @ A ) @ I6 @ ( insert @ A @ I2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( groups1027152243600224163dd_sum @ A @ B @ F3 @ I6 ) ) ) ) ) ) ).
% sum_diff1'
thf(fact_5218_folding__insort__key_Ofinite__set__strict__sorted,axiom,
! [A: $tType,B: $tType,Less_eq: A > A > $o,Less: A > A > $o,S3: set @ B,F3: B > A,A3: set @ B] :
( ( folding_insort_key @ A @ B @ Less_eq @ Less @ S3 @ F3 )
=> ( ( ord_less_eq @ ( set @ B ) @ A3 @ S3 )
=> ( ( finite_finite2 @ B @ A3 )
=> ~ ! [L2: list @ B] :
( ( sorted_wrt @ A @ Less @ ( map @ B @ A @ F3 @ L2 ) )
=> ( ( ( set2 @ B @ L2 )
= A3 )
=> ( ( size_size @ ( list @ B ) @ L2 )
!= ( finite_card @ B @ A3 ) ) ) ) ) ) ) ).
% folding_insort_key.finite_set_strict_sorted
thf(fact_5219_distinct__union,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( distinct @ A @ ( union @ A @ Xs @ Ys ) )
= ( distinct @ A @ Ys ) ) ).
% distinct_union
thf(fact_5220_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_5221_list__update__overwrite,axiom,
! [A: $tType,Xs: list @ A,I2: nat,X: A,Y2: A] :
( ( list_update @ A @ ( list_update @ A @ Xs @ I2 @ X ) @ I2 @ Y2 )
= ( list_update @ A @ Xs @ I2 @ Y2 ) ) ).
% list_update_overwrite
thf(fact_5222_length__list__update,axiom,
! [A: $tType,Xs: list @ A,I2: nat,X: A] :
( ( size_size @ ( list @ A ) @ ( list_update @ A @ Xs @ I2 @ X ) )
= ( size_size @ ( list @ A ) @ Xs ) ) ).
% length_list_update
thf(fact_5223_list__update__id,axiom,
! [A: $tType,Xs: list @ A,I2: nat] :
( ( list_update @ A @ Xs @ I2 @ ( nth @ A @ Xs @ I2 ) )
= Xs ) ).
% list_update_id
thf(fact_5224_nth__list__update__neq,axiom,
! [A: $tType,I2: nat,J3: nat,Xs: list @ A,X: A] :
( ( I2 != J3 )
=> ( ( nth @ A @ ( list_update @ A @ Xs @ I2 @ X ) @ J3 )
= ( nth @ A @ Xs @ J3 ) ) ) ).
% nth_list_update_neq
thf(fact_5225_list__update__beyond,axiom,
! [A: $tType,Xs: list @ A,I2: nat,X: A] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ I2 )
=> ( ( list_update @ A @ Xs @ I2 @ X )
= Xs ) ) ).
% list_update_beyond
thf(fact_5226_take__update__cancel,axiom,
! [A: $tType,N: nat,M: nat,Xs: list @ A,Y2: A] :
( ( ord_less_eq @ nat @ N @ M )
=> ( ( take @ A @ N @ ( list_update @ A @ Xs @ M @ Y2 ) )
= ( take @ A @ N @ Xs ) ) ) ).
% take_update_cancel
thf(fact_5227_nth__list__update__eq,axiom,
! [A: $tType,I2: nat,Xs: list @ A,X: A] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ A @ ( list_update @ A @ Xs @ I2 @ X ) @ I2 )
= X ) ) ).
% nth_list_update_eq
thf(fact_5228_set__swap,axiom,
! [A: $tType,I2: nat,Xs: list @ A,J3: nat] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ord_less @ nat @ J3 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( set2 @ A @ ( list_update @ A @ ( list_update @ A @ Xs @ I2 @ ( nth @ A @ Xs @ J3 ) ) @ J3 @ ( nth @ A @ Xs @ I2 ) ) )
= ( set2 @ A @ Xs ) ) ) ) ).
% set_swap
thf(fact_5229_distinct__swap,axiom,
! [A: $tType,I2: nat,Xs: list @ A,J3: nat] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ord_less @ nat @ J3 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( distinct @ A @ ( list_update @ A @ ( list_update @ A @ Xs @ I2 @ ( nth @ A @ Xs @ J3 ) ) @ J3 @ ( nth @ A @ Xs @ I2 ) ) )
= ( distinct @ A @ Xs ) ) ) ) ).
% distinct_swap
thf(fact_5230_list__update__swap,axiom,
! [A: $tType,I2: nat,I8: nat,Xs: list @ A,X: A,X8: A] :
( ( I2 != I8 )
=> ( ( list_update @ A @ ( list_update @ A @ Xs @ I2 @ X ) @ I8 @ X8 )
= ( list_update @ A @ ( list_update @ A @ Xs @ I8 @ X8 ) @ I2 @ X ) ) ) ).
% list_update_swap
thf(fact_5231_take__update__swap,axiom,
! [A: $tType,M: nat,Xs: list @ A,N: nat,X: A] :
( ( take @ A @ M @ ( list_update @ A @ Xs @ N @ X ) )
= ( list_update @ A @ ( take @ A @ M @ Xs ) @ N @ X ) ) ).
% take_update_swap
thf(fact_5232_map__update,axiom,
! [A: $tType,B: $tType,F3: B > A,Xs: list @ B,K2: nat,Y2: B] :
( ( map @ B @ A @ F3 @ ( list_update @ B @ Xs @ K2 @ Y2 ) )
= ( list_update @ A @ ( map @ B @ A @ F3 @ Xs ) @ K2 @ ( F3 @ Y2 ) ) ) ).
% map_update
thf(fact_5233_set__update__subsetI,axiom,
! [A: $tType,Xs: list @ A,A3: set @ A,X: A,I2: nat] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ A3 )
=> ( ( member @ A @ X @ A3 )
=> ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ ( list_update @ A @ Xs @ I2 @ X ) ) @ A3 ) ) ) ).
% set_update_subsetI
thf(fact_5234_folding__insort__key_Odistinct__if__distinct__map,axiom,
! [A: $tType,B: $tType,Less_eq: A > A > $o,Less: A > A > $o,S3: set @ B,F3: B > A,Xs: list @ B] :
( ( folding_insort_key @ A @ B @ Less_eq @ Less @ S3 @ F3 )
=> ( ( distinct @ A @ ( map @ B @ A @ F3 @ Xs ) )
=> ( distinct @ B @ Xs ) ) ) ).
% folding_insort_key.distinct_if_distinct_map
thf(fact_5235_set__update__subset__insert,axiom,
! [A: $tType,Xs: list @ A,I2: nat,X: A] : ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ ( list_update @ A @ Xs @ I2 @ X ) ) @ ( insert @ A @ X @ ( set2 @ A @ Xs ) ) ) ).
% set_update_subset_insert
thf(fact_5236_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_5237_list__update__same__conv,axiom,
! [A: $tType,I2: nat,Xs: list @ A,X: A] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ( list_update @ A @ Xs @ I2 @ X )
= Xs )
= ( ( nth @ A @ Xs @ I2 )
= X ) ) ) ).
% list_update_same_conv
thf(fact_5238_nth__list__update,axiom,
! [A: $tType,I2: nat,Xs: list @ A,J3: nat,X: A] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ( I2 = J3 )
=> ( ( nth @ A @ ( list_update @ A @ Xs @ I2 @ X ) @ J3 )
= X ) )
& ( ( I2 != J3 )
=> ( ( nth @ A @ ( list_update @ A @ Xs @ I2 @ X ) @ J3 )
= ( nth @ A @ Xs @ J3 ) ) ) ) ) ).
% nth_list_update
thf(fact_5239_sum__list__update,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [K2: nat,Xs: list @ A,X: A] :
( ( ord_less @ nat @ K2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( groups8242544230860333062m_list @ A @ ( list_update @ A @ Xs @ K2 @ X ) )
= ( minus_minus @ A @ ( plus_plus @ A @ ( groups8242544230860333062m_list @ A @ Xs ) @ X ) @ ( nth @ A @ Xs @ K2 ) ) ) ) ) ).
% sum_list_update
thf(fact_5240_distinct__list__update,axiom,
! [A: $tType,Xs: list @ A,A2: A,I2: nat] :
( ( distinct @ A @ Xs )
=> ( ~ ( member @ A @ A2 @ ( minus_minus @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( insert @ A @ ( nth @ A @ Xs @ I2 ) @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( distinct @ A @ ( list_update @ A @ Xs @ I2 @ A2 ) ) ) ) ).
% distinct_list_update
thf(fact_5241_folding__insort__key_Osorted__key__list__of__set__unique,axiom,
! [A: $tType,B: $tType,Less_eq: A > A > $o,Less: A > A > $o,S3: set @ B,F3: B > A,A3: set @ B,L: list @ B] :
( ( folding_insort_key @ A @ B @ Less_eq @ Less @ S3 @ F3 )
=> ( ( ord_less_eq @ ( set @ B ) @ A3 @ S3 )
=> ( ( finite_finite2 @ B @ A3 )
=> ( ( ( sorted_wrt @ A @ Less @ ( map @ B @ A @ F3 @ L ) )
& ( ( set2 @ B @ L )
= A3 )
& ( ( size_size @ ( list @ B ) @ L )
= ( finite_card @ B @ A3 ) ) )
= ( ( sorted8670434370408473282of_set @ A @ B @ Less_eq @ F3 @ A3 )
= L ) ) ) ) ) ).
% folding_insort_key.sorted_key_list_of_set_unique
thf(fact_5242_folding__insort__key_Oidem__if__sorted__distinct,axiom,
! [A: $tType,B: $tType,Less_eq: A > A > $o,Less: A > A > $o,S3: set @ B,F3: B > A,Xs: list @ B] :
( ( folding_insort_key @ A @ B @ Less_eq @ Less @ S3 @ F3 )
=> ( ( ord_less_eq @ ( set @ B ) @ ( set2 @ B @ Xs ) @ S3 )
=> ( ( sorted_wrt @ A @ Less_eq @ ( map @ B @ A @ F3 @ Xs ) )
=> ( ( distinct @ B @ Xs )
=> ( ( sorted8670434370408473282of_set @ A @ B @ Less_eq @ F3 @ ( set2 @ B @ Xs ) )
= Xs ) ) ) ) ) ).
% folding_insort_key.idem_if_sorted_distinct
thf(fact_5243_sorted__list__of__set_Osorted__key__list__of__set__unique,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,L: list @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( ( sorted_wrt @ A @ ( ord_less @ A ) @ L )
& ( ( set2 @ A @ L )
= A3 )
& ( ( size_size @ ( list @ A ) @ L )
= ( finite_card @ A @ A3 ) ) )
= ( ( linord4507533701916653071of_set @ A @ A3 )
= L ) ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_unique
thf(fact_5244_sorted__list__of__set__range,axiom,
! [M: nat,N: nat] :
( ( linord4507533701916653071of_set @ nat @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) )
= ( upt @ M @ N ) ) ).
% sorted_list_of_set_range
thf(fact_5245_sorted__list__of__set_Oset__sorted__key__list__of__set,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( set2 @ A @ ( linord4507533701916653071of_set @ A @ A3 ) )
= A3 ) ) ) ).
% sorted_list_of_set.set_sorted_key_list_of_set
thf(fact_5246_sorted__list__of__set_Olength__sorted__key__list__of__set,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A] :
( ( size_size @ ( list @ A ) @ ( linord4507533701916653071of_set @ A @ A3 ) )
= ( finite_card @ A @ A3 ) ) ) ).
% sorted_list_of_set.length_sorted_key_list_of_set
thf(fact_5247_sorted__list__of__set_Odistinct__sorted__key__list__of__set,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A] : ( distinct @ A @ ( linord4507533701916653071of_set @ A @ A3 ) ) ) ).
% sorted_list_of_set.distinct_sorted_key_list_of_set
thf(fact_5248_linorder_Osorted__key__list__of__set_Ocong,axiom,
! [B: $tType,A: $tType] :
( ( sorted8670434370408473282of_set @ A @ B )
= ( sorted8670434370408473282of_set @ A @ B ) ) ).
% linorder.sorted_key_list_of_set.cong
thf(fact_5249_sorted__list__of__set_Osorted__key__list__of__set__inject,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,B5: set @ A] :
( ( ( linord4507533701916653071of_set @ A @ A3 )
= ( linord4507533701916653071of_set @ A @ B5 ) )
=> ( ( finite_finite2 @ A @ A3 )
=> ( ( finite_finite2 @ A @ B5 )
=> ( A3 = B5 ) ) ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_inject
thf(fact_5250_sorted__list__of__set_Osorted__sorted__key__list__of__set,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A] : ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( linord4507533701916653071of_set @ A @ A3 ) ) ) ).
% sorted_list_of_set.sorted_sorted_key_list_of_set
thf(fact_5251_sorted__list__of__set_Ostrict__sorted__key__list__of__set,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A] : ( sorted_wrt @ A @ ( ord_less @ A ) @ ( linord4507533701916653071of_set @ A @ A3 ) ) ) ).
% sorted_list_of_set.strict_sorted_key_list_of_set
thf(fact_5252_folding__insort__key_Osorted__key__list__of__set__inject,axiom,
! [A: $tType,B: $tType,Less_eq: A > A > $o,Less: A > A > $o,S3: set @ B,F3: B > A,A3: set @ B,B5: set @ B] :
( ( folding_insort_key @ A @ B @ Less_eq @ Less @ S3 @ F3 )
=> ( ( ord_less_eq @ ( set @ B ) @ A3 @ S3 )
=> ( ( ord_less_eq @ ( set @ B ) @ B5 @ S3 )
=> ( ( ( sorted8670434370408473282of_set @ A @ B @ Less_eq @ F3 @ A3 )
= ( sorted8670434370408473282of_set @ A @ B @ Less_eq @ F3 @ B5 ) )
=> ( ( finite_finite2 @ B @ A3 )
=> ( ( finite_finite2 @ B @ B5 )
=> ( A3 = B5 ) ) ) ) ) ) ) ).
% folding_insort_key.sorted_key_list_of_set_inject
thf(fact_5253_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_5254_folding__insort__key_Oset__sorted__key__list__of__set,axiom,
! [A: $tType,B: $tType,Less_eq: A > A > $o,Less: A > A > $o,S3: set @ B,F3: B > A,A3: set @ B] :
( ( folding_insort_key @ A @ B @ Less_eq @ Less @ S3 @ F3 )
=> ( ( ord_less_eq @ ( set @ B ) @ A3 @ S3 )
=> ( ( finite_finite2 @ B @ A3 )
=> ( ( set2 @ B @ ( sorted8670434370408473282of_set @ A @ B @ Less_eq @ F3 @ A3 ) )
= A3 ) ) ) ) ).
% folding_insort_key.set_sorted_key_list_of_set
thf(fact_5255_folding__insort__key_Olength__sorted__key__list__of__set,axiom,
! [A: $tType,B: $tType,Less_eq: A > A > $o,Less: A > A > $o,S3: set @ B,F3: B > A,A3: set @ B] :
( ( folding_insort_key @ A @ B @ Less_eq @ Less @ S3 @ F3 )
=> ( ( ord_less_eq @ ( set @ B ) @ A3 @ S3 )
=> ( ( size_size @ ( list @ B ) @ ( sorted8670434370408473282of_set @ A @ B @ Less_eq @ F3 @ A3 ) )
= ( finite_card @ B @ A3 ) ) ) ) ).
% folding_insort_key.length_sorted_key_list_of_set
thf(fact_5256_folding__insort__key_Odistinct__sorted__key__list__of__set,axiom,
! [A: $tType,B: $tType,Less_eq: A > A > $o,Less: A > A > $o,S3: set @ B,F3: B > A,A3: set @ B] :
( ( folding_insort_key @ A @ B @ Less_eq @ Less @ S3 @ F3 )
=> ( ( ord_less_eq @ ( set @ B ) @ A3 @ S3 )
=> ( distinct @ A @ ( map @ B @ A @ F3 @ ( sorted8670434370408473282of_set @ A @ B @ Less_eq @ F3 @ A3 ) ) ) ) ) ).
% folding_insort_key.distinct_sorted_key_list_of_set
thf(fact_5257_folding__insort__key_Ostrict__sorted__key__list__of__set,axiom,
! [A: $tType,B: $tType,Less_eq: A > A > $o,Less: A > A > $o,S3: set @ B,F3: B > A,A3: set @ B] :
( ( folding_insort_key @ A @ B @ Less_eq @ Less @ S3 @ F3 )
=> ( ( ord_less_eq @ ( set @ B ) @ A3 @ S3 )
=> ( sorted_wrt @ A @ Less @ ( map @ B @ A @ F3 @ ( sorted8670434370408473282of_set @ A @ B @ Less_eq @ F3 @ A3 ) ) ) ) ) ).
% folding_insort_key.strict_sorted_key_list_of_set
thf(fact_5258_folding__insort__key_Osorted__sorted__key__list__of__set,axiom,
! [A: $tType,B: $tType,Less_eq: A > A > $o,Less: A > A > $o,S3: set @ B,F3: B > A,A3: set @ B] :
( ( folding_insort_key @ A @ B @ Less_eq @ Less @ S3 @ F3 )
=> ( ( ord_less_eq @ ( set @ B ) @ A3 @ S3 )
=> ( sorted_wrt @ A @ Less_eq @ ( map @ B @ A @ F3 @ ( sorted8670434370408473282of_set @ A @ B @ Less_eq @ F3 @ A3 ) ) ) ) ) ).
% folding_insort_key.sorted_sorted_key_list_of_set
thf(fact_5259_folding__insort__key_Osorted__key__list__of__set__insert__remove,axiom,
! [A: $tType,B: $tType,Less_eq: A > A > $o,Less: A > A > $o,S3: set @ B,F3: B > A,X: B,A3: set @ B] :
( ( folding_insort_key @ A @ B @ Less_eq @ Less @ S3 @ F3 )
=> ( ( ord_less_eq @ ( set @ B ) @ ( insert @ B @ X @ A3 ) @ S3 )
=> ( ( finite_finite2 @ B @ A3 )
=> ( ( sorted8670434370408473282of_set @ A @ B @ Less_eq @ F3 @ ( insert @ B @ X @ A3 ) )
= ( insort_key @ A @ B @ Less_eq @ F3 @ X @ ( sorted8670434370408473282of_set @ A @ B @ Less_eq @ F3 @ ( minus_minus @ ( set @ B ) @ A3 @ ( insert @ B @ X @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ) ).
% folding_insort_key.sorted_key_list_of_set_insert_remove
thf(fact_5260_folding__insort__key_Osorted__key__list__of__set__insert,axiom,
! [A: $tType,B: $tType,Less_eq: A > A > $o,Less: A > A > $o,S3: set @ B,F3: B > A,X: B,A3: set @ B] :
( ( folding_insort_key @ A @ B @ Less_eq @ Less @ S3 @ F3 )
=> ( ( ord_less_eq @ ( set @ B ) @ ( insert @ B @ X @ A3 ) @ S3 )
=> ( ( finite_finite2 @ B @ A3 )
=> ( ~ ( member @ B @ X @ A3 )
=> ( ( sorted8670434370408473282of_set @ A @ B @ Less_eq @ F3 @ ( insert @ B @ X @ A3 ) )
= ( insort_key @ A @ B @ Less_eq @ F3 @ X @ ( sorted8670434370408473282of_set @ A @ B @ Less_eq @ F3 @ A3 ) ) ) ) ) ) ) ).
% folding_insort_key.sorted_key_list_of_set_insert
thf(fact_5261_folding__insort__key_Osorted__key__list__of__set__remove,axiom,
! [A: $tType,B: $tType,Less_eq: A > A > $o,Less: A > A > $o,S3: set @ B,F3: B > A,X: B,A3: set @ B] :
( ( folding_insort_key @ A @ B @ Less_eq @ Less @ S3 @ F3 )
=> ( ( ord_less_eq @ ( set @ B ) @ ( insert @ B @ X @ A3 ) @ S3 )
=> ( ( finite_finite2 @ B @ A3 )
=> ( ( sorted8670434370408473282of_set @ A @ B @ Less_eq @ F3 @ ( minus_minus @ ( set @ B ) @ A3 @ ( insert @ B @ X @ ( bot_bot @ ( set @ B ) ) ) ) )
= ( remove1 @ B @ X @ ( sorted8670434370408473282of_set @ A @ B @ Less_eq @ F3 @ A3 ) ) ) ) ) ) ).
% folding_insort_key.sorted_key_list_of_set_remove
thf(fact_5262_in__set__remove1,axiom,
! [A: $tType,A2: A,B2: A,Xs: list @ A] :
( ( A2 != B2 )
=> ( ( member @ A @ A2 @ ( set2 @ A @ ( remove1 @ A @ B2 @ Xs ) ) )
= ( member @ A @ A2 @ ( set2 @ A @ Xs ) ) ) ) ).
% in_set_remove1
thf(fact_5263_set__remove1__eq,axiom,
! [A: $tType,Xs: list @ A,X: A] :
( ( distinct @ A @ Xs )
=> ( ( set2 @ A @ ( remove1 @ A @ X @ Xs ) )
= ( minus_minus @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% set_remove1_eq
thf(fact_5264_notin__set__remove1,axiom,
! [A: $tType,X: A,Xs: list @ A,Y2: A] :
( ~ ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ~ ( member @ A @ X @ ( set2 @ A @ ( remove1 @ A @ Y2 @ Xs ) ) ) ) ).
% notin_set_remove1
thf(fact_5265_remove1__idem,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ~ ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ( remove1 @ A @ X @ Xs )
= Xs ) ) ).
% remove1_idem
thf(fact_5266_remove1__commute,axiom,
! [A: $tType,X: A,Y2: A,Zs: list @ A] :
( ( remove1 @ A @ X @ ( remove1 @ A @ Y2 @ Zs ) )
= ( remove1 @ A @ Y2 @ ( remove1 @ A @ X @ Zs ) ) ) ).
% remove1_commute
thf(fact_5267_linorder_Oinsort__key_Ocong,axiom,
! [B: $tType,A: $tType] :
( ( insort_key @ A @ B )
= ( insort_key @ A @ B ) ) ).
% linorder.insort_key.cong
thf(fact_5268_distinct__remove1,axiom,
! [A: $tType,Xs: list @ A,X: A] :
( ( distinct @ A @ Xs )
=> ( distinct @ A @ ( remove1 @ A @ X @ Xs ) ) ) ).
% distinct_remove1
thf(fact_5269_set__remove1__subset,axiom,
! [A: $tType,X: A,Xs: list @ A] : ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ ( remove1 @ A @ X @ Xs ) ) @ ( set2 @ A @ Xs ) ) ).
% set_remove1_subset
thf(fact_5270_sorted__remove1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,A2: A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( remove1 @ A @ A2 @ Xs ) ) ) ) ).
% sorted_remove1
thf(fact_5271_sorted__map__remove1,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F3: B > A,Xs: list @ B,X: B] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F3 @ Xs ) )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F3 @ ( remove1 @ B @ X @ Xs ) ) ) ) ) ).
% sorted_map_remove1
thf(fact_5272_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_5273_sum__list__map__remove1,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [X: B,Xs: list @ B,F3: B > A] :
( ( member @ B @ X @ ( set2 @ B @ Xs ) )
=> ( ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ F3 @ Xs ) )
= ( plus_plus @ A @ ( F3 @ X ) @ ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ F3 @ ( remove1 @ B @ X @ Xs ) ) ) ) ) ) ) ).
% sum_list_map_remove1
thf(fact_5274_sorted__list__of__set_Osorted__key__list__of__set__remove,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,X: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( linord4507533701916653071of_set @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( remove1 @ A @ X @ ( linord4507533701916653071of_set @ A @ A3 ) ) ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_remove
thf(fact_5275_card__Pow,axiom,
! [A: $tType,A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( finite_card @ ( set @ A ) @ ( pow2 @ A @ A3 ) )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( finite_card @ A @ A3 ) ) ) ) ).
% card_Pow
thf(fact_5276_Suc__0__div__numeral,axiom,
! [K2: num] :
( ( divide_divide @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( numeral_numeral @ nat @ K2 ) )
= ( product_fst @ nat @ nat @ ( unique8689654367752047608divmod @ nat @ one2 @ K2 ) ) ) ).
% Suc_0_div_numeral
thf(fact_5277_Suc__0__mod__numeral,axiom,
! [K2: num] :
( ( modulo_modulo @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( numeral_numeral @ nat @ K2 ) )
= ( product_snd @ nat @ nat @ ( unique8689654367752047608divmod @ nat @ one2 @ K2 ) ) ) ).
% Suc_0_mod_numeral
thf(fact_5278_PowI,axiom,
! [A: $tType,A3: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B5 )
=> ( member @ ( set @ A ) @ A3 @ ( pow2 @ A @ B5 ) ) ) ).
% PowI
thf(fact_5279_Pow__iff,axiom,
! [A: $tType,A3: set @ A,B5: set @ A] :
( ( member @ ( set @ A ) @ A3 @ ( pow2 @ A @ B5 ) )
= ( ord_less_eq @ ( set @ A ) @ A3 @ B5 ) ) ).
% Pow_iff
thf(fact_5280_numeral__div__numeral,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [K2: num,L: num] :
( ( divide_divide @ A @ ( numeral_numeral @ A @ K2 ) @ ( numeral_numeral @ A @ L ) )
= ( product_fst @ A @ A @ ( unique8689654367752047608divmod @ A @ K2 @ L ) ) ) ) ).
% numeral_div_numeral
thf(fact_5281_numeral__mod__numeral,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [K2: num,L: num] :
( ( modulo_modulo @ A @ ( numeral_numeral @ A @ K2 ) @ ( numeral_numeral @ A @ L ) )
= ( product_snd @ A @ A @ ( unique8689654367752047608divmod @ A @ K2 @ L ) ) ) ) ).
% numeral_mod_numeral
thf(fact_5282_fst__divmod__nat,axiom,
! [M: nat,N: nat] :
( ( product_fst @ nat @ nat @ ( divmod_nat @ M @ N ) )
= ( divide_divide @ nat @ M @ N ) ) ).
% fst_divmod_nat
thf(fact_5283_snd__divmod__nat,axiom,
! [M: nat,N: nat] :
( ( product_snd @ nat @ nat @ ( divmod_nat @ M @ N ) )
= ( modulo_modulo @ nat @ M @ N ) ) ).
% snd_divmod_nat
thf(fact_5284_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_5285_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_5286_pair__list__eqI,axiom,
! [B: $tType,A: $tType,Xs: list @ ( product_prod @ A @ B ),Ys: list @ ( product_prod @ A @ B )] :
( ( ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xs )
= ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Ys ) )
=> ( ( ( map @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ Xs )
= ( map @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ Ys ) )
=> ( Xs = Ys ) ) ) ).
% pair_list_eqI
thf(fact_5287_Pow__def,axiom,
! [A: $tType] :
( ( pow2 @ A )
= ( ^ [A7: set @ A] :
( collect @ ( set @ A )
@ ^ [B7: set @ A] : ( ord_less_eq @ ( set @ A ) @ B7 @ A7 ) ) ) ) ).
% Pow_def
thf(fact_5288_Eps__case__prod,axiom,
! [B: $tType,A: $tType,P3: A > B > $o] :
( ( fChoice @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ P3 ) )
= ( fChoice @ ( product_prod @ A @ B )
@ ^ [Xy: product_prod @ A @ B] : ( P3 @ ( product_fst @ A @ B @ Xy ) @ ( product_snd @ A @ B @ Xy ) ) ) ) ).
% Eps_case_prod
thf(fact_5289_fst__def,axiom,
! [B: $tType,A: $tType] :
( ( product_fst @ A @ B )
= ( product_case_prod @ A @ B @ A
@ ^ [X1: A,X25: B] : X1 ) ) ).
% fst_def
thf(fact_5290_split__comp__eq,axiom,
! [A: $tType,C: $tType,B: $tType,D: $tType,F3: A > B > C,G: D > A] :
( ( ^ [U2: product_prod @ D @ B] : ( F3 @ ( G @ ( product_fst @ D @ B @ U2 ) ) @ ( product_snd @ D @ B @ U2 ) ) )
= ( product_case_prod @ D @ B @ C
@ ^ [X4: D] : ( F3 @ ( G @ X4 ) ) ) ) ).
% split_comp_eq
thf(fact_5291_case__prod__beta_H,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( product_case_prod @ A @ B @ C )
= ( ^ [F5: A > B > C,X4: product_prod @ A @ B] : ( F5 @ ( product_fst @ A @ B @ X4 ) @ ( product_snd @ A @ B @ X4 ) ) ) ) ).
% case_prod_beta'
thf(fact_5292_case__prod__unfold,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( product_case_prod @ A @ B @ C )
= ( ^ [C3: A > B > C,P4: product_prod @ A @ B] : ( C3 @ ( product_fst @ A @ B @ P4 ) @ ( product_snd @ A @ B @ P4 ) ) ) ) ).
% case_prod_unfold
thf(fact_5293_snd__def,axiom,
! [B: $tType,A: $tType] :
( ( product_snd @ A @ B )
= ( product_case_prod @ A @ B @ B
@ ^ [X1: A,X25: B] : X25 ) ) ).
% snd_def
thf(fact_5294_The__case__prod,axiom,
! [B: $tType,A: $tType,P3: A > B > $o] :
( ( the @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ P3 ) )
= ( the @ ( product_prod @ A @ B )
@ ^ [Xy: product_prod @ A @ B] : ( P3 @ ( product_fst @ A @ B @ Xy ) @ ( product_snd @ A @ B @ Xy ) ) ) ) ).
% The_case_prod
thf(fact_5295_Pow__mono,axiom,
! [A: $tType,A3: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B5 )
=> ( ord_less_eq @ ( set @ ( set @ A ) ) @ ( pow2 @ A @ A3 ) @ ( pow2 @ A @ B5 ) ) ) ).
% Pow_mono
thf(fact_5296_PowD,axiom,
! [A: $tType,A3: set @ A,B5: set @ A] :
( ( member @ ( set @ A ) @ A3 @ ( pow2 @ A @ B5 ) )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ B5 ) ) ).
% PowD
thf(fact_5297_fst__divmod,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M: num,N: num] :
( ( product_fst @ A @ A @ ( unique8689654367752047608divmod @ A @ M @ N ) )
= ( divide_divide @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ N ) ) ) ) ).
% fst_divmod
thf(fact_5298_snd__divmod,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M: num,N: num] :
( ( product_snd @ A @ A @ ( unique8689654367752047608divmod @ A @ M @ N ) )
= ( modulo_modulo @ A @ ( numeral_numeral @ A @ M ) @ ( numeral_numeral @ A @ N ) ) ) ) ).
% snd_divmod
thf(fact_5299_binomial__def,axiom,
( binomial
= ( ^ [N2: 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 ) @ N2 ) ) )
& ( ( finite_card @ nat @ K6 )
= K3 ) ) ) ) ) ) ).
% binomial_def
thf(fact_5300_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_5301_exE__realizer,axiom,
! [C: $tType,A: $tType,B: $tType,P3: A > B > $o,P5: product_prod @ B @ A,Q2: C > $o,F3: B > A > C] :
( ( P3 @ ( product_snd @ B @ A @ P5 ) @ ( product_fst @ B @ A @ P5 ) )
=> ( ! [X5: B,Y7: A] :
( ( P3 @ Y7 @ X5 )
=> ( Q2 @ ( F3 @ X5 @ Y7 ) ) )
=> ( Q2 @ ( product_case_prod @ B @ A @ C @ F3 @ P5 ) ) ) ) ).
% exE_realizer
thf(fact_5302_map__fst__enumerate,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( map @ ( product_prod @ nat @ A ) @ nat @ ( product_fst @ nat @ A ) @ ( enumerate @ A @ N @ Xs ) )
= ( upt @ N @ ( plus_plus @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ).
% map_fst_enumerate
thf(fact_5303_length__enumerate,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( size_size @ ( list @ ( product_prod @ nat @ A ) ) @ ( enumerate @ A @ N @ Xs ) )
= ( size_size @ ( list @ A ) @ Xs ) ) ).
% length_enumerate
thf(fact_5304_snd__divmod__integer,axiom,
! [K2: code_integer,L: code_integer] :
( ( product_snd @ code_integer @ code_integer @ ( code_divmod_integer @ K2 @ L ) )
= ( modulo_modulo @ code_integer @ K2 @ L ) ) ).
% snd_divmod_integer
thf(fact_5305_snd__divmod__abs,axiom,
! [K2: code_integer,L: code_integer] :
( ( product_snd @ code_integer @ code_integer @ ( code_divmod_abs @ K2 @ L ) )
= ( modulo_modulo @ code_integer @ ( abs_abs @ code_integer @ K2 ) @ ( abs_abs @ code_integer @ L ) ) ) ).
% snd_divmod_abs
thf(fact_5306_map__snd__enumerate,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( map @ ( product_prod @ nat @ A ) @ A @ ( product_snd @ nat @ A ) @ ( enumerate @ A @ N @ Xs ) )
= Xs ) ).
% map_snd_enumerate
thf(fact_5307_distinct__enumerate,axiom,
! [A: $tType,N: nat,Xs: list @ A] : ( distinct @ ( product_prod @ nat @ A ) @ ( enumerate @ A @ N @ Xs ) ) ).
% distinct_enumerate
thf(fact_5308_bezw__non__0,axiom,
! [Y2: nat,X: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ Y2 )
=> ( ( bezw @ X @ Y2 )
= ( product_Pair @ int @ int @ ( product_snd @ int @ int @ ( bezw @ Y2 @ ( modulo_modulo @ nat @ X @ Y2 ) ) ) @ ( minus_minus @ int @ ( product_fst @ int @ int @ ( bezw @ Y2 @ ( modulo_modulo @ nat @ X @ Y2 ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ ( bezw @ Y2 @ ( modulo_modulo @ nat @ X @ Y2 ) ) ) @ ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ X @ Y2 ) ) ) ) ) ) ) ).
% bezw_non_0
thf(fact_5309_bezw_Osimps,axiom,
( bezw
= ( ^ [X4: nat,Y: nat] :
( if @ ( product_prod @ int @ int )
@ ( Y
= ( zero_zero @ nat ) )
@ ( product_Pair @ int @ int @ ( one_one @ int ) @ ( zero_zero @ int ) )
@ ( product_Pair @ int @ int @ ( product_snd @ int @ int @ ( bezw @ Y @ ( modulo_modulo @ nat @ X4 @ Y ) ) ) @ ( minus_minus @ int @ ( product_fst @ int @ int @ ( bezw @ Y @ ( modulo_modulo @ nat @ X4 @ Y ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ ( bezw @ Y @ ( modulo_modulo @ nat @ X4 @ Y ) ) ) @ ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ X4 @ Y ) ) ) ) ) ) ) ) ).
% bezw.simps
thf(fact_5310_bezw_Oelims,axiom,
! [X: nat,Xa: nat,Y2: product_prod @ int @ int] :
( ( ( bezw @ X @ Xa )
= Y2 )
=> ( ( ( Xa
= ( zero_zero @ nat ) )
=> ( Y2
= ( product_Pair @ int @ int @ ( one_one @ int ) @ ( zero_zero @ int ) ) ) )
& ( ( Xa
!= ( zero_zero @ nat ) )
=> ( Y2
= ( product_Pair @ int @ int @ ( product_snd @ int @ int @ ( bezw @ Xa @ ( modulo_modulo @ nat @ X @ Xa ) ) ) @ ( minus_minus @ int @ ( product_fst @ int @ int @ ( bezw @ Xa @ ( modulo_modulo @ nat @ X @ Xa ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ ( bezw @ Xa @ ( modulo_modulo @ nat @ X @ Xa ) ) ) @ ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ X @ Xa ) ) ) ) ) ) ) ) ) ).
% bezw.elims
thf(fact_5311_enumerate__map__upt,axiom,
! [A: $tType,N: nat,F3: nat > A,M: nat] :
( ( enumerate @ A @ N @ ( map @ nat @ A @ F3 @ ( upt @ N @ M ) ) )
= ( map @ nat @ ( product_prod @ nat @ A )
@ ^ [K3: nat] : ( product_Pair @ nat @ A @ K3 @ ( F3 @ K3 ) )
@ ( upt @ N @ M ) ) ) ).
% enumerate_map_upt
thf(fact_5312_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_5313_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_5314_enumerate__replicate__eq,axiom,
! [A: $tType,N: nat,M: nat,A2: A] :
( ( enumerate @ A @ N @ ( replicate @ A @ M @ A2 ) )
= ( map @ nat @ ( product_prod @ nat @ A )
@ ^ [Q5: nat] : ( product_Pair @ nat @ A @ Q5 @ A2 )
@ ( upt @ N @ ( plus_plus @ nat @ N @ M ) ) ) ) ).
% enumerate_replicate_eq
thf(fact_5315_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_5316_bezw_Opelims,axiom,
! [X: nat,Xa: nat,Y2: product_prod @ int @ int] :
( ( ( bezw @ X @ Xa )
= Y2 )
=> ( ( accp @ ( product_prod @ nat @ nat ) @ bezw_rel @ ( product_Pair @ nat @ nat @ X @ Xa ) )
=> ~ ( ( ( ( Xa
= ( zero_zero @ nat ) )
=> ( Y2
= ( product_Pair @ int @ int @ ( one_one @ int ) @ ( zero_zero @ int ) ) ) )
& ( ( Xa
!= ( zero_zero @ nat ) )
=> ( Y2
= ( product_Pair @ int @ int @ ( product_snd @ int @ int @ ( bezw @ Xa @ ( modulo_modulo @ nat @ X @ Xa ) ) ) @ ( minus_minus @ int @ ( product_fst @ int @ int @ ( bezw @ Xa @ ( modulo_modulo @ nat @ X @ Xa ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ ( bezw @ Xa @ ( modulo_modulo @ nat @ X @ Xa ) ) ) @ ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ X @ Xa ) ) ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ nat @ nat ) @ bezw_rel @ ( product_Pair @ nat @ nat @ X @ Xa ) ) ) ) ) ).
% bezw.pelims
thf(fact_5317_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_5318_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_5319_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_5320_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_5321_normalize__def,axiom,
( normalize
= ( ^ [P4: product_prod @ int @ int] :
( if @ ( product_prod @ int @ int ) @ ( ord_less @ int @ ( zero_zero @ int ) @ ( product_snd @ int @ int @ P4 ) ) @ ( product_Pair @ int @ int @ ( divide_divide @ int @ ( product_fst @ int @ int @ P4 ) @ ( gcd_gcd @ int @ ( product_fst @ int @ int @ P4 ) @ ( product_snd @ int @ int @ P4 ) ) ) @ ( divide_divide @ int @ ( product_snd @ int @ int @ P4 ) @ ( gcd_gcd @ int @ ( product_fst @ int @ int @ P4 ) @ ( product_snd @ int @ int @ P4 ) ) ) )
@ ( if @ ( product_prod @ int @ int )
@ ( ( product_snd @ int @ int @ P4 )
= ( zero_zero @ int ) )
@ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
@ ( product_Pair @ int @ int @ ( divide_divide @ int @ ( product_fst @ int @ int @ P4 ) @ ( uminus_uminus @ int @ ( gcd_gcd @ int @ ( product_fst @ int @ int @ P4 ) @ ( product_snd @ int @ int @ P4 ) ) ) ) @ ( divide_divide @ int @ ( product_snd @ int @ int @ P4 ) @ ( uminus_uminus @ int @ ( gcd_gcd @ int @ ( product_fst @ int @ int @ P4 ) @ ( product_snd @ int @ int @ P4 ) ) ) ) ) ) ) ) ) ).
% normalize_def
thf(fact_5322_size__prod__simp,axiom,
! [B: $tType,A: $tType] :
( ( basic_BNF_size_prod @ A @ B )
= ( ^ [F5: A > nat,G2: B > nat,P4: product_prod @ A @ B] : ( plus_plus @ nat @ ( plus_plus @ nat @ ( F5 @ ( product_fst @ A @ B @ P4 ) ) @ ( G2 @ ( product_snd @ A @ B @ P4 ) ) ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ).
% size_prod_simp
thf(fact_5323_drop__bit__numeral__minus__bit1,axiom,
! [L: num,K2: num] :
( ( bit_se4197421643247451524op_bit @ int @ ( numeral_numeral @ nat @ L ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ K2 ) ) ) )
= ( bit_se4197421643247451524op_bit @ int @ ( pred_numeral @ L ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( inc @ K2 ) ) ) ) ) ).
% drop_bit_numeral_minus_bit1
thf(fact_5324_gcd_Obottom__right__bottom,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A2: A] :
( ( gcd_gcd @ A @ A2 @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% gcd.bottom_right_bottom
thf(fact_5325_gcd_Obottom__left__bottom,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A2: A] :
( ( gcd_gcd @ A @ ( one_one @ A ) @ A2 )
= ( one_one @ A ) ) ) ).
% gcd.bottom_left_bottom
thf(fact_5326_gcd__add2,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [M: A,N: A] :
( ( gcd_gcd @ A @ M @ ( plus_plus @ A @ M @ N ) )
= ( gcd_gcd @ A @ M @ N ) ) ) ).
% gcd_add2
thf(fact_5327_gcd__add1,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [M: A,N: A] :
( ( gcd_gcd @ A @ ( plus_plus @ A @ M @ N ) @ N )
= ( gcd_gcd @ A @ M @ N ) ) ) ).
% gcd_add1
thf(fact_5328_gcd__exp,axiom,
! [A: $tType] :
( ( semiri6843258321239162965malize @ A )
=> ! [A2: A,N: nat,B2: A] :
( ( gcd_gcd @ A @ ( power_power @ A @ A2 @ N ) @ ( power_power @ A @ B2 @ N ) )
= ( power_power @ A @ ( gcd_gcd @ A @ A2 @ B2 ) @ N ) ) ) ).
% gcd_exp
thf(fact_5329_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_5330_drop__bit__drop__bit,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: nat,N: nat,A2: A] :
( ( bit_se4197421643247451524op_bit @ A @ M @ ( bit_se4197421643247451524op_bit @ A @ N @ A2 ) )
= ( bit_se4197421643247451524op_bit @ A @ ( plus_plus @ nat @ M @ N ) @ A2 ) ) ) ).
% drop_bit_drop_bit
thf(fact_5331_gcd__1__int,axiom,
! [M: int] :
( ( gcd_gcd @ int @ M @ ( one_one @ int ) )
= ( one_one @ int ) ) ).
% gcd_1_int
thf(fact_5332_drop__bit__and,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A2: A,B2: A] :
( ( bit_se4197421643247451524op_bit @ A @ N @ ( bit_se5824344872417868541ns_and @ A @ A2 @ B2 ) )
= ( bit_se5824344872417868541ns_and @ A @ ( bit_se4197421643247451524op_bit @ A @ N @ A2 ) @ ( bit_se4197421643247451524op_bit @ A @ N @ B2 ) ) ) ) ).
% drop_bit_and
thf(fact_5333_drop__bit__or,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A2: A,B2: A] :
( ( bit_se4197421643247451524op_bit @ A @ N @ ( bit_se1065995026697491101ons_or @ A @ A2 @ B2 ) )
= ( bit_se1065995026697491101ons_or @ A @ ( bit_se4197421643247451524op_bit @ A @ N @ A2 ) @ ( bit_se4197421643247451524op_bit @ A @ N @ B2 ) ) ) ) ).
% drop_bit_or
thf(fact_5334_drop__bit__xor,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A2: A,B2: A] :
( ( bit_se4197421643247451524op_bit @ A @ N @ ( bit_se5824344971392196577ns_xor @ A @ A2 @ B2 ) )
= ( bit_se5824344971392196577ns_xor @ A @ ( bit_se4197421643247451524op_bit @ A @ N @ A2 ) @ ( bit_se4197421643247451524op_bit @ A @ N @ B2 ) ) ) ) ).
% drop_bit_xor
thf(fact_5335_gcd__neg__numeral__2,axiom,
! [A: $tType] :
( ( ring_gcd @ A )
=> ! [A2: A,N: num] :
( ( gcd_gcd @ A @ A2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
= ( gcd_gcd @ A @ A2 @ ( numeral_numeral @ A @ N ) ) ) ) ).
% gcd_neg_numeral_2
thf(fact_5336_gcd__neg__numeral__1,axiom,
! [A: $tType] :
( ( ring_gcd @ A )
=> ! [N: num,A2: A] :
( ( gcd_gcd @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) @ A2 )
= ( gcd_gcd @ A @ ( numeral_numeral @ A @ N ) @ A2 ) ) ) ).
% gcd_neg_numeral_1
thf(fact_5337_is__unit__gcd__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A2: A,B2: A] :
( ( dvd_dvd @ A @ ( gcd_gcd @ A @ A2 @ B2 ) @ ( one_one @ A ) )
= ( ( gcd_gcd @ A @ A2 @ B2 )
= ( one_one @ A ) ) ) ) ).
% is_unit_gcd_iff
thf(fact_5338_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_5339_drop__bit__nonnegative__int__iff,axiom,
! [N: nat,K2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se4197421643247451524op_bit @ int @ N @ K2 ) )
= ( ord_less_eq @ int @ ( zero_zero @ int ) @ K2 ) ) ).
% drop_bit_nonnegative_int_iff
thf(fact_5340_drop__bit__negative__int__iff,axiom,
! [N: nat,K2: int] :
( ( ord_less @ int @ ( bit_se4197421643247451524op_bit @ int @ N @ K2 ) @ ( zero_zero @ int ) )
= ( ord_less @ int @ K2 @ ( zero_zero @ int ) ) ) ).
% drop_bit_negative_int_iff
thf(fact_5341_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_5342_drop__bit__Suc__bit0,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [N: nat,K2: num] :
( ( bit_se4197421643247451524op_bit @ A @ ( suc @ N ) @ ( numeral_numeral @ A @ ( bit0 @ K2 ) ) )
= ( bit_se4197421643247451524op_bit @ A @ N @ ( numeral_numeral @ A @ K2 ) ) ) ) ).
% drop_bit_Suc_bit0
thf(fact_5343_drop__bit__Suc__bit1,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [N: nat,K2: num] :
( ( bit_se4197421643247451524op_bit @ A @ ( suc @ N ) @ ( numeral_numeral @ A @ ( bit1 @ K2 ) ) )
= ( bit_se4197421643247451524op_bit @ A @ N @ ( numeral_numeral @ A @ K2 ) ) ) ) ).
% drop_bit_Suc_bit1
thf(fact_5344_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_5345_drop__bit__numeral__bit0,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [L: num,K2: num] :
( ( bit_se4197421643247451524op_bit @ A @ ( numeral_numeral @ nat @ L ) @ ( numeral_numeral @ A @ ( bit0 @ K2 ) ) )
= ( bit_se4197421643247451524op_bit @ A @ ( pred_numeral @ L ) @ ( numeral_numeral @ A @ K2 ) ) ) ) ).
% drop_bit_numeral_bit0
thf(fact_5346_drop__bit__numeral__bit1,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [L: num,K2: num] :
( ( bit_se4197421643247451524op_bit @ A @ ( numeral_numeral @ nat @ L ) @ ( numeral_numeral @ A @ ( bit1 @ K2 ) ) )
= ( bit_se4197421643247451524op_bit @ A @ ( pred_numeral @ L ) @ ( numeral_numeral @ A @ K2 ) ) ) ) ).
% drop_bit_numeral_bit1
thf(fact_5347_drop__bit__Suc__minus__bit0,axiom,
! [N: nat,K2: num] :
( ( bit_se4197421643247451524op_bit @ int @ ( suc @ N ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ K2 ) ) ) )
= ( bit_se4197421643247451524op_bit @ int @ N @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K2 ) ) ) ) ).
% drop_bit_Suc_minus_bit0
thf(fact_5348_drop__bit__numeral__minus__bit0,axiom,
! [L: num,K2: num] :
( ( bit_se4197421643247451524op_bit @ int @ ( numeral_numeral @ nat @ L ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ K2 ) ) ) )
= ( bit_se4197421643247451524op_bit @ int @ ( pred_numeral @ L ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K2 ) ) ) ) ).
% drop_bit_numeral_minus_bit0
thf(fact_5349_drop__bit__Suc__minus__bit1,axiom,
! [N: nat,K2: num] :
( ( bit_se4197421643247451524op_bit @ int @ ( suc @ N ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ K2 ) ) ) )
= ( bit_se4197421643247451524op_bit @ int @ N @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( inc @ K2 ) ) ) ) ) ).
% drop_bit_Suc_minus_bit1
thf(fact_5350_gcd__add__mult,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [M: A,K2: A,N: A] :
( ( gcd_gcd @ A @ M @ ( plus_plus @ A @ ( times_times @ A @ K2 @ M ) @ N ) )
= ( gcd_gcd @ A @ M @ N ) ) ) ).
% gcd_add_mult
thf(fact_5351_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_5352_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_5353_gcd__red__int,axiom,
( ( gcd_gcd @ int )
= ( ^ [X4: int,Y: int] : ( gcd_gcd @ int @ Y @ ( modulo_modulo @ int @ X4 @ Y ) ) ) ) ).
% gcd_red_int
thf(fact_5354_gcd__dvd__prod,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A2: A,B2: A,K2: A] : ( dvd_dvd @ A @ ( gcd_gcd @ A @ A2 @ B2 ) @ ( times_times @ A @ K2 @ B2 ) ) ) ).
% gcd_dvd_prod
thf(fact_5355_gcd__ge__0__int,axiom,
! [X: int,Y2: int] : ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( gcd_gcd @ int @ X @ Y2 ) ) ).
% gcd_ge_0_int
thf(fact_5356_bezout__int,axiom,
! [X: int,Y2: int] :
? [U3: int,V3: int] :
( ( plus_plus @ int @ ( times_times @ int @ U3 @ X ) @ ( times_times @ int @ V3 @ Y2 ) )
= ( gcd_gcd @ int @ X @ Y2 ) ) ).
% bezout_int
thf(fact_5357_gcd__mult__distrib__int,axiom,
! [K2: int,M: int,N: int] :
( ( times_times @ int @ ( abs_abs @ int @ K2 ) @ ( gcd_gcd @ int @ M @ N ) )
= ( gcd_gcd @ int @ ( times_times @ int @ K2 @ M ) @ ( times_times @ int @ K2 @ N ) ) ) ).
% gcd_mult_distrib_int
thf(fact_5358_take__bit__eq__self__iff__drop__bit__eq__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A2: A] :
( ( ( bit_se2584673776208193580ke_bit @ A @ N @ A2 )
= A2 )
= ( ( bit_se4197421643247451524op_bit @ A @ N @ A2 )
= ( zero_zero @ A ) ) ) ) ).
% take_bit_eq_self_iff_drop_bit_eq_0
thf(fact_5359_drop__bit__push__bit__int,axiom,
! [M: nat,N: nat,K2: int] :
( ( bit_se4197421643247451524op_bit @ int @ M @ ( bit_se4730199178511100633sh_bit @ int @ N @ K2 ) )
= ( bit_se4197421643247451524op_bit @ int @ ( minus_minus @ nat @ M @ N ) @ ( bit_se4730199178511100633sh_bit @ int @ ( minus_minus @ nat @ N @ M ) @ K2 ) ) ) ).
% drop_bit_push_bit_int
thf(fact_5360_take__bit__drop__bit,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: nat,N: nat,A2: A] :
( ( bit_se2584673776208193580ke_bit @ A @ M @ ( bit_se4197421643247451524op_bit @ A @ N @ A2 ) )
= ( bit_se4197421643247451524op_bit @ A @ N @ ( bit_se2584673776208193580ke_bit @ A @ ( plus_plus @ nat @ M @ N ) @ A2 ) ) ) ) ).
% take_bit_drop_bit
thf(fact_5361_drop__bit__take__bit,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: nat,N: nat,A2: A] :
( ( bit_se4197421643247451524op_bit @ A @ M @ ( bit_se2584673776208193580ke_bit @ A @ N @ A2 ) )
= ( bit_se2584673776208193580ke_bit @ A @ ( minus_minus @ nat @ N @ M ) @ ( bit_se4197421643247451524op_bit @ A @ M @ A2 ) ) ) ) ).
% drop_bit_take_bit
thf(fact_5362_gcd__mult__unit2,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) )
=> ( ( gcd_gcd @ A @ B2 @ ( times_times @ A @ C2 @ A2 ) )
= ( gcd_gcd @ A @ B2 @ C2 ) ) ) ) ).
% gcd_mult_unit2
thf(fact_5363_gcd__mult__unit1,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) )
=> ( ( gcd_gcd @ A @ ( times_times @ A @ B2 @ A2 ) @ C2 )
= ( gcd_gcd @ A @ B2 @ C2 ) ) ) ) ).
% gcd_mult_unit1
thf(fact_5364_gcd__div__unit1,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) )
=> ( ( gcd_gcd @ A @ ( divide_divide @ A @ B2 @ A2 ) @ C2 )
= ( gcd_gcd @ A @ B2 @ C2 ) ) ) ) ).
% gcd_div_unit1
thf(fact_5365_gcd__div__unit2,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) )
=> ( ( gcd_gcd @ A @ B2 @ ( divide_divide @ A @ C2 @ A2 ) )
= ( gcd_gcd @ A @ B2 @ C2 ) ) ) ) ).
% gcd_div_unit2
thf(fact_5366_gcd__le2__int,axiom,
! [B2: int,A2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
=> ( ord_less_eq @ int @ ( gcd_gcd @ int @ A2 @ B2 ) @ B2 ) ) ).
% gcd_le2_int
thf(fact_5367_gcd__le1__int,axiom,
! [A2: int,B2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ A2 )
=> ( ord_less_eq @ int @ ( gcd_gcd @ int @ A2 @ B2 ) @ A2 ) ) ).
% gcd_le1_int
thf(fact_5368_gcd__cases__int,axiom,
! [X: int,Y2: int,P3: int > $o] :
( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y2 )
=> ( P3 @ ( gcd_gcd @ int @ X @ Y2 ) ) ) )
=> ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
=> ( ( ord_less_eq @ int @ Y2 @ ( zero_zero @ int ) )
=> ( P3 @ ( gcd_gcd @ int @ X @ ( uminus_uminus @ int @ Y2 ) ) ) ) )
=> ( ( ( ord_less_eq @ int @ X @ ( zero_zero @ int ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y2 )
=> ( P3 @ ( gcd_gcd @ int @ ( uminus_uminus @ int @ X ) @ Y2 ) ) ) )
=> ( ( ( ord_less_eq @ int @ X @ ( zero_zero @ int ) )
=> ( ( ord_less_eq @ int @ Y2 @ ( zero_zero @ int ) )
=> ( P3 @ ( gcd_gcd @ int @ ( uminus_uminus @ int @ X ) @ ( uminus_uminus @ int @ Y2 ) ) ) ) )
=> ( P3 @ ( gcd_gcd @ int @ X @ Y2 ) ) ) ) ) ) ).
% gcd_cases_int
thf(fact_5369_gcd__unique__int,axiom,
! [D2: int,A2: int,B2: int] :
( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ D2 )
& ( dvd_dvd @ int @ D2 @ A2 )
& ( dvd_dvd @ int @ D2 @ B2 )
& ! [E3: int] :
( ( ( dvd_dvd @ int @ E3 @ A2 )
& ( dvd_dvd @ int @ E3 @ B2 ) )
=> ( dvd_dvd @ int @ E3 @ D2 ) ) )
= ( D2
= ( gcd_gcd @ int @ A2 @ B2 ) ) ) ).
% gcd_unique_int
thf(fact_5370_gcd__non__0__int,axiom,
! [Y2: int,X: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ Y2 )
=> ( ( gcd_gcd @ int @ X @ Y2 )
= ( gcd_gcd @ int @ Y2 @ ( modulo_modulo @ int @ X @ Y2 ) ) ) ) ).
% gcd_non_0_int
thf(fact_5371_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_5372_div__push__bit__of__1__eq__drop__bit,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A,N: nat] :
( ( divide_divide @ A @ A2 @ ( bit_se4730199178511100633sh_bit @ A @ N @ ( one_one @ A ) ) )
= ( bit_se4197421643247451524op_bit @ A @ N @ A2 ) ) ) ).
% div_push_bit_of_1_eq_drop_bit
thf(fact_5373_bit__iff__and__drop__bit__eq__1,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se5641148757651400278ts_bit @ A )
= ( ^ [A5: A,N2: nat] :
( ( bit_se5824344872417868541ns_and @ A @ ( bit_se4197421643247451524op_bit @ A @ N2 @ A5 ) @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ) ) ).
% bit_iff_and_drop_bit_eq_1
thf(fact_5374_bits__ident,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A2: A] :
( ( plus_plus @ A @ ( bit_se4730199178511100633sh_bit @ A @ N @ ( bit_se4197421643247451524op_bit @ A @ N @ A2 ) ) @ ( bit_se2584673776208193580ke_bit @ A @ N @ A2 ) )
= A2 ) ) ).
% bits_ident
thf(fact_5375_stable__imp__drop__bit__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A,N: nat] :
( ( ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= A2 )
=> ( ( bit_se4197421643247451524op_bit @ A @ N @ A2 )
= A2 ) ) ) ).
% stable_imp_drop_bit_eq
thf(fact_5376_drop__bit__half,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A2: A] :
( ( bit_se4197421643247451524op_bit @ A @ N @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
= ( divide_divide @ A @ ( bit_se4197421643247451524op_bit @ A @ N @ A2 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% drop_bit_half
thf(fact_5377_drop__bit__int__def,axiom,
( ( bit_se4197421643247451524op_bit @ int )
= ( ^ [N2: nat,K3: int] : ( divide_divide @ int @ K3 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ).
% drop_bit_int_def
thf(fact_5378_drop__bit__eq__div,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se4197421643247451524op_bit @ A )
= ( ^ [N2: nat,A5: A] : ( divide_divide @ A @ A5 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ) ).
% drop_bit_eq_div
thf(fact_5379_drop__bit__Suc,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A2: A] :
( ( bit_se4197421643247451524op_bit @ A @ ( suc @ N ) @ A2 )
= ( bit_se4197421643247451524op_bit @ A @ N @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% drop_bit_Suc
thf(fact_5380_even__drop__bit__iff__not__bit,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A2: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se4197421643247451524op_bit @ A @ N @ A2 ) )
= ( ~ ( bit_se5641148757651400278ts_bit @ A @ A2 @ N ) ) ) ) ).
% even_drop_bit_iff_not_bit
thf(fact_5381_bit__iff__odd__drop__bit,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se5641148757651400278ts_bit @ A )
= ( ^ [A5: A,N2: nat] :
~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se4197421643247451524op_bit @ A @ N2 @ A5 ) ) ) ) ) ).
% bit_iff_odd_drop_bit
thf(fact_5382_nat__descend__induct,axiom,
! [N: nat,P3: nat > $o,M: nat] :
( ! [K: nat] :
( ( ord_less @ nat @ N @ K )
=> ( P3 @ K ) )
=> ( ! [K: nat] :
( ( ord_less_eq @ nat @ K @ N )
=> ( ! [I4: nat] :
( ( ord_less @ nat @ K @ I4 )
=> ( P3 @ I4 ) )
=> ( P3 @ K ) ) )
=> ( P3 @ M ) ) ) ).
% nat_descend_induct
thf(fact_5383_slice__eq__mask,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: nat,M: nat,A2: A] :
( ( bit_se4730199178511100633sh_bit @ A @ N @ ( bit_se2584673776208193580ke_bit @ A @ M @ ( bit_se4197421643247451524op_bit @ A @ N @ A2 ) ) )
= ( bit_se5824344872417868541ns_and @ A @ A2 @ ( bit_se5824344872417868541ns_and @ A @ ( bit_se2239418461657761734s_mask @ A @ ( plus_plus @ nat @ M @ N ) ) @ ( bit_ri4277139882892585799ns_not @ A @ ( bit_se2239418461657761734s_mask @ A @ N ) ) ) ) ) ) ).
% slice_eq_mask
thf(fact_5384_drop__bit__rec,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se4197421643247451524op_bit @ A )
= ( ^ [N2: nat,A5: A] :
( if @ A
@ ( N2
= ( zero_zero @ nat ) )
@ A5
@ ( bit_se4197421643247451524op_bit @ A @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) @ ( divide_divide @ A @ A5 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% drop_bit_rec
thf(fact_5385_less__by__empty,axiom,
! [A: $tType,A3: set @ ( product_prod @ A @ A ),B5: set @ ( product_prod @ A @ A )] :
( ( A3
= ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ A3 @ B5 ) ) ).
% less_by_empty
thf(fact_5386_finite__enumerate,axiom,
! [S3: set @ nat] :
( ( finite_finite2 @ nat @ S3 )
=> ? [R3: nat > nat] :
( ( strict_mono_on @ nat @ nat @ R3 @ ( set_ord_lessThan @ nat @ ( finite_card @ nat @ S3 ) ) )
& ! [N9: nat] :
( ( ord_less @ nat @ N9 @ ( finite_card @ nat @ S3 ) )
=> ( member @ nat @ ( R3 @ N9 ) @ S3 ) ) ) ) ).
% finite_enumerate
thf(fact_5387_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,S4: code_integer] :
( if @ ( product_prod @ code_integer @ code_integer )
@ ( S4
= ( 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 ) @ S4 ) ) )
@ ( code_divmod_abs @ K3 @ L3 ) ) ) ) ) ) ) ) ).
% divmod_integer_eq_cases
thf(fact_5388_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_5389_gcd__1__nat,axiom,
! [M: nat] :
( ( gcd_gcd @ nat @ M @ ( one_one @ nat ) )
= ( one_one @ nat ) ) ).
% gcd_1_nat
thf(fact_5390_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_5391_map__comp__map,axiom,
! [B: $tType,C: $tType,A: $tType,F3: C > B,G: A > C] :
( ( comp @ ( list @ C ) @ ( list @ B ) @ ( list @ A ) @ ( map @ C @ B @ F3 ) @ ( map @ A @ C @ G ) )
= ( map @ A @ B @ ( comp @ C @ B @ A @ F3 @ G ) ) ) ).
% map_comp_map
thf(fact_5392_List_Omap_Ocomp,axiom,
! [C: $tType,B: $tType,A: $tType,F3: B > C,G: A > B] :
( ( comp @ ( list @ B ) @ ( list @ C ) @ ( list @ A ) @ ( map @ B @ C @ F3 ) @ ( map @ A @ B @ G ) )
= ( map @ A @ C @ ( comp @ B @ C @ A @ F3 @ G ) ) ) ).
% List.map.comp
thf(fact_5393_list_Omap__comp,axiom,
! [B: $tType,C: $tType,A: $tType,G: B > C,F3: A > B,V2: list @ A] :
( ( map @ B @ C @ G @ ( map @ A @ B @ F3 @ V2 ) )
= ( map @ A @ C @ ( comp @ B @ C @ A @ G @ F3 ) @ V2 ) ) ).
% list.map_comp
thf(fact_5394_List_Omap_Ocompositionality,axiom,
! [B: $tType,C: $tType,A: $tType,F3: B > C,G: A > B,List: list @ A] :
( ( map @ B @ C @ F3 @ ( map @ A @ B @ G @ List ) )
= ( map @ A @ C @ ( comp @ B @ C @ A @ F3 @ G ) @ List ) ) ).
% List.map.compositionality
thf(fact_5395_map__map,axiom,
! [B: $tType,A: $tType,C: $tType,F3: B > A,G: C > B,Xs: list @ C] :
( ( map @ B @ A @ F3 @ ( map @ C @ B @ G @ Xs ) )
= ( map @ C @ A @ ( comp @ B @ A @ C @ F3 @ G ) @ Xs ) ) ).
% map_map
thf(fact_5396_set__rotate1,axiom,
! [A: $tType,Xs: list @ A] :
( ( set2 @ A @ ( rotate1 @ A @ Xs ) )
= ( set2 @ A @ Xs ) ) ).
% set_rotate1
thf(fact_5397_length__rotate1,axiom,
! [A: $tType,Xs: list @ A] :
( ( size_size @ ( list @ A ) @ ( rotate1 @ A @ Xs ) )
= ( size_size @ ( list @ A ) @ Xs ) ) ).
% length_rotate1
thf(fact_5398_distinct1__rotate,axiom,
! [A: $tType,Xs: list @ A] :
( ( distinct @ A @ ( rotate1 @ A @ Xs ) )
= ( distinct @ A @ Xs ) ) ).
% distinct1_rotate
thf(fact_5399_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_5400_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_5401_gcd__mult__distrib__nat,axiom,
! [K2: nat,M: nat,N: nat] :
( ( times_times @ nat @ K2 @ ( gcd_gcd @ nat @ M @ N ) )
= ( gcd_gcd @ nat @ ( times_times @ nat @ K2 @ M ) @ ( times_times @ nat @ K2 @ N ) ) ) ).
% gcd_mult_distrib_nat
thf(fact_5402_gcd__red__nat,axiom,
( ( gcd_gcd @ nat )
= ( ^ [X4: nat,Y: nat] : ( gcd_gcd @ nat @ Y @ ( modulo_modulo @ nat @ X4 @ Y ) ) ) ) ).
% gcd_red_nat
thf(fact_5403_rotate1__map,axiom,
! [A: $tType,B: $tType,F3: B > A,Xs: list @ B] :
( ( rotate1 @ A @ ( map @ B @ A @ F3 @ Xs ) )
= ( map @ B @ A @ F3 @ ( rotate1 @ B @ Xs ) ) ) ).
% rotate1_map
thf(fact_5404_gcd__le1__nat,axiom,
! [A2: nat,B2: nat] :
( ( A2
!= ( zero_zero @ nat ) )
=> ( ord_less_eq @ nat @ ( gcd_gcd @ nat @ A2 @ B2 ) @ A2 ) ) ).
% gcd_le1_nat
thf(fact_5405_gcd__le2__nat,axiom,
! [B2: nat,A2: nat] :
( ( B2
!= ( zero_zero @ nat ) )
=> ( ord_less_eq @ nat @ ( gcd_gcd @ nat @ A2 @ B2 ) @ B2 ) ) ).
% gcd_le2_nat
thf(fact_5406_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_5407_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_5408_gcd__nat_Oelims,axiom,
! [X: nat,Xa: nat,Y2: nat] :
( ( ( gcd_gcd @ nat @ X @ Xa )
= Y2 )
=> ( ( ( Xa
= ( zero_zero @ nat ) )
=> ( Y2 = X ) )
& ( ( Xa
!= ( zero_zero @ nat ) )
=> ( Y2
= ( gcd_gcd @ nat @ Xa @ ( modulo_modulo @ nat @ X @ Xa ) ) ) ) ) ) ).
% gcd_nat.elims
thf(fact_5409_gcd__nat_Osimps,axiom,
( ( gcd_gcd @ nat )
= ( ^ [X4: nat,Y: nat] :
( if @ nat
@ ( Y
= ( zero_zero @ nat ) )
@ X4
@ ( gcd_gcd @ nat @ Y @ ( modulo_modulo @ nat @ X4 @ Y ) ) ) ) ) ).
% gcd_nat.simps
thf(fact_5410_gcd__non__0__nat,axiom,
! [Y2: nat,X: nat] :
( ( Y2
!= ( zero_zero @ nat ) )
=> ( ( gcd_gcd @ nat @ X @ Y2 )
= ( gcd_gcd @ nat @ Y2 @ ( modulo_modulo @ nat @ X @ Y2 ) ) ) ) ).
% gcd_non_0_nat
thf(fact_5411_drop__bit__nat__eq,axiom,
! [N: nat,K2: int] :
( ( bit_se4197421643247451524op_bit @ nat @ N @ ( nat2 @ K2 ) )
= ( nat2 @ ( bit_se4197421643247451524op_bit @ int @ N @ K2 ) ) ) ).
% drop_bit_nat_eq
thf(fact_5412_sum__comp__morphism,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( ( comm_monoid_add @ B )
& ( comm_monoid_add @ A ) )
=> ! [H2: B > A,G: C > B,A3: set @ C] :
( ( ( H2 @ ( zero_zero @ B ) )
= ( zero_zero @ A ) )
=> ( ! [X5: B,Y7: B] :
( ( H2 @ ( plus_plus @ B @ X5 @ Y7 ) )
= ( plus_plus @ A @ ( H2 @ X5 ) @ ( H2 @ Y7 ) ) )
=> ( ( groups7311177749621191930dd_sum @ C @ A @ ( comp @ B @ A @ C @ H2 @ G ) @ A3 )
= ( H2 @ ( groups7311177749621191930dd_sum @ C @ B @ G @ A3 ) ) ) ) ) ) ).
% sum_comp_morphism
thf(fact_5413_bezout__nat,axiom,
! [A2: nat,B2: nat] :
( ( A2
!= ( zero_zero @ nat ) )
=> ? [X5: nat,Y7: nat] :
( ( times_times @ nat @ A2 @ X5 )
= ( plus_plus @ nat @ ( times_times @ nat @ B2 @ Y7 ) @ ( gcd_gcd @ nat @ A2 @ B2 ) ) ) ) ).
% bezout_nat
thf(fact_5414_bezout__gcd__nat_H,axiom,
! [B2: nat,A2: nat] :
? [X5: nat,Y7: nat] :
( ( ( ord_less_eq @ nat @ ( times_times @ nat @ B2 @ Y7 ) @ ( times_times @ nat @ A2 @ X5 ) )
& ( ( minus_minus @ nat @ ( times_times @ nat @ A2 @ X5 ) @ ( times_times @ nat @ B2 @ Y7 ) )
= ( gcd_gcd @ nat @ A2 @ B2 ) ) )
| ( ( ord_less_eq @ nat @ ( times_times @ nat @ A2 @ Y7 ) @ ( times_times @ nat @ B2 @ X5 ) )
& ( ( minus_minus @ nat @ ( times_times @ nat @ B2 @ X5 ) @ ( times_times @ nat @ A2 @ Y7 ) )
= ( gcd_gcd @ nat @ A2 @ B2 ) ) ) ) ).
% bezout_gcd_nat'
thf(fact_5415_uminus__sum__list__map,axiom,
! [A: $tType,B: $tType] :
( ( ab_group_add @ A )
=> ! [F3: B > A,Xs: list @ B] :
( ( uminus_uminus @ A @ ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ F3 @ Xs ) ) )
= ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ ( comp @ A @ A @ B @ ( uminus_uminus @ A ) @ F3 ) @ Xs ) ) ) ) ).
% uminus_sum_list_map
thf(fact_5416_case__prod__comp,axiom,
! [D: $tType,A: $tType,C: $tType,B: $tType,F3: D > C > A,G: B > D,X: product_prod @ B @ C] :
( ( product_case_prod @ B @ C @ A @ ( comp @ D @ ( C > A ) @ B @ F3 @ G ) @ X )
= ( F3 @ ( G @ ( product_fst @ B @ C @ X ) ) @ ( product_snd @ B @ C @ X ) ) ) ).
% case_prod_comp
thf(fact_5417_gcd__code__integer,axiom,
( ( gcd_gcd @ code_integer )
= ( ^ [K3: code_integer,L3: code_integer] :
( abs_abs @ code_integer
@ ( if @ code_integer
@ ( L3
= ( zero_zero @ code_integer ) )
@ K3
@ ( gcd_gcd @ code_integer @ L3 @ ( modulo_modulo @ code_integer @ ( abs_abs @ code_integer @ K3 ) @ ( abs_abs @ code_integer @ L3 ) ) ) ) ) ) ) ).
% gcd_code_integer
thf(fact_5418_drop__bit__nat__def,axiom,
( ( bit_se4197421643247451524op_bit @ nat )
= ( ^ [N2: nat,M2: nat] : ( divide_divide @ nat @ M2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ).
% drop_bit_nat_def
thf(fact_5419_bezw__aux,axiom,
! [X: nat,Y2: nat] :
( ( semiring_1_of_nat @ int @ ( gcd_gcd @ nat @ X @ Y2 ) )
= ( plus_plus @ int @ ( times_times @ int @ ( product_fst @ int @ int @ ( bezw @ X @ Y2 ) ) @ ( semiring_1_of_nat @ int @ X ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ ( bezw @ X @ Y2 ) ) @ ( semiring_1_of_nat @ int @ Y2 ) ) ) ) ).
% bezw_aux
thf(fact_5420_gcd__nat_Opelims,axiom,
! [X: nat,Xa: nat,Y2: nat] :
( ( ( gcd_gcd @ nat @ X @ Xa )
= Y2 )
=> ( ( accp @ ( product_prod @ nat @ nat ) @ gcd_nat_rel @ ( product_Pair @ nat @ nat @ X @ Xa ) )
=> ~ ( ( ( ( Xa
= ( zero_zero @ nat ) )
=> ( Y2 = X ) )
& ( ( Xa
!= ( zero_zero @ nat ) )
=> ( Y2
= ( gcd_gcd @ nat @ Xa @ ( modulo_modulo @ nat @ X @ Xa ) ) ) ) )
=> ~ ( accp @ ( product_prod @ nat @ nat ) @ gcd_nat_rel @ ( product_Pair @ nat @ nat @ X @ Xa ) ) ) ) ) ).
% gcd_nat.pelims
thf(fact_5421_strict__mono__on__def,axiom,
! [B: $tType,A: $tType] :
( ( ( ord @ A )
& ( ord @ B ) )
=> ( ( strict_mono_on @ A @ B )
= ( ^ [F5: A > B,A7: set @ A] :
! [R5: A,S4: A] :
( ( ( member @ A @ R5 @ A7 )
& ( member @ A @ S4 @ A7 )
& ( ord_less @ A @ R5 @ S4 ) )
=> ( ord_less @ B @ ( F5 @ R5 ) @ ( F5 @ S4 ) ) ) ) ) ) ).
% strict_mono_on_def
thf(fact_5422_strict__mono__onI,axiom,
! [B: $tType,A: $tType] :
( ( ( ord @ A )
& ( ord @ B ) )
=> ! [A3: set @ A,F3: A > B] :
( ! [R3: A,S2: A] :
( ( member @ A @ R3 @ A3 )
=> ( ( member @ A @ S2 @ A3 )
=> ( ( ord_less @ A @ R3 @ S2 )
=> ( ord_less @ B @ ( F3 @ R3 ) @ ( F3 @ S2 ) ) ) ) )
=> ( strict_mono_on @ A @ B @ F3 @ A3 ) ) ) ).
% strict_mono_onI
thf(fact_5423_size__list__map,axiom,
! [A: $tType,B: $tType,F3: A > nat,G: B > A,Xs: list @ B] :
( ( size_list @ A @ F3 @ ( map @ B @ A @ G @ Xs ) )
= ( size_list @ B @ ( comp @ A @ nat @ B @ F3 @ G ) @ Xs ) ) ).
% size_list_map
thf(fact_5424_list_Osize__gen__o__map,axiom,
! [B: $tType,A: $tType,F3: B > nat,G: A > B] :
( ( comp @ ( list @ B ) @ nat @ ( list @ A ) @ ( size_list @ B @ F3 ) @ ( map @ A @ B @ G ) )
= ( size_list @ A @ ( comp @ B @ nat @ A @ F3 @ G ) ) ) ).
% list.size_gen_o_map
thf(fact_5425_fst__diag__fst,axiom,
! [B: $tType,A: $tType] :
( ( comp @ ( product_prod @ A @ A ) @ A @ ( product_prod @ A @ B ) @ ( product_fst @ A @ A )
@ ( comp @ A @ ( product_prod @ A @ A ) @ ( product_prod @ A @ B )
@ ^ [X4: A] : ( product_Pair @ A @ A @ X4 @ X4 )
@ ( product_fst @ A @ B ) ) )
= ( product_fst @ A @ B ) ) ).
% fst_diag_fst
thf(fact_5426_fst__diag__snd,axiom,
! [B: $tType,A: $tType] :
( ( comp @ ( product_prod @ B @ B ) @ B @ ( product_prod @ A @ B ) @ ( product_fst @ B @ B )
@ ( comp @ B @ ( product_prod @ B @ B ) @ ( product_prod @ A @ B )
@ ^ [X4: B] : ( product_Pair @ B @ B @ X4 @ X4 )
@ ( product_snd @ A @ B ) ) )
= ( product_snd @ A @ B ) ) ).
% fst_diag_snd
thf(fact_5427_snd__diag__fst,axiom,
! [B: $tType,A: $tType] :
( ( comp @ ( product_prod @ A @ A ) @ A @ ( product_prod @ A @ B ) @ ( product_snd @ A @ A )
@ ( comp @ A @ ( product_prod @ A @ A ) @ ( product_prod @ A @ B )
@ ^ [X4: A] : ( product_Pair @ A @ A @ X4 @ X4 )
@ ( product_fst @ A @ B ) ) )
= ( product_fst @ A @ B ) ) ).
% snd_diag_fst
thf(fact_5428_snd__diag__snd,axiom,
! [B: $tType,A: $tType] :
( ( comp @ ( product_prod @ B @ B ) @ B @ ( product_prod @ A @ B ) @ ( product_snd @ B @ B )
@ ( comp @ B @ ( product_prod @ B @ B ) @ ( product_prod @ A @ B )
@ ^ [X4: B] : ( product_Pair @ B @ B @ X4 @ X4 )
@ ( product_snd @ A @ B ) ) )
= ( product_snd @ A @ B ) ) ).
% snd_diag_snd
thf(fact_5429_folding__insort__key_Oinsort__key__commute,axiom,
! [A: $tType,B: $tType,Less_eq: A > A > $o,Less: A > A > $o,S3: set @ B,F3: B > A,X: B,Y2: B] :
( ( folding_insort_key @ A @ B @ Less_eq @ Less @ S3 @ F3 )
=> ( ( member @ B @ X @ S3 )
=> ( ( member @ B @ Y2 @ S3 )
=> ( ( comp @ ( list @ B ) @ ( list @ B ) @ ( list @ B ) @ ( insort_key @ A @ B @ Less_eq @ F3 @ Y2 ) @ ( insort_key @ A @ B @ Less_eq @ F3 @ X ) )
= ( comp @ ( list @ B ) @ ( list @ B ) @ ( list @ B ) @ ( insort_key @ A @ B @ Less_eq @ F3 @ X ) @ ( insort_key @ A @ B @ Less_eq @ F3 @ Y2 ) ) ) ) ) ) ).
% folding_insort_key.insort_key_commute
thf(fact_5430_sum_OatLeastAtMost__shift__bounds,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,M: nat,K2: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ M @ K2 ) @ ( plus_plus @ nat @ N @ K2 ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ nat @ A @ nat @ G @ ( plus_plus @ nat @ K2 ) ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ).
% sum.atLeastAtMost_shift_bounds
thf(fact_5431_sum_OatLeastLessThan__shift__bounds,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,M: nat,K2: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( plus_plus @ nat @ M @ K2 ) @ ( plus_plus @ nat @ N @ K2 ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ nat @ A @ nat @ G @ ( plus_plus @ nat @ K2 ) ) @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) ) ) ).
% sum.atLeastLessThan_shift_bounds
thf(fact_5432_prod_OatLeastAtMost__shift__bounds,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,M: nat,K2: nat,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ M @ K2 ) @ ( plus_plus @ nat @ N @ K2 ) ) )
= ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G @ ( plus_plus @ nat @ K2 ) ) @ ( set_or1337092689740270186AtMost @ nat @ M @ N ) ) ) ) ).
% prod.atLeastAtMost_shift_bounds
thf(fact_5433_prod_OatLeastLessThan__shift__bounds,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,M: nat,K2: nat,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( plus_plus @ nat @ M @ K2 ) @ ( plus_plus @ nat @ N @ K2 ) ) )
= ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G @ ( plus_plus @ nat @ K2 ) ) @ ( set_or7035219750837199246ssThan @ nat @ M @ N ) ) ) ) ).
% prod.atLeastLessThan_shift_bounds
thf(fact_5434_bit__drop__bit__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A2: A] :
( ( bit_se5641148757651400278ts_bit @ A @ ( bit_se4197421643247451524op_bit @ A @ N @ A2 ) )
= ( comp @ nat @ $o @ nat @ ( bit_se5641148757651400278ts_bit @ A @ A2 ) @ ( plus_plus @ nat @ N ) ) ) ) ).
% bit_drop_bit_eq
thf(fact_5435_summable__inverse__divide,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: nat > A,C2: A] :
( ( summable @ A @ ( comp @ A @ A @ nat @ ( inverse_inverse @ A ) @ F3 ) )
=> ( summable @ A
@ ^ [N2: nat] : ( divide_divide @ A @ C2 @ ( F3 @ N2 ) ) ) ) ) ).
% summable_inverse_divide
thf(fact_5436_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_5437_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_5438_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_5439_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_5440_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_5441_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_5442_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
@ ^ [N2: nat] : ( minus_minus @ nat @ N2 @ ( 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_5443_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
@ ^ [N2: nat] : ( minus_minus @ nat @ N2 @ ( 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_5444_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
@ ^ [N2: nat] : ( minus_minus @ nat @ N2 @ ( 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_5445_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
@ ^ [N2: nat] : ( minus_minus @ nat @ N2 @ ( 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_5446_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_5447_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_5448_strict__mono__on__leD,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( preorder @ B ) )
=> ! [F3: A > B,A3: set @ A,X: A,Y2: A] :
( ( strict_mono_on @ A @ B @ F3 @ A3 )
=> ( ( member @ A @ X @ A3 )
=> ( ( member @ A @ Y2 @ A3 )
=> ( ( ord_less_eq @ A @ X @ Y2 )
=> ( ord_less_eq @ B @ ( F3 @ X ) @ ( F3 @ Y2 ) ) ) ) ) ) ) ).
% strict_mono_on_leD
thf(fact_5449_strict__mono__onD,axiom,
! [B: $tType,A: $tType] :
( ( ( ord @ A )
& ( ord @ B ) )
=> ! [F3: A > B,A3: set @ A,R: A,S: A] :
( ( strict_mono_on @ A @ B @ F3 @ A3 )
=> ( ( member @ A @ R @ A3 )
=> ( ( member @ A @ S @ A3 )
=> ( ( ord_less @ A @ R @ S )
=> ( ord_less @ B @ ( F3 @ R ) @ ( F3 @ S ) ) ) ) ) ) ) ).
% strict_mono_onD
thf(fact_5450_snd__fst__flip,axiom,
! [A: $tType,B: $tType] :
( ( product_snd @ B @ A )
= ( comp @ ( product_prod @ A @ B ) @ A @ ( product_prod @ B @ A ) @ ( product_fst @ A @ B )
@ ( product_case_prod @ B @ A @ ( product_prod @ A @ B )
@ ^ [X4: B,Y: A] : ( product_Pair @ A @ B @ Y @ X4 ) ) ) ) ).
% snd_fst_flip
thf(fact_5451_fst__snd__flip,axiom,
! [B: $tType,A: $tType] :
( ( product_fst @ A @ B )
= ( comp @ ( product_prod @ B @ A ) @ A @ ( product_prod @ A @ B ) @ ( product_snd @ B @ A )
@ ( product_case_prod @ A @ B @ ( product_prod @ B @ A )
@ ^ [X4: A,Y: B] : ( product_Pair @ B @ A @ Y @ X4 ) ) ) ) ).
% fst_snd_flip
thf(fact_5452_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_5453_greaterThanLessThan__iff,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [I2: A,L: A,U: A] :
( ( member @ A @ I2 @ ( set_or5935395276787703475ssThan @ A @ L @ U ) )
= ( ( ord_less @ A @ L @ I2 )
& ( ord_less @ A @ I2 @ U ) ) ) ) ).
% greaterThanLessThan_iff
thf(fact_5454_greaterThanLessThan__empty,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,K2: A] :
( ( ord_less_eq @ A @ L @ K2 )
=> ( ( set_or5935395276787703475ssThan @ A @ K2 @ L )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% greaterThanLessThan_empty
thf(fact_5455_greaterThanLessThan__empty__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A2: A,B2: A] :
( ( ( set_or5935395276787703475ssThan @ A @ A2 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
= ( ord_less_eq @ A @ B2 @ A2 ) ) ) ).
% greaterThanLessThan_empty_iff
thf(fact_5456_greaterThanLessThan__empty__iff2,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A2: A,B2: A] :
( ( ( bot_bot @ ( set @ A ) )
= ( set_or5935395276787703475ssThan @ A @ A2 @ B2 ) )
= ( ord_less_eq @ A @ B2 @ A2 ) ) ) ).
% greaterThanLessThan_empty_iff2
thf(fact_5457_infinite__Ioo__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A2: A,B2: A] :
( ( ~ ( finite_finite2 @ A @ ( set_or5935395276787703475ssThan @ A @ A2 @ B2 ) ) )
= ( ord_less @ A @ A2 @ B2 ) ) ) ).
% infinite_Ioo_iff
thf(fact_5458_infinite__Ioo,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ~ ( finite_finite2 @ A @ ( set_or5935395276787703475ssThan @ A @ A2 @ B2 ) ) ) ) ).
% infinite_Ioo
thf(fact_5459_greaterThanLessThan__subseteq__greaterThanLessThan,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A2: A,B2: A,C2: A,D2: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ A2 @ B2 ) @ ( set_or5935395276787703475ssThan @ A @ C2 @ D2 ) )
= ( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ A2 )
& ( ord_less_eq @ A @ B2 @ D2 ) ) ) ) ) ).
% greaterThanLessThan_subseteq_greaterThanLessThan
thf(fact_5460_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_5461_refl__ge__eq,axiom,
! [A: $tType,R4: A > A > $o] :
( ! [X5: A] : ( R4 @ X5 @ X5 )
=> ( ord_less_eq @ ( A > A > $o )
@ ^ [Y5: A,Z2: A] : Y5 = Z2
@ R4 ) ) ).
% refl_ge_eq
thf(fact_5462_ge__eq__refl,axiom,
! [A: $tType,R4: A > A > $o,X: A] :
( ( ord_less_eq @ ( A > A > $o )
@ ^ [Y5: A,Z2: A] : Y5 = Z2
@ R4 )
=> ( R4 @ X @ X ) ) ).
% ge_eq_refl
thf(fact_5463_greaterThanLessThan__subseteq__atLeastAtMost__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A2: A,B2: A,C2: A,D2: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ A2 @ B2 ) @ ( set_or1337092689740270186AtMost @ A @ C2 @ D2 ) )
= ( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ A2 )
& ( ord_less_eq @ A @ B2 @ D2 ) ) ) ) ) ).
% greaterThanLessThan_subseteq_atLeastAtMost_iff
thf(fact_5464_greaterThanLessThan__subseteq__atLeastLessThan__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A2: A,B2: A,C2: A,D2: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ A2 @ B2 ) @ ( set_or7035219750837199246ssThan @ A @ C2 @ D2 ) )
= ( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ A2 )
& ( ord_less_eq @ A @ B2 @ D2 ) ) ) ) ) ).
% greaterThanLessThan_subseteq_atLeastLessThan_iff
thf(fact_5465_nth__sorted__list__of__set__greaterThanLessThan,axiom,
! [N: nat,J3: nat,I2: nat] :
( ( ord_less @ nat @ N @ ( minus_minus @ nat @ J3 @ ( suc @ I2 ) ) )
=> ( ( nth @ nat @ ( linord4507533701916653071of_set @ nat @ ( set_or5935395276787703475ssThan @ nat @ I2 @ J3 ) ) @ N )
= ( suc @ ( plus_plus @ nat @ I2 @ N ) ) ) ) ).
% nth_sorted_list_of_set_greaterThanLessThan
thf(fact_5466_xor__minus__numerals_I2_J,axiom,
! [K2: int,N: num] :
( ( bit_se5824344971392196577ns_xor @ int @ K2 @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
= ( bit_ri4277139882892585799ns_not @ int @ ( bit_se5824344971392196577ns_xor @ int @ K2 @ ( neg_numeral_sub @ int @ N @ one2 ) ) ) ) ).
% xor_minus_numerals(2)
thf(fact_5467_xor__minus__numerals_I1_J,axiom,
! [N: num,K2: int] :
( ( bit_se5824344971392196577ns_xor @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) @ K2 )
= ( bit_ri4277139882892585799ns_not @ int @ ( bit_se5824344971392196577ns_xor @ int @ ( neg_numeral_sub @ int @ N @ one2 ) @ K2 ) ) ) ).
% xor_minus_numerals(1)
thf(fact_5468_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_5469_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_5470_sub__num__simps_I6_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [K2: num,L: num] :
( ( neg_numeral_sub @ A @ ( bit0 @ K2 ) @ ( bit0 @ L ) )
= ( neg_numeral_dbl @ A @ ( neg_numeral_sub @ A @ K2 @ L ) ) ) ) ).
% sub_num_simps(6)
thf(fact_5471_sub__num__simps_I9_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [K2: num,L: num] :
( ( neg_numeral_sub @ A @ ( bit1 @ K2 ) @ ( bit1 @ L ) )
= ( neg_numeral_dbl @ A @ ( neg_numeral_sub @ A @ K2 @ L ) ) ) ) ).
% sub_num_simps(9)
thf(fact_5472_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_5473_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_5474_semiring__norm_I166_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [V2: num,W: num,Y2: A] :
( ( plus_plus @ A @ ( numeral_numeral @ A @ V2 ) @ ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) @ Y2 ) )
= ( plus_plus @ A @ ( neg_numeral_sub @ A @ V2 @ W ) @ Y2 ) ) ) ).
% semiring_norm(166)
thf(fact_5475_semiring__norm_I167_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [V2: num,W: num,Y2: A] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) @ ( plus_plus @ A @ ( numeral_numeral @ A @ W ) @ Y2 ) )
= ( plus_plus @ A @ ( neg_numeral_sub @ A @ W @ V2 ) @ Y2 ) ) ) ).
% semiring_norm(167)
thf(fact_5476_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_5477_sub__num__simps_I7_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [K2: num,L: num] :
( ( neg_numeral_sub @ A @ ( bit0 @ K2 ) @ ( bit1 @ L ) )
= ( neg_numeral_dbl_dec @ A @ ( neg_numeral_sub @ A @ K2 @ L ) ) ) ) ).
% sub_num_simps(7)
thf(fact_5478_sub__num__simps_I8_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [K2: num,L: num] :
( ( neg_numeral_sub @ A @ ( bit1 @ K2 ) @ ( bit0 @ L ) )
= ( neg_numeral_dbl_inc @ A @ ( neg_numeral_sub @ A @ K2 @ L ) ) ) ) ).
% sub_num_simps(8)
thf(fact_5479_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_5480_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_5481_sub__num__simps_I5_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [K2: num] :
( ( neg_numeral_sub @ A @ ( bit1 @ K2 ) @ one2 )
= ( numeral_numeral @ A @ ( bit0 @ K2 ) ) ) ) ).
% sub_num_simps(5)
thf(fact_5482_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_5483_sub__num__simps_I4_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [K2: num] :
( ( neg_numeral_sub @ A @ ( bit0 @ K2 ) @ one2 )
= ( numeral_numeral @ A @ ( bitM @ K2 ) ) ) ) ).
% sub_num_simps(4)
thf(fact_5484_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_5485_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_5486_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_5487_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_5488_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_5489_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_5490_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_5491_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_5492_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_5493_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_5494_tanh__real__bounds,axiom,
! [X: real] : ( member @ real @ ( tanh @ real @ X ) @ ( set_or5935395276787703475ssThan @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ ( one_one @ real ) ) ) ).
% tanh_real_bounds
thf(fact_5495_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_5496_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_5497_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_5498_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_5499_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_5500_greaterThanLessThan__upt,axiom,
( ( set_or5935395276787703475ssThan @ nat )
= ( ^ [N2: nat,M2: nat] : ( set2 @ nat @ ( upt @ ( suc @ N2 ) @ M2 ) ) ) ) ).
% greaterThanLessThan_upt
thf(fact_5501_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_5502_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_5503_div__add__self2__no__field,axiom,
! [B: $tType,A: $tType] :
( ( ( euclid4440199948858584721cancel @ A )
& ( field @ B ) )
=> ! [X: B,B2: A,A2: A] :
( ( nO_MATCH @ B @ A @ X @ B2 )
=> ( ( B2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ A2 @ B2 ) @ B2 )
= ( plus_plus @ A @ ( divide_divide @ A @ A2 @ B2 ) @ ( one_one @ A ) ) ) ) ) ) ).
% div_add_self2_no_field
thf(fact_5504_div__add__self1__no__field,axiom,
! [B: $tType,A: $tType] :
( ( ( euclid4440199948858584721cancel @ A )
& ( field @ B ) )
=> ! [X: B,B2: A,A2: A] :
( ( nO_MATCH @ B @ A @ X @ B2 )
=> ( ( B2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ B2 @ A2 ) @ B2 )
= ( plus_plus @ A @ ( divide_divide @ A @ A2 @ B2 ) @ ( one_one @ A ) ) ) ) ) ) ).
% div_add_self1_no_field
thf(fact_5505_finite__psubset__def,axiom,
! [A: $tType] :
( ( finite_psubset @ A )
= ( collect @ ( product_prod @ ( set @ A ) @ ( set @ A ) )
@ ( product_case_prod @ ( set @ A ) @ ( set @ A ) @ $o
@ ^ [A7: set @ A,B7: set @ A] :
( ( ord_less @ ( set @ A ) @ A7 @ B7 )
& ( finite_finite2 @ A @ B7 ) ) ) ) ) ).
% finite_psubset_def
thf(fact_5506_scale__right__distrib__NO__MATCH,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [X: A,Y2: A,A2: real] :
( ( nO_MATCH @ A @ real @ ( divide_divide @ A @ X @ Y2 ) @ A2 )
=> ( ( real_V8093663219630862766scaleR @ A @ A2 @ ( plus_plus @ A @ X @ Y2 ) )
= ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ X ) @ ( real_V8093663219630862766scaleR @ A @ A2 @ Y2 ) ) ) ) ) ).
% scale_right_distrib_NO_MATCH
thf(fact_5507_scale__right__diff__distrib__NO__MATCH,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [X: A,Y2: A,A2: real] :
( ( nO_MATCH @ A @ real @ ( divide_divide @ A @ X @ Y2 ) @ A2 )
=> ( ( real_V8093663219630862766scaleR @ A @ A2 @ ( minus_minus @ A @ X @ Y2 ) )
= ( minus_minus @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ X ) @ ( real_V8093663219630862766scaleR @ A @ A2 @ Y2 ) ) ) ) ) ).
% scale_right_diff_distrib_NO_MATCH
thf(fact_5508_distrib__left__NO__MATCH,axiom,
! [B: $tType,A: $tType] :
( ( semiring @ A )
=> ! [X: B,Y2: B,A2: A,B2: A,C2: A] :
( ( nO_MATCH @ B @ A @ ( divide_divide @ B @ X @ Y2 ) @ A2 )
=> ( ( times_times @ A @ A2 @ ( plus_plus @ A @ B2 @ C2 ) )
= ( plus_plus @ A @ ( times_times @ A @ A2 @ B2 ) @ ( times_times @ A @ A2 @ C2 ) ) ) ) ) ).
% distrib_left_NO_MATCH
thf(fact_5509_distrib__right__NO__MATCH,axiom,
! [B: $tType,A: $tType] :
( ( semiring @ A )
=> ! [X: B,Y2: B,C2: A,A2: A,B2: A] :
( ( nO_MATCH @ B @ A @ ( divide_divide @ B @ X @ Y2 ) @ C2 )
=> ( ( times_times @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C2 )
= ( plus_plus @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ) ).
% distrib_right_NO_MATCH
thf(fact_5510_left__diff__distrib__NO__MATCH,axiom,
! [B: $tType,A: $tType] :
( ( ring @ A )
=> ! [X: B,Y2: B,C2: A,A2: A,B2: A] :
( ( nO_MATCH @ B @ A @ ( divide_divide @ B @ X @ Y2 ) @ C2 )
=> ( ( times_times @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C2 )
= ( minus_minus @ A @ ( times_times @ A @ A2 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ) ).
% left_diff_distrib_NO_MATCH
thf(fact_5511_right__diff__distrib__NO__MATCH,axiom,
! [B: $tType,A: $tType] :
( ( ring @ A )
=> ! [X: B,Y2: B,A2: A,B2: A,C2: A] :
( ( nO_MATCH @ B @ A @ ( divide_divide @ B @ X @ Y2 ) @ A2 )
=> ( ( times_times @ A @ A2 @ ( minus_minus @ A @ B2 @ C2 ) )
= ( minus_minus @ A @ ( times_times @ A @ A2 @ B2 ) @ ( times_times @ A @ A2 @ C2 ) ) ) ) ) ).
% right_diff_distrib_NO_MATCH
thf(fact_5512_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_5513_scale__left__distrib__NO__MATCH,axiom,
! [C: $tType,A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [X: A,Y2: A,C2: C,A2: real,B2: real] :
( ( nO_MATCH @ A @ C @ ( divide_divide @ A @ X @ Y2 ) @ C2 )
=> ( ( real_V8093663219630862766scaleR @ A @ ( plus_plus @ real @ A2 @ B2 ) @ X )
= ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ X ) @ ( real_V8093663219630862766scaleR @ A @ B2 @ X ) ) ) ) ) ).
% scale_left_distrib_NO_MATCH
thf(fact_5514_scale__left__diff__distrib__NO__MATCH,axiom,
! [C: $tType,A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [X: A,Y2: A,C2: C,A2: real,B2: real] :
( ( nO_MATCH @ A @ C @ ( divide_divide @ A @ X @ Y2 ) @ C2 )
=> ( ( real_V8093663219630862766scaleR @ A @ ( minus_minus @ real @ A2 @ B2 ) @ X )
= ( minus_minus @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ X ) @ ( real_V8093663219630862766scaleR @ A @ B2 @ X ) ) ) ) ) ).
% scale_left_diff_distrib_NO_MATCH
thf(fact_5515_bounded__linear__axioms_Ointro,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F3: A > B] :
( ? [K8: real] :
! [X5: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( F3 @ X5 ) ) @ ( times_times @ real @ ( real_V7770717601297561774m_norm @ A @ X5 ) @ K8 ) )
=> ( real_V4916620083959148203axioms @ A @ B @ F3 ) ) ) ).
% bounded_linear_axioms.intro
thf(fact_5516_bounded__linear__axioms__def,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ( ( real_V4916620083959148203axioms @ A @ B )
= ( ^ [F5: A > B] :
? [K6: real] :
! [X4: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( F5 @ X4 ) ) @ ( times_times @ real @ ( real_V7770717601297561774m_norm @ A @ X4 ) @ K6 ) ) ) ) ) ).
% bounded_linear_axioms_def
thf(fact_5517_horner__sum__eq__sum__funpow,axiom,
! [A: $tType,B: $tType] :
( ( comm_semiring_0 @ A )
=> ( ( groups4207007520872428315er_sum @ B @ A )
= ( ^ [F5: B > A,A5: A,Xs2: list @ B] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [N2: nat] : ( compow @ ( A > A ) @ N2 @ ( times_times @ A @ A5 ) @ ( F5 @ ( nth @ B @ Xs2 @ N2 ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ B ) @ Xs2 ) ) ) ) ) ) ).
% horner_sum_eq_sum_funpow
thf(fact_5518_Suc__funpow,axiom,
! [N: nat] :
( ( compow @ ( nat > nat ) @ N @ suc )
= ( plus_plus @ nat @ N ) ) ).
% Suc_funpow
thf(fact_5519_funpow__0,axiom,
! [A: $tType,F3: A > A,X: A] :
( ( compow @ ( A > A ) @ ( zero_zero @ nat ) @ F3 @ X )
= X ) ).
% funpow_0
thf(fact_5520_comp__funpow,axiom,
! [B: $tType,A: $tType,N: nat,F3: A > A] :
( ( compow @ ( ( B > A ) > B > A ) @ N @ ( comp @ A @ A @ B @ F3 ) )
= ( comp @ A @ A @ B @ ( compow @ ( A > A ) @ N @ F3 ) ) ) ).
% comp_funpow
thf(fact_5521_funpow_Osimps_I2_J,axiom,
! [A: $tType,N: nat,F3: A > A] :
( ( compow @ ( A > A ) @ ( suc @ N ) @ F3 )
= ( comp @ A @ A @ A @ F3 @ ( compow @ ( A > A ) @ N @ F3 ) ) ) ).
% funpow.simps(2)
thf(fact_5522_funpow__Suc__right,axiom,
! [A: $tType,N: nat,F3: A > A] :
( ( compow @ ( A > A ) @ ( suc @ N ) @ F3 )
= ( comp @ A @ A @ A @ ( compow @ ( A > A ) @ N @ F3 ) @ F3 ) ) ).
% funpow_Suc_right
thf(fact_5523_funpow__add,axiom,
! [A: $tType,M: nat,N: nat,F3: A > A] :
( ( compow @ ( A > A ) @ ( plus_plus @ nat @ M @ N ) @ F3 )
= ( comp @ A @ A @ A @ ( compow @ ( A > A ) @ M @ F3 ) @ ( compow @ ( A > A ) @ N @ F3 ) ) ) ).
% funpow_add
thf(fact_5524_funpow__times__power,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [F3: A > nat,X: A] :
( ( compow @ ( A > A ) @ ( F3 @ X ) @ ( times_times @ A @ X ) )
= ( times_times @ A @ ( power_power @ A @ X @ ( F3 @ X ) ) ) ) ) ).
% funpow_times_power
thf(fact_5525_bij__betw__funpow,axiom,
! [A: $tType,F3: A > A,S3: set @ A,N: nat] :
( ( bij_betw @ A @ A @ F3 @ S3 @ S3 )
=> ( bij_betw @ A @ A @ ( compow @ ( A > A ) @ N @ F3 ) @ S3 @ S3 ) ) ).
% bij_betw_funpow
thf(fact_5526_funpow__mult,axiom,
! [A: $tType,N: nat,M: nat,F3: A > A] :
( ( compow @ ( A > A ) @ N @ ( compow @ ( A > A ) @ M @ F3 ) )
= ( compow @ ( A > A ) @ ( times_times @ nat @ M @ N ) @ F3 ) ) ).
% funpow_mult
thf(fact_5527_funpow__swap1,axiom,
! [A: $tType,F3: A > A,N: nat,X: A] :
( ( F3 @ ( compow @ ( A > A ) @ N @ F3 @ X ) )
= ( compow @ ( A > A ) @ N @ F3 @ ( F3 @ X ) ) ) ).
% funpow_swap1
thf(fact_5528_funpow__mod__eq,axiom,
! [A: $tType,N: nat,F3: A > A,X: A,M: nat] :
( ( ( compow @ ( A > A ) @ N @ F3 @ X )
= X )
=> ( ( compow @ ( A > A ) @ ( modulo_modulo @ nat @ M @ N ) @ F3 @ X )
= ( compow @ ( A > A ) @ M @ F3 @ X ) ) ) ).
% funpow_mod_eq
thf(fact_5529_of__nat__def,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( ( semiring_1_of_nat @ A )
= ( ^ [N2: nat] : ( compow @ ( A > A ) @ N2 @ ( plus_plus @ A @ ( one_one @ A ) ) @ ( zero_zero @ A ) ) ) ) ) ).
% of_nat_def
thf(fact_5530_numeral__add__unfold__funpow,axiom,
! [A: $tType] :
( ( semiring_numeral @ A )
=> ! [K2: num,A2: A] :
( ( plus_plus @ A @ ( numeral_numeral @ A @ K2 ) @ A2 )
= ( compow @ ( A > A ) @ ( numeral_numeral @ nat @ K2 ) @ ( plus_plus @ A @ ( one_one @ A ) ) @ A2 ) ) ) ).
% numeral_add_unfold_funpow
thf(fact_5531_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_5532_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_5533_relpowp__fun__conv,axiom,
! [A: $tType] :
( ( compow @ ( A > A > $o ) )
= ( ^ [N2: nat,P2: A > A > $o,X4: A,Y: A] :
? [F5: nat > A] :
( ( ( F5 @ ( zero_zero @ nat ) )
= X4 )
& ( ( F5 @ N2 )
= Y )
& ! [I: nat] :
( ( ord_less @ nat @ I @ N2 )
=> ( P2 @ ( F5 @ I ) @ ( F5 @ ( suc @ I ) ) ) ) ) ) ) ).
% relpowp_fun_conv
thf(fact_5534_relpowp__1,axiom,
! [A: $tType,P3: A > A > $o] :
( ( compow @ ( A > A > $o ) @ ( one_one @ nat ) @ P3 )
= P3 ) ).
% relpowp_1
thf(fact_5535_Nat_Ofunpow__code__def,axiom,
! [A: $tType] :
( ( funpow @ A )
= ( compow @ ( A > A ) ) ) ).
% Nat.funpow_code_def
thf(fact_5536_sorted__list__of__set_Osorted__key__list__of__set__insert__remove,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,X: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( linord4507533701916653071of_set @ A @ ( insert @ A @ X @ A3 ) )
= ( linorder_insort_key @ A @ A
@ ^ [X4: A] : X4
@ X
@ ( linord4507533701916653071of_set @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_insert_remove
thf(fact_5537_max__nat_Osemilattice__neutr__order__axioms,axiom,
( semila1105856199041335345_order @ nat @ ( ord_max @ nat ) @ ( zero_zero @ nat )
@ ^ [X4: nat,Y: nat] : ( ord_less_eq @ nat @ Y @ X4 )
@ ^ [X4: nat,Y: nat] : ( ord_less @ nat @ Y @ X4 ) ) ).
% max_nat.semilattice_neutr_order_axioms
thf(fact_5538_remove1__insort__key,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [X: B,F3: B > A,Xs: list @ B] :
( ( remove1 @ B @ X @ ( linorder_insort_key @ B @ A @ F3 @ X @ Xs ) )
= Xs ) ) ).
% remove1_insort_key
thf(fact_5539_length__insort,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F3: B > A,X: B,Xs: list @ B] :
( ( size_size @ ( list @ B ) @ ( linorder_insort_key @ B @ A @ F3 @ X @ Xs ) )
= ( suc @ ( size_size @ ( list @ B ) @ Xs ) ) ) ) ).
% length_insort
thf(fact_5540_sorted__list__of__set_Osorted__key__list__of__set__insert,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,X: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ~ ( member @ A @ X @ A3 )
=> ( ( linord4507533701916653071of_set @ A @ ( insert @ A @ X @ A3 ) )
= ( linorder_insort_key @ A @ A
@ ^ [X4: A] : X4
@ X
@ ( linord4507533701916653071of_set @ A @ A3 ) ) ) ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_insert
thf(fact_5541_insort__key__left__comm,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F3: B > A,X: B,Y2: B,Xs: list @ B] :
( ( ( F3 @ X )
!= ( F3 @ Y2 ) )
=> ( ( linorder_insort_key @ B @ A @ F3 @ Y2 @ ( linorder_insort_key @ B @ A @ F3 @ X @ Xs ) )
= ( linorder_insort_key @ B @ A @ F3 @ X @ ( linorder_insort_key @ B @ A @ F3 @ Y2 @ Xs ) ) ) ) ) ).
% insort_key_left_comm
thf(fact_5542_insort__left__comm,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y2: A,Xs: list @ A] :
( ( linorder_insort_key @ A @ A
@ ^ [X4: A] : X4
@ X
@ ( linorder_insort_key @ A @ A
@ ^ [X4: A] : X4
@ Y2
@ Xs ) )
= ( linorder_insort_key @ A @ A
@ ^ [X4: A] : X4
@ Y2
@ ( linorder_insort_key @ A @ A
@ ^ [X4: A] : X4
@ X
@ Xs ) ) ) ) ).
% insort_left_comm
thf(fact_5543_sorted__list__of__set_Ofold__insort__key_Ocomp__fun__commute__on,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Y2: A,X: A] :
( ( comp @ ( list @ A ) @ ( list @ A ) @ ( list @ A )
@ ( linorder_insort_key @ A @ A
@ ^ [X4: A] : X4
@ Y2 )
@ ( linorder_insort_key @ A @ A
@ ^ [X4: A] : X4
@ X ) )
= ( comp @ ( list @ A ) @ ( list @ A ) @ ( list @ A )
@ ( linorder_insort_key @ A @ A
@ ^ [X4: A] : X4
@ X )
@ ( linorder_insort_key @ A @ A
@ ^ [X4: A] : X4
@ Y2 ) ) ) ) ).
% sorted_list_of_set.fold_insort_key.comp_fun_commute_on
thf(fact_5544_set__insort__key,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F3: B > A,X: B,Xs: list @ B] :
( ( set2 @ B @ ( linorder_insort_key @ B @ A @ F3 @ X @ Xs ) )
= ( insert @ B @ X @ ( set2 @ B @ Xs ) ) ) ) ).
% set_insort_key
thf(fact_5545_distinct__insort,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F3: B > A,X: B,Xs: list @ B] :
( ( distinct @ B @ ( linorder_insort_key @ B @ A @ F3 @ X @ Xs ) )
= ( ~ ( member @ B @ X @ ( set2 @ B @ Xs ) )
& ( distinct @ B @ Xs ) ) ) ) ).
% distinct_insort
thf(fact_5546_sorted__insort,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Xs: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A )
@ ( linorder_insort_key @ A @ A
@ ^ [X4: A] : X4
@ X
@ Xs ) )
= ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs ) ) ) ).
% sorted_insort
thf(fact_5547_sorted__insort__key,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F3: B > A,X: B,Xs: list @ B] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F3 @ ( linorder_insort_key @ B @ A @ F3 @ X @ Xs ) ) )
= ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F3 @ Xs ) ) ) ) ).
% sorted_insort_key
thf(fact_5548_insort__remove1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,Xs: list @ A] :
( ( member @ A @ A2 @ ( set2 @ A @ Xs ) )
=> ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( ( linorder_insort_key @ A @ A
@ ^ [X4: A] : X4
@ A2
@ ( remove1 @ A @ A2 @ Xs ) )
= Xs ) ) ) ) ).
% insort_remove1
thf(fact_5549_gcd__nat_Osemilattice__neutr__order__axioms,axiom,
( semila1105856199041335345_order @ nat @ ( gcd_gcd @ nat ) @ ( zero_zero @ nat ) @ ( dvd_dvd @ nat )
@ ^ [M2: nat,N2: nat] :
( ( dvd_dvd @ nat @ M2 @ N2 )
& ( M2 != N2 ) ) ) ).
% gcd_nat.semilattice_neutr_order_axioms
thf(fact_5550_sorted__list__of__set_Ofold__insort__key_Oremove,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A,X: A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( member @ A @ X @ A3 )
=> ( ( linord4507533701916653071of_set @ A @ A3 )
= ( linorder_insort_key @ A @ A
@ ^ [X4: A] : X4
@ X
@ ( linord4507533701916653071of_set @ A @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ).
% sorted_list_of_set.fold_insort_key.remove
thf(fact_5551_nth__sorted__list__of__set__greaterThanAtMost,axiom,
! [N: nat,J3: nat,I2: nat] :
( ( ord_less @ nat @ N @ ( minus_minus @ nat @ J3 @ I2 ) )
=> ( ( nth @ nat @ ( linord4507533701916653071of_set @ nat @ ( set_or3652927894154168847AtMost @ nat @ I2 @ J3 ) ) @ N )
= ( suc @ ( plus_plus @ nat @ I2 @ N ) ) ) ) ).
% nth_sorted_list_of_set_greaterThanAtMost
thf(fact_5552_card__UNION,axiom,
! [A: $tType,A3: set @ ( set @ A )] :
( ( finite_finite2 @ ( set @ A ) @ A3 )
=> ( ! [X5: set @ A] :
( ( member @ ( set @ A ) @ X5 @ A3 )
=> ( finite_finite2 @ A @ X5 ) )
=> ( ( finite_card @ A @ ( complete_Sup_Sup @ ( set @ A ) @ A3 ) )
= ( nat2
@ ( groups7311177749621191930dd_sum @ ( set @ ( set @ A ) ) @ int
@ ^ [I7: set @ ( set @ A )] : ( times_times @ int @ ( power_power @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( plus_plus @ nat @ ( finite_card @ ( set @ A ) @ I7 ) @ ( one_one @ nat ) ) ) @ ( semiring_1_of_nat @ int @ ( finite_card @ A @ ( complete_Inf_Inf @ ( set @ A ) @ I7 ) ) ) )
@ ( collect @ ( set @ ( set @ A ) )
@ ^ [I7: set @ ( set @ A )] :
( ( ord_less_eq @ ( set @ ( set @ A ) ) @ I7 @ A3 )
& ( I7
!= ( bot_bot @ ( set @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ).
% card_UNION
thf(fact_5553_set__removeAll,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( set2 @ A @ ( removeAll @ A @ X @ Xs ) )
= ( minus_minus @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% set_removeAll
thf(fact_5554_removeAll__id,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ~ ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ( removeAll @ A @ X @ Xs )
= Xs ) ) ).
% removeAll_id
thf(fact_5555_Sup__atLeastAtMost,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X: A,Y2: A] :
( ( ord_less_eq @ A @ X @ Y2 )
=> ( ( complete_Sup_Sup @ A @ ( set_or1337092689740270186AtMost @ A @ X @ Y2 ) )
= Y2 ) ) ) ).
% Sup_atLeastAtMost
thf(fact_5556_Inf__atLeastAtMost,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X: A,Y2: A] :
( ( ord_less_eq @ A @ X @ Y2 )
=> ( ( complete_Inf_Inf @ A @ ( set_or1337092689740270186AtMost @ A @ X @ Y2 ) )
= X ) ) ) ).
% Inf_atLeastAtMost
thf(fact_5557_Sup__atLeastLessThan,axiom,
! [A: $tType] :
( ( ( comple6319245703460814977attice @ A )
& ( dense_linorder @ A ) )
=> ! [X: A,Y2: A] :
( ( ord_less @ A @ X @ Y2 )
=> ( ( complete_Sup_Sup @ A @ ( set_or7035219750837199246ssThan @ A @ X @ Y2 ) )
= Y2 ) ) ) ).
% Sup_atLeastLessThan
thf(fact_5558_Inf__atLeastLessThan,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X: A,Y2: A] :
( ( ord_less @ A @ X @ Y2 )
=> ( ( complete_Inf_Inf @ A @ ( set_or7035219750837199246ssThan @ A @ X @ Y2 ) )
= X ) ) ) ).
% Inf_atLeastLessThan
thf(fact_5559_greaterThanAtMost__iff,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [I2: A,L: A,U: A] :
( ( member @ A @ I2 @ ( set_or3652927894154168847AtMost @ A @ L @ U ) )
= ( ( ord_less @ A @ L @ I2 )
& ( ord_less_eq @ A @ I2 @ U ) ) ) ) ).
% greaterThanAtMost_iff
thf(fact_5560_greaterThanAtMost__empty,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,K2: A] :
( ( ord_less_eq @ A @ L @ K2 )
=> ( ( set_or3652927894154168847AtMost @ A @ K2 @ L )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% greaterThanAtMost_empty
thf(fact_5561_greaterThanAtMost__empty__iff2,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [K2: A,L: A] :
( ( ( bot_bot @ ( set @ A ) )
= ( set_or3652927894154168847AtMost @ A @ K2 @ L ) )
= ( ~ ( ord_less @ A @ K2 @ L ) ) ) ) ).
% greaterThanAtMost_empty_iff2
thf(fact_5562_greaterThanAtMost__empty__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [K2: A,L: A] :
( ( ( set_or3652927894154168847AtMost @ A @ K2 @ L )
= ( bot_bot @ ( set @ A ) ) )
= ( ~ ( ord_less @ A @ K2 @ L ) ) ) ) ).
% greaterThanAtMost_empty_iff
thf(fact_5563_infinite__Ioc__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A2: A,B2: A] :
( ( ~ ( finite_finite2 @ A @ ( set_or3652927894154168847AtMost @ A @ A2 @ B2 ) ) )
= ( ord_less @ A @ A2 @ B2 ) ) ) ).
% infinite_Ioc_iff
thf(fact_5564_Sup__greaterThanLessThan,axiom,
! [A: $tType] :
( ( ( comple6319245703460814977attice @ A )
& ( dense_linorder @ A ) )
=> ! [X: A,Y2: A] :
( ( ord_less @ A @ X @ Y2 )
=> ( ( complete_Sup_Sup @ A @ ( set_or5935395276787703475ssThan @ A @ X @ Y2 ) )
= Y2 ) ) ) ).
% Sup_greaterThanLessThan
thf(fact_5565_Sup__greaterThanAtMost,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X: A,Y2: A] :
( ( ord_less @ A @ X @ Y2 )
=> ( ( complete_Sup_Sup @ A @ ( set_or3652927894154168847AtMost @ A @ X @ Y2 ) )
= Y2 ) ) ) ).
% Sup_greaterThanAtMost
thf(fact_5566_Inf__greaterThanLessThan,axiom,
! [A: $tType] :
( ( ( comple6319245703460814977attice @ A )
& ( dense_linorder @ A ) )
=> ! [X: A,Y2: A] :
( ( ord_less @ A @ X @ Y2 )
=> ( ( complete_Inf_Inf @ A @ ( set_or5935395276787703475ssThan @ A @ X @ Y2 ) )
= X ) ) ) ).
% Inf_greaterThanLessThan
thf(fact_5567_Inf__greaterThanAtMost,axiom,
! [A: $tType] :
( ( ( comple6319245703460814977attice @ A )
& ( dense_linorder @ A ) )
=> ! [X: A,Y2: A] :
( ( ord_less @ A @ X @ Y2 )
=> ( ( complete_Inf_Inf @ A @ ( set_or3652927894154168847AtMost @ A @ X @ Y2 ) )
= X ) ) ) ).
% Inf_greaterThanAtMost
thf(fact_5568_Ioc__inj,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A,C2: A,D2: A] :
( ( ( set_or3652927894154168847AtMost @ A @ A2 @ B2 )
= ( set_or3652927894154168847AtMost @ A @ C2 @ D2 ) )
= ( ( ( ord_less_eq @ A @ B2 @ A2 )
& ( ord_less_eq @ A @ D2 @ C2 ) )
| ( ( A2 = C2 )
& ( B2 = D2 ) ) ) ) ) ).
% Ioc_inj
thf(fact_5569_distinct__removeAll,axiom,
! [A: $tType,Xs: list @ A,X: A] :
( ( distinct @ A @ Xs )
=> ( distinct @ A @ ( removeAll @ A @ X @ Xs ) ) ) ).
% distinct_removeAll
thf(fact_5570_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_5571_Ioc__subset__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A,C2: A,D2: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ A2 @ B2 ) @ ( set_or3652927894154168847AtMost @ A @ C2 @ D2 ) )
= ( ( ord_less_eq @ A @ B2 @ A2 )
| ( ( ord_less_eq @ A @ C2 @ A2 )
& ( ord_less_eq @ A @ B2 @ D2 ) ) ) ) ) ).
% Ioc_subset_iff
thf(fact_5572_infinite__Ioc,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ~ ( finite_finite2 @ A @ ( set_or3652927894154168847AtMost @ A @ A2 @ B2 ) ) ) ) ).
% infinite_Ioc
thf(fact_5573_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_5574_cInf__abs__ge,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( linordered_idom @ A ) )
=> ! [S3: set @ A,A2: A] :
( ( S3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ S3 )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ X5 ) @ A2 ) )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ ( complete_Inf_Inf @ A @ S3 ) ) @ A2 ) ) ) ) ).
% cInf_abs_ge
thf(fact_5575_prod_OUnion__comp,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [B5: set @ ( set @ B ),G: B > A] :
( ! [X5: set @ B] :
( ( member @ ( set @ B ) @ X5 @ B5 )
=> ( finite_finite2 @ B @ X5 ) )
=> ( ! [A14: set @ B] :
( ( member @ ( set @ B ) @ A14 @ B5 )
=> ! [A25: set @ B] :
( ( member @ ( set @ B ) @ A25 @ B5 )
=> ( ( A14 != A25 )
=> ! [X5: B] :
( ( member @ B @ X5 @ A14 )
=> ( ( member @ B @ X5 @ A25 )
=> ( ( G @ X5 )
= ( one_one @ A ) ) ) ) ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( complete_Sup_Sup @ ( set @ B ) @ B5 ) )
= ( comp @ ( ( set @ B ) > A ) @ ( ( set @ ( set @ B ) ) > A ) @ ( B > A ) @ ( groups7121269368397514597t_prod @ ( set @ B ) @ A ) @ ( groups7121269368397514597t_prod @ B @ A ) @ G @ B5 ) ) ) ) ) ).
% prod.Union_comp
thf(fact_5576_card__Union__le__sum__card__weak,axiom,
! [A: $tType,U4: set @ ( set @ A )] :
( ! [X5: set @ A] :
( ( member @ ( set @ A ) @ X5 @ U4 )
=> ( finite_finite2 @ A @ X5 ) )
=> ( 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_5577_distinct__remove1__removeAll,axiom,
! [A: $tType,Xs: list @ A,X: A] :
( ( distinct @ A @ Xs )
=> ( ( remove1 @ A @ X @ Xs )
= ( removeAll @ A @ X @ Xs ) ) ) ).
% distinct_remove1_removeAll
thf(fact_5578_greaterThanAtMost__upt,axiom,
( ( set_or3652927894154168847AtMost @ nat )
= ( ^ [N2: nat,M2: nat] : ( set2 @ nat @ ( upt @ ( suc @ N2 ) @ ( suc @ M2 ) ) ) ) ) ).
% greaterThanAtMost_upt
thf(fact_5579_cInf__asclose,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( linordered_idom @ A ) )
=> ! [S3: set @ A,L: A,E: A] :
( ( S3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ S3 )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X5 @ L ) ) @ E ) )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ ( complete_Inf_Inf @ A @ S3 ) @ L ) ) @ E ) ) ) ) ).
% cInf_asclose
thf(fact_5580_cSup__asclose,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( linordered_idom @ A ) )
=> ! [S3: set @ A,L: A,E: A] :
( ( S3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ S3 )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X5 @ L ) ) @ E ) )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ ( complete_Sup_Sup @ A @ S3 ) @ L ) ) @ E ) ) ) ) ).
% cSup_asclose
thf(fact_5581_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_5582_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_5583_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_5584_greaterThanAtMost__subseteq__atLeastAtMost__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A2: A,B2: A,C2: A,D2: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ A2 @ B2 ) @ ( set_or1337092689740270186AtMost @ A @ C2 @ D2 ) )
= ( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ A2 )
& ( ord_less_eq @ A @ B2 @ D2 ) ) ) ) ) ).
% greaterThanAtMost_subseteq_atLeastAtMost_iff
thf(fact_5585_greaterThanAtMost__subseteq__atLeastLessThan__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A2: A,B2: A,C2: A,D2: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ A2 @ B2 ) @ ( set_or7035219750837199246ssThan @ A @ C2 @ D2 ) )
= ( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ A2 )
& ( ord_less @ A @ B2 @ D2 ) ) ) ) ) ).
% greaterThanAtMost_subseteq_atLeastLessThan_iff
thf(fact_5586_greaterThanLessThan__subseteq__greaterThanAtMost__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A2: A,B2: A,C2: A,D2: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ A2 @ B2 ) @ ( set_or3652927894154168847AtMost @ A @ C2 @ D2 ) )
= ( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ A2 )
& ( ord_less_eq @ A @ B2 @ D2 ) ) ) ) ) ).
% greaterThanLessThan_subseteq_greaterThanAtMost_iff
thf(fact_5587_finite__subset__Union,axiom,
! [A: $tType,A3: set @ A,B11: set @ ( set @ A )] :
( ( finite_finite2 @ A @ A3 )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ ( complete_Sup_Sup @ ( set @ A ) @ B11 ) )
=> ~ ! [F8: set @ ( set @ A )] :
( ( finite_finite2 @ ( set @ A ) @ F8 )
=> ( ( ord_less_eq @ ( set @ ( set @ A ) ) @ F8 @ B11 )
=> ~ ( ord_less_eq @ ( set @ A ) @ A3 @ ( complete_Sup_Sup @ ( set @ A ) @ F8 ) ) ) ) ) ) ).
% finite_subset_Union
thf(fact_5588_cInf__greaterThanAtMost,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( dense_linorder @ A ) )
=> ! [Y2: A,X: A] :
( ( ord_less @ A @ Y2 @ X )
=> ( ( complete_Inf_Inf @ A @ ( set_or3652927894154168847AtMost @ A @ Y2 @ X ) )
= Y2 ) ) ) ).
% cInf_greaterThanAtMost
thf(fact_5589_cInf__greaterThanLessThan,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( dense_linorder @ A ) )
=> ! [Y2: A,X: A] :
( ( ord_less @ A @ Y2 @ X )
=> ( ( complete_Inf_Inf @ A @ ( set_or5935395276787703475ssThan @ A @ Y2 @ X ) )
= Y2 ) ) ) ).
% cInf_greaterThanLessThan
thf(fact_5590_cSup__greaterThanAtMost,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [Y2: A,X: A] :
( ( ord_less @ A @ Y2 @ X )
=> ( ( complete_Sup_Sup @ A @ ( set_or3652927894154168847AtMost @ A @ Y2 @ X ) )
= X ) ) ) ).
% cSup_greaterThanAtMost
thf(fact_5591_cSup__atLeastAtMost,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [Y2: A,X: A] :
( ( ord_less_eq @ A @ Y2 @ X )
=> ( ( complete_Sup_Sup @ A @ ( set_or1337092689740270186AtMost @ A @ Y2 @ X ) )
= X ) ) ) ).
% cSup_atLeastAtMost
thf(fact_5592_cInf__atLeastAtMost,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [Y2: A,X: A] :
( ( ord_less_eq @ A @ Y2 @ X )
=> ( ( complete_Inf_Inf @ A @ ( set_or1337092689740270186AtMost @ A @ Y2 @ X ) )
= Y2 ) ) ) ).
% cInf_atLeastAtMost
thf(fact_5593_cSup__atLeastLessThan,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( dense_linorder @ A ) )
=> ! [Y2: A,X: A] :
( ( ord_less @ A @ Y2 @ X )
=> ( ( complete_Sup_Sup @ A @ ( set_or7035219750837199246ssThan @ A @ Y2 @ X ) )
= X ) ) ) ).
% cSup_atLeastLessThan
thf(fact_5594_cInf__atLeastLessThan,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [Y2: A,X: A] :
( ( ord_less @ A @ Y2 @ X )
=> ( ( complete_Inf_Inf @ A @ ( set_or7035219750837199246ssThan @ A @ Y2 @ X ) )
= Y2 ) ) ) ).
% cInf_atLeastLessThan
thf(fact_5595_cSup__greaterThanLessThan,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( dense_linorder @ A ) )
=> ! [Y2: A,X: A] :
( ( ord_less @ A @ Y2 @ X )
=> ( ( complete_Sup_Sup @ A @ ( set_or5935395276787703475ssThan @ A @ Y2 @ X ) )
= X ) ) ) ).
% cSup_greaterThanLessThan
thf(fact_5596_ex__gt__or__lt,axiom,
! [A: $tType] :
( ( condit5016429287641298734tinuum @ A )
=> ! [A2: A] :
? [B3: A] :
( ( ord_less @ A @ A2 @ B3 )
| ( ord_less @ A @ B3 @ A2 ) ) ) ).
% ex_gt_or_lt
thf(fact_5597_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_5598_complete__interval,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [A2: A,B2: A,P3: A > $o] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( P3 @ A2 )
=> ( ~ ( P3 @ B2 )
=> ? [C4: A] :
( ( ord_less_eq @ A @ A2 @ C4 )
& ( ord_less_eq @ A @ C4 @ B2 )
& ! [X2: A] :
( ( ( ord_less_eq @ A @ A2 @ X2 )
& ( ord_less @ A @ X2 @ C4 ) )
=> ( P3 @ X2 ) )
& ! [D6: A] :
( ! [X5: A] :
( ( ( ord_less_eq @ A @ A2 @ X5 )
& ( ord_less @ A @ X5 @ D6 ) )
=> ( P3 @ X5 ) )
=> ( ord_less_eq @ A @ D6 @ C4 ) ) ) ) ) ) ) ).
% complete_interval
thf(fact_5599_cSup__eq__maximum,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [Z3: A,X9: set @ A] :
( ( member @ A @ Z3 @ X9 )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ X9 )
=> ( ord_less_eq @ A @ X5 @ Z3 ) )
=> ( ( complete_Sup_Sup @ A @ X9 )
= Z3 ) ) ) ) ).
% cSup_eq_maximum
thf(fact_5600_cSup__eq,axiom,
! [A: $tType] :
( ( ( condit1219197933456340205attice @ A )
& ( no_bot @ A ) )
=> ! [X9: set @ A,A2: A] :
( ! [X5: A] :
( ( member @ A @ X5 @ X9 )
=> ( ord_less_eq @ A @ X5 @ A2 ) )
=> ( ! [Y7: A] :
( ! [X2: A] :
( ( member @ A @ X2 @ X9 )
=> ( ord_less_eq @ A @ X2 @ Y7 ) )
=> ( ord_less_eq @ A @ A2 @ Y7 ) )
=> ( ( complete_Sup_Sup @ A @ X9 )
= A2 ) ) ) ) ).
% cSup_eq
thf(fact_5601_cInf__eq__minimum,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [Z3: A,X9: set @ A] :
( ( member @ A @ Z3 @ X9 )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ X9 )
=> ( ord_less_eq @ A @ Z3 @ X5 ) )
=> ( ( complete_Inf_Inf @ A @ X9 )
= Z3 ) ) ) ) ).
% cInf_eq_minimum
thf(fact_5602_cInf__eq,axiom,
! [A: $tType] :
( ( ( condit1219197933456340205attice @ A )
& ( no_top @ A ) )
=> ! [X9: set @ A,A2: A] :
( ! [X5: A] :
( ( member @ A @ X5 @ X9 )
=> ( ord_less_eq @ A @ A2 @ X5 ) )
=> ( ! [Y7: A] :
( ! [X2: A] :
( ( member @ A @ X2 @ X9 )
=> ( ord_less_eq @ A @ Y7 @ X2 ) )
=> ( ord_less_eq @ A @ Y7 @ A2 ) )
=> ( ( complete_Inf_Inf @ A @ X9 )
= A2 ) ) ) ) ).
% cInf_eq
thf(fact_5603_cSup__eq__non__empty,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X9: set @ A,A2: A] :
( ( X9
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ X9 )
=> ( ord_less_eq @ A @ X5 @ A2 ) )
=> ( ! [Y7: A] :
( ! [X2: A] :
( ( member @ A @ X2 @ X9 )
=> ( ord_less_eq @ A @ X2 @ Y7 ) )
=> ( ord_less_eq @ A @ A2 @ Y7 ) )
=> ( ( complete_Sup_Sup @ A @ X9 )
= A2 ) ) ) ) ) ).
% cSup_eq_non_empty
thf(fact_5604_cSup__least,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X9: set @ A,Z3: A] :
( ( X9
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ X9 )
=> ( ord_less_eq @ A @ X5 @ Z3 ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ X9 ) @ Z3 ) ) ) ) ).
% cSup_least
thf(fact_5605_le__cSup__finite,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X9: set @ A,X: A] :
( ( finite_finite2 @ A @ X9 )
=> ( ( member @ A @ X @ X9 )
=> ( ord_less_eq @ A @ X @ ( complete_Sup_Sup @ A @ X9 ) ) ) ) ) ).
% le_cSup_finite
thf(fact_5606_less__cSupD,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [X9: set @ A,Z3: A] :
( ( X9
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less @ A @ Z3 @ ( complete_Sup_Sup @ A @ X9 ) )
=> ? [X5: A] :
( ( member @ A @ X5 @ X9 )
& ( ord_less @ A @ Z3 @ X5 ) ) ) ) ) ).
% less_cSupD
thf(fact_5607_less__cSupE,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [Y2: A,X9: set @ A] :
( ( ord_less @ A @ Y2 @ ( complete_Sup_Sup @ A @ X9 ) )
=> ( ( X9
!= ( bot_bot @ ( set @ A ) ) )
=> ~ ! [X5: A] :
( ( member @ A @ X5 @ X9 )
=> ~ ( ord_less @ A @ Y2 @ X5 ) ) ) ) ) ).
% less_cSupE
thf(fact_5608_finite__imp__Sup__less,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [X9: set @ A,X: A,A2: A] :
( ( finite_finite2 @ A @ X9 )
=> ( ( member @ A @ X @ X9 )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ X9 )
=> ( ord_less @ A @ X5 @ A2 ) )
=> ( ord_less @ A @ ( complete_Sup_Sup @ A @ X9 ) @ A2 ) ) ) ) ) ).
% finite_imp_Sup_less
thf(fact_5609_cInf__eq__non__empty,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X9: set @ A,A2: A] :
( ( X9
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ X9 )
=> ( ord_less_eq @ A @ A2 @ X5 ) )
=> ( ! [Y7: A] :
( ! [X2: A] :
( ( member @ A @ X2 @ X9 )
=> ( ord_less_eq @ A @ Y7 @ X2 ) )
=> ( ord_less_eq @ A @ Y7 @ A2 ) )
=> ( ( complete_Inf_Inf @ A @ X9 )
= A2 ) ) ) ) ) ).
% cInf_eq_non_empty
thf(fact_5610_cInf__greatest,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X9: set @ A,Z3: A] :
( ( X9
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ X9 )
=> ( ord_less_eq @ A @ Z3 @ X5 ) )
=> ( ord_less_eq @ A @ Z3 @ ( complete_Inf_Inf @ A @ X9 ) ) ) ) ) ).
% cInf_greatest
thf(fact_5611_cInf__le__finite,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X9: set @ A,X: A] :
( ( finite_finite2 @ A @ X9 )
=> ( ( member @ A @ X @ X9 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ X9 ) @ X ) ) ) ) ).
% cInf_le_finite
thf(fact_5612_cInf__lessD,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [X9: set @ A,Z3: A] :
( ( X9
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less @ A @ ( complete_Inf_Inf @ A @ X9 ) @ Z3 )
=> ? [X5: A] :
( ( member @ A @ X5 @ X9 )
& ( ord_less @ A @ X5 @ Z3 ) ) ) ) ) ).
% cInf_lessD
thf(fact_5613_finite__imp__less__Inf,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [X9: set @ A,X: A,A2: A] :
( ( finite_finite2 @ A @ X9 )
=> ( ( member @ A @ X @ X9 )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ X9 )
=> ( ord_less @ A @ A2 @ X5 ) )
=> ( ord_less @ A @ A2 @ ( complete_Inf_Inf @ A @ X9 ) ) ) ) ) ) ).
% finite_imp_less_Inf
thf(fact_5614_finite__Sup__less__iff,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [X9: set @ A,A2: A] :
( ( finite_finite2 @ A @ X9 )
=> ( ( X9
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less @ A @ ( complete_Sup_Sup @ A @ X9 ) @ A2 )
= ( ! [X4: A] :
( ( member @ A @ X4 @ X9 )
=> ( ord_less @ A @ X4 @ A2 ) ) ) ) ) ) ) ).
% finite_Sup_less_iff
thf(fact_5615_finite__less__Inf__iff,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [X9: set @ A,A2: A] :
( ( finite_finite2 @ A @ X9 )
=> ( ( X9
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less @ A @ A2 @ ( complete_Inf_Inf @ A @ X9 ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ X9 )
=> ( ord_less @ A @ A2 @ X4 ) ) ) ) ) ) ) ).
% finite_less_Inf_iff
thf(fact_5616_cSup__abs__le,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( linordered_idom @ A ) )
=> ! [S3: set @ A,A2: A] :
( ( S3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ S3 )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ X5 ) @ A2 ) )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ ( complete_Sup_Sup @ A @ S3 ) ) @ A2 ) ) ) ) ).
% cSup_abs_le
thf(fact_5617_Inf__eq__bot__iff,axiom,
! [A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [A3: set @ A] :
( ( ( complete_Inf_Inf @ A @ A3 )
= ( bot_bot @ A ) )
= ( ! [X4: A] :
( ( ord_less @ A @ ( bot_bot @ A ) @ X4 )
=> ? [Y: A] :
( ( member @ A @ Y @ A3 )
& ( ord_less @ A @ Y @ X4 ) ) ) ) ) ) ).
% Inf_eq_bot_iff
thf(fact_5618_Inf__le__Sup,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ A] :
( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A3 ) @ ( complete_Sup_Sup @ A @ A3 ) ) ) ) ).
% Inf_le_Sup
thf(fact_5619_subset__Pow__Union,axiom,
! [A: $tType,A3: set @ ( set @ A )] : ( ord_less_eq @ ( set @ ( set @ A ) ) @ A3 @ ( pow2 @ A @ ( complete_Sup_Sup @ ( set @ A ) @ A3 ) ) ) ).
% subset_Pow_Union
thf(fact_5620_Sup__upper2,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [U: A,A3: set @ A,V2: A] :
( ( member @ A @ U @ A3 )
=> ( ( ord_less_eq @ A @ V2 @ U )
=> ( ord_less_eq @ A @ V2 @ ( complete_Sup_Sup @ A @ A3 ) ) ) ) ) ).
% Sup_upper2
thf(fact_5621_Sup__le__iff,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ A,B2: A] :
( ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A3 ) @ B2 )
= ( ! [X4: A] :
( ( member @ A @ X4 @ A3 )
=> ( ord_less_eq @ A @ X4 @ B2 ) ) ) ) ) ).
% Sup_le_iff
thf(fact_5622_Sup__upper,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X: A,A3: set @ A] :
( ( member @ A @ X @ A3 )
=> ( ord_less_eq @ A @ X @ ( complete_Sup_Sup @ A @ A3 ) ) ) ) ).
% Sup_upper
thf(fact_5623_Sup__least,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ A,Z3: A] :
( ! [X5: A] :
( ( member @ A @ X5 @ A3 )
=> ( ord_less_eq @ A @ X5 @ Z3 ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A3 ) @ Z3 ) ) ) ).
% Sup_least
thf(fact_5624_Sup__mono,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ A,B5: set @ A] :
( ! [A4: A] :
( ( member @ A @ A4 @ A3 )
=> ? [X2: A] :
( ( member @ A @ X2 @ B5 )
& ( ord_less_eq @ A @ A4 @ X2 ) ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A3 ) @ ( complete_Sup_Sup @ A @ B5 ) ) ) ) ).
% Sup_mono
thf(fact_5625_Sup__eqI,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ A,X: A] :
( ! [Y7: A] :
( ( member @ A @ Y7 @ A3 )
=> ( ord_less_eq @ A @ Y7 @ X ) )
=> ( ! [Y7: A] :
( ! [Z4: A] :
( ( member @ A @ Z4 @ A3 )
=> ( ord_less_eq @ A @ Z4 @ Y7 ) )
=> ( ord_less_eq @ A @ X @ Y7 ) )
=> ( ( complete_Sup_Sup @ A @ A3 )
= X ) ) ) ) ).
% Sup_eqI
thf(fact_5626_less__Sup__iff,axiom,
! [A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [A2: A,S3: set @ A] :
( ( ord_less @ A @ A2 @ ( complete_Sup_Sup @ A @ S3 ) )
= ( ? [X4: A] :
( ( member @ A @ X4 @ S3 )
& ( ord_less @ A @ A2 @ X4 ) ) ) ) ) ).
% less_Sup_iff
thf(fact_5627_Inf__greatest,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ A,Z3: A] :
( ! [X5: A] :
( ( member @ A @ X5 @ A3 )
=> ( ord_less_eq @ A @ Z3 @ X5 ) )
=> ( ord_less_eq @ A @ Z3 @ ( complete_Inf_Inf @ A @ A3 ) ) ) ) ).
% Inf_greatest
thf(fact_5628_le__Inf__iff,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [B2: A,A3: set @ A] :
( ( ord_less_eq @ A @ B2 @ ( complete_Inf_Inf @ A @ A3 ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ A3 )
=> ( ord_less_eq @ A @ B2 @ X4 ) ) ) ) ) ).
% le_Inf_iff
thf(fact_5629_Inf__lower2,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [U: A,A3: set @ A,V2: A] :
( ( member @ A @ U @ A3 )
=> ( ( ord_less_eq @ A @ U @ V2 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A3 ) @ V2 ) ) ) ) ).
% Inf_lower2
thf(fact_5630_Inf__lower,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X: A,A3: set @ A] :
( ( member @ A @ X @ A3 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A3 ) @ X ) ) ) ).
% Inf_lower
thf(fact_5631_Inf__mono,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [B5: set @ A,A3: set @ A] :
( ! [B3: A] :
( ( member @ A @ B3 @ B5 )
=> ? [X2: A] :
( ( member @ A @ X2 @ A3 )
& ( ord_less_eq @ A @ X2 @ B3 ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A3 ) @ ( complete_Inf_Inf @ A @ B5 ) ) ) ) ).
% Inf_mono
thf(fact_5632_Inf__eqI,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ A,X: A] :
( ! [I3: A] :
( ( member @ A @ I3 @ A3 )
=> ( ord_less_eq @ A @ X @ I3 ) )
=> ( ! [Y7: A] :
( ! [I4: A] :
( ( member @ A @ I4 @ A3 )
=> ( ord_less_eq @ A @ Y7 @ I4 ) )
=> ( ord_less_eq @ A @ Y7 @ X ) )
=> ( ( complete_Inf_Inf @ A @ A3 )
= X ) ) ) ) ).
% Inf_eqI
thf(fact_5633_Inf__less__iff,axiom,
! [A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [S3: set @ A,A2: A] :
( ( ord_less @ A @ ( complete_Inf_Inf @ A @ S3 ) @ A2 )
= ( ? [X4: A] :
( ( member @ A @ X4 @ S3 )
& ( ord_less @ A @ X4 @ A2 ) ) ) ) ) ).
% Inf_less_iff
thf(fact_5634_Union__subsetI,axiom,
! [A: $tType,A3: set @ ( set @ A ),B5: set @ ( set @ A )] :
( ! [X5: set @ A] :
( ( member @ ( set @ A ) @ X5 @ A3 )
=> ? [Y4: set @ A] :
( ( member @ ( set @ A ) @ Y4 @ B5 )
& ( ord_less_eq @ ( set @ A ) @ X5 @ Y4 ) ) )
=> ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ A3 ) @ ( complete_Sup_Sup @ ( set @ A ) @ B5 ) ) ) ).
% Union_subsetI
thf(fact_5635_Union__upper,axiom,
! [A: $tType,B5: set @ A,A3: set @ ( set @ A )] :
( ( member @ ( set @ A ) @ B5 @ A3 )
=> ( ord_less_eq @ ( set @ A ) @ B5 @ ( complete_Sup_Sup @ ( set @ A ) @ A3 ) ) ) ).
% Union_upper
thf(fact_5636_Union__least,axiom,
! [A: $tType,A3: set @ ( set @ A ),C5: set @ A] :
( ! [X10: set @ A] :
( ( member @ ( set @ A ) @ X10 @ A3 )
=> ( ord_less_eq @ ( set @ A ) @ X10 @ C5 ) )
=> ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ A3 ) @ C5 ) ) ).
% Union_least
thf(fact_5637_Inter__greatest,axiom,
! [A: $tType,A3: set @ ( set @ A ),C5: set @ A] :
( ! [X10: set @ A] :
( ( member @ ( set @ A ) @ X10 @ A3 )
=> ( ord_less_eq @ ( set @ A ) @ C5 @ X10 ) )
=> ( ord_less_eq @ ( set @ A ) @ C5 @ ( complete_Inf_Inf @ ( set @ A ) @ A3 ) ) ) ).
% Inter_greatest
thf(fact_5638_Inter__lower,axiom,
! [A: $tType,B5: set @ A,A3: set @ ( set @ A )] :
( ( member @ ( set @ A ) @ B5 @ A3 )
=> ( ord_less_eq @ ( set @ A ) @ ( complete_Inf_Inf @ ( set @ A ) @ A3 ) @ B5 ) ) ).
% Inter_lower
thf(fact_5639_le__Sup__iff,axiom,
! [A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [X: A,A3: set @ A] :
( ( ord_less_eq @ A @ X @ ( complete_Sup_Sup @ A @ A3 ) )
= ( ! [Y: A] :
( ( ord_less @ A @ Y @ X )
=> ? [X4: A] :
( ( member @ A @ X4 @ A3 )
& ( ord_less @ A @ Y @ X4 ) ) ) ) ) ) ).
% le_Sup_iff
thf(fact_5640_Inf__le__iff,axiom,
! [A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [A3: set @ A,X: A] :
( ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A3 ) @ X )
= ( ! [Y: A] :
( ( ord_less @ A @ X @ Y )
=> ? [X4: A] :
( ( member @ A @ X4 @ A3 )
& ( ord_less @ A @ X4 @ Y ) ) ) ) ) ) ).
% Inf_le_iff
thf(fact_5641_less__eq__Sup,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ A,U: A] :
( ! [V3: A] :
( ( member @ A @ V3 @ A3 )
=> ( ord_less_eq @ A @ U @ V3 ) )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ord_less_eq @ A @ U @ ( complete_Sup_Sup @ A @ A3 ) ) ) ) ) ).
% less_eq_Sup
thf(fact_5642_Sup__subset__mono,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B5 )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A3 ) @ ( complete_Sup_Sup @ A @ B5 ) ) ) ) ).
% Sup_subset_mono
thf(fact_5643_Inf__less__eq,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ A,U: A] :
( ! [V3: A] :
( ( member @ A @ V3 @ A3 )
=> ( ord_less_eq @ A @ V3 @ U ) )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A3 ) @ U ) ) ) ) ).
% Inf_less_eq
thf(fact_5644_Inf__superset__mono,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [B5: set @ A,A3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ B5 @ A3 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A3 ) @ ( complete_Inf_Inf @ A @ B5 ) ) ) ) ).
% Inf_superset_mono
thf(fact_5645_Union__mono,axiom,
! [A: $tType,A3: set @ ( set @ A ),B5: set @ ( set @ A )] :
( ( ord_less_eq @ ( set @ ( set @ A ) ) @ A3 @ B5 )
=> ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ A3 ) @ ( complete_Sup_Sup @ ( set @ A ) @ B5 ) ) ) ).
% Union_mono
thf(fact_5646_Inter__anti__mono,axiom,
! [A: $tType,B5: set @ ( set @ A ),A3: set @ ( set @ A )] :
( ( ord_less_eq @ ( set @ ( set @ A ) ) @ B5 @ A3 )
=> ( ord_less_eq @ ( set @ A ) @ ( complete_Inf_Inf @ ( set @ A ) @ A3 ) @ ( complete_Inf_Inf @ ( set @ A ) @ B5 ) ) ) ).
% Inter_anti_mono
thf(fact_5647_Inter__subset,axiom,
! [A: $tType,A3: set @ ( set @ A ),B5: set @ A] :
( ! [X10: set @ A] :
( ( member @ ( set @ A ) @ X10 @ A3 )
=> ( ord_less_eq @ ( set @ A ) @ X10 @ B5 ) )
=> ( ( A3
!= ( bot_bot @ ( set @ ( set @ A ) ) ) )
=> ( ord_less_eq @ ( set @ A ) @ ( complete_Inf_Inf @ ( set @ A ) @ A3 ) @ B5 ) ) ) ).
% Inter_subset
thf(fact_5648_card__partition,axiom,
! [A: $tType,C5: set @ ( set @ A ),K2: nat] :
( ( finite_finite2 @ ( set @ A ) @ C5 )
=> ( ( finite_finite2 @ A @ ( complete_Sup_Sup @ ( set @ A ) @ C5 ) )
=> ( ! [C4: set @ A] :
( ( member @ ( set @ A ) @ C4 @ C5 )
=> ( ( finite_card @ A @ C4 )
= K2 ) )
=> ( ! [C1: set @ A,C22: set @ A] :
( ( member @ ( set @ A ) @ C1 @ C5 )
=> ( ( member @ ( set @ A ) @ C22 @ C5 )
=> ( ( C1 != C22 )
=> ( ( inf_inf @ ( set @ A ) @ C1 @ C22 )
= ( bot_bot @ ( set @ A ) ) ) ) ) )
=> ( ( times_times @ nat @ K2 @ ( finite_card @ ( set @ A ) @ C5 ) )
= ( finite_card @ A @ ( complete_Sup_Sup @ ( set @ A ) @ C5 ) ) ) ) ) ) ) ).
% card_partition
thf(fact_5649_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_5650_times__int_Oabs__eq,axiom,
! [Xa: product_prod @ nat @ nat,X: product_prod @ nat @ nat] :
( ( times_times @ int @ ( abs_Integ @ Xa ) @ ( abs_Integ @ X ) )
= ( abs_Integ
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X4: nat,Y: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V5: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ X4 @ U2 ) @ ( times_times @ nat @ Y @ V5 ) ) @ ( plus_plus @ nat @ ( times_times @ nat @ X4 @ V5 ) @ ( times_times @ nat @ Y @ U2 ) ) ) )
@ Xa
@ X ) ) ) ).
% times_int.abs_eq
thf(fact_5651_inf_Obounded__iff,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ ( inf_inf @ A @ B2 @ C2 ) )
= ( ( ord_less_eq @ A @ A2 @ B2 )
& ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% inf.bounded_iff
thf(fact_5652_le__inf__iff,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X: A,Y2: A,Z3: A] :
( ( ord_less_eq @ A @ X @ ( inf_inf @ A @ Y2 @ Z3 ) )
= ( ( ord_less_eq @ A @ X @ Y2 )
& ( ord_less_eq @ A @ X @ Z3 ) ) ) ) ).
% le_inf_iff
thf(fact_5653_Int__subset__iff,axiom,
! [A: $tType,C5: set @ A,A3: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ C5 @ ( inf_inf @ ( set @ A ) @ A3 @ B5 ) )
= ( ( ord_less_eq @ ( set @ A ) @ C5 @ A3 )
& ( ord_less_eq @ ( set @ A ) @ C5 @ B5 ) ) ) ).
% Int_subset_iff
thf(fact_5654_sum__of__bool__mult__eq,axiom,
! [A: $tType,B: $tType] :
( ( semiring_1 @ A )
=> ! [A3: set @ B,P3: B > $o,F3: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X4: B] : ( times_times @ A @ ( zero_neq_one_of_bool @ A @ ( P3 @ X4 ) ) @ ( F3 @ X4 ) )
@ A3 )
= ( groups7311177749621191930dd_sum @ B @ A @ F3 @ ( inf_inf @ ( set @ B ) @ A3 @ ( collect @ B @ P3 ) ) ) ) ) ) ).
% sum_of_bool_mult_eq
thf(fact_5655_sum__mult__of__bool__eq,axiom,
! [A: $tType,B: $tType] :
( ( semiring_1 @ A )
=> ! [A3: set @ B,F3: B > A,P3: B > $o] :
( ( finite_finite2 @ B @ A3 )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X4: B] : ( times_times @ A @ ( F3 @ X4 ) @ ( zero_neq_one_of_bool @ A @ ( P3 @ X4 ) ) )
@ A3 )
= ( groups7311177749621191930dd_sum @ B @ A @ F3 @ ( inf_inf @ ( set @ B ) @ A3 @ ( collect @ B @ P3 ) ) ) ) ) ) ).
% sum_mult_of_bool_eq
thf(fact_5656_sum__of__bool__eq,axiom,
! [A: $tType,B: $tType] :
( ( semiring_1 @ A )
=> ! [A3: set @ B,P3: B > $o] :
( ( finite_finite2 @ B @ A3 )
=> ( ( finite_finite2 @ B @ A3 )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X4: B] : ( zero_neq_one_of_bool @ A @ ( P3 @ X4 ) )
@ A3 )
= ( semiring_1_of_nat @ A @ ( finite_card @ B @ ( inf_inf @ ( set @ B ) @ A3 @ ( collect @ B @ P3 ) ) ) ) ) ) ) ) ).
% sum_of_bool_eq
thf(fact_5657_Union__Int__subset,axiom,
! [A: $tType,A3: set @ ( set @ A ),B5: set @ ( set @ A )] : ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( inf_inf @ ( set @ ( set @ A ) ) @ A3 @ B5 ) ) @ ( inf_inf @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ A3 ) @ ( complete_Sup_Sup @ ( set @ A ) @ B5 ) ) ) ).
% Union_Int_subset
thf(fact_5658_Sup__inter__less__eq,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ A,B5: set @ A] : ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( inf_inf @ ( set @ A ) @ A3 @ B5 ) ) @ ( inf_inf @ A @ ( complete_Sup_Sup @ A @ A3 ) @ ( complete_Sup_Sup @ A @ B5 ) ) ) ) ).
% Sup_inter_less_eq
thf(fact_5659_less__infI1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A2: A,X: A,B2: A] :
( ( ord_less @ A @ A2 @ X )
=> ( ord_less @ A @ ( inf_inf @ A @ A2 @ B2 ) @ X ) ) ) ).
% less_infI1
thf(fact_5660_less__infI2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [B2: A,X: A,A2: A] :
( ( ord_less @ A @ B2 @ X )
=> ( ord_less @ A @ ( inf_inf @ A @ A2 @ B2 ) @ X ) ) ) ).
% less_infI2
thf(fact_5661_inf_Oabsorb3,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( inf_inf @ A @ A2 @ B2 )
= A2 ) ) ) ).
% inf.absorb3
thf(fact_5662_inf_Oabsorb4,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [B2: A,A2: A] :
( ( ord_less @ A @ B2 @ A2 )
=> ( ( inf_inf @ A @ A2 @ B2 )
= B2 ) ) ) ).
% inf.absorb4
thf(fact_5663_inf_Ostrict__boundedE,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less @ A @ A2 @ ( inf_inf @ A @ B2 @ C2 ) )
=> ~ ( ( ord_less @ A @ A2 @ B2 )
=> ~ ( ord_less @ A @ A2 @ C2 ) ) ) ) ).
% inf.strict_boundedE
thf(fact_5664_inf_Ostrict__order__iff,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( ( ord_less @ A )
= ( ^ [A5: A,B4: A] :
( ( A5
= ( inf_inf @ A @ A5 @ B4 ) )
& ( A5 != B4 ) ) ) ) ) ).
% inf.strict_order_iff
thf(fact_5665_inf_Ostrict__coboundedI1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( ord_less @ A @ A2 @ C2 )
=> ( ord_less @ A @ ( inf_inf @ A @ A2 @ B2 ) @ C2 ) ) ) ).
% inf.strict_coboundedI1
thf(fact_5666_inf_Ostrict__coboundedI2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [B2: A,C2: A,A2: A] :
( ( ord_less @ A @ B2 @ C2 )
=> ( ord_less @ A @ ( inf_inf @ A @ A2 @ B2 ) @ C2 ) ) ) ).
% inf.strict_coboundedI2
thf(fact_5667_Int__def,axiom,
! [A: $tType] :
( ( inf_inf @ ( set @ A ) )
= ( ^ [A7: set @ A,B7: set @ A] :
( collect @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ A7 )
& ( member @ A @ X4 @ B7 ) ) ) ) ) ).
% Int_def
thf(fact_5668_Int__Collect,axiom,
! [A: $tType,X: A,A3: set @ A,P3: A > $o] :
( ( member @ A @ X @ ( inf_inf @ ( set @ A ) @ A3 @ ( collect @ A @ P3 ) ) )
= ( ( member @ A @ X @ A3 )
& ( P3 @ X ) ) ) ).
% Int_Collect
thf(fact_5669_Collect__conj__eq,axiom,
! [A: $tType,P3: A > $o,Q2: A > $o] :
( ( collect @ A
@ ^ [X4: A] :
( ( P3 @ X4 )
& ( Q2 @ X4 ) ) )
= ( inf_inf @ ( set @ A ) @ ( collect @ A @ P3 ) @ ( collect @ A @ Q2 ) ) ) ).
% Collect_conj_eq
thf(fact_5670_Int__Collect__mono,axiom,
! [A: $tType,A3: set @ A,B5: set @ A,P3: A > $o,Q2: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B5 )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ A3 )
=> ( ( P3 @ X5 )
=> ( Q2 @ X5 ) ) )
=> ( ord_less_eq @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A3 @ ( collect @ A @ P3 ) ) @ ( inf_inf @ ( set @ A ) @ B5 @ ( collect @ A @ Q2 ) ) ) ) ) ).
% Int_Collect_mono
thf(fact_5671_Int__greatest,axiom,
! [A: $tType,C5: set @ A,A3: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ C5 @ A3 )
=> ( ( ord_less_eq @ ( set @ A ) @ C5 @ B5 )
=> ( ord_less_eq @ ( set @ A ) @ C5 @ ( inf_inf @ ( set @ A ) @ A3 @ B5 ) ) ) ) ).
% Int_greatest
thf(fact_5672_Int__absorb2,axiom,
! [A: $tType,A3: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B5 )
=> ( ( inf_inf @ ( set @ A ) @ A3 @ B5 )
= A3 ) ) ).
% Int_absorb2
thf(fact_5673_Int__absorb1,axiom,
! [A: $tType,B5: set @ A,A3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ B5 @ A3 )
=> ( ( inf_inf @ ( set @ A ) @ A3 @ B5 )
= B5 ) ) ).
% Int_absorb1
thf(fact_5674_Int__lower2,axiom,
! [A: $tType,A3: set @ A,B5: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A3 @ B5 ) @ B5 ) ).
% Int_lower2
thf(fact_5675_Int__lower1,axiom,
! [A: $tType,A3: set @ A,B5: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A3 @ B5 ) @ A3 ) ).
% Int_lower1
thf(fact_5676_Int__mono,axiom,
! [A: $tType,A3: set @ A,C5: set @ A,B5: set @ A,D4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ C5 )
=> ( ( ord_less_eq @ ( set @ A ) @ B5 @ D4 )
=> ( ord_less_eq @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A3 @ B5 ) @ ( inf_inf @ ( set @ A ) @ C5 @ D4 ) ) ) ) ).
% Int_mono
thf(fact_5677_inf_OcoboundedI2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [B2: A,C2: A,A2: A] :
( ( ord_less_eq @ A @ B2 @ C2 )
=> ( ord_less_eq @ A @ ( inf_inf @ A @ A2 @ B2 ) @ C2 ) ) ) ).
% inf.coboundedI2
thf(fact_5678_inf_OcoboundedI1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A2: A,C2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ C2 )
=> ( ord_less_eq @ A @ ( inf_inf @ A @ A2 @ B2 ) @ C2 ) ) ) ).
% inf.coboundedI1
thf(fact_5679_inf_Oabsorb__iff2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B4: A,A5: A] :
( ( inf_inf @ A @ A5 @ B4 )
= B4 ) ) ) ) ).
% inf.absorb_iff2
thf(fact_5680_inf_Oabsorb__iff1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A5: A,B4: A] :
( ( inf_inf @ A @ A5 @ B4 )
= A5 ) ) ) ) ).
% inf.absorb_iff1
thf(fact_5681_inf_Ocobounded2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A2: A,B2: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ A2 @ B2 ) @ B2 ) ) ).
% inf.cobounded2
thf(fact_5682_inf_Ocobounded1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A2: A,B2: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ A2 @ B2 ) @ A2 ) ) ).
% inf.cobounded1
thf(fact_5683_inf_Oorder__iff,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A5: A,B4: A] :
( A5
= ( inf_inf @ A @ A5 @ B4 ) ) ) ) ) ).
% inf.order_iff
thf(fact_5684_inf__greatest,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X: A,Y2: A,Z3: A] :
( ( ord_less_eq @ A @ X @ Y2 )
=> ( ( ord_less_eq @ A @ X @ Z3 )
=> ( ord_less_eq @ A @ X @ ( inf_inf @ A @ Y2 @ Z3 ) ) ) ) ) ).
% inf_greatest
thf(fact_5685_inf_OboundedI,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ A2 @ C2 )
=> ( ord_less_eq @ A @ A2 @ ( inf_inf @ A @ B2 @ C2 ) ) ) ) ) ).
% inf.boundedI
thf(fact_5686_inf_OboundedE,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A2 @ ( inf_inf @ A @ B2 @ C2 ) )
=> ~ ( ( ord_less_eq @ A @ A2 @ B2 )
=> ~ ( ord_less_eq @ A @ A2 @ C2 ) ) ) ) ).
% inf.boundedE
thf(fact_5687_inf__absorb2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [Y2: A,X: A] :
( ( ord_less_eq @ A @ Y2 @ X )
=> ( ( inf_inf @ A @ X @ Y2 )
= Y2 ) ) ) ).
% inf_absorb2
thf(fact_5688_inf__absorb1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X: A,Y2: A] :
( ( ord_less_eq @ A @ X @ Y2 )
=> ( ( inf_inf @ A @ X @ Y2 )
= X ) ) ) ).
% inf_absorb1
thf(fact_5689_inf_Oabsorb2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [B2: A,A2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( inf_inf @ A @ A2 @ B2 )
= B2 ) ) ) ).
% inf.absorb2
thf(fact_5690_inf_Oabsorb1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( inf_inf @ A @ A2 @ B2 )
= A2 ) ) ) ).
% inf.absorb1
thf(fact_5691_le__iff__inf,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [X4: A,Y: A] :
( ( inf_inf @ A @ X4 @ Y )
= X4 ) ) ) ) ).
% le_iff_inf
thf(fact_5692_inf__unique,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [F3: A > A > A,X: A,Y2: A] :
( ! [X5: A,Y7: A] : ( ord_less_eq @ A @ ( F3 @ X5 @ Y7 ) @ X5 )
=> ( ! [X5: A,Y7: A] : ( ord_less_eq @ A @ ( F3 @ X5 @ Y7 ) @ Y7 )
=> ( ! [X5: A,Y7: A,Z: A] :
( ( ord_less_eq @ A @ X5 @ Y7 )
=> ( ( ord_less_eq @ A @ X5 @ Z )
=> ( ord_less_eq @ A @ X5 @ ( F3 @ Y7 @ Z ) ) ) )
=> ( ( inf_inf @ A @ X @ Y2 )
= ( F3 @ X @ Y2 ) ) ) ) ) ) ).
% inf_unique
thf(fact_5693_inf_OorderI,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A2: A,B2: A] :
( ( A2
= ( inf_inf @ A @ A2 @ B2 ) )
=> ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% inf.orderI
thf(fact_5694_inf_OorderE,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( A2
= ( inf_inf @ A @ A2 @ B2 ) ) ) ) ).
% inf.orderE
thf(fact_5695_le__infI2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [B2: A,X: A,A2: A] :
( ( ord_less_eq @ A @ B2 @ X )
=> ( ord_less_eq @ A @ ( inf_inf @ A @ A2 @ B2 ) @ X ) ) ) ).
% le_infI2
thf(fact_5696_le__infI1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A2: A,X: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ X )
=> ( ord_less_eq @ A @ ( inf_inf @ A @ A2 @ B2 ) @ X ) ) ) ).
% le_infI1
thf(fact_5697_inf__mono,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A2: A,C2: A,B2: A,D2: A] :
( ( ord_less_eq @ A @ A2 @ C2 )
=> ( ( ord_less_eq @ A @ B2 @ D2 )
=> ( ord_less_eq @ A @ ( inf_inf @ A @ A2 @ B2 ) @ ( inf_inf @ A @ C2 @ D2 ) ) ) ) ) ).
% inf_mono
thf(fact_5698_le__infI,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X: A,A2: A,B2: A] :
( ( ord_less_eq @ A @ X @ A2 )
=> ( ( ord_less_eq @ A @ X @ B2 )
=> ( ord_less_eq @ A @ X @ ( inf_inf @ A @ A2 @ B2 ) ) ) ) ) ).
% le_infI
thf(fact_5699_le__infE,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X: A,A2: A,B2: A] :
( ( ord_less_eq @ A @ X @ ( inf_inf @ A @ A2 @ B2 ) )
=> ~ ( ( ord_less_eq @ A @ X @ A2 )
=> ~ ( ord_less_eq @ A @ X @ B2 ) ) ) ) ).
% le_infE
thf(fact_5700_inf__le2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X: A,Y2: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ X @ Y2 ) @ Y2 ) ) ).
% inf_le2
thf(fact_5701_inf__le1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X: A,Y2: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ X @ Y2 ) @ X ) ) ).
% inf_le1
thf(fact_5702_inf__sup__ord_I1_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X: A,Y2: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ X @ Y2 ) @ X ) ) ).
% inf_sup_ord(1)
thf(fact_5703_inf__sup__ord_I2_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X: A,Y2: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ X @ Y2 ) @ Y2 ) ) ).
% inf_sup_ord(2)
thf(fact_5704_inf__shunt,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y2: A] :
( ( ( inf_inf @ A @ X @ Y2 )
= ( bot_bot @ A ) )
= ( ord_less_eq @ A @ X @ ( uminus_uminus @ A @ Y2 ) ) ) ) ).
% inf_shunt
thf(fact_5705_Ioc__disjoint,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A,C2: A,D2: A] :
( ( ( inf_inf @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ A2 @ B2 ) @ ( set_or3652927894154168847AtMost @ A @ C2 @ D2 ) )
= ( bot_bot @ ( set @ A ) ) )
= ( ( ord_less_eq @ A @ B2 @ A2 )
| ( ord_less_eq @ A @ D2 @ C2 )
| ( ord_less_eq @ A @ B2 @ C2 )
| ( ord_less_eq @ A @ D2 @ A2 ) ) ) ) ).
% Ioc_disjoint
thf(fact_5706_disjoint__eq__subset__Compl,axiom,
! [A: $tType,A3: set @ A,B5: set @ A] :
( ( ( inf_inf @ ( set @ A ) @ A3 @ B5 )
= ( bot_bot @ ( set @ A ) ) )
= ( ord_less_eq @ ( set @ A ) @ A3 @ ( uminus_uminus @ ( set @ A ) @ B5 ) ) ) ).
% disjoint_eq_subset_Compl
thf(fact_5707_sum_Ointer__restrict,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A3: set @ B,G: B > A,B5: set @ B] :
( ( finite_finite2 @ B @ A3 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( inf_inf @ ( set @ B ) @ A3 @ B5 ) )
= ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X4: B] : ( if @ A @ ( member @ B @ X4 @ B5 ) @ ( G @ X4 ) @ ( zero_zero @ A ) )
@ A3 ) ) ) ) ).
% sum.inter_restrict
thf(fact_5708_prod_Ointer__restrict,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: set @ B,G: B > A,B5: set @ B] :
( ( finite_finite2 @ B @ A3 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( inf_inf @ ( set @ B ) @ A3 @ B5 ) )
= ( groups7121269368397514597t_prod @ B @ A
@ ^ [X4: B] : ( if @ A @ ( member @ B @ X4 @ B5 ) @ ( G @ X4 ) @ ( one_one @ A ) )
@ A3 ) ) ) ) ).
% prod.inter_restrict
thf(fact_5709_Iio__Int__singleton,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,K2: A] :
( ( ( ord_less @ A @ X @ K2 )
=> ( ( inf_inf @ ( set @ A ) @ ( set_ord_lessThan @ A @ K2 ) @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
& ( ~ ( ord_less @ A @ X @ K2 )
=> ( ( inf_inf @ ( set @ A ) @ ( set_ord_lessThan @ A @ K2 ) @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% Iio_Int_singleton
thf(fact_5710_sum_OInt__Diff,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A3: set @ B,G: B > A,B5: set @ B] :
( ( finite_finite2 @ B @ A3 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ A3 )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G @ ( inf_inf @ ( set @ B ) @ A3 @ B5 ) ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A3 @ B5 ) ) ) ) ) ) ).
% sum.Int_Diff
thf(fact_5711_prod_OInt__Diff,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: set @ B,G: B > A,B5: set @ B] :
( ( finite_finite2 @ B @ A3 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ A3 )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G @ ( inf_inf @ ( set @ B ) @ A3 @ B5 ) ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A3 @ B5 ) ) ) ) ) ) ).
% prod.Int_Diff
thf(fact_5712_prod_Omono__neutral__cong,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [T5: set @ B,S3: set @ B,H2: B > A,G: B > A] :
( ( finite_finite2 @ B @ T5 )
=> ( ( finite_finite2 @ B @ S3 )
=> ( ! [I3: B] :
( ( member @ B @ I3 @ ( minus_minus @ ( set @ B ) @ T5 @ S3 ) )
=> ( ( H2 @ I3 )
= ( one_one @ A ) ) )
=> ( ! [I3: B] :
( ( member @ B @ I3 @ ( minus_minus @ ( set @ B ) @ S3 @ T5 ) )
=> ( ( G @ I3 )
= ( one_one @ A ) ) )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ ( inf_inf @ ( set @ B ) @ S3 @ T5 ) )
=> ( ( G @ X5 )
= ( H2 @ X5 ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ S3 )
= ( groups7121269368397514597t_prod @ B @ A @ H2 @ T5 ) ) ) ) ) ) ) ) ).
% prod.mono_neutral_cong
thf(fact_5713_sum_OIf__cases,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A3: set @ B,P3: B > $o,H2: B > A,G: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X4: B] : ( if @ A @ ( P3 @ X4 ) @ ( H2 @ X4 ) @ ( G @ X4 ) )
@ A3 )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ H2 @ ( inf_inf @ ( set @ B ) @ A3 @ ( collect @ B @ P3 ) ) ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ ( inf_inf @ ( set @ B ) @ A3 @ ( uminus_uminus @ ( set @ B ) @ ( collect @ B @ P3 ) ) ) ) ) ) ) ) ).
% sum.If_cases
thf(fact_5714_prod_OIf__cases,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: set @ B,P3: B > $o,H2: B > A,G: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [X4: B] : ( if @ A @ ( P3 @ X4 ) @ ( H2 @ X4 ) @ ( G @ X4 ) )
@ A3 )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ H2 @ ( inf_inf @ ( set @ B ) @ A3 @ ( collect @ B @ P3 ) ) ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ ( inf_inf @ ( set @ B ) @ A3 @ ( uminus_uminus @ ( set @ B ) @ ( collect @ B @ P3 ) ) ) ) ) ) ) ) ).
% prod.If_cases
thf(fact_5715_uminus__int_Oabs__eq,axiom,
! [X: product_prod @ nat @ nat] :
( ( uminus_uminus @ int @ ( abs_Integ @ X ) )
= ( abs_Integ
@ ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [X4: nat,Y: nat] : ( product_Pair @ nat @ nat @ Y @ X4 )
@ X ) ) ) ).
% uminus_int.abs_eq
thf(fact_5716_one__int__def,axiom,
( ( one_one @ int )
= ( abs_Integ @ ( product_Pair @ nat @ nat @ ( one_one @ nat ) @ ( zero_zero @ nat ) ) ) ) ).
% one_int_def
thf(fact_5717_sum__div__partition,axiom,
! [B: $tType,A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A3: set @ B,F3: B > A,B2: A] :
( ( finite_finite2 @ B @ A3 )
=> ( ( divide_divide @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ A3 ) @ B2 )
= ( plus_plus @ A
@ ( groups7311177749621191930dd_sum @ B @ A
@ ^ [A5: B] : ( divide_divide @ A @ ( F3 @ A5 ) @ B2 )
@ ( inf_inf @ ( set @ B ) @ A3
@ ( collect @ B
@ ^ [A5: B] : ( dvd_dvd @ A @ B2 @ ( F3 @ A5 ) ) ) ) )
@ ( divide_divide @ A
@ ( groups7311177749621191930dd_sum @ B @ A @ F3
@ ( inf_inf @ ( set @ B ) @ A3
@ ( collect @ B
@ ^ [A5: B] :
~ ( dvd_dvd @ A @ B2 @ ( F3 @ A5 ) ) ) ) )
@ B2 ) ) ) ) ) ).
% sum_div_partition
thf(fact_5718_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
@ ^ [I: nat,J4: nat] : ( minus_minus @ A @ ( semiring_1_of_nat @ A @ I ) @ ( semiring_1_of_nat @ A @ J4 ) )
@ X ) ) ) ).
% of_int.abs_eq
thf(fact_5719_distinct__concat,axiom,
! [A: $tType,Xs: list @ ( list @ A )] :
( ( distinct @ ( list @ A ) @ Xs )
=> ( ! [Ys4: list @ A] :
( ( member @ ( list @ A ) @ Ys4 @ ( set2 @ ( list @ A ) @ Xs ) )
=> ( distinct @ A @ Ys4 ) )
=> ( ! [Ys4: list @ A,Zs2: list @ A] :
( ( member @ ( list @ A ) @ Ys4 @ ( set2 @ ( list @ A ) @ Xs ) )
=> ( ( member @ ( list @ A ) @ Zs2 @ ( set2 @ ( list @ A ) @ Xs ) )
=> ( ( Ys4 != Zs2 )
=> ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ Ys4 ) @ ( set2 @ A @ Zs2 ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) )
=> ( distinct @ A @ ( concat @ A @ Xs ) ) ) ) ) ).
% distinct_concat
thf(fact_5720_less__int_Oabs__eq,axiom,
! [Xa: product_prod @ nat @ nat,X: product_prod @ nat @ nat] :
( ( ord_less @ int @ ( abs_Integ @ Xa ) @ ( abs_Integ @ X ) )
= ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [X4: nat,Y: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U2: nat,V5: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ X4 @ V5 ) @ ( plus_plus @ nat @ U2 @ Y ) ) )
@ Xa
@ X ) ) ).
% less_int.abs_eq
thf(fact_5721_less__eq__int_Oabs__eq,axiom,
! [Xa: product_prod @ nat @ nat,X: product_prod @ nat @ nat] :
( ( ord_less_eq @ int @ ( abs_Integ @ Xa ) @ ( abs_Integ @ X ) )
= ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [X4: nat,Y: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U2: nat,V5: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ X4 @ V5 ) @ ( plus_plus @ nat @ U2 @ Y ) ) )
@ Xa
@ X ) ) ).
% less_eq_int.abs_eq
thf(fact_5722_plus__int_Oabs__eq,axiom,
! [Xa: product_prod @ nat @ nat,X: product_prod @ nat @ nat] :
( ( plus_plus @ int @ ( abs_Integ @ Xa ) @ ( abs_Integ @ X ) )
= ( abs_Integ
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X4: nat,Y: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V5: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X4 @ U2 ) @ ( plus_plus @ nat @ Y @ V5 ) ) )
@ Xa
@ X ) ) ) ).
% plus_int.abs_eq
thf(fact_5723_minus__int_Oabs__eq,axiom,
! [Xa: product_prod @ nat @ nat,X: product_prod @ nat @ nat] :
( ( minus_minus @ int @ ( abs_Integ @ Xa ) @ ( abs_Integ @ X ) )
= ( abs_Integ
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X4: nat,Y: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V5: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X4 @ V5 ) @ ( plus_plus @ nat @ Y @ U2 ) ) )
@ Xa
@ X ) ) ) ).
% minus_int.abs_eq
thf(fact_5724_card__disjoint__shuffles,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ Ys ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_card @ ( list @ A ) @ ( shuffles @ A @ Xs @ Ys ) )
= ( binomial @ ( plus_plus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ A ) @ Ys ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ).
% card_disjoint_shuffles
thf(fact_5725_arg__min__if__finite_I2_J,axiom,
! [B: $tType,A: $tType] :
( ( order @ B )
=> ! [S3: set @ A,F3: A > B] :
( ( finite_finite2 @ A @ S3 )
=> ( ( S3
!= ( bot_bot @ ( set @ A ) ) )
=> ~ ? [X2: A] :
( ( member @ A @ X2 @ S3 )
& ( ord_less @ B @ ( F3 @ X2 ) @ ( F3 @ ( lattic7623131987881927897min_on @ A @ B @ F3 @ S3 ) ) ) ) ) ) ) ).
% arg_min_if_finite(2)
thf(fact_5726_arg__min__least,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ! [S3: set @ A,Y2: A,F3: A > B] :
( ( finite_finite2 @ A @ S3 )
=> ( ( S3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( member @ A @ Y2 @ S3 )
=> ( ord_less_eq @ B @ ( F3 @ ( lattic7623131987881927897min_on @ A @ B @ F3 @ S3 ) ) @ ( F3 @ Y2 ) ) ) ) ) ) ).
% arg_min_least
thf(fact_5727_finite__shuffles,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] : ( finite_finite2 @ ( list @ A ) @ ( shuffles @ A @ Xs @ Ys ) ) ).
% finite_shuffles
thf(fact_5728_inf__Int__eq,axiom,
! [A: $tType,R4: set @ A,S3: set @ A] :
( ( inf_inf @ ( A > $o )
@ ^ [X4: A] : ( member @ A @ X4 @ R4 )
@ ^ [X4: A] : ( member @ A @ X4 @ S3 ) )
= ( ^ [X4: A] : ( member @ A @ X4 @ ( inf_inf @ ( set @ A ) @ R4 @ S3 ) ) ) ) ).
% inf_Int_eq
thf(fact_5729_inf__set__def,axiom,
! [A: $tType] :
( ( inf_inf @ ( set @ A ) )
= ( ^ [A7: set @ A,B7: set @ A] :
( collect @ A
@ ( inf_inf @ ( A > $o )
@ ^ [X4: A] : ( member @ A @ X4 @ A7 )
@ ^ [X4: A] : ( member @ A @ X4 @ B7 ) ) ) ) ) ).
% inf_set_def
thf(fact_5730_inf__Int__eq2,axiom,
! [B: $tType,A: $tType,R4: set @ ( product_prod @ A @ B ),S3: set @ ( product_prod @ A @ B )] :
( ( inf_inf @ ( A > B > $o )
@ ^ [X4: A,Y: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y ) @ R4 )
@ ^ [X4: A,Y: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y ) @ S3 ) )
= ( ^ [X4: A,Y: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y ) @ ( inf_inf @ ( set @ ( product_prod @ A @ B ) ) @ R4 @ S3 ) ) ) ) ).
% inf_Int_eq2
thf(fact_5731_shuffles__commutes,axiom,
! [A: $tType] :
( ( shuffles @ A )
= ( ^ [Xs2: list @ A,Ys3: list @ A] : ( shuffles @ A @ Ys3 @ Xs2 ) ) ) ).
% shuffles_commutes
thf(fact_5732_length__shuffles,axiom,
! [A: $tType,Zs: list @ A,Xs: list @ A,Ys: list @ A] :
( ( member @ ( list @ A ) @ Zs @ ( shuffles @ A @ Xs @ Ys ) )
=> ( ( size_size @ ( list @ A ) @ Zs )
= ( plus_plus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ A ) @ Ys ) ) ) ) ).
% length_shuffles
thf(fact_5733_distinct__disjoint__shuffles,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Zs: list @ A] :
( ( distinct @ A @ Xs )
=> ( ( distinct @ A @ Ys )
=> ( ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ Ys ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( member @ ( list @ A ) @ Zs @ ( shuffles @ A @ Xs @ Ys ) )
=> ( distinct @ A @ Zs ) ) ) ) ) ).
% distinct_disjoint_shuffles
thf(fact_5734_distinct__concat__iff,axiom,
! [A: $tType,Xs: list @ ( list @ A )] :
( ( distinct @ A @ ( concat @ A @ Xs ) )
= ( ( distinct @ ( list @ A ) @ ( removeAll @ ( list @ A ) @ ( nil @ A ) @ Xs ) )
& ! [Ys3: list @ A] :
( ( member @ ( list @ A ) @ Ys3 @ ( set2 @ ( list @ A ) @ Xs ) )
=> ( distinct @ A @ Ys3 ) )
& ! [Ys3: list @ A,Zs3: list @ A] :
( ( ( member @ ( list @ A ) @ Ys3 @ ( set2 @ ( list @ A ) @ Xs ) )
& ( member @ ( list @ A ) @ Zs3 @ ( set2 @ ( list @ A ) @ Xs ) )
& ( Ys3 != Zs3 ) )
=> ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ Ys3 ) @ ( set2 @ A @ Zs3 ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% distinct_concat_iff
thf(fact_5735_less__eq__int_Orep__eq,axiom,
( ( ord_less_eq @ int )
= ( ^ [X4: int,Xa4: int] :
( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [Y: nat,Z6: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U2: nat,V5: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ Y @ V5 ) @ ( plus_plus @ nat @ U2 @ Z6 ) ) )
@ ( rep_Integ @ X4 )
@ ( rep_Integ @ Xa4 ) ) ) ) ).
% less_eq_int.rep_eq
thf(fact_5736_less__int_Orep__eq,axiom,
( ( ord_less @ int )
= ( ^ [X4: int,Xa4: int] :
( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [Y: nat,Z6: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U2: nat,V5: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ Y @ V5 ) @ ( plus_plus @ nat @ U2 @ Z6 ) ) )
@ ( rep_Integ @ X4 )
@ ( rep_Integ @ Xa4 ) ) ) ) ).
% less_int.rep_eq
thf(fact_5737_map__is__Nil__conv,axiom,
! [A: $tType,B: $tType,F3: B > A,Xs: list @ B] :
( ( ( map @ B @ A @ F3 @ Xs )
= ( nil @ A ) )
= ( Xs
= ( nil @ B ) ) ) ).
% map_is_Nil_conv
thf(fact_5738_Nil__is__map__conv,axiom,
! [A: $tType,B: $tType,F3: B > A,Xs: list @ B] :
( ( ( nil @ A )
= ( map @ B @ A @ F3 @ Xs ) )
= ( Xs
= ( nil @ B ) ) ) ).
% Nil_is_map_conv
thf(fact_5739_list_Omap__disc__iff,axiom,
! [B: $tType,A: $tType,F3: A > B,A2: list @ A] :
( ( ( map @ A @ B @ F3 @ A2 )
= ( nil @ B ) )
= ( A2
= ( nil @ A ) ) ) ).
% list.map_disc_iff
thf(fact_5740_upt__conv__Nil,axiom,
! [J3: nat,I2: nat] :
( ( ord_less_eq @ nat @ J3 @ I2 )
=> ( ( upt @ I2 @ J3 )
= ( nil @ nat ) ) ) ).
% upt_conv_Nil
thf(fact_5741_list__update__nonempty,axiom,
! [A: $tType,Xs: list @ A,K2: nat,X: A] :
( ( ( list_update @ A @ Xs @ K2 @ X )
= ( nil @ A ) )
= ( Xs
= ( nil @ A ) ) ) ).
% list_update_nonempty
thf(fact_5742_concat__replicate__trivial,axiom,
! [A: $tType,I2: nat] :
( ( concat @ A @ ( replicate @ ( list @ A ) @ I2 @ ( nil @ A ) ) )
= ( nil @ A ) ) ).
% concat_replicate_trivial
thf(fact_5743_Nil__in__shuffles,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( member @ ( list @ A ) @ ( nil @ A ) @ ( shuffles @ A @ Xs @ Ys ) )
= ( ( Xs
= ( nil @ A ) )
& ( Ys
= ( nil @ A ) ) ) ) ).
% Nil_in_shuffles
thf(fact_5744_enumerate__simps_I1_J,axiom,
! [A: $tType,N: nat] :
( ( enumerate @ A @ N @ ( nil @ A ) )
= ( nil @ ( product_prod @ nat @ A ) ) ) ).
% enumerate_simps(1)
thf(fact_5745_rotate1__is__Nil__conv,axiom,
! [A: $tType,Xs: list @ A] :
( ( ( rotate1 @ A @ Xs )
= ( nil @ A ) )
= ( Xs
= ( nil @ A ) ) ) ).
% rotate1_is_Nil_conv
thf(fact_5746_set__empty,axiom,
! [A: $tType,Xs: list @ A] :
( ( ( set2 @ A @ Xs )
= ( bot_bot @ ( set @ A ) ) )
= ( Xs
= ( nil @ A ) ) ) ).
% set_empty
thf(fact_5747_set__empty2,axiom,
! [A: $tType,Xs: list @ A] :
( ( ( bot_bot @ ( set @ A ) )
= ( set2 @ A @ Xs ) )
= ( Xs
= ( nil @ A ) ) ) ).
% set_empty2
thf(fact_5748_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_5749_take0,axiom,
! [A: $tType] :
( ( take @ A @ ( zero_zero @ nat ) )
= ( ^ [Xs2: list @ A] : ( nil @ A ) ) ) ).
% take0
thf(fact_5750_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_5751_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_5752_empty__replicate,axiom,
! [A: $tType,N: nat,X: A] :
( ( ( nil @ A )
= ( replicate @ A @ N @ X ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% empty_replicate
thf(fact_5753_replicate__empty,axiom,
! [A: $tType,N: nat,X: A] :
( ( ( replicate @ A @ N @ X )
= ( nil @ A ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% replicate_empty
thf(fact_5754_upt__eq__Nil__conv,axiom,
! [I2: nat,J3: nat] :
( ( ( upt @ I2 @ J3 )
= ( nil @ nat ) )
= ( ( J3
= ( zero_zero @ nat ) )
| ( ord_less_eq @ nat @ J3 @ I2 ) ) ) ).
% upt_eq_Nil_conv
thf(fact_5755_sorted__list__of__set_Osorted__key__list__of__set__empty,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( linord4507533701916653071of_set @ A @ ( bot_bot @ ( set @ A ) ) )
= ( nil @ A ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_empty
thf(fact_5756_sorted__list__of__set_Ofold__insort__key_Oinfinite,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A] :
( ~ ( finite_finite2 @ A @ A3 )
=> ( ( linord4507533701916653071of_set @ A @ A3 )
= ( nil @ A ) ) ) ) ).
% sorted_list_of_set.fold_insort_key.infinite
thf(fact_5757_concat__eq__Nil__conv,axiom,
! [A: $tType,Xss: list @ ( list @ A )] :
( ( ( concat @ A @ Xss )
= ( nil @ A ) )
= ( ! [X4: list @ A] :
( ( member @ ( list @ A ) @ X4 @ ( set2 @ ( list @ A ) @ Xss ) )
=> ( X4
= ( nil @ A ) ) ) ) ) ).
% concat_eq_Nil_conv
thf(fact_5758_Nil__eq__concat__conv,axiom,
! [A: $tType,Xss: list @ ( list @ A )] :
( ( ( nil @ A )
= ( concat @ A @ Xss ) )
= ( ! [X4: list @ A] :
( ( member @ ( list @ A ) @ X4 @ ( set2 @ ( list @ A ) @ Xss ) )
=> ( X4
= ( nil @ A ) ) ) ) ) ).
% Nil_eq_concat_conv
thf(fact_5759_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_5760_sorted__list__of__set_Osorted__key__list__of__set__eq__Nil__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( ( linord4507533701916653071of_set @ A @ A3 )
= ( nil @ A ) )
= ( A3
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_eq_Nil_iff
thf(fact_5761_Nil__in__shufflesI,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( Xs
= ( nil @ A ) )
=> ( ( Ys
= ( nil @ A ) )
=> ( member @ ( list @ A ) @ ( nil @ A ) @ ( shuffles @ A @ Xs @ Ys ) ) ) ) ).
% Nil_in_shufflesI
thf(fact_5762_distinct_Osimps_I1_J,axiom,
! [A: $tType] : ( distinct @ A @ ( nil @ A ) ) ).
% distinct.simps(1)
thf(fact_5763_shuffles_Osimps_I1_J,axiom,
! [A: $tType,Ys: list @ A] :
( ( shuffles @ A @ ( nil @ A ) @ Ys )
= ( insert @ ( list @ A ) @ Ys @ ( bot_bot @ ( set @ ( list @ A ) ) ) ) ) ).
% shuffles.simps(1)
thf(fact_5764_shuffles_Osimps_I2_J,axiom,
! [A: $tType,Xs: list @ A] :
( ( shuffles @ A @ Xs @ ( nil @ A ) )
= ( insert @ ( list @ A ) @ Xs @ ( bot_bot @ ( set @ ( list @ A ) ) ) ) ) ).
% shuffles.simps(2)
thf(fact_5765_removeAll_Osimps_I1_J,axiom,
! [A: $tType,X: A] :
( ( removeAll @ A @ X @ ( nil @ A ) )
= ( nil @ A ) ) ).
% removeAll.simps(1)
thf(fact_5766_rotate1_Osimps_I1_J,axiom,
! [A: $tType] :
( ( rotate1 @ A @ ( nil @ A ) )
= ( nil @ A ) ) ).
% rotate1.simps(1)
thf(fact_5767_list_Osimps_I8_J,axiom,
! [A: $tType,B: $tType,F3: A > B] :
( ( map @ A @ B @ F3 @ ( nil @ A ) )
= ( nil @ B ) ) ).
% list.simps(8)
thf(fact_5768_product_Osimps_I1_J,axiom,
! [B: $tType,A: $tType,Uu: list @ B] :
( ( product @ A @ B @ ( nil @ A ) @ Uu )
= ( nil @ ( product_prod @ A @ B ) ) ) ).
% product.simps(1)
thf(fact_5769_take__Nil,axiom,
! [A: $tType,N: nat] :
( ( take @ A @ N @ ( nil @ A ) )
= ( nil @ A ) ) ).
% take_Nil
thf(fact_5770_concat_Osimps_I1_J,axiom,
! [A: $tType] :
( ( concat @ A @ ( nil @ ( list @ A ) ) )
= ( nil @ A ) ) ).
% concat.simps(1)
thf(fact_5771_list__update_Osimps_I1_J,axiom,
! [A: $tType,I2: nat,V2: A] :
( ( list_update @ A @ ( nil @ A ) @ I2 @ V2 )
= ( nil @ A ) ) ).
% list_update.simps(1)
thf(fact_5772_list__update__code_I1_J,axiom,
! [A: $tType,I2: nat,Y2: A] :
( ( list_update @ A @ ( nil @ A ) @ I2 @ Y2 )
= ( nil @ A ) ) ).
% list_update_code(1)
thf(fact_5773_sorted__wrt_Osimps_I1_J,axiom,
! [A: $tType,P3: A > A > $o] : ( sorted_wrt @ A @ P3 @ ( nil @ A ) ) ).
% sorted_wrt.simps(1)
thf(fact_5774_remove1_Osimps_I1_J,axiom,
! [A: $tType,X: A] :
( ( remove1 @ A @ X @ ( nil @ A ) )
= ( nil @ A ) ) ).
% remove1.simps(1)
thf(fact_5775_insort__not__Nil,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F3: B > A,A2: B,Xs: list @ B] :
( ( linorder_insort_key @ B @ A @ F3 @ A2 @ Xs )
!= ( nil @ B ) ) ) ).
% insort_not_Nil
thf(fact_5776_empty__set,axiom,
! [A: $tType] :
( ( bot_bot @ ( set @ A ) )
= ( set2 @ A @ ( nil @ A ) ) ) ).
% empty_set
thf(fact_5777_list_Osize_I3_J,axiom,
! [A: $tType] :
( ( size_size @ ( list @ A ) @ ( nil @ A ) )
= ( zero_zero @ nat ) ) ).
% list.size(3)
thf(fact_5778_sorted0,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( nil @ A ) ) ) ).
% sorted0
thf(fact_5779_strict__sorted__simps_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( sorted_wrt @ A @ ( ord_less @ A ) @ ( nil @ A ) ) ) ).
% strict_sorted_simps(1)
thf(fact_5780_take__0,axiom,
! [A: $tType,Xs: list @ A] :
( ( take @ A @ ( zero_zero @ nat ) @ Xs )
= ( nil @ A ) ) ).
% take_0
thf(fact_5781_replicate__0,axiom,
! [A: $tType,X: A] :
( ( replicate @ A @ ( zero_zero @ nat ) @ X )
= ( nil @ A ) ) ).
% replicate_0
thf(fact_5782_upt__0,axiom,
! [I2: nat] :
( ( upt @ I2 @ ( zero_zero @ nat ) )
= ( nil @ nat ) ) ).
% upt_0
thf(fact_5783_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_5784_count__list_Osimps_I1_J,axiom,
! [A: $tType,Y2: A] :
( ( count_list @ A @ ( nil @ A ) @ Y2 )
= ( zero_zero @ nat ) ) ).
% count_list.simps(1)
thf(fact_5785_folding__insort__key_Osorted__key__list__of__set__empty,axiom,
! [A: $tType,B: $tType,Less_eq: A > A > $o,Less: A > A > $o,S3: set @ B,F3: B > A] :
( ( folding_insort_key @ A @ B @ Less_eq @ Less @ S3 @ F3 )
=> ( ( sorted8670434370408473282of_set @ A @ B @ Less_eq @ F3 @ ( bot_bot @ ( set @ B ) ) )
= ( nil @ B ) ) ) ).
% folding_insort_key.sorted_key_list_of_set_empty
thf(fact_5786_sum__list__strict__mono,axiom,
! [B: $tType,A: $tType] :
( ( ( monoid_add @ B )
& ( strict9044650504122735259up_add @ B ) )
=> ! [Xs: list @ A,F3: A > B,G: A > B] :
( ( Xs
!= ( nil @ A ) )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Xs ) )
=> ( ord_less @ B @ ( F3 @ X5 ) @ ( G @ X5 ) ) )
=> ( ord_less @ B @ ( groups8242544230860333062m_list @ B @ ( map @ A @ B @ F3 @ Xs ) ) @ ( groups8242544230860333062m_list @ B @ ( map @ A @ B @ G @ Xs ) ) ) ) ) ) ).
% sum_list_strict_mono
thf(fact_5787_Pow__set_I1_J,axiom,
! [A: $tType] :
( ( pow2 @ A @ ( set2 @ A @ ( nil @ A ) ) )
= ( insert @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ ( bot_bot @ ( set @ ( set @ A ) ) ) ) ) ).
% Pow_set(1)
thf(fact_5788_folding__insort__key_Osorted__key__list__of__set__eq__Nil__iff,axiom,
! [A: $tType,B: $tType,Less_eq: A > A > $o,Less: A > A > $o,S3: set @ B,F3: B > A,A3: set @ B] :
( ( folding_insort_key @ A @ B @ Less_eq @ Less @ S3 @ F3 )
=> ( ( ord_less_eq @ ( set @ B ) @ A3 @ S3 )
=> ( ( finite_finite2 @ B @ A3 )
=> ( ( ( sorted8670434370408473282of_set @ A @ B @ Less_eq @ F3 @ A3 )
= ( nil @ B ) )
= ( A3
= ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ).
% folding_insort_key.sorted_key_list_of_set_eq_Nil_iff
thf(fact_5789_of__int_Orep__eq,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( ( ring_1_of_int @ A )
= ( ^ [X4: int] :
( product_case_prod @ nat @ nat @ A
@ ^ [I: nat,J4: nat] : ( minus_minus @ A @ ( semiring_1_of_nat @ A @ I ) @ ( semiring_1_of_nat @ A @ J4 ) )
@ ( rep_Integ @ X4 ) ) ) ) ) ).
% of_int.rep_eq
thf(fact_5790_transpose__rectangle,axiom,
! [A: $tType,Xs: list @ ( list @ A ),N: nat] :
( ( ( Xs
= ( nil @ ( list @ A ) ) )
=> ( N
= ( zero_zero @ nat ) ) )
=> ( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ ( list @ A ) ) @ Xs ) )
=> ( ( size_size @ ( list @ A ) @ ( nth @ ( list @ A ) @ Xs @ I3 ) )
= N ) )
=> ( ( transpose @ A @ Xs )
= ( map @ nat @ ( list @ A )
@ ^ [I: nat] :
( map @ nat @ A
@ ^ [J4: nat] : ( nth @ A @ ( nth @ ( list @ A ) @ Xs @ J4 ) @ I )
@ ( upt @ ( zero_zero @ nat ) @ ( size_size @ ( list @ ( list @ A ) ) @ Xs ) ) )
@ ( upt @ ( zero_zero @ nat ) @ N ) ) ) ) ) ).
% transpose_rectangle
thf(fact_5791_lex__prod__def,axiom,
! [B: $tType,A: $tType] :
( ( lex_prod @ A @ B )
= ( ^ [Ra: set @ ( product_prod @ A @ A ),Rb: set @ ( product_prod @ B @ B )] :
( collect @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) )
@ ( product_case_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ $o
@ ( product_case_prod @ A @ B @ ( ( product_prod @ A @ B ) > $o )
@ ^ [A5: A,B4: B] :
( product_case_prod @ A @ B @ $o
@ ^ [A9: A,B12: B] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A5 @ A9 ) @ Ra )
| ( ( A5 = A9 )
& ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ B4 @ B12 ) @ Rb ) ) ) ) ) ) ) ) ) ).
% lex_prod_def
thf(fact_5792_distinct__product__lists,axiom,
! [A: $tType,Xss: list @ ( list @ A )] :
( ! [X5: list @ A] :
( ( member @ ( list @ A ) @ X5 @ ( set2 @ ( list @ A ) @ Xss ) )
=> ( distinct @ A @ X5 ) )
=> ( distinct @ ( list @ A ) @ ( product_lists @ A @ Xss ) ) ) ).
% distinct_product_lists
thf(fact_5793_transpose_Osimps_I1_J,axiom,
! [A: $tType] :
( ( transpose @ A @ ( nil @ ( list @ A ) ) )
= ( nil @ ( list @ A ) ) ) ).
% transpose.simps(1)
thf(fact_5794_transpose__empty,axiom,
! [A: $tType,Xs: list @ ( list @ A )] :
( ( ( transpose @ A @ Xs )
= ( nil @ ( list @ A ) ) )
= ( ! [X4: list @ A] :
( ( member @ ( list @ A ) @ X4 @ ( set2 @ ( list @ A ) @ Xs ) )
=> ( X4
= ( nil @ A ) ) ) ) ) ).
% transpose_empty
thf(fact_5795_transpose__map__map,axiom,
! [A: $tType,B: $tType,F3: B > A,Xs: list @ ( list @ B )] :
( ( transpose @ A @ ( map @ ( list @ B ) @ ( list @ A ) @ ( map @ B @ A @ F3 ) @ Xs ) )
= ( map @ ( list @ B ) @ ( list @ A ) @ ( map @ B @ A @ F3 ) @ ( transpose @ B @ Xs ) ) ) ).
% transpose_map_map
thf(fact_5796_in__set__product__lists__length,axiom,
! [A: $tType,Xs: list @ A,Xss: list @ ( list @ A )] :
( ( member @ ( list @ A ) @ Xs @ ( set2 @ ( list @ A ) @ ( product_lists @ A @ Xss ) ) )
=> ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ ( list @ A ) ) @ Xss ) ) ) ).
% in_set_product_lists_length
thf(fact_5797_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_5798_listset_Osimps_I1_J,axiom,
! [A: $tType] :
( ( listset @ A @ ( nil @ ( set @ A ) ) )
= ( insert @ ( list @ A ) @ ( nil @ A ) @ ( bot_bot @ ( set @ ( list @ A ) ) ) ) ) ).
% listset.simps(1)
thf(fact_5799_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_5800_rev__rev__ident,axiom,
! [A: $tType,Xs: list @ A] :
( ( rev @ A @ ( rev @ A @ Xs ) )
= Xs ) ).
% rev_rev_ident
thf(fact_5801_rev__is__rev__conv,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( rev @ A @ Xs )
= ( rev @ A @ Ys ) )
= ( Xs = Ys ) ) ).
% rev_is_rev_conv
thf(fact_5802_Nil__is__rev__conv,axiom,
! [A: $tType,Xs: list @ A] :
( ( ( nil @ A )
= ( rev @ A @ Xs ) )
= ( Xs
= ( nil @ A ) ) ) ).
% Nil_is_rev_conv
thf(fact_5803_rev__is__Nil__conv,axiom,
! [A: $tType,Xs: list @ A] :
( ( ( rev @ A @ Xs )
= ( nil @ A ) )
= ( Xs
= ( nil @ A ) ) ) ).
% rev_is_Nil_conv
thf(fact_5804_set__rev,axiom,
! [A: $tType,Xs: list @ A] :
( ( set2 @ A @ ( rev @ A @ Xs ) )
= ( set2 @ A @ Xs ) ) ).
% set_rev
thf(fact_5805_length__rev,axiom,
! [A: $tType,Xs: list @ A] :
( ( size_size @ ( list @ A ) @ ( rev @ A @ Xs ) )
= ( size_size @ ( list @ A ) @ Xs ) ) ).
% length_rev
thf(fact_5806_distinct__rev,axiom,
! [A: $tType,Xs: list @ A] :
( ( distinct @ A @ ( rev @ A @ Xs ) )
= ( distinct @ A @ Xs ) ) ).
% distinct_rev
thf(fact_5807_rev__replicate,axiom,
! [A: $tType,N: nat,X: A] :
( ( rev @ A @ ( replicate @ A @ N @ X ) )
= ( replicate @ A @ N @ X ) ) ).
% rev_replicate
thf(fact_5808_foldr__replicate,axiom,
! [A: $tType,B: $tType,F3: B > A > A,N: nat,X: B] :
( ( foldr @ B @ A @ F3 @ ( replicate @ B @ N @ X ) )
= ( compow @ ( A > A ) @ N @ ( F3 @ X ) ) ) ).
% foldr_replicate
thf(fact_5809_rev_Osimps_I1_J,axiom,
! [A: $tType] :
( ( rev @ A @ ( nil @ A ) )
= ( nil @ A ) ) ).
% rev.simps(1)
thf(fact_5810_rev__concat,axiom,
! [A: $tType,Xs: list @ ( list @ A )] :
( ( rev @ A @ ( concat @ A @ Xs ) )
= ( concat @ A @ ( map @ ( list @ A ) @ ( list @ A ) @ ( rev @ A ) @ ( rev @ ( list @ A ) @ Xs ) ) ) ) ).
% rev_concat
thf(fact_5811_rev__map,axiom,
! [A: $tType,B: $tType,F3: B > A,Xs: list @ B] :
( ( rev @ A @ ( map @ B @ A @ F3 @ Xs ) )
= ( map @ B @ A @ F3 @ ( rev @ B @ Xs ) ) ) ).
% rev_map
thf(fact_5812_rev__swap,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( rev @ A @ Xs )
= Ys )
= ( Xs
= ( rev @ A @ Ys ) ) ) ).
% rev_swap
thf(fact_5813_sorted__wrt__rev,axiom,
! [A: $tType,P3: A > A > $o,Xs: list @ A] :
( ( sorted_wrt @ A @ P3 @ ( rev @ A @ Xs ) )
= ( sorted_wrt @ A
@ ^ [X4: A,Y: A] : ( P3 @ Y @ X4 )
@ Xs ) ) ).
% sorted_wrt_rev
thf(fact_5814_foldr__cong,axiom,
! [B: $tType,A: $tType,A2: A,B2: A,L: list @ B,K2: list @ B,F3: B > A > A,G: B > A > A] :
( ( A2 = B2 )
=> ( ( L = K2 )
=> ( ! [A4: A,X5: B] :
( ( member @ B @ X5 @ ( set2 @ B @ L ) )
=> ( ( F3 @ X5 @ A4 )
= ( G @ X5 @ A4 ) ) )
=> ( ( foldr @ B @ A @ F3 @ L @ A2 )
= ( foldr @ B @ A @ G @ K2 @ B2 ) ) ) ) ) ).
% foldr_cong
thf(fact_5815_foldr__map,axiom,
! [C: $tType,B: $tType,A: $tType,G: B > A > A,F3: C > B,Xs: list @ C,A2: A] :
( ( foldr @ B @ A @ G @ ( map @ C @ B @ F3 @ Xs ) @ A2 )
= ( foldr @ C @ A @ ( comp @ B @ ( A > A ) @ C @ G @ F3 ) @ Xs @ A2 ) ) ).
% foldr_map
thf(fact_5816_foldr__max__sorted,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,Y2: A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( rev @ A @ Xs ) )
=> ( ( ( Xs
= ( nil @ A ) )
=> ( ( foldr @ A @ A @ ( ord_max @ A ) @ Xs @ Y2 )
= Y2 ) )
& ( ( Xs
!= ( nil @ A ) )
=> ( ( foldr @ A @ A @ ( ord_max @ A ) @ Xs @ Y2 )
= ( ord_max @ A @ ( nth @ A @ Xs @ ( zero_zero @ nat ) ) @ Y2 ) ) ) ) ) ) ).
% foldr_max_sorted
thf(fact_5817_sum__list_Oeq__foldr,axiom,
! [A: $tType] :
( ( monoid_add @ A )
=> ( ( groups8242544230860333062m_list @ A )
= ( ^ [Xs2: list @ A] : ( foldr @ A @ A @ ( plus_plus @ A ) @ Xs2 @ ( zero_zero @ A ) ) ) ) ) ).
% sum_list.eq_foldr
thf(fact_5818_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_5819_rev__update,axiom,
! [A: $tType,K2: nat,Xs: list @ A,Y2: A] :
( ( ord_less @ nat @ K2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( rev @ A @ ( list_update @ A @ Xs @ K2 @ Y2 ) )
= ( list_update @ A @ ( rev @ A @ Xs ) @ ( minus_minus @ nat @ ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ K2 ) @ ( one_one @ nat ) ) @ Y2 ) ) ) ).
% rev_update
thf(fact_5820_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_5821_sorted__rev__iff__nth__Suc,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( rev @ A @ Xs ) )
= ( ! [I: nat] :
( ( ord_less @ nat @ ( suc @ I ) @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ord_less_eq @ A @ ( nth @ A @ Xs @ ( suc @ I ) ) @ ( nth @ A @ Xs @ I ) ) ) ) ) ) ).
% sorted_rev_iff_nth_Suc
thf(fact_5822_horner__sum__foldr,axiom,
! [A: $tType,B: $tType] :
( ( comm_semiring_0 @ A )
=> ( ( groups4207007520872428315er_sum @ B @ A )
= ( ^ [F5: B > A,A5: A,Xs2: list @ B] :
( foldr @ B @ A
@ ^ [X4: B,B4: A] : ( plus_plus @ A @ ( F5 @ X4 ) @ ( times_times @ A @ A5 @ B4 ) )
@ Xs2
@ ( zero_zero @ A ) ) ) ) ) ).
% horner_sum_foldr
thf(fact_5823_sorted__rev__iff__nth__mono,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( rev @ A @ Xs ) )
= ( ! [I: nat,J4: nat] :
( ( ord_less_eq @ nat @ I @ J4 )
=> ( ( ord_less @ nat @ J4 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ord_less_eq @ A @ ( nth @ A @ Xs @ J4 ) @ ( nth @ A @ Xs @ I ) ) ) ) ) ) ) ).
% sorted_rev_iff_nth_mono
thf(fact_5824_sorted__rev__nth__mono,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,I2: nat,J3: nat] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( rev @ A @ Xs ) )
=> ( ( ord_less_eq @ nat @ I2 @ J3 )
=> ( ( ord_less @ nat @ J3 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ord_less_eq @ A @ ( nth @ A @ Xs @ J3 ) @ ( nth @ A @ Xs @ I2 ) ) ) ) ) ) ).
% sorted_rev_nth_mono
thf(fact_5825_nth__nth__transpose__sorted,axiom,
! [A: $tType,Xs: list @ ( list @ A ),I2: nat,J3: nat] :
( ( sorted_wrt @ nat @ ( ord_less_eq @ nat ) @ ( rev @ nat @ ( map @ ( list @ A ) @ nat @ ( size_size @ ( list @ A ) ) @ Xs ) ) )
=> ( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ ( list @ A ) ) @ ( transpose @ A @ Xs ) ) )
=> ( ( ord_less @ nat @ J3
@ ( size_size @ ( list @ ( list @ A ) )
@ ( filter2 @ ( list @ A )
@ ^ [Ys3: list @ A] : ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Ys3 ) )
@ Xs ) ) )
=> ( ( nth @ A @ ( nth @ ( list @ A ) @ ( transpose @ A @ Xs ) @ I2 ) @ J3 )
= ( nth @ A @ ( nth @ ( list @ A ) @ Xs @ J3 ) @ I2 ) ) ) ) ) ).
% nth_nth_transpose_sorted
thf(fact_5826_transpose__column,axiom,
! [A: $tType,Xs: list @ ( list @ A ),I2: nat] :
( ( sorted_wrt @ nat @ ( ord_less_eq @ nat ) @ ( rev @ nat @ ( map @ ( list @ A ) @ nat @ ( size_size @ ( list @ A ) ) @ Xs ) ) )
=> ( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ ( list @ A ) ) @ Xs ) )
=> ( ( map @ ( list @ A ) @ A
@ ^ [Ys3: list @ A] : ( nth @ A @ Ys3 @ I2 )
@ ( filter2 @ ( list @ A )
@ ^ [Ys3: list @ A] : ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Ys3 ) )
@ ( transpose @ A @ Xs ) ) )
= ( nth @ ( list @ A ) @ Xs @ I2 ) ) ) ) ).
% transpose_column
thf(fact_5827_transpose__column__length,axiom,
! [A: $tType,Xs: list @ ( list @ A ),I2: nat] :
( ( sorted_wrt @ nat @ ( ord_less_eq @ nat ) @ ( rev @ nat @ ( map @ ( list @ A ) @ nat @ ( size_size @ ( list @ A ) ) @ Xs ) ) )
=> ( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ ( list @ A ) ) @ Xs ) )
=> ( ( size_size @ ( list @ ( list @ A ) )
@ ( filter2 @ ( list @ A )
@ ^ [Ys3: list @ A] : ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Ys3 ) )
@ ( transpose @ A @ Xs ) ) )
= ( size_size @ ( list @ A ) @ ( nth @ ( list @ A ) @ Xs @ I2 ) ) ) ) ) ).
% transpose_column_length
thf(fact_5828_filter__filter,axiom,
! [A: $tType,P3: A > $o,Q2: A > $o,Xs: list @ A] :
( ( filter2 @ A @ P3 @ ( filter2 @ A @ Q2 @ Xs ) )
= ( filter2 @ A
@ ^ [X4: A] :
( ( Q2 @ X4 )
& ( P3 @ X4 ) )
@ Xs ) ) ).
% filter_filter
thf(fact_5829_filter__True,axiom,
! [A: $tType,Xs: list @ A,P3: A > $o] :
( ! [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Xs ) )
=> ( P3 @ X5 ) )
=> ( ( filter2 @ A @ P3 @ Xs )
= Xs ) ) ).
% filter_True
thf(fact_5830_remove1__filter__not,axiom,
! [A: $tType,P3: A > $o,X: A,Xs: list @ A] :
( ~ ( P3 @ X )
=> ( ( remove1 @ A @ X @ ( filter2 @ A @ P3 @ Xs ) )
= ( filter2 @ A @ P3 @ Xs ) ) ) ).
% remove1_filter_not
thf(fact_5831_removeAll__filter__not,axiom,
! [A: $tType,P3: A > $o,X: A,Xs: list @ A] :
( ~ ( P3 @ X )
=> ( ( removeAll @ A @ X @ ( filter2 @ A @ P3 @ Xs ) )
= ( filter2 @ A @ P3 @ Xs ) ) ) ).
% removeAll_filter_not
thf(fact_5832_set__filter,axiom,
! [A: $tType,P3: A > $o,Xs: list @ A] :
( ( set2 @ A @ ( filter2 @ A @ P3 @ Xs ) )
= ( collect @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
& ( P3 @ X4 ) ) ) ) ).
% set_filter
thf(fact_5833_filter__False,axiom,
! [A: $tType,Xs: list @ A,P3: A > $o] :
( ! [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Xs ) )
=> ~ ( P3 @ X5 ) )
=> ( ( filter2 @ A @ P3 @ Xs )
= ( nil @ A ) ) ) ).
% filter_False
thf(fact_5834_length__concat__rev,axiom,
! [A: $tType,Xs: list @ ( list @ A )] :
( ( size_size @ ( list @ A ) @ ( concat @ A @ ( rev @ ( list @ A ) @ Xs ) ) )
= ( size_size @ ( list @ A ) @ ( concat @ A @ Xs ) ) ) ).
% length_concat_rev
thf(fact_5835_length__filter__map,axiom,
! [A: $tType,B: $tType,P3: A > $o,F3: B > A,Xs: list @ B] :
( ( size_size @ ( list @ A ) @ ( filter2 @ A @ P3 @ ( map @ B @ A @ F3 @ Xs ) ) )
= ( size_size @ ( list @ B ) @ ( filter2 @ B @ ( comp @ A @ $o @ B @ P3 @ F3 ) @ Xs ) ) ) ).
% length_filter_map
thf(fact_5836_filter__replicate,axiom,
! [A: $tType,P3: A > $o,X: A,N: nat] :
( ( ( P3 @ X )
=> ( ( filter2 @ A @ P3 @ ( replicate @ A @ N @ X ) )
= ( replicate @ A @ N @ X ) ) )
& ( ~ ( P3 @ X )
=> ( ( filter2 @ A @ P3 @ ( replicate @ A @ N @ X ) )
= ( nil @ A ) ) ) ) ).
% filter_replicate
thf(fact_5837_filter_Osimps_I1_J,axiom,
! [A: $tType,P3: A > $o] :
( ( filter2 @ A @ P3 @ ( nil @ A ) )
= ( nil @ A ) ) ).
% filter.simps(1)
thf(fact_5838_filter__empty__conv,axiom,
! [A: $tType,P3: A > $o,Xs: list @ A] :
( ( ( filter2 @ A @ P3 @ Xs )
= ( nil @ A ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
=> ~ ( P3 @ X4 ) ) ) ) ).
% filter_empty_conv
thf(fact_5839_empty__filter__conv,axiom,
! [A: $tType,P3: A > $o,Xs: list @ A] :
( ( ( nil @ A )
= ( filter2 @ A @ P3 @ Xs ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
=> ~ ( P3 @ X4 ) ) ) ) ).
% empty_filter_conv
thf(fact_5840_rev__filter,axiom,
! [A: $tType,P3: A > $o,Xs: list @ A] :
( ( rev @ A @ ( filter2 @ A @ P3 @ Xs ) )
= ( filter2 @ A @ P3 @ ( rev @ A @ Xs ) ) ) ).
% rev_filter
thf(fact_5841_removeAll__filter__not__eq,axiom,
! [A: $tType] :
( ( removeAll @ A )
= ( ^ [X4: A] :
( filter2 @ A
@ ^ [Y: A] : X4 != Y ) ) ) ).
% removeAll_filter_not_eq
thf(fact_5842_filter__insort__triv,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [P3: B > $o,X: B,F3: B > A,Xs: list @ B] :
( ~ ( P3 @ X )
=> ( ( filter2 @ B @ P3 @ ( linorder_insort_key @ B @ A @ F3 @ X @ Xs ) )
= ( filter2 @ B @ P3 @ Xs ) ) ) ) ).
% filter_insort_triv
thf(fact_5843_filter__is__subset,axiom,
! [A: $tType,P3: A > $o,Xs: list @ A] : ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ ( filter2 @ A @ P3 @ Xs ) ) @ ( set2 @ A @ Xs ) ) ).
% filter_is_subset
thf(fact_5844_filter__id__conv,axiom,
! [A: $tType,P3: A > $o,Xs: list @ A] :
( ( ( filter2 @ A @ P3 @ Xs )
= Xs )
= ( ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
=> ( P3 @ X4 ) ) ) ) ).
% filter_id_conv
thf(fact_5845_filter__cong,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,P3: A > $o,Q2: A > $o] :
( ( Xs = Ys )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Ys ) )
=> ( ( P3 @ X5 )
= ( Q2 @ X5 ) ) )
=> ( ( filter2 @ A @ P3 @ Xs )
= ( filter2 @ A @ Q2 @ Ys ) ) ) ) ).
% filter_cong
thf(fact_5846_filter__remove1,axiom,
! [A: $tType,Q2: A > $o,X: A,Xs: list @ A] :
( ( filter2 @ A @ Q2 @ ( remove1 @ A @ X @ Xs ) )
= ( remove1 @ A @ X @ ( filter2 @ A @ Q2 @ Xs ) ) ) ).
% filter_remove1
thf(fact_5847_sorted__wrt__filter,axiom,
! [A: $tType,F3: A > A > $o,Xs: list @ A,P3: A > $o] :
( ( sorted_wrt @ A @ F3 @ Xs )
=> ( sorted_wrt @ A @ F3 @ ( filter2 @ A @ P3 @ Xs ) ) ) ).
% sorted_wrt_filter
thf(fact_5848_length__filter__le,axiom,
! [A: $tType,P3: A > $o,Xs: list @ A] : ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ ( filter2 @ A @ P3 @ Xs ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) ).
% length_filter_le
thf(fact_5849_distinct__filter,axiom,
! [A: $tType,Xs: list @ A,P3: A > $o] :
( ( distinct @ A @ Xs )
=> ( distinct @ A @ ( filter2 @ A @ P3 @ Xs ) ) ) ).
% distinct_filter
thf(fact_5850_partition__in__shuffles,axiom,
! [A: $tType,Xs: list @ A,P3: A > $o] :
( member @ ( list @ A ) @ Xs
@ ( shuffles @ A @ ( filter2 @ A @ P3 @ Xs )
@ ( filter2 @ A
@ ^ [X4: A] :
~ ( P3 @ X4 )
@ Xs ) ) ) ).
% partition_in_shuffles
thf(fact_5851_sorted__same,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [G: ( list @ A ) > A,Xs: list @ A] :
( sorted_wrt @ A @ ( ord_less_eq @ A )
@ ( filter2 @ A
@ ^ [X4: A] :
( X4
= ( G @ Xs ) )
@ Xs ) ) ) ).
% sorted_same
thf(fact_5852_inter__set__filter,axiom,
! [A: $tType,A3: set @ A,Xs: list @ A] :
( ( inf_inf @ ( set @ A ) @ A3 @ ( set2 @ A @ Xs ) )
= ( set2 @ A
@ ( filter2 @ A
@ ^ [X4: A] : ( member @ A @ X4 @ A3 )
@ Xs ) ) ) ).
% inter_set_filter
thf(fact_5853_filter__concat,axiom,
! [A: $tType,P5: A > $o,Xs: list @ ( list @ A )] :
( ( filter2 @ A @ P5 @ ( concat @ A @ Xs ) )
= ( concat @ A @ ( map @ ( list @ A ) @ ( list @ A ) @ ( filter2 @ A @ P5 ) @ Xs ) ) ) ).
% filter_concat
thf(fact_5854_sum__length__filter__compl,axiom,
! [A: $tType,P3: A > $o,Xs: list @ A] :
( ( plus_plus @ nat @ ( size_size @ ( list @ A ) @ ( filter2 @ A @ P3 @ Xs ) )
@ ( size_size @ ( list @ A )
@ ( filter2 @ A
@ ^ [X4: A] :
~ ( P3 @ X4 )
@ Xs ) ) )
= ( size_size @ ( list @ A ) @ Xs ) ) ).
% sum_length_filter_compl
thf(fact_5855_distinct__map__filter,axiom,
! [A: $tType,B: $tType,F3: B > A,Xs: list @ B,P3: B > $o] :
( ( distinct @ A @ ( map @ B @ A @ F3 @ Xs ) )
=> ( distinct @ A @ ( map @ B @ A @ F3 @ ( filter2 @ B @ P3 @ Xs ) ) ) ) ).
% distinct_map_filter
thf(fact_5856_filter__map,axiom,
! [A: $tType,B: $tType,P3: A > $o,F3: B > A,Xs: list @ B] :
( ( filter2 @ A @ P3 @ ( map @ B @ A @ F3 @ Xs ) )
= ( map @ B @ A @ F3 @ ( filter2 @ B @ ( comp @ A @ $o @ B @ P3 @ F3 ) @ Xs ) ) ) ).
% filter_map
thf(fact_5857_replicate__length__filter,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( replicate @ A
@ ( size_size @ ( list @ A )
@ ( filter2 @ A
@ ( ^ [Y5: A,Z2: A] : Y5 = Z2
@ X )
@ Xs ) )
@ X )
= ( filter2 @ A
@ ( ^ [Y5: A,Z2: A] : Y5 = Z2
@ X )
@ Xs ) ) ).
% replicate_length_filter
thf(fact_5858_length__filter__less,axiom,
! [A: $tType,X: A,Xs: list @ A,P3: A > $o] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ~ ( P3 @ X )
=> ( ord_less @ nat @ ( size_size @ ( list @ A ) @ ( filter2 @ A @ P3 @ Xs ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ).
% length_filter_less
thf(fact_5859_sorted__filter,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F3: B > A,Xs: list @ B,P3: B > $o] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F3 @ Xs ) )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F3 @ ( filter2 @ B @ P3 @ Xs ) ) ) ) ) ).
% sorted_filter
thf(fact_5860_sorted__map__same,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F3: B > A,G: ( list @ B ) > A,Xs: list @ B] :
( sorted_wrt @ A @ ( ord_less_eq @ A )
@ ( map @ B @ A @ F3
@ ( filter2 @ B
@ ^ [X4: B] :
( ( F3 @ X4 )
= ( G @ Xs ) )
@ Xs ) ) ) ) ).
% sorted_map_same
thf(fact_5861_sum__list__map__filter_H,axiom,
! [A: $tType,B: $tType] :
( ( monoid_add @ A )
=> ! [F3: B > A,P3: B > $o,Xs: list @ B] :
( ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ F3 @ ( filter2 @ B @ P3 @ Xs ) ) )
= ( groups8242544230860333062m_list @ A
@ ( map @ B @ A
@ ^ [X4: B] : ( if @ A @ ( P3 @ X4 ) @ ( F3 @ X4 ) @ ( zero_zero @ A ) )
@ Xs ) ) ) ) ).
% sum_list_map_filter'
thf(fact_5862_sum__list__filter__le__nat,axiom,
! [A: $tType,F3: A > nat,P3: A > $o,Xs: list @ A] : ( ord_less_eq @ nat @ ( groups8242544230860333062m_list @ nat @ ( map @ A @ nat @ F3 @ ( filter2 @ A @ P3 @ Xs ) ) ) @ ( groups8242544230860333062m_list @ nat @ ( map @ A @ nat @ F3 @ Xs ) ) ) ).
% sum_list_filter_le_nat
thf(fact_5863_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
@ ^ [Xs2: list @ B] : ( ord_max @ nat @ ( size_size @ ( list @ B ) @ Xs2 ) )
@ Xss
@ ( zero_zero @ nat ) ) )
= ( suc
@ ( ord_max @ nat @ ( size_size @ ( list @ A ) @ Xs )
@ ( foldr @ ( list @ B ) @ nat
@ ^ [X4: list @ B] : ( ord_max @ nat @ ( minus_minus @ nat @ ( size_size @ ( list @ B ) @ X4 ) @ ( suc @ ( zero_zero @ nat ) ) ) )
@ ( filter2 @ ( list @ B )
@ ^ [Ys3: list @ B] :
( Ys3
!= ( nil @ B ) )
@ Xss )
@ ( zero_zero @ nat ) ) ) ) ) ).
% transpose_aux_max
thf(fact_5864_transpose__max__length,axiom,
! [A: $tType,Xs: list @ ( list @ A )] :
( ( foldr @ ( list @ A ) @ nat
@ ^ [Xs2: list @ A] : ( ord_max @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) )
@ ( transpose @ A @ Xs )
@ ( zero_zero @ nat ) )
= ( size_size @ ( list @ ( list @ A ) )
@ ( filter2 @ ( list @ A )
@ ^ [X4: list @ A] :
( X4
!= ( nil @ A ) )
@ Xs ) ) ) ).
% transpose_max_length
thf(fact_5865_sum__list__map__filter,axiom,
! [A: $tType,B: $tType] :
( ( monoid_add @ A )
=> ! [Xs: list @ B,P3: B > $o,F3: B > A] :
( ! [X5: B] :
( ( member @ B @ X5 @ ( set2 @ B @ Xs ) )
=> ( ~ ( P3 @ X5 )
=> ( ( F3 @ X5 )
= ( zero_zero @ A ) ) ) )
=> ( ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ F3 @ ( filter2 @ B @ P3 @ Xs ) ) )
= ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ F3 @ Xs ) ) ) ) ) ).
% sum_list_map_filter
thf(fact_5866_filter__insort,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F3: B > A,Xs: list @ B,P3: B > $o,X: B] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F3 @ Xs ) )
=> ( ( P3 @ X )
=> ( ( filter2 @ B @ P3 @ ( linorder_insort_key @ B @ A @ F3 @ X @ Xs ) )
= ( linorder_insort_key @ B @ A @ F3 @ X @ ( filter2 @ B @ P3 @ Xs ) ) ) ) ) ) ).
% filter_insort
thf(fact_5867_set__minus__filter__out,axiom,
! [A: $tType,Xs: list @ A,Y2: A] :
( ( minus_minus @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( insert @ A @ Y2 @ ( bot_bot @ ( set @ A ) ) ) )
= ( set2 @ A
@ ( filter2 @ A
@ ^ [X4: A] : X4 != Y2
@ Xs ) ) ) ).
% set_minus_filter_out
thf(fact_5868_filter__shuffles__disjoint2_I1_J,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Zs: list @ A] :
( ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ Ys ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( member @ ( list @ A ) @ Zs @ ( shuffles @ A @ Xs @ Ys ) )
=> ( ( filter2 @ A
@ ^ [X4: A] : ( member @ A @ X4 @ ( set2 @ A @ Ys ) )
@ Zs )
= Ys ) ) ) ).
% filter_shuffles_disjoint2(1)
thf(fact_5869_filter__shuffles__disjoint2_I2_J,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Zs: list @ A] :
( ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ Ys ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( member @ ( list @ A ) @ Zs @ ( shuffles @ A @ Xs @ Ys ) )
=> ( ( filter2 @ A
@ ^ [X4: A] :
~ ( member @ A @ X4 @ ( set2 @ A @ Ys ) )
@ Zs )
= Xs ) ) ) ).
% filter_shuffles_disjoint2(2)
thf(fact_5870_filter__shuffles__disjoint1_I1_J,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Zs: list @ A] :
( ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ Ys ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( member @ ( list @ A ) @ Zs @ ( shuffles @ A @ Xs @ Ys ) )
=> ( ( filter2 @ A
@ ^ [X4: A] : ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
@ Zs )
= Xs ) ) ) ).
% filter_shuffles_disjoint1(1)
thf(fact_5871_filter__shuffles__disjoint1_I2_J,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Zs: list @ A] :
( ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ Ys ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( member @ ( list @ A ) @ Zs @ ( shuffles @ A @ Xs @ Ys ) )
=> ( ( filter2 @ A
@ ^ [X4: A] :
~ ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
@ Zs )
= Ys ) ) ) ).
% filter_shuffles_disjoint1(2)
thf(fact_5872_length__filter__conv__card,axiom,
! [A: $tType,P5: A > $o,Xs: list @ A] :
( ( size_size @ ( list @ A ) @ ( filter2 @ A @ P5 @ Xs ) )
= ( finite_card @ nat
@ ( collect @ nat
@ ^ [I: nat] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
& ( P5 @ ( nth @ A @ Xs @ I ) ) ) ) ) ) ).
% length_filter_conv_card
thf(fact_5873_distinct__length__filter,axiom,
! [A: $tType,Xs: list @ A,P3: A > $o] :
( ( distinct @ A @ Xs )
=> ( ( size_size @ ( list @ A ) @ ( filter2 @ A @ P3 @ Xs ) )
= ( finite_card @ A @ ( inf_inf @ ( set @ A ) @ ( collect @ A @ P3 ) @ ( set2 @ A @ Xs ) ) ) ) ) ).
% distinct_length_filter
thf(fact_5874_length__transpose,axiom,
! [A: $tType,Xs: list @ ( list @ A )] :
( ( size_size @ ( list @ ( list @ A ) ) @ ( transpose @ A @ Xs ) )
= ( foldr @ ( list @ A ) @ nat
@ ^ [Xs2: list @ A] : ( ord_max @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) )
@ Xs
@ ( zero_zero @ nat ) ) ) ).
% length_transpose
thf(fact_5875_nth__transpose,axiom,
! [A: $tType,I2: nat,Xs: list @ ( list @ A )] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ ( list @ A ) ) @ ( transpose @ A @ Xs ) ) )
=> ( ( nth @ ( list @ A ) @ ( transpose @ A @ Xs ) @ I2 )
= ( map @ ( list @ A ) @ A
@ ^ [Xs2: list @ A] : ( nth @ A @ Xs2 @ I2 )
@ ( filter2 @ ( list @ A )
@ ^ [Ys3: list @ A] : ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Ys3 ) )
@ Xs ) ) ) ) ).
% nth_transpose
thf(fact_5876_map__filter__map__filter,axiom,
! [A: $tType,B: $tType,F3: B > A,P3: B > $o,Xs: list @ B] :
( ( map @ B @ A @ F3 @ ( filter2 @ B @ P3 @ Xs ) )
= ( map_filter @ B @ A
@ ^ [X4: B] : ( if @ ( option @ A ) @ ( P3 @ X4 ) @ ( some @ A @ ( F3 @ X4 ) ) @ ( none @ A ) )
@ Xs ) ) ).
% map_filter_map_filter
thf(fact_5877_transpose__transpose,axiom,
! [A: $tType,Xs: list @ ( list @ A )] :
( ( sorted_wrt @ nat @ ( ord_less_eq @ nat ) @ ( rev @ nat @ ( map @ ( list @ A ) @ nat @ ( size_size @ ( list @ A ) ) @ Xs ) ) )
=> ( ( transpose @ A @ ( transpose @ A @ Xs ) )
= ( takeWhile @ ( list @ A )
@ ^ [X4: list @ A] :
( X4
!= ( nil @ A ) )
@ Xs ) ) ) ).
% transpose_transpose
thf(fact_5878_shuffles_Opsimps_I2_J,axiom,
! [A: $tType,Xs: list @ A] :
( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( shuffles_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ ( nil @ A ) ) )
=> ( ( shuffles @ A @ Xs @ ( nil @ A ) )
= ( insert @ ( list @ A ) @ Xs @ ( bot_bot @ ( set @ ( list @ A ) ) ) ) ) ) ).
% shuffles.psimps(2)
thf(fact_5879_takeWhile__idem,axiom,
! [A: $tType,P3: A > $o,Xs: list @ A] :
( ( takeWhile @ A @ P3 @ ( takeWhile @ A @ P3 @ Xs ) )
= ( takeWhile @ A @ P3 @ Xs ) ) ).
% takeWhile_idem
thf(fact_5880_takeWhile__eq__all__conv,axiom,
! [A: $tType,P3: A > $o,Xs: list @ A] :
( ( ( takeWhile @ A @ P3 @ Xs )
= Xs )
= ( ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
=> ( P3 @ X4 ) ) ) ) ).
% takeWhile_eq_all_conv
thf(fact_5881_takeWhile__replicate,axiom,
! [A: $tType,P3: A > $o,X: A,N: nat] :
( ( ( P3 @ X )
=> ( ( takeWhile @ A @ P3 @ ( replicate @ A @ N @ X ) )
= ( replicate @ A @ N @ X ) ) )
& ( ~ ( P3 @ X )
=> ( ( takeWhile @ A @ P3 @ ( replicate @ A @ N @ X ) )
= ( nil @ A ) ) ) ) ).
% takeWhile_replicate
thf(fact_5882_length__takeWhile__le,axiom,
! [A: $tType,P3: A > $o,Xs: list @ A] : ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P3 @ Xs ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) ).
% length_takeWhile_le
thf(fact_5883_distinct__takeWhile,axiom,
! [A: $tType,Xs: list @ A,P3: A > $o] :
( ( distinct @ A @ Xs )
=> ( distinct @ A @ ( takeWhile @ A @ P3 @ Xs ) ) ) ).
% distinct_takeWhile
thf(fact_5884_set__takeWhileD,axiom,
! [A: $tType,X: A,P3: A > $o,Xs: list @ A] :
( ( member @ A @ X @ ( set2 @ A @ ( takeWhile @ A @ P3 @ Xs ) ) )
=> ( ( member @ A @ X @ ( set2 @ A @ Xs ) )
& ( P3 @ X ) ) ) ).
% set_takeWhileD
thf(fact_5885_takeWhile__cong,axiom,
! [A: $tType,L: list @ A,K2: list @ A,P3: A > $o,Q2: A > $o] :
( ( L = K2 )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ L ) )
=> ( ( P3 @ X5 )
= ( Q2 @ X5 ) ) )
=> ( ( takeWhile @ A @ P3 @ L )
= ( takeWhile @ A @ Q2 @ K2 ) ) ) ) ).
% takeWhile_cong
thf(fact_5886_takeWhile_Osimps_I1_J,axiom,
! [A: $tType,P3: A > $o] :
( ( takeWhile @ A @ P3 @ ( nil @ A ) )
= ( nil @ A ) ) ).
% takeWhile.simps(1)
thf(fact_5887_sorted__takeWhile,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,P3: A > $o] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( takeWhile @ A @ P3 @ Xs ) ) ) ) ).
% sorted_takeWhile
thf(fact_5888_takeWhile__eq__take,axiom,
! [A: $tType] :
( ( takeWhile @ A )
= ( ^ [P2: A > $o,Xs2: list @ A] : ( take @ A @ ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P2 @ Xs2 ) ) @ Xs2 ) ) ) ).
% takeWhile_eq_take
thf(fact_5889_takeWhile__map,axiom,
! [A: $tType,B: $tType,P3: A > $o,F3: B > A,Xs: list @ B] :
( ( takeWhile @ A @ P3 @ ( map @ B @ A @ F3 @ Xs ) )
= ( map @ B @ A @ F3 @ ( takeWhile @ B @ ( comp @ A @ $o @ B @ P3 @ F3 ) @ Xs ) ) ) ).
% takeWhile_map
thf(fact_5890_map__filter__simps_I2_J,axiom,
! [B: $tType,A: $tType,F3: B > ( option @ A )] :
( ( map_filter @ B @ A @ F3 @ ( nil @ B ) )
= ( nil @ A ) ) ).
% map_filter_simps(2)
thf(fact_5891_nth__length__takeWhile,axiom,
! [A: $tType,P3: A > $o,Xs: list @ A] :
( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P3 @ Xs ) ) @ ( size_size @ ( list @ A ) @ Xs ) )
=> ~ ( P3 @ ( nth @ A @ Xs @ ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P3 @ Xs ) ) ) ) ) ).
% nth_length_takeWhile
thf(fact_5892_takeWhile__nth,axiom,
! [A: $tType,J3: nat,P3: A > $o,Xs: list @ A] :
( ( ord_less @ nat @ J3 @ ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P3 @ Xs ) ) )
=> ( ( nth @ A @ ( takeWhile @ A @ P3 @ Xs ) @ J3 )
= ( nth @ A @ Xs @ J3 ) ) ) ).
% takeWhile_nth
thf(fact_5893_length__takeWhile__less__P__nth,axiom,
! [A: $tType,J3: nat,P3: A > $o,Xs: list @ A] :
( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ J3 )
=> ( P3 @ ( nth @ A @ Xs @ I3 ) ) )
=> ( ( ord_less_eq @ nat @ J3 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ord_less_eq @ nat @ J3 @ ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P3 @ Xs ) ) ) ) ) ).
% length_takeWhile_less_P_nth
thf(fact_5894_takeWhile__eq__take__P__nth,axiom,
! [A: $tType,N: nat,Xs: list @ A,P3: A > $o] :
( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ N )
=> ( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( P3 @ ( nth @ A @ Xs @ I3 ) ) ) )
=> ( ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ~ ( P3 @ ( nth @ A @ Xs @ N ) ) )
=> ( ( takeWhile @ A @ P3 @ Xs )
= ( take @ A @ N @ Xs ) ) ) ) ).
% takeWhile_eq_take_P_nth
thf(fact_5895_filter__equals__takeWhile__sorted__rev,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F3: B > A,Xs: list @ B,T2: A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( rev @ A @ ( map @ B @ A @ F3 @ Xs ) ) )
=> ( ( filter2 @ B
@ ^ [X4: B] : ( ord_less @ A @ T2 @ ( F3 @ X4 ) )
@ Xs )
= ( takeWhile @ B
@ ^ [X4: B] : ( ord_less @ A @ T2 @ ( F3 @ X4 ) )
@ Xs ) ) ) ) ).
% filter_equals_takeWhile_sorted_rev
thf(fact_5896_shuffles_Opsimps_I1_J,axiom,
! [A: $tType,Ys: list @ A] :
( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( shuffles_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Ys ) )
=> ( ( shuffles @ A @ ( nil @ A ) @ Ys )
= ( insert @ ( list @ A ) @ Ys @ ( bot_bot @ ( set @ ( list @ A ) ) ) ) ) ) ).
% shuffles.psimps(1)
thf(fact_5897_same__fst__def,axiom,
! [B: $tType,A: $tType] :
( ( same_fst @ A @ B )
= ( ^ [P2: A > $o,R6: A > ( set @ ( product_prod @ B @ B ) )] :
( collect @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) )
@ ( product_case_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ $o
@ ( product_case_prod @ A @ B @ ( ( product_prod @ A @ B ) > $o )
@ ^ [X7: A,Y6: B] :
( product_case_prod @ A @ B @ $o
@ ^ [X4: A,Y: B] :
( ( X7 = X4 )
& ( P2 @ X4 )
& ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ Y6 @ Y ) @ ( R6 @ X4 ) ) ) ) ) ) ) ) ) ).
% same_fst_def
thf(fact_5898_lex__take__index,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( lex @ A @ R ) )
=> ~ ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Ys ) )
=> ( ( ( take @ A @ I3 @ Xs )
= ( take @ A @ I3 @ Ys ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( nth @ A @ Xs @ I3 ) @ ( nth @ A @ Ys @ I3 ) ) @ R ) ) ) ) ) ).
% lex_take_index
thf(fact_5899_insort__key__remove1,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [A2: B,Xs: list @ B,F3: B > A] :
( ( member @ B @ A2 @ ( set2 @ B @ Xs ) )
=> ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F3 @ Xs ) )
=> ( ( ( hd @ B
@ ( filter2 @ B
@ ^ [X4: B] :
( ( F3 @ A2 )
= ( F3 @ X4 ) )
@ Xs ) )
= A2 )
=> ( ( linorder_insort_key @ B @ A @ F3 @ A2 @ ( remove1 @ B @ A2 @ Xs ) )
= Xs ) ) ) ) ) ).
% insort_key_remove1
thf(fact_5900_hd__upt,axiom,
! [I2: nat,J3: nat] :
( ( ord_less @ nat @ I2 @ J3 )
=> ( ( hd @ nat @ ( upt @ I2 @ J3 ) )
= I2 ) ) ).
% hd_upt
thf(fact_5901_hd__replicate,axiom,
! [A: $tType,N: nat,X: A] :
( ( N
!= ( zero_zero @ nat ) )
=> ( ( hd @ A @ ( replicate @ A @ N @ X ) )
= X ) ) ).
% hd_replicate
thf(fact_5902_hd__take,axiom,
! [A: $tType,J3: nat,Xs: list @ A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ J3 )
=> ( ( hd @ A @ ( take @ A @ J3 @ Xs ) )
= ( hd @ A @ Xs ) ) ) ).
% hd_take
thf(fact_5903_hd__concat,axiom,
! [A: $tType,Xs: list @ ( list @ A )] :
( ( Xs
!= ( nil @ ( list @ A ) ) )
=> ( ( ( hd @ ( list @ A ) @ Xs )
!= ( nil @ A ) )
=> ( ( hd @ A @ ( concat @ A @ Xs ) )
= ( hd @ A @ ( hd @ ( list @ A ) @ Xs ) ) ) ) ) ).
% hd_concat
thf(fact_5904_list_Oset__sel_I1_J,axiom,
! [A: $tType,A2: list @ A] :
( ( A2
!= ( nil @ A ) )
=> ( member @ A @ ( hd @ A @ A2 ) @ ( set2 @ A @ A2 ) ) ) ).
% list.set_sel(1)
thf(fact_5905_hd__in__set,axiom,
! [A: $tType,Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( member @ A @ ( hd @ A @ Xs ) @ ( set2 @ A @ Xs ) ) ) ).
% hd_in_set
thf(fact_5906_list_Omap__sel_I1_J,axiom,
! [B: $tType,A: $tType,A2: list @ A,F3: A > B] :
( ( A2
!= ( nil @ A ) )
=> ( ( hd @ B @ ( map @ A @ B @ F3 @ A2 ) )
= ( F3 @ ( hd @ A @ A2 ) ) ) ) ).
% list.map_sel(1)
thf(fact_5907_hd__map,axiom,
! [B: $tType,A: $tType,Xs: list @ A,F3: A > B] :
( ( Xs
!= ( nil @ A ) )
=> ( ( hd @ B @ ( map @ A @ B @ F3 @ Xs ) )
= ( F3 @ ( hd @ A @ Xs ) ) ) ) ).
% hd_map
thf(fact_5908_takeWhile__eq__Nil__iff,axiom,
! [A: $tType,P3: A > $o,Xs: list @ A] :
( ( ( takeWhile @ A @ P3 @ Xs )
= ( nil @ A ) )
= ( ( Xs
= ( nil @ A ) )
| ~ ( P3 @ ( hd @ A @ Xs ) ) ) ) ).
% takeWhile_eq_Nil_iff
thf(fact_5909_Nil2__notin__lex,axiom,
! [A: $tType,Xs: list @ A,R: set @ ( product_prod @ A @ A )] :
~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ ( nil @ A ) ) @ ( lex @ A @ R ) ) ).
% Nil2_notin_lex
thf(fact_5910_Nil__notin__lex,axiom,
! [A: $tType,Ys: list @ A,R: set @ ( product_prod @ A @ A )] :
~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Ys ) @ ( lex @ A @ R ) ) ).
% Nil_notin_lex
thf(fact_5911_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_5912_Fpow__Pow__finite,axiom,
! [A: $tType] :
( ( finite_Fpow @ A )
= ( ^ [A7: set @ A] : ( inf_inf @ ( set @ ( set @ A ) ) @ ( pow2 @ A @ A7 ) @ ( collect @ ( set @ A ) @ ( finite_finite2 @ A ) ) ) ) ) ).
% Fpow_Pow_finite
thf(fact_5913_uminus__int__def,axiom,
( ( uminus_uminus @ int )
= ( map_fun @ int @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ int @ rep_Integ @ abs_Integ
@ ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [X4: nat,Y: nat] : ( product_Pair @ nat @ nat @ Y @ X4 ) ) ) ) ).
% uminus_int_def
thf(fact_5914_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_5915_bij__betw__Suc,axiom,
! [M7: set @ nat,N4: set @ nat] :
( ( bij_betw @ nat @ nat @ suc @ M7 @ N4 )
= ( ( image @ nat @ nat @ suc @ M7 )
= N4 ) ) ).
% bij_betw_Suc
thf(fact_5916_image__ident,axiom,
! [A: $tType,Y3: set @ A] :
( ( image @ A @ A
@ ^ [X4: A] : X4
@ Y3 )
= Y3 ) ).
% image_ident
thf(fact_5917_SUP__identity__eq,axiom,
! [A: $tType] :
( ( complete_Sup @ A )
=> ! [A3: set @ A] :
( ( complete_Sup_Sup @ A
@ ( image @ A @ A
@ ^ [X4: A] : X4
@ A3 ) )
= ( complete_Sup_Sup @ A @ A3 ) ) ) ).
% SUP_identity_eq
thf(fact_5918_SUP__apply,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( complete_Sup @ A )
=> ! [F3: C > B > A,A3: set @ C,X: B] :
( ( complete_Sup_Sup @ ( B > A ) @ ( image @ C @ ( B > A ) @ F3 @ A3 ) @ X )
= ( complete_Sup_Sup @ A
@ ( image @ C @ A
@ ^ [Y: C] : ( F3 @ Y @ X )
@ A3 ) ) ) ) ).
% SUP_apply
thf(fact_5919_INF__identity__eq,axiom,
! [A: $tType] :
( ( complete_Inf @ A )
=> ! [A3: set @ A] :
( ( complete_Inf_Inf @ A
@ ( image @ A @ A
@ ^ [X4: A] : X4
@ A3 ) )
= ( complete_Inf_Inf @ A @ A3 ) ) ) ).
% INF_identity_eq
thf(fact_5920_INF__apply,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( complete_Inf @ A )
=> ! [F3: C > B > A,A3: set @ C,X: B] :
( ( complete_Inf_Inf @ ( B > A ) @ ( image @ C @ ( B > A ) @ F3 @ A3 ) @ X )
= ( complete_Inf_Inf @ A
@ ( image @ C @ A
@ ^ [Y: C] : ( F3 @ Y @ X )
@ A3 ) ) ) ) ).
% INF_apply
thf(fact_5921_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_5922_image__add__atLeastAtMost,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [K2: A,I2: A,J3: A] :
( ( image @ A @ A @ ( plus_plus @ A @ K2 ) @ ( set_or1337092689740270186AtMost @ A @ I2 @ J3 ) )
= ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ I2 @ K2 ) @ ( plus_plus @ A @ J3 @ K2 ) ) ) ) ).
% image_add_atLeastAtMost
thf(fact_5923_image__add__atLeastLessThan,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [K2: A,I2: A,J3: A] :
( ( image @ A @ A @ ( plus_plus @ A @ K2 ) @ ( set_or7035219750837199246ssThan @ A @ I2 @ J3 ) )
= ( set_or7035219750837199246ssThan @ A @ ( plus_plus @ A @ I2 @ K2 ) @ ( plus_plus @ A @ J3 @ K2 ) ) ) ) ).
% image_add_atLeastLessThan
thf(fact_5924_image__add__atMost,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [C2: A,A2: A] :
( ( image @ A @ A @ ( plus_plus @ A @ C2 ) @ ( set_ord_atMost @ A @ A2 ) )
= ( set_ord_atMost @ A @ ( plus_plus @ A @ C2 @ A2 ) ) ) ) ).
% image_add_atMost
thf(fact_5925_bij__betw__add,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [A2: A,A3: set @ A,B5: set @ A] :
( ( bij_betw @ A @ A @ ( plus_plus @ A @ A2 ) @ A3 @ B5 )
= ( ( image @ A @ A @ ( plus_plus @ A @ A2 ) @ A3 )
= B5 ) ) ) ).
% bij_betw_add
thf(fact_5926_list_Oset__map,axiom,
! [B: $tType,A: $tType,F3: A > B,V2: list @ A] :
( ( set2 @ B @ ( map @ A @ B @ F3 @ V2 ) )
= ( image @ A @ B @ F3 @ ( set2 @ A @ V2 ) ) ) ).
% list.set_map
thf(fact_5927_image__add__greaterThanAtMost,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( image @ A @ A @ ( plus_plus @ A @ C2 ) @ ( set_or3652927894154168847AtMost @ A @ A2 @ B2 ) )
= ( set_or3652927894154168847AtMost @ A @ ( plus_plus @ A @ C2 @ A2 ) @ ( plus_plus @ A @ C2 @ B2 ) ) ) ) ).
% image_add_greaterThanAtMost
thf(fact_5928_SUP__bot__conv_I2_J,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [B5: B > A,A3: set @ B] :
( ( ( bot_bot @ A )
= ( complete_Sup_Sup @ A @ ( image @ B @ A @ B5 @ A3 ) ) )
= ( ! [X4: B] :
( ( member @ B @ X4 @ A3 )
=> ( ( B5 @ X4 )
= ( bot_bot @ A ) ) ) ) ) ) ).
% SUP_bot_conv(2)
thf(fact_5929_SUP__bot__conv_I1_J,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [B5: B > A,A3: set @ B] :
( ( ( complete_Sup_Sup @ A @ ( image @ B @ A @ B5 @ A3 ) )
= ( bot_bot @ A ) )
= ( ! [X4: B] :
( ( member @ B @ X4 @ A3 )
=> ( ( B5 @ X4 )
= ( bot_bot @ A ) ) ) ) ) ) ).
% SUP_bot_conv(1)
thf(fact_5930_SUP__bot,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ B] :
( ( complete_Sup_Sup @ A
@ ( image @ B @ A
@ ^ [X4: B] : ( bot_bot @ A )
@ A3 ) )
= ( bot_bot @ A ) ) ) ).
% SUP_bot
thf(fact_5931_cSUP__const,axiom,
! [B: $tType,A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A3: set @ B,C2: A] :
( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ A
@ ( image @ B @ A
@ ^ [X4: B] : C2
@ A3 ) )
= C2 ) ) ) ).
% cSUP_const
thf(fact_5932_SUP__const,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ B,F3: A] :
( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ A
@ ( image @ B @ A
@ ^ [I: B] : F3
@ A3 ) )
= F3 ) ) ) ).
% SUP_const
thf(fact_5933_image__add__atLeastAtMost_H,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [K2: A,I2: A,J3: A] :
( ( image @ A @ A
@ ^ [N2: A] : ( plus_plus @ A @ N2 @ K2 )
@ ( set_or1337092689740270186AtMost @ A @ I2 @ J3 ) )
= ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ I2 @ K2 ) @ ( plus_plus @ A @ J3 @ K2 ) ) ) ) ).
% image_add_atLeastAtMost'
thf(fact_5934_cINF__const,axiom,
! [B: $tType,A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A3: set @ B,C2: A] :
( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Inf_Inf @ A
@ ( image @ B @ A
@ ^ [X4: B] : C2
@ A3 ) )
= C2 ) ) ) ).
% cINF_const
thf(fact_5935_INF__const,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ B,F3: A] :
( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Inf_Inf @ A
@ ( image @ B @ A
@ ^ [I: B] : F3
@ A3 ) )
= F3 ) ) ) ).
% INF_const
thf(fact_5936_image__minus__const__atLeastAtMost_H,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [D2: A,A2: A,B2: A] :
( ( image @ A @ A
@ ^ [T3: A] : ( minus_minus @ A @ T3 @ D2 )
@ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ A2 @ D2 ) @ ( minus_minus @ A @ B2 @ D2 ) ) ) ) ).
% image_minus_const_atLeastAtMost'
thf(fact_5937_image__add__atLeastLessThan_H,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [K2: A,I2: A,J3: A] :
( ( image @ A @ A
@ ^ [N2: A] : ( plus_plus @ A @ N2 @ K2 )
@ ( set_or7035219750837199246ssThan @ A @ I2 @ J3 ) )
= ( set_or7035219750837199246ssThan @ A @ ( plus_plus @ A @ I2 @ K2 ) @ ( plus_plus @ A @ J3 @ K2 ) ) ) ) ).
% image_add_atLeastLessThan'
thf(fact_5938_INF__eq__bot__iff,axiom,
! [B: $tType,A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [F3: B > A,A3: set @ B] :
( ( ( complete_Inf_Inf @ A @ ( image @ B @ A @ F3 @ A3 ) )
= ( bot_bot @ A ) )
= ( ! [X4: A] :
( ( ord_less @ A @ ( bot_bot @ A ) @ X4 )
=> ? [Y: B] :
( ( member @ B @ Y @ A3 )
& ( ord_less @ A @ ( F3 @ Y ) @ X4 ) ) ) ) ) ) ).
% INF_eq_bot_iff
thf(fact_5939_image__mult__atLeastAtMost,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [D2: A,A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ D2 )
=> ( ( image @ A @ A @ ( times_times @ A @ D2 ) @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( times_times @ A @ D2 @ A2 ) @ ( times_times @ A @ D2 @ B2 ) ) ) ) ) ).
% image_mult_atLeastAtMost
thf(fact_5940_image__divide__atLeastAtMost,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [D2: A,A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ D2 )
=> ( ( image @ A @ A
@ ^ [C3: A] : ( divide_divide @ A @ C3 @ D2 )
@ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( divide_divide @ A @ A2 @ D2 ) @ ( divide_divide @ A @ B2 @ D2 ) ) ) ) ) ).
% image_divide_atLeastAtMost
thf(fact_5941_image__set,axiom,
! [A: $tType,B: $tType,F3: B > A,Xs: list @ B] :
( ( image @ B @ A @ F3 @ ( set2 @ B @ Xs ) )
= ( set2 @ A @ ( map @ B @ A @ F3 @ Xs ) ) ) ).
% image_set
thf(fact_5942_UNION__singleton__eq__range,axiom,
! [A: $tType,B: $tType,F3: B > A,A3: set @ B] :
( ( complete_Sup_Sup @ ( set @ A )
@ ( image @ B @ ( set @ A )
@ ^ [X4: B] : ( insert @ A @ ( F3 @ X4 ) @ ( bot_bot @ ( set @ A ) ) )
@ A3 ) )
= ( image @ B @ A @ F3 @ A3 ) ) ).
% UNION_singleton_eq_range
thf(fact_5943_SUP__eq,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ B,B5: set @ C,F3: B > A,G: C > A] :
( ! [I3: B] :
( ( member @ B @ I3 @ A3 )
=> ? [X2: C] :
( ( member @ C @ X2 @ B5 )
& ( ord_less_eq @ A @ ( F3 @ I3 ) @ ( G @ X2 ) ) ) )
=> ( ! [J2: C] :
( ( member @ C @ J2 @ B5 )
=> ? [X2: B] :
( ( member @ B @ X2 @ A3 )
& ( ord_less_eq @ A @ ( G @ J2 ) @ ( F3 @ X2 ) ) ) )
=> ( ( complete_Sup_Sup @ A @ ( image @ B @ A @ F3 @ A3 ) )
= ( complete_Sup_Sup @ A @ ( image @ C @ A @ G @ B5 ) ) ) ) ) ) ).
% SUP_eq
thf(fact_5944_INF__eq,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ B,B5: set @ C,G: C > A,F3: B > A] :
( ! [I3: B] :
( ( member @ B @ I3 @ A3 )
=> ? [X2: C] :
( ( member @ C @ X2 @ B5 )
& ( ord_less_eq @ A @ ( G @ X2 ) @ ( F3 @ I3 ) ) ) )
=> ( ! [J2: C] :
( ( member @ C @ J2 @ B5 )
=> ? [X2: B] :
( ( member @ B @ X2 @ A3 )
& ( ord_less_eq @ A @ ( F3 @ X2 ) @ ( G @ J2 ) ) ) )
=> ( ( complete_Inf_Inf @ A @ ( image @ B @ A @ F3 @ A3 ) )
= ( complete_Inf_Inf @ A @ ( image @ C @ A @ G @ B5 ) ) ) ) ) ) ).
% INF_eq
thf(fact_5945_UN__extend__simps_I10_J,axiom,
! [V6: $tType,U5: $tType,T: $tType,B5: U5 > ( set @ V6 ),F3: T > U5,A3: set @ T] :
( ( complete_Sup_Sup @ ( set @ V6 )
@ ( image @ T @ ( set @ V6 )
@ ^ [A5: T] : ( B5 @ ( F3 @ A5 ) )
@ A3 ) )
= ( complete_Sup_Sup @ ( set @ V6 ) @ ( image @ U5 @ ( set @ V6 ) @ B5 @ ( image @ T @ U5 @ F3 @ A3 ) ) ) ) ).
% UN_extend_simps(10)
thf(fact_5946_INT__extend__simps_I10_J,axiom,
! [V6: $tType,U5: $tType,T: $tType,B5: U5 > ( set @ V6 ),F3: T > U5,A3: set @ T] :
( ( complete_Inf_Inf @ ( set @ V6 )
@ ( image @ T @ ( set @ V6 )
@ ^ [A5: T] : ( B5 @ ( F3 @ A5 ) )
@ A3 ) )
= ( complete_Inf_Inf @ ( set @ V6 ) @ ( image @ U5 @ ( set @ V6 ) @ B5 @ ( image @ T @ U5 @ F3 @ A3 ) ) ) ) ).
% INT_extend_simps(10)
thf(fact_5947_INF__commute,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F3: B > C > A,B5: set @ C,A3: set @ B] :
( ( complete_Inf_Inf @ A
@ ( image @ B @ A
@ ^ [I: B] : ( complete_Inf_Inf @ A @ ( image @ C @ A @ ( F3 @ I ) @ B5 ) )
@ A3 ) )
= ( complete_Inf_Inf @ A
@ ( image @ C @ A
@ ^ [J4: C] :
( complete_Inf_Inf @ A
@ ( image @ B @ A
@ ^ [I: B] : ( F3 @ I @ J4 )
@ A3 ) )
@ B5 ) ) ) ) ).
% INF_commute
thf(fact_5948_SUP__commute,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F3: B > C > A,B5: set @ C,A3: set @ B] :
( ( complete_Sup_Sup @ A
@ ( image @ B @ A
@ ^ [I: B] : ( complete_Sup_Sup @ A @ ( image @ C @ A @ ( F3 @ I ) @ B5 ) )
@ A3 ) )
= ( complete_Sup_Sup @ A
@ ( image @ C @ A
@ ^ [J4: C] :
( complete_Sup_Sup @ A
@ ( image @ B @ A
@ ^ [I: B] : ( F3 @ I @ J4 )
@ A3 ) )
@ B5 ) ) ) ) ).
% SUP_commute
thf(fact_5949_image__UN,axiom,
! [A: $tType,B: $tType,C: $tType,F3: B > A,B5: C > ( set @ B ),A3: set @ C] :
( ( image @ B @ A @ F3 @ ( complete_Sup_Sup @ ( set @ B ) @ ( image @ C @ ( set @ B ) @ B5 @ A3 ) ) )
= ( complete_Sup_Sup @ ( set @ A )
@ ( image @ C @ ( set @ A )
@ ^ [X4: C] : ( image @ B @ A @ F3 @ ( B5 @ X4 ) )
@ A3 ) ) ) ).
% image_UN
thf(fact_5950_SUP__UNION,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F3: B > A,G: C > ( set @ B ),A3: set @ C] :
( ( complete_Sup_Sup @ A @ ( image @ B @ A @ F3 @ ( complete_Sup_Sup @ ( set @ B ) @ ( image @ C @ ( set @ B ) @ G @ A3 ) ) ) )
= ( complete_Sup_Sup @ A
@ ( image @ C @ A
@ ^ [Y: C] : ( complete_Sup_Sup @ A @ ( image @ B @ A @ F3 @ ( G @ Y ) ) )
@ A3 ) ) ) ) ).
% SUP_UNION
thf(fact_5951_image__Union,axiom,
! [A: $tType,B: $tType,F3: B > A,S3: set @ ( set @ B )] :
( ( image @ B @ A @ F3 @ ( complete_Sup_Sup @ ( set @ B ) @ S3 ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image @ ( set @ B ) @ ( set @ A ) @ ( image @ B @ A @ F3 ) @ S3 ) ) ) ).
% image_Union
thf(fact_5952_Inf_OINF__identity__eq,axiom,
! [A: $tType,Inf: ( set @ A ) > A,A3: set @ A] :
( ( Inf
@ ( image @ A @ A
@ ^ [X4: A] : X4
@ A3 ) )
= ( Inf @ A3 ) ) ).
% Inf.INF_identity_eq
thf(fact_5953_Sup_OSUP__identity__eq,axiom,
! [A: $tType,Sup: ( set @ A ) > A,A3: set @ A] :
( ( Sup
@ ( image @ A @ A
@ ^ [X4: A] : X4
@ A3 ) )
= ( Sup @ A3 ) ) ).
% Sup.SUP_identity_eq
thf(fact_5954_image__diff__subset,axiom,
! [A: $tType,B: $tType,F3: B > A,A3: set @ B,B5: set @ B] : ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ ( image @ B @ A @ F3 @ A3 ) @ ( image @ B @ A @ F3 @ B5 ) ) @ ( image @ B @ A @ F3 @ ( minus_minus @ ( set @ B ) @ A3 @ B5 ) ) ) ).
% image_diff_subset
thf(fact_5955_pigeonhole__infinite,axiom,
! [B: $tType,A: $tType,A3: set @ A,F3: A > B] :
( ~ ( finite_finite2 @ A @ A3 )
=> ( ( finite_finite2 @ B @ ( image @ A @ B @ F3 @ A3 ) )
=> ? [X5: A] :
( ( member @ A @ X5 @ A3 )
& ~ ( finite_finite2 @ A
@ ( collect @ A
@ ^ [A5: A] :
( ( member @ A @ A5 @ A3 )
& ( ( F3 @ A5 )
= ( F3 @ X5 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_5956_imageE,axiom,
! [A: $tType,B: $tType,B2: A,F3: B > A,A3: set @ B] :
( ( member @ A @ B2 @ ( image @ B @ A @ F3 @ A3 ) )
=> ~ ! [X5: B] :
( ( B2
= ( F3 @ X5 ) )
=> ~ ( member @ B @ X5 @ A3 ) ) ) ).
% imageE
thf(fact_5957_image__image,axiom,
! [A: $tType,B: $tType,C: $tType,F3: B > A,G: C > B,A3: set @ C] :
( ( image @ B @ A @ F3 @ ( image @ C @ B @ G @ A3 ) )
= ( image @ C @ A
@ ^ [X4: C] : ( F3 @ ( G @ X4 ) )
@ A3 ) ) ).
% image_image
thf(fact_5958_Compr__image__eq,axiom,
! [A: $tType,B: $tType,F3: B > A,A3: set @ B,P3: A > $o] :
( ( collect @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ ( image @ B @ A @ F3 @ A3 ) )
& ( P3 @ X4 ) ) )
= ( image @ B @ A @ F3
@ ( collect @ B
@ ^ [X4: B] :
( ( member @ B @ X4 @ A3 )
& ( P3 @ ( F3 @ X4 ) ) ) ) ) ) ).
% Compr_image_eq
thf(fact_5959_image__Fpow__mono,axiom,
! [B: $tType,A: $tType,F3: B > A,A3: set @ B,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ F3 @ A3 ) @ B5 )
=> ( ord_less_eq @ ( set @ ( set @ A ) ) @ ( image @ ( set @ B ) @ ( set @ A ) @ ( image @ B @ A @ F3 ) @ ( finite_Fpow @ B @ A3 ) ) @ ( finite_Fpow @ A @ B5 ) ) ) ).
% image_Fpow_mono
thf(fact_5960_all__subset__image,axiom,
! [A: $tType,B: $tType,F3: B > A,A3: set @ B,P3: ( set @ A ) > $o] :
( ( ! [B7: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ B7 @ ( image @ B @ A @ F3 @ A3 ) )
=> ( P3 @ B7 ) ) )
= ( ! [B7: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ B7 @ A3 )
=> ( P3 @ ( image @ B @ A @ F3 @ B7 ) ) ) ) ) ).
% all_subset_image
thf(fact_5961_subset__image__iff,axiom,
! [A: $tType,B: $tType,B5: set @ A,F3: B > A,A3: set @ B] :
( ( ord_less_eq @ ( set @ A ) @ B5 @ ( image @ B @ A @ F3 @ A3 ) )
= ( ? [AA: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ AA @ A3 )
& ( B5
= ( image @ B @ A @ F3 @ AA ) ) ) ) ) ).
% subset_image_iff
thf(fact_5962_image__subset__iff,axiom,
! [A: $tType,B: $tType,F3: B > A,A3: set @ B,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ F3 @ A3 ) @ B5 )
= ( ! [X4: B] :
( ( member @ B @ X4 @ A3 )
=> ( member @ A @ ( F3 @ X4 ) @ B5 ) ) ) ) ).
% image_subset_iff
thf(fact_5963_subset__imageE,axiom,
! [A: $tType,B: $tType,B5: set @ A,F3: B > A,A3: set @ B] :
( ( ord_less_eq @ ( set @ A ) @ B5 @ ( image @ B @ A @ F3 @ A3 ) )
=> ~ ! [C7: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ C7 @ A3 )
=> ( B5
!= ( image @ B @ A @ F3 @ C7 ) ) ) ) ).
% subset_imageE
thf(fact_5964_image__subsetI,axiom,
! [A: $tType,B: $tType,A3: set @ A,F3: A > B,B5: set @ B] :
( ! [X5: A] :
( ( member @ A @ X5 @ A3 )
=> ( member @ B @ ( F3 @ X5 ) @ B5 ) )
=> ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F3 @ A3 ) @ B5 ) ) ).
% image_subsetI
thf(fact_5965_image__mono,axiom,
! [B: $tType,A: $tType,A3: set @ A,B5: set @ A,F3: A > B] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B5 )
=> ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F3 @ A3 ) @ ( image @ A @ B @ F3 @ B5 ) ) ) ).
% image_mono
thf(fact_5966_image__Pow__mono,axiom,
! [B: $tType,A: $tType,F3: B > A,A3: set @ B,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ F3 @ A3 ) @ B5 )
=> ( ord_less_eq @ ( set @ ( set @ A ) ) @ ( image @ ( set @ B ) @ ( set @ A ) @ ( image @ B @ A @ F3 ) @ ( pow2 @ B @ A3 ) ) @ ( pow2 @ A @ B5 ) ) ) ).
% image_Pow_mono
thf(fact_5967_all__finite__subset__image,axiom,
! [A: $tType,B: $tType,F3: B > A,A3: set @ B,P3: ( set @ A ) > $o] :
( ( ! [B7: set @ A] :
( ( ( finite_finite2 @ A @ B7 )
& ( ord_less_eq @ ( set @ A ) @ B7 @ ( image @ B @ A @ F3 @ A3 ) ) )
=> ( P3 @ B7 ) ) )
= ( ! [B7: set @ B] :
( ( ( finite_finite2 @ B @ B7 )
& ( ord_less_eq @ ( set @ B ) @ B7 @ A3 ) )
=> ( P3 @ ( image @ B @ A @ F3 @ B7 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_5968_ex__finite__subset__image,axiom,
! [A: $tType,B: $tType,F3: B > A,A3: set @ B,P3: ( set @ A ) > $o] :
( ( ? [B7: set @ A] :
( ( finite_finite2 @ A @ B7 )
& ( ord_less_eq @ ( set @ A ) @ B7 @ ( image @ B @ A @ F3 @ A3 ) )
& ( P3 @ B7 ) ) )
= ( ? [B7: set @ B] :
( ( finite_finite2 @ B @ B7 )
& ( ord_less_eq @ ( set @ B ) @ B7 @ A3 )
& ( P3 @ ( image @ B @ A @ F3 @ B7 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_5969_finite__subset__image,axiom,
! [A: $tType,B: $tType,B5: set @ A,F3: B > A,A3: set @ B] :
( ( finite_finite2 @ A @ B5 )
=> ( ( ord_less_eq @ ( set @ A ) @ B5 @ ( image @ B @ A @ F3 @ A3 ) )
=> ? [C7: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ C7 @ A3 )
& ( finite_finite2 @ B @ C7 )
& ( B5
= ( image @ B @ A @ F3 @ C7 ) ) ) ) ) ).
% finite_subset_image
thf(fact_5970_finite__surj,axiom,
! [A: $tType,B: $tType,A3: set @ A,B5: set @ B,F3: A > B] :
( ( finite_finite2 @ A @ A3 )
=> ( ( ord_less_eq @ ( set @ B ) @ B5 @ ( image @ A @ B @ F3 @ A3 ) )
=> ( finite_finite2 @ B @ B5 ) ) ) ).
% finite_surj
thf(fact_5971_translation__diff,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A2: A,S: set @ A,T2: set @ A] :
( ( image @ A @ A @ ( plus_plus @ A @ A2 ) @ ( minus_minus @ ( set @ A ) @ S @ T2 ) )
= ( minus_minus @ ( set @ A ) @ ( image @ A @ A @ ( plus_plus @ A @ A2 ) @ S ) @ ( image @ A @ A @ ( plus_plus @ A @ A2 ) @ T2 ) ) ) ) ).
% translation_diff
thf(fact_5972_bij__betw__byWitness,axiom,
! [A: $tType,B: $tType,A3: set @ A,F9: B > A,F3: A > B,A10: set @ B] :
( ! [X5: A] :
( ( member @ A @ X5 @ A3 )
=> ( ( F9 @ ( F3 @ X5 ) )
= X5 ) )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ A10 )
=> ( ( F3 @ ( F9 @ X5 ) )
= X5 ) )
=> ( ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F3 @ A3 ) @ A10 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ F9 @ A10 ) @ A3 )
=> ( bij_betw @ A @ B @ F3 @ A3 @ A10 ) ) ) ) ) ).
% bij_betw_byWitness
thf(fact_5973_bij__betw__subset,axiom,
! [A: $tType,B: $tType,F3: A > B,A3: set @ A,A10: set @ B,B5: set @ A,B13: set @ B] :
( ( bij_betw @ A @ B @ F3 @ A3 @ A10 )
=> ( ( ord_less_eq @ ( set @ A ) @ B5 @ A3 )
=> ( ( ( image @ A @ B @ F3 @ B5 )
= B13 )
=> ( bij_betw @ A @ B @ F3 @ B5 @ B13 ) ) ) ) ).
% bij_betw_subset
thf(fact_5974_translation__Compl,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A2: A,T2: set @ A] :
( ( image @ A @ A @ ( plus_plus @ A @ A2 ) @ ( uminus_uminus @ ( set @ A ) @ T2 ) )
= ( uminus_uminus @ ( set @ A ) @ ( image @ A @ A @ ( plus_plus @ A @ A2 ) @ T2 ) ) ) ) ).
% translation_Compl
thf(fact_5975_translation__Int,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A2: A,S: set @ A,T2: set @ A] :
( ( image @ A @ A @ ( plus_plus @ A @ A2 ) @ ( inf_inf @ ( set @ A ) @ S @ T2 ) )
= ( inf_inf @ ( set @ A ) @ ( image @ A @ A @ ( plus_plus @ A @ A2 ) @ S ) @ ( image @ A @ A @ ( plus_plus @ A @ A2 ) @ T2 ) ) ) ) ).
% translation_Int
thf(fact_5976_image__Int__subset,axiom,
! [A: $tType,B: $tType,F3: B > A,A3: set @ B,B5: set @ B] : ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ F3 @ ( inf_inf @ ( set @ B ) @ A3 @ B5 ) ) @ ( inf_inf @ ( set @ A ) @ ( image @ B @ A @ F3 @ A3 ) @ ( image @ B @ A @ F3 @ B5 ) ) ) ).
% image_Int_subset
thf(fact_5977_SUP__upper2,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I2: B,A3: set @ B,U: A,F3: B > A] :
( ( member @ B @ I2 @ A3 )
=> ( ( ord_less_eq @ A @ U @ ( F3 @ I2 ) )
=> ( ord_less_eq @ A @ U @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F3 @ A3 ) ) ) ) ) ) ).
% SUP_upper2
thf(fact_5978_SUP__le__iff,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F3: B > A,A3: set @ B,U: A] :
( ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F3 @ A3 ) ) @ U )
= ( ! [X4: B] :
( ( member @ B @ X4 @ A3 )
=> ( ord_less_eq @ A @ ( F3 @ X4 ) @ U ) ) ) ) ) ).
% SUP_le_iff
thf(fact_5979_SUP__upper,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I2: B,A3: set @ B,F3: B > A] :
( ( member @ B @ I2 @ A3 )
=> ( ord_less_eq @ A @ ( F3 @ I2 ) @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F3 @ A3 ) ) ) ) ) ).
% SUP_upper
thf(fact_5980_SUP__mono_H,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F3: B > A,G: B > A,A3: set @ B] :
( ! [X5: B] : ( ord_less_eq @ A @ ( F3 @ X5 ) @ ( G @ X5 ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F3 @ A3 ) ) @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ G @ A3 ) ) ) ) ) ).
% SUP_mono'
thf(fact_5981_SUP__least,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ B,F3: B > A,U: A] :
( ! [I3: B] :
( ( member @ B @ I3 @ A3 )
=> ( ord_less_eq @ A @ ( F3 @ I3 ) @ U ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F3 @ A3 ) ) @ U ) ) ) ).
% SUP_least
thf(fact_5982_SUP__mono,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ B,B5: set @ C,F3: B > A,G: C > A] :
( ! [N3: B] :
( ( member @ B @ N3 @ A3 )
=> ? [X2: C] :
( ( member @ C @ X2 @ B5 )
& ( ord_less_eq @ A @ ( F3 @ N3 ) @ ( G @ X2 ) ) ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F3 @ A3 ) ) @ ( complete_Sup_Sup @ A @ ( image @ C @ A @ G @ B5 ) ) ) ) ) ).
% SUP_mono
thf(fact_5983_SUP__eqI,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ B,F3: B > A,X: A] :
( ! [I3: B] :
( ( member @ B @ I3 @ A3 )
=> ( ord_less_eq @ A @ ( F3 @ I3 ) @ X ) )
=> ( ! [Y7: A] :
( ! [I4: B] :
( ( member @ B @ I4 @ A3 )
=> ( ord_less_eq @ A @ ( F3 @ I4 ) @ Y7 ) )
=> ( ord_less_eq @ A @ X @ Y7 ) )
=> ( ( complete_Sup_Sup @ A @ ( image @ B @ A @ F3 @ A3 ) )
= X ) ) ) ) ).
% SUP_eqI
thf(fact_5984_SUP__lessD,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F3: B > A,A3: set @ B,Y2: A,I2: B] :
( ( ord_less @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F3 @ A3 ) ) @ Y2 )
=> ( ( member @ B @ I2 @ A3 )
=> ( ord_less @ A @ ( F3 @ I2 ) @ Y2 ) ) ) ) ).
% SUP_lessD
thf(fact_5985_less__SUP__iff,axiom,
! [A: $tType,B: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [A2: A,F3: B > A,A3: set @ B] :
( ( ord_less @ A @ A2 @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F3 @ A3 ) ) )
= ( ? [X4: B] :
( ( member @ B @ X4 @ A3 )
& ( ord_less @ A @ A2 @ ( F3 @ X4 ) ) ) ) ) ) ).
% less_SUP_iff
thf(fact_5986_INF__greatest,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ B,U: A,F3: B > A] :
( ! [I3: B] :
( ( member @ B @ I3 @ A3 )
=> ( ord_less_eq @ A @ U @ ( F3 @ I3 ) ) )
=> ( ord_less_eq @ A @ U @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F3 @ A3 ) ) ) ) ) ).
% INF_greatest
thf(fact_5987_le__INF__iff,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [U: A,F3: B > A,A3: set @ B] :
( ( ord_less_eq @ A @ U @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F3 @ A3 ) ) )
= ( ! [X4: B] :
( ( member @ B @ X4 @ A3 )
=> ( ord_less_eq @ A @ U @ ( F3 @ X4 ) ) ) ) ) ) ).
% le_INF_iff
thf(fact_5988_INF__lower2,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I2: B,A3: set @ B,F3: B > A,U: A] :
( ( member @ B @ I2 @ A3 )
=> ( ( ord_less_eq @ A @ ( F3 @ I2 ) @ U )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F3 @ A3 ) ) @ U ) ) ) ) ).
% INF_lower2
thf(fact_5989_INF__mono_H,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F3: B > A,G: B > A,A3: set @ B] :
( ! [X5: B] : ( ord_less_eq @ A @ ( F3 @ X5 ) @ ( G @ X5 ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F3 @ A3 ) ) @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ G @ A3 ) ) ) ) ) ).
% INF_mono'
thf(fact_5990_INF__lower,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I2: B,A3: set @ B,F3: B > A] :
( ( member @ B @ I2 @ A3 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F3 @ A3 ) ) @ ( F3 @ I2 ) ) ) ) ).
% INF_lower
thf(fact_5991_INF__mono,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [B5: set @ B,A3: set @ C,F3: C > A,G: B > A] :
( ! [M4: B] :
( ( member @ B @ M4 @ B5 )
=> ? [X2: C] :
( ( member @ C @ X2 @ A3 )
& ( ord_less_eq @ A @ ( F3 @ X2 ) @ ( G @ M4 ) ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image @ C @ A @ F3 @ A3 ) ) @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ G @ B5 ) ) ) ) ) ).
% INF_mono
thf(fact_5992_INF__eqI,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ B,X: A,F3: B > A] :
( ! [I3: B] :
( ( member @ B @ I3 @ A3 )
=> ( ord_less_eq @ A @ X @ ( F3 @ I3 ) ) )
=> ( ! [Y7: A] :
( ! [I4: B] :
( ( member @ B @ I4 @ A3 )
=> ( ord_less_eq @ A @ Y7 @ ( F3 @ I4 ) ) )
=> ( ord_less_eq @ A @ Y7 @ X ) )
=> ( ( complete_Inf_Inf @ A @ ( image @ B @ A @ F3 @ A3 ) )
= X ) ) ) ) ).
% INF_eqI
thf(fact_5993_less__INF__D,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [Y2: A,F3: B > A,A3: set @ B,I2: B] :
( ( ord_less @ A @ Y2 @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F3 @ A3 ) ) )
=> ( ( member @ B @ I2 @ A3 )
=> ( ord_less @ A @ Y2 @ ( F3 @ I2 ) ) ) ) ) ).
% less_INF_D
thf(fact_5994_INF__less__iff,axiom,
! [A: $tType,B: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [F3: B > A,A3: set @ B,A2: A] :
( ( ord_less @ A @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F3 @ A3 ) ) @ A2 )
= ( ? [X4: B] :
( ( member @ B @ X4 @ A3 )
& ( ord_less @ A @ ( F3 @ X4 ) @ A2 ) ) ) ) ) ).
% INF_less_iff
thf(fact_5995_finite__conv__nat__seg__image,axiom,
! [A: $tType] :
( ( finite_finite2 @ A )
= ( ^ [A7: set @ A] :
? [N2: nat,F5: nat > A] :
( A7
= ( image @ nat @ A @ F5
@ ( collect @ nat
@ ^ [I: nat] : ( ord_less @ nat @ I @ N2 ) ) ) ) ) ) ).
% finite_conv_nat_seg_image
thf(fact_5996_nat__seg__image__imp__finite,axiom,
! [A: $tType,A3: set @ A,F3: nat > A,N: nat] :
( ( A3
= ( image @ nat @ A @ F3
@ ( collect @ nat
@ ^ [I: nat] : ( ord_less @ nat @ I @ N ) ) ) )
=> ( finite_finite2 @ A @ A3 ) ) ).
% nat_seg_image_imp_finite
thf(fact_5997_image__constant,axiom,
! [A: $tType,B: $tType,X: A,A3: set @ A,C2: B] :
( ( member @ A @ X @ A3 )
=> ( ( image @ A @ B
@ ^ [X4: A] : C2
@ A3 )
= ( insert @ B @ C2 @ ( bot_bot @ ( set @ B ) ) ) ) ) ).
% image_constant
thf(fact_5998_image__constant__conv,axiom,
! [B: $tType,A: $tType,A3: set @ B,C2: A] :
( ( ( A3
= ( bot_bot @ ( set @ B ) ) )
=> ( ( image @ B @ A
@ ^ [X4: B] : C2
@ A3 )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( image @ B @ A
@ ^ [X4: B] : C2
@ A3 )
= ( insert @ A @ C2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% image_constant_conv
thf(fact_5999_sum_Oimage__gen,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [S3: set @ B,H2: B > A,G: B > C] :
( ( finite_finite2 @ B @ S3 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ H2 @ S3 )
= ( groups7311177749621191930dd_sum @ C @ A
@ ^ [Y: C] :
( groups7311177749621191930dd_sum @ B @ A @ H2
@ ( collect @ B
@ ^ [X4: B] :
( ( member @ B @ X4 @ S3 )
& ( ( G @ X4 )
= Y ) ) ) )
@ ( image @ B @ C @ G @ S3 ) ) ) ) ) ).
% sum.image_gen
thf(fact_6000_translation__subtract__Int,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A2: A,S: set @ A,T2: set @ A] :
( ( image @ A @ A
@ ^ [X4: A] : ( minus_minus @ A @ X4 @ A2 )
@ ( inf_inf @ ( set @ A ) @ S @ T2 ) )
= ( inf_inf @ ( set @ A )
@ ( image @ A @ A
@ ^ [X4: A] : ( minus_minus @ A @ X4 @ A2 )
@ S )
@ ( image @ A @ A
@ ^ [X4: A] : ( minus_minus @ A @ X4 @ A2 )
@ T2 ) ) ) ) ).
% translation_subtract_Int
thf(fact_6001_SUP__inf,axiom,
! [A: $tType,B: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [F3: B > A,B5: set @ B,A2: A] :
( ( inf_inf @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F3 @ B5 ) ) @ A2 )
= ( complete_Sup_Sup @ A
@ ( image @ B @ A
@ ^ [B4: B] : ( inf_inf @ A @ ( F3 @ B4 ) @ A2 )
@ B5 ) ) ) ) ).
% SUP_inf
thf(fact_6002_Sup__inf,axiom,
! [A: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [B5: set @ A,A2: A] :
( ( inf_inf @ A @ ( complete_Sup_Sup @ A @ B5 ) @ A2 )
= ( complete_Sup_Sup @ A
@ ( image @ A @ A
@ ^ [B4: A] : ( inf_inf @ A @ B4 @ A2 )
@ B5 ) ) ) ) ).
% Sup_inf
thf(fact_6003_inf__SUP,axiom,
! [A: $tType,B: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [A2: A,F3: B > A,B5: set @ B] :
( ( inf_inf @ A @ A2 @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F3 @ B5 ) ) )
= ( complete_Sup_Sup @ A
@ ( image @ B @ A
@ ^ [B4: B] : ( inf_inf @ A @ A2 @ ( F3 @ B4 ) )
@ B5 ) ) ) ) ).
% inf_SUP
thf(fact_6004_SUP__inf__distrib2,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [F3: B > A,A3: set @ B,G: C > A,B5: set @ C] :
( ( inf_inf @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F3 @ A3 ) ) @ ( complete_Sup_Sup @ A @ ( image @ C @ A @ G @ B5 ) ) )
= ( complete_Sup_Sup @ A
@ ( image @ B @ A
@ ^ [A5: B] :
( complete_Sup_Sup @ A
@ ( image @ C @ A
@ ^ [B4: C] : ( inf_inf @ A @ ( F3 @ A5 ) @ ( G @ B4 ) )
@ B5 ) )
@ A3 ) ) ) ) ).
% SUP_inf_distrib2
thf(fact_6005_translation__subtract__diff,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A2: A,S: set @ A,T2: set @ A] :
( ( image @ A @ A
@ ^ [X4: A] : ( minus_minus @ A @ X4 @ A2 )
@ ( minus_minus @ ( set @ A ) @ S @ T2 ) )
= ( minus_minus @ ( set @ A )
@ ( image @ A @ A
@ ^ [X4: A] : ( minus_minus @ A @ X4 @ A2 )
@ S )
@ ( image @ A @ A
@ ^ [X4: A] : ( minus_minus @ A @ X4 @ A2 )
@ T2 ) ) ) ) ).
% translation_subtract_diff
thf(fact_6006_INF__absorb,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [K2: B,I6: set @ B,A3: B > A] :
( ( member @ B @ K2 @ I6 )
=> ( ( inf_inf @ A @ ( A3 @ K2 ) @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ A3 @ I6 ) ) )
= ( complete_Inf_Inf @ A @ ( image @ B @ A @ A3 @ I6 ) ) ) ) ) ).
% INF_absorb
thf(fact_6007_INF__inf__distrib,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F3: B > A,A3: set @ B,G: B > A] :
( ( inf_inf @ A @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F3 @ A3 ) ) @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ G @ A3 ) ) )
= ( complete_Inf_Inf @ A
@ ( image @ B @ A
@ ^ [A5: B] : ( inf_inf @ A @ ( F3 @ A5 ) @ ( G @ A5 ) )
@ A3 ) ) ) ) ).
% INF_inf_distrib
thf(fact_6008_prod_Oimage__gen,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S3: set @ B,H2: B > A,G: B > C] :
( ( finite_finite2 @ B @ S3 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ H2 @ S3 )
= ( groups7121269368397514597t_prod @ C @ A
@ ^ [Y: C] :
( groups7121269368397514597t_prod @ B @ A @ H2
@ ( collect @ B
@ ^ [X4: B] :
( ( member @ B @ X4 @ S3 )
& ( ( G @ X4 )
= Y ) ) ) )
@ ( image @ B @ C @ G @ S3 ) ) ) ) ) ).
% prod.image_gen
thf(fact_6009_translation__subtract__Compl,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A2: A,T2: set @ A] :
( ( image @ A @ A
@ ^ [X4: A] : ( minus_minus @ A @ X4 @ A2 )
@ ( uminus_uminus @ ( set @ A ) @ T2 ) )
= ( uminus_uminus @ ( set @ A )
@ ( image @ A @ A
@ ^ [X4: A] : ( minus_minus @ A @ X4 @ A2 )
@ T2 ) ) ) ) ).
% translation_subtract_Compl
thf(fact_6010_le__SUP__iff,axiom,
! [B: $tType,A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [X: A,F3: B > A,A3: set @ B] :
( ( ord_less_eq @ A @ X @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F3 @ A3 ) ) )
= ( ! [Y: A] :
( ( ord_less @ A @ Y @ X )
=> ? [X4: B] :
( ( member @ B @ X4 @ A3 )
& ( ord_less @ A @ Y @ ( F3 @ X4 ) ) ) ) ) ) ) ).
% le_SUP_iff
thf(fact_6011_INF__le__iff,axiom,
! [B: $tType,A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [F3: B > A,A3: set @ B,X: A] :
( ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F3 @ A3 ) ) @ X )
= ( ! [Y: A] :
( ( ord_less @ A @ X @ Y )
=> ? [X4: B] :
( ( member @ B @ X4 @ A3 )
& ( ord_less @ A @ ( F3 @ X4 ) @ Y ) ) ) ) ) ) ).
% INF_le_iff
thf(fact_6012_cSUP__least,axiom,
! [B: $tType,A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A3: set @ B,F3: B > A,M7: A] :
( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ A3 )
=> ( ord_less_eq @ A @ ( F3 @ X5 ) @ M7 ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F3 @ A3 ) ) @ M7 ) ) ) ) ).
% cSUP_least
thf(fact_6013_SUP__eq__iff,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I6: set @ B,C2: A,F3: B > A] :
( ( I6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [I3: B] :
( ( member @ B @ I3 @ I6 )
=> ( ord_less_eq @ A @ C2 @ ( F3 @ I3 ) ) )
=> ( ( ( complete_Sup_Sup @ A @ ( image @ B @ A @ F3 @ I6 ) )
= C2 )
= ( ! [X4: B] :
( ( member @ B @ X4 @ I6 )
=> ( ( F3 @ X4 )
= C2 ) ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_6014_cINF__greatest,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A3: set @ B,M: A,F3: B > A] :
( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ A3 )
=> ( ord_less_eq @ A @ M @ ( F3 @ X5 ) ) )
=> ( ord_less_eq @ A @ M @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F3 @ A3 ) ) ) ) ) ) ).
% cINF_greatest
thf(fact_6015_INF__eq__iff,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I6: set @ B,F3: B > A,C2: A] :
( ( I6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [I3: B] :
( ( member @ B @ I3 @ I6 )
=> ( ord_less_eq @ A @ ( F3 @ I3 ) @ C2 ) )
=> ( ( ( complete_Inf_Inf @ A @ ( image @ B @ A @ F3 @ I6 ) )
= C2 )
= ( ! [X4: B] :
( ( member @ B @ X4 @ I6 )
=> ( ( F3 @ X4 )
= C2 ) ) ) ) ) ) ) ).
% INF_eq_iff
thf(fact_6016_card__image__le,axiom,
! [B: $tType,A: $tType,A3: set @ A,F3: A > B] :
( ( finite_finite2 @ A @ A3 )
=> ( ord_less_eq @ nat @ ( finite_card @ B @ ( image @ A @ B @ F3 @ A3 ) ) @ ( finite_card @ A @ A3 ) ) ) ).
% card_image_le
thf(fact_6017_bij__betw__comp__iff2,axiom,
! [C: $tType,A: $tType,B: $tType,F9: A > B,A10: set @ A,A11: set @ B,F3: C > A,A3: set @ C] :
( ( bij_betw @ A @ B @ F9 @ A10 @ A11 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image @ C @ A @ F3 @ A3 ) @ A10 )
=> ( ( bij_betw @ C @ A @ F3 @ A3 @ A10 )
= ( bij_betw @ C @ B @ ( comp @ A @ B @ C @ F9 @ F3 ) @ A3 @ A11 ) ) ) ) ).
% bij_betw_comp_iff2
thf(fact_6018_SUP__subset__mono,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ B,B5: set @ B,F3: B > A,G: B > A] :
( ( ord_less_eq @ ( set @ B ) @ A3 @ B5 )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ A3 )
=> ( ord_less_eq @ A @ ( F3 @ X5 ) @ ( G @ X5 ) ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F3 @ A3 ) ) @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ G @ B5 ) ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_6019_INF__superset__mono,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [B5: set @ B,A3: set @ B,F3: B > A,G: B > A] :
( ( ord_less_eq @ ( set @ B ) @ B5 @ A3 )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ B5 )
=> ( ord_less_eq @ A @ ( F3 @ X5 ) @ ( G @ X5 ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F3 @ A3 ) ) @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ G @ B5 ) ) ) ) ) ) ).
% INF_superset_mono
thf(fact_6020_SUP__empty,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F3: B > A] :
( ( complete_Sup_Sup @ A @ ( image @ B @ A @ F3 @ ( bot_bot @ ( set @ B ) ) ) )
= ( bot_bot @ A ) ) ) ).
% SUP_empty
thf(fact_6021_SUP__constant,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ B,C2: A] :
( ( ( A3
= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ A
@ ( image @ B @ A
@ ^ [Y: B] : C2
@ A3 ) )
= ( bot_bot @ A ) ) )
& ( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ A
@ ( image @ B @ A
@ ^ [Y: B] : C2
@ A3 ) )
= C2 ) ) ) ) ).
% SUP_constant
thf(fact_6022_sum_Ogroup,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [S3: set @ B,T5: set @ C,G: B > C,H2: B > A] :
( ( finite_finite2 @ B @ S3 )
=> ( ( finite_finite2 @ C @ T5 )
=> ( ( ord_less_eq @ ( set @ C ) @ ( image @ B @ C @ G @ S3 ) @ T5 )
=> ( ( groups7311177749621191930dd_sum @ C @ A
@ ^ [Y: C] :
( groups7311177749621191930dd_sum @ B @ A @ H2
@ ( collect @ B
@ ^ [X4: B] :
( ( member @ B @ X4 @ S3 )
& ( ( G @ X4 )
= Y ) ) ) )
@ T5 )
= ( groups7311177749621191930dd_sum @ B @ A @ H2 @ S3 ) ) ) ) ) ) ).
% sum.group
thf(fact_6023_uminus__INF,axiom,
! [A: $tType,B: $tType] :
( ( comple489889107523837845lgebra @ A )
=> ! [B5: B > A,A3: set @ B] :
( ( uminus_uminus @ A @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ B5 @ A3 ) ) )
= ( complete_Sup_Sup @ A
@ ( image @ B @ A
@ ^ [X4: B] : ( uminus_uminus @ A @ ( B5 @ X4 ) )
@ A3 ) ) ) ) ).
% uminus_INF
thf(fact_6024_uminus__SUP,axiom,
! [A: $tType,B: $tType] :
( ( comple489889107523837845lgebra @ A )
=> ! [B5: B > A,A3: set @ B] :
( ( uminus_uminus @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ B5 @ A3 ) ) )
= ( complete_Inf_Inf @ A
@ ( image @ B @ A
@ ^ [X4: B] : ( uminus_uminus @ A @ ( B5 @ X4 ) )
@ A3 ) ) ) ) ).
% uminus_SUP
thf(fact_6025_INF__inf__const2,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I6: set @ B,F3: B > A,X: A] :
( ( I6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Inf_Inf @ A
@ ( image @ B @ A
@ ^ [I: B] : ( inf_inf @ A @ ( F3 @ I ) @ X )
@ I6 ) )
= ( inf_inf @ A @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F3 @ I6 ) ) @ X ) ) ) ) ).
% INF_inf_const2
thf(fact_6026_INF__inf__const1,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I6: set @ B,X: A,F3: B > A] :
( ( I6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Inf_Inf @ A
@ ( image @ B @ A
@ ^ [I: B] : ( inf_inf @ A @ X @ ( F3 @ I ) )
@ I6 ) )
= ( inf_inf @ A @ X @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F3 @ I6 ) ) ) ) ) ) ).
% INF_inf_const1
thf(fact_6027_prod_Ogroup,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S3: set @ B,T5: set @ C,G: B > C,H2: B > A] :
( ( finite_finite2 @ B @ S3 )
=> ( ( finite_finite2 @ C @ T5 )
=> ( ( ord_less_eq @ ( set @ C ) @ ( image @ B @ C @ G @ S3 ) @ T5 )
=> ( ( groups7121269368397514597t_prod @ C @ A
@ ^ [Y: C] :
( groups7121269368397514597t_prod @ B @ A @ H2
@ ( collect @ B
@ ^ [X4: B] :
( ( member @ B @ X4 @ S3 )
& ( ( G @ X4 )
= Y ) ) ) )
@ T5 )
= ( groups7121269368397514597t_prod @ B @ A @ H2 @ S3 ) ) ) ) ) ) ).
% prod.group
thf(fact_6028_INF__insert,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F3: B > A,A2: B,A3: set @ B] :
( ( complete_Inf_Inf @ A @ ( image @ B @ A @ F3 @ ( insert @ B @ A2 @ A3 ) ) )
= ( inf_inf @ A @ ( F3 @ A2 ) @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F3 @ A3 ) ) ) ) ) ).
% INF_insert
thf(fact_6029_INF__le__SUP,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ B,F3: B > A] :
( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F3 @ A3 ) ) @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F3 @ A3 ) ) ) ) ) ).
% INF_le_SUP
thf(fact_6030_prod_Oreindex__nontrivial,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: set @ B,H2: B > C,G: C > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ! [X5: B,Y7: B] :
( ( member @ B @ X5 @ A3 )
=> ( ( member @ B @ Y7 @ A3 )
=> ( ( X5 != Y7 )
=> ( ( ( H2 @ X5 )
= ( H2 @ Y7 ) )
=> ( ( G @ ( H2 @ X5 ) )
= ( one_one @ A ) ) ) ) ) )
=> ( ( groups7121269368397514597t_prod @ C @ A @ G @ ( image @ B @ C @ H2 @ A3 ) )
= ( groups7121269368397514597t_prod @ B @ A @ ( comp @ C @ A @ B @ G @ H2 ) @ A3 ) ) ) ) ) ).
% prod.reindex_nontrivial
thf(fact_6031_surj__card__le,axiom,
! [B: $tType,A: $tType,A3: set @ A,B5: set @ B,F3: A > B] :
( ( finite_finite2 @ A @ A3 )
=> ( ( ord_less_eq @ ( set @ B ) @ B5 @ ( image @ A @ B @ F3 @ A3 ) )
=> ( ord_less_eq @ nat @ ( finite_card @ B @ B5 ) @ ( finite_card @ A @ A3 ) ) ) ) ).
% surj_card_le
thf(fact_6032_Fpow__mono,axiom,
! [A: $tType,A3: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B5 )
=> ( ord_less_eq @ ( set @ ( set @ A ) ) @ ( finite_Fpow @ A @ A3 ) @ ( finite_Fpow @ A @ B5 ) ) ) ).
% Fpow_mono
thf(fact_6033_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_6034_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_6035_distinct__insort__key,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F3: B > A,X: B,Xs: list @ B] :
( ( distinct @ A @ ( map @ B @ A @ F3 @ ( linorder_insort_key @ B @ A @ F3 @ X @ Xs ) ) )
= ( ~ ( member @ A @ ( F3 @ X ) @ ( image @ B @ A @ F3 @ ( set2 @ B @ Xs ) ) )
& ( distinct @ A @ ( map @ B @ A @ F3 @ Xs ) ) ) ) ) ).
% distinct_insort_key
thf(fact_6036_Fpow__subset__Pow,axiom,
! [A: $tType,A3: set @ A] : ( ord_less_eq @ ( set @ ( set @ A ) ) @ ( finite_Fpow @ A @ A3 ) @ ( pow2 @ A @ A3 ) ) ).
% Fpow_subset_Pow
thf(fact_6037_Fpow__def,axiom,
! [A: $tType] :
( ( finite_Fpow @ A )
= ( ^ [A7: set @ A] :
( collect @ ( set @ A )
@ ^ [X6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ X6 @ A7 )
& ( finite_finite2 @ A @ X6 ) ) ) ) ) ).
% Fpow_def
thf(fact_6038_UN__le__add__shift__strict,axiom,
! [A: $tType,M7: nat > ( set @ A ),K2: nat,N: nat] :
( ( complete_Sup_Sup @ ( set @ A )
@ ( image @ nat @ ( set @ A )
@ ^ [I: nat] : ( M7 @ ( plus_plus @ nat @ I @ K2 ) )
@ ( set_ord_lessThan @ nat @ N ) ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ M7 @ ( set_or7035219750837199246ssThan @ nat @ K2 @ ( plus_plus @ nat @ N @ K2 ) ) ) ) ) ).
% UN_le_add_shift_strict
thf(fact_6039_UN__le__add__shift,axiom,
! [A: $tType,M7: nat > ( set @ A ),K2: nat,N: nat] :
( ( complete_Sup_Sup @ ( set @ A )
@ ( image @ nat @ ( set @ A )
@ ^ [I: nat] : ( M7 @ ( plus_plus @ nat @ I @ K2 ) )
@ ( set_ord_atMost @ nat @ N ) ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ M7 @ ( set_or1337092689740270186AtMost @ nat @ K2 @ ( plus_plus @ nat @ N @ K2 ) ) ) ) ) ).
% UN_le_add_shift
thf(fact_6040_sum__image__le,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( ordere6911136660526730532id_add @ B )
=> ! [I6: set @ C,G: A > B,F3: C > A] :
( ( finite_finite2 @ C @ I6 )
=> ( ! [I3: C] :
( ( member @ C @ I3 @ I6 )
=> ( ord_less_eq @ B @ ( zero_zero @ B ) @ ( G @ ( F3 @ I3 ) ) ) )
=> ( ord_less_eq @ B @ ( groups7311177749621191930dd_sum @ A @ B @ G @ ( image @ C @ A @ F3 @ I6 ) ) @ ( groups7311177749621191930dd_sum @ C @ B @ ( comp @ A @ B @ C @ G @ F3 ) @ I6 ) ) ) ) ) ).
% sum_image_le
thf(fact_6041_image__mult__atLeastAtMost__if,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,X: A,Y2: A] :
( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( image @ A @ A @ ( times_times @ A @ C2 ) @ ( set_or1337092689740270186AtMost @ A @ X @ Y2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( times_times @ A @ C2 @ X ) @ ( times_times @ A @ C2 @ Y2 ) ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less_eq @ A @ X @ Y2 )
=> ( ( image @ A @ A @ ( times_times @ A @ C2 ) @ ( set_or1337092689740270186AtMost @ A @ X @ Y2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( times_times @ A @ C2 @ Y2 ) @ ( times_times @ A @ C2 @ X ) ) ) )
& ( ~ ( ord_less_eq @ A @ X @ Y2 )
=> ( ( image @ A @ A @ ( times_times @ A @ C2 ) @ ( set_or1337092689740270186AtMost @ A @ X @ Y2 ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% image_mult_atLeastAtMost_if
thf(fact_6042_image__mult__atLeastAtMost__if_H,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y2: A,C2: A] :
( ( ( ord_less_eq @ A @ X @ Y2 )
=> ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( image @ A @ A
@ ^ [X4: A] : ( times_times @ A @ X4 @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ X @ Y2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( times_times @ A @ X @ C2 ) @ ( times_times @ A @ Y2 @ C2 ) ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( image @ A @ A
@ ^ [X4: A] : ( times_times @ A @ X4 @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ X @ Y2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( times_times @ A @ Y2 @ C2 ) @ ( times_times @ A @ X @ C2 ) ) ) ) ) )
& ( ~ ( ord_less_eq @ A @ X @ Y2 )
=> ( ( image @ A @ A
@ ^ [X4: A] : ( times_times @ A @ X4 @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ X @ Y2 ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% image_mult_atLeastAtMost_if'
thf(fact_6043_image__affinity__atLeastAtMost,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A,M: A,C2: A] :
( ( ( ( set_or1337092689740270186AtMost @ A @ A2 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( image @ A @ A
@ ^ [X4: A] : ( plus_plus @ A @ ( times_times @ A @ M @ X4 ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( ( set_or1337092689740270186AtMost @ A @ A2 @ B2 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ M )
=> ( ( image @ A @ A
@ ^ [X4: A] : ( plus_plus @ A @ ( times_times @ A @ M @ X4 ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ ( times_times @ A @ M @ A2 ) @ C2 ) @ ( plus_plus @ A @ ( times_times @ A @ M @ B2 ) @ C2 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ M )
=> ( ( image @ A @ A
@ ^ [X4: A] : ( plus_plus @ A @ ( times_times @ A @ M @ X4 ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ ( times_times @ A @ M @ B2 ) @ C2 ) @ ( plus_plus @ A @ ( times_times @ A @ M @ A2 ) @ C2 ) ) ) ) ) ) ) ) ).
% image_affinity_atLeastAtMost
thf(fact_6044_image__affinity__atLeastAtMost__diff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A,M: A,C2: A] :
( ( ( ( set_or1337092689740270186AtMost @ A @ A2 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( image @ A @ A
@ ^ [X4: A] : ( minus_minus @ A @ ( times_times @ A @ M @ X4 ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( ( set_or1337092689740270186AtMost @ A @ A2 @ B2 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ M )
=> ( ( image @ A @ A
@ ^ [X4: A] : ( minus_minus @ A @ ( times_times @ A @ M @ X4 ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ ( times_times @ A @ M @ A2 ) @ C2 ) @ ( minus_minus @ A @ ( times_times @ A @ M @ B2 ) @ C2 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ M )
=> ( ( image @ A @ A
@ ^ [X4: A] : ( minus_minus @ A @ ( times_times @ A @ M @ X4 ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ ( times_times @ A @ M @ B2 ) @ C2 ) @ ( minus_minus @ A @ ( times_times @ A @ M @ A2 ) @ C2 ) ) ) ) ) ) ) ) ).
% image_affinity_atLeastAtMost_diff
thf(fact_6045_image__affinity__atLeastAtMost__div,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A,M: A,C2: A] :
( ( ( ( set_or1337092689740270186AtMost @ A @ A2 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( image @ A @ A
@ ^ [X4: A] : ( plus_plus @ A @ ( divide_divide @ A @ X4 @ M ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( ( set_or1337092689740270186AtMost @ A @ A2 @ B2 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ M )
=> ( ( image @ A @ A
@ ^ [X4: A] : ( plus_plus @ A @ ( divide_divide @ A @ X4 @ M ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ ( divide_divide @ A @ A2 @ M ) @ C2 ) @ ( plus_plus @ A @ ( divide_divide @ A @ B2 @ M ) @ C2 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ M )
=> ( ( image @ A @ A
@ ^ [X4: A] : ( plus_plus @ A @ ( divide_divide @ A @ X4 @ M ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ ( divide_divide @ A @ B2 @ M ) @ C2 ) @ ( plus_plus @ A @ ( divide_divide @ A @ A2 @ M ) @ C2 ) ) ) ) ) ) ) ) ).
% image_affinity_atLeastAtMost_div
thf(fact_6046_image__affinity__atLeastAtMost__div__diff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A,M: A,C2: A] :
( ( ( ( set_or1337092689740270186AtMost @ A @ A2 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( image @ A @ A
@ ^ [X4: A] : ( minus_minus @ A @ ( divide_divide @ A @ X4 @ M ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( ( set_or1337092689740270186AtMost @ A @ A2 @ B2 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ M )
=> ( ( image @ A @ A
@ ^ [X4: A] : ( minus_minus @ A @ ( divide_divide @ A @ X4 @ M ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ ( divide_divide @ A @ A2 @ M ) @ C2 ) @ ( minus_minus @ A @ ( divide_divide @ A @ B2 @ M ) @ C2 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ M )
=> ( ( image @ A @ A
@ ^ [X4: A] : ( minus_minus @ A @ ( divide_divide @ A @ X4 @ M ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ ( divide_divide @ A @ B2 @ M ) @ C2 ) @ ( minus_minus @ A @ ( divide_divide @ A @ A2 @ M ) @ C2 ) ) ) ) ) ) ) ) ).
% image_affinity_atLeastAtMost_div_diff
thf(fact_6047_sum__fun__comp,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( semiring_1 @ C )
=> ! [S3: set @ A,R4: set @ B,G: A > B,F3: B > C] :
( ( finite_finite2 @ A @ S3 )
=> ( ( finite_finite2 @ B @ R4 )
=> ( ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ G @ S3 ) @ R4 )
=> ( ( groups7311177749621191930dd_sum @ A @ C
@ ^ [X4: A] : ( F3 @ ( G @ X4 ) )
@ S3 )
= ( groups7311177749621191930dd_sum @ B @ C
@ ^ [Y: B] :
( times_times @ C
@ ( semiring_1_of_nat @ C
@ ( finite_card @ A
@ ( collect @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ S3 )
& ( ( G @ X4 )
= Y ) ) ) ) )
@ ( F3 @ Y ) )
@ R4 ) ) ) ) ) ) ).
% sum_fun_comp
thf(fact_6048_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_6049_image__minus__const__atLeastLessThan__nat,axiom,
! [C2: nat,Y2: nat,X: nat] :
( ( ( ord_less @ nat @ C2 @ Y2 )
=> ( ( image @ nat @ nat
@ ^ [I: nat] : ( minus_minus @ nat @ I @ C2 )
@ ( set_or7035219750837199246ssThan @ nat @ X @ Y2 ) )
= ( set_or7035219750837199246ssThan @ nat @ ( minus_minus @ nat @ X @ C2 ) @ ( minus_minus @ nat @ Y2 @ C2 ) ) ) )
& ( ~ ( ord_less @ nat @ C2 @ Y2 )
=> ( ( ( ord_less @ nat @ X @ Y2 )
=> ( ( image @ nat @ nat
@ ^ [I: nat] : ( minus_minus @ nat @ I @ C2 )
@ ( set_or7035219750837199246ssThan @ nat @ X @ Y2 ) )
= ( insert @ nat @ ( zero_zero @ nat ) @ ( bot_bot @ ( set @ nat ) ) ) ) )
& ( ~ ( ord_less @ nat @ X @ Y2 )
=> ( ( image @ nat @ nat
@ ^ [I: nat] : ( minus_minus @ nat @ I @ C2 )
@ ( set_or7035219750837199246ssThan @ nat @ X @ Y2 ) )
= ( bot_bot @ ( set @ nat ) ) ) ) ) ) ) ).
% image_minus_const_atLeastLessThan_nat
thf(fact_6050_ccSUP__empty,axiom,
! [B: $tType,A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [F3: B > A] :
( ( complete_Sup_Sup @ A @ ( image @ B @ A @ F3 @ ( bot_bot @ ( set @ B ) ) ) )
= ( bot_bot @ A ) ) ) ).
% ccSUP_empty
thf(fact_6051_ccINF__const,axiom,
! [B: $tType,A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A3: set @ B,F3: A] :
( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Inf_Inf @ A
@ ( image @ B @ A
@ ^ [I: B] : F3
@ A3 ) )
= F3 ) ) ) ).
% ccINF_const
thf(fact_6052_ccSUP__const,axiom,
! [B: $tType,A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A3: set @ B,F3: A] :
( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ A
@ ( image @ B @ A
@ ^ [I: B] : F3
@ A3 ) )
= F3 ) ) ) ).
% ccSUP_const
thf(fact_6053_pair__imageI,axiom,
! [C: $tType,B: $tType,A: $tType,A2: A,B2: B,A3: set @ ( product_prod @ A @ B ),F3: A > B > C] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ A3 )
=> ( member @ C @ ( F3 @ A2 @ B2 ) @ ( image @ ( product_prod @ A @ B ) @ C @ ( product_case_prod @ A @ B @ C @ F3 ) @ A3 ) ) ) ).
% pair_imageI
thf(fact_6054_UN__I,axiom,
! [B: $tType,A: $tType,A2: A,A3: set @ A,B2: B,B5: A > ( set @ B )] :
( ( member @ A @ A2 @ A3 )
=> ( ( member @ B @ B2 @ ( B5 @ A2 ) )
=> ( member @ B @ B2 @ ( complete_Sup_Sup @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ B5 @ A3 ) ) ) ) ) ).
% UN_I
thf(fact_6055_UN__iff,axiom,
! [A: $tType,B: $tType,B2: A,B5: B > ( set @ A ),A3: set @ B] :
( ( member @ A @ B2 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B5 @ A3 ) ) )
= ( ? [X4: B] :
( ( member @ B @ X4 @ A3 )
& ( member @ A @ B2 @ ( B5 @ X4 ) ) ) ) ) ).
% UN_iff
thf(fact_6056_INT__I,axiom,
! [B: $tType,A: $tType,A3: set @ A,B2: B,B5: A > ( set @ B )] :
( ! [X5: A] :
( ( member @ A @ X5 @ A3 )
=> ( member @ B @ B2 @ ( B5 @ X5 ) ) )
=> ( member @ B @ B2 @ ( complete_Inf_Inf @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ B5 @ A3 ) ) ) ) ).
% INT_I
thf(fact_6057_INT__iff,axiom,
! [A: $tType,B: $tType,B2: A,B5: B > ( set @ A ),A3: set @ B] :
( ( member @ A @ B2 @ ( complete_Inf_Inf @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B5 @ A3 ) ) )
= ( ! [X4: B] :
( ( member @ B @ X4 @ A3 )
=> ( member @ A @ B2 @ ( B5 @ X4 ) ) ) ) ) ).
% INT_iff
thf(fact_6058_Sup__apply,axiom,
! [B: $tType,A: $tType] :
( ( complete_Sup @ B )
=> ( ( complete_Sup_Sup @ ( A > B ) )
= ( ^ [A7: set @ ( A > B ),X4: A] :
( complete_Sup_Sup @ B
@ ( image @ ( A > B ) @ B
@ ^ [F5: A > B] : ( F5 @ X4 )
@ A7 ) ) ) ) ) ).
% Sup_apply
thf(fact_6059_Inf__apply,axiom,
! [B: $tType,A: $tType] :
( ( complete_Inf @ B )
=> ( ( complete_Inf_Inf @ ( A > B ) )
= ( ^ [A7: set @ ( A > B ),X4: A] :
( complete_Inf_Inf @ B
@ ( image @ ( A > B ) @ B
@ ^ [F5: A > B] : ( F5 @ X4 )
@ A7 ) ) ) ) ) ).
% Inf_apply
thf(fact_6060_ccSUP__bot,axiom,
! [B: $tType,A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A3: set @ B] :
( ( complete_Sup_Sup @ A
@ ( image @ B @ A
@ ^ [X4: B] : ( bot_bot @ A )
@ A3 ) )
= ( bot_bot @ A ) ) ) ).
% ccSUP_bot
thf(fact_6061_UN__constant,axiom,
! [B: $tType,A: $tType,A3: set @ B,C2: set @ A] :
( ( ( A3
= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ ( set @ A )
@ ( image @ B @ ( set @ A )
@ ^ [Y: B] : C2
@ A3 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ ( set @ A )
@ ( image @ B @ ( set @ A )
@ ^ [Y: B] : C2
@ A3 ) )
= C2 ) ) ) ).
% UN_constant
thf(fact_6062_finite__UN__I,axiom,
! [B: $tType,A: $tType,A3: set @ A,B5: A > ( set @ B )] :
( ( finite_finite2 @ A @ A3 )
=> ( ! [A4: A] :
( ( member @ A @ A4 @ A3 )
=> ( finite_finite2 @ B @ ( B5 @ A4 ) ) )
=> ( finite_finite2 @ B @ ( complete_Sup_Sup @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ B5 @ A3 ) ) ) ) ) ).
% finite_UN_I
thf(fact_6063_finite__INT,axiom,
! [B: $tType,A: $tType,I6: set @ A,A3: A > ( set @ B )] :
( ? [X2: A] :
( ( member @ A @ X2 @ I6 )
& ( finite_finite2 @ B @ ( A3 @ X2 ) ) )
=> ( finite_finite2 @ B @ ( complete_Inf_Inf @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ A3 @ I6 ) ) ) ) ).
% finite_INT
thf(fact_6064_UN__singleton,axiom,
! [A: $tType,A3: set @ A] :
( ( complete_Sup_Sup @ ( set @ A )
@ ( image @ A @ ( set @ A )
@ ^ [X4: A] : ( insert @ A @ X4 @ ( bot_bot @ ( set @ A ) ) )
@ A3 ) )
= A3 ) ).
% UN_singleton
thf(fact_6065_UN__simps_I1_J,axiom,
! [A: $tType,B: $tType,C5: set @ B,A2: A,B5: B > ( set @ A )] :
( ( ( C5
= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ ( set @ A )
@ ( image @ B @ ( set @ A )
@ ^ [X4: B] : ( insert @ A @ A2 @ ( B5 @ X4 ) )
@ C5 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( C5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ ( set @ A )
@ ( image @ B @ ( set @ A )
@ ^ [X4: B] : ( insert @ A @ A2 @ ( B5 @ X4 ) )
@ C5 ) )
= ( insert @ A @ A2 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B5 @ C5 ) ) ) ) ) ) ).
% UN_simps(1)
thf(fact_6066_INT__insert,axiom,
! [A: $tType,B: $tType,B5: B > ( set @ A ),A2: B,A3: set @ B] :
( ( complete_Inf_Inf @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B5 @ ( insert @ B @ A2 @ A3 ) ) )
= ( inf_inf @ ( set @ A ) @ ( B5 @ A2 ) @ ( complete_Inf_Inf @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B5 @ A3 ) ) ) ) ).
% INT_insert
thf(fact_6067_Compl__INT,axiom,
! [A: $tType,B: $tType,B5: B > ( set @ A ),A3: set @ B] :
( ( uminus_uminus @ ( set @ A ) @ ( complete_Inf_Inf @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B5 @ A3 ) ) )
= ( complete_Sup_Sup @ ( set @ A )
@ ( image @ B @ ( set @ A )
@ ^ [X4: B] : ( uminus_uminus @ ( set @ A ) @ ( B5 @ X4 ) )
@ A3 ) ) ) ).
% Compl_INT
thf(fact_6068_Compl__UN,axiom,
! [A: $tType,B: $tType,B5: B > ( set @ A ),A3: set @ B] :
( ( uminus_uminus @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B5 @ A3 ) ) )
= ( complete_Inf_Inf @ ( set @ A )
@ ( image @ B @ ( set @ A )
@ ^ [X4: B] : ( uminus_uminus @ ( set @ A ) @ ( B5 @ X4 ) )
@ A3 ) ) ) ).
% Compl_UN
thf(fact_6069_set__concat,axiom,
! [A: $tType,Xs: list @ ( list @ A )] :
( ( set2 @ A @ ( concat @ A @ Xs ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image @ ( list @ A ) @ ( set @ A ) @ ( set2 @ A ) @ ( set2 @ ( list @ A ) @ Xs ) ) ) ) ).
% set_concat
thf(fact_6070_UN__Pow__subset,axiom,
! [A: $tType,B: $tType,B5: B > ( set @ A ),A3: set @ B] :
( ord_less_eq @ ( set @ ( set @ A ) )
@ ( complete_Sup_Sup @ ( set @ ( set @ A ) )
@ ( image @ B @ ( set @ ( set @ A ) )
@ ^ [X4: B] : ( pow2 @ A @ ( B5 @ X4 ) )
@ A3 ) )
@ ( pow2 @ A @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B5 @ A3 ) ) ) ) ).
% UN_Pow_subset
thf(fact_6071_Pow__INT__eq,axiom,
! [A: $tType,B: $tType,B5: B > ( set @ A ),A3: set @ B] :
( ( pow2 @ A @ ( complete_Inf_Inf @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B5 @ A3 ) ) )
= ( complete_Inf_Inf @ ( set @ ( set @ A ) )
@ ( image @ B @ ( set @ ( set @ A ) )
@ ^ [X4: B] : ( pow2 @ A @ ( B5 @ X4 ) )
@ A3 ) ) ) ).
% Pow_INT_eq
thf(fact_6072_UN__extend__simps_I9_J,axiom,
! [S8: $tType,R7: $tType,Q7: $tType,C5: R7 > ( set @ S8 ),B5: Q7 > ( set @ R7 ),A3: set @ Q7] :
( ( complete_Sup_Sup @ ( set @ S8 )
@ ( image @ Q7 @ ( set @ S8 )
@ ^ [X4: Q7] : ( complete_Sup_Sup @ ( set @ S8 ) @ ( image @ R7 @ ( set @ S8 ) @ C5 @ ( B5 @ X4 ) ) )
@ A3 ) )
= ( complete_Sup_Sup @ ( set @ S8 ) @ ( image @ R7 @ ( set @ S8 ) @ C5 @ ( complete_Sup_Sup @ ( set @ R7 ) @ ( image @ Q7 @ ( set @ R7 ) @ B5 @ A3 ) ) ) ) ) ).
% UN_extend_simps(9)
thf(fact_6073_UN__extend__simps_I8_J,axiom,
! [P8: $tType,O2: $tType,B5: O2 > ( set @ P8 ),A3: set @ ( set @ O2 )] :
( ( complete_Sup_Sup @ ( set @ P8 )
@ ( image @ ( set @ O2 ) @ ( set @ P8 )
@ ^ [Y: set @ O2] : ( complete_Sup_Sup @ ( set @ P8 ) @ ( image @ O2 @ ( set @ P8 ) @ B5 @ Y ) )
@ A3 ) )
= ( complete_Sup_Sup @ ( set @ P8 ) @ ( image @ O2 @ ( set @ P8 ) @ B5 @ ( complete_Sup_Sup @ ( set @ O2 ) @ A3 ) ) ) ) ).
% UN_extend_simps(8)
thf(fact_6074_INT__extend__simps_I8_J,axiom,
! [P8: $tType,O2: $tType,B5: O2 > ( set @ P8 ),A3: set @ ( set @ O2 )] :
( ( complete_Inf_Inf @ ( set @ P8 )
@ ( image @ ( set @ O2 ) @ ( set @ P8 )
@ ^ [Y: set @ O2] : ( complete_Inf_Inf @ ( set @ P8 ) @ ( image @ O2 @ ( set @ P8 ) @ B5 @ Y ) )
@ A3 ) )
= ( complete_Inf_Inf @ ( set @ P8 ) @ ( image @ O2 @ ( set @ P8 ) @ B5 @ ( complete_Sup_Sup @ ( set @ O2 ) @ A3 ) ) ) ) ).
% INT_extend_simps(8)
thf(fact_6075_UN__E,axiom,
! [A: $tType,B: $tType,B2: A,B5: B > ( set @ A ),A3: set @ B] :
( ( member @ A @ B2 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B5 @ A3 ) ) )
=> ~ ! [X5: B] :
( ( member @ B @ X5 @ A3 )
=> ~ ( member @ A @ B2 @ ( B5 @ X5 ) ) ) ) ).
% UN_E
thf(fact_6076_INT__D,axiom,
! [A: $tType,B: $tType,B2: A,B5: B > ( set @ A ),A3: set @ B,A2: B] :
( ( member @ A @ B2 @ ( complete_Inf_Inf @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B5 @ A3 ) ) )
=> ( ( member @ B @ A2 @ A3 )
=> ( member @ A @ B2 @ ( B5 @ A2 ) ) ) ) ).
% INT_D
thf(fact_6077_INT__E,axiom,
! [A: $tType,B: $tType,B2: A,B5: B > ( set @ A ),A3: set @ B,A2: B] :
( ( member @ A @ B2 @ ( complete_Inf_Inf @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B5 @ A3 ) ) )
=> ( ~ ( member @ A @ B2 @ ( B5 @ A2 ) )
=> ~ ( member @ B @ A2 @ A3 ) ) ) ).
% INT_E
thf(fact_6078_Inf__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( complete_Inf @ B )
=> ( ( complete_Inf_Inf @ ( A > B ) )
= ( ^ [A7: set @ ( A > B ),X4: A] :
( complete_Inf_Inf @ B
@ ( image @ ( A > B ) @ B
@ ^ [F5: A > B] : ( F5 @ X4 )
@ A7 ) ) ) ) ) ).
% Inf_fun_def
thf(fact_6079_Inf__set__def,axiom,
! [A: $tType] :
( ( complete_Inf_Inf @ ( set @ A ) )
= ( ^ [A7: set @ ( set @ A )] :
( collect @ A
@ ^ [X4: A] : ( complete_Inf_Inf @ $o @ ( image @ ( set @ A ) @ $o @ ( member @ A @ X4 ) @ A7 ) ) ) ) ) ).
% Inf_set_def
thf(fact_6080_Sup__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( complete_Sup @ B )
=> ( ( complete_Sup_Sup @ ( A > B ) )
= ( ^ [A7: set @ ( A > B ),X4: A] :
( complete_Sup_Sup @ B
@ ( image @ ( A > B ) @ B
@ ^ [F5: A > B] : ( F5 @ X4 )
@ A7 ) ) ) ) ) ).
% Sup_fun_def
thf(fact_6081_Sup__set__def,axiom,
! [A: $tType] :
( ( complete_Sup_Sup @ ( set @ A ) )
= ( ^ [A7: set @ ( set @ A )] :
( collect @ A
@ ^ [X4: A] : ( complete_Sup_Sup @ $o @ ( image @ ( set @ A ) @ $o @ ( member @ A @ X4 ) @ A7 ) ) ) ) ) ).
% Sup_set_def
thf(fact_6082_UN__UN__flatten,axiom,
! [A: $tType,B: $tType,C: $tType,C5: B > ( set @ A ),B5: C > ( set @ B ),A3: set @ C] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ C5 @ ( complete_Sup_Sup @ ( set @ B ) @ ( image @ C @ ( set @ B ) @ B5 @ A3 ) ) ) )
= ( complete_Sup_Sup @ ( set @ A )
@ ( image @ C @ ( set @ A )
@ ^ [Y: C] : ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ C5 @ ( B5 @ Y ) ) )
@ A3 ) ) ) ).
% UN_UN_flatten
thf(fact_6083_SUP__UN__eq,axiom,
! [B: $tType,A: $tType,R: B > ( set @ A ),S3: set @ B] :
( ( complete_Sup_Sup @ ( A > $o )
@ ( image @ B @ ( A > $o )
@ ^ [I: B,X4: A] : ( member @ A @ X4 @ ( R @ I ) )
@ S3 ) )
= ( ^ [X4: A] : ( member @ A @ X4 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ R @ S3 ) ) ) ) ) ).
% SUP_UN_eq
thf(fact_6084_INF__INT__eq,axiom,
! [B: $tType,A: $tType,R: B > ( set @ A ),S3: set @ B] :
( ( complete_Inf_Inf @ ( A > $o )
@ ( image @ B @ ( A > $o )
@ ^ [I: B,X4: A] : ( member @ A @ X4 @ ( R @ I ) )
@ S3 ) )
= ( ^ [X4: A] : ( member @ A @ X4 @ ( complete_Inf_Inf @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ R @ S3 ) ) ) ) ) ).
% INF_INT_eq
thf(fact_6085_INF__Int__eq,axiom,
! [A: $tType,S3: set @ ( set @ A )] :
( ( complete_Inf_Inf @ ( A > $o )
@ ( image @ ( set @ A ) @ ( A > $o )
@ ^ [I: set @ A,X4: A] : ( member @ A @ X4 @ I )
@ S3 ) )
= ( ^ [X4: A] : ( member @ A @ X4 @ ( complete_Inf_Inf @ ( set @ A ) @ S3 ) ) ) ) ).
% INF_Int_eq
thf(fact_6086_SUP__Sup__eq,axiom,
! [A: $tType,S3: set @ ( set @ A )] :
( ( complete_Sup_Sup @ ( A > $o )
@ ( image @ ( set @ A ) @ ( A > $o )
@ ^ [I: set @ A,X4: A] : ( member @ A @ X4 @ I )
@ S3 ) )
= ( ^ [X4: A] : ( member @ A @ X4 @ ( complete_Sup_Sup @ ( set @ A ) @ S3 ) ) ) ) ).
% SUP_Sup_eq
thf(fact_6087_Inf__real__def,axiom,
( ( complete_Inf_Inf @ real )
= ( ^ [X6: set @ real] : ( uminus_uminus @ real @ ( complete_Sup_Sup @ real @ ( image @ real @ real @ ( uminus_uminus @ real ) @ X6 ) ) ) ) ) ).
% Inf_real_def
thf(fact_6088_None__notin__image__Some,axiom,
! [A: $tType,A3: set @ A] :
~ ( member @ ( option @ A ) @ ( none @ A ) @ ( image @ A @ ( option @ A ) @ ( some @ A ) @ A3 ) ) ).
% None_notin_image_Some
thf(fact_6089_SUP__UN__eq2,axiom,
! [B: $tType,C: $tType,A: $tType,R: C > ( set @ ( product_prod @ A @ B ) ),S3: set @ C] :
( ( complete_Sup_Sup @ ( A > B > $o )
@ ( image @ C @ ( A > B > $o )
@ ^ [I: C,X4: A,Y: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y ) @ ( R @ I ) )
@ S3 ) )
= ( ^ [X4: A,Y: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y ) @ ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) ) @ ( image @ C @ ( set @ ( product_prod @ A @ B ) ) @ R @ S3 ) ) ) ) ) ).
% SUP_UN_eq2
thf(fact_6090_INF__INT__eq2,axiom,
! [B: $tType,C: $tType,A: $tType,R: C > ( set @ ( product_prod @ A @ B ) ),S3: set @ C] :
( ( complete_Inf_Inf @ ( A > B > $o )
@ ( image @ C @ ( A > B > $o )
@ ^ [I: C,X4: A,Y: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y ) @ ( R @ I ) )
@ S3 ) )
= ( ^ [X4: A,Y: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y ) @ ( complete_Inf_Inf @ ( set @ ( product_prod @ A @ B ) ) @ ( image @ C @ ( set @ ( product_prod @ A @ B ) ) @ R @ S3 ) ) ) ) ) ).
% INF_INT_eq2
thf(fact_6091_INF__Int__eq2,axiom,
! [B: $tType,A: $tType,S3: set @ ( set @ ( product_prod @ A @ B ) )] :
( ( complete_Inf_Inf @ ( A > B > $o )
@ ( image @ ( set @ ( product_prod @ A @ B ) ) @ ( A > B > $o )
@ ^ [I: set @ ( product_prod @ A @ B ),X4: A,Y: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y ) @ I )
@ S3 ) )
= ( ^ [X4: A,Y: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y ) @ ( complete_Inf_Inf @ ( set @ ( product_prod @ A @ B ) ) @ S3 ) ) ) ) ).
% INF_Int_eq2
thf(fact_6092_SUP__Sup__eq2,axiom,
! [B: $tType,A: $tType,S3: set @ ( set @ ( product_prod @ A @ B ) )] :
( ( complete_Sup_Sup @ ( A > B > $o )
@ ( image @ ( set @ ( product_prod @ A @ B ) ) @ ( A > B > $o )
@ ^ [I: set @ ( product_prod @ A @ B ),X4: A,Y: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y ) @ I )
@ S3 ) )
= ( ^ [X4: A,Y: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y ) @ ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) ) @ S3 ) ) ) ) ).
% SUP_Sup_eq2
thf(fact_6093_Sup__SUP__eq2,axiom,
! [B: $tType,A: $tType] :
( ( complete_Sup_Sup @ ( A > B > $o ) )
= ( ^ [S6: set @ ( A > B > $o ),X4: A,Y: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y ) @ ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) ) @ ( image @ ( ( product_prod @ A @ B ) > $o ) @ ( set @ ( product_prod @ A @ B ) ) @ ( collect @ ( product_prod @ A @ B ) ) @ ( image @ ( A > B > $o ) @ ( ( product_prod @ A @ B ) > $o ) @ ( product_case_prod @ A @ B @ $o ) @ S6 ) ) ) ) ) ) ).
% Sup_SUP_eq2
thf(fact_6094_Inf__INT__eq2,axiom,
! [B: $tType,A: $tType] :
( ( complete_Inf_Inf @ ( A > B > $o ) )
= ( ^ [S6: set @ ( A > B > $o ),X4: A,Y: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y ) @ ( complete_Inf_Inf @ ( set @ ( product_prod @ A @ B ) ) @ ( image @ ( ( product_prod @ A @ B ) > $o ) @ ( set @ ( product_prod @ A @ B ) ) @ ( collect @ ( product_prod @ A @ B ) ) @ ( image @ ( A > B > $o ) @ ( ( product_prod @ A @ B ) > $o ) @ ( product_case_prod @ A @ B @ $o ) @ S6 ) ) ) ) ) ) ).
% Inf_INT_eq2
thf(fact_6095_filter__shuffles,axiom,
! [A: $tType,P3: A > $o,Xs: list @ A,Ys: list @ A] :
( ( image @ ( list @ A ) @ ( list @ A ) @ ( filter2 @ A @ P3 ) @ ( shuffles @ A @ Xs @ Ys ) )
= ( shuffles @ A @ ( filter2 @ A @ P3 @ Xs ) @ ( filter2 @ A @ P3 @ Ys ) ) ) ).
% filter_shuffles
thf(fact_6096_finite__int__iff__bounded,axiom,
( ( finite_finite2 @ int )
= ( ^ [S6: set @ int] :
? [K3: int] : ( ord_less_eq @ ( set @ int ) @ ( image @ int @ int @ ( abs_abs @ int ) @ S6 ) @ ( set_ord_lessThan @ int @ K3 ) ) ) ) ).
% finite_int_iff_bounded
thf(fact_6097_finite__int__iff__bounded__le,axiom,
( ( finite_finite2 @ int )
= ( ^ [S6: set @ int] :
? [K3: int] : ( ord_less_eq @ ( set @ int ) @ ( image @ int @ int @ ( abs_abs @ int ) @ S6 ) @ ( set_ord_atMost @ int @ K3 ) ) ) ) ).
% finite_int_iff_bounded_le
thf(fact_6098_UN__empty2,axiom,
! [B: $tType,A: $tType,A3: set @ B] :
( ( complete_Sup_Sup @ ( set @ A )
@ ( image @ B @ ( set @ A )
@ ^ [X4: B] : ( bot_bot @ ( set @ A ) )
@ A3 ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% UN_empty2
thf(fact_6099_UN__empty,axiom,
! [B: $tType,A: $tType,B5: B > ( set @ A )] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B5 @ ( bot_bot @ ( set @ B ) ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% UN_empty
thf(fact_6100_UNION__empty__conv_I1_J,axiom,
! [A: $tType,B: $tType,B5: B > ( set @ A ),A3: set @ B] :
( ( ( bot_bot @ ( set @ A ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B5 @ A3 ) ) )
= ( ! [X4: B] :
( ( member @ B @ X4 @ A3 )
=> ( ( B5 @ X4 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% UNION_empty_conv(1)
thf(fact_6101_UNION__empty__conv_I2_J,axiom,
! [A: $tType,B: $tType,B5: B > ( set @ A ),A3: set @ B] :
( ( ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B5 @ A3 ) )
= ( bot_bot @ ( set @ A ) ) )
= ( ! [X4: B] :
( ( member @ B @ X4 @ A3 )
=> ( ( B5 @ X4 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% UNION_empty_conv(2)
thf(fact_6102_UN__mono,axiom,
! [B: $tType,A: $tType,A3: set @ A,B5: set @ A,F3: A > ( set @ B ),G: A > ( set @ B )] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B5 )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ A3 )
=> ( ord_less_eq @ ( set @ B ) @ ( F3 @ X5 ) @ ( G @ X5 ) ) )
=> ( ord_less_eq @ ( set @ B ) @ ( complete_Sup_Sup @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ F3 @ A3 ) ) @ ( complete_Sup_Sup @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ G @ B5 ) ) ) ) ) ).
% UN_mono
thf(fact_6103_UN__least,axiom,
! [A: $tType,B: $tType,A3: set @ A,B5: A > ( set @ B ),C5: set @ B] :
( ! [X5: A] :
( ( member @ A @ X5 @ A3 )
=> ( ord_less_eq @ ( set @ B ) @ ( B5 @ X5 ) @ C5 ) )
=> ( ord_less_eq @ ( set @ B ) @ ( complete_Sup_Sup @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ B5 @ A3 ) ) @ C5 ) ) ).
% UN_least
thf(fact_6104_UN__upper,axiom,
! [B: $tType,A: $tType,A2: A,A3: set @ A,B5: A > ( set @ B )] :
( ( member @ A @ A2 @ A3 )
=> ( ord_less_eq @ ( set @ B ) @ ( B5 @ A2 ) @ ( complete_Sup_Sup @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ B5 @ A3 ) ) ) ) ).
% UN_upper
thf(fact_6105_UN__subset__iff,axiom,
! [A: $tType,B: $tType,A3: B > ( set @ A ),I6: set @ B,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ A3 @ I6 ) ) @ B5 )
= ( ! [X4: B] :
( ( member @ B @ X4 @ I6 )
=> ( ord_less_eq @ ( set @ A ) @ ( A3 @ X4 ) @ B5 ) ) ) ) ).
% UN_subset_iff
thf(fact_6106_UN__insert__distrib,axiom,
! [B: $tType,A: $tType,U: A,A3: set @ A,A2: B,B5: A > ( set @ B )] :
( ( member @ A @ U @ A3 )
=> ( ( complete_Sup_Sup @ ( set @ B )
@ ( image @ A @ ( set @ B )
@ ^ [X4: A] : ( insert @ B @ A2 @ ( B5 @ X4 ) )
@ A3 ) )
= ( insert @ B @ A2 @ ( complete_Sup_Sup @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ B5 @ A3 ) ) ) ) ) ).
% UN_insert_distrib
thf(fact_6107_Int__UN__distrib2,axiom,
! [A: $tType,C: $tType,B: $tType,A3: B > ( set @ A ),I6: set @ B,B5: C > ( set @ A ),J5: set @ C] :
( ( inf_inf @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ A3 @ I6 ) ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ C @ ( set @ A ) @ B5 @ J5 ) ) )
= ( complete_Sup_Sup @ ( set @ A )
@ ( image @ B @ ( set @ A )
@ ^ [I: B] :
( complete_Sup_Sup @ ( set @ A )
@ ( image @ C @ ( set @ A )
@ ^ [J4: C] : ( inf_inf @ ( set @ A ) @ ( A3 @ I ) @ ( B5 @ J4 ) )
@ J5 ) )
@ I6 ) ) ) ).
% Int_UN_distrib2
thf(fact_6108_Int__UN__distrib,axiom,
! [A: $tType,B: $tType,B5: set @ A,A3: B > ( set @ A ),I6: set @ B] :
( ( inf_inf @ ( set @ A ) @ B5 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ A3 @ I6 ) ) )
= ( complete_Sup_Sup @ ( set @ A )
@ ( image @ B @ ( set @ A )
@ ^ [I: B] : ( inf_inf @ ( set @ A ) @ B5 @ ( A3 @ I ) )
@ I6 ) ) ) ).
% Int_UN_distrib
thf(fact_6109_UN__extend__simps_I4_J,axiom,
! [H5: $tType,G4: $tType,A3: G4 > ( set @ H5 ),C5: set @ G4,B5: set @ H5] :
( ( inf_inf @ ( set @ H5 ) @ ( complete_Sup_Sup @ ( set @ H5 ) @ ( image @ G4 @ ( set @ H5 ) @ A3 @ C5 ) ) @ B5 )
= ( complete_Sup_Sup @ ( set @ H5 )
@ ( image @ G4 @ ( set @ H5 )
@ ^ [X4: G4] : ( inf_inf @ ( set @ H5 ) @ ( A3 @ X4 ) @ B5 )
@ C5 ) ) ) ).
% UN_extend_simps(4)
thf(fact_6110_UN__extend__simps_I5_J,axiom,
! [I5: $tType,J6: $tType,A3: set @ I5,B5: J6 > ( set @ I5 ),C5: set @ J6] :
( ( inf_inf @ ( set @ I5 ) @ A3 @ ( complete_Sup_Sup @ ( set @ I5 ) @ ( image @ J6 @ ( set @ I5 ) @ B5 @ C5 ) ) )
= ( complete_Sup_Sup @ ( set @ I5 )
@ ( image @ J6 @ ( set @ I5 )
@ ^ [X4: J6] : ( inf_inf @ ( set @ I5 ) @ A3 @ ( B5 @ X4 ) )
@ C5 ) ) ) ).
% UN_extend_simps(5)
thf(fact_6111_Int__Union2,axiom,
! [A: $tType,B5: set @ ( set @ A ),A3: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ B5 ) @ A3 )
= ( complete_Sup_Sup @ ( set @ A )
@ ( image @ ( set @ A ) @ ( set @ A )
@ ^ [C8: set @ A] : ( inf_inf @ ( set @ A ) @ C8 @ A3 )
@ B5 ) ) ) ).
% Int_Union2
thf(fact_6112_Int__Union,axiom,
! [A: $tType,A3: set @ A,B5: set @ ( set @ A )] :
( ( inf_inf @ ( set @ A ) @ A3 @ ( complete_Sup_Sup @ ( set @ A ) @ B5 ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image @ ( set @ A ) @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A3 ) @ B5 ) ) ) ).
% Int_Union
thf(fact_6113_UN__extend__simps_I6_J,axiom,
! [L5: $tType,K9: $tType,A3: K9 > ( set @ L5 ),C5: set @ K9,B5: set @ L5] :
( ( minus_minus @ ( set @ L5 ) @ ( complete_Sup_Sup @ ( set @ L5 ) @ ( image @ K9 @ ( set @ L5 ) @ A3 @ C5 ) ) @ B5 )
= ( complete_Sup_Sup @ ( set @ L5 )
@ ( image @ K9 @ ( set @ L5 )
@ ^ [X4: K9] : ( minus_minus @ ( set @ L5 ) @ ( A3 @ X4 ) @ B5 )
@ C5 ) ) ) ).
% UN_extend_simps(6)
thf(fact_6114_INT__lower,axiom,
! [B: $tType,A: $tType,A2: A,A3: set @ A,B5: A > ( set @ B )] :
( ( member @ A @ A2 @ A3 )
=> ( ord_less_eq @ ( set @ B ) @ ( complete_Inf_Inf @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ B5 @ A3 ) ) @ ( B5 @ A2 ) ) ) ).
% INT_lower
thf(fact_6115_INT__greatest,axiom,
! [B: $tType,A: $tType,A3: set @ A,C5: set @ B,B5: A > ( set @ B )] :
( ! [X5: A] :
( ( member @ A @ X5 @ A3 )
=> ( ord_less_eq @ ( set @ B ) @ C5 @ ( B5 @ X5 ) ) )
=> ( ord_less_eq @ ( set @ B ) @ C5 @ ( complete_Inf_Inf @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ B5 @ A3 ) ) ) ) ).
% INT_greatest
thf(fact_6116_INT__anti__mono,axiom,
! [B: $tType,A: $tType,A3: set @ A,B5: set @ A,F3: A > ( set @ B ),G: A > ( set @ B )] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B5 )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ A3 )
=> ( ord_less_eq @ ( set @ B ) @ ( F3 @ X5 ) @ ( G @ X5 ) ) )
=> ( ord_less_eq @ ( set @ B ) @ ( complete_Inf_Inf @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ F3 @ B5 ) ) @ ( complete_Inf_Inf @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ G @ A3 ) ) ) ) ) ).
% INT_anti_mono
thf(fact_6117_INT__subset__iff,axiom,
! [A: $tType,B: $tType,B5: set @ A,A3: B > ( set @ A ),I6: set @ B] :
( ( ord_less_eq @ ( set @ A ) @ B5 @ ( complete_Inf_Inf @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ A3 @ I6 ) ) )
= ( ! [X4: B] :
( ( member @ B @ X4 @ I6 )
=> ( ord_less_eq @ ( set @ A ) @ B5 @ ( A3 @ X4 ) ) ) ) ) ).
% INT_subset_iff
thf(fact_6118_INT__insert__distrib,axiom,
! [B: $tType,A: $tType,U: A,A3: set @ A,A2: B,B5: A > ( set @ B )] :
( ( member @ A @ U @ A3 )
=> ( ( complete_Inf_Inf @ ( set @ B )
@ ( image @ A @ ( set @ B )
@ ^ [X4: A] : ( insert @ B @ A2 @ ( B5 @ X4 ) )
@ A3 ) )
= ( insert @ B @ A2 @ ( complete_Inf_Inf @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ B5 @ A3 ) ) ) ) ) ).
% INT_insert_distrib
thf(fact_6119_INT__extend__simps_I5_J,axiom,
! [I5: $tType,J6: $tType,A2: I5,B5: J6 > ( set @ I5 ),C5: set @ J6] :
( ( insert @ I5 @ A2 @ ( complete_Inf_Inf @ ( set @ I5 ) @ ( image @ J6 @ ( set @ I5 ) @ B5 @ C5 ) ) )
= ( complete_Inf_Inf @ ( set @ I5 )
@ ( image @ J6 @ ( set @ I5 )
@ ^ [X4: J6] : ( insert @ I5 @ A2 @ ( B5 @ X4 ) )
@ C5 ) ) ) ).
% INT_extend_simps(5)
thf(fact_6120_Int__Inter__image,axiom,
! [A: $tType,B: $tType,A3: B > ( set @ A ),B5: B > ( set @ A ),C5: set @ B] :
( ( complete_Inf_Inf @ ( set @ A )
@ ( image @ B @ ( set @ A )
@ ^ [X4: B] : ( inf_inf @ ( set @ A ) @ ( A3 @ X4 ) @ ( B5 @ X4 ) )
@ C5 ) )
= ( inf_inf @ ( set @ A ) @ ( complete_Inf_Inf @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ A3 @ C5 ) ) @ ( complete_Inf_Inf @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B5 @ C5 ) ) ) ) ).
% Int_Inter_image
thf(fact_6121_INT__Int__distrib,axiom,
! [A: $tType,B: $tType,A3: B > ( set @ A ),B5: B > ( set @ A ),I6: set @ B] :
( ( complete_Inf_Inf @ ( set @ A )
@ ( image @ B @ ( set @ A )
@ ^ [I: B] : ( inf_inf @ ( set @ A ) @ ( A3 @ I ) @ ( B5 @ I ) )
@ I6 ) )
= ( inf_inf @ ( set @ A ) @ ( complete_Inf_Inf @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ A3 @ I6 ) ) @ ( complete_Inf_Inf @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B5 @ I6 ) ) ) ) ).
% INT_Int_distrib
thf(fact_6122_INT__absorb,axiom,
! [B: $tType,A: $tType,K2: A,I6: set @ A,A3: A > ( set @ B )] :
( ( member @ A @ K2 @ I6 )
=> ( ( inf_inf @ ( set @ B ) @ ( A3 @ K2 ) @ ( complete_Inf_Inf @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ A3 @ I6 ) ) )
= ( complete_Inf_Inf @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ A3 @ I6 ) ) ) ) ).
% INT_absorb
thf(fact_6123_INT__extend__simps_I9_J,axiom,
! [S8: $tType,R7: $tType,Q7: $tType,C5: R7 > ( set @ S8 ),B5: Q7 > ( set @ R7 ),A3: set @ Q7] :
( ( complete_Inf_Inf @ ( set @ S8 )
@ ( image @ Q7 @ ( set @ S8 )
@ ^ [X4: Q7] : ( complete_Inf_Inf @ ( set @ S8 ) @ ( image @ R7 @ ( set @ S8 ) @ C5 @ ( B5 @ X4 ) ) )
@ A3 ) )
= ( complete_Inf_Inf @ ( set @ S8 ) @ ( image @ R7 @ ( set @ S8 ) @ C5 @ ( complete_Sup_Sup @ ( set @ R7 ) @ ( image @ Q7 @ ( set @ R7 ) @ B5 @ A3 ) ) ) ) ) ).
% INT_extend_simps(9)
thf(fact_6124_UN__extend__simps_I1_J,axiom,
! [A: $tType,B: $tType,C5: set @ B,A2: A,B5: B > ( set @ A )] :
( ( ( C5
= ( bot_bot @ ( set @ B ) ) )
=> ( ( insert @ A @ A2 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B5 @ C5 ) ) )
= ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) )
& ( ( C5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( insert @ A @ A2 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B5 @ C5 ) ) )
= ( complete_Sup_Sup @ ( set @ A )
@ ( image @ B @ ( set @ A )
@ ^ [X4: B] : ( insert @ A @ A2 @ ( B5 @ X4 ) )
@ C5 ) ) ) ) ) ).
% UN_extend_simps(1)
thf(fact_6125_INT__extend__simps_I2_J,axiom,
! [C: $tType,D: $tType,C5: set @ D,A3: set @ C,B5: D > ( set @ C )] :
( ( ( C5
= ( bot_bot @ ( set @ D ) ) )
=> ( ( inf_inf @ ( set @ C ) @ A3 @ ( complete_Inf_Inf @ ( set @ C ) @ ( image @ D @ ( set @ C ) @ B5 @ C5 ) ) )
= A3 ) )
& ( ( C5
!= ( bot_bot @ ( set @ D ) ) )
=> ( ( inf_inf @ ( set @ C ) @ A3 @ ( complete_Inf_Inf @ ( set @ C ) @ ( image @ D @ ( set @ C ) @ B5 @ C5 ) ) )
= ( complete_Inf_Inf @ ( set @ C )
@ ( image @ D @ ( set @ C )
@ ^ [X4: D] : ( inf_inf @ ( set @ C ) @ A3 @ ( B5 @ X4 ) )
@ C5 ) ) ) ) ) ).
% INT_extend_simps(2)
thf(fact_6126_INT__extend__simps_I1_J,axiom,
! [B: $tType,A: $tType,C5: set @ A,A3: A > ( set @ B ),B5: set @ B] :
( ( ( C5
= ( bot_bot @ ( set @ A ) ) )
=> ( ( inf_inf @ ( set @ B ) @ ( complete_Inf_Inf @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ A3 @ C5 ) ) @ B5 )
= B5 ) )
& ( ( C5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( inf_inf @ ( set @ B ) @ ( complete_Inf_Inf @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ A3 @ C5 ) ) @ B5 )
= ( complete_Inf_Inf @ ( set @ B )
@ ( image @ A @ ( set @ B )
@ ^ [X4: A] : ( inf_inf @ ( set @ B ) @ ( A3 @ X4 ) @ B5 )
@ C5 ) ) ) ) ) ).
% INT_extend_simps(1)
thf(fact_6127_bij__betw__UNION__chain,axiom,
! [B: $tType,C: $tType,A: $tType,I6: set @ A,A3: A > ( set @ B ),F3: B > C,A10: A > ( set @ C )] :
( ! [I3: A,J2: A] :
( ( member @ A @ I3 @ I6 )
=> ( ( member @ A @ J2 @ I6 )
=> ( ( ord_less_eq @ ( set @ B ) @ ( A3 @ I3 ) @ ( A3 @ J2 ) )
| ( ord_less_eq @ ( set @ B ) @ ( A3 @ J2 ) @ ( A3 @ I3 ) ) ) ) )
=> ( ! [I3: A] :
( ( member @ A @ I3 @ I6 )
=> ( bij_betw @ B @ C @ F3 @ ( A3 @ I3 ) @ ( A10 @ I3 ) ) )
=> ( bij_betw @ B @ C @ F3 @ ( complete_Sup_Sup @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ A3 @ I6 ) ) @ ( complete_Sup_Sup @ ( set @ C ) @ ( image @ A @ ( set @ C ) @ A10 @ I6 ) ) ) ) ) ).
% bij_betw_UNION_chain
thf(fact_6128_UN__extend__simps_I7_J,axiom,
! [M11: $tType,N10: $tType,A3: set @ M11,B5: N10 > ( set @ M11 ),C5: set @ N10] :
( ( minus_minus @ ( set @ M11 ) @ A3 @ ( complete_Inf_Inf @ ( set @ M11 ) @ ( image @ N10 @ ( set @ M11 ) @ B5 @ C5 ) ) )
= ( complete_Sup_Sup @ ( set @ M11 )
@ ( image @ N10 @ ( set @ M11 )
@ ^ [X4: N10] : ( minus_minus @ ( set @ M11 ) @ A3 @ ( B5 @ X4 ) )
@ C5 ) ) ) ).
% UN_extend_simps(7)
thf(fact_6129_Int__Inter__eq_I2_J,axiom,
! [A: $tType,B11: set @ ( set @ A ),A3: set @ A] :
( ( ( B11
= ( bot_bot @ ( set @ ( set @ A ) ) ) )
=> ( ( inf_inf @ ( set @ A ) @ ( complete_Inf_Inf @ ( set @ A ) @ B11 ) @ A3 )
= A3 ) )
& ( ( B11
!= ( bot_bot @ ( set @ ( set @ A ) ) ) )
=> ( ( inf_inf @ ( set @ A ) @ ( complete_Inf_Inf @ ( set @ A ) @ B11 ) @ A3 )
= ( complete_Inf_Inf @ ( set @ A )
@ ( image @ ( set @ A ) @ ( set @ A )
@ ^ [B7: set @ A] : ( inf_inf @ ( set @ A ) @ B7 @ A3 )
@ B11 ) ) ) ) ) ).
% Int_Inter_eq(2)
thf(fact_6130_Int__Inter__eq_I1_J,axiom,
! [A: $tType,B11: set @ ( set @ A ),A3: set @ A] :
( ( ( B11
= ( bot_bot @ ( set @ ( set @ A ) ) ) )
=> ( ( inf_inf @ ( set @ A ) @ A3 @ ( complete_Inf_Inf @ ( set @ A ) @ B11 ) )
= A3 ) )
& ( ( B11
!= ( bot_bot @ ( set @ ( set @ A ) ) ) )
=> ( ( inf_inf @ ( set @ A ) @ A3 @ ( complete_Inf_Inf @ ( set @ A ) @ B11 ) )
= ( complete_Inf_Inf @ ( set @ A ) @ ( image @ ( set @ A ) @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A3 ) @ B11 ) ) ) ) ) ).
% Int_Inter_eq(1)
thf(fact_6131_Collect__split__mono__strong,axiom,
! [B: $tType,A: $tType,X9: set @ A,A3: set @ ( product_prod @ A @ B ),Y3: set @ B,P3: A > B > $o,Q2: A > B > $o] :
( ( X9
= ( image @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ A3 ) )
=> ( ( Y3
= ( image @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ A3 ) )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ X9 )
=> ! [Xa3: B] :
( ( member @ B @ Xa3 @ Y3 )
=> ( ( P3 @ X5 @ Xa3 )
=> ( Q2 @ X5 @ Xa3 ) ) ) )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ A3 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ P3 ) ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ A3 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ Q2 ) ) ) ) ) ) ) ).
% Collect_split_mono_strong
thf(fact_6132_INT__extend__simps_I4_J,axiom,
! [G4: $tType,H5: $tType,C5: set @ H5,A3: set @ G4,B5: H5 > ( set @ G4 )] :
( ( ( C5
= ( bot_bot @ ( set @ H5 ) ) )
=> ( ( minus_minus @ ( set @ G4 ) @ A3 @ ( complete_Sup_Sup @ ( set @ G4 ) @ ( image @ H5 @ ( set @ G4 ) @ B5 @ C5 ) ) )
= A3 ) )
& ( ( C5
!= ( bot_bot @ ( set @ H5 ) ) )
=> ( ( minus_minus @ ( set @ G4 ) @ A3 @ ( complete_Sup_Sup @ ( set @ G4 ) @ ( image @ H5 @ ( set @ G4 ) @ B5 @ C5 ) ) )
= ( complete_Inf_Inf @ ( set @ G4 )
@ ( image @ H5 @ ( set @ G4 )
@ ^ [X4: H5] : ( minus_minus @ ( set @ G4 ) @ A3 @ ( B5 @ X4 ) )
@ C5 ) ) ) ) ) ).
% INT_extend_simps(4)
thf(fact_6133_subset__subseqs,axiom,
! [A: $tType,X9: set @ A,Xs: list @ A] :
( ( ord_less_eq @ ( set @ A ) @ X9 @ ( set2 @ A @ Xs ) )
=> ( member @ ( set @ A ) @ X9 @ ( image @ ( list @ A ) @ ( set @ A ) @ ( set2 @ A ) @ ( set2 @ ( list @ A ) @ ( subseqs @ A @ Xs ) ) ) ) ) ).
% subset_subseqs
thf(fact_6134_subseqs__powset,axiom,
! [A: $tType,Xs: list @ A] :
( ( image @ ( list @ A ) @ ( set @ A ) @ ( set2 @ A ) @ ( set2 @ ( list @ A ) @ ( subseqs @ A @ Xs ) ) )
= ( pow2 @ A @ ( set2 @ A @ Xs ) ) ) ).
% subseqs_powset
thf(fact_6135_image__add__int__atLeastLessThan,axiom,
! [L: int,U: int] :
( ( image @ int @ int
@ ^ [X4: int] : ( plus_plus @ int @ X4 @ L )
@ ( set_or7035219750837199246ssThan @ int @ ( zero_zero @ int ) @ ( minus_minus @ int @ U @ L ) ) )
= ( set_or7035219750837199246ssThan @ int @ L @ U ) ) ).
% image_add_int_atLeastLessThan
thf(fact_6136_sum_OUNION__disjoint,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [I6: set @ B,A3: B > ( set @ C ),G: C > A] :
( ( finite_finite2 @ B @ I6 )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ I6 )
=> ( finite_finite2 @ C @ ( A3 @ X5 ) ) )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ I6 )
=> ! [Xa3: B] :
( ( member @ B @ Xa3 @ I6 )
=> ( ( X5 != Xa3 )
=> ( ( inf_inf @ ( set @ C ) @ ( A3 @ X5 ) @ ( A3 @ Xa3 ) )
= ( bot_bot @ ( set @ C ) ) ) ) ) )
=> ( ( groups7311177749621191930dd_sum @ C @ A @ G @ ( complete_Sup_Sup @ ( set @ C ) @ ( image @ B @ ( set @ C ) @ A3 @ I6 ) ) )
= ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X4: B] : ( groups7311177749621191930dd_sum @ C @ A @ G @ ( A3 @ X4 ) )
@ I6 ) ) ) ) ) ) ).
% sum.UNION_disjoint
thf(fact_6137_prod_OUNION__disjoint,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [I6: set @ B,A3: B > ( set @ C ),G: C > A] :
( ( finite_finite2 @ B @ I6 )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ I6 )
=> ( finite_finite2 @ C @ ( A3 @ X5 ) ) )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ I6 )
=> ! [Xa3: B] :
( ( member @ B @ Xa3 @ I6 )
=> ( ( X5 != Xa3 )
=> ( ( inf_inf @ ( set @ C ) @ ( A3 @ X5 ) @ ( A3 @ Xa3 ) )
= ( bot_bot @ ( set @ C ) ) ) ) ) )
=> ( ( groups7121269368397514597t_prod @ C @ A @ G @ ( complete_Sup_Sup @ ( set @ C ) @ ( image @ B @ ( set @ C ) @ A3 @ I6 ) ) )
= ( groups7121269368397514597t_prod @ B @ A
@ ^ [X4: B] : ( groups7121269368397514597t_prod @ C @ A @ G @ ( A3 @ X4 ) )
@ I6 ) ) ) ) ) ) ).
% prod.UNION_disjoint
thf(fact_6138_card__UN__le,axiom,
! [B: $tType,A: $tType,I6: set @ A,A3: A > ( set @ B )] :
( ( finite_finite2 @ A @ I6 )
=> ( ord_less_eq @ nat @ ( finite_card @ B @ ( complete_Sup_Sup @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ A3 @ I6 ) ) )
@ ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [I: A] : ( finite_card @ B @ ( A3 @ I ) )
@ I6 ) ) ) ).
% card_UN_le
thf(fact_6139_card__UN__disjoint,axiom,
! [B: $tType,A: $tType,I6: set @ A,A3: A > ( set @ B )] :
( ( finite_finite2 @ A @ I6 )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ I6 )
=> ( finite_finite2 @ B @ ( A3 @ X5 ) ) )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ I6 )
=> ! [Xa3: A] :
( ( member @ A @ Xa3 @ I6 )
=> ( ( X5 != Xa3 )
=> ( ( inf_inf @ ( set @ B ) @ ( A3 @ X5 ) @ ( A3 @ Xa3 ) )
= ( bot_bot @ ( set @ B ) ) ) ) ) )
=> ( ( finite_card @ B @ ( complete_Sup_Sup @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ A3 @ I6 ) ) )
= ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [I: A] : ( finite_card @ B @ ( A3 @ I ) )
@ I6 ) ) ) ) ) ).
% card_UN_disjoint
thf(fact_6140_INF__nat__binary,axiom,
! [A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A3: A,B5: A] :
( ( inf_inf @ A @ A3
@ ( complete_Inf_Inf @ A
@ ( image @ nat @ A
@ ^ [X4: nat] : B5
@ ( collect @ nat @ ( ord_less @ nat @ ( zero_zero @ nat ) ) ) ) ) )
= ( inf_inf @ A @ A3 @ B5 ) ) ) ).
% INF_nat_binary
thf(fact_6141_Union__natural,axiom,
! [B: $tType,A: $tType,F3: A > B] :
( ( comp @ ( set @ ( set @ B ) ) @ ( set @ B ) @ ( set @ ( set @ A ) ) @ ( complete_Sup_Sup @ ( set @ B ) ) @ ( image @ ( set @ A ) @ ( set @ B ) @ ( image @ A @ B @ F3 ) ) )
= ( comp @ ( set @ A ) @ ( set @ B ) @ ( set @ ( set @ A ) ) @ ( image @ A @ B @ F3 ) @ ( complete_Sup_Sup @ ( set @ A ) ) ) ) ).
% Union_natural
thf(fact_6142_SUP2__I,axiom,
! [B: $tType,A: $tType,C: $tType,A2: A,A3: set @ A,B5: A > B > C > $o,B2: B,C2: C] :
( ( member @ A @ A2 @ A3 )
=> ( ( B5 @ A2 @ B2 @ C2 )
=> ( complete_Sup_Sup @ ( B > C > $o ) @ ( image @ A @ ( B > C > $o ) @ B5 @ A3 ) @ B2 @ C2 ) ) ) ).
% SUP2_I
thf(fact_6143_INF1__I,axiom,
! [A: $tType,B: $tType,A3: set @ A,B5: A > B > $o,B2: B] :
( ! [X5: A] :
( ( member @ A @ X5 @ A3 )
=> ( B5 @ X5 @ B2 ) )
=> ( complete_Inf_Inf @ ( B > $o ) @ ( image @ A @ ( B > $o ) @ B5 @ A3 ) @ B2 ) ) ).
% INF1_I
thf(fact_6144_INF2__I,axiom,
! [B: $tType,A: $tType,C: $tType,A3: set @ A,B5: A > B > C > $o,B2: B,C2: C] :
( ! [X5: A] :
( ( member @ A @ X5 @ A3 )
=> ( B5 @ X5 @ B2 @ C2 ) )
=> ( complete_Inf_Inf @ ( B > C > $o ) @ ( image @ A @ ( B > C > $o ) @ B5 @ A3 ) @ B2 @ C2 ) ) ).
% INF2_I
thf(fact_6145_SUP1__I,axiom,
! [A: $tType,B: $tType,A2: A,A3: set @ A,B5: A > B > $o,B2: B] :
( ( member @ A @ A2 @ A3 )
=> ( ( B5 @ A2 @ B2 )
=> ( complete_Sup_Sup @ ( B > $o ) @ ( image @ A @ ( B > $o ) @ B5 @ A3 ) @ B2 ) ) ) ).
% SUP1_I
thf(fact_6146_INF1__D,axiom,
! [B: $tType,A: $tType,B5: B > A > $o,A3: set @ B,B2: A,A2: B] :
( ( complete_Inf_Inf @ ( A > $o ) @ ( image @ B @ ( A > $o ) @ B5 @ A3 ) @ B2 )
=> ( ( member @ B @ A2 @ A3 )
=> ( B5 @ A2 @ B2 ) ) ) ).
% INF1_D
thf(fact_6147_INF1__E,axiom,
! [A: $tType,B: $tType,B5: B > A > $o,A3: set @ B,B2: A,A2: B] :
( ( complete_Inf_Inf @ ( A > $o ) @ ( image @ B @ ( A > $o ) @ B5 @ A3 ) @ B2 )
=> ( ~ ( B5 @ A2 @ B2 )
=> ~ ( member @ B @ A2 @ A3 ) ) ) ).
% INF1_E
thf(fact_6148_INF2__D,axiom,
! [A: $tType,C: $tType,B: $tType,B5: C > A > B > $o,A3: set @ C,B2: A,C2: B,A2: C] :
( ( complete_Inf_Inf @ ( A > B > $o ) @ ( image @ C @ ( A > B > $o ) @ B5 @ A3 ) @ B2 @ C2 )
=> ( ( member @ C @ A2 @ A3 )
=> ( B5 @ A2 @ B2 @ C2 ) ) ) ).
% INF2_D
thf(fact_6149_INF2__E,axiom,
! [B: $tType,A: $tType,C: $tType,B5: C > A > B > $o,A3: set @ C,B2: A,C2: B,A2: C] :
( ( complete_Inf_Inf @ ( A > B > $o ) @ ( image @ C @ ( A > B > $o ) @ B5 @ A3 ) @ B2 @ C2 )
=> ( ~ ( B5 @ A2 @ B2 @ C2 )
=> ~ ( member @ C @ A2 @ A3 ) ) ) ).
% INF2_E
thf(fact_6150_SUP1__E,axiom,
! [B: $tType,A: $tType,B5: B > A > $o,A3: set @ B,B2: A] :
( ( complete_Sup_Sup @ ( A > $o ) @ ( image @ B @ ( A > $o ) @ B5 @ A3 ) @ B2 )
=> ~ ! [X5: B] :
( ( member @ B @ X5 @ A3 )
=> ~ ( B5 @ X5 @ B2 ) ) ) ).
% SUP1_E
thf(fact_6151_SUP2__E,axiom,
! [A: $tType,C: $tType,B: $tType,B5: C > A > B > $o,A3: set @ C,B2: A,C2: B] :
( ( complete_Sup_Sup @ ( A > B > $o ) @ ( image @ C @ ( A > B > $o ) @ B5 @ A3 ) @ B2 @ C2 )
=> ~ ! [X5: C] :
( ( member @ C @ X5 @ A3 )
=> ~ ( B5 @ X5 @ B2 @ C2 ) ) ) ).
% SUP2_E
thf(fact_6152_conj__subset__def,axiom,
! [A: $tType,A3: set @ A,P3: A > $o,Q2: A > $o] :
( ( ord_less_eq @ ( set @ A ) @ A3
@ ( collect @ A
@ ^ [X4: A] :
( ( P3 @ X4 )
& ( Q2 @ X4 ) ) ) )
= ( ( ord_less_eq @ ( set @ A ) @ A3 @ ( collect @ A @ P3 ) )
& ( ord_less_eq @ ( set @ A ) @ A3 @ ( collect @ A @ Q2 ) ) ) ) ).
% conj_subset_def
thf(fact_6153_in__Union__o__assoc,axiom,
! [B: $tType,A: $tType,C: $tType,X: A,Gset: B > ( set @ ( set @ A ) ),Gmap: C > B,A3: C] :
( ( member @ A @ X @ ( comp @ B @ ( set @ A ) @ C @ ( comp @ ( set @ ( set @ A ) ) @ ( set @ A ) @ B @ ( complete_Sup_Sup @ ( set @ A ) ) @ Gset ) @ Gmap @ A3 ) )
=> ( member @ A @ X @ ( comp @ ( set @ ( set @ A ) ) @ ( set @ A ) @ C @ ( complete_Sup_Sup @ ( set @ A ) ) @ ( comp @ B @ ( set @ ( set @ A ) ) @ C @ Gset @ Gmap ) @ A3 ) ) ) ).
% in_Union_o_assoc
thf(fact_6154_conj__comp__iff,axiom,
! [B: $tType,A: $tType,P3: B > $o,Q2: B > $o,G: A > B] :
( ( comp @ B @ $o @ A
@ ^ [X4: B] :
( ( P3 @ X4 )
& ( Q2 @ X4 ) )
@ G )
= ( ^ [X4: A] :
( ( comp @ B @ $o @ A @ P3 @ G @ X4 )
& ( comp @ B @ $o @ A @ Q2 @ G @ X4 ) ) ) ) ).
% conj_comp_iff
thf(fact_6155_empty__natural,axiom,
! [C: $tType,B: $tType,D: $tType,A: $tType,F3: A > C,G: D > B] :
( ( comp @ C @ ( set @ B ) @ A
@ ^ [Uu3: C] : ( bot_bot @ ( set @ B ) )
@ F3 )
= ( comp @ ( set @ D ) @ ( set @ B ) @ A @ ( image @ D @ B @ G )
@ ^ [Uu3: A] : ( bot_bot @ ( set @ D ) ) ) ) ).
% empty_natural
thf(fact_6156_UN__image__subset,axiom,
! [C: $tType,A: $tType,B: $tType,F3: B > ( set @ A ),G: C > ( set @ B ),X: C,X9: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ F3 @ ( G @ X ) ) ) @ X9 )
= ( ord_less_eq @ ( set @ B ) @ ( G @ X )
@ ( collect @ B
@ ^ [X4: B] : ( ord_less_eq @ ( set @ A ) @ ( F3 @ X4 ) @ X9 ) ) ) ) ).
% UN_image_subset
thf(fact_6157_times__int__def,axiom,
( ( times_times @ int )
= ( map_fun @ int @ ( product_prod @ nat @ nat ) @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) ) @ ( int > int ) @ rep_Integ @ ( map_fun @ int @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ int @ rep_Integ @ abs_Integ )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X4: nat,Y: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V5: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ X4 @ U2 ) @ ( times_times @ nat @ Y @ V5 ) ) @ ( plus_plus @ nat @ ( times_times @ nat @ X4 @ V5 ) @ ( times_times @ nat @ Y @ U2 ) ) ) ) ) ) ) ).
% times_int_def
thf(fact_6158_minus__int__def,axiom,
( ( minus_minus @ int )
= ( map_fun @ int @ ( product_prod @ nat @ nat ) @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) ) @ ( int > int ) @ rep_Integ @ ( map_fun @ int @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ int @ rep_Integ @ abs_Integ )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X4: nat,Y: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V5: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X4 @ V5 ) @ ( plus_plus @ nat @ Y @ U2 ) ) ) ) ) ) ).
% minus_int_def
thf(fact_6159_plus__int__def,axiom,
( ( plus_plus @ int )
= ( map_fun @ int @ ( product_prod @ nat @ nat ) @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) ) @ ( int > int ) @ rep_Integ @ ( map_fun @ int @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ int @ rep_Integ @ abs_Integ )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X4: nat,Y: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V5: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X4 @ U2 ) @ ( plus_plus @ nat @ Y @ V5 ) ) ) ) ) ) ).
% plus_int_def
thf(fact_6160_length__remdups__concat,axiom,
! [A: $tType,Xss: list @ ( list @ A )] :
( ( size_size @ ( list @ A ) @ ( remdups @ A @ ( concat @ A @ Xss ) ) )
= ( finite_card @ A @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ ( list @ A ) @ ( set @ A ) @ ( set2 @ A ) @ ( set2 @ ( list @ A ) @ Xss ) ) ) ) ) ).
% length_remdups_concat
thf(fact_6161_UN__UN__split__split__eq,axiom,
! [A: $tType,E4: $tType,D: $tType,C: $tType,B: $tType,A3: B > C > D > E4 > ( set @ A ),Y3: set @ ( product_prod @ D @ E4 ),X9: set @ ( product_prod @ B @ C )] :
( ( complete_Sup_Sup @ ( set @ A )
@ ( image @ ( product_prod @ B @ C ) @ ( set @ A )
@ ( product_case_prod @ B @ C @ ( set @ A )
@ ^ [X1: B,X25: C] : ( complete_Sup_Sup @ ( set @ A ) @ ( image @ ( product_prod @ D @ E4 ) @ ( set @ A ) @ ( product_case_prod @ D @ E4 @ ( set @ A ) @ ( A3 @ X1 @ X25 ) ) @ Y3 ) ) )
@ X9 ) )
= ( complete_Sup_Sup @ ( set @ A )
@ ( image @ ( product_prod @ B @ C ) @ ( set @ A )
@ ^ [X4: product_prod @ B @ C] :
( complete_Sup_Sup @ ( set @ A )
@ ( image @ ( product_prod @ D @ E4 ) @ ( set @ A )
@ ^ [Y: product_prod @ D @ E4] :
( product_case_prod @ B @ C @ ( set @ A )
@ ^ [X1: B,X25: C] : ( product_case_prod @ D @ E4 @ ( set @ A ) @ ( A3 @ X1 @ X25 ) @ Y )
@ X4 )
@ Y3 ) )
@ X9 ) ) ) ).
% UN_UN_split_split_eq
thf(fact_6162_remdups__upt,axiom,
! [M: nat,N: nat] :
( ( remdups @ nat @ ( upt @ M @ N ) )
= ( upt @ M @ N ) ) ).
% remdups_upt
thf(fact_6163_remdups__eq__nil__right__iff,axiom,
! [A: $tType,X: list @ A] :
( ( ( nil @ A )
= ( remdups @ A @ X ) )
= ( X
= ( nil @ A ) ) ) ).
% remdups_eq_nil_right_iff
thf(fact_6164_remdups__eq__nil__iff,axiom,
! [A: $tType,X: list @ A] :
( ( ( remdups @ A @ X )
= ( nil @ A ) )
= ( X
= ( nil @ A ) ) ) ).
% remdups_eq_nil_iff
thf(fact_6165_set__remdups,axiom,
! [A: $tType,Xs: list @ A] :
( ( set2 @ A @ ( remdups @ A @ Xs ) )
= ( set2 @ A @ Xs ) ) ).
% set_remdups
thf(fact_6166_length__remdups__eq,axiom,
! [A: $tType,Xs: list @ A] :
( ( ( size_size @ ( list @ A ) @ ( remdups @ A @ Xs ) )
= ( size_size @ ( list @ A ) @ Xs ) )
= ( ( remdups @ A @ Xs )
= Xs ) ) ).
% length_remdups_eq
thf(fact_6167_distinct__remdups,axiom,
! [A: $tType,Xs: list @ A] : ( distinct @ A @ ( remdups @ A @ Xs ) ) ).
% distinct_remdups
thf(fact_6168_remdups__id__iff__distinct,axiom,
! [A: $tType,Xs: list @ A] :
( ( ( remdups @ A @ Xs )
= Xs )
= ( distinct @ A @ Xs ) ) ).
% remdups_id_iff_distinct
thf(fact_6169_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_6170_remdups__map__remdups,axiom,
! [A: $tType,B: $tType,F3: B > A,Xs: list @ B] :
( ( remdups @ A @ ( map @ B @ A @ F3 @ ( remdups @ B @ Xs ) ) )
= ( remdups @ A @ ( map @ B @ A @ F3 @ Xs ) ) ) ).
% remdups_map_remdups
thf(fact_6171_distinct__remdups__id,axiom,
! [A: $tType,Xs: list @ A] :
( ( distinct @ A @ Xs )
=> ( ( remdups @ A @ Xs )
= Xs ) ) ).
% distinct_remdups_id
thf(fact_6172_remdups__remdups,axiom,
! [A: $tType,Xs: list @ A] :
( ( remdups @ A @ ( remdups @ A @ Xs ) )
= ( remdups @ A @ Xs ) ) ).
% remdups_remdups
thf(fact_6173_remdups_Osimps_I1_J,axiom,
! [A: $tType] :
( ( remdups @ A @ ( nil @ A ) )
= ( nil @ A ) ) ).
% remdups.simps(1)
thf(fact_6174_remdups__filter,axiom,
! [A: $tType,P3: A > $o,Xs: list @ A] :
( ( remdups @ A @ ( filter2 @ A @ P3 @ Xs ) )
= ( filter2 @ A @ P3 @ ( remdups @ A @ Xs ) ) ) ).
% remdups_filter
thf(fact_6175_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_6176_remove1__remdups,axiom,
! [A: $tType,Xs: list @ A,X: A] :
( ( distinct @ A @ Xs )
=> ( ( remove1 @ A @ X @ ( remdups @ A @ Xs ) )
= ( remdups @ A @ ( remove1 @ A @ X @ Xs ) ) ) ) ).
% remove1_remdups
thf(fact_6177_length__remdups__card__conv,axiom,
! [A: $tType,Xs: list @ A] :
( ( size_size @ ( list @ A ) @ ( remdups @ A @ Xs ) )
= ( finite_card @ A @ ( set2 @ A @ Xs ) ) ) ).
% length_remdups_card_conv
thf(fact_6178_sum__code,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: B > A,Xs: list @ B] :
( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( set2 @ B @ Xs ) )
= ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ G @ ( remdups @ B @ Xs ) ) ) ) ) ).
% sum_code
thf(fact_6179_UN__constant__eq,axiom,
! [A: $tType,B: $tType,A2: A,A3: set @ A,F3: A > ( set @ B ),C2: set @ B] :
( ( member @ A @ A2 @ A3 )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ A3 )
=> ( ( F3 @ X5 )
= C2 ) )
=> ( ( complete_Sup_Sup @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ F3 @ A3 ) )
= C2 ) ) ) ).
% UN_constant_eq
thf(fact_6180_suminf__eq__SUP__real,axiom,
! [X9: nat > real] :
( ( summable @ real @ X9 )
=> ( ! [I3: nat] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( X9 @ I3 ) )
=> ( ( suminf @ real @ X9 )
= ( complete_Sup_Sup @ real
@ ( image @ nat @ real
@ ^ [I: nat] : ( groups7311177749621191930dd_sum @ nat @ real @ X9 @ ( set_ord_lessThan @ nat @ I ) )
@ ( top_top @ ( set @ nat ) ) ) ) ) ) ) ).
% suminf_eq_SUP_real
thf(fact_6181_finite__mono__strict__prefix__implies__finite__fixpoint,axiom,
! [A: $tType,F3: nat > ( set @ A ),S3: set @ A] :
( ! [I3: nat] : ( ord_less_eq @ ( set @ A ) @ ( F3 @ I3 ) @ S3 )
=> ( ( finite_finite2 @ A @ S3 )
=> ( ? [N7: nat] :
( ! [N3: nat] :
( ( ord_less_eq @ nat @ N3 @ N7 )
=> ! [M4: nat] :
( ( ord_less_eq @ nat @ M4 @ N7 )
=> ( ( ord_less @ nat @ M4 @ N3 )
=> ( ord_less @ ( set @ A ) @ ( F3 @ M4 ) @ ( F3 @ N3 ) ) ) ) )
& ! [N3: nat] :
( ( ord_less_eq @ nat @ N7 @ N3 )
=> ( ( F3 @ N7 )
= ( F3 @ N3 ) ) ) )
=> ( ( F3 @ ( finite_card @ A @ S3 ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ F3 @ ( top_top @ ( set @ nat ) ) ) ) ) ) ) ) ).
% finite_mono_strict_prefix_implies_finite_fixpoint
thf(fact_6182_Collect__const,axiom,
! [A: $tType,P3: $o] :
( ( P3
=> ( ( collect @ A
@ ^ [S4: A] : P3 )
= ( top_top @ ( set @ A ) ) ) )
& ( ~ P3
=> ( ( collect @ A
@ ^ [S4: A] : P3 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Collect_const
thf(fact_6183_finite__Collect__not,axiom,
! [A: $tType,P3: A > $o] :
( ( finite_finite2 @ A @ ( collect @ A @ P3 ) )
=> ( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X4: A] :
~ ( P3 @ X4 ) ) )
= ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) ) ) ).
% finite_Collect_not
thf(fact_6184_surj__plus,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A2: A] :
( ( image @ A @ A @ ( plus_plus @ A @ A2 ) @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) ) ) ).
% surj_plus
thf(fact_6185_range__add,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A] :
( ( image @ A @ A @ ( plus_plus @ A @ A2 ) @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) ) ) ).
% range_add
thf(fact_6186_Sup__eq__top__iff,axiom,
! [A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [A3: set @ A] :
( ( ( complete_Sup_Sup @ A @ A3 )
= ( top_top @ A ) )
= ( ! [X4: A] :
( ( ord_less @ A @ X4 @ ( top_top @ A ) )
=> ? [Y: A] :
( ( member @ A @ Y @ A3 )
& ( ord_less @ A @ X4 @ Y ) ) ) ) ) ) ).
% Sup_eq_top_iff
thf(fact_6187_surj__fn,axiom,
! [A: $tType,F3: A > A,N: nat] :
( ( ( image @ A @ A @ F3 @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) )
=> ( ( image @ A @ A @ ( compow @ ( A > A ) @ N @ F3 ) @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) ) ) ).
% surj_fn
thf(fact_6188_surj__diff__right,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A2: A] :
( ( image @ A @ A
@ ^ [X4: A] : ( minus_minus @ A @ X4 @ A2 )
@ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) ) ) ).
% surj_diff_right
thf(fact_6189_ccINF__top,axiom,
! [B: $tType,A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A3: set @ B] :
( ( complete_Inf_Inf @ A
@ ( image @ B @ A
@ ^ [X4: B] : ( top_top @ A )
@ A3 ) )
= ( top_top @ A ) ) ) ).
% ccINF_top
thf(fact_6190_INF__top,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ B] :
( ( complete_Inf_Inf @ A
@ ( image @ B @ A
@ ^ [X4: B] : ( top_top @ A )
@ A3 ) )
= ( top_top @ A ) ) ) ).
% INF_top
thf(fact_6191_INF__top__conv_I1_J,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [B5: B > A,A3: set @ B] :
( ( ( complete_Inf_Inf @ A @ ( image @ B @ A @ B5 @ A3 ) )
= ( top_top @ A ) )
= ( ! [X4: B] :
( ( member @ B @ X4 @ A3 )
=> ( ( B5 @ X4 )
= ( top_top @ A ) ) ) ) ) ) ).
% INF_top_conv(1)
thf(fact_6192_INF__top__conv_I2_J,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [B5: B > A,A3: set @ B] :
( ( ( top_top @ A )
= ( complete_Inf_Inf @ A @ ( image @ B @ A @ B5 @ A3 ) ) )
= ( ! [X4: B] :
( ( member @ B @ X4 @ A3 )
=> ( ( B5 @ X4 )
= ( top_top @ A ) ) ) ) ) ) ).
% INF_top_conv(2)
thf(fact_6193_SUP__eq__top__iff,axiom,
! [B: $tType,A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [F3: B > A,A3: set @ B] :
( ( ( complete_Sup_Sup @ A @ ( image @ B @ A @ F3 @ A3 ) )
= ( top_top @ A ) )
= ( ! [X4: A] :
( ( ord_less @ A @ X4 @ ( top_top @ A ) )
=> ? [Y: B] :
( ( member @ B @ Y @ A3 )
& ( ord_less @ A @ X4 @ ( F3 @ Y ) ) ) ) ) ) ) ).
% SUP_eq_top_iff
thf(fact_6194_range__constant,axiom,
! [B: $tType,A: $tType,X: A] :
( ( image @ B @ A
@ ^ [Uu3: B] : X
@ ( top_top @ ( set @ B ) ) )
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ).
% range_constant
thf(fact_6195_ccINF__empty,axiom,
! [B: $tType,A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [F3: B > A] :
( ( complete_Inf_Inf @ A @ ( image @ B @ A @ F3 @ ( bot_bot @ ( set @ B ) ) ) )
= ( top_top @ A ) ) ) ).
% ccINF_empty
thf(fact_6196_INT__constant,axiom,
! [B: $tType,A: $tType,A3: set @ B,C2: set @ A] :
( ( ( A3
= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Inf_Inf @ ( set @ A )
@ ( image @ B @ ( set @ A )
@ ^ [Y: B] : C2
@ A3 ) )
= ( top_top @ ( set @ A ) ) ) )
& ( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Inf_Inf @ ( set @ A )
@ ( image @ B @ ( set @ A )
@ ^ [Y: B] : C2
@ A3 ) )
= C2 ) ) ) ).
% INT_constant
thf(fact_6197_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_6198_INT__simps_I1_J,axiom,
! [A: $tType,B: $tType,C5: set @ A,A3: A > ( set @ B ),B5: set @ B] :
( ( ( C5
= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Inf_Inf @ ( set @ B )
@ ( image @ A @ ( set @ B )
@ ^ [X4: A] : ( inf_inf @ ( set @ B ) @ ( A3 @ X4 ) @ B5 )
@ C5 ) )
= ( top_top @ ( set @ B ) ) ) )
& ( ( C5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Inf_Inf @ ( set @ B )
@ ( image @ A @ ( set @ B )
@ ^ [X4: A] : ( inf_inf @ ( set @ B ) @ ( A3 @ X4 ) @ B5 )
@ C5 ) )
= ( inf_inf @ ( set @ B ) @ ( complete_Inf_Inf @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ A3 @ C5 ) ) @ B5 ) ) ) ) ).
% INT_simps(1)
thf(fact_6199_INT__simps_I2_J,axiom,
! [C: $tType,D: $tType,C5: set @ D,A3: set @ C,B5: D > ( set @ C )] :
( ( ( C5
= ( bot_bot @ ( set @ D ) ) )
=> ( ( complete_Inf_Inf @ ( set @ C )
@ ( image @ D @ ( set @ C )
@ ^ [X4: D] : ( inf_inf @ ( set @ C ) @ A3 @ ( B5 @ X4 ) )
@ C5 ) )
= ( top_top @ ( set @ C ) ) ) )
& ( ( C5
!= ( bot_bot @ ( set @ D ) ) )
=> ( ( complete_Inf_Inf @ ( set @ C )
@ ( image @ D @ ( set @ C )
@ ^ [X4: D] : ( inf_inf @ ( set @ C ) @ A3 @ ( B5 @ X4 ) )
@ C5 ) )
= ( inf_inf @ ( set @ C ) @ A3 @ ( complete_Inf_Inf @ ( set @ C ) @ ( image @ D @ ( set @ C ) @ B5 @ C5 ) ) ) ) ) ) ).
% INT_simps(2)
thf(fact_6200_INT__simps_I3_J,axiom,
! [E4: $tType,F: $tType,C5: set @ E4,A3: E4 > ( set @ F ),B5: set @ F] :
( ( ( C5
= ( bot_bot @ ( set @ E4 ) ) )
=> ( ( complete_Inf_Inf @ ( set @ F )
@ ( image @ E4 @ ( set @ F )
@ ^ [X4: E4] : ( minus_minus @ ( set @ F ) @ ( A3 @ X4 ) @ B5 )
@ C5 ) )
= ( top_top @ ( set @ F ) ) ) )
& ( ( C5
!= ( bot_bot @ ( set @ E4 ) ) )
=> ( ( complete_Inf_Inf @ ( set @ F )
@ ( image @ E4 @ ( set @ F )
@ ^ [X4: E4] : ( minus_minus @ ( set @ F ) @ ( A3 @ X4 ) @ B5 )
@ C5 ) )
= ( minus_minus @ ( set @ F ) @ ( complete_Inf_Inf @ ( set @ F ) @ ( image @ E4 @ ( set @ F ) @ A3 @ C5 ) ) @ B5 ) ) ) ) ).
% INT_simps(3)
thf(fact_6201_INT__simps_I4_J,axiom,
! [G4: $tType,H5: $tType,C5: set @ H5,A3: set @ G4,B5: H5 > ( set @ G4 )] :
( ( ( C5
= ( bot_bot @ ( set @ H5 ) ) )
=> ( ( complete_Inf_Inf @ ( set @ G4 )
@ ( image @ H5 @ ( set @ G4 )
@ ^ [X4: H5] : ( minus_minus @ ( set @ G4 ) @ A3 @ ( B5 @ X4 ) )
@ C5 ) )
= ( top_top @ ( set @ G4 ) ) ) )
& ( ( C5
!= ( bot_bot @ ( set @ H5 ) ) )
=> ( ( complete_Inf_Inf @ ( set @ G4 )
@ ( image @ H5 @ ( set @ G4 )
@ ^ [X4: H5] : ( minus_minus @ ( set @ G4 ) @ A3 @ ( B5 @ X4 ) )
@ C5 ) )
= ( minus_minus @ ( set @ G4 ) @ A3 @ ( complete_Sup_Sup @ ( set @ G4 ) @ ( image @ H5 @ ( set @ G4 ) @ B5 @ C5 ) ) ) ) ) ) ).
% INT_simps(4)
thf(fact_6202_sums__SUP,axiom,
! [A: $tType] :
( ( ( comple5582772986160207858norder @ A )
& ( canoni5634975068530333245id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F3: nat > A] :
( sums @ A @ F3
@ ( complete_Sup_Sup @ A
@ ( image @ nat @ A
@ ^ [N2: nat] : ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_ord_lessThan @ nat @ N2 ) )
@ ( top_top @ ( set @ nat ) ) ) ) ) ) ).
% sums_SUP
thf(fact_6203_UN__atMost__UNIV,axiom,
( ( complete_Sup_Sup @ ( set @ nat ) @ ( image @ nat @ ( set @ nat ) @ ( set_ord_atMost @ nat ) @ ( top_top @ ( set @ nat ) ) ) )
= ( top_top @ ( set @ nat ) ) ) ).
% UN_atMost_UNIV
thf(fact_6204_UN__lessThan__UNIV,axiom,
( ( complete_Sup_Sup @ ( set @ nat ) @ ( image @ nat @ ( set @ nat ) @ ( set_ord_lessThan @ nat ) @ ( top_top @ ( set @ nat ) ) ) )
= ( top_top @ ( set @ nat ) ) ) ).
% UN_lessThan_UNIV
thf(fact_6205_range__subsetD,axiom,
! [B: $tType,A: $tType,F3: B > A,B5: set @ A,I2: B] :
( ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) ) @ B5 )
=> ( member @ A @ ( F3 @ I2 ) @ B5 ) ) ).
% range_subsetD
thf(fact_6206_range__composition,axiom,
! [A: $tType,C: $tType,B: $tType,F3: C > A,G: B > C] :
( ( image @ B @ A
@ ^ [X4: B] : ( F3 @ ( G @ X4 ) )
@ ( top_top @ ( set @ B ) ) )
= ( image @ C @ A @ F3 @ ( image @ B @ C @ G @ ( top_top @ ( set @ B ) ) ) ) ) ).
% range_composition
thf(fact_6207_rangeE,axiom,
! [A: $tType,B: $tType,B2: A,F3: B > A] :
( ( member @ A @ B2 @ ( image @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) ) )
=> ~ ! [X5: B] :
( B2
!= ( F3 @ X5 ) ) ) ).
% rangeE
thf(fact_6208_UNIV__option__conv,axiom,
! [A: $tType] :
( ( top_top @ ( set @ ( option @ A ) ) )
= ( insert @ ( option @ A ) @ ( none @ A ) @ ( image @ A @ ( option @ A ) @ ( some @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ).
% UNIV_option_conv
thf(fact_6209_not__UNIV__le__Icc,axiom,
! [A: $tType] :
( ( no_top @ A )
=> ! [L: A,H2: A] :
~ ( ord_less_eq @ ( set @ A ) @ ( top_top @ ( set @ A ) ) @ ( set_or1337092689740270186AtMost @ A @ L @ H2 ) ) ) ).
% not_UNIV_le_Icc
thf(fact_6210_subset__UNIV,axiom,
! [A: $tType,A3: set @ A] : ( ord_less_eq @ ( set @ A ) @ A3 @ ( top_top @ ( set @ A ) ) ) ).
% subset_UNIV
thf(fact_6211_top__greatest,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [A2: A] : ( ord_less_eq @ A @ A2 @ ( top_top @ A ) ) ) ).
% top_greatest
thf(fact_6212_top_Oextremum__unique,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ ( top_top @ A ) @ A2 )
= ( A2
= ( top_top @ A ) ) ) ) ).
% top.extremum_unique
thf(fact_6213_top_Oextremum__uniqueI,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ ( top_top @ A ) @ A2 )
=> ( A2
= ( top_top @ A ) ) ) ) ).
% top.extremum_uniqueI
thf(fact_6214_not__UNIV__le__Iic,axiom,
! [A: $tType] :
( ( no_top @ A )
=> ! [H2: A] :
~ ( ord_less_eq @ ( set @ A ) @ ( top_top @ ( set @ A ) ) @ ( set_ord_atMost @ A @ H2 ) ) ) ).
% not_UNIV_le_Iic
thf(fact_6215_UNIV__def,axiom,
! [A: $tType] :
( ( top_top @ ( set @ A ) )
= ( collect @ A
@ ^ [X4: A] : $true ) ) ).
% UNIV_def
thf(fact_6216_top_Oextremum__strict,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [A2: A] :
~ ( ord_less @ A @ ( top_top @ A ) @ A2 ) ) ).
% top.extremum_strict
thf(fact_6217_top_Onot__eq__extremum,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [A2: A] :
( ( A2
!= ( top_top @ A ) )
= ( ord_less @ A @ A2 @ ( top_top @ A ) ) ) ) ).
% top.not_eq_extremum
thf(fact_6218_bij__fn,axiom,
! [A: $tType,F3: A > A,N: nat] :
( ( bij_betw @ A @ A @ F3 @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ A ) ) )
=> ( bij_betw @ A @ A @ ( compow @ ( A > A ) @ N @ F3 ) @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ A ) ) ) ) ).
% bij_fn
thf(fact_6219_SUP__INF,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [P3: C > B > A] :
( ( complete_Sup_Sup @ A
@ ( image @ B @ A
@ ^ [Y: B] :
( complete_Inf_Inf @ A
@ ( image @ C @ A
@ ^ [X4: C] : ( P3 @ X4 @ Y )
@ ( top_top @ ( set @ C ) ) ) )
@ ( top_top @ ( set @ B ) ) ) )
= ( complete_Inf_Inf @ A
@ ( image @ ( B > C ) @ A
@ ^ [X4: B > C] :
( complete_Sup_Sup @ A
@ ( image @ B @ A
@ ^ [Y: B] : ( P3 @ ( X4 @ Y ) @ Y )
@ ( top_top @ ( set @ B ) ) ) )
@ ( top_top @ ( set @ ( B > C ) ) ) ) ) ) ) ).
% SUP_INF
thf(fact_6220_INF__SUP,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [P3: C > B > A] :
( ( complete_Inf_Inf @ A
@ ( image @ B @ A
@ ^ [Y: B] :
( complete_Sup_Sup @ A
@ ( image @ C @ A
@ ^ [X4: C] : ( P3 @ X4 @ Y )
@ ( top_top @ ( set @ C ) ) ) )
@ ( top_top @ ( set @ B ) ) ) )
= ( complete_Sup_Sup @ A
@ ( image @ ( B > C ) @ A
@ ^ [F5: B > C] :
( complete_Inf_Inf @ A
@ ( image @ B @ A
@ ^ [X4: B] : ( P3 @ ( F5 @ X4 ) @ X4 )
@ ( top_top @ ( set @ B ) ) ) )
@ ( top_top @ ( set @ ( B > C ) ) ) ) ) ) ) ).
% INF_SUP
thf(fact_6221_finite__range__imageI,axiom,
! [C: $tType,A: $tType,B: $tType,G: B > A,F3: A > C] :
( ( finite_finite2 @ A @ ( image @ B @ A @ G @ ( top_top @ ( set @ B ) ) ) )
=> ( finite_finite2 @ C
@ ( image @ B @ C
@ ^ [X4: B] : ( F3 @ ( G @ X4 ) )
@ ( top_top @ ( set @ B ) ) ) ) ) ).
% finite_range_imageI
thf(fact_6222_INTER__UNIV__conv_I2_J,axiom,
! [A: $tType,B: $tType,B5: B > ( set @ A ),A3: set @ B] :
( ( ( complete_Inf_Inf @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B5 @ A3 ) )
= ( top_top @ ( set @ A ) ) )
= ( ! [X4: B] :
( ( member @ B @ X4 @ A3 )
=> ( ( B5 @ X4 )
= ( top_top @ ( set @ A ) ) ) ) ) ) ).
% INTER_UNIV_conv(2)
thf(fact_6223_INTER__UNIV__conv_I1_J,axiom,
! [A: $tType,B: $tType,B5: B > ( set @ A ),A3: set @ B] :
( ( ( top_top @ ( set @ A ) )
= ( complete_Inf_Inf @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B5 @ A3 ) ) )
= ( ! [X4: B] :
( ( member @ B @ X4 @ A3 )
=> ( ( B5 @ X4 )
= ( top_top @ ( set @ A ) ) ) ) ) ) ).
% INTER_UNIV_conv(1)
thf(fact_6224_INF__constant,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ B,C2: A] :
( ( ( A3
= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Inf_Inf @ A
@ ( image @ B @ A
@ ^ [Y: B] : C2
@ A3 ) )
= ( top_top @ A ) ) )
& ( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Inf_Inf @ A
@ ( image @ B @ A
@ ^ [Y: B] : C2
@ A3 ) )
= C2 ) ) ) ) ).
% INF_constant
thf(fact_6225_INF__empty,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F3: B > A] :
( ( complete_Inf_Inf @ A @ ( image @ B @ A @ F3 @ ( bot_bot @ ( set @ B ) ) ) )
= ( top_top @ A ) ) ) ).
% INF_empty
thf(fact_6226_surj__Compl__image__subset,axiom,
! [A: $tType,B: $tType,F3: B > A,A3: set @ B] :
( ( ( image @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
=> ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ ( image @ B @ A @ F3 @ A3 ) ) @ ( image @ B @ A @ F3 @ ( uminus_uminus @ ( set @ B ) @ A3 ) ) ) ) ).
% surj_Compl_image_subset
thf(fact_6227_notin__range__Some,axiom,
! [A: $tType,X: option @ A] :
( ( ~ ( member @ ( option @ A ) @ X @ ( image @ A @ ( option @ A ) @ ( some @ A ) @ ( top_top @ ( set @ A ) ) ) ) )
= ( X
= ( none @ A ) ) ) ).
% notin_range_Some
thf(fact_6228_INT__empty,axiom,
! [B: $tType,A: $tType,B5: B > ( set @ A )] :
( ( complete_Inf_Inf @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B5 @ ( bot_bot @ ( set @ B ) ) ) )
= ( top_top @ ( set @ A ) ) ) ).
% INT_empty
thf(fact_6229_sorted__list__of__set_Ofolding__insort__key__axioms,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( folding_insort_key @ A @ A @ ( ord_less_eq @ A ) @ ( ord_less @ A ) @ ( top_top @ ( set @ A ) )
@ ^ [X4: A] : X4 ) ) ).
% sorted_list_of_set.folding_insort_key_axioms
thf(fact_6230_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_6231_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_6232_INT__extend__simps_I3_J,axiom,
! [F: $tType,E4: $tType,C5: set @ E4,A3: E4 > ( set @ F ),B5: set @ F] :
( ( ( C5
= ( bot_bot @ ( set @ E4 ) ) )
=> ( ( minus_minus @ ( set @ F ) @ ( complete_Inf_Inf @ ( set @ F ) @ ( image @ E4 @ ( set @ F ) @ A3 @ C5 ) ) @ B5 )
= ( minus_minus @ ( set @ F ) @ ( top_top @ ( set @ F ) ) @ B5 ) ) )
& ( ( C5
!= ( bot_bot @ ( set @ E4 ) ) )
=> ( ( minus_minus @ ( set @ F ) @ ( complete_Inf_Inf @ ( set @ F ) @ ( image @ E4 @ ( set @ F ) @ A3 @ C5 ) ) @ B5 )
= ( complete_Inf_Inf @ ( set @ F )
@ ( image @ E4 @ ( set @ F )
@ ^ [X4: E4] : ( minus_minus @ ( set @ F ) @ ( A3 @ X4 ) @ B5 )
@ C5 ) ) ) ) ) ).
% INT_extend_simps(3)
thf(fact_6233_bij__image__INT,axiom,
! [B: $tType,A: $tType,C: $tType,F3: A > B,B5: C > ( set @ A ),A3: set @ C] :
( ( bij_betw @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
=> ( ( image @ A @ B @ F3 @ ( complete_Inf_Inf @ ( set @ A ) @ ( image @ C @ ( set @ A ) @ B5 @ A3 ) ) )
= ( complete_Inf_Inf @ ( set @ B )
@ ( image @ C @ ( set @ B )
@ ^ [X4: C] : ( image @ A @ B @ F3 @ ( B5 @ X4 ) )
@ A3 ) ) ) ) ).
% bij_image_INT
thf(fact_6234_UN__UN__finite__eq,axiom,
! [A: $tType,A3: nat > ( set @ A )] :
( ( complete_Sup_Sup @ ( set @ A )
@ ( image @ nat @ ( set @ A )
@ ^ [N2: nat] : ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ A3 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) ) )
@ ( top_top @ ( set @ nat ) ) ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ A3 @ ( top_top @ ( set @ nat ) ) ) ) ) ).
% UN_UN_finite_eq
thf(fact_6235_card__range__greater__zero,axiom,
! [A: $tType,B: $tType,F3: B > A] :
( ( finite_finite2 @ A @ ( image @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) ) )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ ( finite_card @ A @ ( image @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) ) ) ) ) ).
% card_range_greater_zero
thf(fact_6236_UN__finite__subset,axiom,
! [A: $tType,A3: nat > ( set @ A ),C5: set @ A] :
( ! [N3: nat] : ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ A3 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N3 ) ) ) @ C5 )
=> ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ A3 @ ( top_top @ ( set @ nat ) ) ) ) @ C5 ) ) ).
% UN_finite_subset
thf(fact_6237_UN__finite2__eq,axiom,
! [A: $tType,A3: nat > ( set @ A ),B5: nat > ( set @ A ),K2: nat] :
( ! [N3: nat] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ A3 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N3 ) ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ B5 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( plus_plus @ nat @ N3 @ K2 ) ) ) ) )
=> ( ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ A3 @ ( top_top @ ( set @ nat ) ) ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ B5 @ ( top_top @ ( set @ nat ) ) ) ) ) ) ).
% UN_finite2_eq
thf(fact_6238_suminf__eq__SUP,axiom,
! [A: $tType] :
( ( ( comple5582772986160207858norder @ A )
& ( canoni5634975068530333245id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ( ( suminf @ A )
= ( ^ [F5: nat > A] :
( complete_Sup_Sup @ A
@ ( image @ nat @ A
@ ^ [N2: nat] : ( groups7311177749621191930dd_sum @ nat @ A @ F5 @ ( set_ord_lessThan @ nat @ N2 ) )
@ ( top_top @ ( set @ nat ) ) ) ) ) ) ) ).
% suminf_eq_SUP
thf(fact_6239_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_6240_UN__finite2__subset,axiom,
! [A: $tType,A3: nat > ( set @ A ),B5: nat > ( set @ A ),K2: nat] :
( ! [N3: nat] : ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ A3 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N3 ) ) ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ B5 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( plus_plus @ nat @ N3 @ K2 ) ) ) ) )
=> ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ A3 @ ( top_top @ ( set @ nat ) ) ) ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ B5 @ ( top_top @ ( set @ nat ) ) ) ) ) ) ).
% UN_finite2_subset
thf(fact_6241_cclfp__def,axiom,
! [A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ( ( order_532582986084564980_cclfp @ A )
= ( ^ [F5: A > A] :
( complete_Sup_Sup @ A
@ ( image @ nat @ A
@ ^ [I: nat] : ( compow @ ( A > A ) @ I @ F5 @ ( bot_bot @ A ) )
@ ( top_top @ ( set @ nat ) ) ) ) ) ) ) ).
% cclfp_def
thf(fact_6242_INF__filter__not__bot,axiom,
! [I5: $tType,A: $tType,B5: set @ I5,F4: I5 > ( filter @ A )] :
( ! [X10: set @ I5] :
( ( ord_less_eq @ ( set @ I5 ) @ X10 @ B5 )
=> ( ( finite_finite2 @ I5 @ X10 )
=> ( ( complete_Inf_Inf @ ( filter @ A ) @ ( image @ I5 @ ( filter @ A ) @ F4 @ X10 ) )
!= ( bot_bot @ ( filter @ A ) ) ) ) )
=> ( ( complete_Inf_Inf @ ( filter @ A ) @ ( image @ I5 @ ( filter @ A ) @ F4 @ B5 ) )
!= ( bot_bot @ ( filter @ A ) ) ) ) ).
% INF_filter_not_bot
thf(fact_6243_card__UNIV__unit,axiom,
( ( finite_card @ product_unit @ ( top_top @ ( set @ product_unit ) ) )
= ( one_one @ nat ) ) ).
% card_UNIV_unit
thf(fact_6244_Collect__const__case__prod,axiom,
! [B: $tType,A: $tType,P3: $o] :
( ( P3
=> ( ( collect @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [A5: A,B4: B] : P3 ) )
= ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) )
& ( ~ P3
=> ( ( collect @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [A5: A,B4: B] : P3 ) )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ) ).
% Collect_const_case_prod
thf(fact_6245_card__UNIV__bool,axiom,
( ( finite_card @ $o @ ( top_top @ ( set @ $o ) ) )
= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ).
% card_UNIV_bool
thf(fact_6246_range__mult,axiom,
! [A2: real] :
( ( ( A2
= ( zero_zero @ real ) )
=> ( ( image @ real @ real @ ( times_times @ real @ A2 ) @ ( top_top @ ( set @ real ) ) )
= ( insert @ real @ ( zero_zero @ real ) @ ( bot_bot @ ( set @ real ) ) ) ) )
& ( ( A2
!= ( zero_zero @ real ) )
=> ( ( image @ real @ real @ ( times_times @ real @ A2 ) @ ( top_top @ ( set @ real ) ) )
= ( top_top @ ( set @ real ) ) ) ) ) ).
% range_mult
thf(fact_6247_INF__filter__bot__base,axiom,
! [B: $tType,A: $tType,I6: set @ A,F4: A > ( filter @ B )] :
( ! [I3: A] :
( ( member @ A @ I3 @ I6 )
=> ! [J2: A] :
( ( member @ A @ J2 @ I6 )
=> ? [X2: A] :
( ( member @ A @ X2 @ I6 )
& ( ord_less_eq @ ( filter @ B ) @ ( F4 @ X2 ) @ ( inf_inf @ ( filter @ B ) @ ( F4 @ I3 ) @ ( F4 @ J2 ) ) ) ) ) )
=> ( ( ( complete_Inf_Inf @ ( filter @ B ) @ ( image @ A @ ( filter @ B ) @ F4 @ I6 ) )
= ( bot_bot @ ( filter @ B ) ) )
= ( ? [X4: A] :
( ( member @ A @ X4 @ I6 )
& ( ( F4 @ X4 )
= ( bot_bot @ ( filter @ B ) ) ) ) ) ) ) ).
% INF_filter_bot_base
thf(fact_6248_top__empty__eq2,axiom,
! [B: $tType,A: $tType] :
( ( top_top @ ( A > B > $o ) )
= ( ^ [X4: A,Y: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ) ).
% top_empty_eq2
thf(fact_6249_top__enat__def,axiom,
( ( top_top @ extended_enat )
= ( extend4730790105801354508finity @ extended_enat ) ) ).
% top_enat_def
thf(fact_6250_infinite__UNIV__listI,axiom,
! [A: $tType] :
~ ( finite_finite2 @ ( list @ A ) @ ( top_top @ ( set @ ( list @ A ) ) ) ) ).
% infinite_UNIV_listI
thf(fact_6251_Inf__filter__not__bot,axiom,
! [A: $tType,B5: set @ ( filter @ A )] :
( ! [X10: set @ ( filter @ A )] :
( ( ord_less_eq @ ( set @ ( filter @ A ) ) @ X10 @ B5 )
=> ( ( finite_finite2 @ ( filter @ A ) @ X10 )
=> ( ( complete_Inf_Inf @ ( filter @ A ) @ X10 )
!= ( bot_bot @ ( filter @ A ) ) ) ) )
=> ( ( complete_Inf_Inf @ ( filter @ A ) @ B5 )
!= ( bot_bot @ ( filter @ A ) ) ) ) ).
% Inf_filter_not_bot
thf(fact_6252_INF__UNIV__bool__expand,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: $o > A] :
( ( complete_Inf_Inf @ A @ ( image @ $o @ A @ A3 @ ( top_top @ ( set @ $o ) ) ) )
= ( inf_inf @ A @ ( A3 @ $true ) @ ( A3 @ $false ) ) ) ) ).
% INF_UNIV_bool_expand
thf(fact_6253_INT__bool__eq,axiom,
! [A: $tType,A3: $o > ( set @ A )] :
( ( complete_Inf_Inf @ ( set @ A ) @ ( image @ $o @ ( set @ A ) @ A3 @ ( top_top @ ( set @ $o ) ) ) )
= ( inf_inf @ ( set @ A ) @ ( A3 @ $true ) @ ( A3 @ $false ) ) ) ).
% INT_bool_eq
thf(fact_6254_root__def,axiom,
( root
= ( ^ [N2: nat,X4: real] :
( if @ real
@ ( N2
= ( zero_zero @ nat ) )
@ ( zero_zero @ real )
@ ( the_inv_into @ real @ real @ ( top_top @ ( set @ real ) )
@ ^ [Y: real] : ( times_times @ real @ ( sgn_sgn @ real @ Y ) @ ( power_power @ real @ ( abs_abs @ real @ Y ) @ N2 ) )
@ X4 ) ) ) ) ).
% root_def
thf(fact_6255_card__UNIV__char,axiom,
( ( finite_card @ char @ ( top_top @ ( set @ char ) ) )
= ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% card_UNIV_char
thf(fact_6256_less__filter__def,axiom,
! [A: $tType] :
( ( ord_less @ ( filter @ A ) )
= ( ^ [F10: filter @ A,F11: filter @ A] :
( ( ord_less_eq @ ( filter @ A ) @ F10 @ F11 )
& ~ ( ord_less_eq @ ( filter @ A ) @ F11 @ F10 ) ) ) ) ).
% less_filter_def
thf(fact_6257_the__inv__into__def,axiom,
! [B: $tType,A: $tType] :
( ( the_inv_into @ A @ B )
= ( ^ [A7: set @ A,F5: A > B,X4: B] :
( the @ A
@ ^ [Y: A] :
( ( member @ A @ Y @ A7 )
& ( ( F5 @ Y )
= X4 ) ) ) ) ) ).
% the_inv_into_def
thf(fact_6258_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_6259_char__of__mod__256,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [N: A] :
( ( unique5772411509450598832har_of @ A @ ( modulo_modulo @ A @ N @ ( numeral_numeral @ A @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) )
= ( unique5772411509450598832har_of @ A @ N ) ) ) ).
% char_of_mod_256
thf(fact_6260_char__of__quasi__inj,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [M: A,N: A] :
( ( ( unique5772411509450598832har_of @ A @ M )
= ( unique5772411509450598832har_of @ A @ N ) )
= ( ( modulo_modulo @ A @ M @ ( numeral_numeral @ A @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) )
= ( modulo_modulo @ A @ N @ ( numeral_numeral @ A @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% char_of_quasi_inj
thf(fact_6261_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_6262_of__char__of,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [A2: A] :
( ( comm_s6883823935334413003f_char @ A @ ( unique5772411509450598832har_of @ A @ A2 ) )
= ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ).
% of_char_of
thf(fact_6263_char__of__def,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ( ( unique5772411509450598832har_of @ A )
= ( ^ [N2: A] :
( char2
@ ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N2 )
@ ( bit_se5641148757651400278ts_bit @ A @ N2 @ ( one_one @ nat ) )
@ ( bit_se5641148757651400278ts_bit @ A @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
@ ( bit_se5641148757651400278ts_bit @ A @ N2 @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) )
@ ( bit_se5641148757651400278ts_bit @ A @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) )
@ ( bit_se5641148757651400278ts_bit @ A @ N2 @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ one2 ) ) ) )
@ ( bit_se5641148757651400278ts_bit @ A @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ ( bit1 @ one2 ) ) ) )
@ ( bit_se5641148757651400278ts_bit @ A @ N2 @ ( numeral_numeral @ nat @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ).
% char_of_def
thf(fact_6264_of__char__mod__256,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [C2: char] :
( ( modulo_modulo @ A @ ( comm_s6883823935334413003f_char @ A @ C2 ) @ ( numeral_numeral @ A @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) )
= ( comm_s6883823935334413003f_char @ A @ C2 ) ) ) ).
% of_char_mod_256
thf(fact_6265_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_6266_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_6267_char__of__eq__iff,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [N: A,C2: char] :
( ( ( unique5772411509450598832har_of @ A @ N )
= C2 )
= ( ( bit_se2584673776208193580ke_bit @ A @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) @ N )
= ( comm_s6883823935334413003f_char @ A @ C2 ) ) ) ) ).
% char_of_eq_iff
thf(fact_6268_integer__of__char__code,axiom,
! [B0: $o,B1: $o,B22: $o,B32: $o,B42: $o,B52: $o,B62: $o,B72: $o] :
( ( integer_of_char @ ( char2 @ B0 @ B1 @ B22 @ B32 @ B42 @ B52 @ B62 @ B72 ) )
= ( plus_plus @ code_integer @ ( times_times @ code_integer @ ( plus_plus @ code_integer @ ( times_times @ code_integer @ ( plus_plus @ code_integer @ ( times_times @ code_integer @ ( plus_plus @ code_integer @ ( times_times @ code_integer @ ( plus_plus @ code_integer @ ( times_times @ code_integer @ ( plus_plus @ code_integer @ ( times_times @ code_integer @ ( plus_plus @ code_integer @ ( times_times @ code_integer @ ( zero_neq_one_of_bool @ code_integer @ B72 ) @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) @ ( zero_neq_one_of_bool @ code_integer @ B62 ) ) @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) @ ( zero_neq_one_of_bool @ code_integer @ B52 ) ) @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) @ ( zero_neq_one_of_bool @ code_integer @ B42 ) ) @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) @ ( zero_neq_one_of_bool @ code_integer @ B32 ) ) @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) @ ( zero_neq_one_of_bool @ code_integer @ B22 ) ) @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) @ ( zero_neq_one_of_bool @ code_integer @ B1 ) ) @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) @ ( zero_neq_one_of_bool @ code_integer @ B0 ) ) ) ).
% integer_of_char_code
thf(fact_6269_char__of__integer__code,axiom,
( char_of_integer
= ( ^ [K3: code_integer] :
( product_case_prod @ code_integer @ $o @ char
@ ^ [Q0: code_integer,B02: $o] :
( product_case_prod @ code_integer @ $o @ char
@ ^ [Q1: code_integer,B14: $o] :
( product_case_prod @ code_integer @ $o @ char
@ ^ [Q22: code_integer,B23: $o] :
( product_case_prod @ code_integer @ $o @ char
@ ^ [Q32: code_integer,B33: $o] :
( product_case_prod @ code_integer @ $o @ char
@ ^ [Q42: code_integer,B43: $o] :
( product_case_prod @ code_integer @ $o @ char
@ ^ [Q52: code_integer,B53: $o] :
( product_case_prod @ code_integer @ $o @ char
@ ^ [Q62: code_integer,B63: $o] :
( product_case_prod @ code_integer @ $o @ char
@ ^ [Uu3: code_integer] : ( char2 @ B02 @ B14 @ B23 @ B33 @ B43 @ B53 @ B63 )
@ ( code_bit_cut_integer @ Q62 ) )
@ ( code_bit_cut_integer @ Q52 ) )
@ ( code_bit_cut_integer @ Q42 ) )
@ ( code_bit_cut_integer @ Q32 ) )
@ ( code_bit_cut_integer @ Q22 ) )
@ ( code_bit_cut_integer @ Q1 ) )
@ ( code_bit_cut_integer @ Q0 ) )
@ ( code_bit_cut_integer @ K3 ) ) ) ) ).
% char_of_integer_code
thf(fact_6270_String_Ochar__of__ascii__of,axiom,
! [C2: char] :
( ( comm_s6883823935334413003f_char @ nat @ ( ascii_of @ C2 ) )
= ( bit_se2584673776208193580ke_bit @ nat @ ( numeral_numeral @ nat @ ( bit1 @ ( bit1 @ one2 ) ) ) @ ( comm_s6883823935334413003f_char @ nat @ C2 ) ) ) ).
% String.char_of_ascii_of
thf(fact_6271_of__char__Char,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [B0: $o,B1: $o,B22: $o,B32: $o,B42: $o,B52: $o,B62: $o,B72: $o] :
( ( comm_s6883823935334413003f_char @ A @ ( char2 @ B0 @ B1 @ B22 @ B32 @ B42 @ B52 @ B62 @ B72 ) )
= ( groups4207007520872428315er_sum @ $o @ A @ ( zero_neq_one_of_bool @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( cons @ $o @ B0 @ ( cons @ $o @ B1 @ ( cons @ $o @ B22 @ ( cons @ $o @ B32 @ ( cons @ $o @ B42 @ ( cons @ $o @ B52 @ ( cons @ $o @ B62 @ ( cons @ $o @ B72 @ ( nil @ $o ) ) ) ) ) ) ) ) ) ) ) ) ).
% of_char_Char
thf(fact_6272_list_Oinject,axiom,
! [A: $tType,X21: A,X22: list @ A,Y21: A,Y22: list @ A] :
( ( ( cons @ A @ X21 @ X22 )
= ( cons @ A @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X22 = Y22 ) ) ) ).
% list.inject
thf(fact_6273_list_Osimps_I15_J,axiom,
! [A: $tType,X21: A,X22: list @ A] :
( ( set2 @ A @ ( cons @ A @ X21 @ X22 ) )
= ( insert @ A @ X21 @ ( set2 @ A @ X22 ) ) ) ).
% list.simps(15)
thf(fact_6274_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_6275_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_6276_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_6277_sum__list_OCons,axiom,
! [A: $tType] :
( ( monoid_add @ A )
=> ! [X: A,Xs: list @ A] :
( ( groups8242544230860333062m_list @ A @ ( cons @ A @ X @ Xs ) )
= ( plus_plus @ A @ X @ ( groups8242544230860333062m_list @ A @ Xs ) ) ) ) ).
% sum_list.Cons
thf(fact_6278_singleton__rev__conv,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( ( cons @ A @ X @ ( nil @ A ) )
= ( rev @ A @ Xs ) )
= ( ( cons @ A @ X @ ( nil @ A ) )
= Xs ) ) ).
% singleton_rev_conv
thf(fact_6279_rev__singleton__conv,axiom,
! [A: $tType,Xs: list @ A,X: A] :
( ( ( rev @ A @ Xs )
= ( cons @ A @ X @ ( nil @ A ) ) )
= ( Xs
= ( cons @ A @ X @ ( nil @ A ) ) ) ) ).
% rev_singleton_conv
thf(fact_6280_horner__sum__simps_I2_J,axiom,
! [A: $tType,B: $tType] :
( ( comm_semiring_0 @ A )
=> ! [F3: B > A,A2: A,X: B,Xs: list @ B] :
( ( groups4207007520872428315er_sum @ B @ A @ F3 @ A2 @ ( cons @ B @ X @ Xs ) )
= ( plus_plus @ A @ ( F3 @ X ) @ ( times_times @ A @ A2 @ ( groups4207007520872428315er_sum @ B @ A @ F3 @ A2 @ Xs ) ) ) ) ) ).
% horner_sum_simps(2)
thf(fact_6281_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_6282_nth__Cons__numeral,axiom,
! [A: $tType,X: A,Xs: list @ A,V2: num] :
( ( nth @ A @ ( cons @ A @ X @ Xs ) @ ( numeral_numeral @ nat @ V2 ) )
= ( nth @ A @ Xs @ ( minus_minus @ nat @ ( numeral_numeral @ nat @ V2 ) @ ( one_one @ nat ) ) ) ) ).
% nth_Cons_numeral
thf(fact_6283_take__Cons__numeral,axiom,
! [A: $tType,V2: num,X: A,Xs: list @ A] :
( ( take @ A @ ( numeral_numeral @ nat @ V2 ) @ ( cons @ A @ X @ Xs ) )
= ( cons @ A @ X @ ( take @ A @ ( minus_minus @ nat @ ( numeral_numeral @ nat @ V2 ) @ ( one_one @ nat ) ) @ Xs ) ) ) ).
% take_Cons_numeral
thf(fact_6284_Cons__in__lex,axiom,
! [A: $tType,X: A,Xs: list @ A,Y2: A,Ys: list @ A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y2 @ Ys ) ) @ ( lex @ A @ R ) )
= ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y2 ) @ R )
& ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ A ) @ Ys ) ) )
| ( ( X = Y2 )
& ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( lex @ A @ R ) ) ) ) ) ).
% Cons_in_lex
thf(fact_6285_concat__map__singleton,axiom,
! [A: $tType,B: $tType,F3: B > A,Xs: list @ B] :
( ( concat @ A
@ ( map @ B @ ( list @ A )
@ ^ [X4: B] : ( cons @ A @ ( F3 @ X4 ) @ ( nil @ A ) )
@ Xs ) )
= ( map @ B @ A @ F3 @ Xs ) ) ).
% concat_map_singleton
thf(fact_6286_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_6287_list_Osel_I1_J,axiom,
! [A: $tType,X21: A,X22: list @ A] :
( ( hd @ A @ ( cons @ A @ X21 @ X22 ) )
= X21 ) ).
% list.sel(1)
thf(fact_6288_filter_Osimps_I2_J,axiom,
! [A: $tType,P3: A > $o,X: A,Xs: list @ A] :
( ( ( P3 @ X )
=> ( ( filter2 @ A @ P3 @ ( cons @ A @ X @ Xs ) )
= ( cons @ A @ X @ ( filter2 @ A @ P3 @ Xs ) ) ) )
& ( ~ ( P3 @ X )
=> ( ( filter2 @ A @ P3 @ ( cons @ A @ X @ Xs ) )
= ( filter2 @ A @ P3 @ Xs ) ) ) ) ).
% filter.simps(2)
thf(fact_6289_splice_Ocases,axiom,
! [A: $tType,X: product_prod @ ( list @ A ) @ ( list @ A )] :
( ! [Ys4: list @ A] :
( X
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Ys4 ) )
=> ~ ! [X5: A,Xs3: list @ A,Ys4: list @ A] :
( X
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X5 @ Xs3 ) @ Ys4 ) ) ) ).
% splice.cases
thf(fact_6290_shuffles_Ocases,axiom,
! [A: $tType,X: product_prod @ ( list @ A ) @ ( list @ A )] :
( ! [Ys4: list @ A] :
( X
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Ys4 ) )
=> ( ! [Xs3: list @ A] :
( X
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs3 @ ( nil @ A ) ) )
=> ~ ! [X5: A,Xs3: list @ A,Y7: A,Ys4: list @ A] :
( X
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X5 @ Xs3 ) @ ( cons @ A @ Y7 @ Ys4 ) ) ) ) ) ).
% shuffles.cases
thf(fact_6291_sorted__wrt_Ocases,axiom,
! [A: $tType,X: product_prod @ ( A > A > $o ) @ ( list @ A )] :
( ! [P9: A > A > $o] :
( X
!= ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ P9 @ ( nil @ A ) ) )
=> ~ ! [P9: A > A > $o,X5: A,Ys4: list @ A] :
( X
!= ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ P9 @ ( cons @ A @ X5 @ Ys4 ) ) ) ) ).
% sorted_wrt.cases
thf(fact_6292_arg__min__list_Ocases,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ! [X: product_prod @ ( A > B ) @ ( list @ A )] :
( ! [F6: A > B,X5: A] :
( X
!= ( product_Pair @ ( A > B ) @ ( list @ A ) @ F6 @ ( cons @ A @ X5 @ ( nil @ A ) ) ) )
=> ( ! [F6: A > B,X5: A,Y7: A,Zs2: list @ A] :
( X
!= ( product_Pair @ ( A > B ) @ ( list @ A ) @ F6 @ ( cons @ A @ X5 @ ( cons @ A @ Y7 @ Zs2 ) ) ) )
=> ~ ! [A4: A > B] :
( X
!= ( product_Pair @ ( A > B ) @ ( list @ A ) @ A4 @ ( nil @ A ) ) ) ) ) ) ).
% arg_min_list.cases
thf(fact_6293_successively_Ocases,axiom,
! [A: $tType,X: product_prod @ ( A > A > $o ) @ ( list @ A )] :
( ! [P9: A > A > $o] :
( X
!= ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ P9 @ ( nil @ A ) ) )
=> ( ! [P9: A > A > $o,X5: A] :
( X
!= ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ P9 @ ( cons @ A @ X5 @ ( nil @ A ) ) ) )
=> ~ ! [P9: A > A > $o,X5: A,Y7: A,Xs3: list @ A] :
( X
!= ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ P9 @ ( cons @ A @ X5 @ ( cons @ A @ Y7 @ Xs3 ) ) ) ) ) ) ).
% successively.cases
thf(fact_6294_map__tailrec__rev_Ocases,axiom,
! [A: $tType,B: $tType,X: product_prod @ ( A > B ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) )] :
( ! [F6: A > B,Bs2: list @ B] :
( X
!= ( product_Pair @ ( A > B ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ F6 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( nil @ A ) @ Bs2 ) ) )
=> ~ ! [F6: A > B,A4: A,As: list @ A,Bs2: list @ B] :
( X
!= ( product_Pair @ ( A > B ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ F6 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ A4 @ As ) @ Bs2 ) ) ) ) ).
% map_tailrec_rev.cases
thf(fact_6295_distinct__singleton,axiom,
! [A: $tType,X: A] : ( distinct @ A @ ( cons @ A @ X @ ( nil @ A ) ) ) ).
% distinct_singleton
thf(fact_6296_sorted__wrt1,axiom,
! [A: $tType,P3: A > A > $o,X: A] : ( sorted_wrt @ A @ P3 @ ( cons @ A @ X @ ( nil @ A ) ) ) ).
% sorted_wrt1
thf(fact_6297_distinct_Osimps_I2_J,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( distinct @ A @ ( cons @ A @ X @ Xs ) )
= ( ~ ( member @ A @ X @ ( set2 @ A @ Xs ) )
& ( distinct @ A @ Xs ) ) ) ).
% distinct.simps(2)
thf(fact_6298_list__induct4,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,Xs: list @ A,Ys: list @ B,Zs: list @ C,Ws: list @ D,P3: ( list @ A ) > ( list @ B ) > ( list @ C ) > ( list @ D ) > $o] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( ( size_size @ ( list @ B ) @ Ys )
= ( size_size @ ( list @ C ) @ Zs ) )
=> ( ( ( size_size @ ( list @ C ) @ Zs )
= ( size_size @ ( list @ D ) @ Ws ) )
=> ( ( P3 @ ( nil @ A ) @ ( nil @ B ) @ ( nil @ C ) @ ( nil @ D ) )
=> ( ! [X5: A,Xs3: list @ A,Y7: B,Ys4: list @ B,Z: C,Zs2: list @ C,W2: D,Ws2: list @ D] :
( ( ( size_size @ ( list @ A ) @ Xs3 )
= ( size_size @ ( list @ B ) @ Ys4 ) )
=> ( ( ( size_size @ ( list @ B ) @ Ys4 )
= ( size_size @ ( list @ C ) @ Zs2 ) )
=> ( ( ( size_size @ ( list @ C ) @ Zs2 )
= ( size_size @ ( list @ D ) @ Ws2 ) )
=> ( ( P3 @ Xs3 @ Ys4 @ Zs2 @ Ws2 )
=> ( P3 @ ( cons @ A @ X5 @ Xs3 ) @ ( cons @ B @ Y7 @ Ys4 ) @ ( cons @ C @ Z @ Zs2 ) @ ( cons @ D @ W2 @ Ws2 ) ) ) ) ) )
=> ( P3 @ Xs @ Ys @ Zs @ Ws ) ) ) ) ) ) ).
% list_induct4
thf(fact_6299_list__induct3,axiom,
! [B: $tType,A: $tType,C: $tType,Xs: list @ A,Ys: list @ B,Zs: list @ C,P3: ( list @ A ) > ( list @ B ) > ( list @ C ) > $o] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( ( size_size @ ( list @ B ) @ Ys )
= ( size_size @ ( list @ C ) @ Zs ) )
=> ( ( P3 @ ( nil @ A ) @ ( nil @ B ) @ ( nil @ C ) )
=> ( ! [X5: A,Xs3: list @ A,Y7: B,Ys4: list @ B,Z: C,Zs2: list @ C] :
( ( ( size_size @ ( list @ A ) @ Xs3 )
= ( size_size @ ( list @ B ) @ Ys4 ) )
=> ( ( ( size_size @ ( list @ B ) @ Ys4 )
= ( size_size @ ( list @ C ) @ Zs2 ) )
=> ( ( P3 @ Xs3 @ Ys4 @ Zs2 )
=> ( P3 @ ( cons @ A @ X5 @ Xs3 ) @ ( cons @ B @ Y7 @ Ys4 ) @ ( cons @ C @ Z @ Zs2 ) ) ) ) )
=> ( P3 @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_6300_list__induct2,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B,P3: ( list @ A ) > ( list @ B ) > $o] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( P3 @ ( nil @ A ) @ ( nil @ B ) )
=> ( ! [X5: A,Xs3: list @ A,Y7: B,Ys4: list @ B] :
( ( ( size_size @ ( list @ A ) @ Xs3 )
= ( size_size @ ( list @ B ) @ Ys4 ) )
=> ( ( P3 @ Xs3 @ Ys4 )
=> ( P3 @ ( cons @ A @ X5 @ Xs3 ) @ ( cons @ B @ Y7 @ Ys4 ) ) ) )
=> ( P3 @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_6301_list_Odistinct_I1_J,axiom,
! [A: $tType,X21: A,X22: list @ A] :
( ( nil @ A )
!= ( cons @ A @ X21 @ X22 ) ) ).
% list.distinct(1)
thf(fact_6302_list_OdiscI,axiom,
! [A: $tType,List: list @ A,X21: A,X22: list @ A] :
( ( List
= ( cons @ A @ X21 @ X22 ) )
=> ( List
!= ( nil @ A ) ) ) ).
% list.discI
thf(fact_6303_list_Oexhaust,axiom,
! [A: $tType,Y2: list @ A] :
( ( Y2
!= ( nil @ A ) )
=> ~ ! [X212: A,X222: list @ A] :
( Y2
!= ( cons @ A @ X212 @ X222 ) ) ) ).
% list.exhaust
thf(fact_6304_min__list_Ocases,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X: list @ A] :
( ! [X5: A,Xs3: list @ A] :
( X
!= ( cons @ A @ X5 @ Xs3 ) )
=> ( X
= ( nil @ A ) ) ) ) ).
% min_list.cases
thf(fact_6305_transpose_Ocases,axiom,
! [A: $tType,X: list @ ( list @ A )] :
( ( X
!= ( nil @ ( list @ A ) ) )
=> ( ! [Xss2: list @ ( list @ A )] :
( X
!= ( cons @ ( list @ A ) @ ( nil @ A ) @ Xss2 ) )
=> ~ ! [X5: A,Xs3: list @ A,Xss2: list @ ( list @ A )] :
( X
!= ( cons @ ( list @ A ) @ ( cons @ A @ X5 @ Xs3 ) @ Xss2 ) ) ) ) ).
% transpose.cases
thf(fact_6306_remdups__adj_Ocases,axiom,
! [A: $tType,X: list @ A] :
( ( X
!= ( nil @ A ) )
=> ( ! [X5: A] :
( X
!= ( cons @ A @ X5 @ ( nil @ A ) ) )
=> ~ ! [X5: A,Y7: A,Xs3: list @ A] :
( X
!= ( cons @ A @ X5 @ ( cons @ A @ Y7 @ Xs3 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_6307_neq__Nil__conv,axiom,
! [A: $tType,Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
= ( ? [Y: A,Ys3: list @ A] :
( Xs
= ( cons @ A @ Y @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_6308_list__induct2_H,axiom,
! [A: $tType,B: $tType,P3: ( list @ A ) > ( list @ B ) > $o,Xs: list @ A,Ys: list @ B] :
( ( P3 @ ( nil @ A ) @ ( nil @ B ) )
=> ( ! [X5: A,Xs3: list @ A] : ( P3 @ ( cons @ A @ X5 @ Xs3 ) @ ( nil @ B ) )
=> ( ! [Y7: B,Ys4: list @ B] : ( P3 @ ( nil @ A ) @ ( cons @ B @ Y7 @ Ys4 ) )
=> ( ! [X5: A,Xs3: list @ A,Y7: B,Ys4: list @ B] :
( ( P3 @ Xs3 @ Ys4 )
=> ( P3 @ ( cons @ A @ X5 @ Xs3 ) @ ( cons @ B @ Y7 @ Ys4 ) ) )
=> ( P3 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_6309_list__nonempty__induct,axiom,
! [A: $tType,Xs: list @ A,P3: ( list @ A ) > $o] :
( ( Xs
!= ( nil @ A ) )
=> ( ! [X5: A] : ( P3 @ ( cons @ A @ X5 @ ( nil @ A ) ) )
=> ( ! [X5: A,Xs3: list @ A] :
( ( Xs3
!= ( nil @ A ) )
=> ( ( P3 @ Xs3 )
=> ( P3 @ ( cons @ A @ X5 @ Xs3 ) ) ) )
=> ( P3 @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_6310_removeAll_Osimps_I2_J,axiom,
! [A: $tType,X: A,Y2: A,Xs: list @ A] :
( ( ( X = Y2 )
=> ( ( removeAll @ A @ X @ ( cons @ A @ Y2 @ Xs ) )
= ( removeAll @ A @ X @ Xs ) ) )
& ( ( X != Y2 )
=> ( ( removeAll @ A @ X @ ( cons @ A @ Y2 @ Xs ) )
= ( cons @ A @ Y2 @ ( removeAll @ A @ X @ Xs ) ) ) ) ) ).
% removeAll.simps(2)
thf(fact_6311_set__ConsD,axiom,
! [A: $tType,Y2: A,X: A,Xs: list @ A] :
( ( member @ A @ Y2 @ ( set2 @ A @ ( cons @ A @ X @ Xs ) ) )
=> ( ( Y2 = X )
| ( member @ A @ Y2 @ ( set2 @ A @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_6312_list_Oset__cases,axiom,
! [A: $tType,E: A,A2: list @ A] :
( ( member @ A @ E @ ( set2 @ A @ A2 ) )
=> ( ! [Z23: list @ A] :
( A2
!= ( cons @ A @ E @ Z23 ) )
=> ~ ! [Z12: A,Z23: list @ A] :
( ( A2
= ( cons @ A @ Z12 @ Z23 ) )
=> ~ ( member @ A @ E @ ( set2 @ A @ Z23 ) ) ) ) ) ).
% list.set_cases
thf(fact_6313_list_Oset__intros_I1_J,axiom,
! [A: $tType,X21: A,X22: list @ A] : ( member @ A @ X21 @ ( set2 @ A @ ( cons @ A @ X21 @ X22 ) ) ) ).
% list.set_intros(1)
thf(fact_6314_list_Oset__intros_I2_J,axiom,
! [A: $tType,Y2: A,X22: list @ A,X21: A] :
( ( member @ A @ Y2 @ ( set2 @ A @ X22 ) )
=> ( member @ A @ Y2 @ ( set2 @ A @ ( cons @ A @ X21 @ X22 ) ) ) ) ).
% list.set_intros(2)
thf(fact_6315_remove1_Osimps_I2_J,axiom,
! [A: $tType,X: A,Y2: A,Xs: list @ A] :
( ( ( X = Y2 )
=> ( ( remove1 @ A @ X @ ( cons @ A @ Y2 @ Xs ) )
= Xs ) )
& ( ( X != Y2 )
=> ( ( remove1 @ A @ X @ ( cons @ A @ Y2 @ Xs ) )
= ( cons @ A @ Y2 @ ( remove1 @ A @ X @ Xs ) ) ) ) ) ).
% remove1.simps(2)
thf(fact_6316_list__update_Osimps_I2_J,axiom,
! [A: $tType,X: A,Xs: list @ A,I2: nat,V2: A] :
( ( list_update @ A @ ( cons @ A @ X @ Xs ) @ I2 @ V2 )
= ( case_nat @ ( list @ A ) @ ( cons @ A @ V2 @ Xs )
@ ^ [J4: nat] : ( cons @ A @ X @ ( list_update @ A @ Xs @ J4 @ V2 ) )
@ I2 ) ) ).
% list_update.simps(2)
thf(fact_6317_not__Cons__self2,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( cons @ A @ X @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_6318_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_6319_Suc__length__conv,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( ( suc @ N )
= ( size_size @ ( list @ A ) @ Xs ) )
= ( ? [Y: A,Ys3: list @ A] :
( ( Xs
= ( cons @ A @ Y @ Ys3 ) )
& ( ( size_size @ ( list @ A ) @ Ys3 )
= N ) ) ) ) ).
% Suc_length_conv
thf(fact_6320_length__Suc__conv,axiom,
! [A: $tType,Xs: list @ A,N: nat] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( suc @ N ) )
= ( ? [Y: A,Ys3: list @ A] :
( ( Xs
= ( cons @ A @ Y @ Ys3 ) )
& ( ( size_size @ ( list @ A ) @ Ys3 )
= N ) ) ) ) ).
% length_Suc_conv
thf(fact_6321_distinct__length__2__or__more,axiom,
! [A: $tType,A2: A,B2: A,Xs: list @ A] :
( ( distinct @ A @ ( cons @ A @ A2 @ ( cons @ A @ B2 @ Xs ) ) )
= ( ( A2 != B2 )
& ( distinct @ A @ ( cons @ A @ A2 @ Xs ) )
& ( distinct @ A @ ( cons @ A @ B2 @ Xs ) ) ) ) ).
% distinct_length_2_or_more
thf(fact_6322_Cons__in__shuffles__leftI,axiom,
! [A: $tType,Zs: list @ A,Xs: list @ A,Ys: list @ A,Z3: A] :
( ( member @ ( list @ A ) @ Zs @ ( shuffles @ A @ Xs @ Ys ) )
=> ( member @ ( list @ A ) @ ( cons @ A @ Z3 @ Zs ) @ ( shuffles @ A @ ( cons @ A @ Z3 @ Xs ) @ Ys ) ) ) ).
% Cons_in_shuffles_leftI
thf(fact_6323_Cons__in__shuffles__rightI,axiom,
! [A: $tType,Zs: list @ A,Xs: list @ A,Ys: list @ A,Z3: A] :
( ( member @ ( list @ A ) @ Zs @ ( shuffles @ A @ Xs @ Ys ) )
=> ( member @ ( list @ A ) @ ( cons @ A @ Z3 @ Zs ) @ ( shuffles @ A @ Xs @ ( cons @ A @ Z3 @ Ys ) ) ) ) ).
% Cons_in_shuffles_rightI
thf(fact_6324_product__lists_Osimps_I2_J,axiom,
! [A: $tType,Xs: list @ A,Xss: list @ ( list @ A )] :
( ( product_lists @ A @ ( cons @ ( list @ A ) @ Xs @ Xss ) )
= ( concat @ ( list @ A )
@ ( map @ A @ ( list @ ( list @ A ) )
@ ^ [X4: A] : ( map @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X4 ) @ ( product_lists @ A @ Xss ) )
@ Xs ) ) ) ).
% product_lists.simps(2)
thf(fact_6325_list_Osimps_I9_J,axiom,
! [B: $tType,A: $tType,F3: A > B,X21: A,X22: list @ A] :
( ( map @ A @ B @ F3 @ ( cons @ A @ X21 @ X22 ) )
= ( cons @ B @ ( F3 @ X21 ) @ ( map @ A @ B @ F3 @ X22 ) ) ) ).
% list.simps(9)
thf(fact_6326_Cons__eq__map__D,axiom,
! [A: $tType,B: $tType,X: A,Xs: list @ A,F3: B > A,Ys: list @ B] :
( ( ( cons @ A @ X @ Xs )
= ( map @ B @ A @ F3 @ Ys ) )
=> ? [Z: B,Zs2: list @ B] :
( ( Ys
= ( cons @ B @ Z @ Zs2 ) )
& ( X
= ( F3 @ Z ) )
& ( Xs
= ( map @ B @ A @ F3 @ Zs2 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_6327_map__eq__Cons__D,axiom,
! [B: $tType,A: $tType,F3: B > A,Xs: list @ B,Y2: A,Ys: list @ A] :
( ( ( map @ B @ A @ F3 @ Xs )
= ( cons @ A @ Y2 @ Ys ) )
=> ? [Z: B,Zs2: list @ B] :
( ( Xs
= ( cons @ B @ Z @ Zs2 ) )
& ( ( F3 @ Z )
= Y2 )
& ( ( map @ B @ A @ F3 @ Zs2 )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_6328_Cons__eq__map__conv,axiom,
! [A: $tType,B: $tType,X: A,Xs: list @ A,F3: B > A,Ys: list @ B] :
( ( ( cons @ A @ X @ Xs )
= ( map @ B @ A @ F3 @ Ys ) )
= ( ? [Z6: B,Zs3: list @ B] :
( ( Ys
= ( cons @ B @ Z6 @ Zs3 ) )
& ( X
= ( F3 @ Z6 ) )
& ( Xs
= ( map @ B @ A @ F3 @ Zs3 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_6329_map__eq__Cons__conv,axiom,
! [A: $tType,B: $tType,F3: B > A,Xs: list @ B,Y2: A,Ys: list @ A] :
( ( ( map @ B @ A @ F3 @ Xs )
= ( cons @ A @ Y2 @ Ys ) )
= ( ? [Z6: B,Zs3: list @ B] :
( ( Xs
= ( cons @ B @ Z6 @ Zs3 ) )
& ( ( F3 @ Z6 )
= Y2 )
& ( ( map @ B @ A @ F3 @ Zs3 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_6330_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_6331_insort__key_Osimps_I1_J,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F3: B > A,X: B] :
( ( linorder_insort_key @ B @ A @ F3 @ X @ ( nil @ B ) )
= ( cons @ B @ X @ ( nil @ B ) ) ) ) ).
% insort_key.simps(1)
thf(fact_6332_shufflesE,axiom,
! [A: $tType,Zs: list @ A,Xs: list @ A,Ys: list @ A] :
( ( member @ ( list @ A ) @ Zs @ ( shuffles @ A @ Xs @ Ys ) )
=> ( ( ( Zs = Xs )
=> ( Ys
!= ( nil @ A ) ) )
=> ( ( ( Zs = Ys )
=> ( Xs
!= ( nil @ A ) ) )
=> ( ! [X5: A,Xs4: list @ A] :
( ( Xs
= ( cons @ A @ X5 @ Xs4 ) )
=> ! [Z: A,Zs4: list @ A] :
( ( Zs
= ( cons @ A @ Z @ Zs4 ) )
=> ( ( X5 = Z )
=> ~ ( member @ ( list @ A ) @ Zs4 @ ( shuffles @ A @ Xs4 @ Ys ) ) ) ) )
=> ~ ! [Y7: A,Ys5: list @ A] :
( ( Ys
= ( cons @ A @ Y7 @ Ys5 ) )
=> ! [Z: A,Zs4: list @ A] :
( ( Zs
= ( cons @ A @ Z @ Zs4 ) )
=> ( ( Y7 = Z )
=> ~ ( member @ ( list @ A ) @ Zs4 @ ( shuffles @ A @ Xs @ Ys5 ) ) ) ) ) ) ) ) ) ).
% shufflesE
thf(fact_6333_takeWhile_Osimps_I2_J,axiom,
! [A: $tType,P3: A > $o,X: A,Xs: list @ A] :
( ( ( P3 @ X )
=> ( ( takeWhile @ A @ P3 @ ( cons @ A @ X @ Xs ) )
= ( cons @ A @ X @ ( takeWhile @ A @ P3 @ Xs ) ) ) )
& ( ~ ( P3 @ X )
=> ( ( takeWhile @ A @ P3 @ ( cons @ A @ X @ Xs ) )
= ( nil @ A ) ) ) ) ).
% takeWhile.simps(2)
thf(fact_6334_list__update__code_I2_J,axiom,
! [A: $tType,X: A,Xs: list @ A,Y2: A] :
( ( list_update @ A @ ( cons @ A @ X @ Xs ) @ ( zero_zero @ nat ) @ Y2 )
= ( cons @ A @ Y2 @ Xs ) ) ).
% list_update_code(2)
thf(fact_6335_list__update__code_I3_J,axiom,
! [A: $tType,X: A,Xs: list @ A,I2: nat,Y2: A] :
( ( list_update @ A @ ( cons @ A @ X @ Xs ) @ ( suc @ I2 ) @ Y2 )
= ( cons @ A @ X @ ( list_update @ A @ Xs @ I2 @ Y2 ) ) ) ).
% list_update_code(3)
thf(fact_6336_foldr__Cons,axiom,
! [B: $tType,A: $tType,F3: A > B > B,X: A,Xs: list @ A] :
( ( foldr @ A @ B @ F3 @ ( cons @ A @ X @ Xs ) )
= ( comp @ B @ B @ B @ ( F3 @ X ) @ ( foldr @ A @ B @ F3 @ Xs ) ) ) ).
% foldr_Cons
thf(fact_6337_remdups_Osimps_I2_J,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ( remdups @ A @ ( cons @ A @ X @ Xs ) )
= ( remdups @ A @ Xs ) ) )
& ( ~ ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ( remdups @ A @ ( cons @ A @ X @ Xs ) )
= ( cons @ A @ X @ ( remdups @ A @ Xs ) ) ) ) ) ).
% remdups.simps(2)
thf(fact_6338_sorted2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y2: A,Zs: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( cons @ A @ X @ ( cons @ A @ Y2 @ Zs ) ) )
= ( ( ord_less_eq @ A @ X @ Y2 )
& ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( cons @ A @ Y2 @ Zs ) ) ) ) ) ).
% sorted2
thf(fact_6339_impossible__Cons,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,X: A] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ A ) @ Ys ) )
=> ( Xs
!= ( cons @ A @ X @ Ys ) ) ) ).
% impossible_Cons
thf(fact_6340_set__subset__Cons,axiom,
! [A: $tType,Xs: list @ A,X: A] : ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ ( cons @ A @ X @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_6341_insort__key_Osimps_I2_J,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F3: B > A,X: B,Y2: B,Ys: list @ B] :
( ( ( ord_less_eq @ A @ ( F3 @ X ) @ ( F3 @ Y2 ) )
=> ( ( linorder_insort_key @ B @ A @ F3 @ X @ ( cons @ B @ Y2 @ Ys ) )
= ( cons @ B @ X @ ( cons @ B @ Y2 @ Ys ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( F3 @ X ) @ ( F3 @ Y2 ) )
=> ( ( linorder_insort_key @ B @ A @ F3 @ X @ ( cons @ B @ Y2 @ Ys ) )
= ( cons @ B @ Y2 @ ( linorder_insort_key @ B @ A @ F3 @ X @ Ys ) ) ) ) ) ) ).
% insort_key.simps(2)
thf(fact_6342_take__Cons,axiom,
! [A: $tType,N: nat,X: A,Xs: list @ A] :
( ( take @ A @ N @ ( cons @ A @ X @ Xs ) )
= ( case_nat @ ( list @ A ) @ ( nil @ A )
@ ^ [M2: nat] : ( cons @ A @ X @ ( take @ A @ M2 @ Xs ) )
@ N ) ) ).
% take_Cons
thf(fact_6343_Cons__shuffles__subset2,axiom,
! [A: $tType,Y2: A,Xs: list @ A,Ys: list @ A] : ( ord_less_eq @ ( set @ ( list @ A ) ) @ ( image @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ Y2 ) @ ( shuffles @ A @ Xs @ Ys ) ) @ ( shuffles @ A @ Xs @ ( cons @ A @ Y2 @ Ys ) ) ) ).
% Cons_shuffles_subset2
thf(fact_6344_Cons__shuffles__subset1,axiom,
! [A: $tType,X: A,Xs: list @ A,Ys: list @ A] : ( ord_less_eq @ ( set @ ( list @ A ) ) @ ( image @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X ) @ ( shuffles @ A @ Xs @ Ys ) ) @ ( shuffles @ A @ ( cons @ A @ X @ Xs ) @ Ys ) ) ).
% Cons_shuffles_subset1
thf(fact_6345_Cons__in__subseqsD,axiom,
! [A: $tType,Y2: A,Ys: list @ A,Xs: list @ A] :
( ( member @ ( list @ A ) @ ( cons @ A @ Y2 @ Ys ) @ ( set2 @ ( list @ A ) @ ( subseqs @ A @ Xs ) ) )
=> ( member @ ( list @ A ) @ Ys @ ( set2 @ ( list @ A ) @ ( subseqs @ A @ Xs ) ) ) ) ).
% Cons_in_subseqsD
thf(fact_6346_nth__Cons,axiom,
! [A: $tType,X: A,Xs: list @ A,N: nat] :
( ( nth @ A @ ( cons @ A @ X @ Xs ) @ N )
= ( case_nat @ A @ X @ ( nth @ A @ Xs ) @ N ) ) ).
% nth_Cons
thf(fact_6347_Suc__le__length__iff,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( ord_less_eq @ nat @ ( suc @ N ) @ ( size_size @ ( list @ A ) @ Xs ) )
= ( ? [X4: A,Ys3: list @ A] :
( ( Xs
= ( cons @ A @ X4 @ Ys3 ) )
& ( ord_less_eq @ nat @ N @ ( size_size @ ( list @ A ) @ Ys3 ) ) ) ) ) ).
% Suc_le_length_iff
thf(fact_6348_sorted1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A] : ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( cons @ A @ X @ ( nil @ A ) ) ) ) ).
% sorted1
thf(fact_6349_sorted__simps_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Ys: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( cons @ A @ X @ Ys ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Ys ) )
=> ( ord_less_eq @ A @ X @ X4 ) )
& ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Ys ) ) ) ) ).
% sorted_simps(2)
thf(fact_6350_strict__sorted__simps_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Ys: list @ A] :
( ( sorted_wrt @ A @ ( ord_less @ A ) @ ( cons @ A @ X @ Ys ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Ys ) )
=> ( ord_less @ A @ X @ X4 ) )
& ( sorted_wrt @ A @ ( ord_less @ A ) @ Ys ) ) ) ) ).
% strict_sorted_simps(2)
thf(fact_6351_insort__is__Cons,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ B,F3: B > A,A2: B] :
( ! [X5: B] :
( ( member @ B @ X5 @ ( set2 @ B @ Xs ) )
=> ( ord_less_eq @ A @ ( F3 @ A2 ) @ ( F3 @ X5 ) ) )
=> ( ( linorder_insort_key @ B @ A @ F3 @ A2 @ Xs )
= ( cons @ B @ A2 @ Xs ) ) ) ) ).
% insort_is_Cons
thf(fact_6352_count__list_Osimps_I2_J,axiom,
! [A: $tType,X: A,Y2: A,Xs: list @ A] :
( ( ( X = Y2 )
=> ( ( count_list @ A @ ( cons @ A @ X @ Xs ) @ Y2 )
= ( plus_plus @ nat @ ( count_list @ A @ Xs @ Y2 ) @ ( one_one @ nat ) ) ) )
& ( ( X != Y2 )
=> ( ( count_list @ A @ ( cons @ A @ X @ Xs ) @ Y2 )
= ( count_list @ A @ Xs @ Y2 ) ) ) ) ).
% count_list.simps(2)
thf(fact_6353_the__elem__set,axiom,
! [A: $tType,X: A] :
( ( the_elem @ A @ ( set2 @ A @ ( cons @ A @ X @ ( nil @ A ) ) ) )
= X ) ).
% the_elem_set
thf(fact_6354_list_Osize_I4_J,axiom,
! [A: $tType,X21: A,X22: list @ A] :
( ( size_size @ ( list @ A ) @ ( cons @ A @ X21 @ X22 ) )
= ( plus_plus @ nat @ ( size_size @ ( list @ A ) @ X22 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% list.size(4)
thf(fact_6355_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 )
@ ^ [Y: A] : ( cons @ A @ Y @ Ys3 )
@ Xs )
@ ( n_lists @ A @ N @ Xs ) ) ) ) ).
% n_lists.simps(2)
thf(fact_6356_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_6357_list_Osize__gen_I2_J,axiom,
! [A: $tType,X: A > nat,X21: A,X22: list @ A] :
( ( size_list @ A @ X @ ( cons @ A @ X21 @ X22 ) )
= ( plus_plus @ nat @ ( plus_plus @ nat @ ( X @ X21 ) @ ( size_list @ A @ X @ X22 ) ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% list.size_gen(2)
thf(fact_6358_shuffles_Opinduct,axiom,
! [A: $tType,A0: list @ A,A12: list @ A,P3: ( list @ A ) > ( list @ A ) > $o] :
( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( shuffles_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ A0 @ A12 ) )
=> ( ! [Ys4: list @ A] :
( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( shuffles_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Ys4 ) )
=> ( P3 @ ( nil @ A ) @ Ys4 ) )
=> ( ! [Xs3: list @ A] :
( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( shuffles_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs3 @ ( nil @ A ) ) )
=> ( P3 @ Xs3 @ ( nil @ A ) ) )
=> ( ! [X5: A,Xs3: list @ A,Y7: A,Ys4: list @ A] :
( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( shuffles_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X5 @ Xs3 ) @ ( cons @ A @ Y7 @ Ys4 ) ) )
=> ( ( P3 @ Xs3 @ ( cons @ A @ Y7 @ Ys4 ) )
=> ( ( P3 @ ( cons @ A @ X5 @ Xs3 ) @ Ys4 )
=> ( P3 @ ( cons @ A @ X5 @ Xs3 ) @ ( cons @ A @ Y7 @ Ys4 ) ) ) ) )
=> ( P3 @ A0 @ A12 ) ) ) ) ) ).
% shuffles.pinduct
thf(fact_6359_map__upt__Suc,axiom,
! [A: $tType,F3: nat > A,N: nat] :
( ( map @ nat @ A @ F3 @ ( upt @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
= ( cons @ A @ ( F3 @ ( zero_zero @ nat ) )
@ ( map @ nat @ A
@ ^ [I: nat] : ( F3 @ ( suc @ I ) )
@ ( upt @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% map_upt_Suc
thf(fact_6360_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_6361_nth__non__equal__first__eq,axiom,
! [A: $tType,X: A,Y2: A,Xs: list @ A,N: nat] :
( ( X != Y2 )
=> ( ( ( nth @ A @ ( cons @ A @ X @ Xs ) @ N )
= Y2 )
= ( ( ( nth @ A @ Xs @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) )
= Y2 )
& ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% nth_non_equal_first_eq
thf(fact_6362_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_6363_Cons__replicate__eq,axiom,
! [A: $tType,X: A,Xs: list @ A,N: nat,Y2: A] :
( ( ( cons @ A @ X @ Xs )
= ( replicate @ A @ N @ Y2 ) )
= ( ( X = Y2 )
& ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
& ( Xs
= ( replicate @ A @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ X ) ) ) ) ).
% Cons_replicate_eq
thf(fact_6364_set__Cons__sing__Nil,axiom,
! [A: $tType,A3: set @ A] :
( ( set_Cons @ A @ A3 @ ( insert @ ( list @ A ) @ ( nil @ A ) @ ( bot_bot @ ( set @ ( list @ A ) ) ) ) )
= ( image @ A @ ( list @ A )
@ ^ [X4: A] : ( cons @ A @ X4 @ ( nil @ A ) )
@ A3 ) ) ).
% set_Cons_sing_Nil
thf(fact_6365_transpose__aux__filter__head,axiom,
! [A: $tType,Xss: list @ ( list @ A )] :
( ( concat @ A
@ ( map @ ( list @ A ) @ ( list @ A )
@ ( case_list @ ( list @ A ) @ A @ ( nil @ A )
@ ^ [H: A,T3: list @ A] : ( cons @ A @ H @ ( nil @ A ) ) )
@ Xss ) )
= ( map @ ( list @ A ) @ A @ ( hd @ A )
@ ( filter2 @ ( list @ A )
@ ^ [Ys3: list @ A] :
( Ys3
!= ( nil @ A ) )
@ Xss ) ) ) ).
% transpose_aux_filter_head
thf(fact_6366_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_6367_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_6368_transpose_Osimps_I2_J,axiom,
! [A: $tType,Xss: list @ ( list @ A )] :
( ( transpose @ A @ ( cons @ ( list @ A ) @ ( nil @ A ) @ Xss ) )
= ( transpose @ A @ Xss ) ) ).
% transpose.simps(2)
thf(fact_6369_upt__conv__Cons__Cons,axiom,
! [M: nat,N: nat,Ns: list @ nat,Q: nat] :
( ( ( cons @ nat @ M @ ( cons @ nat @ N @ Ns ) )
= ( upt @ M @ Q ) )
= ( ( cons @ nat @ N @ Ns )
= ( upt @ ( suc @ M ) @ Q ) ) ) ).
% upt_conv_Cons_Cons
thf(fact_6370_list_Osimps_I4_J,axiom,
! [A: $tType,B: $tType,F1: B,F2: A > ( list @ A ) > B] :
( ( case_list @ B @ A @ F1 @ F2 @ ( nil @ A ) )
= F1 ) ).
% list.simps(4)
thf(fact_6371_listset_Osimps_I2_J,axiom,
! [A: $tType,A3: set @ A,As2: list @ ( set @ A )] :
( ( listset @ A @ ( cons @ ( set @ A ) @ A3 @ As2 ) )
= ( set_Cons @ A @ A3 @ ( listset @ A @ As2 ) ) ) ).
% listset.simps(2)
thf(fact_6372_list_Osimps_I5_J,axiom,
! [B: $tType,A: $tType,F1: B,F2: A > ( list @ A ) > B,X21: A,X22: list @ A] :
( ( case_list @ B @ A @ F1 @ F2 @ ( cons @ A @ X21 @ X22 ) )
= ( F2 @ X21 @ X22 ) ) ).
% list.simps(5)
thf(fact_6373_list_Ocase__distrib,axiom,
! [B: $tType,C: $tType,A: $tType,H2: B > C,F1: B,F2: A > ( list @ A ) > B,List: list @ A] :
( ( H2 @ ( case_list @ B @ A @ F1 @ F2 @ List ) )
= ( case_list @ C @ A @ ( H2 @ F1 )
@ ^ [X1: A,X25: list @ A] : ( H2 @ ( F2 @ X1 @ X25 ) )
@ List ) ) ).
% list.case_distrib
thf(fact_6374_upt__conv__Cons,axiom,
! [I2: nat,J3: nat] :
( ( ord_less @ nat @ I2 @ J3 )
=> ( ( upt @ I2 @ J3 )
= ( cons @ nat @ I2 @ ( upt @ ( suc @ I2 ) @ J3 ) ) ) ) ).
% upt_conv_Cons
thf(fact_6375_subseqs_Osimps_I1_J,axiom,
! [A: $tType] :
( ( subseqs @ A @ ( nil @ A ) )
= ( cons @ ( list @ A ) @ ( nil @ A ) @ ( nil @ ( list @ A ) ) ) ) ).
% subseqs.simps(1)
thf(fact_6376_product__lists_Osimps_I1_J,axiom,
! [A: $tType] :
( ( product_lists @ A @ ( nil @ ( list @ A ) ) )
= ( cons @ ( list @ A ) @ ( nil @ A ) @ ( nil @ ( list @ A ) ) ) ) ).
% product_lists.simps(1)
thf(fact_6377_upt__eq__Cons__conv,axiom,
! [I2: nat,J3: nat,X: nat,Xs: list @ nat] :
( ( ( upt @ I2 @ J3 )
= ( cons @ nat @ X @ Xs ) )
= ( ( ord_less @ nat @ I2 @ J3 )
& ( I2 = X )
& ( ( upt @ ( plus_plus @ nat @ I2 @ ( one_one @ nat ) ) @ J3 )
= Xs ) ) ) ).
% upt_eq_Cons_conv
thf(fact_6378_transpose_Osimps_I3_J,axiom,
! [A: $tType,X: A,Xs: list @ A,Xss: list @ ( list @ A )] :
( ( transpose @ A @ ( cons @ ( list @ A ) @ ( cons @ A @ X @ Xs ) @ Xss ) )
= ( cons @ ( list @ A )
@ ( cons @ A @ X
@ ( concat @ A
@ ( map @ ( list @ A ) @ ( list @ A )
@ ( case_list @ ( list @ A ) @ A @ ( nil @ A )
@ ^ [H: A,T3: list @ A] : ( cons @ A @ H @ ( nil @ A ) ) )
@ Xss ) ) )
@ ( transpose @ A
@ ( cons @ ( list @ A ) @ Xs
@ ( concat @ ( list @ A )
@ ( map @ ( list @ A ) @ ( list @ ( list @ A ) )
@ ( case_list @ ( list @ ( list @ A ) ) @ A @ ( nil @ ( list @ A ) )
@ ^ [H: A,T3: list @ A] : ( cons @ ( list @ A ) @ T3 @ ( nil @ ( list @ A ) ) ) )
@ Xss ) ) ) ) ) ) ).
% transpose.simps(3)
thf(fact_6379_transpose_Oelims,axiom,
! [A: $tType,X: list @ ( list @ A ),Y2: list @ ( list @ A )] :
( ( ( transpose @ A @ X )
= Y2 )
=> ( ( ( X
= ( nil @ ( list @ A ) ) )
=> ( Y2
!= ( nil @ ( list @ A ) ) ) )
=> ( ! [Xss2: list @ ( list @ A )] :
( ( X
= ( cons @ ( list @ A ) @ ( nil @ A ) @ Xss2 ) )
=> ( Y2
!= ( transpose @ A @ Xss2 ) ) )
=> ~ ! [X5: A,Xs3: list @ A,Xss2: list @ ( list @ A )] :
( ( X
= ( cons @ ( list @ A ) @ ( cons @ A @ X5 @ Xs3 ) @ Xss2 ) )
=> ( Y2
!= ( cons @ ( list @ A )
@ ( cons @ A @ X5
@ ( concat @ A
@ ( map @ ( list @ A ) @ ( list @ A )
@ ( case_list @ ( list @ A ) @ A @ ( nil @ A )
@ ^ [H: A,T3: list @ A] : ( cons @ A @ H @ ( nil @ A ) ) )
@ Xss2 ) ) )
@ ( transpose @ A
@ ( cons @ ( list @ A ) @ Xs3
@ ( concat @ ( list @ A )
@ ( map @ ( list @ A ) @ ( list @ ( list @ A ) )
@ ( case_list @ ( list @ ( list @ A ) ) @ A @ ( nil @ ( list @ A ) )
@ ^ [H: A,T3: list @ A] : ( cons @ ( list @ A ) @ T3 @ ( nil @ ( list @ A ) ) ) )
@ Xss2 ) ) ) ) ) ) ) ) ) ) ).
% transpose.elims
thf(fact_6380_upt__rec,axiom,
( upt
= ( ^ [I: nat,J4: nat] : ( if @ ( list @ nat ) @ ( ord_less @ nat @ I @ J4 ) @ ( cons @ nat @ I @ ( upt @ ( suc @ I ) @ J4 ) ) @ ( nil @ nat ) ) ) ) ).
% upt_rec
thf(fact_6381_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_6382_sorted__list__of__set__greaterThanAtMost,axiom,
! [I2: nat,J3: nat] :
( ( ord_less_eq @ nat @ ( suc @ I2 ) @ J3 )
=> ( ( linord4507533701916653071of_set @ nat @ ( set_or3652927894154168847AtMost @ nat @ I2 @ J3 ) )
= ( cons @ nat @ ( suc @ I2 ) @ ( linord4507533701916653071of_set @ nat @ ( set_or3652927894154168847AtMost @ nat @ ( suc @ I2 ) @ J3 ) ) ) ) ) ).
% sorted_list_of_set_greaterThanAtMost
thf(fact_6383_sorted__list__of__set__greaterThanLessThan,axiom,
! [I2: nat,J3: nat] :
( ( ord_less @ nat @ ( suc @ I2 ) @ J3 )
=> ( ( linord4507533701916653071of_set @ nat @ ( set_or5935395276787703475ssThan @ nat @ I2 @ J3 ) )
= ( cons @ nat @ ( suc @ I2 ) @ ( linord4507533701916653071of_set @ nat @ ( set_or5935395276787703475ssThan @ nat @ ( suc @ I2 ) @ J3 ) ) ) ) ) ).
% sorted_list_of_set_greaterThanLessThan
thf(fact_6384_transpose_Opsimps_I3_J,axiom,
! [A: $tType,X: A,Xs: list @ A,Xss: list @ ( list @ A )] :
( ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ ( cons @ ( list @ A ) @ ( cons @ A @ X @ Xs ) @ Xss ) )
=> ( ( transpose @ A @ ( cons @ ( list @ A ) @ ( cons @ A @ X @ Xs ) @ Xss ) )
= ( cons @ ( list @ A )
@ ( cons @ A @ X
@ ( concat @ A
@ ( map @ ( list @ A ) @ ( list @ A )
@ ( case_list @ ( list @ A ) @ A @ ( nil @ A )
@ ^ [H: A,T3: list @ A] : ( cons @ A @ H @ ( nil @ A ) ) )
@ Xss ) ) )
@ ( transpose @ A
@ ( cons @ ( list @ A ) @ Xs
@ ( concat @ ( list @ A )
@ ( map @ ( list @ A ) @ ( list @ ( list @ A ) )
@ ( case_list @ ( list @ ( list @ A ) ) @ A @ ( nil @ ( list @ A ) )
@ ^ [H: A,T3: list @ A] : ( cons @ ( list @ A ) @ T3 @ ( nil @ ( list @ A ) ) ) )
@ Xss ) ) ) ) ) ) ) ).
% transpose.psimps(3)
thf(fact_6385_transpose_Opelims,axiom,
! [A: $tType,X: list @ ( list @ A ),Y2: list @ ( list @ A )] :
( ( ( transpose @ A @ X )
= Y2 )
=> ( ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ X )
=> ( ( ( X
= ( nil @ ( list @ A ) ) )
=> ( ( Y2
= ( nil @ ( list @ A ) ) )
=> ~ ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ ( nil @ ( list @ A ) ) ) ) )
=> ( ! [Xss2: list @ ( list @ A )] :
( ( X
= ( cons @ ( list @ A ) @ ( nil @ A ) @ Xss2 ) )
=> ( ( Y2
= ( transpose @ A @ Xss2 ) )
=> ~ ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ ( cons @ ( list @ A ) @ ( nil @ A ) @ Xss2 ) ) ) )
=> ~ ! [X5: A,Xs3: list @ A,Xss2: list @ ( list @ A )] :
( ( X
= ( cons @ ( list @ A ) @ ( cons @ A @ X5 @ Xs3 ) @ Xss2 ) )
=> ( ( Y2
= ( cons @ ( list @ A )
@ ( cons @ A @ X5
@ ( concat @ A
@ ( map @ ( list @ A ) @ ( list @ A )
@ ( case_list @ ( list @ A ) @ A @ ( nil @ A )
@ ^ [H: A,T3: list @ A] : ( cons @ A @ H @ ( nil @ A ) ) )
@ Xss2 ) ) )
@ ( transpose @ A
@ ( cons @ ( list @ A ) @ Xs3
@ ( concat @ ( list @ A )
@ ( map @ ( list @ A ) @ ( list @ ( list @ A ) )
@ ( case_list @ ( list @ ( list @ A ) ) @ A @ ( nil @ ( list @ A ) )
@ ^ [H: A,T3: list @ A] : ( cons @ ( list @ A ) @ T3 @ ( nil @ ( list @ A ) ) ) )
@ Xss2 ) ) ) ) ) )
=> ~ ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ ( cons @ ( list @ A ) @ ( cons @ A @ X5 @ Xs3 ) @ Xss2 ) ) ) ) ) ) ) ) ).
% transpose.pelims
thf(fact_6386_list_Odisc__eq__case_I1_J,axiom,
! [A: $tType,List: list @ A] :
( ( List
= ( nil @ A ) )
= ( case_list @ $o @ A @ $true
@ ^ [Uu3: A,Uv3: list @ A] : $false
@ List ) ) ).
% list.disc_eq_case(1)
thf(fact_6387_list_Odisc__eq__case_I2_J,axiom,
! [A: $tType,List: list @ A] :
( ( List
!= ( nil @ A ) )
= ( case_list @ $o @ A @ $false
@ ^ [Uu3: A,Uv3: list @ A] : $true
@ List ) ) ).
% list.disc_eq_case(2)
thf(fact_6388_transpose_Opsimps_I1_J,axiom,
! [A: $tType] :
( ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ ( nil @ ( list @ A ) ) )
=> ( ( transpose @ A @ ( nil @ ( list @ A ) ) )
= ( nil @ ( list @ A ) ) ) ) ).
% transpose.psimps(1)
thf(fact_6389_transpose_Opsimps_I2_J,axiom,
! [A: $tType,Xss: list @ ( list @ A )] :
( ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ ( cons @ ( list @ A ) @ ( nil @ A ) @ Xss ) )
=> ( ( transpose @ A @ ( cons @ ( list @ A ) @ ( nil @ A ) @ Xss ) )
= ( transpose @ A @ Xss ) ) ) ).
% transpose.psimps(2)
thf(fact_6390_transpose_Opinduct,axiom,
! [A: $tType,A0: list @ ( list @ A ),P3: ( list @ ( list @ A ) ) > $o] :
( ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ A0 )
=> ( ( ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ ( nil @ ( list @ A ) ) )
=> ( P3 @ ( nil @ ( list @ A ) ) ) )
=> ( ! [Xss2: list @ ( list @ A )] :
( ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ ( cons @ ( list @ A ) @ ( nil @ A ) @ Xss2 ) )
=> ( ( P3 @ Xss2 )
=> ( P3 @ ( cons @ ( list @ A ) @ ( nil @ A ) @ Xss2 ) ) ) )
=> ( ! [X5: A,Xs3: list @ A,Xss2: list @ ( list @ A )] :
( ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ ( cons @ ( list @ A ) @ ( cons @ A @ X5 @ Xs3 ) @ Xss2 ) )
=> ( ( P3
@ ( cons @ ( list @ A ) @ Xs3
@ ( concat @ ( list @ A )
@ ( map @ ( list @ A ) @ ( list @ ( list @ A ) )
@ ( case_list @ ( list @ ( list @ A ) ) @ A @ ( nil @ ( list @ A ) )
@ ^ [H: A,T3: list @ A] : ( cons @ ( list @ A ) @ T3 @ ( nil @ ( list @ A ) ) ) )
@ Xss2 ) ) ) )
=> ( P3 @ ( cons @ ( list @ A ) @ ( cons @ A @ X5 @ Xs3 ) @ Xss2 ) ) ) )
=> ( P3 @ A0 ) ) ) ) ) ).
% transpose.pinduct
thf(fact_6391_transpose__aux__filter__tail,axiom,
! [A: $tType,Xss: list @ ( list @ A )] :
( ( concat @ ( list @ A )
@ ( map @ ( list @ A ) @ ( list @ ( list @ A ) )
@ ( case_list @ ( list @ ( list @ A ) ) @ A @ ( nil @ ( list @ A ) )
@ ^ [H: A,T3: list @ A] : ( cons @ ( list @ A ) @ T3 @ ( nil @ ( list @ A ) ) ) )
@ Xss ) )
= ( map @ ( list @ A ) @ ( list @ A ) @ ( tl @ A )
@ ( filter2 @ ( list @ A )
@ ^ [Ys3: list @ A] :
( Ys3
!= ( nil @ A ) )
@ Xss ) ) ) ).
% transpose_aux_filter_tail
thf(fact_6392_shuffles_Opelims,axiom,
! [A: $tType,X: list @ A,Xa: list @ A,Y2: set @ ( list @ A )] :
( ( ( shuffles @ A @ X @ Xa )
= Y2 )
=> ( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( shuffles_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Xa ) )
=> ( ( ( X
= ( nil @ A ) )
=> ( ( Y2
= ( insert @ ( list @ A ) @ Xa @ ( bot_bot @ ( set @ ( list @ A ) ) ) ) )
=> ~ ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( shuffles_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Xa ) ) ) )
=> ( ( ( Xa
= ( nil @ A ) )
=> ( ( Y2
= ( insert @ ( list @ A ) @ X @ ( bot_bot @ ( set @ ( list @ A ) ) ) ) )
=> ~ ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( shuffles_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ ( nil @ A ) ) ) ) )
=> ~ ! [X5: A,Xs3: list @ A] :
( ( X
= ( cons @ A @ X5 @ Xs3 ) )
=> ! [Y7: A,Ys4: list @ A] :
( ( Xa
= ( cons @ A @ Y7 @ Ys4 ) )
=> ( ( Y2
= ( sup_sup @ ( set @ ( list @ A ) ) @ ( image @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X5 ) @ ( shuffles @ A @ Xs3 @ ( cons @ A @ Y7 @ Ys4 ) ) ) @ ( image @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ Y7 ) @ ( shuffles @ A @ ( cons @ A @ X5 @ Xs3 ) @ Ys4 ) ) ) )
=> ~ ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( shuffles_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X5 @ Xs3 ) @ ( cons @ A @ Y7 @ Ys4 ) ) ) ) ) ) ) ) ) ) ).
% shuffles.pelims
thf(fact_6393_le__sup__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A,Y2: A,Z3: A] :
( ( ord_less_eq @ A @ ( sup_sup @ A @ X @ Y2 ) @ Z3 )
= ( ( ord_less_eq @ A @ X @ Z3 )
& ( ord_less_eq @ A @ Y2 @ Z3 ) ) ) ) ).
% le_sup_iff
thf(fact_6394_sup_Obounded__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B2: A,C2: A,A2: A] :
( ( ord_less_eq @ A @ ( sup_sup @ A @ B2 @ C2 ) @ A2 )
= ( ( ord_less_eq @ A @ B2 @ A2 )
& ( ord_less_eq @ A @ C2 @ A2 ) ) ) ) ).
% sup.bounded_iff
thf(fact_6395_Un__subset__iff,axiom,
! [A: $tType,A3: set @ A,B5: set @ A,C5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A3 @ B5 ) @ C5 )
= ( ( ord_less_eq @ ( set @ A ) @ A3 @ C5 )
& ( ord_less_eq @ ( set @ A ) @ B5 @ C5 ) ) ) ).
% Un_subset_iff
thf(fact_6396_tl__upt,axiom,
! [M: nat,N: nat] :
( ( tl @ nat @ ( upt @ M @ N ) )
= ( upt @ ( suc @ M ) @ N ) ) ).
% tl_upt
thf(fact_6397_if__image__distrib,axiom,
! [A: $tType,B: $tType,P3: B > $o,F3: B > A,G: B > A,S3: set @ B] :
( ( image @ B @ A
@ ^ [X4: B] : ( if @ A @ ( P3 @ X4 ) @ ( F3 @ X4 ) @ ( G @ X4 ) )
@ S3 )
= ( sup_sup @ ( set @ A ) @ ( image @ B @ A @ F3 @ ( inf_inf @ ( set @ B ) @ S3 @ ( collect @ B @ P3 ) ) )
@ ( image @ B @ A @ G
@ ( inf_inf @ ( set @ B ) @ S3
@ ( collect @ B
@ ^ [X4: B] :
~ ( P3 @ X4 ) ) ) ) ) ) ).
% if_image_distrib
thf(fact_6398_UN__Un,axiom,
! [A: $tType,B: $tType,M7: B > ( set @ A ),A3: set @ B,B5: set @ B] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ M7 @ ( sup_sup @ ( set @ B ) @ A3 @ B5 ) ) )
= ( sup_sup @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ M7 @ A3 ) ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ M7 @ B5 ) ) ) ) ).
% UN_Un
thf(fact_6399_set__union,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( set2 @ A @ ( union @ A @ Xs @ Ys ) )
= ( sup_sup @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ Ys ) ) ) ).
% set_union
thf(fact_6400_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_6401_hd__Cons__tl,axiom,
! [A: $tType,Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( cons @ A @ ( hd @ A @ Xs ) @ ( tl @ A @ Xs ) )
= Xs ) ) ).
% hd_Cons_tl
thf(fact_6402_list_Ocollapse,axiom,
! [A: $tType,List: list @ A] :
( ( List
!= ( nil @ A ) )
=> ( ( cons @ A @ ( hd @ A @ List ) @ ( tl @ A @ List ) )
= List ) ) ).
% list.collapse
thf(fact_6403_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_6404_UN__simps_I2_J,axiom,
! [C: $tType,D: $tType,C5: set @ C,A3: C > ( set @ D ),B5: set @ D] :
( ( ( C5
= ( bot_bot @ ( set @ C ) ) )
=> ( ( complete_Sup_Sup @ ( set @ D )
@ ( image @ C @ ( set @ D )
@ ^ [X4: C] : ( sup_sup @ ( set @ D ) @ ( A3 @ X4 ) @ B5 )
@ C5 ) )
= ( bot_bot @ ( set @ D ) ) ) )
& ( ( C5
!= ( bot_bot @ ( set @ C ) ) )
=> ( ( complete_Sup_Sup @ ( set @ D )
@ ( image @ C @ ( set @ D )
@ ^ [X4: C] : ( sup_sup @ ( set @ D ) @ ( A3 @ X4 ) @ B5 )
@ C5 ) )
= ( sup_sup @ ( set @ D ) @ ( complete_Sup_Sup @ ( set @ D ) @ ( image @ C @ ( set @ D ) @ A3 @ C5 ) ) @ B5 ) ) ) ) ).
% UN_simps(2)
thf(fact_6405_UN__simps_I3_J,axiom,
! [E4: $tType,F: $tType,C5: set @ F,A3: set @ E4,B5: F > ( set @ E4 )] :
( ( ( C5
= ( bot_bot @ ( set @ F ) ) )
=> ( ( complete_Sup_Sup @ ( set @ E4 )
@ ( image @ F @ ( set @ E4 )
@ ^ [X4: F] : ( sup_sup @ ( set @ E4 ) @ A3 @ ( B5 @ X4 ) )
@ C5 ) )
= ( bot_bot @ ( set @ E4 ) ) ) )
& ( ( C5
!= ( bot_bot @ ( set @ F ) ) )
=> ( ( complete_Sup_Sup @ ( set @ E4 )
@ ( image @ F @ ( set @ E4 )
@ ^ [X4: F] : ( sup_sup @ ( set @ E4 ) @ A3 @ ( B5 @ X4 ) )
@ C5 ) )
= ( sup_sup @ ( set @ E4 ) @ A3 @ ( complete_Sup_Sup @ ( set @ E4 ) @ ( image @ F @ ( set @ E4 ) @ B5 @ C5 ) ) ) ) ) ) ).
% UN_simps(3)
thf(fact_6406_UN__insert,axiom,
! [A: $tType,B: $tType,B5: B > ( set @ A ),A2: B,A3: set @ B] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B5 @ ( insert @ B @ A2 @ A3 ) ) )
= ( sup_sup @ ( set @ A ) @ ( B5 @ A2 ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B5 @ A3 ) ) ) ) ).
% UN_insert
thf(fact_6407_Nil__tl,axiom,
! [A: $tType,Xs: list @ A] :
( ( ( nil @ A )
= ( tl @ A @ Xs ) )
= ( ( Xs
= ( nil @ A ) )
| ? [X4: A] :
( Xs
= ( cons @ A @ X4 @ ( nil @ A ) ) ) ) ) ).
% Nil_tl
thf(fact_6408_tl__Nil,axiom,
! [A: $tType,Xs: list @ A] :
( ( ( tl @ A @ Xs )
= ( nil @ A ) )
= ( ( Xs
= ( nil @ A ) )
| ? [X4: A] :
( Xs
= ( cons @ A @ X4 @ ( nil @ A ) ) ) ) ) ).
% tl_Nil
thf(fact_6409_list_Osel_I3_J,axiom,
! [A: $tType,X21: A,X22: list @ A] :
( ( tl @ A @ ( cons @ A @ X21 @ X22 ) )
= X22 ) ).
% list.sel(3)
thf(fact_6410_list_Oexpand,axiom,
! [A: $tType,List: list @ A,List2: list @ A] :
( ( ( List
= ( nil @ A ) )
= ( List2
= ( nil @ A ) ) )
=> ( ( ( List
!= ( nil @ A ) )
=> ( ( List2
!= ( nil @ A ) )
=> ( ( ( hd @ A @ List )
= ( hd @ A @ List2 ) )
& ( ( tl @ A @ List )
= ( tl @ A @ List2 ) ) ) ) )
=> ( List = List2 ) ) ) ).
% list.expand
thf(fact_6411_list_Osel_I2_J,axiom,
! [A: $tType] :
( ( tl @ A @ ( nil @ A ) )
= ( nil @ A ) ) ).
% list.sel(2)
thf(fact_6412_list_Oset__sel_I2_J,axiom,
! [A: $tType,A2: list @ A,X: A] :
( ( A2
!= ( nil @ A ) )
=> ( ( member @ A @ X @ ( set2 @ A @ ( tl @ A @ A2 ) ) )
=> ( member @ A @ X @ ( set2 @ A @ A2 ) ) ) ) ).
% list.set_sel(2)
thf(fact_6413_insert__def,axiom,
! [A: $tType] :
( ( insert @ A )
= ( ^ [A5: A] :
( sup_sup @ ( set @ A )
@ ( collect @ A
@ ^ [X4: A] : X4 = A5 ) ) ) ) ).
% insert_def
thf(fact_6414_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_6415_set__shuffles,axiom,
! [A: $tType,Zs: list @ A,Xs: list @ A,Ys: list @ A] :
( ( member @ ( list @ A ) @ Zs @ ( shuffles @ A @ Xs @ Ys ) )
=> ( ( set2 @ A @ Zs )
= ( sup_sup @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ Ys ) ) ) ) ).
% set_shuffles
thf(fact_6416_distinct__tl,axiom,
! [A: $tType,Xs: list @ A] :
( ( distinct @ A @ Xs )
=> ( distinct @ A @ ( tl @ A @ Xs ) ) ) ).
% distinct_tl
thf(fact_6417_Un__def,axiom,
! [A: $tType] :
( ( sup_sup @ ( set @ A ) )
= ( ^ [A7: set @ A,B7: set @ A] :
( collect @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ A7 )
| ( member @ A @ X4 @ B7 ) ) ) ) ) ).
% Un_def
thf(fact_6418_Collect__disj__eq,axiom,
! [A: $tType,P3: A > $o,Q2: A > $o] :
( ( collect @ A
@ ^ [X4: A] :
( ( P3 @ X4 )
| ( Q2 @ X4 ) ) )
= ( sup_sup @ ( set @ A ) @ ( collect @ A @ P3 ) @ ( collect @ A @ Q2 ) ) ) ).
% Collect_disj_eq
thf(fact_6419_Collect__imp__eq,axiom,
! [A: $tType,P3: A > $o,Q2: A > $o] :
( ( collect @ A
@ ^ [X4: A] :
( ( P3 @ X4 )
=> ( Q2 @ X4 ) ) )
= ( sup_sup @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ ( collect @ A @ P3 ) ) @ ( collect @ A @ Q2 ) ) ) ).
% Collect_imp_eq
thf(fact_6420_sup_Ostrict__coboundedI2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [C2: A,B2: A,A2: A] :
( ( ord_less @ A @ C2 @ B2 )
=> ( ord_less @ A @ C2 @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ).
% sup.strict_coboundedI2
thf(fact_6421_sup_Ostrict__coboundedI1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less @ A @ C2 @ A2 )
=> ( ord_less @ A @ C2 @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ).
% sup.strict_coboundedI1
thf(fact_6422_sup_Ostrict__order__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( ord_less @ A )
= ( ^ [B4: A,A5: A] :
( ( A5
= ( sup_sup @ A @ A5 @ B4 ) )
& ( A5 != B4 ) ) ) ) ) ).
% sup.strict_order_iff
thf(fact_6423_sup_Ostrict__boundedE,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B2: A,C2: A,A2: A] :
( ( ord_less @ A @ ( sup_sup @ A @ B2 @ C2 ) @ A2 )
=> ~ ( ( ord_less @ A @ B2 @ A2 )
=> ~ ( ord_less @ A @ C2 @ A2 ) ) ) ) ).
% sup.strict_boundedE
thf(fact_6424_sup_Oabsorb4,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( sup_sup @ A @ A2 @ B2 )
= B2 ) ) ) ).
% sup.absorb4
thf(fact_6425_sup_Oabsorb3,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B2: A,A2: A] :
( ( ord_less @ A @ B2 @ A2 )
=> ( ( sup_sup @ A @ A2 @ B2 )
= A2 ) ) ) ).
% sup.absorb3
thf(fact_6426_less__supI2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A,B2: A,A2: A] :
( ( ord_less @ A @ X @ B2 )
=> ( ord_less @ A @ X @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ).
% less_supI2
thf(fact_6427_less__supI1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A,A2: A,B2: A] :
( ( ord_less @ A @ X @ A2 )
=> ( ord_less @ A @ X @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ).
% less_supI1
thf(fact_6428_INF__sup__distrib2,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [F3: B > A,A3: set @ B,G: C > A,B5: set @ C] :
( ( sup_sup @ A @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F3 @ A3 ) ) @ ( complete_Inf_Inf @ A @ ( image @ C @ A @ G @ B5 ) ) )
= ( complete_Inf_Inf @ A
@ ( image @ B @ A
@ ^ [A5: B] :
( complete_Inf_Inf @ A
@ ( image @ C @ A
@ ^ [B4: C] : ( sup_sup @ A @ ( F3 @ A5 ) @ ( G @ B4 ) )
@ B5 ) )
@ A3 ) ) ) ) ).
% INF_sup_distrib2
thf(fact_6429_sup__INF,axiom,
! [A: $tType,B: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [A2: A,F3: B > A,B5: set @ B] :
( ( sup_sup @ A @ A2 @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F3 @ B5 ) ) )
= ( complete_Inf_Inf @ A
@ ( image @ B @ A
@ ^ [B4: B] : ( sup_sup @ A @ A2 @ ( F3 @ B4 ) )
@ B5 ) ) ) ) ).
% sup_INF
thf(fact_6430_Inf__sup,axiom,
! [A: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [B5: set @ A,A2: A] :
( ( sup_sup @ A @ ( complete_Inf_Inf @ A @ B5 ) @ A2 )
= ( complete_Inf_Inf @ A
@ ( image @ A @ A
@ ^ [B4: A] : ( sup_sup @ A @ B4 @ A2 )
@ B5 ) ) ) ) ).
% Inf_sup
thf(fact_6431_INF__sup,axiom,
! [A: $tType,B: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [F3: B > A,B5: set @ B,A2: A] :
( ( sup_sup @ A @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F3 @ B5 ) ) @ A2 )
= ( complete_Inf_Inf @ A
@ ( image @ B @ A
@ ^ [B4: B] : ( sup_sup @ A @ ( F3 @ B4 ) @ A2 )
@ B5 ) ) ) ) ).
% INF_sup
thf(fact_6432_SUP__absorb,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [K2: B,I6: set @ B,A3: B > A] :
( ( member @ B @ K2 @ I6 )
=> ( ( sup_sup @ A @ ( A3 @ K2 ) @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ A3 @ I6 ) ) )
= ( complete_Sup_Sup @ A @ ( image @ B @ A @ A3 @ I6 ) ) ) ) ) ).
% SUP_absorb
thf(fact_6433_complete__lattice__class_OSUP__sup__distrib,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F3: B > A,A3: set @ B,G: B > A] :
( ( sup_sup @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F3 @ A3 ) ) @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ G @ A3 ) ) )
= ( complete_Sup_Sup @ A
@ ( image @ B @ A
@ ^ [A5: B] : ( sup_sup @ A @ ( F3 @ A5 ) @ ( G @ A5 ) )
@ A3 ) ) ) ) ).
% complete_lattice_class.SUP_sup_distrib
thf(fact_6434_SUP__union,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [M7: B > A,A3: set @ B,B5: set @ B] :
( ( complete_Sup_Sup @ A @ ( image @ B @ A @ M7 @ ( sup_sup @ ( set @ B ) @ A3 @ B5 ) ) )
= ( sup_sup @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ M7 @ A3 ) ) @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ M7 @ B5 ) ) ) ) ) ).
% SUP_union
thf(fact_6435_map__tl,axiom,
! [A: $tType,B: $tType,F3: B > A,Xs: list @ B] :
( ( map @ B @ A @ F3 @ ( tl @ B @ Xs ) )
= ( tl @ A @ ( map @ B @ A @ F3 @ Xs ) ) ) ).
% map_tl
thf(fact_6436_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_6437_Un__Int__assoc__eq,axiom,
! [A: $tType,A3: set @ A,B5: set @ A,C5: set @ A] :
( ( ( sup_sup @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A3 @ B5 ) @ C5 )
= ( inf_inf @ ( set @ A ) @ A3 @ ( sup_sup @ ( set @ A ) @ B5 @ C5 ) ) )
= ( ord_less_eq @ ( set @ A ) @ C5 @ A3 ) ) ).
% Un_Int_assoc_eq
thf(fact_6438_distrib__sup__le,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X: A,Y2: A,Z3: A] : ( ord_less_eq @ A @ ( sup_sup @ A @ X @ ( inf_inf @ A @ Y2 @ Z3 ) ) @ ( inf_inf @ A @ ( sup_sup @ A @ X @ Y2 ) @ ( sup_sup @ A @ X @ Z3 ) ) ) ) ).
% distrib_sup_le
thf(fact_6439_distrib__inf__le,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X: A,Y2: A,Z3: A] : ( ord_less_eq @ A @ ( sup_sup @ A @ ( inf_inf @ A @ X @ Y2 ) @ ( inf_inf @ A @ X @ Z3 ) ) @ ( inf_inf @ A @ X @ ( sup_sup @ A @ Y2 @ Z3 ) ) ) ) ).
% distrib_inf_le
thf(fact_6440_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_6441_Un__Pow__subset,axiom,
! [A: $tType,A3: set @ A,B5: set @ A] : ( ord_less_eq @ ( set @ ( set @ A ) ) @ ( sup_sup @ ( set @ ( set @ A ) ) @ ( pow2 @ A @ A3 ) @ ( pow2 @ A @ B5 ) ) @ ( pow2 @ A @ ( sup_sup @ ( set @ A ) @ A3 @ B5 ) ) ) ).
% Un_Pow_subset
thf(fact_6442_inf__sup__ord_I4_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [Y2: A,X: A] : ( ord_less_eq @ A @ Y2 @ ( sup_sup @ A @ X @ Y2 ) ) ) ).
% inf_sup_ord(4)
thf(fact_6443_inf__sup__ord_I3_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X: A,Y2: A] : ( ord_less_eq @ A @ X @ ( sup_sup @ A @ X @ Y2 ) ) ) ).
% inf_sup_ord(3)
thf(fact_6444_le__supE,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A2: A,B2: A,X: A] :
( ( ord_less_eq @ A @ ( sup_sup @ A @ A2 @ B2 ) @ X )
=> ~ ( ( ord_less_eq @ A @ A2 @ X )
=> ~ ( ord_less_eq @ A @ B2 @ X ) ) ) ) ).
% le_supE
thf(fact_6445_le__supI,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A2: A,X: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ X )
=> ( ( ord_less_eq @ A @ B2 @ X )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ A2 @ B2 ) @ X ) ) ) ) ).
% le_supI
thf(fact_6446_sup__ge1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A,Y2: A] : ( ord_less_eq @ A @ X @ ( sup_sup @ A @ X @ Y2 ) ) ) ).
% sup_ge1
thf(fact_6447_sup__ge2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [Y2: A,X: A] : ( ord_less_eq @ A @ Y2 @ ( sup_sup @ A @ X @ Y2 ) ) ) ).
% sup_ge2
thf(fact_6448_le__supI1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A,A2: A,B2: A] :
( ( ord_less_eq @ A @ X @ A2 )
=> ( ord_less_eq @ A @ X @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ).
% le_supI1
thf(fact_6449_le__supI2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A,B2: A,A2: A] :
( ( ord_less_eq @ A @ X @ B2 )
=> ( ord_less_eq @ A @ X @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ).
% le_supI2
thf(fact_6450_sup_Omono,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [C2: A,A2: A,D2: A,B2: A] :
( ( ord_less_eq @ A @ C2 @ A2 )
=> ( ( ord_less_eq @ A @ D2 @ B2 )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ C2 @ D2 ) @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ) ).
% sup.mono
thf(fact_6451_sup__mono,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A2: A,C2: A,B2: A,D2: A] :
( ( ord_less_eq @ A @ A2 @ C2 )
=> ( ( ord_less_eq @ A @ B2 @ D2 )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ A2 @ B2 ) @ ( sup_sup @ A @ C2 @ D2 ) ) ) ) ) ).
% sup_mono
thf(fact_6452_sup__least,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [Y2: A,X: A,Z3: A] :
( ( ord_less_eq @ A @ Y2 @ X )
=> ( ( ord_less_eq @ A @ Z3 @ X )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ Y2 @ Z3 ) @ X ) ) ) ) ).
% sup_least
thf(fact_6453_le__iff__sup,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [X4: A,Y: A] :
( ( sup_sup @ A @ X4 @ Y )
= Y ) ) ) ) ).
% le_iff_sup
thf(fact_6454_sup_OorderE,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B2: A,A2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( A2
= ( sup_sup @ A @ A2 @ B2 ) ) ) ) ).
% sup.orderE
thf(fact_6455_sup_OorderI,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A2: A,B2: A] :
( ( A2
= ( sup_sup @ A @ A2 @ B2 ) )
=> ( ord_less_eq @ A @ B2 @ A2 ) ) ) ).
% sup.orderI
thf(fact_6456_sup__unique,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [F3: A > A > A,X: A,Y2: A] :
( ! [X5: A,Y7: A] : ( ord_less_eq @ A @ X5 @ ( F3 @ X5 @ Y7 ) )
=> ( ! [X5: A,Y7: A] : ( ord_less_eq @ A @ Y7 @ ( F3 @ X5 @ Y7 ) )
=> ( ! [X5: A,Y7: A,Z: A] :
( ( ord_less_eq @ A @ Y7 @ X5 )
=> ( ( ord_less_eq @ A @ Z @ X5 )
=> ( ord_less_eq @ A @ ( F3 @ Y7 @ Z ) @ X5 ) ) )
=> ( ( sup_sup @ A @ X @ Y2 )
= ( F3 @ X @ Y2 ) ) ) ) ) ) ).
% sup_unique
thf(fact_6457_sup_Oabsorb1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B2: A,A2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( sup_sup @ A @ A2 @ B2 )
= A2 ) ) ) ).
% sup.absorb1
thf(fact_6458_sup_Oabsorb2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( sup_sup @ A @ A2 @ B2 )
= B2 ) ) ) ).
% sup.absorb2
thf(fact_6459_sup__absorb1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [Y2: A,X: A] :
( ( ord_less_eq @ A @ Y2 @ X )
=> ( ( sup_sup @ A @ X @ Y2 )
= X ) ) ) ).
% sup_absorb1
thf(fact_6460_sup__absorb2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A,Y2: A] :
( ( ord_less_eq @ A @ X @ Y2 )
=> ( ( sup_sup @ A @ X @ Y2 )
= Y2 ) ) ) ).
% sup_absorb2
thf(fact_6461_sup_OboundedE,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B2: A,C2: A,A2: A] :
( ( ord_less_eq @ A @ ( sup_sup @ A @ B2 @ C2 ) @ A2 )
=> ~ ( ( ord_less_eq @ A @ B2 @ A2 )
=> ~ ( ord_less_eq @ A @ C2 @ A2 ) ) ) ) ).
% sup.boundedE
thf(fact_6462_sup_OboundedI,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_less_eq @ A @ C2 @ A2 )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ B2 @ C2 ) @ A2 ) ) ) ) ).
% sup.boundedI
thf(fact_6463_sup_Oorder__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B4: A,A5: A] :
( A5
= ( sup_sup @ A @ A5 @ B4 ) ) ) ) ) ).
% sup.order_iff
thf(fact_6464_sup_Ocobounded1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A2: A,B2: A] : ( ord_less_eq @ A @ A2 @ ( sup_sup @ A @ A2 @ B2 ) ) ) ).
% sup.cobounded1
thf(fact_6465_sup_Ocobounded2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B2: A,A2: A] : ( ord_less_eq @ A @ B2 @ ( sup_sup @ A @ A2 @ B2 ) ) ) ).
% sup.cobounded2
thf(fact_6466_sup_Oabsorb__iff1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B4: A,A5: A] :
( ( sup_sup @ A @ A5 @ B4 )
= A5 ) ) ) ) ).
% sup.absorb_iff1
thf(fact_6467_sup_Oabsorb__iff2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A5: A,B4: A] :
( ( sup_sup @ A @ A5 @ B4 )
= B4 ) ) ) ) ).
% sup.absorb_iff2
thf(fact_6468_sup_OcoboundedI1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [C2: A,A2: A,B2: A] :
( ( ord_less_eq @ A @ C2 @ A2 )
=> ( ord_less_eq @ A @ C2 @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ).
% sup.coboundedI1
thf(fact_6469_sup_OcoboundedI2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [C2: A,B2: A,A2: A] :
( ( ord_less_eq @ A @ C2 @ B2 )
=> ( ord_less_eq @ A @ C2 @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ).
% sup.coboundedI2
thf(fact_6470_Un__mono,axiom,
! [A: $tType,A3: set @ A,C5: set @ A,B5: set @ A,D4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ C5 )
=> ( ( ord_less_eq @ ( set @ A ) @ B5 @ D4 )
=> ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A3 @ B5 ) @ ( sup_sup @ ( set @ A ) @ C5 @ D4 ) ) ) ) ).
% Un_mono
thf(fact_6471_Un__least,axiom,
! [A: $tType,A3: set @ A,C5: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ C5 )
=> ( ( ord_less_eq @ ( set @ A ) @ B5 @ C5 )
=> ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A3 @ B5 ) @ C5 ) ) ) ).
% Un_least
thf(fact_6472_Un__upper1,axiom,
! [A: $tType,A3: set @ A,B5: set @ A] : ( ord_less_eq @ ( set @ A ) @ A3 @ ( sup_sup @ ( set @ A ) @ A3 @ B5 ) ) ).
% Un_upper1
thf(fact_6473_Un__upper2,axiom,
! [A: $tType,B5: set @ A,A3: set @ A] : ( ord_less_eq @ ( set @ A ) @ B5 @ ( sup_sup @ ( set @ A ) @ A3 @ B5 ) ) ).
% Un_upper2
thf(fact_6474_Un__absorb1,axiom,
! [A: $tType,A3: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B5 )
=> ( ( sup_sup @ ( set @ A ) @ A3 @ B5 )
= B5 ) ) ).
% Un_absorb1
thf(fact_6475_Un__absorb2,axiom,
! [A: $tType,B5: set @ A,A3: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ B5 @ A3 )
=> ( ( sup_sup @ ( set @ A ) @ A3 @ B5 )
= A3 ) ) ).
% Un_absorb2
thf(fact_6476_subset__UnE,axiom,
! [A: $tType,C5: set @ A,A3: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ C5 @ ( sup_sup @ ( set @ A ) @ A3 @ B5 ) )
=> ~ ! [A15: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A15 @ A3 )
=> ! [B15: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ B15 @ B5 )
=> ( C5
!= ( sup_sup @ ( set @ A ) @ A15 @ B15 ) ) ) ) ) ).
% subset_UnE
thf(fact_6477_subset__Un__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A7: set @ A,B7: set @ A] :
( ( sup_sup @ ( set @ A ) @ A7 @ B7 )
= B7 ) ) ) ).
% subset_Un_eq
thf(fact_6478_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_6479_Diff__partition,axiom,
! [A: $tType,A3: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A3 @ B5 )
=> ( ( sup_sup @ ( set @ A ) @ A3 @ ( minus_minus @ ( set @ A ) @ B5 @ A3 ) )
= B5 ) ) ).
% Diff_partition
thf(fact_6480_Diff__subset__conv,axiom,
! [A: $tType,A3: set @ A,B5: set @ A,C5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A3 @ B5 ) @ C5 )
= ( ord_less_eq @ ( set @ A ) @ A3 @ ( sup_sup @ ( set @ A ) @ B5 @ C5 ) ) ) ).
% Diff_subset_conv
thf(fact_6481_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_6482_list_Omap__sel_I2_J,axiom,
! [B: $tType,A: $tType,A2: list @ A,F3: A > B] :
( ( A2
!= ( nil @ A ) )
=> ( ( tl @ B @ ( map @ A @ B @ F3 @ A2 ) )
= ( map @ A @ B @ F3 @ ( tl @ A @ A2 ) ) ) ) ).
% list.map_sel(2)
thf(fact_6483_Un__Union__image,axiom,
! [A: $tType,B: $tType,A3: B > ( set @ A ),B5: B > ( set @ A ),C5: set @ B] :
( ( complete_Sup_Sup @ ( set @ A )
@ ( image @ B @ ( set @ A )
@ ^ [X4: B] : ( sup_sup @ ( set @ A ) @ ( A3 @ X4 ) @ ( B5 @ X4 ) )
@ C5 ) )
= ( sup_sup @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ A3 @ C5 ) ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B5 @ C5 ) ) ) ) ).
% Un_Union_image
thf(fact_6484_UN__Un__distrib,axiom,
! [A: $tType,B: $tType,A3: B > ( set @ A ),B5: B > ( set @ A ),I6: set @ B] :
( ( complete_Sup_Sup @ ( set @ A )
@ ( image @ B @ ( set @ A )
@ ^ [I: B] : ( sup_sup @ ( set @ A ) @ ( A3 @ I ) @ ( B5 @ I ) )
@ I6 ) )
= ( sup_sup @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ A3 @ I6 ) ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B5 @ I6 ) ) ) ) ).
% UN_Un_distrib
thf(fact_6485_UN__absorb,axiom,
! [B: $tType,A: $tType,K2: A,I6: set @ A,A3: A > ( set @ B )] :
( ( member @ A @ K2 @ I6 )
=> ( ( sup_sup @ ( set @ B ) @ ( A3 @ K2 ) @ ( complete_Sup_Sup @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ A3 @ I6 ) ) )
= ( complete_Sup_Sup @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ A3 @ I6 ) ) ) ) ).
% UN_absorb
thf(fact_6486_Un__INT__distrib2,axiom,
! [A: $tType,C: $tType,B: $tType,A3: B > ( set @ A ),I6: set @ B,B5: C > ( set @ A ),J5: set @ C] :
( ( sup_sup @ ( set @ A ) @ ( complete_Inf_Inf @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ A3 @ I6 ) ) @ ( complete_Inf_Inf @ ( set @ A ) @ ( image @ C @ ( set @ A ) @ B5 @ J5 ) ) )
= ( complete_Inf_Inf @ ( set @ A )
@ ( image @ B @ ( set @ A )
@ ^ [I: B] :
( complete_Inf_Inf @ ( set @ A )
@ ( image @ C @ ( set @ A )
@ ^ [J4: C] : ( sup_sup @ ( set @ A ) @ ( A3 @ I ) @ ( B5 @ J4 ) )
@ J5 ) )
@ I6 ) ) ) ).
% Un_INT_distrib2
thf(fact_6487_Un__INT__distrib,axiom,
! [A: $tType,B: $tType,B5: set @ A,A3: B > ( set @ A ),I6: set @ B] :
( ( sup_sup @ ( set @ A ) @ B5 @ ( complete_Inf_Inf @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ A3 @ I6 ) ) )
= ( complete_Inf_Inf @ ( set @ A )
@ ( image @ B @ ( set @ A )
@ ^ [I: B] : ( sup_sup @ ( set @ A ) @ B5 @ ( A3 @ I ) )
@ I6 ) ) ) ).
% Un_INT_distrib
thf(fact_6488_INT__extend__simps_I6_J,axiom,
! [L5: $tType,K9: $tType,A3: K9 > ( set @ L5 ),C5: set @ K9,B5: set @ L5] :
( ( sup_sup @ ( set @ L5 ) @ ( complete_Inf_Inf @ ( set @ L5 ) @ ( image @ K9 @ ( set @ L5 ) @ A3 @ C5 ) ) @ B5 )
= ( complete_Inf_Inf @ ( set @ L5 )
@ ( image @ K9 @ ( set @ L5 )
@ ^ [X4: K9] : ( sup_sup @ ( set @ L5 ) @ ( A3 @ X4 ) @ B5 )
@ C5 ) ) ) ).
% INT_extend_simps(6)
thf(fact_6489_INT__extend__simps_I7_J,axiom,
! [M11: $tType,N10: $tType,A3: set @ M11,B5: N10 > ( set @ M11 ),C5: set @ N10] :
( ( sup_sup @ ( set @ M11 ) @ A3 @ ( complete_Inf_Inf @ ( set @ M11 ) @ ( image @ N10 @ ( set @ M11 ) @ B5 @ C5 ) ) )
= ( complete_Inf_Inf @ ( set @ M11 )
@ ( image @ N10 @ ( set @ M11 )
@ ^ [X4: N10] : ( sup_sup @ ( set @ M11 ) @ A3 @ ( B5 @ X4 ) )
@ C5 ) ) ) ).
% INT_extend_simps(7)
thf(fact_6490_Un__Inter,axiom,
! [A: $tType,A3: set @ A,B5: set @ ( set @ A )] :
( ( sup_sup @ ( set @ A ) @ A3 @ ( complete_Inf_Inf @ ( set @ A ) @ B5 ) )
= ( complete_Inf_Inf @ ( set @ A ) @ ( image @ ( set @ A ) @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A3 ) @ B5 ) ) ) ).
% Un_Inter
thf(fact_6491_tl__def,axiom,
! [A: $tType] :
( ( tl @ A )
= ( case_list @ ( list @ A ) @ A @ ( nil @ A )
@ ^ [X213: A,X223: list @ A] : X223 ) ) ).
% tl_def
thf(fact_6492_sup__shunt,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y2: A] :
( ( ( sup_sup @ A @ X @ Y2 )
= ( top_top @ A ) )
= ( ord_less_eq @ A @ ( uminus_uminus @ A @ X ) @ Y2 ) ) ) ).
% sup_shunt
thf(fact_6493_shunt1,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y2: A,Z3: A] :
( ( ord_less_eq @ A @ ( inf_inf @ A @ X @ Y2 ) @ Z3 )
= ( ord_less_eq @ A @ X @ ( sup_sup @ A @ ( uminus_uminus @ A @ Y2 ) @ Z3 ) ) ) ) ).
% shunt1
thf(fact_6494_shunt2,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y2: A,Z3: A] :
( ( ord_less_eq @ A @ ( inf_inf @ A @ X @ ( uminus_uminus @ A @ Y2 ) ) @ Z3 )
= ( ord_less_eq @ A @ X @ ( sup_sup @ A @ Y2 @ Z3 ) ) ) ) ).
% shunt2
thf(fact_6495_sup__neg__inf,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [P5: A,Q: A,R: A] :
( ( ord_less_eq @ A @ P5 @ ( sup_sup @ A @ Q @ R ) )
= ( ord_less_eq @ A @ ( inf_inf @ A @ P5 @ ( uminus_uminus @ A @ Q ) ) @ R ) ) ) ).
% sup_neg_inf
thf(fact_6496_less__eq__Inf__inter,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ A,B5: set @ A] : ( ord_less_eq @ A @ ( sup_sup @ A @ ( complete_Inf_Inf @ A @ A3 ) @ ( complete_Inf_Inf @ A @ B5 ) ) @ ( complete_Inf_Inf @ A @ ( inf_inf @ ( set @ A ) @ A3 @ B5 ) ) ) ) ).
% less_eq_Inf_inter
thf(fact_6497_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_6498_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_6499_card__Un__le,axiom,
! [A: $tType,A3: set @ A,B5: set @ A] : ( ord_less_eq @ nat @ ( finite_card @ A @ ( sup_sup @ ( set @ A ) @ A3 @ B5 ) ) @ ( plus_plus @ nat @ ( finite_card @ A @ A3 ) @ ( finite_card @ A @ B5 ) ) ) ).
% card_Un_le
thf(fact_6500_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_6501_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_6502_SUP__insert,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F3: B > A,A2: B,A3: set @ B] :
( ( complete_Sup_Sup @ A @ ( image @ B @ A @ F3 @ ( insert @ B @ A2 @ A3 ) ) )
= ( sup_sup @ A @ ( F3 @ A2 ) @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F3 @ A3 ) ) ) ) ) ).
% SUP_insert
thf(fact_6503_list_Oexhaust__sel,axiom,
! [A: $tType,List: list @ A] :
( ( List
!= ( nil @ A ) )
=> ( List
= ( cons @ A @ ( hd @ A @ List ) @ ( tl @ A @ List ) ) ) ) ).
% list.exhaust_sel
thf(fact_6504_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_6505_INF__union,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [M7: B > A,A3: set @ B,B5: set @ B] :
( ( complete_Inf_Inf @ A @ ( image @ B @ A @ M7 @ ( sup_sup @ ( set @ B ) @ A3 @ B5 ) ) )
= ( inf_inf @ A @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ M7 @ A3 ) ) @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ M7 @ B5 ) ) ) ) ) ).
% INF_union
thf(fact_6506_UN__extend__simps_I2_J,axiom,
! [D: $tType,C: $tType,C5: set @ C,A3: C > ( set @ D ),B5: set @ D] :
( ( ( C5
= ( bot_bot @ ( set @ C ) ) )
=> ( ( sup_sup @ ( set @ D ) @ ( complete_Sup_Sup @ ( set @ D ) @ ( image @ C @ ( set @ D ) @ A3 @ C5 ) ) @ B5 )
= B5 ) )
& ( ( C5
!= ( bot_bot @ ( set @ C ) ) )
=> ( ( sup_sup @ ( set @ D ) @ ( complete_Sup_Sup @ ( set @ D ) @ ( image @ C @ ( set @ D ) @ A3 @ C5 ) ) @ B5 )
= ( complete_Sup_Sup @ ( set @ D )
@ ( image @ C @ ( set @ D )
@ ^ [X4: C] : ( sup_sup @ ( set @ D ) @ ( A3 @ X4 ) @ B5 )
@ C5 ) ) ) ) ) ).
% UN_extend_simps(2)
thf(fact_6507_UN__extend__simps_I3_J,axiom,
! [E4: $tType,F: $tType,C5: set @ F,A3: set @ E4,B5: F > ( set @ E4 )] :
( ( ( C5
= ( bot_bot @ ( set @ F ) ) )
=> ( ( sup_sup @ ( set @ E4 ) @ A3 @ ( complete_Sup_Sup @ ( set @ E4 ) @ ( image @ F @ ( set @ E4 ) @ B5 @ C5 ) ) )
= A3 ) )
& ( ( C5
!= ( bot_bot @ ( set @ F ) ) )
=> ( ( sup_sup @ ( set @ E4 ) @ A3 @ ( complete_Sup_Sup @ ( set @ E4 ) @ ( image @ F @ ( set @ E4 ) @ B5 @ C5 ) ) )
= ( complete_Sup_Sup @ ( set @ E4 )
@ ( image @ F @ ( set @ E4 )
@ ^ [X4: F] : ( sup_sup @ ( set @ E4 ) @ A3 @ ( B5 @ X4 ) )
@ C5 ) ) ) ) ) ).
% UN_extend_simps(3)
thf(fact_6508_bij__betw__disjoint__Un,axiom,
! [A: $tType,B: $tType,F3: A > B,A3: set @ A,C5: set @ B,G: A > B,B5: set @ A,D4: set @ B] :
( ( bij_betw @ A @ B @ F3 @ A3 @ C5 )
=> ( ( bij_betw @ A @ B @ G @ B5 @ D4 )
=> ( ( ( inf_inf @ ( set @ A ) @ A3 @ B5 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( inf_inf @ ( set @ B ) @ C5 @ D4 )
= ( bot_bot @ ( set @ B ) ) )
=> ( bij_betw @ A @ B
@ ^ [X4: A] : ( if @ B @ ( member @ A @ X4 @ A3 ) @ ( F3 @ X4 ) @ ( G @ X4 ) )
@ ( sup_sup @ ( set @ A ) @ A3 @ B5 )
@ ( sup_sup @ ( set @ B ) @ C5 @ D4 ) ) ) ) ) ) ).
% bij_betw_disjoint_Un
thf(fact_6509_list_Ocase__eq__if,axiom,
! [A: $tType,B: $tType] :
( ( case_list @ B @ A )
= ( ^ [F12: B,F22: A > ( list @ A ) > B,List3: list @ A] :
( if @ B
@ ( List3
= ( nil @ A ) )
@ F12
@ ( F22 @ ( hd @ A @ List3 ) @ ( tl @ A @ List3 ) ) ) ) ) ).
% list.case_eq_if
thf(fact_6510_INT__Un,axiom,
! [A: $tType,B: $tType,M7: B > ( set @ A ),A3: set @ B,B5: set @ B] :
( ( complete_Inf_Inf @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ M7 @ ( sup_sup @ ( set @ B ) @ A3 @ B5 ) ) )
= ( inf_inf @ ( set @ A ) @ ( complete_Inf_Inf @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ M7 @ A3 ) ) @ ( complete_Inf_Inf @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ M7 @ B5 ) ) ) ) ).
% INT_Un
thf(fact_6511_shuffles_Osimps_I3_J,axiom,
! [A: $tType,X: A,Xs: list @ A,Y2: A,Ys: list @ A] :
( ( shuffles @ A @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y2 @ Ys ) )
= ( sup_sup @ ( set @ ( list @ A ) ) @ ( image @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X ) @ ( shuffles @ A @ Xs @ ( cons @ A @ Y2 @ Ys ) ) ) @ ( image @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ Y2 ) @ ( shuffles @ A @ ( cons @ A @ X @ Xs ) @ Ys ) ) ) ) ).
% shuffles.simps(3)
thf(fact_6512_Inter__Un__subset,axiom,
! [A: $tType,A3: set @ ( set @ A ),B5: set @ ( set @ A )] : ( ord_less_eq @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ ( complete_Inf_Inf @ ( set @ A ) @ A3 ) @ ( complete_Inf_Inf @ ( set @ A ) @ B5 ) ) @ ( complete_Inf_Inf @ ( set @ A ) @ ( inf_inf @ ( set @ ( set @ A ) ) @ A3 @ B5 ) ) ) ).
% Inter_Un_subset
thf(fact_6513_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_6514_sum_Ounion__inter,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A3: set @ B,B5: set @ B,G: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ( finite_finite2 @ B @ B5 )
=> ( ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G @ ( sup_sup @ ( set @ B ) @ A3 @ B5 ) ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ ( inf_inf @ ( set @ B ) @ A3 @ B5 ) ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G @ A3 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ B5 ) ) ) ) ) ) ).
% sum.union_inter
thf(fact_6515_prod_Ounion__inter,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: set @ B,B5: set @ B,G: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ( finite_finite2 @ B @ B5 )
=> ( ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G @ ( sup_sup @ ( set @ B ) @ A3 @ B5 ) ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ ( inf_inf @ ( set @ B ) @ A3 @ B5 ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G @ A3 ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ B5 ) ) ) ) ) ) ).
% prod.union_inter
thf(fact_6516_card__Un__Int,axiom,
! [A: $tType,A3: set @ A,B5: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( finite_finite2 @ A @ B5 )
=> ( ( plus_plus @ nat @ ( finite_card @ A @ A3 ) @ ( finite_card @ A @ B5 ) )
= ( plus_plus @ nat @ ( finite_card @ A @ ( sup_sup @ ( set @ A ) @ A3 @ B5 ) ) @ ( finite_card @ A @ ( inf_inf @ ( set @ A ) @ A3 @ B5 ) ) ) ) ) ) ).
% card_Un_Int
thf(fact_6517_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_6518_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_6519_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_6520_SUP__UNIV__bool__expand,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: $o > A] :
( ( complete_Sup_Sup @ A @ ( image @ $o @ A @ A3 @ ( top_top @ ( set @ $o ) ) ) )
= ( sup_sup @ A @ ( A3 @ $true ) @ ( A3 @ $false ) ) ) ) ).
% SUP_UNIV_bool_expand
thf(fact_6521_Nitpick_Osize__list__simp_I2_J,axiom,
! [A: $tType] :
( ( size_size @ ( list @ A ) )
= ( ^ [Xs2: list @ A] :
( if @ nat
@ ( Xs2
= ( nil @ A ) )
@ ( zero_zero @ nat )
@ ( suc @ ( size_size @ ( list @ A ) @ ( tl @ A @ Xs2 ) ) ) ) ) ) ).
% Nitpick.size_list_simp(2)
thf(fact_6522_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_6523_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_6524_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_6525_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_6526_SUP__nat__binary,axiom,
! [A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A3: A,B5: A] :
( ( sup_sup @ A @ A3
@ ( complete_Sup_Sup @ A
@ ( image @ nat @ A
@ ^ [X4: nat] : B5
@ ( collect @ nat @ ( ord_less @ nat @ ( zero_zero @ nat ) ) ) ) ) )
= ( sup_sup @ A @ A3 @ B5 ) ) ) ).
% SUP_nat_binary
thf(fact_6527_Un__eq__UN,axiom,
! [A: $tType] :
( ( sup_sup @ ( set @ A ) )
= ( ^ [A7: set @ A,B7: set @ A] :
( complete_Sup_Sup @ ( set @ A )
@ ( image @ $o @ ( set @ A )
@ ^ [B4: $o] : ( if @ ( set @ A ) @ B4 @ A7 @ B7 )
@ ( top_top @ ( set @ $o ) ) ) ) ) ) ).
% Un_eq_UN
thf(fact_6528_UN__bool__eq,axiom,
! [A: $tType,A3: $o > ( set @ A )] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( image @ $o @ ( set @ A ) @ A3 @ ( top_top @ ( set @ $o ) ) ) )
= ( sup_sup @ ( set @ A ) @ ( A3 @ $true ) @ ( A3 @ $false ) ) ) ).
% UN_bool_eq
thf(fact_6529_sup__bot_Osemilattice__neutr__order__axioms,axiom,
! [A: $tType] :
( ( bounde4967611905675639751up_bot @ A )
=> ( semila1105856199041335345_order @ A @ ( sup_sup @ A ) @ ( bot_bot @ A )
@ ^ [X4: A,Y: A] : ( ord_less_eq @ A @ Y @ X4 )
@ ^ [X4: A,Y: A] : ( ord_less @ A @ Y @ X4 ) ) ) ).
% sup_bot.semilattice_neutr_order_axioms
thf(fact_6530_Cons__in__shuffles__iff,axiom,
! [A: $tType,Z3: A,Zs: list @ A,Xs: list @ A,Ys: list @ A] :
( ( member @ ( list @ A ) @ ( cons @ A @ Z3 @ Zs ) @ ( shuffles @ A @ Xs @ Ys ) )
= ( ( ( Xs
!= ( nil @ A ) )
& ( ( hd @ A @ Xs )
= Z3 )
& ( member @ ( list @ A ) @ Zs @ ( shuffles @ A @ ( tl @ A @ Xs ) @ Ys ) ) )
| ( ( Ys
!= ( nil @ A ) )
& ( ( hd @ A @ Ys )
= Z3 )
& ( member @ ( list @ A ) @ Zs @ ( shuffles @ A @ Xs @ ( tl @ A @ Ys ) ) ) ) ) ) ).
% Cons_in_shuffles_iff
thf(fact_6531_list_Osplit__sel,axiom,
! [B: $tType,A: $tType,P3: B > $o,F1: B,F2: A > ( list @ A ) > B,List: list @ A] :
( ( P3 @ ( case_list @ B @ A @ F1 @ F2 @ List ) )
= ( ( ( List
= ( nil @ A ) )
=> ( P3 @ F1 ) )
& ( ( List
= ( cons @ A @ ( hd @ A @ List ) @ ( tl @ A @ List ) ) )
=> ( P3 @ ( F2 @ ( hd @ A @ List ) @ ( tl @ A @ List ) ) ) ) ) ) ).
% list.split_sel
thf(fact_6532_list_Osplit__sel__asm,axiom,
! [B: $tType,A: $tType,P3: B > $o,F1: B,F2: A > ( list @ A ) > B,List: list @ A] :
( ( P3 @ ( case_list @ B @ A @ F1 @ F2 @ List ) )
= ( ~ ( ( ( List
= ( nil @ A ) )
& ~ ( P3 @ F1 ) )
| ( ( List
= ( cons @ A @ ( hd @ A @ List ) @ ( tl @ A @ List ) ) )
& ~ ( P3 @ ( F2 @ ( hd @ A @ List ) @ ( tl @ A @ List ) ) ) ) ) ) ) ).
% list.split_sel_asm
thf(fact_6533_sum_Ounion__inter__neutral,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A3: set @ B,B5: set @ B,G: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ( finite_finite2 @ B @ B5 )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ ( inf_inf @ ( set @ B ) @ A3 @ B5 ) )
=> ( ( G @ X5 )
= ( zero_zero @ A ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( sup_sup @ ( set @ B ) @ A3 @ B5 ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G @ A3 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ B5 ) ) ) ) ) ) ) ).
% sum.union_inter_neutral
thf(fact_6534_sum__Un,axiom,
! [A: $tType,B: $tType] :
( ( ab_group_add @ A )
=> ! [A3: set @ B,B5: set @ B,F3: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ( finite_finite2 @ B @ B5 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ F3 @ ( sup_sup @ ( set @ B ) @ A3 @ B5 ) )
= ( minus_minus @ A @ ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ A3 ) @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ B5 ) ) @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ ( inf_inf @ ( set @ B ) @ A3 @ B5 ) ) ) ) ) ) ) ).
% sum_Un
thf(fact_6535_sum_Ounion__disjoint,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A3: set @ B,B5: set @ B,G: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ( finite_finite2 @ B @ B5 )
=> ( ( ( inf_inf @ ( set @ B ) @ A3 @ B5 )
= ( bot_bot @ ( set @ B ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( sup_sup @ ( set @ B ) @ A3 @ B5 ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G @ A3 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ B5 ) ) ) ) ) ) ) ).
% sum.union_disjoint
thf(fact_6536_prod_Ounion__inter__neutral,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: set @ B,B5: set @ B,G: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ( finite_finite2 @ B @ B5 )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ ( inf_inf @ ( set @ B ) @ A3 @ B5 ) )
=> ( ( G @ X5 )
= ( one_one @ A ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( sup_sup @ ( set @ B ) @ A3 @ B5 ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G @ A3 ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ B5 ) ) ) ) ) ) ) ).
% prod.union_inter_neutral
thf(fact_6537_prod_Ounion__disjoint,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: set @ B,B5: set @ B,G: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ( finite_finite2 @ B @ B5 )
=> ( ( ( inf_inf @ ( set @ B ) @ A3 @ B5 )
= ( bot_bot @ ( set @ B ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( sup_sup @ ( set @ B ) @ A3 @ B5 ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G @ A3 ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ B5 ) ) ) ) ) ) ) ).
% prod.union_disjoint
thf(fact_6538_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_6539_sum__Un2,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ B )
=> ! [A3: set @ A,B5: set @ A,F3: A > B] :
( ( finite_finite2 @ A @ ( sup_sup @ ( set @ A ) @ A3 @ B5 ) )
=> ( ( groups7311177749621191930dd_sum @ A @ B @ F3 @ ( sup_sup @ ( set @ A ) @ A3 @ B5 ) )
= ( plus_plus @ B @ ( plus_plus @ B @ ( groups7311177749621191930dd_sum @ A @ B @ F3 @ ( minus_minus @ ( set @ A ) @ A3 @ B5 ) ) @ ( groups7311177749621191930dd_sum @ A @ B @ F3 @ ( minus_minus @ ( set @ A ) @ B5 @ A3 ) ) ) @ ( groups7311177749621191930dd_sum @ A @ B @ F3 @ ( inf_inf @ ( set @ A ) @ A3 @ B5 ) ) ) ) ) ) ).
% sum_Un2
thf(fact_6540_sum_Ounion__diff2,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A3: set @ B,B5: set @ B,G: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ( finite_finite2 @ B @ B5 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( sup_sup @ ( set @ B ) @ A3 @ B5 ) )
= ( plus_plus @ A @ ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A3 @ B5 ) ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ B5 @ A3 ) ) ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ ( inf_inf @ ( set @ B ) @ A3 @ B5 ) ) ) ) ) ) ) ).
% sum.union_diff2
thf(fact_6541_prod_Ounion__diff2,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: set @ B,B5: set @ B,G: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ( finite_finite2 @ B @ B5 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( sup_sup @ ( set @ B ) @ A3 @ B5 ) )
= ( times_times @ A @ ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A3 @ B5 ) ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ B5 @ A3 ) ) ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ ( inf_inf @ ( set @ B ) @ A3 @ B5 ) ) ) ) ) ) ) ).
% prod.union_diff2
thf(fact_6542_card__Un__disjoint,axiom,
! [A: $tType,A3: set @ A,B5: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( finite_finite2 @ A @ B5 )
=> ( ( ( inf_inf @ ( set @ A ) @ A3 @ B5 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_card @ A @ ( sup_sup @ ( set @ A ) @ A3 @ B5 ) )
= ( plus_plus @ nat @ ( finite_card @ A @ A3 ) @ ( finite_card @ A @ B5 ) ) ) ) ) ) ).
% card_Un_disjoint
thf(fact_6543_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_6544_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_6545_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_6546_sum__Un__nat,axiom,
! [A: $tType,A3: set @ A,B5: set @ A,F3: A > nat] :
( ( finite_finite2 @ A @ A3 )
=> ( ( finite_finite2 @ A @ B5 )
=> ( ( groups7311177749621191930dd_sum @ A @ nat @ F3 @ ( sup_sup @ ( set @ A ) @ A3 @ B5 ) )
= ( minus_minus @ nat @ ( plus_plus @ nat @ ( groups7311177749621191930dd_sum @ A @ nat @ F3 @ A3 ) @ ( groups7311177749621191930dd_sum @ A @ nat @ F3 @ B5 ) ) @ ( groups7311177749621191930dd_sum @ A @ nat @ F3 @ ( inf_inf @ ( set @ A ) @ A3 @ B5 ) ) ) ) ) ) ).
% sum_Un_nat
thf(fact_6547_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_6548_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_6549_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_6550_prod__Un,axiom,
! [A: $tType,B: $tType] :
( ( field @ A )
=> ! [A3: set @ B,B5: set @ B,F3: B > A] :
( ( finite_finite2 @ B @ A3 )
=> ( ( finite_finite2 @ B @ B5 )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ ( inf_inf @ ( set @ B ) @ A3 @ B5 ) )
=> ( ( F3 @ X5 )
!= ( zero_zero @ A ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ F3 @ ( sup_sup @ ( set @ B ) @ A3 @ B5 ) )
= ( divide_divide @ A @ ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ F3 @ A3 ) @ ( groups7121269368397514597t_prod @ B @ A @ F3 @ B5 ) ) @ ( groups7121269368397514597t_prod @ B @ A @ F3 @ ( inf_inf @ ( set @ B ) @ A3 @ B5 ) ) ) ) ) ) ) ) ).
% prod_Un
thf(fact_6551_Nitpick_Osize__list__simp_I1_J,axiom,
! [A: $tType] :
( ( size_list @ A )
= ( ^ [F5: A > nat,Xs2: list @ A] :
( if @ nat
@ ( Xs2
= ( nil @ A ) )
@ ( zero_zero @ nat )
@ ( suc @ ( plus_plus @ nat @ ( F5 @ ( hd @ A @ Xs2 ) ) @ ( size_list @ A @ F5 @ ( tl @ A @ Xs2 ) ) ) ) ) ) ) ).
% Nitpick.size_list_simp(1)
thf(fact_6552_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_6553_shuffles_Oelims,axiom,
! [A: $tType,X: list @ A,Xa: list @ A,Y2: set @ ( list @ A )] :
( ( ( shuffles @ A @ X @ Xa )
= Y2 )
=> ( ( ( X
= ( nil @ A ) )
=> ( Y2
!= ( insert @ ( list @ A ) @ Xa @ ( bot_bot @ ( set @ ( list @ A ) ) ) ) ) )
=> ( ( ( Xa
= ( nil @ A ) )
=> ( Y2
!= ( insert @ ( list @ A ) @ X @ ( bot_bot @ ( set @ ( list @ A ) ) ) ) ) )
=> ~ ! [X5: A,Xs3: list @ A] :
( ( X
= ( cons @ A @ X5 @ Xs3 ) )
=> ! [Y7: A,Ys4: list @ A] :
( ( Xa
= ( cons @ A @ Y7 @ Ys4 ) )
=> ( Y2
!= ( sup_sup @ ( set @ ( list @ A ) ) @ ( image @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X5 ) @ ( shuffles @ A @ Xs3 @ ( cons @ A @ Y7 @ Ys4 ) ) ) @ ( image @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ Y7 ) @ ( shuffles @ A @ ( cons @ A @ X5 @ Xs3 ) @ Ys4 ) ) ) ) ) ) ) ) ) ).
% shuffles.elims
thf(fact_6554_shuffles_Opsimps_I3_J,axiom,
! [A: $tType,X: A,Xs: list @ A,Y2: A,Ys: list @ A] :
( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( shuffles_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y2 @ Ys ) ) )
=> ( ( shuffles @ A @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y2 @ Ys ) )
= ( sup_sup @ ( set @ ( list @ A ) ) @ ( image @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X ) @ ( shuffles @ A @ Xs @ ( cons @ A @ Y2 @ Ys ) ) ) @ ( image @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ Y2 ) @ ( shuffles @ A @ ( cons @ A @ X @ Xs ) @ Ys ) ) ) ) ) ).
% shuffles.psimps(3)
thf(fact_6555_concat__inth,axiom,
! [A: $tType,Xs: list @ A,X: A,Ys: list @ A] :
( ( nth @ A @ ( append @ A @ Xs @ ( append @ A @ ( cons @ A @ X @ ( nil @ A ) ) @ Ys ) ) @ ( size_size @ ( list @ A ) @ Xs ) )
= X ) ).
% concat_inth
thf(fact_6556_upto__aux__rec,axiom,
( upto_aux
= ( ^ [I: int,J4: int,Js: list @ int] : ( if @ ( list @ int ) @ ( ord_less @ int @ J4 @ I ) @ Js @ ( upto_aux @ I @ ( minus_minus @ int @ J4 @ ( one_one @ int ) ) @ ( cons @ int @ J4 @ Js ) ) ) ) ) ).
% upto_aux_rec
thf(fact_6557_same__append__eq,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Zs: list @ A] :
( ( ( append @ A @ Xs @ Ys )
= ( append @ A @ Xs @ Zs ) )
= ( Ys = Zs ) ) ).
% same_append_eq
thf(fact_6558_append__same__eq,axiom,
! [A: $tType,Ys: list @ A,Xs: list @ A,Zs: list @ A] :
( ( ( append @ A @ Ys @ Xs )
= ( append @ A @ Zs @ Xs ) )
= ( Ys = Zs ) ) ).
% append_same_eq
thf(fact_6559_append__assoc,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Zs: list @ A] :
( ( append @ A @ ( append @ A @ Xs @ Ys ) @ Zs )
= ( append @ A @ Xs @ ( append @ A @ Ys @ Zs ) ) ) ).
% append_assoc
thf(fact_6560_append_Oassoc,axiom,
! [A: $tType,A2: list @ A,B2: list @ A,C2: list @ A] :
( ( append @ A @ ( append @ A @ A2 @ B2 ) @ C2 )
= ( append @ A @ A2 @ ( append @ A @ B2 @ C2 ) ) ) ).
% append.assoc
thf(fact_6561_append__is__Nil__conv,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( append @ A @ Xs @ Ys )
= ( nil @ A ) )
= ( ( Xs
= ( nil @ A ) )
& ( Ys
= ( nil @ A ) ) ) ) ).
% append_is_Nil_conv
thf(fact_6562_Nil__is__append__conv,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( nil @ A )
= ( append @ A @ Xs @ Ys ) )
= ( ( Xs
= ( nil @ A ) )
& ( Ys
= ( nil @ A ) ) ) ) ).
% Nil_is_append_conv
thf(fact_6563_self__append__conv2,axiom,
! [A: $tType,Y2: list @ A,Xs: list @ A] :
( ( Y2
= ( append @ A @ Xs @ Y2 ) )
= ( Xs
= ( nil @ A ) ) ) ).
% self_append_conv2
thf(fact_6564_append__self__conv2,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( append @ A @ Xs @ Ys )
= Ys )
= ( Xs
= ( nil @ A ) ) ) ).
% append_self_conv2
thf(fact_6565_self__append__conv,axiom,
! [A: $tType,Y2: list @ A,Ys: list @ A] :
( ( Y2
= ( append @ A @ Y2 @ Ys ) )
= ( Ys
= ( nil @ A ) ) ) ).
% self_append_conv
thf(fact_6566_append__self__conv,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( append @ A @ Xs @ Ys )
= Xs )
= ( Ys
= ( nil @ A ) ) ) ).
% append_self_conv
thf(fact_6567_append__Nil2,axiom,
! [A: $tType,Xs: list @ A] :
( ( append @ A @ Xs @ ( nil @ A ) )
= Xs ) ).
% append_Nil2
thf(fact_6568_append_Oright__neutral,axiom,
! [A: $tType,A2: list @ A] :
( ( append @ A @ A2 @ ( nil @ A ) )
= A2 ) ).
% append.right_neutral
thf(fact_6569_append__eq__append__conv,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Us: list @ A,Vs: list @ A] :
( ( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ A ) @ Ys ) )
| ( ( size_size @ ( list @ A ) @ Us )
= ( size_size @ ( list @ A ) @ Vs ) ) )
=> ( ( ( append @ A @ Xs @ Us )
= ( append @ A @ Ys @ Vs ) )
= ( ( Xs = Ys )
& ( Us = Vs ) ) ) ) ).
% append_eq_append_conv
thf(fact_6570_map__append,axiom,
! [A: $tType,B: $tType,F3: B > A,Xs: list @ B,Ys: list @ B] :
( ( map @ B @ A @ F3 @ ( append @ B @ Xs @ Ys ) )
= ( append @ A @ ( map @ B @ A @ F3 @ Xs ) @ ( map @ B @ A @ F3 @ Ys ) ) ) ).
% map_append
thf(fact_6571_filter__append,axiom,
! [A: $tType,P3: A > $o,Xs: list @ A,Ys: list @ A] :
( ( filter2 @ A @ P3 @ ( append @ A @ Xs @ Ys ) )
= ( append @ A @ ( filter2 @ A @ P3 @ Xs ) @ ( filter2 @ A @ P3 @ Ys ) ) ) ).
% filter_append
thf(fact_6572_rev__append,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( rev @ A @ ( append @ A @ Xs @ Ys ) )
= ( append @ A @ ( rev @ A @ Ys ) @ ( rev @ A @ Xs ) ) ) ).
% rev_append
thf(fact_6573_concat__append,axiom,
! [A: $tType,Xs: list @ ( list @ A ),Ys: list @ ( list @ A )] :
( ( concat @ A @ ( append @ ( list @ A ) @ Xs @ Ys ) )
= ( append @ A @ ( concat @ A @ Xs ) @ ( concat @ A @ Ys ) ) ) ).
% concat_append
thf(fact_6574_foldr__append,axiom,
! [B: $tType,A: $tType,F3: B > A > A,Xs: list @ B,Ys: list @ B,A2: A] :
( ( foldr @ B @ A @ F3 @ ( append @ B @ Xs @ Ys ) @ A2 )
= ( foldr @ B @ A @ F3 @ Xs @ ( foldr @ B @ A @ F3 @ Ys @ A2 ) ) ) ).
% foldr_append
thf(fact_6575_removeAll__append,axiom,
! [A: $tType,X: A,Xs: list @ A,Ys: list @ A] :
( ( removeAll @ A @ X @ ( append @ A @ Xs @ Ys ) )
= ( append @ A @ ( removeAll @ A @ X @ Xs ) @ ( removeAll @ A @ X @ Ys ) ) ) ).
% removeAll_append
thf(fact_6576_append1__eq__conv,axiom,
! [A: $tType,Xs: list @ A,X: A,Ys: list @ A,Y2: A] :
( ( ( append @ A @ Xs @ ( cons @ A @ X @ ( nil @ A ) ) )
= ( append @ A @ Ys @ ( cons @ A @ Y2 @ ( nil @ A ) ) ) )
= ( ( Xs = Ys )
& ( X = Y2 ) ) ) ).
% append1_eq_conv
thf(fact_6577_length__append,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( size_size @ ( list @ A ) @ ( append @ A @ Xs @ Ys ) )
= ( plus_plus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ A ) @ Ys ) ) ) ).
% length_append
thf(fact_6578_set__append,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( set2 @ A @ ( append @ A @ Xs @ Ys ) )
= ( sup_sup @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ Ys ) ) ) ).
% set_append
thf(fact_6579_sum__list__append,axiom,
! [A: $tType] :
( ( monoid_add @ A )
=> ! [Xs: list @ A,Ys: list @ A] :
( ( groups8242544230860333062m_list @ A @ ( append @ A @ Xs @ Ys ) )
= ( plus_plus @ A @ ( groups8242544230860333062m_list @ A @ Xs ) @ ( groups8242544230860333062m_list @ A @ Ys ) ) ) ) ).
% sum_list_append
thf(fact_6580_hd__append2,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( hd @ A @ ( append @ A @ Xs @ Ys ) )
= ( hd @ A @ Xs ) ) ) ).
% hd_append2
thf(fact_6581_tl__append2,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( tl @ A @ ( append @ A @ Xs @ Ys ) )
= ( append @ A @ ( tl @ A @ Xs ) @ Ys ) ) ) ).
% tl_append2
thf(fact_6582_takeWhile__append1,axiom,
! [A: $tType,X: A,Xs: list @ A,P3: A > $o,Ys: list @ A] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ~ ( P3 @ X )
=> ( ( takeWhile @ A @ P3 @ ( append @ A @ Xs @ Ys ) )
= ( takeWhile @ A @ P3 @ Xs ) ) ) ) ).
% takeWhile_append1
thf(fact_6583_takeWhile__append2,axiom,
! [A: $tType,Xs: list @ A,P3: A > $o,Ys: list @ A] :
( ! [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Xs ) )
=> ( P3 @ X5 ) )
=> ( ( takeWhile @ A @ P3 @ ( append @ A @ Xs @ Ys ) )
= ( append @ A @ Xs @ ( takeWhile @ A @ P3 @ Ys ) ) ) ) ).
% takeWhile_append2
thf(fact_6584_size__list__append,axiom,
! [A: $tType,F3: A > nat,Xs: list @ A,Ys: list @ A] :
( ( size_list @ A @ F3 @ ( append @ A @ Xs @ Ys ) )
= ( plus_plus @ nat @ ( size_list @ A @ F3 @ Xs ) @ ( size_list @ A @ F3 @ Ys ) ) ) ).
% size_list_append
thf(fact_6585_nth__append__length,axiom,
! [A: $tType,Xs: list @ A,X: A,Ys: list @ A] :
( ( nth @ A @ ( append @ A @ Xs @ ( cons @ A @ X @ Ys ) ) @ ( size_size @ ( list @ A ) @ Xs ) )
= X ) ).
% nth_append_length
thf(fact_6586_nth__append__length__plus,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,N: nat] :
( ( nth @ A @ ( append @ A @ Xs @ Ys ) @ ( plus_plus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N ) )
= ( nth @ A @ Ys @ N ) ) ).
% nth_append_length_plus
thf(fact_6587_rev__eq__Cons__iff,axiom,
! [A: $tType,Xs: list @ A,Y2: A,Ys: list @ A] :
( ( ( rev @ A @ Xs )
= ( cons @ A @ Y2 @ Ys ) )
= ( Xs
= ( append @ A @ ( rev @ A @ Ys ) @ ( cons @ A @ Y2 @ ( nil @ A ) ) ) ) ) ).
% rev_eq_Cons_iff
thf(fact_6588_take__append,axiom,
! [A: $tType,N: nat,Xs: list @ A,Ys: list @ A] :
( ( take @ A @ N @ ( append @ A @ Xs @ Ys ) )
= ( append @ A @ ( take @ A @ N @ Xs ) @ ( take @ A @ ( minus_minus @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) ) @ Ys ) ) ) ).
% take_append
thf(fact_6589_list__update__length,axiom,
! [A: $tType,Xs: list @ A,X: A,Ys: list @ A,Y2: A] :
( ( list_update @ A @ ( append @ A @ Xs @ ( cons @ A @ X @ Ys ) ) @ ( size_size @ ( list @ A ) @ Xs ) @ Y2 )
= ( append @ A @ Xs @ ( cons @ A @ Y2 @ Ys ) ) ) ).
% list_update_length
thf(fact_6590_distinct__append,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( distinct @ A @ ( append @ A @ Xs @ Ys ) )
= ( ( distinct @ A @ Xs )
& ( distinct @ A @ Ys )
& ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ Ys ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% distinct_append
thf(fact_6591_sorted__list__of__set__lessThan__Suc,axiom,
! [K2: nat] :
( ( linord4507533701916653071of_set @ nat @ ( set_ord_lessThan @ nat @ ( suc @ K2 ) ) )
= ( append @ nat @ ( linord4507533701916653071of_set @ nat @ ( set_ord_lessThan @ nat @ K2 ) ) @ ( cons @ nat @ K2 @ ( nil @ nat ) ) ) ) ).
% sorted_list_of_set_lessThan_Suc
thf(fact_6592_sorted__list__of__set__atMost__Suc,axiom,
! [K2: nat] :
( ( linord4507533701916653071of_set @ nat @ ( set_ord_atMost @ nat @ ( suc @ K2 ) ) )
= ( append @ nat @ ( linord4507533701916653071of_set @ nat @ ( set_ord_atMost @ nat @ K2 ) ) @ ( cons @ nat @ ( suc @ K2 ) @ ( nil @ nat ) ) ) ) ).
% sorted_list_of_set_atMost_Suc
thf(fact_6593_sup__Un__eq,axiom,
! [A: $tType,R4: set @ A,S3: set @ A] :
( ( sup_sup @ ( A > $o )
@ ^ [X4: A] : ( member @ A @ X4 @ R4 )
@ ^ [X4: A] : ( member @ A @ X4 @ S3 ) )
= ( ^ [X4: A] : ( member @ A @ X4 @ ( sup_sup @ ( set @ A ) @ R4 @ S3 ) ) ) ) ).
% sup_Un_eq
thf(fact_6594_sup__set__def,axiom,
! [A: $tType] :
( ( sup_sup @ ( set @ A ) )
= ( ^ [A7: set @ A,B7: set @ A] :
( collect @ A
@ ( sup_sup @ ( A > $o )
@ ^ [X4: A] : ( member @ A @ X4 @ A7 )
@ ^ [X4: A] : ( member @ A @ X4 @ B7 ) ) ) ) ) ).
% sup_set_def
thf(fact_6595_sup__enat__def,axiom,
( ( sup_sup @ extended_enat )
= ( ord_max @ extended_enat ) ) ).
% sup_enat_def
thf(fact_6596_sup__nat__def,axiom,
( ( sup_sup @ nat )
= ( ord_max @ nat ) ) ).
% sup_nat_def
thf(fact_6597_sup__Un__eq2,axiom,
! [B: $tType,A: $tType,R4: set @ ( product_prod @ A @ B ),S3: set @ ( product_prod @ A @ B )] :
( ( sup_sup @ ( A > B > $o )
@ ^ [X4: A,Y: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y ) @ R4 )
@ ^ [X4: A,Y: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y ) @ S3 ) )
= ( ^ [X4: A,Y: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y ) @ ( sup_sup @ ( set @ ( product_prod @ A @ B ) ) @ R4 @ S3 ) ) ) ) ).
% sup_Un_eq2
thf(fact_6598_concat__conv__foldr,axiom,
! [A: $tType] :
( ( concat @ A )
= ( ^ [Xss3: list @ ( list @ A )] : ( foldr @ ( list @ A ) @ ( list @ A ) @ ( append @ A ) @ Xss3 @ ( nil @ A ) ) ) ) ).
% concat_conv_foldr
thf(fact_6599_map__eq__append__conv,axiom,
! [A: $tType,B: $tType,F3: B > A,Xs: list @ B,Ys: list @ A,Zs: list @ A] :
( ( ( map @ B @ A @ F3 @ Xs )
= ( append @ A @ Ys @ Zs ) )
= ( ? [Us2: list @ B,Vs2: list @ B] :
( ( Xs
= ( append @ B @ Us2 @ Vs2 ) )
& ( Ys
= ( map @ B @ A @ F3 @ Us2 ) )
& ( Zs
= ( map @ B @ A @ F3 @ Vs2 ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_6600_append__eq__map__conv,axiom,
! [A: $tType,B: $tType,Ys: list @ A,Zs: list @ A,F3: B > A,Xs: list @ B] :
( ( ( append @ A @ Ys @ Zs )
= ( map @ B @ A @ F3 @ Xs ) )
= ( ? [Us2: list @ B,Vs2: list @ B] :
( ( Xs
= ( append @ B @ Us2 @ Vs2 ) )
& ( Ys
= ( map @ B @ A @ F3 @ Us2 ) )
& ( Zs
= ( map @ B @ A @ F3 @ Vs2 ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_6601_lex__append__leftI,axiom,
! [A: $tType,Ys: list @ A,Zs: list @ A,R: set @ ( product_prod @ A @ A ),Xs: list @ A] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ys @ Zs ) @ ( lex @ A @ R ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs @ Ys ) @ ( append @ A @ Xs @ Zs ) ) @ ( lex @ A @ R ) ) ) ).
% lex_append_leftI
thf(fact_6602_append__eq__append__conv2,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Zs: list @ A,Ts: list @ A] :
( ( ( append @ A @ Xs @ Ys )
= ( append @ A @ Zs @ Ts ) )
= ( ? [Us2: list @ A] :
( ( ( Xs
= ( append @ A @ Zs @ Us2 ) )
& ( ( append @ A @ Us2 @ Ys )
= Ts ) )
| ( ( ( append @ A @ Xs @ Us2 )
= Zs )
& ( Ys
= ( append @ A @ Us2 @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_6603_append__eq__appendI,axiom,
! [A: $tType,Xs: list @ A,Xs1: list @ A,Zs: list @ A,Ys: list @ A,Us: list @ A] :
( ( ( append @ A @ Xs @ Xs1 )
= Zs )
=> ( ( Ys
= ( append @ A @ Xs1 @ Us ) )
=> ( ( append @ A @ Xs @ Ys )
= ( append @ A @ Zs @ Us ) ) ) ) ).
% append_eq_appendI
thf(fact_6604_enumerate__append__eq,axiom,
! [A: $tType,N: nat,Xs: list @ A,Ys: list @ A] :
( ( enumerate @ A @ N @ ( append @ A @ Xs @ Ys ) )
= ( append @ ( product_prod @ nat @ A ) @ ( enumerate @ A @ N @ Xs ) @ ( enumerate @ A @ ( plus_plus @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) ) @ Ys ) ) ) ).
% enumerate_append_eq
thf(fact_6605_sorted__wrt__append,axiom,
! [A: $tType,P3: A > A > $o,Xs: list @ A,Ys: list @ A] :
( ( sorted_wrt @ A @ P3 @ ( append @ A @ Xs @ Ys ) )
= ( ( sorted_wrt @ A @ P3 @ Xs )
& ( sorted_wrt @ A @ P3 @ Ys )
& ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
=> ! [Y: A] :
( ( member @ A @ Y @ ( set2 @ A @ Ys ) )
=> ( P3 @ X4 @ Y ) ) ) ) ) ).
% sorted_wrt_append
thf(fact_6606_replicate__add,axiom,
! [A: $tType,N: nat,M: nat,X: A] :
( ( replicate @ A @ ( plus_plus @ nat @ N @ M ) @ X )
= ( append @ A @ ( replicate @ A @ N @ X ) @ ( replicate @ A @ M @ X ) ) ) ).
% replicate_add
thf(fact_6607_append__replicate__commute,axiom,
! [A: $tType,N: nat,X: A,K2: nat] :
( ( append @ A @ ( replicate @ A @ N @ X ) @ ( replicate @ A @ K2 @ X ) )
= ( append @ A @ ( replicate @ A @ K2 @ X ) @ ( replicate @ A @ N @ X ) ) ) ).
% append_replicate_commute
thf(fact_6608_remove1__append,axiom,
! [A: $tType,X: A,Xs: list @ A,Ys: list @ A] :
( ( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ( remove1 @ A @ X @ ( append @ A @ Xs @ Ys ) )
= ( append @ A @ ( remove1 @ A @ X @ Xs ) @ Ys ) ) )
& ( ~ ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ( remove1 @ A @ X @ ( append @ A @ Xs @ Ys ) )
= ( append @ A @ Xs @ ( remove1 @ A @ X @ Ys ) ) ) ) ) ).
% remove1_append
thf(fact_6609_remdups__append2,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( remdups @ A @ ( append @ A @ Xs @ ( remdups @ A @ Ys ) ) )
= ( remdups @ A @ ( append @ A @ Xs @ Ys ) ) ) ).
% remdups_append2
thf(fact_6610_eq__Nil__appendI,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( Xs = Ys )
=> ( Xs
= ( append @ A @ ( nil @ A ) @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_6611_append_Oleft__neutral,axiom,
! [A: $tType,A2: list @ A] :
( ( append @ A @ ( nil @ A ) @ A2 )
= A2 ) ).
% append.left_neutral
thf(fact_6612_append__Nil,axiom,
! [A: $tType,Ys: list @ A] :
( ( append @ A @ ( nil @ A ) @ Ys )
= Ys ) ).
% append_Nil
thf(fact_6613_longest__common__prefix,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
? [Ps: list @ A,Xs4: list @ A,Ys5: list @ A] :
( ( Xs
= ( append @ A @ Ps @ Xs4 ) )
& ( Ys
= ( append @ A @ Ps @ Ys5 ) )
& ( ( Xs4
= ( nil @ A ) )
| ( Ys5
= ( nil @ A ) )
| ( ( hd @ A @ Xs4 )
!= ( hd @ A @ Ys5 ) ) ) ) ).
% longest_common_prefix
thf(fact_6614_hd__append,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( Xs
= ( nil @ A ) )
=> ( ( hd @ A @ ( append @ A @ Xs @ Ys ) )
= ( hd @ A @ Ys ) ) )
& ( ( Xs
!= ( nil @ A ) )
=> ( ( hd @ A @ ( append @ A @ Xs @ Ys ) )
= ( hd @ A @ Xs ) ) ) ) ).
% hd_append
thf(fact_6615_takeWhile__tail,axiom,
! [A: $tType,P3: A > $o,X: A,Xs: list @ A,L: list @ A] :
( ~ ( P3 @ X )
=> ( ( takeWhile @ A @ P3 @ ( append @ A @ Xs @ ( cons @ A @ X @ L ) ) )
= ( takeWhile @ A @ P3 @ Xs ) ) ) ).
% takeWhile_tail
thf(fact_6616_replicate__app__Cons__same,axiom,
! [A: $tType,N: nat,X: A,Xs: list @ A] :
( ( append @ A @ ( replicate @ A @ N @ X ) @ ( cons @ A @ X @ Xs ) )
= ( cons @ A @ X @ ( append @ A @ ( replicate @ A @ N @ X ) @ Xs ) ) ) ).
% replicate_app_Cons_same
thf(fact_6617_append__Cons,axiom,
! [A: $tType,X: A,Xs: list @ A,Ys: list @ A] :
( ( append @ A @ ( cons @ A @ X @ Xs ) @ Ys )
= ( cons @ A @ X @ ( append @ A @ Xs @ Ys ) ) ) ).
% append_Cons
thf(fact_6618_Cons__eq__appendI,axiom,
! [A: $tType,X: A,Xs1: list @ A,Ys: list @ A,Xs: list @ A,Zs: list @ A] :
( ( ( cons @ A @ X @ Xs1 )
= Ys )
=> ( ( Xs
= ( append @ A @ Xs1 @ Zs ) )
=> ( ( cons @ A @ X @ Xs )
= ( append @ A @ Ys @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_6619_concat_Osimps_I2_J,axiom,
! [A: $tType,X: list @ A,Xs: list @ ( list @ A )] :
( ( concat @ A @ ( cons @ ( list @ A ) @ X @ Xs ) )
= ( append @ A @ X @ ( concat @ A @ Xs ) ) ) ).
% concat.simps(2)
thf(fact_6620_split__list__first__prop__iff,axiom,
! [A: $tType,Xs: list @ A,P3: A > $o] :
( ( ? [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
& ( P3 @ X4 ) ) )
= ( ? [Ys3: list @ A,X4: A] :
( ? [Zs3: list @ A] :
( Xs
= ( append @ A @ Ys3 @ ( cons @ A @ X4 @ Zs3 ) ) )
& ( P3 @ X4 )
& ! [Y: A] :
( ( member @ A @ Y @ ( set2 @ A @ Ys3 ) )
=> ~ ( P3 @ Y ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_6621_split__list__last__prop__iff,axiom,
! [A: $tType,Xs: list @ A,P3: A > $o] :
( ( ? [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
& ( P3 @ X4 ) ) )
= ( ? [Ys3: list @ A,X4: A,Zs3: list @ A] :
( ( Xs
= ( append @ A @ Ys3 @ ( cons @ A @ X4 @ Zs3 ) ) )
& ( P3 @ X4 )
& ! [Y: A] :
( ( member @ A @ Y @ ( set2 @ A @ Zs3 ) )
=> ~ ( P3 @ Y ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_6622_in__set__conv__decomp__first,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
= ( ? [Ys3: list @ A,Zs3: list @ A] :
( ( Xs
= ( append @ A @ Ys3 @ ( cons @ A @ X @ Zs3 ) ) )
& ~ ( member @ A @ X @ ( set2 @ A @ Ys3 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_6623_in__set__conv__decomp__last,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
= ( ? [Ys3: list @ A,Zs3: list @ A] :
( ( Xs
= ( append @ A @ Ys3 @ ( cons @ A @ X @ Zs3 ) ) )
& ~ ( member @ A @ X @ ( set2 @ A @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_6624_split__list__first__propE,axiom,
! [A: $tType,Xs: list @ A,P3: A > $o] :
( ? [X2: A] :
( ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
& ( P3 @ X2 ) )
=> ~ ! [Ys4: list @ A,X5: A] :
( ? [Zs2: list @ A] :
( Xs
= ( append @ A @ Ys4 @ ( cons @ A @ X5 @ Zs2 ) ) )
=> ( ( P3 @ X5 )
=> ~ ! [Xa2: A] :
( ( member @ A @ Xa2 @ ( set2 @ A @ Ys4 ) )
=> ~ ( P3 @ Xa2 ) ) ) ) ) ).
% split_list_first_propE
thf(fact_6625_split__list__last__propE,axiom,
! [A: $tType,Xs: list @ A,P3: A > $o] :
( ? [X2: A] :
( ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
& ( P3 @ X2 ) )
=> ~ ! [Ys4: list @ A,X5: A,Zs2: list @ A] :
( ( Xs
= ( append @ A @ Ys4 @ ( cons @ A @ X5 @ Zs2 ) ) )
=> ( ( P3 @ X5 )
=> ~ ! [Xa2: A] :
( ( member @ A @ Xa2 @ ( set2 @ A @ Zs2 ) )
=> ~ ( P3 @ Xa2 ) ) ) ) ) ).
% split_list_last_propE
thf(fact_6626_split__list__first__prop,axiom,
! [A: $tType,Xs: list @ A,P3: A > $o] :
( ? [X2: A] :
( ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
& ( P3 @ X2 ) )
=> ? [Ys4: list @ A,X5: A] :
( ? [Zs2: list @ A] :
( Xs
= ( append @ A @ Ys4 @ ( cons @ A @ X5 @ Zs2 ) ) )
& ( P3 @ X5 )
& ! [Xa2: A] :
( ( member @ A @ Xa2 @ ( set2 @ A @ Ys4 ) )
=> ~ ( P3 @ Xa2 ) ) ) ) ).
% split_list_first_prop
thf(fact_6627_split__list__last__prop,axiom,
! [A: $tType,Xs: list @ A,P3: A > $o] :
( ? [X2: A] :
( ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
& ( P3 @ X2 ) )
=> ? [Ys4: list @ A,X5: A,Zs2: list @ A] :
( ( Xs
= ( append @ A @ Ys4 @ ( cons @ A @ X5 @ Zs2 ) ) )
& ( P3 @ X5 )
& ! [Xa2: A] :
( ( member @ A @ Xa2 @ ( set2 @ A @ Zs2 ) )
=> ~ ( P3 @ Xa2 ) ) ) ) ).
% split_list_last_prop
thf(fact_6628_in__set__conv__decomp,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
= ( ? [Ys3: list @ A,Zs3: list @ A] :
( Xs
= ( append @ A @ Ys3 @ ( cons @ A @ X @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_6629_append__Cons__eq__iff,axiom,
! [A: $tType,X: A,Xs: list @ A,Ys: list @ A,Xs5: list @ A,Ys6: list @ A] :
( ~ ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ~ ( member @ A @ X @ ( set2 @ A @ Ys ) )
=> ( ( ( append @ A @ Xs @ ( cons @ A @ X @ Ys ) )
= ( append @ A @ Xs5 @ ( cons @ A @ X @ Ys6 ) ) )
= ( ( Xs = Xs5 )
& ( Ys = Ys6 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_6630_split__list__propE,axiom,
! [A: $tType,Xs: list @ A,P3: A > $o] :
( ? [X2: A] :
( ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
& ( P3 @ X2 ) )
=> ~ ! [Ys4: list @ A,X5: A] :
( ? [Zs2: list @ A] :
( Xs
= ( append @ A @ Ys4 @ ( cons @ A @ X5 @ Zs2 ) ) )
=> ~ ( P3 @ X5 ) ) ) ).
% split_list_propE
thf(fact_6631_split__list__first,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ? [Ys4: list @ A,Zs2: list @ A] :
( ( Xs
= ( append @ A @ Ys4 @ ( cons @ A @ X @ Zs2 ) ) )
& ~ ( member @ A @ X @ ( set2 @ A @ Ys4 ) ) ) ) ).
% split_list_first
thf(fact_6632_split__list__prop,axiom,
! [A: $tType,Xs: list @ A,P3: A > $o] :
( ? [X2: A] :
( ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
& ( P3 @ X2 ) )
=> ? [Ys4: list @ A,X5: A] :
( ? [Zs2: list @ A] :
( Xs
= ( append @ A @ Ys4 @ ( cons @ A @ X5 @ Zs2 ) ) )
& ( P3 @ X5 ) ) ) ).
% split_list_prop
thf(fact_6633_split__list__last,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ? [Ys4: list @ A,Zs2: list @ A] :
( ( Xs
= ( append @ A @ Ys4 @ ( cons @ A @ X @ Zs2 ) ) )
& ~ ( member @ A @ X @ ( set2 @ A @ Zs2 ) ) ) ) ).
% split_list_last
thf(fact_6634_split__list,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ? [Ys4: list @ A,Zs2: list @ A] :
( Xs
= ( append @ A @ Ys4 @ ( cons @ A @ X @ Zs2 ) ) ) ) ).
% split_list
thf(fact_6635_concat__eq__append__conv,axiom,
! [A: $tType,Xss: list @ ( list @ A ),Ys: list @ A,Zs: list @ A] :
( ( ( concat @ A @ Xss )
= ( append @ A @ Ys @ Zs ) )
= ( ( ( Xss
= ( nil @ ( list @ A ) ) )
=> ( ( Ys
= ( nil @ A ) )
& ( Zs
= ( nil @ A ) ) ) )
& ( ( Xss
!= ( nil @ ( list @ A ) ) )
=> ? [Xss1: list @ ( list @ A ),Xs2: list @ A,Xs6: list @ A,Xss22: list @ ( list @ A )] :
( ( Xss
= ( append @ ( list @ A ) @ Xss1 @ ( cons @ ( list @ A ) @ ( append @ A @ Xs2 @ Xs6 ) @ Xss22 ) ) )
& ( Ys
= ( append @ A @ ( concat @ A @ Xss1 ) @ Xs2 ) )
& ( Zs
= ( append @ A @ Xs6 @ ( concat @ A @ Xss22 ) ) ) ) ) ) ) ).
% concat_eq_append_conv
thf(fact_6636_concat__eq__appendD,axiom,
! [A: $tType,Xss: list @ ( list @ A ),Ys: list @ A,Zs: list @ A] :
( ( ( concat @ A @ Xss )
= ( append @ A @ Ys @ Zs ) )
=> ( ( Xss
!= ( nil @ ( list @ A ) ) )
=> ? [Xss12: list @ ( list @ A ),Xs3: list @ A,Xs4: list @ A,Xss23: list @ ( list @ A )] :
( ( Xss
= ( append @ ( list @ A ) @ Xss12 @ ( cons @ ( list @ A ) @ ( append @ A @ Xs3 @ Xs4 ) @ Xss23 ) ) )
& ( Ys
= ( append @ A @ ( concat @ A @ Xss12 ) @ Xs3 ) )
& ( Zs
= ( append @ A @ Xs4 @ ( concat @ A @ Xss23 ) ) ) ) ) ) ).
% concat_eq_appendD
thf(fact_6637_rev__nonempty__induct,axiom,
! [A: $tType,Xs: list @ A,P3: ( list @ A ) > $o] :
( ( Xs
!= ( nil @ A ) )
=> ( ! [X5: A] : ( P3 @ ( cons @ A @ X5 @ ( nil @ A ) ) )
=> ( ! [X5: A,Xs3: list @ A] :
( ( Xs3
!= ( nil @ A ) )
=> ( ( P3 @ Xs3 )
=> ( P3 @ ( append @ A @ Xs3 @ ( cons @ A @ X5 @ ( nil @ A ) ) ) ) ) )
=> ( P3 @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_6638_append__eq__Cons__conv,axiom,
! [A: $tType,Ys: list @ A,Zs: list @ A,X: A,Xs: list @ A] :
( ( ( append @ A @ Ys @ Zs )
= ( cons @ A @ X @ Xs ) )
= ( ( ( Ys
= ( nil @ A ) )
& ( Zs
= ( cons @ A @ X @ Xs ) ) )
| ? [Ys7: list @ A] :
( ( Ys
= ( cons @ A @ X @ Ys7 ) )
& ( ( append @ A @ Ys7 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_6639_Cons__eq__append__conv,axiom,
! [A: $tType,X: A,Xs: list @ A,Ys: list @ A,Zs: list @ A] :
( ( ( cons @ A @ X @ Xs )
= ( append @ A @ Ys @ Zs ) )
= ( ( ( Ys
= ( nil @ A ) )
& ( ( cons @ A @ X @ Xs )
= Zs ) )
| ? [Ys7: list @ A] :
( ( ( cons @ A @ X @ Ys7 )
= Ys )
& ( Xs
= ( append @ A @ Ys7 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_6640_rev__exhaust,axiom,
! [A: $tType,Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ~ ! [Ys4: list @ A,Y7: A] :
( Xs
!= ( append @ A @ Ys4 @ ( cons @ A @ Y7 @ ( nil @ A ) ) ) ) ) ).
% rev_exhaust
thf(fact_6641_rev__induct,axiom,
! [A: $tType,P3: ( list @ A ) > $o,Xs: list @ A] :
( ( P3 @ ( nil @ A ) )
=> ( ! [X5: A,Xs3: list @ A] :
( ( P3 @ Xs3 )
=> ( P3 @ ( append @ A @ Xs3 @ ( cons @ A @ X5 @ ( nil @ A ) ) ) ) )
=> ( P3 @ Xs ) ) ) ).
% rev_induct
thf(fact_6642_tl__append,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( tl @ A @ ( append @ A @ Xs @ Ys ) )
= ( case_list @ ( list @ A ) @ A @ ( tl @ A @ Ys )
@ ^ [Z6: A,Zs3: list @ A] : ( append @ A @ Zs3 @ Ys )
@ Xs ) ) ).
% tl_append
thf(fact_6643_comm__append__are__replicate,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( append @ A @ Xs @ Ys )
= ( append @ A @ Ys @ Xs ) )
=> ? [M4: nat,N3: nat,Zs2: list @ A] :
( ( ( concat @ A @ ( replicate @ ( list @ A ) @ M4 @ Zs2 ) )
= Xs )
& ( ( concat @ A @ ( replicate @ ( list @ A ) @ N3 @ Zs2 ) )
= Ys ) ) ) ).
% comm_append_are_replicate
thf(fact_6644_same__length__different,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( Xs != Ys )
=> ( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ A ) @ Ys ) )
=> ? [Pre: list @ A,X5: A,Xs4: list @ A,Y7: A,Ys5: list @ A] :
( ( X5 != Y7 )
& ( Xs
= ( append @ A @ Pre @ ( append @ A @ ( cons @ A @ X5 @ ( nil @ A ) ) @ Xs4 ) ) )
& ( Ys
= ( append @ A @ Pre @ ( append @ A @ ( cons @ A @ Y7 @ ( nil @ A ) ) @ Ys5 ) ) ) ) ) ) ).
% same_length_different
thf(fact_6645_sorted__append,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,Ys: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( append @ A @ Xs @ Ys ) )
= ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
& ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Ys )
& ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
=> ! [Y: A] :
( ( member @ A @ Y @ ( set2 @ A @ Ys ) )
=> ( ord_less_eq @ A @ X4 @ Y ) ) ) ) ) ) ).
% sorted_append
thf(fact_6646_not__distinct__decomp,axiom,
! [A: $tType,Ws: list @ A] :
( ~ ( distinct @ A @ Ws )
=> ? [Xs3: list @ A,Ys4: list @ A,Zs2: list @ A,Y7: A] :
( Ws
= ( append @ A @ Xs3 @ ( append @ A @ ( cons @ A @ Y7 @ ( nil @ A ) ) @ ( append @ A @ Ys4 @ ( append @ A @ ( cons @ A @ Y7 @ ( nil @ A ) ) @ Zs2 ) ) ) ) ) ) ).
% not_distinct_decomp
thf(fact_6647_not__distinct__conv__prefix,axiom,
! [A: $tType,As3: list @ A] :
( ( ~ ( distinct @ A @ As3 ) )
= ( ? [Xs2: list @ A,Y: A,Ys3: list @ A] :
( ( member @ A @ Y @ ( set2 @ A @ Xs2 ) )
& ( distinct @ A @ Xs2 )
& ( As3
= ( append @ A @ Xs2 @ ( cons @ A @ Y @ Ys3 ) ) ) ) ) ) ).
% not_distinct_conv_prefix
thf(fact_6648_Cons__eq__filterD,axiom,
! [A: $tType,X: A,Xs: list @ A,P3: A > $o,Ys: list @ A] :
( ( ( cons @ A @ X @ Xs )
= ( filter2 @ A @ P3 @ Ys ) )
=> ? [Us3: list @ A,Vs3: list @ A] :
( ( Ys
= ( append @ A @ Us3 @ ( cons @ A @ X @ Vs3 ) ) )
& ! [X2: A] :
( ( member @ A @ X2 @ ( set2 @ A @ Us3 ) )
=> ~ ( P3 @ X2 ) )
& ( P3 @ X )
& ( Xs
= ( filter2 @ A @ P3 @ Vs3 ) ) ) ) ).
% Cons_eq_filterD
thf(fact_6649_filter__eq__ConsD,axiom,
! [A: $tType,P3: A > $o,Ys: list @ A,X: A,Xs: list @ A] :
( ( ( filter2 @ A @ P3 @ Ys )
= ( cons @ A @ X @ Xs ) )
=> ? [Us3: list @ A,Vs3: list @ A] :
( ( Ys
= ( append @ A @ Us3 @ ( cons @ A @ X @ Vs3 ) ) )
& ! [X2: A] :
( ( member @ A @ X2 @ ( set2 @ A @ Us3 ) )
=> ~ ( P3 @ X2 ) )
& ( P3 @ X )
& ( Xs
= ( filter2 @ A @ P3 @ Vs3 ) ) ) ) ).
% filter_eq_ConsD
thf(fact_6650_Cons__eq__filter__iff,axiom,
! [A: $tType,X: A,Xs: list @ A,P3: A > $o,Ys: list @ A] :
( ( ( cons @ A @ X @ Xs )
= ( filter2 @ A @ P3 @ Ys ) )
= ( ? [Us2: list @ A,Vs2: list @ A] :
( ( Ys
= ( append @ A @ Us2 @ ( cons @ A @ X @ Vs2 ) ) )
& ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Us2 ) )
=> ~ ( P3 @ X4 ) )
& ( P3 @ X )
& ( Xs
= ( filter2 @ A @ P3 @ Vs2 ) ) ) ) ) ).
% Cons_eq_filter_iff
thf(fact_6651_filter__eq__Cons__iff,axiom,
! [A: $tType,P3: A > $o,Ys: list @ A,X: A,Xs: list @ A] :
( ( ( filter2 @ A @ P3 @ Ys )
= ( cons @ A @ X @ Xs ) )
= ( ? [Us2: list @ A,Vs2: list @ A] :
( ( Ys
= ( append @ A @ Us2 @ ( cons @ A @ X @ Vs2 ) ) )
& ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Us2 ) )
=> ~ ( P3 @ X4 ) )
& ( P3 @ X )
& ( Xs
= ( filter2 @ A @ P3 @ Vs2 ) ) ) ) ) ).
% filter_eq_Cons_iff
thf(fact_6652_rev_Osimps_I2_J,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( rev @ A @ ( cons @ A @ X @ Xs ) )
= ( append @ A @ ( rev @ A @ Xs ) @ ( cons @ A @ X @ ( nil @ A ) ) ) ) ).
% rev.simps(2)
thf(fact_6653_replicate__append__same,axiom,
! [A: $tType,I2: nat,X: A] :
( ( append @ A @ ( replicate @ A @ I2 @ X ) @ ( cons @ A @ X @ ( nil @ A ) ) )
= ( cons @ A @ X @ ( replicate @ A @ I2 @ X ) ) ) ).
% replicate_append_same
thf(fact_6654_upt__add__eq__append,axiom,
! [I2: nat,J3: nat,K2: nat] :
( ( ord_less_eq @ nat @ I2 @ J3 )
=> ( ( upt @ I2 @ ( plus_plus @ nat @ J3 @ K2 ) )
= ( append @ nat @ ( upt @ I2 @ J3 ) @ ( upt @ J3 @ ( plus_plus @ nat @ J3 @ K2 ) ) ) ) ) ).
% upt_add_eq_append
thf(fact_6655_list__update__append1,axiom,
! [A: $tType,I2: nat,Xs: list @ A,Ys: list @ A,X: A] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( list_update @ A @ ( append @ A @ Xs @ Ys ) @ I2 @ X )
= ( append @ A @ ( list_update @ A @ Xs @ I2 @ X ) @ Ys ) ) ) ).
% list_update_append1
thf(fact_6656_atLeastLessThan__add__Un,axiom,
! [I2: nat,J3: nat,K2: nat] :
( ( ord_less_eq @ nat @ I2 @ J3 )
=> ( ( set_or7035219750837199246ssThan @ nat @ I2 @ ( plus_plus @ nat @ J3 @ K2 ) )
= ( sup_sup @ ( set @ nat ) @ ( set_or7035219750837199246ssThan @ nat @ I2 @ J3 ) @ ( set_or7035219750837199246ssThan @ nat @ J3 @ ( plus_plus @ nat @ J3 @ K2 ) ) ) ) ) ).
% atLeastLessThan_add_Un
thf(fact_6657_remove1__split,axiom,
! [A: $tType,A2: A,Xs: list @ A,Ys: list @ A] :
( ( member @ A @ A2 @ ( set2 @ A @ Xs ) )
=> ( ( ( remove1 @ A @ A2 @ Xs )
= Ys )
= ( ? [Ls: list @ A,Rs: list @ A] :
( ( Xs
= ( append @ A @ Ls @ ( cons @ A @ A2 @ Rs ) ) )
& ~ ( member @ A @ A2 @ ( set2 @ A @ Ls ) )
& ( Ys
= ( append @ A @ Ls @ Rs ) ) ) ) ) ) ).
% remove1_split
thf(fact_6658_lex__append__left__iff,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),Xs: list @ A,Ys: list @ A,Zs: list @ A] :
( ! [X5: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X5 @ X5 ) @ R )
=> ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs @ Ys ) @ ( append @ A @ Xs @ Zs ) ) @ ( lex @ A @ R ) )
= ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ys @ Zs ) @ ( lex @ A @ R ) ) ) ) ).
% lex_append_left_iff
thf(fact_6659_lex__append__leftD,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),Xs: list @ A,Ys: list @ A,Zs: list @ A] :
( ! [X5: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X5 @ X5 ) @ R )
=> ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs @ Ys ) @ ( append @ A @ Xs @ Zs ) ) @ ( lex @ A @ R ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ys @ Zs ) @ ( lex @ A @ R ) ) ) ) ).
% lex_append_leftD
thf(fact_6660_rotate1_Osimps_I2_J,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( rotate1 @ A @ ( cons @ A @ X @ Xs ) )
= ( append @ A @ Xs @ ( cons @ A @ X @ ( nil @ A ) ) ) ) ).
% rotate1.simps(2)
thf(fact_6661_lex__append__rightI,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,R: set @ ( product_prod @ A @ A ),Vs: list @ A,Us: list @ A] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( lex @ A @ R ) )
=> ( ( ( size_size @ ( list @ A ) @ Vs )
= ( size_size @ ( list @ A ) @ Us ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs @ Us ) @ ( append @ A @ Ys @ Vs ) ) @ ( lex @ A @ R ) ) ) ) ).
% lex_append_rightI
thf(fact_6662_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_6663_length__Suc__conv__rev,axiom,
! [A: $tType,Xs: list @ A,N: nat] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( suc @ N ) )
= ( ? [Y: A,Ys3: list @ A] :
( ( Xs
= ( append @ A @ Ys3 @ ( cons @ A @ Y @ ( nil @ A ) ) ) )
& ( ( size_size @ ( list @ A ) @ Ys3 )
= N ) ) ) ) ).
% length_Suc_conv_rev
thf(fact_6664_subseqs_Osimps_I2_J,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( subseqs @ A @ ( cons @ A @ X @ Xs ) )
= ( append @ ( list @ A ) @ ( map @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X ) @ ( subseqs @ A @ Xs ) ) @ ( subseqs @ A @ Xs ) ) ) ).
% subseqs.simps(2)
thf(fact_6665_nth__append,axiom,
! [A: $tType,N: nat,Xs: list @ A,Ys: list @ A] :
( ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ A @ ( append @ A @ Xs @ Ys ) @ N )
= ( nth @ A @ Xs @ N ) ) )
& ( ~ ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ A @ ( append @ A @ Xs @ Ys ) @ N )
= ( nth @ A @ Ys @ ( minus_minus @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ) ) ).
% nth_append
thf(fact_6666_list__update__append,axiom,
! [A: $tType,N: nat,Xs: list @ A,Ys: list @ A,X: A] :
( ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( list_update @ A @ ( append @ A @ Xs @ Ys ) @ N @ X )
= ( append @ A @ ( list_update @ A @ Xs @ N @ X ) @ Ys ) ) )
& ( ~ ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( list_update @ A @ ( append @ A @ Xs @ Ys ) @ N @ X )
= ( append @ A @ Xs @ ( list_update @ A @ Ys @ ( minus_minus @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) ) @ X ) ) ) ) ) ).
% list_update_append
thf(fact_6667_product_Osimps_I2_J,axiom,
! [A: $tType,B: $tType,X: A,Xs: list @ A,Ys: list @ B] :
( ( product @ A @ B @ ( cons @ A @ X @ Xs ) @ Ys )
= ( append @ ( product_prod @ A @ B ) @ ( map @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X ) @ Ys ) @ ( product @ A @ B @ Xs @ Ys ) ) ) ).
% product.simps(2)
thf(fact_6668_horner__sum__append,axiom,
! [A: $tType,B: $tType] :
( ( comm_semiring_1 @ A )
=> ! [F3: B > A,A2: A,Xs: list @ B,Ys: list @ B] :
( ( groups4207007520872428315er_sum @ B @ A @ F3 @ A2 @ ( append @ B @ Xs @ Ys ) )
= ( plus_plus @ A @ ( groups4207007520872428315er_sum @ B @ A @ F3 @ A2 @ Xs ) @ ( times_times @ A @ ( power_power @ A @ A2 @ ( size_size @ ( list @ B ) @ Xs ) ) @ ( groups4207007520872428315er_sum @ B @ A @ F3 @ A2 @ Ys ) ) ) ) ) ).
% horner_sum_append
thf(fact_6669_rotate1__hd__tl,axiom,
! [A: $tType,Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( rotate1 @ A @ Xs )
= ( append @ A @ ( tl @ A @ Xs ) @ ( cons @ A @ ( hd @ A @ Xs ) @ ( nil @ A ) ) ) ) ) ).
% rotate1_hd_tl
thf(fact_6670_upt__Suc,axiom,
! [I2: nat,J3: nat] :
( ( ( ord_less_eq @ nat @ I2 @ J3 )
=> ( ( upt @ I2 @ ( suc @ J3 ) )
= ( append @ nat @ ( upt @ I2 @ J3 ) @ ( cons @ nat @ J3 @ ( nil @ nat ) ) ) ) )
& ( ~ ( ord_less_eq @ nat @ I2 @ J3 )
=> ( ( upt @ I2 @ ( suc @ J3 ) )
= ( nil @ nat ) ) ) ) ).
% upt_Suc
thf(fact_6671_upt__Suc__append,axiom,
! [I2: nat,J3: nat] :
( ( ord_less_eq @ nat @ I2 @ J3 )
=> ( ( upt @ I2 @ ( suc @ J3 ) )
= ( append @ nat @ ( upt @ I2 @ J3 ) @ ( cons @ nat @ J3 @ ( nil @ nat ) ) ) ) ) ).
% upt_Suc_append
thf(fact_6672_Pow__set_I2_J,axiom,
! [B: $tType,X: B,Xs: list @ B] :
( ( pow2 @ B @ ( set2 @ B @ ( cons @ B @ X @ Xs ) ) )
= ( sup_sup @ ( set @ ( set @ B ) ) @ ( pow2 @ B @ ( set2 @ B @ Xs ) ) @ ( image @ ( set @ B ) @ ( set @ B ) @ ( insert @ B @ X ) @ ( pow2 @ B @ ( set2 @ B @ Xs ) ) ) ) ) ).
% Pow_set(2)
thf(fact_6673_comm__append__is__replicate,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( Ys
!= ( nil @ A ) )
=> ( ( ( append @ A @ Xs @ Ys )
= ( append @ A @ Ys @ Xs ) )
=> ? [N3: nat,Zs2: list @ A] :
( ( ord_less @ nat @ ( one_one @ nat ) @ N3 )
& ( ( concat @ A @ ( replicate @ ( list @ A ) @ N3 @ Zs2 ) )
= ( append @ A @ Xs @ Ys ) ) ) ) ) ) ).
% comm_append_is_replicate
thf(fact_6674_sorted__insort__is__snoc,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,A2: A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Xs ) )
=> ( ord_less_eq @ A @ X5 @ A2 ) )
=> ( ( linorder_insort_key @ A @ A
@ ^ [X4: A] : X4
@ A2
@ Xs )
= ( append @ A @ Xs @ ( cons @ A @ A2 @ ( nil @ A ) ) ) ) ) ) ) ).
% sorted_insort_is_snoc
thf(fact_6675_take__Suc__conv__app__nth,axiom,
! [A: $tType,I2: nat,Xs: list @ A] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( take @ A @ ( suc @ I2 ) @ Xs )
= ( append @ A @ ( take @ A @ I2 @ Xs ) @ ( cons @ A @ ( nth @ A @ Xs @ I2 ) @ ( nil @ A ) ) ) ) ) ).
% take_Suc_conv_app_nth
thf(fact_6676_nth__repl,axiom,
! [A: $tType,M: nat,Xs: list @ A,N: nat,X: A] :
( ( ord_less @ nat @ M @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( M != N )
=> ( ( nth @ A @ ( append @ A @ ( take @ A @ N @ Xs ) @ ( append @ A @ ( cons @ A @ X @ ( nil @ A ) ) @ ( drop @ A @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) @ Xs ) ) ) @ M )
= ( nth @ A @ Xs @ M ) ) ) ) ) ).
% nth_repl
thf(fact_6677_pos__n__replace,axiom,
! [A: $tType,N: nat,Xs: list @ A,Y2: A] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ A ) @ ( append @ A @ ( take @ A @ N @ Xs ) @ ( append @ A @ ( cons @ A @ Y2 @ ( nil @ A ) ) @ ( drop @ A @ ( suc @ N ) @ Xs ) ) ) ) ) ) ).
% pos_n_replace
thf(fact_6678_drop0,axiom,
! [A: $tType] :
( ( drop @ A @ ( zero_zero @ nat ) )
= ( ^ [X4: list @ A] : X4 ) ) ).
% drop0
thf(fact_6679_drop__drop,axiom,
! [A: $tType,N: nat,M: nat,Xs: list @ A] :
( ( drop @ A @ N @ ( drop @ A @ M @ Xs ) )
= ( drop @ A @ ( plus_plus @ nat @ N @ M ) @ Xs ) ) ).
% drop_drop
thf(fact_6680_drop__upt,axiom,
! [M: nat,I2: nat,J3: nat] :
( ( drop @ nat @ M @ ( upt @ I2 @ J3 ) )
= ( upt @ ( plus_plus @ nat @ I2 @ M ) @ J3 ) ) ).
% drop_upt
thf(fact_6681_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_6682_length__drop,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( size_size @ ( list @ A ) @ ( drop @ A @ N @ Xs ) )
= ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N ) ) ).
% length_drop
thf(fact_6683_append__take__drop__id,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( append @ A @ ( take @ A @ N @ Xs ) @ ( drop @ A @ N @ Xs ) )
= Xs ) ).
% append_take_drop_id
thf(fact_6684_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_6685_drop__replicate,axiom,
! [A: $tType,I2: nat,K2: nat,X: A] :
( ( drop @ A @ I2 @ ( replicate @ A @ K2 @ X ) )
= ( replicate @ A @ ( minus_minus @ nat @ K2 @ I2 ) @ X ) ) ).
% drop_replicate
thf(fact_6686_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_6687_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_6688_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_6689_drop__append,axiom,
! [A: $tType,N: nat,Xs: list @ A,Ys: list @ A] :
( ( drop @ A @ N @ ( append @ A @ Xs @ Ys ) )
= ( append @ A @ ( drop @ A @ N @ Xs ) @ ( drop @ A @ ( minus_minus @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) ) @ Ys ) ) ) ).
% drop_append
thf(fact_6690_drop__Cons__numeral,axiom,
! [A: $tType,V2: num,X: A,Xs: list @ A] :
( ( drop @ A @ ( numeral_numeral @ nat @ V2 ) @ ( cons @ A @ X @ Xs ) )
= ( drop @ A @ ( minus_minus @ nat @ ( numeral_numeral @ nat @ V2 ) @ ( one_one @ nat ) ) @ Xs ) ) ).
% drop_Cons_numeral
thf(fact_6691_nth__drop,axiom,
! [A: $tType,N: nat,Xs: list @ A,I2: nat] :
( ( ord_less_eq @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ A @ ( drop @ A @ N @ Xs ) @ I2 )
= ( nth @ A @ Xs @ ( plus_plus @ nat @ N @ I2 ) ) ) ) ).
% nth_drop
thf(fact_6692_tl__drop,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( tl @ A @ ( drop @ A @ N @ Xs ) )
= ( drop @ A @ N @ ( tl @ A @ Xs ) ) ) ).
% tl_drop
thf(fact_6693_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_6694_nth__via__drop,axiom,
! [A: $tType,N: nat,Xs: list @ A,Y2: A,Ys: list @ A] :
( ( ( drop @ A @ N @ Xs )
= ( cons @ A @ Y2 @ Ys ) )
=> ( ( nth @ A @ Xs @ N )
= Y2 ) ) ).
% nth_via_drop
thf(fact_6695_drop__Nil,axiom,
! [A: $tType,N: nat] :
( ( drop @ A @ N @ ( nil @ A ) )
= ( nil @ A ) ) ).
% drop_Nil
thf(fact_6696_distinct__drop,axiom,
! [A: $tType,Xs: list @ A,I2: nat] :
( ( distinct @ A @ Xs )
=> ( distinct @ A @ ( drop @ A @ I2 @ Xs ) ) ) ).
% distinct_drop
thf(fact_6697_take__drop,axiom,
! [A: $tType,N: nat,M: nat,Xs: list @ A] :
( ( take @ A @ N @ ( drop @ A @ M @ Xs ) )
= ( drop @ A @ M @ ( take @ A @ ( plus_plus @ nat @ N @ M ) @ Xs ) ) ) ).
% take_drop
thf(fact_6698_drop__take,axiom,
! [A: $tType,N: nat,M: nat,Xs: list @ A] :
( ( drop @ A @ N @ ( take @ A @ M @ Xs ) )
= ( take @ A @ ( minus_minus @ nat @ M @ N ) @ ( drop @ A @ N @ Xs ) ) ) ).
% drop_take
thf(fact_6699_drop__0,axiom,
! [A: $tType,Xs: list @ A] :
( ( drop @ A @ ( zero_zero @ nat ) @ Xs )
= Xs ) ).
% drop_0
thf(fact_6700_sorted__wrt__drop,axiom,
! [A: $tType,F3: A > A > $o,Xs: list @ A,N: nat] :
( ( sorted_wrt @ A @ F3 @ Xs )
=> ( sorted_wrt @ A @ F3 @ ( drop @ A @ N @ Xs ) ) ) ).
% sorted_wrt_drop
thf(fact_6701_in__set__dropD,axiom,
! [A: $tType,X: A,N: nat,Xs: list @ A] :
( ( member @ A @ X @ ( set2 @ A @ ( drop @ A @ N @ Xs ) ) )
=> ( member @ A @ X @ ( set2 @ A @ Xs ) ) ) ).
% in_set_dropD
thf(fact_6702_drop__map,axiom,
! [A: $tType,B: $tType,N: nat,F3: B > A,Xs: list @ B] :
( ( drop @ A @ N @ ( map @ B @ A @ F3 @ Xs ) )
= ( map @ B @ A @ F3 @ ( drop @ B @ N @ Xs ) ) ) ).
% drop_map
thf(fact_6703_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_6704_set__drop__subset,axiom,
! [A: $tType,N: nat,Xs: list @ A] : ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ ( drop @ A @ N @ Xs ) ) @ ( set2 @ A @ Xs ) ) ).
% set_drop_subset
thf(fact_6705_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_6706_take__add,axiom,
! [A: $tType,I2: nat,J3: nat,Xs: list @ A] :
( ( take @ A @ ( plus_plus @ nat @ I2 @ J3 ) @ Xs )
= ( append @ A @ ( take @ A @ I2 @ Xs ) @ ( take @ A @ J3 @ ( drop @ A @ I2 @ Xs ) ) ) ) ).
% take_add
thf(fact_6707_append__eq__conv__conj,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Zs: list @ A] :
( ( ( append @ A @ Xs @ Ys )
= Zs )
= ( ( Xs
= ( take @ A @ ( size_size @ ( list @ A ) @ Xs ) @ Zs ) )
& ( Ys
= ( drop @ A @ ( size_size @ ( list @ A ) @ Xs ) @ Zs ) ) ) ) ).
% append_eq_conv_conj
thf(fact_6708_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_6709_drop__Cons,axiom,
! [A: $tType,N: nat,X: A,Xs: list @ A] :
( ( drop @ A @ N @ ( cons @ A @ X @ Xs ) )
= ( case_nat @ ( list @ A ) @ ( cons @ A @ X @ Xs )
@ ^ [M2: nat] : ( drop @ A @ M2 @ Xs )
@ N ) ) ).
% drop_Cons
thf(fact_6710_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_6711_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_6712_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_6713_drop__rev,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( drop @ A @ N @ ( rev @ A @ Xs ) )
= ( rev @ A @ ( take @ A @ ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N ) @ Xs ) ) ) ).
% drop_rev
thf(fact_6714_rev__drop,axiom,
! [A: $tType,I2: nat,Xs: list @ A] :
( ( rev @ A @ ( drop @ A @ I2 @ Xs ) )
= ( take @ A @ ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ I2 ) @ ( rev @ A @ Xs ) ) ) ).
% rev_drop
thf(fact_6715_rev__take,axiom,
! [A: $tType,I2: nat,Xs: list @ A] :
( ( rev @ A @ ( take @ A @ I2 @ Xs ) )
= ( drop @ A @ ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ I2 ) @ ( rev @ A @ Xs ) ) ) ).
% rev_take
thf(fact_6716_take__rev,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( take @ A @ N @ ( rev @ A @ Xs ) )
= ( rev @ A @ ( drop @ A @ ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N ) @ Xs ) ) ) ).
% take_rev
thf(fact_6717_Cons__nth__drop__Suc,axiom,
! [A: $tType,I2: nat,Xs: list @ A] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( cons @ A @ ( nth @ A @ Xs @ I2 ) @ ( drop @ A @ ( suc @ I2 ) @ Xs ) )
= ( drop @ A @ I2 @ Xs ) ) ) ).
% Cons_nth_drop_Suc
thf(fact_6718_set__take__disj__set__drop__if__distinct,axiom,
! [A: $tType,Vs: list @ A,I2: nat,J3: nat] :
( ( distinct @ A @ Vs )
=> ( ( ord_less_eq @ nat @ I2 @ J3 )
=> ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ ( take @ A @ I2 @ Vs ) ) @ ( set2 @ A @ ( drop @ A @ J3 @ Vs ) ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% set_take_disj_set_drop_if_distinct
thf(fact_6719_id__take__nth__drop,axiom,
! [A: $tType,I2: nat,Xs: list @ A] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( Xs
= ( append @ A @ ( take @ A @ I2 @ Xs ) @ ( cons @ A @ ( nth @ A @ Xs @ I2 ) @ ( drop @ A @ ( suc @ I2 ) @ Xs ) ) ) ) ) ).
% id_take_nth_drop
thf(fact_6720_upd__conv__take__nth__drop,axiom,
! [A: $tType,I2: nat,Xs: list @ A,A2: A] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( list_update @ A @ Xs @ I2 @ A2 )
= ( append @ A @ ( take @ A @ I2 @ Xs ) @ ( cons @ A @ A2 @ ( drop @ A @ ( suc @ I2 ) @ Xs ) ) ) ) ) ).
% upd_conv_take_nth_drop
thf(fact_6721_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_6722_Pow__fold,axiom,
! [A: $tType,A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( pow2 @ A @ A3 )
= ( finite_fold @ A @ ( set @ ( set @ A ) )
@ ^ [X4: A,A7: set @ ( set @ A )] : ( sup_sup @ ( set @ ( set @ A ) ) @ A7 @ ( image @ ( set @ A ) @ ( set @ A ) @ ( insert @ A @ X4 ) @ A7 ) )
@ ( insert @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ ( bot_bot @ ( set @ ( set @ A ) ) ) )
@ A3 ) ) ) ).
% Pow_fold
thf(fact_6723_upto_Opsimps,axiom,
! [I2: int,J3: int] :
( ( accp @ ( product_prod @ int @ int ) @ upto_rel @ ( product_Pair @ int @ int @ I2 @ J3 ) )
=> ( ( ( ord_less_eq @ int @ I2 @ J3 )
=> ( ( upto @ I2 @ J3 )
= ( cons @ int @ I2 @ ( upto @ ( plus_plus @ int @ I2 @ ( one_one @ int ) ) @ J3 ) ) ) )
& ( ~ ( ord_less_eq @ int @ I2 @ J3 )
=> ( ( upto @ I2 @ J3 )
= ( nil @ int ) ) ) ) ) ).
% upto.psimps
thf(fact_6724_upto__empty,axiom,
! [J3: int,I2: int] :
( ( ord_less @ int @ J3 @ I2 )
=> ( ( upto @ I2 @ J3 )
= ( nil @ int ) ) ) ).
% upto_empty
thf(fact_6725_upto__Nil2,axiom,
! [I2: int,J3: int] :
( ( ( nil @ int )
= ( upto @ I2 @ J3 ) )
= ( ord_less @ int @ J3 @ I2 ) ) ).
% upto_Nil2
thf(fact_6726_upto__Nil,axiom,
! [I2: int,J3: int] :
( ( ( upto @ I2 @ J3 )
= ( nil @ int ) )
= ( ord_less @ int @ J3 @ I2 ) ) ).
% upto_Nil
thf(fact_6727_upto__single,axiom,
! [I2: int] :
( ( upto @ I2 @ I2 )
= ( cons @ int @ I2 @ ( nil @ int ) ) ) ).
% upto_single
thf(fact_6728_nth__upto,axiom,
! [I2: int,K2: nat,J3: int] :
( ( ord_less_eq @ int @ ( plus_plus @ int @ I2 @ ( semiring_1_of_nat @ int @ K2 ) ) @ J3 )
=> ( ( nth @ int @ ( upto @ I2 @ J3 ) @ K2 )
= ( plus_plus @ int @ I2 @ ( semiring_1_of_nat @ int @ K2 ) ) ) ) ).
% nth_upto
thf(fact_6729_length__upto,axiom,
! [I2: int,J3: int] :
( ( size_size @ ( list @ int ) @ ( upto @ I2 @ J3 ) )
= ( nat2 @ ( plus_plus @ int @ ( minus_minus @ int @ J3 @ I2 ) @ ( one_one @ int ) ) ) ) ).
% length_upto
thf(fact_6730_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_6731_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_6732_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_6733_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_6734_upto__aux__def,axiom,
( upto_aux
= ( ^ [I: int,J4: int] : ( append @ int @ ( upto @ I @ J4 ) ) ) ) ).
% upto_aux_def
thf(fact_6735_sorted__upto,axiom,
! [M: int,N: int] : ( sorted_wrt @ int @ ( ord_less_eq @ int ) @ ( upto @ M @ N ) ) ).
% sorted_upto
thf(fact_6736_sorted__wrt__upto,axiom,
! [I2: int,J3: int] : ( sorted_wrt @ int @ ( ord_less @ int ) @ ( upto @ I2 @ J3 ) ) ).
% sorted_wrt_upto
thf(fact_6737_atLeastAtMost__upto,axiom,
( ( set_or1337092689740270186AtMost @ int )
= ( ^ [I: int,J4: int] : ( set2 @ int @ ( upto @ I @ J4 ) ) ) ) ).
% atLeastAtMost_upto
thf(fact_6738_distinct__upto,axiom,
! [I2: int,J3: int] : ( distinct @ int @ ( upto @ I2 @ J3 ) ) ).
% distinct_upto
thf(fact_6739_upto__code,axiom,
( upto
= ( ^ [I: int,J4: int] : ( upto_aux @ I @ J4 @ ( nil @ int ) ) ) ) ).
% upto_code
thf(fact_6740_upto__split2,axiom,
! [I2: int,J3: int,K2: int] :
( ( ord_less_eq @ int @ I2 @ J3 )
=> ( ( ord_less_eq @ int @ J3 @ K2 )
=> ( ( upto @ I2 @ K2 )
= ( append @ int @ ( upto @ I2 @ J3 ) @ ( upto @ ( plus_plus @ int @ J3 @ ( one_one @ int ) ) @ K2 ) ) ) ) ) ).
% upto_split2
thf(fact_6741_upto__split1,axiom,
! [I2: int,J3: int,K2: int] :
( ( ord_less_eq @ int @ I2 @ J3 )
=> ( ( ord_less_eq @ int @ J3 @ K2 )
=> ( ( upto @ I2 @ K2 )
= ( append @ int @ ( upto @ I2 @ ( minus_minus @ int @ J3 @ ( one_one @ int ) ) ) @ ( upto @ J3 @ K2 ) ) ) ) ) ).
% upto_split1
thf(fact_6742_atLeastLessThan__upto,axiom,
( ( set_or7035219750837199246ssThan @ int )
= ( ^ [I: int,J4: int] : ( set2 @ int @ ( upto @ I @ ( minus_minus @ int @ J4 @ ( one_one @ int ) ) ) ) ) ) ).
% atLeastLessThan_upto
thf(fact_6743_greaterThanAtMost__upto,axiom,
( ( set_or3652927894154168847AtMost @ int )
= ( ^ [I: int,J4: int] : ( set2 @ int @ ( upto @ ( plus_plus @ int @ I @ ( one_one @ int ) ) @ J4 ) ) ) ) ).
% greaterThanAtMost_upto
thf(fact_6744_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_6745_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_6746_image__fold__insert,axiom,
! [B: $tType,A: $tType,A3: set @ A,F3: A > B] :
( ( finite_finite2 @ A @ A3 )
=> ( ( image @ A @ B @ F3 @ A3 )
= ( finite_fold @ A @ ( set @ B )
@ ^ [K3: A] : ( insert @ B @ ( F3 @ K3 ) )
@ ( bot_bot @ ( set @ B ) )
@ A3 ) ) ) ).
% image_fold_insert
thf(fact_6747_upto__rec1,axiom,
! [I2: int,J3: int] :
( ( ord_less_eq @ int @ I2 @ J3 )
=> ( ( upto @ I2 @ J3 )
= ( cons @ int @ I2 @ ( upto @ ( plus_plus @ int @ I2 @ ( one_one @ int ) ) @ J3 ) ) ) ) ).
% upto_rec1
thf(fact_6748_upto_Osimps,axiom,
( upto
= ( ^ [I: int,J4: int] : ( if @ ( list @ int ) @ ( ord_less_eq @ int @ I @ J4 ) @ ( cons @ int @ I @ ( upto @ ( plus_plus @ int @ I @ ( one_one @ int ) ) @ J4 ) ) @ ( nil @ int ) ) ) ) ).
% upto.simps
thf(fact_6749_upto_Oelims,axiom,
! [X: int,Xa: int,Y2: list @ int] :
( ( ( upto @ X @ Xa )
= Y2 )
=> ( ( ( ord_less_eq @ int @ X @ Xa )
=> ( Y2
= ( cons @ int @ X @ ( upto @ ( plus_plus @ int @ X @ ( one_one @ int ) ) @ Xa ) ) ) )
& ( ~ ( ord_less_eq @ int @ X @ Xa )
=> ( Y2
= ( nil @ int ) ) ) ) ) ).
% upto.elims
thf(fact_6750_upto__rec2,axiom,
! [I2: int,J3: int] :
( ( ord_less_eq @ int @ I2 @ J3 )
=> ( ( upto @ I2 @ J3 )
= ( append @ int @ ( upto @ I2 @ ( minus_minus @ int @ J3 @ ( one_one @ int ) ) ) @ ( cons @ int @ J3 @ ( nil @ int ) ) ) ) ) ).
% upto_rec2
thf(fact_6751_greaterThanLessThan__upto,axiom,
( ( set_or5935395276787703475ssThan @ int )
= ( ^ [I: int,J4: int] : ( set2 @ int @ ( upto @ ( plus_plus @ int @ I @ ( one_one @ int ) ) @ ( minus_minus @ int @ J4 @ ( one_one @ int ) ) ) ) ) ) ).
% greaterThanLessThan_upto
thf(fact_6752_upto__split3,axiom,
! [I2: int,J3: int,K2: int] :
( ( ord_less_eq @ int @ I2 @ J3 )
=> ( ( ord_less_eq @ int @ J3 @ K2 )
=> ( ( upto @ I2 @ K2 )
= ( append @ int @ ( upto @ I2 @ ( minus_minus @ int @ J3 @ ( one_one @ int ) ) ) @ ( cons @ int @ J3 @ ( upto @ ( plus_plus @ int @ J3 @ ( one_one @ int ) ) @ K2 ) ) ) ) ) ) ).
% upto_split3
thf(fact_6753_upto_Opelims,axiom,
! [X: int,Xa: int,Y2: list @ int] :
( ( ( upto @ X @ Xa )
= Y2 )
=> ( ( accp @ ( product_prod @ int @ int ) @ upto_rel @ ( product_Pair @ int @ int @ X @ Xa ) )
=> ~ ( ( ( ( ord_less_eq @ int @ X @ Xa )
=> ( Y2
= ( cons @ int @ X @ ( upto @ ( plus_plus @ int @ X @ ( one_one @ int ) ) @ Xa ) ) ) )
& ( ~ ( ord_less_eq @ int @ X @ Xa )
=> ( Y2
= ( nil @ int ) ) ) )
=> ~ ( accp @ ( product_prod @ int @ int ) @ upto_rel @ ( product_Pair @ int @ int @ X @ Xa ) ) ) ) ) ).
% upto.pelims
thf(fact_6754_Set__filter__fold,axiom,
! [A: $tType,A3: set @ A,P3: A > $o] :
( ( finite_finite2 @ A @ A3 )
=> ( ( filter3 @ A @ P3 @ A3 )
= ( finite_fold @ A @ ( set @ A )
@ ^ [X4: A,A16: set @ A] : ( if @ ( set @ A ) @ ( P3 @ X4 ) @ ( insert @ A @ X4 @ A16 ) @ A16 )
@ ( bot_bot @ ( set @ A ) )
@ A3 ) ) ) ).
% Set_filter_fold
thf(fact_6755_splice_Opinduct,axiom,
! [A: $tType,A0: list @ A,A12: list @ A,P3: ( list @ A ) > ( list @ A ) > $o] :
( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( splice_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ A0 @ A12 ) )
=> ( ! [Ys4: list @ A] :
( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( splice_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Ys4 ) )
=> ( P3 @ ( nil @ A ) @ Ys4 ) )
=> ( ! [X5: A,Xs3: list @ A,Ys4: list @ A] :
( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( splice_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X5 @ Xs3 ) @ Ys4 ) )
=> ( ( P3 @ Ys4 @ Xs3 )
=> ( P3 @ ( cons @ A @ X5 @ Xs3 ) @ Ys4 ) ) )
=> ( P3 @ A0 @ A12 ) ) ) ) ).
% splice.pinduct
thf(fact_6756_Set_Ofilter__def,axiom,
! [A: $tType] :
( ( filter3 @ A )
= ( ^ [P2: A > $o,A7: set @ A] :
( collect @ A
@ ^ [A5: A] :
( ( member @ A @ A5 @ A7 )
& ( P2 @ A5 ) ) ) ) ) ).
% Set.filter_def
thf(fact_6757_card_Oeq__fold,axiom,
! [A: $tType] :
( ( finite_card @ A )
= ( finite_fold @ A @ nat
@ ^ [Uu3: A] : suc
@ ( zero_zero @ nat ) ) ) ).
% card.eq_fold
thf(fact_6758_sorted__list__of__set_Ofold__insort__key_Oeq__fold,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( linord4507533701916653071of_set @ A )
= ( finite_fold @ A @ ( list @ A )
@ ( linorder_insort_key @ A @ A
@ ^ [X4: A] : X4 )
@ ( nil @ A ) ) ) ) ).
% sorted_list_of_set.fold_insort_key.eq_fold
thf(fact_6759_filter__set,axiom,
! [A: $tType,P3: A > $o,Xs: list @ A] :
( ( filter3 @ A @ P3 @ ( set2 @ A @ Xs ) )
= ( set2 @ A @ ( filter2 @ A @ P3 @ Xs ) ) ) ).
% filter_set
thf(fact_6760_inter__Set__filter,axiom,
! [A: $tType,B5: set @ A,A3: set @ A] :
( ( finite_finite2 @ A @ B5 )
=> ( ( inf_inf @ ( set @ A ) @ A3 @ B5 )
= ( filter3 @ A
@ ^ [X4: A] : ( member @ A @ X4 @ A3 )
@ B5 ) ) ) ).
% inter_Set_filter
thf(fact_6761_fold__union__pair,axiom,
! [B: $tType,A: $tType,B5: set @ A,X: B,A3: set @ ( product_prod @ B @ A )] :
( ( finite_finite2 @ A @ B5 )
=> ( ( sup_sup @ ( set @ ( product_prod @ B @ A ) )
@ ( complete_Sup_Sup @ ( set @ ( product_prod @ B @ A ) )
@ ( image @ A @ ( set @ ( product_prod @ B @ A ) )
@ ^ [Y: A] : ( insert @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X @ Y ) @ ( bot_bot @ ( set @ ( product_prod @ B @ A ) ) ) )
@ B5 ) )
@ A3 )
= ( finite_fold @ A @ ( set @ ( product_prod @ B @ A ) )
@ ^ [Y: A] : ( insert @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X @ Y ) )
@ A3
@ B5 ) ) ) ).
% fold_union_pair
thf(fact_6762_splice_Opelims,axiom,
! [A: $tType,X: list @ A,Xa: list @ A,Y2: list @ A] :
( ( ( splice @ A @ X @ Xa )
= Y2 )
=> ( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( splice_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Xa ) )
=> ( ( ( X
= ( nil @ A ) )
=> ( ( Y2 = Xa )
=> ~ ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( splice_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Xa ) ) ) )
=> ~ ! [X5: A,Xs3: list @ A] :
( ( X
= ( cons @ A @ X5 @ Xs3 ) )
=> ( ( Y2
= ( cons @ A @ X5 @ ( splice @ A @ Xa @ Xs3 ) ) )
=> ~ ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( splice_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X5 @ Xs3 ) @ Xa ) ) ) ) ) ) ) ).
% splice.pelims
thf(fact_6763_set__relcomp,axiom,
! [B: $tType,C: $tType,A: $tType,Xys: list @ ( product_prod @ A @ C ),Yzs: list @ ( product_prod @ C @ B )] :
( ( relcomp @ A @ C @ B @ ( set2 @ ( product_prod @ A @ C ) @ Xys ) @ ( set2 @ ( product_prod @ C @ B ) @ Yzs ) )
= ( set2 @ ( product_prod @ A @ B )
@ ( concat @ ( product_prod @ A @ B )
@ ( map @ ( product_prod @ A @ C ) @ ( list @ ( product_prod @ A @ B ) )
@ ^ [Xy: product_prod @ A @ C] :
( concat @ ( product_prod @ A @ B )
@ ( map @ ( product_prod @ C @ B ) @ ( list @ ( product_prod @ A @ B ) )
@ ^ [Yz: product_prod @ C @ B] :
( if @ ( list @ ( product_prod @ A @ B ) )
@ ( ( product_snd @ A @ C @ Xy )
= ( product_fst @ C @ B @ Yz ) )
@ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ ( product_fst @ A @ C @ Xy ) @ ( product_snd @ C @ B @ Yz ) ) @ ( nil @ ( product_prod @ A @ B ) ) )
@ ( nil @ ( product_prod @ A @ B ) ) )
@ Yzs ) )
@ Xys ) ) ) ) ).
% set_relcomp
thf(fact_6764_splice__Nil2,axiom,
! [A: $tType,Xs: list @ A] :
( ( splice @ A @ Xs @ ( nil @ A ) )
= Xs ) ).
% splice_Nil2
thf(fact_6765_split__Nil__iff,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( splice @ A @ Xs @ Ys )
= ( nil @ A ) )
= ( ( Xs
= ( nil @ A ) )
& ( Ys
= ( nil @ A ) ) ) ) ).
% split_Nil_iff
thf(fact_6766_splice__in__shuffles,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] : ( member @ ( list @ A ) @ ( splice @ A @ Xs @ Ys ) @ ( shuffles @ A @ Xs @ Ys ) ) ).
% splice_in_shuffles
thf(fact_6767_length__splice,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( size_size @ ( list @ A ) @ ( splice @ A @ Xs @ Ys ) )
= ( plus_plus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ A ) @ Ys ) ) ) ).
% length_splice
thf(fact_6768_splice__replicate,axiom,
! [A: $tType,M: nat,X: A,N: nat] :
( ( splice @ A @ ( replicate @ A @ M @ X ) @ ( replicate @ A @ N @ X ) )
= ( replicate @ A @ ( plus_plus @ nat @ M @ N ) @ X ) ) ).
% splice_replicate
thf(fact_6769_relcomp__fold,axiom,
! [C: $tType,B: $tType,A: $tType,R4: set @ ( product_prod @ A @ B ),S3: set @ ( product_prod @ B @ C )] :
( ( finite_finite2 @ ( product_prod @ A @ B ) @ R4 )
=> ( ( finite_finite2 @ ( product_prod @ B @ C ) @ S3 )
=> ( ( relcomp @ A @ B @ C @ R4 @ S3 )
= ( finite_fold @ ( product_prod @ A @ B ) @ ( set @ ( product_prod @ A @ C ) )
@ ( product_case_prod @ A @ B @ ( ( set @ ( product_prod @ A @ C ) ) > ( set @ ( product_prod @ A @ C ) ) )
@ ^ [X4: A,Y: B,A7: set @ ( product_prod @ A @ C )] :
( finite_fold @ ( product_prod @ B @ C ) @ ( set @ ( product_prod @ A @ C ) )
@ ( product_case_prod @ B @ C @ ( ( set @ ( product_prod @ A @ C ) ) > ( set @ ( product_prod @ A @ C ) ) )
@ ^ [W3: B,Z6: C,A16: set @ ( product_prod @ A @ C )] : ( if @ ( set @ ( product_prod @ A @ C ) ) @ ( Y = W3 ) @ ( insert @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ X4 @ Z6 ) @ A16 ) @ A16 ) )
@ A7
@ S3 ) )
@ ( bot_bot @ ( set @ ( product_prod @ A @ C ) ) )
@ R4 ) ) ) ) ).
% relcomp_fold
thf(fact_6770_relcomp__mono,axiom,
! [A: $tType,C: $tType,B: $tType,R2: set @ ( product_prod @ A @ B ),R: set @ ( product_prod @ A @ B ),S9: set @ ( product_prod @ B @ C ),S: set @ ( product_prod @ B @ C )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R2 @ R )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ B @ C ) ) @ S9 @ S )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ C ) ) @ ( relcomp @ A @ B @ C @ R2 @ S9 ) @ ( relcomp @ A @ B @ C @ R @ S ) ) ) ) ).
% relcomp_mono
thf(fact_6771_splice_Osimps_I2_J,axiom,
! [A: $tType,X: A,Xs: list @ A,Ys: list @ A] :
( ( splice @ A @ ( cons @ A @ X @ Xs ) @ Ys )
= ( cons @ A @ X @ ( splice @ A @ Ys @ Xs ) ) ) ).
% splice.simps(2)
thf(fact_6772_relcomp__UNION__distrib,axiom,
! [A: $tType,C: $tType,B: $tType,D: $tType,S: set @ ( product_prod @ A @ C ),R: D > ( set @ ( product_prod @ C @ B ) ),I6: set @ D] :
( ( relcomp @ A @ C @ B @ S @ ( complete_Sup_Sup @ ( set @ ( product_prod @ C @ B ) ) @ ( image @ D @ ( set @ ( product_prod @ C @ B ) ) @ R @ I6 ) ) )
= ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) )
@ ( image @ D @ ( set @ ( product_prod @ A @ B ) )
@ ^ [I: D] : ( relcomp @ A @ C @ B @ S @ ( R @ I ) )
@ I6 ) ) ) ).
% relcomp_UNION_distrib
thf(fact_6773_relcomp__UNION__distrib2,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,R: D > ( set @ ( product_prod @ A @ C ) ),I6: set @ D,S: set @ ( product_prod @ C @ B )] :
( ( relcomp @ A @ C @ B @ ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ C ) ) @ ( image @ D @ ( set @ ( product_prod @ A @ C ) ) @ R @ I6 ) ) @ S )
= ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) )
@ ( image @ D @ ( set @ ( product_prod @ A @ B ) )
@ ^ [I: D] : ( relcomp @ A @ C @ B @ ( R @ I ) @ S )
@ I6 ) ) ) ).
% relcomp_UNION_distrib2
thf(fact_6774_splice_Osimps_I1_J,axiom,
! [A: $tType,Ys: list @ A] :
( ( splice @ A @ ( nil @ A ) @ Ys )
= Ys ) ).
% splice.simps(1)
thf(fact_6775_splice_Oelims,axiom,
! [A: $tType,X: list @ A,Xa: list @ A,Y2: list @ A] :
( ( ( splice @ A @ X @ Xa )
= Y2 )
=> ( ( ( X
= ( nil @ A ) )
=> ( Y2 != Xa ) )
=> ~ ! [X5: A,Xs3: list @ A] :
( ( X
= ( cons @ A @ X5 @ Xs3 ) )
=> ( Y2
!= ( cons @ A @ X5 @ ( splice @ A @ Xa @ Xs3 ) ) ) ) ) ) ).
% splice.elims
thf(fact_6776_insert__relcomp__fold,axiom,
! [C: $tType,B: $tType,A: $tType,S3: set @ ( product_prod @ A @ B ),X: product_prod @ C @ A,R4: set @ ( product_prod @ C @ A )] :
( ( finite_finite2 @ ( product_prod @ A @ B ) @ S3 )
=> ( ( relcomp @ C @ A @ B @ ( insert @ ( product_prod @ C @ A ) @ X @ R4 ) @ S3 )
= ( finite_fold @ ( product_prod @ A @ B ) @ ( set @ ( product_prod @ C @ B ) )
@ ( product_case_prod @ A @ B @ ( ( set @ ( product_prod @ C @ B ) ) > ( set @ ( product_prod @ C @ B ) ) )
@ ^ [W3: A,Z6: B,A16: set @ ( product_prod @ C @ B )] :
( if @ ( set @ ( product_prod @ C @ B ) )
@ ( ( product_snd @ C @ A @ X )
= W3 )
@ ( insert @ ( product_prod @ C @ B ) @ ( product_Pair @ C @ B @ ( product_fst @ C @ A @ X ) @ Z6 ) @ A16 )
@ A16 ) )
@ ( relcomp @ C @ A @ B @ R4 @ S3 )
@ S3 ) ) ) ).
% insert_relcomp_fold
thf(fact_6777_insert__relcomp__union__fold,axiom,
! [C: $tType,B: $tType,A: $tType,S3: set @ ( product_prod @ A @ B ),X: product_prod @ C @ A,X9: set @ ( product_prod @ C @ B )] :
( ( finite_finite2 @ ( product_prod @ A @ B ) @ S3 )
=> ( ( sup_sup @ ( set @ ( product_prod @ C @ B ) ) @ ( relcomp @ C @ A @ B @ ( insert @ ( product_prod @ C @ A ) @ X @ ( bot_bot @ ( set @ ( product_prod @ C @ A ) ) ) ) @ S3 ) @ X9 )
= ( finite_fold @ ( product_prod @ A @ B ) @ ( set @ ( product_prod @ C @ B ) )
@ ( product_case_prod @ A @ B @ ( ( set @ ( product_prod @ C @ B ) ) > ( set @ ( product_prod @ C @ B ) ) )
@ ^ [W3: A,Z6: B,A16: set @ ( product_prod @ C @ B )] :
( if @ ( set @ ( product_prod @ C @ B ) )
@ ( ( product_snd @ C @ A @ X )
= W3 )
@ ( insert @ ( product_prod @ C @ B ) @ ( product_Pair @ C @ B @ ( product_fst @ C @ A @ X ) @ Z6 ) @ A16 )
@ A16 ) )
@ X9
@ S3 ) ) ) ).
% insert_relcomp_union_fold
thf(fact_6778_splice_Opsimps_I2_J,axiom,
! [A: $tType,X: A,Xs: list @ A,Ys: list @ A] :
( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( splice_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Xs ) @ Ys ) )
=> ( ( splice @ A @ ( cons @ A @ X @ Xs ) @ Ys )
= ( cons @ A @ X @ ( splice @ A @ Ys @ Xs ) ) ) ) ).
% splice.psimps(2)
thf(fact_6779_splice_Opsimps_I1_J,axiom,
! [A: $tType,Ys: list @ A] :
( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( splice_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Ys ) )
=> ( ( splice @ A @ ( nil @ A ) @ Ys )
= Ys ) ) ).
% splice.psimps(1)
thf(fact_6780_Id__on__fold,axiom,
! [A: $tType,A3: set @ A] :
( ( finite_finite2 @ A @ A3 )
=> ( ( id_on @ A @ A3 )
= ( finite_fold @ A @ ( set @ ( product_prod @ A @ A ) )
@ ^ [X4: A] : ( insert @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ X4 ) )
@ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) )
@ A3 ) ) ) ).
% Id_on_fold
thf(fact_6781_extract__Some__iff,axiom,
! [A: $tType,P3: A > $o,Xs: list @ A,Ys: list @ A,Y2: A,Zs: list @ A] :
( ( ( extract @ A @ P3 @ Xs )
= ( some @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) @ ( product_Pair @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) @ Ys @ ( product_Pair @ A @ ( list @ A ) @ Y2 @ Zs ) ) ) )
= ( ( Xs
= ( append @ A @ Ys @ ( cons @ A @ Y2 @ Zs ) ) )
& ( P3 @ Y2 )
& ~ ? [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Ys ) )
& ( P3 @ X4 ) ) ) ) ).
% extract_Some_iff
thf(fact_6782_extract__Nil__code,axiom,
! [A: $tType,P3: A > $o] :
( ( extract @ A @ P3 @ ( nil @ A ) )
= ( none @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) ) ) ).
% extract_Nil_code
thf(fact_6783_Id__on__def_H,axiom,
! [A: $tType,A3: A > $o] :
( ( id_on @ A @ ( collect @ A @ A3 ) )
= ( collect @ ( product_prod @ A @ A )
@ ( product_case_prod @ A @ A @ $o
@ ^ [X4: A,Y: A] :
( ( X4 = Y )
& ( A3 @ X4 ) ) ) ) ) ).
% Id_on_def'
thf(fact_6784_extract__None__iff,axiom,
! [A: $tType,P3: A > $o,Xs: list @ A] :
( ( ( extract @ A @ P3 @ Xs )
= ( none @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) ) )
= ( ~ ? [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
& ( P3 @ X4 ) ) ) ) ).
% extract_None_iff
thf(fact_6785_Id__on__set,axiom,
! [A: $tType,Xs: list @ A] :
( ( id_on @ A @ ( set2 @ A @ Xs ) )
= ( set2 @ ( product_prod @ A @ A )
@ ( map @ A @ ( product_prod @ A @ A )
@ ^ [X4: A] : ( product_Pair @ A @ A @ X4 @ X4 )
@ Xs ) ) ) ).
% Id_on_set
thf(fact_6786_Id__on__def,axiom,
! [A: $tType] :
( ( id_on @ A )
= ( ^ [A7: set @ A] :
( complete_Sup_Sup @ ( set @ ( product_prod @ A @ A ) )
@ ( image @ A @ ( set @ ( product_prod @ A @ A ) )
@ ^ [X4: A] : ( insert @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ X4 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
@ A7 ) ) ) ) ).
% Id_on_def
thf(fact_6787_extract__SomeE,axiom,
! [A: $tType,P3: A > $o,Xs: list @ A,Ys: list @ A,Y2: A,Zs: list @ A] :
( ( ( extract @ A @ P3 @ Xs )
= ( some @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) @ ( product_Pair @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) @ Ys @ ( product_Pair @ A @ ( list @ A ) @ Y2 @ Zs ) ) ) )
=> ( ( Xs
= ( append @ A @ Ys @ ( cons @ A @ Y2 @ Zs ) ) )
& ( P3 @ Y2 )
& ~ ? [X2: A] :
( ( member @ A @ X2 @ ( set2 @ A @ Ys ) )
& ( P3 @ X2 ) ) ) ) ).
% extract_SomeE
thf(fact_6788_extract__Cons__code,axiom,
! [A: $tType,P3: A > $o,X: A,Xs: list @ A] :
( ( ( P3 @ X )
=> ( ( extract @ A @ P3 @ ( cons @ A @ X @ Xs ) )
= ( some @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) @ ( product_Pair @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) @ ( nil @ A ) @ ( product_Pair @ A @ ( list @ A ) @ X @ Xs ) ) ) ) )
& ( ~ ( P3 @ X )
=> ( ( extract @ A @ P3 @ ( cons @ A @ X @ Xs ) )
= ( case_option @ ( option @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) ) @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) @ ( none @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) )
@ ( product_case_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) @ ( option @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) )
@ ^ [Ys3: list @ A] :
( product_case_prod @ A @ ( list @ A ) @ ( option @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) )
@ ^ [Y: A,Zs3: list @ A] : ( some @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) @ ( product_Pair @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) @ ( cons @ A @ X @ Ys3 ) @ ( product_Pair @ A @ ( list @ A ) @ Y @ Zs3 ) ) ) ) )
@ ( extract @ A @ P3 @ Xs ) ) ) ) ) ).
% extract_Cons_code
thf(fact_6789_comp__fun__commute__relcomp__fold,axiom,
! [A: $tType,B: $tType,C: $tType,S3: set @ ( product_prod @ A @ B )] :
( ( finite_finite2 @ ( product_prod @ A @ B ) @ S3 )
=> ( finite6289374366891150609ommute @ ( product_prod @ C @ A ) @ ( set @ ( product_prod @ C @ B ) )
@ ( product_case_prod @ C @ A @ ( ( set @ ( product_prod @ C @ B ) ) > ( set @ ( product_prod @ C @ B ) ) )
@ ^ [X4: C,Y: A,A7: set @ ( product_prod @ C @ B )] :
( finite_fold @ ( product_prod @ A @ B ) @ ( set @ ( product_prod @ C @ B ) )
@ ( product_case_prod @ A @ B @ ( ( set @ ( product_prod @ C @ B ) ) > ( set @ ( product_prod @ C @ B ) ) )
@ ^ [W3: A,Z6: B,A16: set @ ( product_prod @ C @ B )] : ( if @ ( set @ ( product_prod @ C @ B ) ) @ ( Y = W3 ) @ ( insert @ ( product_prod @ C @ B ) @ ( product_Pair @ C @ B @ X4 @ Z6 ) @ A16 ) @ A16 ) )
@ A7
@ S3 ) ) ) ) ).
% comp_fun_commute_relcomp_fold
thf(fact_6790_comp__fun__commute_Ocomp__fun__commute__funpow,axiom,
! [B: $tType,A: $tType,F3: A > B > B,G: A > nat] :
( ( finite6289374366891150609ommute @ A @ B @ F3 )
=> ( finite6289374366891150609ommute @ A @ B
@ ^ [X4: A] : ( compow @ ( B > B ) @ ( G @ X4 ) @ ( F3 @ X4 ) ) ) ) ).
% comp_fun_commute.comp_fun_commute_funpow
thf(fact_6791_disjE__realizer2,axiom,
! [B: $tType,A: $tType,P3: $o,Q2: A > $o,X: option @ A,R4: B > $o,F3: B,G: A > B] :
( ( case_option @ $o @ A @ P3 @ Q2 @ X )
=> ( ( P3
=> ( R4 @ F3 ) )
=> ( ! [Q3: A] :
( ( Q2 @ Q3 )
=> ( R4 @ ( G @ Q3 ) ) )
=> ( R4 @ ( case_option @ B @ A @ F3 @ G @ X ) ) ) ) ) ).
% disjE_realizer2
thf(fact_6792_comp__fun__commute__const,axiom,
! [B: $tType,A: $tType,F3: B > B] :
( finite6289374366891150609ommute @ A @ B
@ ^ [Uu3: A] : F3 ) ).
% comp_fun_commute_const
thf(fact_6793_option_Ocase__distrib,axiom,
! [C: $tType,B: $tType,A: $tType,H2: B > C,F1: B,F2: A > B,Option: option @ A] :
( ( H2 @ ( case_option @ B @ A @ F1 @ F2 @ Option ) )
= ( case_option @ C @ A @ ( H2 @ F1 )
@ ^ [X4: A] : ( H2 @ ( F2 @ X4 ) )
@ Option ) ) ).
% option.case_distrib
thf(fact_6794_option_Osimps_I4_J,axiom,
! [A: $tType,B: $tType,F1: B,F2: A > B] :
( ( case_option @ B @ A @ F1 @ F2 @ ( none @ A ) )
= F1 ) ).
% option.simps(4)
thf(fact_6795_comp__fun__commute__filter__fold,axiom,
! [A: $tType,P3: A > $o] :
( finite6289374366891150609ommute @ A @ ( set @ A )
@ ^ [X4: A,A16: set @ A] : ( if @ ( set @ A ) @ ( P3 @ X4 ) @ ( insert @ A @ X4 @ A16 ) @ A16 ) ) ).
% comp_fun_commute_filter_fold
thf(fact_6796_comp__fun__commute__Image__fold,axiom,
! [B: $tType,A: $tType,S3: set @ A] :
( finite6289374366891150609ommute @ ( product_prod @ A @ B ) @ ( set @ B )
@ ( product_case_prod @ A @ B @ ( ( set @ B ) > ( set @ B ) )
@ ^ [X4: A,Y: B,A7: set @ B] : ( if @ ( set @ B ) @ ( member @ A @ X4 @ S3 ) @ ( insert @ B @ Y @ A7 ) @ A7 ) ) ) ).
% comp_fun_commute_Image_fold
thf(fact_6797_comp__fun__commute__product__fold,axiom,
! [A: $tType,B: $tType,B5: set @ A] :
( ( finite_finite2 @ A @ B5 )
=> ( finite6289374366891150609ommute @ B @ ( set @ ( product_prod @ B @ A ) )
@ ^ [X4: B,Z6: set @ ( product_prod @ B @ A )] :
( finite_fold @ A @ ( set @ ( product_prod @ B @ A ) )
@ ^ [Y: A] : ( insert @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X4 @ Y ) )
@ Z6
@ B5 ) ) ) ).
% comp_fun_commute_product_fold
thf(fact_6798_extract__def,axiom,
! [A: $tType] :
( ( extract @ A )
= ( ^ [P2: A > $o,Xs2: list @ A] :
( case_list @ ( option @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) ) @ A @ ( none @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) )
@ ^ [Y: A,Ys3: list @ A] : ( some @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) @ ( product_Pair @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) @ ( takeWhile @ A @ ( comp @ $o @ $o @ A @ (~) @ P2 ) @ Xs2 ) @ ( product_Pair @ A @ ( list @ A ) @ Y @ Ys3 ) ) )
@ ( dropWhile @ A @ ( comp @ $o @ $o @ A @ (~) @ P2 ) @ Xs2 ) ) ) ) ).
% extract_def
thf(fact_6799_image__Collect__subsetI,axiom,
! [A: $tType,B: $tType,P3: A > $o,F3: A > B,B5: set @ B] :
( ! [X5: A] :
( ( P3 @ X5 )
=> ( member @ B @ ( F3 @ X5 ) @ B5 ) )
=> ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F3 @ ( collect @ A @ P3 ) ) @ B5 ) ) ).
% image_Collect_subsetI
thf(fact_6800_dropWhile__idem,axiom,
! [A: $tType,P3: A > $o,Xs: list @ A] :
( ( dropWhile @ A @ P3 @ ( dropWhile @ A @ P3 @ Xs ) )
= ( dropWhile @ A @ P3 @ Xs ) ) ).
% dropWhile_idem
thf(fact_6801_dropWhile__eq__Nil__conv,axiom,
! [A: $tType,P3: A > $o,Xs: list @ A] :
( ( ( dropWhile @ A @ P3 @ Xs )
= ( nil @ A ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
=> ( P3 @ X4 ) ) ) ) ).
% dropWhile_eq_Nil_conv
thf(fact_6802_dropWhile__append2,axiom,
! [A: $tType,Xs: list @ A,P3: A > $o,Ys: list @ A] :
( ! [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Xs ) )
=> ( P3 @ X5 ) )
=> ( ( dropWhile @ A @ P3 @ ( append @ A @ Xs @ Ys ) )
= ( dropWhile @ A @ P3 @ Ys ) ) ) ).
% dropWhile_append2
thf(fact_6803_dropWhile__append1,axiom,
! [A: $tType,X: A,Xs: list @ A,P3: A > $o,Ys: list @ A] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ~ ( P3 @ X )
=> ( ( dropWhile @ A @ P3 @ ( append @ A @ Xs @ Ys ) )
= ( append @ A @ ( dropWhile @ A @ P3 @ Xs ) @ Ys ) ) ) ) ).
% dropWhile_append1
thf(fact_6804_dropWhile__replicate,axiom,
! [A: $tType,P3: A > $o,X: A,N: nat] :
( ( ( P3 @ X )
=> ( ( dropWhile @ A @ P3 @ ( replicate @ A @ N @ X ) )
= ( nil @ A ) ) )
& ( ~ ( P3 @ X )
=> ( ( dropWhile @ A @ P3 @ ( replicate @ A @ N @ X ) )
= ( replicate @ A @ N @ X ) ) ) ) ).
% dropWhile_replicate
thf(fact_6805_takeWhile__dropWhile__id,axiom,
! [A: $tType,P3: A > $o,Xs: list @ A] :
( ( append @ A @ ( takeWhile @ A @ P3 @ Xs ) @ ( dropWhile @ A @ P3 @ Xs ) )
= Xs ) ).
% takeWhile_dropWhile_id
thf(fact_6806_option_Odisc__eq__case_I2_J,axiom,
! [A: $tType,Option: option @ A] :
( ( Option
!= ( none @ A ) )
= ( case_option @ $o @ A @ $false
@ ^ [Uu3: A] : $true
@ Option ) ) ).
% option.disc_eq_case(2)
thf(fact_6807_option_Odisc__eq__case_I1_J,axiom,
! [A: $tType,Option: option @ A] :
( ( Option
= ( none @ A ) )
= ( case_option @ $o @ A @ $true
@ ^ [Uu3: A] : $false
@ Option ) ) ).
% option.disc_eq_case(1)
thf(fact_6808_comp__fun__commute__insort,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( finite6289374366891150609ommute @ A @ ( list @ A )
@ ( linorder_insort_key @ A @ A
@ ^ [X4: A] : X4 ) ) ) ).
% comp_fun_commute_insort
thf(fact_6809_dropWhile__append3,axiom,
! [A: $tType,P3: A > $o,Y2: A,Xs: list @ A,Ys: list @ A] :
( ~ ( P3 @ Y2 )
=> ( ( dropWhile @ A @ P3 @ ( append @ A @ Xs @ ( cons @ A @ Y2 @ Ys ) ) )
= ( append @ A @ ( dropWhile @ A @ P3 @ Xs ) @ ( cons @ A @ Y2 @ Ys ) ) ) ) ).
% dropWhile_append3
thf(fact_6810_dropWhile__map,axiom,
! [A: $tType,B: $tType,P3: A > $o,F3: B > A,Xs: list @ B] :
( ( dropWhile @ A @ P3 @ ( map @ B @ A @ F3 @ Xs ) )
= ( map @ B @ A @ F3 @ ( dropWhile @ B @ ( comp @ A @ $o @ B @ P3 @ F3 ) @ Xs ) ) ) ).
% dropWhile_map
thf(fact_6811_dropWhile_Osimps_I2_J,axiom,
! [A: $tType,P3: A > $o,X: A,Xs: list @ A] :
( ( ( P3 @ X )
=> ( ( dropWhile @ A @ P3 @ ( cons @ A @ X @ Xs ) )
= ( dropWhile @ A @ P3 @ Xs ) ) )
& ( ~ ( P3 @ X )
=> ( ( dropWhile @ A @ P3 @ ( cons @ A @ X @ Xs ) )
= ( cons @ A @ X @ Xs ) ) ) ) ).
% dropWhile.simps(2)
thf(fact_6812_hd__dropWhile,axiom,
! [A: $tType,P3: A > $o,Xs: list @ A] :
( ( ( dropWhile @ A @ P3 @ Xs )
!= ( nil @ A ) )
=> ~ ( P3 @ ( hd @ A @ ( dropWhile @ A @ P3 @ Xs ) ) ) ) ).
% hd_dropWhile
thf(fact_6813_dropWhile__eq__self__iff,axiom,
! [A: $tType,P3: A > $o,Xs: list @ A] :
( ( ( dropWhile @ A @ P3 @ Xs )
= Xs )
= ( ( Xs
= ( nil @ A ) )
| ~ ( P3 @ ( hd @ A @ Xs ) ) ) ) ).
% dropWhile_eq_self_iff
thf(fact_6814_dropWhile_Osimps_I1_J,axiom,
! [A: $tType,P3: A > $o] :
( ( dropWhile @ A @ P3 @ ( nil @ A ) )
= ( nil @ A ) ) ).
% dropWhile.simps(1)
thf(fact_6815_sorted__dropWhile,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,P3: A > $o] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( dropWhile @ A @ P3 @ Xs ) ) ) ) ).
% sorted_dropWhile
thf(fact_6816_length__dropWhile__le,axiom,
! [A: $tType,P3: A > $o,Xs: list @ A] : ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ ( dropWhile @ A @ P3 @ Xs ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) ).
% length_dropWhile_le
thf(fact_6817_set__dropWhileD,axiom,
! [A: $tType,X: A,P3: A > $o,Xs: list @ A] :
( ( member @ A @ X @ ( set2 @ A @ ( dropWhile @ A @ P3 @ Xs ) ) )
=> ( member @ A @ X @ ( set2 @ A @ Xs ) ) ) ).
% set_dropWhileD
thf(fact_6818_dropWhile__cong,axiom,
! [A: $tType,L: list @ A,K2: list @ A,P3: A > $o,Q2: A > $o] :
( ( L = K2 )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ L ) )
=> ( ( P3 @ X5 )
= ( Q2 @ X5 ) ) )
=> ( ( dropWhile @ A @ P3 @ L )
= ( dropWhile @ A @ Q2 @ K2 ) ) ) ) ).
% dropWhile_cong
thf(fact_6819_distinct__dropWhile,axiom,
! [A: $tType,Xs: list @ A,P3: A > $o] :
( ( distinct @ A @ Xs )
=> ( distinct @ A @ ( dropWhile @ A @ P3 @ Xs ) ) ) ).
% distinct_dropWhile
thf(fact_6820_case__optionE,axiom,
! [A: $tType,P3: $o,Q2: A > $o,X: option @ A] :
( ( case_option @ $o @ A @ P3 @ Q2 @ X )
=> ( ( ( X
= ( none @ A ) )
=> ~ P3 )
=> ~ ! [Y7: A] :
( ( X
= ( some @ A @ Y7 ) )
=> ~ ( Q2 @ Y7 ) ) ) ) ).
% case_optionE
thf(fact_6821_dropWhile__eq__Cons__conv,axiom,
! [A: $tType,P3: A > $o,Xs: list @ A,Y2: A,Ys: list @ A] :
( ( ( dropWhile @ A @ P3 @ Xs )
= ( cons @ A @ Y2 @ Ys ) )
= ( ( Xs
= ( append @ A @ ( takeWhile @ A @ P3 @ Xs ) @ ( cons @ A @ Y2 @ Ys ) ) )
& ~ ( P3 @ Y2 ) ) ) ).
% dropWhile_eq_Cons_conv
thf(fact_6822_takeWhile__eq__filter,axiom,
! [A: $tType,P3: A > $o,Xs: list @ A] :
( ! [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ ( dropWhile @ A @ P3 @ Xs ) ) )
=> ~ ( P3 @ X5 ) )
=> ( ( takeWhile @ A @ P3 @ Xs )
= ( filter2 @ A @ P3 @ Xs ) ) ) ).
% takeWhile_eq_filter
thf(fact_6823_dropWhile__eq__drop,axiom,
! [A: $tType] :
( ( dropWhile @ A )
= ( ^ [P2: A > $o,Xs2: list @ A] : ( drop @ A @ ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P2 @ Xs2 ) ) @ Xs2 ) ) ) ).
% dropWhile_eq_drop
thf(fact_6824_map__filter__simps_I1_J,axiom,
! [A: $tType,B: $tType,F3: B > ( option @ A ),X: B,Xs: list @ B] :
( ( map_filter @ B @ A @ F3 @ ( cons @ B @ X @ Xs ) )
= ( case_option @ ( list @ A ) @ A @ ( map_filter @ B @ A @ F3 @ Xs )
@ ^ [Y: A] : ( cons @ A @ Y @ ( map_filter @ B @ A @ F3 @ Xs ) )
@ ( F3 @ X ) ) ) ).
% map_filter_simps(1)
thf(fact_6825_Collect__restrict,axiom,
! [A: $tType,X9: set @ A,P3: A > $o] :
( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ X9 )
& ( P3 @ X4 ) ) )
@ X9 ) ).
% Collect_restrict
thf(fact_6826_prop__restrict,axiom,
! [A: $tType,X: A,Z8: set @ A,X9: set @ A,P3: A > $o] :
( ( member @ A @ X @ Z8 )
=> ( ( ord_less_eq @ ( set @ A ) @ Z8
@ ( collect @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ X9 )
& ( P3 @ X4 ) ) ) )
=> ( P3 @ X ) ) ) ).
% prop_restrict
thf(fact_6827_dropWhile__nth,axiom,
! [A: $tType,J3: nat,P3: A > $o,Xs: list @ A] :
( ( ord_less @ nat @ J3 @ ( size_size @ ( list @ A ) @ ( dropWhile @ A @ P3 @ Xs ) ) )
=> ( ( nth @ A @ ( dropWhile @ A @ P3 @ Xs ) @ J3 )
= ( nth @ A @ Xs @ ( plus_plus @ nat @ J3 @ ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P3 @ Xs ) ) ) ) ) ) ).
% dropWhile_nth
thf(fact_6828_comp__fun__commute__Pow__fold,axiom,
! [A: $tType] :
( finite6289374366891150609ommute @ A @ ( set @ ( set @ A ) )
@ ^ [X4: A,A7: set @ ( set @ A )] : ( sup_sup @ ( set @ ( set @ A ) ) @ A7 @ ( image @ ( set @ A ) @ ( set @ A ) @ ( insert @ A @ X4 ) @ A7 ) ) ) ).
% comp_fun_commute_Pow_fold
thf(fact_6829_dropWhile__neq__rev,axiom,
! [A: $tType,Xs: list @ A,X: A] :
( ( distinct @ A @ Xs )
=> ( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ( dropWhile @ A
@ ^ [Y: A] : Y != X
@ ( rev @ A @ Xs ) )
= ( cons @ A @ X
@ ( rev @ A
@ ( takeWhile @ A
@ ^ [Y: A] : Y != X
@ Xs ) ) ) ) ) ) ).
% dropWhile_neq_rev
thf(fact_6830_takeWhile__neq__rev,axiom,
! [A: $tType,Xs: list @ A,X: A] :
( ( distinct @ A @ Xs )
=> ( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ( takeWhile @ A
@ ^ [Y: A] : Y != X
@ ( rev @ A @ Xs ) )
= ( rev @ A
@ ( tl @ A
@ ( dropWhile @ A
@ ^ [Y: A] : Y != X
@ Xs ) ) ) ) ) ) ).
% takeWhile_neq_rev
thf(fact_6831_subset__emptyI,axiom,
! [A: $tType,A3: set @ A] :
( ! [X5: A] :
~ ( member @ A @ X5 @ A3 )
=> ( ord_less_eq @ ( set @ A ) @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% subset_emptyI
thf(fact_6832_insert__subsetI,axiom,
! [A: $tType,X: A,A3: set @ A,X9: set @ A] :
( ( member @ A @ X @ A3 )
=> ( ( ord_less_eq @ ( set @ A ) @ X9 @ A3 )
=> ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X @ X9 ) @ A3 ) ) ) ).
% insert_subsetI
thf(fact_6833_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 )
@ ^ [Q5: 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 @ Q5 ) ) )
@ ( bit_take_bit_num @ ( numeral_numeral @ nat @ M ) @ N ) ) ) ).
% take_bit_numeral_minus_numeral_int
thf(fact_6834_find__dropWhile,axiom,
! [A: $tType] :
( ( find @ A )
= ( ^ [P2: A > $o,Xs2: list @ A] :
( case_list @ ( option @ A ) @ A @ ( none @ A )
@ ^ [X4: A,Xa4: list @ A] : ( some @ A @ X4 )
@ ( dropWhile @ A @ ( comp @ $o @ $o @ A @ (~) @ P2 ) @ Xs2 ) ) ) ) ).
% find_dropWhile
thf(fact_6835_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_6836_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_6837_take__bit__num__simps_I5_J,axiom,
! [R: num] :
( ( bit_take_bit_num @ ( numeral_numeral @ nat @ R ) @ one2 )
= ( some @ num @ one2 ) ) ).
% take_bit_num_simps(5)
thf(fact_6838_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 )
@ ^ [Q5: num] : ( some @ num @ ( bit0 @ Q5 ) )
@ ( bit_take_bit_num @ N @ M ) ) ) ).
% take_bit_num_simps(3)
thf(fact_6839_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_6840_take__bit__num__simps_I6_J,axiom,
! [R: num,M: num] :
( ( bit_take_bit_num @ ( numeral_numeral @ nat @ R ) @ ( bit0 @ M ) )
= ( case_option @ ( option @ num ) @ num @ ( none @ num )
@ ^ [Q5: num] : ( some @ num @ ( bit0 @ Q5 ) )
@ ( bit_take_bit_num @ ( pred_numeral @ R ) @ M ) ) ) ).
% take_bit_num_simps(6)
thf(fact_6841_take__bit__num__simps_I7_J,axiom,
! [R: num,M: num] :
( ( bit_take_bit_num @ ( numeral_numeral @ nat @ R ) @ ( bit1 @ M ) )
= ( some @ num @ ( case_option @ num @ num @ one2 @ bit1 @ ( bit_take_bit_num @ ( pred_numeral @ R ) @ M ) ) ) ) ).
% take_bit_num_simps(7)
thf(fact_6842_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_6843_Code__Abstract__Nat_Otake__bit__num__code_I2_J,axiom,
! [N: nat,M: num] :
( ( bit_take_bit_num @ N @ ( bit0 @ M ) )
= ( case_nat @ ( option @ num ) @ ( none @ num )
@ ^ [N2: nat] :
( case_option @ ( option @ num ) @ num @ ( none @ num )
@ ^ [Q5: num] : ( some @ num @ ( bit0 @ Q5 ) )
@ ( bit_take_bit_num @ N2 @ M ) )
@ N ) ) ).
% Code_Abstract_Nat.take_bit_num_code(2)
thf(fact_6844_find__cong,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,P3: A > $o,Q2: A > $o] :
( ( Xs = Ys )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Ys ) )
=> ( ( P3 @ X5 )
= ( Q2 @ X5 ) ) )
=> ( ( find @ A @ P3 @ Xs )
= ( find @ A @ Q2 @ Ys ) ) ) ) ).
% find_cong
thf(fact_6845_Code__Abstract__Nat_Otake__bit__num__code_I1_J,axiom,
! [N: nat] :
( ( bit_take_bit_num @ N @ one2 )
= ( case_nat @ ( option @ num ) @ ( none @ num )
@ ^ [N2: nat] : ( some @ num @ one2 )
@ N ) ) ).
% Code_Abstract_Nat.take_bit_num_code(1)
thf(fact_6846_take__bit__num__eq__Some__imp,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M: nat,N: num,Q: num] :
( ( ( bit_take_bit_num @ M @ N )
= ( some @ num @ Q ) )
=> ( ( bit_se2584673776208193580ke_bit @ A @ M @ ( numeral_numeral @ A @ N ) )
= ( numeral_numeral @ A @ Q ) ) ) ) ).
% take_bit_num_eq_Some_imp
thf(fact_6847_Code__Abstract__Nat_Otake__bit__num__code_I3_J,axiom,
! [N: nat,M: num] :
( ( bit_take_bit_num @ N @ ( bit1 @ M ) )
= ( case_nat @ ( option @ num ) @ ( none @ num )
@ ^ [N2: nat] : ( some @ num @ ( case_option @ num @ num @ one2 @ bit1 @ ( bit_take_bit_num @ N2 @ M ) ) )
@ N ) ) ).
% Code_Abstract_Nat.take_bit_num_code(3)
thf(fact_6848_find_Osimps_I2_J,axiom,
! [A: $tType,P3: A > $o,X: A,Xs: list @ A] :
( ( ( P3 @ X )
=> ( ( find @ A @ P3 @ ( cons @ A @ X @ Xs ) )
= ( some @ A @ X ) ) )
& ( ~ ( P3 @ X )
=> ( ( find @ A @ P3 @ ( cons @ A @ X @ Xs ) )
= ( find @ A @ P3 @ Xs ) ) ) ) ).
% find.simps(2)
thf(fact_6849_find_Osimps_I1_J,axiom,
! [A: $tType,Uu: A > $o] :
( ( find @ A @ Uu @ ( nil @ A ) )
= ( none @ A ) ) ).
% find.simps(1)
thf(fact_6850_find__None__iff2,axiom,
! [A: $tType,P3: A > $o,Xs: list @ A] :
( ( ( none @ A )
= ( find @ A @ P3 @ Xs ) )
= ( ~ ? [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
& ( P3 @ X4 ) ) ) ) ).
% find_None_iff2
thf(fact_6851_find__None__iff,axiom,
! [A: $tType,P3: A > $o,Xs: list @ A] :
( ( ( find @ A @ P3 @ Xs )
= ( none @ A ) )
= ( ~ ? [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
& ( P3 @ X4 ) ) ) ) ).
% find_None_iff
thf(fact_6852_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_6853_find__Some__iff,axiom,
! [A: $tType,P3: A > $o,Xs: list @ A,X: A] :
( ( ( find @ A @ P3 @ Xs )
= ( some @ A @ X ) )
= ( ? [I: nat] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
& ( P3 @ ( nth @ A @ Xs @ I ) )
& ( X
= ( nth @ A @ Xs @ I ) )
& ! [J4: nat] :
( ( ord_less @ nat @ J4 @ I )
=> ~ ( P3 @ ( nth @ A @ Xs @ J4 ) ) ) ) ) ) ).
% find_Some_iff
thf(fact_6854_find__Some__iff2,axiom,
! [A: $tType,X: A,P3: A > $o,Xs: list @ A] :
( ( ( some @ A @ X )
= ( find @ A @ P3 @ Xs ) )
= ( ? [I: nat] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
& ( P3 @ ( nth @ A @ Xs @ I ) )
& ( X
= ( nth @ A @ Xs @ I ) )
& ! [J4: nat] :
( ( ord_less @ nat @ J4 @ I )
=> ~ ( P3 @ ( nth @ A @ Xs @ J4 ) ) ) ) ) ) ).
% find_Some_iff2
thf(fact_6855_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_6856_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_6857_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_6858_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_6859_and__not__num_Osimps_I3_J,axiom,
! [N: num] :
( ( bit_and_not_num @ one2 @ ( bit1 @ N ) )
= ( none @ num ) ) ).
% and_not_num.simps(3)
thf(fact_6860_and__not__num_Osimps_I1_J,axiom,
( ( bit_and_not_num @ one2 @ one2 )
= ( none @ num ) ) ).
% and_not_num.simps(1)
thf(fact_6861_and__not__num_Osimps_I2_J,axiom,
! [N: num] :
( ( bit_and_not_num @ one2 @ ( bit0 @ N ) )
= ( some @ num @ one2 ) ) ).
% and_not_num.simps(2)
thf(fact_6862_and__not__num_Osimps_I4_J,axiom,
! [M: num] :
( ( bit_and_not_num @ ( bit0 @ M ) @ one2 )
= ( some @ num @ ( bit0 @ M ) ) ) ).
% and_not_num.simps(4)
thf(fact_6863_and__not__num_Osimps_I7_J,axiom,
! [M: num] :
( ( bit_and_not_num @ ( bit1 @ M ) @ one2 )
= ( some @ num @ ( bit0 @ M ) ) ) ).
% and_not_num.simps(7)
thf(fact_6864_and__not__num__eq__Some__iff,axiom,
! [M: num,N: num,Q: num] :
( ( ( bit_and_not_num @ M @ N )
= ( some @ num @ Q ) )
= ( ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ M ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ N ) ) )
= ( numeral_numeral @ int @ Q ) ) ) ).
% and_not_num_eq_Some_iff
thf(fact_6865_and__not__num_Osimps_I8_J,axiom,
! [M: num,N: num] :
( ( bit_and_not_num @ ( bit1 @ M ) @ ( bit0 @ N ) )
= ( case_option @ ( option @ num ) @ num @ ( some @ num @ one2 )
@ ^ [N11: num] : ( some @ num @ ( bit1 @ N11 ) )
@ ( bit_and_not_num @ M @ N ) ) ) ).
% and_not_num.simps(8)
thf(fact_6866_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_6867_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_6868_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_6869_Bit__Operations_Otake__bit__num__code,axiom,
( bit_take_bit_num
= ( ^ [N2: nat,M2: num] :
( product_case_prod @ nat @ num @ ( option @ num )
@ ^ [A5: nat,X4: num] :
( case_nat @ ( option @ num ) @ ( none @ num )
@ ^ [O: nat] :
( case_num @ ( option @ num ) @ ( some @ num @ one2 )
@ ^ [P4: num] :
( case_option @ ( option @ num ) @ num @ ( none @ num )
@ ^ [Q5: num] : ( some @ num @ ( bit0 @ Q5 ) )
@ ( bit_take_bit_num @ O @ P4 ) )
@ ^ [P4: num] : ( some @ num @ ( case_option @ num @ num @ one2 @ bit1 @ ( bit_take_bit_num @ O @ P4 ) ) )
@ X4 )
@ A5 )
@ ( product_Pair @ nat @ num @ N2 @ M2 ) ) ) ) ).
% Bit_Operations.take_bit_num_code
thf(fact_6870_take__bit__num__def,axiom,
( bit_take_bit_num
= ( ^ [N2: nat,M2: num] :
( if @ ( option @ num )
@ ( ( bit_se2584673776208193580ke_bit @ nat @ N2 @ ( numeral_numeral @ nat @ M2 ) )
= ( zero_zero @ nat ) )
@ ( none @ num )
@ ( some @ num @ ( num_of_nat @ ( bit_se2584673776208193580ke_bit @ nat @ N2 @ ( numeral_numeral @ nat @ M2 ) ) ) ) ) ) ) ).
% take_bit_num_def
thf(fact_6871_num__of__nat__numeral__eq,axiom,
! [Q: num] :
( ( num_of_nat @ ( numeral_numeral @ nat @ Q ) )
= Q ) ).
% num_of_nat_numeral_eq
thf(fact_6872_verit__eq__simplify_I18_J,axiom,
! [A: $tType,F1: A,F2: num > A,F32: num > A,X32: num] :
( ( case_num @ A @ F1 @ F2 @ F32 @ ( bit1 @ X32 ) )
= ( F32 @ X32 ) ) ).
% verit_eq_simplify(18)
thf(fact_6873_verit__eq__simplify_I17_J,axiom,
! [A: $tType,F1: A,F2: num > A,F32: num > A,X23: num] :
( ( case_num @ A @ F1 @ F2 @ F32 @ ( bit0 @ X23 ) )
= ( F2 @ X23 ) ) ).
% verit_eq_simplify(17)
thf(fact_6874_verit__eq__simplify_I16_J,axiom,
! [A: $tType,F1: A,F2: num > A,F32: num > A] :
( ( case_num @ A @ F1 @ F2 @ F32 @ one2 )
= F1 ) ).
% verit_eq_simplify(16)
thf(fact_6875_num_Ocase__distrib,axiom,
! [B: $tType,A: $tType,H2: A > B,F1: A,F2: num > A,F32: num > A,Num: num] :
( ( H2 @ ( case_num @ A @ F1 @ F2 @ F32 @ Num ) )
= ( case_num @ B @ ( H2 @ F1 )
@ ^ [X4: num] : ( H2 @ ( F2 @ X4 ) )
@ ^ [X4: num] : ( H2 @ ( F32 @ X4 ) )
@ Num ) ) ).
% num.case_distrib
thf(fact_6876_num__of__nat_Osimps_I1_J,axiom,
( ( num_of_nat @ ( zero_zero @ nat ) )
= one2 ) ).
% num_of_nat.simps(1)
thf(fact_6877_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_6878_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_6879_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_6880_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_6881_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_6882_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_6883_partition__filter__conv,axiom,
! [A: $tType] :
( ( partition @ A )
= ( ^ [F5: A > $o,Xs2: list @ A] : ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( filter2 @ A @ F5 @ Xs2 ) @ ( filter2 @ A @ ( comp @ $o @ $o @ A @ (~) @ F5 ) @ Xs2 ) ) ) ) ).
% partition_filter_conv
thf(fact_6884_DERIV__real__root__generic,axiom,
! [N: nat,X: real,D4: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( X
!= ( zero_zero @ real ) )
=> ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( D4
= ( inverse_inverse @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ ( power_power @ real @ ( root @ N @ X ) @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ) )
=> ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( ord_less @ real @ X @ ( zero_zero @ real ) )
=> ( D4
= ( uminus_uminus @ real @ ( inverse_inverse @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ ( power_power @ real @ ( root @ N @ X ) @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ) ) )
=> ( ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( D4
= ( inverse_inverse @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ ( power_power @ real @ ( root @ N @ X ) @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) )
=> ( has_field_derivative @ real @ ( root @ N ) @ D4 @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ) ) ).
% DERIV_real_root_generic
thf(fact_6885_DERIV__mirror,axiom,
! [F3: real > real,Y2: real,X: real] :
( ( has_field_derivative @ real @ F3 @ Y2 @ ( topolo174197925503356063within @ real @ ( uminus_uminus @ real @ X ) @ ( top_top @ ( set @ real ) ) ) )
= ( has_field_derivative @ real
@ ^ [X4: real] : ( F3 @ ( uminus_uminus @ real @ X4 ) )
@ ( uminus_uminus @ real @ Y2 )
@ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) ) ) ).
% DERIV_mirror
thf(fact_6886_DERIV__fun__sin,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [G: A > A,M: A,X: A] :
( ( has_field_derivative @ A @ G @ M @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) )
=> ( has_field_derivative @ A
@ ^ [X4: A] : ( sin @ A @ ( G @ X4 ) )
@ ( times_times @ A @ ( cos @ A @ ( G @ X ) ) @ M )
@ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% DERIV_fun_sin
thf(fact_6887_DERIV__chain_H,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,D4: A,X: A,S: set @ A,G: A > A,E5: A] :
( ( has_field_derivative @ A @ F3 @ D4 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( ( has_field_derivative @ A @ G @ E5 @ ( topolo174197925503356063within @ A @ ( F3 @ X ) @ ( top_top @ ( set @ A ) ) ) )
=> ( has_field_derivative @ A
@ ^ [X4: A] : ( G @ ( F3 @ X4 ) )
@ ( times_times @ A @ E5 @ D4 )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ).
% DERIV_chain'
thf(fact_6888_DERIV__chain2,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,Da: A,G: A > A,X: A,Db: A,S: set @ A] :
( ( has_field_derivative @ A @ F3 @ Da @ ( topolo174197925503356063within @ A @ ( G @ X ) @ ( top_top @ ( set @ A ) ) ) )
=> ( ( has_field_derivative @ A @ G @ Db @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( has_field_derivative @ A
@ ^ [X4: A] : ( F3 @ ( G @ X4 ) )
@ ( times_times @ A @ Da @ Db )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ).
% DERIV_chain2
thf(fact_6889_DERIV__chain3,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [G: A > A,G5: A > A,F3: A > A,F9: A,X: A] :
( ! [X5: A] : ( has_field_derivative @ A @ G @ ( G5 @ X5 ) @ ( topolo174197925503356063within @ A @ X5 @ ( top_top @ ( set @ A ) ) ) )
=> ( ( has_field_derivative @ A @ F3 @ F9 @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) )
=> ( has_field_derivative @ A
@ ^ [X4: A] : ( G @ ( F3 @ X4 ) )
@ ( times_times @ A @ F9 @ ( G5 @ ( F3 @ X ) ) )
@ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% DERIV_chain3
thf(fact_6890_DERIV__chain__s,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [S: set @ A,G: A > A,G5: A > A,F3: A > A,F9: A,X: A] :
( ! [X5: A] :
( ( member @ A @ X5 @ S )
=> ( has_field_derivative @ A @ G @ ( G5 @ X5 ) @ ( topolo174197925503356063within @ A @ X5 @ ( top_top @ ( set @ A ) ) ) ) )
=> ( ( has_field_derivative @ A @ F3 @ F9 @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) )
=> ( ( member @ A @ ( F3 @ X ) @ S )
=> ( has_field_derivative @ A
@ ^ [X4: A] : ( G @ ( F3 @ X4 ) )
@ ( times_times @ A @ F9 @ ( G5 @ ( F3 @ X ) ) )
@ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ).
% DERIV_chain_s
thf(fact_6891_DERIV__fun__exp,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [G: A > A,M: A,X: A] :
( ( has_field_derivative @ A @ G @ M @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) )
=> ( has_field_derivative @ A
@ ^ [X4: A] : ( exp @ A @ ( G @ X4 ) )
@ ( times_times @ A @ ( exp @ A @ ( G @ X ) ) @ M )
@ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% DERIV_fun_exp
thf(fact_6892_DERIV__shift,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,Y2: A,X: A,Z3: A] :
( ( has_field_derivative @ A @ F3 @ Y2 @ ( topolo174197925503356063within @ A @ ( plus_plus @ A @ X @ Z3 ) @ ( top_top @ ( set @ A ) ) ) )
= ( has_field_derivative @ A
@ ^ [X4: A] : ( F3 @ ( plus_plus @ A @ X4 @ Z3 ) )
@ Y2
@ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% DERIV_shift
thf(fact_6893_DERIV__const__ratio__const,axiom,
! [A2: real,B2: real,F3: real > real,K2: real] :
( ( A2 != B2 )
=> ( ! [X5: real] : ( has_field_derivative @ real @ F3 @ K2 @ ( topolo174197925503356063within @ real @ X5 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( minus_minus @ real @ ( F3 @ B2 ) @ ( F3 @ A2 ) )
= ( times_times @ real @ ( minus_minus @ real @ B2 @ A2 ) @ K2 ) ) ) ) ).
% DERIV_const_ratio_const
thf(fact_6894_DERIV__cmult__Id,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [C2: A,X: A,S: set @ A] : ( has_field_derivative @ A @ ( times_times @ A @ C2 ) @ C2 @ ( topolo174197925503356063within @ A @ X @ S ) ) ) ).
% DERIV_cmult_Id
thf(fact_6895_DERIV__const,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [K2: A,F4: filter @ A] :
( has_field_derivative @ A
@ ^ [X4: A] : K2
@ ( zero_zero @ A )
@ F4 ) ) ).
% DERIV_const
thf(fact_6896_DERIV__cdivide,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,D4: A,X: A,S: set @ A,C2: A] :
( ( has_field_derivative @ A @ F3 @ D4 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( has_field_derivative @ A
@ ^ [X4: A] : ( divide_divide @ A @ ( F3 @ X4 ) @ C2 )
@ ( divide_divide @ A @ D4 @ C2 )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ).
% DERIV_cdivide
thf(fact_6897_has__field__derivative__scaleR__right,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,D4: A,F4: filter @ A,C2: real] :
( ( has_field_derivative @ A @ F3 @ D4 @ F4 )
=> ( has_field_derivative @ A
@ ^ [X4: A] : ( real_V8093663219630862766scaleR @ A @ C2 @ ( F3 @ X4 ) )
@ ( real_V8093663219630862766scaleR @ A @ C2 @ D4 )
@ F4 ) ) ) ).
% has_field_derivative_scaleR_right
thf(fact_6898_DERIV__add,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,D4: A,X: A,S: set @ A,G: A > A,E5: A] :
( ( has_field_derivative @ A @ F3 @ D4 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( ( has_field_derivative @ A @ G @ E5 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( has_field_derivative @ A
@ ^ [X4: A] : ( plus_plus @ A @ ( F3 @ X4 ) @ ( G @ X4 ) )
@ ( plus_plus @ A @ D4 @ E5 )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ).
% DERIV_add
thf(fact_6899_field__differentiable__add,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,F9: A,F4: filter @ A,G: A > A,G5: A] :
( ( has_field_derivative @ A @ F3 @ F9 @ F4 )
=> ( ( has_field_derivative @ A @ G @ G5 @ F4 )
=> ( has_field_derivative @ A
@ ^ [Z6: A] : ( plus_plus @ A @ ( F3 @ Z6 ) @ ( G @ Z6 ) )
@ ( plus_plus @ A @ F9 @ G5 )
@ F4 ) ) ) ) ).
% field_differentiable_add
thf(fact_6900_DERIV__ident,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F4: filter @ A] :
( has_field_derivative @ A
@ ^ [X4: A] : X4
@ ( one_one @ A )
@ F4 ) ) ).
% DERIV_ident
thf(fact_6901_DERIV__cmult__right,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,D4: A,X: A,S: set @ A,C2: A] :
( ( has_field_derivative @ A @ F3 @ D4 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( has_field_derivative @ A
@ ^ [X4: A] : ( times_times @ A @ ( F3 @ X4 ) @ C2 )
@ ( times_times @ A @ D4 @ C2 )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ).
% DERIV_cmult_right
thf(fact_6902_DERIV__cmult,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,D4: A,X: A,S: set @ A,C2: A] :
( ( has_field_derivative @ A @ F3 @ D4 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( has_field_derivative @ A
@ ^ [X4: A] : ( times_times @ A @ C2 @ ( F3 @ X4 ) )
@ ( times_times @ A @ C2 @ D4 )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ).
% DERIV_cmult
thf(fact_6903_DERIV__mult,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,Da: A,X: A,S: set @ A,G: A > A,Db: A] :
( ( has_field_derivative @ A @ F3 @ Da @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( ( has_field_derivative @ A @ G @ Db @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( has_field_derivative @ A
@ ^ [X4: A] : ( times_times @ A @ ( F3 @ X4 ) @ ( G @ X4 ) )
@ ( plus_plus @ A @ ( times_times @ A @ Da @ ( G @ X ) ) @ ( times_times @ A @ Db @ ( F3 @ X ) ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ).
% DERIV_mult
thf(fact_6904_DERIV__mult_H,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,D4: A,X: A,S: set @ A,G: A > A,E5: A] :
( ( has_field_derivative @ A @ F3 @ D4 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( ( has_field_derivative @ A @ G @ E5 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( has_field_derivative @ A
@ ^ [X4: A] : ( times_times @ A @ ( F3 @ X4 ) @ ( G @ X4 ) )
@ ( plus_plus @ A @ ( times_times @ A @ ( F3 @ X ) @ E5 ) @ ( times_times @ A @ D4 @ ( G @ X ) ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ).
% DERIV_mult'
thf(fact_6905_has__field__derivative__sinh,axiom,
! [A17: $tType] :
( ( ( real_Vector_banach @ A17 )
& ( real_V3459762299906320749_field @ A17 ) )
=> ! [G: A17 > A17,Db: A17,X: A17,S: set @ A17] :
( ( has_field_derivative @ A17 @ G @ Db @ ( topolo174197925503356063within @ A17 @ X @ S ) )
=> ( has_field_derivative @ A17
@ ^ [X4: A17] : ( sinh @ A17 @ ( G @ X4 ) )
@ ( times_times @ A17 @ ( cosh @ A17 @ ( G @ X ) ) @ Db )
@ ( topolo174197925503356063within @ A17 @ X @ S ) ) ) ) ).
% has_field_derivative_sinh
thf(fact_6906_has__field__derivative__cosh,axiom,
! [A17: $tType] :
( ( ( real_Vector_banach @ A17 )
& ( real_V3459762299906320749_field @ A17 ) )
=> ! [G: A17 > A17,Db: A17,X: A17,S: set @ A17] :
( ( has_field_derivative @ A17 @ G @ Db @ ( topolo174197925503356063within @ A17 @ X @ S ) )
=> ( has_field_derivative @ A17
@ ^ [X4: A17] : ( cosh @ A17 @ ( G @ X4 ) )
@ ( times_times @ A17 @ ( sinh @ A17 @ ( G @ X ) ) @ Db )
@ ( topolo174197925503356063within @ A17 @ X @ S ) ) ) ) ).
% has_field_derivative_cosh
thf(fact_6907_field__differentiable__diff,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,F9: A,F4: filter @ A,G: A > A,G5: A] :
( ( has_field_derivative @ A @ F3 @ F9 @ F4 )
=> ( ( has_field_derivative @ A @ G @ G5 @ F4 )
=> ( has_field_derivative @ A
@ ^ [Z6: A] : ( minus_minus @ A @ ( F3 @ Z6 ) @ ( G @ Z6 ) )
@ ( minus_minus @ A @ F9 @ G5 )
@ F4 ) ) ) ) ).
% field_differentiable_diff
thf(fact_6908_DERIV__diff,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,D4: A,X: A,S: set @ A,G: A > A,E5: A] :
( ( has_field_derivative @ A @ F3 @ D4 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( ( has_field_derivative @ A @ G @ E5 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( has_field_derivative @ A
@ ^ [X4: A] : ( minus_minus @ A @ ( F3 @ X4 ) @ ( G @ X4 ) )
@ ( minus_minus @ A @ D4 @ E5 )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ).
% DERIV_diff
thf(fact_6909_field__differentiable__minus,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,F9: A,F4: filter @ A] :
( ( has_field_derivative @ A @ F3 @ F9 @ F4 )
=> ( has_field_derivative @ A
@ ^ [Z6: A] : ( uminus_uminus @ A @ ( F3 @ Z6 ) )
@ ( uminus_uminus @ A @ F9 )
@ F4 ) ) ) ).
% field_differentiable_minus
thf(fact_6910_DERIV__minus,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,D4: A,X: A,S: set @ A] :
( ( has_field_derivative @ A @ F3 @ D4 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( has_field_derivative @ A
@ ^ [X4: A] : ( uminus_uminus @ A @ ( F3 @ X4 ) )
@ ( uminus_uminus @ A @ D4 )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ).
% DERIV_minus
thf(fact_6911_DERIV__divide,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,D4: A,X: A,S: set @ A,G: A > A,E5: A] :
( ( has_field_derivative @ A @ F3 @ D4 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( ( has_field_derivative @ A @ G @ E5 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( ( ( G @ X )
!= ( zero_zero @ A ) )
=> ( has_field_derivative @ A
@ ^ [X4: A] : ( divide_divide @ A @ ( F3 @ X4 ) @ ( G @ X4 ) )
@ ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ D4 @ ( G @ X ) ) @ ( times_times @ A @ ( F3 @ X ) @ E5 ) ) @ ( times_times @ A @ ( G @ X ) @ ( G @ X ) ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ) ).
% DERIV_divide
thf(fact_6912_DERIV__inverse_H,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,D4: A,X: A,S: set @ A] :
( ( has_field_derivative @ A @ F3 @ D4 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( ( ( F3 @ X )
!= ( zero_zero @ A ) )
=> ( has_field_derivative @ A
@ ^ [X4: A] : ( inverse_inverse @ A @ ( F3 @ X4 ) )
@ ( uminus_uminus @ A @ ( times_times @ A @ ( times_times @ A @ ( inverse_inverse @ A @ ( F3 @ X ) ) @ D4 ) @ ( inverse_inverse @ A @ ( F3 @ X ) ) ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ).
% DERIV_inverse'
thf(fact_6913_DERIV__sum,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( real_V3459762299906320749_field @ B )
=> ! [S3: set @ A,F3: B > A > B,F9: C > A > B,X: C,F4: filter @ B] :
( ! [N3: A] :
( ( member @ A @ N3 @ S3 )
=> ( has_field_derivative @ B
@ ^ [X4: B] : ( F3 @ X4 @ N3 )
@ ( F9 @ X @ N3 )
@ F4 ) )
=> ( has_field_derivative @ B
@ ^ [X4: B] : ( groups7311177749621191930dd_sum @ A @ B @ ( F3 @ X4 ) @ S3 )
@ ( groups7311177749621191930dd_sum @ A @ B @ ( F9 @ X ) @ S3 )
@ F4 ) ) ) ).
% DERIV_sum
thf(fact_6914_partition__filter1,axiom,
! [A: $tType,P3: A > $o,Xs: list @ A] :
( ( product_fst @ ( list @ A ) @ ( list @ A ) @ ( partition @ A @ P3 @ Xs ) )
= ( filter2 @ A @ P3 @ Xs ) ) ).
% partition_filter1
thf(fact_6915_DERIV__cos__add,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [K2: A,Xa: A] :
( has_field_derivative @ A
@ ^ [X4: A] : ( cos @ A @ ( plus_plus @ A @ X4 @ K2 ) )
@ ( uminus_uminus @ A @ ( sin @ A @ ( plus_plus @ A @ Xa @ K2 ) ) )
@ ( topolo174197925503356063within @ A @ Xa @ ( top_top @ ( set @ A ) ) ) ) ) ).
% DERIV_cos_add
thf(fact_6916_DERIV__fun__cos,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [G: A > A,M: A,X: A] :
( ( has_field_derivative @ A @ G @ M @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) )
=> ( has_field_derivative @ A
@ ^ [X4: A] : ( cos @ A @ ( G @ X4 ) )
@ ( times_times @ A @ ( uminus_uminus @ A @ ( sin @ A @ ( G @ X ) ) ) @ M )
@ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% DERIV_fun_cos
thf(fact_6917_DERIV__at__within__shift,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,Y2: A,Z3: A,X: A,S3: set @ A] :
( ( has_field_derivative @ A @ F3 @ Y2 @ ( topolo174197925503356063within @ A @ ( plus_plus @ A @ Z3 @ X ) @ ( image @ A @ A @ ( plus_plus @ A @ Z3 ) @ S3 ) ) )
= ( has_field_derivative @ A
@ ^ [X4: A] : ( F3 @ ( plus_plus @ A @ Z3 @ X4 ) )
@ Y2
@ ( topolo174197925503356063within @ A @ X @ S3 ) ) ) ) ).
% DERIV_at_within_shift
thf(fact_6918_DERIV__neg__imp__decreasing,axiom,
! [A2: real,B2: real,F3: real > real] :
( ( ord_less @ real @ A2 @ B2 )
=> ( ! [X5: real] :
( ( ord_less_eq @ real @ A2 @ X5 )
=> ( ( ord_less_eq @ real @ X5 @ B2 )
=> ? [Y4: real] :
( ( has_field_derivative @ real @ F3 @ Y4 @ ( topolo174197925503356063within @ real @ X5 @ ( top_top @ ( set @ real ) ) ) )
& ( ord_less @ real @ Y4 @ ( zero_zero @ real ) ) ) ) )
=> ( ord_less @ real @ ( F3 @ B2 ) @ ( F3 @ A2 ) ) ) ) ).
% DERIV_neg_imp_decreasing
thf(fact_6919_DERIV__pos__imp__increasing,axiom,
! [A2: real,B2: real,F3: real > real] :
( ( ord_less @ real @ A2 @ B2 )
=> ( ! [X5: real] :
( ( ord_less_eq @ real @ A2 @ X5 )
=> ( ( ord_less_eq @ real @ X5 @ B2 )
=> ? [Y4: real] :
( ( has_field_derivative @ real @ F3 @ Y4 @ ( topolo174197925503356063within @ real @ X5 @ ( top_top @ ( set @ real ) ) ) )
& ( ord_less @ real @ ( zero_zero @ real ) @ Y4 ) ) ) )
=> ( ord_less @ real @ ( F3 @ A2 ) @ ( F3 @ B2 ) ) ) ) ).
% DERIV_pos_imp_increasing
thf(fact_6920_MVT2,axiom,
! [A2: real,B2: real,F3: real > real,F9: real > real] :
( ( ord_less @ real @ A2 @ B2 )
=> ( ! [X5: real] :
( ( ord_less_eq @ real @ A2 @ X5 )
=> ( ( ord_less_eq @ real @ X5 @ B2 )
=> ( has_field_derivative @ real @ F3 @ ( F9 @ X5 ) @ ( topolo174197925503356063within @ real @ X5 @ ( top_top @ ( set @ real ) ) ) ) ) )
=> ? [Z: real] :
( ( ord_less @ real @ A2 @ Z )
& ( ord_less @ real @ Z @ B2 )
& ( ( minus_minus @ real @ ( F3 @ B2 ) @ ( F3 @ A2 ) )
= ( times_times @ real @ ( minus_minus @ real @ B2 @ A2 ) @ ( F9 @ Z ) ) ) ) ) ) ).
% MVT2
thf(fact_6921_at__le,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [S: set @ A,T2: set @ A,X: A] :
( ( ord_less_eq @ ( set @ A ) @ S @ T2 )
=> ( ord_less_eq @ ( filter @ A ) @ ( topolo174197925503356063within @ A @ X @ S ) @ ( topolo174197925503356063within @ A @ X @ T2 ) ) ) ) ).
% at_le
thf(fact_6922_has__field__derivative__subset,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,Y2: A,X: A,S: set @ A,T2: set @ A] :
( ( has_field_derivative @ A @ F3 @ Y2 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( ( ord_less_eq @ ( set @ A ) @ T2 @ S )
=> ( has_field_derivative @ A @ F3 @ Y2 @ ( topolo174197925503356063within @ A @ X @ T2 ) ) ) ) ) ).
% has_field_derivative_subset
thf(fact_6923_DERIV__subset,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,F9: A,X: A,S: set @ A,T2: set @ A] :
( ( has_field_derivative @ A @ F3 @ F9 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( ( ord_less_eq @ ( set @ A ) @ T2 @ S )
=> ( has_field_derivative @ A @ F3 @ F9 @ ( topolo174197925503356063within @ A @ X @ T2 ) ) ) ) ) ).
% DERIV_subset
thf(fact_6924_DERIV__nonneg__imp__nondecreasing,axiom,
! [A2: real,B2: real,F3: real > real] :
( ( ord_less_eq @ real @ A2 @ B2 )
=> ( ! [X5: real] :
( ( ord_less_eq @ real @ A2 @ X5 )
=> ( ( ord_less_eq @ real @ X5 @ B2 )
=> ? [Y4: real] :
( ( has_field_derivative @ real @ F3 @ Y4 @ ( topolo174197925503356063within @ real @ X5 @ ( top_top @ ( set @ real ) ) ) )
& ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y4 ) ) ) )
=> ( ord_less_eq @ real @ ( F3 @ A2 ) @ ( F3 @ B2 ) ) ) ) ).
% DERIV_nonneg_imp_nondecreasing
thf(fact_6925_DERIV__nonpos__imp__nonincreasing,axiom,
! [A2: real,B2: real,F3: real > real] :
( ( ord_less_eq @ real @ A2 @ B2 )
=> ( ! [X5: real] :
( ( ord_less_eq @ real @ A2 @ X5 )
=> ( ( ord_less_eq @ real @ X5 @ B2 )
=> ? [Y4: real] :
( ( has_field_derivative @ real @ F3 @ Y4 @ ( topolo174197925503356063within @ real @ X5 @ ( top_top @ ( set @ real ) ) ) )
& ( ord_less_eq @ real @ Y4 @ ( zero_zero @ real ) ) ) ) )
=> ( ord_less_eq @ real @ ( F3 @ B2 ) @ ( F3 @ A2 ) ) ) ) ).
% DERIV_nonpos_imp_nonincreasing
thf(fact_6926_deriv__nonneg__imp__mono,axiom,
! [A2: real,B2: real,G: real > real,G5: real > real] :
( ! [X5: real] :
( ( member @ real @ X5 @ ( set_or1337092689740270186AtMost @ real @ A2 @ B2 ) )
=> ( has_field_derivative @ real @ G @ ( G5 @ X5 ) @ ( topolo174197925503356063within @ real @ X5 @ ( top_top @ ( set @ real ) ) ) ) )
=> ( ! [X5: real] :
( ( member @ real @ X5 @ ( set_or1337092689740270186AtMost @ real @ A2 @ B2 ) )
=> ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( G5 @ X5 ) ) )
=> ( ( ord_less_eq @ real @ A2 @ B2 )
=> ( ord_less_eq @ real @ ( G @ A2 ) @ ( G @ B2 ) ) ) ) ) ).
% deriv_nonneg_imp_mono
thf(fact_6927_DERIV__at__within__shift__lemma,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,Y2: A,Z3: A,X: A,S3: set @ A] :
( ( has_field_derivative @ A @ F3 @ Y2 @ ( topolo174197925503356063within @ A @ ( plus_plus @ A @ Z3 @ X ) @ ( image @ A @ A @ ( plus_plus @ A @ Z3 ) @ S3 ) ) )
=> ( has_field_derivative @ A @ ( comp @ A @ A @ A @ F3 @ ( plus_plus @ A @ Z3 ) ) @ Y2 @ ( topolo174197925503356063within @ A @ X @ S3 ) ) ) ) ).
% DERIV_at_within_shift_lemma
thf(fact_6928_DERIV__image__chain,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,Da: A,G: A > A,X: A,S: set @ A,Db: A] :
( ( has_field_derivative @ A @ F3 @ Da @ ( topolo174197925503356063within @ A @ ( G @ X ) @ ( image @ A @ A @ G @ S ) ) )
=> ( ( has_field_derivative @ A @ G @ Db @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( has_field_derivative @ A @ ( comp @ A @ A @ A @ F3 @ G ) @ ( times_times @ A @ Da @ Db ) @ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ).
% DERIV_image_chain
thf(fact_6929_DERIV__chain,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,Da: A,G: A > A,X: A,Db: A,S: set @ A] :
( ( has_field_derivative @ A @ F3 @ Da @ ( topolo174197925503356063within @ A @ ( G @ X ) @ ( top_top @ ( set @ A ) ) ) )
=> ( ( has_field_derivative @ A @ G @ Db @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( has_field_derivative @ A @ ( comp @ A @ A @ A @ F3 @ G ) @ ( times_times @ A @ Da @ Db ) @ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ).
% DERIV_chain
thf(fact_6930_DERIV__power__Suc,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,D4: A,X: A,S: set @ A,N: nat] :
( ( has_field_derivative @ A @ F3 @ D4 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( has_field_derivative @ A
@ ^ [X4: A] : ( power_power @ A @ ( F3 @ X4 ) @ ( suc @ N ) )
@ ( times_times @ A @ ( plus_plus @ A @ ( one_one @ A ) @ ( semiring_1_of_nat @ A @ N ) ) @ ( times_times @ A @ D4 @ ( power_power @ A @ ( F3 @ X ) @ N ) ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ).
% DERIV_power_Suc
thf(fact_6931_DERIV__const__average,axiom,
! [A2: real,B2: real,V2: real > real,K2: real] :
( ( A2 != B2 )
=> ( ! [X5: real] : ( has_field_derivative @ real @ V2 @ K2 @ ( topolo174197925503356063within @ real @ X5 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( V2 @ ( divide_divide @ real @ ( plus_plus @ real @ A2 @ B2 ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
= ( divide_divide @ real @ ( plus_plus @ real @ ( V2 @ A2 ) @ ( V2 @ B2 ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ).
% DERIV_const_average
thf(fact_6932_DERIV__inverse,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [X: A,S: set @ A] :
( ( X
!= ( zero_zero @ A ) )
=> ( has_field_derivative @ A @ ( inverse_inverse @ A ) @ ( uminus_uminus @ A @ ( power_power @ A @ ( inverse_inverse @ A @ X ) @ ( suc @ ( suc @ ( zero_zero @ nat ) ) ) ) ) @ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ).
% DERIV_inverse
thf(fact_6933_DERIV__power,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,D4: A,X: A,S: set @ A,N: nat] :
( ( has_field_derivative @ A @ F3 @ D4 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( has_field_derivative @ A
@ ^ [X4: A] : ( power_power @ A @ ( F3 @ X4 ) @ N )
@ ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ ( times_times @ A @ D4 @ ( power_power @ A @ ( F3 @ X ) @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) ) ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ).
% DERIV_power
thf(fact_6934_DERIV__local__max,axiom,
! [F3: real > real,L: real,X: real,D2: real] :
( ( has_field_derivative @ real @ F3 @ L @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ D2 )
=> ( ! [Y7: real] :
( ( ord_less @ real @ ( abs_abs @ real @ ( minus_minus @ real @ X @ Y7 ) ) @ D2 )
=> ( ord_less_eq @ real @ ( F3 @ Y7 ) @ ( F3 @ X ) ) )
=> ( L
= ( zero_zero @ real ) ) ) ) ) ).
% DERIV_local_max
thf(fact_6935_DERIV__local__min,axiom,
! [F3: real > real,L: real,X: real,D2: real] :
( ( has_field_derivative @ real @ F3 @ L @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ D2 )
=> ( ! [Y7: real] :
( ( ord_less @ real @ ( abs_abs @ real @ ( minus_minus @ real @ X @ Y7 ) ) @ D2 )
=> ( ord_less_eq @ real @ ( F3 @ X ) @ ( F3 @ Y7 ) ) )
=> ( L
= ( zero_zero @ real ) ) ) ) ) ).
% DERIV_local_min
thf(fact_6936_DERIV__ln__divide,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( has_field_derivative @ real @ ( ln_ln @ real ) @ ( divide_divide @ real @ ( one_one @ real ) @ X ) @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) ) ) ).
% DERIV_ln_divide
thf(fact_6937_DERIV__pow,axiom,
! [N: nat,X: real,S: set @ real] :
( has_field_derivative @ real
@ ^ [X4: real] : ( power_power @ real @ X4 @ N )
@ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ ( power_power @ real @ X @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) ) )
@ ( topolo174197925503356063within @ real @ X @ S ) ) ).
% DERIV_pow
thf(fact_6938_termdiffs__strong__converges__everywhere,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [C2: nat > A,X: A] :
( ! [Y7: A] :
( summable @ A
@ ^ [N2: nat] : ( times_times @ A @ ( C2 @ N2 ) @ ( power_power @ A @ Y7 @ N2 ) ) )
=> ( has_field_derivative @ A
@ ^ [X4: A] :
( suminf @ A
@ ^ [N2: nat] : ( times_times @ A @ ( C2 @ N2 ) @ ( power_power @ A @ X4 @ N2 ) ) )
@ ( suminf @ A
@ ^ [N2: nat] : ( times_times @ A @ ( diffs @ A @ C2 @ N2 ) @ ( power_power @ A @ X @ N2 ) ) )
@ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% termdiffs_strong_converges_everywhere
thf(fact_6939_at__within__Icc__at,axiom,
! [A: $tType] :
( ( topolo2564578578187576103pology @ A )
=> ! [A2: A,X: A,B2: A] :
( ( ord_less @ A @ A2 @ X )
=> ( ( ord_less @ A @ X @ B2 )
=> ( ( topolo174197925503356063within @ A @ X @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
= ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% at_within_Icc_at
thf(fact_6940_DERIV__fun__pow,axiom,
! [G: real > real,M: real,X: real,N: nat] :
( ( has_field_derivative @ real @ G @ M @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( has_field_derivative @ real
@ ^ [X4: real] : ( power_power @ real @ ( G @ X4 ) @ N )
@ ( times_times @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ ( power_power @ real @ ( G @ X ) @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) @ M )
@ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) ) ) ).
% DERIV_fun_pow
thf(fact_6941_at__within__Icc__at__left,axiom,
! [A: $tType] :
( ( topolo2564578578187576103pology @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( topolo174197925503356063within @ A @ B2 @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
= ( topolo174197925503356063within @ A @ B2 @ ( set_ord_lessThan @ A @ B2 ) ) ) ) ) ).
% at_within_Icc_at_left
thf(fact_6942_DERIV__quotient,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,D2: A,X: A,S: set @ A,G: A > A,E: A] :
( ( has_field_derivative @ A @ F3 @ D2 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( ( has_field_derivative @ A @ G @ E @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( ( ( G @ X )
!= ( zero_zero @ A ) )
=> ( has_field_derivative @ A
@ ^ [Y: A] : ( divide_divide @ A @ ( F3 @ Y ) @ ( G @ Y ) )
@ ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ D2 @ ( G @ X ) ) @ ( times_times @ A @ E @ ( F3 @ X ) ) ) @ ( power_power @ A @ ( G @ X ) @ ( suc @ ( suc @ ( zero_zero @ nat ) ) ) ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ) ).
% DERIV_quotient
thf(fact_6943_DERIV__inverse__fun,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,D2: A,X: A,S: set @ A] :
( ( has_field_derivative @ A @ F3 @ D2 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( ( ( F3 @ X )
!= ( zero_zero @ A ) )
=> ( has_field_derivative @ A
@ ^ [X4: A] : ( inverse_inverse @ A @ ( F3 @ X4 ) )
@ ( uminus_uminus @ A @ ( times_times @ A @ D2 @ ( inverse_inverse @ A @ ( power_power @ A @ ( F3 @ X ) @ ( suc @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ).
% DERIV_inverse_fun
thf(fact_6944_termdiffs__sums__strong,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [K5: real,C2: nat > A,F3: A > A,F9: A,Z3: A] :
( ! [Z: A] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ Z ) @ K5 )
=> ( sums @ A
@ ^ [N2: nat] : ( times_times @ A @ ( C2 @ N2 ) @ ( power_power @ A @ Z @ N2 ) )
@ ( F3 @ Z ) ) )
=> ( ( has_field_derivative @ A @ F3 @ F9 @ ( topolo174197925503356063within @ A @ Z3 @ ( top_top @ ( set @ A ) ) ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ Z3 ) @ K5 )
=> ( sums @ A
@ ^ [N2: nat] : ( times_times @ A @ ( diffs @ A @ C2 @ N2 ) @ ( power_power @ A @ Z3 @ N2 ) )
@ F9 ) ) ) ) ) ).
% termdiffs_sums_strong
thf(fact_6945_has__real__derivative__powr,axiom,
! [Z3: real,R: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ Z3 )
=> ( has_field_derivative @ real
@ ^ [Z6: real] : ( powr @ real @ Z6 @ R )
@ ( times_times @ real @ R @ ( powr @ real @ Z3 @ ( minus_minus @ real @ R @ ( one_one @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ Z3 @ ( top_top @ ( set @ real ) ) ) ) ) ).
% has_real_derivative_powr
thf(fact_6946_partition_Osimps_I1_J,axiom,
! [A: $tType,P3: A > $o] :
( ( partition @ A @ P3 @ ( nil @ A ) )
= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) ) ) ).
% partition.simps(1)
thf(fact_6947_partition__P,axiom,
! [A: $tType,P3: A > $o,Xs: list @ A,Yes: list @ A,No: list @ A] :
( ( ( partition @ A @ P3 @ Xs )
= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Yes @ No ) )
=> ( ! [X2: A] :
( ( member @ A @ X2 @ ( set2 @ A @ Yes ) )
=> ( P3 @ X2 ) )
& ! [X2: A] :
( ( member @ A @ X2 @ ( set2 @ A @ No ) )
=> ~ ( P3 @ X2 ) ) ) ) ).
% partition_P
thf(fact_6948_termdiffs,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [C2: nat > A,K5: A,X: A] :
( ( summable @ A
@ ^ [N2: nat] : ( times_times @ A @ ( C2 @ N2 ) @ ( power_power @ A @ K5 @ N2 ) ) )
=> ( ( summable @ A
@ ^ [N2: nat] : ( times_times @ A @ ( diffs @ A @ C2 @ N2 ) @ ( power_power @ A @ K5 @ N2 ) ) )
=> ( ( summable @ A
@ ^ [N2: nat] : ( times_times @ A @ ( diffs @ A @ ( diffs @ A @ C2 ) @ N2 ) @ ( power_power @ A @ K5 @ N2 ) ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ ( real_V7770717601297561774m_norm @ A @ K5 ) )
=> ( has_field_derivative @ A
@ ^ [X4: A] :
( suminf @ A
@ ^ [N2: nat] : ( times_times @ A @ ( C2 @ N2 ) @ ( power_power @ A @ X4 @ N2 ) ) )
@ ( suminf @ A
@ ^ [N2: nat] : ( times_times @ A @ ( diffs @ A @ C2 @ N2 ) @ ( power_power @ A @ X @ N2 ) ) )
@ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% termdiffs
thf(fact_6949_termdiffs__strong,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [C2: nat > A,K5: A,X: A] :
( ( summable @ A
@ ^ [N2: nat] : ( times_times @ A @ ( C2 @ N2 ) @ ( power_power @ A @ K5 @ N2 ) ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ ( real_V7770717601297561774m_norm @ A @ K5 ) )
=> ( has_field_derivative @ A
@ ^ [X4: A] :
( suminf @ A
@ ^ [N2: nat] : ( times_times @ A @ ( C2 @ N2 ) @ ( power_power @ A @ X4 @ N2 ) ) )
@ ( suminf @ A
@ ^ [N2: nat] : ( times_times @ A @ ( diffs @ A @ C2 @ N2 ) @ ( power_power @ A @ X @ N2 ) ) )
@ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% termdiffs_strong
thf(fact_6950_termdiffs__strong_H,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [K5: real,C2: nat > A,Z3: A] :
( ! [Z: A] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ Z ) @ K5 )
=> ( summable @ A
@ ^ [N2: nat] : ( times_times @ A @ ( C2 @ N2 ) @ ( power_power @ A @ Z @ N2 ) ) ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ Z3 ) @ K5 )
=> ( has_field_derivative @ A
@ ^ [Z6: A] :
( suminf @ A
@ ^ [N2: nat] : ( times_times @ A @ ( C2 @ N2 ) @ ( power_power @ A @ Z6 @ N2 ) ) )
@ ( suminf @ A
@ ^ [N2: nat] : ( times_times @ A @ ( diffs @ A @ C2 @ N2 ) @ ( power_power @ A @ Z3 @ N2 ) ) )
@ ( topolo174197925503356063within @ A @ Z3 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% termdiffs_strong'
thf(fact_6951_DERIV__log,axiom,
! [X: real,B2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( has_field_derivative @ real @ ( log @ B2 ) @ ( divide_divide @ real @ ( one_one @ real ) @ ( times_times @ real @ ( ln_ln @ real @ B2 ) @ X ) ) @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) ) ) ).
% DERIV_log
thf(fact_6952_DERIV__fun__powr,axiom,
! [G: real > real,M: real,X: real,R: real] :
( ( has_field_derivative @ real @ G @ M @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( G @ X ) )
=> ( has_field_derivative @ real
@ ^ [X4: real] : ( powr @ real @ ( G @ X4 ) @ R )
@ ( times_times @ real @ ( times_times @ real @ R @ ( powr @ real @ ( G @ X ) @ ( minus_minus @ real @ R @ ( semiring_1_of_nat @ real @ ( one_one @ nat ) ) ) ) ) @ M )
@ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) ) ) ) ).
% DERIV_fun_powr
thf(fact_6953_DERIV__powr,axiom,
! [G: real > real,M: real,X: real,F3: real > real,R: real] :
( ( has_field_derivative @ real @ G @ M @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( G @ X ) )
=> ( ( has_field_derivative @ real @ F3 @ R @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( has_field_derivative @ real
@ ^ [X4: real] : ( powr @ real @ ( G @ X4 ) @ ( F3 @ X4 ) )
@ ( times_times @ real @ ( powr @ real @ ( G @ X ) @ ( F3 @ X ) ) @ ( plus_plus @ real @ ( times_times @ real @ R @ ( ln_ln @ real @ ( G @ X ) ) ) @ ( divide_divide @ real @ ( times_times @ real @ M @ ( F3 @ X ) ) @ ( G @ X ) ) ) )
@ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ).
% DERIV_powr
thf(fact_6954_DERIV__tan,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A] :
( ( ( cos @ A @ X )
!= ( zero_zero @ A ) )
=> ( has_field_derivative @ A @ ( tan @ A ) @ ( inverse_inverse @ A @ ( power_power @ A @ ( cos @ A @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% DERIV_tan
thf(fact_6955_partition_Osimps_I2_J,axiom,
! [A: $tType,P3: A > $o,X: A,Xs: list @ A] :
( ( partition @ A @ P3 @ ( cons @ A @ X @ Xs ) )
= ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) )
@ ^ [Yes2: list @ A,No2: list @ A] : ( if @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( P3 @ X ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Yes2 ) @ No2 ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Yes2 @ ( cons @ A @ X @ No2 ) ) )
@ ( partition @ A @ P3 @ Xs ) ) ) ).
% partition.simps(2)
thf(fact_6956_DERIV__real__sqrt,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( has_field_derivative @ real @ sqrt @ ( divide_divide @ real @ ( inverse_inverse @ real @ ( sqrt @ X ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) ) ) ).
% DERIV_real_sqrt
thf(fact_6957_DERIV__series_H,axiom,
! [F3: real > nat > real,F9: real > nat > real,X0: real,A2: real,B2: real,L6: nat > real] :
( ! [N3: nat] :
( has_field_derivative @ real
@ ^ [X4: real] : ( F3 @ X4 @ N3 )
@ ( F9 @ X0 @ N3 )
@ ( topolo174197925503356063within @ real @ X0 @ ( top_top @ ( set @ real ) ) ) )
=> ( ! [X5: real] :
( ( member @ real @ X5 @ ( set_or5935395276787703475ssThan @ real @ A2 @ B2 ) )
=> ( summable @ real @ ( F3 @ X5 ) ) )
=> ( ( member @ real @ X0 @ ( set_or5935395276787703475ssThan @ real @ A2 @ B2 ) )
=> ( ( summable @ real @ ( F9 @ X0 ) )
=> ( ( summable @ real @ L6 )
=> ( ! [N3: nat,X5: real,Y7: real] :
( ( member @ real @ X5 @ ( set_or5935395276787703475ssThan @ real @ A2 @ B2 ) )
=> ( ( member @ real @ Y7 @ ( set_or5935395276787703475ssThan @ real @ A2 @ B2 ) )
=> ( ord_less_eq @ real @ ( abs_abs @ real @ ( minus_minus @ real @ ( F3 @ X5 @ N3 ) @ ( F3 @ Y7 @ N3 ) ) ) @ ( times_times @ real @ ( L6 @ N3 ) @ ( abs_abs @ real @ ( minus_minus @ real @ X5 @ Y7 ) ) ) ) ) )
=> ( has_field_derivative @ real
@ ^ [X4: real] : ( suminf @ real @ ( F3 @ X4 ) )
@ ( suminf @ real @ ( F9 @ X0 ) )
@ ( topolo174197925503356063within @ real @ X0 @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ) ) ) ).
% DERIV_series'
thf(fact_6958_DERIV__arctan,axiom,
! [X: real] : ( has_field_derivative @ real @ arctan @ ( inverse_inverse @ real @ ( plus_plus @ real @ ( one_one @ real ) @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) ) ).
% DERIV_arctan
thf(fact_6959_arsinh__real__has__field__derivative,axiom,
! [X: real,A3: set @ real] : ( has_field_derivative @ real @ ( arsinh @ real ) @ ( divide_divide @ real @ ( one_one @ real ) @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ real ) ) ) ) @ ( topolo174197925503356063within @ real @ X @ A3 ) ) ).
% arsinh_real_has_field_derivative
thf(fact_6960_partition__filter2,axiom,
! [A: $tType,P3: A > $o,Xs: list @ A] :
( ( product_snd @ ( list @ A ) @ ( list @ A ) @ ( partition @ A @ P3 @ Xs ) )
= ( filter2 @ A @ ( comp @ $o @ $o @ A @ (~) @ P3 ) @ Xs ) ) ).
% partition_filter2
thf(fact_6961_DERIV__cot,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A] :
( ( ( sin @ A @ X )
!= ( zero_zero @ A ) )
=> ( has_field_derivative @ A @ ( cot @ A ) @ ( uminus_uminus @ A @ ( inverse_inverse @ A @ ( power_power @ A @ ( sin @ A @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% DERIV_cot
thf(fact_6962_has__field__derivative__tanh,axiom,
! [A17: $tType] :
( ( ( real_Vector_banach @ A17 )
& ( real_V3459762299906320749_field @ A17 ) )
=> ! [G: A17 > A17,X: A17,Db: A17,S: set @ A17] :
( ( ( cosh @ A17 @ ( G @ X ) )
!= ( zero_zero @ A17 ) )
=> ( ( has_field_derivative @ A17 @ G @ Db @ ( topolo174197925503356063within @ A17 @ X @ S ) )
=> ( has_field_derivative @ A17
@ ^ [X4: A17] : ( tanh @ A17 @ ( G @ X4 ) )
@ ( times_times @ A17 @ ( minus_minus @ A17 @ ( one_one @ A17 ) @ ( power_power @ A17 @ ( tanh @ A17 @ ( G @ X ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ Db )
@ ( topolo174197925503356063within @ A17 @ X @ S ) ) ) ) ) ).
% has_field_derivative_tanh
thf(fact_6963_DERIV__real__sqrt__generic,axiom,
! [X: real,D4: real] :
( ( X
!= ( zero_zero @ real ) )
=> ( ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( D4
= ( divide_divide @ real @ ( inverse_inverse @ real @ ( sqrt @ X ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) )
=> ( ( ( ord_less @ real @ X @ ( zero_zero @ real ) )
=> ( D4
= ( divide_divide @ real @ ( uminus_uminus @ real @ ( inverse_inverse @ real @ ( sqrt @ X ) ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) )
=> ( has_field_derivative @ real @ sqrt @ D4 @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ).
% DERIV_real_sqrt_generic
thf(fact_6964_arcosh__real__has__field__derivative,axiom,
! [X: real,A3: set @ real] :
( ( ord_less @ real @ ( one_one @ real ) @ X )
=> ( has_field_derivative @ real @ ( arcosh @ real ) @ ( divide_divide @ real @ ( one_one @ real ) @ ( sqrt @ ( minus_minus @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ real ) ) ) ) @ ( topolo174197925503356063within @ real @ X @ A3 ) ) ) ).
% arcosh_real_has_field_derivative
thf(fact_6965_artanh__real__has__field__derivative,axiom,
! [X: real,A3: set @ real] :
( ( ord_less @ real @ ( abs_abs @ real @ X ) @ ( one_one @ real ) )
=> ( has_field_derivative @ real @ ( artanh @ real ) @ ( divide_divide @ real @ ( one_one @ real ) @ ( minus_minus @ real @ ( one_one @ real ) @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( topolo174197925503356063within @ real @ X @ A3 ) ) ) ).
% artanh_real_has_field_derivative
thf(fact_6966_partition__set,axiom,
! [A: $tType,P3: A > $o,Xs: list @ A,Yes: list @ A,No: list @ A] :
( ( ( partition @ A @ P3 @ Xs )
= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Yes @ No ) )
=> ( ( sup_sup @ ( set @ A ) @ ( set2 @ A @ Yes ) @ ( set2 @ A @ No ) )
= ( set2 @ A @ Xs ) ) ) ).
% partition_set
thf(fact_6967_DERIV__power__series_H,axiom,
! [R4: real,F3: nat > real,X0: real] :
( ! [X5: real] :
( ( member @ real @ X5 @ ( set_or5935395276787703475ssThan @ real @ ( uminus_uminus @ real @ R4 ) @ R4 ) )
=> ( summable @ real
@ ^ [N2: nat] : ( times_times @ real @ ( times_times @ real @ ( F3 @ N2 ) @ ( semiring_1_of_nat @ real @ ( suc @ N2 ) ) ) @ ( power_power @ real @ X5 @ N2 ) ) ) )
=> ( ( member @ real @ X0 @ ( set_or5935395276787703475ssThan @ real @ ( uminus_uminus @ real @ R4 ) @ R4 ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ R4 )
=> ( has_field_derivative @ real
@ ^ [X4: real] :
( suminf @ real
@ ^ [N2: nat] : ( times_times @ real @ ( F3 @ N2 ) @ ( power_power @ real @ X4 @ ( suc @ N2 ) ) ) )
@ ( suminf @ real
@ ^ [N2: nat] : ( times_times @ real @ ( times_times @ real @ ( F3 @ N2 ) @ ( semiring_1_of_nat @ real @ ( suc @ N2 ) ) ) @ ( power_power @ real @ X0 @ N2 ) ) )
@ ( topolo174197925503356063within @ real @ X0 @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ).
% DERIV_power_series'
thf(fact_6968_DERIV__real__root,axiom,
! [N: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( has_field_derivative @ real @ ( root @ N ) @ ( inverse_inverse @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ ( power_power @ real @ ( root @ N @ X ) @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) ) ) ) ).
% DERIV_real_root
thf(fact_6969_DERIV__arccos,axiom,
! [X: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X )
=> ( ( ord_less @ real @ X @ ( one_one @ real ) )
=> ( has_field_derivative @ real @ arccos @ ( inverse_inverse @ real @ ( uminus_uminus @ real @ ( sqrt @ ( minus_minus @ real @ ( one_one @ real ) @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) ) ) ) ).
% DERIV_arccos
thf(fact_6970_DERIV__arcsin,axiom,
! [X: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X )
=> ( ( ord_less @ real @ X @ ( one_one @ real ) )
=> ( has_field_derivative @ real @ arcsin @ ( inverse_inverse @ real @ ( sqrt @ ( minus_minus @ real @ ( one_one @ real ) @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) ) ) ) ).
% DERIV_arcsin
thf(fact_6971_Maclaurin__all__le,axiom,
! [Diff: nat > real > real,F3: real > real,X: real,N: nat] :
( ( ( Diff @ ( zero_zero @ nat ) )
= F3 )
=> ( ! [M4: nat,X5: real] : ( has_field_derivative @ real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ X5 ) @ ( topolo174197925503356063within @ real @ X5 @ ( top_top @ ( set @ real ) ) ) )
=> ? [T4: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ T4 ) @ ( abs_abs @ real @ X ) )
& ( ( F3 @ X )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M2: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M2 @ ( zero_zero @ real ) ) @ ( semiring_char_0_fact @ real @ M2 ) ) @ ( power_power @ real @ X @ M2 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N @ T4 ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ X @ N ) ) ) ) ) ) ) ).
% Maclaurin_all_le
thf(fact_6972_Maclaurin__all__le__objl,axiom,
! [Diff: nat > real > real,F3: real > real,X: real,N: nat] :
( ( ( ( Diff @ ( zero_zero @ nat ) )
= F3 )
& ! [M4: nat,X5: real] : ( has_field_derivative @ real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ X5 ) @ ( topolo174197925503356063within @ real @ X5 @ ( top_top @ ( set @ real ) ) ) ) )
=> ? [T4: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ T4 ) @ ( abs_abs @ real @ X ) )
& ( ( F3 @ X )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M2: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M2 @ ( zero_zero @ real ) ) @ ( semiring_char_0_fact @ real @ M2 ) ) @ ( power_power @ real @ X @ M2 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N @ T4 ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ X @ N ) ) ) ) ) ) ).
% Maclaurin_all_le_objl
thf(fact_6973_DERIV__odd__real__root,axiom,
! [N: nat,X: real] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( X
!= ( zero_zero @ real ) )
=> ( has_field_derivative @ real @ ( root @ N ) @ ( inverse_inverse @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ ( power_power @ real @ ( root @ N @ X ) @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) ) ) ) ).
% DERIV_odd_real_root
thf(fact_6974_Maclaurin,axiom,
! [H2: real,N: nat,Diff: nat > real > real,F3: real > real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ H2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ( Diff @ ( zero_zero @ nat ) )
= F3 )
=> ( ! [M4: nat,T4: real] :
( ( ( ord_less @ nat @ M4 @ N )
& ( ord_less_eq @ real @ ( zero_zero @ real ) @ T4 )
& ( ord_less_eq @ real @ T4 @ H2 ) )
=> ( has_field_derivative @ real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T4 ) @ ( topolo174197925503356063within @ real @ T4 @ ( top_top @ ( set @ real ) ) ) ) )
=> ? [T4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ T4 )
& ( ord_less @ real @ T4 @ H2 )
& ( ( F3 @ H2 )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M2: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M2 @ ( zero_zero @ real ) ) @ ( semiring_char_0_fact @ real @ M2 ) ) @ ( power_power @ real @ H2 @ M2 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N @ T4 ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ H2 @ N ) ) ) ) ) ) ) ) ) ).
% Maclaurin
thf(fact_6975_Maclaurin2,axiom,
! [H2: real,Diff: nat > real > real,F3: real > real,N: nat] :
( ( ord_less @ real @ ( zero_zero @ real ) @ H2 )
=> ( ( ( Diff @ ( zero_zero @ nat ) )
= F3 )
=> ( ! [M4: nat,T4: real] :
( ( ( ord_less @ nat @ M4 @ N )
& ( ord_less_eq @ real @ ( zero_zero @ real ) @ T4 )
& ( ord_less_eq @ real @ T4 @ H2 ) )
=> ( has_field_derivative @ real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T4 ) @ ( topolo174197925503356063within @ real @ T4 @ ( top_top @ ( set @ real ) ) ) ) )
=> ? [T4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ T4 )
& ( ord_less_eq @ real @ T4 @ H2 )
& ( ( F3 @ H2 )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M2: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M2 @ ( zero_zero @ real ) ) @ ( semiring_char_0_fact @ real @ M2 ) ) @ ( power_power @ real @ H2 @ M2 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N @ T4 ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ H2 @ N ) ) ) ) ) ) ) ) ).
% Maclaurin2
thf(fact_6976_Maclaurin__minus,axiom,
! [H2: real,N: nat,Diff: nat > real > real,F3: real > real] :
( ( ord_less @ real @ H2 @ ( zero_zero @ real ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ( Diff @ ( zero_zero @ nat ) )
= F3 )
=> ( ! [M4: nat,T4: real] :
( ( ( ord_less @ nat @ M4 @ N )
& ( ord_less_eq @ real @ H2 @ T4 )
& ( ord_less_eq @ real @ T4 @ ( zero_zero @ real ) ) )
=> ( has_field_derivative @ real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T4 ) @ ( topolo174197925503356063within @ real @ T4 @ ( top_top @ ( set @ real ) ) ) ) )
=> ? [T4: real] :
( ( ord_less @ real @ H2 @ T4 )
& ( ord_less @ real @ T4 @ ( zero_zero @ real ) )
& ( ( F3 @ H2 )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M2: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M2 @ ( zero_zero @ real ) ) @ ( semiring_char_0_fact @ real @ M2 ) ) @ ( power_power @ real @ H2 @ M2 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N @ T4 ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ H2 @ N ) ) ) ) ) ) ) ) ) ).
% Maclaurin_minus
thf(fact_6977_Maclaurin__all__lt,axiom,
! [Diff: nat > real > real,F3: real > real,N: nat,X: real] :
( ( ( Diff @ ( zero_zero @ nat ) )
= F3 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( X
!= ( zero_zero @ real ) )
=> ( ! [M4: nat,X5: real] : ( has_field_derivative @ real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ X5 ) @ ( topolo174197925503356063within @ real @ X5 @ ( top_top @ ( set @ real ) ) ) )
=> ? [T4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( abs_abs @ real @ T4 ) )
& ( ord_less @ real @ ( abs_abs @ real @ T4 ) @ ( abs_abs @ real @ X ) )
& ( ( F3 @ X )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M2: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M2 @ ( zero_zero @ real ) ) @ ( semiring_char_0_fact @ real @ M2 ) ) @ ( power_power @ real @ X @ M2 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N @ T4 ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ X @ N ) ) ) ) ) ) ) ) ) ).
% Maclaurin_all_lt
thf(fact_6978_Maclaurin__bi__le,axiom,
! [Diff: nat > real > real,F3: real > real,N: nat,X: real] :
( ( ( Diff @ ( zero_zero @ nat ) )
= F3 )
=> ( ! [M4: nat,T4: real] :
( ( ( ord_less @ nat @ M4 @ N )
& ( ord_less_eq @ real @ ( abs_abs @ real @ T4 ) @ ( abs_abs @ real @ X ) ) )
=> ( has_field_derivative @ real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T4 ) @ ( topolo174197925503356063within @ real @ T4 @ ( top_top @ ( set @ real ) ) ) ) )
=> ? [T4: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ T4 ) @ ( abs_abs @ real @ X ) )
& ( ( F3 @ X )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M2: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M2 @ ( zero_zero @ real ) ) @ ( semiring_char_0_fact @ real @ M2 ) ) @ ( power_power @ real @ X @ M2 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N @ T4 ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ X @ N ) ) ) ) ) ) ) ).
% Maclaurin_bi_le
thf(fact_6979_Taylor,axiom,
! [N: nat,Diff: nat > real > real,F3: real > real,A2: real,B2: real,C2: real,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ( Diff @ ( zero_zero @ nat ) )
= F3 )
=> ( ! [M4: nat,T4: real] :
( ( ( ord_less @ nat @ M4 @ N )
& ( ord_less_eq @ real @ A2 @ T4 )
& ( ord_less_eq @ real @ T4 @ B2 ) )
=> ( has_field_derivative @ real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T4 ) @ ( topolo174197925503356063within @ real @ T4 @ ( top_top @ ( set @ real ) ) ) ) )
=> ( ( ord_less_eq @ real @ A2 @ C2 )
=> ( ( ord_less_eq @ real @ C2 @ B2 )
=> ( ( ord_less_eq @ real @ A2 @ X )
=> ( ( ord_less_eq @ real @ X @ B2 )
=> ( ( X != C2 )
=> ? [T4: real] :
( ( ( ord_less @ real @ X @ C2 )
=> ( ( ord_less @ real @ X @ T4 )
& ( ord_less @ real @ T4 @ C2 ) ) )
& ( ~ ( ord_less @ real @ X @ C2 )
=> ( ( ord_less @ real @ C2 @ T4 )
& ( ord_less @ real @ T4 @ X ) ) )
& ( ( F3 @ X )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M2: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M2 @ C2 ) @ ( semiring_char_0_fact @ real @ M2 ) ) @ ( power_power @ real @ ( minus_minus @ real @ X @ C2 ) @ M2 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N @ T4 ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ ( minus_minus @ real @ X @ C2 ) @ N ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% Taylor
thf(fact_6980_Taylor__up,axiom,
! [N: nat,Diff: nat > real > real,F3: real > real,A2: real,B2: real,C2: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ( Diff @ ( zero_zero @ nat ) )
= F3 )
=> ( ! [M4: nat,T4: real] :
( ( ( ord_less @ nat @ M4 @ N )
& ( ord_less_eq @ real @ A2 @ T4 )
& ( ord_less_eq @ real @ T4 @ B2 ) )
=> ( has_field_derivative @ real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T4 ) @ ( topolo174197925503356063within @ real @ T4 @ ( top_top @ ( set @ real ) ) ) ) )
=> ( ( ord_less_eq @ real @ A2 @ C2 )
=> ( ( ord_less @ real @ C2 @ B2 )
=> ? [T4: real] :
( ( ord_less @ real @ C2 @ T4 )
& ( ord_less @ real @ T4 @ B2 )
& ( ( F3 @ B2 )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M2: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M2 @ C2 ) @ ( semiring_char_0_fact @ real @ M2 ) ) @ ( power_power @ real @ ( minus_minus @ real @ B2 @ C2 ) @ M2 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N @ T4 ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ ( minus_minus @ real @ B2 @ C2 ) @ N ) ) ) ) ) ) ) ) ) ) ).
% Taylor_up
thf(fact_6981_Taylor__down,axiom,
! [N: nat,Diff: nat > real > real,F3: real > real,A2: real,B2: real,C2: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ( Diff @ ( zero_zero @ nat ) )
= F3 )
=> ( ! [M4: nat,T4: real] :
( ( ( ord_less @ nat @ M4 @ N )
& ( ord_less_eq @ real @ A2 @ T4 )
& ( ord_less_eq @ real @ T4 @ B2 ) )
=> ( has_field_derivative @ real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T4 ) @ ( topolo174197925503356063within @ real @ T4 @ ( top_top @ ( set @ real ) ) ) ) )
=> ( ( ord_less @ real @ A2 @ C2 )
=> ( ( ord_less_eq @ real @ C2 @ B2 )
=> ? [T4: real] :
( ( ord_less @ real @ A2 @ T4 )
& ( ord_less @ real @ T4 @ C2 )
& ( ( F3 @ A2 )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M2: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M2 @ C2 ) @ ( semiring_char_0_fact @ real @ M2 ) ) @ ( power_power @ real @ ( minus_minus @ real @ A2 @ C2 ) @ M2 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N @ T4 ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ ( minus_minus @ real @ A2 @ C2 ) @ N ) ) ) ) ) ) ) ) ) ) ).
% Taylor_down
thf(fact_6982_Maclaurin__lemma2,axiom,
! [N: nat,H2: real,Diff: nat > real > real,K2: nat,B5: real] :
( ! [M4: nat,T4: real] :
( ( ( ord_less @ nat @ M4 @ N )
& ( ord_less_eq @ real @ ( zero_zero @ real ) @ T4 )
& ( ord_less_eq @ real @ T4 @ H2 ) )
=> ( has_field_derivative @ real @ ( Diff @ M4 ) @ ( Diff @ ( suc @ M4 ) @ T4 ) @ ( topolo174197925503356063within @ real @ T4 @ ( top_top @ ( set @ real ) ) ) ) )
=> ( ( N
= ( suc @ K2 ) )
=> ! [M3: nat,T8: real] :
( ( ( ord_less @ nat @ M3 @ N )
& ( ord_less_eq @ real @ ( zero_zero @ real ) @ T8 )
& ( ord_less_eq @ real @ T8 @ H2 ) )
=> ( has_field_derivative @ real
@ ^ [U2: real] :
( minus_minus @ real @ ( Diff @ M3 @ U2 )
@ ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [P4: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ ( plus_plus @ nat @ M3 @ P4 ) @ ( zero_zero @ real ) ) @ ( semiring_char_0_fact @ real @ P4 ) ) @ ( power_power @ real @ U2 @ P4 ) )
@ ( set_ord_lessThan @ nat @ ( minus_minus @ nat @ N @ M3 ) ) )
@ ( times_times @ real @ B5 @ ( divide_divide @ real @ ( power_power @ real @ U2 @ ( minus_minus @ nat @ N @ M3 ) ) @ ( semiring_char_0_fact @ real @ ( minus_minus @ nat @ N @ M3 ) ) ) ) ) )
@ ( minus_minus @ real @ ( Diff @ ( suc @ M3 ) @ T8 )
@ ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [P4: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ ( plus_plus @ nat @ ( suc @ M3 ) @ P4 ) @ ( zero_zero @ real ) ) @ ( semiring_char_0_fact @ real @ P4 ) ) @ ( power_power @ real @ T8 @ P4 ) )
@ ( set_ord_lessThan @ nat @ ( minus_minus @ nat @ N @ ( suc @ M3 ) ) ) )
@ ( times_times @ real @ B5 @ ( divide_divide @ real @ ( power_power @ real @ T8 @ ( minus_minus @ nat @ N @ ( suc @ M3 ) ) ) @ ( semiring_char_0_fact @ real @ ( minus_minus @ nat @ N @ ( suc @ M3 ) ) ) ) ) ) )
@ ( topolo174197925503356063within @ real @ T8 @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ).
% Maclaurin_lemma2
thf(fact_6983_DERIV__arctan__series,axiom,
! [X: real] :
( ( ord_less @ real @ ( abs_abs @ real @ X ) @ ( one_one @ real ) )
=> ( has_field_derivative @ real
@ ^ [X7: real] :
( suminf @ real
@ ^ [K3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ K3 ) @ ( times_times @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( semiring_1_of_nat @ real @ ( plus_plus @ nat @ ( times_times @ nat @ K3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) @ ( power_power @ real @ X7 @ ( plus_plus @ nat @ ( times_times @ nat @ K3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) ) )
@ ( suminf @ real
@ ^ [K3: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ K3 ) @ ( power_power @ real @ X @ ( times_times @ nat @ K3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) ) ) ).
% DERIV_arctan_series
thf(fact_6984_DERIV__even__real__root,axiom,
! [N: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( ord_less @ real @ X @ ( zero_zero @ real ) )
=> ( has_field_derivative @ real @ ( root @ N ) @ ( inverse_inverse @ real @ ( times_times @ real @ ( uminus_uminus @ real @ ( semiring_1_of_nat @ real @ N ) ) @ ( power_power @ real @ ( root @ N @ X ) @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ).
% DERIV_even_real_root
thf(fact_6985_has__derivative__arcsin,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [G: A > real,X: A,G5: A > real,S: set @ A] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ ( G @ X ) )
=> ( ( ord_less @ real @ ( G @ X ) @ ( one_one @ real ) )
=> ( ( has_derivative @ A @ real @ G @ G5 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( has_derivative @ A @ real
@ ^ [X4: A] : ( arcsin @ ( G @ X4 ) )
@ ^ [X4: A] : ( times_times @ real @ ( G5 @ X4 ) @ ( inverse_inverse @ real @ ( sqrt @ ( minus_minus @ real @ ( one_one @ real ) @ ( power_power @ real @ ( G @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ) ).
% has_derivative_arcsin
thf(fact_6986_has__derivative__arccos,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [G: A > real,X: A,G5: A > real,S: set @ A] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ ( G @ X ) )
=> ( ( ord_less @ real @ ( G @ X ) @ ( one_one @ real ) )
=> ( ( has_derivative @ A @ real @ G @ G5 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( has_derivative @ A @ real
@ ^ [X4: A] : ( arccos @ ( G @ X4 ) )
@ ^ [X4: A] : ( times_times @ real @ ( G5 @ X4 ) @ ( inverse_inverse @ real @ ( uminus_uminus @ real @ ( sqrt @ ( minus_minus @ real @ ( one_one @ real ) @ ( power_power @ real @ ( G @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ) ).
% has_derivative_arccos
thf(fact_6987_has__derivative__compose,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B )
& ( real_V822414075346904944vector @ C ) )
=> ! [F3: A > B,F9: A > B,X: A,S: set @ A,G: B > C,G5: B > C] :
( ( has_derivative @ A @ B @ F3 @ F9 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( ( has_derivative @ B @ C @ G @ G5 @ ( topolo174197925503356063within @ B @ ( F3 @ X ) @ ( top_top @ ( set @ B ) ) ) )
=> ( has_derivative @ A @ C
@ ^ [X4: A] : ( G @ ( F3 @ X4 ) )
@ ^ [X4: A] : ( G5 @ ( F9 @ X4 ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ).
% has_derivative_compose
thf(fact_6988_has__field__derivative__imp__has__derivative,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,D4: A,F4: filter @ A] :
( ( has_field_derivative @ A @ F3 @ D4 @ F4 )
=> ( has_derivative @ A @ A @ F3 @ ( times_times @ A @ D4 ) @ F4 ) ) ) ).
% has_field_derivative_imp_has_derivative
thf(fact_6989_has__derivative__imp__has__field__derivative,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,D4: A > A,F4: filter @ A,D7: A] :
( ( has_derivative @ A @ A @ F3 @ D4 @ F4 )
=> ( ! [X5: A] :
( ( times_times @ A @ X5 @ D7 )
= ( D4 @ X5 ) )
=> ( has_field_derivative @ A @ F3 @ D7 @ F4 ) ) ) ) ).
% has_derivative_imp_has_field_derivative
thf(fact_6990_has__field__derivative__def,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ( ( has_field_derivative @ A )
= ( ^ [F5: A > A,D8: A] : ( has_derivative @ A @ A @ F5 @ ( times_times @ A @ D8 ) ) ) ) ) ).
% has_field_derivative_def
thf(fact_6991_has__derivative__scaleR,axiom,
! [C: $tType,D: $tType] :
( ( ( real_V822414075346904944vector @ D )
& ( real_V822414075346904944vector @ C ) )
=> ! [F3: D > real,F9: D > real,X: D,S: set @ D,G: D > C,G5: D > C] :
( ( has_derivative @ D @ real @ F3 @ F9 @ ( topolo174197925503356063within @ D @ X @ S ) )
=> ( ( has_derivative @ D @ C @ G @ G5 @ ( topolo174197925503356063within @ D @ X @ S ) )
=> ( has_derivative @ D @ C
@ ^ [X4: D] : ( real_V8093663219630862766scaleR @ C @ ( F3 @ X4 ) @ ( G @ X4 ) )
@ ^ [H: D] : ( plus_plus @ C @ ( real_V8093663219630862766scaleR @ C @ ( F3 @ X ) @ ( G5 @ H ) ) @ ( real_V8093663219630862766scaleR @ C @ ( F9 @ H ) @ ( G @ X ) ) )
@ ( topolo174197925503356063within @ D @ X @ S ) ) ) ) ) ).
% has_derivative_scaleR
thf(fact_6992_has__derivative__in__compose,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B )
& ( real_V822414075346904944vector @ C ) )
=> ! [F3: A > B,F9: A > B,X: A,S: set @ A,G: B > C,G5: B > C] :
( ( has_derivative @ A @ B @ F3 @ F9 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( ( has_derivative @ B @ C @ G @ G5 @ ( topolo174197925503356063within @ B @ ( F3 @ X ) @ ( image @ A @ B @ F3 @ S ) ) )
=> ( has_derivative @ A @ C
@ ^ [X4: A] : ( G @ ( F3 @ X4 ) )
@ ^ [X4: A] : ( G5 @ ( F9 @ X4 ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ).
% has_derivative_in_compose
thf(fact_6993_has__derivative__subset,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F3: A > B,F9: A > B,X: A,S: set @ A,T2: set @ A] :
( ( has_derivative @ A @ B @ F3 @ F9 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( ( ord_less_eq @ ( set @ A ) @ T2 @ S )
=> ( has_derivative @ A @ B @ F3 @ F9 @ ( topolo174197925503356063within @ A @ X @ T2 ) ) ) ) ) ).
% has_derivative_subset
thf(fact_6994_has__derivative__sum,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( ( real_V822414075346904944vector @ B )
& ( real_V822414075346904944vector @ C ) )
=> ! [I6: set @ A,F3: A > B > C,F9: A > B > C,F4: filter @ B] :
( ! [I3: A] :
( ( member @ A @ I3 @ I6 )
=> ( has_derivative @ B @ C @ ( F3 @ I3 ) @ ( F9 @ I3 ) @ F4 ) )
=> ( has_derivative @ B @ C
@ ^ [X4: B] :
( groups7311177749621191930dd_sum @ A @ C
@ ^ [I: A] : ( F3 @ I @ X4 )
@ I6 )
@ ^ [X4: B] :
( groups7311177749621191930dd_sum @ A @ C
@ ^ [I: A] : ( F9 @ I @ X4 )
@ I6 )
@ F4 ) ) ) ).
% has_derivative_sum
thf(fact_6995_has__derivative__minus,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F3: A > B,F9: A > B,F4: filter @ A] :
( ( has_derivative @ A @ B @ F3 @ F9 @ F4 )
=> ( has_derivative @ A @ B
@ ^ [X4: A] : ( uminus_uminus @ B @ ( F3 @ X4 ) )
@ ^ [X4: A] : ( uminus_uminus @ B @ ( F9 @ X4 ) )
@ F4 ) ) ) ).
% has_derivative_minus
thf(fact_6996_has__derivative__diff,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F3: A > B,F9: A > B,F4: filter @ A,G: A > B,G5: A > B] :
( ( has_derivative @ A @ B @ F3 @ F9 @ F4 )
=> ( ( has_derivative @ A @ B @ G @ G5 @ F4 )
=> ( has_derivative @ A @ B
@ ^ [X4: A] : ( minus_minus @ B @ ( F3 @ X4 ) @ ( G @ X4 ) )
@ ^ [X4: A] : ( minus_minus @ B @ ( F9 @ X4 ) @ ( G5 @ X4 ) )
@ F4 ) ) ) ) ).
% has_derivative_diff
thf(fact_6997_has__derivative__mult__left,axiom,
! [A: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V4412858255891104859lgebra @ A ) )
=> ! [G: C > A,G5: C > A,F4: filter @ C,Y2: A] :
( ( has_derivative @ C @ A @ G @ G5 @ F4 )
=> ( has_derivative @ C @ A
@ ^ [X4: C] : ( times_times @ A @ ( G @ X4 ) @ Y2 )
@ ^ [X4: C] : ( times_times @ A @ ( G5 @ X4 ) @ Y2 )
@ F4 ) ) ) ).
% has_derivative_mult_left
thf(fact_6998_has__derivative__mult__right,axiom,
! [A: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V4412858255891104859lgebra @ A ) )
=> ! [G: C > A,G5: C > A,F4: filter @ C,X: A] :
( ( has_derivative @ C @ A @ G @ G5 @ F4 )
=> ( has_derivative @ C @ A
@ ^ [X4: C] : ( times_times @ A @ X @ ( G @ X4 ) )
@ ^ [X4: C] : ( times_times @ A @ X @ ( G5 @ X4 ) )
@ F4 ) ) ) ).
% has_derivative_mult_right
thf(fact_6999_has__derivative__add,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F3: A > B,F9: A > B,F4: filter @ A,G: A > B,G5: A > B] :
( ( has_derivative @ A @ B @ F3 @ F9 @ F4 )
=> ( ( has_derivative @ A @ B @ G @ G5 @ F4 )
=> ( has_derivative @ A @ B
@ ^ [X4: A] : ( plus_plus @ B @ ( F3 @ X4 ) @ ( G @ X4 ) )
@ ^ [X4: A] : ( plus_plus @ B @ ( F9 @ X4 ) @ ( G5 @ X4 ) )
@ F4 ) ) ) ) ).
% has_derivative_add
thf(fact_7000_has__derivative__ident,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F4: filter @ A] :
( has_derivative @ A @ A
@ ^ [X4: A] : X4
@ ^ [X4: A] : X4
@ F4 ) ) ).
% has_derivative_ident
thf(fact_7001_has__derivative__of__real,axiom,
! [A: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V2191834092415804123ebra_1 @ A )
& ( real_V822414075346904944vector @ A ) )
=> ! [G: C > real,G5: C > real,F4: filter @ C] :
( ( has_derivative @ C @ real @ G @ G5 @ F4 )
=> ( has_derivative @ C @ A
@ ^ [X4: C] : ( real_Vector_of_real @ A @ ( G @ X4 ) )
@ ^ [X4: C] : ( real_Vector_of_real @ A @ ( G5 @ X4 ) )
@ F4 ) ) ) ).
% has_derivative_of_real
thf(fact_7002_has__derivative__scaleR__right,axiom,
! [B: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V822414075346904944vector @ B ) )
=> ! [G: C > B,G5: C > B,F4: filter @ C,R: real] :
( ( has_derivative @ C @ B @ G @ G5 @ F4 )
=> ( has_derivative @ C @ B
@ ^ [X4: C] : ( real_V8093663219630862766scaleR @ B @ R @ ( G @ X4 ) )
@ ^ [X4: C] : ( real_V8093663219630862766scaleR @ B @ R @ ( G5 @ X4 ) )
@ F4 ) ) ) ).
% has_derivative_scaleR_right
thf(fact_7003_has__derivative__scaleR__left,axiom,
! [B: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V822414075346904944vector @ B ) )
=> ! [G: C > real,G5: C > real,F4: filter @ C,X: B] :
( ( has_derivative @ C @ real @ G @ G5 @ F4 )
=> ( has_derivative @ C @ B
@ ^ [X4: C] : ( real_V8093663219630862766scaleR @ B @ ( G @ X4 ) @ X )
@ ^ [X4: C] : ( real_V8093663219630862766scaleR @ B @ ( G5 @ X4 ) @ X )
@ F4 ) ) ) ).
% has_derivative_scaleR_left
thf(fact_7004_has__derivative__const,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [C2: B,F4: filter @ A] :
( has_derivative @ A @ B
@ ^ [X4: A] : C2
@ ^ [X4: A] : ( zero_zero @ B )
@ F4 ) ) ).
% has_derivative_const
thf(fact_7005_has__derivative__mult,axiom,
! [A: $tType,D: $tType] :
( ( ( real_V822414075346904944vector @ D )
& ( real_V4412858255891104859lgebra @ A ) )
=> ! [F3: D > A,F9: D > A,X: D,S: set @ D,G: D > A,G5: D > A] :
( ( has_derivative @ D @ A @ F3 @ F9 @ ( topolo174197925503356063within @ D @ X @ S ) )
=> ( ( has_derivative @ D @ A @ G @ G5 @ ( topolo174197925503356063within @ D @ X @ S ) )
=> ( has_derivative @ D @ A
@ ^ [X4: D] : ( times_times @ A @ ( F3 @ X4 ) @ ( G @ X4 ) )
@ ^ [H: D] : ( plus_plus @ A @ ( times_times @ A @ ( F3 @ X ) @ ( G5 @ H ) ) @ ( times_times @ A @ ( F9 @ H ) @ ( G @ X ) ) )
@ ( topolo174197925503356063within @ D @ X @ S ) ) ) ) ) ).
% has_derivative_mult
thf(fact_7006_has__derivative__zero__unique,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F4: A > B,X: A] :
( ( has_derivative @ A @ B
@ ^ [X4: A] : ( zero_zero @ B )
@ F4
@ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) )
=> ( F4
= ( ^ [H: A] : ( zero_zero @ B ) ) ) ) ) ).
% has_derivative_zero_unique
thf(fact_7007_has__derivative__in__compose2,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [T2: set @ A,G: A > B,G5: A > A > B,F3: C > A,S: set @ C,X: C,F9: C > A] :
( ! [X5: A] :
( ( member @ A @ X5 @ T2 )
=> ( has_derivative @ A @ B @ G @ ( G5 @ X5 ) @ ( topolo174197925503356063within @ A @ X5 @ T2 ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image @ C @ A @ F3 @ S ) @ T2 )
=> ( ( member @ C @ X @ S )
=> ( ( has_derivative @ C @ A @ F3 @ F9 @ ( topolo174197925503356063within @ C @ X @ S ) )
=> ( has_derivative @ C @ B
@ ^ [X4: C] : ( G @ ( F3 @ X4 ) )
@ ^ [Y: C] : ( G5 @ ( F3 @ X ) @ ( F9 @ Y ) )
@ ( topolo174197925503356063within @ C @ X @ S ) ) ) ) ) ) ) ).
% has_derivative_in_compose2
thf(fact_7008_has__derivative__exp,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [G: A > real,G5: A > real,X: A,S: set @ A] :
( ( has_derivative @ A @ real @ G @ G5 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( has_derivative @ A @ real
@ ^ [X4: A] : ( exp @ real @ ( G @ X4 ) )
@ ^ [X4: A] : ( times_times @ real @ ( G5 @ X4 ) @ ( exp @ real @ ( G @ X ) ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ).
% has_derivative_exp
thf(fact_7009_has__derivative__sin,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [G: A > real,G5: A > real,X: A,S: set @ A] :
( ( has_derivative @ A @ real @ G @ G5 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( has_derivative @ A @ real
@ ^ [X4: A] : ( sin @ real @ ( G @ X4 ) )
@ ^ [X4: A] : ( times_times @ real @ ( G5 @ X4 ) @ ( cos @ real @ ( G @ X ) ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ).
% has_derivative_sin
thf(fact_7010_has__derivative__cosh,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [G: A > A,Db: A,X: A,S: set @ A] :
( ( has_derivative @ A @ A @ G @ ( times_times @ A @ Db ) @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( has_derivative @ A @ A
@ ^ [X4: A] : ( cosh @ A @ ( G @ X4 ) )
@ ( times_times @ A @ ( times_times @ A @ ( sinh @ A @ ( G @ X ) ) @ Db ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ).
% has_derivative_cosh
thf(fact_7011_has__derivative__sinh,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [G: A > A,Db: A,X: A,S: set @ A] :
( ( has_derivative @ A @ A @ G @ ( times_times @ A @ Db ) @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( has_derivative @ A @ A
@ ^ [X4: A] : ( sinh @ A @ ( G @ X4 ) )
@ ( times_times @ A @ ( times_times @ A @ ( cosh @ A @ ( G @ X ) ) @ Db ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ).
% has_derivative_sinh
thf(fact_7012_has__derivative__divide_H,axiom,
! [A: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V3459762299906320749_field @ A ) )
=> ! [F3: C > A,F9: C > A,X: C,S3: set @ C,G: C > A,G5: C > A] :
( ( has_derivative @ C @ A @ F3 @ F9 @ ( topolo174197925503356063within @ C @ X @ S3 ) )
=> ( ( has_derivative @ C @ A @ G @ G5 @ ( topolo174197925503356063within @ C @ X @ S3 ) )
=> ( ( ( G @ X )
!= ( zero_zero @ A ) )
=> ( has_derivative @ C @ A
@ ^ [X4: C] : ( divide_divide @ A @ ( F3 @ X4 ) @ ( G @ X4 ) )
@ ^ [H: C] : ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ ( F9 @ H ) @ ( G @ X ) ) @ ( times_times @ A @ ( F3 @ X ) @ ( G5 @ H ) ) ) @ ( times_times @ A @ ( G @ X ) @ ( G @ X ) ) )
@ ( topolo174197925503356063within @ C @ X @ S3 ) ) ) ) ) ) ).
% has_derivative_divide'
thf(fact_7013_has__derivative__inverse_H,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [X: A,S3: set @ A] :
( ( X
!= ( zero_zero @ A ) )
=> ( has_derivative @ A @ A @ ( inverse_inverse @ A )
@ ^ [H: A] : ( uminus_uminus @ A @ ( times_times @ A @ ( times_times @ A @ ( inverse_inverse @ A @ X ) @ H ) @ ( inverse_inverse @ A @ X ) ) )
@ ( topolo174197925503356063within @ A @ X @ S3 ) ) ) ) ).
% has_derivative_inverse'
thf(fact_7014_has__derivative__inverse,axiom,
! [A: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V8999393235501362500lgebra @ A ) )
=> ! [F3: C > A,X: C,F9: C > A,S3: set @ C] :
( ( ( F3 @ X )
!= ( zero_zero @ A ) )
=> ( ( has_derivative @ C @ A @ F3 @ F9 @ ( topolo174197925503356063within @ C @ X @ S3 ) )
=> ( has_derivative @ C @ A
@ ^ [X4: C] : ( inverse_inverse @ A @ ( F3 @ X4 ) )
@ ^ [H: C] : ( uminus_uminus @ A @ ( times_times @ A @ ( times_times @ A @ ( inverse_inverse @ A @ ( F3 @ X ) ) @ ( F9 @ H ) ) @ ( inverse_inverse @ A @ ( F3 @ X ) ) ) )
@ ( topolo174197925503356063within @ C @ X @ S3 ) ) ) ) ) ).
% has_derivative_inverse
thf(fact_7015_DERIV__compose__FDERIV,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: real > real,F9: real,G: A > real,X: A,G5: A > real,S: set @ A] :
( ( has_field_derivative @ real @ F3 @ F9 @ ( topolo174197925503356063within @ real @ ( G @ X ) @ ( top_top @ ( set @ real ) ) ) )
=> ( ( has_derivative @ A @ real @ G @ G5 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( has_derivative @ A @ real
@ ^ [X4: A] : ( F3 @ ( G @ X4 ) )
@ ^ [X4: A] : ( times_times @ real @ ( G5 @ X4 ) @ F9 )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ).
% DERIV_compose_FDERIV
thf(fact_7016_has__derivative__cos,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [G: A > real,G5: A > real,X: A,S: set @ A] :
( ( has_derivative @ A @ real @ G @ G5 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( has_derivative @ A @ real
@ ^ [X4: A] : ( cos @ real @ ( G @ X4 ) )
@ ^ [X4: A] : ( times_times @ real @ ( G5 @ X4 ) @ ( uminus_uminus @ real @ ( sin @ real @ ( G @ X ) ) ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ).
% has_derivative_cos
thf(fact_7017_has__derivative__power,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V3459762299906320749_field @ B ) )
=> ! [F3: A > B,F9: A > B,X: A,S3: set @ A,N: nat] :
( ( has_derivative @ A @ B @ F3 @ F9 @ ( topolo174197925503356063within @ A @ X @ S3 ) )
=> ( has_derivative @ A @ B
@ ^ [X4: A] : ( power_power @ B @ ( F3 @ X4 ) @ N )
@ ^ [Y: A] : ( times_times @ B @ ( times_times @ B @ ( semiring_1_of_nat @ B @ N ) @ ( F9 @ Y ) ) @ ( power_power @ B @ ( F3 @ X ) @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) )
@ ( topolo174197925503356063within @ A @ X @ S3 ) ) ) ) ).
% has_derivative_power
thf(fact_7018_has__derivative__ln,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [G: A > real,X: A,G5: A > real,S: set @ A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( G @ X ) )
=> ( ( has_derivative @ A @ real @ G @ G5 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( has_derivative @ A @ real
@ ^ [X4: A] : ( ln_ln @ real @ ( G @ X4 ) )
@ ^ [X4: A] : ( times_times @ real @ ( G5 @ X4 ) @ ( inverse_inverse @ real @ ( G @ X ) ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ).
% has_derivative_ln
thf(fact_7019_has__derivative__divide,axiom,
! [A: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V8999393235501362500lgebra @ A ) )
=> ! [F3: C > A,F9: C > A,X: C,S3: set @ C,G: C > A,G5: C > A] :
( ( has_derivative @ C @ A @ F3 @ F9 @ ( topolo174197925503356063within @ C @ X @ S3 ) )
=> ( ( has_derivative @ C @ A @ G @ G5 @ ( topolo174197925503356063within @ C @ X @ S3 ) )
=> ( ( ( G @ X )
!= ( zero_zero @ A ) )
=> ( has_derivative @ C @ A
@ ^ [X4: C] : ( divide_divide @ A @ ( F3 @ X4 ) @ ( G @ X4 ) )
@ ^ [H: C] : ( plus_plus @ A @ ( times_times @ A @ ( uminus_uminus @ A @ ( F3 @ X ) ) @ ( times_times @ A @ ( times_times @ A @ ( inverse_inverse @ A @ ( G @ X ) ) @ ( G5 @ H ) ) @ ( inverse_inverse @ A @ ( G @ X ) ) ) ) @ ( divide_divide @ A @ ( F9 @ H ) @ ( G @ X ) ) )
@ ( topolo174197925503356063within @ C @ X @ S3 ) ) ) ) ) ) ).
% has_derivative_divide
thf(fact_7020_has__derivative__prod,axiom,
! [B: $tType,I5: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V3459762299906320749_field @ B ) )
=> ! [I6: set @ I5,F3: I5 > A > B,F9: I5 > A > B,X: A,S3: set @ A] :
( ! [I3: I5] :
( ( member @ I5 @ I3 @ I6 )
=> ( has_derivative @ A @ B @ ( F3 @ I3 ) @ ( F9 @ I3 ) @ ( topolo174197925503356063within @ A @ X @ S3 ) ) )
=> ( has_derivative @ A @ B
@ ^ [X4: A] :
( groups7121269368397514597t_prod @ I5 @ B
@ ^ [I: I5] : ( F3 @ I @ X4 )
@ I6 )
@ ^ [Y: A] :
( groups7311177749621191930dd_sum @ I5 @ B
@ ^ [I: I5] :
( times_times @ B @ ( F9 @ I @ Y )
@ ( groups7121269368397514597t_prod @ I5 @ B
@ ^ [J4: I5] : ( F3 @ J4 @ X )
@ ( minus_minus @ ( set @ I5 ) @ I6 @ ( insert @ I5 @ I @ ( bot_bot @ ( set @ I5 ) ) ) ) ) )
@ I6 )
@ ( topolo174197925503356063within @ A @ X @ S3 ) ) ) ) ).
% has_derivative_prod
thf(fact_7021_has__derivative__powr,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [G: A > real,G5: A > real,X: A,X9: set @ A,F3: A > real,F9: A > real] :
( ( has_derivative @ A @ real @ G @ G5 @ ( topolo174197925503356063within @ A @ X @ X9 ) )
=> ( ( has_derivative @ A @ real @ F3 @ F9 @ ( topolo174197925503356063within @ A @ X @ X9 ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( G @ X ) )
=> ( ( member @ A @ X @ X9 )
=> ( has_derivative @ A @ real
@ ^ [X4: A] : ( powr @ real @ ( G @ X4 ) @ ( F3 @ X4 ) )
@ ^ [H: A] : ( times_times @ real @ ( powr @ real @ ( G @ X ) @ ( F3 @ X ) ) @ ( plus_plus @ real @ ( times_times @ real @ ( F9 @ H ) @ ( ln_ln @ real @ ( G @ X ) ) ) @ ( divide_divide @ real @ ( times_times @ real @ ( G5 @ H ) @ ( F3 @ X ) ) @ ( G @ X ) ) ) )
@ ( topolo174197925503356063within @ A @ X @ X9 ) ) ) ) ) ) ) ).
% has_derivative_powr
thf(fact_7022_has__derivative__real__sqrt,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [G: A > real,X: A,G5: A > real,S: set @ A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( G @ X ) )
=> ( ( has_derivative @ A @ real @ G @ G5 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( has_derivative @ A @ real
@ ^ [X4: A] : ( sqrt @ ( G @ X4 ) )
@ ^ [X4: A] : ( times_times @ real @ ( G5 @ X4 ) @ ( divide_divide @ real @ ( inverse_inverse @ real @ ( sqrt @ ( G @ X ) ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ).
% has_derivative_real_sqrt
thf(fact_7023_has__derivative__arctan,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [G: A > real,G5: A > real,X: A,S: set @ A] :
( ( has_derivative @ A @ real @ G @ G5 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( has_derivative @ A @ real
@ ^ [X4: A] : ( arctan @ ( G @ X4 ) )
@ ^ [X4: A] : ( times_times @ real @ ( G5 @ X4 ) @ ( inverse_inverse @ real @ ( plus_plus @ real @ ( one_one @ real ) @ ( power_power @ real @ ( G @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ).
% has_derivative_arctan
thf(fact_7024_has__derivative__tan,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [G: A > real,X: A,G5: A > real,S: set @ A] :
( ( ( cos @ real @ ( G @ X ) )
!= ( zero_zero @ real ) )
=> ( ( has_derivative @ A @ real @ G @ G5 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( has_derivative @ A @ real
@ ^ [X4: A] : ( tan @ real @ ( G @ X4 ) )
@ ^ [X4: A] : ( times_times @ real @ ( G5 @ X4 ) @ ( inverse_inverse @ real @ ( power_power @ real @ ( cos @ real @ ( G @ X ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ).
% has_derivative_tan
thf(fact_7025_has__derivative__floor,axiom,
! [Aa: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( archim2362893244070406136eiling @ Aa )
& ( topolo2564578578187576103pology @ Aa ) )
=> ! [G: A > real,X: A,F3: real > Aa,G5: A > real,S: set @ A] :
( ( topolo3448309680560233919inuous @ real @ Aa @ ( topolo174197925503356063within @ real @ ( G @ X ) @ ( top_top @ ( set @ real ) ) ) @ F3 )
=> ( ~ ( member @ Aa @ ( F3 @ ( G @ X ) ) @ ( ring_1_Ints @ Aa ) )
=> ( ( has_derivative @ A @ real @ G @ G5 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( has_derivative @ A @ real
@ ^ [X4: A] : ( ring_1_of_int @ real @ ( archim6421214686448440834_floor @ Aa @ ( F3 @ ( G @ X4 ) ) ) )
@ ^ [X4: A] : ( times_times @ real @ ( G5 @ X4 ) @ ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ) ).
% has_derivative_floor
thf(fact_7026_termdiffs__aux,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [C2: nat > A,K5: A,X: A] :
( ( summable @ A
@ ^ [N2: nat] : ( times_times @ A @ ( diffs @ A @ ( diffs @ A @ C2 ) @ N2 ) @ ( power_power @ A @ K5 @ N2 ) ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ ( real_V7770717601297561774m_norm @ A @ K5 ) )
=> ( filterlim @ A @ A
@ ^ [H: A] :
( suminf @ A
@ ^ [N2: nat] : ( times_times @ A @ ( C2 @ N2 ) @ ( minus_minus @ A @ ( divide_divide @ A @ ( minus_minus @ A @ ( power_power @ A @ ( plus_plus @ A @ X @ H ) @ N2 ) @ ( power_power @ A @ X @ N2 ) ) @ H ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( power_power @ A @ X @ ( minus_minus @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% termdiffs_aux
thf(fact_7027_tendsto__const,axiom,
! [A: $tType,B: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [K2: A,F4: filter @ B] :
( filterlim @ B @ A
@ ^ [X4: B] : K2
@ ( topolo7230453075368039082e_nhds @ A @ K2 )
@ F4 ) ) ).
% tendsto_const
thf(fact_7028_tendsto__ident__at,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [A2: A,S: set @ A] :
( filterlim @ A @ A
@ ^ [X4: A] : X4
@ ( topolo7230453075368039082e_nhds @ A @ A2 )
@ ( topolo174197925503356063within @ A @ A2 @ S ) ) ) ).
% tendsto_ident_at
thf(fact_7029_tendsto__mult__left__iff,axiom,
! [A: $tType,B: $tType] :
( ( ( field @ A )
& ( topolo4211221413907600880p_mult @ A ) )
=> ! [C2: A,F3: B > A,L: A,F4: filter @ B] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( filterlim @ B @ A
@ ^ [X4: B] : ( times_times @ A @ C2 @ ( F3 @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( times_times @ A @ C2 @ L ) )
@ F4 )
= ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F4 ) ) ) ) ).
% tendsto_mult_left_iff
thf(fact_7030_tendsto__mult__right__iff,axiom,
! [A: $tType,B: $tType] :
( ( ( field @ A )
& ( topolo4211221413907600880p_mult @ A ) )
=> ! [C2: A,F3: B > A,L: A,F4: filter @ B] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( filterlim @ B @ A
@ ^ [X4: B] : ( times_times @ A @ ( F3 @ X4 ) @ C2 )
@ ( topolo7230453075368039082e_nhds @ A @ ( times_times @ A @ L @ C2 ) )
@ F4 )
= ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F4 ) ) ) ) ).
% tendsto_mult_right_iff
thf(fact_7031_power__tendsto__0__iff,axiom,
! [A: $tType,N: nat,F3: A > real,F4: filter @ A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( filterlim @ A @ real
@ ^ [X4: A] : ( power_power @ real @ ( F3 @ X4 ) @ N )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ F4 )
= ( filterlim @ A @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ F4 ) ) ) ).
% power_tendsto_0_iff
thf(fact_7032_isCont__Pair,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( topolo4958980785337419405_space @ B )
& ( topolo4958980785337419405_space @ C ) )
=> ! [A2: A,F3: A > B,G: A > C] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) @ F3 )
=> ( ( topolo3448309680560233919inuous @ A @ C @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) @ G )
=> ( topolo3448309680560233919inuous @ A @ ( product_prod @ B @ C ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) )
@ ^ [X4: A] : ( product_Pair @ B @ C @ ( F3 @ X4 ) @ ( G @ X4 ) ) ) ) ) ) ).
% isCont_Pair
thf(fact_7033_LIM__offset__zero__iff,axiom,
! [C: $tType,D: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( topolo4958980785337419405_space @ D )
& ( zero @ C ) )
=> ! [A2: A,F3: A > D,L6: D] :
( ( nO_MATCH @ C @ A @ ( zero_zero @ C ) @ A2 )
=> ( ( filterlim @ A @ D @ F3 @ ( topolo7230453075368039082e_nhds @ D @ L6 ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) )
= ( filterlim @ A @ D
@ ^ [H: A] : ( F3 @ ( plus_plus @ A @ A2 @ H ) )
@ ( topolo7230453075368039082e_nhds @ D @ L6 )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% LIM_offset_zero_iff
thf(fact_7034_LIM__offset,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [F3: A > B,L6: B,A2: A,K2: A] :
( ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ L6 ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) )
=> ( filterlim @ A @ B
@ ^ [X4: A] : ( F3 @ ( plus_plus @ A @ X4 @ K2 ) )
@ ( topolo7230453075368039082e_nhds @ B @ L6 )
@ ( topolo174197925503356063within @ A @ ( minus_minus @ A @ A2 @ K2 ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% LIM_offset
thf(fact_7035_continuous__within__compose3,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( ( topological_t2_space @ C )
& ( topolo4958980785337419405_space @ B )
& ( topological_t2_space @ A ) )
=> ! [F3: C > A,X: C,G: A > B,S: set @ C] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ ( F3 @ X ) @ ( top_top @ ( set @ A ) ) ) @ G )
=> ( ( topolo3448309680560233919inuous @ C @ A @ ( topolo174197925503356063within @ C @ X @ S ) @ F3 )
=> ( topolo3448309680560233919inuous @ C @ B @ ( topolo174197925503356063within @ C @ X @ S )
@ ^ [X4: C] : ( G @ ( F3 @ X4 ) ) ) ) ) ) ).
% continuous_within_compose3
thf(fact_7036_isCont__tendsto__compose,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( ( topolo4958980785337419405_space @ B )
& ( topological_t2_space @ A ) )
=> ! [L: A,G: A > B,F3: C > A,F4: filter @ C] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ L @ ( top_top @ ( set @ A ) ) ) @ G )
=> ( ( filterlim @ C @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F4 )
=> ( filterlim @ C @ B
@ ^ [X4: C] : ( G @ ( F3 @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( G @ L ) )
@ F4 ) ) ) ) ).
% isCont_tendsto_compose
thf(fact_7037_LIM__const__not__eq,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo8386298272705272623_space @ A )
& ( topological_t2_space @ B ) )
=> ! [K2: B,L6: B,A2: A] :
( ( K2 != L6 )
=> ~ ( filterlim @ A @ B
@ ^ [X4: A] : K2
@ ( topolo7230453075368039082e_nhds @ B @ L6 )
@ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% LIM_const_not_eq
thf(fact_7038_tendsto__compose,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( ( topolo4958980785337419405_space @ B )
& ( topolo4958980785337419405_space @ A ) )
=> ! [G: A > B,L: A,F3: C > A,F4: filter @ C] :
( ( filterlim @ A @ B @ G @ ( topolo7230453075368039082e_nhds @ B @ ( G @ L ) ) @ ( topolo174197925503356063within @ A @ L @ ( top_top @ ( set @ A ) ) ) )
=> ( ( filterlim @ C @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F4 )
=> ( filterlim @ C @ B
@ ^ [X4: C] : ( G @ ( F3 @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( G @ L ) )
@ F4 ) ) ) ) ).
% tendsto_compose
thf(fact_7039_LIM__const__eq,axiom,
! [A: $tType,B: $tType] :
( ( ( topological_t2_space @ B )
& ( topolo8386298272705272623_space @ A ) )
=> ! [K2: B,L6: B,A2: A] :
( ( filterlim @ A @ B
@ ^ [X4: A] : K2
@ ( topolo7230453075368039082e_nhds @ B @ L6 )
@ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) )
=> ( K2 = L6 ) ) ) ).
% LIM_const_eq
thf(fact_7040_isCont__o2,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( topolo4958980785337419405_space @ C )
& ( topological_t2_space @ B ) )
=> ! [A2: A,F3: A > B,G: B > C] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) @ F3 )
=> ( ( topolo3448309680560233919inuous @ B @ C @ ( topolo174197925503356063within @ B @ ( F3 @ A2 ) @ ( top_top @ ( set @ B ) ) ) @ G )
=> ( topolo3448309680560233919inuous @ A @ C @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) )
@ ^ [X4: A] : ( G @ ( F3 @ X4 ) ) ) ) ) ) ).
% isCont_o2
thf(fact_7041_filterlim__at__If,axiom,
! [B: $tType,A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F3: A > B,G6: filter @ B,X: A,P3: A > $o,G: A > B] :
( ( filterlim @ A @ B @ F3 @ G6 @ ( topolo174197925503356063within @ A @ X @ ( collect @ A @ P3 ) ) )
=> ( ( filterlim @ A @ B @ G @ G6
@ ( topolo174197925503356063within @ A @ X
@ ( collect @ A
@ ^ [X4: A] :
~ ( P3 @ X4 ) ) ) )
=> ( filterlim @ A @ B
@ ^ [X4: A] : ( if @ B @ ( P3 @ X4 ) @ ( F3 @ X4 ) @ ( G @ X4 ) )
@ G6
@ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% filterlim_at_If
thf(fact_7042_isCont__norm,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [A2: A,F3: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) @ F3 )
=> ( topolo3448309680560233919inuous @ A @ real @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) )
@ ^ [X4: A] : ( real_V7770717601297561774m_norm @ B @ ( F3 @ X4 ) ) ) ) ) ).
% isCont_norm
thf(fact_7043_isCont__of__real,axiom,
! [A: $tType,C: $tType] :
( ( ( topological_t2_space @ C )
& ( real_V2191834092415804123ebra_1 @ A )
& ( real_V822414075346904944vector @ A ) )
=> ! [A2: C,G: C > real] :
( ( topolo3448309680560233919inuous @ C @ real @ ( topolo174197925503356063within @ C @ A2 @ ( top_top @ ( set @ C ) ) ) @ G )
=> ( topolo3448309680560233919inuous @ C @ A @ ( topolo174197925503356063within @ C @ A2 @ ( top_top @ ( set @ C ) ) )
@ ^ [X4: C] : ( real_Vector_of_real @ A @ ( G @ X4 ) ) ) ) ) ).
% isCont_of_real
thf(fact_7044_isCont__scaleR,axiom,
! [C: $tType,D: $tType] :
( ( ( topological_t2_space @ D )
& ( real_V822414075346904944vector @ C ) )
=> ! [A2: D,F3: D > real,G: D > C] :
( ( topolo3448309680560233919inuous @ D @ real @ ( topolo174197925503356063within @ D @ A2 @ ( top_top @ ( set @ D ) ) ) @ F3 )
=> ( ( topolo3448309680560233919inuous @ D @ C @ ( topolo174197925503356063within @ D @ A2 @ ( top_top @ ( set @ D ) ) ) @ G )
=> ( topolo3448309680560233919inuous @ D @ C @ ( topolo174197925503356063within @ D @ A2 @ ( top_top @ ( set @ D ) ) )
@ ^ [X4: D] : ( real_V8093663219630862766scaleR @ C @ ( F3 @ X4 ) @ ( G @ X4 ) ) ) ) ) ) ).
% isCont_scaleR
thf(fact_7045_LIM__not__zero,axiom,
! [Aa: $tType,A: $tType] :
( ( ( topolo8386298272705272623_space @ A )
& ( zero @ Aa )
& ( topological_t2_space @ Aa ) )
=> ! [K2: Aa,A2: A] :
( ( K2
!= ( zero_zero @ Aa ) )
=> ~ ( filterlim @ A @ Aa
@ ^ [X4: A] : K2
@ ( topolo7230453075368039082e_nhds @ Aa @ ( zero_zero @ Aa ) )
@ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% LIM_not_zero
thf(fact_7046_LIM__offset__zero__cancel,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [F3: A > B,A2: A,L6: B] :
( ( filterlim @ A @ B
@ ^ [H: A] : ( F3 @ ( plus_plus @ A @ A2 @ H ) )
@ ( topolo7230453075368039082e_nhds @ B @ L6 )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) )
=> ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ L6 ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% LIM_offset_zero_cancel
thf(fact_7047_LIM__offset__zero,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [F3: A > B,L6: B,A2: A] :
( ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ L6 ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) )
=> ( filterlim @ A @ B
@ ^ [H: A] : ( F3 @ ( plus_plus @ A @ A2 @ H ) )
@ ( topolo7230453075368039082e_nhds @ B @ L6 )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% LIM_offset_zero
thf(fact_7048_LIM__isCont__iff,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [F3: A > B,A2: A] :
( ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ ( F3 @ A2 ) ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) )
= ( filterlim @ A @ B
@ ^ [H: A] : ( F3 @ ( plus_plus @ A @ A2 @ H ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( F3 @ A2 ) )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% LIM_isCont_iff
thf(fact_7049_isCont__iff,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [X: A,F3: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) @ F3 )
= ( filterlim @ A @ B
@ ^ [H: A] : ( F3 @ ( plus_plus @ A @ X @ H ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( F3 @ X ) )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% isCont_iff
thf(fact_7050_isCont__fst,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( topolo4958980785337419405_space @ B )
& ( topolo4958980785337419405_space @ C ) )
=> ! [A2: A,F3: A > ( product_prod @ B @ C )] :
( ( topolo3448309680560233919inuous @ A @ ( product_prod @ B @ C ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) @ F3 )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) )
@ ^ [X4: A] : ( product_fst @ B @ C @ ( F3 @ X4 ) ) ) ) ) ).
% isCont_fst
thf(fact_7051_isCont__snd,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( topolo4958980785337419405_space @ B )
& ( topolo4958980785337419405_space @ C ) )
=> ! [A2: A,F3: A > ( product_prod @ B @ C )] :
( ( topolo3448309680560233919inuous @ A @ ( product_prod @ B @ C ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) @ F3 )
=> ( topolo3448309680560233919inuous @ A @ C @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) )
@ ^ [X4: A] : ( product_snd @ B @ C @ ( F3 @ X4 ) ) ) ) ) ).
% isCont_snd
thf(fact_7052_filterlim__at__within__If,axiom,
! [B: $tType,A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F3: A > B,G6: filter @ B,X: A,A3: set @ A,P3: A > $o,G: A > B] :
( ( filterlim @ A @ B @ F3 @ G6 @ ( topolo174197925503356063within @ A @ X @ ( inf_inf @ ( set @ A ) @ A3 @ ( collect @ A @ P3 ) ) ) )
=> ( ( filterlim @ A @ B @ G @ G6
@ ( topolo174197925503356063within @ A @ X
@ ( inf_inf @ ( set @ A ) @ A3
@ ( collect @ A
@ ^ [X4: A] :
~ ( P3 @ X4 ) ) ) ) )
=> ( filterlim @ A @ B
@ ^ [X4: A] : ( if @ B @ ( P3 @ X4 ) @ ( F3 @ X4 ) @ ( G @ X4 ) )
@ G6
@ ( topolo174197925503356063within @ A @ X @ A3 ) ) ) ) ) ).
% filterlim_at_within_If
thf(fact_7053_continuous__ident,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [X: A,S3: set @ A] :
( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ X @ S3 )
@ ^ [X4: A] : X4 ) ) ).
% continuous_ident
thf(fact_7054_has__field__derivative__iff,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,D4: A,X: A,S3: set @ A] :
( ( has_field_derivative @ A @ F3 @ D4 @ ( topolo174197925503356063within @ A @ X @ S3 ) )
= ( filterlim @ A @ A
@ ^ [Y: A] : ( divide_divide @ A @ ( minus_minus @ A @ ( F3 @ Y ) @ ( F3 @ X ) ) @ ( minus_minus @ A @ Y @ X ) )
@ ( topolo7230453075368039082e_nhds @ A @ D4 )
@ ( topolo174197925503356063within @ A @ X @ S3 ) ) ) ) ).
% has_field_derivative_iff
thf(fact_7055_has__field__derivativeD,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,D4: A,X: A,S3: set @ A] :
( ( has_field_derivative @ A @ F3 @ D4 @ ( topolo174197925503356063within @ A @ X @ S3 ) )
=> ( filterlim @ A @ A
@ ^ [Y: A] : ( divide_divide @ A @ ( minus_minus @ A @ ( F3 @ Y ) @ ( F3 @ X ) ) @ ( minus_minus @ A @ Y @ X ) )
@ ( topolo7230453075368039082e_nhds @ A @ D4 )
@ ( topolo174197925503356063within @ A @ X @ S3 ) ) ) ) ).
% has_field_derivativeD
thf(fact_7056_isCont__LIM__compose2,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( topolo4958980785337419405_space @ B )
& ( topolo4958980785337419405_space @ C ) )
=> ! [A2: A,F3: A > B,G: B > C,L: C] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) @ F3 )
=> ( ( filterlim @ B @ C @ G @ ( topolo7230453075368039082e_nhds @ C @ L ) @ ( topolo174197925503356063within @ B @ ( F3 @ A2 ) @ ( top_top @ ( set @ B ) ) ) )
=> ( ? [D6: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D6 )
& ! [X5: A] :
( ( ( X5 != A2 )
& ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ X5 @ A2 ) ) @ D6 ) )
=> ( ( F3 @ X5 )
!= ( F3 @ A2 ) ) ) )
=> ( filterlim @ A @ C
@ ^ [X4: A] : ( G @ ( F3 @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ C @ L )
@ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ).
% isCont_LIM_compose2
thf(fact_7057_continuous__within__compose2,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( topolo4958980785337419405_space @ C )
& ( topological_t2_space @ B ) )
=> ! [X: A,S: set @ A,F3: A > B,G: B > C] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ X @ S ) @ F3 )
=> ( ( topolo3448309680560233919inuous @ B @ C @ ( topolo174197925503356063within @ B @ ( F3 @ X ) @ ( image @ A @ B @ F3 @ S ) ) @ G )
=> ( topolo3448309680560233919inuous @ A @ C @ ( topolo174197925503356063within @ A @ X @ S )
@ ^ [X4: A] : ( G @ ( F3 @ X4 ) ) ) ) ) ) ).
% continuous_within_compose2
thf(fact_7058_tendsto__within__subset,axiom,
! [B: $tType,A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F3: A > B,L: filter @ B,X: A,S3: set @ A,T5: set @ A] :
( ( filterlim @ A @ B @ F3 @ L @ ( topolo174197925503356063within @ A @ X @ S3 ) )
=> ( ( ord_less_eq @ ( set @ A ) @ T5 @ S3 )
=> ( filterlim @ A @ B @ F3 @ L @ ( topolo174197925503356063within @ A @ X @ T5 ) ) ) ) ) ).
% tendsto_within_subset
thf(fact_7059_LIM__imp__LIM,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ A )
& ( real_V822414075346904944vector @ C )
& ( real_V822414075346904944vector @ B ) )
=> ! [F3: A > B,L: B,A2: A,G: A > C,M: C] :
( ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) )
=> ( ! [X5: A] :
( ( X5 != A2 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ C @ ( minus_minus @ C @ ( G @ X5 ) @ M ) ) @ ( real_V7770717601297561774m_norm @ B @ ( minus_minus @ B @ ( F3 @ X5 ) @ L ) ) ) )
=> ( filterlim @ A @ C @ G @ ( topolo7230453075368039082e_nhds @ C @ M ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% LIM_imp_LIM
thf(fact_7060_IVT,axiom,
! [A: $tType,B: $tType] :
( ( ( topolo1944317154257567458pology @ B )
& ( topolo8458572112393995274pology @ A ) )
=> ! [F3: A > B,A2: A,Y2: B,B2: A] :
( ( ord_less_eq @ B @ ( F3 @ A2 ) @ Y2 )
=> ( ( ord_less_eq @ B @ Y2 @ ( F3 @ B2 ) )
=> ( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ! [X5: A] :
( ( ( ord_less_eq @ A @ A2 @ X5 )
& ( ord_less_eq @ A @ X5 @ B2 ) )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ X5 @ ( top_top @ ( set @ A ) ) ) @ F3 ) )
=> ? [X5: A] :
( ( ord_less_eq @ A @ A2 @ X5 )
& ( ord_less_eq @ A @ X5 @ B2 )
& ( ( F3 @ X5 )
= Y2 ) ) ) ) ) ) ) ).
% IVT
thf(fact_7061_IVT2,axiom,
! [A: $tType,B: $tType] :
( ( ( topolo1944317154257567458pology @ B )
& ( topolo8458572112393995274pology @ A ) )
=> ! [F3: A > B,B2: A,Y2: B,A2: A] :
( ( ord_less_eq @ B @ ( F3 @ B2 ) @ Y2 )
=> ( ( ord_less_eq @ B @ Y2 @ ( F3 @ A2 ) )
=> ( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ! [X5: A] :
( ( ( ord_less_eq @ A @ A2 @ X5 )
& ( ord_less_eq @ A @ X5 @ B2 ) )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ X5 @ ( top_top @ ( set @ A ) ) ) @ F3 ) )
=> ? [X5: A] :
( ( ord_less_eq @ A @ A2 @ X5 )
& ( ord_less_eq @ A @ X5 @ B2 )
& ( ( F3 @ X5 )
= Y2 ) ) ) ) ) ) ) ).
% IVT2
thf(fact_7062_real__LIM__sandwich__zero,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F3: A > real,A2: A,G: A > real] :
( ( filterlim @ A @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) )
=> ( ! [X5: A] :
( ( X5 != A2 )
=> ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( G @ X5 ) ) )
=> ( ! [X5: A] :
( ( X5 != A2 )
=> ( ord_less_eq @ real @ ( G @ X5 ) @ ( F3 @ X5 ) ) )
=> ( filterlim @ A @ real @ G @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ).
% real_LIM_sandwich_zero
thf(fact_7063_has__derivative__Re,axiom,
! [C: $tType] :
( ( real_V822414075346904944vector @ C )
=> ! [G: C > complex,G5: C > complex,F4: filter @ C] :
( ( has_derivative @ C @ complex @ G @ G5 @ F4 )
=> ( has_derivative @ C @ real
@ ^ [X4: C] : ( re @ ( G @ X4 ) )
@ ^ [X4: C] : ( re @ ( G5 @ X4 ) )
@ F4 ) ) ) ).
% has_derivative_Re
thf(fact_7064_has__derivative__Im,axiom,
! [C: $tType] :
( ( real_V822414075346904944vector @ C )
=> ! [G: C > complex,G5: C > complex,F4: filter @ C] :
( ( has_derivative @ C @ complex @ G @ G5 @ F4 )
=> ( has_derivative @ C @ real
@ ^ [X4: C] : ( im @ ( G @ X4 ) )
@ ^ [X4: C] : ( im @ ( G5 @ X4 ) )
@ F4 ) ) ) ).
% has_derivative_Im
thf(fact_7065_has__derivative__cnj,axiom,
! [C: $tType] :
( ( real_V822414075346904944vector @ C )
=> ! [G: C > complex,G5: C > complex,F4: filter @ C] :
( ( has_derivative @ C @ complex @ G @ G5 @ F4 )
=> ( has_derivative @ C @ complex
@ ^ [X4: C] : ( cnj @ ( G @ X4 ) )
@ ^ [X4: C] : ( cnj @ ( G5 @ X4 ) )
@ F4 ) ) ) ).
% has_derivative_cnj
thf(fact_7066_tendsto__fst,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ C )
& ( topolo4958980785337419405_space @ B ) )
=> ! [F3: A > ( product_prod @ B @ C ),A2: product_prod @ B @ C,F4: filter @ A] :
( ( filterlim @ A @ ( product_prod @ B @ C ) @ F3 @ ( topolo7230453075368039082e_nhds @ ( product_prod @ B @ C ) @ A2 ) @ F4 )
=> ( filterlim @ A @ B
@ ^ [X4: A] : ( product_fst @ B @ C @ ( F3 @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( product_fst @ B @ C @ A2 ) )
@ F4 ) ) ) ).
% tendsto_fst
thf(fact_7067_tendsto__snd,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ C )
& ( topolo4958980785337419405_space @ B ) )
=> ! [F3: A > ( product_prod @ B @ C ),A2: product_prod @ B @ C,F4: filter @ A] :
( ( filterlim @ A @ ( product_prod @ B @ C ) @ F3 @ ( topolo7230453075368039082e_nhds @ ( product_prod @ B @ C ) @ A2 ) @ F4 )
=> ( filterlim @ A @ C
@ ^ [X4: A] : ( product_snd @ B @ C @ ( F3 @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ C @ ( product_snd @ B @ C @ A2 ) )
@ F4 ) ) ) ).
% tendsto_snd
thf(fact_7068_continuous__fst,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( topolo4958980785337419405_space @ B )
& ( topolo4958980785337419405_space @ C ) )
=> ! [F4: filter @ A,F3: A > ( product_prod @ B @ C )] :
( ( topolo3448309680560233919inuous @ A @ ( product_prod @ B @ C ) @ F4 @ F3 )
=> ( topolo3448309680560233919inuous @ A @ B @ F4
@ ^ [X4: A] : ( product_fst @ B @ C @ ( F3 @ X4 ) ) ) ) ) ).
% continuous_fst
thf(fact_7069_continuous__snd,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( topolo4958980785337419405_space @ B )
& ( topolo4958980785337419405_space @ C ) )
=> ! [F4: filter @ A,F3: A > ( product_prod @ B @ C )] :
( ( topolo3448309680560233919inuous @ A @ ( product_prod @ B @ C ) @ F4 @ F3 )
=> ( topolo3448309680560233919inuous @ A @ C @ F4
@ ^ [X4: A] : ( product_snd @ B @ C @ ( F3 @ X4 ) ) ) ) ) ).
% continuous_snd
thf(fact_7070_continuous__Pair,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( topolo4958980785337419405_space @ B )
& ( topolo4958980785337419405_space @ C ) )
=> ! [F4: filter @ A,F3: A > B,G: A > C] :
( ( topolo3448309680560233919inuous @ A @ B @ F4 @ F3 )
=> ( ( topolo3448309680560233919inuous @ A @ C @ F4 @ G )
=> ( topolo3448309680560233919inuous @ A @ ( product_prod @ B @ C ) @ F4
@ ^ [X4: A] : ( product_Pair @ B @ C @ ( F3 @ X4 ) @ ( G @ X4 ) ) ) ) ) ) ).
% continuous_Pair
thf(fact_7071_tendsto__Pair,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ B )
& ( topolo4958980785337419405_space @ C ) )
=> ! [F3: A > B,A2: B,F4: filter @ A,G: A > C,B2: C] :
( ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ A2 ) @ F4 )
=> ( ( filterlim @ A @ C @ G @ ( topolo7230453075368039082e_nhds @ C @ B2 ) @ F4 )
=> ( filterlim @ A @ ( product_prod @ B @ C )
@ ^ [X4: A] : ( product_Pair @ B @ C @ ( F3 @ X4 ) @ ( G @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ A2 @ B2 ) )
@ F4 ) ) ) ) ).
% tendsto_Pair
thf(fact_7072_tendsto__const__iff,axiom,
! [B: $tType,A: $tType] :
( ( topological_t2_space @ A )
=> ! [F4: filter @ B,A2: A,B2: A] :
( ( F4
!= ( bot_bot @ ( filter @ B ) ) )
=> ( ( filterlim @ B @ A
@ ^ [X4: B] : A2
@ ( topolo7230453075368039082e_nhds @ A @ B2 )
@ F4 )
= ( A2 = B2 ) ) ) ) ).
% tendsto_const_iff
thf(fact_7073_filterlim__inf,axiom,
! [B: $tType,A: $tType,F3: A > B,F23: filter @ B,F33: filter @ B,F13: filter @ A] :
( ( filterlim @ A @ B @ F3 @ ( inf_inf @ ( filter @ B ) @ F23 @ F33 ) @ F13 )
= ( ( filterlim @ A @ B @ F3 @ F23 @ F13 )
& ( filterlim @ A @ B @ F3 @ F33 @ F13 ) ) ) ).
% filterlim_inf
thf(fact_7074_tendsto__cnj,axiom,
! [C: $tType,G: C > complex,A2: complex,F4: filter @ C] :
( ( filterlim @ C @ complex @ G @ ( topolo7230453075368039082e_nhds @ complex @ A2 ) @ F4 )
=> ( filterlim @ C @ complex
@ ^ [X4: C] : ( cnj @ ( G @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ complex @ ( cnj @ A2 ) )
@ F4 ) ) ).
% tendsto_cnj
thf(fact_7075_lim__cnj,axiom,
! [A: $tType,F3: A > complex,L: complex,F4: filter @ A] :
( ( filterlim @ A @ complex
@ ^ [X4: A] : ( cnj @ ( F3 @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ complex @ ( cnj @ L ) )
@ F4 )
= ( filterlim @ A @ complex @ F3 @ ( topolo7230453075368039082e_nhds @ complex @ L ) @ F4 ) ) ).
% lim_cnj
thf(fact_7076_tendsto__of__int__floor,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( ( ring_1 @ C )
& ( topolo4958980785337419405_space @ C )
& ( archim2362893244070406136eiling @ B )
& ( topolo2564578578187576103pology @ B ) )
=> ! [F3: A > B,L: B,F4: filter @ A] :
( ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ F4 )
=> ( ~ ( member @ B @ L @ ( ring_1_Ints @ B ) )
=> ( filterlim @ A @ C
@ ^ [X4: A] : ( ring_1_of_int @ C @ ( archim6421214686448440834_floor @ B @ ( F3 @ X4 ) ) )
@ ( topolo7230453075368039082e_nhds @ C @ ( ring_1_of_int @ C @ ( archim6421214686448440834_floor @ B @ L ) ) )
@ F4 ) ) ) ) ).
% tendsto_of_int_floor
thf(fact_7077_tendsto__mult__right__zero,axiom,
! [A: $tType,D: $tType] :
( ( real_V4412858255891104859lgebra @ A )
=> ! [F3: D > A,F4: filter @ D,C2: A] :
( ( filterlim @ D @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ F4 )
=> ( filterlim @ D @ A
@ ^ [X4: D] : ( times_times @ A @ C2 @ ( F3 @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ F4 ) ) ) ).
% tendsto_mult_right_zero
thf(fact_7078_tendsto__mult__left__zero,axiom,
! [A: $tType,D: $tType] :
( ( real_V4412858255891104859lgebra @ A )
=> ! [F3: D > A,F4: filter @ D,C2: A] :
( ( filterlim @ D @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ F4 )
=> ( filterlim @ D @ A
@ ^ [X4: D] : ( times_times @ A @ ( F3 @ X4 ) @ C2 )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ F4 ) ) ) ).
% tendsto_mult_left_zero
thf(fact_7079_tendsto__mult__zero,axiom,
! [A: $tType,D: $tType] :
( ( real_V4412858255891104859lgebra @ A )
=> ! [F3: D > A,F4: filter @ D,G: D > A] :
( ( filterlim @ D @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ F4 )
=> ( ( filterlim @ D @ A @ G @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ F4 )
=> ( filterlim @ D @ A
@ ^ [X4: D] : ( times_times @ A @ ( F3 @ X4 ) @ ( G @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ F4 ) ) ) ) ).
% tendsto_mult_zero
thf(fact_7080_tendsto__complex__iff,axiom,
! [A: $tType,F3: A > complex,X: complex,F4: filter @ A] :
( ( filterlim @ A @ complex @ F3 @ ( topolo7230453075368039082e_nhds @ complex @ X ) @ F4 )
= ( ( filterlim @ A @ real
@ ^ [X4: A] : ( re @ ( F3 @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( re @ X ) )
@ F4 )
& ( filterlim @ A @ real
@ ^ [X4: A] : ( im @ ( F3 @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( im @ X ) )
@ F4 ) ) ) ).
% tendsto_complex_iff
thf(fact_7081_tendsto__add__zero,axiom,
! [B: $tType,D: $tType] :
( ( topolo6943815403480290642id_add @ B )
=> ! [F3: D > B,F4: filter @ D,G: D > B] :
( ( filterlim @ D @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) @ F4 )
=> ( ( filterlim @ D @ B @ G @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) @ F4 )
=> ( filterlim @ D @ B
@ ^ [X4: D] : ( plus_plus @ B @ ( F3 @ X4 ) @ ( G @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ F4 ) ) ) ) ).
% tendsto_add_zero
thf(fact_7082_tendsto__inverse,axiom,
! [A: $tType,B: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [F3: B > A,A2: A,F4: filter @ B] :
( ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ F4 )
=> ( ( A2
!= ( zero_zero @ A ) )
=> ( filterlim @ B @ A
@ ^ [X4: B] : ( inverse_inverse @ A @ ( F3 @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( inverse_inverse @ A @ A2 ) )
@ F4 ) ) ) ) ).
% tendsto_inverse
thf(fact_7083_tendsto__sgn,axiom,
! [A: $tType,B: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: B > A,L: A,F4: filter @ B] :
( ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F4 )
=> ( ( L
!= ( zero_zero @ A ) )
=> ( filterlim @ B @ A
@ ^ [X4: B] : ( sgn_sgn @ A @ ( F3 @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( sgn_sgn @ A @ L ) )
@ F4 ) ) ) ) ).
% tendsto_sgn
thf(fact_7084_tendsto__powr,axiom,
! [A: $tType,F3: A > real,A2: real,F4: filter @ A,G: A > real,B2: real] :
( ( filterlim @ A @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ A2 ) @ F4 )
=> ( ( filterlim @ A @ real @ G @ ( topolo7230453075368039082e_nhds @ real @ B2 ) @ F4 )
=> ( ( A2
!= ( zero_zero @ real ) )
=> ( filterlim @ A @ real
@ ^ [X4: A] : ( powr @ real @ ( F3 @ X4 ) @ ( G @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( powr @ real @ A2 @ B2 ) )
@ F4 ) ) ) ) ).
% tendsto_powr
thf(fact_7085_tendsto__ln,axiom,
! [A: $tType,F3: A > real,A2: real,F4: filter @ A] :
( ( filterlim @ A @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ A2 ) @ F4 )
=> ( ( A2
!= ( zero_zero @ real ) )
=> ( filterlim @ A @ real
@ ^ [X4: A] : ( ln_ln @ real @ ( F3 @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( ln_ln @ real @ A2 ) )
@ F4 ) ) ) ).
% tendsto_ln
thf(fact_7086_tendsto__norm__zero__cancel,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F3: A > B,F4: filter @ A] :
( ( filterlim @ A @ real
@ ^ [X4: A] : ( real_V7770717601297561774m_norm @ B @ ( F3 @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ F4 )
=> ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) @ F4 ) ) ) ).
% tendsto_norm_zero_cancel
thf(fact_7087_tendsto__norm__zero__iff,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F3: A > B,F4: filter @ A] :
( ( filterlim @ A @ real
@ ^ [X4: A] : ( real_V7770717601297561774m_norm @ B @ ( F3 @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ F4 )
= ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) @ F4 ) ) ) ).
% tendsto_norm_zero_iff
thf(fact_7088_tendsto__norm__zero,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F3: A > B,F4: filter @ A] :
( ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) @ F4 )
=> ( filterlim @ A @ real
@ ^ [X4: A] : ( real_V7770717601297561774m_norm @ B @ ( F3 @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ F4 ) ) ) ).
% tendsto_norm_zero
thf(fact_7089_tendsto__Im,axiom,
! [C: $tType,G: C > complex,A2: complex,F4: filter @ C] :
( ( filterlim @ C @ complex @ G @ ( topolo7230453075368039082e_nhds @ complex @ A2 ) @ F4 )
=> ( filterlim @ C @ real
@ ^ [X4: C] : ( im @ ( G @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( im @ A2 ) )
@ F4 ) ) ).
% tendsto_Im
thf(fact_7090_tendsto__divide,axiom,
! [A: $tType,B: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: B > A,A2: A,F4: filter @ B,G: B > A,B2: A] :
( ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ F4 )
=> ( ( filterlim @ B @ A @ G @ ( topolo7230453075368039082e_nhds @ A @ B2 ) @ F4 )
=> ( ( B2
!= ( zero_zero @ A ) )
=> ( filterlim @ B @ A
@ ^ [X4: B] : ( divide_divide @ A @ ( F3 @ X4 ) @ ( G @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( divide_divide @ A @ A2 @ B2 ) )
@ F4 ) ) ) ) ) ).
% tendsto_divide
thf(fact_7091_tendsto__divide__zero,axiom,
! [A: $tType,B: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: B > A,F4: filter @ B,C2: A] :
( ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ F4 )
=> ( filterlim @ B @ A
@ ^ [X4: B] : ( divide_divide @ A @ ( F3 @ X4 ) @ C2 )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ F4 ) ) ) ).
% tendsto_divide_zero
thf(fact_7092_tendsto__tan,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [F3: A > A,A2: A,F4: filter @ A] :
( ( filterlim @ A @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ F4 )
=> ( ( ( cos @ A @ A2 )
!= ( zero_zero @ A ) )
=> ( filterlim @ A @ A
@ ^ [X4: A] : ( tan @ A @ ( F3 @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( tan @ A @ A2 ) )
@ F4 ) ) ) ) ).
% tendsto_tan
thf(fact_7093_tendsto__real__root,axiom,
! [A: $tType,F3: A > real,X: real,F4: filter @ A,N: nat] :
( ( filterlim @ A @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ X ) @ F4 )
=> ( filterlim @ A @ real
@ ^ [X4: A] : ( root @ N @ ( F3 @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( root @ N @ X ) )
@ F4 ) ) ).
% tendsto_real_root
thf(fact_7094_continuous__cosh,axiom,
! [A: $tType,C: $tType] :
( ( ( topological_t2_space @ C )
& ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [F4: filter @ C,F3: C > A] :
( ( topolo3448309680560233919inuous @ C @ A @ F4 @ F3 )
=> ( topolo3448309680560233919inuous @ C @ A @ F4
@ ^ [X4: C] : ( cosh @ A @ ( F3 @ X4 ) ) ) ) ) ).
% continuous_cosh
thf(fact_7095_continuous__sinh,axiom,
! [A: $tType,C: $tType] :
( ( ( topological_t2_space @ C )
& ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [F4: filter @ C,F3: C > A] :
( ( topolo3448309680560233919inuous @ C @ A @ F4 @ F3 )
=> ( topolo3448309680560233919inuous @ C @ A @ F4
@ ^ [X4: C] : ( sinh @ A @ ( F3 @ X4 ) ) ) ) ) ).
% continuous_sinh
thf(fact_7096_tendsto__scaleR,axiom,
! [C: $tType,D: $tType] :
( ( real_V822414075346904944vector @ C )
=> ! [F3: D > real,A2: real,F4: filter @ D,G: D > C,B2: C] :
( ( filterlim @ D @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ A2 ) @ F4 )
=> ( ( filterlim @ D @ C @ G @ ( topolo7230453075368039082e_nhds @ C @ B2 ) @ F4 )
=> ( filterlim @ D @ C
@ ^ [X4: D] : ( real_V8093663219630862766scaleR @ C @ ( F3 @ X4 ) @ ( G @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ C @ ( real_V8093663219630862766scaleR @ C @ A2 @ B2 ) )
@ F4 ) ) ) ) ).
% tendsto_scaleR
thf(fact_7097_continuous__scaleR,axiom,
! [C: $tType,D: $tType] :
( ( ( topological_t2_space @ D )
& ( real_V822414075346904944vector @ C ) )
=> ! [F4: filter @ D,F3: D > real,G: D > C] :
( ( topolo3448309680560233919inuous @ D @ real @ F4 @ F3 )
=> ( ( topolo3448309680560233919inuous @ D @ C @ F4 @ G )
=> ( topolo3448309680560233919inuous @ D @ C @ F4
@ ^ [X4: D] : ( real_V8093663219630862766scaleR @ C @ ( F3 @ X4 ) @ ( G @ X4 ) ) ) ) ) ) ).
% continuous_scaleR
thf(fact_7098_tendsto__cosh,axiom,
! [A: $tType,C: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [F3: C > A,A2: A,F4: filter @ C] :
( ( filterlim @ C @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ F4 )
=> ( filterlim @ C @ A
@ ^ [X4: C] : ( cosh @ A @ ( F3 @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( cosh @ A @ A2 ) )
@ F4 ) ) ) ).
% tendsto_cosh
thf(fact_7099_tendsto__sinh,axiom,
! [A: $tType,C: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [F3: C > A,A2: A,F4: filter @ C] :
( ( filterlim @ C @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ F4 )
=> ( filterlim @ C @ A
@ ^ [X4: C] : ( sinh @ A @ ( F3 @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( sinh @ A @ A2 ) )
@ F4 ) ) ) ).
% tendsto_sinh
thf(fact_7100_continuous__cos,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_Vector_banach @ B )
& ( real_V3459762299906320749_field @ B ) )
=> ! [F4: filter @ A,F3: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ F4 @ F3 )
=> ( topolo3448309680560233919inuous @ A @ B @ F4
@ ^ [X4: A] : ( cos @ B @ ( F3 @ X4 ) ) ) ) ) ).
% continuous_cos
thf(fact_7101_continuous__sin,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_Vector_banach @ B )
& ( real_V3459762299906320749_field @ B ) )
=> ! [F4: filter @ A,F3: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ F4 @ F3 )
=> ( topolo3448309680560233919inuous @ A @ B @ F4
@ ^ [X4: A] : ( sin @ B @ ( F3 @ X4 ) ) ) ) ) ).
% continuous_sin
thf(fact_7102_continuous__exp,axiom,
! [A: $tType,C: $tType] :
( ( ( topological_t2_space @ C )
& ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [F4: filter @ C,F3: C > A] :
( ( topolo3448309680560233919inuous @ C @ A @ F4 @ F3 )
=> ( topolo3448309680560233919inuous @ C @ A @ F4
@ ^ [X4: C] : ( exp @ A @ ( F3 @ X4 ) ) ) ) ) ).
% continuous_exp
thf(fact_7103_continuous__power,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( power @ B )
& ( real_V4412858255891104859lgebra @ B ) )
=> ! [F4: filter @ A,F3: A > B,N: nat] :
( ( topolo3448309680560233919inuous @ A @ B @ F4 @ F3 )
=> ( topolo3448309680560233919inuous @ A @ B @ F4
@ ^ [X4: A] : ( power_power @ B @ ( F3 @ X4 ) @ N ) ) ) ) ).
% continuous_power
thf(fact_7104_tendsto__of__real,axiom,
! [A: $tType,C: $tType] :
( ( ( real_V2191834092415804123ebra_1 @ A )
& ( real_V822414075346904944vector @ A ) )
=> ! [G: C > real,A2: real,F4: filter @ C] :
( ( filterlim @ C @ real @ G @ ( topolo7230453075368039082e_nhds @ real @ A2 ) @ F4 )
=> ( filterlim @ C @ A
@ ^ [X4: C] : ( real_Vector_of_real @ A @ ( G @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( real_Vector_of_real @ A @ A2 ) )
@ F4 ) ) ) ).
% tendsto_of_real
thf(fact_7105_continuous__of__real,axiom,
! [A: $tType,C: $tType] :
( ( ( topological_t2_space @ C )
& ( real_V2191834092415804123ebra_1 @ A )
& ( real_V822414075346904944vector @ A ) )
=> ! [F4: filter @ C,G: C > real] :
( ( topolo3448309680560233919inuous @ C @ real @ F4 @ G )
=> ( topolo3448309680560233919inuous @ C @ A @ F4
@ ^ [X4: C] : ( real_Vector_of_real @ A @ ( G @ X4 ) ) ) ) ) ).
% continuous_of_real
thf(fact_7106_tendsto__of__real__iff,axiom,
! [A: $tType,B: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [F3: B > real,C2: real,F4: filter @ B] :
( ( filterlim @ B @ A
@ ^ [X4: B] : ( real_Vector_of_real @ A @ ( F3 @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( real_Vector_of_real @ A @ C2 ) )
@ F4 )
= ( filterlim @ B @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ C2 ) @ F4 ) ) ) ).
% tendsto_of_real_iff
thf(fact_7107_tendsto__exp,axiom,
! [A: $tType,C: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [F3: C > A,A2: A,F4: filter @ C] :
( ( filterlim @ C @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ F4 )
=> ( filterlim @ C @ A
@ ^ [X4: C] : ( exp @ A @ ( F3 @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( exp @ A @ A2 ) )
@ F4 ) ) ) ).
% tendsto_exp
thf(fact_7108_tendsto__cos,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_Vector_banach @ B )
& ( real_V3459762299906320749_field @ B ) )
=> ! [F3: A > B,A2: B,F4: filter @ A] :
( ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ A2 ) @ F4 )
=> ( filterlim @ A @ B
@ ^ [X4: A] : ( cos @ B @ ( F3 @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( cos @ B @ A2 ) )
@ F4 ) ) ) ).
% tendsto_cos
thf(fact_7109_tendsto__sin,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_Vector_banach @ B )
& ( real_V3459762299906320749_field @ B ) )
=> ! [F3: A > B,A2: B,F4: filter @ A] :
( ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ A2 ) @ F4 )
=> ( filterlim @ A @ B
@ ^ [X4: A] : ( sin @ B @ ( F3 @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( sin @ B @ A2 ) )
@ F4 ) ) ) ).
% tendsto_sin
thf(fact_7110_tendsto__Complex,axiom,
! [A: $tType,F3: A > real,A2: real,F4: filter @ A,G: A > real,B2: real] :
( ( filterlim @ A @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ A2 ) @ F4 )
=> ( ( filterlim @ A @ real @ G @ ( topolo7230453075368039082e_nhds @ real @ B2 ) @ F4 )
=> ( filterlim @ A @ complex
@ ^ [X4: A] : ( complex2 @ ( F3 @ X4 ) @ ( G @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ complex @ ( complex2 @ A2 @ B2 ) )
@ F4 ) ) ) ).
% tendsto_Complex
thf(fact_7111_tendsto__norm,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F3: A > B,A2: B,F4: filter @ A] :
( ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ A2 ) @ F4 )
=> ( filterlim @ A @ real
@ ^ [X4: A] : ( real_V7770717601297561774m_norm @ B @ ( F3 @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( real_V7770717601297561774m_norm @ B @ A2 ) )
@ F4 ) ) ) ).
% tendsto_norm
thf(fact_7112_continuous__norm,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F4: filter @ A,F3: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ F4 @ F3 )
=> ( topolo3448309680560233919inuous @ A @ real @ F4
@ ^ [X4: A] : ( real_V7770717601297561774m_norm @ B @ ( F3 @ X4 ) ) ) ) ) ).
% continuous_norm
thf(fact_7113_tendsto__arctan,axiom,
! [A: $tType,F3: A > real,X: real,F4: filter @ A] :
( ( filterlim @ A @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ X ) @ F4 )
=> ( filterlim @ A @ real
@ ^ [X4: A] : ( arctan @ ( F3 @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( arctan @ X ) )
@ F4 ) ) ).
% tendsto_arctan
thf(fact_7114_tendsto__real__sqrt,axiom,
! [A: $tType,F3: A > real,X: real,F4: filter @ A] :
( ( filterlim @ A @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ X ) @ F4 )
=> ( filterlim @ A @ real
@ ^ [X4: A] : ( sqrt @ ( F3 @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( sqrt @ X ) )
@ F4 ) ) ).
% tendsto_real_sqrt
thf(fact_7115_tendsto__arsinh,axiom,
! [B: $tType,F3: B > real,A2: real,F4: filter @ B] :
( ( filterlim @ B @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ A2 ) @ F4 )
=> ( filterlim @ B @ real
@ ^ [X4: B] : ( arsinh @ real @ ( F3 @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( arsinh @ real @ A2 ) )
@ F4 ) ) ).
% tendsto_arsinh
thf(fact_7116_continuous__const,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [F4: filter @ A,C2: B] :
( topolo3448309680560233919inuous @ A @ B @ F4
@ ^ [X4: A] : C2 ) ) ).
% continuous_const
thf(fact_7117_tendsto__power,axiom,
! [B: $tType,A: $tType] :
( ( ( power @ B )
& ( real_V4412858255891104859lgebra @ B ) )
=> ! [F3: A > B,A2: B,F4: filter @ A,N: nat] :
( ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ A2 ) @ F4 )
=> ( filterlim @ A @ B
@ ^ [X4: A] : ( power_power @ B @ ( F3 @ X4 ) @ N )
@ ( topolo7230453075368039082e_nhds @ B @ ( power_power @ B @ A2 @ N ) )
@ F4 ) ) ) ).
% tendsto_power
thf(fact_7118_continuous__power_H,axiom,
! [B: $tType,C: $tType] :
( ( ( topological_t2_space @ C )
& ( topolo1898628316856586783d_mult @ B ) )
=> ! [F4: filter @ C,F3: C > B,G: C > nat] :
( ( topolo3448309680560233919inuous @ C @ B @ F4 @ F3 )
=> ( ( topolo3448309680560233919inuous @ C @ nat @ F4 @ G )
=> ( topolo3448309680560233919inuous @ C @ B @ F4
@ ^ [X4: C] : ( power_power @ B @ ( F3 @ X4 ) @ ( G @ X4 ) ) ) ) ) ) ).
% continuous_power'
thf(fact_7119_tendsto__power__strong,axiom,
! [B: $tType,C: $tType] :
( ( topolo1898628316856586783d_mult @ B )
=> ! [F3: C > B,A2: B,F4: filter @ C,G: C > nat,B2: nat] :
( ( filterlim @ C @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ A2 ) @ F4 )
=> ( ( filterlim @ C @ nat @ G @ ( topolo7230453075368039082e_nhds @ nat @ B2 ) @ F4 )
=> ( filterlim @ C @ B
@ ^ [X4: C] : ( power_power @ B @ ( F3 @ X4 ) @ ( G @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( power_power @ B @ A2 @ B2 ) )
@ F4 ) ) ) ) ).
% tendsto_power_strong
thf(fact_7120_continuous__add,axiom,
! [B: $tType,D: $tType] :
( ( ( topological_t2_space @ D )
& ( topolo6943815403480290642id_add @ B ) )
=> ! [F4: filter @ D,F3: D > B,G: D > B] :
( ( topolo3448309680560233919inuous @ D @ B @ F4 @ F3 )
=> ( ( topolo3448309680560233919inuous @ D @ B @ F4 @ G )
=> ( topolo3448309680560233919inuous @ D @ B @ F4
@ ^ [X4: D] : ( plus_plus @ B @ ( F3 @ X4 ) @ ( G @ X4 ) ) ) ) ) ) ).
% continuous_add
thf(fact_7121_tendsto__add__const__iff,axiom,
! [A: $tType,B: $tType] :
( ( topolo1633459387980952147up_add @ A )
=> ! [C2: A,F3: B > A,D2: A,F4: filter @ B] :
( ( filterlim @ B @ A
@ ^ [X4: B] : ( plus_plus @ A @ C2 @ ( F3 @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( plus_plus @ A @ C2 @ D2 ) )
@ F4 )
= ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ D2 ) @ F4 ) ) ) ).
% tendsto_add_const_iff
thf(fact_7122_tendsto__add,axiom,
! [A: $tType,B: $tType] :
( ( topolo6943815403480290642id_add @ A )
=> ! [F3: B > A,A2: A,F4: filter @ B,G: B > A,B2: A] :
( ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ F4 )
=> ( ( filterlim @ B @ A @ G @ ( topolo7230453075368039082e_nhds @ A @ B2 ) @ F4 )
=> ( filterlim @ B @ A
@ ^ [X4: B] : ( plus_plus @ A @ ( F3 @ X4 ) @ ( G @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( plus_plus @ A @ A2 @ B2 ) )
@ F4 ) ) ) ) ).
% tendsto_add
thf(fact_7123_tendsto__Re,axiom,
! [C: $tType,G: C > complex,A2: complex,F4: filter @ C] :
( ( filterlim @ C @ complex @ G @ ( topolo7230453075368039082e_nhds @ complex @ A2 ) @ F4 )
=> ( filterlim @ C @ real
@ ^ [X4: C] : ( re @ ( G @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( re @ A2 ) )
@ F4 ) ) ).
% tendsto_Re
thf(fact_7124_continuous__mult,axiom,
! [A: $tType,D: $tType] :
( ( ( topological_t2_space @ D )
& ( real_V4412858255891104859lgebra @ A ) )
=> ! [F4: filter @ D,F3: D > A,G: D > A] :
( ( topolo3448309680560233919inuous @ D @ A @ F4 @ F3 )
=> ( ( topolo3448309680560233919inuous @ D @ A @ F4 @ G )
=> ( topolo3448309680560233919inuous @ D @ A @ F4
@ ^ [X4: D] : ( times_times @ A @ ( F3 @ X4 ) @ ( G @ X4 ) ) ) ) ) ) ).
% continuous_mult
thf(fact_7125_continuous__mult_H,axiom,
! [B: $tType,D: $tType] :
( ( ( topological_t2_space @ D )
& ( topolo4211221413907600880p_mult @ B ) )
=> ! [F4: filter @ D,F3: D > B,G: D > B] :
( ( topolo3448309680560233919inuous @ D @ B @ F4 @ F3 )
=> ( ( topolo3448309680560233919inuous @ D @ B @ F4 @ G )
=> ( topolo3448309680560233919inuous @ D @ B @ F4
@ ^ [X4: D] : ( times_times @ B @ ( F3 @ X4 ) @ ( G @ X4 ) ) ) ) ) ) ).
% continuous_mult'
thf(fact_7126_continuous__mult__left,axiom,
! [A: $tType,B: $tType] :
( ( ( topological_t2_space @ B )
& ( real_V4412858255891104859lgebra @ A ) )
=> ! [F4: filter @ B,F3: B > A,C2: A] :
( ( topolo3448309680560233919inuous @ B @ A @ F4 @ F3 )
=> ( topolo3448309680560233919inuous @ B @ A @ F4
@ ^ [X4: B] : ( times_times @ A @ C2 @ ( F3 @ X4 ) ) ) ) ) ).
% continuous_mult_left
thf(fact_7127_continuous__mult__right,axiom,
! [A: $tType,B: $tType] :
( ( ( topological_t2_space @ B )
& ( real_V4412858255891104859lgebra @ A ) )
=> ! [F4: filter @ B,F3: B > A,C2: A] :
( ( topolo3448309680560233919inuous @ B @ A @ F4 @ F3 )
=> ( topolo3448309680560233919inuous @ B @ A @ F4
@ ^ [X4: B] : ( times_times @ A @ ( F3 @ X4 ) @ C2 ) ) ) ) ).
% continuous_mult_right
thf(fact_7128_tendsto__mult__right,axiom,
! [A: $tType,B: $tType] :
( ( topolo4211221413907600880p_mult @ A )
=> ! [F3: B > A,L: A,F4: filter @ B,C2: A] :
( ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F4 )
=> ( filterlim @ B @ A
@ ^ [X4: B] : ( times_times @ A @ ( F3 @ X4 ) @ C2 )
@ ( topolo7230453075368039082e_nhds @ A @ ( times_times @ A @ L @ C2 ) )
@ F4 ) ) ) ).
% tendsto_mult_right
thf(fact_7129_tendsto__mult__left,axiom,
! [A: $tType,B: $tType] :
( ( topolo4211221413907600880p_mult @ A )
=> ! [F3: B > A,L: A,F4: filter @ B,C2: A] :
( ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F4 )
=> ( filterlim @ B @ A
@ ^ [X4: B] : ( times_times @ A @ C2 @ ( F3 @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( times_times @ A @ C2 @ L ) )
@ F4 ) ) ) ).
% tendsto_mult_left
thf(fact_7130_tendsto__mult,axiom,
! [A: $tType,B: $tType] :
( ( topolo4211221413907600880p_mult @ A )
=> ! [F3: B > A,A2: A,F4: filter @ B,G: B > A,B2: A] :
( ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ F4 )
=> ( ( filterlim @ B @ A @ G @ ( topolo7230453075368039082e_nhds @ A @ B2 ) @ F4 )
=> ( filterlim @ B @ A
@ ^ [X4: B] : ( times_times @ A @ ( F3 @ X4 ) @ ( G @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( times_times @ A @ A2 @ B2 ) )
@ F4 ) ) ) ) ).
% tendsto_mult
thf(fact_7131_tendsto__mult__one,axiom,
! [B: $tType,D: $tType] :
( ( topolo1898628316856586783d_mult @ B )
=> ! [F3: D > B,F4: filter @ D,G: D > B] :
( ( filterlim @ D @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ ( one_one @ B ) ) @ F4 )
=> ( ( filterlim @ D @ B @ G @ ( topolo7230453075368039082e_nhds @ B @ ( one_one @ B ) ) @ F4 )
=> ( filterlim @ D @ B
@ ^ [X4: D] : ( times_times @ B @ ( F3 @ X4 ) @ ( G @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( one_one @ B ) )
@ F4 ) ) ) ) ).
% tendsto_mult_one
thf(fact_7132_tendsto__tanh,axiom,
! [A: $tType,C: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [F3: C > A,A2: A,F4: filter @ C] :
( ( filterlim @ C @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ F4 )
=> ( ( ( cosh @ A @ A2 )
!= ( zero_zero @ A ) )
=> ( filterlim @ C @ A
@ ^ [X4: C] : ( tanh @ A @ ( F3 @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( tanh @ A @ A2 ) )
@ F4 ) ) ) ) ).
% tendsto_tanh
thf(fact_7133_tendsto__of__int__ceiling,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( ( ring_1 @ C )
& ( topolo4958980785337419405_space @ C )
& ( archim2362893244070406136eiling @ B )
& ( topolo2564578578187576103pology @ B ) )
=> ! [F3: A > B,L: B,F4: filter @ A] :
( ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ F4 )
=> ( ~ ( member @ B @ L @ ( ring_1_Ints @ B ) )
=> ( filterlim @ A @ C
@ ^ [X4: A] : ( ring_1_of_int @ C @ ( archimedean_ceiling @ B @ ( F3 @ X4 ) ) )
@ ( topolo7230453075368039082e_nhds @ C @ ( ring_1_of_int @ C @ ( archimedean_ceiling @ B @ L ) ) )
@ F4 ) ) ) ) ).
% tendsto_of_int_ceiling
thf(fact_7134_tendsto__cot,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [F3: A > A,A2: A,F4: filter @ A] :
( ( filterlim @ A @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ F4 )
=> ( ( ( sin @ A @ A2 )
!= ( zero_zero @ A ) )
=> ( filterlim @ A @ A
@ ^ [X4: A] : ( cot @ A @ ( F3 @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( cot @ A @ A2 ) )
@ F4 ) ) ) ) ).
% tendsto_cot
thf(fact_7135_tendsto__diff,axiom,
! [A: $tType,B: $tType] :
( ( topolo1633459387980952147up_add @ A )
=> ! [F3: B > A,A2: A,F4: filter @ B,G: B > A,B2: A] :
( ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ F4 )
=> ( ( filterlim @ B @ A @ G @ ( topolo7230453075368039082e_nhds @ A @ B2 ) @ F4 )
=> ( filterlim @ B @ A
@ ^ [X4: B] : ( minus_minus @ A @ ( F3 @ X4 ) @ ( G @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( minus_minus @ A @ A2 @ B2 ) )
@ F4 ) ) ) ) ).
% tendsto_diff
thf(fact_7136_continuous__diff,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( topolo1633459387980952147up_add @ B ) )
=> ! [F4: filter @ A,F3: A > B,G: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ F4 @ F3 )
=> ( ( topolo3448309680560233919inuous @ A @ B @ F4 @ G )
=> ( topolo3448309680560233919inuous @ A @ B @ F4
@ ^ [X4: A] : ( minus_minus @ B @ ( F3 @ X4 ) @ ( G @ X4 ) ) ) ) ) ) ).
% continuous_diff
thf(fact_7137_LIM__zero,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F3: A > B,L: B,F4: filter @ A] :
( ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ F4 )
=> ( filterlim @ A @ B
@ ^ [X4: A] : ( minus_minus @ B @ ( F3 @ X4 ) @ L )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ F4 ) ) ) ).
% LIM_zero
thf(fact_7138_LIM__zero__iff,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F3: A > B,L: B,F4: filter @ A] :
( ( filterlim @ A @ B
@ ^ [X4: A] : ( minus_minus @ B @ ( F3 @ X4 ) @ L )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ F4 )
= ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ F4 ) ) ) ).
% LIM_zero_iff
thf(fact_7139_Lim__transform,axiom,
! [A: $tType,B: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [G: B > A,A2: A,F4: filter @ B,F3: B > A] :
( ( filterlim @ B @ A @ G @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ F4 )
=> ( ( filterlim @ B @ A
@ ^ [X4: B] : ( minus_minus @ A @ ( F3 @ X4 ) @ ( G @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ F4 )
=> ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ F4 ) ) ) ) ).
% Lim_transform
thf(fact_7140_Lim__transform2,axiom,
! [A: $tType,B: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: B > A,A2: A,F4: filter @ B,G: B > A] :
( ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ F4 )
=> ( ( filterlim @ B @ A
@ ^ [X4: B] : ( minus_minus @ A @ ( F3 @ X4 ) @ ( G @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ F4 )
=> ( filterlim @ B @ A @ G @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ F4 ) ) ) ) ).
% Lim_transform2
thf(fact_7141_LIM__zero__cancel,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F3: A > B,L: B,F4: filter @ A] :
( ( filterlim @ A @ B
@ ^ [X4: A] : ( minus_minus @ B @ ( F3 @ X4 ) @ L )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ F4 )
=> ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ F4 ) ) ) ).
% LIM_zero_cancel
thf(fact_7142_Lim__transform__eq,axiom,
! [A: $tType,B: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: B > A,G: B > A,F4: filter @ B,A2: A] :
( ( filterlim @ B @ A
@ ^ [X4: B] : ( minus_minus @ A @ ( F3 @ X4 ) @ ( G @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ F4 )
=> ( ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ F4 )
= ( filterlim @ B @ A @ G @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ F4 ) ) ) ) ).
% Lim_transform_eq
thf(fact_7143_tendsto__rabs,axiom,
! [A: $tType,F3: A > real,L: real,F4: filter @ A] :
( ( filterlim @ A @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ L ) @ F4 )
=> ( filterlim @ A @ real
@ ^ [X4: A] : ( abs_abs @ real @ ( F3 @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( abs_abs @ real @ L ) )
@ F4 ) ) ).
% tendsto_rabs
thf(fact_7144_tendsto__rabs__zero,axiom,
! [A: $tType,F3: A > real,F4: filter @ A] :
( ( filterlim @ A @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ F4 )
=> ( filterlim @ A @ real
@ ^ [X4: A] : ( abs_abs @ real @ ( F3 @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ F4 ) ) ).
% tendsto_rabs_zero
thf(fact_7145_tendsto__rabs__zero__iff,axiom,
! [A: $tType,F3: A > real,F4: filter @ A] :
( ( filterlim @ A @ real
@ ^ [X4: A] : ( abs_abs @ real @ ( F3 @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ F4 )
= ( filterlim @ A @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ F4 ) ) ).
% tendsto_rabs_zero_iff
thf(fact_7146_tendsto__rabs__zero__cancel,axiom,
! [A: $tType,F3: A > real,F4: filter @ A] :
( ( filterlim @ A @ real
@ ^ [X4: A] : ( abs_abs @ real @ ( F3 @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ F4 )
=> ( filterlim @ A @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ F4 ) ) ).
% tendsto_rabs_zero_cancel
thf(fact_7147_continuous__max,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( topolo1944317154257567458pology @ B ) )
=> ! [F4: filter @ A,F3: A > B,G: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ F4 @ F3 )
=> ( ( topolo3448309680560233919inuous @ A @ B @ F4 @ G )
=> ( topolo3448309680560233919inuous @ A @ B @ F4
@ ^ [X4: A] : ( ord_max @ B @ ( F3 @ X4 ) @ ( G @ X4 ) ) ) ) ) ) ).
% continuous_max
thf(fact_7148_tendsto__max,axiom,
! [A: $tType,B: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [X9: B > A,X: A,Net: filter @ B,Y3: B > A,Y2: A] :
( ( filterlim @ B @ A @ X9 @ ( topolo7230453075368039082e_nhds @ A @ X ) @ Net )
=> ( ( filterlim @ B @ A @ Y3 @ ( topolo7230453075368039082e_nhds @ A @ Y2 ) @ Net )
=> ( filterlim @ B @ A
@ ^ [X4: B] : ( ord_max @ A @ ( X9 @ X4 ) @ ( Y3 @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( ord_max @ A @ X @ Y2 ) )
@ Net ) ) ) ) ).
% tendsto_max
thf(fact_7149_tendsto__minus,axiom,
! [A: $tType,B: $tType] :
( ( topolo1633459387980952147up_add @ A )
=> ! [F3: B > A,A2: A,F4: filter @ B] :
( ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ F4 )
=> ( filterlim @ B @ A
@ ^ [X4: B] : ( uminus_uminus @ A @ ( F3 @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( uminus_uminus @ A @ A2 ) )
@ F4 ) ) ) ).
% tendsto_minus
thf(fact_7150_tendsto__minus__cancel,axiom,
! [A: $tType,B: $tType] :
( ( topolo1633459387980952147up_add @ A )
=> ! [F3: B > A,A2: A,F4: filter @ B] :
( ( filterlim @ B @ A
@ ^ [X4: B] : ( uminus_uminus @ A @ ( F3 @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( uminus_uminus @ A @ A2 ) )
@ F4 )
=> ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ F4 ) ) ) ).
% tendsto_minus_cancel
thf(fact_7151_tendsto__minus__cancel__left,axiom,
! [B: $tType,A: $tType] :
( ( topolo1633459387980952147up_add @ B )
=> ! [F3: A > B,Y2: B,F4: filter @ A] :
( ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ ( uminus_uminus @ B @ Y2 ) ) @ F4 )
= ( filterlim @ A @ B
@ ^ [X4: A] : ( uminus_uminus @ B @ ( F3 @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ B @ Y2 )
@ F4 ) ) ) ).
% tendsto_minus_cancel_left
thf(fact_7152_continuous__minus,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( topolo1633459387980952147up_add @ B ) )
=> ! [F4: filter @ A,F3: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ F4 @ F3 )
=> ( topolo3448309680560233919inuous @ A @ B @ F4
@ ^ [X4: A] : ( uminus_uminus @ B @ ( F3 @ X4 ) ) ) ) ) ).
% continuous_minus
thf(fact_7153_continuous__prod_H,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( ( topological_t2_space @ B )
& ( topolo4987421752381908075d_mult @ C ) )
=> ! [I6: set @ A,F4: filter @ B,F3: A > B > C] :
( ! [I3: A] :
( ( member @ A @ I3 @ I6 )
=> ( topolo3448309680560233919inuous @ B @ C @ F4 @ ( F3 @ I3 ) ) )
=> ( topolo3448309680560233919inuous @ B @ C @ F4
@ ^ [X4: B] :
( groups7121269368397514597t_prod @ A @ C
@ ^ [I: A] : ( F3 @ I @ X4 )
@ I6 ) ) ) ) ).
% continuous_prod'
thf(fact_7154_continuous__prod,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( ( topological_t2_space @ B )
& ( real_V4412858255891104859lgebra @ C )
& ( comm_ring_1 @ C ) )
=> ! [S3: set @ A,F4: filter @ B,F3: A > B > C] :
( ! [I3: A] :
( ( member @ A @ I3 @ S3 )
=> ( topolo3448309680560233919inuous @ B @ C @ F4 @ ( F3 @ I3 ) ) )
=> ( topolo3448309680560233919inuous @ B @ C @ F4
@ ^ [X4: B] :
( groups7121269368397514597t_prod @ A @ C
@ ^ [I: A] : ( F3 @ I @ X4 )
@ S3 ) ) ) ) ).
% continuous_prod
thf(fact_7155_tendsto__prod_H,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( topolo4987421752381908075d_mult @ C )
=> ! [I6: set @ A,F3: A > B > C,A2: A > C,F4: filter @ B] :
( ! [I3: A] :
( ( member @ A @ I3 @ I6 )
=> ( filterlim @ B @ C @ ( F3 @ I3 ) @ ( topolo7230453075368039082e_nhds @ C @ ( A2 @ I3 ) ) @ F4 ) )
=> ( filterlim @ B @ C
@ ^ [X4: B] :
( groups7121269368397514597t_prod @ A @ C
@ ^ [I: A] : ( F3 @ I @ X4 )
@ I6 )
@ ( topolo7230453075368039082e_nhds @ C @ ( groups7121269368397514597t_prod @ A @ C @ A2 @ I6 ) )
@ F4 ) ) ) ).
% tendsto_prod'
thf(fact_7156_tendsto__prod,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( ( real_V4412858255891104859lgebra @ C )
& ( comm_ring_1 @ C ) )
=> ! [S3: set @ A,F3: A > B > C,L6: A > C,F4: filter @ B] :
( ! [I3: A] :
( ( member @ A @ I3 @ S3 )
=> ( filterlim @ B @ C @ ( F3 @ I3 ) @ ( topolo7230453075368039082e_nhds @ C @ ( L6 @ I3 ) ) @ F4 ) )
=> ( filterlim @ B @ C
@ ^ [X4: B] :
( groups7121269368397514597t_prod @ A @ C
@ ^ [I: A] : ( F3 @ I @ X4 )
@ S3 )
@ ( topolo7230453075368039082e_nhds @ C @ ( groups7121269368397514597t_prod @ A @ C @ L6 @ S3 ) )
@ F4 ) ) ) ).
% tendsto_prod
thf(fact_7157_tendsto__one__prod_H,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( topolo4987421752381908075d_mult @ C )
=> ! [I6: set @ B,F3: A > B > C,F4: filter @ A] :
( ! [I3: B] :
( ( member @ B @ I3 @ I6 )
=> ( filterlim @ A @ C
@ ^ [X4: A] : ( F3 @ X4 @ I3 )
@ ( topolo7230453075368039082e_nhds @ C @ ( one_one @ C ) )
@ F4 ) )
=> ( filterlim @ A @ C
@ ^ [I: A] : ( groups7121269368397514597t_prod @ B @ C @ ( F3 @ I ) @ I6 )
@ ( topolo7230453075368039082e_nhds @ C @ ( one_one @ C ) )
@ F4 ) ) ) ).
% tendsto_one_prod'
thf(fact_7158_tendsto__null__sum,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( topolo5987344860129210374id_add @ C )
=> ! [I6: set @ B,F3: A > B > C,F4: filter @ A] :
( ! [I3: B] :
( ( member @ B @ I3 @ I6 )
=> ( filterlim @ A @ C
@ ^ [X4: A] : ( F3 @ X4 @ I3 )
@ ( topolo7230453075368039082e_nhds @ C @ ( zero_zero @ C ) )
@ F4 ) )
=> ( filterlim @ A @ C
@ ^ [I: A] : ( groups7311177749621191930dd_sum @ B @ C @ ( F3 @ I ) @ I6 )
@ ( topolo7230453075368039082e_nhds @ C @ ( zero_zero @ C ) )
@ F4 ) ) ) ).
% tendsto_null_sum
thf(fact_7159_tendsto__sum,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( topolo5987344860129210374id_add @ C )
=> ! [I6: set @ A,F3: A > B > C,A2: A > C,F4: filter @ B] :
( ! [I3: A] :
( ( member @ A @ I3 @ I6 )
=> ( filterlim @ B @ C @ ( F3 @ I3 ) @ ( topolo7230453075368039082e_nhds @ C @ ( A2 @ I3 ) ) @ F4 ) )
=> ( filterlim @ B @ C
@ ^ [X4: B] :
( groups7311177749621191930dd_sum @ A @ C
@ ^ [I: A] : ( F3 @ I @ X4 )
@ I6 )
@ ( topolo7230453075368039082e_nhds @ C @ ( groups7311177749621191930dd_sum @ A @ C @ A2 @ I6 ) )
@ F4 ) ) ) ).
% tendsto_sum
thf(fact_7160_continuous__sum,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( ( topological_t2_space @ B )
& ( topolo5987344860129210374id_add @ C ) )
=> ! [I6: set @ A,F4: filter @ B,F3: A > B > C] :
( ! [I3: A] :
( ( member @ A @ I3 @ I6 )
=> ( topolo3448309680560233919inuous @ B @ C @ F4 @ ( F3 @ I3 ) ) )
=> ( topolo3448309680560233919inuous @ B @ C @ F4
@ ^ [X4: B] :
( groups7311177749621191930dd_sum @ A @ C
@ ^ [I: A] : ( F3 @ I @ X4 )
@ I6 ) ) ) ) ).
% continuous_sum
thf(fact_7161_filterlim__ident,axiom,
! [A: $tType,F4: filter @ A] :
( filterlim @ A @ A
@ ^ [X4: A] : X4
@ F4
@ F4 ) ).
% filterlim_ident
thf(fact_7162_filterlim__compose,axiom,
! [B: $tType,A: $tType,C: $tType,G: A > B,F33: filter @ B,F23: filter @ A,F3: C > A,F13: filter @ C] :
( ( filterlim @ A @ B @ G @ F33 @ F23 )
=> ( ( filterlim @ C @ A @ F3 @ F23 @ F13 )
=> ( filterlim @ C @ B
@ ^ [X4: C] : ( G @ ( F3 @ X4 ) )
@ F33
@ F13 ) ) ) ).
% filterlim_compose
thf(fact_7163_tendsto__arcosh,axiom,
! [B: $tType,F3: B > real,A2: real,F4: filter @ B] :
( ( filterlim @ B @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ A2 ) @ F4 )
=> ( ( ord_less @ real @ ( one_one @ real ) @ A2 )
=> ( filterlim @ B @ real
@ ^ [X4: B] : ( arcosh @ real @ ( F3 @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( arcosh @ real @ A2 ) )
@ F4 ) ) ) ).
% tendsto_arcosh
thf(fact_7164_tendsto__null__power,axiom,
! [B: $tType,A: $tType] :
( ( real_V2822296259951069270ebra_1 @ B )
=> ! [F3: A > B,F4: filter @ A,N: nat] :
( ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) @ F4 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( filterlim @ A @ B
@ ^ [X4: A] : ( power_power @ B @ ( F3 @ X4 ) @ N )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ F4 ) ) ) ) ).
% tendsto_null_power
thf(fact_7165_tendsto__log,axiom,
! [A: $tType,F3: A > real,A2: real,F4: filter @ A,G: A > real,B2: real] :
( ( filterlim @ A @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ A2 ) @ F4 )
=> ( ( filterlim @ A @ real @ G @ ( topolo7230453075368039082e_nhds @ real @ B2 ) @ F4 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
=> ( ( A2
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ B2 )
=> ( filterlim @ A @ real
@ ^ [X4: A] : ( log @ ( F3 @ X4 ) @ ( G @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( log @ A2 @ B2 ) )
@ F4 ) ) ) ) ) ) ).
% tendsto_log
thf(fact_7166_tendsto__artanh,axiom,
! [A: $tType,F3: A > real,A2: real,F4: filter @ A] :
( ( filterlim @ A @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ A2 ) @ F4 )
=> ( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ A2 )
=> ( ( ord_less @ real @ A2 @ ( one_one @ real ) )
=> ( filterlim @ A @ real
@ ^ [X4: A] : ( artanh @ real @ ( F3 @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( artanh @ real @ A2 ) )
@ F4 ) ) ) ) ).
% tendsto_artanh
thf(fact_7167_tendsto__mono,axiom,
! [A: $tType,B: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F4: filter @ B,F14: filter @ B,F3: B > A,L: A] :
( ( ord_less_eq @ ( filter @ B ) @ F4 @ F14 )
=> ( ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F14 )
=> ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F4 ) ) ) ) ).
% tendsto_mono
thf(fact_7168_filterlim__mono,axiom,
! [B: $tType,A: $tType,F3: A > B,F23: filter @ B,F13: filter @ A,F24: filter @ B,F15: filter @ A] :
( ( filterlim @ A @ B @ F3 @ F23 @ F13 )
=> ( ( ord_less_eq @ ( filter @ B ) @ F23 @ F24 )
=> ( ( ord_less_eq @ ( filter @ A ) @ F15 @ F13 )
=> ( filterlim @ A @ B @ F3 @ F24 @ F15 ) ) ) ) ).
% filterlim_mono
thf(fact_7169_filterlim__INF_H,axiom,
! [C: $tType,B: $tType,A: $tType,X: A,A3: set @ A,F3: B > C,F4: filter @ C,G6: A > ( filter @ B )] :
( ( member @ A @ X @ A3 )
=> ( ( filterlim @ B @ C @ F3 @ F4 @ ( G6 @ X ) )
=> ( filterlim @ B @ C @ F3 @ F4 @ ( complete_Inf_Inf @ ( filter @ B ) @ ( image @ A @ ( filter @ B ) @ G6 @ A3 ) ) ) ) ) ).
% filterlim_INF'
thf(fact_7170_filterlim__INF,axiom,
! [A: $tType,B: $tType,C: $tType,F3: A > B,G6: C > ( filter @ B ),B5: set @ C,F4: filter @ A] :
( ( filterlim @ A @ B @ F3 @ ( complete_Inf_Inf @ ( filter @ B ) @ ( image @ C @ ( filter @ B ) @ G6 @ B5 ) ) @ F4 )
= ( ! [X4: C] :
( ( member @ C @ X4 @ B5 )
=> ( filterlim @ A @ B @ F3 @ ( G6 @ X4 ) @ F4 ) ) ) ) ).
% filterlim_INF
thf(fact_7171_DERIV__LIM__iff,axiom,
! [A: $tType] :
( ( ( inverse @ A )
& ( real_V822414075346904944vector @ A ) )
=> ! [F3: A > A,A2: A,D4: A] :
( ( filterlim @ A @ A
@ ^ [H: A] : ( divide_divide @ A @ ( minus_minus @ A @ ( F3 @ ( plus_plus @ A @ A2 @ H ) ) @ ( F3 @ A2 ) ) @ H )
@ ( topolo7230453075368039082e_nhds @ A @ D4 )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) )
= ( filterlim @ A @ A
@ ^ [X4: A] : ( divide_divide @ A @ ( minus_minus @ A @ ( F3 @ X4 ) @ ( F3 @ A2 ) ) @ ( minus_minus @ A @ X4 @ A2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ D4 )
@ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% DERIV_LIM_iff
thf(fact_7172_isCont__Lb__Ub,axiom,
! [A2: real,B2: real,F3: real > real] :
( ( ord_less_eq @ real @ A2 @ B2 )
=> ( ! [X5: real] :
( ( ( ord_less_eq @ real @ A2 @ X5 )
& ( ord_less_eq @ real @ X5 @ B2 ) )
=> ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ X5 @ ( top_top @ ( set @ real ) ) ) @ F3 ) )
=> ? [L7: real,M8: real] :
( ! [X2: real] :
( ( ( ord_less_eq @ real @ A2 @ X2 )
& ( ord_less_eq @ real @ X2 @ B2 ) )
=> ( ( ord_less_eq @ real @ L7 @ ( F3 @ X2 ) )
& ( ord_less_eq @ real @ ( F3 @ X2 ) @ M8 ) ) )
& ! [Y4: real] :
( ( ( ord_less_eq @ real @ L7 @ Y4 )
& ( ord_less_eq @ real @ Y4 @ M8 ) )
=> ? [X5: real] :
( ( ord_less_eq @ real @ A2 @ X5 )
& ( ord_less_eq @ real @ X5 @ B2 )
& ( ( F3 @ X5 )
= Y4 ) ) ) ) ) ) ).
% isCont_Lb_Ub
thf(fact_7173_LIM__compose2,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( topolo4958980785337419405_space @ B )
& ( topolo4958980785337419405_space @ C ) )
=> ! [F3: A > B,B2: B,A2: A,G: B > C,C2: C] :
( ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ B2 ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) )
=> ( ( filterlim @ B @ C @ G @ ( topolo7230453075368039082e_nhds @ C @ C2 ) @ ( topolo174197925503356063within @ B @ B2 @ ( top_top @ ( set @ B ) ) ) )
=> ( ? [D6: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D6 )
& ! [X5: A] :
( ( ( X5 != A2 )
& ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ X5 @ A2 ) ) @ D6 ) )
=> ( ( F3 @ X5 )
!= B2 ) ) )
=> ( filterlim @ A @ C
@ ^ [X4: A] : ( G @ ( F3 @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ C @ C2 )
@ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ).
% LIM_compose2
thf(fact_7174_isCont__real__sqrt,axiom,
! [X: real] : ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) @ sqrt ) ).
% isCont_real_sqrt
thf(fact_7175_isCont__real__root,axiom,
! [X: real,N: nat] : ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) @ ( root @ N ) ) ).
% isCont_real_root
thf(fact_7176_continuous__at__within__divide,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_V3459762299906320749_field @ B ) )
=> ! [A2: A,S: set @ A,F3: A > B,G: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ S ) @ F3 )
=> ( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ S ) @ G )
=> ( ( ( G @ A2 )
!= ( zero_zero @ B ) )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ S )
@ ^ [X4: A] : ( divide_divide @ B @ ( F3 @ X4 ) @ ( G @ X4 ) ) ) ) ) ) ) ).
% continuous_at_within_divide
thf(fact_7177_isCont__mult,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_V4412858255891104859lgebra @ B ) )
=> ! [A2: A,F3: A > B,G: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) @ F3 )
=> ( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) @ G )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) )
@ ^ [X4: A] : ( times_times @ B @ ( F3 @ X4 ) @ ( G @ X4 ) ) ) ) ) ) ).
% isCont_mult
thf(fact_7178_isCont__add,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( topolo6943815403480290642id_add @ B ) )
=> ! [A2: A,F3: A > B,G: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) @ F3 )
=> ( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) @ G )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) )
@ ^ [X4: A] : ( plus_plus @ B @ ( F3 @ X4 ) @ ( G @ X4 ) ) ) ) ) ) ).
% isCont_add
thf(fact_7179_isCont__diff,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [A2: A,F3: A > B,G: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) @ F3 )
=> ( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) @ G )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) )
@ ^ [X4: A] : ( minus_minus @ B @ ( F3 @ X4 ) @ ( G @ X4 ) ) ) ) ) ) ).
% isCont_diff
thf(fact_7180_isCont__minus,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [A2: A,F3: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) @ F3 )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) )
@ ^ [X4: A] : ( uminus_uminus @ B @ ( F3 @ X4 ) ) ) ) ) ).
% isCont_minus
thf(fact_7181_isCont__power,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( power @ B )
& ( real_V4412858255891104859lgebra @ B ) )
=> ! [A2: A,F3: A > B,N: nat] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) @ F3 )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) )
@ ^ [X4: A] : ( power_power @ B @ ( F3 @ X4 ) @ N ) ) ) ) ).
% isCont_power
thf(fact_7182_continuous__at__within__inverse,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_V8999393235501362500lgebra @ B ) )
=> ! [A2: A,S: set @ A,F3: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ S ) @ F3 )
=> ( ( ( F3 @ A2 )
!= ( zero_zero @ B ) )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ S )
@ ^ [X4: A] : ( inverse_inverse @ B @ ( F3 @ X4 ) ) ) ) ) ) ).
% continuous_at_within_inverse
thf(fact_7183_isCont__sum,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( ( topological_t2_space @ B )
& ( topolo5987344860129210374id_add @ C ) )
=> ! [A3: set @ A,A2: B,F3: A > B > C] :
( ! [X5: A] :
( ( member @ A @ X5 @ A3 )
=> ( topolo3448309680560233919inuous @ B @ C @ ( topolo174197925503356063within @ B @ A2 @ ( top_top @ ( set @ B ) ) ) @ ( F3 @ X5 ) ) )
=> ( topolo3448309680560233919inuous @ B @ C @ ( topolo174197925503356063within @ B @ A2 @ ( top_top @ ( set @ B ) ) )
@ ^ [X4: B] :
( groups7311177749621191930dd_sum @ A @ C
@ ^ [I: A] : ( F3 @ I @ X4 )
@ A3 ) ) ) ) ).
% isCont_sum
thf(fact_7184_continuous__at__within__sgn,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [A2: A,S: set @ A,F3: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ S ) @ F3 )
=> ( ( ( F3 @ A2 )
!= ( zero_zero @ B ) )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ S )
@ ^ [X4: A] : ( sgn_sgn @ B @ ( F3 @ X4 ) ) ) ) ) ) ).
% continuous_at_within_sgn
thf(fact_7185_isCont__cos_H,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_Vector_banach @ B )
& ( real_V3459762299906320749_field @ B ) )
=> ! [A2: A,F3: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) @ F3 )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) )
@ ^ [X4: A] : ( cos @ B @ ( F3 @ X4 ) ) ) ) ) ).
% isCont_cos'
thf(fact_7186_isCont__sin_H,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_Vector_banach @ B )
& ( real_V3459762299906320749_field @ B ) )
=> ! [A2: A,F3: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) @ F3 )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) )
@ ^ [X4: A] : ( sin @ B @ ( F3 @ X4 ) ) ) ) ) ).
% isCont_sin'
thf(fact_7187_isCont__exp_H,axiom,
! [A: $tType,C: $tType] :
( ( ( topological_t2_space @ C )
& ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [A2: C,F3: C > A] :
( ( topolo3448309680560233919inuous @ C @ A @ ( topolo174197925503356063within @ C @ A2 @ ( top_top @ ( set @ C ) ) ) @ F3 )
=> ( topolo3448309680560233919inuous @ C @ A @ ( topolo174197925503356063within @ C @ A2 @ ( top_top @ ( set @ C ) ) )
@ ^ [X4: C] : ( exp @ A @ ( F3 @ X4 ) ) ) ) ) ).
% isCont_exp'
thf(fact_7188_isCont__pochhammer,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [Z3: A,N: nat] :
( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ Z3 @ ( top_top @ ( set @ A ) ) )
@ ^ [Z6: A] : ( comm_s3205402744901411588hammer @ A @ Z6 @ N ) ) ) ).
% isCont_pochhammer
thf(fact_7189_DERIV__D,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,D4: A,X: A] :
( ( has_field_derivative @ A @ F3 @ D4 @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) )
=> ( filterlim @ A @ A
@ ^ [H: A] : ( divide_divide @ A @ ( minus_minus @ A @ ( F3 @ ( plus_plus @ A @ X @ H ) ) @ ( F3 @ X ) ) @ H )
@ ( topolo7230453075368039082e_nhds @ A @ D4 )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% DERIV_D
thf(fact_7190_DERIV__def,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,D4: A,X: A] :
( ( has_field_derivative @ A @ F3 @ D4 @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) )
= ( filterlim @ A @ A
@ ^ [H: A] : ( divide_divide @ A @ ( minus_minus @ A @ ( F3 @ ( plus_plus @ A @ X @ H ) ) @ ( F3 @ X ) ) @ H )
@ ( topolo7230453075368039082e_nhds @ A @ D4 )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% DERIV_def
thf(fact_7191_lim__exp__minus__1,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ( filterlim @ A @ A
@ ^ [Z6: A] : ( divide_divide @ A @ ( minus_minus @ A @ ( exp @ A @ Z6 ) @ ( one_one @ A ) ) @ Z6 )
@ ( topolo7230453075368039082e_nhds @ A @ ( one_one @ A ) )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ).
% lim_exp_minus_1
thf(fact_7192_lemma__termdiff4,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [K2: real,F3: A > B,K5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K2 )
=> ( ! [H4: A] :
( ( H4
!= ( zero_zero @ A ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ H4 ) @ K2 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( F3 @ H4 ) ) @ ( times_times @ real @ K5 @ ( real_V7770717601297561774m_norm @ A @ H4 ) ) ) ) )
=> ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) @ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% lemma_termdiff4
thf(fact_7193_isCont__bounded,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [A2: real,B2: real,F3: real > A] :
( ( ord_less_eq @ real @ A2 @ B2 )
=> ( ! [X5: real] :
( ( ( ord_less_eq @ real @ A2 @ X5 )
& ( ord_less_eq @ real @ X5 @ B2 ) )
=> ( topolo3448309680560233919inuous @ real @ A @ ( topolo174197925503356063within @ real @ X5 @ ( top_top @ ( set @ real ) ) ) @ F3 ) )
=> ? [M8: A] :
! [X2: real] :
( ( ( ord_less_eq @ real @ A2 @ X2 )
& ( ord_less_eq @ real @ X2 @ B2 ) )
=> ( ord_less_eq @ A @ ( F3 @ X2 ) @ M8 ) ) ) ) ) ).
% isCont_bounded
thf(fact_7194_isCont__eq__Ub,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [A2: real,B2: real,F3: real > A] :
( ( ord_less_eq @ real @ A2 @ B2 )
=> ( ! [X5: real] :
( ( ( ord_less_eq @ real @ A2 @ X5 )
& ( ord_less_eq @ real @ X5 @ B2 ) )
=> ( topolo3448309680560233919inuous @ real @ A @ ( topolo174197925503356063within @ real @ X5 @ ( top_top @ ( set @ real ) ) ) @ F3 ) )
=> ? [M8: A] :
( ! [X2: real] :
( ( ( ord_less_eq @ real @ A2 @ X2 )
& ( ord_less_eq @ real @ X2 @ B2 ) )
=> ( ord_less_eq @ A @ ( F3 @ X2 ) @ M8 ) )
& ? [X5: real] :
( ( ord_less_eq @ real @ A2 @ X5 )
& ( ord_less_eq @ real @ X5 @ B2 )
& ( ( F3 @ X5 )
= M8 ) ) ) ) ) ) ).
% isCont_eq_Ub
thf(fact_7195_isCont__eq__Lb,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [A2: real,B2: real,F3: real > A] :
( ( ord_less_eq @ real @ A2 @ B2 )
=> ( ! [X5: real] :
( ( ( ord_less_eq @ real @ A2 @ X5 )
& ( ord_less_eq @ real @ X5 @ B2 ) )
=> ( topolo3448309680560233919inuous @ real @ A @ ( topolo174197925503356063within @ real @ X5 @ ( top_top @ ( set @ real ) ) ) @ F3 ) )
=> ? [M8: A] :
( ! [X2: real] :
( ( ( ord_less_eq @ real @ A2 @ X2 )
& ( ord_less_eq @ real @ X2 @ B2 ) )
=> ( ord_less_eq @ A @ M8 @ ( F3 @ X2 ) ) )
& ? [X5: real] :
( ( ord_less_eq @ real @ A2 @ X5 )
& ( ord_less_eq @ real @ X5 @ B2 )
& ( ( F3 @ X5 )
= M8 ) ) ) ) ) ) ).
% isCont_eq_Lb
thf(fact_7196_isCont__inverse__function2,axiom,
! [A2: real,X: real,B2: real,G: real > real,F3: real > real] :
( ( ord_less @ real @ A2 @ X )
=> ( ( ord_less @ real @ X @ B2 )
=> ( ! [Z: real] :
( ( ord_less_eq @ real @ A2 @ Z )
=> ( ( ord_less_eq @ real @ Z @ B2 )
=> ( ( G @ ( F3 @ Z ) )
= Z ) ) )
=> ( ! [Z: real] :
( ( ord_less_eq @ real @ A2 @ Z )
=> ( ( ord_less_eq @ real @ Z @ B2 )
=> ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ Z @ ( top_top @ ( set @ real ) ) ) @ F3 ) ) )
=> ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ ( F3 @ X ) @ ( top_top @ ( set @ real ) ) ) @ G ) ) ) ) ) ).
% isCont_inverse_function2
thf(fact_7197_field__has__derivative__at,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,D4: A,X: A] :
( ( has_derivative @ A @ A @ F3 @ ( times_times @ A @ D4 ) @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) )
= ( filterlim @ A @ A
@ ^ [H: A] : ( divide_divide @ A @ ( minus_minus @ A @ ( F3 @ ( plus_plus @ A @ X @ H ) ) @ ( F3 @ X ) ) @ H )
@ ( topolo7230453075368039082e_nhds @ A @ D4 )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% field_has_derivative_at
thf(fact_7198_isCont__divide,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_V3459762299906320749_field @ B ) )
=> ! [A2: A,F3: A > B,G: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) @ F3 )
=> ( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) @ G )
=> ( ( ( G @ A2 )
!= ( zero_zero @ B ) )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) )
@ ^ [X4: A] : ( divide_divide @ B @ ( F3 @ X4 ) @ ( G @ X4 ) ) ) ) ) ) ) ).
% isCont_divide
thf(fact_7199_isCont__sgn,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [A2: A,F3: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) @ F3 )
=> ( ( ( F3 @ A2 )
!= ( zero_zero @ B ) )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) )
@ ^ [X4: A] : ( sgn_sgn @ B @ ( F3 @ X4 ) ) ) ) ) ) ).
% isCont_sgn
thf(fact_7200_filterlim__at__to__0,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: A > B,F4: filter @ B,A2: A] :
( ( filterlim @ A @ B @ F3 @ F4 @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) )
= ( filterlim @ A @ B
@ ^ [X4: A] : ( F3 @ ( plus_plus @ A @ X4 @ A2 ) )
@ F4
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% filterlim_at_to_0
thf(fact_7201_continuous__within__tan,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A,S: set @ A,F3: A > A] :
( ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ X @ S ) @ F3 )
=> ( ( ( cos @ A @ ( F3 @ X ) )
!= ( zero_zero @ A ) )
=> ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ X @ S )
@ ^ [X4: A] : ( tan @ A @ ( F3 @ X4 ) ) ) ) ) ) ).
% continuous_within_tan
thf(fact_7202_continuous__within__cot,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A,S: set @ A,F3: A > A] :
( ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ X @ S ) @ F3 )
=> ( ( ( sin @ A @ ( F3 @ X ) )
!= ( zero_zero @ A ) )
=> ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ X @ S )
@ ^ [X4: A] : ( cot @ A @ ( F3 @ X4 ) ) ) ) ) ) ).
% continuous_within_cot
thf(fact_7203_continuous__at__within__tanh,axiom,
! [A: $tType,C: $tType] :
( ( ( topological_t2_space @ C )
& ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: C,A3: set @ C,F3: C > A] :
( ( topolo3448309680560233919inuous @ C @ A @ ( topolo174197925503356063within @ C @ X @ A3 ) @ F3 )
=> ( ( ( cosh @ A @ ( F3 @ X ) )
!= ( zero_zero @ A ) )
=> ( topolo3448309680560233919inuous @ C @ A @ ( topolo174197925503356063within @ C @ X @ A3 )
@ ^ [X4: C] : ( tanh @ A @ ( F3 @ X4 ) ) ) ) ) ) ).
% continuous_at_within_tanh
thf(fact_7204_CARAT__DERIV,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,L: A,X: A] :
( ( has_field_derivative @ A @ F3 @ L @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) )
= ( ? [G2: A > A] :
( ! [Z6: A] :
( ( minus_minus @ A @ ( F3 @ Z6 ) @ ( F3 @ X ) )
= ( times_times @ A @ ( G2 @ Z6 ) @ ( minus_minus @ A @ Z6 @ X ) ) )
& ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) @ G2 )
& ( ( G2 @ X )
= L ) ) ) ) ) ).
% CARAT_DERIV
thf(fact_7205_isCont__has__Ub,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [A2: real,B2: real,F3: real > A] :
( ( ord_less_eq @ real @ A2 @ B2 )
=> ( ! [X5: real] :
( ( ( ord_less_eq @ real @ A2 @ X5 )
& ( ord_less_eq @ real @ X5 @ B2 ) )
=> ( topolo3448309680560233919inuous @ real @ A @ ( topolo174197925503356063within @ real @ X5 @ ( top_top @ ( set @ real ) ) ) @ F3 ) )
=> ? [M8: A] :
( ! [X2: real] :
( ( ( ord_less_eq @ real @ A2 @ X2 )
& ( ord_less_eq @ real @ X2 @ B2 ) )
=> ( ord_less_eq @ A @ ( F3 @ X2 ) @ M8 ) )
& ! [N7: A] :
( ( ord_less @ A @ N7 @ M8 )
=> ? [X5: real] :
( ( ord_less_eq @ real @ A2 @ X5 )
& ( ord_less_eq @ real @ X5 @ B2 )
& ( ord_less @ A @ N7 @ ( F3 @ X5 ) ) ) ) ) ) ) ) ).
% isCont_has_Ub
thf(fact_7206_filterlim__shift,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: A > B,F4: filter @ B,A2: A,D2: A] :
( ( filterlim @ A @ B @ F3 @ F4 @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) )
=> ( filterlim @ A @ B @ ( comp @ A @ B @ A @ F3 @ ( plus_plus @ A @ D2 ) ) @ F4 @ ( topolo174197925503356063within @ A @ ( minus_minus @ A @ A2 @ D2 ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% filterlim_shift
thf(fact_7207_filterlim__shift__iff,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: A > B,D2: A,F4: filter @ B,A2: A] :
( ( filterlim @ A @ B @ ( comp @ A @ B @ A @ F3 @ ( plus_plus @ A @ D2 ) ) @ F4 @ ( topolo174197925503356063within @ A @ ( minus_minus @ A @ A2 @ D2 ) @ ( top_top @ ( set @ A ) ) ) )
= ( filterlim @ A @ B @ F3 @ F4 @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% filterlim_shift_iff
thf(fact_7208_powser__limit__0,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [S: real,A2: nat > A,F3: A > A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ S )
=> ( ! [X5: A] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X5 ) @ S )
=> ( sums @ A
@ ^ [N2: nat] : ( times_times @ A @ ( A2 @ N2 ) @ ( power_power @ A @ X5 @ N2 ) )
@ ( F3 @ X5 ) ) )
=> ( filterlim @ A @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ ( A2 @ ( zero_zero @ nat ) ) ) @ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% powser_limit_0
thf(fact_7209_powser__limit__0__strong,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [S: real,A2: nat > A,F3: A > A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ S )
=> ( ! [X5: A] :
( ( X5
!= ( zero_zero @ A ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X5 ) @ S )
=> ( sums @ A
@ ^ [N2: nat] : ( times_times @ A @ ( A2 @ N2 ) @ ( power_power @ A @ X5 @ N2 ) )
@ ( F3 @ X5 ) ) ) )
=> ( filterlim @ A @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ ( A2 @ ( zero_zero @ nat ) ) ) @ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% powser_limit_0_strong
thf(fact_7210_lemma__termdiff5,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_Vector_banach @ B ) )
=> ! [K2: real,F3: nat > real,G: A > nat > B] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K2 )
=> ( ( summable @ real @ F3 )
=> ( ! [H4: A,N3: nat] :
( ( H4
!= ( zero_zero @ A ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ H4 ) @ K2 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( G @ H4 @ N3 ) ) @ ( times_times @ real @ ( F3 @ N3 ) @ ( real_V7770717601297561774m_norm @ A @ H4 ) ) ) ) )
=> ( filterlim @ A @ B
@ ^ [H: A] : ( suminf @ B @ ( G @ H ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ).
% lemma_termdiff5
thf(fact_7211_isCont__tan_H,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [A2: A,F3: A > A] :
( ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) @ F3 )
=> ( ( ( cos @ A @ ( F3 @ A2 ) )
!= ( zero_zero @ A ) )
=> ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) )
@ ^ [X4: A] : ( tan @ A @ ( F3 @ X4 ) ) ) ) ) ) ).
% isCont_tan'
thf(fact_7212_isCont__arcosh,axiom,
! [X: real] :
( ( ord_less @ real @ ( one_one @ real ) @ X )
=> ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) @ ( arcosh @ real ) ) ) ).
% isCont_arcosh
thf(fact_7213_LIM__cos__div__sin,axiom,
( filterlim @ real @ real
@ ^ [X4: real] : ( divide_divide @ real @ ( cos @ real @ X4 ) @ ( sin @ real @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( top_top @ ( set @ real ) ) ) ) ).
% LIM_cos_div_sin
thf(fact_7214_isCont__cot_H,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [A2: A,F3: A > A] :
( ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) @ F3 )
=> ( ( ( sin @ A @ ( F3 @ A2 ) )
!= ( zero_zero @ A ) )
=> ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) )
@ ^ [X4: A] : ( cot @ A @ ( F3 @ X4 ) ) ) ) ) ) ).
% isCont_cot'
thf(fact_7215_isCont__polynom,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [A2: A,C2: nat > A,N: nat] :
( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) )
@ ^ [W3: A] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I: nat] : ( times_times @ A @ ( C2 @ I ) @ ( power_power @ A @ W3 @ I ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% isCont_polynom
thf(fact_7216_isCont__arccos,axiom,
! [X: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X )
=> ( ( ord_less @ real @ X @ ( one_one @ real ) )
=> ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) @ arccos ) ) ) ).
% isCont_arccos
thf(fact_7217_isCont__arcsin,axiom,
! [X: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X )
=> ( ( ord_less @ real @ X @ ( one_one @ real ) )
=> ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) @ arcsin ) ) ) ).
% isCont_arcsin
thf(fact_7218_isCont__powser__converges__everywhere,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [C2: nat > A,X: A] :
( ! [Y7: A] :
( summable @ A
@ ^ [N2: nat] : ( times_times @ A @ ( C2 @ N2 ) @ ( power_power @ A @ Y7 @ N2 ) ) )
=> ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) )
@ ^ [X4: A] :
( suminf @ A
@ ^ [N2: nat] : ( times_times @ A @ ( C2 @ N2 ) @ ( power_power @ A @ X4 @ N2 ) ) ) ) ) ) ).
% isCont_powser_converges_everywhere
thf(fact_7219_LIM__less__bound,axiom,
! [B2: real,X: real,F3: real > real] :
( ( ord_less @ real @ B2 @ X )
=> ( ! [X5: real] :
( ( member @ real @ X5 @ ( set_or5935395276787703475ssThan @ real @ B2 @ X ) )
=> ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( F3 @ X5 ) ) )
=> ( ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) @ F3 )
=> ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( F3 @ X ) ) ) ) ) ).
% LIM_less_bound
thf(fact_7220_isCont__artanh,axiom,
! [X: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X )
=> ( ( ord_less @ real @ X @ ( one_one @ real ) )
=> ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) @ ( artanh @ real ) ) ) ) ).
% isCont_artanh
thf(fact_7221_isCont__inverse__function,axiom,
! [D2: real,X: real,G: real > real,F3: real > real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D2 )
=> ( ! [Z: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ ( minus_minus @ real @ Z @ X ) ) @ D2 )
=> ( ( G @ ( F3 @ Z ) )
= Z ) )
=> ( ! [Z: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ ( minus_minus @ real @ Z @ X ) ) @ D2 )
=> ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ Z @ ( top_top @ ( set @ real ) ) ) @ F3 ) )
=> ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ ( F3 @ X ) @ ( top_top @ ( set @ real ) ) ) @ G ) ) ) ) ).
% isCont_inverse_function
thf(fact_7222_GMVT_H,axiom,
! [A2: real,B2: real,F3: real > real,G: real > real,G5: real > real,F9: real > real] :
( ( ord_less @ real @ A2 @ B2 )
=> ( ! [Z: real] :
( ( ord_less_eq @ real @ A2 @ Z )
=> ( ( ord_less_eq @ real @ Z @ B2 )
=> ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ Z @ ( top_top @ ( set @ real ) ) ) @ F3 ) ) )
=> ( ! [Z: real] :
( ( ord_less_eq @ real @ A2 @ Z )
=> ( ( ord_less_eq @ real @ Z @ B2 )
=> ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ Z @ ( top_top @ ( set @ real ) ) ) @ G ) ) )
=> ( ! [Z: real] :
( ( ord_less @ real @ A2 @ Z )
=> ( ( ord_less @ real @ Z @ B2 )
=> ( has_field_derivative @ real @ G @ ( G5 @ Z ) @ ( topolo174197925503356063within @ real @ Z @ ( top_top @ ( set @ real ) ) ) ) ) )
=> ( ! [Z: real] :
( ( ord_less @ real @ A2 @ Z )
=> ( ( ord_less @ real @ Z @ B2 )
=> ( has_field_derivative @ real @ F3 @ ( F9 @ Z ) @ ( topolo174197925503356063within @ real @ Z @ ( top_top @ ( set @ real ) ) ) ) ) )
=> ? [C4: real] :
( ( ord_less @ real @ A2 @ C4 )
& ( ord_less @ real @ C4 @ B2 )
& ( ( times_times @ real @ ( minus_minus @ real @ ( F3 @ B2 ) @ ( F3 @ A2 ) ) @ ( G5 @ C4 ) )
= ( times_times @ real @ ( minus_minus @ real @ ( G @ B2 ) @ ( G @ A2 ) ) @ ( F9 @ C4 ) ) ) ) ) ) ) ) ) ).
% GMVT'
thf(fact_7223_floor__has__real__derivative,axiom,
! [A: $tType] :
( ( ( archim2362893244070406136eiling @ A )
& ( topolo2564578578187576103pology @ A ) )
=> ! [X: real,F3: real > A] :
( ( topolo3448309680560233919inuous @ real @ A @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) @ F3 )
=> ( ~ ( member @ A @ ( F3 @ X ) @ ( ring_1_Ints @ A ) )
=> ( has_field_derivative @ real
@ ^ [X4: real] : ( ring_1_of_int @ real @ ( archim6421214686448440834_floor @ A @ ( F3 @ X4 ) ) )
@ ( zero_zero @ real )
@ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ).
% floor_has_real_derivative
thf(fact_7224_isCont__powser_H,axiom,
! [Aa: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_Vector_banach @ Aa )
& ( real_V3459762299906320749_field @ Aa ) )
=> ! [A2: A,F3: A > Aa,C2: nat > Aa,K5: Aa] :
( ( topolo3448309680560233919inuous @ A @ Aa @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) @ F3 )
=> ( ( summable @ Aa
@ ^ [N2: nat] : ( times_times @ Aa @ ( C2 @ N2 ) @ ( power_power @ Aa @ K5 @ N2 ) ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ Aa @ ( F3 @ A2 ) ) @ ( real_V7770717601297561774m_norm @ Aa @ K5 ) )
=> ( topolo3448309680560233919inuous @ A @ Aa @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) )
@ ^ [X4: A] :
( suminf @ Aa
@ ^ [N2: nat] : ( times_times @ Aa @ ( C2 @ N2 ) @ ( power_power @ Aa @ ( F3 @ X4 ) @ N2 ) ) ) ) ) ) ) ) ).
% isCont_powser'
thf(fact_7225_isCont__powser,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [C2: nat > A,K5: A,X: A] :
( ( summable @ A
@ ^ [N2: nat] : ( times_times @ A @ ( C2 @ N2 ) @ ( power_power @ A @ K5 @ N2 ) ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ ( real_V7770717601297561774m_norm @ A @ K5 ) )
=> ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) )
@ ^ [X4: A] :
( suminf @ A
@ ^ [N2: nat] : ( times_times @ A @ ( C2 @ N2 ) @ ( power_power @ A @ X4 @ N2 ) ) ) ) ) ) ) ).
% isCont_powser
thf(fact_7226_summable__Leibniz_I2_J,axiom,
! [A2: nat > real] :
( ( filterlim @ nat @ real @ A2 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) )
=> ( ( topological_monoseq @ real @ A2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( A2 @ ( zero_zero @ nat ) ) )
=> ! [N9: nat] :
( member @ real
@ ( suminf @ real
@ ^ [I: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I ) @ ( A2 @ I ) ) )
@ ( set_or1337092689740270186AtMost @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I ) @ ( A2 @ I ) )
@ ( set_ord_lessThan @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N9 ) ) )
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I ) @ ( A2 @ I ) )
@ ( set_ord_lessThan @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N9 ) @ ( one_one @ nat ) ) ) ) ) ) ) ) ) ).
% summable_Leibniz(2)
thf(fact_7227_summable__Leibniz_I3_J,axiom,
! [A2: nat > real] :
( ( filterlim @ nat @ real @ A2 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) )
=> ( ( topological_monoseq @ real @ A2 )
=> ( ( ord_less @ real @ ( A2 @ ( zero_zero @ nat ) ) @ ( zero_zero @ real ) )
=> ! [N9: nat] :
( member @ real
@ ( suminf @ real
@ ^ [I: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I ) @ ( A2 @ I ) ) )
@ ( set_or1337092689740270186AtMost @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I ) @ ( A2 @ I ) )
@ ( set_ord_lessThan @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N9 ) @ ( one_one @ nat ) ) ) )
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I ) @ ( A2 @ I ) )
@ ( set_ord_lessThan @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N9 ) ) ) ) ) ) ) ) ).
% summable_Leibniz(3)
thf(fact_7228_tendsto__zero__mult__right__iff,axiom,
! [A: $tType] :
( ( ( field @ A )
& ( topolo4211221413907600880p_mult @ A ) )
=> ! [C2: A,A2: nat > A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( filterlim @ nat @ A
@ ^ [N2: nat] : ( times_times @ A @ ( A2 @ N2 ) @ C2 )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ ( at_top @ nat ) )
= ( filterlim @ nat @ A @ A2 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ ( at_top @ nat ) ) ) ) ) ).
% tendsto_zero_mult_right_iff
thf(fact_7229_tendsto__zero__mult__left__iff,axiom,
! [A: $tType] :
( ( ( field @ A )
& ( topolo4211221413907600880p_mult @ A ) )
=> ! [C2: A,A2: nat > A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( filterlim @ nat @ A
@ ^ [N2: nat] : ( times_times @ A @ C2 @ ( A2 @ N2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ ( at_top @ nat ) )
= ( filterlim @ nat @ A @ A2 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ ( at_top @ nat ) ) ) ) ) ).
% tendsto_zero_mult_left_iff
thf(fact_7230_tendsto__zero__divide__iff,axiom,
! [A: $tType] :
( ( ( field @ A )
& ( topolo4211221413907600880p_mult @ A ) )
=> ! [C2: A,A2: nat > A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( filterlim @ nat @ A
@ ^ [N2: nat] : ( divide_divide @ A @ ( A2 @ N2 ) @ C2 )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ ( at_top @ nat ) )
= ( filterlim @ nat @ A @ A2 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ ( at_top @ nat ) ) ) ) ) ).
% tendsto_zero_divide_iff
thf(fact_7231_isCont__Re,axiom,
! [C: $tType] :
( ( topological_t2_space @ C )
=> ! [A2: C,G: C > complex] :
( ( topolo3448309680560233919inuous @ C @ complex @ ( topolo174197925503356063within @ C @ A2 @ ( top_top @ ( set @ C ) ) ) @ G )
=> ( topolo3448309680560233919inuous @ C @ real @ ( topolo174197925503356063within @ C @ A2 @ ( top_top @ ( set @ C ) ) )
@ ^ [X4: C] : ( re @ ( G @ X4 ) ) ) ) ) ).
% isCont_Re
thf(fact_7232_isCont__Im,axiom,
! [C: $tType] :
( ( topological_t2_space @ C )
=> ! [A2: C,G: C > complex] :
( ( topolo3448309680560233919inuous @ C @ complex @ ( topolo174197925503356063within @ C @ A2 @ ( top_top @ ( set @ C ) ) ) @ G )
=> ( topolo3448309680560233919inuous @ C @ real @ ( topolo174197925503356063within @ C @ A2 @ ( top_top @ ( set @ C ) ) )
@ ^ [X4: C] : ( im @ ( G @ X4 ) ) ) ) ) ).
% isCont_Im
thf(fact_7233_continuous__arsinh,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [F4: filter @ A,F3: A > real] :
( ( topolo3448309680560233919inuous @ A @ real @ F4 @ F3 )
=> ( topolo3448309680560233919inuous @ A @ real @ F4
@ ^ [X4: A] : ( arsinh @ real @ ( F3 @ X4 ) ) ) ) ) ).
% continuous_arsinh
thf(fact_7234_continuous__cnj,axiom,
! [C: $tType] :
( ( topological_t2_space @ C )
=> ! [F4: filter @ C,G: C > complex] :
( ( topolo3448309680560233919inuous @ C @ complex @ F4 @ G )
=> ( topolo3448309680560233919inuous @ C @ complex @ F4
@ ^ [X4: C] : ( cnj @ ( G @ X4 ) ) ) ) ) ).
% continuous_cnj
thf(fact_7235_continuous__complex__iff,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ( ( topolo3448309680560233919inuous @ A @ complex )
= ( ^ [F10: filter @ A,F5: A > complex] :
( ( topolo3448309680560233919inuous @ A @ real @ F10
@ ^ [X4: A] : ( re @ ( F5 @ X4 ) ) )
& ( topolo3448309680560233919inuous @ A @ real @ F10
@ ^ [X4: A] : ( im @ ( F5 @ X4 ) ) ) ) ) ) ) ).
% continuous_complex_iff
thf(fact_7236_continuous__Im,axiom,
! [C: $tType] :
( ( topological_t2_space @ C )
=> ! [F4: filter @ C,G: C > complex] :
( ( topolo3448309680560233919inuous @ C @ complex @ F4 @ G )
=> ( topolo3448309680560233919inuous @ C @ real @ F4
@ ^ [X4: C] : ( im @ ( G @ X4 ) ) ) ) ) ).
% continuous_Im
thf(fact_7237_continuous__real__root,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [F4: filter @ A,F3: A > real,N: nat] :
( ( topolo3448309680560233919inuous @ A @ real @ F4 @ F3 )
=> ( topolo3448309680560233919inuous @ A @ real @ F4
@ ^ [X4: A] : ( root @ N @ ( F3 @ X4 ) ) ) ) ) ).
% continuous_real_root
thf(fact_7238_continuous__arctan,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [F4: filter @ A,F3: A > real] :
( ( topolo3448309680560233919inuous @ A @ real @ F4 @ F3 )
=> ( topolo3448309680560233919inuous @ A @ real @ F4
@ ^ [X4: A] : ( arctan @ ( F3 @ X4 ) ) ) ) ) ).
% continuous_arctan
thf(fact_7239_LIMSEQ__imp__Suc,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F3: nat > A,L: A] :
( ( filterlim @ nat @ A
@ ^ [N2: nat] : ( F3 @ ( suc @ N2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ L )
@ ( at_top @ nat ) )
=> ( filterlim @ nat @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ ( at_top @ nat ) ) ) ) ).
% LIMSEQ_imp_Suc
thf(fact_7240_LIMSEQ__Suc,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F3: nat > A,L: A] :
( ( filterlim @ nat @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ ( at_top @ nat ) )
=> ( filterlim @ nat @ A
@ ^ [N2: nat] : ( F3 @ ( suc @ N2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ L )
@ ( at_top @ nat ) ) ) ) ).
% LIMSEQ_Suc
thf(fact_7241_LIMSEQ__const__iff,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [K2: A,L: A] :
( ( filterlim @ nat @ A
@ ^ [N2: nat] : K2
@ ( topolo7230453075368039082e_nhds @ A @ L )
@ ( at_top @ nat ) )
= ( K2 = L ) ) ) ).
% LIMSEQ_const_iff
thf(fact_7242_continuous__real__sqrt,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [F4: filter @ A,F3: A > real] :
( ( topolo3448309680560233919inuous @ A @ real @ F4 @ F3 )
=> ( topolo3448309680560233919inuous @ A @ real @ F4
@ ^ [X4: A] : ( sqrt @ ( F3 @ X4 ) ) ) ) ) ).
% continuous_real_sqrt
thf(fact_7243_LIMSEQ__ignore__initial__segment,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F3: nat > A,A2: A,K2: nat] :
( ( filterlim @ nat @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ ( at_top @ nat ) )
=> ( filterlim @ nat @ A
@ ^ [N2: nat] : ( F3 @ ( plus_plus @ nat @ N2 @ K2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ A2 )
@ ( at_top @ nat ) ) ) ) ).
% LIMSEQ_ignore_initial_segment
thf(fact_7244_LIMSEQ__offset,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F3: nat > A,K2: nat,A2: A] :
( ( filterlim @ nat @ A
@ ^ [N2: nat] : ( F3 @ ( plus_plus @ nat @ N2 @ K2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ A2 )
@ ( at_top @ nat ) )
=> ( filterlim @ nat @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ ( at_top @ nat ) ) ) ) ).
% LIMSEQ_offset
thf(fact_7245_continuous__Re,axiom,
! [C: $tType] :
( ( topological_t2_space @ C )
=> ! [F4: filter @ C,G: C > complex] :
( ( topolo3448309680560233919inuous @ C @ complex @ F4 @ G )
=> ( topolo3448309680560233919inuous @ C @ real @ F4
@ ^ [X4: C] : ( re @ ( G @ X4 ) ) ) ) ) ).
% continuous_Re
thf(fact_7246_seq__offset__neg,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F3: nat > A,L: A,K2: nat] :
( ( filterlim @ nat @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ ( at_top @ nat ) )
=> ( filterlim @ nat @ A
@ ^ [I: nat] : ( F3 @ ( minus_minus @ nat @ I @ K2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ L )
@ ( at_top @ nat ) ) ) ) ).
% seq_offset_neg
thf(fact_7247_continuous__rabs,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [F4: filter @ A,F3: A > real] :
( ( topolo3448309680560233919inuous @ A @ real @ F4 @ F3 )
=> ( topolo3448309680560233919inuous @ A @ real @ F4
@ ^ [X4: A] : ( abs_abs @ real @ ( F3 @ X4 ) ) ) ) ) ).
% continuous_rabs
thf(fact_7248_filterlim__sequentially__Suc,axiom,
! [A: $tType,F3: nat > A,F4: filter @ A] :
( ( filterlim @ nat @ A
@ ^ [X4: nat] : ( F3 @ ( suc @ X4 ) )
@ F4
@ ( at_top @ nat ) )
= ( filterlim @ nat @ A @ F3 @ F4 @ ( at_top @ nat ) ) ) ).
% filterlim_sequentially_Suc
thf(fact_7249_approx__from__below__dense__linorder,axiom,
! [A: $tType] :
( ( ( dense_linorder @ A )
& ( topolo3112930676232923870pology @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [Y2: A,X: A] :
( ( ord_less @ A @ Y2 @ X )
=> ? [U3: nat > A] :
( ! [N9: nat] : ( ord_less @ A @ ( U3 @ N9 ) @ X )
& ( filterlim @ nat @ A @ U3 @ ( topolo7230453075368039082e_nhds @ A @ X ) @ ( at_top @ nat ) ) ) ) ) ).
% approx_from_below_dense_linorder
thf(fact_7250_approx__from__above__dense__linorder,axiom,
! [A: $tType] :
( ( ( dense_linorder @ A )
& ( topolo3112930676232923870pology @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [X: A,Y2: A] :
( ( ord_less @ A @ X @ Y2 )
=> ? [U3: nat > A] :
( ! [N9: nat] : ( ord_less @ A @ X @ ( U3 @ N9 ) )
& ( filterlim @ nat @ A @ U3 @ ( topolo7230453075368039082e_nhds @ A @ X ) @ ( at_top @ nat ) ) ) ) ) ).
% approx_from_above_dense_linorder
thf(fact_7251_LIMSEQ__le__const2,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [X9: nat > A,X: A,A2: A] :
( ( filterlim @ nat @ A @ X9 @ ( topolo7230453075368039082e_nhds @ A @ X ) @ ( at_top @ nat ) )
=> ( ? [N7: nat] :
! [N3: nat] :
( ( ord_less_eq @ nat @ N7 @ N3 )
=> ( ord_less_eq @ A @ ( X9 @ N3 ) @ A2 ) )
=> ( ord_less_eq @ A @ X @ A2 ) ) ) ) ).
% LIMSEQ_le_const2
thf(fact_7252_LIMSEQ__le__const,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [X9: nat > A,X: A,A2: A] :
( ( filterlim @ nat @ A @ X9 @ ( topolo7230453075368039082e_nhds @ A @ X ) @ ( at_top @ nat ) )
=> ( ? [N7: nat] :
! [N3: nat] :
( ( ord_less_eq @ nat @ N7 @ N3 )
=> ( ord_less_eq @ A @ A2 @ ( X9 @ N3 ) ) )
=> ( ord_less_eq @ A @ A2 @ X ) ) ) ) ).
% LIMSEQ_le_const
thf(fact_7253_Lim__bounded2,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [F3: nat > A,L: A,N4: nat,C5: A] :
( ( filterlim @ nat @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ ( at_top @ nat ) )
=> ( ! [N3: nat] :
( ( ord_less_eq @ nat @ N4 @ N3 )
=> ( ord_less_eq @ A @ C5 @ ( F3 @ N3 ) ) )
=> ( ord_less_eq @ A @ C5 @ L ) ) ) ) ).
% Lim_bounded2
thf(fact_7254_Lim__bounded,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [F3: nat > A,L: A,M7: nat,C5: A] :
( ( filterlim @ nat @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ ( at_top @ nat ) )
=> ( ! [N3: nat] :
( ( ord_less_eq @ nat @ M7 @ N3 )
=> ( ord_less_eq @ A @ ( F3 @ N3 ) @ C5 ) )
=> ( ord_less_eq @ A @ L @ C5 ) ) ) ) ).
% Lim_bounded
thf(fact_7255_LIMSEQ__le,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [X9: nat > A,X: A,Y3: nat > A,Y2: A] :
( ( filterlim @ nat @ A @ X9 @ ( topolo7230453075368039082e_nhds @ A @ X ) @ ( at_top @ nat ) )
=> ( ( filterlim @ nat @ A @ Y3 @ ( topolo7230453075368039082e_nhds @ A @ Y2 ) @ ( at_top @ nat ) )
=> ( ? [N7: nat] :
! [N3: nat] :
( ( ord_less_eq @ nat @ N7 @ N3 )
=> ( ord_less_eq @ A @ ( X9 @ N3 ) @ ( Y3 @ N3 ) ) )
=> ( ord_less_eq @ A @ X @ Y2 ) ) ) ) ) ).
% LIMSEQ_le
thf(fact_7256_lim__mono,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [N4: nat,X9: nat > A,Y3: nat > A,X: A,Y2: A] :
( ! [N3: nat] :
( ( ord_less_eq @ nat @ N4 @ N3 )
=> ( ord_less_eq @ A @ ( X9 @ N3 ) @ ( Y3 @ N3 ) ) )
=> ( ( filterlim @ nat @ A @ X9 @ ( topolo7230453075368039082e_nhds @ A @ X ) @ ( at_top @ nat ) )
=> ( ( filterlim @ nat @ A @ Y3 @ ( topolo7230453075368039082e_nhds @ A @ Y2 ) @ ( at_top @ nat ) )
=> ( ord_less_eq @ A @ X @ Y2 ) ) ) ) ) ).
% lim_mono
thf(fact_7257_Sup__lim,axiom,
! [A: $tType] :
( ( ( comple5582772986160207858norder @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [B2: nat > A,S: set @ A,A2: A] :
( ! [N3: nat] : ( member @ A @ ( B2 @ N3 ) @ S )
=> ( ( filterlim @ nat @ A @ B2 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ ( at_top @ nat ) )
=> ( ord_less_eq @ A @ A2 @ ( complete_Sup_Sup @ A @ S ) ) ) ) ) ).
% Sup_lim
thf(fact_7258_Inf__lim,axiom,
! [A: $tType] :
( ( ( comple5582772986160207858norder @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [B2: nat > A,S: set @ A,A2: A] :
( ! [N3: nat] : ( member @ A @ ( B2 @ N3 ) @ S )
=> ( ( filterlim @ nat @ A @ B2 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ ( at_top @ nat ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ S ) @ A2 ) ) ) ) ).
% Inf_lim
thf(fact_7259_isCont__rabs,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [A2: A,F3: A > real] :
( ( topolo3448309680560233919inuous @ A @ real @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) @ F3 )
=> ( topolo3448309680560233919inuous @ A @ real @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) )
@ ^ [X4: A] : ( abs_abs @ real @ ( F3 @ X4 ) ) ) ) ) ).
% isCont_rabs
thf(fact_7260_isCont__cnj,axiom,
! [C: $tType] :
( ( topological_t2_space @ C )
=> ! [A2: C,G: C > complex] :
( ( topolo3448309680560233919inuous @ C @ complex @ ( topolo174197925503356063within @ C @ A2 @ ( top_top @ ( set @ C ) ) ) @ G )
=> ( topolo3448309680560233919inuous @ C @ complex @ ( topolo174197925503356063within @ C @ A2 @ ( top_top @ ( set @ C ) ) )
@ ^ [X4: C] : ( cnj @ ( G @ X4 ) ) ) ) ) ).
% isCont_cnj
thf(fact_7261_continuous__at__within__powr,axiom,
! [C: $tType] :
( ( topological_t2_space @ C )
=> ! [A2: C,S: set @ C,F3: C > real,G: C > real] :
( ( topolo3448309680560233919inuous @ C @ real @ ( topolo174197925503356063within @ C @ A2 @ S ) @ F3 )
=> ( ( topolo3448309680560233919inuous @ C @ real @ ( topolo174197925503356063within @ C @ A2 @ S ) @ G )
=> ( ( ( F3 @ A2 )
!= ( zero_zero @ real ) )
=> ( topolo3448309680560233919inuous @ C @ real @ ( topolo174197925503356063within @ C @ A2 @ S )
@ ^ [X4: C] : ( powr @ real @ ( F3 @ X4 ) @ ( G @ X4 ) ) ) ) ) ) ) ).
% continuous_at_within_powr
thf(fact_7262_continuous__within__ln,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [X: A,S: set @ A,F3: A > real] :
( ( topolo3448309680560233919inuous @ A @ real @ ( topolo174197925503356063within @ A @ X @ S ) @ F3 )
=> ( ( ( F3 @ X )
!= ( zero_zero @ real ) )
=> ( topolo3448309680560233919inuous @ A @ real @ ( topolo174197925503356063within @ A @ X @ S )
@ ^ [X4: A] : ( ln_ln @ real @ ( F3 @ X4 ) ) ) ) ) ) ).
% continuous_within_ln
thf(fact_7263_mult__nat__right__at__top,axiom,
! [C2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ C2 )
=> ( filterlim @ nat @ nat
@ ^ [X4: nat] : ( times_times @ nat @ X4 @ C2 )
@ ( at_top @ nat )
@ ( at_top @ nat ) ) ) ).
% mult_nat_right_at_top
thf(fact_7264_mult__nat__left__at__top,axiom,
! [C2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ C2 )
=> ( filterlim @ nat @ nat @ ( times_times @ nat @ C2 ) @ ( at_top @ nat ) @ ( at_top @ nat ) ) ) ).
% mult_nat_left_at_top
thf(fact_7265_monoseq__convergent,axiom,
! [X9: nat > real,B5: real] :
( ( topological_monoseq @ real @ X9 )
=> ( ! [I3: nat] : ( ord_less_eq @ real @ ( abs_abs @ real @ ( X9 @ I3 ) ) @ B5 )
=> ~ ! [L7: real] :
~ ( filterlim @ nat @ real @ X9 @ ( topolo7230453075368039082e_nhds @ real @ L7 ) @ ( at_top @ nat ) ) ) ) ).
% monoseq_convergent
thf(fact_7266_LIMSEQ__lessThan__iff__atMost,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F3: ( set @ nat ) > A,X: A] :
( ( filterlim @ nat @ A
@ ^ [N2: nat] : ( F3 @ ( set_ord_lessThan @ nat @ N2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ X )
@ ( at_top @ nat ) )
= ( filterlim @ nat @ A
@ ^ [N2: nat] : ( F3 @ ( set_ord_atMost @ nat @ N2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ X )
@ ( at_top @ nat ) ) ) ) ).
% LIMSEQ_lessThan_iff_atMost
thf(fact_7267_LIMSEQ__root,axiom,
( filterlim @ nat @ real
@ ^ [N2: nat] : ( root @ N2 @ ( semiring_1_of_nat @ real @ N2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( one_one @ real ) )
@ ( at_top @ nat ) ) ).
% LIMSEQ_root
thf(fact_7268_isCont__powr,axiom,
! [C: $tType] :
( ( topological_t2_space @ C )
=> ! [A2: C,F3: C > real,G: C > real] :
( ( topolo3448309680560233919inuous @ C @ real @ ( topolo174197925503356063within @ C @ A2 @ ( top_top @ ( set @ C ) ) ) @ F3 )
=> ( ( topolo3448309680560233919inuous @ C @ real @ ( topolo174197925503356063within @ C @ A2 @ ( top_top @ ( set @ C ) ) ) @ G )
=> ( ( ( F3 @ A2 )
!= ( zero_zero @ real ) )
=> ( topolo3448309680560233919inuous @ C @ real @ ( topolo174197925503356063within @ C @ A2 @ ( top_top @ ( set @ C ) ) )
@ ^ [X4: C] : ( powr @ real @ ( F3 @ X4 ) @ ( G @ X4 ) ) ) ) ) ) ) ).
% isCont_powr
thf(fact_7269_isCont__ln_H,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [X: A,F3: A > real] :
( ( topolo3448309680560233919inuous @ A @ real @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) @ F3 )
=> ( ( ( F3 @ X )
!= ( zero_zero @ real ) )
=> ( topolo3448309680560233919inuous @ A @ real @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) )
@ ^ [X4: A] : ( ln_ln @ real @ ( F3 @ X4 ) ) ) ) ) ) ).
% isCont_ln'
thf(fact_7270_monoseq__le,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [A2: nat > A,X: A] :
( ( topological_monoseq @ A @ A2 )
=> ( ( filterlim @ nat @ A @ A2 @ ( topolo7230453075368039082e_nhds @ A @ X ) @ ( at_top @ nat ) )
=> ( ( ! [N9: nat] : ( ord_less_eq @ A @ ( A2 @ N9 ) @ X )
& ! [M3: nat,N9: nat] :
( ( ord_less_eq @ nat @ M3 @ N9 )
=> ( ord_less_eq @ A @ ( A2 @ M3 ) @ ( A2 @ N9 ) ) ) )
| ( ! [N9: nat] : ( ord_less_eq @ A @ X @ ( A2 @ N9 ) )
& ! [M3: nat,N9: nat] :
( ( ord_less_eq @ nat @ M3 @ N9 )
=> ( ord_less_eq @ A @ ( A2 @ N9 ) @ ( A2 @ M3 ) ) ) ) ) ) ) ) ).
% monoseq_le
thf(fact_7271_lim__const__over__n,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [A2: A] :
( filterlim @ nat @ A
@ ^ [N2: nat] : ( divide_divide @ A @ A2 @ ( semiring_1_of_nat @ A @ N2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ ( at_top @ nat ) ) ) ).
% lim_const_over_n
thf(fact_7272_LIMSEQ__SEQ__conv,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo3112930676232923870pology @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [A2: A,X9: A > B,L6: B] :
( ( ! [S6: nat > A] :
( ( ! [N2: nat] :
( ( S6 @ N2 )
!= A2 )
& ( filterlim @ nat @ A @ S6 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ ( at_top @ nat ) ) )
=> ( filterlim @ nat @ B
@ ^ [N2: nat] : ( X9 @ ( S6 @ N2 ) )
@ ( topolo7230453075368039082e_nhds @ B @ L6 )
@ ( at_top @ nat ) ) ) )
= ( filterlim @ A @ B @ X9 @ ( topolo7230453075368039082e_nhds @ B @ L6 ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% LIMSEQ_SEQ_conv
thf(fact_7273_LIMSEQ__SEQ__conv1,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [F3: A > B,L: B,A2: A] :
( ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) )
=> ! [S10: nat > A] :
( ( ! [N3: nat] :
( ( S10 @ N3 )
!= A2 )
& ( filterlim @ nat @ A @ S10 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ ( at_top @ nat ) ) )
=> ( filterlim @ nat @ B
@ ^ [N2: nat] : ( F3 @ ( S10 @ N2 ) )
@ ( topolo7230453075368039082e_nhds @ B @ L )
@ ( at_top @ nat ) ) ) ) ) ).
% LIMSEQ_SEQ_conv1
thf(fact_7274_LIMSEQ__SEQ__conv2,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo3112930676232923870pology @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [A2: A,F3: A > B,L: B] :
( ! [S7: nat > A] :
( ( ! [N9: nat] :
( ( S7 @ N9 )
!= A2 )
& ( filterlim @ nat @ A @ S7 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ ( at_top @ nat ) ) )
=> ( filterlim @ nat @ B
@ ^ [N2: nat] : ( F3 @ ( S7 @ N2 ) )
@ ( topolo7230453075368039082e_nhds @ B @ L )
@ ( at_top @ nat ) ) )
=> ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% LIMSEQ_SEQ_conv2
thf(fact_7275_lim__inverse__n,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ( filterlim @ nat @ A
@ ^ [N2: nat] : ( inverse_inverse @ A @ ( semiring_1_of_nat @ A @ N2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ ( at_top @ nat ) ) ) ).
% lim_inverse_n
thf(fact_7276_LIMSEQ__linear,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [X9: nat > A,X: A,L: nat] :
( ( filterlim @ nat @ A @ X9 @ ( topolo7230453075368039082e_nhds @ A @ X ) @ ( at_top @ nat ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ L )
=> ( filterlim @ nat @ A
@ ^ [N2: nat] : ( X9 @ ( times_times @ nat @ N2 @ L ) )
@ ( topolo7230453075368039082e_nhds @ A @ X )
@ ( at_top @ nat ) ) ) ) ) ).
% LIMSEQ_linear
thf(fact_7277_telescope__summable,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: nat > A,C2: A] :
( ( filterlim @ nat @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ C2 ) @ ( at_top @ nat ) )
=> ( summable @ A
@ ^ [N2: nat] : ( minus_minus @ A @ ( F3 @ ( suc @ N2 ) ) @ ( F3 @ N2 ) ) ) ) ) ).
% telescope_summable
thf(fact_7278_telescope__summable_H,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: nat > A,C2: A] :
( ( filterlim @ nat @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ C2 ) @ ( at_top @ nat ) )
=> ( summable @ A
@ ^ [N2: nat] : ( minus_minus @ A @ ( F3 @ N2 ) @ ( F3 @ ( suc @ N2 ) ) ) ) ) ) ).
% telescope_summable'
thf(fact_7279_nested__sequence__unique,axiom,
! [F3: nat > real,G: nat > real] :
( ! [N3: nat] : ( ord_less_eq @ real @ ( F3 @ N3 ) @ ( F3 @ ( suc @ N3 ) ) )
=> ( ! [N3: nat] : ( ord_less_eq @ real @ ( G @ ( suc @ N3 ) ) @ ( G @ N3 ) )
=> ( ! [N3: nat] : ( ord_less_eq @ real @ ( F3 @ N3 ) @ ( G @ N3 ) )
=> ( ( filterlim @ nat @ real
@ ^ [N2: nat] : ( minus_minus @ real @ ( F3 @ N2 ) @ ( G @ N2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( at_top @ nat ) )
=> ? [L2: real] :
( ! [N9: nat] : ( ord_less_eq @ real @ ( F3 @ N9 ) @ L2 )
& ( filterlim @ nat @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ L2 ) @ ( at_top @ nat ) )
& ! [N9: nat] : ( ord_less_eq @ real @ L2 @ ( G @ N9 ) )
& ( filterlim @ nat @ real @ G @ ( topolo7230453075368039082e_nhds @ real @ L2 ) @ ( at_top @ nat ) ) ) ) ) ) ) ).
% nested_sequence_unique
thf(fact_7280_LIMSEQ__inverse__zero,axiom,
! [X9: nat > real] :
( ! [R3: real] :
? [N7: nat] :
! [N3: nat] :
( ( ord_less_eq @ nat @ N7 @ N3 )
=> ( ord_less @ real @ R3 @ ( X9 @ N3 ) ) )
=> ( filterlim @ nat @ real
@ ^ [N2: nat] : ( inverse_inverse @ real @ ( X9 @ N2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( at_top @ nat ) ) ) ).
% LIMSEQ_inverse_zero
thf(fact_7281_lim__inverse__n_H,axiom,
( filterlim @ nat @ real
@ ^ [N2: nat] : ( divide_divide @ real @ ( one_one @ real ) @ ( semiring_1_of_nat @ real @ N2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( at_top @ nat ) ) ).
% lim_inverse_n'
thf(fact_7282_LIMSEQ__root__const,axiom,
! [C2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ C2 )
=> ( filterlim @ nat @ real
@ ^ [N2: nat] : ( root @ N2 @ C2 )
@ ( topolo7230453075368039082e_nhds @ real @ ( one_one @ real ) )
@ ( at_top @ nat ) ) ) ).
% LIMSEQ_root_const
thf(fact_7283_LIMSEQ__inverse__real__of__nat,axiom,
( filterlim @ nat @ real
@ ^ [N2: nat] : ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ ( suc @ N2 ) ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( at_top @ nat ) ) ).
% LIMSEQ_inverse_real_of_nat
thf(fact_7284_LIMSEQ__inverse__real__of__nat__add,axiom,
! [R: real] :
( filterlim @ nat @ real
@ ^ [N2: nat] : ( plus_plus @ real @ R @ ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ ( suc @ N2 ) ) ) )
@ ( topolo7230453075368039082e_nhds @ real @ R )
@ ( at_top @ nat ) ) ).
% LIMSEQ_inverse_real_of_nat_add
thf(fact_7285_sums__def,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ( ( sums @ A )
= ( ^ [F5: nat > A,S4: A] :
( filterlim @ nat @ A
@ ^ [N2: nat] : ( groups7311177749621191930dd_sum @ nat @ A @ F5 @ ( set_ord_lessThan @ nat @ N2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ S4 )
@ ( at_top @ nat ) ) ) ) ) ).
% sums_def
thf(fact_7286_sums__def__le,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ( ( sums @ A )
= ( ^ [F5: nat > A,S4: A] :
( filterlim @ nat @ A
@ ^ [N2: nat] : ( groups7311177749621191930dd_sum @ nat @ A @ F5 @ ( set_ord_atMost @ nat @ N2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ S4 )
@ ( at_top @ nat ) ) ) ) ) ).
% sums_def_le
thf(fact_7287_continuous__at__within__log,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [A2: A,S: set @ A,F3: A > real,G: A > real] :
( ( topolo3448309680560233919inuous @ A @ real @ ( topolo174197925503356063within @ A @ A2 @ S ) @ F3 )
=> ( ( topolo3448309680560233919inuous @ A @ real @ ( topolo174197925503356063within @ A @ A2 @ S ) @ G )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( F3 @ A2 ) )
=> ( ( ( F3 @ A2 )
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( G @ A2 ) )
=> ( topolo3448309680560233919inuous @ A @ real @ ( topolo174197925503356063within @ A @ A2 @ S )
@ ^ [X4: A] : ( log @ ( F3 @ X4 ) @ ( G @ X4 ) ) ) ) ) ) ) ) ) ).
% continuous_at_within_log
thf(fact_7288_increasing__LIMSEQ,axiom,
! [F3: nat > real,L: real] :
( ! [N3: nat] : ( ord_less_eq @ real @ ( F3 @ N3 ) @ ( F3 @ ( suc @ N3 ) ) )
=> ( ! [N3: nat] : ( ord_less_eq @ real @ ( F3 @ N3 ) @ L )
=> ( ! [E2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E2 )
=> ? [N9: nat] : ( ord_less_eq @ real @ L @ ( plus_plus @ real @ ( F3 @ N9 ) @ E2 ) ) )
=> ( filterlim @ nat @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ L ) @ ( at_top @ nat ) ) ) ) ) ).
% increasing_LIMSEQ
thf(fact_7289_lim__1__over__n,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ( filterlim @ nat @ A
@ ^ [N2: nat] : ( divide_divide @ A @ ( one_one @ A ) @ ( semiring_1_of_nat @ A @ N2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ ( at_top @ nat ) ) ) ).
% lim_1_over_n
thf(fact_7290_LIMSEQ__Suc__n__over__n,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ( filterlim @ nat @ A
@ ^ [N2: nat] : ( divide_divide @ A @ ( semiring_1_of_nat @ A @ ( suc @ N2 ) ) @ ( semiring_1_of_nat @ A @ N2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( one_one @ A ) )
@ ( at_top @ nat ) ) ) ).
% LIMSEQ_Suc_n_over_n
thf(fact_7291_LIMSEQ__n__over__Suc__n,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ( filterlim @ nat @ A
@ ^ [N2: nat] : ( divide_divide @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( semiring_1_of_nat @ A @ ( suc @ N2 ) ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( one_one @ A ) )
@ ( at_top @ nat ) ) ) ).
% LIMSEQ_n_over_Suc_n
thf(fact_7292_LIMSEQ__realpow__zero,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ X @ ( one_one @ real ) )
=> ( filterlim @ nat @ real @ ( power_power @ real @ X ) @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) ) ) ) ).
% LIMSEQ_realpow_zero
thf(fact_7293_telescope__sums_H,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: nat > A,C2: A] :
( ( filterlim @ nat @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ C2 ) @ ( at_top @ nat ) )
=> ( sums @ A
@ ^ [N2: nat] : ( minus_minus @ A @ ( F3 @ N2 ) @ ( F3 @ ( suc @ N2 ) ) )
@ ( minus_minus @ A @ ( F3 @ ( zero_zero @ nat ) ) @ C2 ) ) ) ) ).
% telescope_sums'
thf(fact_7294_telescope__sums,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: nat > A,C2: A] :
( ( filterlim @ nat @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ C2 ) @ ( at_top @ nat ) )
=> ( sums @ A
@ ^ [N2: nat] : ( minus_minus @ A @ ( F3 @ ( suc @ N2 ) ) @ ( F3 @ N2 ) )
@ ( minus_minus @ A @ C2 @ ( F3 @ ( zero_zero @ nat ) ) ) ) ) ) ).
% telescope_sums
thf(fact_7295_LIMSEQ__divide__realpow__zero,axiom,
! [X: real,A2: real] :
( ( ord_less @ real @ ( one_one @ real ) @ X )
=> ( filterlim @ nat @ real
@ ^ [N2: nat] : ( divide_divide @ real @ A2 @ ( power_power @ real @ X @ N2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( at_top @ nat ) ) ) ).
% LIMSEQ_divide_realpow_zero
thf(fact_7296_LIMSEQ__abs__realpow__zero,axiom,
! [C2: real] :
( ( ord_less @ real @ ( abs_abs @ real @ C2 ) @ ( one_one @ real ) )
=> ( filterlim @ nat @ real @ ( power_power @ real @ ( abs_abs @ real @ C2 ) ) @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) ) ) ).
% LIMSEQ_abs_realpow_zero
thf(fact_7297_LIMSEQ__abs__realpow__zero2,axiom,
! [C2: real] :
( ( ord_less @ real @ ( abs_abs @ real @ C2 ) @ ( one_one @ real ) )
=> ( filterlim @ nat @ real @ ( power_power @ real @ C2 ) @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) ) ) ).
% LIMSEQ_abs_realpow_zero2
thf(fact_7298_LIMSEQ__inverse__realpow__zero,axiom,
! [X: real] :
( ( ord_less @ real @ ( one_one @ real ) @ X )
=> ( filterlim @ nat @ real
@ ^ [N2: nat] : ( inverse_inverse @ real @ ( power_power @ real @ X @ N2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( at_top @ nat ) ) ) ).
% LIMSEQ_inverse_realpow_zero
thf(fact_7299_sums__def_H,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ( ( sums @ A )
= ( ^ [F5: nat > A,S4: A] :
( filterlim @ nat @ A
@ ^ [N2: nat] : ( groups7311177749621191930dd_sum @ nat @ A @ F5 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ S4 )
@ ( at_top @ nat ) ) ) ) ) ).
% sums_def'
thf(fact_7300_root__test__convergence,axiom,
! [A: $tType] :
( ( real_Vector_banach @ A )
=> ! [F3: nat > A,X: real] :
( ( filterlim @ nat @ real
@ ^ [N2: nat] : ( root @ N2 @ ( real_V7770717601297561774m_norm @ A @ ( F3 @ N2 ) ) )
@ ( topolo7230453075368039082e_nhds @ real @ X )
@ ( at_top @ nat ) )
=> ( ( ord_less @ real @ X @ ( one_one @ real ) )
=> ( summable @ A @ F3 ) ) ) ) ).
% root_test_convergence
thf(fact_7301_LIMSEQ__inverse__real__of__nat__add__minus,axiom,
! [R: real] :
( filterlim @ nat @ real
@ ^ [N2: nat] : ( plus_plus @ real @ R @ ( uminus_uminus @ real @ ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ ( suc @ N2 ) ) ) ) )
@ ( topolo7230453075368039082e_nhds @ real @ R )
@ ( at_top @ nat ) ) ).
% LIMSEQ_inverse_real_of_nat_add_minus
thf(fact_7302_summable__LIMSEQ,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topological_t2_space @ A ) )
=> ! [F3: nat > A] :
( ( summable @ A @ F3 )
=> ( filterlim @ nat @ A
@ ^ [N2: nat] : ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_ord_lessThan @ nat @ N2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( suminf @ A @ F3 ) )
@ ( at_top @ nat ) ) ) ) ).
% summable_LIMSEQ
thf(fact_7303_summable__LIMSEQ_H,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topological_t2_space @ A ) )
=> ! [F3: nat > A] :
( ( summable @ A @ F3 )
=> ( filterlim @ nat @ A
@ ^ [N2: nat] : ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_ord_atMost @ nat @ N2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( suminf @ A @ F3 ) )
@ ( at_top @ nat ) ) ) ) ).
% summable_LIMSEQ'
thf(fact_7304_isCont__log,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [A2: A,F3: A > real,G: A > real] :
( ( topolo3448309680560233919inuous @ A @ real @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) @ F3 )
=> ( ( topolo3448309680560233919inuous @ A @ real @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) @ G )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( F3 @ A2 ) )
=> ( ( ( F3 @ A2 )
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( G @ A2 ) )
=> ( topolo3448309680560233919inuous @ A @ real @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) )
@ ^ [X4: A] : ( log @ ( F3 @ X4 ) @ ( G @ X4 ) ) ) ) ) ) ) ) ) ).
% isCont_log
thf(fact_7305_LIMSEQ__D,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X9: nat > A,L6: A,R: real] :
( ( filterlim @ nat @ A @ X9 @ ( topolo7230453075368039082e_nhds @ A @ L6 ) @ ( at_top @ nat ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ R )
=> ? [No3: nat] :
! [N9: nat] :
( ( ord_less_eq @ nat @ No3 @ N9 )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( X9 @ N9 ) @ L6 ) ) @ R ) ) ) ) ) ).
% LIMSEQ_D
thf(fact_7306_LIMSEQ__I,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X9: nat > A,L6: A] :
( ! [R3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R3 )
=> ? [No4: nat] :
! [N3: nat] :
( ( ord_less_eq @ nat @ No4 @ N3 )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( X9 @ N3 ) @ L6 ) ) @ R3 ) ) )
=> ( filterlim @ nat @ A @ X9 @ ( topolo7230453075368039082e_nhds @ A @ L6 ) @ ( at_top @ nat ) ) ) ) ).
% LIMSEQ_I
thf(fact_7307_LIMSEQ__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X9: nat > A,L6: A] :
( ( filterlim @ nat @ A @ X9 @ ( topolo7230453075368039082e_nhds @ A @ L6 ) @ ( at_top @ nat ) )
= ( ! [R5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R5 )
=> ? [No2: nat] :
! [N2: nat] :
( ( ord_less_eq @ nat @ No2 @ N2 )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( X9 @ N2 ) @ L6 ) ) @ R5 ) ) ) ) ) ) ).
% LIMSEQ_iff
thf(fact_7308_LIMSEQ__power__zero,axiom,
! [A: $tType] :
( ( real_V2822296259951069270ebra_1 @ A )
=> ! [X: A] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ ( one_one @ real ) )
=> ( filterlim @ nat @ A @ ( power_power @ A @ X ) @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ ( at_top @ nat ) ) ) ) ).
% LIMSEQ_power_zero
thf(fact_7309_tendsto__exp__limit__sequentially,axiom,
! [X: real] :
( filterlim @ nat @ real
@ ^ [N2: nat] : ( power_power @ real @ ( plus_plus @ real @ ( one_one @ real ) @ ( divide_divide @ real @ X @ ( semiring_1_of_nat @ real @ N2 ) ) ) @ N2 )
@ ( topolo7230453075368039082e_nhds @ real @ ( exp @ real @ X ) )
@ ( at_top @ nat ) ) ).
% tendsto_exp_limit_sequentially
thf(fact_7310_tendsto__power__zero,axiom,
! [A: $tType,B: $tType] :
( ( real_V2822296259951069270ebra_1 @ A )
=> ! [F3: B > nat,F4: filter @ B,X: A] :
( ( filterlim @ B @ nat @ F3 @ ( at_top @ nat ) @ F4 )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ ( one_one @ real ) )
=> ( filterlim @ B @ A
@ ^ [Y: B] : ( power_power @ A @ X @ ( F3 @ Y ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ F4 ) ) ) ) ).
% tendsto_power_zero
thf(fact_7311_LIMSEQ__inverse__real__of__nat__add__minus__mult,axiom,
! [R: real] :
( filterlim @ nat @ real
@ ^ [N2: nat] : ( times_times @ real @ R @ ( plus_plus @ real @ ( one_one @ real ) @ ( uminus_uminus @ real @ ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ ( suc @ N2 ) ) ) ) ) )
@ ( topolo7230453075368039082e_nhds @ real @ R )
@ ( at_top @ nat ) ) ).
% LIMSEQ_inverse_real_of_nat_add_minus_mult
thf(fact_7312_LIMSEQ__norm__0,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: nat > A] :
( ! [N3: nat] : ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( F3 @ N3 ) ) @ ( divide_divide @ real @ ( one_one @ real ) @ ( semiring_1_of_nat @ real @ ( suc @ N3 ) ) ) )
=> ( filterlim @ nat @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ ( at_top @ nat ) ) ) ) ).
% LIMSEQ_norm_0
thf(fact_7313_summable__Leibniz_I1_J,axiom,
! [A2: nat > real] :
( ( filterlim @ nat @ real @ A2 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) )
=> ( ( topological_monoseq @ real @ A2 )
=> ( summable @ real
@ ^ [N2: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ N2 ) @ ( A2 @ N2 ) ) ) ) ) ).
% summable_Leibniz(1)
thf(fact_7314_field__derivative__lim__unique,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,Df: A,Z3: A,S: nat > A,A2: A] :
( ( has_field_derivative @ A @ F3 @ Df @ ( topolo174197925503356063within @ A @ Z3 @ ( top_top @ ( set @ A ) ) ) )
=> ( ( filterlim @ nat @ A @ S @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ ( at_top @ nat ) )
=> ( ! [N3: nat] :
( ( S @ N3 )
!= ( zero_zero @ A ) )
=> ( ( filterlim @ nat @ A
@ ^ [N2: nat] : ( divide_divide @ A @ ( minus_minus @ A @ ( F3 @ ( plus_plus @ A @ Z3 @ ( S @ N2 ) ) ) @ ( F3 @ Z3 ) ) @ ( S @ N2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ A2 )
@ ( at_top @ nat ) )
=> ( Df = A2 ) ) ) ) ) ) ).
% field_derivative_lim_unique
thf(fact_7315_powser__times__n__limit__0,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V8999393235501362500lgebra @ A ) )
=> ! [X: A] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ ( one_one @ real ) )
=> ( filterlim @ nat @ A
@ ^ [N2: nat] : ( times_times @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( power_power @ A @ X @ N2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ ( at_top @ nat ) ) ) ) ).
% powser_times_n_limit_0
thf(fact_7316_lim__n__over__pown,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A] :
( ( ord_less @ real @ ( one_one @ real ) @ ( real_V7770717601297561774m_norm @ A @ X ) )
=> ( filterlim @ nat @ A
@ ^ [N2: nat] : ( divide_divide @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( power_power @ A @ X @ N2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ ( at_top @ nat ) ) ) ) ).
% lim_n_over_pown
thf(fact_7317_summable,axiom,
! [A2: nat > real] :
( ( filterlim @ nat @ real @ A2 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) )
=> ( ! [N3: nat] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( A2 @ N3 ) )
=> ( ! [N3: nat] : ( ord_less_eq @ real @ ( A2 @ ( suc @ N3 ) ) @ ( A2 @ N3 ) )
=> ( summable @ real
@ ^ [N2: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ N2 ) @ ( A2 @ N2 ) ) ) ) ) ) ).
% summable
thf(fact_7318_cos__diff__limit__1,axiom,
! [Theta: nat > real,Theta2: real] :
( ( filterlim @ nat @ real
@ ^ [J4: nat] : ( cos @ real @ ( minus_minus @ real @ ( Theta @ J4 ) @ Theta2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( one_one @ real ) )
@ ( at_top @ nat ) )
=> ~ ! [K: nat > int] :
~ ( filterlim @ nat @ real
@ ^ [J4: nat] : ( minus_minus @ real @ ( Theta @ J4 ) @ ( times_times @ real @ ( ring_1_of_int @ real @ ( K @ J4 ) ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) ) )
@ ( topolo7230453075368039082e_nhds @ real @ Theta2 )
@ ( at_top @ nat ) ) ) ).
% cos_diff_limit_1
thf(fact_7319_cos__limit__1,axiom,
! [Theta: nat > real] :
( ( filterlim @ nat @ real
@ ^ [J4: nat] : ( cos @ real @ ( Theta @ J4 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( one_one @ real ) )
@ ( at_top @ nat ) )
=> ? [K: nat > int] :
( filterlim @ nat @ real
@ ^ [J4: nat] : ( minus_minus @ real @ ( Theta @ J4 ) @ ( times_times @ real @ ( ring_1_of_int @ real @ ( K @ J4 ) ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( at_top @ nat ) ) ) ).
% cos_limit_1
thf(fact_7320_summable__Leibniz_I4_J,axiom,
! [A2: nat > real] :
( ( filterlim @ nat @ real @ A2 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) )
=> ( ( topological_monoseq @ real @ A2 )
=> ( filterlim @ nat @ real
@ ^ [N2: nat] :
( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I ) @ ( A2 @ I ) )
@ ( set_ord_lessThan @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) )
@ ( topolo7230453075368039082e_nhds @ real
@ ( suminf @ real
@ ^ [I: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I ) @ ( A2 @ I ) ) ) )
@ ( at_top @ nat ) ) ) ) ).
% summable_Leibniz(4)
thf(fact_7321_zeroseq__arctan__series,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ X ) @ ( one_one @ real ) )
=> ( filterlim @ nat @ real
@ ^ [N2: nat] : ( times_times @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( semiring_1_of_nat @ real @ ( plus_plus @ nat @ ( times_times @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) @ ( power_power @ real @ X @ ( plus_plus @ nat @ ( times_times @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( at_top @ nat ) ) ) ).
% zeroseq_arctan_series
thf(fact_7322_summable__Leibniz_H_I3_J,axiom,
! [A2: nat > real] :
( ( filterlim @ nat @ real @ A2 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) )
=> ( ! [N3: nat] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( A2 @ N3 ) )
=> ( ! [N3: nat] : ( ord_less_eq @ real @ ( A2 @ ( suc @ N3 ) ) @ ( A2 @ N3 ) )
=> ( filterlim @ nat @ real
@ ^ [N2: nat] :
( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I ) @ ( A2 @ I ) )
@ ( set_ord_lessThan @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) )
@ ( topolo7230453075368039082e_nhds @ real
@ ( suminf @ real
@ ^ [I: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I ) @ ( A2 @ I ) ) ) )
@ ( at_top @ nat ) ) ) ) ) ).
% summable_Leibniz'(3)
thf(fact_7323_summable__Leibniz_H_I2_J,axiom,
! [A2: nat > real,N: nat] :
( ( filterlim @ nat @ real @ A2 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) )
=> ( ! [N3: nat] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( A2 @ N3 ) )
=> ( ! [N3: nat] : ( ord_less_eq @ real @ ( A2 @ ( suc @ N3 ) ) @ ( A2 @ N3 ) )
=> ( ord_less_eq @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I ) @ ( A2 @ I ) )
@ ( set_ord_lessThan @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) )
@ ( suminf @ real
@ ^ [I: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I ) @ ( A2 @ I ) ) ) ) ) ) ) ).
% summable_Leibniz'(2)
thf(fact_7324_sums__alternating__upper__lower,axiom,
! [A2: nat > real] :
( ! [N3: nat] : ( ord_less_eq @ real @ ( A2 @ ( suc @ N3 ) ) @ ( A2 @ N3 ) )
=> ( ! [N3: nat] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( A2 @ N3 ) )
=> ( ( filterlim @ nat @ real @ A2 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) )
=> ? [L2: real] :
( ! [N9: nat] :
( ord_less_eq @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I ) @ ( A2 @ I ) )
@ ( set_ord_lessThan @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N9 ) ) )
@ L2 )
& ( filterlim @ nat @ real
@ ^ [N2: nat] :
( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I ) @ ( A2 @ I ) )
@ ( set_ord_lessThan @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) )
@ ( topolo7230453075368039082e_nhds @ real @ L2 )
@ ( at_top @ nat ) )
& ! [N9: nat] :
( ord_less_eq @ real @ L2
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I ) @ ( A2 @ I ) )
@ ( set_ord_lessThan @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N9 ) @ ( one_one @ nat ) ) ) ) )
& ( filterlim @ nat @ real
@ ^ [N2: nat] :
( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I ) @ ( A2 @ I ) )
@ ( set_ord_lessThan @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) @ ( one_one @ nat ) ) ) )
@ ( topolo7230453075368039082e_nhds @ real @ L2 )
@ ( at_top @ nat ) ) ) ) ) ) ).
% sums_alternating_upper_lower
thf(fact_7325_summable__Leibniz_I5_J,axiom,
! [A2: nat > real] :
( ( filterlim @ nat @ real @ A2 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) )
=> ( ( topological_monoseq @ real @ A2 )
=> ( filterlim @ nat @ real
@ ^ [N2: nat] :
( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I ) @ ( A2 @ I ) )
@ ( set_ord_lessThan @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) @ ( one_one @ nat ) ) ) )
@ ( topolo7230453075368039082e_nhds @ real
@ ( suminf @ real
@ ^ [I: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I ) @ ( A2 @ I ) ) ) )
@ ( at_top @ nat ) ) ) ) ).
% summable_Leibniz(5)
thf(fact_7326_summable__Leibniz_H_I5_J,axiom,
! [A2: nat > real] :
( ( filterlim @ nat @ real @ A2 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) )
=> ( ! [N3: nat] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( A2 @ N3 ) )
=> ( ! [N3: nat] : ( ord_less_eq @ real @ ( A2 @ ( suc @ N3 ) ) @ ( A2 @ N3 ) )
=> ( filterlim @ nat @ real
@ ^ [N2: nat] :
( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I ) @ ( A2 @ I ) )
@ ( set_ord_lessThan @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) @ ( one_one @ nat ) ) ) )
@ ( topolo7230453075368039082e_nhds @ real
@ ( suminf @ real
@ ^ [I: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I ) @ ( A2 @ I ) ) ) )
@ ( at_top @ nat ) ) ) ) ) ).
% summable_Leibniz'(5)
thf(fact_7327_summable__Leibniz_H_I4_J,axiom,
! [A2: nat > real,N: nat] :
( ( filterlim @ nat @ real @ A2 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) )
=> ( ! [N3: nat] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( A2 @ N3 ) )
=> ( ! [N3: nat] : ( ord_less_eq @ real @ ( A2 @ ( suc @ N3 ) ) @ ( A2 @ N3 ) )
=> ( ord_less_eq @ real
@ ( suminf @ real
@ ^ [I: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I ) @ ( A2 @ I ) ) )
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I ) @ ( A2 @ I ) )
@ ( set_ord_lessThan @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% summable_Leibniz'(4)
thf(fact_7328_has__derivative__at2,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F3: A > B,F9: A > B,X: A] :
( ( has_derivative @ A @ B @ F3 @ F9 @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) )
= ( ( real_V3181309239436604168linear @ A @ B @ F9 )
& ( filterlim @ A @ B
@ ^ [Y: A] : ( real_V8093663219630862766scaleR @ B @ ( divide_divide @ real @ ( one_one @ real ) @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ Y @ X ) ) ) @ ( minus_minus @ B @ ( F3 @ Y ) @ ( plus_plus @ B @ ( F3 @ X ) @ ( F9 @ ( minus_minus @ A @ Y @ X ) ) ) ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% has_derivative_at2
thf(fact_7329_has__derivative__at,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F3: A > B,D4: A > B,X: A] :
( ( has_derivative @ A @ B @ F3 @ D4 @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) )
= ( ( real_V3181309239436604168linear @ A @ B @ D4 )
& ( filterlim @ A @ real
@ ^ [H: A] : ( divide_divide @ real @ ( real_V7770717601297561774m_norm @ B @ ( minus_minus @ B @ ( minus_minus @ B @ ( F3 @ ( plus_plus @ A @ X @ H ) ) @ ( F3 @ X ) ) @ ( D4 @ H ) ) ) @ ( real_V7770717601297561774m_norm @ A @ H ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% has_derivative_at
thf(fact_7330_bounded__linear_Ohas__derivative,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F3: A > B,G: C > A,G5: C > A,F4: filter @ C] :
( ( real_V3181309239436604168linear @ A @ B @ F3 )
=> ( ( has_derivative @ C @ A @ G @ G5 @ F4 )
=> ( has_derivative @ C @ B
@ ^ [X4: C] : ( F3 @ ( G @ X4 ) )
@ ^ [X4: C] : ( F3 @ ( G5 @ X4 ) )
@ F4 ) ) ) ) ).
% bounded_linear.has_derivative
thf(fact_7331_bounded__linear_Otendsto,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ B )
& ( real_V822414075346904944vector @ A ) )
=> ! [F3: A > B,G: C > A,A2: A,F4: filter @ C] :
( ( real_V3181309239436604168linear @ A @ B @ F3 )
=> ( ( filterlim @ C @ A @ G @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ F4 )
=> ( filterlim @ C @ B
@ ^ [X4: C] : ( F3 @ ( G @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( F3 @ A2 ) )
@ F4 ) ) ) ) ).
% bounded_linear.tendsto
thf(fact_7332_bounded__linear_Ocontinuous,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( ( topological_t2_space @ C )
& ( real_V822414075346904944vector @ B )
& ( real_V822414075346904944vector @ A ) )
=> ! [F3: A > B,F4: filter @ C,G: C > A] :
( ( real_V3181309239436604168linear @ A @ B @ F3 )
=> ( ( topolo3448309680560233919inuous @ C @ A @ F4 @ G )
=> ( topolo3448309680560233919inuous @ C @ B @ F4
@ ^ [X4: C] : ( F3 @ ( G @ X4 ) ) ) ) ) ) ).
% bounded_linear.continuous
thf(fact_7333_bounded__linear_Osuminf,axiom,
! [A: $tType,B: $tType] :
( ( ( real_V822414075346904944vector @ B )
& ( real_V822414075346904944vector @ A ) )
=> ! [F3: A > B,X9: nat > A] :
( ( real_V3181309239436604168linear @ A @ B @ F3 )
=> ( ( summable @ A @ X9 )
=> ( ( F3 @ ( suminf @ A @ X9 ) )
= ( suminf @ B
@ ^ [N2: nat] : ( F3 @ ( X9 @ N2 ) ) ) ) ) ) ) ).
% bounded_linear.suminf
thf(fact_7334_real__bounded__linear,axiom,
( ( real_V3181309239436604168linear @ real @ real )
= ( ^ [F5: real > real] :
? [C3: real] :
( F5
= ( ^ [X4: real] : ( times_times @ real @ X4 @ C3 ) ) ) ) ) ).
% real_bounded_linear
thf(fact_7335_bounded__linear__zero,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ( real_V3181309239436604168linear @ A @ B
@ ^ [X4: A] : ( zero_zero @ B ) ) ) ).
% bounded_linear_zero
thf(fact_7336_bounded__linear__divide,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [Y2: A] :
( real_V3181309239436604168linear @ A @ A
@ ^ [X4: A] : ( divide_divide @ A @ X4 @ Y2 ) ) ) ).
% bounded_linear_divide
thf(fact_7337_bounded__linear_Osums,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F3: A > B,X9: nat > A,A2: A] :
( ( real_V3181309239436604168linear @ A @ B @ F3 )
=> ( ( sums @ A @ X9 @ A2 )
=> ( sums @ B
@ ^ [N2: nat] : ( F3 @ ( X9 @ N2 ) )
@ ( F3 @ A2 ) ) ) ) ) ).
% bounded_linear.sums
thf(fact_7338_bounded__linear__scaleR__right,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [R: real] : ( real_V3181309239436604168linear @ A @ A @ ( real_V8093663219630862766scaleR @ A @ R ) ) ) ).
% bounded_linear_scaleR_right
thf(fact_7339_bounded__linear__const__scaleR,axiom,
! [B: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V822414075346904944vector @ B ) )
=> ! [G: C > B,R: real] :
( ( real_V3181309239436604168linear @ C @ B @ G )
=> ( real_V3181309239436604168linear @ C @ B
@ ^ [X4: C] : ( real_V8093663219630862766scaleR @ B @ R @ ( G @ X4 ) ) ) ) ) ).
% bounded_linear_const_scaleR
thf(fact_7340_bounded__linear__of__real,axiom,
! [A: $tType] :
( ( ( real_V2191834092415804123ebra_1 @ A )
& ( real_V822414075346904944vector @ A ) )
=> ( real_V3181309239436604168linear @ real @ A @ ( real_Vector_of_real @ A ) ) ) ).
% bounded_linear_of_real
thf(fact_7341_bounded__linear_Osummable,axiom,
! [A: $tType,B: $tType] :
( ( ( real_V822414075346904944vector @ B )
& ( real_V822414075346904944vector @ A ) )
=> ! [F3: A > B,X9: nat > A] :
( ( real_V3181309239436604168linear @ A @ B @ F3 )
=> ( ( summable @ A @ X9 )
=> ( summable @ B
@ ^ [N2: nat] : ( F3 @ ( X9 @ N2 ) ) ) ) ) ) ).
% bounded_linear.summable
thf(fact_7342_bounded__linear__scaleR__const,axiom,
! [B: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V822414075346904944vector @ B ) )
=> ! [G: C > real,X: B] :
( ( real_V3181309239436604168linear @ C @ real @ G )
=> ( real_V3181309239436604168linear @ C @ B
@ ^ [X4: C] : ( real_V8093663219630862766scaleR @ B @ ( G @ X4 ) @ X ) ) ) ) ).
% bounded_linear_scaleR_const
thf(fact_7343_bounded__linear__ident,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ( real_V3181309239436604168linear @ A @ A
@ ^ [X4: A] : X4 ) ) ).
% bounded_linear_ident
thf(fact_7344_bounded__linear__compose,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V822414075346904944vector @ B )
& ( real_V822414075346904944vector @ A ) )
=> ! [F3: A > B,G: C > A] :
( ( real_V3181309239436604168linear @ A @ B @ F3 )
=> ( ( real_V3181309239436604168linear @ C @ A @ G )
=> ( real_V3181309239436604168linear @ C @ B
@ ^ [X4: C] : ( F3 @ ( G @ X4 ) ) ) ) ) ) ).
% bounded_linear_compose
thf(fact_7345_bounded__linear__scaleR__left,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X: A] :
( real_V3181309239436604168linear @ real @ A
@ ^ [R5: real] : ( real_V8093663219630862766scaleR @ A @ R5 @ X ) ) ) ).
% bounded_linear_scaleR_left
thf(fact_7346_bounded__linear__add,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F3: A > B,G: A > B] :
( ( real_V3181309239436604168linear @ A @ B @ F3 )
=> ( ( real_V3181309239436604168linear @ A @ B @ G )
=> ( real_V3181309239436604168linear @ A @ B
@ ^ [X4: A] : ( plus_plus @ B @ ( F3 @ X4 ) @ ( G @ X4 ) ) ) ) ) ) ).
% bounded_linear_add
thf(fact_7347_bounded__linear__mult__right,axiom,
! [A: $tType] :
( ( real_V4412858255891104859lgebra @ A )
=> ! [X: A] : ( real_V3181309239436604168linear @ A @ A @ ( times_times @ A @ X ) ) ) ).
% bounded_linear_mult_right
thf(fact_7348_bounded__linear__mult__const,axiom,
! [A: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V4412858255891104859lgebra @ A ) )
=> ! [G: C > A,Y2: A] :
( ( real_V3181309239436604168linear @ C @ A @ G )
=> ( real_V3181309239436604168linear @ C @ A
@ ^ [X4: C] : ( times_times @ A @ ( G @ X4 ) @ Y2 ) ) ) ) ).
% bounded_linear_mult_const
thf(fact_7349_bounded__linear__const__mult,axiom,
! [A: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V4412858255891104859lgebra @ A ) )
=> ! [G: C > A,X: A] :
( ( real_V3181309239436604168linear @ C @ A @ G )
=> ( real_V3181309239436604168linear @ C @ A
@ ^ [X4: C] : ( times_times @ A @ X @ ( G @ X4 ) ) ) ) ) ).
% bounded_linear_const_mult
thf(fact_7350_bounded__linear__mult__left,axiom,
! [A: $tType] :
( ( real_V4412858255891104859lgebra @ A )
=> ! [Y2: A] :
( real_V3181309239436604168linear @ A @ A
@ ^ [X4: A] : ( times_times @ A @ X4 @ Y2 ) ) ) ).
% bounded_linear_mult_left
thf(fact_7351_bounded__linear_OCauchy,axiom,
! [A: $tType,B: $tType] :
( ( ( real_V822414075346904944vector @ B )
& ( real_V822414075346904944vector @ A ) )
=> ! [F3: A > B,X9: nat > A] :
( ( real_V3181309239436604168linear @ A @ B @ F3 )
=> ( ( topolo3814608138187158403Cauchy @ A @ X9 )
=> ( topolo3814608138187158403Cauchy @ B
@ ^ [N2: nat] : ( F3 @ ( X9 @ N2 ) ) ) ) ) ) ).
% bounded_linear.Cauchy
thf(fact_7352_bounded__linear__sub,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F3: A > B,G: A > B] :
( ( real_V3181309239436604168linear @ A @ B @ F3 )
=> ( ( real_V3181309239436604168linear @ A @ B @ G )
=> ( real_V3181309239436604168linear @ A @ B
@ ^ [X4: A] : ( minus_minus @ B @ ( F3 @ X4 ) @ ( G @ X4 ) ) ) ) ) ) ).
% bounded_linear_sub
thf(fact_7353_bounded__linear__minus,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F3: A > B] :
( ( real_V3181309239436604168linear @ A @ B @ F3 )
=> ( real_V3181309239436604168linear @ A @ B
@ ^ [X4: A] : ( uminus_uminus @ B @ ( F3 @ X4 ) ) ) ) ) ).
% bounded_linear_minus
thf(fact_7354_bounded__linear__sum,axiom,
! [I5: $tType,B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [I6: set @ I5,F3: I5 > A > B] :
( ! [I3: I5] :
( ( member @ I5 @ I3 @ I6 )
=> ( real_V3181309239436604168linear @ A @ B @ ( F3 @ I3 ) ) )
=> ( real_V3181309239436604168linear @ A @ B
@ ^ [X4: A] :
( groups7311177749621191930dd_sum @ I5 @ B
@ ^ [I: I5] : ( F3 @ I @ X4 )
@ I6 ) ) ) ) ).
% bounded_linear_sum
thf(fact_7355_bounded__linear_Obounded,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F3: A > B] :
( ( real_V3181309239436604168linear @ A @ B @ F3 )
=> ? [K10: real] :
! [X2: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( F3 @ X2 ) ) @ ( times_times @ real @ ( real_V7770717601297561774m_norm @ A @ X2 ) @ K10 ) ) ) ) ).
% bounded_linear.bounded
thf(fact_7356_bounded__linear_Otendsto__zero,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F3: A > B,G: C > A,F4: filter @ C] :
( ( real_V3181309239436604168linear @ A @ B @ F3 )
=> ( ( filterlim @ C @ A @ G @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ F4 )
=> ( filterlim @ C @ B
@ ^ [X4: C] : ( F3 @ ( G @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ F4 ) ) ) ) ).
% bounded_linear.tendsto_zero
thf(fact_7357_bounded__linear_OisCont,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( ( topological_t2_space @ C )
& ( real_V822414075346904944vector @ B )
& ( real_V822414075346904944vector @ A ) )
=> ! [F3: A > B,A2: C,G: C > A] :
( ( real_V3181309239436604168linear @ A @ B @ F3 )
=> ( ( topolo3448309680560233919inuous @ C @ A @ ( topolo174197925503356063within @ C @ A2 @ ( top_top @ ( set @ C ) ) ) @ G )
=> ( topolo3448309680560233919inuous @ C @ B @ ( topolo174197925503356063within @ C @ A2 @ ( top_top @ ( set @ C ) ) )
@ ^ [X4: C] : ( F3 @ ( G @ X4 ) ) ) ) ) ) ).
% bounded_linear.isCont
thf(fact_7358_bounded__linear_Ononneg__bounded,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F3: A > B] :
( ( real_V3181309239436604168linear @ A @ B @ F3 )
=> ? [K10: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ K10 )
& ! [X2: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( F3 @ X2 ) ) @ ( times_times @ real @ ( real_V7770717601297561774m_norm @ A @ X2 ) @ K10 ) ) ) ) ) ).
% bounded_linear.nonneg_bounded
thf(fact_7359_bounded__linear_Opos__bounded,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F3: A > B] :
( ( real_V3181309239436604168linear @ A @ B @ F3 )
=> ? [K10: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K10 )
& ! [X2: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( F3 @ X2 ) ) @ ( times_times @ real @ ( real_V7770717601297561774m_norm @ A @ X2 ) @ K10 ) ) ) ) ) ).
% bounded_linear.pos_bounded
thf(fact_7360_bounded__linear__intro,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F3: A > B,K5: real] :
( ! [X5: A,Y7: A] :
( ( F3 @ ( plus_plus @ A @ X5 @ Y7 ) )
= ( plus_plus @ B @ ( F3 @ X5 ) @ ( F3 @ Y7 ) ) )
=> ( ! [R3: real,X5: A] :
( ( F3 @ ( real_V8093663219630862766scaleR @ A @ R3 @ X5 ) )
= ( real_V8093663219630862766scaleR @ B @ R3 @ ( F3 @ X5 ) ) )
=> ( ! [X5: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( F3 @ X5 ) ) @ ( times_times @ real @ ( real_V7770717601297561774m_norm @ A @ X5 ) @ K5 ) )
=> ( real_V3181309239436604168linear @ A @ B @ F3 ) ) ) ) ) ).
% bounded_linear_intro
thf(fact_7361_has__derivative__iff__norm,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F3: A > B,F9: A > B,X: A,S: set @ A] :
( ( has_derivative @ A @ B @ F3 @ F9 @ ( topolo174197925503356063within @ A @ X @ S ) )
= ( ( real_V3181309239436604168linear @ A @ B @ F9 )
& ( filterlim @ A @ real
@ ^ [Y: A] : ( divide_divide @ real @ ( real_V7770717601297561774m_norm @ B @ ( minus_minus @ B @ ( minus_minus @ B @ ( F3 @ Y ) @ ( F3 @ X ) ) @ ( F9 @ ( minus_minus @ A @ Y @ X ) ) ) ) @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ Y @ X ) ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ).
% has_derivative_iff_norm
thf(fact_7362_has__derivative__at__within,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F3: A > B,F9: A > B,X: A,S: set @ A] :
( ( has_derivative @ A @ B @ F3 @ F9 @ ( topolo174197925503356063within @ A @ X @ S ) )
= ( ( real_V3181309239436604168linear @ A @ B @ F9 )
& ( filterlim @ A @ B
@ ^ [Y: A] : ( real_V8093663219630862766scaleR @ B @ ( inverse_inverse @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ Y @ X ) ) ) @ ( minus_minus @ B @ ( minus_minus @ B @ ( F3 @ Y ) @ ( F3 @ X ) ) @ ( F9 @ ( minus_minus @ A @ Y @ X ) ) ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ).
% has_derivative_at_within
thf(fact_7363_has__derivativeI,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F9: A > B,X: A,F3: A > B,S: set @ A] :
( ( real_V3181309239436604168linear @ A @ B @ F9 )
=> ( ( filterlim @ A @ B
@ ^ [Y: A] : ( real_V8093663219630862766scaleR @ B @ ( inverse_inverse @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ Y @ X ) ) ) @ ( minus_minus @ B @ ( minus_minus @ B @ ( F3 @ Y ) @ ( F3 @ X ) ) @ ( F9 @ ( minus_minus @ A @ Y @ X ) ) ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( has_derivative @ A @ B @ F3 @ F9 @ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ).
% has_derivativeI
thf(fact_7364_has__derivative__iff__Ex,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F3: A > B,F9: A > B,X: A] :
( ( has_derivative @ A @ B @ F3 @ F9 @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) )
= ( ( real_V3181309239436604168linear @ A @ B @ F9 )
& ? [E3: A > B] :
( ! [H: A] :
( ( F3 @ ( plus_plus @ A @ X @ H ) )
= ( plus_plus @ B @ ( plus_plus @ B @ ( F3 @ X ) @ ( F9 @ H ) ) @ ( E3 @ H ) ) )
& ( filterlim @ A @ real
@ ^ [H: A] : ( divide_divide @ real @ ( real_V7770717601297561774m_norm @ B @ ( E3 @ H ) ) @ ( real_V7770717601297561774m_norm @ A @ H ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ).
% has_derivative_iff_Ex
thf(fact_7365_has__derivative__within,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F3: A > B,F9: A > B,X: A,S: set @ A] :
( ( has_derivative @ A @ B @ F3 @ F9 @ ( topolo174197925503356063within @ A @ X @ S ) )
= ( ( real_V3181309239436604168linear @ A @ B @ F9 )
& ( filterlim @ A @ B
@ ^ [Y: A] : ( real_V8093663219630862766scaleR @ B @ ( divide_divide @ real @ ( one_one @ real ) @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ Y @ X ) ) ) @ ( minus_minus @ B @ ( F3 @ Y ) @ ( plus_plus @ B @ ( F3 @ X ) @ ( F9 @ ( minus_minus @ A @ Y @ X ) ) ) ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ).
% has_derivative_within
thf(fact_7366_has__derivative__def,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ( ( has_derivative @ A @ B )
= ( ^ [F5: A > B,F16: A > B,F10: filter @ A] :
( ( real_V3181309239436604168linear @ A @ B @ F16 )
& ( filterlim @ A @ B
@ ^ [Y: A] :
( real_V8093663219630862766scaleR @ B
@ ( inverse_inverse @ real
@ ( real_V7770717601297561774m_norm @ A
@ ( minus_minus @ A @ Y
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F10
@ ^ [X4: A] : X4 ) ) ) )
@ ( minus_minus @ B
@ ( minus_minus @ B @ ( F5 @ Y )
@ ( F5
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F10
@ ^ [X4: A] : X4 ) ) )
@ ( F16
@ ( minus_minus @ A @ Y
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F10
@ ^ [X4: A] : X4 ) ) ) ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ F10 ) ) ) ) ) ).
% has_derivative_def
thf(fact_7367_has__derivative__at__within__iff__Ex,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [X: A,S3: set @ A,F3: A > B,F9: A > B] :
( ( member @ A @ X @ S3 )
=> ( ( topolo1002775350975398744n_open @ A @ S3 )
=> ( ( has_derivative @ A @ B @ F3 @ F9 @ ( topolo174197925503356063within @ A @ X @ S3 ) )
= ( ( real_V3181309239436604168linear @ A @ B @ F9 )
& ? [E3: A > B] :
( ! [H: A] :
( ( member @ A @ ( plus_plus @ A @ X @ H ) @ S3 )
=> ( ( F3 @ ( plus_plus @ A @ X @ H ) )
= ( plus_plus @ B @ ( plus_plus @ B @ ( F3 @ X ) @ ( F9 @ H ) ) @ ( E3 @ H ) ) ) )
& ( filterlim @ A @ real
@ ^ [H: A] : ( divide_divide @ real @ ( real_V7770717601297561774m_norm @ B @ ( E3 @ H ) ) @ ( real_V7770717601297561774m_norm @ A @ H ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ) ) ).
% has_derivative_at_within_iff_Ex
thf(fact_7368_lim__const,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [A2: A] :
( ( topolo3827282254853284352ce_Lim @ nat @ A @ ( at_top @ nat )
@ ^ [M2: nat] : A2 )
= A2 ) ) ).
% lim_const
thf(fact_7369_open__UN,axiom,
! [A: $tType,B: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [A3: set @ B,B5: B > ( set @ A )] :
( ! [X5: B] :
( ( member @ B @ X5 @ A3 )
=> ( topolo1002775350975398744n_open @ A @ ( B5 @ X5 ) ) )
=> ( topolo1002775350975398744n_open @ A @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B5 @ A3 ) ) ) ) ) ).
% open_UN
thf(fact_7370_open__INT,axiom,
! [A: $tType,B: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [A3: set @ B,B5: B > ( set @ A )] :
( ( finite_finite2 @ B @ A3 )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ A3 )
=> ( topolo1002775350975398744n_open @ A @ ( B5 @ X5 ) ) )
=> ( topolo1002775350975398744n_open @ A @ ( complete_Inf_Inf @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B5 @ A3 ) ) ) ) ) ) ).
% open_INT
thf(fact_7371_first__countable__basis,axiom,
! [A: $tType] :
( ( topolo3112930676232923870pology @ A )
=> ! [X: A] :
? [A8: nat > ( set @ A )] :
( ! [I4: nat] :
( ( member @ A @ X @ ( A8 @ I4 ) )
& ( topolo1002775350975398744n_open @ A @ ( A8 @ I4 ) ) )
& ! [S10: set @ A] :
( ( ( topolo1002775350975398744n_open @ A @ S10 )
& ( member @ A @ X @ S10 ) )
=> ? [I3: nat] : ( ord_less_eq @ ( set @ A ) @ ( A8 @ I3 ) @ S10 ) ) ) ) ).
% first_countable_basis
thf(fact_7372_open__subopen,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ( ( topolo1002775350975398744n_open @ A )
= ( ^ [S6: set @ A] :
! [X4: A] :
( ( member @ A @ X4 @ S6 )
=> ? [T9: set @ A] :
( ( topolo1002775350975398744n_open @ A @ T9 )
& ( member @ A @ X4 @ T9 )
& ( ord_less_eq @ ( set @ A ) @ T9 @ S6 ) ) ) ) ) ) ).
% open_subopen
thf(fact_7373_openI,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [S3: set @ A] :
( ! [X5: A] :
( ( member @ A @ X5 @ S3 )
=> ? [T10: set @ A] :
( ( topolo1002775350975398744n_open @ A @ T10 )
& ( member @ A @ X5 @ T10 )
& ( ord_less_eq @ ( set @ A ) @ T10 @ S3 ) ) )
=> ( topolo1002775350975398744n_open @ A @ S3 ) ) ) ).
% openI
thf(fact_7374_Sup__notin__open,axiom,
! [A: $tType] :
( ( topolo8458572112393995274pology @ A )
=> ! [A3: set @ A,X: A] :
( ( topolo1002775350975398744n_open @ A @ A3 )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ A3 )
=> ( ord_less @ A @ X5 @ X ) )
=> ~ ( member @ A @ ( complete_Sup_Sup @ A @ A3 ) @ A3 ) ) ) ) ).
% Sup_notin_open
thf(fact_7375_Inf__notin__open,axiom,
! [A: $tType] :
( ( topolo8458572112393995274pology @ A )
=> ! [A3: set @ A,X: A] :
( ( topolo1002775350975398744n_open @ A @ A3 )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ A3 )
=> ( ord_less @ A @ X @ X5 ) )
=> ~ ( member @ A @ ( complete_Inf_Inf @ A @ A3 ) @ A3 ) ) ) ) ).
% Inf_notin_open
thf(fact_7376_open__Collect__const,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [P3: $o] :
( topolo1002775350975398744n_open @ A
@ ( collect @ A
@ ^ [X4: A] : P3 ) ) ) ).
% open_Collect_const
thf(fact_7377_open__Collect__disj,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [P3: A > $o,Q2: A > $o] :
( ( topolo1002775350975398744n_open @ A @ ( collect @ A @ P3 ) )
=> ( ( topolo1002775350975398744n_open @ A @ ( collect @ A @ Q2 ) )
=> ( topolo1002775350975398744n_open @ A
@ ( collect @ A
@ ^ [X4: A] :
( ( P3 @ X4 )
| ( Q2 @ X4 ) ) ) ) ) ) ) ).
% open_Collect_disj
thf(fact_7378_open__Collect__conj,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [P3: A > $o,Q2: A > $o] :
( ( topolo1002775350975398744n_open @ A @ ( collect @ A @ P3 ) )
=> ( ( topolo1002775350975398744n_open @ A @ ( collect @ A @ Q2 ) )
=> ( topolo1002775350975398744n_open @ A
@ ( collect @ A
@ ^ [X4: A] :
( ( P3 @ X4 )
& ( Q2 @ X4 ) ) ) ) ) ) ) ).
% open_Collect_conj
thf(fact_7379_open__left,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [S3: set @ A,X: A,Y2: A] :
( ( topolo1002775350975398744n_open @ A @ S3 )
=> ( ( member @ A @ X @ S3 )
=> ( ( ord_less @ A @ Y2 @ X )
=> ? [B3: A] :
( ( ord_less @ A @ B3 @ X )
& ( ord_less_eq @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ B3 @ X ) @ S3 ) ) ) ) ) ) ).
% open_left
thf(fact_7380_open__right,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [S3: set @ A,X: A,Y2: A] :
( ( topolo1002775350975398744n_open @ A @ S3 )
=> ( ( member @ A @ X @ S3 )
=> ( ( ord_less @ A @ X @ Y2 )
=> ? [B3: A] :
( ( ord_less @ A @ X @ B3 )
& ( ord_less_eq @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ X @ B3 ) @ S3 ) ) ) ) ) ) ).
% open_right
thf(fact_7381_at__within__open__subset,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [A2: A,S3: set @ A,T5: set @ A] :
( ( member @ A @ A2 @ S3 )
=> ( ( topolo1002775350975398744n_open @ A @ S3 )
=> ( ( ord_less_eq @ ( set @ A ) @ S3 @ T5 )
=> ( ( topolo174197925503356063within @ A @ A2 @ T5 )
= ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ).
% at_within_open_subset
thf(fact_7382_Lim__ident__at,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [X: A,S: set @ A] :
( ( ( topolo174197925503356063within @ A @ X @ S )
!= ( bot_bot @ ( filter @ A ) ) )
=> ( ( topolo3827282254853284352ce_Lim @ A @ A @ ( topolo174197925503356063within @ A @ X @ S )
@ ^ [X4: A] : X4 )
= X ) ) ) ).
% Lim_ident_at
thf(fact_7383_lim__explicit,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F3: nat > A,F0: A] :
( ( filterlim @ nat @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ F0 ) @ ( at_top @ nat ) )
= ( ! [S6: set @ A] :
( ( topolo1002775350975398744n_open @ A @ S6 )
=> ( ( member @ A @ F0 @ S6 )
=> ? [N6: nat] :
! [N2: nat] :
( ( ord_less_eq @ nat @ N6 @ N2 )
=> ( member @ A @ ( F3 @ N2 ) @ S6 ) ) ) ) ) ) ) ).
% lim_explicit
thf(fact_7384_continuous__divide,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_V3459762299906320749_field @ B ) )
=> ! [F4: filter @ A,F3: A > B,G: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ F4 @ F3 )
=> ( ( topolo3448309680560233919inuous @ A @ B @ F4 @ G )
=> ( ( ( G
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F4
@ ^ [X4: A] : X4 ) )
!= ( zero_zero @ B ) )
=> ( topolo3448309680560233919inuous @ A @ B @ F4
@ ^ [X4: A] : ( divide_divide @ B @ ( F3 @ X4 ) @ ( G @ X4 ) ) ) ) ) ) ) ).
% continuous_divide
thf(fact_7385_continuous__inverse,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_V8999393235501362500lgebra @ B ) )
=> ! [F4: filter @ A,F3: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ F4 @ F3 )
=> ( ( ( F3
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F4
@ ^ [X4: A] : X4 ) )
!= ( zero_zero @ B ) )
=> ( topolo3448309680560233919inuous @ A @ B @ F4
@ ^ [X4: A] : ( inverse_inverse @ B @ ( F3 @ X4 ) ) ) ) ) ) ).
% continuous_inverse
thf(fact_7386_continuous__def,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ( ( topolo3448309680560233919inuous @ A @ B )
= ( ^ [F10: filter @ A,F5: A > B] :
( filterlim @ A @ B @ F5
@ ( topolo7230453075368039082e_nhds @ B
@ ( F5
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F10
@ ^ [X4: A] : X4 ) ) )
@ F10 ) ) ) ) ).
% continuous_def
thf(fact_7387_continuous__sgn,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F4: filter @ A,F3: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ F4 @ F3 )
=> ( ( ( F3
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F4
@ ^ [X4: A] : X4 ) )
!= ( zero_zero @ B ) )
=> ( topolo3448309680560233919inuous @ A @ B @ F4
@ ^ [X4: A] : ( sgn_sgn @ B @ ( F3 @ X4 ) ) ) ) ) ) ).
% continuous_sgn
thf(fact_7388_t2__space__class_OLim__def,axiom,
! [A: $tType,F: $tType] :
( ( topological_t2_space @ A )
=> ( ( topolo3827282254853284352ce_Lim @ F @ A )
= ( ^ [A7: filter @ F,F5: F > A] :
( the @ A
@ ^ [L3: A] : ( filterlim @ F @ A @ F5 @ ( topolo7230453075368039082e_nhds @ A @ L3 ) @ A7 ) ) ) ) ) ).
% t2_space_class.Lim_def
thf(fact_7389_continuous__powr,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [F4: filter @ A,F3: A > real,G: A > real] :
( ( topolo3448309680560233919inuous @ A @ real @ F4 @ F3 )
=> ( ( topolo3448309680560233919inuous @ A @ real @ F4 @ G )
=> ( ( ( F3
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F4
@ ^ [X4: A] : X4 ) )
!= ( zero_zero @ real ) )
=> ( topolo3448309680560233919inuous @ A @ real @ F4
@ ^ [X4: A] : ( powr @ real @ ( F3 @ X4 ) @ ( G @ X4 ) ) ) ) ) ) ) ).
% continuous_powr
thf(fact_7390_continuous__ln,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [F4: filter @ A,F3: A > real] :
( ( topolo3448309680560233919inuous @ A @ real @ F4 @ F3 )
=> ( ( ( F3
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F4
@ ^ [X4: A] : X4 ) )
!= ( zero_zero @ real ) )
=> ( topolo3448309680560233919inuous @ A @ real @ F4
@ ^ [X4: A] : ( ln_ln @ real @ ( F3 @ X4 ) ) ) ) ) ) ).
% continuous_ln
thf(fact_7391_suminf__eq__lim,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topological_t2_space @ A ) )
=> ( ( suminf @ A )
= ( ^ [F5: nat > A] :
( topolo3827282254853284352ce_Lim @ nat @ A @ ( at_top @ nat )
@ ^ [N2: nat] : ( groups7311177749621191930dd_sum @ nat @ A @ F5 @ ( set_ord_lessThan @ nat @ N2 ) ) ) ) ) ) ).
% suminf_eq_lim
thf(fact_7392_lim__def,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [X9: nat > A] :
( ( topolo3827282254853284352ce_Lim @ nat @ A @ ( at_top @ nat ) @ X9 )
= ( the @ A
@ ^ [L8: A] : ( filterlim @ nat @ A @ X9 @ ( topolo7230453075368039082e_nhds @ A @ L8 ) @ ( at_top @ nat ) ) ) ) ) ).
% lim_def
thf(fact_7393_continuous__tan,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [F4: filter @ A,F3: A > A] :
( ( topolo3448309680560233919inuous @ A @ A @ F4 @ F3 )
=> ( ( ( cos @ A
@ ( F3
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F4
@ ^ [X4: A] : X4 ) ) )
!= ( zero_zero @ A ) )
=> ( topolo3448309680560233919inuous @ A @ A @ F4
@ ^ [X4: A] : ( tan @ A @ ( F3 @ X4 ) ) ) ) ) ) ).
% continuous_tan
thf(fact_7394_continuous__cot,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [F4: filter @ A,F3: A > A] :
( ( topolo3448309680560233919inuous @ A @ A @ F4 @ F3 )
=> ( ( ( sin @ A
@ ( F3
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F4
@ ^ [X4: A] : X4 ) ) )
!= ( zero_zero @ A ) )
=> ( topolo3448309680560233919inuous @ A @ A @ F4
@ ^ [X4: A] : ( cot @ A @ ( F3 @ X4 ) ) ) ) ) ) ).
% continuous_cot
thf(fact_7395_continuous__tanh,axiom,
! [A: $tType,C: $tType] :
( ( ( topological_t2_space @ C )
& ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [F4: filter @ C,F3: C > A] :
( ( topolo3448309680560233919inuous @ C @ A @ F4 @ F3 )
=> ( ( ( cosh @ A
@ ( F3
@ ( topolo3827282254853284352ce_Lim @ C @ C @ F4
@ ^ [X4: C] : X4 ) ) )
!= ( zero_zero @ A ) )
=> ( topolo3448309680560233919inuous @ C @ A @ F4
@ ^ [X4: C] : ( tanh @ A @ ( F3 @ X4 ) ) ) ) ) ) ).
% continuous_tanh
thf(fact_7396_continuous__arcosh,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [F4: filter @ A,F3: A > real] :
( ( topolo3448309680560233919inuous @ A @ real @ F4 @ F3 )
=> ( ( ord_less @ real @ ( one_one @ real )
@ ( F3
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F4
@ ^ [X4: A] : X4 ) ) )
=> ( topolo3448309680560233919inuous @ A @ real @ F4
@ ^ [X4: A] : ( arcosh @ real @ ( F3 @ X4 ) ) ) ) ) ) ).
% continuous_arcosh
thf(fact_7397_tendsto__offset__zero__iff,axiom,
! [C: $tType,D: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( topolo4958980785337419405_space @ D )
& ( zero @ C ) )
=> ! [A2: A,S3: set @ A,F3: A > D,L6: D] :
( ( nO_MATCH @ C @ A @ ( zero_zero @ C ) @ A2 )
=> ( ( member @ A @ A2 @ S3 )
=> ( ( topolo1002775350975398744n_open @ A @ S3 )
=> ( ( filterlim @ A @ D @ F3 @ ( topolo7230453075368039082e_nhds @ D @ L6 ) @ ( topolo174197925503356063within @ A @ A2 @ S3 ) )
= ( filterlim @ A @ D
@ ^ [H: A] : ( F3 @ ( plus_plus @ A @ A2 @ H ) )
@ ( topolo7230453075368039082e_nhds @ D @ L6 )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% tendsto_offset_zero_iff
thf(fact_7398_continuous__log,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [F4: filter @ A,F3: A > real,G: A > real] :
( ( topolo3448309680560233919inuous @ A @ real @ F4 @ F3 )
=> ( ( topolo3448309680560233919inuous @ A @ real @ F4 @ G )
=> ( ( ord_less @ real @ ( zero_zero @ real )
@ ( F3
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F4
@ ^ [X4: A] : X4 ) ) )
=> ( ( ( F3
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F4
@ ^ [X4: A] : X4 ) )
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real )
@ ( G
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F4
@ ^ [X4: A] : X4 ) ) )
=> ( topolo3448309680560233919inuous @ A @ real @ F4
@ ^ [X4: A] : ( log @ ( F3 @ X4 ) @ ( G @ X4 ) ) ) ) ) ) ) ) ) ).
% continuous_log
thf(fact_7399_continuous__artanh,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [F4: filter @ A,F3: A > real] :
( ( topolo3448309680560233919inuous @ A @ real @ F4 @ F3 )
=> ( ( member @ real
@ ( F3
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F4
@ ^ [X4: A] : X4 ) )
@ ( set_or5935395276787703475ssThan @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ ( one_one @ real ) ) )
=> ( topolo3448309680560233919inuous @ A @ real @ F4
@ ^ [X4: A] : ( artanh @ real @ ( F3 @ X4 ) ) ) ) ) ) ).
% continuous_artanh
thf(fact_7400_has__derivativeI__sandwich,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [E: real,F9: A > B,S: set @ A,X: A,F3: A > B,H6: A > real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E )
=> ( ( real_V3181309239436604168linear @ A @ B @ F9 )
=> ( ! [Y7: A] :
( ( member @ A @ Y7 @ S )
=> ( ( Y7 != X )
=> ( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ Y7 @ X ) @ E )
=> ( ord_less_eq @ real @ ( divide_divide @ real @ ( real_V7770717601297561774m_norm @ B @ ( minus_minus @ B @ ( minus_minus @ B @ ( F3 @ Y7 ) @ ( F3 @ X ) ) @ ( F9 @ ( minus_minus @ A @ Y7 @ X ) ) ) ) @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ Y7 @ X ) ) ) @ ( H6 @ Y7 ) ) ) ) )
=> ( ( filterlim @ A @ real @ H6 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( has_derivative @ A @ B @ F3 @ F9 @ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ) ) ).
% has_derivativeI_sandwich
thf(fact_7401_tendsto__exp__limit__at__right,axiom,
! [X: real] :
( filterlim @ real @ real
@ ^ [Y: real] : ( powr @ real @ ( plus_plus @ real @ ( one_one @ real ) @ ( times_times @ real @ X @ Y ) ) @ ( divide_divide @ real @ ( one_one @ real ) @ Y ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( exp @ real @ X ) )
@ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) ) ).
% tendsto_exp_limit_at_right
thf(fact_7402_greaterThan__iff,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [I2: A,K2: A] :
( ( member @ A @ I2 @ ( set_ord_greaterThan @ A @ K2 ) )
= ( ord_less @ A @ K2 @ I2 ) ) ) ).
% greaterThan_iff
thf(fact_7403_dist__add__cancel2,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [B2: A,A2: A,C2: A] :
( ( real_V557655796197034286t_dist @ A @ ( plus_plus @ A @ B2 @ A2 ) @ ( plus_plus @ A @ C2 @ A2 ) )
= ( real_V557655796197034286t_dist @ A @ B2 @ C2 ) ) ) ).
% dist_add_cancel2
thf(fact_7404_dist__add__cancel,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [A2: A,B2: A,C2: A] :
( ( real_V557655796197034286t_dist @ A @ ( plus_plus @ A @ A2 @ B2 ) @ ( plus_plus @ A @ A2 @ C2 ) )
= ( real_V557655796197034286t_dist @ A @ B2 @ C2 ) ) ) ).
% dist_add_cancel
thf(fact_7405_greaterThan__subset__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y2: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_ord_greaterThan @ A @ X ) @ ( set_ord_greaterThan @ A @ Y2 ) )
= ( ord_less_eq @ A @ Y2 @ X ) ) ) ).
% greaterThan_subset_iff
thf(fact_7406_dist__le__zero__iff,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X: A,Y2: A] :
( ( ord_less_eq @ real @ ( real_V557655796197034286t_dist @ A @ X @ Y2 ) @ ( zero_zero @ real ) )
= ( X = Y2 ) ) ) ).
% dist_le_zero_iff
thf(fact_7407_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_7408_dist__scaleR,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X: real,A2: A,Y2: real] :
( ( real_V557655796197034286t_dist @ A @ ( real_V8093663219630862766scaleR @ A @ X @ A2 ) @ ( real_V8093663219630862766scaleR @ A @ Y2 @ A2 ) )
= ( times_times @ real @ ( abs_abs @ real @ ( minus_minus @ real @ X @ Y2 ) ) @ ( real_V7770717601297561774m_norm @ A @ A2 ) ) ) ) ).
% dist_scaleR
thf(fact_7409_open__ball,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X: A,D2: real] :
( topolo1002775350975398744n_open @ A
@ ( collect @ A
@ ^ [Y: A] : ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X @ Y ) @ D2 ) ) ) ) ).
% open_ball
thf(fact_7410_at__within__Icc__at__right,axiom,
! [A: $tType] :
( ( topolo2564578578187576103pology @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( topolo174197925503356063within @ A @ A2 @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
= ( topolo174197925503356063within @ A @ A2 @ ( set_ord_greaterThan @ A @ A2 ) ) ) ) ) ).
% at_within_Icc_at_right
thf(fact_7411_abs__dist__diff__le,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [A2: A,B2: A,C2: A] : ( ord_less_eq @ real @ ( abs_abs @ real @ ( minus_minus @ real @ ( real_V557655796197034286t_dist @ A @ A2 @ B2 ) @ ( real_V557655796197034286t_dist @ A @ B2 @ C2 ) ) ) @ ( real_V557655796197034286t_dist @ A @ A2 @ C2 ) ) ) ).
% abs_dist_diff_le
thf(fact_7412_greaterThan__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( set_ord_greaterThan @ A )
= ( ^ [L3: A] : ( collect @ A @ ( ord_less @ A @ L3 ) ) ) ) ) ).
% greaterThan_def
thf(fact_7413_zero__le__dist,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X: A,Y2: A] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( real_V557655796197034286t_dist @ A @ X @ Y2 ) ) ) ).
% zero_le_dist
thf(fact_7414_dist__triangle__le,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X: A,Z3: A,Y2: A,E: real] :
( ( ord_less_eq @ real @ ( plus_plus @ real @ ( real_V557655796197034286t_dist @ A @ X @ Z3 ) @ ( real_V557655796197034286t_dist @ A @ Y2 @ Z3 ) ) @ E )
=> ( ord_less_eq @ real @ ( real_V557655796197034286t_dist @ A @ X @ Y2 ) @ E ) ) ) ).
% dist_triangle_le
thf(fact_7415_dist__triangle3,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X: A,Y2: A,A2: A] : ( ord_less_eq @ real @ ( real_V557655796197034286t_dist @ A @ X @ Y2 ) @ ( plus_plus @ real @ ( real_V557655796197034286t_dist @ A @ A2 @ X ) @ ( real_V557655796197034286t_dist @ A @ A2 @ Y2 ) ) ) ) ).
% dist_triangle3
thf(fact_7416_dist__triangle2,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X: A,Y2: A,Z3: A] : ( ord_less_eq @ real @ ( real_V557655796197034286t_dist @ A @ X @ Y2 ) @ ( plus_plus @ real @ ( real_V557655796197034286t_dist @ A @ X @ Z3 ) @ ( real_V557655796197034286t_dist @ A @ Y2 @ Z3 ) ) ) ) ).
% dist_triangle2
thf(fact_7417_dist__triangle,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X: A,Z3: A,Y2: A] : ( ord_less_eq @ real @ ( real_V557655796197034286t_dist @ A @ X @ Z3 ) @ ( plus_plus @ real @ ( real_V557655796197034286t_dist @ A @ X @ Y2 ) @ ( real_V557655796197034286t_dist @ A @ Y2 @ Z3 ) ) ) ) ).
% dist_triangle
thf(fact_7418_continuous__dist,axiom,
! [A: $tType,D: $tType] :
( ( ( topological_t2_space @ D )
& ( real_V7819770556892013058_space @ A ) )
=> ! [F4: filter @ D,F3: D > A,G: D > A] :
( ( topolo3448309680560233919inuous @ D @ A @ F4 @ F3 )
=> ( ( topolo3448309680560233919inuous @ D @ A @ F4 @ G )
=> ( topolo3448309680560233919inuous @ D @ real @ F4
@ ^ [X4: D] : ( real_V557655796197034286t_dist @ A @ ( F3 @ X4 ) @ ( G @ X4 ) ) ) ) ) ) ).
% continuous_dist
thf(fact_7419_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_7420_filterlim__at__left__to__right,axiom,
! [A: $tType,F3: real > A,F4: filter @ A,A2: real] :
( ( filterlim @ real @ A @ F3 @ F4 @ ( topolo174197925503356063within @ real @ A2 @ ( set_ord_lessThan @ real @ A2 ) ) )
= ( filterlim @ real @ A
@ ^ [X4: real] : ( F3 @ ( uminus_uminus @ real @ X4 ) )
@ F4
@ ( topolo174197925503356063within @ real @ ( uminus_uminus @ real @ A2 ) @ ( set_ord_greaterThan @ real @ ( uminus_uminus @ real @ A2 ) ) ) ) ) ).
% filterlim_at_left_to_right
thf(fact_7421_metric__CauchyI,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X9: nat > A] :
( ! [E2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E2 )
=> ? [M10: nat] :
! [M4: nat] :
( ( ord_less_eq @ nat @ M10 @ M4 )
=> ! [N3: nat] :
( ( ord_less_eq @ nat @ M10 @ N3 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( X9 @ M4 ) @ ( X9 @ N3 ) ) @ E2 ) ) ) )
=> ( topolo3814608138187158403Cauchy @ A @ X9 ) ) ) ).
% metric_CauchyI
thf(fact_7422_metric__CauchyD,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X9: nat > A,E: real] :
( ( topolo3814608138187158403Cauchy @ A @ X9 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ E )
=> ? [M8: nat] :
! [M3: nat] :
( ( ord_less_eq @ nat @ M8 @ M3 )
=> ! [N9: nat] :
( ( ord_less_eq @ nat @ M8 @ N9 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( X9 @ M3 ) @ ( X9 @ N9 ) ) @ E ) ) ) ) ) ) ).
% metric_CauchyD
thf(fact_7423_Cauchy__altdef2,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ( ( topolo3814608138187158403Cauchy @ A )
= ( ^ [S4: nat > A] :
! [E3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E3 )
=> ? [N6: nat] :
! [N2: nat] :
( ( ord_less_eq @ nat @ N6 @ N2 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( S4 @ N2 ) @ ( S4 @ N6 ) ) @ E3 ) ) ) ) ) ) ).
% Cauchy_altdef2
thf(fact_7424_Cauchy__def,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ( ( topolo3814608138187158403Cauchy @ A )
= ( ^ [X6: nat > A] :
! [E3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E3 )
=> ? [M9: nat] :
! [M2: nat] :
( ( ord_less_eq @ nat @ M9 @ M2 )
=> ! [N2: nat] :
( ( ord_less_eq @ nat @ M9 @ N2 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( X6 @ M2 ) @ ( X6 @ N2 ) ) @ E3 ) ) ) ) ) ) ) ).
% Cauchy_def
thf(fact_7425_less__separate,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,Y2: A] :
( ( ord_less @ A @ X @ Y2 )
=> ? [A4: A,B3: A] :
( ( member @ A @ X @ ( set_ord_lessThan @ A @ A4 ) )
& ( member @ A @ Y2 @ ( set_ord_greaterThan @ A @ B3 ) )
& ( ( inf_inf @ ( set @ A ) @ ( set_ord_lessThan @ A @ A4 ) @ ( set_ord_greaterThan @ A @ B3 ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% less_separate
thf(fact_7426_tendsto__dist,axiom,
! [A: $tType,B: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [F3: B > A,L: A,F4: filter @ B,G: B > A,M: A] :
( ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F4 )
=> ( ( filterlim @ B @ A @ G @ ( topolo7230453075368039082e_nhds @ A @ M ) @ F4 )
=> ( filterlim @ B @ real
@ ^ [X4: B] : ( real_V557655796197034286t_dist @ A @ ( F3 @ X4 ) @ ( G @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( real_V557655796197034286t_dist @ A @ L @ M ) )
@ F4 ) ) ) ) ).
% tendsto_dist
thf(fact_7427_filterlim__at__right__to__0,axiom,
! [A: $tType,F3: real > A,F4: filter @ A,A2: real] :
( ( filterlim @ real @ A @ F3 @ F4 @ ( topolo174197925503356063within @ real @ A2 @ ( set_ord_greaterThan @ real @ A2 ) ) )
= ( filterlim @ real @ A
@ ^ [X4: real] : ( F3 @ ( plus_plus @ real @ X4 @ A2 ) )
@ F4
@ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) ) ) ).
% filterlim_at_right_to_0
thf(fact_7428_metric__LIM__imp__LIM,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( ( topolo4958980785337419405_space @ C )
& ( real_V7819770556892013058_space @ B )
& ( real_V7819770556892013058_space @ A ) )
=> ! [F3: C > A,L: A,A2: C,G: C > B,M: B] :
( ( filterlim @ C @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ ( topolo174197925503356063within @ C @ A2 @ ( top_top @ ( set @ C ) ) ) )
=> ( ! [X5: C] :
( ( X5 != A2 )
=> ( ord_less_eq @ real @ ( real_V557655796197034286t_dist @ B @ ( G @ X5 ) @ M ) @ ( real_V557655796197034286t_dist @ A @ ( F3 @ X5 ) @ L ) ) )
=> ( filterlim @ C @ B @ G @ ( topolo7230453075368039082e_nhds @ B @ M ) @ ( topolo174197925503356063within @ C @ A2 @ ( top_top @ ( set @ C ) ) ) ) ) ) ) ).
% metric_LIM_imp_LIM
thf(fact_7429_dist__triangle__half__l,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X16: A,Y2: A,E: real,X23: A] :
( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X16 @ Y2 ) @ ( divide_divide @ real @ E @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X23 @ Y2 ) @ ( divide_divide @ real @ E @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X16 @ X23 ) @ E ) ) ) ) ).
% dist_triangle_half_l
thf(fact_7430_dist__triangle__half__r,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [Y2: A,X16: A,E: real,X23: A] :
( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ Y2 @ X16 ) @ ( divide_divide @ real @ E @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ Y2 @ X23 ) @ ( divide_divide @ real @ E @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X16 @ X23 ) @ E ) ) ) ) ).
% dist_triangle_half_r
thf(fact_7431_dist__triangle__third,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X16: A,X23: A,E: real,X32: A,X42: A] :
( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X16 @ X23 ) @ ( divide_divide @ real @ E @ ( numeral_numeral @ real @ ( bit1 @ one2 ) ) ) )
=> ( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X23 @ X32 ) @ ( divide_divide @ real @ E @ ( numeral_numeral @ real @ ( bit1 @ one2 ) ) ) )
=> ( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X32 @ X42 ) @ ( divide_divide @ real @ E @ ( numeral_numeral @ real @ ( bit1 @ one2 ) ) ) )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X16 @ X42 ) @ E ) ) ) ) ) ).
% dist_triangle_third
thf(fact_7432_filterlim__transform__within,axiom,
! [B: $tType,A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [G: A > B,G6: filter @ B,X: A,S3: set @ A,F4: filter @ B,D2: real,F3: A > B] :
( ( filterlim @ A @ B @ G @ G6 @ ( topolo174197925503356063within @ A @ X @ S3 ) )
=> ( ( ord_less_eq @ ( filter @ B ) @ G6 @ F4 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ D2 )
=> ( ! [X17: A] :
( ( member @ A @ X17 @ S3 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( real_V557655796197034286t_dist @ A @ X17 @ X ) )
=> ( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X17 @ X ) @ D2 )
=> ( ( F3 @ X17 )
= ( G @ X17 ) ) ) ) )
=> ( filterlim @ A @ B @ F3 @ F4 @ ( topolo174197925503356063within @ A @ X @ S3 ) ) ) ) ) ) ) ).
% filterlim_transform_within
thf(fact_7433_Cauchy__altdef,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ( ( topolo3814608138187158403Cauchy @ A )
= ( ^ [F5: nat > A] :
! [E3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E3 )
=> ? [M9: nat] :
! [M2: nat] :
( ( ord_less_eq @ nat @ M9 @ M2 )
=> ! [N2: nat] :
( ( ord_less @ nat @ M2 @ N2 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( F5 @ M2 ) @ ( F5 @ N2 ) ) @ E3 ) ) ) ) ) ) ) ).
% Cauchy_altdef
thf(fact_7434_CauchyI_H,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X9: nat > A] :
( ! [E2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E2 )
=> ? [M10: nat] :
! [M4: nat] :
( ( ord_less_eq @ nat @ M10 @ M4 )
=> ! [N3: nat] :
( ( ord_less @ nat @ M4 @ N3 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( X9 @ M4 ) @ ( X9 @ N3 ) ) @ E2 ) ) ) )
=> ( topolo3814608138187158403Cauchy @ A @ X9 ) ) ) ).
% CauchyI'
thf(fact_7435_tendsto__dist__iff,axiom,
! [B: $tType,A: $tType] :
( ( real_V7819770556892013058_space @ B )
=> ! [F3: A > B,L: B,F4: filter @ A] :
( ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ F4 )
= ( filterlim @ A @ real
@ ^ [X4: A] : ( real_V557655796197034286t_dist @ B @ ( F3 @ X4 ) @ L )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ F4 ) ) ) ).
% tendsto_dist_iff
thf(fact_7436_filterlim__times__pos,axiom,
! [A: $tType,B: $tType] :
( ( ( linordered_field @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F3: B > A,P5: A,F13: filter @ B,C2: A,L: A] :
( ( filterlim @ B @ A @ F3 @ ( topolo174197925503356063within @ A @ P5 @ ( set_ord_greaterThan @ A @ P5 ) ) @ F13 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( L
= ( times_times @ A @ C2 @ P5 ) )
=> ( filterlim @ B @ A
@ ^ [X4: B] : ( times_times @ A @ C2 @ ( F3 @ X4 ) )
@ ( topolo174197925503356063within @ A @ L @ ( set_ord_greaterThan @ A @ L ) )
@ F13 ) ) ) ) ) ).
% filterlim_times_pos
thf(fact_7437_metric__LIMSEQ__D,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X9: nat > A,L6: A,R: real] :
( ( filterlim @ nat @ A @ X9 @ ( topolo7230453075368039082e_nhds @ A @ L6 ) @ ( at_top @ nat ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ R )
=> ? [No3: nat] :
! [N9: nat] :
( ( ord_less_eq @ nat @ No3 @ N9 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( X9 @ N9 ) @ L6 ) @ R ) ) ) ) ) ).
% metric_LIMSEQ_D
thf(fact_7438_metric__LIMSEQ__I,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X9: nat > A,L6: A] :
( ! [R3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R3 )
=> ? [No4: nat] :
! [N3: nat] :
( ( ord_less_eq @ nat @ No4 @ N3 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( X9 @ N3 ) @ L6 ) @ R3 ) ) )
=> ( filterlim @ nat @ A @ X9 @ ( topolo7230453075368039082e_nhds @ A @ L6 ) @ ( at_top @ nat ) ) ) ) ).
% metric_LIMSEQ_I
thf(fact_7439_lim__sequentially,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X9: nat > A,L6: A] :
( ( filterlim @ nat @ A @ X9 @ ( topolo7230453075368039082e_nhds @ A @ L6 ) @ ( at_top @ nat ) )
= ( ! [R5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R5 )
=> ? [No2: nat] :
! [N2: nat] :
( ( ord_less_eq @ nat @ No2 @ N2 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( X9 @ N2 ) @ L6 ) @ R5 ) ) ) ) ) ) ).
% lim_sequentially
thf(fact_7440_metric__Cauchy__iff2,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ( ( topolo3814608138187158403Cauchy @ A )
= ( ^ [X6: nat > A] :
! [J4: nat] :
? [M9: nat] :
! [M2: nat] :
( ( ord_less_eq @ nat @ M9 @ M2 )
=> ! [N2: nat] :
( ( ord_less_eq @ nat @ M9 @ N2 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( X6 @ M2 ) @ ( X6 @ N2 ) ) @ ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ ( suc @ J4 ) ) ) ) ) ) ) ) ) ).
% metric_Cauchy_iff2
thf(fact_7441_metric__LIM__compose2,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( real_V7819770556892013058_space @ A )
& ( topolo4958980785337419405_space @ B )
& ( topolo4958980785337419405_space @ C ) )
=> ! [F3: A > B,B2: B,A2: A,G: B > C,C2: C] :
( ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ B2 ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) )
=> ( ( filterlim @ B @ C @ G @ ( topolo7230453075368039082e_nhds @ C @ C2 ) @ ( topolo174197925503356063within @ B @ B2 @ ( top_top @ ( set @ B ) ) ) )
=> ( ? [D6: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D6 )
& ! [X5: A] :
( ( ( X5 != A2 )
& ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X5 @ A2 ) @ D6 ) )
=> ( ( F3 @ X5 )
!= B2 ) ) )
=> ( filterlim @ A @ C
@ ^ [X4: A] : ( G @ ( F3 @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ C @ C2 )
@ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ).
% metric_LIM_compose2
thf(fact_7442_tendsto__arcosh__at__left__1,axiom,
filterlim @ real @ real @ ( arcosh @ real ) @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( topolo174197925503356063within @ real @ ( one_one @ real ) @ ( set_ord_greaterThan @ real @ ( one_one @ real ) ) ) ).
% tendsto_arcosh_at_left_1
thf(fact_7443_metric__isCont__LIM__compose2,axiom,
! [D: $tType,C: $tType,A: $tType] :
( ( ( real_V7819770556892013058_space @ A )
& ( topolo4958980785337419405_space @ C )
& ( topolo4958980785337419405_space @ D ) )
=> ! [A2: A,F3: A > C,G: C > D,L: D] :
( ( topolo3448309680560233919inuous @ A @ C @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) @ F3 )
=> ( ( filterlim @ C @ D @ G @ ( topolo7230453075368039082e_nhds @ D @ L ) @ ( topolo174197925503356063within @ C @ ( F3 @ A2 ) @ ( top_top @ ( set @ C ) ) ) )
=> ( ? [D6: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D6 )
& ! [X5: A] :
( ( ( X5 != A2 )
& ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X5 @ A2 ) @ D6 ) )
=> ( ( F3 @ X5 )
!= ( F3 @ A2 ) ) ) )
=> ( filterlim @ A @ D
@ ^ [X4: A] : ( G @ ( F3 @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ D @ L )
@ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ).
% metric_isCont_LIM_compose2
thf(fact_7444_isCont__If__ge,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo1944317154257567458pology @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [A2: A,G: A > B,F3: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ ( set_ord_lessThan @ A @ A2 ) ) @ G )
=> ( ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ ( G @ A2 ) ) @ ( topolo174197925503356063within @ A @ A2 @ ( set_ord_greaterThan @ A @ A2 ) ) )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) )
@ ^ [X4: A] : ( if @ B @ ( ord_less_eq @ A @ X4 @ A2 ) @ ( G @ X4 ) @ ( F3 @ X4 ) ) ) ) ) ) ).
% isCont_If_ge
thf(fact_7445_LIMSEQ__iff__nz,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X9: nat > A,L6: A] :
( ( filterlim @ nat @ A @ X9 @ ( topolo7230453075368039082e_nhds @ A @ L6 ) @ ( at_top @ nat ) )
= ( ! [R5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R5 )
=> ? [No2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ No2 )
& ! [N2: nat] :
( ( ord_less_eq @ nat @ No2 @ N2 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( X9 @ N2 ) @ L6 ) @ R5 ) ) ) ) ) ) ) ).
% LIMSEQ_iff_nz
thf(fact_7446_totally__bounded__metric,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ( ( topolo6688025880775521714ounded @ A )
= ( ^ [S6: set @ A] :
! [E3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E3 )
=> ? [K3: set @ A] :
( ( finite_finite2 @ A @ K3 )
& ( ord_less_eq @ ( set @ A ) @ S6
@ ( complete_Sup_Sup @ ( set @ A )
@ ( image @ A @ ( set @ A )
@ ^ [X4: A] :
( collect @ A
@ ^ [Y: A] : ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X4 @ Y ) @ E3 ) )
@ K3 ) ) ) ) ) ) ) ) ).
% totally_bounded_metric
thf(fact_7447_at__within__order,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [X: A,S: set @ A] :
( ( ( top_top @ ( set @ A ) )
!= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
=> ( ( topolo174197925503356063within @ A @ X @ S )
= ( inf_inf @ ( filter @ A )
@ ( complete_Inf_Inf @ ( filter @ A )
@ ( image @ A @ ( filter @ A )
@ ^ [A5: A] : ( principal @ A @ ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ ( set_ord_lessThan @ A @ A5 ) @ S ) @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
@ ( set_ord_greaterThan @ A @ X ) ) )
@ ( complete_Inf_Inf @ ( filter @ A )
@ ( image @ A @ ( filter @ A )
@ ^ [A5: A] : ( principal @ A @ ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ ( set_ord_greaterThan @ A @ A5 ) @ S ) @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
@ ( set_ord_lessThan @ A @ X ) ) ) ) ) ) ) ).
% at_within_order
thf(fact_7448_principal__le__iff,axiom,
! [A: $tType,A3: set @ A,B5: set @ A] :
( ( ord_less_eq @ ( filter @ A ) @ ( principal @ A @ A3 ) @ ( principal @ A @ B5 ) )
= ( ord_less_eq @ ( set @ A ) @ A3 @ B5 ) ) ).
% principal_le_iff
thf(fact_7449_SUP__principal,axiom,
! [A: $tType,B: $tType,A3: B > ( set @ A ),I6: set @ B] :
( ( complete_Sup_Sup @ ( filter @ A )
@ ( image @ B @ ( filter @ A )
@ ^ [I: B] : ( principal @ A @ ( A3 @ I ) )
@ I6 ) )
= ( principal @ A @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ A3 @ I6 ) ) ) ) ).
% SUP_principal
thf(fact_7450_filterlim__If,axiom,
! [B: $tType,A: $tType,F3: A > B,G6: filter @ B,F4: filter @ A,P3: A > $o,G: A > B] :
( ( filterlim @ A @ B @ F3 @ G6 @ ( inf_inf @ ( filter @ A ) @ F4 @ ( principal @ A @ ( collect @ A @ P3 ) ) ) )
=> ( ( filterlim @ A @ B @ G @ G6
@ ( inf_inf @ ( filter @ A ) @ F4
@ ( principal @ A
@ ( collect @ A
@ ^ [X4: A] :
~ ( P3 @ X4 ) ) ) ) )
=> ( filterlim @ A @ B
@ ^ [X4: A] : ( if @ B @ ( P3 @ X4 ) @ ( F3 @ X4 ) @ ( G @ X4 ) )
@ G6
@ F4 ) ) ) ).
% filterlim_If
thf(fact_7451_totally__bounded__subset,axiom,
! [A: $tType] :
( ( topolo7287701948861334536_space @ A )
=> ! [S3: set @ A,T5: set @ A] :
( ( topolo6688025880775521714ounded @ A @ S3 )
=> ( ( ord_less_eq @ ( set @ A ) @ T5 @ S3 )
=> ( topolo6688025880775521714ounded @ A @ T5 ) ) ) ) ).
% totally_bounded_subset
thf(fact_7452_nhds__def,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ( ( topolo7230453075368039082e_nhds @ A )
= ( ^ [A5: A] :
( complete_Inf_Inf @ ( filter @ A )
@ ( image @ ( set @ A ) @ ( filter @ A ) @ ( principal @ A )
@ ( collect @ ( set @ A )
@ ^ [S6: set @ A] :
( ( topolo1002775350975398744n_open @ A @ S6 )
& ( member @ A @ A5 @ S6 ) ) ) ) ) ) ) ) ).
% nhds_def
thf(fact_7453_filterlim__base,axiom,
! [B: $tType,A: $tType,E4: $tType,D: $tType,C: $tType,J5: set @ A,I2: A > C,I6: set @ C,F4: C > ( set @ D ),F3: D > E4,G6: A > ( set @ E4 )] :
( ! [M4: A,X5: B] :
( ( member @ A @ M4 @ J5 )
=> ( member @ C @ ( I2 @ M4 ) @ I6 ) )
=> ( ! [M4: A,X5: D] :
( ( member @ A @ M4 @ J5 )
=> ( ( member @ D @ X5 @ ( F4 @ ( I2 @ M4 ) ) )
=> ( member @ E4 @ ( F3 @ X5 ) @ ( G6 @ M4 ) ) ) )
=> ( filterlim @ D @ E4 @ F3
@ ( complete_Inf_Inf @ ( filter @ E4 )
@ ( image @ A @ ( filter @ E4 )
@ ^ [J4: A] : ( principal @ E4 @ ( G6 @ J4 ) )
@ J5 ) )
@ ( complete_Inf_Inf @ ( filter @ D )
@ ( image @ C @ ( filter @ D )
@ ^ [I: C] : ( principal @ D @ ( F4 @ I ) )
@ I6 ) ) ) ) ) ).
% filterlim_base
thf(fact_7454_INT__greaterThan__UNIV,axiom,
( ( complete_Inf_Inf @ ( set @ nat ) @ ( image @ nat @ ( set @ nat ) @ ( set_ord_greaterThan @ nat ) @ ( top_top @ ( set @ nat ) ) ) )
= ( bot_bot @ ( set @ nat ) ) ) ).
% INT_greaterThan_UNIV
thf(fact_7455_filterlim__base__iff,axiom,
! [A: $tType,C: $tType,B: $tType,D: $tType,I6: set @ A,F4: A > ( set @ B ),F3: B > C,G6: D > ( set @ C ),J5: set @ D] :
( ( I6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [I3: A] :
( ( member @ A @ I3 @ I6 )
=> ! [J2: A] :
( ( member @ A @ J2 @ I6 )
=> ( ( ord_less_eq @ ( set @ B ) @ ( F4 @ I3 ) @ ( F4 @ J2 ) )
| ( ord_less_eq @ ( set @ B ) @ ( F4 @ J2 ) @ ( F4 @ I3 ) ) ) ) )
=> ( ( filterlim @ B @ C @ F3
@ ( complete_Inf_Inf @ ( filter @ C )
@ ( image @ D @ ( filter @ C )
@ ^ [J4: D] : ( principal @ C @ ( G6 @ J4 ) )
@ J5 ) )
@ ( complete_Inf_Inf @ ( filter @ B )
@ ( image @ A @ ( filter @ B )
@ ^ [I: A] : ( principal @ B @ ( F4 @ I ) )
@ I6 ) ) )
= ( ! [X4: D] :
( ( member @ D @ X4 @ J5 )
=> ? [Y: A] :
( ( member @ A @ Y @ I6 )
& ! [Z6: B] :
( ( member @ B @ Z6 @ ( F4 @ Y ) )
=> ( member @ C @ ( F3 @ Z6 ) @ ( G6 @ X4 ) ) ) ) ) ) ) ) ) ).
% filterlim_base_iff
thf(fact_7456_INF__principal__finite,axiom,
! [B: $tType,A: $tType,X9: set @ A,F3: A > ( set @ B )] :
( ( finite_finite2 @ A @ X9 )
=> ( ( complete_Inf_Inf @ ( filter @ B )
@ ( image @ A @ ( filter @ B )
@ ^ [X4: A] : ( principal @ B @ ( F3 @ X4 ) )
@ X9 ) )
= ( principal @ B @ ( complete_Inf_Inf @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ F3 @ X9 ) ) ) ) ) ).
% INF_principal_finite
thf(fact_7457_nhds__metric,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ( ( topolo7230453075368039082e_nhds @ A )
= ( ^ [X4: A] :
( complete_Inf_Inf @ ( filter @ A )
@ ( image @ real @ ( filter @ A )
@ ^ [E3: real] :
( principal @ A
@ ( collect @ A
@ ^ [Y: A] : ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ Y @ X4 ) @ E3 ) ) )
@ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) ) ) ) ) ).
% nhds_metric
thf(fact_7458_at__left__eq,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [Y2: A,X: A] :
( ( ord_less @ A @ Y2 @ X )
=> ( ( topolo174197925503356063within @ A @ X @ ( set_ord_lessThan @ A @ X ) )
= ( complete_Inf_Inf @ ( filter @ A )
@ ( image @ A @ ( filter @ A )
@ ^ [A5: A] : ( principal @ A @ ( set_or5935395276787703475ssThan @ A @ A5 @ X ) )
@ ( set_ord_lessThan @ A @ X ) ) ) ) ) ) ).
% at_left_eq
thf(fact_7459_at__right__eq,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [X: A,Y2: A] :
( ( ord_less @ A @ X @ Y2 )
=> ( ( topolo174197925503356063within @ A @ X @ ( set_ord_greaterThan @ A @ X ) )
= ( complete_Inf_Inf @ ( filter @ A )
@ ( image @ A @ ( filter @ A )
@ ^ [A5: A] : ( principal @ A @ ( set_or5935395276787703475ssThan @ A @ X @ A5 ) )
@ ( set_ord_greaterThan @ A @ X ) ) ) ) ) ) ).
% at_right_eq
thf(fact_7460_nhds__order,axiom,
! [A: $tType] :
( ( topolo2564578578187576103pology @ A )
=> ( ( topolo7230453075368039082e_nhds @ A )
= ( ^ [X4: A] :
( inf_inf @ ( filter @ A )
@ ( complete_Inf_Inf @ ( filter @ A )
@ ( image @ A @ ( filter @ A )
@ ^ [A5: A] : ( principal @ A @ ( set_ord_lessThan @ A @ A5 ) )
@ ( set_ord_greaterThan @ A @ X4 ) ) )
@ ( complete_Inf_Inf @ ( filter @ A )
@ ( image @ A @ ( filter @ A )
@ ^ [A5: A] : ( principal @ A @ ( set_ord_greaterThan @ A @ A5 ) )
@ ( set_ord_lessThan @ A @ X4 ) ) ) ) ) ) ) ).
% nhds_order
thf(fact_7461_at__within__eq,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ( ( topolo174197925503356063within @ A )
= ( ^ [X4: A,S4: set @ A] :
( complete_Inf_Inf @ ( filter @ A )
@ ( image @ ( set @ A ) @ ( filter @ A )
@ ^ [S6: set @ A] : ( principal @ A @ ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ S6 @ S4 ) @ ( insert @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) ) )
@ ( collect @ ( set @ A )
@ ^ [S6: set @ A] :
( ( topolo1002775350975398744n_open @ A @ S6 )
& ( member @ A @ X4 @ S6 ) ) ) ) ) ) ) ) ).
% at_within_eq
thf(fact_7462_filterlim__tan__at__right,axiom,
filterlim @ real @ real @ ( tan @ real ) @ ( at_bot @ real ) @ ( topolo174197925503356063within @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ ( set_ord_greaterThan @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ).
% filterlim_tan_at_right
thf(fact_7463_interval__cases,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [S3: set @ A] :
( ! [A4: A,B3: A,X5: A] :
( ( member @ A @ A4 @ S3 )
=> ( ( member @ A @ B3 @ S3 )
=> ( ( ord_less_eq @ A @ A4 @ X5 )
=> ( ( ord_less_eq @ A @ X5 @ B3 )
=> ( member @ A @ X5 @ S3 ) ) ) ) )
=> ? [A4: A,B3: A] :
( ( S3
= ( bot_bot @ ( set @ A ) ) )
| ( S3
= ( top_top @ ( set @ A ) ) )
| ( S3
= ( set_ord_lessThan @ A @ B3 ) )
| ( S3
= ( set_ord_atMost @ A @ B3 ) )
| ( S3
= ( set_ord_greaterThan @ A @ A4 ) )
| ( S3
= ( set_ord_atLeast @ A @ A4 ) )
| ( S3
= ( set_or5935395276787703475ssThan @ A @ A4 @ B3 ) )
| ( S3
= ( set_or3652927894154168847AtMost @ A @ A4 @ B3 ) )
| ( S3
= ( set_or7035219750837199246ssThan @ A @ A4 @ B3 ) )
| ( S3
= ( set_or1337092689740270186AtMost @ A @ A4 @ B3 ) ) ) ) ) ).
% interval_cases
thf(fact_7464_atLeast__iff,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [I2: A,K2: A] :
( ( member @ A @ I2 @ ( set_ord_atLeast @ A @ K2 ) )
= ( ord_less_eq @ A @ K2 @ I2 ) ) ) ).
% atLeast_iff
thf(fact_7465_atLeast__subset__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y2: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_ord_atLeast @ A @ X ) @ ( set_ord_atLeast @ A @ Y2 ) )
= ( ord_less_eq @ A @ Y2 @ X ) ) ) ).
% atLeast_subset_iff
thf(fact_7466_image__add__atLeast,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [K2: A,I2: A] :
( ( image @ A @ A @ ( plus_plus @ A @ K2 ) @ ( set_ord_atLeast @ A @ I2 ) )
= ( set_ord_atLeast @ A @ ( plus_plus @ A @ K2 @ I2 ) ) ) ) ).
% image_add_atLeast
thf(fact_7467_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_7468_Ioi__le__Ico,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A2: A] : ( ord_less_eq @ ( set @ A ) @ ( set_ord_greaterThan @ A @ A2 ) @ ( set_ord_atLeast @ A @ A2 ) ) ) ).
% Ioi_le_Ico
thf(fact_7469_not__Ici__le__Icc,axiom,
! [A: $tType] :
( ( no_top @ A )
=> ! [L: A,L4: A,H3: A] :
~ ( ord_less_eq @ ( set @ A ) @ ( set_ord_atLeast @ A @ L ) @ ( set_or1337092689740270186AtMost @ A @ L4 @ H3 ) ) ) ).
% not_Ici_le_Icc
thf(fact_7470_not__Ici__le__Iic,axiom,
! [A: $tType] :
( ( no_top @ A )
=> ! [L: A,H3: A] :
~ ( ord_less_eq @ ( set @ A ) @ ( set_ord_atLeast @ A @ L ) @ ( set_ord_atMost @ A @ H3 ) ) ) ).
% not_Ici_le_Iic
thf(fact_7471_not__Iic__le__Ici,axiom,
! [A: $tType] :
( ( no_bot @ A )
=> ! [H2: A,L4: A] :
~ ( ord_less_eq @ ( set @ A ) @ ( set_ord_atMost @ A @ H2 ) @ ( set_ord_atLeast @ A @ L4 ) ) ) ).
% not_Iic_le_Ici
thf(fact_7472_atLeast__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( set_ord_atLeast @ A )
= ( ^ [L3: A] : ( collect @ A @ ( ord_less_eq @ A @ L3 ) ) ) ) ) ).
% atLeast_def
thf(fact_7473_not__UNIV__le__Ici,axiom,
! [A: $tType] :
( ( no_bot @ A )
=> ! [L: A] :
~ ( ord_less_eq @ ( set @ A ) @ ( top_top @ ( set @ A ) ) @ ( set_ord_atLeast @ A @ L ) ) ) ).
% not_UNIV_le_Ici
thf(fact_7474_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_7475_Ici__subset__Ioi__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_ord_atLeast @ A @ A2 ) @ ( set_ord_greaterThan @ A @ B2 ) )
= ( ord_less @ A @ B2 @ A2 ) ) ) ).
% Ici_subset_Ioi_iff
thf(fact_7476_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_7477_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_7478_UN__atLeast__UNIV,axiom,
( ( complete_Sup_Sup @ ( set @ nat ) @ ( image @ nat @ ( set @ nat ) @ ( set_ord_atLeast @ nat ) @ ( top_top @ ( set @ nat ) ) ) )
= ( top_top @ ( set @ nat ) ) ) ).
% UN_atLeast_UNIV
thf(fact_7479_at__top__sub,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C2: A] :
( ( at_top @ A )
= ( complete_Inf_Inf @ ( filter @ A )
@ ( image @ A @ ( filter @ A )
@ ^ [K3: A] : ( principal @ A @ ( set_ord_atLeast @ A @ K3 ) )
@ ( set_ord_atLeast @ A @ C2 ) ) ) ) ) ).
% at_top_sub
thf(fact_7480_tanh__real__at__bot,axiom,
filterlim @ real @ real @ ( tanh @ real ) @ ( topolo7230453075368039082e_nhds @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) ) @ ( at_bot @ real ) ).
% tanh_real_at_bot
thf(fact_7481_filterlim__tendsto__pos__mult__at__bot,axiom,
! [A: $tType,F3: A > real,C2: real,F4: filter @ A,G: A > real] :
( ( filterlim @ A @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ C2 ) @ F4 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ C2 )
=> ( ( filterlim @ A @ real @ G @ ( at_bot @ real ) @ F4 )
=> ( filterlim @ A @ real
@ ^ [X4: A] : ( times_times @ real @ ( F3 @ X4 ) @ ( G @ X4 ) )
@ ( at_bot @ real )
@ F4 ) ) ) ) ).
% filterlim_tendsto_pos_mult_at_bot
thf(fact_7482_ln__at__0,axiom,
filterlim @ real @ real @ ( ln_ln @ real ) @ ( at_bot @ real ) @ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) ).
% ln_at_0
thf(fact_7483_tendsto__at__botI__sequentially,axiom,
! [B: $tType] :
( ( topolo3112930676232923870pology @ B )
=> ! [F3: real > B,Y2: B] :
( ! [X10: nat > real] :
( ( filterlim @ nat @ real @ X10 @ ( at_bot @ real ) @ ( at_top @ nat ) )
=> ( filterlim @ nat @ B
@ ^ [N2: nat] : ( F3 @ ( X10 @ N2 ) )
@ ( topolo7230453075368039082e_nhds @ B @ Y2 )
@ ( at_top @ nat ) ) )
=> ( filterlim @ real @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ Y2 ) @ ( at_bot @ real ) ) ) ) ).
% tendsto_at_botI_sequentially
thf(fact_7484_at__top__def,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( at_top @ A )
= ( complete_Inf_Inf @ ( filter @ A )
@ ( image @ A @ ( filter @ A )
@ ^ [K3: A] : ( principal @ A @ ( set_ord_atLeast @ A @ K3 ) )
@ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% at_top_def
thf(fact_7485_filterlim__inverse__at__bot__neg,axiom,
filterlim @ real @ real @ ( inverse_inverse @ real ) @ ( at_bot @ real ) @ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_lessThan @ real @ ( zero_zero @ real ) ) ) ).
% filterlim_inverse_at_bot_neg
thf(fact_7486_artanh__real__at__right__1,axiom,
filterlim @ real @ real @ ( artanh @ real ) @ ( at_bot @ real ) @ ( topolo174197925503356063within @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ ( set_ord_greaterThan @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) ) ) ).
% artanh_real_at_right_1
thf(fact_7487_at__bot__sub,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C2: A] :
( ( at_bot @ A )
= ( complete_Inf_Inf @ ( filter @ A )
@ ( image @ A @ ( filter @ A )
@ ^ [K3: A] : ( principal @ A @ ( set_ord_atMost @ A @ K3 ) )
@ ( set_ord_atMost @ A @ C2 ) ) ) ) ) ).
% at_bot_sub
thf(fact_7488_at__bot__def,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( at_bot @ A )
= ( complete_Inf_Inf @ ( filter @ A )
@ ( image @ A @ ( filter @ A )
@ ^ [K3: A] : ( principal @ A @ ( set_ord_atMost @ A @ K3 ) )
@ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% at_bot_def
thf(fact_7489_DERIV__pos__imp__increasing__at__bot,axiom,
! [B2: real,F3: real > real,Flim: real] :
( ! [X5: real] :
( ( ord_less_eq @ real @ X5 @ B2 )
=> ? [Y4: real] :
( ( has_field_derivative @ real @ F3 @ Y4 @ ( topolo174197925503356063within @ real @ X5 @ ( top_top @ ( set @ real ) ) ) )
& ( ord_less @ real @ ( zero_zero @ real ) @ Y4 ) ) )
=> ( ( filterlim @ real @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ Flim ) @ ( at_bot @ real ) )
=> ( ord_less @ real @ Flim @ ( F3 @ B2 ) ) ) ) ).
% DERIV_pos_imp_increasing_at_bot
thf(fact_7490_filterlim__pow__at__bot__odd,axiom,
! [N: nat,F3: real > real,F4: filter @ real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( filterlim @ real @ real @ F3 @ ( at_bot @ real ) @ F4 )
=> ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( filterlim @ real @ real
@ ^ [X4: real] : ( power_power @ real @ ( F3 @ X4 ) @ N )
@ ( at_bot @ real )
@ F4 ) ) ) ) ).
% filterlim_pow_at_bot_odd
thf(fact_7491_tendsto__arctan__at__bot,axiom,
filterlim @ real @ real @ arctan @ ( topolo7230453075368039082e_nhds @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) @ ( at_bot @ real ) ).
% tendsto_arctan_at_bot
thf(fact_7492_filterlim__pow__at__bot__even,axiom,
! [N: nat,F3: real > real,F4: filter @ real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( filterlim @ real @ real @ F3 @ ( at_bot @ real ) @ F4 )
=> ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( filterlim @ real @ real
@ ^ [X4: real] : ( power_power @ real @ ( F3 @ X4 ) @ N )
@ ( at_top @ real )
@ F4 ) ) ) ) ).
% filterlim_pow_at_bot_even
thf(fact_7493_at__infinity__def,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ( ( at_infinity @ A )
= ( complete_Inf_Inf @ ( filter @ A )
@ ( image @ real @ ( filter @ A )
@ ^ [R5: real] :
( principal @ A
@ ( collect @ A
@ ^ [X4: A] : ( ord_less_eq @ real @ R5 @ ( real_V7770717601297561774m_norm @ A @ X4 ) ) ) )
@ ( top_top @ ( set @ real ) ) ) ) ) ) ).
% at_infinity_def
thf(fact_7494_at__bot__le__at__infinity,axiom,
ord_less_eq @ ( filter @ real ) @ ( at_bot @ real ) @ ( at_infinity @ real ) ).
% at_bot_le_at_infinity
thf(fact_7495_at__top__le__at__infinity,axiom,
ord_less_eq @ ( filter @ real ) @ ( at_top @ real ) @ ( at_infinity @ real ) ).
% at_top_le_at_infinity
thf(fact_7496_filterlim__at__top__mult__at__top,axiom,
! [A: $tType,F3: A > real,F4: filter @ A,G: A > real] :
( ( filterlim @ A @ real @ F3 @ ( at_top @ real ) @ F4 )
=> ( ( filterlim @ A @ real @ G @ ( at_top @ real ) @ F4 )
=> ( filterlim @ A @ real
@ ^ [X4: A] : ( times_times @ real @ ( F3 @ X4 ) @ ( G @ X4 ) )
@ ( at_top @ real )
@ F4 ) ) ) ).
% filterlim_at_top_mult_at_top
thf(fact_7497_filterlim__at__top__add__at__top,axiom,
! [A: $tType,F3: A > real,F4: filter @ A,G: A > real] :
( ( filterlim @ A @ real @ F3 @ ( at_top @ real ) @ F4 )
=> ( ( filterlim @ A @ real @ G @ ( at_top @ real ) @ F4 )
=> ( filterlim @ A @ real
@ ^ [X4: A] : ( plus_plus @ real @ ( F3 @ X4 ) @ ( G @ X4 ) )
@ ( at_top @ real )
@ F4 ) ) ) ).
% filterlim_at_top_add_at_top
thf(fact_7498_ln__at__top,axiom,
filterlim @ real @ real @ ( ln_ln @ real ) @ ( at_top @ real ) @ ( at_top @ real ) ).
% ln_at_top
thf(fact_7499_sqrt__at__top,axiom,
filterlim @ real @ real @ sqrt @ ( at_top @ real ) @ ( at_top @ real ) ).
% sqrt_at_top
thf(fact_7500_filterlim__at__infinity__conv__norm__at__top,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F3: A > B,G6: filter @ A] :
( ( filterlim @ A @ B @ F3 @ ( at_infinity @ B ) @ G6 )
= ( filterlim @ A @ real
@ ^ [X4: A] : ( real_V7770717601297561774m_norm @ B @ ( F3 @ X4 ) )
@ ( at_top @ real )
@ G6 ) ) ) ).
% filterlim_at_infinity_conv_norm_at_top
thf(fact_7501_filterlim__norm__at__top__imp__at__infinity,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F3: A > B,F4: filter @ A] :
( ( filterlim @ A @ real
@ ^ [X4: A] : ( real_V7770717601297561774m_norm @ B @ ( F3 @ X4 ) )
@ ( at_top @ real )
@ F4 )
=> ( filterlim @ A @ B @ F3 @ ( at_infinity @ B ) @ F4 ) ) ) ).
% filterlim_norm_at_top_imp_at_infinity
thf(fact_7502_filterlim__at__infinity__imp__norm__at__top,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F3: A > B,F4: filter @ A] :
( ( filterlim @ A @ B @ F3 @ ( at_infinity @ B ) @ F4 )
=> ( filterlim @ A @ real
@ ^ [X4: A] : ( real_V7770717601297561774m_norm @ B @ ( F3 @ X4 ) )
@ ( at_top @ real )
@ F4 ) ) ) ).
% filterlim_at_infinity_imp_norm_at_top
thf(fact_7503_exp__at__top,axiom,
filterlim @ real @ real @ ( exp @ real ) @ ( at_top @ real ) @ ( at_top @ real ) ).
% exp_at_top
thf(fact_7504_filterlim__tendsto__add__at__top,axiom,
! [A: $tType,F3: A > real,C2: real,F4: filter @ A,G: A > real] :
( ( filterlim @ A @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ C2 ) @ F4 )
=> ( ( filterlim @ A @ real @ G @ ( at_top @ real ) @ F4 )
=> ( filterlim @ A @ real
@ ^ [X4: A] : ( plus_plus @ real @ ( F3 @ X4 ) @ ( G @ X4 ) )
@ ( at_top @ real )
@ F4 ) ) ) ).
% filterlim_tendsto_add_at_top
thf(fact_7505_filterlim__real__sequentially,axiom,
filterlim @ nat @ real @ ( semiring_1_of_nat @ real ) @ ( at_top @ real ) @ ( at_top @ nat ) ).
% filterlim_real_sequentially
thf(fact_7506_filterlim__uminus__at__top,axiom,
! [A: $tType,F3: A > real,F4: filter @ A] :
( ( filterlim @ A @ real @ F3 @ ( at_top @ real ) @ F4 )
= ( filterlim @ A @ real
@ ^ [X4: A] : ( uminus_uminus @ real @ ( F3 @ X4 ) )
@ ( at_bot @ real )
@ F4 ) ) ).
% filterlim_uminus_at_top
thf(fact_7507_filterlim__uminus__at__bot,axiom,
! [A: $tType,F3: A > real,F4: filter @ A] :
( ( filterlim @ A @ real @ F3 @ ( at_bot @ real ) @ F4 )
= ( filterlim @ A @ real
@ ^ [X4: A] : ( uminus_uminus @ real @ ( F3 @ X4 ) )
@ ( at_top @ real )
@ F4 ) ) ).
% filterlim_uminus_at_bot
thf(fact_7508_filterlim__at__top__mirror,axiom,
! [A: $tType,F3: real > A,F4: filter @ A] :
( ( filterlim @ real @ A @ F3 @ F4 @ ( at_top @ real ) )
= ( filterlim @ real @ A
@ ^ [X4: real] : ( F3 @ ( uminus_uminus @ real @ X4 ) )
@ F4
@ ( at_bot @ real ) ) ) ).
% filterlim_at_top_mirror
thf(fact_7509_filterlim__at__bot__mirror,axiom,
! [A: $tType,F3: real > A,F4: filter @ A] :
( ( filterlim @ real @ A @ F3 @ F4 @ ( at_bot @ real ) )
= ( filterlim @ real @ A
@ ^ [X4: real] : ( F3 @ ( uminus_uminus @ real @ X4 ) )
@ F4
@ ( at_top @ real ) ) ) ).
% filterlim_at_bot_mirror
thf(fact_7510_filterlim__uminus__at__top__at__bot,axiom,
filterlim @ real @ real @ ( uminus_uminus @ real ) @ ( at_top @ real ) @ ( at_bot @ real ) ).
% filterlim_uminus_at_top_at_bot
thf(fact_7511_filterlim__uminus__at__bot__at__top,axiom,
filterlim @ real @ real @ ( uminus_uminus @ real ) @ ( at_bot @ real ) @ ( at_top @ real ) ).
% filterlim_uminus_at_bot_at_top
thf(fact_7512_filterlim__pow__at__top,axiom,
! [A: $tType,N: nat,F3: A > real,F4: filter @ A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( filterlim @ A @ real @ F3 @ ( at_top @ real ) @ F4 )
=> ( filterlim @ A @ real
@ ^ [X4: A] : ( power_power @ real @ ( F3 @ X4 ) @ N )
@ ( at_top @ real )
@ F4 ) ) ) ).
% filterlim_pow_at_top
thf(fact_7513_tanh__real__at__top,axiom,
filterlim @ real @ real @ ( tanh @ real ) @ ( topolo7230453075368039082e_nhds @ real @ ( one_one @ real ) ) @ ( at_top @ real ) ).
% tanh_real_at_top
thf(fact_7514_tendsto__add__filterlim__at__infinity,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F3: A > B,C2: B,F4: filter @ A,G: A > B] :
( ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ C2 ) @ F4 )
=> ( ( filterlim @ A @ B @ G @ ( at_infinity @ B ) @ F4 )
=> ( filterlim @ A @ B
@ ^ [X4: A] : ( plus_plus @ B @ ( F3 @ X4 ) @ ( G @ X4 ) )
@ ( at_infinity @ B )
@ F4 ) ) ) ) ).
% tendsto_add_filterlim_at_infinity
thf(fact_7515_tendsto__add__filterlim__at__infinity_H,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F3: A > B,F4: filter @ A,G: A > B,C2: B] :
( ( filterlim @ A @ B @ F3 @ ( at_infinity @ B ) @ F4 )
=> ( ( filterlim @ A @ B @ G @ ( topolo7230453075368039082e_nhds @ B @ C2 ) @ F4 )
=> ( filterlim @ A @ B
@ ^ [X4: A] : ( plus_plus @ B @ ( F3 @ X4 ) @ ( G @ X4 ) )
@ ( at_infinity @ B )
@ F4 ) ) ) ) ).
% tendsto_add_filterlim_at_infinity'
thf(fact_7516_real__tendsto__divide__at__top,axiom,
! [A: $tType,F3: A > real,C2: real,F4: filter @ A,G: A > real] :
( ( filterlim @ A @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ C2 ) @ F4 )
=> ( ( filterlim @ A @ real @ G @ ( at_top @ real ) @ F4 )
=> ( filterlim @ A @ real
@ ^ [X4: A] : ( divide_divide @ real @ ( F3 @ X4 ) @ ( G @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ F4 ) ) ) ).
% real_tendsto_divide_at_top
thf(fact_7517_tendsto__inverse__0__at__top,axiom,
! [A: $tType,F3: A > real,F4: filter @ A] :
( ( filterlim @ A @ real @ F3 @ ( at_top @ real ) @ F4 )
=> ( filterlim @ A @ real
@ ^ [X4: A] : ( inverse_inverse @ real @ ( F3 @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ F4 ) ) ).
% tendsto_inverse_0_at_top
thf(fact_7518_filterlim__int__of__nat__at__topD,axiom,
! [A: $tType,F3: int > A,F4: filter @ A] :
( ( filterlim @ nat @ A
@ ^ [X4: nat] : ( F3 @ ( semiring_1_of_nat @ int @ X4 ) )
@ F4
@ ( at_top @ nat ) )
=> ( filterlim @ int @ A @ F3 @ F4 @ ( at_top @ int ) ) ) ).
% filterlim_int_of_nat_at_topD
thf(fact_7519_filterlim__sequentially__iff__filterlim__real,axiom,
! [A: $tType,F3: A > nat,F4: filter @ A] :
( ( filterlim @ A @ nat @ F3 @ ( at_top @ nat ) @ F4 )
= ( filterlim @ A @ real
@ ^ [X4: A] : ( semiring_1_of_nat @ real @ ( F3 @ X4 ) )
@ ( at_top @ real )
@ F4 ) ) ).
% filterlim_sequentially_iff_filterlim_real
thf(fact_7520_filterlim__at__top__mult__tendsto__pos,axiom,
! [A: $tType,F3: A > real,C2: real,F4: filter @ A,G: A > real] :
( ( filterlim @ A @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ C2 ) @ F4 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ C2 )
=> ( ( filterlim @ A @ real @ G @ ( at_top @ real ) @ F4 )
=> ( filterlim @ A @ real
@ ^ [X4: A] : ( times_times @ real @ ( G @ X4 ) @ ( F3 @ X4 ) )
@ ( at_top @ real )
@ F4 ) ) ) ) ).
% filterlim_at_top_mult_tendsto_pos
thf(fact_7521_filterlim__tendsto__pos__mult__at__top,axiom,
! [A: $tType,F3: A > real,C2: real,F4: filter @ A,G: A > real] :
( ( filterlim @ A @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ C2 ) @ F4 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ C2 )
=> ( ( filterlim @ A @ real @ G @ ( at_top @ real ) @ F4 )
=> ( filterlim @ A @ real
@ ^ [X4: A] : ( times_times @ real @ ( F3 @ X4 ) @ ( G @ X4 ) )
@ ( at_top @ real )
@ F4 ) ) ) ) ).
% filterlim_tendsto_pos_mult_at_top
thf(fact_7522_tendsto__neg__powr,axiom,
! [A: $tType,S: real,F3: A > real,F4: filter @ A] :
( ( ord_less @ real @ S @ ( zero_zero @ real ) )
=> ( ( filterlim @ A @ real @ F3 @ ( at_top @ real ) @ F4 )
=> ( filterlim @ A @ real
@ ^ [X4: A] : ( powr @ real @ ( F3 @ X4 ) @ S )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ F4 ) ) ) ).
% tendsto_neg_powr
thf(fact_7523_tendsto__mult__filterlim__at__infinity,axiom,
! [A: $tType,B: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: B > A,C2: A,F4: filter @ B,G: B > A] :
( ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ C2 ) @ F4 )
=> ( ( C2
!= ( zero_zero @ A ) )
=> ( ( filterlim @ B @ A @ G @ ( at_infinity @ A ) @ F4 )
=> ( filterlim @ B @ A
@ ^ [X4: B] : ( times_times @ A @ ( F3 @ X4 ) @ ( G @ X4 ) )
@ ( at_infinity @ A )
@ F4 ) ) ) ) ) ).
% tendsto_mult_filterlim_at_infinity
thf(fact_7524_ln__x__over__x__tendsto__0,axiom,
( filterlim @ real @ real
@ ^ [X4: real] : ( divide_divide @ real @ ( ln_ln @ real @ X4 ) @ X4 )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( at_top @ real ) ) ).
% ln_x_over_x_tendsto_0
thf(fact_7525_tendsto__divide__0,axiom,
! [A: $tType,C: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [F3: C > A,C2: A,F4: filter @ C,G: C > A] :
( ( filterlim @ C @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ C2 ) @ F4 )
=> ( ( filterlim @ C @ A @ G @ ( at_infinity @ A ) @ F4 )
=> ( filterlim @ C @ A
@ ^ [X4: C] : ( divide_divide @ A @ ( F3 @ X4 ) @ ( G @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ F4 ) ) ) ) ).
% tendsto_divide_0
thf(fact_7526_filterlim__at__top__to__right,axiom,
! [A: $tType,F3: real > A,F4: filter @ A] :
( ( filterlim @ real @ A @ F3 @ F4 @ ( at_top @ real ) )
= ( filterlim @ real @ A
@ ^ [X4: real] : ( F3 @ ( inverse_inverse @ real @ X4 ) )
@ F4
@ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) ) ) ).
% filterlim_at_top_to_right
thf(fact_7527_filterlim__at__right__to__top,axiom,
! [A: $tType,F3: real > A,F4: filter @ A] :
( ( filterlim @ real @ A @ F3 @ F4 @ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) )
= ( filterlim @ real @ A
@ ^ [X4: real] : ( F3 @ ( inverse_inverse @ real @ X4 ) )
@ F4
@ ( at_top @ real ) ) ) ).
% filterlim_at_right_to_top
thf(fact_7528_artanh__real__at__left__1,axiom,
filterlim @ real @ real @ ( artanh @ real ) @ ( at_top @ real ) @ ( topolo174197925503356063within @ real @ ( one_one @ real ) @ ( set_ord_lessThan @ real @ ( one_one @ real ) ) ) ).
% artanh_real_at_left_1
thf(fact_7529_filterlim__power__at__infinity,axiom,
! [B: $tType,A: $tType] :
( ( real_V8999393235501362500lgebra @ B )
=> ! [F3: A > B,F4: filter @ A,N: nat] :
( ( filterlim @ A @ B @ F3 @ ( at_infinity @ B ) @ F4 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( filterlim @ A @ B
@ ^ [X4: A] : ( power_power @ B @ ( F3 @ X4 ) @ N )
@ ( at_infinity @ B )
@ F4 ) ) ) ) ).
% filterlim_power_at_infinity
thf(fact_7530_filterlim__inverse__at__top__right,axiom,
filterlim @ real @ real @ ( inverse_inverse @ real ) @ ( at_top @ real ) @ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) ).
% filterlim_inverse_at_top_right
thf(fact_7531_filterlim__inverse__at__right__top,axiom,
filterlim @ real @ real @ ( inverse_inverse @ real ) @ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) @ ( at_top @ real ) ).
% filterlim_inverse_at_right_top
thf(fact_7532_tendsto__at__topI__sequentially,axiom,
! [B: $tType] :
( ( topolo3112930676232923870pology @ B )
=> ! [F3: real > B,Y2: B] :
( ! [X10: nat > real] :
( ( filterlim @ nat @ real @ X10 @ ( at_top @ real ) @ ( at_top @ nat ) )
=> ( filterlim @ nat @ B
@ ^ [N2: nat] : ( F3 @ ( X10 @ N2 ) )
@ ( topolo7230453075368039082e_nhds @ B @ Y2 )
@ ( at_top @ nat ) ) )
=> ( filterlim @ real @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ Y2 ) @ ( at_top @ real ) ) ) ) ).
% tendsto_at_topI_sequentially
thf(fact_7533_filterlim__tendsto__neg__mult__at__bot,axiom,
! [A: $tType,F3: A > real,C2: real,F4: filter @ A,G: A > real] :
( ( filterlim @ A @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ C2 ) @ F4 )
=> ( ( ord_less @ real @ C2 @ ( zero_zero @ real ) )
=> ( ( filterlim @ A @ real @ G @ ( at_top @ real ) @ F4 )
=> ( filterlim @ A @ real
@ ^ [X4: A] : ( times_times @ real @ ( F3 @ X4 ) @ ( G @ X4 ) )
@ ( at_bot @ real )
@ F4 ) ) ) ) ).
% filterlim_tendsto_neg_mult_at_bot
thf(fact_7534_tendsto__power__div__exp__0,axiom,
! [K2: nat] :
( filterlim @ real @ real
@ ^ [X4: real] : ( divide_divide @ real @ ( power_power @ real @ X4 @ K2 ) @ ( exp @ real @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( at_top @ real ) ) ).
% tendsto_power_div_exp_0
thf(fact_7535_lim__infinity__imp__sequentially,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F3: real > A,L: A] :
( ( filterlim @ real @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ ( at_infinity @ real ) )
=> ( filterlim @ nat @ A
@ ^ [N2: nat] : ( F3 @ ( semiring_1_of_nat @ real @ N2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ L )
@ ( at_top @ nat ) ) ) ) ).
% lim_infinity_imp_sequentially
thf(fact_7536_filterlim__inverse__at__iff,axiom,
! [B: $tType,A: $tType] :
( ( real_V8999393235501362500lgebra @ B )
=> ! [G: A > B,F4: filter @ A] :
( ( filterlim @ A @ B
@ ^ [X4: A] : ( inverse_inverse @ B @ ( G @ X4 ) )
@ ( topolo174197925503356063within @ B @ ( zero_zero @ B ) @ ( top_top @ ( set @ B ) ) )
@ F4 )
= ( filterlim @ A @ B @ G @ ( at_infinity @ B ) @ F4 ) ) ) ).
% filterlim_inverse_at_iff
thf(fact_7537_tendsto__exp__limit__at__top,axiom,
! [X: real] :
( filterlim @ real @ real
@ ^ [Y: real] : ( powr @ real @ ( plus_plus @ real @ ( one_one @ real ) @ ( divide_divide @ real @ X @ Y ) ) @ Y )
@ ( topolo7230453075368039082e_nhds @ real @ ( exp @ real @ X ) )
@ ( at_top @ real ) ) ).
% tendsto_exp_limit_at_top
thf(fact_7538_filterlim__divide__at__infinity,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,C2: A,F4: filter @ A,G: A > A] :
( ( filterlim @ A @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ C2 ) @ F4 )
=> ( ( filterlim @ A @ A @ G @ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) @ F4 )
=> ( ( C2
!= ( zero_zero @ A ) )
=> ( filterlim @ A @ A
@ ^ [X4: A] : ( divide_divide @ A @ ( F3 @ X4 ) @ ( G @ X4 ) )
@ ( at_infinity @ A )
@ F4 ) ) ) ) ) ).
% filterlim_divide_at_infinity
thf(fact_7539_DERIV__neg__imp__decreasing__at__top,axiom,
! [B2: real,F3: real > real,Flim: real] :
( ! [X5: real] :
( ( ord_less_eq @ real @ B2 @ X5 )
=> ? [Y4: real] :
( ( has_field_derivative @ real @ F3 @ Y4 @ ( topolo174197925503356063within @ real @ X5 @ ( top_top @ ( set @ real ) ) ) )
& ( ord_less @ real @ Y4 @ ( zero_zero @ real ) ) ) )
=> ( ( filterlim @ real @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ Flim ) @ ( at_top @ real ) )
=> ( ord_less @ real @ Flim @ ( F3 @ B2 ) ) ) ) ).
% DERIV_neg_imp_decreasing_at_top
thf(fact_7540_tendsto__arctan__at__top,axiom,
filterlim @ real @ real @ arctan @ ( topolo7230453075368039082e_nhds @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ ( at_top @ real ) ).
% tendsto_arctan_at_top
thf(fact_7541_filterlim__realpow__sequentially__gt1,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [X: A] :
( ( ord_less @ real @ ( one_one @ real ) @ ( real_V7770717601297561774m_norm @ A @ X ) )
=> ( filterlim @ nat @ A @ ( power_power @ A @ X ) @ ( at_infinity @ A ) @ ( at_top @ nat ) ) ) ) ).
% filterlim_realpow_sequentially_gt1
thf(fact_7542_lim__zero__infinity,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,L: A] :
( ( filterlim @ A @ A
@ ^ [X4: A] : ( F3 @ ( divide_divide @ A @ ( one_one @ A ) @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ A @ L )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) )
=> ( filterlim @ A @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ ( at_infinity @ A ) ) ) ) ).
% lim_zero_infinity
thf(fact_7543_filterlim__tan__at__left,axiom,
filterlim @ real @ real @ ( tan @ real ) @ ( at_top @ real ) @ ( topolo174197925503356063within @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( set_ord_lessThan @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ).
% filterlim_tan_at_left
thf(fact_7544_polyfun__extremal,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [C2: nat > A,K2: nat,N: nat,B5: real] :
( ( ( C2 @ K2 )
!= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ nat @ ( one_one @ nat ) @ K2 )
=> ( ( ord_less_eq @ nat @ K2 @ N )
=> ( eventually @ A
@ ^ [Z6: A] :
( ord_less_eq @ real @ B5
@ ( real_V7770717601297561774m_norm @ A
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I: nat] : ( times_times @ A @ ( C2 @ I ) @ ( power_power @ A @ Z6 @ I ) )
@ ( set_ord_atMost @ nat @ N ) ) ) )
@ ( at_infinity @ A ) ) ) ) ) ) ).
% polyfun_extremal
thf(fact_7545_lhopital__left__at__top,axiom,
! [G: real > real,X: real,G5: real > real,F3: real > real,F9: real > real,Y2: real] :
( ( filterlim @ real @ real @ G @ ( at_top @ real ) @ ( topolo174197925503356063within @ real @ X @ ( set_ord_lessThan @ real @ X ) ) )
=> ( ( eventually @ real
@ ^ [X4: real] :
( ( G5 @ X4 )
!= ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ real @ X @ ( set_ord_lessThan @ real @ X ) ) )
=> ( ( eventually @ real
@ ^ [X4: real] : ( has_field_derivative @ real @ F3 @ ( F9 @ X4 ) @ ( topolo174197925503356063within @ real @ X4 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ X @ ( set_ord_lessThan @ real @ X ) ) )
=> ( ( eventually @ real
@ ^ [X4: real] : ( has_field_derivative @ real @ G @ ( G5 @ X4 ) @ ( topolo174197925503356063within @ real @ X4 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ X @ ( set_ord_lessThan @ real @ X ) ) )
=> ( ( filterlim @ real @ real
@ ^ [X4: real] : ( divide_divide @ real @ ( F9 @ X4 ) @ ( G5 @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ real @ Y2 )
@ ( topolo174197925503356063within @ real @ X @ ( set_ord_lessThan @ real @ X ) ) )
=> ( filterlim @ real @ real
@ ^ [X4: real] : ( divide_divide @ real @ ( F3 @ X4 ) @ ( G @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ real @ Y2 )
@ ( topolo174197925503356063within @ real @ X @ ( set_ord_lessThan @ real @ X ) ) ) ) ) ) ) ) ).
% lhopital_left_at_top
thf(fact_7546_eventually__sequentially__Suc,axiom,
! [P3: nat > $o] :
( ( eventually @ nat
@ ^ [I: nat] : ( P3 @ ( suc @ I ) )
@ ( at_top @ nat ) )
= ( eventually @ nat @ P3 @ ( at_top @ nat ) ) ) ).
% eventually_sequentially_Suc
thf(fact_7547_eventually__sequentially__seg,axiom,
! [P3: nat > $o,K2: nat] :
( ( eventually @ nat
@ ^ [N2: nat] : ( P3 @ ( plus_plus @ nat @ N2 @ K2 ) )
@ ( at_top @ nat ) )
= ( eventually @ nat @ P3 @ ( at_top @ nat ) ) ) ).
% eventually_sequentially_seg
thf(fact_7548_eventually__const,axiom,
! [A: $tType,F4: filter @ A,P3: $o] :
( ( F4
!= ( bot_bot @ ( filter @ A ) ) )
=> ( ( eventually @ A
@ ^ [X4: A] : P3
@ F4 )
= P3 ) ) ).
% eventually_const
thf(fact_7549_eventually__at__bot__not__equal,axiom,
! [A: $tType] :
( ( ( linorder @ A )
& ( no_bot @ A ) )
=> ! [C2: A] :
( eventually @ A
@ ^ [X4: A] : X4 != C2
@ ( at_bot @ A ) ) ) ).
% eventually_at_bot_not_equal
thf(fact_7550_eventually__at__bot__dense,axiom,
! [A: $tType] :
( ( ( linorder @ A )
& ( no_bot @ A ) )
=> ! [P3: A > $o] :
( ( eventually @ A @ P3 @ ( at_bot @ A ) )
= ( ? [N6: A] :
! [N2: A] :
( ( ord_less @ A @ N2 @ N6 )
=> ( P3 @ N2 ) ) ) ) ) ).
% eventually_at_bot_dense
thf(fact_7551_eventually__gt__at__bot,axiom,
! [A: $tType] :
( ( unboun7993243217541854897norder @ A )
=> ! [C2: A] :
( eventually @ A
@ ^ [X4: A] : ( ord_less @ A @ X4 @ C2 )
@ ( at_bot @ A ) ) ) ).
% eventually_gt_at_bot
thf(fact_7552_eventually__le__at__bot,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C2: A] :
( eventually @ A
@ ^ [X4: A] : ( ord_less_eq @ A @ X4 @ C2 )
@ ( at_bot @ A ) ) ) ).
% eventually_le_at_bot
thf(fact_7553_eventually__at__bot__linorder,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P3: A > $o] :
( ( eventually @ A @ P3 @ ( at_bot @ A ) )
= ( ? [N6: A] :
! [N2: A] :
( ( ord_less_eq @ A @ N2 @ N6 )
=> ( P3 @ N2 ) ) ) ) ) ).
% eventually_at_bot_linorder
thf(fact_7554_eventually__at__infinity,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [P3: A > $o] :
( ( eventually @ A @ P3 @ ( at_infinity @ A ) )
= ( ? [B4: real] :
! [X4: A] :
( ( ord_less_eq @ real @ B4 @ ( real_V7770717601297561774m_norm @ A @ X4 ) )
=> ( P3 @ X4 ) ) ) ) ) ).
% eventually_at_infinity
thf(fact_7555_eventually__not__equal__at__infinity,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [A2: A] :
( eventually @ A
@ ^ [X4: A] : X4 != A2
@ ( at_infinity @ A ) ) ) ).
% eventually_not_equal_at_infinity
thf(fact_7556_summable__cong,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: nat > A,G: nat > A] :
( ( eventually @ nat
@ ^ [X4: nat] :
( ( F3 @ X4 )
= ( G @ X4 ) )
@ ( at_top @ nat ) )
=> ( ( summable @ A @ F3 )
= ( summable @ A @ G ) ) ) ) ).
% summable_cong
thf(fact_7557_sequentially__offset,axiom,
! [P3: nat > $o,K2: nat] :
( ( eventually @ nat @ P3 @ ( at_top @ nat ) )
=> ( eventually @ nat
@ ^ [I: nat] : ( P3 @ ( plus_plus @ nat @ I @ K2 ) )
@ ( at_top @ nat ) ) ) ).
% sequentially_offset
thf(fact_7558_eventually__at__top__not__equal,axiom,
! [A: $tType] :
( ( ( linorder @ A )
& ( no_top @ A ) )
=> ! [C2: A] :
( eventually @ A
@ ^ [X4: A] : X4 != C2
@ ( at_top @ A ) ) ) ).
% eventually_at_top_not_equal
thf(fact_7559_eventually__False__sequentially,axiom,
~ ( eventually @ nat
@ ^ [N2: nat] : $false
@ ( at_top @ nat ) ) ).
% eventually_False_sequentially
thf(fact_7560_eventually__at__top__dense,axiom,
! [A: $tType] :
( ( ( linorder @ A )
& ( no_top @ A ) )
=> ! [P3: A > $o] :
( ( eventually @ A @ P3 @ ( at_top @ A ) )
= ( ? [N6: A] :
! [N2: A] :
( ( ord_less @ A @ N6 @ N2 )
=> ( P3 @ N2 ) ) ) ) ) ).
% eventually_at_top_dense
thf(fact_7561_eventually__gt__at__top,axiom,
! [A: $tType] :
( ( ( linorder @ A )
& ( no_top @ A ) )
=> ! [C2: A] : ( eventually @ A @ ( ord_less @ A @ C2 ) @ ( at_top @ A ) ) ) ).
% eventually_gt_at_top
thf(fact_7562_eventually__ge__at__top,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C2: A] : ( eventually @ A @ ( ord_less_eq @ A @ C2 ) @ ( at_top @ A ) ) ) ).
% eventually_ge_at_top
thf(fact_7563_le__sequentially,axiom,
! [F4: filter @ nat] :
( ( ord_less_eq @ ( filter @ nat ) @ F4 @ ( at_top @ nat ) )
= ( ! [N6: nat] : ( eventually @ nat @ ( ord_less_eq @ nat @ N6 ) @ F4 ) ) ) ).
% le_sequentially
thf(fact_7564_eventually__sequentially,axiom,
! [P3: nat > $o] :
( ( eventually @ nat @ P3 @ ( at_top @ nat ) )
= ( ? [N6: nat] :
! [N2: nat] :
( ( ord_less_eq @ nat @ N6 @ N2 )
=> ( P3 @ N2 ) ) ) ) ).
% eventually_sequentially
thf(fact_7565_eventually__sequentiallyI,axiom,
! [C2: nat,P3: nat > $o] :
( ! [X5: nat] :
( ( ord_less_eq @ nat @ C2 @ X5 )
=> ( P3 @ X5 ) )
=> ( eventually @ nat @ P3 @ ( at_top @ nat ) ) ) ).
% eventually_sequentiallyI
thf(fact_7566_eventually__at__top__linorder,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P3: A > $o] :
( ( eventually @ A @ P3 @ ( at_top @ A ) )
= ( ? [N6: A] :
! [N2: A] :
( ( ord_less_eq @ A @ N6 @ N2 )
=> ( P3 @ N2 ) ) ) ) ) ).
% eventually_at_top_linorder
thf(fact_7567_eventually__at__top__linorderI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C2: A,P3: A > $o] :
( ! [X5: A] :
( ( ord_less_eq @ A @ C2 @ X5 )
=> ( P3 @ X5 ) )
=> ( eventually @ A @ P3 @ ( at_top @ A ) ) ) ) ).
% eventually_at_top_linorderI
thf(fact_7568_filterlim__at__top__at__top,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( linorder @ B ) )
=> ! [Q2: A > $o,F3: A > B,P3: B > $o,G: B > A] :
( ! [X5: A,Y7: A] :
( ( Q2 @ X5 )
=> ( ( Q2 @ Y7 )
=> ( ( ord_less_eq @ A @ X5 @ Y7 )
=> ( ord_less_eq @ B @ ( F3 @ X5 ) @ ( F3 @ Y7 ) ) ) ) )
=> ( ! [X5: B] :
( ( P3 @ X5 )
=> ( ( F3 @ ( G @ X5 ) )
= X5 ) )
=> ( ! [X5: B] :
( ( P3 @ X5 )
=> ( Q2 @ ( G @ X5 ) ) )
=> ( ( eventually @ A @ Q2 @ ( at_top @ A ) )
=> ( ( eventually @ B @ P3 @ ( at_top @ B ) )
=> ( filterlim @ A @ B @ F3 @ ( at_top @ B ) @ ( at_top @ A ) ) ) ) ) ) ) ) ).
% filterlim_at_top_at_top
thf(fact_7569_eventually__nhds__iff__sequentially,axiom,
! [A: $tType] :
( ( topolo3112930676232923870pology @ A )
=> ! [P3: A > $o,A2: A] :
( ( eventually @ A @ P3 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) )
= ( ! [F5: nat > A] :
( ( filterlim @ nat @ A @ F5 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ ( at_top @ nat ) )
=> ( eventually @ nat
@ ^ [N2: nat] : ( P3 @ ( F5 @ N2 ) )
@ ( at_top @ nat ) ) ) ) ) ) ).
% eventually_nhds_iff_sequentially
thf(fact_7570_filterlim__at__infinity__imp__eventually__ne,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F3: A > B,F4: filter @ A,C2: B] :
( ( filterlim @ A @ B @ F3 @ ( at_infinity @ B ) @ F4 )
=> ( eventually @ A
@ ^ [Z6: A] :
( ( F3 @ Z6 )
!= C2 )
@ F4 ) ) ) ).
% filterlim_at_infinity_imp_eventually_ne
thf(fact_7571_sequentially__imp__eventually__nhds__within,axiom,
! [A: $tType] :
( ( topolo3112930676232923870pology @ A )
=> ! [S: set @ A,A2: A,P3: A > $o] :
( ! [F6: nat > A] :
( ( ! [N9: nat] : ( member @ A @ ( F6 @ N9 ) @ S )
& ( filterlim @ nat @ A @ F6 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ ( at_top @ nat ) ) )
=> ( eventually @ nat
@ ^ [N2: nat] : ( P3 @ ( F6 @ N2 ) )
@ ( at_top @ nat ) ) )
=> ( eventually @ A @ P3 @ ( inf_inf @ ( filter @ A ) @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ ( principal @ A @ S ) ) ) ) ) ).
% sequentially_imp_eventually_nhds_within
thf(fact_7572_eventually__nhds__within__iff__sequentially,axiom,
! [A: $tType] :
( ( topolo3112930676232923870pology @ A )
=> ! [P3: A > $o,A2: A,S: set @ A] :
( ( eventually @ A @ P3 @ ( inf_inf @ ( filter @ A ) @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ ( principal @ A @ S ) ) )
= ( ! [F5: nat > A] :
( ( ! [N2: nat] : ( member @ A @ ( F5 @ N2 ) @ S )
& ( filterlim @ nat @ A @ F5 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ ( at_top @ nat ) ) )
=> ( eventually @ nat
@ ^ [N2: nat] : ( P3 @ ( F5 @ N2 ) )
@ ( at_top @ nat ) ) ) ) ) ) ).
% eventually_nhds_within_iff_sequentially
thf(fact_7573_sequentially__imp__eventually__within,axiom,
! [A: $tType] :
( ( topolo3112930676232923870pology @ A )
=> ! [S: set @ A,A2: A,P3: A > $o] :
( ! [F6: nat > A] :
( ( ! [N9: nat] :
( ( member @ A @ ( F6 @ N9 ) @ S )
& ( ( F6 @ N9 )
!= A2 ) )
& ( filterlim @ nat @ A @ F6 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ ( at_top @ nat ) ) )
=> ( eventually @ nat
@ ^ [N2: nat] : ( P3 @ ( F6 @ N2 ) )
@ ( at_top @ nat ) ) )
=> ( eventually @ A @ P3 @ ( topolo174197925503356063within @ A @ A2 @ S ) ) ) ) ).
% sequentially_imp_eventually_within
thf(fact_7574_sequentially__imp__eventually__at,axiom,
! [A: $tType] :
( ( topolo3112930676232923870pology @ A )
=> ! [A2: A,P3: A > $o] :
( ! [F6: nat > A] :
( ( ! [N9: nat] :
( ( F6 @ N9 )
!= A2 )
& ( filterlim @ nat @ A @ F6 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ ( at_top @ nat ) ) )
=> ( eventually @ nat
@ ^ [N2: nat] : ( P3 @ ( F6 @ N2 ) )
@ ( at_top @ nat ) ) )
=> ( eventually @ A @ P3 @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% sequentially_imp_eventually_at
thf(fact_7575_filterlim__at__within__not__equal,axiom,
! [B: $tType,A: $tType] :
( ( topological_t2_space @ B )
=> ! [F3: A > B,A2: B,S: set @ B,F4: filter @ A,B2: B] :
( ( filterlim @ A @ B @ F3 @ ( topolo174197925503356063within @ B @ A2 @ S ) @ F4 )
=> ( eventually @ A
@ ^ [W3: A] :
( ( member @ B @ ( F3 @ W3 ) @ S )
& ( ( F3 @ W3 )
!= B2 ) )
@ F4 ) ) ) ).
% filterlim_at_within_not_equal
thf(fact_7576_Lim__transform__eventually,axiom,
! [B: $tType,A: $tType] :
( ( topolo4958980785337419405_space @ B )
=> ! [F3: A > B,L: B,F4: filter @ A,G: A > B] :
( ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ F4 )
=> ( ( eventually @ A
@ ^ [X4: A] :
( ( F3 @ X4 )
= ( G @ X4 ) )
@ F4 )
=> ( filterlim @ A @ B @ G @ ( topolo7230453075368039082e_nhds @ B @ L ) @ F4 ) ) ) ) ).
% Lim_transform_eventually
thf(fact_7577_tendsto__cong,axiom,
! [A: $tType,B: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F3: B > A,G: B > A,F4: filter @ B,C2: A] :
( ( eventually @ B
@ ^ [X4: B] :
( ( F3 @ X4 )
= ( G @ X4 ) )
@ F4 )
=> ( ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ C2 ) @ F4 )
= ( filterlim @ B @ A @ G @ ( topolo7230453075368039082e_nhds @ A @ C2 ) @ F4 ) ) ) ) ).
% tendsto_cong
thf(fact_7578_tendsto__discrete,axiom,
! [A: $tType,B: $tType] :
( ( topolo8865339358273720382pology @ A )
=> ! [F3: B > A,Y2: A,F4: filter @ B] :
( ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ Y2 ) @ F4 )
= ( eventually @ B
@ ^ [X4: B] :
( ( F3 @ X4 )
= Y2 )
@ F4 ) ) ) ).
% tendsto_discrete
thf(fact_7579_tendsto__imp__eventually__ne,axiom,
! [A: $tType,B: $tType] :
( ( topological_t1_space @ A )
=> ! [F3: B > A,C2: A,F4: filter @ B,C9: A] :
( ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ C2 ) @ F4 )
=> ( ( C2 != C9 )
=> ( eventually @ B
@ ^ [Z6: B] :
( ( F3 @ Z6 )
!= C9 )
@ F4 ) ) ) ) ).
% tendsto_imp_eventually_ne
thf(fact_7580_tendsto__eventually,axiom,
! [A: $tType,B: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F3: B > A,L: A,Net: filter @ B] :
( ( eventually @ B
@ ^ [X4: B] :
( ( F3 @ X4 )
= L )
@ Net )
=> ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ Net ) ) ) ).
% tendsto_eventually
thf(fact_7581_filterlim__iff,axiom,
! [B: $tType,A: $tType] :
( ( filterlim @ A @ B )
= ( ^ [F5: A > B,F25: filter @ B,F17: filter @ A] :
! [P2: B > $o] :
( ( eventually @ B @ P2 @ F25 )
=> ( eventually @ A
@ ^ [X4: A] : ( P2 @ ( F5 @ X4 ) )
@ F17 ) ) ) ) ).
% filterlim_iff
thf(fact_7582_filterlim__cong,axiom,
! [A: $tType,B: $tType,F13: filter @ A,F15: filter @ A,F23: filter @ B,F24: filter @ B,F3: B > A,G: B > A] :
( ( F13 = F15 )
=> ( ( F23 = F24 )
=> ( ( eventually @ B
@ ^ [X4: B] :
( ( F3 @ X4 )
= ( G @ X4 ) )
@ F23 )
=> ( ( filterlim @ B @ A @ F3 @ F13 @ F23 )
= ( filterlim @ B @ A @ G @ F15 @ F24 ) ) ) ) ) ).
% filterlim_cong
thf(fact_7583_filterlim__principal,axiom,
! [B: $tType,A: $tType,F3: A > B,S3: set @ B,F4: filter @ A] :
( ( filterlim @ A @ B @ F3 @ ( principal @ B @ S3 ) @ F4 )
= ( eventually @ A
@ ^ [X4: A] : ( member @ B @ ( F3 @ X4 ) @ S3 )
@ F4 ) ) ).
% filterlim_principal
thf(fact_7584_eventually__compose__filterlim,axiom,
! [A: $tType,B: $tType,P3: A > $o,F4: filter @ A,F3: B > A,G6: filter @ B] :
( ( eventually @ A @ P3 @ F4 )
=> ( ( filterlim @ B @ A @ F3 @ F4 @ G6 )
=> ( eventually @ B
@ ^ [X4: B] : ( P3 @ ( F3 @ X4 ) )
@ G6 ) ) ) ).
% eventually_compose_filterlim
thf(fact_7585_filterlim__mono__eventually,axiom,
! [B: $tType,A: $tType,F3: A > B,F4: filter @ B,G6: filter @ A,F14: filter @ B,G7: filter @ A,F9: A > B] :
( ( filterlim @ A @ B @ F3 @ F4 @ G6 )
=> ( ( ord_less_eq @ ( filter @ B ) @ F4 @ F14 )
=> ( ( ord_less_eq @ ( filter @ A ) @ G7 @ G6 )
=> ( ( eventually @ A
@ ^ [X4: A] :
( ( F3 @ X4 )
= ( F9 @ X4 ) )
@ G7 )
=> ( filterlim @ A @ B @ F9 @ F14 @ G7 ) ) ) ) ) ).
% filterlim_mono_eventually
thf(fact_7586_eventually__at__left,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [Y2: A,X: A,P3: A > $o] :
( ( ord_less @ A @ Y2 @ X )
=> ( ( eventually @ A @ P3 @ ( topolo174197925503356063within @ A @ X @ ( set_ord_lessThan @ A @ X ) ) )
= ( ? [B4: A] :
( ( ord_less @ A @ B4 @ X )
& ! [Y: A] :
( ( ord_less @ A @ B4 @ Y )
=> ( ( ord_less @ A @ Y @ X )
=> ( P3 @ Y ) ) ) ) ) ) ) ) ).
% eventually_at_left
thf(fact_7587_eventually__at__left__field,axiom,
! [A: $tType] :
( ( ( linordered_field @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [P3: A > $o,X: A] :
( ( eventually @ A @ P3 @ ( topolo174197925503356063within @ A @ X @ ( set_ord_lessThan @ A @ X ) ) )
= ( ? [B4: A] :
( ( ord_less @ A @ B4 @ X )
& ! [Y: A] :
( ( ord_less @ A @ B4 @ Y )
=> ( ( ord_less @ A @ Y @ X )
=> ( P3 @ Y ) ) ) ) ) ) ) ).
% eventually_at_left_field
thf(fact_7588_le__principal,axiom,
! [A: $tType,F4: filter @ A,A3: set @ A] :
( ( ord_less_eq @ ( filter @ A ) @ F4 @ ( principal @ A @ A3 ) )
= ( eventually @ A
@ ^ [X4: A] : ( member @ A @ X4 @ A3 )
@ F4 ) ) ).
% le_principal
thf(fact_7589_eventually__inf__principal,axiom,
! [A: $tType,P3: A > $o,F4: filter @ A,S: set @ A] :
( ( eventually @ A @ P3 @ ( inf_inf @ ( filter @ A ) @ F4 @ ( principal @ A @ S ) ) )
= ( eventually @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ S )
=> ( P3 @ X4 ) )
@ F4 ) ) ).
% eventually_inf_principal
thf(fact_7590_eventually__nhds__in__open,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [S: set @ A,X: A] :
( ( topolo1002775350975398744n_open @ A @ S )
=> ( ( member @ A @ X @ S )
=> ( eventually @ A
@ ^ [Y: A] : ( member @ A @ Y @ S )
@ ( topolo7230453075368039082e_nhds @ A @ X ) ) ) ) ) ).
% eventually_nhds_in_open
thf(fact_7591_has__derivative__transform__eventually,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F3: A > B,F9: A > B,X: A,S: set @ A,G: A > B] :
( ( has_derivative @ A @ B @ F3 @ F9 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( ( eventually @ A
@ ^ [X7: A] :
( ( F3 @ X7 )
= ( G @ X7 ) )
@ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( ( ( F3 @ X )
= ( G @ X ) )
=> ( ( member @ A @ X @ S )
=> ( has_derivative @ A @ B @ G @ F9 @ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ) ) ).
% has_derivative_transform_eventually
thf(fact_7592_eventually__at__filter,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [P3: A > $o,A2: A,S: set @ A] :
( ( eventually @ A @ P3 @ ( topolo174197925503356063within @ A @ A2 @ S ) )
= ( eventually @ A
@ ^ [X4: A] :
( ( X4 != A2 )
=> ( ( member @ A @ X4 @ S )
=> ( P3 @ X4 ) ) )
@ ( topolo7230453075368039082e_nhds @ A @ A2 ) ) ) ) ).
% eventually_at_filter
thf(fact_7593_t1__space__nhds,axiom,
! [A: $tType] :
( ( topological_t1_space @ A )
=> ! [X: A,Y2: A] :
( ( X != Y2 )
=> ( eventually @ A
@ ^ [X4: A] : X4 != Y2
@ ( topolo7230453075368039082e_nhds @ A @ X ) ) ) ) ).
% t1_space_nhds
thf(fact_7594_eventually__eventually,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [P3: A > $o,X: A] :
( ( eventually @ A
@ ^ [Y: A] : ( eventually @ A @ P3 @ ( topolo7230453075368039082e_nhds @ A @ Y ) )
@ ( topolo7230453075368039082e_nhds @ A @ X ) )
= ( eventually @ A @ P3 @ ( topolo7230453075368039082e_nhds @ A @ X ) ) ) ) ).
% eventually_eventually
thf(fact_7595_eventually__nhds__top,axiom,
! [A: $tType] :
( ( ( order_top @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [B2: A,P3: A > $o] :
( ( ord_less @ A @ B2 @ ( top_top @ A ) )
=> ( ( eventually @ A @ P3 @ ( topolo7230453075368039082e_nhds @ A @ ( top_top @ A ) ) )
= ( ? [B4: A] :
( ( ord_less @ A @ B4 @ ( top_top @ A ) )
& ! [Z6: A] :
( ( ord_less @ A @ B4 @ Z6 )
=> ( P3 @ Z6 ) ) ) ) ) ) ) ).
% eventually_nhds_top
thf(fact_7596_has__field__derivative__cong__eventually,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,G: A > A,X: A,S3: set @ A,U: A] :
( ( eventually @ A
@ ^ [X4: A] :
( ( F3 @ X4 )
= ( G @ X4 ) )
@ ( topolo174197925503356063within @ A @ X @ S3 ) )
=> ( ( ( F3 @ X )
= ( G @ X ) )
=> ( ( has_field_derivative @ A @ F3 @ U @ ( topolo174197925503356063within @ A @ X @ S3 ) )
= ( has_field_derivative @ A @ G @ U @ ( topolo174197925503356063within @ A @ X @ S3 ) ) ) ) ) ) ).
% has_field_derivative_cong_eventually
thf(fact_7597_trivial__limit__def,axiom,
! [A: $tType,F4: filter @ A] :
( ( F4
= ( bot_bot @ ( filter @ A ) ) )
= ( eventually @ A
@ ^ [X4: A] : $false
@ F4 ) ) ).
% trivial_limit_def
thf(fact_7598_eventually__const__iff,axiom,
! [A: $tType,P3: $o,F4: filter @ A] :
( ( eventually @ A
@ ^ [X4: A] : P3
@ F4 )
= ( P3
| ( F4
= ( bot_bot @ ( filter @ A ) ) ) ) ) ).
% eventually_const_iff
thf(fact_7599_False__imp__not__eventually,axiom,
! [A: $tType,P3: A > $o,Net: filter @ A] :
( ! [X5: A] :
~ ( P3 @ X5 )
=> ( ( Net
!= ( bot_bot @ ( filter @ A ) ) )
=> ~ ( eventually @ A @ P3 @ Net ) ) ) ).
% False_imp_not_eventually
thf(fact_7600_eventually__frequently__const__simps_I6_J,axiom,
! [A: $tType,C5: $o,P3: A > $o,F4: filter @ A] :
( ( eventually @ A
@ ^ [X4: A] :
( C5
=> ( P3 @ X4 ) )
@ F4 )
= ( C5
=> ( eventually @ A @ P3 @ F4 ) ) ) ).
% eventually_frequently_const_simps(6)
thf(fact_7601_eventually__frequently__const__simps_I4_J,axiom,
! [A: $tType,C5: $o,P3: A > $o,F4: filter @ A] :
( ( eventually @ A
@ ^ [X4: A] :
( C5
| ( P3 @ X4 ) )
@ F4 )
= ( C5
| ( eventually @ A @ P3 @ F4 ) ) ) ).
% eventually_frequently_const_simps(4)
thf(fact_7602_eventually__frequently__const__simps_I3_J,axiom,
! [A: $tType,P3: A > $o,C5: $o,F4: filter @ A] :
( ( eventually @ A
@ ^ [X4: A] :
( ( P3 @ X4 )
| C5 )
@ F4 )
= ( ( eventually @ A @ P3 @ F4 )
| C5 ) ) ).
% eventually_frequently_const_simps(3)
thf(fact_7603_eventually__mp,axiom,
! [A: $tType,P3: A > $o,Q2: A > $o,F4: filter @ A] :
( ( eventually @ A
@ ^ [X4: A] :
( ( P3 @ X4 )
=> ( Q2 @ X4 ) )
@ F4 )
=> ( ( eventually @ A @ P3 @ F4 )
=> ( eventually @ A @ Q2 @ F4 ) ) ) ).
% eventually_mp
thf(fact_7604_eventually__True,axiom,
! [A: $tType,F4: filter @ A] :
( eventually @ A
@ ^ [X4: A] : $true
@ F4 ) ).
% eventually_True
thf(fact_7605_eventually__conj,axiom,
! [A: $tType,P3: A > $o,F4: filter @ A,Q2: A > $o] :
( ( eventually @ A @ P3 @ F4 )
=> ( ( eventually @ A @ Q2 @ F4 )
=> ( eventually @ A
@ ^ [X4: A] :
( ( P3 @ X4 )
& ( Q2 @ X4 ) )
@ F4 ) ) ) ).
% eventually_conj
thf(fact_7606_eventually__elim2,axiom,
! [A: $tType,P3: A > $o,F4: filter @ A,Q2: A > $o,R4: A > $o] :
( ( eventually @ A @ P3 @ F4 )
=> ( ( eventually @ A @ Q2 @ F4 )
=> ( ! [I3: A] :
( ( P3 @ I3 )
=> ( ( Q2 @ I3 )
=> ( R4 @ I3 ) ) )
=> ( eventually @ A @ R4 @ F4 ) ) ) ) ).
% eventually_elim2
thf(fact_7607_eventually__subst,axiom,
! [A: $tType,P3: A > $o,Q2: A > $o,F4: filter @ A] :
( ( eventually @ A
@ ^ [N2: A] :
( ( P3 @ N2 )
= ( Q2 @ N2 ) )
@ F4 )
=> ( ( eventually @ A @ P3 @ F4 )
= ( eventually @ A @ Q2 @ F4 ) ) ) ).
% eventually_subst
thf(fact_7608_eventually__rev__mp,axiom,
! [A: $tType,P3: A > $o,F4: filter @ A,Q2: A > $o] :
( ( eventually @ A @ P3 @ F4 )
=> ( ( eventually @ A
@ ^ [X4: A] :
( ( P3 @ X4 )
=> ( Q2 @ X4 ) )
@ F4 )
=> ( eventually @ A @ Q2 @ F4 ) ) ) ).
% eventually_rev_mp
thf(fact_7609_eventually__conj__iff,axiom,
! [A: $tType,P3: A > $o,Q2: A > $o,F4: filter @ A] :
( ( eventually @ A
@ ^ [X4: A] :
( ( P3 @ X4 )
& ( Q2 @ X4 ) )
@ F4 )
= ( ( eventually @ A @ P3 @ F4 )
& ( eventually @ A @ Q2 @ F4 ) ) ) ).
% eventually_conj_iff
thf(fact_7610_not__eventually__impI,axiom,
! [A: $tType,P3: A > $o,F4: filter @ A,Q2: A > $o] :
( ( eventually @ A @ P3 @ F4 )
=> ( ~ ( eventually @ A @ Q2 @ F4 )
=> ~ ( eventually @ A
@ ^ [X4: A] :
( ( P3 @ X4 )
=> ( Q2 @ X4 ) )
@ F4 ) ) ) ).
% not_eventually_impI
thf(fact_7611_filter__leD,axiom,
! [A: $tType,F4: filter @ A,F14: filter @ A,P3: A > $o] :
( ( ord_less_eq @ ( filter @ A ) @ F4 @ F14 )
=> ( ( eventually @ A @ P3 @ F14 )
=> ( eventually @ A @ P3 @ F4 ) ) ) ).
% filter_leD
thf(fact_7612_filter__leI,axiom,
! [A: $tType,F14: filter @ A,F4: filter @ A] :
( ! [P9: A > $o] :
( ( eventually @ A @ P9 @ F14 )
=> ( eventually @ A @ P9 @ F4 ) )
=> ( ord_less_eq @ ( filter @ A ) @ F4 @ F14 ) ) ).
% filter_leI
thf(fact_7613_le__filter__def,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( filter @ A ) )
= ( ^ [F10: filter @ A,F11: filter @ A] :
! [P2: A > $o] :
( ( eventually @ A @ P2 @ F11 )
=> ( eventually @ A @ P2 @ F10 ) ) ) ) ).
% le_filter_def
thf(fact_7614_eventually__Lim__ident__at,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [P3: A > A > $o,X: A,X9: set @ A] :
( ( eventually @ A
@ ( P3
@ ( topolo3827282254853284352ce_Lim @ A @ A @ ( topolo174197925503356063within @ A @ X @ X9 )
@ ^ [X4: A] : X4 ) )
@ ( topolo174197925503356063within @ A @ X @ X9 ) )
= ( eventually @ A @ ( P3 @ X ) @ ( topolo174197925503356063within @ A @ X @ X9 ) ) ) ) ).
% eventually_Lim_ident_at
thf(fact_7615_eventually__INF1,axiom,
! [B: $tType,A: $tType,I2: A,I6: set @ A,P3: B > $o,F4: A > ( filter @ B )] :
( ( member @ A @ I2 @ I6 )
=> ( ( eventually @ B @ P3 @ ( F4 @ I2 ) )
=> ( eventually @ B @ P3 @ ( complete_Inf_Inf @ ( filter @ B ) @ ( image @ A @ ( filter @ B ) @ F4 @ I6 ) ) ) ) ) ).
% eventually_INF1
thf(fact_7616_eventually__at__right,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [X: A,Y2: A,P3: A > $o] :
( ( ord_less @ A @ X @ Y2 )
=> ( ( eventually @ A @ P3 @ ( topolo174197925503356063within @ A @ X @ ( set_ord_greaterThan @ A @ X ) ) )
= ( ? [B4: A] :
( ( ord_less @ A @ X @ B4 )
& ! [Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( ( ord_less @ A @ Y @ B4 )
=> ( P3 @ Y ) ) ) ) ) ) ) ) ).
% eventually_at_right
thf(fact_7617_eventually__at__right__field,axiom,
! [A: $tType] :
( ( ( linordered_field @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [P3: A > $o,X: A] :
( ( eventually @ A @ P3 @ ( topolo174197925503356063within @ A @ X @ ( set_ord_greaterThan @ A @ X ) ) )
= ( ? [B4: A] :
( ( ord_less @ A @ X @ B4 )
& ! [Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( ( ord_less @ A @ Y @ B4 )
=> ( P3 @ Y ) ) ) ) ) ) ) ).
% eventually_at_right_field
thf(fact_7618_tendsto__sandwich,axiom,
! [A: $tType,B: $tType] :
( ( topolo2564578578187576103pology @ A )
=> ! [F3: B > A,G: B > A,Net: filter @ B,H2: B > A,C2: A] :
( ( eventually @ B
@ ^ [N2: B] : ( ord_less_eq @ A @ ( F3 @ N2 ) @ ( G @ N2 ) )
@ Net )
=> ( ( eventually @ B
@ ^ [N2: B] : ( ord_less_eq @ A @ ( G @ N2 ) @ ( H2 @ N2 ) )
@ Net )
=> ( ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ C2 ) @ Net )
=> ( ( filterlim @ B @ A @ H2 @ ( topolo7230453075368039082e_nhds @ A @ C2 ) @ Net )
=> ( filterlim @ B @ A @ G @ ( topolo7230453075368039082e_nhds @ A @ C2 ) @ Net ) ) ) ) ) ) ).
% tendsto_sandwich
thf(fact_7619_order__tendstoD_I2_J,axiom,
! [A: $tType,B: $tType] :
( ( topolo2564578578187576103pology @ A )
=> ! [F3: B > A,Y2: A,F4: filter @ B,A2: A] :
( ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ Y2 ) @ F4 )
=> ( ( ord_less @ A @ Y2 @ A2 )
=> ( eventually @ B
@ ^ [X4: B] : ( ord_less @ A @ ( F3 @ X4 ) @ A2 )
@ F4 ) ) ) ) ).
% order_tendstoD(2)
thf(fact_7620_order__tendstoD_I1_J,axiom,
! [A: $tType,B: $tType] :
( ( topolo2564578578187576103pology @ A )
=> ! [F3: B > A,Y2: A,F4: filter @ B,A2: A] :
( ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ Y2 ) @ F4 )
=> ( ( ord_less @ A @ A2 @ Y2 )
=> ( eventually @ B
@ ^ [X4: B] : ( ord_less @ A @ A2 @ ( F3 @ X4 ) )
@ F4 ) ) ) ) ).
% order_tendstoD(1)
thf(fact_7621_order__tendstoI,axiom,
! [A: $tType,B: $tType] :
( ( topolo2564578578187576103pology @ A )
=> ! [Y2: A,F3: B > A,F4: filter @ B] :
( ! [A4: A] :
( ( ord_less @ A @ A4 @ Y2 )
=> ( eventually @ B
@ ^ [X4: B] : ( ord_less @ A @ A4 @ ( F3 @ X4 ) )
@ F4 ) )
=> ( ! [A4: A] :
( ( ord_less @ A @ Y2 @ A4 )
=> ( eventually @ B
@ ^ [X4: B] : ( ord_less @ A @ ( F3 @ X4 ) @ A4 )
@ F4 ) )
=> ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ Y2 ) @ F4 ) ) ) ) ).
% order_tendstoI
thf(fact_7622_order__tendsto__iff,axiom,
! [B: $tType,A: $tType] :
( ( topolo2564578578187576103pology @ A )
=> ! [F3: B > A,X: A,F4: filter @ B] :
( ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ X ) @ F4 )
= ( ! [L3: A] :
( ( ord_less @ A @ L3 @ X )
=> ( eventually @ B
@ ^ [X4: B] : ( ord_less @ A @ L3 @ ( F3 @ X4 ) )
@ F4 ) )
& ! [U2: A] :
( ( ord_less @ A @ X @ U2 )
=> ( eventually @ B
@ ^ [X4: B] : ( ord_less @ A @ ( F3 @ X4 ) @ U2 )
@ F4 ) ) ) ) ) ).
% order_tendsto_iff
thf(fact_7623_filterlim__at__top,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ B )
=> ! [F3: A > B,F4: filter @ A] :
( ( filterlim @ A @ B @ F3 @ ( at_top @ B ) @ F4 )
= ( ! [Z9: B] :
( eventually @ A
@ ^ [X4: A] : ( ord_less_eq @ B @ Z9 @ ( F3 @ X4 ) )
@ F4 ) ) ) ) ).
% filterlim_at_top
thf(fact_7624_filterlim__at__top__ge,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ B )
=> ! [F3: A > B,F4: filter @ A,C2: B] :
( ( filterlim @ A @ B @ F3 @ ( at_top @ B ) @ F4 )
= ( ! [Z9: B] :
( ( ord_less_eq @ B @ C2 @ Z9 )
=> ( eventually @ A
@ ^ [X4: A] : ( ord_less_eq @ B @ Z9 @ ( F3 @ X4 ) )
@ F4 ) ) ) ) ) ).
% filterlim_at_top_ge
thf(fact_7625_filterlim__at__top__mono,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F3: B > A,F4: filter @ B,G: B > A] :
( ( filterlim @ B @ A @ F3 @ ( at_top @ A ) @ F4 )
=> ( ( eventually @ B
@ ^ [X4: B] : ( ord_less_eq @ A @ ( F3 @ X4 ) @ ( G @ X4 ) )
@ F4 )
=> ( filterlim @ B @ A @ G @ ( at_top @ A ) @ F4 ) ) ) ) ).
% filterlim_at_top_mono
thf(fact_7626_filterlim__at__top__dense,axiom,
! [A: $tType,B: $tType] :
( ( unboun7993243217541854897norder @ B )
=> ! [F3: A > B,F4: filter @ A] :
( ( filterlim @ A @ B @ F3 @ ( at_top @ B ) @ F4 )
= ( ! [Z9: B] :
( eventually @ A
@ ^ [X4: A] : ( ord_less @ B @ Z9 @ ( F3 @ X4 ) )
@ F4 ) ) ) ) ).
% filterlim_at_top_dense
thf(fact_7627_filterlim__at,axiom,
! [A: $tType,B: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F3: B > A,B2: A,S: set @ A,F4: filter @ B] :
( ( filterlim @ B @ A @ F3 @ ( topolo174197925503356063within @ A @ B2 @ S ) @ F4 )
= ( ( eventually @ B
@ ^ [X4: B] :
( ( member @ A @ ( F3 @ X4 ) @ S )
& ( ( F3 @ X4 )
!= B2 ) )
@ F4 )
& ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ B2 ) @ F4 ) ) ) ) ).
% filterlim_at
thf(fact_7628_eventually__at__right__less,axiom,
! [A: $tType] :
( ( ( no_top @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [X: A] : ( eventually @ A @ ( ord_less @ A @ X ) @ ( topolo174197925503356063within @ A @ X @ ( set_ord_greaterThan @ A @ X ) ) ) ) ).
% eventually_at_right_less
thf(fact_7629_has__field__derivative__cong__ev,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [X: A,Y2: A,S3: set @ A,F3: A > A,G: A > A,U: A,V2: A,T2: set @ A] :
( ( X = Y2 )
=> ( ( eventually @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ S3 )
=> ( ( F3 @ X4 )
= ( G @ X4 ) ) )
@ ( topolo7230453075368039082e_nhds @ A @ X ) )
=> ( ( U = V2 )
=> ( ( S3 = T2 )
=> ( ( member @ A @ X @ S3 )
=> ( ( has_field_derivative @ A @ F3 @ U @ ( topolo174197925503356063within @ A @ X @ S3 ) )
= ( has_field_derivative @ A @ G @ V2 @ ( topolo174197925503356063within @ A @ Y2 @ T2 ) ) ) ) ) ) ) ) ) ).
% has_field_derivative_cong_ev
thf(fact_7630_topological__tendstoI,axiom,
! [A: $tType,B: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [L: A,F3: B > A,F4: filter @ B] :
( ! [S7: set @ A] :
( ( topolo1002775350975398744n_open @ A @ S7 )
=> ( ( member @ A @ L @ S7 )
=> ( eventually @ B
@ ^ [X4: B] : ( member @ A @ ( F3 @ X4 ) @ S7 )
@ F4 ) ) )
=> ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F4 ) ) ) ).
% topological_tendstoI
thf(fact_7631_topological__tendstoD,axiom,
! [A: $tType,B: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F3: B > A,L: A,F4: filter @ B,S3: set @ A] :
( ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F4 )
=> ( ( topolo1002775350975398744n_open @ A @ S3 )
=> ( ( member @ A @ L @ S3 )
=> ( eventually @ B
@ ^ [X4: B] : ( member @ A @ ( F3 @ X4 ) @ S3 )
@ F4 ) ) ) ) ) ).
% topological_tendstoD
thf(fact_7632_tendsto__def,axiom,
! [B: $tType,A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F3: B > A,L: A,F4: filter @ B] :
( ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F4 )
= ( ! [S6: set @ A] :
( ( topolo1002775350975398744n_open @ A @ S6 )
=> ( ( member @ A @ L @ S6 )
=> ( eventually @ B
@ ^ [X4: B] : ( member @ A @ ( F3 @ X4 ) @ S6 )
@ F4 ) ) ) ) ) ) ).
% tendsto_def
thf(fact_7633_filterlim__at__bot__le,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ B )
=> ! [F3: A > B,F4: filter @ A,C2: B] :
( ( filterlim @ A @ B @ F3 @ ( at_bot @ B ) @ F4 )
= ( ! [Z9: B] :
( ( ord_less_eq @ B @ Z9 @ C2 )
=> ( eventually @ A
@ ^ [X4: A] : ( ord_less_eq @ B @ ( F3 @ X4 ) @ Z9 )
@ F4 ) ) ) ) ) ).
% filterlim_at_bot_le
thf(fact_7634_filterlim__at__bot,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ B )
=> ! [F3: A > B,F4: filter @ A] :
( ( filterlim @ A @ B @ F3 @ ( at_bot @ B ) @ F4 )
= ( ! [Z9: B] :
( eventually @ A
@ ^ [X4: A] : ( ord_less_eq @ B @ ( F3 @ X4 ) @ Z9 )
@ F4 ) ) ) ) ).
% filterlim_at_bot
thf(fact_7635_filterlim__at__bot__dense,axiom,
! [A: $tType,B: $tType] :
( ( ( dense_linorder @ B )
& ( no_bot @ B ) )
=> ! [F3: A > B,F4: filter @ A] :
( ( filterlim @ A @ B @ F3 @ ( at_bot @ B ) @ F4 )
= ( ! [Z9: B] :
( eventually @ A
@ ^ [X4: A] : ( ord_less @ B @ ( F3 @ X4 ) @ Z9 )
@ F4 ) ) ) ) ).
% filterlim_at_bot_dense
thf(fact_7636_real__tendsto__sandwich,axiom,
! [B: $tType,F3: B > real,G: B > real,Net: filter @ B,H2: B > real,C2: real] :
( ( eventually @ B
@ ^ [N2: B] : ( ord_less_eq @ real @ ( F3 @ N2 ) @ ( G @ N2 ) )
@ Net )
=> ( ( eventually @ B
@ ^ [N2: B] : ( ord_less_eq @ real @ ( G @ N2 ) @ ( H2 @ N2 ) )
@ Net )
=> ( ( filterlim @ B @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ C2 ) @ Net )
=> ( ( filterlim @ B @ real @ H2 @ ( topolo7230453075368039082e_nhds @ real @ C2 ) @ Net )
=> ( filterlim @ B @ real @ G @ ( topolo7230453075368039082e_nhds @ real @ C2 ) @ Net ) ) ) ) ) ).
% real_tendsto_sandwich
thf(fact_7637_countable__basis__at__decseq,axiom,
! [A: $tType] :
( ( topolo3112930676232923870pology @ A )
=> ! [X: A] :
~ ! [A8: nat > ( set @ A )] :
( ! [I4: nat] : ( topolo1002775350975398744n_open @ A @ ( A8 @ I4 ) )
=> ( ! [I4: nat] : ( member @ A @ X @ ( A8 @ I4 ) )
=> ~ ! [S10: set @ A] :
( ( topolo1002775350975398744n_open @ A @ S10 )
=> ( ( member @ A @ X @ S10 )
=> ( eventually @ nat
@ ^ [I: nat] : ( ord_less_eq @ ( set @ A ) @ ( A8 @ I ) @ S10 )
@ ( at_top @ nat ) ) ) ) ) ) ) ).
% countable_basis_at_decseq
thf(fact_7638_eventually__Inf__base,axiom,
! [A: $tType,B5: set @ ( filter @ A ),P3: A > $o] :
( ( B5
!= ( bot_bot @ ( set @ ( filter @ A ) ) ) )
=> ( ! [F7: filter @ A] :
( ( member @ ( filter @ A ) @ F7 @ B5 )
=> ! [G3: filter @ A] :
( ( member @ ( filter @ A ) @ G3 @ B5 )
=> ? [X2: filter @ A] :
( ( member @ ( filter @ A ) @ X2 @ B5 )
& ( ord_less_eq @ ( filter @ A ) @ X2 @ ( inf_inf @ ( filter @ A ) @ F7 @ G3 ) ) ) ) )
=> ( ( eventually @ A @ P3 @ ( complete_Inf_Inf @ ( filter @ A ) @ B5 ) )
= ( ? [X4: filter @ A] :
( ( member @ ( filter @ A ) @ X4 @ B5 )
& ( eventually @ A @ P3 @ X4 ) ) ) ) ) ) ).
% eventually_Inf_base
thf(fact_7639_eventually__INF__finite,axiom,
! [B: $tType,A: $tType,A3: set @ A,P3: B > $o,F4: A > ( filter @ B )] :
( ( finite_finite2 @ A @ A3 )
=> ( ( eventually @ B @ P3 @ ( complete_Inf_Inf @ ( filter @ B ) @ ( image @ A @ ( filter @ B ) @ F4 @ A3 ) ) )
= ( ? [Q8: A > B > $o] :
( ! [X4: A] :
( ( member @ A @ X4 @ A3 )
=> ( eventually @ B @ ( Q8 @ X4 ) @ ( F4 @ X4 ) ) )
& ! [Y: B] :
( ! [X4: A] :
( ( member @ A @ X4 @ A3 )
=> ( Q8 @ X4 @ Y ) )
=> ( P3 @ Y ) ) ) ) ) ) ).
% eventually_INF_finite
thf(fact_7640_eventually__at__leftI,axiom,
! [A: $tType] :
( ( topolo2564578578187576103pology @ A )
=> ! [A2: A,B2: A,P3: A > $o] :
( ! [X5: A] :
( ( member @ A @ X5 @ ( set_or5935395276787703475ssThan @ A @ A2 @ B2 ) )
=> ( P3 @ X5 ) )
=> ( ( ord_less @ A @ A2 @ B2 )
=> ( eventually @ A @ P3 @ ( topolo174197925503356063within @ A @ B2 @ ( set_ord_lessThan @ A @ B2 ) ) ) ) ) ) ).
% eventually_at_leftI
thf(fact_7641_eventually__at__rightI,axiom,
! [A: $tType] :
( ( topolo2564578578187576103pology @ A )
=> ! [A2: A,B2: A,P3: A > $o] :
( ! [X5: A] :
( ( member @ A @ X5 @ ( set_or5935395276787703475ssThan @ A @ A2 @ B2 ) )
=> ( P3 @ X5 ) )
=> ( ( ord_less @ A @ A2 @ B2 )
=> ( eventually @ A @ P3 @ ( topolo174197925503356063within @ A @ A2 @ ( set_ord_greaterThan @ A @ A2 ) ) ) ) ) ) ).
% eventually_at_rightI
thf(fact_7642_eventually__at__to__0,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [P3: A > $o,A2: A] :
( ( eventually @ A @ P3 @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) )
= ( eventually @ A
@ ^ [X4: A] : ( P3 @ ( plus_plus @ A @ X4 @ A2 ) )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% eventually_at_to_0
thf(fact_7643_increasing__tendsto,axiom,
! [A: $tType,B: $tType] :
( ( topolo2564578578187576103pology @ A )
=> ! [F3: B > A,L: A,F4: filter @ B] :
( ( eventually @ B
@ ^ [N2: B] : ( ord_less_eq @ A @ ( F3 @ N2 ) @ L )
@ F4 )
=> ( ! [X5: A] :
( ( ord_less @ A @ X5 @ L )
=> ( eventually @ B
@ ^ [N2: B] : ( ord_less @ A @ X5 @ ( F3 @ N2 ) )
@ F4 ) )
=> ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F4 ) ) ) ) ).
% increasing_tendsto
thf(fact_7644_decreasing__tendsto,axiom,
! [A: $tType,B: $tType] :
( ( topolo2564578578187576103pology @ A )
=> ! [L: A,F3: B > A,F4: filter @ B] :
( ( eventually @ B
@ ^ [N2: B] : ( ord_less_eq @ A @ L @ ( F3 @ N2 ) )
@ F4 )
=> ( ! [X5: A] :
( ( ord_less @ A @ L @ X5 )
=> ( eventually @ B
@ ^ [N2: B] : ( ord_less @ A @ ( F3 @ N2 ) @ X5 )
@ F4 ) )
=> ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F4 ) ) ) ) ).
% decreasing_tendsto
thf(fact_7645_filterlim__at__top__gt,axiom,
! [A: $tType,B: $tType] :
( ( unboun7993243217541854897norder @ B )
=> ! [F3: A > B,F4: filter @ A,C2: B] :
( ( filterlim @ A @ B @ F3 @ ( at_top @ B ) @ F4 )
= ( ! [Z9: B] :
( ( ord_less @ B @ C2 @ Z9 )
=> ( eventually @ A
@ ^ [X4: A] : ( ord_less_eq @ B @ Z9 @ ( F3 @ X4 ) )
@ F4 ) ) ) ) ) ).
% filterlim_at_top_gt
thf(fact_7646_tendsto__compose__eventually,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( ( topolo4958980785337419405_space @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [G: A > B,M: B,L: A,F3: C > A,F4: filter @ C] :
( ( filterlim @ A @ B @ G @ ( topolo7230453075368039082e_nhds @ B @ M ) @ ( topolo174197925503356063within @ A @ L @ ( top_top @ ( set @ A ) ) ) )
=> ( ( filterlim @ C @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F4 )
=> ( ( eventually @ C
@ ^ [X4: C] :
( ( F3 @ X4 )
!= L )
@ F4 )
=> ( filterlim @ C @ B
@ ^ [X4: C] : ( G @ ( F3 @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ B @ M )
@ F4 ) ) ) ) ) ).
% tendsto_compose_eventually
thf(fact_7647_LIM__compose__eventually,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ A )
& ( topolo4958980785337419405_space @ B )
& ( topolo4958980785337419405_space @ C ) )
=> ! [F3: A > B,B2: B,A2: A,G: B > C,C2: C] :
( ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ B2 ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) )
=> ( ( filterlim @ B @ C @ G @ ( topolo7230453075368039082e_nhds @ C @ C2 ) @ ( topolo174197925503356063within @ B @ B2 @ ( top_top @ ( set @ B ) ) ) )
=> ( ( eventually @ A
@ ^ [X4: A] :
( ( F3 @ X4 )
!= B2 )
@ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) )
=> ( filterlim @ A @ C
@ ^ [X4: A] : ( G @ ( F3 @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ C @ C2 )
@ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ).
% LIM_compose_eventually
thf(fact_7648_filterlim__atI,axiom,
! [A: $tType,B: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F3: B > A,C2: A,F4: filter @ B] :
( ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ C2 ) @ F4 )
=> ( ( eventually @ B
@ ^ [X4: B] :
( ( F3 @ X4 )
!= C2 )
@ F4 )
=> ( filterlim @ B @ A @ F3 @ ( topolo174197925503356063within @ A @ C2 @ ( top_top @ ( set @ A ) ) ) @ F4 ) ) ) ) ).
% filterlim_atI
thf(fact_7649_isCont__cong,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [F3: A > B,G: A > B,X: A] :
( ( eventually @ A
@ ^ [X4: A] :
( ( F3 @ X4 )
= ( G @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ A @ X ) )
=> ( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) @ F3 )
= ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) @ G ) ) ) ) ).
% isCont_cong
thf(fact_7650_DERIV__cong__ev,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [X: A,Y2: A,F3: A > A,G: A > A,U: A,V2: A] :
( ( X = Y2 )
=> ( ( eventually @ A
@ ^ [X4: A] :
( ( F3 @ X4 )
= ( G @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ A @ X ) )
=> ( ( U = V2 )
=> ( ( has_field_derivative @ A @ F3 @ U @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) )
= ( has_field_derivative @ A @ G @ V2 @ ( topolo174197925503356063within @ A @ Y2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% DERIV_cong_ev
thf(fact_7651_filterlim__at__bot__lt,axiom,
! [A: $tType,B: $tType] :
( ( unboun7993243217541854897norder @ B )
=> ! [F3: A > B,F4: filter @ A,C2: B] :
( ( filterlim @ A @ B @ F3 @ ( at_bot @ B ) @ F4 )
= ( ! [Z9: B] :
( ( ord_less @ B @ Z9 @ C2 )
=> ( eventually @ A
@ ^ [X4: A] : ( ord_less_eq @ B @ ( F3 @ X4 ) @ Z9 )
@ F4 ) ) ) ) ) ).
% filterlim_at_bot_lt
thf(fact_7652_tendsto__upperbound,axiom,
! [B: $tType,A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [F3: B > A,X: A,F4: filter @ B,A2: A] :
( ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ X ) @ F4 )
=> ( ( eventually @ B
@ ^ [I: B] : ( ord_less_eq @ A @ ( F3 @ I ) @ A2 )
@ F4 )
=> ( ( F4
!= ( bot_bot @ ( filter @ B ) ) )
=> ( ord_less_eq @ A @ X @ A2 ) ) ) ) ) ).
% tendsto_upperbound
thf(fact_7653_tendsto__lowerbound,axiom,
! [B: $tType,A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [F3: B > A,X: A,F4: filter @ B,A2: A] :
( ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ X ) @ F4 )
=> ( ( eventually @ B
@ ^ [I: B] : ( ord_less_eq @ A @ A2 @ ( F3 @ I ) )
@ F4 )
=> ( ( F4
!= ( bot_bot @ ( filter @ B ) ) )
=> ( ord_less_eq @ A @ A2 @ X ) ) ) ) ) ).
% tendsto_lowerbound
thf(fact_7654_tendsto__le,axiom,
! [B: $tType,A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [F4: filter @ B,F3: B > A,X: A,G: B > A,Y2: A] :
( ( F4
!= ( bot_bot @ ( filter @ B ) ) )
=> ( ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ X ) @ F4 )
=> ( ( filterlim @ B @ A @ G @ ( topolo7230453075368039082e_nhds @ A @ Y2 ) @ F4 )
=> ( ( eventually @ B
@ ^ [X4: B] : ( ord_less_eq @ A @ ( G @ X4 ) @ ( F3 @ X4 ) )
@ F4 )
=> ( ord_less_eq @ A @ Y2 @ X ) ) ) ) ) ) ).
% tendsto_le
thf(fact_7655_metric__tendsto__imp__tendsto,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( ( real_V7819770556892013058_space @ B )
& ( real_V7819770556892013058_space @ A ) )
=> ! [F3: C > A,A2: A,F4: filter @ C,G: C > B,B2: B] :
( ( filterlim @ C @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ F4 )
=> ( ( eventually @ C
@ ^ [X4: C] : ( ord_less_eq @ real @ ( real_V557655796197034286t_dist @ B @ ( G @ X4 ) @ B2 ) @ ( real_V557655796197034286t_dist @ A @ ( F3 @ X4 ) @ A2 ) )
@ F4 )
=> ( filterlim @ C @ B @ G @ ( topolo7230453075368039082e_nhds @ B @ B2 ) @ F4 ) ) ) ) ).
% metric_tendsto_imp_tendsto
thf(fact_7656_filterlim__at__infinity__imp__filterlim__at__top,axiom,
! [A: $tType,F3: A > real,F4: filter @ A] :
( ( filterlim @ A @ real @ F3 @ ( at_infinity @ real ) @ F4 )
=> ( ( eventually @ A
@ ^ [X4: A] : ( ord_less @ real @ ( zero_zero @ real ) @ ( F3 @ X4 ) )
@ F4 )
=> ( filterlim @ A @ real @ F3 @ ( at_top @ real ) @ F4 ) ) ) ).
% filterlim_at_infinity_imp_filterlim_at_top
thf(fact_7657_filterlim__at__infinity__imp__filterlim__at__bot,axiom,
! [A: $tType,F3: A > real,F4: filter @ A] :
( ( filterlim @ A @ real @ F3 @ ( at_infinity @ real ) @ F4 )
=> ( ( eventually @ A
@ ^ [X4: A] : ( ord_less @ real @ ( F3 @ X4 ) @ ( zero_zero @ real ) )
@ F4 )
=> ( filterlim @ A @ real @ F3 @ ( at_bot @ real ) @ F4 ) ) ) ).
% filterlim_at_infinity_imp_filterlim_at_bot
thf(fact_7658_eventually__floor__eq,axiom,
! [B: $tType,A: $tType] :
( ( ( archim2362893244070406136eiling @ B )
& ( topolo2564578578187576103pology @ B ) )
=> ! [F3: A > B,L: B,F4: filter @ A] :
( ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ F4 )
=> ( ~ ( member @ B @ L @ ( ring_1_Ints @ B ) )
=> ( eventually @ A
@ ^ [X4: A] :
( ( archim6421214686448440834_floor @ B @ ( F3 @ X4 ) )
= ( archim6421214686448440834_floor @ B @ L ) )
@ F4 ) ) ) ) ).
% eventually_floor_eq
thf(fact_7659_eventually__ceiling__eq,axiom,
! [B: $tType,A: $tType] :
( ( ( archim2362893244070406136eiling @ B )
& ( topolo2564578578187576103pology @ B ) )
=> ! [F3: A > B,L: B,F4: filter @ A] :
( ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ F4 )
=> ( ~ ( member @ B @ L @ ( ring_1_Ints @ B ) )
=> ( eventually @ A
@ ^ [X4: A] :
( ( archimedean_ceiling @ B @ ( F3 @ X4 ) )
= ( archimedean_ceiling @ B @ L ) )
@ F4 ) ) ) ) ).
% eventually_ceiling_eq
thf(fact_7660_eventually__at__right__to__0,axiom,
! [P3: real > $o,A2: real] :
( ( eventually @ real @ P3 @ ( topolo174197925503356063within @ real @ A2 @ ( set_ord_greaterThan @ real @ A2 ) ) )
= ( eventually @ real
@ ^ [X4: real] : ( P3 @ ( plus_plus @ real @ X4 @ A2 ) )
@ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) ) ) ).
% eventually_at_right_to_0
thf(fact_7661_eventually__INF,axiom,
! [A: $tType,B: $tType,P3: A > $o,F4: B > ( filter @ A ),B5: set @ B] :
( ( eventually @ A @ P3 @ ( complete_Inf_Inf @ ( filter @ A ) @ ( image @ B @ ( filter @ A ) @ F4 @ B5 ) ) )
= ( ? [X6: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ X6 @ B5 )
& ( finite_finite2 @ B @ X6 )
& ( eventually @ A @ P3 @ ( complete_Inf_Inf @ ( filter @ A ) @ ( image @ B @ ( filter @ A ) @ F4 @ X6 ) ) ) ) ) ) ).
% eventually_INF
thf(fact_7662_eventually__at__left__to__right,axiom,
! [P3: real > $o,A2: real] :
( ( eventually @ real @ P3 @ ( topolo174197925503356063within @ real @ A2 @ ( set_ord_lessThan @ real @ A2 ) ) )
= ( eventually @ real
@ ^ [X4: real] : ( P3 @ ( uminus_uminus @ real @ X4 ) )
@ ( topolo174197925503356063within @ real @ ( uminus_uminus @ real @ A2 ) @ ( set_ord_greaterThan @ real @ ( uminus_uminus @ real @ A2 ) ) ) ) ) ).
% eventually_at_left_to_right
thf(fact_7663_continuous__arcosh__strong,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [F4: filter @ A,F3: A > real] :
( ( topolo3448309680560233919inuous @ A @ real @ F4 @ F3 )
=> ( ( eventually @ A
@ ^ [X4: A] : ( ord_less_eq @ real @ ( one_one @ real ) @ ( F3 @ X4 ) )
@ F4 )
=> ( topolo3448309680560233919inuous @ A @ real @ F4
@ ^ [X4: A] : ( arcosh @ real @ ( F3 @ X4 ) ) ) ) ) ) ).
% continuous_arcosh_strong
thf(fact_7664_eventually__at__right__real,axiom,
! [A2: real,B2: real] :
( ( ord_less @ real @ A2 @ B2 )
=> ( eventually @ real
@ ^ [X4: real] : ( member @ real @ X4 @ ( set_or5935395276787703475ssThan @ real @ A2 @ B2 ) )
@ ( topolo174197925503356063within @ real @ A2 @ ( set_ord_greaterThan @ real @ A2 ) ) ) ) ).
% eventually_at_right_real
thf(fact_7665_eventually__at__left__real,axiom,
! [B2: real,A2: real] :
( ( ord_less @ real @ B2 @ A2 )
=> ( eventually @ real
@ ^ [X4: real] : ( member @ real @ X4 @ ( set_or5935395276787703475ssThan @ real @ B2 @ A2 ) )
@ ( topolo174197925503356063within @ real @ A2 @ ( set_ord_lessThan @ real @ A2 ) ) ) ) ).
% eventually_at_left_real
thf(fact_7666_eventually__at__le,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [P3: A > $o,A2: A,S3: set @ A] :
( ( eventually @ A @ P3 @ ( topolo174197925503356063within @ A @ A2 @ S3 ) )
= ( ? [D5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D5 )
& ! [X4: A] :
( ( member @ A @ X4 @ S3 )
=> ( ( ( X4 != A2 )
& ( ord_less_eq @ real @ ( real_V557655796197034286t_dist @ A @ X4 @ A2 ) @ D5 ) )
=> ( P3 @ X4 ) ) ) ) ) ) ) ).
% eventually_at_le
thf(fact_7667_eventually__at__infinity__pos,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [P5: A > $o] :
( ( eventually @ A @ P5 @ ( at_infinity @ A ) )
= ( ? [B4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ B4 )
& ! [X4: A] :
( ( ord_less_eq @ real @ B4 @ ( real_V7770717601297561774m_norm @ A @ X4 ) )
=> ( P5 @ X4 ) ) ) ) ) ) ).
% eventually_at_infinity_pos
thf(fact_7668_tendsto__compose__at,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ B )
& ( topolo4958980785337419405_space @ C ) )
=> ! [F3: A > B,Y2: B,F4: filter @ A,G: B > C,Z3: C] :
( ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ Y2 ) @ F4 )
=> ( ( filterlim @ B @ C @ G @ ( topolo7230453075368039082e_nhds @ C @ Z3 ) @ ( topolo174197925503356063within @ B @ Y2 @ ( top_top @ ( set @ B ) ) ) )
=> ( ( eventually @ A
@ ^ [W3: A] :
( ( ( F3 @ W3 )
= Y2 )
=> ( ( G @ Y2 )
= Z3 ) )
@ F4 )
=> ( filterlim @ A @ C @ ( comp @ B @ C @ A @ G @ F3 ) @ ( topolo7230453075368039082e_nhds @ C @ Z3 ) @ F4 ) ) ) ) ) ).
% tendsto_compose_at
thf(fact_7669_tendsto__imp__filterlim__at__left,axiom,
! [B: $tType,A: $tType] :
( ( topolo2564578578187576103pology @ B )
=> ! [F3: A > B,L6: B,F4: filter @ A] :
( ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ L6 ) @ F4 )
=> ( ( eventually @ A
@ ^ [X4: A] : ( ord_less @ B @ ( F3 @ X4 ) @ L6 )
@ F4 )
=> ( filterlim @ A @ B @ F3 @ ( topolo174197925503356063within @ B @ L6 @ ( set_ord_lessThan @ B @ L6 ) ) @ F4 ) ) ) ) ).
% tendsto_imp_filterlim_at_left
thf(fact_7670_tendsto__imp__filterlim__at__right,axiom,
! [B: $tType,A: $tType] :
( ( topolo2564578578187576103pology @ B )
=> ! [F3: A > B,L6: B,F4: filter @ A] :
( ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ L6 ) @ F4 )
=> ( ( eventually @ A
@ ^ [X4: A] : ( ord_less @ B @ L6 @ ( F3 @ X4 ) )
@ F4 )
=> ( filterlim @ A @ B @ F3 @ ( topolo174197925503356063within @ B @ L6 @ ( set_ord_greaterThan @ B @ L6 ) ) @ F4 ) ) ) ) ).
% tendsto_imp_filterlim_at_right
thf(fact_7671_tendstoD,axiom,
! [A: $tType,B: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [F3: B > A,L: A,F4: filter @ B,E: real] :
( ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F4 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ E )
=> ( eventually @ B
@ ^ [X4: B] : ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( F3 @ X4 ) @ L ) @ E )
@ F4 ) ) ) ) ).
% tendstoD
thf(fact_7672_tendstoI,axiom,
! [A: $tType,B: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [F3: B > A,L: A,F4: filter @ B] :
( ! [E2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E2 )
=> ( eventually @ B
@ ^ [X4: B] : ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( F3 @ X4 ) @ L ) @ E2 )
@ F4 ) )
=> ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F4 ) ) ) ).
% tendstoI
thf(fact_7673_tendsto__iff,axiom,
! [B: $tType,A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [F3: B > A,L: A,F4: filter @ B] :
( ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F4 )
= ( ! [E3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E3 )
=> ( eventually @ B
@ ^ [X4: B] : ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( F3 @ X4 ) @ L ) @ E3 )
@ F4 ) ) ) ) ) ).
% tendsto_iff
thf(fact_7674_eventually__Inf,axiom,
! [A: $tType,P3: A > $o,B5: set @ ( filter @ A )] :
( ( eventually @ A @ P3 @ ( complete_Inf_Inf @ ( filter @ A ) @ B5 ) )
= ( ? [X6: set @ ( filter @ A )] :
( ( ord_less_eq @ ( set @ ( filter @ A ) ) @ X6 @ B5 )
& ( finite_finite2 @ ( filter @ A ) @ X6 )
& ( eventually @ A @ P3 @ ( complete_Inf_Inf @ ( filter @ A ) @ X6 ) ) ) ) ) ).
% eventually_Inf
thf(fact_7675_summable__comparison__test__ev,axiom,
! [A: $tType] :
( ( real_Vector_banach @ A )
=> ! [F3: nat > A,G: nat > real] :
( ( eventually @ nat
@ ^ [N2: nat] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( F3 @ N2 ) ) @ ( G @ N2 ) )
@ ( at_top @ nat ) )
=> ( ( summable @ real @ G )
=> ( summable @ A @ F3 ) ) ) ) ).
% summable_comparison_test_ev
thf(fact_7676_eventually__at__top__to__right,axiom,
! [P3: real > $o] :
( ( eventually @ real @ P3 @ ( at_top @ real ) )
= ( eventually @ real
@ ^ [X4: real] : ( P3 @ ( inverse_inverse @ real @ X4 ) )
@ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) ) ) ).
% eventually_at_top_to_right
thf(fact_7677_eventually__at__right__to__top,axiom,
! [P3: real > $o] :
( ( eventually @ real @ P3 @ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) )
= ( eventually @ real
@ ^ [X4: real] : ( P3 @ ( inverse_inverse @ real @ X4 ) )
@ ( at_top @ real ) ) ) ).
% eventually_at_right_to_top
thf(fact_7678_tendsto__arcosh__strong,axiom,
! [B: $tType,F3: B > real,A2: real,F4: filter @ B] :
( ( filterlim @ B @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ A2 ) @ F4 )
=> ( ( ord_less_eq @ real @ ( one_one @ real ) @ A2 )
=> ( ( eventually @ B
@ ^ [X4: B] : ( ord_less_eq @ real @ ( one_one @ real ) @ ( F3 @ X4 ) )
@ F4 )
=> ( filterlim @ B @ real
@ ^ [X4: B] : ( arcosh @ real @ ( F3 @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( arcosh @ real @ A2 ) )
@ F4 ) ) ) ) ).
% tendsto_arcosh_strong
thf(fact_7679_filterlim__at__top__at__left,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo1944317154257567458pology @ A )
& ( linorder @ B ) )
=> ! [Q2: A > $o,F3: A > B,P3: B > $o,G: B > A,A2: A] :
( ! [X5: A,Y7: A] :
( ( Q2 @ X5 )
=> ( ( Q2 @ Y7 )
=> ( ( ord_less_eq @ A @ X5 @ Y7 )
=> ( ord_less_eq @ B @ ( F3 @ X5 ) @ ( F3 @ Y7 ) ) ) ) )
=> ( ! [X5: B] :
( ( P3 @ X5 )
=> ( ( F3 @ ( G @ X5 ) )
= X5 ) )
=> ( ! [X5: B] :
( ( P3 @ X5 )
=> ( Q2 @ ( G @ X5 ) ) )
=> ( ( eventually @ A @ Q2 @ ( topolo174197925503356063within @ A @ A2 @ ( set_ord_lessThan @ A @ A2 ) ) )
=> ( ! [B3: A] :
( ( Q2 @ B3 )
=> ( ord_less @ A @ B3 @ A2 ) )
=> ( ( eventually @ B @ P3 @ ( at_top @ B ) )
=> ( filterlim @ A @ B @ F3 @ ( at_top @ B ) @ ( topolo174197925503356063within @ A @ A2 @ ( set_ord_lessThan @ A @ A2 ) ) ) ) ) ) ) ) ) ) ).
% filterlim_at_top_at_left
thf(fact_7680_eventually__INF__base,axiom,
! [B: $tType,A: $tType,B5: set @ A,F4: A > ( filter @ B ),P3: B > $o] :
( ( B5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [A4: A] :
( ( member @ A @ A4 @ B5 )
=> ! [B3: A] :
( ( member @ A @ B3 @ B5 )
=> ? [X2: A] :
( ( member @ A @ X2 @ B5 )
& ( ord_less_eq @ ( filter @ B ) @ ( F4 @ X2 ) @ ( inf_inf @ ( filter @ B ) @ ( F4 @ A4 ) @ ( F4 @ B3 ) ) ) ) ) )
=> ( ( eventually @ B @ P3 @ ( complete_Inf_Inf @ ( filter @ B ) @ ( image @ A @ ( filter @ B ) @ F4 @ B5 ) ) )
= ( ? [X4: A] :
( ( member @ A @ X4 @ B5 )
& ( eventually @ B @ P3 @ ( F4 @ X4 ) ) ) ) ) ) ) ).
% eventually_INF_base
thf(fact_7681_filterlim__at__bot__at__right,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo1944317154257567458pology @ A )
& ( linorder @ B ) )
=> ! [Q2: A > $o,F3: A > B,P3: B > $o,G: B > A,A2: A] :
( ! [X5: A,Y7: A] :
( ( Q2 @ X5 )
=> ( ( Q2 @ Y7 )
=> ( ( ord_less_eq @ A @ X5 @ Y7 )
=> ( ord_less_eq @ B @ ( F3 @ X5 ) @ ( F3 @ Y7 ) ) ) ) )
=> ( ! [X5: B] :
( ( P3 @ X5 )
=> ( ( F3 @ ( G @ X5 ) )
= X5 ) )
=> ( ! [X5: B] :
( ( P3 @ X5 )
=> ( Q2 @ ( G @ X5 ) ) )
=> ( ( eventually @ A @ Q2 @ ( topolo174197925503356063within @ A @ A2 @ ( set_ord_greaterThan @ A @ A2 ) ) )
=> ( ! [B3: A] :
( ( Q2 @ B3 )
=> ( ord_less @ A @ A2 @ B3 ) )
=> ( ( eventually @ B @ P3 @ ( at_bot @ B ) )
=> ( filterlim @ A @ B @ F3 @ ( at_bot @ B ) @ ( topolo174197925503356063within @ A @ A2 @ ( set_ord_greaterThan @ A @ A2 ) ) ) ) ) ) ) ) ) ) ).
% filterlim_at_bot_at_right
thf(fact_7682_tendsto__0__le,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V822414075346904944vector @ B ) )
=> ! [F3: A > B,F4: filter @ A,G: A > C,K5: real] :
( ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) @ F4 )
=> ( ( eventually @ A
@ ^ [X4: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ C @ ( G @ X4 ) ) @ ( times_times @ real @ ( real_V7770717601297561774m_norm @ B @ ( F3 @ X4 ) ) @ K5 ) )
@ F4 )
=> ( filterlim @ A @ C @ G @ ( topolo7230453075368039082e_nhds @ C @ ( zero_zero @ C ) ) @ F4 ) ) ) ) ).
% tendsto_0_le
thf(fact_7683_filterlim__at__withinI,axiom,
! [A: $tType,B: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F3: B > A,C2: A,F4: filter @ B,A3: set @ A] :
( ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ C2 ) @ F4 )
=> ( ( eventually @ B
@ ^ [X4: B] : ( member @ A @ ( F3 @ X4 ) @ ( minus_minus @ ( set @ A ) @ A3 @ ( insert @ A @ C2 @ ( bot_bot @ ( set @ A ) ) ) ) )
@ F4 )
=> ( filterlim @ B @ A @ F3 @ ( topolo174197925503356063within @ A @ C2 @ A3 ) @ F4 ) ) ) ) ).
% filterlim_at_withinI
thf(fact_7684_filterlim__at__infinity,axiom,
! [C: $tType,A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [C2: real,F3: C > A,F4: filter @ C] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ C2 )
=> ( ( filterlim @ C @ A @ F3 @ ( at_infinity @ A ) @ F4 )
= ( ! [R5: real] :
( ( ord_less @ real @ C2 @ R5 )
=> ( eventually @ C
@ ^ [X4: C] : ( ord_less_eq @ real @ R5 @ ( real_V7770717601297561774m_norm @ A @ ( F3 @ X4 ) ) )
@ F4 ) ) ) ) ) ) ).
% filterlim_at_infinity
thf(fact_7685_tendsto__zero__powrI,axiom,
! [A: $tType,F3: A > real,F4: filter @ A,G: A > real,B2: real] :
( ( filterlim @ A @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ F4 )
=> ( ( filterlim @ A @ real @ G @ ( topolo7230453075368039082e_nhds @ real @ B2 ) @ F4 )
=> ( ( eventually @ A
@ ^ [X4: A] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( F3 @ X4 ) )
@ F4 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ B2 )
=> ( filterlim @ A @ real
@ ^ [X4: A] : ( powr @ real @ ( F3 @ X4 ) @ ( G @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ F4 ) ) ) ) ) ).
% tendsto_zero_powrI
thf(fact_7686_tendsto__powr2,axiom,
! [A: $tType,F3: A > real,A2: real,F4: filter @ A,G: A > real,B2: real] :
( ( filterlim @ A @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ A2 ) @ F4 )
=> ( ( filterlim @ A @ real @ G @ ( topolo7230453075368039082e_nhds @ real @ B2 ) @ F4 )
=> ( ( eventually @ A
@ ^ [X4: A] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( F3 @ X4 ) )
@ F4 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ B2 )
=> ( filterlim @ A @ real
@ ^ [X4: A] : ( powr @ real @ ( F3 @ X4 ) @ ( G @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( powr @ real @ A2 @ B2 ) )
@ F4 ) ) ) ) ) ).
% tendsto_powr2
thf(fact_7687_tendsto__powr_H,axiom,
! [A: $tType,F3: A > real,A2: real,F4: filter @ A,G: A > real,B2: real] :
( ( filterlim @ A @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ A2 ) @ F4 )
=> ( ( filterlim @ A @ real @ G @ ( topolo7230453075368039082e_nhds @ real @ B2 ) @ F4 )
=> ( ( ( A2
!= ( zero_zero @ real ) )
| ( ( ord_less @ real @ ( zero_zero @ real ) @ B2 )
& ( eventually @ A
@ ^ [X4: A] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( F3 @ X4 ) )
@ F4 ) ) )
=> ( filterlim @ A @ real
@ ^ [X4: A] : ( powr @ real @ ( F3 @ X4 ) @ ( G @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( powr @ real @ A2 @ B2 ) )
@ F4 ) ) ) ) ).
% tendsto_powr'
thf(fact_7688_eventually__floor__less,axiom,
! [B: $tType,A: $tType] :
( ( ( archim2362893244070406136eiling @ B )
& ( topolo2564578578187576103pology @ B ) )
=> ! [F3: A > B,L: B,F4: filter @ A] :
( ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ F4 )
=> ( ~ ( member @ B @ L @ ( ring_1_Ints @ B ) )
=> ( eventually @ A
@ ^ [X4: A] : ( ord_less @ B @ ( ring_1_of_int @ B @ ( archim6421214686448440834_floor @ B @ L ) ) @ ( F3 @ X4 ) )
@ F4 ) ) ) ) ).
% eventually_floor_less
thf(fact_7689_LIM__at__top__divide,axiom,
! [A: $tType,F3: A > real,A2: real,F4: filter @ A,G: A > real] :
( ( filterlim @ A @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ A2 ) @ F4 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
=> ( ( filterlim @ A @ real @ G @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ F4 )
=> ( ( eventually @ A
@ ^ [X4: A] : ( ord_less @ real @ ( zero_zero @ real ) @ ( G @ X4 ) )
@ F4 )
=> ( filterlim @ A @ real
@ ^ [X4: A] : ( divide_divide @ real @ ( F3 @ X4 ) @ ( G @ X4 ) )
@ ( at_top @ real )
@ F4 ) ) ) ) ) ).
% LIM_at_top_divide
thf(fact_7690_eventually__less__ceiling,axiom,
! [B: $tType,A: $tType] :
( ( ( archim2362893244070406136eiling @ B )
& ( topolo2564578578187576103pology @ B ) )
=> ! [F3: A > B,L: B,F4: filter @ A] :
( ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ F4 )
=> ( ~ ( member @ B @ L @ ( ring_1_Ints @ B ) )
=> ( eventually @ A
@ ^ [X4: A] : ( ord_less @ B @ ( F3 @ X4 ) @ ( ring_1_of_int @ B @ ( archimedean_ceiling @ B @ L ) ) )
@ F4 ) ) ) ) ).
% eventually_less_ceiling
thf(fact_7691_filterlim__inverse__at__top__iff,axiom,
! [A: $tType,F3: A > real,F4: filter @ A] :
( ( eventually @ A
@ ^ [X4: A] : ( ord_less @ real @ ( zero_zero @ real ) @ ( F3 @ X4 ) )
@ F4 )
=> ( ( filterlim @ A @ real
@ ^ [X4: A] : ( inverse_inverse @ real @ ( F3 @ X4 ) )
@ ( at_top @ real )
@ F4 )
= ( filterlim @ A @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ F4 ) ) ) ).
% filterlim_inverse_at_top_iff
thf(fact_7692_filterlim__inverse__at__top,axiom,
! [A: $tType,F3: A > real,F4: filter @ A] :
( ( filterlim @ A @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ F4 )
=> ( ( eventually @ A
@ ^ [X4: A] : ( ord_less @ real @ ( zero_zero @ real ) @ ( F3 @ X4 ) )
@ F4 )
=> ( filterlim @ A @ real
@ ^ [X4: A] : ( inverse_inverse @ real @ ( F3 @ X4 ) )
@ ( at_top @ real )
@ F4 ) ) ) ).
% filterlim_inverse_at_top
thf(fact_7693_filterlim__at__top__iff__inverse__0,axiom,
! [A: $tType,F3: A > real,F4: filter @ A] :
( ( eventually @ A
@ ^ [X4: A] : ( ord_less @ real @ ( zero_zero @ real ) @ ( F3 @ X4 ) )
@ F4 )
=> ( ( filterlim @ A @ real @ F3 @ ( at_top @ real ) @ F4 )
= ( filterlim @ A @ real @ ( comp @ real @ real @ A @ ( inverse_inverse @ real ) @ F3 ) @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ F4 ) ) ) ).
% filterlim_at_top_iff_inverse_0
thf(fact_7694_filterlim__inverse__at__bot,axiom,
! [A: $tType,F3: A > real,F4: filter @ A] :
( ( filterlim @ A @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ F4 )
=> ( ( eventually @ A
@ ^ [X4: A] : ( ord_less @ real @ ( F3 @ X4 ) @ ( zero_zero @ real ) )
@ F4 )
=> ( filterlim @ A @ real
@ ^ [X4: A] : ( inverse_inverse @ real @ ( F3 @ X4 ) )
@ ( at_bot @ real )
@ F4 ) ) ) ).
% filterlim_inverse_at_bot
thf(fact_7695_lhopital__at__top__at__top,axiom,
! [F3: real > real,A2: real,G: real > real,F9: real > real,G5: real > real] :
( ( filterlim @ real @ real @ F3 @ ( at_top @ real ) @ ( topolo174197925503356063within @ real @ A2 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( filterlim @ real @ real @ G @ ( at_top @ real ) @ ( topolo174197925503356063within @ real @ A2 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( eventually @ real
@ ^ [X4: real] : ( has_field_derivative @ real @ F3 @ ( F9 @ X4 ) @ ( topolo174197925503356063within @ real @ X4 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ A2 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( eventually @ real
@ ^ [X4: real] : ( has_field_derivative @ real @ G @ ( G5 @ X4 ) @ ( topolo174197925503356063within @ real @ X4 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ A2 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( filterlim @ real @ real
@ ^ [X4: real] : ( divide_divide @ real @ ( F9 @ X4 ) @ ( G5 @ X4 ) )
@ ( at_top @ real )
@ ( topolo174197925503356063within @ real @ A2 @ ( top_top @ ( set @ real ) ) ) )
=> ( filterlim @ real @ real
@ ^ [X4: real] : ( divide_divide @ real @ ( F3 @ X4 ) @ ( G @ X4 ) )
@ ( at_top @ real )
@ ( topolo174197925503356063within @ real @ A2 @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ) ) ).
% lhopital_at_top_at_top
thf(fact_7696_lhopital,axiom,
! [F3: real > real,X: real,G: real > real,G5: real > real,F9: real > real,F4: filter @ real] :
( ( filterlim @ real @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( ( filterlim @ real @ real @ G @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( ( eventually @ real
@ ^ [X4: real] :
( ( G @ X4 )
!= ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( ( eventually @ real
@ ^ [X4: real] :
( ( G5 @ X4 )
!= ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( ( eventually @ real
@ ^ [X4: real] : ( has_field_derivative @ real @ F3 @ ( F9 @ X4 ) @ ( topolo174197925503356063within @ real @ X4 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( ( eventually @ real
@ ^ [X4: real] : ( has_field_derivative @ real @ G @ ( G5 @ X4 ) @ ( topolo174197925503356063within @ real @ X4 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( ( filterlim @ real @ real
@ ^ [X4: real] : ( divide_divide @ real @ ( F9 @ X4 ) @ ( G5 @ X4 ) )
@ F4
@ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( filterlim @ real @ real
@ ^ [X4: real] : ( divide_divide @ real @ ( F3 @ X4 ) @ ( G @ X4 ) )
@ F4
@ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ) ) ) ) ).
% lhopital
thf(fact_7697_lhopital__right__at__top__at__top,axiom,
! [F3: real > real,A2: real,G: real > real,F9: real > real,G5: real > real] :
( ( filterlim @ real @ real @ F3 @ ( at_top @ real ) @ ( topolo174197925503356063within @ real @ A2 @ ( set_ord_greaterThan @ real @ A2 ) ) )
=> ( ( filterlim @ real @ real @ G @ ( at_top @ real ) @ ( topolo174197925503356063within @ real @ A2 @ ( set_ord_greaterThan @ real @ A2 ) ) )
=> ( ( eventually @ real
@ ^ [X4: real] : ( has_field_derivative @ real @ F3 @ ( F9 @ X4 ) @ ( topolo174197925503356063within @ real @ X4 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ A2 @ ( set_ord_greaterThan @ real @ A2 ) ) )
=> ( ( eventually @ real
@ ^ [X4: real] : ( has_field_derivative @ real @ G @ ( G5 @ X4 ) @ ( topolo174197925503356063within @ real @ X4 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ A2 @ ( set_ord_greaterThan @ real @ A2 ) ) )
=> ( ( filterlim @ real @ real
@ ^ [X4: real] : ( divide_divide @ real @ ( F9 @ X4 ) @ ( G5 @ X4 ) )
@ ( at_top @ real )
@ ( topolo174197925503356063within @ real @ A2 @ ( set_ord_greaterThan @ real @ A2 ) ) )
=> ( filterlim @ real @ real
@ ^ [X4: real] : ( divide_divide @ real @ ( F3 @ X4 ) @ ( G @ X4 ) )
@ ( at_top @ real )
@ ( topolo174197925503356063within @ real @ A2 @ ( set_ord_greaterThan @ real @ A2 ) ) ) ) ) ) ) ) ).
% lhopital_right_at_top_at_top
thf(fact_7698_lhopital__at__top__at__bot,axiom,
! [F3: real > real,A2: real,G: real > real,F9: real > real,G5: real > real] :
( ( filterlim @ real @ real @ F3 @ ( at_top @ real ) @ ( topolo174197925503356063within @ real @ A2 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( filterlim @ real @ real @ G @ ( at_bot @ real ) @ ( topolo174197925503356063within @ real @ A2 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( eventually @ real
@ ^ [X4: real] : ( has_field_derivative @ real @ F3 @ ( F9 @ X4 ) @ ( topolo174197925503356063within @ real @ X4 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ A2 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( eventually @ real
@ ^ [X4: real] : ( has_field_derivative @ real @ G @ ( G5 @ X4 ) @ ( topolo174197925503356063within @ real @ X4 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ A2 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( filterlim @ real @ real
@ ^ [X4: real] : ( divide_divide @ real @ ( F9 @ X4 ) @ ( G5 @ X4 ) )
@ ( at_bot @ real )
@ ( topolo174197925503356063within @ real @ A2 @ ( top_top @ ( set @ real ) ) ) )
=> ( filterlim @ real @ real
@ ^ [X4: real] : ( divide_divide @ real @ ( F3 @ X4 ) @ ( G @ X4 ) )
@ ( at_bot @ real )
@ ( topolo174197925503356063within @ real @ A2 @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ) ) ).
% lhopital_at_top_at_bot
thf(fact_7699_lhopital__left__at__top__at__top,axiom,
! [F3: real > real,A2: real,G: real > real,F9: real > real,G5: real > real] :
( ( filterlim @ real @ real @ F3 @ ( at_top @ real ) @ ( topolo174197925503356063within @ real @ A2 @ ( set_ord_lessThan @ real @ A2 ) ) )
=> ( ( filterlim @ real @ real @ G @ ( at_top @ real ) @ ( topolo174197925503356063within @ real @ A2 @ ( set_ord_lessThan @ real @ A2 ) ) )
=> ( ( eventually @ real
@ ^ [X4: real] : ( has_field_derivative @ real @ F3 @ ( F9 @ X4 ) @ ( topolo174197925503356063within @ real @ X4 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ A2 @ ( set_ord_lessThan @ real @ A2 ) ) )
=> ( ( eventually @ real
@ ^ [X4: real] : ( has_field_derivative @ real @ G @ ( G5 @ X4 ) @ ( topolo174197925503356063within @ real @ X4 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ A2 @ ( set_ord_lessThan @ real @ A2 ) ) )
=> ( ( filterlim @ real @ real
@ ^ [X4: real] : ( divide_divide @ real @ ( F9 @ X4 ) @ ( G5 @ X4 ) )
@ ( at_top @ real )
@ ( topolo174197925503356063within @ real @ A2 @ ( set_ord_lessThan @ real @ A2 ) ) )
=> ( filterlim @ real @ real
@ ^ [X4: real] : ( divide_divide @ real @ ( F3 @ X4 ) @ ( G @ X4 ) )
@ ( at_top @ real )
@ ( topolo174197925503356063within @ real @ A2 @ ( set_ord_lessThan @ real @ A2 ) ) ) ) ) ) ) ) ).
% lhopital_left_at_top_at_top
thf(fact_7700_lhopital__at__top,axiom,
! [G: real > real,X: real,G5: real > real,F3: real > real,F9: real > real,Y2: real] :
( ( filterlim @ real @ real @ G @ ( at_top @ real ) @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( ( eventually @ real
@ ^ [X4: real] :
( ( G5 @ X4 )
!= ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( ( eventually @ real
@ ^ [X4: real] : ( has_field_derivative @ real @ F3 @ ( F9 @ X4 ) @ ( topolo174197925503356063within @ real @ X4 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( ( eventually @ real
@ ^ [X4: real] : ( has_field_derivative @ real @ G @ ( G5 @ X4 ) @ ( topolo174197925503356063within @ real @ X4 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( ( filterlim @ real @ real
@ ^ [X4: real] : ( divide_divide @ real @ ( F9 @ X4 ) @ ( G5 @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ real @ Y2 )
@ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( filterlim @ real @ real
@ ^ [X4: real] : ( divide_divide @ real @ ( F3 @ X4 ) @ ( G @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ real @ Y2 )
@ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ) ) ).
% lhopital_at_top
thf(fact_7701_lhospital__at__top__at__top,axiom,
! [G: real > real,G5: real > real,F3: real > real,F9: real > real,X: real] :
( ( filterlim @ real @ real @ G @ ( at_top @ real ) @ ( at_top @ real ) )
=> ( ( eventually @ real
@ ^ [X4: real] :
( ( G5 @ X4 )
!= ( zero_zero @ real ) )
@ ( at_top @ real ) )
=> ( ( eventually @ real
@ ^ [X4: real] : ( has_field_derivative @ real @ F3 @ ( F9 @ X4 ) @ ( topolo174197925503356063within @ real @ X4 @ ( top_top @ ( set @ real ) ) ) )
@ ( at_top @ real ) )
=> ( ( eventually @ real
@ ^ [X4: real] : ( has_field_derivative @ real @ G @ ( G5 @ X4 ) @ ( topolo174197925503356063within @ real @ X4 @ ( top_top @ ( set @ real ) ) ) )
@ ( at_top @ real ) )
=> ( ( filterlim @ real @ real
@ ^ [X4: real] : ( divide_divide @ real @ ( F9 @ X4 ) @ ( G5 @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ real @ X )
@ ( at_top @ real ) )
=> ( filterlim @ real @ real
@ ^ [X4: real] : ( divide_divide @ real @ ( F3 @ X4 ) @ ( G @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ real @ X )
@ ( at_top @ real ) ) ) ) ) ) ) ).
% lhospital_at_top_at_top
thf(fact_7702_lhopital__right,axiom,
! [F3: real > real,X: real,G: real > real,G5: real > real,F9: real > real,F4: filter @ real] :
( ( filterlim @ real @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( topolo174197925503356063within @ real @ X @ ( set_ord_greaterThan @ real @ X ) ) )
=> ( ( filterlim @ real @ real @ G @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( topolo174197925503356063within @ real @ X @ ( set_ord_greaterThan @ real @ X ) ) )
=> ( ( eventually @ real
@ ^ [X4: real] :
( ( G @ X4 )
!= ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ real @ X @ ( set_ord_greaterThan @ real @ X ) ) )
=> ( ( eventually @ real
@ ^ [X4: real] :
( ( G5 @ X4 )
!= ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ real @ X @ ( set_ord_greaterThan @ real @ X ) ) )
=> ( ( eventually @ real
@ ^ [X4: real] : ( has_field_derivative @ real @ F3 @ ( F9 @ X4 ) @ ( topolo174197925503356063within @ real @ X4 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ X @ ( set_ord_greaterThan @ real @ X ) ) )
=> ( ( eventually @ real
@ ^ [X4: real] : ( has_field_derivative @ real @ G @ ( G5 @ X4 ) @ ( topolo174197925503356063within @ real @ X4 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ X @ ( set_ord_greaterThan @ real @ X ) ) )
=> ( ( filterlim @ real @ real
@ ^ [X4: real] : ( divide_divide @ real @ ( F9 @ X4 ) @ ( G5 @ X4 ) )
@ F4
@ ( topolo174197925503356063within @ real @ X @ ( set_ord_greaterThan @ real @ X ) ) )
=> ( filterlim @ real @ real
@ ^ [X4: real] : ( divide_divide @ real @ ( F3 @ X4 ) @ ( G @ X4 ) )
@ F4
@ ( topolo174197925503356063within @ real @ X @ ( set_ord_greaterThan @ real @ X ) ) ) ) ) ) ) ) ) ) ).
% lhopital_right
thf(fact_7703_lhopital__right__0,axiom,
! [F0: real > real,G0: real > real,G5: real > real,F9: real > real,F4: filter @ real] :
( ( filterlim @ real @ real @ F0 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) )
=> ( ( filterlim @ real @ real @ G0 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) )
=> ( ( eventually @ real
@ ^ [X4: real] :
( ( G0 @ X4 )
!= ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) )
=> ( ( eventually @ real
@ ^ [X4: real] :
( ( G5 @ X4 )
!= ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) )
=> ( ( eventually @ real
@ ^ [X4: real] : ( has_field_derivative @ real @ F0 @ ( F9 @ X4 ) @ ( topolo174197925503356063within @ real @ X4 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) )
=> ( ( eventually @ real
@ ^ [X4: real] : ( has_field_derivative @ real @ G0 @ ( G5 @ X4 ) @ ( topolo174197925503356063within @ real @ X4 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) )
=> ( ( filterlim @ real @ real
@ ^ [X4: real] : ( divide_divide @ real @ ( F9 @ X4 ) @ ( G5 @ X4 ) )
@ F4
@ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) )
=> ( filterlim @ real @ real
@ ^ [X4: real] : ( divide_divide @ real @ ( F0 @ X4 ) @ ( G0 @ X4 ) )
@ F4
@ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) ) ) ) ) ) ) ) ) ).
% lhopital_right_0
thf(fact_7704_lhopital__left,axiom,
! [F3: real > real,X: real,G: real > real,G5: real > real,F9: real > real,F4: filter @ real] :
( ( filterlim @ real @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( topolo174197925503356063within @ real @ X @ ( set_ord_lessThan @ real @ X ) ) )
=> ( ( filterlim @ real @ real @ G @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( topolo174197925503356063within @ real @ X @ ( set_ord_lessThan @ real @ X ) ) )
=> ( ( eventually @ real
@ ^ [X4: real] :
( ( G @ X4 )
!= ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ real @ X @ ( set_ord_lessThan @ real @ X ) ) )
=> ( ( eventually @ real
@ ^ [X4: real] :
( ( G5 @ X4 )
!= ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ real @ X @ ( set_ord_lessThan @ real @ X ) ) )
=> ( ( eventually @ real
@ ^ [X4: real] : ( has_field_derivative @ real @ F3 @ ( F9 @ X4 ) @ ( topolo174197925503356063within @ real @ X4 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ X @ ( set_ord_lessThan @ real @ X ) ) )
=> ( ( eventually @ real
@ ^ [X4: real] : ( has_field_derivative @ real @ G @ ( G5 @ X4 ) @ ( topolo174197925503356063within @ real @ X4 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ X @ ( set_ord_lessThan @ real @ X ) ) )
=> ( ( filterlim @ real @ real
@ ^ [X4: real] : ( divide_divide @ real @ ( F9 @ X4 ) @ ( G5 @ X4 ) )
@ F4
@ ( topolo174197925503356063within @ real @ X @ ( set_ord_lessThan @ real @ X ) ) )
=> ( filterlim @ real @ real
@ ^ [X4: real] : ( divide_divide @ real @ ( F3 @ X4 ) @ ( G @ X4 ) )
@ F4
@ ( topolo174197925503356063within @ real @ X @ ( set_ord_lessThan @ real @ X ) ) ) ) ) ) ) ) ) ) ).
% lhopital_left
thf(fact_7705_lhopital__right__at__top__at__bot,axiom,
! [F3: real > real,A2: real,G: real > real,F9: real > real,G5: real > real] :
( ( filterlim @ real @ real @ F3 @ ( at_top @ real ) @ ( topolo174197925503356063within @ real @ A2 @ ( set_ord_greaterThan @ real @ A2 ) ) )
=> ( ( filterlim @ real @ real @ G @ ( at_bot @ real ) @ ( topolo174197925503356063within @ real @ A2 @ ( set_ord_greaterThan @ real @ A2 ) ) )
=> ( ( eventually @ real
@ ^ [X4: real] : ( has_field_derivative @ real @ F3 @ ( F9 @ X4 ) @ ( topolo174197925503356063within @ real @ X4 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ A2 @ ( set_ord_greaterThan @ real @ A2 ) ) )
=> ( ( eventually @ real
@ ^ [X4: real] : ( has_field_derivative @ real @ G @ ( G5 @ X4 ) @ ( topolo174197925503356063within @ real @ X4 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ A2 @ ( set_ord_greaterThan @ real @ A2 ) ) )
=> ( ( filterlim @ real @ real
@ ^ [X4: real] : ( divide_divide @ real @ ( F9 @ X4 ) @ ( G5 @ X4 ) )
@ ( at_bot @ real )
@ ( topolo174197925503356063within @ real @ A2 @ ( set_ord_greaterThan @ real @ A2 ) ) )
=> ( filterlim @ real @ real
@ ^ [X4: real] : ( divide_divide @ real @ ( F3 @ X4 ) @ ( G @ X4 ) )
@ ( at_bot @ real )
@ ( topolo174197925503356063within @ real @ A2 @ ( set_ord_greaterThan @ real @ A2 ) ) ) ) ) ) ) ) ).
% lhopital_right_at_top_at_bot
thf(fact_7706_lhopital__left__at__top__at__bot,axiom,
! [F3: real > real,A2: real,G: real > real,F9: real > real,G5: real > real] :
( ( filterlim @ real @ real @ F3 @ ( at_top @ real ) @ ( topolo174197925503356063within @ real @ A2 @ ( set_ord_lessThan @ real @ A2 ) ) )
=> ( ( filterlim @ real @ real @ G @ ( at_bot @ real ) @ ( topolo174197925503356063within @ real @ A2 @ ( set_ord_lessThan @ real @ A2 ) ) )
=> ( ( eventually @ real
@ ^ [X4: real] : ( has_field_derivative @ real @ F3 @ ( F9 @ X4 ) @ ( topolo174197925503356063within @ real @ X4 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ A2 @ ( set_ord_lessThan @ real @ A2 ) ) )
=> ( ( eventually @ real
@ ^ [X4: real] : ( has_field_derivative @ real @ G @ ( G5 @ X4 ) @ ( topolo174197925503356063within @ real @ X4 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ A2 @ ( set_ord_lessThan @ real @ A2 ) ) )
=> ( ( filterlim @ real @ real
@ ^ [X4: real] : ( divide_divide @ real @ ( F9 @ X4 ) @ ( G5 @ X4 ) )
@ ( at_bot @ real )
@ ( topolo174197925503356063within @ real @ A2 @ ( set_ord_lessThan @ real @ A2 ) ) )
=> ( filterlim @ real @ real
@ ^ [X4: real] : ( divide_divide @ real @ ( F3 @ X4 ) @ ( G @ X4 ) )
@ ( at_bot @ real )
@ ( topolo174197925503356063within @ real @ A2 @ ( set_ord_lessThan @ real @ A2 ) ) ) ) ) ) ) ) ).
% lhopital_left_at_top_at_bot
thf(fact_7707_lhopital__right__at__top,axiom,
! [G: real > real,X: real,G5: real > real,F3: real > real,F9: real > real,Y2: real] :
( ( filterlim @ real @ real @ G @ ( at_top @ real ) @ ( topolo174197925503356063within @ real @ X @ ( set_ord_greaterThan @ real @ X ) ) )
=> ( ( eventually @ real
@ ^ [X4: real] :
( ( G5 @ X4 )
!= ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ real @ X @ ( set_ord_greaterThan @ real @ X ) ) )
=> ( ( eventually @ real
@ ^ [X4: real] : ( has_field_derivative @ real @ F3 @ ( F9 @ X4 ) @ ( topolo174197925503356063within @ real @ X4 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ X @ ( set_ord_greaterThan @ real @ X ) ) )
=> ( ( eventually @ real
@ ^ [X4: real] : ( has_field_derivative @ real @ G @ ( G5 @ X4 ) @ ( topolo174197925503356063within @ real @ X4 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ X @ ( set_ord_greaterThan @ real @ X ) ) )
=> ( ( filterlim @ real @ real
@ ^ [X4: real] : ( divide_divide @ real @ ( F9 @ X4 ) @ ( G5 @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ real @ Y2 )
@ ( topolo174197925503356063within @ real @ X @ ( set_ord_greaterThan @ real @ X ) ) )
=> ( filterlim @ real @ real
@ ^ [X4: real] : ( divide_divide @ real @ ( F3 @ X4 ) @ ( G @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ real @ Y2 )
@ ( topolo174197925503356063within @ real @ X @ ( set_ord_greaterThan @ real @ X ) ) ) ) ) ) ) ) ).
% lhopital_right_at_top
thf(fact_7708_lhopital__right__0__at__top,axiom,
! [G: real > real,G5: real > real,F3: real > real,F9: real > real,X: real] :
( ( filterlim @ real @ real @ G @ ( at_top @ real ) @ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) )
=> ( ( eventually @ real
@ ^ [X4: real] :
( ( G5 @ X4 )
!= ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) )
=> ( ( eventually @ real
@ ^ [X4: real] : ( has_field_derivative @ real @ F3 @ ( F9 @ X4 ) @ ( topolo174197925503356063within @ real @ X4 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) )
=> ( ( eventually @ real
@ ^ [X4: real] : ( has_field_derivative @ real @ G @ ( G5 @ X4 ) @ ( topolo174197925503356063within @ real @ X4 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) )
=> ( ( filterlim @ real @ real
@ ^ [X4: real] : ( divide_divide @ real @ ( F9 @ X4 ) @ ( G5 @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ real @ X )
@ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) )
=> ( filterlim @ real @ real
@ ^ [X4: real] : ( divide_divide @ real @ ( F3 @ X4 ) @ ( G @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ real @ X )
@ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) ) ) ) ) ) ) ).
% lhopital_right_0_at_top
thf(fact_7709_summable__bounded__partials,axiom,
! [A: $tType] :
( ( ( real_V8037385150606011577_space @ A )
& ( real_V822414075346904944vector @ A ) )
=> ! [F3: nat > A,G: nat > real] :
( ( eventually @ nat
@ ^ [X02: nat] :
! [A5: nat] :
( ( ord_less_eq @ nat @ X02 @ A5 )
=> ! [B4: nat] :
( ( ord_less @ nat @ A5 @ B4 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_or3652927894154168847AtMost @ nat @ A5 @ B4 ) ) ) @ ( G @ A5 ) ) ) )
@ ( at_top @ nat ) )
=> ( ( filterlim @ nat @ real @ G @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) )
=> ( summable @ A @ F3 ) ) ) ) ).
% summable_bounded_partials
thf(fact_7710_summable__Cauchy_H,axiom,
! [A: $tType] :
( ( real_Vector_banach @ A )
=> ! [F3: nat > A,G: nat > real] :
( ( eventually @ nat
@ ^ [M2: nat] :
! [N2: nat] :
( ( ord_less_eq @ nat @ M2 @ N2 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N2 ) ) ) @ ( G @ M2 ) ) )
@ ( at_top @ nat ) )
=> ( ( filterlim @ nat @ real @ G @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) )
=> ( summable @ A @ F3 ) ) ) ) ).
% summable_Cauchy'
thf(fact_7711_eventually__all__finite,axiom,
! [B: $tType,A: $tType] :
( ( finite_finite @ B )
=> ! [P3: A > B > $o,Net: filter @ A] :
( ! [Y7: B] :
( eventually @ A
@ ^ [X4: A] : ( P3 @ X4 @ Y7 )
@ Net )
=> ( eventually @ A
@ ^ [X4: A] :
! [X6: B] : ( P3 @ X4 @ X6 )
@ Net ) ) ) ).
% eventually_all_finite
thf(fact_7712_finite__set__of__finite__funs,axiom,
! [B: $tType,A: $tType,A3: set @ A,B5: set @ B,D2: B] :
( ( finite_finite2 @ A @ A3 )
=> ( ( finite_finite2 @ B @ B5 )
=> ( finite_finite2 @ ( A > B )
@ ( collect @ ( A > B )
@ ^ [F5: A > B] :
! [X4: A] :
( ( ( member @ A @ X4 @ A3 )
=> ( member @ B @ ( F5 @ X4 ) @ B5 ) )
& ( ~ ( member @ A @ X4 @ A3 )
=> ( ( F5 @ X4 )
= D2 ) ) ) ) ) ) ) ).
% finite_set_of_finite_funs
thf(fact_7713_eventually__all__ge__at__top,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P3: A > $o] :
( ( eventually @ A @ P3 @ ( at_top @ A ) )
=> ( eventually @ A
@ ^ [X4: A] :
! [Y: A] :
( ( ord_less_eq @ A @ X4 @ Y )
=> ( P3 @ Y ) )
@ ( at_top @ A ) ) ) ) ).
% eventually_all_ge_at_top
thf(fact_7714_Collect__all__eq,axiom,
! [A: $tType,B: $tType,P3: A > B > $o] :
( ( collect @ A
@ ^ [X4: A] :
! [X6: B] : ( P3 @ X4 @ X6 ) )
= ( complete_Inf_Inf @ ( set @ A )
@ ( image @ B @ ( set @ A )
@ ^ [Y: B] :
( collect @ A
@ ^ [X4: A] : ( P3 @ X4 @ Y ) )
@ ( top_top @ ( set @ B ) ) ) ) ) ).
% Collect_all_eq
thf(fact_7715_Greatest__def,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( order_Greatest @ A )
= ( ^ [P2: A > $o] :
( the @ A
@ ^ [X4: A] :
( ( P2 @ X4 )
& ! [Y: A] :
( ( P2 @ Y )
=> ( ord_less_eq @ A @ Y @ X4 ) ) ) ) ) ) ) ).
% Greatest_def
thf(fact_7716_Bfun__metric__def,axiom,
! [B: $tType,A: $tType] :
( ( real_V7819770556892013058_space @ B )
=> ( ( bfun @ A @ B )
= ( ^ [F5: A > B,F10: filter @ A] :
? [Y: B,K6: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K6 )
& ( eventually @ A
@ ^ [X4: A] : ( ord_less_eq @ real @ ( real_V557655796197034286t_dist @ B @ ( F5 @ X4 ) @ Y ) @ K6 )
@ F10 ) ) ) ) ) ).
% Bfun_metric_def
thf(fact_7717_GreatestI__nat,axiom,
! [P3: nat > $o,K2: nat,B2: nat] :
( ( P3 @ K2 )
=> ( ! [Y7: nat] :
( ( P3 @ Y7 )
=> ( ord_less_eq @ nat @ Y7 @ B2 ) )
=> ( P3 @ ( order_Greatest @ nat @ P3 ) ) ) ) ).
% GreatestI_nat
thf(fact_7718_Greatest__le__nat,axiom,
! [P3: nat > $o,K2: nat,B2: nat] :
( ( P3 @ K2 )
=> ( ! [Y7: nat] :
( ( P3 @ Y7 )
=> ( ord_less_eq @ nat @ Y7 @ B2 ) )
=> ( ord_less_eq @ nat @ K2 @ ( order_Greatest @ nat @ P3 ) ) ) ) ).
% Greatest_le_nat
thf(fact_7719_GreatestI__ex__nat,axiom,
! [P3: nat > $o,B2: nat] :
( ? [X_1: nat] : ( P3 @ X_1 )
=> ( ! [Y7: nat] :
( ( P3 @ Y7 )
=> ( ord_less_eq @ nat @ Y7 @ B2 ) )
=> ( P3 @ ( order_Greatest @ nat @ P3 ) ) ) ) ).
% GreatestI_ex_nat
thf(fact_7720_Bfun__const,axiom,
! [B: $tType,A: $tType] :
( ( real_V7819770556892013058_space @ B )
=> ! [C2: B,F4: filter @ A] :
( bfun @ A @ B
@ ^ [Uu3: A] : C2
@ F4 ) ) ).
% Bfun_const
thf(fact_7721_Bseq__minus__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X9: nat > A] :
( ( bfun @ nat @ A
@ ^ [N2: nat] : ( uminus_uminus @ A @ ( X9 @ N2 ) )
@ ( at_top @ nat ) )
= ( bfun @ nat @ A @ X9 @ ( at_top @ nat ) ) ) ) ).
% Bseq_minus_iff
thf(fact_7722_Bseq__mult,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: nat > A,G: nat > A] :
( ( bfun @ nat @ A @ F3 @ ( at_top @ nat ) )
=> ( ( bfun @ nat @ A @ G @ ( at_top @ nat ) )
=> ( bfun @ nat @ A
@ ^ [X4: nat] : ( times_times @ A @ ( F3 @ X4 ) @ ( G @ X4 ) )
@ ( at_top @ nat ) ) ) ) ) ).
% Bseq_mult
thf(fact_7723_Bseq__add,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: nat > A,C2: A] :
( ( bfun @ nat @ A @ F3 @ ( at_top @ nat ) )
=> ( bfun @ nat @ A
@ ^ [X4: nat] : ( plus_plus @ A @ ( F3 @ X4 ) @ C2 )
@ ( at_top @ nat ) ) ) ) ).
% Bseq_add
thf(fact_7724_Bseq__add__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: nat > A,C2: A] :
( ( bfun @ nat @ A
@ ^ [X4: nat] : ( plus_plus @ A @ ( F3 @ X4 ) @ C2 )
@ ( at_top @ nat ) )
= ( bfun @ nat @ A @ F3 @ ( at_top @ nat ) ) ) ) ).
% Bseq_add_iff
thf(fact_7725_Bseq__offset,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X9: nat > A,K2: nat] :
( ( bfun @ nat @ A
@ ^ [N2: nat] : ( X9 @ ( plus_plus @ nat @ N2 @ K2 ) )
@ ( at_top @ nat ) )
=> ( bfun @ nat @ A @ X9 @ ( at_top @ nat ) ) ) ) ).
% Bseq_offset
thf(fact_7726_Bseq__ignore__initial__segment,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X9: nat > A,K2: nat] :
( ( bfun @ nat @ A @ X9 @ ( at_top @ nat ) )
=> ( bfun @ nat @ A
@ ^ [N2: nat] : ( X9 @ ( plus_plus @ nat @ N2 @ K2 ) )
@ ( at_top @ nat ) ) ) ) ).
% Bseq_ignore_initial_segment
thf(fact_7727_Bseq__subseq,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: nat > A,G: nat > nat] :
( ( bfun @ nat @ A @ F3 @ ( at_top @ nat ) )
=> ( bfun @ nat @ A
@ ^ [X4: nat] : ( F3 @ ( G @ X4 ) )
@ ( at_top @ nat ) ) ) ) ).
% Bseq_subseq
thf(fact_7728_Bseq__Suc__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: nat > A] :
( ( bfun @ nat @ A
@ ^ [N2: nat] : ( F3 @ ( suc @ N2 ) )
@ ( at_top @ nat ) )
= ( bfun @ nat @ A @ F3 @ ( at_top @ nat ) ) ) ) ).
% Bseq_Suc_iff
thf(fact_7729_BseqI_H,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X9: nat > A,K5: real] :
( ! [N3: nat] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( X9 @ N3 ) ) @ K5 )
=> ( bfun @ nat @ A @ X9 @ ( at_top @ nat ) ) ) ) ).
% BseqI'
thf(fact_7730_Greatest__equality,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [P3: A > $o,X: A] :
( ( P3 @ X )
=> ( ! [Y7: A] :
( ( P3 @ Y7 )
=> ( ord_less_eq @ A @ Y7 @ X ) )
=> ( ( order_Greatest @ A @ P3 )
= X ) ) ) ) ).
% Greatest_equality
thf(fact_7731_GreatestI2__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [P3: A > $o,X: A,Q2: A > $o] :
( ( P3 @ X )
=> ( ! [Y7: A] :
( ( P3 @ Y7 )
=> ( ord_less_eq @ A @ Y7 @ X ) )
=> ( ! [X5: A] :
( ( P3 @ X5 )
=> ( ! [Y4: A] :
( ( P3 @ Y4 )
=> ( ord_less_eq @ A @ Y4 @ X5 ) )
=> ( Q2 @ X5 ) ) )
=> ( Q2 @ ( order_Greatest @ A @ P3 ) ) ) ) ) ) ).
% GreatestI2_order
thf(fact_7732_Bseq__cmult__iff,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [C2: A,F3: nat > A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( bfun @ nat @ A
@ ^ [X4: nat] : ( times_times @ A @ C2 @ ( F3 @ X4 ) )
@ ( at_top @ nat ) )
= ( bfun @ nat @ A @ F3 @ ( at_top @ nat ) ) ) ) ) ).
% Bseq_cmult_iff
thf(fact_7733_Bseq__eventually__mono,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F3: nat > A,G: nat > B] :
( ( eventually @ nat
@ ^ [N2: nat] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( F3 @ N2 ) ) @ ( real_V7770717601297561774m_norm @ B @ ( G @ N2 ) ) )
@ ( at_top @ nat ) )
=> ( ( bfun @ nat @ B @ G @ ( at_top @ nat ) )
=> ( bfun @ nat @ A @ F3 @ ( at_top @ nat ) ) ) ) ) ).
% Bseq_eventually_mono
thf(fact_7734_Bseq__eq__bounded,axiom,
! [F3: nat > real,A2: real,B2: real] :
( ( ord_less_eq @ ( set @ real ) @ ( image @ nat @ real @ F3 @ ( top_top @ ( set @ nat ) ) ) @ ( set_or1337092689740270186AtMost @ real @ A2 @ B2 ) )
=> ( bfun @ nat @ real @ F3 @ ( at_top @ nat ) ) ) ).
% Bseq_eq_bounded
thf(fact_7735_BseqD,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X9: nat > A] :
( ( bfun @ nat @ A @ X9 @ ( at_top @ nat ) )
=> ? [K10: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K10 )
& ! [N9: nat] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( X9 @ N9 ) ) @ K10 ) ) ) ) ).
% BseqD
thf(fact_7736_BseqE,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X9: nat > A] :
( ( bfun @ nat @ A @ X9 @ ( at_top @ nat ) )
=> ~ ! [K10: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K10 )
=> ~ ! [N9: nat] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( X9 @ N9 ) ) @ K10 ) ) ) ) ).
% BseqE
thf(fact_7737_BseqI,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [K5: real,X9: nat > A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K5 )
=> ( ! [N3: nat] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( X9 @ N3 ) ) @ K5 )
=> ( bfun @ nat @ A @ X9 @ ( at_top @ nat ) ) ) ) ) ).
% BseqI
thf(fact_7738_Bseq__def,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X9: nat > A] :
( ( bfun @ nat @ A @ X9 @ ( at_top @ nat ) )
= ( ? [K6: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K6 )
& ! [N2: nat] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( X9 @ N2 ) ) @ K6 ) ) ) ) ) ).
% Bseq_def
thf(fact_7739_Bseq__realpow,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ X @ ( one_one @ real ) )
=> ( bfun @ nat @ real @ ( power_power @ real @ X ) @ ( at_top @ nat ) ) ) ) ).
% Bseq_realpow
thf(fact_7740_Bseq__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X9: nat > A] :
( ( bfun @ nat @ A @ X9 @ ( at_top @ nat ) )
= ( ? [N6: nat] :
! [N2: nat] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( X9 @ N2 ) ) @ ( semiring_1_of_nat @ real @ ( suc @ N6 ) ) ) ) ) ) ).
% Bseq_iff
thf(fact_7741_BfunI,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F3: A > B,K5: real,F4: filter @ A] :
( ( eventually @ A
@ ^ [X4: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( F3 @ X4 ) ) @ K5 )
@ F4 )
=> ( bfun @ A @ B @ F3 @ F4 ) ) ) ).
% BfunI
thf(fact_7742_Bseq__iff3,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X9: nat > A] :
( ( bfun @ nat @ A @ X9 @ ( at_top @ nat ) )
= ( ? [K3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K3 )
& ? [N6: nat] :
! [N2: nat] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ ( X9 @ N2 ) @ ( uminus_uminus @ A @ ( X9 @ N6 ) ) ) ) @ K3 ) ) ) ) ) ).
% Bseq_iff3
thf(fact_7743_Bseq__iff2,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X9: nat > A] :
( ( bfun @ nat @ A @ X9 @ ( at_top @ nat ) )
= ( ? [K3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K3 )
& ? [X4: A] :
! [N2: nat] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ ( X9 @ N2 ) @ ( uminus_uminus @ A @ X4 ) ) ) @ K3 ) ) ) ) ) ).
% Bseq_iff2
thf(fact_7744_Bfun__inverse,axiom,
! [A: $tType,B: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [F3: B > A,A2: A,F4: filter @ B] :
( ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ F4 )
=> ( ( A2
!= ( zero_zero @ A ) )
=> ( bfun @ B @ A
@ ^ [X4: B] : ( inverse_inverse @ A @ ( F3 @ X4 ) )
@ F4 ) ) ) ) ).
% Bfun_inverse
thf(fact_7745_Bfun__def,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ( ( bfun @ A @ B )
= ( ^ [F5: A > B,F10: filter @ A] :
? [K6: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K6 )
& ( eventually @ A
@ ^ [X4: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( F5 @ X4 ) ) @ K6 )
@ F10 ) ) ) ) ) ).
% Bfun_def
thf(fact_7746_BfunE,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F3: A > B,F4: filter @ A] :
( ( bfun @ A @ B @ F3 @ F4 )
=> ~ ! [B9: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ B9 )
=> ~ ( eventually @ A
@ ^ [X4: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( F3 @ X4 ) ) @ B9 )
@ F4 ) ) ) ) ).
% BfunE
thf(fact_7747_sequentially__imp__eventually__at__right,axiom,
! [A: $tType] :
( ( ( topolo3112930676232923870pology @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [A2: A,B2: A,P3: A > $o] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ! [F6: nat > A] :
( ! [N9: nat] : ( ord_less @ A @ A2 @ ( F6 @ N9 ) )
=> ( ! [N9: nat] : ( ord_less @ A @ ( F6 @ N9 ) @ B2 )
=> ( ( order_antimono @ nat @ A @ F6 )
=> ( ( filterlim @ nat @ A @ F6 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ ( at_top @ nat ) )
=> ( eventually @ nat
@ ^ [N2: nat] : ( P3 @ ( F6 @ N2 ) )
@ ( at_top @ nat ) ) ) ) ) )
=> ( eventually @ A @ P3 @ ( topolo174197925503356063within @ A @ A2 @ ( set_ord_greaterThan @ A @ A2 ) ) ) ) ) ) ).
% sequentially_imp_eventually_at_right
thf(fact_7748_VEBT__internal_Ovalid_H_Oelims_I3_J,axiom,
! [X: vEBT_VEBT,Xa: nat] :
( ~ ( vEBT_VEBT_valid @ X @ Xa )
=> ( ( ? [Uu2: $o,Uv2: $o] :
( X
= ( vEBT_Leaf @ Uu2 @ Uv2 ) )
=> ( Xa
= ( one_one @ nat ) ) )
=> ~ ! [Mima: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( ( Deg2 = Xa )
& ! [X5: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X5 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ( vEBT_VEBT_valid @ X5 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
& ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
& ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( case_option @ $o @ ( product_prod @ nat @ nat )
@ ( ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X6 )
& ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X6 ) ) )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [Mi2: nat,Ma2: nat] :
( ( ord_less_eq @ nat @ Mi2 @ Ma2 )
& ( ord_less @ nat @ Ma2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
& ! [I: nat] :
( ( ord_less @ nat @ I @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
=> ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I ) @ X6 ) )
= ( vEBT_V8194947554948674370ptions @ Summary2 @ I ) ) )
& ( ( Mi2 = Ma2 )
=> ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X6 ) ) )
& ( ( Mi2 != Ma2 )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList2 @ Ma2 )
& ! [X4: nat] :
( ( ord_less @ nat @ X4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList2 @ X4 )
=> ( ( ord_less @ nat @ Mi2 @ X4 )
& ( ord_less_eq @ nat @ X4 @ Ma2 ) ) ) ) ) ) ) )
@ Mima ) ) ) ) ) ).
% VEBT_internal.valid'.elims(3)
thf(fact_7749_decseq__const,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [K2: A] :
( order_antimono @ nat @ A
@ ^ [X4: nat] : K2 ) ) ).
% decseq_const
thf(fact_7750_finite__Collect__bounded__ex,axiom,
! [B: $tType,A: $tType,P3: A > $o,Q2: B > A > $o] :
( ( finite_finite2 @ A @ ( collect @ A @ P3 ) )
=> ( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X4: B] :
? [Y: A] :
( ( P3 @ Y )
& ( Q2 @ X4 @ Y ) ) ) )
= ( ! [Y: A] :
( ( P3 @ Y )
=> ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X4: B] : ( Q2 @ X4 @ Y ) ) ) ) ) ) ) ).
% finite_Collect_bounded_ex
thf(fact_7751_INF__bool__eq,axiom,
! [A: $tType] :
( ( ^ [A7: set @ A,F5: A > $o] : ( complete_Inf_Inf @ $o @ ( image @ A @ $o @ F5 @ A7 ) ) )
= ( ball @ A ) ) ).
% INF_bool_eq
thf(fact_7752_decseq__bounded,axiom,
! [X9: nat > real,B5: real] :
( ( order_antimono @ nat @ real @ X9 )
=> ( ! [I3: nat] : ( ord_less_eq @ real @ B5 @ ( X9 @ I3 ) )
=> ( bfun @ nat @ real @ X9 @ ( at_top @ nat ) ) ) ) ).
% decseq_bounded
thf(fact_7753_eventually__ball__finite__distrib,axiom,
! [B: $tType,A: $tType,A3: set @ A,P3: B > A > $o,Net: filter @ B] :
( ( finite_finite2 @ A @ A3 )
=> ( ( eventually @ B
@ ^ [X4: B] :
! [Y: A] :
( ( member @ A @ Y @ A3 )
=> ( P3 @ X4 @ Y ) )
@ Net )
= ( ! [X4: A] :
( ( member @ A @ X4 @ A3 )
=> ( eventually @ B
@ ^ [Y: B] : ( P3 @ Y @ X4 )
@ Net ) ) ) ) ) ).
% eventually_ball_finite_distrib
thf(fact_7754_eventually__ball__finite,axiom,
! [A: $tType,B: $tType,A3: set @ A,P3: B > A > $o,Net: filter @ B] :
( ( finite_finite2 @ A @ A3 )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ A3 )
=> ( eventually @ B
@ ^ [Y: B] : ( P3 @ Y @ X5 )
@ Net ) )
=> ( eventually @ B
@ ^ [X4: B] :
! [Y: A] :
( ( member @ A @ Y @ A3 )
=> ( P3 @ X4 @ Y ) )
@ Net ) ) ) ).
% eventually_ball_finite
thf(fact_7755_Ball__def__raw,axiom,
! [A: $tType] :
( ( ball @ A )
= ( ^ [A7: set @ A,P2: A > $o] :
! [X4: A] :
( ( member @ A @ X4 @ A7 )
=> ( P2 @ X4 ) ) ) ) ).
% Ball_def_raw
thf(fact_7756_finite__image__set,axiom,
! [A: $tType,B: $tType,P3: A > $o,F3: A > B] :
( ( finite_finite2 @ A @ ( collect @ A @ P3 ) )
=> ( finite_finite2 @ B
@ ( collect @ B
@ ^ [Uu3: B] :
? [X4: A] :
( ( Uu3
= ( F3 @ X4 ) )
& ( P3 @ X4 ) ) ) ) ) ).
% finite_image_set
thf(fact_7757_finite__image__set2,axiom,
! [A: $tType,B: $tType,C: $tType,P3: A > $o,Q2: B > $o,F3: A > B > C] :
( ( finite_finite2 @ A @ ( collect @ A @ P3 ) )
=> ( ( finite_finite2 @ B @ ( collect @ B @ Q2 ) )
=> ( finite_finite2 @ C
@ ( collect @ C
@ ^ [Uu3: C] :
? [X4: A,Y: B] :
( ( Uu3
= ( F3 @ X4 @ Y ) )
& ( P3 @ X4 )
& ( Q2 @ Y ) ) ) ) ) ) ).
% finite_image_set2
thf(fact_7758_Ball__fold,axiom,
! [A: $tType,A3: set @ A,P3: A > $o] :
( ( finite_finite2 @ A @ A3 )
=> ( ( ! [X4: A] :
( ( member @ A @ X4 @ A3 )
=> ( P3 @ X4 ) ) )
= ( finite_fold @ A @ $o
@ ^ [K3: A,S4: $o] :
( S4
& ( P3 @ K3 ) )
@ $true
@ A3 ) ) ) ).
% Ball_fold
thf(fact_7759_Union__SetCompr__eq,axiom,
! [B: $tType,A: $tType,F3: B > ( set @ A ),P3: B > $o] :
( ( complete_Sup_Sup @ ( set @ A )
@ ( collect @ ( set @ A )
@ ^ [Uu3: set @ A] :
? [X4: B] :
( ( Uu3
= ( F3 @ X4 ) )
& ( P3 @ X4 ) ) ) )
= ( collect @ A
@ ^ [A5: A] :
? [X4: B] :
( ( P3 @ X4 )
& ( member @ A @ A5 @ ( F3 @ X4 ) ) ) ) ) ).
% Union_SetCompr_eq
thf(fact_7760_full__SetCompr__eq,axiom,
! [A: $tType,B: $tType,F3: B > A] :
( ( collect @ A
@ ^ [U2: A] :
? [X4: B] :
( U2
= ( F3 @ X4 ) ) )
= ( image @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) ) ) ).
% full_SetCompr_eq
thf(fact_7761_Setcompr__eq__image,axiom,
! [A: $tType,B: $tType,F3: B > A,A3: set @ B] :
( ( collect @ A
@ ^ [Uu3: A] :
? [X4: B] :
( ( Uu3
= ( F3 @ X4 ) )
& ( member @ B @ X4 @ A3 ) ) )
= ( image @ B @ A @ F3 @ A3 ) ) ).
% Setcompr_eq_image
thf(fact_7762_setcompr__eq__image,axiom,
! [A: $tType,B: $tType,F3: B > A,P3: B > $o] :
( ( collect @ A
@ ^ [Uu3: A] :
? [X4: B] :
( ( Uu3
= ( F3 @ X4 ) )
& ( P3 @ X4 ) ) )
= ( image @ B @ A @ F3 @ ( collect @ B @ P3 ) ) ) ).
% setcompr_eq_image
thf(fact_7763_Collect__ball__eq,axiom,
! [A: $tType,B: $tType,A3: set @ B,P3: A > B > $o] :
( ( collect @ A
@ ^ [X4: A] :
! [Y: B] :
( ( member @ B @ Y @ A3 )
=> ( P3 @ X4 @ Y ) ) )
= ( complete_Inf_Inf @ ( set @ A )
@ ( image @ B @ ( set @ A )
@ ^ [Y: B] :
( collect @ A
@ ^ [X4: A] : ( P3 @ X4 @ Y ) )
@ A3 ) ) ) ).
% Collect_ball_eq
thf(fact_7764_INTER__eq,axiom,
! [B: $tType,A: $tType,B5: B > ( set @ A ),A3: set @ B] :
( ( complete_Inf_Inf @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B5 @ A3 ) )
= ( collect @ A
@ ^ [Y: A] :
! [X4: B] :
( ( member @ B @ X4 @ A3 )
=> ( member @ A @ Y @ ( B5 @ X4 ) ) ) ) ) ).
% INTER_eq
thf(fact_7765_sorted__wrt_Osimps_I2_J,axiom,
! [A: $tType,P3: A > A > $o,X: A,Ys: list @ A] :
( ( sorted_wrt @ A @ P3 @ ( cons @ A @ X @ Ys ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Ys ) )
=> ( P3 @ X @ X4 ) )
& ( sorted_wrt @ A @ P3 @ Ys ) ) ) ).
% sorted_wrt.simps(2)
thf(fact_7766_sorted__wrt_Oelims_I3_J,axiom,
! [A: $tType,X: A > A > $o,Xa: list @ A] :
( ~ ( sorted_wrt @ A @ X @ Xa )
=> ~ ! [X5: A,Ys4: list @ A] :
( ( Xa
= ( cons @ A @ X5 @ Ys4 ) )
=> ( ! [Xa3: A] :
( ( member @ A @ Xa3 @ ( set2 @ A @ Ys4 ) )
=> ( X @ X5 @ Xa3 ) )
& ( sorted_wrt @ A @ X @ Ys4 ) ) ) ) ).
% sorted_wrt.elims(3)
thf(fact_7767_set__Cons__def,axiom,
! [A: $tType] :
( ( set_Cons @ A )
= ( ^ [A7: set @ A,XS: set @ ( list @ A )] :
( collect @ ( list @ A )
@ ^ [Z6: list @ A] :
? [X4: A,Xs2: list @ A] :
( ( Z6
= ( cons @ A @ X4 @ Xs2 ) )
& ( member @ A @ X4 @ A7 )
& ( member @ ( list @ A ) @ Xs2 @ XS ) ) ) ) ) ).
% set_Cons_def
thf(fact_7768_lex__conv,axiom,
! [A: $tType] :
( ( lex @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ ( list @ A ) @ ( list @ A ) )
@ ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ $o
@ ^ [Xs2: list @ A,Ys3: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ A ) @ Ys3 ) )
& ? [Xys2: list @ A,X4: A,Y: A,Xs6: list @ A,Ys7: list @ A] :
( ( Xs2
= ( append @ A @ Xys2 @ ( cons @ A @ X4 @ Xs6 ) ) )
& ( Ys3
= ( append @ A @ Xys2 @ ( cons @ A @ Y @ Ys7 ) ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y ) @ R5 ) ) ) ) ) ) ) ).
% lex_conv
thf(fact_7769_open__subdiagonal,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ( topolo1002775350975398744n_open @ ( product_prod @ A @ A )
@ ( collect @ ( product_prod @ A @ A )
@ ^ [Uu3: product_prod @ A @ A] :
? [X4: A,Y: A] :
( ( Uu3
= ( product_Pair @ A @ A @ X4 @ Y ) )
& ( ord_less @ A @ X4 @ Y ) ) ) ) ) ).
% open_subdiagonal
thf(fact_7770_open__superdiagonal,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ( topolo1002775350975398744n_open @ ( product_prod @ A @ A )
@ ( collect @ ( product_prod @ A @ A )
@ ^ [Uu3: product_prod @ A @ A] :
? [X4: A,Y: A] :
( ( Uu3
= ( product_Pair @ A @ A @ X4 @ Y ) )
& ( ord_less @ A @ Y @ X4 ) ) ) ) ) ).
% open_superdiagonal
thf(fact_7771_open__diagonal__complement,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ( topolo1002775350975398744n_open @ ( product_prod @ A @ A )
@ ( collect @ ( product_prod @ A @ A )
@ ^ [Uu3: product_prod @ A @ A] :
? [X4: A,Y: A] :
( ( Uu3
= ( product_Pair @ A @ A @ X4 @ Y ) )
& ( X4 != Y ) ) ) ) ) ).
% open_diagonal_complement
thf(fact_7772_open__Collect__ex,axiom,
! [B: $tType,A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [P3: B > A > $o] :
( ! [I3: B] : ( topolo1002775350975398744n_open @ A @ ( collect @ A @ ( P3 @ I3 ) ) )
=> ( topolo1002775350975398744n_open @ A
@ ( collect @ A
@ ^ [X4: A] :
? [I: B] : ( P3 @ I @ X4 ) ) ) ) ) ).
% open_Collect_ex
thf(fact_7773_Sup__eq__Inf,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ( ( complete_Sup_Sup @ A )
= ( ^ [A7: set @ A] :
( complete_Inf_Inf @ A
@ ( collect @ A
@ ^ [B4: A] :
! [X4: A] :
( ( member @ A @ X4 @ A7 )
=> ( ord_less_eq @ A @ X4 @ B4 ) ) ) ) ) ) ) ).
% Sup_eq_Inf
thf(fact_7774_Inf__eq__Sup,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ( ( complete_Inf_Inf @ A )
= ( ^ [A7: set @ A] :
( complete_Sup_Sup @ A
@ ( collect @ A
@ ^ [B4: A] :
! [X4: A] :
( ( member @ A @ X4 @ A7 )
=> ( ord_less_eq @ A @ B4 @ X4 ) ) ) ) ) ) ) ).
% Inf_eq_Sup
thf(fact_7775_Ball__comp__iff,axiom,
! [C: $tType,B: $tType,A: $tType,A3: B > ( set @ C ),F3: C > $o,G: A > B] :
( ( comp @ B @ $o @ A
@ ^ [X4: B] :
! [Y: C] :
( ( member @ C @ Y @ ( A3 @ X4 ) )
=> ( F3 @ Y ) )
@ G )
= ( ^ [X4: A] :
! [Y: C] :
( ( member @ C @ Y @ ( comp @ B @ ( set @ C ) @ A @ A3 @ G @ X4 ) )
=> ( F3 @ Y ) ) ) ) ).
% Ball_comp_iff
thf(fact_7776_Ball__Collect,axiom,
! [A: $tType] :
( ( ball @ A )
= ( ^ [A7: set @ A,P2: A > $o] : ( ord_less_eq @ ( set @ A ) @ A7 @ ( collect @ A @ P2 ) ) ) ) ).
% Ball_Collect
thf(fact_7777_antimonoD,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F3: A > B,X: A,Y2: A] :
( ( order_antimono @ A @ B @ F3 )
=> ( ( ord_less_eq @ A @ X @ Y2 )
=> ( ord_less_eq @ B @ ( F3 @ Y2 ) @ ( F3 @ X ) ) ) ) ) ).
% antimonoD
thf(fact_7778_antimonoE,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F3: A > B,X: A,Y2: A] :
( ( order_antimono @ A @ B @ F3 )
=> ( ( ord_less_eq @ A @ X @ Y2 )
=> ( ord_less_eq @ B @ ( F3 @ Y2 ) @ ( F3 @ X ) ) ) ) ) ).
% antimonoE
thf(fact_7779_antimonoI,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F3: A > B] :
( ! [X5: A,Y7: A] :
( ( ord_less_eq @ A @ X5 @ Y7 )
=> ( ord_less_eq @ B @ ( F3 @ Y7 ) @ ( F3 @ X5 ) ) )
=> ( order_antimono @ A @ B @ F3 ) ) ) ).
% antimonoI
thf(fact_7780_antimono__def,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ( ( order_antimono @ A @ B )
= ( ^ [F5: A > B] :
! [X4: A,Y: A] :
( ( ord_less_eq @ A @ X4 @ Y )
=> ( ord_less_eq @ B @ ( F5 @ Y ) @ ( F5 @ X4 ) ) ) ) ) ) ).
% antimono_def
thf(fact_7781_decseq__Suc__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( order_antimono @ nat @ A )
= ( ^ [F5: nat > A] :
! [N2: nat] : ( ord_less_eq @ A @ ( F5 @ ( suc @ N2 ) ) @ ( F5 @ N2 ) ) ) ) ) ).
% decseq_Suc_iff
thf(fact_7782_decseq__SucI,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X9: nat > A] :
( ! [N3: nat] : ( ord_less_eq @ A @ ( X9 @ ( suc @ N3 ) ) @ ( X9 @ N3 ) )
=> ( order_antimono @ nat @ A @ X9 ) ) ) ).
% decseq_SucI
thf(fact_7783_decseq__SucD,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: nat > A,I2: nat] :
( ( order_antimono @ nat @ A @ A3 )
=> ( ord_less_eq @ A @ ( A3 @ ( suc @ I2 ) ) @ ( A3 @ I2 ) ) ) ) ).
% decseq_SucD
thf(fact_7784_decseq__def,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( order_antimono @ nat @ A )
= ( ^ [X6: nat > A] :
! [M2: nat,N2: nat] :
( ( ord_less_eq @ nat @ M2 @ N2 )
=> ( ord_less_eq @ A @ ( X6 @ N2 ) @ ( X6 @ M2 ) ) ) ) ) ) ).
% decseq_def
thf(fact_7785_decseqD,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [F3: nat > A,I2: nat,J3: nat] :
( ( order_antimono @ nat @ A @ F3 )
=> ( ( ord_less_eq @ nat @ I2 @ J3 )
=> ( ord_less_eq @ A @ ( F3 @ J3 ) @ ( F3 @ I2 ) ) ) ) ) ).
% decseqD
thf(fact_7786_eventually__ex,axiom,
! [B: $tType,A: $tType,P3: A > B > $o,F4: filter @ A] :
( ( eventually @ A
@ ^ [X4: A] :
? [X6: B] : ( P3 @ X4 @ X6 )
@ F4 )
= ( ? [Y8: A > B] :
( eventually @ A
@ ^ [X4: A] : ( P3 @ X4 @ ( Y8 @ X4 ) )
@ F4 ) ) ) ).
% eventually_ex
thf(fact_7787_Collect__ex__eq,axiom,
! [A: $tType,B: $tType,P3: A > B > $o] :
( ( collect @ A
@ ^ [X4: A] :
? [X6: B] : ( P3 @ X4 @ X6 ) )
= ( complete_Sup_Sup @ ( set @ A )
@ ( image @ B @ ( set @ A )
@ ^ [Y: B] :
( collect @ A
@ ^ [X4: A] : ( P3 @ X4 @ Y ) )
@ ( top_top @ ( set @ B ) ) ) ) ) ).
% Collect_ex_eq
thf(fact_7788_relcomp__unfold,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( relcomp @ A @ C @ B )
= ( ^ [R5: set @ ( product_prod @ A @ C ),S4: set @ ( product_prod @ C @ B )] :
( collect @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [X4: A,Z6: B] :
? [Y: C] :
( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ X4 @ Y ) @ R5 )
& ( member @ ( product_prod @ C @ B ) @ ( product_Pair @ C @ B @ Y @ Z6 ) @ S4 ) ) ) ) ) ) ).
% relcomp_unfold
thf(fact_7789_takeWhile__append,axiom,
! [A: $tType,Xs: list @ A,P3: A > $o,Ys: list @ A] :
( ( ! [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Xs ) )
=> ( P3 @ X5 ) )
=> ( ( takeWhile @ A @ P3 @ ( append @ A @ Xs @ Ys ) )
= ( append @ A @ Xs @ ( takeWhile @ A @ P3 @ Ys ) ) ) )
& ( ~ ! [X2: A] :
( ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
=> ( P3 @ X2 ) )
=> ( ( takeWhile @ A @ P3 @ ( append @ A @ Xs @ Ys ) )
= ( takeWhile @ A @ P3 @ Xs ) ) ) ) ).
% takeWhile_append
thf(fact_7790_dropWhile__append,axiom,
! [A: $tType,Xs: list @ A,P3: A > $o,Ys: list @ A] :
( ( ! [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Xs ) )
=> ( P3 @ X5 ) )
=> ( ( dropWhile @ A @ P3 @ ( append @ A @ Xs @ Ys ) )
= ( dropWhile @ A @ P3 @ Ys ) ) )
& ( ~ ! [X2: A] :
( ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
=> ( P3 @ X2 ) )
=> ( ( dropWhile @ A @ P3 @ ( append @ A @ Xs @ Ys ) )
= ( append @ A @ ( dropWhile @ A @ P3 @ Xs ) @ Ys ) ) ) ) ).
% dropWhile_append
thf(fact_7791_sorted__wrt_Oelims_I1_J,axiom,
! [A: $tType,X: A > A > $o,Xa: list @ A,Y2: $o] :
( ( ( sorted_wrt @ A @ X @ Xa )
= Y2 )
=> ( ( ( Xa
= ( nil @ A ) )
=> ~ Y2 )
=> ~ ! [X5: A,Ys4: list @ A] :
( ( Xa
= ( cons @ A @ X5 @ Ys4 ) )
=> ( Y2
= ( ~ ( ! [Y: A] :
( ( member @ A @ Y @ ( set2 @ A @ Ys4 ) )
=> ( X @ X5 @ Y ) )
& ( sorted_wrt @ A @ X @ Ys4 ) ) ) ) ) ) ) ).
% sorted_wrt.elims(1)
thf(fact_7792_sorted__wrt_Oelims_I2_J,axiom,
! [A: $tType,X: A > A > $o,Xa: list @ A] :
( ( sorted_wrt @ A @ X @ Xa )
=> ( ( Xa
!= ( nil @ A ) )
=> ~ ! [X5: A,Ys4: list @ A] :
( ( Xa
= ( cons @ A @ X5 @ Ys4 ) )
=> ~ ( ! [Xa2: A] :
( ( member @ A @ Xa2 @ ( set2 @ A @ Ys4 ) )
=> ( X @ X5 @ Xa2 ) )
& ( sorted_wrt @ A @ X @ Ys4 ) ) ) ) ) ).
% sorted_wrt.elims(2)
thf(fact_7793_set__conv__nth,axiom,
! [A: $tType] :
( ( set2 @ A )
= ( ^ [Xs2: list @ A] :
( collect @ A
@ ^ [Uu3: A] :
? [I: nat] :
( ( Uu3
= ( nth @ A @ Xs2 @ I ) )
& ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs2 ) ) ) ) ) ) ).
% set_conv_nth
thf(fact_7794_decseq__ge,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [X9: nat > A,L6: A,N: nat] :
( ( order_antimono @ nat @ A @ X9 )
=> ( ( filterlim @ nat @ A @ X9 @ ( topolo7230453075368039082e_nhds @ A @ L6 ) @ ( at_top @ nat ) )
=> ( ord_less_eq @ A @ L6 @ ( X9 @ N ) ) ) ) ) ).
% decseq_ge
thf(fact_7795_decseq__convergent,axiom,
! [X9: nat > real,B5: real] :
( ( order_antimono @ nat @ real @ X9 )
=> ( ! [I3: nat] : ( ord_less_eq @ real @ B5 @ ( X9 @ I3 ) )
=> ~ ! [L7: real] :
( ( filterlim @ nat @ real @ X9 @ ( topolo7230453075368039082e_nhds @ real @ L7 ) @ ( at_top @ nat ) )
=> ~ ! [I4: nat] : ( ord_less_eq @ real @ L7 @ ( X9 @ I4 ) ) ) ) ) ).
% decseq_convergent
thf(fact_7796_INT__decseq__offset,axiom,
! [A: $tType,F4: nat > ( set @ A ),N: nat] :
( ( order_antimono @ nat @ ( set @ A ) @ F4 )
=> ( ( complete_Inf_Inf @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ F4 @ ( top_top @ ( set @ nat ) ) ) )
= ( complete_Inf_Inf @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ F4 @ ( set_ord_atLeast @ nat @ N ) ) ) ) ) ).
% INT_decseq_offset
thf(fact_7797_nhds__countable,axiom,
! [A: $tType] :
( ( topolo3112930676232923870pology @ A )
=> ! [X: A] :
~ ! [X10: nat > ( set @ A )] :
( ( order_antimono @ nat @ ( set @ A ) @ X10 )
=> ( ! [N9: nat] : ( topolo1002775350975398744n_open @ A @ ( X10 @ N9 ) )
=> ( ! [N9: nat] : ( member @ A @ X @ ( X10 @ N9 ) )
=> ( ( topolo7230453075368039082e_nhds @ A @ X )
!= ( complete_Inf_Inf @ ( filter @ A )
@ ( image @ nat @ ( filter @ A )
@ ^ [N2: nat] : ( principal @ A @ ( X10 @ N2 ) )
@ ( top_top @ ( set @ nat ) ) ) ) ) ) ) ) ) ).
% nhds_countable
thf(fact_7798_INF__Lim,axiom,
! [A: $tType] :
( ( ( comple5582772986160207858norder @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [X9: nat > A,L: A] :
( ( order_antimono @ nat @ A @ X9 )
=> ( ( filterlim @ nat @ A @ X9 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ ( at_top @ nat ) )
=> ( ( complete_Inf_Inf @ A @ ( image @ nat @ A @ X9 @ ( top_top @ ( set @ nat ) ) ) )
= L ) ) ) ) ).
% INF_Lim
thf(fact_7799_LIMSEQ__INF,axiom,
! [A: $tType] :
( ( ( comple5582772986160207858norder @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [X9: nat > A] :
( ( order_antimono @ nat @ A @ X9 )
=> ( filterlim @ nat @ A @ X9 @ ( topolo7230453075368039082e_nhds @ A @ ( complete_Inf_Inf @ A @ ( image @ nat @ A @ X9 @ ( top_top @ ( set @ nat ) ) ) ) ) @ ( at_top @ nat ) ) ) ) ).
% LIMSEQ_INF
thf(fact_7800_VEBT__internal_Ovalid_H_Osimps_I2_J,axiom,
! [Mima2: option @ ( product_prod @ nat @ nat ),Deg: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,Deg4: nat] :
( ( vEBT_VEBT_valid @ ( vEBT_Node @ Mima2 @ Deg @ TreeList @ Summary ) @ Deg4 )
= ( ( Deg = Deg4 )
& ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ( vEBT_VEBT_valid @ X4 @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
& ( vEBT_VEBT_valid @ Summary @ ( minus_minus @ nat @ Deg @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
& ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( case_option @ $o @ ( product_prod @ nat @ nat )
@ ( ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X6 )
& ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X6 ) ) )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [Mi2: nat,Ma2: nat] :
( ( ord_less_eq @ nat @ Mi2 @ Ma2 )
& ( ord_less @ nat @ Ma2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg ) )
& ! [I: nat] :
( ( ord_less @ nat @ I @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
=> ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ I ) @ X6 ) )
= ( vEBT_V8194947554948674370ptions @ Summary @ I ) ) )
& ( ( Mi2 = Ma2 )
=> ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X6 ) ) )
& ( ( Mi2 != Ma2 )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList @ Ma2 )
& ! [X4: nat] :
( ( ord_less @ nat @ X4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg ) )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList @ X4 )
=> ( ( ord_less @ nat @ Mi2 @ X4 )
& ( ord_less_eq @ nat @ X4 @ Ma2 ) ) ) ) ) ) ) )
@ Mima2 ) ) ) ).
% VEBT_internal.valid'.simps(2)
thf(fact_7801_tendsto__at__right__sequentially,axiom,
! [C: $tType,B: $tType] :
( ( ( topolo3112930676232923870pology @ B )
& ( topolo1944317154257567458pology @ B )
& ( topolo4958980785337419405_space @ C ) )
=> ! [A2: B,B2: B,X9: B > C,L6: C] :
( ( ord_less @ B @ A2 @ B2 )
=> ( ! [S7: nat > B] :
( ! [N9: nat] : ( ord_less @ B @ A2 @ ( S7 @ N9 ) )
=> ( ! [N9: nat] : ( ord_less @ B @ ( S7 @ N9 ) @ B2 )
=> ( ( order_antimono @ nat @ B @ S7 )
=> ( ( filterlim @ nat @ B @ S7 @ ( topolo7230453075368039082e_nhds @ B @ A2 ) @ ( at_top @ nat ) )
=> ( filterlim @ nat @ C
@ ^ [N2: nat] : ( X9 @ ( S7 @ N2 ) )
@ ( topolo7230453075368039082e_nhds @ C @ L6 )
@ ( at_top @ nat ) ) ) ) ) )
=> ( filterlim @ B @ C @ X9 @ ( topolo7230453075368039082e_nhds @ C @ L6 ) @ ( topolo174197925503356063within @ B @ A2 @ ( set_ord_greaterThan @ B @ A2 ) ) ) ) ) ) ).
% tendsto_at_right_sequentially
thf(fact_7802_VEBT__internal_Ovalid_H_Oelims_I1_J,axiom,
! [X: vEBT_VEBT,Xa: nat,Y2: $o] :
( ( ( vEBT_VEBT_valid @ X @ Xa )
= Y2 )
=> ( ( ? [Uu2: $o,Uv2: $o] :
( X
= ( vEBT_Leaf @ Uu2 @ Uv2 ) )
=> ( Y2
= ( Xa
!= ( one_one @ nat ) ) ) )
=> ~ ! [Mima: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( Y2
= ( ~ ( ( Deg2 = Xa )
& ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ( vEBT_VEBT_valid @ X4 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
& ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
& ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( case_option @ $o @ ( product_prod @ nat @ nat )
@ ( ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X6 )
& ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X6 ) ) )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [Mi2: nat,Ma2: nat] :
( ( ord_less_eq @ nat @ Mi2 @ Ma2 )
& ( ord_less @ nat @ Ma2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
& ! [I: nat] :
( ( ord_less @ nat @ I @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
=> ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I ) @ X6 ) )
= ( vEBT_V8194947554948674370ptions @ Summary2 @ I ) ) )
& ( ( Mi2 = Ma2 )
=> ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X6 ) ) )
& ( ( Mi2 != Ma2 )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList2 @ Ma2 )
& ! [X4: nat] :
( ( ord_less @ nat @ X4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList2 @ X4 )
=> ( ( ord_less @ nat @ Mi2 @ X4 )
& ( ord_less_eq @ nat @ X4 @ Ma2 ) ) ) ) ) ) ) )
@ Mima ) ) ) ) ) ) ) ).
% VEBT_internal.valid'.elims(1)
thf(fact_7803_VEBT__internal_Ovalid_H_Oelims_I2_J,axiom,
! [X: vEBT_VEBT,Xa: nat] :
( ( vEBT_VEBT_valid @ X @ Xa )
=> ( ( ? [Uu2: $o,Uv2: $o] :
( X
= ( vEBT_Leaf @ Uu2 @ Uv2 ) )
=> ( Xa
!= ( one_one @ nat ) ) )
=> ~ ! [Mima: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary2 ) )
=> ~ ( ( Deg2 = Xa )
& ! [X2: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ( vEBT_VEBT_valid @ X2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
& ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
& ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( case_option @ $o @ ( product_prod @ nat @ nat )
@ ( ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X6 )
& ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X6 ) ) )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [Mi2: nat,Ma2: nat] :
( ( ord_less_eq @ nat @ Mi2 @ Ma2 )
& ( ord_less @ nat @ Ma2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
& ! [I: nat] :
( ( ord_less @ nat @ I @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
=> ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I ) @ X6 ) )
= ( vEBT_V8194947554948674370ptions @ Summary2 @ I ) ) )
& ( ( Mi2 = Ma2 )
=> ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X6 ) ) )
& ( ( Mi2 != Ma2 )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList2 @ Ma2 )
& ! [X4: nat] :
( ( ord_less @ nat @ X4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList2 @ X4 )
=> ( ( ord_less @ nat @ Mi2 @ X4 )
& ( ord_less_eq @ nat @ X4 @ Ma2 ) ) ) ) ) ) ) )
@ Mima ) ) ) ) ) ).
% VEBT_internal.valid'.elims(2)
thf(fact_7804_VEBT__internal_Ovalid_H_Opelims_I1_J,axiom,
! [X: vEBT_VEBT,Xa: nat,Y2: $o] :
( ( ( vEBT_VEBT_valid @ X @ Xa )
= Y2 )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_VEBT_valid_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa ) )
=> ( ! [Uu2: $o,Uv2: $o] :
( ( X
= ( vEBT_Leaf @ Uu2 @ Uv2 ) )
=> ( ( Y2
= ( Xa
= ( one_one @ nat ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_VEBT_valid_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa ) ) ) )
=> ~ ! [Mima: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( ( Y2
= ( ( Deg2 = Xa )
& ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ( vEBT_VEBT_valid @ X4 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
& ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
& ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( case_option @ $o @ ( product_prod @ nat @ nat )
@ ( ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X6 )
& ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X6 ) ) )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [Mi2: nat,Ma2: nat] :
( ( ord_less_eq @ nat @ Mi2 @ Ma2 )
& ( ord_less @ nat @ Ma2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
& ! [I: nat] :
( ( ord_less @ nat @ I @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
=> ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I ) @ X6 ) )
= ( vEBT_V8194947554948674370ptions @ Summary2 @ I ) ) )
& ( ( Mi2 = Ma2 )
=> ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X6 ) ) )
& ( ( Mi2 != Ma2 )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList2 @ Ma2 )
& ! [X4: nat] :
( ( ord_less @ nat @ X4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList2 @ X4 )
=> ( ( ord_less @ nat @ Mi2 @ X4 )
& ( ord_less_eq @ nat @ X4 @ Ma2 ) ) ) ) ) ) ) )
@ Mima ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_VEBT_valid_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ).
% VEBT_internal.valid'.pelims(1)
thf(fact_7805_VEBT__internal_Ovalid_H_Opelims_I2_J,axiom,
! [X: vEBT_VEBT,Xa: nat] :
( ( vEBT_VEBT_valid @ X @ Xa )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_VEBT_valid_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa ) )
=> ( ! [Uu2: $o,Uv2: $o] :
( ( X
= ( vEBT_Leaf @ Uu2 @ Uv2 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_VEBT_valid_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa ) )
=> ( Xa
!= ( one_one @ nat ) ) ) )
=> ~ ! [Mima: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_VEBT_valid_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary2 ) @ Xa ) )
=> ~ ( ( Deg2 = Xa )
& ! [X2: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ( vEBT_VEBT_valid @ X2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
& ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
& ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( case_option @ $o @ ( product_prod @ nat @ nat )
@ ( ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X6 )
& ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X6 ) ) )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [Mi2: nat,Ma2: nat] :
( ( ord_less_eq @ nat @ Mi2 @ Ma2 )
& ( ord_less @ nat @ Ma2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
& ! [I: nat] :
( ( ord_less @ nat @ I @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
=> ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I ) @ X6 ) )
= ( vEBT_V8194947554948674370ptions @ Summary2 @ I ) ) )
& ( ( Mi2 = Ma2 )
=> ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X6 ) ) )
& ( ( Mi2 != Ma2 )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList2 @ Ma2 )
& ! [X4: nat] :
( ( ord_less @ nat @ X4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList2 @ X4 )
=> ( ( ord_less @ nat @ Mi2 @ X4 )
& ( ord_less_eq @ nat @ X4 @ Ma2 ) ) ) ) ) ) ) )
@ Mima ) ) ) ) ) ) ) ).
% VEBT_internal.valid'.pelims(2)
thf(fact_7806_Inter__eq,axiom,
! [A: $tType] :
( ( complete_Inf_Inf @ ( set @ A ) )
= ( ^ [A7: set @ ( set @ A )] :
( collect @ A
@ ^ [X4: A] :
! [Y: set @ A] :
( ( member @ ( set @ A ) @ Y @ A7 )
=> ( member @ A @ X4 @ Y ) ) ) ) ) ).
% Inter_eq
thf(fact_7807_Inf__Sup,axiom,
! [A: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [A3: set @ ( set @ A )] :
( ( complete_Inf_Inf @ A @ ( image @ ( set @ A ) @ A @ ( complete_Sup_Sup @ A ) @ A3 ) )
= ( complete_Sup_Sup @ A
@ ( image @ ( set @ A ) @ A @ ( complete_Inf_Inf @ A )
@ ( collect @ ( set @ A )
@ ^ [Uu3: set @ A] :
? [F5: ( set @ A ) > A] :
( ( Uu3
= ( image @ ( set @ A ) @ A @ F5 @ A3 ) )
& ! [X4: set @ A] :
( ( member @ ( set @ A ) @ X4 @ A3 )
=> ( member @ A @ ( F5 @ X4 ) @ X4 ) ) ) ) ) ) ) ) ).
% Inf_Sup
thf(fact_7808_Sup__Inf,axiom,
! [A: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [A3: set @ ( set @ A )] :
( ( complete_Sup_Sup @ A @ ( image @ ( set @ A ) @ A @ ( complete_Inf_Inf @ A ) @ A3 ) )
= ( complete_Inf_Inf @ A
@ ( image @ ( set @ A ) @ A @ ( complete_Sup_Sup @ A )
@ ( collect @ ( set @ A )
@ ^ [Uu3: set @ A] :
? [F5: ( set @ A ) > A] :
( ( Uu3
= ( image @ ( set @ A ) @ A @ F5 @ A3 ) )
& ! [X4: set @ A] :
( ( member @ ( set @ A ) @ X4 @ A3 )
=> ( member @ A @ ( F5 @ X4 ) @ X4 ) ) ) ) ) ) ) ) ).
% Sup_Inf
thf(fact_7809_INF__SUP__set,axiom,
! [A: $tType,B: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [G: B > A,A3: set @ ( set @ B )] :
( ( complete_Inf_Inf @ A
@ ( image @ ( set @ B ) @ A
@ ^ [B7: set @ B] : ( complete_Sup_Sup @ A @ ( image @ B @ A @ G @ B7 ) )
@ A3 ) )
= ( complete_Sup_Sup @ A
@ ( image @ ( set @ B ) @ A
@ ^ [B7: set @ B] : ( complete_Inf_Inf @ A @ ( image @ B @ A @ G @ B7 ) )
@ ( collect @ ( set @ B )
@ ^ [Uu3: set @ B] :
? [F5: ( set @ B ) > B] :
( ( Uu3
= ( image @ ( set @ B ) @ B @ F5 @ A3 ) )
& ! [X4: set @ B] :
( ( member @ ( set @ B ) @ X4 @ A3 )
=> ( member @ B @ ( F5 @ X4 ) @ X4 ) ) ) ) ) ) ) ) ).
% INF_SUP_set
thf(fact_7810_SUP__INF__set,axiom,
! [A: $tType,B: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [G: B > A,A3: set @ ( set @ B )] :
( ( complete_Sup_Sup @ A
@ ( image @ ( set @ B ) @ A
@ ^ [X4: set @ B] : ( complete_Inf_Inf @ A @ ( image @ B @ A @ G @ X4 ) )
@ A3 ) )
= ( complete_Inf_Inf @ A
@ ( image @ ( set @ B ) @ A
@ ^ [X4: set @ B] : ( complete_Sup_Sup @ A @ ( image @ B @ A @ G @ X4 ) )
@ ( collect @ ( set @ B )
@ ^ [Uu3: set @ B] :
? [F5: ( set @ B ) > B] :
( ( Uu3
= ( image @ ( set @ B ) @ B @ F5 @ A3 ) )
& ! [X4: set @ B] :
( ( member @ ( set @ B ) @ X4 @ A3 )
=> ( member @ B @ ( F5 @ X4 ) @ X4 ) ) ) ) ) ) ) ) ).
% SUP_INF_set
thf(fact_7811_Pow__Compl,axiom,
! [A: $tType,A3: set @ A] :
( ( pow2 @ A @ ( uminus_uminus @ ( set @ A ) @ A3 ) )
= ( collect @ ( set @ A )
@ ^ [Uu3: set @ A] :
? [B7: set @ A] :
( ( Uu3
= ( uminus_uminus @ ( set @ A ) @ B7 ) )
& ( member @ ( set @ A ) @ A3 @ ( pow2 @ A @ B7 ) ) ) ) ) ).
% Pow_Compl
thf(fact_7812_Sup__int__def,axiom,
( ( complete_Sup_Sup @ int )
= ( ^ [X6: set @ int] :
( the @ int
@ ^ [X4: int] :
( ( member @ int @ X4 @ X6 )
& ! [Y: int] :
( ( member @ int @ Y @ X6 )
=> ( ord_less_eq @ int @ Y @ X4 ) ) ) ) ) ) ).
% Sup_int_def
thf(fact_7813_Sup__Inf__le,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A3: set @ ( set @ A )] :
( ord_less_eq @ A
@ ( complete_Sup_Sup @ A
@ ( image @ ( set @ A ) @ A @ ( complete_Inf_Inf @ A )
@ ( collect @ ( set @ A )
@ ^ [Uu3: set @ A] :
? [F5: ( set @ A ) > A] :
( ( Uu3
= ( image @ ( set @ A ) @ A @ F5 @ A3 ) )
& ! [X4: set @ A] :
( ( member @ ( set @ A ) @ X4 @ A3 )
=> ( member @ A @ ( F5 @ X4 ) @ X4 ) ) ) ) ) )
@ ( complete_Inf_Inf @ A @ ( image @ ( set @ A ) @ A @ ( complete_Sup_Sup @ A ) @ A3 ) ) ) ) ).
% Sup_Inf_le
thf(fact_7814_Inf__Sup__le,axiom,
! [A: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [A3: set @ ( set @ A )] :
( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image @ ( set @ A ) @ A @ ( complete_Sup_Sup @ A ) @ A3 ) )
@ ( complete_Sup_Sup @ A
@ ( image @ ( set @ A ) @ A @ ( complete_Inf_Inf @ A )
@ ( collect @ ( set @ A )
@ ^ [Uu3: set @ A] :
? [F5: ( set @ A ) > A] :
( ( Uu3
= ( image @ ( set @ A ) @ A @ F5 @ A3 ) )
& ! [X4: set @ A] :
( ( member @ ( set @ A ) @ X4 @ A3 )
=> ( member @ A @ ( F5 @ X4 ) @ X4 ) ) ) ) ) ) ) ) ).
% Inf_Sup_le
thf(fact_7815_Union__maximal__sets,axiom,
! [A: $tType,F18: set @ ( set @ A )] :
( ( finite_finite2 @ ( set @ A ) @ F18 )
=> ( ( complete_Sup_Sup @ ( set @ A )
@ ( collect @ ( set @ A )
@ ^ [T9: set @ A] :
( ( member @ ( set @ A ) @ T9 @ F18 )
& ! [X4: set @ A] :
( ( member @ ( set @ A ) @ X4 @ F18 )
=> ~ ( ord_less @ ( set @ A ) @ T9 @ X4 ) ) ) ) )
= ( complete_Sup_Sup @ ( set @ A ) @ F18 ) ) ) ).
% Union_maximal_sets
thf(fact_7816_Inf__filter__def,axiom,
! [A: $tType] :
( ( complete_Inf_Inf @ ( filter @ A ) )
= ( ^ [S6: set @ ( filter @ A )] :
( complete_Sup_Sup @ ( filter @ A )
@ ( collect @ ( filter @ A )
@ ^ [F10: filter @ A] :
! [X4: filter @ A] :
( ( member @ ( filter @ A ) @ X4 @ S6 )
=> ( ord_less_eq @ ( filter @ A ) @ F10 @ X4 ) ) ) ) ) ) ).
% Inf_filter_def
thf(fact_7817_VEBT__internal_Ovalid_H_Opelims_I3_J,axiom,
! [X: vEBT_VEBT,Xa: nat] :
( ~ ( vEBT_VEBT_valid @ X @ Xa )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_VEBT_valid_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa ) )
=> ( ! [Uu2: $o,Uv2: $o] :
( ( X
= ( vEBT_Leaf @ Uu2 @ Uv2 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_VEBT_valid_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa ) )
=> ( Xa
= ( one_one @ nat ) ) ) )
=> ~ ! [Mima: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_VEBT_valid_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary2 ) @ Xa ) )
=> ( ( Deg2 = Xa )
& ! [X5: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X5 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ( vEBT_VEBT_valid @ X5 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
& ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
& ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( case_option @ $o @ ( product_prod @ nat @ nat )
@ ( ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X6 )
& ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X6 ) ) )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [Mi2: nat,Ma2: nat] :
( ( ord_less_eq @ nat @ Mi2 @ Ma2 )
& ( ord_less @ nat @ Ma2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
& ! [I: nat] :
( ( ord_less @ nat @ I @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
=> ( ( ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I ) @ X6 ) )
= ( vEBT_V8194947554948674370ptions @ Summary2 @ I ) ) )
& ( ( Mi2 = Ma2 )
=> ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X6: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X6 ) ) )
& ( ( Mi2 != Ma2 )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList2 @ Ma2 )
& ! [X4: nat] :
( ( ord_less @ nat @ X4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList2 @ X4 )
=> ( ( ord_less @ nat @ Mi2 @ X4 )
& ( ord_less_eq @ nat @ X4 @ Ma2 ) ) ) ) ) ) ) )
@ Mima ) ) ) ) ) ) ) ).
% VEBT_internal.valid'.pelims(3)
thf(fact_7818_finite__Inf__Sup,axiom,
! [A: $tType] :
( ( finite8700451911770168679attice @ A )
=> ! [A3: set @ ( set @ A )] :
( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image @ ( set @ A ) @ A @ ( complete_Sup_Sup @ A ) @ A3 ) )
@ ( complete_Sup_Sup @ A
@ ( image @ ( set @ A ) @ A @ ( complete_Inf_Inf @ A )
@ ( collect @ ( set @ A )
@ ^ [Uu3: set @ A] :
? [F5: ( set @ A ) > A] :
( ( Uu3
= ( image @ ( set @ A ) @ A @ F5 @ A3 ) )
& ! [X4: set @ A] :
( ( member @ ( set @ A ) @ X4 @ A3 )
=> ( member @ A @ ( F5 @ X4 ) @ X4 ) ) ) ) ) ) ) ) ).
% finite_Inf_Sup
thf(fact_7819_lexn__conv,axiom,
! [A: $tType] :
( ( lexn @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A ),N2: nat] :
( collect @ ( product_prod @ ( list @ A ) @ ( list @ A ) )
@ ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ $o
@ ^ [Xs2: list @ A,Ys3: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= N2 )
& ( ( size_size @ ( list @ A ) @ Ys3 )
= N2 )
& ? [Xys2: list @ A,X4: A,Y: A,Xs6: list @ A,Ys7: list @ A] :
( ( Xs2
= ( append @ A @ Xys2 @ ( cons @ A @ X4 @ Xs6 ) ) )
& ( Ys3
= ( append @ A @ Xys2 @ ( cons @ A @ Y @ Ys7 ) ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y ) @ R5 ) ) ) ) ) ) ) ).
% lexn_conv
thf(fact_7820_lexn_Osimps_I1_J,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( lexn @ A @ R @ ( zero_zero @ nat ) )
= ( bot_bot @ ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ) ) ).
% lexn.simps(1)
thf(fact_7821_lexn__length,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,R: set @ ( product_prod @ A @ A ),N: nat] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( lexn @ A @ R @ N ) )
=> ( ( ( size_size @ ( list @ A ) @ Xs )
= N )
& ( ( size_size @ ( list @ A ) @ Ys )
= N ) ) ) ).
% lexn_length
thf(fact_7822_lex__def,axiom,
! [A: $tType] :
( ( lex @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A )] : ( complete_Sup_Sup @ ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) @ ( image @ nat @ ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) @ ( lexn @ A @ R5 ) @ ( top_top @ ( set @ nat ) ) ) ) ) ) ).
% lex_def
thf(fact_7823_finite__inf__Sup,axiom,
! [A: $tType] :
( ( finite8700451911770168679attice @ A )
=> ! [A2: A,A3: set @ A] :
( ( inf_inf @ A @ A2 @ ( complete_Sup_Sup @ A @ A3 ) )
= ( complete_Sup_Sup @ A
@ ( collect @ A
@ ^ [Uu3: A] :
? [B4: A] :
( ( Uu3
= ( inf_inf @ A @ A2 @ B4 ) )
& ( member @ A @ B4 @ A3 ) ) ) ) ) ) ).
% finite_inf_Sup
thf(fact_7824_sorted__wrt_Opelims_I2_J,axiom,
! [A: $tType,X: A > A > $o,Xa: list @ A] :
( ( sorted_wrt @ A @ X @ Xa )
=> ( ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( sorted_wrt_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X @ Xa ) )
=> ( ( ( Xa
= ( nil @ A ) )
=> ~ ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( sorted_wrt_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X @ ( nil @ A ) ) ) )
=> ~ ! [X5: A,Ys4: list @ A] :
( ( Xa
= ( cons @ A @ X5 @ Ys4 ) )
=> ( ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( sorted_wrt_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X @ ( cons @ A @ X5 @ Ys4 ) ) )
=> ~ ( ! [Xa2: A] :
( ( member @ A @ Xa2 @ ( set2 @ A @ Ys4 ) )
=> ( X @ X5 @ Xa2 ) )
& ( sorted_wrt @ A @ X @ Ys4 ) ) ) ) ) ) ) ).
% sorted_wrt.pelims(2)
thf(fact_7825_sorted__wrt_Opelims_I1_J,axiom,
! [A: $tType,X: A > A > $o,Xa: list @ A,Y2: $o] :
( ( ( sorted_wrt @ A @ X @ Xa )
= Y2 )
=> ( ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( sorted_wrt_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X @ Xa ) )
=> ( ( ( Xa
= ( nil @ A ) )
=> ( Y2
=> ~ ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( sorted_wrt_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X @ ( nil @ A ) ) ) ) )
=> ~ ! [X5: A,Ys4: list @ A] :
( ( Xa
= ( cons @ A @ X5 @ Ys4 ) )
=> ( ( Y2
= ( ! [Y: A] :
( ( member @ A @ Y @ ( set2 @ A @ Ys4 ) )
=> ( X @ X5 @ Y ) )
& ( sorted_wrt @ A @ X @ Ys4 ) ) )
=> ~ ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( sorted_wrt_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X @ ( cons @ A @ X5 @ Ys4 ) ) ) ) ) ) ) ) ).
% sorted_wrt.pelims(1)
thf(fact_7826_sorted__wrt_Opelims_I3_J,axiom,
! [A: $tType,X: A > A > $o,Xa: list @ A] :
( ~ ( sorted_wrt @ A @ X @ Xa )
=> ( ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( sorted_wrt_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X @ Xa ) )
=> ~ ! [X5: A,Ys4: list @ A] :
( ( Xa
= ( cons @ A @ X5 @ Ys4 ) )
=> ( ( accp @ ( product_prod @ ( A > A > $o ) @ ( list @ A ) ) @ ( sorted_wrt_rel @ A ) @ ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ X @ ( cons @ A @ X5 @ Ys4 ) ) )
=> ( ! [Xa3: A] :
( ( member @ A @ Xa3 @ ( set2 @ A @ Ys4 ) )
=> ( X @ X5 @ Xa3 ) )
& ( sorted_wrt @ A @ X @ Ys4 ) ) ) ) ) ) ).
% sorted_wrt.pelims(3)
thf(fact_7827_mlex__eq,axiom,
! [A: $tType] :
( ( mlex_prod @ A )
= ( ^ [F5: A > nat,R6: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ A @ A )
@ ( product_case_prod @ A @ A @ $o
@ ^ [X4: A,Y: A] :
( ( ord_less @ nat @ ( F5 @ X4 ) @ ( F5 @ Y ) )
| ( ( ord_less_eq @ nat @ ( F5 @ X4 ) @ ( F5 @ Y ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y ) @ R6 ) ) ) ) ) ) ) ).
% mlex_eq
thf(fact_7828_map__filter__on__comp,axiom,
! [A: $tType,C: $tType,B: $tType,G: B > A,Y3: set @ B,X9: set @ A,F4: filter @ B,F3: A > C] :
( ( ord_less_eq @ ( set @ A ) @ ( image @ B @ A @ G @ Y3 ) @ X9 )
=> ( ( eventually @ B
@ ^ [X4: B] : ( member @ B @ X4 @ Y3 )
@ F4 )
=> ( ( map_filter_on @ A @ C @ X9 @ F3 @ ( map_filter_on @ B @ A @ Y3 @ G @ F4 ) )
= ( map_filter_on @ B @ C @ Y3 @ ( comp @ A @ C @ B @ F3 @ G ) @ F4 ) ) ) ) ).
% map_filter_on_comp
thf(fact_7829_eventually__map__filter__on,axiom,
! [B: $tType,A: $tType,X9: set @ A,F4: filter @ A,P3: B > $o,F3: A > B] :
( ( eventually @ A
@ ^ [X4: A] : ( member @ A @ X4 @ X9 )
@ F4 )
=> ( ( eventually @ B @ P3 @ ( map_filter_on @ A @ B @ X9 @ F3 @ F4 ) )
= ( eventually @ A
@ ^ [X4: A] :
( ( P3 @ ( F3 @ X4 ) )
& ( member @ A @ X4 @ X9 ) )
@ F4 ) ) ) ).
% eventually_map_filter_on
thf(fact_7830_mlex__less,axiom,
! [A: $tType,F3: A > nat,X: A,Y2: A,R4: set @ ( product_prod @ A @ A )] :
( ( ord_less @ nat @ ( F3 @ X ) @ ( F3 @ Y2 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y2 ) @ ( mlex_prod @ A @ F3 @ R4 ) ) ) ).
% mlex_less
thf(fact_7831_mlex__iff,axiom,
! [A: $tType,X: A,Y2: A,F3: A > nat,R4: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y2 ) @ ( mlex_prod @ A @ F3 @ R4 ) )
= ( ( ord_less @ nat @ ( F3 @ X ) @ ( F3 @ Y2 ) )
| ( ( ( F3 @ X )
= ( F3 @ Y2 ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y2 ) @ R4 ) ) ) ) ).
% mlex_iff
thf(fact_7832_mlex__leq,axiom,
! [A: $tType,F3: A > nat,X: A,Y2: A,R4: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ nat @ ( F3 @ X ) @ ( F3 @ Y2 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y2 ) @ R4 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y2 ) @ ( mlex_prod @ A @ F3 @ R4 ) ) ) ) ).
% mlex_leq
thf(fact_7833_open__generated__order,axiom,
! [A: $tType] :
( ( topolo2564578578187576103pology @ A )
=> ( ( topolo1002775350975398744n_open @ A )
= ( topolo8378437560675496660pology @ A @ ( sup_sup @ ( set @ ( set @ A ) ) @ ( image @ A @ ( set @ A ) @ ( set_ord_lessThan @ A ) @ ( top_top @ ( set @ A ) ) ) @ ( image @ A @ ( set @ A ) @ ( set_ord_greaterThan @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% open_generated_order
thf(fact_7834_relpow__finite__bounded1,axiom,
! [A: $tType,R4: set @ ( product_prod @ A @ A ),K2: nat] :
( ( finite_finite2 @ ( product_prod @ A @ A ) @ R4 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K2 )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ K2 @ R4 )
@ ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ A ) )
@ ( image @ nat @ ( set @ ( product_prod @ A @ A ) )
@ ^ [N2: nat] : ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N2 @ R4 )
@ ( collect @ nat
@ ^ [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
& ( ord_less_eq @ nat @ N2 @ ( finite_card @ ( product_prod @ A @ A ) @ R4 ) ) ) ) ) ) ) ) ) ).
% relpow_finite_bounded1
thf(fact_7835_relpow__1,axiom,
! [A: $tType,R4: set @ ( product_prod @ A @ A )] :
( ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( one_one @ nat ) @ R4 )
= R4 ) ).
% relpow_1
thf(fact_7836_finite__relpow,axiom,
! [A: $tType,R4: set @ ( product_prod @ A @ A ),N: nat] :
( ( finite_finite2 @ ( product_prod @ A @ A ) @ R4 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( finite_finite2 @ ( product_prod @ A @ A ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R4 ) ) ) ) ).
% finite_relpow
thf(fact_7837_relpowp__relpow__eq,axiom,
! [A: $tType,N: nat,R4: set @ ( product_prod @ A @ A )] :
( ( compow @ ( A > A > $o ) @ N
@ ^ [X4: A,Y: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y ) @ R4 ) )
= ( ^ [X4: A,Y: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R4 ) ) ) ) ).
% relpowp_relpow_eq
thf(fact_7838_relpow__add,axiom,
! [A: $tType,M: nat,N: nat,R4: set @ ( product_prod @ A @ A )] :
( ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( plus_plus @ nat @ M @ N ) @ R4 )
= ( relcomp @ A @ A @ A @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ M @ R4 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R4 ) ) ) ).
% relpow_add
thf(fact_7839_open__bool__def,axiom,
( ( topolo1002775350975398744n_open @ $o )
= ( topolo8378437560675496660pology @ $o @ ( sup_sup @ ( set @ ( set @ $o ) ) @ ( image @ $o @ ( set @ $o ) @ ( set_ord_lessThan @ $o ) @ ( top_top @ ( set @ $o ) ) ) @ ( image @ $o @ ( set @ $o ) @ ( set_ord_greaterThan @ $o ) @ ( top_top @ ( set @ $o ) ) ) ) ) ) ).
% open_bool_def
thf(fact_7840_generate__topology__Union,axiom,
! [B: $tType,A: $tType,I6: set @ A,S3: set @ ( set @ B ),K5: A > ( set @ B )] :
( ! [K: A] :
( ( member @ A @ K @ I6 )
=> ( topolo8378437560675496660pology @ B @ S3 @ ( K5 @ K ) ) )
=> ( topolo8378437560675496660pology @ B @ S3 @ ( complete_Sup_Sup @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ K5 @ I6 ) ) ) ) ).
% generate_topology_Union
thf(fact_7841_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_7842_open__int__def,axiom,
( ( topolo1002775350975398744n_open @ int )
= ( topolo8378437560675496660pology @ int @ ( sup_sup @ ( set @ ( set @ int ) ) @ ( image @ int @ ( set @ int ) @ ( set_ord_lessThan @ int ) @ ( top_top @ ( set @ int ) ) ) @ ( image @ int @ ( set @ int ) @ ( set_ord_greaterThan @ int ) @ ( top_top @ ( set @ int ) ) ) ) ) ) ).
% open_int_def
thf(fact_7843_relpow__fun__conv,axiom,
! [A: $tType,A2: A,B2: A,N: nat,R4: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R4 ) )
= ( ? [F5: nat > A] :
( ( ( F5 @ ( zero_zero @ nat ) )
= A2 )
& ( ( F5 @ N )
= B2 )
& ! [I: nat] :
( ( ord_less @ nat @ I @ N )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( F5 @ I ) @ ( F5 @ ( suc @ I ) ) ) @ R4 ) ) ) ) ) ).
% relpow_fun_conv
thf(fact_7844_relpow__finite__bounded,axiom,
! [A: $tType,R4: set @ ( product_prod @ A @ A ),K2: nat] :
( ( finite_finite2 @ ( product_prod @ A @ A ) @ R4 )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ K2 @ R4 )
@ ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ A ) )
@ ( image @ nat @ ( set @ ( product_prod @ A @ A ) )
@ ^ [N2: nat] : ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N2 @ R4 )
@ ( collect @ nat
@ ^ [N2: nat] : ( ord_less_eq @ nat @ N2 @ ( finite_card @ ( product_prod @ A @ A ) @ R4 ) ) ) ) ) ) ) ).
% relpow_finite_bounded
thf(fact_7845_open__nat__def,axiom,
( ( topolo1002775350975398744n_open @ nat )
= ( topolo8378437560675496660pology @ nat @ ( sup_sup @ ( set @ ( set @ nat ) ) @ ( image @ nat @ ( set @ nat ) @ ( set_ord_lessThan @ nat ) @ ( top_top @ ( set @ nat ) ) ) @ ( image @ nat @ ( set @ nat ) @ ( set_ord_greaterThan @ nat ) @ ( top_top @ ( set @ nat ) ) ) ) ) ) ).
% open_nat_def
thf(fact_7846_nhds__generated__topology,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [T5: set @ ( set @ A ),X: A] :
( ( ( topolo1002775350975398744n_open @ A )
= ( topolo8378437560675496660pology @ A @ T5 ) )
=> ( ( topolo7230453075368039082e_nhds @ A @ X )
= ( complete_Inf_Inf @ ( filter @ A )
@ ( image @ ( set @ A ) @ ( filter @ A ) @ ( principal @ A )
@ ( collect @ ( set @ A )
@ ^ [S6: set @ A] :
( ( member @ ( set @ A ) @ S6 @ T5 )
& ( member @ A @ X @ S6 ) ) ) ) ) ) ) ) ).
% nhds_generated_topology
thf(fact_7847_ntrancl__def,axiom,
! [A: $tType] :
( ( transitive_ntrancl @ A )
= ( ^ [N2: nat,R6: set @ ( product_prod @ A @ A )] :
( complete_Sup_Sup @ ( set @ ( product_prod @ A @ A ) )
@ ( image @ nat @ ( set @ ( product_prod @ A @ A ) )
@ ^ [I: nat] : ( compow @ ( set @ ( product_prod @ A @ A ) ) @ I @ R6 )
@ ( collect @ nat
@ ^ [I: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ I )
& ( ord_less_eq @ nat @ I @ ( suc @ N2 ) ) ) ) ) ) ) ) ).
% ntrancl_def
thf(fact_7848_trancl__finite__eq__relpow,axiom,
! [A: $tType,R4: set @ ( product_prod @ A @ A )] :
( ( finite_finite2 @ ( product_prod @ A @ A ) @ R4 )
=> ( ( transitive_trancl @ A @ R4 )
= ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ A ) )
@ ( image @ nat @ ( set @ ( product_prod @ A @ A ) )
@ ^ [N2: nat] : ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N2 @ R4 )
@ ( collect @ nat
@ ^ [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
& ( ord_less_eq @ nat @ N2 @ ( finite_card @ ( product_prod @ A @ A ) @ R4 ) ) ) ) ) ) ) ) ).
% trancl_finite_eq_relpow
thf(fact_7849_trancl__power,axiom,
! [A: $tType,P5: product_prod @ A @ A,R4: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ P5 @ ( transitive_trancl @ A @ R4 ) )
= ( ? [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
& ( member @ ( product_prod @ A @ A ) @ P5 @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N2 @ R4 ) ) ) ) ) ).
% trancl_power
thf(fact_7850_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_7851_finite__trancl__ntranl,axiom,
! [A: $tType,R4: set @ ( product_prod @ A @ A )] :
( ( finite_finite2 @ ( product_prod @ A @ A ) @ R4 )
=> ( ( transitive_trancl @ A @ R4 )
= ( transitive_ntrancl @ A @ ( minus_minus @ nat @ ( finite_card @ ( product_prod @ A @ A ) @ R4 ) @ ( one_one @ nat ) ) @ R4 ) ) ) ).
% finite_trancl_ntranl
thf(fact_7852_trancl__mono,axiom,
! [A: $tType,P5: product_prod @ A @ A,R: set @ ( product_prod @ A @ A ),S: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ P5 @ ( transitive_trancl @ A @ R ) )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R @ S )
=> ( member @ ( product_prod @ A @ A ) @ P5 @ ( transitive_trancl @ A @ S ) ) ) ) ).
% trancl_mono
thf(fact_7853_trancl__Int__subset,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),S: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R @ S )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( relcomp @ A @ A @ A @ ( inf_inf @ ( set @ ( product_prod @ A @ A ) ) @ ( transitive_trancl @ A @ R ) @ S ) @ R ) @ S )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( transitive_trancl @ A @ R ) @ S ) ) ) ).
% trancl_Int_subset
thf(fact_7854_trancl__insert2,axiom,
! [A: $tType,A2: A,B2: A,R: set @ ( product_prod @ A @ A )] :
( ( transitive_trancl @ A @ ( insert @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ R ) )
= ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ ( transitive_trancl @ A @ R )
@ ( collect @ ( product_prod @ A @ A )
@ ( product_case_prod @ A @ A @ $o
@ ^ [X4: A,Y: A] :
( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ A2 ) @ ( transitive_trancl @ A @ R ) )
| ( X4 = A2 ) )
& ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ Y ) @ ( transitive_trancl @ A @ R ) )
| ( Y = B2 ) ) ) ) ) ) ) ).
% trancl_insert2
thf(fact_7855_GMVT,axiom,
! [A2: real,B2: real,F3: real > real,G: real > real] :
( ( ord_less @ real @ A2 @ B2 )
=> ( ! [X5: real] :
( ( ( ord_less_eq @ real @ A2 @ X5 )
& ( ord_less_eq @ real @ X5 @ B2 ) )
=> ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ X5 @ ( top_top @ ( set @ real ) ) ) @ F3 ) )
=> ( ! [X5: real] :
( ( ( ord_less @ real @ A2 @ X5 )
& ( ord_less @ real @ X5 @ B2 ) )
=> ( differentiable @ real @ real @ F3 @ ( topolo174197925503356063within @ real @ X5 @ ( top_top @ ( set @ real ) ) ) ) )
=> ( ! [X5: real] :
( ( ( ord_less_eq @ real @ A2 @ X5 )
& ( ord_less_eq @ real @ X5 @ B2 ) )
=> ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ X5 @ ( top_top @ ( set @ real ) ) ) @ G ) )
=> ( ! [X5: real] :
( ( ( ord_less @ real @ A2 @ X5 )
& ( ord_less @ real @ X5 @ B2 ) )
=> ( differentiable @ real @ real @ G @ ( topolo174197925503356063within @ real @ X5 @ ( top_top @ ( set @ real ) ) ) ) )
=> ? [G_c: real,F_c: real,C4: real] :
( ( has_field_derivative @ real @ G @ G_c @ ( topolo174197925503356063within @ real @ C4 @ ( top_top @ ( set @ real ) ) ) )
& ( has_field_derivative @ real @ F3 @ F_c @ ( topolo174197925503356063within @ real @ C4 @ ( top_top @ ( set @ real ) ) ) )
& ( ord_less @ real @ A2 @ C4 )
& ( ord_less @ real @ C4 @ B2 )
& ( ( times_times @ real @ ( minus_minus @ real @ ( F3 @ B2 ) @ ( F3 @ A2 ) ) @ G_c )
= ( times_times @ real @ ( minus_minus @ real @ ( G @ B2 ) @ ( G @ A2 ) ) @ F_c ) ) ) ) ) ) ) ) ).
% GMVT
thf(fact_7856_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
@ ^ [Xs2: list @ A,Ys3: list @ A] :
( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Xs2 ) @ ( size_size @ ( list @ A ) @ Ys3 ) )
| ( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ A ) @ Ys3 ) )
& ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ Ys3 ) @ ( lex @ A @ R5 ) ) ) ) ) ) ) ) ).
% lenlex_conv
thf(fact_7857_Nil__lenlex__iff1,axiom,
! [A: $tType,Ns: list @ A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Ns ) @ ( lenlex @ A @ R ) )
= ( Ns
!= ( nil @ A ) ) ) ).
% Nil_lenlex_iff1
thf(fact_7858_differentiable__cmult__right__iff,axiom,
! [A: $tType,B: $tType] :
( ( ( real_V822414075346904944vector @ B )
& ( real_V3459762299906320749_field @ A ) )
=> ! [Q: B > A,C2: A,T2: B] :
( ( differentiable @ B @ A
@ ^ [T3: B] : ( times_times @ A @ ( Q @ T3 ) @ C2 )
@ ( topolo174197925503356063within @ B @ T2 @ ( top_top @ ( set @ B ) ) ) )
= ( ( C2
= ( zero_zero @ A ) )
| ( differentiable @ B @ A @ Q @ ( topolo174197925503356063within @ B @ T2 @ ( top_top @ ( set @ B ) ) ) ) ) ) ) ).
% differentiable_cmult_right_iff
thf(fact_7859_differentiable__cmult__left__iff,axiom,
! [A: $tType,B: $tType] :
( ( ( real_V822414075346904944vector @ B )
& ( real_V3459762299906320749_field @ A ) )
=> ! [C2: A,Q: B > A,T2: B] :
( ( differentiable @ B @ A
@ ^ [T3: B] : ( times_times @ A @ C2 @ ( Q @ T3 ) )
@ ( topolo174197925503356063within @ B @ T2 @ ( top_top @ ( set @ B ) ) ) )
= ( ( C2
= ( zero_zero @ A ) )
| ( differentiable @ B @ A @ Q @ ( topolo174197925503356063within @ B @ T2 @ ( top_top @ ( set @ B ) ) ) ) ) ) ) ).
% differentiable_cmult_left_iff
thf(fact_7860_differentiable__in__compose,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F3: A > B,G: C > A,X: C,S: set @ C] :
( ( differentiable @ A @ B @ F3 @ ( topolo174197925503356063within @ A @ ( G @ X ) @ ( image @ C @ A @ G @ S ) ) )
=> ( ( differentiable @ C @ A @ G @ ( topolo174197925503356063within @ C @ X @ S ) )
=> ( differentiable @ C @ B
@ ^ [X4: C] : ( F3 @ ( G @ X4 ) )
@ ( topolo174197925503356063within @ C @ X @ S ) ) ) ) ) ).
% differentiable_in_compose
thf(fact_7861_differentiable__compose,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F3: A > B,G: C > A,X: C,S: set @ C] :
( ( differentiable @ A @ B @ F3 @ ( topolo174197925503356063within @ A @ ( G @ X ) @ ( top_top @ ( set @ A ) ) ) )
=> ( ( differentiable @ C @ A @ G @ ( topolo174197925503356063within @ C @ X @ S ) )
=> ( differentiable @ C @ B
@ ^ [X4: C] : ( F3 @ ( G @ X4 ) )
@ ( topolo174197925503356063within @ C @ X @ S ) ) ) ) ) ).
% differentiable_compose
thf(fact_7862_differentiable__within__subset,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F3: A > B,X: A,S: set @ A,T2: set @ A] :
( ( differentiable @ A @ B @ F3 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( ( ord_less_eq @ ( set @ A ) @ T2 @ S )
=> ( differentiable @ A @ B @ F3 @ ( topolo174197925503356063within @ A @ X @ T2 ) ) ) ) ) ).
% differentiable_within_subset
thf(fact_7863_lenlex__irreflexive,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),Xs: list @ A] :
( ! [X5: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X5 @ X5 ) @ R )
=> ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Xs ) @ ( lenlex @ A @ R ) ) ) ).
% lenlex_irreflexive
thf(fact_7864_differentiable__scaleR,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F3: A > real,X: A,S: set @ A,G: A > B] :
( ( differentiable @ A @ real @ F3 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( ( differentiable @ A @ B @ G @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( differentiable @ A @ B
@ ^ [X4: A] : ( real_V8093663219630862766scaleR @ B @ ( F3 @ X4 ) @ ( G @ X4 ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ).
% differentiable_scaleR
thf(fact_7865_differentiable__power,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V3459762299906320749_field @ B ) )
=> ! [F3: A > B,X: A,S: set @ A,N: nat] :
( ( differentiable @ A @ B @ F3 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( differentiable @ A @ B
@ ^ [X4: A] : ( power_power @ B @ ( F3 @ X4 ) @ N )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ).
% differentiable_power
thf(fact_7866_differentiable__mult,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V4412858255891104859lgebra @ B ) )
=> ! [F3: A > B,X: A,S: set @ A,G: A > B] :
( ( differentiable @ A @ B @ F3 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( ( differentiable @ A @ B @ G @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( differentiable @ A @ B
@ ^ [X4: A] : ( times_times @ B @ ( F3 @ X4 ) @ ( G @ X4 ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ).
% differentiable_mult
thf(fact_7867_differentiable__const,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [A2: B,F4: filter @ A] :
( differentiable @ A @ B
@ ^ [Z6: A] : A2
@ F4 ) ) ).
% differentiable_const
thf(fact_7868_differentiable__ident,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F4: filter @ A] :
( differentiable @ A @ A
@ ^ [X4: A] : X4
@ F4 ) ) ).
% differentiable_ident
thf(fact_7869_differentiable__add,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F3: A > B,F4: filter @ A,G: A > B] :
( ( differentiable @ A @ B @ F3 @ F4 )
=> ( ( differentiable @ A @ B @ G @ F4 )
=> ( differentiable @ A @ B
@ ^ [X4: A] : ( plus_plus @ B @ ( F3 @ X4 ) @ ( G @ X4 ) )
@ F4 ) ) ) ) ).
% differentiable_add
thf(fact_7870_differentiable__diff,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F3: A > B,F4: filter @ A,G: A > B] :
( ( differentiable @ A @ B @ F3 @ F4 )
=> ( ( differentiable @ A @ B @ G @ F4 )
=> ( differentiable @ A @ B
@ ^ [X4: A] : ( minus_minus @ B @ ( F3 @ X4 ) @ ( G @ X4 ) )
@ F4 ) ) ) ) ).
% differentiable_diff
thf(fact_7871_differentiable__minus,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F3: A > B,F4: filter @ A] :
( ( differentiable @ A @ B @ F3 @ F4 )
=> ( differentiable @ A @ B
@ ^ [X4: A] : ( uminus_uminus @ B @ ( F3 @ X4 ) )
@ F4 ) ) ) ).
% differentiable_minus
thf(fact_7872_Nil__lenlex__iff2,axiom,
! [A: $tType,Ns: list @ A,R: set @ ( product_prod @ A @ A )] :
~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ns @ ( nil @ A ) ) @ ( lenlex @ A @ R ) ) ).
% Nil_lenlex_iff2
thf(fact_7873_differentiable__sum,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( ( real_V822414075346904944vector @ B )
& ( real_V822414075346904944vector @ C ) )
=> ! [S: set @ A,F3: A > B > C,Net: filter @ B] :
( ( finite_finite2 @ A @ S )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ S )
=> ( differentiable @ B @ C @ ( F3 @ X5 ) @ Net ) )
=> ( differentiable @ B @ C
@ ^ [X4: B] :
( groups7311177749621191930dd_sum @ A @ C
@ ^ [A5: A] : ( F3 @ A5 @ X4 )
@ S )
@ Net ) ) ) ) ).
% differentiable_sum
thf(fact_7874_differentiable__divide,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V3459762299906320749_field @ B ) )
=> ! [F3: A > B,X: A,S: set @ A,G: A > B] :
( ( differentiable @ A @ B @ F3 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( ( differentiable @ A @ B @ G @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( ( ( G @ X )
!= ( zero_zero @ B ) )
=> ( differentiable @ A @ B
@ ^ [X4: A] : ( divide_divide @ B @ ( F3 @ X4 ) @ ( G @ X4 ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ) ).
% differentiable_divide
thf(fact_7875_differentiable__inverse,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V3459762299906320749_field @ B ) )
=> ! [F3: A > B,X: A,S: set @ A] :
( ( differentiable @ A @ B @ F3 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( ( ( F3 @ X )
!= ( zero_zero @ B ) )
=> ( differentiable @ A @ B
@ ^ [X4: A] : ( inverse_inverse @ B @ ( F3 @ X4 ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ).
% differentiable_inverse
thf(fact_7876_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_7877_lenlex__append1,axiom,
! [A: $tType,Us: list @ A,Xs: list @ A,R4: set @ ( product_prod @ A @ A ),Vs: list @ A,Ys: list @ A] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Us @ Xs ) @ ( lenlex @ A @ R4 ) )
=> ( ( ( size_size @ ( list @ A ) @ Vs )
= ( size_size @ ( list @ A ) @ Ys ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Us @ Vs ) @ ( append @ A @ Xs @ Ys ) ) @ ( lenlex @ A @ R4 ) ) ) ) ).
% lenlex_append1
thf(fact_7878_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_7879_continuous__at__Sup__antimono,axiom,
! [B: $tType,A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( topolo1944317154257567458pology @ A )
& ( condit6923001295902523014norder @ B )
& ( topolo1944317154257567458pology @ B ) )
=> ! [F3: A > B,S3: set @ A] :
( ( order_antimono @ A @ B @ F3 )
=> ( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ ( complete_Sup_Sup @ A @ S3 ) @ ( set_ord_lessThan @ A @ ( complete_Sup_Sup @ A @ S3 ) ) ) @ F3 )
=> ( ( S3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit941137186595557371_above @ A @ S3 )
=> ( ( F3 @ ( complete_Sup_Sup @ A @ S3 ) )
= ( complete_Inf_Inf @ B @ ( image @ A @ B @ F3 @ S3 ) ) ) ) ) ) ) ) ).
% continuous_at_Sup_antimono
thf(fact_7880_continuous__at__Inf__antimono,axiom,
! [B: $tType,A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( topolo1944317154257567458pology @ A )
& ( condit6923001295902523014norder @ B )
& ( topolo1944317154257567458pology @ B ) )
=> ! [F3: A > B,S3: set @ A] :
( ( order_antimono @ A @ B @ F3 )
=> ( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ ( complete_Inf_Inf @ A @ S3 ) @ ( set_ord_greaterThan @ A @ ( complete_Inf_Inf @ A @ S3 ) ) ) @ F3 )
=> ( ( S3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit1013018076250108175_below @ A @ S3 )
=> ( ( F3 @ ( complete_Inf_Inf @ A @ S3 ) )
= ( complete_Sup_Sup @ B @ ( image @ A @ B @ F3 @ S3 ) ) ) ) ) ) ) ) ).
% continuous_at_Inf_antimono
thf(fact_7881_bdd__belowI,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: set @ A,M: A] :
( ! [X5: A] :
( ( member @ A @ X5 @ A3 )
=> ( ord_less_eq @ A @ M @ X5 ) )
=> ( condit1013018076250108175_below @ A @ A3 ) ) ) ).
% bdd_belowI
thf(fact_7882_bdd__below_OI,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: set @ A,M7: A] :
( ! [X5: A] :
( ( member @ A @ X5 @ A3 )
=> ( ord_less_eq @ A @ M7 @ X5 ) )
=> ( condit1013018076250108175_below @ A @ A3 ) ) ) ).
% bdd_below.I
thf(fact_7883_bdd__above_OI,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: set @ A,M7: A] :
( ! [X5: A] :
( ( member @ A @ X5 @ A3 )
=> ( ord_less_eq @ A @ X5 @ M7 ) )
=> ( condit941137186595557371_above @ A @ A3 ) ) ) ).
% bdd_above.I
thf(fact_7884_bdd__below__image__inf,axiom,
! [A: $tType,B: $tType] :
( ( lattice @ A )
=> ! [F3: B > A,G: B > A,A3: set @ B] :
( ( condit1013018076250108175_below @ A
@ ( image @ B @ A
@ ^ [X4: B] : ( inf_inf @ A @ ( F3 @ X4 ) @ ( G @ X4 ) )
@ A3 ) )
= ( ( condit1013018076250108175_below @ A @ ( image @ B @ A @ F3 @ A3 ) )
& ( condit1013018076250108175_below @ A @ ( image @ B @ A @ G @ A3 ) ) ) ) ) ).
% bdd_below_image_inf
thf(fact_7885_bdd__above__image__sup,axiom,
! [A: $tType,B: $tType] :
( ( lattice @ A )
=> ! [F3: B > A,G: B > A,A3: set @ B] :
( ( condit941137186595557371_above @ A
@ ( image @ B @ A
@ ^ [X4: B] : ( sup_sup @ A @ ( F3 @ X4 ) @ ( G @ X4 ) )
@ A3 ) )
= ( ( condit941137186595557371_above @ A @ ( image @ B @ A @ F3 @ A3 ) )
& ( condit941137186595557371_above @ A @ ( image @ B @ A @ G @ A3 ) ) ) ) ) ).
% bdd_above_image_sup
thf(fact_7886_bdd__below__UN,axiom,
! [A: $tType,B: $tType] :
( ( lattice @ A )
=> ! [I6: set @ B,A3: B > ( set @ A )] :
( ( finite_finite2 @ B @ I6 )
=> ( ( condit1013018076250108175_below @ A @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ A3 @ I6 ) ) )
= ( ! [X4: B] :
( ( member @ B @ X4 @ I6 )
=> ( condit1013018076250108175_below @ A @ ( A3 @ X4 ) ) ) ) ) ) ) ).
% bdd_below_UN
thf(fact_7887_bdd__above__UN,axiom,
! [A: $tType,B: $tType] :
( ( lattice @ A )
=> ! [I6: set @ B,A3: B > ( set @ A )] :
( ( finite_finite2 @ B @ I6 )
=> ( ( condit941137186595557371_above @ A @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ A3 @ I6 ) ) )
= ( ! [X4: B] :
( ( member @ B @ X4 @ I6 )
=> ( condit941137186595557371_above @ A @ ( A3 @ X4 ) ) ) ) ) ) ) ).
% bdd_above_UN
thf(fact_7888_differentiable__cnj__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: A > complex,X: A,A3: set @ A] :
( ( differentiable @ A @ complex
@ ^ [Z6: A] : ( cnj @ ( F3 @ Z6 ) )
@ ( topolo174197925503356063within @ A @ X @ A3 ) )
= ( differentiable @ A @ complex @ F3 @ ( topolo174197925503356063within @ A @ X @ A3 ) ) ) ) ).
% differentiable_cnj_iff
thf(fact_7889_less__cSup__iff,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [X9: set @ A,Y2: A] :
( ( X9
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit941137186595557371_above @ A @ X9 )
=> ( ( ord_less @ A @ Y2 @ ( complete_Sup_Sup @ A @ X9 ) )
= ( ? [X4: A] :
( ( member @ A @ X4 @ X9 )
& ( ord_less @ A @ Y2 @ X4 ) ) ) ) ) ) ) ).
% less_cSup_iff
thf(fact_7890_cSup__mono,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [B5: set @ A,A3: set @ A] :
( ( B5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit941137186595557371_above @ A @ A3 )
=> ( ! [B3: A] :
( ( member @ A @ B3 @ B5 )
=> ? [X2: A] :
( ( member @ A @ X2 @ A3 )
& ( ord_less_eq @ A @ B3 @ X2 ) ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ B5 ) @ ( complete_Sup_Sup @ A @ A3 ) ) ) ) ) ) ).
% cSup_mono
thf(fact_7891_cSup__le__iff,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [S3: set @ A,A2: A] :
( ( S3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit941137186595557371_above @ A @ S3 )
=> ( ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ S3 ) @ A2 )
= ( ! [X4: A] :
( ( member @ A @ X4 @ S3 )
=> ( ord_less_eq @ A @ X4 @ A2 ) ) ) ) ) ) ) ).
% cSup_le_iff
thf(fact_7892_cInf__le__cSup,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A3: set @ A] :
( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit941137186595557371_above @ A @ A3 )
=> ( ( condit1013018076250108175_below @ A @ A3 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A3 ) @ ( complete_Sup_Sup @ A @ A3 ) ) ) ) ) ) ).
% cInf_le_cSup
thf(fact_7893_cSup__upper,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X: A,X9: set @ A] :
( ( member @ A @ X @ X9 )
=> ( ( condit941137186595557371_above @ A @ X9 )
=> ( ord_less_eq @ A @ X @ ( complete_Sup_Sup @ A @ X9 ) ) ) ) ) ).
% cSup_upper
thf(fact_7894_cSup__upper2,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X: A,X9: set @ A,Y2: A] :
( ( member @ A @ X @ X9 )
=> ( ( ord_less_eq @ A @ Y2 @ X )
=> ( ( condit941137186595557371_above @ A @ X9 )
=> ( ord_less_eq @ A @ Y2 @ ( complete_Sup_Sup @ A @ X9 ) ) ) ) ) ) ).
% cSup_upper2
thf(fact_7895_le__cInf__iff,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [S3: set @ A,A2: A] :
( ( S3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit1013018076250108175_below @ A @ S3 )
=> ( ( ord_less_eq @ A @ A2 @ ( complete_Inf_Inf @ A @ S3 ) )
= ( ! [X4: A] :
( ( member @ A @ X4 @ S3 )
=> ( ord_less_eq @ A @ A2 @ X4 ) ) ) ) ) ) ) ).
% le_cInf_iff
thf(fact_7896_cInf__mono,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [B5: set @ A,A3: set @ A] :
( ( B5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit1013018076250108175_below @ A @ A3 )
=> ( ! [B3: A] :
( ( member @ A @ B3 @ B5 )
=> ? [X2: A] :
( ( member @ A @ X2 @ A3 )
& ( ord_less_eq @ A @ X2 @ B3 ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A3 ) @ ( complete_Inf_Inf @ A @ B5 ) ) ) ) ) ) ).
% cInf_mono
thf(fact_7897_cInf__less__iff,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [X9: set @ A,Y2: A] :
( ( X9
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit1013018076250108175_below @ A @ X9 )
=> ( ( ord_less @ A @ ( complete_Inf_Inf @ A @ X9 ) @ Y2 )
= ( ? [X4: A] :
( ( member @ A @ X4 @ X9 )
& ( ord_less @ A @ X4 @ Y2 ) ) ) ) ) ) ) ).
% cInf_less_iff
thf(fact_7898_bdd__above__mono,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [B5: set @ A,A3: set @ A] :
( ( condit941137186595557371_above @ A @ B5 )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ B5 )
=> ( condit941137186595557371_above @ A @ A3 ) ) ) ) ).
% bdd_above_mono
thf(fact_7899_bdd__below__mono,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [B5: set @ A,A3: set @ A] :
( ( condit1013018076250108175_below @ A @ B5 )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ B5 )
=> ( condit1013018076250108175_below @ A @ A3 ) ) ) ) ).
% bdd_below_mono
thf(fact_7900_bdd__above_OE,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: set @ A] :
( ( condit941137186595557371_above @ A @ A3 )
=> ~ ! [M8: A] :
~ ! [X2: A] :
( ( member @ A @ X2 @ A3 )
=> ( ord_less_eq @ A @ X2 @ M8 ) ) ) ) ).
% bdd_above.E
thf(fact_7901_bdd__above_Ounfold,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( ( condit941137186595557371_above @ A )
= ( ^ [A7: set @ A] :
? [M9: A] :
! [X4: A] :
( ( member @ A @ X4 @ A7 )
=> ( ord_less_eq @ A @ X4 @ M9 ) ) ) ) ) ).
% bdd_above.unfold
thf(fact_7902_bdd__below_OE,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: set @ A] :
( ( condit1013018076250108175_below @ A @ A3 )
=> ~ ! [M8: A] :
~ ! [X2: A] :
( ( member @ A @ X2 @ A3 )
=> ( ord_less_eq @ A @ M8 @ X2 ) ) ) ) ).
% bdd_below.E
thf(fact_7903_bdd__below_Ounfold,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( ( condit1013018076250108175_below @ A )
= ( ^ [A7: set @ A] :
? [M9: A] :
! [X4: A] :
( ( member @ A @ X4 @ A7 )
=> ( ord_less_eq @ A @ M9 @ X4 ) ) ) ) ) ).
% bdd_below.unfold
thf(fact_7904_cInf__lower,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X: A,X9: set @ A] :
( ( member @ A @ X @ X9 )
=> ( ( condit1013018076250108175_below @ A @ X9 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ X9 ) @ X ) ) ) ) ).
% cInf_lower
thf(fact_7905_cInf__lower2,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X: A,X9: set @ A,Y2: A] :
( ( member @ A @ X @ X9 )
=> ( ( ord_less_eq @ A @ X @ Y2 )
=> ( ( condit1013018076250108175_below @ A @ X9 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ X9 ) @ Y2 ) ) ) ) ) ).
% cInf_lower2
thf(fact_7906_cSUP__lessD,axiom,
! [B: $tType,A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [F3: B > A,A3: set @ B,Y2: A,I2: B] :
( ( condit941137186595557371_above @ A @ ( image @ B @ A @ F3 @ A3 ) )
=> ( ( ord_less @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F3 @ A3 ) ) @ Y2 )
=> ( ( member @ B @ I2 @ A3 )
=> ( ord_less @ A @ ( F3 @ I2 ) @ Y2 ) ) ) ) ) ).
% cSUP_lessD
thf(fact_7907_bdd__below_OI2,axiom,
! [A: $tType,B: $tType] :
( ( preorder @ A )
=> ! [A3: set @ B,M7: A,F3: B > A] :
( ! [X5: B] :
( ( member @ B @ X5 @ A3 )
=> ( ord_less_eq @ A @ M7 @ ( F3 @ X5 ) ) )
=> ( condit1013018076250108175_below @ A @ ( image @ B @ A @ F3 @ A3 ) ) ) ) ).
% bdd_below.I2
thf(fact_7908_bdd__above_OI2,axiom,
! [A: $tType,B: $tType] :
( ( preorder @ A )
=> ! [A3: set @ B,F3: B > A,M7: A] :
( ! [X5: B] :
( ( member @ B @ X5 @ A3 )
=> ( ord_less_eq @ A @ ( F3 @ X5 ) @ M7 ) )
=> ( condit941137186595557371_above @ A @ ( image @ B @ A @ F3 @ A3 ) ) ) ) ).
% bdd_above.I2
thf(fact_7909_bdd__belowI2,axiom,
! [A: $tType,B: $tType] :
( ( preorder @ A )
=> ! [A3: set @ B,M: A,F3: B > A] :
( ! [X5: B] :
( ( member @ B @ X5 @ A3 )
=> ( ord_less_eq @ A @ M @ ( F3 @ X5 ) ) )
=> ( condit1013018076250108175_below @ A @ ( image @ B @ A @ F3 @ A3 ) ) ) ) ).
% bdd_belowI2
thf(fact_7910_cSUP__upper2,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [F3: B > A,A3: set @ B,X: B,U: A] :
( ( condit941137186595557371_above @ A @ ( image @ B @ A @ F3 @ A3 ) )
=> ( ( member @ B @ X @ A3 )
=> ( ( ord_less_eq @ A @ U @ ( F3 @ X ) )
=> ( ord_less_eq @ A @ U @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F3 @ A3 ) ) ) ) ) ) ) ).
% cSUP_upper2
thf(fact_7911_cSUP__upper,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X: B,A3: set @ B,F3: B > A] :
( ( member @ B @ X @ A3 )
=> ( ( condit941137186595557371_above @ A @ ( image @ B @ A @ F3 @ A3 ) )
=> ( ord_less_eq @ A @ ( F3 @ X ) @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F3 @ A3 ) ) ) ) ) ) ).
% cSUP_upper
thf(fact_7912_cINF__lower2,axiom,
! [B: $tType,A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [F3: B > A,A3: set @ B,X: B,U: A] :
( ( condit1013018076250108175_below @ A @ ( image @ B @ A @ F3 @ A3 ) )
=> ( ( member @ B @ X @ A3 )
=> ( ( ord_less_eq @ A @ ( F3 @ X ) @ U )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F3 @ A3 ) ) @ U ) ) ) ) ) ).
% cINF_lower2
thf(fact_7913_cINF__lower,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [F3: B > A,A3: set @ B,X: B] :
( ( condit1013018076250108175_below @ A @ ( image @ B @ A @ F3 @ A3 ) )
=> ( ( member @ B @ X @ A3 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F3 @ A3 ) ) @ ( F3 @ X ) ) ) ) ) ).
% cINF_lower
thf(fact_7914_cSUP__eq__cINF__D,axiom,
! [B: $tType,C: $tType] :
( ( condit1219197933456340205attice @ B )
=> ! [F3: C > B,A3: set @ C,A2: C] :
( ( ( complete_Sup_Sup @ B @ ( image @ C @ B @ F3 @ A3 ) )
= ( complete_Inf_Inf @ B @ ( image @ C @ B @ F3 @ A3 ) ) )
=> ( ( condit941137186595557371_above @ B @ ( image @ C @ B @ F3 @ A3 ) )
=> ( ( condit1013018076250108175_below @ B @ ( image @ C @ B @ F3 @ A3 ) )
=> ( ( member @ C @ A2 @ A3 )
=> ( ( F3 @ A2 )
= ( complete_Inf_Inf @ B @ ( image @ C @ B @ F3 @ A3 ) ) ) ) ) ) ) ) ).
% cSUP_eq_cINF_D
thf(fact_7915_less__cINF__D,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [F3: B > A,A3: set @ B,Y2: A,I2: B] :
( ( condit1013018076250108175_below @ A @ ( image @ B @ A @ F3 @ A3 ) )
=> ( ( ord_less @ A @ Y2 @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F3 @ A3 ) ) )
=> ( ( member @ B @ I2 @ A3 )
=> ( ord_less @ A @ Y2 @ ( F3 @ I2 ) ) ) ) ) ) ).
% less_cINF_D
thf(fact_7916_le__cINF__iff,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A3: set @ B,F3: B > A,U: A] :
( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit1013018076250108175_below @ A @ ( image @ B @ A @ F3 @ A3 ) )
=> ( ( ord_less_eq @ A @ U @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F3 @ A3 ) ) )
= ( ! [X4: B] :
( ( member @ B @ X4 @ A3 )
=> ( ord_less_eq @ A @ U @ ( F3 @ X4 ) ) ) ) ) ) ) ) ).
% le_cINF_iff
thf(fact_7917_cINF__mono,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [B5: set @ B,F3: C > A,A3: set @ C,G: B > A] :
( ( B5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit1013018076250108175_below @ A @ ( image @ C @ A @ F3 @ A3 ) )
=> ( ! [M4: B] :
( ( member @ B @ M4 @ B5 )
=> ? [X2: C] :
( ( member @ C @ X2 @ A3 )
& ( ord_less_eq @ A @ ( F3 @ X2 ) @ ( G @ M4 ) ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image @ C @ A @ F3 @ A3 ) ) @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ G @ B5 ) ) ) ) ) ) ) ).
% cINF_mono
thf(fact_7918_cInf__superset__mono,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A3: set @ A,B5: set @ A] :
( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit1013018076250108175_below @ A @ B5 )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ B5 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ B5 ) @ ( complete_Inf_Inf @ A @ A3 ) ) ) ) ) ) ).
% cInf_superset_mono
thf(fact_7919_cSUP__le__iff,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A3: set @ B,F3: B > A,U: A] :
( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit941137186595557371_above @ A @ ( image @ B @ A @ F3 @ A3 ) )
=> ( ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F3 @ A3 ) ) @ U )
= ( ! [X4: B] :
( ( member @ B @ X4 @ A3 )
=> ( ord_less_eq @ A @ ( F3 @ X4 ) @ U ) ) ) ) ) ) ) ).
% cSUP_le_iff
thf(fact_7920_cSUP__mono,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A3: set @ B,G: C > A,B5: set @ C,F3: B > A] :
( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit941137186595557371_above @ A @ ( image @ C @ A @ G @ B5 ) )
=> ( ! [N3: B] :
( ( member @ B @ N3 @ A3 )
=> ? [X2: C] :
( ( member @ C @ X2 @ B5 )
& ( ord_less_eq @ A @ ( F3 @ N3 ) @ ( G @ X2 ) ) ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F3 @ A3 ) ) @ ( complete_Sup_Sup @ A @ ( image @ C @ A @ G @ B5 ) ) ) ) ) ) ) ).
% cSUP_mono
thf(fact_7921_cSup__subset__mono,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A3: set @ A,B5: set @ A] :
( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit941137186595557371_above @ A @ B5 )
=> ( ( ord_less_eq @ ( set @ A ) @ A3 @ B5 )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A3 ) @ ( complete_Sup_Sup @ A @ B5 ) ) ) ) ) ) ).
% cSup_subset_mono
thf(fact_7922_cINF__less__iff,axiom,
! [A: $tType,B: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [A3: set @ B,F3: B > A,A2: A] :
( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit1013018076250108175_below @ A @ ( image @ B @ A @ F3 @ A3 ) )
=> ( ( ord_less @ A @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F3 @ A3 ) ) @ A2 )
= ( ? [X4: B] :
( ( member @ B @ X4 @ A3 )
& ( ord_less @ A @ ( F3 @ X4 ) @ A2 ) ) ) ) ) ) ) ).
% cINF_less_iff
thf(fact_7923_less__cSUP__iff,axiom,
! [A: $tType,B: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [A3: set @ B,F3: B > A,A2: A] :
( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit941137186595557371_above @ A @ ( image @ B @ A @ F3 @ A3 ) )
=> ( ( ord_less @ A @ A2 @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F3 @ A3 ) ) )
= ( ? [X4: B] :
( ( member @ B @ X4 @ A3 )
& ( ord_less @ A @ A2 @ ( F3 @ X4 ) ) ) ) ) ) ) ) ).
% less_cSUP_iff
thf(fact_7924_cINF__inf__distrib,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A3: set @ B,F3: B > A,G: B > A] :
( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit1013018076250108175_below @ A @ ( image @ B @ A @ F3 @ A3 ) )
=> ( ( condit1013018076250108175_below @ A @ ( image @ B @ A @ G @ A3 ) )
=> ( ( inf_inf @ A @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F3 @ A3 ) ) @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ G @ A3 ) ) )
= ( complete_Inf_Inf @ A
@ ( image @ B @ A
@ ^ [A5: B] : ( inf_inf @ A @ ( F3 @ A5 ) @ ( G @ A5 ) )
@ A3 ) ) ) ) ) ) ) ).
% cINF_inf_distrib
thf(fact_7925_conditionally__complete__lattice__class_OSUP__sup__distrib,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A3: set @ B,F3: B > A,G: B > A] :
( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit941137186595557371_above @ A @ ( image @ B @ A @ F3 @ A3 ) )
=> ( ( condit941137186595557371_above @ A @ ( image @ B @ A @ G @ A3 ) )
=> ( ( sup_sup @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F3 @ A3 ) ) @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ G @ A3 ) ) )
= ( complete_Sup_Sup @ A
@ ( image @ B @ A
@ ^ [A5: B] : ( sup_sup @ A @ ( F3 @ A5 ) @ ( G @ A5 ) )
@ A3 ) ) ) ) ) ) ) ).
% conditionally_complete_lattice_class.SUP_sup_distrib
thf(fact_7926_cINF__superset__mono,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A3: set @ B,G: B > A,B5: set @ B,F3: B > A] :
( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit1013018076250108175_below @ A @ ( image @ B @ A @ G @ B5 ) )
=> ( ( ord_less_eq @ ( set @ B ) @ A3 @ B5 )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ B5 )
=> ( ord_less_eq @ A @ ( G @ X5 ) @ ( F3 @ X5 ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ G @ B5 ) ) @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F3 @ A3 ) ) ) ) ) ) ) ) ).
% cINF_superset_mono
thf(fact_7927_cSUP__subset__mono,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A3: set @ B,G: B > A,B5: set @ B,F3: B > A] :
( ( A3
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit941137186595557371_above @ A @ ( image @ B @ A @ G @ B5 ) )
=> ( ( ord_less_eq @ ( set @ B ) @ A3 @ B5 )
=> ( ! [X5: B] :
( ( member @ B @ X5 @ A3 )
=> ( ord_less_eq @ A @ ( F3 @ X5 ) @ ( G @ X5 ) ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F3 @ A3 ) ) @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ G @ B5 ) ) ) ) ) ) ) ) ).
% cSUP_subset_mono
thf(fact_7928_less__eq__cInf__inter,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A3: set @ A,B5: set @ A] :
( ( condit1013018076250108175_below @ A @ A3 )
=> ( ( condit1013018076250108175_below @ A @ B5 )
=> ( ( ( inf_inf @ ( set @ A ) @ A3 @ B5 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ord_less_eq @ A @ ( inf_inf @ A @ ( complete_Inf_Inf @ A @ A3 ) @ ( complete_Inf_Inf @ A @ B5 ) ) @ ( complete_Inf_Inf @ A @ ( inf_inf @ ( set @ A ) @ A3 @ B5 ) ) ) ) ) ) ) ).
% less_eq_cInf_inter
thf(fact_7929_cSup__inter__less__eq,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A3: set @ A,B5: set @ A] :
( ( condit941137186595557371_above @ A @ A3 )
=> ( ( condit941137186595557371_above @ A @ B5 )
=> ( ( ( inf_inf @ ( set @ A ) @ A3 @ B5 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( inf_inf @ ( set @ A ) @ A3 @ B5 ) ) @ ( sup_sup @ A @ ( complete_Sup_Sup @ A @ A3 ) @ ( complete_Sup_Sup @ A @ B5 ) ) ) ) ) ) ) ).
% cSup_inter_less_eq
thf(fact_7930_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
@ ^ [X4: A] :
! [Y: A] :
( ( member @ A @ Y @ S3 )
=> ( ord_less_eq @ A @ X4 @ Y ) ) ) ) ) ) ) ) ).
% cInf_cSup
thf(fact_7931_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
@ ^ [X4: A] :
! [Y: A] :
( ( member @ A @ Y @ S3 )
=> ( ord_less_eq @ A @ Y @ X4 ) ) ) ) ) ) ) ) ).
% cSup_cInf
thf(fact_7932_cINF__UNION,axiom,
! [B: $tType,D: $tType,C: $tType] :
( ( condit1219197933456340205attice @ B )
=> ! [A3: set @ C,B5: C > ( set @ D ),F3: D > B] :
( ( A3
!= ( bot_bot @ ( set @ C ) ) )
=> ( ! [X5: C] :
( ( member @ C @ X5 @ A3 )
=> ( ( B5 @ X5 )
!= ( bot_bot @ ( set @ D ) ) ) )
=> ( ( condit1013018076250108175_below @ B
@ ( complete_Sup_Sup @ ( set @ B )
@ ( image @ C @ ( set @ B )
@ ^ [X4: C] : ( image @ D @ B @ F3 @ ( B5 @ X4 ) )
@ A3 ) ) )
=> ( ( complete_Inf_Inf @ B @ ( image @ D @ B @ F3 @ ( complete_Sup_Sup @ ( set @ D ) @ ( image @ C @ ( set @ D ) @ B5 @ A3 ) ) ) )
= ( complete_Inf_Inf @ B
@ ( image @ C @ B
@ ^ [X4: C] : ( complete_Inf_Inf @ B @ ( image @ D @ B @ F3 @ ( B5 @ X4 ) ) )
@ A3 ) ) ) ) ) ) ) ).
% cINF_UNION
thf(fact_7933_cSUP__UNION,axiom,
! [B: $tType,D: $tType,C: $tType] :
( ( condit1219197933456340205attice @ B )
=> ! [A3: set @ C,B5: C > ( set @ D ),F3: D > B] :
( ( A3
!= ( bot_bot @ ( set @ C ) ) )
=> ( ! [X5: C] :
( ( member @ C @ X5 @ A3 )
=> ( ( B5 @ X5 )
!= ( bot_bot @ ( set @ D ) ) ) )
=> ( ( condit941137186595557371_above @ B
@ ( complete_Sup_Sup @ ( set @ B )
@ ( image @ C @ ( set @ B )
@ ^ [X4: C] : ( image @ D @ B @ F3 @ ( B5 @ X4 ) )
@ A3 ) ) )
=> ( ( complete_Sup_Sup @ B @ ( image @ D @ B @ F3 @ ( complete_Sup_Sup @ ( set @ D ) @ ( image @ C @ ( set @ D ) @ B5 @ A3 ) ) ) )
= ( complete_Sup_Sup @ B
@ ( image @ C @ B
@ ^ [X4: C] : ( complete_Sup_Sup @ B @ ( image @ D @ B @ F3 @ ( B5 @ X4 ) ) )
@ A3 ) ) ) ) ) ) ) ).
% cSUP_UNION
thf(fact_7934_Bseq__bdd__above_H,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X9: nat > A] :
( ( bfun @ nat @ A @ X9 @ ( at_top @ nat ) )
=> ( condit941137186595557371_above @ real
@ ( image @ nat @ real
@ ^ [N2: nat] : ( real_V7770717601297561774m_norm @ A @ ( X9 @ N2 ) )
@ ( top_top @ ( set @ nat ) ) ) ) ) ) ).
% Bseq_bdd_above'
thf(fact_7935_LIMSEQ__decseq__INF,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [X9: nat > A] :
( ( condit1013018076250108175_below @ A @ ( image @ nat @ A @ X9 @ ( top_top @ ( set @ nat ) ) ) )
=> ( ( order_antimono @ nat @ A @ X9 )
=> ( filterlim @ nat @ A @ X9 @ ( topolo7230453075368039082e_nhds @ A @ ( complete_Inf_Inf @ A @ ( image @ nat @ A @ X9 @ ( top_top @ ( set @ nat ) ) ) ) ) @ ( at_top @ nat ) ) ) ) ) ).
% LIMSEQ_decseq_INF
thf(fact_7936_MVT,axiom,
! [A2: real,B2: real,F3: real > real] :
( ( ord_less @ real @ A2 @ B2 )
=> ( ( topolo81223032696312382ous_on @ real @ real @ ( set_or1337092689740270186AtMost @ real @ A2 @ B2 ) @ F3 )
=> ( ! [X5: real] :
( ( ord_less @ real @ A2 @ X5 )
=> ( ( ord_less @ real @ X5 @ B2 )
=> ( differentiable @ real @ real @ F3 @ ( topolo174197925503356063within @ real @ X5 @ ( top_top @ ( set @ real ) ) ) ) ) )
=> ? [L2: real,Z: real] :
( ( ord_less @ real @ A2 @ Z )
& ( ord_less @ real @ Z @ B2 )
& ( has_field_derivative @ real @ F3 @ L2 @ ( topolo174197925503356063within @ real @ Z @ ( top_top @ ( set @ real ) ) ) )
& ( ( minus_minus @ real @ ( F3 @ B2 ) @ ( F3 @ A2 ) )
= ( times_times @ real @ ( minus_minus @ real @ B2 @ A2 ) @ L2 ) ) ) ) ) ) ).
% MVT
thf(fact_7937_ord_OLeast__def,axiom,
! [A: $tType] :
( ( least @ A )
= ( ^ [Less_eq2: A > A > $o,P2: A > $o] :
( the @ A
@ ^ [X4: A] :
( ( P2 @ X4 )
& ! [Y: A] :
( ( P2 @ Y )
=> ( Less_eq2 @ X4 @ Y ) ) ) ) ) ) ).
% ord.Least_def
thf(fact_7938_continuous__on__open__UN,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( ( topolo4958980785337419405_space @ B )
& ( topolo4958980785337419405_space @ C ) )
=> ! [S3: set @ A,A3: A > ( set @ B ),F3: B > C] :
( ! [S2: A] :
( ( member @ A @ S2 @ S3 )
=> ( topolo1002775350975398744n_open @ B @ ( A3 @ S2 ) ) )
=> ( ! [S2: A] :
( ( member @ A @ S2 @ S3 )
=> ( topolo81223032696312382ous_on @ B @ C @ ( A3 @ S2 ) @ F3 ) )
=> ( topolo81223032696312382ous_on @ B @ C @ ( complete_Sup_Sup @ ( set @ B ) @ ( image @ A @ ( set @ B ) @ A3 @ S3 ) ) @ F3 ) ) ) ) ).
% continuous_on_open_UN
thf(fact_7939_continuous__on__compose2,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( ( topolo4958980785337419405_space @ C )
& ( topolo4958980785337419405_space @ B )
& ( topolo4958980785337419405_space @ A ) )
=> ! [T2: set @ A,G: A > B,S: set @ C,F3: C > A] :
( ( topolo81223032696312382ous_on @ A @ B @ T2 @ G )
=> ( ( topolo81223032696312382ous_on @ C @ A @ S @ F3 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image @ C @ A @ F3 @ S ) @ T2 )
=> ( topolo81223032696312382ous_on @ C @ B @ S
@ ^ [X4: C] : ( G @ ( F3 @ X4 ) ) ) ) ) ) ) ).
% continuous_on_compose2
thf(fact_7940_continuous__onI__mono,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo1944317154257567458pology @ A )
& ( dense_order @ B )
& ( topolo1944317154257567458pology @ B ) )
=> ! [F3: A > B,A3: set @ A] :
( ( topolo1002775350975398744n_open @ B @ ( image @ A @ B @ F3 @ A3 ) )
=> ( ! [X5: A,Y7: A] :
( ( member @ A @ X5 @ A3 )
=> ( ( member @ A @ Y7 @ A3 )
=> ( ( ord_less_eq @ A @ X5 @ Y7 )
=> ( ord_less_eq @ B @ ( F3 @ X5 ) @ ( F3 @ Y7 ) ) ) ) )
=> ( topolo81223032696312382ous_on @ A @ B @ A3 @ F3 ) ) ) ) ).
% continuous_onI_mono
thf(fact_7941_open__Collect__less__Int,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [S: set @ A,F3: A > real,G: A > real] :
( ( topolo81223032696312382ous_on @ A @ real @ S @ F3 )
=> ( ( topolo81223032696312382ous_on @ A @ real @ S @ G )
=> ? [A8: set @ A] :
( ( topolo1002775350975398744n_open @ A @ A8 )
& ( ( inf_inf @ ( set @ A ) @ A8 @ S )
= ( collect @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ S )
& ( ord_less @ real @ ( F3 @ X4 ) @ ( G @ X4 ) ) ) ) ) ) ) ) ) ).
% open_Collect_less_Int
thf(fact_7942_open__Collect__less,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ A )
& ( topolo1944317154257567458pology @ B ) )
=> ! [F3: A > B,G: A > B] :
( ( topolo81223032696312382ous_on @ A @ B @ ( top_top @ ( set @ A ) ) @ F3 )
=> ( ( topolo81223032696312382ous_on @ A @ B @ ( top_top @ ( set @ A ) ) @ G )
=> ( topolo1002775350975398744n_open @ A
@ ( collect @ A
@ ^ [X4: A] : ( ord_less @ B @ ( F3 @ X4 ) @ ( G @ X4 ) ) ) ) ) ) ) ).
% open_Collect_less
thf(fact_7943_open__Collect__neq,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ A )
& ( topological_t2_space @ B ) )
=> ! [F3: A > B,G: A > B] :
( ( topolo81223032696312382ous_on @ A @ B @ ( top_top @ ( set @ A ) ) @ F3 )
=> ( ( topolo81223032696312382ous_on @ A @ B @ ( top_top @ ( set @ A ) ) @ G )
=> ( topolo1002775350975398744n_open @ A
@ ( collect @ A
@ ^ [X4: A] :
( ( F3 @ X4 )
!= ( G @ X4 ) ) ) ) ) ) ) ).
% open_Collect_neq
thf(fact_7944_continuous__on__subset,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [S: set @ A,F3: A > B,T2: set @ A] :
( ( topolo81223032696312382ous_on @ A @ B @ S @ F3 )
=> ( ( ord_less_eq @ ( set @ A ) @ T2 @ S )
=> ( topolo81223032696312382ous_on @ A @ B @ T2 @ F3 ) ) ) ) ).
% continuous_on_subset
thf(fact_7945_IVT_H,axiom,
! [A: $tType,B: $tType] :
( ( ( topolo1944317154257567458pology @ B )
& ( topolo8458572112393995274pology @ A ) )
=> ! [F3: A > B,A2: A,Y2: B,B2: A] :
( ( ord_less_eq @ B @ ( F3 @ A2 ) @ Y2 )
=> ( ( ord_less_eq @ B @ Y2 @ ( F3 @ B2 ) )
=> ( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( topolo81223032696312382ous_on @ A @ B @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) @ F3 )
=> ? [X5: A] :
( ( ord_less_eq @ A @ A2 @ X5 )
& ( ord_less_eq @ A @ X5 @ B2 )
& ( ( F3 @ X5 )
= Y2 ) ) ) ) ) ) ) ).
% IVT'
thf(fact_7946_IVT2_H,axiom,
! [A: $tType,B: $tType] :
( ( ( topolo1944317154257567458pology @ B )
& ( topolo8458572112393995274pology @ A ) )
=> ! [F3: A > B,B2: A,Y2: B,A2: A] :
( ( ord_less_eq @ B @ ( F3 @ B2 ) @ Y2 )
=> ( ( ord_less_eq @ B @ Y2 @ ( F3 @ A2 ) )
=> ( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( topolo81223032696312382ous_on @ A @ B @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) @ F3 )
=> ? [X5: A] :
( ( ord_less_eq @ A @ A2 @ X5 )
& ( ord_less_eq @ A @ X5 @ B2 )
& ( ( F3 @ X5 )
= Y2 ) ) ) ) ) ) ) ).
% IVT2'
thf(fact_7947_continuous__on__tendsto__compose,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( ( topolo4958980785337419405_space @ B )
& ( topolo4958980785337419405_space @ A ) )
=> ! [S: set @ A,F3: A > B,G: C > A,L: A,F4: filter @ C] :
( ( topolo81223032696312382ous_on @ A @ B @ S @ F3 )
=> ( ( filterlim @ C @ A @ G @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F4 )
=> ( ( member @ A @ L @ S )
=> ( ( eventually @ C
@ ^ [X4: C] : ( member @ A @ ( G @ X4 ) @ S )
@ F4 )
=> ( filterlim @ C @ B
@ ^ [X4: C] : ( F3 @ ( G @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( F3 @ L ) )
@ F4 ) ) ) ) ) ) ).
% continuous_on_tendsto_compose
thf(fact_7948_continuous__on__tan,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [S: set @ A,F3: A > A] :
( ( topolo81223032696312382ous_on @ A @ A @ S @ F3 )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ S )
=> ( ( cos @ A @ ( F3 @ X5 ) )
!= ( zero_zero @ A ) ) )
=> ( topolo81223032696312382ous_on @ A @ A @ S
@ ^ [X4: A] : ( tan @ A @ ( F3 @ X4 ) ) ) ) ) ) ).
% continuous_on_tan
thf(fact_7949_bounded__linear_Ocontinuous__on,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( ( topolo4958980785337419405_space @ C )
& ( real_V822414075346904944vector @ B )
& ( real_V822414075346904944vector @ A ) )
=> ! [F3: A > B,S: set @ C,G: C > A] :
( ( real_V3181309239436604168linear @ A @ B @ F3 )
=> ( ( topolo81223032696312382ous_on @ C @ A @ S @ G )
=> ( topolo81223032696312382ous_on @ C @ B @ S
@ ^ [X4: C] : ( F3 @ ( G @ X4 ) ) ) ) ) ) ).
% bounded_linear.continuous_on
thf(fact_7950_continuous__on__fst,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ A )
& ( topolo4958980785337419405_space @ B )
& ( topolo4958980785337419405_space @ C ) )
=> ! [S: set @ A,F3: A > ( product_prod @ B @ C )] :
( ( topolo81223032696312382ous_on @ A @ ( product_prod @ B @ C ) @ S @ F3 )
=> ( topolo81223032696312382ous_on @ A @ B @ S
@ ^ [X4: A] : ( product_fst @ B @ C @ ( F3 @ X4 ) ) ) ) ) ).
% continuous_on_fst
thf(fact_7951_continuous__on__snd,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ A )
& ( topolo4958980785337419405_space @ B )
& ( topolo4958980785337419405_space @ C ) )
=> ! [S: set @ A,F3: A > ( product_prod @ B @ C )] :
( ( topolo81223032696312382ous_on @ A @ ( product_prod @ B @ C ) @ S @ F3 )
=> ( topolo81223032696312382ous_on @ A @ C @ S
@ ^ [X4: A] : ( product_snd @ B @ C @ ( F3 @ X4 ) ) ) ) ) ).
% continuous_on_snd
thf(fact_7952_continuous__on__Pair,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ A )
& ( topolo4958980785337419405_space @ B )
& ( topolo4958980785337419405_space @ C ) )
=> ! [S: set @ A,F3: A > B,G: A > C] :
( ( topolo81223032696312382ous_on @ A @ B @ S @ F3 )
=> ( ( topolo81223032696312382ous_on @ A @ C @ S @ G )
=> ( topolo81223032696312382ous_on @ A @ ( product_prod @ B @ C ) @ S
@ ^ [X4: A] : ( product_Pair @ B @ C @ ( F3 @ X4 ) @ ( G @ X4 ) ) ) ) ) ) ).
% continuous_on_Pair
thf(fact_7953_continuous__on__powr,axiom,
! [C: $tType] :
( ( topolo4958980785337419405_space @ C )
=> ! [S: set @ C,F3: C > real,G: C > real] :
( ( topolo81223032696312382ous_on @ C @ real @ S @ F3 )
=> ( ( topolo81223032696312382ous_on @ C @ real @ S @ G )
=> ( ! [X5: C] :
( ( member @ C @ X5 @ S )
=> ( ( F3 @ X5 )
!= ( zero_zero @ real ) ) )
=> ( topolo81223032696312382ous_on @ C @ real @ S
@ ^ [X4: C] : ( powr @ real @ ( F3 @ X4 ) @ ( G @ X4 ) ) ) ) ) ) ) ).
% continuous_on_powr
thf(fact_7954_continuous__on__inverse,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ A )
& ( real_V8999393235501362500lgebra @ B ) )
=> ! [S: set @ A,F3: A > B] :
( ( topolo81223032696312382ous_on @ A @ B @ S @ F3 )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ S )
=> ( ( F3 @ X5 )
!= ( zero_zero @ B ) ) )
=> ( topolo81223032696312382ous_on @ A @ B @ S
@ ^ [X4: A] : ( inverse_inverse @ B @ ( F3 @ X4 ) ) ) ) ) ) ).
% continuous_on_inverse
thf(fact_7955_continuous__on__divide,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ A )
& ( real_V3459762299906320749_field @ B ) )
=> ! [S: set @ A,F3: A > B,G: A > B] :
( ( topolo81223032696312382ous_on @ A @ B @ S @ F3 )
=> ( ( topolo81223032696312382ous_on @ A @ B @ S @ G )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ S )
=> ( ( G @ X5 )
!= ( zero_zero @ B ) ) )
=> ( topolo81223032696312382ous_on @ A @ B @ S
@ ^ [X4: A] : ( divide_divide @ B @ ( F3 @ X4 ) @ ( G @ X4 ) ) ) ) ) ) ) ).
% continuous_on_divide
thf(fact_7956_continuous__on__mult__const,axiom,
! [A: $tType] :
( ( real_V4412858255891104859lgebra @ A )
=> ! [S: set @ A,C2: A] : ( topolo81223032696312382ous_on @ A @ A @ S @ ( times_times @ A @ C2 ) ) ) ).
% continuous_on_mult_const
thf(fact_7957_continuous__on__arsinh_H,axiom,
! [A3: set @ real,F3: real > real] :
( ( topolo81223032696312382ous_on @ real @ real @ A3 @ F3 )
=> ( topolo81223032696312382ous_on @ real @ real @ A3
@ ^ [X4: real] : ( arsinh @ real @ ( F3 @ X4 ) ) ) ) ).
% continuous_on_arsinh'
thf(fact_7958_continuous__on__real__root,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [S: set @ A,F3: A > real,N: nat] :
( ( topolo81223032696312382ous_on @ A @ real @ S @ F3 )
=> ( topolo81223032696312382ous_on @ A @ real @ S
@ ^ [X4: A] : ( root @ N @ ( F3 @ X4 ) ) ) ) ) ).
% continuous_on_real_root
thf(fact_7959_continuous__on__sinh,axiom,
! [A: $tType,C: $tType] :
( ( ( topolo4958980785337419405_space @ C )
& ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [A3: set @ C,F3: C > A] :
( ( topolo81223032696312382ous_on @ C @ A @ A3 @ F3 )
=> ( topolo81223032696312382ous_on @ C @ A @ A3
@ ^ [X4: C] : ( sinh @ A @ ( F3 @ X4 ) ) ) ) ) ).
% continuous_on_sinh
thf(fact_7960_continuous__on__arctan,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [S: set @ A,F3: A > real] :
( ( topolo81223032696312382ous_on @ A @ real @ S @ F3 )
=> ( topolo81223032696312382ous_on @ A @ real @ S
@ ^ [X4: A] : ( arctan @ ( F3 @ X4 ) ) ) ) ) ).
% continuous_on_arctan
thf(fact_7961_continuous__on__scaleR,axiom,
! [C: $tType,D: $tType] :
( ( ( topolo4958980785337419405_space @ D )
& ( real_V822414075346904944vector @ C ) )
=> ! [S: set @ D,F3: D > real,G: D > C] :
( ( topolo81223032696312382ous_on @ D @ real @ S @ F3 )
=> ( ( topolo81223032696312382ous_on @ D @ C @ S @ G )
=> ( topolo81223032696312382ous_on @ D @ C @ S
@ ^ [X4: D] : ( real_V8093663219630862766scaleR @ C @ ( F3 @ X4 ) @ ( G @ X4 ) ) ) ) ) ) ).
% continuous_on_scaleR
thf(fact_7962_continuous__on__exp,axiom,
! [A: $tType,C: $tType] :
( ( ( topolo4958980785337419405_space @ C )
& ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [S: set @ C,F3: C > A] :
( ( topolo81223032696312382ous_on @ C @ A @ S @ F3 )
=> ( topolo81223032696312382ous_on @ C @ A @ S
@ ^ [X4: C] : ( exp @ A @ ( F3 @ X4 ) ) ) ) ) ).
% continuous_on_exp
thf(fact_7963_continuous__on__real__sqrt,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [S: set @ A,F3: A > real] :
( ( topolo81223032696312382ous_on @ A @ real @ S @ F3 )
=> ( topolo81223032696312382ous_on @ A @ real @ S
@ ^ [X4: A] : ( sqrt @ ( F3 @ X4 ) ) ) ) ) ).
% continuous_on_real_sqrt
thf(fact_7964_continuous__on__power,axiom,
! [B: $tType,C: $tType] :
( ( ( topolo4958980785337419405_space @ C )
& ( power @ B )
& ( real_V4412858255891104859lgebra @ B ) )
=> ! [S: set @ C,F3: C > B,N: nat] :
( ( topolo81223032696312382ous_on @ C @ B @ S @ F3 )
=> ( topolo81223032696312382ous_on @ C @ B @ S
@ ^ [X4: C] : ( power_power @ B @ ( F3 @ X4 ) @ N ) ) ) ) ).
% continuous_on_power
thf(fact_7965_continuous__on__of__real,axiom,
! [A: $tType,C: $tType] :
( ( ( topolo4958980785337419405_space @ C )
& ( real_V2191834092415804123ebra_1 @ A )
& ( real_V822414075346904944vector @ A ) )
=> ! [S: set @ C,G: C > real] :
( ( topolo81223032696312382ous_on @ C @ real @ S @ G )
=> ( topolo81223032696312382ous_on @ C @ A @ S
@ ^ [X4: C] : ( real_Vector_of_real @ A @ ( G @ X4 ) ) ) ) ) ).
% continuous_on_of_real
thf(fact_7966_continuous__on__norm,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [S: set @ A,F3: A > B] :
( ( topolo81223032696312382ous_on @ A @ B @ S @ F3 )
=> ( topolo81223032696312382ous_on @ A @ real @ S
@ ^ [X4: A] : ( real_V7770717601297561774m_norm @ B @ ( F3 @ X4 ) ) ) ) ) ).
% continuous_on_norm
thf(fact_7967_continuous__on__id,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [S: set @ A] :
( topolo81223032696312382ous_on @ A @ A @ S
@ ^ [X4: A] : X4 ) ) ).
% continuous_on_id
thf(fact_7968_continuous__on__const,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [S: set @ A,C2: B] :
( topolo81223032696312382ous_on @ A @ B @ S
@ ^ [X4: A] : C2 ) ) ).
% continuous_on_const
thf(fact_7969_continuous__on__power_H,axiom,
! [B: $tType,C: $tType] :
( ( ( topolo4958980785337419405_space @ C )
& ( topolo1898628316856586783d_mult @ B ) )
=> ! [A3: set @ C,F3: C > B,G: C > nat] :
( ( topolo81223032696312382ous_on @ C @ B @ A3 @ F3 )
=> ( ( topolo81223032696312382ous_on @ C @ nat @ A3 @ G )
=> ( topolo81223032696312382ous_on @ C @ B @ A3
@ ^ [X4: C] : ( power_power @ B @ ( F3 @ X4 ) @ ( G @ X4 ) ) ) ) ) ) ).
% continuous_on_power'
thf(fact_7970_continuous__on__sin,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_Vector_banach @ B )
& ( real_V3459762299906320749_field @ B ) )
=> ! [S: set @ A,F3: A > B] :
( ( topolo81223032696312382ous_on @ A @ B @ S @ F3 )
=> ( topolo81223032696312382ous_on @ A @ B @ S
@ ^ [X4: A] : ( sin @ B @ ( F3 @ X4 ) ) ) ) ) ).
% continuous_on_sin
thf(fact_7971_continuous__on__cos,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_Vector_banach @ B )
& ( real_V3459762299906320749_field @ B ) )
=> ! [S: set @ A,F3: A > B] :
( ( topolo81223032696312382ous_on @ A @ B @ S @ F3 )
=> ( topolo81223032696312382ous_on @ A @ B @ S
@ ^ [X4: A] : ( cos @ B @ ( F3 @ X4 ) ) ) ) ) ).
% continuous_on_cos
thf(fact_7972_continuous__on__pochhammer_H,axiom,
! [A: $tType,C: $tType] :
( ( ( topolo4958980785337419405_space @ C )
& ( real_V3459762299906320749_field @ A ) )
=> ! [S: set @ C,F3: C > A,N: nat] :
( ( topolo81223032696312382ous_on @ C @ A @ S @ F3 )
=> ( topolo81223032696312382ous_on @ C @ A @ S
@ ^ [X4: C] : ( comm_s3205402744901411588hammer @ A @ ( F3 @ X4 ) @ N ) ) ) ) ).
% continuous_on_pochhammer'
thf(fact_7973_continuous__on__pochhammer,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [A3: set @ A,N: nat] :
( topolo81223032696312382ous_on @ A @ A @ A3
@ ^ [Z6: A] : ( comm_s3205402744901411588hammer @ A @ Z6 @ N ) ) ) ).
% continuous_on_pochhammer
thf(fact_7974_continuous__on__cosh,axiom,
! [A: $tType,C: $tType] :
( ( ( topolo4958980785337419405_space @ C )
& ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [A3: set @ C,F3: C > A] :
( ( topolo81223032696312382ous_on @ C @ A @ A3 @ F3 )
=> ( topolo81223032696312382ous_on @ C @ A @ A3
@ ^ [X4: C] : ( cosh @ A @ ( F3 @ X4 ) ) ) ) ) ).
% continuous_on_cosh
thf(fact_7975_continuous__on__add,axiom,
! [B: $tType,D: $tType] :
( ( ( topolo4958980785337419405_space @ D )
& ( topolo6943815403480290642id_add @ B ) )
=> ! [S: set @ D,F3: D > B,G: D > B] :
( ( topolo81223032696312382ous_on @ D @ B @ S @ F3 )
=> ( ( topolo81223032696312382ous_on @ D @ B @ S @ G )
=> ( topolo81223032696312382ous_on @ D @ B @ S
@ ^ [X4: D] : ( plus_plus @ B @ ( F3 @ X4 ) @ ( G @ X4 ) ) ) ) ) ) ).
% continuous_on_add
thf(fact_7976_continuous__on__mult,axiom,
! [A: $tType,D: $tType] :
( ( ( topolo4958980785337419405_space @ D )
& ( real_V4412858255891104859lgebra @ A ) )
=> ! [S: set @ D,F3: D > A,G: D > A] :
( ( topolo81223032696312382ous_on @ D @ A @ S @ F3 )
=> ( ( topolo81223032696312382ous_on @ D @ A @ S @ G )
=> ( topolo81223032696312382ous_on @ D @ A @ S
@ ^ [X4: D] : ( times_times @ A @ ( F3 @ X4 ) @ ( G @ X4 ) ) ) ) ) ) ).
% continuous_on_mult
thf(fact_7977_continuous__on__mult_H,axiom,
! [B: $tType,D: $tType] :
( ( ( topolo4958980785337419405_space @ D )
& ( topolo4211221413907600880p_mult @ B ) )
=> ! [A3: set @ D,F3: D > B,G: D > B] :
( ( topolo81223032696312382ous_on @ D @ B @ A3 @ F3 )
=> ( ( topolo81223032696312382ous_on @ D @ B @ A3 @ G )
=> ( topolo81223032696312382ous_on @ D @ B @ A3
@ ^ [X4: D] : ( times_times @ B @ ( F3 @ X4 ) @ ( G @ X4 ) ) ) ) ) ) ).
% continuous_on_mult'
thf(fact_7978_continuous__on__mult__left,axiom,
! [A: $tType,B: $tType] :
( ( ( topolo4958980785337419405_space @ B )
& ( real_V4412858255891104859lgebra @ A ) )
=> ! [S: set @ B,F3: B > A,C2: A] :
( ( topolo81223032696312382ous_on @ B @ A @ S @ F3 )
=> ( topolo81223032696312382ous_on @ B @ A @ S
@ ^ [X4: B] : ( times_times @ A @ C2 @ ( F3 @ X4 ) ) ) ) ) ).
% continuous_on_mult_left
thf(fact_7979_continuous__on__mult__right,axiom,
! [A: $tType,B: $tType] :
( ( ( topolo4958980785337419405_space @ B )
& ( real_V4412858255891104859lgebra @ A ) )
=> ! [S: set @ B,F3: B > A,C2: A] :
( ( topolo81223032696312382ous_on @ B @ A @ S @ F3 )
=> ( topolo81223032696312382ous_on @ B @ A @ S
@ ^ [X4: B] : ( times_times @ A @ ( F3 @ X4 ) @ C2 ) ) ) ) ).
% continuous_on_mult_right
thf(fact_7980_continuous__on__diff,axiom,
! [B: $tType,D: $tType] :
( ( ( topolo4958980785337419405_space @ D )
& ( topolo1633459387980952147up_add @ B ) )
=> ! [S: set @ D,F3: D > B,G: D > B] :
( ( topolo81223032696312382ous_on @ D @ B @ S @ F3 )
=> ( ( topolo81223032696312382ous_on @ D @ B @ S @ G )
=> ( topolo81223032696312382ous_on @ D @ B @ S
@ ^ [X4: D] : ( minus_minus @ B @ ( F3 @ X4 ) @ ( G @ X4 ) ) ) ) ) ) ).
% continuous_on_diff
thf(fact_7981_continuous__on__rabs,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [S: set @ A,F3: A > real] :
( ( topolo81223032696312382ous_on @ A @ real @ S @ F3 )
=> ( topolo81223032696312382ous_on @ A @ real @ S
@ ^ [X4: A] : ( abs_abs @ real @ ( F3 @ X4 ) ) ) ) ) ).
% continuous_on_rabs
thf(fact_7982_continuous__on__max,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ A )
& ( topolo1944317154257567458pology @ B ) )
=> ! [A3: set @ A,F3: A > B,G: A > B] :
( ( topolo81223032696312382ous_on @ A @ B @ A3 @ F3 )
=> ( ( topolo81223032696312382ous_on @ A @ B @ A3 @ G )
=> ( topolo81223032696312382ous_on @ A @ B @ A3
@ ^ [X4: A] : ( ord_max @ B @ ( F3 @ X4 ) @ ( G @ X4 ) ) ) ) ) ) ).
% continuous_on_max
thf(fact_7983_continuous__on__minus,axiom,
! [B: $tType,C: $tType] :
( ( ( topolo4958980785337419405_space @ C )
& ( topolo1633459387980952147up_add @ B ) )
=> ! [S: set @ C,F3: C > B] :
( ( topolo81223032696312382ous_on @ C @ B @ S @ F3 )
=> ( topolo81223032696312382ous_on @ C @ B @ S
@ ^ [X4: C] : ( uminus_uminus @ B @ ( F3 @ X4 ) ) ) ) ) ).
% continuous_on_minus
thf(fact_7984_continuous__on__prod_H,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( ( topolo4958980785337419405_space @ B )
& ( topolo4987421752381908075d_mult @ C ) )
=> ! [I6: set @ A,S3: set @ B,F3: A > B > C] :
( ! [I3: A] :
( ( member @ A @ I3 @ I6 )
=> ( topolo81223032696312382ous_on @ B @ C @ S3 @ ( F3 @ I3 ) ) )
=> ( topolo81223032696312382ous_on @ B @ C @ S3
@ ^ [X4: B] :
( groups7121269368397514597t_prod @ A @ C
@ ^ [I: A] : ( F3 @ I @ X4 )
@ I6 ) ) ) ) ).
% continuous_on_prod'
thf(fact_7985_continuous__on__prod,axiom,
! [A: $tType,C: $tType,D: $tType] :
( ( ( topolo4958980785337419405_space @ D )
& ( real_V4412858255891104859lgebra @ C )
& ( comm_ring_1 @ C ) )
=> ! [S3: set @ A,S: set @ D,F3: A > D > C] :
( ! [I3: A] :
( ( member @ A @ I3 @ S3 )
=> ( topolo81223032696312382ous_on @ D @ C @ S @ ( F3 @ I3 ) ) )
=> ( topolo81223032696312382ous_on @ D @ C @ S
@ ^ [X4: D] :
( groups7121269368397514597t_prod @ A @ C
@ ^ [I: A] : ( F3 @ I @ X4 )
@ S3 ) ) ) ) ).
% continuous_on_prod
thf(fact_7986_continuous__on__sum,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( ( topolo4958980785337419405_space @ B )
& ( topolo5987344860129210374id_add @ C ) )
=> ! [I6: set @ A,S3: set @ B,F3: A > B > C] :
( ! [I3: A] :
( ( member @ A @ I3 @ I6 )
=> ( topolo81223032696312382ous_on @ B @ C @ S3 @ ( F3 @ I3 ) ) )
=> ( topolo81223032696312382ous_on @ B @ C @ S3
@ ^ [X4: B] :
( groups7311177749621191930dd_sum @ A @ C
@ ^ [I: A] : ( F3 @ I @ X4 )
@ I6 ) ) ) ) ).
% continuous_on_sum
thf(fact_7987_continuous__on__sgn,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [S: set @ A,F3: A > B] :
( ( topolo81223032696312382ous_on @ A @ B @ S @ F3 )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ S )
=> ( ( F3 @ X5 )
!= ( zero_zero @ B ) ) )
=> ( topolo81223032696312382ous_on @ A @ B @ S
@ ^ [X4: A] : ( sgn_sgn @ B @ ( F3 @ X4 ) ) ) ) ) ) ).
% continuous_on_sgn
thf(fact_7988_continuous__on__ln,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [S: set @ A,F3: A > real] :
( ( topolo81223032696312382ous_on @ A @ real @ S @ F3 )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ S )
=> ( ( F3 @ X5 )
!= ( zero_zero @ real ) ) )
=> ( topolo81223032696312382ous_on @ A @ real @ S
@ ^ [X4: A] : ( ln_ln @ real @ ( F3 @ X4 ) ) ) ) ) ) ).
% continuous_on_ln
thf(fact_7989_continuous__on__dist,axiom,
! [A: $tType,D: $tType] :
( ( ( topolo4958980785337419405_space @ D )
& ( real_V7819770556892013058_space @ A ) )
=> ! [S: set @ D,F3: D > A,G: D > A] :
( ( topolo81223032696312382ous_on @ D @ A @ S @ F3 )
=> ( ( topolo81223032696312382ous_on @ D @ A @ S @ G )
=> ( topolo81223032696312382ous_on @ D @ real @ S
@ ^ [X4: D] : ( real_V557655796197034286t_dist @ A @ ( F3 @ X4 ) @ ( G @ X4 ) ) ) ) ) ) ).
% continuous_on_dist
thf(fact_7990_continuous__on__tanh,axiom,
! [A: $tType,C: $tType] :
( ( ( topolo4958980785337419405_space @ C )
& ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [A3: set @ C,F3: C > A] :
( ( topolo81223032696312382ous_on @ C @ A @ A3 @ F3 )
=> ( ! [X5: C] :
( ( member @ C @ X5 @ A3 )
=> ( ( cosh @ A @ ( F3 @ X5 ) )
!= ( zero_zero @ A ) ) )
=> ( topolo81223032696312382ous_on @ C @ A @ A3
@ ^ [X4: C] : ( tanh @ A @ ( F3 @ X4 ) ) ) ) ) ) ).
% continuous_on_tanh
thf(fact_7991_continuous__on__cot,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [S: set @ A,F3: A > A] :
( ( topolo81223032696312382ous_on @ A @ A @ S @ F3 )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ S )
=> ( ( sin @ A @ ( F3 @ X5 ) )
!= ( zero_zero @ A ) ) )
=> ( topolo81223032696312382ous_on @ A @ A @ S
@ ^ [X4: A] : ( cot @ A @ ( F3 @ X4 ) ) ) ) ) ) ).
% continuous_on_cot
thf(fact_7992_continuous__on__arcosh_H,axiom,
! [A3: set @ real,F3: real > real] :
( ( topolo81223032696312382ous_on @ real @ real @ A3 @ F3 )
=> ( ! [X5: real] :
( ( member @ real @ X5 @ A3 )
=> ( ord_less_eq @ real @ ( one_one @ real ) @ ( F3 @ X5 ) ) )
=> ( topolo81223032696312382ous_on @ real @ real @ A3
@ ^ [X4: real] : ( arcosh @ real @ ( F3 @ X4 ) ) ) ) ) ).
% continuous_on_arcosh'
thf(fact_7993_continuous__image__closed__interval,axiom,
! [A2: real,B2: real,F3: real > real] :
( ( ord_less_eq @ real @ A2 @ B2 )
=> ( ( topolo81223032696312382ous_on @ real @ real @ ( set_or1337092689740270186AtMost @ real @ A2 @ B2 ) @ F3 )
=> ? [C4: real,D3: real] :
( ( ( image @ real @ real @ F3 @ ( set_or1337092689740270186AtMost @ real @ A2 @ B2 ) )
= ( set_or1337092689740270186AtMost @ real @ C4 @ D3 ) )
& ( ord_less_eq @ real @ C4 @ D3 ) ) ) ) ).
% continuous_image_closed_interval
thf(fact_7994_continuous__on__arcosh,axiom,
! [A3: set @ real] :
( ( ord_less_eq @ ( set @ real ) @ A3 @ ( set_ord_atLeast @ real @ ( one_one @ real ) ) )
=> ( topolo81223032696312382ous_on @ real @ real @ A3 @ ( arcosh @ real ) ) ) ).
% continuous_on_arcosh
thf(fact_7995_open__Collect__positive,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [S: set @ A,F3: A > real] :
( ( topolo81223032696312382ous_on @ A @ real @ S @ F3 )
=> ? [A8: set @ A] :
( ( topolo1002775350975398744n_open @ A @ A8 )
& ( ( inf_inf @ ( set @ A ) @ A8 @ S )
= ( collect @ A
@ ^ [X4: A] :
( ( member @ A @ X4 @ S )
& ( ord_less @ real @ ( zero_zero @ real ) @ ( F3 @ X4 ) ) ) ) ) ) ) ) ).
% open_Collect_positive
thf(fact_7996_continuous__on__powr_H,axiom,
! [C: $tType] :
( ( topolo4958980785337419405_space @ C )
=> ! [S: set @ C,F3: C > real,G: C > real] :
( ( topolo81223032696312382ous_on @ C @ real @ S @ F3 )
=> ( ( topolo81223032696312382ous_on @ C @ real @ S @ G )
=> ( ! [X5: C] :
( ( member @ C @ X5 @ S )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( F3 @ X5 ) )
& ( ( ( F3 @ X5 )
= ( zero_zero @ real ) )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( G @ X5 ) ) ) ) )
=> ( topolo81223032696312382ous_on @ C @ real @ S
@ ^ [X4: C] : ( powr @ real @ ( F3 @ X4 ) @ ( G @ X4 ) ) ) ) ) ) ) ).
% continuous_on_powr'
thf(fact_7997_continuous__on__log,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [S: set @ A,F3: A > real,G: A > real] :
( ( topolo81223032696312382ous_on @ A @ real @ S @ F3 )
=> ( ( topolo81223032696312382ous_on @ A @ real @ S @ G )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ S )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( F3 @ X5 ) ) )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ S )
=> ( ( F3 @ X5 )
!= ( one_one @ real ) ) )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ S )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( G @ X5 ) ) )
=> ( topolo81223032696312382ous_on @ A @ real @ S
@ ^ [X4: A] : ( log @ ( F3 @ X4 ) @ ( G @ X4 ) ) ) ) ) ) ) ) ) ).
% continuous_on_log
thf(fact_7998_continuous__on__arccos_H,axiom,
topolo81223032696312382ous_on @ real @ real @ ( set_or1337092689740270186AtMost @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ ( one_one @ real ) ) @ arccos ).
% continuous_on_arccos'
thf(fact_7999_continuous__on__arcsin_H,axiom,
topolo81223032696312382ous_on @ real @ real @ ( set_or1337092689740270186AtMost @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ ( one_one @ real ) ) @ arcsin ).
% continuous_on_arcsin'
thf(fact_8000_continuous__on__arccos,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [S: set @ A,F3: A > real] :
( ( topolo81223032696312382ous_on @ A @ real @ S @ F3 )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ S )
=> ( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ ( F3 @ X5 ) )
& ( ord_less_eq @ real @ ( F3 @ X5 ) @ ( one_one @ real ) ) ) )
=> ( topolo81223032696312382ous_on @ A @ real @ S
@ ^ [X4: A] : ( arccos @ ( F3 @ X4 ) ) ) ) ) ) ).
% continuous_on_arccos
thf(fact_8001_continuous__on__arcsin,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [S: set @ A,F3: A > real] :
( ( topolo81223032696312382ous_on @ A @ real @ S @ F3 )
=> ( ! [X5: A] :
( ( member @ A @ X5 @ S )
=> ( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ ( F3 @ X5 ) )
& ( ord_less_eq @ real @ ( F3 @ X5 ) @ ( one_one @ real ) ) ) )
=> ( topolo81223032696312382ous_on @ A @ real @ S
@ ^ [X4: A] : ( arcsin @ ( F3 @ X4 ) ) ) ) ) ) ).
% continuous_on_arcsin
thf(fact_8002_DERIV__atLeastAtMost__imp__continuous__on,axiom,
! [A: $tType] :
( ( ( ord @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [A2: A,B2: A,F3: A > A] :
( ! [X5: A] :
( ( ord_less_eq @ A @ A2 @ X5 )
=> ( ( ord_less_eq @ A @ X5 @ B2 )
=> ? [Y4: A] : ( has_field_derivative @ A @ F3 @ Y4 @ ( topolo174197925503356063within @ A @ X5 @ ( top_top @ ( set @ A ) ) ) ) ) )
=> ( topolo81223032696312382ous_on @ A @ A @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) @ F3 ) ) ) ).
% DERIV_atLeastAtMost_imp_continuous_on
thf(fact_8003_continuous__on__artanh_H,axiom,
! [A3: set @ real,F3: real > real] :
( ( topolo81223032696312382ous_on @ real @ real @ A3 @ F3 )
=> ( ! [X5: real] :
( ( member @ real @ X5 @ A3 )
=> ( member @ real @ ( F3 @ X5 ) @ ( set_or5935395276787703475ssThan @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ ( one_one @ real ) ) ) )
=> ( topolo81223032696312382ous_on @ real @ real @ A3
@ ^ [X4: real] : ( artanh @ real @ ( F3 @ X4 ) ) ) ) ) ).
% continuous_on_artanh'
thf(fact_8004_continuous__on__of__int__floor,axiom,
! [B: $tType,A: $tType] :
( ( ( archim2362893244070406136eiling @ A )
& ( topolo2564578578187576103pology @ A )
& ( ring_1 @ B )
& ( topolo4958980785337419405_space @ B ) )
=> ( topolo81223032696312382ous_on @ A @ B @ ( minus_minus @ ( set @ A ) @ ( top_top @ ( set @ A ) ) @ ( ring_1_Ints @ A ) )
@ ^ [X4: A] : ( ring_1_of_int @ B @ ( archim6421214686448440834_floor @ A @ X4 ) ) ) ) ).
% continuous_on_of_int_floor
thf(fact_8005_continuous__on__of__int__ceiling,axiom,
! [B: $tType,A: $tType] :
( ( ( archim2362893244070406136eiling @ A )
& ( topolo2564578578187576103pology @ A )
& ( ring_1 @ B )
& ( topolo4958980785337419405_space @ B ) )
=> ( topolo81223032696312382ous_on @ A @ B @ ( minus_minus @ ( set @ A ) @ ( top_top @ ( set @ A ) ) @ ( ring_1_Ints @ A ) )
@ ^ [X4: A] : ( ring_1_of_int @ B @ ( archimedean_ceiling @ A @ X4 ) ) ) ) ).
% continuous_on_of_int_ceiling
thf(fact_8006_continuous__on__Icc__at__leftD,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo1944317154257567458pology @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [A2: A,B2: A,F3: A > B] :
( ( topolo81223032696312382ous_on @ A @ B @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) @ F3 )
=> ( ( ord_less @ A @ A2 @ B2 )
=> ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ ( F3 @ B2 ) ) @ ( topolo174197925503356063within @ A @ B2 @ ( set_ord_lessThan @ A @ B2 ) ) ) ) ) ) ).
% continuous_on_Icc_at_leftD
thf(fact_8007_continuous__on__Icc__at__rightD,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo1944317154257567458pology @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [A2: A,B2: A,F3: A > B] :
( ( topolo81223032696312382ous_on @ A @ B @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) @ F3 )
=> ( ( ord_less @ A @ A2 @ B2 )
=> ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ ( F3 @ A2 ) ) @ ( topolo174197925503356063within @ A @ A2 @ ( set_ord_greaterThan @ A @ A2 ) ) ) ) ) ) ).
% continuous_on_Icc_at_rightD
thf(fact_8008_continuous__on__artanh,axiom,
! [A3: set @ real] :
( ( ord_less_eq @ ( set @ real ) @ A3 @ ( set_or5935395276787703475ssThan @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ ( one_one @ real ) ) )
=> ( topolo81223032696312382ous_on @ real @ real @ A3 @ ( artanh @ real ) ) ) ).
% continuous_on_artanh
thf(fact_8009_DERIV__isconst2,axiom,
! [A2: real,B2: real,F3: real > real,X: real] :
( ( ord_less @ real @ A2 @ B2 )
=> ( ( topolo81223032696312382ous_on @ real @ real @ ( set_or1337092689740270186AtMost @ real @ A2 @ B2 ) @ F3 )
=> ( ! [X5: real] :
( ( ord_less @ real @ A2 @ X5 )
=> ( ( ord_less @ real @ X5 @ B2 )
=> ( has_field_derivative @ real @ F3 @ ( zero_zero @ real ) @ ( topolo174197925503356063within @ real @ X5 @ ( top_top @ ( set @ real ) ) ) ) ) )
=> ( ( ord_less_eq @ real @ A2 @ X )
=> ( ( ord_less_eq @ real @ X @ B2 )
=> ( ( F3 @ X )
= ( F3 @ A2 ) ) ) ) ) ) ) ).
% DERIV_isconst2
thf(fact_8010_continuous__on__IccI,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo1944317154257567458pology @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [F3: A > B,A2: A,B2: A] :
( ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ ( F3 @ A2 ) ) @ ( topolo174197925503356063within @ A @ A2 @ ( set_ord_greaterThan @ A @ A2 ) ) )
=> ( ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ ( F3 @ B2 ) ) @ ( topolo174197925503356063within @ A @ B2 @ ( set_ord_lessThan @ A @ B2 ) ) )
=> ( ! [X5: A] :
( ( ord_less @ A @ A2 @ X5 )
=> ( ( ord_less @ A @ X5 @ B2 )
=> ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ ( F3 @ X5 ) ) @ ( topolo174197925503356063within @ A @ X5 @ ( top_top @ ( set @ A ) ) ) ) ) )
=> ( ( ord_less @ A @ A2 @ B2 )
=> ( topolo81223032696312382ous_on @ A @ B @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) @ F3 ) ) ) ) ) ) ).
% continuous_on_IccI
thf(fact_8011_min__ext__compat,axiom,
! [A: $tType,R4: set @ ( product_prod @ A @ A ),S3: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( relcomp @ A @ A @ A @ R4 @ S3 ) @ R4 )
=> ( ord_less_eq @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( relcomp @ ( set @ A ) @ ( set @ A ) @ ( set @ A ) @ ( min_ext @ A @ R4 ) @ ( 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 @ R4 ) ) ) ).
% min_ext_compat
thf(fact_8012_lexord__def,axiom,
! [A: $tType] :
( ( lexord @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ ( list @ A ) @ ( list @ A ) )
@ ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ $o
@ ^ [X4: list @ A,Y: list @ A] :
? [A5: A,V5: list @ A] :
( ( Y
= ( append @ A @ X4 @ ( cons @ A @ A5 @ V5 ) ) )
| ? [U2: list @ A,B4: A,C3: A,W3: list @ A,Z6: list @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B4 @ C3 ) @ R5 )
& ( X4
= ( append @ A @ U2 @ ( cons @ A @ B4 @ W3 ) ) )
& ( Y
= ( append @ A @ U2 @ ( cons @ A @ C3 @ Z6 ) ) ) ) ) ) ) ) ) ).
% lexord_def
thf(fact_8013_continuous__on__of__real__o__iff,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [S3: set @ A,G: A > real] :
( ( topolo81223032696312382ous_on @ A @ complex @ S3
@ ^ [X4: A] : ( real_Vector_of_real @ complex @ ( G @ X4 ) ) )
= ( topolo81223032696312382ous_on @ A @ real @ S3 @ G ) ) ) ).
% continuous_on_of_real_o_iff
thf(fact_8014_lexord__cons__cons,axiom,
! [A: $tType,A2: A,X: list @ A,B2: A,Y2: list @ A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ A2 @ X ) @ ( cons @ A @ B2 @ Y2 ) ) @ ( lexord @ A @ R ) )
= ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ R )
| ( ( A2 = B2 )
& ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Y2 ) @ ( lexord @ A @ R ) ) ) ) ) ).
% lexord_cons_cons
thf(fact_8015_lexord__Nil__left,axiom,
! [A: $tType,Y2: list @ A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Y2 ) @ ( lexord @ A @ R ) )
= ( ? [A5: A,X4: list @ A] :
( Y2
= ( cons @ A @ A5 @ X4 ) ) ) ) ).
% lexord_Nil_left
thf(fact_8016_continuous__on__cnj,axiom,
! [C: $tType] :
( ( topolo4958980785337419405_space @ C )
=> ! [S: set @ C,G: C > complex] :
( ( topolo81223032696312382ous_on @ C @ complex @ S @ G )
=> ( topolo81223032696312382ous_on @ C @ complex @ S
@ ^ [X4: C] : ( cnj @ ( G @ X4 ) ) ) ) ) ).
% continuous_on_cnj
thf(fact_8017_continuous__on__Im,axiom,
! [C: $tType] :
( ( topolo4958980785337419405_space @ C )
=> ! [S: set @ C,G: C > complex] :
( ( topolo81223032696312382ous_on @ C @ complex @ S @ G )
=> ( topolo81223032696312382ous_on @ C @ real @ S
@ ^ [X4: C] : ( im @ ( G @ X4 ) ) ) ) ) ).
% continuous_on_Im
thf(fact_8018_continuous__on__cis,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [A3: set @ A,F3: A > real] :
( ( topolo81223032696312382ous_on @ A @ real @ A3 @ F3 )
=> ( topolo81223032696312382ous_on @ A @ complex @ A3
@ ^ [X4: A] : ( cis @ ( F3 @ X4 ) ) ) ) ) ).
% continuous_on_cis
thf(fact_8019_continuous__on__Re,axiom,
! [C: $tType] :
( ( topolo4958980785337419405_space @ C )
=> ! [S: set @ C,G: C > complex] :
( ( topolo81223032696312382ous_on @ C @ complex @ S @ G )
=> ( topolo81223032696312382ous_on @ C @ real @ S
@ ^ [X4: C] : ( re @ ( G @ X4 ) ) ) ) ) ).
% continuous_on_Re
thf(fact_8020_lexord__append__leftI,axiom,
! [A: $tType,U: list @ A,V2: list @ A,R: set @ ( product_prod @ A @ A ),X: list @ A] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ U @ V2 ) @ ( lexord @ A @ R ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ X @ U ) @ ( append @ A @ X @ V2 ) ) @ ( lexord @ A @ R ) ) ) ).
% lexord_append_leftI
thf(fact_8021_lexord__irreflexive,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),Xs: list @ A] :
( ! [X5: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X5 @ X5 ) @ R )
=> ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Xs ) @ ( lexord @ A @ R ) ) ) ).
% lexord_irreflexive
thf(fact_8022_lexord__linear,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),X: list @ A,Y2: list @ A] :
( ! [A4: A,B3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B3 ) @ R )
| ( A4 = B3 )
| ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B3 @ A4 ) @ R ) )
=> ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Y2 ) @ ( lexord @ A @ R ) )
| ( X = Y2 )
| ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Y2 @ X ) @ ( lexord @ A @ R ) ) ) ) ).
% lexord_linear
thf(fact_8023_lexord__Nil__right,axiom,
! [A: $tType,X: list @ A,R: set @ ( product_prod @ A @ A )] :
~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ ( nil @ A ) ) @ ( lexord @ A @ R ) ) ).
% lexord_Nil_right
thf(fact_8024_lexord__partial__trans,axiom,
! [A: $tType,Xs: list @ A,R: set @ ( product_prod @ A @ A ),Ys: list @ A,Zs: list @ A] :
( ! [X5: A,Y7: A,Z: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Xs ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X5 @ Y7 ) @ R )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y7 @ Z ) @ R )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X5 @ Z ) @ R ) ) ) )
=> ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( lexord @ A @ R ) )
=> ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ys @ Zs ) @ ( lexord @ A @ R ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Zs ) @ ( lexord @ A @ R ) ) ) ) ) ).
% lexord_partial_trans
thf(fact_8025_lexord__append__leftD,axiom,
! [A: $tType,X: list @ A,U: list @ A,V2: list @ A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ X @ U ) @ ( append @ A @ X @ V2 ) ) @ ( lexord @ A @ R ) )
=> ( ! [A4: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ A4 ) @ R )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ U @ V2 ) @ ( lexord @ A @ R ) ) ) ) ).
% lexord_append_leftD
thf(fact_8026_lexord__append__rightI,axiom,
! [A: $tType,Y2: list @ A,X: list @ A,R: set @ ( product_prod @ A @ A )] :
( ? [B10: A,Z4: list @ A] :
( Y2
= ( cons @ A @ B10 @ Z4 ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ ( append @ A @ X @ Y2 ) ) @ ( lexord @ A @ R ) ) ) ).
% lexord_append_rightI
thf(fact_8027_lexord__sufE,axiom,
! [A: $tType,Xs: list @ A,Zs: list @ A,Ys: list @ A,Qs: list @ A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs @ Zs ) @ ( append @ A @ Ys @ Qs ) ) @ ( lexord @ A @ R ) )
=> ( ( Xs != Ys )
=> ( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ A ) @ Ys ) )
=> ( ( ( size_size @ ( list @ A ) @ Zs )
= ( size_size @ ( list @ A ) @ Qs ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( lexord @ A @ R ) ) ) ) ) ) ).
% lexord_sufE
thf(fact_8028_lexord__lex,axiom,
! [A: $tType,X: list @ A,Y2: list @ A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Y2 ) @ ( lex @ A @ R ) )
= ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Y2 ) @ ( lexord @ A @ R ) )
& ( ( size_size @ ( list @ A ) @ X )
= ( size_size @ ( list @ A ) @ Y2 ) ) ) ) ).
% lexord_lex
thf(fact_8029_lexord__append__left__rightI,axiom,
! [A: $tType,A2: A,B2: A,R: set @ ( product_prod @ A @ A ),U: list @ A,X: list @ A,Y2: list @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ R )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ U @ ( cons @ A @ A2 @ X ) ) @ ( append @ A @ U @ ( cons @ A @ B2 @ Y2 ) ) ) @ ( lexord @ A @ R ) ) ) ).
% lexord_append_left_rightI
thf(fact_8030_lexord__same__pref__iff,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Zs: list @ A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs @ Ys ) @ ( append @ A @ Xs @ Zs ) ) @ ( lexord @ A @ R ) )
= ( ? [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ X4 ) @ R ) )
| ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ys @ Zs ) @ ( lexord @ A @ R ) ) ) ) ).
% lexord_same_pref_iff
thf(fact_8031_lexord__sufI,axiom,
! [A: $tType,U: list @ A,W: list @ A,R: set @ ( product_prod @ A @ A ),V2: list @ A,Z3: list @ A] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ U @ W ) @ ( lexord @ A @ R ) )
=> ( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ W ) @ ( size_size @ ( list @ A ) @ U ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ U @ V2 ) @ ( append @ A @ W @ Z3 ) ) @ ( lexord @ A @ R ) ) ) ) ).
% lexord_sufI
thf(fact_8032_List_Olexordp__def,axiom,
! [A: $tType] :
( ( lexordp @ A )
= ( ^ [R5: A > A > $o,Xs2: list @ A,Ys3: list @ A] : ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ Ys3 ) @ ( lexord @ A @ ( collect @ ( product_prod @ A @ A ) @ ( product_case_prod @ A @ A @ $o @ R5 ) ) ) ) ) ) ).
% List.lexordp_def
thf(fact_8033_max__ext__compat,axiom,
! [A: $tType,R4: set @ ( product_prod @ A @ A ),S3: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( relcomp @ A @ A @ A @ R4 @ S3 ) @ R4 )
=> ( ord_less_eq @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( relcomp @ ( set @ A ) @ ( set @ A ) @ ( set @ A ) @ ( max_ext @ A @ R4 ) @ ( 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 @ R4 ) ) ) ).
% max_ext_compat
thf(fact_8034_max__ext__def,axiom,
! [A: $tType] :
( ( max_ext @ A )
= ( ^ [R6: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ ( set @ A ) @ ( set @ A ) )
@ ( product_case_prod @ ( set @ A ) @ ( set @ A ) @ $o
@ ( max_extp @ A
@ ^ [X4: A,Y: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y ) @ R6 ) ) ) ) ) ) ).
% max_ext_def
thf(fact_8035_Succ__def,axiom,
! [A: $tType] :
( ( bNF_Greatest_Succ @ A )
= ( ^ [Kl: set @ ( list @ A ),Kl2: list @ A] :
( collect @ A
@ ^ [K3: A] : ( member @ ( list @ A ) @ ( append @ A @ Kl2 @ ( cons @ A @ K3 @ ( nil @ A ) ) ) @ Kl ) ) ) ) ).
% Succ_def
thf(fact_8036_max__extp__max__ext__eq,axiom,
! [A: $tType,R4: set @ ( product_prod @ A @ A )] :
( ( max_extp @ A
@ ^ [X4: A,Y: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y ) @ R4 ) )
= ( ^ [X4: set @ A,Y: set @ A] : ( member @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ X4 @ Y ) @ ( max_ext @ A @ R4 ) ) ) ) ).
% max_extp_max_ext_eq
thf(fact_8037_max__extp__eq,axiom,
! [A: $tType] :
( ( max_extp @ A )
= ( ^ [R5: A > A > $o,X4: set @ A,Y: set @ A] : ( member @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ X4 @ Y ) @ ( max_ext @ A @ ( collect @ ( product_prod @ A @ A ) @ ( product_case_prod @ A @ A @ $o @ R5 ) ) ) ) ) ) ).
% max_extp_eq
thf(fact_8038_uniformity__dist,axiom,
! [A: $tType] :
( ( real_V768167426530841204y_dist @ A )
=> ( ( topolo7806501430040627800ormity @ A )
= ( complete_Inf_Inf @ ( filter @ ( product_prod @ A @ A ) )
@ ( image @ real @ ( filter @ ( product_prod @ A @ A ) )
@ ^ [E3: real] :
( principal @ ( product_prod @ A @ A )
@ ( collect @ ( product_prod @ A @ A )
@ ( product_case_prod @ A @ A @ $o
@ ^ [X4: A,Y: A] : ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X4 @ Y ) @ E3 ) ) ) )
@ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) ) ) ) ).
% uniformity_dist
thf(fact_8039_compactE__image,axiom,
! [A: $tType,B: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [S3: set @ A,C5: set @ B,F3: B > ( set @ A )] :
( ( topolo2193935891317330818ompact @ A @ S3 )
=> ( ! [T7: B] :
( ( member @ B @ T7 @ C5 )
=> ( topolo1002775350975398744n_open @ A @ ( F3 @ T7 ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ S3 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ F3 @ C5 ) ) )
=> ~ ! [C10: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ C10 @ C5 )
=> ( ( finite_finite2 @ B @ C10 )
=> ~ ( ord_less_eq @ ( set @ A ) @ S3 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ F3 @ C10 ) ) ) ) ) ) ) ) ) ).
% compactE_image
thf(fact_8040_compact__attains__inf,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [S3: set @ A] :
( ( topolo2193935891317330818ompact @ A @ S3 )
=> ( ( S3
!= ( bot_bot @ ( set @ A ) ) )
=> ? [X5: A] :
( ( member @ A @ X5 @ S3 )
& ! [Xa2: A] :
( ( member @ A @ Xa2 @ S3 )
=> ( ord_less_eq @ A @ X5 @ Xa2 ) ) ) ) ) ) ).
% compact_attains_inf
thf(fact_8041_compact__attains__sup,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [S3: set @ A] :
( ( topolo2193935891317330818ompact @ A @ S3 )
=> ( ( S3
!= ( bot_bot @ ( set @ A ) ) )
=> ? [X5: A] :
( ( member @ A @ X5 @ S3 )
& ! [Xa2: A] :
( ( member @ A @ Xa2 @ S3 )
=> ( ord_less_eq @ A @ Xa2 @ X5 ) ) ) ) ) ) ).
% compact_attains_sup
thf(fact_8042_uniformity__sym,axiom,
! [A: $tType] :
( ( topolo7287701948861334536_space @ A )
=> ! [E5: ( product_prod @ A @ A ) > $o] :
( ( eventually @ ( product_prod @ A @ A ) @ E5 @ ( topolo7806501430040627800ormity @ A ) )
=> ( eventually @ ( product_prod @ A @ A )
@ ( product_case_prod @ A @ A @ $o
@ ^ [X4: A,Y: A] : ( E5 @ ( product_Pair @ A @ A @ Y @ X4 ) ) )
@ ( topolo7806501430040627800ormity @ A ) ) ) ) ).
% uniformity_sym
thf(fact_8043_open__uniformity,axiom,
! [A: $tType] :
( ( topolo569519726778239578ormity @ A )
=> ( ( topolo1002775350975398744n_open @ A )
= ( ^ [U6: set @ A] :
! [X4: A] :
( ( member @ A @ X4 @ U6 )
=> ( eventually @ ( product_prod @ A @ A )
@ ( product_case_prod @ A @ A @ $o
@ ^ [X7: A,Y: A] :
( ( X7 = X4 )
=> ( member @ A @ Y @ U6 ) ) )
@ ( topolo7806501430040627800ormity @ A ) ) ) ) ) ) ).
% open_uniformity
thf(fact_8044_continuous__attains__inf,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ A )
& ( topolo1944317154257567458pology @ B ) )
=> ! [S: set @ A,F3: A > B] :
( ( topolo2193935891317330818ompact @ A @ S )
=> ( ( S
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( topolo81223032696312382ous_on @ A @ B @ S @ F3 )
=> ? [X5: A] :
( ( member @ A @ X5 @ S )
& ! [Xa2: A] :
( ( member @ A @ Xa2 @ S )
=> ( ord_less_eq @ B @ ( F3 @ X5 ) @ ( F3 @ Xa2 ) ) ) ) ) ) ) ) ).
% continuous_attains_inf
thf(fact_8045_continuous__attains__sup,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ A )
& ( topolo1944317154257567458pology @ B ) )
=> ! [S: set @ A,F3: A > B] :
( ( topolo2193935891317330818ompact @ A @ S )
=> ( ( S
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( topolo81223032696312382ous_on @ A @ B @ S @ F3 )
=> ? [X5: A] :
( ( member @ A @ X5 @ S )
& ! [Xa2: A] :
( ( member @ A @ Xa2 @ S )
=> ( ord_less_eq @ B @ ( F3 @ Xa2 ) @ ( F3 @ X5 ) ) ) ) ) ) ) ) ).
% continuous_attains_sup
thf(fact_8046_Cauchy__uniform__iff,axiom,
! [A: $tType] :
( ( topolo7287701948861334536_space @ A )
=> ( ( topolo3814608138187158403Cauchy @ A )
= ( ^ [X6: nat > A] :
! [P2: ( product_prod @ A @ A ) > $o] :
( ( eventually @ ( product_prod @ A @ A ) @ P2 @ ( topolo7806501430040627800ormity @ A ) )
=> ? [N6: nat] :
! [N2: nat] :
( ( ord_less_eq @ nat @ N6 @ N2 )
=> ! [M2: nat] :
( ( ord_less_eq @ nat @ N6 @ M2 )
=> ( P2 @ ( product_Pair @ A @ A @ ( X6 @ N2 ) @ ( X6 @ M2 ) ) ) ) ) ) ) ) ) ).
% Cauchy_uniform_iff
thf(fact_8047_uniformity__complex__def,axiom,
( ( topolo7806501430040627800ormity @ complex )
= ( complete_Inf_Inf @ ( filter @ ( product_prod @ complex @ complex ) )
@ ( image @ real @ ( filter @ ( product_prod @ complex @ complex ) )
@ ^ [E3: real] :
( principal @ ( product_prod @ complex @ complex )
@ ( collect @ ( product_prod @ complex @ complex )
@ ( product_case_prod @ complex @ complex @ $o
@ ^ [X4: complex,Y: complex] : ( ord_less @ real @ ( real_V557655796197034286t_dist @ complex @ X4 @ Y ) @ E3 ) ) ) )
@ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) ) ) ).
% uniformity_complex_def
thf(fact_8048_uniformity__real__def,axiom,
( ( topolo7806501430040627800ormity @ real )
= ( complete_Inf_Inf @ ( filter @ ( product_prod @ real @ real ) )
@ ( image @ real @ ( filter @ ( product_prod @ real @ real ) )
@ ^ [E3: real] :
( principal @ ( product_prod @ real @ real )
@ ( collect @ ( product_prod @ real @ real )
@ ( product_case_prod @ real @ real @ $o
@ ^ [X4: real,Y: real] : ( ord_less @ real @ ( real_V557655796197034286t_dist @ real @ X4 @ Y ) @ E3 ) ) ) )
@ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) ) ) ).
% uniformity_real_def
thf(fact_8049_eventually__nhds__uniformity,axiom,
! [A: $tType] :
( ( topolo7287701948861334536_space @ A )
=> ! [P3: A > $o,X: A] :
( ( eventually @ A @ P3 @ ( topolo7230453075368039082e_nhds @ A @ X ) )
= ( eventually @ ( product_prod @ A @ A )
@ ( product_case_prod @ A @ A @ $o
@ ^ [X7: A,Y: A] :
( ( X7 = X )
=> ( P3 @ Y ) ) )
@ ( topolo7806501430040627800ormity @ A ) ) ) ) ).
% eventually_nhds_uniformity
thf(fact_8050_tendsto__iff__uniformity,axiom,
! [A: $tType,B: $tType] :
( ( topolo7287701948861334536_space @ B )
=> ! [F3: A > B,L: B,F4: filter @ A] :
( ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ F4 )
= ( ! [E6: ( product_prod @ B @ B ) > $o] :
( ( eventually @ ( product_prod @ B @ B ) @ E6 @ ( topolo7806501430040627800ormity @ B ) )
=> ( eventually @ A
@ ^ [X4: A] : ( E6 @ ( product_Pair @ B @ B @ ( F3 @ X4 ) @ L ) )
@ F4 ) ) ) ) ) ).
% tendsto_iff_uniformity
thf(fact_8051_compactE,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [S3: set @ A,T11: set @ ( set @ A )] :
( ( topolo2193935891317330818ompact @ A @ S3 )
=> ( ( ord_less_eq @ ( set @ A ) @ S3 @ ( complete_Sup_Sup @ ( set @ A ) @ T11 ) )
=> ( ! [B9: set @ A] :
( ( member @ ( set @ A ) @ B9 @ T11 )
=> ( topolo1002775350975398744n_open @ A @ B9 ) )
=> ~ ! [T12: set @ ( set @ A )] :
( ( ord_less_eq @ ( set @ ( set @ A ) ) @ T12 @ T11 )
=> ( ( finite_finite2 @ ( set @ A ) @ T12 )
=> ~ ( ord_less_eq @ ( set @ A ) @ S3 @ ( complete_Sup_Sup @ ( set @ A ) @ T12 ) ) ) ) ) ) ) ) ).
% compactE
thf(fact_8052_compactI,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [S: set @ A] :
( ! [C7: set @ ( set @ A )] :
( ! [X2: set @ A] :
( ( member @ ( set @ A ) @ X2 @ C7 )
=> ( topolo1002775350975398744n_open @ A @ X2 ) )
=> ( ( ord_less_eq @ ( set @ A ) @ S @ ( complete_Sup_Sup @ ( set @ A ) @ C7 ) )
=> ? [C11: set @ ( set @ A )] :
( ( ord_less_eq @ ( set @ ( set @ A ) ) @ C11 @ C7 )
& ( finite_finite2 @ ( set @ A ) @ C11 )
& ( ord_less_eq @ ( set @ A ) @ S @ ( complete_Sup_Sup @ ( set @ A ) @ C11 ) ) ) ) )
=> ( topolo2193935891317330818ompact @ A @ S ) ) ) ).
% compactI
thf(fact_8053_compact__eq__Heine__Borel,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ( ( topolo2193935891317330818ompact @ A )
= ( ^ [S6: set @ A] :
! [C8: set @ ( set @ A )] :
( ( ! [X4: set @ A] :
( ( member @ ( set @ A ) @ X4 @ C8 )
=> ( topolo1002775350975398744n_open @ A @ X4 ) )
& ( ord_less_eq @ ( set @ A ) @ S6 @ ( complete_Sup_Sup @ ( set @ A ) @ C8 ) ) )
=> ? [D8: set @ ( set @ A )] :
( ( ord_less_eq @ ( set @ ( set @ A ) ) @ D8 @ C8 )
& ( finite_finite2 @ ( set @ A ) @ D8 )
& ( ord_less_eq @ ( set @ A ) @ S6 @ ( complete_Sup_Sup @ ( set @ A ) @ D8 ) ) ) ) ) ) ) ).
% compact_eq_Heine_Borel
thf(fact_8054_listrel1__def,axiom,
! [A: $tType] :
( ( listrel1 @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ ( list @ A ) @ ( list @ A ) )
@ ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ $o
@ ^ [Xs2: list @ A,Ys3: list @ A] :
? [Us2: list @ A,Z6: A,Z5: A,Vs2: list @ A] :
( ( Xs2
= ( append @ A @ Us2 @ ( cons @ A @ Z6 @ Vs2 ) ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z6 @ Z5 ) @ R5 )
& ( Ys3
= ( append @ A @ Us2 @ ( cons @ A @ Z5 @ Vs2 ) ) ) ) ) ) ) ) ).
% listrel1_def
thf(fact_8055_prod__filter__INF,axiom,
! [C: $tType,D: $tType,B: $tType,A: $tType,I6: set @ A,J5: set @ B,A3: A > ( filter @ C ),B5: B > ( filter @ D )] :
( ( I6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( J5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( prod_filter @ C @ D @ ( complete_Inf_Inf @ ( filter @ C ) @ ( image @ A @ ( filter @ C ) @ A3 @ I6 ) ) @ ( complete_Inf_Inf @ ( filter @ D ) @ ( image @ B @ ( filter @ D ) @ B5 @ J5 ) ) )
= ( complete_Inf_Inf @ ( filter @ ( product_prod @ C @ D ) )
@ ( image @ A @ ( filter @ ( product_prod @ C @ D ) )
@ ^ [I: A] :
( complete_Inf_Inf @ ( filter @ ( product_prod @ C @ D ) )
@ ( image @ B @ ( filter @ ( product_prod @ C @ D ) )
@ ^ [J4: B] : ( prod_filter @ C @ D @ ( A3 @ I ) @ ( B5 @ J4 ) )
@ J5 ) )
@ I6 ) ) ) ) ) ).
% prod_filter_INF
thf(fact_8056_Cons__listrel1__Cons,axiom,
! [A: $tType,X: A,Xs: list @ A,Y2: A,Ys: list @ A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y2 @ Ys ) ) @ ( listrel1 @ A @ R ) )
= ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y2 ) @ R )
& ( Xs = Ys ) )
| ( ( X = Y2 )
& ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( listrel1 @ A @ R ) ) ) ) ) ).
% Cons_listrel1_Cons
thf(fact_8057_open__real__def,axiom,
( ( topolo1002775350975398744n_open @ real )
= ( ^ [U6: set @ real] :
! [X4: real] :
( ( member @ real @ X4 @ U6 )
=> ( eventually @ ( product_prod @ real @ real )
@ ( product_case_prod @ real @ real @ $o
@ ^ [X7: real,Y: real] :
( ( X7 = X4 )
=> ( member @ real @ Y @ U6 ) ) )
@ ( topolo7806501430040627800ormity @ real ) ) ) ) ) ).
% open_real_def
thf(fact_8058_open__complex__def,axiom,
( ( topolo1002775350975398744n_open @ complex )
= ( ^ [U6: set @ complex] :
! [X4: complex] :
( ( member @ complex @ X4 @ U6 )
=> ( eventually @ ( product_prod @ complex @ complex )
@ ( product_case_prod @ complex @ complex @ $o
@ ^ [X7: complex,Y: complex] :
( ( X7 = X4 )
=> ( member @ complex @ Y @ U6 ) ) )
@ ( topolo7806501430040627800ormity @ complex ) ) ) ) ) ).
% open_complex_def
thf(fact_8059_listrel1I1,axiom,
! [A: $tType,X: A,Y2: A,R: set @ ( product_prod @ A @ A ),Xs: list @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y2 ) @ R )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y2 @ Xs ) ) @ ( listrel1 @ A @ R ) ) ) ).
% listrel1I1
thf(fact_8060_Cons__listrel1E1,axiom,
! [A: $tType,X: A,Xs: list @ A,Ys: list @ A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Xs ) @ Ys ) @ ( listrel1 @ A @ R ) )
=> ( ! [Y7: A] :
( ( Ys
= ( cons @ A @ Y7 @ Xs ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y7 ) @ R ) )
=> ~ ! [Zs2: list @ A] :
( ( Ys
= ( cons @ A @ X @ Zs2 ) )
=> ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Zs2 ) @ ( listrel1 @ A @ R ) ) ) ) ) ).
% Cons_listrel1E1
thf(fact_8061_Cons__listrel1E2,axiom,
! [A: $tType,Xs: list @ A,Y2: A,Ys: list @ A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ ( cons @ A @ Y2 @ Ys ) ) @ ( listrel1 @ A @ R ) )
=> ( ! [X5: A] :
( ( Xs
= ( cons @ A @ X5 @ Ys ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X5 @ Y2 ) @ R ) )
=> ~ ! [Zs2: list @ A] :
( ( Xs
= ( cons @ A @ Y2 @ Zs2 ) )
=> ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Zs2 @ Ys ) @ ( listrel1 @ A @ R ) ) ) ) ) ).
% Cons_listrel1E2
thf(fact_8062_prod__filter__mono,axiom,
! [A: $tType,B: $tType,F4: filter @ A,F14: filter @ A,G6: filter @ B,G7: filter @ B] :
( ( ord_less_eq @ ( filter @ A ) @ F4 @ F14 )
=> ( ( ord_less_eq @ ( filter @ B ) @ G6 @ G7 )
=> ( ord_less_eq @ ( filter @ ( product_prod @ A @ B ) ) @ ( prod_filter @ A @ B @ F4 @ G6 ) @ ( prod_filter @ A @ B @ F14 @ G7 ) ) ) ) ).
% prod_filter_mono
thf(fact_8063_listrel1__mono,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),S: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R @ S )
=> ( ord_less_eq @ ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) @ ( listrel1 @ A @ R ) @ ( listrel1 @ A @ S ) ) ) ).
% listrel1_mono
thf(fact_8064_listrel1I2,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,R: set @ ( product_prod @ A @ A ),X: A] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( listrel1 @ A @ R ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ X @ Ys ) ) @ ( listrel1 @ A @ R ) ) ) ).
% listrel1I2
thf(fact_8065_not__listrel1__Nil,axiom,
! [A: $tType,Xs: list @ A,R: set @ ( product_prod @ A @ A )] :
~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ ( nil @ A ) ) @ ( listrel1 @ A @ R ) ) ).
% not_listrel1_Nil
thf(fact_8066_not__Nil__listrel1,axiom,
! [A: $tType,Xs: list @ A,R: set @ ( product_prod @ A @ A )] :
~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Xs ) @ ( listrel1 @ A @ R ) ) ).
% not_Nil_listrel1
thf(fact_8067_append__listrel1I,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,R: set @ ( product_prod @ A @ A ),Us: list @ A,Vs: list @ A] :
( ( ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( listrel1 @ A @ R ) )
& ( Us = Vs ) )
| ( ( Xs = Ys )
& ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Us @ Vs ) @ ( listrel1 @ A @ R ) ) ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs @ Us ) @ ( append @ A @ Ys @ Vs ) ) @ ( listrel1 @ A @ R ) ) ) ).
% append_listrel1I
thf(fact_8068_listrel1__eq__len,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( listrel1 @ A @ R ) )
=> ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ A ) @ Ys ) ) ) ).
% listrel1_eq_len
thf(fact_8069_prod__filter__mono__iff,axiom,
! [A: $tType,B: $tType,A3: filter @ A,B5: filter @ B,C5: filter @ A,D4: filter @ B] :
( ( A3
!= ( bot_bot @ ( filter @ A ) ) )
=> ( ( B5
!= ( bot_bot @ ( filter @ B ) ) )
=> ( ( ord_less_eq @ ( filter @ ( product_prod @ A @ B ) ) @ ( prod_filter @ A @ B @ A3 @ B5 ) @ ( prod_filter @ A @ B @ C5 @ D4 ) )
= ( ( ord_less_eq @ ( filter @ A ) @ A3 @ C5 )
& ( ord_less_eq @ ( filter @ B ) @ B5 @ D4 ) ) ) ) ) ).
% prod_filter_mono_iff
thf(fact_8070_eventually__prod__sequentially,axiom,
! [P3: ( product_prod @ nat @ nat ) > $o] :
( ( eventually @ ( product_prod @ nat @ nat ) @ P3 @ ( prod_filter @ nat @ nat @ ( at_top @ nat ) @ ( at_top @ nat ) ) )
= ( ? [N6: nat] :
! [M2: nat] :
( ( ord_less_eq @ nat @ N6 @ M2 )
=> ! [N2: nat] :
( ( ord_less_eq @ nat @ N6 @ N2 )
=> ( P3 @ ( product_Pair @ nat @ nat @ N2 @ M2 ) ) ) ) ) ) ).
% eventually_prod_sequentially
thf(fact_8071_eventually__prodI,axiom,
! [A: $tType,B: $tType,P3: A > $o,F4: filter @ A,Q2: B > $o,G6: filter @ B] :
( ( eventually @ A @ P3 @ F4 )
=> ( ( eventually @ B @ Q2 @ G6 )
=> ( eventually @ ( product_prod @ A @ B )
@ ^ [X4: product_prod @ A @ B] :
( ( P3 @ ( product_fst @ A @ B @ X4 ) )
& ( Q2 @ ( product_snd @ A @ B @ X4 ) ) )
@ ( prod_filter @ A @ B @ F4 @ G6 ) ) ) ) ).
% eventually_prodI
thf(fact_8072_eventually__prod1,axiom,
! [A: $tType,B: $tType,B5: filter @ A,P3: B > $o,A3: filter @ B] :
( ( B5
!= ( bot_bot @ ( filter @ A ) ) )
=> ( ( eventually @ ( product_prod @ B @ A )
@ ( product_case_prod @ B @ A @ $o
@ ^ [X4: B,Y: A] : ( P3 @ X4 ) )
@ ( prod_filter @ B @ A @ A3 @ B5 ) )
= ( eventually @ B @ P3 @ A3 ) ) ) ).
% eventually_prod1
thf(fact_8073_eventually__prod2,axiom,
! [A: $tType,B: $tType,A3: filter @ A,P3: B > $o,B5: filter @ B] :
( ( A3
!= ( bot_bot @ ( filter @ A ) ) )
=> ( ( eventually @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [X4: A] : P3 )
@ ( prod_filter @ A @ B @ A3 @ B5 ) )
= ( eventually @ B @ P3 @ B5 ) ) ) ).
% eventually_prod2
thf(fact_8074_listrel1E,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( listrel1 @ A @ R ) )
=> ~ ! [X5: A,Y7: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X5 @ Y7 ) @ R )
=> ! [Us3: list @ A,Vs3: list @ A] :
( ( Xs
= ( append @ A @ Us3 @ ( cons @ A @ X5 @ Vs3 ) ) )
=> ( Ys
!= ( append @ A @ Us3 @ ( cons @ A @ Y7 @ Vs3 ) ) ) ) ) ) ).
% listrel1E
thf(fact_8075_listrel1I,axiom,
! [A: $tType,X: A,Y2: A,R: set @ ( product_prod @ A @ A ),Xs: list @ A,Us: list @ A,Vs: list @ A,Ys: list @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y2 ) @ R )
=> ( ( Xs
= ( append @ A @ Us @ ( cons @ A @ X @ Vs ) ) )
=> ( ( Ys
= ( append @ A @ Us @ ( cons @ A @ Y2 @ Vs ) ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( listrel1 @ A @ R ) ) ) ) ) ).
% listrel1I
thf(fact_8076_filterlim__Pair,axiom,
! [C: $tType,B: $tType,A: $tType,F3: A > B,G6: filter @ B,F4: filter @ A,G: A > C,H6: filter @ C] :
( ( filterlim @ A @ B @ F3 @ G6 @ F4 )
=> ( ( filterlim @ A @ C @ G @ H6 @ F4 )
=> ( filterlim @ A @ ( product_prod @ B @ C )
@ ^ [X4: A] : ( product_Pair @ B @ C @ ( F3 @ X4 ) @ ( G @ X4 ) )
@ ( prod_filter @ B @ C @ G6 @ H6 )
@ F4 ) ) ) ).
% filterlim_Pair
thf(fact_8077_tendsto__mult__Pair,axiom,
! [A: $tType] :
( ( topolo4211221413907600880p_mult @ A )
=> ! [A2: A,B2: A] :
( filterlim @ ( product_prod @ A @ A ) @ A
@ ^ [X4: product_prod @ A @ A] : ( times_times @ A @ ( product_fst @ A @ A @ X4 ) @ ( product_snd @ A @ A @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( times_times @ A @ A2 @ B2 ) )
@ ( prod_filter @ A @ A @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ ( topolo7230453075368039082e_nhds @ A @ B2 ) ) ) ) ).
% tendsto_mult_Pair
thf(fact_8078_tendsto__add__Pair,axiom,
! [A: $tType] :
( ( topolo6943815403480290642id_add @ A )
=> ! [A2: A,B2: A] :
( filterlim @ ( product_prod @ A @ A ) @ A
@ ^ [X4: product_prod @ A @ A] : ( plus_plus @ A @ ( product_fst @ A @ A @ X4 ) @ ( product_snd @ A @ A @ X4 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( plus_plus @ A @ A2 @ B2 ) )
@ ( prod_filter @ A @ A @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ ( topolo7230453075368039082e_nhds @ A @ B2 ) ) ) ) ).
% tendsto_add_Pair
thf(fact_8079_snoc__listrel1__snoc__iff,axiom,
! [A: $tType,Xs: list @ A,X: A,Ys: list @ A,Y2: A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs @ ( cons @ A @ X @ ( nil @ A ) ) ) @ ( append @ A @ Ys @ ( cons @ A @ Y2 @ ( nil @ A ) ) ) ) @ ( listrel1 @ A @ R ) )
= ( ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( listrel1 @ A @ R ) )
& ( X = Y2 ) )
| ( ( Xs = Ys )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y2 ) @ R ) ) ) ) ).
% snoc_listrel1_snoc_iff
thf(fact_8080_prod__filter__def,axiom,
! [B: $tType,A: $tType] :
( ( prod_filter @ A @ B )
= ( ^ [F10: filter @ A,G8: filter @ B] :
( complete_Inf_Inf @ ( filter @ ( product_prod @ A @ B ) )
@ ( image @ ( product_prod @ ( A > $o ) @ ( B > $o ) ) @ ( filter @ ( product_prod @ A @ B ) )
@ ( product_case_prod @ ( A > $o ) @ ( B > $o ) @ ( filter @ ( product_prod @ A @ B ) )
@ ^ [P2: A > $o,Q8: B > $o] :
( principal @ ( product_prod @ A @ B )
@ ( collect @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [X4: A,Y: B] :
( ( P2 @ X4 )
& ( Q8 @ Y ) ) ) ) ) )
@ ( collect @ ( product_prod @ ( A > $o ) @ ( B > $o ) )
@ ( product_case_prod @ ( A > $o ) @ ( B > $o ) @ $o
@ ^ [P2: A > $o,Q8: B > $o] :
( ( eventually @ A @ P2 @ F10 )
& ( eventually @ B @ Q8 @ G8 ) ) ) ) ) ) ) ) ).
% prod_filter_def
thf(fact_8081_listrel1__iff__update,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( listrel1 @ A @ R ) )
= ( ? [Y: A,N2: nat] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( nth @ A @ Xs @ N2 ) @ Y ) @ R )
& ( ord_less @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs ) )
& ( Ys
= ( list_update @ A @ Xs @ N2 @ Y ) ) ) ) ) ).
% listrel1_iff_update
thf(fact_8082_uniformity__trans_H,axiom,
! [A: $tType] :
( ( topolo7287701948861334536_space @ A )
=> ! [E5: ( product_prod @ A @ A ) > $o] :
( ( eventually @ ( product_prod @ A @ A ) @ E5 @ ( topolo7806501430040627800ormity @ A ) )
=> ( eventually @ ( product_prod @ ( product_prod @ A @ A ) @ ( product_prod @ A @ A ) )
@ ( product_case_prod @ ( product_prod @ A @ A ) @ ( product_prod @ A @ A ) @ $o
@ ( product_case_prod @ A @ A @ ( ( product_prod @ A @ A ) > $o )
@ ^ [X4: A,Y: A] :
( product_case_prod @ A @ A @ $o
@ ^ [Y6: A,Z6: A] :
( ( Y = Y6 )
=> ( E5 @ ( product_Pair @ A @ A @ X4 @ Z6 ) ) ) ) ) )
@ ( prod_filter @ ( product_prod @ A @ A ) @ ( product_prod @ A @ A ) @ ( topolo7806501430040627800ormity @ A ) @ ( topolo7806501430040627800ormity @ A ) ) ) ) ) ).
% uniformity_trans'
thf(fact_8083_prod__filter__INF2,axiom,
! [B: $tType,C: $tType,A: $tType,J5: set @ A,A3: filter @ B,B5: A > ( filter @ C )] :
( ( J5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( prod_filter @ B @ C @ A3 @ ( complete_Inf_Inf @ ( filter @ C ) @ ( image @ A @ ( filter @ C ) @ B5 @ J5 ) ) )
= ( complete_Inf_Inf @ ( filter @ ( product_prod @ B @ C ) )
@ ( image @ A @ ( filter @ ( product_prod @ B @ C ) )
@ ^ [I: A] : ( prod_filter @ B @ C @ A3 @ ( B5 @ I ) )
@ J5 ) ) ) ) ).
% prod_filter_INF2
thf(fact_8084_prod__filter__INF1,axiom,
! [B: $tType,C: $tType,A: $tType,I6: set @ A,A3: A > ( filter @ B ),B5: filter @ C] :
( ( I6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( prod_filter @ B @ C @ ( complete_Inf_Inf @ ( filter @ B ) @ ( image @ A @ ( filter @ B ) @ A3 @ I6 ) ) @ B5 )
= ( complete_Inf_Inf @ ( filter @ ( product_prod @ B @ C ) )
@ ( image @ A @ ( filter @ ( product_prod @ B @ C ) )
@ ^ [I: A] : ( prod_filter @ B @ C @ ( A3 @ I ) @ B5 )
@ I6 ) ) ) ) ).
% prod_filter_INF1
thf(fact_8085_listrel1p__def,axiom,
! [A: $tType] :
( ( listrel1p @ A )
= ( ^ [R5: A > A > $o,Xs2: list @ A,Ys3: list @ A] : ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ Ys3 ) @ ( listrel1 @ A @ ( collect @ ( product_prod @ A @ A ) @ ( product_case_prod @ A @ A @ $o @ R5 ) ) ) ) ) ) ).
% listrel1p_def
thf(fact_8086_sequentially__imp__eventually__at__left,axiom,
! [A: $tType] :
( ( ( topolo3112930676232923870pology @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [B2: A,A2: A,P3: A > $o] :
( ( ord_less @ A @ B2 @ A2 )
=> ( ! [F6: nat > A] :
( ! [N9: nat] : ( ord_less @ A @ B2 @ ( F6 @ N9 ) )
=> ( ! [N9: nat] : ( ord_less @ A @ ( F6 @ N9 ) @ A2 )
=> ( ( order_mono @ nat @ A @ F6 )
=> ( ( filterlim @ nat @ A @ F6 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ ( at_top @ nat ) )
=> ( eventually @ nat
@ ^ [N2: nat] : ( P3 @ ( F6 @ N2 ) )
@ ( at_top @ nat ) ) ) ) ) )
=> ( eventually @ A @ P3 @ ( topolo174197925503356063within @ A @ A2 @ ( set_ord_lessThan @ A @ A2 ) ) ) ) ) ) ).
% sequentially_imp_eventually_at_left
thf(fact_8087_incseq__const,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [K2: A] :
( order_mono @ nat @ A
@ ^ [X4: nat] : K2 ) ) ).
% incseq_const
thf(fact_8088_mono__Sup,axiom,
! [B: $tType,A: $tType] :
( ( ( comple6319245703460814977attice @ A )
& ( comple6319245703460814977attice @ B ) )
=> ! [F3: A > B,A3: set @ A] :
( ( order_mono @ A @ B @ F3 )
=> ( ord_less_eq @ B @ ( complete_Sup_Sup @ B @ ( image @ A @ B @ F3 @ A3 ) ) @ ( F3 @ ( complete_Sup_Sup @ A @ A3 ) ) ) ) ) ).
% mono_Sup
thf(fact_8089_mono__SUP,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( ( comple6319245703460814977attice @ A )
& ( comple6319245703460814977attice @ B ) )
=> ! [F3: A > B,A3: C > A,I6: set @ C] :
( ( order_mono @ A @ B @ F3 )
=> ( ord_less_eq @ B
@ ( complete_Sup_Sup @ B
@ ( image @ C @ B
@ ^ [X4: C] : ( F3 @ ( A3 @ X4 ) )
@ I6 ) )
@ ( F3 @ ( complete_Sup_Sup @ A @ ( image @ C @ A @ A3 @ I6 ) ) ) ) ) ) ).
% mono_SUP
thf(fact_8090_mono__INF,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( ( comple6319245703460814977attice @ A )
& ( comple6319245703460814977attice @ B ) )
=> ! [F3: A > B,A3: C > A,I6: set @ C] :
( ( order_mono @ A @ B @ F3 )
=> ( ord_less_eq @ B @ ( F3 @ ( complete_Inf_Inf @ A @ ( image @ C @ A @ A3 @ I6 ) ) )
@ ( complete_Inf_Inf @ B
@ ( image @ C @ B
@ ^ [X4: C] : ( F3 @ ( A3 @ X4 ) )
@ I6 ) ) ) ) ) ).
% mono_INF
thf(fact_8091_mono__Inf,axiom,
! [B: $tType,A: $tType] :
( ( ( comple6319245703460814977attice @ A )
& ( comple6319245703460814977attice @ B ) )
=> ! [F3: A > B,A3: set @ A] :
( ( order_mono @ A @ B @ F3 )
=> ( ord_less_eq @ B @ ( F3 @ ( complete_Inf_Inf @ A @ A3 ) ) @ ( complete_Inf_Inf @ B @ ( image @ A @ B @ F3 @ A3 ) ) ) ) ) ).
% mono_Inf
thf(fact_8092_mono__image__least,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [F3: A > B,M: A,N: A,M6: B,N5: B] :
( ( order_mono @ A @ B @ F3 )
=> ( ( ( image @ A @ B @ F3 @ ( set_or7035219750837199246ssThan @ A @ M @ N ) )
= ( set_or7035219750837199246ssThan @ B @ M6 @ N5 ) )
=> ( ( ord_less @ A @ M @ N )
=> ( ( F3 @ M )
= M6 ) ) ) ) ) ).
% mono_image_least
thf(fact_8093_funpow__mono,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [F3: A > A,A3: A,B5: A,N: nat] :
( ( order_mono @ A @ A @ F3 )
=> ( ( ord_less_eq @ A @ A3 @ B5 )
=> ( ord_less_eq @ A @ ( compow @ ( A > A ) @ N @ F3 @ A3 ) @ ( compow @ ( A > A ) @ N @ F3 @ B5 ) ) ) ) ) ).
% funpow_mono
thf(fact_8094_funpow__mono2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [F3: A > A,I2: nat,J3: nat,X: A,Y2: A] :
( ( order_mono @ A @ A @ F3 )
=> ( ( ord_less_eq @ nat @ I2 @ J3 )
=> ( ( ord_less_eq @ A @ X @ Y2 )
=> ( ( ord_less_eq @ A @ X @ ( F3 @ X ) )
=> ( ord_less_eq @ A @ ( compow @ ( A > A ) @ I2 @ F3 @ X ) @ ( compow @ ( A > A ) @ J3 @ F3 @ Y2 ) ) ) ) ) ) ) ).
% funpow_mono2
thf(fact_8095_mono__funpow,axiom,
! [A: $tType] :
( ( ( lattice @ A )
& ( order_bot @ A ) )
=> ! [Q2: A > A] :
( ( order_mono @ A @ A @ Q2 )
=> ( order_mono @ nat @ A
@ ^ [I: nat] : ( compow @ ( A > A ) @ I @ Q2 @ ( bot_bot @ A ) ) ) ) ) ).
% mono_funpow
thf(fact_8096_Kleene__iter__lpfp,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [F3: A > A,P5: A,K2: nat] :
( ( order_mono @ A @ A @ F3 )
=> ( ( ord_less_eq @ A @ ( F3 @ P5 ) @ P5 )
=> ( ord_less_eq @ A @ ( compow @ ( A > A ) @ K2 @ F3 @ ( bot_bot @ A ) ) @ P5 ) ) ) ) ).
% Kleene_iter_lpfp
thf(fact_8097_mono__mult,axiom,
! [A: $tType] :
( ( ordered_semiring @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( order_mono @ A @ A @ ( times_times @ A @ A2 ) ) ) ) ).
% mono_mult
thf(fact_8098_mono__invE,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( order @ B ) )
=> ! [F3: A > B,X: A,Y2: A] :
( ( order_mono @ A @ B @ F3 )
=> ( ( ord_less @ B @ ( F3 @ X ) @ ( F3 @ Y2 ) )
=> ( ord_less_eq @ A @ X @ Y2 ) ) ) ) ).
% mono_invE
thf(fact_8099_mono__inf,axiom,
! [B: $tType,A: $tType] :
( ( ( semilattice_inf @ A )
& ( semilattice_inf @ B ) )
=> ! [F3: A > B,A3: A,B5: A] :
( ( order_mono @ A @ B @ F3 )
=> ( ord_less_eq @ B @ ( F3 @ ( inf_inf @ A @ A3 @ B5 ) ) @ ( inf_inf @ B @ ( F3 @ A3 ) @ ( F3 @ B5 ) ) ) ) ) ).
% mono_inf
thf(fact_8100_mono__def,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ( ( order_mono @ A @ B )
= ( ^ [F5: A > B] :
! [X4: A,Y: A] :
( ( ord_less_eq @ A @ X4 @ Y )
=> ( ord_less_eq @ B @ ( F5 @ X4 ) @ ( F5 @ Y ) ) ) ) ) ) ).
% mono_def
thf(fact_8101_monoI,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F3: A > B] :
( ! [X5: A,Y7: A] :
( ( ord_less_eq @ A @ X5 @ Y7 )
=> ( ord_less_eq @ B @ ( F3 @ X5 ) @ ( F3 @ Y7 ) ) )
=> ( order_mono @ A @ B @ F3 ) ) ) ).
% monoI
thf(fact_8102_monoE,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F3: A > B,X: A,Y2: A] :
( ( order_mono @ A @ B @ F3 )
=> ( ( ord_less_eq @ A @ X @ Y2 )
=> ( ord_less_eq @ B @ ( F3 @ X ) @ ( F3 @ Y2 ) ) ) ) ) ).
% monoE
thf(fact_8103_monoD,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F3: A > B,X: A,Y2: A] :
( ( order_mono @ A @ B @ F3 )
=> ( ( ord_less_eq @ A @ X @ Y2 )
=> ( ord_less_eq @ B @ ( F3 @ X ) @ ( F3 @ Y2 ) ) ) ) ) ).
% monoD
thf(fact_8104_incseq__Suc__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( order_mono @ nat @ A )
= ( ^ [F5: nat > A] :
! [N2: nat] : ( ord_less_eq @ A @ ( F5 @ N2 ) @ ( F5 @ ( suc @ N2 ) ) ) ) ) ) ).
% incseq_Suc_iff
thf(fact_8105_incseq__SucI,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X9: nat > A] :
( ! [N3: nat] : ( ord_less_eq @ A @ ( X9 @ N3 ) @ ( X9 @ ( suc @ N3 ) ) )
=> ( order_mono @ nat @ A @ X9 ) ) ) ).
% incseq_SucI
thf(fact_8106_incseq__SucD,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: nat > A,I2: nat] :
( ( order_mono @ nat @ A @ A3 )
=> ( ord_less_eq @ A @ ( A3 @ I2 ) @ ( A3 @ ( suc @ I2 ) ) ) ) ) ).
% incseq_SucD
thf(fact_8107_incseq__def,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( order_mono @ nat @ A )
= ( ^ [X6: nat > A] :
! [M2: nat,N2: nat] :
( ( ord_less_eq @ nat @ M2 @ N2 )
=> ( ord_less_eq @ A @ ( X6 @ M2 ) @ ( X6 @ N2 ) ) ) ) ) ) ).
% incseq_def
thf(fact_8108_incseqD,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [F3: nat > A,I2: nat,J3: nat] :
( ( order_mono @ nat @ A @ F3 )
=> ( ( ord_less_eq @ nat @ I2 @ J3 )
=> ( ord_less_eq @ A @ ( F3 @ I2 ) @ ( F3 @ J3 ) ) ) ) ) ).
% incseqD
thf(fact_8109_cclfp__lowerbound,axiom,
! [A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [F3: A > A,A3: A] :
( ( order_mono @ A @ A @ F3 )
=> ( ( ord_less_eq @ A @ ( F3 @ A3 ) @ A3 )
=> ( ord_less_eq @ A @ ( order_532582986084564980_cclfp @ A @ F3 ) @ A3 ) ) ) ) ).
% cclfp_lowerbound
thf(fact_8110_incseq__bounded,axiom,
! [X9: nat > real,B5: real] :
( ( order_mono @ nat @ real @ X9 )
=> ( ! [I3: nat] : ( ord_less_eq @ real @ ( X9 @ I3 ) @ B5 )
=> ( bfun @ nat @ real @ X9 @ ( at_top @ nat ) ) ) ) ).
% incseq_bounded
thf(fact_8111_mono__sup,axiom,
! [B: $tType,A: $tType] :
( ( ( semilattice_sup @ A )
& ( semilattice_sup @ B ) )
=> ! [F3: A > B,A3: A,B5: A] :
( ( order_mono @ A @ B @ F3 )
=> ( ord_less_eq @ B @ ( sup_sup @ B @ ( F3 @ A3 ) @ ( F3 @ B5 ) ) @ ( F3 @ ( sup_sup @ A @ A3 @ B5 ) ) ) ) ) ).
% mono_sup
thf(fact_8112_Kleene__iter__gpfp,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [F3: A > A,P5: A,K2: nat] :
( ( order_mono @ A @ A @ F3 )
=> ( ( ord_less_eq @ A @ P5 @ ( F3 @ P5 ) )
=> ( ord_less_eq @ A @ P5 @ ( compow @ ( A > A ) @ K2 @ F3 @ ( top_top @ A ) ) ) ) ) ) ).
% Kleene_iter_gpfp
thf(fact_8113_antimono__funpow,axiom,
! [A: $tType] :
( ( ( lattice @ A )
& ( order_top @ A ) )
=> ! [Q2: A > A] :
( ( order_mono @ A @ A @ Q2 )
=> ( order_antimono @ nat @ A
@ ^ [I: nat] : ( compow @ ( A > A ) @ I @ Q2 @ ( top_top @ A ) ) ) ) ) ).
% antimono_funpow
thf(fact_8114_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_8115_mono__add,axiom,
! [A: $tType] :
( ( ordere6658533253407199908up_add @ A )
=> ! [A2: A] : ( order_mono @ A @ A @ ( plus_plus @ A @ A2 ) ) ) ).
% mono_add
thf(fact_8116_mono__Suc,axiom,
order_mono @ nat @ nat @ suc ).
% mono_Suc
thf(fact_8117_mono__strict__invE,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( order @ B ) )
=> ! [F3: A > B,X: A,Y2: A] :
( ( order_mono @ A @ B @ F3 )
=> ( ( ord_less @ B @ ( F3 @ X ) @ ( F3 @ Y2 ) )
=> ( ord_less @ A @ X @ Y2 ) ) ) ) ).
% mono_strict_invE
thf(fact_8118_decseq__eq__incseq,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ( ( order_antimono @ nat @ A )
= ( ^ [X6: nat > A] :
( order_mono @ nat @ A
@ ^ [N2: nat] : ( uminus_uminus @ A @ ( X6 @ N2 ) ) ) ) ) ) ).
% decseq_eq_incseq
thf(fact_8119_mono__pow,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F3: A > A,N: nat] :
( ( order_mono @ A @ A @ F3 )
=> ( order_mono @ A @ A @ ( compow @ ( A > A ) @ N @ F3 ) ) ) ) ).
% mono_pow
thf(fact_8120_incseq__le,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [X9: nat > A,L6: A,N: nat] :
( ( order_mono @ nat @ A @ X9 )
=> ( ( filterlim @ nat @ A @ X9 @ ( topolo7230453075368039082e_nhds @ A @ L6 ) @ ( at_top @ nat ) )
=> ( ord_less_eq @ A @ ( X9 @ N ) @ L6 ) ) ) ) ).
% incseq_le
thf(fact_8121_funpow__increasing,axiom,
! [A: $tType] :
( ( ( lattice @ A )
& ( order_top @ A ) )
=> ! [M: nat,N: nat,F3: A > A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( order_mono @ A @ A @ F3 )
=> ( ord_less_eq @ A @ ( compow @ ( A > A ) @ N @ F3 @ ( top_top @ A ) ) @ ( compow @ ( A > A ) @ M @ F3 @ ( top_top @ A ) ) ) ) ) ) ).
% funpow_increasing
thf(fact_8122_funpow__decreasing,axiom,
! [A: $tType] :
( ( ( lattice @ A )
& ( order_bot @ A ) )
=> ! [M: nat,N: nat,F3: A > A] :
( ( ord_less_eq @ nat @ M @ N )
=> ( ( order_mono @ A @ A @ F3 )
=> ( ord_less_eq @ A @ ( compow @ ( A > A ) @ M @ F3 @ ( bot_bot @ A ) ) @ ( compow @ ( A > A ) @ N @ F3 @ ( bot_bot @ A ) ) ) ) ) ) ).
% funpow_decreasing
thf(fact_8123_incseq__convergent,axiom,
! [X9: nat > real,B5: real] :
( ( order_mono @ nat @ real @ X9 )
=> ( ! [I3: nat] : ( ord_less_eq @ real @ ( X9 @ I3 ) @ B5 )
=> ~ ! [L7: real] :
( ( filterlim @ nat @ real @ X9 @ ( topolo7230453075368039082e_nhds @ real @ L7 ) @ ( at_top @ nat ) )
=> ~ ! [I4: nat] : ( ord_less_eq @ real @ ( X9 @ I4 ) @ L7 ) ) ) ) ).
% incseq_convergent
thf(fact_8124_mono__cSUP,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( ( condit1219197933456340205attice @ A )
& ( condit1219197933456340205attice @ B ) )
=> ! [F3: A > B,A3: C > A,I6: set @ C] :
( ( order_mono @ A @ B @ F3 )
=> ( ( condit941137186595557371_above @ A @ ( image @ C @ A @ A3 @ I6 ) )
=> ( ( I6
!= ( bot_bot @ ( set @ C ) ) )
=> ( ord_less_eq @ B
@ ( complete_Sup_Sup @ B
@ ( image @ C @ B
@ ^ [X4: C] : ( F3 @ ( A3 @ X4 ) )
@ I6 ) )
@ ( F3 @ ( complete_Sup_Sup @ A @ ( image @ C @ A @ A3 @ I6 ) ) ) ) ) ) ) ) ).
% mono_cSUP
thf(fact_8125_mono__cSup,axiom,
! [B: $tType,A: $tType] :
( ( ( condit1219197933456340205attice @ A )
& ( condit1219197933456340205attice @ B ) )
=> ! [F3: A > B,A3: set @ A] :
( ( order_mono @ A @ B @ F3 )
=> ( ( condit941137186595557371_above @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ord_less_eq @ B @ ( complete_Sup_Sup @ B @ ( image @ A @ B @ F3 @ A3 ) ) @ ( F3 @ ( complete_Sup_Sup @ A @ A3 ) ) ) ) ) ) ) ).
% mono_cSup
thf(fact_8126_mono__cINF,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( ( condit1219197933456340205attice @ A )
& ( condit1219197933456340205attice @ B ) )
=> ! [F3: A > B,A3: C > A,I6: set @ C] :
( ( order_mono @ A @ B @ F3 )
=> ( ( condit1013018076250108175_below @ A @ ( image @ C @ A @ A3 @ I6 ) )
=> ( ( I6
!= ( bot_bot @ ( set @ C ) ) )
=> ( ord_less_eq @ B @ ( F3 @ ( complete_Inf_Inf @ A @ ( image @ C @ A @ A3 @ I6 ) ) )
@ ( complete_Inf_Inf @ B
@ ( image @ C @ B
@ ^ [X4: C] : ( F3 @ ( A3 @ X4 ) )
@ I6 ) ) ) ) ) ) ) ).
% mono_cINF
thf(fact_8127_mono__cInf,axiom,
! [B: $tType,A: $tType] :
( ( ( condit1219197933456340205attice @ A )
& ( condit1219197933456340205attice @ B ) )
=> ! [F3: A > B,A3: set @ A] :
( ( order_mono @ A @ B @ F3 )
=> ( ( condit1013018076250108175_below @ A @ A3 )
=> ( ( A3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ord_less_eq @ B @ ( F3 @ ( complete_Inf_Inf @ A @ A3 ) ) @ ( complete_Inf_Inf @ B @ ( image @ A @ B @ F3 @ A3 ) ) ) ) ) ) ) ).
% mono_cInf
thf(fact_8128_mono__ge2__power__minus__self,axiom,
! [K2: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K2 )
=> ( order_mono @ nat @ nat
@ ^ [M2: nat] : ( minus_minus @ nat @ ( power_power @ nat @ K2 @ M2 ) @ M2 ) ) ) ).
% mono_ge2_power_minus_self
thf(fact_8129_LIMSEQ__SUP,axiom,
! [A: $tType] :
( ( ( comple5582772986160207858norder @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [X9: nat > A] :
( ( order_mono @ nat @ A @ X9 )
=> ( filterlim @ nat @ A @ X9 @ ( topolo7230453075368039082e_nhds @ A @ ( complete_Sup_Sup @ A @ ( image @ nat @ A @ X9 @ ( top_top @ ( set @ nat ) ) ) ) ) @ ( at_top @ nat ) ) ) ) ).
% LIMSEQ_SUP
thf(fact_8130_SUP__Lim,axiom,
! [A: $tType] :
( ( ( comple5582772986160207858norder @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [X9: nat > A,L: A] :
( ( order_mono @ nat @ A @ X9 )
=> ( ( filterlim @ nat @ A @ X9 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ ( at_top @ nat ) )
=> ( ( complete_Sup_Sup @ A @ ( image @ nat @ A @ X9 @ ( top_top @ ( set @ nat ) ) ) )
= L ) ) ) ) ).
% SUP_Lim
thf(fact_8131_finite__mono__remains__stable__implies__strict__prefix,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [F3: nat > A] :
( ( finite_finite2 @ A @ ( image @ nat @ A @ F3 @ ( top_top @ ( set @ nat ) ) ) )
=> ( ( order_mono @ nat @ A @ F3 )
=> ( ! [N3: nat] :
( ( ( F3 @ N3 )
= ( F3 @ ( suc @ N3 ) ) )
=> ( ( F3 @ ( suc @ N3 ) )
= ( F3 @ ( suc @ ( suc @ N3 ) ) ) ) )
=> ? [N8: nat] :
( ! [N9: nat] :
( ( ord_less_eq @ nat @ N9 @ N8 )
=> ! [M3: nat] :
( ( ord_less_eq @ nat @ M3 @ N8 )
=> ( ( ord_less @ nat @ M3 @ N9 )
=> ( ord_less @ A @ ( F3 @ M3 ) @ ( F3 @ N9 ) ) ) ) )
& ! [N9: nat] :
( ( ord_less_eq @ nat @ N8 @ N9 )
=> ( ( F3 @ N8 )
= ( F3 @ N9 ) ) ) ) ) ) ) ) ).
% finite_mono_remains_stable_implies_strict_prefix
thf(fact_8132_tendsto__at__left__sequentially,axiom,
! [A: $tType,B: $tType] :
( ( ( topolo3112930676232923870pology @ B )
& ( topolo1944317154257567458pology @ B )
& ( topolo4958980785337419405_space @ A ) )
=> ! [B2: B,A2: B,X9: B > A,L6: A] :
( ( ord_less @ B @ B2 @ A2 )
=> ( ! [S7: nat > B] :
( ! [N9: nat] : ( ord_less @ B @ ( S7 @ N9 ) @ A2 )
=> ( ! [N9: nat] : ( ord_less @ B @ B2 @ ( S7 @ N9 ) )
=> ( ( order_mono @ nat @ B @ S7 )
=> ( ( filterlim @ nat @ B @ S7 @ ( topolo7230453075368039082e_nhds @ B @ A2 ) @ ( at_top @ nat ) )
=> ( filterlim @ nat @ A
@ ^ [N2: nat] : ( X9 @ ( S7 @ N2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ L6 )
@ ( at_top @ nat ) ) ) ) ) )
=> ( filterlim @ B @ A @ X9 @ ( topolo7230453075368039082e_nhds @ A @ L6 ) @ ( topolo174197925503356063within @ B @ A2 @ ( set_ord_lessThan @ B @ A2 ) ) ) ) ) ) ).
% tendsto_at_left_sequentially
thf(fact_8133_LIMSEQ__incseq__SUP,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [X9: nat > A] :
( ( condit941137186595557371_above @ A @ ( image @ nat @ A @ X9 @ ( top_top @ ( set @ nat ) ) ) )
=> ( ( order_mono @ nat @ A @ X9 )
=> ( filterlim @ nat @ A @ X9 @ ( topolo7230453075368039082e_nhds @ A @ ( complete_Sup_Sup @ A @ ( image @ nat @ A @ X9 @ ( top_top @ ( set @ nat ) ) ) ) ) @ ( at_top @ nat ) ) ) ) ) ).
% LIMSEQ_incseq_SUP
thf(fact_8134_continuous__at__Sup__mono,axiom,
! [B: $tType,A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( topolo1944317154257567458pology @ A )
& ( condit6923001295902523014norder @ B )
& ( topolo1944317154257567458pology @ B ) )
=> ! [F3: A > B,S3: set @ A] :
( ( order_mono @ A @ B @ F3 )
=> ( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ ( complete_Sup_Sup @ A @ S3 ) @ ( set_ord_lessThan @ A @ ( complete_Sup_Sup @ A @ S3 ) ) ) @ F3 )
=> ( ( S3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit941137186595557371_above @ A @ S3 )
=> ( ( F3 @ ( complete_Sup_Sup @ A @ S3 ) )
= ( complete_Sup_Sup @ B @ ( image @ A @ B @ F3 @ S3 ) ) ) ) ) ) ) ) ).
% continuous_at_Sup_mono
thf(fact_8135_continuous__at__Inf__mono,axiom,
! [B: $tType,A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( topolo1944317154257567458pology @ A )
& ( condit6923001295902523014norder @ B )
& ( topolo1944317154257567458pology @ B ) )
=> ! [F3: A > B,S3: set @ A] :
( ( order_mono @ A @ B @ F3 )
=> ( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ ( complete_Inf_Inf @ A @ S3 ) @ ( set_ord_greaterThan @ A @ ( complete_Inf_Inf @ A @ S3 ) ) ) @ F3 )
=> ( ( S3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit1013018076250108175_below @ A @ S3 )
=> ( ( F3 @ ( complete_Inf_Inf @ A @ S3 ) )
= ( complete_Inf_Inf @ B @ ( image @ A @ B @ F3 @ S3 ) ) ) ) ) ) ) ) ).
% continuous_at_Inf_mono
thf(fact_8136_cauchy__filter__def,axiom,
! [A: $tType] :
( ( topolo7287701948861334536_space @ A )
=> ( ( topolo6773858410816713723filter @ A )
= ( ^ [F10: filter @ A] : ( ord_less_eq @ ( filter @ ( product_prod @ A @ A ) ) @ ( prod_filter @ A @ A @ F10 @ F10 ) @ ( topolo7806501430040627800ormity @ A ) ) ) ) ) ).
% cauchy_filter_def
thf(fact_8137_remdups__adj__altdef,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( remdups_adj @ A @ Xs )
= Ys )
= ( ? [F5: nat > nat] :
( ( order_mono @ nat @ nat @ F5 )
& ( ( image @ nat @ nat @ F5 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Xs ) ) )
= ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Ys ) ) )
& ! [I: nat] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ A @ Xs @ I )
= ( nth @ A @ Ys @ ( F5 @ I ) ) ) )
& ! [I: nat] :
( ( ord_less @ nat @ ( plus_plus @ nat @ I @ ( one_one @ nat ) ) @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ( nth @ A @ Xs @ I )
= ( nth @ A @ Xs @ ( plus_plus @ nat @ I @ ( one_one @ nat ) ) ) )
= ( ( F5 @ I )
= ( F5 @ ( plus_plus @ nat @ I @ ( one_one @ nat ) ) ) ) ) ) ) ) ) ).
% remdups_adj_altdef
thf(fact_8138_remdups__adj__Nil__iff,axiom,
! [A: $tType,Xs: list @ A] :
( ( ( remdups_adj @ A @ Xs )
= ( nil @ A ) )
= ( Xs
= ( nil @ A ) ) ) ).
% remdups_adj_Nil_iff
thf(fact_8139_remdups__adj__set,axiom,
! [A: $tType,Xs: list @ A] :
( ( set2 @ A @ ( remdups_adj @ A @ Xs ) )
= ( set2 @ A @ Xs ) ) ).
% remdups_adj_set
thf(fact_8140_remdups__adj__rev,axiom,
! [A: $tType,Xs: list @ A] :
( ( remdups_adj @ A @ ( rev @ A @ Xs ) )
= ( rev @ A @ ( remdups_adj @ A @ Xs ) ) ) ).
% remdups_adj_rev
thf(fact_8141_hd__remdups__adj,axiom,
! [A: $tType,Xs: list @ A] :
( ( hd @ A @ ( remdups_adj @ A @ Xs ) )
= ( hd @ A @ Xs ) ) ).
% hd_remdups_adj
thf(fact_8142_remdups__adj__Cons__alt,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( cons @ A @ X @ ( tl @ A @ ( remdups_adj @ A @ ( cons @ A @ X @ Xs ) ) ) )
= ( remdups_adj @ A @ ( cons @ A @ X @ Xs ) ) ) ).
% remdups_adj_Cons_alt
thf(fact_8143_mono__Int,axiom,
! [B: $tType,A: $tType,F3: ( set @ A ) > ( set @ B ),A3: set @ A,B5: set @ A] :
( ( order_mono @ ( set @ A ) @ ( set @ B ) @ F3 )
=> ( ord_less_eq @ ( set @ B ) @ ( F3 @ ( inf_inf @ ( set @ A ) @ A3 @ B5 ) ) @ ( inf_inf @ ( set @ B ) @ ( F3 @ A3 ) @ ( F3 @ B5 ) ) ) ) ).
% mono_Int
thf(fact_8144_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_8145_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_8146_mono__Un,axiom,
! [B: $tType,A: $tType,F3: ( set @ A ) > ( set @ B ),A3: set @ A,B5: set @ A] :
( ( order_mono @ ( set @ A ) @ ( set @ B ) @ F3 )
=> ( ord_less_eq @ ( set @ B ) @ ( sup_sup @ ( set @ B ) @ ( F3 @ A3 ) @ ( F3 @ B5 ) ) @ ( F3 @ ( sup_sup @ ( set @ A ) @ A3 @ B5 ) ) ) ) ).
% mono_Un
thf(fact_8147_rtranclp_Omono,axiom,
! [A: $tType,R: A > A > $o] :
( order_mono @ ( A > A > $o ) @ ( A > A > $o )
@ ^ [P4: A > A > $o,X1: A,X25: A] :
( ? [A5: A] :
( ( X1 = A5 )
& ( X25 = A5 ) )
| ? [A5: A,B4: A,C3: A] :
( ( X1 = A5 )
& ( X25 = C3 )
& ( P4 @ A5 @ B4 )
& ( R @ B4 @ C3 ) ) ) ) ).
% rtranclp.mono
thf(fact_8148_tranclp_Omono,axiom,
! [A: $tType,R: A > A > $o] :
( order_mono @ ( A > A > $o ) @ ( A > A > $o )
@ ^ [P4: A > A > $o,X1: A,X25: A] :
( ? [A5: A,B4: A] :
( ( X1 = A5 )
& ( X25 = B4 )
& ( R @ A5 @ B4 ) )
| ? [A5: A,B4: A,C3: A] :
( ( X1 = A5 )
& ( X25 = C3 )
& ( P4 @ A5 @ B4 )
& ( R @ B4 @ C3 ) ) ) ) ).
% tranclp.mono
thf(fact_8149_remdups__adj__Cons_H,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( remdups_adj @ A @ ( cons @ A @ X @ Xs ) )
= ( cons @ A @ X
@ ( remdups_adj @ A
@ ( dropWhile @ A
@ ^ [Y: A] : Y = X
@ Xs ) ) ) ) ).
% remdups_adj_Cons'
thf(fact_8150_remdups__adj__distinct,axiom,
! [A: $tType,Xs: list @ A] :
( ( distinct @ A @ Xs )
=> ( ( remdups_adj @ A @ Xs )
= Xs ) ) ).
% remdups_adj_distinct
thf(fact_8151_remdups__adj_Osimps_I1_J,axiom,
! [A: $tType] :
( ( remdups_adj @ A @ ( nil @ A ) )
= ( nil @ A ) ) ).
% remdups_adj.simps(1)
thf(fact_8152_remdups__adj_Osimps_I2_J,axiom,
! [A: $tType,X: A] :
( ( remdups_adj @ A @ ( cons @ A @ X @ ( nil @ A ) ) )
= ( cons @ A @ X @ ( nil @ A ) ) ) ).
% remdups_adj.simps(2)
thf(fact_8153_remdups__adj_Oelims,axiom,
! [A: $tType,X: list @ A,Y2: list @ A] :
( ( ( remdups_adj @ A @ X )
= Y2 )
=> ( ( ( X
= ( nil @ A ) )
=> ( Y2
!= ( nil @ A ) ) )
=> ( ! [X5: A] :
( ( X
= ( cons @ A @ X5 @ ( nil @ A ) ) )
=> ( Y2
!= ( cons @ A @ X5 @ ( nil @ A ) ) ) )
=> ~ ! [X5: A,Y7: A,Xs3: list @ A] :
( ( X
= ( cons @ A @ X5 @ ( cons @ A @ Y7 @ Xs3 ) ) )
=> ~ ( ( ( X5 = Y7 )
=> ( Y2
= ( remdups_adj @ A @ ( cons @ A @ X5 @ Xs3 ) ) ) )
& ( ( X5 != Y7 )
=> ( Y2
= ( cons @ A @ X5 @ ( remdups_adj @ A @ ( cons @ A @ Y7 @ Xs3 ) ) ) ) ) ) ) ) ) ) ).
% remdups_adj.elims
thf(fact_8154_remdups__adj_Osimps_I3_J,axiom,
! [A: $tType,X: A,Y2: A,Xs: list @ A] :
( ( ( X = Y2 )
=> ( ( remdups_adj @ A @ ( cons @ A @ X @ ( cons @ A @ Y2 @ Xs ) ) )
= ( remdups_adj @ A @ ( cons @ A @ X @ Xs ) ) ) )
& ( ( X != Y2 )
=> ( ( remdups_adj @ A @ ( cons @ A @ X @ ( cons @ A @ Y2 @ Xs ) ) )
= ( cons @ A @ X @ ( remdups_adj @ A @ ( cons @ A @ Y2 @ Xs ) ) ) ) ) ) ).
% remdups_adj.simps(3)
thf(fact_8155_remdups__adj__append__two,axiom,
! [A: $tType,Xs: list @ A,X: A,Y2: A] :
( ( remdups_adj @ A @ ( append @ A @ Xs @ ( cons @ A @ X @ ( cons @ A @ Y2 @ ( nil @ A ) ) ) ) )
= ( append @ A @ ( remdups_adj @ A @ ( append @ A @ Xs @ ( cons @ A @ X @ ( nil @ A ) ) ) ) @ ( if @ ( list @ A ) @ ( X = Y2 ) @ ( nil @ A ) @ ( cons @ A @ Y2 @ ( nil @ A ) ) ) ) ) ).
% remdups_adj_append_two
thf(fact_8156_remdups__adj__Cons,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( remdups_adj @ A @ ( cons @ A @ X @ Xs ) )
= ( case_list @ ( list @ A ) @ A @ ( cons @ A @ X @ ( nil @ A ) )
@ ^ [Y: A,Xs2: list @ A] : ( if @ ( list @ A ) @ ( X = Y ) @ ( cons @ A @ Y @ Xs2 ) @ ( cons @ A @ X @ ( cons @ A @ Y @ Xs2 ) ) )
@ ( remdups_adj @ A @ Xs ) ) ) ).
% remdups_adj_Cons
thf(fact_8157_ord_Olexordp_Omono,axiom,
! [A: $tType,Less: A > A > $o] :
( order_mono @ ( ( list @ A ) > ( list @ A ) > $o ) @ ( ( list @ A ) > ( list @ A ) > $o )
@ ^ [P4: ( list @ A ) > ( list @ A ) > $o,X1: list @ A,X25: list @ A] :
( ? [Y: A,Ys3: list @ A] :
( ( X1
= ( nil @ A ) )
& ( X25
= ( cons @ A @ Y @ Ys3 ) ) )
| ? [X4: A,Y: A,Xs2: list @ A,Ys3: list @ A] :
( ( X1
= ( cons @ A @ X4 @ Xs2 ) )
& ( X25
= ( cons @ A @ Y @ Ys3 ) )
& ( Less @ X4 @ Y ) )
| ? [X4: A,Y: A,Xs2: list @ A,Ys3: list @ A] :
( ( X1
= ( cons @ A @ X4 @ Xs2 ) )
& ( X25
= ( cons @ A @ Y @ Ys3 ) )
& ~ ( Less @ X4 @ Y )
& ~ ( Less @ Y @ X4 )
& ( P4 @ Xs2 @ Ys3 ) ) ) ) ).
% ord.lexordp.mono
thf(fact_8158_finite_Omono,axiom,
! [A: $tType] :
( order_mono @ ( ( set @ A ) > $o ) @ ( ( set @ A ) > $o )
@ ^ [P4: ( set @ A ) > $o,X4: set @ A] :
( ( X4
= ( bot_bot @ ( set @ A ) ) )
| ? [A7: set @ A,A5: A] :
( ( X4
= ( insert @ A @ A5 @ A7 ) )
& ( P4 @ A7 ) ) ) ) ).
% finite.mono
thf(fact_8159_remdups__adj__adjacent,axiom,
! [A: $tType,I2: nat,Xs: list @ A] :
( ( ord_less @ nat @ ( suc @ I2 ) @ ( size_size @ ( list @ A ) @ ( remdups_adj @ A @ Xs ) ) )
=> ( ( nth @ A @ ( remdups_adj @ A @ Xs ) @ I2 )
!= ( nth @ A @ ( remdups_adj @ A @ Xs ) @ ( suc @ I2 ) ) ) ) ).
% remdups_adj_adjacent
thf(fact_8160_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_8161_remdups__adj__singleton,axiom,
! [A: $tType,Xs: list @ A,X: A] :
( ( ( remdups_adj @ A @ Xs )
= ( cons @ A @ X @ ( nil @ A ) ) )
=> ( Xs
= ( replicate @ A @ ( size_size @ ( list @ A ) @ Xs ) @ X ) ) ) ).
% remdups_adj_singleton
thf(fact_8162_remdups__adj__append,axiom,
! [A: $tType,Xs_1: list @ A,X: A,Xs_2: list @ A] :
( ( remdups_adj @ A @ ( append @ A @ Xs_1 @ ( cons @ A @ X @ Xs_2 ) ) )
= ( append @ A @ ( remdups_adj @ A @ ( append @ A @ Xs_1 @ ( cons @ A @ X @ ( nil @ A ) ) ) ) @ ( tl @ A @ ( remdups_adj @ A @ ( cons @ A @ X @ Xs_2 ) ) ) ) ) ).
% remdups_adj_append
thf(fact_8163_remdups__adj__append__dropWhile,axiom,
! [A: $tType,Xs: list @ A,Y2: A,Ys: list @ A] :
( ( remdups_adj @ A @ ( append @ A @ Xs @ ( cons @ A @ Y2 @ Ys ) ) )
= ( append @ A @ ( remdups_adj @ A @ ( append @ A @ Xs @ ( cons @ A @ Y2 @ ( nil @ A ) ) ) )
@ ( remdups_adj @ A
@ ( dropWhile @ A
@ ^ [X4: A] : X4 = Y2
@ Ys ) ) ) ) ).
% remdups_adj_append_dropWhile
thf(fact_8164_lexordp_Omono,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( order_mono @ ( ( list @ A ) > ( list @ A ) > $o ) @ ( ( list @ A ) > ( list @ A ) > $o )
@ ^ [P4: ( list @ A ) > ( list @ A ) > $o,X1: list @ A,X25: list @ A] :
( ? [Y: A,Ys3: list @ A] :
( ( X1
= ( nil @ A ) )
& ( X25
= ( cons @ A @ Y @ Ys3 ) ) )
| ? [X4: A,Y: A,Xs2: list @ A,Ys3: list @ A] :
( ( X1
= ( cons @ A @ X4 @ Xs2 ) )
& ( X25
= ( cons @ A @ Y @ Ys3 ) )
& ( ord_less @ A @ X4 @ Y ) )
| ? [X4: A,Y: A,Xs2: list @ A,Ys3: list @ A] :
( ( X1
= ( cons @ A @ X4 @ Xs2 ) )
& ( X25
= ( cons @ A @ Y @ Ys3 ) )
& ~ ( ord_less @ A @ X4 @ Y )
& ~ ( ord_less @ A @ Y @ X4 )
& ( P4 @ Xs2 @ Ys3 ) ) ) ) ) ).
% lexordp.mono
thf(fact_8165_tl__remdups__adj,axiom,
! [A: $tType,Ys: list @ A] :
( ( Ys
!= ( nil @ A ) )
=> ( ( tl @ A @ ( remdups_adj @ A @ Ys ) )
= ( remdups_adj @ A
@ ( dropWhile @ A
@ ^ [X4: A] :
( X4
= ( hd @ A @ Ys ) )
@ ( tl @ A @ Ys ) ) ) ) ) ).
% tl_remdups_adj
thf(fact_8166_nhds__imp__cauchy__filter,axiom,
! [A: $tType] :
( ( topolo7287701948861334536_space @ A )
=> ! [F4: filter @ A,X: A] :
( ( ord_less_eq @ ( filter @ A ) @ F4 @ ( topolo7230453075368039082e_nhds @ A @ X ) )
=> ( topolo6773858410816713723filter @ A @ F4 ) ) ) ).
% nhds_imp_cauchy_filter
thf(fact_8167_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_8168_tendsto__at__topI__sequentially__real,axiom,
! [F3: real > real,Y2: real] :
( ( order_mono @ real @ real @ F3 )
=> ( ( filterlim @ nat @ real
@ ^ [N2: nat] : ( F3 @ ( semiring_1_of_nat @ real @ N2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ Y2 )
@ ( at_top @ nat ) )
=> ( filterlim @ real @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ Y2 ) @ ( at_top @ real ) ) ) ) ).
% tendsto_at_topI_sequentially_real
thf(fact_8169_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_8170_complete__uniform,axiom,
! [A: $tType] :
( ( topolo7287701948861334536_space @ A )
=> ( ( topolo2479028161051973599mplete @ A )
= ( ^ [S6: set @ A] :
! [F10: filter @ A] :
( ( ord_less_eq @ ( filter @ A ) @ F10 @ ( principal @ A @ S6 ) )
=> ( ( F10
!= ( bot_bot @ ( filter @ A ) ) )
=> ( ( topolo6773858410816713723filter @ A @ F10 )
=> ? [X4: A] :
( ( member @ A @ X4 @ S6 )
& ( ord_less_eq @ ( filter @ A ) @ F10 @ ( topolo7230453075368039082e_nhds @ A @ X4 ) ) ) ) ) ) ) ) ) ).
% complete_uniform
thf(fact_8171_remdups__adj_Opelims,axiom,
! [A: $tType,X: list @ A,Y2: list @ A] :
( ( ( remdups_adj @ A @ X )
= Y2 )
=> ( ( accp @ ( list @ A ) @ ( remdups_adj_rel @ A ) @ X )
=> ( ( ( X
= ( nil @ A ) )
=> ( ( Y2
= ( nil @ A ) )
=> ~ ( accp @ ( list @ A ) @ ( remdups_adj_rel @ A ) @ ( nil @ A ) ) ) )
=> ( ! [X5: A] :
( ( X
= ( cons @ A @ X5 @ ( nil @ A ) ) )
=> ( ( Y2
= ( cons @ A @ X5 @ ( nil @ A ) ) )
=> ~ ( accp @ ( list @ A ) @ ( remdups_adj_rel @ A ) @ ( cons @ A @ X5 @ ( nil @ A ) ) ) ) )
=> ~ ! [X5: A,Y7: A,Xs3: list @ A] :
( ( X
= ( cons @ A @ X5 @ ( cons @ A @ Y7 @ Xs3 ) ) )
=> ( ( ( ( X5 = Y7 )
=> ( Y2
= ( remdups_adj @ A @ ( cons @ A @ X5 @ Xs3 ) ) ) )
& ( ( X5 != Y7 )
=> ( Y2
= ( cons @ A @ X5 @ ( remdups_adj @ A @ ( cons @ A @ Y7 @ Xs3 ) ) ) ) ) )
=> ~ ( accp @ ( list @ A ) @ ( remdups_adj_rel @ A ) @ ( cons @ A @ X5 @ ( cons @ A @ Y7 @ Xs3 ) ) ) ) ) ) ) ) ) ).
% remdups_adj.pelims
thf(fact_8172_mono__compose,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ C ) )
=> ! [Q2: A > B > C,F3: D > B] :
( ( order_mono @ A @ ( B > C ) @ Q2 )
=> ( order_mono @ A @ ( D > C )
@ ^ [I: A,X4: D] : ( Q2 @ I @ ( F3 @ X4 ) ) ) ) ) ).
% mono_compose
thf(fact_8173_coinduct3__mono__lemma,axiom,
! [B: $tType,A: $tType] :
( ( order @ A )
=> ! [F3: A > ( set @ B ),X9: set @ B,B5: set @ B] :
( ( order_mono @ A @ ( set @ B ) @ F3 )
=> ( order_mono @ A @ ( set @ B )
@ ^ [X4: A] : ( sup_sup @ ( set @ B ) @ ( sup_sup @ ( set @ B ) @ ( F3 @ X4 ) @ X9 ) @ B5 ) ) ) ) ).
% coinduct3_mono_lemma
thf(fact_8174_image2__def,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( bNF_Greatest_image2 @ C @ A @ B )
= ( ^ [A7: set @ C,F5: C > A,G2: C > B] :
( collect @ ( product_prod @ A @ B )
@ ^ [Uu3: product_prod @ A @ B] :
? [A5: C] :
( ( Uu3
= ( product_Pair @ A @ B @ ( F5 @ A5 ) @ ( G2 @ A5 ) ) )
& ( member @ C @ A5 @ A7 ) ) ) ) ) ).
% image2_def
thf(fact_8175_le__rel__bool__arg__iff,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( ord_less_eq @ ( $o > A ) )
= ( ^ [X6: $o > A,Y8: $o > A] :
( ( ord_less_eq @ A @ ( X6 @ $false ) @ ( Y8 @ $false ) )
& ( ord_less_eq @ A @ ( X6 @ $true ) @ ( Y8 @ $true ) ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_8176_and__not__num_Oelims,axiom,
! [X: num,Xa: num,Y2: option @ num] :
( ( ( bit_and_not_num @ X @ Xa )
= Y2 )
=> ( ( ( X = one2 )
=> ( ( Xa = one2 )
=> ( Y2
!= ( none @ num ) ) ) )
=> ( ( ( X = one2 )
=> ( ? [N3: num] :
( Xa
= ( bit0 @ N3 ) )
=> ( Y2
!= ( some @ num @ one2 ) ) ) )
=> ( ( ( X = one2 )
=> ( ? [N3: num] :
( Xa
= ( bit1 @ N3 ) )
=> ( Y2
!= ( none @ num ) ) ) )
=> ( ! [M4: num] :
( ( X
= ( bit0 @ M4 ) )
=> ( ( Xa = one2 )
=> ( Y2
!= ( some @ num @ ( bit0 @ M4 ) ) ) ) )
=> ( ! [M4: num] :
( ( X
= ( bit0 @ M4 ) )
=> ! [N3: num] :
( ( Xa
= ( bit0 @ N3 ) )
=> ( Y2
!= ( map_option @ num @ num @ bit0 @ ( bit_and_not_num @ M4 @ N3 ) ) ) ) )
=> ( ! [M4: num] :
( ( X
= ( bit0 @ M4 ) )
=> ! [N3: num] :
( ( Xa
= ( bit1 @ N3 ) )
=> ( Y2
!= ( map_option @ num @ num @ bit0 @ ( bit_and_not_num @ M4 @ N3 ) ) ) ) )
=> ( ! [M4: num] :
( ( X
= ( bit1 @ M4 ) )
=> ( ( Xa = one2 )
=> ( Y2
!= ( some @ num @ ( bit0 @ M4 ) ) ) ) )
=> ( ! [M4: num] :
( ( X
= ( bit1 @ M4 ) )
=> ! [N3: num] :
( ( Xa
= ( bit0 @ N3 ) )
=> ( Y2
!= ( case_option @ ( option @ num ) @ num @ ( some @ num @ one2 )
@ ^ [N11: num] : ( some @ num @ ( bit1 @ N11 ) )
@ ( bit_and_not_num @ M4 @ N3 ) ) ) ) )
=> ~ ! [M4: num] :
( ( X
= ( bit1 @ M4 ) )
=> ! [N3: num] :
( ( Xa
= ( bit1 @ N3 ) )
=> ( Y2
!= ( map_option @ num @ num @ bit0 @ ( bit_and_not_num @ M4 @ N3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% and_not_num.elims
thf(fact_8177_nonneg__incseq__Bseq__subseq__iff,axiom,
! [F3: nat > real,G: nat > nat] :
( ! [X5: nat] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( F3 @ X5 ) )
=> ( ( order_mono @ nat @ real @ F3 )
=> ( ( order_strict_mono @ nat @ nat @ G )
=> ( ( bfun @ nat @ real
@ ^ [X4: nat] : ( F3 @ ( G @ X4 ) )
@ ( at_top @ nat ) )
= ( bfun @ nat @ real @ F3 @ ( at_top @ nat ) ) ) ) ) ) ).
% nonneg_incseq_Bseq_subseq_iff
thf(fact_8178_option_Omap__disc__iff,axiom,
! [B: $tType,A: $tType,F3: A > B,A2: option @ A] :
( ( ( map_option @ A @ B @ F3 @ A2 )
= ( none @ B ) )
= ( A2
= ( none @ A ) ) ) ).
% option.map_disc_iff
thf(fact_8179_map__option__is__None,axiom,
! [A: $tType,B: $tType,F3: B > A,Opt: option @ B] :
( ( ( map_option @ B @ A @ F3 @ Opt )
= ( none @ A ) )
= ( Opt
= ( none @ B ) ) ) ).
% map_option_is_None
% Type constructors (798)
thf(tcon_Product__Type_Ounit___Lattices_Obounded__lattice,axiom,
bounded_lattice @ product_unit ).
thf(tcon_Extended__Nat_Oenat___Lattices_Obounded__lattice_1,axiom,
bounded_lattice @ extended_enat ).
thf(tcon_Filter_Ofilter___Lattices_Obounded__lattice_2,axiom,
! [A17: $tType] : ( bounded_lattice @ ( filter @ A17 ) ) ).
thf(tcon_HOL_Obool___Lattices_Obounded__lattice_3,axiom,
bounded_lattice @ $o ).
thf(tcon_Set_Oset___Lattices_Obounded__lattice_4,axiom,
! [A17: $tType] : ( bounded_lattice @ ( set @ A17 ) ) ).
thf(tcon_fun___Lattices_Obounded__lattice_5,axiom,
! [A17: $tType,A18: $tType] :
( ( bounded_lattice @ A18 )
=> ( bounded_lattice @ ( A17 > A18 ) ) ) ).
thf(tcon_fun___Conditionally__Complete__Lattices_Oconditionally__complete__lattice,axiom,
! [A17: $tType,A18: $tType] :
( ( comple6319245703460814977attice @ A18 )
=> ( condit1219197933456340205attice @ ( A17 > A18 ) ) ) ).
thf(tcon_fun___Countable__Complete__Lattices_Ocountable__complete__lattice,axiom,
! [A17: $tType,A18: $tType] :
( ( counta3822494911875563373attice @ A18 )
=> ( counta3822494911875563373attice @ ( 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_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___Complete__Lattices_OSup,axiom,
! [A17: $tType,A18: $tType] :
( ( complete_Sup @ A18 )
=> ( complete_Sup @ ( A17 > A18 ) ) ) ).
thf(tcon_fun___Complete__Lattices_OInf,axiom,
! [A17: $tType,A18: $tType] :
( ( complete_Inf @ A18 )
=> ( complete_Inf @ ( 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___Orderings_Opreorder,axiom,
! [A17: $tType,A18: $tType] :
( ( preorder @ A18 )
=> ( preorder @ ( A17 > A18 ) ) ) ).
thf(tcon_fun___Finite__Set_Ofinite,axiom,
! [A17: $tType,A18: $tType] :
( ( ( finite_finite @ A17 )
& ( finite_finite @ A18 ) )
=> ( finite_finite @ ( A17 > A18 ) ) ) ).
thf(tcon_fun___Lattices_Olattice,axiom,
! [A17: $tType,A18: $tType] :
( ( lattice @ A18 )
=> ( lattice @ ( A17 > A18 ) ) ) ).
thf(tcon_fun___Orderings_Oorder,axiom,
! [A17: $tType,A18: $tType] :
( ( order @ A18 )
=> ( order @ ( A17 > A18 ) ) ) ).
thf(tcon_fun___Orderings_Oord,axiom,
! [A17: $tType,A18: $tType] :
( ( ord @ A18 )
=> ( ord @ ( A17 > A18 ) ) ) ).
thf(tcon_fun___Groups_Ouminus,axiom,
! [A17: $tType,A18: $tType] :
( ( uminus @ A18 )
=> ( uminus @ ( 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_6,axiom,
condit1219197933456340205attice @ int ).
thf(tcon_Int_Oint___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations,axiom,
bit_un5681908812861735899ations @ int ).
thf(tcon_Int_Oint___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct,axiom,
semiri1453513574482234551roduct @ int ).
thf(tcon_Int_Oint___Euclidean__Division_Ounique__euclidean__semiring__with__nat,axiom,
euclid5411537665997757685th_nat @ int ).
thf(tcon_Int_Oint___Euclidean__Division_Ounique__euclidean__ring__with__nat,axiom,
euclid8789492081693882211th_nat @ int ).
thf(tcon_Int_Oint___Groups_Oordered__ab__semigroup__monoid__add__imp__le,axiom,
ordere1937475149494474687imp_le @ int ).
thf(tcon_Int_Oint___Euclidean__Division_Ounique__euclidean__semiring,axiom,
euclid3128863361964157862miring @ int ).
thf(tcon_Int_Oint___Euclidean__Division_Oeuclidean__semiring__cancel,axiom,
euclid4440199948858584721cancel @ int ).
thf(tcon_Int_Oint___Divides_Ounique__euclidean__semiring__numeral,axiom,
unique1627219031080169319umeral @ int ).
thf(tcon_Int_Oint___Euclidean__Division_Oeuclidean__ring__cancel,axiom,
euclid8851590272496341667cancel @ int ).
thf(tcon_Int_Oint___Rings_Osemiring__no__zero__divisors__cancel,axiom,
semiri6575147826004484403cancel @ int ).
thf(tcon_Int_Oint___Groups_Ostrict__ordered__ab__semigroup__add,axiom,
strict9044650504122735259up_add @ int ).
thf(tcon_Int_Oint___Groups_Oordered__cancel__ab__semigroup__add,axiom,
ordere580206878836729694up_add @ int ).
thf(tcon_Int_Oint___Groups_Oordered__ab__semigroup__add__imp__le,axiom,
ordere2412721322843649153imp_le @ int ).
thf(tcon_Int_Oint___Bit__Operations_Osemiring__bit__operations,axiom,
bit_se359711467146920520ations @ int ).
thf(tcon_Int_Oint___Rings_Olinordered__comm__semiring__strict,axiom,
linord2810124833399127020strict @ int ).
thf(tcon_Int_Oint___Groups_Ostrict__ordered__comm__monoid__add,axiom,
strict7427464778891057005id_add @ int ).
thf(tcon_Int_Oint___Groups_Oordered__cancel__comm__monoid__add,axiom,
ordere8940638589300402666id_add @ int ).
thf(tcon_Int_Oint___Euclidean__Division_Oeuclidean__semiring,axiom,
euclid3725896446679973847miring @ int ).
thf(tcon_Int_Oint___Topological__Spaces_Otopological__space,axiom,
topolo4958980785337419405_space @ int ).
thf(tcon_Int_Oint___Topological__Spaces_Olinorder__topology,axiom,
topolo1944317154257567458pology @ int ).
thf(tcon_Int_Oint___Topological__Spaces_Odiscrete__topology,axiom,
topolo8865339358273720382pology @ int ).
thf(tcon_Int_Oint___Limits_Otopological__comm__monoid__mult,axiom,
topolo4987421752381908075d_mult @ int ).
thf(tcon_Int_Oint___Rings_Olinordered__semiring__1__strict,axiom,
linord715952674999750819strict @ int ).
thf(tcon_Int_Oint___Limits_Otopological__comm__monoid__add,axiom,
topolo5987344860129210374id_add @ int ).
thf(tcon_Int_Oint___Bit__Operations_Oring__bit__operations,axiom,
bit_ri3973907225187159222ations @ int ).
thf(tcon_Int_Oint___Topological__Spaces_Oorder__topology,axiom,
topolo2564578578187576103pology @ 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___Limits_Otopological__semigroup__mult,axiom,
topolo4211221413907600880p_mult @ int ).
thf(tcon_Int_Oint___Euclidean__Division_Oeuclidean__ring,axiom,
euclid5891614535332579305n_ring @ 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___Limits_Otopological__monoid__mult,axiom,
topolo1898628316856586783d_mult @ 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___Limits_Otopological__monoid__add,axiom,
topolo6943815403480290642id_add @ 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___Topological__Spaces_Ot2__space,axiom,
topological_t2_space @ int ).
thf(tcon_Int_Oint___Topological__Spaces_Ot1__space,axiom,
topological_t1_space @ int ).
thf(tcon_Int_Oint___Rings_Oordered__comm__semiring,axiom,
ordere2520102378445227354miring @ int ).
thf(tcon_Int_Oint___Rings_Olinordered__semiring__1,axiom,
linord6961819062388156250ring_1 @ int ).
thf(tcon_Int_Oint___Groups_Oordered__ab__group__add,axiom,
ordered_ab_group_add @ int ).
thf(tcon_Int_Oint___Groups_Ocancel__semigroup__add,axiom,
cancel_semigroup_add @ int ).
thf(tcon_Int_Oint___Rings_Olinordered__semiring,axiom,
linordered_semiring @ int ).
thf(tcon_Int_Oint___Rings_Oordered__semiring__0,axiom,
ordered_semiring_0 @ int ).
thf(tcon_Int_Oint___Rings_Olinordered__semidom,axiom,
linordered_semidom @ int ).
thf(tcon_Int_Oint___Lattices_Osemilattice__sup_7,axiom,
semilattice_sup @ int ).
thf(tcon_Int_Oint___Lattices_Osemilattice__inf_8,axiom,
semilattice_inf @ int ).
thf(tcon_Int_Oint___Groups_Oab__semigroup__mult,axiom,
ab_semigroup_mult @ int ).
thf(tcon_Int_Oint___Rings_Osemiring__1__cancel,axiom,
semiring_1_cancel @ int ).
thf(tcon_Int_Oint___Rings_Oalgebraic__semidom,axiom,
algebraic_semidom @ int ).
thf(tcon_Int_Oint___Groups_Ocomm__monoid__mult,axiom,
comm_monoid_mult @ int ).
thf(tcon_Int_Oint___Groups_Oab__semigroup__add,axiom,
ab_semigroup_add @ int ).
thf(tcon_Int_Oint___Rings_Oordered__semiring,axiom,
ordered_semiring @ int ).
thf(tcon_Int_Oint___Rings_Oordered__ring__abs,axiom,
ordered_ring_abs @ int ).
thf(tcon_Int_Oint___Parity_Osemiring__parity,axiom,
semiring_parity @ int ).
thf(tcon_Int_Oint___Groups_Ocomm__monoid__add,axiom,
comm_monoid_add @ int ).
thf(tcon_Int_Oint___Rings_Osemiring__modulo,axiom,
semiring_modulo @ int ).
thf(tcon_Int_Oint___Rings_Olinordered__ring,axiom,
linordered_ring @ int ).
thf(tcon_Int_Oint___Rings_Olinordered__idom,axiom,
linordered_idom @ int ).
thf(tcon_Int_Oint___Rings_Ocomm__semiring__1,axiom,
comm_semiring_1 @ int ).
thf(tcon_Int_Oint___Rings_Ocomm__semiring__0,axiom,
comm_semiring_0 @ int ).
thf(tcon_Int_Oint___Groups_Osemigroup__mult,axiom,
semigroup_mult @ int ).
thf(tcon_Int_Oint___Complete__Lattices_OSup_9,axiom,
complete_Sup @ int ).
thf(tcon_Int_Oint___Complete__Lattices_OInf_10,axiom,
complete_Inf @ int ).
thf(tcon_Int_Oint___Rings_Osemidom__modulo,axiom,
semidom_modulo @ int ).
thf(tcon_Int_Oint___Rings_Osemidom__divide,axiom,
semidom_divide @ int ).
thf(tcon_Int_Oint___Num_Osemiring__numeral,axiom,
semiring_numeral @ int ).
thf(tcon_Int_Oint___Groups_Osemigroup__add,axiom,
semigroup_add @ int ).
thf(tcon_Int_Oint___Rings_Ozero__less__one,axiom,
zero_less_one @ int ).
thf(tcon_Int_Oint___Rings_Ocomm__semiring,axiom,
comm_semiring @ int ).
thf(tcon_Int_Oint___Nat_Osemiring__char__0,axiom,
semiring_char_0 @ int ).
thf(tcon_Int_Oint___Groups_Oab__group__add,axiom,
ab_group_add @ int ).
thf(tcon_Int_Oint___Rings_Ozero__neq__one,axiom,
zero_neq_one @ int ).
thf(tcon_Int_Oint___Rings_Oordered__ring,axiom,
ordered_ring @ int ).
thf(tcon_Int_Oint___Rings_Oidom__abs__sgn,axiom,
idom_abs_sgn @ int ).
thf(tcon_Int_Oint___Parity_Oring__parity,axiom,
ring_parity @ int ).
thf(tcon_Int_Oint___Orderings_Opreorder_11,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_12,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___Rings_Omult__zero,axiom,
mult_zero @ int ).
thf(tcon_Int_Oint___Rings_Ocomm__ring,axiom,
comm_ring @ int ).
thf(tcon_Int_Oint___Orderings_Oorder_13,axiom,
order @ int ).
thf(tcon_Int_Oint___Num_Oneg__numeral,axiom,
neg_numeral @ int ).
thf(tcon_Int_Oint___Nat_Oring__char__0,axiom,
ring_char_0 @ int ).
thf(tcon_Int_Oint___Rings_Osemiring,axiom,
semiring @ int ).
thf(tcon_Int_Oint___Orderings_Oord_14,axiom,
ord @ int ).
thf(tcon_Int_Oint___Groups_Ouminus_15,axiom,
uminus @ int ).
thf(tcon_Int_Oint___Rings_Oring__1,axiom,
ring_1 @ int ).
thf(tcon_Int_Oint___Rings_Oabs__if,axiom,
abs_if @ int ).
thf(tcon_Int_Oint___GCD_Oring__gcd,axiom,
ring_gcd @ int ).
thf(tcon_Int_Oint___Power_Opower,axiom,
power @ int ).
thf(tcon_Int_Oint___Num_Onumeral,axiom,
numeral @ int ).
thf(tcon_Int_Oint___Groups_Ozero,axiom,
zero @ int ).
thf(tcon_Int_Oint___Groups_Oplus,axiom,
plus @ int ).
thf(tcon_Int_Oint___Rings_Oring,axiom,
ring @ int ).
thf(tcon_Int_Oint___Rings_Oidom,axiom,
idom @ int ).
thf(tcon_Int_Oint___Groups_Oone,axiom,
one @ int ).
thf(tcon_Int_Oint___Rings_Odvd,axiom,
dvd @ int ).
thf(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_16,axiom,
condit6923001295902523014norder @ nat ).
thf(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_17,axiom,
condit1219197933456340205attice @ nat ).
thf(tcon_Nat_Onat___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_18,axiom,
bit_un5681908812861735899ations @ nat ).
thf(tcon_Nat_Onat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_19,axiom,
semiri1453513574482234551roduct @ nat ).
thf(tcon_Nat_Onat___Euclidean__Division_Ounique__euclidean__semiring__with__nat_20,axiom,
euclid5411537665997757685th_nat @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__monoid__add__imp__le_21,axiom,
ordere1937475149494474687imp_le @ nat ).
thf(tcon_Nat_Onat___Euclidean__Division_Ounique__euclidean__semiring_22,axiom,
euclid3128863361964157862miring @ nat ).
thf(tcon_Nat_Onat___Euclidean__Division_Oeuclidean__semiring__cancel_23,axiom,
euclid4440199948858584721cancel @ nat ).
thf(tcon_Nat_Onat___Divides_Ounique__euclidean__semiring__numeral_24,axiom,
unique1627219031080169319umeral @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors__cancel_25,axiom,
semiri6575147826004484403cancel @ nat ).
thf(tcon_Nat_Onat___Groups_Ostrict__ordered__ab__semigroup__add_26,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_27,axiom,
ordere580206878836729694up_add @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le_28,axiom,
ordere2412721322843649153imp_le @ nat ).
thf(tcon_Nat_Onat___Bit__Operations_Osemiring__bit__operations_29,axiom,
bit_se359711467146920520ations @ nat ).
thf(tcon_Nat_Onat___Rings_Olinordered__comm__semiring__strict_30,axiom,
linord2810124833399127020strict @ nat ).
thf(tcon_Nat_Onat___Groups_Ostrict__ordered__comm__monoid__add_31,axiom,
strict7427464778891057005id_add @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__cancel__comm__monoid__add_32,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_33,axiom,
euclid3725896446679973847miring @ nat ).
thf(tcon_Nat_Onat___Topological__Spaces_Otopological__space_34,axiom,
topolo4958980785337419405_space @ nat ).
thf(tcon_Nat_Onat___Topological__Spaces_Olinorder__topology_35,axiom,
topolo1944317154257567458pology @ nat ).
thf(tcon_Nat_Onat___Topological__Spaces_Odiscrete__topology_36,axiom,
topolo8865339358273720382pology @ nat ).
thf(tcon_Nat_Onat___Limits_Otopological__comm__monoid__mult_37,axiom,
topolo4987421752381908075d_mult @ nat ).
thf(tcon_Nat_Onat___Limits_Otopological__comm__monoid__add_38,axiom,
topolo5987344860129210374id_add @ nat ).
thf(tcon_Nat_Onat___Topological__Spaces_Oorder__topology_39,axiom,
topolo2564578578187576103pology @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__1__no__zero__divisors_40,axiom,
semiri2026040879449505780visors @ nat ).
thf(tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring_41,axiom,
linord181362715937106298miring @ nat ).
thf(tcon_Nat_Onat___Limits_Otopological__semigroup__mult_42,axiom,
topolo4211221413907600880p_mult @ nat ).
thf(tcon_Nat_Onat___Rings_Olinordered__semiring__strict_43,axiom,
linord8928482502909563296strict @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors_44,axiom,
semiri3467727345109120633visors @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add_45,axiom,
ordere6658533253407199908up_add @ nat ).
thf(tcon_Nat_Onat___GCD_Osemiring__gcd__mult__normalize_46,axiom,
semiri6843258321239162965malize @ nat ).
thf(tcon_Nat_Onat___Limits_Otopological__monoid__mult_47,axiom,
topolo1898628316856586783d_mult @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__comm__monoid__add_48,axiom,
ordere6911136660526730532id_add @ nat ).
thf(tcon_Nat_Onat___Groups_Ocancel__ab__semigroup__add_49,axiom,
cancel2418104881723323429up_add @ nat ).
thf(tcon_Nat_Onat___Limits_Otopological__monoid__add_50,axiom,
topolo6943815403480290642id_add @ nat ).
thf(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add_51,axiom,
cancel1802427076303600483id_add @ nat ).
thf(tcon_Nat_Onat___Rings_Ocomm__semiring__1__cancel_52,axiom,
comm_s4317794764714335236cancel @ nat ).
thf(tcon_Nat_Onat___Bit__Operations_Osemiring__bits_53,axiom,
bit_semiring_bits @ nat ).
thf(tcon_Nat_Onat___Topological__Spaces_Ot2__space_54,axiom,
topological_t2_space @ nat ).
thf(tcon_Nat_Onat___Topological__Spaces_Ot1__space_55,axiom,
topological_t1_space @ nat ).
thf(tcon_Nat_Onat___Rings_Oordered__comm__semiring_56,axiom,
ordere2520102378445227354miring @ 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___Groups_Oab__semigroup__mult_63,axiom,
ab_semigroup_mult @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__1__cancel_64,axiom,
semiring_1_cancel @ nat ).
thf(tcon_Nat_Onat___Rings_Oalgebraic__semidom_65,axiom,
algebraic_semidom @ nat ).
thf(tcon_Nat_Onat___Groups_Ocomm__monoid__mult_66,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_67,axiom,
ab_semigroup_add @ nat ).
thf(tcon_Nat_Onat___Rings_Oordered__semiring_68,axiom,
ordered_semiring @ nat ).
thf(tcon_Nat_Onat___Parity_Osemiring__parity_69,axiom,
semiring_parity @ nat ).
thf(tcon_Nat_Onat___Groups_Ocomm__monoid__add_70,axiom,
comm_monoid_add @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__modulo_71,axiom,
semiring_modulo @ nat ).
thf(tcon_Nat_Onat___Rings_Ocomm__semiring__1_72,axiom,
comm_semiring_1 @ nat ).
thf(tcon_Nat_Onat___Rings_Ocomm__semiring__0_73,axiom,
comm_semiring_0 @ nat ).
thf(tcon_Nat_Onat___Groups_Osemigroup__mult_74,axiom,
semigroup_mult @ nat ).
thf(tcon_Nat_Onat___Complete__Lattices_OSup_75,axiom,
complete_Sup @ nat ).
thf(tcon_Nat_Onat___Complete__Lattices_OInf_76,axiom,
complete_Inf @ nat ).
thf(tcon_Nat_Onat___Rings_Osemidom__modulo_77,axiom,
semidom_modulo @ nat ).
thf(tcon_Nat_Onat___Rings_Osemidom__divide_78,axiom,
semidom_divide @ nat ).
thf(tcon_Nat_Onat___Num_Osemiring__numeral_79,axiom,
semiring_numeral @ nat ).
thf(tcon_Nat_Onat___Groups_Osemigroup__add_80,axiom,
semigroup_add @ nat ).
thf(tcon_Nat_Onat___Rings_Ozero__less__one_81,axiom,
zero_less_one @ nat ).
thf(tcon_Nat_Onat___Rings_Ocomm__semiring_82,axiom,
comm_semiring @ nat ).
thf(tcon_Nat_Onat___Orderings_Owellorder,axiom,
wellorder @ nat ).
thf(tcon_Nat_Onat___Orderings_Oorder__bot_83,axiom,
order_bot @ nat ).
thf(tcon_Nat_Onat___Nat_Osemiring__char__0_84,axiom,
semiring_char_0 @ 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___Rings_Omult__zero_95,axiom,
mult_zero @ nat ).
thf(tcon_Nat_Onat___Orderings_Oorder_96,axiom,
order @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring_97,axiom,
semiring @ nat ).
thf(tcon_Nat_Onat___Orderings_Oord_98,axiom,
ord @ nat ).
thf(tcon_Nat_Onat___Power_Opower_99,axiom,
power @ nat ).
thf(tcon_Nat_Onat___Num_Onumeral_100,axiom,
numeral @ nat ).
thf(tcon_Nat_Onat___Groups_Ozero_101,axiom,
zero @ nat ).
thf(tcon_Nat_Onat___Groups_Oplus_102,axiom,
plus @ nat ).
thf(tcon_Nat_Onat___Groups_Oone_103,axiom,
one @ nat ).
thf(tcon_Nat_Onat___Rings_Odvd_104,axiom,
dvd @ nat ).
thf(tcon_Nat_Onat___Nat_Osize,axiom,
size @ nat ).
thf(tcon_Num_Onum___Orderings_Opreorder_105,axiom,
preorder @ num ).
thf(tcon_Num_Onum___Orderings_Olinorder_106,axiom,
linorder @ num ).
thf(tcon_Num_Onum___Orderings_Oorder_107,axiom,
order @ num ).
thf(tcon_Num_Onum___Orderings_Oord_108,axiom,
ord @ num ).
thf(tcon_Num_Onum___Groups_Oplus_109,axiom,
plus @ num ).
thf(tcon_Num_Onum___Nat_Osize_110,axiom,
size @ num ).
thf(tcon_Rat_Orat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_111,axiom,
semiri1453513574482234551roduct @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__monoid__add__imp__le_112,axiom,
ordere1937475149494474687imp_le @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring__no__zero__divisors__cancel_113,axiom,
semiri6575147826004484403cancel @ rat ).
thf(tcon_Rat_Orat___Groups_Ostrict__ordered__ab__semigroup__add_114,axiom,
strict9044650504122735259up_add @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__cancel__ab__semigroup__add_115,axiom,
ordere580206878836729694up_add @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__add__imp__le_116,axiom,
ordere2412721322843649153imp_le @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__comm__semiring__strict_117,axiom,
linord2810124833399127020strict @ rat ).
thf(tcon_Rat_Orat___Groups_Ostrict__ordered__comm__monoid__add_118,axiom,
strict7427464778891057005id_add @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__cancel__comm__monoid__add_119,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_120,axiom,
linord715952674999750819strict @ rat ).
thf(tcon_Rat_Orat___Orderings_Ounbounded__dense__linorder,axiom,
unboun7993243217541854897norder @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring__1__no__zero__divisors_121,axiom,
semiri2026040879449505780visors @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__nonzero__semiring_122,axiom,
linord181362715937106298miring @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__semiring__strict_123,axiom,
linord8928482502909563296strict @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring__no__zero__divisors_124,axiom,
semiri3467727345109120633visors @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__add_125,axiom,
ordere6658533253407199908up_add @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__ab__group__add__abs_126,axiom,
ordere166539214618696060dd_abs @ rat ).
thf(tcon_Rat_Orat___Archimedean__Field_Ofloor__ceiling,axiom,
archim2362893244070406136eiling @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__comm__monoid__add_127,axiom,
ordere6911136660526730532id_add @ rat ).
thf(tcon_Rat_Orat___Groups_Olinordered__ab__group__add_128,axiom,
linord5086331880401160121up_add @ rat ).
thf(tcon_Rat_Orat___Groups_Ocancel__ab__semigroup__add_129,axiom,
cancel2418104881723323429up_add @ rat ).
thf(tcon_Rat_Orat___Rings_Oring__1__no__zero__divisors_130,axiom,
ring_15535105094025558882visors @ rat ).
thf(tcon_Rat_Orat___Groups_Ocancel__comm__monoid__add_131,axiom,
cancel1802427076303600483id_add @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__ring__strict_132,axiom,
linord4710134922213307826strict @ rat ).
thf(tcon_Rat_Orat___Rings_Ocomm__semiring__1__cancel_133,axiom,
comm_s4317794764714335236cancel @ rat ).
thf(tcon_Rat_Orat___Rings_Oordered__comm__semiring_134,axiom,
ordere2520102378445227354miring @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__semiring__1_135,axiom,
linord6961819062388156250ring_1 @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__ab__group__add_136,axiom,
ordered_ab_group_add @ rat ).
thf(tcon_Rat_Orat___Groups_Ocancel__semigroup__add_137,axiom,
cancel_semigroup_add @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__semiring_138,axiom,
linordered_semiring @ rat ).
thf(tcon_Rat_Orat___Rings_Oordered__semiring__0_139,axiom,
ordered_semiring_0 @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__semidom_140,axiom,
linordered_semidom @ rat ).
thf(tcon_Rat_Orat___Orderings_Odense__linorder,axiom,
dense_linorder @ rat ).
thf(tcon_Rat_Orat___Lattices_Osemilattice__sup_141,axiom,
semilattice_sup @ rat ).
thf(tcon_Rat_Orat___Lattices_Osemilattice__inf_142,axiom,
semilattice_inf @ rat ).
thf(tcon_Rat_Orat___Groups_Oab__semigroup__mult_143,axiom,
ab_semigroup_mult @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring__1__cancel_144,axiom,
semiring_1_cancel @ rat ).
thf(tcon_Rat_Orat___Groups_Ocomm__monoid__mult_145,axiom,
comm_monoid_mult @ rat ).
thf(tcon_Rat_Orat___Groups_Oab__semigroup__add_146,axiom,
ab_semigroup_add @ rat ).
thf(tcon_Rat_Orat___Fields_Olinordered__field,axiom,
linordered_field @ rat ).
thf(tcon_Rat_Orat___Rings_Oordered__semiring_147,axiom,
ordered_semiring @ rat ).
thf(tcon_Rat_Orat___Rings_Oordered__ring__abs_148,axiom,
ordered_ring_abs @ rat ).
thf(tcon_Rat_Orat___Groups_Ocomm__monoid__add_149,axiom,
comm_monoid_add @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__ring_150,axiom,
linordered_ring @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__idom_151,axiom,
linordered_idom @ rat ).
thf(tcon_Rat_Orat___Rings_Ocomm__semiring__1_152,axiom,
comm_semiring_1 @ rat ).
thf(tcon_Rat_Orat___Rings_Ocomm__semiring__0_153,axiom,
comm_semiring_0 @ rat ).
thf(tcon_Rat_Orat___Orderings_Odense__order,axiom,
dense_order @ rat ).
thf(tcon_Rat_Orat___Groups_Osemigroup__mult_154,axiom,
semigroup_mult @ rat ).
thf(tcon_Rat_Orat___Rings_Osemidom__divide_155,axiom,
semidom_divide @ rat ).
thf(tcon_Rat_Orat___Num_Osemiring__numeral_156,axiom,
semiring_numeral @ rat ).
thf(tcon_Rat_Orat___Groups_Osemigroup__add_157,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_158,axiom,
zero_less_one @ rat ).
thf(tcon_Rat_Orat___Rings_Ocomm__semiring_159,axiom,
comm_semiring @ rat ).
thf(tcon_Rat_Orat___Nat_Osemiring__char__0_160,axiom,
semiring_char_0 @ rat ).
thf(tcon_Rat_Orat___Groups_Oab__group__add_161,axiom,
ab_group_add @ rat ).
thf(tcon_Rat_Orat___Fields_Ofield__char__0,axiom,
field_char_0 @ rat ).
thf(tcon_Rat_Orat___Rings_Ozero__neq__one_162,axiom,
zero_neq_one @ rat ).
thf(tcon_Rat_Orat___Rings_Oordered__ring_163,axiom,
ordered_ring @ rat ).
thf(tcon_Rat_Orat___Rings_Oidom__abs__sgn_164,axiom,
idom_abs_sgn @ rat ).
thf(tcon_Rat_Orat___Orderings_Opreorder_165,axiom,
preorder @ rat ).
thf(tcon_Rat_Orat___Orderings_Olinorder_166,axiom,
linorder @ rat ).
thf(tcon_Rat_Orat___Groups_Omonoid__mult_167,axiom,
monoid_mult @ rat ).
thf(tcon_Rat_Orat___Rings_Oidom__divide_168,axiom,
idom_divide @ rat ).
thf(tcon_Rat_Orat___Rings_Ocomm__ring__1_169,axiom,
comm_ring_1 @ rat ).
thf(tcon_Rat_Orat___Groups_Omonoid__add_170,axiom,
monoid_add @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring__1_171,axiom,
semiring_1 @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring__0_172,axiom,
semiring_0 @ rat ).
thf(tcon_Rat_Orat___Orderings_Ono__top_173,axiom,
no_top @ rat ).
thf(tcon_Rat_Orat___Orderings_Ono__bot_174,axiom,
no_bot @ rat ).
thf(tcon_Rat_Orat___Lattices_Olattice_175,axiom,
lattice @ rat ).
thf(tcon_Rat_Orat___Groups_Ogroup__add_176,axiom,
group_add @ rat ).
thf(tcon_Rat_Orat___Rings_Omult__zero_177,axiom,
mult_zero @ rat ).
thf(tcon_Rat_Orat___Rings_Ocomm__ring_178,axiom,
comm_ring @ rat ).
thf(tcon_Rat_Orat___Orderings_Oorder_179,axiom,
order @ rat ).
thf(tcon_Rat_Orat___Num_Oneg__numeral_180,axiom,
neg_numeral @ rat ).
thf(tcon_Rat_Orat___Nat_Oring__char__0_181,axiom,
ring_char_0 @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring_182,axiom,
semiring @ rat ).
thf(tcon_Rat_Orat___Fields_Oinverse,axiom,
inverse @ rat ).
thf(tcon_Rat_Orat___Orderings_Oord_183,axiom,
ord @ rat ).
thf(tcon_Rat_Orat___Groups_Ouminus_184,axiom,
uminus @ rat ).
thf(tcon_Rat_Orat___Rings_Oring__1_185,axiom,
ring_1 @ rat ).
thf(tcon_Rat_Orat___Rings_Oabs__if_186,axiom,
abs_if @ rat ).
thf(tcon_Rat_Orat___Fields_Ofield,axiom,
field @ rat ).
thf(tcon_Rat_Orat___Power_Opower_187,axiom,
power @ rat ).
thf(tcon_Rat_Orat___Num_Onumeral_188,axiom,
numeral @ rat ).
thf(tcon_Rat_Orat___Groups_Ozero_189,axiom,
zero @ rat ).
thf(tcon_Rat_Orat___Groups_Oplus_190,axiom,
plus @ rat ).
thf(tcon_Rat_Orat___Rings_Oring_191,axiom,
ring @ rat ).
thf(tcon_Rat_Orat___Rings_Oidom_192,axiom,
idom @ rat ).
thf(tcon_Rat_Orat___Groups_Oone_193,axiom,
one @ rat ).
thf(tcon_Rat_Orat___Rings_Odvd_194,axiom,
dvd @ rat ).
thf(tcon_Set_Oset___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_195,axiom,
! [A17: $tType] : ( condit1219197933456340205attice @ ( set @ A17 ) ) ).
thf(tcon_Set_Oset___Countable__Complete__Lattices_Ocountable__complete__lattice_196,axiom,
! [A17: $tType] : ( counta3822494911875563373attice @ ( set @ A17 ) ) ).
thf(tcon_Set_Oset___Complete__Lattices_Ocomplete__distrib__lattice_197,axiom,
! [A17: $tType] : ( comple592849572758109894attice @ ( set @ A17 ) ) ).
thf(tcon_Set_Oset___Complete__Lattices_Ocomplete__boolean__algebra_198,axiom,
! [A17: $tType] : ( comple489889107523837845lgebra @ ( set @ A17 ) ) ).
thf(tcon_Set_Oset___Lattices_Obounded__semilattice__sup__bot_199,axiom,
! [A17: $tType] : ( bounde4967611905675639751up_bot @ ( set @ A17 ) ) ).
thf(tcon_Set_Oset___Lattices_Obounded__semilattice__inf__top_200,axiom,
! [A17: $tType] : ( bounde4346867609351753570nf_top @ ( set @ A17 ) ) ).
thf(tcon_Set_Oset___Complete__Lattices_Ocomplete__lattice_201,axiom,
! [A17: $tType] : ( comple6319245703460814977attice @ ( set @ A17 ) ) ).
thf(tcon_Set_Oset___Boolean__Algebras_Oboolean__algebra_202,axiom,
! [A17: $tType] : ( boolea8198339166811842893lgebra @ ( set @ A17 ) ) ).
thf(tcon_Set_Oset___Lattices_Osemilattice__sup_203,axiom,
! [A17: $tType] : ( semilattice_sup @ ( set @ A17 ) ) ).
thf(tcon_Set_Oset___Lattices_Osemilattice__inf_204,axiom,
! [A17: $tType] : ( semilattice_inf @ ( set @ A17 ) ) ).
thf(tcon_Set_Oset___Complete__Lattices_OSup_205,axiom,
! [A17: $tType] : ( complete_Sup @ ( set @ A17 ) ) ).
thf(tcon_Set_Oset___Complete__Lattices_OInf_206,axiom,
! [A17: $tType] : ( complete_Inf @ ( set @ A17 ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder__top_207,axiom,
! [A17: $tType] : ( order_top @ ( set @ A17 ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder__bot_208,axiom,
! [A17: $tType] : ( order_bot @ ( set @ A17 ) ) ).
thf(tcon_Set_Oset___Orderings_Opreorder_209,axiom,
! [A17: $tType] : ( preorder @ ( set @ A17 ) ) ).
thf(tcon_Set_Oset___Finite__Set_Ofinite_210,axiom,
! [A17: $tType] :
( ( finite_finite @ A17 )
=> ( finite_finite @ ( set @ A17 ) ) ) ).
thf(tcon_Set_Oset___Lattices_Olattice_211,axiom,
! [A17: $tType] : ( lattice @ ( set @ A17 ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder_212,axiom,
! [A17: $tType] : ( order @ ( set @ A17 ) ) ).
thf(tcon_Set_Oset___Orderings_Oord_213,axiom,
! [A17: $tType] : ( ord @ ( set @ A17 ) ) ).
thf(tcon_Set_Oset___Groups_Ouminus_214,axiom,
! [A17: $tType] : ( uminus @ ( set @ A17 ) ) ).
thf(tcon_HOL_Obool___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_215,axiom,
condit1219197933456340205attice @ $o ).
thf(tcon_HOL_Obool___Countable__Complete__Lattices_Ocountable__complete__lattice_216,axiom,
counta3822494911875563373attice @ $o ).
thf(tcon_HOL_Obool___Complete__Lattices_Ocomplete__distrib__lattice_217,axiom,
comple592849572758109894attice @ $o ).
thf(tcon_HOL_Obool___Complete__Lattices_Ocomplete__boolean__algebra_218,axiom,
comple489889107523837845lgebra @ $o ).
thf(tcon_HOL_Obool___Topological__Spaces_Otopological__space_219,axiom,
topolo4958980785337419405_space @ $o ).
thf(tcon_HOL_Obool___Topological__Spaces_Olinorder__topology_220,axiom,
topolo1944317154257567458pology @ $o ).
thf(tcon_HOL_Obool___Topological__Spaces_Odiscrete__topology_221,axiom,
topolo8865339358273720382pology @ $o ).
thf(tcon_HOL_Obool___Lattices_Obounded__semilattice__sup__bot_222,axiom,
bounde4967611905675639751up_bot @ $o ).
thf(tcon_HOL_Obool___Lattices_Obounded__semilattice__inf__top_223,axiom,
bounde4346867609351753570nf_top @ $o ).
thf(tcon_HOL_Obool___Complete__Lattices_Ocomplete__lattice_224,axiom,
comple6319245703460814977attice @ $o ).
thf(tcon_HOL_Obool___Topological__Spaces_Oorder__topology_225,axiom,
topolo2564578578187576103pology @ $o ).
thf(tcon_HOL_Obool___Boolean__Algebras_Oboolean__algebra_226,axiom,
boolea8198339166811842893lgebra @ $o ).
thf(tcon_HOL_Obool___Topological__Spaces_Ot2__space_227,axiom,
topological_t2_space @ $o ).
thf(tcon_HOL_Obool___Topological__Spaces_Ot1__space_228,axiom,
topological_t1_space @ $o ).
thf(tcon_HOL_Obool___Lattices_Osemilattice__sup_229,axiom,
semilattice_sup @ $o ).
thf(tcon_HOL_Obool___Lattices_Osemilattice__inf_230,axiom,
semilattice_inf @ $o ).
thf(tcon_HOL_Obool___Complete__Lattices_OSup_231,axiom,
complete_Sup @ $o ).
thf(tcon_HOL_Obool___Complete__Lattices_OInf_232,axiom,
complete_Inf @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder__top_233,axiom,
order_top @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder__bot_234,axiom,
order_bot @ $o ).
thf(tcon_HOL_Obool___Orderings_Opreorder_235,axiom,
preorder @ $o ).
thf(tcon_HOL_Obool___Orderings_Olinorder_236,axiom,
linorder @ $o ).
thf(tcon_HOL_Obool___Finite__Set_Ofinite_237,axiom,
finite_finite @ $o ).
thf(tcon_HOL_Obool___Lattices_Olattice_238,axiom,
lattice @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder_239,axiom,
order @ $o ).
thf(tcon_HOL_Obool___Orderings_Oord_240,axiom,
ord @ $o ).
thf(tcon_HOL_Obool___Groups_Ouminus_241,axiom,
uminus @ $o ).
thf(tcon_List_Olist___Nat_Osize_242,axiom,
! [A17: $tType] : ( size @ ( list @ A17 ) ) ).
thf(tcon_Real_Oreal___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_243,axiom,
condit6923001295902523014norder @ real ).
thf(tcon_Real_Oreal___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_244,axiom,
condit1219197933456340205attice @ real ).
thf(tcon_Real_Oreal___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_245,axiom,
semiri1453513574482234551roduct @ real ).
thf(tcon_Real_Oreal___Conditionally__Complete__Lattices_Olinear__continuum,axiom,
condit5016429287641298734tinuum @ real ).
thf(tcon_Real_Oreal___Groups_Oordered__ab__semigroup__monoid__add__imp__le_246,axiom,
ordere1937475149494474687imp_le @ real ).
thf(tcon_Real_Oreal___Topological__Spaces_Olinear__continuum__topology,axiom,
topolo8458572112393995274pology @ real ).
thf(tcon_Real_Oreal___Topological__Spaces_Ofirst__countable__topology,axiom,
topolo3112930676232923870pology @ real ).
thf(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__normed__div__algebra,axiom,
real_V8999393235501362500lgebra @ real ).
thf(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__normed__algebra__1,axiom,
real_V2822296259951069270ebra_1 @ real ).
thf(tcon_Real_Oreal___Rings_Osemiring__no__zero__divisors__cancel_247,axiom,
semiri6575147826004484403cancel @ real ).
thf(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__normed__algebra,axiom,
real_V4412858255891104859lgebra @ real ).
thf(tcon_Real_Oreal___Real__Vector__Spaces_Oordered__real__vector,axiom,
real_V5355595471888546746vector @ real ).
thf(tcon_Real_Oreal___Groups_Ostrict__ordered__ab__semigroup__add_248,axiom,
strict9044650504122735259up_add @ real ).
thf(tcon_Real_Oreal___Groups_Oordered__cancel__ab__semigroup__add_249,axiom,
ordere580206878836729694up_add @ real ).
thf(tcon_Real_Oreal___Groups_Oordered__ab__semigroup__add__imp__le_250,axiom,
ordere2412721322843649153imp_le @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__comm__semiring__strict_251,axiom,
linord2810124833399127020strict @ real ).
thf(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__normed__vector,axiom,
real_V822414075346904944vector @ real ).
thf(tcon_Real_Oreal___Groups_Ostrict__ordered__comm__monoid__add_252,axiom,
strict7427464778891057005id_add @ real ).
thf(tcon_Real_Oreal___Groups_Oordered__cancel__comm__monoid__add_253,axiom,
ordere8940638589300402666id_add @ real ).
thf(tcon_Real_Oreal___Topological__Spaces_Otopological__space_254,axiom,
topolo4958980785337419405_space @ real ).
thf(tcon_Real_Oreal___Topological__Spaces_Olinorder__topology_255,axiom,
topolo1944317154257567458pology @ real ).
thf(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__normed__field,axiom,
real_V3459762299906320749_field @ real ).
thf(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__div__algebra,axiom,
real_V5047593784448816457lgebra @ real ).
thf(tcon_Real_Oreal___Archimedean__Field_Oarchimedean__field_256,axiom,
archim462609752435547400_field @ real ).
thf(tcon_Real_Oreal___Topological__Spaces_Oopen__uniformity,axiom,
topolo569519726778239578ormity @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__semiring__1__strict_257,axiom,
linord715952674999750819strict @ real ).
thf(tcon_Real_Oreal___Real__Vector__Spaces_Ouniformity__dist,axiom,
real_V768167426530841204y_dist @ real ).
thf(tcon_Real_Oreal___Orderings_Ounbounded__dense__linorder_258,axiom,
unboun7993243217541854897norder @ real ).
thf(tcon_Real_Oreal___Limits_Otopological__comm__monoid__add_259,axiom,
topolo5987344860129210374id_add @ real ).
thf(tcon_Real_Oreal___Topological__Spaces_Oorder__topology_260,axiom,
topolo2564578578187576103pology @ real ).
thf(tcon_Real_Oreal___Rings_Osemiring__1__no__zero__divisors_261,axiom,
semiri2026040879449505780visors @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__nonzero__semiring_262,axiom,
linord181362715937106298miring @ real ).
thf(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__algebra__1,axiom,
real_V2191834092415804123ebra_1 @ real ).
thf(tcon_Real_Oreal___Real__Vector__Spaces_Ocomplete__space,axiom,
real_V8037385150606011577_space @ real ).
thf(tcon_Real_Oreal___Limits_Otopological__semigroup__mult_263,axiom,
topolo4211221413907600880p_mult @ real ).
thf(tcon_Real_Oreal___Topological__Spaces_Ouniform__space,axiom,
topolo7287701948861334536_space @ real ).
thf(tcon_Real_Oreal___Topological__Spaces_Operfect__space,axiom,
topolo8386298272705272623_space @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__semiring__strict_264,axiom,
linord8928482502909563296strict @ real ).
thf(tcon_Real_Oreal___Rings_Osemiring__no__zero__divisors_265,axiom,
semiri3467727345109120633visors @ real ).
thf(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__algebra,axiom,
real_V6157519004096292374lgebra @ real ).
thf(tcon_Real_Oreal___Real__Vector__Spaces_Ometric__space,axiom,
real_V7819770556892013058_space @ real ).
thf(tcon_Real_Oreal___Limits_Otopological__ab__group__add,axiom,
topolo1287966508704411220up_add @ real ).
thf(tcon_Real_Oreal___Groups_Oordered__ab__semigroup__add_266,axiom,
ordere6658533253407199908up_add @ real ).
thf(tcon_Real_Oreal___Groups_Oordered__ab__group__add__abs_267,axiom,
ordere166539214618696060dd_abs @ real ).
thf(tcon_Real_Oreal___Archimedean__Field_Ofloor__ceiling_268,axiom,
archim2362893244070406136eiling @ real ).
thf(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__vector,axiom,
real_V4867850818363320053vector @ real ).
thf(tcon_Real_Oreal___Groups_Oordered__comm__monoid__add_269,axiom,
ordere6911136660526730532id_add @ real ).
thf(tcon_Real_Oreal___Groups_Olinordered__ab__group__add_270,axiom,
linord5086331880401160121up_add @ real ).
thf(tcon_Real_Oreal___Groups_Ocancel__ab__semigroup__add_271,axiom,
cancel2418104881723323429up_add @ real ).
thf(tcon_Real_Oreal___Rings_Oring__1__no__zero__divisors_272,axiom,
ring_15535105094025558882visors @ real ).
thf(tcon_Real_Oreal___Real__Vector__Spaces_Oreal__field,axiom,
real_V7773925162809079976_field @ real ).
thf(tcon_Real_Oreal___Limits_Otopological__monoid__add_273,axiom,
topolo6943815403480290642id_add @ real ).
thf(tcon_Real_Oreal___Groups_Ocancel__comm__monoid__add_274,axiom,
cancel1802427076303600483id_add @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__ring__strict_275,axiom,
linord4710134922213307826strict @ real ).
thf(tcon_Real_Oreal___Rings_Ocomm__semiring__1__cancel_276,axiom,
comm_s4317794764714335236cancel @ real ).
thf(tcon_Real_Oreal___Limits_Otopological__group__add,axiom,
topolo1633459387980952147up_add @ real ).
thf(tcon_Real_Oreal___Topological__Spaces_Ot2__space_277,axiom,
topological_t2_space @ real ).
thf(tcon_Real_Oreal___Topological__Spaces_Ot1__space_278,axiom,
topological_t1_space @ real ).
thf(tcon_Real_Oreal___Rings_Oordered__comm__semiring_279,axiom,
ordere2520102378445227354miring @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__semiring__1_280,axiom,
linord6961819062388156250ring_1 @ real ).
thf(tcon_Real_Oreal___Groups_Oordered__ab__group__add_281,axiom,
ordered_ab_group_add @ real ).
thf(tcon_Real_Oreal___Groups_Ocancel__semigroup__add_282,axiom,
cancel_semigroup_add @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__semiring_283,axiom,
linordered_semiring @ real ).
thf(tcon_Real_Oreal___Real__Vector__Spaces_Obanach,axiom,
real_Vector_banach @ real ).
thf(tcon_Real_Oreal___Rings_Oordered__semiring__0_284,axiom,
ordered_semiring_0 @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__semidom_285,axiom,
linordered_semidom @ real ).
thf(tcon_Real_Oreal___Orderings_Odense__linorder_286,axiom,
dense_linorder @ real ).
thf(tcon_Real_Oreal___Lattices_Osemilattice__sup_287,axiom,
semilattice_sup @ real ).
thf(tcon_Real_Oreal___Lattices_Osemilattice__inf_288,axiom,
semilattice_inf @ real ).
thf(tcon_Real_Oreal___Groups_Oab__semigroup__mult_289,axiom,
ab_semigroup_mult @ real ).
thf(tcon_Real_Oreal___Rings_Osemiring__1__cancel_290,axiom,
semiring_1_cancel @ real ).
thf(tcon_Real_Oreal___Groups_Ocomm__monoid__mult_291,axiom,
comm_monoid_mult @ real ).
thf(tcon_Real_Oreal___Groups_Oab__semigroup__add_292,axiom,
ab_semigroup_add @ real ).
thf(tcon_Real_Oreal___Fields_Olinordered__field_293,axiom,
linordered_field @ real ).
thf(tcon_Real_Oreal___Rings_Oordered__semiring_294,axiom,
ordered_semiring @ real ).
thf(tcon_Real_Oreal___Rings_Oordered__ring__abs_295,axiom,
ordered_ring_abs @ real ).
thf(tcon_Real_Oreal___Groups_Ocomm__monoid__add_296,axiom,
comm_monoid_add @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__ring_297,axiom,
linordered_ring @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__idom_298,axiom,
linordered_idom @ real ).
thf(tcon_Real_Oreal___Rings_Ocomm__semiring__1_299,axiom,
comm_semiring_1 @ real ).
thf(tcon_Real_Oreal___Rings_Ocomm__semiring__0_300,axiom,
comm_semiring_0 @ real ).
thf(tcon_Real_Oreal___Orderings_Odense__order_301,axiom,
dense_order @ real ).
thf(tcon_Real_Oreal___Groups_Osemigroup__mult_302,axiom,
semigroup_mult @ real ).
thf(tcon_Real_Oreal___Complete__Lattices_OSup_303,axiom,
complete_Sup @ real ).
thf(tcon_Real_Oreal___Complete__Lattices_OInf_304,axiom,
complete_Inf @ real ).
thf(tcon_Real_Oreal___Rings_Osemidom__divide_305,axiom,
semidom_divide @ real ).
thf(tcon_Real_Oreal___Num_Osemiring__numeral_306,axiom,
semiring_numeral @ real ).
thf(tcon_Real_Oreal___Groups_Osemigroup__add_307,axiom,
semigroup_add @ real ).
thf(tcon_Real_Oreal___Fields_Ofield__abs__sgn_308,axiom,
field_abs_sgn @ real ).
thf(tcon_Real_Oreal___Fields_Odivision__ring_309,axiom,
division_ring @ real ).
thf(tcon_Real_Oreal___Rings_Ozero__less__one_310,axiom,
zero_less_one @ real ).
thf(tcon_Real_Oreal___Rings_Ocomm__semiring_311,axiom,
comm_semiring @ real ).
thf(tcon_Real_Oreal___Nat_Osemiring__char__0_312,axiom,
semiring_char_0 @ real ).
thf(tcon_Real_Oreal___Groups_Oab__group__add_313,axiom,
ab_group_add @ real ).
thf(tcon_Real_Oreal___Fields_Ofield__char__0_314,axiom,
field_char_0 @ real ).
thf(tcon_Real_Oreal___Rings_Ozero__neq__one_315,axiom,
zero_neq_one @ real ).
thf(tcon_Real_Oreal___Rings_Oordered__ring_316,axiom,
ordered_ring @ real ).
thf(tcon_Real_Oreal___Rings_Oidom__abs__sgn_317,axiom,
idom_abs_sgn @ real ).
thf(tcon_Real_Oreal___Orderings_Opreorder_318,axiom,
preorder @ real ).
thf(tcon_Real_Oreal___Orderings_Olinorder_319,axiom,
linorder @ real ).
thf(tcon_Real_Oreal___Groups_Omonoid__mult_320,axiom,
monoid_mult @ real ).
thf(tcon_Real_Oreal___Transcendental_Oln,axiom,
ln @ real ).
thf(tcon_Real_Oreal___Rings_Oidom__divide_321,axiom,
idom_divide @ real ).
thf(tcon_Real_Oreal___Rings_Ocomm__ring__1_322,axiom,
comm_ring_1 @ real ).
thf(tcon_Real_Oreal___Groups_Omonoid__add_323,axiom,
monoid_add @ real ).
thf(tcon_Real_Oreal___Rings_Osemiring__1_324,axiom,
semiring_1 @ real ).
thf(tcon_Real_Oreal___Rings_Osemiring__0_325,axiom,
semiring_0 @ real ).
thf(tcon_Real_Oreal___Orderings_Ono__top_326,axiom,
no_top @ real ).
thf(tcon_Real_Oreal___Orderings_Ono__bot_327,axiom,
no_bot @ real ).
thf(tcon_Real_Oreal___Lattices_Olattice_328,axiom,
lattice @ real ).
thf(tcon_Real_Oreal___Groups_Ogroup__add_329,axiom,
group_add @ real ).
thf(tcon_Real_Oreal___Rings_Omult__zero_330,axiom,
mult_zero @ real ).
thf(tcon_Real_Oreal___Rings_Ocomm__ring_331,axiom,
comm_ring @ real ).
thf(tcon_Real_Oreal___Orderings_Oorder_332,axiom,
order @ real ).
thf(tcon_Real_Oreal___Num_Oneg__numeral_333,axiom,
neg_numeral @ real ).
thf(tcon_Real_Oreal___Nat_Oring__char__0_334,axiom,
ring_char_0 @ real ).
thf(tcon_Real_Oreal___Rings_Osemiring_335,axiom,
semiring @ real ).
thf(tcon_Real_Oreal___Fields_Oinverse_336,axiom,
inverse @ real ).
thf(tcon_Real_Oreal___Orderings_Oord_337,axiom,
ord @ real ).
thf(tcon_Real_Oreal___Groups_Ouminus_338,axiom,
uminus @ real ).
thf(tcon_Real_Oreal___Rings_Oring__1_339,axiom,
ring_1 @ real ).
thf(tcon_Real_Oreal___Rings_Oabs__if_340,axiom,
abs_if @ real ).
thf(tcon_Real_Oreal___Fields_Ofield_341,axiom,
field @ real ).
thf(tcon_Real_Oreal___Power_Opower_342,axiom,
power @ real ).
thf(tcon_Real_Oreal___Num_Onumeral_343,axiom,
numeral @ real ).
thf(tcon_Real_Oreal___Groups_Ozero_344,axiom,
zero @ real ).
thf(tcon_Real_Oreal___Groups_Oplus_345,axiom,
plus @ real ).
thf(tcon_Real_Oreal___Rings_Oring_346,axiom,
ring @ real ).
thf(tcon_Real_Oreal___Rings_Oidom_347,axiom,
idom @ real ).
thf(tcon_Real_Oreal___Groups_Oone_348,axiom,
one @ real ).
thf(tcon_Real_Oreal___Rings_Odvd_349,axiom,
dvd @ real ).
thf(tcon_String_Ochar___Finite__Set_Ofinite_350,axiom,
finite_finite @ char ).
thf(tcon_String_Ochar___Nat_Osize_351,axiom,
size @ char ).
thf(tcon_Filter_Ofilter___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_352,axiom,
! [A17: $tType] : ( condit1219197933456340205attice @ ( filter @ A17 ) ) ).
thf(tcon_Filter_Ofilter___Countable__Complete__Lattices_Ocountable__complete__lattice_353,axiom,
! [A17: $tType] : ( counta3822494911875563373attice @ ( filter @ A17 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Obounded__semilattice__sup__bot_354,axiom,
! [A17: $tType] : ( bounde4967611905675639751up_bot @ ( filter @ A17 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Obounded__semilattice__inf__top_355,axiom,
! [A17: $tType] : ( bounde4346867609351753570nf_top @ ( filter @ A17 ) ) ).
thf(tcon_Filter_Ofilter___Complete__Lattices_Ocomplete__lattice_356,axiom,
! [A17: $tType] : ( comple6319245703460814977attice @ ( filter @ A17 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Osemilattice__sup_357,axiom,
! [A17: $tType] : ( semilattice_sup @ ( filter @ A17 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Osemilattice__inf_358,axiom,
! [A17: $tType] : ( semilattice_inf @ ( filter @ A17 ) ) ).
thf(tcon_Filter_Ofilter___Complete__Lattices_OSup_359,axiom,
! [A17: $tType] : ( complete_Sup @ ( filter @ A17 ) ) ).
thf(tcon_Filter_Ofilter___Complete__Lattices_OInf_360,axiom,
! [A17: $tType] : ( complete_Inf @ ( filter @ A17 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Oorder__top_361,axiom,
! [A17: $tType] : ( order_top @ ( filter @ A17 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Oorder__bot_362,axiom,
! [A17: $tType] : ( order_bot @ ( filter @ A17 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Opreorder_363,axiom,
! [A17: $tType] : ( preorder @ ( filter @ A17 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Olattice_364,axiom,
! [A17: $tType] : ( lattice @ ( filter @ A17 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Oorder_365,axiom,
! [A17: $tType] : ( order @ ( filter @ A17 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Oord_366,axiom,
! [A17: $tType] : ( ord @ ( filter @ A17 ) ) ).
thf(tcon_Option_Ooption___Finite__Set_Ofinite_367,axiom,
! [A17: $tType] :
( ( finite_finite @ A17 )
=> ( finite_finite @ ( option @ A17 ) ) ) ).
thf(tcon_Option_Ooption___Nat_Osize_368,axiom,
! [A17: $tType] : ( size @ ( option @ A17 ) ) ).
thf(tcon_Complex_Ocomplex___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_369,axiom,
semiri1453513574482234551roduct @ complex ).
thf(tcon_Complex_Ocomplex___Topological__Spaces_Ofirst__countable__topology_370,axiom,
topolo3112930676232923870pology @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__div__algebra_371,axiom,
real_V8999393235501362500lgebra @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__algebra__1_372,axiom,
real_V2822296259951069270ebra_1 @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Osemiring__no__zero__divisors__cancel_373,axiom,
semiri6575147826004484403cancel @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__algebra_374,axiom,
real_V4412858255891104859lgebra @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__vector_375,axiom,
real_V822414075346904944vector @ complex ).
thf(tcon_Complex_Ocomplex___Topological__Spaces_Otopological__space_376,axiom,
topolo4958980785337419405_space @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__field_377,axiom,
real_V3459762299906320749_field @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__div__algebra_378,axiom,
real_V5047593784448816457lgebra @ complex ).
thf(tcon_Complex_Ocomplex___Topological__Spaces_Oopen__uniformity_379,axiom,
topolo569519726778239578ormity @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Ouniformity__dist_380,axiom,
real_V768167426530841204y_dist @ complex ).
thf(tcon_Complex_Ocomplex___Limits_Otopological__comm__monoid__add_381,axiom,
topolo5987344860129210374id_add @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Osemiring__1__no__zero__divisors_382,axiom,
semiri2026040879449505780visors @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__algebra__1_383,axiom,
real_V2191834092415804123ebra_1 @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Ocomplete__space_384,axiom,
real_V8037385150606011577_space @ complex ).
thf(tcon_Complex_Ocomplex___Limits_Otopological__semigroup__mult_385,axiom,
topolo4211221413907600880p_mult @ complex ).
thf(tcon_Complex_Ocomplex___Topological__Spaces_Ouniform__space_386,axiom,
topolo7287701948861334536_space @ complex ).
thf(tcon_Complex_Ocomplex___Topological__Spaces_Operfect__space_387,axiom,
topolo8386298272705272623_space @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Osemiring__no__zero__divisors_388,axiom,
semiri3467727345109120633visors @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__algebra_389,axiom,
real_V6157519004096292374lgebra @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Ometric__space_390,axiom,
real_V7819770556892013058_space @ complex ).
thf(tcon_Complex_Ocomplex___Limits_Otopological__ab__group__add_391,axiom,
topolo1287966508704411220up_add @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__vector_392,axiom,
real_V4867850818363320053vector @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Ocancel__ab__semigroup__add_393,axiom,
cancel2418104881723323429up_add @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Oring__1__no__zero__divisors_394,axiom,
ring_15535105094025558882visors @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__field_395,axiom,
real_V7773925162809079976_field @ complex ).
thf(tcon_Complex_Ocomplex___Limits_Otopological__monoid__add_396,axiom,
topolo6943815403480290642id_add @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Ocancel__comm__monoid__add_397,axiom,
cancel1802427076303600483id_add @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Ocomm__semiring__1__cancel_398,axiom,
comm_s4317794764714335236cancel @ complex ).
thf(tcon_Complex_Ocomplex___Limits_Otopological__group__add_399,axiom,
topolo1633459387980952147up_add @ complex ).
thf(tcon_Complex_Ocomplex___Topological__Spaces_Ot2__space_400,axiom,
topological_t2_space @ complex ).
thf(tcon_Complex_Ocomplex___Topological__Spaces_Ot1__space_401,axiom,
topological_t1_space @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Ocancel__semigroup__add_402,axiom,
cancel_semigroup_add @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Obanach_403,axiom,
real_Vector_banach @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Oab__semigroup__mult_404,axiom,
ab_semigroup_mult @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Osemiring__1__cancel_405,axiom,
semiring_1_cancel @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Ocomm__monoid__mult_406,axiom,
comm_monoid_mult @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Oab__semigroup__add_407,axiom,
ab_semigroup_add @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Ocomm__monoid__add_408,axiom,
comm_monoid_add @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Ocomm__semiring__1_409,axiom,
comm_semiring_1 @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Ocomm__semiring__0_410,axiom,
comm_semiring_0 @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Osemigroup__mult_411,axiom,
semigroup_mult @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Osemidom__divide_412,axiom,
semidom_divide @ complex ).
thf(tcon_Complex_Ocomplex___Num_Osemiring__numeral_413,axiom,
semiring_numeral @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Osemigroup__add_414,axiom,
semigroup_add @ complex ).
thf(tcon_Complex_Ocomplex___Fields_Ofield__abs__sgn_415,axiom,
field_abs_sgn @ complex ).
thf(tcon_Complex_Ocomplex___Fields_Odivision__ring_416,axiom,
division_ring @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Ocomm__semiring_417,axiom,
comm_semiring @ complex ).
thf(tcon_Complex_Ocomplex___Nat_Osemiring__char__0_418,axiom,
semiring_char_0 @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Oab__group__add_419,axiom,
ab_group_add @ complex ).
thf(tcon_Complex_Ocomplex___Fields_Ofield__char__0_420,axiom,
field_char_0 @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Ozero__neq__one_421,axiom,
zero_neq_one @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Oidom__abs__sgn_422,axiom,
idom_abs_sgn @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Omonoid__mult_423,axiom,
monoid_mult @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Oidom__divide_424,axiom,
idom_divide @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Ocomm__ring__1_425,axiom,
comm_ring_1 @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Omonoid__add_426,axiom,
monoid_add @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Osemiring__1_427,axiom,
semiring_1 @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Osemiring__0_428,axiom,
semiring_0 @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Ogroup__add_429,axiom,
group_add @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Omult__zero_430,axiom,
mult_zero @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Ocomm__ring_431,axiom,
comm_ring @ complex ).
thf(tcon_Complex_Ocomplex___Num_Oneg__numeral_432,axiom,
neg_numeral @ complex ).
thf(tcon_Complex_Ocomplex___Nat_Oring__char__0_433,axiom,
ring_char_0 @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Osemiring_434,axiom,
semiring @ complex ).
thf(tcon_Complex_Ocomplex___Fields_Oinverse_435,axiom,
inverse @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Ouminus_436,axiom,
uminus @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Oring__1_437,axiom,
ring_1 @ complex ).
thf(tcon_Complex_Ocomplex___Fields_Ofield_438,axiom,
field @ complex ).
thf(tcon_Complex_Ocomplex___Power_Opower_439,axiom,
power @ complex ).
thf(tcon_Complex_Ocomplex___Num_Onumeral_440,axiom,
numeral @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Ozero_441,axiom,
zero @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Oplus_442,axiom,
plus @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Oring_443,axiom,
ring @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Oidom_444,axiom,
idom @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Oone_445,axiom,
one @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Odvd_446,axiom,
dvd @ complex ).
thf(tcon_Extended__Nat_Oenat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_447,axiom,
condit6923001295902523014norder @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_448,axiom,
condit1219197933456340205attice @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Countable__Complete__Lattices_Ocountable__complete__lattice_449,axiom,
counta3822494911875563373attice @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Complete__Lattices_Ocomplete__distrib__lattice_450,axiom,
comple592849572758109894attice @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Ostrict__ordered__ab__semigroup__add_451,axiom,
strict9044650504122735259up_add @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Ostrict__ordered__comm__monoid__add_452,axiom,
strict7427464778891057005id_add @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Ocanonically__ordered__monoid__add_453,axiom,
canoni5634975068530333245id_add @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Lattices_Obounded__semilattice__sup__bot_454,axiom,
bounde4967611905675639751up_bot @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Lattices_Obounded__semilattice__inf__top_455,axiom,
bounde4346867609351753570nf_top @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Complete__Lattices_Ocomplete__linorder,axiom,
comple5582772986160207858norder @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Complete__Lattices_Ocomplete__lattice_456,axiom,
comple6319245703460814977attice @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Olinordered__nonzero__semiring_457,axiom,
linord181362715937106298miring @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Osemiring__no__zero__divisors_458,axiom,
semiri3467727345109120633visors @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Oordered__ab__semigroup__add_459,axiom,
ordere6658533253407199908up_add @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Oordered__comm__monoid__add_460,axiom,
ordere6911136660526730532id_add @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Oordered__comm__semiring_461,axiom,
ordere2520102378445227354miring @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Lattices_Osemilattice__sup_462,axiom,
semilattice_sup @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Lattices_Osemilattice__inf_463,axiom,
semilattice_inf @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Oab__semigroup__mult_464,axiom,
ab_semigroup_mult @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Ocomm__monoid__mult_465,axiom,
comm_monoid_mult @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Oab__semigroup__add_466,axiom,
ab_semigroup_add @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Oordered__semiring_467,axiom,
ordered_semiring @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Ocomm__monoid__add_468,axiom,
comm_monoid_add @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Ocomm__semiring__1_469,axiom,
comm_semiring_1 @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Ocomm__semiring__0_470,axiom,
comm_semiring_0 @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Osemigroup__mult_471,axiom,
semigroup_mult @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Complete__Lattices_OSup_472,axiom,
complete_Sup @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Complete__Lattices_OInf_473,axiom,
complete_Inf @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Num_Osemiring__numeral_474,axiom,
semiring_numeral @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Osemigroup__add_475,axiom,
semigroup_add @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Ozero__less__one_476,axiom,
zero_less_one @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Ocomm__semiring_477,axiom,
comm_semiring @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Orderings_Owellorder_478,axiom,
wellorder @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Orderings_Oorder__top_479,axiom,
order_top @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Orderings_Oorder__bot_480,axiom,
order_bot @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Nat_Osemiring__char__0_481,axiom,
semiring_char_0 @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Ozero__neq__one_482,axiom,
zero_neq_one @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Orderings_Opreorder_483,axiom,
preorder @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Orderings_Olinorder_484,axiom,
linorder @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Omonoid__mult_485,axiom,
monoid_mult @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Omonoid__add_486,axiom,
monoid_add @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Osemiring__1_487,axiom,
semiring_1 @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Osemiring__0_488,axiom,
semiring_0 @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Lattices_Olattice_489,axiom,
lattice @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Omult__zero_490,axiom,
mult_zero @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Orderings_Oorder_491,axiom,
order @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Osemiring_492,axiom,
semiring @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Orderings_Oord_493,axiom,
ord @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Power_Opower_494,axiom,
power @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Num_Onumeral_495,axiom,
numeral @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Ozero_496,axiom,
zero @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Oplus_497,axiom,
plus @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Oone_498,axiom,
one @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Odvd_499,axiom,
dvd @ extended_enat ).
thf(tcon_Product__Type_Oprod___Topological__Spaces_Otopological__space_500,axiom,
! [A17: $tType,A18: $tType] :
( ( ( topolo4958980785337419405_space @ A17 )
& ( topolo4958980785337419405_space @ A18 ) )
=> ( topolo4958980785337419405_space @ ( product_prod @ A17 @ A18 ) ) ) ).
thf(tcon_Product__Type_Oprod___Topological__Spaces_Ot2__space_501,axiom,
! [A17: $tType,A18: $tType] :
( ( ( topological_t2_space @ A17 )
& ( topological_t2_space @ A18 ) )
=> ( topological_t2_space @ ( product_prod @ A17 @ A18 ) ) ) ).
thf(tcon_Product__Type_Oprod___Topological__Spaces_Ot1__space_502,axiom,
! [A17: $tType,A18: $tType] :
( ( ( topological_t1_space @ A17 )
& ( topological_t1_space @ A18 ) )
=> ( topological_t1_space @ ( product_prod @ A17 @ A18 ) ) ) ).
thf(tcon_Product__Type_Oprod___Finite__Set_Ofinite_503,axiom,
! [A17: $tType,A18: $tType] :
( ( ( finite_finite @ A17 )
& ( finite_finite @ A18 ) )
=> ( finite_finite @ ( product_prod @ A17 @ A18 ) ) ) ).
thf(tcon_Product__Type_Oprod___Nat_Osize_504,axiom,
! [A17: $tType,A18: $tType] : ( size @ ( product_prod @ A17 @ A18 ) ) ).
thf(tcon_Product__Type_Ounit___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_505,axiom,
condit6923001295902523014norder @ product_unit ).
thf(tcon_Product__Type_Ounit___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_506,axiom,
condit1219197933456340205attice @ product_unit ).
thf(tcon_Product__Type_Ounit___Countable__Complete__Lattices_Ocountable__complete__lattice_507,axiom,
counta3822494911875563373attice @ product_unit ).
thf(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__distrib__lattice_508,axiom,
comple592849572758109894attice @ product_unit ).
thf(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__boolean__algebra_509,axiom,
comple489889107523837845lgebra @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Obounded__semilattice__sup__bot_510,axiom,
bounde4967611905675639751up_bot @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Obounded__semilattice__inf__top_511,axiom,
bounde4346867609351753570nf_top @ product_unit ).
thf(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__linorder_512,axiom,
comple5582772986160207858norder @ product_unit ).
thf(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__lattice_513,axiom,
comple6319245703460814977attice @ product_unit ).
thf(tcon_Product__Type_Ounit___Boolean__Algebras_Oboolean__algebra_514,axiom,
boolea8198339166811842893lgebra @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Osemilattice__sup_515,axiom,
semilattice_sup @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Osemilattice__inf_516,axiom,
semilattice_inf @ product_unit ).
thf(tcon_Product__Type_Ounit___Complete__Lattices_OSup_517,axiom,
complete_Sup @ product_unit ).
thf(tcon_Product__Type_Ounit___Complete__Lattices_OInf_518,axiom,
complete_Inf @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Owellorder_519,axiom,
wellorder @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Oorder__top_520,axiom,
order_top @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Oorder__bot_521,axiom,
order_bot @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Opreorder_522,axiom,
preorder @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Olinorder_523,axiom,
linorder @ product_unit ).
thf(tcon_Product__Type_Ounit___Finite__Set_Ofinite_524,axiom,
finite_finite @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Olattice_525,axiom,
lattice @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Oorder_526,axiom,
order @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Oord_527,axiom,
ord @ product_unit ).
thf(tcon_Product__Type_Ounit___Groups_Ouminus_528,axiom,
uminus @ product_unit ).
thf(tcon_Code__Numeral_Ointeger___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_529,axiom,
bit_un5681908812861735899ations @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_530,axiom,
semiri1453513574482234551roduct @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__semiring__with__nat_531,axiom,
euclid5411537665997757685th_nat @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__ring__with__nat_532,axiom,
euclid8789492081693882211th_nat @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__monoid__add__imp__le_533,axiom,
ordere1937475149494474687imp_le @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__semiring_534,axiom,
euclid3128863361964157862miring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__semiring__cancel_535,axiom,
euclid4440199948858584721cancel @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Divides_Ounique__euclidean__semiring__numeral_536,axiom,
unique1627219031080169319umeral @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__ring__cancel_537,axiom,
euclid8851590272496341667cancel @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__no__zero__divisors__cancel_538,axiom,
semiri6575147826004484403cancel @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ostrict__ordered__ab__semigroup__add_539,axiom,
strict9044650504122735259up_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__cancel__ab__semigroup__add_540,axiom,
ordere580206878836729694up_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__add__imp__le_541,axiom,
ordere2412721322843649153imp_le @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Bit__Operations_Osemiring__bit__operations_542,axiom,
bit_se359711467146920520ations @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__comm__semiring__strict_543,axiom,
linord2810124833399127020strict @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ostrict__ordered__comm__monoid__add_544,axiom,
strict7427464778891057005id_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__cancel__comm__monoid__add_545,axiom,
ordere8940638589300402666id_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__semiring_546,axiom,
euclid3725896446679973847miring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__1__strict_547,axiom,
linord715952674999750819strict @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Bit__Operations_Oring__bit__operations_548,axiom,
bit_ri3973907225187159222ations @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1__no__zero__divisors_549,axiom,
semiri2026040879449505780visors @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__nonzero__semiring_550,axiom,
linord181362715937106298miring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__ring_551,axiom,
euclid5891614535332579305n_ring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__strict_552,axiom,
linord8928482502909563296strict @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__no__zero__divisors_553,axiom,
semiri3467727345109120633visors @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__add_554,axiom,
ordere6658533253407199908up_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__group__add__abs_555,axiom,
ordere166539214618696060dd_abs @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__comm__monoid__add_556,axiom,
ordere6911136660526730532id_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Olinordered__ab__group__add_557,axiom,
linord5086331880401160121up_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ocancel__ab__semigroup__add_558,axiom,
cancel2418104881723323429up_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oring__1__no__zero__divisors_559,axiom,
ring_15535105094025558882visors @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ocancel__comm__monoid__add_560,axiom,
cancel1802427076303600483id_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__ring__strict_561,axiom,
linord4710134922213307826strict @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__1__cancel_562,axiom,
comm_s4317794764714335236cancel @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Bit__Operations_Osemiring__bits_563,axiom,
bit_semiring_bits @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oordered__comm__semiring_564,axiom,
ordere2520102378445227354miring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__1_565,axiom,
linord6961819062388156250ring_1 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__group__add_566,axiom,
ordered_ab_group_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ocancel__semigroup__add_567,axiom,
cancel_semigroup_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring_568,axiom,
linordered_semiring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oordered__semiring__0_569,axiom,
ordered_semiring_0 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semidom_570,axiom,
linordered_semidom @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oab__semigroup__mult_571,axiom,
ab_semigroup_mult @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1__cancel_572,axiom,
semiring_1_cancel @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oalgebraic__semidom_573,axiom,
algebraic_semidom @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ocomm__monoid__mult_574,axiom,
comm_monoid_mult @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oab__semigroup__add_575,axiom,
ab_semigroup_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oordered__semiring_576,axiom,
ordered_semiring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oordered__ring__abs_577,axiom,
ordered_ring_abs @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Parity_Osemiring__parity_578,axiom,
semiring_parity @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ocomm__monoid__add_579,axiom,
comm_monoid_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__modulo_580,axiom,
semiring_modulo @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__ring_581,axiom,
linordered_ring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__idom_582,axiom,
linordered_idom @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__1_583,axiom,
comm_semiring_1 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__0_584,axiom,
comm_semiring_0 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Osemigroup__mult_585,axiom,
semigroup_mult @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemidom__modulo_586,axiom,
semidom_modulo @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemidom__divide_587,axiom,
semidom_divide @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Num_Osemiring__numeral_588,axiom,
semiring_numeral @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Osemigroup__add_589,axiom,
semigroup_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ozero__less__one_590,axiom,
zero_less_one @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring_591,axiom,
comm_semiring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Nat_Osemiring__char__0_592,axiom,
semiring_char_0 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oab__group__add_593,axiom,
ab_group_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ozero__neq__one_594,axiom,
zero_neq_one @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oordered__ring_595,axiom,
ordered_ring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oidom__abs__sgn_596,axiom,
idom_abs_sgn @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Parity_Oring__parity_597,axiom,
ring_parity @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Orderings_Opreorder_598,axiom,
preorder @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Orderings_Olinorder_599,axiom,
linorder @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Omonoid__mult_600,axiom,
monoid_mult @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oidom__modulo_601,axiom,
idom_modulo @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oidom__divide_602,axiom,
idom_divide @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__ring__1_603,axiom,
comm_ring_1 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Omonoid__add_604,axiom,
monoid_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1_605,axiom,
semiring_1 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__0_606,axiom,
semiring_0 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ogroup__add_607,axiom,
group_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Omult__zero_608,axiom,
mult_zero @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__ring_609,axiom,
comm_ring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Orderings_Oorder_610,axiom,
order @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Num_Oneg__numeral_611,axiom,
neg_numeral @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Nat_Oring__char__0_612,axiom,
ring_char_0 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring_613,axiom,
semiring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Orderings_Oord_614,axiom,
ord @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ouminus_615,axiom,
uminus @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oring__1_616,axiom,
ring_1 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oabs__if_617,axiom,
abs_if @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Power_Opower_618,axiom,
power @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Num_Onumeral_619,axiom,
numeral @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ozero_620,axiom,
zero @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oplus_621,axiom,
plus @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oring_622,axiom,
ring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oidom_623,axiom,
idom @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oone_624,axiom,
one @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Odvd_625,axiom,
dvd @ code_integer ).
thf(tcon_VEBT__Definitions_OVEBT___Nat_Osize_626,axiom,
size @ vEBT_VEBT ).
% Helper facts (4)
thf(help_If_3_1_T,axiom,
! [P3: $o] :
( ( P3 = $true )
| ( P3 = $false ) ) ).
thf(help_If_2_1_T,axiom,
! [A: $tType,X: A,Y2: A] :
( ( if @ A @ $false @ X @ Y2 )
= Y2 ) ).
thf(help_If_1_1_T,axiom,
! [A: $tType,X: A,Y2: A] :
( ( if @ A @ $true @ X @ Y2 )
= X ) ).
thf(help_fChoice_1_1_T,axiom,
! [A: $tType,P3: A > $o] :
( ( P3 @ ( fChoice @ A @ P3 ) )
= ( ? [X6: A] : ( P3 @ X6 ) ) ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( vEBT_VEBT_elim_dead @ summarya @ ( extend4730790105801354508finity @ extended_enat ) )
= summarya ) ).
%------------------------------------------------------------------------------