TPTP Problem File: ITP226^4.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : ITP226^4 : TPTP v8.2.0. Released v8.0.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer problem VEBT_Member 00468_022002
% 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 : 0065_VEBT_Member_00468_022002 [Des22]
% Status : Theorem
% Rating : 1.00 v8.2.0, 0.67 v8.1.0
% Syntax : Number of formulae : 9741 (3160 unt; 730 typ; 0 def)
% Number of atoms : 27680 (9637 equ; 3 cnn)
% Maximal formula atoms : 71 ( 3 avg)
% Number of connectives : 161810 (2041 ~; 298 |;2100 &;145452 @)
% ( 0 <=>;11919 =>; 0 <=; 0 <~>)
% Maximal formula depth : 40 ( 7 avg)
% Number of types : 14 ( 13 usr)
% Number of type conns : 4329 (4329 >; 0 *; 0 +; 0 <<)
% Number of symbols : 721 ( 717 usr; 15 con; 0-8 aty)
% Number of variables : 26861 (2904 ^;22614 !; 687 ?;26861 :)
% ( 656 !>; 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 19:06:37.232
%------------------------------------------------------------------------------
% Could-be-implicit typings (20)
thf(ty_t_VEBT__Definitions_OVEBT,type,
vEBT_VEBT: $tType ).
thf(ty_t_Code__Numeral_Onatural,type,
code_natural: $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_String_Oliteral,type,
literal: $tType ).
thf(ty_t_Sum__Type_Osum,type,
sum_sum: $tType > $tType > $tType ).
thf(ty_t_Option_Ooption,type,
option: $tType > $tType ).
thf(ty_t_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 ).
thf(ty_t_itself,type,
itself: $tType > $tType ).
% Explicit typings (710)
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_Groups_Ominus,type,
minus:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oabs__if,type,
abs_if:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oring__1,type,
ring_1:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ouminus,type,
uminus:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oord,type,
ord:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemidom,type,
semidom:
!>[A: $tType] : $o ).
thf(sy_cl_Fields_Oinverse,type,
inverse:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemiring,type,
semiring:
!>[A: $tType] : $o ).
thf(sy_cl_Nat_Oring__char__0,type,
ring_char_0:
!>[A: $tType] : $o ).
thf(sy_cl_Num_Oneg__numeral,type,
neg_numeral:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oorder,type,
order:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Ocomm__ring,type,
comm_ring:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Omult__zero,type,
mult_zero:
!>[A: $tType] : $o ).
thf(sy_cl_GCD_Osemiring__Gcd,type,
semiring_Gcd:
!>[A: $tType] : $o ).
thf(sy_cl_GCD_Osemiring__gcd,type,
semiring_gcd:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ogroup__add,type,
group_add:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Olattice,type,
lattice:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Ono__bot,type,
no_bot:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Ono__top,type,
no_top:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemiring__0,type,
semiring_0:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemiring__1,type,
semiring_1:
!>[A: $tType] : $o ).
thf(sy_cl_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_Groups_Osemigroup__mult,type,
semigroup_mult:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Odense__order,type,
dense_order:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Ocomm__semiring__0,type,
comm_semiring_0:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Ocomm__semiring__1,type,
comm_semiring_1:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__idom,type,
linordered_idom:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__ring,type,
linordered_ring:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemiring__modulo,type,
semiring_modulo:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocomm__monoid__add,type,
comm_monoid_add:
!>[A: $tType] : $o ).
thf(sy_cl_Parity_Osemiring__parity,type,
semiring_parity:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oordered__ring__abs,type,
ordered_ring_abs:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oordered__semiring,type,
ordered_semiring:
!>[A: $tType] : $o ).
thf(sy_cl_Fields_Olinordered__field,type,
linordered_field:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oab__semigroup__add,type,
ab_semigroup_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocomm__monoid__diff,type,
comm_monoid_diff:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocomm__monoid__mult,type,
comm_monoid_mult:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oalgebraic__semidom,type,
algebraic_semidom:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemiring__1__cancel,type,
semiring_1_cancel:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oab__semigroup__mult,type,
ab_semigroup_mult:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Odistrib__lattice,type,
distrib_lattice:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Osemilattice__inf,type,
semilattice_inf:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Osemilattice__sup,type,
semilattice_sup:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Odense__linorder,type,
dense_linorder:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__semidom,type,
linordered_semidom:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oordered__semiring__0,type,
ordered_semiring_0:
!>[A: $tType] : $o ).
thf(sy_cl_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_Groups_Ocancel__semigroup__add,type,
cancel_semigroup_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__ab__group__add,type,
ordered_ab_group_add:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__semiring__1,type,
linord6961819062388156250ring_1:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Onormalization__semidom,type,
normal8620421768224518004emidom:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oordered__comm__semiring,type,
ordere2520102378445227354miring:
!>[A: $tType] : $o ).
thf(sy_cl_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_Lattices_Obounded__lattice__bot,type,
bounded_lattice_bot:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Obounded__lattice__top,type,
bounded_lattice_top:
!>[A: $tType] : $o ).
thf(sy_cl_Limits_Otopological__group__add,type,
topolo1633459387980952147up_add:
!>[A: $tType] : $o ).
thf(sy_cl_Real__Vector__Spaces_Odist__norm,type,
real_V6936659425649961206t_norm:
!>[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_Rings_Osemiring__no__zero__divisors,type,
semiri3467727345109120633visors:
!>[A: $tType] : $o ).
thf(sy_cl_Boolean__Algebras_Oboolean__algebra,type,
boolea8198339166811842893lgebra:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__semiring__strict,type,
linord8928482502909563296strict:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemidom__divide__unit__factor,type,
semido2269285787275462019factor:
!>[A: $tType] : $o ).
thf(sy_cl_Topological__Spaces_Operfect__space,type,
topolo8386298272705272623_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_Rings_Olinordered__semiring__1__strict,type,
linord715952674999750819strict:
!>[A: $tType] : $o ).
thf(sy_cl_Archimedean__Field_Oarchimedean__field,type,
archim462609752435547400_field:
!>[A: $tType] : $o ).
thf(sy_cl_Complete__Lattices_Ocomplete__linorder,type,
comple5582772986160207858norder:
!>[A: $tType] : $o ).
thf(sy_cl_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_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_Rings_Onormalization__semidom__multiplicative,type,
normal6328177297339901930cative:
!>[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__Cardinal__Order__Relation_OnatLeq,type,
bNF_Ca8665028551170535155natLeq: set @ ( product_prod @ nat @ nat ) ).
thf(sy_c_BNF__Cardinal__Order__Relation_OnatLess,type,
bNF_Ca8459412986667044542atLess: set @ ( product_prod @ nat @ nat ) ).
thf(sy_c_BNF__Def_Orel__fun,type,
bNF_rel_fun:
!>[A: $tType,C: $tType,B: $tType,D: $tType] : ( ( A > C > $o ) > ( B > D > $o ) > ( A > B ) > ( C > D ) > $o ) ).
thf(sy_c_BNF__Wellorder__Constructions_Obsqr,type,
bNF_Wellorder_bsqr:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( product_prod @ A @ A ) @ ( product_prod @ A @ A ) ) ) ) ).
thf(sy_c_BNF__Wellorder__Constructions_Odir__image,type,
bNF_We2720479622203943262_image:
!>[A: $tType,A2: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( A > A2 ) > ( set @ ( product_prod @ A2 @ A2 ) ) ) ).
thf(sy_c_BNF__Wellorder__Relation_Owo__rel,type,
bNF_Wellorder_wo_rel:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_BNF__Wellorder__Relation_Owo__rel_Omax2,type,
bNF_We1388413361240627857o_max2:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > A > A > A ) ).
thf(sy_c_Basic__BNF__LFPs_Oprod_Osize__prod,type,
basic_BNF_size_prod:
!>[A: $tType,B: $tType] : ( ( A > nat ) > ( B > nat ) > ( product_prod @ A @ B ) > nat ) ).
thf(sy_c_Basic__BNFs_Orel__prod,type,
basic_rel_prod:
!>[A: $tType,B: $tType,C: $tType,D: $tType] : ( ( A > B > $o ) > ( C > D > $o ) > ( product_prod @ A @ C ) > ( product_prod @ B @ D ) > $o ) ).
thf(sy_c_Binomial_Obinomial,type,
binomial: nat > nat > nat ).
thf(sy_c_Binomial_Ogbinomial,type,
gbinomial:
!>[A: $tType] : ( A > nat > A ) ).
thf(sy_c_Bit__Operations_Oand__int__rel,type,
bit_and_int_rel: ( product_prod @ int @ int ) > ( product_prod @ int @ int ) > $o ).
thf(sy_c_Bit__Operations_Oand__not__num,type,
bit_and_not_num: num > num > ( option @ num ) ).
thf(sy_c_Bit__Operations_Oand__not__num__rel,type,
bit_and_not_num_rel: ( product_prod @ num @ num ) > ( product_prod @ num @ num ) > $o ).
thf(sy_c_Bit__Operations_Oconcat__bit,type,
bit_concat_bit: nat > int > int > int ).
thf(sy_c_Bit__Operations_Oor__not__num__neg,type,
bit_or_not_num_neg: num > num > num ).
thf(sy_c_Bit__Operations_Oring__bit__operations__class_Onot,type,
bit_ri4277139882892585799ns_not:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Bit__Operations_Oring__bit__operations__class_Osigned__take__bit,type,
bit_ri4674362597316999326ke_bit:
!>[A: $tType] : ( nat > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oand,type,
bit_se5824344872417868541ns_and:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Odrop__bit,type,
bit_se4197421643247451524op_bit:
!>[A: $tType] : ( nat > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oflip__bit,type,
bit_se8732182000553998342ip_bit:
!>[A: $tType] : ( nat > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Omask,type,
bit_se2239418461657761734s_mask:
!>[A: $tType] : ( nat > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oor,type,
bit_se1065995026697491101ons_or:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Opush__bit,type,
bit_se4730199178511100633sh_bit:
!>[A: $tType] : ( nat > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oset__bit,type,
bit_se5668285175392031749et_bit:
!>[A: $tType] : ( nat > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Otake__bit,type,
bit_se2584673776208193580ke_bit:
!>[A: $tType] : ( nat > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Ounset__bit,type,
bit_se2638667681897837118et_bit:
!>[A: $tType] : ( nat > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oxor,type,
bit_se5824344971392196577ns_xor:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bits__class_Obit,type,
bit_se5641148757651400278ts_bit:
!>[A: $tType] : ( A > nat > $o ) ).
thf(sy_c_Bit__Operations_Osemiring__bits__class_Opossible__bit,type,
bit_se6407376104438227557le_bit:
!>[A: $tType] : ( ( itself @ A ) > nat > $o ) ).
thf(sy_c_Bit__Operations_Otake__bit__num,type,
bit_take_bit_num: nat > num > ( option @ num ) ).
thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oand__num,type,
bit_un7362597486090784418nd_num: num > num > ( option @ num ) ).
thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oand__num__rel,type,
bit_un4731106466462545111um_rel: ( product_prod @ num @ num ) > ( product_prod @ num @ num ) > $o ).
thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oxor__num,type,
bit_un2480387367778600638or_num: num > num > ( option @ num ) ).
thf(sy_c_Bit__Operations_Ounique__euclidean__semiring__with__bit__operations__class_Oxor__num__rel,type,
bit_un2901131394128224187um_rel: ( product_prod @ num @ num ) > ( product_prod @ num @ num ) > $o ).
thf(sy_c_Boolean__Algebras_Oabstract__boolean__algebra,type,
boolea2506097494486148201lgebra:
!>[A: $tType] : ( ( A > A > A ) > ( A > A > A ) > ( A > A ) > A > A > $o ) ).
thf(sy_c_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_OSuc,type,
code_Suc: code_natural > code_natural ).
thf(sy_c_Code__Numeral_Obit__cut__integer,type,
code_bit_cut_integer: code_integer > ( product_prod @ code_integer @ $o ) ).
thf(sy_c_Code__Numeral_Odivmod__abs,type,
code_divmod_abs: code_integer > code_integer > ( product_prod @ code_integer @ code_integer ) ).
thf(sy_c_Code__Numeral_Odivmod__integer,type,
code_divmod_integer: code_integer > code_integer > ( product_prod @ code_integer @ code_integer ) ).
thf(sy_c_Code__Numeral_Odup,type,
code_dup: code_integer > code_integer ).
thf(sy_c_Code__Numeral_Ointeger_Oint__of__integer,type,
code_int_of_integer: code_integer > int ).
thf(sy_c_Code__Numeral_Ointeger_Ointeger__of__int,type,
code_integer_of_int: int > code_integer ).
thf(sy_c_Code__Numeral_Ointeger__of__nat,type,
code_integer_of_nat: nat > code_integer ).
thf(sy_c_Code__Numeral_Ointeger__of__natural,type,
code_i5400310926305786745atural: code_natural > code_integer ).
thf(sy_c_Code__Numeral_Ointeger__of__num,type,
code_integer_of_num: num > code_integer ).
thf(sy_c_Code__Numeral_Onat__of__integer,type,
code_nat_of_integer: code_integer > nat ).
thf(sy_c_Code__Numeral_Onatural_Onat__of__natural,type,
code_nat_of_natural: code_natural > nat ).
thf(sy_c_Code__Numeral_Onatural_Onatural__of__nat,type,
code_natural_of_nat: nat > code_natural ).
thf(sy_c_Code__Numeral_Onegative,type,
code_negative: num > code_integer ).
thf(sy_c_Code__Numeral_Onum__of__integer,type,
code_num_of_integer: code_integer > num ).
thf(sy_c_Code__Numeral_Opcr__integer,type,
code_pcr_integer: int > code_integer > $o ).
thf(sy_c_Code__Numeral_Opcr__natural,type,
code_pcr_natural: nat > code_natural > $o ).
thf(sy_c_Code__Numeral_Osub,type,
code_sub: num > num > code_integer ).
thf(sy_c_Code__Target__Int_Onegative,type,
code_Target_negative: num > int ).
thf(sy_c_Code__Target__Nat_Oint__of__nat,type,
code_T6385005292777649522of_nat: nat > int ).
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_Complex_Orcis,type,
rcis: real > real > 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_Conditionally__Complete__Lattices_Opreordering__bdd,type,
condit622319405099724424ng_bdd:
!>[A: $tType] : ( ( A > A > $o ) > ( A > A > $o ) > $o ) ).
thf(sy_c_Countable__Set_Ocountable,type,
countable_countable:
!>[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_Euclidean__Division_Oeuclidean__semiring__class_Oeuclidean__size,type,
euclid6346220572633701492n_size:
!>[A: $tType] : ( A > nat ) ).
thf(sy_c_Euclidean__Division_Ounique__euclidean__semiring__class_Odivision__segment,type,
euclid7384307370059645450egment:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Extended__Nat_OeSuc,type,
extended_eSuc: extended_enat > extended_enat ).
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_Oinfinity__class_Oinfinity,type,
extend4730790105801354508finity:
!>[A: $tType] : A ).
thf(sy_c_Factorial_Ocomm__semiring__1__class_Opochhammer,type,
comm_s3205402744901411588hammer:
!>[A: $tType] : ( A > nat > A ) ).
thf(sy_c_Factorial_Osemiring__char__0__class_Ofact,type,
semiring_char_0_fact:
!>[A: $tType] : ( nat > A ) ).
thf(sy_c_Fields_Oinverse__class_Oinverse,type,
inverse_inverse:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Filter_Oat__bot,type,
at_bot:
!>[A: $tType] : ( filter @ A ) ).
thf(sy_c_Filter_Oat__top,type,
at_top:
!>[A: $tType] : ( filter @ A ) ).
thf(sy_c_Filter_Oeventually,type,
eventually:
!>[A: $tType] : ( ( A > $o ) > ( filter @ A ) > $o ) ).
thf(sy_c_Filter_Ofiltercomap,type,
filtercomap:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( filter @ B ) > ( filter @ A ) ) ).
thf(sy_c_Filter_Ofilterlim,type,
filterlim:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( filter @ B ) > ( filter @ A ) > $o ) ).
thf(sy_c_Filter_Ofiltermap,type,
filtermap:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( filter @ A ) > ( filter @ B ) ) ).
thf(sy_c_Filter_Oprincipal,type,
principal:
!>[A: $tType] : ( ( set @ A ) > ( filter @ A ) ) ).
thf(sy_c_Finite__Set_Ocard,type,
finite_card:
!>[B: $tType] : ( ( set @ B ) > nat ) ).
thf(sy_c_Finite__Set_Ocomp__fun__commute__on,type,
finite4664212375090638736ute_on:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B > B ) > $o ) ).
thf(sy_c_Finite__Set_Ofinite,type,
finite_finite:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Finite__Set_Ofold,type,
finite_fold:
!>[A: $tType,B: $tType] : ( ( A > B > B ) > B > ( set @ A ) > B ) ).
thf(sy_c_Finite__Set_Ofold__graph,type,
finite_fold_graph:
!>[A: $tType,B: $tType] : ( ( A > B > B ) > B > ( set @ A ) > B > $o ) ).
thf(sy_c_Finite__Set_Ofolding__on,type,
finite_folding_on:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B > B ) > $o ) ).
thf(sy_c_Finite__Set_Ofolding__on_OF,type,
finite_folding_F:
!>[A: $tType,B: $tType] : ( ( A > B > B ) > B > ( set @ A ) > B ) ).
thf(sy_c_Fun_Obij__betw,type,
bij_betw:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > ( set @ B ) > $o ) ).
thf(sy_c_Fun_Ocomp,type,
comp:
!>[B: $tType,C: $tType,A: $tType] : ( ( B > C ) > ( A > B ) > A > C ) ).
thf(sy_c_Fun_Ofun__upd,type,
fun_upd:
!>[A: $tType,B: $tType] : ( ( A > B ) > A > B > A > B ) ).
thf(sy_c_Fun_Oid,type,
id:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Fun_Oinj__on,type,
inj_on:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > $o ) ).
thf(sy_c_Fun_Omap__fun,type,
map_fun:
!>[C: $tType,A: $tType,B: $tType,D: $tType] : ( ( C > A ) > ( B > D ) > ( A > B ) > C > D ) ).
thf(sy_c_Fun_Ooverride__on,type,
override_on:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( A > B ) > ( set @ A ) > A > B ) ).
thf(sy_c_Fun_Ostrict__mono__on,type,
strict_mono_on:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > $o ) ).
thf(sy_c_Fun_Othe__inv__into,type,
the_inv_into:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B ) > B > A ) ).
thf(sy_c_Fun__Def_Ois__measure,type,
fun_is_measure:
!>[A: $tType] : ( ( A > nat ) > $o ) ).
thf(sy_c_Fun__Def_Opair__leq,type,
fun_pair_leq: set @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) ).
thf(sy_c_Fun__Def_Opair__less,type,
fun_pair_less: set @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) ).
thf(sy_c_GCD_OGcd__class_OGcd,type,
gcd_Gcd:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_GCD_OGcd__class_OLcm,type,
gcd_Lcm:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_GCD_Obezw,type,
bezw: nat > nat > ( product_prod @ int @ int ) ).
thf(sy_c_GCD_Obezw__rel,type,
bezw_rel: ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) > $o ).
thf(sy_c_GCD_Obounded__quasi__semilattice,type,
bounde8507323023520639062attice:
!>[A: $tType] : ( ( A > A > A ) > A > A > ( A > A ) > $o ) ).
thf(sy_c_GCD_Obounded__quasi__semilattice__set,type,
bounde6485984586167503788ce_set:
!>[A: $tType] : ( ( A > A > A ) > A > A > ( A > A ) > $o ) ).
thf(sy_c_GCD_Obounded__quasi__semilattice__set_OF,type,
bounde2362111253966948842tice_F:
!>[A: $tType] : ( ( A > A > A ) > A > A > ( set @ A ) > A ) ).
thf(sy_c_GCD_Ogcd__class_Ogcd,type,
gcd_gcd:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_GCD_Ogcd__class_Olcm,type,
gcd_lcm:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_GCD_Ogcd__nat__rel,type,
gcd_nat_rel: ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) > $o ).
thf(sy_c_GCD_Osemiring__1__class_Osemiring__char,type,
semiri4206861660011772517g_char:
!>[A: $tType] : ( ( itself @ A ) > nat ) ).
thf(sy_c_GCD_Osemiring__gcd__class_OGcd__fin,type,
semiring_gcd_Gcd_fin:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_GCD_Osemiring__gcd__class_OLcm__fin,type,
semiring_gcd_Lcm_fin:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_Groups_Oabs__class_Oabs,type,
abs_abs:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Groups_Ocomm__monoid,type,
comm_monoid:
!>[A: $tType] : ( ( A > A > A ) > A > $o ) ).
thf(sy_c_Groups_Ogroup,type,
group:
!>[A: $tType] : ( ( A > A > A ) > A > ( A > A ) > $o ) ).
thf(sy_c_Groups_Ogroup__axioms,type,
group_axioms:
!>[A: $tType] : ( ( A > A > A ) > A > ( A > A ) > $o ) ).
thf(sy_c_Groups_Ominus__class_Ominus,type,
minus_minus:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Groups_Omonoid,type,
monoid:
!>[A: $tType] : ( ( A > A > A ) > A > $o ) ).
thf(sy_c_Groups_Omonoid__axioms,type,
monoid_axioms:
!>[A: $tType] : ( ( A > A > A ) > A > $o ) ).
thf(sy_c_Groups_Oone__class_Oone,type,
one_one:
!>[A: $tType] : A ).
thf(sy_c_Groups_Oplus__class_Oplus,type,
plus_plus:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Groups_Osemigroup,type,
semigroup:
!>[A: $tType] : ( ( A > A > A ) > $o ) ).
thf(sy_c_Groups_Osgn__class_Osgn,type,
sgn_sgn:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Groups_Otimes__class_Otimes,type,
times_times:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Groups_Ouminus__class_Ouminus,type,
uminus_uminus:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Groups_Ozero__class_Ozero,type,
zero_zero:
!>[A: $tType] : A ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum,type,
groups7311177749621191930dd_sum:
!>[B: $tType,A: $tType] : ( ( B > A ) > ( set @ B ) > A ) ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_H,type,
groups1027152243600224163dd_sum:
!>[C: $tType,A: $tType] : ( ( C > A ) > ( set @ C ) > A ) ).
thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod,type,
groups7121269368397514597t_prod:
!>[B: $tType,A: $tType] : ( ( B > A ) > ( set @ B ) > A ) ).
thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_H,type,
groups1962203154675924110t_prod:
!>[C: $tType,A: $tType] : ( ( C > A ) > ( set @ C ) > A ) ).
thf(sy_c_Groups__Big_Ocomm__monoid__set_OG,type,
groups_comm_monoid_G:
!>[A: $tType,B: $tType] : ( ( A > A > A ) > A > ( B > A ) > ( set @ B ) > A ) ).
thf(sy_c_Groups__List_Ocomm__monoid__list,type,
groups1828464146339083142d_list:
!>[A: $tType] : ( ( A > A > A ) > A > $o ) ).
thf(sy_c_Groups__List_Ocomm__monoid__list__set,type,
groups4802862169904069756st_set:
!>[A: $tType] : ( ( A > A > A ) > A > $o ) ).
thf(sy_c_Groups__List_Ocomm__semiring__0__class_Ohorner__sum,type,
groups4207007520872428315er_sum:
!>[B: $tType,A: $tType] : ( ( B > A ) > A > ( list @ B ) > A ) ).
thf(sy_c_Groups__List_Omonoid__add__class_Osum__list,type,
groups8242544230860333062m_list:
!>[A: $tType] : ( ( list @ A ) > A ) ).
thf(sy_c_Groups__List_Omonoid__list_OF,type,
groups_monoid_F:
!>[A: $tType] : ( ( A > A > A ) > A > ( list @ A ) > A ) ).
thf(sy_c_Groups__List_Omonoid__mult__class_Oprod__list,type,
groups5270119922927024881d_list:
!>[A: $tType] : ( ( list @ A ) > A ) ).
thf(sy_c_HOL_ONO__MATCH,type,
nO_MATCH:
!>[A: $tType,B: $tType] : ( A > B > $o ) ).
thf(sy_c_HOL_OThe,type,
the:
!>[A: $tType] : ( ( A > $o ) > A ) ).
thf(sy_c_HOL_OUniq,type,
uniq:
!>[A: $tType] : ( ( A > $o ) > $o ) ).
thf(sy_c_Hilbert__Choice_Obijection,type,
hilbert_bijection:
!>[A: $tType] : ( ( A > A ) > $o ) ).
thf(sy_c_Hilbert__Choice_Oinv__into,type,
hilbert_inv_into:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B ) > B > A ) ).
thf(sy_c_If,type,
if:
!>[A: $tType] : ( $o > A > A > A ) ).
thf(sy_c_Inductive_Ocomplete__lattice__class_Ogfp,type,
complete_lattice_gfp:
!>[A: $tType] : ( ( A > A ) > A ) ).
thf(sy_c_Inductive_Ocomplete__lattice__class_Olfp,type,
complete_lattice_lfp:
!>[A: $tType] : ( ( A > A ) > A ) ).
thf(sy_c_Infinite__Set_Owellorder__class_Oenumerate,type,
infini527867602293511546merate:
!>[A: $tType] : ( ( set @ A ) > nat > A ) ).
thf(sy_c_Int_OAbs__Integ,type,
abs_Integ: ( product_prod @ nat @ nat ) > int ).
thf(sy_c_Int_ONeg,type,
neg: num > int ).
thf(sy_c_Int_OPos,type,
pos: num > int ).
thf(sy_c_Int_ORep__Integ,type,
rep_Integ: int > ( product_prod @ nat @ nat ) ).
thf(sy_c_Int_Ocr__int,type,
cr_int: ( product_prod @ nat @ nat ) > int > $o ).
thf(sy_c_Int_Odup,type,
dup: int > int ).
thf(sy_c_Int_Oint__ge__less__than,type,
int_ge_less_than: int > ( set @ ( product_prod @ int @ int ) ) ).
thf(sy_c_Int_Oint__ge__less__than2,type,
int_ge_less_than2: int > ( set @ ( product_prod @ int @ int ) ) ).
thf(sy_c_Int_Ointrel,type,
intrel: ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) > $o ).
thf(sy_c_Int_Onat,type,
nat2: int > nat ).
thf(sy_c_Int_Opcr__int,type,
pcr_int: ( product_prod @ nat @ nat ) > int > $o ).
thf(sy_c_Int_Opower__int,type,
power_int:
!>[A: $tType] : ( A > int > A ) ).
thf(sy_c_Int_Oring__1__class_OInts,type,
ring_1_Ints:
!>[A: $tType] : ( set @ A ) ).
thf(sy_c_Int_Oring__1__class_Oof__int,type,
ring_1_of_int:
!>[A: $tType] : ( int > A ) ).
thf(sy_c_Int_Osub,type,
sub: num > num > int ).
thf(sy_c_Lattices_Oinf__class_Oinf,type,
inf_inf:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Lattices_Osemilattice__neutr,type,
semilattice_neutr:
!>[A: $tType] : ( ( A > A > A ) > A > $o ) ).
thf(sy_c_Lattices_Osemilattice__neutr__order,type,
semila1105856199041335345_order:
!>[A: $tType] : ( ( A > A > A ) > A > ( A > A > $o ) > ( A > A > $o ) > $o ) ).
thf(sy_c_Lattices_Osemilattice__order,type,
semilattice_order:
!>[A: $tType] : ( ( A > A > A ) > ( A > A > $o ) > ( A > A > $o ) > $o ) ).
thf(sy_c_Lattices_Osup__class_Osup,type,
sup_sup:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Lattices__Big_Olinorder__class_OMax,type,
lattic643756798349783984er_Max:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_Lattices__Big_Olinorder__class_OMin,type,
lattic643756798350308766er_Min:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_Lattices__Big_Oord__class_Oarg__max,type,
lattices_ord_arg_max:
!>[B: $tType,A: $tType] : ( ( B > A ) > ( B > $o ) > B ) ).
thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on,type,
lattic7623131987881927897min_on:
!>[B: $tType,A: $tType] : ( ( B > A ) > ( set @ B ) > B ) ).
thf(sy_c_Lattices__Big_Oord__class_Ois__arg__max,type,
lattic501386751176901750rg_max:
!>[B: $tType,A: $tType] : ( ( B > A ) > ( B > $o ) > B > $o ) ).
thf(sy_c_Lattices__Big_Osemilattice__inf__class_OInf__fin,type,
lattic7752659483105999362nf_fin:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_Lattices__Big_Osemilattice__neutr__set,type,
lattic5652469242046573047tr_set:
!>[A: $tType] : ( ( A > A > A ) > A > $o ) ).
thf(sy_c_Lattices__Big_Osemilattice__neutr__set_OF,type,
lattic5214292709420241887eutr_F:
!>[A: $tType] : ( ( A > A > A ) > A > ( set @ A ) > A ) ).
thf(sy_c_Lattices__Big_Osemilattice__order__set,type,
lattic4895041142388067077er_set:
!>[A: $tType] : ( ( A > A > A ) > ( A > A > $o ) > ( A > A > $o ) > $o ) ).
thf(sy_c_Lattices__Big_Osemilattice__set,type,
lattic149705377957585745ce_set:
!>[A: $tType] : ( ( A > A > A ) > $o ) ).
thf(sy_c_Lattices__Big_Osemilattice__set_OF,type,
lattic1715443433743089157tice_F:
!>[A: $tType] : ( ( A > A > A ) > ( set @ A ) > A ) ).
thf(sy_c_Lattices__Big_Osemilattice__sup__class_OSup__fin,type,
lattic5882676163264333800up_fin:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_Lifting_OQuotient,type,
quotient:
!>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( A > B ) > ( B > A ) > ( A > B > $o ) > $o ) ).
thf(sy_c_Limits_OBfun,type,
bfun:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( filter @ A ) > $o ) ).
thf(sy_c_Limits_OZfun,type,
zfun:
!>[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_Obutlast,type,
butlast:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Oconcat,type,
concat:
!>[A: $tType] : ( ( list @ ( list @ A ) ) > ( list @ A ) ) ).
thf(sy_c_List_Ocoset,type,
coset:
!>[A: $tType] : ( ( list @ A ) > ( set @ A ) ) ).
thf(sy_c_List_Ocount__list,type,
count_list:
!>[A: $tType] : ( ( list @ A ) > A > nat ) ).
thf(sy_c_List_Odistinct,type,
distinct:
!>[A: $tType] : ( ( list @ A ) > $o ) ).
thf(sy_c_List_Odrop,type,
drop:
!>[A: $tType] : ( nat > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_OdropWhile,type,
dropWhile:
!>[A: $tType] : ( ( A > $o ) > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Oenumerate,type,
enumerate:
!>[A: $tType] : ( nat > ( list @ A ) > ( list @ ( product_prod @ nat @ A ) ) ) ).
thf(sy_c_List_Oextract,type,
extract:
!>[A: $tType] : ( ( A > $o ) > ( list @ A ) > ( option @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) ) ) ).
thf(sy_c_List_Ofilter,type,
filter2:
!>[A: $tType] : ( ( A > $o ) > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Ofind,type,
find:
!>[A: $tType] : ( ( A > $o ) > ( list @ A ) > ( option @ A ) ) ).
thf(sy_c_List_Ofold,type,
fold:
!>[A: $tType,B: $tType] : ( ( A > B > B ) > ( list @ A ) > B > B ) ).
thf(sy_c_List_Ofolding__insort__key,type,
folding_insort_key:
!>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( A > A > $o ) > ( set @ B ) > ( B > A ) > $o ) ).
thf(sy_c_List_Ofoldr,type,
foldr:
!>[A: $tType,B: $tType] : ( ( A > B > B ) > ( list @ A ) > B > B ) ).
thf(sy_c_List_Ogen__length,type,
gen_length:
!>[A: $tType] : ( nat > ( list @ A ) > nat ) ).
thf(sy_c_List_Olast,type,
last:
!>[A: $tType] : ( ( list @ A ) > A ) ).
thf(sy_c_List_Olenlex,type,
lenlex:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ) ).
thf(sy_c_List_Olex,type,
lex:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ) ).
thf(sy_c_List_Olexn,type,
lexn:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > nat > ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ) ).
thf(sy_c_List_Olexord,type,
lexord:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ) ).
thf(sy_c_List_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_Olist__all,type,
list_all:
!>[A: $tType] : ( ( A > $o ) > ( list @ A ) > $o ) ).
thf(sy_c_List_Olist_Olist__all2,type,
list_all2:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > ( list @ A ) > ( list @ B ) > $o ) ).
thf(sy_c_List_Olist_Omap,type,
map:
!>[A: $tType,Aa: $tType] : ( ( A > Aa ) > ( list @ A ) > ( list @ Aa ) ) ).
thf(sy_c_List_Olist_Oset,type,
set2:
!>[A: $tType] : ( ( list @ A ) > ( set @ A ) ) ).
thf(sy_c_List_Olist_Osize__list,type,
size_list:
!>[A: $tType] : ( ( A > nat ) > ( list @ A ) > nat ) ).
thf(sy_c_List_Olist_Otl,type,
tl:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Olist__ex,type,
list_ex:
!>[A: $tType] : ( ( A > $o ) > ( list @ A ) > $o ) ).
thf(sy_c_List_Olist__update,type,
list_update:
!>[A: $tType] : ( ( list @ A ) > nat > A > ( list @ A ) ) ).
thf(sy_c_List_Olistrel,type,
listrel:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( set @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) ) ).
thf(sy_c_List_Olistrel1,type,
listrel1:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ) ).
thf(sy_c_List_Omap__filter,type,
map_filter:
!>[A: $tType,B: $tType] : ( ( A > ( option @ B ) ) > ( list @ A ) > ( list @ B ) ) ).
thf(sy_c_List_Omeasures,type,
measures:
!>[A: $tType] : ( ( list @ ( A > nat ) ) > ( set @ ( product_prod @ A @ A ) ) ) ).
thf(sy_c_List_On__lists,type,
n_lists:
!>[A: $tType] : ( nat > ( list @ A ) > ( list @ ( list @ A ) ) ) ).
thf(sy_c_List_Onth,type,
nth:
!>[A: $tType] : ( ( list @ A ) > nat > A ) ).
thf(sy_c_List_Onths,type,
nths:
!>[A: $tType] : ( ( list @ A ) > ( set @ nat ) > ( list @ A ) ) ).
thf(sy_c_List_Oord__class_Olexordp,type,
ord_lexordp:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) > $o ) ).
thf(sy_c_List_Oord__class_Olexordp__eq,type,
ord_lexordp_eq:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) > $o ) ).
thf(sy_c_List_Oproduct,type,
product:
!>[A: $tType,B: $tType] : ( ( list @ A ) > ( list @ B ) > ( list @ ( product_prod @ A @ B ) ) ) ).
thf(sy_c_List_Oproduct__lists,type,
product_lists:
!>[A: $tType] : ( ( list @ ( list @ A ) ) > ( list @ ( list @ A ) ) ) ).
thf(sy_c_List_Oremdups,type,
remdups:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Oremdups__adj,type,
remdups_adj:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Oremove1,type,
remove1:
!>[A: $tType] : ( A > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_OremoveAll,type,
removeAll:
!>[A: $tType] : ( A > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Oreplicate,type,
replicate:
!>[A: $tType] : ( nat > A > ( list @ A ) ) ).
thf(sy_c_List_Orev,type,
rev:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Orotate,type,
rotate:
!>[A: $tType] : ( nat > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Orotate1,type,
rotate1:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Oshuffles,type,
shuffles:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) > ( set @ ( list @ A ) ) ) ).
thf(sy_c_List_Osorted__wrt,type,
sorted_wrt:
!>[A: $tType] : ( ( A > A > $o ) > ( list @ A ) > $o ) ).
thf(sy_c_List_Osplice,type,
splice:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Osubseqs,type,
subseqs:
!>[A: $tType] : ( ( list @ A ) > ( list @ ( list @ A ) ) ) ).
thf(sy_c_List_Otake,type,
take:
!>[A: $tType] : ( nat > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_OtakeWhile,type,
takeWhile:
!>[A: $tType] : ( ( A > $o ) > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Othose,type,
those:
!>[A: $tType] : ( ( list @ ( option @ A ) ) > ( option @ ( list @ A ) ) ) ).
thf(sy_c_List_Otranspose,type,
transpose:
!>[A: $tType] : ( ( list @ ( list @ A ) ) > ( list @ ( list @ A ) ) ) ).
thf(sy_c_List_Oupt,type,
upt: nat > nat > ( list @ nat ) ).
thf(sy_c_List_Oupto,type,
upto: int > int > ( list @ int ) ).
thf(sy_c_List_Oupto__aux,type,
upto_aux: int > int > ( list @ int ) > ( list @ int ) ).
thf(sy_c_List_Oupto__rel,type,
upto_rel: ( product_prod @ int @ int ) > ( product_prod @ int @ int ) > $o ).
thf(sy_c_List_Ozip,type,
zip:
!>[A: $tType,B: $tType] : ( ( list @ A ) > ( list @ B ) > ( list @ ( product_prod @ A @ B ) ) ) ).
thf(sy_c_Map_Odom,type,
dom:
!>[A: $tType,B: $tType] : ( ( A > ( option @ B ) ) > ( set @ A ) ) ).
thf(sy_c_Map_Ograph,type,
graph:
!>[A: $tType,B: $tType] : ( ( A > ( option @ B ) ) > ( set @ ( product_prod @ A @ B ) ) ) ).
thf(sy_c_Map_Omap__comp,type,
map_comp:
!>[B: $tType,C: $tType,A: $tType] : ( ( B > ( option @ C ) ) > ( A > ( option @ B ) ) > A > ( option @ C ) ) ).
thf(sy_c_Map_Omap__le,type,
map_le:
!>[A: $tType,B: $tType] : ( ( A > ( option @ B ) ) > ( A > ( option @ B ) ) > $o ) ).
thf(sy_c_Map_Omap__of,type,
map_of:
!>[A: $tType,B: $tType] : ( ( list @ ( product_prod @ A @ B ) ) > A > ( option @ B ) ) ).
thf(sy_c_Map_Omap__upds,type,
map_upds:
!>[A: $tType,B: $tType] : ( ( A > ( option @ B ) ) > ( list @ A ) > ( list @ B ) > A > ( option @ B ) ) ).
thf(sy_c_Map_Oran,type,
ran:
!>[A: $tType,B: $tType] : ( ( A > ( option @ B ) ) > ( set @ B ) ) ).
thf(sy_c_Map_Orestrict__map,type,
restrict_map:
!>[A: $tType,B: $tType] : ( ( A > ( option @ B ) ) > ( set @ A ) > A > ( option @ B ) ) ).
thf(sy_c_Nat_OSuc,type,
suc: nat > nat ).
thf(sy_c_Nat_Ocompow,type,
compow:
!>[A: $tType] : ( nat > A > A ) ).
thf(sy_c_Nat_Ofunpow,type,
funpow:
!>[A: $tType] : ( nat > ( A > A ) > A > A ) ).
thf(sy_c_Nat_Onat_Ocase__nat,type,
case_nat:
!>[A: $tType] : ( A > ( nat > A ) > nat > A ) ).
thf(sy_c_Nat_Onat_Opred,type,
pred: nat > nat ).
thf(sy_c_Nat_Oold_Onat_Orec__nat,type,
rec_nat:
!>[T: $tType] : ( T > ( nat > T > T ) > nat > T ) ).
thf(sy_c_Nat_Oold_Onat_Orec__set__nat,type,
rec_set_nat:
!>[T: $tType] : ( T > ( nat > T > T ) > nat > T > $o ) ).
thf(sy_c_Nat_Osemiring__1__class_ONats,type,
semiring_1_Nats:
!>[A: $tType] : ( set @ A ) ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat,type,
semiring_1_of_nat:
!>[A: $tType] : ( nat > A ) ).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat__aux,type,
semiri8178284476397505188at_aux:
!>[A: $tType] : ( ( A > A ) > nat > A > A ) ).
thf(sy_c_Nat_Osize__class_Osize,type,
size_size:
!>[A: $tType] : ( A > nat ) ).
thf(sy_c_Nat__Bijection_Oint__decode,type,
nat_int_decode: nat > int ).
thf(sy_c_Nat__Bijection_Oint__encode,type,
nat_int_encode: int > nat ).
thf(sy_c_Nat__Bijection_Olist__decode,type,
nat_list_decode: nat > ( list @ nat ) ).
thf(sy_c_Nat__Bijection_Olist__decode__rel,type,
nat_list_decode_rel: nat > nat > $o ).
thf(sy_c_Nat__Bijection_Olist__encode,type,
nat_list_encode: ( list @ nat ) > nat ).
thf(sy_c_Nat__Bijection_Olist__encode__rel,type,
nat_list_encode_rel: ( list @ nat ) > ( list @ nat ) > $o ).
thf(sy_c_Nat__Bijection_Oprod__decode,type,
nat_prod_decode: nat > ( product_prod @ nat @ nat ) ).
thf(sy_c_Nat__Bijection_Oprod__decode__aux,type,
nat_prod_decode_aux: nat > nat > ( product_prod @ nat @ nat ) ).
thf(sy_c_Nat__Bijection_Oprod__decode__aux__rel,type,
nat_pr5047031295181774490ux_rel: ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) > $o ).
thf(sy_c_Nat__Bijection_Oprod__encode,type,
nat_prod_encode: ( product_prod @ nat @ nat ) > nat ).
thf(sy_c_Nat__Bijection_Oset__decode,type,
nat_set_decode: nat > ( set @ nat ) ).
thf(sy_c_Nat__Bijection_Oset__encode,type,
nat_set_encode: ( set @ nat ) > nat ).
thf(sy_c_Nat__Bijection_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_Onat__of__num,type,
nat_of_num: num > nat ).
thf(sy_c_Num_Oneg__numeral__class_Odbl,type,
neg_numeral_dbl:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__dec,type,
neg_numeral_dbl_dec:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Num_Oneg__numeral__class_Odbl__inc,type,
neg_numeral_dbl_inc:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Num_Oneg__numeral__class_Ois__num,type,
neg_numeral_is_num:
!>[A: $tType] : ( A > $o ) ).
thf(sy_c_Num_Oneg__numeral__class_Osub,type,
neg_numeral_sub:
!>[A: $tType] : ( num > num > A ) ).
thf(sy_c_Num_Onum_OBit0,type,
bit0: num > num ).
thf(sy_c_Num_Onum_OBit1,type,
bit1: num > num ).
thf(sy_c_Num_Onum_OOne,type,
one2: num ).
thf(sy_c_Num_Onum_Ocase__num,type,
case_num:
!>[A: $tType] : ( A > ( num > A ) > ( num > A ) > num > A ) ).
thf(sy_c_Num_Onum_Orec__num,type,
rec_num:
!>[A: $tType] : ( A > ( num > A > A ) > ( num > A > A ) > num > A ) ).
thf(sy_c_Num_Onum_Osize__num,type,
size_num: num > nat ).
thf(sy_c_Num_Onum__of__nat,type,
num_of_nat: nat > num ).
thf(sy_c_Num_Onumeral__class_Onumeral,type,
numeral_numeral:
!>[A: $tType] : ( num > A ) ).
thf(sy_c_Num_Opow,type,
pow: num > num > num ).
thf(sy_c_Num_Opred__numeral,type,
pred_numeral: num > nat ).
thf(sy_c_Num_Oring__1__class_Oiszero,type,
ring_1_iszero:
!>[A: $tType] : ( A > $o ) ).
thf(sy_c_Num_Osqr,type,
sqr: num > num ).
thf(sy_c_Option_Obind,type,
bind:
!>[A: $tType,B: $tType] : ( ( option @ A ) > ( A > ( option @ B ) ) > ( option @ B ) ) ).
thf(sy_c_Option_Ocombine__options,type,
combine_options:
!>[A: $tType] : ( ( A > A > A ) > ( option @ A ) > ( option @ A ) > ( option @ A ) ) ).
thf(sy_c_Option_Ois__none,type,
is_none:
!>[A: $tType] : ( ( option @ A ) > $o ) ).
thf(sy_c_Option_Ooption_ONone,type,
none:
!>[A: $tType] : ( option @ A ) ).
thf(sy_c_Option_Ooption_OSome,type,
some:
!>[A: $tType] : ( A > ( option @ A ) ) ).
thf(sy_c_Option_Ooption_Ocase__option,type,
case_option:
!>[B: $tType,A: $tType] : ( B > ( A > B ) > ( option @ A ) > B ) ).
thf(sy_c_Option_Ooption_Omap__option,type,
map_option:
!>[A: $tType,Aa: $tType] : ( ( A > Aa ) > ( option @ A ) > ( option @ Aa ) ) ).
thf(sy_c_Option_Ooption_Opred__option,type,
pred_option:
!>[A: $tType] : ( ( A > $o ) > ( option @ A ) > $o ) ).
thf(sy_c_Option_Ooption_Orec__option,type,
rec_option:
!>[C: $tType,A: $tType] : ( C > ( A > C ) > ( option @ A ) > C ) ).
thf(sy_c_Option_Ooption_Orel__option,type,
rel_option:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > ( option @ A ) > ( option @ B ) > $o ) ).
thf(sy_c_Option_Ooption_Oset__option,type,
set_option:
!>[A: $tType] : ( ( option @ A ) > ( set @ A ) ) ).
thf(sy_c_Option_Ooption_Osize__option,type,
size_option:
!>[A: $tType] : ( ( A > nat ) > ( option @ A ) > nat ) ).
thf(sy_c_Option_Ooption_Othe,type,
the2:
!>[A: $tType] : ( ( option @ A ) > A ) ).
thf(sy_c_Option_Othese,type,
these:
!>[A: $tType] : ( ( set @ ( option @ A ) ) > ( set @ A ) ) ).
thf(sy_c_Order__Relation_Olinear__order__on,type,
order_679001287576687338der_on:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Order__Relation_Opartial__order__on,type,
order_7125193373082350890der_on:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Order__Relation_Ostrict__linear__order__on,type,
order_5396836661320670305der_on:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Order__Relation_OunderS,type,
order_underS:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > A > ( set @ A ) ) ).
thf(sy_c_Order__Relation_Owell__order__on,type,
order_well_order_on:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Orderings_Obot__class_Obot,type,
bot_bot:
!>[A: $tType] : A ).
thf(sy_c_Orderings_Oord__class_OLeast,type,
ord_Least:
!>[A: $tType] : ( ( A > $o ) > A ) ).
thf(sy_c_Orderings_Oord__class_Oless,type,
ord_less:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Orderings_Oord__class_Oless__eq,type,
ord_less_eq:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Orderings_Oord__class_Omax,type,
ord_max:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Orderings_Oord__class_Omin,type,
ord_min:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Orderings_Oorder__class_Oantimono,type,
order_antimono:
!>[A: $tType,B: $tType] : ( ( A > B ) > $o ) ).
thf(sy_c_Orderings_Oorder__class_Omono,type,
order_mono:
!>[A: $tType,B: $tType] : ( ( A > B ) > $o ) ).
thf(sy_c_Orderings_Oorder__class_Ostrict__mono,type,
order_strict_mono:
!>[A: $tType,B: $tType] : ( ( A > B ) > $o ) ).
thf(sy_c_Orderings_Oordering__top,type,
ordering_top:
!>[A: $tType] : ( ( A > A > $o ) > ( A > A > $o ) > A > $o ) ).
thf(sy_c_Orderings_Otop__class_Otop,type,
top_top:
!>[A: $tType] : A ).
thf(sy_c_Partial__Function_Oflat__lub,type,
partial_flat_lub:
!>[A: $tType] : ( A > ( set @ A ) > A ) ).
thf(sy_c_Power_Opower_Opower,type,
power2:
!>[A: $tType] : ( A > ( A > A > A ) > A > nat > A ) ).
thf(sy_c_Power_Opower__class_Opower,type,
power_power:
!>[A: $tType] : ( A > nat > A ) ).
thf(sy_c_Product__Type_OPair,type,
product_Pair:
!>[A: $tType,B: $tType] : ( A > B > ( product_prod @ A @ B ) ) ).
thf(sy_c_Product__Type_OSigma,type,
product_Sigma:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > ( set @ B ) ) > ( set @ ( product_prod @ A @ B ) ) ) ).
thf(sy_c_Product__Type_Oapsnd,type,
product_apsnd:
!>[B: $tType,C: $tType,A: $tType] : ( ( B > C ) > ( product_prod @ A @ B ) > ( product_prod @ A @ C ) ) ).
thf(sy_c_Product__Type_Omap__prod,type,
product_map_prod:
!>[A: $tType,C: $tType,B: $tType,D: $tType] : ( ( A > C ) > ( B > D ) > ( product_prod @ A @ B ) > ( product_prod @ C @ D ) ) ).
thf(sy_c_Product__Type_Oprod_Ocase__prod,type,
product_case_prod:
!>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
thf(sy_c_Product__Type_Oprod_Ofst,type,
product_fst:
!>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > A ) ).
thf(sy_c_Product__Type_Oprod_Osnd,type,
product_snd:
!>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > B ) ).
thf(sy_c_Product__Type_Oscomp,type,
product_scomp:
!>[A: $tType,B: $tType,C: $tType,D: $tType] : ( ( A > ( product_prod @ B @ C ) ) > ( B > C > D ) > A > D ) ).
thf(sy_c_Pure_Otype,type,
type2:
!>[A: $tType] : ( itself @ A ) ).
thf(sy_c_Quotient_OQuotient3,type,
quotient3:
!>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( A > B ) > ( B > A ) > $o ) ).
thf(sy_c_Random_Oinc__shift,type,
inc_shift: code_natural > code_natural > code_natural ).
thf(sy_c_Random_Oiterate,type,
iterate:
!>[B: $tType,A: $tType] : ( code_natural > ( B > A > ( product_prod @ B @ A ) ) > B > A > ( product_prod @ B @ A ) ) ).
thf(sy_c_Random_Oiterate__rel,type,
iterate_rel:
!>[B: $tType,A: $tType] : ( ( product_prod @ code_natural @ ( product_prod @ ( B > A > ( product_prod @ B @ A ) ) @ B ) ) > ( product_prod @ code_natural @ ( product_prod @ ( B > A > ( product_prod @ B @ A ) ) @ B ) ) > $o ) ).
thf(sy_c_Random_Olog,type,
log: code_natural > code_natural > code_natural ).
thf(sy_c_Random_Olog__rel,type,
log_rel: ( product_prod @ code_natural @ code_natural ) > ( product_prod @ code_natural @ code_natural ) > $o ).
thf(sy_c_Random_Ominus__shift,type,
minus_shift: code_natural > code_natural > code_natural > code_natural ).
thf(sy_c_Random_Onext,type,
next: ( product_prod @ code_natural @ code_natural ) > ( product_prod @ code_natural @ ( product_prod @ code_natural @ code_natural ) ) ).
thf(sy_c_Random_Opick,type,
pick:
!>[A: $tType] : ( ( list @ ( product_prod @ code_natural @ A ) ) > code_natural > A ) ).
thf(sy_c_Random_Orange,type,
range: code_natural > ( product_prod @ code_natural @ code_natural ) > ( product_prod @ code_natural @ ( product_prod @ code_natural @ code_natural ) ) ).
thf(sy_c_Random_Oselect,type,
select:
!>[A: $tType] : ( ( list @ A ) > ( product_prod @ code_natural @ code_natural ) > ( product_prod @ A @ ( product_prod @ code_natural @ code_natural ) ) ) ).
thf(sy_c_Random_Oselect__weight,type,
select_weight:
!>[A: $tType] : ( ( list @ ( product_prod @ code_natural @ A ) ) > ( product_prod @ code_natural @ code_natural ) > ( product_prod @ A @ ( product_prod @ code_natural @ code_natural ) ) ) ).
thf(sy_c_Rat_OAbs__Rat,type,
abs_Rat: ( product_prod @ int @ int ) > rat ).
thf(sy_c_Rat_OFract,type,
fract: int > int > rat ).
thf(sy_c_Rat_OFrct,type,
frct: ( product_prod @ int @ int ) > rat ).
thf(sy_c_Rat_ORep__Rat,type,
rep_Rat: rat > ( product_prod @ int @ int ) ).
thf(sy_c_Rat_Ofield__char__0__class_ORats,type,
field_char_0_Rats:
!>[A: $tType] : ( set @ A ) ).
thf(sy_c_Rat_Ofield__char__0__class_Oof__rat,type,
field_char_0_of_rat:
!>[A: $tType] : ( rat > A ) ).
thf(sy_c_Rat_Onormalize,type,
normalize: ( product_prod @ int @ int ) > ( product_prod @ int @ int ) ).
thf(sy_c_Rat_Oof__int,type,
of_int: int > rat ).
thf(sy_c_Rat_Opcr__rat,type,
pcr_rat: ( product_prod @ int @ int ) > rat > $o ).
thf(sy_c_Rat_Opositive,type,
positive: rat > $o ).
thf(sy_c_Rat_Oquotient__of,type,
quotient_of: rat > ( product_prod @ int @ int ) ).
thf(sy_c_Rat_Oratrel,type,
ratrel: ( product_prod @ int @ int ) > ( product_prod @ int @ int ) > $o ).
thf(sy_c_Real_ORatreal,type,
ratreal: rat > real ).
thf(sy_c_Real_OReal,type,
real2: ( nat > rat ) > real ).
thf(sy_c_Real_Ocauchy,type,
cauchy: ( nat > rat ) > $o ).
thf(sy_c_Real_Ocr__real,type,
cr_real: ( nat > rat ) > real > $o ).
thf(sy_c_Real_Opcr__real,type,
pcr_real: ( nat > rat ) > real > $o ).
thf(sy_c_Real_Opositive,type,
positive2: real > $o ).
thf(sy_c_Real_Orealrel,type,
realrel: ( nat > rat ) > ( nat > rat ) > $o ).
thf(sy_c_Real_Orep__real,type,
rep_real: real > nat > rat ).
thf(sy_c_Real_Ovanishes,type,
vanishes: ( nat > rat ) > $o ).
thf(sy_c_Real__Vector__Spaces_OReals,type,
real_Vector_Reals:
!>[A: $tType] : ( set @ A ) ).
thf(sy_c_Real__Vector__Spaces_Obounded__bilinear,type,
real_V2442710119149674383linear:
!>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > $o ) ).
thf(sy_c_Real__Vector__Spaces_Obounded__linear,type,
real_V3181309239436604168linear:
!>[A: $tType,B: $tType] : ( ( A > B ) > $o ) ).
thf(sy_c_Real__Vector__Spaces_Oconstruct,type,
real_V4425403222259421789struct:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B ) > A > B ) ).
thf(sy_c_Real__Vector__Spaces_Odependent,type,
real_V358717886546972837endent:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Real__Vector__Spaces_Odim,type,
real_Vector_dim:
!>[A: $tType] : ( ( set @ A ) > nat ) ).
thf(sy_c_Real__Vector__Spaces_Odist__class_Odist,type,
real_V557655796197034286t_dist:
!>[A: $tType] : ( A > A > real ) ).
thf(sy_c_Real__Vector__Spaces_Oextend__basis,type,
real_V4986007116245087402_basis:
!>[A: $tType] : ( ( set @ A ) > ( set @ A ) ) ).
thf(sy_c_Real__Vector__Spaces_Olinear,type,
real_Vector_linear:
!>[A: $tType,B: $tType] : ( ( A > B ) > $o ) ).
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_Orepresentation,type,
real_V7696804695334737415tation:
!>[A: $tType] : ( ( set @ A ) > A > A > real ) ).
thf(sy_c_Real__Vector__Spaces_OscaleR__class_OscaleR,type,
real_V8093663219630862766scaleR:
!>[A: $tType] : ( real > A > A ) ).
thf(sy_c_Real__Vector__Spaces_Ospan,type,
real_Vector_span:
!>[A: $tType] : ( ( set @ A ) > ( set @ A ) ) ).
thf(sy_c_Real__Vector__Spaces_Osubspace,type,
real_Vector_subspace:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Relation_ODomain,type,
domain:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( set @ A ) ) ).
thf(sy_c_Relation_OField,type,
field2:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ A ) ) ).
thf(sy_c_Relation_OId,type,
id2:
!>[A: $tType] : ( set @ ( product_prod @ A @ A ) ) ).
thf(sy_c_Relation_ORange,type,
range2:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( set @ B ) ) ).
thf(sy_c_Relation_Oinv__image,type,
inv_image:
!>[B: $tType,A: $tType] : ( ( set @ ( product_prod @ B @ B ) ) > ( A > B ) > ( set @ ( product_prod @ A @ A ) ) ) ).
thf(sy_c_Relation_Oirrefl,type,
irrefl:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Relation_Orelcomp,type,
relcomp:
!>[A: $tType,B: $tType,C: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( set @ ( product_prod @ B @ C ) ) > ( set @ ( product_prod @ A @ C ) ) ) ).
thf(sy_c_Relation_Ototal__on,type,
total_on:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Relation_Otransp,type,
transp:
!>[A: $tType] : ( ( A > A > $o ) > $o ) ).
thf(sy_c_Rings_Oalgebraic__semidom__class_Ocoprime,type,
algebr8660921524188924756oprime:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Rings_Odivide__class_Odivide,type,
divide_divide:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Rings_Odvd__class_Odvd,type,
dvd_dvd:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Rings_Omodulo__class_Omodulo,type,
modulo_modulo:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Rings_Onormalization__semidom__class_Onormalize,type,
normal6383669964737779283malize:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Rings_Ounit__factor__class_Ounit__factor,type,
unit_f5069060285200089521factor:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool,type,
zero_neq_one_of_bool:
!>[A: $tType] : ( $o > A ) ).
thf(sy_c_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_Oimage,type,
image:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > ( set @ B ) ) ).
thf(sy_c_Set_Oinsert,type,
insert:
!>[A: $tType] : ( A > ( set @ A ) > ( set @ A ) ) ).
thf(sy_c_Set_Ois__singleton,type,
is_singleton:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Set_Opairwise,type,
pairwise:
!>[A: $tType] : ( ( A > A > $o ) > ( set @ A ) > $o ) ).
thf(sy_c_Set_Oremove,type,
remove:
!>[A: $tType] : ( A > ( set @ A ) > ( set @ A ) ) ).
thf(sy_c_Set_Ovimage,type,
vimage:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ B ) > ( set @ A ) ) ).
thf(sy_c_Set__Interval_Ofold__atLeastAtMost__nat,type,
set_fo6178422350223883121st_nat:
!>[A: $tType] : ( ( nat > A > A ) > nat > nat > A > A ) ).
thf(sy_c_Set__Interval_Ofold__atLeastAtMost__nat__rel,type,
set_fo1817059534552279752at_rel:
!>[A: $tType] : ( ( product_prod @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) ) > ( product_prod @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) ) > $o ) ).
thf(sy_c_Set__Interval_Oord__class_OatLeast,type,
set_ord_atLeast:
!>[A: $tType] : ( A > ( set @ A ) ) ).
thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost,type,
set_or1337092689740270186AtMost:
!>[A: $tType] : ( A > A > ( set @ A ) ) ).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan,type,
set_or7035219750837199246ssThan:
!>[A: $tType] : ( A > A > ( set @ A ) ) ).
thf(sy_c_Set__Interval_Oord__class_OatMost,type,
set_ord_atMost:
!>[A: $tType] : ( A > ( set @ A ) ) ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThan,type,
set_ord_greaterThan:
!>[A: $tType] : ( A > ( set @ A ) ) ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanAtMost,type,
set_or3652927894154168847AtMost:
!>[A: $tType] : ( A > A > ( set @ A ) ) ).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan,type,
set_or5935395276787703475ssThan:
!>[A: $tType] : ( A > A > ( set @ A ) ) ).
thf(sy_c_Set__Interval_Oord__class_OlessThan,type,
set_ord_lessThan:
!>[A: $tType] : ( A > ( set @ A ) ) ).
thf(sy_c_String_OCode_Oabort,type,
abort:
!>[A: $tType] : ( literal > ( product_unit > A ) > A ) ).
thf(sy_c_String_OLiteral,type,
literal2: $o > $o > $o > $o > $o > $o > $o > literal > literal ).
thf(sy_c_String_Ochar_OChar,type,
char2: $o > $o > $o > $o > $o > $o > $o > $o > char ).
thf(sy_c_String_Ochar_Osize__char,type,
size_char: char > nat ).
thf(sy_c_String_Ocomm__semiring__1__class_Oof__char,type,
comm_s6883823935334413003f_char:
!>[A: $tType] : ( char > A ) ).
thf(sy_c_String_Ounique__euclidean__semiring__with__bit__operations__class_Ochar__of,type,
unique5772411509450598832har_of:
!>[A: $tType] : ( A > char ) ).
thf(sy_c_Sum__Type_OPlus,type,
sum_Plus:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( set @ B ) > ( set @ ( sum_sum @ A @ B ) ) ) ).
thf(sy_c_Sum__Type_Osum_Ocase__sum,type,
sum_case_sum:
!>[A: $tType,C: $tType,B: $tType] : ( ( A > C ) > ( B > C ) > ( sum_sum @ A @ B ) > C ) ).
thf(sy_c_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_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_Oconvergent,type,
topolo6863149650580417670ergent:
!>[A: $tType] : ( ( nat > 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_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,
log2: 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_Opowr__real,type,
powr_real: real > real > real ).
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_Oacyclic,type,
transitive_acyclic:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Transitive__Closure_Ontrancl,type,
transitive_ntrancl:
!>[A: $tType] : ( nat > ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ A @ A ) ) ) ).
thf(sy_c_Transitive__Closure_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_Ocase__VEBT,type,
vEBT_case_VEBT:
!>[A: $tType] : ( ( ( option @ ( product_prod @ nat @ nat ) ) > nat > ( list @ vEBT_VEBT ) > vEBT_VEBT > A ) > ( $o > $o > A ) > vEBT_VEBT > A ) ).
thf(sy_c_VEBT__Definitions_OVEBT_Orec__VEBT,type,
vEBT_rec_VEBT:
!>[A: $tType] : ( ( ( option @ ( product_prod @ nat @ nat ) ) > nat > ( list @ ( product_prod @ vEBT_VEBT @ A ) ) > vEBT_VEBT > A > A ) > ( $o > $o > A ) > vEBT_VEBT > A ) ).
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_VEBT__Member_OVEBT__internal_Obit__concat,type,
vEBT_VEBT_bit_concat: nat > nat > nat > nat ).
thf(sy_c_VEBT__Member_OVEBT__internal_OminNull,type,
vEBT_VEBT_minNull: vEBT_VEBT > $o ).
thf(sy_c_VEBT__Member_OVEBT__internal_OminNull__rel,type,
vEBT_V6963167321098673237ll_rel: vEBT_VEBT > vEBT_VEBT > $o ).
thf(sy_c_VEBT__Member_OVEBT__internal_Oset__vebt_H,type,
vEBT_VEBT_set_vebt: vEBT_VEBT > ( set @ nat ) ).
thf(sy_c_VEBT__Member_Ovebt__member,type,
vEBT_vebt_member: vEBT_VEBT > nat > $o ).
thf(sy_c_VEBT__Member_Ovebt__member__rel,type,
vEBT_vebt_member_rel: ( product_prod @ vEBT_VEBT @ nat ) > ( product_prod @ vEBT_VEBT @ nat ) > $o ).
thf(sy_c_Wellfounded_Oaccp,type,
accp:
!>[A: $tType] : ( ( A > A > $o ) > A > $o ) ).
thf(sy_c_Wellfounded_Oless__than,type,
less_than: set @ ( product_prod @ nat @ nat ) ).
thf(sy_c_Wellfounded_Olex__prod,type,
lex_prod:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ B @ B ) ) > ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) ) ).
thf(sy_c_Wellfounded_Omeasure,type,
measure:
!>[A: $tType] : ( ( A > nat ) > ( set @ ( product_prod @ A @ A ) ) ) ).
thf(sy_c_Wellfounded_Omlex__prod,type,
mlex_prod:
!>[A: $tType] : ( ( A > nat ) > ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ A @ A ) ) ) ).
thf(sy_c_Wellfounded_Opred__nat,type,
pred_nat: set @ ( product_prod @ nat @ nat ) ).
thf(sy_c_Wellfounded_Owf,type,
wf:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_fChoice,type,
fChoice:
!>[A: $tType] : ( ( A > $o ) > A ) ).
thf(sy_c_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_n,type,
n: nat ).
thf(sy_v_t,type,
t: vEBT_VEBT ).
thf(sy_v_x,type,
x: nat ).
% Relevant facts (8121)
thf(fact_0_min__Null__member,axiom,
! [T2: vEBT_VEBT,X: nat] :
( ( vEBT_VEBT_minNull @ T2 )
=> ~ ( vEBT_vebt_member @ T2 @ X ) ) ).
% min_Null_member
thf(fact_1_valid__eq,axiom,
vEBT_VEBT_valid = vEBT_invar_vebt ).
% valid_eq
thf(fact_2_valid__eq1,axiom,
! [T2: vEBT_VEBT,D2: nat] :
( ( vEBT_invar_vebt @ T2 @ D2 )
=> ( vEBT_VEBT_valid @ T2 @ D2 ) ) ).
% valid_eq1
thf(fact_3_valid__eq2,axiom,
! [T2: vEBT_VEBT,D2: nat] :
( ( vEBT_VEBT_valid @ T2 @ D2 )
=> ( vEBT_invar_vebt @ T2 @ D2 ) ) ).
% valid_eq2
thf(fact_4_both__member__options__equiv__member,axiom,
! [T2: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( vEBT_V8194947554948674370ptions @ T2 @ X )
= ( vEBT_vebt_member @ T2 @ X ) ) ) ).
% both_member_options_equiv_member
thf(fact_5_valid__member__both__member__options,axiom,
! [T2: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( vEBT_V8194947554948674370ptions @ T2 @ X )
=> ( vEBT_vebt_member @ T2 @ X ) ) ) ).
% valid_member_both_member_options
thf(fact_6_valid__0__not,axiom,
! [T2: vEBT_VEBT] :
~ ( vEBT_invar_vebt @ T2 @ ( zero_zero @ nat ) ) ).
% valid_0_not
thf(fact_7_valid__tree__deg__neq__0,axiom,
! [T2: vEBT_VEBT] :
~ ( vEBT_invar_vebt @ T2 @ ( zero_zero @ nat ) ) ).
% valid_tree_deg_neq_0
thf(fact_8_deg__deg__n,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 )
=> ( Deg = N ) ) ).
% deg_deg_n
thf(fact_9_member__valid__both__member__options,axiom,
! [Tree: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ Tree @ N )
=> ( ( vEBT_vebt_member @ Tree @ X )
=> ( ( vEBT_V5719532721284313246member @ Tree @ X )
| ( vEBT_VEBT_membermima @ Tree @ X ) ) ) ) ).
% member_valid_both_member_options
thf(fact_10_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_11_Leaf__0__not,axiom,
! [A3: $o,B2: $o] :
~ ( vEBT_invar_vebt @ ( vEBT_Leaf @ A3 @ B2 ) @ ( zero_zero @ nat ) ) ).
% Leaf_0_not
thf(fact_12_not__min__Null__member,axiom,
! [T2: vEBT_VEBT] :
( ~ ( vEBT_VEBT_minNull @ T2 )
=> ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ T2 @ X_1 ) ) ).
% not_min_Null_member
thf(fact_13_both__member__options__def,axiom,
( vEBT_V8194947554948674370ptions
= ( ^ [T3: vEBT_VEBT,X2: nat] :
( ( vEBT_V5719532721284313246member @ T3 @ X2 )
| ( vEBT_VEBT_membermima @ T3 @ X2 ) ) ) ) ).
% both_member_options_def
thf(fact_14_VEBT__internal_OminNull_Osimps_I3_J,axiom,
! [Uu: $o] :
~ ( vEBT_VEBT_minNull @ ( vEBT_Leaf @ Uu @ $true ) ) ).
% VEBT_internal.minNull.simps(3)
thf(fact_15_VEBT__internal_OminNull_Osimps_I2_J,axiom,
! [Uv: $o] :
~ ( vEBT_VEBT_minNull @ ( vEBT_Leaf @ $true @ Uv ) ) ).
% VEBT_internal.minNull.simps(2)
thf(fact_16_VEBT__internal_OminNull_Osimps_I1_J,axiom,
vEBT_VEBT_minNull @ ( vEBT_Leaf @ $false @ $false ) ).
% VEBT_internal.minNull.simps(1)
thf(fact_17_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_18_bot__nat__0_Onot__eq__extremum,axiom,
! [A3: nat] :
( ( A3
!= ( zero_zero @ nat ) )
= ( ord_less @ nat @ ( zero_zero @ nat ) @ A3 ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_19_neq0__conv,axiom,
! [N: nat] :
( ( N
!= ( zero_zero @ nat ) )
= ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ).
% neq0_conv
thf(fact_20_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% less_nat_zero_code
thf(fact_21_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_22_buildup__nothing__in__leaf,axiom,
! [N: nat,X: nat] :
~ ( vEBT_V5719532721284313246member @ ( vEBT_vebt_buildup @ N ) @ X ) ).
% buildup_nothing_in_leaf
thf(fact_23_VEBT__internal_Onaive__member_Osimps_I2_J,axiom,
! [Uu: option @ ( product_prod @ nat @ nat ),Uv: list @ vEBT_VEBT,Uw: vEBT_VEBT,Ux: nat] :
~ ( vEBT_V5719532721284313246member @ ( vEBT_Node @ Uu @ ( zero_zero @ nat ) @ Uv @ Uw ) @ Ux ) ).
% VEBT_internal.naive_member.simps(2)
thf(fact_24_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_25_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_26_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_27_VEBT__internal_Omembermima_Osimps_I1_J,axiom,
! [Uu: $o,Uv: $o,Uw: nat] :
~ ( vEBT_VEBT_membermima @ ( vEBT_Leaf @ Uu @ Uv ) @ Uw ) ).
% VEBT_internal.membermima.simps(1)
thf(fact_28_vebt__buildup_Osimps_I1_J,axiom,
( ( vEBT_vebt_buildup @ ( zero_zero @ nat ) )
= ( vEBT_Leaf @ $false @ $false ) ) ).
% vebt_buildup.simps(1)
thf(fact_29_zero__reorient,axiom,
! [A: $tType] :
( ( zero @ A )
=> ! [X: A] :
( ( ( zero_zero @ A )
= X )
= ( X
= ( zero_zero @ A ) ) ) ) ).
% zero_reorient
thf(fact_30_measure__induct__rule,axiom,
! [B: $tType,A: $tType] :
( ( wellorder @ B )
=> ! [F2: A > B,P: A > $o,A3: A] :
( ! [X3: A] :
( ! [Y: A] :
( ( ord_less @ B @ ( F2 @ Y ) @ ( F2 @ X3 ) )
=> ( P @ Y ) )
=> ( P @ X3 ) )
=> ( P @ A3 ) ) ) ).
% measure_induct_rule
thf(fact_31_measure__induct,axiom,
! [B: $tType,A: $tType] :
( ( wellorder @ B )
=> ! [F2: A > B,P: A > $o,A3: A] :
( ! [X3: A] :
( ! [Y: A] :
( ( ord_less @ B @ ( F2 @ Y ) @ ( F2 @ X3 ) )
=> ( P @ Y ) )
=> ( P @ X3 ) )
=> ( P @ A3 ) ) ) ).
% measure_induct
thf(fact_32_infinite__descent__measure,axiom,
! [A: $tType,P: A > $o,V: A > nat,X: A] :
( ! [X3: A] :
( ~ ( P @ X3 )
=> ? [Y: A] :
( ( ord_less @ nat @ ( V @ Y ) @ ( V @ X3 ) )
& ~ ( P @ Y ) ) )
=> ( P @ X ) ) ).
% infinite_descent_measure
thf(fact_33_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_34_infinite__descent,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ~ ( P @ N2 )
=> ? [M: nat] :
( ( ord_less @ nat @ M @ N2 )
& ~ ( P @ M ) ) )
=> ( P @ N ) ) ).
% infinite_descent
thf(fact_35_nat__less__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M: nat] :
( ( ord_less @ nat @ M @ N2 )
=> ( P @ M ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% nat_less_induct
thf(fact_36_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_37_less__not__refl3,axiom,
! [S: nat,T2: nat] :
( ( ord_less @ nat @ S @ T2 )
=> ( S != T2 ) ) ).
% less_not_refl3
thf(fact_38_less__not__refl2,axiom,
! [N: nat,M2: nat] :
( ( ord_less @ nat @ N @ M2 )
=> ( M2 != N ) ) ).
% less_not_refl2
thf(fact_39_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ N ) ).
% less_not_refl
thf(fact_40_nat__neq__iff,axiom,
! [M2: nat,N: nat] :
( ( M2 != N )
= ( ( ord_less @ nat @ M2 @ N )
| ( ord_less @ nat @ N @ M2 ) ) ) ).
% nat_neq_iff
thf(fact_41_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_42_gr__implies__not__zero,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [M2: A,N: A] :
( ( ord_less @ A @ M2 @ N )
=> ( N
!= ( zero_zero @ A ) ) ) ) ).
% gr_implies_not_zero
thf(fact_43_mem__Collect__eq,axiom,
! [A: $tType,A3: A,P: A > $o] :
( ( member @ A @ A3 @ ( collect @ A @ P ) )
= ( P @ A3 ) ) ).
% mem_Collect_eq
thf(fact_44_Collect__mem__eq,axiom,
! [A: $tType,A4: set @ A] :
( ( collect @ A
@ ^ [X2: A] : ( member @ A @ X2 @ A4 ) )
= A4 ) ).
% Collect_mem_eq
thf(fact_45_Collect__cong,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X3: A] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( collect @ A @ P )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_46_ext,axiom,
! [B: $tType,A: $tType,F2: A > B,G: A > B] :
( ! [X3: A] :
( ( F2 @ X3 )
= ( G @ X3 ) )
=> ( F2 = G ) ) ).
% ext
thf(fact_47_not__less__zero,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [N: A] :
~ ( ord_less @ A @ N @ ( zero_zero @ A ) ) ) ).
% not_less_zero
thf(fact_48_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_49_infinite__descent0__measure,axiom,
! [A: $tType,V: A > nat,P: A > $o,X: A] :
( ! [X3: A] :
( ( ( V @ X3 )
= ( zero_zero @ nat ) )
=> ( P @ X3 ) )
=> ( ! [X3: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( V @ X3 ) )
=> ( ~ ( P @ X3 )
=> ? [Y: A] :
( ( ord_less @ nat @ ( V @ Y ) @ ( V @ X3 ) )
& ~ ( P @ Y ) ) ) )
=> ( P @ X ) ) ) ).
% infinite_descent0_measure
thf(fact_50_infinite__descent0,axiom,
! [P: nat > $o,N: nat] :
( ( P @ ( zero_zero @ nat ) )
=> ( ! [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ~ ( P @ N2 )
=> ? [M: nat] :
( ( ord_less @ nat @ M @ N2 )
& ~ ( P @ M ) ) ) )
=> ( P @ N ) ) ) ).
% infinite_descent0
thf(fact_51_gr__implies__not0,axiom,
! [M2: nat,N: nat] :
( ( ord_less @ nat @ M2 @ N )
=> ( N
!= ( zero_zero @ nat ) ) ) ).
% gr_implies_not0
thf(fact_52_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% less_zeroE
thf(fact_53_not__less0,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% not_less0
thf(fact_54_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% not_gr0
thf(fact_55_gr0I,axiom,
! [N: nat] :
( ( N
!= ( zero_zero @ nat ) )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ).
% gr0I
thf(fact_56_bot__nat__0_Oextremum__strict,axiom,
! [A3: nat] :
~ ( ord_less @ nat @ A3 @ ( zero_zero @ nat ) ) ).
% bot_nat_0.extremum_strict
thf(fact_57_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_58_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_59_deg1Leaf,axiom,
! [T2: vEBT_VEBT] :
( ( vEBT_invar_vebt @ T2 @ ( one_one @ nat ) )
= ( ? [A5: $o,B3: $o] :
( T2
= ( vEBT_Leaf @ A5 @ B3 ) ) ) ) ).
% deg1Leaf
thf(fact_60_deg__1__Leaf,axiom,
! [T2: vEBT_VEBT] :
( ( vEBT_invar_vebt @ T2 @ ( one_one @ nat ) )
=> ? [A6: $o,B4: $o] :
( T2
= ( vEBT_Leaf @ A6 @ B4 ) ) ) ).
% deg_1_Leaf
thf(fact_61_deg__1__Leafy,axiom,
! [T2: vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( N
= ( one_one @ nat ) )
=> ? [A6: $o,B4: $o] :
( T2
= ( vEBT_Leaf @ A6 @ B4 ) ) ) ) ).
% deg_1_Leafy
thf(fact_62_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 )
=> ? [E: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ E )
& ( ord_less @ A @ E @ D1 )
& ( ord_less @ A @ E @ D22 ) ) ) ) ) ).
% field_lbound_gt_zero
thf(fact_63_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_64_deg__SUcn__Node,axiom,
! [Tree: vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ Tree @ ( suc @ ( suc @ N ) ) )
=> ? [Info2: option @ ( product_prod @ nat @ nat ),TreeList2: list @ vEBT_VEBT,S2: vEBT_VEBT] :
( Tree
= ( vEBT_Node @ Info2 @ ( suc @ ( suc @ N ) ) @ TreeList2 @ S2 ) ) ) ).
% deg_SUcn_Node
thf(fact_65_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_66_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_67_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_68_VEBT__internal_Onaive__member_Osimps_I1_J,axiom,
! [A3: $o,B2: $o,X: nat] :
( ( vEBT_V5719532721284313246member @ ( vEBT_Leaf @ A3 @ B2 ) @ X )
= ( ( ( X
= ( zero_zero @ nat ) )
=> A3 )
& ( ( X
!= ( zero_zero @ nat ) )
=> ( ( ( X
= ( one_one @ nat ) )
=> B2 )
& ( X
= ( one_one @ nat ) ) ) ) ) ) ).
% VEBT_internal.naive_member.simps(1)
thf(fact_69_vebt__member_Osimps_I1_J,axiom,
! [A3: $o,B2: $o,X: nat] :
( ( vEBT_vebt_member @ ( vEBT_Leaf @ A3 @ B2 ) @ X )
= ( ( ( X
= ( zero_zero @ nat ) )
=> A3 )
& ( ( X
!= ( zero_zero @ nat ) )
=> ( ( ( X
= ( one_one @ nat ) )
=> B2 )
& ( X
= ( one_one @ nat ) ) ) ) ) ) ).
% vebt_member.simps(1)
thf(fact_70_VEBT__internal_OminNull_Oelims_I2_J,axiom,
! [X: vEBT_VEBT] :
( ( vEBT_VEBT_minNull @ X )
=> ( ( X
!= ( vEBT_Leaf @ $false @ $false ) )
=> ~ ! [Uw2: nat,Ux2: list @ vEBT_VEBT,Uy: vEBT_VEBT] :
( X
!= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uw2 @ Ux2 @ Uy ) ) ) ) ).
% VEBT_internal.minNull.elims(2)
thf(fact_71_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_72_nat_Oinject,axiom,
! [X23: nat,Y23: nat] :
( ( ( suc @ X23 )
= ( suc @ Y23 ) )
= ( X23 = Y23 ) ) ).
% nat.inject
thf(fact_73_of__nat__eq__iff,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [M2: nat,N: nat] :
( ( ( semiring_1_of_nat @ A @ M2 )
= ( semiring_1_of_nat @ A @ N ) )
= ( M2 = N ) ) ) ).
% of_nat_eq_iff
thf(fact_74_lessI,axiom,
! [N: nat] : ( ord_less @ nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_75_Suc__mono,axiom,
! [M2: nat,N: nat] :
( ( ord_less @ nat @ M2 @ N )
=> ( ord_less @ nat @ ( suc @ M2 ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_76_Suc__less__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less @ nat @ ( suc @ M2 ) @ ( suc @ N ) )
= ( ord_less @ nat @ M2 @ N ) ) ).
% Suc_less_eq
thf(fact_77_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_78_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_79_of__nat__eq__0__iff,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [M2: nat] :
( ( ( semiring_1_of_nat @ A @ M2 )
= ( zero_zero @ A ) )
= ( M2
= ( zero_zero @ nat ) ) ) ) ).
% of_nat_eq_0_iff
thf(fact_80_of__nat__less__iff,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [M2: nat,N: nat] :
( ( ord_less @ A @ ( semiring_1_of_nat @ A @ M2 ) @ ( semiring_1_of_nat @ A @ N ) )
= ( ord_less @ nat @ M2 @ N ) ) ) ).
% of_nat_less_iff
thf(fact_81_less__Suc0,axiom,
! [N: nat] :
( ( ord_less @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% less_Suc0
thf(fact_82_zero__less__Suc,axiom,
! [N: nat] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_83_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_84_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_85_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_86_less__one,axiom,
! [N: nat] :
( ( ord_less @ nat @ N @ ( one_one @ nat ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% less_one
thf(fact_87_one__reorient,axiom,
! [A: $tType] :
( ( one @ A )
=> ! [X: A] :
( ( ( one_one @ A )
= X )
= ( X
= ( one_one @ A ) ) ) ) ).
% one_reorient
thf(fact_88_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_89_Suc__inject,axiom,
! [X: nat,Y2: nat] :
( ( ( suc @ X )
= ( suc @ Y2 ) )
=> ( X = Y2 ) ) ).
% Suc_inject
thf(fact_90_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_91_One__nat__def,axiom,
( ( one_one @ nat )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% One_nat_def
thf(fact_92_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_93_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_94_nat__induct__non__zero,axiom,
! [N: nat,P: nat > $o] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( P @ ( one_one @ nat ) )
=> ( ! [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_non_zero
thf(fact_95_nat_Odistinct_I1_J,axiom,
! [X23: nat] :
( ( zero_zero @ nat )
!= ( suc @ X23 ) ) ).
% nat.distinct(1)
thf(fact_96_old_Onat_Odistinct_I2_J,axiom,
! [Nat3: nat] :
( ( suc @ Nat3 )
!= ( zero_zero @ nat ) ) ).
% old.nat.distinct(2)
thf(fact_97_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( ( zero_zero @ nat )
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_98_nat_OdiscI,axiom,
! [Nat: nat,X23: nat] :
( ( Nat
= ( suc @ X23 ) )
=> ( Nat
!= ( zero_zero @ nat ) ) ) ).
% nat.discI
thf(fact_99_old_Onat_Oexhaust,axiom,
! [Y2: nat] :
( ( Y2
!= ( zero_zero @ nat ) )
=> ~ ! [Nat4: nat] :
( Y2
!= ( suc @ Nat4 ) ) ) ).
% old.nat.exhaust
thf(fact_100_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_101_nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ( P @ ( zero_zero @ nat ) )
=> ( ! [N2: nat] :
( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) )
=> ( P @ N ) ) ) ).
% nat_induct
thf(fact_102_diff__induct,axiom,
! [P: nat > nat > $o,M2: nat,N: nat] :
( ! [X3: nat] : ( P @ X3 @ ( zero_zero @ nat ) )
=> ( ! [Y3: nat] : ( P @ ( zero_zero @ nat ) @ ( suc @ Y3 ) )
=> ( ! [X3: nat,Y3: nat] :
( ( P @ X3 @ Y3 )
=> ( P @ ( suc @ X3 ) @ ( suc @ Y3 ) ) )
=> ( P @ M2 @ N ) ) ) ) ).
% diff_induct
thf(fact_103_zero__induct,axiom,
! [P: nat > $o,K: nat] :
( ( P @ K )
=> ( ! [N2: nat] :
( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) )
=> ( P @ ( zero_zero @ nat ) ) ) ) ).
% zero_induct
thf(fact_104_Suc__neq__Zero,axiom,
! [M2: nat] :
( ( suc @ M2 )
!= ( zero_zero @ nat ) ) ).
% Suc_neq_Zero
thf(fact_105_Zero__neq__Suc,axiom,
! [M2: nat] :
( ( zero_zero @ nat )
!= ( suc @ M2 ) ) ).
% Zero_neq_Suc
thf(fact_106_Zero__not__Suc,axiom,
! [M2: nat] :
( ( zero_zero @ nat )
!= ( suc @ M2 ) ) ).
% Zero_not_Suc
thf(fact_107_not0__implies__Suc,axiom,
! [N: nat] :
( ( N
!= ( zero_zero @ nat ) )
=> ? [M3: nat] :
( N
= ( suc @ M3 ) ) ) ).
% not0_implies_Suc
thf(fact_108_Nat_OlessE,axiom,
! [I: nat,K: nat] :
( ( ord_less @ nat @ I @ K )
=> ( ( K
!= ( suc @ I ) )
=> ~ ! [J: nat] :
( ( ord_less @ nat @ I @ J )
=> ( K
!= ( suc @ J ) ) ) ) ) ).
% Nat.lessE
thf(fact_109_Suc__lessD,axiom,
! [M2: nat,N: nat] :
( ( ord_less @ nat @ ( suc @ M2 ) @ N )
=> ( ord_less @ nat @ M2 @ N ) ) ).
% Suc_lessD
thf(fact_110_Suc__lessE,axiom,
! [I: nat,K: nat] :
( ( ord_less @ nat @ ( suc @ I ) @ K )
=> ~ ! [J: nat] :
( ( ord_less @ nat @ I @ J )
=> ( K
!= ( suc @ J ) ) ) ) ).
% Suc_lessE
thf(fact_111_Suc__lessI,axiom,
! [M2: nat,N: nat] :
( ( ord_less @ nat @ M2 @ N )
=> ( ( ( suc @ M2 )
!= N )
=> ( ord_less @ nat @ ( suc @ M2 ) @ N ) ) ) ).
% Suc_lessI
thf(fact_112_less__SucE,axiom,
! [M2: nat,N: nat] :
( ( ord_less @ nat @ M2 @ ( suc @ N ) )
=> ( ~ ( ord_less @ nat @ M2 @ N )
=> ( M2 = N ) ) ) ).
% less_SucE
thf(fact_113_less__SucI,axiom,
! [M2: nat,N: nat] :
( ( ord_less @ nat @ M2 @ N )
=> ( ord_less @ nat @ M2 @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_114_Ex__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I2: nat] :
( ( ord_less @ nat @ I2 @ ( suc @ N ) )
& ( P @ I2 ) ) )
= ( ( P @ N )
| ? [I2: nat] :
( ( ord_less @ nat @ I2 @ N )
& ( P @ I2 ) ) ) ) ).
% Ex_less_Suc
thf(fact_115_less__Suc__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less @ nat @ M2 @ ( suc @ N ) )
= ( ( ord_less @ nat @ M2 @ N )
| ( M2 = N ) ) ) ).
% less_Suc_eq
thf(fact_116_not__less__eq,axiom,
! [M2: nat,N: nat] :
( ( ~ ( ord_less @ nat @ M2 @ N ) )
= ( ord_less @ nat @ N @ ( suc @ M2 ) ) ) ).
% not_less_eq
thf(fact_117_All__less__Suc,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( suc @ N ) )
=> ( P @ I2 ) ) )
= ( ( P @ N )
& ! [I2: nat] :
( ( ord_less @ nat @ I2 @ N )
=> ( P @ I2 ) ) ) ) ).
% All_less_Suc
thf(fact_118_Suc__less__eq2,axiom,
! [N: nat,M2: nat] :
( ( ord_less @ nat @ ( suc @ N ) @ M2 )
= ( ? [M4: nat] :
( ( M2
= ( suc @ M4 ) )
& ( ord_less @ nat @ N @ M4 ) ) ) ) ).
% Suc_less_eq2
thf(fact_119_less__antisym,axiom,
! [N: nat,M2: nat] :
( ~ ( ord_less @ nat @ N @ M2 )
=> ( ( ord_less @ nat @ N @ ( suc @ M2 ) )
=> ( M2 = N ) ) ) ).
% less_antisym
thf(fact_120_Suc__less__SucD,axiom,
! [M2: nat,N: nat] :
( ( ord_less @ nat @ ( suc @ M2 ) @ ( suc @ N ) )
=> ( ord_less @ nat @ M2 @ N ) ) ).
% Suc_less_SucD
thf(fact_121_less__trans__Suc,axiom,
! [I: nat,J2: nat,K: nat] :
( ( ord_less @ nat @ I @ J2 )
=> ( ( ord_less @ nat @ J2 @ K )
=> ( ord_less @ nat @ ( suc @ I ) @ K ) ) ) ).
% less_trans_Suc
thf(fact_122_less__Suc__induct,axiom,
! [I: nat,J2: nat,P: nat > nat > $o] :
( ( ord_less @ nat @ I @ J2 )
=> ( ! [I3: nat] : ( P @ I3 @ ( suc @ I3 ) )
=> ( ! [I3: nat,J: nat,K2: nat] :
( ( ord_less @ nat @ I3 @ J )
=> ( ( ord_less @ nat @ J @ K2 )
=> ( ( P @ I3 @ J )
=> ( ( P @ J @ K2 )
=> ( P @ I3 @ K2 ) ) ) ) )
=> ( P @ I @ J2 ) ) ) ) ).
% less_Suc_induct
thf(fact_123_strict__inc__induct,axiom,
! [I: nat,J2: nat,P: nat > $o] :
( ( ord_less @ nat @ I @ J2 )
=> ( ! [I3: nat] :
( ( J2
= ( suc @ I3 ) )
=> ( P @ I3 ) )
=> ( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ J2 )
=> ( ( P @ ( suc @ I3 ) )
=> ( P @ I3 ) ) )
=> ( P @ I ) ) ) ) ).
% strict_inc_induct
thf(fact_124_not__less__less__Suc__eq,axiom,
! [N: nat,M2: nat] :
( ~ ( ord_less @ nat @ N @ M2 )
=> ( ( ord_less @ nat @ N @ ( suc @ M2 ) )
= ( N = M2 ) ) ) ).
% not_less_less_Suc_eq
thf(fact_125_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_126_of__nat__less__0__iff,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [M2: nat] :
~ ( ord_less @ A @ ( semiring_1_of_nat @ A @ M2 ) @ ( zero_zero @ A ) ) ) ).
% of_nat_less_0_iff
thf(fact_127_less__imp__of__nat__less,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [M2: nat,N: nat] :
( ( ord_less @ nat @ M2 @ N )
=> ( ord_less @ A @ ( semiring_1_of_nat @ A @ M2 ) @ ( semiring_1_of_nat @ A @ N ) ) ) ) ).
% less_imp_of_nat_less
thf(fact_128_of__nat__less__imp__less,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [M2: nat,N: nat] :
( ( ord_less @ A @ ( semiring_1_of_nat @ A @ M2 ) @ ( semiring_1_of_nat @ A @ N ) )
=> ( ord_less @ nat @ M2 @ N ) ) ) ).
% of_nat_less_imp_less
thf(fact_129_lift__Suc__mono__less,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [F2: nat > A,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less @ A @ ( F2 @ N2 ) @ ( F2 @ ( suc @ N2 ) ) )
=> ( ( ord_less @ nat @ N @ N3 )
=> ( ord_less @ A @ ( F2 @ N ) @ ( F2 @ N3 ) ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_130_lift__Suc__mono__less__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [F2: nat > A,N: nat,M2: nat] :
( ! [N2: nat] : ( ord_less @ A @ ( F2 @ N2 ) @ ( F2 @ ( suc @ N2 ) ) )
=> ( ( ord_less @ A @ ( F2 @ N ) @ ( F2 @ M2 ) )
= ( ord_less @ nat @ N @ M2 ) ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_131_Ex__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ? [I2: nat] :
( ( ord_less @ nat @ I2 @ ( suc @ N ) )
& ( P @ I2 ) ) )
= ( ( P @ ( zero_zero @ nat ) )
| ? [I2: nat] :
( ( ord_less @ nat @ I2 @ N )
& ( P @ ( suc @ I2 ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_132_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
= ( ? [M5: nat] :
( N
= ( suc @ M5 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_133_All__less__Suc2,axiom,
! [N: nat,P: nat > $o] :
( ( ! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( suc @ N ) )
=> ( P @ I2 ) ) )
= ( ( P @ ( zero_zero @ nat ) )
& ! [I2: nat] :
( ( ord_less @ nat @ I2 @ N )
=> ( P @ ( suc @ I2 ) ) ) ) ) ).
% All_less_Suc2
thf(fact_134_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ? [M3: nat] :
( N
= ( suc @ M3 ) ) ) ).
% gr0_implies_Suc
thf(fact_135_less__Suc__eq__0__disj,axiom,
! [M2: nat,N: nat] :
( ( ord_less @ nat @ M2 @ ( suc @ N ) )
= ( ( M2
= ( zero_zero @ nat ) )
| ? [J3: nat] :
( ( M2
= ( suc @ J3 ) )
& ( ord_less @ nat @ J3 @ N ) ) ) ) ).
% less_Suc_eq_0_disj
thf(fact_136_vebt__member_Osimps_I2_J,axiom,
! [Uu: nat,Uv: list @ vEBT_VEBT,Uw: vEBT_VEBT,X: nat] :
~ ( vEBT_vebt_member @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu @ Uv @ Uw ) @ X ) ).
% vebt_member.simps(2)
thf(fact_137_VEBT__internal_OminNull_Osimps_I4_J,axiom,
! [Uw: nat,Ux: list @ vEBT_VEBT,Uy2: vEBT_VEBT] : ( vEBT_VEBT_minNull @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uw @ Ux @ Uy2 ) ) ).
% VEBT_internal.minNull.simps(4)
thf(fact_138_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_139_VEBT__internal_Omembermima_Osimps_I2_J,axiom,
! [Ux: list @ vEBT_VEBT,Uy2: vEBT_VEBT,Uz: nat] :
~ ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( zero_zero @ nat ) @ Ux @ Uy2 ) @ Uz ) ).
% VEBT_internal.membermima.simps(2)
thf(fact_140_invar__vebt_Ointros_I1_J,axiom,
! [A3: $o,B2: $o] : ( vEBT_invar_vebt @ ( vEBT_Leaf @ A3 @ B2 ) @ ( suc @ ( zero_zero @ nat ) ) ) ).
% invar_vebt.intros(1)
thf(fact_141_vebt__buildup_Osimps_I2_J,axiom,
( ( vEBT_vebt_buildup @ ( suc @ ( zero_zero @ nat ) ) )
= ( vEBT_Leaf @ $false @ $false ) ) ).
% vebt_buildup.simps(2)
thf(fact_142_zero__less__one,axiom,
! [A: $tType] :
( ( zero_less_one @ A )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ).
% zero_less_one
thf(fact_143_not__one__less__zero,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ~ ( ord_less @ A @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ).
% not_one_less_zero
thf(fact_144_pos__int__cases,axiom,
! [K: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ K )
=> ~ ! [N2: nat] :
( ( K
= ( semiring_1_of_nat @ int @ N2 ) )
=> ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ).
% pos_int_cases
thf(fact_145_zero__less__imp__eq__int,axiom,
! [K: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ K )
=> ? [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
& ( K
= ( semiring_1_of_nat @ int @ N2 ) ) ) ) ).
% zero_less_imp_eq_int
thf(fact_146_list__decode_Ocases,axiom,
! [X: nat] :
( ( X
!= ( zero_zero @ nat ) )
=> ~ ! [N2: nat] :
( X
!= ( suc @ N2 ) ) ) ).
% list_decode.cases
thf(fact_147_dependent__nat__choice,axiom,
! [A: $tType,P: nat > A > $o,Q: nat > A > A > $o] :
( ? [X_12: A] : ( P @ ( zero_zero @ nat ) @ X_12 )
=> ( ! [X3: A,N2: nat] :
( ( P @ N2 @ X3 )
=> ? [Y: A] :
( ( P @ ( suc @ N2 ) @ Y )
& ( Q @ N2 @ X3 @ Y ) ) )
=> ? [F3: nat > A] :
! [N4: nat] :
( ( P @ N4 @ ( F3 @ N4 ) )
& ( Q @ N4 @ ( F3 @ N4 ) @ ( F3 @ ( suc @ N4 ) ) ) ) ) ) ).
% dependent_nat_choice
thf(fact_148_exists__least__lemma,axiom,
! [P: nat > $o] :
( ~ ( P @ ( zero_zero @ nat ) )
=> ( ? [X_12: nat] : ( P @ X_12 )
=> ? [N2: nat] :
( ~ ( P @ N2 )
& ( P @ ( suc @ N2 ) ) ) ) ) ).
% exists_least_lemma
thf(fact_149_reals__Archimedean2,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X: A] :
? [N2: nat] : ( ord_less @ A @ X @ ( semiring_1_of_nat @ A @ N2 ) ) ) ).
% reals_Archimedean2
thf(fact_150_zero__neq__one,axiom,
! [A: $tType] :
( ( zero_neq_one @ A )
=> ( ( zero_zero @ A )
!= ( one_one @ A ) ) ) ).
% zero_neq_one
thf(fact_151_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_152_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_153_arcosh__1,axiom,
! [A: $tType] :
( ( ln @ A )
=> ( ( arcosh @ A @ ( one_one @ A ) )
= ( zero_zero @ A ) ) ) ).
% arcosh_1
thf(fact_154_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add @ A )
=> ! [A3: A] :
( ( minus_minus @ A @ A3 @ A3 )
= ( zero_zero @ A ) ) ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_155_diff__zero,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add @ A )
=> ! [A3: A] :
( ( minus_minus @ A @ A3 @ ( zero_zero @ A ) )
= A3 ) ) ).
% diff_zero
thf(fact_156_zero__diff,axiom,
! [A: $tType] :
( ( comm_monoid_diff @ A )
=> ! [A3: A] :
( ( minus_minus @ A @ ( zero_zero @ A ) @ A3 )
= ( zero_zero @ A ) ) ) ).
% zero_diff
thf(fact_157_diff__0__right,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A] :
( ( minus_minus @ A @ A3 @ ( zero_zero @ A ) )
= A3 ) ) ).
% diff_0_right
thf(fact_158_diff__self,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A] :
( ( minus_minus @ A @ A3 @ A3 )
= ( zero_zero @ A ) ) ) ).
% diff_self
thf(fact_159_diff__Suc__Suc,axiom,
! [M2: nat,N: nat] :
( ( minus_minus @ nat @ ( suc @ M2 ) @ ( suc @ N ) )
= ( minus_minus @ nat @ M2 @ N ) ) ).
% diff_Suc_Suc
thf(fact_160_Suc__diff__diff,axiom,
! [M2: nat,N: nat,K: nat] :
( ( minus_minus @ nat @ ( minus_minus @ nat @ ( suc @ M2 ) @ N ) @ ( suc @ K ) )
= ( minus_minus @ nat @ ( minus_minus @ nat @ M2 @ N ) @ K ) ) ).
% Suc_diff_diff
thf(fact_161_diff__self__eq__0,axiom,
! [M2: nat] :
( ( minus_minus @ nat @ M2 @ M2 )
= ( zero_zero @ nat ) ) ).
% diff_self_eq_0
thf(fact_162_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus @ nat @ ( zero_zero @ nat ) @ N )
= ( zero_zero @ nat ) ) ).
% diff_0_eq_0
thf(fact_163_diff__gt__0__iff__gt,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( minus_minus @ A @ A3 @ B2 ) )
= ( ord_less @ A @ B2 @ A3 ) ) ) ).
% diff_gt_0_iff_gt
thf(fact_164_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_165_zero__less__diff,axiom,
! [N: nat,M2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ N @ M2 ) )
= ( ord_less @ nat @ M2 @ N ) ) ).
% zero_less_diff
thf(fact_166_diff__Suc__1,axiom,
! [N: nat] :
( ( minus_minus @ nat @ ( suc @ N ) @ ( one_one @ nat ) )
= N ) ).
% diff_Suc_1
thf(fact_167_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_168_int__int__eq,axiom,
! [M2: nat,N: nat] :
( ( ( semiring_1_of_nat @ int @ M2 )
= ( semiring_1_of_nat @ int @ N ) )
= ( M2 = N ) ) ).
% int_int_eq
thf(fact_169_diff__eq__diff__eq,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B2: A,C2: A,D2: A] :
( ( ( minus_minus @ A @ A3 @ B2 )
= ( minus_minus @ A @ C2 @ D2 ) )
=> ( ( A3 = B2 )
= ( C2 = D2 ) ) ) ) ).
% diff_eq_diff_eq
thf(fact_170_diff__right__commute,axiom,
! [A: $tType] :
( ( cancel2418104881723323429up_add @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( minus_minus @ A @ ( minus_minus @ A @ A3 @ C2 ) @ B2 )
= ( minus_minus @ A @ ( minus_minus @ A @ A3 @ B2 ) @ C2 ) ) ) ).
% diff_right_commute
thf(fact_171_diff__commute,axiom,
! [I: nat,J2: nat,K: nat] :
( ( minus_minus @ nat @ ( minus_minus @ nat @ I @ J2 ) @ K )
= ( minus_minus @ nat @ ( minus_minus @ nat @ I @ K ) @ J2 ) ) ).
% diff_commute
thf(fact_172_less__int__code_I1_J,axiom,
~ ( ord_less @ int @ ( zero_zero @ int ) @ ( zero_zero @ int ) ) ).
% less_int_code(1)
thf(fact_173_eq__iff__diff__eq__0,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ( ( ^ [Y4: A,Z: A] : Y4 = Z )
= ( ^ [A5: A,B3: A] :
( ( minus_minus @ A @ A5 @ B3 )
= ( zero_zero @ A ) ) ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_174_diff__strict__right__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ord_less @ A @ ( minus_minus @ A @ A3 @ C2 ) @ ( minus_minus @ A @ B2 @ C2 ) ) ) ) ).
% diff_strict_right_mono
thf(fact_175_diff__strict__left__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( ord_less @ A @ B2 @ A3 )
=> ( ord_less @ A @ ( minus_minus @ A @ C2 @ A3 ) @ ( minus_minus @ A @ C2 @ B2 ) ) ) ) ).
% diff_strict_left_mono
thf(fact_176_diff__eq__diff__less,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B2: A,C2: A,D2: A] :
( ( ( minus_minus @ A @ A3 @ B2 )
= ( minus_minus @ A @ C2 @ D2 ) )
=> ( ( ord_less @ A @ A3 @ B2 )
= ( ord_less @ A @ C2 @ D2 ) ) ) ) ).
% diff_eq_diff_less
thf(fact_177_diff__strict__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B2: A,D2: A,C2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less @ A @ D2 @ C2 )
=> ( ord_less @ A @ ( minus_minus @ A @ A3 @ C2 ) @ ( minus_minus @ A @ B2 @ D2 ) ) ) ) ) ).
% diff_strict_mono
thf(fact_178_zero__induct__lemma,axiom,
! [P: nat > $o,K: nat,I: nat] :
( ( P @ K )
=> ( ! [N2: nat] :
( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) )
=> ( P @ ( minus_minus @ nat @ K @ I ) ) ) ) ).
% zero_induct_lemma
thf(fact_179_diffs0__imp__equal,axiom,
! [M2: nat,N: nat] :
( ( ( minus_minus @ nat @ M2 @ N )
= ( zero_zero @ nat ) )
=> ( ( ( minus_minus @ nat @ N @ M2 )
= ( zero_zero @ nat ) )
=> ( M2 = N ) ) ) ).
% diffs0_imp_equal
thf(fact_180_minus__nat_Odiff__0,axiom,
! [M2: nat] :
( ( minus_minus @ nat @ M2 @ ( zero_zero @ nat ) )
= M2 ) ).
% minus_nat.diff_0
thf(fact_181_less__imp__diff__less,axiom,
! [J2: nat,K: nat,N: nat] :
( ( ord_less @ nat @ J2 @ K )
=> ( ord_less @ nat @ ( minus_minus @ nat @ J2 @ N ) @ K ) ) ).
% less_imp_diff_less
thf(fact_182_diff__less__mono2,axiom,
! [M2: nat,N: nat,L: nat] :
( ( ord_less @ nat @ M2 @ N )
=> ( ( ord_less @ nat @ M2 @ L )
=> ( ord_less @ nat @ ( minus_minus @ nat @ L @ N ) @ ( minus_minus @ nat @ L @ M2 ) ) ) ) ).
% diff_less_mono2
thf(fact_183_less__iff__diff__less__0,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ( ( ord_less @ A )
= ( ^ [A5: A,B3: A] : ( ord_less @ A @ ( minus_minus @ A @ A5 @ B3 ) @ ( zero_zero @ A ) ) ) ) ) ).
% less_iff_diff_less_0
thf(fact_184_Suc__diff__Suc,axiom,
! [N: nat,M2: nat] :
( ( ord_less @ nat @ N @ M2 )
=> ( ( suc @ ( minus_minus @ nat @ M2 @ ( suc @ N ) ) )
= ( minus_minus @ nat @ M2 @ N ) ) ) ).
% Suc_diff_Suc
thf(fact_185_diff__less__Suc,axiom,
! [M2: nat,N: nat] : ( ord_less @ nat @ ( minus_minus @ nat @ M2 @ N ) @ ( suc @ M2 ) ) ).
% diff_less_Suc
thf(fact_186_diff__less,axiom,
! [N: nat,M2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M2 )
=> ( ord_less @ nat @ ( minus_minus @ nat @ M2 @ N ) @ M2 ) ) ) ).
% diff_less
thf(fact_187_diff__Suc__eq__diff__pred,axiom,
! [M2: nat,N: nat] :
( ( minus_minus @ nat @ M2 @ ( suc @ N ) )
= ( minus_minus @ nat @ ( minus_minus @ nat @ M2 @ ( one_one @ nat ) ) @ N ) ) ).
% diff_Suc_eq_diff_pred
thf(fact_188_diff__Suc__less,axiom,
! [N: nat,I: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ord_less @ nat @ ( minus_minus @ nat @ N @ ( suc @ I ) ) @ N ) ) ).
% diff_Suc_less
thf(fact_189_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_190_Suc__diff__eq__diff__pred,axiom,
! [N: nat,M2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( minus_minus @ nat @ ( suc @ M2 ) @ N )
= ( minus_minus @ nat @ M2 @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ).
% Suc_diff_eq_diff_pred
thf(fact_191_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_192_Suc__if__eq,axiom,
! [A: $tType,F2: nat > A,H: nat > A,G: A,N: nat] :
( ! [N2: nat] :
( ( F2 @ ( suc @ N2 ) )
= ( H @ N2 ) )
=> ( ( ( F2 @ ( zero_zero @ nat ) )
= G )
=> ( ( ( N
= ( zero_zero @ nat ) )
=> ( ( F2 @ N )
= G ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ( ( F2 @ N )
= ( H @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% Suc_if_eq
thf(fact_193_artanh__0,axiom,
! [A: $tType] :
( ( ( real_V3459762299906320749_field @ A )
& ( ln @ A ) )
=> ( ( artanh @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% artanh_0
thf(fact_194_arsinh__0,axiom,
! [A: $tType] :
( ( ln @ A )
=> ( ( arsinh @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% arsinh_0
thf(fact_195_nat__int__comparison_I2_J,axiom,
( ( ord_less @ nat )
= ( ^ [A5: nat,B3: nat] : ( ord_less @ int @ ( semiring_1_of_nat @ int @ A5 ) @ ( semiring_1_of_nat @ int @ B3 ) ) ) ) ).
% nat_int_comparison(2)
thf(fact_196_int__ops_I1_J,axiom,
( ( semiring_1_of_nat @ int @ ( zero_zero @ nat ) )
= ( zero_zero @ int ) ) ).
% int_ops(1)
thf(fact_197_option_Osize_I3_J,axiom,
! [A: $tType] :
( ( size_size @ ( option @ A ) @ ( none @ A ) )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% option.size(3)
thf(fact_198_minus__apply,axiom,
! [B: $tType,A: $tType] :
( ( minus @ B )
=> ( ( minus_minus @ ( A > B ) )
= ( ^ [A7: A > B,B5: A > B,X2: A] : ( minus_minus @ B @ ( A7 @ X2 ) @ ( B5 @ X2 ) ) ) ) ) ).
% minus_apply
thf(fact_199_ln__one,axiom,
! [A: $tType] :
( ( ln @ A )
=> ( ( ln_ln @ A @ ( one_one @ A ) )
= ( zero_zero @ A ) ) ) ).
% ln_one
thf(fact_200_int__ops_I2_J,axiom,
( ( semiring_1_of_nat @ int @ ( one_one @ nat ) )
= ( one_one @ int ) ) ).
% int_ops(2)
thf(fact_201_neg__int__cases,axiom,
! [K: int] :
( ( ord_less @ int @ K @ ( zero_zero @ int ) )
=> ~ ! [N2: nat] :
( ( K
= ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N2 ) ) )
=> ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ).
% neg_int_cases
thf(fact_202_nat__approx__posE,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [E2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ E2 )
=> ~ ! [N2: nat] :
~ ( ord_less @ A @ ( divide_divide @ A @ ( one_one @ A ) @ ( semiring_1_of_nat @ A @ ( suc @ N2 ) ) ) @ E2 ) ) ) ).
% nat_approx_posE
thf(fact_203_neg__equal__iff__equal,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B2: A] :
( ( ( uminus_uminus @ A @ A3 )
= ( uminus_uminus @ A @ B2 ) )
= ( A3 = B2 ) ) ) ).
% neg_equal_iff_equal
thf(fact_204_add_Oinverse__inverse,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A] :
( ( uminus_uminus @ A @ ( uminus_uminus @ A @ A3 ) )
= A3 ) ) ).
% add.inverse_inverse
thf(fact_205_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_206_uminus__apply,axiom,
! [B: $tType,A: $tType] :
( ( uminus @ B )
=> ( ( uminus_uminus @ ( A > B ) )
= ( ^ [A7: A > B,X2: A] : ( uminus_uminus @ B @ ( A7 @ X2 ) ) ) ) ) ).
% uminus_apply
thf(fact_207_div__0,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [A3: A] :
( ( divide_divide @ A @ ( zero_zero @ A ) @ A3 )
= ( zero_zero @ A ) ) ) ).
% div_0
thf(fact_208_div__by__0,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [A3: A] :
( ( divide_divide @ A @ A3 @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% div_by_0
thf(fact_209_add_Oinverse__neutral,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ( ( uminus_uminus @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% add.inverse_neutral
thf(fact_210_neg__0__equal__iff__equal,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A] :
( ( ( zero_zero @ A )
= ( uminus_uminus @ A @ A3 ) )
= ( ( zero_zero @ A )
= A3 ) ) ) ).
% neg_0_equal_iff_equal
thf(fact_211_neg__equal__0__iff__equal,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A] :
( ( ( uminus_uminus @ A @ A3 )
= ( zero_zero @ A ) )
= ( A3
= ( zero_zero @ A ) ) ) ) ).
% neg_equal_0_iff_equal
thf(fact_212_equal__neg__zero,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A3: A] :
( ( A3
= ( uminus_uminus @ A @ A3 ) )
= ( A3
= ( zero_zero @ A ) ) ) ) ).
% equal_neg_zero
thf(fact_213_neg__equal__zero,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A3: A] :
( ( ( uminus_uminus @ A @ A3 )
= A3 )
= ( A3
= ( zero_zero @ A ) ) ) ) ).
% neg_equal_zero
thf(fact_214_neg__less__iff__less,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [B2: A,A3: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A3 ) )
= ( ord_less @ A @ A3 @ B2 ) ) ) ).
% neg_less_iff_less
thf(fact_215_div__by__1,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [A3: A] :
( ( divide_divide @ A @ A3 @ ( one_one @ A ) )
= A3 ) ) ).
% div_by_1
thf(fact_216_minus__diff__eq,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B2: A] :
( ( uminus_uminus @ A @ ( minus_minus @ A @ A3 @ B2 ) )
= ( minus_minus @ A @ B2 @ A3 ) ) ) ).
% minus_diff_eq
thf(fact_217_less__neg__neg,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A3: A] :
( ( ord_less @ A @ A3 @ ( uminus_uminus @ A @ A3 ) )
= ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% less_neg_neg
thf(fact_218_neg__less__pos,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ A3 ) @ A3 )
= ( ord_less @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% neg_less_pos
thf(fact_219_neg__0__less__iff__less,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ A3 ) )
= ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% neg_0_less_iff_less
thf(fact_220_neg__less__0__iff__less,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ A3 ) @ ( zero_zero @ A ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% neg_less_0_iff_less
thf(fact_221_div__self,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [A3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ A3 @ A3 )
= ( one_one @ A ) ) ) ) ).
% div_self
thf(fact_222_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_223_diff__0,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A] :
( ( minus_minus @ A @ ( zero_zero @ A ) @ A3 )
= ( uminus_uminus @ A @ A3 ) ) ) ).
% diff_0
thf(fact_224_negative__eq__positive,axiom,
! [N: nat,M2: nat] :
( ( ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N ) )
= ( semiring_1_of_nat @ int @ M2 ) )
= ( ( N
= ( zero_zero @ nat ) )
& ( M2
= ( zero_zero @ nat ) ) ) ) ).
% negative_eq_positive
thf(fact_225_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_226_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_227_negative__zless,axiom,
! [N: nat,M2: nat] : ( ord_less @ int @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ ( suc @ N ) ) ) @ ( semiring_1_of_nat @ int @ M2 ) ) ).
% negative_zless
thf(fact_228_minus__equation__iff,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B2: A] :
( ( ( uminus_uminus @ A @ A3 )
= B2 )
= ( ( uminus_uminus @ A @ B2 )
= A3 ) ) ) ).
% minus_equation_iff
thf(fact_229_equation__minus__iff,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B2: A] :
( ( A3
= ( uminus_uminus @ A @ B2 ) )
= ( B2
= ( uminus_uminus @ A @ A3 ) ) ) ) ).
% equation_minus_iff
thf(fact_230_verit__negate__coefficient_I3_J,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B2: A] :
( ( A3 = B2 )
=> ( ( uminus_uminus @ A @ A3 )
= ( uminus_uminus @ A @ B2 ) ) ) ) ).
% verit_negate_coefficient(3)
thf(fact_231_fun__Compl__def,axiom,
! [B: $tType,A: $tType] :
( ( uminus @ B )
=> ( ( uminus_uminus @ ( A > B ) )
= ( ^ [A7: A > B,X2: A] : ( uminus_uminus @ B @ ( A7 @ X2 ) ) ) ) ) ).
% fun_Compl_def
thf(fact_232_minus__int__code_I2_J,axiom,
! [L: int] :
( ( minus_minus @ int @ ( zero_zero @ int ) @ L )
= ( uminus_uminus @ int @ L ) ) ).
% minus_int_code(2)
thf(fact_233_verit__negate__coefficient_I2_J,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A3 ) ) ) ) ).
% verit_negate_coefficient(2)
thf(fact_234_less__minus__iff,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ ( uminus_uminus @ A @ B2 ) )
= ( ord_less @ A @ B2 @ ( uminus_uminus @ A @ A3 ) ) ) ) ).
% less_minus_iff
thf(fact_235_minus__less__iff,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ A3 ) @ B2 )
= ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ A3 ) ) ) ).
% minus_less_iff
thf(fact_236_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_237_minus__diff__commute,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [B2: A,A3: A] :
( ( minus_minus @ A @ ( uminus_uminus @ A @ B2 ) @ A3 )
= ( minus_minus @ A @ ( uminus_uminus @ A @ A3 ) @ B2 ) ) ) ).
% minus_diff_commute
thf(fact_238_minus__diff__minus,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A3: A,B2: A] :
( ( minus_minus @ A @ ( uminus_uminus @ A @ A3 ) @ ( uminus_uminus @ A @ B2 ) )
= ( uminus_uminus @ A @ ( minus_minus @ A @ A3 @ B2 ) ) ) ) ).
% minus_diff_minus
thf(fact_239_minus__int__code_I1_J,axiom,
! [K: int] :
( ( minus_minus @ int @ K @ ( zero_zero @ int ) )
= K ) ).
% minus_int_code(1)
thf(fact_240_uminus__int__code_I1_J,axiom,
( ( uminus_uminus @ int @ ( zero_zero @ int ) )
= ( zero_zero @ int ) ) ).
% uminus_int_code(1)
thf(fact_241_int__less__induct,axiom,
! [I: int,K: int,P: int > $o] :
( ( ord_less @ int @ I @ K )
=> ( ( P @ ( minus_minus @ int @ K @ ( one_one @ int ) ) )
=> ( ! [I3: int] :
( ( ord_less @ int @ I3 @ K )
=> ( ( P @ I3 )
=> ( P @ ( minus_minus @ int @ I3 @ ( one_one @ int ) ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_less_induct
thf(fact_242_int__diff__cases,axiom,
! [Z2: int] :
~ ! [M3: nat,N2: nat] :
( Z2
!= ( minus_minus @ int @ ( semiring_1_of_nat @ int @ M3 ) @ ( semiring_1_of_nat @ int @ N2 ) ) ) ).
% int_diff_cases
thf(fact_243_int__cases2,axiom,
! [Z2: int] :
( ! [N2: nat] :
( Z2
!= ( semiring_1_of_nat @ int @ N2 ) )
=> ~ ! [N2: nat] :
( Z2
!= ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N2 ) ) ) ) ).
% int_cases2
thf(fact_244_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_245_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_246_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_247_int__cases,axiom,
! [Z2: int] :
( ! [N2: nat] :
( Z2
!= ( semiring_1_of_nat @ int @ N2 ) )
=> ~ ! [N2: nat] :
( Z2
!= ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ ( suc @ N2 ) ) ) ) ) ).
% int_cases
thf(fact_248_int__of__nat__induct,axiom,
! [P: int > $o,Z2: int] :
( ! [N2: nat] : ( P @ ( semiring_1_of_nat @ int @ N2 ) )
=> ( ! [N2: nat] : ( P @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ ( suc @ N2 ) ) ) )
=> ( P @ Z2 ) ) ) ).
% int_of_nat_induct
thf(fact_249_not__int__zless__negative,axiom,
! [N: nat,M2: nat] :
~ ( ord_less @ int @ ( semiring_1_of_nat @ int @ N ) @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ M2 ) ) ) ).
% not_int_zless_negative
thf(fact_250_int__ops_I6_J,axiom,
! [A3: nat,B2: nat] :
( ( ( ord_less @ int @ ( semiring_1_of_nat @ int @ A3 ) @ ( semiring_1_of_nat @ int @ B2 ) )
=> ( ( semiring_1_of_nat @ int @ ( minus_minus @ nat @ A3 @ B2 ) )
= ( zero_zero @ int ) ) )
& ( ~ ( ord_less @ int @ ( semiring_1_of_nat @ int @ A3 ) @ ( semiring_1_of_nat @ int @ B2 ) )
=> ( ( semiring_1_of_nat @ int @ ( minus_minus @ nat @ A3 @ B2 ) )
= ( minus_minus @ int @ ( semiring_1_of_nat @ int @ A3 ) @ ( semiring_1_of_nat @ int @ B2 ) ) ) ) ) ).
% int_ops(6)
thf(fact_251_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_252_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_253_int__cases4,axiom,
! [M2: int] :
( ! [N2: nat] :
( M2
!= ( semiring_1_of_nat @ int @ N2 ) )
=> ~ ! [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( M2
!= ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N2 ) ) ) ) ) ).
% int_cases4
thf(fact_254_verit__comp__simplify1_I1_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A] :
~ ( ord_less @ A @ A3 @ A3 ) ) ).
% verit_comp_simplify1(1)
thf(fact_255_fun__diff__def,axiom,
! [B: $tType,A: $tType] :
( ( minus @ B )
=> ( ( minus_minus @ ( A > B ) )
= ( ^ [A7: A > B,B5: A > B,X2: A] : ( minus_minus @ B @ ( A7 @ X2 ) @ ( B5 @ X2 ) ) ) ) ) ).
% fun_diff_def
thf(fact_256_option_Osize__neq,axiom,
! [A: $tType,X: option @ A] :
( ( size_size @ ( option @ A ) @ X )
!= ( zero_zero @ nat ) ) ).
% option.size_neq
thf(fact_257_int__if,axiom,
! [P: $o,A3: nat,B2: nat] :
( ( P
=> ( ( semiring_1_of_nat @ int @ ( if @ nat @ P @ A3 @ B2 ) )
= ( semiring_1_of_nat @ int @ A3 ) ) )
& ( ~ P
=> ( ( semiring_1_of_nat @ int @ ( if @ nat @ P @ A3 @ B2 ) )
= ( semiring_1_of_nat @ int @ B2 ) ) ) ) ).
% int_if
thf(fact_258_nat__int__comparison_I1_J,axiom,
( ( ^ [Y4: nat,Z: nat] : Y4 = Z )
= ( ^ [A5: nat,B3: nat] :
( ( semiring_1_of_nat @ int @ A5 )
= ( semiring_1_of_nat @ int @ B3 ) ) ) ) ).
% nat_int_comparison(1)
thf(fact_259_int__cases3,axiom,
! [K: int] :
( ( K
!= ( zero_zero @ int ) )
=> ( ! [N2: nat] :
( ( K
= ( semiring_1_of_nat @ int @ N2 ) )
=> ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) )
=> ~ ! [N2: nat] :
( ( K
= ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N2 ) ) )
=> ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ) ).
% int_cases3
thf(fact_260_negD,axiom,
! [X: int] :
( ( ord_less @ int @ X @ ( zero_zero @ int ) )
=> ? [N2: nat] :
( X
= ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ ( suc @ N2 ) ) ) ) ) ).
% negD
thf(fact_261_negative__zless__0,axiom,
! [N: nat] : ( ord_less @ int @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ ( suc @ N ) ) ) @ ( zero_zero @ int ) ) ).
% negative_zless_0
thf(fact_262_divide__less__0__1__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( divide_divide @ A @ ( one_one @ A ) @ A3 ) @ ( zero_zero @ A ) )
= ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% divide_less_0_1_iff
thf(fact_263_divide__less__eq__1__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( divide_divide @ A @ B2 @ A3 ) @ ( one_one @ A ) )
= ( ord_less @ A @ A3 @ B2 ) ) ) ) ).
% divide_less_eq_1_neg
thf(fact_264_divide__less__eq__1__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ ( divide_divide @ A @ B2 @ A3 ) @ ( one_one @ A ) )
= ( ord_less @ A @ B2 @ A3 ) ) ) ) ).
% divide_less_eq_1_pos
thf(fact_265_less__divide__eq__1__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B2 @ A3 ) )
= ( ord_less @ A @ B2 @ A3 ) ) ) ) ).
% less_divide_eq_1_neg
thf(fact_266_less__divide__eq__1__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B2 @ A3 ) )
= ( ord_less @ A @ A3 @ B2 ) ) ) ) ).
% less_divide_eq_1_pos
thf(fact_267_zero__less__divide__1__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ ( one_one @ A ) @ A3 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% zero_less_divide_1_iff
thf(fact_268_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_269_div__minus1__right,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel @ A )
=> ! [A3: A] :
( ( divide_divide @ A @ A3 @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( uminus_uminus @ A @ A3 ) ) ) ).
% div_minus1_right
thf(fact_270_divide__eq__1__iff,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A3: A,B2: A] :
( ( ( divide_divide @ A @ A3 @ B2 )
= ( one_one @ A ) )
= ( ( B2
!= ( zero_zero @ A ) )
& ( A3 = B2 ) ) ) ) ).
% divide_eq_1_iff
thf(fact_271_one__eq__divide__iff,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A3: A,B2: A] :
( ( ( one_one @ A )
= ( divide_divide @ A @ A3 @ B2 ) )
= ( ( B2
!= ( zero_zero @ A ) )
& ( A3 = B2 ) ) ) ) ).
% one_eq_divide_iff
thf(fact_272_divide__self,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ A3 @ A3 )
= ( one_one @ A ) ) ) ) ).
% divide_self
thf(fact_273_division__ring__divide__zero,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A] :
( ( divide_divide @ A @ A3 @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% division_ring_divide_zero
thf(fact_274_divide__cancel__right,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( ( divide_divide @ A @ A3 @ C2 )
= ( divide_divide @ A @ B2 @ C2 ) )
= ( ( C2
= ( zero_zero @ A ) )
| ( A3 = B2 ) ) ) ) ).
% divide_cancel_right
thf(fact_275_divide__cancel__left,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( ( divide_divide @ A @ C2 @ A3 )
= ( divide_divide @ A @ C2 @ B2 ) )
= ( ( C2
= ( zero_zero @ A ) )
| ( A3 = B2 ) ) ) ) ).
% divide_cancel_left
thf(fact_276_divide__eq__0__iff,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A3: A,B2: A] :
( ( ( divide_divide @ A @ A3 @ B2 )
= ( zero_zero @ A ) )
= ( ( A3
= ( zero_zero @ A ) )
| ( B2
= ( zero_zero @ A ) ) ) ) ) ).
% divide_eq_0_iff
thf(fact_277_div__minus__minus,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel @ A )
=> ! [A3: A,B2: A] :
( ( divide_divide @ A @ ( uminus_uminus @ A @ A3 ) @ ( uminus_uminus @ A @ B2 ) )
= ( divide_divide @ A @ A3 @ B2 ) ) ) ).
% div_minus_minus
thf(fact_278_div__by__Suc__0,axiom,
! [M2: nat] :
( ( divide_divide @ nat @ M2 @ ( suc @ ( zero_zero @ nat ) ) )
= M2 ) ).
% div_by_Suc_0
thf(fact_279_div__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less @ nat @ M2 @ N )
=> ( ( divide_divide @ nat @ M2 @ N )
= ( zero_zero @ nat ) ) ) ).
% div_less
thf(fact_280_zero__eq__1__divide__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A] :
( ( ( zero_zero @ A )
= ( divide_divide @ A @ ( one_one @ A ) @ A3 ) )
= ( A3
= ( zero_zero @ A ) ) ) ) ).
% zero_eq_1_divide_iff
thf(fact_281_one__divide__eq__0__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A] :
( ( ( divide_divide @ A @ ( one_one @ A ) @ A3 )
= ( zero_zero @ A ) )
= ( A3
= ( zero_zero @ A ) ) ) ) ).
% one_divide_eq_0_iff
thf(fact_282_eq__divide__eq__1,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,A3: A] :
( ( ( one_one @ A )
= ( divide_divide @ A @ B2 @ A3 ) )
= ( ( A3
!= ( zero_zero @ A ) )
& ( A3 = B2 ) ) ) ) ).
% eq_divide_eq_1
thf(fact_283_divide__eq__eq__1,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,A3: A] :
( ( ( divide_divide @ A @ B2 @ A3 )
= ( one_one @ A ) )
= ( ( A3
!= ( zero_zero @ A ) )
& ( A3 = B2 ) ) ) ) ).
% divide_eq_eq_1
thf(fact_284_divide__self__if,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A] :
( ( ( A3
= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ A3 @ A3 )
= ( zero_zero @ A ) ) )
& ( ( A3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ A3 @ A3 )
= ( one_one @ A ) ) ) ) ) ).
% divide_self_if
thf(fact_285_linordered__field__no__ub,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X4: A] :
? [X_1: A] : ( ord_less @ A @ X4 @ X_1 ) ) ).
% linordered_field_no_ub
thf(fact_286_linordered__field__no__lb,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X4: A] :
? [Y3: A] : ( ord_less @ A @ Y3 @ X4 ) ) ).
% linordered_field_no_lb
thf(fact_287_diff__divide__distrib,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( divide_divide @ A @ ( minus_minus @ A @ A3 @ B2 ) @ C2 )
= ( minus_minus @ A @ ( divide_divide @ A @ A3 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) ) ) ) ).
% diff_divide_distrib
thf(fact_288_div__minus__right,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel @ A )
=> ! [A3: A,B2: A] :
( ( divide_divide @ A @ A3 @ ( uminus_uminus @ A @ B2 ) )
= ( divide_divide @ A @ ( uminus_uminus @ A @ A3 ) @ B2 ) ) ) ).
% div_minus_right
thf(fact_289_minus__divide__left,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A,B2: A] :
( ( uminus_uminus @ A @ ( divide_divide @ A @ A3 @ B2 ) )
= ( divide_divide @ A @ ( uminus_uminus @ A @ A3 ) @ B2 ) ) ) ).
% minus_divide_left
thf(fact_290_minus__divide__divide,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A3: A,B2: A] :
( ( divide_divide @ A @ ( uminus_uminus @ A @ A3 ) @ ( uminus_uminus @ A @ B2 ) )
= ( divide_divide @ A @ A3 @ B2 ) ) ) ).
% minus_divide_divide
thf(fact_291_minus__divide__right,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A3: A,B2: A] :
( ( uminus_uminus @ A @ ( divide_divide @ A @ A3 @ B2 ) )
= ( divide_divide @ A @ A3 @ ( uminus_uminus @ A @ B2 ) ) ) ) ).
% minus_divide_right
thf(fact_292_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [M2: nat,N: nat] :
( ( semiring_1_of_nat @ A @ ( divide_divide @ nat @ M2 @ N ) )
= ( divide_divide @ A @ ( semiring_1_of_nat @ A @ M2 ) @ ( semiring_1_of_nat @ A @ N ) ) ) ) ).
% unique_euclidean_semiring_with_nat_class.of_nat_div
thf(fact_293_Euclidean__Division_Odiv__eq__0__iff,axiom,
! [M2: nat,N: nat] :
( ( ( divide_divide @ nat @ M2 @ N )
= ( zero_zero @ nat ) )
= ( ( ord_less @ nat @ M2 @ N )
| ( N
= ( zero_zero @ nat ) ) ) ) ).
% Euclidean_Division.div_eq_0_iff
thf(fact_294_divide__strict__right__mono__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( ord_less @ A @ B2 @ A3 )
=> ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( divide_divide @ A @ A3 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) ) ) ) ) ).
% divide_strict_right_mono_neg
thf(fact_295_divide__strict__right__mono,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( divide_divide @ A @ A3 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) ) ) ) ) ).
% divide_strict_right_mono
thf(fact_296_zero__less__divide__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ A3 @ B2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
& ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) )
| ( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
& ( ord_less @ A @ B2 @ ( zero_zero @ A ) ) ) ) ) ) ).
% zero_less_divide_iff
thf(fact_297_divide__less__cancel,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( ord_less @ A @ ( divide_divide @ A @ A3 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ A3 @ B2 ) )
& ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B2 @ A3 ) )
& ( C2
!= ( zero_zero @ A ) ) ) ) ) ).
% divide_less_cancel
thf(fact_298_divide__less__0__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ ( divide_divide @ A @ A3 @ B2 ) @ ( zero_zero @ A ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
& ( ord_less @ A @ B2 @ ( zero_zero @ A ) ) )
| ( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
& ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) ) ) ) ) ).
% divide_less_0_iff
thf(fact_299_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_300_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_301_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_302_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_303_right__inverse__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B2: A,A3: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( ( divide_divide @ A @ A3 @ B2 )
= ( one_one @ A ) )
= ( A3 = B2 ) ) ) ) ).
% right_inverse_eq
thf(fact_304_nonzero__minus__divide__right,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B2: A,A3: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( uminus_uminus @ A @ ( divide_divide @ A @ A3 @ B2 ) )
= ( divide_divide @ A @ A3 @ ( uminus_uminus @ A @ B2 ) ) ) ) ) ).
% nonzero_minus_divide_right
thf(fact_305_nonzero__minus__divide__divide,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B2: A,A3: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( uminus_uminus @ A @ A3 ) @ ( uminus_uminus @ A @ B2 ) )
= ( divide_divide @ A @ A3 @ B2 ) ) ) ) ).
% nonzero_minus_divide_divide
thf(fact_306_div__eq__dividend__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M2 )
=> ( ( ( divide_divide @ nat @ M2 @ N )
= M2 )
= ( N
= ( one_one @ nat ) ) ) ) ).
% div_eq_dividend_iff
thf(fact_307_div__less__dividend,axiom,
! [N: nat,M2: nat] :
( ( ord_less @ nat @ ( one_one @ nat ) @ N )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M2 )
=> ( ord_less @ nat @ ( divide_divide @ nat @ M2 @ N ) @ M2 ) ) ) ).
% div_less_dividend
thf(fact_308_less__divide__eq__1,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,A3: A] :
( ( ord_less @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B2 @ A3 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
& ( ord_less @ A @ A3 @ B2 ) )
| ( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
& ( ord_less @ A @ B2 @ A3 ) ) ) ) ) ).
% less_divide_eq_1
thf(fact_309_divide__less__eq__1,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,A3: A] :
( ( ord_less @ A @ ( divide_divide @ A @ B2 @ A3 ) @ ( one_one @ A ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
& ( ord_less @ A @ B2 @ A3 ) )
| ( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
& ( ord_less @ A @ A3 @ B2 ) )
| ( A3
= ( zero_zero @ A ) ) ) ) ) ).
% divide_less_eq_1
thf(fact_310_divide__eq__minus__1__iff,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A3: A,B2: A] :
( ( ( divide_divide @ A @ A3 @ B2 )
= ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( ( B2
!= ( zero_zero @ A ) )
& ( A3
= ( uminus_uminus @ A @ B2 ) ) ) ) ) ).
% divide_eq_minus_1_iff
thf(fact_311_div__if,axiom,
( ( divide_divide @ nat )
= ( ^ [M5: nat,N5: nat] :
( if @ nat
@ ( ( ord_less @ nat @ M5 @ N5 )
| ( N5
= ( zero_zero @ nat ) ) )
@ ( zero_zero @ nat )
@ ( suc @ ( divide_divide @ nat @ ( minus_minus @ nat @ M5 @ N5 ) @ N5 ) ) ) ) ) ).
% div_if
thf(fact_312_bits__div__by__1,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A] :
( ( divide_divide @ A @ A3 @ ( one_one @ A ) )
= A3 ) ) ).
% bits_div_by_1
thf(fact_313_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_314_bits__div__by__0,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A] :
( ( divide_divide @ A @ A3 @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% bits_div_by_0
thf(fact_315_bits__div__0,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A] :
( ( divide_divide @ A @ ( zero_zero @ A ) @ A3 )
= ( zero_zero @ A ) ) ) ).
% bits_div_0
thf(fact_316_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_317_div__geq,axiom,
! [N: nat,M2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ~ ( ord_less @ nat @ M2 @ N )
=> ( ( divide_divide @ nat @ M2 @ N )
= ( suc @ ( divide_divide @ nat @ ( minus_minus @ nat @ M2 @ N ) @ N ) ) ) ) ) ).
% div_geq
thf(fact_318_int__div__less__self,axiom,
! [X: int,K: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ X )
=> ( ( ord_less @ int @ ( one_one @ int ) @ K )
=> ( ord_less @ int @ ( divide_divide @ int @ X @ K ) @ X ) ) ) ).
% int_div_less_self
thf(fact_319_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_320_boolean__algebra__class_Oboolean__algebra_Ocompl__eq__compl__iff,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y2: A] :
( ( ( uminus_uminus @ A @ X )
= ( uminus_uminus @ A @ Y2 ) )
= ( X = Y2 ) ) ) ).
% boolean_algebra_class.boolean_algebra.compl_eq_compl_iff
thf(fact_321_boolean__algebra__class_Oboolean__algebra_Odouble__compl,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A] :
( ( uminus_uminus @ A @ ( uminus_uminus @ A @ X ) )
= X ) ) ).
% boolean_algebra_class.boolean_algebra.double_compl
thf(fact_322_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_323_zdiv__int,axiom,
! [A3: nat,B2: nat] :
( ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ A3 @ B2 ) )
= ( divide_divide @ int @ ( semiring_1_of_nat @ int @ A3 ) @ ( semiring_1_of_nat @ int @ B2 ) ) ) ).
% zdiv_int
thf(fact_324_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_325_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_326_div__neg__pos__less0,axiom,
! [A3: int,B2: int] :
( ( ord_less @ int @ A3 @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
=> ( ord_less @ int @ ( divide_divide @ int @ A3 @ B2 ) @ ( zero_zero @ int ) ) ) ) ).
% div_neg_pos_less0
thf(fact_327_neg__imp__zdiv__neg__iff,axiom,
! [B2: int,A3: int] :
( ( ord_less @ int @ B2 @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ ( divide_divide @ int @ A3 @ B2 ) @ ( zero_zero @ int ) )
= ( ord_less @ int @ ( zero_zero @ int ) @ A3 ) ) ) ).
% neg_imp_zdiv_neg_iff
thf(fact_328_pos__imp__zdiv__neg__iff,axiom,
! [B2: int,A3: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
=> ( ( ord_less @ int @ ( divide_divide @ int @ A3 @ B2 ) @ ( zero_zero @ int ) )
= ( ord_less @ int @ A3 @ ( zero_zero @ int ) ) ) ) ).
% pos_imp_zdiv_neg_iff
thf(fact_329_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_330_int__power__div__base,axiom,
! [M2: nat,K: int] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M2 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ K )
=> ( ( divide_divide @ int @ ( power_power @ int @ K @ M2 ) @ K )
= ( power_power @ int @ K @ ( minus_minus @ nat @ M2 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ).
% int_power_div_base
thf(fact_331_one__less__nat__eq,axiom,
! [Z2: int] :
( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( nat2 @ Z2 ) )
= ( ord_less @ int @ ( one_one @ int ) @ Z2 ) ) ).
% one_less_nat_eq
thf(fact_332_Divides_Oadjust__mod__def,axiom,
( adjust_mod
= ( ^ [L2: int,R: int] :
( if @ int
@ ( R
= ( zero_zero @ int ) )
@ ( zero_zero @ int )
@ ( minus_minus @ int @ L2 @ R ) ) ) ) ).
% Divides.adjust_mod_def
thf(fact_333_le__div__geq,axiom,
! [N: nat,M2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ nat @ N @ M2 )
=> ( ( divide_divide @ nat @ M2 @ N )
= ( suc @ ( divide_divide @ nat @ ( minus_minus @ nat @ M2 @ N ) @ N ) ) ) ) ) ).
% le_div_geq
thf(fact_334_zdiv__zminus1__eq__if,axiom,
! [B2: int,A3: int] :
( ( B2
!= ( zero_zero @ int ) )
=> ( ( ( ( modulo_modulo @ int @ A3 @ B2 )
= ( zero_zero @ int ) )
=> ( ( divide_divide @ int @ ( uminus_uminus @ int @ A3 ) @ B2 )
= ( uminus_uminus @ int @ ( divide_divide @ int @ A3 @ B2 ) ) ) )
& ( ( ( modulo_modulo @ int @ A3 @ B2 )
!= ( zero_zero @ int ) )
=> ( ( divide_divide @ int @ ( uminus_uminus @ int @ A3 ) @ B2 )
= ( minus_minus @ int @ ( uminus_uminus @ int @ ( divide_divide @ int @ A3 @ B2 ) ) @ ( one_one @ int ) ) ) ) ) ) ).
% zdiv_zminus1_eq_if
thf(fact_335_zdiv__zminus2__eq__if,axiom,
! [B2: int,A3: int] :
( ( B2
!= ( zero_zero @ int ) )
=> ( ( ( ( modulo_modulo @ int @ A3 @ B2 )
= ( zero_zero @ int ) )
=> ( ( divide_divide @ int @ A3 @ ( uminus_uminus @ int @ B2 ) )
= ( uminus_uminus @ int @ ( divide_divide @ int @ A3 @ B2 ) ) ) )
& ( ( ( modulo_modulo @ int @ A3 @ B2 )
!= ( zero_zero @ int ) )
=> ( ( divide_divide @ int @ A3 @ ( uminus_uminus @ int @ B2 ) )
= ( minus_minus @ int @ ( uminus_uminus @ int @ ( divide_divide @ int @ A3 @ B2 ) ) @ ( one_one @ int ) ) ) ) ) ) ).
% zdiv_zminus2_eq_if
thf(fact_336_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_337_Nitpick_Ocase__nat__unfold,axiom,
! [A: $tType] :
( ( case_nat @ A )
= ( ^ [X2: A,F4: nat > A,N5: nat] :
( if @ A
@ ( N5
= ( zero_zero @ nat ) )
@ X2
@ ( F4 @ ( minus_minus @ nat @ N5 @ ( one_one @ nat ) ) ) ) ) ) ).
% Nitpick.case_nat_unfold
thf(fact_338_divide__le__eq__1__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ A3 ) @ ( one_one @ A ) )
= ( ord_less_eq @ A @ A3 @ B2 ) ) ) ) ).
% divide_le_eq_1_neg
thf(fact_339_divide__le__eq__1__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ A3 ) @ ( one_one @ A ) )
= ( ord_less_eq @ A @ B2 @ A3 ) ) ) ) ).
% divide_le_eq_1_pos
thf(fact_340_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_341_neg__le__iff__le,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [B2: A,A3: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A3 ) )
= ( ord_less_eq @ A @ A3 @ B2 ) ) ) ).
% neg_le_iff_le
thf(fact_342_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_343_mod__self,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [A3: A] :
( ( modulo_modulo @ A @ A3 @ A3 )
= ( zero_zero @ A ) ) ) ).
% mod_self
thf(fact_344_mod__by__0,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [A3: A] :
( ( modulo_modulo @ A @ A3 @ ( zero_zero @ A ) )
= A3 ) ) ).
% mod_by_0
thf(fact_345_mod__0,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [A3: A] :
( ( modulo_modulo @ A @ ( zero_zero @ A ) @ A3 )
= ( zero_zero @ A ) ) ) ).
% mod_0
thf(fact_346_bits__mod__0,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A] :
( ( modulo_modulo @ A @ ( zero_zero @ A ) @ A3 )
= ( zero_zero @ A ) ) ) ).
% bits_mod_0
thf(fact_347_Suc__le__mono,axiom,
! [N: nat,M2: nat] :
( ( ord_less_eq @ nat @ ( suc @ N ) @ ( suc @ M2 ) )
= ( ord_less_eq @ nat @ N @ M2 ) ) ).
% Suc_le_mono
thf(fact_348_le0,axiom,
! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% le0
thf(fact_349_bot__nat__0_Oextremum,axiom,
! [A3: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ A3 ) ).
% bot_nat_0.extremum
thf(fact_350_minus__mod__self2,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel @ A )
=> ! [A3: A,B2: A] :
( ( modulo_modulo @ A @ ( minus_minus @ A @ A3 @ B2 ) @ B2 )
= ( modulo_modulo @ A @ A3 @ B2 ) ) ) ).
% minus_mod_self2
thf(fact_351_mod__minus__minus,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel @ A )
=> ! [A3: A,B2: A] :
( ( modulo_modulo @ A @ ( uminus_uminus @ A @ A3 ) @ ( uminus_uminus @ A @ B2 ) )
= ( uminus_uminus @ A @ ( modulo_modulo @ A @ A3 @ B2 ) ) ) ) ).
% mod_minus_minus
thf(fact_352_diff__diff__cancel,axiom,
! [I: nat,N: nat] :
( ( ord_less_eq @ nat @ I @ N )
=> ( ( minus_minus @ nat @ N @ ( minus_minus @ nat @ N @ I ) )
= I ) ) ).
% diff_diff_cancel
thf(fact_353_negative__zle,axiom,
! [N: nat,M2: nat] : ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N ) ) @ ( semiring_1_of_nat @ int @ M2 ) ) ).
% negative_zle
thf(fact_354_nat__int,axiom,
! [N: nat] :
( ( nat2 @ ( semiring_1_of_nat @ int @ N ) )
= N ) ).
% nat_int
thf(fact_355_diff__ge__0__iff__ge,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( minus_minus @ A @ A3 @ B2 ) )
= ( ord_less_eq @ A @ B2 @ A3 ) ) ) ).
% diff_ge_0_iff_ge
thf(fact_356_neg__0__le__iff__le,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ A3 ) )
= ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% neg_0_le_iff_le
thf(fact_357_neg__le__0__iff__le,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A3 ) @ ( zero_zero @ A ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% neg_le_0_iff_le
thf(fact_358_less__eq__neg__nonpos,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ A3 @ ( uminus_uminus @ A @ A3 ) )
= ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% less_eq_neg_nonpos
thf(fact_359_neg__less__eq__nonneg,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A3 ) @ A3 )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% neg_less_eq_nonneg
thf(fact_360_mod__by__1,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [A3: A] :
( ( modulo_modulo @ A @ A3 @ ( one_one @ A ) )
= ( zero_zero @ A ) ) ) ).
% mod_by_1
thf(fact_361_bits__mod__by__1,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A] :
( ( modulo_modulo @ A @ A3 @ ( one_one @ A ) )
= ( zero_zero @ A ) ) ) ).
% bits_mod_by_1
thf(fact_362_mod__div__trivial,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A3: A,B2: A] :
( ( divide_divide @ A @ ( modulo_modulo @ A @ A3 @ B2 ) @ B2 )
= ( zero_zero @ A ) ) ) ).
% mod_div_trivial
thf(fact_363_bits__mod__div__trivial,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A,B2: A] :
( ( divide_divide @ A @ ( modulo_modulo @ A @ A3 @ B2 ) @ B2 )
= ( zero_zero @ A ) ) ) ).
% bits_mod_div_trivial
thf(fact_364_of__nat__le__iff,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [M2: nat,N: nat] :
( ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ M2 ) @ ( semiring_1_of_nat @ A @ N ) )
= ( ord_less_eq @ nat @ M2 @ N ) ) ) ).
% of_nat_le_iff
thf(fact_365_minus__mod__self1,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel @ A )
=> ! [B2: A,A3: A] :
( ( modulo_modulo @ A @ ( minus_minus @ A @ B2 @ A3 ) @ B2 )
= ( modulo_modulo @ A @ ( uminus_uminus @ A @ A3 ) @ B2 ) ) ) ).
% minus_mod_self1
thf(fact_366_diff__is__0__eq,axiom,
! [M2: nat,N: nat] :
( ( ( minus_minus @ nat @ M2 @ N )
= ( zero_zero @ nat ) )
= ( ord_less_eq @ nat @ M2 @ N ) ) ).
% diff_is_0_eq
thf(fact_367_diff__is__0__eq_H,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( minus_minus @ nat @ M2 @ N )
= ( zero_zero @ nat ) ) ) ).
% diff_is_0_eq'
thf(fact_368_nat__0__iff,axiom,
! [I: int] :
( ( ( nat2 @ I )
= ( zero_zero @ nat ) )
= ( ord_less_eq @ int @ I @ ( zero_zero @ int ) ) ) ).
% nat_0_iff
thf(fact_369_nat__le__0,axiom,
! [Z2: int] :
( ( ord_less_eq @ int @ Z2 @ ( zero_zero @ int ) )
=> ( ( nat2 @ Z2 )
= ( zero_zero @ nat ) ) ) ).
% nat_le_0
thf(fact_370_int__nat__eq,axiom,
! [Z2: int] :
( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z2 )
=> ( ( semiring_1_of_nat @ int @ ( nat2 @ Z2 ) )
= Z2 ) )
& ( ~ ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z2 )
=> ( ( semiring_1_of_nat @ int @ ( nat2 @ Z2 ) )
= ( zero_zero @ int ) ) ) ) ).
% int_nat_eq
thf(fact_371_div__pos__pos__trivial,axiom,
! [K: int,L: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
=> ( ( ord_less @ int @ K @ L )
=> ( ( divide_divide @ int @ K @ L )
= ( zero_zero @ int ) ) ) ) ).
% div_pos_pos_trivial
thf(fact_372_div__neg__neg__trivial,axiom,
! [K: int,L: int] :
( ( ord_less_eq @ int @ K @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ L @ K )
=> ( ( divide_divide @ int @ K @ L )
= ( zero_zero @ int ) ) ) ) ).
% div_neg_neg_trivial
thf(fact_373_zle__diff1__eq,axiom,
! [W: int,Z2: int] :
( ( ord_less_eq @ int @ W @ ( minus_minus @ int @ Z2 @ ( one_one @ int ) ) )
= ( ord_less @ int @ W @ Z2 ) ) ).
% zle_diff1_eq
thf(fact_374_mod__pos__pos__trivial,axiom,
! [K: int,L: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
=> ( ( ord_less @ int @ K @ L )
=> ( ( modulo_modulo @ int @ K @ L )
= K ) ) ) ).
% mod_pos_pos_trivial
thf(fact_375_mod__neg__neg__trivial,axiom,
! [K: int,L: int] :
( ( ord_less_eq @ int @ K @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ L @ K )
=> ( ( modulo_modulo @ int @ K @ L )
= K ) ) ) ).
% mod_neg_neg_trivial
thf(fact_376_zero__le__divide__1__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ ( one_one @ A ) @ A3 ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% zero_le_divide_1_iff
thf(fact_377_divide__le__0__1__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ ( one_one @ A ) @ A3 ) @ ( zero_zero @ A ) )
= ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% divide_le_0_1_iff
thf(fact_378_of__nat__le__0__iff,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [M2: nat] :
( ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ M2 ) @ ( zero_zero @ A ) )
= ( M2
= ( zero_zero @ nat ) ) ) ) ).
% of_nat_le_0_iff
thf(fact_379_mod__minus1__right,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel @ A )
=> ! [A3: A] :
( ( modulo_modulo @ A @ A3 @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( zero_zero @ A ) ) ) ).
% mod_minus1_right
thf(fact_380_nat__1,axiom,
( ( nat2 @ ( one_one @ int ) )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% nat_1
thf(fact_381_zless__nat__conj,axiom,
! [W: int,Z2: int] :
( ( ord_less @ nat @ ( nat2 @ W ) @ ( nat2 @ Z2 ) )
= ( ( ord_less @ int @ ( zero_zero @ int ) @ Z2 )
& ( ord_less @ int @ W @ Z2 ) ) ) ).
% zless_nat_conj
thf(fact_382_nat__zminus__int,axiom,
! [N: nat] :
( ( nat2 @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N ) ) )
= ( zero_zero @ nat ) ) ).
% nat_zminus_int
thf(fact_383_le__divide__eq__1__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B2 @ A3 ) )
= ( ord_less_eq @ A @ A3 @ B2 ) ) ) ) ).
% le_divide_eq_1_pos
thf(fact_384_le__divide__eq__1__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B2 @ A3 ) )
= ( ord_less_eq @ A @ B2 @ A3 ) ) ) ) ).
% le_divide_eq_1_neg
thf(fact_385_zero__less__nat__eq,axiom,
! [Z2: int] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( nat2 @ Z2 ) )
= ( ord_less @ int @ ( zero_zero @ int ) @ Z2 ) ) ).
% zero_less_nat_eq
thf(fact_386_verit__comp__simplify1_I2_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A] : ( ord_less_eq @ A @ A3 @ A3 ) ) ).
% verit_comp_simplify1(2)
thf(fact_387_verit__la__disequality,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A] :
( ( A3 = B2 )
| ~ ( ord_less_eq @ A @ A3 @ B2 )
| ~ ( ord_less_eq @ A @ B2 @ A3 ) ) ) ).
% verit_la_disequality
thf(fact_388_le__refl,axiom,
! [N: nat] : ( ord_less_eq @ nat @ N @ N ) ).
% le_refl
thf(fact_389_le__trans,axiom,
! [I: nat,J2: nat,K: nat] :
( ( ord_less_eq @ nat @ I @ J2 )
=> ( ( ord_less_eq @ nat @ J2 @ K )
=> ( ord_less_eq @ nat @ I @ K ) ) ) ).
% le_trans
thf(fact_390_eq__imp__le,axiom,
! [M2: nat,N: nat] :
( ( M2 = N )
=> ( ord_less_eq @ nat @ M2 @ N ) ) ).
% eq_imp_le
thf(fact_391_le__antisym,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( ord_less_eq @ nat @ N @ M2 )
=> ( M2 = N ) ) ) ).
% le_antisym
thf(fact_392_nat__le__linear,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq @ nat @ M2 @ N )
| ( ord_less_eq @ nat @ N @ M2 ) ) ).
% nat_le_linear
thf(fact_393_Nat_Oex__has__greatest__nat,axiom,
! [P: nat > $o,K: nat,B2: nat] :
( ( P @ K )
=> ( ! [Y3: nat] :
( ( P @ Y3 )
=> ( ord_less_eq @ nat @ Y3 @ B2 ) )
=> ? [X3: nat] :
( ( P @ X3 )
& ! [Y: nat] :
( ( P @ Y )
=> ( ord_less_eq @ nat @ Y @ X3 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_394_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_395_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_396_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ord_less_eq @ A @ ( modulo_modulo @ A @ A3 @ B2 ) @ A3 ) ) ) ).
% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
thf(fact_397_of__nat__mod,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [M2: nat,N: nat] :
( ( semiring_1_of_nat @ A @ ( modulo_modulo @ nat @ M2 @ N ) )
= ( modulo_modulo @ A @ ( semiring_1_of_nat @ A @ M2 ) @ ( semiring_1_of_nat @ A @ N ) ) ) ) ).
% of_nat_mod
thf(fact_398_eq__nat__nat__iff,axiom,
! [Z2: int,Z3: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z2 )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z3 )
=> ( ( ( nat2 @ Z2 )
= ( nat2 @ Z3 ) )
= ( Z2 = Z3 ) ) ) ) ).
% eq_nat_nat_iff
thf(fact_399_all__nat,axiom,
( ( ^ [P2: nat > $o] :
! [X5: nat] : ( P2 @ X5 ) )
= ( ^ [P3: nat > $o] :
! [X2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X2 )
=> ( P3 @ ( nat2 @ X2 ) ) ) ) ) ).
% all_nat
thf(fact_400_ex__nat,axiom,
( ( ^ [P2: nat > $o] :
? [X5: nat] : ( P2 @ X5 ) )
= ( ^ [P3: nat > $o] :
? [X2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X2 )
& ( P3 @ ( nat2 @ X2 ) ) ) ) ) ).
% ex_nat
thf(fact_401_zmod__le__nonneg__dividend,axiom,
! [M2: int,K: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ M2 )
=> ( ord_less_eq @ int @ ( modulo_modulo @ int @ M2 @ K ) @ M2 ) ) ).
% zmod_le_nonneg_dividend
thf(fact_402_lift__Suc__mono__le,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [F2: nat > A,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_eq @ A @ ( F2 @ N2 ) @ ( F2 @ ( suc @ N2 ) ) )
=> ( ( ord_less_eq @ nat @ N @ N3 )
=> ( ord_less_eq @ A @ ( F2 @ N ) @ ( F2 @ N3 ) ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_403_lift__Suc__antimono__le,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [F2: nat > A,N: nat,N3: nat] :
( ! [N2: nat] : ( ord_less_eq @ A @ ( F2 @ ( suc @ N2 ) ) @ ( F2 @ N2 ) )
=> ( ( ord_less_eq @ nat @ N @ N3 )
=> ( ord_less_eq @ A @ ( F2 @ N3 ) @ ( F2 @ N ) ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_404_of__nat__mono,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [I: nat,J2: nat] :
( ( ord_less_eq @ nat @ I @ J2 )
=> ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ I ) @ ( semiring_1_of_nat @ A @ J2 ) ) ) ) ).
% of_nat_mono
thf(fact_405_zle__int,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ M2 ) @ ( semiring_1_of_nat @ int @ N ) )
= ( ord_less_eq @ nat @ M2 @ N ) ) ).
% zle_int
thf(fact_406_nat__int__comparison_I3_J,axiom,
( ( ord_less_eq @ nat )
= ( ^ [A5: nat,B3: nat] : ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ A5 ) @ ( semiring_1_of_nat @ int @ B3 ) ) ) ) ).
% nat_int_comparison(3)
thf(fact_407_nat__le__eq__zle,axiom,
! [W: int,Z2: int] :
( ( ( ord_less @ int @ ( zero_zero @ int ) @ W )
| ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z2 ) )
=> ( ( ord_less_eq @ nat @ ( nat2 @ W ) @ ( nat2 @ Z2 ) )
= ( ord_less_eq @ int @ W @ Z2 ) ) ) ).
% nat_le_eq_zle
thf(fact_408_le__nat__iff,axiom,
! [K: int,N: nat] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
=> ( ( ord_less_eq @ nat @ N @ ( nat2 @ K ) )
= ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ N ) @ K ) ) ) ).
% le_nat_iff
thf(fact_409_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [B2: A,A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( modulo_modulo @ A @ A3 @ B2 ) ) ) ) ).
% unique_euclidean_semiring_numeral_class.pos_mod_sign
thf(fact_410_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ A3 @ B2 )
=> ( ( modulo_modulo @ A @ A3 @ B2 )
= A3 ) ) ) ) ).
% unique_euclidean_semiring_numeral_class.mod_less
thf(fact_411_nat__0__le,axiom,
! [Z2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z2 )
=> ( ( semiring_1_of_nat @ int @ ( nat2 @ Z2 ) )
= Z2 ) ) ).
% nat_0_le
thf(fact_412_int__eq__iff,axiom,
! [M2: nat,Z2: int] :
( ( ( semiring_1_of_nat @ int @ M2 )
= Z2 )
= ( ( M2
= ( nat2 @ Z2 ) )
& ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z2 ) ) ) ).
% int_eq_iff
thf(fact_413_Euclidean__Division_Opos__mod__sign,axiom,
! [L: int,K: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ L )
=> ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( modulo_modulo @ int @ K @ L ) ) ) ).
% Euclidean_Division.pos_mod_sign
thf(fact_414_neg__mod__sign,axiom,
! [L: int,K: int] :
( ( ord_less @ int @ L @ ( zero_zero @ int ) )
=> ( ord_less_eq @ int @ ( modulo_modulo @ int @ K @ L ) @ ( zero_zero @ int ) ) ) ).
% neg_mod_sign
thf(fact_415_zmod__trivial__iff,axiom,
! [I: int,K: int] :
( ( ( modulo_modulo @ int @ I @ K )
= I )
= ( ( K
= ( zero_zero @ int ) )
| ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ I )
& ( ord_less @ int @ I @ K ) )
| ( ( ord_less_eq @ int @ I @ ( zero_zero @ int ) )
& ( ord_less @ int @ K @ I ) ) ) ) ).
% zmod_trivial_iff
thf(fact_416_pos__mod__conj,axiom,
! [B2: int,A3: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( modulo_modulo @ int @ A3 @ B2 ) )
& ( ord_less @ int @ ( modulo_modulo @ int @ A3 @ B2 ) @ B2 ) ) ) ).
% pos_mod_conj
thf(fact_417_neg__mod__conj,axiom,
! [B2: int,A3: int] :
( ( ord_less @ int @ B2 @ ( zero_zero @ int ) )
=> ( ( ord_less_eq @ int @ ( modulo_modulo @ int @ A3 @ B2 ) @ ( zero_zero @ int ) )
& ( ord_less @ int @ B2 @ ( modulo_modulo @ int @ A3 @ B2 ) ) ) ) ).
% neg_mod_conj
thf(fact_418_mod__diff__right__eq,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( modulo_modulo @ A @ ( minus_minus @ A @ A3 @ ( modulo_modulo @ A @ B2 @ C2 ) ) @ C2 )
= ( modulo_modulo @ A @ ( minus_minus @ A @ A3 @ B2 ) @ C2 ) ) ) ).
% mod_diff_right_eq
thf(fact_419_mod__diff__left__eq,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( modulo_modulo @ A @ ( minus_minus @ A @ ( modulo_modulo @ A @ A3 @ C2 ) @ B2 ) @ C2 )
= ( modulo_modulo @ A @ ( minus_minus @ A @ A3 @ B2 ) @ C2 ) ) ) ).
% mod_diff_left_eq
thf(fact_420_mod__diff__cong,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel @ A )
=> ! [A3: A,C2: A,A8: A,B2: A,B6: A] :
( ( ( modulo_modulo @ A @ A3 @ C2 )
= ( modulo_modulo @ A @ A8 @ C2 ) )
=> ( ( ( modulo_modulo @ A @ B2 @ C2 )
= ( modulo_modulo @ A @ B6 @ C2 ) )
=> ( ( modulo_modulo @ A @ ( minus_minus @ A @ A3 @ B2 ) @ C2 )
= ( modulo_modulo @ A @ ( minus_minus @ A @ A8 @ B6 ) @ C2 ) ) ) ) ) ).
% mod_diff_cong
thf(fact_421_mod__diff__eq,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( modulo_modulo @ A @ ( minus_minus @ A @ ( modulo_modulo @ A @ A3 @ C2 ) @ ( modulo_modulo @ A @ B2 @ C2 ) ) @ C2 )
= ( modulo_modulo @ A @ ( minus_minus @ A @ A3 @ B2 ) @ C2 ) ) ) ).
% mod_diff_eq
thf(fact_422_mod__minus__right,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel @ A )
=> ! [A3: A,B2: A] :
( ( modulo_modulo @ A @ A3 @ ( uminus_uminus @ A @ B2 ) )
= ( uminus_uminus @ A @ ( modulo_modulo @ A @ ( uminus_uminus @ A @ A3 ) @ B2 ) ) ) ) ).
% mod_minus_right
thf(fact_423_mod__minus__cong,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel @ A )
=> ! [A3: A,B2: A,A8: A] :
( ( ( modulo_modulo @ A @ A3 @ B2 )
= ( modulo_modulo @ A @ A8 @ B2 ) )
=> ( ( modulo_modulo @ A @ ( uminus_uminus @ A @ A3 ) @ B2 )
= ( modulo_modulo @ A @ ( uminus_uminus @ A @ A8 ) @ B2 ) ) ) ) ).
% mod_minus_cong
thf(fact_424_mod__minus__eq,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel @ A )
=> ! [A3: A,B2: A] :
( ( modulo_modulo @ A @ ( uminus_uminus @ A @ ( modulo_modulo @ A @ A3 @ B2 ) ) @ B2 )
= ( modulo_modulo @ A @ ( uminus_uminus @ A @ A3 ) @ B2 ) ) ) ).
% mod_minus_eq
thf(fact_425_zero__le,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [X: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ X ) ) ).
% zero_le
thf(fact_426_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_427_verit__comp__simplify1_I3_J,axiom,
! [B: $tType] :
( ( linorder @ B )
=> ! [B6: B,A8: B] :
( ( ~ ( ord_less_eq @ B @ B6 @ A8 ) )
= ( ord_less @ B @ A8 @ B6 ) ) ) ).
% verit_comp_simplify1(3)
thf(fact_428_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_429_diff__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B2: A,D2: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ D2 @ C2 )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ A3 @ C2 ) @ ( minus_minus @ A @ B2 @ D2 ) ) ) ) ) ).
% diff_mono
thf(fact_430_diff__left__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( ord_less_eq @ A @ B2 @ A3 )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ C2 @ A3 ) @ ( minus_minus @ A @ C2 @ B2 ) ) ) ) ).
% diff_left_mono
thf(fact_431_diff__right__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ A3 @ C2 ) @ ( minus_minus @ A @ B2 @ C2 ) ) ) ) ).
% diff_right_mono
thf(fact_432_diff__eq__diff__less__eq,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B2: A,C2: A,D2: A] :
( ( ( minus_minus @ A @ A3 @ B2 )
= ( minus_minus @ A @ C2 @ D2 ) )
=> ( ( ord_less_eq @ A @ A3 @ B2 )
= ( ord_less_eq @ A @ C2 @ D2 ) ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_433_le__minus__iff,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ ( uminus_uminus @ A @ B2 ) )
= ( ord_less_eq @ A @ B2 @ ( uminus_uminus @ A @ A3 ) ) ) ) ).
% le_minus_iff
thf(fact_434_minus__le__iff,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A3 ) @ B2 )
= ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ A3 ) ) ) ).
% minus_le_iff
thf(fact_435_le__imp__neg__le,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A3 ) ) ) ) ).
% le_imp_neg_le
thf(fact_436_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_437_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_438_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_439_Suc__leD,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq @ nat @ ( suc @ M2 ) @ N )
=> ( ord_less_eq @ nat @ M2 @ N ) ) ).
% Suc_leD
thf(fact_440_le__SucE,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq @ nat @ M2 @ ( suc @ N ) )
=> ( ~ ( ord_less_eq @ nat @ M2 @ N )
=> ( M2
= ( suc @ N ) ) ) ) ).
% le_SucE
thf(fact_441_le__SucI,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ord_less_eq @ nat @ M2 @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_442_Suc__le__D,axiom,
! [N: nat,M6: nat] :
( ( ord_less_eq @ nat @ ( suc @ N ) @ M6 )
=> ? [M3: nat] :
( M6
= ( suc @ M3 ) ) ) ).
% Suc_le_D
thf(fact_443_le__Suc__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq @ nat @ M2 @ ( suc @ N ) )
= ( ( ord_less_eq @ nat @ M2 @ N )
| ( M2
= ( suc @ N ) ) ) ) ).
% le_Suc_eq
thf(fact_444_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq @ nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_445_not__less__eq__eq,axiom,
! [M2: nat,N: nat] :
( ( ~ ( ord_less_eq @ nat @ M2 @ N ) )
= ( ord_less_eq @ nat @ ( suc @ N ) @ M2 ) ) ).
% not_less_eq_eq
thf(fact_446_full__nat__induct,axiom,
! [P: nat > $o,N: nat] :
( ! [N2: nat] :
( ! [M: nat] :
( ( ord_less_eq @ nat @ ( suc @ M ) @ N2 )
=> ( P @ M ) )
=> ( P @ N2 ) )
=> ( P @ N ) ) ).
% full_nat_induct
thf(fact_447_nat__induct__at__least,axiom,
! [M2: nat,N: nat,P: nat > $o] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( P @ M2 )
=> ( ! [N2: nat] :
( ( ord_less_eq @ nat @ M2 @ N2 )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct_at_least
thf(fact_448_transitive__stepwise__le,axiom,
! [M2: nat,N: nat,R2: nat > nat > $o] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ! [X3: nat] : ( R2 @ X3 @ X3 )
=> ( ! [X3: nat,Y3: nat,Z4: nat] :
( ( R2 @ X3 @ Y3 )
=> ( ( R2 @ Y3 @ Z4 )
=> ( R2 @ X3 @ Z4 ) ) )
=> ( ! [N2: nat] : ( R2 @ N2 @ ( suc @ N2 ) )
=> ( R2 @ M2 @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_449_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq @ nat @ N @ ( zero_zero @ nat ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% le_0_eq
thf(fact_450_bot__nat__0_Oextremum__uniqueI,axiom,
! [A3: nat] :
( ( ord_less_eq @ nat @ A3 @ ( zero_zero @ nat ) )
=> ( A3
= ( zero_zero @ nat ) ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_451_bot__nat__0_Oextremum__unique,axiom,
! [A3: nat] :
( ( ord_less_eq @ nat @ A3 @ ( zero_zero @ nat ) )
= ( A3
= ( zero_zero @ nat ) ) ) ).
% bot_nat_0.extremum_unique
thf(fact_452_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% less_eq_nat.simps(1)
thf(fact_453_real__arch__simple,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X: A] :
? [N2: nat] : ( ord_less_eq @ A @ X @ ( semiring_1_of_nat @ A @ N2 ) ) ) ).
% real_arch_simple
thf(fact_454_less__mono__imp__le__mono,axiom,
! [F2: nat > nat,I: nat,J2: nat] :
( ! [I3: nat,J: nat] :
( ( ord_less @ nat @ I3 @ J )
=> ( ord_less @ nat @ ( F2 @ I3 ) @ ( F2 @ J ) ) )
=> ( ( ord_less_eq @ nat @ I @ J2 )
=> ( ord_less_eq @ nat @ ( F2 @ I ) @ ( F2 @ J2 ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_455_le__neq__implies__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( M2 != N )
=> ( ord_less @ nat @ M2 @ N ) ) ) ).
% le_neq_implies_less
thf(fact_456_less__or__eq__imp__le,axiom,
! [M2: nat,N: nat] :
( ( ( ord_less @ nat @ M2 @ N )
| ( M2 = N ) )
=> ( ord_less_eq @ nat @ M2 @ N ) ) ).
% less_or_eq_imp_le
thf(fact_457_le__eq__less__or__eq,axiom,
( ( ord_less_eq @ nat )
= ( ^ [M5: nat,N5: nat] :
( ( ord_less @ nat @ M5 @ N5 )
| ( M5 = N5 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_458_less__imp__le__nat,axiom,
! [M2: nat,N: nat] :
( ( ord_less @ nat @ M2 @ N )
=> ( ord_less_eq @ nat @ M2 @ N ) ) ).
% less_imp_le_nat
thf(fact_459_nat__less__le,axiom,
( ( ord_less @ nat )
= ( ^ [M5: nat,N5: nat] :
( ( ord_less_eq @ nat @ M5 @ N5 )
& ( M5 != N5 ) ) ) ) ).
% nat_less_le
thf(fact_460_less__eq__int__code_I1_J,axiom,
ord_less_eq @ int @ ( zero_zero @ int ) @ ( zero_zero @ int ) ).
% less_eq_int_code(1)
thf(fact_461_eq__diff__iff,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq @ nat @ K @ M2 )
=> ( ( ord_less_eq @ nat @ K @ N )
=> ( ( ( minus_minus @ nat @ M2 @ K )
= ( minus_minus @ nat @ N @ K ) )
= ( M2 = N ) ) ) ) ).
% eq_diff_iff
thf(fact_462_le__diff__iff,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq @ nat @ K @ M2 )
=> ( ( ord_less_eq @ nat @ K @ N )
=> ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ M2 @ K ) @ ( minus_minus @ nat @ N @ K ) )
= ( ord_less_eq @ nat @ M2 @ N ) ) ) ) ).
% le_diff_iff
thf(fact_463_Nat_Odiff__diff__eq,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq @ nat @ K @ M2 )
=> ( ( ord_less_eq @ nat @ K @ N )
=> ( ( minus_minus @ nat @ ( minus_minus @ nat @ M2 @ K ) @ ( minus_minus @ nat @ N @ K ) )
= ( minus_minus @ nat @ M2 @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_464_diff__le__mono,axiom,
! [M2: nat,N: nat,L: nat] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ord_less_eq @ nat @ ( minus_minus @ nat @ M2 @ L ) @ ( minus_minus @ nat @ N @ L ) ) ) ).
% diff_le_mono
thf(fact_465_diff__le__self,axiom,
! [M2: nat,N: nat] : ( ord_less_eq @ nat @ ( minus_minus @ nat @ M2 @ N ) @ M2 ) ).
% diff_le_self
thf(fact_466_le__diff__iff_H,axiom,
! [A3: nat,C2: nat,B2: nat] :
( ( ord_less_eq @ nat @ A3 @ C2 )
=> ( ( ord_less_eq @ nat @ B2 @ C2 )
=> ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ C2 @ A3 ) @ ( minus_minus @ nat @ C2 @ B2 ) )
= ( ord_less_eq @ nat @ B2 @ A3 ) ) ) ) ).
% le_diff_iff'
thf(fact_467_diff__le__mono2,axiom,
! [M2: nat,N: nat,L: nat] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ord_less_eq @ nat @ ( minus_minus @ nat @ L @ N ) @ ( minus_minus @ nat @ L @ M2 ) ) ) ).
% diff_le_mono2
thf(fact_468_div__le__dividend,axiom,
! [M2: nat,N: nat] : ( ord_less_eq @ nat @ ( divide_divide @ nat @ M2 @ N ) @ M2 ) ).
% div_le_dividend
thf(fact_469_div__le__mono,axiom,
! [M2: nat,N: nat,K: nat] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ord_less_eq @ nat @ ( divide_divide @ nat @ M2 @ K ) @ ( divide_divide @ nat @ N @ K ) ) ) ).
% div_le_mono
thf(fact_470_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_471_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_472_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_473_nat__less__eq__zless,axiom,
! [W: int,Z2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ W )
=> ( ( ord_less @ nat @ ( nat2 @ W ) @ ( nat2 @ Z2 ) )
= ( ord_less @ int @ W @ Z2 ) ) ) ).
% nat_less_eq_zless
thf(fact_474_nat__eq__iff,axiom,
! [W: int,M2: nat] :
( ( ( nat2 @ W )
= M2 )
= ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ W )
=> ( W
= ( semiring_1_of_nat @ int @ M2 ) ) )
& ( ~ ( ord_less_eq @ int @ ( zero_zero @ int ) @ W )
=> ( M2
= ( zero_zero @ nat ) ) ) ) ) ).
% nat_eq_iff
thf(fact_475_nat__eq__iff2,axiom,
! [M2: nat,W: int] :
( ( M2
= ( nat2 @ W ) )
= ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ W )
=> ( W
= ( semiring_1_of_nat @ int @ M2 ) ) )
& ( ~ ( ord_less_eq @ int @ ( zero_zero @ int ) @ W )
=> ( M2
= ( zero_zero @ nat ) ) ) ) ) ).
% nat_eq_iff2
thf(fact_476_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_477_nat__diff__distrib,axiom,
! [Z3: int,Z2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z3 )
=> ( ( ord_less_eq @ int @ Z3 @ Z2 )
=> ( ( nat2 @ ( minus_minus @ int @ Z2 @ Z3 ) )
= ( minus_minus @ nat @ ( nat2 @ Z2 ) @ ( nat2 @ Z3 ) ) ) ) ) ).
% nat_diff_distrib
thf(fact_478_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_479_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_480_mod__pos__geq,axiom,
! [L: int,K: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ L )
=> ( ( ord_less_eq @ int @ L @ K )
=> ( ( modulo_modulo @ int @ K @ L )
= ( modulo_modulo @ int @ ( minus_minus @ int @ K @ L ) @ L ) ) ) ) ).
% mod_pos_geq
thf(fact_481_old_Onat_Osimps_I5_J,axiom,
! [A: $tType,F1: A,F22: nat > A,X23: nat] :
( ( case_nat @ A @ F1 @ F22 @ ( suc @ X23 ) )
= ( F22 @ X23 ) ) ).
% old.nat.simps(5)
thf(fact_482_old_Onat_Osimps_I4_J,axiom,
! [A: $tType,F1: A,F22: nat > A] :
( ( case_nat @ A @ F1 @ F22 @ ( zero_zero @ nat ) )
= F1 ) ).
% old.nat.simps(4)
thf(fact_483_power__diff__power__eq,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A3: A,N: nat,M2: nat] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( ( ord_less_eq @ nat @ N @ M2 )
=> ( ( divide_divide @ A @ ( power_power @ A @ A3 @ M2 ) @ ( power_power @ A @ A3 @ N ) )
= ( power_power @ A @ A3 @ ( minus_minus @ nat @ M2 @ N ) ) ) )
& ( ~ ( ord_less_eq @ nat @ N @ M2 )
=> ( ( divide_divide @ A @ ( power_power @ A @ A3 @ M2 ) @ ( power_power @ A @ A3 @ N ) )
= ( divide_divide @ A @ ( one_one @ A ) @ ( power_power @ A @ A3 @ ( minus_minus @ nat @ N @ M2 ) ) ) ) ) ) ) ) ).
% power_diff_power_eq
thf(fact_484_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [B2: A,A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less @ A @ ( modulo_modulo @ A @ A3 @ B2 ) @ B2 ) ) ) ).
% unique_euclidean_semiring_numeral_class.pos_mod_bound
thf(fact_485_nat__less__iff,axiom,
! [W: int,M2: nat] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ W )
=> ( ( ord_less @ nat @ ( nat2 @ W ) @ M2 )
= ( ord_less @ int @ W @ ( semiring_1_of_nat @ int @ M2 ) ) ) ) ).
% nat_less_iff
thf(fact_486_mod__eq__self__iff__div__eq__0,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ! [A3: A,B2: A] :
( ( ( modulo_modulo @ A @ A3 @ B2 )
= A3 )
= ( ( divide_divide @ A @ A3 @ B2 )
= ( zero_zero @ A ) ) ) ) ).
% mod_eq_self_iff_div_eq_0
thf(fact_487_inverse__of__nat__le,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [N: nat,M2: nat] :
( ( ord_less_eq @ nat @ N @ M2 )
=> ( ( N
!= ( zero_zero @ nat ) )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ ( one_one @ A ) @ ( semiring_1_of_nat @ A @ M2 ) ) @ ( divide_divide @ A @ ( one_one @ A ) @ ( semiring_1_of_nat @ A @ N ) ) ) ) ) ) ).
% inverse_of_nat_le
thf(fact_488_nat__zero__as__int,axiom,
( ( zero_zero @ nat )
= ( nat2 @ ( zero_zero @ int ) ) ) ).
% nat_zero_as_int
thf(fact_489_minus__mod__int__eq,axiom,
! [L: int,K: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ L )
=> ( ( modulo_modulo @ int @ ( uminus_uminus @ int @ K ) @ L )
= ( minus_minus @ int @ ( minus_minus @ int @ L @ ( one_one @ int ) ) @ ( modulo_modulo @ int @ ( minus_minus @ int @ K @ ( one_one @ int ) ) @ L ) ) ) ) ).
% minus_mod_int_eq
thf(fact_490_nat__one__as__int,axiom,
( ( one_one @ nat )
= ( nat2 @ ( one_one @ int ) ) ) ).
% nat_one_as_int
thf(fact_491_Euclidean__Division_Opos__mod__bound,axiom,
! [L: int,K: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ L )
=> ( ord_less @ int @ ( modulo_modulo @ int @ K @ L ) @ L ) ) ).
% Euclidean_Division.pos_mod_bound
thf(fact_492_neg__mod__bound,axiom,
! [L: int,K: int] :
( ( ord_less @ int @ L @ ( zero_zero @ int ) )
=> ( ord_less @ int @ L @ ( modulo_modulo @ int @ K @ L ) ) ) ).
% neg_mod_bound
thf(fact_493_zmod__zminus1__not__zero,axiom,
! [K: int,L: int] :
( ( ( modulo_modulo @ int @ ( uminus_uminus @ int @ K ) @ L )
!= ( zero_zero @ int ) )
=> ( ( modulo_modulo @ int @ K @ L )
!= ( zero_zero @ int ) ) ) ).
% zmod_zminus1_not_zero
thf(fact_494_zmod__zminus2__not__zero,axiom,
! [K: int,L: int] :
( ( ( modulo_modulo @ int @ K @ ( uminus_uminus @ int @ L ) )
!= ( zero_zero @ int ) )
=> ( ( modulo_modulo @ int @ K @ L )
!= ( zero_zero @ int ) ) ) ).
% zmod_zminus2_not_zero
thf(fact_495_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_496_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_497_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_498_le__iff__diff__le__0,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A5: A,B3: A] : ( ord_less_eq @ A @ ( minus_minus @ A @ A5 @ B3 ) @ ( zero_zero @ A ) ) ) ) ) ).
% le_iff_diff_le_0
thf(fact_499_divide__le__0__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ A3 @ B2 ) @ ( zero_zero @ A ) )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
& ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) )
| ( ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) ) ) ) ) ).
% divide_le_0_iff
thf(fact_500_divide__right__mono,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ A3 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) ) ) ) ) ).
% divide_right_mono
thf(fact_501_zero__le__divide__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ A3 @ B2 ) )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) )
| ( ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) )
& ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) ) ) ) ) ).
% zero_le_divide_iff
thf(fact_502_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_503_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_504_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_505_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_506_divide__right__mono__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ C2 ) @ ( divide_divide @ A @ A3 @ C2 ) ) ) ) ) ).
% divide_right_mono_neg
thf(fact_507_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_508_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_509_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_510_Suc__leI,axiom,
! [M2: nat,N: nat] :
( ( ord_less @ nat @ M2 @ N )
=> ( ord_less_eq @ nat @ ( suc @ M2 ) @ N ) ) ).
% Suc_leI
thf(fact_511_Suc__le__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq @ nat @ ( suc @ M2 ) @ N )
= ( ord_less @ nat @ M2 @ N ) ) ).
% Suc_le_eq
thf(fact_512_dec__induct,axiom,
! [I: nat,J2: nat,P: nat > $o] :
( ( ord_less_eq @ nat @ I @ J2 )
=> ( ( P @ I )
=> ( ! [N2: nat] :
( ( ord_less_eq @ nat @ I @ N2 )
=> ( ( ord_less @ nat @ N2 @ J2 )
=> ( ( P @ N2 )
=> ( P @ ( suc @ N2 ) ) ) ) )
=> ( P @ J2 ) ) ) ) ).
% dec_induct
thf(fact_513_inc__induct,axiom,
! [I: nat,J2: nat,P: nat > $o] :
( ( ord_less_eq @ nat @ I @ J2 )
=> ( ( P @ J2 )
=> ( ! [N2: nat] :
( ( ord_less_eq @ nat @ I @ N2 )
=> ( ( ord_less @ nat @ N2 @ J2 )
=> ( ( P @ ( suc @ N2 ) )
=> ( P @ N2 ) ) ) )
=> ( P @ I ) ) ) ) ).
% inc_induct
thf(fact_514_Suc__le__lessD,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq @ nat @ ( suc @ M2 ) @ N )
=> ( ord_less @ nat @ M2 @ N ) ) ).
% Suc_le_lessD
thf(fact_515_le__less__Suc__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( ord_less @ nat @ N @ ( suc @ M2 ) )
= ( N = M2 ) ) ) ).
% le_less_Suc_eq
thf(fact_516_less__Suc__eq__le,axiom,
! [M2: nat,N: nat] :
( ( ord_less @ nat @ M2 @ ( suc @ N ) )
= ( ord_less_eq @ nat @ M2 @ N ) ) ).
% less_Suc_eq_le
thf(fact_517_less__eq__Suc__le,axiom,
( ( ord_less @ nat )
= ( ^ [N5: nat] : ( ord_less_eq @ nat @ ( suc @ N5 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_518_le__imp__less__Suc,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ord_less @ nat @ M2 @ ( suc @ N ) ) ) ).
% le_imp_less_Suc
thf(fact_519_ex__least__nat__le,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ ( zero_zero @ nat ) )
=> ? [K2: nat] :
( ( ord_less_eq @ nat @ K2 @ N )
& ! [I4: nat] :
( ( ord_less @ nat @ I4 @ K2 )
=> ~ ( P @ I4 ) )
& ( P @ K2 ) ) ) ) ).
% ex_least_nat_le
thf(fact_520_Suc__diff__le,axiom,
! [N: nat,M2: nat] :
( ( ord_less_eq @ nat @ N @ M2 )
=> ( ( minus_minus @ nat @ ( suc @ M2 ) @ N )
= ( suc @ ( minus_minus @ nat @ M2 @ N ) ) ) ) ).
% Suc_diff_le
thf(fact_521_less__diff__iff,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq @ nat @ K @ M2 )
=> ( ( ord_less_eq @ nat @ K @ N )
=> ( ( ord_less @ nat @ ( minus_minus @ nat @ M2 @ K ) @ ( minus_minus @ nat @ N @ K ) )
= ( ord_less @ nat @ M2 @ N ) ) ) ) ).
% less_diff_iff
thf(fact_522_diff__less__mono,axiom,
! [A3: nat,B2: nat,C2: nat] :
( ( ord_less @ nat @ A3 @ B2 )
=> ( ( ord_less_eq @ nat @ C2 @ A3 )
=> ( ord_less @ nat @ ( minus_minus @ nat @ A3 @ C2 ) @ ( minus_minus @ nat @ B2 @ C2 ) ) ) ) ).
% diff_less_mono
thf(fact_523_Suc__div__le__mono,axiom,
! [M2: nat,N: nat] : ( ord_less_eq @ nat @ ( divide_divide @ nat @ M2 @ N ) @ ( divide_divide @ nat @ ( suc @ M2 ) @ N ) ) ).
% Suc_div_le_mono
thf(fact_524_nonneg__int__cases,axiom,
! [K: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
=> ~ ! [N2: nat] :
( K
!= ( semiring_1_of_nat @ int @ N2 ) ) ) ).
% nonneg_int_cases
thf(fact_525_zero__le__imp__eq__int,axiom,
! [K: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
=> ? [N2: nat] :
( K
= ( semiring_1_of_nat @ int @ N2 ) ) ) ).
% zero_le_imp_eq_int
thf(fact_526_int__le__induct,axiom,
! [I: int,K: int,P: int > $o] :
( ( ord_less_eq @ int @ I @ K )
=> ( ( P @ K )
=> ( ! [I3: int] :
( ( ord_less_eq @ int @ I3 @ K )
=> ( ( P @ I3 )
=> ( P @ ( minus_minus @ int @ I3 @ ( one_one @ int ) ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_le_induct
thf(fact_527_nat__mono__iff,axiom,
! [Z2: int,W: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ Z2 )
=> ( ( ord_less @ nat @ ( nat2 @ W ) @ ( nat2 @ Z2 ) )
= ( ord_less @ int @ W @ Z2 ) ) ) ).
% nat_mono_iff
thf(fact_528_zless__nat__eq__int__zless,axiom,
! [M2: nat,Z2: int] :
( ( ord_less @ nat @ M2 @ ( nat2 @ Z2 ) )
= ( ord_less @ int @ ( semiring_1_of_nat @ int @ M2 ) @ Z2 ) ) ).
% zless_nat_eq_int_zless
thf(fact_529_int__minus,axiom,
! [N: nat,M2: nat] :
( ( semiring_1_of_nat @ int @ ( minus_minus @ nat @ N @ M2 ) )
= ( semiring_1_of_nat @ int @ ( nat2 @ ( minus_minus @ int @ ( semiring_1_of_nat @ int @ N ) @ ( semiring_1_of_nat @ int @ M2 ) ) ) ) ) ).
% int_minus
thf(fact_530_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_531_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_532_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_533_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_534_divide__le__cancel,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ A3 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ A3 @ B2 ) )
& ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B2 @ A3 ) ) ) ) ) ).
% divide_le_cancel
thf(fact_535_frac__less2,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y2: A,W: A,Z2: 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 @ Z2 )
=> ( ord_less @ A @ ( divide_divide @ A @ X @ Z2 ) @ ( divide_divide @ A @ Y2 @ W ) ) ) ) ) ) ) ).
% frac_less2
thf(fact_536_frac__less,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y2: A,W: A,Z2: 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 @ Z2 )
=> ( ord_less @ A @ ( divide_divide @ A @ X @ Z2 ) @ ( divide_divide @ A @ Y2 @ W ) ) ) ) ) ) ) ).
% frac_less
thf(fact_537_frac__le,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y2: A,X: A,W: A,Z2: 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 @ Z2 )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ X @ Z2 ) @ ( divide_divide @ A @ Y2 @ W ) ) ) ) ) ) ) ).
% frac_le
thf(fact_538_div__positive,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [B2: A,A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ A3 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ A3 @ B2 ) ) ) ) ) ).
% div_positive
thf(fact_539_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ A3 @ B2 )
=> ( ( divide_divide @ A @ A3 @ B2 )
= ( zero_zero @ A ) ) ) ) ) ).
% unique_euclidean_semiring_numeral_class.div_less
thf(fact_540_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_541_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_542_zdiv__mono__strict,axiom,
! [A4: int,B7: int,N: int] :
( ( ord_less @ int @ A4 @ B7 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ N )
=> ( ( ( modulo_modulo @ int @ A4 @ N )
= ( zero_zero @ int ) )
=> ( ( ( modulo_modulo @ int @ B7 @ N )
= ( zero_zero @ int ) )
=> ( ord_less @ int @ ( divide_divide @ int @ A4 @ N ) @ ( divide_divide @ int @ B7 @ N ) ) ) ) ) ) ).
% zdiv_mono_strict
thf(fact_543_zmod__zminus1__eq__if,axiom,
! [A3: int,B2: int] :
( ( ( ( modulo_modulo @ int @ A3 @ B2 )
= ( zero_zero @ int ) )
=> ( ( modulo_modulo @ int @ ( uminus_uminus @ int @ A3 ) @ B2 )
= ( zero_zero @ int ) ) )
& ( ( ( modulo_modulo @ int @ A3 @ B2 )
!= ( zero_zero @ int ) )
=> ( ( modulo_modulo @ int @ ( uminus_uminus @ int @ A3 ) @ B2 )
= ( minus_minus @ int @ B2 @ ( modulo_modulo @ int @ A3 @ B2 ) ) ) ) ) ).
% zmod_zminus1_eq_if
thf(fact_544_zmod__zminus2__eq__if,axiom,
! [A3: int,B2: int] :
( ( ( ( modulo_modulo @ int @ A3 @ B2 )
= ( zero_zero @ int ) )
=> ( ( modulo_modulo @ int @ A3 @ ( uminus_uminus @ int @ B2 ) )
= ( zero_zero @ int ) ) )
& ( ( ( modulo_modulo @ int @ A3 @ B2 )
!= ( zero_zero @ int ) )
=> ( ( modulo_modulo @ int @ A3 @ ( uminus_uminus @ int @ B2 ) )
= ( minus_minus @ int @ ( modulo_modulo @ int @ A3 @ B2 ) @ B2 ) ) ) ) ).
% zmod_zminus2_eq_if
thf(fact_545_ex__least__nat__less,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ ( zero_zero @ nat ) )
=> ? [K2: nat] :
( ( ord_less @ nat @ K2 @ N )
& ! [I4: nat] :
( ( ord_less_eq @ nat @ I4 @ K2 )
=> ~ ( P @ I4 ) )
& ( P @ ( suc @ K2 ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_546_of__nat__diff,axiom,
! [A: $tType] :
( ( semiring_1_cancel @ A )
=> ! [N: nat,M2: nat] :
( ( ord_less_eq @ nat @ N @ M2 )
=> ( ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ M2 @ N ) )
= ( minus_minus @ A @ ( semiring_1_of_nat @ A @ M2 ) @ ( semiring_1_of_nat @ A @ N ) ) ) ) ) ).
% of_nat_diff
thf(fact_547_div__greater__zero__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( divide_divide @ nat @ M2 @ N ) )
= ( ( ord_less_eq @ nat @ N @ M2 )
& ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ).
% div_greater_zero_iff
thf(fact_548_div__le__mono2,axiom,
! [M2: nat,N: nat,K: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M2 )
=> ( ( ord_less_eq @ nat @ M2 @ N )
=> ( ord_less_eq @ nat @ ( divide_divide @ nat @ K @ N ) @ ( divide_divide @ nat @ K @ M2 ) ) ) ) ).
% div_le_mono2
thf(fact_549_int__one__le__iff__zero__less,axiom,
! [Z2: int] :
( ( ord_less_eq @ int @ ( one_one @ int ) @ Z2 )
= ( ord_less @ int @ ( zero_zero @ int ) @ Z2 ) ) ).
% int_one_le_iff_zero_less
thf(fact_550_int__zle__neg,axiom,
! [N: nat,M2: nat] :
( ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ N ) @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ M2 ) ) )
= ( ( N
= ( zero_zero @ nat ) )
& ( M2
= ( zero_zero @ nat ) ) ) ) ).
% int_zle_neg
thf(fact_551_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_552_nonpos__int__cases,axiom,
! [K: int] :
( ( ord_less_eq @ int @ K @ ( zero_zero @ int ) )
=> ~ ! [N2: nat] :
( K
!= ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N2 ) ) ) ) ).
% nonpos_int_cases
thf(fact_553_nonneg1__imp__zdiv__pos__iff,axiom,
! [A3: int,B2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A3 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ ( divide_divide @ int @ A3 @ B2 ) )
= ( ( ord_less_eq @ int @ B2 @ A3 )
& ( ord_less @ int @ ( zero_zero @ int ) @ B2 ) ) ) ) ).
% nonneg1_imp_zdiv_pos_iff
thf(fact_554_pos__imp__zdiv__nonneg__iff,axiom,
! [B2: int,A3: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( divide_divide @ int @ A3 @ B2 ) )
= ( ord_less_eq @ int @ ( zero_zero @ int ) @ A3 ) ) ) ).
% pos_imp_zdiv_nonneg_iff
thf(fact_555_neg__imp__zdiv__nonneg__iff,axiom,
! [B2: int,A3: int] :
( ( ord_less @ int @ B2 @ ( zero_zero @ int ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( divide_divide @ int @ A3 @ B2 ) )
= ( ord_less_eq @ int @ A3 @ ( zero_zero @ int ) ) ) ) ).
% neg_imp_zdiv_nonneg_iff
thf(fact_556_pos__imp__zdiv__pos__iff,axiom,
! [K: int,I: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ K )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ ( divide_divide @ int @ I @ K ) )
= ( ord_less_eq @ int @ K @ I ) ) ) ).
% pos_imp_zdiv_pos_iff
thf(fact_557_div__nonpos__pos__le0,axiom,
! [A3: int,B2: int] :
( ( ord_less_eq @ int @ A3 @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
=> ( ord_less_eq @ int @ ( divide_divide @ int @ A3 @ B2 ) @ ( zero_zero @ int ) ) ) ) ).
% div_nonpos_pos_le0
thf(fact_558_div__nonneg__neg__le0,axiom,
! [A3: int,B2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A3 )
=> ( ( ord_less @ int @ B2 @ ( zero_zero @ int ) )
=> ( ord_less_eq @ int @ ( divide_divide @ int @ A3 @ B2 ) @ ( zero_zero @ int ) ) ) ) ).
% div_nonneg_neg_le0
thf(fact_559_div__positive__int,axiom,
! [L: int,K: int] :
( ( ord_less_eq @ int @ L @ K )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ L )
=> ( ord_less @ int @ ( zero_zero @ int ) @ ( divide_divide @ int @ K @ L ) ) ) ) ).
% div_positive_int
thf(fact_560_div__int__pos__iff,axiom,
! [K: int,L: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( divide_divide @ int @ K @ L ) )
= ( ( K
= ( zero_zero @ int ) )
| ( L
= ( zero_zero @ int ) )
| ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
& ( ord_less_eq @ int @ ( zero_zero @ int ) @ L ) )
| ( ( ord_less @ int @ K @ ( zero_zero @ int ) )
& ( ord_less @ int @ L @ ( zero_zero @ int ) ) ) ) ) ).
% div_int_pos_iff
thf(fact_561_zdiv__mono2__neg,axiom,
! [A3: int,B6: int,B2: int] :
( ( ord_less @ int @ A3 @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B6 )
=> ( ( ord_less_eq @ int @ B6 @ B2 )
=> ( ord_less_eq @ int @ ( divide_divide @ int @ A3 @ B6 ) @ ( divide_divide @ int @ A3 @ B2 ) ) ) ) ) ).
% zdiv_mono2_neg
thf(fact_562_zdiv__mono1__neg,axiom,
! [A3: int,A8: int,B2: int] :
( ( ord_less_eq @ int @ A3 @ A8 )
=> ( ( ord_less @ int @ B2 @ ( zero_zero @ int ) )
=> ( ord_less_eq @ int @ ( divide_divide @ int @ A8 @ B2 ) @ ( divide_divide @ int @ A3 @ B2 ) ) ) ) ).
% zdiv_mono1_neg
thf(fact_563_zdiv__eq__0__iff,axiom,
! [I: int,K: int] :
( ( ( divide_divide @ int @ I @ K )
= ( zero_zero @ int ) )
= ( ( K
= ( zero_zero @ int ) )
| ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ I )
& ( ord_less @ int @ I @ K ) )
| ( ( ord_less_eq @ int @ I @ ( zero_zero @ int ) )
& ( ord_less @ int @ K @ I ) ) ) ) ).
% zdiv_eq_0_iff
thf(fact_564_zdiv__mono2,axiom,
! [A3: int,B6: int,B2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A3 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B6 )
=> ( ( ord_less_eq @ int @ B6 @ B2 )
=> ( ord_less_eq @ int @ ( divide_divide @ int @ A3 @ B2 ) @ ( divide_divide @ int @ A3 @ B6 ) ) ) ) ) ).
% zdiv_mono2
thf(fact_565_zdiv__mono1,axiom,
! [A3: int,A8: int,B2: int] :
( ( ord_less_eq @ int @ A3 @ A8 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
=> ( ord_less_eq @ int @ ( divide_divide @ int @ A3 @ B2 ) @ ( divide_divide @ int @ A8 @ B2 ) ) ) ) ).
% zdiv_mono1
thf(fact_566_split__nat,axiom,
! [P: nat > $o,I: int] :
( ( P @ ( nat2 @ I ) )
= ( ! [N5: nat] :
( ( I
= ( semiring_1_of_nat @ int @ N5 ) )
=> ( P @ N5 ) )
& ( ( ord_less @ int @ I @ ( zero_zero @ int ) )
=> ( P @ ( zero_zero @ nat ) ) ) ) ) ).
% split_nat
thf(fact_567_le__divide__eq__1,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,A3: A] :
( ( ord_less_eq @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B2 @ A3 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
& ( ord_less_eq @ A @ A3 @ B2 ) )
| ( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
& ( ord_less_eq @ A @ B2 @ A3 ) ) ) ) ) ).
% le_divide_eq_1
thf(fact_568_divide__le__eq__1,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,A3: A] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ A3 ) @ ( one_one @ A ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
& ( ord_less_eq @ A @ B2 @ A3 ) )
| ( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
& ( ord_less_eq @ A @ A3 @ B2 ) )
| ( A3
= ( zero_zero @ A ) ) ) ) ) ).
% divide_le_eq_1
thf(fact_569_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_570_verit__less__mono__div__int2,axiom,
! [A4: int,B7: int,N: int] :
( ( ord_less_eq @ int @ A4 @ B7 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ ( uminus_uminus @ int @ N ) )
=> ( ord_less_eq @ int @ ( divide_divide @ int @ B7 @ N ) @ ( divide_divide @ int @ A4 @ N ) ) ) ) ).
% verit_less_mono_div_int2
thf(fact_571_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_572_power__decreasing__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [B2: A,M2: 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 @ M2 ) @ ( power_power @ A @ B2 @ N ) )
= ( ord_less_eq @ nat @ N @ M2 ) ) ) ) ) ).
% power_decreasing_iff
thf(fact_573_power__mono__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,B2: A,N: nat] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ A @ ( power_power @ A @ A3 @ N ) @ ( power_power @ A @ B2 @ N ) )
= ( ord_less_eq @ A @ A3 @ B2 ) ) ) ) ) ) ).
% power_mono_iff
thf(fact_574_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_575_power__strict__decreasing__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [B2: A,M2: nat,N: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less @ A @ B2 @ ( one_one @ A ) )
=> ( ( ord_less @ A @ ( power_power @ A @ B2 @ M2 ) @ ( power_power @ A @ B2 @ N ) )
= ( ord_less @ nat @ N @ M2 ) ) ) ) ) ).
% power_strict_decreasing_iff
thf(fact_576_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_577_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_578_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_579_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_580_power__eq__0__iff,axiom,
! [A: $tType] :
( ( semiri2026040879449505780visors @ A )
=> ! [A3: A,N: nat] :
( ( ( power_power @ A @ A3 @ N )
= ( zero_zero @ A ) )
= ( ( A3
= ( zero_zero @ A ) )
& ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% power_eq_0_iff
thf(fact_581_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_582_ln__inj__iff,axiom,
! [X: real,Y2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ Y2 )
=> ( ( ( ln_ln @ real @ X )
= ( ln_ln @ real @ Y2 ) )
= ( X = Y2 ) ) ) ) ).
% ln_inj_iff
thf(fact_583_ln__less__cancel__iff,axiom,
! [X: real,Y2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ Y2 )
=> ( ( ord_less @ real @ ( ln_ln @ real @ X ) @ ( ln_ln @ real @ Y2 ) )
= ( ord_less @ real @ X @ Y2 ) ) ) ) ).
% ln_less_cancel_iff
thf(fact_584_power__one,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [N: nat] :
( ( power_power @ A @ ( one_one @ A ) @ N )
= ( one_one @ A ) ) ) ).
% power_one
thf(fact_585_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_586_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_587_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_588_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_589_power__one__right,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A3: A] :
( ( power_power @ A @ A3 @ ( one_one @ nat ) )
= A3 ) ) ).
% power_one_right
thf(fact_590_mod__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less @ nat @ M2 @ N )
=> ( ( modulo_modulo @ nat @ M2 @ N )
= M2 ) ) ).
% mod_less
thf(fact_591_power__inject__exp,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,M2: nat,N: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ A3 )
=> ( ( ( power_power @ A @ A3 @ M2 )
= ( power_power @ A @ A3 @ N ) )
= ( M2 = N ) ) ) ) ).
% power_inject_exp
thf(fact_592_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_593_power__Suc0__right,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A3: A] :
( ( power_power @ A @ A3 @ ( suc @ ( zero_zero @ nat ) ) )
= A3 ) ) ).
% power_Suc0_right
thf(fact_594_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_595_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_596_power__Suc__0,axiom,
! [N: nat] :
( ( power_power @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% power_Suc_0
thf(fact_597_nat__power__eq__Suc__0__iff,axiom,
! [X: nat,M2: nat] :
( ( ( power_power @ nat @ X @ M2 )
= ( suc @ ( zero_zero @ nat ) ) )
= ( ( M2
= ( zero_zero @ nat ) )
| ( X
= ( suc @ ( zero_zero @ nat ) ) ) ) ) ).
% nat_power_eq_Suc_0_iff
thf(fact_598_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_599_of__nat__power,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [M2: nat,N: nat] :
( ( semiring_1_of_nat @ A @ ( power_power @ nat @ M2 @ N ) )
= ( power_power @ A @ ( semiring_1_of_nat @ A @ M2 ) @ N ) ) ) ).
% of_nat_power
thf(fact_600_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_601_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_602_mod__by__Suc__0,axiom,
! [M2: nat] :
( ( modulo_modulo @ nat @ M2 @ ( suc @ ( zero_zero @ nat ) ) )
= ( zero_zero @ nat ) ) ).
% mod_by_Suc_0
thf(fact_603_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_604_real__arch__pow,axiom,
! [X: real,Y2: real] :
( ( ord_less @ real @ ( one_one @ real ) @ X )
=> ? [N2: nat] : ( ord_less @ real @ Y2 @ ( power_power @ real @ X @ N2 ) ) ) ).
% real_arch_pow
thf(fact_605_real__arch__pow__inv,axiom,
! [Y2: real,X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ Y2 )
=> ( ( ord_less @ real @ X @ ( one_one @ real ) )
=> ? [N2: nat] : ( ord_less @ real @ ( power_power @ real @ X @ N2 ) @ Y2 ) ) ) ).
% real_arch_pow_inv
thf(fact_606_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_607_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_608_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_609_verit__la__generic,axiom,
! [A3: int,X: int] :
( ( ord_less_eq @ int @ A3 @ X )
| ( A3 = X )
| ( ord_less_eq @ int @ X @ A3 ) ) ).
% verit_la_generic
thf(fact_610_complete__real,axiom,
! [S3: set @ real] :
( ? [X4: real] : ( member @ real @ X4 @ S3 )
=> ( ? [Z5: real] :
! [X3: real] :
( ( member @ real @ X3 @ S3 )
=> ( ord_less_eq @ real @ X3 @ Z5 ) )
=> ? [Y3: real] :
( ! [X4: real] :
( ( member @ real @ X4 @ S3 )
=> ( ord_less_eq @ real @ X4 @ Y3 ) )
& ! [Z5: real] :
( ! [X3: real] :
( ( member @ real @ X3 @ S3 )
=> ( ord_less_eq @ real @ X3 @ Z5 ) )
=> ( ord_less_eq @ real @ Y3 @ Z5 ) ) ) ) ) ).
% complete_real
thf(fact_611_less__eq__real__def,axiom,
( ( ord_less_eq @ real )
= ( ^ [X2: real,Y5: real] :
( ( ord_less @ real @ X2 @ Y5 )
| ( X2 = Y5 ) ) ) ) ).
% less_eq_real_def
thf(fact_612_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_613_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_614_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_615_ln__less__self,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ord_less @ real @ ( ln_ln @ real @ X ) @ X ) ) ).
% ln_less_self
thf(fact_616_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_617_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_618_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_619_ln__div,axiom,
! [X: real,Y2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ Y2 )
=> ( ( ln_ln @ real @ ( divide_divide @ real @ X @ Y2 ) )
= ( minus_minus @ real @ ( ln_ln @ real @ X ) @ ( ln_ln @ real @ Y2 ) ) ) ) ) ).
% ln_div
thf(fact_620_mod__Suc__eq,axiom,
! [M2: nat,N: nat] :
( ( modulo_modulo @ nat @ ( suc @ ( modulo_modulo @ nat @ M2 @ N ) ) @ N )
= ( modulo_modulo @ nat @ ( suc @ M2 ) @ N ) ) ).
% mod_Suc_eq
thf(fact_621_mod__Suc__Suc__eq,axiom,
! [M2: nat,N: nat] :
( ( modulo_modulo @ nat @ ( suc @ ( suc @ ( modulo_modulo @ nat @ M2 @ N ) ) ) @ N )
= ( modulo_modulo @ nat @ ( suc @ ( suc @ M2 ) ) @ N ) ) ).
% mod_Suc_Suc_eq
thf(fact_622_nat__power__less__imp__less,axiom,
! [I: nat,M2: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ I )
=> ( ( ord_less @ nat @ ( power_power @ nat @ I @ M2 ) @ ( power_power @ nat @ I @ N ) )
=> ( ord_less @ nat @ M2 @ N ) ) ) ).
% nat_power_less_imp_less
thf(fact_623_mod__Suc,axiom,
! [M2: nat,N: nat] :
( ( ( ( suc @ ( modulo_modulo @ nat @ M2 @ N ) )
= N )
=> ( ( modulo_modulo @ nat @ ( suc @ M2 ) @ N )
= ( zero_zero @ nat ) ) )
& ( ( ( suc @ ( modulo_modulo @ nat @ M2 @ N ) )
!= N )
=> ( ( modulo_modulo @ nat @ ( suc @ M2 ) @ N )
= ( suc @ ( modulo_modulo @ nat @ M2 @ N ) ) ) ) ) ).
% mod_Suc
thf(fact_624_mod__induct,axiom,
! [P: nat > $o,N: nat,P4: nat,M2: nat] :
( ( P @ N )
=> ( ( ord_less @ nat @ N @ P4 )
=> ( ( ord_less @ nat @ M2 @ P4 )
=> ( ! [N2: nat] :
( ( ord_less @ nat @ N2 @ P4 )
=> ( ( P @ N2 )
=> ( P @ ( modulo_modulo @ nat @ ( suc @ N2 ) @ P4 ) ) ) )
=> ( P @ M2 ) ) ) ) ) ).
% mod_induct
thf(fact_625_mod__less__divisor,axiom,
! [N: nat,M2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ord_less @ nat @ ( modulo_modulo @ nat @ M2 @ N ) @ N ) ) ).
% mod_less_divisor
thf(fact_626_mod__Suc__le__divisor,axiom,
! [M2: nat,N: nat] : ( ord_less_eq @ nat @ ( modulo_modulo @ nat @ M2 @ ( suc @ N ) ) @ N ) ).
% mod_Suc_le_divisor
thf(fact_627_power__gt__expt,axiom,
! [N: nat,K: nat] :
( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N )
=> ( ord_less @ nat @ K @ ( power_power @ nat @ N @ K ) ) ) ).
% power_gt_expt
thf(fact_628_mod__geq,axiom,
! [M2: nat,N: nat] :
( ~ ( ord_less @ nat @ M2 @ N )
=> ( ( modulo_modulo @ nat @ M2 @ N )
= ( modulo_modulo @ nat @ ( minus_minus @ nat @ M2 @ N ) @ N ) ) ) ).
% mod_geq
thf(fact_629_mod__if,axiom,
( ( modulo_modulo @ nat )
= ( ^ [M5: nat,N5: nat] : ( if @ nat @ ( ord_less @ nat @ M5 @ N5 ) @ M5 @ ( modulo_modulo @ nat @ ( minus_minus @ nat @ M5 @ N5 ) @ N5 ) ) ) ) ).
% mod_if
thf(fact_630_nat__one__le__power,axiom,
! [I: nat,N: nat] :
( ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ I )
=> ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( power_power @ nat @ I @ N ) ) ) ).
% nat_one_le_power
thf(fact_631_le__mod__geq,axiom,
! [N: nat,M2: nat] :
( ( ord_less_eq @ nat @ N @ M2 )
=> ( ( modulo_modulo @ nat @ M2 @ N )
= ( modulo_modulo @ nat @ ( minus_minus @ nat @ M2 @ N ) @ N ) ) ) ).
% le_mod_geq
thf(fact_632_zmod__int,axiom,
! [A3: nat,B2: nat] :
( ( semiring_1_of_nat @ int @ ( modulo_modulo @ nat @ A3 @ B2 ) )
= ( modulo_modulo @ int @ ( semiring_1_of_nat @ int @ A3 ) @ ( semiring_1_of_nat @ int @ B2 ) ) ) ).
% zmod_int
thf(fact_633_mod__le__divisor,axiom,
! [N: nat,M2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ord_less_eq @ nat @ ( modulo_modulo @ nat @ M2 @ N ) @ N ) ) ).
% mod_le_divisor
thf(fact_634_div__less__mono,axiom,
! [A4: nat,B7: nat,N: nat] :
( ( ord_less @ nat @ A4 @ B7 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ( modulo_modulo @ nat @ A4 @ N )
= ( zero_zero @ nat ) )
=> ( ( ( modulo_modulo @ nat @ B7 @ N )
= ( zero_zero @ nat ) )
=> ( ord_less @ nat @ ( divide_divide @ nat @ A4 @ N ) @ ( divide_divide @ nat @ B7 @ N ) ) ) ) ) ) ).
% div_less_mono
thf(fact_635_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_636_nat__power__eq,axiom,
! [Z2: int,N: nat] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z2 )
=> ( ( nat2 @ ( power_power @ int @ Z2 @ N ) )
= ( power_power @ nat @ ( nat2 @ Z2 ) @ N ) ) ) ).
% nat_power_eq
thf(fact_637_field__char__0__class_Oof__nat__div,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [M2: nat,N: nat] :
( ( semiring_1_of_nat @ A @ ( divide_divide @ nat @ M2 @ N ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( semiring_1_of_nat @ A @ M2 ) @ ( semiring_1_of_nat @ A @ ( modulo_modulo @ nat @ M2 @ N ) ) ) @ ( semiring_1_of_nat @ A @ N ) ) ) ) ).
% field_char_0_class.of_nat_div
thf(fact_638_power__not__zero,axiom,
! [A: $tType] :
( ( semiri2026040879449505780visors @ A )
=> ! [A3: A,N: nat] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( power_power @ A @ A3 @ N )
!= ( zero_zero @ A ) ) ) ) ).
% power_not_zero
thf(fact_639_power__divide,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A3: A,B2: A,N: nat] :
( ( power_power @ A @ ( divide_divide @ A @ A3 @ B2 ) @ N )
= ( divide_divide @ A @ ( power_power @ A @ A3 @ N ) @ ( power_power @ A @ B2 @ N ) ) ) ) ).
% power_divide
thf(fact_640_zero__le__power,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,N: nat] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A3 @ N ) ) ) ) ).
% zero_le_power
thf(fact_641_power__mono,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,B2: A,N: nat] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ord_less_eq @ A @ ( power_power @ A @ A3 @ N ) @ ( power_power @ A @ B2 @ N ) ) ) ) ) ).
% power_mono
thf(fact_642_zero__less__power,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,N: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A3 @ N ) ) ) ) ).
% zero_less_power
thf(fact_643_one__le__power,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,N: nat] :
( ( ord_less_eq @ A @ ( one_one @ A ) @ A3 )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ ( power_power @ A @ A3 @ N ) ) ) ) ).
% one_le_power
thf(fact_644_power__one__over,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A,N: nat] :
( ( power_power @ A @ ( divide_divide @ A @ ( one_one @ A ) @ A3 ) @ N )
= ( divide_divide @ A @ ( one_one @ A ) @ ( power_power @ A @ A3 @ N ) ) ) ) ).
% power_one_over
thf(fact_645_power__0,axiom,
! [A: $tType] :
( ( power @ A )
=> ! [A3: A] :
( ( power_power @ A @ A3 @ ( zero_zero @ nat ) )
= ( one_one @ A ) ) ) ).
% power_0
thf(fact_646_power__less__imp__less__base,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,N: nat,B2: A] :
( ( ord_less @ A @ ( power_power @ A @ A3 @ N ) @ ( power_power @ A @ B2 @ N ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less @ A @ A3 @ B2 ) ) ) ) ).
% power_less_imp_less_base
thf(fact_647_power__le__one,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,N: nat] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ A3 @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( power_power @ A @ A3 @ N ) @ ( one_one @ A ) ) ) ) ) ).
% power_le_one
thf(fact_648_power__inject__base,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,N: nat,B2: A] :
( ( ( power_power @ A @ A3 @ ( suc @ N ) )
= ( power_power @ A @ B2 @ ( suc @ N ) ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( A3 = B2 ) ) ) ) ) ).
% power_inject_base
thf(fact_649_power__le__imp__le__base,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,N: nat,B2: A] :
( ( ord_less_eq @ A @ ( power_power @ A @ A3 @ ( suc @ N ) ) @ ( power_power @ A @ B2 @ ( suc @ N ) ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less_eq @ A @ A3 @ B2 ) ) ) ) ).
% power_le_imp_le_base
thf(fact_650_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_651_power__gt1,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,N: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ A3 )
=> ( ord_less @ A @ ( one_one @ A ) @ ( power_power @ A @ A3 @ ( suc @ N ) ) ) ) ) ).
% power_gt1
thf(fact_652_power__less__imp__less__exp,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,M2: nat,N: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ A3 )
=> ( ( ord_less @ A @ ( power_power @ A @ A3 @ M2 ) @ ( power_power @ A @ A3 @ N ) )
=> ( ord_less @ nat @ M2 @ N ) ) ) ) ).
% power_less_imp_less_exp
thf(fact_653_power__strict__increasing,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N: nat,N6: nat,A3: A] :
( ( ord_less @ nat @ N @ N6 )
=> ( ( ord_less @ A @ ( one_one @ A ) @ A3 )
=> ( ord_less @ A @ ( power_power @ A @ A3 @ N ) @ ( power_power @ A @ A3 @ N6 ) ) ) ) ) ).
% power_strict_increasing
thf(fact_654_power__increasing,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N: nat,N6: nat,A3: A] :
( ( ord_less_eq @ nat @ N @ N6 )
=> ( ( ord_less_eq @ A @ ( one_one @ A ) @ A3 )
=> ( ord_less_eq @ A @ ( power_power @ A @ A3 @ N ) @ ( power_power @ A @ A3 @ N6 ) ) ) ) ) ).
% power_increasing
thf(fact_655_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_656_power__Suc__le__self,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,N: nat] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ A3 @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( power_power @ A @ A3 @ ( suc @ N ) ) @ A3 ) ) ) ) ).
% power_Suc_le_self
thf(fact_657_power__Suc__less__one,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,N: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ A3 @ ( one_one @ A ) )
=> ( ord_less @ A @ ( power_power @ A @ A3 @ ( suc @ N ) ) @ ( one_one @ A ) ) ) ) ) ).
% power_Suc_less_one
thf(fact_658_power__strict__decreasing,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N: nat,N6: nat,A3: A] :
( ( ord_less @ nat @ N @ N6 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ A3 @ ( one_one @ A ) )
=> ( ord_less @ A @ ( power_power @ A @ A3 @ N6 ) @ ( power_power @ A @ A3 @ N ) ) ) ) ) ) ).
% power_strict_decreasing
thf(fact_659_power__decreasing,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N: nat,N6: nat,A3: A] :
( ( ord_less_eq @ nat @ N @ N6 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ A3 @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( power_power @ A @ A3 @ N6 ) @ ( power_power @ A @ A3 @ N ) ) ) ) ) ) ).
% power_decreasing
thf(fact_660_power__le__imp__le__exp,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,M2: nat,N: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ ( power_power @ A @ A3 @ M2 ) @ ( power_power @ A @ A3 @ N ) )
=> ( ord_less_eq @ nat @ M2 @ N ) ) ) ) ).
% power_le_imp_le_exp
thf(fact_661_power__eq__iff__eq__base,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N: nat,A3: A,B2: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ( power_power @ A @ A3 @ N )
= ( power_power @ A @ B2 @ N ) )
= ( A3 = B2 ) ) ) ) ) ) ).
% power_eq_iff_eq_base
thf(fact_662_power__eq__imp__eq__base,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,N: nat,B2: A] :
( ( ( power_power @ A @ A3 @ N )
= ( power_power @ A @ B2 @ N ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( A3 = B2 ) ) ) ) ) ) ).
% power_eq_imp_eq_base
thf(fact_663_self__le__power,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,N: nat] :
( ( ord_less_eq @ A @ ( one_one @ A ) @ A3 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ord_less_eq @ A @ A3 @ ( power_power @ A @ A3 @ N ) ) ) ) ) ).
% self_le_power
thf(fact_664_one__less__power,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,N: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ A3 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ord_less @ A @ ( one_one @ A ) @ ( power_power @ A @ A3 @ N ) ) ) ) ) ).
% one_less_power
thf(fact_665_power__diff,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [A3: A,N: nat,M2: nat] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ nat @ N @ M2 )
=> ( ( power_power @ A @ A3 @ ( minus_minus @ nat @ M2 @ N ) )
= ( divide_divide @ A @ ( power_power @ A @ A3 @ M2 ) @ ( power_power @ A @ A3 @ N ) ) ) ) ) ) ).
% power_diff
thf(fact_666_power__strict__mono,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,B2: A,N: nat] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ord_less @ A @ ( power_power @ A @ A3 @ N ) @ ( power_power @ A @ B2 @ N ) ) ) ) ) ) ).
% power_strict_mono
thf(fact_667_zdiff__int__split,axiom,
! [P: int > $o,X: nat,Y2: nat] :
( ( P @ ( semiring_1_of_nat @ int @ ( minus_minus @ nat @ X @ Y2 ) ) )
= ( ( ( ord_less_eq @ nat @ Y2 @ X )
=> ( P @ ( minus_minus @ int @ ( semiring_1_of_nat @ int @ X ) @ ( semiring_1_of_nat @ int @ Y2 ) ) ) )
& ( ( ord_less @ nat @ X @ Y2 )
=> ( P @ ( zero_zero @ int ) ) ) ) ) ).
% zdiff_int_split
thf(fact_668_Bolzano,axiom,
! [A3: real,B2: real,P: real > real > $o] :
( ( ord_less_eq @ real @ A3 @ B2 )
=> ( ! [A6: real,B4: real,C3: real] :
( ( P @ A6 @ B4 )
=> ( ( P @ B4 @ C3 )
=> ( ( ord_less_eq @ real @ A6 @ B4 )
=> ( ( ord_less_eq @ real @ B4 @ C3 )
=> ( P @ A6 @ C3 ) ) ) ) )
=> ( ! [X3: real] :
( ( ord_less_eq @ real @ A3 @ X3 )
=> ( ( ord_less_eq @ real @ X3 @ B2 )
=> ? [D3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D3 )
& ! [A6: real,B4: real] :
( ( ( ord_less_eq @ real @ A6 @ X3 )
& ( ord_less_eq @ real @ X3 @ B4 )
& ( ord_less @ real @ ( minus_minus @ real @ B4 @ A6 ) @ D3 ) )
=> ( P @ A6 @ B4 ) ) ) ) )
=> ( P @ A3 @ B2 ) ) ) ) ).
% Bolzano
thf(fact_669_gcd__nat__induct,axiom,
! [P: nat > nat > $o,M2: nat,N: nat] :
( ! [M3: nat] : ( P @ M3 @ ( zero_zero @ nat ) )
=> ( ! [M3: nat,N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( P @ N2 @ ( modulo_modulo @ nat @ M3 @ N2 ) )
=> ( P @ M3 @ N2 ) ) )
=> ( P @ M2 @ N ) ) ) ).
% gcd_nat_induct
thf(fact_670_realpow__pos__nth,axiom,
! [N: nat,A3: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ A3 )
=> ? [R3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R3 )
& ( ( power_power @ real @ R3 @ N )
= A3 ) ) ) ) ).
% realpow_pos_nth
thf(fact_671_realpow__pos__nth__unique,axiom,
! [N: nat,A3: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ A3 )
=> ? [X3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
& ( ( power_power @ real @ X3 @ N )
= A3 )
& ! [Y: real] :
( ( ( ord_less @ real @ ( zero_zero @ real ) @ Y )
& ( ( power_power @ real @ Y @ N )
= A3 ) )
=> ( Y = X3 ) ) ) ) ) ).
% realpow_pos_nth_unique
thf(fact_672_nat_Osplit__sels_I1_J,axiom,
! [A: $tType,P: A > $o,F1: A,F22: nat > A,Nat: nat] :
( ( P @ ( case_nat @ A @ F1 @ F22 @ Nat ) )
= ( ( ( Nat
= ( zero_zero @ nat ) )
=> ( P @ F1 ) )
& ( ( Nat
= ( suc @ ( pred @ Nat ) ) )
=> ( P @ ( F22 @ ( pred @ Nat ) ) ) ) ) ) ).
% nat.split_sels(1)
thf(fact_673_nat_Osplit__sels_I2_J,axiom,
! [A: $tType,P: A > $o,F1: A,F22: nat > A,Nat: nat] :
( ( P @ ( case_nat @ A @ F1 @ F22 @ Nat ) )
= ( ~ ( ( ( Nat
= ( zero_zero @ nat ) )
& ~ ( P @ F1 ) )
| ( ( Nat
= ( suc @ ( pred @ Nat ) ) )
& ~ ( P @ ( F22 @ ( pred @ Nat ) ) ) ) ) ) ) ).
% nat.split_sels(2)
thf(fact_674_imp__le__cong,axiom,
! [X: int,X6: int,P: $o,P5: $o] :
( ( X = X6 )
=> ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X6 )
=> ( P = P5 ) )
=> ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
=> P )
= ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X6 )
=> P5 ) ) ) ) ).
% imp_le_cong
thf(fact_675_conj__le__cong,axiom,
! [X: int,X6: int,P: $o,P5: $o] :
( ( X = X6 )
=> ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X6 )
=> ( P = P5 ) )
=> ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
& P )
= ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X6 )
& P5 ) ) ) ) ).
% conj_le_cong
thf(fact_676_arsinh__minus__real,axiom,
! [X: real] :
( ( arsinh @ real @ ( uminus_uminus @ real @ X ) )
= ( uminus_uminus @ real @ ( arsinh @ real @ X ) ) ) ).
% arsinh_minus_real
thf(fact_677_pinf_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o,P5: A > $o,Q: A > $o,Q2: A > $o] :
( ? [Z5: A] :
! [X3: A] :
( ( ord_less @ A @ Z5 @ X3 )
=> ( ( P @ X3 )
= ( P5 @ X3 ) ) )
=> ( ? [Z5: A] :
! [X3: A] :
( ( ord_less @ A @ Z5 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z4: A] :
! [X4: A] :
( ( ord_less @ A @ Z4 @ X4 )
=> ( ( ( P @ X4 )
& ( Q @ X4 ) )
= ( ( P5 @ X4 )
& ( Q2 @ X4 ) ) ) ) ) ) ) ).
% pinf(1)
thf(fact_678_pinf_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o,P5: A > $o,Q: A > $o,Q2: A > $o] :
( ? [Z5: A] :
! [X3: A] :
( ( ord_less @ A @ Z5 @ X3 )
=> ( ( P @ X3 )
= ( P5 @ X3 ) ) )
=> ( ? [Z5: A] :
! [X3: A] :
( ( ord_less @ A @ Z5 @ X3 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z4: A] :
! [X4: A] :
( ( ord_less @ A @ Z4 @ X4 )
=> ( ( ( P @ X4 )
| ( Q @ X4 ) )
= ( ( P5 @ X4 )
| ( Q2 @ X4 ) ) ) ) ) ) ) ).
% pinf(2)
thf(fact_679_pinf_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z4: A] :
! [X4: A] :
( ( ord_less @ A @ Z4 @ X4 )
=> ( X4 != T2 ) ) ) ).
% pinf(3)
thf(fact_680_pinf_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z4: A] :
! [X4: A] :
( ( ord_less @ A @ Z4 @ X4 )
=> ( X4 != T2 ) ) ) ).
% pinf(4)
thf(fact_681_pinf_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z4: A] :
! [X4: A] :
( ( ord_less @ A @ Z4 @ X4 )
=> ~ ( ord_less @ A @ X4 @ T2 ) ) ) ).
% pinf(5)
thf(fact_682_pinf_I7_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z4: A] :
! [X4: A] :
( ( ord_less @ A @ Z4 @ X4 )
=> ( ord_less @ A @ T2 @ X4 ) ) ) ).
% pinf(7)
thf(fact_683_pinf_I11_J,axiom,
! [C: $tType,D: $tType] :
( ( ord @ C )
=> ! [F5: D] :
? [Z4: C] :
! [X4: C] :
( ( ord_less @ C @ Z4 @ X4 )
=> ( F5 = F5 ) ) ) ).
% pinf(11)
thf(fact_684_minf_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o,P5: A > $o,Q: A > $o,Q2: A > $o] :
( ? [Z5: A] :
! [X3: A] :
( ( ord_less @ A @ X3 @ Z5 )
=> ( ( P @ X3 )
= ( P5 @ X3 ) ) )
=> ( ? [Z5: A] :
! [X3: A] :
( ( ord_less @ A @ X3 @ Z5 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z4: A] :
! [X4: A] :
( ( ord_less @ A @ X4 @ Z4 )
=> ( ( ( P @ X4 )
& ( Q @ X4 ) )
= ( ( P5 @ X4 )
& ( Q2 @ X4 ) ) ) ) ) ) ) ).
% minf(1)
thf(fact_685_minf_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > $o,P5: A > $o,Q: A > $o,Q2: A > $o] :
( ? [Z5: A] :
! [X3: A] :
( ( ord_less @ A @ X3 @ Z5 )
=> ( ( P @ X3 )
= ( P5 @ X3 ) ) )
=> ( ? [Z5: A] :
! [X3: A] :
( ( ord_less @ A @ X3 @ Z5 )
=> ( ( Q @ X3 )
= ( Q2 @ X3 ) ) )
=> ? [Z4: A] :
! [X4: A] :
( ( ord_less @ A @ X4 @ Z4 )
=> ( ( ( P @ X4 )
| ( Q @ X4 ) )
= ( ( P5 @ X4 )
| ( Q2 @ X4 ) ) ) ) ) ) ) ).
% minf(2)
thf(fact_686_minf_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z4: A] :
! [X4: A] :
( ( ord_less @ A @ X4 @ Z4 )
=> ( X4 != T2 ) ) ) ).
% minf(3)
thf(fact_687_minf_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z4: A] :
! [X4: A] :
( ( ord_less @ A @ X4 @ Z4 )
=> ( X4 != T2 ) ) ) ).
% minf(4)
thf(fact_688_minf_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z4: A] :
! [X4: A] :
( ( ord_less @ A @ X4 @ Z4 )
=> ( ord_less @ A @ X4 @ T2 ) ) ) ).
% minf(5)
thf(fact_689_minf_I7_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z4: A] :
! [X4: A] :
( ( ord_less @ A @ X4 @ Z4 )
=> ~ ( ord_less @ A @ T2 @ X4 ) ) ) ).
% minf(7)
thf(fact_690_minf_I11_J,axiom,
! [C: $tType,D: $tType] :
( ( ord @ C )
=> ! [F5: D] :
? [Z4: C] :
! [X4: C] :
( ( ord_less @ C @ X4 @ Z4 )
=> ( F5 = F5 ) ) ) ).
% minf(11)
thf(fact_691_pinf_I6_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z4: A] :
! [X4: A] :
( ( ord_less @ A @ Z4 @ X4 )
=> ~ ( ord_less_eq @ A @ X4 @ T2 ) ) ) ).
% pinf(6)
thf(fact_692_pinf_I8_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z4: A] :
! [X4: A] :
( ( ord_less @ A @ Z4 @ X4 )
=> ( ord_less_eq @ A @ T2 @ X4 ) ) ) ).
% pinf(8)
thf(fact_693_minf_I6_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z4: A] :
! [X4: A] :
( ( ord_less @ A @ X4 @ Z4 )
=> ( ord_less_eq @ A @ X4 @ T2 ) ) ) ).
% minf(6)
thf(fact_694_minf_I8_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z4: A] :
! [X4: A] :
( ( ord_less @ A @ X4 @ Z4 )
=> ~ ( ord_less_eq @ A @ T2 @ X4 ) ) ) ).
% minf(8)
thf(fact_695_realpow__pos__nth2,axiom,
! [A3: real,N: nat] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A3 )
=> ? [R3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R3 )
& ( ( power_power @ real @ R3 @ ( suc @ N ) )
= A3 ) ) ) ).
% realpow_pos_nth2
thf(fact_696_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_697_Lattices__Big_Oex__has__greatest__nat,axiom,
! [A: $tType,P: A > $o,K: A,F2: A > nat,B2: nat] :
( ( P @ K )
=> ( ! [Y3: A] :
( ( P @ Y3 )
=> ( ord_less @ nat @ ( F2 @ Y3 ) @ B2 ) )
=> ? [X3: A] :
( ( P @ X3 )
& ! [Y: A] :
( ( P @ Y )
=> ( ord_less_eq @ nat @ ( F2 @ Y ) @ ( F2 @ X3 ) ) ) ) ) ) ).
% Lattices_Big.ex_has_greatest_nat
thf(fact_698_nat__descend__induct,axiom,
! [N: nat,P: nat > $o,M2: nat] :
( ! [K2: nat] :
( ( ord_less @ nat @ N @ K2 )
=> ( P @ K2 ) )
=> ( ! [K2: nat] :
( ( ord_less_eq @ nat @ K2 @ N )
=> ( ! [I4: nat] :
( ( ord_less @ nat @ K2 @ I4 )
=> ( P @ I4 ) )
=> ( P @ K2 ) ) )
=> ( P @ M2 ) ) ) ).
% nat_descend_induct
thf(fact_699_nat__ivt__aux,axiom,
! [N: nat,F2: nat > int,K: int] :
( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ N )
=> ( ord_less_eq @ int @ ( abs_abs @ int @ ( minus_minus @ int @ ( F2 @ ( suc @ I3 ) ) @ ( F2 @ I3 ) ) ) @ ( one_one @ int ) ) )
=> ( ( ord_less_eq @ int @ ( F2 @ ( zero_zero @ nat ) ) @ K )
=> ( ( ord_less_eq @ int @ K @ ( F2 @ N ) )
=> ? [I3: nat] :
( ( ord_less_eq @ nat @ I3 @ N )
& ( ( F2 @ I3 )
= K ) ) ) ) ) ).
% nat_ivt_aux
thf(fact_700_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_701_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_702_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_703_order__less__le__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C )
& ( order @ A ) )
=> ! [A3: A,B2: A,F2: A > C,C2: C] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ C @ ( F2 @ B2 ) @ C2 )
=> ( ! [X3: A,Y3: A] :
( ( ord_less @ A @ X3 @ Y3 )
=> ( ord_less @ C @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less @ C @ ( F2 @ A3 ) @ C2 ) ) ) ) ) ).
% order_less_le_subst2
thf(fact_704_order__less__le__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A3: A,F2: B > A,B2: B,C2: B] :
( ( ord_less @ A @ A3 @ ( F2 @ B2 ) )
=> ( ( ord_less_eq @ B @ B2 @ C2 )
=> ( ! [X3: B,Y3: B] :
( ( ord_less_eq @ B @ X3 @ Y3 )
=> ( ord_less_eq @ A @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less @ A @ A3 @ ( F2 @ C2 ) ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_705_order__le__less__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C )
& ( order @ A ) )
=> ! [A3: A,B2: A,F2: A > C,C2: C] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less @ C @ ( F2 @ B2 ) @ C2 )
=> ( ! [X3: A,Y3: A] :
( ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ord_less_eq @ C @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less @ C @ ( F2 @ A3 ) @ C2 ) ) ) ) ) ).
% order_le_less_subst2
thf(fact_706_abs__abs,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ! [A3: A] :
( ( abs_abs @ A @ ( abs_abs @ A @ A3 ) )
= ( abs_abs @ A @ A3 ) ) ) ).
% abs_abs
thf(fact_707_abs__idempotent,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A] :
( ( abs_abs @ A @ ( abs_abs @ A @ A3 ) )
= ( abs_abs @ A @ A3 ) ) ) ).
% abs_idempotent
thf(fact_708_exp__inj__iff,axiom,
! [X: real,Y2: real] :
( ( ( exp @ real @ X )
= ( exp @ real @ Y2 ) )
= ( X = Y2 ) ) ).
% exp_inj_iff
thf(fact_709_abs__0__eq,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A] :
( ( ( zero_zero @ A )
= ( abs_abs @ A @ A3 ) )
= ( A3
= ( zero_zero @ A ) ) ) ) ).
% abs_0_eq
thf(fact_710_abs__eq__0,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A] :
( ( ( abs_abs @ A @ A3 )
= ( zero_zero @ A ) )
= ( A3
= ( zero_zero @ A ) ) ) ) ).
% abs_eq_0
thf(fact_711_abs__zero,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ( ( abs_abs @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% abs_zero
thf(fact_712_abs__0,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ( ( abs_abs @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% abs_0
thf(fact_713_abs__1,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ( ( abs_abs @ A @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% abs_1
thf(fact_714_abs__divide,axiom,
! [A: $tType] :
( ( field_abs_sgn @ A )
=> ! [A3: A,B2: A] :
( ( abs_abs @ A @ ( divide_divide @ A @ A3 @ B2 ) )
= ( divide_divide @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ B2 ) ) ) ) ).
% abs_divide
thf(fact_715_abs__minus,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ! [A3: A] :
( ( abs_abs @ A @ ( uminus_uminus @ A @ A3 ) )
= ( abs_abs @ A @ A3 ) ) ) ).
% abs_minus
thf(fact_716_abs__minus__cancel,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A] :
( ( abs_abs @ A @ ( uminus_uminus @ A @ A3 ) )
= ( abs_abs @ A @ A3 ) ) ) ).
% abs_minus_cancel
thf(fact_717_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_718_exp__less__cancel__iff,axiom,
! [X: real,Y2: real] :
( ( ord_less @ real @ ( exp @ real @ X ) @ ( exp @ real @ Y2 ) )
= ( ord_less @ real @ X @ Y2 ) ) ).
% exp_less_cancel_iff
thf(fact_719_exp__less__mono,axiom,
! [X: real,Y2: real] :
( ( ord_less @ real @ X @ Y2 )
=> ( ord_less @ real @ ( exp @ real @ X ) @ ( exp @ real @ Y2 ) ) ) ).
% exp_less_mono
thf(fact_720_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_721_ln__exp,axiom,
! [X: real] :
( ( ln_ln @ real @ ( exp @ real @ X ) )
= X ) ).
% ln_exp
thf(fact_722_abs__le__zero__iff,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( abs_abs @ A @ A3 ) @ ( zero_zero @ A ) )
= ( A3
= ( zero_zero @ A ) ) ) ) ).
% abs_le_zero_iff
thf(fact_723_abs__le__self__iff,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( abs_abs @ A @ A3 ) @ A3 )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% abs_le_self_iff
thf(fact_724_abs__of__nonneg,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( abs_abs @ A @ A3 )
= A3 ) ) ) ).
% abs_of_nonneg
thf(fact_725_zero__less__abs__iff,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( abs_abs @ A @ A3 ) )
= ( A3
!= ( zero_zero @ A ) ) ) ) ).
% zero_less_abs_iff
thf(fact_726_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_727_abs__power__minus,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,N: nat] :
( ( abs_abs @ A @ ( power_power @ A @ ( uminus_uminus @ A @ A3 ) @ N ) )
= ( abs_abs @ A @ ( power_power @ A @ A3 @ N ) ) ) ) ).
% abs_power_minus
thf(fact_728_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_729_real__root__Suc__0,axiom,
! [X: real] :
( ( root @ ( suc @ ( zero_zero @ nat ) ) @ X )
= X ) ).
% real_root_Suc_0
thf(fact_730_root__0,axiom,
! [X: real] :
( ( root @ ( zero_zero @ nat ) @ X )
= ( zero_zero @ real ) ) ).
% root_0
thf(fact_731_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_732_exp__eq__one__iff,axiom,
! [X: real] :
( ( ( exp @ real @ X )
= ( one_one @ real ) )
= ( X
= ( zero_zero @ real ) ) ) ).
% exp_eq_one_iff
thf(fact_733_zero__le__divide__abs__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ A3 @ ( abs_abs @ A @ B2 ) ) )
= ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
| ( B2
= ( zero_zero @ A ) ) ) ) ) ).
% zero_le_divide_abs_iff
thf(fact_734_divide__le__0__abs__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ A3 @ ( abs_abs @ A @ B2 ) ) @ ( zero_zero @ A ) )
= ( ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) )
| ( B2
= ( zero_zero @ A ) ) ) ) ) ).
% divide_le_0_abs_iff
thf(fact_735_abs__of__nonpos,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( abs_abs @ A @ A3 )
= ( uminus_uminus @ A @ A3 ) ) ) ) ).
% abs_of_nonpos
thf(fact_736_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_737_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_738_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_739_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_740_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_741_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_742_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_743_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_744_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_745_zabs__less__one__iff,axiom,
! [Z2: int] :
( ( ord_less @ int @ ( abs_abs @ int @ Z2 ) @ ( one_one @ int ) )
= ( Z2
= ( zero_zero @ int ) ) ) ).
% zabs_less_one_iff
thf(fact_746_exp__ln,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( exp @ real @ ( ln_ln @ real @ X ) )
= X ) ) ).
% exp_ln
thf(fact_747_exp__ln__iff,axiom,
! [X: real] :
( ( ( exp @ real @ ( ln_ln @ real @ X ) )
= X )
= ( ord_less @ real @ ( zero_zero @ real ) @ X ) ) ).
% exp_ln_iff
thf(fact_748_zero__less__power__abs__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,N: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( power_power @ A @ ( abs_abs @ A @ A3 ) @ N ) )
= ( ( A3
!= ( zero_zero @ A ) )
| ( N
= ( zero_zero @ nat ) ) ) ) ) ).
% zero_less_power_abs_iff
thf(fact_749_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_750_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_751_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_752_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_753_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_754_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_755_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_756_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_757_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_758_abs__ge__self,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A] : ( ord_less_eq @ A @ A3 @ ( abs_abs @ A @ A3 ) ) ) ).
% abs_ge_self
thf(fact_759_abs__le__D1,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( abs_abs @ A @ A3 ) @ B2 )
=> ( ord_less_eq @ A @ A3 @ B2 ) ) ) ).
% abs_le_D1
thf(fact_760_abs__eq__0__iff,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ! [A3: A] :
( ( ( abs_abs @ A @ A3 )
= ( zero_zero @ A ) )
= ( A3
= ( zero_zero @ A ) ) ) ) ).
% abs_eq_0_iff
thf(fact_761_abs__one,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( ( abs_abs @ A @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% abs_one
thf(fact_762_abs__minus__commute,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A,B2: A] :
( ( abs_abs @ A @ ( minus_minus @ A @ A3 @ B2 ) )
= ( abs_abs @ A @ ( minus_minus @ A @ B2 @ A3 ) ) ) ) ).
% abs_minus_commute
thf(fact_763_exp__less__cancel,axiom,
! [X: real,Y2: real] :
( ( ord_less @ real @ ( exp @ real @ X ) @ ( exp @ real @ Y2 ) )
=> ( ord_less @ real @ X @ Y2 ) ) ).
% exp_less_cancel
thf(fact_764_abs__eq__iff,axiom,
! [A: $tType] :
( ( linordered_ring @ A )
=> ! [X: A,Y2: A] :
( ( ( abs_abs @ A @ X )
= ( abs_abs @ A @ Y2 ) )
= ( ( X = Y2 )
| ( X
= ( uminus_uminus @ A @ Y2 ) ) ) ) ) ).
% abs_eq_iff
thf(fact_765_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_766_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_767_exp__not__eq__zero,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X: A] :
( ( exp @ A @ X )
!= ( zero_zero @ A ) ) ) ).
% exp_not_eq_zero
thf(fact_768_ln__unique,axiom,
! [Y2: real,X: real] :
( ( ( exp @ real @ Y2 )
= X )
=> ( ( ln_ln @ real @ X )
= Y2 ) ) ).
% ln_unique
thf(fact_769_abs__ge__zero,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( abs_abs @ A @ A3 ) ) ) ).
% abs_ge_zero
thf(fact_770_abs__of__pos,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( abs_abs @ A @ A3 )
= A3 ) ) ) ).
% abs_of_pos
thf(fact_771_abs__not__less__zero,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A] :
~ ( ord_less @ A @ ( abs_abs @ A @ A3 ) @ ( zero_zero @ A ) ) ) ).
% abs_not_less_zero
thf(fact_772_abs__triangle__ineq2__sym,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A,B2: A] : ( ord_less_eq @ A @ ( minus_minus @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ B2 ) ) @ ( abs_abs @ A @ ( minus_minus @ A @ B2 @ A3 ) ) ) ) ).
% abs_triangle_ineq2_sym
thf(fact_773_abs__triangle__ineq3,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A,B2: A] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ B2 ) ) ) @ ( abs_abs @ A @ ( minus_minus @ A @ A3 @ B2 ) ) ) ) ).
% abs_triangle_ineq3
thf(fact_774_abs__triangle__ineq2,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A,B2: A] : ( ord_less_eq @ A @ ( minus_minus @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ B2 ) ) @ ( abs_abs @ A @ ( minus_minus @ A @ A3 @ B2 ) ) ) ) ).
% abs_triangle_ineq2
thf(fact_775_nonzero__abs__divide,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,A3: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( abs_abs @ A @ ( divide_divide @ A @ A3 @ B2 ) )
= ( divide_divide @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ B2 ) ) ) ) ) ).
% nonzero_abs_divide
thf(fact_776_abs__ge__minus__self,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A] : ( ord_less_eq @ A @ ( uminus_uminus @ A @ A3 ) @ ( abs_abs @ A @ A3 ) ) ) ).
% abs_ge_minus_self
thf(fact_777_abs__le__iff,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( abs_abs @ A @ A3 ) @ B2 )
= ( ( ord_less_eq @ A @ A3 @ B2 )
& ( ord_less_eq @ A @ ( uminus_uminus @ A @ A3 ) @ B2 ) ) ) ) ).
% abs_le_iff
thf(fact_778_abs__le__D2,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( abs_abs @ A @ A3 ) @ B2 )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ A3 ) @ B2 ) ) ) ).
% abs_le_D2
thf(fact_779_abs__leI,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A3 ) @ B2 )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ A3 ) @ B2 ) ) ) ) ).
% abs_leI
thf(fact_780_abs__less__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ ( abs_abs @ A @ A3 ) @ B2 )
= ( ( ord_less @ A @ A3 @ B2 )
& ( ord_less @ A @ ( uminus_uminus @ A @ A3 ) @ B2 ) ) ) ) ).
% abs_less_iff
thf(fact_781_not__exp__less__zero,axiom,
! [X: real] :
~ ( ord_less @ real @ ( exp @ real @ X ) @ ( zero_zero @ real ) ) ).
% not_exp_less_zero
thf(fact_782_exp__gt__zero,axiom,
! [X: real] : ( ord_less @ real @ ( zero_zero @ real ) @ ( exp @ real @ X ) ) ).
% exp_gt_zero
thf(fact_783_exp__total,axiom,
! [Y2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ Y2 )
=> ? [X3: real] :
( ( exp @ real @ X3 )
= Y2 ) ) ).
% exp_total
thf(fact_784_exp__ge__zero,axiom,
! [X: real] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( exp @ real @ X ) ) ).
% exp_ge_zero
thf(fact_785_not__exp__le__zero,axiom,
! [X: real] :
~ ( ord_less_eq @ real @ ( exp @ real @ X ) @ ( zero_zero @ real ) ) ).
% not_exp_le_zero
thf(fact_786_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_787_dense__eq0__I,axiom,
! [A: $tType] :
( ( ( ordere166539214618696060dd_abs @ A )
& ( dense_linorder @ A ) )
=> ! [X: A] :
( ! [E: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ E )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ X ) @ E ) )
=> ( X
= ( zero_zero @ A ) ) ) ) ).
% dense_eq0_I
thf(fact_788_abs__minus__le__zero,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A] : ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( abs_abs @ A @ A3 ) ) @ ( zero_zero @ A ) ) ) ).
% abs_minus_le_zero
thf(fact_789_eq__abs__iff_H,axiom,
! [A: $tType] :
( ( linordered_ring @ A )
=> ! [A3: A,B2: A] :
( ( A3
= ( abs_abs @ A @ B2 ) )
= ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
& ( ( B2 = A3 )
| ( B2
= ( uminus_uminus @ A @ A3 ) ) ) ) ) ) ).
% eq_abs_iff'
thf(fact_790_abs__eq__iff_H,axiom,
! [A: $tType] :
( ( linordered_ring @ A )
=> ! [A3: A,B2: A] :
( ( ( abs_abs @ A @ A3 )
= B2 )
= ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
& ( ( A3 = B2 )
| ( A3
= ( uminus_uminus @ A @ B2 ) ) ) ) ) ) ).
% abs_eq_iff'
thf(fact_791_zero__le__power__abs,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,N: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ ( abs_abs @ A @ A3 ) @ N ) ) ) ).
% zero_le_power_abs
thf(fact_792_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_793_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_794_abs__of__neg,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A] :
( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( abs_abs @ A @ A3 )
= ( uminus_uminus @ A @ A3 ) ) ) ) ).
% abs_of_neg
thf(fact_795_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_796_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_797_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_798_real__root__power,axiom,
! [N: nat,X: real,K: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( root @ N @ ( power_power @ real @ X @ K ) )
= ( power_power @ real @ ( root @ N @ X ) @ K ) ) ) ).
% real_root_power
thf(fact_799_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_800_zabs__def,axiom,
( ( abs_abs @ int )
= ( ^ [I2: int] : ( if @ int @ ( ord_less @ int @ I2 @ ( zero_zero @ int ) ) @ ( uminus_uminus @ int @ I2 ) @ I2 ) ) ) ).
% zabs_def
thf(fact_801_abs__mod__less,axiom,
! [L: int,K: int] :
( ( L
!= ( zero_zero @ int ) )
=> ( ord_less @ int @ ( abs_abs @ int @ ( modulo_modulo @ int @ K @ L ) ) @ ( abs_abs @ int @ L ) ) ) ).
% abs_mod_less
thf(fact_802_lt__ex,axiom,
! [A: $tType] :
( ( no_bot @ A )
=> ! [X: A] :
? [Y3: A] : ( ord_less @ A @ Y3 @ X ) ) ).
% lt_ex
thf(fact_803_gt__ex,axiom,
! [A: $tType] :
( ( no_top @ A )
=> ! [X: A] :
? [X_1: A] : ( ord_less @ A @ X @ X_1 ) ) ).
% gt_ex
thf(fact_804_dense,axiom,
! [A: $tType] :
( ( dense_order @ A )
=> ! [X: A,Y2: A] :
( ( ord_less @ A @ X @ Y2 )
=> ? [Z4: A] :
( ( ord_less @ A @ X @ Z4 )
& ( ord_less @ A @ Z4 @ Y2 ) ) ) ) ).
% dense
thf(fact_805_less__imp__neq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,Y2: A] :
( ( ord_less @ A @ X @ Y2 )
=> ( X != Y2 ) ) ) ).
% less_imp_neq
thf(fact_806_order_Oasym,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ~ ( ord_less @ A @ B2 @ A3 ) ) ) ).
% order.asym
thf(fact_807_ord__eq__less__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( A3 = B2 )
=> ( ( ord_less @ A @ B2 @ C2 )
=> ( ord_less @ A @ A3 @ C2 ) ) ) ) ).
% ord_eq_less_trans
thf(fact_808_ord__less__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( B2 = C2 )
=> ( ord_less @ A @ A3 @ C2 ) ) ) ) ).
% ord_less_eq_trans
thf(fact_809_less__induct,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [P: A > $o,A3: A] :
( ! [X3: A] :
( ! [Y: A] :
( ( ord_less @ A @ Y @ X3 )
=> ( P @ Y ) )
=> ( P @ X3 ) )
=> ( P @ A3 ) ) ) ).
% less_induct
thf(fact_810_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_811_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_812_dual__order_Oasym,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [B2: A,A3: A] :
( ( ord_less @ A @ B2 @ A3 )
=> ~ ( ord_less @ A @ A3 @ B2 ) ) ) ).
% dual_order.asym
thf(fact_813_dual__order_Oirrefl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A] :
~ ( ord_less @ A @ A3 @ A3 ) ) ).
% dual_order.irrefl
thf(fact_814_exists__least__iff,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ( ( ^ [P2: A > $o] :
? [X5: A] : ( P2 @ X5 ) )
= ( ^ [P3: A > $o] :
? [N5: A] :
( ( P3 @ N5 )
& ! [M5: A] :
( ( ord_less @ A @ M5 @ N5 )
=> ~ ( P3 @ M5 ) ) ) ) ) ) ).
% exists_least_iff
thf(fact_815_linorder__less__wlog,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P: A > A > $o,A3: A,B2: A] :
( ! [A6: A,B4: A] :
( ( ord_less @ A @ A6 @ B4 )
=> ( P @ A6 @ B4 ) )
=> ( ! [A6: A] : ( P @ A6 @ A6 )
=> ( ! [A6: A,B4: A] :
( ( P @ B4 @ A6 )
=> ( P @ A6 @ B4 ) )
=> ( P @ A3 @ B2 ) ) ) ) ) ).
% linorder_less_wlog
thf(fact_816_order_Ostrict__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less @ A @ B2 @ C2 )
=> ( ord_less @ A @ A3 @ C2 ) ) ) ) ).
% order.strict_trans
thf(fact_817_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_818_dual__order_Ostrict__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( ord_less @ A @ B2 @ A3 )
=> ( ( ord_less @ A @ C2 @ B2 )
=> ( ord_less @ A @ C2 @ A3 ) ) ) ) ).
% dual_order.strict_trans
thf(fact_819_order_Ostrict__implies__not__eq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( A3 != B2 ) ) ) ).
% order.strict_implies_not_eq
thf(fact_820_dual__order_Ostrict__implies__not__eq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B2: A,A3: A] :
( ( ord_less @ A @ B2 @ A3 )
=> ( A3 != B2 ) ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_821_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_822_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_823_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_824_order__less__asym_H,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ~ ( ord_less @ A @ B2 @ A3 ) ) ) ).
% order_less_asym'
thf(fact_825_order__less__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y2: A,Z2: A] :
( ( ord_less @ A @ X @ Y2 )
=> ( ( ord_less @ A @ Y2 @ Z2 )
=> ( ord_less @ A @ X @ Z2 ) ) ) ) ).
% order_less_trans
thf(fact_826_ord__eq__less__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A3: A,F2: B > A,B2: B,C2: B] :
( ( A3
= ( F2 @ B2 ) )
=> ( ( ord_less @ B @ B2 @ C2 )
=> ( ! [X3: B,Y3: B] :
( ( ord_less @ B @ X3 @ Y3 )
=> ( ord_less @ A @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less @ A @ A3 @ ( F2 @ C2 ) ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_827_ord__less__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A3: A,B2: A,F2: A > B,C2: B] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ( F2 @ B2 )
= C2 )
=> ( ! [X3: A,Y3: A] :
( ( ord_less @ A @ X3 @ Y3 )
=> ( ord_less @ B @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less @ B @ ( F2 @ A3 ) @ C2 ) ) ) ) ) ).
% ord_less_eq_subst
thf(fact_828_order__less__irrefl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A] :
~ ( ord_less @ A @ X @ X ) ) ).
% order_less_irrefl
thf(fact_829_order__less__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A3: A,F2: B > A,B2: B,C2: B] :
( ( ord_less @ A @ A3 @ ( F2 @ B2 ) )
=> ( ( ord_less @ B @ B2 @ C2 )
=> ( ! [X3: B,Y3: B] :
( ( ord_less @ B @ X3 @ Y3 )
=> ( ord_less @ A @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less @ A @ A3 @ ( F2 @ C2 ) ) ) ) ) ) ).
% order_less_subst1
thf(fact_830_order__less__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C )
& ( order @ A ) )
=> ! [A3: A,B2: A,F2: A > C,C2: C] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less @ C @ ( F2 @ B2 ) @ C2 )
=> ( ! [X3: A,Y3: A] :
( ( ord_less @ A @ X3 @ Y3 )
=> ( ord_less @ C @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less @ C @ ( F2 @ A3 ) @ C2 ) ) ) ) ) ).
% order_less_subst2
thf(fact_831_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_832_order__less__imp__triv,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y2: A,P: $o] :
( ( ord_less @ A @ X @ Y2 )
=> ( ( ord_less @ A @ Y2 @ X )
=> P ) ) ) ).
% order_less_imp_triv
thf(fact_833_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_834_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_835_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_836_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_837_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_838_real__root__strict__decreasing,axiom,
! [N: nat,N6: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ nat @ N @ N6 )
=> ( ( ord_less @ real @ ( one_one @ real ) @ X )
=> ( ord_less @ real @ ( root @ N6 @ X ) @ ( root @ N @ X ) ) ) ) ) ).
% real_root_strict_decreasing
thf(fact_839_lemma__exp__total,axiom,
! [Y2: real] :
( ( ord_less_eq @ real @ ( one_one @ real ) @ Y2 )
=> ? [X3: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 )
& ( ord_less_eq @ real @ X3 @ ( minus_minus @ real @ Y2 @ ( one_one @ real ) ) )
& ( ( exp @ real @ X3 )
= Y2 ) ) ) ).
% lemma_exp_total
thf(fact_840_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_841_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_842_div__abs__eq__div__nat,axiom,
! [K: int,L: int] :
( ( divide_divide @ int @ ( abs_abs @ int @ K ) @ ( abs_abs @ int @ L ) )
= ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ ( nat2 @ ( abs_abs @ int @ K ) ) @ ( nat2 @ ( abs_abs @ int @ L ) ) ) ) ) ).
% div_abs_eq_div_nat
thf(fact_843_mod__abs__eq__div__nat,axiom,
! [K: int,L: int] :
( ( modulo_modulo @ int @ ( abs_abs @ int @ K ) @ ( abs_abs @ int @ L ) )
= ( semiring_1_of_nat @ int @ ( modulo_modulo @ nat @ ( nat2 @ ( abs_abs @ int @ K ) ) @ ( nat2 @ ( abs_abs @ int @ L ) ) ) ) ) ).
% mod_abs_eq_div_nat
thf(fact_844_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_845_real__root__strict__increasing,axiom,
! [N: nat,N6: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ nat @ N @ N6 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ X @ ( one_one @ real ) )
=> ( ord_less @ real @ ( root @ N @ X ) @ ( root @ N6 @ X ) ) ) ) ) ) ).
% real_root_strict_increasing
thf(fact_846_real__root__decreasing,axiom,
! [N: nat,N6: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ nat @ N @ N6 )
=> ( ( ord_less_eq @ real @ ( one_one @ real ) @ X )
=> ( ord_less_eq @ real @ ( root @ N6 @ X ) @ ( root @ N @ X ) ) ) ) ) ).
% real_root_decreasing
thf(fact_847_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_848_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_849_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_850_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_851_nat__abs__int__diff,axiom,
! [A3: nat,B2: nat] :
( ( ( ord_less_eq @ nat @ A3 @ B2 )
=> ( ( nat2 @ ( abs_abs @ int @ ( minus_minus @ int @ ( semiring_1_of_nat @ int @ A3 ) @ ( semiring_1_of_nat @ int @ B2 ) ) ) )
= ( minus_minus @ nat @ B2 @ A3 ) ) )
& ( ~ ( ord_less_eq @ nat @ A3 @ B2 )
=> ( ( nat2 @ ( abs_abs @ int @ ( minus_minus @ int @ ( semiring_1_of_nat @ int @ A3 ) @ ( semiring_1_of_nat @ int @ B2 ) ) ) )
= ( minus_minus @ nat @ A3 @ B2 ) ) ) ) ).
% nat_abs_int_diff
thf(fact_852_real__root__increasing,axiom,
! [N: nat,N6: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ nat @ N @ N6 )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ X @ ( one_one @ real ) )
=> ( ord_less_eq @ real @ ( root @ N @ X ) @ ( root @ N6 @ X ) ) ) ) ) ) ).
% real_root_increasing
thf(fact_853_nat__intermed__int__val,axiom,
! [M2: nat,N: nat,F2: nat > int,K: int] :
( ! [I3: nat] :
( ( ( ord_less_eq @ nat @ M2 @ I3 )
& ( ord_less @ nat @ I3 @ N ) )
=> ( ord_less_eq @ int @ ( abs_abs @ int @ ( minus_minus @ int @ ( F2 @ ( suc @ I3 ) ) @ ( F2 @ I3 ) ) ) @ ( one_one @ int ) ) )
=> ( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( ord_less_eq @ int @ ( F2 @ M2 ) @ K )
=> ( ( ord_less_eq @ int @ K @ ( F2 @ N ) )
=> ? [I3: nat] :
( ( ord_less_eq @ nat @ M2 @ I3 )
& ( ord_less_eq @ nat @ I3 @ N )
& ( ( F2 @ I3 )
= K ) ) ) ) ) ) ).
% nat_intermed_int_val
thf(fact_854_leD,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Y2: A,X: A] :
( ( ord_less_eq @ A @ Y2 @ X )
=> ~ ( ord_less @ A @ X @ Y2 ) ) ) ).
% leD
thf(fact_855_leI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y2: A] :
( ~ ( ord_less @ A @ X @ Y2 )
=> ( ord_less_eq @ A @ Y2 @ X ) ) ) ).
% leI
thf(fact_856_nless__le,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A,B2: A] :
( ( ~ ( ord_less @ A @ A3 @ B2 ) )
= ( ~ ( ord_less_eq @ A @ A3 @ B2 )
| ( A3 = B2 ) ) ) ) ).
% nless_le
thf(fact_857_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_858_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_859_dense__ge,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [Z2: A,Y2: A] :
( ! [X3: A] :
( ( ord_less @ A @ Z2 @ X3 )
=> ( ord_less_eq @ A @ Y2 @ X3 ) )
=> ( ord_less_eq @ A @ Y2 @ Z2 ) ) ) ).
% dense_ge
thf(fact_860_dense__le,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [Y2: A,Z2: A] :
( ! [X3: A] :
( ( ord_less @ A @ X3 @ Y2 )
=> ( ord_less_eq @ A @ X3 @ Z2 ) )
=> ( ord_less_eq @ A @ Y2 @ Z2 ) ) ) ).
% dense_le
thf(fact_861_less__le__not__le,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( ( ord_less @ A )
= ( ^ [X2: A,Y5: A] :
( ( ord_less_eq @ A @ X2 @ Y5 )
& ~ ( ord_less_eq @ A @ Y5 @ X2 ) ) ) ) ) ).
% less_le_not_le
thf(fact_862_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_863_order_Oorder__iff__strict,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A5: A,B3: A] :
( ( ord_less @ A @ A5 @ B3 )
| ( A5 = B3 ) ) ) ) ) ).
% order.order_iff_strict
thf(fact_864_order_Ostrict__iff__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less @ A )
= ( ^ [A5: A,B3: A] :
( ( ord_less_eq @ A @ A5 @ B3 )
& ( A5 != B3 ) ) ) ) ) ).
% order.strict_iff_order
thf(fact_865_order_Ostrict__trans1,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less @ A @ B2 @ C2 )
=> ( ord_less @ A @ A3 @ C2 ) ) ) ) ).
% order.strict_trans1
thf(fact_866_order_Ostrict__trans2,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ C2 )
=> ( ord_less @ A @ A3 @ C2 ) ) ) ) ).
% order.strict_trans2
thf(fact_867_order_Ostrict__iff__not,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( ( ord_less @ A )
= ( ^ [A5: A,B3: A] :
( ( ord_less_eq @ A @ A5 @ B3 )
& ~ ( ord_less_eq @ A @ B3 @ A5 ) ) ) ) ) ).
% order.strict_iff_not
thf(fact_868_dense__ge__bounded,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [Z2: A,X: A,Y2: A] :
( ( ord_less @ A @ Z2 @ X )
=> ( ! [W2: A] :
( ( ord_less @ A @ Z2 @ W2 )
=> ( ( ord_less @ A @ W2 @ X )
=> ( ord_less_eq @ A @ Y2 @ W2 ) ) )
=> ( ord_less_eq @ A @ Y2 @ Z2 ) ) ) ) ).
% dense_ge_bounded
thf(fact_869_dense__le__bounded,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [X: A,Y2: A,Z2: A] :
( ( ord_less @ A @ X @ Y2 )
=> ( ! [W2: A] :
( ( ord_less @ A @ X @ W2 )
=> ( ( ord_less @ A @ W2 @ Y2 )
=> ( ord_less_eq @ A @ W2 @ Z2 ) ) )
=> ( ord_less_eq @ A @ Y2 @ Z2 ) ) ) ) ).
% dense_le_bounded
thf(fact_870_dual__order_Oorder__iff__strict,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B3: A,A5: A] :
( ( ord_less @ A @ B3 @ A5 )
| ( A5 = B3 ) ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_871_dual__order_Ostrict__iff__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less @ A )
= ( ^ [B3: A,A5: A] :
( ( ord_less_eq @ A @ B3 @ A5 )
& ( A5 != B3 ) ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_872_dual__order_Ostrict__trans1,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( ord_less_eq @ A @ B2 @ A3 )
=> ( ( ord_less @ A @ C2 @ B2 )
=> ( ord_less @ A @ C2 @ A3 ) ) ) ) ).
% dual_order.strict_trans1
thf(fact_873_dual__order_Ostrict__trans2,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( ord_less @ A @ B2 @ A3 )
=> ( ( ord_less_eq @ A @ C2 @ B2 )
=> ( ord_less @ A @ C2 @ A3 ) ) ) ) ).
% dual_order.strict_trans2
thf(fact_874_dual__order_Ostrict__iff__not,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( ( ord_less @ A )
= ( ^ [B3: A,A5: A] :
( ( ord_less_eq @ A @ B3 @ A5 )
& ~ ( ord_less_eq @ A @ A5 @ B3 ) ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_875_order_Ostrict__implies__order,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ord_less_eq @ A @ A3 @ B2 ) ) ) ).
% order.strict_implies_order
thf(fact_876_dual__order_Ostrict__implies__order,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [B2: A,A3: A] :
( ( ord_less @ A @ B2 @ A3 )
=> ( ord_less_eq @ A @ B2 @ A3 ) ) ) ).
% dual_order.strict_implies_order
thf(fact_877_order__le__less,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [X2: A,Y5: A] :
( ( ord_less @ A @ X2 @ Y5 )
| ( X2 = Y5 ) ) ) ) ) ).
% order_le_less
thf(fact_878_order__less__le,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less @ A )
= ( ^ [X2: A,Y5: A] :
( ( ord_less_eq @ A @ X2 @ Y5 )
& ( X2 != Y5 ) ) ) ) ) ).
% order_less_le
thf(fact_879_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_880_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_881_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_882_order__le__neq__trans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( A3 != B2 )
=> ( ord_less @ A @ A3 @ B2 ) ) ) ) ).
% order_le_neq_trans
thf(fact_883_order__neq__le__trans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A3: A,B2: A] :
( ( A3 != B2 )
=> ( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ord_less @ A @ A3 @ B2 ) ) ) ) ).
% order_neq_le_trans
thf(fact_884_order__le__less__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y2: A,Z2: A] :
( ( ord_less_eq @ A @ X @ Y2 )
=> ( ( ord_less @ A @ Y2 @ Z2 )
=> ( ord_less @ A @ X @ Z2 ) ) ) ) ).
% order_le_less_trans
thf(fact_885_order__less__le__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y2: A,Z2: A] :
( ( ord_less @ A @ X @ Y2 )
=> ( ( ord_less_eq @ A @ Y2 @ Z2 )
=> ( ord_less @ A @ X @ Z2 ) ) ) ) ).
% order_less_le_trans
thf(fact_886_order__le__less__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A3: A,F2: B > A,B2: B,C2: B] :
( ( ord_less_eq @ A @ A3 @ ( F2 @ B2 ) )
=> ( ( ord_less @ B @ B2 @ C2 )
=> ( ! [X3: B,Y3: B] :
( ( ord_less @ B @ X3 @ Y3 )
=> ( ord_less @ A @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ord_less @ A @ A3 @ ( F2 @ C2 ) ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_887_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_888_nat0__intermed__int__val,axiom,
! [N: nat,F2: nat > int,K: int] :
( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ N )
=> ( ord_less_eq @ int @ ( abs_abs @ int @ ( minus_minus @ int @ ( F2 @ ( plus_plus @ nat @ I3 @ ( one_one @ nat ) ) ) @ ( F2 @ I3 ) ) ) @ ( one_one @ int ) ) )
=> ( ( ord_less_eq @ int @ ( F2 @ ( zero_zero @ nat ) ) @ K )
=> ( ( ord_less_eq @ int @ K @ ( F2 @ N ) )
=> ? [I3: nat] :
( ( ord_less_eq @ nat @ I3 @ N )
& ( ( F2 @ I3 )
= K ) ) ) ) ) ).
% nat0_intermed_int_val
thf(fact_889_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_890_log__root,axiom,
! [N: nat,A3: real,B2: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ A3 )
=> ( ( log2 @ B2 @ ( root @ N @ A3 ) )
= ( divide_divide @ real @ ( log2 @ B2 @ A3 ) @ ( semiring_1_of_nat @ real @ N ) ) ) ) ) ).
% log_root
thf(fact_891_complete__interval,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [A3: A,B2: A,P: A > $o] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( P @ A3 )
=> ( ~ ( P @ B2 )
=> ? [C3: A] :
( ( ord_less_eq @ A @ A3 @ C3 )
& ( ord_less_eq @ A @ C3 @ B2 )
& ! [X4: A] :
( ( ( ord_less_eq @ A @ A3 @ X4 )
& ( ord_less @ A @ X4 @ C3 ) )
=> ( P @ X4 ) )
& ! [D3: A] :
( ! [X3: A] :
( ( ( ord_less_eq @ A @ A3 @ X3 )
& ( ord_less @ A @ X3 @ D3 ) )
=> ( P @ X3 ) )
=> ( ord_less_eq @ A @ D3 @ C3 ) ) ) ) ) ) ) ).
% complete_interval
thf(fact_892_log__of__power__le,axiom,
! [M2: nat,B2: real,N: nat] :
( ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ M2 ) @ ( power_power @ real @ B2 @ N ) )
=> ( ( ord_less @ real @ ( one_one @ real ) @ B2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M2 )
=> ( ord_less_eq @ real @ ( log2 @ B2 @ ( semiring_1_of_nat @ real @ M2 ) ) @ ( semiring_1_of_nat @ real @ N ) ) ) ) ) ).
% log_of_power_le
thf(fact_893_verit__le__mono__div__int,axiom,
! [A4: int,B7: int,N: int] :
( ( ord_less @ int @ A4 @ B7 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ N )
=> ( ord_less_eq @ int
@ ( plus_plus @ int @ ( divide_divide @ int @ A4 @ N )
@ ( if @ int
@ ( ( modulo_modulo @ int @ B7 @ N )
= ( zero_zero @ int ) )
@ ( one_one @ int )
@ ( zero_zero @ int ) ) )
@ ( divide_divide @ int @ B7 @ N ) ) ) ) ).
% verit_le_mono_div_int
thf(fact_894_div__pos__geq,axiom,
! [L: int,K: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ L )
=> ( ( ord_less_eq @ int @ L @ K )
=> ( ( divide_divide @ int @ K @ L )
= ( plus_plus @ int @ ( divide_divide @ int @ ( minus_minus @ int @ K @ L ) @ L ) @ ( one_one @ int ) ) ) ) ) ).
% div_pos_geq
thf(fact_895_div__pos__neg__trivial,axiom,
! [K: int,L: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ K )
=> ( ( ord_less_eq @ int @ ( plus_plus @ int @ K @ L ) @ ( zero_zero @ int ) )
=> ( ( divide_divide @ int @ K @ L )
= ( uminus_uminus @ int @ ( one_one @ int ) ) ) ) ) ).
% div_pos_neg_trivial
thf(fact_896_verit__le__mono__div,axiom,
! [A4: nat,B7: nat,N: nat] :
( ( ord_less @ nat @ A4 @ B7 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ord_less_eq @ nat
@ ( plus_plus @ nat @ ( divide_divide @ nat @ A4 @ N )
@ ( if @ nat
@ ( ( modulo_modulo @ nat @ B7 @ N )
= ( zero_zero @ nat ) )
@ ( one_one @ nat )
@ ( zero_zero @ nat ) ) )
@ ( divide_divide @ nat @ B7 @ N ) ) ) ) ).
% verit_le_mono_div
thf(fact_897_even__odd__cases,axiom,
! [X: nat] :
( ! [N2: nat] :
( X
!= ( plus_plus @ nat @ N2 @ N2 ) )
=> ~ ! [N2: nat] :
( X
!= ( plus_plus @ nat @ N2 @ ( suc @ N2 ) ) ) ) ).
% even_odd_cases
thf(fact_898_add__left__cancel,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ( plus_plus @ A @ A3 @ B2 )
= ( plus_plus @ A @ A3 @ C2 ) )
= ( B2 = C2 ) ) ) ).
% add_left_cancel
thf(fact_899_add__right__cancel,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( ( plus_plus @ A @ B2 @ A3 )
= ( plus_plus @ A @ C2 @ A3 ) )
= ( B2 = C2 ) ) ) ).
% add_right_cancel
thf(fact_900_abs__exp__cancel,axiom,
! [X: real] :
( ( abs_abs @ real @ ( exp @ real @ X ) )
= ( exp @ real @ X ) ) ).
% abs_exp_cancel
thf(fact_901_add__le__cancel__left,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ C2 @ A3 ) @ ( plus_plus @ A @ C2 @ B2 ) )
= ( ord_less_eq @ A @ A3 @ B2 ) ) ) ).
% add_le_cancel_left
thf(fact_902_add__le__cancel__right,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ A3 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) )
= ( ord_less_eq @ A @ A3 @ B2 ) ) ) ).
% add_le_cancel_right
thf(fact_903_double__eq__0__iff,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A3: A] :
( ( ( plus_plus @ A @ A3 @ A3 )
= ( zero_zero @ A ) )
= ( A3
= ( zero_zero @ A ) ) ) ) ).
% double_eq_0_iff
thf(fact_904_add__0,axiom,
! [A: $tType] :
( ( monoid_add @ A )
=> ! [A3: A] :
( ( plus_plus @ A @ ( zero_zero @ A ) @ A3 )
= A3 ) ) ).
% add_0
thf(fact_905_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_906_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_907_add__cancel__right__right,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add @ A )
=> ! [A3: A,B2: A] :
( ( A3
= ( plus_plus @ A @ A3 @ B2 ) )
= ( B2
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_right_right
thf(fact_908_add__cancel__right__left,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add @ A )
=> ! [A3: A,B2: A] :
( ( A3
= ( plus_plus @ A @ B2 @ A3 ) )
= ( B2
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_right_left
thf(fact_909_add__cancel__left__right,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add @ A )
=> ! [A3: A,B2: A] :
( ( ( plus_plus @ A @ A3 @ B2 )
= A3 )
= ( B2
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_left_right
thf(fact_910_add__cancel__left__left,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add @ A )
=> ! [B2: A,A3: A] :
( ( ( plus_plus @ A @ B2 @ A3 )
= A3 )
= ( B2
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_left_left
thf(fact_911_double__zero__sym,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A3: A] :
( ( ( zero_zero @ A )
= ( plus_plus @ A @ A3 @ A3 ) )
= ( A3
= ( zero_zero @ A ) ) ) ) ).
% double_zero_sym
thf(fact_912_add_Oright__neutral,axiom,
! [A: $tType] :
( ( monoid_add @ A )
=> ! [A3: A] :
( ( plus_plus @ A @ A3 @ ( zero_zero @ A ) )
= A3 ) ) ).
% add.right_neutral
thf(fact_913_add__less__cancel__right,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ A3 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) )
= ( ord_less @ A @ A3 @ B2 ) ) ) ).
% add_less_cancel_right
thf(fact_914_add__less__cancel__left,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ C2 @ A3 ) @ ( plus_plus @ A @ C2 @ B2 ) )
= ( ord_less @ A @ A3 @ B2 ) ) ) ).
% add_less_cancel_left
thf(fact_915_add__diff__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B2: A] :
( ( minus_minus @ A @ ( plus_plus @ A @ A3 @ B2 ) @ B2 )
= A3 ) ) ).
% add_diff_cancel
thf(fact_916_diff__add__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B2: A] :
( ( plus_plus @ A @ ( minus_minus @ A @ A3 @ B2 ) @ B2 )
= A3 ) ) ).
% diff_add_cancel
thf(fact_917_add__diff__cancel__left,axiom,
! [A: $tType] :
( ( cancel2418104881723323429up_add @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( minus_minus @ A @ ( plus_plus @ A @ C2 @ A3 ) @ ( plus_plus @ A @ C2 @ B2 ) )
= ( minus_minus @ A @ A3 @ B2 ) ) ) ).
% add_diff_cancel_left
thf(fact_918_add__diff__cancel__left_H,axiom,
! [A: $tType] :
( ( cancel2418104881723323429up_add @ A )
=> ! [A3: A,B2: A] :
( ( minus_minus @ A @ ( plus_plus @ A @ A3 @ B2 ) @ A3 )
= B2 ) ) ).
% add_diff_cancel_left'
thf(fact_919_add__diff__cancel__right,axiom,
! [A: $tType] :
( ( cancel2418104881723323429up_add @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( minus_minus @ A @ ( plus_plus @ A @ A3 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) )
= ( minus_minus @ A @ A3 @ B2 ) ) ) ).
% add_diff_cancel_right
thf(fact_920_add__diff__cancel__right_H,axiom,
! [A: $tType] :
( ( cancel2418104881723323429up_add @ A )
=> ! [A3: A,B2: A] :
( ( minus_minus @ A @ ( plus_plus @ A @ A3 @ B2 ) @ B2 )
= A3 ) ) ).
% add_diff_cancel_right'
thf(fact_921_add__minus__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B2: A] :
( ( plus_plus @ A @ A3 @ ( plus_plus @ A @ ( uminus_uminus @ A @ A3 ) @ B2 ) )
= B2 ) ) ).
% add_minus_cancel
thf(fact_922_minus__add__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B2: A] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ A3 ) @ ( plus_plus @ A @ A3 @ B2 ) )
= B2 ) ) ).
% minus_add_cancel
thf(fact_923_minus__add__distrib,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A3: A,B2: A] :
( ( uminus_uminus @ A @ ( plus_plus @ A @ A3 @ B2 ) )
= ( plus_plus @ A @ ( uminus_uminus @ A @ A3 ) @ ( uminus_uminus @ A @ B2 ) ) ) ) ).
% minus_add_distrib
thf(fact_924_abs__add__abs,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A,B2: A] :
( ( abs_abs @ A @ ( plus_plus @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ B2 ) ) )
= ( plus_plus @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ B2 ) ) ) ) ).
% abs_add_abs
thf(fact_925_add__Suc__right,axiom,
! [M2: nat,N: nat] :
( ( plus_plus @ nat @ M2 @ ( suc @ N ) )
= ( suc @ ( plus_plus @ nat @ M2 @ N ) ) ) ).
% add_Suc_right
thf(fact_926_Nat_Oadd__0__right,axiom,
! [M2: nat] :
( ( plus_plus @ nat @ M2 @ ( zero_zero @ nat ) )
= M2 ) ).
% Nat.add_0_right
thf(fact_927_add__is__0,axiom,
! [M2: nat,N: nat] :
( ( ( plus_plus @ nat @ M2 @ N )
= ( zero_zero @ nat ) )
= ( ( M2
= ( zero_zero @ nat ) )
& ( N
= ( zero_zero @ nat ) ) ) ) ).
% add_is_0
thf(fact_928_nat__add__left__cancel__less,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less @ nat @ ( plus_plus @ nat @ K @ M2 ) @ ( plus_plus @ nat @ K @ N ) )
= ( ord_less @ nat @ M2 @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_929_nat__add__left__cancel__le,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq @ nat @ ( plus_plus @ nat @ K @ M2 ) @ ( plus_plus @ nat @ K @ N ) )
= ( ord_less_eq @ nat @ M2 @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_930_diff__diff__left,axiom,
! [I: nat,J2: nat,K: nat] :
( ( minus_minus @ nat @ ( minus_minus @ nat @ I @ J2 ) @ K )
= ( minus_minus @ nat @ I @ ( plus_plus @ nat @ J2 @ K ) ) ) ).
% diff_diff_left
thf(fact_931_powr__eq__0__iff,axiom,
! [A: $tType] :
( ( ln @ A )
=> ! [W: A,Z2: A] :
( ( ( powr @ A @ W @ Z2 )
= ( zero_zero @ A ) )
= ( W
= ( zero_zero @ A ) ) ) ) ).
% powr_eq_0_iff
thf(fact_932_powr__0,axiom,
! [A: $tType] :
( ( ln @ A )
=> ! [Z2: A] :
( ( powr @ A @ ( zero_zero @ A ) @ Z2 )
= ( zero_zero @ A ) ) ) ).
% powr_0
thf(fact_933_powr__one__eq__one,axiom,
! [A: $tType] :
( ( ln @ A )
=> ! [A3: A] :
( ( powr @ A @ ( one_one @ A ) @ A3 )
= ( one_one @ A ) ) ) ).
% powr_one_eq_one
thf(fact_934_add__le__same__cancel1,axiom,
! [A: $tType] :
( ( ordere1937475149494474687imp_le @ A )
=> ! [B2: A,A3: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ B2 @ A3 ) @ B2 )
= ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% add_le_same_cancel1
thf(fact_935_add__le__same__cancel2,axiom,
! [A: $tType] :
( ( ordere1937475149494474687imp_le @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ A3 @ B2 ) @ B2 )
= ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% add_le_same_cancel2
thf(fact_936_le__add__same__cancel1,axiom,
! [A: $tType] :
( ( ordere1937475149494474687imp_le @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ ( plus_plus @ A @ A3 @ B2 ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) ) ) ).
% le_add_same_cancel1
thf(fact_937_le__add__same__cancel2,axiom,
! [A: $tType] :
( ( ordere1937475149494474687imp_le @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ ( plus_plus @ A @ B2 @ A3 ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) ) ) ).
% le_add_same_cancel2
thf(fact_938_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ A3 @ A3 ) @ ( zero_zero @ A ) )
= ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_939_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A3 @ A3 ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_940_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A3 @ A3 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_941_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( plus_plus @ A @ A3 @ A3 ) @ ( zero_zero @ A ) )
= ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_942_less__add__same__cancel2,axiom,
! [A: $tType] :
( ( ordere1937475149494474687imp_le @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ ( plus_plus @ A @ B2 @ A3 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) ) ) ).
% less_add_same_cancel2
thf(fact_943_less__add__same__cancel1,axiom,
! [A: $tType] :
( ( ordere1937475149494474687imp_le @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ ( plus_plus @ A @ A3 @ B2 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) ) ) ).
% less_add_same_cancel1
thf(fact_944_add__less__same__cancel2,axiom,
! [A: $tType] :
( ( ordere1937475149494474687imp_le @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ A3 @ B2 ) @ B2 )
= ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% add_less_same_cancel2
thf(fact_945_add__less__same__cancel1,axiom,
! [A: $tType] :
( ( ordere1937475149494474687imp_le @ A )
=> ! [B2: A,A3: A] :
( ( ord_less @ A @ ( plus_plus @ A @ B2 @ A3 ) @ B2 )
= ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% add_less_same_cancel1
thf(fact_946_le__add__diff__inverse2,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [B2: A,A3: A] :
( ( ord_less_eq @ A @ B2 @ A3 )
=> ( ( plus_plus @ A @ ( minus_minus @ A @ A3 @ B2 ) @ B2 )
= A3 ) ) ) ).
% le_add_diff_inverse2
thf(fact_947_le__add__diff__inverse,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [B2: A,A3: A] :
( ( ord_less_eq @ A @ B2 @ A3 )
=> ( ( plus_plus @ A @ B2 @ ( minus_minus @ A @ A3 @ B2 ) )
= A3 ) ) ) ).
% le_add_diff_inverse
thf(fact_948_diff__add__zero,axiom,
! [A: $tType] :
( ( comm_monoid_diff @ A )
=> ! [A3: A,B2: A] :
( ( minus_minus @ A @ A3 @ ( plus_plus @ A @ A3 @ B2 ) )
= ( zero_zero @ A ) ) ) ).
% diff_add_zero
thf(fact_949_add_Oright__inverse,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A] :
( ( plus_plus @ A @ A3 @ ( uminus_uminus @ A @ A3 ) )
= ( zero_zero @ A ) ) ) ).
% add.right_inverse
thf(fact_950_ab__left__minus,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ A3 ) @ A3 )
= ( zero_zero @ A ) ) ) ).
% ab_left_minus
thf(fact_951_diff__minus__eq__add,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B2: A] :
( ( minus_minus @ A @ A3 @ ( uminus_uminus @ A @ B2 ) )
= ( plus_plus @ A @ A3 @ B2 ) ) ) ).
% diff_minus_eq_add
thf(fact_952_uminus__add__conv__diff,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A3: A,B2: A] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ A3 ) @ B2 )
= ( minus_minus @ A @ B2 @ A3 ) ) ) ).
% uminus_add_conv_diff
thf(fact_953_of__nat__add,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [M2: nat,N: nat] :
( ( semiring_1_of_nat @ A @ ( plus_plus @ nat @ M2 @ N ) )
= ( plus_plus @ A @ ( semiring_1_of_nat @ A @ M2 ) @ ( semiring_1_of_nat @ A @ N ) ) ) ) ).
% of_nat_add
thf(fact_954_add__gr__0,axiom,
! [M2: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( plus_plus @ nat @ M2 @ N ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M2 )
| ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ).
% add_gr_0
thf(fact_955_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_956_Nat_Oadd__diff__assoc,axiom,
! [K: nat,J2: nat,I: nat] :
( ( ord_less_eq @ nat @ K @ J2 )
=> ( ( plus_plus @ nat @ I @ ( minus_minus @ nat @ J2 @ K ) )
= ( minus_minus @ nat @ ( plus_plus @ nat @ I @ J2 ) @ K ) ) ) ).
% Nat.add_diff_assoc
thf(fact_957_Nat_Oadd__diff__assoc2,axiom,
! [K: nat,J2: nat,I: nat] :
( ( ord_less_eq @ nat @ K @ J2 )
=> ( ( plus_plus @ nat @ ( minus_minus @ nat @ J2 @ K ) @ I )
= ( minus_minus @ nat @ ( plus_plus @ nat @ J2 @ I ) @ K ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_958_Nat_Odiff__diff__right,axiom,
! [K: nat,J2: nat,I: nat] :
( ( ord_less_eq @ nat @ K @ J2 )
=> ( ( minus_minus @ nat @ I @ ( minus_minus @ nat @ J2 @ K ) )
= ( minus_minus @ nat @ ( plus_plus @ nat @ I @ K ) @ J2 ) ) ) ).
% Nat.diff_diff_right
thf(fact_959_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_960_powr__gt__zero,axiom,
! [X: real,A3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( powr @ real @ X @ A3 ) )
= ( X
!= ( zero_zero @ real ) ) ) ).
% powr_gt_zero
thf(fact_961_real__add__minus__iff,axiom,
! [X: real,A3: real] :
( ( ( plus_plus @ real @ X @ ( uminus_uminus @ real @ A3 ) )
= ( zero_zero @ real ) )
= ( X = A3 ) ) ).
% real_add_minus_iff
thf(fact_962_powr__nonneg__iff,axiom,
! [A3: real,X: real] :
( ( ord_less_eq @ real @ ( powr @ real @ A3 @ X ) @ ( zero_zero @ real ) )
= ( A3
= ( zero_zero @ real ) ) ) ).
% powr_nonneg_iff
thf(fact_963_powr__less__cancel__iff,axiom,
! [X: real,A3: real,B2: real] :
( ( ord_less @ real @ ( one_one @ real ) @ X )
=> ( ( ord_less @ real @ ( powr @ real @ X @ A3 ) @ ( powr @ real @ X @ B2 ) )
= ( ord_less @ real @ A3 @ B2 ) ) ) ).
% powr_less_cancel_iff
thf(fact_964_log__one,axiom,
! [A3: real] :
( ( log2 @ A3 @ ( one_one @ real ) )
= ( zero_zero @ real ) ) ).
% log_one
thf(fact_965_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_966_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_967_of__nat__Suc,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [M2: nat] :
( ( semiring_1_of_nat @ A @ ( suc @ M2 ) )
= ( plus_plus @ A @ ( one_one @ A ) @ ( semiring_1_of_nat @ A @ M2 ) ) ) ) ).
% of_nat_Suc
thf(fact_968_diff__Suc__diff__eq2,axiom,
! [K: nat,J2: nat,I: nat] :
( ( ord_less_eq @ nat @ K @ J2 )
=> ( ( minus_minus @ nat @ ( suc @ ( minus_minus @ nat @ J2 @ K ) ) @ I )
= ( minus_minus @ nat @ ( suc @ J2 ) @ ( plus_plus @ nat @ K @ I ) ) ) ) ).
% diff_Suc_diff_eq2
thf(fact_969_diff__Suc__diff__eq1,axiom,
! [K: nat,J2: nat,I: nat] :
( ( ord_less_eq @ nat @ K @ J2 )
=> ( ( minus_minus @ nat @ I @ ( suc @ ( minus_minus @ nat @ J2 @ K ) ) )
= ( minus_minus @ nat @ ( plus_plus @ nat @ I @ K ) @ ( suc @ J2 ) ) ) ) ).
% diff_Suc_diff_eq1
thf(fact_970_powr__eq__one__iff,axiom,
! [A3: real,X: real] :
( ( ord_less @ real @ ( one_one @ real ) @ A3 )
=> ( ( ( powr @ real @ A3 @ X )
= ( one_one @ real ) )
= ( X
= ( zero_zero @ real ) ) ) ) ).
% powr_eq_one_iff
thf(fact_971_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_972_powr__one,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( powr @ real @ X @ ( one_one @ real ) )
= X ) ) ).
% powr_one
thf(fact_973_powr__le__cancel__iff,axiom,
! [X: real,A3: real,B2: real] :
( ( ord_less @ real @ ( one_one @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( powr @ real @ X @ A3 ) @ ( powr @ real @ X @ B2 ) )
= ( ord_less_eq @ real @ A3 @ B2 ) ) ) ).
% powr_le_cancel_iff
thf(fact_974_log__eq__one,axiom,
! [A3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A3 )
=> ( ( A3
!= ( one_one @ real ) )
=> ( ( log2 @ A3 @ A3 )
= ( one_one @ real ) ) ) ) ).
% log_eq_one
thf(fact_975_log__less__cancel__iff,axiom,
! [A3: real,X: real,Y2: real] :
( ( ord_less @ real @ ( one_one @ real ) @ A3 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ Y2 )
=> ( ( ord_less @ real @ ( log2 @ A3 @ X ) @ ( log2 @ A3 @ Y2 ) )
= ( ord_less @ real @ X @ Y2 ) ) ) ) ) ).
% log_less_cancel_iff
thf(fact_976_log__less__one__cancel__iff,axiom,
! [A3: real,X: real] :
( ( ord_less @ real @ ( one_one @ real ) @ A3 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( log2 @ A3 @ X ) @ ( one_one @ real ) )
= ( ord_less @ real @ X @ A3 ) ) ) ) ).
% log_less_one_cancel_iff
thf(fact_977_one__less__log__cancel__iff,axiom,
! [A3: real,X: real] :
( ( ord_less @ real @ ( one_one @ real ) @ A3 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( one_one @ real ) @ ( log2 @ A3 @ X ) )
= ( ord_less @ real @ A3 @ X ) ) ) ) ).
% one_less_log_cancel_iff
thf(fact_978_log__less__zero__cancel__iff,axiom,
! [A3: real,X: real] :
( ( ord_less @ real @ ( one_one @ real ) @ A3 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( log2 @ A3 @ X ) @ ( zero_zero @ real ) )
= ( ord_less @ real @ X @ ( one_one @ real ) ) ) ) ) ).
% log_less_zero_cancel_iff
thf(fact_979_zero__less__log__cancel__iff,axiom,
! [A3: real,X: real] :
( ( ord_less @ real @ ( one_one @ real ) @ A3 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( log2 @ A3 @ X ) )
= ( ord_less @ real @ ( one_one @ real ) @ X ) ) ) ) ).
% zero_less_log_cancel_iff
thf(fact_980_zle__add1__eq__le,axiom,
! [W: int,Z2: int] :
( ( ord_less @ int @ W @ ( plus_plus @ int @ Z2 @ ( one_one @ int ) ) )
= ( ord_less_eq @ int @ W @ Z2 ) ) ).
% zle_add1_eq_le
thf(fact_981_zero__le__log__cancel__iff,axiom,
! [A3: real,X: real] :
( ( ord_less @ real @ ( one_one @ real ) @ A3 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( log2 @ A3 @ X ) )
= ( ord_less_eq @ real @ ( one_one @ real ) @ X ) ) ) ) ).
% zero_le_log_cancel_iff
thf(fact_982_log__le__zero__cancel__iff,axiom,
! [A3: real,X: real] :
( ( ord_less @ real @ ( one_one @ real ) @ A3 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( log2 @ A3 @ X ) @ ( zero_zero @ real ) )
= ( ord_less_eq @ real @ X @ ( one_one @ real ) ) ) ) ) ).
% log_le_zero_cancel_iff
thf(fact_983_one__le__log__cancel__iff,axiom,
! [A3: real,X: real] :
( ( ord_less @ real @ ( one_one @ real ) @ A3 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( one_one @ real ) @ ( log2 @ A3 @ X ) )
= ( ord_less_eq @ real @ A3 @ X ) ) ) ) ).
% one_le_log_cancel_iff
thf(fact_984_log__le__one__cancel__iff,axiom,
! [A3: real,X: real] :
( ( ord_less @ real @ ( one_one @ real ) @ A3 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( log2 @ A3 @ X ) @ ( one_one @ real ) )
= ( ord_less_eq @ real @ X @ A3 ) ) ) ) ).
% log_le_one_cancel_iff
thf(fact_985_log__le__cancel__iff,axiom,
! [A3: real,X: real,Y2: real] :
( ( ord_less @ real @ ( one_one @ real ) @ A3 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ Y2 )
=> ( ( ord_less_eq @ real @ ( log2 @ A3 @ X ) @ ( log2 @ A3 @ Y2 ) )
= ( ord_less_eq @ real @ X @ Y2 ) ) ) ) ) ).
% log_le_cancel_iff
thf(fact_986_log__powr__cancel,axiom,
! [A3: real,Y2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A3 )
=> ( ( A3
!= ( one_one @ real ) )
=> ( ( log2 @ A3 @ ( powr @ real @ A3 @ Y2 ) )
= Y2 ) ) ) ).
% log_powr_cancel
thf(fact_987_powr__log__cancel,axiom,
! [A3: real,X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A3 )
=> ( ( A3
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( powr @ real @ A3 @ ( log2 @ A3 @ X ) )
= X ) ) ) ) ).
% powr_log_cancel
thf(fact_988_log__pow__cancel,axiom,
! [A3: real,B2: nat] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A3 )
=> ( ( A3
!= ( one_one @ real ) )
=> ( ( log2 @ A3 @ ( power_power @ real @ A3 @ B2 ) )
= ( semiring_1_of_nat @ real @ B2 ) ) ) ) ).
% log_pow_cancel
thf(fact_989_is__num__normalize_I1_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C2 )
= ( plus_plus @ A @ A3 @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% is_num_normalize(1)
thf(fact_990_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C2 )
= ( plus_plus @ A @ A3 @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_991_add__mono__thms__linordered__semiring_I4_J,axiom,
! [A: $tType] :
( ( ordere6658533253407199908up_add @ A )
=> ! [I: A,J2: A,K: A,L: A] :
( ( ( I = J2 )
& ( K = L ) )
=> ( ( plus_plus @ A @ I @ K )
= ( plus_plus @ A @ J2 @ L ) ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_992_group__cancel_Oadd1,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A4: A,K: A,A3: A,B2: A] :
( ( A4
= ( plus_plus @ A @ K @ A3 ) )
=> ( ( plus_plus @ A @ A4 @ B2 )
= ( plus_plus @ A @ K @ ( plus_plus @ A @ A3 @ B2 ) ) ) ) ) ).
% group_cancel.add1
thf(fact_993_group__cancel_Oadd2,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [B7: A,K: A,B2: A,A3: A] :
( ( B7
= ( plus_plus @ A @ K @ B2 ) )
=> ( ( plus_plus @ A @ A3 @ B7 )
= ( plus_plus @ A @ K @ ( plus_plus @ A @ A3 @ B2 ) ) ) ) ) ).
% group_cancel.add2
thf(fact_994_add_Oassoc,axiom,
! [A: $tType] :
( ( semigroup_add @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C2 )
= ( plus_plus @ A @ A3 @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% add.assoc
thf(fact_995_add_Oleft__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ( plus_plus @ A @ A3 @ B2 )
= ( plus_plus @ A @ A3 @ C2 ) )
= ( B2 = C2 ) ) ) ).
% add.left_cancel
thf(fact_996_add_Oright__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( ( plus_plus @ A @ B2 @ A3 )
= ( plus_plus @ A @ C2 @ A3 ) )
= ( B2 = C2 ) ) ) ).
% add.right_cancel
thf(fact_997_add_Ocommute,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A )
=> ( ( plus_plus @ A )
= ( ^ [A5: A,B3: A] : ( plus_plus @ A @ B3 @ A5 ) ) ) ) ).
% add.commute
thf(fact_998_add_Oleft__commute,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( plus_plus @ A @ B2 @ ( plus_plus @ A @ A3 @ C2 ) )
= ( plus_plus @ A @ A3 @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% add.left_commute
thf(fact_999_add__left__imp__eq,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ( plus_plus @ A @ A3 @ B2 )
= ( plus_plus @ A @ A3 @ C2 ) )
=> ( B2 = C2 ) ) ) ).
% add_left_imp_eq
thf(fact_1000_add__right__imp__eq,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( ( plus_plus @ A @ B2 @ A3 )
= ( plus_plus @ A @ C2 @ A3 ) )
=> ( B2 = C2 ) ) ) ).
% add_right_imp_eq
thf(fact_1001_powr__powr__swap,axiom,
! [X: real,A3: real,B2: real] :
( ( powr @ real @ ( powr @ real @ X @ A3 ) @ B2 )
= ( powr @ real @ ( powr @ real @ X @ B2 ) @ A3 ) ) ).
% powr_powr_swap
thf(fact_1002_zadd__int__left,axiom,
! [M2: nat,N: nat,Z2: int] :
( ( plus_plus @ int @ ( semiring_1_of_nat @ int @ M2 ) @ ( plus_plus @ int @ ( semiring_1_of_nat @ int @ N ) @ Z2 ) )
= ( plus_plus @ int @ ( semiring_1_of_nat @ int @ ( plus_plus @ nat @ M2 @ N ) ) @ Z2 ) ) ).
% zadd_int_left
thf(fact_1003_int__plus,axiom,
! [N: nat,M2: nat] :
( ( semiring_1_of_nat @ int @ ( plus_plus @ nat @ N @ M2 ) )
= ( plus_plus @ int @ ( semiring_1_of_nat @ int @ N ) @ ( semiring_1_of_nat @ int @ M2 ) ) ) ).
% int_plus
thf(fact_1004_int__ops_I5_J,axiom,
! [A3: nat,B2: nat] :
( ( semiring_1_of_nat @ int @ ( plus_plus @ nat @ A3 @ B2 ) )
= ( plus_plus @ int @ ( semiring_1_of_nat @ int @ A3 ) @ ( semiring_1_of_nat @ int @ B2 ) ) ) ).
% int_ops(5)
thf(fact_1005_log__base__powr,axiom,
! [A3: real,B2: real,X: real] :
( ( A3
!= ( zero_zero @ real ) )
=> ( ( log2 @ ( powr @ real @ A3 @ B2 ) @ X )
= ( divide_divide @ real @ ( log2 @ A3 @ X ) @ B2 ) ) ) ).
% log_base_powr
thf(fact_1006_nat__int__add,axiom,
! [A3: nat,B2: nat] :
( ( nat2 @ ( plus_plus @ int @ ( semiring_1_of_nat @ int @ A3 ) @ ( semiring_1_of_nat @ int @ B2 ) ) )
= ( plus_plus @ nat @ A3 @ B2 ) ) ).
% nat_int_add
thf(fact_1007_add__mono__thms__linordered__semiring_I3_J,axiom,
! [A: $tType] :
( ( ordere6658533253407199908up_add @ A )
=> ! [I: A,J2: A,K: A,L: A] :
( ( ( ord_less_eq @ A @ I @ J2 )
& ( K = L ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J2 @ L ) ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_1008_add__mono__thms__linordered__semiring_I2_J,axiom,
! [A: $tType] :
( ( ordere6658533253407199908up_add @ A )
=> ! [I: A,J2: A,K: A,L: A] :
( ( ( I = J2 )
& ( ord_less_eq @ A @ K @ L ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J2 @ L ) ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_1009_add__mono__thms__linordered__semiring_I1_J,axiom,
! [A: $tType] :
( ( ordere6658533253407199908up_add @ A )
=> ! [I: A,J2: A,K: A,L: A] :
( ( ( ord_less_eq @ A @ I @ J2 )
& ( ord_less_eq @ A @ K @ L ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J2 @ L ) ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_1010_add__mono,axiom,
! [A: $tType] :
( ( ordere6658533253407199908up_add @ A )
=> ! [A3: A,B2: A,C2: A,D2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ D2 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A3 @ C2 ) @ ( plus_plus @ A @ B2 @ D2 ) ) ) ) ) ).
% add_mono
thf(fact_1011_add__left__mono,axiom,
! [A: $tType] :
( ( ordere6658533253407199908up_add @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ C2 @ A3 ) @ ( plus_plus @ A @ C2 @ B2 ) ) ) ) ).
% add_left_mono
thf(fact_1012_less__eqE,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ~ ! [C3: A] :
( B2
!= ( plus_plus @ A @ A3 @ C3 ) ) ) ) ).
% less_eqE
thf(fact_1013_add__right__mono,axiom,
! [A: $tType] :
( ( ordere6658533253407199908up_add @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A3 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% add_right_mono
thf(fact_1014_le__iff__add,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A5: A,B3: A] :
? [C4: A] :
( B3
= ( plus_plus @ A @ A5 @ C4 ) ) ) ) ) ).
% le_iff_add
thf(fact_1015_add__le__imp__le__left,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ C2 @ A3 ) @ ( plus_plus @ A @ C2 @ B2 ) )
=> ( ord_less_eq @ A @ A3 @ B2 ) ) ) ).
% add_le_imp_le_left
thf(fact_1016_add__le__imp__le__right,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ A3 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) )
=> ( ord_less_eq @ A @ A3 @ B2 ) ) ) ).
% add_le_imp_le_right
thf(fact_1017_add_Ogroup__left__neutral,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A] :
( ( plus_plus @ A @ ( zero_zero @ A ) @ A3 )
= A3 ) ) ).
% add.group_left_neutral
thf(fact_1018_add_Ocomm__neutral,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A3: A] :
( ( plus_plus @ A @ A3 @ ( zero_zero @ A ) )
= A3 ) ) ).
% add.comm_neutral
thf(fact_1019_comm__monoid__add__class_Oadd__0,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A3: A] :
( ( plus_plus @ A @ ( zero_zero @ A ) @ A3 )
= A3 ) ) ).
% comm_monoid_add_class.add_0
thf(fact_1020_verit__sum__simplify,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add @ A )
=> ! [A3: A] :
( ( plus_plus @ A @ A3 @ ( zero_zero @ A ) )
= A3 ) ) ).
% verit_sum_simplify
thf(fact_1021_add__less__imp__less__right,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ A3 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) )
=> ( ord_less @ A @ A3 @ B2 ) ) ) ).
% add_less_imp_less_right
thf(fact_1022_add__less__imp__less__left,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ C2 @ A3 ) @ ( plus_plus @ A @ C2 @ B2 ) )
=> ( ord_less @ A @ A3 @ B2 ) ) ) ).
% add_less_imp_less_left
thf(fact_1023_add__strict__right__mono,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ord_less @ A @ ( plus_plus @ A @ A3 @ C2 ) @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% add_strict_right_mono
thf(fact_1024_add__strict__left__mono,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ord_less @ A @ ( plus_plus @ A @ C2 @ A3 ) @ ( plus_plus @ A @ C2 @ B2 ) ) ) ) ).
% add_strict_left_mono
thf(fact_1025_add__strict__mono,axiom,
! [A: $tType] :
( ( strict9044650504122735259up_add @ A )
=> ! [A3: A,B2: A,C2: A,D2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less @ A @ C2 @ D2 )
=> ( ord_less @ A @ ( plus_plus @ A @ A3 @ C2 ) @ ( plus_plus @ A @ B2 @ D2 ) ) ) ) ) ).
% add_strict_mono
thf(fact_1026_add__mono__thms__linordered__field_I1_J,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [I: A,J2: A,K: A,L: A] :
( ( ( ord_less @ A @ I @ J2 )
& ( K = L ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J2 @ L ) ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_1027_add__mono__thms__linordered__field_I2_J,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [I: A,J2: A,K: A,L: A] :
( ( ( I = J2 )
& ( ord_less @ A @ K @ L ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J2 @ L ) ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_1028_add__mono__thms__linordered__field_I5_J,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [I: A,J2: A,K: A,L: A] :
( ( ( ord_less @ A @ I @ J2 )
& ( ord_less @ A @ K @ L ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J2 @ L ) ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_1029_add__diff__add,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A3: A,C2: A,B2: A,D2: A] :
( ( minus_minus @ A @ ( plus_plus @ A @ A3 @ C2 ) @ ( plus_plus @ A @ B2 @ D2 ) )
= ( plus_plus @ A @ ( minus_minus @ A @ A3 @ B2 ) @ ( minus_minus @ A @ C2 @ D2 ) ) ) ) ).
% add_diff_add
thf(fact_1030_group__cancel_Osub1,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A4: A,K: A,A3: A,B2: A] :
( ( A4
= ( plus_plus @ A @ K @ A3 ) )
=> ( ( minus_minus @ A @ A4 @ B2 )
= ( plus_plus @ A @ K @ ( minus_minus @ A @ A3 @ B2 ) ) ) ) ) ).
% group_cancel.sub1
thf(fact_1031_diff__eq__eq,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ( minus_minus @ A @ A3 @ B2 )
= C2 )
= ( A3
= ( plus_plus @ A @ C2 @ B2 ) ) ) ) ).
% diff_eq_eq
thf(fact_1032_eq__diff__eq,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( A3
= ( minus_minus @ A @ C2 @ B2 ) )
= ( ( plus_plus @ A @ A3 @ B2 )
= C2 ) ) ) ).
% eq_diff_eq
thf(fact_1033_add__diff__eq,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( plus_plus @ A @ A3 @ ( minus_minus @ A @ B2 @ C2 ) )
= ( minus_minus @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C2 ) ) ) ).
% add_diff_eq
thf(fact_1034_diff__diff__eq2,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( minus_minus @ A @ A3 @ ( minus_minus @ A @ B2 @ C2 ) )
= ( minus_minus @ A @ ( plus_plus @ A @ A3 @ C2 ) @ B2 ) ) ) ).
% diff_diff_eq2
thf(fact_1035_diff__add__eq,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( plus_plus @ A @ ( minus_minus @ A @ A3 @ B2 ) @ C2 )
= ( minus_minus @ A @ ( plus_plus @ A @ A3 @ C2 ) @ B2 ) ) ) ).
% diff_add_eq
thf(fact_1036_diff__add__eq__diff__diff__swap,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( minus_minus @ A @ A3 @ ( plus_plus @ A @ B2 @ C2 ) )
= ( minus_minus @ A @ ( minus_minus @ A @ A3 @ C2 ) @ B2 ) ) ) ).
% diff_add_eq_diff_diff_swap
thf(fact_1037_add__implies__diff,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add @ A )
=> ! [C2: A,B2: A,A3: A] :
( ( ( plus_plus @ A @ C2 @ B2 )
= A3 )
=> ( C2
= ( minus_minus @ A @ A3 @ B2 ) ) ) ) ).
% add_implies_diff
thf(fact_1038_diff__diff__eq,axiom,
! [A: $tType] :
( ( cancel2418104881723323429up_add @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( minus_minus @ A @ ( minus_minus @ A @ A3 @ B2 ) @ C2 )
= ( minus_minus @ A @ A3 @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ).
% diff_diff_eq
thf(fact_1039_add__divide__distrib,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( divide_divide @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C2 )
= ( plus_plus @ A @ ( divide_divide @ A @ A3 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) ) ) ) ).
% add_divide_distrib
thf(fact_1040_is__num__normalize_I8_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [A3: A,B2: A] :
( ( uminus_uminus @ A @ ( plus_plus @ A @ A3 @ B2 ) )
= ( plus_plus @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A3 ) ) ) ) ).
% is_num_normalize(8)
thf(fact_1041_group__cancel_Oneg1,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A4: A,K: A,A3: A] :
( ( A4
= ( plus_plus @ A @ K @ A3 ) )
=> ( ( uminus_uminus @ A @ A4 )
= ( plus_plus @ A @ ( uminus_uminus @ A @ K ) @ ( uminus_uminus @ A @ A3 ) ) ) ) ) ).
% group_cancel.neg1
thf(fact_1042_add_Oinverse__distrib__swap,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B2: A] :
( ( uminus_uminus @ A @ ( plus_plus @ A @ A3 @ B2 ) )
= ( plus_plus @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A3 ) ) ) ) ).
% add.inverse_distrib_swap
thf(fact_1043_nat__arith_Osuc1,axiom,
! [A4: nat,K: nat,A3: nat] :
( ( A4
= ( plus_plus @ nat @ K @ A3 ) )
=> ( ( suc @ A4 )
= ( plus_plus @ nat @ K @ ( suc @ A3 ) ) ) ) ).
% nat_arith.suc1
thf(fact_1044_add__Suc,axiom,
! [M2: nat,N: nat] :
( ( plus_plus @ nat @ ( suc @ M2 ) @ N )
= ( suc @ ( plus_plus @ nat @ M2 @ N ) ) ) ).
% add_Suc
thf(fact_1045_add__Suc__shift,axiom,
! [M2: nat,N: nat] :
( ( plus_plus @ nat @ ( suc @ M2 ) @ N )
= ( plus_plus @ nat @ M2 @ ( suc @ N ) ) ) ).
% add_Suc_shift
thf(fact_1046_Euclid__induct,axiom,
! [P: nat > nat > $o,A3: nat,B2: nat] :
( ! [A6: nat,B4: nat] :
( ( P @ A6 @ B4 )
= ( P @ B4 @ A6 ) )
=> ( ! [A6: nat] : ( P @ A6 @ ( zero_zero @ nat ) )
=> ( ! [A6: nat,B4: nat] :
( ( P @ A6 @ B4 )
=> ( P @ A6 @ ( plus_plus @ nat @ A6 @ B4 ) ) )
=> ( P @ A3 @ B2 ) ) ) ) ).
% Euclid_induct
thf(fact_1047_add__eq__self__zero,axiom,
! [M2: nat,N: nat] :
( ( ( plus_plus @ nat @ M2 @ N )
= M2 )
=> ( N
= ( zero_zero @ nat ) ) ) ).
% add_eq_self_zero
thf(fact_1048_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus @ nat @ ( zero_zero @ nat ) @ N )
= N ) ).
% plus_nat.add_0
thf(fact_1049_less__add__eq__less,axiom,
! [K: nat,L: nat,M2: nat,N: nat] :
( ( ord_less @ nat @ K @ L )
=> ( ( ( plus_plus @ nat @ M2 @ L )
= ( plus_plus @ nat @ K @ N ) )
=> ( ord_less @ nat @ M2 @ N ) ) ) ).
% less_add_eq_less
thf(fact_1050_trans__less__add2,axiom,
! [I: nat,J2: nat,M2: nat] :
( ( ord_less @ nat @ I @ J2 )
=> ( ord_less @ nat @ I @ ( plus_plus @ nat @ M2 @ J2 ) ) ) ).
% trans_less_add2
thf(fact_1051_trans__less__add1,axiom,
! [I: nat,J2: nat,M2: nat] :
( ( ord_less @ nat @ I @ J2 )
=> ( ord_less @ nat @ I @ ( plus_plus @ nat @ J2 @ M2 ) ) ) ).
% trans_less_add1
thf(fact_1052_add__less__mono1,axiom,
! [I: nat,J2: nat,K: nat] :
( ( ord_less @ nat @ I @ J2 )
=> ( ord_less @ nat @ ( plus_plus @ nat @ I @ K ) @ ( plus_plus @ nat @ J2 @ K ) ) ) ).
% add_less_mono1
thf(fact_1053_not__add__less2,axiom,
! [J2: nat,I: nat] :
~ ( ord_less @ nat @ ( plus_plus @ nat @ J2 @ I ) @ I ) ).
% not_add_less2
thf(fact_1054_not__add__less1,axiom,
! [I: nat,J2: nat] :
~ ( ord_less @ nat @ ( plus_plus @ nat @ I @ J2 ) @ I ) ).
% not_add_less1
thf(fact_1055_add__less__mono,axiom,
! [I: nat,J2: nat,K: nat,L: nat] :
( ( ord_less @ nat @ I @ J2 )
=> ( ( ord_less @ nat @ K @ L )
=> ( ord_less @ nat @ ( plus_plus @ nat @ I @ K ) @ ( plus_plus @ nat @ J2 @ L ) ) ) ) ).
% add_less_mono
thf(fact_1056_add__lessD1,axiom,
! [I: nat,J2: nat,K: nat] :
( ( ord_less @ nat @ ( plus_plus @ nat @ I @ J2 ) @ K )
=> ( ord_less @ nat @ I @ K ) ) ).
% add_lessD1
thf(fact_1057_plus__int__code_I1_J,axiom,
! [K: int] :
( ( plus_plus @ int @ K @ ( zero_zero @ int ) )
= K ) ).
% plus_int_code(1)
thf(fact_1058_plus__int__code_I2_J,axiom,
! [L: int] :
( ( plus_plus @ int @ ( zero_zero @ int ) @ L )
= L ) ).
% plus_int_code(2)
thf(fact_1059_add__leE,axiom,
! [M2: nat,K: nat,N: nat] :
( ( ord_less_eq @ nat @ ( plus_plus @ nat @ M2 @ K ) @ N )
=> ~ ( ( ord_less_eq @ nat @ M2 @ N )
=> ~ ( ord_less_eq @ nat @ K @ N ) ) ) ).
% add_leE
thf(fact_1060_le__add1,axiom,
! [N: nat,M2: nat] : ( ord_less_eq @ nat @ N @ ( plus_plus @ nat @ N @ M2 ) ) ).
% le_add1
thf(fact_1061_le__add2,axiom,
! [N: nat,M2: nat] : ( ord_less_eq @ nat @ N @ ( plus_plus @ nat @ M2 @ N ) ) ).
% le_add2
thf(fact_1062_add__leD1,axiom,
! [M2: nat,K: nat,N: nat] :
( ( ord_less_eq @ nat @ ( plus_plus @ nat @ M2 @ K ) @ N )
=> ( ord_less_eq @ nat @ M2 @ N ) ) ).
% add_leD1
thf(fact_1063_add__leD2,axiom,
! [M2: nat,K: nat,N: nat] :
( ( ord_less_eq @ nat @ ( plus_plus @ nat @ M2 @ K ) @ N )
=> ( ord_less_eq @ nat @ K @ N ) ) ).
% add_leD2
thf(fact_1064_le__Suc__ex,axiom,
! [K: nat,L: nat] :
( ( ord_less_eq @ nat @ K @ L )
=> ? [N2: nat] :
( L
= ( plus_plus @ nat @ K @ N2 ) ) ) ).
% le_Suc_ex
thf(fact_1065_add__le__mono,axiom,
! [I: nat,J2: nat,K: nat,L: nat] :
( ( ord_less_eq @ nat @ I @ J2 )
=> ( ( ord_less_eq @ nat @ K @ L )
=> ( ord_less_eq @ nat @ ( plus_plus @ nat @ I @ K ) @ ( plus_plus @ nat @ J2 @ L ) ) ) ) ).
% add_le_mono
thf(fact_1066_add__le__mono1,axiom,
! [I: nat,J2: nat,K: nat] :
( ( ord_less_eq @ nat @ I @ J2 )
=> ( ord_less_eq @ nat @ ( plus_plus @ nat @ I @ K ) @ ( plus_plus @ nat @ J2 @ K ) ) ) ).
% add_le_mono1
thf(fact_1067_trans__le__add1,axiom,
! [I: nat,J2: nat,M2: nat] :
( ( ord_less_eq @ nat @ I @ J2 )
=> ( ord_less_eq @ nat @ I @ ( plus_plus @ nat @ J2 @ M2 ) ) ) ).
% trans_le_add1
thf(fact_1068_trans__le__add2,axiom,
! [I: nat,J2: nat,M2: nat] :
( ( ord_less_eq @ nat @ I @ J2 )
=> ( ord_less_eq @ nat @ I @ ( plus_plus @ nat @ M2 @ J2 ) ) ) ).
% trans_le_add2
thf(fact_1069_nat__le__iff__add,axiom,
( ( ord_less_eq @ nat )
= ( ^ [M5: nat,N5: nat] :
? [K3: nat] :
( N5
= ( plus_plus @ nat @ M5 @ K3 ) ) ) ) ).
% nat_le_iff_add
thf(fact_1070_Nat_Odiff__cancel,axiom,
! [K: nat,M2: nat,N: nat] :
( ( minus_minus @ nat @ ( plus_plus @ nat @ K @ M2 ) @ ( plus_plus @ nat @ K @ N ) )
= ( minus_minus @ nat @ M2 @ N ) ) ).
% Nat.diff_cancel
thf(fact_1071_diff__cancel2,axiom,
! [M2: nat,K: nat,N: nat] :
( ( minus_minus @ nat @ ( plus_plus @ nat @ M2 @ K ) @ ( plus_plus @ nat @ N @ K ) )
= ( minus_minus @ nat @ M2 @ N ) ) ).
% diff_cancel2
thf(fact_1072_diff__add__inverse,axiom,
! [N: nat,M2: nat] :
( ( minus_minus @ nat @ ( plus_plus @ nat @ N @ M2 ) @ N )
= M2 ) ).
% diff_add_inverse
thf(fact_1073_diff__add__inverse2,axiom,
! [M2: nat,N: nat] :
( ( minus_minus @ nat @ ( plus_plus @ nat @ M2 @ N ) @ N )
= M2 ) ).
% diff_add_inverse2
thf(fact_1074_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 @ ( log2 @ B2 @ X ) )
= ( ord_less @ real @ ( powr @ real @ B2 @ Y2 ) @ X ) ) ) ) ).
% less_log_iff
thf(fact_1075_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 @ ( log2 @ B2 @ X ) @ Y2 )
= ( ord_less @ real @ X @ ( powr @ real @ B2 @ Y2 ) ) ) ) ) ).
% log_less_iff
thf(fact_1076_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 @ ( log2 @ B2 @ X ) @ Y2 ) ) ) ) ).
% less_powr_iff
thf(fact_1077_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 @ ( log2 @ B2 @ X ) ) ) ) ) ).
% powr_less_iff
thf(fact_1078_nat__add__distrib,axiom,
! [Z2: int,Z3: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z2 )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z3 )
=> ( ( nat2 @ ( plus_plus @ int @ Z2 @ Z3 ) )
= ( plus_plus @ nat @ ( nat2 @ Z2 ) @ ( nat2 @ Z3 ) ) ) ) ) ).
% nat_add_distrib
thf(fact_1079_nat__abs__triangle__ineq,axiom,
! [K: int,L: int] : ( ord_less_eq @ nat @ ( nat2 @ ( abs_abs @ int @ ( plus_plus @ int @ K @ L ) ) ) @ ( plus_plus @ nat @ ( nat2 @ ( abs_abs @ int @ K ) ) @ ( nat2 @ ( abs_abs @ int @ L ) ) ) ) ).
% nat_abs_triangle_ineq
thf(fact_1080_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 @ ( log2 @ B2 @ X ) ) ) ) ) ).
% powr_le_iff
thf(fact_1081_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 @ ( log2 @ B2 @ X ) @ Y2 ) ) ) ) ).
% le_powr_iff
thf(fact_1082_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 @ ( log2 @ B2 @ X ) @ Y2 )
= ( ord_less_eq @ real @ X @ ( powr @ real @ B2 @ Y2 ) ) ) ) ) ).
% log_le_iff
thf(fact_1083_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 @ ( log2 @ B2 @ X ) )
= ( ord_less_eq @ real @ ( powr @ real @ B2 @ Y2 ) @ X ) ) ) ) ).
% le_log_iff
thf(fact_1084_powr__non__neg,axiom,
! [A3: real,X: real] :
~ ( ord_less @ real @ ( powr @ real @ A3 @ X ) @ ( zero_zero @ real ) ) ).
% powr_non_neg
thf(fact_1085_powr__less__mono2__neg,axiom,
! [A3: real,X: real,Y2: real] :
( ( ord_less @ real @ A3 @ ( zero_zero @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ X @ Y2 )
=> ( ord_less @ real @ ( powr @ real @ Y2 @ A3 ) @ ( powr @ real @ X @ A3 ) ) ) ) ) ).
% powr_less_mono2_neg
thf(fact_1086_powr__ge__pzero,axiom,
! [X: real,Y2: real] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( powr @ real @ X @ Y2 ) ) ).
% powr_ge_pzero
thf(fact_1087_powr__mono2,axiom,
! [A3: real,X: real,Y2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ A3 )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ X @ Y2 )
=> ( ord_less_eq @ real @ ( powr @ real @ X @ A3 ) @ ( powr @ real @ Y2 @ A3 ) ) ) ) ) ).
% powr_mono2
thf(fact_1088_powr__less__mono,axiom,
! [A3: real,B2: real,X: real] :
( ( ord_less @ real @ A3 @ B2 )
=> ( ( ord_less @ real @ ( one_one @ real ) @ X )
=> ( ord_less @ real @ ( powr @ real @ X @ A3 ) @ ( powr @ real @ X @ B2 ) ) ) ) ).
% powr_less_mono
thf(fact_1089_powr__less__cancel,axiom,
! [X: real,A3: real,B2: real] :
( ( ord_less @ real @ ( powr @ real @ X @ A3 ) @ ( powr @ real @ X @ B2 ) )
=> ( ( ord_less @ real @ ( one_one @ real ) @ X )
=> ( ord_less @ real @ A3 @ B2 ) ) ) ).
% powr_less_cancel
thf(fact_1090_powr__mono,axiom,
! [A3: real,B2: real,X: real] :
( ( ord_less_eq @ real @ A3 @ B2 )
=> ( ( ord_less_eq @ real @ ( one_one @ real ) @ X )
=> ( ord_less_eq @ real @ ( powr @ real @ X @ A3 ) @ ( powr @ real @ X @ B2 ) ) ) ) ).
% powr_mono
thf(fact_1091_add__decreasing,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ C2 @ B2 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A3 @ C2 ) @ B2 ) ) ) ) ).
% add_decreasing
thf(fact_1092_add__increasing,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ B2 @ C2 )
=> ( ord_less_eq @ A @ B2 @ ( plus_plus @ A @ A3 @ C2 ) ) ) ) ) ).
% add_increasing
thf(fact_1093_add__decreasing2,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A3 @ C2 ) @ B2 ) ) ) ) ).
% add_decreasing2
thf(fact_1094_add__increasing2,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [C2: A,B2: A,A3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less_eq @ A @ B2 @ A3 )
=> ( ord_less_eq @ A @ B2 @ ( plus_plus @ A @ A3 @ C2 ) ) ) ) ) ).
% add_increasing2
thf(fact_1095_add__nonneg__nonneg,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A3 @ B2 ) ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_1096_add__nonpos__nonpos,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A3 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% add_nonpos_nonpos
thf(fact_1097_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_1098_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_1099_add__mono__thms__linordered__field_I4_J,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [I: A,J2: A,K: A,L: A] :
( ( ( ord_less_eq @ A @ I @ J2 )
& ( ord_less @ A @ K @ L ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J2 @ L ) ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_1100_add__mono__thms__linordered__field_I3_J,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [I: A,J2: A,K: A,L: A] :
( ( ( ord_less @ A @ I @ J2 )
& ( ord_less_eq @ A @ K @ L ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I @ K ) @ ( plus_plus @ A @ J2 @ L ) ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_1101_add__le__less__mono,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [A3: A,B2: A,C2: A,D2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less @ A @ C2 @ D2 )
=> ( ord_less @ A @ ( plus_plus @ A @ A3 @ C2 ) @ ( plus_plus @ A @ B2 @ D2 ) ) ) ) ) ).
% add_le_less_mono
thf(fact_1102_add__less__le__mono,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [A3: A,B2: A,C2: A,D2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ D2 )
=> ( ord_less @ A @ ( plus_plus @ A @ A3 @ C2 ) @ ( plus_plus @ A @ B2 @ D2 ) ) ) ) ) ).
% add_less_le_mono
thf(fact_1103_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_1104_pos__add__strict,axiom,
! [A: $tType] :
( ( strict7427464778891057005id_add @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ B2 @ C2 )
=> ( ord_less @ A @ B2 @ ( plus_plus @ A @ A3 @ C2 ) ) ) ) ) ).
% pos_add_strict
thf(fact_1105_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ~ ! [C3: A] :
( ( B2
= ( plus_plus @ A @ A3 @ C3 ) )
=> ( C3
= ( zero_zero @ A ) ) ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
thf(fact_1106_add__pos__pos,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A3 @ B2 ) ) ) ) ) ).
% add_pos_pos
thf(fact_1107_add__neg__neg,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( plus_plus @ A @ A3 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% add_neg_neg
thf(fact_1108_log__def,axiom,
( log2
= ( ^ [A5: real,X2: real] : ( divide_divide @ real @ ( ln_ln @ real @ X2 ) @ ( ln_ln @ real @ A5 ) ) ) ) ).
% log_def
thf(fact_1109_add__le__add__imp__diff__le,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [I: A,K: A,N: A,J2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ N )
=> ( ( ord_less_eq @ A @ N @ ( plus_plus @ A @ J2 @ K ) )
=> ( ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ N )
=> ( ( ord_less_eq @ A @ N @ ( plus_plus @ A @ J2 @ K ) )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ N @ K ) @ J2 ) ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_1110_add__le__imp__le__diff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [I: A,K: A,N: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ I @ K ) @ N )
=> ( ord_less_eq @ A @ I @ ( minus_minus @ A @ N @ K ) ) ) ) ).
% add_le_imp_le_diff
thf(fact_1111_diff__le__eq,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ ( minus_minus @ A @ A3 @ B2 ) @ C2 )
= ( ord_less_eq @ A @ A3 @ ( plus_plus @ A @ C2 @ B2 ) ) ) ) ).
% diff_le_eq
thf(fact_1112_le__diff__eq,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ ( minus_minus @ A @ C2 @ B2 ) )
= ( ord_less_eq @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C2 ) ) ) ).
% le_diff_eq
thf(fact_1113_diff__add,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( plus_plus @ A @ ( minus_minus @ A @ B2 @ A3 ) @ A3 )
= B2 ) ) ) ).
% diff_add
thf(fact_1114_le__add__diff,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ord_less_eq @ A @ C2 @ ( minus_minus @ A @ ( plus_plus @ A @ B2 @ C2 ) @ A3 ) ) ) ) ).
% le_add_diff
thf(fact_1115_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ ( minus_minus @ A @ B2 @ A3 ) )
= ( ord_less_eq @ A @ ( plus_plus @ A @ C2 @ A3 ) @ B2 ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_1116_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( plus_plus @ A @ C2 @ ( minus_minus @ A @ B2 @ A3 ) )
= ( minus_minus @ A @ ( plus_plus @ A @ C2 @ B2 ) @ A3 ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_1117_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( minus_minus @ A @ ( plus_plus @ A @ C2 @ B2 ) @ A3 )
= ( plus_plus @ A @ C2 @ ( minus_minus @ A @ B2 @ A3 ) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_1118_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( plus_plus @ A @ ( minus_minus @ A @ B2 @ A3 ) @ C2 )
= ( minus_minus @ A @ ( plus_plus @ A @ B2 @ C2 ) @ A3 ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_1119_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( minus_minus @ A @ ( plus_plus @ A @ B2 @ C2 ) @ A3 )
= ( plus_plus @ A @ ( minus_minus @ A @ B2 @ A3 ) @ C2 ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_1120_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( minus_minus @ A @ C2 @ ( minus_minus @ A @ B2 @ A3 ) )
= ( minus_minus @ A @ ( plus_plus @ A @ C2 @ A3 ) @ B2 ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_1121_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( plus_plus @ A @ A3 @ ( minus_minus @ A @ B2 @ A3 ) )
= B2 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
thf(fact_1122_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ( minus_minus @ A @ B2 @ A3 )
= C2 )
= ( B2
= ( plus_plus @ A @ C2 @ A3 ) ) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_1123_add__mono1,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ord_less @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) @ ( plus_plus @ A @ B2 @ ( one_one @ A ) ) ) ) ) ).
% add_mono1
thf(fact_1124_less__add__one,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A] : ( ord_less @ A @ A3 @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) ) ) ).
% less_add_one
thf(fact_1125_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,B2: A] :
( ~ ( ord_less @ A @ A3 @ B2 )
=> ( ( plus_plus @ A @ B2 @ ( minus_minus @ A @ A3 @ B2 ) )
= A3 ) ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_1126_diff__less__eq,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less @ A @ ( minus_minus @ A @ A3 @ B2 ) @ C2 )
= ( ord_less @ A @ A3 @ ( plus_plus @ A @ C2 @ B2 ) ) ) ) ).
% diff_less_eq
thf(fact_1127_less__diff__eq,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( ord_less @ A @ A3 @ ( minus_minus @ A @ C2 @ B2 ) )
= ( ord_less @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C2 ) ) ) ).
% less_diff_eq
thf(fact_1128_add__eq__0__iff,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B2: A] :
( ( ( plus_plus @ A @ A3 @ B2 )
= ( zero_zero @ A ) )
= ( B2
= ( uminus_uminus @ A @ A3 ) ) ) ) ).
% add_eq_0_iff
thf(fact_1129_ab__group__add__class_Oab__left__minus,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A3: A] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ A3 ) @ A3 )
= ( zero_zero @ A ) ) ) ).
% ab_group_add_class.ab_left_minus
thf(fact_1130_add_Oinverse__unique,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B2: A] :
( ( ( plus_plus @ A @ A3 @ B2 )
= ( zero_zero @ A ) )
=> ( ( uminus_uminus @ A @ A3 )
= B2 ) ) ) ).
% add.inverse_unique
thf(fact_1131_eq__neg__iff__add__eq__0,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B2: A] :
( ( A3
= ( uminus_uminus @ A @ B2 ) )
= ( ( plus_plus @ A @ A3 @ B2 )
= ( zero_zero @ A ) ) ) ) ).
% eq_neg_iff_add_eq_0
thf(fact_1132_neg__eq__iff__add__eq__0,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A,B2: A] :
( ( ( uminus_uminus @ A @ A3 )
= B2 )
= ( ( plus_plus @ A @ A3 @ B2 )
= ( zero_zero @ A ) ) ) ) ).
% neg_eq_iff_add_eq_0
thf(fact_1133_group__cancel_Osub2,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [B7: A,K: A,B2: A,A3: A] :
( ( B7
= ( plus_plus @ A @ K @ B2 ) )
=> ( ( minus_minus @ A @ A3 @ B7 )
= ( plus_plus @ A @ ( uminus_uminus @ A @ K ) @ ( minus_minus @ A @ A3 @ B2 ) ) ) ) ) ).
% group_cancel.sub2
thf(fact_1134_diff__conv__add__uminus,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ( ( minus_minus @ A )
= ( ^ [A5: A,B3: A] : ( plus_plus @ A @ A5 @ ( uminus_uminus @ A @ B3 ) ) ) ) ) ).
% diff_conv_add_uminus
thf(fact_1135_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ( ( minus_minus @ A )
= ( ^ [A5: A,B3: A] : ( plus_plus @ A @ A5 @ ( uminus_uminus @ A @ B3 ) ) ) ) ) ).
% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_1136_abs__triangle__ineq,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A,B2: A] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( plus_plus @ A @ A3 @ B2 ) ) @ ( plus_plus @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ B2 ) ) ) ) ).
% abs_triangle_ineq
thf(fact_1137_div__add1__eq,axiom,
! [A: $tType] :
( ( euclid3128863361964157862miring @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( divide_divide @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C2 )
= ( plus_plus @ A @ ( plus_plus @ A @ ( divide_divide @ A @ A3 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) ) @ ( divide_divide @ A @ ( plus_plus @ A @ ( modulo_modulo @ A @ A3 @ C2 ) @ ( modulo_modulo @ A @ B2 @ C2 ) ) @ C2 ) ) ) ) ).
% div_add1_eq
thf(fact_1138_add__is__1,axiom,
! [M2: nat,N: nat] :
( ( ( plus_plus @ nat @ M2 @ N )
= ( suc @ ( zero_zero @ nat ) ) )
= ( ( ( M2
= ( suc @ ( zero_zero @ nat ) ) )
& ( N
= ( zero_zero @ nat ) ) )
| ( ( M2
= ( zero_zero @ nat ) )
& ( N
= ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ).
% add_is_1
thf(fact_1139_one__is__add,axiom,
! [M2: nat,N: nat] :
( ( ( suc @ ( zero_zero @ nat ) )
= ( plus_plus @ nat @ M2 @ N ) )
= ( ( ( M2
= ( suc @ ( zero_zero @ nat ) ) )
& ( N
= ( zero_zero @ nat ) ) )
| ( ( M2
= ( zero_zero @ nat ) )
& ( N
= ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ).
% one_is_add
thf(fact_1140_less__natE,axiom,
! [M2: nat,N: nat] :
( ( ord_less @ nat @ M2 @ N )
=> ~ ! [Q3: nat] :
( N
!= ( suc @ ( plus_plus @ nat @ M2 @ Q3 ) ) ) ) ).
% less_natE
thf(fact_1141_less__add__Suc1,axiom,
! [I: nat,M2: nat] : ( ord_less @ nat @ I @ ( suc @ ( plus_plus @ nat @ I @ M2 ) ) ) ).
% less_add_Suc1
thf(fact_1142_less__add__Suc2,axiom,
! [I: nat,M2: nat] : ( ord_less @ nat @ I @ ( suc @ ( plus_plus @ nat @ M2 @ I ) ) ) ).
% less_add_Suc2
thf(fact_1143_less__iff__Suc__add,axiom,
( ( ord_less @ nat )
= ( ^ [M5: nat,N5: nat] :
? [K3: nat] :
( N5
= ( suc @ ( plus_plus @ nat @ M5 @ K3 ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_1144_less__imp__Suc__add,axiom,
! [M2: nat,N: nat] :
( ( ord_less @ nat @ M2 @ N )
=> ? [K2: nat] :
( N
= ( suc @ ( plus_plus @ nat @ M2 @ K2 ) ) ) ) ).
% less_imp_Suc_add
thf(fact_1145_less__imp__add__positive,axiom,
! [I: nat,J2: nat] :
( ( ord_less @ nat @ I @ J2 )
=> ? [K2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K2 )
& ( ( plus_plus @ nat @ I @ K2 )
= J2 ) ) ) ).
% less_imp_add_positive
thf(fact_1146_mono__nat__linear__lb,axiom,
! [F2: nat > nat,M2: nat,K: nat] :
( ! [M3: nat,N2: nat] :
( ( ord_less @ nat @ M3 @ N2 )
=> ( ord_less @ nat @ ( F2 @ M3 ) @ ( F2 @ N2 ) ) )
=> ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( F2 @ M2 ) @ K ) @ ( F2 @ ( plus_plus @ nat @ M2 @ K ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_1147_ex__has__greatest__nat__lemma,axiom,
! [A: $tType,P: A > $o,K: A,F2: A > nat,N: nat] :
( ( P @ K )
=> ( ! [X3: A] :
( ( P @ X3 )
=> ? [Y: A] :
( ( P @ Y )
& ~ ( ord_less_eq @ nat @ ( F2 @ Y ) @ ( F2 @ X3 ) ) ) )
=> ? [Y3: A] :
( ( P @ Y3 )
& ~ ( ord_less @ nat @ ( F2 @ Y3 ) @ ( plus_plus @ nat @ ( F2 @ K ) @ N ) ) ) ) ) ).
% ex_has_greatest_nat_lemma
thf(fact_1148_diff__add__0,axiom,
! [N: nat,M2: nat] :
( ( minus_minus @ nat @ N @ ( plus_plus @ nat @ N @ M2 ) )
= ( zero_zero @ nat ) ) ).
% diff_add_0
thf(fact_1149_less__diff__conv,axiom,
! [I: nat,J2: nat,K: nat] :
( ( ord_less @ nat @ I @ ( minus_minus @ nat @ J2 @ K ) )
= ( ord_less @ nat @ ( plus_plus @ nat @ I @ K ) @ J2 ) ) ).
% less_diff_conv
thf(fact_1150_add__diff__inverse__nat,axiom,
! [M2: nat,N: nat] :
( ~ ( ord_less @ nat @ M2 @ N )
=> ( ( plus_plus @ nat @ N @ ( minus_minus @ nat @ M2 @ N ) )
= M2 ) ) ).
% add_diff_inverse_nat
thf(fact_1151_Suc__eq__plus1,axiom,
( suc
= ( ^ [N5: nat] : ( plus_plus @ nat @ N5 @ ( one_one @ nat ) ) ) ) ).
% Suc_eq_plus1
thf(fact_1152_plus__1__eq__Suc,axiom,
( ( plus_plus @ nat @ ( one_one @ nat ) )
= suc ) ).
% plus_1_eq_Suc
thf(fact_1153_Suc__eq__plus1__left,axiom,
( suc
= ( plus_plus @ nat @ ( one_one @ nat ) ) ) ).
% Suc_eq_plus1_left
thf(fact_1154_odd__nonzero,axiom,
! [Z2: int] :
( ( plus_plus @ int @ ( plus_plus @ int @ ( one_one @ int ) @ Z2 ) @ Z2 )
!= ( zero_zero @ int ) ) ).
% odd_nonzero
thf(fact_1155_le__diff__conv,axiom,
! [J2: nat,K: nat,I: nat] :
( ( ord_less_eq @ nat @ ( minus_minus @ nat @ J2 @ K ) @ I )
= ( ord_less_eq @ nat @ J2 @ ( plus_plus @ nat @ I @ K ) ) ) ).
% le_diff_conv
thf(fact_1156_Nat_Ole__diff__conv2,axiom,
! [K: nat,J2: nat,I: nat] :
( ( ord_less_eq @ nat @ K @ J2 )
=> ( ( ord_less_eq @ nat @ I @ ( minus_minus @ nat @ J2 @ K ) )
= ( ord_less_eq @ nat @ ( plus_plus @ nat @ I @ K ) @ J2 ) ) ) ).
% Nat.le_diff_conv2
thf(fact_1157_Nat_Odiff__add__assoc,axiom,
! [K: nat,J2: nat,I: nat] :
( ( ord_less_eq @ nat @ K @ J2 )
=> ( ( minus_minus @ nat @ ( plus_plus @ nat @ I @ J2 ) @ K )
= ( plus_plus @ nat @ I @ ( minus_minus @ nat @ J2 @ K ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_1158_Nat_Odiff__add__assoc2,axiom,
! [K: nat,J2: nat,I: nat] :
( ( ord_less_eq @ nat @ K @ J2 )
=> ( ( minus_minus @ nat @ ( plus_plus @ nat @ J2 @ I ) @ K )
= ( plus_plus @ nat @ ( minus_minus @ nat @ J2 @ K ) @ I ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_1159_Nat_Ole__imp__diff__is__add,axiom,
! [I: nat,J2: nat,K: nat] :
( ( ord_less_eq @ nat @ I @ J2 )
=> ( ( ( minus_minus @ nat @ J2 @ I )
= K )
= ( J2
= ( plus_plus @ nat @ K @ I ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_1160_int__ge__induct,axiom,
! [K: int,I: int,P: int > $o] :
( ( ord_less_eq @ int @ K @ I )
=> ( ( P @ K )
=> ( ! [I3: int] :
( ( ord_less_eq @ int @ K @ I3 )
=> ( ( P @ I3 )
=> ( P @ ( plus_plus @ int @ I3 @ ( one_one @ int ) ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_ge_induct
thf(fact_1161_int__gr__induct,axiom,
! [K: int,I: int,P: int > $o] :
( ( ord_less @ int @ K @ I )
=> ( ( P @ ( plus_plus @ int @ K @ ( one_one @ int ) ) )
=> ( ! [I3: int] :
( ( ord_less @ int @ K @ I3 )
=> ( ( P @ I3 )
=> ( P @ ( plus_plus @ int @ I3 @ ( one_one @ int ) ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_gr_induct
thf(fact_1162_zless__add1__eq,axiom,
! [W: int,Z2: int] :
( ( ord_less @ int @ W @ ( plus_plus @ int @ Z2 @ ( one_one @ int ) ) )
= ( ( ord_less @ int @ W @ Z2 )
| ( W = Z2 ) ) ) ).
% zless_add1_eq
thf(fact_1163_zle__iff__zadd,axiom,
( ( ord_less_eq @ int )
= ( ^ [W3: int,Z6: int] :
? [N5: nat] :
( Z6
= ( plus_plus @ int @ W3 @ ( semiring_1_of_nat @ int @ N5 ) ) ) ) ) ).
% zle_iff_zadd
thf(fact_1164_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 @ ( log2 @ B2 @ X ) )
= ( log2 @ B2 @ ( divide_divide @ real @ ( powr @ real @ B2 @ Y2 ) @ X ) ) ) ) ) ) ).
% minus_log_eq_powr
thf(fact_1165_minus__real__def,axiom,
( ( minus_minus @ real )
= ( ^ [X2: real,Y5: real] : ( plus_plus @ real @ X2 @ ( uminus_uminus @ real @ Y5 ) ) ) ) ).
% minus_real_def
thf(fact_1166_dbl__inc__def,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( neg_numeral_dbl_inc @ A )
= ( ^ [X2: A] : ( plus_plus @ A @ ( plus_plus @ A @ X2 @ X2 ) @ ( one_one @ A ) ) ) ) ) ).
% dbl_inc_def
thf(fact_1167_powr__less__mono2,axiom,
! [A3: real,X: real,Y2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A3 )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ X @ Y2 )
=> ( ord_less @ real @ ( powr @ real @ X @ A3 ) @ ( powr @ real @ Y2 @ A3 ) ) ) ) ) ).
% powr_less_mono2
thf(fact_1168_powr__mono2_H,axiom,
! [A3: real,X: real,Y2: real] :
( ( ord_less_eq @ real @ A3 @ ( zero_zero @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ X @ Y2 )
=> ( ord_less_eq @ real @ ( powr @ real @ Y2 @ A3 ) @ ( powr @ real @ X @ A3 ) ) ) ) ) ).
% powr_mono2'
thf(fact_1169_powr__inj,axiom,
! [A3: real,X: real,Y2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A3 )
=> ( ( A3
!= ( one_one @ real ) )
=> ( ( ( powr @ real @ A3 @ X )
= ( powr @ real @ A3 @ Y2 ) )
= ( X = Y2 ) ) ) ) ).
% powr_inj
thf(fact_1170_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_1171_ge__one__powr__ge__zero,axiom,
! [X: real,A3: real] :
( ( ord_less_eq @ real @ ( one_one @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ A3 )
=> ( ord_less_eq @ real @ ( one_one @ real ) @ ( powr @ real @ X @ A3 ) ) ) ) ).
% ge_one_powr_ge_zero
thf(fact_1172_powr__mono__both,axiom,
! [A3: real,B2: real,X: real,Y2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ A3 )
=> ( ( ord_less_eq @ real @ A3 @ B2 )
=> ( ( ord_less_eq @ real @ ( one_one @ real ) @ X )
=> ( ( ord_less_eq @ real @ X @ Y2 )
=> ( ord_less_eq @ real @ ( powr @ real @ X @ A3 ) @ ( powr @ real @ Y2 @ B2 ) ) ) ) ) ) ).
% powr_mono_both
thf(fact_1173_powr__le1,axiom,
! [A3: real,X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ A3 )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ X @ ( one_one @ real ) )
=> ( ord_less_eq @ real @ ( powr @ real @ X @ A3 ) @ ( one_one @ real ) ) ) ) ) ).
% powr_le1
thf(fact_1174_powr__divide,axiom,
! [X: real,Y2: real,A3: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y2 )
=> ( ( powr @ real @ ( divide_divide @ real @ X @ Y2 ) @ A3 )
= ( divide_divide @ real @ ( powr @ real @ X @ A3 ) @ ( powr @ real @ Y2 @ A3 ) ) ) ) ) ).
% powr_divide
thf(fact_1175_field__le__epsilon,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y2: A] :
( ! [E: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ E )
=> ( ord_less_eq @ A @ X @ ( plus_plus @ A @ Y2 @ E ) ) )
=> ( ord_less_eq @ A @ X @ Y2 ) ) ) ).
% field_le_epsilon
thf(fact_1176_add__strict__increasing2,axiom,
! [A: $tType] :
( ( ordere8940638589300402666id_add @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ B2 @ C2 )
=> ( ord_less @ A @ B2 @ ( plus_plus @ A @ A3 @ C2 ) ) ) ) ) ).
% add_strict_increasing2
thf(fact_1177_add__strict__increasing,axiom,
! [A: $tType] :
( ( ordere8940638589300402666id_add @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ B2 @ C2 )
=> ( ord_less @ A @ B2 @ ( plus_plus @ A @ A3 @ C2 ) ) ) ) ) ).
% add_strict_increasing
thf(fact_1178_add__pos__nonneg,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A3 @ B2 ) ) ) ) ) ).
% add_pos_nonneg
thf(fact_1179_add__nonpos__neg,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( plus_plus @ A @ A3 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% add_nonpos_neg
thf(fact_1180_add__nonneg__pos,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A3 @ B2 ) ) ) ) ) ).
% add_nonneg_pos
thf(fact_1181_add__neg__nonpos,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( plus_plus @ A @ A3 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% add_neg_nonpos
thf(fact_1182_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_1183_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_1184_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_1185_div__add__self2,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [B2: A,A3: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ A3 @ B2 ) @ B2 )
= ( plus_plus @ A @ ( divide_divide @ A @ A3 @ B2 ) @ ( one_one @ A ) ) ) ) ) ).
% div_add_self2
thf(fact_1186_div__add__self1,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [B2: A,A3: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ B2 @ A3 ) @ B2 )
= ( plus_plus @ A @ ( divide_divide @ A @ A3 @ B2 ) @ ( one_one @ A ) ) ) ) ) ).
% div_add_self1
thf(fact_1187_less__half__sum,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ord_less @ A @ A3 @ ( divide_divide @ A @ ( plus_plus @ A @ A3 @ B2 ) @ ( plus_plus @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) ) ) ) ).
% less_half_sum
thf(fact_1188_gt__half__sum,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ord_less @ A @ ( divide_divide @ A @ ( plus_plus @ A @ A3 @ B2 ) @ ( plus_plus @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) @ B2 ) ) ) ).
% gt_half_sum
thf(fact_1189_abs__diff__triangle__ineq,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A,B2: A,C2: A,D2: A] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ ( plus_plus @ A @ A3 @ B2 ) @ ( plus_plus @ A @ C2 @ D2 ) ) ) @ ( plus_plus @ A @ ( abs_abs @ A @ ( minus_minus @ A @ A3 @ C2 ) ) @ ( abs_abs @ A @ ( minus_minus @ A @ B2 @ D2 ) ) ) ) ) ).
% abs_diff_triangle_ineq
thf(fact_1190_abs__triangle__ineq4,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A3: A,B2: A] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ A3 @ B2 ) ) @ ( plus_plus @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ B2 ) ) ) ) ).
% abs_triangle_ineq4
thf(fact_1191_abs__diff__le__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,A3: A,R4: A] :
( ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X @ A3 ) ) @ R4 )
= ( ( ord_less_eq @ A @ ( minus_minus @ A @ A3 @ R4 ) @ X )
& ( ord_less_eq @ A @ X @ ( plus_plus @ A @ A3 @ R4 ) ) ) ) ) ).
% abs_diff_le_iff
thf(fact_1192_log__ln,axiom,
( ( ln_ln @ real )
= ( log2 @ ( exp @ real @ ( one_one @ real ) ) ) ) ).
% log_ln
thf(fact_1193_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_1194_abs__diff__less__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,A3: A,R4: A] :
( ( ord_less @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X @ A3 ) ) @ R4 )
= ( ( ord_less @ A @ ( minus_minus @ A @ A3 @ R4 ) @ X )
& ( ord_less @ A @ X @ ( plus_plus @ A @ A3 @ R4 ) ) ) ) ) ).
% abs_diff_less_iff
thf(fact_1195_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_1196_ex__gt__or__lt,axiom,
! [A: $tType] :
( ( condit5016429287641298734tinuum @ A )
=> ! [A3: A] :
? [B4: A] :
( ( ord_less @ A @ A3 @ B4 )
| ( ord_less @ A @ B4 @ A3 ) ) ) ).
% ex_gt_or_lt
thf(fact_1197_nat__diff__split__asm,axiom,
! [P: nat > $o,A3: nat,B2: nat] :
( ( P @ ( minus_minus @ nat @ A3 @ B2 ) )
= ( ~ ( ( ( ord_less @ nat @ A3 @ B2 )
& ~ ( P @ ( zero_zero @ nat ) ) )
| ? [D4: nat] :
( ( A3
= ( plus_plus @ nat @ B2 @ D4 ) )
& ~ ( P @ D4 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_1198_nat__diff__split,axiom,
! [P: nat > $o,A3: nat,B2: nat] :
( ( P @ ( minus_minus @ nat @ A3 @ B2 ) )
= ( ( ( ord_less @ nat @ A3 @ B2 )
=> ( P @ ( zero_zero @ nat ) ) )
& ! [D4: nat] :
( ( A3
= ( plus_plus @ nat @ B2 @ D4 ) )
=> ( P @ D4 ) ) ) ) ).
% nat_diff_split
thf(fact_1199_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_1200_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_1201_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_1202_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_1203_less__diff__conv2,axiom,
! [K: nat,J2: nat,I: nat] :
( ( ord_less_eq @ nat @ K @ J2 )
=> ( ( ord_less @ nat @ ( minus_minus @ nat @ J2 @ K ) @ I )
= ( ord_less @ nat @ J2 @ ( plus_plus @ nat @ I @ K ) ) ) ) ).
% less_diff_conv2
thf(fact_1204_int__ops_I4_J,axiom,
! [A3: nat] :
( ( semiring_1_of_nat @ int @ ( suc @ A3 ) )
= ( plus_plus @ int @ ( semiring_1_of_nat @ int @ A3 ) @ ( one_one @ int ) ) ) ).
% int_ops(4)
thf(fact_1205_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_1206_zless__iff__Suc__zadd,axiom,
( ( ord_less @ int )
= ( ^ [W3: int,Z6: int] :
? [N5: nat] :
( Z6
= ( plus_plus @ int @ W3 @ ( semiring_1_of_nat @ int @ ( suc @ N5 ) ) ) ) ) ) ).
% zless_iff_Suc_zadd
thf(fact_1207_odd__less__0__iff,axiom,
! [Z2: int] :
( ( ord_less @ int @ ( plus_plus @ int @ ( plus_plus @ int @ ( one_one @ int ) @ Z2 ) @ Z2 ) @ ( zero_zero @ int ) )
= ( ord_less @ int @ Z2 @ ( zero_zero @ int ) ) ) ).
% odd_less_0_iff
thf(fact_1208_add1__zle__eq,axiom,
! [W: int,Z2: int] :
( ( ord_less_eq @ int @ ( plus_plus @ int @ W @ ( one_one @ int ) ) @ Z2 )
= ( ord_less @ int @ W @ Z2 ) ) ).
% add1_zle_eq
thf(fact_1209_zless__imp__add1__zle,axiom,
! [W: int,Z2: int] :
( ( ord_less @ int @ W @ Z2 )
=> ( ord_less_eq @ int @ ( plus_plus @ int @ W @ ( one_one @ int ) ) @ Z2 ) ) ).
% zless_imp_add1_zle
thf(fact_1210_int__induct,axiom,
! [P: int > $o,K: int,I: int] :
( ( P @ K )
=> ( ! [I3: int] :
( ( ord_less_eq @ int @ K @ I3 )
=> ( ( P @ I3 )
=> ( P @ ( plus_plus @ int @ I3 @ ( one_one @ int ) ) ) ) )
=> ( ! [I3: int] :
( ( ord_less_eq @ int @ I3 @ K )
=> ( ( P @ I3 )
=> ( P @ ( minus_minus @ int @ I3 @ ( one_one @ int ) ) ) ) )
=> ( P @ I ) ) ) ) ).
% int_induct
thf(fact_1211_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_1212_dbl__dec__def,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( neg_numeral_dbl_dec @ A )
= ( ^ [X2: A] : ( minus_minus @ A @ ( plus_plus @ A @ X2 @ X2 ) @ ( one_one @ A ) ) ) ) ) ).
% dbl_dec_def
thf(fact_1213_log__base__change,axiom,
! [A3: real,B2: real,X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A3 )
=> ( ( A3
!= ( one_one @ real ) )
=> ( ( log2 @ B2 @ X )
= ( divide_divide @ real @ ( log2 @ A3 @ X ) @ ( log2 @ A3 @ B2 ) ) ) ) ) ).
% log_base_change
thf(fact_1214_powr__realpow,axiom,
! [X: real,N: nat] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( powr @ real @ X @ ( semiring_1_of_nat @ real @ N ) )
= ( power_power @ real @ X @ N ) ) ) ).
% powr_realpow
thf(fact_1215_less__log__of__power,axiom,
! [B2: real,N: nat,M2: real] :
( ( ord_less @ real @ ( power_power @ real @ B2 @ N ) @ M2 )
=> ( ( ord_less @ real @ ( one_one @ real ) @ B2 )
=> ( ord_less @ real @ ( semiring_1_of_nat @ real @ N ) @ ( log2 @ B2 @ M2 ) ) ) ) ).
% less_log_of_power
thf(fact_1216_log__of__power__eq,axiom,
! [M2: nat,B2: real,N: nat] :
( ( ( semiring_1_of_nat @ real @ M2 )
= ( power_power @ real @ B2 @ N ) )
=> ( ( ord_less @ real @ ( one_one @ real ) @ B2 )
=> ( ( semiring_1_of_nat @ real @ N )
= ( log2 @ B2 @ ( semiring_1_of_nat @ real @ M2 ) ) ) ) ) ).
% log_of_power_eq
thf(fact_1217_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_1218_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_1219_add__eq__if,axiom,
( ( plus_plus @ nat )
= ( ^ [M5: nat,N5: nat] :
( if @ nat
@ ( M5
= ( zero_zero @ nat ) )
@ N5
@ ( suc @ ( plus_plus @ nat @ ( minus_minus @ nat @ M5 @ ( one_one @ nat ) ) @ N5 ) ) ) ) ) ).
% add_eq_if
thf(fact_1220_nat__less__real__le,axiom,
( ( ord_less @ nat )
= ( ^ [N5: nat,M5: nat] : ( ord_less_eq @ real @ ( plus_plus @ real @ ( semiring_1_of_nat @ real @ N5 ) @ ( one_one @ real ) ) @ ( semiring_1_of_nat @ real @ M5 ) ) ) ) ).
% nat_less_real_le
thf(fact_1221_nat__le__real__less,axiom,
( ( ord_less_eq @ nat )
= ( ^ [N5: nat,M5: nat] : ( ord_less @ real @ ( semiring_1_of_nat @ real @ N5 ) @ ( plus_plus @ real @ ( semiring_1_of_nat @ real @ M5 ) @ ( one_one @ real ) ) ) ) ) ).
% nat_le_real_less
thf(fact_1222_le__imp__0__less,axiom,
! [Z2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z2 )
=> ( ord_less @ int @ ( zero_zero @ int ) @ ( plus_plus @ int @ ( one_one @ int ) @ Z2 ) ) ) ).
% le_imp_0_less
thf(fact_1223_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_1224_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_1225_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_1226_mod__pos__neg__trivial,axiom,
! [K: int,L: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ K )
=> ( ( ord_less_eq @ int @ ( plus_plus @ int @ K @ L ) @ ( zero_zero @ int ) )
=> ( ( modulo_modulo @ int @ K @ L )
= ( plus_plus @ int @ K @ L ) ) ) ) ).
% mod_pos_neg_trivial
thf(fact_1227_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_1228_powr__minus__divide,axiom,
! [A: $tType] :
( ( ( real_V3459762299906320749_field @ A )
& ( ln @ A ) )
=> ! [X: A,A3: A] :
( ( powr @ A @ X @ ( uminus_uminus @ A @ A3 ) )
= ( divide_divide @ A @ ( one_one @ A ) @ ( powr @ A @ X @ A3 ) ) ) ) ).
% powr_minus_divide
thf(fact_1229_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_1230_log__divide,axiom,
! [A3: real,X: real,Y2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A3 )
=> ( ( A3
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ Y2 )
=> ( ( log2 @ A3 @ ( divide_divide @ real @ X @ Y2 ) )
= ( minus_minus @ real @ ( log2 @ A3 @ X ) @ ( log2 @ A3 @ Y2 ) ) ) ) ) ) ) ).
% log_divide
thf(fact_1231_le__log__of__power,axiom,
! [B2: real,N: nat,M2: real] :
( ( ord_less_eq @ real @ ( power_power @ real @ B2 @ N ) @ M2 )
=> ( ( ord_less @ real @ ( one_one @ real ) @ B2 )
=> ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ N ) @ ( log2 @ B2 @ M2 ) ) ) ) ).
% le_log_of_power
thf(fact_1232_log__base__pow,axiom,
! [A3: real,N: nat,X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A3 )
=> ( ( log2 @ ( power_power @ real @ A3 @ N ) @ X )
= ( divide_divide @ real @ ( log2 @ A3 @ X ) @ ( semiring_1_of_nat @ real @ N ) ) ) ) ).
% log_base_pow
thf(fact_1233_Suc__nat__eq__nat__zadd1,axiom,
! [Z2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z2 )
=> ( ( suc @ ( nat2 @ Z2 ) )
= ( nat2 @ ( plus_plus @ int @ ( one_one @ int ) @ Z2 ) ) ) ) ).
% Suc_nat_eq_nat_zadd1
thf(fact_1234_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_1235_log__of__power__less,axiom,
! [M2: nat,B2: real,N: nat] :
( ( ord_less @ real @ ( semiring_1_of_nat @ real @ M2 ) @ ( power_power @ real @ B2 @ N ) )
=> ( ( ord_less @ real @ ( one_one @ real ) @ B2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M2 )
=> ( ord_less @ real @ ( log2 @ B2 @ ( semiring_1_of_nat @ real @ M2 ) ) @ ( semiring_1_of_nat @ real @ N ) ) ) ) ) ).
% log_of_power_less
thf(fact_1236_ln__powr__bound,axiom,
! [X: real,A3: real] :
( ( ord_less_eq @ real @ ( one_one @ real ) @ X )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ A3 )
=> ( ord_less_eq @ real @ ( ln_ln @ real @ X ) @ ( divide_divide @ real @ ( powr @ real @ X @ A3 ) @ A3 ) ) ) ) ).
% ln_powr_bound
thf(fact_1237_neg__one__power__add__eq__neg__one__power__diff,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [K: nat,N: nat] :
( ( ord_less_eq @ nat @ K @ N )
=> ( ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( plus_plus @ nat @ N @ K ) )
= ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( minus_minus @ nat @ N @ K ) ) ) ) ) ).
% neg_one_power_add_eq_neg_one_power_diff
thf(fact_1238_lemma__interval,axiom,
! [A3: real,X: real,B2: real] :
( ( ord_less @ real @ A3 @ X )
=> ( ( ord_less @ real @ X @ B2 )
=> ? [D5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D5 )
& ! [Y: real] :
( ( ord_less @ real @ ( abs_abs @ real @ ( minus_minus @ real @ X @ Y ) ) @ D5 )
=> ( ( ord_less_eq @ real @ A3 @ Y )
& ( ord_less_eq @ real @ Y @ B2 ) ) ) ) ) ) ).
% lemma_interval
thf(fact_1239_ceiling__log__eq__powr__iff,axiom,
! [X: real,B2: real,K: nat] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( one_one @ real ) @ B2 )
=> ( ( ( archimedean_ceiling @ real @ ( log2 @ B2 @ X ) )
= ( plus_plus @ int @ ( semiring_1_of_nat @ int @ K ) @ ( one_one @ int ) ) )
= ( ( ord_less @ real @ ( powr @ real @ B2 @ ( semiring_1_of_nat @ real @ K ) ) @ X )
& ( ord_less_eq @ real @ X @ ( powr @ real @ B2 @ ( semiring_1_of_nat @ real @ ( plus_plus @ nat @ K @ ( one_one @ nat ) ) ) ) ) ) ) ) ) ).
% ceiling_log_eq_powr_iff
thf(fact_1240_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_1241_lemma__interval__lt,axiom,
! [A3: real,X: real,B2: real] :
( ( ord_less @ real @ A3 @ X )
=> ( ( ord_less @ real @ X @ B2 )
=> ? [D5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D5 )
& ! [Y: real] :
( ( ord_less @ real @ ( abs_abs @ real @ ( minus_minus @ real @ X @ Y ) ) @ D5 )
=> ( ( ord_less @ real @ A3 @ Y )
& ( ord_less @ real @ Y @ B2 ) ) ) ) ) ) ).
% lemma_interval_lt
thf(fact_1242_ceiling__zero,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archimedean_ceiling @ A @ ( zero_zero @ A ) )
= ( zero_zero @ int ) ) ) ).
% ceiling_zero
thf(fact_1243_ceiling__one,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archimedean_ceiling @ A @ ( one_one @ A ) )
= ( one_one @ int ) ) ) ).
% ceiling_one
thf(fact_1244_ceiling__of__nat,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [N: nat] :
( ( archimedean_ceiling @ A @ ( semiring_1_of_nat @ A @ N ) )
= ( semiring_1_of_nat @ int @ N ) ) ) ).
% ceiling_of_nat
thf(fact_1245_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_1246_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_1247_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_1248_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_1249_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_1250_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_1251_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_1252_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_1253_nat__ceiling__le__eq,axiom,
! [X: real,A3: nat] :
( ( ord_less_eq @ nat @ ( nat2 @ ( archimedean_ceiling @ real @ X ) ) @ A3 )
= ( ord_less_eq @ real @ X @ ( semiring_1_of_nat @ real @ A3 ) ) ) ).
% nat_ceiling_le_eq
thf(fact_1254_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_1255_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_1256_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_1257_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_1258_of__nat__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [R4: A] : ( ord_less_eq @ A @ R4 @ ( semiring_1_of_nat @ A @ ( nat2 @ ( archimedean_ceiling @ A @ R4 ) ) ) ) ) ).
% of_nat_ceiling
thf(fact_1259_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_1260_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_1261_length__induct,axiom,
! [A: $tType,P: ( list @ A ) > $o,Xs: list @ A] :
( ! [Xs2: list @ A] :
( ! [Ys: list @ A] :
( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Ys ) @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( P @ Ys ) )
=> ( P @ Xs2 ) )
=> ( P @ Xs ) ) ).
% length_induct
thf(fact_1262_eq__diff__eq_H,axiom,
! [X: real,Y2: real,Z2: real] :
( ( X
= ( minus_minus @ real @ Y2 @ Z2 ) )
= ( Y2
= ( plus_plus @ real @ X @ Z2 ) ) ) ).
% eq_diff_eq'
thf(fact_1263_powr__int,axiom,
! [X: real,I: int] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ I )
=> ( ( powr @ real @ X @ ( ring_1_of_int @ real @ I ) )
= ( power_power @ real @ X @ ( nat2 @ I ) ) ) )
& ( ~ ( ord_less_eq @ int @ ( zero_zero @ int ) @ I )
=> ( ( powr @ real @ X @ ( ring_1_of_int @ real @ I ) )
= ( divide_divide @ real @ ( one_one @ real ) @ ( power_power @ real @ X @ ( nat2 @ ( uminus_uminus @ int @ I ) ) ) ) ) ) ) ) ).
% powr_int
thf(fact_1264_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 @ ( log2 @ B2 @ X ) @ Y2 )
= ( log2 @ B2 @ ( times_times @ real @ X @ ( powr @ real @ B2 @ ( uminus_uminus @ real @ Y2 ) ) ) ) ) ) ) ) ).
% log_minus_eq_powr
thf(fact_1265_log__base__root,axiom,
! [N: nat,B2: real,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ B2 )
=> ( ( log2 @ ( root @ N @ B2 ) @ X )
= ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ ( log2 @ B2 @ X ) ) ) ) ) ).
% log_base_root
thf(fact_1266_tanh__altdef,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ( ( tanh @ A )
= ( ^ [X2: A] : ( divide_divide @ A @ ( minus_minus @ A @ ( exp @ A @ X2 ) @ ( exp @ A @ ( uminus_uminus @ A @ X2 ) ) ) @ ( plus_plus @ A @ ( exp @ A @ X2 ) @ ( exp @ A @ ( uminus_uminus @ A @ X2 ) ) ) ) ) ) ) ).
% tanh_altdef
thf(fact_1267_triangle__Suc,axiom,
! [N: nat] :
( ( nat_triangle @ ( suc @ N ) )
= ( plus_plus @ nat @ ( nat_triangle @ N ) @ ( suc @ N ) ) ) ).
% triangle_Suc
thf(fact_1268_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_1269_add__0__iff,axiom,
! [A: $tType] :
( ( semiri1453513574482234551roduct @ A )
=> ! [B2: A,A3: A] :
( ( B2
= ( plus_plus @ A @ B2 @ A3 ) )
= ( A3
= ( zero_zero @ A ) ) ) ) ).
% add_0_iff
thf(fact_1270_split__neg__lemma,axiom,
! [K: int,P: int > int > $o,N: int] :
( ( ord_less @ int @ K @ ( zero_zero @ int ) )
=> ( ( P @ ( divide_divide @ int @ N @ K ) @ ( modulo_modulo @ int @ N @ K ) )
= ( ! [I2: int,J3: int] :
( ( ( ord_less @ int @ K @ J3 )
& ( ord_less_eq @ int @ J3 @ ( zero_zero @ int ) )
& ( N
= ( plus_plus @ int @ ( times_times @ int @ K @ I2 ) @ J3 ) ) )
=> ( P @ I2 @ J3 ) ) ) ) ) ).
% split_neg_lemma
thf(fact_1271_tanh__real__zero__iff,axiom,
! [X: real] :
( ( ( tanh @ real @ X )
= ( zero_zero @ real ) )
= ( X
= ( zero_zero @ real ) ) ) ).
% tanh_real_zero_iff
thf(fact_1272_tanh__real__less__iff,axiom,
! [X: real,Y2: real] :
( ( ord_less @ real @ ( tanh @ real @ X ) @ ( tanh @ real @ Y2 ) )
= ( ord_less @ real @ X @ Y2 ) ) ).
% tanh_real_less_iff
thf(fact_1273_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_1274_of__int__eq__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [W: int,Z2: int] :
( ( ( ring_1_of_int @ A @ W )
= ( ring_1_of_int @ A @ Z2 ) )
= ( W = Z2 ) ) ) ).
% of_int_eq_iff
thf(fact_1275_tanh__real__abs,axiom,
! [X: real] :
( ( tanh @ real @ ( abs_abs @ real @ X ) )
= ( abs_abs @ real @ ( tanh @ real @ X ) ) ) ).
% tanh_real_abs
thf(fact_1276_mult__cancel__right,axiom,
! [A: $tType] :
( ( semiri6575147826004484403cancel @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( ( times_times @ A @ A3 @ C2 )
= ( times_times @ A @ B2 @ C2 ) )
= ( ( C2
= ( zero_zero @ A ) )
| ( A3 = B2 ) ) ) ) ).
% mult_cancel_right
thf(fact_1277_mult__cancel__left,axiom,
! [A: $tType] :
( ( semiri6575147826004484403cancel @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( ( times_times @ A @ C2 @ A3 )
= ( times_times @ A @ C2 @ B2 ) )
= ( ( C2
= ( zero_zero @ A ) )
| ( A3 = B2 ) ) ) ) ).
% mult_cancel_left
thf(fact_1278_mult__eq__0__iff,axiom,
! [A: $tType] :
( ( semiri3467727345109120633visors @ A )
=> ! [A3: A,B2: A] :
( ( ( times_times @ A @ A3 @ B2 )
= ( zero_zero @ A ) )
= ( ( A3
= ( zero_zero @ A ) )
| ( B2
= ( zero_zero @ A ) ) ) ) ) ).
% mult_eq_0_iff
thf(fact_1279_mult__zero__right,axiom,
! [A: $tType] :
( ( mult_zero @ A )
=> ! [A3: A] :
( ( times_times @ A @ A3 @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% mult_zero_right
thf(fact_1280_mult__zero__left,axiom,
! [A: $tType] :
( ( mult_zero @ A )
=> ! [A3: A] :
( ( times_times @ A @ ( zero_zero @ A ) @ A3 )
= ( zero_zero @ A ) ) ) ).
% mult_zero_left
thf(fact_1281_mult_Oright__neutral,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A3: A] :
( ( times_times @ A @ A3 @ ( one_one @ A ) )
= A3 ) ) ).
% mult.right_neutral
thf(fact_1282_mult__1,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A3: A] :
( ( times_times @ A @ ( one_one @ A ) @ A3 )
= A3 ) ) ).
% mult_1
thf(fact_1283_times__divide__eq__right,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( times_times @ A @ A3 @ ( divide_divide @ A @ B2 @ C2 ) )
= ( divide_divide @ A @ ( times_times @ A @ A3 @ B2 ) @ C2 ) ) ) ).
% times_divide_eq_right
thf(fact_1284_divide__divide__eq__right,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( divide_divide @ A @ A3 @ ( divide_divide @ A @ B2 @ C2 ) )
= ( divide_divide @ A @ ( times_times @ A @ A3 @ C2 ) @ B2 ) ) ) ).
% divide_divide_eq_right
thf(fact_1285_divide__divide__eq__left,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( divide_divide @ A @ ( divide_divide @ A @ A3 @ B2 ) @ C2 )
= ( divide_divide @ A @ A3 @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% divide_divide_eq_left
thf(fact_1286_times__divide__eq__left,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [B2: A,C2: A,A3: A] :
( ( times_times @ A @ ( divide_divide @ A @ B2 @ C2 ) @ A3 )
= ( divide_divide @ A @ ( times_times @ A @ B2 @ A3 ) @ C2 ) ) ) ).
% times_divide_eq_left
thf(fact_1287_mult__minus__left,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A3: A,B2: A] :
( ( times_times @ A @ ( uminus_uminus @ A @ A3 ) @ B2 )
= ( uminus_uminus @ A @ ( times_times @ A @ A3 @ B2 ) ) ) ) ).
% mult_minus_left
thf(fact_1288_minus__mult__minus,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A3: A,B2: A] :
( ( times_times @ A @ ( uminus_uminus @ A @ A3 ) @ ( uminus_uminus @ A @ B2 ) )
= ( times_times @ A @ A3 @ B2 ) ) ) ).
% minus_mult_minus
thf(fact_1289_mult__minus__right,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A3: A,B2: A] :
( ( times_times @ A @ A3 @ ( uminus_uminus @ A @ B2 ) )
= ( uminus_uminus @ A @ ( times_times @ A @ A3 @ B2 ) ) ) ) ).
% mult_minus_right
thf(fact_1290_of__nat__mult,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [M2: nat,N: nat] :
( ( semiring_1_of_nat @ A @ ( times_times @ nat @ M2 @ N ) )
= ( times_times @ A @ ( semiring_1_of_nat @ A @ M2 ) @ ( semiring_1_of_nat @ A @ N ) ) ) ) ).
% of_nat_mult
thf(fact_1291_abs__mult__self__eq,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A] :
( ( times_times @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ A3 ) )
= ( times_times @ A @ A3 @ A3 ) ) ) ).
% abs_mult_self_eq
thf(fact_1292_tanh__real__neg__iff,axiom,
! [X: real] :
( ( ord_less @ real @ ( tanh @ real @ X ) @ ( zero_zero @ real ) )
= ( ord_less @ real @ X @ ( zero_zero @ real ) ) ) ).
% tanh_real_neg_iff
thf(fact_1293_tanh__real__pos__iff,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( tanh @ real @ X ) )
= ( ord_less @ real @ ( zero_zero @ real ) @ X ) ) ).
% tanh_real_pos_iff
thf(fact_1294_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_1295_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_1296_real__divide__square__eq,axiom,
! [R4: real,A3: real] :
( ( divide_divide @ real @ ( times_times @ real @ R4 @ A3 ) @ ( times_times @ real @ R4 @ R4 ) )
= ( divide_divide @ real @ A3 @ R4 ) ) ).
% real_divide_square_eq
thf(fact_1297_tanh__0,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ( ( tanh @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% tanh_0
thf(fact_1298_of__int__ceiling__cancel,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ X ) )
= X )
= ( ? [N5: int] :
( X
= ( ring_1_of_int @ A @ N5 ) ) ) ) ) ).
% of_int_ceiling_cancel
thf(fact_1299_ceiling__of__int,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z2: int] :
( ( archimedean_ceiling @ A @ ( ring_1_of_int @ A @ Z2 ) )
= Z2 ) ) ).
% ceiling_of_int
thf(fact_1300_tanh__minus,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A] :
( ( tanh @ A @ ( uminus_uminus @ A @ X ) )
= ( uminus_uminus @ A @ ( tanh @ A @ X ) ) ) ) ).
% tanh_minus
thf(fact_1301_triangle__0,axiom,
( ( nat_triangle @ ( zero_zero @ nat ) )
= ( zero_zero @ nat ) ) ).
% triangle_0
thf(fact_1302_mult__cancel__right2,axiom,
! [A: $tType] :
( ( ring_15535105094025558882visors @ A )
=> ! [A3: A,C2: A] :
( ( ( times_times @ A @ A3 @ C2 )
= C2 )
= ( ( C2
= ( zero_zero @ A ) )
| ( A3
= ( one_one @ A ) ) ) ) ) ).
% mult_cancel_right2
thf(fact_1303_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_1304_mult__cancel__left2,axiom,
! [A: $tType] :
( ( ring_15535105094025558882visors @ A )
=> ! [C2: A,A3: A] :
( ( ( times_times @ A @ C2 @ A3 )
= C2 )
= ( ( C2
= ( zero_zero @ A ) )
| ( A3
= ( one_one @ A ) ) ) ) ) ).
% mult_cancel_left2
thf(fact_1305_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_1306_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_1307_mult__divide__mult__cancel__left__if,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( ( C2
= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) )
= ( zero_zero @ A ) ) )
& ( ( C2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) )
= ( divide_divide @ A @ A3 @ B2 ) ) ) ) ) ).
% mult_divide_mult_cancel_left_if
thf(fact_1308_nonzero__mult__divide__mult__cancel__left,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) )
= ( divide_divide @ A @ A3 @ B2 ) ) ) ) ).
% nonzero_mult_divide_mult_cancel_left
thf(fact_1309_nonzero__mult__divide__mult__cancel__left2,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ B2 @ C2 ) )
= ( divide_divide @ A @ A3 @ B2 ) ) ) ) ).
% nonzero_mult_divide_mult_cancel_left2
thf(fact_1310_nonzero__mult__divide__mult__cancel__right,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) )
= ( divide_divide @ A @ A3 @ B2 ) ) ) ) ).
% nonzero_mult_divide_mult_cancel_right
thf(fact_1311_nonzero__mult__divide__mult__cancel__right2,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ C2 @ B2 ) )
= ( divide_divide @ A @ A3 @ B2 ) ) ) ) ).
% nonzero_mult_divide_mult_cancel_right2
thf(fact_1312_div__mult__mult1,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) )
= ( divide_divide @ A @ A3 @ B2 ) ) ) ) ).
% div_mult_mult1
thf(fact_1313_div__mult__mult2,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) )
= ( divide_divide @ A @ A3 @ B2 ) ) ) ) ).
% div_mult_mult2
thf(fact_1314_div__mult__mult1__if,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( ( C2
= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) )
= ( zero_zero @ A ) ) )
& ( ( C2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) )
= ( divide_divide @ A @ A3 @ B2 ) ) ) ) ) ).
% div_mult_mult1_if
thf(fact_1315_nonzero__mult__div__cancel__right,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [B2: A,A3: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ A3 @ B2 ) @ B2 )
= A3 ) ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_1316_nonzero__mult__div__cancel__left,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [A3: A,B2: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ A3 @ B2 ) @ A3 )
= B2 ) ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_1317_mult__minus1__right,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [Z2: A] :
( ( times_times @ A @ Z2 @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( uminus_uminus @ A @ Z2 ) ) ) ).
% mult_minus1_right
thf(fact_1318_mult__minus1,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [Z2: A] :
( ( times_times @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ Z2 )
= ( uminus_uminus @ A @ Z2 ) ) ) ).
% mult_minus1
thf(fact_1319_mod__mult__self1__is__0,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [B2: A,A3: A] :
( ( modulo_modulo @ A @ ( times_times @ A @ B2 @ A3 ) @ B2 )
= ( zero_zero @ A ) ) ) ).
% mod_mult_self1_is_0
thf(fact_1320_mod__mult__self2__is__0,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A3: A,B2: A] :
( ( modulo_modulo @ A @ ( times_times @ A @ A3 @ B2 ) @ B2 )
= ( zero_zero @ A ) ) ) ).
% mod_mult_self2_is_0
thf(fact_1321_of__int__eq__0__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [Z2: int] :
( ( ( ring_1_of_int @ A @ Z2 )
= ( zero_zero @ A ) )
= ( Z2
= ( zero_zero @ int ) ) ) ) ).
% of_int_eq_0_iff
thf(fact_1322_of__int__0__eq__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [Z2: int] :
( ( ( zero_zero @ A )
= ( ring_1_of_int @ A @ Z2 ) )
= ( Z2
= ( zero_zero @ int ) ) ) ) ).
% of_int_0_eq_iff
thf(fact_1323_of__int__0,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( ( ring_1_of_int @ A @ ( zero_zero @ int ) )
= ( zero_zero @ A ) ) ) ).
% of_int_0
thf(fact_1324_of__int__le__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [W: int,Z2: int] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ W ) @ ( ring_1_of_int @ A @ Z2 ) )
= ( ord_less_eq @ int @ W @ Z2 ) ) ) ).
% of_int_le_iff
thf(fact_1325_of__int__less__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [W: int,Z2: int] :
( ( ord_less @ A @ ( ring_1_of_int @ A @ W ) @ ( ring_1_of_int @ A @ Z2 ) )
= ( ord_less @ int @ W @ Z2 ) ) ) ).
% of_int_less_iff
thf(fact_1326_of__int__eq__1__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [Z2: int] :
( ( ( ring_1_of_int @ A @ Z2 )
= ( one_one @ A ) )
= ( Z2
= ( one_one @ int ) ) ) ) ).
% of_int_eq_1_iff
thf(fact_1327_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_1328_of__int__mult,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [W: int,Z2: int] :
( ( ring_1_of_int @ A @ ( times_times @ int @ W @ Z2 ) )
= ( times_times @ A @ ( ring_1_of_int @ A @ W ) @ ( ring_1_of_int @ A @ Z2 ) ) ) ) ).
% of_int_mult
thf(fact_1329_of__int__add,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [W: int,Z2: int] :
( ( ring_1_of_int @ A @ ( plus_plus @ int @ W @ Z2 ) )
= ( plus_plus @ A @ ( ring_1_of_int @ A @ W ) @ ( ring_1_of_int @ A @ Z2 ) ) ) ) ).
% of_int_add
thf(fact_1330_of__int__minus,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [Z2: int] :
( ( ring_1_of_int @ A @ ( uminus_uminus @ int @ Z2 ) )
= ( uminus_uminus @ A @ ( ring_1_of_int @ A @ Z2 ) ) ) ) ).
% of_int_minus
thf(fact_1331_of__int__diff,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [W: int,Z2: int] :
( ( ring_1_of_int @ A @ ( minus_minus @ int @ W @ Z2 ) )
= ( minus_minus @ A @ ( ring_1_of_int @ A @ W ) @ ( ring_1_of_int @ A @ Z2 ) ) ) ) ).
% of_int_diff
thf(fact_1332_of__int__of__nat__eq,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [N: nat] :
( ( ring_1_of_int @ A @ ( semiring_1_of_nat @ int @ N ) )
= ( semiring_1_of_nat @ A @ N ) ) ) ).
% of_int_of_nat_eq
thf(fact_1333_of__int__abs,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: int] :
( ( ring_1_of_int @ A @ ( abs_abs @ int @ X ) )
= ( abs_abs @ A @ ( ring_1_of_int @ A @ X ) ) ) ) ).
% of_int_abs
thf(fact_1334_of__int__power,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [Z2: int,N: nat] :
( ( ring_1_of_int @ A @ ( power_power @ int @ Z2 @ N ) )
= ( power_power @ A @ ( ring_1_of_int @ A @ Z2 ) @ N ) ) ) ).
% of_int_power
thf(fact_1335_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_1336_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_1337_nonzero__divide__mult__cancel__right,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [B2: A,A3: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ B2 @ ( times_times @ A @ A3 @ B2 ) )
= ( divide_divide @ A @ ( one_one @ A ) @ A3 ) ) ) ) ).
% nonzero_divide_mult_cancel_right
thf(fact_1338_nonzero__divide__mult__cancel__left,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A3: A,B2: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ A3 @ ( times_times @ A @ A3 @ B2 ) )
= ( divide_divide @ A @ ( one_one @ A ) @ B2 ) ) ) ) ).
% nonzero_divide_mult_cancel_left
thf(fact_1339_div__mult__self1,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ A3 @ ( times_times @ A @ C2 @ B2 ) ) @ B2 )
= ( plus_plus @ A @ C2 @ ( divide_divide @ A @ A3 @ B2 ) ) ) ) ) ).
% div_mult_self1
thf(fact_1340_div__mult__self2,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ A3 @ ( times_times @ A @ B2 @ C2 ) ) @ B2 )
= ( plus_plus @ A @ C2 @ ( divide_divide @ A @ A3 @ B2 ) ) ) ) ) ).
% div_mult_self2
thf(fact_1341_div__mult__self3,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [B2: A,C2: A,A3: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ C2 @ B2 ) @ A3 ) @ B2 )
= ( plus_plus @ A @ C2 @ ( divide_divide @ A @ A3 @ B2 ) ) ) ) ) ).
% div_mult_self3
thf(fact_1342_div__mult__self4,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [B2: A,C2: A,A3: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ B2 @ C2 ) @ A3 ) @ B2 )
= ( plus_plus @ A @ C2 @ ( divide_divide @ A @ A3 @ B2 ) ) ) ) ) ).
% div_mult_self4
thf(fact_1343_left__minus__one__mult__self,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [N: nat,A3: A] :
( ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N ) @ ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N ) @ A3 ) )
= A3 ) ) ).
% left_minus_one_mult_self
thf(fact_1344_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_1345_ceiling__add__of__int,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Z2: int] :
( ( archimedean_ceiling @ A @ ( plus_plus @ A @ X @ ( ring_1_of_int @ A @ Z2 ) ) )
= ( plus_plus @ int @ ( archimedean_ceiling @ A @ X ) @ Z2 ) ) ) ).
% ceiling_add_of_int
thf(fact_1346_ceiling__diff__of__int,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Z2: int] :
( ( archimedean_ceiling @ A @ ( minus_minus @ A @ X @ ( ring_1_of_int @ A @ Z2 ) ) )
= ( minus_minus @ int @ ( archimedean_ceiling @ A @ X ) @ Z2 ) ) ) ).
% ceiling_diff_of_int
thf(fact_1347_of__int__le__0__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z2: int] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Z2 ) @ ( zero_zero @ A ) )
= ( ord_less_eq @ int @ Z2 @ ( zero_zero @ int ) ) ) ) ).
% of_int_le_0_iff
thf(fact_1348_of__int__0__le__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z2: int] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( ring_1_of_int @ A @ Z2 ) )
= ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z2 ) ) ) ).
% of_int_0_le_iff
thf(fact_1349_of__int__less__0__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z2: int] :
( ( ord_less @ A @ ( ring_1_of_int @ A @ Z2 ) @ ( zero_zero @ A ) )
= ( ord_less @ int @ Z2 @ ( zero_zero @ int ) ) ) ) ).
% of_int_less_0_iff
thf(fact_1350_of__int__0__less__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z2: int] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( ring_1_of_int @ A @ Z2 ) )
= ( ord_less @ int @ ( zero_zero @ int ) @ Z2 ) ) ) ).
% of_int_0_less_iff
thf(fact_1351_of__int__le__1__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z2: int] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Z2 ) @ ( one_one @ A ) )
= ( ord_less_eq @ int @ Z2 @ ( one_one @ int ) ) ) ) ).
% of_int_le_1_iff
thf(fact_1352_of__int__1__le__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z2: int] :
( ( ord_less_eq @ A @ ( one_one @ A ) @ ( ring_1_of_int @ A @ Z2 ) )
= ( ord_less_eq @ int @ ( one_one @ int ) @ Z2 ) ) ) ).
% of_int_1_le_iff
thf(fact_1353_of__int__less__1__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z2: int] :
( ( ord_less @ A @ ( ring_1_of_int @ A @ Z2 ) @ ( one_one @ A ) )
= ( ord_less @ int @ Z2 @ ( one_one @ int ) ) ) ) ).
% of_int_less_1_iff
thf(fact_1354_of__int__1__less__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z2: int] :
( ( ord_less @ A @ ( one_one @ A ) @ ( ring_1_of_int @ A @ Z2 ) )
= ( ord_less @ int @ ( one_one @ int ) @ Z2 ) ) ) ).
% of_int_1_less_iff
thf(fact_1355_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_1356_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_1357_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_1358_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_1359_of__nat__nat,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [Z2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z2 )
=> ( ( semiring_1_of_nat @ A @ ( nat2 @ Z2 ) )
= ( ring_1_of_int @ A @ Z2 ) ) ) ) ).
% of_nat_nat
thf(fact_1360_neq__if__length__neq,axiom,
! [A: $tType,Xs: list @ A,Ys2: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs )
!= ( size_size @ ( list @ A ) @ Ys2 ) )
=> ( Xs != Ys2 ) ) ).
% neq_if_length_neq
thf(fact_1361_Ex__list__of__length,axiom,
! [A: $tType,N: nat] :
? [Xs2: list @ A] :
( ( size_size @ ( list @ A ) @ Xs2 )
= N ) ).
% Ex_list_of_length
thf(fact_1362_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: $tType] :
( ( ab_semigroup_mult @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( times_times @ A @ ( times_times @ A @ A3 @ B2 ) @ C2 )
= ( times_times @ A @ A3 @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_1363_mult_Oassoc,axiom,
! [A: $tType] :
( ( semigroup_mult @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( times_times @ A @ ( times_times @ A @ A3 @ B2 ) @ C2 )
= ( times_times @ A @ A3 @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% mult.assoc
thf(fact_1364_mult_Ocommute,axiom,
! [A: $tType] :
( ( ab_semigroup_mult @ A )
=> ( ( times_times @ A )
= ( ^ [A5: A,B3: A] : ( times_times @ A @ B3 @ A5 ) ) ) ) ).
% mult.commute
thf(fact_1365_mult_Oleft__commute,axiom,
! [A: $tType] :
( ( ab_semigroup_mult @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( times_times @ A @ B2 @ ( times_times @ A @ A3 @ C2 ) )
= ( times_times @ A @ A3 @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% mult.left_commute
thf(fact_1366_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_1367_ex__le__of__int,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X: A] :
? [Z4: int] : ( ord_less_eq @ A @ X @ ( ring_1_of_int @ A @ Z4 ) ) ) ).
% ex_le_of_int
thf(fact_1368_ex__of__int__less,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X: A] :
? [Z4: int] : ( ord_less @ A @ ( ring_1_of_int @ A @ Z4 ) @ X ) ) ).
% ex_of_int_less
thf(fact_1369_ex__less__of__int,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X: A] :
? [Z4: int] : ( ord_less @ A @ X @ ( ring_1_of_int @ A @ Z4 ) ) ) ).
% ex_less_of_int
thf(fact_1370_mult__right__cancel,axiom,
! [A: $tType] :
( ( semiri6575147826004484403cancel @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( ( times_times @ A @ A3 @ C2 )
= ( times_times @ A @ B2 @ C2 ) )
= ( A3 = B2 ) ) ) ) ).
% mult_right_cancel
thf(fact_1371_mult__left__cancel,axiom,
! [A: $tType] :
( ( semiri6575147826004484403cancel @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( ( times_times @ A @ C2 @ A3 )
= ( times_times @ A @ C2 @ B2 ) )
= ( A3 = B2 ) ) ) ) ).
% mult_left_cancel
thf(fact_1372_no__zero__divisors,axiom,
! [A: $tType] :
( ( semiri3467727345109120633visors @ A )
=> ! [A3: A,B2: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( B2
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ A3 @ B2 )
!= ( zero_zero @ A ) ) ) ) ) ).
% no_zero_divisors
thf(fact_1373_divisors__zero,axiom,
! [A: $tType] :
( ( semiri3467727345109120633visors @ A )
=> ! [A3: A,B2: A] :
( ( ( times_times @ A @ A3 @ B2 )
= ( zero_zero @ A ) )
=> ( ( A3
= ( zero_zero @ A ) )
| ( B2
= ( zero_zero @ A ) ) ) ) ) ).
% divisors_zero
thf(fact_1374_mult__not__zero,axiom,
! [A: $tType] :
( ( mult_zero @ A )
=> ! [A3: A,B2: A] :
( ( ( times_times @ A @ A3 @ B2 )
!= ( zero_zero @ A ) )
=> ( ( A3
!= ( zero_zero @ A ) )
& ( B2
!= ( zero_zero @ A ) ) ) ) ) ).
% mult_not_zero
thf(fact_1375_comm__monoid__mult__class_Omult__1,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: A] :
( ( times_times @ A @ ( one_one @ A ) @ A3 )
= A3 ) ) ).
% comm_monoid_mult_class.mult_1
thf(fact_1376_mult_Ocomm__neutral,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: A] :
( ( times_times @ A @ A3 @ ( one_one @ A ) )
= A3 ) ) ).
% mult.comm_neutral
thf(fact_1377_ring__class_Oring__distribs_I2_J,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( times_times @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C2 )
= ( plus_plus @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% ring_class.ring_distribs(2)
thf(fact_1378_ring__class_Oring__distribs_I1_J,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( times_times @ A @ A3 @ ( plus_plus @ A @ B2 @ C2 ) )
= ( plus_plus @ A @ ( times_times @ A @ A3 @ B2 ) @ ( times_times @ A @ A3 @ C2 ) ) ) ) ).
% ring_class.ring_distribs(1)
thf(fact_1379_comm__semiring__class_Odistrib,axiom,
! [A: $tType] :
( ( comm_semiring @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( times_times @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C2 )
= ( plus_plus @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% comm_semiring_class.distrib
thf(fact_1380_distrib__left,axiom,
! [A: $tType] :
( ( semiring @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( times_times @ A @ A3 @ ( plus_plus @ A @ B2 @ C2 ) )
= ( plus_plus @ A @ ( times_times @ A @ A3 @ B2 ) @ ( times_times @ A @ A3 @ C2 ) ) ) ) ).
% distrib_left
thf(fact_1381_distrib__right,axiom,
! [A: $tType] :
( ( semiring @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( times_times @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C2 )
= ( plus_plus @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% distrib_right
thf(fact_1382_combine__common__factor,axiom,
! [A: $tType] :
( ( semiring @ A )
=> ! [A3: A,E2: A,B2: A,C2: A] :
( ( plus_plus @ A @ ( times_times @ A @ A3 @ E2 ) @ ( plus_plus @ A @ ( times_times @ A @ B2 @ E2 ) @ C2 ) )
= ( plus_plus @ A @ ( times_times @ A @ ( plus_plus @ A @ A3 @ B2 ) @ E2 ) @ C2 ) ) ) ).
% combine_common_factor
thf(fact_1383_inf__period_I1_J,axiom,
! [A: $tType] :
( ( ( comm_ring @ A )
& ( dvd @ A ) )
=> ! [P: A > $o,D6: A,Q: A > $o] :
( ! [X3: A,K2: A] :
( ( P @ X3 )
= ( P @ ( minus_minus @ A @ X3 @ ( times_times @ A @ K2 @ D6 ) ) ) )
=> ( ! [X3: A,K2: A] :
( ( Q @ X3 )
= ( Q @ ( minus_minus @ A @ X3 @ ( times_times @ A @ K2 @ D6 ) ) ) )
=> ! [X4: A,K4: A] :
( ( ( P @ X4 )
& ( Q @ X4 ) )
= ( ( P @ ( minus_minus @ A @ X4 @ ( times_times @ A @ K4 @ D6 ) ) )
& ( Q @ ( minus_minus @ A @ X4 @ ( times_times @ A @ K4 @ D6 ) ) ) ) ) ) ) ) ).
% inf_period(1)
thf(fact_1384_inf__period_I2_J,axiom,
! [A: $tType] :
( ( ( comm_ring @ A )
& ( dvd @ A ) )
=> ! [P: A > $o,D6: A,Q: A > $o] :
( ! [X3: A,K2: A] :
( ( P @ X3 )
= ( P @ ( minus_minus @ A @ X3 @ ( times_times @ A @ K2 @ D6 ) ) ) )
=> ( ! [X3: A,K2: A] :
( ( Q @ X3 )
= ( Q @ ( minus_minus @ A @ X3 @ ( times_times @ A @ K2 @ D6 ) ) ) )
=> ! [X4: A,K4: A] :
( ( ( P @ X4 )
| ( Q @ X4 ) )
= ( ( P @ ( minus_minus @ A @ X4 @ ( times_times @ A @ K4 @ D6 ) ) )
| ( Q @ ( minus_minus @ A @ X4 @ ( times_times @ A @ K4 @ D6 ) ) ) ) ) ) ) ) ).
% inf_period(2)
thf(fact_1385_right__diff__distrib_H,axiom,
! [A: $tType] :
( ( comm_s4317794764714335236cancel @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( times_times @ A @ A3 @ ( minus_minus @ A @ B2 @ C2 ) )
= ( minus_minus @ A @ ( times_times @ A @ A3 @ B2 ) @ ( times_times @ A @ A3 @ C2 ) ) ) ) ).
% right_diff_distrib'
thf(fact_1386_left__diff__distrib_H,axiom,
! [A: $tType] :
( ( comm_s4317794764714335236cancel @ A )
=> ! [B2: A,C2: A,A3: A] :
( ( times_times @ A @ ( minus_minus @ A @ B2 @ C2 ) @ A3 )
= ( minus_minus @ A @ ( times_times @ A @ B2 @ A3 ) @ ( times_times @ A @ C2 @ A3 ) ) ) ) ).
% left_diff_distrib'
thf(fact_1387_right__diff__distrib,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( times_times @ A @ A3 @ ( minus_minus @ A @ B2 @ C2 ) )
= ( minus_minus @ A @ ( times_times @ A @ A3 @ B2 ) @ ( times_times @ A @ A3 @ C2 ) ) ) ) ).
% right_diff_distrib
thf(fact_1388_left__diff__distrib,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( times_times @ A @ ( minus_minus @ A @ A3 @ B2 ) @ C2 )
= ( minus_minus @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% left_diff_distrib
thf(fact_1389_divide__divide__eq__left_H,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( divide_divide @ A @ ( divide_divide @ A @ A3 @ B2 ) @ C2 )
= ( divide_divide @ A @ A3 @ ( times_times @ A @ C2 @ B2 ) ) ) ) ).
% divide_divide_eq_left'
thf(fact_1390_divide__divide__times__eq,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [X: A,Y2: A,Z2: A,W: A] :
( ( divide_divide @ A @ ( divide_divide @ A @ X @ Y2 ) @ ( divide_divide @ A @ Z2 @ W ) )
= ( divide_divide @ A @ ( times_times @ A @ X @ W ) @ ( times_times @ A @ Y2 @ Z2 ) ) ) ) ).
% divide_divide_times_eq
thf(fact_1391_times__divide__times__eq,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [X: A,Y2: A,Z2: A,W: A] :
( ( times_times @ A @ ( divide_divide @ A @ X @ Y2 ) @ ( divide_divide @ A @ Z2 @ W ) )
= ( divide_divide @ A @ ( times_times @ A @ X @ Z2 ) @ ( times_times @ A @ Y2 @ W ) ) ) ) ).
% times_divide_times_eq
thf(fact_1392_square__eq__iff,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [A3: A,B2: A] :
( ( ( times_times @ A @ A3 @ A3 )
= ( times_times @ A @ B2 @ B2 ) )
= ( ( A3 = B2 )
| ( A3
= ( uminus_uminus @ A @ B2 ) ) ) ) ) ).
% square_eq_iff
thf(fact_1393_minus__mult__commute,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A3: A,B2: A] :
( ( times_times @ A @ ( uminus_uminus @ A @ A3 ) @ B2 )
= ( times_times @ A @ A3 @ ( uminus_uminus @ A @ B2 ) ) ) ) ).
% minus_mult_commute
thf(fact_1394_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_1395_times__int__code_I1_J,axiom,
! [K: int] :
( ( times_times @ int @ K @ ( zero_zero @ int ) )
= ( zero_zero @ int ) ) ).
% times_int_code(1)
thf(fact_1396_times__int__code_I2_J,axiom,
! [L: int] :
( ( times_times @ int @ ( zero_zero @ int ) @ L )
= ( zero_zero @ int ) ) ).
% times_int_code(2)
thf(fact_1397_abs__mult,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ! [A3: A,B2: A] :
( ( abs_abs @ A @ ( times_times @ A @ A3 @ B2 ) )
= ( times_times @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ B2 ) ) ) ) ).
% abs_mult
thf(fact_1398_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_1399_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_1400_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_1401_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_1402_exp__times__arg__commute,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [A4: A] :
( ( times_times @ A @ ( exp @ A @ A4 ) @ A4 )
= ( times_times @ A @ A4 @ ( exp @ A @ A4 ) ) ) ) ).
% exp_times_arg_commute
thf(fact_1403_powr__powr,axiom,
! [X: real,A3: real,B2: real] :
( ( powr @ real @ ( powr @ real @ X @ A3 ) @ B2 )
= ( powr @ real @ X @ ( times_times @ real @ A3 @ B2 ) ) ) ).
% powr_powr
thf(fact_1404_add__scale__eq__noteq,axiom,
! [A: $tType] :
( ( semiri1453513574482234551roduct @ A )
=> ! [R4: A,A3: A,B2: A,C2: A,D2: A] :
( ( R4
!= ( zero_zero @ A ) )
=> ( ( ( A3 = B2 )
& ( C2 != D2 ) )
=> ( ( plus_plus @ A @ A3 @ ( times_times @ A @ R4 @ C2 ) )
!= ( plus_plus @ A @ B2 @ ( times_times @ A @ R4 @ D2 ) ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_1405_artanh__tanh__real,axiom,
! [X: real] :
( ( artanh @ real @ ( tanh @ real @ X ) )
= X ) ).
% artanh_tanh_real
thf(fact_1406_mult__ceiling__le,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ ( times_times @ A @ A3 @ B2 ) ) @ ( times_times @ int @ ( archimedean_ceiling @ A @ A3 ) @ ( archimedean_ceiling @ A @ B2 ) ) ) ) ) ) ).
% mult_ceiling_le
thf(fact_1407_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_1408_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: $tType] :
( ( ordere2520102378445227354miring @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_1409_zero__le__mult__iff,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B2 ) )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) )
| ( ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) )
& ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) ) ) ) ) ).
% zero_le_mult_iff
thf(fact_1410_mult__nonneg__nonpos2,axiom,
! [A: $tType] :
( ( ordered_semiring_0 @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ B2 @ A3 ) @ ( zero_zero @ A ) ) ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_1411_mult__nonpos__nonneg,axiom,
! [A: $tType] :
( ( ordered_semiring_0 @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ A3 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% mult_nonpos_nonneg
thf(fact_1412_mult__nonneg__nonpos,axiom,
! [A: $tType] :
( ( ordered_semiring_0 @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ A3 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% mult_nonneg_nonpos
thf(fact_1413_mult__nonneg__nonneg,axiom,
! [A: $tType] :
( ( ordered_semiring_0 @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B2 ) ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_1414_split__mult__neg__le,axiom,
! [A: $tType] :
( ( ordered_semiring_0 @ A )
=> ! [A3: A,B2: A] :
( ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
& ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) )
| ( ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ A3 @ B2 ) @ ( zero_zero @ A ) ) ) ) ).
% split_mult_neg_le
thf(fact_1415_mult__le__0__iff,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( times_times @ A @ A3 @ B2 ) @ ( zero_zero @ A ) )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
& ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) )
| ( ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) ) ) ) ) ).
% mult_le_0_iff
thf(fact_1416_mult__right__mono,axiom,
! [A: $tType] :
( ( ordered_semiring @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ) ).
% mult_right_mono
thf(fact_1417_mult__right__mono__neg,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( ord_less_eq @ A @ B2 @ A3 )
=> ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ) ).
% mult_right_mono_neg
thf(fact_1418_mult__left__mono,axiom,
! [A: $tType] :
( ( ordered_semiring @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) ) ) ) ) ).
% mult_left_mono
thf(fact_1419_mult__nonpos__nonpos,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B2 ) ) ) ) ) ).
% mult_nonpos_nonpos
thf(fact_1420_mult__left__mono__neg,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( ord_less_eq @ A @ B2 @ A3 )
=> ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) ) ) ) ) ).
% mult_left_mono_neg
thf(fact_1421_split__mult__pos__le,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [A3: A,B2: A] :
( ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) )
| ( ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) )
& ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B2 ) ) ) ) ).
% split_mult_pos_le
thf(fact_1422_zero__le__square,axiom,
! [A: $tType] :
( ( linordered_ring @ A )
=> ! [A3: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ A3 ) ) ) ).
% zero_le_square
thf(fact_1423_mult__mono_H,axiom,
! [A: $tType] :
( ( ordered_semiring @ A )
=> ! [A3: A,B2: A,C2: A,D2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ D2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ D2 ) ) ) ) ) ) ) ).
% mult_mono'
thf(fact_1424_mult__mono,axiom,
! [A: $tType] :
( ( ordered_semiring @ A )
=> ! [A3: A,B2: A,C2: A,D2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ D2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ D2 ) ) ) ) ) ) ) ).
% mult_mono
thf(fact_1425_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: $tType] :
( ( linord2810124833399127020strict @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_1426_mult__less__cancel__right__disj,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( ord_less @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
& ( ord_less @ A @ A3 @ B2 ) )
| ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
& ( ord_less @ A @ B2 @ A3 ) ) ) ) ) ).
% mult_less_cancel_right_disj
thf(fact_1427_mult__strict__right__mono,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ) ).
% mult_strict_right_mono
thf(fact_1428_mult__strict__right__mono__neg,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( ord_less @ A @ B2 @ A3 )
=> ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ) ).
% mult_strict_right_mono_neg
thf(fact_1429_mult__less__cancel__left__disj,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( ord_less @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
& ( ord_less @ A @ A3 @ B2 ) )
| ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
& ( ord_less @ A @ B2 @ A3 ) ) ) ) ) ).
% mult_less_cancel_left_disj
thf(fact_1430_mult__strict__left__mono,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) ) ) ) ) ).
% mult_strict_left_mono
thf(fact_1431_mult__strict__left__mono__neg,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( ord_less @ A @ B2 @ A3 )
=> ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) ) ) ) ) ).
% mult_strict_left_mono_neg
thf(fact_1432_mult__less__cancel__left__pos,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) )
= ( ord_less @ A @ A3 @ B2 ) ) ) ) ).
% mult_less_cancel_left_pos
thf(fact_1433_mult__less__cancel__left__neg,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) )
= ( ord_less @ A @ B2 @ A3 ) ) ) ) ).
% mult_less_cancel_left_neg
thf(fact_1434_zero__less__mult__pos2,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [B2: A,A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ B2 @ A3 ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) ) ) ) ).
% zero_less_mult_pos2
thf(fact_1435_zero__less__mult__pos,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B2 ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) ) ) ) ).
% zero_less_mult_pos
thf(fact_1436_zero__less__mult__iff,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
& ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) )
| ( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
& ( ord_less @ A @ B2 @ ( zero_zero @ A ) ) ) ) ) ) ).
% zero_less_mult_iff
thf(fact_1437_mult__pos__neg2,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ B2 @ A3 ) @ ( zero_zero @ A ) ) ) ) ) ).
% mult_pos_neg2
thf(fact_1438_mult__pos__pos,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B2 ) ) ) ) ) ).
% mult_pos_pos
thf(fact_1439_mult__pos__neg,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ A3 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% mult_pos_neg
thf(fact_1440_mult__neg__pos,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less @ A @ ( times_times @ A @ A3 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% mult_neg_pos
thf(fact_1441_mult__less__0__iff,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ ( times_times @ A @ A3 @ B2 ) @ ( zero_zero @ A ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
& ( ord_less @ A @ B2 @ ( zero_zero @ A ) ) )
| ( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
& ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) ) ) ) ) ).
% mult_less_0_iff
thf(fact_1442_not__square__less__zero,axiom,
! [A: $tType] :
( ( linordered_ring @ A )
=> ! [A3: A] :
~ ( ord_less @ A @ ( times_times @ A @ A3 @ A3 ) @ ( zero_zero @ A ) ) ) ).
% not_square_less_zero
thf(fact_1443_mult__neg__neg,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B2 ) ) ) ) ) ).
% mult_neg_neg
thf(fact_1444_less__1__mult,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [M2: A,N: A] :
( ( ord_less @ A @ ( one_one @ A ) @ M2 )
=> ( ( ord_less @ A @ ( one_one @ A ) @ N )
=> ( ord_less @ A @ ( one_one @ A ) @ ( times_times @ A @ M2 @ N ) ) ) ) ) ).
% less_1_mult
thf(fact_1445_frac__eq__eq,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [Y2: A,Z2: A,X: A,W: A] :
( ( Y2
!= ( zero_zero @ A ) )
=> ( ( Z2
!= ( zero_zero @ A ) )
=> ( ( ( divide_divide @ A @ X @ Y2 )
= ( divide_divide @ A @ W @ Z2 ) )
= ( ( times_times @ A @ X @ Z2 )
= ( times_times @ A @ W @ Y2 ) ) ) ) ) ) ).
% frac_eq_eq
thf(fact_1446_divide__eq__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B2: A,C2: A,A3: A] :
( ( ( divide_divide @ A @ B2 @ C2 )
= A3 )
= ( ( ( C2
!= ( zero_zero @ A ) )
=> ( B2
= ( times_times @ A @ A3 @ C2 ) ) )
& ( ( C2
= ( zero_zero @ A ) )
=> ( A3
= ( zero_zero @ A ) ) ) ) ) ) ).
% divide_eq_eq
thf(fact_1447_eq__divide__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( A3
= ( divide_divide @ A @ B2 @ C2 ) )
= ( ( ( C2
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ A3 @ C2 )
= B2 ) )
& ( ( C2
= ( zero_zero @ A ) )
=> ( A3
= ( zero_zero @ A ) ) ) ) ) ) ).
% eq_divide_eq
thf(fact_1448_divide__eq__imp,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [C2: A,B2: A,A3: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( B2
= ( times_times @ A @ A3 @ C2 ) )
=> ( ( divide_divide @ A @ B2 @ C2 )
= A3 ) ) ) ) ).
% divide_eq_imp
thf(fact_1449_eq__divide__imp,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( ( times_times @ A @ A3 @ C2 )
= B2 )
=> ( A3
= ( divide_divide @ A @ B2 @ C2 ) ) ) ) ) ).
% eq_divide_imp
thf(fact_1450_nonzero__divide__eq__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [C2: A,B2: A,A3: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( ( divide_divide @ A @ B2 @ C2 )
= A3 )
= ( B2
= ( times_times @ A @ A3 @ C2 ) ) ) ) ) ).
% nonzero_divide_eq_eq
thf(fact_1451_nonzero__eq__divide__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( A3
= ( divide_divide @ A @ B2 @ C2 ) )
= ( ( times_times @ A @ A3 @ C2 )
= B2 ) ) ) ) ).
% nonzero_eq_divide_eq
thf(fact_1452_eq__add__iff1,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A3: A,E2: A,C2: A,B2: A,D2: A] :
( ( ( plus_plus @ A @ ( times_times @ A @ A3 @ E2 ) @ C2 )
= ( plus_plus @ A @ ( times_times @ A @ B2 @ E2 ) @ D2 ) )
= ( ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ A3 @ B2 ) @ E2 ) @ C2 )
= D2 ) ) ) ).
% eq_add_iff1
thf(fact_1453_eq__add__iff2,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A3: A,E2: A,C2: A,B2: A,D2: A] :
( ( ( plus_plus @ A @ ( times_times @ A @ A3 @ E2 ) @ C2 )
= ( plus_plus @ A @ ( times_times @ A @ B2 @ E2 ) @ D2 ) )
= ( C2
= ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ B2 @ A3 ) @ E2 ) @ D2 ) ) ) ) ).
% eq_add_iff2
thf(fact_1454_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_1455_mult__diff__mult,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [X: A,Y2: A,A3: A,B2: A] :
( ( minus_minus @ A @ ( times_times @ A @ X @ Y2 ) @ ( times_times @ A @ A3 @ B2 ) )
= ( plus_plus @ A @ ( times_times @ A @ X @ ( minus_minus @ A @ Y2 @ B2 ) ) @ ( times_times @ A @ ( minus_minus @ A @ X @ A3 ) @ B2 ) ) ) ) ).
% mult_diff_mult
thf(fact_1456_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_1457_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_1458_power__Suc,axiom,
! [A: $tType] :
( ( power @ A )
=> ! [A3: A,N: nat] :
( ( power_power @ A @ A3 @ ( suc @ N ) )
= ( times_times @ A @ A3 @ ( power_power @ A @ A3 @ N ) ) ) ) ).
% power_Suc
thf(fact_1459_power__Suc2,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A3: A,N: nat] :
( ( power_power @ A @ A3 @ ( suc @ N ) )
= ( times_times @ A @ ( power_power @ A @ A3 @ N ) @ A3 ) ) ) ).
% power_Suc2
thf(fact_1460_abs__mult__less,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,C2: A,B2: A,D2: A] :
( ( ord_less @ A @ ( abs_abs @ A @ A3 ) @ C2 )
=> ( ( ord_less @ A @ ( abs_abs @ A @ B2 ) @ D2 )
=> ( ord_less @ A @ ( times_times @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ B2 ) ) @ ( times_times @ A @ C2 @ D2 ) ) ) ) ) ).
% abs_mult_less
thf(fact_1461_div__mult2__eq_H,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [A3: A,M2: nat,N: nat] :
( ( divide_divide @ A @ A3 @ ( times_times @ A @ ( semiring_1_of_nat @ A @ M2 ) @ ( semiring_1_of_nat @ A @ N ) ) )
= ( divide_divide @ A @ ( divide_divide @ A @ A3 @ ( semiring_1_of_nat @ A @ M2 ) ) @ ( semiring_1_of_nat @ A @ N ) ) ) ) ).
% div_mult2_eq'
thf(fact_1462_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_1463_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_1464_zmult__zless__mono2,axiom,
! [I: int,J2: int,K: int] :
( ( ord_less @ int @ I @ J2 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ K )
=> ( ord_less @ int @ ( times_times @ int @ K @ I ) @ ( times_times @ int @ K @ J2 ) ) ) ) ).
% zmult_zless_mono2
thf(fact_1465_powr__add,axiom,
! [A: $tType] :
( ( ( real_V3459762299906320749_field @ A )
& ( ln @ A ) )
=> ! [X: A,A3: A,B2: A] :
( ( powr @ A @ X @ ( plus_plus @ A @ A3 @ B2 ) )
= ( times_times @ A @ ( powr @ A @ X @ A3 ) @ ( powr @ A @ X @ B2 ) ) ) ) ).
% powr_add
thf(fact_1466_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_1467_pos__zmult__eq__1__iff__lemma,axiom,
! [M2: int,N: int] :
( ( ( times_times @ int @ M2 @ N )
= ( one_one @ int ) )
=> ( ( M2
= ( one_one @ int ) )
| ( M2
= ( uminus_uminus @ int @ ( one_one @ int ) ) ) ) ) ).
% pos_zmult_eq_1_iff_lemma
thf(fact_1468_zmult__eq__1__iff,axiom,
! [M2: int,N: int] :
( ( ( times_times @ int @ M2 @ N )
= ( one_one @ int ) )
= ( ( ( M2
= ( one_one @ int ) )
& ( N
= ( one_one @ int ) ) )
| ( ( M2
= ( uminus_uminus @ int @ ( one_one @ int ) ) )
& ( N
= ( uminus_uminus @ int @ ( one_one @ int ) ) ) ) ) ) ).
% zmult_eq_1_iff
thf(fact_1469_abs__zmult__eq__1,axiom,
! [M2: int,N: int] :
( ( ( abs_abs @ int @ ( times_times @ int @ M2 @ N ) )
= ( one_one @ int ) )
=> ( ( abs_abs @ int @ M2 )
= ( one_one @ int ) ) ) ).
% abs_zmult_eq_1
thf(fact_1470_zmod__eq__0D,axiom,
! [M2: int,D2: int] :
( ( ( modulo_modulo @ int @ M2 @ D2 )
= ( zero_zero @ int ) )
=> ? [Q3: int] :
( M2
= ( times_times @ int @ D2 @ Q3 ) ) ) ).
% zmod_eq_0D
thf(fact_1471_zmod__eq__0__iff,axiom,
! [M2: int,D2: int] :
( ( ( modulo_modulo @ int @ M2 @ D2 )
= ( zero_zero @ int ) )
= ( ? [Q4: int] :
( M2
= ( times_times @ int @ D2 @ Q4 ) ) ) ) ).
% zmod_eq_0_iff
thf(fact_1472_ceiling__divide__upper,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Q5: A,P4: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Q5 )
=> ( ord_less_eq @ A @ P4 @ ( times_times @ A @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ ( divide_divide @ A @ P4 @ Q5 ) ) ) @ Q5 ) ) ) ) ).
% ceiling_divide_upper
thf(fact_1473_tanh__real__lt__1,axiom,
! [X: real] : ( ord_less @ real @ ( tanh @ real @ X ) @ ( one_one @ real ) ) ).
% tanh_real_lt_1
thf(fact_1474_ceiling__divide__lower,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Q5: A,P4: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Q5 )
=> ( ord_less @ A @ ( times_times @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ ( divide_divide @ A @ P4 @ Q5 ) ) ) @ ( one_one @ A ) ) @ Q5 ) @ P4 ) ) ) ).
% ceiling_divide_lower
thf(fact_1475_mult__less__le__imp__less,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A3: A,B2: A,C2: A,D2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ D2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ D2 ) ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_1476_mult__le__less__imp__less,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A3: A,B2: A,C2: A,D2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less @ A @ C2 @ D2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ D2 ) ) ) ) ) ) ) ).
% mult_le_less_imp_less
thf(fact_1477_mult__right__le__imp__le,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ A3 @ B2 ) ) ) ) ).
% mult_right_le_imp_le
thf(fact_1478_mult__left__le__imp__le,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ A3 @ B2 ) ) ) ) ).
% mult_left_le_imp_le
thf(fact_1479_mult__le__cancel__left__pos,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) )
= ( ord_less_eq @ A @ A3 @ B2 ) ) ) ) ).
% mult_le_cancel_left_pos
thf(fact_1480_mult__le__cancel__left__neg,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) )
= ( ord_less_eq @ A @ B2 @ A3 ) ) ) ) ).
% mult_le_cancel_left_neg
thf(fact_1481_mult__less__cancel__right,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( ord_less @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ A3 @ B2 ) )
& ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B2 @ A3 ) ) ) ) ) ).
% mult_less_cancel_right
thf(fact_1482_mult__strict__mono_H,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A3: A,B2: A,C2: A,D2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less @ A @ C2 @ D2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ D2 ) ) ) ) ) ) ) ).
% mult_strict_mono'
thf(fact_1483_mult__right__less__imp__less,axiom,
! [A: $tType] :
( ( linordered_semiring @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( ord_less @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ A3 @ B2 ) ) ) ) ).
% mult_right_less_imp_less
thf(fact_1484_mult__less__cancel__left,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( ord_less @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ A3 @ B2 ) )
& ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B2 @ A3 ) ) ) ) ) ).
% mult_less_cancel_left
thf(fact_1485_mult__strict__mono,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A3: A,B2: A,C2: A,D2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less @ A @ C2 @ D2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ D2 ) ) ) ) ) ) ) ).
% mult_strict_mono
thf(fact_1486_mult__left__less__imp__less,axiom,
! [A: $tType] :
( ( linordered_semiring @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( ord_less @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ A3 @ B2 ) ) ) ) ).
% mult_left_less_imp_less
thf(fact_1487_mult__le__cancel__right,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ A3 @ B2 ) )
& ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B2 @ A3 ) ) ) ) ) ).
% mult_le_cancel_right
thf(fact_1488_mult__le__cancel__left,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ A3 @ B2 ) )
& ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B2 @ A3 ) ) ) ) ) ).
% mult_le_cancel_left
thf(fact_1489_ceiling__le__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Z2: int] :
( ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ X ) @ Z2 )
= ( ord_less_eq @ A @ X @ ( ring_1_of_int @ A @ Z2 ) ) ) ) ).
% ceiling_le_iff
thf(fact_1490_ceiling__le,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,A3: int] :
( ( ord_less_eq @ A @ X @ ( ring_1_of_int @ A @ A3 ) )
=> ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ X ) @ A3 ) ) ) ).
% ceiling_le
thf(fact_1491_less__ceiling__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z2: int,X: A] :
( ( ord_less @ int @ Z2 @ ( archimedean_ceiling @ A @ X ) )
= ( ord_less @ A @ ( ring_1_of_int @ A @ Z2 ) @ X ) ) ) ).
% less_ceiling_iff
thf(fact_1492_mult__left__le,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [C2: A,A3: A] :
( ( ord_less_eq @ A @ C2 @ ( one_one @ A ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C2 ) @ A3 ) ) ) ) ).
% mult_left_le
thf(fact_1493_mult__le__one,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ ( one_one @ A ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ A3 @ B2 ) @ ( one_one @ A ) ) ) ) ) ) ).
% mult_le_one
thf(fact_1494_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_1495_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_1496_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_1497_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_1498_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_1499_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_1500_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( divide_divide @ A @ A3 @ ( times_times @ A @ B2 @ C2 ) )
= ( divide_divide @ A @ ( divide_divide @ A @ A3 @ B2 ) @ C2 ) ) ) ) ).
% unique_euclidean_semiring_numeral_class.div_mult2_eq
thf(fact_1501_divide__strict__left__mono__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B2 ) )
=> ( ord_less @ A @ ( divide_divide @ A @ C2 @ A3 ) @ ( divide_divide @ A @ C2 @ B2 ) ) ) ) ) ) ).
% divide_strict_left_mono_neg
thf(fact_1502_divide__strict__left__mono,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( ord_less @ A @ B2 @ A3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B2 ) )
=> ( ord_less @ A @ ( divide_divide @ A @ C2 @ A3 ) @ ( divide_divide @ A @ C2 @ B2 ) ) ) ) ) ) ).
% divide_strict_left_mono
thf(fact_1503_mult__imp__less__div__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y2: A,Z2: A,X: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Y2 )
=> ( ( ord_less @ A @ ( times_times @ A @ Z2 @ Y2 ) @ X )
=> ( ord_less @ A @ Z2 @ ( divide_divide @ A @ X @ Y2 ) ) ) ) ) ).
% mult_imp_less_div_pos
thf(fact_1504_mult__imp__div__pos__less,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y2: A,X: A,Z2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Y2 )
=> ( ( ord_less @ A @ X @ ( times_times @ A @ Z2 @ Y2 ) )
=> ( ord_less @ A @ ( divide_divide @ A @ X @ Y2 ) @ Z2 ) ) ) ) ).
% mult_imp_div_pos_less
thf(fact_1505_pos__less__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less @ A @ A3 @ ( divide_divide @ A @ B2 @ C2 ) )
= ( ord_less @ A @ ( times_times @ A @ A3 @ C2 ) @ B2 ) ) ) ) ).
% pos_less_divide_eq
thf(fact_1506_pos__divide__less__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,B2: A,A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less @ A @ ( divide_divide @ A @ B2 @ C2 ) @ A3 )
= ( ord_less @ A @ B2 @ ( times_times @ A @ A3 @ C2 ) ) ) ) ) ).
% pos_divide_less_eq
thf(fact_1507_neg__less__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ A3 @ ( divide_divide @ A @ B2 @ C2 ) )
= ( ord_less @ A @ B2 @ ( times_times @ A @ A3 @ C2 ) ) ) ) ) ).
% neg_less_divide_eq
thf(fact_1508_neg__divide__less__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,B2: A,A3: A] :
( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( divide_divide @ A @ B2 @ C2 ) @ A3 )
= ( ord_less @ A @ ( times_times @ A @ A3 @ C2 ) @ B2 ) ) ) ) ).
% neg_divide_less_eq
thf(fact_1509_less__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less @ A @ A3 @ ( divide_divide @ A @ B2 @ C2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( times_times @ A @ A3 @ C2 ) @ B2 ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B2 @ ( times_times @ A @ A3 @ C2 ) ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% less_divide_eq
thf(fact_1510_divide__less__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,C2: A,A3: A] :
( ( ord_less @ A @ ( divide_divide @ A @ B2 @ C2 ) @ A3 )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ B2 @ ( times_times @ A @ A3 @ C2 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ A3 @ C2 ) @ B2 ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ A3 ) ) ) ) ) ) ) ).
% divide_less_eq
thf(fact_1511_ordered__ring__class_Ole__add__iff1,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [A3: A,E2: A,C2: A,B2: A,D2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ ( times_times @ A @ A3 @ E2 ) @ C2 ) @ ( plus_plus @ A @ ( times_times @ A @ B2 @ E2 ) @ D2 ) )
= ( ord_less_eq @ A @ ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ A3 @ B2 ) @ E2 ) @ C2 ) @ D2 ) ) ) ).
% ordered_ring_class.le_add_iff1
thf(fact_1512_ordered__ring__class_Ole__add__iff2,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [A3: A,E2: A,C2: A,B2: A,D2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ ( times_times @ A @ A3 @ E2 ) @ C2 ) @ ( plus_plus @ A @ ( times_times @ A @ B2 @ E2 ) @ D2 ) )
= ( ord_less_eq @ A @ C2 @ ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ B2 @ A3 ) @ E2 ) @ D2 ) ) ) ) ).
% ordered_ring_class.le_add_iff2
thf(fact_1513_add__divide__eq__if__simps_I2_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z2: A,A3: A,B2: A] :
( ( ( Z2
= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( divide_divide @ A @ A3 @ Z2 ) @ B2 )
= B2 ) )
& ( ( Z2
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( divide_divide @ A @ A3 @ Z2 ) @ B2 )
= ( divide_divide @ A @ ( plus_plus @ A @ A3 @ ( times_times @ A @ B2 @ Z2 ) ) @ Z2 ) ) ) ) ) ).
% add_divide_eq_if_simps(2)
thf(fact_1514_add__divide__eq__if__simps_I1_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z2: A,A3: A,B2: A] :
( ( ( Z2
= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ A3 @ ( divide_divide @ A @ B2 @ Z2 ) )
= A3 ) )
& ( ( Z2
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ A3 @ ( divide_divide @ A @ B2 @ Z2 ) )
= ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ A3 @ Z2 ) @ B2 ) @ Z2 ) ) ) ) ) ).
% add_divide_eq_if_simps(1)
thf(fact_1515_add__frac__eq,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [Y2: A,Z2: A,X: A,W: A] :
( ( Y2
!= ( zero_zero @ A ) )
=> ( ( Z2
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( divide_divide @ A @ X @ Y2 ) @ ( divide_divide @ A @ W @ Z2 ) )
= ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ X @ Z2 ) @ ( times_times @ A @ W @ Y2 ) ) @ ( times_times @ A @ Y2 @ Z2 ) ) ) ) ) ) ).
% add_frac_eq
thf(fact_1516_add__frac__num,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [Y2: A,X: A,Z2: A] :
( ( Y2
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( divide_divide @ A @ X @ Y2 ) @ Z2 )
= ( divide_divide @ A @ ( plus_plus @ A @ X @ ( times_times @ A @ Z2 @ Y2 ) ) @ Y2 ) ) ) ) ).
% add_frac_num
thf(fact_1517_add__num__frac,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [Y2: A,Z2: A,X: A] :
( ( Y2
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ Z2 @ ( divide_divide @ A @ X @ Y2 ) )
= ( divide_divide @ A @ ( plus_plus @ A @ X @ ( times_times @ A @ Z2 @ Y2 ) ) @ Y2 ) ) ) ) ).
% add_num_frac
thf(fact_1518_add__divide__eq__iff,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z2: A,X: A,Y2: A] :
( ( Z2
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ X @ ( divide_divide @ A @ Y2 @ Z2 ) )
= ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ X @ Z2 ) @ Y2 ) @ Z2 ) ) ) ) ).
% add_divide_eq_iff
thf(fact_1519_divide__add__eq__iff,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z2: A,X: A,Y2: A] :
( ( Z2
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( divide_divide @ A @ X @ Z2 ) @ Y2 )
= ( divide_divide @ A @ ( plus_plus @ A @ X @ ( times_times @ A @ Y2 @ Z2 ) ) @ Z2 ) ) ) ) ).
% divide_add_eq_iff
thf(fact_1520_less__add__iff2,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [A3: A,E2: A,C2: A,B2: A,D2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ ( times_times @ A @ A3 @ E2 ) @ C2 ) @ ( plus_plus @ A @ ( times_times @ A @ B2 @ E2 ) @ D2 ) )
= ( ord_less @ A @ C2 @ ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ B2 @ A3 ) @ E2 ) @ D2 ) ) ) ) ).
% less_add_iff2
thf(fact_1521_less__add__iff1,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [A3: A,E2: A,C2: A,B2: A,D2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ ( times_times @ A @ A3 @ E2 ) @ C2 ) @ ( plus_plus @ A @ ( times_times @ A @ B2 @ E2 ) @ D2 ) )
= ( ord_less @ A @ ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ A3 @ B2 ) @ E2 ) @ C2 ) @ D2 ) ) ) ).
% less_add_iff1
thf(fact_1522_divide__diff__eq__iff,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z2: A,X: A,Y2: A] :
( ( Z2
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( divide_divide @ A @ X @ Z2 ) @ Y2 )
= ( divide_divide @ A @ ( minus_minus @ A @ X @ ( times_times @ A @ Y2 @ Z2 ) ) @ Z2 ) ) ) ) ).
% divide_diff_eq_iff
thf(fact_1523_diff__divide__eq__iff,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z2: A,X: A,Y2: A] :
( ( Z2
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ X @ ( divide_divide @ A @ Y2 @ Z2 ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ X @ Z2 ) @ Y2 ) @ Z2 ) ) ) ) ).
% diff_divide_eq_iff
thf(fact_1524_diff__frac__eq,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [Y2: A,Z2: A,X: A,W: A] :
( ( Y2
!= ( zero_zero @ A ) )
=> ( ( Z2
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( divide_divide @ A @ X @ Y2 ) @ ( divide_divide @ A @ W @ Z2 ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ X @ Z2 ) @ ( times_times @ A @ W @ Y2 ) ) @ ( times_times @ A @ Y2 @ Z2 ) ) ) ) ) ) ).
% diff_frac_eq
thf(fact_1525_add__divide__eq__if__simps_I4_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z2: A,A3: A,B2: A] :
( ( ( Z2
= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ A3 @ ( divide_divide @ A @ B2 @ Z2 ) )
= A3 ) )
& ( ( Z2
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ A3 @ ( divide_divide @ A @ B2 @ Z2 ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ A3 @ Z2 ) @ B2 ) @ Z2 ) ) ) ) ) ).
% add_divide_eq_if_simps(4)
thf(fact_1526_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_1527_ex__less__of__nat__mult,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X: A,Y2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X )
=> ? [N2: nat] : ( ord_less @ A @ Y2 @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N2 ) @ X ) ) ) ) ).
% ex_less_of_nat_mult
thf(fact_1528_eq__minus__divide__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( A3
= ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) )
= ( ( ( C2
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ A3 @ C2 )
= ( uminus_uminus @ A @ B2 ) ) )
& ( ( C2
= ( zero_zero @ A ) )
=> ( A3
= ( zero_zero @ A ) ) ) ) ) ) ).
% eq_minus_divide_eq
thf(fact_1529_minus__divide__eq__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B2: A,C2: A,A3: A] :
( ( ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) )
= A3 )
= ( ( ( C2
!= ( zero_zero @ A ) )
=> ( ( uminus_uminus @ A @ B2 )
= ( times_times @ A @ A3 @ C2 ) ) )
& ( ( C2
= ( zero_zero @ A ) )
=> ( A3
= ( zero_zero @ A ) ) ) ) ) ) ).
% minus_divide_eq_eq
thf(fact_1530_nonzero__neg__divide__eq__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( ( uminus_uminus @ A @ ( divide_divide @ A @ A3 @ B2 ) )
= C2 )
= ( ( uminus_uminus @ A @ A3 )
= ( times_times @ A @ C2 @ B2 ) ) ) ) ) ).
% nonzero_neg_divide_eq_eq
thf(fact_1531_nonzero__neg__divide__eq__eq2,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B2: A,C2: A,A3: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( C2
= ( uminus_uminus @ A @ ( divide_divide @ A @ A3 @ B2 ) ) )
= ( ( times_times @ A @ C2 @ B2 )
= ( uminus_uminus @ A @ A3 ) ) ) ) ) ).
% nonzero_neg_divide_eq_eq2
thf(fact_1532_power__less__power__Suc,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,N: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ A3 )
=> ( ord_less @ A @ ( power_power @ A @ A3 @ N ) @ ( times_times @ A @ A3 @ ( power_power @ A @ A3 @ N ) ) ) ) ) ).
% power_less_power_Suc
thf(fact_1533_power__gt1__lemma,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,N: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ A3 )
=> ( ord_less @ A @ ( one_one @ A ) @ ( times_times @ A @ A3 @ ( power_power @ A @ A3 @ N ) ) ) ) ) ).
% power_gt1_lemma
thf(fact_1534_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_1535_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_1536_abs__eq__mult,axiom,
! [A: $tType] :
( ( ordered_ring_abs @ A )
=> ! [A3: A,B2: A] :
( ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
| ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) ) )
& ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
| ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) ) )
=> ( ( abs_abs @ A @ ( times_times @ A @ A3 @ B2 ) )
= ( times_times @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ B2 ) ) ) ) ) ).
% abs_eq_mult
thf(fact_1537_power__minus,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [A3: A,N: nat] :
( ( power_power @ A @ ( uminus_uminus @ A @ A3 ) @ N )
= ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N ) @ ( power_power @ A @ A3 @ N ) ) ) ) ).
% power_minus
thf(fact_1538_div__mult1__eq,axiom,
! [A: $tType] :
( ( euclid3128863361964157862miring @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( divide_divide @ A @ ( times_times @ A @ A3 @ B2 ) @ C2 )
= ( plus_plus @ A @ ( times_times @ A @ A3 @ ( divide_divide @ A @ B2 @ C2 ) ) @ ( divide_divide @ A @ ( times_times @ A @ A3 @ ( modulo_modulo @ A @ B2 @ C2 ) ) @ C2 ) ) ) ) ).
% div_mult1_eq
thf(fact_1539_cancel__div__mod__rules_I2_J,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ ( times_times @ A @ B2 @ ( divide_divide @ A @ A3 @ B2 ) ) @ ( modulo_modulo @ A @ A3 @ B2 ) ) @ C2 )
= ( plus_plus @ A @ A3 @ C2 ) ) ) ).
% cancel_div_mod_rules(2)
thf(fact_1540_cancel__div__mod__rules_I1_J,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ ( times_times @ A @ ( divide_divide @ A @ A3 @ B2 ) @ B2 ) @ ( modulo_modulo @ A @ A3 @ B2 ) ) @ C2 )
= ( plus_plus @ A @ A3 @ C2 ) ) ) ).
% cancel_div_mod_rules(1)
thf(fact_1541_mod__div__decomp,axiom,
! [A: $tType] :
( ( semiring_modulo @ A )
=> ! [A3: A,B2: A] :
( A3
= ( plus_plus @ A @ ( times_times @ A @ ( divide_divide @ A @ A3 @ B2 ) @ B2 ) @ ( modulo_modulo @ A @ A3 @ B2 ) ) ) ) ).
% mod_div_decomp
thf(fact_1542_div__mult__mod__eq,axiom,
! [A: $tType] :
( ( semiring_modulo @ A )
=> ! [A3: A,B2: A] :
( ( plus_plus @ A @ ( times_times @ A @ ( divide_divide @ A @ A3 @ B2 ) @ B2 ) @ ( modulo_modulo @ A @ A3 @ B2 ) )
= A3 ) ) ).
% div_mult_mod_eq
thf(fact_1543_mod__div__mult__eq,axiom,
! [A: $tType] :
( ( semiring_modulo @ A )
=> ! [A3: A,B2: A] :
( ( plus_plus @ A @ ( modulo_modulo @ A @ A3 @ B2 ) @ ( times_times @ A @ ( divide_divide @ A @ A3 @ B2 ) @ B2 ) )
= A3 ) ) ).
% mod_div_mult_eq
thf(fact_1544_mod__mult__div__eq,axiom,
! [A: $tType] :
( ( semiring_modulo @ A )
=> ! [A3: A,B2: A] :
( ( plus_plus @ A @ ( modulo_modulo @ A @ A3 @ B2 ) @ ( times_times @ A @ B2 @ ( divide_divide @ A @ A3 @ B2 ) ) )
= A3 ) ) ).
% mod_mult_div_eq
thf(fact_1545_mult__div__mod__eq,axiom,
! [A: $tType] :
( ( semiring_modulo @ A )
=> ! [B2: A,A3: A] :
( ( plus_plus @ A @ ( times_times @ A @ B2 @ ( divide_divide @ A @ A3 @ B2 ) ) @ ( modulo_modulo @ A @ A3 @ B2 ) )
= A3 ) ) ).
% mult_div_mod_eq
thf(fact_1546_minus__div__mult__eq__mod,axiom,
! [A: $tType] :
( ( semiring_modulo @ A )
=> ! [A3: A,B2: A] :
( ( minus_minus @ A @ A3 @ ( times_times @ A @ ( divide_divide @ A @ A3 @ B2 ) @ B2 ) )
= ( modulo_modulo @ A @ A3 @ B2 ) ) ) ).
% minus_div_mult_eq_mod
thf(fact_1547_minus__mod__eq__div__mult,axiom,
! [A: $tType] :
( ( semiring_modulo @ A )
=> ! [A3: A,B2: A] :
( ( minus_minus @ A @ A3 @ ( modulo_modulo @ A @ A3 @ B2 ) )
= ( times_times @ A @ ( divide_divide @ A @ A3 @ B2 ) @ B2 ) ) ) ).
% minus_mod_eq_div_mult
thf(fact_1548_minus__mod__eq__mult__div,axiom,
! [A: $tType] :
( ( semiring_modulo @ A )
=> ! [A3: A,B2: A] :
( ( minus_minus @ A @ A3 @ ( modulo_modulo @ A @ A3 @ B2 ) )
= ( times_times @ A @ B2 @ ( divide_divide @ A @ A3 @ B2 ) ) ) ) ).
% minus_mod_eq_mult_div
thf(fact_1549_minus__mult__div__eq__mod,axiom,
! [A: $tType] :
( ( semiring_modulo @ A )
=> ! [A3: A,B2: A] :
( ( minus_minus @ A @ A3 @ ( times_times @ A @ B2 @ ( divide_divide @ A @ A3 @ B2 ) ) )
= ( modulo_modulo @ A @ A3 @ B2 ) ) ) ).
% minus_mult_div_eq_mod
thf(fact_1550_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_1551_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_1552_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_1553_reals__Archimedean3,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ! [Y: real] :
? [N2: nat] : ( ord_less @ real @ Y @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N2 ) @ X ) ) ) ).
% reals_Archimedean3
thf(fact_1554_pos__zmult__eq__1__iff,axiom,
! [M2: int,N: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ M2 )
=> ( ( ( times_times @ int @ M2 @ N )
= ( one_one @ int ) )
= ( ( M2
= ( one_one @ int ) )
& ( N
= ( one_one @ int ) ) ) ) ) ).
% pos_zmult_eq_1_iff
thf(fact_1555_powr__mult,axiom,
! [X: real,Y2: real,A3: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y2 )
=> ( ( powr @ real @ ( times_times @ real @ X @ Y2 ) @ A3 )
= ( times_times @ real @ ( powr @ real @ X @ A3 ) @ ( powr @ real @ Y2 @ A3 ) ) ) ) ) ).
% powr_mult
thf(fact_1556_minusinfinity,axiom,
! [D2: int,P1: int > $o,P: int > $o] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D2 )
=> ( ! [X3: int,K2: int] :
( ( P1 @ X3 )
= ( P1 @ ( minus_minus @ int @ X3 @ ( times_times @ int @ K2 @ D2 ) ) ) )
=> ( ? [Z5: int] :
! [X3: int] :
( ( ord_less @ int @ X3 @ Z5 )
=> ( ( P @ X3 )
= ( P1 @ X3 ) ) )
=> ( ? [X_12: int] : ( P1 @ X_12 )
=> ? [X_1: int] : ( P @ X_1 ) ) ) ) ) ).
% minusinfinity
thf(fact_1557_plusinfinity,axiom,
! [D2: int,P5: int > $o,P: int > $o] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D2 )
=> ( ! [X3: int,K2: int] :
( ( P5 @ X3 )
= ( P5 @ ( minus_minus @ int @ X3 @ ( times_times @ int @ K2 @ D2 ) ) ) )
=> ( ? [Z5: int] :
! [X3: int] :
( ( ord_less @ int @ Z5 @ X3 )
=> ( ( P @ X3 )
= ( P5 @ X3 ) ) )
=> ( ? [X_12: int] : ( P5 @ X_12 )
=> ? [X_1: int] : ( P @ X_1 ) ) ) ) ) ).
% plusinfinity
thf(fact_1558_zdiv__zmult2__eq,axiom,
! [C2: int,A3: int,B2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ C2 )
=> ( ( divide_divide @ int @ A3 @ ( times_times @ int @ B2 @ C2 ) )
= ( divide_divide @ int @ ( divide_divide @ int @ A3 @ B2 ) @ C2 ) ) ) ).
% zdiv_zmult2_eq
thf(fact_1559_divide__powr__uminus,axiom,
! [A3: real,B2: real,C2: real] :
( ( divide_divide @ real @ A3 @ ( powr @ real @ B2 @ C2 ) )
= ( times_times @ real @ A3 @ ( powr @ real @ B2 @ ( uminus_uminus @ real @ C2 ) ) ) ) ).
% divide_powr_uminus
thf(fact_1560_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_1561_log__powr,axiom,
! [X: real,B2: real,Y2: real] :
( ( X
!= ( zero_zero @ real ) )
=> ( ( log2 @ B2 @ ( powr @ real @ X @ Y2 ) )
= ( times_times @ real @ Y2 @ ( log2 @ B2 @ X ) ) ) ) ).
% log_powr
thf(fact_1562_div__mod__decomp__int,axiom,
! [A4: int,N: int] :
( A4
= ( plus_plus @ int @ ( times_times @ int @ ( divide_divide @ int @ A4 @ N ) @ N ) @ ( modulo_modulo @ int @ A4 @ N ) ) ) ).
% div_mod_decomp_int
thf(fact_1563_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_1564_of__int__nonneg,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z2 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( ring_1_of_int @ A @ Z2 ) ) ) ) ).
% of_int_nonneg
thf(fact_1565_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_1566_of__int__pos,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ Z2 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( ring_1_of_int @ A @ Z2 ) ) ) ) ).
% of_int_pos
thf(fact_1567_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_1568_floor__exists1,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X: A] :
? [X3: int] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ X3 ) @ X )
& ( ord_less @ A @ X @ ( ring_1_of_int @ A @ ( plus_plus @ int @ X3 @ ( one_one @ int ) ) ) )
& ! [Y: int] :
( ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Y ) @ X )
& ( ord_less @ A @ X @ ( ring_1_of_int @ A @ ( plus_plus @ int @ Y @ ( one_one @ int ) ) ) ) )
=> ( Y = X3 ) ) ) ) ).
% floor_exists1
thf(fact_1569_floor__exists,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X: A] :
? [Z4: int] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Z4 ) @ X )
& ( ord_less @ A @ X @ ( ring_1_of_int @ A @ ( plus_plus @ int @ Z4 @ ( one_one @ int ) ) ) ) ) ) ).
% floor_exists
thf(fact_1570_of__int__ceiling__le__add__one,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [R4: A] : ( ord_less_eq @ A @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ R4 ) ) @ ( plus_plus @ A @ R4 @ ( one_one @ A ) ) ) ) ).
% of_int_ceiling_le_add_one
thf(fact_1571_of__int__ceiling__diff__one__le,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [R4: A] : ( ord_less_eq @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ R4 ) ) @ ( one_one @ A ) ) @ R4 ) ) ).
% of_int_ceiling_diff_one_le
thf(fact_1572_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_1573_field__le__mult__one__interval,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y2: A] :
( ! [Z4: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Z4 )
=> ( ( ord_less @ A @ Z4 @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ Z4 @ X ) @ Y2 ) ) )
=> ( ord_less_eq @ A @ X @ Y2 ) ) ) ).
% field_le_mult_one_interval
thf(fact_1574_mult__less__cancel__right2,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,C2: A] :
( ( ord_less @ A @ ( times_times @ A @ A3 @ C2 ) @ C2 )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ A3 @ ( one_one @ A ) ) )
& ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( one_one @ A ) @ A3 ) ) ) ) ) ).
% mult_less_cancel_right2
thf(fact_1575_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_1576_mult__less__cancel__left2,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [C2: A,A3: A] :
( ( ord_less @ A @ ( times_times @ A @ C2 @ A3 ) @ C2 )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ A3 @ ( one_one @ A ) ) )
& ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( one_one @ A ) @ A3 ) ) ) ) ) ).
% mult_less_cancel_left2
thf(fact_1577_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_1578_mult__le__cancel__right2,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,C2: A] :
( ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C2 ) @ C2 )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ A3 @ ( one_one @ A ) ) )
& ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ A3 ) ) ) ) ) ).
% mult_le_cancel_right2
thf(fact_1579_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_1580_mult__le__cancel__left2,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [C2: A,A3: A] :
( ( ord_less_eq @ A @ ( times_times @ A @ C2 @ A3 ) @ C2 )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ A3 @ ( one_one @ A ) ) )
& ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ A3 ) ) ) ) ) ).
% mult_le_cancel_left2
thf(fact_1581_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_1582_divide__le__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,C2: A,A3: A] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ C2 ) @ A3 )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ B2 @ ( times_times @ A @ A3 @ C2 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C2 ) @ B2 ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 ) ) ) ) ) ) ) ).
% divide_le_eq
thf(fact_1583_le__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ ( divide_divide @ A @ B2 @ C2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C2 ) @ B2 ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B2 @ ( times_times @ A @ A3 @ C2 ) ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% le_divide_eq
thf(fact_1584_divide__left__mono,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( ord_less_eq @ A @ B2 @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B2 ) )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ C2 @ A3 ) @ ( divide_divide @ A @ C2 @ B2 ) ) ) ) ) ) ).
% divide_left_mono
thf(fact_1585_neg__divide__le__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,B2: A,A3: A] :
( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ C2 ) @ A3 )
= ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C2 ) @ B2 ) ) ) ) ).
% neg_divide_le_eq
thf(fact_1586_neg__le__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ A3 @ ( divide_divide @ A @ B2 @ C2 ) )
= ( ord_less_eq @ A @ B2 @ ( times_times @ A @ A3 @ C2 ) ) ) ) ) ).
% neg_le_divide_eq
thf(fact_1587_pos__divide__le__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,B2: A,A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ C2 ) @ A3 )
= ( ord_less_eq @ A @ B2 @ ( times_times @ A @ A3 @ C2 ) ) ) ) ) ).
% pos_divide_le_eq
thf(fact_1588_pos__le__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less_eq @ A @ A3 @ ( divide_divide @ A @ B2 @ C2 ) )
= ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C2 ) @ B2 ) ) ) ) ).
% pos_le_divide_eq
thf(fact_1589_mult__imp__div__pos__le,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y2: A,X: A,Z2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Y2 )
=> ( ( ord_less_eq @ A @ X @ ( times_times @ A @ Z2 @ Y2 ) )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ X @ Y2 ) @ Z2 ) ) ) ) ).
% mult_imp_div_pos_le
thf(fact_1590_mult__imp__le__div__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y2: A,Z2: A,X: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Y2 )
=> ( ( ord_less_eq @ A @ ( times_times @ A @ Z2 @ Y2 ) @ X )
=> ( ord_less_eq @ A @ Z2 @ ( divide_divide @ A @ X @ Y2 ) ) ) ) ) ).
% mult_imp_le_div_pos
thf(fact_1591_divide__left__mono__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B2 ) )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ C2 @ A3 ) @ ( divide_divide @ A @ C2 @ B2 ) ) ) ) ) ) ).
% divide_left_mono_neg
thf(fact_1592_convex__bound__le,axiom,
! [A: $tType] :
( ( linord6961819062388156250ring_1 @ A )
=> ! [X: A,A3: A,Y2: A,U: A,V2: A] :
( ( ord_less_eq @ A @ X @ A3 )
=> ( ( ord_less_eq @ A @ Y2 @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ U )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ V2 )
=> ( ( ( plus_plus @ A @ U @ V2 )
= ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ ( times_times @ A @ U @ X ) @ ( times_times @ A @ V2 @ Y2 ) ) @ A3 ) ) ) ) ) ) ) ).
% convex_bound_le
thf(fact_1593_int__le__real__less,axiom,
( ( ord_less_eq @ int )
= ( ^ [N5: int,M5: int] : ( ord_less @ real @ ( ring_1_of_int @ real @ N5 ) @ ( plus_plus @ real @ ( ring_1_of_int @ real @ M5 ) @ ( one_one @ real ) ) ) ) ) ).
% int_le_real_less
thf(fact_1594_int__less__real__le,axiom,
( ( ord_less @ int )
= ( ^ [N5: int,M5: int] : ( ord_less_eq @ real @ ( plus_plus @ real @ ( ring_1_of_int @ real @ N5 ) @ ( one_one @ real ) ) @ ( ring_1_of_int @ real @ M5 ) ) ) ) ).
% int_less_real_le
thf(fact_1595_frac__le__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y2: A,Z2: A,X: A,W: A] :
( ( Y2
!= ( zero_zero @ A ) )
=> ( ( Z2
!= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( divide_divide @ A @ X @ Y2 ) @ ( divide_divide @ A @ W @ Z2 ) )
= ( ord_less_eq @ A @ ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ X @ Z2 ) @ ( times_times @ A @ W @ Y2 ) ) @ ( times_times @ A @ Y2 @ Z2 ) ) @ ( zero_zero @ A ) ) ) ) ) ) ).
% frac_le_eq
thf(fact_1596_frac__less__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y2: A,Z2: A,X: A,W: A] :
( ( Y2
!= ( zero_zero @ A ) )
=> ( ( Z2
!= ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( divide_divide @ A @ X @ Y2 ) @ ( divide_divide @ A @ W @ Z2 ) )
= ( ord_less @ A @ ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ X @ Z2 ) @ ( times_times @ A @ W @ Y2 ) ) @ ( times_times @ A @ Y2 @ Z2 ) ) @ ( zero_zero @ A ) ) ) ) ) ) ).
% frac_less_eq
thf(fact_1597_power__Suc__less,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A3: A,N: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ A3 @ ( one_one @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ A3 @ ( power_power @ A @ A3 @ N ) ) @ ( power_power @ A @ A3 @ N ) ) ) ) ) ).
% power_Suc_less
thf(fact_1598_pos__minus__divide__less__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,B2: A,A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) @ A3 )
= ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ ( times_times @ A @ A3 @ C2 ) ) ) ) ) ).
% pos_minus_divide_less_eq
thf(fact_1599_pos__less__minus__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less @ A @ A3 @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) )
= ( ord_less @ A @ ( times_times @ A @ A3 @ C2 ) @ ( uminus_uminus @ A @ B2 ) ) ) ) ) ).
% pos_less_minus_divide_eq
thf(fact_1600_neg__minus__divide__less__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,B2: A,A3: A] :
( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) @ A3 )
= ( ord_less @ A @ ( times_times @ A @ A3 @ C2 ) @ ( uminus_uminus @ A @ B2 ) ) ) ) ) ).
% neg_minus_divide_less_eq
thf(fact_1601_neg__less__minus__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ A3 @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) )
= ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ ( times_times @ A @ A3 @ C2 ) ) ) ) ) ).
% neg_less_minus_divide_eq
thf(fact_1602_minus__divide__less__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,C2: A,A3: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) @ A3 )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ ( times_times @ A @ A3 @ C2 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ A3 @ C2 ) @ ( uminus_uminus @ A @ B2 ) ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ A3 ) ) ) ) ) ) ) ).
% minus_divide_less_eq
thf(fact_1603_less__minus__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less @ A @ A3 @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less @ A @ ( times_times @ A @ A3 @ C2 ) @ ( uminus_uminus @ A @ B2 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ ( times_times @ A @ A3 @ C2 ) ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% less_minus_divide_eq
thf(fact_1604_add__divide__eq__if__simps_I3_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z2: A,A3: A,B2: A] :
( ( ( Z2
= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ A3 @ Z2 ) ) @ B2 )
= B2 ) )
& ( ( Z2
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ A3 @ Z2 ) ) @ B2 )
= ( divide_divide @ A @ ( plus_plus @ A @ ( uminus_uminus @ A @ A3 ) @ ( times_times @ A @ B2 @ Z2 ) ) @ Z2 ) ) ) ) ) ).
% add_divide_eq_if_simps(3)
thf(fact_1605_minus__divide__add__eq__iff,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z2: A,X: A,Y2: A] :
( ( Z2
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ X @ Z2 ) ) @ Y2 )
= ( divide_divide @ A @ ( plus_plus @ A @ ( uminus_uminus @ A @ X ) @ ( times_times @ A @ Y2 @ Z2 ) ) @ Z2 ) ) ) ) ).
% minus_divide_add_eq_iff
thf(fact_1606_add__divide__eq__if__simps_I6_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z2: A,A3: A,B2: A] :
( ( ( Z2
= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ A3 @ Z2 ) ) @ B2 )
= ( uminus_uminus @ A @ B2 ) ) )
& ( ( Z2
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ A3 @ Z2 ) ) @ B2 )
= ( divide_divide @ A @ ( minus_minus @ A @ ( uminus_uminus @ A @ A3 ) @ ( times_times @ A @ B2 @ Z2 ) ) @ Z2 ) ) ) ) ) ).
% add_divide_eq_if_simps(6)
thf(fact_1607_add__divide__eq__if__simps_I5_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z2: A,A3: A,B2: A] :
( ( ( Z2
= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( divide_divide @ A @ A3 @ Z2 ) @ B2 )
= ( uminus_uminus @ A @ B2 ) ) )
& ( ( Z2
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( divide_divide @ A @ A3 @ Z2 ) @ B2 )
= ( divide_divide @ A @ ( minus_minus @ A @ A3 @ ( times_times @ A @ B2 @ Z2 ) ) @ Z2 ) ) ) ) ) ).
% add_divide_eq_if_simps(5)
thf(fact_1608_minus__divide__diff__eq__iff,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z2: A,X: A,Y2: A] :
( ( Z2
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ X @ Z2 ) ) @ Y2 )
= ( divide_divide @ A @ ( minus_minus @ A @ ( uminus_uminus @ A @ X ) @ ( times_times @ A @ Y2 @ Z2 ) ) @ Z2 ) ) ) ) ).
% minus_divide_diff_eq_iff
thf(fact_1609_ceiling__divide__eq__div,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [A3: int,B2: int] :
( ( archimedean_ceiling @ A @ ( divide_divide @ A @ ( ring_1_of_int @ A @ A3 ) @ ( ring_1_of_int @ A @ B2 ) ) )
= ( uminus_uminus @ int @ ( divide_divide @ int @ ( uminus_uminus @ int @ A3 ) @ B2 ) ) ) ) ).
% ceiling_divide_eq_div
thf(fact_1610_mod__mult2__eq_H,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [A3: A,M2: nat,N: nat] :
( ( modulo_modulo @ A @ A3 @ ( times_times @ A @ ( semiring_1_of_nat @ A @ M2 ) @ ( semiring_1_of_nat @ A @ N ) ) )
= ( plus_plus @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ M2 ) @ ( modulo_modulo @ A @ ( divide_divide @ A @ A3 @ ( semiring_1_of_nat @ A @ M2 ) ) @ ( semiring_1_of_nat @ A @ N ) ) ) @ ( modulo_modulo @ A @ A3 @ ( semiring_1_of_nat @ A @ M2 ) ) ) ) ) ).
% mod_mult2_eq'
thf(fact_1611_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_1612_zmult__zless__mono2__lemma,axiom,
! [I: int,J2: int,K: nat] :
( ( ord_less @ int @ I @ J2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ord_less @ int @ ( times_times @ int @ ( semiring_1_of_nat @ int @ K ) @ I ) @ ( times_times @ int @ ( semiring_1_of_nat @ int @ K ) @ J2 ) ) ) ) ).
% zmult_zless_mono2_lemma
thf(fact_1613_incr__mult__lemma,axiom,
! [D2: int,P: int > $o,K: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D2 )
=> ( ! [X3: int] :
( ( P @ X3 )
=> ( P @ ( plus_plus @ int @ X3 @ D2 ) ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
=> ! [X4: int] :
( ( P @ X4 )
=> ( P @ ( plus_plus @ int @ X4 @ ( times_times @ int @ K @ D2 ) ) ) ) ) ) ) ).
% incr_mult_lemma
thf(fact_1614_unique__quotient__lemma__neg,axiom,
! [B2: int,Q6: int,R5: int,Q5: int,R4: int] :
( ( ord_less_eq @ int @ ( plus_plus @ int @ ( times_times @ int @ B2 @ Q6 ) @ R5 ) @ ( plus_plus @ int @ ( times_times @ int @ B2 @ Q5 ) @ R4 ) )
=> ( ( ord_less_eq @ int @ R4 @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ B2 @ R4 )
=> ( ( ord_less @ int @ B2 @ R5 )
=> ( ord_less_eq @ int @ Q5 @ Q6 ) ) ) ) ) ).
% unique_quotient_lemma_neg
thf(fact_1615_unique__quotient__lemma,axiom,
! [B2: int,Q6: int,R5: int,Q5: int,R4: int] :
( ( ord_less_eq @ int @ ( plus_plus @ int @ ( times_times @ int @ B2 @ Q6 ) @ R5 ) @ ( plus_plus @ int @ ( times_times @ int @ B2 @ Q5 ) @ R4 ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ R5 )
=> ( ( ord_less @ int @ R5 @ B2 )
=> ( ( ord_less @ int @ R4 @ B2 )
=> ( ord_less_eq @ int @ Q6 @ Q5 ) ) ) ) ) ).
% unique_quotient_lemma
thf(fact_1616_zdiv__mono2__neg__lemma,axiom,
! [B2: int,Q5: int,R4: int,B6: int,Q6: int,R5: int] :
( ( ( plus_plus @ int @ ( times_times @ int @ B2 @ Q5 ) @ R4 )
= ( plus_plus @ int @ ( times_times @ int @ B6 @ Q6 ) @ R5 ) )
=> ( ( ord_less @ int @ ( plus_plus @ int @ ( times_times @ int @ B6 @ Q6 ) @ R5 ) @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ R4 @ B2 )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ R5 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B6 )
=> ( ( ord_less_eq @ int @ B6 @ B2 )
=> ( ord_less_eq @ int @ Q6 @ Q5 ) ) ) ) ) ) ) ).
% zdiv_mono2_neg_lemma
thf(fact_1617_zdiv__mono2__lemma,axiom,
! [B2: int,Q5: int,R4: int,B6: int,Q6: int,R5: int] :
( ( ( plus_plus @ int @ ( times_times @ int @ B2 @ Q5 ) @ R4 )
= ( plus_plus @ int @ ( times_times @ int @ B6 @ Q6 ) @ R5 ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( plus_plus @ int @ ( times_times @ int @ B6 @ Q6 ) @ R5 ) )
=> ( ( ord_less @ int @ R5 @ B6 )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ R4 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B6 )
=> ( ( ord_less_eq @ int @ B6 @ B2 )
=> ( ord_less_eq @ int @ Q5 @ Q6 ) ) ) ) ) ) ) ).
% zdiv_mono2_lemma
thf(fact_1618_q__pos__lemma,axiom,
! [B6: int,Q6: int,R5: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( plus_plus @ int @ ( times_times @ int @ B6 @ Q6 ) @ R5 ) )
=> ( ( ord_less @ int @ R5 @ B6 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B6 )
=> ( ord_less_eq @ int @ ( zero_zero @ int ) @ Q6 ) ) ) ) ).
% q_pos_lemma
thf(fact_1619_decr__mult__lemma,axiom,
! [D2: int,P: int > $o,K: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D2 )
=> ( ! [X3: int] :
( ( P @ X3 )
=> ( P @ ( minus_minus @ int @ X3 @ D2 ) ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
=> ! [X4: int] :
( ( P @ X4 )
=> ( P @ ( minus_minus @ int @ X4 @ ( times_times @ int @ K @ D2 ) ) ) ) ) ) ) ).
% decr_mult_lemma
thf(fact_1620_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_1621_powr__def,axiom,
! [A: $tType] :
( ( ln @ A )
=> ( ( powr @ A )
= ( ^ [X2: A,A5: A] :
( if @ A
@ ( X2
= ( zero_zero @ A ) )
@ ( zero_zero @ A )
@ ( exp @ A @ ( times_times @ A @ A5 @ ( ln_ln @ A @ X2 ) ) ) ) ) ) ) ).
% powr_def
thf(fact_1622_ceiling__split,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [P: int > $o,T2: A] :
( ( P @ ( archimedean_ceiling @ A @ T2 ) )
= ( ! [I2: int] :
( ( ( ord_less @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ I2 ) @ ( one_one @ A ) ) @ T2 )
& ( ord_less_eq @ A @ T2 @ ( ring_1_of_int @ A @ I2 ) ) )
=> ( P @ I2 ) ) ) ) ) ).
% ceiling_split
thf(fact_1623_ceiling__eq__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,A3: int] :
( ( ( archimedean_ceiling @ A @ X )
= A3 )
= ( ( ord_less @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ A3 ) @ ( one_one @ A ) ) @ X )
& ( ord_less_eq @ A @ X @ ( ring_1_of_int @ A @ A3 ) ) ) ) ) ).
% ceiling_eq_iff
thf(fact_1624_ceiling__unique,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z2: int,X: A] :
( ( ord_less @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ Z2 ) @ ( one_one @ A ) ) @ X )
=> ( ( ord_less_eq @ A @ X @ ( ring_1_of_int @ A @ Z2 ) )
=> ( ( archimedean_ceiling @ A @ X )
= Z2 ) ) ) ) ).
% ceiling_unique
thf(fact_1625_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_1626_ceiling__less__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Z2: int] :
( ( ord_less @ int @ ( archimedean_ceiling @ A @ X ) @ Z2 )
= ( ord_less_eq @ A @ X @ ( minus_minus @ A @ ( ring_1_of_int @ A @ Z2 ) @ ( one_one @ A ) ) ) ) ) ).
% ceiling_less_iff
thf(fact_1627_le__ceiling__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z2: int,X: A] :
( ( ord_less_eq @ int @ Z2 @ ( archimedean_ceiling @ A @ X ) )
= ( ord_less @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ Z2 ) @ ( one_one @ A ) ) @ X ) ) ) ).
% le_ceiling_iff
thf(fact_1628_convex__bound__lt,axiom,
! [A: $tType] :
( ( linord715952674999750819strict @ A )
=> ! [X: A,A3: A,Y2: A,U: A,V2: A] :
( ( ord_less @ A @ X @ A3 )
=> ( ( ord_less @ A @ Y2 @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ U )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ V2 )
=> ( ( ( plus_plus @ A @ U @ V2 )
= ( one_one @ A ) )
=> ( ord_less @ A @ ( plus_plus @ A @ ( times_times @ A @ U @ X ) @ ( times_times @ A @ V2 @ Y2 ) ) @ A3 ) ) ) ) ) ) ) ).
% convex_bound_lt
thf(fact_1629_le__minus__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C2 ) @ ( uminus_uminus @ A @ B2 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( times_times @ A @ A3 @ C2 ) ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% le_minus_divide_eq
thf(fact_1630_minus__divide__le__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,C2: A,A3: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) @ A3 )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( times_times @ A @ A3 @ C2 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C2 ) @ ( uminus_uminus @ A @ B2 ) ) )
& ( ~ ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 ) ) ) ) ) ) ) ).
% minus_divide_le_eq
thf(fact_1631_neg__le__minus__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ A3 @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) )
= ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( times_times @ A @ A3 @ C2 ) ) ) ) ) ).
% neg_le_minus_divide_eq
thf(fact_1632_neg__minus__divide__le__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,B2: A,A3: A] :
( ( ord_less @ A @ C2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) @ A3 )
= ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C2 ) @ ( uminus_uminus @ A @ B2 ) ) ) ) ) ).
% neg_minus_divide_le_eq
thf(fact_1633_pos__le__minus__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less_eq @ A @ A3 @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) )
= ( ord_less_eq @ A @ ( times_times @ A @ A3 @ C2 ) @ ( uminus_uminus @ A @ B2 ) ) ) ) ) ).
% pos_le_minus_divide_eq
thf(fact_1634_pos__minus__divide__le__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C2: A,B2: A,A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C2 ) ) @ A3 )
= ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( times_times @ A @ A3 @ C2 ) ) ) ) ) ).
% pos_minus_divide_le_eq
thf(fact_1635_scaling__mono,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [U: A,V2: A,R4: A,S: A] :
( ( ord_less_eq @ A @ U @ V2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ R4 )
=> ( ( ord_less_eq @ A @ R4 @ S )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ U @ ( divide_divide @ A @ ( times_times @ A @ R4 @ ( minus_minus @ A @ V2 @ U ) ) @ S ) ) @ V2 ) ) ) ) ) ).
% scaling_mono
thf(fact_1636_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_1637_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( modulo_modulo @ A @ A3 @ ( times_times @ A @ B2 @ C2 ) )
= ( plus_plus @ A @ ( times_times @ A @ B2 @ ( modulo_modulo @ A @ ( divide_divide @ A @ A3 @ B2 ) @ C2 ) ) @ ( modulo_modulo @ A @ A3 @ B2 ) ) ) ) ) ).
% unique_euclidean_semiring_numeral_class.mod_mult2_eq
thf(fact_1638_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_1639_power__eq__if,axiom,
! [A: $tType] :
( ( power @ A )
=> ( ( power_power @ A )
= ( ^ [P6: A,M5: nat] :
( if @ A
@ ( M5
= ( zero_zero @ nat ) )
@ ( one_one @ A )
@ ( times_times @ A @ P6 @ ( power_power @ A @ P6 @ ( minus_minus @ nat @ M5 @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% power_eq_if
thf(fact_1640_power__minus__mult,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [N: nat,A3: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( times_times @ A @ ( power_power @ A @ A3 @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) @ A3 )
= ( power_power @ A @ A3 @ N ) ) ) ) ).
% power_minus_mult
thf(fact_1641_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 )
=> ( ! [M3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M3 )
=> ( ord_less_eq @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ M3 ) @ X ) @ C2 ) )
=> ( X
= ( zero_zero @ real ) ) ) ) ) ).
% real_archimedian_rdiv_eq_0
thf(fact_1642_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_1643_log__mult,axiom,
! [A3: real,X: real,Y2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A3 )
=> ( ( A3
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ Y2 )
=> ( ( log2 @ A3 @ ( times_times @ real @ X @ Y2 ) )
= ( plus_plus @ real @ ( log2 @ A3 @ X ) @ ( log2 @ A3 @ Y2 ) ) ) ) ) ) ) ).
% log_mult
thf(fact_1644_split__zdiv,axiom,
! [P: int > $o,N: int,K: int] :
( ( P @ ( divide_divide @ int @ N @ K ) )
= ( ( ( K
= ( zero_zero @ int ) )
=> ( P @ ( zero_zero @ int ) ) )
& ( ( ord_less @ int @ ( zero_zero @ int ) @ K )
=> ! [I2: int,J3: int] :
( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ J3 )
& ( ord_less @ int @ J3 @ K )
& ( N
= ( plus_plus @ int @ ( times_times @ int @ K @ I2 ) @ J3 ) ) )
=> ( P @ I2 ) ) )
& ( ( ord_less @ int @ K @ ( zero_zero @ int ) )
=> ! [I2: int,J3: int] :
( ( ( ord_less @ int @ K @ J3 )
& ( ord_less_eq @ int @ J3 @ ( zero_zero @ int ) )
& ( N
= ( plus_plus @ int @ ( times_times @ int @ K @ I2 ) @ J3 ) ) )
=> ( P @ I2 ) ) ) ) ) ).
% split_zdiv
thf(fact_1645_int__div__neg__eq,axiom,
! [A3: int,B2: int,Q5: int,R4: int] :
( ( A3
= ( plus_plus @ int @ ( times_times @ int @ B2 @ Q5 ) @ R4 ) )
=> ( ( ord_less_eq @ int @ R4 @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ B2 @ R4 )
=> ( ( divide_divide @ int @ A3 @ B2 )
= Q5 ) ) ) ) ).
% int_div_neg_eq
thf(fact_1646_int__div__pos__eq,axiom,
! [A3: int,B2: int,Q5: int,R4: int] :
( ( A3
= ( plus_plus @ int @ ( times_times @ int @ B2 @ Q5 ) @ R4 ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ R4 )
=> ( ( ord_less @ int @ R4 @ B2 )
=> ( ( divide_divide @ int @ A3 @ B2 )
= Q5 ) ) ) ) ).
% int_div_pos_eq
thf(fact_1647_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_1648_log__nat__power,axiom,
! [X: real,B2: real,N: nat] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( log2 @ B2 @ ( power_power @ real @ X @ N ) )
= ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ ( log2 @ B2 @ X ) ) ) ) ).
% log_nat_power
thf(fact_1649_int__mod__pos__eq,axiom,
! [A3: int,B2: int,Q5: int,R4: int] :
( ( A3
= ( plus_plus @ int @ ( times_times @ int @ B2 @ Q5 ) @ R4 ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ R4 )
=> ( ( ord_less @ int @ R4 @ B2 )
=> ( ( modulo_modulo @ int @ A3 @ B2 )
= R4 ) ) ) ) ).
% int_mod_pos_eq
thf(fact_1650_int__mod__neg__eq,axiom,
! [A3: int,B2: int,Q5: int,R4: int] :
( ( A3
= ( plus_plus @ int @ ( times_times @ int @ B2 @ Q5 ) @ R4 ) )
=> ( ( ord_less_eq @ int @ R4 @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ B2 @ R4 )
=> ( ( modulo_modulo @ int @ A3 @ B2 )
= R4 ) ) ) ) ).
% int_mod_neg_eq
thf(fact_1651_split__zmod,axiom,
! [P: int > $o,N: int,K: int] :
( ( P @ ( modulo_modulo @ int @ N @ K ) )
= ( ( ( K
= ( zero_zero @ int ) )
=> ( P @ N ) )
& ( ( ord_less @ int @ ( zero_zero @ int ) @ K )
=> ! [I2: int,J3: int] :
( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ J3 )
& ( ord_less @ int @ J3 @ K )
& ( N
= ( plus_plus @ int @ ( times_times @ int @ K @ I2 ) @ J3 ) ) )
=> ( P @ J3 ) ) )
& ( ( ord_less @ int @ K @ ( zero_zero @ int ) )
=> ! [I2: int,J3: int] :
( ( ( ord_less @ int @ K @ J3 )
& ( ord_less_eq @ int @ J3 @ ( zero_zero @ int ) )
& ( N
= ( plus_plus @ int @ ( times_times @ int @ K @ I2 ) @ J3 ) ) )
=> ( P @ J3 ) ) ) ) ) ).
% split_zmod
thf(fact_1652_zmod__zmult2__eq,axiom,
! [C2: int,A3: int,B2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ C2 )
=> ( ( modulo_modulo @ int @ A3 @ ( times_times @ int @ B2 @ C2 ) )
= ( plus_plus @ int @ ( times_times @ int @ B2 @ ( modulo_modulo @ int @ ( divide_divide @ int @ A3 @ B2 ) @ C2 ) ) @ ( modulo_modulo @ int @ A3 @ B2 ) ) ) ) ).
% zmod_zmult2_eq
thf(fact_1653_of__int__of__nat,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( ( ring_1_of_int @ A )
= ( ^ [K3: int] : ( if @ A @ ( ord_less @ int @ K3 @ ( zero_zero @ int ) ) @ ( uminus_uminus @ A @ ( semiring_1_of_nat @ A @ ( nat2 @ ( uminus_uminus @ int @ K3 ) ) ) ) @ ( semiring_1_of_nat @ A @ ( nat2 @ K3 ) ) ) ) ) ) ).
% of_int_of_nat
thf(fact_1654_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_1655_ln__powr__bound2,axiom,
! [X: real,A3: real] :
( ( ord_less @ real @ ( one_one @ real ) @ X )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ A3 )
=> ( ord_less_eq @ real @ ( powr @ real @ ( ln_ln @ real @ X ) @ A3 ) @ ( times_times @ real @ ( powr @ real @ A3 @ A3 ) @ X ) ) ) ) ).
% ln_powr_bound2
thf(fact_1656_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_1657_log__eq__div__ln__mult__log,axiom,
! [A3: real,B2: real,X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A3 )
=> ( ( A3
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ B2 )
=> ( ( B2
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( log2 @ A3 @ X )
= ( times_times @ real @ ( divide_divide @ real @ ( ln_ln @ real @ B2 ) @ ( ln_ln @ real @ A3 ) ) @ ( log2 @ B2 @ X ) ) ) ) ) ) ) ) ).
% log_eq_div_ln_mult_log
thf(fact_1658_incr__lemma,axiom,
! [D2: int,Z2: int,X: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D2 )
=> ( ord_less @ int @ Z2 @ ( plus_plus @ int @ X @ ( times_times @ int @ ( plus_plus @ int @ ( abs_abs @ int @ ( minus_minus @ int @ X @ Z2 ) ) @ ( one_one @ int ) ) @ D2 ) ) ) ) ).
% incr_lemma
thf(fact_1659_decr__lemma,axiom,
! [D2: int,X: int,Z2: 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 @ Z2 ) ) @ ( one_one @ int ) ) @ D2 ) ) @ Z2 ) ) ).
% decr_lemma
thf(fact_1660_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 @ ( log2 @ B2 @ X ) @ Y2 )
= ( log2 @ B2 @ ( times_times @ real @ X @ ( powr @ real @ B2 @ Y2 ) ) ) ) ) ) ) ).
% log_add_eq_powr
thf(fact_1661_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 @ ( log2 @ B2 @ X ) )
= ( log2 @ B2 @ ( times_times @ real @ ( powr @ real @ B2 @ Y2 ) @ X ) ) ) ) ) ) ).
% add_log_eq_powr
thf(fact_1662_split__pos__lemma,axiom,
! [K: int,P: int > int > $o,N: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ K )
=> ( ( P @ ( divide_divide @ int @ N @ K ) @ ( modulo_modulo @ int @ N @ K ) )
= ( ! [I2: int,J3: int] :
( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ J3 )
& ( ord_less @ int @ J3 @ K )
& ( N
= ( plus_plus @ int @ ( times_times @ int @ K @ I2 ) @ J3 ) ) )
=> ( P @ I2 @ J3 ) ) ) ) ) ).
% split_pos_lemma
thf(fact_1663_mult__le__cancel__iff2,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z2: A,X: A,Y2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Z2 )
=> ( ( ord_less_eq @ A @ ( times_times @ A @ Z2 @ X ) @ ( times_times @ A @ Z2 @ Y2 ) )
= ( ord_less_eq @ A @ X @ Y2 ) ) ) ) ).
% mult_le_cancel_iff2
thf(fact_1664_mult__le__cancel__iff1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z2: A,X: A,Y2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Z2 )
=> ( ( ord_less_eq @ A @ ( times_times @ A @ X @ Z2 ) @ ( times_times @ A @ Y2 @ Z2 ) )
= ( ord_less_eq @ A @ X @ Y2 ) ) ) ) ).
% mult_le_cancel_iff1
thf(fact_1665_mult__less__iff1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z2: A,X: A,Y2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Z2 )
=> ( ( ord_less @ A @ ( times_times @ A @ X @ Z2 ) @ ( times_times @ A @ Y2 @ Z2 ) )
= ( ord_less @ A @ X @ Y2 ) ) ) ) ).
% mult_less_iff1
thf(fact_1666_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_1667_floor__log__eq__powr__iff,axiom,
! [X: real,B2: real,K: int] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( one_one @ real ) @ B2 )
=> ( ( ( archim6421214686448440834_floor @ real @ ( log2 @ B2 @ X ) )
= K )
= ( ( ord_less_eq @ real @ ( powr @ real @ B2 @ ( ring_1_of_int @ real @ K ) ) @ X )
& ( ord_less @ real @ X @ ( powr @ real @ B2 @ ( ring_1_of_int @ real @ ( plus_plus @ int @ K @ ( one_one @ int ) ) ) ) ) ) ) ) ) ).
% floor_log_eq_powr_iff
thf(fact_1668_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_1669_split__root,axiom,
! [P: real > $o,N: nat,X: real] :
( ( P @ ( root @ N @ X ) )
= ( ( ( N
= ( zero_zero @ nat ) )
=> ( P @ ( zero_zero @ real ) ) )
& ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ! [Y5: real] :
( ( ( times_times @ real @ ( sgn_sgn @ real @ Y5 ) @ ( power_power @ real @ ( abs_abs @ real @ Y5 ) @ N ) )
= X )
=> ( P @ Y5 ) ) ) ) ) ).
% split_root
thf(fact_1670_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_1671_sgn__sgn,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ! [A3: A] :
( ( sgn_sgn @ A @ ( sgn_sgn @ A @ A3 ) )
= ( sgn_sgn @ A @ A3 ) ) ) ).
% sgn_sgn
thf(fact_1672_cosh__real__eq__iff,axiom,
! [X: real,Y2: real] :
( ( ( cosh @ real @ X )
= ( cosh @ real @ Y2 ) )
= ( ( abs_abs @ real @ X )
= ( abs_abs @ real @ Y2 ) ) ) ).
% cosh_real_eq_iff
thf(fact_1673_cosh__real__abs,axiom,
! [X: real] :
( ( cosh @ real @ ( abs_abs @ real @ X ) )
= ( cosh @ real @ X ) ) ).
% cosh_real_abs
thf(fact_1674_tanh__real__eq__iff,axiom,
! [X: real,Y2: real] :
( ( ( tanh @ real @ X )
= ( tanh @ real @ Y2 ) )
= ( X = Y2 ) ) ).
% tanh_real_eq_iff
thf(fact_1675_inverse__zero,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ( ( inverse_inverse @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% inverse_zero
thf(fact_1676_inverse__nonzero__iff__nonzero,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A] :
( ( ( inverse_inverse @ A @ A3 )
= ( zero_zero @ A ) )
= ( A3
= ( zero_zero @ A ) ) ) ) ).
% inverse_nonzero_iff_nonzero
thf(fact_1677_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_1678_inverse__1,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ( ( inverse_inverse @ A @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% inverse_1
thf(fact_1679_mult__is__0,axiom,
! [M2: nat,N: nat] :
( ( ( times_times @ nat @ M2 @ N )
= ( zero_zero @ nat ) )
= ( ( M2
= ( zero_zero @ nat ) )
| ( N
= ( zero_zero @ nat ) ) ) ) ).
% mult_is_0
thf(fact_1680_mult__0__right,axiom,
! [M2: nat] :
( ( times_times @ nat @ M2 @ ( zero_zero @ nat ) )
= ( zero_zero @ nat ) ) ).
% mult_0_right
thf(fact_1681_mult__cancel1,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ( times_times @ nat @ K @ M2 )
= ( times_times @ nat @ K @ N ) )
= ( ( M2 = N )
| ( K
= ( zero_zero @ nat ) ) ) ) ).
% mult_cancel1
thf(fact_1682_mult__cancel2,axiom,
! [M2: nat,K: nat,N: nat] :
( ( ( times_times @ nat @ M2 @ K )
= ( times_times @ nat @ N @ K ) )
= ( ( M2 = N )
| ( K
= ( zero_zero @ nat ) ) ) ) ).
% mult_cancel2
thf(fact_1683_inverse__divide,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A3: A,B2: A] :
( ( inverse_inverse @ A @ ( divide_divide @ A @ A3 @ B2 ) )
= ( divide_divide @ A @ B2 @ A3 ) ) ) ).
% inverse_divide
thf(fact_1684_inverse__minus__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A] :
( ( inverse_inverse @ A @ ( uminus_uminus @ A @ A3 ) )
= ( uminus_uminus @ A @ ( inverse_inverse @ A @ A3 ) ) ) ) ).
% inverse_minus_eq
thf(fact_1685_sgn__0,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ( ( sgn_sgn @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% sgn_0
thf(fact_1686_sgn__1,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ( ( sgn_sgn @ A @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% sgn_1
thf(fact_1687_sgn__divide,axiom,
! [A: $tType] :
( ( field_abs_sgn @ A )
=> ! [A3: A,B2: A] :
( ( sgn_sgn @ A @ ( divide_divide @ A @ A3 @ B2 ) )
= ( divide_divide @ A @ ( sgn_sgn @ A @ A3 ) @ ( sgn_sgn @ A @ B2 ) ) ) ) ).
% sgn_divide
thf(fact_1688_idom__abs__sgn__class_Osgn__minus,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ! [A3: A] :
( ( sgn_sgn @ A @ ( uminus_uminus @ A @ A3 ) )
= ( uminus_uminus @ A @ ( sgn_sgn @ A @ A3 ) ) ) ) ).
% idom_abs_sgn_class.sgn_minus
thf(fact_1689_nat__mult__eq__1__iff,axiom,
! [M2: nat,N: nat] :
( ( ( times_times @ nat @ M2 @ N )
= ( one_one @ nat ) )
= ( ( M2
= ( one_one @ nat ) )
& ( N
= ( one_one @ nat ) ) ) ) ).
% nat_mult_eq_1_iff
thf(fact_1690_nat__1__eq__mult__iff,axiom,
! [M2: nat,N: nat] :
( ( ( one_one @ nat )
= ( times_times @ nat @ M2 @ N ) )
= ( ( M2
= ( one_one @ nat ) )
& ( N
= ( one_one @ nat ) ) ) ) ).
% nat_1_eq_mult_iff
thf(fact_1691_cosh__minus,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X: A] :
( ( cosh @ A @ ( uminus_uminus @ A @ X ) )
= ( cosh @ A @ X ) ) ) ).
% cosh_minus
thf(fact_1692_floor__of__int,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z2: int] :
( ( archim6421214686448440834_floor @ A @ ( ring_1_of_int @ A @ Z2 ) )
= Z2 ) ) ).
% floor_of_int
thf(fact_1693_of__int__floor__cancel,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ A @ X ) )
= X )
= ( ? [N5: int] :
( X
= ( ring_1_of_int @ A @ N5 ) ) ) ) ) ).
% of_int_floor_cancel
thf(fact_1694_arctan__eq__zero__iff,axiom,
! [X: real] :
( ( ( arctan @ X )
= ( zero_zero @ real ) )
= ( X
= ( zero_zero @ real ) ) ) ).
% arctan_eq_zero_iff
thf(fact_1695_arctan__zero__zero,axiom,
( ( arctan @ ( zero_zero @ real ) )
= ( zero_zero @ real ) ) ).
% arctan_zero_zero
thf(fact_1696_inverse__nonpositive__iff__nonpositive,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( inverse_inverse @ A @ A3 ) @ ( zero_zero @ A ) )
= ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% inverse_nonpositive_iff_nonpositive
thf(fact_1697_inverse__nonnegative__iff__nonnegative,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( inverse_inverse @ A @ A3 ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% inverse_nonnegative_iff_nonnegative
thf(fact_1698_inverse__less__iff__less,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less @ A @ ( inverse_inverse @ A @ A3 ) @ ( inverse_inverse @ A @ B2 ) )
= ( ord_less @ A @ B2 @ A3 ) ) ) ) ) ).
% inverse_less_iff_less
thf(fact_1699_inverse__less__iff__less__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( inverse_inverse @ A @ A3 ) @ ( inverse_inverse @ A @ B2 ) )
= ( ord_less @ A @ B2 @ A3 ) ) ) ) ) ).
% inverse_less_iff_less_neg
thf(fact_1700_inverse__negative__iff__negative,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( inverse_inverse @ A @ A3 ) @ ( zero_zero @ A ) )
= ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% inverse_negative_iff_negative
thf(fact_1701_inverse__positive__iff__positive,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( inverse_inverse @ A @ A3 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% inverse_positive_iff_positive
thf(fact_1702_sgn__less,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( sgn_sgn @ A @ A3 ) @ ( zero_zero @ A ) )
= ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% sgn_less
thf(fact_1703_sgn__greater,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( sgn_sgn @ A @ A3 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% sgn_greater
thf(fact_1704_one__eq__mult__iff,axiom,
! [M2: nat,N: nat] :
( ( ( suc @ ( zero_zero @ nat ) )
= ( times_times @ nat @ M2 @ N ) )
= ( ( M2
= ( suc @ ( zero_zero @ nat ) ) )
& ( N
= ( suc @ ( zero_zero @ nat ) ) ) ) ) ).
% one_eq_mult_iff
thf(fact_1705_mult__eq__1__iff,axiom,
! [M2: nat,N: nat] :
( ( ( times_times @ nat @ M2 @ N )
= ( suc @ ( zero_zero @ nat ) ) )
= ( ( M2
= ( suc @ ( zero_zero @ nat ) ) )
& ( N
= ( suc @ ( zero_zero @ nat ) ) ) ) ) ).
% mult_eq_1_iff
thf(fact_1706_mult__less__cancel2,axiom,
! [M2: nat,K: nat,N: nat] :
( ( ord_less @ nat @ ( times_times @ nat @ M2 @ K ) @ ( times_times @ nat @ N @ K ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
& ( ord_less @ nat @ M2 @ N ) ) ) ).
% mult_less_cancel2
thf(fact_1707_nat__0__less__mult__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( times_times @ nat @ M2 @ N ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M2 )
& ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ).
% nat_0_less_mult_iff
thf(fact_1708_divide__sgn,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( divide_divide @ A @ A3 @ ( sgn_sgn @ A @ B2 ) )
= ( times_times @ A @ A3 @ ( sgn_sgn @ A @ B2 ) ) ) ) ).
% divide_sgn
thf(fact_1709_mult__Suc__right,axiom,
! [M2: nat,N: nat] :
( ( times_times @ nat @ M2 @ ( suc @ N ) )
= ( plus_plus @ nat @ M2 @ ( times_times @ nat @ M2 @ N ) ) ) ).
% mult_Suc_right
thf(fact_1710_floor__zero,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archim6421214686448440834_floor @ A @ ( zero_zero @ A ) )
= ( zero_zero @ int ) ) ) ).
% floor_zero
thf(fact_1711_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_1712_floor__one,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archim6421214686448440834_floor @ A @ ( one_one @ A ) )
= ( one_one @ int ) ) ) ).
% floor_one
thf(fact_1713_floor__of__nat,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [N: nat] :
( ( archim6421214686448440834_floor @ A @ ( semiring_1_of_nat @ A @ N ) )
= ( semiring_1_of_nat @ int @ N ) ) ) ).
% floor_of_nat
thf(fact_1714_zero__less__arctan__iff,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( arctan @ X ) )
= ( ord_less @ real @ ( zero_zero @ real ) @ X ) ) ).
% zero_less_arctan_iff
thf(fact_1715_arctan__less__zero__iff,axiom,
! [X: real] :
( ( ord_less @ real @ ( arctan @ X ) @ ( zero_zero @ real ) )
= ( ord_less @ real @ X @ ( zero_zero @ real ) ) ) ).
% arctan_less_zero_iff
thf(fact_1716_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_1717_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_1718_inverse__le__iff__le,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less_eq @ A @ ( inverse_inverse @ A @ A3 ) @ ( inverse_inverse @ A @ B2 ) )
= ( ord_less_eq @ A @ B2 @ A3 ) ) ) ) ) ).
% inverse_le_iff_le
thf(fact_1719_inverse__le__iff__le__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( inverse_inverse @ A @ A3 ) @ ( inverse_inverse @ A @ B2 ) )
= ( ord_less_eq @ A @ B2 @ A3 ) ) ) ) ) ).
% inverse_le_iff_le_neg
thf(fact_1720_left__inverse,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ ( inverse_inverse @ A @ A3 ) @ A3 )
= ( one_one @ A ) ) ) ) ).
% left_inverse
thf(fact_1721_right__inverse,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ A3 @ ( inverse_inverse @ A @ A3 ) )
= ( one_one @ A ) ) ) ) ).
% right_inverse
thf(fact_1722_sgn__pos,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( sgn_sgn @ A @ A3 )
= ( one_one @ A ) ) ) ) ).
% sgn_pos
thf(fact_1723_one__le__mult__iff,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( times_times @ nat @ M2 @ N ) )
= ( ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ M2 )
& ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N ) ) ) ).
% one_le_mult_iff
thf(fact_1724_abs__sgn__eq__1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( abs_abs @ A @ ( sgn_sgn @ A @ A3 ) )
= ( one_one @ A ) ) ) ) ).
% abs_sgn_eq_1
thf(fact_1725_mult__le__cancel2,axiom,
! [M2: nat,K: nat,N: nat] :
( ( ord_less_eq @ nat @ ( times_times @ nat @ M2 @ K ) @ ( times_times @ nat @ N @ K ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ord_less_eq @ nat @ M2 @ N ) ) ) ).
% mult_le_cancel2
thf(fact_1726_div__mult__self__is__m,axiom,
! [N: nat,M2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( divide_divide @ nat @ ( times_times @ nat @ M2 @ N ) @ N )
= M2 ) ) ).
% div_mult_self_is_m
thf(fact_1727_div__mult__self1__is__m,axiom,
! [N: nat,M2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( divide_divide @ nat @ ( times_times @ nat @ N @ M2 ) @ N )
= M2 ) ) ).
% div_mult_self1_is_m
thf(fact_1728_Suc__mod__mult__self4,axiom,
! [N: nat,K: nat,M2: nat] :
( ( modulo_modulo @ nat @ ( suc @ ( plus_plus @ nat @ ( times_times @ nat @ N @ K ) @ M2 ) ) @ N )
= ( modulo_modulo @ nat @ ( suc @ M2 ) @ N ) ) ).
% Suc_mod_mult_self4
thf(fact_1729_Suc__mod__mult__self3,axiom,
! [K: nat,N: nat,M2: nat] :
( ( modulo_modulo @ nat @ ( suc @ ( plus_plus @ nat @ ( times_times @ nat @ K @ N ) @ M2 ) ) @ N )
= ( modulo_modulo @ nat @ ( suc @ M2 ) @ N ) ) ).
% Suc_mod_mult_self3
thf(fact_1730_Suc__mod__mult__self2,axiom,
! [M2: nat,N: nat,K: nat] :
( ( modulo_modulo @ nat @ ( suc @ ( plus_plus @ nat @ M2 @ ( times_times @ nat @ N @ K ) ) ) @ N )
= ( modulo_modulo @ nat @ ( suc @ M2 ) @ N ) ) ).
% Suc_mod_mult_self2
thf(fact_1731_Suc__mod__mult__self1,axiom,
! [M2: nat,K: nat,N: nat] :
( ( modulo_modulo @ nat @ ( suc @ ( plus_plus @ nat @ M2 @ ( times_times @ nat @ K @ N ) ) ) @ N )
= ( modulo_modulo @ nat @ ( suc @ M2 ) @ N ) ) ).
% Suc_mod_mult_self1
thf(fact_1732_floor__uminus__of__int,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z2: int] :
( ( archim6421214686448440834_floor @ A @ ( uminus_uminus @ A @ ( ring_1_of_int @ A @ Z2 ) ) )
= ( uminus_uminus @ int @ Z2 ) ) ) ).
% floor_uminus_of_int
thf(fact_1733_floor__diff__of__int,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Z2: int] :
( ( archim6421214686448440834_floor @ A @ ( minus_minus @ A @ X @ ( ring_1_of_int @ A @ Z2 ) ) )
= ( minus_minus @ int @ ( archim6421214686448440834_floor @ A @ X ) @ Z2 ) ) ) ).
% floor_diff_of_int
thf(fact_1734_sgn__neg,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A] :
( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
=> ( ( sgn_sgn @ A @ A3 )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ).
% sgn_neg
thf(fact_1735_zero__le__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( archim6421214686448440834_floor @ A @ X ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ X ) ) ) ).
% zero_le_floor
thf(fact_1736_floor__less__zero,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( zero_zero @ int ) )
= ( ord_less @ A @ X @ ( zero_zero @ A ) ) ) ) ).
% floor_less_zero
thf(fact_1737_zero__less__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less @ int @ ( zero_zero @ int ) @ ( archim6421214686448440834_floor @ A @ X ) )
= ( ord_less_eq @ A @ ( one_one @ A ) @ X ) ) ) ).
% zero_less_floor
thf(fact_1738_floor__le__zero,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( zero_zero @ int ) )
= ( ord_less @ A @ X @ ( one_one @ A ) ) ) ) ).
% floor_le_zero
thf(fact_1739_one__le__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less_eq @ int @ ( one_one @ int ) @ ( archim6421214686448440834_floor @ A @ X ) )
= ( ord_less_eq @ A @ ( one_one @ A ) @ X ) ) ) ).
% one_le_floor
thf(fact_1740_floor__less__one,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ord_less @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( one_one @ int ) )
= ( ord_less @ A @ X @ ( one_one @ A ) ) ) ) ).
% floor_less_one
thf(fact_1741_floor__diff__one,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( archim6421214686448440834_floor @ A @ ( minus_minus @ A @ X @ ( one_one @ A ) ) )
= ( minus_minus @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( one_one @ int ) ) ) ) ).
% floor_diff_one
thf(fact_1742_arctan__eq__iff,axiom,
! [X: real,Y2: real] :
( ( ( arctan @ X )
= ( arctan @ Y2 ) )
= ( X = Y2 ) ) ).
% arctan_eq_iff
thf(fact_1743_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_1744_inverse__zero__imp__zero,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A] :
( ( ( inverse_inverse @ A @ A3 )
= ( zero_zero @ A ) )
=> ( A3
= ( zero_zero @ A ) ) ) ) ).
% inverse_zero_imp_zero
thf(fact_1745_nonzero__inverse__eq__imp__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A,B2: A] :
( ( ( inverse_inverse @ A @ A3 )
= ( inverse_inverse @ A @ B2 ) )
=> ( ( A3
!= ( zero_zero @ A ) )
=> ( ( B2
!= ( zero_zero @ A ) )
=> ( A3 = B2 ) ) ) ) ) ).
% nonzero_inverse_eq_imp_eq
thf(fact_1746_nonzero__inverse__inverse__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( inverse_inverse @ A @ ( inverse_inverse @ A @ A3 ) )
= A3 ) ) ) ).
% nonzero_inverse_inverse_eq
thf(fact_1747_nonzero__imp__inverse__nonzero,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( inverse_inverse @ A @ A3 )
!= ( zero_zero @ A ) ) ) ) ).
% nonzero_imp_inverse_nonzero
thf(fact_1748_sgn__0__0,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A] :
( ( ( sgn_sgn @ A @ A3 )
= ( zero_zero @ A ) )
= ( A3
= ( zero_zero @ A ) ) ) ) ).
% sgn_0_0
thf(fact_1749_sgn__eq__0__iff,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ! [A3: A] :
( ( ( sgn_sgn @ A @ A3 )
= ( zero_zero @ A ) )
= ( A3
= ( zero_zero @ A ) ) ) ) ).
% sgn_eq_0_iff
thf(fact_1750_sgn__mult,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ! [A3: A,B2: A] :
( ( sgn_sgn @ A @ ( times_times @ A @ A3 @ B2 ) )
= ( times_times @ A @ ( sgn_sgn @ A @ A3 ) @ ( sgn_sgn @ A @ B2 ) ) ) ) ).
% sgn_mult
thf(fact_1751_same__sgn__sgn__add,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [B2: A,A3: A] :
( ( ( sgn_sgn @ A @ B2 )
= ( sgn_sgn @ A @ A3 ) )
=> ( ( sgn_sgn @ A @ ( plus_plus @ A @ A3 @ B2 ) )
= ( sgn_sgn @ A @ A3 ) ) ) ) ).
% same_sgn_sgn_add
thf(fact_1752_cosh__real__nonzero,axiom,
! [X: real] :
( ( cosh @ real @ X )
!= ( zero_zero @ real ) ) ).
% cosh_real_nonzero
thf(fact_1753_Suc__mult__cancel1,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ( times_times @ nat @ ( suc @ K ) @ M2 )
= ( times_times @ nat @ ( suc @ K ) @ N ) )
= ( M2 = N ) ) ).
% Suc_mult_cancel1
thf(fact_1754_mult__0,axiom,
! [N: nat] :
( ( times_times @ nat @ ( zero_zero @ nat ) @ N )
= ( zero_zero @ nat ) ) ).
% mult_0
thf(fact_1755_arctan__monotone,axiom,
! [X: real,Y2: real] :
( ( ord_less @ real @ X @ Y2 )
=> ( ord_less @ real @ ( arctan @ X ) @ ( arctan @ Y2 ) ) ) ).
% arctan_monotone
thf(fact_1756_arctan__less__iff,axiom,
! [X: real,Y2: real] :
( ( ord_less @ real @ ( arctan @ X ) @ ( arctan @ Y2 ) )
= ( ord_less @ real @ X @ Y2 ) ) ).
% arctan_less_iff
thf(fact_1757_mult__le__mono2,axiom,
! [I: nat,J2: nat,K: nat] :
( ( ord_less_eq @ nat @ I @ J2 )
=> ( ord_less_eq @ nat @ ( times_times @ nat @ K @ I ) @ ( times_times @ nat @ K @ J2 ) ) ) ).
% mult_le_mono2
thf(fact_1758_mult__le__mono1,axiom,
! [I: nat,J2: nat,K: nat] :
( ( ord_less_eq @ nat @ I @ J2 )
=> ( ord_less_eq @ nat @ ( times_times @ nat @ I @ K ) @ ( times_times @ nat @ J2 @ K ) ) ) ).
% mult_le_mono1
thf(fact_1759_mult__le__mono,axiom,
! [I: nat,J2: nat,K: nat,L: nat] :
( ( ord_less_eq @ nat @ I @ J2 )
=> ( ( ord_less_eq @ nat @ K @ L )
=> ( ord_less_eq @ nat @ ( times_times @ nat @ I @ K ) @ ( times_times @ nat @ J2 @ L ) ) ) ) ).
% mult_le_mono
thf(fact_1760_le__square,axiom,
! [M2: nat] : ( ord_less_eq @ nat @ M2 @ ( times_times @ nat @ M2 @ M2 ) ) ).
% le_square
thf(fact_1761_le__cube,axiom,
! [M2: nat] : ( ord_less_eq @ nat @ M2 @ ( times_times @ nat @ M2 @ ( times_times @ nat @ M2 @ M2 ) ) ) ).
% le_cube
thf(fact_1762_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_1763_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_1764_add__mult__distrib2,axiom,
! [K: nat,M2: nat,N: nat] :
( ( times_times @ nat @ K @ ( plus_plus @ nat @ M2 @ N ) )
= ( plus_plus @ nat @ ( times_times @ nat @ K @ M2 ) @ ( times_times @ nat @ K @ N ) ) ) ).
% add_mult_distrib2
thf(fact_1765_add__mult__distrib,axiom,
! [M2: nat,N: nat,K: nat] :
( ( times_times @ nat @ ( plus_plus @ nat @ M2 @ N ) @ K )
= ( plus_plus @ nat @ ( times_times @ nat @ M2 @ K ) @ ( times_times @ nat @ N @ K ) ) ) ).
% add_mult_distrib
thf(fact_1766_diff__mult__distrib2,axiom,
! [K: nat,M2: nat,N: nat] :
( ( times_times @ nat @ K @ ( minus_minus @ nat @ M2 @ N ) )
= ( minus_minus @ nat @ ( times_times @ nat @ K @ M2 ) @ ( times_times @ nat @ K @ N ) ) ) ).
% diff_mult_distrib2
thf(fact_1767_diff__mult__distrib,axiom,
! [M2: nat,N: nat,K: nat] :
( ( times_times @ nat @ ( minus_minus @ nat @ M2 @ N ) @ K )
= ( minus_minus @ nat @ ( times_times @ nat @ M2 @ K ) @ ( times_times @ nat @ N @ K ) ) ) ).
% diff_mult_distrib
thf(fact_1768_nat__mult__1__right,axiom,
! [N: nat] :
( ( times_times @ nat @ N @ ( one_one @ nat ) )
= N ) ).
% nat_mult_1_right
thf(fact_1769_nat__mult__1,axiom,
! [N: nat] :
( ( times_times @ nat @ ( one_one @ nat ) @ N )
= N ) ).
% nat_mult_1
thf(fact_1770_div__mult2__eq,axiom,
! [M2: nat,N: nat,Q5: nat] :
( ( divide_divide @ nat @ M2 @ ( times_times @ nat @ N @ Q5 ) )
= ( divide_divide @ nat @ ( divide_divide @ nat @ M2 @ N ) @ Q5 ) ) ).
% div_mult2_eq
thf(fact_1771_arctan__minus,axiom,
! [X: real] :
( ( arctan @ ( uminus_uminus @ real @ X ) )
= ( uminus_uminus @ real @ ( arctan @ X ) ) ) ).
% arctan_minus
thf(fact_1772_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_1773_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_1774_inverse__less__imp__less,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ ( inverse_inverse @ A @ A3 ) @ ( inverse_inverse @ A @ B2 ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ord_less @ A @ B2 @ A3 ) ) ) ) ).
% inverse_less_imp_less
thf(fact_1775_less__imp__inverse__less,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ord_less @ A @ ( inverse_inverse @ A @ B2 ) @ ( inverse_inverse @ A @ A3 ) ) ) ) ) ).
% less_imp_inverse_less
thf(fact_1776_inverse__less__imp__less__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ ( inverse_inverse @ A @ A3 ) @ ( inverse_inverse @ A @ B2 ) )
=> ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B2 @ A3 ) ) ) ) ).
% inverse_less_imp_less_neg
thf(fact_1777_less__imp__inverse__less__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( inverse_inverse @ A @ B2 ) @ ( inverse_inverse @ A @ A3 ) ) ) ) ) ).
% less_imp_inverse_less_neg
thf(fact_1778_inverse__negative__imp__negative,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( inverse_inverse @ A @ A3 ) @ ( zero_zero @ A ) )
=> ( ( A3
!= ( zero_zero @ A ) )
=> ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ) ).
% inverse_negative_imp_negative
thf(fact_1779_inverse__positive__imp__positive,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( inverse_inverse @ A @ A3 ) )
=> ( ( A3
!= ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ A3 ) ) ) ) ).
% inverse_positive_imp_positive
thf(fact_1780_negative__imp__inverse__negative,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A] :
( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( inverse_inverse @ A @ A3 ) @ ( zero_zero @ A ) ) ) ) ).
% negative_imp_inverse_negative
thf(fact_1781_positive__imp__inverse__positive,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( inverse_inverse @ A @ A3 ) ) ) ) ).
% positive_imp_inverse_positive
thf(fact_1782_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_1783_nonzero__inverse__mult__distrib,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A,B2: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( B2
!= ( zero_zero @ A ) )
=> ( ( inverse_inverse @ A @ ( times_times @ A @ A3 @ B2 ) )
= ( times_times @ A @ ( inverse_inverse @ A @ B2 ) @ ( inverse_inverse @ A @ A3 ) ) ) ) ) ) ).
% nonzero_inverse_mult_distrib
thf(fact_1784_nonzero__inverse__minus__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( inverse_inverse @ A @ ( uminus_uminus @ A @ A3 ) )
= ( uminus_uminus @ A @ ( inverse_inverse @ A @ A3 ) ) ) ) ) ).
% nonzero_inverse_minus_eq
thf(fact_1785_inverse__unique,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A,B2: A] :
( ( ( times_times @ A @ A3 @ B2 )
= ( one_one @ A ) )
=> ( ( inverse_inverse @ A @ A3 )
= B2 ) ) ) ).
% inverse_unique
thf(fact_1786_field__class_Ofield__divide__inverse,axiom,
! [A: $tType] :
( ( field @ A )
=> ( ( divide_divide @ A )
= ( ^ [A5: A,B3: A] : ( times_times @ A @ A5 @ ( inverse_inverse @ A @ B3 ) ) ) ) ) ).
% field_class.field_divide_inverse
thf(fact_1787_divide__inverse,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ( ( divide_divide @ A )
= ( ^ [A5: A,B3: A] : ( times_times @ A @ A5 @ ( inverse_inverse @ A @ B3 ) ) ) ) ) ).
% divide_inverse
thf(fact_1788_divide__inverse__commute,axiom,
! [A: $tType] :
( ( field @ A )
=> ( ( divide_divide @ A )
= ( ^ [A5: A,B3: A] : ( times_times @ A @ ( inverse_inverse @ A @ B3 ) @ A5 ) ) ) ) ).
% divide_inverse_commute
thf(fact_1789_inverse__eq__divide,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ( ( inverse_inverse @ A )
= ( divide_divide @ A @ ( one_one @ A ) ) ) ) ).
% inverse_eq_divide
thf(fact_1790_power__mult__power__inverse__commute,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X: A,M2: nat,N: nat] :
( ( times_times @ A @ ( power_power @ A @ X @ M2 ) @ ( power_power @ A @ ( inverse_inverse @ A @ X ) @ N ) )
= ( times_times @ A @ ( power_power @ A @ ( inverse_inverse @ A @ X ) @ N ) @ ( power_power @ A @ X @ M2 ) ) ) ) ).
% power_mult_power_inverse_commute
thf(fact_1791_power__mult__inverse__distrib,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X: A,M2: nat] :
( ( times_times @ A @ ( power_power @ A @ X @ M2 ) @ ( inverse_inverse @ A @ X ) )
= ( times_times @ A @ ( inverse_inverse @ A @ X ) @ ( power_power @ A @ X @ M2 ) ) ) ) ).
% power_mult_inverse_distrib
thf(fact_1792_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_1793_nonzero__abs__inverse,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( abs_abs @ A @ ( inverse_inverse @ A @ A3 ) )
= ( inverse_inverse @ A @ ( abs_abs @ A @ A3 ) ) ) ) ) ).
% nonzero_abs_inverse
thf(fact_1794_sgn__not__eq__imp,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [B2: A,A3: A] :
( ( ( sgn_sgn @ A @ B2 )
!= ( sgn_sgn @ A @ A3 ) )
=> ( ( ( sgn_sgn @ A @ A3 )
!= ( zero_zero @ A ) )
=> ( ( ( sgn_sgn @ A @ B2 )
!= ( zero_zero @ A ) )
=> ( ( sgn_sgn @ A @ A3 )
= ( uminus_uminus @ A @ ( sgn_sgn @ A @ B2 ) ) ) ) ) ) ) ).
% sgn_not_eq_imp
thf(fact_1795_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_1796_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_1797_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_1798_abs__mult__sgn,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ! [A3: A] :
( ( times_times @ A @ ( abs_abs @ A @ A3 ) @ ( sgn_sgn @ A @ A3 ) )
= A3 ) ) ).
% abs_mult_sgn
thf(fact_1799_sgn__mult__abs,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ! [A3: A] :
( ( times_times @ A @ ( sgn_sgn @ A @ A3 ) @ ( abs_abs @ A @ A3 ) )
= A3 ) ) ).
% sgn_mult_abs
thf(fact_1800_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_1801_same__sgn__abs__add,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [B2: A,A3: A] :
( ( ( sgn_sgn @ A @ B2 )
= ( sgn_sgn @ A @ A3 ) )
=> ( ( abs_abs @ A @ ( plus_plus @ A @ A3 @ B2 ) )
= ( plus_plus @ A @ ( abs_abs @ A @ A3 ) @ ( abs_abs @ A @ B2 ) ) ) ) ) ).
% same_sgn_abs_add
thf(fact_1802_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_1803_exp__minus,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A] :
( ( exp @ A @ ( uminus_uminus @ A @ X ) )
= ( inverse_inverse @ A @ ( exp @ A @ X ) ) ) ) ).
% exp_minus
thf(fact_1804_powr__minus,axiom,
! [A: $tType] :
( ( ( real_V3459762299906320749_field @ A )
& ( ln @ A ) )
=> ! [X: A,A3: A] :
( ( powr @ A @ X @ ( uminus_uminus @ A @ A3 ) )
= ( inverse_inverse @ A @ ( powr @ A @ X @ A3 ) ) ) ) ).
% powr_minus
thf(fact_1805_cosh__real__pos,axiom,
! [X: real] : ( ord_less @ real @ ( zero_zero @ real ) @ ( cosh @ real @ X ) ) ).
% cosh_real_pos
thf(fact_1806_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_1807_cosh__real__nonneg,axiom,
! [X: real] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( cosh @ real @ X ) ) ).
% cosh_real_nonneg
thf(fact_1808_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_1809_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_1810_divide__real__def,axiom,
( ( divide_divide @ real )
= ( ^ [X2: real,Y5: real] : ( times_times @ real @ X2 @ ( inverse_inverse @ real @ Y5 ) ) ) ) ).
% divide_real_def
thf(fact_1811_cosh__real__ge__1,axiom,
! [X: real] : ( ord_less_eq @ real @ ( one_one @ real ) @ ( cosh @ real @ X ) ) ).
% cosh_real_ge_1
thf(fact_1812_Suc__mult__less__cancel1,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less @ nat @ ( times_times @ nat @ ( suc @ K ) @ M2 ) @ ( times_times @ nat @ ( suc @ K ) @ N ) )
= ( ord_less @ nat @ M2 @ N ) ) ).
% Suc_mult_less_cancel1
thf(fact_1813_mult__less__mono1,axiom,
! [I: nat,J2: nat,K: nat] :
( ( ord_less @ nat @ I @ J2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ord_less @ nat @ ( times_times @ nat @ I @ K ) @ ( times_times @ nat @ J2 @ K ) ) ) ) ).
% mult_less_mono1
thf(fact_1814_mult__less__mono2,axiom,
! [I: nat,J2: nat,K: nat] :
( ( ord_less @ nat @ I @ J2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ord_less @ nat @ ( times_times @ nat @ K @ I ) @ ( times_times @ nat @ K @ J2 ) ) ) ) ).
% mult_less_mono2
thf(fact_1815_Suc__mult__le__cancel1,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq @ nat @ ( times_times @ nat @ ( suc @ K ) @ M2 ) @ ( times_times @ nat @ ( suc @ K ) @ N ) )
= ( ord_less_eq @ nat @ M2 @ N ) ) ).
% Suc_mult_le_cancel1
thf(fact_1816_mult__Suc,axiom,
! [M2: nat,N: nat] :
( ( times_times @ nat @ ( suc @ M2 ) @ N )
= ( plus_plus @ nat @ N @ ( times_times @ nat @ M2 @ N ) ) ) ).
% mult_Suc
thf(fact_1817_mult__eq__self__implies__10,axiom,
! [M2: nat,N: nat] :
( ( M2
= ( times_times @ nat @ M2 @ N ) )
=> ( ( N
= ( one_one @ nat ) )
| ( M2
= ( zero_zero @ nat ) ) ) ) ).
% mult_eq_self_implies_10
thf(fact_1818_less__mult__imp__div__less,axiom,
! [M2: nat,I: nat,N: nat] :
( ( ord_less @ nat @ M2 @ ( times_times @ nat @ I @ N ) )
=> ( ord_less @ nat @ ( divide_divide @ nat @ M2 @ N ) @ I ) ) ).
% less_mult_imp_div_less
thf(fact_1819_div__times__less__eq__dividend,axiom,
! [M2: nat,N: nat] : ( ord_less_eq @ nat @ ( times_times @ nat @ ( divide_divide @ nat @ M2 @ N ) @ N ) @ M2 ) ).
% div_times_less_eq_dividend
thf(fact_1820_times__div__less__eq__dividend,axiom,
! [N: nat,M2: nat] : ( ord_less_eq @ nat @ ( times_times @ nat @ N @ ( divide_divide @ nat @ M2 @ N ) ) @ M2 ) ).
% times_div_less_eq_dividend
thf(fact_1821_mod__eq__0D,axiom,
! [M2: nat,D2: nat] :
( ( ( modulo_modulo @ nat @ M2 @ D2 )
= ( zero_zero @ nat ) )
=> ? [Q3: nat] :
( M2
= ( times_times @ nat @ D2 @ Q3 ) ) ) ).
% mod_eq_0D
thf(fact_1822_int__ops_I7_J,axiom,
! [A3: nat,B2: nat] :
( ( semiring_1_of_nat @ int @ ( times_times @ nat @ A3 @ B2 ) )
= ( times_times @ int @ ( semiring_1_of_nat @ int @ A3 ) @ ( semiring_1_of_nat @ int @ B2 ) ) ) ).
% int_ops(7)
thf(fact_1823_inverse__le__imp__le,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( inverse_inverse @ A @ A3 ) @ ( inverse_inverse @ A @ B2 ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ord_less_eq @ A @ B2 @ A3 ) ) ) ) ).
% inverse_le_imp_le
thf(fact_1824_le__imp__inverse__le,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ord_less_eq @ A @ ( inverse_inverse @ A @ B2 ) @ ( inverse_inverse @ A @ A3 ) ) ) ) ) ).
% le_imp_inverse_le
thf(fact_1825_inverse__le__imp__le__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( inverse_inverse @ A @ A3 ) @ ( inverse_inverse @ A @ B2 ) )
=> ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B2 @ A3 ) ) ) ) ).
% inverse_le_imp_le_neg
thf(fact_1826_le__imp__inverse__le__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( inverse_inverse @ A @ B2 ) @ ( inverse_inverse @ A @ A3 ) ) ) ) ) ).
% le_imp_inverse_le_neg
thf(fact_1827_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_1828_one__less__inverse,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ A3 @ ( one_one @ A ) )
=> ( ord_less @ A @ ( one_one @ A ) @ ( inverse_inverse @ A @ A3 ) ) ) ) ) ).
% one_less_inverse
thf(fact_1829_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_1830_field__class_Ofield__inverse,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ ( inverse_inverse @ A @ A3 ) @ A3 )
= ( one_one @ A ) ) ) ) ).
% field_class.field_inverse
thf(fact_1831_division__ring__inverse__add,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A,B2: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( B2
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( inverse_inverse @ A @ A3 ) @ ( inverse_inverse @ A @ B2 ) )
= ( times_times @ A @ ( times_times @ A @ ( inverse_inverse @ A @ A3 ) @ ( plus_plus @ A @ A3 @ B2 ) ) @ ( inverse_inverse @ A @ B2 ) ) ) ) ) ) ).
% division_ring_inverse_add
thf(fact_1832_inverse__add,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A3: A,B2: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( B2
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( inverse_inverse @ A @ A3 ) @ ( inverse_inverse @ A @ B2 ) )
= ( times_times @ A @ ( times_times @ A @ ( plus_plus @ A @ A3 @ B2 ) @ ( inverse_inverse @ A @ A3 ) ) @ ( inverse_inverse @ A @ B2 ) ) ) ) ) ) ).
% inverse_add
thf(fact_1833_division__ring__inverse__diff,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A,B2: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( B2
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( inverse_inverse @ A @ A3 ) @ ( inverse_inverse @ A @ B2 ) )
= ( times_times @ A @ ( times_times @ A @ ( inverse_inverse @ A @ A3 ) @ ( minus_minus @ A @ B2 @ A3 ) ) @ ( inverse_inverse @ A @ B2 ) ) ) ) ) ) ).
% division_ring_inverse_diff
thf(fact_1834_nonzero__inverse__eq__divide,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( inverse_inverse @ A @ A3 )
= ( divide_divide @ A @ ( one_one @ A ) @ A3 ) ) ) ) ).
% nonzero_inverse_eq_divide
thf(fact_1835_le__floor__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z2: int,X: A] :
( ( ord_less_eq @ int @ Z2 @ ( archim6421214686448440834_floor @ A @ X ) )
= ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Z2 ) @ X ) ) ) ).
% le_floor_iff
thf(fact_1836_floor__less__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Z2: int] :
( ( ord_less @ int @ ( archim6421214686448440834_floor @ A @ X ) @ Z2 )
= ( ord_less @ A @ X @ ( ring_1_of_int @ A @ Z2 ) ) ) ) ).
% floor_less_iff
thf(fact_1837_le__mult__nat__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [A3: A,B2: A] : ( ord_less_eq @ nat @ ( times_times @ nat @ ( nat2 @ ( archim6421214686448440834_floor @ A @ A3 ) ) @ ( nat2 @ ( archim6421214686448440834_floor @ A @ B2 ) ) ) @ ( nat2 @ ( archim6421214686448440834_floor @ A @ ( times_times @ A @ A3 @ B2 ) ) ) ) ) ).
% le_mult_nat_floor
thf(fact_1838_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_1839_int__add__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z2: int,X: A] :
( ( plus_plus @ int @ Z2 @ ( archim6421214686448440834_floor @ A @ X ) )
= ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ ( ring_1_of_int @ A @ Z2 ) @ X ) ) ) ) ).
% int_add_floor
thf(fact_1840_floor__add__int,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Z2: int] :
( ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X ) @ Z2 )
= ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X @ ( ring_1_of_int @ A @ Z2 ) ) ) ) ) ).
% floor_add_int
thf(fact_1841_sgn__1__pos,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A] :
( ( ( sgn_sgn @ A @ A3 )
= ( one_one @ A ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% sgn_1_pos
thf(fact_1842_floor__divide__of__int__eq,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [K: int,L: int] :
( ( archim6421214686448440834_floor @ A @ ( divide_divide @ A @ ( ring_1_of_int @ A @ K ) @ ( ring_1_of_int @ A @ L ) ) )
= ( divide_divide @ int @ K @ L ) ) ) ).
% floor_divide_of_int_eq
thf(fact_1843_ceiling__minus,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( archimedean_ceiling @ A @ ( uminus_uminus @ A @ X ) )
= ( uminus_uminus @ int @ ( archim6421214686448440834_floor @ A @ X ) ) ) ) ).
% ceiling_minus
thf(fact_1844_floor__minus,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( archim6421214686448440834_floor @ A @ ( uminus_uminus @ A @ X ) )
= ( uminus_uminus @ int @ ( archimedean_ceiling @ A @ X ) ) ) ) ).
% floor_minus
thf(fact_1845_ceiling__def,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archimedean_ceiling @ A )
= ( ^ [X2: A] : ( uminus_uminus @ int @ ( archim6421214686448440834_floor @ A @ ( uminus_uminus @ A @ X2 ) ) ) ) ) ) ).
% ceiling_def
thf(fact_1846_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_1847_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_1848_abs__sgn__eq,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A] :
( ( ( A3
= ( zero_zero @ A ) )
=> ( ( abs_abs @ A @ ( sgn_sgn @ A @ A3 ) )
= ( zero_zero @ A ) ) )
& ( ( A3
!= ( zero_zero @ A ) )
=> ( ( abs_abs @ A @ ( sgn_sgn @ A @ A3 ) )
= ( one_one @ A ) ) ) ) ) ).
% abs_sgn_eq
thf(fact_1849_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_1850_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_1851_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_1852_inverse__powr,axiom,
! [Y2: real,A3: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y2 )
=> ( ( powr @ real @ ( inverse_inverse @ real @ Y2 ) @ A3 )
= ( inverse_inverse @ real @ ( powr @ real @ Y2 @ A3 ) ) ) ) ).
% inverse_powr
thf(fact_1853_one__less__mult,axiom,
! [N: nat,M2: nat] :
( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N )
=> ( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ M2 )
=> ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( times_times @ nat @ M2 @ N ) ) ) ) ).
% one_less_mult
thf(fact_1854_n__less__m__mult__n,axiom,
! [N: nat,M2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ M2 )
=> ( ord_less @ nat @ N @ ( times_times @ nat @ M2 @ N ) ) ) ) ).
% n_less_m_mult_n
thf(fact_1855_n__less__n__mult__m,axiom,
! [N: nat,M2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ M2 )
=> ( ord_less @ nat @ N @ ( times_times @ nat @ N @ M2 ) ) ) ) ).
% n_less_n_mult_m
thf(fact_1856_div__less__iff__less__mult,axiom,
! [Q5: nat,M2: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ Q5 )
=> ( ( ord_less @ nat @ ( divide_divide @ nat @ M2 @ Q5 ) @ N )
= ( ord_less @ nat @ M2 @ ( times_times @ nat @ N @ Q5 ) ) ) ) ).
% div_less_iff_less_mult
thf(fact_1857_mod__mult2__eq,axiom,
! [M2: nat,N: nat,Q5: nat] :
( ( modulo_modulo @ nat @ M2 @ ( times_times @ nat @ N @ Q5 ) )
= ( plus_plus @ nat @ ( times_times @ nat @ N @ ( modulo_modulo @ nat @ ( divide_divide @ nat @ M2 @ N ) @ Q5 ) ) @ ( modulo_modulo @ nat @ M2 @ N ) ) ) ).
% mod_mult2_eq
thf(fact_1858_div__mod__decomp,axiom,
! [A4: nat,N: nat] :
( A4
= ( plus_plus @ nat @ ( times_times @ nat @ ( divide_divide @ nat @ A4 @ N ) @ N ) @ ( modulo_modulo @ nat @ A4 @ N ) ) ) ).
% div_mod_decomp
thf(fact_1859_modulo__nat__def,axiom,
( ( modulo_modulo @ nat )
= ( ^ [M5: nat,N5: nat] : ( minus_minus @ nat @ M5 @ ( times_times @ nat @ ( divide_divide @ nat @ M5 @ N5 ) @ N5 ) ) ) ) ).
% modulo_nat_def
thf(fact_1860_nat__abs__mult__distrib,axiom,
! [W: int,Z2: int] :
( ( nat2 @ ( abs_abs @ int @ ( times_times @ int @ W @ Z2 ) ) )
= ( times_times @ nat @ ( nat2 @ ( abs_abs @ int @ W ) ) @ ( nat2 @ ( abs_abs @ int @ Z2 ) ) ) ) ).
% nat_abs_mult_distrib
thf(fact_1861_of__nat__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [R4: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ R4 )
=> ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ ( nat2 @ ( archim6421214686448440834_floor @ A @ R4 ) ) ) @ R4 ) ) ) ).
% of_nat_floor
thf(fact_1862_inverse__le__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( inverse_inverse @ A @ A3 ) @ ( inverse_inverse @ A @ B2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B2 ) )
=> ( ord_less_eq @ A @ B2 @ A3 ) )
& ( ( ord_less_eq @ A @ ( times_times @ A @ A3 @ B2 ) @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ A3 @ B2 ) ) ) ) ) ).
% inverse_le_iff
thf(fact_1863_inverse__less__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ ( inverse_inverse @ A @ A3 ) @ ( inverse_inverse @ A @ B2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A3 @ B2 ) )
=> ( ord_less @ A @ B2 @ A3 ) )
& ( ( ord_less_eq @ A @ ( times_times @ A @ A3 @ B2 ) @ ( zero_zero @ A ) )
=> ( ord_less @ A @ A3 @ B2 ) ) ) ) ) ).
% inverse_less_iff
thf(fact_1864_one__le__inverse,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ A3 @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ ( inverse_inverse @ A @ A3 ) ) ) ) ) ).
% one_le_inverse
thf(fact_1865_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_1866_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_1867_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_1868_inverse__diff__inverse,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A,B2: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( B2
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( inverse_inverse @ A @ A3 ) @ ( inverse_inverse @ A @ B2 ) )
= ( uminus_uminus @ A @ ( times_times @ A @ ( times_times @ A @ ( inverse_inverse @ A @ A3 ) @ ( minus_minus @ A @ A3 @ B2 ) ) @ ( inverse_inverse @ A @ B2 ) ) ) ) ) ) ) ).
% inverse_diff_inverse
thf(fact_1869_reals__Archimedean,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X )
=> ? [N2: nat] : ( ord_less @ A @ ( inverse_inverse @ A @ ( semiring_1_of_nat @ A @ ( suc @ N2 ) ) ) @ X ) ) ) ).
% reals_Archimedean
thf(fact_1870_floor__divide__of__nat__eq,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [M2: nat,N: nat] :
( ( archim6421214686448440834_floor @ A @ ( divide_divide @ A @ ( semiring_1_of_nat @ A @ M2 ) @ ( semiring_1_of_nat @ A @ N ) ) )
= ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ M2 @ N ) ) ) ) ).
% floor_divide_of_nat_eq
thf(fact_1871_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_1872_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_1873_sgn__if,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( ( sgn_sgn @ A )
= ( ^ [X2: A] :
( if @ A
@ ( X2
= ( zero_zero @ A ) )
@ ( zero_zero @ A )
@ ( if @ A @ ( ord_less @ A @ ( zero_zero @ A ) @ X2 ) @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ) ) ).
% sgn_if
thf(fact_1874_sgn__1__neg,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A] :
( ( ( sgn_sgn @ A @ A3 )
= ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ).
% sgn_1_neg
thf(fact_1875_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_1876_le__nat__floor,axiom,
! [X: nat,A3: real] :
( ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ X ) @ A3 )
=> ( ord_less_eq @ nat @ X @ ( nat2 @ ( archim6421214686448440834_floor @ real @ A3 ) ) ) ) ).
% le_nat_floor
thf(fact_1877_ceiling__altdef,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archimedean_ceiling @ A )
= ( ^ [X2: A] :
( if @ int
@ ( X2
= ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ A @ X2 ) ) )
@ ( archim6421214686448440834_floor @ A @ X2 )
@ ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X2 ) @ ( one_one @ int ) ) ) ) ) ) ).
% ceiling_altdef
thf(fact_1878_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_1879_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_1880_real__of__int__floor__add__one__gt,axiom,
! [R4: real] : ( ord_less @ real @ R4 @ ( plus_plus @ real @ ( ring_1_of_int @ real @ ( archim6421214686448440834_floor @ real @ R4 ) ) @ ( one_one @ real ) ) ) ).
% real_of_int_floor_add_one_gt
thf(fact_1881_real__of__int__floor__add__one__ge,axiom,
! [R4: real] : ( ord_less_eq @ real @ R4 @ ( plus_plus @ real @ ( ring_1_of_int @ real @ ( archim6421214686448440834_floor @ real @ R4 ) ) @ ( one_one @ real ) ) ) ).
% real_of_int_floor_add_one_ge
thf(fact_1882_real__of__int__floor__gt__diff__one,axiom,
! [R4: real] : ( ord_less @ real @ ( minus_minus @ real @ R4 @ ( one_one @ real ) ) @ ( ring_1_of_int @ real @ ( archim6421214686448440834_floor @ real @ R4 ) ) ) ).
% real_of_int_floor_gt_diff_one
thf(fact_1883_real__of__int__floor__ge__diff__one,axiom,
! [R4: real] : ( ord_less_eq @ real @ ( minus_minus @ real @ R4 @ ( one_one @ real ) ) @ ( ring_1_of_int @ real @ ( archim6421214686448440834_floor @ real @ R4 ) ) ) ).
% real_of_int_floor_ge_diff_one
thf(fact_1884_forall__pos__mono__1,axiom,
! [P: real > $o,E2: real] :
( ! [D5: real,E: real] :
( ( ord_less @ real @ D5 @ E )
=> ( ( P @ D5 )
=> ( P @ E ) ) )
=> ( ! [N2: nat] : ( P @ ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ ( suc @ N2 ) ) ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ E2 )
=> ( P @ E2 ) ) ) ) ).
% forall_pos_mono_1
thf(fact_1885_forall__pos__mono,axiom,
! [P: real > $o,E2: real] :
( ! [D5: real,E: real] :
( ( ord_less @ real @ D5 @ E )
=> ( ( P @ D5 )
=> ( P @ E ) ) )
=> ( ! [N2: nat] :
( ( N2
!= ( zero_zero @ nat ) )
=> ( P @ ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ N2 ) ) ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ E2 )
=> ( P @ E2 ) ) ) ) ).
% forall_pos_mono
thf(fact_1886_real__arch__inverse,axiom,
! [E2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E2 )
= ( ? [N5: nat] :
( ( N5
!= ( zero_zero @ nat ) )
& ( ord_less @ real @ ( zero_zero @ real ) @ ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ N5 ) ) )
& ( ord_less @ real @ ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ N5 ) ) @ E2 ) ) ) ) ).
% real_arch_inverse
thf(fact_1887_ln__inverse,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ln_ln @ real @ ( inverse_inverse @ real @ X ) )
= ( uminus_uminus @ real @ ( ln_ln @ real @ X ) ) ) ) ).
% ln_inverse
thf(fact_1888_div__nat__eqI,axiom,
! [N: nat,Q5: nat,M2: nat] :
( ( ord_less_eq @ nat @ ( times_times @ nat @ N @ Q5 ) @ M2 )
=> ( ( ord_less @ nat @ M2 @ ( times_times @ nat @ N @ ( suc @ Q5 ) ) )
=> ( ( divide_divide @ nat @ M2 @ N )
= Q5 ) ) ) ).
% div_nat_eqI
thf(fact_1889_less__eq__div__iff__mult__less__eq,axiom,
! [Q5: nat,M2: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ Q5 )
=> ( ( ord_less_eq @ nat @ M2 @ ( divide_divide @ nat @ N @ Q5 ) )
= ( ord_less_eq @ nat @ ( times_times @ nat @ M2 @ Q5 ) @ N ) ) ) ).
% less_eq_div_iff_mult_less_eq
thf(fact_1890_split__div,axiom,
! [P: nat > $o,M2: nat,N: nat] :
( ( P @ ( divide_divide @ nat @ M2 @ N ) )
= ( ( ( N
= ( zero_zero @ nat ) )
=> ( P @ ( zero_zero @ nat ) ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ! [I2: nat,J3: nat] :
( ( ord_less @ nat @ J3 @ N )
=> ( ( M2
= ( plus_plus @ nat @ ( times_times @ nat @ N @ I2 ) @ J3 ) )
=> ( P @ I2 ) ) ) ) ) ) ).
% split_div
thf(fact_1891_dividend__less__div__times,axiom,
! [N: nat,M2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ord_less @ nat @ M2 @ ( plus_plus @ nat @ N @ ( times_times @ nat @ ( divide_divide @ nat @ M2 @ N ) @ N ) ) ) ) ).
% dividend_less_div_times
thf(fact_1892_dividend__less__times__div,axiom,
! [N: nat,M2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ord_less @ nat @ M2 @ ( plus_plus @ nat @ N @ ( times_times @ nat @ N @ ( divide_divide @ nat @ M2 @ N ) ) ) ) ) ).
% dividend_less_times_div
thf(fact_1893_mult__eq__if,axiom,
( ( times_times @ nat )
= ( ^ [M5: nat,N5: nat] :
( if @ nat
@ ( M5
= ( zero_zero @ nat ) )
@ ( zero_zero @ nat )
@ ( plus_plus @ nat @ N5 @ ( times_times @ nat @ ( minus_minus @ nat @ M5 @ ( one_one @ nat ) ) @ N5 ) ) ) ) ) ).
% mult_eq_if
thf(fact_1894_split__mod,axiom,
! [P: nat > $o,M2: nat,N: nat] :
( ( P @ ( modulo_modulo @ nat @ M2 @ N ) )
= ( ( ( N
= ( zero_zero @ nat ) )
=> ( P @ M2 ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ! [I2: nat,J3: nat] :
( ( ord_less @ nat @ J3 @ N )
=> ( ( M2
= ( plus_plus @ nat @ ( times_times @ nat @ N @ I2 ) @ J3 ) )
=> ( P @ J3 ) ) ) ) ) ) ).
% split_mod
thf(fact_1895_nat__mult__distrib,axiom,
! [Z2: int,Z3: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z2 )
=> ( ( nat2 @ ( times_times @ int @ Z2 @ Z3 ) )
= ( times_times @ nat @ ( nat2 @ Z2 ) @ ( nat2 @ Z3 ) ) ) ) ).
% nat_mult_distrib
thf(fact_1896_floor__unique,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z2: int,X: A] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Z2 ) @ X )
=> ( ( ord_less @ A @ X @ ( plus_plus @ A @ ( ring_1_of_int @ A @ Z2 ) @ ( one_one @ A ) ) )
=> ( ( archim6421214686448440834_floor @ A @ X )
= Z2 ) ) ) ) ).
% floor_unique
thf(fact_1897_floor__eq__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,A3: int] :
( ( ( archim6421214686448440834_floor @ A @ X )
= A3 )
= ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ A3 ) @ X )
& ( ord_less @ A @ X @ ( plus_plus @ A @ ( ring_1_of_int @ A @ A3 ) @ ( one_one @ A ) ) ) ) ) ) ).
% floor_eq_iff
thf(fact_1898_floor__split,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [P: int > $o,T2: A] :
( ( P @ ( archim6421214686448440834_floor @ A @ T2 ) )
= ( ! [I2: int] :
( ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ I2 ) @ T2 )
& ( ord_less @ A @ T2 @ ( plus_plus @ A @ ( ring_1_of_int @ A @ I2 ) @ ( one_one @ A ) ) ) )
=> ( P @ I2 ) ) ) ) ) ).
% floor_split
thf(fact_1899_le__mult__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less_eq @ int @ ( times_times @ int @ ( archim6421214686448440834_floor @ A @ A3 ) @ ( archim6421214686448440834_floor @ A @ B2 ) ) @ ( archim6421214686448440834_floor @ A @ ( times_times @ A @ A3 @ B2 ) ) ) ) ) ) ).
% le_mult_floor
thf(fact_1900_less__floor__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z2: int,X: A] :
( ( ord_less @ int @ Z2 @ ( archim6421214686448440834_floor @ A @ X ) )
= ( ord_less_eq @ A @ ( plus_plus @ A @ ( ring_1_of_int @ A @ Z2 ) @ ( one_one @ A ) ) @ X ) ) ) ).
% less_floor_iff
thf(fact_1901_floor__le__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Z2: int] :
( ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X ) @ Z2 )
= ( ord_less @ A @ X @ ( plus_plus @ A @ ( ring_1_of_int @ A @ Z2 ) @ ( one_one @ A ) ) ) ) ) ).
% floor_le_iff
thf(fact_1902_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_1903_ex__inverse__of__nat__less,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X )
=> ? [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
& ( ord_less @ A @ ( inverse_inverse @ A @ ( semiring_1_of_nat @ A @ N2 ) ) @ X ) ) ) ) ).
% ex_inverse_of_nat_less
thf(fact_1904_power__diff__conv__inverse,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X: A,M2: nat,N: nat] :
( ( X
!= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( power_power @ A @ X @ ( minus_minus @ nat @ N @ M2 ) )
= ( times_times @ A @ ( power_power @ A @ X @ N ) @ ( power_power @ A @ ( inverse_inverse @ A @ X ) @ M2 ) ) ) ) ) ) ).
% power_diff_conv_inverse
thf(fact_1905_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_1906_sgn__power__injE,axiom,
! [A3: real,N: nat,X: real,B2: real] :
( ( ( times_times @ real @ ( sgn_sgn @ real @ A3 ) @ ( power_power @ real @ ( abs_abs @ real @ A3 ) @ 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 )
=> ( A3 = B2 ) ) ) ) ).
% sgn_power_injE
thf(fact_1907_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_1908_floor__divide__real__eq__div,axiom,
! [B2: int,A3: real] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ B2 )
=> ( ( archim6421214686448440834_floor @ real @ ( divide_divide @ real @ A3 @ ( ring_1_of_int @ real @ B2 ) ) )
= ( divide_divide @ int @ ( archim6421214686448440834_floor @ real @ A3 ) @ B2 ) ) ) ).
% floor_divide_real_eq_div
thf(fact_1909_log__inverse,axiom,
! [A3: real,X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A3 )
=> ( ( A3
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( log2 @ A3 @ ( inverse_inverse @ real @ X ) )
= ( uminus_uminus @ real @ ( log2 @ A3 @ X ) ) ) ) ) ) ).
% log_inverse
thf(fact_1910_split__div_H,axiom,
! [P: nat > $o,M2: nat,N: nat] :
( ( P @ ( divide_divide @ nat @ M2 @ N ) )
= ( ( ( N
= ( zero_zero @ nat ) )
& ( P @ ( zero_zero @ nat ) ) )
| ? [Q4: nat] :
( ( ord_less_eq @ nat @ ( times_times @ nat @ N @ Q4 ) @ M2 )
& ( ord_less @ nat @ M2 @ ( times_times @ nat @ N @ ( suc @ Q4 ) ) )
& ( P @ Q4 ) ) ) ) ).
% split_div'
thf(fact_1911_Suc__times__mod__eq,axiom,
! [M2: nat,N: nat] :
( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ M2 )
=> ( ( modulo_modulo @ nat @ ( suc @ ( times_times @ nat @ M2 @ N ) ) @ M2 )
= ( one_one @ nat ) ) ) ).
% Suc_times_mod_eq
thf(fact_1912_floor__divide__lower,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Q5: A,P4: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Q5 )
=> ( ord_less_eq @ A @ ( times_times @ A @ ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ A @ ( divide_divide @ A @ P4 @ Q5 ) ) ) @ Q5 ) @ P4 ) ) ) ).
% floor_divide_lower
thf(fact_1913_nat__mult__distrib__neg,axiom,
! [Z2: int,Z3: int] :
( ( ord_less_eq @ int @ Z2 @ ( zero_zero @ int ) )
=> ( ( nat2 @ ( times_times @ int @ Z2 @ Z3 ) )
= ( times_times @ nat @ ( nat2 @ ( uminus_uminus @ int @ Z2 ) ) @ ( nat2 @ ( uminus_uminus @ int @ Z3 ) ) ) ) ) ).
% nat_mult_distrib_neg
thf(fact_1914_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_1915_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_1916_floor__divide__upper,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Q5: A,P4: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Q5 )
=> ( ord_less @ A @ P4 @ ( times_times @ A @ ( plus_plus @ A @ ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ A @ ( divide_divide @ A @ P4 @ Q5 ) ) ) @ ( one_one @ A ) ) @ Q5 ) ) ) ) ).
% floor_divide_upper
thf(fact_1917_nat__mult__le__cancel__disj,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq @ nat @ ( times_times @ nat @ K @ M2 ) @ ( times_times @ nat @ K @ N ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ord_less_eq @ nat @ M2 @ N ) ) ) ).
% nat_mult_le_cancel_disj
thf(fact_1918_nat__mult__div__cancel__disj,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ( K
= ( zero_zero @ nat ) )
=> ( ( divide_divide @ nat @ ( times_times @ nat @ K @ M2 ) @ ( times_times @ nat @ K @ N ) )
= ( zero_zero @ nat ) ) )
& ( ( K
!= ( zero_zero @ nat ) )
=> ( ( divide_divide @ nat @ ( times_times @ nat @ K @ M2 ) @ ( times_times @ nat @ K @ N ) )
= ( divide_divide @ nat @ M2 @ N ) ) ) ) ).
% nat_mult_div_cancel_disj
thf(fact_1919_nat__mult__less__cancel__disj,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less @ nat @ ( times_times @ nat @ K @ M2 ) @ ( times_times @ nat @ K @ N ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
& ( ord_less @ nat @ M2 @ N ) ) ) ).
% nat_mult_less_cancel_disj
thf(fact_1920_sgn__one,axiom,
! [A: $tType] :
( ( real_V2822296259951069270ebra_1 @ A )
=> ( ( sgn_sgn @ A @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% sgn_one
thf(fact_1921_nat__less__add__iff2,axiom,
! [I: nat,J2: nat,U: nat,M2: nat,N: nat] :
( ( ord_less_eq @ nat @ I @ J2 )
=> ( ( ord_less @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ M2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ J2 @ U ) @ N ) )
= ( ord_less @ nat @ M2 @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ J2 @ I ) @ U ) @ N ) ) ) ) ).
% nat_less_add_iff2
thf(fact_1922_nat__less__add__iff1,axiom,
! [J2: nat,I: nat,U: nat,M2: nat,N: nat] :
( ( ord_less_eq @ nat @ J2 @ I )
=> ( ( ord_less @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ M2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ J2 @ U ) @ N ) )
= ( ord_less @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ I @ J2 ) @ U ) @ M2 ) @ N ) ) ) ).
% nat_less_add_iff1
thf(fact_1923_sgn__zero,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ( ( sgn_sgn @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% sgn_zero
thf(fact_1924_nat__diff__add__eq2,axiom,
! [I: nat,J2: nat,U: nat,M2: nat,N: nat] :
( ( ord_less_eq @ nat @ I @ J2 )
=> ( ( minus_minus @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ M2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ J2 @ U ) @ N ) )
= ( minus_minus @ nat @ M2 @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ J2 @ I ) @ U ) @ N ) ) ) ) ).
% nat_diff_add_eq2
thf(fact_1925_int__sgnE,axiom,
! [K: int] :
~ ! [N2: nat,L3: int] :
( K
!= ( times_times @ int @ ( sgn_sgn @ int @ L3 ) @ ( semiring_1_of_nat @ int @ N2 ) ) ) ).
% int_sgnE
thf(fact_1926_div__eq__sgn__abs,axiom,
! [K: int,L: int] :
( ( ( sgn_sgn @ int @ K )
= ( sgn_sgn @ int @ L ) )
=> ( ( divide_divide @ int @ K @ L )
= ( divide_divide @ int @ ( abs_abs @ int @ K ) @ ( abs_abs @ int @ L ) ) ) ) ).
% div_eq_sgn_abs
thf(fact_1927_zsgn__def,axiom,
( ( sgn_sgn @ int )
= ( ^ [I2: int] :
( if @ int
@ ( I2
= ( zero_zero @ int ) )
@ ( zero_zero @ int )
@ ( if @ int @ ( ord_less @ int @ ( zero_zero @ int ) @ I2 ) @ ( one_one @ int ) @ ( uminus_uminus @ int @ ( one_one @ int ) ) ) ) ) ) ).
% zsgn_def
thf(fact_1928_div__sgn__abs__cancel,axiom,
! [V2: int,K: int,L: int] :
( ( V2
!= ( zero_zero @ int ) )
=> ( ( divide_divide @ int @ ( times_times @ int @ ( sgn_sgn @ int @ V2 ) @ ( abs_abs @ int @ K ) ) @ ( times_times @ int @ ( sgn_sgn @ int @ V2 ) @ ( abs_abs @ int @ L ) ) )
= ( divide_divide @ int @ ( abs_abs @ int @ K ) @ ( abs_abs @ int @ L ) ) ) ) ).
% div_sgn_abs_cancel
thf(fact_1929_nat__mult__eq__cancel__disj,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ( times_times @ nat @ K @ M2 )
= ( times_times @ nat @ K @ N ) )
= ( ( K
= ( zero_zero @ nat ) )
| ( M2 = N ) ) ) ).
% nat_mult_eq_cancel_disj
thf(fact_1930_sgn__zero__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X: A] :
( ( ( sgn_sgn @ A @ X )
= ( zero_zero @ A ) )
= ( X
= ( zero_zero @ A ) ) ) ) ).
% sgn_zero_iff
thf(fact_1931_Real__Vector__Spaces_Osgn__minus,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X: A] :
( ( sgn_sgn @ A @ ( uminus_uminus @ A @ X ) )
= ( uminus_uminus @ A @ ( sgn_sgn @ A @ X ) ) ) ) ).
% Real_Vector_Spaces.sgn_minus
thf(fact_1932_nat__mult__less__cancel1,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ( ord_less @ nat @ ( times_times @ nat @ K @ M2 ) @ ( times_times @ nat @ K @ N ) )
= ( ord_less @ nat @ M2 @ N ) ) ) ).
% nat_mult_less_cancel1
thf(fact_1933_nat__mult__eq__cancel1,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ( ( times_times @ nat @ K @ M2 )
= ( times_times @ nat @ K @ N ) )
= ( M2 = N ) ) ) ).
% nat_mult_eq_cancel1
thf(fact_1934_real__sgn__eq,axiom,
( ( sgn_sgn @ real )
= ( ^ [X2: real] : ( divide_divide @ real @ X2 @ ( abs_abs @ real @ X2 ) ) ) ) ).
% real_sgn_eq
thf(fact_1935_nat__mult__le__cancel1,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ( ord_less_eq @ nat @ ( times_times @ nat @ K @ M2 ) @ ( times_times @ nat @ K @ N ) )
= ( ord_less_eq @ nat @ M2 @ N ) ) ) ).
% nat_mult_le_cancel1
thf(fact_1936_nat__mult__div__cancel1,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ( divide_divide @ nat @ ( times_times @ nat @ K @ M2 ) @ ( times_times @ nat @ K @ N ) )
= ( divide_divide @ nat @ M2 @ N ) ) ) ).
% nat_mult_div_cancel1
thf(fact_1937_nat__eq__add__iff1,axiom,
! [J2: nat,I: nat,U: nat,M2: nat,N: nat] :
( ( ord_less_eq @ nat @ J2 @ I )
=> ( ( ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ M2 )
= ( plus_plus @ nat @ ( times_times @ nat @ J2 @ U ) @ N ) )
= ( ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ I @ J2 ) @ U ) @ M2 )
= N ) ) ) ).
% nat_eq_add_iff1
thf(fact_1938_nat__eq__add__iff2,axiom,
! [I: nat,J2: nat,U: nat,M2: nat,N: nat] :
( ( ord_less_eq @ nat @ I @ J2 )
=> ( ( ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ M2 )
= ( plus_plus @ nat @ ( times_times @ nat @ J2 @ U ) @ N ) )
= ( M2
= ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ J2 @ I ) @ U ) @ N ) ) ) ) ).
% nat_eq_add_iff2
thf(fact_1939_nat__le__add__iff1,axiom,
! [J2: nat,I: nat,U: nat,M2: nat,N: nat] :
( ( ord_less_eq @ nat @ J2 @ I )
=> ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ M2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ J2 @ U ) @ N ) )
= ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ I @ J2 ) @ U ) @ M2 ) @ N ) ) ) ).
% nat_le_add_iff1
thf(fact_1940_nat__le__add__iff2,axiom,
! [I: nat,J2: nat,U: nat,M2: nat,N: nat] :
( ( ord_less_eq @ nat @ I @ J2 )
=> ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ M2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ J2 @ U ) @ N ) )
= ( ord_less_eq @ nat @ M2 @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ J2 @ I ) @ U ) @ N ) ) ) ) ).
% nat_le_add_iff2
thf(fact_1941_nat__diff__add__eq1,axiom,
! [J2: nat,I: nat,U: nat,M2: nat,N: nat] :
( ( ord_less_eq @ nat @ J2 @ I )
=> ( ( minus_minus @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I @ U ) @ M2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ J2 @ U ) @ N ) )
= ( minus_minus @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ I @ J2 ) @ U ) @ M2 ) @ N ) ) ) ).
% nat_diff_add_eq1
thf(fact_1942_Cauchy__iff2,axiom,
( ( topolo3814608138187158403Cauchy @ real )
= ( ^ [X7: nat > real] :
! [J3: nat] :
? [M7: nat] :
! [M5: nat] :
( ( ord_less_eq @ nat @ M7 @ M5 )
=> ! [N5: nat] :
( ( ord_less_eq @ nat @ M7 @ N5 )
=> ( ord_less @ real @ ( abs_abs @ real @ ( minus_minus @ real @ ( X7 @ M5 ) @ ( X7 @ N5 ) ) ) @ ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ ( suc @ J3 ) ) ) ) ) ) ) ) ).
% Cauchy_iff2
thf(fact_1943_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_1944_pochhammer__minus_H,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [B2: A,K: nat] :
( ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ ( minus_minus @ A @ B2 @ ( semiring_1_of_nat @ A @ K ) ) @ ( one_one @ A ) ) @ K )
= ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ K ) @ ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ B2 ) @ K ) ) ) ) ).
% pochhammer_minus'
thf(fact_1945_pochhammer__minus,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [B2: A,K: nat] :
( ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ B2 ) @ K )
= ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ K ) @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ ( minus_minus @ A @ B2 @ ( semiring_1_of_nat @ A @ K ) ) @ ( one_one @ A ) ) @ K ) ) ) ) ).
% pochhammer_minus
thf(fact_1946_gbinomial__absorption_H,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,A3: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ( gbinomial @ A @ A3 @ K )
= ( times_times @ A @ ( divide_divide @ A @ A3 @ ( semiring_1_of_nat @ A @ K ) ) @ ( gbinomial @ A @ ( minus_minus @ A @ A3 @ ( one_one @ A ) ) @ ( minus_minus @ nat @ K @ ( one_one @ nat ) ) ) ) ) ) ) ).
% gbinomial_absorption'
thf(fact_1947_mult__ceiling__le__Ints,axiom,
! [A: $tType,B: $tType] :
( ( ( archim2362893244070406136eiling @ B )
& ( linordered_idom @ A ) )
=> ! [A3: B,B2: B] :
( ( ord_less_eq @ B @ ( zero_zero @ B ) @ A3 )
=> ( ( member @ B @ A3 @ ( ring_1_Ints @ B ) )
=> ( ord_less_eq @ A @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ B @ ( times_times @ B @ A3 @ B2 ) ) ) @ ( ring_1_of_int @ A @ ( times_times @ int @ ( archimedean_ceiling @ B @ A3 ) @ ( archimedean_ceiling @ B @ B2 ) ) ) ) ) ) ) ).
% mult_ceiling_le_Ints
thf(fact_1948_le__mult__floor__Ints,axiom,
! [A: $tType,B: $tType] :
( ( ( archim2362893244070406136eiling @ B )
& ( linordered_idom @ A ) )
=> ! [A3: B,B2: B] :
( ( ord_less_eq @ B @ ( zero_zero @ B ) @ A3 )
=> ( ( member @ B @ A3 @ ( ring_1_Ints @ B ) )
=> ( ord_less_eq @ A @ ( ring_1_of_int @ A @ ( times_times @ int @ ( archim6421214686448440834_floor @ B @ A3 ) @ ( archim6421214686448440834_floor @ B @ B2 ) ) ) @ ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ B @ ( times_times @ B @ A3 @ B2 ) ) ) ) ) ) ) ).
% le_mult_floor_Ints
thf(fact_1949_frac__frac,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( archimedean_frac @ A @ ( archimedean_frac @ A @ X ) )
= ( archimedean_frac @ A @ X ) ) ) ).
% frac_frac
thf(fact_1950_gbinomial__1,axiom,
! [A: $tType] :
( ( ( semiring_char_0 @ A )
& ( semidom_divide @ A ) )
=> ! [A3: A] :
( ( gbinomial @ A @ A3 @ ( one_one @ nat ) )
= A3 ) ) ).
% gbinomial_1
thf(fact_1951_pochhammer__1,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A3: A] :
( ( comm_s3205402744901411588hammer @ A @ A3 @ ( one_one @ nat ) )
= A3 ) ) ).
% pochhammer_1
thf(fact_1952_frac__in__Ints__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( member @ A @ ( archimedean_frac @ A @ X ) @ ( ring_1_Ints @ A ) )
= ( member @ A @ X @ ( ring_1_Ints @ A ) ) ) ) ).
% frac_in_Ints_iff
thf(fact_1953_gbinomial__0_I2_J,axiom,
! [B: $tType] :
( ( ( semiring_char_0 @ B )
& ( semidom_divide @ B ) )
=> ! [K: nat] :
( ( gbinomial @ B @ ( zero_zero @ B ) @ ( suc @ K ) )
= ( zero_zero @ B ) ) ) ).
% gbinomial_0(2)
thf(fact_1954_gbinomial__0_I1_J,axiom,
! [A: $tType] :
( ( ( semiring_char_0 @ A )
& ( semidom_divide @ A ) )
=> ! [A3: A] :
( ( gbinomial @ A @ A3 @ ( zero_zero @ nat ) )
= ( one_one @ A ) ) ) ).
% gbinomial_0(1)
thf(fact_1955_pochhammer__0,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A3: A] :
( ( comm_s3205402744901411588hammer @ A @ A3 @ ( zero_zero @ nat ) )
= ( one_one @ A ) ) ) ).
% pochhammer_0
thf(fact_1956_gbinomial__Suc0,axiom,
! [A: $tType] :
( ( ( semiring_char_0 @ A )
& ( semidom_divide @ A ) )
=> ! [A3: A] :
( ( gbinomial @ A @ A3 @ ( suc @ ( zero_zero @ nat ) ) )
= A3 ) ) ).
% gbinomial_Suc0
thf(fact_1957_pochhammer__Suc0,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A3: A] :
( ( comm_s3205402744901411588hammer @ A @ A3 @ ( suc @ ( zero_zero @ nat ) ) )
= A3 ) ) ).
% pochhammer_Suc0
thf(fact_1958_frac__of__int,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z2: int] :
( ( archimedean_frac @ A @ ( ring_1_of_int @ A @ Z2 ) )
= ( zero_zero @ A ) ) ) ).
% frac_of_int
thf(fact_1959_frac__eq__0__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A] :
( ( ( archimedean_frac @ A @ X )
= ( zero_zero @ A ) )
= ( member @ A @ X @ ( ring_1_Ints @ A ) ) ) ) ).
% frac_eq_0_iff
thf(fact_1960_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_1961_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_1962_of__nat__gbinomial,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [N: nat,K: nat] :
( ( semiring_1_of_nat @ A @ ( gbinomial @ nat @ N @ K ) )
= ( gbinomial @ A @ ( semiring_1_of_nat @ A @ N ) @ K ) ) ) ).
% of_nat_gbinomial
thf(fact_1963_pochhammer__of__nat,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [X: nat,N: nat] :
( ( comm_s3205402744901411588hammer @ A @ ( semiring_1_of_nat @ A @ X ) @ N )
= ( semiring_1_of_nat @ A @ ( comm_s3205402744901411588hammer @ nat @ X @ N ) ) ) ) ).
% pochhammer_of_nat
thf(fact_1964_Ints__0,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( member @ A @ ( zero_zero @ A ) @ ( ring_1_Ints @ A ) ) ) ).
% Ints_0
thf(fact_1965_Ints__mult,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [A3: A,B2: A] :
( ( member @ A @ A3 @ ( ring_1_Ints @ A ) )
=> ( ( member @ A @ B2 @ ( ring_1_Ints @ A ) )
=> ( member @ A @ ( times_times @ A @ A3 @ B2 ) @ ( ring_1_Ints @ A ) ) ) ) ) ).
% Ints_mult
thf(fact_1966_Ints__1,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( member @ A @ ( one_one @ A ) @ ( ring_1_Ints @ A ) ) ) ).
% Ints_1
thf(fact_1967_Ints__add,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [A3: A,B2: A] :
( ( member @ A @ A3 @ ( ring_1_Ints @ A ) )
=> ( ( member @ A @ B2 @ ( ring_1_Ints @ A ) )
=> ( member @ A @ ( plus_plus @ A @ A3 @ B2 ) @ ( ring_1_Ints @ A ) ) ) ) ) ).
% Ints_add
thf(fact_1968_Ints__diff,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [A3: A,B2: A] :
( ( member @ A @ A3 @ ( ring_1_Ints @ A ) )
=> ( ( member @ A @ B2 @ ( ring_1_Ints @ A ) )
=> ( member @ A @ ( minus_minus @ A @ A3 @ B2 ) @ ( ring_1_Ints @ A ) ) ) ) ) ).
% Ints_diff
thf(fact_1969_minus__in__Ints__iff,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X: A] :
( ( member @ A @ ( uminus_uminus @ A @ X ) @ ( ring_1_Ints @ A ) )
= ( member @ A @ X @ ( ring_1_Ints @ A ) ) ) ) ).
% minus_in_Ints_iff
thf(fact_1970_Ints__minus,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [A3: A] :
( ( member @ A @ A3 @ ( ring_1_Ints @ A ) )
=> ( member @ A @ ( uminus_uminus @ A @ A3 ) @ ( ring_1_Ints @ A ) ) ) ) ).
% Ints_minus
thf(fact_1971_Ints__power,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [A3: A,N: nat] :
( ( member @ A @ A3 @ ( ring_1_Ints @ A ) )
=> ( member @ A @ ( power_power @ A @ A3 @ N ) @ ( ring_1_Ints @ A ) ) ) ) ).
% Ints_power
thf(fact_1972_Ints__of__nat,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [N: nat] : ( member @ A @ ( semiring_1_of_nat @ A @ N ) @ ( ring_1_Ints @ A ) ) ) ).
% Ints_of_nat
thf(fact_1973_Ints__abs,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A] :
( ( member @ A @ A3 @ ( ring_1_Ints @ A ) )
=> ( member @ A @ ( abs_abs @ A @ A3 ) @ ( ring_1_Ints @ A ) ) ) ) ).
% Ints_abs
thf(fact_1974_Ints__of__int,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [Z2: int] : ( member @ A @ ( ring_1_of_int @ A @ Z2 ) @ ( ring_1_Ints @ A ) ) ) ).
% Ints_of_int
thf(fact_1975_Ints__induct,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [Q5: A,P: A > $o] :
( ( member @ A @ Q5 @ ( ring_1_Ints @ A ) )
=> ( ! [Z4: int] : ( P @ ( ring_1_of_int @ A @ Z4 ) )
=> ( P @ Q5 ) ) ) ) ).
% Ints_induct
thf(fact_1976_Ints__cases,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [Q5: A] :
( ( member @ A @ Q5 @ ( ring_1_Ints @ A ) )
=> ~ ! [Z4: int] :
( Q5
!= ( ring_1_of_int @ A @ Z4 ) ) ) ) ).
% Ints_cases
thf(fact_1977_gbinomial__Suc__Suc,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: A,K: nat] :
( ( gbinomial @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) @ ( suc @ K ) )
= ( plus_plus @ A @ ( gbinomial @ A @ A3 @ K ) @ ( gbinomial @ A @ A3 @ ( suc @ K ) ) ) ) ) ).
% gbinomial_Suc_Suc
thf(fact_1978_gbinomial__of__nat__symmetric,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,N: nat] :
( ( ord_less_eq @ nat @ K @ N )
=> ( ( gbinomial @ A @ ( semiring_1_of_nat @ A @ N ) @ K )
= ( gbinomial @ A @ ( semiring_1_of_nat @ A @ N ) @ ( minus_minus @ nat @ N @ K ) ) ) ) ) ).
% gbinomial_of_nat_symmetric
thf(fact_1979_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_1980_Ints__double__eq__0__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [A3: A] :
( ( member @ A @ A3 @ ( ring_1_Ints @ A ) )
=> ( ( ( plus_plus @ A @ A3 @ A3 )
= ( zero_zero @ A ) )
= ( A3
= ( zero_zero @ A ) ) ) ) ) ).
% Ints_double_eq_0_iff
thf(fact_1981_pochhammer__neq__0__mono,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: A,M2: nat,N: nat] :
( ( ( comm_s3205402744901411588hammer @ A @ A3 @ M2 )
!= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ nat @ N @ M2 )
=> ( ( comm_s3205402744901411588hammer @ A @ A3 @ N )
!= ( zero_zero @ A ) ) ) ) ) ).
% pochhammer_neq_0_mono
thf(fact_1982_pochhammer__eq__0__mono,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: A,N: nat,M2: nat] :
( ( ( comm_s3205402744901411588hammer @ A @ A3 @ N )
= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ nat @ N @ M2 )
=> ( ( comm_s3205402744901411588hammer @ A @ A3 @ M2 )
= ( zero_zero @ A ) ) ) ) ) ).
% pochhammer_eq_0_mono
thf(fact_1983_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_1984_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_1985_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_1986_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_1987_frac__unique__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,A3: A] :
( ( ( archimedean_frac @ A @ X )
= A3 )
= ( ( member @ A @ ( minus_minus @ A @ X @ A3 ) @ ( ring_1_Ints @ A ) )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
& ( ord_less @ A @ A3 @ ( one_one @ A ) ) ) ) ) ).
% frac_unique_iff
thf(fact_1988_gbinomial__addition__formula,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: A,K: nat] :
( ( gbinomial @ A @ A3 @ ( suc @ K ) )
= ( plus_plus @ A @ ( gbinomial @ A @ ( minus_minus @ A @ A3 @ ( one_one @ A ) ) @ ( suc @ K ) ) @ ( gbinomial @ A @ ( minus_minus @ A @ A3 @ ( one_one @ A ) ) @ K ) ) ) ) ).
% gbinomial_addition_formula
thf(fact_1989_gbinomial__absorb__comp,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: A,K: nat] :
( ( times_times @ A @ ( minus_minus @ A @ A3 @ ( semiring_1_of_nat @ A @ K ) ) @ ( gbinomial @ A @ A3 @ K ) )
= ( times_times @ A @ A3 @ ( gbinomial @ A @ ( minus_minus @ A @ A3 @ ( one_one @ A ) ) @ K ) ) ) ) ).
% gbinomial_absorb_comp
thf(fact_1990_gbinomial__ge__n__over__k__pow__k,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [K: nat,A3: A] :
( ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ K ) @ A3 )
=> ( ord_less_eq @ A @ ( power_power @ A @ ( divide_divide @ A @ A3 @ ( semiring_1_of_nat @ A @ K ) ) @ K ) @ ( gbinomial @ A @ A3 @ K ) ) ) ) ).
% gbinomial_ge_n_over_k_pow_k
thf(fact_1991_gbinomial__mult__1,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: A,K: nat] :
( ( times_times @ A @ A3 @ ( gbinomial @ A @ A3 @ K ) )
= ( plus_plus @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ K ) @ ( gbinomial @ A @ A3 @ K ) ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( suc @ K ) ) @ ( gbinomial @ A @ A3 @ ( suc @ K ) ) ) ) ) ) ).
% gbinomial_mult_1
thf(fact_1992_gbinomial__mult__1_H,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: A,K: nat] :
( ( times_times @ A @ ( gbinomial @ A @ A3 @ K ) @ A3 )
= ( plus_plus @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ K ) @ ( gbinomial @ A @ A3 @ K ) ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( suc @ K ) ) @ ( gbinomial @ A @ A3 @ ( suc @ K ) ) ) ) ) ) ).
% gbinomial_mult_1'
thf(fact_1993_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_1994_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_1995_Ints__odd__nonzero,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [A3: A] :
( ( member @ A @ A3 @ ( ring_1_Ints @ A ) )
=> ( ( plus_plus @ A @ ( plus_plus @ A @ ( one_one @ A ) @ A3 ) @ A3 )
!= ( zero_zero @ A ) ) ) ) ).
% Ints_odd_nonzero
thf(fact_1996_Suc__times__gbinomial,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,A3: A] :
( ( times_times @ A @ ( semiring_1_of_nat @ A @ ( suc @ K ) ) @ ( gbinomial @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) @ ( suc @ K ) ) )
= ( times_times @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) @ ( gbinomial @ A @ A3 @ K ) ) ) ) ).
% Suc_times_gbinomial
thf(fact_1997_gbinomial__absorption,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,A3: A] :
( ( times_times @ A @ ( semiring_1_of_nat @ A @ ( suc @ K ) ) @ ( gbinomial @ A @ A3 @ ( suc @ K ) ) )
= ( times_times @ A @ A3 @ ( gbinomial @ A @ ( minus_minus @ A @ A3 @ ( one_one @ A ) ) @ K ) ) ) ) ).
% gbinomial_absorption
thf(fact_1998_gbinomial__trinomial__revision,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,M2: nat,A3: A] :
( ( ord_less_eq @ nat @ K @ M2 )
=> ( ( times_times @ A @ ( gbinomial @ A @ A3 @ M2 ) @ ( gbinomial @ A @ ( semiring_1_of_nat @ A @ M2 ) @ K ) )
= ( times_times @ A @ ( gbinomial @ A @ A3 @ K ) @ ( gbinomial @ A @ ( minus_minus @ A @ A3 @ ( semiring_1_of_nat @ A @ K ) ) @ ( minus_minus @ nat @ M2 @ K ) ) ) ) ) ) ).
% gbinomial_trinomial_revision
thf(fact_1999_Ints__odd__less__0,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A] :
( ( member @ A @ A3 @ ( ring_1_Ints @ A ) )
=> ( ( ord_less @ A @ ( plus_plus @ A @ ( plus_plus @ A @ ( one_one @ A ) @ A3 ) @ A3 ) @ ( zero_zero @ A ) )
= ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ) ).
% Ints_odd_less_0
thf(fact_2000_frac__def,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archimedean_frac @ A )
= ( ^ [X2: A] : ( minus_minus @ A @ X2 @ ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ A @ X2 ) ) ) ) ) ) ).
% frac_def
thf(fact_2001_pochhammer__rec,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A3: A,N: nat] :
( ( comm_s3205402744901411588hammer @ A @ A3 @ ( suc @ N ) )
= ( times_times @ A @ A3 @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) @ N ) ) ) ) ).
% pochhammer_rec
thf(fact_2002_pochhammer__Suc,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A3: A,N: nat] :
( ( comm_s3205402744901411588hammer @ A @ A3 @ ( suc @ N ) )
= ( times_times @ A @ ( comm_s3205402744901411588hammer @ A @ A3 @ N ) @ ( plus_plus @ A @ A3 @ ( semiring_1_of_nat @ A @ N ) ) ) ) ) ).
% pochhammer_Suc
thf(fact_2003_pochhammer__rec_H,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [Z2: A,N: nat] :
( ( comm_s3205402744901411588hammer @ A @ Z2 @ ( suc @ N ) )
= ( times_times @ A @ ( plus_plus @ A @ Z2 @ ( semiring_1_of_nat @ A @ N ) ) @ ( comm_s3205402744901411588hammer @ A @ Z2 @ N ) ) ) ) ).
% pochhammer_rec'
thf(fact_2004_pochhammer__eq__0__iff,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: A,N: nat] :
( ( ( comm_s3205402744901411588hammer @ A @ A3 @ N )
= ( zero_zero @ A ) )
= ( ? [K3: nat] :
( ( ord_less @ nat @ K3 @ N )
& ( A3
= ( uminus_uminus @ A @ ( semiring_1_of_nat @ A @ K3 ) ) ) ) ) ) ) ).
% pochhammer_eq_0_iff
thf(fact_2005_pochhammer__of__nat__eq__0__iff,axiom,
! [A: $tType] :
( ( ( ring_char_0 @ A )
& ( idom @ A ) )
=> ! [N: nat,K: nat] :
( ( ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ ( semiring_1_of_nat @ A @ N ) ) @ K )
= ( zero_zero @ A ) )
= ( ord_less @ nat @ N @ K ) ) ) ).
% pochhammer_of_nat_eq_0_iff
thf(fact_2006_pochhammer__of__nat__eq__0__lemma,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [N: nat,K: nat] :
( ( ord_less @ nat @ N @ K )
=> ( ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ ( semiring_1_of_nat @ A @ N ) ) @ K )
= ( zero_zero @ A ) ) ) ) ).
% pochhammer_of_nat_eq_0_lemma
thf(fact_2007_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_2008_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_2009_pochhammer__of__nat__eq__0__lemma_H,axiom,
! [A: $tType] :
( ( ( ring_char_0 @ A )
& ( idom @ A ) )
=> ! [K: nat,N: nat] :
( ( ord_less_eq @ nat @ K @ N )
=> ( ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ ( semiring_1_of_nat @ A @ N ) ) @ K )
!= ( zero_zero @ A ) ) ) ) ).
% pochhammer_of_nat_eq_0_lemma'
thf(fact_2010_pochhammer__product_H,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [Z2: A,N: nat,M2: nat] :
( ( comm_s3205402744901411588hammer @ A @ Z2 @ ( plus_plus @ nat @ N @ M2 ) )
= ( times_times @ A @ ( comm_s3205402744901411588hammer @ A @ Z2 @ N ) @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ Z2 @ ( semiring_1_of_nat @ A @ N ) ) @ M2 ) ) ) ) ).
% pochhammer_product'
thf(fact_2011_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_2012_gbinomial__rec,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: A,K: nat] :
( ( gbinomial @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) @ ( suc @ K ) )
= ( times_times @ A @ ( gbinomial @ A @ A3 @ K ) @ ( divide_divide @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) @ ( semiring_1_of_nat @ A @ ( suc @ K ) ) ) ) ) ) ).
% gbinomial_rec
thf(fact_2013_gbinomial__factors,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: A,K: nat] :
( ( gbinomial @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) @ ( suc @ K ) )
= ( times_times @ A @ ( divide_divide @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) @ ( semiring_1_of_nat @ A @ ( suc @ K ) ) ) @ ( gbinomial @ A @ A3 @ K ) ) ) ) ).
% gbinomial_factors
thf(fact_2014_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_2015_gbinomial__index__swap,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,N: nat] :
( ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ K ) @ ( gbinomial @ A @ ( minus_minus @ A @ ( uminus_uminus @ A @ ( semiring_1_of_nat @ A @ N ) ) @ ( one_one @ A ) ) @ K ) )
= ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N ) @ ( gbinomial @ A @ ( minus_minus @ A @ ( uminus_uminus @ A @ ( semiring_1_of_nat @ A @ K ) ) @ ( one_one @ A ) ) @ N ) ) ) ) ).
% gbinomial_index_swap
thf(fact_2016_gbinomial__minus,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: A,K: nat] :
( ( gbinomial @ A @ ( uminus_uminus @ A @ A3 ) @ K )
= ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ K ) @ ( gbinomial @ A @ ( minus_minus @ A @ ( plus_plus @ A @ A3 @ ( semiring_1_of_nat @ A @ K ) ) @ ( one_one @ A ) ) @ K ) ) ) ) ).
% gbinomial_minus
thf(fact_2017_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_2018_gbinomial__reduce__nat,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,A3: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ( gbinomial @ A @ A3 @ K )
= ( plus_plus @ A @ ( gbinomial @ A @ ( minus_minus @ A @ A3 @ ( one_one @ A ) ) @ ( minus_minus @ nat @ K @ ( one_one @ nat ) ) ) @ ( gbinomial @ A @ ( minus_minus @ A @ A3 @ ( one_one @ A ) ) @ K ) ) ) ) ) ).
% gbinomial_reduce_nat
thf(fact_2019_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_2020_pochhammer__product,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [M2: nat,N: nat,Z2: A] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( comm_s3205402744901411588hammer @ A @ Z2 @ N )
= ( times_times @ A @ ( comm_s3205402744901411588hammer @ A @ Z2 @ M2 ) @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ Z2 @ ( semiring_1_of_nat @ A @ M2 ) ) @ ( minus_minus @ nat @ N @ M2 ) ) ) ) ) ) ).
% pochhammer_product
thf(fact_2021_pochhammer__absorb__comp,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [R4: A,K: nat] :
( ( times_times @ A @ ( minus_minus @ A @ R4 @ ( semiring_1_of_nat @ A @ K ) ) @ ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ R4 ) @ K ) )
= ( times_times @ A @ R4 @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ ( uminus_uminus @ A @ R4 ) @ ( one_one @ A ) ) @ K ) ) ) ) ).
% pochhammer_absorb_comp
thf(fact_2022_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_2023_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_2024_power_Opower__eq__if,axiom,
! [A: $tType] :
( ( power2 @ A )
= ( ^ [One: A,Times: A > A > A,P6: A,M5: nat] :
( if @ A
@ ( M5
= ( zero_zero @ nat ) )
@ One
@ ( Times @ P6 @ ( power2 @ A @ One @ Times @ P6 @ ( minus_minus @ nat @ M5 @ ( one_one @ nat ) ) ) ) ) ) ) ).
% power.power_eq_if
thf(fact_2025_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_2026_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_2027_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_2028_fact__num__eq__if,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ( ( semiring_char_0_fact @ A )
= ( ^ [M5: nat] :
( if @ A
@ ( M5
= ( zero_zero @ nat ) )
@ ( one_one @ A )
@ ( times_times @ A @ ( semiring_1_of_nat @ A @ M5 ) @ ( semiring_char_0_fact @ A @ ( minus_minus @ nat @ M5 @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% fact_num_eq_if
thf(fact_2029_of__nat__fact,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [N: nat] :
( ( semiring_1_of_nat @ A @ ( semiring_char_0_fact @ nat @ N ) )
= ( semiring_char_0_fact @ A @ N ) ) ) ).
% of_nat_fact
thf(fact_2030_length__rotate1,axiom,
! [A: $tType,Xs: list @ A] :
( ( size_size @ ( list @ A ) @ ( rotate1 @ A @ Xs ) )
= ( size_size @ ( list @ A ) @ Xs ) ) ).
% length_rotate1
thf(fact_2031_fact__0,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ( ( semiring_char_0_fact @ A @ ( zero_zero @ nat ) )
= ( one_one @ A ) ) ) ).
% fact_0
thf(fact_2032_fact__1,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ( ( semiring_char_0_fact @ A @ ( one_one @ nat ) )
= ( one_one @ A ) ) ) ).
% fact_1
thf(fact_2033_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_2034_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_2035_fact__nonzero,axiom,
! [A: $tType] :
( ( ( semiring_char_0 @ A )
& ( semiri3467727345109120633visors @ A ) )
=> ! [N: nat] :
( ( semiring_char_0_fact @ A @ N )
!= ( zero_zero @ A ) ) ) ).
% fact_nonzero
thf(fact_2036_fact__less__mono__nat,axiom,
! [M2: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M2 )
=> ( ( ord_less @ nat @ M2 @ N )
=> ( ord_less @ nat @ ( semiring_char_0_fact @ nat @ M2 ) @ ( semiring_char_0_fact @ nat @ N ) ) ) ) ).
% fact_less_mono_nat
thf(fact_2037_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_2038_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_2039_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_2040_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_2041_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_2042_power_Opower_Opower__Suc,axiom,
! [A: $tType,One2: A,Times2: A > A > A,A3: A,N: nat] :
( ( power2 @ A @ One2 @ Times2 @ A3 @ ( suc @ N ) )
= ( Times2 @ A3 @ ( power2 @ A @ One2 @ Times2 @ A3 @ N ) ) ) ).
% power.power.power_Suc
thf(fact_2043_power_Opower_Opower__0,axiom,
! [A: $tType,One2: A,Times2: A > A > A,A3: A] :
( ( power2 @ A @ One2 @ Times2 @ A3 @ ( zero_zero @ nat ) )
= One2 ) ).
% power.power.power_0
thf(fact_2044_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_2045_fact__less__mono,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [M2: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M2 )
=> ( ( ord_less @ nat @ M2 @ N )
=> ( ord_less @ A @ ( semiring_char_0_fact @ A @ M2 ) @ ( semiring_char_0_fact @ A @ N ) ) ) ) ) ).
% fact_less_mono
thf(fact_2046_fact__mod,axiom,
! [A: $tType] :
( ( ( linordered_semidom @ A )
& ( semidom_modulo @ A ) )
=> ! [M2: nat,N: nat] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( modulo_modulo @ A @ ( semiring_char_0_fact @ A @ N ) @ ( semiring_char_0_fact @ A @ M2 ) )
= ( zero_zero @ A ) ) ) ) ).
% fact_mod
thf(fact_2047_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_2048_fact__diff__Suc,axiom,
! [N: nat,M2: nat] :
( ( ord_less @ nat @ N @ ( suc @ M2 ) )
=> ( ( semiring_char_0_fact @ nat @ ( minus_minus @ nat @ ( suc @ M2 ) @ N ) )
= ( times_times @ nat @ ( minus_minus @ nat @ ( suc @ M2 ) @ N ) @ ( semiring_char_0_fact @ nat @ ( minus_minus @ nat @ M2 @ N ) ) ) ) ) ).
% fact_diff_Suc
thf(fact_2049_fact__div__fact__le__pow,axiom,
! [R4: nat,N: nat] :
( ( ord_less_eq @ nat @ R4 @ N )
=> ( ord_less_eq @ nat @ ( divide_divide @ nat @ ( semiring_char_0_fact @ nat @ N ) @ ( semiring_char_0_fact @ nat @ ( minus_minus @ nat @ N @ R4 ) ) ) @ ( power_power @ nat @ N @ R4 ) ) ) ).
% fact_div_fact_le_pow
thf(fact_2050_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_2051_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_2052_norm__power__diff,axiom,
! [A: $tType] :
( ( ( comm_monoid_mult @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [Z2: A,W: A,M2: nat] :
( ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ Z2 ) @ ( 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 @ Z2 @ M2 ) @ ( power_power @ A @ W @ M2 ) ) ) @ ( times_times @ real @ ( semiring_1_of_nat @ real @ M2 ) @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ Z2 @ W ) ) ) ) ) ) ) ).
% norm_power_diff
thf(fact_2053_cppi,axiom,
! [D6: int,P: int > $o,P5: int > $o,A4: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D6 )
=> ( ? [Z5: int] :
! [X3: int] :
( ( ord_less @ int @ Z5 @ X3 )
=> ( ( P @ X3 )
= ( P5 @ X3 ) ) )
=> ( ! [X3: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D6 ) )
=> ! [Xb: int] :
( ( member @ int @ Xb @ A4 )
=> ( X3
!= ( minus_minus @ int @ Xb @ Xa2 ) ) ) )
=> ( ( P @ X3 )
=> ( P @ ( plus_plus @ int @ X3 @ D6 ) ) ) )
=> ( ! [X3: int,K2: int] :
( ( P5 @ X3 )
= ( P5 @ ( minus_minus @ int @ X3 @ ( times_times @ int @ K2 @ D6 ) ) ) )
=> ( ( ? [X7: int] : ( P @ X7 ) )
= ( ? [X2: int] :
( ( member @ int @ X2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D6 ) )
& ( P5 @ X2 ) )
| ? [X2: int] :
( ( member @ int @ X2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D6 ) )
& ? [Y5: int] :
( ( member @ int @ Y5 @ A4 )
& ( P @ ( minus_minus @ int @ Y5 @ X2 ) ) ) ) ) ) ) ) ) ) ).
% cppi
thf(fact_2054_cpmi,axiom,
! [D6: int,P: int > $o,P5: int > $o,B7: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D6 )
=> ( ? [Z5: int] :
! [X3: int] :
( ( ord_less @ int @ X3 @ Z5 )
=> ( ( P @ X3 )
= ( P5 @ X3 ) ) )
=> ( ! [X3: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D6 ) )
=> ! [Xb: int] :
( ( member @ int @ Xb @ B7 )
=> ( X3
!= ( plus_plus @ int @ Xb @ Xa2 ) ) ) )
=> ( ( P @ X3 )
=> ( P @ ( minus_minus @ int @ X3 @ D6 ) ) ) )
=> ( ! [X3: int,K2: int] :
( ( P5 @ X3 )
= ( P5 @ ( minus_minus @ int @ X3 @ ( times_times @ int @ K2 @ D6 ) ) ) )
=> ( ( ? [X7: int] : ( P @ X7 ) )
= ( ? [X2: int] :
( ( member @ int @ X2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D6 ) )
& ( P5 @ X2 ) )
| ? [X2: int] :
( ( member @ int @ X2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D6 ) )
& ? [Y5: int] :
( ( member @ int @ Y5 @ B7 )
& ( P @ ( plus_plus @ int @ Y5 @ X2 ) ) ) ) ) ) ) ) ) ) ).
% cpmi
thf(fact_2055_bset_I6_J,axiom,
! [D6: int,B7: set @ int,T2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D6 )
=> ! [X4: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D6 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ B7 )
=> ( X4
!= ( plus_plus @ int @ Xb2 @ Xa3 ) ) ) )
=> ( ( ord_less_eq @ int @ X4 @ T2 )
=> ( ord_less_eq @ int @ ( minus_minus @ int @ X4 @ D6 ) @ T2 ) ) ) ) ).
% bset(6)
thf(fact_2056_bset_I8_J,axiom,
! [D6: int,T2: int,B7: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D6 )
=> ( ( member @ int @ ( minus_minus @ int @ T2 @ ( one_one @ int ) ) @ B7 )
=> ! [X4: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D6 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ B7 )
=> ( X4
!= ( plus_plus @ int @ Xb2 @ Xa3 ) ) ) )
=> ( ( ord_less_eq @ int @ T2 @ X4 )
=> ( ord_less_eq @ int @ T2 @ ( minus_minus @ int @ X4 @ D6 ) ) ) ) ) ) ).
% bset(8)
thf(fact_2057_numeral__eq__iff,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [M2: num,N: num] :
( ( ( numeral_numeral @ A @ M2 )
= ( numeral_numeral @ A @ N ) )
= ( M2 = N ) ) ) ).
% numeral_eq_iff
thf(fact_2058_int__eq__iff__numeral,axiom,
! [M2: nat,V2: num] :
( ( ( semiring_1_of_nat @ int @ M2 )
= ( numeral_numeral @ int @ V2 ) )
= ( M2
= ( numeral_numeral @ nat @ V2 ) ) ) ).
% int_eq_iff_numeral
thf(fact_2059_nat__numeral,axiom,
! [K: num] :
( ( nat2 @ ( numeral_numeral @ int @ K ) )
= ( numeral_numeral @ nat @ K ) ) ).
% nat_numeral
thf(fact_2060_numeral__le__iff,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [M2: num,N: num] :
( ( ord_less_eq @ A @ ( numeral_numeral @ A @ M2 ) @ ( numeral_numeral @ A @ N ) )
= ( ord_less_eq @ num @ M2 @ N ) ) ) ).
% numeral_le_iff
thf(fact_2061_numeral__less__iff,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [M2: num,N: num] :
( ( ord_less @ A @ ( numeral_numeral @ A @ M2 ) @ ( numeral_numeral @ A @ N ) )
= ( ord_less @ num @ M2 @ N ) ) ) ).
% numeral_less_iff
thf(fact_2062_mult__numeral__left__semiring__numeral,axiom,
! [A: $tType] :
( ( semiring_numeral @ A )
=> ! [V2: num,W: num,Z2: A] :
( ( times_times @ A @ ( numeral_numeral @ A @ V2 ) @ ( times_times @ A @ ( numeral_numeral @ A @ W ) @ Z2 ) )
= ( times_times @ A @ ( numeral_numeral @ A @ ( times_times @ num @ V2 @ W ) ) @ Z2 ) ) ) ).
% mult_numeral_left_semiring_numeral
thf(fact_2063_numeral__times__numeral,axiom,
! [A: $tType] :
( ( semiring_numeral @ A )
=> ! [M2: num,N: num] :
( ( times_times @ A @ ( numeral_numeral @ A @ M2 ) @ ( numeral_numeral @ A @ N ) )
= ( numeral_numeral @ A @ ( times_times @ num @ M2 @ N ) ) ) ) ).
% numeral_times_numeral
thf(fact_2064_add__numeral__left,axiom,
! [A: $tType] :
( ( numeral @ A )
=> ! [V2: num,W: num,Z2: A] :
( ( plus_plus @ A @ ( numeral_numeral @ A @ V2 ) @ ( plus_plus @ A @ ( numeral_numeral @ A @ W ) @ Z2 ) )
= ( plus_plus @ A @ ( numeral_numeral @ A @ ( plus_plus @ num @ V2 @ W ) ) @ Z2 ) ) ) ).
% add_numeral_left
thf(fact_2065_numeral__plus__numeral,axiom,
! [A: $tType] :
( ( numeral @ A )
=> ! [M2: num,N: num] :
( ( plus_plus @ A @ ( numeral_numeral @ A @ M2 ) @ ( numeral_numeral @ A @ N ) )
= ( numeral_numeral @ A @ ( plus_plus @ num @ M2 @ N ) ) ) ) ).
% numeral_plus_numeral
thf(fact_2066_power__zero__numeral,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [K: num] :
( ( power_power @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ nat @ K ) )
= ( zero_zero @ A ) ) ) ).
% power_zero_numeral
thf(fact_2067_neg__numeral__eq__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [M2: num,N: num] :
( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
= ( M2 = N ) ) ) ).
% neg_numeral_eq_iff
thf(fact_2068_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_2069_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_2070_of__int__numeral,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [K: num] :
( ( ring_1_of_int @ A @ ( numeral_numeral @ int @ K ) )
= ( numeral_numeral @ A @ K ) ) ) ).
% of_int_numeral
thf(fact_2071_of__int__eq__numeral__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [Z2: int,N: num] :
( ( ( ring_1_of_int @ A @ Z2 )
= ( numeral_numeral @ A @ N ) )
= ( Z2
= ( numeral_numeral @ int @ N ) ) ) ) ).
% of_int_eq_numeral_iff
thf(fact_2072_norm__minus__cancel,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X: A] :
( ( real_V7770717601297561774m_norm @ A @ ( uminus_uminus @ A @ X ) )
= ( real_V7770717601297561774m_norm @ A @ X ) ) ) ).
% norm_minus_cancel
thf(fact_2073_floor__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V2: num] :
( ( archim6421214686448440834_floor @ A @ ( numeral_numeral @ A @ V2 ) )
= ( numeral_numeral @ int @ V2 ) ) ) ).
% floor_numeral
thf(fact_2074_ceiling__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V2: num] :
( ( archimedean_ceiling @ A @ ( numeral_numeral @ A @ V2 ) )
= ( numeral_numeral @ int @ V2 ) ) ) ).
% ceiling_numeral
thf(fact_2075_norm__fact,axiom,
! [A: $tType] :
( ( real_V2822296259951069270ebra_1 @ A )
=> ! [N: nat] :
( ( real_V7770717601297561774m_norm @ A @ ( semiring_char_0_fact @ A @ N ) )
= ( semiring_char_0_fact @ real @ N ) ) ) ).
% norm_fact
thf(fact_2076_numeral__powr__numeral__real,axiom,
! [M2: num,N: num] :
( ( powr @ real @ ( numeral_numeral @ real @ M2 ) @ ( numeral_numeral @ real @ N ) )
= ( power_power @ real @ ( numeral_numeral @ real @ M2 ) @ ( numeral_numeral @ nat @ N ) ) ) ).
% numeral_powr_numeral_real
thf(fact_2077_neg__numeral__le__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M2: num,N: num] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
= ( ord_less_eq @ num @ N @ M2 ) ) ) ).
% neg_numeral_le_iff
thf(fact_2078_distrib__right__numeral,axiom,
! [A: $tType] :
( ( ( numeral @ A )
& ( semiring @ A ) )
=> ! [A3: A,B2: A,V2: num] :
( ( times_times @ A @ ( plus_plus @ A @ A3 @ B2 ) @ ( numeral_numeral @ A @ V2 ) )
= ( plus_plus @ A @ ( times_times @ A @ A3 @ ( numeral_numeral @ A @ V2 ) ) @ ( times_times @ A @ B2 @ ( numeral_numeral @ A @ V2 ) ) ) ) ) ).
% distrib_right_numeral
thf(fact_2079_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_2080_neg__numeral__less__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M2: num,N: num] :
( ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
= ( ord_less @ num @ N @ M2 ) ) ) ).
% neg_numeral_less_iff
thf(fact_2081_left__diff__distrib__numeral,axiom,
! [A: $tType] :
( ( ( numeral @ A )
& ( ring @ A ) )
=> ! [A3: A,B2: A,V2: num] :
( ( times_times @ A @ ( minus_minus @ A @ A3 @ B2 ) @ ( numeral_numeral @ A @ V2 ) )
= ( minus_minus @ A @ ( times_times @ A @ A3 @ ( numeral_numeral @ A @ V2 ) ) @ ( times_times @ A @ B2 @ ( numeral_numeral @ A @ V2 ) ) ) ) ) ).
% left_diff_distrib_numeral
thf(fact_2082_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_2083_mult__neg__numeral__simps_I3_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [M2: num,N: num] :
( ( times_times @ A @ ( numeral_numeral @ A @ M2 ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( times_times @ num @ M2 @ N ) ) ) ) ) ).
% mult_neg_numeral_simps(3)
thf(fact_2084_mult__neg__numeral__simps_I2_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [M2: num,N: num] :
( ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) @ ( numeral_numeral @ A @ N ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( times_times @ num @ M2 @ N ) ) ) ) ) ).
% mult_neg_numeral_simps(2)
thf(fact_2085_mult__neg__numeral__simps_I1_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [M2: num,N: num] :
( ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
= ( numeral_numeral @ A @ ( times_times @ num @ M2 @ N ) ) ) ) ).
% mult_neg_numeral_simps(1)
thf(fact_2086_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_2087_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_2088_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_2089_add__neg__numeral__simps_I3_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M2: num,N: num] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
= ( uminus_uminus @ A @ ( plus_plus @ A @ ( numeral_numeral @ A @ M2 ) @ ( numeral_numeral @ A @ N ) ) ) ) ) ).
% add_neg_numeral_simps(3)
thf(fact_2090_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_2091_diff__numeral__simps_I3_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M2: num,N: num] :
( ( minus_minus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) @ ( numeral_numeral @ A @ N ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( plus_plus @ num @ M2 @ N ) ) ) ) ) ).
% diff_numeral_simps(3)
thf(fact_2092_diff__numeral__simps_I2_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M2: num,N: num] :
( ( minus_minus @ A @ ( numeral_numeral @ A @ M2 ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
= ( numeral_numeral @ A @ ( plus_plus @ num @ M2 @ N ) ) ) ) ).
% diff_numeral_simps(2)
thf(fact_2093_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_2094_norm__zero,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ( ( real_V7770717601297561774m_norm @ A @ ( zero_zero @ A ) )
= ( zero_zero @ real ) ) ) ).
% norm_zero
thf(fact_2095_norm__eq__zero,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X: A] :
( ( ( real_V7770717601297561774m_norm @ A @ X )
= ( zero_zero @ real ) )
= ( X
= ( zero_zero @ A ) ) ) ) ).
% norm_eq_zero
thf(fact_2096_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_2097_norm__one,axiom,
! [A: $tType] :
( ( real_V2822296259951069270ebra_1 @ A )
=> ( ( real_V7770717601297561774m_norm @ A @ ( one_one @ A ) )
= ( one_one @ real ) ) ) ).
% norm_one
thf(fact_2098_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_2099_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_2100_numeral__le__real__of__nat__iff,axiom,
! [N: num,M2: nat] :
( ( ord_less_eq @ real @ ( numeral_numeral @ real @ N ) @ ( semiring_1_of_nat @ real @ M2 ) )
= ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ N ) @ M2 ) ) ).
% numeral_le_real_of_nat_iff
thf(fact_2101_nat__neg__numeral,axiom,
! [K: num] :
( ( nat2 @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) )
= ( zero_zero @ nat ) ) ).
% nat_neg_numeral
thf(fact_2102_norm__of__nat,axiom,
! [A: $tType] :
( ( real_V2822296259951069270ebra_1 @ A )
=> ! [N: nat] :
( ( real_V7770717601297561774m_norm @ A @ ( semiring_1_of_nat @ A @ N ) )
= ( semiring_1_of_nat @ real @ N ) ) ) ).
% norm_of_nat
thf(fact_2103_diff__nat__numeral,axiom,
! [V2: num,V3: num] :
( ( minus_minus @ nat @ ( numeral_numeral @ nat @ V2 ) @ ( numeral_numeral @ nat @ V3 ) )
= ( nat2 @ ( minus_minus @ int @ ( numeral_numeral @ int @ V2 ) @ ( numeral_numeral @ int @ V3 ) ) ) ) ).
% diff_nat_numeral
thf(fact_2104_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_2105_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_2106_floor__divide__eq__div__numeral,axiom,
! [A3: num,B2: num] :
( ( archim6421214686448440834_floor @ real @ ( divide_divide @ real @ ( numeral_numeral @ real @ A3 ) @ ( numeral_numeral @ real @ B2 ) ) )
= ( divide_divide @ int @ ( numeral_numeral @ int @ A3 ) @ ( numeral_numeral @ int @ B2 ) ) ) ).
% floor_divide_eq_div_numeral
thf(fact_2107_le__divide__eq__numeral1_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A,W: num] :
( ( ord_less_eq @ A @ A3 @ ( divide_divide @ A @ B2 @ ( numeral_numeral @ A @ W ) ) )
= ( ord_less_eq @ A @ ( times_times @ A @ A3 @ ( numeral_numeral @ A @ W ) ) @ B2 ) ) ) ).
% le_divide_eq_numeral1(1)
thf(fact_2108_divide__le__eq__numeral1_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,W: num,A3: A] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ ( numeral_numeral @ A @ W ) ) @ A3 )
= ( ord_less_eq @ A @ B2 @ ( times_times @ A @ A3 @ ( numeral_numeral @ A @ W ) ) ) ) ) ).
% divide_le_eq_numeral1(1)
thf(fact_2109_divide__eq__eq__numeral1_I1_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B2: A,W: num,A3: A] :
( ( ( divide_divide @ A @ B2 @ ( numeral_numeral @ A @ W ) )
= A3 )
= ( ( ( ( numeral_numeral @ A @ W )
!= ( zero_zero @ A ) )
=> ( B2
= ( times_times @ A @ A3 @ ( numeral_numeral @ A @ W ) ) ) )
& ( ( ( numeral_numeral @ A @ W )
= ( zero_zero @ A ) )
=> ( A3
= ( zero_zero @ A ) ) ) ) ) ) ).
% divide_eq_eq_numeral1(1)
thf(fact_2110_eq__divide__eq__numeral1_I1_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A,B2: A,W: num] :
( ( A3
= ( divide_divide @ A @ B2 @ ( numeral_numeral @ A @ W ) ) )
= ( ( ( ( numeral_numeral @ A @ W )
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ A3 @ ( numeral_numeral @ A @ W ) )
= B2 ) )
& ( ( ( numeral_numeral @ A @ W )
= ( zero_zero @ A ) )
=> ( A3
= ( zero_zero @ A ) ) ) ) ) ) ).
% eq_divide_eq_numeral1(1)
thf(fact_2111_divide__less__eq__numeral1_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,W: num,A3: A] :
( ( ord_less @ A @ ( divide_divide @ A @ B2 @ ( numeral_numeral @ A @ W ) ) @ A3 )
= ( ord_less @ A @ B2 @ ( times_times @ A @ A3 @ ( numeral_numeral @ A @ W ) ) ) ) ) ).
% divide_less_eq_numeral1(1)
thf(fact_2112_less__divide__eq__numeral1_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A,W: num] :
( ( ord_less @ A @ A3 @ ( divide_divide @ A @ B2 @ ( numeral_numeral @ A @ W ) ) )
= ( ord_less @ A @ ( times_times @ A @ A3 @ ( numeral_numeral @ A @ W ) ) @ B2 ) ) ) ).
% less_divide_eq_numeral1(1)
thf(fact_2113_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_2114_zero__less__norm__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X: A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( real_V7770717601297561774m_norm @ A @ X ) )
= ( X
!= ( zero_zero @ A ) ) ) ) ).
% zero_less_norm_iff
thf(fact_2115_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_2116_of__int__numeral__le__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: num,Z2: int] :
( ( ord_less_eq @ A @ ( numeral_numeral @ A @ N ) @ ( ring_1_of_int @ A @ Z2 ) )
= ( ord_less_eq @ int @ ( numeral_numeral @ int @ N ) @ Z2 ) ) ) ).
% of_int_numeral_le_iff
thf(fact_2117_of__int__le__numeral__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z2: int,N: num] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Z2 ) @ ( numeral_numeral @ A @ N ) )
= ( ord_less_eq @ int @ Z2 @ ( numeral_numeral @ int @ N ) ) ) ) ).
% of_int_le_numeral_iff
thf(fact_2118_of__int__numeral__less__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: num,Z2: int] :
( ( ord_less @ A @ ( numeral_numeral @ A @ N ) @ ( ring_1_of_int @ A @ Z2 ) )
= ( ord_less @ int @ ( numeral_numeral @ int @ N ) @ Z2 ) ) ) ).
% of_int_numeral_less_iff
thf(fact_2119_of__int__less__numeral__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z2: int,N: num] :
( ( ord_less @ A @ ( ring_1_of_int @ A @ Z2 ) @ ( numeral_numeral @ A @ N ) )
= ( ord_less @ int @ Z2 @ ( numeral_numeral @ int @ N ) ) ) ) ).
% of_int_less_numeral_iff
thf(fact_2120_norm__divide__numeral,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [A3: A,W: num] :
( ( real_V7770717601297561774m_norm @ A @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ W ) ) )
= ( divide_divide @ real @ ( real_V7770717601297561774m_norm @ A @ A3 ) @ ( numeral_numeral @ real @ W ) ) ) ) ).
% norm_divide_numeral
thf(fact_2121_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_2122_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_2123_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_2124_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_2125_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_2126_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_2127_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_2128_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_2129_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_2130_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_2131_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_2132_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_2133_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_2134_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_2135_Suc__times__numeral__mod__eq,axiom,
! [K: num,N: nat] :
( ( ( numeral_numeral @ nat @ K )
!= ( one_one @ nat ) )
=> ( ( modulo_modulo @ nat @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ K ) @ N ) ) @ ( numeral_numeral @ nat @ K ) )
= ( one_one @ nat ) ) ) ).
% Suc_times_numeral_mod_eq
thf(fact_2136_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_2137_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_2138_ceiling__divide__eq__div__numeral,axiom,
! [A3: num,B2: num] :
( ( archimedean_ceiling @ real @ ( divide_divide @ real @ ( numeral_numeral @ real @ A3 ) @ ( numeral_numeral @ real @ B2 ) ) )
= ( uminus_uminus @ int @ ( divide_divide @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ A3 ) ) @ ( numeral_numeral @ int @ B2 ) ) ) ) ).
% ceiling_divide_eq_div_numeral
thf(fact_2139_le__divide__eq__numeral1_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A,W: num] :
( ( ord_less_eq @ A @ A3 @ ( divide_divide @ A @ B2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) ) )
= ( ord_less_eq @ A @ B2 @ ( times_times @ A @ A3 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) ) ) ) ) ).
% le_divide_eq_numeral1(2)
thf(fact_2140_divide__le__eq__numeral1_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,W: num,A3: A] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) ) @ A3 )
= ( ord_less_eq @ A @ ( times_times @ A @ A3 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) ) @ B2 ) ) ) ).
% divide_le_eq_numeral1(2)
thf(fact_2141_divide__eq__eq__numeral1_I2_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B2: A,W: num,A3: A] :
( ( ( divide_divide @ A @ B2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) )
= A3 )
= ( ( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) )
!= ( zero_zero @ A ) )
=> ( B2
= ( times_times @ A @ A3 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) ) ) )
& ( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) )
= ( zero_zero @ A ) )
=> ( A3
= ( zero_zero @ A ) ) ) ) ) ) ).
% divide_eq_eq_numeral1(2)
thf(fact_2142_eq__divide__eq__numeral1_I2_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: A,B2: A,W: num] :
( ( A3
= ( divide_divide @ A @ B2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) ) )
= ( ( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) )
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ A3 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) )
= B2 ) )
& ( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) )
= ( zero_zero @ A ) )
=> ( A3
= ( zero_zero @ A ) ) ) ) ) ) ).
% eq_divide_eq_numeral1(2)
thf(fact_2143_divide__less__eq__numeral1_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,W: num,A3: A] :
( ( ord_less @ A @ ( divide_divide @ A @ B2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) ) @ A3 )
= ( ord_less @ A @ ( times_times @ A @ A3 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) ) @ B2 ) ) ) ).
% divide_less_eq_numeral1(2)
thf(fact_2144_less__divide__eq__numeral1_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A,W: num] :
( ( ord_less @ A @ A3 @ ( divide_divide @ A @ B2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) ) )
= ( ord_less @ A @ B2 @ ( times_times @ A @ A3 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) ) ) ) ) ).
% less_divide_eq_numeral1(2)
thf(fact_2145_dbl__inc__simps_I1_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [K: num] :
( ( neg_numeral_dbl_inc @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ K ) ) )
= ( uminus_uminus @ A @ ( neg_numeral_dbl_dec @ A @ ( numeral_numeral @ A @ K ) ) ) ) ) ).
% dbl_inc_simps(1)
thf(fact_2146_dbl__dec__simps_I1_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [K: num] :
( ( neg_numeral_dbl_dec @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ K ) ) )
= ( uminus_uminus @ A @ ( neg_numeral_dbl_inc @ A @ ( numeral_numeral @ A @ K ) ) ) ) ) ).
% dbl_dec_simps(1)
thf(fact_2147_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_2148_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_2149_numeral__power__less__nat__cancel__iff,axiom,
! [X: num,N: nat,A3: int] :
( ( ord_less @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ X ) @ N ) @ ( nat2 @ A3 ) )
= ( ord_less @ int @ ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N ) @ A3 ) ) ).
% numeral_power_less_nat_cancel_iff
thf(fact_2150_nat__less__numeral__power__cancel__iff,axiom,
! [A3: int,X: num,N: nat] :
( ( ord_less @ nat @ ( nat2 @ A3 ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ X ) @ N ) )
= ( ord_less @ int @ A3 @ ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N ) ) ) ).
% nat_less_numeral_power_cancel_iff
thf(fact_2151_numeral__power__le__nat__cancel__iff,axiom,
! [X: num,N: nat,A3: int] :
( ( ord_less_eq @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ X ) @ N ) @ ( nat2 @ A3 ) )
= ( ord_less_eq @ int @ ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N ) @ A3 ) ) ).
% numeral_power_le_nat_cancel_iff
thf(fact_2152_nat__le__numeral__power__cancel__iff,axiom,
! [A3: int,X: num,N: nat] :
( ( ord_less_eq @ nat @ ( nat2 @ A3 ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ X ) @ N ) )
= ( ord_less_eq @ int @ A3 @ ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N ) ) ) ).
% nat_le_numeral_power_cancel_iff
thf(fact_2153_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_2154_floor__minus__divide__eq__div__numeral,axiom,
! [A3: num,B2: num] :
( ( archim6421214686448440834_floor @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ ( numeral_numeral @ real @ A3 ) @ ( numeral_numeral @ real @ B2 ) ) ) )
= ( divide_divide @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ A3 ) ) @ ( numeral_numeral @ int @ B2 ) ) ) ).
% floor_minus_divide_eq_div_numeral
thf(fact_2155_ceiling__minus__divide__eq__div__numeral,axiom,
! [A3: num,B2: num] :
( ( archimedean_ceiling @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ ( numeral_numeral @ real @ A3 ) @ ( numeral_numeral @ real @ B2 ) ) ) )
= ( uminus_uminus @ int @ ( divide_divide @ int @ ( numeral_numeral @ int @ A3 ) @ ( numeral_numeral @ int @ B2 ) ) ) ) ).
% ceiling_minus_divide_eq_div_numeral
thf(fact_2156_numeral__power__less__of__nat__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [I: num,N: nat,X: nat] :
( ( ord_less @ A @ ( power_power @ A @ ( numeral_numeral @ A @ I ) @ N ) @ ( semiring_1_of_nat @ A @ X ) )
= ( ord_less @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ I ) @ N ) @ X ) ) ) ).
% numeral_power_less_of_nat_cancel_iff
thf(fact_2157_of__nat__less__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X: nat,I: num,N: nat] :
( ( ord_less @ A @ ( semiring_1_of_nat @ A @ X ) @ ( power_power @ A @ ( numeral_numeral @ A @ I ) @ N ) )
= ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ I ) @ N ) ) ) ) ).
% of_nat_less_numeral_power_cancel_iff
thf(fact_2158_numeral__power__le__of__nat__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [I: num,N: nat,X: nat] :
( ( ord_less_eq @ A @ ( power_power @ A @ ( numeral_numeral @ A @ I ) @ N ) @ ( semiring_1_of_nat @ A @ X ) )
= ( ord_less_eq @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ I ) @ N ) @ X ) ) ) ).
% numeral_power_le_of_nat_cancel_iff
thf(fact_2159_of__nat__le__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X: nat,I: num,N: nat] :
( ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ X ) @ ( power_power @ A @ ( numeral_numeral @ A @ I ) @ N ) )
= ( ord_less_eq @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ I ) @ N ) ) ) ) ).
% of_nat_le_numeral_power_cancel_iff
thf(fact_2160_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_2161_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_2162_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_2163_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_2164_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_2165_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_2166_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_2167_of__int__le__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: int,X: num,N: nat] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ A3 ) @ ( power_power @ A @ ( numeral_numeral @ A @ X ) @ N ) )
= ( ord_less_eq @ int @ A3 @ ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N ) ) ) ) ).
% of_int_le_numeral_power_cancel_iff
thf(fact_2168_numeral__power__le__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: num,N: nat,A3: int] :
( ( ord_less_eq @ A @ ( power_power @ A @ ( numeral_numeral @ A @ X ) @ N ) @ ( ring_1_of_int @ A @ A3 ) )
= ( ord_less_eq @ int @ ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N ) @ A3 ) ) ) ).
% numeral_power_le_of_int_cancel_iff
thf(fact_2169_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_2170_of__int__less__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: int,X: num,N: nat] :
( ( ord_less @ A @ ( ring_1_of_int @ A @ A3 ) @ ( power_power @ A @ ( numeral_numeral @ A @ X ) @ N ) )
= ( ord_less @ int @ A3 @ ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N ) ) ) ) ).
% of_int_less_numeral_power_cancel_iff
thf(fact_2171_numeral__power__less__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: num,N: nat,A3: int] :
( ( ord_less @ A @ ( power_power @ A @ ( numeral_numeral @ A @ X ) @ N ) @ ( ring_1_of_int @ A @ A3 ) )
= ( ord_less @ int @ ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N ) @ A3 ) ) ) ).
% numeral_power_less_of_int_cancel_iff
thf(fact_2172_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_2173_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_2174_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_2175_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_2176_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_2177_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_2178_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_2179_of__int__le__neg__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: int,X: num,N: nat] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ A3 ) @ ( power_power @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ X ) ) @ N ) )
= ( ord_less_eq @ int @ A3 @ ( power_power @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ X ) ) @ N ) ) ) ) ).
% of_int_le_neg_numeral_power_cancel_iff
thf(fact_2180_neg__numeral__power__le__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: num,N: nat,A3: int] :
( ( ord_less_eq @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ X ) ) @ N ) @ ( ring_1_of_int @ A @ A3 ) )
= ( ord_less_eq @ int @ ( power_power @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ X ) ) @ N ) @ A3 ) ) ) ).
% neg_numeral_power_le_of_int_cancel_iff
thf(fact_2181_of__int__less__neg__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: int,X: num,N: nat] :
( ( ord_less @ A @ ( ring_1_of_int @ A @ A3 ) @ ( power_power @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ X ) ) @ N ) )
= ( ord_less @ int @ A3 @ ( power_power @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ X ) ) @ N ) ) ) ) ).
% of_int_less_neg_numeral_power_cancel_iff
thf(fact_2182_neg__numeral__power__less__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: num,N: nat,A3: int] :
( ( ord_less @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ X ) ) @ N ) @ ( ring_1_of_int @ A @ A3 ) )
= ( ord_less @ int @ ( power_power @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ X ) ) @ N ) @ A3 ) ) ) ).
% neg_numeral_power_less_of_int_cancel_iff
thf(fact_2183_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_2184_nat__numeral__as__int,axiom,
( ( numeral_numeral @ nat )
= ( ^ [I2: num] : ( nat2 @ ( numeral_numeral @ int @ I2 ) ) ) ) ).
% nat_numeral_as_int
thf(fact_2185_norm__minus__commute,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [A3: A,B2: A] :
( ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ A3 @ B2 ) )
= ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ B2 @ A3 ) ) ) ) ).
% norm_minus_commute
thf(fact_2186_zero__neq__numeral,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [N: num] :
( ( zero_zero @ A )
!= ( numeral_numeral @ A @ N ) ) ) ).
% zero_neq_numeral
thf(fact_2187_div__mult2__numeral__eq,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [A3: A,K: num,L: num] :
( ( divide_divide @ A @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ K ) ) @ ( numeral_numeral @ A @ L ) )
= ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( times_times @ num @ K @ L ) ) ) ) ) ).
% div_mult2_numeral_eq
thf(fact_2188_numeral__neq__neg__numeral,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [M2: num,N: num] :
( ( numeral_numeral @ A @ M2 )
!= ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) ) ) ).
% numeral_neq_neg_numeral
thf(fact_2189_neg__numeral__neq__numeral,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [M2: num,N: num] :
( ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) )
!= ( numeral_numeral @ A @ N ) ) ) ).
% neg_numeral_neq_numeral
thf(fact_2190_Ints__numeral,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [N: num] : ( member @ A @ ( numeral_numeral @ A @ N ) @ ( ring_1_Ints @ A ) ) ) ).
% Ints_numeral
thf(fact_2191_of__int__neg__numeral,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [K: num] :
( ( ring_1_of_int @ A @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ K ) ) ) ) ).
% of_int_neg_numeral
thf(fact_2192_atLeastatMost__psubset__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A,B2: A,C2: A,D2: A] :
( ( ord_less @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) @ ( set_or1337092689740270186AtMost @ A @ C2 @ D2 ) )
= ( ( ~ ( ord_less_eq @ A @ A3 @ B2 )
| ( ( ord_less_eq @ A @ C2 @ A3 )
& ( ord_less_eq @ A @ B2 @ D2 )
& ( ( ord_less @ A @ C2 @ A3 )
| ( ord_less @ A @ B2 @ D2 ) ) ) )
& ( ord_less_eq @ A @ C2 @ D2 ) ) ) ) ).
% atLeastatMost_psubset_iff
thf(fact_2193_norm__divide,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [A3: A,B2: A] :
( ( real_V7770717601297561774m_norm @ A @ ( divide_divide @ A @ A3 @ B2 ) )
= ( divide_divide @ real @ ( real_V7770717601297561774m_norm @ A @ A3 ) @ ( real_V7770717601297561774m_norm @ A @ B2 ) ) ) ) ).
% norm_divide
thf(fact_2194_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_2195_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_2196_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_2197_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_2198_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_2199_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_2200_not__numeral__le__neg__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M2: num,N: num] :
~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ M2 ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) ) ) ).
% not_numeral_le_neg_numeral
thf(fact_2201_neg__numeral__le__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M2: num,N: num] : ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) @ ( numeral_numeral @ A @ N ) ) ) ).
% neg_numeral_le_numeral
thf(fact_2202_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_2203_neg__numeral__less__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M2: num,N: num] : ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) @ ( numeral_numeral @ A @ N ) ) ) ).
% neg_numeral_less_numeral
thf(fact_2204_not__numeral__less__neg__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M2: num,N: num] :
~ ( ord_less @ A @ ( numeral_numeral @ A @ M2 ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) ) ) ).
% not_numeral_less_neg_numeral
thf(fact_2205_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_2206_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_2207_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_2208_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_2209_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_2210_nonzero__norm__divide,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [B2: A,A3: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( real_V7770717601297561774m_norm @ A @ ( divide_divide @ A @ A3 @ B2 ) )
= ( divide_divide @ real @ ( real_V7770717601297561774m_norm @ A @ A3 ) @ ( real_V7770717601297561774m_norm @ A @ B2 ) ) ) ) ) ).
% nonzero_norm_divide
thf(fact_2211_power__eq__imp__eq__norm,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [W: A,N: nat,Z2: A] :
( ( ( power_power @ A @ W @ N )
= ( power_power @ A @ Z2 @ N ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( real_V7770717601297561774m_norm @ A @ W )
= ( real_V7770717601297561774m_norm @ A @ Z2 ) ) ) ) ) ).
% power_eq_imp_eq_norm
thf(fact_2212_norm__diff__triangle__less,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X: A,Y2: A,E1: real,Z2: A,E22: real] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ X @ Y2 ) ) @ E1 )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ Y2 @ Z2 ) ) @ E22 )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ X @ Z2 ) ) @ ( plus_plus @ real @ E1 @ E22 ) ) ) ) ) ).
% norm_diff_triangle_less
thf(fact_2213_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_2214_norm__triangle__ineq4,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [A3: A,B2: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ A3 @ B2 ) ) @ ( plus_plus @ real @ ( real_V7770717601297561774m_norm @ A @ A3 ) @ ( real_V7770717601297561774m_norm @ A @ B2 ) ) ) ) ).
% norm_triangle_ineq4
thf(fact_2215_norm__diff__triangle__le,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X: A,Y2: A,E1: real,Z2: 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 @ Z2 ) ) @ E22 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ X @ Z2 ) ) @ ( plus_plus @ real @ E1 @ E22 ) ) ) ) ) ).
% norm_diff_triangle_le
thf(fact_2216_norm__triangle__le__diff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X: A,Y2: A,E2: real] :
( ( ord_less_eq @ real @ ( plus_plus @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ ( real_V7770717601297561774m_norm @ A @ Y2 ) ) @ E2 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ X @ Y2 ) ) @ E2 ) ) ) ).
% norm_triangle_le_diff
thf(fact_2217_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_2218_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_2219_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_2220_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_2221_bset_I1_J,axiom,
! [D6: int,B7: set @ int,P: int > $o,Q: int > $o] :
( ! [X3: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D6 ) )
=> ! [Xb: int] :
( ( member @ int @ Xb @ B7 )
=> ( X3
!= ( plus_plus @ int @ Xb @ Xa2 ) ) ) )
=> ( ( P @ X3 )
=> ( P @ ( minus_minus @ int @ X3 @ D6 ) ) ) )
=> ( ! [X3: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D6 ) )
=> ! [Xb: int] :
( ( member @ int @ Xb @ B7 )
=> ( X3
!= ( plus_plus @ int @ Xb @ Xa2 ) ) ) )
=> ( ( Q @ X3 )
=> ( Q @ ( minus_minus @ int @ X3 @ D6 ) ) ) )
=> ! [X4: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D6 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ B7 )
=> ( X4
!= ( plus_plus @ int @ Xb2 @ Xa3 ) ) ) )
=> ( ( ( P @ X4 )
& ( Q @ X4 ) )
=> ( ( P @ ( minus_minus @ int @ X4 @ D6 ) )
& ( Q @ ( minus_minus @ int @ X4 @ D6 ) ) ) ) ) ) ) ).
% bset(1)
thf(fact_2222_bset_I2_J,axiom,
! [D6: int,B7: set @ int,P: int > $o,Q: int > $o] :
( ! [X3: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D6 ) )
=> ! [Xb: int] :
( ( member @ int @ Xb @ B7 )
=> ( X3
!= ( plus_plus @ int @ Xb @ Xa2 ) ) ) )
=> ( ( P @ X3 )
=> ( P @ ( minus_minus @ int @ X3 @ D6 ) ) ) )
=> ( ! [X3: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D6 ) )
=> ! [Xb: int] :
( ( member @ int @ Xb @ B7 )
=> ( X3
!= ( plus_plus @ int @ Xb @ Xa2 ) ) ) )
=> ( ( Q @ X3 )
=> ( Q @ ( minus_minus @ int @ X3 @ D6 ) ) ) )
=> ! [X4: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D6 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ B7 )
=> ( X4
!= ( plus_plus @ int @ Xb2 @ Xa3 ) ) ) )
=> ( ( ( P @ X4 )
| ( Q @ X4 ) )
=> ( ( P @ ( minus_minus @ int @ X4 @ D6 ) )
| ( Q @ ( minus_minus @ int @ X4 @ D6 ) ) ) ) ) ) ) ).
% bset(2)
thf(fact_2223_aset_I1_J,axiom,
! [D6: int,A4: set @ int,P: int > $o,Q: int > $o] :
( ! [X3: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D6 ) )
=> ! [Xb: int] :
( ( member @ int @ Xb @ A4 )
=> ( X3
!= ( minus_minus @ int @ Xb @ Xa2 ) ) ) )
=> ( ( P @ X3 )
=> ( P @ ( plus_plus @ int @ X3 @ D6 ) ) ) )
=> ( ! [X3: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D6 ) )
=> ! [Xb: int] :
( ( member @ int @ Xb @ A4 )
=> ( X3
!= ( minus_minus @ int @ Xb @ Xa2 ) ) ) )
=> ( ( Q @ X3 )
=> ( Q @ ( plus_plus @ int @ X3 @ D6 ) ) ) )
=> ! [X4: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D6 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ A4 )
=> ( X4
!= ( minus_minus @ int @ Xb2 @ Xa3 ) ) ) )
=> ( ( ( P @ X4 )
& ( Q @ X4 ) )
=> ( ( P @ ( plus_plus @ int @ X4 @ D6 ) )
& ( Q @ ( plus_plus @ int @ X4 @ D6 ) ) ) ) ) ) ) ).
% aset(1)
thf(fact_2224_aset_I2_J,axiom,
! [D6: int,A4: set @ int,P: int > $o,Q: int > $o] :
( ! [X3: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D6 ) )
=> ! [Xb: int] :
( ( member @ int @ Xb @ A4 )
=> ( X3
!= ( minus_minus @ int @ Xb @ Xa2 ) ) ) )
=> ( ( P @ X3 )
=> ( P @ ( plus_plus @ int @ X3 @ D6 ) ) ) )
=> ( ! [X3: int] :
( ! [Xa2: int] :
( ( member @ int @ Xa2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D6 ) )
=> ! [Xb: int] :
( ( member @ int @ Xb @ A4 )
=> ( X3
!= ( minus_minus @ int @ Xb @ Xa2 ) ) ) )
=> ( ( Q @ X3 )
=> ( Q @ ( plus_plus @ int @ X3 @ D6 ) ) ) )
=> ! [X4: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D6 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ A4 )
=> ( X4
!= ( minus_minus @ int @ Xb2 @ Xa3 ) ) ) )
=> ( ( ( P @ X4 )
| ( Q @ X4 ) )
=> ( ( P @ ( plus_plus @ int @ X4 @ D6 ) )
| ( Q @ ( plus_plus @ int @ X4 @ D6 ) ) ) ) ) ) ) ).
% aset(2)
thf(fact_2225_norm__diff__ineq,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [A3: A,B2: A] : ( ord_less_eq @ real @ ( minus_minus @ real @ ( real_V7770717601297561774m_norm @ A @ A3 ) @ ( real_V7770717601297561774m_norm @ A @ B2 ) ) @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ A3 @ B2 ) ) ) ) ).
% norm_diff_ineq
thf(fact_2226_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_2227_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_2228_norm__triangle__ineq2,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [A3: A,B2: A] : ( ord_less_eq @ real @ ( minus_minus @ real @ ( real_V7770717601297561774m_norm @ A @ A3 ) @ ( real_V7770717601297561774m_norm @ A @ B2 ) ) @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ A3 @ B2 ) ) ) ) ).
% norm_triangle_ineq2
thf(fact_2229_neg__numeral__le__one,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M2: num] : ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) @ ( one_one @ A ) ) ) ).
% neg_numeral_le_one
thf(fact_2230_neg__one__le__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M2: num] : ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ M2 ) ) ) ).
% neg_one_le_numeral
thf(fact_2231_neg__numeral__le__neg__one,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M2: num] : ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% neg_numeral_le_neg_one
thf(fact_2232_not__numeral__le__neg__one,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M2: num] :
~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ M2 ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% not_numeral_le_neg_one
thf(fact_2233_not__one__le__neg__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M2: num] :
~ ( ord_less_eq @ A @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) ) ) ).
% not_one_le_neg_numeral
thf(fact_2234_neg__numeral__less__one,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M2: num] : ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) @ ( one_one @ A ) ) ) ).
% neg_numeral_less_one
thf(fact_2235_neg__one__less__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M2: num] : ( ord_less @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ M2 ) ) ) ).
% neg_one_less_numeral
thf(fact_2236_not__numeral__less__neg__one,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M2: num] :
~ ( ord_less @ A @ ( numeral_numeral @ A @ M2 ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% not_numeral_less_neg_one
thf(fact_2237_not__one__less__neg__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M2: num] :
~ ( ord_less @ A @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) ) ) ).
% not_one_less_neg_numeral
thf(fact_2238_not__neg__one__less__neg__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M2: num] :
~ ( ord_less @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) ) ) ).
% not_neg_one_less_neg_numeral
thf(fact_2239_nonzero__norm__inverse,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [A3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( real_V7770717601297561774m_norm @ A @ ( inverse_inverse @ A @ A3 ) )
= ( inverse_inverse @ real @ ( real_V7770717601297561774m_norm @ A @ A3 ) ) ) ) ) ).
% nonzero_norm_inverse
thf(fact_2240_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_2241_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_2242_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_2243_norm__diff__triangle__ineq,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [A3: A,B2: A,C2: A,D2: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( plus_plus @ A @ A3 @ B2 ) @ ( plus_plus @ A @ C2 @ D2 ) ) ) @ ( plus_plus @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ A3 @ C2 ) ) @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ B2 @ D2 ) ) ) ) ) ).
% norm_diff_triangle_ineq
thf(fact_2244_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_2245_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_2246_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_2247_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_2248_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_2249_norm__triangle__ineq3,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [A3: A,B2: A] : ( ord_less_eq @ real @ ( abs_abs @ real @ ( minus_minus @ real @ ( real_V7770717601297561774m_norm @ A @ A3 ) @ ( real_V7770717601297561774m_norm @ A @ B2 ) ) ) @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ A3 @ B2 ) ) ) ) ).
% norm_triangle_ineq3
thf(fact_2250_periodic__finite__ex,axiom,
! [D2: int,P: int > $o] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D2 )
=> ( ! [X3: int,K2: int] :
( ( P @ X3 )
= ( P @ ( minus_minus @ int @ X3 @ ( times_times @ int @ K2 @ D2 ) ) ) )
=> ( ( ? [X7: int] : ( P @ X7 ) )
= ( ? [X2: int] :
( ( member @ int @ X2 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D2 ) )
& ( P @ X2 ) ) ) ) ) ) ).
% periodic_finite_ex
thf(fact_2251_bset_I3_J,axiom,
! [D6: int,T2: int,B7: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D6 )
=> ( ( member @ int @ ( minus_minus @ int @ T2 @ ( one_one @ int ) ) @ B7 )
=> ! [X4: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D6 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ B7 )
=> ( X4
!= ( plus_plus @ int @ Xb2 @ Xa3 ) ) ) )
=> ( ( X4 = T2 )
=> ( ( minus_minus @ int @ X4 @ D6 )
= T2 ) ) ) ) ) ).
% bset(3)
thf(fact_2252_bset_I4_J,axiom,
! [D6: int,T2: int,B7: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D6 )
=> ( ( member @ int @ T2 @ B7 )
=> ! [X4: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D6 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ B7 )
=> ( X4
!= ( plus_plus @ int @ Xb2 @ Xa3 ) ) ) )
=> ( ( X4 != T2 )
=> ( ( minus_minus @ int @ X4 @ D6 )
!= T2 ) ) ) ) ) ).
% bset(4)
thf(fact_2253_bset_I5_J,axiom,
! [D6: int,B7: set @ int,T2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D6 )
=> ! [X4: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D6 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ B7 )
=> ( X4
!= ( plus_plus @ int @ Xb2 @ Xa3 ) ) ) )
=> ( ( ord_less @ int @ X4 @ T2 )
=> ( ord_less @ int @ ( minus_minus @ int @ X4 @ D6 ) @ T2 ) ) ) ) ).
% bset(5)
thf(fact_2254_bset_I7_J,axiom,
! [D6: int,T2: int,B7: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D6 )
=> ( ( member @ int @ T2 @ B7 )
=> ! [X4: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D6 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ B7 )
=> ( X4
!= ( plus_plus @ int @ Xb2 @ Xa3 ) ) ) )
=> ( ( ord_less @ int @ T2 @ X4 )
=> ( ord_less @ int @ T2 @ ( minus_minus @ int @ X4 @ D6 ) ) ) ) ) ) ).
% bset(7)
thf(fact_2255_aset_I3_J,axiom,
! [D6: int,T2: int,A4: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D6 )
=> ( ( member @ int @ ( plus_plus @ int @ T2 @ ( one_one @ int ) ) @ A4 )
=> ! [X4: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D6 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ A4 )
=> ( X4
!= ( minus_minus @ int @ Xb2 @ Xa3 ) ) ) )
=> ( ( X4 = T2 )
=> ( ( plus_plus @ int @ X4 @ D6 )
= T2 ) ) ) ) ) ).
% aset(3)
thf(fact_2256_aset_I4_J,axiom,
! [D6: int,T2: int,A4: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D6 )
=> ( ( member @ int @ T2 @ A4 )
=> ! [X4: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D6 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ A4 )
=> ( X4
!= ( minus_minus @ int @ Xb2 @ Xa3 ) ) ) )
=> ( ( X4 != T2 )
=> ( ( plus_plus @ int @ X4 @ D6 )
!= T2 ) ) ) ) ) ).
% aset(4)
thf(fact_2257_aset_I5_J,axiom,
! [D6: int,T2: int,A4: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D6 )
=> ( ( member @ int @ T2 @ A4 )
=> ! [X4: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D6 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ A4 )
=> ( X4
!= ( minus_minus @ int @ Xb2 @ Xa3 ) ) ) )
=> ( ( ord_less @ int @ X4 @ T2 )
=> ( ord_less @ int @ ( plus_plus @ int @ X4 @ D6 ) @ T2 ) ) ) ) ) ).
% aset(5)
thf(fact_2258_aset_I7_J,axiom,
! [D6: int,A4: set @ int,T2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D6 )
=> ! [X4: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D6 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ A4 )
=> ( X4
!= ( minus_minus @ int @ Xb2 @ Xa3 ) ) ) )
=> ( ( ord_less @ int @ T2 @ X4 )
=> ( ord_less @ int @ T2 @ ( plus_plus @ int @ X4 @ D6 ) ) ) ) ) ).
% aset(7)
thf(fact_2259_lemma__NBseq__def,axiom,
! [A: $tType,B: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [X8: A > B] :
( ( ? [K5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K5 )
& ! [N5: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( X8 @ N5 ) ) @ K5 ) ) )
= ( ? [N7: nat] :
! [N5: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( X8 @ N5 ) ) @ ( semiring_1_of_nat @ real @ ( suc @ N7 ) ) ) ) ) ) ).
% lemma_NBseq_def
thf(fact_2260_lemma__NBseq__def2,axiom,
! [A: $tType,B: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [X8: A > B] :
( ( ? [K5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K5 )
& ! [N5: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( X8 @ N5 ) ) @ K5 ) ) )
= ( ? [N7: nat] :
! [N5: A] : ( ord_less @ real @ ( real_V7770717601297561774m_norm @ B @ ( X8 @ N5 ) ) @ ( semiring_1_of_nat @ real @ ( suc @ N7 ) ) ) ) ) ) ).
% lemma_NBseq_def2
thf(fact_2261_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_2262_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_2263_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_2264_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_2265_Cauchy__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ( ( topolo3814608138187158403Cauchy @ A )
= ( ^ [X7: nat > A] :
! [E3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E3 )
=> ? [M7: nat] :
! [M5: nat] :
( ( ord_less_eq @ nat @ M7 @ M5 )
=> ! [N5: nat] :
( ( ord_less_eq @ nat @ M7 @ N5 )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( X7 @ M5 ) @ ( X7 @ N5 ) ) ) @ E3 ) ) ) ) ) ) ) ).
% Cauchy_iff
thf(fact_2266_CauchyI,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X8: nat > A] :
( ! [E: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E )
=> ? [M8: nat] :
! [M3: nat] :
( ( ord_less_eq @ nat @ M8 @ M3 )
=> ! [N2: nat] :
( ( ord_less_eq @ nat @ M8 @ N2 )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( X8 @ M3 ) @ ( X8 @ N2 ) ) ) @ E ) ) ) )
=> ( topolo3814608138187158403Cauchy @ A @ X8 ) ) ) ).
% CauchyI
thf(fact_2267_CauchyD,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X8: nat > A,E2: real] :
( ( topolo3814608138187158403Cauchy @ A @ X8 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ E2 )
=> ? [M9: nat] :
! [M: nat] :
( ( ord_less_eq @ nat @ M9 @ M )
=> ! [N4: nat] :
( ( ord_less_eq @ nat @ M9 @ N4 )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( X8 @ M ) @ ( X8 @ N4 ) ) ) @ E2 ) ) ) ) ) ) ).
% CauchyD
thf(fact_2268_aset_I8_J,axiom,
! [D6: int,A4: set @ int,T2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D6 )
=> ! [X4: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D6 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ A4 )
=> ( X4
!= ( minus_minus @ int @ Xb2 @ Xa3 ) ) ) )
=> ( ( ord_less_eq @ int @ T2 @ X4 )
=> ( ord_less_eq @ int @ T2 @ ( plus_plus @ int @ X4 @ D6 ) ) ) ) ) ).
% aset(8)
thf(fact_2269_aset_I6_J,axiom,
! [D6: int,T2: int,A4: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D6 )
=> ( ( member @ int @ ( plus_plus @ int @ T2 @ ( one_one @ int ) ) @ A4 )
=> ! [X4: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D6 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ A4 )
=> ( X4
!= ( minus_minus @ int @ Xb2 @ Xa3 ) ) ) )
=> ( ( ord_less_eq @ int @ X4 @ T2 )
=> ( ord_less_eq @ int @ ( plus_plus @ int @ X4 @ D6 ) @ T2 ) ) ) ) ) ).
% aset(6)
thf(fact_2270_enat__ord__number_I2_J,axiom,
! [M2: num,N: num] :
( ( ord_less @ extended_enat @ ( numeral_numeral @ extended_enat @ M2 ) @ ( numeral_numeral @ extended_enat @ N ) )
= ( ord_less @ nat @ ( numeral_numeral @ nat @ M2 ) @ ( numeral_numeral @ nat @ N ) ) ) ).
% enat_ord_number(2)
thf(fact_2271_lemma__termdiff3,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [H: A,Z2: A,K6: real,N: nat] :
( ( H
!= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ Z2 ) @ K6 )
=> ( ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ Z2 @ H ) ) @ K6 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( divide_divide @ A @ ( minus_minus @ A @ ( power_power @ A @ ( plus_plus @ A @ Z2 @ H ) @ N ) @ ( power_power @ A @ Z2 @ N ) ) @ H ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ ( power_power @ A @ Z2 @ ( 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 @ K6 @ ( minus_minus @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( real_V7770717601297561774m_norm @ A @ H ) ) ) ) ) ) ) ).
% lemma_termdiff3
thf(fact_2272_complex__mod__triangle__ineq2,axiom,
! [B2: complex,A3: complex] : ( ord_less_eq @ real @ ( minus_minus @ real @ ( real_V7770717601297561774m_norm @ complex @ ( plus_plus @ complex @ B2 @ A3 ) ) @ ( real_V7770717601297561774m_norm @ complex @ B2 ) ) @ ( real_V7770717601297561774m_norm @ complex @ A3 ) ) ).
% complex_mod_triangle_ineq2
thf(fact_2273_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_2274_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_2275_ceiling__log__nat__eq__powr__iff,axiom,
! [B2: nat,K: nat,N: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ B2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ( ( archimedean_ceiling @ real @ ( log2 @ ( semiring_1_of_nat @ real @ B2 ) @ ( semiring_1_of_nat @ real @ K ) ) )
= ( plus_plus @ int @ ( semiring_1_of_nat @ int @ N ) @ ( one_one @ int ) ) )
= ( ( ord_less @ nat @ ( power_power @ nat @ B2 @ N ) @ K )
& ( ord_less_eq @ nat @ K @ ( power_power @ nat @ B2 @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% ceiling_log_nat_eq_powr_iff
thf(fact_2276_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_2277_verit__eq__simplify_I8_J,axiom,
! [X23: num,Y23: num] :
( ( ( bit0 @ X23 )
= ( bit0 @ Y23 ) )
= ( X23 = Y23 ) ) ).
% verit_eq_simplify(8)
thf(fact_2278_pow__sum,axiom,
! [A3: nat,B2: nat] :
( ( divide_divide @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( plus_plus @ nat @ A3 @ B2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ A3 ) )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ B2 ) ) ).
% pow_sum
thf(fact_2279_member__bound,axiom,
! [Tree: vEBT_VEBT,X: nat,N: nat] :
( ( vEBT_vebt_member @ Tree @ X )
=> ( ( vEBT_invar_vebt @ Tree @ N )
=> ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% member_bound
thf(fact_2280_bit__concat__def,axiom,
( vEBT_VEBT_bit_concat
= ( ^ [H2: nat,L2: nat,D4: nat] : ( plus_plus @ nat @ ( times_times @ nat @ H2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ D4 ) ) @ L2 ) ) ) ).
% bit_concat_def
thf(fact_2281_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_2282_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_2283_of__real__eq__0__iff,axiom,
! [A: $tType] :
( ( real_V2191834092415804123ebra_1 @ A )
=> ! [X: real] :
( ( ( real_Vector_of_real @ A @ X )
= ( zero_zero @ A ) )
= ( X
= ( zero_zero @ real ) ) ) ) ).
% of_real_eq_0_iff
thf(fact_2284_of__real__0,axiom,
! [A: $tType] :
( ( real_V2191834092415804123ebra_1 @ A )
=> ( ( real_Vector_of_real @ A @ ( zero_zero @ real ) )
= ( zero_zero @ A ) ) ) ).
% of_real_0
thf(fact_2285_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_2286_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_2287_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_2288_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_2289_of__real__eq__minus__of__real__iff,axiom,
! [A: $tType] :
( ( real_V2191834092415804123ebra_1 @ A )
=> ! [X: real,Y2: real] :
( ( ( real_Vector_of_real @ A @ X )
= ( uminus_uminus @ A @ ( real_Vector_of_real @ A @ Y2 ) ) )
= ( X
= ( uminus_uminus @ real @ Y2 ) ) ) ) ).
% of_real_eq_minus_of_real_iff
thf(fact_2290_minus__of__real__eq__of__real__iff,axiom,
! [A: $tType] :
( ( real_V2191834092415804123ebra_1 @ A )
=> ! [X: real,Y2: real] :
( ( ( uminus_uminus @ A @ ( real_Vector_of_real @ A @ X ) )
= ( real_Vector_of_real @ A @ Y2 ) )
= ( ( uminus_uminus @ real @ X )
= Y2 ) ) ) ).
% minus_of_real_eq_of_real_iff
thf(fact_2291_of__real__minus,axiom,
! [A: $tType] :
( ( real_V2191834092415804123ebra_1 @ A )
=> ! [X: real] :
( ( real_Vector_of_real @ A @ ( uminus_uminus @ real @ X ) )
= ( uminus_uminus @ A @ ( real_Vector_of_real @ A @ X ) ) ) ) ).
% of_real_minus
thf(fact_2292_of__real__diff,axiom,
! [A: $tType] :
( ( real_V2191834092415804123ebra_1 @ A )
=> ! [X: real,Y2: real] :
( ( real_Vector_of_real @ A @ ( minus_minus @ real @ X @ Y2 ) )
= ( minus_minus @ A @ ( real_Vector_of_real @ A @ X ) @ ( real_Vector_of_real @ A @ Y2 ) ) ) ) ).
% of_real_diff
thf(fact_2293_of__real__of__nat__eq,axiom,
! [A: $tType] :
( ( real_V2191834092415804123ebra_1 @ A )
=> ! [N: nat] :
( ( real_Vector_of_real @ A @ ( semiring_1_of_nat @ real @ N ) )
= ( semiring_1_of_nat @ A @ N ) ) ) ).
% of_real_of_nat_eq
thf(fact_2294_num__double,axiom,
! [N: num] :
( ( times_times @ num @ ( bit0 @ one2 ) @ N )
= ( bit0 @ N ) ) ).
% num_double
thf(fact_2295_of__real__fact,axiom,
! [A: $tType] :
( ( real_V2191834092415804123ebra_1 @ A )
=> ! [N: nat] :
( ( real_Vector_of_real @ A @ ( semiring_char_0_fact @ real @ N ) )
= ( semiring_char_0_fact @ A @ N ) ) ) ).
% of_real_fact
thf(fact_2296_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_2297_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_2298_Suc__numeral,axiom,
! [N: num] :
( ( suc @ ( numeral_numeral @ nat @ N ) )
= ( numeral_numeral @ nat @ ( plus_plus @ num @ N @ one2 ) ) ) ).
% Suc_numeral
thf(fact_2299_inrange,axiom,
! [T2: vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ord_less_eq @ ( set @ nat ) @ ( vEBT_VEBT_set_vebt @ T2 ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ nat ) ) ) ) ) ).
% inrange
thf(fact_2300_not__neg__one__le__neg__numeral__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M2: num] :
( ( ~ ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) ) )
= ( M2 != one2 ) ) ) ).
% not_neg_one_le_neg_numeral_iff
thf(fact_2301_neg__numeral__less__neg__one__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M2: num] :
( ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( M2 != one2 ) ) ) ).
% neg_numeral_less_neg_one_iff
thf(fact_2302_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_2303_zero__eq__power2,axiom,
! [A: $tType] :
( ( semiri2026040879449505780visors @ A )
=> ! [A3: A] :
( ( ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( zero_zero @ A ) )
= ( A3
= ( zero_zero @ A ) ) ) ) ).
% zero_eq_power2
thf(fact_2304_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_2305_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_2306_power2__minus,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [A3: A] :
( ( power_power @ A @ ( uminus_uminus @ A @ A3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% power2_minus
thf(fact_2307_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_2308_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_2309_Suc__1,axiom,
( ( suc @ ( one_one @ nat ) )
= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ).
% Suc_1
thf(fact_2310_div2__Suc__Suc,axiom,
! [M2: nat] :
( ( divide_divide @ nat @ ( suc @ ( suc @ M2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( suc @ ( divide_divide @ nat @ M2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% div2_Suc_Suc
thf(fact_2311_add__self__div__2,axiom,
! [M2: nat] :
( ( divide_divide @ nat @ ( plus_plus @ nat @ M2 @ M2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= M2 ) ).
% add_self_div_2
thf(fact_2312_mod2__Suc__Suc,axiom,
! [M2: nat] :
( ( modulo_modulo @ nat @ ( suc @ ( suc @ M2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( modulo_modulo @ nat @ M2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% mod2_Suc_Suc
thf(fact_2313_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_2314_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_2315_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_2316_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_2317_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_2318_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_2319_add__neg__numeral__special_I6_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M2: num] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( inc @ M2 ) ) ) ) ) ).
% add_neg_numeral_special(6)
thf(fact_2320_diff__numeral__special_I6_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M2: num] :
( ( minus_minus @ A @ ( numeral_numeral @ A @ M2 ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( numeral_numeral @ A @ ( inc @ M2 ) ) ) ) ).
% diff_numeral_special(6)
thf(fact_2321_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_2322_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_2323_power2__less__eq__zero__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( zero_zero @ A ) )
= ( A3
= ( zero_zero @ A ) ) ) ) ).
% power2_less_eq_zero_iff
thf(fact_2324_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_2325_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_2326_zero__less__power2,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( A3
!= ( zero_zero @ A ) ) ) ) ).
% zero_less_power2
thf(fact_2327_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_2328_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_2329_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_2330_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_2331_not__mod__2__eq__0__eq__1,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A3: A] :
( ( ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
!= ( zero_zero @ A ) )
= ( ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( one_one @ A ) ) ) ) ).
% not_mod_2_eq_0_eq_1
thf(fact_2332_not__mod__2__eq__1__eq__0,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A3: A] :
( ( ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
!= ( one_one @ A ) )
= ( ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( zero_zero @ A ) ) ) ) ).
% not_mod_2_eq_1_eq_0
thf(fact_2333_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_2334_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_2335_Power_Oring__1__class_Opower__minus__even,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [A3: A,N: nat] :
( ( power_power @ A @ ( uminus_uminus @ A @ A3 ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
= ( power_power @ A @ A3 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% Power.ring_1_class.power_minus_even
thf(fact_2336_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_2337_diff__numeral__special_I4_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M2: num] :
( ( minus_minus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) @ ( one_one @ A ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( plus_plus @ num @ M2 @ one2 ) ) ) ) ) ).
% diff_numeral_special(4)
thf(fact_2338_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_2339_add__self__mod__2,axiom,
! [M2: nat] :
( ( modulo_modulo @ nat @ ( plus_plus @ nat @ M2 @ M2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( zero_zero @ nat ) ) ).
% add_self_mod_2
thf(fact_2340_half__nonnegative__int__iff,axiom,
! [K: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( divide_divide @ int @ K @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) )
= ( ord_less_eq @ int @ ( zero_zero @ int ) @ K ) ) ).
% half_nonnegative_int_iff
thf(fact_2341_half__negative__int__iff,axiom,
! [K: int] :
( ( ord_less @ int @ ( divide_divide @ int @ K @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( zero_zero @ int ) )
= ( ord_less @ int @ K @ ( zero_zero @ int ) ) ) ).
% half_negative_int_iff
thf(fact_2342_real__average__minus__first,axiom,
! [A3: real,B2: real] :
( ( minus_minus @ real @ ( divide_divide @ real @ ( plus_plus @ real @ A3 @ B2 ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ A3 )
= ( divide_divide @ real @ ( minus_minus @ real @ B2 @ A3 ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ).
% real_average_minus_first
thf(fact_2343_real__average__minus__second,axiom,
! [B2: real,A3: real] :
( ( minus_minus @ real @ ( divide_divide @ real @ ( plus_plus @ real @ B2 @ A3 ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ A3 )
= ( divide_divide @ real @ ( minus_minus @ real @ B2 @ A3 ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ).
% real_average_minus_second
thf(fact_2344_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_2345_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_2346_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_2347_mod2__gr__0,axiom,
! [M2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( modulo_modulo @ nat @ M2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( ( modulo_modulo @ nat @ M2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( one_one @ nat ) ) ) ).
% mod2_gr_0
thf(fact_2348_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_2349_sgn__eq,axiom,
( ( sgn_sgn @ complex )
= ( ^ [Z6: complex] : ( divide_divide @ complex @ Z6 @ ( real_Vector_of_real @ complex @ ( real_V7770717601297561774m_norm @ complex @ Z6 ) ) ) ) ) ).
% sgn_eq
thf(fact_2350_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_2351_add__One,axiom,
! [X: num] :
( ( plus_plus @ num @ X @ one2 )
= ( inc @ X ) ) ).
% add_One
thf(fact_2352_add__inc,axiom,
! [X: num,Y2: num] :
( ( plus_plus @ num @ X @ ( inc @ Y2 ) )
= ( inc @ ( plus_plus @ num @ X @ Y2 ) ) ) ).
% add_inc
thf(fact_2353_add__One__commute,axiom,
! [N: num] :
( ( plus_plus @ num @ one2 @ N )
= ( plus_plus @ num @ N @ one2 ) ) ).
% add_One_commute
thf(fact_2354_le__num__One__iff,axiom,
! [X: num] :
( ( ord_less_eq @ num @ X @ one2 )
= ( X = one2 ) ) ).
% le_num_One_iff
thf(fact_2355_verit__eq__simplify_I10_J,axiom,
! [X23: num] :
( one2
!= ( bit0 @ X23 ) ) ).
% verit_eq_simplify(10)
thf(fact_2356_inc_Osimps_I1_J,axiom,
( ( inc @ one2 )
= ( bit0 @ one2 ) ) ).
% inc.simps(1)
thf(fact_2357_num__induct,axiom,
! [P: num > $o,X: num] :
( ( P @ one2 )
=> ( ! [X3: num] :
( ( P @ X3 )
=> ( P @ ( inc @ X3 ) ) )
=> ( P @ X ) ) ) ).
% num_induct
thf(fact_2358_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_2359_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_2360_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_2361_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_2362_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_2363_numeral__2__eq__2,axiom,
( ( numeral_numeral @ nat @ ( bit0 @ one2 ) )
= ( suc @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% numeral_2_eq_2
thf(fact_2364_pos2,axiom,
ord_less @ nat @ ( zero_zero @ nat ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ).
% pos2
thf(fact_2365_double__not__eq__Suc__double,axiom,
! [M2: nat,N: nat] :
( ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M2 )
!= ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ).
% double_not_eq_Suc_double
thf(fact_2366_Suc__double__not__eq__double,axiom,
! [M2: nat,N: nat] :
( ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M2 ) )
!= ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ).
% Suc_double_not_eq_double
thf(fact_2367_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_2368_less__exp,axiom,
! [N: nat] : ( ord_less @ nat @ N @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ).
% less_exp
thf(fact_2369_pochhammer__of__real,axiom,
! [A: $tType] :
( ( ( real_V2191834092415804123ebra_1 @ A )
& ( comm_semiring_1 @ A ) )
=> ! [X: real,N: nat] :
( ( comm_s3205402744901411588hammer @ A @ ( real_Vector_of_real @ A @ X ) @ N )
= ( real_Vector_of_real @ A @ ( comm_s3205402744901411588hammer @ real @ X @ N ) ) ) ) ).
% pochhammer_of_real
thf(fact_2370_num_Osize_I4_J,axiom,
( ( size_size @ num @ one2 )
= ( zero_zero @ nat ) ) ).
% num.size(4)
thf(fact_2371_numerals_I1_J,axiom,
( ( numeral_numeral @ nat @ one2 )
= ( one_one @ nat ) ) ).
% numerals(1)
thf(fact_2372_zero__le__power2,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% zero_le_power2
thf(fact_2373_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_2374_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_2375_power2__less__0,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A] :
~ ( ord_less @ A @ ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( zero_zero @ A ) ) ) ).
% power2_less_0
thf(fact_2376_mult__2,axiom,
! [A: $tType] :
( ( semiring_numeral @ A )
=> ! [Z2: A] :
( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Z2 )
= ( plus_plus @ A @ Z2 @ Z2 ) ) ) ).
% mult_2
thf(fact_2377_mult__2__right,axiom,
! [A: $tType] :
( ( semiring_numeral @ A )
=> ! [Z2: A] :
( ( times_times @ A @ Z2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( plus_plus @ A @ Z2 @ Z2 ) ) ) ).
% mult_2_right
thf(fact_2378_left__add__twice,axiom,
! [A: $tType] :
( ( semiring_numeral @ A )
=> ! [A3: A,B2: A] :
( ( plus_plus @ A @ A3 @ ( plus_plus @ A @ A3 @ B2 ) )
= ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) @ B2 ) ) ) ).
% left_add_twice
thf(fact_2379_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_2380_power2__eq__1__iff,axiom,
! [A: $tType] :
( ( ring_15535105094025558882visors @ A )
=> ! [A3: A] :
( ( ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( one_one @ A ) )
= ( ( A3
= ( one_one @ A ) )
| ( A3
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ) ).
% power2_eq_1_iff
thf(fact_2381_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_2382_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_2383_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_2384_abs__sqrt__wlog,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [P: A > A > $o,X: A] :
( ! [X3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 )
=> ( P @ X3 @ ( power_power @ A @ X3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
=> ( P @ ( abs_abs @ A @ X ) @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% abs_sqrt_wlog
thf(fact_2385_nat__2,axiom,
( ( nat2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) )
= ( suc @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% nat_2
thf(fact_2386_nat__induct2,axiom,
! [P: nat > $o,N: nat] :
( ( P @ ( zero_zero @ nat ) )
=> ( ( P @ ( one_one @ nat ) )
=> ( ! [N2: nat] :
( ( P @ N2 )
=> ( P @ ( plus_plus @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_induct2
thf(fact_2387_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_2388_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_2389_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_2390_diff__le__diff__pow,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K )
=> ( ord_less_eq @ nat @ ( minus_minus @ nat @ M2 @ N ) @ ( minus_minus @ nat @ ( power_power @ nat @ K @ M2 ) @ ( power_power @ nat @ K @ N ) ) ) ) ).
% diff_le_diff_pow
thf(fact_2391_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_2392_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_2393_log2__of__power__eq,axiom,
! [M2: nat,N: nat] :
( ( M2
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( semiring_1_of_nat @ real @ N )
= ( log2 @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ M2 ) ) ) ) ).
% log2_of_power_eq
thf(fact_2394_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_2395_half__gt__zero,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% half_gt_zero
thf(fact_2396_half__gt__zero__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% half_gt_zero_iff
thf(fact_2397_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_2398_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_2399_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_2400_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_2401_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_2402_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_2403_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_2404_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_2405_exp__add__not__zero__imp__left,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [M2: nat,N: nat] :
( ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( plus_plus @ nat @ M2 @ N ) )
!= ( zero_zero @ A ) )
=> ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M2 )
!= ( zero_zero @ A ) ) ) ) ).
% exp_add_not_zero_imp_left
thf(fact_2406_exp__add__not__zero__imp__right,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [M2: nat,N: nat] :
( ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( plus_plus @ nat @ M2 @ N ) )
!= ( zero_zero @ A ) )
=> ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N )
!= ( zero_zero @ A ) ) ) ) ).
% exp_add_not_zero_imp_right
thf(fact_2407_zero__le__even__power_H,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,N: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A3 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% zero_le_even_power'
thf(fact_2408_exp__not__zero__imp__exp__diff__not__zero,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [N: nat,M2: 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 @ M2 ) )
!= ( zero_zero @ A ) ) ) ) ).
% exp_not_zero_imp_exp_diff_not_zero
thf(fact_2409_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_2410_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_2411_div__exp__eq,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A,M2: nat,N: nat] :
( ( divide_divide @ A @ ( divide_divide @ A @ A3 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M2 ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
= ( divide_divide @ A @ A3 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( plus_plus @ nat @ M2 @ N ) ) ) ) ) ).
% div_exp_eq
thf(fact_2412_ex__nat__less,axiom,
! [N: nat,P: nat > $o] :
( ( ? [M5: nat] :
( ( ord_less_eq @ nat @ M5 @ N )
& ( P @ M5 ) ) )
= ( ? [X2: nat] :
( ( member @ nat @ X2 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
& ( P @ X2 ) ) ) ) ).
% ex_nat_less
thf(fact_2413_all__nat__less,axiom,
! [N: nat,P: nat > $o] :
( ( ! [M5: nat] :
( ( ord_less_eq @ nat @ M5 @ N )
=> ( P @ M5 ) ) )
= ( ! [X2: nat] :
( ( member @ nat @ X2 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
=> ( P @ X2 ) ) ) ) ).
% all_nat_less
thf(fact_2414_minus__power__mult__self,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [A3: A,N: nat] :
( ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ A3 ) @ N ) @ ( power_power @ A @ ( uminus_uminus @ A @ A3 ) @ N ) )
= ( power_power @ A @ A3 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% minus_power_mult_self
thf(fact_2415_power__odd__eq,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A3: A,N: nat] :
( ( power_power @ A @ A3 @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) )
= ( times_times @ A @ A3 @ ( power_power @ A @ ( power_power @ A @ A3 @ N ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% power_odd_eq
thf(fact_2416_exp__double,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [Z2: A] :
( ( exp @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Z2 ) )
= ( power_power @ A @ ( exp @ A @ Z2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% exp_double
thf(fact_2417_nat__bit__induct,axiom,
! [P: nat > $o,N: nat] :
( ( P @ ( zero_zero @ nat ) )
=> ( ! [N2: nat] :
( ( P @ N2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( P @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) )
=> ( ! [N2: nat] :
( ( P @ N2 )
=> ( P @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) )
=> ( P @ N ) ) ) ) ).
% nat_bit_induct
thf(fact_2418_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_2419_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_2420_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_2421_of__real__exp,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X: real] :
( ( real_Vector_of_real @ A @ ( exp @ real @ X ) )
= ( exp @ A @ ( real_Vector_of_real @ A @ X ) ) ) ) ).
% of_real_exp
thf(fact_2422_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_2423_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_2424_power__minus__Bit0,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X: A,K: num] :
( ( power_power @ A @ ( uminus_uminus @ A @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ K ) ) )
= ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ K ) ) ) ) ) ).
% power_minus_Bit0
thf(fact_2425_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_2426_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_2427_mult__numeral__1,axiom,
! [A: $tType] :
( ( semiring_numeral @ A )
=> ! [A3: A] :
( ( times_times @ A @ ( numeral_numeral @ A @ one2 ) @ A3 )
= A3 ) ) ).
% mult_numeral_1
thf(fact_2428_mult__numeral__1__right,axiom,
! [A: $tType] :
( ( semiring_numeral @ A )
=> ! [A3: A] :
( ( times_times @ A @ A3 @ ( numeral_numeral @ A @ one2 ) )
= A3 ) ) ).
% mult_numeral_1_right
thf(fact_2429_numeral__One,axiom,
! [A: $tType] :
( ( numeral @ A )
=> ( ( numeral_numeral @ A @ one2 )
= ( one_one @ A ) ) ) ).
% numeral_One
thf(fact_2430_divide__numeral__1,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A3: A] :
( ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ one2 ) )
= A3 ) ) ).
% divide_numeral_1
thf(fact_2431_numeral__1__eq__Suc__0,axiom,
( ( numeral_numeral @ nat @ one2 )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% numeral_1_eq_Suc_0
thf(fact_2432_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_2433_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_2434_triangle__def,axiom,
( nat_triangle
= ( ^ [N5: nat] : ( divide_divide @ nat @ ( times_times @ nat @ N5 @ ( suc @ N5 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% triangle_def
thf(fact_2435_divmod__digit__0_I2_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [B2: A,A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less @ A @ ( modulo_modulo @ A @ A3 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) @ B2 )
=> ( ( modulo_modulo @ A @ A3 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) )
= ( modulo_modulo @ A @ A3 @ B2 ) ) ) ) ) ).
% divmod_digit_0(2)
thf(fact_2436_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_2437_bits__stable__imp__add__self,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A] :
( ( ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= A3 )
=> ( ( plus_plus @ A @ A3 @ ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
= ( zero_zero @ A ) ) ) ) ).
% bits_stable_imp_add_self
thf(fact_2438_odd__0__le__power__imp__0__le,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,N: nat] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A3 @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 ) ) ) ).
% odd_0_le_power_imp_0_le
thf(fact_2439_odd__power__less__zero,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,N: nat] :
( ( ord_less @ A @ A3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( power_power @ A @ A3 @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) @ ( zero_zero @ A ) ) ) ) ).
% odd_power_less_zero
thf(fact_2440_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_2441_div__exp__mod__exp__eq,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A,N: nat,M2: nat] :
( ( modulo_modulo @ A @ ( divide_divide @ A @ A3 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M2 ) )
= ( divide_divide @ A @ ( modulo_modulo @ A @ A3 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( plus_plus @ nat @ N @ M2 ) ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% div_exp_mod_exp_eq
thf(fact_2442_ex__power__ivl2,axiom,
! [B2: nat,K: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ B2 )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K )
=> ? [N2: nat] :
( ( ord_less @ nat @ ( power_power @ nat @ B2 @ N2 ) @ K )
& ( ord_less_eq @ nat @ K @ ( power_power @ nat @ B2 @ ( plus_plus @ nat @ N2 @ ( one_one @ nat ) ) ) ) ) ) ) ).
% ex_power_ivl2
thf(fact_2443_ex__power__ivl1,axiom,
! [B2: nat,K: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ B2 )
=> ( ( ord_less_eq @ nat @ ( one_one @ nat ) @ K )
=> ? [N2: nat] :
( ( ord_less_eq @ nat @ ( power_power @ nat @ B2 @ N2 ) @ K )
& ( ord_less @ nat @ K @ ( power_power @ nat @ B2 @ ( plus_plus @ nat @ N2 @ ( one_one @ nat ) ) ) ) ) ) ) ).
% ex_power_ivl1
thf(fact_2444_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_2445_exp__bound__half,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [Z2: A] :
( ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ Z2 ) @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( exp @ A @ Z2 ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ).
% exp_bound_half
thf(fact_2446_int__bit__induct,axiom,
! [P: int > $o,K: int] :
( ( P @ ( zero_zero @ int ) )
=> ( ( P @ ( uminus_uminus @ int @ ( one_one @ int ) ) )
=> ( ! [K2: int] :
( ( P @ K2 )
=> ( ( K2
!= ( zero_zero @ int ) )
=> ( P @ ( times_times @ int @ K2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) )
=> ( ! [K2: int] :
( ( P @ K2 )
=> ( ( K2
!= ( uminus_uminus @ int @ ( one_one @ int ) ) )
=> ( P @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ K2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) )
=> ( P @ K ) ) ) ) ) ).
% int_bit_induct
thf(fact_2447_less__log2__of__power,axiom,
! [N: nat,M2: nat] :
( ( ord_less @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ M2 )
=> ( ord_less @ real @ ( semiring_1_of_nat @ real @ N ) @ ( log2 @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ M2 ) ) ) ) ).
% less_log2_of_power
thf(fact_2448_le__log2__of__power,axiom,
! [N: nat,M2: nat] :
( ( ord_less_eq @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ M2 )
=> ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ N ) @ ( log2 @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ M2 ) ) ) ) ).
% le_log2_of_power
thf(fact_2449_arsinh__def,axiom,
! [A: $tType] :
( ( ln @ A )
=> ( ( arsinh @ A )
= ( ^ [X2: A] : ( ln_ln @ A @ ( plus_plus @ A @ X2 @ ( powr @ A @ ( plus_plus @ A @ ( power_power @ A @ X2 @ ( 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_2450_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_2451_divmod__digit__0_I1_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [B2: A,A3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less @ A @ ( modulo_modulo @ A @ A3 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) @ B2 )
=> ( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A3 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) )
= ( divide_divide @ A @ A3 @ B2 ) ) ) ) ) ).
% divmod_digit_0(1)
thf(fact_2452_arcosh__def,axiom,
! [A: $tType] :
( ( ln @ A )
=> ( ( arcosh @ A )
= ( ^ [X2: A] : ( ln_ln @ A @ ( plus_plus @ A @ X2 @ ( powr @ A @ ( minus_minus @ A @ ( power_power @ A @ X2 @ ( 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_2453_mult__exp__mod__exp__eq,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [M2: nat,N: nat,A3: A] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( modulo_modulo @ A @ ( times_times @ A @ A3 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M2 ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
= ( times_times @ A @ ( modulo_modulo @ A @ A3 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ N @ M2 ) ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M2 ) ) ) ) ) ).
% mult_exp_mod_exp_eq
thf(fact_2454_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_2455_cong__exp__iff__simps_I2_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [N: num,Q5: num] :
( ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit0 @ N ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q5 ) ) )
= ( zero_zero @ A ) )
= ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ N ) @ ( numeral_numeral @ A @ Q5 ) )
= ( zero_zero @ A ) ) ) ) ).
% cong_exp_iff_simps(2)
thf(fact_2456_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_2457_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_2458_log2__of__power__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less @ nat @ M2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M2 )
=> ( ord_less @ real @ ( log2 @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ M2 ) ) @ ( semiring_1_of_nat @ real @ N ) ) ) ) ).
% log2_of_power_less
thf(fact_2459_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_2460_pos__zdiv__mult__2,axiom,
! [A3: int,B2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A3 )
=> ( ( divide_divide @ int @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ B2 ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ A3 ) )
= ( divide_divide @ int @ B2 @ A3 ) ) ) ).
% pos_zdiv_mult_2
thf(fact_2461_neg__zdiv__mult__2,axiom,
! [A3: int,B2: int] :
( ( ord_less_eq @ int @ A3 @ ( zero_zero @ int ) )
=> ( ( divide_divide @ int @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ B2 ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ A3 ) )
= ( divide_divide @ int @ ( plus_plus @ int @ B2 @ ( one_one @ int ) ) @ A3 ) ) ) ).
% neg_zdiv_mult_2
thf(fact_2462_pos__zmod__mult__2,axiom,
! [A3: int,B2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A3 )
=> ( ( modulo_modulo @ int @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ B2 ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ A3 ) )
= ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( modulo_modulo @ int @ B2 @ A3 ) ) ) ) ) ).
% pos_zmod_mult_2
thf(fact_2463_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_2464_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_2465_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_2466_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_2467_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_2468_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_2469_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_2470_mod__double__modulus,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M2: A,X: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ M2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ( modulo_modulo @ A @ X @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M2 ) )
= ( modulo_modulo @ A @ X @ M2 ) )
| ( ( modulo_modulo @ A @ X @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M2 ) )
= ( plus_plus @ A @ ( modulo_modulo @ A @ X @ M2 ) @ M2 ) ) ) ) ) ) ).
% mod_double_modulus
thf(fact_2471_divmod__digit__1_I2_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ ( modulo_modulo @ A @ A3 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) )
=> ( ( minus_minus @ A @ ( modulo_modulo @ A @ A3 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) @ B2 )
= ( modulo_modulo @ A @ A3 @ B2 ) ) ) ) ) ) ).
% divmod_digit_1(2)
thf(fact_2472_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_2473_log2__of__power__le,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq @ nat @ M2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M2 )
=> ( ord_less_eq @ real @ ( log2 @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ M2 ) ) @ ( semiring_1_of_nat @ real @ N ) ) ) ) ).
% log2_of_power_le
thf(fact_2474_exp__bound__lemma,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [Z2: A] :
( ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ Z2 ) @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( exp @ A @ Z2 ) ) @ ( plus_plus @ real @ ( one_one @ real ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( real_V7770717601297561774m_norm @ A @ Z2 ) ) ) ) ) ) ).
% exp_bound_lemma
thf(fact_2475_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_2476_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_2477_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_2478_artanh__def,axiom,
! [A: $tType] :
( ( ( real_V3459762299906320749_field @ A )
& ( ln @ A ) )
=> ( ( artanh @ A )
= ( ^ [X2: A] : ( divide_divide @ A @ ( ln_ln @ A @ ( divide_divide @ A @ ( plus_plus @ A @ ( one_one @ A ) @ X2 ) @ ( minus_minus @ A @ ( one_one @ A ) @ X2 ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% artanh_def
thf(fact_2479_neg__zmod__mult__2,axiom,
! [A3: int,B2: int] :
( ( ord_less_eq @ int @ A3 @ ( zero_zero @ int ) )
=> ( ( modulo_modulo @ int @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ B2 ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ A3 ) )
= ( minus_minus @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( modulo_modulo @ int @ ( plus_plus @ int @ B2 @ ( one_one @ int ) ) @ A3 ) ) @ ( one_one @ int ) ) ) ) ).
% neg_zmod_mult_2
thf(fact_2480_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_2481_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_2482_floor__log2__div2,axiom,
! [N: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( archim6421214686448440834_floor @ real @ ( log2 @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ N ) ) )
= ( plus_plus @ int @ ( archim6421214686448440834_floor @ real @ ( log2 @ ( 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_2483_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_2484_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_2485_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_2486_tanh__real__altdef,axiom,
( ( tanh @ real )
= ( ^ [X2: real] : ( divide_divide @ real @ ( minus_minus @ real @ ( one_one @ real ) @ ( exp @ real @ ( times_times @ real @ ( uminus_uminus @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ X2 ) ) ) @ ( plus_plus @ real @ ( one_one @ real ) @ ( exp @ real @ ( times_times @ real @ ( uminus_uminus @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ X2 ) ) ) ) ) ) ).
% tanh_real_altdef
thf(fact_2487_divmod__digit__1_I1_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ ( modulo_modulo @ A @ A3 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) )
=> ( ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A3 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) ) @ ( one_one @ A ) )
= ( divide_divide @ A @ A3 @ B2 ) ) ) ) ) ) ).
% divmod_digit_1(1)
thf(fact_2488_pochhammer__double,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [Z2: A,N: nat] :
( ( comm_s3205402744901411588hammer @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Z2 ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
= ( times_times @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) @ ( comm_s3205402744901411588hammer @ A @ Z2 @ N ) ) @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ Z2 @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ N ) ) ) ) ).
% pochhammer_double
thf(fact_2489_norm__of__real__diff,axiom,
! [A: $tType] :
( ( real_V2822296259951069270ebra_1 @ A )
=> ! [B2: real,A3: real] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( real_Vector_of_real @ A @ B2 ) @ ( real_Vector_of_real @ A @ A3 ) ) ) @ ( abs_abs @ real @ ( minus_minus @ real @ B2 @ A3 ) ) ) ) ).
% norm_of_real_diff
thf(fact_2490_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_2491_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_2492_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_2493_floor__log__nat__eq__if,axiom,
! [B2: nat,N: nat,K: nat] :
( ( ord_less_eq @ nat @ ( power_power @ nat @ B2 @ N ) @ K )
=> ( ( ord_less @ nat @ K @ ( power_power @ nat @ B2 @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ B2 )
=> ( ( archim6421214686448440834_floor @ real @ ( log2 @ ( semiring_1_of_nat @ real @ B2 ) @ ( semiring_1_of_nat @ real @ K ) ) )
= ( semiring_1_of_nat @ int @ N ) ) ) ) ) ).
% floor_log_nat_eq_if
thf(fact_2494_floor__log__nat__eq__powr__iff,axiom,
! [B2: nat,K: nat,N: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ B2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ( ( archim6421214686448440834_floor @ real @ ( log2 @ ( semiring_1_of_nat @ real @ B2 ) @ ( semiring_1_of_nat @ real @ K ) ) )
= ( semiring_1_of_nat @ int @ N ) )
= ( ( ord_less_eq @ nat @ ( power_power @ nat @ B2 @ N ) @ K )
& ( ord_less @ nat @ K @ ( power_power @ nat @ B2 @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% floor_log_nat_eq_powr_iff
thf(fact_2495_ceiling__log2__div2,axiom,
! [N: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( archimedean_ceiling @ real @ ( log2 @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ N ) ) )
= ( plus_plus @ int @ ( archimedean_ceiling @ real @ ( log2 @ ( 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_2496_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_2497_ceiling__log__nat__eq__if,axiom,
! [B2: nat,N: nat,K: nat] :
( ( ord_less @ nat @ ( power_power @ nat @ B2 @ N ) @ K )
=> ( ( ord_less_eq @ nat @ K @ ( power_power @ nat @ B2 @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ B2 )
=> ( ( archimedean_ceiling @ real @ ( log2 @ ( semiring_1_of_nat @ real @ B2 ) @ ( semiring_1_of_nat @ real @ K ) ) )
= ( plus_plus @ int @ ( semiring_1_of_nat @ int @ N ) @ ( one_one @ int ) ) ) ) ) ) ).
% ceiling_log_nat_eq_if
thf(fact_2498_set__bit__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A] :
( ( bit_se5668285175392031749et_bit @ A @ ( zero_zero @ nat ) @ A3 )
= ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% set_bit_0
thf(fact_2499_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_2500_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_2501_set__n__deg__not__0,axiom,
! [TreeList: list @ vEBT_VEBT,N: nat,M2: nat] :
( ! [X3: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X3 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ( vEBT_invar_vebt @ X3 @ N ) )
=> ( ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M2 ) )
=> ( ord_less_eq @ nat @ ( one_one @ nat ) @ N ) ) ) ).
% set_n_deg_not_0
thf(fact_2502_high__bound__aux,axiom,
! [Ma: nat,N: nat,M2: nat] :
( ( ord_less @ nat @ Ma @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( plus_plus @ nat @ N @ M2 ) ) )
=> ( ord_less @ nat @ ( vEBT_VEBT_high @ Ma @ N ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M2 ) ) ) ).
% high_bound_aux
thf(fact_2503_unset__bit__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A] :
( ( bit_se2638667681897837118et_bit @ A @ ( zero_zero @ nat ) @ A3 )
= ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% unset_bit_0
thf(fact_2504_bit__split__inv,axiom,
! [X: nat,D2: nat] :
( ( vEBT_VEBT_bit_concat @ ( vEBT_VEBT_high @ X @ D2 ) @ ( vEBT_VEBT_low @ X @ D2 ) @ D2 )
= X ) ).
% bit_split_inv
thf(fact_2505_high__def,axiom,
( vEBT_VEBT_high
= ( ^ [X2: nat,N5: nat] : ( divide_divide @ nat @ X2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N5 ) ) ) ) ).
% high_def
thf(fact_2506_low__def,axiom,
( vEBT_VEBT_low
= ( ^ [X2: nat,N5: nat] : ( modulo_modulo @ nat @ X2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N5 ) ) ) ) ).
% low_def
thf(fact_2507_unset__bit__nonnegative__int__iff,axiom,
! [N: nat,K: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se2638667681897837118et_bit @ int @ N @ K ) )
= ( ord_less_eq @ int @ ( zero_zero @ int ) @ K ) ) ).
% unset_bit_nonnegative_int_iff
thf(fact_2508_set__bit__nonnegative__int__iff,axiom,
! [N: nat,K: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se5668285175392031749et_bit @ int @ N @ K ) )
= ( ord_less_eq @ int @ ( zero_zero @ int ) @ K ) ) ).
% set_bit_nonnegative_int_iff
thf(fact_2509_unset__bit__negative__int__iff,axiom,
! [N: nat,K: int] :
( ( ord_less @ int @ ( bit_se2638667681897837118et_bit @ int @ N @ K ) @ ( zero_zero @ int ) )
= ( ord_less @ int @ K @ ( zero_zero @ int ) ) ) ).
% unset_bit_negative_int_iff
thf(fact_2510_set__bit__negative__int__iff,axiom,
! [N: nat,K: int] :
( ( ord_less @ int @ ( bit_se5668285175392031749et_bit @ int @ N @ K ) @ ( zero_zero @ int ) )
= ( ord_less @ int @ K @ ( zero_zero @ int ) ) ) ).
% set_bit_negative_int_iff
thf(fact_2511_divide__complex__def,axiom,
( ( divide_divide @ complex )
= ( ^ [X2: complex,Y5: complex] : ( times_times @ complex @ X2 @ ( inverse_inverse @ complex @ Y5 ) ) ) ) ).
% divide_complex_def
thf(fact_2512_unset__bit__nat__def,axiom,
( ( bit_se2638667681897837118et_bit @ nat )
= ( ^ [M5: nat,N5: nat] : ( nat2 @ ( bit_se2638667681897837118et_bit @ int @ M5 @ ( semiring_1_of_nat @ int @ N5 ) ) ) ) ) ).
% unset_bit_nat_def
thf(fact_2513_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_2514_VEBT__internal_Oexp__split__high__low_I1_J,axiom,
! [X: nat,N: nat,M2: nat] :
( ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( plus_plus @ nat @ N @ M2 ) ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M2 )
=> ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ N ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M2 ) ) ) ) ) ).
% VEBT_internal.exp_split_high_low(1)
thf(fact_2515_VEBT__internal_Oexp__split__high__low_I2_J,axiom,
! [X: nat,N: nat,M2: nat] :
( ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( plus_plus @ nat @ N @ M2 ) ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ M2 )
=> ( 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_2516_invar__vebt_Ointros_I2_J,axiom,
! [TreeList: list @ vEBT_VEBT,N: nat,Summary: vEBT_VEBT,M2: nat,Deg: nat] :
( ! [X3: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X3 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ( vEBT_invar_vebt @ X3 @ N ) )
=> ( ( vEBT_invar_vebt @ Summary @ M2 )
=> ( ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M2 ) )
=> ( ( M2 = N )
=> ( ( Deg
= ( plus_plus @ nat @ N @ M2 ) )
=> ( ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_1 )
=> ( ! [X3: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X3 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X_1 ) )
=> ( vEBT_invar_vebt @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg @ TreeList @ Summary ) @ Deg ) ) ) ) ) ) ) ) ).
% invar_vebt.intros(2)
thf(fact_2517_invar__vebt_Ointros_I3_J,axiom,
! [TreeList: list @ vEBT_VEBT,N: nat,Summary: vEBT_VEBT,M2: nat,Deg: nat] :
( ! [X3: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X3 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ( vEBT_invar_vebt @ X3 @ N ) )
=> ( ( vEBT_invar_vebt @ Summary @ M2 )
=> ( ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M2 ) )
=> ( ( M2
= ( suc @ N ) )
=> ( ( Deg
= ( plus_plus @ nat @ N @ M2 ) )
=> ( ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_1 )
=> ( ! [X3: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X3 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X_1 ) )
=> ( vEBT_invar_vebt @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg @ TreeList @ Summary ) @ Deg ) ) ) ) ) ) ) ) ).
% invar_vebt.intros(3)
thf(fact_2518_unset__bit__Suc,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A3: A] :
( ( bit_se2638667681897837118et_bit @ A @ ( suc @ N ) @ A3 )
= ( plus_plus @ A @ ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se2638667681897837118et_bit @ A @ N @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% unset_bit_Suc
thf(fact_2519_set__bit__Suc,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A3: A] :
( ( bit_se5668285175392031749et_bit @ A @ ( suc @ N ) @ A3 )
= ( plus_plus @ A @ ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5668285175392031749et_bit @ A @ N @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% set_bit_Suc
thf(fact_2520_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_2521_flip__bit__Suc,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A3: A] :
( ( bit_se8732182000553998342ip_bit @ A @ ( suc @ N ) @ A3 )
= ( plus_plus @ A @ ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se8732182000553998342ip_bit @ A @ N @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% flip_bit_Suc
thf(fact_2522_signed__take__bit__rec,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( ( bit_ri4674362597316999326ke_bit @ A )
= ( ^ [N5: nat,A5: A] :
( if @ A
@ ( N5
= ( 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 @ N5 @ ( one_one @ nat ) ) @ ( divide_divide @ A @ A5 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% signed_take_bit_rec
thf(fact_2523_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_2524_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_2525_round__altdef,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archimedean_round @ A )
= ( ^ [X2: A] : ( if @ int @ ( ord_less_eq @ A @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( archimedean_frac @ A @ X2 ) ) @ ( archimedean_ceiling @ A @ X2 ) @ ( archim6421214686448440834_floor @ A @ X2 ) ) ) ) ) ).
% round_altdef
thf(fact_2526_inthall,axiom,
! [A: $tType,Xs: list @ A,P: A > $o,N: nat] :
( ! [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( P @ X3 ) )
=> ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( P @ ( nth @ A @ Xs @ N ) ) ) ) ).
% inthall
thf(fact_2527_flip__bit__nonnegative__int__iff,axiom,
! [N: nat,K: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se8732182000553998342ip_bit @ int @ N @ K ) )
= ( ord_less_eq @ int @ ( zero_zero @ int ) @ K ) ) ).
% flip_bit_nonnegative_int_iff
thf(fact_2528_flip__bit__negative__int__iff,axiom,
! [N: nat,K: int] :
( ( ord_less @ int @ ( bit_se8732182000553998342ip_bit @ int @ N @ K ) @ ( zero_zero @ int ) )
= ( ord_less @ int @ K @ ( zero_zero @ int ) ) ) ).
% flip_bit_negative_int_iff
thf(fact_2529_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_2530_round__of__int,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [N: int] :
( ( archimedean_round @ A @ ( ring_1_of_int @ A @ N ) )
= N ) ) ).
% round_of_int
thf(fact_2531_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_2532_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_2533_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_2534_signed__take__bit__numeral__of__1,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [K: num] :
( ( bit_ri4674362597316999326ke_bit @ A @ ( numeral_numeral @ nat @ K ) @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% signed_take_bit_numeral_of_1
thf(fact_2535_round__0,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archimedean_round @ A @ ( zero_zero @ A ) )
= ( zero_zero @ int ) ) ) ).
% round_0
thf(fact_2536_round__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [N: num] :
( ( archimedean_round @ A @ ( numeral_numeral @ A @ N ) )
= ( numeral_numeral @ int @ N ) ) ) ).
% round_numeral
thf(fact_2537_round__1,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archimedean_round @ A @ ( one_one @ A ) )
= ( one_one @ int ) ) ) ).
% round_1
thf(fact_2538_round__of__nat,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [N: nat] :
( ( archimedean_round @ A @ ( semiring_1_of_nat @ A @ N ) )
= ( semiring_1_of_nat @ int @ N ) ) ) ).
% round_of_nat
thf(fact_2539_dbl__simps_I5_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [K: num] :
( ( neg_numeral_dbl @ A @ ( numeral_numeral @ A @ K ) )
= ( numeral_numeral @ A @ ( bit0 @ K ) ) ) ) ).
% dbl_simps(5)
thf(fact_2540_dbl__simps_I1_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [K: num] :
( ( neg_numeral_dbl @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ K ) ) )
= ( uminus_uminus @ A @ ( neg_numeral_dbl @ A @ ( numeral_numeral @ A @ K ) ) ) ) ) ).
% dbl_simps(1)
thf(fact_2541_signed__take__bit__Suc__bit0,axiom,
! [N: nat,K: num] :
( ( bit_ri4674362597316999326ke_bit @ int @ ( suc @ N ) @ ( numeral_numeral @ int @ ( bit0 @ K ) ) )
= ( times_times @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N @ ( numeral_numeral @ int @ K ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ).
% signed_take_bit_Suc_bit0
thf(fact_2542_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_2543_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_2544_signed__take__bit__Suc__minus__bit0,axiom,
! [N: nat,K: num] :
( ( bit_ri4674362597316999326ke_bit @ int @ ( suc @ N ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ K ) ) ) )
= ( times_times @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ).
% signed_take_bit_Suc_minus_bit0
thf(fact_2545_signed__take__bit__0,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [A3: A] :
( ( bit_ri4674362597316999326ke_bit @ A @ ( zero_zero @ nat ) @ A3 )
= ( uminus_uminus @ A @ ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% signed_take_bit_0
thf(fact_2546_signed__take__bit__minus,axiom,
! [N: nat,K: int] :
( ( bit_ri4674362597316999326ke_bit @ int @ N @ ( uminus_uminus @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N @ K ) ) )
= ( bit_ri4674362597316999326ke_bit @ int @ N @ ( uminus_uminus @ int @ K ) ) ) ).
% signed_take_bit_minus
thf(fact_2547_signed__take__bit__diff,axiom,
! [N: nat,K: int,L: int] :
( ( bit_ri4674362597316999326ke_bit @ int @ N @ ( minus_minus @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N @ K ) @ ( bit_ri4674362597316999326ke_bit @ int @ N @ L ) ) )
= ( bit_ri4674362597316999326ke_bit @ int @ N @ ( minus_minus @ int @ K @ L ) ) ) ).
% signed_take_bit_diff
thf(fact_2548_list__eq__iff__nth__eq,axiom,
! [A: $tType] :
( ( ^ [Y4: list @ A,Z: list @ A] : Y4 = Z )
= ( ^ [Xs3: list @ A,Ys3: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs3 )
= ( size_size @ ( list @ A ) @ Ys3 ) )
& ! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs3 ) )
=> ( ( nth @ A @ Xs3 @ I2 )
= ( nth @ A @ Ys3 @ I2 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_2549_Skolem__list__nth,axiom,
! [A: $tType,K: nat,P: nat > A > $o] :
( ( ! [I2: nat] :
( ( ord_less @ nat @ I2 @ K )
=> ? [X7: A] : ( P @ I2 @ X7 ) ) )
= ( ? [Xs3: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs3 )
= K )
& ! [I2: nat] :
( ( ord_less @ nat @ I2 @ K )
=> ( P @ I2 @ ( nth @ A @ Xs3 @ I2 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_2550_nth__equalityI,axiom,
! [A: $tType,Xs: list @ A,Ys2: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ A ) @ Ys2 ) )
=> ( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ A @ Xs @ I3 )
= ( nth @ A @ Ys2 @ I3 ) ) )
=> ( Xs = Ys2 ) ) ) ).
% nth_equalityI
thf(fact_2551_dbl__def,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( neg_numeral_dbl @ A )
= ( ^ [X2: A] : ( plus_plus @ A @ X2 @ X2 ) ) ) ) ).
% dbl_def
thf(fact_2552_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_2553_list__ball__nth,axiom,
! [A: $tType,N: nat,Xs: list @ A,P: A > $o] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( P @ X3 ) )
=> ( P @ ( nth @ A @ Xs @ N ) ) ) ) ).
% list_ball_nth
thf(fact_2554_in__set__conv__nth,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
= ( ? [I2: nat] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs ) )
& ( ( nth @ A @ Xs @ I2 )
= X ) ) ) ) ).
% in_set_conv_nth
thf(fact_2555_all__nth__imp__all__set,axiom,
! [A: $tType,Xs: list @ A,P: A > $o,X: A] :
( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( P @ ( nth @ A @ Xs @ I3 ) ) )
=> ( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( P @ X ) ) ) ).
% all_nth_imp_all_set
thf(fact_2556_all__set__conv__all__nth,axiom,
! [A: $tType,Xs: list @ A,P: A > $o] :
( ( ! [X2: A] :
( ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
=> ( P @ X2 ) ) )
= ( ! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( P @ ( nth @ A @ Xs @ I2 ) ) ) ) ) ).
% all_set_conv_all_nth
thf(fact_2557_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_2558_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_2559_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_2560_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_2561_signed__take__bit__int__less__exp,axiom,
! [N: nat,K: int] : ( ord_less @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N @ K ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ).
% signed_take_bit_int_less_exp
thf(fact_2562_round__diff__minimal,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z2: A,M2: int] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ Z2 @ ( ring_1_of_int @ A @ ( archimedean_round @ A @ Z2 ) ) ) ) @ ( abs_abs @ A @ ( minus_minus @ A @ Z2 @ ( ring_1_of_int @ A @ M2 ) ) ) ) ) ).
% round_diff_minimal
thf(fact_2563_signed__take__bit__int__less__self__iff,axiom,
! [N: nat,K: int] :
( ( ord_less @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N @ K ) @ K )
= ( ord_less_eq @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) @ K ) ) ).
% signed_take_bit_int_less_self_iff
thf(fact_2564_signed__take__bit__int__greater__eq__self__iff,axiom,
! [K: int,N: nat] :
( ( ord_less_eq @ int @ K @ ( bit_ri4674362597316999326ke_bit @ int @ N @ K ) )
= ( ord_less @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ) ).
% signed_take_bit_int_greater_eq_self_iff
thf(fact_2565_signed__take__bit__int__greater__eq__minus__exp,axiom,
! [N: nat,K: int] : ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) @ ( bit_ri4674362597316999326ke_bit @ int @ N @ K ) ) ).
% signed_take_bit_int_greater_eq_minus_exp
thf(fact_2566_signed__take__bit__int__less__eq__self__iff,axiom,
! [N: nat,K: int] :
( ( ord_less_eq @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N @ K ) @ K )
= ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) @ K ) ) ).
% signed_take_bit_int_less_eq_self_iff
thf(fact_2567_signed__take__bit__int__greater__self__iff,axiom,
! [K: int,N: nat] :
( ( ord_less @ int @ K @ ( bit_ri4674362597316999326ke_bit @ int @ N @ K ) )
= ( ord_less @ int @ K @ ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% signed_take_bit_int_greater_self_iff
thf(fact_2568_signed__take__bit__int__less__eq,axiom,
! [N: nat,K: int] :
( ( ord_less_eq @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) @ K )
=> ( ord_less_eq @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N @ K ) @ ( minus_minus @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( suc @ N ) ) ) ) ) ).
% signed_take_bit_int_less_eq
thf(fact_2569_signed__take__bit__int__eq__self__iff,axiom,
! [N: nat,K: int] :
( ( ( bit_ri4674362597316999326ke_bit @ int @ N @ K )
= K )
= ( ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) @ K )
& ( ord_less @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% signed_take_bit_int_eq_self_iff
thf(fact_2570_signed__take__bit__int__eq__self,axiom,
! [N: nat,K: int] :
( ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) @ K )
=> ( ( ord_less @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( bit_ri4674362597316999326ke_bit @ int @ N @ K )
= K ) ) ) ).
% signed_take_bit_int_eq_self
thf(fact_2571_signed__take__bit__int__greater__eq,axiom,
! [K: int,N: nat] :
( ( ord_less @ int @ K @ ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) )
=> ( ord_less_eq @ int @ ( plus_plus @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( suc @ N ) ) ) @ ( bit_ri4674362597316999326ke_bit @ int @ N @ K ) ) ) ).
% signed_take_bit_int_greater_eq
thf(fact_2572_round__def,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archimedean_round @ A )
= ( ^ [X2: A] : ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X2 @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% round_def
thf(fact_2573_signed__take__bit__Suc,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: nat,A3: A] :
( ( bit_ri4674362597316999326ke_bit @ A @ ( suc @ N ) @ A3 )
= ( plus_plus @ A @ ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_ri4674362597316999326ke_bit @ A @ N @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% signed_take_bit_Suc
thf(fact_2574_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_2575_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_2576_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_2577_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_2578_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_2579_in__children__def,axiom,
( vEBT_V5917875025757280293ildren
= ( ^ [N5: nat,TreeList3: list @ vEBT_VEBT,X2: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ X2 @ N5 ) ) @ ( vEBT_VEBT_low @ X2 @ N5 ) ) ) ) ).
% in_children_def
thf(fact_2580_log__base__10__eq1,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( log2 @ ( 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_2581_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_2582_signed__take__bit__Suc__minus__bit1,axiom,
! [N: nat,K: num] :
( ( bit_ri4674362597316999326ke_bit @ int @ ( suc @ N ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ K ) ) ) )
= ( plus_plus @ int @ ( times_times @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N @ ( minus_minus @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) @ ( one_one @ int ) ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( one_one @ int ) ) ) ).
% signed_take_bit_Suc_minus_bit1
thf(fact_2583_concat__bit__Suc,axiom,
! [N: nat,K: int,L: int] :
( ( bit_concat_bit @ ( suc @ N ) @ K @ L )
= ( plus_plus @ int @ ( modulo_modulo @ int @ K @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_concat_bit @ N @ ( divide_divide @ int @ K @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ L ) ) ) ) ).
% concat_bit_Suc
thf(fact_2584_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_2585_verit__eq__simplify_I9_J,axiom,
! [X32: num,Y32: num] :
( ( ( bit1 @ X32 )
= ( bit1 @ Y32 ) )
= ( X32 = Y32 ) ) ).
% verit_eq_simplify(9)
thf(fact_2586_binomial__Suc__n,axiom,
! [N: nat] :
( ( binomial @ ( suc @ N ) @ N )
= ( suc @ N ) ) ).
% binomial_Suc_n
thf(fact_2587_binomial__n__n,axiom,
! [N: nat] :
( ( binomial @ N @ N )
= ( one_one @ nat ) ) ).
% binomial_n_n
thf(fact_2588_concat__bit__0,axiom,
! [K: int,L: int] :
( ( bit_concat_bit @ ( zero_zero @ nat ) @ K @ L )
= L ) ).
% concat_bit_0
thf(fact_2589_binomial__1,axiom,
! [N: nat] :
( ( binomial @ N @ ( suc @ ( zero_zero @ nat ) ) )
= N ) ).
% binomial_1
thf(fact_2590_binomial__0__Suc,axiom,
! [K: nat] :
( ( binomial @ ( zero_zero @ nat ) @ ( suc @ K ) )
= ( zero_zero @ nat ) ) ).
% binomial_0_Suc
thf(fact_2591_binomial__eq__0__iff,axiom,
! [N: nat,K: nat] :
( ( ( binomial @ N @ K )
= ( zero_zero @ nat ) )
= ( ord_less @ nat @ N @ K ) ) ).
% binomial_eq_0_iff
thf(fact_2592_binomial__Suc__Suc,axiom,
! [N: nat,K: nat] :
( ( binomial @ ( suc @ N ) @ ( suc @ K ) )
= ( plus_plus @ nat @ ( binomial @ N @ K ) @ ( binomial @ N @ ( suc @ K ) ) ) ) ).
% binomial_Suc_Suc
thf(fact_2593_binomial__n__0,axiom,
! [N: nat] :
( ( binomial @ N @ ( zero_zero @ nat ) )
= ( one_one @ nat ) ) ).
% binomial_n_0
thf(fact_2594_concat__bit__nonnegative__iff,axiom,
! [N: nat,K: int,L: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_concat_bit @ N @ K @ L ) )
= ( ord_less_eq @ int @ ( zero_zero @ int ) @ L ) ) ).
% concat_bit_nonnegative_iff
thf(fact_2595_concat__bit__negative__iff,axiom,
! [N: nat,K: int,L: int] :
( ( ord_less @ int @ ( bit_concat_bit @ N @ K @ L ) @ ( zero_zero @ int ) )
= ( ord_less @ int @ L @ ( zero_zero @ int ) ) ) ).
% concat_bit_negative_iff
thf(fact_2596_dbl__inc__simps_I5_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [K: num] :
( ( neg_numeral_dbl_inc @ A @ ( numeral_numeral @ A @ K ) )
= ( numeral_numeral @ A @ ( bit1 @ K ) ) ) ) ).
% dbl_inc_simps(5)
thf(fact_2597_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_2598_zero__less__binomial__iff,axiom,
! [N: nat,K: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( binomial @ N @ K ) )
= ( ord_less_eq @ nat @ K @ N ) ) ).
% zero_less_binomial_iff
thf(fact_2599_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_2600_div__Suc__eq__div__add3,axiom,
! [M2: nat,N: nat] :
( ( divide_divide @ nat @ M2 @ ( suc @ ( suc @ ( suc @ N ) ) ) )
= ( divide_divide @ nat @ M2 @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) @ N ) ) ) ).
% div_Suc_eq_div_add3
thf(fact_2601_Suc__div__eq__add3__div__numeral,axiom,
! [M2: nat,V2: num] :
( ( divide_divide @ nat @ ( suc @ ( suc @ ( suc @ M2 ) ) ) @ ( numeral_numeral @ nat @ V2 ) )
= ( divide_divide @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) @ M2 ) @ ( numeral_numeral @ nat @ V2 ) ) ) ).
% Suc_div_eq_add3_div_numeral
thf(fact_2602_mod__Suc__eq__mod__add3,axiom,
! [M2: nat,N: nat] :
( ( modulo_modulo @ nat @ M2 @ ( suc @ ( suc @ ( suc @ N ) ) ) )
= ( modulo_modulo @ nat @ M2 @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) @ N ) ) ) ).
% mod_Suc_eq_mod_add3
thf(fact_2603_Suc__mod__eq__add3__mod__numeral,axiom,
! [M2: nat,V2: num] :
( ( modulo_modulo @ nat @ ( suc @ ( suc @ ( suc @ M2 ) ) ) @ ( numeral_numeral @ nat @ V2 ) )
= ( modulo_modulo @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) @ M2 ) @ ( numeral_numeral @ nat @ V2 ) ) ) ).
% Suc_mod_eq_add3_mod_numeral
thf(fact_2604_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_2605_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_2606_signed__take__bit__Suc__bit1,axiom,
! [N: nat,K: num] :
( ( bit_ri4674362597316999326ke_bit @ int @ ( suc @ N ) @ ( numeral_numeral @ int @ ( bit1 @ K ) ) )
= ( plus_plus @ int @ ( times_times @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N @ ( numeral_numeral @ int @ K ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( one_one @ int ) ) ) ).
% signed_take_bit_Suc_bit1
thf(fact_2607_choose__one,axiom,
! [N: nat] :
( ( binomial @ N @ ( one_one @ nat ) )
= N ) ).
% choose_one
thf(fact_2608_verit__eq__simplify_I14_J,axiom,
! [X23: num,X32: num] :
( ( bit0 @ X23 )
!= ( bit1 @ X32 ) ) ).
% verit_eq_simplify(14)
thf(fact_2609_verit__eq__simplify_I12_J,axiom,
! [X32: num] :
( one2
!= ( bit1 @ X32 ) ) ).
% verit_eq_simplify(12)
thf(fact_2610_pi__neq__zero,axiom,
( pi
!= ( zero_zero @ real ) ) ).
% pi_neq_zero
thf(fact_2611_binomial__eq__0,axiom,
! [N: nat,K: nat] :
( ( ord_less @ nat @ N @ K )
=> ( ( binomial @ N @ K )
= ( zero_zero @ nat ) ) ) ).
% binomial_eq_0
thf(fact_2612_Suc__times__binomial,axiom,
! [K: nat,N: nat] :
( ( times_times @ nat @ ( suc @ K ) @ ( binomial @ ( suc @ N ) @ ( suc @ K ) ) )
= ( times_times @ nat @ ( suc @ N ) @ ( binomial @ N @ K ) ) ) ).
% Suc_times_binomial
thf(fact_2613_Suc__times__binomial__eq,axiom,
! [N: nat,K: nat] :
( ( times_times @ nat @ ( suc @ N ) @ ( binomial @ N @ K ) )
= ( times_times @ nat @ ( binomial @ ( suc @ N ) @ ( suc @ K ) ) @ ( suc @ K ) ) ) ).
% Suc_times_binomial_eq
thf(fact_2614_binomial__symmetric,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq @ nat @ K @ N )
=> ( ( binomial @ N @ K )
= ( binomial @ N @ ( minus_minus @ nat @ N @ K ) ) ) ) ).
% binomial_symmetric
thf(fact_2615_binomial__gbinomial,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [N: nat,K: nat] :
( ( semiring_1_of_nat @ A @ ( binomial @ N @ K ) )
= ( gbinomial @ A @ ( semiring_1_of_nat @ A @ N ) @ K ) ) ) ).
% binomial_gbinomial
thf(fact_2616_num_Oexhaust,axiom,
! [Y2: num] :
( ( Y2 != one2 )
=> ( ! [X24: num] :
( Y2
!= ( bit0 @ X24 ) )
=> ~ ! [X33: num] :
( Y2
!= ( bit1 @ X33 ) ) ) ) ).
% num.exhaust
thf(fact_2617_pi__not__less__zero,axiom,
~ ( ord_less @ real @ pi @ ( zero_zero @ real ) ) ).
% pi_not_less_zero
thf(fact_2618_pi__gt__zero,axiom,
ord_less @ real @ ( zero_zero @ real ) @ pi ).
% pi_gt_zero
thf(fact_2619_pi__ge__zero,axiom,
ord_less_eq @ real @ ( zero_zero @ real ) @ pi ).
% pi_ge_zero
thf(fact_2620_inc_Osimps_I2_J,axiom,
! [X: num] :
( ( inc @ ( bit0 @ X ) )
= ( bit1 @ X ) ) ).
% inc.simps(2)
thf(fact_2621_inc_Osimps_I3_J,axiom,
! [X: num] :
( ( inc @ ( bit1 @ X ) )
= ( bit0 @ ( inc @ X ) ) ) ).
% inc.simps(3)
thf(fact_2622_zero__less__binomial,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq @ nat @ K @ N )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ ( binomial @ N @ K ) ) ) ).
% zero_less_binomial
thf(fact_2623_Suc__times__binomial__add,axiom,
! [A3: nat,B2: nat] :
( ( times_times @ nat @ ( suc @ A3 ) @ ( binomial @ ( suc @ ( plus_plus @ nat @ A3 @ B2 ) ) @ ( suc @ A3 ) ) )
= ( times_times @ nat @ ( suc @ B2 ) @ ( binomial @ ( suc @ ( plus_plus @ nat @ A3 @ B2 ) ) @ A3 ) ) ) ).
% Suc_times_binomial_add
thf(fact_2624_choose__mult,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less_eq @ nat @ K @ M2 )
=> ( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( times_times @ nat @ ( binomial @ N @ M2 ) @ ( binomial @ M2 @ K ) )
= ( times_times @ nat @ ( binomial @ N @ K ) @ ( binomial @ ( minus_minus @ nat @ N @ K ) @ ( minus_minus @ nat @ M2 @ K ) ) ) ) ) ) ).
% choose_mult
thf(fact_2625_binomial__Suc__Suc__eq__times,axiom,
! [N: nat,K: nat] :
( ( binomial @ ( suc @ N ) @ ( suc @ K ) )
= ( divide_divide @ nat @ ( times_times @ nat @ ( suc @ N ) @ ( binomial @ N @ K ) ) @ ( suc @ K ) ) ) ).
% binomial_Suc_Suc_eq_times
thf(fact_2626_binomial__absorb__comp,axiom,
! [N: nat,K: nat] :
( ( times_times @ nat @ ( minus_minus @ nat @ N @ K ) @ ( binomial @ N @ K ) )
= ( times_times @ nat @ N @ ( binomial @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ K ) ) ) ).
% binomial_absorb_comp
thf(fact_2627_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_2628_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_2629_power__minus__Bit1,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X: A,K: num] :
( ( power_power @ A @ ( uminus_uminus @ A @ X ) @ ( numeral_numeral @ nat @ ( bit1 @ K ) ) )
= ( uminus_uminus @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit1 @ K ) ) ) ) ) ) ).
% power_minus_Bit1
thf(fact_2630_binomial__absorption,axiom,
! [K: nat,N: nat] :
( ( times_times @ nat @ ( suc @ K ) @ ( binomial @ N @ ( suc @ K ) ) )
= ( times_times @ nat @ N @ ( binomial @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ K ) ) ) ).
% binomial_absorption
thf(fact_2631_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_2632_binomial__fact__lemma,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq @ nat @ K @ N )
=> ( ( times_times @ nat @ ( times_times @ nat @ ( semiring_char_0_fact @ nat @ K ) @ ( semiring_char_0_fact @ nat @ ( minus_minus @ nat @ N @ K ) ) ) @ ( binomial @ N @ K ) )
= ( semiring_char_0_fact @ nat @ N ) ) ) ).
% binomial_fact_lemma
thf(fact_2633_cong__exp__iff__simps_I3_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [N: num,Q5: num] :
( ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit1 @ N ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q5 ) ) )
!= ( zero_zero @ A ) ) ) ).
% cong_exp_iff_simps(3)
thf(fact_2634_numeral__3__eq__3,axiom,
( ( numeral_numeral @ nat @ ( bit1 @ one2 ) )
= ( suc @ ( suc @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ).
% numeral_3_eq_3
thf(fact_2635_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_2636_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_2637_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_2638_pi__less__4,axiom,
ord_less @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ one2 ) ) ) ).
% pi_less_4
thf(fact_2639_pi__ge__two,axiom,
ord_less_eq @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ).
% pi_ge_two
thf(fact_2640_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_2641_binomial__ge__n__over__k__pow__k,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [K: nat,N: nat] :
( ( ord_less_eq @ nat @ K @ N )
=> ( ord_less_eq @ A @ ( power_power @ A @ ( divide_divide @ A @ ( semiring_1_of_nat @ A @ N ) @ ( semiring_1_of_nat @ A @ K ) ) @ K ) @ ( semiring_1_of_nat @ A @ ( binomial @ N @ K ) ) ) ) ) ).
% binomial_ge_n_over_k_pow_k
thf(fact_2642_binomial__mono,axiom,
! [K: nat,K7: nat,N: nat] :
( ( ord_less_eq @ nat @ K @ K7 )
=> ( ( ord_less_eq @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K7 ) @ N )
=> ( ord_less_eq @ nat @ ( binomial @ N @ K ) @ ( binomial @ N @ K7 ) ) ) ) ).
% binomial_mono
thf(fact_2643_binomial__maximum_H,axiom,
! [N: nat,K: nat] : ( ord_less_eq @ nat @ ( binomial @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ K ) @ ( binomial @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ N ) ) ).
% binomial_maximum'
thf(fact_2644_binomial__maximum,axiom,
! [N: nat,K: nat] : ( ord_less_eq @ nat @ ( binomial @ N @ K ) @ ( binomial @ N @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% binomial_maximum
thf(fact_2645_binomial__antimono,axiom,
! [K: nat,K7: nat,N: nat] :
( ( ord_less_eq @ nat @ K @ K7 )
=> ( ( ord_less_eq @ nat @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ K )
=> ( ( ord_less_eq @ nat @ K7 @ N )
=> ( ord_less_eq @ nat @ ( binomial @ N @ K7 ) @ ( binomial @ N @ K ) ) ) ) ) ).
% binomial_antimono
thf(fact_2646_choose__reduce__nat,axiom,
! [N: nat,K: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ( binomial @ N @ K )
= ( plus_plus @ nat @ ( binomial @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ ( minus_minus @ nat @ K @ ( one_one @ nat ) ) ) @ ( binomial @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ K ) ) ) ) ) ).
% choose_reduce_nat
thf(fact_2647_times__binomial__minus1__eq,axiom,
! [K: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ( times_times @ nat @ K @ ( binomial @ N @ K ) )
= ( times_times @ nat @ N @ ( binomial @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ ( minus_minus @ nat @ K @ ( one_one @ nat ) ) ) ) ) ) ).
% times_binomial_minus1_eq
thf(fact_2648_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_2649_binomial__altdef__nat,axiom,
! [K: nat,N: nat] :
( ( ord_less_eq @ nat @ K @ N )
=> ( ( binomial @ N @ K )
= ( divide_divide @ nat @ ( semiring_char_0_fact @ nat @ N ) @ ( times_times @ nat @ ( semiring_char_0_fact @ nat @ K ) @ ( semiring_char_0_fact @ nat @ ( minus_minus @ nat @ N @ K ) ) ) ) ) ) ).
% binomial_altdef_nat
thf(fact_2650_cong__exp__iff__simps_I11_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M2: num,Q5: num] :
( ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit1 @ M2 ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q5 ) ) )
= ( modulo_modulo @ A @ ( numeral_numeral @ A @ one2 ) @ ( numeral_numeral @ A @ ( bit0 @ Q5 ) ) ) )
= ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ M2 ) @ ( numeral_numeral @ A @ Q5 ) )
= ( zero_zero @ A ) ) ) ) ).
% cong_exp_iff_simps(11)
thf(fact_2651_cong__exp__iff__simps_I7_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [Q5: num,N: num] :
( ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ one2 ) @ ( numeral_numeral @ A @ ( bit0 @ Q5 ) ) )
= ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit1 @ N ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q5 ) ) ) )
= ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ N ) @ ( numeral_numeral @ A @ Q5 ) )
= ( zero_zero @ A ) ) ) ) ).
% cong_exp_iff_simps(7)
thf(fact_2652_Suc__div__eq__add3__div,axiom,
! [M2: nat,N: nat] :
( ( divide_divide @ nat @ ( suc @ ( suc @ ( suc @ M2 ) ) ) @ N )
= ( divide_divide @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) @ M2 ) @ N ) ) ).
% Suc_div_eq_add3_div
thf(fact_2653_Suc__mod__eq__add3__mod,axiom,
! [M2: nat,N: nat] :
( ( modulo_modulo @ nat @ ( suc @ ( suc @ ( suc @ M2 ) ) ) @ N )
= ( modulo_modulo @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) @ M2 ) @ N ) ) ).
% Suc_mod_eq_add3_mod
thf(fact_2654_pi__half__neq__zero,axiom,
( ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) )
!= ( zero_zero @ real ) ) ).
% pi_half_neq_zero
thf(fact_2655_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_2656_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_2657_exp__le,axiom,
ord_less_eq @ real @ ( exp @ real @ ( one_one @ real ) ) @ ( numeral_numeral @ real @ ( bit1 @ one2 ) ) ).
% exp_le
thf(fact_2658_binomial__strict__mono,axiom,
! [K: nat,K7: nat,N: nat] :
( ( ord_less @ nat @ K @ K7 )
=> ( ( ord_less_eq @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K7 ) @ N )
=> ( ord_less @ nat @ ( binomial @ N @ K ) @ ( binomial @ N @ K7 ) ) ) ) ).
% binomial_strict_mono
thf(fact_2659_binomial__strict__antimono,axiom,
! [K: nat,K7: nat,N: nat] :
( ( ord_less @ nat @ K @ K7 )
=> ( ( ord_less_eq @ nat @ N @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K ) )
=> ( ( ord_less_eq @ nat @ K7 @ N )
=> ( ord_less @ nat @ ( binomial @ N @ K7 ) @ ( binomial @ N @ K ) ) ) ) ) ).
% binomial_strict_antimono
thf(fact_2660_binomial__less__binomial__Suc,axiom,
! [K: nat,N: nat] :
( ( ord_less @ nat @ K @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
=> ( ord_less @ nat @ ( binomial @ N @ K ) @ ( binomial @ N @ ( suc @ K ) ) ) ) ).
% binomial_less_binomial_Suc
thf(fact_2661_binomial__addition__formula,axiom,
! [N: nat,K: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( binomial @ N @ ( suc @ K ) )
= ( plus_plus @ nat @ ( binomial @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ ( suc @ K ) ) @ ( binomial @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ K ) ) ) ) ).
% binomial_addition_formula
thf(fact_2662_binomial__fact,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,N: nat] :
( ( ord_less_eq @ nat @ K @ N )
=> ( ( semiring_1_of_nat @ A @ ( binomial @ N @ K ) )
= ( divide_divide @ A @ ( semiring_char_0_fact @ A @ N ) @ ( times_times @ A @ ( semiring_char_0_fact @ A @ K ) @ ( semiring_char_0_fact @ A @ ( minus_minus @ nat @ N @ K ) ) ) ) ) ) ) ).
% binomial_fact
thf(fact_2663_fact__binomial,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,N: nat] :
( ( ord_less_eq @ nat @ K @ N )
=> ( ( times_times @ A @ ( semiring_char_0_fact @ A @ K ) @ ( semiring_1_of_nat @ A @ ( binomial @ N @ K ) ) )
= ( divide_divide @ A @ ( semiring_char_0_fact @ A @ N ) @ ( semiring_char_0_fact @ A @ ( minus_minus @ nat @ N @ K ) ) ) ) ) ) ).
% fact_binomial
thf(fact_2664_mod__exhaust__less__4,axiom,
! [M2: nat] :
( ( ( modulo_modulo @ nat @ M2 @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) )
= ( zero_zero @ nat ) )
| ( ( modulo_modulo @ nat @ M2 @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) )
= ( one_one @ nat ) )
| ( ( modulo_modulo @ nat @ M2 @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) )
= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
| ( ( modulo_modulo @ nat @ M2 @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) )
= ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) ) ) ).
% mod_exhaust_less_4
thf(fact_2665_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_2666_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_2667_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_2668_arctan__ubound,axiom,
! [Y2: real] : ( ord_less @ real @ ( arctan @ Y2 ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ).
% arctan_ubound
thf(fact_2669_arctan__one,axiom,
( ( arctan @ ( one_one @ real ) )
= ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ).
% arctan_one
thf(fact_2670_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_2671_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_2672_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_2673_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_2674_log__base__10__eq2,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( log2 @ ( numeral_numeral @ real @ ( bit0 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) @ X )
= ( times_times @ real @ ( log2 @ ( numeral_numeral @ real @ ( bit0 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) @ ( exp @ real @ ( one_one @ real ) ) ) @ ( ln_ln @ real @ X ) ) ) ) ).
% log_base_10_eq2
thf(fact_2675_binomial__code,axiom,
( binomial
= ( ^ [N5: nat,K3: nat] : ( if @ nat @ ( ord_less @ nat @ N5 @ K3 ) @ ( zero_zero @ nat ) @ ( if @ nat @ ( ord_less @ nat @ N5 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K3 ) ) @ ( binomial @ N5 @ ( minus_minus @ nat @ N5 @ K3 ) ) @ ( divide_divide @ nat @ ( set_fo6178422350223883121st_nat @ nat @ ( times_times @ nat ) @ ( plus_plus @ nat @ ( minus_minus @ nat @ N5 @ K3 ) @ ( one_one @ nat ) ) @ N5 @ ( one_one @ nat ) ) @ ( semiring_char_0_fact @ nat @ K3 ) ) ) ) ) ) ).
% binomial_code
thf(fact_2676_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_2677_signed__take__bit__numeral__minus__bit1,axiom,
! [L: num,K: num] :
( ( bit_ri4674362597316999326ke_bit @ int @ ( numeral_numeral @ nat @ L ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ K ) ) ) )
= ( plus_plus @ int @ ( times_times @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( pred_numeral @ L ) @ ( minus_minus @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) @ ( one_one @ int ) ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( one_one @ int ) ) ) ).
% signed_take_bit_numeral_minus_bit1
thf(fact_2678_cos__pi__eq__zero,axiom,
! [M2: nat] :
( ( cos @ real @ ( divide_divide @ real @ ( times_times @ real @ pi @ ( semiring_1_of_nat @ real @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M2 ) ) ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
= ( zero_zero @ real ) ) ).
% cos_pi_eq_zero
thf(fact_2679_signed__take__bit__numeral__bit1,axiom,
! [L: num,K: num] :
( ( bit_ri4674362597316999326ke_bit @ int @ ( numeral_numeral @ nat @ L ) @ ( numeral_numeral @ int @ ( bit1 @ K ) ) )
= ( plus_plus @ int @ ( times_times @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( pred_numeral @ L ) @ ( numeral_numeral @ int @ K ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( one_one @ int ) ) ) ).
% signed_take_bit_numeral_bit1
thf(fact_2680_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_2681_sin__zero,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ( ( sin @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% sin_zero
thf(fact_2682_cos__minus,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X: A] :
( ( cos @ A @ ( uminus_uminus @ A @ X ) )
= ( cos @ A @ X ) ) ) ).
% cos_minus
thf(fact_2683_sin__minus,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X: A] :
( ( sin @ A @ ( uminus_uminus @ A @ X ) )
= ( uminus_uminus @ A @ ( sin @ A @ X ) ) ) ) ).
% sin_minus
thf(fact_2684_cot__zero,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ( ( cot @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% cot_zero
thf(fact_2685_cot__minus,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A] :
( ( cot @ A @ ( uminus_uminus @ A @ X ) )
= ( uminus_uminus @ A @ ( cot @ A @ X ) ) ) ) ).
% cot_minus
thf(fact_2686_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_2687_pred__numeral__simps_I1_J,axiom,
( ( pred_numeral @ one2 )
= ( zero_zero @ nat ) ) ).
% pred_numeral_simps(1)
thf(fact_2688_sin__pi,axiom,
( ( sin @ real @ pi )
= ( zero_zero @ real ) ) ).
% sin_pi
thf(fact_2689_Suc__eq__numeral,axiom,
! [N: nat,K: num] :
( ( ( suc @ N )
= ( numeral_numeral @ nat @ K ) )
= ( N
= ( pred_numeral @ K ) ) ) ).
% Suc_eq_numeral
thf(fact_2690_eq__numeral__Suc,axiom,
! [K: num,N: nat] :
( ( ( numeral_numeral @ nat @ K )
= ( suc @ N ) )
= ( ( pred_numeral @ K )
= N ) ) ).
% eq_numeral_Suc
thf(fact_2691_sin__pi__minus,axiom,
! [X: real] :
( ( sin @ real @ ( minus_minus @ real @ pi @ X ) )
= ( sin @ real @ X ) ) ).
% sin_pi_minus
thf(fact_2692_pred__numeral__inc,axiom,
! [K: num] :
( ( pred_numeral @ ( inc @ K ) )
= ( numeral_numeral @ nat @ K ) ) ).
% pred_numeral_inc
thf(fact_2693_cot__pi,axiom,
( ( cot @ real @ pi )
= ( zero_zero @ real ) ) ).
% cot_pi
thf(fact_2694_sin__of__real__pi,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ( ( sin @ A @ ( real_Vector_of_real @ A @ pi ) )
= ( zero_zero @ A ) ) ) ).
% sin_of_real_pi
thf(fact_2695_pred__numeral__simps_I3_J,axiom,
! [K: num] :
( ( pred_numeral @ ( bit1 @ K ) )
= ( numeral_numeral @ nat @ ( bit0 @ K ) ) ) ).
% pred_numeral_simps(3)
thf(fact_2696_less__numeral__Suc,axiom,
! [K: num,N: nat] :
( ( ord_less @ nat @ ( numeral_numeral @ nat @ K ) @ ( suc @ N ) )
= ( ord_less @ nat @ ( pred_numeral @ K ) @ N ) ) ).
% less_numeral_Suc
thf(fact_2697_less__Suc__numeral,axiom,
! [N: nat,K: num] :
( ( ord_less @ nat @ ( suc @ N ) @ ( numeral_numeral @ nat @ K ) )
= ( ord_less @ nat @ N @ ( pred_numeral @ K ) ) ) ).
% less_Suc_numeral
thf(fact_2698_le__Suc__numeral,axiom,
! [N: nat,K: num] :
( ( ord_less_eq @ nat @ ( suc @ N ) @ ( numeral_numeral @ nat @ K ) )
= ( ord_less_eq @ nat @ N @ ( pred_numeral @ K ) ) ) ).
% le_Suc_numeral
thf(fact_2699_le__numeral__Suc,axiom,
! [K: num,N: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ K ) @ ( suc @ N ) )
= ( ord_less_eq @ nat @ ( pred_numeral @ K ) @ N ) ) ).
% le_numeral_Suc
thf(fact_2700_diff__Suc__numeral,axiom,
! [N: nat,K: num] :
( ( minus_minus @ nat @ ( suc @ N ) @ ( numeral_numeral @ nat @ K ) )
= ( minus_minus @ nat @ N @ ( pred_numeral @ K ) ) ) ).
% diff_Suc_numeral
thf(fact_2701_diff__numeral__Suc,axiom,
! [K: num,N: nat] :
( ( minus_minus @ nat @ ( numeral_numeral @ nat @ K ) @ ( suc @ N ) )
= ( minus_minus @ nat @ ( pred_numeral @ K ) @ N ) ) ).
% diff_numeral_Suc
thf(fact_2702_cos__pi,axiom,
( ( cos @ real @ pi )
= ( uminus_uminus @ real @ ( one_one @ real ) ) ) ).
% cos_pi
thf(fact_2703_cos__periodic__pi2,axiom,
! [X: real] :
( ( cos @ real @ ( plus_plus @ real @ pi @ X ) )
= ( uminus_uminus @ real @ ( cos @ real @ X ) ) ) ).
% cos_periodic_pi2
thf(fact_2704_cos__periodic__pi,axiom,
! [X: real] :
( ( cos @ real @ ( plus_plus @ real @ X @ pi ) )
= ( uminus_uminus @ real @ ( cos @ real @ X ) ) ) ).
% cos_periodic_pi
thf(fact_2705_sin__periodic__pi2,axiom,
! [X: real] :
( ( sin @ real @ ( plus_plus @ real @ pi @ X ) )
= ( uminus_uminus @ real @ ( sin @ real @ X ) ) ) ).
% sin_periodic_pi2
thf(fact_2706_sin__periodic__pi,axiom,
! [X: real] :
( ( sin @ real @ ( plus_plus @ real @ X @ pi ) )
= ( uminus_uminus @ real @ ( sin @ real @ X ) ) ) ).
% sin_periodic_pi
thf(fact_2707_cos__minus__pi,axiom,
! [X: real] :
( ( cos @ real @ ( minus_minus @ real @ X @ pi ) )
= ( uminus_uminus @ real @ ( cos @ real @ X ) ) ) ).
% cos_minus_pi
thf(fact_2708_cos__pi__minus,axiom,
! [X: real] :
( ( cos @ real @ ( minus_minus @ real @ pi @ X ) )
= ( uminus_uminus @ real @ ( cos @ real @ X ) ) ) ).
% cos_pi_minus
thf(fact_2709_sin__minus__pi,axiom,
! [X: real] :
( ( sin @ real @ ( minus_minus @ real @ X @ pi ) )
= ( uminus_uminus @ real @ ( sin @ real @ X ) ) ) ).
% sin_minus_pi
thf(fact_2710_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_2711_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_2712_sin__npi2,axiom,
! [N: nat] :
( ( sin @ real @ ( times_times @ real @ pi @ ( semiring_1_of_nat @ real @ N ) ) )
= ( zero_zero @ real ) ) ).
% sin_npi2
thf(fact_2713_sin__npi,axiom,
! [N: nat] :
( ( sin @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ pi ) )
= ( zero_zero @ real ) ) ).
% sin_npi
thf(fact_2714_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_2715_cot__npi,axiom,
! [N: nat] :
( ( cot @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ pi ) )
= ( zero_zero @ real ) ) ).
% cot_npi
thf(fact_2716_cos__pi__half,axiom,
( ( cos @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
= ( zero_zero @ real ) ) ).
% cos_pi_half
thf(fact_2717_sin__two__pi,axiom,
( ( sin @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) )
= ( zero_zero @ real ) ) ).
% sin_two_pi
thf(fact_2718_sin__pi__half,axiom,
( ( sin @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
= ( one_one @ real ) ) ).
% sin_pi_half
thf(fact_2719_cos__two__pi,axiom,
( ( cos @ real @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) )
= ( one_one @ real ) ) ).
% cos_two_pi
thf(fact_2720_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_2721_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_2722_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_2723_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_2724_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_2725_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_2726_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_2727_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_2728_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_2729_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_2730_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_2731_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_2732_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_2733_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_2734_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_2735_signed__take__bit__numeral__minus__bit0,axiom,
! [L: num,K: num] :
( ( bit_ri4674362597316999326ke_bit @ int @ ( numeral_numeral @ nat @ L ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ K ) ) ) )
= ( times_times @ int @ ( bit_ri4674362597316999326ke_bit @ int @ ( pred_numeral @ L ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ).
% signed_take_bit_numeral_minus_bit0
thf(fact_2736_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_2737_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_2738_cos__of__real,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X: real] :
( ( cos @ A @ ( real_Vector_of_real @ A @ X ) )
= ( real_Vector_of_real @ A @ ( cos @ real @ X ) ) ) ) ).
% cos_of_real
thf(fact_2739_cot__of__real,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: real] :
( ( real_Vector_of_real @ A @ ( cot @ real @ X ) )
= ( cot @ A @ ( real_Vector_of_real @ A @ X ) ) ) ) ).
% cot_of_real
thf(fact_2740_sin__of__real,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X: real] :
( ( sin @ A @ ( real_Vector_of_real @ A @ X ) )
= ( real_Vector_of_real @ A @ ( sin @ real @ X ) ) ) ) ).
% sin_of_real
thf(fact_2741_cot__def,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ( ( cot @ A )
= ( ^ [X2: A] : ( divide_divide @ A @ ( cos @ A @ X2 ) @ ( sin @ A @ X2 ) ) ) ) ) ).
% cot_def
thf(fact_2742_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_2743_polar__Ex,axiom,
! [X: real,Y2: real] :
? [R3: real,A6: real] :
( ( X
= ( times_times @ real @ R3 @ ( cos @ real @ A6 ) ) )
& ( Y2
= ( times_times @ real @ R3 @ ( sin @ real @ A6 ) ) ) ) ).
% polar_Ex
thf(fact_2744_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_2745_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_2746_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_2747_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_2748_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_2749_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_2750_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_2751_sincos__principal__value,axiom,
! [X: real] :
? [Y3: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ pi ) @ Y3 )
& ( ord_less_eq @ real @ Y3 @ pi )
& ( ( sin @ real @ Y3 )
= ( sin @ real @ X ) )
& ( ( cos @ real @ Y3 )
= ( cos @ real @ X ) ) ) ).
% sincos_principal_value
thf(fact_2752_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_2753_sin__le__one,axiom,
! [X: real] : ( ord_less_eq @ real @ ( sin @ real @ X ) @ ( one_one @ real ) ) ).
% sin_le_one
thf(fact_2754_cos__le__one,axiom,
! [X: real] : ( ord_less_eq @ real @ ( cos @ real @ X ) @ ( one_one @ real ) ) ).
% cos_le_one
thf(fact_2755_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_2756_cos__arctan__not__zero,axiom,
! [X: real] :
( ( cos @ real @ ( arctan @ X ) )
!= ( zero_zero @ real ) ) ).
% cos_arctan_not_zero
thf(fact_2757_numeral__eq__Suc,axiom,
( ( numeral_numeral @ nat )
= ( ^ [K3: num] : ( suc @ ( pred_numeral @ K3 ) ) ) ) ).
% numeral_eq_Suc
thf(fact_2758_cos__int__times__real,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [M2: int,X: real] :
( ( cos @ A @ ( times_times @ A @ ( ring_1_of_int @ A @ M2 ) @ ( real_Vector_of_real @ A @ X ) ) )
= ( real_Vector_of_real @ A @ ( cos @ real @ ( times_times @ real @ ( ring_1_of_int @ real @ M2 ) @ X ) ) ) ) ) ).
% cos_int_times_real
thf(fact_2759_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_2760_sin__int__times__real,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [M2: int,X: real] :
( ( sin @ A @ ( times_times @ A @ ( ring_1_of_int @ A @ M2 ) @ ( real_Vector_of_real @ A @ X ) ) )
= ( real_Vector_of_real @ A @ ( sin @ real @ ( times_times @ real @ ( ring_1_of_int @ real @ M2 ) @ X ) ) ) ) ) ).
% sin_int_times_real
thf(fact_2761_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_2762_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_2763_sin__gt__zero,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ X @ pi )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( sin @ real @ X ) ) ) ) ).
% sin_gt_zero
thf(fact_2764_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_2765_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_2766_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_2767_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_2768_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_2769_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_2770_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_2771_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_2772_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_2773_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_2774_cos__diff__cos,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [W: A,Z2: A] :
( ( minus_minus @ A @ ( cos @ A @ W ) @ ( cos @ A @ Z2 ) )
= ( times_times @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( sin @ A @ ( divide_divide @ A @ ( plus_plus @ A @ W @ Z2 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) @ ( sin @ A @ ( divide_divide @ A @ ( minus_minus @ A @ Z2 @ W ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% cos_diff_cos
thf(fact_2775_sin__diff__sin,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [W: A,Z2: A] :
( ( minus_minus @ A @ ( sin @ A @ W ) @ ( sin @ A @ Z2 ) )
= ( times_times @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( sin @ A @ ( divide_divide @ A @ ( minus_minus @ A @ W @ Z2 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) @ ( cos @ A @ ( divide_divide @ A @ ( plus_plus @ A @ W @ Z2 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% sin_diff_sin
thf(fact_2776_sin__plus__sin,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [W: A,Z2: A] :
( ( plus_plus @ A @ ( sin @ A @ W ) @ ( sin @ A @ Z2 ) )
= ( times_times @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( sin @ A @ ( divide_divide @ A @ ( plus_plus @ A @ W @ Z2 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) @ ( cos @ A @ ( divide_divide @ A @ ( minus_minus @ A @ W @ Z2 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% sin_plus_sin
thf(fact_2777_cos__times__sin,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [W: A,Z2: A] :
( ( times_times @ A @ ( cos @ A @ W ) @ ( sin @ A @ Z2 ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( sin @ A @ ( plus_plus @ A @ W @ Z2 ) ) @ ( sin @ A @ ( minus_minus @ A @ W @ Z2 ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% cos_times_sin
thf(fact_2778_sin__times__cos,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [W: A,Z2: A] :
( ( times_times @ A @ ( sin @ A @ W ) @ ( cos @ A @ Z2 ) )
= ( divide_divide @ A @ ( plus_plus @ A @ ( sin @ A @ ( plus_plus @ A @ W @ Z2 ) ) @ ( sin @ A @ ( minus_minus @ A @ W @ Z2 ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% sin_times_cos
thf(fact_2779_sin__times__sin,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [W: A,Z2: A] :
( ( times_times @ A @ ( sin @ A @ W ) @ ( sin @ A @ Z2 ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( cos @ A @ ( minus_minus @ A @ W @ Z2 ) ) @ ( cos @ A @ ( plus_plus @ A @ W @ Z2 ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% sin_times_sin
thf(fact_2780_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_2781_sin__cos__eq,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ( ( sin @ A )
= ( ^ [X2: A] : ( cos @ A @ ( minus_minus @ A @ ( divide_divide @ A @ ( real_Vector_of_real @ A @ pi ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ X2 ) ) ) ) ) ).
% sin_cos_eq
thf(fact_2782_cos__sin__eq,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ( ( cos @ A )
= ( ^ [X2: A] : ( sin @ A @ ( minus_minus @ A @ ( divide_divide @ A @ ( real_Vector_of_real @ A @ pi ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ X2 ) ) ) ) ) ).
% cos_sin_eq
thf(fact_2783_pred__numeral__def,axiom,
( pred_numeral
= ( ^ [K3: num] : ( minus_minus @ nat @ ( numeral_numeral @ nat @ K3 ) @ ( one_one @ nat ) ) ) ) ).
% pred_numeral_def
thf(fact_2784_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_2785_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_2786_cos__two__neq__zero,axiom,
( ( cos @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) )
!= ( zero_zero @ real ) ) ).
% cos_two_neq_zero
thf(fact_2787_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_2788_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_2789_sin__eq__0__pi,axiom,
! [X: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ pi ) @ X )
=> ( ( ord_less @ real @ X @ pi )
=> ( ( ( sin @ real @ X )
= ( zero_zero @ real ) )
=> ( X
= ( zero_zero @ real ) ) ) ) ) ).
% sin_eq_0_pi
thf(fact_2790_sin__zero__pi__iff,axiom,
! [X: real] :
( ( ord_less @ real @ ( abs_abs @ real @ X ) @ pi )
=> ( ( ( sin @ real @ X )
= ( zero_zero @ real ) )
= ( X
= ( zero_zero @ real ) ) ) ) ).
% sin_zero_pi_iff
thf(fact_2791_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_2792_sin__zero__iff__int2,axiom,
! [X: real] :
( ( ( sin @ real @ X )
= ( zero_zero @ real ) )
= ( ? [I2: int] :
( X
= ( times_times @ real @ ( ring_1_of_int @ real @ I2 ) @ pi ) ) ) ) ).
% sin_zero_iff_int2
thf(fact_2793_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_2794_sin__expansion__lemma,axiom,
! [X: real,M2: nat] :
( ( sin @ real @ ( plus_plus @ real @ X @ ( divide_divide @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ ( suc @ M2 ) ) @ pi ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) )
= ( cos @ real @ ( plus_plus @ real @ X @ ( divide_divide @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ M2 ) @ pi ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ).
% sin_expansion_lemma
thf(fact_2795_cos__expansion__lemma,axiom,
! [X: real,M2: nat] :
( ( cos @ real @ ( plus_plus @ real @ X @ ( divide_divide @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ ( suc @ M2 ) ) @ 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 @ M2 ) @ pi ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% cos_expansion_lemma
thf(fact_2796_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_2797_cos__two__less__zero,axiom,
ord_less @ real @ ( cos @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( zero_zero @ real ) ).
% cos_two_less_zero
thf(fact_2798_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_2799_cos__is__zero,axiom,
? [X3: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 )
& ( ord_less_eq @ real @ X3 @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) )
& ( ( cos @ real @ X3 )
= ( zero_zero @ real ) )
& ! [Y: real] :
( ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y )
& ( ord_less_eq @ real @ Y @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) )
& ( ( cos @ real @ Y )
= ( zero_zero @ real ) ) )
=> ( Y = X3 ) ) ) ).
% cos_is_zero
thf(fact_2800_fold__atLeastAtMost__nat_Osimps,axiom,
! [A: $tType] :
( ( set_fo6178422350223883121st_nat @ A )
= ( ^ [F4: nat > A > A,A5: nat,B3: nat,Acc: A] : ( if @ A @ ( ord_less @ nat @ B3 @ A5 ) @ Acc @ ( set_fo6178422350223883121st_nat @ A @ F4 @ ( plus_plus @ nat @ A5 @ ( one_one @ nat ) ) @ B3 @ ( F4 @ A5 @ Acc ) ) ) ) ) ).
% fold_atLeastAtMost_nat.simps
thf(fact_2801_fold__atLeastAtMost__nat_Oelims,axiom,
! [A: $tType,X: nat > A > A,Xa: nat,Xb3: nat,Xc: A,Y2: A] :
( ( ( set_fo6178422350223883121st_nat @ A @ X @ Xa @ Xb3 @ Xc )
= Y2 )
=> ( ( ( ord_less @ nat @ Xb3 @ Xa )
=> ( Y2 = Xc ) )
& ( ~ ( ord_less @ nat @ Xb3 @ Xa )
=> ( Y2
= ( set_fo6178422350223883121st_nat @ A @ X @ ( plus_plus @ nat @ Xa @ ( one_one @ nat ) ) @ Xb3 @ ( X @ Xa @ Xc ) ) ) ) ) ) ).
% fold_atLeastAtMost_nat.elims
thf(fact_2802_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_2803_cos__total,axiom,
! [Y2: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y2 )
=> ( ( ord_less_eq @ real @ Y2 @ ( one_one @ real ) )
=> ? [X3: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 )
& ( ord_less_eq @ real @ X3 @ pi )
& ( ( cos @ real @ X3 )
= Y2 )
& ! [Y: real] :
( ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y )
& ( ord_less_eq @ real @ Y @ pi )
& ( ( cos @ real @ Y )
= Y2 ) )
=> ( Y = X3 ) ) ) ) ) ).
% cos_total
thf(fact_2804_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_2805_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_2806_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_2807_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_2808_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_2809_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_2810_cos__plus__cos,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [W: A,Z2: A] :
( ( plus_plus @ A @ ( cos @ A @ W ) @ ( cos @ A @ Z2 ) )
= ( times_times @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( cos @ A @ ( divide_divide @ A @ ( plus_plus @ A @ W @ Z2 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) @ ( cos @ A @ ( divide_divide @ A @ ( minus_minus @ A @ W @ Z2 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% cos_plus_cos
thf(fact_2811_cos__times__cos,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [W: A,Z2: A] :
( ( times_times @ A @ ( cos @ A @ W ) @ ( cos @ A @ Z2 ) )
= ( divide_divide @ A @ ( plus_plus @ A @ ( cos @ A @ ( minus_minus @ A @ W @ Z2 ) ) @ ( cos @ A @ ( plus_plus @ A @ W @ Z2 ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% cos_times_cos
thf(fact_2812_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_2813_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_2814_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_2815_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_2816_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_2817_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_2818_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_2819_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_2820_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_2821_cos__one__2pi__int,axiom,
! [X: real] :
( ( ( cos @ real @ X )
= ( one_one @ real ) )
= ( ? [X2: int] :
( X
= ( times_times @ real @ ( times_times @ real @ ( ring_1_of_int @ real @ X2 ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ pi ) ) ) ) ).
% cos_one_2pi_int
thf(fact_2822_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_2823_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_2824_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_2825_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_2826_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_2827_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_2828_sin__total,axiom,
! [Y2: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y2 )
=> ( ( ord_less_eq @ real @ Y2 @ ( one_one @ real ) )
=> ? [X3: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X3 )
& ( ord_less_eq @ real @ X3 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
& ( ( sin @ real @ X3 )
= Y2 )
& ! [Y: real] :
( ( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ Y )
& ( ord_less_eq @ real @ Y @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
& ( ( sin @ real @ Y )
= Y2 ) )
=> ( Y = X3 ) ) ) ) ) ).
% sin_total
thf(fact_2829_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_2830_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_2831_cos__one__2pi,axiom,
! [X: real] :
( ( ( cos @ real @ X )
= ( one_one @ real ) )
= ( ? [X2: nat] :
( X
= ( times_times @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ X2 ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ pi ) )
| ? [X2: nat] :
( X
= ( uminus_uminus @ real @ ( times_times @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ X2 ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ pi ) ) ) ) ) ).
% cos_one_2pi
thf(fact_2832_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_2833_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_2834_fact__code,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ( ( semiring_char_0_fact @ A )
= ( ^ [N5: nat] : ( semiring_1_of_nat @ A @ ( set_fo6178422350223883121st_nat @ nat @ ( times_times @ nat ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N5 @ ( one_one @ nat ) ) ) ) ) ) ).
% fact_code
thf(fact_2835_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_2836_complex__unimodular__polar,axiom,
! [Z2: complex] :
( ( ( real_V7770717601297561774m_norm @ complex @ Z2 )
= ( 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 ) )
=> ( Z2
!= ( complex2 @ ( cos @ real @ T4 ) @ ( sin @ real @ T4 ) ) ) ) ) ) ).
% complex_unimodular_polar
thf(fact_2837_sin__zero__iff,axiom,
! [X: real] :
( ( ( sin @ real @ X )
= ( zero_zero @ real ) )
= ( ? [N5: nat] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N5 )
& ( X
= ( times_times @ real @ ( semiring_1_of_nat @ real @ N5 ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) )
| ? [N5: nat] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N5 )
& ( X
= ( uminus_uminus @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N5 ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ).
% sin_zero_iff
thf(fact_2838_cos__zero__iff,axiom,
! [X: real] :
( ( ( cos @ real @ X )
= ( zero_zero @ real ) )
= ( ? [N5: nat] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N5 )
& ( X
= ( times_times @ real @ ( semiring_1_of_nat @ real @ N5 ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) )
| ? [N5: nat] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N5 )
& ( X
= ( uminus_uminus @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N5 ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ).
% cos_zero_iff
thf(fact_2839_sin__zero__lemma,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ( 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 ) ) ) ) ) ) ) ) ).
% sin_zero_lemma
thf(fact_2840_cos__zero__lemma,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ( 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 ) ) ) ) ) ) ) ) ).
% cos_zero_lemma
thf(fact_2841_dvd__0__right,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A3: A] : ( dvd_dvd @ A @ A3 @ ( zero_zero @ A ) ) ) ).
% dvd_0_right
thf(fact_2842_dvd__0__left__iff,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A3: A] :
( ( dvd_dvd @ A @ ( zero_zero @ A ) @ A3 )
= ( A3
= ( zero_zero @ A ) ) ) ) ).
% dvd_0_left_iff
thf(fact_2843_dvd__add__triv__right__iff,axiom,
! [A: $tType] :
( ( comm_s4317794764714335236cancel @ A )
=> ! [A3: A,B2: A] :
( ( dvd_dvd @ A @ A3 @ ( plus_plus @ A @ B2 @ A3 ) )
= ( dvd_dvd @ A @ A3 @ B2 ) ) ) ).
% dvd_add_triv_right_iff
thf(fact_2844_dvd__add__triv__left__iff,axiom,
! [A: $tType] :
( ( comm_s4317794764714335236cancel @ A )
=> ! [A3: A,B2: A] :
( ( dvd_dvd @ A @ A3 @ ( plus_plus @ A @ A3 @ B2 ) )
= ( dvd_dvd @ A @ A3 @ B2 ) ) ) ).
% dvd_add_triv_left_iff
thf(fact_2845_div__dvd__div,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A3 @ B2 )
=> ( ( dvd_dvd @ A @ A3 @ C2 )
=> ( ( dvd_dvd @ A @ ( divide_divide @ A @ B2 @ A3 ) @ ( divide_divide @ A @ C2 @ A3 ) )
= ( dvd_dvd @ A @ B2 @ C2 ) ) ) ) ) ).
% div_dvd_div
thf(fact_2846_minus__dvd__iff,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [X: A,Y2: A] :
( ( dvd_dvd @ A @ ( uminus_uminus @ A @ X ) @ Y2 )
= ( dvd_dvd @ A @ X @ Y2 ) ) ) ).
% minus_dvd_iff
thf(fact_2847_dvd__minus__iff,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [X: A,Y2: A] :
( ( dvd_dvd @ A @ X @ ( uminus_uminus @ A @ Y2 ) )
= ( dvd_dvd @ A @ X @ Y2 ) ) ) ).
% dvd_minus_iff
thf(fact_2848_dvd__abs__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M2: A,K: A] :
( ( dvd_dvd @ A @ M2 @ ( abs_abs @ A @ K ) )
= ( dvd_dvd @ A @ M2 @ K ) ) ) ).
% dvd_abs_iff
thf(fact_2849_abs__dvd__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M2: A,K: A] :
( ( dvd_dvd @ A @ ( abs_abs @ A @ M2 ) @ K )
= ( dvd_dvd @ A @ M2 @ K ) ) ) ).
% abs_dvd_iff
thf(fact_2850_tan__pi,axiom,
( ( tan @ real @ pi )
= ( zero_zero @ real ) ) ).
% tan_pi
thf(fact_2851_tan__zero,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ( ( tan @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% tan_zero
thf(fact_2852_nat__dvd__1__iff__1,axiom,
! [M2: nat] :
( ( dvd_dvd @ nat @ M2 @ ( one_one @ nat ) )
= ( M2
= ( one_one @ nat ) ) ) ).
% nat_dvd_1_iff_1
thf(fact_2853_tan__minus,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A] :
( ( tan @ A @ ( uminus_uminus @ A @ X ) )
= ( uminus_uminus @ A @ ( tan @ A @ X ) ) ) ) ).
% tan_minus
thf(fact_2854_tan__periodic__pi,axiom,
! [X: real] :
( ( tan @ real @ ( plus_plus @ real @ X @ pi ) )
= ( tan @ real @ X ) ) ).
% tan_periodic_pi
thf(fact_2855_dvd__times__right__cancel__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ ( times_times @ A @ B2 @ A3 ) @ ( times_times @ A @ C2 @ A3 ) )
= ( dvd_dvd @ A @ B2 @ C2 ) ) ) ) ).
% dvd_times_right_cancel_iff
thf(fact_2856_dvd__times__left__cancel__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ ( times_times @ A @ A3 @ B2 ) @ ( times_times @ A @ A3 @ C2 ) )
= ( dvd_dvd @ A @ B2 @ C2 ) ) ) ) ).
% dvd_times_left_cancel_iff
thf(fact_2857_dvd__mult__cancel__right,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( dvd_dvd @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) )
= ( ( C2
= ( zero_zero @ A ) )
| ( dvd_dvd @ A @ A3 @ B2 ) ) ) ) ).
% dvd_mult_cancel_right
thf(fact_2858_dvd__mult__cancel__left,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( dvd_dvd @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) )
= ( ( C2
= ( zero_zero @ A ) )
| ( dvd_dvd @ A @ A3 @ B2 ) ) ) ) ).
% dvd_mult_cancel_left
thf(fact_2859_unit__prod,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B2: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( dvd_dvd @ A @ ( times_times @ A @ A3 @ B2 ) @ ( one_one @ A ) ) ) ) ) ).
% unit_prod
thf(fact_2860_dvd__add__times__triv__right__iff,axiom,
! [A: $tType] :
( ( comm_s4317794764714335236cancel @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A3 @ ( plus_plus @ A @ B2 @ ( times_times @ A @ C2 @ A3 ) ) )
= ( dvd_dvd @ A @ A3 @ B2 ) ) ) ).
% dvd_add_times_triv_right_iff
thf(fact_2861_dvd__add__times__triv__left__iff,axiom,
! [A: $tType] :
( ( comm_s4317794764714335236cancel @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( dvd_dvd @ A @ A3 @ ( plus_plus @ A @ ( times_times @ A @ C2 @ A3 ) @ B2 ) )
= ( dvd_dvd @ A @ A3 @ B2 ) ) ) ).
% dvd_add_times_triv_left_iff
thf(fact_2862_dvd__div__mult__self,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B2: A] :
( ( dvd_dvd @ A @ A3 @ B2 )
=> ( ( times_times @ A @ ( divide_divide @ A @ B2 @ A3 ) @ A3 )
= B2 ) ) ) ).
% dvd_div_mult_self
thf(fact_2863_dvd__mult__div__cancel,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B2: A] :
( ( dvd_dvd @ A @ A3 @ B2 )
=> ( ( times_times @ A @ A3 @ ( divide_divide @ A @ B2 @ A3 ) )
= B2 ) ) ) ).
% dvd_mult_div_cancel
thf(fact_2864_unit__div__1__div__1,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( divide_divide @ A @ ( one_one @ A ) @ ( divide_divide @ A @ ( one_one @ A ) @ A3 ) )
= A3 ) ) ) ).
% unit_div_1_div_1
thf(fact_2865_unit__div__1__unit,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( dvd_dvd @ A @ ( divide_divide @ A @ ( one_one @ A ) @ A3 ) @ ( one_one @ A ) ) ) ) ).
% unit_div_1_unit
thf(fact_2866_unit__div,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B2: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( dvd_dvd @ A @ ( divide_divide @ A @ A3 @ B2 ) @ ( one_one @ A ) ) ) ) ) ).
% unit_div
thf(fact_2867_div__add,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( dvd_dvd @ A @ C2 @ A3 )
=> ( ( dvd_dvd @ A @ C2 @ B2 )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C2 )
= ( plus_plus @ A @ ( divide_divide @ A @ A3 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) ) ) ) ) ) ).
% div_add
thf(fact_2868_div__diff,axiom,
! [A: $tType] :
( ( idom_modulo @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( dvd_dvd @ A @ C2 @ A3 )
=> ( ( dvd_dvd @ A @ C2 @ B2 )
=> ( ( divide_divide @ A @ ( minus_minus @ A @ A3 @ B2 ) @ C2 )
= ( minus_minus @ A @ ( divide_divide @ A @ A3 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) ) ) ) ) ) ).
% div_diff
thf(fact_2869_dvd__imp__mod__0,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [A3: A,B2: A] :
( ( dvd_dvd @ A @ A3 @ B2 )
=> ( ( modulo_modulo @ A @ B2 @ A3 )
= ( zero_zero @ A ) ) ) ) ).
% dvd_imp_mod_0
thf(fact_2870_dvd__1__iff__1,axiom,
! [M2: nat] :
( ( dvd_dvd @ nat @ M2 @ ( suc @ ( zero_zero @ nat ) ) )
= ( M2
= ( suc @ ( zero_zero @ nat ) ) ) ) ).
% dvd_1_iff_1
thf(fact_2871_dvd__1__left,axiom,
! [K: nat] : ( dvd_dvd @ nat @ ( suc @ ( zero_zero @ nat ) ) @ K ) ).
% dvd_1_left
thf(fact_2872_nat__mult__dvd__cancel__disj,axiom,
! [K: nat,M2: nat,N: nat] :
( ( dvd_dvd @ nat @ ( times_times @ nat @ K @ M2 ) @ ( times_times @ nat @ K @ N ) )
= ( ( K
= ( zero_zero @ nat ) )
| ( dvd_dvd @ nat @ M2 @ N ) ) ) ).
% nat_mult_dvd_cancel_disj
thf(fact_2873_unit__mult__div__div,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B2: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( times_times @ A @ B2 @ ( divide_divide @ A @ ( one_one @ A ) @ A3 ) )
= ( divide_divide @ A @ B2 @ A3 ) ) ) ) ).
% unit_mult_div_div
thf(fact_2874_unit__div__mult__self,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B2: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( times_times @ A @ ( divide_divide @ A @ B2 @ A3 ) @ A3 )
= B2 ) ) ) ).
% unit_div_mult_self
thf(fact_2875_pow__divides__pow__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [N: nat,A3: A,B2: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( dvd_dvd @ A @ ( power_power @ A @ A3 @ N ) @ ( power_power @ A @ B2 @ N ) )
= ( dvd_dvd @ A @ A3 @ B2 ) ) ) ) ).
% pow_divides_pow_iff
thf(fact_2876_tan__npi,axiom,
! [N: nat] :
( ( tan @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ pi ) )
= ( zero_zero @ real ) ) ).
% tan_npi
thf(fact_2877_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_2878_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_2879_tan__periodic__int,axiom,
! [X: real,I: int] :
( ( tan @ real @ ( plus_plus @ real @ X @ ( times_times @ real @ ( ring_1_of_int @ real @ I ) @ pi ) ) )
= ( tan @ real @ X ) ) ).
% tan_periodic_int
thf(fact_2880_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_2881_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_2882_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_2883_even__plus__one__iff,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A3: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) )
= ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) ) ) ) ).
% even_plus_one_iff
thf(fact_2884_even__diff,axiom,
! [A: $tType] :
( ( ring_parity @ A )
=> ! [A3: A,B2: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( minus_minus @ A @ A3 @ B2 ) )
= ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( plus_plus @ A @ A3 @ B2 ) ) ) ) ).
% even_diff
thf(fact_2885_Parity_Oring__1__class_Opower__minus__even,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [N: nat,A3: A] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( power_power @ A @ ( uminus_uminus @ A @ A3 ) @ N )
= ( power_power @ A @ A3 @ N ) ) ) ) ).
% Parity.ring_1_class.power_minus_even
thf(fact_2886_power__minus__odd,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [N: nat,A3: A] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( power_power @ A @ ( uminus_uminus @ A @ A3 ) @ N )
= ( uminus_uminus @ A @ ( power_power @ A @ A3 @ N ) ) ) ) ) ).
% power_minus_odd
thf(fact_2887_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_2888_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_2889_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_2890_even__succ__div__2,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ ( one_one @ A ) @ A3 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% even_succ_div_2
thf(fact_2891_even__succ__div__two,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [A3: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% even_succ_div_two
thf(fact_2892_odd__succ__div__two,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [A3: A] :
( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( plus_plus @ A @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) ) ) ) ) ).
% odd_succ_div_two
thf(fact_2893_zero__le__power__eq__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,W: num] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ W ) ) )
= ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W ) )
| ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W ) )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 ) ) ) ) ) ).
% zero_le_power_eq_numeral
thf(fact_2894_even__power,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A3: A,N: nat] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( power_power @ A @ A3 @ N ) )
= ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
& ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% even_power
thf(fact_2895_power__less__zero__eq__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,W: num] :
( ( ord_less @ A @ ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ W ) ) @ ( zero_zero @ A ) )
= ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W ) )
& ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ) ).
% power_less_zero_eq_numeral
thf(fact_2896_power__less__zero__eq,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,N: nat] :
( ( ord_less @ A @ ( power_power @ A @ A3 @ N ) @ ( zero_zero @ A ) )
= ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
& ( ord_less @ A @ A3 @ ( zero_zero @ A ) ) ) ) ) ).
% power_less_zero_eq
thf(fact_2897_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_2898_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_2899_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_2900_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_2901_even__diff__nat,axiom,
! [M2: nat,N: nat] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ M2 @ N ) )
= ( ( ord_less @ nat @ M2 @ N )
| ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( plus_plus @ nat @ M2 @ N ) ) ) ) ).
% even_diff_nat
thf(fact_2902_odd__two__times__div__two__succ,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [A3: A] :
( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
=> ( ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( one_one @ A ) )
= A3 ) ) ) ).
% odd_two_times_div_two_succ
thf(fact_2903_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_2904_zero__less__power__eq__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,W: num] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ W ) ) )
= ( ( ( numeral_numeral @ nat @ W )
= ( zero_zero @ nat ) )
| ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W ) )
& ( A3
!= ( zero_zero @ A ) ) )
| ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W ) )
& ( ord_less @ A @ ( zero_zero @ A ) @ A3 ) ) ) ) ) ).
% zero_less_power_eq_numeral
thf(fact_2905_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_2906_power__le__zero__eq__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,W: num] :
( ( ord_less_eq @ A @ ( power_power @ A @ A3 @ ( numeral_numeral @ nat @ W ) ) @ ( zero_zero @ A ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( numeral_numeral @ nat @ W ) )
& ( ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W ) )
& ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) ) )
| ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W ) )
& ( A3
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% power_le_zero_eq_numeral
thf(fact_2907_even__succ__div__exp,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A,N: nat] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ ( one_one @ A ) @ A3 ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
= ( divide_divide @ A @ A3 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) ) ) ) ) ).
% even_succ_div_exp
thf(fact_2908_even__succ__mod__exp,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A,N: nat] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( modulo_modulo @ A @ ( plus_plus @ A @ ( one_one @ A ) @ A3 ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
= ( plus_plus @ A @ ( one_one @ A ) @ ( modulo_modulo @ A @ A3 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) ) ) ) ) ) ).
% even_succ_mod_exp
thf(fact_2909_dvd__trans,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A3 @ B2 )
=> ( ( dvd_dvd @ A @ B2 @ C2 )
=> ( dvd_dvd @ A @ A3 @ C2 ) ) ) ) ).
% dvd_trans
thf(fact_2910_dvd__refl,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: A] : ( dvd_dvd @ A @ A3 @ A3 ) ) ).
% dvd_refl
thf(fact_2911_dvd__antisym,axiom,
! [M2: nat,N: nat] :
( ( dvd_dvd @ nat @ M2 @ N )
=> ( ( dvd_dvd @ nat @ N @ M2 )
=> ( M2 = N ) ) ) ).
% dvd_antisym
thf(fact_2912_of__nat__dvd__iff,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [M2: nat,N: nat] :
( ( dvd_dvd @ A @ ( semiring_1_of_nat @ A @ M2 ) @ ( semiring_1_of_nat @ A @ N ) )
= ( dvd_dvd @ nat @ M2 @ N ) ) ) ).
% of_nat_dvd_iff
thf(fact_2913_tan__of__real,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: real] :
( ( real_Vector_of_real @ A @ ( tan @ real @ X ) )
= ( tan @ A @ ( real_Vector_of_real @ A @ X ) ) ) ) ).
% tan_of_real
thf(fact_2914_dvd__0__left,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A3: A] :
( ( dvd_dvd @ A @ ( zero_zero @ A ) @ A3 )
=> ( A3
= ( zero_zero @ A ) ) ) ) ).
% dvd_0_left
thf(fact_2915_dvd__field__iff,axiom,
! [A: $tType] :
( ( field @ A )
=> ( ( dvd_dvd @ A )
= ( ^ [A5: A,B3: A] :
( ( A5
= ( zero_zero @ A ) )
=> ( B3
= ( zero_zero @ A ) ) ) ) ) ) ).
% dvd_field_iff
thf(fact_2916_dvd__triv__right,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: A,B2: A] : ( dvd_dvd @ A @ A3 @ ( times_times @ A @ B2 @ A3 ) ) ) ).
% dvd_triv_right
thf(fact_2917_dvd__mult__right,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ ( times_times @ A @ A3 @ B2 ) @ C2 )
=> ( dvd_dvd @ A @ B2 @ C2 ) ) ) ).
% dvd_mult_right
thf(fact_2918_mult__dvd__mono,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: A,B2: A,C2: A,D2: A] :
( ( dvd_dvd @ A @ A3 @ B2 )
=> ( ( dvd_dvd @ A @ C2 @ D2 )
=> ( dvd_dvd @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ D2 ) ) ) ) ) ).
% mult_dvd_mono
thf(fact_2919_dvd__triv__left,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: A,B2: A] : ( dvd_dvd @ A @ A3 @ ( times_times @ A @ A3 @ B2 ) ) ) ).
% dvd_triv_left
thf(fact_2920_dvd__mult__left,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ ( times_times @ A @ A3 @ B2 ) @ C2 )
=> ( dvd_dvd @ A @ A3 @ C2 ) ) ) ).
% dvd_mult_left
thf(fact_2921_dvd__mult2,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A3 @ B2 )
=> ( dvd_dvd @ A @ A3 @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% dvd_mult2
thf(fact_2922_dvd__mult,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( dvd_dvd @ A @ A3 @ C2 )
=> ( dvd_dvd @ A @ A3 @ ( times_times @ A @ B2 @ C2 ) ) ) ) ).
% dvd_mult
thf(fact_2923_dvd__def,axiom,
! [A: $tType] :
( ( dvd @ A )
=> ( ( dvd_dvd @ A )
= ( ^ [B3: A,A5: A] :
? [K3: A] :
( A5
= ( times_times @ A @ B3 @ K3 ) ) ) ) ) ).
% dvd_def
thf(fact_2924_dvdI,axiom,
! [A: $tType] :
( ( dvd @ A )
=> ! [A3: A,B2: A,K: A] :
( ( A3
= ( times_times @ A @ B2 @ K ) )
=> ( dvd_dvd @ A @ B2 @ A3 ) ) ) ).
% dvdI
thf(fact_2925_dvdE,axiom,
! [A: $tType] :
( ( dvd @ A )
=> ! [B2: A,A3: A] :
( ( dvd_dvd @ A @ B2 @ A3 )
=> ~ ! [K2: A] :
( A3
!= ( times_times @ A @ B2 @ K2 ) ) ) ) ).
% dvdE
thf(fact_2926_one__dvd,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: A] : ( dvd_dvd @ A @ ( one_one @ A ) @ A3 ) ) ).
% one_dvd
thf(fact_2927_unit__imp__dvd,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,A3: A] :
( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( dvd_dvd @ A @ B2 @ A3 ) ) ) ).
% unit_imp_dvd
thf(fact_2928_dvd__unit__imp__unit,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B2: A] :
( ( dvd_dvd @ A @ A3 @ B2 )
=> ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( dvd_dvd @ A @ A3 @ ( one_one @ A ) ) ) ) ) ).
% dvd_unit_imp_unit
thf(fact_2929_dvd__add__right__iff,axiom,
! [A: $tType] :
( ( comm_s4317794764714335236cancel @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A3 @ B2 )
=> ( ( dvd_dvd @ A @ A3 @ ( plus_plus @ A @ B2 @ C2 ) )
= ( dvd_dvd @ A @ A3 @ C2 ) ) ) ) ).
% dvd_add_right_iff
thf(fact_2930_dvd__add__left__iff,axiom,
! [A: $tType] :
( ( comm_s4317794764714335236cancel @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( dvd_dvd @ A @ A3 @ C2 )
=> ( ( dvd_dvd @ A @ A3 @ ( plus_plus @ A @ B2 @ C2 ) )
= ( dvd_dvd @ A @ A3 @ B2 ) ) ) ) ).
% dvd_add_left_iff
thf(fact_2931_dvd__add,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A3 @ B2 )
=> ( ( dvd_dvd @ A @ A3 @ C2 )
=> ( dvd_dvd @ A @ A3 @ ( plus_plus @ A @ B2 @ C2 ) ) ) ) ) ).
% dvd_add
thf(fact_2932_dvd__diff,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [X: A,Y2: A,Z2: A] :
( ( dvd_dvd @ A @ X @ Y2 )
=> ( ( dvd_dvd @ A @ X @ Z2 )
=> ( dvd_dvd @ A @ X @ ( minus_minus @ A @ Y2 @ Z2 ) ) ) ) ) ).
% dvd_diff
thf(fact_2933_dvd__diff__commute,axiom,
! [A: $tType] :
( ( euclid5891614535332579305n_ring @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( dvd_dvd @ A @ A3 @ ( minus_minus @ A @ C2 @ B2 ) )
= ( dvd_dvd @ A @ A3 @ ( minus_minus @ A @ B2 @ C2 ) ) ) ) ).
% dvd_diff_commute
thf(fact_2934_div__div__div__same,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [D2: A,B2: A,A3: A] :
( ( dvd_dvd @ A @ D2 @ B2 )
=> ( ( dvd_dvd @ A @ B2 @ A3 )
=> ( ( divide_divide @ A @ ( divide_divide @ A @ A3 @ D2 ) @ ( divide_divide @ A @ B2 @ D2 ) )
= ( divide_divide @ A @ A3 @ B2 ) ) ) ) ) ).
% div_div_div_same
thf(fact_2935_dvd__div__eq__cancel,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( ( divide_divide @ A @ A3 @ C2 )
= ( divide_divide @ A @ B2 @ C2 ) )
=> ( ( dvd_dvd @ A @ C2 @ A3 )
=> ( ( dvd_dvd @ A @ C2 @ B2 )
=> ( A3 = B2 ) ) ) ) ) ).
% dvd_div_eq_cancel
thf(fact_2936_dvd__div__eq__iff,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( dvd_dvd @ A @ C2 @ A3 )
=> ( ( dvd_dvd @ A @ C2 @ B2 )
=> ( ( ( divide_divide @ A @ A3 @ C2 )
= ( divide_divide @ A @ B2 @ C2 ) )
= ( A3 = B2 ) ) ) ) ) ).
% dvd_div_eq_iff
thf(fact_2937_gcd__nat_Oextremum,axiom,
! [A3: nat] : ( dvd_dvd @ nat @ A3 @ ( zero_zero @ nat ) ) ).
% gcd_nat.extremum
thf(fact_2938_gcd__nat_Oextremum__strict,axiom,
! [A3: nat] :
~ ( ( dvd_dvd @ nat @ ( zero_zero @ nat ) @ A3 )
& ( ( zero_zero @ nat )
!= A3 ) ) ).
% gcd_nat.extremum_strict
thf(fact_2939_gcd__nat_Oextremum__unique,axiom,
! [A3: nat] :
( ( dvd_dvd @ nat @ ( zero_zero @ nat ) @ A3 )
= ( A3
= ( zero_zero @ nat ) ) ) ).
% gcd_nat.extremum_unique
thf(fact_2940_gcd__nat_Onot__eq__extremum,axiom,
! [A3: nat] :
( ( A3
!= ( zero_zero @ nat ) )
= ( ( dvd_dvd @ nat @ A3 @ ( zero_zero @ nat ) )
& ( A3
!= ( zero_zero @ nat ) ) ) ) ).
% gcd_nat.not_eq_extremum
thf(fact_2941_gcd__nat_Oextremum__uniqueI,axiom,
! [A3: nat] :
( ( dvd_dvd @ nat @ ( zero_zero @ nat ) @ A3 )
=> ( A3
= ( zero_zero @ nat ) ) ) ).
% gcd_nat.extremum_uniqueI
thf(fact_2942_complex__minus,axiom,
! [A3: real,B2: real] :
( ( uminus_uminus @ complex @ ( complex2 @ A3 @ B2 ) )
= ( complex2 @ ( uminus_uminus @ real @ A3 ) @ ( uminus_uminus @ real @ B2 ) ) ) ).
% complex_minus
thf(fact_2943_dvd__if__abs__eq,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [L: A,K: A] :
( ( ( abs_abs @ A @ L )
= ( abs_abs @ A @ K ) )
=> ( dvd_dvd @ A @ L @ K ) ) ) ).
% dvd_if_abs_eq
thf(fact_2944_dvd__mod__imp__dvd,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( dvd_dvd @ A @ C2 @ ( modulo_modulo @ A @ A3 @ B2 ) )
=> ( ( dvd_dvd @ A @ C2 @ B2 )
=> ( dvd_dvd @ A @ C2 @ A3 ) ) ) ) ).
% dvd_mod_imp_dvd
thf(fact_2945_dvd__mod__iff,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [C2: A,B2: A,A3: A] :
( ( dvd_dvd @ A @ C2 @ B2 )
=> ( ( dvd_dvd @ A @ C2 @ ( modulo_modulo @ A @ A3 @ B2 ) )
= ( dvd_dvd @ A @ C2 @ A3 ) ) ) ) ).
% dvd_mod_iff
thf(fact_2946_dvd__diff__nat,axiom,
! [K: nat,M2: nat,N: nat] :
( ( dvd_dvd @ nat @ K @ M2 )
=> ( ( dvd_dvd @ nat @ K @ N )
=> ( dvd_dvd @ nat @ K @ ( minus_minus @ nat @ M2 @ N ) ) ) ) ).
% dvd_diff_nat
thf(fact_2947_complex__diff,axiom,
! [A3: real,B2: real,C2: real,D2: real] :
( ( minus_minus @ complex @ ( complex2 @ A3 @ B2 ) @ ( complex2 @ C2 @ D2 ) )
= ( complex2 @ ( minus_minus @ real @ A3 @ C2 ) @ ( minus_minus @ real @ B2 @ D2 ) ) ) ).
% complex_diff
thf(fact_2948_tan__arctan,axiom,
! [Y2: real] :
( ( tan @ real @ ( arctan @ Y2 ) )
= Y2 ) ).
% tan_arctan
thf(fact_2949_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_2950_minf_I10_J,axiom,
! [B: $tType] :
( ( ( plus @ B )
& ( linorder @ B )
& ( dvd @ B ) )
=> ! [D2: B,S: B] :
? [Z4: B] :
! [X4: B] :
( ( ord_less @ B @ X4 @ Z4 )
=> ( ( ~ ( dvd_dvd @ B @ D2 @ ( plus_plus @ B @ X4 @ S ) ) )
= ( ~ ( dvd_dvd @ B @ D2 @ ( plus_plus @ B @ X4 @ S ) ) ) ) ) ) ).
% minf(10)
thf(fact_2951_minf_I9_J,axiom,
! [B: $tType] :
( ( ( plus @ B )
& ( linorder @ B )
& ( dvd @ B ) )
=> ! [D2: B,S: B] :
? [Z4: B] :
! [X4: B] :
( ( ord_less @ B @ X4 @ Z4 )
=> ( ( dvd_dvd @ B @ D2 @ ( plus_plus @ B @ X4 @ S ) )
= ( dvd_dvd @ B @ D2 @ ( plus_plus @ B @ X4 @ S ) ) ) ) ) ).
% minf(9)
thf(fact_2952_pinf_I10_J,axiom,
! [B: $tType] :
( ( ( plus @ B )
& ( linorder @ B )
& ( dvd @ B ) )
=> ! [D2: B,S: B] :
? [Z4: B] :
! [X4: B] :
( ( ord_less @ B @ Z4 @ X4 )
=> ( ( ~ ( dvd_dvd @ B @ D2 @ ( plus_plus @ B @ X4 @ S ) ) )
= ( ~ ( dvd_dvd @ B @ D2 @ ( plus_plus @ B @ X4 @ S ) ) ) ) ) ) ).
% pinf(10)
thf(fact_2953_pinf_I9_J,axiom,
! [B: $tType] :
( ( ( plus @ B )
& ( linorder @ B )
& ( dvd @ B ) )
=> ! [D2: B,S: B] :
? [Z4: B] :
! [X4: B] :
( ( ord_less @ B @ Z4 @ X4 )
=> ( ( dvd_dvd @ B @ D2 @ ( plus_plus @ B @ X4 @ S ) )
= ( dvd_dvd @ B @ D2 @ ( plus_plus @ B @ X4 @ S ) ) ) ) ) ).
% pinf(9)
thf(fact_2954_dvd__div__eq__0__iff,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [B2: A,A3: A] :
( ( dvd_dvd @ A @ B2 @ A3 )
=> ( ( ( divide_divide @ A @ A3 @ B2 )
= ( zero_zero @ A ) )
= ( A3
= ( zero_zero @ A ) ) ) ) ) ).
% dvd_div_eq_0_iff
thf(fact_2955_is__unit__mult__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B2: A] :
( ( dvd_dvd @ A @ ( times_times @ A @ A3 @ B2 ) @ ( one_one @ A ) )
= ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
& ( dvd_dvd @ A @ B2 @ ( one_one @ A ) ) ) ) ) ).
% is_unit_mult_iff
thf(fact_2956_dvd__mult__unit__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ A3 @ ( times_times @ A @ C2 @ B2 ) )
= ( dvd_dvd @ A @ A3 @ C2 ) ) ) ) ).
% dvd_mult_unit_iff
thf(fact_2957_mult__unit__dvd__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ ( times_times @ A @ A3 @ B2 ) @ C2 )
= ( dvd_dvd @ A @ A3 @ C2 ) ) ) ) ).
% mult_unit_dvd_iff
thf(fact_2958_dvd__mult__unit__iff_H,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ A3 @ ( times_times @ A @ B2 @ C2 ) )
= ( dvd_dvd @ A @ A3 @ C2 ) ) ) ) ).
% dvd_mult_unit_iff'
thf(fact_2959_mult__unit__dvd__iff_H,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ ( times_times @ A @ A3 @ B2 ) @ C2 )
= ( dvd_dvd @ A @ B2 @ C2 ) ) ) ) ).
% mult_unit_dvd_iff'
thf(fact_2960_unit__mult__left__cancel,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( ( times_times @ A @ A3 @ B2 )
= ( times_times @ A @ A3 @ C2 ) )
= ( B2 = C2 ) ) ) ) ).
% unit_mult_left_cancel
thf(fact_2961_unit__mult__right__cancel,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( ( times_times @ A @ B2 @ A3 )
= ( times_times @ A @ C2 @ A3 ) )
= ( B2 = C2 ) ) ) ) ).
% unit_mult_right_cancel
thf(fact_2962_dvd__div__mult,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [C2: A,B2: A,A3: A] :
( ( dvd_dvd @ A @ C2 @ B2 )
=> ( ( times_times @ A @ ( divide_divide @ A @ B2 @ C2 ) @ A3 )
= ( divide_divide @ A @ ( times_times @ A @ B2 @ A3 ) @ C2 ) ) ) ) ).
% dvd_div_mult
thf(fact_2963_div__mult__swap,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [C2: A,B2: A,A3: A] :
( ( dvd_dvd @ A @ C2 @ B2 )
=> ( ( times_times @ A @ A3 @ ( divide_divide @ A @ B2 @ C2 ) )
= ( divide_divide @ A @ ( times_times @ A @ A3 @ B2 ) @ C2 ) ) ) ) ).
% div_mult_swap
thf(fact_2964_div__div__eq__right,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [C2: A,B2: A,A3: A] :
( ( dvd_dvd @ A @ C2 @ B2 )
=> ( ( dvd_dvd @ A @ B2 @ A3 )
=> ( ( divide_divide @ A @ A3 @ ( divide_divide @ A @ B2 @ C2 ) )
= ( times_times @ A @ ( divide_divide @ A @ A3 @ B2 ) @ C2 ) ) ) ) ) ).
% div_div_eq_right
thf(fact_2965_dvd__div__mult2__eq,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,C2: A,A3: A] :
( ( dvd_dvd @ A @ ( times_times @ A @ B2 @ C2 ) @ A3 )
=> ( ( divide_divide @ A @ A3 @ ( times_times @ A @ B2 @ C2 ) )
= ( divide_divide @ A @ ( divide_divide @ A @ A3 @ B2 ) @ C2 ) ) ) ) ).
% dvd_div_mult2_eq
thf(fact_2966_dvd__mult__imp__div,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( dvd_dvd @ A @ ( times_times @ A @ A3 @ C2 ) @ B2 )
=> ( dvd_dvd @ A @ A3 @ ( divide_divide @ A @ B2 @ C2 ) ) ) ) ).
% dvd_mult_imp_div
thf(fact_2967_div__mult__div__if__dvd,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,A3: A,D2: A,C2: A] :
( ( dvd_dvd @ A @ B2 @ A3 )
=> ( ( dvd_dvd @ A @ D2 @ C2 )
=> ( ( times_times @ A @ ( divide_divide @ A @ A3 @ B2 ) @ ( divide_divide @ A @ C2 @ D2 ) )
= ( divide_divide @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ D2 ) ) ) ) ) ) ).
% div_mult_div_if_dvd
thf(fact_2968_dvd__div__unit__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ A3 @ ( divide_divide @ A @ C2 @ B2 ) )
= ( dvd_dvd @ A @ A3 @ C2 ) ) ) ) ).
% dvd_div_unit_iff
thf(fact_2969_div__unit__dvd__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ ( divide_divide @ A @ A3 @ B2 ) @ C2 )
= ( dvd_dvd @ A @ A3 @ C2 ) ) ) ) ).
% div_unit_dvd_iff
thf(fact_2970_unit__div__cancel,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( ( divide_divide @ A @ B2 @ A3 )
= ( divide_divide @ A @ C2 @ A3 ) )
= ( B2 = C2 ) ) ) ) ).
% unit_div_cancel
thf(fact_2971_div__plus__div__distrib__dvd__left,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( dvd_dvd @ A @ C2 @ A3 )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C2 )
= ( plus_plus @ A @ ( divide_divide @ A @ A3 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) ) ) ) ) ).
% div_plus_div_distrib_dvd_left
thf(fact_2972_div__plus__div__distrib__dvd__right,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [C2: A,B2: A,A3: A] :
( ( dvd_dvd @ A @ C2 @ B2 )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C2 )
= ( plus_plus @ A @ ( divide_divide @ A @ A3 @ C2 ) @ ( divide_divide @ A @ B2 @ C2 ) ) ) ) ) ).
% div_plus_div_distrib_dvd_right
thf(fact_2973_Complex__eq__1,axiom,
! [A3: real,B2: real] :
( ( ( complex2 @ A3 @ B2 )
= ( one_one @ complex ) )
= ( ( A3
= ( one_one @ real ) )
& ( B2
= ( zero_zero @ real ) ) ) ) ).
% Complex_eq_1
thf(fact_2974_one__complex_Ocode,axiom,
( ( one_one @ complex )
= ( complex2 @ ( one_one @ real ) @ ( zero_zero @ real ) ) ) ).
% one_complex.code
thf(fact_2975_dvd__neg__div,axiom,
! [A: $tType] :
( ( idom_divide @ A )
=> ! [B2: A,A3: A] :
( ( dvd_dvd @ A @ B2 @ A3 )
=> ( ( divide_divide @ A @ ( uminus_uminus @ A @ A3 ) @ B2 )
= ( uminus_uminus @ A @ ( divide_divide @ A @ A3 @ B2 ) ) ) ) ) ).
% dvd_neg_div
thf(fact_2976_dvd__div__neg,axiom,
! [A: $tType] :
( ( idom_divide @ A )
=> ! [B2: A,A3: A] :
( ( dvd_dvd @ A @ B2 @ A3 )
=> ( ( divide_divide @ A @ A3 @ ( uminus_uminus @ A @ B2 ) )
= ( uminus_uminus @ A @ ( divide_divide @ A @ A3 @ B2 ) ) ) ) ) ).
% dvd_div_neg
thf(fact_2977_div__power,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,A3: A,N: nat] :
( ( dvd_dvd @ A @ B2 @ A3 )
=> ( ( power_power @ A @ ( divide_divide @ A @ A3 @ B2 ) @ N )
= ( divide_divide @ A @ ( power_power @ A @ A3 @ N ) @ ( power_power @ A @ B2 @ N ) ) ) ) ) ).
% div_power
thf(fact_2978_mod__0__imp__dvd,axiom,
! [A: $tType] :
( ( semiring_modulo @ A )
=> ! [A3: A,B2: A] :
( ( ( modulo_modulo @ A @ A3 @ B2 )
= ( zero_zero @ A ) )
=> ( dvd_dvd @ A @ B2 @ A3 ) ) ) ).
% mod_0_imp_dvd
thf(fact_2979_dvd__eq__mod__eq__0,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ( ( dvd_dvd @ A )
= ( ^ [A5: A,B3: A] :
( ( modulo_modulo @ A @ B3 @ A5 )
= ( zero_zero @ A ) ) ) ) ) ).
% dvd_eq_mod_eq_0
thf(fact_2980_mod__eq__0__iff__dvd,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [A3: A,B2: A] :
( ( ( modulo_modulo @ A @ A3 @ B2 )
= ( zero_zero @ A ) )
= ( dvd_dvd @ A @ B2 @ A3 ) ) ) ).
% mod_eq_0_iff_dvd
thf(fact_2981_dvd__minus__mod,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [B2: A,A3: A] : ( dvd_dvd @ A @ B2 @ ( minus_minus @ A @ A3 @ ( modulo_modulo @ A @ A3 @ B2 ) ) ) ) ).
% dvd_minus_mod
thf(fact_2982_mod__eq__dvd__iff,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( ( modulo_modulo @ A @ A3 @ C2 )
= ( modulo_modulo @ A @ B2 @ C2 ) )
= ( dvd_dvd @ A @ C2 @ ( minus_minus @ A @ A3 @ B2 ) ) ) ) ).
% mod_eq_dvd_iff
thf(fact_2983_nat__dvd__not__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M2 )
=> ( ( ord_less @ nat @ M2 @ N )
=> ~ ( dvd_dvd @ nat @ N @ M2 ) ) ) ).
% nat_dvd_not_less
thf(fact_2984_dvd__pos__nat,axiom,
! [N: nat,M2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( dvd_dvd @ nat @ M2 @ N )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ M2 ) ) ) ).
% dvd_pos_nat
thf(fact_2985_dvd__minus__self,axiom,
! [M2: nat,N: nat] :
( ( dvd_dvd @ nat @ M2 @ ( minus_minus @ nat @ N @ M2 ) )
= ( ( ord_less @ nat @ N @ M2 )
| ( dvd_dvd @ nat @ M2 @ N ) ) ) ).
% dvd_minus_self
thf(fact_2986_less__eq__dvd__minus,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( dvd_dvd @ nat @ M2 @ N )
= ( dvd_dvd @ nat @ M2 @ ( minus_minus @ nat @ N @ M2 ) ) ) ) ).
% less_eq_dvd_minus
thf(fact_2987_dvd__diffD1,axiom,
! [K: nat,M2: nat,N: nat] :
( ( dvd_dvd @ nat @ K @ ( minus_minus @ nat @ M2 @ N ) )
=> ( ( dvd_dvd @ nat @ K @ M2 )
=> ( ( ord_less_eq @ nat @ N @ M2 )
=> ( dvd_dvd @ nat @ K @ N ) ) ) ) ).
% dvd_diffD1
thf(fact_2988_dvd__diffD,axiom,
! [K: nat,M2: nat,N: nat] :
( ( dvd_dvd @ nat @ K @ ( minus_minus @ nat @ M2 @ N ) )
=> ( ( dvd_dvd @ nat @ K @ N )
=> ( ( ord_less_eq @ nat @ N @ M2 )
=> ( dvd_dvd @ nat @ K @ M2 ) ) ) ) ).
% dvd_diffD
thf(fact_2989_bezout1__nat,axiom,
! [A3: nat,B2: nat] :
? [D5: nat,X3: nat,Y3: nat] :
( ( dvd_dvd @ nat @ D5 @ A3 )
& ( dvd_dvd @ nat @ D5 @ B2 )
& ( ( ( minus_minus @ nat @ ( times_times @ nat @ A3 @ X3 ) @ ( times_times @ nat @ B2 @ Y3 ) )
= D5 )
| ( ( minus_minus @ nat @ ( times_times @ nat @ B2 @ X3 ) @ ( times_times @ nat @ A3 @ Y3 ) )
= D5 ) ) ) ).
% bezout1_nat
thf(fact_2990_cot__altdef,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ( ( cot @ A )
= ( ^ [X2: A] : ( inverse_inverse @ A @ ( tan @ A @ X2 ) ) ) ) ) ).
% cot_altdef
thf(fact_2991_tan__altdef,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ( ( tan @ A )
= ( ^ [X2: A] : ( inverse_inverse @ A @ ( cot @ A @ X2 ) ) ) ) ) ).
% tan_altdef
thf(fact_2992_unit__dvdE,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B2: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ~ ( ( A3
!= ( zero_zero @ A ) )
=> ! [C3: A] :
( B2
!= ( times_times @ A @ A3 @ C3 ) ) ) ) ) ).
% unit_dvdE
thf(fact_2993_unity__coeff__ex,axiom,
! [A: $tType] :
( ( ( dvd @ A )
& ( semiring_0 @ A ) )
=> ! [P: A > $o,L: A] :
( ( ? [X2: A] : ( P @ ( times_times @ A @ L @ X2 ) ) )
= ( ? [X2: A] :
( ( dvd_dvd @ A @ L @ ( plus_plus @ A @ X2 @ ( zero_zero @ A ) ) )
& ( P @ X2 ) ) ) ) ) ).
% unity_coeff_ex
thf(fact_2994_dvd__div__div__eq__mult,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,C2: A,B2: A,D2: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( C2
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ A3 @ B2 )
=> ( ( dvd_dvd @ A @ C2 @ D2 )
=> ( ( ( divide_divide @ A @ B2 @ A3 )
= ( divide_divide @ A @ D2 @ C2 ) )
= ( ( times_times @ A @ B2 @ C2 )
= ( times_times @ A @ A3 @ D2 ) ) ) ) ) ) ) ) ).
% dvd_div_div_eq_mult
thf(fact_2995_dvd__div__iff__mult,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [C2: A,B2: A,A3: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ C2 @ B2 )
=> ( ( dvd_dvd @ A @ A3 @ ( divide_divide @ A @ B2 @ C2 ) )
= ( dvd_dvd @ A @ ( times_times @ A @ A3 @ C2 ) @ B2 ) ) ) ) ) ).
% dvd_div_iff_mult
thf(fact_2996_div__dvd__iff__mult,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ B2 @ A3 )
=> ( ( dvd_dvd @ A @ ( divide_divide @ A @ A3 @ B2 ) @ C2 )
= ( dvd_dvd @ A @ A3 @ ( times_times @ A @ C2 @ B2 ) ) ) ) ) ) ).
% div_dvd_iff_mult
thf(fact_2997_dvd__div__eq__mult,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ A3 @ B2 )
=> ( ( ( divide_divide @ A @ B2 @ A3 )
= C2 )
= ( B2
= ( times_times @ A @ C2 @ A3 ) ) ) ) ) ) ).
% dvd_div_eq_mult
thf(fact_2998_unit__div__eq__0__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,A3: A] :
( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( ( divide_divide @ A @ A3 @ B2 )
= ( zero_zero @ A ) )
= ( A3
= ( zero_zero @ A ) ) ) ) ) ).
% unit_div_eq_0_iff
thf(fact_2999_inf__period_I3_J,axiom,
! [A: $tType] :
( ( ( comm_ring @ A )
& ( dvd @ A ) )
=> ! [D2: A,D6: A,T2: A] :
( ( dvd_dvd @ A @ D2 @ D6 )
=> ! [X4: A,K4: A] :
( ( dvd_dvd @ A @ D2 @ ( plus_plus @ A @ X4 @ T2 ) )
= ( dvd_dvd @ A @ D2 @ ( plus_plus @ A @ ( minus_minus @ A @ X4 @ ( times_times @ A @ K4 @ D6 ) ) @ T2 ) ) ) ) ) ).
% inf_period(3)
thf(fact_3000_inf__period_I4_J,axiom,
! [A: $tType] :
( ( ( comm_ring @ A )
& ( dvd @ A ) )
=> ! [D2: A,D6: A,T2: A] :
( ( dvd_dvd @ A @ D2 @ D6 )
=> ! [X4: A,K4: A] :
( ( ~ ( dvd_dvd @ A @ D2 @ ( plus_plus @ A @ X4 @ T2 ) ) )
= ( ~ ( dvd_dvd @ A @ D2 @ ( plus_plus @ A @ ( minus_minus @ A @ X4 @ ( times_times @ A @ K4 @ D6 ) ) @ T2 ) ) ) ) ) ) ).
% inf_period(4)
thf(fact_3001_unit__eq__div1,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( ( divide_divide @ A @ A3 @ B2 )
= C2 )
= ( A3
= ( times_times @ A @ C2 @ B2 ) ) ) ) ) ).
% unit_eq_div1
thf(fact_3002_unit__eq__div2,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( A3
= ( divide_divide @ A @ C2 @ B2 ) )
= ( ( times_times @ A @ A3 @ B2 )
= C2 ) ) ) ) ).
% unit_eq_div2
thf(fact_3003_div__mult__unit2,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [C2: A,B2: A,A3: A] :
( ( dvd_dvd @ A @ C2 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ B2 @ A3 )
=> ( ( divide_divide @ A @ A3 @ ( times_times @ A @ B2 @ C2 ) )
= ( divide_divide @ A @ ( divide_divide @ A @ A3 @ B2 ) @ C2 ) ) ) ) ) ).
% div_mult_unit2
thf(fact_3004_unit__div__commute,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( times_times @ A @ ( divide_divide @ A @ A3 @ B2 ) @ C2 )
= ( divide_divide @ A @ ( times_times @ A @ A3 @ C2 ) @ B2 ) ) ) ) ).
% unit_div_commute
thf(fact_3005_unit__div__mult__swap,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( dvd_dvd @ A @ C2 @ ( one_one @ A ) )
=> ( ( times_times @ A @ A3 @ ( divide_divide @ A @ B2 @ C2 ) )
= ( divide_divide @ A @ ( times_times @ A @ A3 @ B2 ) @ C2 ) ) ) ) ).
% unit_div_mult_swap
thf(fact_3006_is__unit__div__mult2__eq,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,C2: A,A3: A] :
( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ C2 @ ( one_one @ A ) )
=> ( ( divide_divide @ A @ A3 @ ( times_times @ A @ B2 @ C2 ) )
= ( divide_divide @ A @ ( divide_divide @ A @ A3 @ B2 ) @ C2 ) ) ) ) ) ).
% is_unit_div_mult2_eq
thf(fact_3007_unit__imp__mod__eq__0,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ! [B2: A,A3: A] :
( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( modulo_modulo @ A @ A3 @ B2 )
= ( zero_zero @ A ) ) ) ) ).
% unit_imp_mod_eq_0
thf(fact_3008_is__unit__power__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,N: nat] :
( ( dvd_dvd @ A @ ( power_power @ A @ A3 @ N ) @ ( one_one @ A ) )
= ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
| ( N
= ( zero_zero @ nat ) ) ) ) ) ).
% is_unit_power_iff
thf(fact_3009_Complex__eq__neg__1,axiom,
! [A3: real,B2: real] :
( ( ( complex2 @ A3 @ B2 )
= ( uminus_uminus @ complex @ ( one_one @ complex ) ) )
= ( ( A3
= ( uminus_uminus @ real @ ( one_one @ real ) ) )
& ( B2
= ( zero_zero @ real ) ) ) ) ).
% Complex_eq_neg_1
thf(fact_3010_Complex__eq__neg__numeral,axiom,
! [A3: real,B2: real,W: num] :
( ( ( complex2 @ A3 @ B2 )
= ( uminus_uminus @ complex @ ( numeral_numeral @ complex @ W ) ) )
= ( ( A3
= ( uminus_uminus @ real @ ( numeral_numeral @ real @ W ) ) )
& ( B2
= ( zero_zero @ real ) ) ) ) ).
% Complex_eq_neg_numeral
thf(fact_3011_dvd__imp__le,axiom,
! [K: nat,N: nat] :
( ( dvd_dvd @ nat @ K @ N )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ord_less_eq @ nat @ K @ N ) ) ) ).
% dvd_imp_le
thf(fact_3012_dvd__mult__cancel,axiom,
! [K: nat,M2: nat,N: nat] :
( ( dvd_dvd @ nat @ ( times_times @ nat @ K @ M2 ) @ ( times_times @ nat @ K @ N ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( dvd_dvd @ nat @ M2 @ N ) ) ) ).
% dvd_mult_cancel
thf(fact_3013_nat__mult__dvd__cancel1,axiom,
! [K: nat,M2: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ( dvd_dvd @ nat @ ( times_times @ nat @ K @ M2 ) @ ( times_times @ nat @ K @ N ) )
= ( dvd_dvd @ nat @ M2 @ N ) ) ) ).
% nat_mult_dvd_cancel1
thf(fact_3014_bezout__add__strong__nat,axiom,
! [A3: nat,B2: nat] :
( ( A3
!= ( zero_zero @ nat ) )
=> ? [D5: nat,X3: nat,Y3: nat] :
( ( dvd_dvd @ nat @ D5 @ A3 )
& ( dvd_dvd @ nat @ D5 @ B2 )
& ( ( times_times @ nat @ A3 @ X3 )
= ( plus_plus @ nat @ ( times_times @ nat @ B2 @ Y3 ) @ D5 ) ) ) ) ).
% bezout_add_strong_nat
thf(fact_3015_mod__greater__zero__iff__not__dvd,axiom,
! [M2: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( modulo_modulo @ nat @ M2 @ N ) )
= ( ~ ( dvd_dvd @ nat @ N @ M2 ) ) ) ).
% mod_greater_zero_iff_not_dvd
thf(fact_3016_mod__eq__dvd__iff__nat,axiom,
! [N: nat,M2: nat,Q5: nat] :
( ( ord_less_eq @ nat @ N @ M2 )
=> ( ( ( modulo_modulo @ nat @ M2 @ Q5 )
= ( modulo_modulo @ nat @ N @ Q5 ) )
= ( dvd_dvd @ nat @ Q5 @ ( minus_minus @ nat @ M2 @ N ) ) ) ) ).
% mod_eq_dvd_iff_nat
thf(fact_3017_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_3018_dvd__fact,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ M2 )
=> ( ( ord_less_eq @ nat @ M2 @ N )
=> ( dvd_dvd @ nat @ M2 @ ( semiring_char_0_fact @ nat @ N ) ) ) ) ).
% dvd_fact
thf(fact_3019_tan__def,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ( ( tan @ A )
= ( ^ [X2: A] : ( divide_divide @ A @ ( sin @ A @ X2 ) @ ( cos @ A @ X2 ) ) ) ) ) ).
% tan_def
thf(fact_3020_even__zero,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( zero_zero @ A ) ) ) ).
% even_zero
thf(fact_3021_is__unitE,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,C2: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ~ ( ( A3
!= ( zero_zero @ A ) )
=> ! [B4: A] :
( ( B4
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ B4 @ ( one_one @ A ) )
=> ( ( ( divide_divide @ A @ ( one_one @ A ) @ A3 )
= B4 )
=> ( ( ( divide_divide @ A @ ( one_one @ A ) @ B4 )
= A3 )
=> ( ( ( times_times @ A @ A3 @ B4 )
= ( one_one @ A ) )
=> ( ( divide_divide @ A @ C2 @ A3 )
!= ( times_times @ A @ C2 @ B4 ) ) ) ) ) ) ) ) ) ) ).
% is_unitE
thf(fact_3022_is__unit__div__mult__cancel__left,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B2: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( divide_divide @ A @ A3 @ ( times_times @ A @ A3 @ B2 ) )
= ( divide_divide @ A @ ( one_one @ A ) @ B2 ) ) ) ) ) ).
% is_unit_div_mult_cancel_left
thf(fact_3023_is__unit__div__mult__cancel__right,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B2: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( divide_divide @ A @ A3 @ ( times_times @ A @ B2 @ A3 ) )
= ( divide_divide @ A @ ( one_one @ A ) @ B2 ) ) ) ) ) ).
% is_unit_div_mult_cancel_right
thf(fact_3024_odd__one,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( one_one @ A ) ) ) ).
% odd_one
thf(fact_3025_bit__eq__rec,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ( ( ^ [Y4: A,Z: A] : Y4 = Z )
= ( ^ [A5: A,B3: A] :
( ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A5 )
= ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B3 ) )
& ( ( divide_divide @ A @ A5 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( divide_divide @ A @ B3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% bit_eq_rec
thf(fact_3026_even__minus,axiom,
! [A: $tType] :
( ( ring_parity @ A )
=> ! [A3: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( uminus_uminus @ A @ A3 ) )
= ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) ) ) ).
% even_minus
thf(fact_3027_dvd__power__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [X: A,M2: nat,N: nat] :
( ( X
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ ( power_power @ A @ X @ M2 ) @ ( power_power @ A @ X @ N ) )
= ( ( dvd_dvd @ A @ X @ ( one_one @ A ) )
| ( ord_less_eq @ nat @ M2 @ N ) ) ) ) ) ).
% dvd_power_iff
thf(fact_3028_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_3029_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_3030_choose__dvd,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [K: nat,N: nat] :
( ( ord_less_eq @ nat @ K @ N )
=> ( dvd_dvd @ A @ ( times_times @ A @ ( semiring_char_0_fact @ A @ K ) @ ( semiring_char_0_fact @ A @ ( minus_minus @ nat @ N @ K ) ) ) @ ( semiring_char_0_fact @ A @ N ) ) ) ) ).
% choose_dvd
thf(fact_3031_dvd__mult__cancel1,axiom,
! [M2: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M2 )
=> ( ( dvd_dvd @ nat @ ( times_times @ nat @ M2 @ N ) @ M2 )
= ( N
= ( one_one @ nat ) ) ) ) ).
% dvd_mult_cancel1
thf(fact_3032_dvd__mult__cancel2,axiom,
! [M2: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M2 )
=> ( ( dvd_dvd @ nat @ ( times_times @ nat @ N @ M2 ) @ M2 )
= ( N
= ( one_one @ nat ) ) ) ) ).
% dvd_mult_cancel2
thf(fact_3033_dvd__minus__add,axiom,
! [Q5: nat,N: nat,R4: nat,M2: nat] :
( ( ord_less_eq @ nat @ Q5 @ N )
=> ( ( ord_less_eq @ nat @ Q5 @ ( times_times @ nat @ R4 @ M2 ) )
=> ( ( dvd_dvd @ nat @ M2 @ ( minus_minus @ nat @ N @ Q5 ) )
= ( dvd_dvd @ nat @ M2 @ ( plus_plus @ nat @ N @ ( minus_minus @ nat @ ( times_times @ nat @ R4 @ M2 ) @ Q5 ) ) ) ) ) ) ).
% dvd_minus_add
thf(fact_3034_power__dvd__imp__le,axiom,
! [I: nat,M2: nat,N: nat] :
( ( dvd_dvd @ nat @ ( power_power @ nat @ I @ M2 ) @ ( power_power @ nat @ I @ N ) )
=> ( ( ord_less @ nat @ ( one_one @ nat ) @ I )
=> ( ord_less_eq @ nat @ M2 @ N ) ) ) ).
% power_dvd_imp_le
thf(fact_3035_mod__nat__eqI,axiom,
! [R4: nat,N: nat,M2: nat] :
( ( ord_less @ nat @ R4 @ N )
=> ( ( ord_less_eq @ nat @ R4 @ M2 )
=> ( ( dvd_dvd @ nat @ N @ ( minus_minus @ nat @ M2 @ R4 ) )
=> ( ( modulo_modulo @ nat @ M2 @ N )
= R4 ) ) ) ) ).
% mod_nat_eqI
thf(fact_3036_complex__mult,axiom,
! [A3: real,B2: real,C2: real,D2: real] :
( ( times_times @ complex @ ( complex2 @ A3 @ B2 ) @ ( complex2 @ C2 @ D2 ) )
= ( complex2 @ ( minus_minus @ real @ ( times_times @ real @ A3 @ C2 ) @ ( times_times @ real @ B2 @ D2 ) ) @ ( plus_plus @ real @ ( times_times @ real @ A3 @ D2 ) @ ( times_times @ real @ B2 @ C2 ) ) ) ) ).
% complex_mult
thf(fact_3037_even__two__times__div__two,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [A3: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
=> ( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
= A3 ) ) ) ).
% even_two_times_div_two
thf(fact_3038_even__iff__mod__2__eq__zero,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A3: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
= ( ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( zero_zero @ A ) ) ) ) ).
% even_iff_mod_2_eq_zero
thf(fact_3039_odd__iff__mod__2__eq__one,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A3: A] :
( ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) )
= ( ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( one_one @ A ) ) ) ) ).
% odd_iff_mod_2_eq_one
thf(fact_3040_uminus__power__if,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [N: nat,A3: A] :
( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( power_power @ A @ ( uminus_uminus @ A @ A3 ) @ N )
= ( power_power @ A @ A3 @ N ) ) )
& ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( power_power @ A @ ( uminus_uminus @ A @ A3 ) @ N )
= ( uminus_uminus @ A @ ( power_power @ A @ A3 @ N ) ) ) ) ) ) ).
% uminus_power_if
thf(fact_3041_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_3042_even__unset__bit__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M2: nat,A3: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se2638667681897837118et_bit @ A @ M2 @ A3 ) )
= ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
| ( M2
= ( zero_zero @ nat ) ) ) ) ) ).
% even_unset_bit_iff
thf(fact_3043_even__set__bit__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M2: nat,A3: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5668285175392031749et_bit @ A @ M2 @ A3 ) )
= ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
& ( M2
!= ( zero_zero @ nat ) ) ) ) ) ).
% even_set_bit_iff
thf(fact_3044_even__flip__bit__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M2: nat,A3: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se8732182000553998342ip_bit @ A @ M2 @ A3 ) )
= ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
!= ( M2
= ( zero_zero @ nat ) ) ) ) ) ).
% even_flip_bit_iff
thf(fact_3045_tan__45,axiom,
( ( tan @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ one2 ) ) ) ) )
= ( one_one @ real ) ) ).
% tan_45
thf(fact_3046_oddE,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A3: A] :
( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
=> ~ ! [B4: A] :
( A3
!= ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B4 ) @ ( one_one @ A ) ) ) ) ) ).
% oddE
thf(fact_3047_parity__cases,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A3: A] :
( ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
=> ( ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
!= ( zero_zero @ A ) ) )
=> ~ ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
=> ( ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
!= ( one_one @ A ) ) ) ) ) ).
% parity_cases
thf(fact_3048_mod2__eq__if,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A3: A] :
( ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
=> ( ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( zero_zero @ A ) ) )
& ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
=> ( ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( one_one @ A ) ) ) ) ) ).
% mod2_eq_if
thf(fact_3049_zero__le__even__power,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: nat,A3: A] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A3 @ N ) ) ) ) ).
% zero_le_even_power
thf(fact_3050_zero__le__odd__power,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: nat,A3: A] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A3 @ N ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 ) ) ) ) ).
% zero_le_odd_power
thf(fact_3051_zero__le__power__eq,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,N: nat] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A3 @ N ) )
= ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
| ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 ) ) ) ) ) ).
% zero_le_power_eq
thf(fact_3052_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_3053_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_3054_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_3055_lemma__tan__total,axiom,
! [Y2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ Y2 )
=> ? [X3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X3 )
& ( ord_less @ real @ X3 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
& ( ord_less @ real @ Y2 @ ( tan @ real @ X3 ) ) ) ) ).
% lemma_tan_total
thf(fact_3056_lemma__tan__total1,axiom,
! [Y2: real] :
? [X3: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X3 )
& ( ord_less @ real @ X3 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
& ( ( tan @ real @ X3 )
= Y2 ) ) ).
% lemma_tan_total1
thf(fact_3057_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_3058_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_3059_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_3060_tan__total,axiom,
! [Y2: real] :
? [X3: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X3 )
& ( ord_less @ real @ X3 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
& ( ( tan @ real @ X3 )
= Y2 )
& ! [Y: real] :
( ( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ Y )
& ( ord_less @ real @ Y @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
& ( ( tan @ real @ Y )
= Y2 ) )
=> ( Y = X3 ) ) ) ).
% tan_total
thf(fact_3061_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_3062_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_3063_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_3064_zero__less__power__eq,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,N: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A3 @ N ) )
= ( ( N
= ( zero_zero @ nat ) )
| ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
& ( A3
!= ( zero_zero @ A ) ) )
| ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
& ( ord_less @ A @ ( zero_zero @ A ) @ A3 ) ) ) ) ) ).
% zero_less_power_eq
thf(fact_3065_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_3066_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_3067_tan__total__pos,axiom,
! [Y2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y2 )
=> ? [X3: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X3 )
& ( ord_less @ real @ X3 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
& ( ( tan @ real @ X3 )
= Y2 ) ) ) ).
% tan_total_pos
thf(fact_3068_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_3069_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_3070_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_3071_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_3072_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_3073_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_3074_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_3075_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_3076_even__mask__div__iff_H,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [M2: 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 ) ) @ M2 ) @ ( one_one @ A ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) )
= ( ord_less_eq @ nat @ M2 @ N ) ) ) ).
% even_mask_div_iff'
thf(fact_3077_power__le__zero__eq,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,N: nat] :
( ( ord_less_eq @ A @ ( power_power @ A @ A3 @ N ) @ ( zero_zero @ A ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
& ( ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
& ( ord_less_eq @ A @ A3 @ ( zero_zero @ A ) ) )
| ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
& ( A3
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% power_le_zero_eq
thf(fact_3078_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_3079_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_3080_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_3081_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_3082_complex__inverse,axiom,
! [A3: real,B2: real] :
( ( inverse_inverse @ complex @ ( complex2 @ A3 @ B2 ) )
= ( complex2 @ ( divide_divide @ real @ A3 @ ( plus_plus @ real @ ( power_power @ real @ A3 @ ( 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 @ A3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ B2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% complex_inverse
thf(fact_3083_tan__total__pi4,axiom,
! [X: real] :
( ( ord_less @ real @ ( abs_abs @ real @ X ) @ ( one_one @ real ) )
=> ? [Z4: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) @ Z4 )
& ( ord_less @ real @ Z4 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ one2 ) ) ) ) )
& ( ( tan @ real @ Z4 )
= X ) ) ) ).
% tan_total_pi4
thf(fact_3084_even__mask__div__iff,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [M2: 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 ) ) @ M2 ) @ ( 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 @ M2 @ N ) ) ) ) ).
% even_mask_div_iff
thf(fact_3085_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_3086_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_3087_even__mult__exp__div__exp__iff,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A,M2: nat,N: nat] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ ( times_times @ A @ A3 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M2 ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) )
= ( ( ord_less @ nat @ N @ M2 )
| ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N )
= ( zero_zero @ A ) )
| ( ( ord_less_eq @ nat @ M2 @ N )
& ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A3 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ N @ M2 ) ) ) ) ) ) ) ) ).
% even_mult_exp_div_exp_iff
thf(fact_3088_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_3089_tan__half,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ( ( tan @ A )
= ( ^ [X2: A] : ( divide_divide @ A @ ( sin @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X2 ) ) @ ( plus_plus @ A @ ( cos @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X2 ) ) @ ( one_one @ A ) ) ) ) ) ) ).
% tan_half
thf(fact_3090_sin__coeff__def,axiom,
( sin_coeff
= ( ^ [N5: nat] : ( if @ real @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N5 ) @ ( zero_zero @ real ) @ ( divide_divide @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ ( divide_divide @ nat @ ( minus_minus @ nat @ N5 @ ( suc @ ( zero_zero @ nat ) ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( semiring_char_0_fact @ real @ N5 ) ) ) ) ) ).
% sin_coeff_def
thf(fact_3091_cos__coeff__def,axiom,
( cos_coeff
= ( ^ [N5: nat] : ( if @ real @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N5 ) @ ( divide_divide @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ ( divide_divide @ nat @ N5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( semiring_char_0_fact @ real @ N5 ) ) @ ( zero_zero @ real ) ) ) ) ).
% cos_coeff_def
thf(fact_3092_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_3093_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_3094_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_3095_real__sqrt__one,axiom,
( ( sqrt @ ( one_one @ real ) )
= ( one_one @ real ) ) ).
% real_sqrt_one
thf(fact_3096_real__sqrt__eq__1__iff,axiom,
! [X: real] :
( ( ( sqrt @ X )
= ( one_one @ real ) )
= ( X
= ( one_one @ real ) ) ) ).
% real_sqrt_eq_1_iff
thf(fact_3097_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_3098_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_3099_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_3100_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_3101_int__dvd__int__iff,axiom,
! [M2: nat,N: nat] :
( ( dvd_dvd @ int @ ( semiring_1_of_nat @ int @ M2 ) @ ( semiring_1_of_nat @ int @ N ) )
= ( dvd_dvd @ nat @ M2 @ N ) ) ).
% int_dvd_int_iff
thf(fact_3102_zdvd1__eq,axiom,
! [X: int] :
( ( dvd_dvd @ int @ X @ ( one_one @ int ) )
= ( ( abs_abs @ int @ X )
= ( one_one @ int ) ) ) ).
% zdvd1_eq
thf(fact_3103_sin__coeff__0,axiom,
( ( sin_coeff @ ( zero_zero @ nat ) )
= ( zero_zero @ real ) ) ).
% sin_coeff_0
thf(fact_3104_cos__coeff__0,axiom,
( ( cos_coeff @ ( zero_zero @ nat ) )
= ( one_one @ real ) ) ).
% cos_coeff_0
thf(fact_3105_sgn__mult__dvd__iff,axiom,
! [R4: int,L: int,K: int] :
( ( dvd_dvd @ int @ ( times_times @ int @ ( sgn_sgn @ int @ R4 ) @ L ) @ K )
= ( ( dvd_dvd @ int @ L @ K )
& ( ( R4
= ( zero_zero @ int ) )
=> ( K
= ( zero_zero @ int ) ) ) ) ) ).
% sgn_mult_dvd_iff
thf(fact_3106_mult__sgn__dvd__iff,axiom,
! [L: int,R4: int,K: int] :
( ( dvd_dvd @ int @ ( times_times @ int @ L @ ( sgn_sgn @ int @ R4 ) ) @ K )
= ( ( dvd_dvd @ int @ L @ K )
& ( ( R4
= ( zero_zero @ int ) )
=> ( K
= ( zero_zero @ int ) ) ) ) ) ).
% mult_sgn_dvd_iff
thf(fact_3107_dvd__sgn__mult__iff,axiom,
! [L: int,R4: int,K: int] :
( ( dvd_dvd @ int @ L @ ( times_times @ int @ ( sgn_sgn @ int @ R4 ) @ K ) )
= ( ( dvd_dvd @ int @ L @ K )
| ( R4
= ( zero_zero @ int ) ) ) ) ).
% dvd_sgn_mult_iff
thf(fact_3108_dvd__mult__sgn__iff,axiom,
! [L: int,K: int,R4: int] :
( ( dvd_dvd @ int @ L @ ( times_times @ int @ K @ ( sgn_sgn @ int @ R4 ) ) )
= ( ( dvd_dvd @ int @ L @ K )
| ( R4
= ( zero_zero @ int ) ) ) ) ).
% dvd_mult_sgn_iff
thf(fact_3109_nat__abs__dvd__iff,axiom,
! [K: int,N: nat] :
( ( dvd_dvd @ nat @ ( nat2 @ ( abs_abs @ int @ K ) ) @ N )
= ( dvd_dvd @ int @ K @ ( semiring_1_of_nat @ int @ N ) ) ) ).
% nat_abs_dvd_iff
thf(fact_3110_dvd__nat__abs__iff,axiom,
! [N: nat,K: int] :
( ( dvd_dvd @ nat @ N @ ( nat2 @ ( abs_abs @ int @ K ) ) )
= ( dvd_dvd @ int @ ( semiring_1_of_nat @ int @ N ) @ K ) ) ).
% dvd_nat_abs_iff
thf(fact_3111_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_3112_real__sqrt__minus,axiom,
! [X: real] :
( ( sqrt @ ( uminus_uminus @ real @ X ) )
= ( uminus_uminus @ real @ ( sqrt @ X ) ) ) ).
% real_sqrt_minus
thf(fact_3113_uminus__dvd__conv_I1_J,axiom,
( ( dvd_dvd @ int )
= ( ^ [D4: int] : ( dvd_dvd @ int @ ( uminus_uminus @ int @ D4 ) ) ) ) ).
% uminus_dvd_conv(1)
thf(fact_3114_uminus__dvd__conv_I2_J,axiom,
( ( dvd_dvd @ int )
= ( ^ [D4: int,T3: int] : ( dvd_dvd @ int @ D4 @ ( uminus_uminus @ int @ T3 ) ) ) ) ).
% uminus_dvd_conv(2)
thf(fact_3115_zdvd__zdiffD,axiom,
! [K: int,M2: int,N: int] :
( ( dvd_dvd @ int @ K @ ( minus_minus @ int @ M2 @ N ) )
=> ( ( dvd_dvd @ int @ K @ N )
=> ( dvd_dvd @ int @ K @ M2 ) ) ) ).
% zdvd_zdiffD
thf(fact_3116_zdvd__antisym__abs,axiom,
! [A3: int,B2: int] :
( ( dvd_dvd @ int @ A3 @ B2 )
=> ( ( dvd_dvd @ int @ B2 @ A3 )
=> ( ( abs_abs @ int @ A3 )
= ( abs_abs @ int @ B2 ) ) ) ) ).
% zdvd_antisym_abs
thf(fact_3117_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_3118_zdvd__antisym__nonneg,axiom,
! [M2: int,N: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ M2 )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ N )
=> ( ( dvd_dvd @ int @ M2 @ N )
=> ( ( dvd_dvd @ int @ N @ M2 )
=> ( M2 = N ) ) ) ) ) ).
% zdvd_antisym_nonneg
thf(fact_3119_zdvd__not__zless,axiom,
! [M2: int,N: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ M2 )
=> ( ( ord_less @ int @ M2 @ N )
=> ~ ( dvd_dvd @ int @ N @ M2 ) ) ) ).
% zdvd_not_zless
thf(fact_3120_zdvd__mult__cancel,axiom,
! [K: int,M2: int,N: int] :
( ( dvd_dvd @ int @ ( times_times @ int @ K @ M2 ) @ ( times_times @ int @ K @ N ) )
=> ( ( K
!= ( zero_zero @ int ) )
=> ( dvd_dvd @ int @ M2 @ N ) ) ) ).
% zdvd_mult_cancel
thf(fact_3121_zdvd__mono,axiom,
! [K: int,M2: int,T2: int] :
( ( K
!= ( zero_zero @ int ) )
=> ( ( dvd_dvd @ int @ M2 @ T2 )
= ( dvd_dvd @ int @ ( times_times @ int @ K @ M2 ) @ ( times_times @ int @ K @ T2 ) ) ) ) ).
% zdvd_mono
thf(fact_3122_zdvd__period,axiom,
! [A3: int,D2: int,X: int,T2: int,C2: int] :
( ( dvd_dvd @ int @ A3 @ D2 )
=> ( ( dvd_dvd @ int @ A3 @ ( plus_plus @ int @ X @ T2 ) )
= ( dvd_dvd @ int @ A3 @ ( plus_plus @ int @ ( plus_plus @ int @ X @ ( times_times @ int @ C2 @ D2 ) ) @ T2 ) ) ) ) ).
% zdvd_period
thf(fact_3123_zdvd__reduce,axiom,
! [K: int,N: int,M2: int] :
( ( dvd_dvd @ int @ K @ ( plus_plus @ int @ N @ ( times_times @ int @ K @ M2 ) ) )
= ( dvd_dvd @ int @ K @ N ) ) ).
% zdvd_reduce
thf(fact_3124_abs__div,axiom,
! [Y2: int,X: int] :
( ( dvd_dvd @ int @ Y2 @ X )
=> ( ( abs_abs @ int @ ( divide_divide @ int @ X @ Y2 ) )
= ( divide_divide @ int @ ( abs_abs @ int @ X ) @ ( abs_abs @ int @ Y2 ) ) ) ) ).
% abs_div
thf(fact_3125_sin__coeff__Suc,axiom,
! [N: nat] :
( ( sin_coeff @ ( suc @ N ) )
= ( divide_divide @ real @ ( cos_coeff @ N ) @ ( semiring_1_of_nat @ real @ ( suc @ N ) ) ) ) ).
% sin_coeff_Suc
thf(fact_3126_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_3127_cos__coeff__Suc,axiom,
! [N: nat] :
( ( cos_coeff @ ( suc @ N ) )
= ( divide_divide @ real @ ( uminus_uminus @ real @ ( sin_coeff @ N ) ) @ ( semiring_1_of_nat @ real @ ( suc @ N ) ) ) ) ).
% cos_coeff_Suc
thf(fact_3128_zdvd__imp__le,axiom,
! [Z2: int,N: int] :
( ( dvd_dvd @ int @ Z2 @ N )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ N )
=> ( ord_less_eq @ int @ Z2 @ N ) ) ) ).
% zdvd_imp_le
thf(fact_3129_dvd__imp__le__int,axiom,
! [I: int,D2: int] :
( ( I
!= ( zero_zero @ int ) )
=> ( ( dvd_dvd @ int @ D2 @ I )
=> ( ord_less_eq @ int @ ( abs_abs @ int @ D2 ) @ ( abs_abs @ int @ I ) ) ) ) ).
% dvd_imp_le_int
thf(fact_3130_of__int__divide__in__Ints,axiom,
! [A: $tType] :
( ( idom_divide @ A )
=> ! [B2: int,A3: int] :
( ( dvd_dvd @ int @ B2 @ A3 )
=> ( member @ A @ ( divide_divide @ A @ ( ring_1_of_int @ A @ A3 ) @ ( ring_1_of_int @ A @ B2 ) ) @ ( ring_1_Ints @ A ) ) ) ) ).
% of_int_divide_in_Ints
thf(fact_3131_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_3132_sgn__mod,axiom,
! [L: int,K: int] :
( ( L
!= ( zero_zero @ int ) )
=> ( ~ ( dvd_dvd @ int @ L @ K )
=> ( ( sgn_sgn @ int @ ( modulo_modulo @ int @ K @ L ) )
= ( sgn_sgn @ int @ L ) ) ) ) ).
% sgn_mod
thf(fact_3133_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_3134_zdvd__mult__cancel1,axiom,
! [M2: int,N: int] :
( ( M2
!= ( zero_zero @ int ) )
=> ( ( dvd_dvd @ int @ ( times_times @ int @ M2 @ N ) @ M2 )
= ( ( abs_abs @ int @ N )
= ( one_one @ int ) ) ) ) ).
% zdvd_mult_cancel1
thf(fact_3135_mod__int__pos__iff,axiom,
! [K: int,L: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( modulo_modulo @ int @ K @ L ) )
= ( ( dvd_dvd @ int @ L @ K )
| ( ( L
= ( zero_zero @ int ) )
& ( ord_less_eq @ int @ ( zero_zero @ int ) @ K ) )
| ( ord_less @ int @ ( zero_zero @ int ) @ L ) ) ) ).
% mod_int_pos_iff
thf(fact_3136_aset_I10_J,axiom,
! [D2: int,D6: int,A4: set @ int,T2: int] :
( ( dvd_dvd @ int @ D2 @ D6 )
=> ! [X4: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D6 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ A4 )
=> ( X4
!= ( minus_minus @ int @ Xb2 @ Xa3 ) ) ) )
=> ( ~ ( dvd_dvd @ int @ D2 @ ( plus_plus @ int @ X4 @ T2 ) )
=> ~ ( dvd_dvd @ int @ D2 @ ( plus_plus @ int @ ( plus_plus @ int @ X4 @ D6 ) @ T2 ) ) ) ) ) ).
% aset(10)
thf(fact_3137_aset_I9_J,axiom,
! [D2: int,D6: int,A4: set @ int,T2: int] :
( ( dvd_dvd @ int @ D2 @ D6 )
=> ! [X4: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D6 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ A4 )
=> ( X4
!= ( minus_minus @ int @ Xb2 @ Xa3 ) ) ) )
=> ( ( dvd_dvd @ int @ D2 @ ( plus_plus @ int @ X4 @ T2 ) )
=> ( dvd_dvd @ int @ D2 @ ( plus_plus @ int @ ( plus_plus @ int @ X4 @ D6 ) @ T2 ) ) ) ) ) ).
% aset(9)
thf(fact_3138_bset_I10_J,axiom,
! [D2: int,D6: int,B7: set @ int,T2: int] :
( ( dvd_dvd @ int @ D2 @ D6 )
=> ! [X4: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D6 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ B7 )
=> ( X4
!= ( plus_plus @ int @ Xb2 @ Xa3 ) ) ) )
=> ( ~ ( dvd_dvd @ int @ D2 @ ( plus_plus @ int @ X4 @ T2 ) )
=> ~ ( dvd_dvd @ int @ D2 @ ( plus_plus @ int @ ( minus_minus @ int @ X4 @ D6 ) @ T2 ) ) ) ) ) ).
% bset(10)
thf(fact_3139_bset_I9_J,axiom,
! [D2: int,D6: int,B7: set @ int,T2: int] :
( ( dvd_dvd @ int @ D2 @ D6 )
=> ! [X4: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D6 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ B7 )
=> ( X4
!= ( plus_plus @ int @ Xb2 @ Xa3 ) ) ) )
=> ( ( dvd_dvd @ int @ D2 @ ( plus_plus @ int @ X4 @ T2 ) )
=> ( dvd_dvd @ int @ D2 @ ( plus_plus @ int @ ( minus_minus @ int @ X4 @ D6 ) @ T2 ) ) ) ) ) ).
% bset(9)
thf(fact_3140_div__dvd__sgn__abs,axiom,
! [L: int,K: int] :
( ( dvd_dvd @ int @ L @ K )
=> ( ( divide_divide @ int @ K @ L )
= ( times_times @ int @ ( times_times @ int @ ( sgn_sgn @ int @ K ) @ ( sgn_sgn @ int @ L ) ) @ ( divide_divide @ int @ ( abs_abs @ int @ K ) @ ( abs_abs @ int @ L ) ) ) ) ) ).
% div_dvd_sgn_abs
thf(fact_3141_tan__60,axiom,
( ( tan @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit1 @ one2 ) ) ) )
= ( sqrt @ ( numeral_numeral @ real @ ( bit1 @ one2 ) ) ) ) ).
% tan_60
thf(fact_3142_even__diff__iff,axiom,
! [K: int,L: int] :
( ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( minus_minus @ int @ K @ L ) )
= ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( plus_plus @ int @ K @ L ) ) ) ).
% even_diff_iff
thf(fact_3143_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_3144_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_3145_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_3146_nat__dvd__iff,axiom,
! [Z2: int,M2: nat] :
( ( dvd_dvd @ nat @ ( nat2 @ Z2 ) @ M2 )
= ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z2 )
=> ( dvd_dvd @ int @ Z2 @ ( semiring_1_of_nat @ int @ M2 ) ) )
& ( ~ ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z2 )
=> ( M2
= ( zero_zero @ nat ) ) ) ) ) ).
% nat_dvd_iff
thf(fact_3147_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_3148_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_3149_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_3150_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_3151_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_3152_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_3153_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_3154_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_3155_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_3156_arsinh__real__def,axiom,
( ( arsinh @ real )
= ( ^ [X2: real] : ( ln_ln @ real @ ( plus_plus @ real @ X2 @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ real ) ) ) ) ) ) ) ).
% arsinh_real_def
thf(fact_3157_even__nat__iff,axiom,
! [K: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
=> ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( nat2 @ K ) )
= ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K ) ) ) ).
% even_nat_iff
thf(fact_3158_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_3159_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_3160_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_3161_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_3162_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_3163_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_3164_arctan__half,axiom,
( arctan
= ( ^ [X2: real] : ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( arctan @ ( divide_divide @ real @ X2 @ ( plus_plus @ real @ ( one_one @ real ) @ ( sqrt @ ( plus_plus @ real @ ( one_one @ real ) @ ( power_power @ real @ X2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ).
% arctan_half
thf(fact_3165_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_3166_cos__zero__iff__int,axiom,
! [X: real] :
( ( ( cos @ real @ X )
= ( zero_zero @ real ) )
= ( ? [I2: int] :
( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ I2 )
& ( X
= ( times_times @ real @ ( ring_1_of_int @ real @ I2 ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% cos_zero_iff_int
thf(fact_3167_sin__zero__iff__int,axiom,
! [X: real] :
( ( ( sin @ real @ X )
= ( zero_zero @ real ) )
= ( ? [I2: int] :
( ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ I2 )
& ( X
= ( times_times @ real @ ( ring_1_of_int @ real @ I2 ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% sin_zero_iff_int
thf(fact_3168_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_3169_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_3170_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_3171_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_3172_take__bit__numeral__minus__bit1,axiom,
! [L: num,K: num] :
( ( bit_se2584673776208193580ke_bit @ int @ ( numeral_numeral @ nat @ L ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ K ) ) ) )
= ( plus_plus @ int @ ( times_times @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( pred_numeral @ L ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( inc @ K ) ) ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( one_one @ int ) ) ) ).
% take_bit_numeral_minus_bit1
thf(fact_3173_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_3174_arcsin__0,axiom,
( ( arcsin @ ( zero_zero @ real ) )
= ( zero_zero @ real ) ) ).
% arcsin_0
thf(fact_3175_take__bit__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A] :
( ( bit_se2584673776208193580ke_bit @ A @ ( zero_zero @ nat ) @ A3 )
= ( zero_zero @ A ) ) ) ).
% take_bit_0
thf(fact_3176_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_3177_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_3178_arccos__1,axiom,
( ( arccos @ ( one_one @ real ) )
= ( zero_zero @ real ) ) ).
% arccos_1
thf(fact_3179_norm__cis,axiom,
! [A3: real] :
( ( real_V7770717601297561774m_norm @ complex @ ( cis @ A3 ) )
= ( one_one @ real ) ) ).
% norm_cis
thf(fact_3180_concat__bit__of__zero__2,axiom,
! [N: nat,K: int] :
( ( bit_concat_bit @ N @ K @ ( zero_zero @ int ) )
= ( bit_se2584673776208193580ke_bit @ int @ N @ K ) ) ).
% concat_bit_of_zero_2
thf(fact_3181_cis__zero,axiom,
( ( cis @ ( zero_zero @ real ) )
= ( one_one @ complex ) ) ).
% cis_zero
thf(fact_3182_cis__inverse,axiom,
! [A3: real] :
( ( inverse_inverse @ complex @ ( cis @ A3 ) )
= ( cis @ ( uminus_uminus @ real @ A3 ) ) ) ).
% cis_inverse
thf(fact_3183_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_3184_of__nat__nat__take__bit__eq,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [N: nat,K: int] :
( ( semiring_1_of_nat @ A @ ( nat2 @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) ) )
= ( ring_1_of_int @ A @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) ) ) ) ).
% of_nat_nat_take_bit_eq
thf(fact_3185_arccos__minus__1,axiom,
( ( arccos @ ( uminus_uminus @ real @ ( one_one @ real ) ) )
= pi ) ).
% arccos_minus_1
thf(fact_3186_cis__pi,axiom,
( ( cis @ pi )
= ( uminus_uminus @ complex @ ( one_one @ complex ) ) ) ).
% cis_pi
thf(fact_3187_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_3188_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_3189_even__take__bit__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A3: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se2584673776208193580ke_bit @ A @ N @ A3 ) )
= ( ( N
= ( zero_zero @ nat ) )
| ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) ) ) ) ).
% even_take_bit_eq
thf(fact_3190_take__bit__Suc__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A] :
( ( bit_se2584673776208193580ke_bit @ A @ ( suc @ ( zero_zero @ nat ) ) @ A3 )
= ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% take_bit_Suc_0
thf(fact_3191_arccos__0,axiom,
( ( arccos @ ( zero_zero @ real ) )
= ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ).
% arccos_0
thf(fact_3192_arcsin__1,axiom,
( ( arcsin @ ( one_one @ real ) )
= ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ).
% arcsin_1
thf(fact_3193_cis__2pi,axiom,
( ( cis @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) )
= ( one_one @ complex ) ) ).
% cis_2pi
thf(fact_3194_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_3195_take__bit__minus,axiom,
! [N: nat,K: int] :
( ( bit_se2584673776208193580ke_bit @ int @ N @ ( uminus_uminus @ int @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) ) )
= ( bit_se2584673776208193580ke_bit @ int @ N @ ( uminus_uminus @ int @ K ) ) ) ).
% take_bit_minus
thf(fact_3196_take__bit__diff,axiom,
! [N: nat,K: int,L: int] :
( ( bit_se2584673776208193580ke_bit @ int @ N @ ( minus_minus @ int @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) @ ( bit_se2584673776208193580ke_bit @ int @ N @ L ) ) )
= ( bit_se2584673776208193580ke_bit @ int @ N @ ( minus_minus @ int @ K @ L ) ) ) ).
% take_bit_diff
thf(fact_3197_take__bit__of__nat,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,M2: nat] :
( ( bit_se2584673776208193580ke_bit @ A @ N @ ( semiring_1_of_nat @ A @ M2 ) )
= ( semiring_1_of_nat @ A @ ( bit_se2584673776208193580ke_bit @ nat @ N @ M2 ) ) ) ) ).
% take_bit_of_nat
thf(fact_3198_take__bit__int__less__eq__self__iff,axiom,
! [N: nat,K: int] :
( ( ord_less_eq @ int @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) @ K )
= ( ord_less_eq @ int @ ( zero_zero @ int ) @ K ) ) ).
% take_bit_int_less_eq_self_iff
thf(fact_3199_take__bit__nonnegative,axiom,
! [N: nat,K: int] : ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) ) ).
% take_bit_nonnegative
thf(fact_3200_not__take__bit__negative,axiom,
! [N: nat,K: int] :
~ ( ord_less @ int @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) @ ( zero_zero @ int ) ) ).
% not_take_bit_negative
thf(fact_3201_take__bit__int__greater__self__iff,axiom,
! [K: int,N: nat] :
( ( ord_less @ int @ K @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) )
= ( ord_less @ int @ K @ ( zero_zero @ int ) ) ) ).
% take_bit_int_greater_self_iff
thf(fact_3202_signed__take__bit__eq__iff__take__bit__eq,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: nat,A3: A,B2: A] :
( ( ( bit_ri4674362597316999326ke_bit @ A @ N @ A3 )
= ( bit_ri4674362597316999326ke_bit @ A @ N @ B2 ) )
= ( ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N ) @ A3 )
= ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N ) @ B2 ) ) ) ) ).
% signed_take_bit_eq_iff_take_bit_eq
thf(fact_3203_cis__divide,axiom,
! [A3: real,B2: real] :
( ( divide_divide @ complex @ ( cis @ A3 ) @ ( cis @ B2 ) )
= ( cis @ ( minus_minus @ real @ A3 @ B2 ) ) ) ).
% cis_divide
thf(fact_3204_take__bit__signed__take__bit,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [M2: nat,N: nat,A3: A] :
( ( ord_less_eq @ nat @ M2 @ ( suc @ N ) )
=> ( ( bit_se2584673776208193580ke_bit @ A @ M2 @ ( bit_ri4674362597316999326ke_bit @ A @ N @ A3 ) )
= ( bit_se2584673776208193580ke_bit @ A @ M2 @ A3 ) ) ) ) ).
% take_bit_signed_take_bit
thf(fact_3205_take__bit__decr__eq,axiom,
! [N: nat,K: int] :
( ( ( bit_se2584673776208193580ke_bit @ int @ N @ K )
!= ( zero_zero @ int ) )
=> ( ( bit_se2584673776208193580ke_bit @ int @ N @ ( minus_minus @ int @ K @ ( one_one @ int ) ) )
= ( minus_minus @ int @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) @ ( one_one @ int ) ) ) ) ).
% take_bit_decr_eq
thf(fact_3206_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_3207_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_3208_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_3209_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_3210_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_3211_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_3212_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_3213_take__bit__Suc__bit0,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [N: nat,K: num] :
( ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N ) @ ( numeral_numeral @ A @ ( bit0 @ K ) ) )
= ( times_times @ A @ ( bit_se2584673776208193580ke_bit @ A @ N @ ( numeral_numeral @ A @ K ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% take_bit_Suc_bit0
thf(fact_3214_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_3215_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_3216_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_3217_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_3218_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_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_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_3221_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_3222_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_3223_take__bit__int__less__exp,axiom,
! [N: nat,K: int] : ( ord_less @ int @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ).
% take_bit_int_less_exp
thf(fact_3224_take__bit__eq__0__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A3: A] :
( ( ( bit_se2584673776208193580ke_bit @ A @ N @ A3 )
= ( zero_zero @ A ) )
= ( dvd_dvd @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) @ A3 ) ) ) ).
% take_bit_eq_0_iff
thf(fact_3225_take__bit__Suc__minus__bit0,axiom,
! [N: nat,K: num] :
( ( bit_se2584673776208193580ke_bit @ int @ ( suc @ N ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ K ) ) ) )
= ( times_times @ int @ ( bit_se2584673776208193580ke_bit @ int @ N @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ).
% take_bit_Suc_minus_bit0
thf(fact_3226_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_3227_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_3228_take__bit__int__greater__eq__self__iff,axiom,
! [K: int,N: nat] :
( ( ord_less_eq @ int @ K @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) )
= ( ord_less @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ) ).
% take_bit_int_greater_eq_self_iff
thf(fact_3229_take__bit__int__less__self__iff,axiom,
! [N: nat,K: int] :
( ( ord_less @ int @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) @ K )
= ( ord_less_eq @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) @ K ) ) ).
% take_bit_int_less_self_iff
thf(fact_3230_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_3231_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_3232_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_3233_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_3234_take__bit__int__eq__self,axiom,
! [K: int,N: nat] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
=> ( ( ord_less @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( bit_se2584673776208193580ke_bit @ int @ N @ K )
= K ) ) ) ).
% take_bit_int_eq_self
thf(fact_3235_take__bit__int__eq__self__iff,axiom,
! [N: nat,K: int] :
( ( ( bit_se2584673776208193580ke_bit @ int @ N @ K )
= K )
= ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
& ( ord_less @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% take_bit_int_eq_self_iff
thf(fact_3236_take__bit__numeral__minus__bit0,axiom,
! [L: num,K: num] :
( ( bit_se2584673776208193580ke_bit @ int @ ( numeral_numeral @ nat @ L ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ K ) ) ) )
= ( times_times @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( pred_numeral @ L ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ).
% take_bit_numeral_minus_bit0
thf(fact_3237_take__bit__incr__eq,axiom,
! [N: nat,K: int] :
( ( ( bit_se2584673776208193580ke_bit @ int @ N @ K )
!= ( minus_minus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ int ) ) )
=> ( ( bit_se2584673776208193580ke_bit @ int @ N @ ( plus_plus @ int @ K @ ( one_one @ int ) ) )
= ( plus_plus @ int @ ( one_one @ int ) @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) ) ) ) ).
% take_bit_incr_eq
thf(fact_3238_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_3239_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_3240_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_3241_take__bit__Suc__bit1,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [N: nat,K: num] :
( ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N ) @ ( numeral_numeral @ A @ ( bit1 @ K ) ) )
= ( plus_plus @ A @ ( times_times @ A @ ( bit_se2584673776208193580ke_bit @ A @ N @ ( numeral_numeral @ A @ K ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) ) ) ) ).
% take_bit_Suc_bit1
thf(fact_3242_take__bit__numeral__minus__1__eq,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [K: num] :
( ( bit_se2584673776208193580ke_bit @ A @ ( numeral_numeral @ nat @ K ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ K ) ) @ ( one_one @ A ) ) ) ) ).
% take_bit_numeral_minus_1_eq
thf(fact_3243_take__bit__Suc,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A3: A] :
( ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N ) @ A3 )
= ( plus_plus @ A @ ( times_times @ A @ ( bit_se2584673776208193580ke_bit @ A @ N @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% take_bit_Suc
thf(fact_3244_take__bit__int__less__eq,axiom,
! [N: nat,K: int] :
( ( ord_less_eq @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) @ K )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ord_less_eq @ int @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) @ ( minus_minus @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ) ) ) ).
% take_bit_int_less_eq
thf(fact_3245_take__bit__int__greater__eq,axiom,
! [K: int,N: nat] :
( ( ord_less @ int @ K @ ( zero_zero @ int ) )
=> ( ord_less_eq @ int @ ( plus_plus @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) ) ) ).
% take_bit_int_greater_eq
thf(fact_3246_signed__take__bit__eq__take__bit__shift,axiom,
( ( bit_ri4674362597316999326ke_bit @ int )
= ( ^ [N5: nat,K3: int] : ( minus_minus @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( suc @ N5 ) @ ( plus_plus @ int @ K3 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N5 ) ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N5 ) ) ) ) ).
% signed_take_bit_eq_take_bit_shift
thf(fact_3247_stable__imp__take__bit__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A,N: nat] :
( ( ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= A3 )
=> ( ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
=> ( ( bit_se2584673776208193580ke_bit @ A @ N @ A3 )
= ( zero_zero @ A ) ) )
& ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
=> ( ( bit_se2584673776208193580ke_bit @ A @ N @ A3 )
= ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ A ) ) ) ) ) ) ) ).
% stable_imp_take_bit_eq
thf(fact_3248_take__bit__numeral__bit1,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [L: num,K: num] :
( ( bit_se2584673776208193580ke_bit @ A @ ( numeral_numeral @ nat @ L ) @ ( numeral_numeral @ A @ ( bit1 @ K ) ) )
= ( plus_plus @ A @ ( times_times @ A @ ( bit_se2584673776208193580ke_bit @ A @ ( pred_numeral @ L ) @ ( numeral_numeral @ A @ K ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) ) ) ) ).
% take_bit_numeral_bit1
thf(fact_3249_take__bit__minus__small__eq,axiom,
! [K: int,N: nat] :
( ( ord_less @ int @ ( zero_zero @ int ) @ K )
=> ( ( ord_less_eq @ int @ K @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( bit_se2584673776208193580ke_bit @ int @ N @ ( uminus_uminus @ int @ K ) )
= ( minus_minus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) @ K ) ) ) ) ).
% take_bit_minus_small_eq
thf(fact_3250_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_3251_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_3252_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_3253_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_3254_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_3255_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_3256_take__bit__Suc__minus__bit1,axiom,
! [N: nat,K: num] :
( ( bit_se2584673776208193580ke_bit @ int @ ( suc @ N ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ K ) ) ) )
= ( plus_plus @ int @ ( times_times @ int @ ( bit_se2584673776208193580ke_bit @ int @ N @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( inc @ K ) ) ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( one_one @ int ) ) ) ).
% take_bit_Suc_minus_bit1
thf(fact_3257_take__bit__rec,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se2584673776208193580ke_bit @ A )
= ( ^ [N5: nat,A5: A] :
( if @ A
@ ( N5
= ( zero_zero @ nat ) )
@ ( zero_zero @ A )
@ ( plus_plus @ A @ ( times_times @ A @ ( bit_se2584673776208193580ke_bit @ A @ ( minus_minus @ nat @ N5 @ ( one_one @ nat ) ) @ ( divide_divide @ A @ 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_3258_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_3259_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_3260_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_3261_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_3262_arccos__cos__eq__abs__2pi,axiom,
! [Theta: real] :
~ ! [K2: int] :
( ( arccos @ ( cos @ real @ Theta ) )
!= ( abs_abs @ real @ ( minus_minus @ real @ Theta @ ( times_times @ real @ ( ring_1_of_int @ real @ K2 ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) ) ) ) ) ).
% arccos_cos_eq_abs_2pi
thf(fact_3263_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_3264_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_3265_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_3266_flip__bit__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A] :
( ( bit_se8732182000553998342ip_bit @ A @ ( zero_zero @ nat ) @ A3 )
= ( plus_plus @ A @ ( zero_neq_one_of_bool @ A @ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% flip_bit_0
thf(fact_3267_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_3268_nat__of__bool,axiom,
! [P: $o] :
( ( nat2 @ ( zero_neq_one_of_bool @ int @ P ) )
= ( zero_neq_one_of_bool @ nat @ P ) ) ).
% nat_of_bool
thf(fact_3269_of__bool__less__eq__iff,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [P: $o,Q: $o] :
( ( ord_less_eq @ A @ ( zero_neq_one_of_bool @ A @ P ) @ ( zero_neq_one_of_bool @ A @ Q ) )
= ( P
=> Q ) ) ) ).
% of_bool_less_eq_iff
thf(fact_3270_of__bool__eq__0__iff,axiom,
! [A: $tType] :
( ( zero_neq_one @ A )
=> ! [P: $o] :
( ( ( zero_neq_one_of_bool @ A @ P )
= ( zero_zero @ A ) )
= ~ P ) ) ).
% of_bool_eq_0_iff
thf(fact_3271_of__bool__eq_I1_J,axiom,
! [A: $tType] :
( ( zero_neq_one @ A )
=> ( ( zero_neq_one_of_bool @ A @ $false )
= ( zero_zero @ A ) ) ) ).
% of_bool_eq(1)
thf(fact_3272_of__bool__less__iff,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [P: $o,Q: $o] :
( ( ord_less @ A @ ( zero_neq_one_of_bool @ A @ P ) @ ( zero_neq_one_of_bool @ A @ Q ) )
= ( ~ P
& Q ) ) ) ).
% of_bool_less_iff
thf(fact_3273_of__bool__eq__1__iff,axiom,
! [A: $tType] :
( ( zero_neq_one @ A )
=> ! [P: $o] :
( ( ( zero_neq_one_of_bool @ A @ P )
= ( one_one @ A ) )
= P ) ) ).
% of_bool_eq_1_iff
thf(fact_3274_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_3275_of__nat__of__bool,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [P: $o] :
( ( semiring_1_of_nat @ A @ ( zero_neq_one_of_bool @ nat @ P ) )
= ( zero_neq_one_of_bool @ A @ P ) ) ) ).
% of_nat_of_bool
thf(fact_3276_abs__bool__eq,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [P: $o] :
( ( abs_abs @ A @ ( zero_neq_one_of_bool @ A @ P ) )
= ( zero_neq_one_of_bool @ A @ P ) ) ) ).
% abs_bool_eq
thf(fact_3277_of__int__of__bool,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [P: $o] :
( ( ring_1_of_int @ A @ ( zero_neq_one_of_bool @ int @ P ) )
= ( zero_neq_one_of_bool @ A @ P ) ) ) ).
% of_int_of_bool
thf(fact_3278_zero__less__of__bool__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [P: $o] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( zero_neq_one_of_bool @ A @ P ) )
= P ) ) ).
% zero_less_of_bool_iff
thf(fact_3279_of__bool__less__one__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [P: $o] :
( ( ord_less @ A @ ( zero_neq_one_of_bool @ A @ P ) @ ( one_one @ A ) )
= ~ P ) ) ).
% of_bool_less_one_iff
thf(fact_3280_of__bool__not__iff,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [P: $o] :
( ( zero_neq_one_of_bool @ A @ ~ P )
= ( minus_minus @ A @ ( one_one @ A ) @ ( zero_neq_one_of_bool @ A @ P ) ) ) ) ).
% of_bool_not_iff
thf(fact_3281_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_3282_norm__ii,axiom,
( ( real_V7770717601297561774m_norm @ complex @ imaginary_unit )
= ( one_one @ real ) ) ).
% norm_ii
thf(fact_3283_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_3284_inverse__i,axiom,
( ( inverse_inverse @ complex @ imaginary_unit )
= ( uminus_uminus @ complex @ imaginary_unit ) ) ).
% inverse_i
thf(fact_3285_sgn__mult__self__eq,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A] :
( ( times_times @ A @ ( sgn_sgn @ A @ A3 ) @ ( sgn_sgn @ A @ A3 ) )
= ( zero_neq_one_of_bool @ A
@ ( A3
!= ( zero_zero @ A ) ) ) ) ) ).
% sgn_mult_self_eq
thf(fact_3286_sgn__abs,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ! [A3: A] :
( ( abs_abs @ A @ ( sgn_sgn @ A @ A3 ) )
= ( zero_neq_one_of_bool @ A
@ ( A3
!= ( zero_zero @ A ) ) ) ) ) ).
% sgn_abs
thf(fact_3287_idom__abs__sgn__class_Oabs__sgn,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ! [A3: A] :
( ( sgn_sgn @ A @ ( abs_abs @ A @ A3 ) )
= ( zero_neq_one_of_bool @ A
@ ( A3
!= ( zero_zero @ A ) ) ) ) ) ).
% idom_abs_sgn_class.abs_sgn
thf(fact_3288_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_3289_divide__i,axiom,
! [X: complex] :
( ( divide_divide @ complex @ X @ imaginary_unit )
= ( times_times @ complex @ ( uminus_uminus @ complex @ imaginary_unit ) @ X ) ) ).
% divide_i
thf(fact_3290_i__squared,axiom,
( ( times_times @ complex @ imaginary_unit @ imaginary_unit )
= ( uminus_uminus @ complex @ ( one_one @ complex ) ) ) ).
% i_squared
thf(fact_3291_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_3292_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_3293_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_3294_divide__numeral__i,axiom,
! [Z2: complex,N: num] :
( ( divide_divide @ complex @ Z2 @ ( times_times @ complex @ ( numeral_numeral @ complex @ N ) @ imaginary_unit ) )
= ( divide_divide @ complex @ ( uminus_uminus @ complex @ ( times_times @ complex @ imaginary_unit @ Z2 ) ) @ ( numeral_numeral @ complex @ N ) ) ) ).
% divide_numeral_i
thf(fact_3295_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_3296_power2__i,axiom,
( ( power_power @ complex @ imaginary_unit @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( uminus_uminus @ complex @ ( one_one @ complex ) ) ) ).
% power2_i
thf(fact_3297_cis__pi__half,axiom,
( ( cis @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
= imaginary_unit ) ).
% cis_pi_half
thf(fact_3298_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_3299_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_3300_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_3301_take__bit__of__exp,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [M2: nat,N: nat] :
( ( bit_se2584673776208193580ke_bit @ A @ M2 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
= ( times_times @ A @ ( zero_neq_one_of_bool @ A @ ( ord_less @ nat @ N @ M2 ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% take_bit_of_exp
thf(fact_3302_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_3303_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_3304_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_3305_of__bool__eq__iff,axiom,
! [A: $tType] :
( ( zero_neq_one @ A )
=> ! [P4: $o,Q5: $o] :
( ( ( zero_neq_one_of_bool @ A @ P4 )
= ( zero_neq_one_of_bool @ A @ Q5 ) )
= ( P4 = Q5 ) ) ) ).
% of_bool_eq_iff
thf(fact_3306_of__bool__conj,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [P: $o,Q: $o] :
( ( zero_neq_one_of_bool @ A
@ ( P
& Q ) )
= ( times_times @ A @ ( zero_neq_one_of_bool @ A @ P ) @ ( zero_neq_one_of_bool @ A @ Q ) ) ) ) ).
% of_bool_conj
thf(fact_3307_complex__i__not__one,axiom,
( imaginary_unit
!= ( one_one @ complex ) ) ).
% complex_i_not_one
thf(fact_3308_zero__less__eq__of__bool,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [P: $o] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( zero_neq_one_of_bool @ A @ P ) ) ) ).
% zero_less_eq_of_bool
thf(fact_3309_of__bool__less__eq__one,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [P: $o] : ( ord_less_eq @ A @ ( zero_neq_one_of_bool @ A @ P ) @ ( one_one @ A ) ) ) ).
% of_bool_less_eq_one
thf(fact_3310_split__of__bool__asm,axiom,
! [A: $tType] :
( ( zero_neq_one @ A )
=> ! [P: A > $o,P4: $o] :
( ( P @ ( zero_neq_one_of_bool @ A @ P4 ) )
= ( ~ ( ( P4
& ~ ( P @ ( one_one @ A ) ) )
| ( ~ P4
& ~ ( P @ ( zero_zero @ A ) ) ) ) ) ) ) ).
% split_of_bool_asm
thf(fact_3311_split__of__bool,axiom,
! [A: $tType] :
( ( zero_neq_one @ A )
=> ! [P: A > $o,P4: $o] :
( ( P @ ( zero_neq_one_of_bool @ A @ P4 ) )
= ( ( P4
=> ( P @ ( one_one @ A ) ) )
& ( ~ P4
=> ( P @ ( zero_zero @ A ) ) ) ) ) ) ).
% split_of_bool
thf(fact_3312_of__bool__def,axiom,
! [A: $tType] :
( ( zero_neq_one @ A )
=> ( ( zero_neq_one_of_bool @ A )
= ( ^ [P6: $o] : ( if @ A @ P6 @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ) ) ).
% of_bool_def
thf(fact_3313_i__times__eq__iff,axiom,
! [W: complex,Z2: complex] :
( ( ( times_times @ complex @ imaginary_unit @ W )
= Z2 )
= ( W
= ( uminus_uminus @ complex @ ( times_times @ complex @ imaginary_unit @ Z2 ) ) ) ) ).
% i_times_eq_iff
thf(fact_3314_complex__i__not__neg__numeral,axiom,
! [W: num] :
( imaginary_unit
!= ( uminus_uminus @ complex @ ( numeral_numeral @ complex @ W ) ) ) ).
% complex_i_not_neg_numeral
thf(fact_3315_imaginary__unit_Ocode,axiom,
( imaginary_unit
= ( complex2 @ ( zero_zero @ real ) @ ( one_one @ real ) ) ) ).
% imaginary_unit.code
thf(fact_3316_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_3317_i__mult__Complex,axiom,
! [A3: real,B2: real] :
( ( times_times @ complex @ imaginary_unit @ ( complex2 @ A3 @ B2 ) )
= ( complex2 @ ( uminus_uminus @ real @ B2 ) @ A3 ) ) ).
% i_mult_Complex
thf(fact_3318_Complex__mult__i,axiom,
! [A3: real,B2: real] :
( ( times_times @ complex @ ( complex2 @ A3 @ B2 ) @ imaginary_unit )
= ( complex2 @ ( uminus_uminus @ real @ B2 ) @ A3 ) ) ).
% Complex_mult_i
thf(fact_3319_take__bit__nat__eq,axiom,
! [K: int,N: nat] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
=> ( ( bit_se2584673776208193580ke_bit @ nat @ N @ ( nat2 @ K ) )
= ( nat2 @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) ) ) ) ).
% take_bit_nat_eq
thf(fact_3320_nat__take__bit__eq,axiom,
! [K: int,N: nat] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
=> ( ( nat2 @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) )
= ( bit_se2584673776208193580ke_bit @ nat @ N @ ( nat2 @ K ) ) ) ) ).
% nat_take_bit_eq
thf(fact_3321_subset__decode__imp__le,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq @ ( set @ nat ) @ ( nat_set_decode @ M2 ) @ ( nat_set_decode @ N ) )
=> ( ord_less_eq @ nat @ M2 @ N ) ) ).
% subset_decode_imp_le
thf(fact_3322_take__bit__nat__eq__self__iff,axiom,
! [N: nat,M2: nat] :
( ( ( bit_se2584673776208193580ke_bit @ nat @ N @ M2 )
= M2 )
= ( ord_less @ nat @ M2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ).
% take_bit_nat_eq_self_iff
thf(fact_3323_take__bit__nat__less__exp,axiom,
! [N: nat,M2: nat] : ( ord_less @ nat @ ( bit_se2584673776208193580ke_bit @ nat @ N @ M2 ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ).
% take_bit_nat_less_exp
thf(fact_3324_take__bit__nat__eq__self,axiom,
! [M2: nat,N: nat] :
( ( ord_less @ nat @ M2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( bit_se2584673776208193580ke_bit @ nat @ N @ M2 )
= M2 ) ) ).
% take_bit_nat_eq_self
thf(fact_3325_take__bit__nat__less__self__iff,axiom,
! [N: nat,M2: nat] :
( ( ord_less @ nat @ ( bit_se2584673776208193580ke_bit @ nat @ N @ M2 ) @ M2 )
= ( ord_less_eq @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ M2 ) ) ).
% take_bit_nat_less_self_iff
thf(fact_3326_bits__induct,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [P: A > $o,A3: A] :
( ! [A6: A] :
( ( ( divide_divide @ A @ A6 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= A6 )
=> ( P @ A6 ) )
=> ( ! [A6: A,B4: $o] :
( ( P @ A6 )
=> ( ( ( divide_divide @ A @ ( plus_plus @ A @ ( zero_neq_one_of_bool @ A @ B4 ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A6 ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= A6 )
=> ( P @ ( plus_plus @ A @ ( zero_neq_one_of_bool @ A @ B4 ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A6 ) ) ) ) )
=> ( P @ A3 ) ) ) ) ).
% bits_induct
thf(fact_3327_exp__mod__exp,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [M2: nat,N: nat] :
( ( modulo_modulo @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M2 ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
= ( times_times @ A @ ( zero_neq_one_of_bool @ A @ ( ord_less @ nat @ M2 @ N ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M2 ) ) ) ) ).
% exp_mod_exp
thf(fact_3328_div__noneq__sgn__abs,axiom,
! [L: int,K: int] :
( ( L
!= ( zero_zero @ int ) )
=> ( ( ( sgn_sgn @ int @ K )
!= ( sgn_sgn @ int @ L ) )
=> ( ( divide_divide @ int @ K @ L )
= ( minus_minus @ int @ ( uminus_uminus @ int @ ( divide_divide @ int @ ( abs_abs @ int @ K ) @ ( abs_abs @ int @ L ) ) )
@ ( zero_neq_one_of_bool @ int
@ ~ ( dvd_dvd @ int @ L @ K ) ) ) ) ) ) ).
% div_noneq_sgn_abs
thf(fact_3329_exp__div__exp__eq,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [M2: nat,N: nat] :
( ( divide_divide @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M2 ) @ ( 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 ) ) @ M2 )
!= ( zero_zero @ A ) )
& ( ord_less_eq @ nat @ N @ M2 ) ) )
@ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ M2 @ N ) ) ) ) ) ).
% exp_div_exp_eq
thf(fact_3330_cmod__unit__one,axiom,
! [A3: real] :
( ( real_V7770717601297561774m_norm @ complex @ ( plus_plus @ complex @ ( real_Vector_of_real @ complex @ ( cos @ real @ A3 ) ) @ ( times_times @ complex @ imaginary_unit @ ( real_Vector_of_real @ complex @ ( sin @ real @ A3 ) ) ) ) )
= ( one_one @ real ) ) ).
% cmod_unit_one
thf(fact_3331_divide__int__unfold,axiom,
! [L: int,K: int,N: nat,M2: nat] :
( ( ( ( ( sgn_sgn @ int @ L )
= ( zero_zero @ int ) )
| ( ( sgn_sgn @ int @ K )
= ( zero_zero @ int ) )
| ( N
= ( zero_zero @ nat ) ) )
=> ( ( divide_divide @ int @ ( times_times @ int @ ( sgn_sgn @ int @ K ) @ ( semiring_1_of_nat @ int @ M2 ) ) @ ( times_times @ int @ ( sgn_sgn @ int @ L ) @ ( semiring_1_of_nat @ int @ N ) ) )
= ( zero_zero @ int ) ) )
& ( ~ ( ( ( sgn_sgn @ int @ L )
= ( zero_zero @ int ) )
| ( ( sgn_sgn @ int @ K )
= ( zero_zero @ int ) )
| ( N
= ( zero_zero @ nat ) ) )
=> ( ( ( ( sgn_sgn @ int @ K )
= ( sgn_sgn @ int @ L ) )
=> ( ( divide_divide @ int @ ( times_times @ int @ ( sgn_sgn @ int @ K ) @ ( semiring_1_of_nat @ int @ M2 ) ) @ ( times_times @ int @ ( sgn_sgn @ int @ L ) @ ( semiring_1_of_nat @ int @ N ) ) )
= ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ M2 @ N ) ) ) )
& ( ( ( sgn_sgn @ int @ K )
!= ( sgn_sgn @ int @ L ) )
=> ( ( divide_divide @ int @ ( times_times @ int @ ( sgn_sgn @ int @ K ) @ ( semiring_1_of_nat @ int @ M2 ) ) @ ( 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 @ M2 @ N )
@ ( zero_neq_one_of_bool @ nat
@ ~ ( dvd_dvd @ nat @ N @ M2 ) ) ) ) ) ) ) ) ) ) ).
% divide_int_unfold
thf(fact_3332_modulo__int__def,axiom,
( ( modulo_modulo @ int )
= ( ^ [K3: int,L2: int] :
( if @ int
@ ( L2
= ( zero_zero @ int ) )
@ K3
@ ( if @ int
@ ( ( sgn_sgn @ int @ K3 )
= ( sgn_sgn @ int @ L2 ) )
@ ( times_times @ int @ ( sgn_sgn @ int @ L2 ) @ ( semiring_1_of_nat @ int @ ( modulo_modulo @ nat @ ( nat2 @ ( abs_abs @ int @ K3 ) ) @ ( nat2 @ ( abs_abs @ int @ L2 ) ) ) ) )
@ ( times_times @ int @ ( sgn_sgn @ int @ L2 )
@ ( minus_minus @ int
@ ( times_times @ int @ ( abs_abs @ int @ L2 )
@ ( zero_neq_one_of_bool @ int
@ ~ ( dvd_dvd @ int @ L2 @ K3 ) ) )
@ ( semiring_1_of_nat @ int @ ( modulo_modulo @ nat @ ( nat2 @ ( abs_abs @ int @ K3 ) ) @ ( nat2 @ ( abs_abs @ int @ L2 ) ) ) ) ) ) ) ) ) ) ).
% modulo_int_def
thf(fact_3333_modulo__int__unfold,axiom,
! [L: int,K: int,N: nat,M2: nat] :
( ( ( ( ( sgn_sgn @ int @ L )
= ( zero_zero @ int ) )
| ( ( sgn_sgn @ int @ K )
= ( zero_zero @ int ) )
| ( N
= ( zero_zero @ nat ) ) )
=> ( ( modulo_modulo @ int @ ( times_times @ int @ ( sgn_sgn @ int @ K ) @ ( semiring_1_of_nat @ int @ M2 ) ) @ ( times_times @ int @ ( sgn_sgn @ int @ L ) @ ( semiring_1_of_nat @ int @ N ) ) )
= ( times_times @ int @ ( sgn_sgn @ int @ K ) @ ( semiring_1_of_nat @ int @ M2 ) ) ) )
& ( ~ ( ( ( sgn_sgn @ int @ L )
= ( zero_zero @ int ) )
| ( ( sgn_sgn @ int @ K )
= ( zero_zero @ int ) )
| ( N
= ( zero_zero @ nat ) ) )
=> ( ( ( ( sgn_sgn @ int @ K )
= ( sgn_sgn @ int @ L ) )
=> ( ( modulo_modulo @ int @ ( times_times @ int @ ( sgn_sgn @ int @ K ) @ ( semiring_1_of_nat @ int @ M2 ) ) @ ( 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 @ M2 @ N ) ) ) ) )
& ( ( ( sgn_sgn @ int @ K )
!= ( sgn_sgn @ int @ L ) )
=> ( ( modulo_modulo @ int @ ( times_times @ int @ ( sgn_sgn @ int @ K ) @ ( semiring_1_of_nat @ int @ M2 ) ) @ ( 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 @ M2 ) ) ) )
@ ( semiring_1_of_nat @ int @ ( modulo_modulo @ nat @ M2 @ N ) ) ) ) ) ) ) ) ) ).
% modulo_int_unfold
thf(fact_3334_divide__int__def,axiom,
( ( divide_divide @ int )
= ( ^ [K3: int,L2: int] :
( if @ int
@ ( L2
= ( zero_zero @ int ) )
@ ( zero_zero @ int )
@ ( if @ int
@ ( ( sgn_sgn @ int @ K3 )
= ( sgn_sgn @ int @ L2 ) )
@ ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ ( nat2 @ ( abs_abs @ int @ K3 ) ) @ ( nat2 @ ( abs_abs @ int @ L2 ) ) ) )
@ ( uminus_uminus @ int
@ ( semiring_1_of_nat @ int
@ ( plus_plus @ nat @ ( divide_divide @ nat @ ( nat2 @ ( abs_abs @ int @ K3 ) ) @ ( nat2 @ ( abs_abs @ int @ L2 ) ) )
@ ( zero_neq_one_of_bool @ nat
@ ~ ( dvd_dvd @ int @ L2 @ K3 ) ) ) ) ) ) ) ) ) ).
% divide_int_def
thf(fact_3335_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_3336_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_3337_Arg__ii,axiom,
( ( arg @ imaginary_unit )
= ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ).
% Arg_ii
thf(fact_3338_Arg__correct,axiom,
! [Z2: complex] :
( ( Z2
!= ( zero_zero @ complex ) )
=> ( ( ( sgn_sgn @ complex @ Z2 )
= ( cis @ ( arg @ Z2 ) ) )
& ( ord_less @ real @ ( uminus_uminus @ real @ pi ) @ ( arg @ Z2 ) )
& ( ord_less_eq @ real @ ( arg @ Z2 ) @ pi ) ) ) ).
% Arg_correct
thf(fact_3339_cis__Arg__unique,axiom,
! [Z2: complex,X: real] :
( ( ( sgn_sgn @ complex @ Z2 )
= ( cis @ X ) )
=> ( ( ord_less @ real @ ( uminus_uminus @ real @ pi ) @ X )
=> ( ( ord_less_eq @ real @ X @ pi )
=> ( ( arg @ Z2 )
= X ) ) ) ) ).
% cis_Arg_unique
thf(fact_3340_csqrt__eq__1,axiom,
! [Z2: complex] :
( ( ( csqrt @ Z2 )
= ( one_one @ complex ) )
= ( Z2
= ( one_one @ complex ) ) ) ).
% csqrt_eq_1
thf(fact_3341_csqrt__1,axiom,
( ( csqrt @ ( one_one @ complex ) )
= ( one_one @ complex ) ) ).
% csqrt_1
thf(fact_3342_Arg__bounded,axiom,
! [Z2: complex] :
( ( ord_less @ real @ ( uminus_uminus @ real @ pi ) @ ( arg @ Z2 ) )
& ( ord_less_eq @ real @ ( arg @ Z2 ) @ pi ) ) ).
% Arg_bounded
thf(fact_3343_buildup__gives__empty,axiom,
! [N: nat] :
( ( vEBT_VEBT_set_vebt @ ( vEBT_vebt_buildup @ N ) )
= ( bot_bot @ ( set @ nat ) ) ) ).
% buildup_gives_empty
thf(fact_3344_signed__take__bit__eq__take__bit__minus,axiom,
( ( bit_ri4674362597316999326ke_bit @ int )
= ( ^ [N5: nat,K3: int] : ( minus_minus @ int @ ( bit_se2584673776208193580ke_bit @ int @ ( suc @ N5 ) @ K3 ) @ ( times_times @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( suc @ N5 ) ) @ ( zero_neq_one_of_bool @ int @ ( bit_se5641148757651400278ts_bit @ int @ K3 @ N5 ) ) ) ) ) ) ).
% signed_take_bit_eq_take_bit_minus
thf(fact_3345_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_3346_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_3347_invar__vebt_Ocases,axiom,
! [A1: vEBT_VEBT,A22: nat] :
( ( vEBT_invar_vebt @ A1 @ A22 )
=> ( ( ? [A6: $o,B4: $o] :
( A1
= ( vEBT_Leaf @ A6 @ B4 ) )
=> ( A22
!= ( suc @ ( zero_zero @ nat ) ) ) )
=> ( ! [TreeList2: list @ vEBT_VEBT,N2: nat,Summary2: vEBT_VEBT,M3: nat,Deg2: nat] :
( ( A1
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( ( A22 = Deg2 )
=> ( ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ( vEBT_invar_vebt @ X4 @ N2 ) )
=> ( ( vEBT_invar_vebt @ Summary2 @ M3 )
=> ( ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M3 ) )
=> ( ( M3 = N2 )
=> ( ( Deg2
= ( plus_plus @ nat @ N2 @ M3 ) )
=> ( ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X_12 )
=> ~ ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X_12 ) ) ) ) ) ) ) ) ) )
=> ( ! [TreeList2: list @ vEBT_VEBT,N2: nat,Summary2: vEBT_VEBT,M3: nat,Deg2: nat] :
( ( A1
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( ( A22 = Deg2 )
=> ( ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ( vEBT_invar_vebt @ X4 @ N2 ) )
=> ( ( vEBT_invar_vebt @ Summary2 @ M3 )
=> ( ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M3 ) )
=> ( ( M3
= ( suc @ N2 ) )
=> ( ( Deg2
= ( plus_plus @ nat @ N2 @ M3 ) )
=> ( ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X_12 )
=> ~ ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X_12 ) ) ) ) ) ) ) ) ) )
=> ( ! [TreeList2: list @ vEBT_VEBT,N2: nat,Summary2: vEBT_VEBT,M3: nat,Deg2: nat,Mi: nat,Ma2: nat] :
( ( A1
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma2 ) ) @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( ( A22 = Deg2 )
=> ( ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ( vEBT_invar_vebt @ X4 @ N2 ) )
=> ( ( vEBT_invar_vebt @ Summary2 @ M3 )
=> ( ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M3 ) )
=> ( ( M3 = N2 )
=> ( ( Deg2
= ( plus_plus @ nat @ N2 @ M3 ) )
=> ( ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M3 ) )
=> ( ( ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I4 ) @ X7 ) )
= ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
=> ( ( ( Mi = Ma2 )
=> ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X_12 ) ) )
=> ( ( ord_less_eq @ nat @ Mi @ Ma2 )
=> ( ( ord_less @ nat @ Ma2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
=> ~ ( ( Mi != Ma2 )
=> ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M3 ) )
=> ( ( ( ( vEBT_VEBT_high @ Ma2 @ N2 )
= I4 )
=> ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I4 ) @ ( vEBT_VEBT_low @ Ma2 @ N2 ) ) )
& ! [X4: nat] :
( ( ( ( vEBT_VEBT_high @ X4 @ N2 )
= I4 )
& ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I4 ) @ ( vEBT_VEBT_low @ X4 @ N2 ) ) )
=> ( ( ord_less @ nat @ Mi @ X4 )
& ( ord_less_eq @ nat @ X4 @ Ma2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
=> ~ ! [TreeList2: list @ vEBT_VEBT,N2: nat,Summary2: vEBT_VEBT,M3: nat,Deg2: nat,Mi: nat,Ma2: nat] :
( ( A1
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma2 ) ) @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( ( A22 = Deg2 )
=> ( ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ( vEBT_invar_vebt @ X4 @ N2 ) )
=> ( ( vEBT_invar_vebt @ Summary2 @ M3 )
=> ( ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M3 ) )
=> ( ( M3
= ( suc @ N2 ) )
=> ( ( Deg2
= ( plus_plus @ nat @ N2 @ M3 ) )
=> ( ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M3 ) )
=> ( ( ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I4 ) @ X7 ) )
= ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
=> ( ( ( Mi = Ma2 )
=> ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X_12 ) ) )
=> ( ( ord_less_eq @ nat @ Mi @ Ma2 )
=> ( ( ord_less @ nat @ Ma2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
=> ~ ( ( Mi != Ma2 )
=> ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M3 ) )
=> ( ( ( ( vEBT_VEBT_high @ Ma2 @ N2 )
= I4 )
=> ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I4 ) @ ( vEBT_VEBT_low @ Ma2 @ N2 ) ) )
& ! [X4: nat] :
( ( ( ( vEBT_VEBT_high @ X4 @ N2 )
= I4 )
& ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I4 ) @ ( vEBT_VEBT_low @ X4 @ N2 ) ) )
=> ( ( ord_less @ nat @ Mi @ X4 )
& ( ord_less_eq @ nat @ X4 @ Ma2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% invar_vebt.cases
thf(fact_3348_option_Oinject,axiom,
! [A: $tType,X23: A,Y23: A] :
( ( ( some @ A @ X23 )
= ( some @ A @ Y23 ) )
= ( X23 = Y23 ) ) ).
% option.inject
thf(fact_3349_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_3350_not__Some__eq,axiom,
! [A: $tType,X: option @ A] :
( ( ! [Y5: A] :
( X
!= ( some @ A @ Y5 ) ) )
= ( X
= ( none @ A ) ) ) ).
% not_Some_eq
thf(fact_3351_not__None__eq,axiom,
! [A: $tType,X: option @ A] :
( ( X
!= ( none @ A ) )
= ( ? [Y5: A] :
( X
= ( some @ A @ Y5 ) ) ) ) ).
% not_None_eq
thf(fact_3352_mi__ma__2__deg,axiom,
! [Mi2: 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 @ Mi2 @ Ma ) ) @ Deg @ TreeList @ Summary ) @ N )
=> ( ( ord_less_eq @ nat @ Mi2 @ Ma )
& ( ord_less @ nat @ Ma @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg ) ) ) ) ).
% mi_ma_2_deg
thf(fact_3353_atLeastatMost__empty,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B2: A,A3: A] :
( ( ord_less @ A @ B2 @ A3 )
=> ( ( set_or1337092689740270186AtMost @ A @ A3 @ B2 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% atLeastatMost_empty
thf(fact_3354_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_3355_mask__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se2239418461657761734s_mask @ A @ ( zero_zero @ nat ) )
= ( zero_zero @ A ) ) ) ).
% mask_0
thf(fact_3356_both__member__options__from__complete__tree__to__child,axiom,
! [Deg: nat,Mi2: 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 @ Mi2 @ 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 = Mi2 )
| ( X = Ma ) ) ) ) ).
% both_member_options_from_complete_tree_to_child
thf(fact_3357_set__decode__zero,axiom,
( ( nat_set_decode @ ( zero_zero @ nat ) )
= ( bot_bot @ ( set @ nat ) ) ) ).
% set_decode_zero
thf(fact_3358_member__inv,axiom,
! [Mi2: nat,Ma: nat,Deg: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
( ( vEBT_vebt_member @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
& ( ( X = Mi2 )
| ( X = Ma )
| ( ( ord_less @ nat @ X @ Ma )
& ( ord_less @ nat @ Mi2 @ X )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
& ( vEBT_vebt_member @ ( 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 ) ) ) ) ) ) ) ) ) ).
% member_inv
thf(fact_3359_bit__numeral__Bit0__Suc__iff,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [M2: num,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( numeral_numeral @ A @ ( bit0 @ M2 ) ) @ ( suc @ N ) )
= ( bit_se5641148757651400278ts_bit @ A @ ( numeral_numeral @ A @ M2 ) @ N ) ) ) ).
% bit_numeral_Bit0_Suc_iff
thf(fact_3360_bit__numeral__Bit1__Suc__iff,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [M2: num,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( numeral_numeral @ A @ ( bit1 @ M2 ) ) @ ( suc @ N ) )
= ( bit_se5641148757651400278ts_bit @ A @ ( numeral_numeral @ A @ M2 ) @ N ) ) ) ).
% bit_numeral_Bit1_Suc_iff
thf(fact_3361_both__member__options__from__chilf__to__complete__tree,axiom,
! [X: nat,Deg: nat,TreeList: list @ vEBT_VEBT,Mi2: 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 @ Mi2 @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X ) ) ) ) ).
% both_member_options_from_chilf_to_complete_tree
thf(fact_3362_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_3363_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_3364_signed__take__bit__nonnegative__iff,axiom,
! [N: nat,K: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_ri4674362597316999326ke_bit @ int @ N @ K ) )
= ( ~ ( bit_se5641148757651400278ts_bit @ int @ K @ N ) ) ) ).
% signed_take_bit_nonnegative_iff
thf(fact_3365_signed__take__bit__negative__iff,axiom,
! [N: nat,K: int] :
( ( ord_less @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N @ K ) @ ( zero_zero @ int ) )
= ( bit_se5641148757651400278ts_bit @ int @ K @ N ) ) ).
% signed_take_bit_negative_iff
thf(fact_3366_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_3367_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_3368_bit__0,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A] :
( ( bit_se5641148757651400278ts_bit @ A @ A3 @ ( zero_zero @ nat ) )
= ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) ) ) ) ).
% bit_0
thf(fact_3369_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_3370_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_3371_bit__mod__2__iff,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ N )
= ( ( N
= ( zero_zero @ nat ) )
& ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) ) ) ) ).
% bit_mod_2_iff
thf(fact_3372_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_3373_bit__of__nat__iff__bit,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [M2: nat,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( semiring_1_of_nat @ A @ M2 ) @ N )
= ( bit_se5641148757651400278ts_bit @ nat @ M2 @ N ) ) ) ).
% bit_of_nat_iff_bit
thf(fact_3374_prod__decode__aux_Ocases,axiom,
! [X: product_prod @ nat @ nat] :
~ ! [K2: nat,M3: nat] :
( X
!= ( product_Pair @ nat @ nat @ K2 @ M3 ) ) ).
% prod_decode_aux.cases
thf(fact_3375_option_Odistinct_I1_J,axiom,
! [A: $tType,X23: A] :
( ( none @ A )
!= ( some @ A @ X23 ) ) ).
% option.distinct(1)
thf(fact_3376_option_OdiscI,axiom,
! [A: $tType,Option: option @ A,X23: A] :
( ( Option
= ( some @ A @ X23 ) )
=> ( Option
!= ( none @ A ) ) ) ).
% option.discI
thf(fact_3377_option_Oexhaust,axiom,
! [A: $tType,Y2: option @ A] :
( ( Y2
!= ( none @ A ) )
=> ~ ! [X24: A] :
( Y2
!= ( some @ A @ X24 ) ) ) ).
% option.exhaust
thf(fact_3378_split__option__ex,axiom,
! [A: $tType] :
( ( ^ [P2: ( option @ A ) > $o] :
? [X5: option @ A] : ( P2 @ X5 ) )
= ( ^ [P3: ( option @ A ) > $o] :
( ( P3 @ ( none @ A ) )
| ? [X2: A] : ( P3 @ ( some @ A @ X2 ) ) ) ) ) ).
% split_option_ex
thf(fact_3379_split__option__all,axiom,
! [A: $tType] :
( ( ^ [P2: ( option @ A ) > $o] :
! [X5: option @ A] : ( P2 @ X5 ) )
= ( ^ [P3: ( option @ A ) > $o] :
( ( P3 @ ( none @ A ) )
& ! [X2: A] : ( P3 @ ( some @ A @ X2 ) ) ) ) ) ).
% split_option_all
thf(fact_3380_combine__options__cases,axiom,
! [A: $tType,B: $tType,X: option @ A,P: ( option @ A ) > ( option @ B ) > $o,Y2: option @ B] :
( ( ( X
= ( none @ A ) )
=> ( P @ X @ Y2 ) )
=> ( ( ( Y2
= ( none @ B ) )
=> ( P @ X @ Y2 ) )
=> ( ! [A6: A,B4: B] :
( ( X
= ( some @ A @ A6 ) )
=> ( ( Y2
= ( some @ B @ B4 ) )
=> ( P @ X @ Y2 ) ) )
=> ( P @ X @ Y2 ) ) ) ) ).
% combine_options_cases
thf(fact_3381_bot_Oextremum__strict,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [A3: A] :
~ ( ord_less @ A @ A3 @ ( bot_bot @ A ) ) ) ).
% bot.extremum_strict
thf(fact_3382_bot_Onot__eq__extremum,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [A3: A] :
( ( A3
!= ( bot_bot @ A ) )
= ( ord_less @ A @ ( bot_bot @ A ) @ A3 ) ) ) ).
% bot.not_eq_extremum
thf(fact_3383_VEBT__internal_Omembermima_Osimps_I3_J,axiom,
! [Mi2: nat,Ma: nat,Va2: list @ vEBT_VEBT,Vb: vEBT_VEBT,X: nat] :
( ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma ) ) @ ( zero_zero @ nat ) @ Va2 @ Vb ) @ X )
= ( ( X = Mi2 )
| ( X = Ma ) ) ) ).
% VEBT_internal.membermima.simps(3)
thf(fact_3384_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_3385_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_3386_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_3387_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_3388_bit__take__bit__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M2: nat,A3: A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( bit_se2584673776208193580ke_bit @ A @ M2 @ A3 ) @ N )
= ( ( ord_less @ nat @ N @ M2 )
& ( bit_se5641148757651400278ts_bit @ A @ A3 @ N ) ) ) ) ).
% bit_take_bit_iff
thf(fact_3389_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_3390_mask__nonnegative__int,axiom,
! [N: nat] : ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se2239418461657761734s_mask @ int @ N ) ) ).
% mask_nonnegative_int
thf(fact_3391_not__mask__negative__int,axiom,
! [N: nat] :
~ ( ord_less @ int @ ( bit_se2239418461657761734s_mask @ int @ N ) @ ( zero_zero @ int ) ) ).
% not_mask_negative_int
thf(fact_3392_VEBT__internal_OminNull_Osimps_I5_J,axiom,
! [Uz: product_prod @ nat @ nat,Va2: nat,Vb: list @ vEBT_VEBT,Vc: vEBT_VEBT] :
~ ( vEBT_VEBT_minNull @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ Uz ) @ Va2 @ Vb @ Vc ) ) ).
% VEBT_internal.minNull.simps(5)
thf(fact_3393_bit__not__int__iff_H,axiom,
! [K: int,N: nat] :
( ( bit_se5641148757651400278ts_bit @ int @ ( minus_minus @ int @ ( uminus_uminus @ int @ K ) @ ( one_one @ int ) ) @ N )
= ( ~ ( bit_se5641148757651400278ts_bit @ int @ K @ N ) ) ) ).
% bit_not_int_iff'
thf(fact_3394_vebt__member_Osimps_I3_J,axiom,
! [V2: product_prod @ nat @ nat,Uy2: list @ vEBT_VEBT,Uz: vEBT_VEBT,X: nat] :
~ ( vEBT_vebt_member @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V2 ) @ ( zero_zero @ nat ) @ Uy2 @ Uz ) @ X ) ).
% vebt_member.simps(3)
thf(fact_3395_VEBT__internal_OminNull_Ocases,axiom,
! [X: vEBT_VEBT] :
( ( X
!= ( vEBT_Leaf @ $false @ $false ) )
=> ( ! [Uv2: $o] :
( X
!= ( vEBT_Leaf @ $true @ Uv2 ) )
=> ( ! [Uu2: $o] :
( X
!= ( vEBT_Leaf @ Uu2 @ $true ) )
=> ( ! [Uw2: nat,Ux2: list @ vEBT_VEBT,Uy: vEBT_VEBT] :
( X
!= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uw2 @ Ux2 @ Uy ) )
=> ~ ! [Uz2: product_prod @ nat @ nat,Va3: nat,Vb2: list @ vEBT_VEBT,Vc2: vEBT_VEBT] :
( X
!= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) ) ) ) ) ) ).
% VEBT_internal.minNull.cases
thf(fact_3396_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_3397_VEBT__internal_OminNull_Oelims_I3_J,axiom,
! [X: vEBT_VEBT] :
( ~ ( vEBT_VEBT_minNull @ X )
=> ( ! [Uv2: $o] :
( X
!= ( vEBT_Leaf @ $true @ Uv2 ) )
=> ( ! [Uu2: $o] :
( X
!= ( vEBT_Leaf @ Uu2 @ $true ) )
=> ~ ! [Uz2: product_prod @ nat @ nat,Va3: nat,Vb2: list @ vEBT_VEBT,Vc2: vEBT_VEBT] :
( X
!= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) ) ) ) ) ).
% VEBT_internal.minNull.elims(3)
thf(fact_3398_option_Osize_I4_J,axiom,
! [A: $tType,X23: A] :
( ( size_size @ ( option @ A ) @ ( some @ A @ X23 ) )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% option.size(4)
thf(fact_3399_bit__imp__take__bit__positive,axiom,
! [N: nat,M2: nat,K: int] :
( ( ord_less @ nat @ N @ M2 )
=> ( ( bit_se5641148757651400278ts_bit @ int @ K @ N )
=> ( ord_less @ int @ ( zero_zero @ int ) @ ( bit_se2584673776208193580ke_bit @ int @ M2 @ K ) ) ) ) ).
% bit_imp_take_bit_positive
thf(fact_3400_bit__concat__bit__iff,axiom,
! [M2: nat,K: int,L: int,N: nat] :
( ( bit_se5641148757651400278ts_bit @ int @ ( bit_concat_bit @ M2 @ K @ L ) @ N )
= ( ( ( ord_less @ nat @ N @ M2 )
& ( bit_se5641148757651400278ts_bit @ int @ K @ N ) )
| ( ( ord_less_eq @ nat @ M2 @ N )
& ( bit_se5641148757651400278ts_bit @ int @ L @ ( minus_minus @ nat @ N @ M2 ) ) ) ) ) ).
% bit_concat_bit_iff
thf(fact_3401_vebt__member_Osimps_I4_J,axiom,
! [V2: product_prod @ nat @ nat,Vb: list @ vEBT_VEBT,Vc: vEBT_VEBT,X: nat] :
~ ( vEBT_vebt_member @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V2 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb @ Vc ) @ X ) ).
% vebt_member.simps(4)
thf(fact_3402_signed__take__bit__eq__concat__bit,axiom,
( ( bit_ri4674362597316999326ke_bit @ int )
= ( ^ [N5: nat,K3: int] : ( bit_concat_bit @ N5 @ K3 @ ( uminus_uminus @ int @ ( zero_neq_one_of_bool @ int @ ( bit_se5641148757651400278ts_bit @ int @ K3 @ N5 ) ) ) ) ) ) ).
% signed_take_bit_eq_concat_bit
thf(fact_3403_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_3404_exp__eq__0__imp__not__bit,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [N: nat,A3: A] :
( ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N )
= ( zero_zero @ A ) )
=> ~ ( bit_se5641148757651400278ts_bit @ A @ A3 @ N ) ) ) ).
% exp_eq_0_imp_not_bit
thf(fact_3405_bit__Suc,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ A3 @ ( suc @ N ) )
= ( bit_se5641148757651400278ts_bit @ A @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ N ) ) ) ).
% bit_Suc
thf(fact_3406_stable__imp__bit__iff__odd,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A,N: nat] :
( ( ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= A3 )
=> ( ( bit_se5641148757651400278ts_bit @ A @ A3 @ N )
= ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) ) ) ) ) ).
% stable_imp_bit_iff_odd
thf(fact_3407_bit__iff__idd__imp__stable,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A] :
( ! [N2: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ A3 @ N2 )
= ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) ) )
=> ( ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= A3 ) ) ) ).
% bit_iff_idd_imp_stable
thf(fact_3408_VEBT__internal_OminNull_Oelims_I1_J,axiom,
! [X: vEBT_VEBT,Y2: $o] :
( ( ( vEBT_VEBT_minNull @ X )
= Y2 )
=> ( ( ( X
= ( vEBT_Leaf @ $false @ $false ) )
=> ~ Y2 )
=> ( ( ? [Uv2: $o] :
( X
= ( vEBT_Leaf @ $true @ Uv2 ) )
=> Y2 )
=> ( ( ? [Uu2: $o] :
( X
= ( vEBT_Leaf @ Uu2 @ $true ) )
=> Y2 )
=> ( ( ? [Uw2: nat,Ux2: list @ vEBT_VEBT,Uy: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uw2 @ Ux2 @ Uy ) )
=> ~ Y2 )
=> ~ ( ? [Uz2: product_prod @ nat @ nat,Va3: nat,Vb2: list @ vEBT_VEBT,Vc2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) )
=> Y2 ) ) ) ) ) ) ).
% VEBT_internal.minNull.elims(1)
thf(fact_3409_take__bit__eq__mask__iff,axiom,
! [N: nat,K: int] :
( ( ( bit_se2584673776208193580ke_bit @ int @ N @ K )
= ( bit_se2239418461657761734s_mask @ int @ N ) )
= ( ( bit_se2584673776208193580ke_bit @ int @ N @ ( plus_plus @ int @ K @ ( one_one @ int ) ) )
= ( zero_zero @ int ) ) ) ).
% take_bit_eq_mask_iff
thf(fact_3410_int__bit__bound,axiom,
! [K: int] :
~ ! [N2: nat] :
( ! [M: nat] :
( ( ord_less_eq @ nat @ N2 @ M )
=> ( ( bit_se5641148757651400278ts_bit @ int @ K @ M )
= ( bit_se5641148757651400278ts_bit @ int @ K @ N2 ) ) )
=> ~ ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( bit_se5641148757651400278ts_bit @ int @ K @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) )
= ( ~ ( bit_se5641148757651400278ts_bit @ int @ K @ N2 ) ) ) ) ) ).
% int_bit_bound
thf(fact_3411_num_Osize__gen_I1_J,axiom,
( ( size_num @ one2 )
= ( zero_zero @ nat ) ) ).
% num.size_gen(1)
thf(fact_3412_bit__iff__odd,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ( ( bit_se5641148757651400278ts_bit @ A )
= ( ^ [A5: A,N5: nat] :
~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A5 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N5 ) ) ) ) ) ) ).
% bit_iff_odd
thf(fact_3413_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_3414_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_3415_bit__int__def,axiom,
( ( bit_se5641148757651400278ts_bit @ int )
= ( ^ [K3: int,N5: nat] :
~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( divide_divide @ int @ K3 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N5 ) ) ) ) ) ).
% bit_int_def
thf(fact_3416_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_3417_even__bit__succ__iff,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A,N: nat] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
=> ( ( bit_se5641148757651400278ts_bit @ A @ ( plus_plus @ A @ ( one_one @ A ) @ A3 ) @ N )
= ( ( bit_se5641148757651400278ts_bit @ A @ A3 @ N )
| ( N
= ( zero_zero @ nat ) ) ) ) ) ) ).
% even_bit_succ_iff
thf(fact_3418_odd__bit__iff__bit__pred,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A,N: nat] :
( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 )
=> ( ( bit_se5641148757651400278ts_bit @ A @ A3 @ N )
= ( ( bit_se5641148757651400278ts_bit @ A @ ( minus_minus @ A @ A3 @ ( one_one @ A ) ) @ N )
| ( N
= ( zero_zero @ nat ) ) ) ) ) ) ).
% odd_bit_iff_bit_pred
thf(fact_3419_mask__nat__def,axiom,
( ( bit_se2239418461657761734s_mask @ nat )
= ( ^ [N5: nat] : ( minus_minus @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N5 ) @ ( one_one @ nat ) ) ) ) ).
% mask_nat_def
thf(fact_3420_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_3421_mask__int__def,axiom,
( ( bit_se2239418461657761734s_mask @ int )
= ( ^ [N5: nat] : ( minus_minus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N5 ) @ ( one_one @ int ) ) ) ) ).
% mask_int_def
thf(fact_3422_bit__sum__mult__2__cases,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A,B2: A,N: nat] :
( ! [J: nat] :
~ ( bit_se5641148757651400278ts_bit @ A @ A3 @ ( suc @ J ) )
=> ( ( bit_se5641148757651400278ts_bit @ A @ ( plus_plus @ A @ A3 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) @ N )
= ( ( ( N
= ( zero_zero @ nat ) )
=> ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ( bit_se5641148757651400278ts_bit @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) @ N ) ) ) ) ) ) ).
% bit_sum_mult_2_cases
thf(fact_3423_bit__rec,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ( ( bit_se5641148757651400278ts_bit @ A )
= ( ^ [A5: A,N5: nat] :
( ( ( N5
= ( zero_zero @ nat ) )
=> ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A5 ) )
& ( ( N5
!= ( zero_zero @ nat ) )
=> ( bit_se5641148757651400278ts_bit @ A @ ( divide_divide @ A @ A5 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( minus_minus @ nat @ N5 @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% bit_rec
thf(fact_3424_mask__eq__exp__minus__1,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se2239418461657761734s_mask @ A )
= ( ^ [N5: nat] : ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N5 ) @ ( one_one @ A ) ) ) ) ) ).
% mask_eq_exp_minus_1
thf(fact_3425_invar__vebt_Ointros_I4_J,axiom,
! [TreeList: list @ vEBT_VEBT,N: nat,Summary: vEBT_VEBT,M2: nat,Deg: nat,Mi2: nat,Ma: nat] :
( ! [X3: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X3 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ( vEBT_invar_vebt @ X3 @ N ) )
=> ( ( vEBT_invar_vebt @ Summary @ M2 )
=> ( ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M2 ) )
=> ( ( M2 = N )
=> ( ( Deg
= ( plus_plus @ nat @ N @ M2 ) )
=> ( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M2 ) )
=> ( ( ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ I3 ) @ X7 ) )
= ( vEBT_V8194947554948674370ptions @ Summary @ I3 ) ) )
=> ( ( ( Mi2 = Ma )
=> ! [X3: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X3 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X_1 ) ) )
=> ( ( ord_less_eq @ nat @ Mi2 @ Ma )
=> ( ( ord_less @ nat @ Ma @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg ) )
=> ( ( ( Mi2 != Ma )
=> ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M2 ) )
=> ( ( ( ( vEBT_VEBT_high @ Ma @ N )
= I3 )
=> ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ I3 ) @ ( vEBT_VEBT_low @ Ma @ N ) ) )
& ! [X3: nat] :
( ( ( ( vEBT_VEBT_high @ X3 @ N )
= I3 )
& ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ I3 ) @ ( vEBT_VEBT_low @ X3 @ N ) ) )
=> ( ( ord_less @ nat @ Mi2 @ X3 )
& ( ord_less_eq @ nat @ X3 @ Ma ) ) ) ) ) )
=> ( vEBT_invar_vebt @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma ) ) @ Deg @ TreeList @ Summary ) @ Deg ) ) ) ) ) ) ) ) ) ) ) ).
% invar_vebt.intros(4)
thf(fact_3426_invar__vebt_Ointros_I5_J,axiom,
! [TreeList: list @ vEBT_VEBT,N: nat,Summary: vEBT_VEBT,M2: nat,Deg: nat,Mi2: nat,Ma: nat] :
( ! [X3: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X3 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ( vEBT_invar_vebt @ X3 @ N ) )
=> ( ( vEBT_invar_vebt @ Summary @ M2 )
=> ( ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M2 ) )
=> ( ( M2
= ( suc @ N ) )
=> ( ( Deg
= ( plus_plus @ nat @ N @ M2 ) )
=> ( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M2 ) )
=> ( ( ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ I3 ) @ X7 ) )
= ( vEBT_V8194947554948674370ptions @ Summary @ I3 ) ) )
=> ( ( ( Mi2 = Ma )
=> ! [X3: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X3 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X_1 ) ) )
=> ( ( ord_less_eq @ nat @ Mi2 @ Ma )
=> ( ( ord_less @ nat @ Ma @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg ) )
=> ( ( ( Mi2 != Ma )
=> ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M2 ) )
=> ( ( ( ( vEBT_VEBT_high @ Ma @ N )
= I3 )
=> ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ I3 ) @ ( vEBT_VEBT_low @ Ma @ N ) ) )
& ! [X3: nat] :
( ( ( ( vEBT_VEBT_high @ X3 @ N )
= I3 )
& ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ I3 ) @ ( vEBT_VEBT_low @ X3 @ N ) ) )
=> ( ( ord_less @ nat @ Mi2 @ X3 )
& ( ord_less_eq @ nat @ X3 @ Ma ) ) ) ) ) )
=> ( vEBT_invar_vebt @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma ) ) @ Deg @ TreeList @ Summary ) @ Deg ) ) ) ) ) ) ) ) ) ) ) ).
% invar_vebt.intros(5)
thf(fact_3427_unset__bit__eq,axiom,
( ( bit_se2638667681897837118et_bit @ int )
= ( ^ [N5: nat,K3: int] : ( minus_minus @ int @ K3 @ ( times_times @ int @ ( zero_neq_one_of_bool @ int @ ( bit_se5641148757651400278ts_bit @ int @ K3 @ N5 ) ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N5 ) ) ) ) ) ).
% unset_bit_eq
thf(fact_3428_take__bit__Suc__from__most,axiom,
! [N: nat,K: int] :
( ( bit_se2584673776208193580ke_bit @ int @ ( suc @ N ) @ K )
= ( plus_plus @ int @ ( times_times @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) @ ( zero_neq_one_of_bool @ int @ ( bit_se5641148757651400278ts_bit @ int @ K @ N ) ) ) @ ( bit_se2584673776208193580ke_bit @ int @ N @ K ) ) ) ).
% take_bit_Suc_from_most
thf(fact_3429_take__bit__eq__mask__iff__exp__dvd,axiom,
! [N: nat,K: int] :
( ( ( bit_se2584673776208193580ke_bit @ int @ N @ K )
= ( bit_se2239418461657761734s_mask @ int @ N ) )
= ( dvd_dvd @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) @ ( plus_plus @ int @ K @ ( one_one @ int ) ) ) ) ).
% take_bit_eq_mask_iff_exp_dvd
thf(fact_3430_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_3431_invar__vebt_Osimps,axiom,
( vEBT_invar_vebt
= ( ^ [A12: vEBT_VEBT,A23: nat] :
( ( ? [A5: $o,B3: $o] :
( A12
= ( vEBT_Leaf @ A5 @ B3 ) )
& ( A23
= ( suc @ ( zero_zero @ nat ) ) ) )
| ? [TreeList3: list @ vEBT_VEBT,N5: nat,Summary3: vEBT_VEBT] :
( ( A12
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ A23 @ TreeList3 @ Summary3 ) )
& ! [X2: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
=> ( vEBT_invar_vebt @ X2 @ N5 ) )
& ( vEBT_invar_vebt @ Summary3 @ N5 )
& ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N5 ) )
& ( A23
= ( plus_plus @ nat @ N5 @ N5 ) )
& ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X7 )
& ! [X2: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
=> ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X7 ) ) )
| ? [TreeList3: list @ vEBT_VEBT,N5: nat,Summary3: vEBT_VEBT] :
( ( A12
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ A23 @ TreeList3 @ Summary3 ) )
& ! [X2: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
=> ( vEBT_invar_vebt @ X2 @ N5 ) )
& ( vEBT_invar_vebt @ Summary3 @ ( suc @ N5 ) )
& ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ N5 ) ) )
& ( A23
= ( plus_plus @ nat @ N5 @ ( suc @ N5 ) ) )
& ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X7 )
& ! [X2: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
=> ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X7 ) ) )
| ? [TreeList3: list @ vEBT_VEBT,N5: nat,Summary3: vEBT_VEBT,Mi3: nat,Ma3: nat] :
( ( A12
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ Ma3 ) ) @ A23 @ TreeList3 @ Summary3 ) )
& ! [X2: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
=> ( vEBT_invar_vebt @ X2 @ N5 ) )
& ( vEBT_invar_vebt @ Summary3 @ N5 )
& ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N5 ) )
& ( A23
= ( plus_plus @ nat @ N5 @ N5 ) )
& ! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N5 ) )
=> ( ( ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList3 @ I2 ) @ X7 ) )
= ( vEBT_V8194947554948674370ptions @ Summary3 @ I2 ) ) )
& ( ( Mi3 = Ma3 )
=> ! [X2: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
=> ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X7 ) ) )
& ( ord_less_eq @ nat @ Mi3 @ Ma3 )
& ( ord_less @ nat @ Ma3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ A23 ) )
& ( ( Mi3 != Ma3 )
=> ! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N5 ) )
=> ( ( ( ( vEBT_VEBT_high @ Ma3 @ N5 )
= I2 )
=> ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList3 @ I2 ) @ ( vEBT_VEBT_low @ Ma3 @ N5 ) ) )
& ! [X2: nat] :
( ( ( ( vEBT_VEBT_high @ X2 @ N5 )
= I2 )
& ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList3 @ I2 ) @ ( vEBT_VEBT_low @ X2 @ N5 ) ) )
=> ( ( ord_less @ nat @ Mi3 @ X2 )
& ( ord_less_eq @ nat @ X2 @ Ma3 ) ) ) ) ) ) )
| ? [TreeList3: list @ vEBT_VEBT,N5: nat,Summary3: vEBT_VEBT,Mi3: nat,Ma3: nat] :
( ( A12
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi3 @ Ma3 ) ) @ A23 @ TreeList3 @ Summary3 ) )
& ! [X2: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
=> ( vEBT_invar_vebt @ X2 @ N5 ) )
& ( vEBT_invar_vebt @ Summary3 @ ( suc @ N5 ) )
& ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ N5 ) ) )
& ( A23
= ( plus_plus @ nat @ N5 @ ( suc @ N5 ) ) )
& ! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ N5 ) ) )
=> ( ( ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList3 @ I2 ) @ X7 ) )
= ( vEBT_V8194947554948674370ptions @ Summary3 @ I2 ) ) )
& ( ( Mi3 = Ma3 )
=> ! [X2: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
=> ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X7 ) ) )
& ( ord_less_eq @ nat @ Mi3 @ Ma3 )
& ( ord_less @ nat @ Ma3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ A23 ) )
& ( ( Mi3 != Ma3 )
=> ! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ N5 ) ) )
=> ( ( ( ( vEBT_VEBT_high @ Ma3 @ N5 )
= I2 )
=> ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList3 @ I2 ) @ ( vEBT_VEBT_low @ Ma3 @ N5 ) ) )
& ! [X2: nat] :
( ( ( ( vEBT_VEBT_high @ X2 @ N5 )
= I2 )
& ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList3 @ I2 ) @ ( vEBT_VEBT_low @ X2 @ N5 ) ) )
=> ( ( ord_less @ nat @ Mi3 @ X2 )
& ( ord_less_eq @ nat @ X2 @ Ma3 ) ) ) ) ) ) ) ) ) ) ).
% invar_vebt.simps
thf(fact_3432_divmod__step__eq,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [L: num,R4: A,Q5: A] :
( ( ( ord_less_eq @ A @ ( numeral_numeral @ A @ L ) @ R4 )
=> ( ( unique1321980374590559556d_step @ A @ L @ ( product_Pair @ A @ A @ Q5 @ R4 ) )
= ( product_Pair @ A @ A @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Q5 ) @ ( one_one @ A ) ) @ ( minus_minus @ A @ R4 @ ( numeral_numeral @ A @ L ) ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ L ) @ R4 )
=> ( ( unique1321980374590559556d_step @ A @ L @ ( product_Pair @ A @ A @ Q5 @ R4 ) )
= ( product_Pair @ A @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Q5 ) @ R4 ) ) ) ) ) ).
% divmod_step_eq
thf(fact_3433_Diff__eq__empty__iff,axiom,
! [A: $tType,A4: set @ A,B7: set @ A] :
( ( ( minus_minus @ ( set @ A ) @ A4 @ B7 )
= ( bot_bot @ ( set @ A ) ) )
= ( ord_less_eq @ ( set @ A ) @ A4 @ B7 ) ) ).
% Diff_eq_empty_iff
thf(fact_3434_divides__aux__eq,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [Q5: A,R4: A] :
( ( unique5940410009612947441es_aux @ A @ ( product_Pair @ A @ A @ Q5 @ R4 ) )
= ( R4
= ( zero_zero @ A ) ) ) ) ).
% divides_aux_eq
thf(fact_3435_product__nth,axiom,
! [A: $tType,B: $tType,N: nat,Xs: list @ A,Ys2: list @ B] :
( ( ord_less @ nat @ N @ ( times_times @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ B ) @ Ys2 ) ) )
=> ( ( nth @ ( product_prod @ A @ B ) @ ( product @ A @ B @ Xs @ Ys2 ) @ N )
= ( product_Pair @ A @ B @ ( nth @ A @ Xs @ ( divide_divide @ nat @ N @ ( size_size @ ( list @ B ) @ Ys2 ) ) ) @ ( nth @ B @ Ys2 @ ( modulo_modulo @ nat @ N @ ( size_size @ ( list @ B ) @ Ys2 ) ) ) ) ) ) ).
% product_nth
thf(fact_3436_and__int__unfold,axiom,
( ( bit_se5824344872417868541ns_and @ int )
= ( ^ [K3: int,L2: int] :
( if @ int
@ ( ( K3
= ( zero_zero @ int ) )
| ( L2
= ( zero_zero @ int ) ) )
@ ( zero_zero @ int )
@ ( if @ int
@ ( K3
= ( uminus_uminus @ int @ ( one_one @ int ) ) )
@ L2
@ ( if @ int
@ ( L2
= ( 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 @ L2 @ ( 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 @ L2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ).
% and_int_unfold
thf(fact_3437_Compl__anti__mono,axiom,
! [A: $tType,A4: set @ A,B7: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B7 )
=> ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ B7 ) @ ( uminus_uminus @ ( set @ A ) @ A4 ) ) ) ).
% Compl_anti_mono
thf(fact_3438_Compl__subset__Compl__iff,axiom,
! [A: $tType,A4: set @ A,B7: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A4 ) @ ( uminus_uminus @ ( set @ A ) @ B7 ) )
= ( ord_less_eq @ ( set @ A ) @ B7 @ A4 ) ) ).
% Compl_subset_Compl_iff
thf(fact_3439_Diff__idemp,axiom,
! [A: $tType,A4: set @ A,B7: set @ A] :
( ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A4 @ B7 ) @ B7 )
= ( minus_minus @ ( set @ A ) @ A4 @ B7 ) ) ).
% Diff_idemp
thf(fact_3440_Diff__iff,axiom,
! [A: $tType,C2: A,A4: set @ A,B7: set @ A] :
( ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A4 @ B7 ) )
= ( ( member @ A @ C2 @ A4 )
& ~ ( member @ A @ C2 @ B7 ) ) ) ).
% Diff_iff
thf(fact_3441_DiffI,axiom,
! [A: $tType,C2: A,A4: set @ A,B7: set @ A] :
( ( member @ A @ C2 @ A4 )
=> ( ~ ( member @ A @ C2 @ B7 )
=> ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A4 @ B7 ) ) ) ) ).
% DiffI
thf(fact_3442_Diff__cancel,axiom,
! [A: $tType,A4: set @ A] :
( ( minus_minus @ ( set @ A ) @ A4 @ A4 )
= ( bot_bot @ ( set @ A ) ) ) ).
% Diff_cancel
thf(fact_3443_empty__Diff,axiom,
! [A: $tType,A4: set @ A] :
( ( minus_minus @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A4 )
= ( bot_bot @ ( set @ A ) ) ) ).
% empty_Diff
thf(fact_3444_Diff__empty,axiom,
! [A: $tType,A4: set @ A] :
( ( minus_minus @ ( set @ A ) @ A4 @ ( bot_bot @ ( set @ A ) ) )
= A4 ) ).
% Diff_empty
thf(fact_3445_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_3446_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_3447_zero__and__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A] :
( ( bit_se5824344872417868541ns_and @ A @ ( zero_zero @ A ) @ A3 )
= ( zero_zero @ A ) ) ) ).
% zero_and_eq
thf(fact_3448_and__zero__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A] :
( ( bit_se5824344872417868541ns_and @ A @ A3 @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% and_zero_eq
thf(fact_3449_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_3450_and_Oleft__neutral,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [A3: A] :
( ( bit_se5824344872417868541ns_and @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ A3 )
= A3 ) ) ).
% and.left_neutral
thf(fact_3451_and_Oright__neutral,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [A3: A] :
( ( bit_se5824344872417868541ns_and @ A @ A3 @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= A3 ) ) ).
% and.right_neutral
thf(fact_3452_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_3453_and__nonnegative__int__iff,axiom,
! [K: int,L: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se5824344872417868541ns_and @ int @ K @ L ) )
= ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
| ( ord_less_eq @ int @ ( zero_zero @ int ) @ L ) ) ) ).
% and_nonnegative_int_iff
thf(fact_3454_and__negative__int__iff,axiom,
! [K: int,L: int] :
( ( ord_less @ int @ ( bit_se5824344872417868541ns_and @ int @ K @ L ) @ ( zero_zero @ int ) )
= ( ( ord_less @ int @ K @ ( zero_zero @ int ) )
& ( ord_less @ int @ L @ ( zero_zero @ int ) ) ) ) ).
% and_negative_int_iff
thf(fact_3455_length__product,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys2: list @ B] :
( ( size_size @ ( list @ ( product_prod @ A @ B ) ) @ ( product @ A @ B @ Xs @ Ys2 ) )
= ( times_times @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ B ) @ Ys2 ) ) ) ).
% length_product
thf(fact_3456_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_3457_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_3458_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_3459_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_3460_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_3461_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_3462_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_3463_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_3464_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_3465_bot__nat__def,axiom,
( ( bot_bot @ nat )
= ( zero_zero @ nat ) ) ).
% bot_nat_def
thf(fact_3466_DiffD2,axiom,
! [A: $tType,C2: A,A4: set @ A,B7: set @ A] :
( ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A4 @ B7 ) )
=> ~ ( member @ A @ C2 @ B7 ) ) ).
% DiffD2
thf(fact_3467_DiffD1,axiom,
! [A: $tType,C2: A,A4: set @ A,B7: set @ A] :
( ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A4 @ B7 ) )
=> ( member @ A @ C2 @ A4 ) ) ).
% DiffD1
thf(fact_3468_DiffE,axiom,
! [A: $tType,C2: A,A4: set @ A,B7: set @ A] :
( ( member @ A @ C2 @ ( minus_minus @ ( set @ A ) @ A4 @ B7 ) )
=> ~ ( ( member @ A @ C2 @ A4 )
=> ( member @ A @ C2 @ B7 ) ) ) ).
% DiffE
thf(fact_3469_of__nat__and__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M2: nat,N: nat] :
( ( semiring_1_of_nat @ A @ ( bit_se5824344872417868541ns_and @ nat @ M2 @ N ) )
= ( bit_se5824344872417868541ns_and @ A @ ( semiring_1_of_nat @ A @ M2 ) @ ( semiring_1_of_nat @ A @ N ) ) ) ) ).
% of_nat_and_eq
thf(fact_3470_and__eq__minus__1__iff,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [A3: A,B2: A] :
( ( ( bit_se5824344872417868541ns_and @ A @ A3 @ B2 )
= ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( ( A3
= ( uminus_uminus @ A @ ( one_one @ A ) ) )
& ( B2
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ) ).
% and_eq_minus_1_iff
thf(fact_3471_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_3472_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_3473_AND__upper2_H,axiom,
! [Y2: int,Z2: int,X: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y2 )
=> ( ( ord_less_eq @ int @ Y2 @ Z2 )
=> ( ord_less_eq @ int @ ( bit_se5824344872417868541ns_and @ int @ X @ Y2 ) @ Z2 ) ) ) ).
% AND_upper2'
thf(fact_3474_AND__upper1_H,axiom,
! [Y2: int,Z2: int,Ya: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y2 )
=> ( ( ord_less_eq @ int @ Y2 @ Z2 )
=> ( ord_less_eq @ int @ ( bit_se5824344872417868541ns_and @ int @ Y2 @ Ya ) @ Z2 ) ) ) ).
% AND_upper1'
thf(fact_3475_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_3476_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_3477_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_3478_VEBT__internal_Ovalid_H_Ocases,axiom,
! [X: product_prod @ vEBT_VEBT @ nat] :
( ! [Uu2: $o,Uv2: $o,D5: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ D5 ) )
=> ~ ! [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_3479_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_3480_and__less__eq,axiom,
! [L: int,K: int] :
( ( ord_less @ int @ L @ ( zero_zero @ int ) )
=> ( ord_less_eq @ int @ ( bit_se5824344872417868541ns_and @ int @ K @ L ) @ K ) ) ).
% and_less_eq
thf(fact_3481_AND__upper1_H_H,axiom,
! [Y2: int,Z2: int,Ya: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y2 )
=> ( ( ord_less @ int @ Y2 @ Z2 )
=> ( ord_less @ int @ ( bit_se5824344872417868541ns_and @ int @ Y2 @ Ya ) @ Z2 ) ) ) ).
% AND_upper1''
thf(fact_3482_AND__upper2_H_H,axiom,
! [Y2: int,Z2: int,X: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y2 )
=> ( ( ord_less @ int @ Y2 @ Z2 )
=> ( ord_less @ int @ ( bit_se5824344872417868541ns_and @ int @ X @ Y2 ) @ Z2 ) ) ) ).
% AND_upper2''
thf(fact_3483_Diff__mono,axiom,
! [A: $tType,A4: set @ A,C5: set @ A,D6: set @ A,B7: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ C5 )
=> ( ( ord_less_eq @ ( set @ A ) @ D6 @ B7 )
=> ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A4 @ B7 ) @ ( minus_minus @ ( set @ A ) @ C5 @ D6 ) ) ) ) ).
% Diff_mono
thf(fact_3484_Diff__subset,axiom,
! [A: $tType,A4: set @ A,B7: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A4 @ B7 ) @ A4 ) ).
% Diff_subset
thf(fact_3485_double__diff,axiom,
! [A: $tType,A4: set @ A,B7: set @ A,C5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B7 )
=> ( ( ord_less_eq @ ( set @ A ) @ B7 @ C5 )
=> ( ( minus_minus @ ( set @ A ) @ B7 @ ( minus_minus @ ( set @ A ) @ C5 @ A4 ) )
= A4 ) ) ) ).
% double_diff
thf(fact_3486_psubset__imp__ex__mem,axiom,
! [A: $tType,A4: set @ A,B7: set @ A] :
( ( ord_less @ ( set @ A ) @ A4 @ B7 )
=> ? [B4: A] : ( member @ A @ B4 @ ( minus_minus @ ( set @ A ) @ B7 @ A4 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_3487_bit__nat__iff,axiom,
! [K: int,N: nat] :
( ( bit_se5641148757651400278ts_bit @ nat @ ( nat2 @ K ) @ N )
= ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
& ( bit_se5641148757651400278ts_bit @ int @ K @ N ) ) ) ).
% bit_nat_iff
thf(fact_3488_VEBT__internal_Onaive__member_Ocases,axiom,
! [X: product_prod @ vEBT_VEBT @ nat] :
( ! [A6: $o,B4: $o,X3: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A6 @ B4 ) @ X3 ) )
=> ( ! [Uu2: option @ ( product_prod @ nat @ nat ),Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT,Ux2: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Uu2 @ ( zero_zero @ nat ) @ Uv2 @ Uw2 ) @ Ux2 ) )
=> ~ ! [Uy: option @ ( product_prod @ nat @ nat ),V4: nat,TreeList2: list @ vEBT_VEBT,S2: vEBT_VEBT,X3: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Uy @ ( suc @ V4 ) @ TreeList2 @ S2 ) @ X3 ) ) ) ) ).
% VEBT_internal.naive_member.cases
thf(fact_3489_and__one__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A] :
( ( bit_se5824344872417868541ns_and @ A @ A3 @ ( one_one @ A ) )
= ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% and_one_eq
thf(fact_3490_one__and__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A] :
( ( bit_se5824344872417868541ns_and @ A @ ( one_one @ A ) @ A3 )
= ( modulo_modulo @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% one_and_eq
thf(fact_3491_and__exp__eq__0__iff__not__bit,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A,N: nat] :
( ( ( bit_se5824344872417868541ns_and @ A @ A3 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
= ( zero_zero @ A ) )
= ( ~ ( bit_se5641148757651400278ts_bit @ A @ A3 @ N ) ) ) ) ).
% and_exp_eq_0_iff_not_bit
thf(fact_3492_bit__nat__def,axiom,
( ( bit_se5641148757651400278ts_bit @ nat )
= ( ^ [M5: nat,N5: nat] :
~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ M5 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N5 ) ) ) ) ) ).
% bit_nat_def
thf(fact_3493_subset__Compl__self__eq,axiom,
! [A: $tType,A4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ ( uminus_uminus @ ( set @ A ) @ A4 ) )
= ( A4
= ( bot_bot @ ( set @ A ) ) ) ) ).
% subset_Compl_self_eq
thf(fact_3494_vebt__member_Ocases,axiom,
! [X: product_prod @ vEBT_VEBT @ nat] :
( ! [A6: $o,B4: $o,X3: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A6 @ B4 ) @ X3 ) )
=> ( ! [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT,X3: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) @ X3 ) )
=> ( ! [V4: product_prod @ nat @ nat,Uy: list @ vEBT_VEBT,Uz2: vEBT_VEBT,X3: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( zero_zero @ nat ) @ Uy @ Uz2 ) @ X3 ) )
=> ( ! [V4: product_prod @ nat @ nat,Vb2: list @ vEBT_VEBT,Vc2: vEBT_VEBT,X3: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb2 @ Vc2 ) @ X3 ) )
=> ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT,X3: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) @ X3 ) ) ) ) ) ) ).
% vebt_member.cases
thf(fact_3495_VEBT__internal_Omembermima_Ocases,axiom,
! [X: product_prod @ vEBT_VEBT @ nat] :
( ! [Uu2: $o,Uv2: $o,Uw2: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Uw2 ) )
=> ( ! [Ux2: list @ vEBT_VEBT,Uy: vEBT_VEBT,Uz2: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( zero_zero @ nat ) @ Ux2 @ Uy ) @ Uz2 ) )
=> ( ! [Mi: nat,Ma2: nat,Va3: list @ vEBT_VEBT,Vb2: vEBT_VEBT,X3: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma2 ) ) @ ( zero_zero @ nat ) @ Va3 @ Vb2 ) @ X3 ) )
=> ( ! [Mi: nat,Ma2: nat,V4: nat,TreeList2: list @ vEBT_VEBT,Vc2: vEBT_VEBT,X3: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma2 ) ) @ ( suc @ V4 ) @ TreeList2 @ Vc2 ) @ X3 ) )
=> ~ ! [V4: nat,TreeList2: list @ vEBT_VEBT,Vd: vEBT_VEBT,X3: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V4 ) @ TreeList2 @ Vd ) @ X3 ) ) ) ) ) ) ).
% VEBT_internal.membermima.cases
thf(fact_3496_and__int__rec,axiom,
( ( bit_se5824344872417868541ns_and @ int )
= ( ^ [K3: int,L2: 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 ) ) @ L2 ) ) )
@ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( divide_divide @ int @ K3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% and_int_rec
thf(fact_3497_divmod__algorithm__code_I7_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M2: num,N: num] :
( ( ( ord_less_eq @ num @ M2 @ N )
=> ( ( unique8689654367752047608divmod @ A @ ( bit0 @ M2 ) @ ( bit1 @ N ) )
= ( product_Pair @ A @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ ( bit0 @ M2 ) ) ) ) )
& ( ~ ( ord_less_eq @ num @ M2 @ N )
=> ( ( unique8689654367752047608divmod @ A @ ( bit0 @ M2 ) @ ( bit1 @ N ) )
= ( unique1321980374590559556d_step @ A @ ( bit1 @ N ) @ ( unique8689654367752047608divmod @ A @ ( bit0 @ M2 ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ) ).
% divmod_algorithm_code(7)
thf(fact_3498_divmod__algorithm__code_I8_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M2: num,N: num] :
( ( ( ord_less @ num @ M2 @ N )
=> ( ( unique8689654367752047608divmod @ A @ ( bit1 @ M2 ) @ ( bit1 @ N ) )
= ( product_Pair @ A @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ ( bit1 @ M2 ) ) ) ) )
& ( ~ ( ord_less @ num @ M2 @ N )
=> ( ( unique8689654367752047608divmod @ A @ ( bit1 @ M2 ) @ ( bit1 @ N ) )
= ( unique1321980374590559556d_step @ A @ ( bit1 @ N ) @ ( unique8689654367752047608divmod @ A @ ( bit1 @ M2 ) @ ( bit0 @ ( bit1 @ N ) ) ) ) ) ) ) ) ).
% divmod_algorithm_code(8)
thf(fact_3499_neg__eucl__rel__int__mult__2,axiom,
! [B2: int,A3: int,Q5: int,R4: int] :
( ( ord_less_eq @ int @ B2 @ ( zero_zero @ int ) )
=> ( ( eucl_rel_int @ ( plus_plus @ int @ A3 @ ( one_one @ int ) ) @ B2 @ ( product_Pair @ int @ int @ Q5 @ R4 ) )
=> ( eucl_rel_int @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ A3 ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ B2 ) @ ( product_Pair @ int @ int @ Q5 @ ( minus_minus @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ R4 ) @ ( one_one @ int ) ) ) ) ) ) ).
% neg_eucl_rel_int_mult_2
thf(fact_3500_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_3501_and__int_Osimps,axiom,
( ( bit_se5824344872417868541ns_and @ int )
= ( ^ [K3: int,L2: int] :
( if @ int
@ ( ( member @ int @ K3 @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
& ( member @ int @ L2 @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) )
@ ( uminus_uminus @ int
@ ( zero_neq_one_of_bool @ int
@ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K3 )
& ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ L2 ) ) ) )
@ ( plus_plus @ int
@ ( zero_neq_one_of_bool @ int
@ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K3 )
& ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ L2 ) ) )
@ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( divide_divide @ int @ K3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ).
% and_int.simps
thf(fact_3502_ComplI,axiom,
! [A: $tType,C2: A,A4: set @ A] :
( ~ ( member @ A @ C2 @ A4 )
=> ( member @ A @ C2 @ ( uminus_uminus @ ( set @ A ) @ A4 ) ) ) ).
% ComplI
thf(fact_3503_Compl__iff,axiom,
! [A: $tType,C2: A,A4: set @ A] :
( ( member @ A @ C2 @ ( uminus_uminus @ ( set @ A ) @ A4 ) )
= ( ~ ( member @ A @ C2 @ A4 ) ) ) ).
% Compl_iff
thf(fact_3504_Compl__eq__Compl__iff,axiom,
! [A: $tType,A4: set @ A,B7: set @ A] :
( ( ( uminus_uminus @ ( set @ A ) @ A4 )
= ( uminus_uminus @ ( set @ A ) @ B7 ) )
= ( A4 = B7 ) ) ).
% Compl_eq_Compl_iff
thf(fact_3505_Diff__insert0,axiom,
! [A: $tType,X: A,A4: set @ A,B7: set @ A] :
( ~ ( member @ A @ X @ A4 )
=> ( ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ X @ B7 ) )
= ( minus_minus @ ( set @ A ) @ A4 @ B7 ) ) ) ).
% Diff_insert0
thf(fact_3506_insert__Diff1,axiom,
! [A: $tType,X: A,B7: set @ A,A4: set @ A] :
( ( member @ A @ X @ B7 )
=> ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X @ A4 ) @ B7 )
= ( minus_minus @ ( set @ A ) @ A4 @ B7 ) ) ) ).
% insert_Diff1
thf(fact_3507_insert__Diff__single,axiom,
! [A: $tType,A3: A,A4: set @ A] :
( ( insert @ A @ A3 @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( insert @ A @ A3 @ A4 ) ) ).
% insert_Diff_single
thf(fact_3508_subset__Compl__singleton,axiom,
! [A: $tType,A4: set @ A,B2: A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ ( uminus_uminus @ ( set @ A ) @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( ~ ( member @ A @ B2 @ A4 ) ) ) ).
% subset_Compl_singleton
thf(fact_3509_divmod__algorithm__code_I2_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M2: num] :
( ( unique8689654367752047608divmod @ A @ M2 @ one2 )
= ( product_Pair @ A @ A @ ( numeral_numeral @ A @ M2 ) @ ( zero_zero @ A ) ) ) ) ).
% divmod_algorithm_code(2)
thf(fact_3510_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_3511_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_3512_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_3513_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_3514_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_3515_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_3516_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_3517_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_3518_insert__Diff__if,axiom,
! [A: $tType,X: A,B7: set @ A,A4: set @ A] :
( ( ( member @ A @ X @ B7 )
=> ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X @ A4 ) @ B7 )
= ( minus_minus @ ( set @ A ) @ A4 @ B7 ) ) )
& ( ~ ( member @ A @ X @ B7 )
=> ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X @ A4 ) @ B7 )
= ( insert @ A @ X @ ( minus_minus @ ( set @ A ) @ A4 @ B7 ) ) ) ) ) ).
% insert_Diff_if
thf(fact_3519_ComplD,axiom,
! [A: $tType,C2: A,A4: set @ A] :
( ( member @ A @ C2 @ ( uminus_uminus @ ( set @ A ) @ A4 ) )
=> ~ ( member @ A @ C2 @ A4 ) ) ).
% ComplD
thf(fact_3520_double__complement,axiom,
! [A: $tType,A4: set @ A] :
( ( uminus_uminus @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A4 ) )
= A4 ) ).
% double_complement
thf(fact_3521_eucl__rel__int__by0,axiom,
! [K: int] : ( eucl_rel_int @ K @ ( zero_zero @ int ) @ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ K ) ) ).
% eucl_rel_int_by0
thf(fact_3522_div__int__unique,axiom,
! [K: int,L: int,Q5: int,R4: int] :
( ( eucl_rel_int @ K @ L @ ( product_Pair @ int @ int @ Q5 @ R4 ) )
=> ( ( divide_divide @ int @ K @ L )
= Q5 ) ) ).
% div_int_unique
thf(fact_3523_Diff__insert__absorb,axiom,
! [A: $tType,X: A,A4: set @ A] :
( ~ ( member @ A @ X @ A4 )
=> ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X @ A4 ) @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= A4 ) ) ).
% Diff_insert_absorb
thf(fact_3524_Diff__insert2,axiom,
! [A: $tType,A4: set @ A,A3: A,B7: set @ A] :
( ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ A3 @ B7 ) )
= ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) @ B7 ) ) ).
% Diff_insert2
thf(fact_3525_insert__Diff,axiom,
! [A: $tType,A3: A,A4: set @ A] :
( ( member @ A @ A3 @ A4 )
=> ( ( insert @ A @ A3 @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) )
= A4 ) ) ).
% insert_Diff
thf(fact_3526_Diff__insert,axiom,
! [A: $tType,A4: set @ A,A3: A,B7: set @ A] :
( ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ A3 @ B7 ) )
= ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A4 @ B7 ) @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Diff_insert
thf(fact_3527_Compl__insert,axiom,
! [A: $tType,X: A,A4: set @ A] :
( ( uminus_uminus @ ( set @ A ) @ ( insert @ A @ X @ A4 ) )
= ( minus_minus @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A4 ) @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Compl_insert
thf(fact_3528_subset__Diff__insert,axiom,
! [A: $tType,A4: set @ A,B7: set @ A,X: A,C5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ ( minus_minus @ ( set @ A ) @ B7 @ ( insert @ A @ X @ C5 ) ) )
= ( ( ord_less_eq @ ( set @ A ) @ A4 @ ( minus_minus @ ( set @ A ) @ B7 @ C5 ) )
& ~ ( member @ A @ X @ A4 ) ) ) ).
% subset_Diff_insert
thf(fact_3529_and__nat__def,axiom,
( ( bit_se5824344872417868541ns_and @ nat )
= ( ^ [M5: nat,N5: nat] : ( nat2 @ ( bit_se5824344872417868541ns_and @ int @ ( semiring_1_of_nat @ int @ M5 ) @ ( semiring_1_of_nat @ int @ N5 ) ) ) ) ) ).
% and_nat_def
thf(fact_3530_eucl__rel__int__dividesI,axiom,
! [L: int,K: int,Q5: int] :
( ( L
!= ( zero_zero @ int ) )
=> ( ( K
= ( times_times @ int @ Q5 @ L ) )
=> ( eucl_rel_int @ K @ L @ ( product_Pair @ int @ int @ Q5 @ ( zero_zero @ int ) ) ) ) ) ).
% eucl_rel_int_dividesI
thf(fact_3531_Diff__single__insert,axiom,
! [A: $tType,A4: set @ A,X: A,B7: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ B7 )
=> ( ord_less_eq @ ( set @ A ) @ A4 @ ( insert @ A @ X @ B7 ) ) ) ).
% Diff_single_insert
thf(fact_3532_subset__insert__iff,axiom,
! [A: $tType,A4: set @ A,X: A,B7: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ ( insert @ A @ X @ B7 ) )
= ( ( ( member @ A @ X @ A4 )
=> ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ B7 ) )
& ( ~ ( member @ A @ X @ A4 )
=> ( ord_less_eq @ ( set @ A ) @ A4 @ B7 ) ) ) ) ).
% subset_insert_iff
thf(fact_3533_eucl__rel__int,axiom,
! [K: int,L: int] : ( eucl_rel_int @ K @ L @ ( product_Pair @ int @ int @ ( divide_divide @ int @ K @ L ) @ ( modulo_modulo @ int @ K @ L ) ) ) ).
% eucl_rel_int
thf(fact_3534_divmod__int__def,axiom,
( ( unique8689654367752047608divmod @ int )
= ( ^ [M5: num,N5: num] : ( product_Pair @ int @ int @ ( divide_divide @ int @ ( numeral_numeral @ int @ M5 ) @ ( numeral_numeral @ int @ N5 ) ) @ ( modulo_modulo @ int @ ( numeral_numeral @ int @ M5 ) @ ( numeral_numeral @ int @ N5 ) ) ) ) ) ).
% divmod_int_def
thf(fact_3535_psubset__insert__iff,axiom,
! [A: $tType,A4: set @ A,X: A,B7: set @ A] :
( ( ord_less @ ( set @ A ) @ A4 @ ( insert @ A @ X @ B7 ) )
= ( ( ( member @ A @ X @ B7 )
=> ( ord_less @ ( set @ A ) @ A4 @ B7 ) )
& ( ~ ( member @ A @ X @ B7 )
=> ( ( ( member @ A @ X @ A4 )
=> ( ord_less @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ B7 ) )
& ( ~ ( member @ A @ X @ A4 )
=> ( ord_less_eq @ ( set @ A ) @ A4 @ B7 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_3536_atLeastAtMostPlus1__int__conv,axiom,
! [M2: int,N: int] :
( ( ord_less_eq @ int @ M2 @ ( plus_plus @ int @ ( one_one @ int ) @ N ) )
=> ( ( set_or1337092689740270186AtMost @ int @ M2 @ ( plus_plus @ int @ ( one_one @ int ) @ N ) )
= ( insert @ int @ ( plus_plus @ int @ ( one_one @ int ) @ N ) @ ( set_or1337092689740270186AtMost @ int @ M2 @ N ) ) ) ) ).
% atLeastAtMostPlus1_int_conv
thf(fact_3537_simp__from__to,axiom,
( ( set_or1337092689740270186AtMost @ int )
= ( ^ [I2: int,J3: int] : ( if @ ( set @ int ) @ ( ord_less @ int @ J3 @ I2 ) @ ( bot_bot @ ( set @ int ) ) @ ( insert @ int @ I2 @ ( set_or1337092689740270186AtMost @ int @ ( plus_plus @ int @ I2 @ ( one_one @ int ) ) @ J3 ) ) ) ) ) ).
% simp_from_to
thf(fact_3538_divmod__def,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ( ( unique8689654367752047608divmod @ A )
= ( ^ [M5: num,N5: num] : ( product_Pair @ A @ A @ ( divide_divide @ A @ ( numeral_numeral @ A @ M5 ) @ ( numeral_numeral @ A @ N5 ) ) @ ( modulo_modulo @ A @ ( numeral_numeral @ A @ M5 ) @ ( numeral_numeral @ A @ N5 ) ) ) ) ) ) ).
% divmod_def
thf(fact_3539_divmod_H__nat__def,axiom,
( ( unique8689654367752047608divmod @ nat )
= ( ^ [M5: num,N5: num] : ( product_Pair @ nat @ nat @ ( divide_divide @ nat @ ( numeral_numeral @ nat @ M5 ) @ ( numeral_numeral @ nat @ N5 ) ) @ ( modulo_modulo @ nat @ ( numeral_numeral @ nat @ M5 ) @ ( numeral_numeral @ nat @ N5 ) ) ) ) ) ).
% divmod'_nat_def
thf(fact_3540_zminus1__lemma,axiom,
! [A3: int,B2: int,Q5: int,R4: int] :
( ( eucl_rel_int @ A3 @ B2 @ ( product_Pair @ int @ int @ Q5 @ R4 ) )
=> ( ( B2
!= ( zero_zero @ int ) )
=> ( eucl_rel_int @ ( uminus_uminus @ int @ A3 ) @ B2
@ ( product_Pair @ int @ int
@ ( if @ int
@ ( R4
= ( zero_zero @ int ) )
@ ( uminus_uminus @ int @ Q5 )
@ ( minus_minus @ int @ ( uminus_uminus @ int @ Q5 ) @ ( one_one @ int ) ) )
@ ( if @ int
@ ( R4
= ( zero_zero @ int ) )
@ ( zero_zero @ int )
@ ( minus_minus @ int @ B2 @ R4 ) ) ) ) ) ) ).
% zminus1_lemma
thf(fact_3541_eucl__rel__int__iff,axiom,
! [K: int,L: int,Q5: int,R4: int] :
( ( eucl_rel_int @ K @ L @ ( product_Pair @ int @ int @ Q5 @ R4 ) )
= ( ( K
= ( plus_plus @ int @ ( times_times @ int @ L @ Q5 ) @ R4 ) )
& ( ( ord_less @ int @ ( zero_zero @ int ) @ L )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ R4 )
& ( ord_less @ int @ R4 @ L ) ) )
& ( ~ ( ord_less @ int @ ( zero_zero @ int ) @ L )
=> ( ( ( ord_less @ int @ L @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ L @ R4 )
& ( ord_less_eq @ int @ R4 @ ( zero_zero @ int ) ) ) )
& ( ~ ( ord_less @ int @ L @ ( zero_zero @ int ) )
=> ( Q5
= ( zero_zero @ int ) ) ) ) ) ) ) ).
% eucl_rel_int_iff
thf(fact_3542_and__nat__unfold,axiom,
( ( bit_se5824344872417868541ns_and @ nat )
= ( ^ [M5: nat,N5: nat] :
( if @ nat
@ ( ( M5
= ( zero_zero @ nat ) )
| ( N5
= ( zero_zero @ nat ) ) )
@ ( zero_zero @ nat )
@ ( plus_plus @ nat @ ( times_times @ nat @ ( modulo_modulo @ nat @ M5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ nat @ N5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ nat @ ( divide_divide @ nat @ M5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ nat @ N5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ).
% and_nat_unfold
thf(fact_3543_eucl__rel__int__remainderI,axiom,
! [R4: int,L: int,K: int,Q5: int] :
( ( ( sgn_sgn @ int @ R4 )
= ( sgn_sgn @ int @ L ) )
=> ( ( ord_less @ int @ ( abs_abs @ int @ R4 ) @ ( abs_abs @ int @ L ) )
=> ( ( K
= ( plus_plus @ int @ ( times_times @ int @ Q5 @ L ) @ R4 ) )
=> ( eucl_rel_int @ K @ L @ ( product_Pair @ int @ int @ Q5 @ R4 ) ) ) ) ) ).
% eucl_rel_int_remainderI
thf(fact_3544_and__nat__rec,axiom,
( ( bit_se5824344872417868541ns_and @ nat )
= ( ^ [M5: nat,N5: nat] :
( plus_plus @ nat
@ ( zero_neq_one_of_bool @ nat
@ ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M5 )
& ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N5 ) ) )
@ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ nat @ ( divide_divide @ nat @ M5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ nat @ N5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% and_nat_rec
thf(fact_3545_eucl__rel__int_Osimps,axiom,
( eucl_rel_int
= ( ^ [A12: int,A23: int,A32: product_prod @ int @ int] :
( ? [K3: int] :
( ( A12 = K3 )
& ( A23
= ( zero_zero @ int ) )
& ( A32
= ( product_Pair @ int @ int @ ( zero_zero @ int ) @ K3 ) ) )
| ? [L2: int,K3: int,Q4: int] :
( ( A12 = K3 )
& ( A23 = L2 )
& ( A32
= ( product_Pair @ int @ int @ Q4 @ ( zero_zero @ int ) ) )
& ( L2
!= ( zero_zero @ int ) )
& ( K3
= ( times_times @ int @ Q4 @ L2 ) ) )
| ? [R: int,L2: int,K3: int,Q4: int] :
( ( A12 = K3 )
& ( A23 = L2 )
& ( A32
= ( product_Pair @ int @ int @ Q4 @ R ) )
& ( ( sgn_sgn @ int @ R )
= ( sgn_sgn @ int @ L2 ) )
& ( ord_less @ int @ ( abs_abs @ int @ R ) @ ( abs_abs @ int @ L2 ) )
& ( K3
= ( plus_plus @ int @ ( times_times @ int @ Q4 @ L2 ) @ R ) ) ) ) ) ) ).
% eucl_rel_int.simps
thf(fact_3546_eucl__rel__int_Ocases,axiom,
! [A1: int,A22: int,A33: product_prod @ int @ int] :
( ( eucl_rel_int @ A1 @ A22 @ A33 )
=> ( ( ( A22
= ( zero_zero @ int ) )
=> ( A33
!= ( product_Pair @ int @ int @ ( zero_zero @ int ) @ A1 ) ) )
=> ( ! [Q3: int] :
( ( A33
= ( product_Pair @ int @ int @ Q3 @ ( zero_zero @ int ) ) )
=> ( ( A22
!= ( zero_zero @ int ) )
=> ( A1
!= ( times_times @ int @ Q3 @ A22 ) ) ) )
=> ~ ! [R3: int,Q3: int] :
( ( A33
= ( product_Pair @ int @ int @ Q3 @ R3 ) )
=> ( ( ( sgn_sgn @ int @ R3 )
= ( sgn_sgn @ int @ A22 ) )
=> ( ( ord_less @ int @ ( abs_abs @ int @ R3 ) @ ( abs_abs @ int @ A22 ) )
=> ( A1
!= ( plus_plus @ int @ ( times_times @ int @ Q3 @ A22 ) @ R3 ) ) ) ) ) ) ) ) ).
% eucl_rel_int.cases
thf(fact_3547_divmod__divmod__step,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ( ( unique8689654367752047608divmod @ A )
= ( ^ [M5: num,N5: num] : ( if @ ( product_prod @ A @ A ) @ ( ord_less @ num @ M5 @ N5 ) @ ( product_Pair @ A @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ M5 ) ) @ ( unique1321980374590559556d_step @ A @ N5 @ ( unique8689654367752047608divmod @ A @ M5 @ ( bit0 @ N5 ) ) ) ) ) ) ) ).
% divmod_divmod_step
thf(fact_3548_pos__eucl__rel__int__mult__2,axiom,
! [B2: int,A3: int,Q5: int,R4: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ B2 )
=> ( ( eucl_rel_int @ A3 @ B2 @ ( product_Pair @ int @ int @ Q5 @ R4 ) )
=> ( eucl_rel_int @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ A3 ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ B2 ) @ ( product_Pair @ int @ int @ Q5 @ ( plus_plus @ int @ ( one_one @ int ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ R4 ) ) ) ) ) ) ).
% pos_eucl_rel_int_mult_2
thf(fact_3549_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_3550_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_3551_Divides_Oadjust__div__eq,axiom,
! [Q5: int,R4: int] :
( ( adjust_div @ ( product_Pair @ int @ int @ Q5 @ R4 ) )
= ( plus_plus @ int @ Q5
@ ( zero_neq_one_of_bool @ int
@ ( R4
!= ( zero_zero @ int ) ) ) ) ) ).
% Divides.adjust_div_eq
thf(fact_3552_and__int_Opsimps,axiom,
! [K: int,L: int] :
( ( accp @ ( product_prod @ int @ int ) @ bit_and_int_rel @ ( product_Pair @ int @ int @ K @ L ) )
=> ( ( ( ( member @ int @ K @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
& ( member @ int @ L @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) )
=> ( ( bit_se5824344872417868541ns_and @ int @ K @ L )
= ( uminus_uminus @ int
@ ( zero_neq_one_of_bool @ int
@ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K )
& ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ L ) ) ) ) ) )
& ( ~ ( ( member @ int @ K @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
& ( member @ int @ L @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) )
=> ( ( bit_se5824344872417868541ns_and @ int @ K @ L )
= ( plus_plus @ int
@ ( zero_neq_one_of_bool @ int
@ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K )
& ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ L ) ) )
@ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( divide_divide @ int @ K @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% and_int.psimps
thf(fact_3553_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_3554_minus__numeral__div__numeral,axiom,
! [M2: num,N: num] :
( ( divide_divide @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M2 ) ) @ ( numeral_numeral @ int @ N ) )
= ( uminus_uminus @ int @ ( adjust_div @ ( unique8689654367752047608divmod @ int @ M2 @ N ) ) ) ) ).
% minus_numeral_div_numeral
thf(fact_3555_numeral__div__minus__numeral,axiom,
! [M2: num,N: num] :
( ( divide_divide @ int @ ( numeral_numeral @ int @ M2 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
= ( uminus_uminus @ int @ ( adjust_div @ ( unique8689654367752047608divmod @ int @ M2 @ N ) ) ) ) ).
% numeral_div_minus_numeral
thf(fact_3556_atLeast0__atMost__Suc,axiom,
! [N: nat] :
( ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) )
= ( insert @ nat @ ( suc @ N ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) ) ).
% atLeast0_atMost_Suc
thf(fact_3557_atLeastAtMost__insertL,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( insert @ nat @ M2 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M2 ) @ N ) )
= ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ).
% atLeastAtMost_insertL
thf(fact_3558_atLeastAtMostSuc__conv,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq @ nat @ M2 @ ( suc @ N ) )
=> ( ( set_or1337092689740270186AtMost @ nat @ M2 @ ( suc @ N ) )
= ( insert @ nat @ ( suc @ N ) @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ).
% atLeastAtMostSuc_conv
thf(fact_3559_Icc__eq__insert__lb__nat,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( set_or1337092689740270186AtMost @ nat @ M2 @ N )
= ( insert @ nat @ M2 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M2 ) @ N ) ) ) ) ).
% Icc_eq_insert_lb_nat
thf(fact_3560_set__decode__plus__power__2,axiom,
! [N: nat,Z2: nat] :
( ~ ( member @ nat @ N @ ( nat_set_decode @ Z2 ) )
=> ( ( nat_set_decode @ ( plus_plus @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ Z2 ) )
= ( insert @ nat @ N @ ( nat_set_decode @ Z2 ) ) ) ) ).
% set_decode_plus_power_2
thf(fact_3561_and__int_Opinduct,axiom,
! [A0: int,A1: int,P: int > int > $o] :
( ( accp @ ( product_prod @ int @ int ) @ bit_and_int_rel @ ( product_Pair @ int @ int @ A0 @ A1 ) )
=> ( ! [K2: int,L3: int] :
( ( accp @ ( product_prod @ int @ int ) @ bit_and_int_rel @ ( product_Pair @ int @ int @ K2 @ L3 ) )
=> ( ( ~ ( ( member @ int @ K2 @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
& ( member @ int @ L3 @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) )
=> ( P @ ( divide_divide @ int @ K2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) )
=> ( P @ K2 @ L3 ) ) )
=> ( P @ A0 @ A1 ) ) ) ).
% and_int.pinduct
thf(fact_3562_upto_Opinduct,axiom,
! [A0: int,A1: int,P: int > int > $o] :
( ( accp @ ( product_prod @ int @ int ) @ upto_rel @ ( product_Pair @ int @ int @ A0 @ A1 ) )
=> ( ! [I3: int,J: int] :
( ( accp @ ( product_prod @ int @ int ) @ upto_rel @ ( product_Pair @ int @ int @ I3 @ J ) )
=> ( ( ( ord_less_eq @ int @ I3 @ J )
=> ( P @ ( plus_plus @ int @ I3 @ ( one_one @ int ) ) @ J ) )
=> ( P @ I3 @ J ) ) )
=> ( P @ A0 @ A1 ) ) ) ).
% upto.pinduct
thf(fact_3563_divmod__BitM__2__eq,axiom,
! [M2: num] :
( ( unique8689654367752047608divmod @ int @ ( bitM @ M2 ) @ ( bit0 @ one2 ) )
= ( product_Pair @ int @ int @ ( minus_minus @ int @ ( numeral_numeral @ int @ M2 ) @ ( one_one @ int ) ) @ ( one_one @ int ) ) ) ).
% divmod_BitM_2_eq
thf(fact_3564_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_3565_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_3566_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_3567_sinh__real__eq__iff,axiom,
! [X: real,Y2: real] :
( ( ( sinh @ real @ X )
= ( sinh @ real @ Y2 ) )
= ( X = Y2 ) ) ).
% sinh_real_eq_iff
thf(fact_3568_sinh__0,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ( ( sinh @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% sinh_0
thf(fact_3569_sinh__minus,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X: A] :
( ( sinh @ A @ ( uminus_uminus @ A @ X ) )
= ( uminus_uminus @ A @ ( sinh @ A @ X ) ) ) ) ).
% sinh_minus
thf(fact_3570_sinh__real__zero__iff,axiom,
! [X: real] :
( ( ( sinh @ real @ X )
= ( zero_zero @ real ) )
= ( X
= ( zero_zero @ real ) ) ) ).
% sinh_real_zero_iff
thf(fact_3571_sinh__real__less__iff,axiom,
! [X: real,Y2: real] :
( ( ord_less @ real @ ( sinh @ real @ X ) @ ( sinh @ real @ Y2 ) )
= ( ord_less @ real @ X @ Y2 ) ) ).
% sinh_real_less_iff
thf(fact_3572_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_3573_sinh__real__abs,axiom,
! [X: real] :
( ( sinh @ real @ ( abs_abs @ real @ X ) )
= ( abs_abs @ real @ ( sinh @ real @ X ) ) ) ).
% sinh_real_abs
thf(fact_3574_sinh__real__neg__iff,axiom,
! [X: real] :
( ( ord_less @ real @ ( sinh @ real @ X ) @ ( zero_zero @ real ) )
= ( ord_less @ real @ X @ ( zero_zero @ real ) ) ) ).
% sinh_real_neg_iff
thf(fact_3575_sinh__real__pos__iff,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( sinh @ real @ X ) )
= ( ord_less @ real @ ( zero_zero @ real ) @ X ) ) ).
% sinh_real_pos_iff
thf(fact_3576_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_3577_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_3578_dbl__dec__simps_I5_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [K: num] :
( ( neg_numeral_dbl_dec @ A @ ( numeral_numeral @ A @ K ) )
= ( numeral_numeral @ A @ ( bitM @ K ) ) ) ) ).
% dbl_dec_simps(5)
thf(fact_3579_pred__numeral__simps_I2_J,axiom,
! [K: num] :
( ( pred_numeral @ ( bit0 @ K ) )
= ( numeral_numeral @ nat @ ( bitM @ K ) ) ) ).
% pred_numeral_simps(2)
thf(fact_3580_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_3581_minus__numeral__mod__numeral,axiom,
! [M2: num,N: num] :
( ( modulo_modulo @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M2 ) ) @ ( numeral_numeral @ int @ N ) )
= ( adjust_mod @ ( numeral_numeral @ int @ N ) @ ( product_snd @ int @ int @ ( unique8689654367752047608divmod @ int @ M2 @ N ) ) ) ) ).
% minus_numeral_mod_numeral
thf(fact_3582_numeral__mod__minus__numeral,axiom,
! [M2: num,N: num] :
( ( modulo_modulo @ int @ ( numeral_numeral @ int @ M2 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
= ( uminus_uminus @ int @ ( adjust_mod @ ( numeral_numeral @ int @ N ) @ ( product_snd @ int @ int @ ( unique8689654367752047608divmod @ int @ M2 @ N ) ) ) ) ) ).
% numeral_mod_minus_numeral
thf(fact_3583_arsinh__sinh__real,axiom,
! [X: real] :
( ( arsinh @ real @ ( sinh @ real @ X ) )
= X ) ).
% arsinh_sinh_real
thf(fact_3584_sinh__less__cosh__real,axiom,
! [X: real] : ( ord_less @ real @ ( sinh @ real @ X ) @ ( cosh @ real @ X ) ) ).
% sinh_less_cosh_real
thf(fact_3585_sinh__le__cosh__real,axiom,
! [X: real] : ( ord_less_eq @ real @ ( sinh @ real @ X ) @ ( cosh @ real @ X ) ) ).
% sinh_le_cosh_real
thf(fact_3586_inc__BitM__eq,axiom,
! [N: num] :
( ( inc @ ( bitM @ N ) )
= ( bit0 @ N ) ) ).
% inc_BitM_eq
thf(fact_3587_divides__aux__def,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ( ( unique5940410009612947441es_aux @ A )
= ( ^ [Qr: product_prod @ A @ A] :
( ( product_snd @ A @ A @ Qr )
= ( zero_zero @ A ) ) ) ) ) ).
% divides_aux_def
thf(fact_3588_BitM__inc__eq,axiom,
! [N: num] :
( ( bitM @ ( inc @ N ) )
= ( bit1 @ N ) ) ).
% BitM_inc_eq
thf(fact_3589_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_3590_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_3591_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_3592_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_3593_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_3594_BitM__plus__one,axiom,
! [N: num] :
( ( plus_plus @ num @ ( bitM @ N ) @ one2 )
= ( bit0 @ N ) ) ).
% BitM_plus_one
thf(fact_3595_one__plus__BitM,axiom,
! [N: num] :
( ( plus_plus @ num @ one2 @ ( bitM @ N ) )
= ( bit0 @ N ) ) ).
% one_plus_BitM
thf(fact_3596_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_3597_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_3598_tanh__def,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ( ( tanh @ A )
= ( ^ [X2: A] : ( divide_divide @ A @ ( sinh @ A @ X2 ) @ ( cosh @ A @ X2 ) ) ) ) ) ).
% tanh_def
thf(fact_3599_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_3600_cosh__minus__sinh,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X: A] :
( ( minus_minus @ A @ ( cosh @ A @ X ) @ ( sinh @ A @ X ) )
= ( exp @ A @ ( uminus_uminus @ A @ X ) ) ) ) ).
% cosh_minus_sinh
thf(fact_3601_sinh__minus__cosh,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X: A] :
( ( minus_minus @ A @ ( sinh @ A @ X ) @ ( cosh @ A @ X ) )
= ( uminus_uminus @ A @ ( exp @ A @ ( uminus_uminus @ A @ X ) ) ) ) ) ).
% sinh_minus_cosh
thf(fact_3602_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_3603_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_3604_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_3605_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_3606_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_3607_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_3608_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_3609_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_3610_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_3611_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_3612_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_3613_vebt__member_Oelims_I3_J,axiom,
! [X: vEBT_VEBT,Xa: nat] :
( ~ ( vEBT_vebt_member @ X @ Xa )
=> ( ! [A6: $o,B4: $o] :
( ( X
= ( vEBT_Leaf @ A6 @ B4 ) )
=> ( ( ( Xa
= ( zero_zero @ nat ) )
=> A6 )
& ( ( Xa
!= ( zero_zero @ nat ) )
=> ( ( ( Xa
= ( one_one @ nat ) )
=> B4 )
& ( Xa
= ( one_one @ nat ) ) ) ) ) )
=> ( ! [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] :
( X
!= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) )
=> ( ! [V4: product_prod @ nat @ nat,Uy: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( X
!= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( zero_zero @ nat ) @ Uy @ Uz2 ) )
=> ( ! [V4: product_prod @ nat @ nat,Vb2: list @ vEBT_VEBT,Vc2: vEBT_VEBT] :
( X
!= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb2 @ Vc2 ) )
=> ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList2: list @ vEBT_VEBT] :
( ? [Summary2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
=> ( ( Xa != Mi )
=> ( ( Xa != Ma2 )
=> ( ~ ( ord_less @ nat @ Xa @ Mi )
& ( ~ ( ord_less @ nat @ Xa @ Mi )
=> ( ~ ( ord_less @ nat @ Ma2 @ Xa )
& ( ~ ( ord_less @ nat @ Ma2 @ Xa )
=> ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_vebt_member @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_member.elims(3)
thf(fact_3614_set__vebt_H__def,axiom,
( vEBT_VEBT_set_vebt
= ( ^ [T3: vEBT_VEBT] : ( collect @ nat @ ( vEBT_vebt_member @ T3 ) ) ) ) ).
% set_vebt'_def
thf(fact_3615_push__bit__nonnegative__int__iff,axiom,
! [N: nat,K: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se4730199178511100633sh_bit @ int @ N @ K ) )
= ( ord_less_eq @ int @ ( zero_zero @ int ) @ K ) ) ).
% push_bit_nonnegative_int_iff
thf(fact_3616_push__bit__negative__int__iff,axiom,
! [N: nat,K: int] :
( ( ord_less @ int @ ( bit_se4730199178511100633sh_bit @ int @ N @ K ) @ ( zero_zero @ int ) )
= ( ord_less @ int @ K @ ( zero_zero @ int ) ) ) ).
% push_bit_negative_int_iff
thf(fact_3617_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_3618_xor__self__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A] :
( ( bit_se5824344971392196577ns_xor @ A @ A3 @ A3 )
= ( zero_zero @ A ) ) ) ).
% xor_self_eq
thf(fact_3619_xor_Oleft__neutral,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A] :
( ( bit_se5824344971392196577ns_xor @ A @ ( zero_zero @ A ) @ A3 )
= A3 ) ) ).
% xor.left_neutral
thf(fact_3620_xor_Oright__neutral,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A] :
( ( bit_se5824344971392196577ns_xor @ A @ A3 @ ( zero_zero @ A ) )
= A3 ) ) ).
% xor.right_neutral
thf(fact_3621_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_3622_push__bit__eq__0__iff,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [N: nat,A3: A] :
( ( ( bit_se4730199178511100633sh_bit @ A @ N @ A3 )
= ( zero_zero @ A ) )
= ( A3
= ( zero_zero @ A ) ) ) ) ).
% push_bit_eq_0_iff
thf(fact_3623_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_3624_case__nat__numeral,axiom,
! [A: $tType,A3: A,F2: nat > A,V2: num] :
( ( case_nat @ A @ A3 @ F2 @ ( numeral_numeral @ nat @ V2 ) )
= ( F2 @ ( pred_numeral @ V2 ) ) ) ).
% case_nat_numeral
thf(fact_3625_push__bit__Suc__numeral,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,K: num] :
( ( bit_se4730199178511100633sh_bit @ A @ ( suc @ N ) @ ( numeral_numeral @ A @ K ) )
= ( bit_se4730199178511100633sh_bit @ A @ N @ ( numeral_numeral @ A @ ( bit0 @ K ) ) ) ) ) ).
% push_bit_Suc_numeral
thf(fact_3626_case__nat__add__eq__if,axiom,
! [A: $tType,A3: A,F2: nat > A,V2: num,N: nat] :
( ( case_nat @ A @ A3 @ F2 @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ V2 ) @ N ) )
= ( F2 @ ( plus_plus @ nat @ ( pred_numeral @ V2 ) @ N ) ) ) ).
% case_nat_add_eq_if
thf(fact_3627_push__bit__Suc__minus__numeral,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: nat,K: num] :
( ( bit_se4730199178511100633sh_bit @ A @ ( suc @ N ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ K ) ) )
= ( bit_se4730199178511100633sh_bit @ A @ N @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( bit0 @ K ) ) ) ) ) ) ).
% push_bit_Suc_minus_numeral
thf(fact_3628_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_3629_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_3630_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_3631_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_3632_push__bit__Suc,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A3: A] :
( ( bit_se4730199178511100633sh_bit @ A @ ( suc @ N ) @ A3 )
= ( bit_se4730199178511100633sh_bit @ A @ N @ ( times_times @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% push_bit_Suc
thf(fact_3633_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_3634_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_3635_even__push__bit__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A3: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se4730199178511100633sh_bit @ A @ N @ A3 ) )
= ( ( N
!= ( zero_zero @ nat ) )
| ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) ) ) ) ).
% even_push_bit_iff
thf(fact_3636_push__bit__minus__numeral,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [L: num,K: num] :
( ( bit_se4730199178511100633sh_bit @ A @ ( numeral_numeral @ nat @ L ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ K ) ) )
= ( bit_se4730199178511100633sh_bit @ A @ ( pred_numeral @ L ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( bit0 @ K ) ) ) ) ) ) ).
% push_bit_minus_numeral
thf(fact_3637_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_3638_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_3639_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_3640_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_3641_Suc__0__mod__numeral,axiom,
! [K: num] :
( ( modulo_modulo @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( numeral_numeral @ nat @ K ) )
= ( product_snd @ nat @ nat @ ( unique8689654367752047608divmod @ nat @ one2 @ K ) ) ) ).
% Suc_0_mod_numeral
thf(fact_3642_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_3643_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_3644_Compl__eq,axiom,
! [A: $tType] :
( ( uminus_uminus @ ( set @ A ) )
= ( ^ [A7: set @ A] :
( collect @ A
@ ^ [X2: A] :
~ ( member @ A @ X2 @ A7 ) ) ) ) ).
% Compl_eq
thf(fact_3645_Collect__neg__eq,axiom,
! [A: $tType,P: A > $o] :
( ( collect @ A
@ ^ [X2: A] :
~ ( P @ X2 ) )
= ( uminus_uminus @ ( set @ A ) @ ( collect @ A @ P ) ) ) ).
% Collect_neg_eq
thf(fact_3646_uminus__set__def,axiom,
! [A: $tType] :
( ( uminus_uminus @ ( set @ A ) )
= ( ^ [A7: set @ A] :
( collect @ A
@ ( uminus_uminus @ ( A > $o )
@ ^ [X2: A] : ( member @ A @ X2 @ A7 ) ) ) ) ) ).
% uminus_set_def
thf(fact_3647_push__bit__minus,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: nat,A3: A] :
( ( bit_se4730199178511100633sh_bit @ A @ N @ ( uminus_uminus @ A @ A3 ) )
= ( uminus_uminus @ A @ ( bit_se4730199178511100633sh_bit @ A @ N @ A3 ) ) ) ) ).
% push_bit_minus
thf(fact_3648_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_3649_strict__subset__divisors__dvd,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ ( set @ A )
@ ( collect @ A
@ ^ [C4: A] : ( dvd_dvd @ A @ C4 @ A3 ) )
@ ( collect @ A
@ ^ [C4: A] : ( dvd_dvd @ A @ C4 @ B2 ) ) )
= ( ( dvd_dvd @ A @ A3 @ B2 )
& ~ ( dvd_dvd @ A @ B2 @ A3 ) ) ) ) ).
% strict_subset_divisors_dvd
thf(fact_3650_flip__bit__nat__def,axiom,
( ( bit_se8732182000553998342ip_bit @ nat )
= ( ^ [M5: nat,N5: nat] : ( bit_se5824344971392196577ns_xor @ nat @ N5 @ ( bit_se4730199178511100633sh_bit @ nat @ M5 @ ( one_one @ nat ) ) ) ) ) ).
% flip_bit_nat_def
thf(fact_3651_lambda__one,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ( ( ^ [X2: A] : X2 )
= ( times_times @ A @ ( one_one @ A ) ) ) ) ).
% lambda_one
thf(fact_3652_flip__bit__eq__xor,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se8732182000553998342ip_bit @ A )
= ( ^ [N5: nat,A5: A] : ( bit_se5824344971392196577ns_xor @ A @ A5 @ ( bit_se4730199178511100633sh_bit @ A @ N5 @ ( one_one @ A ) ) ) ) ) ) ).
% flip_bit_eq_xor
thf(fact_3653_nat_Ocase__distrib,axiom,
! [B: $tType,A: $tType,H: A > B,F1: A,F22: nat > A,Nat: nat] :
( ( H @ ( case_nat @ A @ F1 @ F22 @ Nat ) )
= ( case_nat @ B @ ( H @ F1 )
@ ^ [X2: nat] : ( H @ ( F22 @ X2 ) )
@ Nat ) ) ).
% nat.case_distrib
thf(fact_3654_push__bit__of__nat,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,M2: nat] :
( ( bit_se4730199178511100633sh_bit @ A @ N @ ( semiring_1_of_nat @ A @ M2 ) )
= ( semiring_1_of_nat @ A @ ( bit_se4730199178511100633sh_bit @ nat @ N @ M2 ) ) ) ) ).
% push_bit_of_nat
thf(fact_3655_of__nat__push__bit,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M2: nat,N: nat] :
( ( semiring_1_of_nat @ A @ ( bit_se4730199178511100633sh_bit @ nat @ M2 @ N ) )
= ( bit_se4730199178511100633sh_bit @ A @ M2 @ ( semiring_1_of_nat @ A @ N ) ) ) ) ).
% of_nat_push_bit
thf(fact_3656_of__nat__xor__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M2: nat,N: nat] :
( ( semiring_1_of_nat @ A @ ( bit_se5824344971392196577ns_xor @ nat @ M2 @ N ) )
= ( bit_se5824344971392196577ns_xor @ A @ ( semiring_1_of_nat @ A @ M2 ) @ ( semiring_1_of_nat @ A @ N ) ) ) ) ).
% of_nat_xor_eq
thf(fact_3657_nat_Odisc__eq__case_I2_J,axiom,
! [Nat: nat] :
( ( Nat
!= ( zero_zero @ nat ) )
= ( case_nat @ $o @ $false
@ ^ [Uu3: nat] : $true
@ Nat ) ) ).
% nat.disc_eq_case(2)
thf(fact_3658_nat_Odisc__eq__case_I1_J,axiom,
! [Nat: nat] :
( ( Nat
= ( zero_zero @ nat ) )
= ( case_nat @ $o @ $true
@ ^ [Uu3: nat] : $false
@ Nat ) ) ).
% nat.disc_eq_case(1)
thf(fact_3659_lambda__zero,axiom,
! [A: $tType] :
( ( mult_zero @ A )
=> ( ( ^ [H2: A] : ( zero_zero @ A ) )
= ( times_times @ A @ ( zero_zero @ A ) ) ) ) ).
% lambda_zero
thf(fact_3660_pred__def,axiom,
( pred
= ( case_nat @ nat @ ( zero_zero @ nat )
@ ^ [X25: nat] : X25 ) ) ).
% pred_def
thf(fact_3661_less__eq__nat_Osimps_I2_J,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq @ nat @ ( suc @ M2 ) @ N )
= ( case_nat @ $o @ $false @ ( ord_less_eq @ nat @ M2 ) @ N ) ) ).
% less_eq_nat.simps(2)
thf(fact_3662_set__diff__eq,axiom,
! [A: $tType] :
( ( minus_minus @ ( set @ A ) )
= ( ^ [A7: set @ A,B5: set @ A] :
( collect @ A
@ ^ [X2: A] :
( ( member @ A @ X2 @ A7 )
& ~ ( member @ A @ X2 @ B5 ) ) ) ) ) ).
% set_diff_eq
thf(fact_3663_minus__set__def,axiom,
! [A: $tType] :
( ( minus_minus @ ( set @ A ) )
= ( ^ [A7: set @ A,B5: set @ A] :
( collect @ A
@ ( minus_minus @ ( A > $o )
@ ^ [X2: A] : ( member @ A @ X2 @ A7 )
@ ^ [X2: A] : ( member @ A @ X2 @ B5 ) ) ) ) ) ).
% minus_set_def
thf(fact_3664_subset__divisors__dvd,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [C4: A] : ( dvd_dvd @ A @ C4 @ A3 ) )
@ ( collect @ A
@ ^ [C4: A] : ( dvd_dvd @ A @ C4 @ B2 ) ) )
= ( dvd_dvd @ A @ A3 @ B2 ) ) ) ).
% subset_divisors_dvd
thf(fact_3665_diff__Suc,axiom,
! [M2: nat,N: nat] :
( ( minus_minus @ nat @ M2 @ ( suc @ N ) )
= ( case_nat @ nat @ ( zero_zero @ nat )
@ ^ [K3: nat] : K3
@ ( minus_minus @ nat @ M2 @ N ) ) ) ).
% diff_Suc
thf(fact_3666_nat__less__as__int,axiom,
( ( ord_less @ nat )
= ( ^ [A5: nat,B3: nat] : ( ord_less @ int @ ( semiring_1_of_nat @ int @ A5 ) @ ( semiring_1_of_nat @ int @ B3 ) ) ) ) ).
% nat_less_as_int
thf(fact_3667_nat__leq__as__int,axiom,
( ( ord_less_eq @ nat )
= ( ^ [A5: nat,B3: nat] : ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ A5 ) @ ( semiring_1_of_nat @ int @ B3 ) ) ) ) ).
% nat_leq_as_int
thf(fact_3668_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_3669_set__vebt__def,axiom,
( vEBT_set_vebt
= ( ^ [T3: vEBT_VEBT] : ( collect @ nat @ ( vEBT_V8194947554948674370ptions @ T3 ) ) ) ) ).
% set_vebt_def
thf(fact_3670_nat__plus__as__int,axiom,
( ( plus_plus @ nat )
= ( ^ [A5: nat,B3: nat] : ( nat2 @ ( plus_plus @ int @ ( semiring_1_of_nat @ int @ A5 ) @ ( semiring_1_of_nat @ int @ B3 ) ) ) ) ) ).
% nat_plus_as_int
thf(fact_3671_nat__times__as__int,axiom,
( ( times_times @ nat )
= ( ^ [A5: nat,B3: nat] : ( nat2 @ ( times_times @ int @ ( semiring_1_of_nat @ int @ A5 ) @ ( semiring_1_of_nat @ int @ B3 ) ) ) ) ) ).
% nat_times_as_int
thf(fact_3672_nat__minus__as__int,axiom,
( ( minus_minus @ nat )
= ( ^ [A5: nat,B3: nat] : ( nat2 @ ( minus_minus @ int @ ( semiring_1_of_nat @ int @ A5 ) @ ( semiring_1_of_nat @ int @ B3 ) ) ) ) ) ).
% nat_minus_as_int
thf(fact_3673_nat__div__as__int,axiom,
( ( divide_divide @ nat )
= ( ^ [A5: nat,B3: nat] : ( nat2 @ ( divide_divide @ int @ ( semiring_1_of_nat @ int @ A5 ) @ ( semiring_1_of_nat @ int @ B3 ) ) ) ) ) ).
% nat_div_as_int
thf(fact_3674_nat__mod__as__int,axiom,
( ( modulo_modulo @ nat )
= ( ^ [A5: nat,B3: nat] : ( nat2 @ ( modulo_modulo @ int @ ( semiring_1_of_nat @ int @ A5 ) @ ( semiring_1_of_nat @ int @ B3 ) ) ) ) ) ).
% nat_mod_as_int
thf(fact_3675_signed__take__bit__code,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( ( bit_ri4674362597316999326ke_bit @ A )
= ( ^ [N5: nat,A5: A] : ( if @ A @ ( bit_se5641148757651400278ts_bit @ A @ ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N5 ) @ A5 ) @ N5 ) @ ( plus_plus @ A @ ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N5 ) @ A5 ) @ ( bit_se4730199178511100633sh_bit @ A @ ( suc @ N5 ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) @ ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N5 ) @ A5 ) ) ) ) ) ).
% signed_take_bit_code
thf(fact_3676_take__bit__push__bit,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M2: nat,N: nat,A3: A] :
( ( bit_se2584673776208193580ke_bit @ A @ M2 @ ( bit_se4730199178511100633sh_bit @ A @ N @ A3 ) )
= ( bit_se4730199178511100633sh_bit @ A @ N @ ( bit_se2584673776208193580ke_bit @ A @ ( minus_minus @ nat @ M2 @ N ) @ A3 ) ) ) ) ).
% take_bit_push_bit
thf(fact_3677_diff__nat__eq__if,axiom,
! [Z3: int,Z2: int] :
( ( ( ord_less @ int @ Z3 @ ( zero_zero @ int ) )
=> ( ( minus_minus @ nat @ ( nat2 @ Z2 ) @ ( nat2 @ Z3 ) )
= ( nat2 @ Z2 ) ) )
& ( ~ ( ord_less @ int @ Z3 @ ( zero_zero @ int ) )
=> ( ( minus_minus @ nat @ ( nat2 @ Z2 ) @ ( nat2 @ Z3 ) )
= ( if @ nat @ ( ord_less @ int @ ( minus_minus @ int @ Z2 @ Z3 ) @ ( zero_zero @ int ) ) @ ( zero_zero @ nat ) @ ( nat2 @ ( minus_minus @ int @ Z2 @ Z3 ) ) ) ) ) ) ).
% diff_nat_eq_if
thf(fact_3678_bit__push__bit__iff__int,axiom,
! [M2: nat,K: int,N: nat] :
( ( bit_se5641148757651400278ts_bit @ int @ ( bit_se4730199178511100633sh_bit @ int @ M2 @ K ) @ N )
= ( ( ord_less_eq @ nat @ M2 @ N )
& ( bit_se5641148757651400278ts_bit @ int @ K @ ( minus_minus @ nat @ N @ M2 ) ) ) ) ).
% bit_push_bit_iff_int
thf(fact_3679_set__decode__def,axiom,
( nat_set_decode
= ( ^ [X2: nat] :
( collect @ nat
@ ^ [N5: nat] :
~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ X2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N5 ) ) ) ) ) ) ).
% set_decode_def
thf(fact_3680_bit__push__bit__iff__nat,axiom,
! [M2: nat,Q5: nat,N: nat] :
( ( bit_se5641148757651400278ts_bit @ nat @ ( bit_se4730199178511100633sh_bit @ nat @ M2 @ Q5 ) @ N )
= ( ( ord_less_eq @ nat @ M2 @ N )
& ( bit_se5641148757651400278ts_bit @ nat @ Q5 @ ( minus_minus @ nat @ N @ M2 ) ) ) ) ).
% bit_push_bit_iff_nat
thf(fact_3681_pochhammer__code,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ( ( comm_s3205402744901411588hammer @ A )
= ( ^ [A5: A,N5: nat] :
( if @ A
@ ( N5
= ( 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 @ N5 @ ( one_one @ nat ) )
@ ( one_one @ A ) ) ) ) ) ) ).
% pochhammer_code
thf(fact_3682_bit__iff__and__push__bit__not__eq__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se5641148757651400278ts_bit @ A )
= ( ^ [A5: A,N5: nat] :
( ( bit_se5824344872417868541ns_and @ A @ A5 @ ( bit_se4730199178511100633sh_bit @ A @ N5 @ ( one_one @ A ) ) )
!= ( zero_zero @ A ) ) ) ) ) ).
% bit_iff_and_push_bit_not_eq_0
thf(fact_3683_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
@ ^ [L2: nat] : ( times_times @ A @ ( minus_minus @ A @ A5 @ ( semiring_1_of_nat @ A @ L2 ) ) )
@ ( zero_zero @ nat )
@ ( minus_minus @ nat @ K3 @ ( one_one @ nat ) )
@ ( one_one @ A ) )
@ ( semiring_char_0_fact @ A @ K3 ) ) ) ) ) ) ).
% gbinomial_code
thf(fact_3684_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_3685_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_3686_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_3687_VEBT__internal_Omembermima_Osimps_I4_J,axiom,
! [Mi2: nat,Ma: nat,V2: nat,TreeList: list @ vEBT_VEBT,Vc: vEBT_VEBT,X: nat] :
( ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma ) ) @ ( suc @ V2 ) @ TreeList @ Vc ) @ X )
= ( ( X = Mi2 )
| ( 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_3688_vebt__member_Osimps_I5_J,axiom,
! [Mi2: nat,Ma: nat,Va2: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
( ( vEBT_vebt_member @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList @ Summary ) @ X )
= ( ( X != Mi2 )
=> ( ( X != Ma )
=> ( ~ ( ord_less @ nat @ X @ Mi2 )
& ( ~ ( ord_less @ nat @ X @ Mi2 )
=> ( ~ ( ord_less @ nat @ Ma @ X )
& ( ~ ( ord_less @ nat @ Ma @ X )
=> ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
=> ( vEBT_vebt_member @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) ) ) ) ) ) ) ) ) ) ).
% vebt_member.simps(5)
thf(fact_3689_VEBT__internal_Onaive__member_Oelims_I3_J,axiom,
! [X: vEBT_VEBT,Xa: nat] :
( ~ ( vEBT_V5719532721284313246member @ X @ Xa )
=> ( ! [A6: $o,B4: $o] :
( ( X
= ( vEBT_Leaf @ A6 @ B4 ) )
=> ( ( ( Xa
= ( zero_zero @ nat ) )
=> A6 )
& ( ( Xa
!= ( zero_zero @ nat ) )
=> ( ( ( Xa
= ( one_one @ nat ) )
=> B4 )
& ( Xa
= ( one_one @ nat ) ) ) ) ) )
=> ( ! [Uu2: option @ ( product_prod @ nat @ nat ),Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] :
( X
!= ( vEBT_Node @ Uu2 @ ( zero_zero @ nat ) @ Uv2 @ Uw2 ) )
=> ~ ! [Uy: option @ ( product_prod @ nat @ nat ),V4: nat,TreeList2: list @ vEBT_VEBT] :
( ? [S2: vEBT_VEBT] :
( X
= ( vEBT_Node @ Uy @ ( suc @ V4 ) @ TreeList2 @ S2 ) )
=> ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( 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 @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ).
% VEBT_internal.naive_member.elims(3)
thf(fact_3690_VEBT__internal_Onaive__member_Oelims_I2_J,axiom,
! [X: vEBT_VEBT,Xa: nat] :
( ( vEBT_V5719532721284313246member @ X @ Xa )
=> ( ! [A6: $o,B4: $o] :
( ( X
= ( vEBT_Leaf @ A6 @ B4 ) )
=> ~ ( ( ( Xa
= ( zero_zero @ nat ) )
=> A6 )
& ( ( Xa
!= ( zero_zero @ nat ) )
=> ( ( ( Xa
= ( one_one @ nat ) )
=> B4 )
& ( Xa
= ( one_one @ nat ) ) ) ) ) )
=> ~ ! [Uy: option @ ( product_prod @ nat @ nat ),V4: nat,TreeList2: list @ vEBT_VEBT] :
( ? [S2: vEBT_VEBT] :
( X
= ( vEBT_Node @ Uy @ ( suc @ V4 ) @ TreeList2 @ S2 ) )
=> ~ ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( 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 @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ).
% VEBT_internal.naive_member.elims(2)
thf(fact_3691_VEBT__internal_Onaive__member_Oelims_I1_J,axiom,
! [X: vEBT_VEBT,Xa: nat,Y2: $o] :
( ( ( vEBT_V5719532721284313246member @ X @ Xa )
= Y2 )
=> ( ! [A6: $o,B4: $o] :
( ( X
= ( vEBT_Leaf @ A6 @ B4 ) )
=> ( Y2
= ( ~ ( ( ( Xa
= ( zero_zero @ nat ) )
=> A6 )
& ( ( Xa
!= ( zero_zero @ nat ) )
=> ( ( ( Xa
= ( one_one @ nat ) )
=> B4 )
& ( Xa
= ( one_one @ nat ) ) ) ) ) ) ) )
=> ( ( ? [Uu2: option @ ( product_prod @ nat @ nat ),Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] :
( X
= ( vEBT_Node @ Uu2 @ ( zero_zero @ nat ) @ Uv2 @ Uw2 ) )
=> Y2 )
=> ~ ! [Uy: option @ ( product_prod @ nat @ nat ),V4: nat,TreeList2: list @ vEBT_VEBT] :
( ? [S2: vEBT_VEBT] :
( X
= ( vEBT_Node @ Uy @ ( suc @ V4 ) @ TreeList2 @ S2 ) )
=> ( Y2
= ( ~ ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( 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 @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.naive_member.elims(1)
thf(fact_3692_VEBT__internal_Omembermima_Oelims_I2_J,axiom,
! [X: vEBT_VEBT,Xa: nat] :
( ( vEBT_VEBT_membermima @ X @ Xa )
=> ( ! [Mi: nat,Ma2: nat] :
( ? [Va3: list @ vEBT_VEBT,Vb2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma2 ) ) @ ( zero_zero @ nat ) @ Va3 @ Vb2 ) )
=> ~ ( ( Xa = Mi )
| ( Xa = Ma2 ) ) )
=> ( ! [Mi: nat,Ma2: nat,V4: nat,TreeList2: list @ vEBT_VEBT] :
( ? [Vc2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma2 ) ) @ ( suc @ V4 ) @ TreeList2 @ Vc2 ) )
=> ~ ( ( Xa = Mi )
| ( Xa = Ma2 )
| ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( 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 @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) )
=> ~ ! [V4: nat,TreeList2: list @ vEBT_VEBT] :
( ? [Vd: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V4 ) @ TreeList2 @ Vd ) )
=> ~ ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( 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 @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ).
% VEBT_internal.membermima.elims(2)
thf(fact_3693_xor__nat__unfold,axiom,
( ( bit_se5824344971392196577ns_xor @ nat )
= ( ^ [M5: nat,N5: nat] :
( if @ nat
@ ( M5
= ( zero_zero @ nat ) )
@ N5
@ ( if @ nat
@ ( N5
= ( zero_zero @ nat ) )
@ M5
@ ( plus_plus @ nat @ ( modulo_modulo @ nat @ ( plus_plus @ nat @ ( modulo_modulo @ nat @ M5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ nat @ N5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( bit_se5824344971392196577ns_xor @ nat @ ( divide_divide @ nat @ M5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ nat @ N5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% xor_nat_unfold
thf(fact_3694_xor__nat__rec,axiom,
( ( bit_se5824344971392196577ns_xor @ nat )
= ( ^ [M5: nat,N5: nat] :
( plus_plus @ nat
@ ( zero_neq_one_of_bool @ nat
@ ( ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M5 ) )
!= ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N5 ) ) ) )
@ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( bit_se5824344971392196577ns_xor @ nat @ ( divide_divide @ nat @ M5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ nat @ N5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% xor_nat_rec
thf(fact_3695_xor__one__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A] :
( ( bit_se5824344971392196577ns_xor @ A @ A3 @ ( one_one @ A ) )
= ( minus_minus @ A @ ( plus_plus @ A @ A3 @ ( zero_neq_one_of_bool @ A @ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) ) )
@ ( zero_neq_one_of_bool @ A
@ ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) ) ) ) ) ).
% xor_one_eq
thf(fact_3696_one__xor__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A] :
( ( bit_se5824344971392196577ns_xor @ A @ ( one_one @ A ) @ A3 )
= ( minus_minus @ A @ ( plus_plus @ A @ A3 @ ( zero_neq_one_of_bool @ A @ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) ) )
@ ( zero_neq_one_of_bool @ A
@ ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) ) ) ) ) ).
% one_xor_eq
thf(fact_3697_vebt__member_Oelims_I2_J,axiom,
! [X: vEBT_VEBT,Xa: nat] :
( ( vEBT_vebt_member @ X @ Xa )
=> ( ! [A6: $o,B4: $o] :
( ( X
= ( vEBT_Leaf @ A6 @ B4 ) )
=> ~ ( ( ( Xa
= ( zero_zero @ nat ) )
=> A6 )
& ( ( Xa
!= ( zero_zero @ nat ) )
=> ( ( ( Xa
= ( one_one @ nat ) )
=> B4 )
& ( Xa
= ( one_one @ nat ) ) ) ) ) )
=> ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList2: list @ vEBT_VEBT] :
( ? [Summary2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
=> ~ ( ( Xa != Mi )
=> ( ( Xa != Ma2 )
=> ( ~ ( ord_less @ nat @ Xa @ Mi )
& ( ~ ( ord_less @ nat @ Xa @ Mi )
=> ( ~ ( ord_less @ nat @ Ma2 @ Xa )
& ( ~ ( ord_less @ nat @ Ma2 @ Xa )
=> ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_vebt_member @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_member.elims(2)
thf(fact_3698_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 ) )
=> ( ! [Ux2: list @ vEBT_VEBT,Uy: vEBT_VEBT] :
( X
!= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( zero_zero @ nat ) @ Ux2 @ Uy ) )
=> ( ! [Mi: nat,Ma2: nat] :
( ? [Va3: list @ vEBT_VEBT,Vb2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma2 ) ) @ ( zero_zero @ nat ) @ Va3 @ Vb2 ) )
=> ( ( Xa = Mi )
| ( Xa = Ma2 ) ) )
=> ( ! [Mi: nat,Ma2: nat,V4: nat,TreeList2: list @ vEBT_VEBT] :
( ? [Vc2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma2 ) ) @ ( suc @ V4 ) @ TreeList2 @ Vc2 ) )
=> ( ( Xa = Mi )
| ( Xa = Ma2 )
| ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( 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 @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) )
=> ~ ! [V4: nat,TreeList2: list @ vEBT_VEBT] :
( ? [Vd: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V4 ) @ TreeList2 @ Vd ) )
=> ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( 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 @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.membermima.elims(3)
thf(fact_3699_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 )
=> ( ( ? [Ux2: list @ vEBT_VEBT,Uy: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( zero_zero @ nat ) @ Ux2 @ Uy ) )
=> Y2 )
=> ( ! [Mi: nat,Ma2: nat] :
( ? [Va3: list @ vEBT_VEBT,Vb2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma2 ) ) @ ( zero_zero @ nat ) @ Va3 @ Vb2 ) )
=> ( Y2
= ( ~ ( ( Xa = Mi )
| ( Xa = Ma2 ) ) ) ) )
=> ( ! [Mi: nat,Ma2: nat,V4: nat,TreeList2: list @ vEBT_VEBT] :
( ? [Vc2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma2 ) ) @ ( suc @ V4 ) @ TreeList2 @ Vc2 ) )
=> ( Y2
= ( ~ ( ( Xa = Mi )
| ( Xa = Ma2 )
| ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( 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 @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) )
=> ~ ! [V4: nat,TreeList2: list @ vEBT_VEBT] :
( ? [Vd: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V4 ) @ TreeList2 @ Vd ) )
=> ( Y2
= ( ~ ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( 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 @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.membermima.elims(1)
thf(fact_3700_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_3701_vebt__member_Oelims_I1_J,axiom,
! [X: vEBT_VEBT,Xa: nat,Y2: $o] :
( ( ( vEBT_vebt_member @ X @ Xa )
= Y2 )
=> ( ! [A6: $o,B4: $o] :
( ( X
= ( vEBT_Leaf @ A6 @ B4 ) )
=> ( Y2
= ( ~ ( ( ( Xa
= ( zero_zero @ nat ) )
=> A6 )
& ( ( Xa
!= ( zero_zero @ nat ) )
=> ( ( ( Xa
= ( one_one @ nat ) )
=> B4 )
& ( Xa
= ( one_one @ nat ) ) ) ) ) ) ) )
=> ( ( ? [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) )
=> Y2 )
=> ( ( ? [V4: product_prod @ nat @ nat,Uy: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( zero_zero @ nat ) @ Uy @ Uz2 ) )
=> Y2 )
=> ( ( ? [V4: product_prod @ nat @ nat,Vb2: list @ vEBT_VEBT,Vc2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb2 @ Vc2 ) )
=> Y2 )
=> ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList2: list @ vEBT_VEBT] :
( ? [Summary2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
=> ( Y2
= ( ~ ( ( Xa != Mi )
=> ( ( Xa != Ma2 )
=> ( ~ ( ord_less @ nat @ Xa @ Mi )
& ( ~ ( ord_less @ nat @ Xa @ Mi )
=> ( ~ ( ord_less @ nat @ Ma2 @ Xa )
& ( ~ ( ord_less @ nat @ Ma2 @ Xa )
=> ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_vebt_member @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_member.elims(1)
thf(fact_3702_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_3703_monoseq__arctan__series,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ X ) @ ( one_one @ real ) )
=> ( topological_monoseq @ real
@ ^ [N5: nat] : ( times_times @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( semiring_1_of_nat @ real @ ( plus_plus @ nat @ ( times_times @ nat @ N5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) @ ( power_power @ real @ X @ ( plus_plus @ nat @ ( times_times @ nat @ N5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) ) ) ).
% monoseq_arctan_series
thf(fact_3704_pochhammer__times__pochhammer__half,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [Z2: A,N: nat] :
( ( times_times @ A @ ( comm_s3205402744901411588hammer @ A @ Z2 @ ( suc @ N ) ) @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ Z2 @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( suc @ N ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [K3: nat] : ( plus_plus @ A @ Z2 @ ( divide_divide @ A @ ( semiring_1_of_nat @ A @ K3 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ nat ) ) ) ) ) ) ).
% pochhammer_times_pochhammer_half
thf(fact_3705_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
@ ^ [N5: nat] : ( times_times @ real @ ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ N5 ) @ ( divide_divide @ real @ ( one_one @ real ) @ ( semiring_1_of_nat @ real @ ( plus_plus @ nat @ N5 @ ( one_one @ nat ) ) ) ) ) @ ( power_power @ real @ ( minus_minus @ real @ X @ ( one_one @ real ) ) @ ( suc @ N5 ) ) ) ) ) ) ) ).
% ln_series
thf(fact_3706_xor__nonnegative__int__iff,axiom,
! [K: int,L: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se5824344971392196577ns_xor @ int @ K @ L ) )
= ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
= ( ord_less_eq @ int @ ( zero_zero @ int ) @ L ) ) ) ).
% xor_nonnegative_int_iff
thf(fact_3707_xor__negative__int__iff,axiom,
! [K: int,L: int] :
( ( ord_less @ int @ ( bit_se5824344971392196577ns_xor @ int @ K @ L ) @ ( zero_zero @ int ) )
= ( ( ord_less @ int @ K @ ( zero_zero @ int ) )
!= ( ord_less @ int @ L @ ( zero_zero @ int ) ) ) ) ).
% xor_negative_int_iff
thf(fact_3708_Ints__prod,axiom,
! [A: $tType,B: $tType] :
( ( ( comm_monoid_mult @ B )
& ( ring_1 @ B ) )
=> ! [A4: set @ A,F2: A > B] :
( ! [X3: A] :
( ( member @ A @ X3 @ A4 )
=> ( member @ B @ ( F2 @ X3 ) @ ( ring_1_Ints @ B ) ) )
=> ( member @ B @ ( groups7121269368397514597t_prod @ A @ B @ F2 @ A4 ) @ ( ring_1_Ints @ B ) ) ) ) ).
% Ints_prod
thf(fact_3709_of__nat__prod,axiom,
! [A: $tType,B: $tType] :
( ( comm_semiring_1 @ A )
=> ! [F2: B > nat,A4: set @ B] :
( ( semiring_1_of_nat @ A @ ( groups7121269368397514597t_prod @ B @ nat @ F2 @ A4 ) )
= ( groups7121269368397514597t_prod @ B @ A
@ ^ [X2: B] : ( semiring_1_of_nat @ A @ ( F2 @ X2 ) )
@ A4 ) ) ) ).
% of_nat_prod
thf(fact_3710_of__int__prod,axiom,
! [A: $tType,B: $tType] :
( ( comm_ring_1 @ A )
=> ! [F2: B > int,A4: set @ B] :
( ( ring_1_of_int @ A @ ( groups7121269368397514597t_prod @ B @ int @ F2 @ A4 ) )
= ( groups7121269368397514597t_prod @ B @ A
@ ^ [X2: B] : ( ring_1_of_int @ A @ ( F2 @ X2 ) )
@ A4 ) ) ) ).
% of_int_prod
thf(fact_3711_powser__zero,axiom,
! [A: $tType] :
( ( real_V2822296259951069270ebra_1 @ A )
=> ! [F2: nat > A] :
( ( suminf @ A
@ ^ [N5: nat] : ( times_times @ A @ ( F2 @ N5 ) @ ( power_power @ A @ ( zero_zero @ A ) @ N5 ) ) )
= ( F2 @ ( zero_zero @ nat ) ) ) ) ).
% powser_zero
thf(fact_3712_prod_Ocl__ivl__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [N: nat,M2: nat,G: nat > A] :
( ( ( ord_less @ nat @ ( suc @ N ) @ M2 )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ ( suc @ N ) ) )
= ( one_one @ A ) ) )
& ( ~ ( ord_less @ nat @ ( suc @ N ) @ M2 )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ ( suc @ N ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ) ).
% prod.cl_ivl_Suc
thf(fact_3713_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_3714_prod_Oshift__bounds__cl__Suc__ivl,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,M2: nat,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M2 ) @ ( suc @ N ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
@ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ).
% prod.shift_bounds_cl_Suc_ivl
thf(fact_3715_flip__bit__int__def,axiom,
( ( bit_se8732182000553998342ip_bit @ int )
= ( ^ [N5: nat,K3: int] : ( bit_se5824344971392196577ns_xor @ int @ K3 @ ( bit_se4730199178511100633sh_bit @ int @ N5 @ ( one_one @ int ) ) ) ) ) ).
% flip_bit_int_def
thf(fact_3716_xor__nat__def,axiom,
( ( bit_se5824344971392196577ns_xor @ nat )
= ( ^ [M5: nat,N5: nat] : ( nat2 @ ( bit_se5824344971392196577ns_xor @ int @ ( semiring_1_of_nat @ int @ M5 ) @ ( semiring_1_of_nat @ int @ N5 ) ) ) ) ) ).
% xor_nat_def
thf(fact_3717_prod_OatLeastAtMost__rev,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,N: nat,M2: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ N @ M2 ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I2: nat] : ( G @ ( minus_minus @ nat @ ( plus_plus @ nat @ M2 @ N ) @ I2 ) )
@ ( set_or1337092689740270186AtMost @ nat @ N @ M2 ) ) ) ) ).
% prod.atLeastAtMost_rev
thf(fact_3718_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_3719_prod_OatLeast__Suc__atMost,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M2: nat,N: nat,G: nat > A] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
= ( times_times @ A @ ( G @ M2 ) @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M2 ) @ N ) ) ) ) ) ) ).
% prod.atLeast_Suc_atMost
thf(fact_3720_prod_Onat__ivl__Suc_H,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M2: nat,N: nat,G: nat > A] :
( ( ord_less_eq @ nat @ M2 @ ( suc @ N ) )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ ( suc @ N ) ) )
= ( times_times @ A @ ( G @ ( suc @ N ) ) @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ) ) ).
% prod.nat_ivl_Suc'
thf(fact_3721_prod_OSuc__reindex__ivl,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M2: nat,N: nat,G: nat > A] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) @ ( G @ ( suc @ N ) ) )
= ( times_times @ A @ ( G @ M2 )
@ ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
@ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ) ) ).
% prod.Suc_reindex_ivl
thf(fact_3722_fact__prod,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ( ( semiring_char_0_fact @ A )
= ( ^ [N5: nat] :
( semiring_1_of_nat @ A
@ ( groups7121269368397514597t_prod @ nat @ nat
@ ^ [X2: nat] : X2
@ ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ N5 ) ) ) ) ) ) ).
% fact_prod
thf(fact_3723_prod__atLeastAtMost__code,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [F2: nat > A,A3: nat,B2: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ F2 @ ( set_or1337092689740270186AtMost @ nat @ A3 @ B2 ) )
= ( set_fo6178422350223883121st_nat @ A
@ ^ [A5: nat] : ( times_times @ A @ ( F2 @ A5 ) )
@ A3
@ B2
@ ( one_one @ A ) ) ) ) ).
% prod_atLeastAtMost_code
thf(fact_3724_prod_Oub__add__nat,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M2: nat,N: nat,G: nat > A,P4: nat] :
( ( ord_less_eq @ nat @ M2 @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ ( plus_plus @ nat @ N @ P4 ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) @ ( plus_plus @ nat @ N @ P4 ) ) ) ) ) ) ) ).
% prod.ub_add_nat
thf(fact_3725_fact__eq__fact__times,axiom,
! [N: nat,M2: nat] :
( ( ord_less_eq @ nat @ N @ M2 )
=> ( ( semiring_char_0_fact @ nat @ M2 )
= ( times_times @ nat @ ( semiring_char_0_fact @ nat @ N )
@ ( groups7121269368397514597t_prod @ nat @ nat
@ ^ [X2: nat] : X2
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ N ) @ M2 ) ) ) ) ) ).
% fact_eq_fact_times
thf(fact_3726_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_3727_pochhammer__Suc__prod,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A3: A,N: nat] :
( ( comm_s3205402744901411588hammer @ A @ A3 @ ( suc @ N ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I2: nat] : ( plus_plus @ A @ A3 @ ( semiring_1_of_nat @ A @ I2 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% pochhammer_Suc_prod
thf(fact_3728_pochhammer__prod__rev,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ( ( comm_s3205402744901411588hammer @ A )
= ( ^ [A5: A,N5: nat] :
( groups7121269368397514597t_prod @ nat @ A
@ ^ [I2: nat] : ( plus_plus @ A @ A5 @ ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ N5 @ I2 ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ N5 ) ) ) ) ) ).
% pochhammer_prod_rev
thf(fact_3729_fact__div__fact,axiom,
! [N: nat,M2: nat] :
( ( ord_less_eq @ nat @ N @ M2 )
=> ( ( divide_divide @ nat @ ( semiring_char_0_fact @ nat @ M2 ) @ ( semiring_char_0_fact @ nat @ N ) )
= ( groups7121269368397514597t_prod @ nat @ nat
@ ^ [X2: nat] : X2
@ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) @ M2 ) ) ) ) ).
% fact_div_fact
thf(fact_3730_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_3731_prod_Oin__pairs,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,M2: nat,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M2 ) @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I2: nat] : ( times_times @ A @ ( G @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I2 ) ) @ ( G @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I2 ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ).
% prod.in_pairs
thf(fact_3732_pochhammer__Suc__prod__rev,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A3: A,N: nat] :
( ( comm_s3205402744901411588hammer @ A @ A3 @ ( suc @ N ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I2: nat] : ( plus_plus @ A @ A3 @ ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ N @ I2 ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% pochhammer_Suc_prod_rev
thf(fact_3733_gbinomial__Suc,axiom,
! [A: $tType] :
( ( ( semiring_char_0 @ A )
& ( semidom_divide @ A ) )
=> ! [A3: A,K: nat] :
( ( gbinomial @ A @ A3 @ ( suc @ K ) )
= ( divide_divide @ A
@ ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I2: nat] : ( minus_minus @ A @ A3 @ ( semiring_1_of_nat @ A @ I2 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ K ) )
@ ( semiring_char_0_fact @ A @ ( suc @ K ) ) ) ) ) ).
% gbinomial_Suc
thf(fact_3734_xor__int__rec,axiom,
( ( bit_se5824344971392196577ns_xor @ int )
= ( ^ [K3: int,L2: 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 ) ) @ L2 ) ) ) )
@ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344971392196577ns_xor @ int @ ( divide_divide @ int @ K3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% xor_int_rec
thf(fact_3735_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_3736_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_3737_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_3738_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_3739_suminf__zero,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topological_t2_space @ A ) )
=> ( ( suminf @ A
@ ^ [N5: nat] : ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% suminf_zero
thf(fact_3740_prod_Oneutral__const,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A4: set @ B] :
( ( groups7121269368397514597t_prod @ B @ A
@ ^ [Uu3: B] : ( one_one @ A )
@ A4 )
= ( one_one @ A ) ) ) ).
% prod.neutral_const
thf(fact_3741_int__prod,axiom,
! [B: $tType,F2: B > nat,A4: set @ B] :
( ( semiring_1_of_nat @ int @ ( groups7121269368397514597t_prod @ B @ nat @ F2 @ A4 ) )
= ( groups7121269368397514597t_prod @ B @ int
@ ^ [X2: B] : ( semiring_1_of_nat @ int @ ( F2 @ X2 ) )
@ A4 ) ) ).
% int_prod
thf(fact_3742_prod__int__eq,axiom,
! [I: nat,J2: nat] :
( ( groups7121269368397514597t_prod @ nat @ int @ ( semiring_1_of_nat @ int ) @ ( set_or1337092689740270186AtMost @ nat @ I @ J2 ) )
= ( groups7121269368397514597t_prod @ int @ int
@ ^ [X2: int] : X2
@ ( set_or1337092689740270186AtMost @ int @ ( semiring_1_of_nat @ int @ I ) @ ( semiring_1_of_nat @ int @ J2 ) ) ) ) ).
% prod_int_eq
thf(fact_3743_prod__int__plus__eq,axiom,
! [I: nat,J2: nat] :
( ( groups7121269368397514597t_prod @ nat @ int @ ( semiring_1_of_nat @ int ) @ ( set_or1337092689740270186AtMost @ nat @ I @ ( plus_plus @ nat @ I @ J2 ) ) )
= ( groups7121269368397514597t_prod @ int @ int
@ ^ [X2: int] : X2
@ ( set_or1337092689740270186AtMost @ int @ ( semiring_1_of_nat @ int @ I ) @ ( semiring_1_of_nat @ int @ ( plus_plus @ nat @ I @ J2 ) ) ) ) ) ).
% prod_int_plus_eq
thf(fact_3744_prod_Oneutral,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A4: set @ B,G: B > A] :
( ! [X3: B] :
( ( member @ B @ X3 @ A4 )
=> ( ( G @ X3 )
= ( one_one @ A ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ A4 )
= ( one_one @ A ) ) ) ) ).
% prod.neutral
thf(fact_3745_prod_Onot__neutral__contains__not__neutral,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: B > A,A4: set @ B] :
( ( ( groups7121269368397514597t_prod @ B @ A @ G @ A4 )
!= ( one_one @ A ) )
=> ~ ! [A6: B] :
( ( member @ B @ A6 @ A4 )
=> ( ( G @ A6 )
= ( one_one @ A ) ) ) ) ) ).
% prod.not_neutral_contains_not_neutral
thf(fact_3746_prod__dividef,axiom,
! [A: $tType,B: $tType] :
( ( field @ A )
=> ! [F2: B > A,G: B > A,A4: set @ B] :
( ( groups7121269368397514597t_prod @ B @ A
@ ^ [X2: B] : ( divide_divide @ A @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ A4 )
= ( divide_divide @ A @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A4 ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ A4 ) ) ) ) ).
% prod_dividef
thf(fact_3747_prod__mono,axiom,
! [A: $tType,B: $tType] :
( ( linordered_semidom @ A )
=> ! [A4: set @ B,F2: B > A,G: B > A] :
( ! [I3: B] :
( ( member @ B @ I3 @ A4 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ I3 ) )
& ( ord_less_eq @ A @ ( F2 @ I3 ) @ ( G @ I3 ) ) ) )
=> ( ord_less_eq @ A @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A4 ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ A4 ) ) ) ) ).
% prod_mono
thf(fact_3748_prod__nonneg,axiom,
! [A: $tType,B: $tType] :
( ( linordered_semidom @ A )
=> ! [A4: set @ B,F2: B > A] :
( ! [X3: B] :
( ( member @ B @ X3 @ A4 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ X3 ) ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A4 ) ) ) ) ).
% prod_nonneg
thf(fact_3749_prod__pos,axiom,
! [A: $tType,B: $tType] :
( ( linordered_semidom @ A )
=> ! [A4: set @ B,F2: B > A] :
( ! [X3: B] :
( ( member @ B @ X3 @ A4 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( F2 @ X3 ) ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A4 ) ) ) ) ).
% prod_pos
thf(fact_3750_prod__ge__1,axiom,
! [A: $tType,B: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [A4: set @ B,F2: B > A] :
( ! [X3: B] :
( ( member @ B @ X3 @ A4 )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ ( F2 @ X3 ) ) )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A4 ) ) ) ) ).
% prod_ge_1
thf(fact_3751_prod__le__1,axiom,
! [B: $tType,A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [A4: set @ B,F2: B > A] :
( ! [X3: B] :
( ( member @ B @ X3 @ A4 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ X3 ) )
& ( ord_less_eq @ A @ ( F2 @ X3 ) @ ( one_one @ A ) ) ) )
=> ( ord_less_eq @ A @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A4 ) @ ( one_one @ A ) ) ) ) ).
% prod_le_1
thf(fact_3752_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_3753_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_3754_xor__int__unfold,axiom,
( ( bit_se5824344971392196577ns_xor @ int )
= ( ^ [K3: int,L2: int] :
( if @ int
@ ( K3
= ( uminus_uminus @ int @ ( one_one @ int ) ) )
@ ( bit_ri4277139882892585799ns_not @ int @ L2 )
@ ( if @ int
@ ( L2
= ( uminus_uminus @ int @ ( one_one @ int ) ) )
@ ( bit_ri4277139882892585799ns_not @ int @ K3 )
@ ( if @ int
@ ( K3
= ( zero_zero @ int ) )
@ L2
@ ( if @ int
@ ( L2
= ( 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 @ L2 @ ( 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 @ L2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ).
% xor_int_unfold
thf(fact_3755_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_3756_intind,axiom,
! [A: $tType,I: nat,N: nat,P: A > $o,X: A] :
( ( ord_less @ nat @ I @ N )
=> ( ( P @ X )
=> ( P @ ( nth @ A @ ( replicate @ A @ N @ X ) @ I ) ) ) ) ).
% intind
thf(fact_3757_replicate__eq__replicate,axiom,
! [A: $tType,M2: nat,X: A,N: nat,Y2: A] :
( ( ( replicate @ A @ M2 @ X )
= ( replicate @ A @ N @ Y2 ) )
= ( ( M2 = N )
& ( ( M2
!= ( zero_zero @ nat ) )
=> ( X = Y2 ) ) ) ) ).
% replicate_eq_replicate
thf(fact_3758_length__replicate,axiom,
! [A: $tType,N: nat,X: A] :
( ( size_size @ ( list @ A ) @ ( replicate @ A @ N @ X ) )
= N ) ).
% length_replicate
thf(fact_3759_summable__single,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [I: nat,F2: nat > A] :
( summable @ A
@ ^ [R: nat] : ( if @ A @ ( R = I ) @ ( F2 @ R ) @ ( zero_zero @ A ) ) ) ) ).
% summable_single
thf(fact_3760_summable__zero,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ( summable @ A
@ ^ [N5: nat] : ( zero_zero @ A ) ) ) ).
% summable_zero
thf(fact_3761_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_3762_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_3763_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_3764_Bex__set__replicate,axiom,
! [A: $tType,N: nat,A3: A,P: A > $o] :
( ( ? [X2: A] :
( ( member @ A @ X2 @ ( set2 @ A @ ( replicate @ A @ N @ A3 ) ) )
& ( P @ X2 ) ) )
= ( ( P @ A3 )
& ( N
!= ( zero_zero @ nat ) ) ) ) ).
% Bex_set_replicate
thf(fact_3765_Ball__set__replicate,axiom,
! [A: $tType,N: nat,A3: A,P: A > $o] :
( ( ! [X2: A] :
( ( member @ A @ X2 @ ( set2 @ A @ ( replicate @ A @ N @ A3 ) ) )
=> ( P @ X2 ) ) )
= ( ( P @ A3 )
| ( N
= ( zero_zero @ nat ) ) ) ) ).
% Ball_set_replicate
thf(fact_3766_nth__replicate,axiom,
! [A: $tType,I: nat,N: nat,X: A] :
( ( ord_less @ nat @ I @ N )
=> ( ( nth @ A @ ( replicate @ A @ N @ X ) @ I )
= X ) ) ).
% nth_replicate
thf(fact_3767_summable__cmult__iff,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [C2: A,F2: nat > A] :
( ( summable @ A
@ ^ [N5: nat] : ( times_times @ A @ C2 @ ( F2 @ N5 ) ) )
= ( ( C2
= ( zero_zero @ A ) )
| ( summable @ A @ F2 ) ) ) ) ).
% summable_cmult_iff
thf(fact_3768_summable__divide__iff,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: nat > A,C2: A] :
( ( summable @ A
@ ^ [N5: nat] : ( divide_divide @ A @ ( F2 @ N5 ) @ C2 ) )
= ( ( C2
= ( zero_zero @ A ) )
| ( summable @ A @ F2 ) ) ) ) ).
% summable_divide_iff
thf(fact_3769_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_3770_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_3771_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_3772_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_3773_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_3774_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_3775_not__nonnegative__int__iff,axiom,
! [K: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_ri4277139882892585799ns_not @ int @ K ) )
= ( ord_less @ int @ K @ ( zero_zero @ int ) ) ) ).
% not_nonnegative_int_iff
thf(fact_3776_not__negative__int__iff,axiom,
! [K: int] :
( ( ord_less @ int @ ( bit_ri4277139882892585799ns_not @ int @ K ) @ ( zero_zero @ int ) )
= ( ord_less_eq @ int @ ( zero_zero @ int ) @ K ) ) ).
% not_negative_int_iff
thf(fact_3777_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_3778_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_3779_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_3780_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_3781_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_3782_summable__minus__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A] :
( ( summable @ A
@ ^ [N5: nat] : ( uminus_uminus @ A @ ( F2 @ N5 ) ) )
= ( summable @ A @ F2 ) ) ) ).
% summable_minus_iff
thf(fact_3783_summable__minus,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A] :
( ( summable @ A @ F2 )
=> ( summable @ A
@ ^ [N5: nat] : ( uminus_uminus @ A @ ( F2 @ N5 ) ) ) ) ) ).
% summable_minus
thf(fact_3784_summable__diff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A,G: nat > A] :
( ( summable @ A @ F2 )
=> ( ( summable @ A @ G )
=> ( summable @ A
@ ^ [N5: nat] : ( minus_minus @ A @ ( F2 @ N5 ) @ ( G @ N5 ) ) ) ) ) ) ).
% summable_diff
thf(fact_3785_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_3786_summable__divide,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: nat > A,C2: A] :
( ( summable @ A @ F2 )
=> ( summable @ A
@ ^ [N5: nat] : ( divide_divide @ A @ ( F2 @ N5 ) @ C2 ) ) ) ) ).
% summable_divide
thf(fact_3787_summable__Suc__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A] :
( ( summable @ A
@ ^ [N5: nat] : ( F2 @ ( suc @ N5 ) ) )
= ( summable @ A @ F2 ) ) ) ).
% summable_Suc_iff
thf(fact_3788_powser__insidea,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [F2: nat > A,X: A,Z2: A] :
( ( summable @ A
@ ^ [N5: nat] : ( times_times @ A @ ( F2 @ N5 ) @ ( power_power @ A @ X @ N5 ) ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ Z2 ) @ ( real_V7770717601297561774m_norm @ A @ X ) )
=> ( summable @ real
@ ^ [N5: nat] : ( real_V7770717601297561774m_norm @ A @ ( times_times @ A @ ( F2 @ N5 ) @ ( power_power @ A @ Z2 @ N5 ) ) ) ) ) ) ) ).
% powser_insidea
thf(fact_3789_not__diff__distrib,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [A3: A,B2: A] :
( ( bit_ri4277139882892585799ns_not @ A @ ( minus_minus @ A @ A3 @ B2 ) )
= ( plus_plus @ A @ ( bit_ri4277139882892585799ns_not @ A @ A3 ) @ B2 ) ) ) ).
% not_diff_distrib
thf(fact_3790_not__add__distrib,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [A3: A,B2: A] :
( ( bit_ri4277139882892585799ns_not @ A @ ( plus_plus @ A @ A3 @ B2 ) )
= ( minus_minus @ A @ ( bit_ri4277139882892585799ns_not @ A @ A3 ) @ B2 ) ) ) ).
% not_add_distrib
thf(fact_3791_replicate__eqI,axiom,
! [A: $tType,Xs: list @ A,N: nat,X: A] :
( ( ( size_size @ ( list @ A ) @ Xs )
= N )
=> ( ! [Y3: A] :
( ( member @ A @ Y3 @ ( set2 @ A @ Xs ) )
=> ( Y3 = X ) )
=> ( Xs
= ( replicate @ A @ N @ X ) ) ) ) ).
% replicate_eqI
thf(fact_3792_replicate__length__same,axiom,
! [A: $tType,Xs: list @ A,X: A] :
( ! [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( X3 = X ) )
=> ( ( replicate @ A @ ( size_size @ ( list @ A ) @ Xs ) @ X )
= Xs ) ) ).
% replicate_length_same
thf(fact_3793_summable__mult__D,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [C2: A,F2: nat > A] :
( ( summable @ A
@ ^ [N5: nat] : ( times_times @ A @ C2 @ ( F2 @ N5 ) ) )
=> ( ( C2
!= ( zero_zero @ A ) )
=> ( summable @ A @ F2 ) ) ) ) ).
% summable_mult_D
thf(fact_3794_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_3795_suminf__diff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A,G: nat > A] :
( ( summable @ A @ F2 )
=> ( ( summable @ A @ G )
=> ( ( minus_minus @ A @ ( suminf @ A @ F2 ) @ ( suminf @ A @ G ) )
= ( suminf @ A
@ ^ [N5: nat] : ( minus_minus @ A @ ( F2 @ N5 ) @ ( G @ N5 ) ) ) ) ) ) ) ).
% suminf_diff
thf(fact_3796_suminf__divide,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: nat > A,C2: A] :
( ( summable @ A @ F2 )
=> ( ( suminf @ A
@ ^ [N5: nat] : ( divide_divide @ A @ ( F2 @ N5 ) @ C2 ) )
= ( divide_divide @ A @ ( suminf @ A @ F2 ) @ C2 ) ) ) ) ).
% suminf_divide
thf(fact_3797_suminf__minus,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A] :
( ( summable @ A @ F2 )
=> ( ( suminf @ A
@ ^ [N5: nat] : ( uminus_uminus @ A @ ( F2 @ N5 ) ) )
= ( uminus_uminus @ A @ ( suminf @ A @ F2 ) ) ) ) ) ).
% suminf_minus
thf(fact_3798_suminf__eq__zero__iff,axiom,
! [A: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F2: nat > A] :
( ( summable @ A @ F2 )
=> ( ! [N2: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ N2 ) )
=> ( ( ( suminf @ A @ F2 )
= ( zero_zero @ A ) )
= ( ! [N5: nat] :
( ( F2 @ N5 )
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% suminf_eq_zero_iff
thf(fact_3799_suminf__nonneg,axiom,
! [A: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F2: nat > A] :
( ( summable @ A @ F2 )
=> ( ! [N2: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ N2 ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( suminf @ A @ F2 ) ) ) ) ) ).
% suminf_nonneg
thf(fact_3800_suminf__pos,axiom,
! [A: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F2: nat > A] :
( ( summable @ A @ F2 )
=> ( ! [N2: nat] : ( ord_less @ A @ ( zero_zero @ A ) @ ( F2 @ N2 ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( suminf @ A @ F2 ) ) ) ) ) ).
% suminf_pos
thf(fact_3801_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_3802_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_3803_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_3804_summable__0__powser,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [F2: nat > A] :
( summable @ A
@ ^ [N5: nat] : ( times_times @ A @ ( F2 @ N5 ) @ ( power_power @ A @ ( zero_zero @ A ) @ N5 ) ) ) ) ).
% summable_0_powser
thf(fact_3805_summable__zero__power_H,axiom,
! [A: $tType] :
( ( ( ring_1 @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [F2: nat > A] :
( summable @ A
@ ^ [N5: nat] : ( times_times @ A @ ( F2 @ N5 ) @ ( power_power @ A @ ( zero_zero @ A ) @ N5 ) ) ) ) ).
% summable_zero_power'
thf(fact_3806_powser__split__head_I3_J,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V8999393235501362500lgebra @ A ) )
=> ! [F2: nat > A,Z2: A] :
( ( summable @ A
@ ^ [N5: nat] : ( times_times @ A @ ( F2 @ N5 ) @ ( power_power @ A @ Z2 @ N5 ) ) )
=> ( summable @ A
@ ^ [N5: nat] : ( times_times @ A @ ( F2 @ ( suc @ N5 ) ) @ ( power_power @ A @ Z2 @ N5 ) ) ) ) ) ).
% powser_split_head(3)
thf(fact_3807_summable__powser__split__head,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [F2: nat > A,Z2: A] :
( ( summable @ A
@ ^ [N5: nat] : ( times_times @ A @ ( F2 @ ( suc @ N5 ) ) @ ( power_power @ A @ Z2 @ N5 ) ) )
= ( summable @ A
@ ^ [N5: nat] : ( times_times @ A @ ( F2 @ N5 ) @ ( power_power @ A @ Z2 @ N5 ) ) ) ) ) ).
% summable_powser_split_head
thf(fact_3808_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_3809_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_3810_disjunctive__diff,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [B2: A,A3: A] :
( ! [N2: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ B2 @ N2 )
=> ( bit_se5641148757651400278ts_bit @ A @ A3 @ N2 ) )
=> ( ( minus_minus @ A @ A3 @ B2 )
= ( bit_se5824344872417868541ns_and @ A @ A3 @ ( bit_ri4277139882892585799ns_not @ A @ B2 ) ) ) ) ) ).
% disjunctive_diff
thf(fact_3811_take__bit__not__eq__mask__diff,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: nat,A3: A] :
( ( bit_se2584673776208193580ke_bit @ A @ N @ ( bit_ri4277139882892585799ns_not @ A @ A3 ) )
= ( minus_minus @ A @ ( bit_se2239418461657761734s_mask @ A @ N ) @ ( bit_se2584673776208193580ke_bit @ A @ N @ A3 ) ) ) ) ).
% take_bit_not_eq_mask_diff
thf(fact_3812_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_3813_suminf__pos__iff,axiom,
! [A: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F2: nat > A] :
( ( summable @ A @ F2 )
=> ( ! [N2: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ N2 ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ ( suminf @ A @ F2 ) )
= ( ? [I2: nat] : ( ord_less @ A @ ( zero_zero @ A ) @ ( F2 @ I2 ) ) ) ) ) ) ) ).
% suminf_pos_iff
thf(fact_3814_suminf__pos2,axiom,
! [A: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F2: nat > A,I: nat] :
( ( summable @ A @ F2 )
=> ( ! [N2: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ N2 ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ ( F2 @ I ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( suminf @ A @ F2 ) ) ) ) ) ) ).
% suminf_pos2
thf(fact_3815_not__int__div__2,axiom,
! [K: int] :
( ( divide_divide @ int @ ( bit_ri4277139882892585799ns_not @ int @ K ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) )
= ( bit_ri4277139882892585799ns_not @ int @ ( divide_divide @ int @ K @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ).
% not_int_div_2
thf(fact_3816_powser__inside,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V8999393235501362500lgebra @ A ) )
=> ! [F2: nat > A,X: A,Z2: A] :
( ( summable @ A
@ ^ [N5: nat] : ( times_times @ A @ ( F2 @ N5 ) @ ( power_power @ A @ X @ N5 ) ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ Z2 ) @ ( real_V7770717601297561774m_norm @ A @ X ) )
=> ( summable @ A
@ ^ [N5: nat] : ( times_times @ A @ ( F2 @ N5 ) @ ( power_power @ A @ Z2 @ N5 ) ) ) ) ) ) ).
% powser_inside
thf(fact_3817_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_3818_and__not__numerals_I4_J,axiom,
! [M2: num] :
( ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ ( bit0 @ M2 ) ) @ ( bit_ri4277139882892585799ns_not @ int @ ( one_one @ int ) ) )
= ( numeral_numeral @ int @ ( bit0 @ M2 ) ) ) ).
% and_not_numerals(4)
thf(fact_3819_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_3820_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_3821_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_3822_suminf__split__head,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A] :
( ( summable @ A @ F2 )
=> ( ( suminf @ A
@ ^ [N5: nat] : ( F2 @ ( suc @ N5 ) ) )
= ( minus_minus @ A @ ( suminf @ A @ F2 ) @ ( F2 @ ( zero_zero @ nat ) ) ) ) ) ) ).
% suminf_split_head
thf(fact_3823_summable__exp,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A] :
( summable @ A
@ ^ [N5: nat] : ( times_times @ A @ ( inverse_inverse @ A @ ( semiring_char_0_fact @ A @ N5 ) ) @ ( power_power @ A @ X @ N5 ) ) ) ) ).
% summable_exp
thf(fact_3824_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_3825_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_3826_bit__minus__int__iff,axiom,
! [K: int,N: nat] :
( ( bit_se5641148757651400278ts_bit @ int @ ( uminus_uminus @ int @ K ) @ N )
= ( bit_se5641148757651400278ts_bit @ int @ ( bit_ri4277139882892585799ns_not @ int @ ( minus_minus @ int @ K @ ( one_one @ int ) ) ) @ N ) ) ).
% bit_minus_int_iff
thf(fact_3827_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_3828_take__bit__not__mask__eq__0,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [M2: nat,N: nat] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( bit_se2584673776208193580ke_bit @ A @ M2 @ ( bit_ri4277139882892585799ns_not @ A @ ( bit_se2239418461657761734s_mask @ A @ N ) ) )
= ( zero_zero @ A ) ) ) ) ).
% take_bit_not_mask_eq_0
thf(fact_3829_unset__bit__eq__and__not,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( ( bit_se2638667681897837118et_bit @ A )
= ( ^ [N5: nat,A5: A] : ( bit_se5824344872417868541ns_and @ A @ A5 @ ( bit_ri4277139882892585799ns_not @ A @ ( bit_se4730199178511100633sh_bit @ A @ N5 @ ( one_one @ A ) ) ) ) ) ) ) ).
% unset_bit_eq_and_not
thf(fact_3830_unset__bit__int__def,axiom,
( ( bit_se2638667681897837118et_bit @ int )
= ( ^ [N5: nat,K3: int] : ( bit_se5824344872417868541ns_and @ int @ K3 @ ( bit_ri4277139882892585799ns_not @ int @ ( bit_se4730199178511100633sh_bit @ int @ N5 @ ( one_one @ int ) ) ) ) ) ) ).
% unset_bit_int_def
thf(fact_3831_and__not__numerals_I7_J,axiom,
! [M2: num] :
( ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ ( bit1 @ M2 ) ) @ ( bit_ri4277139882892585799ns_not @ int @ ( one_one @ int ) ) )
= ( numeral_numeral @ int @ ( bit0 @ M2 ) ) ) ).
% and_not_numerals(7)
thf(fact_3832_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_3833_powser__split__head_I1_J,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V8999393235501362500lgebra @ A ) )
=> ! [F2: nat > A,Z2: A] :
( ( summable @ A
@ ^ [N5: nat] : ( times_times @ A @ ( F2 @ N5 ) @ ( power_power @ A @ Z2 @ N5 ) ) )
=> ( ( suminf @ A
@ ^ [N5: nat] : ( times_times @ A @ ( F2 @ N5 ) @ ( power_power @ A @ Z2 @ N5 ) ) )
= ( plus_plus @ A @ ( F2 @ ( zero_zero @ nat ) )
@ ( times_times @ A
@ ( suminf @ A
@ ^ [N5: nat] : ( times_times @ A @ ( F2 @ ( suc @ N5 ) ) @ ( power_power @ A @ Z2 @ N5 ) ) )
@ Z2 ) ) ) ) ) ).
% powser_split_head(1)
thf(fact_3834_powser__split__head_I2_J,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V8999393235501362500lgebra @ A ) )
=> ! [F2: nat > A,Z2: A] :
( ( summable @ A
@ ^ [N5: nat] : ( times_times @ A @ ( F2 @ N5 ) @ ( power_power @ A @ Z2 @ N5 ) ) )
=> ( ( times_times @ A
@ ( suminf @ A
@ ^ [N5: nat] : ( times_times @ A @ ( F2 @ ( suc @ N5 ) ) @ ( power_power @ A @ Z2 @ N5 ) ) )
@ Z2 )
= ( minus_minus @ A
@ ( suminf @ A
@ ^ [N5: nat] : ( times_times @ A @ ( F2 @ N5 ) @ ( power_power @ A @ Z2 @ N5 ) ) )
@ ( F2 @ ( zero_zero @ nat ) ) ) ) ) ) ).
% powser_split_head(2)
thf(fact_3835_summable__power__series,axiom,
! [F2: nat > real,Z2: real] :
( ! [I3: nat] : ( ord_less_eq @ real @ ( F2 @ I3 ) @ ( one_one @ real ) )
=> ( ! [I3: nat] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( F2 @ I3 ) )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Z2 )
=> ( ( ord_less @ real @ Z2 @ ( one_one @ real ) )
=> ( summable @ real
@ ^ [I2: nat] : ( times_times @ real @ ( F2 @ I2 ) @ ( power_power @ real @ Z2 @ I2 ) ) ) ) ) ) ) ).
% summable_power_series
thf(fact_3836_bit__not__iff__eq,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [A3: A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( bit_ri4277139882892585799ns_not @ A @ A3 ) @ N )
= ( ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N )
!= ( zero_zero @ A ) )
& ~ ( bit_se5641148757651400278ts_bit @ A @ A3 @ N ) ) ) ) ).
% bit_not_iff_eq
thf(fact_3837_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_3838_summable__ratio__test,axiom,
! [A: $tType] :
( ( real_Vector_banach @ A )
=> ! [C2: real,N6: nat,F2: nat > A] :
( ( ord_less @ real @ C2 @ ( one_one @ real ) )
=> ( ! [N2: nat] :
( ( ord_less_eq @ nat @ N6 @ N2 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( F2 @ ( suc @ N2 ) ) ) @ ( times_times @ real @ C2 @ ( real_V7770717601297561774m_norm @ A @ ( F2 @ N2 ) ) ) ) )
=> ( summable @ A @ F2 ) ) ) ) ).
% summable_ratio_test
thf(fact_3839_and__not__numerals_I8_J,axiom,
! [M2: num,N: num] :
( ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ ( bit1 @ M2 ) ) @ ( 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 @ M2 ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ N ) ) ) ) ) ) ).
% and_not_numerals(8)
thf(fact_3840_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_3841_vebt__buildup_Osimps_I3_J,axiom,
! [Va2: nat] :
( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va2 ) ) )
=> ( ( vEBT_vebt_buildup @ ( suc @ ( suc @ Va2 ) ) )
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ Va2 ) ) @ ( replicate @ vEBT_VEBT @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) )
& ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va2 ) ) )
=> ( ( vEBT_vebt_buildup @ ( suc @ ( suc @ Va2 ) ) )
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ Va2 ) ) @ ( replicate @ vEBT_VEBT @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_vebt_buildup @ ( suc @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ).
% vebt_buildup.simps(3)
thf(fact_3842_sin__paired,axiom,
! [X: real] :
( sums @ real
@ ^ [N5: nat] : ( times_times @ real @ ( divide_divide @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ N5 ) @ ( semiring_char_0_fact @ real @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N5 ) @ ( one_one @ nat ) ) ) ) @ ( power_power @ real @ X @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N5 ) @ ( one_one @ nat ) ) ) )
@ ( sin @ real @ X ) ) ).
% sin_paired
thf(fact_3843_vebt__member_Opelims_I3_J,axiom,
! [X: vEBT_VEBT,Xa: nat] :
( ~ ( vEBT_vebt_member @ X @ Xa )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa ) )
=> ( ! [A6: $o,B4: $o] :
( ( X
= ( vEBT_Leaf @ A6 @ B4 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A6 @ B4 ) @ Xa ) )
=> ( ( ( Xa
= ( zero_zero @ nat ) )
=> A6 )
& ( ( Xa
!= ( zero_zero @ nat ) )
=> ( ( ( Xa
= ( one_one @ nat ) )
=> B4 )
& ( Xa
= ( one_one @ nat ) ) ) ) ) ) )
=> ( ! [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) @ Xa ) ) )
=> ( ! [V4: product_prod @ nat @ nat,Uy: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( zero_zero @ nat ) @ Uy @ Uz2 ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( zero_zero @ nat ) @ Uy @ Uz2 ) @ Xa ) ) )
=> ( ! [V4: product_prod @ nat @ nat,Vb2: list @ vEBT_VEBT,Vc2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb2 @ Vc2 ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb2 @ Vc2 ) @ Xa ) ) )
=> ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) @ Xa ) )
=> ( ( Xa != Mi )
=> ( ( Xa != Ma2 )
=> ( ~ ( ord_less @ nat @ Xa @ Mi )
& ( ~ ( ord_less @ nat @ Xa @ Mi )
=> ( ~ ( ord_less @ nat @ Ma2 @ Xa )
& ( ~ ( ord_less @ nat @ Ma2 @ Xa )
=> ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_vebt_member @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_member.pelims(3)
thf(fact_3844_vebt__member_Opelims_I1_J,axiom,
! [X: vEBT_VEBT,Xa: nat,Y2: $o] :
( ( ( vEBT_vebt_member @ X @ Xa )
= Y2 )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa ) )
=> ( ! [A6: $o,B4: $o] :
( ( X
= ( vEBT_Leaf @ A6 @ B4 ) )
=> ( ( Y2
= ( ( ( Xa
= ( zero_zero @ nat ) )
=> A6 )
& ( ( Xa
!= ( zero_zero @ nat ) )
=> ( ( ( Xa
= ( one_one @ nat ) )
=> B4 )
& ( Xa
= ( one_one @ nat ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A6 @ B4 ) @ Xa ) ) ) )
=> ( ! [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) )
=> ( ~ Y2
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) @ Xa ) ) ) )
=> ( ! [V4: product_prod @ nat @ nat,Uy: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( zero_zero @ nat ) @ Uy @ Uz2 ) )
=> ( ~ Y2
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( zero_zero @ nat ) @ Uy @ Uz2 ) @ Xa ) ) ) )
=> ( ! [V4: product_prod @ nat @ nat,Vb2: list @ vEBT_VEBT,Vc2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb2 @ Vc2 ) )
=> ( ~ Y2
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V4 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb2 @ Vc2 ) @ Xa ) ) ) )
=> ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
=> ( ( Y2
= ( ( Xa != Mi )
=> ( ( Xa != Ma2 )
=> ( ~ ( ord_less @ nat @ Xa @ Mi )
& ( ~ ( ord_less @ nat @ Xa @ Mi )
=> ( ~ ( ord_less @ nat @ Ma2 @ Xa )
& ( ~ ( ord_less @ nat @ Ma2 @ Xa )
=> ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_vebt_member @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) @ Xa ) ) ) ) ) ) ) ) ) ) ).
% vebt_member.pelims(1)
thf(fact_3845_sum__gp,axiom,
! [A: $tType] :
( ( ( division_ring @ A )
& ( comm_ring @ A ) )
=> ! [N: nat,M2: nat,X: A] :
( ( ( ord_less @ nat @ N @ M2 )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
= ( zero_zero @ A ) ) )
& ( ~ ( ord_less @ nat @ N @ M2 )
=> ( ( ( X
= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
= ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) @ M2 ) ) ) )
& ( ( X
!= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( power_power @ A @ X @ M2 ) @ ( power_power @ A @ X @ ( suc @ N ) ) ) @ ( minus_minus @ A @ ( one_one @ A ) @ X ) ) ) ) ) ) ) ) ).
% sum_gp
thf(fact_3846_Ints__sum,axiom,
! [A: $tType,B: $tType] :
( ( ring_1 @ B )
=> ! [A4: set @ A,F2: A > B] :
( ! [X3: A] :
( ( member @ A @ X3 @ A4 )
=> ( member @ B @ ( F2 @ X3 ) @ ( ring_1_Ints @ B ) ) )
=> ( member @ B @ ( groups7311177749621191930dd_sum @ A @ B @ F2 @ A4 ) @ ( ring_1_Ints @ B ) ) ) ) ).
% Ints_sum
thf(fact_3847_sum_Oneutral__const,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A4: set @ B] :
( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [Uu3: B] : ( zero_zero @ A )
@ A4 )
= ( zero_zero @ A ) ) ) ).
% sum.neutral_const
thf(fact_3848_of__nat__sum,axiom,
! [A: $tType,B: $tType] :
( ( semiring_1 @ A )
=> ! [F2: B > nat,A4: set @ B] :
( ( semiring_1_of_nat @ A @ ( groups7311177749621191930dd_sum @ B @ nat @ F2 @ A4 ) )
= ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X2: B] : ( semiring_1_of_nat @ A @ ( F2 @ X2 ) )
@ A4 ) ) ) ).
% of_nat_sum
thf(fact_3849_of__int__sum,axiom,
! [A: $tType,B: $tType] :
( ( ring_1 @ A )
=> ! [F2: B > int,A4: set @ B] :
( ( ring_1_of_int @ A @ ( groups7311177749621191930dd_sum @ B @ int @ F2 @ A4 ) )
= ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X2: B] : ( ring_1_of_int @ A @ ( F2 @ X2 ) )
@ A4 ) ) ) ).
% of_int_sum
thf(fact_3850_sums__zero,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ( sums @ A
@ ^ [N5: nat] : ( zero_zero @ A )
@ ( zero_zero @ A ) ) ) ).
% sums_zero
thf(fact_3851_sum_Oempty,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: B > A] :
( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( bot_bot @ ( set @ B ) ) )
= ( zero_zero @ A ) ) ) ).
% sum.empty
thf(fact_3852_sum__abs__ge__zero,axiom,
! [B: $tType,A: $tType] :
( ( ordere166539214618696060dd_abs @ B )
=> ! [F2: A > B,A4: set @ A] :
( ord_less_eq @ B @ ( zero_zero @ B )
@ ( groups7311177749621191930dd_sum @ A @ B
@ ^ [I2: A] : ( abs_abs @ B @ ( F2 @ I2 ) )
@ A4 ) ) ) ).
% sum_abs_ge_zero
thf(fact_3853_powser__sums__zero__iff,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [A3: nat > A,X: A] :
( ( sums @ A
@ ^ [N5: nat] : ( times_times @ A @ ( A3 @ N5 ) @ ( power_power @ A @ ( zero_zero @ A ) @ N5 ) )
@ X )
= ( ( A3 @ ( zero_zero @ nat ) )
= X ) ) ) ).
% powser_sums_zero_iff
thf(fact_3854_sum_Ocl__ivl__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [N: nat,M2: nat,G: nat > A] :
( ( ( ord_less @ nat @ ( suc @ N ) @ M2 )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ ( suc @ N ) ) )
= ( zero_zero @ A ) ) )
& ( ~ ( ord_less @ nat @ ( suc @ N ) @ M2 )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ ( suc @ N ) ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) @ ( G @ ( suc @ N ) ) ) ) ) ) ) ).
% sum.cl_ivl_Suc
thf(fact_3855_sums__divide,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: nat > A,A3: A,C2: A] :
( ( sums @ A @ F2 @ A3 )
=> ( sums @ A
@ ^ [N5: nat] : ( divide_divide @ A @ ( F2 @ N5 ) @ C2 )
@ ( divide_divide @ A @ A3 @ C2 ) ) ) ) ).
% sums_divide
thf(fact_3856_sums__single,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [I: nat,F2: nat > A] :
( sums @ A
@ ^ [R: nat] : ( if @ A @ ( R = I ) @ ( F2 @ R ) @ ( zero_zero @ A ) )
@ ( F2 @ I ) ) ) ).
% sums_single
thf(fact_3857_sum_Onot__neutral__contains__not__neutral,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: B > A,A4: set @ B] :
( ( ( groups7311177749621191930dd_sum @ B @ A @ G @ A4 )
!= ( zero_zero @ A ) )
=> ~ ! [A6: B] :
( ( member @ B @ A6 @ A4 )
=> ( ( G @ A6 )
= ( zero_zero @ A ) ) ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_3858_sum_Oneutral,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A4: set @ B,G: B > A] :
( ! [X3: B] :
( ( member @ B @ X3 @ A4 )
=> ( ( G @ X3 )
= ( zero_zero @ A ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ A4 )
= ( zero_zero @ A ) ) ) ) ).
% sum.neutral
thf(fact_3859_sums__0,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [F2: nat > A] :
( ! [N2: nat] :
( ( F2 @ N2 )
= ( zero_zero @ A ) )
=> ( sums @ A @ F2 @ ( zero_zero @ A ) ) ) ) ).
% sums_0
thf(fact_3860_sums__diff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A,A3: A,G: nat > A,B2: A] :
( ( sums @ A @ F2 @ A3 )
=> ( ( sums @ A @ G @ B2 )
=> ( sums @ A
@ ^ [N5: nat] : ( minus_minus @ A @ ( F2 @ N5 ) @ ( G @ N5 ) )
@ ( minus_minus @ A @ A3 @ B2 ) ) ) ) ) ).
% sums_diff
thf(fact_3861_sums__minus,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A,A3: A] :
( ( sums @ A @ F2 @ A3 )
=> ( sums @ A
@ ^ [N5: nat] : ( uminus_uminus @ A @ ( F2 @ N5 ) )
@ ( uminus_uminus @ A @ A3 ) ) ) ) ).
% sums_minus
thf(fact_3862_sum__subtractf,axiom,
! [A: $tType,B: $tType] :
( ( ab_group_add @ A )
=> ! [F2: B > A,G: B > A,A4: set @ B] :
( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X2: B] : ( minus_minus @ A @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ A4 )
= ( minus_minus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A4 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ A4 ) ) ) ) ).
% sum_subtractf
thf(fact_3863_sum__divide__distrib,axiom,
! [A: $tType,B: $tType] :
( ( field @ A )
=> ! [F2: B > A,A4: set @ B,R4: A] :
( ( divide_divide @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A4 ) @ R4 )
= ( groups7311177749621191930dd_sum @ B @ A
@ ^ [N5: B] : ( divide_divide @ A @ ( F2 @ N5 ) @ R4 )
@ A4 ) ) ) ).
% sum_divide_distrib
thf(fact_3864_sum__negf,axiom,
! [A: $tType,B: $tType] :
( ( ab_group_add @ A )
=> ! [F2: B > A,A4: set @ B] :
( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X2: B] : ( uminus_uminus @ A @ ( F2 @ X2 ) )
@ A4 )
= ( uminus_uminus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A4 ) ) ) ) ).
% sum_negf
thf(fact_3865_sum__nonneg,axiom,
! [A: $tType,B: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A4: set @ B,F2: B > A] :
( ! [X3: B] :
( ( member @ B @ X3 @ A4 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ X3 ) ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A4 ) ) ) ) ).
% sum_nonneg
thf(fact_3866_sum__nonpos,axiom,
! [B: $tType,A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A4: set @ B,F2: B > A] :
( ! [X3: B] :
( ( member @ B @ X3 @ A4 )
=> ( ord_less_eq @ A @ ( F2 @ X3 ) @ ( zero_zero @ A ) ) )
=> ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A4 ) @ ( zero_zero @ A ) ) ) ) ).
% sum_nonpos
thf(fact_3867_sum__cong__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A4: set @ nat,F2: nat > A,G: nat > A] :
( ~ ( member @ nat @ ( zero_zero @ nat ) @ A4 )
=> ( ! [X3: nat] :
( ( member @ nat @ ( suc @ X3 ) @ A4 )
=> ( ( F2 @ ( suc @ X3 ) )
= ( G @ ( suc @ X3 ) ) ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ A4 )
= ( groups7311177749621191930dd_sum @ nat @ A @ G @ A4 ) ) ) ) ) ).
% sum_cong_Suc
thf(fact_3868_sums__mult2__iff,axiom,
! [A: $tType] :
( ( ( field @ A )
& ( real_V4412858255891104859lgebra @ A ) )
=> ! [C2: A,F2: nat > A,D2: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( sums @ A
@ ^ [N5: nat] : ( times_times @ A @ ( F2 @ N5 ) @ C2 )
@ ( times_times @ A @ D2 @ C2 ) )
= ( sums @ A @ F2 @ D2 ) ) ) ) ).
% sums_mult2_iff
thf(fact_3869_sums__mult__iff,axiom,
! [A: $tType] :
( ( ( field @ A )
& ( real_V4412858255891104859lgebra @ A ) )
=> ! [C2: A,F2: nat > A,D2: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( sums @ A
@ ^ [N5: nat] : ( times_times @ A @ C2 @ ( F2 @ N5 ) )
@ ( times_times @ A @ C2 @ D2 ) )
= ( sums @ A @ F2 @ D2 ) ) ) ) ).
% sums_mult_iff
thf(fact_3870_sum_Oshift__bounds__cl__Suc__ivl,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,M2: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M2 ) @ ( suc @ N ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
@ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ).
% sum.shift_bounds_cl_Suc_ivl
thf(fact_3871_sums__mult__D,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [C2: A,F2: nat > A,A3: A] :
( ( sums @ A
@ ^ [N5: nat] : ( times_times @ A @ C2 @ ( F2 @ N5 ) )
@ A3 )
=> ( ( C2
!= ( zero_zero @ A ) )
=> ( sums @ A @ F2 @ ( divide_divide @ A @ A3 @ C2 ) ) ) ) ) ).
% sums_mult_D
thf(fact_3872_sums__Suc__imp,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A,S: A] :
( ( ( F2 @ ( zero_zero @ nat ) )
= ( zero_zero @ A ) )
=> ( ( sums @ A
@ ^ [N5: nat] : ( F2 @ ( suc @ N5 ) )
@ S )
=> ( sums @ A @ F2 @ S ) ) ) ) ).
% sums_Suc_imp
thf(fact_3873_sums__Suc,axiom,
! [A: $tType] :
( ( ( topolo5987344860129210374id_add @ A )
& ( topological_t2_space @ A ) )
=> ! [F2: nat > A,L: A] :
( ( sums @ A
@ ^ [N5: nat] : ( F2 @ ( suc @ N5 ) )
@ L )
=> ( sums @ A @ F2 @ ( plus_plus @ A @ L @ ( F2 @ ( zero_zero @ nat ) ) ) ) ) ) ).
% sums_Suc
thf(fact_3874_sums__Suc__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A,S: A] :
( ( sums @ A
@ ^ [N5: nat] : ( F2 @ ( suc @ N5 ) )
@ S )
= ( sums @ A @ F2 @ ( plus_plus @ A @ S @ ( F2 @ ( zero_zero @ nat ) ) ) ) ) ) ).
% sums_Suc_iff
thf(fact_3875_sums__zero__iff__shift,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [N: nat,F2: nat > A,S: A] :
( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ N )
=> ( ( F2 @ I3 )
= ( zero_zero @ A ) ) )
=> ( ( sums @ A
@ ^ [I2: nat] : ( F2 @ ( plus_plus @ nat @ I2 @ N ) )
@ S )
= ( sums @ A @ F2 @ S ) ) ) ) ).
% sums_zero_iff_shift
thf(fact_3876_sum_OatLeastAtMost__rev,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,N: nat,M2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ N @ M2 ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( G @ ( minus_minus @ nat @ ( plus_plus @ nat @ M2 @ N ) @ I2 ) )
@ ( set_or1337092689740270186AtMost @ nat @ N @ M2 ) ) ) ) ).
% sum.atLeastAtMost_rev
thf(fact_3877_powser__sums__if,axiom,
! [A: $tType] :
( ( ( ring_1 @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [M2: nat,Z2: A] :
( sums @ A
@ ^ [N5: nat] : ( times_times @ A @ ( if @ A @ ( N5 = M2 ) @ ( one_one @ A ) @ ( zero_zero @ A ) ) @ ( power_power @ A @ Z2 @ N5 ) )
@ ( power_power @ A @ Z2 @ M2 ) ) ) ).
% powser_sums_if
thf(fact_3878_powser__sums__zero,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [A3: nat > A] :
( sums @ A
@ ^ [N5: nat] : ( times_times @ A @ ( A3 @ N5 ) @ ( power_power @ A @ ( zero_zero @ A ) @ N5 ) )
@ ( A3 @ ( zero_zero @ nat ) ) ) ) ).
% powser_sums_zero
thf(fact_3879_sum__shift__lb__Suc0__0,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [F2: nat > A,K: nat] :
( ( ( F2 @ ( zero_zero @ nat ) )
= ( zero_zero @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ ( zero_zero @ nat ) ) @ K ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ K ) ) ) ) ) ).
% sum_shift_lb_Suc0_0
thf(fact_3880_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_3881_sum_OatLeast__Suc__atMost,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [M2: nat,N: nat,G: nat > A] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
= ( plus_plus @ A @ ( G @ M2 ) @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M2 ) @ N ) ) ) ) ) ) ).
% sum.atLeast_Suc_atMost
thf(fact_3882_sum_Onat__ivl__Suc_H,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [M2: nat,N: nat,G: nat > A] :
( ( ord_less_eq @ nat @ M2 @ ( suc @ N ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ ( suc @ N ) ) )
= ( plus_plus @ A @ ( G @ ( suc @ N ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ) ) ).
% sum.nat_ivl_Suc'
thf(fact_3883_sum_OSuc__reindex__ivl,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [M2: nat,N: nat,G: nat > A] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) @ ( G @ ( suc @ N ) ) )
= ( plus_plus @ A @ ( G @ M2 )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
@ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ) ) ).
% sum.Suc_reindex_ivl
thf(fact_3884_sum__Suc__diff,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [M2: nat,N: nat,F2: nat > A] :
( ( ord_less_eq @ nat @ M2 @ ( suc @ N ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( minus_minus @ A @ ( F2 @ ( suc @ I2 ) ) @ ( F2 @ I2 ) )
@ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
= ( minus_minus @ A @ ( F2 @ ( suc @ N ) ) @ ( F2 @ M2 ) ) ) ) ) ).
% sum_Suc_diff
thf(fact_3885_sum__atLeastAtMost__code,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [F2: nat > A,A3: nat,B2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_or1337092689740270186AtMost @ nat @ A3 @ B2 ) )
= ( set_fo6178422350223883121st_nat @ A
@ ^ [A5: nat] : ( plus_plus @ A @ ( F2 @ A5 ) )
@ A3
@ B2
@ ( zero_zero @ A ) ) ) ) ).
% sum_atLeastAtMost_code
thf(fact_3886_sum_Oub__add__nat,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [M2: nat,N: nat,G: nat > A,P4: nat] :
( ( ord_less_eq @ nat @ M2 @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ ( plus_plus @ nat @ N @ P4 ) ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) @ ( plus_plus @ nat @ N @ P4 ) ) ) ) ) ) ) ).
% sum.ub_add_nat
thf(fact_3887_convex__sum__bound__le,axiom,
! [A: $tType,B: $tType] :
( ( linordered_idom @ B )
=> ! [I5: set @ A,X: A > B,A3: A > B,B2: B,Delta: B] :
( ! [I3: A] :
( ( member @ A @ I3 @ I5 )
=> ( ord_less_eq @ B @ ( zero_zero @ B ) @ ( X @ I3 ) ) )
=> ( ( ( groups7311177749621191930dd_sum @ A @ B @ X @ I5 )
= ( one_one @ B ) )
=> ( ! [I3: A] :
( ( member @ A @ I3 @ I5 )
=> ( ord_less_eq @ B @ ( abs_abs @ B @ ( minus_minus @ B @ ( A3 @ I3 ) @ B2 ) ) @ Delta ) )
=> ( ord_less_eq @ B
@ ( abs_abs @ B
@ ( minus_minus @ B
@ ( groups7311177749621191930dd_sum @ A @ B
@ ^ [I2: A] : ( times_times @ B @ ( A3 @ I2 ) @ ( X @ I2 ) )
@ I5 )
@ B2 ) )
@ Delta ) ) ) ) ) ).
% convex_sum_bound_le
thf(fact_3888_sum__natinterval__diff,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [M2: nat,N: nat,F2: nat > A] :
( ( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( minus_minus @ A @ ( F2 @ K3 ) @ ( F2 @ ( plus_plus @ nat @ K3 @ ( one_one @ nat ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
= ( minus_minus @ A @ ( F2 @ M2 ) @ ( F2 @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) ) ) )
& ( ~ ( ord_less_eq @ nat @ M2 @ N )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( minus_minus @ A @ ( F2 @ K3 ) @ ( F2 @ ( plus_plus @ nat @ K3 @ ( one_one @ nat ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
= ( zero_zero @ A ) ) ) ) ) ).
% sum_natinterval_diff
thf(fact_3889_sum__telescope_H_H,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [M2: nat,N: nat,F2: nat > A] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( minus_minus @ A @ ( F2 @ K3 ) @ ( F2 @ ( minus_minus @ nat @ K3 @ ( one_one @ nat ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M2 ) @ N ) )
= ( minus_minus @ A @ ( F2 @ N ) @ ( F2 @ M2 ) ) ) ) ) ).
% sum_telescope''
thf(fact_3890_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_3891_power__half__series,axiom,
( sums @ real
@ ^ [N5: nat] : ( power_power @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( suc @ N5 ) )
@ ( one_one @ real ) ) ).
% power_half_series
thf(fact_3892_mask__eq__sum__exp,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [N: nat] :
( ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ A ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
@ ( collect @ nat
@ ^ [Q4: nat] : ( ord_less @ nat @ Q4 @ N ) ) ) ) ) ).
% mask_eq_sum_exp
thf(fact_3893_sum__gp__multiplied,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [M2: nat,N: nat,X: A] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( times_times @ A @ ( minus_minus @ A @ ( one_one @ A ) @ X ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) )
= ( minus_minus @ A @ ( power_power @ A @ X @ M2 ) @ ( power_power @ A @ X @ ( suc @ N ) ) ) ) ) ) ).
% sum_gp_multiplied
thf(fact_3894_sum_Oin__pairs,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,M2: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M2 ) @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( plus_plus @ A @ ( G @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I2 ) ) @ ( G @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I2 ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ).
% sum.in_pairs
thf(fact_3895_sums__if_H,axiom,
! [G: nat > real,X: real] :
( ( sums @ real @ G @ X )
=> ( sums @ real
@ ^ [N5: nat] : ( if @ real @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N5 ) @ ( zero_zero @ real ) @ ( G @ ( divide_divide @ nat @ ( minus_minus @ nat @ N5 @ ( one_one @ nat ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
@ X ) ) ).
% sums_if'
thf(fact_3896_sums__if,axiom,
! [G: nat > real,X: real,F2: nat > real,Y2: real] :
( ( sums @ real @ G @ X )
=> ( ( sums @ real @ F2 @ Y2 )
=> ( sums @ real
@ ^ [N5: nat] : ( if @ real @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N5 ) @ ( F2 @ ( divide_divide @ nat @ N5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( G @ ( divide_divide @ nat @ ( minus_minus @ nat @ N5 @ ( one_one @ nat ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
@ ( plus_plus @ real @ X @ Y2 ) ) ) ) ).
% sums_if
thf(fact_3897_mask__eq__sum__exp__nat,axiom,
! [N: nat] :
( ( minus_minus @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ ( suc @ ( zero_zero @ nat ) ) )
= ( groups7311177749621191930dd_sum @ nat @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
@ ( collect @ nat
@ ^ [Q4: nat] : ( ord_less @ nat @ Q4 @ N ) ) ) ) ).
% mask_eq_sum_exp_nat
thf(fact_3898_gauss__sum__nat,axiom,
! [N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [X2: nat] : X2
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
= ( divide_divide @ nat @ ( times_times @ nat @ N @ ( suc @ N ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% gauss_sum_nat
thf(fact_3899_gbinomial__sum__up__index,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [J3: nat] : ( gbinomial @ A @ ( semiring_1_of_nat @ A @ J3 ) @ K )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
= ( gbinomial @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N ) @ ( one_one @ A ) ) @ ( plus_plus @ nat @ K @ ( one_one @ nat ) ) ) ) ) ).
% gbinomial_sum_up_index
thf(fact_3900_double__arith__series,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A3: A,D2: A,N: nat] :
( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( plus_plus @ A @ A3 @ ( times_times @ A @ ( semiring_1_of_nat @ A @ I2 ) @ D2 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) )
= ( times_times @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N ) @ ( one_one @ A ) ) @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ D2 ) ) ) ) ) ).
% double_arith_series
thf(fact_3901_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_3902_arith__series__nat,axiom,
! [A3: nat,D2: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [I2: nat] : ( plus_plus @ nat @ A3 @ ( times_times @ nat @ I2 @ D2 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
= ( divide_divide @ nat @ ( times_times @ nat @ ( suc @ N ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ A3 ) @ ( times_times @ nat @ N @ D2 ) ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% arith_series_nat
thf(fact_3903_Sum__Icc__nat,axiom,
! [M2: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [X2: nat] : X2
@ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
= ( divide_divide @ nat @ ( minus_minus @ nat @ ( times_times @ nat @ N @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) @ ( times_times @ nat @ M2 @ ( minus_minus @ nat @ M2 @ ( one_one @ nat ) ) ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% Sum_Icc_nat
thf(fact_3904_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_3905_arith__series,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [A3: A,D2: A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( plus_plus @ A @ A3 @ ( times_times @ A @ ( semiring_1_of_nat @ A @ I2 ) @ D2 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
= ( divide_divide @ A @ ( times_times @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N ) @ ( one_one @ A ) ) @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ D2 ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% arith_series
thf(fact_3906_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_3907_sum__gp__offset,axiom,
! [A: $tType] :
( ( ( division_ring @ A )
& ( comm_ring @ A ) )
=> ! [X: A,M2: nat,N: nat] :
( ( ( X
= ( one_one @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_or1337092689740270186AtMost @ nat @ M2 @ ( plus_plus @ nat @ M2 @ 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 @ M2 @ ( plus_plus @ nat @ M2 @ N ) ) )
= ( divide_divide @ A @ ( times_times @ A @ ( power_power @ A @ X @ M2 ) @ ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ X @ ( suc @ N ) ) ) ) @ ( minus_minus @ A @ ( one_one @ A ) @ X ) ) ) ) ) ) ).
% sum_gp_offset
thf(fact_3908_cos__paired,axiom,
! [X: real] :
( sums @ real
@ ^ [N5: nat] : ( times_times @ real @ ( divide_divide @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ N5 ) @ ( semiring_char_0_fact @ real @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N5 ) ) ) @ ( power_power @ real @ X @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N5 ) ) )
@ ( cos @ real @ X ) ) ).
% cos_paired
thf(fact_3909_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_3910_gchoose__row__sum__weighted,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [R4: A,M2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( times_times @ A @ ( gbinomial @ A @ R4 @ K3 ) @ ( minus_minus @ A @ ( divide_divide @ A @ R4 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( semiring_1_of_nat @ A @ K3 ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ M2 ) )
= ( times_times @ A @ ( divide_divide @ A @ ( semiring_1_of_nat @ A @ ( suc @ M2 ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( gbinomial @ A @ R4 @ ( suc @ M2 ) ) ) ) ) ).
% gchoose_row_sum_weighted
thf(fact_3911_vebt__member_Opelims_I2_J,axiom,
! [X: vEBT_VEBT,Xa: nat] :
( ( vEBT_vebt_member @ X @ Xa )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa ) )
=> ( ! [A6: $o,B4: $o] :
( ( X
= ( vEBT_Leaf @ A6 @ B4 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A6 @ B4 ) @ Xa ) )
=> ~ ( ( ( Xa
= ( zero_zero @ nat ) )
=> A6 )
& ( ( Xa
!= ( zero_zero @ nat ) )
=> ( ( ( Xa
= ( one_one @ nat ) )
=> B4 )
& ( Xa
= ( one_one @ nat ) ) ) ) ) ) )
=> ~ ! [Mi: nat,Ma2: nat,Va: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma2 ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary2 ) @ Xa ) )
=> ~ ( ( Xa != Mi )
=> ( ( Xa != Ma2 )
=> ( ~ ( ord_less @ nat @ Xa @ Mi )
& ( ~ ( ord_less @ nat @ Xa @ Mi )
=> ( ~ ( ord_less @ nat @ Ma2 @ Xa )
& ( ~ ( ord_less @ nat @ Ma2 @ Xa )
=> ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_vebt_member @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_member.pelims(2)
thf(fact_3912_geometric__deriv__sums,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [Z2: A] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ Z2 ) @ ( one_one @ real ) )
=> ( sums @ A
@ ^ [N5: nat] : ( times_times @ A @ ( semiring_1_of_nat @ A @ ( suc @ N5 ) ) @ ( power_power @ A @ Z2 @ N5 ) )
@ ( divide_divide @ A @ ( one_one @ A ) @ ( power_power @ A @ ( minus_minus @ A @ ( one_one @ A ) @ Z2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% geometric_deriv_sums
thf(fact_3913_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 ) )
=> ( ! [A6: $o,B4: $o] :
( ( X
= ( vEBT_Leaf @ A6 @ B4 ) )
=> ( ( Y2
= ( ( ( Xa
= ( zero_zero @ nat ) )
=> A6 )
& ( ( Xa
!= ( zero_zero @ nat ) )
=> ( ( ( Xa
= ( one_one @ nat ) )
=> B4 )
& ( Xa
= ( one_one @ nat ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A6 @ B4 ) @ Xa ) ) ) )
=> ( ! [Uu2: option @ ( product_prod @ nat @ nat ),Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Uu2 @ ( zero_zero @ nat ) @ Uv2 @ Uw2 ) )
=> ( ~ Y2
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Uu2 @ ( zero_zero @ nat ) @ Uv2 @ Uw2 ) @ Xa ) ) ) )
=> ~ ! [Uy: option @ ( product_prod @ nat @ nat ),V4: nat,TreeList2: list @ vEBT_VEBT,S2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Uy @ ( suc @ V4 ) @ TreeList2 @ S2 ) )
=> ( ( Y2
= ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( 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 @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( 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 @ V4 ) @ TreeList2 @ S2 ) @ Xa ) ) ) ) ) ) ) ) ).
% VEBT_internal.naive_member.pelims(1)
thf(fact_3914_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 ) )
=> ( ! [A6: $o,B4: $o] :
( ( X
= ( vEBT_Leaf @ A6 @ B4 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A6 @ B4 ) @ Xa ) )
=> ~ ( ( ( Xa
= ( zero_zero @ nat ) )
=> A6 )
& ( ( Xa
!= ( zero_zero @ nat ) )
=> ( ( ( Xa
= ( one_one @ nat ) )
=> B4 )
& ( Xa
= ( one_one @ nat ) ) ) ) ) ) )
=> ~ ! [Uy: option @ ( product_prod @ nat @ nat ),V4: nat,TreeList2: list @ vEBT_VEBT,S2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Uy @ ( suc @ V4 ) @ TreeList2 @ S2 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Uy @ ( suc @ V4 ) @ TreeList2 @ S2 ) @ Xa ) )
=> ~ ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( 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 @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ) ).
% VEBT_internal.naive_member.pelims(2)
thf(fact_3915_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 ) )
=> ( ! [A6: $o,B4: $o] :
( ( X
= ( vEBT_Leaf @ A6 @ B4 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A6 @ B4 ) @ Xa ) )
=> ( ( ( Xa
= ( zero_zero @ nat ) )
=> A6 )
& ( ( Xa
!= ( zero_zero @ nat ) )
=> ( ( ( Xa
= ( one_one @ nat ) )
=> B4 )
& ( Xa
= ( one_one @ nat ) ) ) ) ) ) )
=> ( ! [Uu2: option @ ( product_prod @ nat @ nat ),Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Uu2 @ ( zero_zero @ nat ) @ Uv2 @ Uw2 ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Uu2 @ ( zero_zero @ nat ) @ Uv2 @ Uw2 ) @ Xa ) ) )
=> ~ ! [Uy: option @ ( product_prod @ nat @ nat ),V4: nat,TreeList2: list @ vEBT_VEBT,S2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Uy @ ( suc @ V4 ) @ TreeList2 @ S2 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Uy @ ( suc @ V4 ) @ TreeList2 @ S2 ) @ Xa ) )
=> ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( 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 @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.naive_member.pelims(3)
thf(fact_3916_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 ) ) )
=> ( ! [Ux2: list @ vEBT_VEBT,Uy: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( zero_zero @ nat ) @ Ux2 @ 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 ) @ Ux2 @ Uy ) @ Xa ) ) )
=> ( ! [Mi: nat,Ma2: nat,Va3: list @ vEBT_VEBT,Vb2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma2 ) ) @ ( zero_zero @ nat ) @ Va3 @ Vb2 ) )
=> ( ( 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 @ Mi @ Ma2 ) ) @ ( zero_zero @ nat ) @ Va3 @ Vb2 ) @ Xa ) )
=> ( ( Xa = Mi )
| ( Xa = Ma2 ) ) ) )
=> ( ! [Mi: nat,Ma2: nat,V4: nat,TreeList2: list @ vEBT_VEBT,Vc2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma2 ) ) @ ( suc @ V4 ) @ TreeList2 @ Vc2 ) )
=> ( ( 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 @ Mi @ Ma2 ) ) @ ( suc @ V4 ) @ TreeList2 @ Vc2 ) @ Xa ) )
=> ( ( Xa = Mi )
| ( Xa = Ma2 )
| ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( 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 @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) )
=> ~ ! [V4: nat,TreeList2: list @ vEBT_VEBT,Vd: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V4 ) @ TreeList2 @ Vd ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V4 ) @ TreeList2 @ Vd ) @ Xa ) )
=> ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( 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 @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.membermima.pelims(3)
thf(fact_3917_int__sum,axiom,
! [B: $tType,F2: B > nat,A4: set @ B] :
( ( semiring_1_of_nat @ int @ ( groups7311177749621191930dd_sum @ B @ nat @ F2 @ A4 ) )
= ( groups7311177749621191930dd_sum @ B @ int
@ ^ [X2: B] : ( semiring_1_of_nat @ int @ ( F2 @ X2 ) )
@ A4 ) ) ).
% int_sum
thf(fact_3918_sum__subtractf__nat,axiom,
! [A: $tType,A4: set @ A,G: A > nat,F2: A > nat] :
( ! [X3: A] :
( ( member @ A @ X3 @ A4 )
=> ( ord_less_eq @ nat @ ( G @ X3 ) @ ( F2 @ X3 ) ) )
=> ( ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [X2: A] : ( minus_minus @ nat @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ A4 )
= ( minus_minus @ nat @ ( groups7311177749621191930dd_sum @ A @ nat @ F2 @ A4 ) @ ( groups7311177749621191930dd_sum @ A @ nat @ G @ A4 ) ) ) ) ).
% sum_subtractf_nat
thf(fact_3919_sum__SucD,axiom,
! [A: $tType,F2: A > nat,A4: set @ A,N: nat] :
( ( ( groups7311177749621191930dd_sum @ A @ nat @ F2 @ A4 )
= ( suc @ N ) )
=> ? [X3: A] :
( ( member @ A @ X3 @ A4 )
& ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F2 @ X3 ) ) ) ) ).
% sum_SucD
thf(fact_3920_sum__diff1__nat,axiom,
! [A: $tType,A3: A,A4: set @ A,F2: A > nat] :
( ( ( member @ A @ A3 @ A4 )
=> ( ( groups7311177749621191930dd_sum @ A @ nat @ F2 @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( minus_minus @ nat @ ( groups7311177749621191930dd_sum @ A @ nat @ F2 @ A4 ) @ ( F2 @ A3 ) ) ) )
& ( ~ ( member @ A @ A3 @ A4 )
=> ( ( groups7311177749621191930dd_sum @ A @ nat @ F2 @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( groups7311177749621191930dd_sum @ A @ nat @ F2 @ A4 ) ) ) ) ).
% sum_diff1_nat
thf(fact_3921_sum__nth__roots,axiom,
! [N: nat,C2: complex] :
( ( ord_less @ nat @ ( one_one @ nat ) @ N )
=> ( ( groups7311177749621191930dd_sum @ complex @ complex
@ ^ [X2: complex] : X2
@ ( collect @ complex
@ ^ [Z6: complex] :
( ( power_power @ complex @ Z6 @ N )
= C2 ) ) )
= ( zero_zero @ complex ) ) ) ).
% sum_nth_roots
thf(fact_3922_norm__prod__diff,axiom,
! [A: $tType,I6: $tType] :
( ( ( comm_monoid_mult @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [I5: set @ I6,Z2: I6 > A,W: I6 > A] :
( ! [I3: I6] :
( ( member @ I6 @ I3 @ I5 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( Z2 @ I3 ) ) @ ( one_one @ real ) ) )
=> ( ! [I3: I6] :
( ( member @ I6 @ I3 @ I5 )
=> ( 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 @ I6 @ A @ Z2 @ I5 ) @ ( groups7121269368397514597t_prod @ I6 @ A @ W @ I5 ) ) )
@ ( groups7311177749621191930dd_sum @ I6 @ real
@ ^ [I2: I6] : ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( Z2 @ I2 ) @ ( W @ I2 ) ) )
@ I5 ) ) ) ) ) ).
% norm_prod_diff
thf(fact_3923_sum__roots__unity,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( one_one @ nat ) @ N )
=> ( ( groups7311177749621191930dd_sum @ complex @ complex
@ ^ [X2: complex] : X2
@ ( collect @ complex
@ ^ [Z6: complex] :
( ( power_power @ complex @ Z6 @ N )
= ( one_one @ complex ) ) ) )
= ( zero_zero @ complex ) ) ) ).
% sum_roots_unity
thf(fact_3924_Sum__Icc__int,axiom,
! [M2: int,N: int] :
( ( ord_less_eq @ int @ M2 @ N )
=> ( ( groups7311177749621191930dd_sum @ int @ int
@ ^ [X2: int] : X2
@ ( set_or1337092689740270186AtMost @ int @ M2 @ N ) )
= ( divide_divide @ int @ ( minus_minus @ int @ ( times_times @ int @ N @ ( plus_plus @ int @ N @ ( one_one @ int ) ) ) @ ( times_times @ int @ M2 @ ( minus_minus @ int @ M2 @ ( one_one @ int ) ) ) ) @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ).
% Sum_Icc_int
thf(fact_3925_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 ) )
=> ( ! [Mi: nat,Ma2: nat,Va3: list @ vEBT_VEBT,Vb2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma2 ) ) @ ( zero_zero @ nat ) @ Va3 @ Vb2 ) )
=> ( ( 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 @ Mi @ Ma2 ) ) @ ( zero_zero @ nat ) @ Va3 @ Vb2 ) @ Xa ) )
=> ~ ( ( Xa = Mi )
| ( Xa = Ma2 ) ) ) )
=> ( ! [Mi: nat,Ma2: nat,V4: nat,TreeList2: list @ vEBT_VEBT,Vc2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma2 ) ) @ ( suc @ V4 ) @ TreeList2 @ Vc2 ) )
=> ( ( 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 @ Mi @ Ma2 ) ) @ ( suc @ V4 ) @ TreeList2 @ Vc2 ) @ Xa ) )
=> ~ ( ( Xa = Mi )
| ( Xa = Ma2 )
| ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( 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 @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) )
=> ~ ! [V4: nat,TreeList2: list @ vEBT_VEBT,Vd: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V4 ) @ TreeList2 @ Vd ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V4 ) @ TreeList2 @ Vd ) @ Xa ) )
=> ~ ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( 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 @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.membermima.pelims(2)
thf(fact_3926_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 ) ) ) )
=> ( ! [Ux2: list @ vEBT_VEBT,Uy: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( zero_zero @ nat ) @ Ux2 @ 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 ) @ Ux2 @ Uy ) @ Xa ) ) ) )
=> ( ! [Mi: nat,Ma2: nat,Va3: list @ vEBT_VEBT,Vb2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma2 ) ) @ ( zero_zero @ nat ) @ Va3 @ Vb2 ) )
=> ( ( Y2
= ( ( Xa = Mi )
| ( Xa = Ma2 ) ) )
=> ~ ( 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 @ Mi @ Ma2 ) ) @ ( zero_zero @ nat ) @ Va3 @ Vb2 ) @ Xa ) ) ) )
=> ( ! [Mi: nat,Ma2: nat,V4: nat,TreeList2: list @ vEBT_VEBT,Vc2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma2 ) ) @ ( suc @ V4 ) @ TreeList2 @ Vc2 ) )
=> ( ( Y2
= ( ( Xa = Mi )
| ( Xa = Ma2 )
| ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( 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 @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( 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 @ Mi @ Ma2 ) ) @ ( suc @ V4 ) @ TreeList2 @ Vc2 ) @ Xa ) ) ) )
=> ~ ! [V4: nat,TreeList2: list @ vEBT_VEBT,Vd: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V4 ) @ TreeList2 @ Vd ) )
=> ( ( Y2
= ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( 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 @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide @ nat @ ( suc @ V4 ) @ ( 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 @ V4 ) @ TreeList2 @ Vd ) @ Xa ) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.membermima.pelims(1)
thf(fact_3927_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
@ ^ [M5: nat] : ( times_times @ real @ ( cos_coeff @ M5 ) @ ( power_power @ real @ X @ M5 ) )
@ ( 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_3928_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
@ ^ [M5: nat] : ( times_times @ real @ ( cos_coeff @ M5 ) @ ( power_power @ real @ X @ M5 ) )
@ ( 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_3929_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
@ ^ [M5: nat] : ( times_times @ real @ ( sin_coeff @ M5 ) @ ( power_power @ real @ X @ M5 ) )
@ ( 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_3930_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
@ ^ [M5: nat] : ( times_times @ real @ ( sin_coeff @ M5 ) @ ( power_power @ real @ X @ M5 ) )
@ ( 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_3931_lessThan__iff,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [I: A,K: A] :
( ( member @ A @ I @ ( set_ord_lessThan @ A @ K ) )
= ( ord_less @ A @ I @ K ) ) ) ).
% lessThan_iff
thf(fact_3932_lessThan__0,axiom,
( ( set_ord_lessThan @ nat @ ( zero_zero @ nat ) )
= ( bot_bot @ ( set @ nat ) ) ) ).
% lessThan_0
thf(fact_3933_single__Diff__lessThan,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [K: A] :
( ( minus_minus @ ( set @ A ) @ ( insert @ A @ K @ ( bot_bot @ ( set @ A ) ) ) @ ( set_ord_lessThan @ A @ K ) )
= ( insert @ A @ K @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% single_Diff_lessThan
thf(fact_3934_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_3935_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_3936_sumr__cos__zero__one,axiom,
! [N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M5: nat] : ( times_times @ real @ ( cos_coeff @ M5 ) @ ( power_power @ real @ ( zero_zero @ real ) @ M5 ) )
@ ( set_ord_lessThan @ nat @ ( suc @ N ) ) )
= ( one_one @ real ) ) ).
% sumr_cos_zero_one
thf(fact_3937_lessThan__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( set_ord_lessThan @ A )
= ( ^ [U2: A] :
( collect @ A
@ ^ [X2: A] : ( ord_less @ A @ X2 @ U2 ) ) ) ) ) ).
% lessThan_def
thf(fact_3938_lessThan__strict__subset__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [M2: A,N: A] :
( ( ord_less @ ( set @ A ) @ ( set_ord_lessThan @ A @ M2 ) @ ( set_ord_lessThan @ A @ N ) )
= ( ord_less @ A @ M2 @ N ) ) ) ).
% lessThan_strict_subset_iff
thf(fact_3939_lessThan__Suc,axiom,
! [K: nat] :
( ( set_ord_lessThan @ nat @ ( suc @ K ) )
= ( insert @ nat @ K @ ( set_ord_lessThan @ nat @ K ) ) ) ).
% lessThan_Suc
thf(fact_3940_lessThan__empty__iff,axiom,
! [N: nat] :
( ( ( set_ord_lessThan @ nat @ N )
= ( bot_bot @ ( set @ nat ) ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% lessThan_empty_iff
thf(fact_3941_sum_Onat__diff__reindex,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( G @ ( minus_minus @ nat @ N @ ( suc @ I2 ) ) )
@ ( set_ord_lessThan @ nat @ N ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% sum.nat_diff_reindex
thf(fact_3942_prod_Onat__diff__reindex,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I2: nat] : ( G @ ( minus_minus @ nat @ N @ ( suc @ I2 ) ) )
@ ( set_ord_lessThan @ nat @ N ) )
= ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% prod.nat_diff_reindex
thf(fact_3943_sum__diff__distrib,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [Q: A > nat,P: A > nat,N: A] :
( ! [X3: A] : ( ord_less_eq @ nat @ ( Q @ X3 ) @ ( P @ X3 ) )
=> ( ( minus_minus @ nat @ ( groups7311177749621191930dd_sum @ A @ nat @ P @ ( set_ord_lessThan @ A @ N ) ) @ ( groups7311177749621191930dd_sum @ A @ nat @ Q @ ( set_ord_lessThan @ A @ N ) ) )
= ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [X2: A] : ( minus_minus @ nat @ ( P @ X2 ) @ ( Q @ X2 ) )
@ ( set_ord_lessThan @ A @ N ) ) ) ) ) ).
% sum_diff_distrib
thf(fact_3944_sum_OlessThan__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_lessThan @ nat @ ( suc @ N ) ) )
= ( plus_plus @ A @ ( G @ ( zero_zero @ nat ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% sum.lessThan_Suc_shift
thf(fact_3945_sumr__diff__mult__const2,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [F2: nat > A,N: nat,R4: A] :
( ( minus_minus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_ord_lessThan @ nat @ N ) ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ R4 ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( minus_minus @ A @ ( F2 @ I2 ) @ R4 )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% sumr_diff_mult_const2
thf(fact_3946_sum__lessThan__telescope_H,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [F2: nat > A,M2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [N5: nat] : ( minus_minus @ A @ ( F2 @ N5 ) @ ( F2 @ ( suc @ N5 ) ) )
@ ( set_ord_lessThan @ nat @ M2 ) )
= ( minus_minus @ A @ ( F2 @ ( zero_zero @ nat ) ) @ ( F2 @ M2 ) ) ) ) ).
% sum_lessThan_telescope'
thf(fact_3947_sum__lessThan__telescope,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [F2: nat > A,M2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [N5: nat] : ( minus_minus @ A @ ( F2 @ ( suc @ N5 ) ) @ ( F2 @ N5 ) )
@ ( set_ord_lessThan @ nat @ M2 ) )
= ( minus_minus @ A @ ( F2 @ M2 ) @ ( F2 @ ( zero_zero @ nat ) ) ) ) ) ).
% sum_lessThan_telescope
thf(fact_3948_summableI__nonneg__bounded,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( ordere6911136660526730532id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F2: nat > A,X: A] :
( ! [N2: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ N2 ) )
=> ( ! [N2: nat] : ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_ord_lessThan @ nat @ N2 ) ) @ X )
=> ( summable @ A @ F2 ) ) ) ) ).
% summableI_nonneg_bounded
thf(fact_3949_prod_OlessThan__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_lessThan @ nat @ ( suc @ N ) ) )
= ( times_times @ A @ ( G @ ( zero_zero @ nat ) )
@ ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% prod.lessThan_Suc_shift
thf(fact_3950_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_3951_sums__iff__shift_H,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A,N: nat,S: A] :
( ( sums @ A
@ ^ [I2: nat] : ( F2 @ ( plus_plus @ nat @ I2 @ N ) )
@ ( minus_minus @ A @ S @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_ord_lessThan @ nat @ N ) ) ) )
= ( sums @ A @ F2 @ S ) ) ) ).
% sums_iff_shift'
thf(fact_3952_sums__split__initial__segment,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A,S: A,N: nat] :
( ( sums @ A @ F2 @ S )
=> ( sums @ A
@ ^ [I2: nat] : ( F2 @ ( plus_plus @ nat @ I2 @ N ) )
@ ( minus_minus @ A @ S @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ) ).
% sums_split_initial_segment
thf(fact_3953_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_3954_sum__bounds__lt__plus1,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [F2: nat > A,Mm: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( F2 @ ( suc @ K3 ) )
@ ( set_ord_lessThan @ nat @ Mm ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ Mm ) ) ) ) ).
% sum_bounds_lt_plus1
thf(fact_3955_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_3956_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_3957_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_3958_suminf__minus__initial__segment,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A,K: nat] :
( ( summable @ A @ F2 )
=> ( ( suminf @ A
@ ^ [N5: nat] : ( F2 @ ( plus_plus @ nat @ N5 @ K ) ) )
= ( minus_minus @ A @ ( suminf @ A @ F2 ) @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_ord_lessThan @ nat @ K ) ) ) ) ) ) ).
% suminf_minus_initial_segment
thf(fact_3959_sum__less__suminf,axiom,
! [A: $tType] :
( ( ( ordere8940638589300402666id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F2: nat > A,N: nat] :
( ( summable @ A @ F2 )
=> ( ! [M3: nat] :
( ( ord_less_eq @ nat @ N @ M3 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( F2 @ M3 ) ) )
=> ( ord_less @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_ord_lessThan @ nat @ N ) ) @ ( suminf @ A @ F2 ) ) ) ) ) ).
% sum_less_suminf
thf(fact_3960_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_3961_lemma__termdiff1,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [Z2: A,H: A,M2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [P6: nat] : ( minus_minus @ A @ ( times_times @ A @ ( power_power @ A @ ( plus_plus @ A @ Z2 @ H ) @ ( minus_minus @ nat @ M2 @ P6 ) ) @ ( power_power @ A @ Z2 @ P6 ) ) @ ( power_power @ A @ Z2 @ M2 ) )
@ ( set_ord_lessThan @ nat @ M2 ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [P6: nat] : ( times_times @ A @ ( power_power @ A @ Z2 @ P6 ) @ ( minus_minus @ A @ ( power_power @ A @ ( plus_plus @ A @ Z2 @ H ) @ ( minus_minus @ nat @ M2 @ P6 ) ) @ ( power_power @ A @ Z2 @ ( minus_minus @ nat @ M2 @ P6 ) ) ) )
@ ( set_ord_lessThan @ nat @ M2 ) ) ) ) ).
% lemma_termdiff1
thf(fact_3962_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
@ ^ [I2: nat] : ( times_times @ A @ ( power_power @ A @ Y2 @ ( minus_minus @ nat @ N @ ( suc @ I2 ) ) ) @ ( power_power @ A @ X @ I2 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% power_diff_sumr2
thf(fact_3963_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
@ ^ [P6: nat] : ( times_times @ A @ ( power_power @ A @ X @ P6 ) @ ( power_power @ A @ Y2 @ ( minus_minus @ nat @ N @ P6 ) ) )
@ ( set_ord_lessThan @ nat @ ( suc @ N ) ) ) ) ) ) ).
% diff_power_eq_sum
thf(fact_3964_real__sum__nat__ivl__bounded2,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N: nat,F2: nat > A,K6: A,K: nat] :
( ! [P7: nat] :
( ( ord_less @ nat @ P7 @ N )
=> ( ord_less_eq @ A @ ( F2 @ P7 ) @ K6 ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ K6 )
=> ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_ord_lessThan @ nat @ ( minus_minus @ nat @ N @ K ) ) ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ K6 ) ) ) ) ) ).
% real_sum_nat_ivl_bounded2
thf(fact_3965_sum__less__suminf2,axiom,
! [A: $tType] :
( ( ( ordere8940638589300402666id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F2: nat > A,N: nat,I: nat] :
( ( summable @ A @ F2 )
=> ( ! [M3: nat] :
( ( ord_less_eq @ nat @ N @ M3 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ M3 ) ) )
=> ( ( ord_less_eq @ nat @ N @ I )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ ( F2 @ I ) )
=> ( ord_less @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_ord_lessThan @ nat @ N ) ) @ ( suminf @ A @ F2 ) ) ) ) ) ) ) ).
% sum_less_suminf2
thf(fact_3966_one__diff__power__eq_H,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [X: A,N: nat] :
( ( minus_minus @ A @ ( one_one @ A ) @ ( power_power @ A @ X @ N ) )
= ( times_times @ A @ ( minus_minus @ A @ ( one_one @ A ) @ X )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( power_power @ A @ X @ ( minus_minus @ nat @ N @ ( suc @ I2 ) ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% one_diff_power_eq'
thf(fact_3967_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
@ ^ [M5: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M5 @ ( zero_zero @ A ) ) @ ( semiring_char_0_fact @ real @ M5 ) ) @ ( power_power @ real @ X @ M5 ) )
@ ( set_ord_lessThan @ nat @ N ) )
= ( Diff @ ( zero_zero @ nat ) @ ( zero_zero @ A ) ) ) ) ) ) ).
% Maclaurin_zero
thf(fact_3968_Maclaurin__lemma,axiom,
! [H: real,F2: real > real,J2: nat > real,N: nat] :
( ( ord_less @ real @ ( zero_zero @ real ) @ H )
=> ? [B8: real] :
( ( F2 @ H )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M5: nat] : ( times_times @ real @ ( divide_divide @ real @ ( J2 @ M5 ) @ ( semiring_char_0_fact @ real @ M5 ) ) @ ( power_power @ real @ H @ M5 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ B8 @ ( divide_divide @ real @ ( power_power @ real @ H @ N ) @ ( semiring_char_0_fact @ real @ N ) ) ) ) ) ) ).
% Maclaurin_lemma
thf(fact_3969_sum__split__even__odd,axiom,
! [F2: nat > real,G: nat > real,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I2: nat] : ( if @ real @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I2 ) @ ( F2 @ I2 ) @ ( G @ I2 ) )
@ ( set_ord_lessThan @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I2: nat] : ( F2 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I2 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I2: nat] : ( G @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I2 ) @ ( one_one @ nat ) ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% sum_split_even_odd
thf(fact_3970_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
@ ^ [M5: nat] : ( divide_divide @ real @ ( power_power @ real @ X @ M5 ) @ ( semiring_char_0_fact @ real @ M5 ) )
@ ( 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_3971_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
@ ^ [M5: nat] : ( times_times @ real @ ( sin_coeff @ M5 ) @ ( power_power @ real @ X @ M5 ) )
@ ( 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_3972_sum__pos__lt__pair,axiom,
! [F2: nat > real,K: nat] :
( ( summable @ real @ F2 )
=> ( ! [D5: nat] : ( ord_less @ real @ ( zero_zero @ real ) @ ( plus_plus @ real @ ( F2 @ ( plus_plus @ nat @ K @ ( times_times @ nat @ ( suc @ ( suc @ ( zero_zero @ nat ) ) ) @ D5 ) ) ) @ ( F2 @ ( plus_plus @ nat @ K @ ( plus_plus @ nat @ ( times_times @ nat @ ( suc @ ( suc @ ( zero_zero @ nat ) ) ) @ D5 ) @ ( one_one @ nat ) ) ) ) ) )
=> ( ord_less @ real @ ( groups7311177749621191930dd_sum @ nat @ real @ F2 @ ( set_ord_lessThan @ nat @ K ) ) @ ( suminf @ real @ F2 ) ) ) ) ).
% sum_pos_lt_pair
thf(fact_3973_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
@ ^ [M5: nat] : ( divide_divide @ real @ ( power_power @ real @ X @ M5 ) @ ( semiring_char_0_fact @ real @ M5 ) )
@ ( 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_3974_lemma__termdiff2,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [H: A,Z2: A,N: nat] :
( ( H
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( divide_divide @ A @ ( minus_minus @ A @ ( power_power @ A @ ( plus_plus @ A @ Z2 @ H ) @ N ) @ ( power_power @ A @ Z2 @ N ) ) @ H ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ ( power_power @ A @ Z2 @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) ) ) )
= ( times_times @ A @ H
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [P6: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [Q4: nat] : ( times_times @ A @ ( power_power @ A @ ( plus_plus @ A @ Z2 @ H ) @ Q4 ) @ ( power_power @ A @ Z2 @ ( minus_minus @ nat @ ( minus_minus @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ Q4 ) ) )
@ ( set_ord_lessThan @ nat @ ( minus_minus @ nat @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) @ P6 ) ) )
@ ( set_ord_lessThan @ nat @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ) ) ).
% lemma_termdiff2
thf(fact_3975_Maclaurin__sin__expansion,axiom,
! [X: real,N: nat] :
? [T4: real] :
( ( sin @ real @ X )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M5: nat] : ( times_times @ real @ ( sin_coeff @ M5 ) @ ( power_power @ real @ X @ M5 ) )
@ ( 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_3976_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
@ ^ [M5: nat] : ( times_times @ real @ ( sin_coeff @ M5 ) @ ( power_power @ real @ X @ M5 ) )
@ ( 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_3977_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
@ ^ [M5: nat] : ( times_times @ real @ ( cos_coeff @ M5 ) @ ( power_power @ real @ X @ M5 ) )
@ ( 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_3978_diffs__equiv,axiom,
! [A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( ring_1 @ A ) )
=> ! [C2: nat > A,X: A] :
( ( summable @ A
@ ^ [N5: nat] : ( times_times @ A @ ( diffs @ A @ C2 @ N5 ) @ ( power_power @ A @ X @ N5 ) ) )
=> ( sums @ A
@ ^ [N5: nat] : ( times_times @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N5 ) @ ( C2 @ N5 ) ) @ ( power_power @ A @ X @ ( minus_minus @ nat @ N5 @ ( suc @ ( zero_zero @ nat ) ) ) ) )
@ ( suminf @ A
@ ^ [N5: nat] : ( times_times @ A @ ( diffs @ A @ C2 @ N5 ) @ ( power_power @ A @ X @ N5 ) ) ) ) ) ) ).
% diffs_equiv
thf(fact_3979_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_3980_gbinomial__partial__row__sum,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: A,M2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( times_times @ A @ ( gbinomial @ A @ A3 @ K3 ) @ ( minus_minus @ A @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( semiring_1_of_nat @ A @ K3 ) ) )
@ ( set_ord_atMost @ nat @ M2 ) )
= ( times_times @ A @ ( divide_divide @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ M2 ) @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( gbinomial @ A @ A3 @ ( plus_plus @ nat @ M2 @ ( one_one @ nat ) ) ) ) ) ) ).
% gbinomial_partial_row_sum
thf(fact_3981_choose__even__sum,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( if @ A @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I2 ) @ ( semiring_1_of_nat @ A @ ( binomial @ N @ I2 ) ) @ ( zero_zero @ A ) )
@ ( set_ord_atMost @ nat @ N ) ) )
= ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% choose_even_sum
thf(fact_3982_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_3983_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_3984_atMost__0,axiom,
( ( set_ord_atMost @ nat @ ( zero_zero @ nat ) )
= ( insert @ nat @ ( zero_zero @ nat ) @ ( bot_bot @ ( set @ nat ) ) ) ) ).
% atMost_0
thf(fact_3985_diffs__of__real,axiom,
! [A: $tType] :
( ( real_V2191834092415804123ebra_1 @ A )
=> ! [F2: nat > real] :
( ( diffs @ A
@ ^ [N5: nat] : ( real_Vector_of_real @ A @ ( F2 @ N5 ) ) )
= ( ^ [N5: nat] : ( real_Vector_of_real @ A @ ( diffs @ real @ F2 @ N5 ) ) ) ) ) ).
% diffs_of_real
thf(fact_3986_atMost__atLeast0,axiom,
( ( set_ord_atMost @ nat )
= ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) ) ) ).
% atMost_atLeast0
thf(fact_3987_lessThan__Suc__atMost,axiom,
! [K: nat] :
( ( set_ord_lessThan @ nat @ ( suc @ K ) )
= ( set_ord_atMost @ nat @ K ) ) ).
% lessThan_Suc_atMost
thf(fact_3988_diffs__minus,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [C2: nat > A] :
( ( diffs @ A
@ ^ [N5: nat] : ( uminus_uminus @ A @ ( C2 @ N5 ) ) )
= ( ^ [N5: nat] : ( uminus_uminus @ A @ ( diffs @ A @ C2 @ N5 ) ) ) ) ) ).
% diffs_minus
thf(fact_3989_atMost__Suc,axiom,
! [K: nat] :
( ( set_ord_atMost @ nat @ ( suc @ K ) )
= ( insert @ nat @ ( suc @ K ) @ ( set_ord_atMost @ nat @ K ) ) ) ).
% atMost_Suc
thf(fact_3990_Iic__subset__Iio__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_ord_atMost @ A @ A3 ) @ ( set_ord_lessThan @ A @ B2 ) )
= ( ord_less @ A @ A3 @ B2 ) ) ) ).
% Iic_subset_Iio_iff
thf(fact_3991_exp__fdiffs,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ( ( diffs @ A
@ ^ [N5: nat] : ( inverse_inverse @ A @ ( semiring_char_0_fact @ A @ N5 ) ) )
= ( ^ [N5: nat] : ( inverse_inverse @ A @ ( semiring_char_0_fact @ A @ N5 ) ) ) ) ) ).
% exp_fdiffs
thf(fact_3992_sum__choose__upper,axiom,
! [M2: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [K3: nat] : ( binomial @ K3 @ M2 )
@ ( set_ord_atMost @ nat @ N ) )
= ( binomial @ ( suc @ N ) @ ( suc @ M2 ) ) ) ).
% sum_choose_upper
thf(fact_3993_diffs__sin__coeff,axiom,
( ( diffs @ real @ sin_coeff )
= cos_coeff ) ).
% diffs_sin_coeff
thf(fact_3994_diffs__def,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( ( diffs @ A )
= ( ^ [C4: nat > A,N5: nat] : ( times_times @ A @ ( semiring_1_of_nat @ A @ ( suc @ N5 ) ) @ ( C4 @ ( suc @ N5 ) ) ) ) ) ) ).
% diffs_def
thf(fact_3995_sum_OatMost__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_atMost @ nat @ ( suc @ N ) ) )
= ( plus_plus @ A @ ( G @ ( zero_zero @ nat ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ) ).
% sum.atMost_Suc_shift
thf(fact_3996_sum__telescope,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [F2: nat > A,I: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( minus_minus @ A @ ( F2 @ I2 ) @ ( F2 @ ( suc @ I2 ) ) )
@ ( set_ord_atMost @ nat @ I ) )
= ( minus_minus @ A @ ( F2 @ ( zero_zero @ nat ) ) @ ( F2 @ ( suc @ I ) ) ) ) ) ).
% sum_telescope
thf(fact_3997_polyfun__eq__coeffs,axiom,
! [A: $tType] :
( ( ( real_V8999393235501362500lgebra @ A )
& ( idom @ A ) )
=> ! [C2: nat > A,N: nat,D2: nat > A] :
( ( ! [X2: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( times_times @ A @ ( C2 @ I2 ) @ ( power_power @ A @ X2 @ I2 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( times_times @ A @ ( D2 @ I2 ) @ ( power_power @ A @ X2 @ I2 ) )
@ ( set_ord_atMost @ nat @ N ) ) ) )
= ( ! [I2: nat] :
( ( ord_less_eq @ nat @ I2 @ N )
=> ( ( C2 @ I2 )
= ( D2 @ I2 ) ) ) ) ) ) ).
% polyfun_eq_coeffs
thf(fact_3998_bounded__imp__summable,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( linord2810124833399127020strict @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [A3: nat > A,B7: A] :
( ! [N2: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( A3 @ N2 ) )
=> ( ! [N2: nat] : ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ A3 @ ( set_ord_atMost @ nat @ N2 ) ) @ B7 )
=> ( summable @ A @ A3 ) ) ) ) ).
% bounded_imp_summable
thf(fact_3999_prod_OatMost__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_atMost @ nat @ ( suc @ N ) ) )
= ( times_times @ A @ ( G @ ( zero_zero @ nat ) )
@ ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ) ).
% prod.atMost_Suc_shift
thf(fact_4000_sum_Onested__swap_H,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A3: nat > nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( groups7311177749621191930dd_sum @ nat @ A @ ( A3 @ I2 ) @ ( set_ord_lessThan @ nat @ I2 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [J3: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( A3 @ I2 @ J3 )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ J3 ) @ N ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% sum.nested_swap'
thf(fact_4001_prod_Onested__swap_H,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: nat > nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I2: nat] : ( groups7121269368397514597t_prod @ nat @ A @ ( A3 @ I2 ) @ ( set_ord_lessThan @ nat @ I2 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [J3: nat] :
( groups7121269368397514597t_prod @ nat @ A
@ ^ [I2: nat] : ( A3 @ I2 @ J3 )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ J3 ) @ N ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% prod.nested_swap'
thf(fact_4002_sum__choose__lower,axiom,
! [R4: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [K3: nat] : ( binomial @ ( plus_plus @ nat @ R4 @ K3 ) @ K3 )
@ ( set_ord_atMost @ nat @ N ) )
= ( binomial @ ( suc @ ( plus_plus @ nat @ R4 @ N ) ) @ N ) ) ).
% sum_choose_lower
thf(fact_4003_choose__rising__sum_I1_J,axiom,
! [N: nat,M2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [J3: nat] : ( binomial @ ( plus_plus @ nat @ N @ J3 ) @ N )
@ ( set_ord_atMost @ nat @ M2 ) )
= ( binomial @ ( plus_plus @ nat @ ( plus_plus @ nat @ N @ M2 ) @ ( one_one @ nat ) ) @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) ) ).
% choose_rising_sum(1)
thf(fact_4004_choose__rising__sum_I2_J,axiom,
! [N: nat,M2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [J3: nat] : ( binomial @ ( plus_plus @ nat @ N @ J3 ) @ N )
@ ( set_ord_atMost @ nat @ M2 ) )
= ( binomial @ ( plus_plus @ nat @ ( plus_plus @ nat @ N @ M2 ) @ ( one_one @ nat ) ) @ M2 ) ) ).
% choose_rising_sum(2)
thf(fact_4005_termdiff__converges__all,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [C2: nat > A,X: A] :
( ! [X3: A] :
( summable @ A
@ ^ [N5: nat] : ( times_times @ A @ ( C2 @ N5 ) @ ( power_power @ A @ X3 @ N5 ) ) )
=> ( summable @ A
@ ^ [N5: nat] : ( times_times @ A @ ( diffs @ A @ C2 @ N5 ) @ ( power_power @ A @ X @ N5 ) ) ) ) ) ).
% termdiff_converges_all
thf(fact_4006_diffs__cos__coeff,axiom,
( ( diffs @ real @ cos_coeff )
= ( ^ [N5: nat] : ( uminus_uminus @ real @ ( sin_coeff @ N5 ) ) ) ) ).
% diffs_cos_coeff
thf(fact_4007_zero__polynom__imp__zero__coeffs,axiom,
! [A: $tType] :
( ( ( ab_semigroup_mult @ A )
& ( real_V8999393235501362500lgebra @ A ) )
=> ! [C2: nat > A,N: nat,K: nat] :
( ! [W2: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( times_times @ A @ ( C2 @ I2 ) @ ( power_power @ A @ W2 @ I2 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ nat @ K @ N )
=> ( ( C2 @ K )
= ( zero_zero @ A ) ) ) ) ) ).
% zero_polynom_imp_zero_coeffs
thf(fact_4008_polyfun__eq__0,axiom,
! [A: $tType] :
( ( ( real_V8999393235501362500lgebra @ A )
& ( idom @ A ) )
=> ! [C2: nat > A,N: nat] :
( ( ! [X2: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( times_times @ A @ ( C2 @ I2 ) @ ( power_power @ A @ X2 @ I2 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( zero_zero @ A ) ) )
= ( ! [I2: nat] :
( ( ord_less_eq @ nat @ I2 @ N )
=> ( ( C2 @ I2 )
= ( zero_zero @ A ) ) ) ) ) ) ).
% polyfun_eq_0
thf(fact_4009_sum_OatMost__shift,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_atMost @ nat @ N ) )
= ( plus_plus @ A @ ( G @ ( zero_zero @ nat ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% sum.atMost_shift
thf(fact_4010_sum__up__index__split,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [F2: nat > A,M2: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_ord_atMost @ nat @ ( plus_plus @ nat @ M2 @ N ) ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_ord_atMost @ nat @ M2 ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M2 ) @ ( plus_plus @ nat @ M2 @ N ) ) ) ) ) ) ).
% sum_up_index_split
thf(fact_4011_prod_OatMost__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_atMost @ nat @ N ) )
= ( times_times @ A @ ( G @ ( zero_zero @ nat ) )
@ ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% prod.atMost_shift
thf(fact_4012_atLeast1__atMost__eq__remove0,axiom,
! [N: nat] :
( ( set_or1337092689740270186AtMost @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N )
= ( minus_minus @ ( set @ nat ) @ ( set_ord_atMost @ nat @ N ) @ ( insert @ nat @ ( zero_zero @ nat ) @ ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% atLeast1_atMost_eq_remove0
thf(fact_4013_gbinomial__parallel__sum,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( gbinomial @ A @ ( plus_plus @ A @ A3 @ ( semiring_1_of_nat @ A @ K3 ) ) @ K3 )
@ ( set_ord_atMost @ nat @ N ) )
= ( gbinomial @ A @ ( plus_plus @ A @ ( plus_plus @ A @ A3 @ ( semiring_1_of_nat @ A @ N ) ) @ ( one_one @ A ) ) @ N ) ) ) ).
% gbinomial_parallel_sum
thf(fact_4014_sum__choose__diagonal,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [K3: nat] : ( binomial @ ( minus_minus @ nat @ N @ K3 ) @ ( minus_minus @ nat @ M2 @ K3 ) )
@ ( set_ord_atMost @ nat @ M2 ) )
= ( binomial @ ( suc @ N ) @ M2 ) ) ) ).
% sum_choose_diagonal
thf(fact_4015_vandermonde,axiom,
! [M2: nat,N: nat,R4: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [K3: nat] : ( times_times @ nat @ ( binomial @ M2 @ K3 ) @ ( binomial @ N @ ( minus_minus @ nat @ R4 @ K3 ) ) )
@ ( set_ord_atMost @ nat @ R4 ) )
= ( binomial @ ( plus_plus @ nat @ M2 @ N ) @ R4 ) ) ).
% vandermonde
thf(fact_4016_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_4017_polyfun__linear__factor__root,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [C2: nat > A,A3: A,N: nat] :
( ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( times_times @ A @ ( C2 @ I2 ) @ ( power_power @ A @ A3 @ I2 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( zero_zero @ A ) )
=> ~ ! [B4: nat > A] :
~ ! [Z5: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( times_times @ A @ ( C2 @ I2 ) @ ( power_power @ A @ Z5 @ I2 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( times_times @ A @ ( minus_minus @ A @ Z5 @ A3 )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( times_times @ A @ ( B4 @ I2 ) @ ( power_power @ A @ Z5 @ I2 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ) ).
% polyfun_linear_factor_root
thf(fact_4018_polyfun__linear__factor,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [C2: nat > A,N: nat,A3: A] :
? [B4: nat > A] :
! [Z5: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( times_times @ A @ ( C2 @ I2 ) @ ( power_power @ A @ Z5 @ I2 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( plus_plus @ A
@ ( times_times @ A @ ( minus_minus @ A @ Z5 @ A3 )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( times_times @ A @ ( B4 @ I2 ) @ ( power_power @ A @ Z5 @ I2 ) )
@ ( set_ord_lessThan @ nat @ N ) ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( times_times @ A @ ( C2 @ I2 ) @ ( power_power @ A @ A3 @ I2 ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ) ).
% polyfun_linear_factor
thf(fact_4019_sum__power__shift,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [M2: nat,N: nat,X: A] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
= ( times_times @ A @ ( power_power @ A @ X @ M2 ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ ( set_ord_atMost @ nat @ ( minus_minus @ nat @ N @ M2 ) ) ) ) ) ) ) ).
% sum_power_shift
thf(fact_4020_summable__Cauchy__product,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V4412858255891104859lgebra @ A ) )
=> ! [A3: nat > A,B2: nat > A] :
( ( summable @ real
@ ^ [K3: nat] : ( real_V7770717601297561774m_norm @ A @ ( A3 @ K3 ) ) )
=> ( ( summable @ real
@ ^ [K3: nat] : ( real_V7770717601297561774m_norm @ A @ ( B2 @ K3 ) ) )
=> ( summable @ A
@ ^ [K3: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( times_times @ A @ ( A3 @ I2 ) @ ( B2 @ ( minus_minus @ nat @ K3 @ I2 ) ) )
@ ( set_ord_atMost @ nat @ K3 ) ) ) ) ) ) ).
% summable_Cauchy_product
thf(fact_4021_Cauchy__product,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V4412858255891104859lgebra @ A ) )
=> ! [A3: nat > A,B2: nat > A] :
( ( summable @ real
@ ^ [K3: nat] : ( real_V7770717601297561774m_norm @ A @ ( A3 @ K3 ) ) )
=> ( ( summable @ real
@ ^ [K3: nat] : ( real_V7770717601297561774m_norm @ A @ ( B2 @ K3 ) ) )
=> ( ( times_times @ A @ ( suminf @ A @ A3 ) @ ( suminf @ A @ B2 ) )
= ( suminf @ A
@ ^ [K3: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( times_times @ A @ ( A3 @ I2 ) @ ( B2 @ ( minus_minus @ nat @ K3 @ I2 ) ) )
@ ( set_ord_atMost @ nat @ K3 ) ) ) ) ) ) ) ).
% Cauchy_product
thf(fact_4022_binomial,axiom,
! [A3: nat,B2: nat,N: nat] :
( ( power_power @ nat @ ( plus_plus @ nat @ A3 @ B2 ) @ N )
= ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [K3: nat] : ( times_times @ nat @ ( times_times @ nat @ ( semiring_1_of_nat @ nat @ ( binomial @ N @ K3 ) ) @ ( power_power @ nat @ A3 @ K3 ) ) @ ( power_power @ nat @ B2 @ ( minus_minus @ nat @ N @ K3 ) ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ).
% binomial
thf(fact_4023_sum_Oin__pairs__0,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_ord_atMost @ nat @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( plus_plus @ A @ ( G @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I2 ) ) @ ( G @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I2 ) ) ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% sum.in_pairs_0
thf(fact_4024_polynomial__product,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [M2: nat,A3: nat > A,N: nat,B2: nat > A,X: A] :
( ! [I3: nat] :
( ( ord_less @ nat @ M2 @ I3 )
=> ( ( A3 @ I3 )
= ( zero_zero @ A ) ) )
=> ( ! [J: nat] :
( ( ord_less @ nat @ N @ J )
=> ( ( B2 @ J )
= ( zero_zero @ A ) ) )
=> ( ( times_times @ A
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( times_times @ A @ ( A3 @ I2 ) @ ( power_power @ A @ X @ I2 ) )
@ ( set_ord_atMost @ nat @ M2 ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [J3: nat] : ( times_times @ A @ ( B2 @ J3 ) @ ( power_power @ A @ X @ J3 ) )
@ ( set_ord_atMost @ nat @ N ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [R: nat] :
( times_times @ A
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( times_times @ A @ ( A3 @ K3 ) @ ( B2 @ ( minus_minus @ nat @ R @ K3 ) ) )
@ ( set_ord_atMost @ nat @ R ) )
@ ( power_power @ A @ X @ R ) )
@ ( set_ord_atMost @ nat @ ( plus_plus @ nat @ M2 @ N ) ) ) ) ) ) ) ).
% polynomial_product
thf(fact_4025_prod_Oin__pairs__0,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_ord_atMost @ nat @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I2: nat] : ( times_times @ A @ ( G @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I2 ) ) @ ( G @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I2 ) ) ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% prod.in_pairs_0
thf(fact_4026_polyfun__eq__const,axiom,
! [A: $tType] :
( ( ( real_V8999393235501362500lgebra @ A )
& ( idom @ A ) )
=> ! [C2: nat > A,N: nat,K: A] :
( ( ! [X2: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( times_times @ A @ ( C2 @ I2 ) @ ( power_power @ A @ X2 @ I2 ) )
@ ( set_ord_atMost @ nat @ N ) )
= K ) )
= ( ( ( C2 @ ( zero_zero @ nat ) )
= K )
& ! [X2: nat] :
( ( member @ nat @ X2 @ ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ N ) )
=> ( ( C2 @ X2 )
= ( zero_zero @ A ) ) ) ) ) ) ).
% polyfun_eq_const
thf(fact_4027_gbinomial__sum__lower__neg,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: A,M2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( times_times @ A @ ( gbinomial @ A @ A3 @ K3 ) @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ K3 ) )
@ ( set_ord_atMost @ nat @ M2 ) )
= ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ M2 ) @ ( gbinomial @ A @ ( minus_minus @ A @ A3 @ ( one_one @ A ) ) @ M2 ) ) ) ) ).
% gbinomial_sum_lower_neg
thf(fact_4028_binomial__ring,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A3: A,B2: A,N: nat] :
( ( power_power @ A @ ( plus_plus @ A @ A3 @ B2 ) @ N )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( times_times @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( binomial @ N @ K3 ) ) @ ( power_power @ A @ A3 @ K3 ) ) @ ( power_power @ A @ B2 @ ( minus_minus @ nat @ N @ K3 ) ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% binomial_ring
thf(fact_4029_pochhammer__binomial__sum,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [A3: A,B2: A,N: nat] :
( ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ A3 @ B2 ) @ N )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( times_times @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( binomial @ N @ K3 ) ) @ ( comm_s3205402744901411588hammer @ A @ A3 @ K3 ) ) @ ( comm_s3205402744901411588hammer @ A @ B2 @ ( minus_minus @ nat @ N @ K3 ) ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% pochhammer_binomial_sum
thf(fact_4030_polynomial__product__nat,axiom,
! [M2: nat,A3: nat > nat,N: nat,B2: nat > nat,X: nat] :
( ! [I3: nat] :
( ( ord_less @ nat @ M2 @ I3 )
=> ( ( A3 @ I3 )
= ( zero_zero @ nat ) ) )
=> ( ! [J: nat] :
( ( ord_less @ nat @ N @ J )
=> ( ( B2 @ J )
= ( zero_zero @ nat ) ) )
=> ( ( times_times @ nat
@ ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [I2: nat] : ( times_times @ nat @ ( A3 @ I2 ) @ ( power_power @ nat @ X @ I2 ) )
@ ( set_ord_atMost @ nat @ M2 ) )
@ ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [J3: nat] : ( times_times @ nat @ ( B2 @ J3 ) @ ( power_power @ nat @ X @ J3 ) )
@ ( set_ord_atMost @ nat @ N ) ) )
= ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [R: nat] :
( times_times @ nat
@ ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [K3: nat] : ( times_times @ nat @ ( A3 @ K3 ) @ ( B2 @ ( minus_minus @ nat @ R @ K3 ) ) )
@ ( set_ord_atMost @ nat @ R ) )
@ ( power_power @ nat @ X @ R ) )
@ ( set_ord_atMost @ nat @ ( plus_plus @ nat @ M2 @ N ) ) ) ) ) ) ).
% polynomial_product_nat
thf(fact_4031_Cauchy__product__sums,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V4412858255891104859lgebra @ A ) )
=> ! [A3: nat > A,B2: nat > A] :
( ( summable @ real
@ ^ [K3: nat] : ( real_V7770717601297561774m_norm @ A @ ( A3 @ K3 ) ) )
=> ( ( summable @ real
@ ^ [K3: nat] : ( real_V7770717601297561774m_norm @ A @ ( B2 @ K3 ) ) )
=> ( sums @ A
@ ^ [K3: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( times_times @ A @ ( A3 @ I2 ) @ ( B2 @ ( minus_minus @ nat @ K3 @ I2 ) ) )
@ ( set_ord_atMost @ nat @ K3 ) )
@ ( times_times @ A @ ( suminf @ A @ A3 ) @ ( suminf @ A @ B2 ) ) ) ) ) ) ).
% Cauchy_product_sums
thf(fact_4032_termdiff__converges,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A,K6: real,C2: nat > A] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ K6 )
=> ( ! [X3: A] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X3 ) @ K6 )
=> ( summable @ A
@ ^ [N5: nat] : ( times_times @ A @ ( C2 @ N5 ) @ ( power_power @ A @ X3 @ N5 ) ) ) )
=> ( summable @ A
@ ^ [N5: nat] : ( times_times @ A @ ( diffs @ A @ C2 @ N5 ) @ ( power_power @ A @ X @ N5 ) ) ) ) ) ) ).
% termdiff_converges
thf(fact_4033_sum_Ozero__middle,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [P4: nat,K: nat,G: nat > A,H: nat > A] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ P4 )
=> ( ( ord_less_eq @ nat @ K @ P4 )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [J3: nat] : ( if @ A @ ( ord_less @ nat @ J3 @ K ) @ ( G @ J3 ) @ ( if @ A @ ( J3 = K ) @ ( zero_zero @ A ) @ ( H @ ( minus_minus @ nat @ J3 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) )
@ ( set_ord_atMost @ nat @ P4 ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [J3: nat] : ( if @ A @ ( ord_less @ nat @ J3 @ K ) @ ( G @ J3 ) @ ( H @ J3 ) )
@ ( set_ord_atMost @ nat @ ( minus_minus @ nat @ P4 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ) ) ).
% sum.zero_middle
thf(fact_4034_prod_Ozero__middle,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [P4: nat,K: nat,G: nat > A,H: nat > A] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ P4 )
=> ( ( ord_less_eq @ nat @ K @ P4 )
=> ( ( groups7121269368397514597t_prod @ nat @ A
@ ^ [J3: nat] : ( if @ A @ ( ord_less @ nat @ J3 @ K ) @ ( G @ J3 ) @ ( if @ A @ ( J3 = K ) @ ( one_one @ A ) @ ( H @ ( minus_minus @ nat @ J3 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) )
@ ( set_ord_atMost @ nat @ P4 ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [J3: nat] : ( if @ A @ ( ord_less @ nat @ J3 @ K ) @ ( G @ J3 ) @ ( H @ J3 ) )
@ ( set_ord_atMost @ nat @ ( minus_minus @ nat @ P4 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ) ) ).
% prod.zero_middle
thf(fact_4035_gbinomial__partial__sum__poly,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [M2: nat,A3: 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 @ M2 ) @ A3 ) @ K3 ) @ ( power_power @ A @ X @ K3 ) ) @ ( power_power @ A @ Y2 @ ( minus_minus @ nat @ M2 @ K3 ) ) )
@ ( set_ord_atMost @ nat @ M2 ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( times_times @ A @ ( times_times @ A @ ( gbinomial @ A @ ( uminus_uminus @ A @ A3 ) @ K3 ) @ ( power_power @ A @ ( uminus_uminus @ A @ X ) @ K3 ) ) @ ( power_power @ A @ ( plus_plus @ A @ X @ Y2 ) @ ( minus_minus @ nat @ M2 @ K3 ) ) )
@ ( set_ord_atMost @ nat @ M2 ) ) ) ) ).
% gbinomial_partial_sum_poly
thf(fact_4036_root__polyfun,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [N: nat,Z2: A,A3: A] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ N )
=> ( ( ( power_power @ A @ Z2 @ N )
= A3 )
= ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] :
( times_times @ A
@ ( if @ A
@ ( I2
= ( zero_zero @ nat ) )
@ ( uminus_uminus @ A @ A3 )
@ ( if @ A @ ( I2 = N ) @ ( one_one @ A ) @ ( zero_zero @ A ) ) )
@ ( power_power @ A @ Z2 @ I2 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( zero_zero @ A ) ) ) ) ) ).
% root_polyfun
thf(fact_4037_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_4038_choose__alternating__linear__sum,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [N: nat] :
( ( N
!= ( one_one @ nat ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( times_times @ A @ ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ I2 ) @ ( semiring_1_of_nat @ A @ I2 ) ) @ ( semiring_1_of_nat @ A @ ( binomial @ N @ I2 ) ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( zero_zero @ A ) ) ) ) ).
% choose_alternating_linear_sum
thf(fact_4039_gbinomial__sum__nat__pow2,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [M2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( divide_divide @ A @ ( gbinomial @ A @ ( semiring_1_of_nat @ A @ ( plus_plus @ nat @ M2 @ K3 ) ) @ K3 ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ K3 ) )
@ ( set_ord_atMost @ nat @ M2 ) )
= ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M2 ) ) ) ).
% gbinomial_sum_nat_pow2
thf(fact_4040_gbinomial__partial__sum__poly__xpos,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [M2: nat,A3: 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 @ M2 ) @ A3 ) @ K3 ) @ ( power_power @ A @ X @ K3 ) ) @ ( power_power @ A @ Y2 @ ( minus_minus @ nat @ M2 @ K3 ) ) )
@ ( set_ord_atMost @ nat @ M2 ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( times_times @ A @ ( times_times @ A @ ( gbinomial @ A @ ( minus_minus @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ K3 ) @ A3 ) @ ( one_one @ A ) ) @ K3 ) @ ( power_power @ A @ X @ K3 ) ) @ ( power_power @ A @ ( plus_plus @ A @ X @ Y2 ) @ ( minus_minus @ nat @ M2 @ K3 ) ) )
@ ( set_ord_atMost @ nat @ M2 ) ) ) ) ).
% gbinomial_partial_sum_poly_xpos
thf(fact_4041_polyfun__diff__alt,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [N: nat,A3: nat > A,X: A,Y2: A] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ N )
=> ( ( minus_minus @ A
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( times_times @ A @ ( A3 @ I2 ) @ ( power_power @ A @ X @ I2 ) )
@ ( set_ord_atMost @ nat @ N ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( times_times @ A @ ( A3 @ I2 ) @ ( power_power @ A @ Y2 @ I2 ) )
@ ( set_ord_atMost @ nat @ N ) ) )
= ( times_times @ A @ ( minus_minus @ A @ X @ Y2 )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [J3: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( times_times @ A @ ( times_times @ A @ ( A3 @ ( plus_plus @ nat @ ( plus_plus @ nat @ J3 @ K3 ) @ ( one_one @ nat ) ) ) @ ( power_power @ A @ Y2 @ K3 ) ) @ ( power_power @ A @ X @ J3 ) )
@ ( set_ord_lessThan @ nat @ ( minus_minus @ nat @ N @ J3 ) ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ) ).
% polyfun_diff_alt
thf(fact_4042_binomial__r__part__sum,axiom,
! [M2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat @ ( binomial @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M2 ) @ ( one_one @ nat ) ) ) @ ( set_ord_atMost @ nat @ M2 ) )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M2 ) ) ) ).
% binomial_r_part_sum
thf(fact_4043_choose__linear__sum,axiom,
! [N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [I2: nat] : ( times_times @ nat @ I2 @ ( binomial @ N @ I2 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( times_times @ nat @ N @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ).
% choose_linear_sum
thf(fact_4044_choose__alternating__sum,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ I2 ) @ ( semiring_1_of_nat @ A @ ( binomial @ N @ I2 ) ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( zero_zero @ A ) ) ) ) ).
% choose_alternating_sum
thf(fact_4045_polyfun__extremal__lemma,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [E2: real,C2: nat > A,N: nat] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E2 )
=> ? [M9: real] :
! [Z5: A] :
( ( ord_less_eq @ real @ M9 @ ( real_V7770717601297561774m_norm @ A @ Z5 ) )
=> ( ord_less_eq @ real
@ ( real_V7770717601297561774m_norm @ A
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( times_times @ A @ ( C2 @ I2 ) @ ( power_power @ A @ Z5 @ I2 ) )
@ ( set_ord_atMost @ nat @ N ) ) )
@ ( times_times @ real @ E2 @ ( power_power @ real @ ( real_V7770717601297561774m_norm @ A @ Z5 ) @ ( suc @ N ) ) ) ) ) ) ) ).
% polyfun_extremal_lemma
thf(fact_4046_polyfun__diff,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [N: nat,A3: nat > A,X: A,Y2: A] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ N )
=> ( ( minus_minus @ A
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( times_times @ A @ ( A3 @ I2 ) @ ( power_power @ A @ X @ I2 ) )
@ ( set_ord_atMost @ nat @ N ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( times_times @ A @ ( A3 @ I2 ) @ ( power_power @ A @ Y2 @ I2 ) )
@ ( set_ord_atMost @ nat @ N ) ) )
= ( times_times @ A @ ( minus_minus @ A @ X @ Y2 )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [J3: nat] :
( times_times @ A
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( times_times @ A @ ( A3 @ I2 ) @ ( power_power @ A @ Y2 @ ( minus_minus @ nat @ ( minus_minus @ nat @ I2 @ J3 ) @ ( one_one @ nat ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ J3 ) @ N ) )
@ ( power_power @ A @ X @ J3 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ) ).
% polyfun_diff
thf(fact_4047_gbinomial__r__part__sum,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [M2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ ( gbinomial @ A @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ A @ M2 ) ) @ ( one_one @ A ) ) ) @ ( set_ord_atMost @ nat @ M2 ) )
= ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M2 ) ) ) ) ).
% gbinomial_r_part_sum
thf(fact_4048_choose__odd__sum,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] :
( if @ A
@ ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I2 )
@ ( semiring_1_of_nat @ A @ ( binomial @ N @ I2 ) )
@ ( zero_zero @ A ) )
@ ( set_ord_atMost @ nat @ N ) ) )
= ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% choose_odd_sum
thf(fact_4049_sin__x__sin__y,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A,Y2: A] :
( sums @ A
@ ^ [P6: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [N5: nat] :
( if @ A
@ ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ P6 )
& ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N5 ) )
@ ( 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 @ P6 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( semiring_1_of_nat @ int @ ( binomial @ P6 @ N5 ) ) ) ) @ ( semiring_char_0_fact @ real @ P6 ) ) ) @ ( power_power @ A @ X @ N5 ) ) @ ( power_power @ A @ Y2 @ ( minus_minus @ nat @ P6 @ N5 ) ) )
@ ( zero_zero @ A ) )
@ ( set_ord_atMost @ nat @ P6 ) )
@ ( times_times @ A @ ( sin @ A @ X ) @ ( sin @ A @ Y2 ) ) ) ) ).
% sin_x_sin_y
thf(fact_4050_sums__cos__x__plus__y,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A,Y2: A] :
( sums @ A
@ ^ [P6: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [N5: nat] : ( if @ A @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ P6 ) @ ( 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 @ P6 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( semiring_1_of_nat @ int @ ( binomial @ P6 @ N5 ) ) ) ) @ ( semiring_char_0_fact @ real @ P6 ) ) @ ( power_power @ A @ X @ N5 ) ) @ ( power_power @ A @ Y2 @ ( minus_minus @ nat @ P6 @ N5 ) ) ) @ ( zero_zero @ A ) )
@ ( set_ord_atMost @ nat @ P6 ) )
@ ( cos @ A @ ( plus_plus @ A @ X @ Y2 ) ) ) ) ).
% sums_cos_x_plus_y
thf(fact_4051_cos__x__cos__y,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A,Y2: A] :
( sums @ A
@ ^ [P6: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [N5: nat] :
( if @ A
@ ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ P6 )
& ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N5 ) )
@ ( 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 @ P6 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( semiring_1_of_nat @ int @ ( binomial @ P6 @ N5 ) ) ) ) @ ( semiring_char_0_fact @ real @ P6 ) ) @ ( power_power @ A @ X @ N5 ) ) @ ( power_power @ A @ Y2 @ ( minus_minus @ nat @ P6 @ N5 ) ) )
@ ( zero_zero @ A ) )
@ ( set_ord_atMost @ nat @ P6 ) )
@ ( times_times @ A @ ( cos @ A @ X ) @ ( cos @ A @ Y2 ) ) ) ) ).
% cos_x_cos_y
thf(fact_4052_exp__first__two__terms,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ( ( exp @ A )
= ( ^ [X2: A] :
( plus_plus @ A @ ( plus_plus @ A @ ( one_one @ A ) @ X2 )
@ ( suminf @ A
@ ^ [N5: nat] : ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ ( plus_plus @ nat @ N5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( power_power @ A @ X2 @ ( plus_plus @ nat @ N5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% exp_first_two_terms
thf(fact_4053_scaleR__zero__right,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [A3: real] :
( ( real_V8093663219630862766scaleR @ A @ A3 @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% scaleR_zero_right
thf(fact_4054_scaleR__cancel__right,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [A3: real,X: A,B2: real] :
( ( ( real_V8093663219630862766scaleR @ A @ A3 @ X )
= ( real_V8093663219630862766scaleR @ A @ B2 @ X ) )
= ( ( A3 = B2 )
| ( X
= ( zero_zero @ A ) ) ) ) ) ).
% scaleR_cancel_right
thf(fact_4055_scaleR__minus__right,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [A3: real,X: A] :
( ( real_V8093663219630862766scaleR @ A @ A3 @ ( uminus_uminus @ A @ X ) )
= ( uminus_uminus @ A @ ( real_V8093663219630862766scaleR @ A @ A3 @ X ) ) ) ) ).
% scaleR_minus_right
thf(fact_4056_scaleR__one,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [X: A] :
( ( real_V8093663219630862766scaleR @ A @ ( one_one @ real ) @ X )
= X ) ) ).
% scaleR_one
thf(fact_4057_scaleR__eq__0__iff,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [A3: real,X: A] :
( ( ( real_V8093663219630862766scaleR @ A @ A3 @ X )
= ( zero_zero @ A ) )
= ( ( A3
= ( zero_zero @ real ) )
| ( X
= ( zero_zero @ A ) ) ) ) ) ).
% scaleR_eq_0_iff
thf(fact_4058_scaleR__zero__left,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [X: A] :
( ( real_V8093663219630862766scaleR @ A @ ( zero_zero @ real ) @ X )
= ( zero_zero @ A ) ) ) ).
% scaleR_zero_left
thf(fact_4059_scaleR__eq__iff,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [B2: A,U: real,A3: A] :
( ( ( plus_plus @ A @ B2 @ ( real_V8093663219630862766scaleR @ A @ U @ A3 ) )
= ( plus_plus @ A @ A3 @ ( real_V8093663219630862766scaleR @ A @ U @ B2 ) ) )
= ( ( A3 = B2 )
| ( U
= ( one_one @ real ) ) ) ) ) ).
% scaleR_eq_iff
thf(fact_4060_scaleR__left_Ominus,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [X: real,Xa: A] :
( ( real_V8093663219630862766scaleR @ A @ ( uminus_uminus @ real @ X ) @ Xa )
= ( uminus_uminus @ A @ ( real_V8093663219630862766scaleR @ A @ X @ Xa ) ) ) ) ).
% scaleR_left.minus
thf(fact_4061_scaleR__minus__left,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [A3: real,X: A] :
( ( real_V8093663219630862766scaleR @ A @ ( uminus_uminus @ real @ A3 ) @ X )
= ( uminus_uminus @ A @ ( real_V8093663219630862766scaleR @ A @ A3 @ X ) ) ) ) ).
% scaleR_minus_left
thf(fact_4062_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_4063_scaleR__collapse,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [U: real,A3: A] :
( ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ ( minus_minus @ real @ ( one_one @ real ) @ U ) @ A3 ) @ ( real_V8093663219630862766scaleR @ A @ U @ A3 ) )
= A3 ) ) ).
% scaleR_collapse
thf(fact_4064_inverse__scaleR__times,axiom,
! [A: $tType] :
( ( real_V2191834092415804123ebra_1 @ A )
=> ! [V2: num,W: num,A3: A] :
( ( real_V8093663219630862766scaleR @ A @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ V2 ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ W ) @ A3 ) )
= ( real_V8093663219630862766scaleR @ A @ ( divide_divide @ real @ ( numeral_numeral @ real @ W ) @ ( numeral_numeral @ real @ V2 ) ) @ A3 ) ) ) ).
% inverse_scaleR_times
thf(fact_4065_fraction__scaleR__times,axiom,
! [A: $tType] :
( ( real_V2191834092415804123ebra_1 @ A )
=> ! [U: num,V2: num,W: num,A3: A] :
( ( real_V8093663219630862766scaleR @ A @ ( divide_divide @ real @ ( numeral_numeral @ real @ U ) @ ( numeral_numeral @ real @ V2 ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ W ) @ A3 ) )
= ( real_V8093663219630862766scaleR @ A @ ( divide_divide @ real @ ( times_times @ real @ ( numeral_numeral @ real @ U ) @ ( numeral_numeral @ real @ W ) ) @ ( numeral_numeral @ real @ V2 ) ) @ A3 ) ) ) ).
% fraction_scaleR_times
thf(fact_4066_scaleR__half__double,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [A3: A] :
( ( real_V8093663219630862766scaleR @ A @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( plus_plus @ A @ A3 @ A3 ) )
= A3 ) ) ).
% scaleR_half_double
thf(fact_4067_scaleR__right__diff__distrib,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [A3: real,X: A,Y2: A] :
( ( real_V8093663219630862766scaleR @ A @ A3 @ ( minus_minus @ A @ X @ Y2 ) )
= ( minus_minus @ A @ ( real_V8093663219630862766scaleR @ A @ A3 @ X ) @ ( real_V8093663219630862766scaleR @ A @ A3 @ Y2 ) ) ) ) ).
% scaleR_right_diff_distrib
thf(fact_4068_scaleR__right__imp__eq,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [X: A,A3: real,B2: real] :
( ( X
!= ( zero_zero @ A ) )
=> ( ( ( real_V8093663219630862766scaleR @ A @ A3 @ X )
= ( real_V8093663219630862766scaleR @ A @ B2 @ X ) )
=> ( A3 = B2 ) ) ) ) ).
% scaleR_right_imp_eq
thf(fact_4069_of__real__def,axiom,
! [A: $tType] :
( ( real_V2191834092415804123ebra_1 @ A )
=> ( ( real_Vector_of_real @ A )
= ( ^ [R: real] : ( real_V8093663219630862766scaleR @ A @ R @ ( one_one @ A ) ) ) ) ) ).
% of_real_def
thf(fact_4070_scaleR__left_Odiff,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [X: real,Y2: real,Xa: A] :
( ( real_V8093663219630862766scaleR @ A @ ( minus_minus @ real @ X @ Y2 ) @ Xa )
= ( minus_minus @ A @ ( real_V8093663219630862766scaleR @ A @ X @ Xa ) @ ( real_V8093663219630862766scaleR @ A @ Y2 @ Xa ) ) ) ) ).
% scaleR_left.diff
thf(fact_4071_scaleR__left__diff__distrib,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [A3: real,B2: real,X: A] :
( ( real_V8093663219630862766scaleR @ A @ ( minus_minus @ real @ A3 @ B2 ) @ X )
= ( minus_minus @ A @ ( real_V8093663219630862766scaleR @ A @ A3 @ X ) @ ( real_V8093663219630862766scaleR @ A @ B2 @ X ) ) ) ) ).
% scaleR_left_diff_distrib
thf(fact_4072_scaleR__right__mono__neg,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [B2: real,A3: real,C2: A] :
( ( ord_less_eq @ real @ B2 @ A3 )
=> ( ( ord_less_eq @ A @ C2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A3 @ C2 ) @ ( real_V8093663219630862766scaleR @ A @ B2 @ C2 ) ) ) ) ) ).
% scaleR_right_mono_neg
thf(fact_4073_scaleR__right__mono,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A3: real,B2: real,X: A] :
( ( ord_less_eq @ real @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A3 @ X ) @ ( real_V8093663219630862766scaleR @ A @ B2 @ X ) ) ) ) ) ).
% scaleR_right_mono
thf(fact_4074_vector__fraction__eq__iff,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [U: real,V2: real,A3: A,X: A] :
( ( ( real_V8093663219630862766scaleR @ A @ ( divide_divide @ real @ U @ V2 ) @ A3 )
= X )
= ( ( ( V2
= ( zero_zero @ real ) )
=> ( X
= ( zero_zero @ A ) ) )
& ( ( V2
!= ( zero_zero @ real ) )
=> ( ( real_V8093663219630862766scaleR @ A @ U @ A3 )
= ( real_V8093663219630862766scaleR @ A @ V2 @ X ) ) ) ) ) ) ).
% vector_fraction_eq_iff
thf(fact_4075_eq__vector__fraction__iff,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [X: A,U: real,V2: real,A3: A] :
( ( X
= ( real_V8093663219630862766scaleR @ A @ ( divide_divide @ real @ U @ V2 ) @ A3 ) )
= ( ( ( V2
= ( zero_zero @ real ) )
=> ( X
= ( zero_zero @ A ) ) )
& ( ( V2
!= ( zero_zero @ real ) )
=> ( ( real_V8093663219630862766scaleR @ A @ V2 @ X )
= ( real_V8093663219630862766scaleR @ A @ U @ A3 ) ) ) ) ) ) ).
% eq_vector_fraction_iff
thf(fact_4076_Real__Vector__Spaces_Ole__add__iff1,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A3: real,E2: A,C2: A,B2: real,D2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ A3 @ E2 ) @ C2 ) @ ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ B2 @ E2 ) @ D2 ) )
= ( ord_less_eq @ A @ ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ ( minus_minus @ real @ A3 @ B2 ) @ E2 ) @ C2 ) @ D2 ) ) ) ).
% Real_Vector_Spaces.le_add_iff1
thf(fact_4077_Real__Vector__Spaces_Ole__add__iff2,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A3: real,E2: A,C2: A,B2: real,D2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ A3 @ E2 ) @ C2 ) @ ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ B2 @ E2 ) @ D2 ) )
= ( ord_less_eq @ A @ C2 @ ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ ( minus_minus @ real @ B2 @ A3 ) @ E2 ) @ D2 ) ) ) ) ).
% Real_Vector_Spaces.le_add_iff2
thf(fact_4078_zero__le__scaleR__iff,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A3: real,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( real_V8093663219630862766scaleR @ A @ A3 @ B2 ) )
= ( ( ( ord_less @ real @ ( zero_zero @ real ) @ A3 )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) )
| ( ( ord_less @ real @ A3 @ ( zero_zero @ real ) )
& ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) )
| ( A3
= ( zero_zero @ real ) ) ) ) ) ).
% zero_le_scaleR_iff
thf(fact_4079_scaleR__le__0__iff,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A3: real,B2: A] :
( ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A3 @ B2 ) @ ( zero_zero @ A ) )
= ( ( ( ord_less @ real @ ( zero_zero @ real ) @ A3 )
& ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) )
| ( ( ord_less @ real @ A3 @ ( zero_zero @ real ) )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) )
| ( A3
= ( zero_zero @ real ) ) ) ) ) ).
% scaleR_le_0_iff
thf(fact_4080_scaleR__nonpos__nonpos,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A3: real,B2: A] :
( ( ord_less_eq @ real @ A3 @ ( zero_zero @ real ) )
=> ( ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( real_V8093663219630862766scaleR @ A @ A3 @ B2 ) ) ) ) ) ).
% scaleR_nonpos_nonpos
thf(fact_4081_scaleR__nonpos__nonneg,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A3: real,X: A] :
( ( ord_less_eq @ real @ A3 @ ( zero_zero @ real ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A3 @ X ) @ ( zero_zero @ A ) ) ) ) ) ).
% scaleR_nonpos_nonneg
thf(fact_4082_scaleR__nonneg__nonpos,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A3: real,X: A] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ A3 )
=> ( ( ord_less_eq @ A @ X @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A3 @ X ) @ ( zero_zero @ A ) ) ) ) ) ).
% scaleR_nonneg_nonpos
thf(fact_4083_scaleR__nonneg__nonneg,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A3: real,X: A] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( real_V8093663219630862766scaleR @ A @ A3 @ X ) ) ) ) ) ).
% scaleR_nonneg_nonneg
thf(fact_4084_split__scaleR__pos__le,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A3: real,B2: A] :
( ( ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ A3 )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) )
| ( ( ord_less_eq @ real @ A3 @ ( zero_zero @ real ) )
& ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( real_V8093663219630862766scaleR @ A @ A3 @ B2 ) ) ) ) ).
% split_scaleR_pos_le
thf(fact_4085_split__scaleR__neg__le,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A3: real,X: A] :
( ( ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ A3 )
& ( ord_less_eq @ A @ X @ ( zero_zero @ A ) ) )
| ( ( ord_less_eq @ real @ A3 @ ( zero_zero @ real ) )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ X ) ) )
=> ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A3 @ X ) @ ( zero_zero @ A ) ) ) ) ).
% split_scaleR_neg_le
thf(fact_4086_scaleR__mono_H,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A3: real,B2: real,C2: A,D2: A] :
( ( ord_less_eq @ real @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ D2 )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C2 )
=> ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A3 @ C2 ) @ ( real_V8093663219630862766scaleR @ A @ B2 @ D2 ) ) ) ) ) ) ) ).
% scaleR_mono'
thf(fact_4087_scaleR__mono,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A3: real,B2: real,X: A,Y2: A] :
( ( ord_less_eq @ real @ A3 @ 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 @ A3 @ X ) @ ( real_V8093663219630862766scaleR @ A @ B2 @ Y2 ) ) ) ) ) ) ) ).
% scaleR_mono
thf(fact_4088_scaleR__left__le__one__le,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [X: A,A3: real] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less_eq @ real @ A3 @ ( one_one @ real ) )
=> ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A3 @ X ) @ X ) ) ) ) ).
% scaleR_left_le_one_le
thf(fact_4089_real__vector__affinity__eq,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [M2: real,X: A,C2: A,Y2: A] :
( ( M2
!= ( zero_zero @ real ) )
=> ( ( ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ M2 @ X ) @ C2 )
= Y2 )
= ( X
= ( minus_minus @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ M2 ) @ Y2 ) @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ M2 ) @ C2 ) ) ) ) ) ) ).
% real_vector_affinity_eq
thf(fact_4090_real__vector__eq__affinity,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [M2: real,Y2: A,X: A,C2: A] :
( ( M2
!= ( zero_zero @ real ) )
=> ( ( Y2
= ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ M2 @ X ) @ C2 ) )
= ( ( minus_minus @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ M2 ) @ Y2 ) @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ M2 ) @ C2 ) )
= X ) ) ) ) ).
% real_vector_eq_affinity
thf(fact_4091_neg__less__divideR__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C2: real,A3: A,B2: A] :
( ( ord_less @ real @ C2 @ ( zero_zero @ real ) )
=> ( ( ord_less @ A @ A3 @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C2 ) @ B2 ) )
= ( ord_less @ A @ B2 @ ( real_V8093663219630862766scaleR @ A @ C2 @ A3 ) ) ) ) ) ).
% neg_less_divideR_eq
thf(fact_4092_neg__divideR__less__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C2: real,B2: A,A3: A] :
( ( ord_less @ real @ C2 @ ( zero_zero @ real ) )
=> ( ( ord_less @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C2 ) @ B2 ) @ A3 )
= ( ord_less @ A @ ( real_V8093663219630862766scaleR @ A @ C2 @ A3 ) @ B2 ) ) ) ) ).
% neg_divideR_less_eq
thf(fact_4093_pos__less__divideR__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C2: real,A3: A,B2: A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ C2 )
=> ( ( ord_less @ A @ A3 @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C2 ) @ B2 ) )
= ( ord_less @ A @ ( real_V8093663219630862766scaleR @ A @ C2 @ A3 ) @ B2 ) ) ) ) ).
% pos_less_divideR_eq
thf(fact_4094_pos__divideR__less__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C2: real,B2: A,A3: A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ C2 )
=> ( ( ord_less @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C2 ) @ B2 ) @ A3 )
= ( ord_less @ A @ B2 @ ( real_V8093663219630862766scaleR @ A @ C2 @ A3 ) ) ) ) ) ).
% pos_divideR_less_eq
thf(fact_4095_nonzero__inverse__scaleR__distrib,axiom,
! [A: $tType] :
( ( real_V5047593784448816457lgebra @ A )
=> ! [A3: real,X: A] :
( ( A3
!= ( zero_zero @ real ) )
=> ( ( X
!= ( zero_zero @ A ) )
=> ( ( inverse_inverse @ A @ ( real_V8093663219630862766scaleR @ A @ A3 @ X ) )
= ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ A3 ) @ ( inverse_inverse @ A @ X ) ) ) ) ) ) ).
% nonzero_inverse_scaleR_distrib
thf(fact_4096_summable__exp__generic,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X: A] :
( summable @ A
@ ^ [N5: nat] : ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ N5 ) ) @ ( power_power @ A @ X @ N5 ) ) ) ) ).
% summable_exp_generic
thf(fact_4097_sin__converges,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X: A] :
( sums @ A
@ ^ [N5: nat] : ( real_V8093663219630862766scaleR @ A @ ( sin_coeff @ N5 ) @ ( power_power @ A @ X @ N5 ) )
@ ( sin @ A @ X ) ) ) ).
% sin_converges
thf(fact_4098_sin__def,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ( ( sin @ A )
= ( ^ [X2: A] :
( suminf @ A
@ ^ [N5: nat] : ( real_V8093663219630862766scaleR @ A @ ( sin_coeff @ N5 ) @ ( power_power @ A @ X2 @ N5 ) ) ) ) ) ) ).
% sin_def
thf(fact_4099_cos__converges,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X: A] :
( sums @ A
@ ^ [N5: nat] : ( real_V8093663219630862766scaleR @ A @ ( cos_coeff @ N5 ) @ ( power_power @ A @ X @ N5 ) )
@ ( cos @ A @ X ) ) ) ).
% cos_converges
thf(fact_4100_cos__def,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ( ( cos @ A )
= ( ^ [X2: A] :
( suminf @ A
@ ^ [N5: nat] : ( real_V8093663219630862766scaleR @ A @ ( cos_coeff @ N5 ) @ ( power_power @ A @ X2 @ N5 ) ) ) ) ) ) ).
% cos_def
thf(fact_4101_summable__norm__sin,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X: A] :
( summable @ real
@ ^ [N5: nat] : ( real_V7770717601297561774m_norm @ A @ ( real_V8093663219630862766scaleR @ A @ ( sin_coeff @ N5 ) @ ( power_power @ A @ X @ N5 ) ) ) ) ) ).
% summable_norm_sin
thf(fact_4102_summable__norm__cos,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X: A] :
( summable @ real
@ ^ [N5: nat] : ( real_V7770717601297561774m_norm @ A @ ( real_V8093663219630862766scaleR @ A @ ( cos_coeff @ N5 ) @ ( power_power @ A @ X @ N5 ) ) ) ) ) ).
% summable_norm_cos
thf(fact_4103_neg__minus__divideR__le__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C2: real,B2: A,A3: A] :
( ( ord_less @ real @ C2 @ ( zero_zero @ real ) )
=> ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C2 ) @ B2 ) ) @ A3 )
= ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ C2 @ A3 ) @ ( uminus_uminus @ A @ B2 ) ) ) ) ) ).
% neg_minus_divideR_le_eq
thf(fact_4104_neg__le__minus__divideR__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C2: real,A3: A,B2: A] :
( ( ord_less @ real @ C2 @ ( zero_zero @ real ) )
=> ( ( ord_less_eq @ A @ A3 @ ( uminus_uminus @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C2 ) @ B2 ) ) )
= ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( real_V8093663219630862766scaleR @ A @ C2 @ A3 ) ) ) ) ) ).
% neg_le_minus_divideR_eq
thf(fact_4105_pos__minus__divideR__le__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C2: real,B2: A,A3: A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ C2 )
=> ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C2 ) @ B2 ) ) @ A3 )
= ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( real_V8093663219630862766scaleR @ A @ C2 @ A3 ) ) ) ) ) ).
% pos_minus_divideR_le_eq
thf(fact_4106_pos__le__minus__divideR__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C2: real,A3: A,B2: A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ C2 )
=> ( ( ord_less_eq @ A @ A3 @ ( uminus_uminus @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C2 ) @ B2 ) ) )
= ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ C2 @ A3 ) @ ( uminus_uminus @ A @ B2 ) ) ) ) ) ).
% pos_le_minus_divideR_eq
thf(fact_4107_neg__minus__divideR__less__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C2: real,B2: A,A3: A] :
( ( ord_less @ real @ C2 @ ( zero_zero @ real ) )
=> ( ( ord_less @ A @ ( uminus_uminus @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C2 ) @ B2 ) ) @ A3 )
= ( ord_less @ A @ ( real_V8093663219630862766scaleR @ A @ C2 @ A3 ) @ ( uminus_uminus @ A @ B2 ) ) ) ) ) ).
% neg_minus_divideR_less_eq
thf(fact_4108_neg__less__minus__divideR__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C2: real,A3: A,B2: A] :
( ( ord_less @ real @ C2 @ ( zero_zero @ real ) )
=> ( ( ord_less @ A @ A3 @ ( uminus_uminus @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C2 ) @ B2 ) ) )
= ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ ( real_V8093663219630862766scaleR @ A @ C2 @ A3 ) ) ) ) ) ).
% neg_less_minus_divideR_eq
thf(fact_4109_pos__minus__divideR__less__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C2: real,B2: A,A3: A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ C2 )
=> ( ( ord_less @ A @ ( uminus_uminus @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C2 ) @ B2 ) ) @ A3 )
= ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ ( real_V8093663219630862766scaleR @ A @ C2 @ A3 ) ) ) ) ) ).
% pos_minus_divideR_less_eq
thf(fact_4110_pos__less__minus__divideR__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C2: real,A3: A,B2: A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ C2 )
=> ( ( ord_less @ A @ A3 @ ( uminus_uminus @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C2 ) @ B2 ) ) )
= ( ord_less @ A @ ( real_V8093663219630862766scaleR @ A @ C2 @ A3 ) @ ( uminus_uminus @ A @ B2 ) ) ) ) ) ).
% pos_less_minus_divideR_eq
thf(fact_4111_exp__converges,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X: A] :
( sums @ A
@ ^ [N5: nat] : ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ N5 ) ) @ ( power_power @ A @ X @ N5 ) )
@ ( exp @ A @ X ) ) ) ).
% exp_converges
thf(fact_4112_exp__def,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ( ( exp @ A )
= ( ^ [X2: A] :
( suminf @ A
@ ^ [N5: nat] : ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ N5 ) ) @ ( power_power @ A @ X2 @ N5 ) ) ) ) ) ) ).
% exp_def
thf(fact_4113_summable__norm__exp,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X: A] :
( summable @ real
@ ^ [N5: nat] : ( real_V7770717601297561774m_norm @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ N5 ) ) @ ( power_power @ A @ X @ N5 ) ) ) ) ) ).
% summable_norm_exp
thf(fact_4114_sin__minus__converges,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X: A] :
( sums @ A
@ ^ [N5: nat] : ( uminus_uminus @ A @ ( real_V8093663219630862766scaleR @ A @ ( sin_coeff @ N5 ) @ ( power_power @ A @ ( uminus_uminus @ A @ X ) @ N5 ) ) )
@ ( sin @ A @ X ) ) ) ).
% sin_minus_converges
thf(fact_4115_cos__minus__converges,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X: A] :
( sums @ A
@ ^ [N5: nat] : ( real_V8093663219630862766scaleR @ A @ ( cos_coeff @ N5 ) @ ( power_power @ A @ ( uminus_uminus @ A @ X ) @ N5 ) )
@ ( cos @ A @ X ) ) ) ).
% cos_minus_converges
thf(fact_4116_cosh__def,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ( ( cosh @ A )
= ( ^ [X2: A] : ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( plus_plus @ A @ ( exp @ A @ X2 ) @ ( exp @ A @ ( uminus_uminus @ A @ X2 ) ) ) ) ) ) ) ).
% cosh_def
thf(fact_4117_sinh__def,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ( ( sinh @ A )
= ( ^ [X2: A] : ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( minus_minus @ A @ ( exp @ A @ X2 ) @ ( exp @ A @ ( uminus_uminus @ A @ X2 ) ) ) ) ) ) ) ).
% sinh_def
thf(fact_4118_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
@ ^ [I2: nat] : ( times_times @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ I2 ) ) @ ( power_power @ A @ X @ I2 ) ) @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ ( minus_minus @ nat @ N @ I2 ) ) ) @ ( power_power @ A @ Y2 @ ( minus_minus @ nat @ N @ I2 ) ) ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ) ).
% exp_series_add_commuting
thf(fact_4119_exp__first__term,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ( ( exp @ A )
= ( ^ [X2: A] :
( plus_plus @ A @ ( one_one @ A )
@ ( suminf @ A
@ ^ [N5: nat] : ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ ( suc @ N5 ) ) ) @ ( power_power @ A @ X2 @ ( suc @ N5 ) ) ) ) ) ) ) ) ).
% exp_first_term
thf(fact_4120_exp__first__terms,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [K: nat] :
( ( exp @ A )
= ( ^ [X2: A] :
( plus_plus @ A
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [N5: nat] : ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ N5 ) ) @ ( power_power @ A @ X2 @ N5 ) )
@ ( set_ord_lessThan @ nat @ K ) )
@ ( suminf @ A
@ ^ [N5: nat] : ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ ( plus_plus @ nat @ N5 @ K ) ) ) @ ( power_power @ A @ X2 @ ( plus_plus @ nat @ N5 @ K ) ) ) ) ) ) ) ) ).
% exp_first_terms
thf(fact_4121_cosh__converges,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X: A] :
( sums @ A
@ ^ [N5: nat] : ( if @ A @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N5 ) @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ N5 ) ) @ ( power_power @ A @ X @ N5 ) ) @ ( zero_zero @ A ) )
@ ( cosh @ A @ X ) ) ) ).
% cosh_converges
thf(fact_4122_sinh__converges,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X: A] :
( sums @ A
@ ^ [N5: nat] : ( if @ A @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N5 ) @ ( zero_zero @ A ) @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ N5 ) ) @ ( power_power @ A @ X @ N5 ) ) )
@ ( sinh @ A @ X ) ) ) ).
% sinh_converges
thf(fact_4123_monoseq__Suc,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( topological_monoseq @ A )
= ( ^ [X7: nat > A] :
( ! [N5: nat] : ( ord_less_eq @ A @ ( X7 @ N5 ) @ ( X7 @ ( suc @ N5 ) ) )
| ! [N5: nat] : ( ord_less_eq @ A @ ( X7 @ ( suc @ N5 ) ) @ ( X7 @ N5 ) ) ) ) ) ) ).
% monoseq_Suc
thf(fact_4124_mono__SucI2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X8: nat > A] :
( ! [N2: nat] : ( ord_less_eq @ A @ ( X8 @ ( suc @ N2 ) ) @ ( X8 @ N2 ) )
=> ( topological_monoseq @ A @ X8 ) ) ) ).
% mono_SucI2
thf(fact_4125_mono__SucI1,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X8: nat > A] :
( ! [N2: nat] : ( ord_less_eq @ A @ ( X8 @ N2 ) @ ( X8 @ ( suc @ N2 ) ) )
=> ( topological_monoseq @ A @ X8 ) ) ) ).
% mono_SucI1
thf(fact_4126_of__nat__code,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( ( semiring_1_of_nat @ A )
= ( ^ [N5: nat] :
( semiri8178284476397505188at_aux @ A
@ ^ [I2: A] : ( plus_plus @ A @ I2 @ ( one_one @ A ) )
@ N5
@ ( zero_zero @ A ) ) ) ) ) ).
% of_nat_code
thf(fact_4127_of__nat__aux_Osimps_I2_J,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [Inc: A > A,N: nat,I: A] :
( ( semiri8178284476397505188at_aux @ A @ Inc @ ( suc @ N ) @ I )
= ( semiri8178284476397505188at_aux @ A @ Inc @ N @ ( Inc @ I ) ) ) ) ).
% of_nat_aux.simps(2)
thf(fact_4128_of__nat__aux_Osimps_I1_J,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [Inc: A > A,I: A] :
( ( semiri8178284476397505188at_aux @ A @ Inc @ ( zero_zero @ nat ) @ I )
= I ) ) ).
% of_nat_aux.simps(1)
thf(fact_4129_monoseq__minus,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A3: nat > A] :
( ( topological_monoseq @ A @ A3 )
=> ( topological_monoseq @ A
@ ^ [N5: nat] : ( uminus_uminus @ A @ ( A3 @ N5 ) ) ) ) ) ).
% monoseq_minus
thf(fact_4130_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_4131_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_4132_divmod__algorithm__code_I6_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M2: num,N: num] :
( ( unique8689654367752047608divmod @ A @ ( bit1 @ M2 ) @ ( bit0 @ N ) )
= ( product_case_prod @ A @ A @ ( product_prod @ A @ A )
@ ^ [Q4: A,R: A] : ( product_Pair @ A @ A @ Q4 @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ R ) @ ( one_one @ A ) ) )
@ ( unique8689654367752047608divmod @ A @ M2 @ N ) ) ) ) ).
% divmod_algorithm_code(6)
thf(fact_4133_arctan__def,axiom,
( arctan
= ( ^ [Y5: real] :
( the @ real
@ ^ [X2: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X2 )
& ( ord_less @ real @ X2 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
& ( ( tan @ real @ X2 )
= Y5 ) ) ) ) ) ).
% arctan_def
thf(fact_4134_some__equality,axiom,
! [A: $tType,P: A > $o,A3: A] :
( ( P @ A3 )
=> ( ! [X3: A] :
( ( P @ X3 )
=> ( X3 = A3 ) )
=> ( ( fChoice @ A @ P )
= A3 ) ) ) ).
% some_equality
thf(fact_4135_some__eq__trivial,axiom,
! [A: $tType,X: A] :
( ( fChoice @ A
@ ^ [Y5: A] : Y5 = X )
= X ) ).
% some_eq_trivial
thf(fact_4136_some__sym__eq__trivial,axiom,
! [A: $tType,X: A] :
( ( fChoice @ A
@ ( ^ [Y4: A,Z: A] : Y4 = Z
@ X ) )
= X ) ).
% some_sym_eq_trivial
thf(fact_4137_some__eq__imp,axiom,
! [A: $tType,P: A > $o,A3: A,B2: A] :
( ( ( fChoice @ A @ P )
= A3 )
=> ( ( P @ B2 )
=> ( P @ A3 ) ) ) ).
% some_eq_imp
thf(fact_4138_tfl__some,axiom,
! [A: $tType,P8: A > $o,X4: A] :
( ( P8 @ X4 )
=> ( P8 @ ( fChoice @ A @ P8 ) ) ) ).
% tfl_some
thf(fact_4139_Eps__cong,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ! [X3: A] :
( ( P @ X3 )
= ( Q @ X3 ) )
=> ( ( fChoice @ A @ P )
= ( fChoice @ A @ Q ) ) ) ).
% Eps_cong
thf(fact_4140_someI,axiom,
! [A: $tType,P: A > $o,X: A] :
( ( P @ X )
=> ( P @ ( fChoice @ A @ P ) ) ) ).
% someI
thf(fact_4141_verit__sko__ex_H,axiom,
! [A: $tType,P: A > $o,A4: $o] :
( ( ( P @ ( fChoice @ A @ P ) )
= A4 )
=> ( ( ? [X7: A] : ( P @ X7 ) )
= A4 ) ) ).
% verit_sko_ex'
thf(fact_4142_verit__sko__forall,axiom,
! [A: $tType] :
( ( ^ [P2: A > $o] :
! [X5: A] : ( P2 @ X5 ) )
= ( ^ [P3: A > $o] :
( P3
@ ( fChoice @ A
@ ^ [X2: A] :
~ ( P3 @ X2 ) ) ) ) ) ).
% verit_sko_forall
thf(fact_4143_verit__sko__forall_H,axiom,
! [A: $tType,P: A > $o,A4: $o] :
( ( ( P
@ ( fChoice @ A
@ ^ [X2: A] :
~ ( P @ X2 ) ) )
= A4 )
=> ( ( ! [X7: A] : ( P @ X7 ) )
= A4 ) ) ).
% verit_sko_forall'
thf(fact_4144_verit__sko__forall_H_H,axiom,
! [A: $tType,B7: A,A4: A,P: A > $o] :
( ( B7 = A4 )
=> ( ( ( fChoice @ A @ P )
= A4 )
= ( ( fChoice @ A @ P )
= B7 ) ) ) ).
% verit_sko_forall''
thf(fact_4145_verit__sko__ex__indirect,axiom,
! [A: $tType,X: A,P: A > $o] :
( ( X
= ( fChoice @ A @ P ) )
=> ( ( ? [X7: A] : ( P @ X7 ) )
= ( P @ X ) ) ) ).
% verit_sko_ex_indirect
thf(fact_4146_verit__sko__ex__indirect2,axiom,
! [A: $tType,X: A,P: A > $o,P5: A > $o] :
( ( X
= ( fChoice @ A @ P ) )
=> ( ! [X3: A] :
( ( P @ X3 )
= ( P5 @ X3 ) )
=> ( ( ? [X7: A] : ( P5 @ X7 ) )
= ( P @ X ) ) ) ) ).
% verit_sko_ex_indirect2
thf(fact_4147_verit__sko__forall__indirect,axiom,
! [A: $tType,X: A,P: A > $o] :
( ( X
= ( fChoice @ A
@ ^ [X2: A] :
~ ( P @ X2 ) ) )
=> ( ( ! [X7: A] : ( P @ X7 ) )
= ( P @ X ) ) ) ).
% verit_sko_forall_indirect
thf(fact_4148_verit__sko__forall__indirect2,axiom,
! [A: $tType,X: A,P: A > $o,P5: A > $o] :
( ( X
= ( fChoice @ A
@ ^ [X2: A] :
~ ( P @ X2 ) ) )
=> ( ! [X3: A] :
( ( P @ X3 )
= ( P5 @ X3 ) )
=> ( ( ! [X7: A] : ( P5 @ X7 ) )
= ( P @ X ) ) ) ) ).
% verit_sko_forall_indirect2
thf(fact_4149_someI2,axiom,
! [A: $tType,P: A > $o,A3: A,Q: A > $o] :
( ( P @ A3 )
=> ( ! [X3: A] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( Q @ ( fChoice @ A @ P ) ) ) ) ).
% someI2
thf(fact_4150_someI__ex,axiom,
! [A: $tType,P: A > $o] :
( ? [X_12: A] : ( P @ X_12 )
=> ( P @ ( fChoice @ A @ P ) ) ) ).
% someI_ex
thf(fact_4151_someI2__ex,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ? [X_12: A] : ( P @ X_12 )
=> ( ! [X3: A] :
( ( P @ X3 )
=> ( Q @ X3 ) )
=> ( Q @ ( fChoice @ A @ P ) ) ) ) ).
% someI2_ex
thf(fact_4152_someI2__bex,axiom,
! [A: $tType,A4: set @ A,P: A > $o,Q: A > $o] :
( ? [X4: A] :
( ( member @ A @ X4 @ A4 )
& ( P @ X4 ) )
=> ( ! [X3: A] :
( ( ( member @ A @ X3 @ A4 )
& ( P @ X3 ) )
=> ( Q @ X3 ) )
=> ( Q
@ ( fChoice @ A
@ ^ [X2: A] :
( ( member @ A @ X2 @ A4 )
& ( P @ X2 ) ) ) ) ) ) ).
% someI2_bex
thf(fact_4153_some__eq__ex,axiom,
! [A: $tType,P: A > $o] :
( ( P @ ( fChoice @ A @ P ) )
= ( ? [X7: A] : ( P @ X7 ) ) ) ).
% some_eq_ex
thf(fact_4154_some1__equality,axiom,
! [A: $tType,P: A > $o,A3: A] :
( ? [X4: A] :
( ( P @ X4 )
& ! [Y3: A] :
( ( P @ Y3 )
=> ( Y3 = X4 ) ) )
=> ( ( P @ A3 )
=> ( ( fChoice @ A @ P )
= A3 ) ) ) ).
% some1_equality
thf(fact_4155_some__in__eq,axiom,
! [A: $tType,A4: set @ A] :
( ( member @ A
@ ( fChoice @ A
@ ^ [X2: A] : ( member @ A @ X2 @ A4 ) )
@ A4 )
= ( A4
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% some_in_eq
thf(fact_4156_ln__real__def,axiom,
( ( ln_ln @ real )
= ( ^ [X2: real] :
( the @ real
@ ^ [U2: real] :
( ( exp @ real @ U2 )
= X2 ) ) ) ) ).
% ln_real_def
thf(fact_4157_ln__neg__is__const,axiom,
! [X: real] :
( ( ord_less_eq @ real @ X @ ( zero_zero @ real ) )
=> ( ( ln_ln @ real @ X )
= ( the @ real
@ ^ [X2: real] : $false ) ) ) ).
% ln_neg_is_const
thf(fact_4158_arccos__def,axiom,
( arccos
= ( ^ [Y5: real] :
( the @ real
@ ^ [X2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
& ( ord_less_eq @ real @ X2 @ pi )
& ( ( cos @ real @ X2 )
= Y5 ) ) ) ) ) ).
% arccos_def
thf(fact_4159_divmod__step__nat__def,axiom,
( ( unique1321980374590559556d_step @ nat )
= ( ^ [L2: num] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [Q4: nat,R: nat] : ( if @ ( product_prod @ nat @ nat ) @ ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ L2 ) @ R ) @ ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Q4 ) @ ( one_one @ nat ) ) @ ( minus_minus @ nat @ R @ ( numeral_numeral @ nat @ L2 ) ) ) @ ( product_Pair @ nat @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Q4 ) @ R ) ) ) ) ) ).
% divmod_step_nat_def
thf(fact_4160_divmod__step__int__def,axiom,
( ( unique1321980374590559556d_step @ int )
= ( ^ [L2: num] :
( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [Q4: int,R: int] : ( if @ ( product_prod @ int @ int ) @ ( ord_less_eq @ int @ ( numeral_numeral @ int @ L2 ) @ R ) @ ( product_Pair @ int @ int @ ( plus_plus @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Q4 ) @ ( one_one @ int ) ) @ ( minus_minus @ int @ R @ ( numeral_numeral @ int @ L2 ) ) ) @ ( product_Pair @ int @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Q4 ) @ R ) ) ) ) ) ).
% divmod_step_int_def
thf(fact_4161_pi__half,axiom,
( ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) )
= ( the @ real
@ ^ [X2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
& ( ord_less_eq @ real @ X2 @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) )
& ( ( cos @ real @ X2 )
= ( zero_zero @ real ) ) ) ) ) ).
% pi_half
thf(fact_4162_pi__def,axiom,
( pi
= ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) )
@ ( the @ real
@ ^ [X2: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X2 )
& ( ord_less_eq @ real @ X2 @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) )
& ( ( cos @ real @ X2 )
= ( zero_zero @ real ) ) ) ) ) ) ).
% pi_def
thf(fact_4163_divmod__step__def,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ( ( unique1321980374590559556d_step @ A )
= ( ^ [L2: num] :
( product_case_prod @ A @ A @ ( product_prod @ A @ A )
@ ^ [Q4: A,R: A] : ( if @ ( product_prod @ A @ A ) @ ( ord_less_eq @ A @ ( numeral_numeral @ A @ L2 ) @ R ) @ ( product_Pair @ A @ A @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Q4 ) @ ( one_one @ A ) ) @ ( minus_minus @ A @ R @ ( numeral_numeral @ A @ L2 ) ) ) @ ( product_Pair @ A @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Q4 ) @ R ) ) ) ) ) ) ).
% divmod_step_def
thf(fact_4164_arcsin__def,axiom,
( arcsin
= ( ^ [Y5: real] :
( the @ real
@ ^ [X2: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X2 )
& ( ord_less_eq @ real @ X2 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
& ( ( sin @ real @ X2 )
= Y5 ) ) ) ) ) ).
% arcsin_def
thf(fact_4165_divmod__nat__if,axiom,
( divmod_nat
= ( ^ [M5: nat,N5: nat] :
( if @ ( product_prod @ nat @ nat )
@ ( ( N5
= ( zero_zero @ nat ) )
| ( ord_less @ nat @ M5 @ N5 ) )
@ ( product_Pair @ nat @ nat @ ( zero_zero @ nat ) @ M5 )
@ ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [Q4: nat] : ( product_Pair @ nat @ nat @ ( suc @ Q4 ) )
@ ( divmod_nat @ ( minus_minus @ nat @ M5 @ N5 ) @ N5 ) ) ) ) ) ).
% divmod_nat_if
thf(fact_4166_set__encode__def,axiom,
( nat_set_encode
= ( groups7311177749621191930dd_sum @ nat @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% set_encode_def
thf(fact_4167_Sum__Ico__nat,axiom,
! [M2: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [X2: nat] : X2
@ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) )
= ( divide_divide @ nat @ ( minus_minus @ nat @ ( times_times @ nat @ N @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) @ ( times_times @ nat @ M2 @ ( minus_minus @ nat @ M2 @ ( one_one @ nat ) ) ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% Sum_Ico_nat
thf(fact_4168_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_4169_set__decode__inverse,axiom,
! [N: nat] :
( ( nat_set_encode @ ( nat_set_decode @ N ) )
= N ) ).
% set_decode_inverse
thf(fact_4170_atLeastLessThan__iff,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [I: A,L: A,U: A] :
( ( member @ A @ I @ ( set_or7035219750837199246ssThan @ A @ L @ U ) )
= ( ( ord_less_eq @ A @ L @ I )
& ( ord_less @ A @ I @ U ) ) ) ) ).
% atLeastLessThan_iff
thf(fact_4171_atLeastLessThan__empty__iff2,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A,B2: A] :
( ( ( bot_bot @ ( set @ A ) )
= ( set_or7035219750837199246ssThan @ A @ A3 @ B2 ) )
= ( ~ ( ord_less @ A @ A3 @ B2 ) ) ) ) ).
% atLeastLessThan_empty_iff2
thf(fact_4172_atLeastLessThan__empty__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A,B2: A] :
( ( ( set_or7035219750837199246ssThan @ A @ A3 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
= ( ~ ( ord_less @ A @ A3 @ B2 ) ) ) ) ).
% atLeastLessThan_empty_iff
thf(fact_4173_ivl__diff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [I: A,N: A,M2: A] :
( ( ord_less_eq @ A @ I @ N )
=> ( ( minus_minus @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ I @ M2 ) @ ( set_or7035219750837199246ssThan @ A @ I @ N ) )
= ( set_or7035219750837199246ssThan @ A @ N @ M2 ) ) ) ) ).
% ivl_diff
thf(fact_4174_lessThan__minus__lessThan,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [N: A,M2: A] :
( ( minus_minus @ ( set @ A ) @ ( set_ord_lessThan @ A @ N ) @ ( set_ord_lessThan @ A @ M2 ) )
= ( set_or7035219750837199246ssThan @ A @ M2 @ N ) ) ) ).
% lessThan_minus_lessThan
thf(fact_4175_set__encode__empty,axiom,
( ( nat_set_encode @ ( bot_bot @ ( set @ nat ) ) )
= ( zero_zero @ nat ) ) ).
% set_encode_empty
thf(fact_4176_atLeastLessThan__singleton,axiom,
! [M2: nat] :
( ( set_or7035219750837199246ssThan @ nat @ M2 @ ( suc @ M2 ) )
= ( insert @ nat @ M2 @ ( bot_bot @ ( set @ nat ) ) ) ) ).
% atLeastLessThan_singleton
thf(fact_4177_sum_Oop__ivl__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [N: nat,M2: nat,G: nat > A] :
( ( ( ord_less @ nat @ N @ M2 )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M2 @ ( suc @ N ) ) )
= ( zero_zero @ A ) ) )
& ( ~ ( ord_less @ nat @ N @ M2 )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M2 @ ( suc @ N ) ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) @ ( G @ N ) ) ) ) ) ) ).
% sum.op_ivl_Suc
thf(fact_4178_prod_Oop__ivl__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [N: nat,M2: nat,G: nat > A] :
( ( ( ord_less @ nat @ N @ M2 )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M2 @ ( suc @ N ) ) )
= ( one_one @ A ) ) )
& ( ~ ( ord_less @ nat @ N @ M2 )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M2 @ ( suc @ N ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) @ ( G @ N ) ) ) ) ) ) ).
% prod.op_ivl_Suc
thf(fact_4179_atLeastLessThan__eq__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A,C2: A,D2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less @ A @ C2 @ D2 )
=> ( ( ( set_or7035219750837199246ssThan @ A @ A3 @ B2 )
= ( set_or7035219750837199246ssThan @ A @ C2 @ D2 ) )
= ( ( A3 = C2 )
& ( B2 = D2 ) ) ) ) ) ) ).
% atLeastLessThan_eq_iff
thf(fact_4180_atLeastLessThan__inj_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A,C2: A,D2: A] :
( ( ( set_or7035219750837199246ssThan @ A @ A3 @ B2 )
= ( set_or7035219750837199246ssThan @ A @ C2 @ D2 ) )
=> ( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less @ A @ C2 @ D2 )
=> ( A3 = C2 ) ) ) ) ) ).
% atLeastLessThan_inj(1)
thf(fact_4181_atLeastLessThan__inj_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A,C2: A,D2: A] :
( ( ( set_or7035219750837199246ssThan @ A @ A3 @ B2 )
= ( set_or7035219750837199246ssThan @ A @ C2 @ D2 ) )
=> ( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less @ A @ C2 @ D2 )
=> ( B2 = D2 ) ) ) ) ) ).
% atLeastLessThan_inj(2)
thf(fact_4182_all__nat__less__eq,axiom,
! [N: nat,P: nat > $o] :
( ( ! [M5: nat] :
( ( ord_less @ nat @ M5 @ N )
=> ( P @ M5 ) ) )
= ( ! [X2: nat] :
( ( member @ nat @ X2 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) )
=> ( P @ X2 ) ) ) ) ).
% all_nat_less_eq
thf(fact_4183_ex__nat__less__eq,axiom,
! [N: nat,P: nat > $o] :
( ( ? [M5: nat] :
( ( ord_less @ nat @ M5 @ N )
& ( P @ M5 ) ) )
= ( ? [X2: nat] :
( ( member @ nat @ X2 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) )
& ( P @ X2 ) ) ) ) ).
% ex_nat_less_eq
thf(fact_4184_atLeastLessThanSuc__atLeastAtMost,axiom,
! [L: nat,U: nat] :
( ( set_or7035219750837199246ssThan @ nat @ L @ ( suc @ U ) )
= ( set_or1337092689740270186AtMost @ nat @ L @ U ) ) ).
% atLeastLessThanSuc_atLeastAtMost
thf(fact_4185_lessThan__atLeast0,axiom,
( ( set_ord_lessThan @ nat )
= ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) ) ) ).
% lessThan_atLeast0
thf(fact_4186_atLeastLessThan0,axiom,
! [M2: nat] :
( ( set_or7035219750837199246ssThan @ nat @ M2 @ ( zero_zero @ nat ) )
= ( bot_bot @ ( set @ nat ) ) ) ).
% atLeastLessThan0
thf(fact_4187_sum_Oshift__bounds__Suc__ivl,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,M2: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M2 ) @ ( suc @ N ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
@ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) ) ) ).
% sum.shift_bounds_Suc_ivl
thf(fact_4188_prod_Oshift__bounds__Suc__ivl,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,M2: nat,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M2 ) @ ( suc @ N ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
@ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) ) ) ).
% prod.shift_bounds_Suc_ivl
thf(fact_4189_sum_Otriangle__reindex__eq,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ ( product_prod @ nat @ nat ) @ A @ ( product_case_prod @ nat @ nat @ A @ G )
@ ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [I2: nat,J3: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ I2 @ J3 ) @ N ) ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( G @ I2 @ ( minus_minus @ nat @ K3 @ I2 ) )
@ ( set_ord_atMost @ nat @ K3 ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% sum.triangle_reindex_eq
thf(fact_4190_prod_Otriangle__reindex__eq,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ ( product_prod @ nat @ nat ) @ A @ ( product_case_prod @ nat @ nat @ A @ G )
@ ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [I2: nat,J3: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ I2 @ J3 ) @ N ) ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [K3: nat] :
( groups7121269368397514597t_prod @ nat @ A
@ ^ [I2: nat] : ( G @ I2 @ ( minus_minus @ nat @ K3 @ I2 ) )
@ ( set_ord_atMost @ nat @ K3 ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% prod.triangle_reindex_eq
thf(fact_4191_sum_Oivl__cong,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( comm_monoid_add @ A ) )
=> ! [A3: B,C2: B,B2: B,D2: B,G: B > A,H: B > A] :
( ( A3 = C2 )
=> ( ( B2 = D2 )
=> ( ! [X3: B] :
( ( ord_less_eq @ B @ C2 @ X3 )
=> ( ( ord_less @ B @ X3 @ D2 )
=> ( ( G @ X3 )
= ( H @ X3 ) ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( set_or7035219750837199246ssThan @ B @ A3 @ B2 ) )
= ( groups7311177749621191930dd_sum @ B @ A @ H @ ( set_or7035219750837199246ssThan @ B @ C2 @ D2 ) ) ) ) ) ) ) ).
% sum.ivl_cong
thf(fact_4192_prod_Oivl__cong,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( comm_monoid_mult @ A ) )
=> ! [A3: B,C2: B,B2: B,D2: B,G: B > A,H: B > A] :
( ( A3 = C2 )
=> ( ( B2 = D2 )
=> ( ! [X3: B] :
( ( ord_less_eq @ B @ C2 @ X3 )
=> ( ( ord_less @ B @ X3 @ D2 )
=> ( ( G @ X3 )
= ( H @ X3 ) ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( set_or7035219750837199246ssThan @ B @ A3 @ B2 ) )
= ( groups7121269368397514597t_prod @ B @ A @ H @ ( set_or7035219750837199246ssThan @ B @ C2 @ D2 ) ) ) ) ) ) ) ).
% prod.ivl_cong
thf(fact_4193_sum__diff__nat__ivl,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [M2: nat,N: nat,P4: nat,F2: nat > A] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( ord_less_eq @ nat @ N @ P4 )
=> ( ( minus_minus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_or7035219750837199246ssThan @ nat @ M2 @ P4 ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_or7035219750837199246ssThan @ nat @ N @ P4 ) ) ) ) ) ) ).
% sum_diff_nat_ivl
thf(fact_4194_sum_Otriangle__reindex,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ ( product_prod @ nat @ nat ) @ A @ ( product_case_prod @ nat @ nat @ A @ G )
@ ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [I2: nat,J3: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ I2 @ J3 ) @ N ) ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( G @ I2 @ ( minus_minus @ nat @ K3 @ I2 ) )
@ ( set_ord_atMost @ nat @ K3 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% sum.triangle_reindex
thf(fact_4195_prod_Otriangle__reindex,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ ( product_prod @ nat @ nat ) @ A @ ( product_case_prod @ nat @ nat @ A @ G )
@ ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [I2: nat,J3: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ I2 @ J3 ) @ N ) ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [K3: nat] :
( groups7121269368397514597t_prod @ nat @ A
@ ^ [I2: nat] : ( G @ I2 @ ( minus_minus @ nat @ K3 @ I2 ) )
@ ( set_ord_atMost @ nat @ K3 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% prod.triangle_reindex
thf(fact_4196_size__list__estimation,axiom,
! [A: $tType,X: A,Xs: list @ A,Y2: nat,F2: A > nat] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ( ord_less @ nat @ Y2 @ ( F2 @ X ) )
=> ( ord_less @ nat @ Y2 @ ( size_list @ A @ F2 @ Xs ) ) ) ) ).
% size_list_estimation
thf(fact_4197_atLeast0__lessThan__Suc,axiom,
! [N: nat] :
( ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) )
= ( insert @ nat @ N @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ).
% atLeast0_lessThan_Suc
thf(fact_4198_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A3: A,B2: A,C2: A,D2: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) @ ( set_or7035219750837199246ssThan @ A @ C2 @ D2 ) )
= ( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ A3 )
& ( ord_less @ A @ B2 @ D2 ) ) ) ) ) ).
% atLeastAtMost_subseteq_atLeastLessThan_iff
thf(fact_4199_atLeastLessThan__subseteq__atLeastAtMost__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A3: A,B2: A,C2: A,D2: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ A3 @ B2 ) @ ( set_or1337092689740270186AtMost @ A @ C2 @ D2 ) )
= ( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ A3 )
& ( ord_less_eq @ A @ B2 @ D2 ) ) ) ) ) ).
% atLeastLessThan_subseteq_atLeastAtMost_iff
thf(fact_4200_sum__shift__lb__Suc0__0__upt,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [F2: nat > A,K: nat] :
( ( ( F2 @ ( zero_zero @ nat ) )
= ( zero_zero @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ ( zero_zero @ nat ) ) @ K ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K ) ) ) ) ) ).
% sum_shift_lb_Suc0_0_upt
thf(fact_4201_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_4202_sum_OatLeast__Suc__lessThan,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [M2: nat,N: nat,G: nat > A] :
( ( ord_less @ nat @ M2 @ N )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) )
= ( plus_plus @ A @ ( G @ M2 ) @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M2 ) @ N ) ) ) ) ) ) ).
% sum.atLeast_Suc_lessThan
thf(fact_4203_sum_OatLeastLessThan__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A3: nat,B2: nat,G: nat > A] :
( ( ord_less_eq @ nat @ A3 @ B2 )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ A3 @ ( suc @ B2 ) ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ A3 @ B2 ) ) @ ( G @ B2 ) ) ) ) ) ).
% sum.atLeastLessThan_Suc
thf(fact_4204_atLeastLessThan__eq__atLeastAtMost__diff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( set_or7035219750837199246ssThan @ A )
= ( ^ [A5: A,B3: A] : ( minus_minus @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A5 @ B3 ) @ ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% atLeastLessThan_eq_atLeastAtMost_diff
thf(fact_4205_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_4206_prod_OatLeast__Suc__lessThan,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M2: nat,N: nat,G: nat > A] :
( ( ord_less @ nat @ M2 @ N )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) )
= ( times_times @ A @ ( G @ M2 ) @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M2 ) @ N ) ) ) ) ) ) ).
% prod.atLeast_Suc_lessThan
thf(fact_4207_prod_OatLeastLessThan__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: nat,B2: nat,G: nat > A] :
( ( ord_less_eq @ nat @ A3 @ B2 )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ A3 @ ( suc @ B2 ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ A3 @ B2 ) ) @ ( G @ B2 ) ) ) ) ) ).
% prod.atLeastLessThan_Suc
thf(fact_4208_sum__Suc__diff_H,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [M2: nat,N: nat,F2: nat > A] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( minus_minus @ A @ ( F2 @ ( suc @ I2 ) ) @ ( F2 @ I2 ) )
@ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) )
= ( minus_minus @ A @ ( F2 @ N ) @ ( F2 @ M2 ) ) ) ) ) ).
% sum_Suc_diff'
thf(fact_4209_sum_OatLeastLessThan__rev,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,N: nat,M2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ N @ M2 ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( G @ ( minus_minus @ nat @ ( plus_plus @ nat @ M2 @ N ) @ ( suc @ I2 ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ N @ M2 ) ) ) ) ).
% sum.atLeastLessThan_rev
thf(fact_4210_atLeastLessThanSuc,axiom,
! [M2: nat,N: nat] :
( ( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( set_or7035219750837199246ssThan @ nat @ M2 @ ( suc @ N ) )
= ( insert @ nat @ N @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) ) )
& ( ~ ( ord_less_eq @ nat @ M2 @ N )
=> ( ( set_or7035219750837199246ssThan @ nat @ M2 @ ( suc @ N ) )
= ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% atLeastLessThanSuc
thf(fact_4211_sum_Onested__swap,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A3: nat > nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( groups7311177749621191930dd_sum @ nat @ A @ ( A3 @ I2 ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ I2 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [J3: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( A3 @ I2 @ J3 )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ J3 ) @ N ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% sum.nested_swap
thf(fact_4212_prod_OatLeastLessThan__rev,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,N: nat,M2: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ N @ M2 ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I2: nat] : ( G @ ( minus_minus @ nat @ ( plus_plus @ nat @ M2 @ N ) @ ( suc @ I2 ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ N @ M2 ) ) ) ) ).
% prod.atLeastLessThan_rev
thf(fact_4213_prod_Onested__swap,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A3: nat > nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I2: nat] : ( groups7121269368397514597t_prod @ nat @ A @ ( A3 @ I2 ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ I2 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [J3: nat] :
( groups7121269368397514597t_prod @ nat @ A
@ ^ [I2: nat] : ( A3 @ I2 @ J3 )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ J3 ) @ N ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% prod.nested_swap
thf(fact_4214_prod__Suc__Suc__fact,axiom,
! [N: nat] :
( ( groups7121269368397514597t_prod @ nat @ nat @ suc @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N ) )
= ( semiring_char_0_fact @ nat @ N ) ) ).
% prod_Suc_Suc_fact
thf(fact_4215_prod__Suc__fact,axiom,
! [N: nat] :
( ( groups7121269368397514597t_prod @ nat @ nat @ suc @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) )
= ( semiring_char_0_fact @ nat @ N ) ) ).
% prod_Suc_fact
thf(fact_4216_sum_Ohead__if,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [N: nat,M2: nat,G: nat > A] :
( ( ( ord_less @ nat @ N @ M2 )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
= ( zero_zero @ A ) ) )
& ( ~ ( ord_less @ nat @ N @ M2 )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) @ ( G @ N ) ) ) ) ) ) ).
% sum.head_if
thf(fact_4217_prod_Ohead__if,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [N: nat,M2: nat,G: nat > A] :
( ( ( ord_less @ nat @ N @ M2 )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
= ( one_one @ A ) ) )
& ( ~ ( ord_less @ nat @ N @ M2 )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) @ ( G @ N ) ) ) ) ) ) ).
% prod.head_if
thf(fact_4218_fact__prod__Suc,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ( ( semiring_char_0_fact @ A )
= ( ^ [N5: nat] : ( semiring_1_of_nat @ A @ ( groups7121269368397514597t_prod @ nat @ nat @ suc @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N5 ) ) ) ) ) ) ).
% fact_prod_Suc
thf(fact_4219_sum_OatLeastLessThan__rev__at__least__Suc__atMost,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,N: nat,M2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ N @ M2 ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( G @ ( minus_minus @ nat @ ( plus_plus @ nat @ M2 @ N ) @ I2 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ N ) @ M2 ) ) ) ) ).
% sum.atLeastLessThan_rev_at_least_Suc_atMost
thf(fact_4220_prod_OatLeastLessThan__rev__at__least__Suc__atMost,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,N: nat,M2: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ N @ M2 ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I2: nat] : ( G @ ( minus_minus @ nat @ ( plus_plus @ nat @ M2 @ N ) @ I2 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ N ) @ M2 ) ) ) ) ).
% prod.atLeastLessThan_rev_at_least_Suc_atMost
thf(fact_4221_pochhammer__prod,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ( ( comm_s3205402744901411588hammer @ A )
= ( ^ [A5: A,N5: nat] :
( groups7121269368397514597t_prod @ nat @ A
@ ^ [I2: nat] : ( plus_plus @ A @ A5 @ ( semiring_1_of_nat @ A @ I2 ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N5 ) ) ) ) ) ).
% pochhammer_prod
thf(fact_4222_fact__prod__rev,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ( ( semiring_char_0_fact @ A )
= ( ^ [N5: nat] : ( semiring_1_of_nat @ A @ ( groups7121269368397514597t_prod @ nat @ nat @ ( minus_minus @ nat @ N5 ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N5 ) ) ) ) ) ) ).
% fact_prod_rev
thf(fact_4223_sums__group,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [F2: nat > A,S: A,K: nat] :
( ( sums @ A @ F2 @ S )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( sums @ A
@ ^ [N5: nat] : ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_or7035219750837199246ssThan @ nat @ ( times_times @ nat @ N5 @ K ) @ ( plus_plus @ nat @ ( times_times @ nat @ N5 @ K ) @ K ) ) )
@ S ) ) ) ) ).
% sums_group
thf(fact_4224_take__bit__sum,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ( ( bit_se2584673776208193580ke_bit @ A )
= ( ^ [N5: 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 ) @ N5 ) ) ) ) ) ).
% take_bit_sum
thf(fact_4225_atLeast1__lessThan__eq__remove0,axiom,
! [N: nat] :
( ( set_or7035219750837199246ssThan @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N )
= ( minus_minus @ ( set @ nat ) @ ( set_ord_lessThan @ nat @ N ) @ ( insert @ nat @ ( zero_zero @ nat ) @ ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% atLeast1_lessThan_eq_remove0
thf(fact_4226_divmod__nat__def,axiom,
( divmod_nat
= ( ^ [M5: nat,N5: nat] : ( product_Pair @ nat @ nat @ ( divide_divide @ nat @ M5 @ N5 ) @ ( modulo_modulo @ nat @ M5 @ N5 ) ) ) ) ).
% divmod_nat_def
thf(fact_4227_fact__split,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [K: nat,N: nat] :
( ( ord_less_eq @ nat @ K @ N )
=> ( ( semiring_char_0_fact @ A @ N )
= ( times_times @ A @ ( semiring_1_of_nat @ A @ ( groups7121269368397514597t_prod @ nat @ nat @ suc @ ( set_or7035219750837199246ssThan @ nat @ ( minus_minus @ nat @ N @ K ) @ N ) ) ) @ ( semiring_char_0_fact @ A @ ( minus_minus @ nat @ N @ K ) ) ) ) ) ) ).
% fact_split
thf(fact_4228_binomial__altdef__of__nat,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,N: nat] :
( ( ord_less_eq @ nat @ K @ N )
=> ( ( semiring_1_of_nat @ A @ ( binomial @ N @ K ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I2: nat] : ( divide_divide @ A @ ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ N @ I2 ) ) @ ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ K @ I2 ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K ) ) ) ) ) ).
% binomial_altdef_of_nat
thf(fact_4229_gbinomial__altdef__of__nat,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( ( gbinomial @ A )
= ( ^ [A5: A,K3: nat] :
( groups7121269368397514597t_prod @ nat @ A
@ ^ [I2: nat] : ( divide_divide @ A @ ( minus_minus @ A @ A5 @ ( semiring_1_of_nat @ A @ I2 ) ) @ ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ K3 @ I2 ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K3 ) ) ) ) ) ).
% gbinomial_altdef_of_nat
thf(fact_4230_gbinomial__mult__fact,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K: nat,A3: A] :
( ( times_times @ A @ ( semiring_char_0_fact @ A @ K ) @ ( gbinomial @ A @ A3 @ K ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I2: nat] : ( minus_minus @ A @ A3 @ ( semiring_1_of_nat @ A @ I2 ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K ) ) ) ) ).
% gbinomial_mult_fact
thf(fact_4231_gbinomial__mult__fact_H,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: A,K: nat] :
( ( times_times @ A @ ( gbinomial @ A @ A3 @ K ) @ ( semiring_char_0_fact @ A @ K ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I2: nat] : ( minus_minus @ A @ A3 @ ( semiring_1_of_nat @ A @ I2 ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K ) ) ) ) ).
% gbinomial_mult_fact'
thf(fact_4232_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
@ ^ [I2: nat] : ( minus_minus @ A @ A5 @ ( semiring_1_of_nat @ A @ I2 ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K3 ) )
@ ( semiring_char_0_fact @ A @ K3 ) ) ) ) ) ).
% gbinomial_prod_rev
thf(fact_4233_sum__power2,axiom,
! [K: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K ) )
= ( minus_minus @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K ) @ ( one_one @ nat ) ) ) ).
% sum_power2
thf(fact_4234_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_4235_of__nat__code__if,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( ( semiring_1_of_nat @ A )
= ( ^ [N5: nat] :
( if @ A
@ ( N5
= ( zero_zero @ nat ) )
@ ( zero_zero @ A )
@ ( product_case_prod @ nat @ nat @ A
@ ^ [M5: nat,Q4: nat] :
( if @ A
@ ( Q4
= ( zero_zero @ nat ) )
@ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ A @ M5 ) )
@ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ A @ M5 ) ) @ ( one_one @ A ) ) )
@ ( divmod_nat @ N5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% of_nat_code_if
thf(fact_4236_horner__sum__eq__sum,axiom,
! [A: $tType,B: $tType] :
( ( comm_semiring_1 @ A )
=> ( ( groups4207007520872428315er_sum @ B @ A )
= ( ^ [F4: B > A,A5: A,Xs3: list @ B] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [N5: nat] : ( times_times @ A @ ( F4 @ ( nth @ B @ Xs3 @ N5 ) ) @ ( power_power @ A @ A5 @ N5 ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ B ) @ Xs3 ) ) ) ) ) ) ).
% horner_sum_eq_sum
thf(fact_4237_Chebyshev__sum__upper,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: nat,A3: nat > A,B2: nat > A] :
( ! [I3: nat,J: nat] :
( ( ord_less_eq @ nat @ I3 @ J )
=> ( ( ord_less @ nat @ J @ N )
=> ( ord_less_eq @ A @ ( A3 @ I3 ) @ ( A3 @ J ) ) ) )
=> ( ! [I3: nat,J: nat] :
( ( ord_less_eq @ nat @ I3 @ J )
=> ( ( ord_less @ nat @ J @ N )
=> ( ord_less_eq @ A @ ( B2 @ J ) @ ( B2 @ I3 ) ) ) )
=> ( ord_less_eq @ A
@ ( times_times @ A @ ( semiring_1_of_nat @ A @ N )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( times_times @ A @ ( A3 @ K3 ) @ ( B2 @ K3 ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) )
@ ( times_times @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ A3 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ B2 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ) ) ) ).
% Chebyshev_sum_upper
thf(fact_4238_Chebyshev__sum__upper__nat,axiom,
! [N: nat,A3: nat > nat,B2: nat > nat] :
( ! [I3: nat,J: nat] :
( ( ord_less_eq @ nat @ I3 @ J )
=> ( ( ord_less @ nat @ J @ N )
=> ( ord_less_eq @ nat @ ( A3 @ I3 ) @ ( A3 @ J ) ) ) )
=> ( ! [I3: nat,J: nat] :
( ( ord_less_eq @ nat @ I3 @ J )
=> ( ( ord_less @ nat @ J @ N )
=> ( ord_less_eq @ nat @ ( B2 @ J ) @ ( B2 @ I3 ) ) ) )
=> ( ord_less_eq @ nat
@ ( times_times @ nat @ N
@ ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [I2: nat] : ( times_times @ nat @ ( A3 @ I2 ) @ ( B2 @ I2 ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) )
@ ( times_times @ nat @ ( groups7311177749621191930dd_sum @ nat @ nat @ A3 @ ( 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_4239_Eps__case__prod__eq,axiom,
! [A: $tType,B: $tType,X: A,Y2: B] :
( ( fChoice @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [X9: A,Y6: B] :
( ( X = X9 )
& ( Y2 = Y6 ) ) ) )
= ( product_Pair @ A @ B @ X @ Y2 ) ) ).
% Eps_case_prod_eq
thf(fact_4240_split__paired__Eps,axiom,
! [B: $tType,A: $tType] :
( ( fChoice @ ( product_prod @ A @ B ) )
= ( ^ [P3: ( product_prod @ A @ B ) > $o] :
( fChoice @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [A5: A,B3: B] : ( P3 @ ( product_Pair @ A @ B @ A5 @ B3 ) ) ) ) ) ) ).
% split_paired_Eps
thf(fact_4241_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_4242_set__encode__insert,axiom,
! [A4: set @ nat,N: nat] :
( ( finite_finite @ nat @ A4 )
=> ( ~ ( member @ nat @ N @ A4 )
=> ( ( nat_set_encode @ ( insert @ nat @ N @ A4 ) )
= ( plus_plus @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ ( nat_set_encode @ A4 ) ) ) ) ) ).
% set_encode_insert
thf(fact_4243_int__of__nat__def,axiom,
( code_T6385005292777649522of_nat
= ( semiring_1_of_nat @ int ) ) ).
% int_of_nat_def
thf(fact_4244_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_4245_divmod__step__integer__def,axiom,
( ( unique1321980374590559556d_step @ code_integer )
= ( ^ [L2: num] :
( product_case_prod @ code_integer @ code_integer @ ( product_prod @ code_integer @ code_integer )
@ ^ [Q4: code_integer,R: code_integer] : ( if @ ( product_prod @ code_integer @ code_integer ) @ ( ord_less_eq @ code_integer @ ( numeral_numeral @ code_integer @ L2 ) @ R ) @ ( product_Pair @ code_integer @ code_integer @ ( plus_plus @ code_integer @ ( times_times @ code_integer @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) @ Q4 ) @ ( one_one @ code_integer ) ) @ ( minus_minus @ code_integer @ R @ ( numeral_numeral @ code_integer @ L2 ) ) ) @ ( product_Pair @ code_integer @ code_integer @ ( times_times @ code_integer @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) @ Q4 ) @ R ) ) ) ) ) ).
% divmod_step_integer_def
thf(fact_4246_sum__eq__0__iff,axiom,
! [A: $tType,B: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [F5: set @ B,F2: B > A] :
( ( finite_finite @ B @ F5 )
=> ( ( ( groups7311177749621191930dd_sum @ B @ A @ F2 @ F5 )
= ( zero_zero @ A ) )
= ( ! [X2: B] :
( ( member @ B @ X2 @ F5 )
=> ( ( F2 @ X2 )
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% sum_eq_0_iff
thf(fact_4247_sum_Oinfinite,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A4: set @ B,G: B > A] :
( ~ ( finite_finite @ B @ A4 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ A4 )
= ( zero_zero @ A ) ) ) ) ).
% sum.infinite
thf(fact_4248_prod__zero__iff,axiom,
! [A: $tType,B: $tType] :
( ( semidom @ A )
=> ! [A4: set @ B,F2: B > A] :
( ( finite_finite @ B @ A4 )
=> ( ( ( groups7121269368397514597t_prod @ B @ A @ F2 @ A4 )
= ( zero_zero @ A ) )
= ( ? [X2: B] :
( ( member @ B @ X2 @ A4 )
& ( ( F2 @ X2 )
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% prod_zero_iff
thf(fact_4249_infinite__Icc__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A3: A,B2: A] :
( ( ~ ( finite_finite @ A @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) ) )
= ( ord_less @ A @ A3 @ B2 ) ) ) ).
% infinite_Icc_iff
thf(fact_4250_infinite__Ico__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A3: A,B2: A] :
( ( ~ ( finite_finite @ A @ ( set_or7035219750837199246ssThan @ A @ A3 @ B2 ) ) )
= ( ord_less @ A @ A3 @ B2 ) ) ) ).
% infinite_Ico_iff
thf(fact_4251_prod_Oinfinite,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A4: set @ B,G: B > A] :
( ~ ( finite_finite @ B @ A4 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ A4 )
= ( one_one @ A ) ) ) ) ).
% prod.infinite
thf(fact_4252_sum_Odelta,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ A )
=> ! [S3: set @ B,A3: B,B2: B > A] :
( ( finite_finite @ B @ S3 )
=> ( ( ( member @ B @ A3 @ S3 )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [K3: B] : ( if @ A @ ( K3 = A3 ) @ ( B2 @ K3 ) @ ( zero_zero @ A ) )
@ S3 )
= ( B2 @ A3 ) ) )
& ( ~ ( member @ B @ A3 @ S3 )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [K3: B] : ( if @ A @ ( K3 = A3 ) @ ( B2 @ K3 ) @ ( zero_zero @ A ) )
@ S3 )
= ( zero_zero @ A ) ) ) ) ) ) ).
% sum.delta
thf(fact_4253_sum_Odelta_H,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ A )
=> ! [S3: set @ B,A3: B,B2: B > A] :
( ( finite_finite @ B @ S3 )
=> ( ( ( member @ B @ A3 @ S3 )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [K3: B] : ( if @ A @ ( A3 = K3 ) @ ( B2 @ K3 ) @ ( zero_zero @ A ) )
@ S3 )
= ( B2 @ A3 ) ) )
& ( ~ ( member @ B @ A3 @ S3 )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [K3: B] : ( if @ A @ ( A3 = K3 ) @ ( B2 @ K3 ) @ ( zero_zero @ A ) )
@ S3 )
= ( zero_zero @ A ) ) ) ) ) ) ).
% sum.delta'
thf(fact_4254_prod__eq__1__iff,axiom,
! [A: $tType,A4: set @ A,F2: A > nat] :
( ( finite_finite @ A @ A4 )
=> ( ( ( groups7121269368397514597t_prod @ A @ nat @ F2 @ A4 )
= ( one_one @ nat ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ A4 )
=> ( ( F2 @ X2 )
= ( one_one @ nat ) ) ) ) ) ) ).
% prod_eq_1_iff
thf(fact_4255_prod_Odelta_H,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S3: set @ B,A3: B,B2: B > A] :
( ( finite_finite @ B @ S3 )
=> ( ( ( member @ B @ A3 @ S3 )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K3: B] : ( if @ A @ ( A3 = K3 ) @ ( B2 @ K3 ) @ ( one_one @ A ) )
@ S3 )
= ( B2 @ A3 ) ) )
& ( ~ ( member @ B @ A3 @ S3 )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K3: B] : ( if @ A @ ( A3 = K3 ) @ ( B2 @ K3 ) @ ( one_one @ A ) )
@ S3 )
= ( one_one @ A ) ) ) ) ) ) ).
% prod.delta'
thf(fact_4256_prod_Odelta,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S3: set @ B,A3: B,B2: B > A] :
( ( finite_finite @ B @ S3 )
=> ( ( ( member @ B @ A3 @ S3 )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K3: B] : ( if @ A @ ( K3 = A3 ) @ ( B2 @ K3 ) @ ( one_one @ A ) )
@ S3 )
= ( B2 @ A3 ) ) )
& ( ~ ( member @ B @ A3 @ S3 )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K3: B] : ( if @ A @ ( K3 = A3 ) @ ( B2 @ K3 ) @ ( one_one @ A ) )
@ S3 )
= ( one_one @ A ) ) ) ) ) ) ).
% prod.delta
thf(fact_4257_summable__If__finite,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [P: nat > $o,F2: nat > A] :
( ( finite_finite @ nat @ ( collect @ nat @ P ) )
=> ( summable @ A
@ ^ [R: nat] : ( if @ A @ ( P @ R ) @ ( F2 @ R ) @ ( zero_zero @ A ) ) ) ) ) ).
% summable_If_finite
thf(fact_4258_summable__If__finite__set,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [A4: set @ nat,F2: nat > A] :
( ( finite_finite @ nat @ A4 )
=> ( summable @ A
@ ^ [R: nat] : ( if @ A @ ( member @ nat @ R @ A4 ) @ ( F2 @ R ) @ ( zero_zero @ A ) ) ) ) ) ).
% summable_If_finite_set
thf(fact_4259_set__encode__inverse,axiom,
! [A4: set @ nat] :
( ( finite_finite @ nat @ A4 )
=> ( ( nat_set_decode @ ( nat_set_encode @ A4 ) )
= A4 ) ) ).
% set_encode_inverse
thf(fact_4260_prod__pos__nat__iff,axiom,
! [A: $tType,A4: set @ A,F2: A > nat] :
( ( finite_finite @ A @ A4 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( groups7121269368397514597t_prod @ A @ nat @ F2 @ A4 ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ A4 )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F2 @ X2 ) ) ) ) ) ) ).
% prod_pos_nat_iff
thf(fact_4261_sum__zero__power,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A4: set @ nat,C2: nat > A] :
( ( ( ( finite_finite @ nat @ A4 )
& ( member @ nat @ ( zero_zero @ nat ) @ A4 ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( times_times @ A @ ( C2 @ I2 ) @ ( power_power @ A @ ( zero_zero @ A ) @ I2 ) )
@ A4 )
= ( C2 @ ( zero_zero @ nat ) ) ) )
& ( ~ ( ( finite_finite @ nat @ A4 )
& ( member @ nat @ ( zero_zero @ nat ) @ A4 ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( times_times @ A @ ( C2 @ I2 ) @ ( power_power @ A @ ( zero_zero @ A ) @ I2 ) )
@ A4 )
= ( zero_zero @ A ) ) ) ) ) ).
% sum_zero_power
thf(fact_4262_sum__zero__power_H,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A4: set @ nat,C2: nat > A,D2: nat > A] :
( ( ( ( finite_finite @ nat @ A4 )
& ( member @ nat @ ( zero_zero @ nat ) @ A4 ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( divide_divide @ A @ ( times_times @ A @ ( C2 @ I2 ) @ ( power_power @ A @ ( zero_zero @ A ) @ I2 ) ) @ ( D2 @ I2 ) )
@ A4 )
= ( divide_divide @ A @ ( C2 @ ( zero_zero @ nat ) ) @ ( D2 @ ( zero_zero @ nat ) ) ) ) )
& ( ~ ( ( finite_finite @ nat @ A4 )
& ( member @ nat @ ( zero_zero @ nat ) @ A4 ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( divide_divide @ A @ ( times_times @ A @ ( C2 @ I2 ) @ ( power_power @ A @ ( zero_zero @ A ) @ I2 ) ) @ ( D2 @ I2 ) )
@ A4 )
= ( zero_zero @ A ) ) ) ) ) ).
% sum_zero_power'
thf(fact_4263_set__encode__eq,axiom,
! [A4: set @ nat,B7: set @ nat] :
( ( finite_finite @ nat @ A4 )
=> ( ( finite_finite @ nat @ B7 )
=> ( ( ( nat_set_encode @ A4 )
= ( nat_set_encode @ B7 ) )
= ( A4 = B7 ) ) ) ) ).
% set_encode_eq
thf(fact_4264_minus__integer__code_I1_J,axiom,
! [K: code_integer] :
( ( minus_minus @ code_integer @ K @ ( zero_zero @ code_integer ) )
= K ) ).
% minus_integer_code(1)
thf(fact_4265_minus__integer__code_I2_J,axiom,
! [L: code_integer] :
( ( minus_minus @ code_integer @ ( zero_zero @ code_integer ) @ L )
= ( uminus_uminus @ code_integer @ L ) ) ).
% minus_integer_code(2)
thf(fact_4266_divmod__integer_H__def,axiom,
( ( unique8689654367752047608divmod @ code_integer )
= ( ^ [M5: num,N5: num] : ( product_Pair @ code_integer @ code_integer @ ( divide_divide @ code_integer @ ( numeral_numeral @ code_integer @ M5 ) @ ( numeral_numeral @ code_integer @ N5 ) ) @ ( modulo_modulo @ code_integer @ ( numeral_numeral @ code_integer @ M5 ) @ ( numeral_numeral @ code_integer @ N5 ) ) ) ) ) ).
% divmod_integer'_def
thf(fact_4267_bounded__nat__set__is__finite,axiom,
! [N6: set @ nat,N: nat] :
( ! [X3: nat] :
( ( member @ nat @ X3 @ N6 )
=> ( ord_less @ nat @ X3 @ N ) )
=> ( finite_finite @ nat @ N6 ) ) ).
% bounded_nat_set_is_finite
thf(fact_4268_finite__nat__set__iff__bounded,axiom,
( ( finite_finite @ nat )
= ( ^ [N7: set @ nat] :
? [M5: nat] :
! [X2: nat] :
( ( member @ nat @ X2 @ N7 )
=> ( ord_less @ nat @ X2 @ M5 ) ) ) ) ).
% finite_nat_set_iff_bounded
thf(fact_4269_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_4270_finite__set__decode,axiom,
! [N: nat] : ( finite_finite @ nat @ ( nat_set_decode @ N ) ) ).
% finite_set_decode
thf(fact_4271_finite__M__bounded__by__nat,axiom,
! [P: nat > $o,I: nat] :
( finite_finite @ nat
@ ( collect @ nat
@ ^ [K3: nat] :
( ( P @ K3 )
& ( ord_less @ nat @ K3 @ I ) ) ) ) ).
% finite_M_bounded_by_nat
thf(fact_4272_finite__lists__length__eq,axiom,
! [A: $tType,A4: set @ A,N: nat] :
( ( finite_finite @ A @ A4 )
=> ( finite_finite @ ( list @ A )
@ ( collect @ ( list @ A )
@ ^ [Xs3: list @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs3 ) @ A4 )
& ( ( size_size @ ( list @ A ) @ Xs3 )
= N ) ) ) ) ) ).
% finite_lists_length_eq
thf(fact_4273_ex__min__if__finite,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [S3: set @ A] :
( ( finite_finite @ A @ S3 )
=> ( ( S3
!= ( bot_bot @ ( set @ A ) ) )
=> ? [X3: A] :
( ( member @ A @ X3 @ S3 )
& ~ ? [Xa2: A] :
( ( member @ A @ Xa2 @ S3 )
& ( ord_less @ A @ Xa2 @ X3 ) ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_4274_infinite__growing,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X8: set @ A] :
( ( X8
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ X8 )
=> ? [Xa2: A] :
( ( member @ A @ Xa2 @ X8 )
& ( ord_less @ A @ X3 @ Xa2 ) ) )
=> ~ ( finite_finite @ A @ X8 ) ) ) ) ).
% infinite_growing
thf(fact_4275_prod__zero,axiom,
! [B: $tType,A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A4: set @ B,F2: B > A] :
( ( finite_finite @ B @ A4 )
=> ( ? [X4: B] :
( ( member @ B @ X4 @ A4 )
& ( ( F2 @ X4 )
= ( zero_zero @ A ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ F2 @ A4 )
= ( zero_zero @ A ) ) ) ) ) ).
% prod_zero
thf(fact_4276_infinite__Icc,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ~ ( finite_finite @ A @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) ) ) ) ).
% infinite_Icc
thf(fact_4277_infinite__Ico,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ~ ( finite_finite @ A @ ( set_or7035219750837199246ssThan @ A @ A3 @ B2 ) ) ) ) ).
% infinite_Ico
thf(fact_4278_finite__lists__length__le,axiom,
! [A: $tType,A4: set @ A,N: nat] :
( ( finite_finite @ A @ A4 )
=> ( finite_finite @ ( list @ A )
@ ( collect @ ( list @ A )
@ ^ [Xs3: list @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs3 ) @ A4 )
& ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs3 ) @ N ) ) ) ) ) ).
% finite_lists_length_le
thf(fact_4279_summable__finite,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [N6: set @ nat,F2: nat > A] :
( ( finite_finite @ nat @ N6 )
=> ( ! [N2: nat] :
( ~ ( member @ nat @ N2 @ N6 )
=> ( ( F2 @ N2 )
= ( zero_zero @ A ) ) )
=> ( summable @ A @ F2 ) ) ) ) ).
% summable_finite
thf(fact_4280_sum_Ofinite__Collect__op,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [I5: set @ B,X: B > A,Y2: B > A] :
( ( finite_finite @ B
@ ( collect @ B
@ ^ [I2: B] :
( ( member @ B @ I2 @ I5 )
& ( ( X @ I2 )
!= ( zero_zero @ A ) ) ) ) )
=> ( ( finite_finite @ B
@ ( collect @ B
@ ^ [I2: B] :
( ( member @ B @ I2 @ I5 )
& ( ( Y2 @ I2 )
!= ( zero_zero @ A ) ) ) ) )
=> ( finite_finite @ B
@ ( collect @ B
@ ^ [I2: B] :
( ( member @ B @ I2 @ I5 )
& ( ( plus_plus @ A @ ( X @ I2 ) @ ( Y2 @ I2 ) )
!= ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_4281_prod_Ofinite__Collect__op,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [I5: set @ B,X: B > A,Y2: B > A] :
( ( finite_finite @ B
@ ( collect @ B
@ ^ [I2: B] :
( ( member @ B @ I2 @ I5 )
& ( ( X @ I2 )
!= ( one_one @ A ) ) ) ) )
=> ( ( finite_finite @ B
@ ( collect @ B
@ ^ [I2: B] :
( ( member @ B @ I2 @ I5 )
& ( ( Y2 @ I2 )
!= ( one_one @ A ) ) ) ) )
=> ( finite_finite @ B
@ ( collect @ B
@ ^ [I2: B] :
( ( member @ B @ I2 @ I5 )
& ( ( times_times @ A @ ( X @ I2 ) @ ( Y2 @ I2 ) )
!= ( one_one @ A ) ) ) ) ) ) ) ) ).
% prod.finite_Collect_op
thf(fact_4282_sum_Ointer__filter,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A4: set @ B,G: B > A,P: B > $o] :
( ( finite_finite @ B @ A4 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G
@ ( collect @ B
@ ^ [X2: B] :
( ( member @ B @ X2 @ A4 )
& ( P @ X2 ) ) ) )
= ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X2: B] : ( if @ A @ ( P @ X2 ) @ ( G @ X2 ) @ ( zero_zero @ A ) )
@ A4 ) ) ) ) ).
% sum.inter_filter
thf(fact_4283_set__encode__inf,axiom,
! [A4: set @ nat] :
( ~ ( finite_finite @ nat @ A4 )
=> ( ( nat_set_encode @ A4 )
= ( zero_zero @ nat ) ) ) ).
% set_encode_inf
thf(fact_4284_prod_Ointer__filter,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A4: set @ B,G: B > A,P: B > $o] :
( ( finite_finite @ B @ A4 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G
@ ( collect @ B
@ ^ [X2: B] :
( ( member @ B @ X2 @ A4 )
& ( P @ X2 ) ) ) )
= ( groups7121269368397514597t_prod @ B @ A
@ ^ [X2: B] : ( if @ A @ ( P @ X2 ) @ ( G @ X2 ) @ ( one_one @ A ) )
@ A4 ) ) ) ) ).
% prod.inter_filter
thf(fact_4285_finite__int__segment,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [A3: A,B2: A] :
( finite_finite @ A
@ ( collect @ A
@ ^ [X2: A] :
( ( member @ A @ X2 @ ( ring_1_Ints @ A ) )
& ( ord_less_eq @ A @ A3 @ X2 )
& ( ord_less_eq @ A @ X2 @ B2 ) ) ) ) ) ).
% finite_int_segment
thf(fact_4286_sum__le__included,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [S: set @ B,T2: set @ C,G: C > A,I: C > B,F2: B > A] :
( ( finite_finite @ B @ S )
=> ( ( finite_finite @ C @ T2 )
=> ( ! [X3: C] :
( ( member @ C @ X3 @ T2 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( G @ X3 ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ S )
=> ? [Xa2: C] :
( ( member @ C @ Xa2 @ T2 )
& ( ( I @ Xa2 )
= X3 )
& ( ord_less_eq @ A @ ( F2 @ X3 ) @ ( G @ Xa2 ) ) ) )
=> ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ S ) @ ( groups7311177749621191930dd_sum @ C @ A @ G @ T2 ) ) ) ) ) ) ) ).
% sum_le_included
thf(fact_4287_sum__nonneg__eq__0__iff,axiom,
! [A: $tType,B: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A4: set @ B,F2: B > A] :
( ( finite_finite @ B @ A4 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ A4 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ X3 ) ) )
=> ( ( ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A4 )
= ( zero_zero @ A ) )
= ( ! [X2: B] :
( ( member @ B @ X2 @ A4 )
=> ( ( F2 @ X2 )
= ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
thf(fact_4288_sum__strict__mono__ex1,axiom,
! [A: $tType,I6: $tType] :
( ( ordere8940638589300402666id_add @ A )
=> ! [A4: set @ I6,F2: I6 > A,G: I6 > A] :
( ( finite_finite @ I6 @ A4 )
=> ( ! [X3: I6] :
( ( member @ I6 @ X3 @ A4 )
=> ( ord_less_eq @ A @ ( F2 @ X3 ) @ ( G @ X3 ) ) )
=> ( ? [X4: I6] :
( ( member @ I6 @ X4 @ A4 )
& ( ord_less @ A @ ( F2 @ X4 ) @ ( G @ X4 ) ) )
=> ( ord_less @ A @ ( groups7311177749621191930dd_sum @ I6 @ A @ F2 @ A4 ) @ ( groups7311177749621191930dd_sum @ I6 @ A @ G @ A4 ) ) ) ) ) ) ).
% sum_strict_mono_ex1
thf(fact_4289_sum_Orelated,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [R2: A > A > $o,S3: set @ B,H: B > A,G: B > A] :
( ( R2 @ ( zero_zero @ A ) @ ( zero_zero @ A ) )
=> ( ! [X1: A,Y1: A,X24: A,Y24: A] :
( ( ( R2 @ X1 @ X24 )
& ( R2 @ Y1 @ Y24 ) )
=> ( R2 @ ( plus_plus @ A @ X1 @ Y1 ) @ ( plus_plus @ A @ X24 @ Y24 ) ) )
=> ( ( finite_finite @ B @ S3 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ S3 )
=> ( R2 @ ( H @ X3 ) @ ( G @ X3 ) ) )
=> ( R2 @ ( groups7311177749621191930dd_sum @ B @ A @ H @ S3 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ S3 ) ) ) ) ) ) ) ).
% sum.related
thf(fact_4290_finite__linorder__min__induct,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite @ A @ A4 )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [B4: A,A9: set @ A] :
( ( finite_finite @ A @ A9 )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ A9 )
=> ( ord_less @ A @ B4 @ X4 ) )
=> ( ( P @ A9 )
=> ( P @ ( insert @ A @ B4 @ A9 ) ) ) ) )
=> ( P @ A4 ) ) ) ) ) ).
% finite_linorder_min_induct
thf(fact_4291_finite__linorder__max__induct,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite @ A @ A4 )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [B4: A,A9: set @ A] :
( ( finite_finite @ A @ A9 )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ A9 )
=> ( ord_less @ A @ X4 @ B4 ) )
=> ( ( P @ A9 )
=> ( P @ ( insert @ A @ B4 @ A9 ) ) ) ) )
=> ( P @ A4 ) ) ) ) ) ).
% finite_linorder_max_induct
thf(fact_4292_sum__strict__mono,axiom,
! [A: $tType,B: $tType] :
( ( strict7427464778891057005id_add @ A )
=> ! [A4: set @ B,F2: B > A,G: B > A] :
( ( finite_finite @ B @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ A4 )
=> ( ord_less @ A @ ( F2 @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A4 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ A4 ) ) ) ) ) ) ).
% sum_strict_mono
thf(fact_4293_prod_Orelated,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [R2: A > A > $o,S3: set @ B,H: B > A,G: B > A] :
( ( R2 @ ( one_one @ A ) @ ( one_one @ A ) )
=> ( ! [X1: A,Y1: A,X24: A,Y24: A] :
( ( ( R2 @ X1 @ X24 )
& ( R2 @ Y1 @ Y24 ) )
=> ( R2 @ ( times_times @ A @ X1 @ Y1 ) @ ( times_times @ A @ X24 @ Y24 ) ) )
=> ( ( finite_finite @ B @ S3 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ S3 )
=> ( R2 @ ( H @ X3 ) @ ( G @ X3 ) ) )
=> ( R2 @ ( groups7121269368397514597t_prod @ B @ A @ H @ S3 ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ S3 ) ) ) ) ) ) ) ).
% prod.related
thf(fact_4294_sum_Oreindex__bij__witness__not__neutral,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comm_monoid_add @ A )
=> ! [S4: set @ B,T5: set @ C,S3: set @ B,I: C > B,J2: B > C,T6: set @ C,G: B > A,H: C > A] :
( ( finite_finite @ B @ S4 )
=> ( ( finite_finite @ C @ T5 )
=> ( ! [A6: B] :
( ( member @ B @ A6 @ ( minus_minus @ ( set @ B ) @ S3 @ S4 ) )
=> ( ( I @ ( J2 @ A6 ) )
= A6 ) )
=> ( ! [A6: B] :
( ( member @ B @ A6 @ ( minus_minus @ ( set @ B ) @ S3 @ S4 ) )
=> ( member @ C @ ( J2 @ A6 ) @ ( minus_minus @ ( set @ C ) @ T6 @ T5 ) ) )
=> ( ! [B4: C] :
( ( member @ C @ B4 @ ( minus_minus @ ( set @ C ) @ T6 @ T5 ) )
=> ( ( J2 @ ( I @ B4 ) )
= B4 ) )
=> ( ! [B4: C] :
( ( member @ C @ B4 @ ( minus_minus @ ( set @ C ) @ T6 @ T5 ) )
=> ( member @ B @ ( I @ B4 ) @ ( minus_minus @ ( set @ B ) @ S3 @ S4 ) ) )
=> ( ! [A6: B] :
( ( member @ B @ A6 @ S4 )
=> ( ( G @ A6 )
= ( zero_zero @ A ) ) )
=> ( ! [B4: C] :
( ( member @ C @ B4 @ T5 )
=> ( ( H @ B4 )
= ( zero_zero @ A ) ) )
=> ( ! [A6: B] :
( ( member @ B @ A6 @ S3 )
=> ( ( H @ ( J2 @ A6 ) )
= ( G @ A6 ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ S3 )
= ( groups7311177749621191930dd_sum @ C @ A @ H @ T6 ) ) ) ) ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness_not_neutral
thf(fact_4295_prod_Oreindex__bij__witness__not__neutral,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S4: set @ B,T5: set @ C,S3: set @ B,I: C > B,J2: B > C,T6: set @ C,G: B > A,H: C > A] :
( ( finite_finite @ B @ S4 )
=> ( ( finite_finite @ C @ T5 )
=> ( ! [A6: B] :
( ( member @ B @ A6 @ ( minus_minus @ ( set @ B ) @ S3 @ S4 ) )
=> ( ( I @ ( J2 @ A6 ) )
= A6 ) )
=> ( ! [A6: B] :
( ( member @ B @ A6 @ ( minus_minus @ ( set @ B ) @ S3 @ S4 ) )
=> ( member @ C @ ( J2 @ A6 ) @ ( minus_minus @ ( set @ C ) @ T6 @ T5 ) ) )
=> ( ! [B4: C] :
( ( member @ C @ B4 @ ( minus_minus @ ( set @ C ) @ T6 @ T5 ) )
=> ( ( J2 @ ( I @ B4 ) )
= B4 ) )
=> ( ! [B4: C] :
( ( member @ C @ B4 @ ( minus_minus @ ( set @ C ) @ T6 @ T5 ) )
=> ( member @ B @ ( I @ B4 ) @ ( minus_minus @ ( set @ B ) @ S3 @ S4 ) ) )
=> ( ! [A6: B] :
( ( member @ B @ A6 @ S4 )
=> ( ( G @ A6 )
= ( one_one @ A ) ) )
=> ( ! [B4: C] :
( ( member @ C @ B4 @ T5 )
=> ( ( H @ B4 )
= ( one_one @ A ) ) )
=> ( ! [A6: B] :
( ( member @ B @ A6 @ S3 )
=> ( ( H @ ( J2 @ A6 ) )
= ( G @ A6 ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ S3 )
= ( groups7121269368397514597t_prod @ C @ A @ H @ T6 ) ) ) ) ) ) ) ) ) ) ) ) ).
% prod.reindex_bij_witness_not_neutral
thf(fact_4296_sum__eq__Suc0__iff,axiom,
! [A: $tType,A4: set @ A,F2: A > nat] :
( ( finite_finite @ A @ A4 )
=> ( ( ( groups7311177749621191930dd_sum @ A @ nat @ F2 @ A4 )
= ( suc @ ( zero_zero @ nat ) ) )
= ( ? [X2: A] :
( ( member @ A @ X2 @ A4 )
& ( ( F2 @ X2 )
= ( suc @ ( zero_zero @ nat ) ) )
& ! [Y5: A] :
( ( member @ A @ Y5 @ A4 )
=> ( ( X2 != Y5 )
=> ( ( F2 @ Y5 )
= ( zero_zero @ nat ) ) ) ) ) ) ) ) ).
% sum_eq_Suc0_iff
thf(fact_4297_zero__natural_Orsp,axiom,
( ( zero_zero @ nat )
= ( zero_zero @ nat ) ) ).
% zero_natural.rsp
thf(fact_4298_sum__eq__1__iff,axiom,
! [A: $tType,A4: set @ A,F2: A > nat] :
( ( finite_finite @ A @ A4 )
=> ( ( ( groups7311177749621191930dd_sum @ A @ nat @ F2 @ A4 )
= ( one_one @ nat ) )
= ( ? [X2: A] :
( ( member @ A @ X2 @ A4 )
& ( ( F2 @ X2 )
= ( one_one @ nat ) )
& ! [Y5: A] :
( ( member @ A @ Y5 @ A4 )
=> ( ( X2 != Y5 )
=> ( ( F2 @ Y5 )
= ( zero_zero @ nat ) ) ) ) ) ) ) ) ).
% sum_eq_1_iff
thf(fact_4299_zero__integer_Orsp,axiom,
( ( zero_zero @ int )
= ( zero_zero @ int ) ) ).
% zero_integer.rsp
thf(fact_4300_sum__nonneg__0,axiom,
! [B: $tType,A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [S: set @ B,F2: B > A,I: B] :
( ( finite_finite @ B @ S )
=> ( ! [I3: B] :
( ( member @ B @ I3 @ S )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ I3 ) ) )
=> ( ( ( groups7311177749621191930dd_sum @ B @ A @ F2 @ S )
= ( zero_zero @ A ) )
=> ( ( member @ B @ I @ S )
=> ( ( F2 @ I )
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_4301_sum__nonneg__leq__bound,axiom,
! [B: $tType,A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [S: set @ B,F2: B > A,B7: A,I: B] :
( ( finite_finite @ B @ S )
=> ( ! [I3: B] :
( ( member @ B @ I3 @ S )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ I3 ) ) )
=> ( ( ( groups7311177749621191930dd_sum @ B @ A @ F2 @ S )
= B7 )
=> ( ( member @ B @ I @ S )
=> ( ord_less_eq @ A @ ( F2 @ I ) @ B7 ) ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_4302_one__integer_Orsp,axiom,
( ( one_one @ int )
= ( one_one @ int ) ) ).
% one_integer.rsp
thf(fact_4303_one__natural_Orsp,axiom,
( ( one_one @ nat )
= ( one_one @ nat ) ) ).
% one_natural.rsp
thf(fact_4304_sum_Osetdiff__irrelevant,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A4: set @ B,G: B > A] :
( ( finite_finite @ B @ A4 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G
@ ( minus_minus @ ( set @ B ) @ A4
@ ( collect @ B
@ ^ [X2: B] :
( ( G @ X2 )
= ( zero_zero @ A ) ) ) ) )
= ( groups7311177749621191930dd_sum @ B @ A @ G @ A4 ) ) ) ) ).
% sum.setdiff_irrelevant
thf(fact_4305_prod_Osetdiff__irrelevant,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A4: set @ B,G: B > A] :
( ( finite_finite @ B @ A4 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G
@ ( minus_minus @ ( set @ B ) @ A4
@ ( collect @ B
@ ^ [X2: B] :
( ( G @ X2 )
= ( one_one @ A ) ) ) ) )
= ( groups7121269368397514597t_prod @ B @ A @ G @ A4 ) ) ) ) ).
% prod.setdiff_irrelevant
thf(fact_4306_finite__divisors__nat,axiom,
! [M2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M2 )
=> ( finite_finite @ nat
@ ( collect @ nat
@ ^ [D4: nat] : ( dvd_dvd @ nat @ D4 @ M2 ) ) ) ) ).
% finite_divisors_nat
thf(fact_4307_sums__finite,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [N6: set @ nat,F2: nat > A] :
( ( finite_finite @ nat @ N6 )
=> ( ! [N2: nat] :
( ~ ( member @ nat @ N2 @ N6 )
=> ( ( F2 @ N2 )
= ( zero_zero @ A ) ) )
=> ( sums @ A @ F2 @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ N6 ) ) ) ) ) ).
% sums_finite
thf(fact_4308_sums__If__finite,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [P: nat > $o,F2: nat > A] :
( ( finite_finite @ nat @ ( collect @ nat @ P ) )
=> ( sums @ A
@ ^ [R: nat] : ( if @ A @ ( P @ R ) @ ( F2 @ R ) @ ( zero_zero @ A ) )
@ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( collect @ nat @ P ) ) ) ) ) ).
% sums_If_finite
thf(fact_4309_sums__If__finite__set,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [A4: set @ nat,F2: nat > A] :
( ( finite_finite @ nat @ A4 )
=> ( sums @ A
@ ^ [R: nat] : ( if @ A @ ( member @ nat @ R @ A4 ) @ ( F2 @ R ) @ ( zero_zero @ A ) )
@ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ A4 ) ) ) ) ).
% sums_If_finite_set
thf(fact_4310_suminf__finite,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topological_t2_space @ A ) )
=> ! [N6: set @ nat,F2: nat > A] :
( ( finite_finite @ nat @ N6 )
=> ( ! [N2: nat] :
( ~ ( member @ nat @ N2 @ N6 )
=> ( ( F2 @ N2 )
= ( zero_zero @ A ) ) )
=> ( ( suminf @ A @ F2 )
= ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ N6 ) ) ) ) ) ).
% suminf_finite
thf(fact_4311_finite__abs__int__segment,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [A3: A] :
( finite_finite @ A
@ ( collect @ A
@ ^ [K3: A] :
( ( member @ A @ K3 @ ( ring_1_Ints @ A ) )
& ( ord_less_eq @ A @ ( abs_abs @ A @ K3 ) @ A3 ) ) ) ) ) ).
% finite_abs_int_segment
thf(fact_4312_subset__eq__atLeast0__atMost__finite,axiom,
! [N6: set @ nat,N: nat] :
( ( ord_less_eq @ ( set @ nat ) @ N6 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
=> ( finite_finite @ nat @ N6 ) ) ).
% subset_eq_atLeast0_atMost_finite
thf(fact_4313_subset__eq__atLeast0__lessThan__finite,axiom,
! [N6: set @ nat,N: nat] :
( ( ord_less_eq @ ( set @ nat ) @ N6 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) )
=> ( finite_finite @ nat @ N6 ) ) ).
% subset_eq_atLeast0_lessThan_finite
thf(fact_4314_exp__sum,axiom,
! [B: $tType,A: $tType] :
( ( ( comm_monoid_mult @ B )
& ( real_Vector_banach @ B )
& ( real_V2822296259951069270ebra_1 @ B ) )
=> ! [I5: set @ A,F2: A > B] :
( ( finite_finite @ A @ I5 )
=> ( ( exp @ B @ ( groups7311177749621191930dd_sum @ A @ B @ F2 @ I5 ) )
= ( groups7121269368397514597t_prod @ A @ B
@ ^ [X2: A] : ( exp @ B @ ( F2 @ X2 ) )
@ I5 ) ) ) ) ).
% exp_sum
thf(fact_4315_sum__pos2,axiom,
! [A: $tType,B: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [I5: set @ B,I: B,F2: B > A] :
( ( finite_finite @ B @ I5 )
=> ( ( member @ B @ I @ I5 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ ( F2 @ I ) )
=> ( ! [I3: B] :
( ( member @ B @ I3 @ I5 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ I3 ) ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ I5 ) ) ) ) ) ) ) ).
% sum_pos2
thf(fact_4316_sum__pos,axiom,
! [A: $tType,B: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [I5: set @ B,F2: B > A] :
( ( finite_finite @ B @ I5 )
=> ( ( I5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [I3: B] :
( ( member @ B @ I3 @ I5 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( F2 @ I3 ) ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ I5 ) ) ) ) ) ) ).
% sum_pos
thf(fact_4317_less__1__prod2,axiom,
! [B: $tType,A: $tType] :
( ( linordered_idom @ B )
=> ! [I5: set @ A,I: A,F2: A > B] :
( ( finite_finite @ A @ I5 )
=> ( ( member @ A @ I @ I5 )
=> ( ( ord_less @ B @ ( one_one @ B ) @ ( F2 @ I ) )
=> ( ! [I3: A] :
( ( member @ A @ I3 @ I5 )
=> ( ord_less_eq @ B @ ( one_one @ B ) @ ( F2 @ I3 ) ) )
=> ( ord_less @ B @ ( one_one @ B ) @ ( groups7121269368397514597t_prod @ A @ B @ F2 @ I5 ) ) ) ) ) ) ) ).
% less_1_prod2
thf(fact_4318_less__1__prod,axiom,
! [B: $tType,A: $tType] :
( ( linordered_idom @ B )
=> ! [I5: set @ A,F2: A > B] :
( ( finite_finite @ A @ I5 )
=> ( ( I5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [I3: A] :
( ( member @ A @ I3 @ I5 )
=> ( ord_less @ B @ ( one_one @ B ) @ ( F2 @ I3 ) ) )
=> ( ord_less @ B @ ( one_one @ B ) @ ( groups7121269368397514597t_prod @ A @ B @ F2 @ I5 ) ) ) ) ) ) ).
% less_1_prod
thf(fact_4319_sum_Osame__carrier,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [C5: set @ B,A4: set @ B,B7: set @ B,G: B > A,H: B > A] :
( ( finite_finite @ B @ C5 )
=> ( ( ord_less_eq @ ( set @ B ) @ A4 @ C5 )
=> ( ( ord_less_eq @ ( set @ B ) @ B7 @ C5 )
=> ( ! [A6: B] :
( ( member @ B @ A6 @ ( minus_minus @ ( set @ B ) @ C5 @ A4 ) )
=> ( ( G @ A6 )
= ( zero_zero @ A ) ) )
=> ( ! [B4: B] :
( ( member @ B @ B4 @ ( minus_minus @ ( set @ B ) @ C5 @ B7 ) )
=> ( ( H @ B4 )
= ( zero_zero @ A ) ) )
=> ( ( ( groups7311177749621191930dd_sum @ B @ A @ G @ A4 )
= ( groups7311177749621191930dd_sum @ B @ A @ H @ B7 ) )
= ( ( groups7311177749621191930dd_sum @ B @ A @ G @ C5 )
= ( groups7311177749621191930dd_sum @ B @ A @ H @ C5 ) ) ) ) ) ) ) ) ) ).
% sum.same_carrier
thf(fact_4320_sum_Osame__carrierI,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [C5: set @ B,A4: set @ B,B7: set @ B,G: B > A,H: B > A] :
( ( finite_finite @ B @ C5 )
=> ( ( ord_less_eq @ ( set @ B ) @ A4 @ C5 )
=> ( ( ord_less_eq @ ( set @ B ) @ B7 @ C5 )
=> ( ! [A6: B] :
( ( member @ B @ A6 @ ( minus_minus @ ( set @ B ) @ C5 @ A4 ) )
=> ( ( G @ A6 )
= ( zero_zero @ A ) ) )
=> ( ! [B4: B] :
( ( member @ B @ B4 @ ( minus_minus @ ( set @ B ) @ C5 @ B7 ) )
=> ( ( H @ B4 )
= ( zero_zero @ A ) ) )
=> ( ( ( groups7311177749621191930dd_sum @ B @ A @ G @ C5 )
= ( groups7311177749621191930dd_sum @ B @ A @ H @ C5 ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ A4 )
= ( groups7311177749621191930dd_sum @ B @ A @ H @ B7 ) ) ) ) ) ) ) ) ) ).
% sum.same_carrierI
thf(fact_4321_sum_Omono__neutral__left,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [T6: set @ B,S3: set @ B,G: B > A] :
( ( finite_finite @ B @ T6 )
=> ( ( ord_less_eq @ ( set @ B ) @ S3 @ T6 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T6 @ S3 ) )
=> ( ( G @ X3 )
= ( zero_zero @ A ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ S3 )
= ( groups7311177749621191930dd_sum @ B @ A @ G @ T6 ) ) ) ) ) ) ).
% sum.mono_neutral_left
thf(fact_4322_sum_Omono__neutral__right,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [T6: set @ B,S3: set @ B,G: B > A] :
( ( finite_finite @ B @ T6 )
=> ( ( ord_less_eq @ ( set @ B ) @ S3 @ T6 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T6 @ S3 ) )
=> ( ( G @ X3 )
= ( zero_zero @ A ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ T6 )
= ( groups7311177749621191930dd_sum @ B @ A @ G @ S3 ) ) ) ) ) ) ).
% sum.mono_neutral_right
thf(fact_4323_sum_Omono__neutral__cong__left,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [T6: set @ B,S3: set @ B,H: B > A,G: B > A] :
( ( finite_finite @ B @ T6 )
=> ( ( ord_less_eq @ ( set @ B ) @ S3 @ T6 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T6 @ S3 ) )
=> ( ( H @ X3 )
= ( zero_zero @ A ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ S3 )
=> ( ( G @ X3 )
= ( H @ X3 ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ S3 )
= ( groups7311177749621191930dd_sum @ B @ A @ H @ T6 ) ) ) ) ) ) ) ).
% sum.mono_neutral_cong_left
thf(fact_4324_sum_Omono__neutral__cong__right,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [T6: set @ B,S3: set @ B,G: B > A,H: B > A] :
( ( finite_finite @ B @ T6 )
=> ( ( ord_less_eq @ ( set @ B ) @ S3 @ T6 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T6 @ S3 ) )
=> ( ( G @ X3 )
= ( zero_zero @ A ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ S3 )
=> ( ( G @ X3 )
= ( H @ X3 ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ T6 )
= ( groups7311177749621191930dd_sum @ B @ A @ H @ S3 ) ) ) ) ) ) ) ).
% sum.mono_neutral_cong_right
thf(fact_4325_sum_Osubset__diff,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [B7: set @ B,A4: set @ B,G: B > A] :
( ( ord_less_eq @ ( set @ B ) @ B7 @ A4 )
=> ( ( finite_finite @ B @ A4 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ A4 )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A4 @ B7 ) ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ B7 ) ) ) ) ) ) ).
% sum.subset_diff
thf(fact_4326_sum__diff,axiom,
! [A: $tType,B: $tType] :
( ( ab_group_add @ A )
=> ! [A4: set @ B,B7: set @ B,F2: B > A] :
( ( finite_finite @ B @ A4 )
=> ( ( ord_less_eq @ ( set @ B ) @ B7 @ A4 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ F2 @ ( minus_minus @ ( set @ B ) @ A4 @ B7 ) )
= ( minus_minus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A4 ) @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ B7 ) ) ) ) ) ) ).
% sum_diff
thf(fact_4327_prod_Osubset__diff,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [B7: set @ B,A4: set @ B,G: B > A] :
( ( ord_less_eq @ ( set @ B ) @ B7 @ A4 )
=> ( ( finite_finite @ B @ A4 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ A4 )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A4 @ B7 ) ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ B7 ) ) ) ) ) ) ).
% prod.subset_diff
thf(fact_4328_prod_Omono__neutral__cong__right,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [T6: set @ B,S3: set @ B,G: B > A,H: B > A] :
( ( finite_finite @ B @ T6 )
=> ( ( ord_less_eq @ ( set @ B ) @ S3 @ T6 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T6 @ S3 ) )
=> ( ( G @ X3 )
= ( one_one @ A ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ S3 )
=> ( ( G @ X3 )
= ( H @ X3 ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ T6 )
= ( groups7121269368397514597t_prod @ B @ A @ H @ S3 ) ) ) ) ) ) ) ).
% prod.mono_neutral_cong_right
thf(fact_4329_prod_Omono__neutral__cong__left,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [T6: set @ B,S3: set @ B,H: B > A,G: B > A] :
( ( finite_finite @ B @ T6 )
=> ( ( ord_less_eq @ ( set @ B ) @ S3 @ T6 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T6 @ S3 ) )
=> ( ( H @ X3 )
= ( one_one @ A ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ S3 )
=> ( ( G @ X3 )
= ( H @ X3 ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ S3 )
= ( groups7121269368397514597t_prod @ B @ A @ H @ T6 ) ) ) ) ) ) ) ).
% prod.mono_neutral_cong_left
thf(fact_4330_prod_Omono__neutral__right,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [T6: set @ B,S3: set @ B,G: B > A] :
( ( finite_finite @ B @ T6 )
=> ( ( ord_less_eq @ ( set @ B ) @ S3 @ T6 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T6 @ S3 ) )
=> ( ( G @ X3 )
= ( one_one @ A ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ T6 )
= ( groups7121269368397514597t_prod @ B @ A @ G @ S3 ) ) ) ) ) ) ).
% prod.mono_neutral_right
thf(fact_4331_prod_Omono__neutral__left,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [T6: set @ B,S3: set @ B,G: B > A] :
( ( finite_finite @ B @ T6 )
=> ( ( ord_less_eq @ ( set @ B ) @ S3 @ T6 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T6 @ S3 ) )
=> ( ( G @ X3 )
= ( one_one @ A ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ S3 )
= ( groups7121269368397514597t_prod @ B @ A @ G @ T6 ) ) ) ) ) ) ).
% prod.mono_neutral_left
thf(fact_4332_prod_Osame__carrierI,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [C5: set @ B,A4: set @ B,B7: set @ B,G: B > A,H: B > A] :
( ( finite_finite @ B @ C5 )
=> ( ( ord_less_eq @ ( set @ B ) @ A4 @ C5 )
=> ( ( ord_less_eq @ ( set @ B ) @ B7 @ C5 )
=> ( ! [A6: B] :
( ( member @ B @ A6 @ ( minus_minus @ ( set @ B ) @ C5 @ A4 ) )
=> ( ( G @ A6 )
= ( one_one @ A ) ) )
=> ( ! [B4: B] :
( ( member @ B @ B4 @ ( minus_minus @ ( set @ B ) @ C5 @ B7 ) )
=> ( ( H @ B4 )
= ( one_one @ A ) ) )
=> ( ( ( groups7121269368397514597t_prod @ B @ A @ G @ C5 )
= ( groups7121269368397514597t_prod @ B @ A @ H @ C5 ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ A4 )
= ( groups7121269368397514597t_prod @ B @ A @ H @ B7 ) ) ) ) ) ) ) ) ) ).
% prod.same_carrierI
thf(fact_4333_prod_Osame__carrier,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [C5: set @ B,A4: set @ B,B7: set @ B,G: B > A,H: B > A] :
( ( finite_finite @ B @ C5 )
=> ( ( ord_less_eq @ ( set @ B ) @ A4 @ C5 )
=> ( ( ord_less_eq @ ( set @ B ) @ B7 @ C5 )
=> ( ! [A6: B] :
( ( member @ B @ A6 @ ( minus_minus @ ( set @ B ) @ C5 @ A4 ) )
=> ( ( G @ A6 )
= ( one_one @ A ) ) )
=> ( ! [B4: B] :
( ( member @ B @ B4 @ ( minus_minus @ ( set @ B ) @ C5 @ B7 ) )
=> ( ( H @ B4 )
= ( one_one @ A ) ) )
=> ( ( ( groups7121269368397514597t_prod @ B @ A @ G @ A4 )
= ( groups7121269368397514597t_prod @ B @ A @ H @ B7 ) )
= ( ( groups7121269368397514597t_prod @ B @ A @ G @ C5 )
= ( groups7121269368397514597t_prod @ B @ A @ H @ C5 ) ) ) ) ) ) ) ) ) ).
% prod.same_carrier
thf(fact_4334_infinite__imp__bij__betw,axiom,
! [A: $tType,A4: set @ A,A3: A] :
( ~ ( finite_finite @ A @ A4 )
=> ? [H3: A > A] : ( bij_betw @ A @ A @ H3 @ A4 @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% infinite_imp_bij_betw
thf(fact_4335_sum__diff__nat,axiom,
! [A: $tType,B7: set @ A,A4: set @ A,F2: A > nat] :
( ( finite_finite @ A @ B7 )
=> ( ( ord_less_eq @ ( set @ A ) @ B7 @ A4 )
=> ( ( groups7311177749621191930dd_sum @ A @ nat @ F2 @ ( minus_minus @ ( set @ A ) @ A4 @ B7 ) )
= ( minus_minus @ nat @ ( groups7311177749621191930dd_sum @ A @ nat @ F2 @ A4 ) @ ( groups7311177749621191930dd_sum @ A @ nat @ F2 @ B7 ) ) ) ) ) ).
% sum_diff_nat
thf(fact_4336_finite__roots__unity,axiom,
! [A: $tType] :
( ( ( real_V8999393235501362500lgebra @ A )
& ( idom @ A ) )
=> ! [N: nat] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ N )
=> ( finite_finite @ A
@ ( collect @ A
@ ^ [Z6: A] :
( ( power_power @ A @ Z6 @ N )
= ( one_one @ A ) ) ) ) ) ) ).
% finite_roots_unity
thf(fact_4337_sum_Oreindex__bij__betw__not__neutral,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comm_monoid_add @ A )
=> ! [S4: set @ B,T5: set @ C,H: B > C,S3: set @ B,T6: set @ C,G: C > A] :
( ( finite_finite @ B @ S4 )
=> ( ( finite_finite @ C @ T5 )
=> ( ( bij_betw @ B @ C @ H @ ( minus_minus @ ( set @ B ) @ S3 @ S4 ) @ ( minus_minus @ ( set @ C ) @ T6 @ T5 ) )
=> ( ! [A6: B] :
( ( member @ B @ A6 @ S4 )
=> ( ( G @ ( H @ A6 ) )
= ( zero_zero @ A ) ) )
=> ( ! [B4: C] :
( ( member @ C @ B4 @ T5 )
=> ( ( G @ B4 )
= ( zero_zero @ A ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X2: B] : ( G @ ( H @ X2 ) )
@ S3 )
= ( groups7311177749621191930dd_sum @ C @ A @ G @ T6 ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_betw_not_neutral
thf(fact_4338_sums__If__finite__set_H,axiom,
! [A: $tType] :
( ( ( topolo1287966508704411220up_add @ A )
& ( topological_t2_space @ A ) )
=> ! [G: nat > A,S3: A,A4: set @ nat,S4: A,F2: nat > A] :
( ( sums @ A @ G @ S3 )
=> ( ( finite_finite @ nat @ A4 )
=> ( ( S4
= ( plus_plus @ A @ S3
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [N5: nat] : ( minus_minus @ A @ ( F2 @ N5 ) @ ( G @ N5 ) )
@ A4 ) ) )
=> ( sums @ A
@ ^ [N5: nat] : ( if @ A @ ( member @ nat @ N5 @ A4 ) @ ( F2 @ N5 ) @ ( G @ N5 ) )
@ S4 ) ) ) ) ) ).
% sums_If_finite_set'
thf(fact_4339_prod_Oreindex__bij__betw__not__neutral,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S4: set @ B,T5: set @ C,H: B > C,S3: set @ B,T6: set @ C,G: C > A] :
( ( finite_finite @ B @ S4 )
=> ( ( finite_finite @ C @ T5 )
=> ( ( bij_betw @ B @ C @ H @ ( minus_minus @ ( set @ B ) @ S3 @ S4 ) @ ( minus_minus @ ( set @ C ) @ T6 @ T5 ) )
=> ( ! [A6: B] :
( ( member @ B @ A6 @ S4 )
=> ( ( G @ ( H @ A6 ) )
= ( one_one @ A ) ) )
=> ( ! [B4: C] :
( ( member @ C @ B4 @ T5 )
=> ( ( G @ B4 )
= ( one_one @ A ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [X2: B] : ( G @ ( H @ X2 ) )
@ S3 )
= ( groups7121269368397514597t_prod @ C @ A @ G @ T6 ) ) ) ) ) ) ) ) ).
% prod.reindex_bij_betw_not_neutral
thf(fact_4340_ln__prod,axiom,
! [A: $tType,I5: set @ A,F2: A > real] :
( ( finite_finite @ A @ I5 )
=> ( ! [I3: A] :
( ( member @ A @ I3 @ I5 )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( F2 @ I3 ) ) )
=> ( ( ln_ln @ real @ ( groups7121269368397514597t_prod @ A @ real @ F2 @ I5 ) )
= ( groups7311177749621191930dd_sum @ A @ real
@ ^ [X2: A] : ( ln_ln @ real @ ( F2 @ X2 ) )
@ I5 ) ) ) ) ).
% ln_prod
thf(fact_4341_prod__mono__strict,axiom,
! [A: $tType,B: $tType] :
( ( linordered_semidom @ A )
=> ! [A4: set @ B,F2: B > A,G: B > A] :
( ( finite_finite @ B @ A4 )
=> ( ! [I3: B] :
( ( member @ B @ I3 @ A4 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ I3 ) )
& ( ord_less @ A @ ( F2 @ I3 ) @ ( G @ I3 ) ) ) )
=> ( ( A4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ord_less @ A @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A4 ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ A4 ) ) ) ) ) ) ).
% prod_mono_strict
thf(fact_4342_sum__mono2,axiom,
! [A: $tType,B: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [B7: set @ B,A4: set @ B,F2: B > A] :
( ( finite_finite @ B @ B7 )
=> ( ( ord_less_eq @ ( set @ B ) @ A4 @ B7 )
=> ( ! [B4: B] :
( ( member @ B @ B4 @ ( minus_minus @ ( set @ B ) @ B7 @ A4 ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ B4 ) ) )
=> ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A4 ) @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ B7 ) ) ) ) ) ) ).
% sum_mono2
thf(fact_4343_sum_Oinsert__remove,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A4: set @ B,G: B > A,X: B] :
( ( finite_finite @ B @ A4 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( insert @ B @ X @ A4 ) )
= ( plus_plus @ A @ ( G @ X ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A4 @ ( insert @ B @ X @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ).
% sum.insert_remove
thf(fact_4344_sum_Oremove,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A4: set @ B,X: B,G: B > A] :
( ( finite_finite @ B @ A4 )
=> ( ( member @ B @ X @ A4 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ A4 )
= ( plus_plus @ A @ ( G @ X ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A4 @ ( insert @ B @ X @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ) ).
% sum.remove
thf(fact_4345_sum__diff1,axiom,
! [A: $tType,B: $tType] :
( ( ab_group_add @ A )
=> ! [A4: set @ B,A3: B,F2: B > A] :
( ( finite_finite @ B @ A4 )
=> ( ( ( member @ B @ A3 @ A4 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ F2 @ ( minus_minus @ ( set @ B ) @ A4 @ ( insert @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) ) )
= ( minus_minus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A4 ) @ ( F2 @ A3 ) ) ) )
& ( ~ ( member @ B @ A3 @ A4 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ F2 @ ( minus_minus @ ( set @ B ) @ A4 @ ( insert @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) ) )
= ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A4 ) ) ) ) ) ) ).
% sum_diff1
thf(fact_4346_prod_Oremove,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A4: set @ B,X: B,G: B > A] :
( ( finite_finite @ B @ A4 )
=> ( ( member @ B @ X @ A4 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ A4 )
= ( times_times @ A @ ( G @ X ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A4 @ ( insert @ B @ X @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ) ).
% prod.remove
thf(fact_4347_prod_Oinsert__remove,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A4: set @ B,G: B > A,X: B] :
( ( finite_finite @ B @ A4 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( insert @ B @ X @ A4 ) )
= ( times_times @ A @ ( G @ X ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A4 @ ( insert @ B @ X @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ).
% prod.insert_remove
thf(fact_4348_sum__le__suminf,axiom,
! [A: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F2: nat > A,I5: set @ nat] :
( ( summable @ A @ F2 )
=> ( ( finite_finite @ nat @ I5 )
=> ( ! [N2: nat] :
( ( member @ nat @ N2 @ ( uminus_uminus @ ( set @ nat ) @ I5 ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ N2 ) ) )
=> ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ I5 ) @ ( suminf @ A @ F2 ) ) ) ) ) ) ).
% sum_le_suminf
thf(fact_4349_sum_Odelta__remove,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [S3: set @ B,A3: B,B2: B > A,C2: B > A] :
( ( finite_finite @ B @ S3 )
=> ( ( ( member @ B @ A3 @ S3 )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [K3: B] : ( if @ A @ ( K3 = A3 ) @ ( B2 @ K3 ) @ ( C2 @ K3 ) )
@ S3 )
= ( plus_plus @ A @ ( B2 @ A3 ) @ ( groups7311177749621191930dd_sum @ B @ A @ C2 @ ( minus_minus @ ( set @ B ) @ S3 @ ( insert @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) )
& ( ~ ( member @ B @ A3 @ S3 )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [K3: B] : ( if @ A @ ( K3 = A3 ) @ ( B2 @ K3 ) @ ( C2 @ K3 ) )
@ S3 )
= ( groups7311177749621191930dd_sum @ B @ A @ C2 @ ( minus_minus @ ( set @ B ) @ S3 @ ( insert @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ) ).
% sum.delta_remove
thf(fact_4350_prod_Odelta__remove,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S3: set @ B,A3: B,B2: B > A,C2: B > A] :
( ( finite_finite @ B @ S3 )
=> ( ( ( member @ B @ A3 @ S3 )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K3: B] : ( if @ A @ ( K3 = A3 ) @ ( B2 @ K3 ) @ ( C2 @ K3 ) )
@ S3 )
= ( times_times @ A @ ( B2 @ A3 ) @ ( groups7121269368397514597t_prod @ B @ A @ C2 @ ( minus_minus @ ( set @ B ) @ S3 @ ( insert @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) )
& ( ~ ( member @ B @ A3 @ S3 )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K3: B] : ( if @ A @ ( K3 = A3 ) @ ( B2 @ K3 ) @ ( C2 @ K3 ) )
@ S3 )
= ( groups7121269368397514597t_prod @ B @ A @ C2 @ ( minus_minus @ ( set @ B ) @ S3 @ ( insert @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ) ).
% prod.delta_remove
thf(fact_4351_sum__strict__mono2,axiom,
! [B: $tType,A: $tType] :
( ( ordere8940638589300402666id_add @ B )
=> ! [B7: set @ A,A4: set @ A,B2: A,F2: A > B] :
( ( finite_finite @ A @ B7 )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ B7 )
=> ( ( member @ A @ B2 @ ( minus_minus @ ( set @ A ) @ B7 @ A4 ) )
=> ( ( ord_less @ B @ ( zero_zero @ B ) @ ( F2 @ B2 ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ B7 )
=> ( ord_less_eq @ B @ ( zero_zero @ B ) @ ( F2 @ X3 ) ) )
=> ( ord_less @ B @ ( groups7311177749621191930dd_sum @ A @ B @ F2 @ A4 ) @ ( groups7311177749621191930dd_sum @ A @ B @ F2 @ B7 ) ) ) ) ) ) ) ) ).
% sum_strict_mono2
thf(fact_4352_member__le__sum,axiom,
! [B: $tType,C: $tType] :
( ( ( ordere6911136660526730532id_add @ B )
& ( semiring_1 @ B ) )
=> ! [I: C,A4: set @ C,F2: C > B] :
( ( member @ C @ I @ A4 )
=> ( ! [X3: C] :
( ( member @ C @ X3 @ ( minus_minus @ ( set @ C ) @ A4 @ ( insert @ C @ I @ ( bot_bot @ ( set @ C ) ) ) ) )
=> ( ord_less_eq @ B @ ( zero_zero @ B ) @ ( F2 @ X3 ) ) )
=> ( ( finite_finite @ C @ A4 )
=> ( ord_less_eq @ B @ ( F2 @ I ) @ ( groups7311177749621191930dd_sum @ C @ B @ F2 @ A4 ) ) ) ) ) ) ).
% member_le_sum
thf(fact_4353_prod__mono2,axiom,
! [B: $tType,A: $tType] :
( ( linordered_idom @ B )
=> ! [B7: set @ A,A4: set @ A,F2: A > B] :
( ( finite_finite @ A @ B7 )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ B7 )
=> ( ! [B4: A] :
( ( member @ A @ B4 @ ( minus_minus @ ( set @ A ) @ B7 @ A4 ) )
=> ( ord_less_eq @ B @ ( one_one @ B ) @ ( F2 @ B4 ) ) )
=> ( ! [A6: A] :
( ( member @ A @ A6 @ A4 )
=> ( ord_less_eq @ B @ ( zero_zero @ B ) @ ( F2 @ A6 ) ) )
=> ( ord_less_eq @ B @ ( groups7121269368397514597t_prod @ A @ B @ F2 @ A4 ) @ ( groups7121269368397514597t_prod @ A @ B @ F2 @ B7 ) ) ) ) ) ) ) ).
% prod_mono2
thf(fact_4354_prod__diff1,axiom,
! [A: $tType,B: $tType] :
( ( semidom_divide @ A )
=> ! [A4: set @ B,F2: B > A,A3: B] :
( ( finite_finite @ B @ A4 )
=> ( ( ( F2 @ A3 )
!= ( zero_zero @ A ) )
=> ( ( ( member @ B @ A3 @ A4 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ F2 @ ( minus_minus @ ( set @ B ) @ A4 @ ( insert @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) ) )
= ( divide_divide @ A @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A4 ) @ ( F2 @ A3 ) ) ) )
& ( ~ ( member @ B @ A3 @ A4 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ F2 @ ( minus_minus @ ( set @ B ) @ A4 @ ( insert @ B @ A3 @ ( bot_bot @ ( set @ B ) ) ) ) )
= ( groups7121269368397514597t_prod @ B @ A @ F2 @ A4 ) ) ) ) ) ) ) ).
% prod_diff1
thf(fact_4355_even__set__encode__iff,axiom,
! [A4: set @ nat] :
( ( finite_finite @ nat @ A4 )
=> ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( nat_set_encode @ A4 ) )
= ( ~ ( member @ nat @ ( zero_zero @ nat ) @ A4 ) ) ) ) ).
% even_set_encode_iff
thf(fact_4356_polyfun__roots__finite,axiom,
! [A: $tType] :
( ( ( real_V8999393235501362500lgebra @ A )
& ( idom @ A ) )
=> ! [C2: nat > A,K: nat,N: nat] :
( ( ( C2 @ K )
!= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ nat @ K @ N )
=> ( finite_finite @ A
@ ( collect @ A
@ ^ [Z6: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( times_times @ A @ ( C2 @ I2 ) @ ( power_power @ A @ Z6 @ I2 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% polyfun_roots_finite
thf(fact_4357_polyfun__finite__roots,axiom,
! [A: $tType] :
( ( ( real_V8999393235501362500lgebra @ A )
& ( idom @ A ) )
=> ! [C2: nat > A,N: nat] :
( ( finite_finite @ A
@ ( collect @ A
@ ^ [X2: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( times_times @ A @ ( C2 @ I2 ) @ ( power_power @ A @ X2 @ I2 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( zero_zero @ A ) ) ) )
= ( ? [I2: nat] :
( ( ord_less_eq @ nat @ I2 @ N )
& ( ( C2 @ I2 )
!= ( zero_zero @ A ) ) ) ) ) ) ).
% polyfun_finite_roots
thf(fact_4358_finite__Diff__insert,axiom,
! [A: $tType,A4: set @ A,A3: A,B7: set @ A] :
( ( finite_finite @ A @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ A3 @ B7 ) ) )
= ( finite_finite @ A @ ( minus_minus @ ( set @ A ) @ A4 @ B7 ) ) ) ).
% finite_Diff_insert
thf(fact_4359_finite__Collect__less__nat,axiom,
! [K: nat] :
( finite_finite @ nat
@ ( collect @ nat
@ ^ [N5: nat] : ( ord_less @ nat @ N5 @ K ) ) ) ).
% finite_Collect_less_nat
thf(fact_4360_finite__induct__select,axiom,
! [A: $tType,S3: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite @ A @ S3 )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [T7: set @ A] :
( ( ord_less @ ( set @ A ) @ T7 @ S3 )
=> ( ( P @ T7 )
=> ? [X4: A] :
( ( member @ A @ X4 @ ( minus_minus @ ( set @ A ) @ S3 @ T7 ) )
& ( P @ ( insert @ A @ X4 @ T7 ) ) ) ) )
=> ( P @ S3 ) ) ) ) ).
% finite_induct_select
thf(fact_4361_finite__Diff2,axiom,
! [A: $tType,B7: set @ A,A4: set @ A] :
( ( finite_finite @ A @ B7 )
=> ( ( finite_finite @ A @ ( minus_minus @ ( set @ A ) @ A4 @ B7 ) )
= ( finite_finite @ A @ A4 ) ) ) ).
% finite_Diff2
thf(fact_4362_finite__interval__int1,axiom,
! [A3: int,B2: int] :
( finite_finite @ int
@ ( collect @ int
@ ^ [I2: int] :
( ( ord_less_eq @ int @ A3 @ I2 )
& ( ord_less_eq @ int @ I2 @ B2 ) ) ) ) ).
% finite_interval_int1
thf(fact_4363_finite__interval__int4,axiom,
! [A3: int,B2: int] :
( finite_finite @ int
@ ( collect @ int
@ ^ [I2: int] :
( ( ord_less @ int @ A3 @ I2 )
& ( ord_less @ int @ I2 @ B2 ) ) ) ) ).
% finite_interval_int4
thf(fact_4364_finite__Diff,axiom,
! [A: $tType,A4: set @ A,B7: set @ A] :
( ( finite_finite @ A @ A4 )
=> ( finite_finite @ A @ ( minus_minus @ ( set @ A ) @ A4 @ B7 ) ) ) ).
% finite_Diff
thf(fact_4365_finite__interval__int2,axiom,
! [A3: int,B2: int] :
( finite_finite @ int
@ ( collect @ int
@ ^ [I2: int] :
( ( ord_less_eq @ int @ A3 @ I2 )
& ( ord_less @ int @ I2 @ B2 ) ) ) ) ).
% finite_interval_int2
thf(fact_4366_finite__interval__int3,axiom,
! [A3: int,B2: int] :
( finite_finite @ int
@ ( collect @ int
@ ^ [I2: int] :
( ( ord_less @ int @ A3 @ I2 )
& ( ord_less_eq @ int @ I2 @ B2 ) ) ) ) ).
% finite_interval_int3
thf(fact_4367_finite__nth__roots,axiom,
! [N: nat,C2: complex] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( finite_finite @ complex
@ ( collect @ complex
@ ^ [Z6: complex] :
( ( power_power @ complex @ Z6 @ N )
= C2 ) ) ) ) ).
% finite_nth_roots
thf(fact_4368_uminus__integer__code_I1_J,axiom,
( ( uminus_uminus @ code_integer @ ( zero_zero @ code_integer ) )
= ( zero_zero @ code_integer ) ) ).
% uminus_integer_code(1)
thf(fact_4369_abs__integer__code,axiom,
( ( abs_abs @ code_integer )
= ( ^ [K3: code_integer] : ( if @ code_integer @ ( ord_less @ code_integer @ K3 @ ( zero_zero @ code_integer ) ) @ ( uminus_uminus @ code_integer @ K3 ) @ K3 ) ) ) ).
% abs_integer_code
thf(fact_4370_finite__maxlen,axiom,
! [A: $tType,M10: set @ ( list @ A )] :
( ( finite_finite @ ( list @ A ) @ M10 )
=> ? [N2: nat] :
! [X4: list @ A] :
( ( member @ ( list @ A ) @ X4 @ M10 )
=> ( ord_less @ nat @ ( size_size @ ( list @ A ) @ X4 ) @ N2 ) ) ) ).
% finite_maxlen
thf(fact_4371_finite__atLeastZeroLessThan__int,axiom,
! [U: int] : ( finite_finite @ int @ ( set_or7035219750837199246ssThan @ int @ ( zero_zero @ int ) @ U ) ) ).
% finite_atLeastZeroLessThan_int
thf(fact_4372_finite__divisors__int,axiom,
! [I: int] :
( ( I
!= ( zero_zero @ int ) )
=> ( finite_finite @ int
@ ( collect @ int
@ ^ [D4: int] : ( dvd_dvd @ int @ D4 @ I ) ) ) ) ).
% finite_divisors_int
thf(fact_4373_Diff__infinite__finite,axiom,
! [A: $tType,T6: set @ A,S3: set @ A] :
( ( finite_finite @ A @ T6 )
=> ( ~ ( finite_finite @ A @ S3 )
=> ~ ( finite_finite @ A @ ( minus_minus @ ( set @ A ) @ S3 @ T6 ) ) ) ) ).
% Diff_infinite_finite
thf(fact_4374_finite__empty__induct,axiom,
! [A: $tType,A4: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite @ A @ A4 )
=> ( ( P @ A4 )
=> ( ! [A6: A,A9: set @ A] :
( ( finite_finite @ A @ A9 )
=> ( ( member @ A @ A6 @ A9 )
=> ( ( P @ A9 )
=> ( P @ ( minus_minus @ ( set @ A ) @ A9 @ ( insert @ A @ A6 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) )
=> ( P @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% finite_empty_induct
thf(fact_4375_infinite__coinduct,axiom,
! [A: $tType,X8: ( set @ A ) > $o,A4: set @ A] :
( ( X8 @ A4 )
=> ( ! [A9: set @ A] :
( ( X8 @ A9 )
=> ? [X4: A] :
( ( member @ A @ X4 @ A9 )
& ( ( X8 @ ( minus_minus @ ( set @ A ) @ A9 @ ( insert @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) ) )
| ~ ( finite_finite @ A @ ( minus_minus @ ( set @ A ) @ A9 @ ( insert @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) )
=> ~ ( finite_finite @ A @ A4 ) ) ) ).
% infinite_coinduct
thf(fact_4376_infinite__remove,axiom,
! [A: $tType,S3: set @ A,A3: A] :
( ~ ( finite_finite @ A @ S3 )
=> ~ ( finite_finite @ A @ ( minus_minus @ ( set @ A ) @ S3 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% infinite_remove
thf(fact_4377_remove__induct,axiom,
! [A: $tType,P: ( set @ A ) > $o,B7: set @ A] :
( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ( ~ ( finite_finite @ A @ B7 )
=> ( P @ B7 ) )
=> ( ! [A9: set @ A] :
( ( finite_finite @ A @ A9 )
=> ( ( A9
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ A9 @ B7 )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ A9 )
=> ( P @ ( minus_minus @ ( set @ A ) @ A9 @ ( insert @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) ) ) )
=> ( P @ A9 ) ) ) ) )
=> ( P @ B7 ) ) ) ) ).
% remove_induct
thf(fact_4378_finite__remove__induct,axiom,
! [A: $tType,B7: set @ A,P: ( set @ A ) > $o] :
( ( finite_finite @ A @ B7 )
=> ( ( P @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [A9: set @ A] :
( ( finite_finite @ A @ A9 )
=> ( ( A9
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ A9 @ B7 )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ A9 )
=> ( P @ ( minus_minus @ ( set @ A ) @ A9 @ ( insert @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) ) ) )
=> ( P @ A9 ) ) ) ) )
=> ( P @ B7 ) ) ) ) ).
% finite_remove_induct
thf(fact_4379_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_4380_length__mul__elem,axiom,
! [A: $tType,Xs: list @ ( list @ A ),N: nat] :
( ! [X3: list @ A] :
( ( member @ ( list @ A ) @ X3 @ ( set2 @ ( list @ A ) @ Xs ) )
=> ( ( size_size @ ( list @ A ) @ X3 )
= N ) )
=> ( ( size_size @ ( list @ A ) @ ( concat @ A @ Xs ) )
= ( times_times @ nat @ ( size_size @ ( list @ ( list @ A ) ) @ Xs ) @ N ) ) ) ).
% length_mul_elem
thf(fact_4381_int__ge__less__than2__def,axiom,
( int_ge_less_than2
= ( ^ [D4: int] :
( collect @ ( product_prod @ int @ int )
@ ( product_case_prod @ int @ int @ $o
@ ^ [Z7: int,Z6: int] :
( ( ord_less_eq @ int @ D4 @ Z6 )
& ( ord_less @ int @ Z7 @ Z6 ) ) ) ) ) ) ).
% int_ge_less_than2_def
thf(fact_4382_int__ge__less__than__def,axiom,
( int_ge_less_than
= ( ^ [D4: int] :
( collect @ ( product_prod @ int @ int )
@ ( product_case_prod @ int @ int @ $o
@ ^ [Z7: int,Z6: int] :
( ( ord_less_eq @ int @ D4 @ Z7 )
& ( ord_less @ int @ Z7 @ Z6 ) ) ) ) ) ) ).
% int_ge_less_than_def
thf(fact_4383_zero__integer__def,axiom,
( ( zero_zero @ code_integer )
= ( code_integer_of_int @ ( zero_zero @ int ) ) ) ).
% zero_integer_def
thf(fact_4384_less__integer_Oabs__eq,axiom,
! [Xa: int,X: int] :
( ( ord_less @ code_integer @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X ) )
= ( ord_less @ int @ Xa @ X ) ) ).
% less_integer.abs_eq
thf(fact_4385_uminus__integer_Oabs__eq,axiom,
! [X: int] :
( ( uminus_uminus @ code_integer @ ( code_integer_of_int @ X ) )
= ( code_integer_of_int @ ( uminus_uminus @ int @ X ) ) ) ).
% uminus_integer.abs_eq
thf(fact_4386_one__integer__def,axiom,
( ( one_one @ code_integer )
= ( code_integer_of_int @ ( one_one @ int ) ) ) ).
% one_integer_def
thf(fact_4387_minus__integer_Oabs__eq,axiom,
! [Xa: int,X: int] :
( ( minus_minus @ code_integer @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X ) )
= ( code_integer_of_int @ ( minus_minus @ int @ Xa @ X ) ) ) ).
% minus_integer.abs_eq
thf(fact_4388_divide__integer_Oabs__eq,axiom,
! [Xa: int,X: int] :
( ( divide_divide @ code_integer @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X ) )
= ( code_integer_of_int @ ( divide_divide @ int @ Xa @ X ) ) ) ).
% divide_integer.abs_eq
thf(fact_4389_infinite__int__iff__unbounded,axiom,
! [S3: set @ int] :
( ( ~ ( finite_finite @ int @ S3 ) )
= ( ! [M5: int] :
? [N5: int] :
( ( ord_less @ int @ M5 @ ( abs_abs @ int @ N5 ) )
& ( member @ int @ N5 @ S3 ) ) ) ) ).
% infinite_int_iff_unbounded
thf(fact_4390_sum__count__set,axiom,
! [A: $tType,Xs: list @ A,X8: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ X8 )
=> ( ( finite_finite @ A @ X8 )
=> ( ( groups7311177749621191930dd_sum @ A @ nat @ ( count_list @ A @ Xs ) @ X8 )
= ( size_size @ ( list @ A ) @ Xs ) ) ) ) ).
% sum_count_set
thf(fact_4391_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_4392_sum__diff1_H__aux,axiom,
! [B: $tType,A: $tType] :
( ( ab_group_add @ B )
=> ! [F5: set @ A,I5: set @ A,F2: A > B,I: A] :
( ( finite_finite @ A @ F5 )
=> ( ( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [I2: A] :
( ( member @ A @ I2 @ I5 )
& ( ( F2 @ I2 )
!= ( zero_zero @ B ) ) ) )
@ F5 )
=> ( ( ( member @ A @ I @ I5 )
=> ( ( groups1027152243600224163dd_sum @ A @ B @ F2 @ ( minus_minus @ ( set @ A ) @ I5 @ ( insert @ A @ I @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( minus_minus @ B @ ( groups1027152243600224163dd_sum @ A @ B @ F2 @ I5 ) @ ( F2 @ I ) ) ) )
& ( ~ ( member @ A @ I @ I5 )
=> ( ( groups1027152243600224163dd_sum @ A @ B @ F2 @ ( minus_minus @ ( set @ A ) @ I5 @ ( insert @ A @ I @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( groups1027152243600224163dd_sum @ A @ B @ F2 @ I5 ) ) ) ) ) ) ) ).
% sum_diff1'_aux
thf(fact_4393_sum_Oempty_H,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ A )
=> ! [P4: B > A] :
( ( groups1027152243600224163dd_sum @ B @ A @ P4 @ ( bot_bot @ ( set @ B ) ) )
= ( zero_zero @ A ) ) ) ).
% sum.empty'
thf(fact_4394_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_4395_sum_Oinsert_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [I5: set @ B,P4: B > A,I: B] :
( ( finite_finite @ B
@ ( collect @ B
@ ^ [X2: B] :
( ( member @ B @ X2 @ I5 )
& ( ( P4 @ X2 )
!= ( zero_zero @ A ) ) ) ) )
=> ( ( ( member @ B @ I @ I5 )
=> ( ( groups1027152243600224163dd_sum @ B @ A @ P4 @ ( insert @ B @ I @ I5 ) )
= ( groups1027152243600224163dd_sum @ B @ A @ P4 @ I5 ) ) )
& ( ~ ( member @ B @ I @ I5 )
=> ( ( groups1027152243600224163dd_sum @ B @ A @ P4 @ ( insert @ B @ I @ I5 ) )
= ( plus_plus @ A @ ( P4 @ I ) @ ( groups1027152243600224163dd_sum @ B @ A @ P4 @ I5 ) ) ) ) ) ) ) ).
% sum.insert'
thf(fact_4396_sum_Onon__neutral_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: B > A,I5: set @ B] :
( ( groups1027152243600224163dd_sum @ B @ A @ G
@ ( collect @ B
@ ^ [X2: B] :
( ( member @ B @ X2 @ I5 )
& ( ( G @ X2 )
!= ( zero_zero @ A ) ) ) ) )
= ( groups1027152243600224163dd_sum @ B @ A @ G @ I5 ) ) ) ).
% sum.non_neutral'
thf(fact_4397_sum_Omono__neutral__cong__right_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [S3: set @ B,T6: set @ B,G: B > A,H: B > A] :
( ( ord_less_eq @ ( set @ B ) @ S3 @ T6 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T6 @ S3 ) )
=> ( ( G @ X3 )
= ( zero_zero @ A ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ S3 )
=> ( ( G @ X3 )
= ( H @ X3 ) ) )
=> ( ( groups1027152243600224163dd_sum @ B @ A @ G @ T6 )
= ( groups1027152243600224163dd_sum @ B @ A @ H @ S3 ) ) ) ) ) ) ).
% sum.mono_neutral_cong_right'
thf(fact_4398_sum_Omono__neutral__cong__left_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [S3: set @ B,T6: set @ B,H: B > A,G: B > A] :
( ( ord_less_eq @ ( set @ B ) @ S3 @ T6 )
=> ( ! [I3: B] :
( ( member @ B @ I3 @ ( minus_minus @ ( set @ B ) @ T6 @ S3 ) )
=> ( ( H @ I3 )
= ( zero_zero @ A ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ S3 )
=> ( ( G @ X3 )
= ( H @ X3 ) ) )
=> ( ( groups1027152243600224163dd_sum @ B @ A @ G @ S3 )
= ( groups1027152243600224163dd_sum @ B @ A @ H @ T6 ) ) ) ) ) ) ).
% sum.mono_neutral_cong_left'
thf(fact_4399_sum_Omono__neutral__right_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [S3: set @ B,T6: set @ B,G: B > A] :
( ( ord_less_eq @ ( set @ B ) @ S3 @ T6 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T6 @ S3 ) )
=> ( ( G @ X3 )
= ( zero_zero @ A ) ) )
=> ( ( groups1027152243600224163dd_sum @ B @ A @ G @ T6 )
= ( groups1027152243600224163dd_sum @ B @ A @ G @ S3 ) ) ) ) ) ).
% sum.mono_neutral_right'
thf(fact_4400_sum_Omono__neutral__left_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [S3: set @ B,T6: set @ B,G: B > A] :
( ( ord_less_eq @ ( set @ B ) @ S3 @ T6 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T6 @ S3 ) )
=> ( ( G @ X3 )
= ( zero_zero @ A ) ) )
=> ( ( groups1027152243600224163dd_sum @ B @ A @ G @ S3 )
= ( groups1027152243600224163dd_sum @ B @ A @ G @ T6 ) ) ) ) ) ).
% sum.mono_neutral_left'
thf(fact_4401_sum_Odistrib_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [I5: set @ B,G: B > A,H: B > A] :
( ( finite_finite @ B
@ ( collect @ B
@ ^ [X2: B] :
( ( member @ B @ X2 @ I5 )
& ( ( G @ X2 )
!= ( zero_zero @ A ) ) ) ) )
=> ( ( finite_finite @ B
@ ( collect @ B
@ ^ [X2: B] :
( ( member @ B @ X2 @ I5 )
& ( ( H @ X2 )
!= ( zero_zero @ A ) ) ) ) )
=> ( ( groups1027152243600224163dd_sum @ B @ A
@ ^ [I2: B] : ( plus_plus @ A @ ( G @ I2 ) @ ( H @ I2 ) )
@ I5 )
= ( plus_plus @ A @ ( groups1027152243600224163dd_sum @ B @ A @ G @ I5 ) @ ( groups1027152243600224163dd_sum @ B @ A @ H @ I5 ) ) ) ) ) ) ).
% sum.distrib'
thf(fact_4402_sum_OG__def,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ( ( groups1027152243600224163dd_sum @ B @ A )
= ( ^ [P6: B > A,I7: set @ B] :
( if @ A
@ ( finite_finite @ B
@ ( collect @ B
@ ^ [X2: B] :
( ( member @ B @ X2 @ I7 )
& ( ( P6 @ X2 )
!= ( zero_zero @ A ) ) ) ) )
@ ( groups7311177749621191930dd_sum @ B @ A @ P6
@ ( collect @ B
@ ^ [X2: B] :
( ( member @ B @ X2 @ I7 )
& ( ( P6 @ X2 )
!= ( zero_zero @ A ) ) ) ) )
@ ( zero_zero @ A ) ) ) ) ) ).
% sum.G_def
thf(fact_4403_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_4404_length__n__lists__elem,axiom,
! [A: $tType,Ys2: list @ A,N: nat,Xs: list @ A] :
( ( member @ ( list @ A ) @ Ys2 @ ( set2 @ ( list @ A ) @ ( n_lists @ A @ N @ Xs ) ) )
=> ( ( size_size @ ( list @ A ) @ Ys2 )
= N ) ) ).
% length_n_lists_elem
thf(fact_4405_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_4406_sum__diff1_H,axiom,
! [B: $tType,A: $tType] :
( ( ab_group_add @ B )
=> ! [I5: set @ A,F2: A > B,I: A] :
( ( finite_finite @ A
@ ( collect @ A
@ ^ [I2: A] :
( ( member @ A @ I2 @ I5 )
& ( ( F2 @ I2 )
!= ( zero_zero @ B ) ) ) ) )
=> ( ( ( member @ A @ I @ I5 )
=> ( ( groups1027152243600224163dd_sum @ A @ B @ F2 @ ( minus_minus @ ( set @ A ) @ I5 @ ( insert @ A @ I @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( minus_minus @ B @ ( groups1027152243600224163dd_sum @ A @ B @ F2 @ I5 ) @ ( F2 @ I ) ) ) )
& ( ~ ( member @ A @ I @ I5 )
=> ( ( groups1027152243600224163dd_sum @ A @ B @ F2 @ ( minus_minus @ ( set @ A ) @ I5 @ ( insert @ A @ I @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( groups1027152243600224163dd_sum @ A @ B @ F2 @ I5 ) ) ) ) ) ) ).
% sum_diff1'
thf(fact_4407_unbounded__k__infinite,axiom,
! [K: nat,S3: set @ nat] :
( ! [M3: nat] :
( ( ord_less @ nat @ K @ M3 )
=> ? [N4: nat] :
( ( ord_less @ nat @ M3 @ N4 )
& ( member @ nat @ N4 @ S3 ) ) )
=> ~ ( finite_finite @ nat @ S3 ) ) ).
% unbounded_k_infinite
thf(fact_4408_infinite__nat__iff__unbounded,axiom,
! [S3: set @ nat] :
( ( ~ ( finite_finite @ nat @ S3 ) )
= ( ! [M5: nat] :
? [N5: nat] :
( ( ord_less @ nat @ M5 @ N5 )
& ( member @ nat @ N5 @ S3 ) ) ) ) ).
% infinite_nat_iff_unbounded
thf(fact_4409_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_4410_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_4411_divmod__integer__def,axiom,
( code_divmod_integer
= ( ^ [K3: code_integer,L2: code_integer] : ( product_Pair @ code_integer @ code_integer @ ( divide_divide @ code_integer @ K3 @ L2 ) @ ( modulo_modulo @ code_integer @ K3 @ L2 ) ) ) ) ).
% divmod_integer_def
thf(fact_4412_csqrt_Osimps_I1_J,axiom,
! [Z2: complex] :
( ( re @ ( csqrt @ Z2 ) )
= ( sqrt @ ( divide_divide @ real @ ( plus_plus @ real @ ( real_V7770717601297561774m_norm @ complex @ Z2 ) @ ( re @ Z2 ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ).
% csqrt.simps(1)
thf(fact_4413_Re__divide__of__nat,axiom,
! [Z2: complex,N: nat] :
( ( re @ ( divide_divide @ complex @ Z2 @ ( semiring_1_of_nat @ complex @ N ) ) )
= ( divide_divide @ real @ ( re @ Z2 ) @ ( semiring_1_of_nat @ real @ N ) ) ) ).
% Re_divide_of_nat
thf(fact_4414_Re__divide__of__real,axiom,
! [Z2: complex,R4: real] :
( ( re @ ( divide_divide @ complex @ Z2 @ ( real_Vector_of_real @ complex @ R4 ) ) )
= ( divide_divide @ real @ ( re @ Z2 ) @ R4 ) ) ).
% Re_divide_of_real
thf(fact_4415_Re__sgn,axiom,
! [Z2: complex] :
( ( re @ ( sgn_sgn @ complex @ Z2 ) )
= ( divide_divide @ real @ ( re @ Z2 ) @ ( real_V7770717601297561774m_norm @ complex @ Z2 ) ) ) ).
% Re_sgn
thf(fact_4416_Re__divide__numeral,axiom,
! [Z2: complex,W: num] :
( ( re @ ( divide_divide @ complex @ Z2 @ ( numeral_numeral @ complex @ W ) ) )
= ( divide_divide @ real @ ( re @ Z2 ) @ ( numeral_numeral @ real @ W ) ) ) ).
% Re_divide_numeral
thf(fact_4417_one__complex_Osimps_I1_J,axiom,
( ( re @ ( one_one @ complex ) )
= ( one_one @ real ) ) ).
% one_complex.simps(1)
thf(fact_4418_uminus__complex_Osimps_I1_J,axiom,
! [X: complex] :
( ( re @ ( uminus_uminus @ complex @ X ) )
= ( uminus_uminus @ real @ ( re @ X ) ) ) ).
% uminus_complex.simps(1)
thf(fact_4419_minus__complex_Osimps_I1_J,axiom,
! [X: complex,Y2: complex] :
( ( re @ ( minus_minus @ complex @ X @ Y2 ) )
= ( minus_minus @ real @ ( re @ X ) @ ( re @ Y2 ) ) ) ).
% minus_complex.simps(1)
thf(fact_4420_integer__of__num__triv_I1_J,axiom,
( ( code_integer_of_num @ one2 )
= ( one_one @ code_integer ) ) ).
% integer_of_num_triv(1)
thf(fact_4421_cmod__plus__Re__le__0__iff,axiom,
! [Z2: complex] :
( ( ord_less_eq @ real @ ( plus_plus @ real @ ( real_V7770717601297561774m_norm @ complex @ Z2 ) @ ( re @ Z2 ) ) @ ( zero_zero @ real ) )
= ( ( re @ Z2 )
= ( uminus_uminus @ real @ ( real_V7770717601297561774m_norm @ complex @ Z2 ) ) ) ) ).
% cmod_plus_Re_le_0_iff
thf(fact_4422_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 )
@ ^ [R: code_integer,S5: code_integer] :
( product_Pair @ code_integer @ $o @ ( if @ code_integer @ ( ord_less @ code_integer @ ( zero_zero @ code_integer ) @ K3 ) @ R @ ( minus_minus @ code_integer @ ( uminus_uminus @ code_integer @ R ) @ S5 ) )
@ ( S5
= ( one_one @ code_integer ) ) )
@ ( code_divmod_abs @ K3 @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% bit_cut_integer_code
thf(fact_4423_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_4424_csqrt_Osimps_I2_J,axiom,
! [Z2: complex] :
( ( im @ ( csqrt @ Z2 ) )
= ( times_times @ real
@ ( if @ real
@ ( ( im @ Z2 )
= ( zero_zero @ real ) )
@ ( one_one @ real )
@ ( sgn_sgn @ real @ ( im @ Z2 ) ) )
@ ( sqrt @ ( divide_divide @ real @ ( minus_minus @ real @ ( real_V7770717601297561774m_norm @ complex @ Z2 ) @ ( re @ Z2 ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ).
% csqrt.simps(2)
thf(fact_4425_Complex__divide,axiom,
( ( divide_divide @ complex )
= ( ^ [X2: complex,Y5: complex] : ( complex2 @ ( divide_divide @ real @ ( plus_plus @ real @ ( times_times @ real @ ( re @ X2 ) @ ( re @ Y5 ) ) @ ( times_times @ real @ ( im @ X2 ) @ ( im @ Y5 ) ) ) @ ( plus_plus @ real @ ( power_power @ real @ ( re @ Y5 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ ( im @ Y5 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( divide_divide @ real @ ( minus_minus @ real @ ( times_times @ real @ ( im @ X2 ) @ ( re @ Y5 ) ) @ ( times_times @ real @ ( re @ X2 ) @ ( im @ Y5 ) ) ) @ ( plus_plus @ real @ ( power_power @ real @ ( re @ Y5 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ ( im @ Y5 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% Complex_divide
thf(fact_4426_Im__divide__of__real,axiom,
! [Z2: complex,R4: real] :
( ( im @ ( divide_divide @ complex @ Z2 @ ( real_Vector_of_real @ complex @ R4 ) ) )
= ( divide_divide @ real @ ( im @ Z2 ) @ R4 ) ) ).
% Im_divide_of_real
thf(fact_4427_Im__sgn,axiom,
! [Z2: complex] :
( ( im @ ( sgn_sgn @ complex @ Z2 ) )
= ( divide_divide @ real @ ( im @ Z2 ) @ ( real_V7770717601297561774m_norm @ complex @ Z2 ) ) ) ).
% Im_sgn
thf(fact_4428_Re__i__times,axiom,
! [Z2: complex] :
( ( re @ ( times_times @ complex @ imaginary_unit @ Z2 ) )
= ( uminus_uminus @ real @ ( im @ Z2 ) ) ) ).
% Re_i_times
thf(fact_4429_Im__divide__numeral,axiom,
! [Z2: complex,W: num] :
( ( im @ ( divide_divide @ complex @ Z2 @ ( numeral_numeral @ complex @ W ) ) )
= ( divide_divide @ real @ ( im @ Z2 ) @ ( numeral_numeral @ real @ W ) ) ) ).
% Im_divide_numeral
thf(fact_4430_Im__divide__of__nat,axiom,
! [Z2: complex,N: nat] :
( ( im @ ( divide_divide @ complex @ Z2 @ ( semiring_1_of_nat @ complex @ N ) ) )
= ( divide_divide @ real @ ( im @ Z2 ) @ ( semiring_1_of_nat @ real @ N ) ) ) ).
% Im_divide_of_nat
thf(fact_4431_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_4432_imaginary__unit_Osimps_I2_J,axiom,
( ( im @ imaginary_unit )
= ( one_one @ real ) ) ).
% imaginary_unit.simps(2)
thf(fact_4433_one__complex_Osimps_I2_J,axiom,
( ( im @ ( one_one @ complex ) )
= ( zero_zero @ real ) ) ).
% one_complex.simps(2)
thf(fact_4434_uminus__complex_Osimps_I2_J,axiom,
! [X: complex] :
( ( im @ ( uminus_uminus @ complex @ X ) )
= ( uminus_uminus @ real @ ( im @ X ) ) ) ).
% uminus_complex.simps(2)
thf(fact_4435_minus__complex_Osimps_I2_J,axiom,
! [X: complex,Y2: complex] :
( ( im @ ( minus_minus @ complex @ X @ Y2 ) )
= ( minus_minus @ real @ ( im @ X ) @ ( im @ Y2 ) ) ) ).
% minus_complex.simps(2)
thf(fact_4436_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_4437_uminus__complex_Ocode,axiom,
( ( uminus_uminus @ complex )
= ( ^ [X2: complex] : ( complex2 @ ( uminus_uminus @ real @ ( re @ X2 ) ) @ ( uminus_uminus @ real @ ( im @ X2 ) ) ) ) ) ).
% uminus_complex.code
thf(fact_4438_minus__complex_Ocode,axiom,
( ( minus_minus @ complex )
= ( ^ [X2: complex,Y5: complex] : ( complex2 @ ( minus_minus @ real @ ( re @ X2 ) @ ( re @ Y5 ) ) @ ( minus_minus @ real @ ( im @ X2 ) @ ( im @ Y5 ) ) ) ) ) ).
% minus_complex.code
thf(fact_4439_times__complex_Ocode,axiom,
( ( times_times @ complex )
= ( ^ [X2: complex,Y5: complex] : ( complex2 @ ( minus_minus @ real @ ( times_times @ real @ ( re @ X2 ) @ ( re @ Y5 ) ) @ ( times_times @ real @ ( im @ X2 ) @ ( im @ Y5 ) ) ) @ ( plus_plus @ real @ ( times_times @ real @ ( re @ X2 ) @ ( im @ Y5 ) ) @ ( times_times @ real @ ( im @ X2 ) @ ( re @ Y5 ) ) ) ) ) ) ).
% times_complex.code
thf(fact_4440_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_4441_divmod__abs__def,axiom,
( code_divmod_abs
= ( ^ [K3: code_integer,L2: code_integer] : ( product_Pair @ code_integer @ code_integer @ ( divide_divide @ code_integer @ ( abs_abs @ code_integer @ K3 ) @ ( abs_abs @ code_integer @ L2 ) ) @ ( modulo_modulo @ code_integer @ ( abs_abs @ code_integer @ K3 ) @ ( abs_abs @ code_integer @ L2 ) ) ) ) ) ).
% divmod_abs_def
thf(fact_4442_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_4443_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_4444_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_4445_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_4446_complex__unit__circle,axiom,
! [Z2: complex] :
( ( Z2
!= ( zero_zero @ complex ) )
=> ( ( plus_plus @ real @ ( power_power @ real @ ( divide_divide @ real @ ( re @ Z2 ) @ ( real_V7770717601297561774m_norm @ complex @ Z2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ ( divide_divide @ real @ ( im @ Z2 ) @ ( real_V7770717601297561774m_norm @ complex @ Z2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( one_one @ real ) ) ) ).
% complex_unit_circle
thf(fact_4447_inverse__complex_Ocode,axiom,
( ( inverse_inverse @ complex )
= ( ^ [X2: complex] : ( complex2 @ ( divide_divide @ real @ ( re @ X2 ) @ ( plus_plus @ real @ ( power_power @ real @ ( re @ X2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ ( im @ X2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( divide_divide @ real @ ( uminus_uminus @ real @ ( im @ X2 ) ) @ ( plus_plus @ real @ ( power_power @ real @ ( re @ X2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ ( im @ X2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% inverse_complex.code
thf(fact_4448_divmod__integer__code,axiom,
( code_divmod_integer
= ( ^ [K3: code_integer,L2: code_integer] :
( if @ ( product_prod @ code_integer @ code_integer )
@ ( K3
= ( zero_zero @ code_integer ) )
@ ( product_Pair @ code_integer @ code_integer @ ( zero_zero @ code_integer ) @ ( zero_zero @ code_integer ) )
@ ( if @ ( product_prod @ code_integer @ code_integer ) @ ( ord_less @ code_integer @ ( zero_zero @ code_integer ) @ L2 )
@ ( if @ ( product_prod @ code_integer @ code_integer ) @ ( ord_less @ code_integer @ ( zero_zero @ code_integer ) @ K3 ) @ ( code_divmod_abs @ K3 @ L2 )
@ ( product_case_prod @ code_integer @ code_integer @ ( product_prod @ code_integer @ code_integer )
@ ^ [R: code_integer,S5: code_integer] :
( if @ ( product_prod @ code_integer @ code_integer )
@ ( S5
= ( zero_zero @ code_integer ) )
@ ( product_Pair @ code_integer @ code_integer @ ( uminus_uminus @ code_integer @ R ) @ ( zero_zero @ code_integer ) )
@ ( product_Pair @ code_integer @ code_integer @ ( minus_minus @ code_integer @ ( uminus_uminus @ code_integer @ R ) @ ( one_one @ code_integer ) ) @ ( minus_minus @ code_integer @ L2 @ S5 ) ) )
@ ( code_divmod_abs @ K3 @ L2 ) ) )
@ ( if @ ( product_prod @ code_integer @ code_integer )
@ ( L2
= ( zero_zero @ code_integer ) )
@ ( product_Pair @ code_integer @ code_integer @ ( zero_zero @ code_integer ) @ K3 )
@ ( product_apsnd @ code_integer @ code_integer @ code_integer @ ( uminus_uminus @ code_integer )
@ ( if @ ( product_prod @ code_integer @ code_integer ) @ ( ord_less @ code_integer @ K3 @ ( zero_zero @ code_integer ) ) @ ( code_divmod_abs @ K3 @ L2 )
@ ( product_case_prod @ code_integer @ code_integer @ ( product_prod @ code_integer @ code_integer )
@ ^ [R: code_integer,S5: code_integer] :
( if @ ( product_prod @ code_integer @ code_integer )
@ ( S5
= ( zero_zero @ code_integer ) )
@ ( product_Pair @ code_integer @ code_integer @ ( uminus_uminus @ code_integer @ R ) @ ( zero_zero @ code_integer ) )
@ ( product_Pair @ code_integer @ code_integer @ ( minus_minus @ code_integer @ ( uminus_uminus @ code_integer @ R ) @ ( one_one @ code_integer ) ) @ ( minus_minus @ code_integer @ ( uminus_uminus @ code_integer @ L2 ) @ S5 ) ) )
@ ( code_divmod_abs @ K3 @ L2 ) ) ) ) ) ) ) ) ) ).
% divmod_integer_code
thf(fact_4449_Im__Reals__divide,axiom,
! [R4: complex,Z2: complex] :
( ( member @ complex @ R4 @ ( real_Vector_Reals @ complex ) )
=> ( ( im @ ( divide_divide @ complex @ R4 @ Z2 ) )
= ( divide_divide @ real @ ( times_times @ real @ ( uminus_uminus @ real @ ( re @ R4 ) ) @ ( im @ Z2 ) ) @ ( power_power @ real @ ( real_V7770717601297561774m_norm @ complex @ Z2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% Im_Reals_divide
thf(fact_4450_Re__Reals__divide,axiom,
! [R4: complex,Z2: complex] :
( ( member @ complex @ R4 @ ( real_Vector_Reals @ complex ) )
=> ( ( re @ ( divide_divide @ complex @ R4 @ Z2 ) )
= ( divide_divide @ real @ ( times_times @ real @ ( re @ R4 ) @ ( re @ Z2 ) ) @ ( power_power @ real @ ( real_V7770717601297561774m_norm @ complex @ Z2 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% Re_Reals_divide
thf(fact_4451_complex__diff__cnj,axiom,
! [Z2: complex] :
( ( minus_minus @ complex @ Z2 @ ( cnj @ Z2 ) )
= ( times_times @ complex @ ( real_Vector_of_real @ complex @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( im @ Z2 ) ) ) @ imaginary_unit ) ) ).
% complex_diff_cnj
thf(fact_4452_Reals__minus__iff,axiom,
! [A: $tType] :
( ( real_V2191834092415804123ebra_1 @ A )
=> ! [A3: A] :
( ( member @ A @ ( uminus_uminus @ A @ A3 ) @ ( real_Vector_Reals @ A ) )
= ( member @ A @ A3 @ ( real_Vector_Reals @ A ) ) ) ) ).
% Reals_minus_iff
thf(fact_4453_complex__cnj__divide,axiom,
! [X: complex,Y2: complex] :
( ( cnj @ ( divide_divide @ complex @ X @ Y2 ) )
= ( divide_divide @ complex @ ( cnj @ X ) @ ( cnj @ Y2 ) ) ) ).
% complex_cnj_divide
thf(fact_4454_complex__cnj__one,axiom,
( ( cnj @ ( one_one @ complex ) )
= ( one_one @ complex ) ) ).
% complex_cnj_one
thf(fact_4455_complex__cnj__one__iff,axiom,
! [Z2: complex] :
( ( ( cnj @ Z2 )
= ( one_one @ complex ) )
= ( Z2
= ( one_one @ complex ) ) ) ).
% complex_cnj_one_iff
thf(fact_4456_complex__cnj__minus,axiom,
! [X: complex] :
( ( cnj @ ( uminus_uminus @ complex @ X ) )
= ( uminus_uminus @ complex @ ( cnj @ X ) ) ) ).
% complex_cnj_minus
thf(fact_4457_complex__cnj__diff,axiom,
! [X: complex,Y2: complex] :
( ( cnj @ ( minus_minus @ complex @ X @ Y2 ) )
= ( minus_minus @ complex @ ( cnj @ X ) @ ( cnj @ Y2 ) ) ) ).
% complex_cnj_diff
thf(fact_4458_complex__cnj__i,axiom,
( ( cnj @ imaginary_unit )
= ( uminus_uminus @ complex @ imaginary_unit ) ) ).
% complex_cnj_i
thf(fact_4459_complex__cnj__neg__numeral,axiom,
! [W: num] :
( ( cnj @ ( uminus_uminus @ complex @ ( numeral_numeral @ complex @ W ) ) )
= ( uminus_uminus @ complex @ ( numeral_numeral @ complex @ W ) ) ) ).
% complex_cnj_neg_numeral
thf(fact_4460_Re__divide__Reals,axiom,
! [R4: complex,Z2: complex] :
( ( member @ complex @ R4 @ ( real_Vector_Reals @ complex ) )
=> ( ( re @ ( divide_divide @ complex @ Z2 @ R4 ) )
= ( divide_divide @ real @ ( re @ Z2 ) @ ( re @ R4 ) ) ) ) ).
% Re_divide_Reals
thf(fact_4461_Im__divide__Reals,axiom,
! [R4: complex,Z2: complex] :
( ( member @ complex @ R4 @ ( real_Vector_Reals @ complex ) )
=> ( ( im @ ( divide_divide @ complex @ Z2 @ R4 ) )
= ( divide_divide @ real @ ( im @ Z2 ) @ ( re @ R4 ) ) ) ) ).
% Im_divide_Reals
thf(fact_4462_Reals__of__nat,axiom,
! [A: $tType] :
( ( real_V2191834092415804123ebra_1 @ A )
=> ! [N: nat] : ( member @ A @ ( semiring_1_of_nat @ A @ N ) @ ( real_Vector_Reals @ A ) ) ) ).
% Reals_of_nat
thf(fact_4463_Reals__minus,axiom,
! [A: $tType] :
( ( real_V2191834092415804123ebra_1 @ A )
=> ! [A3: A] :
( ( member @ A @ A3 @ ( real_Vector_Reals @ A ) )
=> ( member @ A @ ( uminus_uminus @ A @ A3 ) @ ( real_Vector_Reals @ A ) ) ) ) ).
% Reals_minus
thf(fact_4464_Reals__0,axiom,
! [A: $tType] :
( ( real_V2191834092415804123ebra_1 @ A )
=> ( member @ A @ ( zero_zero @ A ) @ ( real_Vector_Reals @ A ) ) ) ).
% Reals_0
thf(fact_4465_Reals__divide,axiom,
! [A: $tType] :
( ( real_V7773925162809079976_field @ A )
=> ! [A3: A,B2: A] :
( ( member @ A @ A3 @ ( real_Vector_Reals @ A ) )
=> ( ( member @ A @ B2 @ ( real_Vector_Reals @ A ) )
=> ( member @ A @ ( divide_divide @ A @ A3 @ B2 ) @ ( real_Vector_Reals @ A ) ) ) ) ) ).
% Reals_divide
thf(fact_4466_cot__in__Reals,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [Z2: A] :
( ( member @ A @ Z2 @ ( real_Vector_Reals @ A ) )
=> ( member @ A @ ( cot @ A @ Z2 ) @ ( real_Vector_Reals @ A ) ) ) ) ).
% cot_in_Reals
thf(fact_4467_cos__in__Reals,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [Z2: A] :
( ( member @ A @ Z2 @ ( real_Vector_Reals @ A ) )
=> ( member @ A @ ( cos @ A @ Z2 ) @ ( real_Vector_Reals @ A ) ) ) ) ).
% cos_in_Reals
thf(fact_4468_sin__in__Reals,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [Z2: A] :
( ( member @ A @ Z2 @ ( real_Vector_Reals @ A ) )
=> ( member @ A @ ( sin @ A @ Z2 ) @ ( real_Vector_Reals @ A ) ) ) ) ).
% sin_in_Reals
thf(fact_4469_fact__in__Reals,axiom,
! [A: $tType] :
( ( real_V2191834092415804123ebra_1 @ A )
=> ! [N: nat] : ( member @ A @ ( semiring_char_0_fact @ A @ N ) @ ( real_Vector_Reals @ A ) ) ) ).
% fact_in_Reals
thf(fact_4470_exp__in__Reals,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [Z2: A] :
( ( member @ A @ Z2 @ ( real_Vector_Reals @ A ) )
=> ( member @ A @ ( exp @ A @ Z2 ) @ ( real_Vector_Reals @ A ) ) ) ) ).
% exp_in_Reals
thf(fact_4471_tan__in__Reals,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [Z2: A] :
( ( member @ A @ Z2 @ ( real_Vector_Reals @ A ) )
=> ( member @ A @ ( tan @ A @ Z2 ) @ ( real_Vector_Reals @ A ) ) ) ) ).
% tan_in_Reals
thf(fact_4472_Reals__1,axiom,
! [B: $tType] :
( ( real_V2191834092415804123ebra_1 @ B )
=> ( member @ B @ ( one_one @ B ) @ ( real_Vector_Reals @ B ) ) ) ).
% Reals_1
thf(fact_4473_Reals__diff,axiom,
! [A: $tType] :
( ( real_V2191834092415804123ebra_1 @ A )
=> ! [A3: A,B2: A] :
( ( member @ A @ A3 @ ( real_Vector_Reals @ A ) )
=> ( ( member @ A @ B2 @ ( real_Vector_Reals @ A ) )
=> ( member @ A @ ( minus_minus @ A @ A3 @ B2 ) @ ( real_Vector_Reals @ A ) ) ) ) ) ).
% Reals_diff
thf(fact_4474_cnj_Osimps_I2_J,axiom,
! [Z2: complex] :
( ( im @ ( cnj @ Z2 ) )
= ( uminus_uminus @ real @ ( im @ Z2 ) ) ) ).
% cnj.simps(2)
thf(fact_4475_complex__cnj,axiom,
! [A3: real,B2: real] :
( ( cnj @ ( complex2 @ A3 @ B2 ) )
= ( complex2 @ A3 @ ( uminus_uminus @ real @ B2 ) ) ) ).
% complex_cnj
thf(fact_4476_cis__cnj,axiom,
! [T2: real] :
( ( cnj @ ( cis @ T2 ) )
= ( cis @ ( uminus_uminus @ real @ T2 ) ) ) ).
% cis_cnj
thf(fact_4477_nonzero__Reals__divide,axiom,
! [A: $tType] :
( ( real_V7773925162809079976_field @ A )
=> ! [A3: A,B2: A] :
( ( member @ A @ A3 @ ( real_Vector_Reals @ A ) )
=> ( ( member @ A @ B2 @ ( real_Vector_Reals @ A ) )
=> ( ( B2
!= ( zero_zero @ A ) )
=> ( member @ A @ ( divide_divide @ A @ A3 @ B2 ) @ ( real_Vector_Reals @ A ) ) ) ) ) ) ).
% nonzero_Reals_divide
thf(fact_4478_nonzero__Reals__inverse,axiom,
! [A: $tType] :
( ( real_V5047593784448816457lgebra @ A )
=> ! [A3: A] :
( ( member @ A @ A3 @ ( real_Vector_Reals @ A ) )
=> ( ( A3
!= ( zero_zero @ A ) )
=> ( member @ A @ ( inverse_inverse @ A @ A3 ) @ ( real_Vector_Reals @ A ) ) ) ) ) ).
% nonzero_Reals_inverse
thf(fact_4479_cnj_Ocode,axiom,
( cnj
= ( ^ [Z6: complex] : ( complex2 @ ( re @ Z6 ) @ ( uminus_uminus @ real @ ( im @ Z6 ) ) ) ) ) ).
% cnj.code
thf(fact_4480_Re__complex__div__eq__0,axiom,
! [A3: complex,B2: complex] :
( ( ( re @ ( divide_divide @ complex @ A3 @ B2 ) )
= ( zero_zero @ real ) )
= ( ( re @ ( times_times @ complex @ A3 @ ( cnj @ B2 ) ) )
= ( zero_zero @ real ) ) ) ).
% Re_complex_div_eq_0
thf(fact_4481_Im__complex__div__eq__0,axiom,
! [A3: complex,B2: complex] :
( ( ( im @ ( divide_divide @ complex @ A3 @ B2 ) )
= ( zero_zero @ real ) )
= ( ( im @ ( times_times @ complex @ A3 @ ( cnj @ B2 ) ) )
= ( zero_zero @ real ) ) ) ).
% Im_complex_div_eq_0
thf(fact_4482_Re__complex__div__lt__0,axiom,
! [A3: complex,B2: complex] :
( ( ord_less @ real @ ( re @ ( divide_divide @ complex @ A3 @ B2 ) ) @ ( zero_zero @ real ) )
= ( ord_less @ real @ ( re @ ( times_times @ complex @ A3 @ ( cnj @ B2 ) ) ) @ ( zero_zero @ real ) ) ) ).
% Re_complex_div_lt_0
thf(fact_4483_Re__complex__div__gt__0,axiom,
! [A3: complex,B2: complex] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( re @ ( divide_divide @ complex @ A3 @ B2 ) ) )
= ( ord_less @ real @ ( zero_zero @ real ) @ ( re @ ( times_times @ complex @ A3 @ ( cnj @ B2 ) ) ) ) ) ).
% Re_complex_div_gt_0
thf(fact_4484_Re__complex__div__le__0,axiom,
! [A3: complex,B2: complex] :
( ( ord_less_eq @ real @ ( re @ ( divide_divide @ complex @ A3 @ B2 ) ) @ ( zero_zero @ real ) )
= ( ord_less_eq @ real @ ( re @ ( times_times @ complex @ A3 @ ( cnj @ B2 ) ) ) @ ( zero_zero @ real ) ) ) ).
% Re_complex_div_le_0
thf(fact_4485_Re__complex__div__ge__0,axiom,
! [A3: complex,B2: complex] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( re @ ( divide_divide @ complex @ A3 @ B2 ) ) )
= ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( re @ ( times_times @ complex @ A3 @ ( cnj @ B2 ) ) ) ) ) ).
% Re_complex_div_ge_0
thf(fact_4486_Im__complex__div__lt__0,axiom,
! [A3: complex,B2: complex] :
( ( ord_less @ real @ ( im @ ( divide_divide @ complex @ A3 @ B2 ) ) @ ( zero_zero @ real ) )
= ( ord_less @ real @ ( im @ ( times_times @ complex @ A3 @ ( cnj @ B2 ) ) ) @ ( zero_zero @ real ) ) ) ).
% Im_complex_div_lt_0
thf(fact_4487_Im__complex__div__gt__0,axiom,
! [A3: complex,B2: complex] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( im @ ( divide_divide @ complex @ A3 @ B2 ) ) )
= ( ord_less @ real @ ( zero_zero @ real ) @ ( im @ ( times_times @ complex @ A3 @ ( cnj @ B2 ) ) ) ) ) ).
% Im_complex_div_gt_0
thf(fact_4488_Im__complex__div__le__0,axiom,
! [A3: complex,B2: complex] :
( ( ord_less_eq @ real @ ( im @ ( divide_divide @ complex @ A3 @ B2 ) ) @ ( zero_zero @ real ) )
= ( ord_less_eq @ real @ ( im @ ( times_times @ complex @ A3 @ ( cnj @ B2 ) ) ) @ ( zero_zero @ real ) ) ) ).
% Im_complex_div_le_0
thf(fact_4489_Im__complex__div__ge__0,axiom,
! [A3: complex,B2: complex] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( im @ ( divide_divide @ complex @ A3 @ B2 ) ) )
= ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( im @ ( times_times @ complex @ A3 @ ( cnj @ B2 ) ) ) ) ) ).
% Im_complex_div_ge_0
thf(fact_4490_complex__div__gt__0,axiom,
! [A3: complex,B2: complex] :
( ( ( ord_less @ real @ ( zero_zero @ real ) @ ( re @ ( divide_divide @ complex @ A3 @ B2 ) ) )
= ( ord_less @ real @ ( zero_zero @ real ) @ ( re @ ( times_times @ complex @ A3 @ ( cnj @ B2 ) ) ) ) )
& ( ( ord_less @ real @ ( zero_zero @ real ) @ ( im @ ( divide_divide @ complex @ A3 @ B2 ) ) )
= ( ord_less @ real @ ( zero_zero @ real ) @ ( im @ ( times_times @ complex @ A3 @ ( cnj @ B2 ) ) ) ) ) ) ).
% complex_div_gt_0
thf(fact_4491_complex__div__cnj,axiom,
( ( divide_divide @ complex )
= ( ^ [A5: complex,B3: complex] : ( divide_divide @ complex @ ( times_times @ complex @ A5 @ ( cnj @ B3 ) ) @ ( real_Vector_of_real @ complex @ ( power_power @ real @ ( real_V7770717601297561774m_norm @ complex @ B3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% complex_div_cnj
thf(fact_4492_rec__nat__add__eq__if,axiom,
! [A: $tType,A3: A,F2: nat > A > A,V2: num,N: nat] :
( ( rec_nat @ A @ A3 @ F2 @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ V2 ) @ N ) )
= ( F2 @ ( plus_plus @ nat @ ( pred_numeral @ V2 ) @ N ) @ ( rec_nat @ A @ A3 @ F2 @ ( plus_plus @ nat @ ( pred_numeral @ V2 ) @ N ) ) ) ) ).
% rec_nat_add_eq_if
thf(fact_4493_or__int__rec,axiom,
( ( bit_se1065995026697491101ons_or @ int )
= ( ^ [K3: int,L2: 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 ) ) @ L2 ) ) )
@ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ int @ ( divide_divide @ int @ K3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% or_int_rec
thf(fact_4494_bezw__0,axiom,
! [X: nat] :
( ( bezw @ X @ ( zero_zero @ nat ) )
= ( product_Pair @ int @ int @ ( one_one @ int ) @ ( zero_zero @ int ) ) ) ).
% bezw_0
thf(fact_4495_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_4496_or_Oleft__neutral,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A] :
( ( bit_se1065995026697491101ons_or @ A @ ( zero_zero @ A ) @ A3 )
= A3 ) ) ).
% or.left_neutral
thf(fact_4497_or_Oright__neutral,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A] :
( ( bit_se1065995026697491101ons_or @ A @ A3 @ ( zero_zero @ A ) )
= A3 ) ) ).
% or.right_neutral
thf(fact_4498_old_Onat_Osimps_I7_J,axiom,
! [T: $tType,F1: T,F22: nat > T > T,Nat: nat] :
( ( rec_nat @ T @ F1 @ F22 @ ( suc @ Nat ) )
= ( F22 @ Nat @ ( rec_nat @ T @ F1 @ F22 @ Nat ) ) ) ).
% old.nat.simps(7)
thf(fact_4499_old_Onat_Osimps_I6_J,axiom,
! [T: $tType,F1: T,F22: nat > T > T] :
( ( rec_nat @ T @ F1 @ F22 @ ( zero_zero @ nat ) )
= F1 ) ).
% old.nat.simps(6)
thf(fact_4500_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_4501_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_4502_or__nonnegative__int__iff,axiom,
! [K: int,L: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se1065995026697491101ons_or @ int @ K @ L ) )
= ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K )
& ( ord_less_eq @ int @ ( zero_zero @ int ) @ L ) ) ) ).
% or_nonnegative_int_iff
thf(fact_4503_or__negative__int__iff,axiom,
! [K: int,L: int] :
( ( ord_less @ int @ ( bit_se1065995026697491101ons_or @ int @ K @ L ) @ ( zero_zero @ int ) )
= ( ( ord_less @ int @ K @ ( zero_zero @ int ) )
| ( ord_less @ int @ L @ ( zero_zero @ int ) ) ) ) ).
% or_negative_int_iff
thf(fact_4504_rec__nat__numeral,axiom,
! [A: $tType,A3: A,F2: nat > A > A,V2: num] :
( ( rec_nat @ A @ A3 @ F2 @ ( numeral_numeral @ nat @ V2 ) )
= ( F2 @ ( pred_numeral @ V2 ) @ ( rec_nat @ A @ A3 @ F2 @ ( pred_numeral @ V2 ) ) ) ) ).
% rec_nat_numeral
thf(fact_4505_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_4506_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_4507_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_4508_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_4509_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_4510_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_4511_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_4512_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_4513_or__minus__minus__numerals,axiom,
! [M2: num,N: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M2 ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
= ( bit_ri4277139882892585799ns_not @ int @ ( bit_se5824344872417868541ns_and @ int @ ( minus_minus @ int @ ( numeral_numeral @ int @ M2 ) @ ( one_one @ int ) ) @ ( minus_minus @ int @ ( numeral_numeral @ int @ N ) @ ( one_one @ int ) ) ) ) ) ).
% or_minus_minus_numerals
thf(fact_4514_and__minus__minus__numerals,axiom,
! [M2: num,N: num] :
( ( bit_se5824344872417868541ns_and @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M2 ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
= ( bit_ri4277139882892585799ns_not @ int @ ( bit_se1065995026697491101ons_or @ int @ ( minus_minus @ int @ ( numeral_numeral @ int @ M2 ) @ ( one_one @ int ) ) @ ( minus_minus @ int @ ( numeral_numeral @ int @ N ) @ ( one_one @ int ) ) ) ) ) ).
% and_minus_minus_numerals
thf(fact_4515_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_4516_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_4517_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_4518_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_4519_or__eq__0__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A,B2: A] :
( ( ( bit_se1065995026697491101ons_or @ A @ A3 @ B2 )
= ( zero_zero @ A ) )
= ( ( A3
= ( zero_zero @ A ) )
& ( B2
= ( zero_zero @ A ) ) ) ) ) ).
% or_eq_0_iff
thf(fact_4520_of__nat__or__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M2: nat,N: nat] :
( ( semiring_1_of_nat @ A @ ( bit_se1065995026697491101ons_or @ nat @ M2 @ N ) )
= ( bit_se1065995026697491101ons_or @ A @ ( semiring_1_of_nat @ A @ M2 ) @ ( semiring_1_of_nat @ A @ N ) ) ) ) ).
% of_nat_or_eq
thf(fact_4521_or__greater__eq,axiom,
! [L: int,K: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ L )
=> ( ord_less_eq @ int @ K @ ( bit_se1065995026697491101ons_or @ int @ K @ L ) ) ) ).
% or_greater_eq
thf(fact_4522_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_4523_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_4524_set__bit__eq__or,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se5668285175392031749et_bit @ A )
= ( ^ [N5: nat,A5: A] : ( bit_se1065995026697491101ons_or @ A @ A5 @ ( bit_se4730199178511100633sh_bit @ A @ N5 @ ( one_one @ A ) ) ) ) ) ) ).
% set_bit_eq_or
thf(fact_4525_set__bit__int__def,axiom,
( ( bit_se5668285175392031749et_bit @ int )
= ( ^ [N5: nat,K3: int] : ( bit_se1065995026697491101ons_or @ int @ K3 @ ( bit_se4730199178511100633sh_bit @ int @ N5 @ ( one_one @ int ) ) ) ) ) ).
% set_bit_int_def
thf(fact_4526_bit_Ocomplement__unique,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [A3: A,X: A,Y2: A] :
( ( ( bit_se5824344872417868541ns_and @ A @ A3 @ X )
= ( zero_zero @ A ) )
=> ( ( ( bit_se1065995026697491101ons_or @ A @ A3 @ X )
= ( uminus_uminus @ A @ ( one_one @ A ) ) )
=> ( ( ( bit_se5824344872417868541ns_and @ A @ A3 @ Y2 )
= ( zero_zero @ A ) )
=> ( ( ( bit_se1065995026697491101ons_or @ A @ A3 @ Y2 )
= ( uminus_uminus @ A @ ( one_one @ A ) ) )
=> ( X = Y2 ) ) ) ) ) ) ).
% bit.complement_unique
thf(fact_4527_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_4528_or__not__numerals_I4_J,axiom,
! [M2: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ ( bit0 @ M2 ) ) @ ( bit_ri4277139882892585799ns_not @ int @ ( one_one @ int ) ) )
= ( bit_ri4277139882892585799ns_not @ int @ ( one_one @ int ) ) ) ).
% or_not_numerals(4)
thf(fact_4529_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_4530_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_4531_or__not__numerals_I7_J,axiom,
! [M2: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ ( bit1 @ M2 ) ) @ ( bit_ri4277139882892585799ns_not @ int @ ( one_one @ int ) ) )
= ( bit_ri4277139882892585799ns_not @ int @ ( zero_zero @ int ) ) ) ).
% or_not_numerals(7)
thf(fact_4532_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_4533_one__or__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A] :
( ( bit_se1065995026697491101ons_or @ A @ ( one_one @ A ) @ A3 )
= ( plus_plus @ A @ A3 @ ( zero_neq_one_of_bool @ A @ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) ) ) ) ) ).
% one_or_eq
thf(fact_4534_or__one__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A] :
( ( bit_se1065995026697491101ons_or @ A @ A3 @ ( one_one @ A ) )
= ( plus_plus @ A @ A3 @ ( zero_neq_one_of_bool @ A @ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) ) ) ) ) ).
% or_one_eq
thf(fact_4535_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_4536_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_4537_or__not__numerals_I5_J,axiom,
! [M2: num,N: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ ( bit0 @ M2 ) ) @ ( 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 @ M2 ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ N ) ) ) ) ) ) ).
% or_not_numerals(5)
thf(fact_4538_or__not__numerals_I9_J,axiom,
! [M2: num,N: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ ( bit1 @ M2 ) ) @ ( 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 @ M2 ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ N ) ) ) ) ) ) ).
% or_not_numerals(9)
thf(fact_4539_or__not__numerals_I8_J,axiom,
! [M2: num,N: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ ( bit1 @ M2 ) ) @ ( 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 @ M2 ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ N ) ) ) ) ) ) ).
% or_not_numerals(8)
thf(fact_4540_prod__decode__aux_Osimps,axiom,
( nat_prod_decode_aux
= ( ^ [K3: nat,M5: nat] : ( if @ ( product_prod @ nat @ nat ) @ ( ord_less_eq @ nat @ M5 @ K3 ) @ ( product_Pair @ nat @ nat @ M5 @ ( minus_minus @ nat @ K3 @ M5 ) ) @ ( nat_prod_decode_aux @ ( suc @ K3 ) @ ( minus_minus @ nat @ M5 @ ( suc @ K3 ) ) ) ) ) ) ).
% prod_decode_aux.simps
thf(fact_4541_old_Orec__nat__def,axiom,
! [T: $tType] :
( ( rec_nat @ T )
= ( ^ [F12: T,F23: nat > T > T,X2: nat] : ( the @ T @ ( rec_set_nat @ T @ F12 @ F23 @ X2 ) ) ) ) ).
% old.rec_nat_def
thf(fact_4542_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_4543_bezw_Osimps,axiom,
( bezw
= ( ^ [X2: nat,Y5: nat] :
( if @ ( product_prod @ int @ int )
@ ( Y5
= ( zero_zero @ nat ) )
@ ( product_Pair @ int @ int @ ( one_one @ int ) @ ( zero_zero @ int ) )
@ ( product_Pair @ int @ int @ ( product_snd @ int @ int @ ( bezw @ Y5 @ ( modulo_modulo @ nat @ X2 @ Y5 ) ) ) @ ( minus_minus @ int @ ( product_fst @ int @ int @ ( bezw @ Y5 @ ( modulo_modulo @ nat @ X2 @ Y5 ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ ( bezw @ Y5 @ ( modulo_modulo @ nat @ X2 @ Y5 ) ) ) @ ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ X2 @ Y5 ) ) ) ) ) ) ) ) ).
% bezw.simps
thf(fact_4544_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_4545_numeral__div__numeral,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [K: num,L: num] :
( ( divide_divide @ A @ ( numeral_numeral @ A @ K ) @ ( numeral_numeral @ A @ L ) )
= ( product_fst @ A @ A @ ( unique8689654367752047608divmod @ A @ K @ L ) ) ) ) ).
% numeral_div_numeral
thf(fact_4546_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_4547_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_4548_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_4549_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_4550_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_4551_or__minus__numerals_I8_J,axiom,
! [N: num,M2: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ N ) ) ) @ ( numeral_numeral @ int @ M2 ) )
= ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit_or_not_num_neg @ M2 @ ( bit0 @ N ) ) ) ) ) ).
% or_minus_numerals(8)
thf(fact_4552_or__minus__numerals_I4_J,axiom,
! [M2: num,N: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ M2 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ N ) ) ) )
= ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit_or_not_num_neg @ M2 @ ( bit0 @ N ) ) ) ) ) ).
% or_minus_numerals(4)
thf(fact_4553_or__minus__numerals_I7_J,axiom,
! [N: num,M2: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ N ) ) ) @ ( numeral_numeral @ int @ M2 ) )
= ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit_or_not_num_neg @ M2 @ ( bitM @ N ) ) ) ) ) ).
% or_minus_numerals(7)
thf(fact_4554_or__minus__numerals_I3_J,axiom,
! [M2: num,N: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ M2 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ N ) ) ) )
= ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit_or_not_num_neg @ M2 @ ( bitM @ N ) ) ) ) ) ).
% or_minus_numerals(3)
thf(fact_4555_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_4556_set__bit__nat__def,axiom,
( ( bit_se5668285175392031749et_bit @ nat )
= ( ^ [M5: nat,N5: nat] : ( bit_se1065995026697491101ons_or @ nat @ N5 @ ( bit_se4730199178511100633sh_bit @ nat @ M5 @ ( one_one @ nat ) ) ) ) ) ).
% set_bit_nat_def
thf(fact_4557_fst__divmod,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M2: num,N: num] :
( ( product_fst @ A @ A @ ( unique8689654367752047608divmod @ A @ M2 @ N ) )
= ( divide_divide @ A @ ( numeral_numeral @ A @ M2 ) @ ( numeral_numeral @ A @ N ) ) ) ) ).
% fst_divmod
thf(fact_4558_or__nat__def,axiom,
( ( bit_se1065995026697491101ons_or @ nat )
= ( ^ [M5: nat,N5: nat] : ( nat2 @ ( bit_se1065995026697491101ons_or @ int @ ( semiring_1_of_nat @ int @ M5 ) @ ( semiring_1_of_nat @ int @ N5 ) ) ) ) ) ).
% or_nat_def
thf(fact_4559_Eps__case__prod,axiom,
! [B: $tType,A: $tType,P: A > B > $o] :
( ( fChoice @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ P ) )
= ( fChoice @ ( product_prod @ A @ B )
@ ^ [Xy: product_prod @ A @ B] : ( P @ ( product_fst @ A @ B @ Xy ) @ ( product_snd @ A @ B @ Xy ) ) ) ) ).
% Eps_case_prod
thf(fact_4560_numeral__or__not__num__eq,axiom,
! [M2: num,N: num] :
( ( numeral_numeral @ int @ ( bit_or_not_num_neg @ M2 @ N ) )
= ( uminus_uminus @ int @ ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ M2 ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ N ) ) ) ) ) ).
% numeral_or_not_num_eq
thf(fact_4561_int__numeral__not__or__num__neg,axiom,
! [M2: num,N: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ M2 ) ) @ ( numeral_numeral @ int @ N ) )
= ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit_or_not_num_neg @ N @ M2 ) ) ) ) ).
% int_numeral_not_or_num_neg
thf(fact_4562_int__numeral__or__not__num__neg,axiom,
! [M2: num,N: num] :
( ( bit_se1065995026697491101ons_or @ int @ ( numeral_numeral @ int @ M2 ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ N ) ) )
= ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit_or_not_num_neg @ M2 @ N ) ) ) ) ).
% int_numeral_or_not_num_neg
thf(fact_4563_floor__real__def,axiom,
( ( archim6421214686448440834_floor @ real )
= ( ^ [X2: real] :
( the @ int
@ ^ [Z6: int] :
( ( ord_less_eq @ real @ ( ring_1_of_int @ real @ Z6 ) @ X2 )
& ( ord_less @ real @ X2 @ ( ring_1_of_int @ real @ ( plus_plus @ int @ Z6 @ ( one_one @ int ) ) ) ) ) ) ) ) ).
% floor_real_def
thf(fact_4564_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_4565_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_4566_or__nat__rec,axiom,
( ( bit_se1065995026697491101ons_or @ nat )
= ( ^ [M5: nat,N5: nat] :
( plus_plus @ nat
@ ( zero_neq_one_of_bool @ nat
@ ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M5 )
| ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N5 ) ) )
@ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ nat @ ( divide_divide @ nat @ M5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ nat @ N5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% or_nat_rec
thf(fact_4567_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_4568_size__prod__simp,axiom,
! [B: $tType,A: $tType] :
( ( basic_BNF_size_prod @ A @ B )
= ( ^ [F4: A > nat,G2: B > nat,P6: product_prod @ A @ B] : ( plus_plus @ nat @ ( plus_plus @ nat @ ( F4 @ ( product_fst @ A @ B @ P6 ) ) @ ( G2 @ ( product_snd @ A @ B @ P6 ) ) ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ).
% size_prod_simp
thf(fact_4569_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_4570_or__int__unfold,axiom,
( ( bit_se1065995026697491101ons_or @ int )
= ( ^ [K3: int,L2: int] :
( if @ int
@ ( ( K3
= ( uminus_uminus @ int @ ( one_one @ int ) ) )
| ( L2
= ( uminus_uminus @ int @ ( one_one @ int ) ) ) )
@ ( uminus_uminus @ int @ ( one_one @ int ) )
@ ( if @ int
@ ( K3
= ( zero_zero @ int ) )
@ L2
@ ( if @ int
@ ( L2
= ( zero_zero @ int ) )
@ K3
@ ( plus_plus @ int @ ( ord_max @ int @ ( modulo_modulo @ int @ K3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ int @ L2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ int @ ( divide_divide @ int @ K3 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ).
% or_int_unfold
thf(fact_4571_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_4572_max_Oright__idem,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A] :
( ( ord_max @ A @ ( ord_max @ A @ A3 @ B2 ) @ B2 )
= ( ord_max @ A @ A3 @ B2 ) ) ) ).
% max.right_idem
thf(fact_4573_max_Oleft__idem,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A] :
( ( ord_max @ A @ A3 @ ( ord_max @ A @ A3 @ B2 ) )
= ( ord_max @ A @ A3 @ B2 ) ) ) ).
% max.left_idem
thf(fact_4574_max_Oidem,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A] :
( ( ord_max @ A @ A3 @ A3 )
= A3 ) ) ).
% max.idem
thf(fact_4575_max_Obounded__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,C2: A,A3: A] :
( ( ord_less_eq @ A @ ( ord_max @ A @ B2 @ C2 ) @ A3 )
= ( ( ord_less_eq @ A @ B2 @ A3 )
& ( ord_less_eq @ A @ C2 @ A3 ) ) ) ) ).
% max.bounded_iff
thf(fact_4576_max_Oabsorb2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_max @ A @ A3 @ B2 )
= B2 ) ) ) ).
% max.absorb2
thf(fact_4577_max_Oabsorb1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,A3: A] :
( ( ord_less_eq @ A @ B2 @ A3 )
=> ( ( ord_max @ A @ A3 @ B2 )
= A3 ) ) ) ).
% max.absorb1
thf(fact_4578_max_Oabsorb3,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,A3: A] :
( ( ord_less @ A @ B2 @ A3 )
=> ( ( ord_max @ A @ A3 @ B2 )
= A3 ) ) ) ).
% max.absorb3
thf(fact_4579_max_Oabsorb4,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_max @ A @ A3 @ B2 )
= B2 ) ) ) ).
% max.absorb4
thf(fact_4580_max__less__iff__conj,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y2: A,Z2: A] :
( ( ord_less @ A @ ( ord_max @ A @ X @ Y2 ) @ Z2 )
= ( ( ord_less @ A @ X @ Z2 )
& ( ord_less @ A @ Y2 @ Z2 ) ) ) ) ).
% max_less_iff_conj
thf(fact_4581_of__bool__or__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [P: $o,Q: $o] :
( ( zero_neq_one_of_bool @ A
@ ( P
| Q ) )
= ( ord_max @ A @ ( zero_neq_one_of_bool @ A @ P ) @ ( zero_neq_one_of_bool @ A @ Q ) ) ) ) ).
% of_bool_or_iff
thf(fact_4582_fst__divmod__nat,axiom,
! [M2: nat,N: nat] :
( ( product_fst @ nat @ nat @ ( divmod_nat @ M2 @ N ) )
= ( divide_divide @ nat @ M2 @ N ) ) ).
% fst_divmod_nat
thf(fact_4583_fst__divmod__integer,axiom,
! [K: code_integer,L: code_integer] :
( ( product_fst @ code_integer @ code_integer @ ( code_divmod_integer @ K @ L ) )
= ( divide_divide @ code_integer @ K @ L ) ) ).
% fst_divmod_integer
thf(fact_4584_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_4585_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_4586_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_4587_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_4588_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_4589_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_4590_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_4591_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_4592_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_4593_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_4594_fst__divmod__abs,axiom,
! [K: code_integer,L: code_integer] :
( ( product_fst @ code_integer @ code_integer @ ( code_divmod_abs @ K @ L ) )
= ( divide_divide @ code_integer @ ( abs_abs @ code_integer @ K ) @ ( abs_abs @ code_integer @ L ) ) ) ).
% fst_divmod_abs
thf(fact_4595_Suc__0__div__numeral,axiom,
! [K: num] :
( ( divide_divide @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( numeral_numeral @ nat @ K ) )
= ( product_fst @ nat @ nat @ ( unique8689654367752047608divmod @ nat @ one2 @ K ) ) ) ).
% Suc_0_div_numeral
thf(fact_4596_max__def__raw,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( ord_max @ A )
= ( ^ [A5: A,B3: A] : ( if @ A @ ( ord_less_eq @ A @ A5 @ B3 ) @ B3 @ A5 ) ) ) ) ).
% max_def_raw
thf(fact_4597_max__diff__distrib__left,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [X: A,Y2: A,Z2: A] :
( ( minus_minus @ A @ ( ord_max @ A @ X @ Y2 ) @ Z2 )
= ( ord_max @ A @ ( minus_minus @ A @ X @ Z2 ) @ ( minus_minus @ A @ Y2 @ Z2 ) ) ) ) ).
% max_diff_distrib_left
thf(fact_4598_max_Omono,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C2: A,A3: A,D2: A,B2: A] :
( ( ord_less_eq @ A @ C2 @ A3 )
=> ( ( ord_less_eq @ A @ D2 @ B2 )
=> ( ord_less_eq @ A @ ( ord_max @ A @ C2 @ D2 ) @ ( ord_max @ A @ A3 @ B2 ) ) ) ) ) ).
% max.mono
thf(fact_4599_max_OorderE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,A3: A] :
( ( ord_less_eq @ A @ B2 @ A3 )
=> ( A3
= ( ord_max @ A @ A3 @ B2 ) ) ) ) ).
% max.orderE
thf(fact_4600_max_OorderI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A] :
( ( A3
= ( ord_max @ A @ A3 @ B2 ) )
=> ( ord_less_eq @ A @ B2 @ A3 ) ) ) ).
% max.orderI
thf(fact_4601_max_OboundedE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,C2: A,A3: A] :
( ( ord_less_eq @ A @ ( ord_max @ A @ B2 @ C2 ) @ A3 )
=> ~ ( ( ord_less_eq @ A @ B2 @ A3 )
=> ~ ( ord_less_eq @ A @ C2 @ A3 ) ) ) ) ).
% max.boundedE
thf(fact_4602_max_OboundedI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( ord_less_eq @ A @ B2 @ A3 )
=> ( ( ord_less_eq @ A @ C2 @ A3 )
=> ( ord_less_eq @ A @ ( ord_max @ A @ B2 @ C2 ) @ A3 ) ) ) ) ).
% max.boundedI
thf(fact_4603_max_Oorder__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B3: A,A5: A] :
( A5
= ( ord_max @ A @ A5 @ B3 ) ) ) ) ) ).
% max.order_iff
thf(fact_4604_max_Ocobounded1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A] : ( ord_less_eq @ A @ A3 @ ( ord_max @ A @ A3 @ B2 ) ) ) ).
% max.cobounded1
thf(fact_4605_max_Ocobounded2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,A3: A] : ( ord_less_eq @ A @ B2 @ ( ord_max @ A @ A3 @ B2 ) ) ) ).
% max.cobounded2
thf(fact_4606_le__max__iff__disj,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Z2: A,X: A,Y2: A] :
( ( ord_less_eq @ A @ Z2 @ ( ord_max @ A @ X @ Y2 ) )
= ( ( ord_less_eq @ A @ Z2 @ X )
| ( ord_less_eq @ A @ Z2 @ Y2 ) ) ) ) ).
% le_max_iff_disj
thf(fact_4607_max_Oabsorb__iff1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B3: A,A5: A] :
( ( ord_max @ A @ A5 @ B3 )
= A5 ) ) ) ) ).
% max.absorb_iff1
thf(fact_4608_max_Oabsorb__iff2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A5: A,B3: A] :
( ( ord_max @ A @ A5 @ B3 )
= B3 ) ) ) ) ).
% max.absorb_iff2
thf(fact_4609_max_OcoboundedI1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( ord_less_eq @ A @ C2 @ A3 )
=> ( ord_less_eq @ A @ C2 @ ( ord_max @ A @ A3 @ B2 ) ) ) ) ).
% max.coboundedI1
thf(fact_4610_max_OcoboundedI2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C2: A,B2: A,A3: A] :
( ( ord_less_eq @ A @ C2 @ B2 )
=> ( ord_less_eq @ A @ C2 @ ( ord_max @ A @ A3 @ B2 ) ) ) ) ).
% max.coboundedI2
thf(fact_4611_max__add__distrib__right,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [X: A,Y2: A,Z2: A] :
( ( plus_plus @ A @ X @ ( ord_max @ A @ Y2 @ Z2 ) )
= ( ord_max @ A @ ( plus_plus @ A @ X @ Y2 ) @ ( plus_plus @ A @ X @ Z2 ) ) ) ) ).
% max_add_distrib_right
thf(fact_4612_max__add__distrib__left,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [X: A,Y2: A,Z2: A] :
( ( plus_plus @ A @ ( ord_max @ A @ X @ Y2 ) @ Z2 )
= ( ord_max @ A @ ( plus_plus @ A @ X @ Z2 ) @ ( plus_plus @ A @ Y2 @ Z2 ) ) ) ) ).
% max_add_distrib_left
thf(fact_4613_of__int__max,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: int,Y2: int] :
( ( ring_1_of_int @ A @ ( ord_max @ int @ X @ Y2 ) )
= ( ord_max @ A @ ( ring_1_of_int @ A @ X ) @ ( ring_1_of_int @ A @ Y2 ) ) ) ) ).
% of_int_max
thf(fact_4614_max_Oleft__commute,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( ord_max @ A @ B2 @ ( ord_max @ A @ A3 @ C2 ) )
= ( ord_max @ A @ A3 @ ( ord_max @ A @ B2 @ C2 ) ) ) ) ).
% max.left_commute
thf(fact_4615_max_Ocommute,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_max @ A )
= ( ^ [A5: A,B3: A] : ( ord_max @ A @ B3 @ A5 ) ) ) ) ).
% max.commute
thf(fact_4616_max_Oassoc,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_max @ A @ ( ord_max @ A @ A3 @ B2 ) @ C2 )
= ( ord_max @ A @ A3 @ ( ord_max @ A @ B2 @ C2 ) ) ) ) ).
% max.assoc
thf(fact_4617_less__max__iff__disj,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Z2: A,X: A,Y2: A] :
( ( ord_less @ A @ Z2 @ ( ord_max @ A @ X @ Y2 ) )
= ( ( ord_less @ A @ Z2 @ X )
| ( ord_less @ A @ Z2 @ Y2 ) ) ) ) ).
% less_max_iff_disj
thf(fact_4618_max_Ostrict__boundedE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,C2: A,A3: A] :
( ( ord_less @ A @ ( ord_max @ A @ B2 @ C2 ) @ A3 )
=> ~ ( ( ord_less @ A @ B2 @ A3 )
=> ~ ( ord_less @ A @ C2 @ A3 ) ) ) ) ).
% max.strict_boundedE
thf(fact_4619_max_Ostrict__order__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less @ A )
= ( ^ [B3: A,A5: A] :
( ( A5
= ( ord_max @ A @ A5 @ B3 ) )
& ( A5 != B3 ) ) ) ) ) ).
% max.strict_order_iff
thf(fact_4620_max_Ostrict__coboundedI1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( ord_less @ A @ C2 @ A3 )
=> ( ord_less @ A @ C2 @ ( ord_max @ A @ A3 @ B2 ) ) ) ) ).
% max.strict_coboundedI1
thf(fact_4621_max_Ostrict__coboundedI2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C2: A,B2: A,A3: A] :
( ( ord_less @ A @ C2 @ B2 )
=> ( ord_less @ A @ C2 @ ( ord_max @ A @ A3 @ B2 ) ) ) ) ).
% max.strict_coboundedI2
thf(fact_4622_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_4623_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_4624_floor__rat__def,axiom,
( ( archim6421214686448440834_floor @ rat )
= ( ^ [X2: rat] :
( the @ int
@ ^ [Z6: int] :
( ( ord_less_eq @ rat @ ( ring_1_of_int @ rat @ Z6 ) @ X2 )
& ( ord_less @ rat @ X2 @ ( ring_1_of_int @ rat @ ( plus_plus @ int @ Z6 @ ( one_one @ int ) ) ) ) ) ) ) ) ).
% floor_rat_def
thf(fact_4625_in__set__enumerate__eq,axiom,
! [A: $tType,P4: product_prod @ nat @ A,N: nat,Xs: list @ A] :
( ( member @ ( product_prod @ nat @ A ) @ P4 @ ( set2 @ ( product_prod @ nat @ A ) @ ( enumerate @ A @ N @ Xs ) ) )
= ( ( ord_less_eq @ nat @ N @ ( product_fst @ nat @ A @ P4 ) )
& ( ord_less @ nat @ ( product_fst @ nat @ A @ P4 ) @ ( plus_plus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N ) )
& ( ( nth @ A @ Xs @ ( minus_minus @ nat @ ( product_fst @ nat @ A @ P4 ) @ N ) )
= ( product_snd @ nat @ A @ P4 ) ) ) ) ).
% in_set_enumerate_eq
thf(fact_4626_max__Suc__Suc,axiom,
! [M2: nat,N: nat] :
( ( ord_max @ nat @ ( suc @ M2 ) @ ( suc @ N ) )
= ( suc @ ( ord_max @ nat @ M2 @ N ) ) ) ).
% max_Suc_Suc
thf(fact_4627_max__nat_Oeq__neutr__iff,axiom,
! [A3: nat,B2: nat] :
( ( ( ord_max @ nat @ A3 @ B2 )
= ( zero_zero @ nat ) )
= ( ( A3
= ( zero_zero @ nat ) )
& ( B2
= ( zero_zero @ nat ) ) ) ) ).
% max_nat.eq_neutr_iff
thf(fact_4628_max__nat_Oleft__neutral,axiom,
! [A3: nat] :
( ( ord_max @ nat @ ( zero_zero @ nat ) @ A3 )
= A3 ) ).
% max_nat.left_neutral
thf(fact_4629_max__nat_Oneutr__eq__iff,axiom,
! [A3: nat,B2: nat] :
( ( ( zero_zero @ nat )
= ( ord_max @ nat @ A3 @ B2 ) )
= ( ( A3
= ( zero_zero @ nat ) )
& ( B2
= ( zero_zero @ nat ) ) ) ) ).
% max_nat.neutr_eq_iff
thf(fact_4630_max__nat_Oright__neutral,axiom,
! [A3: nat] :
( ( ord_max @ nat @ A3 @ ( zero_zero @ nat ) )
= A3 ) ).
% max_nat.right_neutral
thf(fact_4631_max__0L,axiom,
! [N: nat] :
( ( ord_max @ nat @ ( zero_zero @ nat ) @ N )
= N ) ).
% max_0L
thf(fact_4632_max__0R,axiom,
! [N: nat] :
( ( ord_max @ nat @ N @ ( zero_zero @ nat ) )
= N ) ).
% max_0R
thf(fact_4633_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_4634_max__numeral__Suc,axiom,
! [K: num,N: nat] :
( ( ord_max @ nat @ ( numeral_numeral @ nat @ K ) @ ( suc @ N ) )
= ( suc @ ( ord_max @ nat @ ( pred_numeral @ K ) @ N ) ) ) ).
% max_numeral_Suc
thf(fact_4635_max__Suc__numeral,axiom,
! [N: nat,K: num] :
( ( ord_max @ nat @ ( suc @ N ) @ ( numeral_numeral @ nat @ K ) )
= ( suc @ ( ord_max @ nat @ N @ ( pred_numeral @ K ) ) ) ) ).
% max_Suc_numeral
thf(fact_4636_nat__mult__max__left,axiom,
! [M2: nat,N: nat,Q5: nat] :
( ( times_times @ nat @ ( ord_max @ nat @ M2 @ N ) @ Q5 )
= ( ord_max @ nat @ ( times_times @ nat @ M2 @ Q5 ) @ ( times_times @ nat @ N @ Q5 ) ) ) ).
% nat_mult_max_left
thf(fact_4637_nat__mult__max__right,axiom,
! [M2: nat,N: nat,Q5: nat] :
( ( times_times @ nat @ M2 @ ( ord_max @ nat @ N @ Q5 ) )
= ( ord_max @ nat @ ( times_times @ nat @ M2 @ N ) @ ( times_times @ nat @ M2 @ Q5 ) ) ) ).
% nat_mult_max_right
thf(fact_4638_nat__add__max__right,axiom,
! [M2: nat,N: nat,Q5: nat] :
( ( plus_plus @ nat @ M2 @ ( ord_max @ nat @ N @ Q5 ) )
= ( ord_max @ nat @ ( plus_plus @ nat @ M2 @ N ) @ ( plus_plus @ nat @ M2 @ Q5 ) ) ) ).
% nat_add_max_right
thf(fact_4639_nat__add__max__left,axiom,
! [M2: nat,N: nat,Q5: nat] :
( ( plus_plus @ nat @ ( ord_max @ nat @ M2 @ N ) @ Q5 )
= ( ord_max @ nat @ ( plus_plus @ nat @ M2 @ Q5 ) @ ( plus_plus @ nat @ N @ Q5 ) ) ) ).
% nat_add_max_left
thf(fact_4640_obtain__pos__sum,axiom,
! [R4: rat] :
( ( ord_less @ rat @ ( zero_zero @ rat ) @ R4 )
=> ~ ! [S2: rat] :
( ( ord_less @ rat @ ( zero_zero @ rat ) @ S2 )
=> ! [T4: rat] :
( ( ord_less @ rat @ ( zero_zero @ rat ) @ T4 )
=> ( R4
!= ( plus_plus @ rat @ S2 @ T4 ) ) ) ) ) ).
% obtain_pos_sum
thf(fact_4641_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_4642_abs__rat__def,axiom,
( ( abs_abs @ rat )
= ( ^ [A5: rat] : ( if @ rat @ ( ord_less @ rat @ A5 @ ( zero_zero @ rat ) ) @ ( uminus_uminus @ rat @ A5 ) @ A5 ) ) ) ).
% abs_rat_def
thf(fact_4643_nat__minus__add__max,axiom,
! [N: nat,M2: nat] :
( ( plus_plus @ nat @ ( minus_minus @ nat @ N @ M2 ) @ M2 )
= ( ord_max @ nat @ N @ M2 ) ) ).
% nat_minus_add_max
thf(fact_4644_max__Suc1,axiom,
! [N: nat,M2: nat] :
( ( ord_max @ nat @ ( suc @ N ) @ M2 )
= ( case_nat @ nat @ ( suc @ N )
@ ^ [M4: nat] : ( suc @ ( ord_max @ nat @ N @ M4 ) )
@ M2 ) ) ).
% max_Suc1
thf(fact_4645_max__Suc2,axiom,
! [M2: nat,N: nat] :
( ( ord_max @ nat @ M2 @ ( suc @ N ) )
= ( case_nat @ nat @ ( suc @ N )
@ ^ [M4: nat] : ( suc @ ( ord_max @ nat @ M4 @ N ) )
@ M2 ) ) ).
% max_Suc2
thf(fact_4646_nth__enumerate__eq,axiom,
! [A: $tType,M2: nat,Xs: list @ A,N: nat] :
( ( ord_less @ nat @ M2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ ( product_prod @ nat @ A ) @ ( enumerate @ A @ N @ Xs ) @ M2 )
= ( product_Pair @ nat @ A @ ( plus_plus @ nat @ N @ M2 ) @ ( nth @ A @ Xs @ M2 ) ) ) ) ).
% nth_enumerate_eq
thf(fact_4647_or__nat__unfold,axiom,
( ( bit_se1065995026697491101ons_or @ nat )
= ( ^ [M5: nat,N5: nat] :
( if @ nat
@ ( M5
= ( zero_zero @ nat ) )
@ N5
@ ( if @ nat
@ ( N5
= ( zero_zero @ nat ) )
@ M5
@ ( plus_plus @ nat @ ( ord_max @ nat @ ( modulo_modulo @ nat @ M5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ nat @ N5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ nat @ ( divide_divide @ nat @ M5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ nat @ N5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% or_nat_unfold
thf(fact_4648_rec__nat__0__imp,axiom,
! [A: $tType,F2: nat > A,F1: A,F22: nat > A > A] :
( ( F2
= ( rec_nat @ A @ F1 @ F22 ) )
=> ( ( F2 @ ( zero_zero @ nat ) )
= F1 ) ) ).
% rec_nat_0_imp
thf(fact_4649_rec__nat__Suc__imp,axiom,
! [A: $tType,F2: nat > A,F1: A,F22: nat > A > A,N: nat] :
( ( F2
= ( rec_nat @ A @ F1 @ F22 ) )
=> ( ( F2 @ ( suc @ N ) )
= ( F22 @ N @ ( F2 @ N ) ) ) ) ).
% rec_nat_Suc_imp
thf(fact_4650_rat__inverse__code,axiom,
! [P4: rat] :
( ( quotient_of @ ( inverse_inverse @ rat @ P4 ) )
= ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [A5: int,B3: int] :
( if @ ( product_prod @ int @ int )
@ ( A5
= ( zero_zero @ int ) )
@ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
@ ( product_Pair @ int @ int @ ( times_times @ int @ ( sgn_sgn @ int @ A5 ) @ B3 ) @ ( abs_abs @ int @ A5 ) ) )
@ ( quotient_of @ P4 ) ) ) ).
% rat_inverse_code
thf(fact_4651_quotient__of__number_I3_J,axiom,
! [K: num] :
( ( quotient_of @ ( numeral_numeral @ rat @ K ) )
= ( product_Pair @ int @ int @ ( numeral_numeral @ int @ K ) @ ( one_one @ int ) ) ) ).
% quotient_of_number(3)
thf(fact_4652_rat__one__code,axiom,
( ( quotient_of @ ( one_one @ rat ) )
= ( product_Pair @ int @ int @ ( one_one @ int ) @ ( one_one @ int ) ) ) ).
% rat_one_code
thf(fact_4653_rat__zero__code,axiom,
( ( quotient_of @ ( zero_zero @ rat ) )
= ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) ) ) ).
% rat_zero_code
thf(fact_4654_quotient__of__number_I5_J,axiom,
! [K: num] :
( ( quotient_of @ ( uminus_uminus @ rat @ ( numeral_numeral @ rat @ K ) ) )
= ( product_Pair @ int @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) @ ( one_one @ int ) ) ) ).
% quotient_of_number(5)
thf(fact_4655_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_4656_diff__rat__def,axiom,
( ( minus_minus @ rat )
= ( ^ [Q4: rat,R: rat] : ( plus_plus @ rat @ Q4 @ ( uminus_uminus @ rat @ R ) ) ) ) ).
% diff_rat_def
thf(fact_4657_divide__rat__def,axiom,
( ( divide_divide @ rat )
= ( ^ [Q4: rat,R: rat] : ( times_times @ rat @ Q4 @ ( inverse_inverse @ rat @ R ) ) ) ) ).
% divide_rat_def
thf(fact_4658_quotient__of__div,axiom,
! [R4: rat,N: int,D2: int] :
( ( ( quotient_of @ R4 )
= ( product_Pair @ int @ int @ N @ D2 ) )
=> ( R4
= ( divide_divide @ rat @ ( ring_1_of_int @ rat @ N ) @ ( ring_1_of_int @ rat @ D2 ) ) ) ) ).
% quotient_of_div
thf(fact_4659_quotient__of__denom__pos,axiom,
! [R4: rat,P4: int,Q5: int] :
( ( ( quotient_of @ R4 )
= ( product_Pair @ int @ int @ P4 @ Q5 ) )
=> ( ord_less @ int @ ( zero_zero @ int ) @ Q5 ) ) ).
% quotient_of_denom_pos
thf(fact_4660_quotient__of__denom__pos_H,axiom,
! [R4: rat] : ( ord_less @ int @ ( zero_zero @ int ) @ ( product_snd @ int @ int @ ( quotient_of @ R4 ) ) ) ).
% quotient_of_denom_pos'
thf(fact_4661_rat__uminus__code,axiom,
! [P4: rat] :
( ( quotient_of @ ( uminus_uminus @ rat @ P4 ) )
= ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [A5: int] : ( product_Pair @ int @ int @ ( uminus_uminus @ int @ A5 ) )
@ ( quotient_of @ P4 ) ) ) ).
% rat_uminus_code
thf(fact_4662_rat__sgn__code,axiom,
! [P4: rat] :
( ( quotient_of @ ( sgn_sgn @ rat @ P4 ) )
= ( product_Pair @ int @ int @ ( sgn_sgn @ int @ ( product_fst @ int @ int @ ( quotient_of @ P4 ) ) ) @ ( one_one @ int ) ) ) ).
% rat_sgn_code
thf(fact_4663_rat__less__code,axiom,
( ( ord_less @ rat )
= ( ^ [P6: rat,Q4: rat] :
( product_case_prod @ int @ int @ $o
@ ^ [A5: int,C4: int] :
( product_case_prod @ int @ int @ $o
@ ^ [B3: int,D4: int] : ( ord_less @ int @ ( times_times @ int @ A5 @ D4 ) @ ( times_times @ int @ C4 @ B3 ) )
@ ( quotient_of @ Q4 ) )
@ ( quotient_of @ P6 ) ) ) ) ).
% rat_less_code
thf(fact_4664_quotient__of__int,axiom,
! [A3: int] :
( ( quotient_of @ ( of_int @ A3 ) )
= ( product_Pair @ int @ int @ A3 @ ( one_one @ int ) ) ) ).
% quotient_of_int
thf(fact_4665_rat__minus__code,axiom,
! [P4: rat,Q5: rat] :
( ( quotient_of @ ( minus_minus @ rat @ P4 @ Q5 ) )
= ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [A5: int,C4: int] :
( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [B3: int,D4: int] : ( normalize @ ( product_Pair @ int @ int @ ( minus_minus @ int @ ( times_times @ int @ A5 @ D4 ) @ ( times_times @ int @ B3 @ C4 ) ) @ ( times_times @ int @ C4 @ D4 ) ) )
@ ( quotient_of @ Q5 ) )
@ ( quotient_of @ P4 ) ) ) ).
% rat_minus_code
thf(fact_4666_normalize__negative,axiom,
! [Q5: int,P4: int] :
( ( ord_less @ int @ Q5 @ ( zero_zero @ int ) )
=> ( ( normalize @ ( product_Pair @ int @ int @ P4 @ Q5 ) )
= ( normalize @ ( product_Pair @ int @ int @ ( uminus_uminus @ int @ P4 ) @ ( uminus_uminus @ int @ Q5 ) ) ) ) ) ).
% normalize_negative
thf(fact_4667_normalize__denom__zero,axiom,
! [P4: int] :
( ( normalize @ ( product_Pair @ int @ int @ P4 @ ( zero_zero @ int ) ) )
= ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) ) ) ).
% normalize_denom_zero
thf(fact_4668_normalize__denom__pos,axiom,
! [R4: product_prod @ int @ int,P4: int,Q5: int] :
( ( ( normalize @ R4 )
= ( product_Pair @ int @ int @ P4 @ Q5 ) )
=> ( ord_less @ int @ ( zero_zero @ int ) @ Q5 ) ) ).
% normalize_denom_pos
thf(fact_4669_normalize__crossproduct,axiom,
! [Q5: int,S: int,P4: int,R4: int] :
( ( Q5
!= ( zero_zero @ int ) )
=> ( ( S
!= ( zero_zero @ int ) )
=> ( ( ( normalize @ ( product_Pair @ int @ int @ P4 @ Q5 ) )
= ( normalize @ ( product_Pair @ int @ int @ R4 @ S ) ) )
=> ( ( times_times @ int @ P4 @ S )
= ( times_times @ int @ R4 @ Q5 ) ) ) ) ) ).
% normalize_crossproduct
thf(fact_4670_rat__divide__code,axiom,
! [P4: rat,Q5: rat] :
( ( quotient_of @ ( divide_divide @ rat @ P4 @ Q5 ) )
= ( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [A5: int,C4: int] :
( product_case_prod @ int @ int @ ( product_prod @ int @ int )
@ ^ [B3: int,D4: int] : ( normalize @ ( product_Pair @ int @ int @ ( times_times @ int @ A5 @ D4 ) @ ( times_times @ int @ C4 @ B3 ) ) )
@ ( quotient_of @ Q5 ) )
@ ( quotient_of @ P4 ) ) ) ).
% rat_divide_code
thf(fact_4671_Frct__code__post_I5_J,axiom,
! [K: num] :
( ( frct @ ( product_Pair @ int @ int @ ( one_one @ int ) @ ( numeral_numeral @ int @ K ) ) )
= ( divide_divide @ rat @ ( one_one @ rat ) @ ( numeral_numeral @ rat @ K ) ) ) ).
% Frct_code_post(5)
thf(fact_4672_normalize__def,axiom,
( normalize
= ( ^ [P6: product_prod @ int @ int] :
( if @ ( product_prod @ int @ int ) @ ( ord_less @ int @ ( zero_zero @ int ) @ ( product_snd @ int @ int @ P6 ) ) @ ( product_Pair @ int @ int @ ( divide_divide @ int @ ( product_fst @ int @ int @ P6 ) @ ( gcd_gcd @ int @ ( product_fst @ int @ int @ P6 ) @ ( product_snd @ int @ int @ P6 ) ) ) @ ( divide_divide @ int @ ( product_snd @ int @ int @ P6 ) @ ( gcd_gcd @ int @ ( product_fst @ int @ int @ P6 ) @ ( product_snd @ int @ int @ P6 ) ) ) )
@ ( if @ ( product_prod @ int @ int )
@ ( ( product_snd @ int @ int @ P6 )
= ( zero_zero @ int ) )
@ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
@ ( product_Pair @ int @ int @ ( divide_divide @ int @ ( product_fst @ int @ int @ P6 ) @ ( uminus_uminus @ int @ ( gcd_gcd @ int @ ( product_fst @ int @ int @ P6 ) @ ( product_snd @ int @ int @ P6 ) ) ) ) @ ( divide_divide @ int @ ( product_snd @ int @ int @ P6 ) @ ( uminus_uminus @ int @ ( gcd_gcd @ int @ ( product_fst @ int @ int @ P6 ) @ ( product_snd @ int @ int @ P6 ) ) ) ) ) ) ) ) ) ).
% normalize_def
thf(fact_4673_Frct__code__post_I6_J,axiom,
! [K: num,L: num] :
( ( frct @ ( product_Pair @ int @ int @ ( numeral_numeral @ int @ K ) @ ( numeral_numeral @ int @ L ) ) )
= ( divide_divide @ rat @ ( numeral_numeral @ rat @ K ) @ ( numeral_numeral @ rat @ L ) ) ) ).
% Frct_code_post(6)
thf(fact_4674_gcd__eq__0__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A] :
( ( ( gcd_gcd @ A @ A3 @ B2 )
= ( zero_zero @ A ) )
= ( ( A3
= ( zero_zero @ A ) )
& ( B2
= ( zero_zero @ A ) ) ) ) ) ).
% gcd_eq_0_iff
thf(fact_4675_gcd_Obottom__right__bottom,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A] :
( ( gcd_gcd @ A @ A3 @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% gcd.bottom_right_bottom
thf(fact_4676_gcd_Obottom__left__bottom,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A] :
( ( gcd_gcd @ A @ ( one_one @ A ) @ A3 )
= ( one_one @ A ) ) ) ).
% gcd.bottom_left_bottom
thf(fact_4677_gcd__neg1,axiom,
! [A: $tType] :
( ( ring_gcd @ A )
=> ! [A3: A,B2: A] :
( ( gcd_gcd @ A @ ( uminus_uminus @ A @ A3 ) @ B2 )
= ( gcd_gcd @ A @ A3 @ B2 ) ) ) ).
% gcd_neg1
thf(fact_4678_gcd__neg2,axiom,
! [A: $tType] :
( ( ring_gcd @ A )
=> ! [A3: A,B2: A] :
( ( gcd_gcd @ A @ A3 @ ( uminus_uminus @ A @ B2 ) )
= ( gcd_gcd @ A @ A3 @ B2 ) ) ) ).
% gcd_neg2
thf(fact_4679_gcd__1__int,axiom,
! [M2: int] :
( ( gcd_gcd @ int @ M2 @ ( one_one @ int ) )
= ( one_one @ int ) ) ).
% gcd_1_int
thf(fact_4680_gcd__neg2__int,axiom,
! [X: int,Y2: int] :
( ( gcd_gcd @ int @ X @ ( uminus_uminus @ int @ Y2 ) )
= ( gcd_gcd @ int @ X @ Y2 ) ) ).
% gcd_neg2_int
thf(fact_4681_gcd__neg1__int,axiom,
! [X: int,Y2: int] :
( ( gcd_gcd @ int @ ( uminus_uminus @ int @ X ) @ Y2 )
= ( gcd_gcd @ int @ X @ Y2 ) ) ).
% gcd_neg1_int
thf(fact_4682_gcd__neg__numeral__2,axiom,
! [A: $tType] :
( ( ring_gcd @ A )
=> ! [A3: A,N: num] :
( ( gcd_gcd @ A @ A3 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
= ( gcd_gcd @ A @ A3 @ ( numeral_numeral @ A @ N ) ) ) ) ).
% gcd_neg_numeral_2
thf(fact_4683_gcd__neg__numeral__1,axiom,
! [A: $tType] :
( ( ring_gcd @ A )
=> ! [N: num,A3: A] :
( ( gcd_gcd @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) @ A3 )
= ( gcd_gcd @ A @ ( numeral_numeral @ A @ N ) @ A3 ) ) ) ).
% gcd_neg_numeral_1
thf(fact_4684_is__unit__gcd__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A] :
( ( dvd_dvd @ A @ ( gcd_gcd @ A @ A3 @ B2 ) @ ( one_one @ A ) )
= ( ( gcd_gcd @ A @ A3 @ B2 )
= ( one_one @ A ) ) ) ) ).
% is_unit_gcd_iff
thf(fact_4685_gcd__pos__int,axiom,
! [M2: int,N: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ ( gcd_gcd @ int @ M2 @ N ) )
= ( ( M2
!= ( zero_zero @ int ) )
| ( N
!= ( zero_zero @ int ) ) ) ) ).
% gcd_pos_int
thf(fact_4686_gcd__neg__numeral__2__int,axiom,
! [X: int,N: num] :
( ( gcd_gcd @ int @ X @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
= ( gcd_gcd @ int @ X @ ( numeral_numeral @ int @ N ) ) ) ).
% gcd_neg_numeral_2_int
thf(fact_4687_gcd__neg__numeral__1__int,axiom,
! [N: num,X: int] :
( ( gcd_gcd @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) @ X )
= ( gcd_gcd @ int @ ( numeral_numeral @ int @ N ) @ X ) ) ).
% gcd_neg_numeral_1_int
thf(fact_4688_gcd__0__int,axiom,
! [X: int] :
( ( gcd_gcd @ int @ X @ ( zero_zero @ int ) )
= ( abs_abs @ int @ X ) ) ).
% gcd_0_int
thf(fact_4689_gcd__0__left__int,axiom,
! [X: int] :
( ( gcd_gcd @ int @ ( zero_zero @ int ) @ X )
= ( abs_abs @ int @ X ) ) ).
% gcd_0_left_int
thf(fact_4690_gcd__diff2,axiom,
! [A: $tType] :
( ( ring_gcd @ A )
=> ! [N: A,M2: A] :
( ( gcd_gcd @ A @ ( minus_minus @ A @ N @ M2 ) @ N )
= ( gcd_gcd @ A @ M2 @ N ) ) ) ).
% gcd_diff2
thf(fact_4691_gcd__diff1,axiom,
! [A: $tType] :
( ( ring_gcd @ A )
=> ! [M2: A,N: A] :
( ( gcd_gcd @ A @ ( minus_minus @ A @ M2 @ N ) @ N )
= ( gcd_gcd @ A @ M2 @ N ) ) ) ).
% gcd_diff1
thf(fact_4692_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_4693_Frct__code__post_I9_J,axiom,
! [Q5: product_prod @ int @ int] :
( ( uminus_uminus @ rat @ ( uminus_uminus @ rat @ ( frct @ Q5 ) ) )
= ( frct @ Q5 ) ) ).
% Frct_code_post(9)
thf(fact_4694_gcd__mult__unit2,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( gcd_gcd @ A @ B2 @ ( times_times @ A @ C2 @ A3 ) )
= ( gcd_gcd @ A @ B2 @ C2 ) ) ) ) ).
% gcd_mult_unit2
thf(fact_4695_gcd__mult__unit1,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( gcd_gcd @ A @ ( times_times @ A @ B2 @ A3 ) @ C2 )
= ( gcd_gcd @ A @ B2 @ C2 ) ) ) ) ).
% gcd_mult_unit1
thf(fact_4696_gcd__div__unit1,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( gcd_gcd @ A @ ( divide_divide @ A @ B2 @ A3 ) @ C2 )
= ( gcd_gcd @ A @ B2 @ C2 ) ) ) ) ).
% gcd_div_unit1
thf(fact_4697_gcd__div__unit2,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( gcd_gcd @ A @ B2 @ ( divide_divide @ A @ C2 @ A3 ) )
= ( gcd_gcd @ A @ B2 @ C2 ) ) ) ) ).
% gcd_div_unit2
thf(fact_4698_gcd__le1__int,axiom,
! [A3: int,B2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ A3 )
=> ( ord_less_eq @ int @ ( gcd_gcd @ int @ A3 @ B2 ) @ A3 ) ) ).
% gcd_le1_int
thf(fact_4699_gcd__le2__int,axiom,
! [B2: int,A3: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
=> ( ord_less_eq @ int @ ( gcd_gcd @ int @ A3 @ B2 ) @ B2 ) ) ).
% gcd_le2_int
thf(fact_4700_gcd__cases__int,axiom,
! [X: int,Y2: int,P: int > $o] :
( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y2 )
=> ( P @ ( gcd_gcd @ int @ X @ Y2 ) ) ) )
=> ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
=> ( ( ord_less_eq @ int @ Y2 @ ( zero_zero @ int ) )
=> ( P @ ( gcd_gcd @ int @ X @ ( uminus_uminus @ int @ Y2 ) ) ) ) )
=> ( ( ( ord_less_eq @ int @ X @ ( zero_zero @ int ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y2 )
=> ( P @ ( gcd_gcd @ int @ ( uminus_uminus @ int @ X ) @ Y2 ) ) ) )
=> ( ( ( ord_less_eq @ int @ X @ ( zero_zero @ int ) )
=> ( ( ord_less_eq @ int @ Y2 @ ( zero_zero @ int ) )
=> ( P @ ( gcd_gcd @ int @ ( uminus_uminus @ int @ X ) @ ( uminus_uminus @ int @ Y2 ) ) ) ) )
=> ( P @ ( gcd_gcd @ int @ X @ Y2 ) ) ) ) ) ) ).
% gcd_cases_int
thf(fact_4701_gcd__unique__int,axiom,
! [D2: int,A3: int,B2: int] :
( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ D2 )
& ( dvd_dvd @ int @ D2 @ A3 )
& ( dvd_dvd @ int @ D2 @ B2 )
& ! [E3: int] :
( ( ( dvd_dvd @ int @ E3 @ A3 )
& ( dvd_dvd @ int @ E3 @ B2 ) )
=> ( dvd_dvd @ int @ E3 @ D2 ) ) )
= ( D2
= ( gcd_gcd @ int @ A3 @ B2 ) ) ) ).
% gcd_unique_int
thf(fact_4702_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_4703_gcd__code__int,axiom,
( ( gcd_gcd @ int )
= ( ^ [K3: int,L2: int] :
( abs_abs @ int
@ ( if @ int
@ ( L2
= ( zero_zero @ int ) )
@ K3
@ ( gcd_gcd @ int @ L2 @ ( modulo_modulo @ int @ ( abs_abs @ int @ K3 ) @ ( abs_abs @ int @ L2 ) ) ) ) ) ) ) ).
% gcd_code_int
thf(fact_4704_Frct__code__post_I1_J,axiom,
! [A3: int] :
( ( frct @ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ A3 ) )
= ( zero_zero @ rat ) ) ).
% Frct_code_post(1)
thf(fact_4705_Frct__code__post_I2_J,axiom,
! [A3: int] :
( ( frct @ ( product_Pair @ int @ int @ A3 @ ( zero_zero @ int ) ) )
= ( zero_zero @ rat ) ) ).
% Frct_code_post(2)
thf(fact_4706_Frct__code__post_I8_J,axiom,
! [A3: int,B2: int] :
( ( frct @ ( product_Pair @ int @ int @ A3 @ ( uminus_uminus @ int @ B2 ) ) )
= ( uminus_uminus @ rat @ ( frct @ ( product_Pair @ int @ int @ A3 @ B2 ) ) ) ) ).
% Frct_code_post(8)
thf(fact_4707_Frct__code__post_I7_J,axiom,
! [A3: int,B2: int] :
( ( frct @ ( product_Pair @ int @ int @ ( uminus_uminus @ int @ A3 ) @ B2 ) )
= ( uminus_uminus @ rat @ ( frct @ ( product_Pair @ int @ int @ A3 @ B2 ) ) ) ) ).
% Frct_code_post(7)
thf(fact_4708_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_4709_Frct__code__post_I4_J,axiom,
! [K: num] :
( ( frct @ ( product_Pair @ int @ int @ ( numeral_numeral @ int @ K ) @ ( one_one @ int ) ) )
= ( numeral_numeral @ rat @ K ) ) ).
% Frct_code_post(4)
thf(fact_4710_drop__bit__numeral__minus__bit1,axiom,
! [L: num,K: num] :
( ( bit_se4197421643247451524op_bit @ int @ ( numeral_numeral @ nat @ L ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ K ) ) ) )
= ( bit_se4197421643247451524op_bit @ int @ ( pred_numeral @ L ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( inc @ K ) ) ) ) ) ).
% drop_bit_numeral_minus_bit1
thf(fact_4711_divmod__integer__eq__cases,axiom,
( code_divmod_integer
= ( ^ [K3: code_integer,L2: code_integer] :
( if @ ( product_prod @ code_integer @ code_integer )
@ ( K3
= ( zero_zero @ code_integer ) )
@ ( product_Pair @ code_integer @ code_integer @ ( zero_zero @ code_integer ) @ ( zero_zero @ code_integer ) )
@ ( if @ ( product_prod @ code_integer @ code_integer )
@ ( L2
= ( zero_zero @ code_integer ) )
@ ( product_Pair @ code_integer @ code_integer @ ( zero_zero @ code_integer ) @ K3 )
@ ( comp @ code_integer @ ( ( product_prod @ code_integer @ code_integer ) > ( product_prod @ code_integer @ code_integer ) ) @ code_integer @ ( comp @ ( code_integer > code_integer ) @ ( ( product_prod @ code_integer @ code_integer ) > ( product_prod @ code_integer @ code_integer ) ) @ code_integer @ ( product_apsnd @ code_integer @ code_integer @ code_integer ) @ ( times_times @ code_integer ) ) @ ( sgn_sgn @ code_integer ) @ L2
@ ( if @ ( product_prod @ code_integer @ code_integer )
@ ( ( sgn_sgn @ code_integer @ K3 )
= ( sgn_sgn @ code_integer @ L2 ) )
@ ( code_divmod_abs @ K3 @ L2 )
@ ( product_case_prod @ code_integer @ code_integer @ ( product_prod @ code_integer @ code_integer )
@ ^ [R: code_integer,S5: code_integer] :
( if @ ( product_prod @ code_integer @ code_integer )
@ ( S5
= ( zero_zero @ code_integer ) )
@ ( product_Pair @ code_integer @ code_integer @ ( uminus_uminus @ code_integer @ R ) @ ( zero_zero @ code_integer ) )
@ ( product_Pair @ code_integer @ code_integer @ ( minus_minus @ code_integer @ ( uminus_uminus @ code_integer @ R ) @ ( one_one @ code_integer ) ) @ ( minus_minus @ code_integer @ ( abs_abs @ code_integer @ L2 ) @ S5 ) ) )
@ ( code_divmod_abs @ K3 @ L2 ) ) ) ) ) ) ) ) ).
% divmod_integer_eq_cases
thf(fact_4712_drop__bit__rec,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se4197421643247451524op_bit @ A )
= ( ^ [N5: nat,A5: A] :
( if @ A
@ ( N5
= ( zero_zero @ nat ) )
@ A5
@ ( bit_se4197421643247451524op_bit @ A @ ( minus_minus @ nat @ N5 @ ( one_one @ nat ) ) @ ( divide_divide @ A @ A5 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% drop_bit_rec
thf(fact_4713_gcd__0__left__nat,axiom,
! [X: nat] :
( ( gcd_gcd @ nat @ ( zero_zero @ nat ) @ X )
= X ) ).
% gcd_0_left_nat
thf(fact_4714_gcd__0__nat,axiom,
! [X: nat] :
( ( gcd_gcd @ nat @ X @ ( zero_zero @ nat ) )
= X ) ).
% gcd_0_nat
thf(fact_4715_gcd__nat_Oright__neutral,axiom,
! [A3: nat] :
( ( gcd_gcd @ nat @ A3 @ ( zero_zero @ nat ) )
= A3 ) ).
% gcd_nat.right_neutral
thf(fact_4716_gcd__nat_Oneutr__eq__iff,axiom,
! [A3: nat,B2: nat] :
( ( ( zero_zero @ nat )
= ( gcd_gcd @ nat @ A3 @ B2 ) )
= ( ( A3
= ( zero_zero @ nat ) )
& ( B2
= ( zero_zero @ nat ) ) ) ) ).
% gcd_nat.neutr_eq_iff
thf(fact_4717_gcd__nat_Oleft__neutral,axiom,
! [A3: nat] :
( ( gcd_gcd @ nat @ ( zero_zero @ nat ) @ A3 )
= A3 ) ).
% gcd_nat.left_neutral
thf(fact_4718_gcd__nat_Oeq__neutr__iff,axiom,
! [A3: nat,B2: nat] :
( ( ( gcd_gcd @ nat @ A3 @ B2 )
= ( zero_zero @ nat ) )
= ( ( A3
= ( zero_zero @ nat ) )
& ( B2
= ( zero_zero @ nat ) ) ) ) ).
% gcd_nat.eq_neutr_iff
thf(fact_4719_gcd__1__nat,axiom,
! [M2: nat] :
( ( gcd_gcd @ nat @ M2 @ ( one_one @ nat ) )
= ( one_one @ nat ) ) ).
% gcd_1_nat
thf(fact_4720_gcd__Suc__0,axiom,
! [M2: nat] :
( ( gcd_gcd @ nat @ M2 @ ( suc @ ( zero_zero @ nat ) ) )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% gcd_Suc_0
thf(fact_4721_gcd__pos__nat,axiom,
! [M2: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( gcd_gcd @ nat @ M2 @ N ) )
= ( ( M2
!= ( zero_zero @ nat ) )
| ( N
!= ( zero_zero @ nat ) ) ) ) ).
% gcd_pos_nat
thf(fact_4722_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_4723_gcd__int__int__eq,axiom,
! [M2: nat,N: nat] :
( ( gcd_gcd @ int @ ( semiring_1_of_nat @ int @ M2 ) @ ( semiring_1_of_nat @ int @ N ) )
= ( semiring_1_of_nat @ int @ ( gcd_gcd @ nat @ M2 @ N ) ) ) ).
% gcd_int_int_eq
thf(fact_4724_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_4725_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_4726_drop__bit__nonnegative__int__iff,axiom,
! [N: nat,K: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_se4197421643247451524op_bit @ int @ N @ K ) )
= ( ord_less_eq @ int @ ( zero_zero @ int ) @ K ) ) ).
% drop_bit_nonnegative_int_iff
thf(fact_4727_drop__bit__negative__int__iff,axiom,
! [N: nat,K: int] :
( ( ord_less @ int @ ( bit_se4197421643247451524op_bit @ int @ N @ K ) @ ( zero_zero @ int ) )
= ( ord_less @ int @ K @ ( zero_zero @ int ) ) ) ).
% drop_bit_negative_int_iff
thf(fact_4728_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_4729_drop__bit__Suc__bit0,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [N: nat,K: num] :
( ( bit_se4197421643247451524op_bit @ A @ ( suc @ N ) @ ( numeral_numeral @ A @ ( bit0 @ K ) ) )
= ( bit_se4197421643247451524op_bit @ A @ N @ ( numeral_numeral @ A @ K ) ) ) ) ).
% drop_bit_Suc_bit0
thf(fact_4730_drop__bit__Suc__bit1,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [N: nat,K: num] :
( ( bit_se4197421643247451524op_bit @ A @ ( suc @ N ) @ ( numeral_numeral @ A @ ( bit1 @ K ) ) )
= ( bit_se4197421643247451524op_bit @ A @ N @ ( numeral_numeral @ A @ K ) ) ) ) ).
% drop_bit_Suc_bit1
thf(fact_4731_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_4732_gcd__nat__abs__right__eq,axiom,
! [N: nat,K: int] :
( ( gcd_gcd @ nat @ N @ ( nat2 @ ( abs_abs @ int @ K ) ) )
= ( nat2 @ ( gcd_gcd @ int @ ( semiring_1_of_nat @ int @ N ) @ K ) ) ) ).
% gcd_nat_abs_right_eq
thf(fact_4733_gcd__nat__abs__left__eq,axiom,
! [K: int,N: nat] :
( ( gcd_gcd @ nat @ ( nat2 @ ( abs_abs @ int @ K ) ) @ N )
= ( nat2 @ ( gcd_gcd @ int @ K @ ( semiring_1_of_nat @ int @ N ) ) ) ) ).
% gcd_nat_abs_left_eq
thf(fact_4734_drop__bit__Suc__minus__bit0,axiom,
! [N: nat,K: num] :
( ( bit_se4197421643247451524op_bit @ int @ ( suc @ N ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ K ) ) ) )
= ( bit_se4197421643247451524op_bit @ int @ N @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) ) ) ).
% drop_bit_Suc_minus_bit0
thf(fact_4735_drop__bit__numeral__minus__bit0,axiom,
! [L: num,K: num] :
( ( bit_se4197421643247451524op_bit @ int @ ( numeral_numeral @ nat @ L ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ K ) ) ) )
= ( bit_se4197421643247451524op_bit @ int @ ( pred_numeral @ L ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) ) ) ).
% drop_bit_numeral_minus_bit0
thf(fact_4736_drop__bit__Suc__minus__bit1,axiom,
! [N: nat,K: num] :
( ( bit_se4197421643247451524op_bit @ int @ ( suc @ N ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ K ) ) ) )
= ( bit_se4197421643247451524op_bit @ int @ N @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( inc @ K ) ) ) ) ) ).
% drop_bit_Suc_minus_bit1
thf(fact_4737_drop__bit__of__nat,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [N: nat,M2: nat] :
( ( bit_se4197421643247451524op_bit @ A @ N @ ( semiring_1_of_nat @ A @ M2 ) )
= ( semiring_1_of_nat @ A @ ( bit_se4197421643247451524op_bit @ nat @ N @ M2 ) ) ) ) ).
% drop_bit_of_nat
thf(fact_4738_of__nat__drop__bit,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [M2: nat,N: nat] :
( ( semiring_1_of_nat @ A @ ( bit_se4197421643247451524op_bit @ nat @ M2 @ N ) )
= ( bit_se4197421643247451524op_bit @ A @ M2 @ ( semiring_1_of_nat @ A @ N ) ) ) ) ).
% of_nat_drop_bit
thf(fact_4739_gcd__le1__nat,axiom,
! [A3: nat,B2: nat] :
( ( A3
!= ( zero_zero @ nat ) )
=> ( ord_less_eq @ nat @ ( gcd_gcd @ nat @ A3 @ B2 ) @ A3 ) ) ).
% gcd_le1_nat
thf(fact_4740_gcd__le2__nat,axiom,
! [B2: nat,A3: nat] :
( ( B2
!= ( zero_zero @ nat ) )
=> ( ord_less_eq @ nat @ ( gcd_gcd @ nat @ A3 @ B2 ) @ B2 ) ) ).
% gcd_le2_nat
thf(fact_4741_gcd__diff1__nat,axiom,
! [N: nat,M2: nat] :
( ( ord_less_eq @ nat @ N @ M2 )
=> ( ( gcd_gcd @ nat @ ( minus_minus @ nat @ M2 @ N ) @ N )
= ( gcd_gcd @ nat @ M2 @ N ) ) ) ).
% gcd_diff1_nat
thf(fact_4742_gcd__diff2__nat,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( gcd_gcd @ nat @ ( minus_minus @ nat @ N @ M2 ) @ N )
= ( gcd_gcd @ nat @ M2 @ N ) ) ) ).
% gcd_diff2_nat
thf(fact_4743_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_4744_gcd__nat_Osimps,axiom,
( ( gcd_gcd @ nat )
= ( ^ [X2: nat,Y5: nat] :
( if @ nat
@ ( Y5
= ( zero_zero @ nat ) )
@ X2
@ ( gcd_gcd @ nat @ Y5 @ ( modulo_modulo @ nat @ X2 @ Y5 ) ) ) ) ) ).
% gcd_nat.simps
thf(fact_4745_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_4746_take__bit__eq__self__iff__drop__bit__eq__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A3: A] :
( ( ( bit_se2584673776208193580ke_bit @ A @ N @ A3 )
= A3 )
= ( ( bit_se4197421643247451524op_bit @ A @ N @ A3 )
= ( zero_zero @ A ) ) ) ) ).
% take_bit_eq_self_iff_drop_bit_eq_0
thf(fact_4747_drop__bit__push__bit__int,axiom,
! [M2: nat,N: nat,K: int] :
( ( bit_se4197421643247451524op_bit @ int @ M2 @ ( bit_se4730199178511100633sh_bit @ int @ N @ K ) )
= ( bit_se4197421643247451524op_bit @ int @ ( minus_minus @ nat @ M2 @ N ) @ ( bit_se4730199178511100633sh_bit @ int @ ( minus_minus @ nat @ N @ M2 ) @ K ) ) ) ).
% drop_bit_push_bit_int
thf(fact_4748_drop__bit__take__bit,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M2: nat,N: nat,A3: A] :
( ( bit_se4197421643247451524op_bit @ A @ M2 @ ( bit_se2584673776208193580ke_bit @ A @ N @ A3 ) )
= ( bit_se2584673776208193580ke_bit @ A @ ( minus_minus @ nat @ N @ M2 ) @ ( bit_se4197421643247451524op_bit @ A @ M2 @ A3 ) ) ) ) ).
% drop_bit_take_bit
thf(fact_4749_sum__comp__morphism,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( ( comm_monoid_add @ B )
& ( comm_monoid_add @ A ) )
=> ! [H: B > A,G: C > B,A4: set @ C] :
( ( ( H @ ( zero_zero @ B ) )
= ( zero_zero @ A ) )
=> ( ! [X3: B,Y3: B] :
( ( H @ ( plus_plus @ B @ X3 @ Y3 ) )
= ( plus_plus @ A @ ( H @ X3 ) @ ( H @ Y3 ) ) )
=> ( ( groups7311177749621191930dd_sum @ C @ A @ ( comp @ B @ A @ C @ H @ G ) @ A4 )
= ( H @ ( groups7311177749621191930dd_sum @ C @ B @ G @ A4 ) ) ) ) ) ) ).
% sum_comp_morphism
thf(fact_4750_bezout__nat,axiom,
! [A3: nat,B2: nat] :
( ( A3
!= ( zero_zero @ nat ) )
=> ? [X3: nat,Y3: nat] :
( ( times_times @ nat @ A3 @ X3 )
= ( plus_plus @ nat @ ( times_times @ nat @ B2 @ Y3 ) @ ( gcd_gcd @ nat @ A3 @ B2 ) ) ) ) ).
% bezout_nat
thf(fact_4751_bezout__gcd__nat_H,axiom,
! [B2: nat,A3: nat] :
? [X3: nat,Y3: nat] :
( ( ( ord_less_eq @ nat @ ( times_times @ nat @ B2 @ Y3 ) @ ( times_times @ nat @ A3 @ X3 ) )
& ( ( minus_minus @ nat @ ( times_times @ nat @ A3 @ X3 ) @ ( times_times @ nat @ B2 @ Y3 ) )
= ( gcd_gcd @ nat @ A3 @ B2 ) ) )
| ( ( ord_less_eq @ nat @ ( times_times @ nat @ A3 @ Y3 ) @ ( times_times @ nat @ B2 @ X3 ) )
& ( ( minus_minus @ nat @ ( times_times @ nat @ B2 @ X3 ) @ ( times_times @ nat @ A3 @ Y3 ) )
= ( gcd_gcd @ nat @ A3 @ B2 ) ) ) ) ).
% bezout_gcd_nat'
thf(fact_4752_div__push__bit__of__1__eq__drop__bit,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A,N: nat] :
( ( divide_divide @ A @ A3 @ ( bit_se4730199178511100633sh_bit @ A @ N @ ( one_one @ A ) ) )
= ( bit_se4197421643247451524op_bit @ A @ N @ A3 ) ) ) ).
% div_push_bit_of_1_eq_drop_bit
thf(fact_4753_bit__iff__and__drop__bit__eq__1,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se5641148757651400278ts_bit @ A )
= ( ^ [A5: A,N5: nat] :
( ( bit_se5824344872417868541ns_and @ A @ ( bit_se4197421643247451524op_bit @ A @ N5 @ A5 ) @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ) ) ).
% bit_iff_and_drop_bit_eq_1
thf(fact_4754_gcd__int__def,axiom,
( ( gcd_gcd @ int )
= ( ^ [X2: int,Y5: int] : ( semiring_1_of_nat @ int @ ( gcd_gcd @ nat @ ( nat2 @ ( abs_abs @ int @ X2 ) ) @ ( nat2 @ ( abs_abs @ int @ Y5 ) ) ) ) ) ) ).
% gcd_int_def
thf(fact_4755_stable__imp__drop__bit__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A,N: nat] :
( ( ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= A3 )
=> ( ( bit_se4197421643247451524op_bit @ A @ N @ A3 )
= A3 ) ) ) ).
% stable_imp_drop_bit_eq
thf(fact_4756_drop__bit__half,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A3: A] :
( ( bit_se4197421643247451524op_bit @ A @ N @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
= ( divide_divide @ A @ ( bit_se4197421643247451524op_bit @ A @ N @ A3 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% drop_bit_half
thf(fact_4757_drop__bit__int__def,axiom,
( ( bit_se4197421643247451524op_bit @ int )
= ( ^ [N5: nat,K3: int] : ( divide_divide @ int @ K3 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N5 ) ) ) ) ).
% drop_bit_int_def
thf(fact_4758_drop__bit__nat__def,axiom,
( ( bit_se4197421643247451524op_bit @ nat )
= ( ^ [N5: nat,M5: nat] : ( divide_divide @ nat @ M5 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N5 ) ) ) ) ).
% drop_bit_nat_def
thf(fact_4759_drop__bit__Suc,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A3: A] :
( ( bit_se4197421643247451524op_bit @ A @ ( suc @ N ) @ A3 )
= ( bit_se4197421643247451524op_bit @ A @ N @ ( divide_divide @ A @ A3 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% drop_bit_Suc
thf(fact_4760_drop__bit__eq__div,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se4197421643247451524op_bit @ A )
= ( ^ [N5: nat,A5: A] : ( divide_divide @ A @ A5 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N5 ) ) ) ) ) ).
% drop_bit_eq_div
thf(fact_4761_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_4762_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_4763_xor__minus__numerals_I2_J,axiom,
! [K: int,N: num] :
( ( bit_se5824344971392196577ns_xor @ int @ K @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
= ( bit_ri4277139882892585799ns_not @ int @ ( bit_se5824344971392196577ns_xor @ int @ K @ ( neg_numeral_sub @ int @ N @ one2 ) ) ) ) ).
% xor_minus_numerals(2)
thf(fact_4764_xor__minus__numerals_I1_J,axiom,
! [N: num,K: int] :
( ( bit_se5824344971392196577ns_xor @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) @ K )
= ( bit_ri4277139882892585799ns_not @ int @ ( bit_se5824344971392196577ns_xor @ int @ ( neg_numeral_sub @ int @ N @ one2 ) @ K ) ) ) ).
% xor_minus_numerals(1)
thf(fact_4765_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_4766_diff__numeral__simps_I1_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M2: num,N: num] :
( ( minus_minus @ A @ ( numeral_numeral @ A @ M2 ) @ ( numeral_numeral @ A @ N ) )
= ( neg_numeral_sub @ A @ M2 @ N ) ) ) ).
% diff_numeral_simps(1)
thf(fact_4767_sub__num__simps_I6_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [K: num,L: num] :
( ( neg_numeral_sub @ A @ ( bit0 @ K ) @ ( bit0 @ L ) )
= ( neg_numeral_dbl @ A @ ( neg_numeral_sub @ A @ K @ L ) ) ) ) ).
% sub_num_simps(6)
thf(fact_4768_sub__num__simps_I9_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [K: num,L: num] :
( ( neg_numeral_sub @ A @ ( bit1 @ K ) @ ( bit1 @ L ) )
= ( neg_numeral_dbl @ A @ ( neg_numeral_sub @ A @ K @ L ) ) ) ) ).
% sub_num_simps(9)
thf(fact_4769_add__neg__numeral__simps_I2_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M2: num,N: num] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) @ ( numeral_numeral @ A @ N ) )
= ( neg_numeral_sub @ A @ N @ M2 ) ) ) ).
% add_neg_numeral_simps(2)
thf(fact_4770_add__neg__numeral__simps_I1_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M2: num,N: num] :
( ( plus_plus @ A @ ( numeral_numeral @ A @ M2 ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
= ( neg_numeral_sub @ A @ M2 @ N ) ) ) ).
% add_neg_numeral_simps(1)
thf(fact_4771_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_4772_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_4773_diff__numeral__simps_I4_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M2: num,N: num] :
( ( minus_minus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
= ( neg_numeral_sub @ A @ N @ M2 ) ) ) ).
% diff_numeral_simps(4)
thf(fact_4774_sub__num__simps_I8_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [K: num,L: num] :
( ( neg_numeral_sub @ A @ ( bit1 @ K ) @ ( bit0 @ L ) )
= ( neg_numeral_dbl_inc @ A @ ( neg_numeral_sub @ A @ K @ L ) ) ) ) ).
% sub_num_simps(8)
thf(fact_4775_sub__num__simps_I7_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [K: num,L: num] :
( ( neg_numeral_sub @ A @ ( bit0 @ K ) @ ( bit1 @ L ) )
= ( neg_numeral_dbl_dec @ A @ ( neg_numeral_sub @ A @ K @ L ) ) ) ) ).
% sub_num_simps(7)
thf(fact_4776_diff__numeral__special_I2_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M2: num] :
( ( minus_minus @ A @ ( numeral_numeral @ A @ M2 ) @ ( one_one @ A ) )
= ( neg_numeral_sub @ A @ M2 @ one2 ) ) ) ).
% diff_numeral_special(2)
thf(fact_4777_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_4778_sub__num__simps_I5_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [K: num] :
( ( neg_numeral_sub @ A @ ( bit1 @ K ) @ one2 )
= ( numeral_numeral @ A @ ( bit0 @ K ) ) ) ) ).
% sub_num_simps(5)
thf(fact_4779_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_4780_sub__num__simps_I4_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [K: num] :
( ( neg_numeral_sub @ A @ ( bit0 @ K ) @ one2 )
= ( numeral_numeral @ A @ ( bitM @ K ) ) ) ) ).
% sub_num_simps(4)
thf(fact_4781_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_4782_add__neg__numeral__special_I3_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M2: num] :
( ( plus_plus @ A @ ( numeral_numeral @ A @ M2 ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( neg_numeral_sub @ A @ M2 @ one2 ) ) ) ).
% add_neg_numeral_special(3)
thf(fact_4783_add__neg__numeral__special_I2_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M2: num] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) @ ( one_one @ A ) )
= ( neg_numeral_sub @ A @ one2 @ M2 ) ) ) ).
% add_neg_numeral_special(2)
thf(fact_4784_add__neg__numeral__special_I1_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M2: num] :
( ( plus_plus @ A @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) )
= ( neg_numeral_sub @ A @ one2 @ M2 ) ) ) ).
% add_neg_numeral_special(1)
thf(fact_4785_minus__sub__one__diff__one,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [M2: num] :
( ( minus_minus @ A @ ( uminus_uminus @ A @ ( neg_numeral_sub @ A @ M2 @ one2 ) ) @ ( one_one @ A ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) ) ) ).
% minus_sub_one_diff_one
thf(fact_4786_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_4787_diff__numeral__special_I8_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [M2: num] :
( ( minus_minus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ M2 ) ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( neg_numeral_sub @ A @ one2 @ M2 ) ) ) ).
% diff_numeral_special(8)
thf(fact_4788_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_4789_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_4790_card_Ocomp__fun__commute__on,axiom,
( ( comp @ nat @ nat @ nat @ suc @ suc )
= ( comp @ nat @ nat @ nat @ suc @ suc ) ) ).
% card.comp_fun_commute_on
thf(fact_4791_neg__numeral__class_Osub__def,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( neg_numeral_sub @ A )
= ( ^ [K3: num,L2: num] : ( minus_minus @ A @ ( numeral_numeral @ A @ K3 ) @ ( numeral_numeral @ A @ L2 ) ) ) ) ) ).
% neg_numeral_class.sub_def
thf(fact_4792_sum_OatLeast__Suc__atMost__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,M2: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M2 ) @ ( suc @ N ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ).
% sum.atLeast_Suc_atMost_Suc_shift
thf(fact_4793_sum_OatLeast__Suc__lessThan__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,M2: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M2 ) @ ( suc @ N ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) ) ) ).
% sum.atLeast_Suc_lessThan_Suc_shift
thf(fact_4794_prod_OatLeast__Suc__atMost__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,M2: nat,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M2 ) @ ( suc @ N ) ) )
= ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ).
% prod.atLeast_Suc_atMost_Suc_shift
thf(fact_4795_prod_OatLeast__Suc__lessThan__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,M2: nat,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M2 ) @ ( suc @ N ) ) )
= ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G @ suc ) @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) ) ) ).
% prod.atLeast_Suc_lessThan_Suc_shift
thf(fact_4796_summable__inverse__divide,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: nat > A,C2: A] :
( ( summable @ A @ ( comp @ A @ A @ nat @ ( inverse_inverse @ A ) @ F2 ) )
=> ( summable @ A
@ ^ [N5: nat] : ( divide_divide @ A @ C2 @ ( F2 @ N5 ) ) ) ) ) ).
% summable_inverse_divide
thf(fact_4797_sub__non__negative,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: num,M2: num] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( neg_numeral_sub @ A @ N @ M2 ) )
= ( ord_less_eq @ num @ M2 @ N ) ) ) ).
% sub_non_negative
thf(fact_4798_sub__non__positive,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: num,M2: num] :
( ( ord_less_eq @ A @ ( neg_numeral_sub @ A @ N @ M2 ) @ ( zero_zero @ A ) )
= ( ord_less_eq @ num @ N @ M2 ) ) ) ).
% sub_non_positive
thf(fact_4799_sub__negative,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: num,M2: num] :
( ( ord_less @ A @ ( neg_numeral_sub @ A @ N @ M2 ) @ ( zero_zero @ A ) )
= ( ord_less @ num @ N @ M2 ) ) ) ).
% sub_negative
thf(fact_4800_sub__positive,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: num,M2: num] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( neg_numeral_sub @ A @ N @ M2 ) )
= ( ord_less @ num @ M2 @ N ) ) ) ).
% sub_positive
thf(fact_4801_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_4802_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_4803_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_4804_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_4805_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_4806_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_4807_sum_OatLeastLessThan__shift__0,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,M2: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ nat @ A @ nat @ G @ ( plus_plus @ nat @ M2 ) ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ N @ M2 ) ) ) ) ) ).
% sum.atLeastLessThan_shift_0
thf(fact_4808_prod_OatLeastLessThan__shift__0,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,M2: nat,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) )
= ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G @ ( plus_plus @ nat @ M2 ) ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ N @ M2 ) ) ) ) ) ).
% prod.atLeastLessThan_shift_0
thf(fact_4809_sum_OatLeast__atMost__pred__shift,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,M2: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ( comp @ nat @ A @ nat @ G
@ ^ [N5: nat] : ( minus_minus @ nat @ N5 @ ( suc @ ( zero_zero @ nat ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M2 ) @ ( suc @ N ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ).
% sum.atLeast_atMost_pred_shift
thf(fact_4810_sum_OatLeast__lessThan__pred__shift,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: nat > A,M2: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ( comp @ nat @ A @ nat @ G
@ ^ [N5: nat] : ( minus_minus @ nat @ N5 @ ( suc @ ( zero_zero @ nat ) ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M2 ) @ ( suc @ N ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) ) ) ).
% sum.atLeast_lessThan_pred_shift
thf(fact_4811_prod_OatLeast__atMost__pred__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,M2: nat,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A
@ ( comp @ nat @ A @ nat @ G
@ ^ [N5: nat] : ( minus_minus @ nat @ N5 @ ( suc @ ( zero_zero @ nat ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M2 ) @ ( suc @ N ) ) )
= ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ).
% prod.atLeast_atMost_pred_shift
thf(fact_4812_prod_OatLeast__lessThan__pred__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: nat > A,M2: nat,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A
@ ( comp @ nat @ A @ nat @ G
@ ^ [N5: nat] : ( minus_minus @ nat @ N5 @ ( suc @ ( zero_zero @ nat ) ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M2 ) @ ( suc @ N ) ) )
= ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) ) ) ).
% prod.atLeast_lessThan_pred_shift
thf(fact_4813_sum_OatLeast__int__atMost__int__shift,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: int > A,M2: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ int @ A @ G @ ( set_or1337092689740270186AtMost @ int @ ( semiring_1_of_nat @ int @ M2 ) @ ( semiring_1_of_nat @ int @ N ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ int @ A @ nat @ G @ ( semiring_1_of_nat @ int ) ) @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ).
% sum.atLeast_int_atMost_int_shift
thf(fact_4814_prod_OatLeast__int__atMost__int__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: int > A,M2: nat,N: nat] :
( ( groups7121269368397514597t_prod @ int @ A @ G @ ( set_or1337092689740270186AtMost @ int @ ( semiring_1_of_nat @ int @ M2 ) @ ( semiring_1_of_nat @ int @ N ) ) )
= ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ int @ A @ nat @ G @ ( semiring_1_of_nat @ int ) ) @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ).
% prod.atLeast_int_atMost_int_shift
thf(fact_4815_sum_OatLeast__int__lessThan__int__shift,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G: int > A,M2: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ int @ A @ G @ ( set_or7035219750837199246ssThan @ int @ ( semiring_1_of_nat @ int @ M2 ) @ ( semiring_1_of_nat @ int @ N ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ int @ A @ nat @ G @ ( semiring_1_of_nat @ int ) ) @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) ) ) ).
% sum.atLeast_int_lessThan_int_shift
thf(fact_4816_sum_OatLeastAtMost__shift__0,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [M2: nat,N: nat,G: nat > A] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ nat @ A @ nat @ G @ ( plus_plus @ nat @ M2 ) ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ N @ M2 ) ) ) ) ) ) ).
% sum.atLeastAtMost_shift_0
thf(fact_4817_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_4818_prod_OatLeastAtMost__shift__0,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M2: nat,N: nat,G: nat > A] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
= ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G @ ( plus_plus @ nat @ M2 ) ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ N @ M2 ) ) ) ) ) ) ).
% prod.atLeastAtMost_shift_0
thf(fact_4819_prod_OatLeast__int__lessThan__int__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: int > A,M2: nat,N: nat] :
( ( groups7121269368397514597t_prod @ int @ A @ G @ ( set_or7035219750837199246ssThan @ int @ ( semiring_1_of_nat @ int @ M2 ) @ ( semiring_1_of_nat @ int @ N ) ) )
= ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ int @ A @ nat @ G @ ( semiring_1_of_nat @ int ) ) @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) ) ) ).
% prod.atLeast_int_lessThan_int_shift
thf(fact_4820_Code__Numeral_Onegative__def,axiom,
( code_negative
= ( comp @ code_integer @ code_integer @ num @ ( uminus_uminus @ code_integer ) @ ( numeral_numeral @ code_integer ) ) ) ).
% Code_Numeral.negative_def
thf(fact_4821_Code__Target__Int_Onegative__def,axiom,
( code_Target_negative
= ( comp @ int @ int @ num @ ( uminus_uminus @ int ) @ ( numeral_numeral @ int ) ) ) ).
% Code_Target_Int.negative_def
thf(fact_4822_div__add__self1__no__field,axiom,
! [B: $tType,A: $tType] :
( ( ( euclid4440199948858584721cancel @ A )
& ( field @ B ) )
=> ! [X: B,B2: A,A3: A] :
( ( nO_MATCH @ B @ A @ X @ B2 )
=> ( ( B2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ B2 @ A3 ) @ B2 )
= ( plus_plus @ A @ ( divide_divide @ A @ A3 @ B2 ) @ ( one_one @ A ) ) ) ) ) ) ).
% div_add_self1_no_field
thf(fact_4823_div__add__self2__no__field,axiom,
! [B: $tType,A: $tType] :
( ( ( euclid4440199948858584721cancel @ A )
& ( field @ B ) )
=> ! [X: B,B2: A,A3: A] :
( ( nO_MATCH @ B @ A @ X @ B2 )
=> ( ( B2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ A3 @ B2 ) @ B2 )
= ( plus_plus @ A @ ( divide_divide @ A @ A3 @ B2 ) @ ( one_one @ A ) ) ) ) ) ) ).
% div_add_self2_no_field
thf(fact_4824_scale__right__distrib__NO__MATCH,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [X: A,Y2: A,A3: real] :
( ( nO_MATCH @ A @ real @ ( divide_divide @ A @ X @ Y2 ) @ A3 )
=> ( ( real_V8093663219630862766scaleR @ A @ A3 @ ( plus_plus @ A @ X @ Y2 ) )
= ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ A3 @ X ) @ ( real_V8093663219630862766scaleR @ A @ A3 @ Y2 ) ) ) ) ) ).
% scale_right_distrib_NO_MATCH
thf(fact_4825_scale__right__diff__distrib__NO__MATCH,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [X: A,Y2: A,A3: real] :
( ( nO_MATCH @ A @ real @ ( divide_divide @ A @ X @ Y2 ) @ A3 )
=> ( ( real_V8093663219630862766scaleR @ A @ A3 @ ( minus_minus @ A @ X @ Y2 ) )
= ( minus_minus @ A @ ( real_V8093663219630862766scaleR @ A @ A3 @ X ) @ ( real_V8093663219630862766scaleR @ A @ A3 @ Y2 ) ) ) ) ) ).
% scale_right_diff_distrib_NO_MATCH
thf(fact_4826_distrib__right__NO__MATCH,axiom,
! [B: $tType,A: $tType] :
( ( semiring @ A )
=> ! [X: B,Y2: B,C2: A,A3: A,B2: A] :
( ( nO_MATCH @ B @ A @ ( divide_divide @ B @ X @ Y2 ) @ C2 )
=> ( ( times_times @ A @ ( plus_plus @ A @ A3 @ B2 ) @ C2 )
= ( plus_plus @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ) ).
% distrib_right_NO_MATCH
thf(fact_4827_distrib__left__NO__MATCH,axiom,
! [B: $tType,A: $tType] :
( ( semiring @ A )
=> ! [X: B,Y2: B,A3: A,B2: A,C2: A] :
( ( nO_MATCH @ B @ A @ ( divide_divide @ B @ X @ Y2 ) @ A3 )
=> ( ( times_times @ A @ A3 @ ( plus_plus @ A @ B2 @ C2 ) )
= ( plus_plus @ A @ ( times_times @ A @ A3 @ B2 ) @ ( times_times @ A @ A3 @ C2 ) ) ) ) ) ).
% distrib_left_NO_MATCH
thf(fact_4828_left__diff__distrib__NO__MATCH,axiom,
! [B: $tType,A: $tType] :
( ( ring @ A )
=> ! [X: B,Y2: B,C2: A,A3: A,B2: A] :
( ( nO_MATCH @ B @ A @ ( divide_divide @ B @ X @ Y2 ) @ C2 )
=> ( ( times_times @ A @ ( minus_minus @ A @ A3 @ B2 ) @ C2 )
= ( minus_minus @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) ) ) ) ) ).
% left_diff_distrib_NO_MATCH
thf(fact_4829_right__diff__distrib__NO__MATCH,axiom,
! [B: $tType,A: $tType] :
( ( ring @ A )
=> ! [X: B,Y2: B,A3: A,B2: A,C2: A] :
( ( nO_MATCH @ B @ A @ ( divide_divide @ B @ X @ Y2 ) @ A3 )
=> ( ( times_times @ A @ A3 @ ( minus_minus @ A @ B2 @ C2 ) )
= ( minus_minus @ A @ ( times_times @ A @ A3 @ B2 ) @ ( times_times @ A @ A3 @ C2 ) ) ) ) ) ).
% right_diff_distrib_NO_MATCH
thf(fact_4830_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_4831_scale__left__distrib__NO__MATCH,axiom,
! [C: $tType,A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [X: A,Y2: A,C2: C,A3: real,B2: real] :
( ( nO_MATCH @ A @ C @ ( divide_divide @ A @ X @ Y2 ) @ C2 )
=> ( ( real_V8093663219630862766scaleR @ A @ ( plus_plus @ real @ A3 @ B2 ) @ X )
= ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ A3 @ X ) @ ( real_V8093663219630862766scaleR @ A @ B2 @ X ) ) ) ) ) ).
% scale_left_distrib_NO_MATCH
thf(fact_4832_scale__left__diff__distrib__NO__MATCH,axiom,
! [C: $tType,A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [X: A,Y2: A,C2: C,A3: real,B2: real] :
( ( nO_MATCH @ A @ C @ ( divide_divide @ A @ X @ Y2 ) @ C2 )
=> ( ( real_V8093663219630862766scaleR @ A @ ( minus_minus @ real @ A3 @ B2 ) @ X )
= ( minus_minus @ A @ ( real_V8093663219630862766scaleR @ A @ A3 @ X ) @ ( real_V8093663219630862766scaleR @ A @ B2 @ X ) ) ) ) ) ).
% scale_left_diff_distrib_NO_MATCH
thf(fact_4833_horner__sum__eq__sum__funpow,axiom,
! [A: $tType,B: $tType] :
( ( comm_semiring_0 @ A )
=> ( ( groups4207007520872428315er_sum @ B @ A )
= ( ^ [F4: B > A,A5: A,Xs3: list @ B] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [N5: nat] : ( compow @ ( A > A ) @ N5 @ ( times_times @ A @ A5 ) @ ( F4 @ ( nth @ B @ Xs3 @ N5 ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ B ) @ Xs3 ) ) ) ) ) ) ).
% horner_sum_eq_sum_funpow
thf(fact_4834_nat__of__integer__non__positive,axiom,
! [K: code_integer] :
( ( ord_less_eq @ code_integer @ K @ ( zero_zero @ code_integer ) )
=> ( ( code_nat_of_integer @ K )
= ( zero_zero @ nat ) ) ) ).
% nat_of_integer_non_positive
thf(fact_4835_polyfun__rootbound,axiom,
! [A: $tType] :
( ( ( real_V8999393235501362500lgebra @ A )
& ( idom @ A ) )
=> ! [C2: nat > A,K: nat,N: nat] :
( ( ( C2 @ K )
!= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ nat @ K @ N )
=> ( ( finite_finite @ A
@ ( collect @ A
@ ^ [Z6: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( times_times @ A @ ( C2 @ I2 ) @ ( power_power @ A @ Z6 @ I2 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( zero_zero @ A ) ) ) )
& ( ord_less_eq @ nat
@ ( finite_card @ A
@ ( collect @ A
@ ^ [Z6: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( times_times @ A @ ( C2 @ I2 ) @ ( power_power @ A @ Z6 @ I2 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( zero_zero @ A ) ) ) )
@ N ) ) ) ) ) ).
% polyfun_rootbound
thf(fact_4836_card__Collect__less__nat,axiom,
! [N: nat] :
( ( finite_card @ nat
@ ( collect @ nat
@ ^ [I2: nat] : ( ord_less @ nat @ I2 @ N ) ) )
= N ) ).
% card_Collect_less_nat
thf(fact_4837_Suc__funpow,axiom,
! [N: nat] :
( ( compow @ ( nat > nat ) @ N @ suc )
= ( plus_plus @ nat @ N ) ) ).
% Suc_funpow
thf(fact_4838_card__atMost,axiom,
! [U: nat] :
( ( finite_card @ nat @ ( set_ord_atMost @ nat @ U ) )
= ( suc @ U ) ) ).
% card_atMost
thf(fact_4839_funpow__0,axiom,
! [A: $tType,F2: A > A,X: A] :
( ( compow @ ( A > A ) @ ( zero_zero @ nat ) @ F2 @ X )
= X ) ).
% funpow_0
thf(fact_4840_card__atLeastLessThan,axiom,
! [L: nat,U: nat] :
( ( finite_card @ nat @ ( set_or7035219750837199246ssThan @ nat @ L @ U ) )
= ( minus_minus @ nat @ U @ L ) ) ).
% card_atLeastLessThan
thf(fact_4841_card__Collect__le__nat,axiom,
! [N: nat] :
( ( finite_card @ nat
@ ( collect @ nat
@ ^ [I2: nat] : ( ord_less_eq @ nat @ I2 @ N ) ) )
= ( suc @ N ) ) ).
% card_Collect_le_nat
thf(fact_4842_card_Oempty,axiom,
! [A: $tType] :
( ( finite_card @ A @ ( bot_bot @ ( set @ A ) ) )
= ( zero_zero @ nat ) ) ).
% card.empty
thf(fact_4843_card_Oinfinite,axiom,
! [A: $tType,A4: set @ A] :
( ~ ( finite_finite @ A @ A4 )
=> ( ( finite_card @ A @ A4 )
= ( zero_zero @ nat ) ) ) ).
% card.infinite
thf(fact_4844_card__atLeastAtMost,axiom,
! [L: nat,U: nat] :
( ( finite_card @ nat @ ( set_or1337092689740270186AtMost @ nat @ L @ U ) )
= ( minus_minus @ nat @ ( suc @ U ) @ L ) ) ).
% card_atLeastAtMost
thf(fact_4845_card__atLeastLessThan__int,axiom,
! [L: int,U: int] :
( ( finite_card @ int @ ( set_or7035219750837199246ssThan @ int @ L @ U ) )
= ( nat2 @ ( minus_minus @ int @ U @ L ) ) ) ).
% card_atLeastLessThan_int
thf(fact_4846_card__0__eq,axiom,
! [A: $tType,A4: set @ A] :
( ( finite_finite @ A @ A4 )
=> ( ( ( finite_card @ A @ A4 )
= ( zero_zero @ nat ) )
= ( A4
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% card_0_eq
thf(fact_4847_card__insert__disjoint,axiom,
! [A: $tType,A4: set @ A,X: A] :
( ( finite_finite @ A @ A4 )
=> ( ~ ( member @ A @ X @ A4 )
=> ( ( finite_card @ A @ ( insert @ A @ X @ A4 ) )
= ( suc @ ( finite_card @ A @ A4 ) ) ) ) ) ).
% card_insert_disjoint
thf(fact_4848_sum__constant,axiom,
! [B: $tType,A: $tType] :
( ( semiring_1 @ A )
=> ! [Y2: A,A4: set @ B] :
( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X2: B] : Y2
@ A4 )
= ( times_times @ A @ ( semiring_1_of_nat @ A @ ( finite_card @ B @ A4 ) ) @ Y2 ) ) ) ).
% sum_constant
thf(fact_4849_card__Diff__insert,axiom,
! [A: $tType,A3: A,A4: set @ A,B7: set @ A] :
( ( member @ A @ A3 @ A4 )
=> ( ~ ( member @ A @ A3 @ B7 )
=> ( ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ A3 @ B7 ) ) )
= ( minus_minus @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A4 @ B7 ) ) @ ( one_one @ nat ) ) ) ) ) ).
% card_Diff_insert
thf(fact_4850_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_4851_comp__funpow,axiom,
! [B: $tType,A: $tType,N: nat,F2: A > A] :
( ( compow @ ( ( B > A ) > B > A ) @ N @ ( comp @ A @ A @ B @ F2 ) )
= ( comp @ A @ A @ B @ ( compow @ ( A > A ) @ N @ F2 ) ) ) ).
% comp_funpow
thf(fact_4852_funpow__mult,axiom,
! [A: $tType,N: nat,M2: nat,F2: A > A] :
( ( compow @ ( A > A ) @ N @ ( compow @ ( A > A ) @ M2 @ F2 ) )
= ( compow @ ( A > A ) @ ( times_times @ nat @ M2 @ N ) @ F2 ) ) ).
% funpow_mult
thf(fact_4853_funpow__swap1,axiom,
! [A: $tType,F2: A > A,N: nat,X: A] :
( ( F2 @ ( compow @ ( A > A ) @ N @ F2 @ X ) )
= ( compow @ ( A > A ) @ N @ F2 @ ( F2 @ X ) ) ) ).
% funpow_swap1
thf(fact_4854_bij__betw__funpow,axiom,
! [A: $tType,F2: A > A,S3: set @ A,N: nat] :
( ( bij_betw @ A @ A @ F2 @ S3 @ S3 )
=> ( bij_betw @ A @ A @ ( compow @ ( A > A ) @ N @ F2 ) @ S3 @ S3 ) ) ).
% bij_betw_funpow
thf(fact_4855_funpow_Osimps_I2_J,axiom,
! [A: $tType,N: nat,F2: A > A] :
( ( compow @ ( A > A ) @ ( suc @ N ) @ F2 )
= ( comp @ A @ A @ A @ F2 @ ( compow @ ( A > A ) @ N @ F2 ) ) ) ).
% funpow.simps(2)
thf(fact_4856_funpow__Suc__right,axiom,
! [A: $tType,N: nat,F2: A > A] :
( ( compow @ ( A > A ) @ ( suc @ N ) @ F2 )
= ( comp @ A @ A @ A @ ( compow @ ( A > A ) @ N @ F2 ) @ F2 ) ) ).
% funpow_Suc_right
thf(fact_4857_funpow__add,axiom,
! [A: $tType,M2: nat,N: nat,F2: A > A] :
( ( compow @ ( A > A ) @ ( plus_plus @ nat @ M2 @ N ) @ F2 )
= ( comp @ A @ A @ A @ ( compow @ ( A > A ) @ M2 @ F2 ) @ ( compow @ ( A > A ) @ N @ F2 ) ) ) ).
% funpow_add
thf(fact_4858_card__lists__length__eq,axiom,
! [A: $tType,A4: set @ A,N: nat] :
( ( finite_finite @ A @ A4 )
=> ( ( finite_card @ ( list @ A )
@ ( collect @ ( list @ A )
@ ^ [Xs3: list @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs3 ) @ A4 )
& ( ( size_size @ ( list @ A ) @ Xs3 )
= N ) ) ) )
= ( power_power @ nat @ ( finite_card @ A @ A4 ) @ N ) ) ) ).
% card_lists_length_eq
thf(fact_4859_card__eq__sum,axiom,
! [A: $tType] :
( ( finite_card @ A )
= ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [X2: A] : ( one_one @ nat ) ) ) ).
% card_eq_sum
thf(fact_4860_card__eq__0__iff,axiom,
! [A: $tType,A4: set @ A] :
( ( ( finite_card @ A @ A4 )
= ( zero_zero @ nat ) )
= ( ( A4
= ( bot_bot @ ( set @ A ) ) )
| ~ ( finite_finite @ A @ A4 ) ) ) ).
% card_eq_0_iff
thf(fact_4861_card__ge__0__finite,axiom,
! [A: $tType,A4: set @ A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( finite_card @ A @ A4 ) )
=> ( finite_finite @ A @ A4 ) ) ).
% card_ge_0_finite
thf(fact_4862_card__insert__if,axiom,
! [A: $tType,A4: set @ A,X: A] :
( ( finite_finite @ A @ A4 )
=> ( ( ( member @ A @ X @ A4 )
=> ( ( finite_card @ A @ ( insert @ A @ X @ A4 ) )
= ( finite_card @ A @ A4 ) ) )
& ( ~ ( member @ A @ X @ A4 )
=> ( ( finite_card @ A @ ( insert @ A @ X @ A4 ) )
= ( suc @ ( finite_card @ A @ A4 ) ) ) ) ) ) ).
% card_insert_if
thf(fact_4863_card__Suc__eq__finite,axiom,
! [A: $tType,A4: set @ A,K: nat] :
( ( ( finite_card @ A @ A4 )
= ( suc @ K ) )
= ( ? [B3: A,B5: set @ A] :
( ( A4
= ( insert @ A @ B3 @ B5 ) )
& ~ ( member @ A @ B3 @ B5 )
& ( ( finite_card @ A @ B5 )
= K )
& ( finite_finite @ A @ B5 ) ) ) ) ).
% card_Suc_eq_finite
thf(fact_4864_card__less__sym__Diff,axiom,
! [A: $tType,A4: set @ A,B7: set @ A] :
( ( finite_finite @ A @ A4 )
=> ( ( finite_finite @ A @ B7 )
=> ( ( ord_less @ nat @ ( finite_card @ A @ A4 ) @ ( finite_card @ A @ B7 ) )
=> ( ord_less @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A4 @ B7 ) ) @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ B7 @ A4 ) ) ) ) ) ) ).
% card_less_sym_Diff
thf(fact_4865_card__le__sym__Diff,axiom,
! [A: $tType,A4: set @ A,B7: set @ A] :
( ( finite_finite @ A @ A4 )
=> ( ( finite_finite @ A @ B7 )
=> ( ( ord_less_eq @ nat @ ( finite_card @ A @ A4 ) @ ( finite_card @ A @ B7 ) )
=> ( ord_less_eq @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A4 @ B7 ) ) @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ B7 @ A4 ) ) ) ) ) ) ).
% card_le_sym_Diff
thf(fact_4866_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_4867_card__1__singletonE,axiom,
! [A: $tType,A4: set @ A] :
( ( ( finite_card @ A @ A4 )
= ( one_one @ nat ) )
=> ~ ! [X3: A] :
( A4
!= ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% card_1_singletonE
thf(fact_4868_ex__bij__betw__finite__nat,axiom,
! [A: $tType,M10: set @ A] :
( ( finite_finite @ A @ M10 )
=> ? [H3: A > nat] : ( bij_betw @ A @ nat @ H3 @ M10 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( finite_card @ A @ M10 ) ) ) ) ).
% ex_bij_betw_finite_nat
thf(fact_4869_psubset__card__mono,axiom,
! [A: $tType,B7: set @ A,A4: set @ A] :
( ( finite_finite @ A @ B7 )
=> ( ( ord_less @ ( set @ A ) @ A4 @ B7 )
=> ( ord_less @ nat @ ( finite_card @ A @ A4 ) @ ( finite_card @ A @ B7 ) ) ) ) ).
% psubset_card_mono
thf(fact_4870_card__less__Suc2,axiom,
! [M10: set @ nat,I: nat] :
( ~ ( member @ nat @ ( zero_zero @ nat ) @ M10 )
=> ( ( finite_card @ nat
@ ( collect @ nat
@ ^ [K3: nat] :
( ( member @ nat @ ( suc @ K3 ) @ M10 )
& ( ord_less @ nat @ K3 @ I ) ) ) )
= ( finite_card @ nat
@ ( collect @ nat
@ ^ [K3: nat] :
( ( member @ nat @ K3 @ M10 )
& ( ord_less @ nat @ K3 @ ( suc @ I ) ) ) ) ) ) ) ).
% card_less_Suc2
thf(fact_4871_card__less__Suc,axiom,
! [M10: set @ nat,I: nat] :
( ( member @ nat @ ( zero_zero @ nat ) @ M10 )
=> ( ( suc
@ ( finite_card @ nat
@ ( collect @ nat
@ ^ [K3: nat] :
( ( member @ nat @ ( suc @ K3 ) @ M10 )
& ( ord_less @ nat @ K3 @ I ) ) ) ) )
= ( finite_card @ nat
@ ( collect @ nat
@ ^ [K3: nat] :
( ( member @ nat @ K3 @ M10 )
& ( ord_less @ nat @ K3 @ ( suc @ I ) ) ) ) ) ) ) ).
% card_less_Suc
thf(fact_4872_card__less,axiom,
! [M10: set @ nat,I: nat] :
( ( member @ nat @ ( zero_zero @ nat ) @ M10 )
=> ( ( finite_card @ nat
@ ( collect @ nat
@ ^ [K3: nat] :
( ( member @ nat @ K3 @ M10 )
& ( ord_less @ nat @ K3 @ ( suc @ I ) ) ) ) )
!= ( zero_zero @ nat ) ) ) ).
% card_less
thf(fact_4873_card__atLeastZeroLessThan__int,axiom,
! [U: int] :
( ( finite_card @ int @ ( set_or7035219750837199246ssThan @ int @ ( zero_zero @ int ) @ U ) )
= ( nat2 @ U ) ) ).
% card_atLeastZeroLessThan_int
thf(fact_4874_sum__Suc,axiom,
! [A: $tType,F2: A > nat,A4: set @ A] :
( ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [X2: A] : ( suc @ ( F2 @ X2 ) )
@ A4 )
= ( plus_plus @ nat @ ( groups7311177749621191930dd_sum @ A @ nat @ F2 @ A4 ) @ ( finite_card @ A @ A4 ) ) ) ).
% sum_Suc
thf(fact_4875_nat__of__integer__code__post_I1_J,axiom,
( ( code_nat_of_integer @ ( zero_zero @ code_integer ) )
= ( zero_zero @ nat ) ) ).
% nat_of_integer_code_post(1)
thf(fact_4876_real__of__card,axiom,
! [A: $tType,A4: set @ A] :
( ( semiring_1_of_nat @ real @ ( finite_card @ A @ A4 ) )
= ( groups7311177749621191930dd_sum @ A @ real
@ ^ [X2: A] : ( one_one @ real )
@ A4 ) ) ).
% real_of_card
thf(fact_4877_of__nat__def,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( ( semiring_1_of_nat @ A )
= ( ^ [N5: nat] : ( compow @ ( A > A ) @ N5 @ ( plus_plus @ A @ ( one_one @ A ) ) @ ( zero_zero @ A ) ) ) ) ) ).
% of_nat_def
thf(fact_4878_numeral__add__unfold__funpow,axiom,
! [A: $tType] :
( ( semiring_numeral @ A )
=> ! [K: num,A3: A] :
( ( plus_plus @ A @ ( numeral_numeral @ A @ K ) @ A3 )
= ( compow @ ( A > A ) @ ( numeral_numeral @ nat @ K ) @ ( plus_plus @ A @ ( one_one @ A ) ) @ A3 ) ) ) ).
% numeral_add_unfold_funpow
thf(fact_4879_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_4880_sum__bounded__below,axiom,
! [A: $tType,B: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( semiring_1 @ A ) )
=> ! [A4: set @ B,K6: A,F2: B > A] :
( ! [I3: B] :
( ( member @ B @ I3 @ A4 )
=> ( ord_less_eq @ A @ K6 @ ( F2 @ I3 ) ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( finite_card @ B @ A4 ) ) @ K6 ) @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A4 ) ) ) ) ).
% sum_bounded_below
thf(fact_4881_sum__bounded__above,axiom,
! [B: $tType,A: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( semiring_1 @ A ) )
=> ! [A4: set @ B,F2: B > A,K6: A] :
( ! [I3: B] :
( ( member @ B @ I3 @ A4 )
=> ( ord_less_eq @ A @ ( F2 @ I3 ) @ K6 ) )
=> ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A4 ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( finite_card @ B @ A4 ) ) @ K6 ) ) ) ) ).
% sum_bounded_above
thf(fact_4882_card__gt__0__iff,axiom,
! [A: $tType,A4: set @ A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( finite_card @ A @ A4 ) )
= ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
& ( finite_finite @ A @ A4 ) ) ) ).
% card_gt_0_iff
thf(fact_4883_card__1__singleton__iff,axiom,
! [A: $tType,A4: set @ A] :
( ( ( finite_card @ A @ A4 )
= ( suc @ ( zero_zero @ nat ) ) )
= ( ? [X2: A] :
( A4
= ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% card_1_singleton_iff
thf(fact_4884_card__eq__SucD,axiom,
! [A: $tType,A4: set @ A,K: nat] :
( ( ( finite_card @ A @ A4 )
= ( suc @ K ) )
=> ? [B4: A,B8: set @ A] :
( ( A4
= ( insert @ A @ B4 @ B8 ) )
& ~ ( member @ A @ B4 @ B8 )
& ( ( finite_card @ A @ B8 )
= K )
& ( ( K
= ( zero_zero @ nat ) )
=> ( B8
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% card_eq_SucD
thf(fact_4885_card__Suc__eq,axiom,
! [A: $tType,A4: set @ A,K: nat] :
( ( ( finite_card @ A @ A4 )
= ( suc @ K ) )
= ( ? [B3: A,B5: set @ A] :
( ( A4
= ( insert @ A @ B3 @ B5 ) )
& ~ ( member @ A @ B3 @ B5 )
& ( ( finite_card @ A @ B5 )
= K )
& ( ( K
= ( zero_zero @ nat ) )
=> ( B5
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% card_Suc_eq
thf(fact_4886_card__le__Suc0__iff__eq,axiom,
! [A: $tType,A4: set @ A] :
( ( finite_finite @ A @ A4 )
=> ( ( ord_less_eq @ nat @ ( finite_card @ A @ A4 ) @ ( suc @ ( zero_zero @ nat ) ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ A4 )
=> ! [Y5: A] :
( ( member @ A @ Y5 @ A4 )
=> ( X2 = Y5 ) ) ) ) ) ) ).
% card_le_Suc0_iff_eq
thf(fact_4887_card__le__Suc__iff,axiom,
! [A: $tType,N: nat,A4: set @ A] :
( ( ord_less_eq @ nat @ ( suc @ N ) @ ( finite_card @ A @ A4 ) )
= ( ? [A5: A,B5: set @ A] :
( ( A4
= ( insert @ A @ A5 @ B5 ) )
& ~ ( member @ A @ A5 @ B5 )
& ( ord_less_eq @ nat @ N @ ( finite_card @ A @ B5 ) )
& ( finite_finite @ A @ B5 ) ) ) ) ).
% card_le_Suc_iff
thf(fact_4888_card__Diff1__le,axiom,
! [A: $tType,A4: set @ A,X: A] : ( ord_less_eq @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) @ ( finite_card @ A @ A4 ) ) ).
% card_Diff1_le
thf(fact_4889_card__Diff__subset,axiom,
! [A: $tType,B7: set @ A,A4: set @ A] :
( ( finite_finite @ A @ B7 )
=> ( ( ord_less_eq @ ( set @ A ) @ B7 @ A4 )
=> ( ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A4 @ B7 ) )
= ( minus_minus @ nat @ ( finite_card @ A @ A4 ) @ ( finite_card @ A @ B7 ) ) ) ) ) ).
% card_Diff_subset
thf(fact_4890_card__psubset,axiom,
! [A: $tType,B7: set @ A,A4: set @ A] :
( ( finite_finite @ A @ B7 )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ B7 )
=> ( ( ord_less @ nat @ ( finite_card @ A @ A4 ) @ ( finite_card @ A @ B7 ) )
=> ( ord_less @ ( set @ A ) @ A4 @ B7 ) ) ) ) ).
% card_psubset
thf(fact_4891_diff__card__le__card__Diff,axiom,
! [A: $tType,B7: set @ A,A4: set @ A] :
( ( finite_finite @ A @ B7 )
=> ( ord_less_eq @ nat @ ( minus_minus @ nat @ ( finite_card @ A @ A4 ) @ ( finite_card @ A @ B7 ) ) @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A4 @ B7 ) ) ) ) ).
% diff_card_le_card_Diff
thf(fact_4892_card__lists__length__le,axiom,
! [A: $tType,A4: set @ A,N: nat] :
( ( finite_finite @ A @ A4 )
=> ( ( finite_card @ ( list @ A )
@ ( collect @ ( list @ A )
@ ^ [Xs3: list @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs3 ) @ A4 )
& ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs3 ) @ N ) ) ) )
= ( groups7311177749621191930dd_sum @ nat @ nat @ ( power_power @ nat @ ( finite_card @ A @ A4 ) ) @ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% card_lists_length_le
thf(fact_4893_ex__bij__betw__nat__finite,axiom,
! [A: $tType,M10: set @ A] :
( ( finite_finite @ A @ M10 )
=> ? [H3: nat > A] : ( bij_betw @ nat @ A @ H3 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( finite_card @ A @ M10 ) ) @ M10 ) ) ).
% ex_bij_betw_nat_finite
thf(fact_4894_ex__bij__betw__nat__finite__1,axiom,
! [A: $tType,M10: set @ A] :
( ( finite_finite @ A @ M10 )
=> ? [H3: nat > A] : ( bij_betw @ nat @ A @ H3 @ ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ ( finite_card @ A @ M10 ) ) @ M10 ) ) ).
% ex_bij_betw_nat_finite_1
thf(fact_4895_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_4896_subset__eq__atLeast0__lessThan__card,axiom,
! [N6: set @ nat,N: nat] :
( ( ord_less_eq @ ( set @ nat ) @ N6 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) )
=> ( ord_less_eq @ nat @ ( finite_card @ nat @ N6 ) @ N ) ) ).
% subset_eq_atLeast0_lessThan_card
thf(fact_4897_card__sum__le__nat__sum,axiom,
! [S3: set @ nat] :
( ord_less_eq @ nat
@ ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [X2: nat] : X2
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( finite_card @ nat @ S3 ) ) )
@ ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [X2: nat] : X2
@ S3 ) ) ).
% card_sum_le_nat_sum
thf(fact_4898_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_4899_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_4900_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_4901_card__Suc__Diff1,axiom,
! [A: $tType,A4: set @ A,X: A] :
( ( finite_finite @ A @ A4 )
=> ( ( member @ A @ X @ A4 )
=> ( ( suc @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) )
= ( finite_card @ A @ A4 ) ) ) ) ).
% card_Suc_Diff1
thf(fact_4902_card_Oinsert__remove,axiom,
! [A: $tType,A4: set @ A,X: A] :
( ( finite_finite @ A @ A4 )
=> ( ( finite_card @ A @ ( insert @ A @ X @ A4 ) )
= ( suc @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% card.insert_remove
thf(fact_4903_card_Oremove,axiom,
! [A: $tType,A4: set @ A,X: A] :
( ( finite_finite @ A @ A4 )
=> ( ( member @ A @ X @ A4 )
=> ( ( finite_card @ A @ A4 )
= ( suc @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% card.remove
thf(fact_4904_card__Diff1__less,axiom,
! [A: $tType,A4: set @ A,X: A] :
( ( finite_finite @ A @ A4 )
=> ( ( member @ A @ X @ A4 )
=> ( ord_less @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) @ ( finite_card @ A @ A4 ) ) ) ) ).
% card_Diff1_less
thf(fact_4905_card__Diff2__less,axiom,
! [A: $tType,A4: set @ A,X: A,Y2: A] :
( ( finite_finite @ A @ A4 )
=> ( ( member @ A @ X @ A4 )
=> ( ( member @ A @ Y2 @ A4 )
=> ( ord_less @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ ( insert @ A @ Y2 @ ( bot_bot @ ( set @ A ) ) ) ) ) @ ( finite_card @ A @ A4 ) ) ) ) ) ).
% card_Diff2_less
thf(fact_4906_card__Diff1__less__iff,axiom,
! [A: $tType,A4: set @ A,X: A] :
( ( ord_less @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) @ ( finite_card @ A @ A4 ) )
= ( ( finite_finite @ A @ A4 )
& ( member @ A @ X @ A4 ) ) ) ).
% card_Diff1_less_iff
thf(fact_4907_card__Diff__singleton,axiom,
! [A: $tType,X: A,A4: set @ A] :
( ( member @ A @ X @ A4 )
=> ( ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( minus_minus @ nat @ ( finite_card @ A @ A4 ) @ ( one_one @ nat ) ) ) ) ).
% card_Diff_singleton
thf(fact_4908_card__Diff__singleton__if,axiom,
! [A: $tType,X: A,A4: set @ A] :
( ( ( member @ A @ X @ A4 )
=> ( ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( minus_minus @ nat @ ( finite_card @ A @ A4 ) @ ( one_one @ nat ) ) ) )
& ( ~ ( member @ A @ X @ A4 )
=> ( ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( finite_card @ A @ A4 ) ) ) ) ).
% card_Diff_singleton_if
thf(fact_4909_prod__le__power,axiom,
! [B: $tType,A: $tType] :
( ( linordered_semidom @ A )
=> ! [A4: set @ B,F2: B > A,N: A,K: nat] :
( ! [I3: B] :
( ( member @ B @ I3 @ A4 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F2 @ I3 ) )
& ( ord_less_eq @ A @ ( F2 @ I3 ) @ N ) ) )
=> ( ( ord_less_eq @ nat @ ( finite_card @ B @ A4 ) @ K )
=> ( ( ord_less_eq @ A @ ( one_one @ A ) @ N )
=> ( ord_less_eq @ A @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A4 ) @ ( power_power @ A @ N @ K ) ) ) ) ) ) ).
% prod_le_power
thf(fact_4910_sum__bounded__above__divide,axiom,
! [B: $tType,A: $tType] :
( ( linordered_field @ A )
=> ! [A4: set @ B,F2: B > A,K6: A] :
( ! [I3: B] :
( ( member @ B @ I3 @ A4 )
=> ( ord_less_eq @ A @ ( F2 @ I3 ) @ ( divide_divide @ A @ K6 @ ( semiring_1_of_nat @ A @ ( finite_card @ B @ A4 ) ) ) ) )
=> ( ( finite_finite @ B @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A4 ) @ K6 ) ) ) ) ) ).
% sum_bounded_above_divide
thf(fact_4911_sum__bounded__above__strict,axiom,
! [B: $tType,A: $tType] :
( ( ( ordere8940638589300402666id_add @ A )
& ( semiring_1 @ A ) )
=> ! [A4: set @ B,F2: B > A,K6: A] :
( ! [I3: B] :
( ( member @ B @ I3 @ A4 )
=> ( ord_less @ A @ ( F2 @ I3 ) @ K6 ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( finite_card @ B @ A4 ) )
=> ( ord_less @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A4 ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( finite_card @ B @ A4 ) ) @ K6 ) ) ) ) ) ).
% sum_bounded_above_strict
thf(fact_4912_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_4913_polyfun__roots__card,axiom,
! [A: $tType] :
( ( ( real_V8999393235501362500lgebra @ A )
& ( idom @ A ) )
=> ! [C2: nat > A,K: nat,N: nat] :
( ( ( C2 @ K )
!= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ nat @ K @ N )
=> ( ord_less_eq @ nat
@ ( finite_card @ A
@ ( collect @ A
@ ^ [Z6: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( times_times @ A @ ( C2 @ I2 ) @ ( power_power @ A @ Z6 @ I2 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( zero_zero @ A ) ) ) )
@ N ) ) ) ) ).
% polyfun_roots_card
thf(fact_4914_prod__gen__delta,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S3: set @ B,A3: B,B2: B > A,C2: A] :
( ( finite_finite @ B @ S3 )
=> ( ( ( member @ B @ A3 @ S3 )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K3: B] : ( if @ A @ ( K3 = A3 ) @ ( B2 @ K3 ) @ C2 )
@ S3 )
= ( times_times @ A @ ( B2 @ A3 ) @ ( power_power @ A @ C2 @ ( minus_minus @ nat @ ( finite_card @ B @ S3 ) @ ( one_one @ nat ) ) ) ) ) )
& ( ~ ( member @ B @ A3 @ S3 )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K3: B] : ( if @ A @ ( K3 = A3 ) @ ( B2 @ K3 ) @ C2 )
@ S3 )
= ( power_power @ A @ C2 @ ( finite_card @ B @ S3 ) ) ) ) ) ) ) ).
% prod_gen_delta
thf(fact_4915_nat__of__integer__code,axiom,
( code_nat_of_integer
= ( ^ [K3: code_integer] :
( if @ nat @ ( ord_less_eq @ code_integer @ K3 @ ( zero_zero @ code_integer ) ) @ ( zero_zero @ nat )
@ ( product_case_prod @ code_integer @ code_integer @ nat
@ ^ [L2: code_integer,J3: code_integer] :
( if @ nat
@ ( J3
= ( zero_zero @ code_integer ) )
@ ( plus_plus @ nat @ ( code_nat_of_integer @ L2 ) @ ( code_nat_of_integer @ L2 ) )
@ ( plus_plus @ nat @ ( plus_plus @ nat @ ( code_nat_of_integer @ L2 ) @ ( code_nat_of_integer @ L2 ) ) @ ( one_one @ nat ) ) )
@ ( code_divmod_integer @ K3 @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% nat_of_integer_code
thf(fact_4916_card__lists__distinct__length__eq,axiom,
! [A: $tType,A4: set @ A,K: nat] :
( ( finite_finite @ A @ A4 )
=> ( ( ord_less_eq @ nat @ K @ ( finite_card @ A @ A4 ) )
=> ( ( finite_card @ ( list @ A )
@ ( collect @ ( list @ A )
@ ^ [Xs3: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs3 )
= K )
& ( distinct @ A @ Xs3 )
& ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs3 ) @ A4 ) ) ) )
= ( groups7121269368397514597t_prod @ nat @ nat
@ ^ [X2: nat] : X2
@ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ ( minus_minus @ nat @ ( finite_card @ A @ A4 ) @ K ) @ ( one_one @ nat ) ) @ ( finite_card @ A @ A4 ) ) ) ) ) ) ).
% card_lists_distinct_length_eq
thf(fact_4917_relpowp__fun__conv,axiom,
! [A: $tType] :
( ( compow @ ( A > A > $o ) )
= ( ^ [N5: nat,P3: A > A > $o,X2: A,Y5: A] :
? [F4: nat > A] :
( ( ( F4 @ ( zero_zero @ nat ) )
= X2 )
& ( ( F4 @ N5 )
= Y5 )
& ! [I2: nat] :
( ( ord_less @ nat @ I2 @ N5 )
=> ( P3 @ ( F4 @ I2 ) @ ( F4 @ ( suc @ I2 ) ) ) ) ) ) ) ).
% relpowp_fun_conv
thf(fact_4918_relpowp__1,axiom,
! [A: $tType,P: A > A > $o] :
( ( compow @ ( A > A > $o ) @ ( one_one @ nat ) @ P )
= P ) ).
% relpowp_1
thf(fact_4919_finite__lists__distinct__length__eq,axiom,
! [A: $tType,A4: set @ A,N: nat] :
( ( finite_finite @ A @ A4 )
=> ( finite_finite @ ( list @ A )
@ ( collect @ ( list @ A )
@ ^ [Xs3: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs3 )
= N )
& ( distinct @ A @ Xs3 )
& ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs3 ) @ A4 ) ) ) ) ) ).
% finite_lists_distinct_length_eq
thf(fact_4920_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_4921_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_4922_distinct__conv__nth,axiom,
! [A: $tType] :
( ( distinct @ A )
= ( ^ [Xs3: list @ A] :
! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs3 ) )
=> ! [J3: nat] :
( ( ord_less @ nat @ J3 @ ( size_size @ ( list @ A ) @ Xs3 ) )
=> ( ( I2 != J3 )
=> ( ( nth @ A @ Xs3 @ I2 )
!= ( nth @ A @ Xs3 @ J3 ) ) ) ) ) ) ) ).
% distinct_conv_nth
thf(fact_4923_nth__eq__iff__index__eq,axiom,
! [A: $tType,Xs: list @ A,I: nat,J2: nat] :
( ( distinct @ A @ Xs )
=> ( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ord_less @ nat @ J2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ( nth @ A @ Xs @ I )
= ( nth @ A @ Xs @ J2 ) )
= ( I = J2 ) ) ) ) ) ).
% nth_eq_iff_index_eq
thf(fact_4924_relpowp__Suc__E,axiom,
! [A: $tType,N: nat,P: A > A > $o,X: A,Z2: A] :
( ( compow @ ( A > A > $o ) @ ( suc @ N ) @ P @ X @ Z2 )
=> ~ ! [Y3: A] :
( ( compow @ ( A > A > $o ) @ N @ P @ X @ Y3 )
=> ~ ( P @ Y3 @ Z2 ) ) ) ).
% relpowp_Suc_E
thf(fact_4925_relpowp__Suc__I,axiom,
! [A: $tType,N: nat,P: A > A > $o,X: A,Y2: A,Z2: A] :
( ( compow @ ( A > A > $o ) @ N @ P @ X @ Y2 )
=> ( ( P @ Y2 @ Z2 )
=> ( compow @ ( A > A > $o ) @ ( suc @ N ) @ P @ X @ Z2 ) ) ) ).
% relpowp_Suc_I
thf(fact_4926_relpowp__Suc__D2,axiom,
! [A: $tType,N: nat,P: A > A > $o,X: A,Z2: A] :
( ( compow @ ( A > A > $o ) @ ( suc @ N ) @ P @ X @ Z2 )
=> ? [Y3: A] :
( ( P @ X @ Y3 )
& ( compow @ ( A > A > $o ) @ N @ P @ Y3 @ Z2 ) ) ) ).
% relpowp_Suc_D2
thf(fact_4927_relpowp__Suc__E2,axiom,
! [A: $tType,N: nat,P: A > A > $o,X: A,Z2: A] :
( ( compow @ ( A > A > $o ) @ ( suc @ N ) @ P @ X @ Z2 )
=> ~ ! [Y3: A] :
( ( P @ X @ Y3 )
=> ~ ( compow @ ( A > A > $o ) @ N @ P @ Y3 @ Z2 ) ) ) ).
% relpowp_Suc_E2
thf(fact_4928_relpowp__Suc__I2,axiom,
! [A: $tType,P: A > A > $o,X: A,Y2: A,N: nat,Z2: A] :
( ( P @ X @ Y2 )
=> ( ( compow @ ( A > A > $o ) @ N @ P @ Y2 @ Z2 )
=> ( compow @ ( A > A > $o ) @ ( suc @ N ) @ P @ X @ Z2 ) ) ) ).
% relpowp_Suc_I2
thf(fact_4929_relpowp__0__I,axiom,
! [A: $tType,P: A > A > $o,X: A] : ( compow @ ( A > A > $o ) @ ( zero_zero @ nat ) @ P @ X @ X ) ).
% relpowp_0_I
thf(fact_4930_relpowp__0__E,axiom,
! [A: $tType,P: A > A > $o,X: A,Y2: A] :
( ( compow @ ( A > A > $o ) @ ( zero_zero @ nat ) @ P @ X @ Y2 )
=> ( X = Y2 ) ) ).
% relpowp_0_E
thf(fact_4931_relpowp_Osimps_I1_J,axiom,
! [A: $tType,R2: A > A > $o] :
( ( compow @ ( A > A > $o ) @ ( zero_zero @ nat ) @ R2 )
= ( ^ [Y4: A,Z: A] : Y4 = Z ) ) ).
% relpowp.simps(1)
thf(fact_4932_distinct__Ex1,axiom,
! [A: $tType,Xs: list @ A,X: A] :
( ( distinct @ A @ Xs )
=> ( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ? [X3: nat] :
( ( ord_less @ nat @ X3 @ ( size_size @ ( list @ A ) @ Xs ) )
& ( ( nth @ A @ Xs @ X3 )
= X )
& ! [Y: nat] :
( ( ( ord_less @ nat @ Y @ ( size_size @ ( list @ A ) @ Xs ) )
& ( ( nth @ A @ Xs @ Y )
= X ) )
=> ( Y = X3 ) ) ) ) ) ).
% distinct_Ex1
thf(fact_4933_bij__betw__nth,axiom,
! [A: $tType,Xs: list @ A,A4: set @ nat,B7: set @ A] :
( ( distinct @ A @ Xs )
=> ( ( A4
= ( set_ord_lessThan @ nat @ ( size_size @ ( list @ A ) @ Xs ) ) )
=> ( ( B7
= ( set2 @ A @ Xs ) )
=> ( bij_betw @ nat @ A @ ( nth @ A @ Xs ) @ A4 @ B7 ) ) ) ) ).
% bij_betw_nth
thf(fact_4934_relpowp__E2,axiom,
! [A: $tType,N: nat,P: A > A > $o,X: A,Z2: A] :
( ( compow @ ( A > A > $o ) @ N @ P @ X @ Z2 )
=> ( ( ( N
= ( zero_zero @ nat ) )
=> ( X != Z2 ) )
=> ~ ! [Y3: A,M3: nat] :
( ( N
= ( suc @ M3 ) )
=> ( ( P @ X @ Y3 )
=> ~ ( compow @ ( A > A > $o ) @ M3 @ P @ Y3 @ Z2 ) ) ) ) ) ).
% relpowp_E2
thf(fact_4935_relpowp__E,axiom,
! [A: $tType,N: nat,P: A > A > $o,X: A,Z2: A] :
( ( compow @ ( A > A > $o ) @ N @ P @ X @ Z2 )
=> ( ( ( N
= ( zero_zero @ nat ) )
=> ( X != Z2 ) )
=> ~ ! [Y3: A,M3: nat] :
( ( N
= ( suc @ M3 ) )
=> ( ( compow @ ( A > A > $o ) @ M3 @ P @ X @ Y3 )
=> ~ ( P @ Y3 @ Z2 ) ) ) ) ) ).
% relpowp_E
thf(fact_4936_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_4937_card__lists__distinct__length__eq_H,axiom,
! [A: $tType,K: nat,A4: set @ A] :
( ( ord_less @ nat @ K @ ( finite_card @ A @ A4 ) )
=> ( ( finite_card @ ( list @ A )
@ ( collect @ ( list @ A )
@ ^ [Xs3: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs3 )
= K )
& ( distinct @ A @ Xs3 )
& ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs3 ) @ A4 ) ) ) )
= ( groups7121269368397514597t_prod @ nat @ nat
@ ^ [X2: nat] : X2
@ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ ( minus_minus @ nat @ ( finite_card @ A @ A4 ) @ K ) @ ( one_one @ nat ) ) @ ( finite_card @ A @ A4 ) ) ) ) ) ).
% card_lists_distinct_length_eq'
thf(fact_4938_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_4939_Nat_Ofunpow__code__def,axiom,
! [A: $tType] :
( ( funpow @ A )
= ( compow @ ( A > A ) ) ) ).
% Nat.funpow_code_def
thf(fact_4940_finite__enumerate,axiom,
! [S3: set @ nat] :
( ( finite_finite @ nat @ S3 )
=> ? [R3: nat > nat] :
( ( strict_mono_on @ nat @ nat @ R3 @ ( set_ord_lessThan @ nat @ ( finite_card @ nat @ S3 ) ) )
& ! [N4: nat] :
( ( ord_less @ nat @ N4 @ ( finite_card @ nat @ S3 ) )
=> ( member @ nat @ ( R3 @ N4 ) @ S3 ) ) ) ) ).
% finite_enumerate
thf(fact_4941_length__list__update,axiom,
! [A: $tType,Xs: list @ A,I: nat,X: A] :
( ( size_size @ ( list @ A ) @ ( list_update @ A @ Xs @ I @ X ) )
= ( size_size @ ( list @ A ) @ Xs ) ) ).
% length_list_update
thf(fact_4942_list__update__beyond,axiom,
! [A: $tType,Xs: list @ A,I: nat,X: A] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ I )
=> ( ( list_update @ A @ Xs @ I @ X )
= Xs ) ) ).
% list_update_beyond
thf(fact_4943_nth__list__update__eq,axiom,
! [A: $tType,I: nat,Xs: list @ A,X: A] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ A @ ( list_update @ A @ Xs @ I @ X ) @ I )
= X ) ) ).
% nth_list_update_eq
thf(fact_4944_set__swap,axiom,
! [A: $tType,I: nat,Xs: list @ A,J2: nat] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ord_less @ nat @ J2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( set2 @ A @ ( list_update @ A @ ( list_update @ A @ Xs @ I @ ( nth @ A @ Xs @ J2 ) ) @ J2 @ ( nth @ A @ Xs @ I ) ) )
= ( set2 @ A @ Xs ) ) ) ) ).
% set_swap
thf(fact_4945_distinct__swap,axiom,
! [A: $tType,I: nat,Xs: list @ A,J2: nat] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ord_less @ nat @ J2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( distinct @ A @ ( list_update @ A @ ( list_update @ A @ Xs @ I @ ( nth @ A @ Xs @ J2 ) ) @ J2 @ ( nth @ A @ Xs @ I ) ) )
= ( distinct @ A @ Xs ) ) ) ) ).
% distinct_swap
thf(fact_4946_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_4947_nth__list__update,axiom,
! [A: $tType,I: nat,Xs: list @ A,J2: nat,X: A] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ( I = J2 )
=> ( ( nth @ A @ ( list_update @ A @ Xs @ I @ X ) @ J2 )
= X ) )
& ( ( I != J2 )
=> ( ( nth @ A @ ( list_update @ A @ Xs @ I @ X ) @ J2 )
= ( nth @ A @ Xs @ J2 ) ) ) ) ) ).
% nth_list_update
thf(fact_4948_list__update__same__conv,axiom,
! [A: $tType,I: nat,Xs: list @ A,X: A] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ( list_update @ A @ Xs @ I @ X )
= Xs )
= ( ( nth @ A @ Xs @ I )
= X ) ) ) ).
% list_update_same_conv
thf(fact_4949_distinct__list__update,axiom,
! [A: $tType,Xs: list @ A,A3: A,I: nat] :
( ( distinct @ A @ Xs )
=> ( ~ ( member @ A @ A3 @ ( minus_minus @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( insert @ A @ ( nth @ A @ Xs @ I ) @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( distinct @ A @ ( list_update @ A @ Xs @ I @ A3 ) ) ) ) ).
% distinct_list_update
thf(fact_4950_strict__mono__on__def,axiom,
! [B: $tType,A: $tType] :
( ( ( ord @ A )
& ( ord @ B ) )
=> ( ( strict_mono_on @ A @ B )
= ( ^ [F4: A > B,A7: set @ A] :
! [R: A,S5: A] :
( ( ( member @ A @ R @ A7 )
& ( member @ A @ S5 @ A7 )
& ( ord_less @ A @ R @ S5 ) )
=> ( ord_less @ B @ ( F4 @ R ) @ ( F4 @ S5 ) ) ) ) ) ) ).
% strict_mono_on_def
thf(fact_4951_strict__mono__onI,axiom,
! [B: $tType,A: $tType] :
( ( ( ord @ A )
& ( ord @ B ) )
=> ! [A4: set @ A,F2: A > B] :
( ! [R3: A,S2: A] :
( ( member @ A @ R3 @ A4 )
=> ( ( member @ A @ S2 @ A4 )
=> ( ( ord_less @ A @ R3 @ S2 )
=> ( ord_less @ B @ ( F2 @ R3 ) @ ( F2 @ S2 ) ) ) ) )
=> ( strict_mono_on @ A @ B @ F2 @ A4 ) ) ) ).
% strict_mono_onI
thf(fact_4952_strict__mono__onD,axiom,
! [B: $tType,A: $tType] :
( ( ( ord @ A )
& ( ord @ B ) )
=> ! [F2: A > B,A4: set @ A,R4: A,S: A] :
( ( strict_mono_on @ A @ B @ F2 @ A4 )
=> ( ( member @ A @ R4 @ A4 )
=> ( ( member @ A @ S @ A4 )
=> ( ( ord_less @ A @ R4 @ S )
=> ( ord_less @ B @ ( F2 @ R4 ) @ ( F2 @ S ) ) ) ) ) ) ) ).
% strict_mono_onD
thf(fact_4953_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_4954_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_4955_max__nat_Osemilattice__neutr__order__axioms,axiom,
( semila1105856199041335345_order @ nat @ ( ord_max @ nat ) @ ( zero_zero @ nat )
@ ^ [X2: nat,Y5: nat] : ( ord_less_eq @ nat @ Y5 @ X2 )
@ ^ [X2: nat,Y5: nat] : ( ord_less @ nat @ Y5 @ X2 ) ) ).
% max_nat.semilattice_neutr_order_axioms
thf(fact_4956_greaterThanLessThan__iff,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [I: A,L: A,U: A] :
( ( member @ A @ I @ ( set_or5935395276787703475ssThan @ A @ L @ U ) )
= ( ( ord_less @ A @ L @ I )
& ( ord_less @ A @ I @ U ) ) ) ) ).
% greaterThanLessThan_iff
thf(fact_4957_infinite__Ioo__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A3: A,B2: A] :
( ( ~ ( finite_finite @ A @ ( set_or5935395276787703475ssThan @ A @ A3 @ B2 ) ) )
= ( ord_less @ A @ A3 @ B2 ) ) ) ).
% infinite_Ioo_iff
thf(fact_4958_semilattice__neutr__order_Oneutr__eq__iff,axiom,
! [A: $tType,F2: A > A > A,Z2: A,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B2: A] :
( ( semila1105856199041335345_order @ A @ F2 @ Z2 @ Less_eq @ Less )
=> ( ( Z2
= ( F2 @ A3 @ B2 ) )
= ( ( A3 = Z2 )
& ( B2 = Z2 ) ) ) ) ).
% semilattice_neutr_order.neutr_eq_iff
thf(fact_4959_semilattice__neutr__order_Oeq__neutr__iff,axiom,
! [A: $tType,F2: A > A > A,Z2: A,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B2: A] :
( ( semila1105856199041335345_order @ A @ F2 @ Z2 @ Less_eq @ Less )
=> ( ( ( F2 @ A3 @ B2 )
= Z2 )
= ( ( A3 = Z2 )
& ( B2 = Z2 ) ) ) ) ).
% semilattice_neutr_order.eq_neutr_iff
thf(fact_4960_infinite__Ioo,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ~ ( finite_finite @ A @ ( set_or5935395276787703475ssThan @ A @ A3 @ B2 ) ) ) ) ).
% infinite_Ioo
thf(fact_4961_greaterThanLessThan__subseteq__greaterThanLessThan,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A3: A,B2: A,C2: A,D2: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ A3 @ B2 ) @ ( set_or5935395276787703475ssThan @ A @ C2 @ D2 ) )
= ( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ A3 )
& ( ord_less_eq @ A @ B2 @ D2 ) ) ) ) ) ).
% greaterThanLessThan_subseteq_greaterThanLessThan
thf(fact_4962_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_4963_greaterThanLessThan__subseteq__atLeastAtMost__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A3: A,B2: A,C2: A,D2: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ A3 @ B2 ) @ ( set_or1337092689740270186AtMost @ A @ C2 @ D2 ) )
= ( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ A3 )
& ( ord_less_eq @ A @ B2 @ D2 ) ) ) ) ) ).
% greaterThanLessThan_subseteq_atLeastAtMost_iff
thf(fact_4964_greaterThanLessThan__subseteq__atLeastLessThan__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A3: A,B2: A,C2: A,D2: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ A3 @ B2 ) @ ( set_or7035219750837199246ssThan @ A @ C2 @ D2 ) )
= ( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ A3 )
& ( ord_less_eq @ A @ B2 @ D2 ) ) ) ) ) ).
% greaterThanLessThan_subseteq_atLeastLessThan_iff
thf(fact_4965_atLeastAtMost__diff__ends,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A] :
( ( minus_minus @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) @ ( insert @ A @ A3 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( set_or5935395276787703475ssThan @ A @ A3 @ B2 ) ) ) ).
% atLeastAtMost_diff_ends
thf(fact_4966_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_4967_gcd__nat_Osemilattice__neutr__order__axioms,axiom,
( semila1105856199041335345_order @ nat @ ( gcd_gcd @ nat ) @ ( zero_zero @ nat ) @ ( dvd_dvd @ nat )
@ ^ [M5: nat,N5: nat] :
( ( dvd_dvd @ nat @ M5 @ N5 )
& ( M5 != N5 ) ) ) ).
% gcd_nat.semilattice_neutr_order_axioms
thf(fact_4968_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_4969_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_4970_int__of__integer__code,axiom,
( code_int_of_integer
= ( ^ [K3: code_integer] :
( if @ int @ ( ord_less @ code_integer @ K3 @ ( zero_zero @ code_integer ) ) @ ( uminus_uminus @ int @ ( code_int_of_integer @ ( uminus_uminus @ code_integer @ K3 ) ) )
@ ( if @ int
@ ( K3
= ( zero_zero @ code_integer ) )
@ ( zero_zero @ int )
@ ( product_case_prod @ code_integer @ code_integer @ int
@ ^ [L2: code_integer,J3: code_integer] :
( if @ int
@ ( J3
= ( zero_zero @ code_integer ) )
@ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( code_int_of_integer @ L2 ) )
@ ( plus_plus @ int @ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( code_int_of_integer @ L2 ) ) @ ( one_one @ int ) ) )
@ ( code_divmod_integer @ K3 @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% int_of_integer_code
thf(fact_4971_int__of__integer__of__nat,axiom,
! [N: nat] :
( ( code_int_of_integer @ ( semiring_1_of_nat @ code_integer @ N ) )
= ( semiring_1_of_nat @ int @ N ) ) ).
% int_of_integer_of_nat
thf(fact_4972_zero__integer_Orep__eq,axiom,
( ( code_int_of_integer @ ( zero_zero @ code_integer ) )
= ( zero_zero @ int ) ) ).
% zero_integer.rep_eq
thf(fact_4973_card__greaterThanLessThan,axiom,
! [L: nat,U: nat] :
( ( finite_card @ nat @ ( set_or5935395276787703475ssThan @ nat @ L @ U ) )
= ( minus_minus @ nat @ U @ ( suc @ L ) ) ) ).
% card_greaterThanLessThan
thf(fact_4974_uminus__integer_Orep__eq,axiom,
! [X: code_integer] :
( ( code_int_of_integer @ ( uminus_uminus @ code_integer @ X ) )
= ( uminus_uminus @ int @ ( code_int_of_integer @ X ) ) ) ).
% uminus_integer.rep_eq
thf(fact_4975_one__integer_Orep__eq,axiom,
( ( code_int_of_integer @ ( one_one @ code_integer ) )
= ( one_one @ int ) ) ).
% one_integer.rep_eq
thf(fact_4976_minus__integer_Orep__eq,axiom,
! [X: code_integer,Xa: code_integer] :
( ( code_int_of_integer @ ( minus_minus @ code_integer @ X @ Xa ) )
= ( minus_minus @ int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa ) ) ) ).
% minus_integer.rep_eq
thf(fact_4977_divide__integer_Orep__eq,axiom,
! [X: code_integer,Xa: code_integer] :
( ( code_int_of_integer @ ( divide_divide @ code_integer @ X @ Xa ) )
= ( divide_divide @ int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa ) ) ) ).
% divide_integer.rep_eq
thf(fact_4978_less__integer_Orep__eq,axiom,
( ( ord_less @ code_integer )
= ( ^ [X2: code_integer,Xa4: code_integer] : ( ord_less @ int @ ( code_int_of_integer @ X2 ) @ ( code_int_of_integer @ Xa4 ) ) ) ) ).
% less_integer.rep_eq
thf(fact_4979_integer__less__iff,axiom,
( ( ord_less @ code_integer )
= ( ^ [K3: code_integer,L2: code_integer] : ( ord_less @ int @ ( code_int_of_integer @ K3 ) @ ( code_int_of_integer @ L2 ) ) ) ) ).
% integer_less_iff
thf(fact_4980_atLeastSucLessThan__greaterThanLessThan,axiom,
! [L: nat,U: nat] :
( ( set_or7035219750837199246ssThan @ nat @ ( suc @ L ) @ U )
= ( set_or5935395276787703475ssThan @ nat @ L @ U ) ) ).
% atLeastSucLessThan_greaterThanLessThan
thf(fact_4981_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_4982_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_4983_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_4984_arg__min__if__finite_I2_J,axiom,
! [B: $tType,A: $tType] :
( ( order @ B )
=> ! [S3: set @ A,F2: A > B] :
( ( finite_finite @ A @ S3 )
=> ( ( S3
!= ( bot_bot @ ( set @ A ) ) )
=> ~ ? [X4: A] :
( ( member @ A @ X4 @ S3 )
& ( ord_less @ B @ ( F2 @ X4 ) @ ( F2 @ ( lattic7623131987881927897min_on @ A @ B @ F2 @ S3 ) ) ) ) ) ) ) ).
% arg_min_if_finite(2)
thf(fact_4985_times__int_Oabs__eq,axiom,
! [Xa: product_prod @ nat @ nat,X: product_prod @ nat @ nat] :
( ( times_times @ int @ ( abs_Integ @ Xa ) @ ( abs_Integ @ X ) )
= ( abs_Integ
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X2: nat,Y5: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V5: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ X2 @ U2 ) @ ( times_times @ nat @ Y5 @ V5 ) ) @ ( plus_plus @ nat @ ( times_times @ nat @ X2 @ V5 ) @ ( times_times @ nat @ Y5 @ U2 ) ) ) )
@ Xa
@ X ) ) ) ).
% times_int.abs_eq
thf(fact_4986_Gcd__remove0__nat,axiom,
! [M10: set @ nat] :
( ( finite_finite @ nat @ M10 )
=> ( ( gcd_Gcd @ nat @ M10 )
= ( gcd_Gcd @ nat @ ( minus_minus @ ( set @ nat ) @ M10 @ ( insert @ nat @ ( zero_zero @ nat ) @ ( bot_bot @ ( set @ nat ) ) ) ) ) ) ) ).
% Gcd_remove0_nat
thf(fact_4987_Gcd__empty,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ( ( gcd_Gcd @ A @ ( bot_bot @ ( set @ A ) ) )
= ( zero_zero @ A ) ) ) ).
% Gcd_empty
thf(fact_4988_Gcd__0__iff,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A4: set @ A] :
( ( ( gcd_Gcd @ A @ A4 )
= ( zero_zero @ A ) )
= ( ord_less_eq @ ( set @ A ) @ A4 @ ( insert @ A @ ( zero_zero @ A ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% Gcd_0_iff
thf(fact_4989_int_Oabs__induct,axiom,
! [P: int > $o,X: int] :
( ! [Y3: product_prod @ nat @ nat] : ( P @ ( abs_Integ @ Y3 ) )
=> ( P @ X ) ) ).
% int.abs_induct
thf(fact_4990_Gcd__1,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A4: set @ A] :
( ( member @ A @ ( one_one @ A ) @ A4 )
=> ( ( gcd_Gcd @ A @ A4 )
= ( one_one @ A ) ) ) ) ).
% Gcd_1
thf(fact_4991_Gcd__nat__eq__one,axiom,
! [N6: set @ nat] :
( ( member @ nat @ ( one_one @ nat ) @ N6 )
=> ( ( gcd_Gcd @ nat @ N6 )
= ( one_one @ nat ) ) ) ).
% Gcd_nat_eq_one
thf(fact_4992_eq__Abs__Integ,axiom,
! [Z2: int] :
~ ! [X3: nat,Y3: nat] :
( Z2
!= ( abs_Integ @ ( product_Pair @ nat @ nat @ X3 @ Y3 ) ) ) ).
% eq_Abs_Integ
thf(fact_4993_Gcd__eq__1__I,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A3: A,A4: set @ A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( member @ A @ A3 @ A4 )
=> ( ( gcd_Gcd @ A @ A4 )
= ( one_one @ A ) ) ) ) ) ).
% Gcd_eq_1_I
thf(fact_4994_nat_Oabs__eq,axiom,
! [X: product_prod @ nat @ nat] :
( ( nat2 @ ( abs_Integ @ X ) )
= ( product_case_prod @ nat @ nat @ nat @ ( minus_minus @ nat ) @ X ) ) ).
% nat.abs_eq
thf(fact_4995_zero__int__def,axiom,
( ( zero_zero @ int )
= ( abs_Integ @ ( product_Pair @ nat @ nat @ ( zero_zero @ nat ) @ ( zero_zero @ nat ) ) ) ) ).
% zero_int_def
thf(fact_4996_int__def,axiom,
( ( semiring_1_of_nat @ int )
= ( ^ [N5: nat] : ( abs_Integ @ ( product_Pair @ nat @ nat @ N5 @ ( zero_zero @ nat ) ) ) ) ) ).
% int_def
thf(fact_4997_uminus__int_Oabs__eq,axiom,
! [X: product_prod @ nat @ nat] :
( ( uminus_uminus @ int @ ( abs_Integ @ X ) )
= ( abs_Integ
@ ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [X2: nat,Y5: nat] : ( product_Pair @ nat @ nat @ Y5 @ X2 )
@ X ) ) ) ).
% uminus_int.abs_eq
thf(fact_4998_one__int__def,axiom,
( ( one_one @ int )
= ( abs_Integ @ ( product_Pair @ nat @ nat @ ( one_one @ nat ) @ ( zero_zero @ nat ) ) ) ) ).
% one_int_def
thf(fact_4999_of__int_Oabs__eq,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X: product_prod @ nat @ nat] :
( ( ring_1_of_int @ A @ ( abs_Integ @ X ) )
= ( product_case_prod @ nat @ nat @ A
@ ^ [I2: nat,J3: nat] : ( minus_minus @ A @ ( semiring_1_of_nat @ A @ I2 ) @ ( semiring_1_of_nat @ A @ J3 ) )
@ X ) ) ) ).
% of_int.abs_eq
thf(fact_5000_less__int_Oabs__eq,axiom,
! [Xa: product_prod @ nat @ nat,X: product_prod @ nat @ nat] :
( ( ord_less @ int @ ( abs_Integ @ Xa ) @ ( abs_Integ @ X ) )
= ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [X2: nat,Y5: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U2: nat,V5: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ X2 @ V5 ) @ ( plus_plus @ nat @ U2 @ Y5 ) ) )
@ Xa
@ X ) ) ).
% less_int.abs_eq
thf(fact_5001_less__eq__int_Oabs__eq,axiom,
! [Xa: product_prod @ nat @ nat,X: product_prod @ nat @ nat] :
( ( ord_less_eq @ int @ ( abs_Integ @ Xa ) @ ( abs_Integ @ X ) )
= ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [X2: nat,Y5: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U2: nat,V5: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ X2 @ V5 ) @ ( plus_plus @ nat @ U2 @ Y5 ) ) )
@ Xa
@ X ) ) ).
% less_eq_int.abs_eq
thf(fact_5002_plus__int_Oabs__eq,axiom,
! [Xa: product_prod @ nat @ nat,X: product_prod @ nat @ nat] :
( ( plus_plus @ int @ ( abs_Integ @ Xa ) @ ( abs_Integ @ X ) )
= ( abs_Integ
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X2: nat,Y5: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V5: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X2 @ U2 ) @ ( plus_plus @ nat @ Y5 @ V5 ) ) )
@ Xa
@ X ) ) ) ).
% plus_int.abs_eq
thf(fact_5003_minus__int_Oabs__eq,axiom,
! [Xa: product_prod @ nat @ nat,X: product_prod @ nat @ nat] :
( ( minus_minus @ int @ ( abs_Integ @ Xa ) @ ( abs_Integ @ X ) )
= ( abs_Integ
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X2: nat,Y5: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V5: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X2 @ V5 ) @ ( plus_plus @ nat @ Y5 @ U2 ) ) )
@ Xa
@ X ) ) ) ).
% minus_int.abs_eq
thf(fact_5004_semiring__char__def,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( ( semiri4206861660011772517g_char @ A )
= ( ^ [Uu4: itself @ A] :
( gcd_Gcd @ nat
@ ( collect @ nat
@ ^ [N5: nat] :
( ( semiring_1_of_nat @ A @ N5 )
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% semiring_char_def
thf(fact_5005_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_5006_eq__numeral__iff__iszero_I7_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X: num] :
( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ X ) )
= ( one_one @ A ) )
= ( ring_1_iszero @ A @ ( numeral_numeral @ A @ ( plus_plus @ num @ X @ one2 ) ) ) ) ) ).
% eq_numeral_iff_iszero(7)
thf(fact_5007_num__of__nat__numeral__eq,axiom,
! [Q5: num] :
( ( num_of_nat @ ( numeral_numeral @ nat @ Q5 ) )
= Q5 ) ).
% num_of_nat_numeral_eq
thf(fact_5008_iszero__neg__numeral,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [W: num] :
( ( ring_1_iszero @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) )
= ( ring_1_iszero @ A @ ( numeral_numeral @ A @ W ) ) ) ) ).
% iszero_neg_numeral
thf(fact_5009_not__iszero__1,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ~ ( ring_1_iszero @ A @ ( one_one @ A ) ) ) ).
% not_iszero_1
thf(fact_5010_iszero__def,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( ( ring_1_iszero @ A )
= ( ^ [Z6: A] :
( Z6
= ( zero_zero @ A ) ) ) ) ) ).
% iszero_def
thf(fact_5011_iszero__0,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( ring_1_iszero @ A @ ( zero_zero @ A ) ) ) ).
% iszero_0
thf(fact_5012_eq__iff__iszero__diff,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( ( ^ [Y4: A,Z: A] : Y4 = Z )
= ( ^ [X2: A,Y5: A] : ( ring_1_iszero @ A @ ( minus_minus @ A @ X2 @ Y5 ) ) ) ) ) ).
% eq_iff_iszero_diff
thf(fact_5013_not__iszero__numeral,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [W: num] :
~ ( ring_1_iszero @ A @ ( numeral_numeral @ A @ W ) ) ) ).
% not_iszero_numeral
thf(fact_5014_Gcd__int__greater__eq__0,axiom,
! [K6: set @ int] : ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( gcd_Gcd @ int @ K6 ) ) ).
% Gcd_int_greater_eq_0
thf(fact_5015_num__of__nat_Osimps_I1_J,axiom,
( ( num_of_nat @ ( zero_zero @ nat ) )
= one2 ) ).
% num_of_nat.simps(1)
thf(fact_5016_eq__numeral__iff__iszero_I10_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [Y2: num] :
( ( ( zero_zero @ A )
= ( numeral_numeral @ A @ Y2 ) )
= ( ring_1_iszero @ A @ ( numeral_numeral @ A @ Y2 ) ) ) ) ).
% eq_numeral_iff_iszero(10)
thf(fact_5017_eq__numeral__iff__iszero_I9_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X: num] :
( ( ( numeral_numeral @ A @ X )
= ( zero_zero @ A ) )
= ( ring_1_iszero @ A @ ( numeral_numeral @ A @ X ) ) ) ) ).
% eq_numeral_iff_iszero(9)
thf(fact_5018_not__iszero__Numeral1,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ~ ( ring_1_iszero @ A @ ( numeral_numeral @ A @ one2 ) ) ) ).
% not_iszero_Numeral1
thf(fact_5019_not__iszero__neg__1,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ~ ( ring_1_iszero @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% not_iszero_neg_1
thf(fact_5020_eq__numeral__iff__iszero_I1_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X: num,Y2: num] :
( ( ( numeral_numeral @ A @ X )
= ( numeral_numeral @ A @ Y2 ) )
= ( ring_1_iszero @ A @ ( neg_numeral_sub @ A @ X @ Y2 ) ) ) ) ).
% eq_numeral_iff_iszero(1)
thf(fact_5021_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_5022_eq__numeral__iff__iszero_I11_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X: num] :
( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ X ) )
= ( zero_zero @ A ) )
= ( ring_1_iszero @ A @ ( numeral_numeral @ A @ X ) ) ) ) ).
% eq_numeral_iff_iszero(11)
thf(fact_5023_eq__numeral__iff__iszero_I12_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [Y2: num] :
( ( ( zero_zero @ A )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ Y2 ) ) )
= ( ring_1_iszero @ A @ ( numeral_numeral @ A @ Y2 ) ) ) ) ).
% eq_numeral_iff_iszero(12)
thf(fact_5024_not__iszero__neg__Numeral1,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ~ ( ring_1_iszero @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ one2 ) ) ) ) ).
% not_iszero_neg_Numeral1
thf(fact_5025_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_5026_eq__numeral__iff__iszero_I2_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X: num,Y2: num] :
( ( ( numeral_numeral @ A @ X )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ Y2 ) ) )
= ( ring_1_iszero @ A @ ( numeral_numeral @ A @ ( plus_plus @ num @ X @ Y2 ) ) ) ) ) ).
% eq_numeral_iff_iszero(2)
thf(fact_5027_eq__numeral__iff__iszero_I3_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X: num,Y2: num] :
( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ X ) )
= ( numeral_numeral @ A @ Y2 ) )
= ( ring_1_iszero @ A @ ( numeral_numeral @ A @ ( plus_plus @ num @ X @ Y2 ) ) ) ) ) ).
% eq_numeral_iff_iszero(3)
thf(fact_5028_eq__numeral__iff__iszero_I4_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X: num,Y2: num] :
( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ X ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ Y2 ) ) )
= ( ring_1_iszero @ A @ ( neg_numeral_sub @ A @ Y2 @ X ) ) ) ) ).
% eq_numeral_iff_iszero(4)
thf(fact_5029_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_5030_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_5031_num__of__nat__plus__distrib,axiom,
! [M2: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( num_of_nat @ ( plus_plus @ nat @ M2 @ N ) )
= ( plus_plus @ num @ ( num_of_nat @ M2 ) @ ( num_of_nat @ N ) ) ) ) ) ).
% num_of_nat_plus_distrib
thf(fact_5032_eq__numeral__iff__iszero_I6_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [Y2: num] :
( ( ( one_one @ A )
= ( numeral_numeral @ A @ Y2 ) )
= ( ring_1_iszero @ A @ ( neg_numeral_sub @ A @ one2 @ Y2 ) ) ) ) ).
% eq_numeral_iff_iszero(6)
thf(fact_5033_eq__numeral__iff__iszero_I5_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X: num] :
( ( ( numeral_numeral @ A @ X )
= ( one_one @ A ) )
= ( ring_1_iszero @ A @ ( neg_numeral_sub @ A @ X @ one2 ) ) ) ) ).
% eq_numeral_iff_iszero(5)
thf(fact_5034_eq__numeral__iff__iszero_I8_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [Y2: num] :
( ( ( one_one @ A )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ Y2 ) ) )
= ( ring_1_iszero @ A @ ( numeral_numeral @ A @ ( plus_plus @ num @ one2 @ Y2 ) ) ) ) ) ).
% eq_numeral_iff_iszero(8)
thf(fact_5035_less__eq__int_Orep__eq,axiom,
( ( ord_less_eq @ int )
= ( ^ [X2: int,Xa4: int] :
( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [Y5: nat,Z6: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U2: nat,V5: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ Y5 @ V5 ) @ ( plus_plus @ nat @ U2 @ Z6 ) ) )
@ ( rep_Integ @ X2 )
@ ( rep_Integ @ Xa4 ) ) ) ) ).
% less_eq_int.rep_eq
thf(fact_5036_less__int_Orep__eq,axiom,
( ( ord_less @ int )
= ( ^ [X2: int,Xa4: int] :
( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [Y5: nat,Z6: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U2: nat,V5: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ Y5 @ V5 ) @ ( plus_plus @ nat @ U2 @ Z6 ) ) )
@ ( rep_Integ @ X2 )
@ ( rep_Integ @ Xa4 ) ) ) ) ).
% less_int.rep_eq
thf(fact_5037_prod__encode__def,axiom,
( nat_prod_encode
= ( product_case_prod @ nat @ nat @ nat
@ ^ [M5: nat,N5: nat] : ( plus_plus @ nat @ ( nat_triangle @ ( plus_plus @ nat @ M5 @ N5 ) ) @ M5 ) ) ) ).
% prod_encode_def
thf(fact_5038_prod__encode__eq,axiom,
! [X: product_prod @ nat @ nat,Y2: product_prod @ nat @ nat] :
( ( ( nat_prod_encode @ X )
= ( nat_prod_encode @ Y2 ) )
= ( X = Y2 ) ) ).
% prod_encode_eq
thf(fact_5039_le__prod__encode__1,axiom,
! [A3: nat,B2: nat] : ( ord_less_eq @ nat @ A3 @ ( nat_prod_encode @ ( product_Pair @ nat @ nat @ A3 @ B2 ) ) ) ).
% le_prod_encode_1
thf(fact_5040_le__prod__encode__2,axiom,
! [B2: nat,A3: nat] : ( ord_less_eq @ nat @ B2 @ ( nat_prod_encode @ ( product_Pair @ nat @ nat @ A3 @ B2 ) ) ) ).
% le_prod_encode_2
thf(fact_5041_nat_Orep__eq,axiom,
( nat2
= ( ^ [X2: int] : ( product_case_prod @ nat @ nat @ nat @ ( minus_minus @ nat ) @ ( rep_Integ @ X2 ) ) ) ) ).
% nat.rep_eq
thf(fact_5042_of__int_Orep__eq,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( ( ring_1_of_int @ A )
= ( ^ [X2: int] :
( product_case_prod @ nat @ nat @ A
@ ^ [I2: nat,J3: nat] : ( minus_minus @ A @ ( semiring_1_of_nat @ A @ I2 ) @ ( semiring_1_of_nat @ A @ J3 ) )
@ ( rep_Integ @ X2 ) ) ) ) ) ).
% of_int.rep_eq
thf(fact_5043_prod__encode__prod__decode__aux,axiom,
! [K: nat,M2: nat] :
( ( nat_prod_encode @ ( nat_prod_decode_aux @ K @ M2 ) )
= ( plus_plus @ nat @ ( nat_triangle @ K ) @ M2 ) ) ).
% prod_encode_prod_decode_aux
thf(fact_5044_uminus__int__def,axiom,
( ( uminus_uminus @ int )
= ( map_fun @ int @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ int @ rep_Integ @ abs_Integ
@ ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [X2: nat,Y5: nat] : ( product_Pair @ nat @ nat @ Y5 @ X2 ) ) ) ) ).
% uminus_int_def
thf(fact_5045_prod_Oinsert_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [I5: set @ B,P4: B > A,I: B] :
( ( finite_finite @ B
@ ( collect @ B
@ ^ [X2: B] :
( ( member @ B @ X2 @ I5 )
& ( ( P4 @ X2 )
!= ( one_one @ A ) ) ) ) )
=> ( ( ( member @ B @ I @ I5 )
=> ( ( groups1962203154675924110t_prod @ B @ A @ P4 @ ( insert @ B @ I @ I5 ) )
= ( groups1962203154675924110t_prod @ B @ A @ P4 @ I5 ) ) )
& ( ~ ( member @ B @ I @ I5 )
=> ( ( groups1962203154675924110t_prod @ B @ A @ P4 @ ( insert @ B @ I @ I5 ) )
= ( times_times @ A @ ( P4 @ I ) @ ( groups1962203154675924110t_prod @ B @ A @ P4 @ I5 ) ) ) ) ) ) ) ).
% prod.insert'
thf(fact_5046_sorted__list__of__set_Osorted__key__list__of__set__remove,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite @ A @ A4 )
=> ( ( linord4507533701916653071of_set @ A @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( remove1 @ A @ X @ ( linord4507533701916653071of_set @ A @ A4 ) ) ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_remove
thf(fact_5047_prod_Oempty_H,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [P4: B > A] :
( ( groups1962203154675924110t_prod @ B @ A @ P4 @ ( bot_bot @ ( set @ B ) ) )
= ( one_one @ A ) ) ) ).
% prod.empty'
thf(fact_5048_sorted__list__of__set_Olength__sorted__key__list__of__set,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A] :
( ( size_size @ ( list @ A ) @ ( linord4507533701916653071of_set @ A @ A4 ) )
= ( finite_card @ A @ A4 ) ) ) ).
% sorted_list_of_set.length_sorted_key_list_of_set
thf(fact_5049_prod_Onon__neutral_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G: B > A,I5: set @ B] :
( ( groups1962203154675924110t_prod @ B @ A @ G
@ ( collect @ B
@ ^ [X2: B] :
( ( member @ B @ X2 @ I5 )
& ( ( G @ X2 )
!= ( one_one @ A ) ) ) ) )
= ( groups1962203154675924110t_prod @ B @ A @ G @ I5 ) ) ) ).
% prod.non_neutral'
thf(fact_5050_prod_Omono__neutral__cong__right_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S3: set @ B,T6: set @ B,G: B > A,H: B > A] :
( ( ord_less_eq @ ( set @ B ) @ S3 @ T6 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T6 @ S3 ) )
=> ( ( G @ X3 )
= ( one_one @ A ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ S3 )
=> ( ( G @ X3 )
= ( H @ X3 ) ) )
=> ( ( groups1962203154675924110t_prod @ B @ A @ G @ T6 )
= ( groups1962203154675924110t_prod @ B @ A @ H @ S3 ) ) ) ) ) ) ).
% prod.mono_neutral_cong_right'
thf(fact_5051_prod_Omono__neutral__cong__left_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S3: set @ B,T6: set @ B,H: B > A,G: B > A] :
( ( ord_less_eq @ ( set @ B ) @ S3 @ T6 )
=> ( ! [I3: B] :
( ( member @ B @ I3 @ ( minus_minus @ ( set @ B ) @ T6 @ S3 ) )
=> ( ( H @ I3 )
= ( one_one @ A ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ S3 )
=> ( ( G @ X3 )
= ( H @ X3 ) ) )
=> ( ( groups1962203154675924110t_prod @ B @ A @ G @ S3 )
= ( groups1962203154675924110t_prod @ B @ A @ H @ T6 ) ) ) ) ) ) ).
% prod.mono_neutral_cong_left'
thf(fact_5052_prod_Omono__neutral__right_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S3: set @ B,T6: set @ B,G: B > A] :
( ( ord_less_eq @ ( set @ B ) @ S3 @ T6 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T6 @ S3 ) )
=> ( ( G @ X3 )
= ( one_one @ A ) ) )
=> ( ( groups1962203154675924110t_prod @ B @ A @ G @ T6 )
= ( groups1962203154675924110t_prod @ B @ A @ G @ S3 ) ) ) ) ) ).
% prod.mono_neutral_right'
thf(fact_5053_prod_Omono__neutral__left_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S3: set @ B,T6: set @ B,G: B > A] :
( ( ord_less_eq @ ( set @ B ) @ S3 @ T6 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( minus_minus @ ( set @ B ) @ T6 @ S3 ) )
=> ( ( G @ X3 )
= ( one_one @ A ) ) )
=> ( ( groups1962203154675924110t_prod @ B @ A @ G @ S3 )
= ( groups1962203154675924110t_prod @ B @ A @ G @ T6 ) ) ) ) ) ).
% prod.mono_neutral_left'
thf(fact_5054_prod_Odistrib_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [I5: set @ B,G: B > A,H: B > A] :
( ( finite_finite @ B
@ ( collect @ B
@ ^ [X2: B] :
( ( member @ B @ X2 @ I5 )
& ( ( G @ X2 )
!= ( one_one @ A ) ) ) ) )
=> ( ( finite_finite @ B
@ ( collect @ B
@ ^ [X2: B] :
( ( member @ B @ X2 @ I5 )
& ( ( H @ X2 )
!= ( one_one @ A ) ) ) ) )
=> ( ( groups1962203154675924110t_prod @ B @ A
@ ^ [I2: B] : ( times_times @ A @ ( G @ I2 ) @ ( H @ I2 ) )
@ I5 )
= ( times_times @ A @ ( groups1962203154675924110t_prod @ B @ A @ G @ I5 ) @ ( groups1962203154675924110t_prod @ B @ A @ H @ I5 ) ) ) ) ) ) ).
% prod.distrib'
thf(fact_5055_prod_OG__def,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ( ( groups1962203154675924110t_prod @ B @ A )
= ( ^ [P6: B > A,I7: set @ B] :
( if @ A
@ ( finite_finite @ B
@ ( collect @ B
@ ^ [X2: B] :
( ( member @ B @ X2 @ I7 )
& ( ( P6 @ X2 )
!= ( one_one @ A ) ) ) ) )
@ ( groups7121269368397514597t_prod @ B @ A @ P6
@ ( collect @ B
@ ^ [X2: B] :
( ( member @ B @ X2 @ I7 )
& ( ( P6 @ X2 )
!= ( one_one @ A ) ) ) ) )
@ ( one_one @ A ) ) ) ) ) ).
% prod.G_def
thf(fact_5056_nth__sorted__list__of__set__greaterThanLessThan,axiom,
! [N: nat,J2: nat,I: nat] :
( ( ord_less @ nat @ N @ ( minus_minus @ nat @ J2 @ ( suc @ I ) ) )
=> ( ( nth @ nat @ ( linord4507533701916653071of_set @ nat @ ( set_or5935395276787703475ssThan @ nat @ I @ J2 ) ) @ N )
= ( suc @ ( plus_plus @ nat @ I @ N ) ) ) ) ).
% nth_sorted_list_of_set_greaterThanLessThan
thf(fact_5057_times__int__def,axiom,
( ( times_times @ int )
= ( map_fun @ int @ ( product_prod @ nat @ nat ) @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) ) @ ( int > int ) @ rep_Integ @ ( map_fun @ int @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ int @ rep_Integ @ abs_Integ )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X2: nat,Y5: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V5: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ X2 @ U2 ) @ ( times_times @ nat @ Y5 @ V5 ) ) @ ( plus_plus @ nat @ ( times_times @ nat @ X2 @ V5 ) @ ( times_times @ nat @ Y5 @ U2 ) ) ) ) ) ) ) ).
% times_int_def
thf(fact_5058_minus__int__def,axiom,
( ( minus_minus @ int )
= ( map_fun @ int @ ( product_prod @ nat @ nat ) @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) ) @ ( int > int ) @ rep_Integ @ ( map_fun @ int @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ int @ rep_Integ @ abs_Integ )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X2: nat,Y5: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V5: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X2 @ V5 ) @ ( plus_plus @ nat @ Y5 @ U2 ) ) ) ) ) ) ).
% minus_int_def
thf(fact_5059_plus__int__def,axiom,
( ( plus_plus @ int )
= ( map_fun @ int @ ( product_prod @ nat @ nat ) @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) ) @ ( int > int ) @ rep_Integ @ ( map_fun @ int @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ int @ rep_Integ @ abs_Integ )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X2: nat,Y5: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V5: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X2 @ U2 ) @ ( plus_plus @ nat @ Y5 @ V5 ) ) ) ) ) ) ).
% plus_int_def
thf(fact_5060_sorted__list__of__set_Osorted__key__list__of__set__insert__remove,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite @ A @ A4 )
=> ( ( linord4507533701916653071of_set @ A @ ( insert @ A @ X @ A4 ) )
= ( linorder_insort_key @ A @ A
@ ^ [X2: A] : X2
@ X
@ ( linord4507533701916653071of_set @ A @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_insert_remove
thf(fact_5061_nth__sorted__list__of__set__greaterThanAtMost,axiom,
! [N: nat,J2: nat,I: nat] :
( ( ord_less @ nat @ N @ ( minus_minus @ nat @ J2 @ I ) )
=> ( ( nth @ nat @ ( linord4507533701916653071of_set @ nat @ ( set_or3652927894154168847AtMost @ nat @ I @ J2 ) ) @ N )
= ( suc @ ( plus_plus @ nat @ I @ N ) ) ) ) ).
% nth_sorted_list_of_set_greaterThanAtMost
thf(fact_5062_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_5063_greaterThanAtMost__iff,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [I: A,L: A,U: A] :
( ( member @ A @ I @ ( set_or3652927894154168847AtMost @ A @ L @ U ) )
= ( ( ord_less @ A @ L @ I )
& ( ord_less_eq @ A @ I @ U ) ) ) ) ).
% greaterThanAtMost_iff
thf(fact_5064_greaterThanAtMost__empty__iff2,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [K: A,L: A] :
( ( ( bot_bot @ ( set @ A ) )
= ( set_or3652927894154168847AtMost @ A @ K @ L ) )
= ( ~ ( ord_less @ A @ K @ L ) ) ) ) ).
% greaterThanAtMost_empty_iff2
thf(fact_5065_greaterThanAtMost__empty__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [K: A,L: A] :
( ( ( set_or3652927894154168847AtMost @ A @ K @ L )
= ( bot_bot @ ( set @ A ) ) )
= ( ~ ( ord_less @ A @ K @ L ) ) ) ) ).
% greaterThanAtMost_empty_iff
thf(fact_5066_infinite__Ioc__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A3: A,B2: A] :
( ( ~ ( finite_finite @ A @ ( set_or3652927894154168847AtMost @ A @ A3 @ B2 ) ) )
= ( ord_less @ A @ A3 @ B2 ) ) ) ).
% infinite_Ioc_iff
thf(fact_5067_length__insort,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F2: B > A,X: B,Xs: list @ B] :
( ( size_size @ ( list @ B ) @ ( linorder_insort_key @ B @ A @ F2 @ X @ Xs ) )
= ( suc @ ( size_size @ ( list @ B ) @ Xs ) ) ) ) ).
% length_insort
thf(fact_5068_card__greaterThanAtMost,axiom,
! [L: nat,U: nat] :
( ( finite_card @ nat @ ( set_or3652927894154168847AtMost @ nat @ L @ U ) )
= ( minus_minus @ nat @ U @ L ) ) ).
% card_greaterThanAtMost
thf(fact_5069_atLeastSucAtMost__greaterThanAtMost,axiom,
! [L: nat,U: nat] :
( ( set_or1337092689740270186AtMost @ nat @ ( suc @ L ) @ U )
= ( set_or3652927894154168847AtMost @ nat @ L @ U ) ) ).
% atLeastSucAtMost_greaterThanAtMost
thf(fact_5070_infinite__Ioc,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ~ ( finite_finite @ A @ ( set_or3652927894154168847AtMost @ A @ A3 @ B2 ) ) ) ) ).
% infinite_Ioc
thf(fact_5071_greaterThanAtMost__subseteq__atLeastAtMost__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A3: A,B2: A,C2: A,D2: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ A3 @ B2 ) @ ( set_or1337092689740270186AtMost @ A @ C2 @ D2 ) )
= ( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ A3 )
& ( ord_less_eq @ A @ B2 @ D2 ) ) ) ) ) ).
% greaterThanAtMost_subseteq_atLeastAtMost_iff
thf(fact_5072_greaterThanAtMost__subseteq__atLeastLessThan__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A3: A,B2: A,C2: A,D2: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ A3 @ B2 ) @ ( set_or7035219750837199246ssThan @ A @ C2 @ D2 ) )
= ( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ A3 )
& ( ord_less @ A @ B2 @ D2 ) ) ) ) ) ).
% greaterThanAtMost_subseteq_atLeastLessThan_iff
thf(fact_5073_greaterThanLessThan__subseteq__greaterThanAtMost__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A3: A,B2: A,C2: A,D2: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ A3 @ B2 ) @ ( set_or3652927894154168847AtMost @ A @ C2 @ D2 ) )
= ( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ C2 @ A3 )
& ( ord_less_eq @ A @ B2 @ D2 ) ) ) ) ) ).
% greaterThanLessThan_subseteq_greaterThanAtMost_iff
thf(fact_5074_greaterThanAtMost__eq__atLeastAtMost__diff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( set_or3652927894154168847AtMost @ A )
= ( ^ [A5: A,B3: A] : ( minus_minus @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A5 @ B3 ) @ ( insert @ A @ A5 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% greaterThanAtMost_eq_atLeastAtMost_diff
thf(fact_5075_sorted__list__of__set_Ofold__insort__key_Oremove,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite @ A @ A4 )
=> ( ( member @ A @ X @ A4 )
=> ( ( linord4507533701916653071of_set @ A @ A4 )
= ( linorder_insort_key @ A @ A
@ ^ [X2: A] : X2
@ X
@ ( linord4507533701916653071of_set @ A @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ).
% sorted_list_of_set.fold_insort_key.remove
thf(fact_5076_sum__of__bool__eq,axiom,
! [A: $tType,B: $tType] :
( ( semiring_1 @ A )
=> ! [A4: set @ B,P: B > $o] :
( ( finite_finite @ B @ A4 )
=> ( ( finite_finite @ B @ A4 )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X2: B] : ( zero_neq_one_of_bool @ A @ ( P @ X2 ) )
@ A4 )
= ( semiring_1_of_nat @ A @ ( finite_card @ B @ ( inf_inf @ ( set @ B ) @ A4 @ ( collect @ B @ P ) ) ) ) ) ) ) ) ).
% sum_of_bool_eq
thf(fact_5077_image__minus__const__atLeastLessThan__nat,axiom,
! [C2: nat,Y2: nat,X: nat] :
( ( ( ord_less @ nat @ C2 @ Y2 )
=> ( ( image @ nat @ nat
@ ^ [I2: nat] : ( minus_minus @ nat @ I2 @ 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
@ ^ [I2: nat] : ( minus_minus @ nat @ I2 @ C2 )
@ ( set_or7035219750837199246ssThan @ nat @ X @ Y2 ) )
= ( insert @ nat @ ( zero_zero @ nat ) @ ( bot_bot @ ( set @ nat ) ) ) ) )
& ( ~ ( ord_less @ nat @ X @ Y2 )
=> ( ( image @ nat @ nat
@ ^ [I2: nat] : ( minus_minus @ nat @ I2 @ C2 )
@ ( set_or7035219750837199246ssThan @ nat @ X @ Y2 ) )
= ( bot_bot @ ( set @ nat ) ) ) ) ) ) ) ).
% image_minus_const_atLeastLessThan_nat
thf(fact_5078_rat__floor__lemma,axiom,
! [A3: int,B2: int] :
( ( ord_less_eq @ rat @ ( ring_1_of_int @ rat @ ( divide_divide @ int @ A3 @ B2 ) ) @ ( fract @ A3 @ B2 ) )
& ( ord_less @ rat @ ( fract @ A3 @ B2 ) @ ( ring_1_of_int @ rat @ ( plus_plus @ int @ ( divide_divide @ int @ A3 @ B2 ) @ ( one_one @ int ) ) ) ) ) ).
% rat_floor_lemma
thf(fact_5079_inf__apply,axiom,
! [B: $tType,A: $tType] :
( ( semilattice_inf @ B )
=> ( ( inf_inf @ ( A > B ) )
= ( ^ [F4: A > B,G2: A > B,X2: A] : ( inf_inf @ B @ ( F4 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ).
% inf_apply
thf(fact_5080_inf__right__idem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X: A,Y2: A] :
( ( inf_inf @ A @ ( inf_inf @ A @ X @ Y2 ) @ Y2 )
= ( inf_inf @ A @ X @ Y2 ) ) ) ).
% inf_right_idem
thf(fact_5081_inf_Oright__idem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,B2: A] :
( ( inf_inf @ A @ ( inf_inf @ A @ A3 @ B2 ) @ B2 )
= ( inf_inf @ A @ A3 @ B2 ) ) ) ).
% inf.right_idem
thf(fact_5082_inf__left__idem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X: A,Y2: A] :
( ( inf_inf @ A @ X @ ( inf_inf @ A @ X @ Y2 ) )
= ( inf_inf @ A @ X @ Y2 ) ) ) ).
% inf_left_idem
thf(fact_5083_inf_Oleft__idem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,B2: A] :
( ( inf_inf @ A @ A3 @ ( inf_inf @ A @ A3 @ B2 ) )
= ( inf_inf @ A @ A3 @ B2 ) ) ) ).
% inf.left_idem
thf(fact_5084_inf__idem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X: A] :
( ( inf_inf @ A @ X @ X )
= X ) ) ).
% inf_idem
thf(fact_5085_inf_Oidem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A] :
( ( inf_inf @ A @ A3 @ A3 )
= A3 ) ) ).
% inf.idem
thf(fact_5086_le__inf__iff,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X: A,Y2: A,Z2: A] :
( ( ord_less_eq @ A @ X @ ( inf_inf @ A @ Y2 @ Z2 ) )
= ( ( ord_less_eq @ A @ X @ Y2 )
& ( ord_less_eq @ A @ X @ Z2 ) ) ) ) ).
% le_inf_iff
thf(fact_5087_inf_Obounded__iff,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ ( inf_inf @ A @ B2 @ C2 ) )
= ( ( ord_less_eq @ A @ A3 @ B2 )
& ( ord_less_eq @ A @ A3 @ C2 ) ) ) ) ).
% inf.bounded_iff
thf(fact_5088_inf__bot__right,axiom,
! [A: $tType] :
( ( bounded_lattice_bot @ A )
=> ! [X: A] :
( ( inf_inf @ A @ X @ ( bot_bot @ A ) )
= ( bot_bot @ A ) ) ) ).
% inf_bot_right
thf(fact_5089_inf__bot__left,axiom,
! [A: $tType] :
( ( bounded_lattice_bot @ A )
=> ! [X: A] :
( ( inf_inf @ A @ ( bot_bot @ A ) @ X )
= ( bot_bot @ A ) ) ) ).
% inf_bot_left
thf(fact_5090_minus__rat__cancel,axiom,
! [A3: int,B2: int] :
( ( fract @ ( uminus_uminus @ int @ A3 ) @ ( uminus_uminus @ int @ B2 ) )
= ( fract @ A3 @ B2 ) ) ).
% minus_rat_cancel
thf(fact_5091_bij__betw__Suc,axiom,
! [M10: set @ nat,N6: set @ nat] :
( ( bij_betw @ nat @ nat @ suc @ M10 @ N6 )
= ( ( image @ nat @ nat @ suc @ M10 )
= N6 ) ) ).
% bij_betw_Suc
thf(fact_5092_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_5093_inf__compl__bot__left1,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y2: A] :
( ( inf_inf @ A @ ( uminus_uminus @ A @ X ) @ ( inf_inf @ A @ X @ Y2 ) )
= ( bot_bot @ A ) ) ) ).
% inf_compl_bot_left1
thf(fact_5094_inf__compl__bot__left2,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y2: A] :
( ( inf_inf @ A @ X @ ( inf_inf @ A @ ( uminus_uminus @ A @ X ) @ Y2 ) )
= ( bot_bot @ A ) ) ) ).
% inf_compl_bot_left2
thf(fact_5095_inf__compl__bot__right,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y2: A] :
( ( inf_inf @ A @ X @ ( inf_inf @ A @ Y2 @ ( uminus_uminus @ A @ X ) ) )
= ( bot_bot @ A ) ) ) ).
% inf_compl_bot_right
thf(fact_5096_boolean__algebra_Oconj__cancel__left,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A] :
( ( inf_inf @ A @ ( uminus_uminus @ A @ X ) @ X )
= ( bot_bot @ A ) ) ) ).
% boolean_algebra.conj_cancel_left
thf(fact_5097_boolean__algebra_Oconj__cancel__right,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A] :
( ( inf_inf @ A @ X @ ( uminus_uminus @ A @ X ) )
= ( bot_bot @ A ) ) ) ).
% boolean_algebra.conj_cancel_right
thf(fact_5098_image__diff__atLeastAtMost,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [D2: A,A3: A,B2: A] :
( ( image @ A @ A @ ( minus_minus @ A @ D2 ) @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ D2 @ B2 ) @ ( minus_minus @ A @ D2 @ A3 ) ) ) ) ).
% image_diff_atLeastAtMost
thf(fact_5099_image__uminus__atLeastAtMost,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [X: A,Y2: A] :
( ( image @ A @ A @ ( uminus_uminus @ A ) @ ( set_or1337092689740270186AtMost @ A @ X @ Y2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( uminus_uminus @ A @ Y2 ) @ ( uminus_uminus @ A @ X ) ) ) ) ).
% image_uminus_atLeastAtMost
thf(fact_5100_Diff__disjoint,axiom,
! [A: $tType,A4: set @ A,B7: set @ A] :
( ( inf_inf @ ( set @ A ) @ A4 @ ( minus_minus @ ( set @ A ) @ B7 @ A4 ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Diff_disjoint
thf(fact_5101_image__uminus__greaterThanLessThan,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [X: A,Y2: A] :
( ( image @ A @ A @ ( uminus_uminus @ A ) @ ( set_or5935395276787703475ssThan @ A @ X @ Y2 ) )
= ( set_or5935395276787703475ssThan @ A @ ( uminus_uminus @ A @ Y2 ) @ ( uminus_uminus @ A @ X ) ) ) ) ).
% image_uminus_greaterThanLessThan
thf(fact_5102_Compl__disjoint2,axiom,
! [A: $tType,A4: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A4 ) @ A4 )
= ( bot_bot @ ( set @ A ) ) ) ).
% Compl_disjoint2
thf(fact_5103_Compl__disjoint,axiom,
! [A: $tType,A4: set @ A] :
( ( inf_inf @ ( set @ A ) @ A4 @ ( uminus_uminus @ ( set @ A ) @ A4 ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Compl_disjoint
thf(fact_5104_Diff__Compl,axiom,
! [A: $tType,A4: set @ A,B7: set @ A] :
( ( minus_minus @ ( set @ A ) @ A4 @ ( uminus_uminus @ ( set @ A ) @ B7 ) )
= ( inf_inf @ ( set @ A ) @ A4 @ B7 ) ) ).
% Diff_Compl
thf(fact_5105_image__Suc__atLeastAtMost,axiom,
! [I: nat,J2: nat] :
( ( image @ nat @ nat @ suc @ ( set_or1337092689740270186AtMost @ nat @ I @ J2 ) )
= ( set_or1337092689740270186AtMost @ nat @ ( suc @ I ) @ ( suc @ J2 ) ) ) ).
% image_Suc_atLeastAtMost
thf(fact_5106_image__Suc__atLeastLessThan,axiom,
! [I: nat,J2: nat] :
( ( image @ nat @ nat @ suc @ ( set_or7035219750837199246ssThan @ nat @ I @ J2 ) )
= ( set_or7035219750837199246ssThan @ nat @ ( suc @ I ) @ ( suc @ J2 ) ) ) ).
% image_Suc_atLeastLessThan
thf(fact_5107_bij__betw__of__nat,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [N6: set @ nat,A4: set @ A] :
( ( bij_betw @ nat @ A @ ( semiring_1_of_nat @ A ) @ N6 @ A4 )
= ( ( image @ nat @ A @ ( semiring_1_of_nat @ A ) @ N6 )
= A4 ) ) ) ).
% bij_betw_of_nat
thf(fact_5108_image__minus__const__atLeastAtMost_H,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [D2: A,A3: A,B2: A] :
( ( image @ A @ A
@ ^ [T3: A] : ( minus_minus @ A @ T3 @ D2 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ A3 @ D2 ) @ ( minus_minus @ A @ B2 @ D2 ) ) ) ) ).
% image_minus_const_atLeastAtMost'
thf(fact_5109_minus__rat,axiom,
! [A3: int,B2: int] :
( ( uminus_uminus @ rat @ ( fract @ A3 @ B2 ) )
= ( fract @ ( uminus_uminus @ int @ A3 ) @ B2 ) ) ).
% minus_rat
thf(fact_5110_divide__rat,axiom,
! [A3: int,B2: int,C2: int,D2: int] :
( ( divide_divide @ rat @ ( fract @ A3 @ B2 ) @ ( fract @ C2 @ D2 ) )
= ( fract @ ( times_times @ int @ A3 @ D2 ) @ ( times_times @ int @ B2 @ C2 ) ) ) ).
% divide_rat
thf(fact_5111_card__greaterThanAtMost__int,axiom,
! [L: int,U: int] :
( ( finite_card @ int @ ( set_or3652927894154168847AtMost @ int @ L @ U ) )
= ( nat2 @ ( minus_minus @ int @ U @ L ) ) ) ).
% card_greaterThanAtMost_int
thf(fact_5112_floor__Fract,axiom,
! [A3: int,B2: int] :
( ( archim6421214686448440834_floor @ rat @ ( fract @ A3 @ B2 ) )
= ( divide_divide @ int @ A3 @ B2 ) ) ).
% floor_Fract
thf(fact_5113_image__minus__const__greaterThanAtMost,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( image @ A @ A @ ( minus_minus @ A @ C2 ) @ ( set_or3652927894154168847AtMost @ A @ A3 @ B2 ) )
= ( set_or7035219750837199246ssThan @ A @ ( minus_minus @ A @ C2 @ B2 ) @ ( minus_minus @ A @ C2 @ A3 ) ) ) ) ).
% image_minus_const_greaterThanAtMost
thf(fact_5114_image__diff__atLeastLessThan,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( image @ A @ A @ ( minus_minus @ A @ C2 ) @ ( set_or7035219750837199246ssThan @ A @ A3 @ B2 ) )
= ( set_or3652927894154168847AtMost @ A @ ( minus_minus @ A @ C2 @ B2 ) @ ( minus_minus @ A @ C2 @ A3 ) ) ) ) ).
% image_diff_atLeastLessThan
thf(fact_5115_image__uminus__atLeastLessThan,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [X: A,Y2: A] :
( ( image @ A @ A @ ( uminus_uminus @ A ) @ ( set_or7035219750837199246ssThan @ A @ X @ Y2 ) )
= ( set_or3652927894154168847AtMost @ A @ ( uminus_uminus @ A @ Y2 ) @ ( uminus_uminus @ A @ X ) ) ) ) ).
% image_uminus_atLeastLessThan
thf(fact_5116_image__uminus__greaterThanAtMost,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [X: A,Y2: A] :
( ( image @ A @ A @ ( uminus_uminus @ A ) @ ( set_or3652927894154168847AtMost @ A @ X @ Y2 ) )
= ( set_or7035219750837199246ssThan @ A @ ( uminus_uminus @ A @ Y2 ) @ ( uminus_uminus @ A @ X ) ) ) ) ).
% image_uminus_greaterThanAtMost
thf(fact_5117_image__mult__atLeastAtMost,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [D2: A,A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ D2 )
=> ( ( image @ A @ A @ ( times_times @ A @ D2 ) @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( times_times @ A @ D2 @ A3 ) @ ( times_times @ A @ D2 @ B2 ) ) ) ) ) ).
% image_mult_atLeastAtMost
thf(fact_5118_image__divide__atLeastAtMost,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [D2: A,A3: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ D2 )
=> ( ( image @ A @ A
@ ^ [C4: A] : ( divide_divide @ A @ C4 @ D2 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( divide_divide @ A @ A3 @ D2 ) @ ( divide_divide @ A @ B2 @ D2 ) ) ) ) ) ).
% image_divide_atLeastAtMost
thf(fact_5119_less__rat,axiom,
! [B2: int,D2: int,A3: int,C2: int] :
( ( B2
!= ( zero_zero @ int ) )
=> ( ( D2
!= ( zero_zero @ int ) )
=> ( ( ord_less @ rat @ ( fract @ A3 @ B2 ) @ ( fract @ C2 @ D2 ) )
= ( ord_less @ int @ ( times_times @ int @ ( times_times @ int @ A3 @ D2 ) @ ( times_times @ int @ B2 @ D2 ) ) @ ( times_times @ int @ ( times_times @ int @ C2 @ B2 ) @ ( times_times @ int @ B2 @ D2 ) ) ) ) ) ) ).
% less_rat
thf(fact_5120_add__rat,axiom,
! [B2: int,D2: int,A3: int,C2: int] :
( ( B2
!= ( zero_zero @ int ) )
=> ( ( D2
!= ( zero_zero @ int ) )
=> ( ( plus_plus @ rat @ ( fract @ A3 @ B2 ) @ ( fract @ C2 @ D2 ) )
= ( fract @ ( plus_plus @ int @ ( times_times @ int @ A3 @ D2 ) @ ( times_times @ int @ C2 @ B2 ) ) @ ( times_times @ int @ B2 @ D2 ) ) ) ) ) ).
% add_rat
thf(fact_5121_le__rat,axiom,
! [B2: int,D2: int,A3: int,C2: int] :
( ( B2
!= ( zero_zero @ int ) )
=> ( ( D2
!= ( zero_zero @ int ) )
=> ( ( ord_less_eq @ rat @ ( fract @ A3 @ B2 ) @ ( fract @ C2 @ D2 ) )
= ( ord_less_eq @ int @ ( times_times @ int @ ( times_times @ int @ A3 @ D2 ) @ ( times_times @ int @ B2 @ D2 ) ) @ ( times_times @ int @ ( times_times @ int @ C2 @ B2 ) @ ( times_times @ int @ B2 @ D2 ) ) ) ) ) ) ).
% le_rat
thf(fact_5122_diff__rat,axiom,
! [B2: int,D2: int,A3: int,C2: int] :
( ( B2
!= ( zero_zero @ int ) )
=> ( ( D2
!= ( zero_zero @ int ) )
=> ( ( minus_minus @ rat @ ( fract @ A3 @ B2 ) @ ( fract @ C2 @ D2 ) )
= ( fract @ ( minus_minus @ int @ ( times_times @ int @ A3 @ D2 ) @ ( times_times @ int @ C2 @ B2 ) ) @ ( times_times @ int @ B2 @ D2 ) ) ) ) ) ).
% diff_rat
thf(fact_5123_translation__subtract__Int,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A3: A,S: set @ A,T2: set @ A] :
( ( image @ A @ A
@ ^ [X2: A] : ( minus_minus @ A @ X2 @ A3 )
@ ( inf_inf @ ( set @ A ) @ S @ T2 ) )
= ( inf_inf @ ( set @ A )
@ ( image @ A @ A
@ ^ [X2: A] : ( minus_minus @ A @ X2 @ A3 )
@ S )
@ ( image @ A @ A
@ ^ [X2: A] : ( minus_minus @ A @ X2 @ A3 )
@ T2 ) ) ) ) ).
% translation_subtract_Int
thf(fact_5124_Int__Diff,axiom,
! [A: $tType,A4: set @ A,B7: set @ A,C5: set @ A] :
( ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A4 @ B7 ) @ C5 )
= ( inf_inf @ ( set @ A ) @ A4 @ ( minus_minus @ ( set @ A ) @ B7 @ C5 ) ) ) ).
% Int_Diff
thf(fact_5125_Diff__Int2,axiom,
! [A: $tType,A4: set @ A,C5: set @ A,B7: set @ A] :
( ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A4 @ C5 ) @ ( inf_inf @ ( set @ A ) @ B7 @ C5 ) )
= ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A4 @ C5 ) @ B7 ) ) ).
% Diff_Int2
thf(fact_5126_Diff__Diff__Int,axiom,
! [A: $tType,A4: set @ A,B7: set @ A] :
( ( minus_minus @ ( set @ A ) @ A4 @ ( minus_minus @ ( set @ A ) @ A4 @ B7 ) )
= ( inf_inf @ ( set @ A ) @ A4 @ B7 ) ) ).
% Diff_Diff_Int
thf(fact_5127_Diff__Int__distrib,axiom,
! [A: $tType,C5: set @ A,A4: set @ A,B7: set @ A] :
( ( inf_inf @ ( set @ A ) @ C5 @ ( minus_minus @ ( set @ A ) @ A4 @ B7 ) )
= ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ C5 @ A4 ) @ ( inf_inf @ ( set @ A ) @ C5 @ B7 ) ) ) ).
% Diff_Int_distrib
thf(fact_5128_Diff__Int__distrib2,axiom,
! [A: $tType,A4: set @ A,B7: set @ A,C5: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A4 @ B7 ) @ C5 )
= ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A4 @ C5 ) @ ( inf_inf @ ( set @ A ) @ B7 @ C5 ) ) ) ).
% Diff_Int_distrib2
thf(fact_5129_inf_Ostrict__coboundedI2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [B2: A,C2: A,A3: A] :
( ( ord_less @ A @ B2 @ C2 )
=> ( ord_less @ A @ ( inf_inf @ A @ A3 @ B2 ) @ C2 ) ) ) ).
% inf.strict_coboundedI2
thf(fact_5130_inf_Ostrict__coboundedI1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( ord_less @ A @ A3 @ C2 )
=> ( ord_less @ A @ ( inf_inf @ A @ A3 @ B2 ) @ C2 ) ) ) ).
% inf.strict_coboundedI1
thf(fact_5131_inf_Ostrict__order__iff,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( ( ord_less @ A )
= ( ^ [A5: A,B3: A] :
( ( A5
= ( inf_inf @ A @ A5 @ B3 ) )
& ( A5 != B3 ) ) ) ) ) ).
% inf.strict_order_iff
thf(fact_5132_inf_Ostrict__boundedE,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less @ A @ A3 @ ( inf_inf @ A @ B2 @ C2 ) )
=> ~ ( ( ord_less @ A @ A3 @ B2 )
=> ~ ( ord_less @ A @ A3 @ C2 ) ) ) ) ).
% inf.strict_boundedE
thf(fact_5133_inf_Oabsorb4,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [B2: A,A3: A] :
( ( ord_less @ A @ B2 @ A3 )
=> ( ( inf_inf @ A @ A3 @ B2 )
= B2 ) ) ) ).
% inf.absorb4
thf(fact_5134_inf_Oabsorb3,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( inf_inf @ A @ A3 @ B2 )
= A3 ) ) ) ).
% inf.absorb3
thf(fact_5135_less__infI2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [B2: A,X: A,A3: A] :
( ( ord_less @ A @ B2 @ X )
=> ( ord_less @ A @ ( inf_inf @ A @ A3 @ B2 ) @ X ) ) ) ).
% less_infI2
thf(fact_5136_less__infI1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,X: A,B2: A] :
( ( ord_less @ A @ A3 @ X )
=> ( ord_less @ A @ ( inf_inf @ A @ A3 @ B2 ) @ X ) ) ) ).
% less_infI1
thf(fact_5137_inf_OcoboundedI2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [B2: A,C2: A,A3: A] :
( ( ord_less_eq @ A @ B2 @ C2 )
=> ( ord_less_eq @ A @ ( inf_inf @ A @ A3 @ B2 ) @ C2 ) ) ) ).
% inf.coboundedI2
thf(fact_5138_inf_OcoboundedI1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ C2 )
=> ( ord_less_eq @ A @ ( inf_inf @ A @ A3 @ B2 ) @ C2 ) ) ) ).
% inf.coboundedI1
thf(fact_5139_inf_Oabsorb__iff2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B3: A,A5: A] :
( ( inf_inf @ A @ A5 @ B3 )
= B3 ) ) ) ) ).
% inf.absorb_iff2
thf(fact_5140_inf_Oabsorb__iff1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A5: A,B3: A] :
( ( inf_inf @ A @ A5 @ B3 )
= A5 ) ) ) ) ).
% inf.absorb_iff1
thf(fact_5141_inf_Ocobounded2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,B2: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ A3 @ B2 ) @ B2 ) ) ).
% inf.cobounded2
thf(fact_5142_inf_Ocobounded1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,B2: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ A3 @ B2 ) @ A3 ) ) ).
% inf.cobounded1
thf(fact_5143_inf_Oorder__iff,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A5: A,B3: A] :
( A5
= ( inf_inf @ A @ A5 @ B3 ) ) ) ) ) ).
% inf.order_iff
thf(fact_5144_inf__greatest,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X: A,Y2: A,Z2: A] :
( ( ord_less_eq @ A @ X @ Y2 )
=> ( ( ord_less_eq @ A @ X @ Z2 )
=> ( ord_less_eq @ A @ X @ ( inf_inf @ A @ Y2 @ Z2 ) ) ) ) ) ).
% inf_greatest
thf(fact_5145_inf_OboundedI,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ A3 @ C2 )
=> ( ord_less_eq @ A @ A3 @ ( inf_inf @ A @ B2 @ C2 ) ) ) ) ) ).
% inf.boundedI
thf(fact_5146_inf_OboundedE,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ ( inf_inf @ A @ B2 @ C2 ) )
=> ~ ( ( ord_less_eq @ A @ A3 @ B2 )
=> ~ ( ord_less_eq @ A @ A3 @ C2 ) ) ) ) ).
% inf.boundedE
thf(fact_5147_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_5148_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_5149_inf_Oabsorb2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [B2: A,A3: A] :
( ( ord_less_eq @ A @ B2 @ A3 )
=> ( ( inf_inf @ A @ A3 @ B2 )
= B2 ) ) ) ).
% inf.absorb2
thf(fact_5150_inf_Oabsorb1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( inf_inf @ A @ A3 @ B2 )
= A3 ) ) ) ).
% inf.absorb1
thf(fact_5151_le__iff__inf,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [X2: A,Y5: A] :
( ( inf_inf @ A @ X2 @ Y5 )
= X2 ) ) ) ) ).
% le_iff_inf
thf(fact_5152_inf__unique,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [F2: A > A > A,X: A,Y2: A] :
( ! [X3: A,Y3: A] : ( ord_less_eq @ A @ ( F2 @ X3 @ Y3 ) @ X3 )
=> ( ! [X3: A,Y3: A] : ( ord_less_eq @ A @ ( F2 @ X3 @ Y3 ) @ Y3 )
=> ( ! [X3: A,Y3: A,Z4: A] :
( ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ( ord_less_eq @ A @ X3 @ Z4 )
=> ( ord_less_eq @ A @ X3 @ ( F2 @ Y3 @ Z4 ) ) ) )
=> ( ( inf_inf @ A @ X @ Y2 )
= ( F2 @ X @ Y2 ) ) ) ) ) ) ).
% inf_unique
thf(fact_5153_inf_OorderI,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,B2: A] :
( ( A3
= ( inf_inf @ A @ A3 @ B2 ) )
=> ( ord_less_eq @ A @ A3 @ B2 ) ) ) ).
% inf.orderI
thf(fact_5154_inf_OorderE,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( A3
= ( inf_inf @ A @ A3 @ B2 ) ) ) ) ).
% inf.orderE
thf(fact_5155_le__infI2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [B2: A,X: A,A3: A] :
( ( ord_less_eq @ A @ B2 @ X )
=> ( ord_less_eq @ A @ ( inf_inf @ A @ A3 @ B2 ) @ X ) ) ) ).
% le_infI2
thf(fact_5156_le__infI1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,X: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ X )
=> ( ord_less_eq @ A @ ( inf_inf @ A @ A3 @ B2 ) @ X ) ) ) ).
% le_infI1
thf(fact_5157_inf__mono,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,C2: A,B2: A,D2: A] :
( ( ord_less_eq @ A @ A3 @ C2 )
=> ( ( ord_less_eq @ A @ B2 @ D2 )
=> ( ord_less_eq @ A @ ( inf_inf @ A @ A3 @ B2 ) @ ( inf_inf @ A @ C2 @ D2 ) ) ) ) ) ).
% inf_mono
thf(fact_5158_le__infI,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X: A,A3: A,B2: A] :
( ( ord_less_eq @ A @ X @ A3 )
=> ( ( ord_less_eq @ A @ X @ B2 )
=> ( ord_less_eq @ A @ X @ ( inf_inf @ A @ A3 @ B2 ) ) ) ) ) ).
% le_infI
thf(fact_5159_le__infE,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X: A,A3: A,B2: A] :
( ( ord_less_eq @ A @ X @ ( inf_inf @ A @ A3 @ B2 ) )
=> ~ ( ( ord_less_eq @ A @ X @ A3 )
=> ~ ( ord_less_eq @ A @ X @ B2 ) ) ) ) ).
% le_infE
thf(fact_5160_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_5161_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_5162_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_5163_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_5164_inf__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( semilattice_inf @ B )
=> ( ( inf_inf @ ( A > B ) )
= ( ^ [F4: A > B,G2: A > B,X2: A] : ( inf_inf @ B @ ( F4 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ).
% inf_fun_def
thf(fact_5165_inf__left__commute,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X: A,Y2: A,Z2: A] :
( ( inf_inf @ A @ X @ ( inf_inf @ A @ Y2 @ Z2 ) )
= ( inf_inf @ A @ Y2 @ ( inf_inf @ A @ X @ Z2 ) ) ) ) ).
% inf_left_commute
thf(fact_5166_inf_Oleft__commute,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( inf_inf @ A @ B2 @ ( inf_inf @ A @ A3 @ C2 ) )
= ( inf_inf @ A @ A3 @ ( inf_inf @ A @ B2 @ C2 ) ) ) ) ).
% inf.left_commute
thf(fact_5167_inf__commute,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( ( inf_inf @ A )
= ( ^ [X2: A,Y5: A] : ( inf_inf @ A @ Y5 @ X2 ) ) ) ) ).
% inf_commute
thf(fact_5168_inf_Ocommute,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( ( inf_inf @ A )
= ( ^ [A5: A,B3: A] : ( inf_inf @ A @ B3 @ A5 ) ) ) ) ).
% inf.commute
thf(fact_5169_inf__assoc,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X: A,Y2: A,Z2: A] :
( ( inf_inf @ A @ ( inf_inf @ A @ X @ Y2 ) @ Z2 )
= ( inf_inf @ A @ X @ ( inf_inf @ A @ Y2 @ Z2 ) ) ) ) ).
% inf_assoc
thf(fact_5170_inf_Oassoc,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( inf_inf @ A @ ( inf_inf @ A @ A3 @ B2 ) @ C2 )
= ( inf_inf @ A @ A3 @ ( inf_inf @ A @ B2 @ C2 ) ) ) ) ).
% inf.assoc
thf(fact_5171_inf__sup__aci_I1_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ( ( inf_inf @ A )
= ( ^ [X2: A,Y5: A] : ( inf_inf @ A @ Y5 @ X2 ) ) ) ) ).
% inf_sup_aci(1)
thf(fact_5172_inf__sup__aci_I2_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X: A,Y2: A,Z2: A] :
( ( inf_inf @ A @ ( inf_inf @ A @ X @ Y2 ) @ Z2 )
= ( inf_inf @ A @ X @ ( inf_inf @ A @ Y2 @ Z2 ) ) ) ) ).
% inf_sup_aci(2)
thf(fact_5173_inf__sup__aci_I3_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X: A,Y2: A,Z2: A] :
( ( inf_inf @ A @ X @ ( inf_inf @ A @ Y2 @ Z2 ) )
= ( inf_inf @ A @ Y2 @ ( inf_inf @ A @ X @ Z2 ) ) ) ) ).
% inf_sup_aci(3)
thf(fact_5174_inf__sup__aci_I4_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X: A,Y2: A] :
( ( inf_inf @ A @ X @ ( inf_inf @ A @ X @ Y2 ) )
= ( inf_inf @ A @ X @ Y2 ) ) ) ).
% inf_sup_aci(4)
thf(fact_5175_eq__rat_I3_J,axiom,
! [A3: int,C2: int] :
( ( fract @ ( zero_zero @ int ) @ A3 )
= ( fract @ ( zero_zero @ int ) @ C2 ) ) ).
% eq_rat(3)
thf(fact_5176_diff__eq,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ( ( minus_minus @ A )
= ( ^ [X2: A,Y5: A] : ( inf_inf @ A @ X2 @ ( uminus_uminus @ A @ Y5 ) ) ) ) ) ).
% diff_eq
thf(fact_5177_inf__cancel__left2,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,A3: A,B2: A] :
( ( inf_inf @ A @ ( inf_inf @ A @ ( uminus_uminus @ A @ X ) @ A3 ) @ ( inf_inf @ A @ X @ B2 ) )
= ( bot_bot @ A ) ) ) ).
% inf_cancel_left2
thf(fact_5178_inf__cancel__left1,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,A3: A,B2: A] :
( ( inf_inf @ A @ ( inf_inf @ A @ X @ A3 ) @ ( inf_inf @ A @ ( uminus_uminus @ A @ X ) @ B2 ) )
= ( bot_bot @ A ) ) ) ).
% inf_cancel_left1
thf(fact_5179_zero__notin__Suc__image,axiom,
! [A4: set @ nat] :
~ ( member @ nat @ ( zero_zero @ nat ) @ ( image @ nat @ nat @ suc @ A4 ) ) ).
% zero_notin_Suc_image
thf(fact_5180_translation__diff,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A3: A,S: set @ A,T2: set @ A] :
( ( image @ A @ A @ ( plus_plus @ A @ A3 ) @ ( minus_minus @ ( set @ A ) @ S @ T2 ) )
= ( minus_minus @ ( set @ A ) @ ( image @ A @ A @ ( plus_plus @ A @ A3 ) @ S ) @ ( image @ A @ A @ ( plus_plus @ A @ A3 ) @ T2 ) ) ) ) ).
% translation_diff
thf(fact_5181_image__diff__subset,axiom,
! [A: $tType,B: $tType,F2: B > A,A4: set @ B,B7: set @ B] : ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ ( image @ B @ A @ F2 @ A4 ) @ ( image @ B @ A @ F2 @ B7 ) ) @ ( image @ B @ A @ F2 @ ( minus_minus @ ( set @ B ) @ A4 @ B7 ) ) ) ).
% image_diff_subset
thf(fact_5182_eq__rat_I2_J,axiom,
! [A3: int] :
( ( fract @ A3 @ ( zero_zero @ int ) )
= ( fract @ ( zero_zero @ int ) @ ( one_one @ int ) ) ) ).
% eq_rat(2)
thf(fact_5183_Rat__induct__pos,axiom,
! [P: rat > $o,Q5: rat] :
( ! [A6: int,B4: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ B4 )
=> ( P @ ( fract @ A6 @ B4 ) ) )
=> ( P @ Q5 ) ) ).
% Rat_induct_pos
thf(fact_5184_mult__rat__cancel,axiom,
! [C2: int,A3: int,B2: int] :
( ( C2
!= ( zero_zero @ int ) )
=> ( ( fract @ ( times_times @ int @ C2 @ A3 ) @ ( times_times @ int @ C2 @ B2 ) )
= ( fract @ A3 @ B2 ) ) ) ).
% mult_rat_cancel
thf(fact_5185_eq__rat_I1_J,axiom,
! [B2: int,D2: int,A3: int,C2: int] :
( ( B2
!= ( zero_zero @ int ) )
=> ( ( D2
!= ( zero_zero @ int ) )
=> ( ( ( fract @ A3 @ B2 )
= ( fract @ C2 @ D2 ) )
= ( ( times_times @ int @ A3 @ D2 )
= ( times_times @ int @ C2 @ B2 ) ) ) ) ) ).
% eq_rat(1)
thf(fact_5186_Fract__of__nat__eq,axiom,
! [K: nat] :
( ( fract @ ( semiring_1_of_nat @ int @ K ) @ ( one_one @ int ) )
= ( semiring_1_of_nat @ rat @ K ) ) ).
% Fract_of_nat_eq
thf(fact_5187_translation__Compl,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A3: A,T2: set @ A] :
( ( image @ A @ A @ ( plus_plus @ A @ A3 ) @ ( uminus_uminus @ ( set @ A ) @ T2 ) )
= ( uminus_uminus @ ( set @ A ) @ ( image @ A @ A @ ( plus_plus @ A @ A3 ) @ T2 ) ) ) ) ).
% translation_Compl
thf(fact_5188_Diff__triv,axiom,
! [A: $tType,A4: set @ A,B7: set @ A] :
( ( ( inf_inf @ ( set @ A ) @ A4 @ B7 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( minus_minus @ ( set @ A ) @ A4 @ B7 )
= A4 ) ) ).
% Diff_triv
thf(fact_5189_Int__Diff__disjoint,axiom,
! [A: $tType,A4: set @ A,B7: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A4 @ B7 ) @ ( minus_minus @ ( set @ A ) @ A4 @ B7 ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Int_Diff_disjoint
thf(fact_5190_rat__number__collapse_I6_J,axiom,
! [K: int] :
( ( fract @ K @ ( zero_zero @ int ) )
= ( zero_zero @ rat ) ) ).
% rat_number_collapse(6)
thf(fact_5191_rat__number__collapse_I1_J,axiom,
! [K: int] :
( ( fract @ ( zero_zero @ int ) @ K )
= ( zero_zero @ rat ) ) ).
% rat_number_collapse(1)
thf(fact_5192_Fract__coprime,axiom,
! [A3: int,B2: int] :
( ( fract @ ( divide_divide @ int @ A3 @ ( gcd_gcd @ int @ A3 @ B2 ) ) @ ( divide_divide @ int @ B2 @ ( gcd_gcd @ int @ A3 @ B2 ) ) )
= ( fract @ A3 @ B2 ) ) ).
% Fract_coprime
thf(fact_5193_Diff__eq,axiom,
! [A: $tType] :
( ( minus_minus @ ( set @ A ) )
= ( ^ [A7: set @ A,B5: set @ A] : ( inf_inf @ ( set @ A ) @ A7 @ ( uminus_uminus @ ( set @ A ) @ B5 ) ) ) ) ).
% Diff_eq
thf(fact_5194_One__rat__def,axiom,
( ( one_one @ rat )
= ( fract @ ( one_one @ int ) @ ( one_one @ int ) ) ) ).
% One_rat_def
thf(fact_5195_Fract__of__int__eq,axiom,
! [K: int] :
( ( fract @ K @ ( one_one @ int ) )
= ( ring_1_of_int @ rat @ K ) ) ).
% Fract_of_int_eq
thf(fact_5196_nat__seg__image__imp__finite,axiom,
! [A: $tType,A4: set @ A,F2: nat > A,N: nat] :
( ( A4
= ( image @ nat @ A @ F2
@ ( collect @ nat
@ ^ [I2: nat] : ( ord_less @ nat @ I2 @ N ) ) ) )
=> ( finite_finite @ A @ A4 ) ) ).
% nat_seg_image_imp_finite
thf(fact_5197_finite__conv__nat__seg__image,axiom,
! [A: $tType] :
( ( finite_finite @ A )
= ( ^ [A7: set @ A] :
? [N5: nat,F4: nat > A] :
( A7
= ( image @ nat @ A @ F4
@ ( collect @ nat
@ ^ [I2: nat] : ( ord_less @ nat @ I2 @ N5 ) ) ) ) ) ) ).
% finite_conv_nat_seg_image
thf(fact_5198_translation__subtract__diff,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A3: A,S: set @ A,T2: set @ A] :
( ( image @ A @ A
@ ^ [X2: A] : ( minus_minus @ A @ X2 @ A3 )
@ ( minus_minus @ ( set @ A ) @ S @ T2 ) )
= ( minus_minus @ ( set @ A )
@ ( image @ A @ A
@ ^ [X2: A] : ( minus_minus @ A @ X2 @ A3 )
@ S )
@ ( image @ A @ A
@ ^ [X2: A] : ( minus_minus @ A @ X2 @ A3 )
@ T2 ) ) ) ) ).
% translation_subtract_diff
thf(fact_5199_translation__subtract__Compl,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A3: A,T2: set @ A] :
( ( image @ A @ A
@ ^ [X2: A] : ( minus_minus @ A @ X2 @ A3 )
@ ( uminus_uminus @ ( set @ A ) @ T2 ) )
= ( uminus_uminus @ ( set @ A )
@ ( image @ A @ A
@ ^ [X2: A] : ( minus_minus @ A @ X2 @ A3 )
@ T2 ) ) ) ) ).
% translation_subtract_Compl
thf(fact_5200_Fract__of__int__quotient,axiom,
( fract
= ( ^ [K3: int,L2: int] : ( divide_divide @ rat @ ( ring_1_of_int @ rat @ K3 ) @ ( ring_1_of_int @ rat @ L2 ) ) ) ) ).
% Fract_of_int_quotient
thf(fact_5201_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_5202_Zero__rat__def,axiom,
( ( zero_zero @ rat )
= ( fract @ ( zero_zero @ int ) @ ( one_one @ int ) ) ) ).
% Zero_rat_def
thf(fact_5203_disjoint__eq__subset__Compl,axiom,
! [A: $tType,A4: set @ A,B7: set @ A] :
( ( ( inf_inf @ ( set @ A ) @ A4 @ B7 )
= ( bot_bot @ ( set @ A ) ) )
= ( ord_less_eq @ ( set @ A ) @ A4 @ ( uminus_uminus @ ( set @ A ) @ B7 ) ) ) ).
% disjoint_eq_subset_Compl
thf(fact_5204_rat__number__collapse_I3_J,axiom,
! [W: num] :
( ( fract @ ( numeral_numeral @ int @ W ) @ ( one_one @ int ) )
= ( numeral_numeral @ rat @ W ) ) ).
% rat_number_collapse(3)
thf(fact_5205_rat__number__expand_I3_J,axiom,
( ( numeral_numeral @ rat )
= ( ^ [K3: num] : ( fract @ ( numeral_numeral @ int @ K3 ) @ ( one_one @ int ) ) ) ) ).
% rat_number_expand(3)
thf(fact_5206_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_5207_sum_Ointer__restrict,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A4: set @ B,G: B > A,B7: set @ B] :
( ( finite_finite @ B @ A4 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( inf_inf @ ( set @ B ) @ A4 @ B7 ) )
= ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X2: B] : ( if @ A @ ( member @ B @ X2 @ B7 ) @ ( G @ X2 ) @ ( zero_zero @ A ) )
@ A4 ) ) ) ) ).
% sum.inter_restrict
thf(fact_5208_prod_Ointer__restrict,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A4: set @ B,G: B > A,B7: set @ B] :
( ( finite_finite @ B @ A4 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( inf_inf @ ( set @ B ) @ A4 @ B7 ) )
= ( groups7121269368397514597t_prod @ B @ A
@ ^ [X2: B] : ( if @ A @ ( member @ B @ X2 @ B7 ) @ ( G @ X2 ) @ ( one_one @ A ) )
@ A4 ) ) ) ) ).
% prod.inter_restrict
thf(fact_5209_sum_Oreindex__nontrivial,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A4: set @ B,H: B > C,G: C > A] :
( ( finite_finite @ B @ A4 )
=> ( ! [X3: B,Y3: B] :
( ( member @ B @ X3 @ A4 )
=> ( ( member @ B @ Y3 @ A4 )
=> ( ( X3 != Y3 )
=> ( ( ( H @ X3 )
= ( H @ Y3 ) )
=> ( ( G @ ( H @ X3 ) )
= ( zero_zero @ A ) ) ) ) ) )
=> ( ( groups7311177749621191930dd_sum @ C @ A @ G @ ( image @ B @ C @ H @ A4 ) )
= ( groups7311177749621191930dd_sum @ B @ A @ ( comp @ C @ A @ B @ G @ H ) @ A4 ) ) ) ) ) ).
% sum.reindex_nontrivial
thf(fact_5210_prod_Oreindex__nontrivial,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A4: set @ B,H: B > C,G: C > A] :
( ( finite_finite @ B @ A4 )
=> ( ! [X3: B,Y3: B] :
( ( member @ B @ X3 @ A4 )
=> ( ( member @ B @ Y3 @ A4 )
=> ( ( X3 != Y3 )
=> ( ( ( H @ X3 )
= ( H @ Y3 ) )
=> ( ( G @ ( H @ X3 ) )
= ( one_one @ A ) ) ) ) ) )
=> ( ( groups7121269368397514597t_prod @ C @ A @ G @ ( image @ B @ C @ H @ A4 ) )
= ( groups7121269368397514597t_prod @ B @ A @ ( comp @ C @ A @ B @ G @ H ) @ A4 ) ) ) ) ) ).
% prod.reindex_nontrivial
thf(fact_5211_sum_Omono__neutral__cong,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [T6: set @ B,S3: set @ B,H: B > A,G: B > A] :
( ( finite_finite @ B @ T6 )
=> ( ( finite_finite @ B @ S3 )
=> ( ! [I3: B] :
( ( member @ B @ I3 @ ( minus_minus @ ( set @ B ) @ T6 @ S3 ) )
=> ( ( H @ I3 )
= ( zero_zero @ A ) ) )
=> ( ! [I3: B] :
( ( member @ B @ I3 @ ( minus_minus @ ( set @ B ) @ S3 @ T6 ) )
=> ( ( G @ I3 )
= ( zero_zero @ A ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( inf_inf @ ( set @ B ) @ S3 @ T6 ) )
=> ( ( G @ X3 )
= ( H @ X3 ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ S3 )
= ( groups7311177749621191930dd_sum @ B @ A @ H @ T6 ) ) ) ) ) ) ) ) ).
% sum.mono_neutral_cong
thf(fact_5212_Iio__Int__singleton,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,K: A] :
( ( ( ord_less @ A @ X @ K )
=> ( ( inf_inf @ ( set @ A ) @ ( set_ord_lessThan @ A @ K ) @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
& ( ~ ( ord_less @ A @ X @ K )
=> ( ( inf_inf @ ( set @ A ) @ ( set_ord_lessThan @ A @ K ) @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% Iio_Int_singleton
thf(fact_5213_sum_OInt__Diff,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A4: set @ B,G: B > A,B7: set @ B] :
( ( finite_finite @ B @ A4 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ A4 )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G @ ( inf_inf @ ( set @ B ) @ A4 @ B7 ) ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A4 @ B7 ) ) ) ) ) ) ).
% sum.Int_Diff
thf(fact_5214_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_5215_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_5216_prod_OInt__Diff,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A4: set @ B,G: B > A,B7: set @ B] :
( ( finite_finite @ B @ A4 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ A4 )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G @ ( inf_inf @ ( set @ B ) @ A4 @ B7 ) ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A4 @ B7 ) ) ) ) ) ) ).
% prod.Int_Diff
thf(fact_5217_prod_Omono__neutral__cong,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [T6: set @ B,S3: set @ B,H: B > A,G: B > A] :
( ( finite_finite @ B @ T6 )
=> ( ( finite_finite @ B @ S3 )
=> ( ! [I3: B] :
( ( member @ B @ I3 @ ( minus_minus @ ( set @ B ) @ T6 @ S3 ) )
=> ( ( H @ I3 )
= ( one_one @ A ) ) )
=> ( ! [I3: B] :
( ( member @ B @ I3 @ ( minus_minus @ ( set @ B ) @ S3 @ T6 ) )
=> ( ( G @ I3 )
= ( one_one @ A ) ) )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( inf_inf @ ( set @ B ) @ S3 @ T6 ) )
=> ( ( G @ X3 )
= ( H @ X3 ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ S3 )
= ( groups7121269368397514597t_prod @ B @ A @ H @ T6 ) ) ) ) ) ) ) ) ).
% prod.mono_neutral_cong
thf(fact_5218_card__Diff__subset__Int,axiom,
! [A: $tType,A4: set @ A,B7: set @ A] :
( ( finite_finite @ A @ ( inf_inf @ ( set @ A ) @ A4 @ B7 ) )
=> ( ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A4 @ B7 ) )
= ( minus_minus @ nat @ ( finite_card @ A @ A4 ) @ ( finite_card @ A @ ( inf_inf @ ( set @ A ) @ A4 @ B7 ) ) ) ) ) ).
% card_Diff_subset_Int
thf(fact_5219_atLeast0__atMost__Suc__eq__insert__0,axiom,
! [N: nat] :
( ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) )
= ( insert @ nat @ ( zero_zero @ nat ) @ ( image @ nat @ nat @ suc @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% atLeast0_atMost_Suc_eq_insert_0
thf(fact_5220_atLeast0__lessThan__Suc__eq__insert__0,axiom,
! [N: nat] :
( ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) )
= ( insert @ nat @ ( zero_zero @ nat ) @ ( image @ nat @ nat @ suc @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% atLeast0_lessThan_Suc_eq_insert_0
thf(fact_5221_lessThan__Suc__eq__insert__0,axiom,
! [N: nat] :
( ( set_ord_lessThan @ nat @ ( suc @ N ) )
= ( insert @ nat @ ( zero_zero @ nat ) @ ( image @ nat @ nat @ suc @ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% lessThan_Suc_eq_insert_0
thf(fact_5222_atMost__Suc__eq__insert__0,axiom,
! [N: nat] :
( ( set_ord_atMost @ nat @ ( suc @ N ) )
= ( insert @ nat @ ( zero_zero @ nat ) @ ( image @ nat @ nat @ suc @ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% atMost_Suc_eq_insert_0
thf(fact_5223_Fract__less__zero__iff,axiom,
! [B2: int,A3: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
=> ( ( ord_less @ rat @ ( fract @ A3 @ B2 ) @ ( zero_zero @ rat ) )
= ( ord_less @ int @ A3 @ ( zero_zero @ int ) ) ) ) ).
% Fract_less_zero_iff
thf(fact_5224_zero__less__Fract__iff,axiom,
! [B2: int,A3: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
=> ( ( ord_less @ rat @ ( zero_zero @ rat ) @ ( fract @ A3 @ B2 ) )
= ( ord_less @ int @ ( zero_zero @ int ) @ A3 ) ) ) ).
% zero_less_Fract_iff
thf(fact_5225_Fract__less__one__iff,axiom,
! [B2: int,A3: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
=> ( ( ord_less @ rat @ ( fract @ A3 @ B2 ) @ ( one_one @ rat ) )
= ( ord_less @ int @ A3 @ B2 ) ) ) ).
% Fract_less_one_iff
thf(fact_5226_one__less__Fract__iff,axiom,
! [B2: int,A3: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
=> ( ( ord_less @ rat @ ( one_one @ rat ) @ ( fract @ A3 @ B2 ) )
= ( ord_less @ int @ B2 @ A3 ) ) ) ).
% one_less_Fract_iff
thf(fact_5227_rat__number__collapse_I5_J,axiom,
( ( fract @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( one_one @ int ) )
= ( uminus_uminus @ rat @ ( one_one @ rat ) ) ) ).
% rat_number_collapse(5)
thf(fact_5228_Fract__add__one,axiom,
! [N: int,M2: int] :
( ( N
!= ( zero_zero @ int ) )
=> ( ( fract @ ( plus_plus @ int @ M2 @ N ) @ N )
= ( plus_plus @ rat @ ( fract @ M2 @ N ) @ ( one_one @ rat ) ) ) ) ).
% Fract_add_one
thf(fact_5229_sum_OIf__cases,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A4: set @ B,P: B > $o,H: B > A,G: B > A] :
( ( finite_finite @ B @ A4 )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X2: B] : ( if @ A @ ( P @ X2 ) @ ( H @ X2 ) @ ( G @ X2 ) )
@ A4 )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ H @ ( inf_inf @ ( set @ B ) @ A4 @ ( collect @ B @ P ) ) ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ ( inf_inf @ ( set @ B ) @ A4 @ ( uminus_uminus @ ( set @ B ) @ ( collect @ B @ P ) ) ) ) ) ) ) ) ).
% sum.If_cases
thf(fact_5230_prod_OIf__cases,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A4: set @ B,P: B > $o,H: B > A,G: B > A] :
( ( finite_finite @ B @ A4 )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [X2: B] : ( if @ A @ ( P @ X2 ) @ ( H @ X2 ) @ ( G @ X2 ) )
@ A4 )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ H @ ( inf_inf @ ( set @ B ) @ A4 @ ( collect @ B @ P ) ) ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ ( inf_inf @ ( set @ B ) @ A4 @ ( uminus_uminus @ ( set @ B ) @ ( collect @ B @ P ) ) ) ) ) ) ) ) ).
% prod.If_cases
thf(fact_5231_sum__image__le,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( ordere6911136660526730532id_add @ B )
=> ! [I5: set @ C,G: A > B,F2: C > A] :
( ( finite_finite @ C @ I5 )
=> ( ! [I3: C] :
( ( member @ C @ I3 @ I5 )
=> ( ord_less_eq @ B @ ( zero_zero @ B ) @ ( G @ ( F2 @ I3 ) ) ) )
=> ( ord_less_eq @ B @ ( groups7311177749621191930dd_sum @ A @ B @ G @ ( image @ C @ A @ F2 @ I5 ) ) @ ( groups7311177749621191930dd_sum @ C @ B @ ( comp @ A @ B @ C @ G @ F2 ) @ I5 ) ) ) ) ) ).
% sum_image_le
thf(fact_5232_zero__le__Fract__iff,axiom,
! [B2: int,A3: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
=> ( ( ord_less_eq @ rat @ ( zero_zero @ rat ) @ ( fract @ A3 @ B2 ) )
= ( ord_less_eq @ int @ ( zero_zero @ int ) @ A3 ) ) ) ).
% zero_le_Fract_iff
thf(fact_5233_Fract__le__zero__iff,axiom,
! [B2: int,A3: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
=> ( ( ord_less_eq @ rat @ ( fract @ A3 @ B2 ) @ ( zero_zero @ rat ) )
= ( ord_less_eq @ int @ A3 @ ( zero_zero @ int ) ) ) ) ).
% Fract_le_zero_iff
thf(fact_5234_sum__div__partition,axiom,
! [B: $tType,A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A4: set @ B,F2: B > A,B2: A] :
( ( finite_finite @ B @ A4 )
=> ( ( divide_divide @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A4 ) @ B2 )
= ( plus_plus @ A
@ ( groups7311177749621191930dd_sum @ B @ A
@ ^ [A5: B] : ( divide_divide @ A @ ( F2 @ A5 ) @ B2 )
@ ( inf_inf @ ( set @ B ) @ A4
@ ( collect @ B
@ ^ [A5: B] : ( dvd_dvd @ A @ B2 @ ( F2 @ A5 ) ) ) ) )
@ ( divide_divide @ A
@ ( groups7311177749621191930dd_sum @ B @ A @ F2
@ ( inf_inf @ ( set @ B ) @ A4
@ ( collect @ B
@ ^ [A5: B] :
~ ( dvd_dvd @ A @ B2 @ ( F2 @ A5 ) ) ) ) )
@ B2 ) ) ) ) ) ).
% sum_div_partition
thf(fact_5235_Fract__le__one__iff,axiom,
! [B2: int,A3: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
=> ( ( ord_less_eq @ rat @ ( fract @ A3 @ B2 ) @ ( one_one @ rat ) )
= ( ord_less_eq @ int @ A3 @ B2 ) ) ) ).
% Fract_le_one_iff
thf(fact_5236_one__le__Fract__iff,axiom,
! [B2: int,A3: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
=> ( ( ord_less_eq @ rat @ ( one_one @ rat ) @ ( fract @ A3 @ B2 ) )
= ( ord_less_eq @ int @ B2 @ A3 ) ) ) ).
% one_le_Fract_iff
thf(fact_5237_rat__number__expand_I5_J,axiom,
! [K: num] :
( ( uminus_uminus @ rat @ ( numeral_numeral @ rat @ K ) )
= ( fract @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K ) ) @ ( one_one @ int ) ) ) ).
% rat_number_expand(5)
thf(fact_5238_rat__number__collapse_I4_J,axiom,
! [W: num] :
( ( fract @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ W ) ) @ ( one_one @ int ) )
= ( uminus_uminus @ rat @ ( numeral_numeral @ rat @ W ) ) ) ).
% rat_number_collapse(4)
thf(fact_5239_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_5240_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
@ ^ [X2: A] : ( times_times @ A @ X2 @ 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
@ ^ [X2: A] : ( times_times @ A @ X2 @ 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
@ ^ [X2: A] : ( times_times @ A @ X2 @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ X @ Y2 ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% image_mult_atLeastAtMost_if'
thf(fact_5241_image__affinity__atLeastAtMost,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A,M2: A,C2: A] :
( ( ( ( set_or1337092689740270186AtMost @ A @ A3 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( image @ A @ A
@ ^ [X2: A] : ( plus_plus @ A @ ( times_times @ A @ M2 @ X2 ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( ( set_or1337092689740270186AtMost @ A @ A3 @ B2 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ M2 )
=> ( ( image @ A @ A
@ ^ [X2: A] : ( plus_plus @ A @ ( times_times @ A @ M2 @ X2 ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ ( times_times @ A @ M2 @ A3 ) @ C2 ) @ ( plus_plus @ A @ ( times_times @ A @ M2 @ B2 ) @ C2 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ M2 )
=> ( ( image @ A @ A
@ ^ [X2: A] : ( plus_plus @ A @ ( times_times @ A @ M2 @ X2 ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ ( times_times @ A @ M2 @ B2 ) @ C2 ) @ ( plus_plus @ A @ ( times_times @ A @ M2 @ A3 ) @ C2 ) ) ) ) ) ) ) ) ).
% image_affinity_atLeastAtMost
thf(fact_5242_image__affinity__atLeastAtMost__diff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A,M2: A,C2: A] :
( ( ( ( set_or1337092689740270186AtMost @ A @ A3 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( image @ A @ A
@ ^ [X2: A] : ( minus_minus @ A @ ( times_times @ A @ M2 @ X2 ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( ( set_or1337092689740270186AtMost @ A @ A3 @ B2 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ M2 )
=> ( ( image @ A @ A
@ ^ [X2: A] : ( minus_minus @ A @ ( times_times @ A @ M2 @ X2 ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ ( times_times @ A @ M2 @ A3 ) @ C2 ) @ ( minus_minus @ A @ ( times_times @ A @ M2 @ B2 ) @ C2 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ M2 )
=> ( ( image @ A @ A
@ ^ [X2: A] : ( minus_minus @ A @ ( times_times @ A @ M2 @ X2 ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ ( times_times @ A @ M2 @ B2 ) @ C2 ) @ ( minus_minus @ A @ ( times_times @ A @ M2 @ A3 ) @ C2 ) ) ) ) ) ) ) ) ).
% image_affinity_atLeastAtMost_diff
thf(fact_5243_image__affinity__atLeastAtMost__div,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A,M2: A,C2: A] :
( ( ( ( set_or1337092689740270186AtMost @ A @ A3 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( image @ A @ A
@ ^ [X2: A] : ( plus_plus @ A @ ( divide_divide @ A @ X2 @ M2 ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( ( set_or1337092689740270186AtMost @ A @ A3 @ B2 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ M2 )
=> ( ( image @ A @ A
@ ^ [X2: A] : ( plus_plus @ A @ ( divide_divide @ A @ X2 @ M2 ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ ( divide_divide @ A @ A3 @ M2 ) @ C2 ) @ ( plus_plus @ A @ ( divide_divide @ A @ B2 @ M2 ) @ C2 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ M2 )
=> ( ( image @ A @ A
@ ^ [X2: A] : ( plus_plus @ A @ ( divide_divide @ A @ X2 @ M2 ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ ( divide_divide @ A @ B2 @ M2 ) @ C2 ) @ ( plus_plus @ A @ ( divide_divide @ A @ A3 @ M2 ) @ C2 ) ) ) ) ) ) ) ) ).
% image_affinity_atLeastAtMost_div
thf(fact_5244_image__affinity__atLeastAtMost__div__diff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A,M2: A,C2: A] :
( ( ( ( set_or1337092689740270186AtMost @ A @ A3 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( image @ A @ A
@ ^ [X2: A] : ( minus_minus @ A @ ( divide_divide @ A @ X2 @ M2 ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( ( set_or1337092689740270186AtMost @ A @ A3 @ B2 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ M2 )
=> ( ( image @ A @ A
@ ^ [X2: A] : ( minus_minus @ A @ ( divide_divide @ A @ X2 @ M2 ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ ( divide_divide @ A @ A3 @ M2 ) @ C2 ) @ ( minus_minus @ A @ ( divide_divide @ A @ B2 @ M2 ) @ C2 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ M2 )
=> ( ( image @ A @ A
@ ^ [X2: A] : ( minus_minus @ A @ ( divide_divide @ A @ X2 @ M2 ) @ C2 )
@ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ ( divide_divide @ A @ B2 @ M2 ) @ C2 ) @ ( minus_minus @ A @ ( divide_divide @ A @ A3 @ M2 ) @ C2 ) ) ) ) ) ) ) ) ).
% image_affinity_atLeastAtMost_div_diff
thf(fact_5245_sum__fun__comp,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( semiring_1 @ C )
=> ! [S3: set @ A,R2: set @ B,G: A > B,F2: B > C] :
( ( finite_finite @ A @ S3 )
=> ( ( finite_finite @ B @ R2 )
=> ( ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ G @ S3 ) @ R2 )
=> ( ( groups7311177749621191930dd_sum @ A @ C
@ ^ [X2: A] : ( F2 @ ( G @ X2 ) )
@ S3 )
= ( groups7311177749621191930dd_sum @ B @ C
@ ^ [Y5: B] :
( times_times @ C
@ ( semiring_1_of_nat @ C
@ ( finite_card @ A
@ ( collect @ A
@ ^ [X2: A] :
( ( member @ A @ X2 @ S3 )
& ( ( G @ X2 )
= Y5 ) ) ) ) )
@ ( F2 @ Y5 ) )
@ R2 ) ) ) ) ) ) ).
% sum_fun_comp
thf(fact_5246_card__disjoint__shuffles,axiom,
! [A: $tType,Xs: list @ A,Ys2: list @ A] :
( ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ Ys2 ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_card @ ( list @ A ) @ ( shuffles @ A @ Xs @ Ys2 ) )
= ( binomial @ ( plus_plus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ A ) @ Ys2 ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ).
% card_disjoint_shuffles
thf(fact_5247_ring__1__class_Oof__int__def,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( ( ring_1_of_int @ A )
= ( map_fun @ int @ ( product_prod @ nat @ nat ) @ A @ A @ rep_Integ @ ( id @ A )
@ ( product_case_prod @ nat @ nat @ A
@ ^ [I2: nat,J3: nat] : ( minus_minus @ A @ ( semiring_1_of_nat @ A @ I2 ) @ ( semiring_1_of_nat @ A @ J3 ) ) ) ) ) ) ).
% ring_1_class.of_int_def
thf(fact_5248_take__bit__numeral__minus__numeral__int,axiom,
! [M2: num,N: num] :
( ( bit_se2584673776208193580ke_bit @ int @ ( numeral_numeral @ nat @ M2 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
= ( case_option @ int @ num @ ( zero_zero @ int )
@ ^ [Q4: num] : ( bit_se2584673776208193580ke_bit @ int @ ( numeral_numeral @ nat @ M2 ) @ ( minus_minus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ M2 ) ) @ ( numeral_numeral @ int @ Q4 ) ) )
@ ( bit_take_bit_num @ ( numeral_numeral @ nat @ M2 ) @ N ) ) ) ).
% take_bit_numeral_minus_numeral_int
thf(fact_5249_of__nat__eq__id,axiom,
( ( semiring_1_of_nat @ nat )
= ( id @ nat ) ) ).
% of_nat_eq_id
thf(fact_5250_id__funpow,axiom,
! [A: $tType,N: nat] :
( ( compow @ ( A > A ) @ N @ ( id @ A ) )
= ( id @ A ) ) ).
% id_funpow
thf(fact_5251_take__bit__num__simps_I1_J,axiom,
! [M2: num] :
( ( bit_take_bit_num @ ( zero_zero @ nat ) @ M2 )
= ( none @ num ) ) ).
% take_bit_num_simps(1)
thf(fact_5252_group__add__class_Ominus__comp__minus,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ( ( comp @ A @ A @ A @ ( uminus_uminus @ A ) @ ( uminus_uminus @ A ) )
= ( id @ A ) ) ) ).
% group_add_class.minus_comp_minus
thf(fact_5253_boolean__algebra__class_Ominus__comp__minus,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ( ( comp @ A @ A @ A @ ( uminus_uminus @ A ) @ ( uminus_uminus @ A ) )
= ( id @ A ) ) ) ).
% boolean_algebra_class.minus_comp_minus
thf(fact_5254_push__bit__0__id,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se4730199178511100633sh_bit @ A @ ( zero_zero @ nat ) )
= ( id @ A ) ) ) ).
% push_bit_0_id
thf(fact_5255_drop__bit__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se4197421643247451524op_bit @ A @ ( zero_zero @ nat ) )
= ( id @ A ) ) ) ).
% drop_bit_0
thf(fact_5256_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_5257_Gcd__int__eq,axiom,
! [N6: set @ nat] :
( ( gcd_Gcd @ int @ ( image @ nat @ int @ ( semiring_1_of_nat @ int ) @ N6 ) )
= ( semiring_1_of_nat @ int @ ( gcd_Gcd @ nat @ N6 ) ) ) ).
% Gcd_int_eq
thf(fact_5258_take__bit__numeral__numeral,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M2: num,N: num] :
( ( bit_se2584673776208193580ke_bit @ A @ ( numeral_numeral @ nat @ M2 ) @ ( numeral_numeral @ A @ N ) )
= ( case_option @ A @ num @ ( zero_zero @ A ) @ ( numeral_numeral @ A ) @ ( bit_take_bit_num @ ( numeral_numeral @ nat @ M2 ) @ N ) ) ) ) ).
% take_bit_numeral_numeral
thf(fact_5259_option_Osimps_I5_J,axiom,
! [B: $tType,A: $tType,F1: B,F22: A > B,X23: A] :
( ( case_option @ B @ A @ F1 @ F22 @ ( some @ A @ X23 ) )
= ( F22 @ X23 ) ) ).
% option.simps(5)
thf(fact_5260_option_Osimps_I4_J,axiom,
! [A: $tType,B: $tType,F1: B,F22: A > B] :
( ( case_option @ B @ A @ F1 @ F22 @ ( none @ A ) )
= F1 ) ).
% option.simps(4)
thf(fact_5261_option_Ocase__distrib,axiom,
! [C: $tType,B: $tType,A: $tType,H: B > C,F1: B,F22: A > B,Option: option @ A] :
( ( H @ ( case_option @ B @ A @ F1 @ F22 @ Option ) )
= ( case_option @ C @ A @ ( H @ F1 )
@ ^ [X2: A] : ( H @ ( F22 @ X2 ) )
@ Option ) ) ).
% option.case_distrib
thf(fact_5262_Code__Abstract__Nat_Otake__bit__num__code_I1_J,axiom,
! [N: nat] :
( ( bit_take_bit_num @ N @ one2 )
= ( case_nat @ ( option @ num ) @ ( none @ num )
@ ^ [N5: nat] : ( some @ num @ one2 )
@ N ) ) ).
% Code_Abstract_Nat.take_bit_num_code(1)
thf(fact_5263_funpow__simps__right_I1_J,axiom,
! [A: $tType,F2: A > A] :
( ( compow @ ( A > A ) @ ( zero_zero @ nat ) @ F2 )
= ( id @ A ) ) ).
% funpow_simps_right(1)
thf(fact_5264_None__notin__image__Some,axiom,
! [A: $tType,A4: set @ A] :
~ ( member @ ( option @ A ) @ ( none @ A ) @ ( image @ A @ ( option @ A ) @ ( some @ A ) @ A4 ) ) ).
% None_notin_image_Some
thf(fact_5265_length__shuffles,axiom,
! [A: $tType,Zs: list @ A,Xs: list @ A,Ys2: list @ A] :
( ( member @ ( list @ A ) @ Zs @ ( shuffles @ A @ Xs @ Ys2 ) )
=> ( ( size_size @ ( list @ A ) @ Zs )
= ( plus_plus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ A ) @ Ys2 ) ) ) ) ).
% length_shuffles
thf(fact_5266_less__int__def,axiom,
( ( ord_less @ int )
= ( map_fun @ int @ ( product_prod @ nat @ nat ) @ ( ( product_prod @ nat @ nat ) > $o ) @ ( int > $o ) @ rep_Integ @ ( map_fun @ int @ ( product_prod @ nat @ nat ) @ $o @ $o @ rep_Integ @ ( id @ $o ) )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [X2: nat,Y5: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U2: nat,V5: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ X2 @ V5 ) @ ( plus_plus @ nat @ U2 @ Y5 ) ) ) ) ) ) ).
% less_int_def
thf(fact_5267_less__eq__int__def,axiom,
( ( ord_less_eq @ int )
= ( map_fun @ int @ ( product_prod @ nat @ nat ) @ ( ( product_prod @ nat @ nat ) > $o ) @ ( int > $o ) @ rep_Integ @ ( map_fun @ int @ ( product_prod @ nat @ nat ) @ $o @ $o @ rep_Integ @ ( id @ $o ) )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [X2: nat,Y5: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U2: nat,V5: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ X2 @ V5 ) @ ( plus_plus @ nat @ U2 @ Y5 ) ) ) ) ) ) ).
% less_eq_int_def
thf(fact_5268_nat__def,axiom,
( nat2
= ( map_fun @ int @ ( product_prod @ nat @ nat ) @ nat @ nat @ rep_Integ @ ( id @ nat ) @ ( product_case_prod @ nat @ nat @ nat @ ( minus_minus @ nat ) ) ) ) ).
% nat_def
thf(fact_5269_in__image__insert__iff,axiom,
! [A: $tType,B7: set @ ( set @ A ),X: A,A4: set @ A] :
( ! [C6: set @ A] :
( ( member @ ( set @ A ) @ C6 @ B7 )
=> ~ ( member @ A @ X @ C6 ) )
=> ( ( member @ ( set @ A ) @ A4 @ ( image @ ( set @ A ) @ ( set @ A ) @ ( insert @ A @ X ) @ B7 ) )
= ( ( member @ A @ X @ A4 )
& ( member @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ B7 ) ) ) ) ).
% in_image_insert_iff
thf(fact_5270_image__int__atLeastAtMost,axiom,
! [A3: nat,B2: nat] :
( ( image @ nat @ int @ ( semiring_1_of_nat @ int ) @ ( set_or1337092689740270186AtMost @ nat @ A3 @ B2 ) )
= ( set_or1337092689740270186AtMost @ int @ ( semiring_1_of_nat @ int @ A3 ) @ ( semiring_1_of_nat @ int @ B2 ) ) ) ).
% image_int_atLeastAtMost
thf(fact_5271_image__int__atLeastLessThan,axiom,
! [A3: nat,B2: nat] :
( ( image @ nat @ int @ ( semiring_1_of_nat @ int ) @ ( set_or7035219750837199246ssThan @ nat @ A3 @ B2 ) )
= ( set_or7035219750837199246ssThan @ int @ ( semiring_1_of_nat @ int @ A3 ) @ ( semiring_1_of_nat @ int @ B2 ) ) ) ).
% image_int_atLeastLessThan
thf(fact_5272_take__bit__num__eq__None__imp,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M2: nat,N: num] :
( ( ( bit_take_bit_num @ M2 @ N )
= ( none @ num ) )
=> ( ( bit_se2584673776208193580ke_bit @ A @ M2 @ ( numeral_numeral @ A @ N ) )
= ( zero_zero @ A ) ) ) ) ).
% take_bit_num_eq_None_imp
thf(fact_5273_image__add__int__atLeastLessThan,axiom,
! [L: int,U: int] :
( ( image @ int @ int
@ ^ [X2: int] : ( plus_plus @ int @ X2 @ L )
@ ( set_or7035219750837199246ssThan @ int @ ( zero_zero @ int ) @ ( minus_minus @ int @ U @ L ) ) )
= ( set_or7035219750837199246ssThan @ int @ L @ U ) ) ).
% image_add_int_atLeastLessThan
thf(fact_5274_take__bit__num__def,axiom,
( bit_take_bit_num
= ( ^ [N5: nat,M5: num] :
( if @ ( option @ num )
@ ( ( bit_se2584673776208193580ke_bit @ nat @ N5 @ ( numeral_numeral @ nat @ M5 ) )
= ( zero_zero @ nat ) )
@ ( none @ num )
@ ( some @ num @ ( num_of_nat @ ( bit_se2584673776208193580ke_bit @ nat @ N5 @ ( numeral_numeral @ nat @ M5 ) ) ) ) ) ) ) ).
% take_bit_num_def
thf(fact_5275_Gcd__int__def,axiom,
( ( gcd_Gcd @ int )
= ( ^ [K5: set @ int] : ( semiring_1_of_nat @ int @ ( gcd_Gcd @ nat @ ( image @ int @ nat @ ( comp @ int @ nat @ int @ nat2 @ ( abs_abs @ int ) ) @ K5 ) ) ) ) ) ).
% Gcd_int_def
thf(fact_5276_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_5277_and__minus__numerals_I3_J,axiom,
! [M2: num,N: num] :
( ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ M2 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ N ) ) ) )
= ( case_option @ int @ num @ ( zero_zero @ int ) @ ( numeral_numeral @ int ) @ ( bit_and_not_num @ M2 @ ( bitM @ N ) ) ) ) ).
% and_minus_numerals(3)
thf(fact_5278_and__minus__numerals_I7_J,axiom,
! [N: num,M2: num] :
( ( bit_se5824344872417868541ns_and @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit0 @ N ) ) ) @ ( numeral_numeral @ int @ M2 ) )
= ( case_option @ int @ num @ ( zero_zero @ int ) @ ( numeral_numeral @ int ) @ ( bit_and_not_num @ M2 @ ( bitM @ N ) ) ) ) ).
% and_minus_numerals(7)
thf(fact_5279_and__minus__numerals_I8_J,axiom,
! [N: num,M2: num] :
( ( bit_se5824344872417868541ns_and @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ N ) ) ) @ ( numeral_numeral @ int @ M2 ) )
= ( case_option @ int @ num @ ( zero_zero @ int ) @ ( numeral_numeral @ int ) @ ( bit_and_not_num @ M2 @ ( bit0 @ N ) ) ) ) ).
% and_minus_numerals(8)
thf(fact_5280_take__bit__num__simps_I4_J,axiom,
! [N: nat,M2: num] :
( ( bit_take_bit_num @ ( suc @ N ) @ ( bit1 @ M2 ) )
= ( some @ num @ ( case_option @ num @ num @ one2 @ bit1 @ ( bit_take_bit_num @ N @ M2 ) ) ) ) ).
% take_bit_num_simps(4)
thf(fact_5281_take__bit__num__simps_I3_J,axiom,
! [N: nat,M2: num] :
( ( bit_take_bit_num @ ( suc @ N ) @ ( bit0 @ M2 ) )
= ( case_option @ ( option @ num ) @ num @ ( none @ num )
@ ^ [Q4: num] : ( some @ num @ ( bit0 @ Q4 ) )
@ ( bit_take_bit_num @ N @ M2 ) ) ) ).
% take_bit_num_simps(3)
thf(fact_5282_take__bit__num__simps_I6_J,axiom,
! [R4: num,M2: num] :
( ( bit_take_bit_num @ ( numeral_numeral @ nat @ R4 ) @ ( bit0 @ M2 ) )
= ( case_option @ ( option @ num ) @ num @ ( none @ num )
@ ^ [Q4: num] : ( some @ num @ ( bit0 @ Q4 ) )
@ ( bit_take_bit_num @ ( pred_numeral @ R4 ) @ M2 ) ) ) ).
% take_bit_num_simps(6)
thf(fact_5283_and__minus__numerals_I4_J,axiom,
! [M2: num,N: num] :
( ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ M2 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ ( bit1 @ N ) ) ) )
= ( case_option @ int @ num @ ( zero_zero @ int ) @ ( numeral_numeral @ int ) @ ( bit_and_not_num @ M2 @ ( bit0 @ N ) ) ) ) ).
% and_minus_numerals(4)
thf(fact_5284_and__not__num_Osimps_I1_J,axiom,
( ( bit_and_not_num @ one2 @ one2 )
= ( none @ num ) ) ).
% and_not_num.simps(1)
thf(fact_5285_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_5286_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_5287_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_5288_Code__Abstract__Nat_Otake__bit__num__code_I2_J,axiom,
! [N: nat,M2: num] :
( ( bit_take_bit_num @ N @ ( bit0 @ M2 ) )
= ( case_nat @ ( option @ num ) @ ( none @ num )
@ ^ [N5: nat] :
( case_option @ ( option @ num ) @ num @ ( none @ num )
@ ^ [Q4: num] : ( some @ num @ ( bit0 @ Q4 ) )
@ ( bit_take_bit_num @ N5 @ M2 ) )
@ N ) ) ).
% Code_Abstract_Nat.take_bit_num_code(2)
thf(fact_5289_case__optionE,axiom,
! [A: $tType,P: $o,Q: A > $o,X: option @ A] :
( ( case_option @ $o @ A @ P @ Q @ X )
=> ( ( ( X
= ( none @ A ) )
=> ~ P )
=> ~ ! [Y3: A] :
( ( X
= ( some @ A @ Y3 ) )
=> ~ ( Q @ Y3 ) ) ) ) ).
% case_optionE
thf(fact_5290_Code__Abstract__Nat_Otake__bit__num__code_I3_J,axiom,
! [N: nat,M2: num] :
( ( bit_take_bit_num @ N @ ( bit1 @ M2 ) )
= ( case_nat @ ( option @ num ) @ ( none @ num )
@ ^ [N5: nat] : ( some @ num @ ( case_option @ num @ num @ one2 @ bit1 @ ( bit_take_bit_num @ N5 @ M2 ) ) )
@ N ) ) ).
% Code_Abstract_Nat.take_bit_num_code(3)
thf(fact_5291_and__not__num__eq__None__iff,axiom,
! [M2: num,N: num] :
( ( ( bit_and_not_num @ M2 @ N )
= ( none @ num ) )
= ( ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ M2 ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ N ) ) )
= ( zero_zero @ int ) ) ) ).
% and_not_num_eq_None_iff
thf(fact_5292_int__numeral__and__not__num,axiom,
! [M2: num,N: num] :
( ( bit_se5824344872417868541ns_and @ int @ ( numeral_numeral @ int @ M2 ) @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ N ) ) )
= ( case_option @ int @ num @ ( zero_zero @ int ) @ ( numeral_numeral @ int ) @ ( bit_and_not_num @ M2 @ N ) ) ) ).
% int_numeral_and_not_num
thf(fact_5293_int__numeral__not__and__num,axiom,
! [M2: num,N: num] :
( ( bit_se5824344872417868541ns_and @ int @ ( bit_ri4277139882892585799ns_not @ int @ ( numeral_numeral @ int @ M2 ) ) @ ( numeral_numeral @ int @ N ) )
= ( case_option @ int @ num @ ( zero_zero @ int ) @ ( numeral_numeral @ int ) @ ( bit_and_not_num @ N @ M2 ) ) ) ).
% int_numeral_not_and_num
thf(fact_5294_measure__function__int,axiom,
fun_is_measure @ int @ ( comp @ int @ nat @ int @ nat2 @ ( abs_abs @ int ) ) ).
% measure_function_int
thf(fact_5295_Some__image__these__eq,axiom,
! [A: $tType,A4: set @ ( option @ A )] :
( ( image @ A @ ( option @ A ) @ ( some @ A ) @ ( these @ A @ A4 ) )
= ( collect @ ( option @ A )
@ ^ [X2: option @ A] :
( ( member @ ( option @ A ) @ X2 @ A4 )
& ( X2
!= ( none @ A ) ) ) ) ) ).
% Some_image_these_eq
thf(fact_5296_positive__rat,axiom,
! [A3: int,B2: int] :
( ( positive @ ( fract @ A3 @ B2 ) )
= ( ord_less @ int @ ( zero_zero @ int ) @ ( times_times @ int @ A3 @ B2 ) ) ) ).
% positive_rat
thf(fact_5297_these__empty,axiom,
! [A: $tType] :
( ( these @ A @ ( bot_bot @ ( set @ ( option @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% these_empty
thf(fact_5298_these__insert__None,axiom,
! [A: $tType,A4: set @ ( option @ A )] :
( ( these @ A @ ( insert @ ( option @ A ) @ ( none @ A ) @ A4 ) )
= ( these @ A @ A4 ) ) ).
% these_insert_None
thf(fact_5299_these__insert__Some,axiom,
! [A: $tType,X: A,A4: set @ ( option @ A )] :
( ( these @ A @ ( insert @ ( option @ A ) @ ( some @ A @ X ) @ A4 ) )
= ( insert @ A @ X @ ( these @ A @ A4 ) ) ) ).
% these_insert_Some
thf(fact_5300_these__image__Some__eq,axiom,
! [A: $tType,A4: set @ A] :
( ( these @ A @ ( image @ A @ ( option @ A ) @ ( some @ A ) @ A4 ) )
= A4 ) ).
% these_image_Some_eq
thf(fact_5301_in__these__eq,axiom,
! [A: $tType,X: A,A4: set @ ( option @ A )] :
( ( member @ A @ X @ ( these @ A @ A4 ) )
= ( member @ ( option @ A ) @ ( some @ A @ X ) @ A4 ) ) ).
% in_these_eq
thf(fact_5302_measure__size,axiom,
! [A: $tType] :
( ( size @ A )
=> ( fun_is_measure @ A @ ( size_size @ A ) ) ) ).
% measure_size
thf(fact_5303_these__not__empty__eq,axiom,
! [A: $tType,B7: set @ ( option @ A )] :
( ( ( these @ A @ B7 )
!= ( bot_bot @ ( set @ A ) ) )
= ( ( B7
!= ( bot_bot @ ( set @ ( option @ A ) ) ) )
& ( B7
!= ( insert @ ( option @ A ) @ ( none @ A ) @ ( bot_bot @ ( set @ ( option @ A ) ) ) ) ) ) ) ).
% these_not_empty_eq
thf(fact_5304_these__empty__eq,axiom,
! [A: $tType,B7: set @ ( option @ A )] :
( ( ( these @ A @ B7 )
= ( bot_bot @ ( set @ A ) ) )
= ( ( B7
= ( bot_bot @ ( set @ ( option @ A ) ) ) )
| ( B7
= ( insert @ ( option @ A ) @ ( none @ A ) @ ( bot_bot @ ( set @ ( option @ A ) ) ) ) ) ) ) ).
% these_empty_eq
thf(fact_5305_Rat_Opositive__minus,axiom,
! [X: rat] :
( ~ ( positive @ X )
=> ( ( X
!= ( zero_zero @ rat ) )
=> ( positive @ ( uminus_uminus @ rat @ X ) ) ) ) ).
% Rat.positive_minus
thf(fact_5306_less__rat__def,axiom,
( ( ord_less @ rat )
= ( ^ [X2: rat,Y5: rat] : ( positive @ ( minus_minus @ rat @ Y5 @ X2 ) ) ) ) ).
% less_rat_def
thf(fact_5307_Rat_Opositive_Orep__eq,axiom,
( positive
= ( ^ [X2: rat] : ( ord_less @ int @ ( zero_zero @ int ) @ ( times_times @ int @ ( product_fst @ int @ int @ ( rep_Rat @ X2 ) ) @ ( product_snd @ int @ int @ ( rep_Rat @ X2 ) ) ) ) ) ) ).
% Rat.positive.rep_eq
thf(fact_5308_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 )
=> ( ? [N2: num] :
( Xa
= ( bit0 @ N2 ) )
=> ( Y2
!= ( some @ num @ one2 ) ) ) )
=> ( ( ( X = one2 )
=> ( ? [N2: num] :
( Xa
= ( bit1 @ N2 ) )
=> ( Y2
!= ( none @ num ) ) ) )
=> ( ! [M3: num] :
( ( X
= ( bit0 @ M3 ) )
=> ( ( Xa = one2 )
=> ( Y2
!= ( some @ num @ ( bit0 @ M3 ) ) ) ) )
=> ( ! [M3: num] :
( ( X
= ( bit0 @ M3 ) )
=> ! [N2: num] :
( ( Xa
= ( bit0 @ N2 ) )
=> ( Y2
!= ( map_option @ num @ num @ bit0 @ ( bit_and_not_num @ M3 @ N2 ) ) ) ) )
=> ( ! [M3: num] :
( ( X
= ( bit0 @ M3 ) )
=> ! [N2: num] :
( ( Xa
= ( bit1 @ N2 ) )
=> ( Y2
!= ( map_option @ num @ num @ bit0 @ ( bit_and_not_num @ M3 @ N2 ) ) ) ) )
=> ( ! [M3: num] :
( ( X
= ( bit1 @ M3 ) )
=> ( ( Xa = one2 )
=> ( Y2
!= ( some @ num @ ( bit0 @ M3 ) ) ) ) )
=> ( ! [M3: num] :
( ( X
= ( bit1 @ M3 ) )
=> ! [N2: num] :
( ( Xa
= ( bit0 @ N2 ) )
=> ( Y2
!= ( case_option @ ( option @ num ) @ num @ ( some @ num @ one2 )
@ ^ [N8: num] : ( some @ num @ ( bit1 @ N8 ) )
@ ( bit_and_not_num @ M3 @ N2 ) ) ) ) )
=> ~ ! [M3: num] :
( ( X
= ( bit1 @ M3 ) )
=> ! [N2: num] :
( ( Xa
= ( bit1 @ N2 ) )
=> ( Y2
!= ( map_option @ num @ num @ bit0 @ ( bit_and_not_num @ M3 @ N2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% and_not_num.elims
thf(fact_5309_Bit__Operations_Otake__bit__num__code,axiom,
( bit_take_bit_num
= ( ^ [N5: nat,M5: num] :
( product_case_prod @ nat @ num @ ( option @ num )
@ ^ [A5: nat,X2: num] :
( case_nat @ ( option @ num ) @ ( none @ num )
@ ^ [O: nat] :
( case_num @ ( option @ num ) @ ( some @ num @ one2 )
@ ^ [P6: num] :
( case_option @ ( option @ num ) @ num @ ( none @ num )
@ ^ [Q4: num] : ( some @ num @ ( bit0 @ Q4 ) )
@ ( bit_take_bit_num @ O @ P6 ) )
@ ^ [P6: num] : ( some @ num @ ( case_option @ num @ num @ one2 @ bit1 @ ( bit_take_bit_num @ O @ P6 ) ) )
@ X2 )
@ A5 )
@ ( product_Pair @ nat @ num @ N5 @ M5 ) ) ) ) ).
% Bit_Operations.take_bit_num_code
thf(fact_5310_map__option__eq__Some,axiom,
! [A: $tType,B: $tType,F2: B > A,Xo: option @ B,Y2: A] :
( ( ( map_option @ B @ A @ F2 @ Xo )
= ( some @ A @ Y2 ) )
= ( ? [Z6: B] :
( ( Xo
= ( some @ B @ Z6 ) )
& ( ( F2 @ Z6 )
= Y2 ) ) ) ) ).
% map_option_eq_Some
thf(fact_5311_option_Omap__disc__iff,axiom,
! [B: $tType,A: $tType,F2: A > B,A3: option @ A] :
( ( ( map_option @ A @ B @ F2 @ A3 )
= ( none @ B ) )
= ( A3
= ( none @ A ) ) ) ).
% option.map_disc_iff
thf(fact_5312_map__option__is__None,axiom,
! [A: $tType,B: $tType,F2: B > A,Opt: option @ B] :
( ( ( map_option @ B @ A @ F2 @ Opt )
= ( none @ A ) )
= ( Opt
= ( none @ B ) ) ) ).
% map_option_is_None
thf(fact_5313_None__eq__map__option__iff,axiom,
! [A: $tType,B: $tType,F2: B > A,X: option @ B] :
( ( ( none @ A )
= ( map_option @ B @ A @ F2 @ X ) )
= ( X
= ( none @ B ) ) ) ).
% None_eq_map_option_iff
thf(fact_5314_case__map__option,axiom,
! [B: $tType,A: $tType,C: $tType,G: A,H: B > A,F2: C > B,X: option @ C] :
( ( case_option @ A @ B @ G @ H @ ( map_option @ C @ B @ F2 @ X ) )
= ( case_option @ A @ C @ G @ ( comp @ B @ A @ C @ H @ F2 ) @ X ) ) ).
% case_map_option
thf(fact_5315_num_Ocase__distrib,axiom,
! [B: $tType,A: $tType,H: A > B,F1: A,F22: num > A,F32: num > A,Num: num] :
( ( H @ ( case_num @ A @ F1 @ F22 @ F32 @ Num ) )
= ( case_num @ B @ ( H @ F1 )
@ ^ [X2: num] : ( H @ ( F22 @ X2 ) )
@ ^ [X2: num] : ( H @ ( F32 @ X2 ) )
@ Num ) ) ).
% num.case_distrib
thf(fact_5316_option_Omap__ident,axiom,
! [A: $tType,T2: option @ A] :
( ( map_option @ A @ A
@ ^ [X2: A] : X2
@ T2 )
= T2 ) ).
% option.map_ident
thf(fact_5317_option_Osimps_I9_J,axiom,
! [B: $tType,A: $tType,F2: A > B,X23: A] :
( ( map_option @ A @ B @ F2 @ ( some @ A @ X23 ) )
= ( some @ B @ ( F2 @ X23 ) ) ) ).
% option.simps(9)
thf(fact_5318_map__option__cong,axiom,
! [B: $tType,A: $tType,X: option @ A,Y2: option @ A,F2: A > B,G: A > B] :
( ( X = Y2 )
=> ( ! [A6: A] :
( ( Y2
= ( some @ A @ A6 ) )
=> ( ( F2 @ A6 )
= ( G @ A6 ) ) )
=> ( ( map_option @ A @ B @ F2 @ X )
= ( map_option @ A @ B @ G @ Y2 ) ) ) ) ).
% map_option_cong
thf(fact_5319_option_Osimps_I8_J,axiom,
! [A: $tType,B: $tType,F2: A > B] :
( ( map_option @ A @ B @ F2 @ ( none @ A ) )
= ( none @ B ) ) ).
% option.simps(8)
thf(fact_5320_map__option_Ocompositionality,axiom,
! [B: $tType,C: $tType,A: $tType,F2: B > C,G: A > B,Option: option @ A] :
( ( map_option @ B @ C @ F2 @ ( map_option @ A @ B @ G @ Option ) )
= ( map_option @ A @ C @ ( comp @ B @ C @ A @ F2 @ G ) @ Option ) ) ).
% map_option.compositionality
thf(fact_5321_option_Omap__comp,axiom,
! [B: $tType,C: $tType,A: $tType,G: B > C,F2: A > B,V2: option @ A] :
( ( map_option @ B @ C @ G @ ( map_option @ A @ B @ F2 @ V2 ) )
= ( map_option @ A @ C @ ( comp @ B @ C @ A @ G @ F2 ) @ V2 ) ) ).
% option.map_comp
thf(fact_5322_map__option_Ocomp,axiom,
! [C: $tType,B: $tType,A: $tType,F2: B > C,G: A > B] :
( ( comp @ ( option @ B ) @ ( option @ C ) @ ( option @ A ) @ ( map_option @ B @ C @ F2 ) @ ( map_option @ A @ B @ G ) )
= ( map_option @ A @ C @ ( comp @ B @ C @ A @ F2 @ G ) ) ) ).
% map_option.comp
thf(fact_5323_map__option_Oidentity,axiom,
! [A: $tType] :
( ( map_option @ A @ A
@ ^ [X2: A] : X2 )
= ( id @ ( option @ A ) ) ) ).
% map_option.identity
thf(fact_5324_option_Omap__id0,axiom,
! [A: $tType] :
( ( map_option @ A @ A @ ( id @ A ) )
= ( id @ ( option @ A ) ) ) ).
% option.map_id0
thf(fact_5325_option_Omap__id,axiom,
! [A: $tType,T2: option @ A] :
( ( map_option @ A @ A @ ( id @ A ) @ T2 )
= T2 ) ).
% option.map_id
thf(fact_5326_verit__eq__simplify_I17_J,axiom,
! [A: $tType,F1: A,F22: num > A,F32: num > A,X23: num] :
( ( case_num @ A @ F1 @ F22 @ F32 @ ( bit0 @ X23 ) )
= ( F22 @ X23 ) ) ).
% verit_eq_simplify(17)
thf(fact_5327_verit__eq__simplify_I16_J,axiom,
! [A: $tType,F1: A,F22: num > A,F32: num > A] :
( ( case_num @ A @ F1 @ F22 @ F32 @ one2 )
= F1 ) ).
% verit_eq_simplify(16)
thf(fact_5328_verit__eq__simplify_I18_J,axiom,
! [A: $tType,F1: A,F22: num > A,F32: num > A,X32: num] :
( ( case_num @ A @ F1 @ F22 @ F32 @ ( bit1 @ X32 ) )
= ( F32 @ X32 ) ) ).
% verit_eq_simplify(18)
thf(fact_5329_option_Osize__gen__o__map,axiom,
! [B: $tType,A: $tType,F2: B > nat,G: A > B] :
( ( comp @ ( option @ B ) @ nat @ ( option @ A ) @ ( size_option @ B @ F2 ) @ ( map_option @ A @ B @ G ) )
= ( size_option @ A @ ( comp @ B @ nat @ A @ F2 @ G ) ) ) ).
% option.size_gen_o_map
thf(fact_5330_map__option__case,axiom,
! [A: $tType,B: $tType] :
( ( map_option @ B @ A )
= ( ^ [F4: B > A] :
( case_option @ ( option @ A ) @ B @ ( none @ A )
@ ^ [X2: B] : ( some @ A @ ( F4 @ X2 ) ) ) ) ) ).
% map_option_case
thf(fact_5331_map__option__o__empty,axiom,
! [C: $tType,B: $tType,A: $tType,F2: C > B] :
( ( comp @ ( option @ C ) @ ( option @ B ) @ A @ ( map_option @ C @ B @ F2 )
@ ^ [X2: A] : ( none @ C ) )
= ( ^ [X2: A] : ( none @ B ) ) ) ).
% map_option_o_empty
thf(fact_5332_Rat_Opositive__def,axiom,
( positive
= ( map_fun @ rat @ ( product_prod @ int @ int ) @ $o @ $o @ rep_Rat @ ( id @ $o )
@ ^ [X2: product_prod @ int @ int] : ( ord_less @ int @ ( zero_zero @ int ) @ ( times_times @ int @ ( product_fst @ int @ int @ X2 ) @ ( product_snd @ int @ int @ X2 ) ) ) ) ) ).
% Rat.positive_def
thf(fact_5333_and__num_Oelims,axiom,
! [X: num,Xa: num,Y2: option @ num] :
( ( ( bit_un7362597486090784418nd_num @ X @ Xa )
= Y2 )
=> ( ( ( X = one2 )
=> ( ( Xa = one2 )
=> ( Y2
!= ( some @ num @ one2 ) ) ) )
=> ( ( ( X = one2 )
=> ( ? [N2: num] :
( Xa
= ( bit0 @ N2 ) )
=> ( Y2
!= ( none @ num ) ) ) )
=> ( ( ( X = one2 )
=> ( ? [N2: num] :
( Xa
= ( bit1 @ N2 ) )
=> ( Y2
!= ( some @ num @ one2 ) ) ) )
=> ( ( ? [M3: num] :
( X
= ( bit0 @ M3 ) )
=> ( ( Xa = one2 )
=> ( Y2
!= ( none @ num ) ) ) )
=> ( ! [M3: num] :
( ( X
= ( bit0 @ M3 ) )
=> ! [N2: num] :
( ( Xa
= ( bit0 @ N2 ) )
=> ( Y2
!= ( map_option @ num @ num @ bit0 @ ( bit_un7362597486090784418nd_num @ M3 @ N2 ) ) ) ) )
=> ( ! [M3: num] :
( ( X
= ( bit0 @ M3 ) )
=> ! [N2: num] :
( ( Xa
= ( bit1 @ N2 ) )
=> ( Y2
!= ( map_option @ num @ num @ bit0 @ ( bit_un7362597486090784418nd_num @ M3 @ N2 ) ) ) ) )
=> ( ( ? [M3: num] :
( X
= ( bit1 @ M3 ) )
=> ( ( Xa = one2 )
=> ( Y2
!= ( some @ num @ one2 ) ) ) )
=> ( ! [M3: num] :
( ( X
= ( bit1 @ M3 ) )
=> ! [N2: num] :
( ( Xa
= ( bit0 @ N2 ) )
=> ( Y2
!= ( map_option @ num @ num @ bit0 @ ( bit_un7362597486090784418nd_num @ M3 @ N2 ) ) ) ) )
=> ~ ! [M3: num] :
( ( X
= ( bit1 @ M3 ) )
=> ! [N2: num] :
( ( Xa
= ( bit1 @ N2 ) )
=> ( Y2
!= ( case_option @ ( option @ num ) @ num @ ( some @ num @ one2 )
@ ^ [N8: num] : ( some @ num @ ( bit1 @ N8 ) )
@ ( bit_un7362597486090784418nd_num @ M3 @ N2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% and_num.elims
thf(fact_5334_and__num_Osimps_I4_J,axiom,
! [M2: num] :
( ( bit_un7362597486090784418nd_num @ ( bit0 @ M2 ) @ one2 )
= ( none @ num ) ) ).
% and_num.simps(4)
thf(fact_5335_and__num_Osimps_I2_J,axiom,
! [N: num] :
( ( bit_un7362597486090784418nd_num @ one2 @ ( bit0 @ N ) )
= ( none @ num ) ) ).
% and_num.simps(2)
thf(fact_5336_and__num__eq__None__iff,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [M2: num,N: num] :
( ( ( bit_un7362597486090784418nd_num @ M2 @ N )
= ( none @ num ) )
= ( ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ M2 ) @ ( numeral_numeral @ A @ N ) )
= ( zero_zero @ A ) ) ) ) ).
% and_num_eq_None_iff
thf(fact_5337_numeral__and__num,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [M2: num,N: num] :
( ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ M2 ) @ ( numeral_numeral @ A @ N ) )
= ( case_option @ A @ num @ ( zero_zero @ A ) @ ( numeral_numeral @ A ) @ ( bit_un7362597486090784418nd_num @ M2 @ N ) ) ) ) ).
% numeral_and_num
thf(fact_5338_xor__num_Oelims,axiom,
! [X: num,Xa: num,Y2: option @ num] :
( ( ( bit_un2480387367778600638or_num @ X @ Xa )
= Y2 )
=> ( ( ( X = one2 )
=> ( ( Xa = one2 )
=> ( Y2
!= ( none @ num ) ) ) )
=> ( ( ( X = one2 )
=> ! [N2: num] :
( ( Xa
= ( bit0 @ N2 ) )
=> ( Y2
!= ( some @ num @ ( bit1 @ N2 ) ) ) ) )
=> ( ( ( X = one2 )
=> ! [N2: num] :
( ( Xa
= ( bit1 @ N2 ) )
=> ( Y2
!= ( some @ num @ ( bit0 @ N2 ) ) ) ) )
=> ( ! [M3: num] :
( ( X
= ( bit0 @ M3 ) )
=> ( ( Xa = one2 )
=> ( Y2
!= ( some @ num @ ( bit1 @ M3 ) ) ) ) )
=> ( ! [M3: num] :
( ( X
= ( bit0 @ M3 ) )
=> ! [N2: num] :
( ( Xa
= ( bit0 @ N2 ) )
=> ( Y2
!= ( map_option @ num @ num @ bit0 @ ( bit_un2480387367778600638or_num @ M3 @ N2 ) ) ) ) )
=> ( ! [M3: num] :
( ( X
= ( bit0 @ M3 ) )
=> ! [N2: num] :
( ( Xa
= ( bit1 @ N2 ) )
=> ( Y2
!= ( some @ num @ ( case_option @ num @ num @ one2 @ bit1 @ ( bit_un2480387367778600638or_num @ M3 @ N2 ) ) ) ) ) )
=> ( ! [M3: num] :
( ( X
= ( bit1 @ M3 ) )
=> ( ( Xa = one2 )
=> ( Y2
!= ( some @ num @ ( bit0 @ M3 ) ) ) ) )
=> ( ! [M3: num] :
( ( X
= ( bit1 @ M3 ) )
=> ! [N2: num] :
( ( Xa
= ( bit0 @ N2 ) )
=> ( Y2
!= ( some @ num @ ( case_option @ num @ num @ one2 @ bit1 @ ( bit_un2480387367778600638or_num @ M3 @ N2 ) ) ) ) ) )
=> ~ ! [M3: num] :
( ( X
= ( bit1 @ M3 ) )
=> ! [N2: num] :
( ( Xa
= ( bit1 @ N2 ) )
=> ( Y2
!= ( map_option @ num @ num @ bit0 @ ( bit_un2480387367778600638or_num @ M3 @ N2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% xor_num.elims
thf(fact_5339_of__rat__def,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( ( field_char_0_of_rat @ A )
= ( map_fun @ rat @ ( product_prod @ int @ int ) @ A @ A @ rep_Rat @ ( id @ A )
@ ^ [X2: product_prod @ int @ int] : ( divide_divide @ A @ ( ring_1_of_int @ A @ ( product_fst @ int @ int @ X2 ) ) @ ( ring_1_of_int @ A @ ( product_snd @ int @ int @ X2 ) ) ) ) ) ) ).
% of_rat_def
thf(fact_5340_option_Orec__o__map,axiom,
! [B: $tType,C: $tType,A: $tType,G: C,Ga: B > C,F2: A > B] :
( ( comp @ ( option @ B ) @ C @ ( option @ A ) @ ( rec_option @ C @ B @ G @ Ga ) @ ( map_option @ A @ B @ F2 ) )
= ( rec_option @ C @ A @ G
@ ^ [X2: A] : ( Ga @ ( F2 @ X2 ) ) ) ) ).
% option.rec_o_map
thf(fact_5341_of__rat__0,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( ( field_char_0_of_rat @ A @ ( zero_zero @ rat ) )
= ( zero_zero @ A ) ) ) ).
% of_rat_0
thf(fact_5342_of__rat__eq__0__iff,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: rat] :
( ( ( field_char_0_of_rat @ A @ A3 )
= ( zero_zero @ A ) )
= ( A3
= ( zero_zero @ rat ) ) ) ) ).
% of_rat_eq_0_iff
thf(fact_5343_zero__eq__of__rat__iff,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: rat] :
( ( ( zero_zero @ A )
= ( field_char_0_of_rat @ A @ A3 ) )
= ( ( zero_zero @ rat )
= A3 ) ) ) ).
% zero_eq_of_rat_iff
thf(fact_5344_of__rat__1,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( ( field_char_0_of_rat @ A @ ( one_one @ rat ) )
= ( one_one @ A ) ) ) ).
% of_rat_1
thf(fact_5345_of__rat__eq__1__iff,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: rat] :
( ( ( field_char_0_of_rat @ A @ A3 )
= ( one_one @ A ) )
= ( A3
= ( one_one @ rat ) ) ) ) ).
% of_rat_eq_1_iff
thf(fact_5346_one__eq__of__rat__iff,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: rat] :
( ( ( one_one @ A )
= ( field_char_0_of_rat @ A @ A3 ) )
= ( ( one_one @ rat )
= A3 ) ) ) ).
% one_eq_of_rat_iff
thf(fact_5347_of__rat__of__nat__eq,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [N: nat] :
( ( field_char_0_of_rat @ A @ ( semiring_1_of_nat @ rat @ N ) )
= ( semiring_1_of_nat @ A @ N ) ) ) ).
% of_rat_of_nat_eq
thf(fact_5348_of__rat__less__0__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [R4: rat] :
( ( ord_less @ A @ ( field_char_0_of_rat @ A @ R4 ) @ ( zero_zero @ A ) )
= ( ord_less @ rat @ R4 @ ( zero_zero @ rat ) ) ) ) ).
% of_rat_less_0_iff
thf(fact_5349_zero__less__of__rat__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [R4: rat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( field_char_0_of_rat @ A @ R4 ) )
= ( ord_less @ rat @ ( zero_zero @ rat ) @ R4 ) ) ) ).
% zero_less_of_rat_iff
thf(fact_5350_one__less__of__rat__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [R4: rat] :
( ( ord_less @ A @ ( one_one @ A ) @ ( field_char_0_of_rat @ A @ R4 ) )
= ( ord_less @ rat @ ( one_one @ rat ) @ R4 ) ) ) ).
% one_less_of_rat_iff
thf(fact_5351_of__rat__less__1__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [R4: rat] :
( ( ord_less @ A @ ( field_char_0_of_rat @ A @ R4 ) @ ( one_one @ A ) )
= ( ord_less @ rat @ R4 @ ( one_one @ rat ) ) ) ) ).
% of_rat_less_1_iff
thf(fact_5352_of__rat__le__0__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [R4: rat] :
( ( ord_less_eq @ A @ ( field_char_0_of_rat @ A @ R4 ) @ ( zero_zero @ A ) )
= ( ord_less_eq @ rat @ R4 @ ( zero_zero @ rat ) ) ) ) ).
% of_rat_le_0_iff
thf(fact_5353_zero__le__of__rat__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [R4: rat] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( field_char_0_of_rat @ A @ R4 ) )
= ( ord_less_eq @ rat @ ( zero_zero @ rat ) @ R4 ) ) ) ).
% zero_le_of_rat_iff
thf(fact_5354_of__rat__neg__one,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( ( field_char_0_of_rat @ A @ ( uminus_uminus @ rat @ ( one_one @ rat ) ) )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% of_rat_neg_one
thf(fact_5355_one__le__of__rat__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [R4: rat] :
( ( ord_less_eq @ A @ ( one_one @ A ) @ ( field_char_0_of_rat @ A @ R4 ) )
= ( ord_less_eq @ rat @ ( one_one @ rat ) @ R4 ) ) ) ).
% one_le_of_rat_iff
thf(fact_5356_of__rat__le__1__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [R4: rat] :
( ( ord_less_eq @ A @ ( field_char_0_of_rat @ A @ R4 ) @ ( one_one @ A ) )
= ( ord_less_eq @ rat @ R4 @ ( one_one @ rat ) ) ) ) ).
% of_rat_le_1_iff
thf(fact_5357_of__rat__neg__numeral__eq,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [W: num] :
( ( field_char_0_of_rat @ A @ ( uminus_uminus @ rat @ ( numeral_numeral @ rat @ W ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ W ) ) ) ) ).
% of_rat_neg_numeral_eq
thf(fact_5358_of__rat__dense,axiom,
! [X: real,Y2: real] :
( ( ord_less @ real @ X @ Y2 )
=> ? [Q3: rat] :
( ( ord_less @ real @ X @ ( field_char_0_of_rat @ real @ Q3 ) )
& ( ord_less @ real @ ( field_char_0_of_rat @ real @ Q3 ) @ Y2 ) ) ) ).
% of_rat_dense
thf(fact_5359_of__rat__less,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [R4: rat,S: rat] :
( ( ord_less @ A @ ( field_char_0_of_rat @ A @ R4 ) @ ( field_char_0_of_rat @ A @ S ) )
= ( ord_less @ rat @ R4 @ S ) ) ) ).
% of_rat_less
thf(fact_5360_of__rat__minus,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: rat] :
( ( field_char_0_of_rat @ A @ ( uminus_uminus @ rat @ A3 ) )
= ( uminus_uminus @ A @ ( field_char_0_of_rat @ A @ A3 ) ) ) ) ).
% of_rat_minus
thf(fact_5361_of__rat__diff,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: rat,B2: rat] :
( ( field_char_0_of_rat @ A @ ( minus_minus @ rat @ A3 @ B2 ) )
= ( minus_minus @ A @ ( field_char_0_of_rat @ A @ A3 ) @ ( field_char_0_of_rat @ A @ B2 ) ) ) ) ).
% of_rat_diff
thf(fact_5362_of__rat__divide,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: rat,B2: rat] :
( ( field_char_0_of_rat @ A @ ( divide_divide @ rat @ A3 @ B2 ) )
= ( divide_divide @ A @ ( field_char_0_of_rat @ A @ A3 ) @ ( field_char_0_of_rat @ A @ B2 ) ) ) ) ).
% of_rat_divide
thf(fact_5363_option_Osimps_I7_J,axiom,
! [C: $tType,A: $tType,F1: C,F22: A > C,X23: A] :
( ( rec_option @ C @ A @ F1 @ F22 @ ( some @ A @ X23 ) )
= ( F22 @ X23 ) ) ).
% option.simps(7)
thf(fact_5364_option_Osimps_I6_J,axiom,
! [A: $tType,C: $tType,F1: C,F22: A > C] :
( ( rec_option @ C @ A @ F1 @ F22 @ ( none @ A ) )
= F1 ) ).
% option.simps(6)
thf(fact_5365_xor__num_Osimps_I1_J,axiom,
( ( bit_un2480387367778600638or_num @ one2 @ one2 )
= ( none @ num ) ) ).
% xor_num.simps(1)
thf(fact_5366_nonzero__of__rat__divide,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [B2: rat,A3: rat] :
( ( B2
!= ( zero_zero @ rat ) )
=> ( ( field_char_0_of_rat @ A @ ( divide_divide @ rat @ A3 @ B2 ) )
= ( divide_divide @ A @ ( field_char_0_of_rat @ A @ A3 ) @ ( field_char_0_of_rat @ A @ B2 ) ) ) ) ) ).
% nonzero_of_rat_divide
thf(fact_5367_of__rat__rat,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [B2: int,A3: int] :
( ( B2
!= ( zero_zero @ int ) )
=> ( ( field_char_0_of_rat @ A @ ( fract @ A3 @ B2 ) )
= ( divide_divide @ A @ ( ring_1_of_int @ A @ A3 ) @ ( ring_1_of_int @ A @ B2 ) ) ) ) ) ).
% of_rat_rat
thf(fact_5368_xor__num__eq__None__iff,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [M2: num,N: num] :
( ( ( bit_un2480387367778600638or_num @ M2 @ N )
= ( none @ num ) )
= ( ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ M2 ) @ ( numeral_numeral @ A @ N ) )
= ( zero_zero @ A ) ) ) ) ).
% xor_num_eq_None_iff
thf(fact_5369_of__rat_Orep__eq,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( ( field_char_0_of_rat @ A )
= ( ^ [X2: rat] : ( divide_divide @ A @ ( ring_1_of_int @ A @ ( product_fst @ int @ int @ ( rep_Rat @ X2 ) ) ) @ ( ring_1_of_int @ A @ ( product_snd @ int @ int @ ( rep_Rat @ X2 ) ) ) ) ) ) ) ).
% of_rat.rep_eq
thf(fact_5370_numeral__xor__num,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [M2: num,N: num] :
( ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ M2 ) @ ( numeral_numeral @ A @ N ) )
= ( case_option @ A @ num @ ( zero_zero @ A ) @ ( numeral_numeral @ A ) @ ( bit_un2480387367778600638or_num @ M2 @ N ) ) ) ) ).
% numeral_xor_num
thf(fact_5371_inverse__rat__def,axiom,
( ( inverse_inverse @ rat )
= ( map_fun @ rat @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ rat @ rep_Rat @ abs_Rat
@ ^ [X2: product_prod @ int @ int] :
( if @ ( product_prod @ int @ int )
@ ( ( product_fst @ int @ int @ X2 )
= ( zero_zero @ int ) )
@ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
@ ( product_Pair @ int @ int @ ( product_snd @ int @ int @ X2 ) @ ( product_fst @ int @ int @ X2 ) ) ) ) ) ).
% inverse_rat_def
thf(fact_5372_uminus__rat__def,axiom,
( ( uminus_uminus @ rat )
= ( map_fun @ rat @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ rat @ rep_Rat @ abs_Rat
@ ^ [X2: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( uminus_uminus @ int @ ( product_fst @ int @ int @ X2 ) ) @ ( product_snd @ int @ int @ X2 ) ) ) ) ).
% uminus_rat_def
thf(fact_5373_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_5374_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_5375_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_5376_nth__take,axiom,
! [A: $tType,I: nat,N: nat,Xs: list @ A] :
( ( ord_less @ nat @ I @ N )
=> ( ( nth @ A @ ( take @ A @ N @ Xs ) @ I )
= ( nth @ A @ Xs @ I ) ) ) ).
% nth_take
thf(fact_5377_nth__take__lemma,axiom,
! [A: $tType,K: nat,Xs: list @ A,Ys2: list @ A] :
( ( ord_less_eq @ nat @ K @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ord_less_eq @ nat @ K @ ( size_size @ ( list @ A ) @ Ys2 ) )
=> ( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ K )
=> ( ( nth @ A @ Xs @ I3 )
= ( nth @ A @ Ys2 @ I3 ) ) )
=> ( ( take @ A @ K @ Xs )
= ( take @ A @ K @ Ys2 ) ) ) ) ) ).
% nth_take_lemma
thf(fact_5378_one__rat__def,axiom,
( ( one_one @ rat )
= ( abs_Rat @ ( product_Pair @ int @ int @ ( one_one @ int ) @ ( one_one @ int ) ) ) ) ).
% one_rat_def
thf(fact_5379_Fract_Oabs__eq,axiom,
( fract
= ( ^ [Xa4: int,X2: int] :
( abs_Rat
@ ( if @ ( product_prod @ int @ int )
@ ( X2
= ( zero_zero @ int ) )
@ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
@ ( product_Pair @ int @ int @ Xa4 @ X2 ) ) ) ) ) ).
% Fract.abs_eq
thf(fact_5380_zero__rat__def,axiom,
( ( zero_zero @ rat )
= ( abs_Rat @ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) ) ) ) ).
% zero_rat_def
thf(fact_5381_inverse__rat_Oabs__eq,axiom,
! [X: product_prod @ int @ int] :
( ( ratrel @ X @ X )
=> ( ( inverse_inverse @ rat @ ( abs_Rat @ X ) )
= ( abs_Rat
@ ( if @ ( product_prod @ int @ int )
@ ( ( product_fst @ int @ int @ X )
= ( zero_zero @ int ) )
@ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
@ ( product_Pair @ int @ int @ ( product_snd @ int @ int @ X ) @ ( product_fst @ int @ int @ X ) ) ) ) ) ) ).
% inverse_rat.abs_eq
thf(fact_5382_Rat_Opositive_Oabs__eq,axiom,
! [X: product_prod @ int @ int] :
( ( ratrel @ X @ X )
=> ( ( positive @ ( abs_Rat @ X ) )
= ( ord_less @ int @ ( zero_zero @ int ) @ ( times_times @ int @ ( product_fst @ int @ int @ X ) @ ( product_snd @ int @ int @ X ) ) ) ) ) ).
% Rat.positive.abs_eq
thf(fact_5383_uminus__rat_Oabs__eq,axiom,
! [X: product_prod @ int @ int] :
( ( ratrel @ X @ X )
=> ( ( uminus_uminus @ rat @ ( abs_Rat @ X ) )
= ( abs_Rat @ ( product_Pair @ int @ int @ ( uminus_uminus @ int @ ( product_fst @ int @ int @ X ) ) @ ( product_snd @ int @ int @ X ) ) ) ) ) ).
% uminus_rat.abs_eq
thf(fact_5384_ratrel__iff,axiom,
( ratrel
= ( ^ [X2: product_prod @ int @ int,Y5: product_prod @ int @ int] :
( ( ( product_snd @ int @ int @ X2 )
!= ( zero_zero @ int ) )
& ( ( product_snd @ int @ int @ Y5 )
!= ( zero_zero @ int ) )
& ( ( times_times @ int @ ( product_fst @ int @ int @ X2 ) @ ( product_snd @ int @ int @ Y5 ) )
= ( times_times @ int @ ( product_fst @ int @ int @ Y5 ) @ ( product_snd @ int @ int @ X2 ) ) ) ) ) ) ).
% ratrel_iff
thf(fact_5385_one__rat_Orsp,axiom,
ratrel @ ( product_Pair @ int @ int @ ( one_one @ int ) @ ( one_one @ int ) ) @ ( product_Pair @ int @ int @ ( one_one @ int ) @ ( one_one @ int ) ) ).
% one_rat.rsp
thf(fact_5386_zero__rat_Orsp,axiom,
ratrel @ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) ) @ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) ) ).
% zero_rat.rsp
thf(fact_5387_ratrel__def,axiom,
( ratrel
= ( ^ [X2: product_prod @ int @ int,Y5: product_prod @ int @ int] :
( ( ( product_snd @ int @ int @ X2 )
!= ( zero_zero @ int ) )
& ( ( product_snd @ int @ int @ Y5 )
!= ( zero_zero @ int ) )
& ( ( times_times @ int @ ( product_fst @ int @ int @ X2 ) @ ( product_snd @ int @ int @ Y5 ) )
= ( times_times @ int @ ( product_fst @ int @ int @ Y5 ) @ ( product_snd @ int @ int @ X2 ) ) ) ) ) ) ).
% ratrel_def
thf(fact_5388_of__rat_Oabs__eq,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [X: product_prod @ int @ int] :
( ( ratrel @ X @ X )
=> ( ( field_char_0_of_rat @ A @ ( abs_Rat @ X ) )
= ( divide_divide @ A @ ( ring_1_of_int @ A @ ( product_fst @ int @ int @ X ) ) @ ( ring_1_of_int @ A @ ( product_snd @ int @ int @ X ) ) ) ) ) ) ).
% of_rat.abs_eq
thf(fact_5389_lex__take__index,axiom,
! [A: $tType,Xs: list @ A,Ys2: list @ A,R4: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys2 ) @ ( lex @ A @ R4 ) )
=> ~ ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Ys2 ) )
=> ( ( ( take @ A @ I3 @ Xs )
= ( take @ A @ I3 @ Ys2 ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( nth @ A @ Xs @ I3 ) @ ( nth @ A @ Ys2 @ I3 ) ) @ R4 ) ) ) ) ) ).
% lex_take_index
thf(fact_5390_num__of__integer__code,axiom,
( code_num_of_integer
= ( ^ [K3: code_integer] :
( if @ num @ ( ord_less_eq @ code_integer @ K3 @ ( one_one @ code_integer ) ) @ one2
@ ( product_case_prod @ code_integer @ code_integer @ num
@ ^ [L2: code_integer,J3: code_integer] :
( if @ num
@ ( J3
= ( zero_zero @ code_integer ) )
@ ( plus_plus @ num @ ( code_num_of_integer @ L2 ) @ ( code_num_of_integer @ L2 ) )
@ ( plus_plus @ num @ ( plus_plus @ num @ ( code_num_of_integer @ L2 ) @ ( code_num_of_integer @ L2 ) ) @ one2 ) )
@ ( code_divmod_integer @ K3 @ ( numeral_numeral @ code_integer @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% num_of_integer_code
thf(fact_5391_mask__mod__exp,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [N: nat,M2: nat] :
( ( modulo_modulo @ A @ ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ A ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M2 ) )
= ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( ord_min @ nat @ M2 @ N ) ) @ ( one_one @ A ) ) ) ) ).
% mask_mod_exp
thf(fact_5392_min_Oidem,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A] :
( ( ord_min @ A @ A3 @ A3 )
= A3 ) ) ).
% min.idem
thf(fact_5393_min_Oleft__idem,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A] :
( ( ord_min @ A @ A3 @ ( ord_min @ A @ A3 @ B2 ) )
= ( ord_min @ A @ A3 @ B2 ) ) ) ).
% min.left_idem
thf(fact_5394_min_Oright__idem,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A] :
( ( ord_min @ A @ ( ord_min @ A @ A3 @ B2 ) @ B2 )
= ( ord_min @ A @ A3 @ B2 ) ) ) ).
% min.right_idem
thf(fact_5395_min_Obounded__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ ( ord_min @ A @ B2 @ C2 ) )
= ( ( ord_less_eq @ A @ A3 @ B2 )
& ( ord_less_eq @ A @ A3 @ C2 ) ) ) ) ).
% min.bounded_iff
thf(fact_5396_min_Oabsorb2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,A3: A] :
( ( ord_less_eq @ A @ B2 @ A3 )
=> ( ( ord_min @ A @ A3 @ B2 )
= B2 ) ) ) ).
% min.absorb2
thf(fact_5397_min_Oabsorb1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_min @ A @ A3 @ B2 )
= A3 ) ) ) ).
% min.absorb1
thf(fact_5398_min_Oabsorb3,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_min @ A @ A3 @ B2 )
= A3 ) ) ) ).
% min.absorb3
thf(fact_5399_min_Oabsorb4,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,A3: A] :
( ( ord_less @ A @ B2 @ A3 )
=> ( ( ord_min @ A @ A3 @ B2 )
= B2 ) ) ) ).
% min.absorb4
thf(fact_5400_min__less__iff__conj,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Z2: A,X: A,Y2: A] :
( ( ord_less @ A @ Z2 @ ( ord_min @ A @ X @ Y2 ) )
= ( ( ord_less @ A @ Z2 @ X )
& ( ord_less @ A @ Z2 @ Y2 ) ) ) ) ).
% min_less_iff_conj
thf(fact_5401_min__Suc__Suc,axiom,
! [M2: nat,N: nat] :
( ( ord_min @ nat @ ( suc @ M2 ) @ ( suc @ N ) )
= ( suc @ ( ord_min @ nat @ M2 @ N ) ) ) ).
% min_Suc_Suc
thf(fact_5402_min__0L,axiom,
! [N: nat] :
( ( ord_min @ nat @ ( zero_zero @ nat ) @ N )
= ( zero_zero @ nat ) ) ).
% min_0L
thf(fact_5403_min__0R,axiom,
! [N: nat] :
( ( ord_min @ nat @ N @ ( zero_zero @ nat ) )
= ( zero_zero @ nat ) ) ).
% min_0R
thf(fact_5404_min__number__of_I1_J,axiom,
! [A: $tType] :
( ( ( numeral @ A )
& ( ord @ A ) )
=> ! [U: num,V2: num] :
( ( ( ord_less_eq @ A @ ( numeral_numeral @ A @ U ) @ ( numeral_numeral @ A @ V2 ) )
=> ( ( ord_min @ A @ ( numeral_numeral @ A @ U ) @ ( numeral_numeral @ A @ V2 ) )
= ( numeral_numeral @ A @ U ) ) )
& ( ~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ U ) @ ( numeral_numeral @ A @ V2 ) )
=> ( ( ord_min @ A @ ( numeral_numeral @ A @ U ) @ ( numeral_numeral @ A @ V2 ) )
= ( numeral_numeral @ A @ V2 ) ) ) ) ) ).
% min_number_of(1)
thf(fact_5405_min__0__1_I4_J,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [X: num] :
( ( ord_min @ A @ ( numeral_numeral @ A @ X ) @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% min_0_1(4)
thf(fact_5406_min__0__1_I3_J,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [X: num] :
( ( ord_min @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ X ) )
= ( zero_zero @ A ) ) ) ).
% min_0_1(3)
thf(fact_5407_min__0__1_I1_J,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ( ( ord_min @ A @ ( zero_zero @ A ) @ ( one_one @ A ) )
= ( zero_zero @ A ) ) ) ).
% min_0_1(1)
thf(fact_5408_min__0__1_I2_J,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ( ( ord_min @ A @ ( one_one @ A ) @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% min_0_1(2)
thf(fact_5409_min__0__1_I6_J,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [X: num] :
( ( ord_min @ A @ ( numeral_numeral @ A @ X ) @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% min_0_1(6)
thf(fact_5410_min__0__1_I5_J,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [X: num] :
( ( ord_min @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ X ) )
= ( one_one @ A ) ) ) ).
% min_0_1(5)
thf(fact_5411_length__take,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( size_size @ ( list @ A ) @ ( take @ A @ N @ Xs ) )
= ( ord_min @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N ) ) ).
% length_take
thf(fact_5412_min__number__of_I4_J,axiom,
! [A: $tType] :
( ( ( uminus @ A )
& ( numeral @ A )
& ( ord @ A ) )
=> ! [U: num,V2: num] :
( ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) )
=> ( ( ord_min @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) )
=> ( ( ord_min @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) ) ) ) ) ).
% min_number_of(4)
thf(fact_5413_min__number__of_I3_J,axiom,
! [A: $tType] :
( ( ( uminus @ A )
& ( numeral @ A )
& ( ord @ A ) )
=> ! [U: num,V2: num] :
( ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( numeral_numeral @ A @ V2 ) )
=> ( ( ord_min @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( numeral_numeral @ A @ V2 ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( numeral_numeral @ A @ V2 ) )
=> ( ( ord_min @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( numeral_numeral @ A @ V2 ) )
= ( numeral_numeral @ A @ V2 ) ) ) ) ) ).
% min_number_of(3)
thf(fact_5414_min__number__of_I2_J,axiom,
! [A: $tType] :
( ( ( uminus @ A )
& ( numeral @ A )
& ( ord @ A ) )
=> ! [U: num,V2: num] :
( ( ( ord_less_eq @ A @ ( numeral_numeral @ A @ U ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) )
=> ( ( ord_min @ A @ ( numeral_numeral @ A @ U ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) )
= ( numeral_numeral @ A @ U ) ) )
& ( ~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ U ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) )
=> ( ( ord_min @ A @ ( numeral_numeral @ A @ U ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ V2 ) ) ) ) ) ) ).
% min_number_of(2)
thf(fact_5415_min__numeral__Suc,axiom,
! [K: num,N: nat] :
( ( ord_min @ nat @ ( numeral_numeral @ nat @ K ) @ ( suc @ N ) )
= ( suc @ ( ord_min @ nat @ ( pred_numeral @ K ) @ N ) ) ) ).
% min_numeral_Suc
thf(fact_5416_min__Suc__numeral,axiom,
! [N: nat,K: num] :
( ( ord_min @ nat @ ( suc @ N ) @ ( numeral_numeral @ nat @ K ) )
= ( suc @ ( ord_min @ nat @ N @ ( pred_numeral @ K ) ) ) ) ).
% min_Suc_numeral
thf(fact_5417_min__add__distrib__right,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [X: A,Y2: A,Z2: A] :
( ( plus_plus @ A @ X @ ( ord_min @ A @ Y2 @ Z2 ) )
= ( ord_min @ A @ ( plus_plus @ A @ X @ Y2 ) @ ( plus_plus @ A @ X @ Z2 ) ) ) ) ).
% min_add_distrib_right
thf(fact_5418_min__add__distrib__left,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [X: A,Y2: A,Z2: A] :
( ( plus_plus @ A @ ( ord_min @ A @ X @ Y2 ) @ Z2 )
= ( ord_min @ A @ ( plus_plus @ A @ X @ Z2 ) @ ( plus_plus @ A @ Y2 @ Z2 ) ) ) ) ).
% min_add_distrib_left
thf(fact_5419_min_Oassoc,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_min @ A @ ( ord_min @ A @ A3 @ B2 ) @ C2 )
= ( ord_min @ A @ A3 @ ( ord_min @ A @ B2 @ C2 ) ) ) ) ).
% min.assoc
thf(fact_5420_min_Ocommute,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_min @ A )
= ( ^ [A5: A,B3: A] : ( ord_min @ A @ B3 @ A5 ) ) ) ) ).
% min.commute
thf(fact_5421_min_Oleft__commute,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( ord_min @ A @ B2 @ ( ord_min @ A @ A3 @ C2 ) )
= ( ord_min @ A @ A3 @ ( ord_min @ A @ B2 @ C2 ) ) ) ) ).
% min.left_commute
thf(fact_5422_min__def__raw,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( ord_min @ A )
= ( ^ [A5: A,B3: A] : ( if @ A @ ( ord_less_eq @ A @ A5 @ B3 ) @ A5 @ B3 ) ) ) ) ).
% min_def_raw
thf(fact_5423_min_Omono,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,C2: A,B2: A,D2: A] :
( ( ord_less_eq @ A @ A3 @ C2 )
=> ( ( ord_less_eq @ A @ B2 @ D2 )
=> ( ord_less_eq @ A @ ( ord_min @ A @ A3 @ B2 ) @ ( ord_min @ A @ C2 @ D2 ) ) ) ) ) ).
% min.mono
thf(fact_5424_min_OorderE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( A3
= ( ord_min @ A @ A3 @ B2 ) ) ) ) ).
% min.orderE
thf(fact_5425_min_OorderI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A] :
( ( A3
= ( ord_min @ A @ A3 @ B2 ) )
=> ( ord_less_eq @ A @ A3 @ B2 ) ) ) ).
% min.orderI
thf(fact_5426_min_OboundedE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ ( ord_min @ A @ B2 @ C2 ) )
=> ~ ( ( ord_less_eq @ A @ A3 @ B2 )
=> ~ ( ord_less_eq @ A @ A3 @ C2 ) ) ) ) ).
% min.boundedE
thf(fact_5427_min_OboundedI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ A3 @ C2 )
=> ( ord_less_eq @ A @ A3 @ ( ord_min @ A @ B2 @ C2 ) ) ) ) ) ).
% min.boundedI
thf(fact_5428_min_Oorder__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A5: A,B3: A] :
( A5
= ( ord_min @ A @ A5 @ B3 ) ) ) ) ) ).
% min.order_iff
thf(fact_5429_min_Ocobounded1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A] : ( ord_less_eq @ A @ ( ord_min @ A @ A3 @ B2 ) @ A3 ) ) ).
% min.cobounded1
thf(fact_5430_min_Ocobounded2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A] : ( ord_less_eq @ A @ ( ord_min @ A @ A3 @ B2 ) @ B2 ) ) ).
% min.cobounded2
thf(fact_5431_min_Oabsorb__iff1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A5: A,B3: A] :
( ( ord_min @ A @ A5 @ B3 )
= A5 ) ) ) ) ).
% min.absorb_iff1
thf(fact_5432_min_Oabsorb__iff2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B3: A,A5: A] :
( ( ord_min @ A @ A5 @ B3 )
= B3 ) ) ) ) ).
% min.absorb_iff2
thf(fact_5433_min_OcoboundedI1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ C2 )
=> ( ord_less_eq @ A @ ( ord_min @ A @ A3 @ B2 ) @ C2 ) ) ) ).
% min.coboundedI1
thf(fact_5434_min_OcoboundedI2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,C2: A,A3: A] :
( ( ord_less_eq @ A @ B2 @ C2 )
=> ( ord_less_eq @ A @ ( ord_min @ A @ A3 @ B2 ) @ C2 ) ) ) ).
% min.coboundedI2
thf(fact_5435_min__le__iff__disj,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y2: A,Z2: A] :
( ( ord_less_eq @ A @ ( ord_min @ A @ X @ Y2 ) @ Z2 )
= ( ( ord_less_eq @ A @ X @ Z2 )
| ( ord_less_eq @ A @ Y2 @ Z2 ) ) ) ) ).
% min_le_iff_disj
thf(fact_5436_of__int__min,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: int,Y2: int] :
( ( ring_1_of_int @ A @ ( ord_min @ int @ X @ Y2 ) )
= ( ord_min @ A @ ( ring_1_of_int @ A @ X ) @ ( ring_1_of_int @ A @ Y2 ) ) ) ) ).
% of_int_min
thf(fact_5437_min__less__iff__disj,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y2: A,Z2: A] :
( ( ord_less @ A @ ( ord_min @ A @ X @ Y2 ) @ Z2 )
= ( ( ord_less @ A @ X @ Z2 )
| ( ord_less @ A @ Y2 @ Z2 ) ) ) ) ).
% min_less_iff_disj
thf(fact_5438_min_Ostrict__boundedE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_less @ A @ A3 @ ( ord_min @ A @ B2 @ C2 ) )
=> ~ ( ( ord_less @ A @ A3 @ B2 )
=> ~ ( ord_less @ A @ A3 @ C2 ) ) ) ) ).
% min.strict_boundedE
thf(fact_5439_min_Ostrict__order__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less @ A )
= ( ^ [A5: A,B3: A] :
( ( A5
= ( ord_min @ A @ A5 @ B3 ) )
& ( A5 != B3 ) ) ) ) ) ).
% min.strict_order_iff
thf(fact_5440_min_Ostrict__coboundedI1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( ord_less @ A @ A3 @ C2 )
=> ( ord_less @ A @ ( ord_min @ A @ A3 @ B2 ) @ C2 ) ) ) ).
% min.strict_coboundedI1
thf(fact_5441_min_Ostrict__coboundedI2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,C2: A,A3: A] :
( ( ord_less @ A @ B2 @ C2 )
=> ( ord_less @ A @ ( ord_min @ A @ A3 @ B2 ) @ C2 ) ) ) ).
% min.strict_coboundedI2
thf(fact_5442_max__min__distrib1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,C2: A,A3: A] :
( ( ord_max @ A @ ( ord_min @ A @ B2 @ C2 ) @ A3 )
= ( ord_min @ A @ ( ord_max @ A @ B2 @ A3 ) @ ( ord_max @ A @ C2 @ A3 ) ) ) ) ).
% max_min_distrib1
thf(fact_5443_max__min__distrib2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_max @ A @ A3 @ ( ord_min @ A @ B2 @ C2 ) )
= ( ord_min @ A @ ( ord_max @ A @ A3 @ B2 ) @ ( ord_max @ A @ A3 @ C2 ) ) ) ) ).
% max_min_distrib2
thf(fact_5444_min__max__distrib1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,C2: A,A3: A] :
( ( ord_min @ A @ ( ord_max @ A @ B2 @ C2 ) @ A3 )
= ( ord_max @ A @ ( ord_min @ A @ B2 @ A3 ) @ ( ord_min @ A @ C2 @ A3 ) ) ) ) ).
% min_max_distrib1
thf(fact_5445_min__max__distrib2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ord_min @ A @ A3 @ ( ord_max @ A @ B2 @ C2 ) )
= ( ord_max @ A @ ( ord_min @ A @ A3 @ B2 ) @ ( ord_min @ A @ A3 @ C2 ) ) ) ) ).
% min_max_distrib2
thf(fact_5446_min__diff__distrib__left,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [X: A,Y2: A,Z2: A] :
( ( minus_minus @ A @ ( ord_min @ A @ X @ Y2 ) @ Z2 )
= ( ord_min @ A @ ( minus_minus @ A @ X @ Z2 ) @ ( minus_minus @ A @ Y2 @ Z2 ) ) ) ) ).
% min_diff_distrib_left
thf(fact_5447_min__diff,axiom,
! [M2: nat,I: nat,N: nat] :
( ( ord_min @ nat @ ( minus_minus @ nat @ M2 @ I ) @ ( minus_minus @ nat @ N @ I ) )
= ( minus_minus @ nat @ ( ord_min @ nat @ M2 @ N ) @ I ) ) ).
% min_diff
thf(fact_5448_nat__mult__min__left,axiom,
! [M2: nat,N: nat,Q5: nat] :
( ( times_times @ nat @ ( ord_min @ nat @ M2 @ N ) @ Q5 )
= ( ord_min @ nat @ ( times_times @ nat @ M2 @ Q5 ) @ ( times_times @ nat @ N @ Q5 ) ) ) ).
% nat_mult_min_left
thf(fact_5449_nat__mult__min__right,axiom,
! [M2: nat,N: nat,Q5: nat] :
( ( times_times @ nat @ M2 @ ( ord_min @ nat @ N @ Q5 ) )
= ( ord_min @ nat @ ( times_times @ nat @ M2 @ N ) @ ( times_times @ nat @ M2 @ Q5 ) ) ) ).
% nat_mult_min_right
thf(fact_5450_of__nat__min,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [X: nat,Y2: nat] :
( ( semiring_1_of_nat @ A @ ( ord_min @ nat @ X @ Y2 ) )
= ( ord_min @ A @ ( semiring_1_of_nat @ A @ X ) @ ( semiring_1_of_nat @ A @ Y2 ) ) ) ) ).
% of_nat_min
thf(fact_5451_inf__min,axiom,
! [A: $tType] :
( ( ( semilattice_inf @ A )
& ( linorder @ A ) )
=> ( ( inf_inf @ A )
= ( ord_min @ A ) ) ) ).
% inf_min
thf(fact_5452_inf__nat__def,axiom,
( ( inf_inf @ nat )
= ( ord_min @ nat ) ) ).
% inf_nat_def
thf(fact_5453_minus__min__eq__max,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [X: A,Y2: A] :
( ( uminus_uminus @ A @ ( ord_min @ A @ X @ Y2 ) )
= ( ord_max @ A @ ( uminus_uminus @ A @ X ) @ ( uminus_uminus @ A @ Y2 ) ) ) ) ).
% minus_min_eq_max
thf(fact_5454_minus__max__eq__min,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [X: A,Y2: A] :
( ( uminus_uminus @ A @ ( ord_max @ A @ X @ Y2 ) )
= ( ord_min @ A @ ( uminus_uminus @ A @ X ) @ ( uminus_uminus @ A @ Y2 ) ) ) ) ).
% minus_max_eq_min
thf(fact_5455_concat__bit__assoc__sym,axiom,
! [M2: nat,N: nat,K: int,L: int,R4: int] :
( ( bit_concat_bit @ M2 @ ( bit_concat_bit @ N @ K @ L ) @ R4 )
= ( bit_concat_bit @ ( ord_min @ nat @ M2 @ N ) @ K @ ( bit_concat_bit @ ( minus_minus @ nat @ M2 @ N ) @ L @ R4 ) ) ) ).
% concat_bit_assoc_sym
thf(fact_5456_take__bit__concat__bit__eq,axiom,
! [M2: nat,N: nat,K: int,L: int] :
( ( bit_se2584673776208193580ke_bit @ int @ M2 @ ( bit_concat_bit @ N @ K @ L ) )
= ( bit_concat_bit @ ( ord_min @ nat @ M2 @ N ) @ K @ ( bit_se2584673776208193580ke_bit @ int @ ( minus_minus @ nat @ M2 @ N ) @ L ) ) ) ).
% take_bit_concat_bit_eq
thf(fact_5457_min__mult__distrib__right,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [P4: A,X: A,Y2: A] :
( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ P4 )
=> ( ( times_times @ A @ ( ord_min @ A @ X @ Y2 ) @ P4 )
= ( ord_min @ A @ ( times_times @ A @ X @ P4 ) @ ( times_times @ A @ Y2 @ P4 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ P4 )
=> ( ( times_times @ A @ ( ord_min @ A @ X @ Y2 ) @ P4 )
= ( ord_max @ A @ ( times_times @ A @ X @ P4 ) @ ( times_times @ A @ Y2 @ P4 ) ) ) ) ) ) ).
% min_mult_distrib_right
thf(fact_5458_max__mult__distrib__right,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [P4: A,X: A,Y2: A] :
( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ P4 )
=> ( ( times_times @ A @ ( ord_max @ A @ X @ Y2 ) @ P4 )
= ( ord_max @ A @ ( times_times @ A @ X @ P4 ) @ ( times_times @ A @ Y2 @ P4 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ P4 )
=> ( ( times_times @ A @ ( ord_max @ A @ X @ Y2 ) @ P4 )
= ( ord_min @ A @ ( times_times @ A @ X @ P4 ) @ ( times_times @ A @ Y2 @ P4 ) ) ) ) ) ) ).
% max_mult_distrib_right
thf(fact_5459_min__mult__distrib__left,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [P4: A,X: A,Y2: A] :
( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ P4 )
=> ( ( times_times @ A @ P4 @ ( ord_min @ A @ X @ Y2 ) )
= ( ord_min @ A @ ( times_times @ A @ P4 @ X ) @ ( times_times @ A @ P4 @ Y2 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ P4 )
=> ( ( times_times @ A @ P4 @ ( ord_min @ A @ X @ Y2 ) )
= ( ord_max @ A @ ( times_times @ A @ P4 @ X ) @ ( times_times @ A @ P4 @ Y2 ) ) ) ) ) ) ).
% min_mult_distrib_left
thf(fact_5460_max__mult__distrib__left,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [P4: A,X: A,Y2: A] :
( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ P4 )
=> ( ( times_times @ A @ P4 @ ( ord_max @ A @ X @ Y2 ) )
= ( ord_max @ A @ ( times_times @ A @ P4 @ X ) @ ( times_times @ A @ P4 @ Y2 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ P4 )
=> ( ( times_times @ A @ P4 @ ( ord_max @ A @ X @ Y2 ) )
= ( ord_min @ A @ ( times_times @ A @ P4 @ X ) @ ( times_times @ A @ P4 @ Y2 ) ) ) ) ) ) ).
% max_mult_distrib_left
thf(fact_5461_max__divide__distrib__right,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [P4: A,X: A,Y2: A] :
( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ P4 )
=> ( ( divide_divide @ A @ ( ord_max @ A @ X @ Y2 ) @ P4 )
= ( ord_max @ A @ ( divide_divide @ A @ X @ P4 ) @ ( divide_divide @ A @ Y2 @ P4 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ P4 )
=> ( ( divide_divide @ A @ ( ord_max @ A @ X @ Y2 ) @ P4 )
= ( ord_min @ A @ ( divide_divide @ A @ X @ P4 ) @ ( divide_divide @ A @ Y2 @ P4 ) ) ) ) ) ) ).
% max_divide_distrib_right
thf(fact_5462_min__divide__distrib__right,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [P4: A,X: A,Y2: A] :
( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ P4 )
=> ( ( divide_divide @ A @ ( ord_min @ A @ X @ Y2 ) @ P4 )
= ( ord_min @ A @ ( divide_divide @ A @ X @ P4 ) @ ( divide_divide @ A @ Y2 @ P4 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ P4 )
=> ( ( divide_divide @ A @ ( ord_min @ A @ X @ Y2 ) @ P4 )
= ( ord_max @ A @ ( divide_divide @ A @ X @ P4 ) @ ( divide_divide @ A @ Y2 @ P4 ) ) ) ) ) ) ).
% min_divide_distrib_right
thf(fact_5463_min__Suc1,axiom,
! [N: nat,M2: nat] :
( ( ord_min @ nat @ ( suc @ N ) @ M2 )
= ( case_nat @ nat @ ( zero_zero @ nat )
@ ^ [M4: nat] : ( suc @ ( ord_min @ nat @ N @ M4 ) )
@ M2 ) ) ).
% min_Suc1
thf(fact_5464_min__Suc2,axiom,
! [M2: nat,N: nat] :
( ( ord_min @ nat @ M2 @ ( suc @ N ) )
= ( case_nat @ nat @ ( zero_zero @ nat )
@ ^ [M4: nat] : ( suc @ ( ord_min @ nat @ M4 @ N ) )
@ M2 ) ) ).
% min_Suc2
thf(fact_5465_lexord__take__index__conv,axiom,
! [A: $tType,X: list @ A,Y2: list @ A,R4: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Y2 ) @ ( lexord @ A @ R4 ) )
= ( ( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ X ) @ ( size_size @ ( list @ A ) @ Y2 ) )
& ( ( take @ A @ ( size_size @ ( list @ A ) @ X ) @ Y2 )
= X ) )
| ? [I2: nat] :
( ( ord_less @ nat @ I2 @ ( ord_min @ nat @ ( size_size @ ( list @ A ) @ X ) @ ( size_size @ ( list @ A ) @ Y2 ) ) )
& ( ( take @ A @ I2 @ X )
= ( take @ A @ I2 @ Y2 ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( nth @ A @ X @ I2 ) @ ( nth @ A @ Y2 @ I2 ) ) @ R4 ) ) ) ) ).
% lexord_take_index_conv
thf(fact_5466_listrel1__iff__update,axiom,
! [A: $tType,Xs: list @ A,Ys2: list @ A,R4: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys2 ) @ ( listrel1 @ A @ R4 ) )
= ( ? [Y5: A,N5: nat] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( nth @ A @ Xs @ N5 ) @ Y5 ) @ R4 )
& ( ord_less @ nat @ N5 @ ( size_size @ ( list @ A ) @ Xs ) )
& ( Ys2
= ( list_update @ A @ Xs @ N5 @ Y5 ) ) ) ) ) ).
% listrel1_iff_update
thf(fact_5467_lenlex__conv,axiom,
! [A: $tType] :
( ( lenlex @ A )
= ( ^ [R: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ ( list @ A ) @ ( list @ A ) )
@ ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ $o
@ ^ [Xs3: list @ A,Ys3: list @ A] :
( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Xs3 ) @ ( size_size @ ( list @ A ) @ Ys3 ) )
| ( ( ( size_size @ ( list @ A ) @ Xs3 )
= ( size_size @ ( list @ A ) @ Ys3 ) )
& ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs3 @ Ys3 ) @ ( lex @ A @ R ) ) ) ) ) ) ) ) ).
% lenlex_conv
thf(fact_5468_inf__int__def,axiom,
( ( inf_inf @ int )
= ( ord_min @ int ) ) ).
% inf_int_def
thf(fact_5469_listrel1__eq__len,axiom,
! [A: $tType,Xs: list @ A,Ys2: list @ A,R4: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys2 ) @ ( listrel1 @ A @ R4 ) )
=> ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ A ) @ Ys2 ) ) ) ).
% listrel1_eq_len
thf(fact_5470_lexord__lex,axiom,
! [A: $tType,X: list @ A,Y2: list @ A,R4: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Y2 ) @ ( lex @ A @ R4 ) )
= ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Y2 ) @ ( lexord @ A @ R4 ) )
& ( ( size_size @ ( list @ A ) @ X )
= ( size_size @ ( list @ A ) @ Y2 ) ) ) ) ).
% lexord_lex
thf(fact_5471_lenlex__length,axiom,
! [A: $tType,Ms: list @ A,Ns: list @ A,R4: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ms @ Ns ) @ ( lenlex @ A @ R4 ) )
=> ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Ms ) @ ( size_size @ ( list @ A ) @ Ns ) ) ) ).
% lenlex_length
thf(fact_5472_rcis__cnj,axiom,
( cnj
= ( ^ [A5: complex] : ( rcis @ ( real_V7770717601297561774m_norm @ complex @ A5 ) @ ( uminus_uminus @ real @ ( arg @ A5 ) ) ) ) ) ).
% rcis_cnj
thf(fact_5473_find__Some__iff,axiom,
! [A: $tType,P: A > $o,Xs: list @ A,X: A] :
( ( ( find @ A @ P @ Xs )
= ( some @ A @ X ) )
= ( ? [I2: nat] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs ) )
& ( P @ ( nth @ A @ Xs @ I2 ) )
& ( X
= ( nth @ A @ Xs @ I2 ) )
& ! [J3: nat] :
( ( ord_less @ nat @ J3 @ I2 )
=> ~ ( P @ ( nth @ A @ Xs @ J3 ) ) ) ) ) ) ).
% find_Some_iff
thf(fact_5474_find__Some__iff2,axiom,
! [A: $tType,X: A,P: A > $o,Xs: list @ A] :
( ( ( some @ A @ X )
= ( find @ A @ P @ Xs ) )
= ( ? [I2: nat] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs ) )
& ( P @ ( nth @ A @ Xs @ I2 ) )
& ( X
= ( nth @ A @ Xs @ I2 ) )
& ! [J3: nat] :
( ( ord_less @ nat @ J3 @ I2 )
=> ~ ( P @ ( nth @ A @ Xs @ J3 ) ) ) ) ) ) ).
% find_Some_iff2
thf(fact_5475_cis__rcis__eq,axiom,
( cis
= ( rcis @ ( one_one @ real ) ) ) ).
% cis_rcis_eq
thf(fact_5476_find__None__iff,axiom,
! [A: $tType,P: A > $o,Xs: list @ A] :
( ( ( find @ A @ P @ Xs )
= ( none @ A ) )
= ( ~ ? [X2: A] :
( ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
& ( P @ X2 ) ) ) ) ).
% find_None_iff
thf(fact_5477_find__None__iff2,axiom,
! [A: $tType,P: A > $o,Xs: list @ A] :
( ( ( none @ A )
= ( find @ A @ P @ Xs ) )
= ( ~ ? [X2: A] :
( ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
& ( P @ X2 ) ) ) ) ).
% find_None_iff2
thf(fact_5478_rcis__divide,axiom,
! [R1: real,A3: real,R22: real,B2: real] :
( ( divide_divide @ complex @ ( rcis @ R1 @ A3 ) @ ( rcis @ R22 @ B2 ) )
= ( rcis @ ( divide_divide @ real @ R1 @ R22 ) @ ( minus_minus @ real @ A3 @ B2 ) ) ) ).
% rcis_divide
thf(fact_5479_rcis__inverse,axiom,
! [R4: real,A3: real] :
( ( inverse_inverse @ complex @ ( rcis @ R4 @ A3 ) )
= ( rcis @ ( divide_divide @ real @ ( one_one @ real ) @ R4 ) @ ( uminus_uminus @ real @ A3 ) ) ) ).
% rcis_inverse
thf(fact_5480_nth__rotate,axiom,
! [A: $tType,N: nat,Xs: list @ A,M2: nat] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ A @ ( rotate @ A @ M2 @ Xs ) @ N )
= ( nth @ A @ Xs @ ( modulo_modulo @ nat @ ( plus_plus @ nat @ M2 @ N ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ) ).
% nth_rotate
thf(fact_5481_nth__zip,axiom,
! [A: $tType,B: $tType,I: nat,Xs: list @ A,Ys2: list @ B] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ord_less @ nat @ I @ ( size_size @ ( list @ B ) @ Ys2 ) )
=> ( ( nth @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys2 ) @ I )
= ( product_Pair @ A @ B @ ( nth @ A @ Xs @ I ) @ ( nth @ B @ Ys2 @ I ) ) ) ) ) ).
% nth_zip
thf(fact_5482_quotient__of__def,axiom,
( quotient_of
= ( ^ [X2: rat] :
( the @ ( product_prod @ int @ int )
@ ^ [Pair: product_prod @ int @ int] :
( ( X2
= ( fract @ ( product_fst @ int @ int @ Pair ) @ ( product_snd @ int @ int @ Pair ) ) )
& ( ord_less @ int @ ( zero_zero @ int ) @ ( product_snd @ int @ int @ Pair ) )
& ( algebr8660921524188924756oprime @ int @ ( product_fst @ int @ int @ Pair ) @ ( product_snd @ int @ int @ Pair ) ) ) ) ) ) ).
% quotient_of_def
thf(fact_5483_coprime__minus__left__iff,axiom,
! [A: $tType] :
( ( ring_gcd @ A )
=> ! [A3: A,B2: A] :
( ( algebr8660921524188924756oprime @ A @ ( uminus_uminus @ A @ A3 ) @ B2 )
= ( algebr8660921524188924756oprime @ A @ A3 @ B2 ) ) ) ).
% coprime_minus_left_iff
thf(fact_5484_coprime__minus__right__iff,axiom,
! [A: $tType] :
( ( ring_gcd @ A )
=> ! [A3: A,B2: A] :
( ( algebr8660921524188924756oprime @ A @ A3 @ ( uminus_uminus @ A @ B2 ) )
= ( algebr8660921524188924756oprime @ A @ A3 @ B2 ) ) ) ).
% coprime_minus_right_iff
thf(fact_5485_rotate0,axiom,
! [A: $tType] :
( ( rotate @ A @ ( zero_zero @ nat ) )
= ( id @ ( list @ A ) ) ) ).
% rotate0
thf(fact_5486_length__rotate,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( size_size @ ( list @ A ) @ ( rotate @ A @ N @ Xs ) )
= ( size_size @ ( list @ A ) @ Xs ) ) ).
% length_rotate
thf(fact_5487_coprime__self,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A] :
( ( algebr8660921524188924756oprime @ A @ A3 @ A3 )
= ( dvd_dvd @ A @ A3 @ ( one_one @ A ) ) ) ) ).
% coprime_self
thf(fact_5488_coprime__mod__right__iff,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ! [A3: A,B2: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( algebr8660921524188924756oprime @ A @ A3 @ ( modulo_modulo @ A @ B2 @ A3 ) )
= ( algebr8660921524188924756oprime @ A @ A3 @ B2 ) ) ) ) ).
% coprime_mod_right_iff
thf(fact_5489_coprime__mod__left__iff,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ! [B2: A,A3: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( algebr8660921524188924756oprime @ A @ ( modulo_modulo @ A @ A3 @ B2 ) @ B2 )
= ( algebr8660921524188924756oprime @ A @ A3 @ B2 ) ) ) ) ).
% coprime_mod_left_iff
thf(fact_5490_coprime__power__right__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A,N: nat] :
( ( algebr8660921524188924756oprime @ A @ A3 @ ( power_power @ A @ B2 @ N ) )
= ( ( algebr8660921524188924756oprime @ A @ A3 @ B2 )
| ( N
= ( zero_zero @ nat ) ) ) ) ) ).
% coprime_power_right_iff
thf(fact_5491_coprime__power__left__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,N: nat,B2: A] :
( ( algebr8660921524188924756oprime @ A @ ( power_power @ A @ A3 @ N ) @ B2 )
= ( ( algebr8660921524188924756oprime @ A @ A3 @ B2 )
| ( N
= ( zero_zero @ nat ) ) ) ) ) ).
% coprime_power_left_iff
thf(fact_5492_coprime__imp__gcd__eq__1,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A] :
( ( algebr8660921524188924756oprime @ A @ A3 @ B2 )
=> ( ( gcd_gcd @ A @ A3 @ B2 )
= ( one_one @ A ) ) ) ) ).
% coprime_imp_gcd_eq_1
thf(fact_5493_rotate__Suc,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( rotate @ A @ ( suc @ N ) @ Xs )
= ( rotate1 @ A @ ( rotate @ A @ N @ Xs ) ) ) ).
% rotate_Suc
thf(fact_5494_coprime__0__right__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A] :
( ( algebr8660921524188924756oprime @ A @ A3 @ ( zero_zero @ A ) )
= ( dvd_dvd @ A @ A3 @ ( one_one @ A ) ) ) ) ).
% coprime_0_right_iff
thf(fact_5495_coprime__0__left__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A] :
( ( algebr8660921524188924756oprime @ A @ ( zero_zero @ A ) @ A3 )
= ( dvd_dvd @ A @ A3 @ ( one_one @ A ) ) ) ) ).
% coprime_0_left_iff
thf(fact_5496_coprime__mult__self__right__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,C2: A,B2: A] :
( ( algebr8660921524188924756oprime @ A @ ( times_times @ A @ A3 @ C2 ) @ ( times_times @ A @ B2 @ C2 ) )
= ( ( dvd_dvd @ A @ C2 @ ( one_one @ A ) )
& ( algebr8660921524188924756oprime @ A @ A3 @ B2 ) ) ) ) ).
% coprime_mult_self_right_iff
thf(fact_5497_coprime__mult__self__left__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( algebr8660921524188924756oprime @ A @ ( times_times @ A @ C2 @ A3 ) @ ( times_times @ A @ C2 @ B2 ) )
= ( ( dvd_dvd @ A @ C2 @ ( one_one @ A ) )
& ( algebr8660921524188924756oprime @ A @ A3 @ B2 ) ) ) ) ).
% coprime_mult_self_left_iff
thf(fact_5498_is__unit__gcd,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A] :
( ( dvd_dvd @ A @ ( gcd_gcd @ A @ A3 @ B2 ) @ ( one_one @ A ) )
= ( algebr8660921524188924756oprime @ A @ A3 @ B2 ) ) ) ).
% is_unit_gcd
thf(fact_5499_rotate__length01,axiom,
! [A: $tType,Xs: list @ A,N: nat] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) )
=> ( ( rotate @ A @ N @ Xs )
= Xs ) ) ).
% rotate_length01
thf(fact_5500_rotate__id,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( ( modulo_modulo @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
= ( zero_zero @ nat ) )
=> ( ( rotate @ A @ N @ Xs )
= Xs ) ) ).
% rotate_id
thf(fact_5501_length__zip,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys2: list @ B] :
( ( size_size @ ( list @ ( product_prod @ A @ B ) ) @ ( zip @ A @ B @ Xs @ Ys2 ) )
= ( ord_min @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ B ) @ Ys2 ) ) ) ).
% length_zip
thf(fact_5502_normalize__stable,axiom,
! [Q5: int,P4: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ Q5 )
=> ( ( algebr8660921524188924756oprime @ int @ P4 @ Q5 )
=> ( ( normalize @ ( product_Pair @ int @ int @ P4 @ Q5 ) )
= ( product_Pair @ int @ int @ P4 @ Q5 ) ) ) ) ).
% normalize_stable
thf(fact_5503_is__unit__right__imp__coprime,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,A3: A] :
( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( algebr8660921524188924756oprime @ A @ A3 @ B2 ) ) ) ).
% is_unit_right_imp_coprime
thf(fact_5504_is__unit__left__imp__coprime,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B2: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( algebr8660921524188924756oprime @ A @ A3 @ B2 ) ) ) ).
% is_unit_left_imp_coprime
thf(fact_5505_coprime__common__divisor,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( algebr8660921524188924756oprime @ A @ A3 @ B2 )
=> ( ( dvd_dvd @ A @ C2 @ A3 )
=> ( ( dvd_dvd @ A @ C2 @ B2 )
=> ( dvd_dvd @ A @ C2 @ ( one_one @ A ) ) ) ) ) ) ).
% coprime_common_divisor
thf(fact_5506_coprime__absorb__right,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [Y2: A,X: A] :
( ( dvd_dvd @ A @ Y2 @ X )
=> ( ( algebr8660921524188924756oprime @ A @ X @ Y2 )
= ( dvd_dvd @ A @ Y2 @ ( one_one @ A ) ) ) ) ) ).
% coprime_absorb_right
thf(fact_5507_coprime__imp__coprime,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [C2: A,D2: A,A3: A,B2: A] :
( ( algebr8660921524188924756oprime @ A @ C2 @ D2 )
=> ( ! [E: A] :
( ~ ( dvd_dvd @ A @ E @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ E @ A3 )
=> ( ( dvd_dvd @ A @ E @ B2 )
=> ( dvd_dvd @ A @ E @ C2 ) ) ) )
=> ( ! [E: A] :
( ~ ( dvd_dvd @ A @ E @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ E @ A3 )
=> ( ( dvd_dvd @ A @ E @ B2 )
=> ( dvd_dvd @ A @ E @ D2 ) ) ) )
=> ( algebr8660921524188924756oprime @ A @ A3 @ B2 ) ) ) ) ) ).
% coprime_imp_coprime
thf(fact_5508_coprime__absorb__left,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [X: A,Y2: A] :
( ( dvd_dvd @ A @ X @ Y2 )
=> ( ( algebr8660921524188924756oprime @ A @ X @ Y2 )
= ( dvd_dvd @ A @ X @ ( one_one @ A ) ) ) ) ) ).
% coprime_absorb_left
thf(fact_5509_not__coprimeI,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( dvd_dvd @ A @ C2 @ A3 )
=> ( ( dvd_dvd @ A @ C2 @ B2 )
=> ( ~ ( dvd_dvd @ A @ C2 @ ( one_one @ A ) )
=> ~ ( algebr8660921524188924756oprime @ A @ A3 @ B2 ) ) ) ) ) ).
% not_coprimeI
thf(fact_5510_not__coprimeE,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B2: A] :
( ~ ( algebr8660921524188924756oprime @ A @ A3 @ B2 )
=> ~ ! [C3: A] :
( ( dvd_dvd @ A @ C3 @ A3 )
=> ( ( dvd_dvd @ A @ C3 @ B2 )
=> ( dvd_dvd @ A @ C3 @ ( one_one @ A ) ) ) ) ) ) ).
% not_coprimeE
thf(fact_5511_coprime__def,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ( ( algebr8660921524188924756oprime @ A )
= ( ^ [A5: A,B3: A] :
! [C4: A] :
( ( dvd_dvd @ A @ C4 @ A5 )
=> ( ( dvd_dvd @ A @ C4 @ B3 )
=> ( dvd_dvd @ A @ C4 @ ( one_one @ A ) ) ) ) ) ) ) ).
% coprime_def
thf(fact_5512_coprimeI,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,B2: A] :
( ! [C3: A] :
( ( dvd_dvd @ A @ C3 @ A3 )
=> ( ( dvd_dvd @ A @ C3 @ B2 )
=> ( dvd_dvd @ A @ C3 @ ( one_one @ A ) ) ) )
=> ( algebr8660921524188924756oprime @ A @ A3 @ B2 ) ) ) ).
% coprimeI
thf(fact_5513_coprime__add__one__left,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A] : ( algebr8660921524188924756oprime @ A @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) @ A3 ) ) ).
% coprime_add_one_left
thf(fact_5514_coprime__add__one__right,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A] : ( algebr8660921524188924756oprime @ A @ A3 @ ( plus_plus @ A @ A3 @ ( one_one @ A ) ) ) ) ).
% coprime_add_one_right
thf(fact_5515_coprime__divisors,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A,C2: A,B2: A,D2: A] :
( ( dvd_dvd @ A @ A3 @ C2 )
=> ( ( dvd_dvd @ A @ B2 @ D2 )
=> ( ( algebr8660921524188924756oprime @ A @ C2 @ D2 )
=> ( algebr8660921524188924756oprime @ A @ A3 @ B2 ) ) ) ) ) ).
% coprime_divisors
thf(fact_5516_coprime__commute,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ( ( algebr8660921524188924756oprime @ A )
= ( ^ [B3: A,A5: A] : ( algebr8660921524188924756oprime @ A @ A5 @ B3 ) ) ) ) ).
% coprime_commute
thf(fact_5517_coprime__1__left,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A] : ( algebr8660921524188924756oprime @ A @ ( one_one @ A ) @ A3 ) ) ).
% coprime_1_left
thf(fact_5518_coprime__1__right,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A3: A] : ( algebr8660921524188924756oprime @ A @ A3 @ ( one_one @ A ) ) ) ).
% coprime_1_right
thf(fact_5519_coprime__diff__one__left,axiom,
! [A: $tType] :
( ( ring_gcd @ A )
=> ! [A3: A] : ( algebr8660921524188924756oprime @ A @ ( minus_minus @ A @ A3 @ ( one_one @ A ) ) @ A3 ) ) ).
% coprime_diff_one_left
thf(fact_5520_coprime__doff__one__right,axiom,
! [A: $tType] :
( ( ring_gcd @ A )
=> ! [A3: A] : ( algebr8660921524188924756oprime @ A @ A3 @ ( minus_minus @ A @ A3 @ ( one_one @ A ) ) ) ) ).
% coprime_doff_one_right
thf(fact_5521_gcd__eq__1__imp__coprime,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A] :
( ( ( gcd_gcd @ A @ A3 @ B2 )
= ( one_one @ A ) )
=> ( algebr8660921524188924756oprime @ A @ A3 @ B2 ) ) ) ).
% gcd_eq_1_imp_coprime
thf(fact_5522_coprime__iff__gcd__eq__1,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ( ( algebr8660921524188924756oprime @ A )
= ( ^ [A5: A,B3: A] :
( ( gcd_gcd @ A @ A5 @ B3 )
= ( one_one @ A ) ) ) ) ) ).
% coprime_iff_gcd_eq_1
thf(fact_5523_rotate__conv__mod,axiom,
! [A: $tType] :
( ( rotate @ A )
= ( ^ [N5: nat,Xs3: list @ A] : ( rotate @ A @ ( modulo_modulo @ nat @ N5 @ ( size_size @ ( list @ A ) @ Xs3 ) ) @ Xs3 ) ) ) ).
% rotate_conv_mod
thf(fact_5524_invertible__coprime,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( ( modulo_modulo @ A @ ( times_times @ A @ A3 @ B2 ) @ C2 )
= ( one_one @ A ) )
=> ( algebr8660921524188924756oprime @ A @ A3 @ C2 ) ) ) ).
% invertible_coprime
thf(fact_5525_gcd__coprime,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A,A8: A,B6: A] :
( ( ( gcd_gcd @ A @ A3 @ B2 )
!= ( zero_zero @ A ) )
=> ( ( A3
= ( times_times @ A @ A8 @ ( gcd_gcd @ A @ A3 @ B2 ) ) )
=> ( ( B2
= ( times_times @ A @ B6 @ ( gcd_gcd @ A @ A3 @ B2 ) ) )
=> ( algebr8660921524188924756oprime @ A @ A8 @ B6 ) ) ) ) ) ).
% gcd_coprime
thf(fact_5526_gcd__coprime__exists,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A] :
( ( ( gcd_gcd @ A @ A3 @ B2 )
!= ( zero_zero @ A ) )
=> ? [A10: A,B9: A] :
( ( A3
= ( times_times @ A @ A10 @ ( gcd_gcd @ A @ A3 @ B2 ) ) )
& ( B2
= ( times_times @ A @ B9 @ ( gcd_gcd @ A @ A3 @ B2 ) ) )
& ( algebr8660921524188924756oprime @ A @ A10 @ B9 ) ) ) ) ).
% gcd_coprime_exists
thf(fact_5527_div__gcd__coprime,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A] :
( ( ( A3
!= ( zero_zero @ A ) )
| ( B2
!= ( zero_zero @ A ) ) )
=> ( algebr8660921524188924756oprime @ A @ ( divide_divide @ A @ A3 @ ( gcd_gcd @ A @ A3 @ B2 ) ) @ ( divide_divide @ A @ B2 @ ( gcd_gcd @ A @ A3 @ B2 ) ) ) ) ) ).
% div_gcd_coprime
thf(fact_5528_Rat__cases,axiom,
! [Q5: rat] :
~ ! [A6: int,B4: int] :
( ( Q5
= ( fract @ A6 @ B4 ) )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B4 )
=> ~ ( algebr8660921524188924756oprime @ int @ A6 @ B4 ) ) ) ).
% Rat_cases
thf(fact_5529_Rat__induct,axiom,
! [P: rat > $o,Q5: rat] :
( ! [A6: int,B4: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ B4 )
=> ( ( algebr8660921524188924756oprime @ int @ A6 @ B4 )
=> ( P @ ( fract @ A6 @ B4 ) ) ) )
=> ( P @ Q5 ) ) ).
% Rat_induct
thf(fact_5530_coprime__common__divisor__int,axiom,
! [A3: int,B2: int,X: int] :
( ( algebr8660921524188924756oprime @ int @ A3 @ B2 )
=> ( ( dvd_dvd @ int @ X @ A3 )
=> ( ( dvd_dvd @ int @ X @ B2 )
=> ( ( abs_abs @ int @ X )
= ( one_one @ int ) ) ) ) ) ).
% coprime_common_divisor_int
thf(fact_5531_zip__obtain__same__length,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys2: list @ B,P: ( list @ ( product_prod @ A @ B ) ) > $o] :
( ! [Zs2: list @ A,Ws: list @ B,N2: nat] :
( ( ( size_size @ ( list @ A ) @ Zs2 )
= ( size_size @ ( list @ B ) @ Ws ) )
=> ( ( N2
= ( ord_min @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ B ) @ Ys2 ) ) )
=> ( ( Zs2
= ( take @ A @ N2 @ Xs ) )
=> ( ( Ws
= ( take @ B @ N2 @ Ys2 ) )
=> ( P @ ( zip @ A @ B @ Zs2 @ Ws ) ) ) ) ) )
=> ( P @ ( zip @ A @ B @ Xs @ Ys2 ) ) ) ).
% zip_obtain_same_length
thf(fact_5532_in__set__impl__in__set__zip2,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys2: list @ B,Y2: B] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys2 ) )
=> ( ( member @ B @ Y2 @ ( set2 @ B @ Ys2 ) )
=> ~ ! [X3: A] :
~ ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X3 @ Y2 ) @ ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys2 ) ) ) ) ) ).
% in_set_impl_in_set_zip2
thf(fact_5533_in__set__impl__in__set__zip1,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys2: list @ B,X: A] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys2 ) )
=> ( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ~ ! [Y3: B] :
~ ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y3 ) @ ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys2 ) ) ) ) ) ).
% in_set_impl_in_set_zip1
thf(fact_5534_Rat__cases__nonzero,axiom,
! [Q5: rat] :
( ! [A6: int,B4: int] :
( ( Q5
= ( fract @ A6 @ B4 ) )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B4 )
=> ( ( A6
!= ( zero_zero @ int ) )
=> ~ ( algebr8660921524188924756oprime @ int @ A6 @ B4 ) ) ) )
=> ( Q5
= ( zero_zero @ rat ) ) ) ).
% Rat_cases_nonzero
thf(fact_5535_quotient__of__unique,axiom,
! [R4: rat] :
? [X3: product_prod @ int @ int] :
( ( R4
= ( fract @ ( product_fst @ int @ int @ X3 ) @ ( product_snd @ int @ int @ X3 ) ) )
& ( ord_less @ int @ ( zero_zero @ int ) @ ( product_snd @ int @ int @ X3 ) )
& ( algebr8660921524188924756oprime @ int @ ( product_fst @ int @ int @ X3 ) @ ( product_snd @ int @ int @ X3 ) )
& ! [Y: product_prod @ int @ int] :
( ( ( R4
= ( fract @ ( product_fst @ int @ int @ Y ) @ ( product_snd @ int @ int @ Y ) ) )
& ( ord_less @ int @ ( zero_zero @ int ) @ ( product_snd @ int @ int @ Y ) )
& ( algebr8660921524188924756oprime @ int @ ( product_fst @ int @ int @ Y ) @ ( product_snd @ int @ int @ Y ) ) )
=> ( Y = X3 ) ) ) ).
% quotient_of_unique
thf(fact_5536_in__set__zip,axiom,
! [B: $tType,A: $tType,P4: product_prod @ A @ B,Xs: list @ A,Ys2: list @ B] :
( ( member @ ( product_prod @ A @ B ) @ P4 @ ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys2 ) ) )
= ( ? [N5: nat] :
( ( ( nth @ A @ Xs @ N5 )
= ( product_fst @ A @ B @ P4 ) )
& ( ( nth @ B @ Ys2 @ N5 )
= ( product_snd @ A @ B @ P4 ) )
& ( ord_less @ nat @ N5 @ ( size_size @ ( list @ A ) @ Xs ) )
& ( ord_less @ nat @ N5 @ ( size_size @ ( list @ B ) @ Ys2 ) ) ) ) ) ).
% in_set_zip
thf(fact_5537_set__zip,axiom,
! [B: $tType,A: $tType,Xs: list @ A,Ys2: list @ B] :
( ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys2 ) )
= ( collect @ ( product_prod @ A @ B )
@ ^ [Uu3: product_prod @ A @ B] :
? [I2: nat] :
( ( Uu3
= ( product_Pair @ A @ B @ ( nth @ A @ Xs @ I2 ) @ ( nth @ B @ Ys2 @ I2 ) ) )
& ( ord_less @ nat @ I2 @ ( ord_min @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ B ) @ Ys2 ) ) ) ) ) ) ).
% set_zip
thf(fact_5538_Rats__cases_H,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [X: A] :
( ( member @ A @ X @ ( field_char_0_Rats @ A ) )
=> ~ ! [A6: int,B4: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ B4 )
=> ( ( algebr8660921524188924756oprime @ int @ A6 @ B4 )
=> ( X
!= ( divide_divide @ A @ ( ring_1_of_int @ A @ A6 ) @ ( ring_1_of_int @ A @ B4 ) ) ) ) ) ) ) ).
% Rats_cases'
thf(fact_5539_map__of__zip__nth,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys2: list @ B,I: nat] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys2 ) )
=> ( ( distinct @ A @ Xs )
=> ( ( ord_less @ nat @ I @ ( size_size @ ( list @ B ) @ Ys2 ) )
=> ( ( map_of @ A @ B @ ( zip @ A @ B @ Xs @ Ys2 ) @ ( nth @ A @ Xs @ I ) )
= ( some @ B @ ( nth @ B @ Ys2 @ I ) ) ) ) ) ) ).
% map_of_zip_nth
thf(fact_5540_Rats__abs__iff,axiom,
! [X: real] :
( ( member @ real @ ( abs_abs @ real @ X ) @ ( field_char_0_Rats @ real ) )
= ( member @ real @ X @ ( field_char_0_Rats @ real ) ) ) ).
% Rats_abs_iff
thf(fact_5541_Rats__minus__iff,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: A] :
( ( member @ A @ ( uminus_uminus @ A @ A3 ) @ ( field_char_0_Rats @ A ) )
= ( member @ A @ A3 @ ( field_char_0_Rats @ A ) ) ) ) ).
% Rats_minus_iff
thf(fact_5542_coprime__int__iff,axiom,
! [M2: nat,N: nat] :
( ( algebr8660921524188924756oprime @ int @ ( semiring_1_of_nat @ int @ M2 ) @ ( semiring_1_of_nat @ int @ N ) )
= ( algebr8660921524188924756oprime @ nat @ M2 @ N ) ) ).
% coprime_int_iff
thf(fact_5543_coprime__nat__abs__left__iff,axiom,
! [K: int,N: nat] :
( ( algebr8660921524188924756oprime @ nat @ ( nat2 @ ( abs_abs @ int @ K ) ) @ N )
= ( algebr8660921524188924756oprime @ int @ K @ ( semiring_1_of_nat @ int @ N ) ) ) ).
% coprime_nat_abs_left_iff
thf(fact_5544_coprime__nat__abs__right__iff,axiom,
! [N: nat,K: int] :
( ( algebr8660921524188924756oprime @ nat @ N @ ( nat2 @ ( abs_abs @ int @ K ) ) )
= ( algebr8660921524188924756oprime @ int @ ( semiring_1_of_nat @ int @ N ) @ K ) ) ).
% coprime_nat_abs_right_iff
thf(fact_5545_map__of__zip__is__None,axiom,
! [B: $tType,A: $tType,Xs: list @ A,Ys2: list @ B,X: A] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys2 ) )
=> ( ( ( map_of @ A @ B @ ( zip @ A @ B @ Xs @ Ys2 ) @ X )
= ( none @ B ) )
= ( ~ ( member @ A @ X @ ( set2 @ A @ Xs ) ) ) ) ) ).
% map_of_zip_is_None
thf(fact_5546_coprime__common__divisor__nat,axiom,
! [A3: nat,B2: nat,X: nat] :
( ( algebr8660921524188924756oprime @ nat @ A3 @ B2 )
=> ( ( dvd_dvd @ nat @ X @ A3 )
=> ( ( dvd_dvd @ nat @ X @ B2 )
=> ( X
= ( one_one @ nat ) ) ) ) ) ).
% coprime_common_divisor_nat
thf(fact_5547_coprime__Suc__0__right,axiom,
! [N: nat] : ( algebr8660921524188924756oprime @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) ).
% coprime_Suc_0_right
thf(fact_5548_coprime__Suc__0__left,axiom,
! [N: nat] : ( algebr8660921524188924756oprime @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N ) ).
% coprime_Suc_0_left
thf(fact_5549_coprime__Suc__right__nat,axiom,
! [N: nat] : ( algebr8660921524188924756oprime @ nat @ N @ ( suc @ N ) ) ).
% coprime_Suc_right_nat
thf(fact_5550_coprime__Suc__left__nat,axiom,
! [N: nat] : ( algebr8660921524188924756oprime @ nat @ ( suc @ N ) @ N ) ).
% coprime_Suc_left_nat
thf(fact_5551_Rats__0,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( member @ A @ ( zero_zero @ A ) @ ( field_char_0_Rats @ A ) ) ) ).
% Rats_0
thf(fact_5552_Rats__1,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( member @ A @ ( one_one @ A ) @ ( field_char_0_Rats @ A ) ) ) ).
% Rats_1
thf(fact_5553_exE__some,axiom,
! [A: $tType,P: A > $o,C2: A] :
( ? [X_12: A] : ( P @ X_12 )
=> ( ( C2
= ( fChoice @ A @ P ) )
=> ( P @ C2 ) ) ) ).
% exE_some
thf(fact_5554_Rats__no__top__le,axiom,
! [X: real] :
? [X3: real] :
( ( member @ real @ X3 @ ( field_char_0_Rats @ real ) )
& ( ord_less_eq @ real @ X @ X3 ) ) ).
% Rats_no_top_le
thf(fact_5555_Rats__no__bot__less,axiom,
! [X: real] :
? [X3: real] :
( ( member @ real @ X3 @ ( field_char_0_Rats @ real ) )
& ( ord_less @ real @ X3 @ X ) ) ).
% Rats_no_bot_less
thf(fact_5556_Rats__dense__in__real,axiom,
! [X: real,Y2: real] :
( ( ord_less @ real @ X @ Y2 )
=> ? [X3: real] :
( ( member @ real @ X3 @ ( field_char_0_Rats @ real ) )
& ( ord_less @ real @ X @ X3 )
& ( ord_less @ real @ X3 @ Y2 ) ) ) ).
% Rats_dense_in_real
thf(fact_5557_Rats__divide,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: A,B2: A] :
( ( member @ A @ A3 @ ( field_char_0_Rats @ A ) )
=> ( ( member @ A @ B2 @ ( field_char_0_Rats @ A ) )
=> ( member @ A @ ( divide_divide @ A @ A3 @ B2 ) @ ( field_char_0_Rats @ A ) ) ) ) ) ).
% Rats_divide
thf(fact_5558_Rats__diff,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A3: A,B2: A] :
( ( member @ A @ A3 @ ( field_char_0_Rats @ A ) )
=> ( ( member @ A @ B2 @ ( field_char_0_Rats @ A ) )
=> ( member @ A @ ( minus_minus @ A @ A3 @ B2 ) @ ( field_char_0_Rats @ A ) ) ) ) ) ).
% Rats_diff
thf(fact_5559_Rats__of__nat,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [N: nat] : ( member @ A @ ( semiring_1_of_nat @ A @ N ) @ ( field_char_0_Rats @ A ) ) ) ).
% Rats_of_nat
thf(fact_5560_Rats__abs__nat__div__natE,axiom,
! [X: real] :
( ( member @ real @ X @ ( field_char_0_Rats @ real ) )
=> ~ ! [M3: nat,N2: nat] :
( ( N2
!= ( zero_zero @ nat ) )
=> ( ( ( abs_abs @ real @ X )
= ( divide_divide @ real @ ( semiring_1_of_nat @ real @ M3 ) @ ( semiring_1_of_nat @ real @ N2 ) ) )
=> ~ ( algebr8660921524188924756oprime @ nat @ M3 @ N2 ) ) ) ) ).
% Rats_abs_nat_div_natE
thf(fact_5561_map__of__zip__inject,axiom,
! [B: $tType,A: $tType,Ys2: list @ A,Xs: list @ B,Zs: list @ A] :
( ( ( size_size @ ( list @ A ) @ Ys2 )
= ( size_size @ ( list @ B ) @ Xs ) )
=> ( ( ( size_size @ ( list @ A ) @ Zs )
= ( size_size @ ( list @ B ) @ Xs ) )
=> ( ( distinct @ B @ Xs )
=> ( ( ( map_of @ B @ A @ ( zip @ B @ A @ Xs @ Ys2 ) )
= ( map_of @ B @ A @ ( zip @ B @ A @ Xs @ Zs ) ) )
=> ( Ys2 = Zs ) ) ) ) ) ).
% map_of_zip_inject
thf(fact_5562_map__of__eq__None__iff,axiom,
! [A: $tType,B: $tType,Xys: list @ ( product_prod @ B @ A ),X: B] :
( ( ( map_of @ B @ A @ Xys @ X )
= ( none @ A ) )
= ( ~ ( member @ B @ X @ ( image @ ( product_prod @ B @ A ) @ B @ ( product_fst @ B @ A ) @ ( set2 @ ( product_prod @ B @ A ) @ Xys ) ) ) ) ) ).
% map_of_eq_None_iff
thf(fact_5563_coprime__diff__one__right__nat,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( algebr8660921524188924756oprime @ nat @ N @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ).
% coprime_diff_one_right_nat
thf(fact_5564_coprime__diff__one__left__nat,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( algebr8660921524188924756oprime @ nat @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ N ) ) ).
% coprime_diff_one_left_nat
thf(fact_5565_set__conv__nth,axiom,
! [A: $tType] :
( ( set2 @ A )
= ( ^ [Xs3: list @ A] :
( collect @ A
@ ^ [Uu3: A] :
? [I2: nat] :
( ( Uu3
= ( nth @ A @ Xs3 @ I2 ) )
& ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs3 ) ) ) ) ) ) ).
% set_conv_nth
thf(fact_5566_map__of__zip__is__Some,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys2: list @ B,X: A] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys2 ) )
=> ( ( member @ A @ X @ ( set2 @ A @ Xs ) )
= ( ? [Y5: B] :
( ( map_of @ A @ B @ ( zip @ A @ B @ Xs @ Ys2 ) @ X )
= ( some @ B @ Y5 ) ) ) ) ) ).
% map_of_zip_is_Some
thf(fact_5567_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 )
& ! [X3: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X3 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ( vEBT_VEBT_valid @ X3 @ ( 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 )
@ ( ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X7 )
& ! [X2: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X7 ) ) )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [Mi3: nat,Ma3: nat] :
( ( ord_less_eq @ nat @ Mi3 @ Ma3 )
& ( ord_less @ nat @ Ma3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
& ! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
=> ( ( ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I2 ) @ X7 ) )
= ( vEBT_V8194947554948674370ptions @ Summary2 @ I2 ) ) )
& ( ( Mi3 = Ma3 )
=> ! [X2: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X7 ) ) )
& ( ( Mi3 != Ma3 )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList2 @ Ma3 )
& ! [X2: nat] :
( ( ord_less @ nat @ X2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList2 @ X2 )
=> ( ( ord_less @ nat @ Mi3 @ X2 )
& ( ord_less_eq @ nat @ X2 @ Ma3 ) ) ) ) ) ) ) )
@ Mima ) ) ) ) ) ).
% VEBT_internal.valid'.elims(3)
thf(fact_5568_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 )
& ! [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 )
@ ( ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X7 )
& ! [X2: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X7 ) ) )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [Mi3: nat,Ma3: nat] :
( ( ord_less_eq @ nat @ Mi3 @ Ma3 )
& ( ord_less @ nat @ Ma3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
& ! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
=> ( ( ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I2 ) @ X7 ) )
= ( vEBT_V8194947554948674370ptions @ Summary2 @ I2 ) ) )
& ( ( Mi3 = Ma3 )
=> ! [X2: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X7 ) ) )
& ( ( Mi3 != Ma3 )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList2 @ Ma3 )
& ! [X2: nat] :
( ( ord_less @ nat @ X2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList2 @ X2 )
=> ( ( ord_less @ nat @ Mi3 @ X2 )
& ( ord_less_eq @ nat @ X2 @ Ma3 ) ) ) ) ) ) ) )
@ Mima ) ) ) ) ) ).
% VEBT_internal.valid'.elims(2)
thf(fact_5569_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 )
& ! [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 )
@ ( ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X7 )
& ! [X2: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X7 ) ) )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [Mi3: nat,Ma3: nat] :
( ( ord_less_eq @ nat @ Mi3 @ Ma3 )
& ( ord_less @ nat @ Ma3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
& ! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
=> ( ( ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I2 ) @ X7 ) )
= ( vEBT_V8194947554948674370ptions @ Summary2 @ I2 ) ) )
& ( ( Mi3 = Ma3 )
=> ! [X2: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X7 ) ) )
& ( ( Mi3 != Ma3 )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList2 @ Ma3 )
& ! [X2: nat] :
( ( ord_less @ nat @ X2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList2 @ X2 )
=> ( ( ord_less @ nat @ Mi3 @ X2 )
& ( ord_less_eq @ nat @ X2 @ Ma3 ) ) ) ) ) ) ) )
@ Mima ) ) ) ) ) ) ) ).
% VEBT_internal.valid'.elims(1)
thf(fact_5570_Ball__def__raw,axiom,
! [A: $tType] :
( ( ball @ A )
= ( ^ [A7: set @ A,P3: A > $o] :
! [X2: A] :
( ( member @ A @ X2 @ A7 )
=> ( P3 @ X2 ) ) ) ) ).
% Ball_def_raw
thf(fact_5571_Rats__eq__int__div__int,axiom,
( ( field_char_0_Rats @ real )
= ( collect @ real
@ ^ [Uu3: real] :
? [I2: int,J3: int] :
( ( Uu3
= ( divide_divide @ real @ ( ring_1_of_int @ real @ I2 ) @ ( ring_1_of_int @ real @ J3 ) ) )
& ( J3
!= ( zero_zero @ int ) ) ) ) ) ).
% Rats_eq_int_div_int
thf(fact_5572_Rats__eq__int__div__nat,axiom,
( ( field_char_0_Rats @ real )
= ( collect @ real
@ ^ [Uu3: real] :
? [I2: int,N5: nat] :
( ( Uu3
= ( divide_divide @ real @ ( ring_1_of_int @ real @ I2 ) @ ( semiring_1_of_nat @ real @ N5 ) ) )
& ( N5
!= ( zero_zero @ nat ) ) ) ) ) ).
% Rats_eq_int_div_nat
thf(fact_5573_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 )
& ! [X2: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ( vEBT_VEBT_valid @ X2 @ ( 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 )
@ ( ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X7 )
& ! [X2: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X7 ) ) )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [Mi3: nat,Ma3: nat] :
( ( ord_less_eq @ nat @ Mi3 @ Ma3 )
& ( ord_less @ nat @ Ma3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg ) )
& ! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
=> ( ( ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ I2 ) @ X7 ) )
= ( vEBT_V8194947554948674370ptions @ Summary @ I2 ) ) )
& ( ( Mi3 = Ma3 )
=> ! [X2: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X7 ) ) )
& ( ( Mi3 != Ma3 )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList @ Ma3 )
& ! [X2: nat] :
( ( ord_less @ nat @ X2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg ) )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList @ X2 )
=> ( ( ord_less @ nat @ Mi3 @ X2 )
& ( ord_less_eq @ nat @ X2 @ Ma3 ) ) ) ) ) ) ) )
@ Mima2 ) ) ) ).
% VEBT_internal.valid'.simps(2)
thf(fact_5574_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 )
& ! [X3: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X3 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ( vEBT_VEBT_valid @ X3 @ ( 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 )
@ ( ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X7 )
& ! [X2: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X7 ) ) )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [Mi3: nat,Ma3: nat] :
( ( ord_less_eq @ nat @ Mi3 @ Ma3 )
& ( ord_less @ nat @ Ma3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
& ! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
=> ( ( ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I2 ) @ X7 ) )
= ( vEBT_V8194947554948674370ptions @ Summary2 @ I2 ) ) )
& ( ( Mi3 = Ma3 )
=> ! [X2: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X7 ) ) )
& ( ( Mi3 != Ma3 )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList2 @ Ma3 )
& ! [X2: nat] :
( ( ord_less @ nat @ X2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList2 @ X2 )
=> ( ( ord_less @ nat @ Mi3 @ X2 )
& ( ord_less_eq @ nat @ X2 @ Ma3 ) ) ) ) ) ) ) )
@ Mima ) ) ) ) ) ) ) ).
% VEBT_internal.valid'.pelims(3)
thf(fact_5575_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 )
& ! [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 )
@ ( ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X7 )
& ! [X2: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X7 ) ) )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [Mi3: nat,Ma3: nat] :
( ( ord_less_eq @ nat @ Mi3 @ Ma3 )
& ( ord_less @ nat @ Ma3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
& ! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
=> ( ( ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I2 ) @ X7 ) )
= ( vEBT_V8194947554948674370ptions @ Summary2 @ I2 ) ) )
& ( ( Mi3 = Ma3 )
=> ! [X2: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X7 ) ) )
& ( ( Mi3 != Ma3 )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList2 @ Ma3 )
& ! [X2: nat] :
( ( ord_less @ nat @ X2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList2 @ X2 )
=> ( ( ord_less @ nat @ Mi3 @ X2 )
& ( ord_less_eq @ nat @ X2 @ Ma3 ) ) ) ) ) ) ) )
@ Mima ) ) ) ) ) ) ) ).
% VEBT_internal.valid'.pelims(2)
thf(fact_5576_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 )
& ! [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 )
@ ( ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X7 )
& ! [X2: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X7 ) ) )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [Mi3: nat,Ma3: nat] :
( ( ord_less_eq @ nat @ Mi3 @ Ma3 )
& ( ord_less @ nat @ Ma3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
& ! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
=> ( ( ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I2 ) @ X7 ) )
= ( vEBT_V8194947554948674370ptions @ Summary2 @ I2 ) ) )
& ( ( Mi3 = Ma3 )
=> ! [X2: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X2 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X7: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X7 ) ) )
& ( ( Mi3 != Ma3 )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList2 @ Ma3 )
& ! [X2: nat] :
( ( ord_less @ nat @ X2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList2 @ X2 )
=> ( ( ord_less @ nat @ Mi3 @ X2 )
& ( ord_less_eq @ nat @ X2 @ Ma3 ) ) ) ) ) ) ) )
@ 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_5577_list__eq__iff__zip__eq,axiom,
! [A: $tType] :
( ( ^ [Y4: list @ A,Z: list @ A] : Y4 = Z )
= ( ^ [Xs3: list @ A,Ys3: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs3 )
= ( size_size @ ( list @ A ) @ Ys3 ) )
& ! [X2: product_prod @ A @ A] :
( ( member @ ( product_prod @ A @ A ) @ X2 @ ( set2 @ ( product_prod @ A @ A ) @ ( zip @ A @ A @ Xs3 @ Ys3 ) ) )
=> ( product_case_prod @ A @ A @ $o
@ ^ [Y4: A,Z: A] : Y4 = Z
@ X2 ) ) ) ) ) ).
% list_eq_iff_zip_eq
thf(fact_5578_concat__eq__concat__iff,axiom,
! [A: $tType,Xs: list @ ( list @ A ),Ys2: list @ ( list @ A )] :
( ! [X3: product_prod @ ( list @ A ) @ ( list @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ X3 @ ( set2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( zip @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys2 ) ) )
=> ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ $o
@ ^ [Y5: list @ A,Z6: list @ A] :
( ( size_size @ ( list @ A ) @ Y5 )
= ( size_size @ ( list @ A ) @ Z6 ) )
@ X3 ) )
=> ( ( ( size_size @ ( list @ ( list @ A ) ) @ Xs )
= ( size_size @ ( list @ ( list @ A ) ) @ Ys2 ) )
=> ( ( ( concat @ A @ Xs )
= ( concat @ A @ Ys2 ) )
= ( Xs = Ys2 ) ) ) ) ).
% concat_eq_concat_iff
thf(fact_5579_concat__injective,axiom,
! [A: $tType,Xs: list @ ( list @ A ),Ys2: list @ ( list @ A )] :
( ( ( concat @ A @ Xs )
= ( concat @ A @ Ys2 ) )
=> ( ( ( size_size @ ( list @ ( list @ A ) ) @ Xs )
= ( size_size @ ( list @ ( list @ A ) ) @ Ys2 ) )
=> ( ! [X3: product_prod @ ( list @ A ) @ ( list @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ X3 @ ( set2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( zip @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys2 ) ) )
=> ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ $o
@ ^ [Y5: list @ A,Z6: list @ A] :
( ( size_size @ ( list @ A ) @ Y5 )
= ( size_size @ ( list @ A ) @ Z6 ) )
@ X3 ) )
=> ( Xs = Ys2 ) ) ) ) ).
% concat_injective
thf(fact_5580_listrel__iff__zip,axiom,
! [B: $tType,A: $tType,Xs: list @ A,Ys2: list @ B,R4: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs @ Ys2 ) @ ( listrel @ A @ B @ R4 ) )
= ( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys2 ) )
& ! [X2: product_prod @ A @ B] :
( ( member @ ( product_prod @ A @ B ) @ X2 @ ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys2 ) ) )
=> ( product_case_prod @ A @ B @ $o
@ ^ [Y5: A,Z6: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Y5 @ Z6 ) @ R4 )
@ X2 ) ) ) ) ).
% listrel_iff_zip
thf(fact_5581_ran__map__of__zip,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys2: list @ B] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys2 ) )
=> ( ( distinct @ A @ Xs )
=> ( ( ran @ A @ B @ ( map_of @ A @ B @ ( zip @ A @ B @ Xs @ Ys2 ) ) )
= ( set2 @ B @ Ys2 ) ) ) ) ).
% ran_map_of_zip
thf(fact_5582_Nats__altdef1,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( ( semiring_1_Nats @ A )
= ( collect @ A
@ ^ [Uu3: A] :
? [N5: int] :
( ( Uu3
= ( ring_1_of_int @ A @ N5 ) )
& ( ord_less_eq @ int @ ( zero_zero @ int ) @ N5 ) ) ) ) ) ).
% Nats_altdef1
thf(fact_5583_ran__empty,axiom,
! [B: $tType,A: $tType] :
( ( ran @ B @ A
@ ^ [X2: B] : ( none @ A ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% ran_empty
thf(fact_5584_Nats__cases,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [X: A] :
( ( member @ A @ X @ ( semiring_1_Nats @ A ) )
=> ~ ! [N2: nat] :
( X
!= ( semiring_1_of_nat @ A @ N2 ) ) ) ) ).
% Nats_cases
thf(fact_5585_Nats__induct,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [X: A,P: A > $o] :
( ( member @ A @ X @ ( semiring_1_Nats @ A ) )
=> ( ! [N2: nat] : ( P @ ( semiring_1_of_nat @ A @ N2 ) )
=> ( P @ X ) ) ) ) ).
% Nats_induct
thf(fact_5586_of__nat__in__Nats,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [N: nat] : ( member @ A @ ( semiring_1_of_nat @ A @ N ) @ ( semiring_1_Nats @ A ) ) ) ).
% of_nat_in_Nats
thf(fact_5587_Nats__mult,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [A3: A,B2: A] :
( ( member @ A @ A3 @ ( semiring_1_Nats @ A ) )
=> ( ( member @ A @ B2 @ ( semiring_1_Nats @ A ) )
=> ( member @ A @ ( times_times @ A @ A3 @ B2 ) @ ( semiring_1_Nats @ A ) ) ) ) ) ).
% Nats_mult
thf(fact_5588_Nats__numeral,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [W: num] : ( member @ A @ ( numeral_numeral @ A @ W ) @ ( semiring_1_Nats @ A ) ) ) ).
% Nats_numeral
thf(fact_5589_Nats__add,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [A3: A,B2: A] :
( ( member @ A @ A3 @ ( semiring_1_Nats @ A ) )
=> ( ( member @ A @ B2 @ ( semiring_1_Nats @ A ) )
=> ( member @ A @ ( plus_plus @ A @ A3 @ B2 ) @ ( semiring_1_Nats @ A ) ) ) ) ) ).
% Nats_add
thf(fact_5590_Nats__0,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( member @ A @ ( zero_zero @ A ) @ ( semiring_1_Nats @ A ) ) ) ).
% Nats_0
thf(fact_5591_Nats__1,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( member @ A @ ( one_one @ A ) @ ( semiring_1_Nats @ A ) ) ) ).
% Nats_1
thf(fact_5592_listrel__eq__len,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys2: list @ B,R4: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs @ Ys2 ) @ ( listrel @ A @ B @ R4 ) )
=> ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys2 ) ) ) ).
% listrel_eq_len
thf(fact_5593_Nats__diff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A3: A,B2: A] :
( ( member @ A @ A3 @ ( semiring_1_Nats @ A ) )
=> ( ( member @ A @ B2 @ ( semiring_1_Nats @ A ) )
=> ( ( ord_less_eq @ A @ B2 @ A3 )
=> ( member @ A @ ( minus_minus @ A @ A3 @ B2 ) @ ( semiring_1_Nats @ A ) ) ) ) ) ) ).
% Nats_diff
thf(fact_5594_Nats__subset__Ints,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( ord_less_eq @ ( set @ A ) @ ( semiring_1_Nats @ A ) @ ( ring_1_Ints @ A ) ) ) ).
% Nats_subset_Ints
thf(fact_5595_Nats__altdef2,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( ( semiring_1_Nats @ A )
= ( collect @ A
@ ^ [N5: A] :
( ( member @ A @ N5 @ ( ring_1_Ints @ A ) )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ N5 ) ) ) ) ) ).
% Nats_altdef2
thf(fact_5596_listrel__iff__nth,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys2: list @ B,R4: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs @ Ys2 ) @ ( listrel @ A @ B @ R4 ) )
= ( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys2 ) )
& ! [N5: nat] :
( ( ord_less @ nat @ N5 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ ( nth @ A @ Xs @ N5 ) @ ( nth @ B @ Ys2 @ N5 ) ) @ R4 ) ) ) ) ).
% listrel_iff_nth
thf(fact_5597_set__nths,axiom,
! [A: $tType,Xs: list @ A,I5: set @ nat] :
( ( set2 @ A @ ( nths @ A @ Xs @ I5 ) )
= ( collect @ A
@ ^ [Uu3: A] :
? [I2: nat] :
( ( Uu3
= ( nth @ A @ Xs @ I2 ) )
& ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs ) )
& ( member @ nat @ I2 @ I5 ) ) ) ) ).
% set_nths
thf(fact_5598_sum__list__update,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [K: nat,Xs: list @ A,X: A] :
( ( ord_less @ nat @ K @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( groups8242544230860333062m_list @ A @ ( list_update @ A @ Xs @ K @ X ) )
= ( minus_minus @ A @ ( plus_plus @ A @ ( groups8242544230860333062m_list @ A @ Xs ) @ X ) @ ( nth @ A @ Xs @ K ) ) ) ) ) ).
% sum_list_update
thf(fact_5599_Rat_Opositive_Orsp,axiom,
( bNF_rel_fun @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ $o @ $o @ ratrel
@ ^ [Y4: $o,Z: $o] : Y4 = Z
@ ^ [X2: product_prod @ int @ int] : ( ord_less @ int @ ( zero_zero @ int ) @ ( times_times @ int @ ( product_fst @ int @ int @ X2 ) @ ( product_snd @ int @ int @ X2 ) ) )
@ ^ [X2: product_prod @ int @ int] : ( ord_less @ int @ ( zero_zero @ int ) @ ( times_times @ int @ ( product_fst @ int @ int @ X2 ) @ ( product_snd @ int @ int @ X2 ) ) ) ) ).
% Rat.positive.rsp
thf(fact_5600_sum__list__eq__0__iff,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [Ns: list @ A] :
( ( ( groups8242544230860333062m_list @ A @ Ns )
= ( zero_zero @ A ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ ( set2 @ A @ Ns ) )
=> ( X2
= ( zero_zero @ A ) ) ) ) ) ) ).
% sum_list_eq_0_iff
thf(fact_5601_transfer__rule__of__int,axiom,
! [A: $tType,B: $tType] :
( ( ( ring_1 @ B )
& ( ring_1 @ A ) )
=> ! [R2: A > B > $o] :
( ( R2 @ ( zero_zero @ A ) @ ( zero_zero @ B ) )
=> ( ( R2 @ ( one_one @ A ) @ ( one_one @ B ) )
=> ( ( bNF_rel_fun @ A @ B @ ( A > A ) @ ( B > B ) @ R2 @ ( bNF_rel_fun @ A @ B @ A @ B @ R2 @ R2 ) @ ( plus_plus @ A ) @ ( plus_plus @ B ) )
=> ( ( bNF_rel_fun @ A @ B @ A @ B @ R2 @ R2 @ ( uminus_uminus @ A ) @ ( uminus_uminus @ B ) )
=> ( bNF_rel_fun @ int @ int @ A @ B
@ ^ [Y4: int,Z: int] : Y4 = Z
@ R2
@ ( ring_1_of_int @ A )
@ ( ring_1_of_int @ B ) ) ) ) ) ) ) ).
% transfer_rule_of_int
thf(fact_5602_transfer__rule__numeral,axiom,
! [A: $tType,B: $tType] :
( ( ( monoid_add @ B )
& ( semiring_numeral @ B )
& ( monoid_add @ A )
& ( semiring_numeral @ A ) )
=> ! [R2: A > B > $o] :
( ( R2 @ ( zero_zero @ A ) @ ( zero_zero @ B ) )
=> ( ( R2 @ ( one_one @ A ) @ ( one_one @ B ) )
=> ( ( bNF_rel_fun @ A @ B @ ( A > A ) @ ( B > B ) @ R2 @ ( bNF_rel_fun @ A @ B @ A @ B @ R2 @ R2 ) @ ( plus_plus @ A ) @ ( plus_plus @ B ) )
=> ( bNF_rel_fun @ num @ num @ A @ B
@ ^ [Y4: num,Z: num] : Y4 = Z
@ R2
@ ( numeral_numeral @ A )
@ ( numeral_numeral @ B ) ) ) ) ) ) ).
% transfer_rule_numeral
thf(fact_5603_power__transfer,axiom,
! [A: $tType,B: $tType] :
( ( ( power @ B )
& ( power @ A ) )
=> ! [R2: A > B > $o] :
( ( R2 @ ( one_one @ A ) @ ( one_one @ B ) )
=> ( ( bNF_rel_fun @ A @ B @ ( A > A ) @ ( B > B ) @ R2 @ ( bNF_rel_fun @ A @ B @ A @ B @ R2 @ R2 ) @ ( times_times @ A ) @ ( times_times @ B ) )
=> ( bNF_rel_fun @ A @ B @ ( nat > A ) @ ( nat > B ) @ R2
@ ( bNF_rel_fun @ nat @ nat @ A @ B
@ ^ [Y4: nat,Z: nat] : Y4 = Z
@ R2 )
@ ( power_power @ A )
@ ( power_power @ B ) ) ) ) ) ).
% power_transfer
thf(fact_5604_nths__all,axiom,
! [A: $tType,Xs: list @ A,I5: set @ nat] :
( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( member @ nat @ I3 @ I5 ) )
=> ( ( nths @ A @ Xs @ I5 )
= Xs ) ) ).
% nths_all
thf(fact_5605_sum__list__nonpos,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [Xs: list @ A] :
( ! [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( ord_less_eq @ A @ X3 @ ( zero_zero @ A ) ) )
=> ( ord_less_eq @ A @ ( groups8242544230860333062m_list @ A @ Xs ) @ ( zero_zero @ A ) ) ) ) ).
% sum_list_nonpos
thf(fact_5606_sum__list__nonneg__eq__0__iff,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [Xs: list @ A] :
( ! [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 ) )
=> ( ( ( groups8242544230860333062m_list @ A @ Xs )
= ( zero_zero @ A ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
=> ( X2
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% sum_list_nonneg_eq_0_iff
thf(fact_5607_Groups__List_Osum__list__nonneg,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [Xs: list @ A] :
( ! [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ X3 ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( groups8242544230860333062m_list @ A @ Xs ) ) ) ) ).
% Groups_List.sum_list_nonneg
thf(fact_5608_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_5609_elem__le__sum__list,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [K: nat,Ns: list @ A] :
( ( ord_less @ nat @ K @ ( size_size @ ( list @ A ) @ Ns ) )
=> ( ord_less_eq @ A @ ( nth @ A @ Ns @ K ) @ ( groups8242544230860333062m_list @ A @ Ns ) ) ) ) ).
% elem_le_sum_list
thf(fact_5610_card__length__sum__list__rec,axiom,
! [M2: nat,N6: nat] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ M2 )
=> ( ( finite_card @ ( list @ nat )
@ ( collect @ ( list @ nat )
@ ^ [L2: list @ nat] :
( ( ( size_size @ ( list @ nat ) @ L2 )
= M2 )
& ( ( groups8242544230860333062m_list @ nat @ L2 )
= N6 ) ) ) )
= ( plus_plus @ nat
@ ( finite_card @ ( list @ nat )
@ ( collect @ ( list @ nat )
@ ^ [L2: list @ nat] :
( ( ( size_size @ ( list @ nat ) @ L2 )
= ( minus_minus @ nat @ M2 @ ( one_one @ nat ) ) )
& ( ( groups8242544230860333062m_list @ nat @ L2 )
= N6 ) ) ) )
@ ( finite_card @ ( list @ nat )
@ ( collect @ ( list @ nat )
@ ^ [L2: list @ nat] :
( ( ( size_size @ ( list @ nat ) @ L2 )
= M2 )
& ( ( plus_plus @ nat @ ( groups8242544230860333062m_list @ nat @ L2 ) @ ( one_one @ nat ) )
= N6 ) ) ) ) ) ) ) ).
% card_length_sum_list_rec
thf(fact_5611_card__length__sum__list,axiom,
! [M2: nat,N6: nat] :
( ( finite_card @ ( list @ nat )
@ ( collect @ ( list @ nat )
@ ^ [L2: list @ nat] :
( ( ( size_size @ ( list @ nat ) @ L2 )
= M2 )
& ( ( groups8242544230860333062m_list @ nat @ L2 )
= N6 ) ) ) )
= ( binomial @ ( minus_minus @ nat @ ( plus_plus @ nat @ N6 @ M2 ) @ ( one_one @ nat ) ) @ N6 ) ) ).
% card_length_sum_list
thf(fact_5612_length__nths,axiom,
! [A: $tType,Xs: list @ A,I5: set @ nat] :
( ( size_size @ ( list @ A ) @ ( nths @ A @ Xs @ I5 ) )
= ( finite_card @ nat
@ ( collect @ nat
@ ^ [I2: nat] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs ) )
& ( member @ nat @ I2 @ I5 ) ) ) ) ) ).
% length_nths
thf(fact_5613_of__rat_Orsp,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( bNF_rel_fun @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ A @ A @ ratrel
@ ^ [Y4: A,Z: A] : Y4 = Z
@ ^ [X2: product_prod @ int @ int] : ( divide_divide @ A @ ( ring_1_of_int @ A @ ( product_fst @ int @ int @ X2 ) ) @ ( ring_1_of_int @ A @ ( product_snd @ int @ int @ X2 ) ) )
@ ^ [X2: product_prod @ int @ int] : ( divide_divide @ A @ ( ring_1_of_int @ A @ ( product_fst @ int @ int @ X2 ) ) @ ( ring_1_of_int @ A @ ( product_snd @ int @ int @ X2 ) ) ) ) ) ).
% of_rat.rsp
thf(fact_5614_sum__list__sum__nth,axiom,
! [B: $tType] :
( ( comm_monoid_add @ B )
=> ( ( groups8242544230860333062m_list @ B )
= ( ^ [Xs3: list @ B] : ( groups7311177749621191930dd_sum @ nat @ B @ ( nth @ B @ Xs3 ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ B ) @ Xs3 ) ) ) ) ) ) ).
% sum_list_sum_nth
thf(fact_5615_transfer__rule__of__nat,axiom,
! [A: $tType,B: $tType] :
( ( ( semiring_1 @ B )
& ( semiring_1 @ A ) )
=> ! [R2: A > B > $o] :
( ( R2 @ ( zero_zero @ A ) @ ( zero_zero @ B ) )
=> ( ( R2 @ ( one_one @ A ) @ ( one_one @ B ) )
=> ( ( bNF_rel_fun @ A @ B @ ( A > A ) @ ( B > B ) @ R2 @ ( bNF_rel_fun @ A @ B @ A @ B @ R2 @ R2 ) @ ( plus_plus @ A ) @ ( plus_plus @ B ) )
=> ( bNF_rel_fun @ nat @ nat @ A @ B
@ ^ [Y4: nat,Z: nat] : Y4 = Z
@ R2
@ ( semiring_1_of_nat @ A )
@ ( semiring_1_of_nat @ B ) ) ) ) ) ) ).
% transfer_rule_of_nat
thf(fact_5616_vanishes__mult__bounded,axiom,
! [X8: nat > rat,Y7: nat > rat] :
( ? [A11: rat] :
( ( ord_less @ rat @ ( zero_zero @ rat ) @ A11 )
& ! [N2: nat] : ( ord_less @ rat @ ( abs_abs @ rat @ ( X8 @ N2 ) ) @ A11 ) )
=> ( ( vanishes @ Y7 )
=> ( vanishes
@ ^ [N5: nat] : ( times_times @ rat @ ( X8 @ N5 ) @ ( Y7 @ N5 ) ) ) ) ) ).
% vanishes_mult_bounded
thf(fact_5617_range__mod,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( image @ nat @ nat
@ ^ [M5: nat] : ( modulo_modulo @ nat @ M5 @ N )
@ ( top_top @ ( set @ nat ) ) )
= ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ).
% range_mod
thf(fact_5618_finite__option__UNIV,axiom,
! [A: $tType] :
( ( finite_finite @ ( option @ A ) @ ( top_top @ ( set @ ( option @ A ) ) ) )
= ( finite_finite @ A @ ( top_top @ ( set @ A ) ) ) ) ).
% finite_option_UNIV
thf(fact_5619_inf__top_Oright__neutral,axiom,
! [A: $tType] :
( ( bounde4346867609351753570nf_top @ A )
=> ! [A3: A] :
( ( inf_inf @ A @ A3 @ ( top_top @ A ) )
= A3 ) ) ).
% inf_top.right_neutral
thf(fact_5620_inf__top_Oneutr__eq__iff,axiom,
! [A: $tType] :
( ( bounde4346867609351753570nf_top @ A )
=> ! [A3: A,B2: A] :
( ( ( top_top @ A )
= ( inf_inf @ A @ A3 @ B2 ) )
= ( ( A3
= ( top_top @ A ) )
& ( B2
= ( top_top @ A ) ) ) ) ) ).
% inf_top.neutr_eq_iff
thf(fact_5621_inf__top_Oleft__neutral,axiom,
! [A: $tType] :
( ( bounde4346867609351753570nf_top @ A )
=> ! [A3: A] :
( ( inf_inf @ A @ ( top_top @ A ) @ A3 )
= A3 ) ) ).
% inf_top.left_neutral
thf(fact_5622_inf__top_Oeq__neutr__iff,axiom,
! [A: $tType] :
( ( bounde4346867609351753570nf_top @ A )
=> ! [A3: A,B2: A] :
( ( ( inf_inf @ A @ A3 @ B2 )
= ( top_top @ A ) )
= ( ( A3
= ( top_top @ A ) )
& ( B2
= ( top_top @ A ) ) ) ) ) ).
% inf_top.eq_neutr_iff
thf(fact_5623_top__eq__inf__iff,axiom,
! [A: $tType] :
( ( bounde4346867609351753570nf_top @ A )
=> ! [X: A,Y2: A] :
( ( ( top_top @ A )
= ( inf_inf @ A @ X @ Y2 ) )
= ( ( X
= ( top_top @ A ) )
& ( Y2
= ( top_top @ A ) ) ) ) ) ).
% top_eq_inf_iff
thf(fact_5624_inf__eq__top__iff,axiom,
! [A: $tType] :
( ( bounde4346867609351753570nf_top @ A )
=> ! [X: A,Y2: A] :
( ( ( inf_inf @ A @ X @ Y2 )
= ( top_top @ A ) )
= ( ( X
= ( top_top @ A ) )
& ( Y2
= ( top_top @ A ) ) ) ) ) ).
% inf_eq_top_iff
thf(fact_5625_inf__top__right,axiom,
! [A: $tType] :
( ( bounde4346867609351753570nf_top @ A )
=> ! [X: A] :
( ( inf_inf @ A @ X @ ( top_top @ A ) )
= X ) ) ).
% inf_top_right
thf(fact_5626_inf__top__left,axiom,
! [A: $tType] :
( ( bounde4346867609351753570nf_top @ A )
=> ! [X: A] :
( ( inf_inf @ A @ ( top_top @ A ) @ X )
= X ) ) ).
% inf_top_left
thf(fact_5627_vanishes__const,axiom,
! [C2: rat] :
( ( vanishes
@ ^ [N5: nat] : C2 )
= ( C2
= ( zero_zero @ rat ) ) ) ).
% vanishes_const
thf(fact_5628_range__diff,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A3: A] :
( ( image @ A @ A @ ( minus_minus @ A @ A3 ) @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) ) ) ).
% range_diff
thf(fact_5629_boolean__algebra_Ocompl__one,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ( ( uminus_uminus @ A @ ( top_top @ A ) )
= ( bot_bot @ A ) ) ) ).
% boolean_algebra.compl_one
thf(fact_5630_boolean__algebra_Ocompl__zero,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ( ( uminus_uminus @ A @ ( bot_bot @ A ) )
= ( top_top @ A ) ) ) ).
% boolean_algebra.compl_zero
thf(fact_5631_Diff__UNIV,axiom,
! [A: $tType,A4: set @ A] :
( ( minus_minus @ ( set @ A ) @ A4 @ ( top_top @ ( set @ A ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Diff_UNIV
thf(fact_5632_surj__fn,axiom,
! [A: $tType,F2: A > A,N: nat] :
( ( ( image @ A @ A @ F2 @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) )
=> ( ( image @ A @ A @ ( compow @ ( A > A ) @ N @ F2 ) @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) ) ) ).
% surj_fn
thf(fact_5633_finite__compl,axiom,
! [A: $tType,A4: set @ A] :
( ( finite_finite @ A @ A4 )
=> ( ( finite_finite @ A @ ( uminus_uminus @ ( set @ A ) @ A4 ) )
= ( finite_finite @ A @ ( top_top @ ( set @ A ) ) ) ) ) ).
% finite_compl
thf(fact_5634_Gcd__UNIV,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ( ( gcd_Gcd @ A @ ( top_top @ ( set @ A ) ) )
= ( one_one @ A ) ) ) ).
% Gcd_UNIV
thf(fact_5635_surj__diff__right,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A3: A] :
( ( image @ A @ A
@ ^ [X2: A] : ( minus_minus @ A @ X2 @ A3 )
@ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) ) ) ).
% surj_diff_right
thf(fact_5636_uminus__integer_Orsp,axiom,
( bNF_rel_fun @ int @ int @ int @ int
@ ^ [Y4: int,Z: int] : Y4 = Z
@ ^ [Y4: int,Z: int] : Y4 = Z
@ ( uminus_uminus @ int )
@ ( uminus_uminus @ int ) ) ).
% uminus_integer.rsp
thf(fact_5637_Suc_Orsp,axiom,
( bNF_rel_fun @ nat @ nat @ nat @ nat
@ ^ [Y4: nat,Z: nat] : Y4 = Z
@ ^ [Y4: nat,Z: nat] : Y4 = Z
@ suc
@ suc ) ).
% Suc.rsp
thf(fact_5638_less__natural_Orsp,axiom,
( bNF_rel_fun @ nat @ nat @ ( nat > $o ) @ ( nat > $o )
@ ^ [Y4: nat,Z: nat] : Y4 = Z
@ ( bNF_rel_fun @ nat @ nat @ $o @ $o
@ ^ [Y4: nat,Z: nat] : Y4 = Z
@ ^ [Y4: $o,Z: $o] : Y4 = Z )
@ ( ord_less @ nat )
@ ( ord_less @ nat ) ) ).
% less_natural.rsp
thf(fact_5639_less__integer_Orsp,axiom,
( bNF_rel_fun @ int @ int @ ( int > $o ) @ ( int > $o )
@ ^ [Y4: int,Z: int] : Y4 = Z
@ ( bNF_rel_fun @ int @ int @ $o @ $o
@ ^ [Y4: int,Z: int] : Y4 = Z
@ ^ [Y4: $o,Z: $o] : Y4 = Z )
@ ( ord_less @ int )
@ ( ord_less @ int ) ) ).
% less_integer.rsp
thf(fact_5640_sub_Orsp,axiom,
( bNF_rel_fun @ num @ num @ ( num > int ) @ ( num > int )
@ ^ [Y4: num,Z: num] : Y4 = Z
@ ( bNF_rel_fun @ num @ num @ int @ int
@ ^ [Y4: num,Z: num] : Y4 = Z
@ ^ [Y4: int,Z: int] : Y4 = Z )
@ ^ [M5: num,N5: num] : ( minus_minus @ int @ ( numeral_numeral @ int @ M5 ) @ ( numeral_numeral @ int @ N5 ) )
@ ^ [M5: num,N5: num] : ( minus_minus @ int @ ( numeral_numeral @ int @ M5 ) @ ( numeral_numeral @ int @ N5 ) ) ) ).
% sub.rsp
thf(fact_5641_minus__natural_Orsp,axiom,
( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( nat > nat )
@ ^ [Y4: nat,Z: nat] : Y4 = Z
@ ( bNF_rel_fun @ nat @ nat @ nat @ nat
@ ^ [Y4: nat,Z: nat] : Y4 = Z
@ ^ [Y4: nat,Z: nat] : Y4 = Z )
@ ( minus_minus @ nat )
@ ( minus_minus @ nat ) ) ).
% minus_natural.rsp
thf(fact_5642_minus__integer_Orsp,axiom,
( bNF_rel_fun @ int @ int @ ( int > int ) @ ( int > int )
@ ^ [Y4: int,Z: int] : Y4 = Z
@ ( bNF_rel_fun @ int @ int @ int @ int
@ ^ [Y4: int,Z: int] : Y4 = Z
@ ^ [Y4: int,Z: int] : Y4 = Z )
@ ( minus_minus @ int )
@ ( minus_minus @ int ) ) ).
% minus_integer.rsp
thf(fact_5643_divide__integer_Orsp,axiom,
( bNF_rel_fun @ int @ int @ ( int > int ) @ ( int > int )
@ ^ [Y4: int,Z: int] : Y4 = Z
@ ( bNF_rel_fun @ int @ int @ int @ int
@ ^ [Y4: int,Z: int] : Y4 = Z
@ ^ [Y4: int,Z: int] : Y4 = Z )
@ ( divide_divide @ int )
@ ( divide_divide @ int ) ) ).
% divide_integer.rsp
thf(fact_5644_divide__natural_Orsp,axiom,
( bNF_rel_fun @ nat @ nat @ ( nat > nat ) @ ( nat > nat )
@ ^ [Y4: nat,Z: nat] : Y4 = Z
@ ( bNF_rel_fun @ nat @ nat @ nat @ nat
@ ^ [Y4: nat,Z: nat] : Y4 = Z
@ ^ [Y4: nat,Z: nat] : Y4 = Z )
@ ( divide_divide @ nat )
@ ( divide_divide @ nat ) ) ).
% divide_natural.rsp
thf(fact_5645_integer__of__natural_Orsp,axiom,
( bNF_rel_fun @ nat @ nat @ int @ int
@ ^ [Y4: nat,Z: nat] : Y4 = Z
@ ^ [Y4: int,Z: int] : Y4 = Z
@ ( semiring_1_of_nat @ int )
@ ( semiring_1_of_nat @ int ) ) ).
% integer_of_natural.rsp
thf(fact_5646_num_Ocase__transfer,axiom,
! [A: $tType,B: $tType,S3: A > B > $o] :
( bNF_rel_fun @ A @ B @ ( ( num > A ) > ( num > A ) > num > A ) @ ( ( num > B ) > ( num > B ) > num > B ) @ S3
@ ( bNF_rel_fun @ ( num > A ) @ ( num > B ) @ ( ( num > A ) > num > A ) @ ( ( num > B ) > num > B )
@ ( bNF_rel_fun @ num @ num @ A @ B
@ ^ [Y4: num,Z: num] : Y4 = Z
@ S3 )
@ ( bNF_rel_fun @ ( num > A ) @ ( num > B ) @ ( num > A ) @ ( num > B )
@ ( bNF_rel_fun @ num @ num @ A @ B
@ ^ [Y4: num,Z: num] : Y4 = Z
@ S3 )
@ ( bNF_rel_fun @ num @ num @ A @ B
@ ^ [Y4: num,Z: num] : Y4 = Z
@ S3 ) ) )
@ ( case_num @ A )
@ ( case_num @ B ) ) ).
% num.case_transfer
thf(fact_5647_Fract_Orsp,axiom,
( bNF_rel_fun @ int @ int @ ( int > ( product_prod @ int @ int ) ) @ ( int > ( product_prod @ int @ int ) )
@ ^ [Y4: int,Z: int] : Y4 = Z
@ ( bNF_rel_fun @ int @ int @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int )
@ ^ [Y4: int,Z: int] : Y4 = Z
@ ratrel )
@ ^ [A5: int,B3: int] :
( if @ ( product_prod @ int @ int )
@ ( B3
= ( zero_zero @ int ) )
@ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
@ ( product_Pair @ int @ int @ A5 @ B3 ) )
@ ^ [A5: int,B3: int] :
( if @ ( product_prod @ int @ int )
@ ( B3
= ( zero_zero @ int ) )
@ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
@ ( product_Pair @ int @ int @ A5 @ B3 ) ) ) ).
% Fract.rsp
thf(fact_5648_vanishes__minus,axiom,
! [X8: nat > rat] :
( ( vanishes @ X8 )
=> ( vanishes
@ ^ [N5: nat] : ( uminus_uminus @ rat @ ( X8 @ N5 ) ) ) ) ).
% vanishes_minus
thf(fact_5649_top_Onot__eq__extremum,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [A3: A] :
( ( A3
!= ( top_top @ A ) )
= ( ord_less @ A @ A3 @ ( top_top @ A ) ) ) ) ).
% top.not_eq_extremum
thf(fact_5650_top_Oextremum__strict,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [A3: A] :
~ ( ord_less @ A @ ( top_top @ A ) @ A3 ) ) ).
% top.extremum_strict
thf(fact_5651_finite__fun__UNIVD1,axiom,
! [B: $tType,A: $tType] :
( ( finite_finite @ ( A > B ) @ ( top_top @ ( set @ ( A > B ) ) ) )
=> ( ( ( finite_card @ B @ ( top_top @ ( set @ B ) ) )
!= ( suc @ ( zero_zero @ nat ) ) )
=> ( finite_finite @ A @ ( top_top @ ( set @ A ) ) ) ) ) ).
% finite_fun_UNIVD1
thf(fact_5652_vanishes__add,axiom,
! [X8: nat > rat,Y7: nat > rat] :
( ( vanishes @ X8 )
=> ( ( vanishes @ Y7 )
=> ( vanishes
@ ^ [N5: nat] : ( plus_plus @ rat @ ( X8 @ N5 ) @ ( Y7 @ N5 ) ) ) ) ) ).
% vanishes_add
thf(fact_5653_vanishes__diff,axiom,
! [X8: nat > rat,Y7: nat > rat] :
( ( vanishes @ X8 )
=> ( ( vanishes @ Y7 )
=> ( vanishes
@ ^ [N5: nat] : ( minus_minus @ rat @ ( X8 @ N5 ) @ ( Y7 @ N5 ) ) ) ) ) ).
% vanishes_diff
thf(fact_5654_transfer__rule__of__bool,axiom,
! [A: $tType,B: $tType] :
( ( ( zero_neq_one @ B )
& ( zero_neq_one @ A ) )
=> ! [R2: A > B > $o] :
( ( R2 @ ( zero_zero @ A ) @ ( zero_zero @ B ) )
=> ( ( R2 @ ( one_one @ A ) @ ( one_one @ B ) )
=> ( bNF_rel_fun @ $o @ $o @ A @ B
@ ^ [Y4: $o,Z: $o] : Y4 = Z
@ R2
@ ( zero_neq_one_of_bool @ A )
@ ( zero_neq_one_of_bool @ B ) ) ) ) ) ).
% transfer_rule_of_bool
thf(fact_5655_bij__uminus,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ( bij_betw @ A @ A @ ( uminus_uminus @ A ) @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ A ) ) ) ) ).
% bij_uminus
thf(fact_5656_Compl__UNIV__eq,axiom,
! [A: $tType] :
( ( uminus_uminus @ ( set @ A ) @ ( top_top @ ( set @ A ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Compl_UNIV_eq
thf(fact_5657_Compl__empty__eq,axiom,
! [A: $tType] :
( ( uminus_uminus @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) ) ).
% Compl_empty_eq
thf(fact_5658_Compl__eq__Diff__UNIV,axiom,
! [A: $tType] :
( ( uminus_uminus @ ( set @ A ) )
= ( minus_minus @ ( set @ A ) @ ( top_top @ ( set @ A ) ) ) ) ).
% Compl_eq_Diff_UNIV
thf(fact_5659_bij__fn,axiom,
! [A: $tType,F2: A > A,N: nat] :
( ( bij_betw @ A @ A @ F2 @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ A ) ) )
=> ( bij_betw @ A @ A @ ( compow @ ( A > A ) @ N @ F2 ) @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ A ) ) ) ) ).
% bij_fn
thf(fact_5660_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_5661_surj__Compl__image__subset,axiom,
! [A: $tType,B: $tType,F2: B > A,A4: set @ B] :
( ( ( image @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
=> ( ord_less_eq @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ ( image @ B @ A @ F2 @ A4 ) ) @ ( image @ B @ A @ F2 @ ( uminus_uminus @ ( set @ B ) @ A4 ) ) ) ) ).
% surj_Compl_image_subset
thf(fact_5662_bij__image__Compl__eq,axiom,
! [B: $tType,A: $tType,F2: A > B,A4: set @ A] :
( ( bij_betw @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
=> ( ( image @ A @ B @ F2 @ ( uminus_uminus @ ( set @ A ) @ A4 ) )
= ( uminus_uminus @ ( set @ B ) @ ( image @ A @ B @ F2 @ A4 ) ) ) ) ).
% bij_image_Compl_eq
thf(fact_5663_Nats__def,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( ( semiring_1_Nats @ A )
= ( image @ nat @ A @ ( semiring_1_of_nat @ A ) @ ( top_top @ ( set @ nat ) ) ) ) ) ).
% Nats_def
thf(fact_5664_finite__range__Some,axiom,
! [A: $tType] :
( ( finite_finite @ ( option @ A ) @ ( image @ A @ ( option @ A ) @ ( some @ A ) @ ( top_top @ ( set @ A ) ) ) )
= ( finite_finite @ A @ ( top_top @ ( set @ A ) ) ) ) ).
% finite_range_Some
thf(fact_5665_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_5666_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_5667_finite__UNIV__card__ge__0,axiom,
! [A: $tType] :
( ( finite_finite @ 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_5668_uminus__rat_Orsp,axiom,
( bNF_rel_fun @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ ratrel @ ratrel
@ ^ [X2: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( uminus_uminus @ int @ ( product_fst @ int @ int @ X2 ) ) @ ( product_snd @ int @ int @ X2 ) )
@ ^ [X2: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( uminus_uminus @ int @ ( product_fst @ int @ int @ X2 ) ) @ ( product_snd @ int @ int @ X2 ) ) ) ).
% uminus_rat.rsp
thf(fact_5669_card__range__greater__zero,axiom,
! [A: $tType,B: $tType,F2: B > A] :
( ( finite_finite @ A @ ( image @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) ) )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ ( finite_card @ A @ ( image @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) ) ) ) ) ).
% card_range_greater_zero
thf(fact_5670_vanishes__def,axiom,
( vanishes
= ( ^ [X7: nat > rat] :
! [R: rat] :
( ( ord_less @ rat @ ( zero_zero @ rat ) @ R )
=> ? [K3: nat] :
! [N5: nat] :
( ( ord_less_eq @ nat @ K3 @ N5 )
=> ( ord_less @ rat @ ( abs_abs @ rat @ ( X7 @ N5 ) ) @ R ) ) ) ) ) ).
% vanishes_def
thf(fact_5671_vanishesI,axiom,
! [X8: nat > rat] :
( ! [R3: rat] :
( ( ord_less @ rat @ ( zero_zero @ rat ) @ R3 )
=> ? [K4: nat] :
! [N2: nat] :
( ( ord_less_eq @ nat @ K4 @ N2 )
=> ( ord_less @ rat @ ( abs_abs @ rat @ ( X8 @ N2 ) ) @ R3 ) ) )
=> ( vanishes @ X8 ) ) ).
% vanishesI
thf(fact_5672_vanishesD,axiom,
! [X8: nat > rat,R4: rat] :
( ( vanishes @ X8 )
=> ( ( ord_less @ rat @ ( zero_zero @ rat ) @ R4 )
=> ? [K2: nat] :
! [N4: nat] :
( ( ord_less_eq @ nat @ K2 @ N4 )
=> ( ord_less @ rat @ ( abs_abs @ rat @ ( X8 @ N4 ) ) @ R4 ) ) ) ) ).
% vanishesD
thf(fact_5673_inverse__rat_Orsp,axiom,
( bNF_rel_fun @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ ( product_prod @ int @ int ) @ ratrel @ ratrel
@ ^ [X2: product_prod @ int @ int] :
( if @ ( product_prod @ int @ int )
@ ( ( product_fst @ int @ int @ X2 )
= ( zero_zero @ int ) )
@ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
@ ( product_Pair @ int @ int @ ( product_snd @ int @ int @ X2 ) @ ( product_fst @ int @ int @ X2 ) ) )
@ ^ [X2: product_prod @ int @ int] :
( if @ ( product_prod @ int @ int )
@ ( ( product_fst @ int @ int @ X2 )
= ( zero_zero @ int ) )
@ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
@ ( product_Pair @ int @ int @ ( product_snd @ int @ int @ X2 ) @ ( product_fst @ int @ int @ X2 ) ) ) ) ).
% inverse_rat.rsp
thf(fact_5674_UNIV__nat__eq,axiom,
( ( top_top @ ( set @ nat ) )
= ( insert @ nat @ ( zero_zero @ nat ) @ ( image @ nat @ nat @ suc @ ( top_top @ ( set @ nat ) ) ) ) ) ).
% UNIV_nat_eq
thf(fact_5675_inverse__rat_Otransfer,axiom,
( bNF_rel_fun @ ( product_prod @ int @ int ) @ rat @ ( product_prod @ int @ int ) @ rat @ pcr_rat @ pcr_rat
@ ^ [X2: product_prod @ int @ int] :
( if @ ( product_prod @ int @ int )
@ ( ( product_fst @ int @ int @ X2 )
= ( zero_zero @ int ) )
@ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
@ ( product_Pair @ int @ int @ ( product_snd @ int @ int @ X2 ) @ ( product_fst @ int @ int @ X2 ) ) )
@ ( inverse_inverse @ rat ) ) ).
% inverse_rat.transfer
thf(fact_5676_Rat_Opositive_Otransfer,axiom,
( bNF_rel_fun @ ( product_prod @ int @ int ) @ rat @ $o @ $o @ pcr_rat
@ ^ [Y4: $o,Z: $o] : Y4 = Z
@ ^ [X2: product_prod @ int @ int] : ( ord_less @ int @ ( zero_zero @ int ) @ ( times_times @ int @ ( product_fst @ int @ int @ X2 ) @ ( product_snd @ int @ int @ X2 ) ) )
@ positive ) ).
% Rat.positive.transfer
thf(fact_5677_card__UNIV__unit,axiom,
( ( finite_card @ product_unit @ ( top_top @ ( set @ product_unit ) ) )
= ( one_one @ nat ) ) ).
% card_UNIV_unit
thf(fact_5678_range__abs__Nats,axiom,
( ( image @ int @ int @ ( abs_abs @ int ) @ ( top_top @ ( set @ int ) ) )
= ( semiring_1_Nats @ int ) ) ).
% range_abs_Nats
thf(fact_5679_infinite__UNIV__int,axiom,
~ ( finite_finite @ int @ ( top_top @ ( set @ int ) ) ) ).
% infinite_UNIV_int
thf(fact_5680_surj__prod__encode,axiom,
( ( image @ ( product_prod @ nat @ nat ) @ nat @ nat_prod_encode @ ( top_top @ ( set @ ( product_prod @ nat @ nat ) ) ) )
= ( top_top @ ( set @ nat ) ) ) ).
% surj_prod_encode
thf(fact_5681_bij__prod__encode,axiom,
bij_betw @ ( product_prod @ nat @ nat ) @ nat @ nat_prod_encode @ ( top_top @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( top_top @ ( set @ nat ) ) ).
% bij_prod_encode
thf(fact_5682_Ints__def,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( ( ring_1_Ints @ A )
= ( image @ int @ A @ ( ring_1_of_int @ A ) @ ( top_top @ ( set @ int ) ) ) ) ) ).
% Ints_def
thf(fact_5683_int__in__range__abs,axiom,
! [N: nat] : ( member @ int @ ( semiring_1_of_nat @ int @ N ) @ ( image @ int @ int @ ( abs_abs @ int ) @ ( top_top @ ( set @ int ) ) ) ) ).
% int_in_range_abs
thf(fact_5684_one__rat_Otransfer,axiom,
pcr_rat @ ( product_Pair @ int @ int @ ( one_one @ int ) @ ( one_one @ int ) ) @ ( one_one @ rat ) ).
% one_rat.transfer
thf(fact_5685_zero__rat_Otransfer,axiom,
pcr_rat @ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) ) @ ( zero_zero @ rat ) ).
% zero_rat.transfer
thf(fact_5686_Fract_Otransfer,axiom,
( bNF_rel_fun @ int @ int @ ( int > ( product_prod @ int @ int ) ) @ ( int > rat )
@ ^ [Y4: int,Z: int] : Y4 = Z
@ ( bNF_rel_fun @ int @ int @ ( product_prod @ int @ int ) @ rat
@ ^ [Y4: int,Z: int] : Y4 = Z
@ pcr_rat )
@ ^ [A5: int,B3: int] :
( if @ ( product_prod @ int @ int )
@ ( B3
= ( zero_zero @ int ) )
@ ( product_Pair @ int @ int @ ( zero_zero @ int ) @ ( one_one @ int ) )
@ ( product_Pair @ int @ int @ A5 @ B3 ) )
@ fract ) ).
% Fract.transfer
thf(fact_5687_of__rat_Otransfer,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( bNF_rel_fun @ ( product_prod @ int @ int ) @ rat @ A @ A @ pcr_rat
@ ^ [Y4: A,Z: A] : Y4 = Z
@ ^ [X2: product_prod @ int @ int] : ( divide_divide @ A @ ( ring_1_of_int @ A @ ( product_fst @ int @ int @ X2 ) ) @ ( ring_1_of_int @ A @ ( product_snd @ int @ int @ X2 ) ) )
@ ( field_char_0_of_rat @ A ) ) ) ).
% of_rat.transfer
thf(fact_5688_uminus__rat_Otransfer,axiom,
( bNF_rel_fun @ ( product_prod @ int @ int ) @ rat @ ( product_prod @ int @ int ) @ rat @ pcr_rat @ pcr_rat
@ ^ [X2: product_prod @ int @ int] : ( product_Pair @ int @ int @ ( uminus_uminus @ int @ ( product_fst @ int @ int @ X2 ) ) @ ( product_snd @ int @ int @ X2 ) )
@ ( uminus_uminus @ rat ) ) ).
% uminus_rat.transfer
thf(fact_5689_root__def,axiom,
( root
= ( ^ [N5: nat,X2: real] :
( if @ real
@ ( N5
= ( zero_zero @ nat ) )
@ ( zero_zero @ real )
@ ( the_inv_into @ real @ real @ ( top_top @ ( set @ real ) )
@ ^ [Y5: real] : ( times_times @ real @ ( sgn_sgn @ real @ Y5 ) @ ( power_power @ real @ ( abs_abs @ real @ Y5 ) @ N5 ) )
@ X2 ) ) ) ) ).
% root_def
thf(fact_5690_times__int_Otransfer,axiom,
( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ int @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) ) @ ( int > int ) @ pcr_int @ ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ int @ ( product_prod @ nat @ nat ) @ int @ pcr_int @ pcr_int )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X2: nat,Y5: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V5: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ X2 @ U2 ) @ ( times_times @ nat @ Y5 @ V5 ) ) @ ( plus_plus @ nat @ ( times_times @ nat @ X2 @ V5 ) @ ( times_times @ nat @ Y5 @ U2 ) ) ) ) )
@ ( times_times @ int ) ) ).
% times_int.transfer
thf(fact_5691_num_Orec__transfer,axiom,
! [A: $tType,B: $tType,S3: A > B > $o] :
( bNF_rel_fun @ A @ B @ ( ( num > A > A ) > ( num > A > A ) > num > A ) @ ( ( num > B > B ) > ( num > B > B ) > num > B ) @ S3
@ ( bNF_rel_fun @ ( num > A > A ) @ ( num > B > B ) @ ( ( num > A > A ) > num > A ) @ ( ( num > B > B ) > num > B )
@ ( bNF_rel_fun @ num @ num @ ( A > A ) @ ( B > B )
@ ^ [Y4: num,Z: num] : Y4 = Z
@ ( bNF_rel_fun @ A @ B @ A @ B @ S3 @ S3 ) )
@ ( bNF_rel_fun @ ( num > A > A ) @ ( num > B > B ) @ ( num > A ) @ ( num > B )
@ ( bNF_rel_fun @ num @ num @ ( A > A ) @ ( B > B )
@ ^ [Y4: num,Z: num] : Y4 = Z
@ ( bNF_rel_fun @ A @ B @ A @ B @ S3 @ S3 ) )
@ ( bNF_rel_fun @ num @ num @ A @ B
@ ^ [Y4: num,Z: num] : Y4 = Z
@ S3 ) ) )
@ ( rec_num @ A )
@ ( rec_num @ B ) ) ).
% num.rec_transfer
thf(fact_5692_verit__eq__simplify_I20_J,axiom,
! [A: $tType,F1: A,F22: num > A > A,F32: num > A > A,X23: num] :
( ( rec_num @ A @ F1 @ F22 @ F32 @ ( bit0 @ X23 ) )
= ( F22 @ X23 @ ( rec_num @ A @ F1 @ F22 @ F32 @ X23 ) ) ) ).
% verit_eq_simplify(20)
thf(fact_5693_verit__eq__simplify_I19_J,axiom,
! [A: $tType,F1: A,F22: num > A > A,F32: num > A > A] :
( ( rec_num @ A @ F1 @ F22 @ F32 @ one2 )
= F1 ) ).
% verit_eq_simplify(19)
thf(fact_5694_verit__eq__simplify_I21_J,axiom,
! [A: $tType,F1: A,F22: num > A > A,F32: num > A > A,X32: num] :
( ( rec_num @ A @ F1 @ F22 @ F32 @ ( bit1 @ X32 ) )
= ( F32 @ X32 @ ( rec_num @ A @ F1 @ F22 @ F32 @ X32 ) ) ) ).
% verit_eq_simplify(21)
thf(fact_5695_zero__int_Otransfer,axiom,
pcr_int @ ( product_Pair @ nat @ nat @ ( zero_zero @ nat ) @ ( zero_zero @ nat ) ) @ ( zero_zero @ int ) ).
% zero_int.transfer
thf(fact_5696_int__transfer,axiom,
( bNF_rel_fun @ nat @ nat @ ( product_prod @ nat @ nat ) @ int
@ ^ [Y4: nat,Z: nat] : Y4 = Z
@ pcr_int
@ ^ [N5: nat] : ( product_Pair @ nat @ nat @ N5 @ ( zero_zero @ nat ) )
@ ( semiring_1_of_nat @ int ) ) ).
% int_transfer
thf(fact_5697_uminus__int_Otransfer,axiom,
( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ int @ ( product_prod @ nat @ nat ) @ int @ pcr_int @ pcr_int
@ ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [X2: nat,Y5: nat] : ( product_Pair @ nat @ nat @ Y5 @ X2 ) )
@ ( uminus_uminus @ int ) ) ).
% uminus_int.transfer
thf(fact_5698_nat_Otransfer,axiom,
( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ int @ nat @ nat @ pcr_int
@ ^ [Y4: nat,Z: nat] : Y4 = Z
@ ( product_case_prod @ nat @ nat @ nat @ ( minus_minus @ nat ) )
@ nat2 ) ).
% nat.transfer
thf(fact_5699_one__int_Otransfer,axiom,
pcr_int @ ( product_Pair @ nat @ nat @ ( one_one @ nat ) @ ( zero_zero @ nat ) ) @ ( one_one @ int ) ).
% one_int.transfer
thf(fact_5700_of__int_Otransfer,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ int @ A @ A @ pcr_int
@ ^ [Y4: A,Z: A] : Y4 = Z
@ ( product_case_prod @ nat @ nat @ A
@ ^ [I2: nat,J3: nat] : ( minus_minus @ A @ ( semiring_1_of_nat @ A @ I2 ) @ ( semiring_1_of_nat @ A @ J3 ) ) )
@ ( ring_1_of_int @ A ) ) ) ).
% of_int.transfer
thf(fact_5701_less__int_Otransfer,axiom,
( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ int @ ( ( product_prod @ nat @ nat ) > $o ) @ ( int > $o ) @ pcr_int
@ ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ int @ $o @ $o @ pcr_int
@ ^ [Y4: $o,Z: $o] : Y4 = Z )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [X2: nat,Y5: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U2: nat,V5: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ X2 @ V5 ) @ ( plus_plus @ nat @ U2 @ Y5 ) ) ) )
@ ( ord_less @ int ) ) ).
% less_int.transfer
thf(fact_5702_less__eq__int_Otransfer,axiom,
( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ int @ ( ( product_prod @ nat @ nat ) > $o ) @ ( int > $o ) @ pcr_int
@ ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ int @ $o @ $o @ pcr_int
@ ^ [Y4: $o,Z: $o] : Y4 = Z )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [X2: nat,Y5: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U2: nat,V5: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ X2 @ V5 ) @ ( plus_plus @ nat @ U2 @ Y5 ) ) ) )
@ ( ord_less_eq @ int ) ) ).
% less_eq_int.transfer
thf(fact_5703_plus__int_Otransfer,axiom,
( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ int @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) ) @ ( int > int ) @ pcr_int @ ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ int @ ( product_prod @ nat @ nat ) @ int @ pcr_int @ pcr_int )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X2: nat,Y5: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V5: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X2 @ U2 ) @ ( plus_plus @ nat @ Y5 @ V5 ) ) ) )
@ ( plus_plus @ int ) ) ).
% plus_int.transfer
thf(fact_5704_minus__int_Otransfer,axiom,
( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ int @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) ) @ ( int > int ) @ pcr_int @ ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ int @ ( product_prod @ nat @ nat ) @ int @ pcr_int @ pcr_int )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X2: nat,Y5: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V5: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X2 @ V5 ) @ ( plus_plus @ nat @ Y5 @ U2 ) ) ) )
@ ( minus_minus @ int ) ) ).
% minus_int.transfer
thf(fact_5705_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_5706_DERIV__real__root__generic,axiom,
! [N: nat,X: real,D6: 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 )
=> ( D6
= ( 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 ) )
=> ( D6
= ( 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 )
=> ( D6
= ( 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 ) @ D6 @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ) ) ).
% DERIV_real_root_generic
thf(fact_5707_DERIV__arctan__series,axiom,
! [X: real] :
( ( ord_less @ real @ ( abs_abs @ real @ X ) @ ( one_one @ real ) )
=> ( has_field_derivative @ real
@ ^ [X9: 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 @ X9 @ ( 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_5708_DERIV__pos__inc__left,axiom,
! [F2: real > real,L: real,X: real] :
( ( has_field_derivative @ real @ F2 @ L @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ L )
=> ? [D5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D5 )
& ! [H4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ H4 )
=> ( ( ord_less @ real @ H4 @ D5 )
=> ( ord_less @ real @ ( F2 @ ( minus_minus @ real @ X @ H4 ) ) @ ( F2 @ X ) ) ) ) ) ) ) ).
% DERIV_pos_inc_left
thf(fact_5709_DERIV__neg__dec__left,axiom,
! [F2: real > real,L: real,X: real] :
( ( has_field_derivative @ real @ F2 @ L @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( ( ord_less @ real @ L @ ( zero_zero @ real ) )
=> ? [D5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D5 )
& ! [H4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ H4 )
=> ( ( ord_less @ real @ H4 @ D5 )
=> ( ord_less @ real @ ( F2 @ X ) @ ( F2 @ ( minus_minus @ real @ X @ H4 ) ) ) ) ) ) ) ) ).
% DERIV_neg_dec_left
thf(fact_5710_DERIV__const__ratio__const2,axiom,
! [A3: real,B2: real,F2: real > real,K: real] :
( ( A3 != B2 )
=> ( ! [X3: real] : ( has_field_derivative @ real @ F2 @ K @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( divide_divide @ real @ ( minus_minus @ real @ ( F2 @ B2 ) @ ( F2 @ A3 ) ) @ ( minus_minus @ real @ B2 @ A3 ) )
= K ) ) ) ).
% DERIV_const_ratio_const2
thf(fact_5711_DERIV__const__ratio__const,axiom,
! [A3: real,B2: real,F2: real > real,K: real] :
( ( A3 != B2 )
=> ( ! [X3: real] : ( has_field_derivative @ real @ F2 @ K @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( minus_minus @ real @ ( F2 @ B2 ) @ ( F2 @ A3 ) )
= ( times_times @ real @ ( minus_minus @ real @ B2 @ A3 ) @ K ) ) ) ) ).
% DERIV_const_ratio_const
thf(fact_5712_DERIV__fun__exp,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [G: A > A,M2: A,X: A] :
( ( has_field_derivative @ A @ G @ M2 @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) )
=> ( has_field_derivative @ A
@ ^ [X2: A] : ( exp @ A @ ( G @ X2 ) )
@ ( times_times @ A @ ( exp @ A @ ( G @ X ) ) @ M2 )
@ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% DERIV_fun_exp
thf(fact_5713_DERIV__fun__sin,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [G: A > A,M2: A,X: A] :
( ( has_field_derivative @ A @ G @ M2 @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) )
=> ( has_field_derivative @ A
@ ^ [X2: A] : ( sin @ A @ ( G @ X2 ) )
@ ( times_times @ A @ ( cos @ A @ ( G @ X ) ) @ M2 )
@ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% DERIV_fun_sin
thf(fact_5714_DERIV__exp,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A] : ( has_field_derivative @ A @ ( exp @ A ) @ ( exp @ A @ X ) @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) ) ) ).
% DERIV_exp
thf(fact_5715_DERIV__sin,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A] : ( has_field_derivative @ A @ ( sin @ A ) @ ( cos @ A @ X ) @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) ) ) ).
% DERIV_sin
thf(fact_5716_DERIV__cos,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A] : ( has_field_derivative @ A @ ( cos @ A ) @ ( uminus_uminus @ A @ ( sin @ A @ X ) ) @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) ) ) ).
% DERIV_cos
thf(fact_5717_DERIV__mirror,axiom,
! [F2: real > real,Y2: real,X: real] :
( ( has_field_derivative @ real @ F2 @ Y2 @ ( topolo174197925503356063within @ real @ ( uminus_uminus @ real @ X ) @ ( top_top @ ( set @ real ) ) ) )
= ( has_field_derivative @ real
@ ^ [X2: real] : ( F2 @ ( uminus_uminus @ real @ X2 ) )
@ ( uminus_uminus @ real @ Y2 )
@ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) ) ) ).
% DERIV_mirror
thf(fact_5718_DERIV__inverse_H,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D6: A,X: A,S: set @ A] :
( ( has_field_derivative @ A @ F2 @ D6 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( ( ( F2 @ X )
!= ( zero_zero @ A ) )
=> ( has_field_derivative @ A
@ ^ [X2: A] : ( inverse_inverse @ A @ ( F2 @ X2 ) )
@ ( uminus_uminus @ A @ ( times_times @ A @ ( times_times @ A @ ( inverse_inverse @ A @ ( F2 @ X ) ) @ D6 ) @ ( inverse_inverse @ A @ ( F2 @ X ) ) ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ).
% DERIV_inverse'
thf(fact_5719_DERIV__minus,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D6: A,X: A,S: set @ A] :
( ( has_field_derivative @ A @ F2 @ D6 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( has_field_derivative @ A
@ ^ [X2: A] : ( uminus_uminus @ A @ ( F2 @ X2 ) )
@ ( uminus_uminus @ A @ D6 )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ).
% DERIV_minus
thf(fact_5720_field__differentiable__minus,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,F6: A,F5: filter @ A] :
( ( has_field_derivative @ A @ F2 @ F6 @ F5 )
=> ( has_field_derivative @ A
@ ^ [Z6: A] : ( uminus_uminus @ A @ ( F2 @ Z6 ) )
@ ( uminus_uminus @ A @ F6 )
@ F5 ) ) ) ).
% field_differentiable_minus
thf(fact_5721_DERIV__const,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [K: A,F5: filter @ A] :
( has_field_derivative @ A
@ ^ [X2: A] : K
@ ( zero_zero @ A )
@ F5 ) ) ).
% DERIV_const
thf(fact_5722_DERIV__ident,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F5: filter @ A] :
( has_field_derivative @ A
@ ^ [X2: A] : X2
@ ( one_one @ A )
@ F5 ) ) ).
% DERIV_ident
thf(fact_5723_has__field__derivative__scaleR__right,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D6: A,F5: filter @ A,C2: real] :
( ( has_field_derivative @ A @ F2 @ D6 @ F5 )
=> ( has_field_derivative @ A
@ ^ [X2: A] : ( real_V8093663219630862766scaleR @ A @ C2 @ ( F2 @ X2 ) )
@ ( real_V8093663219630862766scaleR @ A @ C2 @ D6 )
@ F5 ) ) ) ).
% has_field_derivative_scaleR_right
thf(fact_5724_DERIV__cdivide,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D6: A,X: A,S: set @ A,C2: A] :
( ( has_field_derivative @ A @ F2 @ D6 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( has_field_derivative @ A
@ ^ [X2: A] : ( divide_divide @ A @ ( F2 @ X2 ) @ C2 )
@ ( divide_divide @ A @ D6 @ C2 )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ).
% DERIV_cdivide
thf(fact_5725_DERIV__diff,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D6: A,X: A,S: set @ A,G: A > A,E4: A] :
( ( has_field_derivative @ A @ F2 @ D6 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( ( has_field_derivative @ A @ G @ E4 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( has_field_derivative @ A
@ ^ [X2: A] : ( minus_minus @ A @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( minus_minus @ A @ D6 @ E4 )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ).
% DERIV_diff
thf(fact_5726_field__differentiable__diff,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,F6: A,F5: filter @ A,G: A > A,G3: A] :
( ( has_field_derivative @ A @ F2 @ F6 @ F5 )
=> ( ( has_field_derivative @ A @ G @ G3 @ F5 )
=> ( has_field_derivative @ A
@ ^ [Z6: A] : ( minus_minus @ A @ ( F2 @ Z6 ) @ ( G @ Z6 ) )
@ ( minus_minus @ A @ F6 @ G3 )
@ F5 ) ) ) ) ).
% field_differentiable_diff
thf(fact_5727_has__field__derivative__cosh,axiom,
! [A13: $tType] :
( ( ( real_Vector_banach @ A13 )
& ( real_V3459762299906320749_field @ A13 ) )
=> ! [G: A13 > A13,Db: A13,X: A13,S: set @ A13] :
( ( has_field_derivative @ A13 @ G @ Db @ ( topolo174197925503356063within @ A13 @ X @ S ) )
=> ( has_field_derivative @ A13
@ ^ [X2: A13] : ( cosh @ A13 @ ( G @ X2 ) )
@ ( times_times @ A13 @ ( sinh @ A13 @ ( G @ X ) ) @ Db )
@ ( topolo174197925503356063within @ A13 @ X @ S ) ) ) ) ).
% has_field_derivative_cosh
thf(fact_5728_has__field__derivative__sinh,axiom,
! [A13: $tType] :
( ( ( real_Vector_banach @ A13 )
& ( real_V3459762299906320749_field @ A13 ) )
=> ! [G: A13 > A13,Db: A13,X: A13,S: set @ A13] :
( ( has_field_derivative @ A13 @ G @ Db @ ( topolo174197925503356063within @ A13 @ X @ S ) )
=> ( has_field_derivative @ A13
@ ^ [X2: A13] : ( sinh @ A13 @ ( G @ X2 ) )
@ ( times_times @ A13 @ ( cosh @ A13 @ ( G @ X ) ) @ Db )
@ ( topolo174197925503356063within @ A13 @ X @ S ) ) ) ) ).
% has_field_derivative_sinh
thf(fact_5729_has__real__derivative__pos__inc__left,axiom,
! [F2: real > real,L: real,X: real,S3: set @ real] :
( ( has_field_derivative @ real @ F2 @ L @ ( topolo174197925503356063within @ real @ X @ S3 ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ L )
=> ? [D5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D5 )
& ! [H4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ H4 )
=> ( ( member @ real @ ( minus_minus @ real @ X @ H4 ) @ S3 )
=> ( ( ord_less @ real @ H4 @ D5 )
=> ( ord_less @ real @ ( F2 @ ( minus_minus @ real @ X @ H4 ) ) @ ( F2 @ X ) ) ) ) ) ) ) ) ).
% has_real_derivative_pos_inc_left
thf(fact_5730_has__real__derivative__neg__dec__left,axiom,
! [F2: real > real,L: real,X: real,S3: set @ real] :
( ( has_field_derivative @ real @ F2 @ L @ ( topolo174197925503356063within @ real @ X @ S3 ) )
=> ( ( ord_less @ real @ L @ ( zero_zero @ real ) )
=> ? [D5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D5 )
& ! [H4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ H4 )
=> ( ( member @ real @ ( minus_minus @ real @ X @ H4 ) @ S3 )
=> ( ( ord_less @ real @ H4 @ D5 )
=> ( ord_less @ real @ ( F2 @ X ) @ ( F2 @ ( minus_minus @ real @ X @ H4 ) ) ) ) ) ) ) ) ) ).
% has_real_derivative_neg_dec_left
thf(fact_5731_DERIV__divide,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D6: A,X: A,S: set @ A,G: A > A,E4: A] :
( ( has_field_derivative @ A @ F2 @ D6 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( ( has_field_derivative @ A @ G @ E4 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( ( ( G @ X )
!= ( zero_zero @ A ) )
=> ( has_field_derivative @ A
@ ^ [X2: A] : ( divide_divide @ A @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ D6 @ ( G @ X ) ) @ ( times_times @ A @ ( F2 @ X ) @ E4 ) ) @ ( times_times @ A @ ( G @ X ) @ ( G @ X ) ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ) ).
% DERIV_divide
thf(fact_5732_MVT2,axiom,
! [A3: real,B2: real,F2: real > real,F6: real > real] :
( ( ord_less @ real @ A3 @ B2 )
=> ( ! [X3: real] :
( ( ord_less_eq @ real @ A3 @ X3 )
=> ( ( ord_less_eq @ real @ X3 @ B2 )
=> ( has_field_derivative @ real @ F2 @ ( F6 @ X3 ) @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) ) )
=> ? [Z4: real] :
( ( ord_less @ real @ A3 @ Z4 )
& ( ord_less @ real @ Z4 @ B2 )
& ( ( minus_minus @ real @ ( F2 @ B2 ) @ ( F2 @ A3 ) )
= ( times_times @ real @ ( minus_minus @ real @ B2 @ A3 ) @ ( F6 @ Z4 ) ) ) ) ) ) ).
% MVT2
thf(fact_5733_DERIV__local__const,axiom,
! [F2: real > real,L: real,X: real,D2: real] :
( ( has_field_derivative @ real @ F2 @ L @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ D2 )
=> ( ! [Y3: real] :
( ( ord_less @ real @ ( abs_abs @ real @ ( minus_minus @ real @ X @ Y3 ) ) @ D2 )
=> ( ( F2 @ X )
= ( F2 @ Y3 ) ) )
=> ( L
= ( zero_zero @ real ) ) ) ) ) ).
% DERIV_local_const
thf(fact_5734_DERIV__ln,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( has_field_derivative @ real @ ( ln_ln @ real ) @ ( inverse_inverse @ real @ X ) @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) ) ) ).
% DERIV_ln
thf(fact_5735_DERIV__fun__cos,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [G: A > A,M2: A,X: A] :
( ( has_field_derivative @ A @ G @ M2 @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) )
=> ( has_field_derivative @ A
@ ^ [X2: A] : ( cos @ A @ ( G @ X2 ) )
@ ( times_times @ A @ ( uminus_uminus @ A @ ( sin @ A @ ( G @ X ) ) ) @ M2 )
@ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% DERIV_fun_cos
thf(fact_5736_DERIV__cos__add,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [K: A,Xa: A] :
( has_field_derivative @ A
@ ^ [X2: A] : ( cos @ A @ ( plus_plus @ A @ X2 @ K ) )
@ ( uminus_uminus @ A @ ( sin @ A @ ( plus_plus @ A @ Xa @ K ) ) )
@ ( topolo174197925503356063within @ A @ Xa @ ( top_top @ ( set @ A ) ) ) ) ) ).
% DERIV_cos_add
thf(fact_5737_DERIV__power__Suc,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D6: A,X: A,S: set @ A,N: nat] :
( ( has_field_derivative @ A @ F2 @ D6 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( has_field_derivative @ A
@ ^ [X2: A] : ( power_power @ A @ ( F2 @ X2 ) @ ( suc @ N ) )
@ ( times_times @ A @ ( plus_plus @ A @ ( one_one @ A ) @ ( semiring_1_of_nat @ A @ N ) ) @ ( times_times @ A @ D6 @ ( power_power @ A @ ( F2 @ X ) @ N ) ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ).
% DERIV_power_Suc
thf(fact_5738_DERIV__const__average,axiom,
! [A3: real,B2: real,V2: real > real,K: real] :
( ( A3 != B2 )
=> ( ! [X3: real] : ( has_field_derivative @ real @ V2 @ K @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( V2 @ ( divide_divide @ real @ ( plus_plus @ real @ A3 @ B2 ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
= ( divide_divide @ real @ ( plus_plus @ real @ ( V2 @ A3 ) @ ( V2 @ B2 ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ).
% DERIV_const_average
thf(fact_5739_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_5740_DERIV__power,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D6: A,X: A,S: set @ A,N: nat] :
( ( has_field_derivative @ A @ F2 @ D6 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( has_field_derivative @ A
@ ^ [X2: A] : ( power_power @ A @ ( F2 @ X2 ) @ N )
@ ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ ( times_times @ A @ D6 @ ( power_power @ A @ ( F2 @ X ) @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) ) ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ).
% DERIV_power
thf(fact_5741_DERIV__local__max,axiom,
! [F2: real > real,L: real,X: real,D2: real] :
( ( has_field_derivative @ real @ F2 @ L @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ D2 )
=> ( ! [Y3: real] :
( ( ord_less @ real @ ( abs_abs @ real @ ( minus_minus @ real @ X @ Y3 ) ) @ D2 )
=> ( ord_less_eq @ real @ ( F2 @ Y3 ) @ ( F2 @ X ) ) )
=> ( L
= ( zero_zero @ real ) ) ) ) ) ).
% DERIV_local_max
thf(fact_5742_DERIV__local__min,axiom,
! [F2: real > real,L: real,X: real,D2: real] :
( ( has_field_derivative @ real @ F2 @ L @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ D2 )
=> ( ! [Y3: real] :
( ( ord_less @ real @ ( abs_abs @ real @ ( minus_minus @ real @ X @ Y3 ) ) @ D2 )
=> ( ord_less_eq @ real @ ( F2 @ X ) @ ( F2 @ Y3 ) ) )
=> ( L
= ( zero_zero @ real ) ) ) ) ) ).
% DERIV_local_min
thf(fact_5743_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_5744_DERIV__pow,axiom,
! [N: nat,X: real,S: set @ real] :
( has_field_derivative @ real
@ ^ [X2: real] : ( power_power @ real @ X2 @ 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_5745_termdiffs__strong__converges__everywhere,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [C2: nat > A,X: A] :
( ! [Y3: A] :
( summable @ A
@ ^ [N5: nat] : ( times_times @ A @ ( C2 @ N5 ) @ ( power_power @ A @ Y3 @ N5 ) ) )
=> ( has_field_derivative @ A
@ ^ [X2: A] :
( suminf @ A
@ ^ [N5: nat] : ( times_times @ A @ ( C2 @ N5 ) @ ( power_power @ A @ X2 @ N5 ) ) )
@ ( suminf @ A
@ ^ [N5: nat] : ( times_times @ A @ ( diffs @ A @ C2 @ N5 ) @ ( power_power @ A @ X @ N5 ) ) )
@ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% termdiffs_strong_converges_everywhere
thf(fact_5746_at__within__Icc__at,axiom,
! [A: $tType] :
( ( topolo2564578578187576103pology @ A )
=> ! [A3: A,X: A,B2: A] :
( ( ord_less @ A @ A3 @ X )
=> ( ( ord_less @ A @ X @ B2 )
=> ( ( topolo174197925503356063within @ A @ X @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
= ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% at_within_Icc_at
thf(fact_5747_DERIV__fun__pow,axiom,
! [G: real > real,M2: real,X: real,N: nat] :
( ( has_field_derivative @ real @ G @ M2 @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( has_field_derivative @ real
@ ^ [X2: real] : ( power_power @ real @ ( G @ X2 ) @ 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 ) ) ) ) @ M2 )
@ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) ) ) ).
% DERIV_fun_pow
thf(fact_5748_at__within__Icc__at__left,axiom,
! [A: $tType] :
( ( topolo2564578578187576103pology @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( topolo174197925503356063within @ A @ B2 @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
= ( topolo174197925503356063within @ A @ B2 @ ( set_ord_lessThan @ A @ B2 ) ) ) ) ) ).
% at_within_Icc_at_left
thf(fact_5749_DERIV__quotient,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D2: A,X: A,S: set @ A,G: A > A,E2: A] :
( ( has_field_derivative @ A @ F2 @ D2 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( ( has_field_derivative @ A @ G @ E2 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( ( ( G @ X )
!= ( zero_zero @ A ) )
=> ( has_field_derivative @ A
@ ^ [Y5: A] : ( divide_divide @ A @ ( F2 @ Y5 ) @ ( G @ Y5 ) )
@ ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ D2 @ ( G @ X ) ) @ ( times_times @ A @ E2 @ ( F2 @ X ) ) ) @ ( power_power @ A @ ( G @ X ) @ ( suc @ ( suc @ ( zero_zero @ nat ) ) ) ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ) ).
% DERIV_quotient
thf(fact_5750_DERIV__inverse__fun,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D2: A,X: A,S: set @ A] :
( ( has_field_derivative @ A @ F2 @ D2 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( ( ( F2 @ X )
!= ( zero_zero @ A ) )
=> ( has_field_derivative @ A
@ ^ [X2: A] : ( inverse_inverse @ A @ ( F2 @ X2 ) )
@ ( uminus_uminus @ A @ ( times_times @ A @ D2 @ ( inverse_inverse @ A @ ( power_power @ A @ ( F2 @ X ) @ ( suc @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ).
% DERIV_inverse_fun
thf(fact_5751_termdiffs__sums__strong,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [K6: real,C2: nat > A,F2: A > A,F6: A,Z2: A] :
( ! [Z4: A] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ Z4 ) @ K6 )
=> ( sums @ A
@ ^ [N5: nat] : ( times_times @ A @ ( C2 @ N5 ) @ ( power_power @ A @ Z4 @ N5 ) )
@ ( F2 @ Z4 ) ) )
=> ( ( has_field_derivative @ A @ F2 @ F6 @ ( topolo174197925503356063within @ A @ Z2 @ ( top_top @ ( set @ A ) ) ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ Z2 ) @ K6 )
=> ( sums @ A
@ ^ [N5: nat] : ( times_times @ A @ ( diffs @ A @ C2 @ N5 ) @ ( power_power @ A @ Z2 @ N5 ) )
@ F6 ) ) ) ) ) ).
% termdiffs_sums_strong
thf(fact_5752_has__real__derivative__powr,axiom,
! [Z2: real,R4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ Z2 )
=> ( has_field_derivative @ real
@ ^ [Z6: real] : ( powr @ real @ Z6 @ R4 )
@ ( times_times @ real @ R4 @ ( powr @ real @ Z2 @ ( minus_minus @ real @ R4 @ ( one_one @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ Z2 @ ( top_top @ ( set @ real ) ) ) ) ) ).
% has_real_derivative_powr
thf(fact_5753_termdiffs,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [C2: nat > A,K6: A,X: A] :
( ( summable @ A
@ ^ [N5: nat] : ( times_times @ A @ ( C2 @ N5 ) @ ( power_power @ A @ K6 @ N5 ) ) )
=> ( ( summable @ A
@ ^ [N5: nat] : ( times_times @ A @ ( diffs @ A @ C2 @ N5 ) @ ( power_power @ A @ K6 @ N5 ) ) )
=> ( ( summable @ A
@ ^ [N5: nat] : ( times_times @ A @ ( diffs @ A @ ( diffs @ A @ C2 ) @ N5 ) @ ( power_power @ A @ K6 @ N5 ) ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ ( real_V7770717601297561774m_norm @ A @ K6 ) )
=> ( has_field_derivative @ A
@ ^ [X2: A] :
( suminf @ A
@ ^ [N5: nat] : ( times_times @ A @ ( C2 @ N5 ) @ ( power_power @ A @ X2 @ N5 ) ) )
@ ( suminf @ A
@ ^ [N5: nat] : ( times_times @ A @ ( diffs @ A @ C2 @ N5 ) @ ( power_power @ A @ X @ N5 ) ) )
@ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% termdiffs
thf(fact_5754_termdiffs__strong,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [C2: nat > A,K6: A,X: A] :
( ( summable @ A
@ ^ [N5: nat] : ( times_times @ A @ ( C2 @ N5 ) @ ( power_power @ A @ K6 @ N5 ) ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ ( real_V7770717601297561774m_norm @ A @ K6 ) )
=> ( has_field_derivative @ A
@ ^ [X2: A] :
( suminf @ A
@ ^ [N5: nat] : ( times_times @ A @ ( C2 @ N5 ) @ ( power_power @ A @ X2 @ N5 ) ) )
@ ( suminf @ A
@ ^ [N5: nat] : ( times_times @ A @ ( diffs @ A @ C2 @ N5 ) @ ( power_power @ A @ X @ N5 ) ) )
@ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% termdiffs_strong
thf(fact_5755_termdiffs__strong_H,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [K6: real,C2: nat > A,Z2: A] :
( ! [Z4: A] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ Z4 ) @ K6 )
=> ( summable @ A
@ ^ [N5: nat] : ( times_times @ A @ ( C2 @ N5 ) @ ( power_power @ A @ Z4 @ N5 ) ) ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ Z2 ) @ K6 )
=> ( has_field_derivative @ A
@ ^ [Z6: A] :
( suminf @ A
@ ^ [N5: nat] : ( times_times @ A @ ( C2 @ N5 ) @ ( power_power @ A @ Z6 @ N5 ) ) )
@ ( suminf @ A
@ ^ [N5: nat] : ( times_times @ A @ ( diffs @ A @ C2 @ N5 ) @ ( power_power @ A @ Z2 @ N5 ) ) )
@ ( topolo174197925503356063within @ A @ Z2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% termdiffs_strong'
thf(fact_5756_DERIV__log,axiom,
! [X: real,B2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( has_field_derivative @ real @ ( log2 @ 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_5757_DERIV__fun__powr,axiom,
! [G: real > real,M2: real,X: real,R4: real] :
( ( has_field_derivative @ real @ G @ M2 @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( G @ X ) )
=> ( has_field_derivative @ real
@ ^ [X2: real] : ( powr @ real @ ( G @ X2 ) @ R4 )
@ ( times_times @ real @ ( times_times @ real @ R4 @ ( powr @ real @ ( G @ X ) @ ( minus_minus @ real @ R4 @ ( semiring_1_of_nat @ real @ ( one_one @ nat ) ) ) ) ) @ M2 )
@ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) ) ) ) ).
% DERIV_fun_powr
thf(fact_5758_DERIV__powr,axiom,
! [G: real > real,M2: real,X: real,F2: real > real,R4: real] :
( ( has_field_derivative @ real @ G @ M2 @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( G @ X ) )
=> ( ( has_field_derivative @ real @ F2 @ R4 @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( has_field_derivative @ real
@ ^ [X2: real] : ( powr @ real @ ( G @ X2 ) @ ( F2 @ X2 ) )
@ ( times_times @ real @ ( powr @ real @ ( G @ X ) @ ( F2 @ X ) ) @ ( plus_plus @ real @ ( times_times @ real @ R4 @ ( ln_ln @ real @ ( G @ X ) ) ) @ ( divide_divide @ real @ ( times_times @ real @ M2 @ ( F2 @ X ) ) @ ( G @ X ) ) ) )
@ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ).
% DERIV_powr
thf(fact_5759_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_5760_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_5761_DERIV__series_H,axiom,
! [F2: real > nat > real,F6: real > nat > real,X0: real,A3: real,B2: real,L4: nat > real] :
( ! [N2: nat] :
( has_field_derivative @ real
@ ^ [X2: real] : ( F2 @ X2 @ N2 )
@ ( F6 @ X0 @ N2 )
@ ( topolo174197925503356063within @ real @ X0 @ ( top_top @ ( set @ real ) ) ) )
=> ( ! [X3: real] :
( ( member @ real @ X3 @ ( set_or5935395276787703475ssThan @ real @ A3 @ B2 ) )
=> ( summable @ real @ ( F2 @ X3 ) ) )
=> ( ( member @ real @ X0 @ ( set_or5935395276787703475ssThan @ real @ A3 @ B2 ) )
=> ( ( summable @ real @ ( F6 @ X0 ) )
=> ( ( summable @ real @ L4 )
=> ( ! [N2: nat,X3: real,Y3: real] :
( ( member @ real @ X3 @ ( set_or5935395276787703475ssThan @ real @ A3 @ B2 ) )
=> ( ( member @ real @ Y3 @ ( set_or5935395276787703475ssThan @ real @ A3 @ B2 ) )
=> ( ord_less_eq @ real @ ( abs_abs @ real @ ( minus_minus @ real @ ( F2 @ X3 @ N2 ) @ ( F2 @ Y3 @ N2 ) ) ) @ ( times_times @ real @ ( L4 @ N2 ) @ ( abs_abs @ real @ ( minus_minus @ real @ X3 @ Y3 ) ) ) ) ) )
=> ( has_field_derivative @ real
@ ^ [X2: real] : ( suminf @ real @ ( F2 @ X2 ) )
@ ( suminf @ real @ ( F6 @ X0 ) )
@ ( topolo174197925503356063within @ real @ X0 @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ) ) ) ).
% DERIV_series'
thf(fact_5762_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_5763_arsinh__real__has__field__derivative,axiom,
! [X: real,A4: 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 @ A4 ) ) ).
% arsinh_real_has_field_derivative
thf(fact_5764_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_5765_has__field__derivative__tanh,axiom,
! [A13: $tType] :
( ( ( real_Vector_banach @ A13 )
& ( real_V3459762299906320749_field @ A13 ) )
=> ! [G: A13 > A13,X: A13,Db: A13,S: set @ A13] :
( ( ( cosh @ A13 @ ( G @ X ) )
!= ( zero_zero @ A13 ) )
=> ( ( has_field_derivative @ A13 @ G @ Db @ ( topolo174197925503356063within @ A13 @ X @ S ) )
=> ( has_field_derivative @ A13
@ ^ [X2: A13] : ( tanh @ A13 @ ( G @ X2 ) )
@ ( times_times @ A13 @ ( minus_minus @ A13 @ ( one_one @ A13 ) @ ( power_power @ A13 @ ( tanh @ A13 @ ( G @ X ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ Db )
@ ( topolo174197925503356063within @ A13 @ X @ S ) ) ) ) ) ).
% has_field_derivative_tanh
thf(fact_5766_DERIV__real__sqrt__generic,axiom,
! [X: real,D6: real] :
( ( X
!= ( zero_zero @ real ) )
=> ( ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( D6
= ( divide_divide @ real @ ( inverse_inverse @ real @ ( sqrt @ X ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) )
=> ( ( ( ord_less @ real @ X @ ( zero_zero @ real ) )
=> ( D6
= ( divide_divide @ real @ ( uminus_uminus @ real @ ( inverse_inverse @ real @ ( sqrt @ X ) ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) )
=> ( has_field_derivative @ real @ sqrt @ D6 @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ).
% DERIV_real_sqrt_generic
thf(fact_5767_arcosh__real__has__field__derivative,axiom,
! [X: real,A4: 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 @ A4 ) ) ) ).
% arcosh_real_has_field_derivative
thf(fact_5768_artanh__real__has__field__derivative,axiom,
! [X: real,A4: 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 @ A4 ) ) ) ).
% artanh_real_has_field_derivative
thf(fact_5769_DERIV__power__series_H,axiom,
! [R2: real,F2: nat > real,X0: real] :
( ! [X3: real] :
( ( member @ real @ X3 @ ( set_or5935395276787703475ssThan @ real @ ( uminus_uminus @ real @ R2 ) @ R2 ) )
=> ( summable @ real
@ ^ [N5: nat] : ( times_times @ real @ ( times_times @ real @ ( F2 @ N5 ) @ ( semiring_1_of_nat @ real @ ( suc @ N5 ) ) ) @ ( power_power @ real @ X3 @ N5 ) ) ) )
=> ( ( member @ real @ X0 @ ( set_or5935395276787703475ssThan @ real @ ( uminus_uminus @ real @ R2 ) @ R2 ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ R2 )
=> ( has_field_derivative @ real
@ ^ [X2: real] :
( suminf @ real
@ ^ [N5: nat] : ( times_times @ real @ ( F2 @ N5 ) @ ( power_power @ real @ X2 @ ( suc @ N5 ) ) ) )
@ ( suminf @ real
@ ^ [N5: nat] : ( times_times @ real @ ( times_times @ real @ ( F2 @ N5 ) @ ( semiring_1_of_nat @ real @ ( suc @ N5 ) ) ) @ ( power_power @ real @ X0 @ N5 ) ) )
@ ( topolo174197925503356063within @ real @ X0 @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ).
% DERIV_power_series'
thf(fact_5770_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_5771_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_5772_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_5773_Maclaurin__all__le__objl,axiom,
! [Diff: nat > real > real,F2: real > real,X: real,N: nat] :
( ( ( ( Diff @ ( zero_zero @ nat ) )
= F2 )
& ! [M3: nat,X3: real] : ( has_field_derivative @ real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ X3 ) @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) )
=> ? [T4: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ T4 ) @ ( abs_abs @ real @ X ) )
& ( ( F2 @ X )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M5: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M5 @ ( zero_zero @ real ) ) @ ( semiring_char_0_fact @ real @ M5 ) ) @ ( power_power @ real @ X @ M5 ) )
@ ( 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_5774_Maclaurin__all__le,axiom,
! [Diff: nat > real > real,F2: real > real,X: real,N: nat] :
( ( ( Diff @ ( zero_zero @ nat ) )
= F2 )
=> ( ! [M3: nat,X3: real] : ( has_field_derivative @ real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ X3 ) @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) )
=> ? [T4: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ T4 ) @ ( abs_abs @ real @ X ) )
& ( ( F2 @ X )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M5: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M5 @ ( zero_zero @ real ) ) @ ( semiring_char_0_fact @ real @ M5 ) ) @ ( power_power @ real @ X @ M5 ) )
@ ( 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_5775_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_5776_Maclaurin__minus,axiom,
! [H: real,N: nat,Diff: nat > real > real,F2: real > real] :
( ( ord_less @ real @ H @ ( zero_zero @ real ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ( Diff @ ( zero_zero @ nat ) )
= F2 )
=> ( ! [M3: nat,T4: real] :
( ( ( ord_less @ nat @ M3 @ N )
& ( ord_less_eq @ real @ H @ T4 )
& ( ord_less_eq @ real @ T4 @ ( zero_zero @ real ) ) )
=> ( has_field_derivative @ real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ T4 ) @ ( topolo174197925503356063within @ real @ T4 @ ( top_top @ ( set @ real ) ) ) ) )
=> ? [T4: real] :
( ( ord_less @ real @ H @ T4 )
& ( ord_less @ real @ T4 @ ( zero_zero @ real ) )
& ( ( F2 @ H )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M5: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M5 @ ( zero_zero @ real ) ) @ ( semiring_char_0_fact @ real @ M5 ) ) @ ( power_power @ real @ H @ M5 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N @ T4 ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ H @ N ) ) ) ) ) ) ) ) ) ).
% Maclaurin_minus
thf(fact_5777_Maclaurin2,axiom,
! [H: real,Diff: nat > real > real,F2: real > real,N: nat] :
( ( ord_less @ real @ ( zero_zero @ real ) @ H )
=> ( ( ( Diff @ ( zero_zero @ nat ) )
= F2 )
=> ( ! [M3: nat,T4: real] :
( ( ( ord_less @ nat @ M3 @ N )
& ( ord_less_eq @ real @ ( zero_zero @ real ) @ T4 )
& ( ord_less_eq @ real @ T4 @ H ) )
=> ( has_field_derivative @ real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ T4 ) @ ( topolo174197925503356063within @ real @ T4 @ ( top_top @ ( set @ real ) ) ) ) )
=> ? [T4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ T4 )
& ( ord_less_eq @ real @ T4 @ H )
& ( ( F2 @ H )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M5: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M5 @ ( zero_zero @ real ) ) @ ( semiring_char_0_fact @ real @ M5 ) ) @ ( power_power @ real @ H @ M5 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N @ T4 ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ H @ N ) ) ) ) ) ) ) ) ).
% Maclaurin2
thf(fact_5778_Maclaurin,axiom,
! [H: real,N: nat,Diff: nat > real > real,F2: real > real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ H )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ( Diff @ ( zero_zero @ nat ) )
= F2 )
=> ( ! [M3: nat,T4: real] :
( ( ( ord_less @ nat @ M3 @ N )
& ( ord_less_eq @ real @ ( zero_zero @ real ) @ T4 )
& ( ord_less_eq @ real @ T4 @ H ) )
=> ( has_field_derivative @ real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ T4 ) @ ( topolo174197925503356063within @ real @ T4 @ ( top_top @ ( set @ real ) ) ) ) )
=> ? [T4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ T4 )
& ( ord_less @ real @ T4 @ H )
& ( ( F2 @ H )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M5: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M5 @ ( zero_zero @ real ) ) @ ( semiring_char_0_fact @ real @ M5 ) ) @ ( power_power @ real @ H @ M5 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N @ T4 ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ H @ N ) ) ) ) ) ) ) ) ) ).
% Maclaurin
thf(fact_5779_Maclaurin__all__lt,axiom,
! [Diff: nat > real > real,F2: real > real,N: nat,X: real] :
( ( ( Diff @ ( zero_zero @ nat ) )
= F2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( X
!= ( zero_zero @ real ) )
=> ( ! [M3: nat,X3: real] : ( has_field_derivative @ real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ X3 ) @ ( topolo174197925503356063within @ real @ X3 @ ( 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 ) )
& ( ( F2 @ X )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M5: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M5 @ ( zero_zero @ real ) ) @ ( semiring_char_0_fact @ real @ M5 ) ) @ ( power_power @ real @ X @ M5 ) )
@ ( 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_5780_Maclaurin__bi__le,axiom,
! [Diff: nat > real > real,F2: real > real,N: nat,X: real] :
( ( ( Diff @ ( zero_zero @ nat ) )
= F2 )
=> ( ! [M3: nat,T4: real] :
( ( ( ord_less @ nat @ M3 @ N )
& ( ord_less_eq @ real @ ( abs_abs @ real @ T4 ) @ ( abs_abs @ real @ X ) ) )
=> ( has_field_derivative @ real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ T4 ) @ ( topolo174197925503356063within @ real @ T4 @ ( top_top @ ( set @ real ) ) ) ) )
=> ? [T4: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ T4 ) @ ( abs_abs @ real @ X ) )
& ( ( F2 @ X )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M5: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M5 @ ( zero_zero @ real ) ) @ ( semiring_char_0_fact @ real @ M5 ) ) @ ( power_power @ real @ X @ M5 ) )
@ ( 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_5781_Taylor,axiom,
! [N: nat,Diff: nat > real > real,F2: real > real,A3: real,B2: real,C2: real,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ( Diff @ ( zero_zero @ nat ) )
= F2 )
=> ( ! [M3: nat,T4: real] :
( ( ( ord_less @ nat @ M3 @ N )
& ( ord_less_eq @ real @ A3 @ T4 )
& ( ord_less_eq @ real @ T4 @ B2 ) )
=> ( has_field_derivative @ real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ T4 ) @ ( topolo174197925503356063within @ real @ T4 @ ( top_top @ ( set @ real ) ) ) ) )
=> ( ( ord_less_eq @ real @ A3 @ C2 )
=> ( ( ord_less_eq @ real @ C2 @ B2 )
=> ( ( ord_less_eq @ real @ A3 @ 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 ) ) )
& ( ( F2 @ X )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M5: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M5 @ C2 ) @ ( semiring_char_0_fact @ real @ M5 ) ) @ ( power_power @ real @ ( minus_minus @ real @ X @ C2 ) @ M5 ) )
@ ( 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_5782_Taylor__up,axiom,
! [N: nat,Diff: nat > real > real,F2: real > real,A3: real,B2: real,C2: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ( Diff @ ( zero_zero @ nat ) )
= F2 )
=> ( ! [M3: nat,T4: real] :
( ( ( ord_less @ nat @ M3 @ N )
& ( ord_less_eq @ real @ A3 @ T4 )
& ( ord_less_eq @ real @ T4 @ B2 ) )
=> ( has_field_derivative @ real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ T4 ) @ ( topolo174197925503356063within @ real @ T4 @ ( top_top @ ( set @ real ) ) ) ) )
=> ( ( ord_less_eq @ real @ A3 @ C2 )
=> ( ( ord_less @ real @ C2 @ B2 )
=> ? [T4: real] :
( ( ord_less @ real @ C2 @ T4 )
& ( ord_less @ real @ T4 @ B2 )
& ( ( F2 @ B2 )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M5: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M5 @ C2 ) @ ( semiring_char_0_fact @ real @ M5 ) ) @ ( power_power @ real @ ( minus_minus @ real @ B2 @ C2 ) @ M5 ) )
@ ( 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_5783_Taylor__down,axiom,
! [N: nat,Diff: nat > real > real,F2: real > real,A3: real,B2: real,C2: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ( Diff @ ( zero_zero @ nat ) )
= F2 )
=> ( ! [M3: nat,T4: real] :
( ( ( ord_less @ nat @ M3 @ N )
& ( ord_less_eq @ real @ A3 @ T4 )
& ( ord_less_eq @ real @ T4 @ B2 ) )
=> ( has_field_derivative @ real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ T4 ) @ ( topolo174197925503356063within @ real @ T4 @ ( top_top @ ( set @ real ) ) ) ) )
=> ( ( ord_less @ real @ A3 @ C2 )
=> ( ( ord_less_eq @ real @ C2 @ B2 )
=> ? [T4: real] :
( ( ord_less @ real @ A3 @ T4 )
& ( ord_less @ real @ T4 @ C2 )
& ( ( F2 @ A3 )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M5: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M5 @ C2 ) @ ( semiring_char_0_fact @ real @ M5 ) ) @ ( power_power @ real @ ( minus_minus @ real @ A3 @ C2 ) @ M5 ) )
@ ( 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 @ A3 @ C2 ) @ N ) ) ) ) ) ) ) ) ) ) ).
% Taylor_down
thf(fact_5784_Maclaurin__lemma2,axiom,
! [N: nat,H: real,Diff: nat > real > real,K: nat,B7: real] :
( ! [M3: nat,T4: real] :
( ( ( ord_less @ nat @ M3 @ N )
& ( ord_less_eq @ real @ ( zero_zero @ real ) @ T4 )
& ( ord_less_eq @ real @ T4 @ H ) )
=> ( has_field_derivative @ real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ T4 ) @ ( topolo174197925503356063within @ real @ T4 @ ( top_top @ ( set @ real ) ) ) ) )
=> ( ( N
= ( suc @ K ) )
=> ! [M: nat,T8: real] :
( ( ( ord_less @ nat @ M @ N )
& ( ord_less_eq @ real @ ( zero_zero @ real ) @ T8 )
& ( ord_less_eq @ real @ T8 @ H ) )
=> ( has_field_derivative @ real
@ ^ [U2: real] :
( minus_minus @ real @ ( Diff @ M @ U2 )
@ ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [P6: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ ( plus_plus @ nat @ M @ P6 ) @ ( zero_zero @ real ) ) @ ( semiring_char_0_fact @ real @ P6 ) ) @ ( power_power @ real @ U2 @ P6 ) )
@ ( set_ord_lessThan @ nat @ ( minus_minus @ nat @ N @ M ) ) )
@ ( times_times @ real @ B7 @ ( divide_divide @ real @ ( power_power @ real @ U2 @ ( minus_minus @ nat @ N @ M ) ) @ ( semiring_char_0_fact @ real @ ( minus_minus @ nat @ N @ M ) ) ) ) ) )
@ ( minus_minus @ real @ ( Diff @ ( suc @ M ) @ T8 )
@ ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [P6: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ ( plus_plus @ nat @ ( suc @ M ) @ P6 ) @ ( zero_zero @ real ) ) @ ( semiring_char_0_fact @ real @ P6 ) ) @ ( power_power @ real @ T8 @ P6 ) )
@ ( set_ord_lessThan @ nat @ ( minus_minus @ nat @ N @ ( suc @ M ) ) ) )
@ ( times_times @ real @ B7 @ ( divide_divide @ real @ ( power_power @ real @ T8 @ ( minus_minus @ nat @ N @ ( suc @ M ) ) ) @ ( semiring_char_0_fact @ real @ ( minus_minus @ nat @ N @ ( suc @ M ) ) ) ) ) ) )
@ ( topolo174197925503356063within @ real @ T8 @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ).
% Maclaurin_lemma2
thf(fact_5785_has__derivative__arcsin,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [G: A > real,X: A,G3: 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 @ G3 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( has_derivative @ A @ real
@ ^ [X2: A] : ( arcsin @ ( G @ X2 ) )
@ ^ [X2: A] : ( times_times @ real @ ( G3 @ X2 ) @ ( 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_5786_has__derivative__arccos,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [G: A > real,X: A,G3: 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 @ G3 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( has_derivative @ A @ real
@ ^ [X2: A] : ( arccos @ ( G @ X2 ) )
@ ^ [X2: A] : ( times_times @ real @ ( G3 @ X2 ) @ ( 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_5787_has__derivative__tan,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [G: A > real,X: A,G3: A > real,S: set @ A] :
( ( ( cos @ real @ ( G @ X ) )
!= ( zero_zero @ real ) )
=> ( ( has_derivative @ A @ real @ G @ G3 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( has_derivative @ A @ real
@ ^ [X2: A] : ( tan @ real @ ( G @ X2 ) )
@ ^ [X2: A] : ( times_times @ real @ ( G3 @ X2 ) @ ( inverse_inverse @ real @ ( power_power @ real @ ( cos @ real @ ( G @ X ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ).
% has_derivative_tan
thf(fact_5788_has__derivative__diff,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,F6: A > B,F5: filter @ A,G: A > B,G3: A > B] :
( ( has_derivative @ A @ B @ F2 @ F6 @ F5 )
=> ( ( has_derivative @ A @ B @ G @ G3 @ F5 )
=> ( has_derivative @ A @ B
@ ^ [X2: A] : ( minus_minus @ B @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ^ [X2: A] : ( minus_minus @ B @ ( F6 @ X2 ) @ ( G3 @ X2 ) )
@ F5 ) ) ) ) ).
% has_derivative_diff
thf(fact_5789_has__derivative__const,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [C2: B,F5: filter @ A] :
( has_derivative @ A @ B
@ ^ [X2: A] : C2
@ ^ [X2: A] : ( zero_zero @ B )
@ F5 ) ) ).
% has_derivative_const
thf(fact_5790_has__derivative__minus,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,F6: A > B,F5: filter @ A] :
( ( has_derivative @ A @ B @ F2 @ F6 @ F5 )
=> ( has_derivative @ A @ B
@ ^ [X2: A] : ( uminus_uminus @ B @ ( F2 @ X2 ) )
@ ^ [X2: A] : ( uminus_uminus @ B @ ( F6 @ X2 ) )
@ F5 ) ) ) ).
% has_derivative_minus
thf(fact_5791_has__derivative__zero__unique,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F5: A > B,X: A] :
( ( has_derivative @ A @ B
@ ^ [X2: A] : ( zero_zero @ B )
@ F5
@ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) )
=> ( F5
= ( ^ [H2: A] : ( zero_zero @ B ) ) ) ) ) ).
% has_derivative_zero_unique
thf(fact_5792_has__derivative__exp,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [G: A > real,G3: A > real,X: A,S: set @ A] :
( ( has_derivative @ A @ real @ G @ G3 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( has_derivative @ A @ real
@ ^ [X2: A] : ( exp @ real @ ( G @ X2 ) )
@ ^ [X2: A] : ( times_times @ real @ ( G3 @ X2 ) @ ( exp @ real @ ( G @ X ) ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ).
% has_derivative_exp
thf(fact_5793_has__derivative__sin,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [G: A > real,G3: A > real,X: A,S: set @ A] :
( ( has_derivative @ A @ real @ G @ G3 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( has_derivative @ A @ real
@ ^ [X2: A] : ( sin @ real @ ( G @ X2 ) )
@ ^ [X2: A] : ( times_times @ real @ ( G3 @ X2 ) @ ( cos @ real @ ( G @ X ) ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ).
% has_derivative_sin
thf(fact_5794_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
@ ^ [X2: A] : ( cosh @ A @ ( G @ X2 ) )
@ ( times_times @ A @ ( times_times @ A @ ( sinh @ A @ ( G @ X ) ) @ Db ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ).
% has_derivative_cosh
thf(fact_5795_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
@ ^ [X2: A] : ( sinh @ A @ ( G @ X2 ) )
@ ( times_times @ A @ ( times_times @ A @ ( cosh @ A @ ( G @ X ) ) @ Db ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ).
% has_derivative_sinh
thf(fact_5796_has__derivative__divide_H,axiom,
! [A: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V3459762299906320749_field @ A ) )
=> ! [F2: C > A,F6: C > A,X: C,S3: set @ C,G: C > A,G3: C > A] :
( ( has_derivative @ C @ A @ F2 @ F6 @ ( topolo174197925503356063within @ C @ X @ S3 ) )
=> ( ( has_derivative @ C @ A @ G @ G3 @ ( topolo174197925503356063within @ C @ X @ S3 ) )
=> ( ( ( G @ X )
!= ( zero_zero @ A ) )
=> ( has_derivative @ C @ A
@ ^ [X2: C] : ( divide_divide @ A @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ^ [H2: C] : ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ ( F6 @ H2 ) @ ( G @ X ) ) @ ( times_times @ A @ ( F2 @ X ) @ ( G3 @ H2 ) ) ) @ ( times_times @ A @ ( G @ X ) @ ( G @ X ) ) )
@ ( topolo174197925503356063within @ C @ X @ S3 ) ) ) ) ) ) ).
% has_derivative_divide'
thf(fact_5797_has__derivative__inverse,axiom,
! [A: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V8999393235501362500lgebra @ A ) )
=> ! [F2: C > A,X: C,F6: C > A,S3: set @ C] :
( ( ( F2 @ X )
!= ( zero_zero @ A ) )
=> ( ( has_derivative @ C @ A @ F2 @ F6 @ ( topolo174197925503356063within @ C @ X @ S3 ) )
=> ( has_derivative @ C @ A
@ ^ [X2: C] : ( inverse_inverse @ A @ ( F2 @ X2 ) )
@ ^ [H2: C] : ( uminus_uminus @ A @ ( times_times @ A @ ( times_times @ A @ ( inverse_inverse @ A @ ( F2 @ X ) ) @ ( F6 @ H2 ) ) @ ( inverse_inverse @ A @ ( F2 @ X ) ) ) )
@ ( topolo174197925503356063within @ C @ X @ S3 ) ) ) ) ) ).
% has_derivative_inverse
thf(fact_5798_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 )
@ ^ [H2: A] : ( uminus_uminus @ A @ ( times_times @ A @ ( times_times @ A @ ( inverse_inverse @ A @ X ) @ H2 ) @ ( inverse_inverse @ A @ X ) ) )
@ ( topolo174197925503356063within @ A @ X @ S3 ) ) ) ) ).
% has_derivative_inverse'
thf(fact_5799_has__derivative__cos,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [G: A > real,G3: A > real,X: A,S: set @ A] :
( ( has_derivative @ A @ real @ G @ G3 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( has_derivative @ A @ real
@ ^ [X2: A] : ( cos @ real @ ( G @ X2 ) )
@ ^ [X2: A] : ( times_times @ real @ ( G3 @ X2 ) @ ( uminus_uminus @ real @ ( sin @ real @ ( G @ X ) ) ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ).
% has_derivative_cos
thf(fact_5800_has__derivative__power,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V3459762299906320749_field @ B ) )
=> ! [F2: A > B,F6: A > B,X: A,S3: set @ A,N: nat] :
( ( has_derivative @ A @ B @ F2 @ F6 @ ( topolo174197925503356063within @ A @ X @ S3 ) )
=> ( has_derivative @ A @ B
@ ^ [X2: A] : ( power_power @ B @ ( F2 @ X2 ) @ N )
@ ^ [Y5: A] : ( times_times @ B @ ( times_times @ B @ ( semiring_1_of_nat @ B @ N ) @ ( F6 @ Y5 ) ) @ ( power_power @ B @ ( F2 @ X ) @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) )
@ ( topolo174197925503356063within @ A @ X @ S3 ) ) ) ) ).
% has_derivative_power
thf(fact_5801_has__derivative__ln,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [G: A > real,X: A,G3: A > real,S: set @ A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( G @ X ) )
=> ( ( has_derivative @ A @ real @ G @ G3 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( has_derivative @ A @ real
@ ^ [X2: A] : ( ln_ln @ real @ ( G @ X2 ) )
@ ^ [X2: A] : ( times_times @ real @ ( G3 @ X2 ) @ ( inverse_inverse @ real @ ( G @ X ) ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ).
% has_derivative_ln
thf(fact_5802_has__derivative__divide,axiom,
! [A: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V8999393235501362500lgebra @ A ) )
=> ! [F2: C > A,F6: C > A,X: C,S3: set @ C,G: C > A,G3: C > A] :
( ( has_derivative @ C @ A @ F2 @ F6 @ ( topolo174197925503356063within @ C @ X @ S3 ) )
=> ( ( has_derivative @ C @ A @ G @ G3 @ ( topolo174197925503356063within @ C @ X @ S3 ) )
=> ( ( ( G @ X )
!= ( zero_zero @ A ) )
=> ( has_derivative @ C @ A
@ ^ [X2: C] : ( divide_divide @ A @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ^ [H2: C] : ( plus_plus @ A @ ( times_times @ A @ ( uminus_uminus @ A @ ( F2 @ X ) ) @ ( times_times @ A @ ( times_times @ A @ ( inverse_inverse @ A @ ( G @ X ) ) @ ( G3 @ H2 ) ) @ ( inverse_inverse @ A @ ( G @ X ) ) ) ) @ ( divide_divide @ A @ ( F6 @ H2 ) @ ( G @ X ) ) )
@ ( topolo174197925503356063within @ C @ X @ S3 ) ) ) ) ) ) ).
% has_derivative_divide
thf(fact_5803_has__derivative__prod,axiom,
! [B: $tType,I6: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V3459762299906320749_field @ B ) )
=> ! [I5: set @ I6,F2: I6 > A > B,F6: I6 > A > B,X: A,S3: set @ A] :
( ! [I3: I6] :
( ( member @ I6 @ I3 @ I5 )
=> ( has_derivative @ A @ B @ ( F2 @ I3 ) @ ( F6 @ I3 ) @ ( topolo174197925503356063within @ A @ X @ S3 ) ) )
=> ( has_derivative @ A @ B
@ ^ [X2: A] :
( groups7121269368397514597t_prod @ I6 @ B
@ ^ [I2: I6] : ( F2 @ I2 @ X2 )
@ I5 )
@ ^ [Y5: A] :
( groups7311177749621191930dd_sum @ I6 @ B
@ ^ [I2: I6] :
( times_times @ B @ ( F6 @ I2 @ Y5 )
@ ( groups7121269368397514597t_prod @ I6 @ B
@ ^ [J3: I6] : ( F2 @ J3 @ X )
@ ( minus_minus @ ( set @ I6 ) @ I5 @ ( insert @ I6 @ I2 @ ( bot_bot @ ( set @ I6 ) ) ) ) ) )
@ I5 )
@ ( topolo174197925503356063within @ A @ X @ S3 ) ) ) ) ).
% has_derivative_prod
thf(fact_5804_has__derivative__powr,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [G: A > real,G3: A > real,X: A,X8: set @ A,F2: A > real,F6: A > real] :
( ( has_derivative @ A @ real @ G @ G3 @ ( topolo174197925503356063within @ A @ X @ X8 ) )
=> ( ( has_derivative @ A @ real @ F2 @ F6 @ ( topolo174197925503356063within @ A @ X @ X8 ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( G @ X ) )
=> ( ( member @ A @ X @ X8 )
=> ( has_derivative @ A @ real
@ ^ [X2: A] : ( powr @ real @ ( G @ X2 ) @ ( F2 @ X2 ) )
@ ^ [H2: A] : ( times_times @ real @ ( powr @ real @ ( G @ X ) @ ( F2 @ X ) ) @ ( plus_plus @ real @ ( times_times @ real @ ( F6 @ H2 ) @ ( ln_ln @ real @ ( G @ X ) ) ) @ ( divide_divide @ real @ ( times_times @ real @ ( G3 @ H2 ) @ ( F2 @ X ) ) @ ( G @ X ) ) ) )
@ ( topolo174197925503356063within @ A @ X @ X8 ) ) ) ) ) ) ) ).
% has_derivative_powr
thf(fact_5805_has__derivative__real__sqrt,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [G: A > real,X: A,G3: A > real,S: set @ A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( G @ X ) )
=> ( ( has_derivative @ A @ real @ G @ G3 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( has_derivative @ A @ real
@ ^ [X2: A] : ( sqrt @ ( G @ X2 ) )
@ ^ [X2: A] : ( times_times @ real @ ( G3 @ X2 ) @ ( divide_divide @ real @ ( inverse_inverse @ real @ ( sqrt @ ( G @ X ) ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ).
% has_derivative_real_sqrt
thf(fact_5806_has__derivative__arctan,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [G: A > real,G3: A > real,X: A,S: set @ A] :
( ( has_derivative @ A @ real @ G @ G3 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( has_derivative @ A @ real
@ ^ [X2: A] : ( arctan @ ( G @ X2 ) )
@ ^ [X2: A] : ( times_times @ real @ ( G3 @ X2 ) @ ( 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_5807_termdiffs__aux,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [C2: nat > A,K6: A,X: A] :
( ( summable @ A
@ ^ [N5: nat] : ( times_times @ A @ ( diffs @ A @ ( diffs @ A @ C2 ) @ N5 ) @ ( power_power @ A @ K6 @ N5 ) ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ ( real_V7770717601297561774m_norm @ A @ K6 ) )
=> ( filterlim @ A @ A
@ ^ [H2: A] :
( suminf @ A
@ ^ [N5: nat] : ( times_times @ A @ ( C2 @ N5 ) @ ( minus_minus @ A @ ( divide_divide @ A @ ( minus_minus @ A @ ( power_power @ A @ ( plus_plus @ A @ X @ H2 ) @ N5 ) @ ( power_power @ A @ X @ N5 ) ) @ H2 ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N5 ) @ ( power_power @ A @ X @ ( minus_minus @ nat @ N5 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% termdiffs_aux
thf(fact_5808_surj__int__encode,axiom,
( ( image @ int @ nat @ nat_int_encode @ ( top_top @ ( set @ int ) ) )
= ( top_top @ ( set @ nat ) ) ) ).
% surj_int_encode
thf(fact_5809_isCont__powser,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [C2: nat > A,K6: A,X: A] :
( ( summable @ A
@ ^ [N5: nat] : ( times_times @ A @ ( C2 @ N5 ) @ ( power_power @ A @ K6 @ N5 ) ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ ( real_V7770717601297561774m_norm @ A @ K6 ) )
=> ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) )
@ ^ [X2: A] :
( suminf @ A
@ ^ [N5: nat] : ( times_times @ A @ ( C2 @ N5 ) @ ( power_power @ A @ X2 @ N5 ) ) ) ) ) ) ) ).
% isCont_powser
thf(fact_5810_tendsto__mult__right__iff,axiom,
! [A: $tType,B: $tType] :
( ( ( field @ A )
& ( topolo4211221413907600880p_mult @ A ) )
=> ! [C2: A,F2: B > A,L: A,F5: filter @ B] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( filterlim @ B @ A
@ ^ [X2: B] : ( times_times @ A @ ( F2 @ X2 ) @ C2 )
@ ( topolo7230453075368039082e_nhds @ A @ ( times_times @ A @ L @ C2 ) )
@ F5 )
= ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F5 ) ) ) ) ).
% tendsto_mult_right_iff
thf(fact_5811_tendsto__mult__left__iff,axiom,
! [A: $tType,B: $tType] :
( ( ( field @ A )
& ( topolo4211221413907600880p_mult @ A ) )
=> ! [C2: A,F2: B > A,L: A,F5: filter @ B] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( filterlim @ B @ A
@ ^ [X2: B] : ( times_times @ A @ C2 @ ( F2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( times_times @ A @ C2 @ L ) )
@ F5 )
= ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F5 ) ) ) ) ).
% tendsto_mult_left_iff
thf(fact_5812_power__tendsto__0__iff,axiom,
! [A: $tType,N: nat,F2: A > real,F5: filter @ A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( filterlim @ A @ real
@ ^ [X2: A] : ( power_power @ real @ ( F2 @ X2 ) @ N )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ F5 )
= ( filterlim @ A @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ F5 ) ) ) ).
% power_tendsto_0_iff
thf(fact_5813_isCont__LIM__compose2,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( topolo4958980785337419405_space @ B )
& ( topolo4958980785337419405_space @ C ) )
=> ! [A3: A,F2: A > B,G: B > C,L: C] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) @ F2 )
=> ( ( filterlim @ B @ C @ G @ ( topolo7230453075368039082e_nhds @ C @ L ) @ ( topolo174197925503356063within @ B @ ( F2 @ A3 ) @ ( top_top @ ( set @ B ) ) ) )
=> ( ? [D3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D3 )
& ! [X3: A] :
( ( ( X3 != A3 )
& ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ X3 @ A3 ) ) @ D3 ) )
=> ( ( F2 @ X3 )
!= ( F2 @ A3 ) ) ) )
=> ( filterlim @ A @ C
@ ^ [X2: A] : ( G @ ( F2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ C @ L )
@ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ).
% isCont_LIM_compose2
thf(fact_5814_LIM__offset,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [F2: A > B,L4: B,A3: A,K: A] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L4 ) @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) )
=> ( filterlim @ A @ B
@ ^ [X2: A] : ( F2 @ ( plus_plus @ A @ X2 @ K ) )
@ ( topolo7230453075368039082e_nhds @ B @ L4 )
@ ( topolo174197925503356063within @ A @ ( minus_minus @ A @ A3 @ K ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% LIM_offset
thf(fact_5815_LIM__not__zero,axiom,
! [Aa: $tType,A: $tType] :
( ( ( topolo8386298272705272623_space @ A )
& ( zero @ Aa )
& ( topological_t2_space @ Aa ) )
=> ! [K: Aa,A3: A] :
( ( K
!= ( zero_zero @ Aa ) )
=> ~ ( filterlim @ A @ Aa
@ ^ [X2: A] : K
@ ( topolo7230453075368039082e_nhds @ Aa @ ( zero_zero @ Aa ) )
@ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% LIM_not_zero
thf(fact_5816_isCont__iff,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [X: A,F2: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) @ F2 )
= ( filterlim @ A @ B
@ ^ [H2: A] : ( F2 @ ( plus_plus @ A @ X @ H2 ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( F2 @ X ) )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% isCont_iff
thf(fact_5817_LIM__isCont__iff,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [F2: A > B,A3: A] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ ( F2 @ A3 ) ) @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) )
= ( filterlim @ A @ B
@ ^ [H2: A] : ( F2 @ ( plus_plus @ A @ A3 @ H2 ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( F2 @ A3 ) )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% LIM_isCont_iff
thf(fact_5818_LIM__offset__zero,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [F2: A > B,L4: B,A3: A] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L4 ) @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) )
=> ( filterlim @ A @ B
@ ^ [H2: A] : ( F2 @ ( plus_plus @ A @ A3 @ H2 ) )
@ ( topolo7230453075368039082e_nhds @ B @ L4 )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% LIM_offset_zero
thf(fact_5819_LIM__offset__zero__cancel,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [F2: A > B,A3: A,L4: B] :
( ( filterlim @ A @ B
@ ^ [H2: A] : ( F2 @ ( plus_plus @ A @ A3 @ H2 ) )
@ ( topolo7230453075368039082e_nhds @ B @ L4 )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) )
=> ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L4 ) @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% LIM_offset_zero_cancel
thf(fact_5820_continuous__within__cos,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [Z2: A,S: set @ A] : ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ Z2 @ S ) @ ( cos @ A ) ) ) ).
% continuous_within_cos
thf(fact_5821_continuous__within__sin,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [Z2: A,S: set @ A] : ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ Z2 @ S ) @ ( sin @ A ) ) ) ).
% continuous_within_sin
thf(fact_5822_has__field__derivative__iff,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D6: A,X: A,S3: set @ A] :
( ( has_field_derivative @ A @ F2 @ D6 @ ( topolo174197925503356063within @ A @ X @ S3 ) )
= ( filterlim @ A @ A
@ ^ [Y5: A] : ( divide_divide @ A @ ( minus_minus @ A @ ( F2 @ Y5 ) @ ( F2 @ X ) ) @ ( minus_minus @ A @ Y5 @ X ) )
@ ( topolo7230453075368039082e_nhds @ A @ D6 )
@ ( topolo174197925503356063within @ A @ X @ S3 ) ) ) ) ).
% has_field_derivative_iff
thf(fact_5823_has__field__derivativeD,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D6: A,X: A,S3: set @ A] :
( ( has_field_derivative @ A @ F2 @ D6 @ ( topolo174197925503356063within @ A @ X @ S3 ) )
=> ( filterlim @ A @ A
@ ^ [Y5: A] : ( divide_divide @ A @ ( minus_minus @ A @ ( F2 @ Y5 ) @ ( F2 @ X ) ) @ ( minus_minus @ A @ Y5 @ X ) )
@ ( topolo7230453075368039082e_nhds @ A @ D6 )
@ ( topolo174197925503356063within @ A @ X @ S3 ) ) ) ) ).
% has_field_derivativeD
thf(fact_5824_tendsto__tan,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [F2: A > A,A3: A,F5: filter @ A] :
( ( filterlim @ A @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ A3 ) @ F5 )
=> ( ( ( cos @ A @ A3 )
!= ( zero_zero @ A ) )
=> ( filterlim @ A @ A
@ ^ [X2: A] : ( tan @ A @ ( F2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( tan @ A @ A3 ) )
@ F5 ) ) ) ) ).
% tendsto_tan
thf(fact_5825_tendsto__add__zero,axiom,
! [B: $tType,D: $tType] :
( ( topolo6943815403480290642id_add @ B )
=> ! [F2: D > B,F5: filter @ D,G: D > B] :
( ( filterlim @ D @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) @ F5 )
=> ( ( filterlim @ D @ B @ G @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) @ F5 )
=> ( filterlim @ D @ B
@ ^ [X2: D] : ( plus_plus @ B @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ F5 ) ) ) ) ).
% tendsto_add_zero
thf(fact_5826_tendsto__minus__cancel__left,axiom,
! [B: $tType,A: $tType] :
( ( topolo1633459387980952147up_add @ B )
=> ! [F2: A > B,Y2: B,F5: filter @ A] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ ( uminus_uminus @ B @ Y2 ) ) @ F5 )
= ( filterlim @ A @ B
@ ^ [X2: A] : ( uminus_uminus @ B @ ( F2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ B @ Y2 )
@ F5 ) ) ) ).
% tendsto_minus_cancel_left
thf(fact_5827_tendsto__minus__cancel,axiom,
! [A: $tType,B: $tType] :
( ( topolo1633459387980952147up_add @ A )
=> ! [F2: B > A,A3: A,F5: filter @ B] :
( ( filterlim @ B @ A
@ ^ [X2: B] : ( uminus_uminus @ A @ ( F2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( uminus_uminus @ A @ A3 ) )
@ F5 )
=> ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ A3 ) @ F5 ) ) ) ).
% tendsto_minus_cancel
thf(fact_5828_tendsto__minus,axiom,
! [A: $tType,B: $tType] :
( ( topolo1633459387980952147up_add @ A )
=> ! [F2: B > A,A3: A,F5: filter @ B] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ A3 ) @ F5 )
=> ( filterlim @ B @ A
@ ^ [X2: B] : ( uminus_uminus @ A @ ( F2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( uminus_uminus @ A @ A3 ) )
@ F5 ) ) ) ).
% tendsto_minus
thf(fact_5829_tendsto__inverse,axiom,
! [A: $tType,B: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [F2: B > A,A3: A,F5: filter @ B] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ A3 ) @ F5 )
=> ( ( A3
!= ( zero_zero @ A ) )
=> ( filterlim @ B @ A
@ ^ [X2: B] : ( inverse_inverse @ A @ ( F2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( inverse_inverse @ A @ A3 ) )
@ F5 ) ) ) ) ).
% tendsto_inverse
thf(fact_5830_tendsto__norm__zero__cancel,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F2: A > B,F5: filter @ A] :
( ( filterlim @ A @ real
@ ^ [X2: A] : ( real_V7770717601297561774m_norm @ B @ ( F2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ F5 )
=> ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) @ F5 ) ) ) ).
% tendsto_norm_zero_cancel
thf(fact_5831_tendsto__norm__zero__iff,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F2: A > B,F5: filter @ A] :
( ( filterlim @ A @ real
@ ^ [X2: A] : ( real_V7770717601297561774m_norm @ B @ ( F2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ F5 )
= ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) @ F5 ) ) ) ).
% tendsto_norm_zero_iff
thf(fact_5832_tendsto__norm__zero,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F2: A > B,F5: filter @ A] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) @ F5 )
=> ( filterlim @ A @ real
@ ^ [X2: A] : ( real_V7770717601297561774m_norm @ B @ ( F2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ F5 ) ) ) ).
% tendsto_norm_zero
thf(fact_5833_tendsto__ln,axiom,
! [A: $tType,F2: A > real,A3: real,F5: filter @ A] :
( ( filterlim @ A @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ A3 ) @ F5 )
=> ( ( A3
!= ( zero_zero @ real ) )
=> ( filterlim @ A @ real
@ ^ [X2: A] : ( ln_ln @ real @ ( F2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( ln_ln @ real @ A3 ) )
@ F5 ) ) ) ).
% tendsto_ln
thf(fact_5834_tendsto__powr,axiom,
! [A: $tType,F2: A > real,A3: real,F5: filter @ A,G: A > real,B2: real] :
( ( filterlim @ A @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ A3 ) @ F5 )
=> ( ( filterlim @ A @ real @ G @ ( topolo7230453075368039082e_nhds @ real @ B2 ) @ F5 )
=> ( ( A3
!= ( zero_zero @ real ) )
=> ( filterlim @ A @ real
@ ^ [X2: A] : ( powr @ real @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( powr @ real @ A3 @ B2 ) )
@ F5 ) ) ) ) ).
% tendsto_powr
thf(fact_5835_tendsto__sgn,axiom,
! [A: $tType,B: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: B > A,L: A,F5: filter @ B] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F5 )
=> ( ( L
!= ( zero_zero @ A ) )
=> ( filterlim @ B @ A
@ ^ [X2: B] : ( sgn_sgn @ A @ ( F2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( sgn_sgn @ A @ L ) )
@ F5 ) ) ) ) ).
% tendsto_sgn
thf(fact_5836_continuous__minus,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( topolo1633459387980952147up_add @ B ) )
=> ! [F5: filter @ A,F2: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ F5 @ F2 )
=> ( topolo3448309680560233919inuous @ A @ B @ F5
@ ^ [X2: A] : ( uminus_uminus @ B @ ( F2 @ X2 ) ) ) ) ) ).
% continuous_minus
thf(fact_5837_tendsto__uminus__nhds,axiom,
! [A: $tType] :
( ( topolo1633459387980952147up_add @ A )
=> ! [A3: A] : ( filterlim @ A @ A @ ( uminus_uminus @ A ) @ ( topolo7230453075368039082e_nhds @ A @ ( uminus_uminus @ A @ A3 ) ) @ ( topolo7230453075368039082e_nhds @ A @ A3 ) ) ) ).
% tendsto_uminus_nhds
thf(fact_5838_tendsto__cot,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [F2: A > A,A3: A,F5: filter @ A] :
( ( filterlim @ A @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ A3 ) @ F5 )
=> ( ( ( sin @ A @ A3 )
!= ( zero_zero @ A ) )
=> ( filterlim @ A @ A
@ ^ [X2: A] : ( cot @ A @ ( F2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( cot @ A @ A3 ) )
@ F5 ) ) ) ) ).
% tendsto_cot
thf(fact_5839_int__encode__eq,axiom,
! [X: int,Y2: int] :
( ( ( nat_int_encode @ X )
= ( nat_int_encode @ Y2 ) )
= ( X = Y2 ) ) ).
% int_encode_eq
thf(fact_5840_continuous__sinh,axiom,
! [A: $tType,C: $tType] :
( ( ( topological_t2_space @ C )
& ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [F5: filter @ C,F2: C > A] :
( ( topolo3448309680560233919inuous @ C @ A @ F5 @ F2 )
=> ( topolo3448309680560233919inuous @ C @ A @ F5
@ ^ [X2: C] : ( sinh @ A @ ( F2 @ X2 ) ) ) ) ) ).
% continuous_sinh
thf(fact_5841_continuous__cosh,axiom,
! [A: $tType,C: $tType] :
( ( ( topological_t2_space @ C )
& ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [F5: filter @ C,F2: C > A] :
( ( topolo3448309680560233919inuous @ C @ A @ F5 @ F2 )
=> ( topolo3448309680560233919inuous @ C @ A @ F5
@ ^ [X2: C] : ( cosh @ A @ ( F2 @ X2 ) ) ) ) ) ).
% continuous_cosh
thf(fact_5842_continuous__exp,axiom,
! [A: $tType,C: $tType] :
( ( ( topological_t2_space @ C )
& ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [F5: filter @ C,F2: C > A] :
( ( topolo3448309680560233919inuous @ C @ A @ F5 @ F2 )
=> ( topolo3448309680560233919inuous @ C @ A @ F5
@ ^ [X2: C] : ( exp @ A @ ( F2 @ X2 ) ) ) ) ) ).
% continuous_exp
thf(fact_5843_continuous__cos,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_Vector_banach @ B )
& ( real_V3459762299906320749_field @ B ) )
=> ! [F5: filter @ A,F2: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ F5 @ F2 )
=> ( topolo3448309680560233919inuous @ A @ B @ F5
@ ^ [X2: A] : ( cos @ B @ ( F2 @ X2 ) ) ) ) ) ).
% continuous_cos
thf(fact_5844_tendsto__cos,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_Vector_banach @ B )
& ( real_V3459762299906320749_field @ B ) )
=> ! [F2: A > B,A3: B,F5: filter @ A] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ A3 ) @ F5 )
=> ( filterlim @ A @ B
@ ^ [X2: A] : ( cos @ B @ ( F2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( cos @ B @ A3 ) )
@ F5 ) ) ) ).
% tendsto_cos
thf(fact_5845_tendsto__sin,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_Vector_banach @ B )
& ( real_V3459762299906320749_field @ B ) )
=> ! [F2: A > B,A3: B,F5: filter @ A] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ A3 ) @ F5 )
=> ( filterlim @ A @ B
@ ^ [X2: A] : ( sin @ B @ ( F2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( sin @ B @ A3 ) )
@ F5 ) ) ) ).
% tendsto_sin
thf(fact_5846_tendsto__arctan,axiom,
! [A: $tType,F2: A > real,X: real,F5: filter @ A] :
( ( filterlim @ A @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ X ) @ F5 )
=> ( filterlim @ A @ real
@ ^ [X2: A] : ( arctan @ ( F2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( arctan @ X ) )
@ F5 ) ) ).
% tendsto_arctan
thf(fact_5847_tendsto__arsinh,axiom,
! [B: $tType,F2: B > real,A3: real,F5: filter @ B] :
( ( filterlim @ B @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ A3 ) @ F5 )
=> ( filterlim @ B @ real
@ ^ [X2: B] : ( arsinh @ real @ ( F2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( arsinh @ real @ A3 ) )
@ F5 ) ) ).
% tendsto_arsinh
thf(fact_5848_tendsto__cosh,axiom,
! [A: $tType,C: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [F2: C > A,A3: A,F5: filter @ C] :
( ( filterlim @ C @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ A3 ) @ F5 )
=> ( filterlim @ C @ A
@ ^ [X2: C] : ( cosh @ A @ ( F2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( cosh @ A @ A3 ) )
@ F5 ) ) ) ).
% tendsto_cosh
thf(fact_5849_tendsto__exp,axiom,
! [A: $tType,C: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [F2: C > A,A3: A,F5: filter @ C] :
( ( filterlim @ C @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ A3 ) @ F5 )
=> ( filterlim @ C @ A
@ ^ [X2: C] : ( exp @ A @ ( F2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( exp @ A @ A3 ) )
@ F5 ) ) ) ).
% tendsto_exp
thf(fact_5850_tendsto__sinh,axiom,
! [A: $tType,C: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [F2: C > A,A3: A,F5: filter @ C] :
( ( filterlim @ C @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ A3 ) @ F5 )
=> ( filterlim @ C @ A
@ ^ [X2: C] : ( sinh @ A @ ( F2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( sinh @ A @ A3 ) )
@ F5 ) ) ) ).
% tendsto_sinh
thf(fact_5851_continuous__sin,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_Vector_banach @ B )
& ( real_V3459762299906320749_field @ B ) )
=> ! [F5: filter @ A,F2: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ F5 @ F2 )
=> ( topolo3448309680560233919inuous @ A @ B @ F5
@ ^ [X2: A] : ( sin @ B @ ( F2 @ X2 ) ) ) ) ) ).
% continuous_sin
thf(fact_5852_continuous__arctan,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [F5: filter @ A,F2: A > real] :
( ( topolo3448309680560233919inuous @ A @ real @ F5 @ F2 )
=> ( topolo3448309680560233919inuous @ A @ real @ F5
@ ^ [X2: A] : ( arctan @ ( F2 @ X2 ) ) ) ) ) ).
% continuous_arctan
thf(fact_5853_continuous__arsinh,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [F5: filter @ A,F2: A > real] :
( ( topolo3448309680560233919inuous @ A @ real @ F5 @ F2 )
=> ( topolo3448309680560233919inuous @ A @ real @ F5
@ ^ [X2: A] : ( arsinh @ real @ ( F2 @ X2 ) ) ) ) ) ).
% continuous_arsinh
thf(fact_5854_tendsto__tanh,axiom,
! [A: $tType,C: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [F2: C > A,A3: A,F5: filter @ C] :
( ( filterlim @ C @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ A3 ) @ F5 )
=> ( ( ( cosh @ A @ A3 )
!= ( zero_zero @ A ) )
=> ( filterlim @ C @ A
@ ^ [X2: C] : ( tanh @ A @ ( F2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( tanh @ A @ A3 ) )
@ F5 ) ) ) ) ).
% tendsto_tanh
thf(fact_5855_continuous__diff,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( topolo1633459387980952147up_add @ B ) )
=> ! [F5: filter @ A,F2: A > B,G: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ F5 @ F2 )
=> ( ( topolo3448309680560233919inuous @ A @ B @ F5 @ G )
=> ( topolo3448309680560233919inuous @ A @ B @ F5
@ ^ [X2: A] : ( minus_minus @ B @ ( F2 @ X2 ) @ ( G @ X2 ) ) ) ) ) ) ).
% continuous_diff
thf(fact_5856_tendsto__diff,axiom,
! [A: $tType,B: $tType] :
( ( topolo1633459387980952147up_add @ A )
=> ! [F2: B > A,A3: A,F5: filter @ B,G: B > A,B2: A] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ A3 ) @ F5 )
=> ( ( filterlim @ B @ A @ G @ ( topolo7230453075368039082e_nhds @ A @ B2 ) @ F5 )
=> ( filterlim @ B @ A
@ ^ [X2: B] : ( minus_minus @ A @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( minus_minus @ A @ A3 @ B2 ) )
@ F5 ) ) ) ) ).
% tendsto_diff
thf(fact_5857_tendsto__mult__one,axiom,
! [B: $tType,D: $tType] :
( ( topolo1898628316856586783d_mult @ B )
=> ! [F2: D > B,F5: filter @ D,G: D > B] :
( ( filterlim @ D @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ ( one_one @ B ) ) @ F5 )
=> ( ( filterlim @ D @ B @ G @ ( topolo7230453075368039082e_nhds @ B @ ( one_one @ B ) ) @ F5 )
=> ( filterlim @ D @ B
@ ^ [X2: D] : ( times_times @ B @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( one_one @ B ) )
@ F5 ) ) ) ) ).
% tendsto_mult_one
thf(fact_5858_tendsto__divide__zero,axiom,
! [A: $tType,B: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: B > A,F5: filter @ B,C2: A] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ F5 )
=> ( filterlim @ B @ A
@ ^ [X2: B] : ( divide_divide @ A @ ( F2 @ X2 ) @ C2 )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ F5 ) ) ) ).
% tendsto_divide_zero
thf(fact_5859_tendsto__divide,axiom,
! [A: $tType,B: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: B > A,A3: A,F5: filter @ B,G: B > A,B2: A] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ A3 ) @ F5 )
=> ( ( filterlim @ B @ A @ G @ ( topolo7230453075368039082e_nhds @ A @ B2 ) @ F5 )
=> ( ( B2
!= ( zero_zero @ A ) )
=> ( filterlim @ B @ A
@ ^ [X2: B] : ( divide_divide @ A @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( divide_divide @ A @ A3 @ B2 ) )
@ F5 ) ) ) ) ) ).
% tendsto_divide
thf(fact_5860_tendsto__mult__right__zero,axiom,
! [A: $tType,D: $tType] :
( ( real_V4412858255891104859lgebra @ A )
=> ! [F2: D > A,F5: filter @ D,C2: A] :
( ( filterlim @ D @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ F5 )
=> ( filterlim @ D @ A
@ ^ [X2: D] : ( times_times @ A @ C2 @ ( F2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ F5 ) ) ) ).
% tendsto_mult_right_zero
thf(fact_5861_tendsto__mult__left__zero,axiom,
! [A: $tType,D: $tType] :
( ( real_V4412858255891104859lgebra @ A )
=> ! [F2: D > A,F5: filter @ D,C2: A] :
( ( filterlim @ D @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ F5 )
=> ( filterlim @ D @ A
@ ^ [X2: D] : ( times_times @ A @ ( F2 @ X2 ) @ C2 )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ F5 ) ) ) ).
% tendsto_mult_left_zero
thf(fact_5862_tendsto__mult__zero,axiom,
! [A: $tType,D: $tType] :
( ( real_V4412858255891104859lgebra @ A )
=> ! [F2: D > A,F5: filter @ D,G: D > A] :
( ( filterlim @ D @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ F5 )
=> ( ( filterlim @ D @ A @ G @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ F5 )
=> ( filterlim @ D @ A
@ ^ [X2: D] : ( times_times @ A @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ F5 ) ) ) ) ).
% tendsto_mult_zero
thf(fact_5863_Lim__transform__eq,axiom,
! [A: $tType,B: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: B > A,G: B > A,F5: filter @ B,A3: A] :
( ( filterlim @ B @ A
@ ^ [X2: B] : ( minus_minus @ A @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ F5 )
=> ( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ A3 ) @ F5 )
= ( filterlim @ B @ A @ G @ ( topolo7230453075368039082e_nhds @ A @ A3 ) @ F5 ) ) ) ) ).
% Lim_transform_eq
thf(fact_5864_LIM__zero__cancel,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F2: A > B,L: B,F5: filter @ A] :
( ( filterlim @ A @ B
@ ^ [X2: A] : ( minus_minus @ B @ ( F2 @ X2 ) @ L )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ F5 )
=> ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ F5 ) ) ) ).
% LIM_zero_cancel
thf(fact_5865_Lim__transform2,axiom,
! [A: $tType,B: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: B > A,A3: A,F5: filter @ B,G: B > A] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ A3 ) @ F5 )
=> ( ( filterlim @ B @ A
@ ^ [X2: B] : ( minus_minus @ A @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ F5 )
=> ( filterlim @ B @ A @ G @ ( topolo7230453075368039082e_nhds @ A @ A3 ) @ F5 ) ) ) ) ).
% Lim_transform2
thf(fact_5866_Lim__transform,axiom,
! [A: $tType,B: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [G: B > A,A3: A,F5: filter @ B,F2: B > A] :
( ( filterlim @ B @ A @ G @ ( topolo7230453075368039082e_nhds @ A @ A3 ) @ F5 )
=> ( ( filterlim @ B @ A
@ ^ [X2: B] : ( minus_minus @ A @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ F5 )
=> ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ A3 ) @ F5 ) ) ) ) ).
% Lim_transform
thf(fact_5867_LIM__zero__iff,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F2: A > B,L: B,F5: filter @ A] :
( ( filterlim @ A @ B
@ ^ [X2: A] : ( minus_minus @ B @ ( F2 @ X2 ) @ L )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ F5 )
= ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ F5 ) ) ) ).
% LIM_zero_iff
thf(fact_5868_LIM__zero,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F2: A > B,L: B,F5: filter @ A] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ F5 )
=> ( filterlim @ A @ B
@ ^ [X2: A] : ( minus_minus @ B @ ( F2 @ X2 ) @ L )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ F5 ) ) ) ).
% LIM_zero
thf(fact_5869_tendsto__arcosh,axiom,
! [B: $tType,F2: B > real,A3: real,F5: filter @ B] :
( ( filterlim @ B @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ A3 ) @ F5 )
=> ( ( ord_less @ real @ ( one_one @ real ) @ A3 )
=> ( filterlim @ B @ real
@ ^ [X2: B] : ( arcosh @ real @ ( F2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( arcosh @ real @ A3 ) )
@ F5 ) ) ) ).
% tendsto_arcosh
thf(fact_5870_tendsto__null__sum,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( topolo5987344860129210374id_add @ C )
=> ! [I5: set @ B,F2: A > B > C,F5: filter @ A] :
( ! [I3: B] :
( ( member @ B @ I3 @ I5 )
=> ( filterlim @ A @ C
@ ^ [X2: A] : ( F2 @ X2 @ I3 )
@ ( topolo7230453075368039082e_nhds @ C @ ( zero_zero @ C ) )
@ F5 ) )
=> ( filterlim @ A @ C
@ ^ [I2: A] : ( groups7311177749621191930dd_sum @ B @ C @ ( F2 @ I2 ) @ I5 )
@ ( topolo7230453075368039082e_nhds @ C @ ( zero_zero @ C ) )
@ F5 ) ) ) ).
% tendsto_null_sum
thf(fact_5871_tendsto__one__prod_H,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( topolo4987421752381908075d_mult @ C )
=> ! [I5: set @ B,F2: A > B > C,F5: filter @ A] :
( ! [I3: B] :
( ( member @ B @ I3 @ I5 )
=> ( filterlim @ A @ C
@ ^ [X2: A] : ( F2 @ X2 @ I3 )
@ ( topolo7230453075368039082e_nhds @ C @ ( one_one @ C ) )
@ F5 ) )
=> ( filterlim @ A @ C
@ ^ [I2: A] : ( groups7121269368397514597t_prod @ B @ C @ ( F2 @ I2 ) @ I5 )
@ ( topolo7230453075368039082e_nhds @ C @ ( one_one @ C ) )
@ F5 ) ) ) ).
% tendsto_one_prod'
thf(fact_5872_tendsto__null__power,axiom,
! [B: $tType,A: $tType] :
( ( real_V2822296259951069270ebra_1 @ B )
=> ! [F2: A > B,F5: filter @ A,N: nat] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) @ F5 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( filterlim @ A @ B
@ ^ [X2: A] : ( power_power @ B @ ( F2 @ X2 ) @ N )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ F5 ) ) ) ) ).
% tendsto_null_power
thf(fact_5873_tendsto__log,axiom,
! [A: $tType,F2: A > real,A3: real,F5: filter @ A,G: A > real,B2: real] :
( ( filterlim @ A @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ A3 ) @ F5 )
=> ( ( filterlim @ A @ real @ G @ ( topolo7230453075368039082e_nhds @ real @ B2 ) @ F5 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ A3 )
=> ( ( A3
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ B2 )
=> ( filterlim @ A @ real
@ ^ [X2: A] : ( log2 @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( log2 @ A3 @ B2 ) )
@ F5 ) ) ) ) ) ) ).
% tendsto_log
thf(fact_5874_tendsto__artanh,axiom,
! [A: $tType,F2: A > real,A3: real,F5: filter @ A] :
( ( filterlim @ A @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ A3 ) @ F5 )
=> ( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ A3 )
=> ( ( ord_less @ real @ A3 @ ( one_one @ real ) )
=> ( filterlim @ A @ real
@ ^ [X2: A] : ( artanh @ real @ ( F2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( artanh @ real @ A3 ) )
@ F5 ) ) ) ) ).
% tendsto_artanh
thf(fact_5875_LIM__imp__LIM,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ A )
& ( real_V822414075346904944vector @ C )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,L: B,A3: A,G: A > C,M2: C] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) )
=> ( ! [X3: A] :
( ( X3 != A3 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ C @ ( minus_minus @ C @ ( G @ X3 ) @ M2 ) ) @ ( real_V7770717601297561774m_norm @ B @ ( minus_minus @ B @ ( F2 @ X3 ) @ L ) ) ) )
=> ( filterlim @ A @ C @ G @ ( topolo7230453075368039082e_nhds @ C @ M2 ) @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% LIM_imp_LIM
thf(fact_5876_LIM__offset__zero__iff,axiom,
! [C: $tType,D: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( topolo4958980785337419405_space @ D )
& ( zero @ C ) )
=> ! [A3: A,F2: A > D,L4: D] :
( ( nO_MATCH @ C @ A @ ( zero_zero @ C ) @ A3 )
=> ( ( filterlim @ A @ D @ F2 @ ( topolo7230453075368039082e_nhds @ D @ L4 ) @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) )
= ( filterlim @ A @ D
@ ^ [H2: A] : ( F2 @ ( plus_plus @ A @ A3 @ H2 ) )
@ ( topolo7230453075368039082e_nhds @ D @ L4 )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% LIM_offset_zero_iff
thf(fact_5877_LIM__D,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,L4: B,A3: A,R4: real] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L4 ) @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ R4 )
=> ? [S2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ S2 )
& ! [X4: A] :
( ( ( X4 != A3 )
& ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ X4 @ A3 ) ) @ S2 ) )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ B @ ( minus_minus @ B @ ( F2 @ X4 ) @ L4 ) ) @ R4 ) ) ) ) ) ) ).
% LIM_D
thf(fact_5878_LIM__I,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [A3: A,F2: A > B,L4: B] :
( ! [R3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R3 )
=> ? [S6: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ S6 )
& ! [X3: A] :
( ( ( X3 != A3 )
& ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ X3 @ A3 ) ) @ S6 ) )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ B @ ( minus_minus @ B @ ( F2 @ X3 ) @ L4 ) ) @ R3 ) ) ) )
=> ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L4 ) @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% LIM_I
thf(fact_5879_LIM__eq,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,L4: B,A3: A] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L4 ) @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) )
= ( ! [R: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R )
=> ? [S5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ S5 )
& ! [X2: A] :
( ( ( X2 != A3 )
& ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ X2 @ A3 ) ) @ S5 ) )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ B @ ( minus_minus @ B @ ( F2 @ X2 ) @ L4 ) ) @ R ) ) ) ) ) ) ) ).
% LIM_eq
thf(fact_5880_LIM__equal2,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [R2: real,A3: A,F2: A > B,G: A > B,L: B] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R2 )
=> ( ! [X3: A] :
( ( X3 != A3 )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ X3 @ A3 ) ) @ R2 )
=> ( ( F2 @ X3 )
= ( G @ X3 ) ) ) )
=> ( ( filterlim @ A @ B @ G @ ( topolo7230453075368039082e_nhds @ B @ L ) @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) )
=> ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ).
% LIM_equal2
thf(fact_5881_isCont__cos,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A] : ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) @ ( cos @ A ) ) ) ).
% isCont_cos
thf(fact_5882_isCont__sin,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A] : ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) @ ( sin @ A ) ) ) ).
% isCont_sin
thf(fact_5883_isCont__exp,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A] : ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) @ ( exp @ A ) ) ) ).
% isCont_exp
thf(fact_5884_isCont__cosh,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A] : ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) @ ( cosh @ A ) ) ) ).
% isCont_cosh
thf(fact_5885_isCont__sinh,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A] : ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) @ ( sinh @ A ) ) ) ).
% isCont_sinh
thf(fact_5886_DERIV__LIM__iff,axiom,
! [A: $tType] :
( ( ( inverse @ A )
& ( real_V822414075346904944vector @ A ) )
=> ! [F2: A > A,A3: A,D6: A] :
( ( filterlim @ A @ A
@ ^ [H2: A] : ( divide_divide @ A @ ( minus_minus @ A @ ( F2 @ ( plus_plus @ A @ A3 @ H2 ) ) @ ( F2 @ A3 ) ) @ H2 )
@ ( topolo7230453075368039082e_nhds @ A @ D6 )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) )
= ( filterlim @ A @ A
@ ^ [X2: A] : ( divide_divide @ A @ ( minus_minus @ A @ ( F2 @ X2 ) @ ( F2 @ A3 ) ) @ ( minus_minus @ A @ X2 @ A3 ) )
@ ( topolo7230453075368039082e_nhds @ A @ D6 )
@ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% DERIV_LIM_iff
thf(fact_5887_LIM__fun__gt__zero,axiom,
! [F2: real > real,L: real,C2: real] :
( ( filterlim @ real @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ L ) @ ( topolo174197925503356063within @ real @ C2 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ L )
=> ? [R3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R3 )
& ! [X4: real] :
( ( ( X4 != C2 )
& ( ord_less @ real @ ( abs_abs @ real @ ( minus_minus @ real @ C2 @ X4 ) ) @ R3 ) )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( F2 @ X4 ) ) ) ) ) ) ).
% LIM_fun_gt_zero
thf(fact_5888_LIM__fun__not__zero,axiom,
! [F2: real > real,L: real,C2: real] :
( ( filterlim @ real @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ L ) @ ( topolo174197925503356063within @ real @ C2 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( L
!= ( zero_zero @ real ) )
=> ? [R3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R3 )
& ! [X4: real] :
( ( ( X4 != C2 )
& ( ord_less @ real @ ( abs_abs @ real @ ( minus_minus @ real @ C2 @ X4 ) ) @ R3 ) )
=> ( ( F2 @ X4 )
!= ( zero_zero @ real ) ) ) ) ) ) ).
% LIM_fun_not_zero
thf(fact_5889_LIM__fun__less__zero,axiom,
! [F2: real > real,L: real,C2: real] :
( ( filterlim @ real @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ L ) @ ( topolo174197925503356063within @ real @ C2 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( ord_less @ real @ L @ ( zero_zero @ real ) )
=> ? [R3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R3 )
& ! [X4: real] :
( ( ( X4 != C2 )
& ( ord_less @ real @ ( abs_abs @ real @ ( minus_minus @ real @ C2 @ X4 ) ) @ R3 ) )
=> ( ord_less @ real @ ( F2 @ X4 ) @ ( zero_zero @ real ) ) ) ) ) ) ).
% LIM_fun_less_zero
thf(fact_5890_LIM__compose2,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( topolo4958980785337419405_space @ B )
& ( topolo4958980785337419405_space @ C ) )
=> ! [F2: A > B,B2: B,A3: A,G: B > C,C2: C] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ B2 ) @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) )
=> ( ( filterlim @ B @ C @ G @ ( topolo7230453075368039082e_nhds @ C @ C2 ) @ ( topolo174197925503356063within @ B @ B2 @ ( top_top @ ( set @ B ) ) ) )
=> ( ? [D3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D3 )
& ! [X3: A] :
( ( ( X3 != A3 )
& ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ X3 @ A3 ) ) @ D3 ) )
=> ( ( F2 @ X3 )
!= B2 ) ) )
=> ( filterlim @ A @ C
@ ^ [X2: A] : ( G @ ( F2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ C @ C2 )
@ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ).
% LIM_compose2
thf(fact_5891_continuous__at__within__divide,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_V3459762299906320749_field @ B ) )
=> ! [A3: A,S: set @ A,F2: A > B,G: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A3 @ S ) @ F2 )
=> ( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A3 @ S ) @ G )
=> ( ( ( G @ A3 )
!= ( zero_zero @ B ) )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A3 @ S )
@ ^ [X2: A] : ( divide_divide @ B @ ( F2 @ X2 ) @ ( G @ X2 ) ) ) ) ) ) ) ).
% continuous_at_within_divide
thf(fact_5892_isCont__arctan,axiom,
! [X: real] : ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) @ arctan ) ).
% isCont_arctan
thf(fact_5893_isCont__diff,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [A3: A,F2: A > B,G: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) @ F2 )
=> ( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) @ G )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) )
@ ^ [X2: A] : ( minus_minus @ B @ ( F2 @ X2 ) @ ( G @ X2 ) ) ) ) ) ) ).
% isCont_diff
thf(fact_5894_isCont__minus,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [A3: A,F2: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) @ F2 )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) )
@ ^ [X2: A] : ( uminus_uminus @ B @ ( F2 @ X2 ) ) ) ) ) ).
% isCont_minus
thf(fact_5895_continuous__at__within__inverse,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_V8999393235501362500lgebra @ B ) )
=> ! [A3: A,S: set @ A,F2: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A3 @ S ) @ F2 )
=> ( ( ( F2 @ A3 )
!= ( zero_zero @ B ) )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A3 @ S )
@ ^ [X2: A] : ( inverse_inverse @ B @ ( F2 @ X2 ) ) ) ) ) ) ).
% continuous_at_within_inverse
thf(fact_5896_continuous__at__within__sgn,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [A3: A,S: set @ A,F2: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A3 @ S ) @ F2 )
=> ( ( ( F2 @ A3 )
!= ( zero_zero @ B ) )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A3 @ S )
@ ^ [X2: A] : ( sgn_sgn @ B @ ( F2 @ X2 ) ) ) ) ) ) ).
% continuous_at_within_sgn
thf(fact_5897_isCont__cos_H,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_Vector_banach @ B )
& ( real_V3459762299906320749_field @ B ) )
=> ! [A3: A,F2: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) @ F2 )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) )
@ ^ [X2: A] : ( cos @ B @ ( F2 @ X2 ) ) ) ) ) ).
% isCont_cos'
thf(fact_5898_isCont__sin_H,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_Vector_banach @ B )
& ( real_V3459762299906320749_field @ B ) )
=> ! [A3: A,F2: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) @ F2 )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) )
@ ^ [X2: A] : ( sin @ B @ ( F2 @ X2 ) ) ) ) ) ).
% isCont_sin'
thf(fact_5899_isCont__exp_H,axiom,
! [A: $tType,C: $tType] :
( ( ( topological_t2_space @ C )
& ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [A3: C,F2: C > A] :
( ( topolo3448309680560233919inuous @ C @ A @ ( topolo174197925503356063within @ C @ A3 @ ( top_top @ ( set @ C ) ) ) @ F2 )
=> ( topolo3448309680560233919inuous @ C @ A @ ( topolo174197925503356063within @ C @ A3 @ ( top_top @ ( set @ C ) ) )
@ ^ [X2: C] : ( exp @ A @ ( F2 @ X2 ) ) ) ) ) ).
% isCont_exp'
thf(fact_5900_isCont__pochhammer,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [Z2: A,N: nat] :
( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ Z2 @ ( top_top @ ( set @ A ) ) )
@ ^ [Z6: A] : ( comm_s3205402744901411588hammer @ A @ Z6 @ N ) ) ) ).
% isCont_pochhammer
thf(fact_5901_continuous__at__within__powr,axiom,
! [C: $tType] :
( ( topological_t2_space @ C )
=> ! [A3: C,S: set @ C,F2: C > real,G: C > real] :
( ( topolo3448309680560233919inuous @ C @ real @ ( topolo174197925503356063within @ C @ A3 @ S ) @ F2 )
=> ( ( topolo3448309680560233919inuous @ C @ real @ ( topolo174197925503356063within @ C @ A3 @ S ) @ G )
=> ( ( ( F2 @ A3 )
!= ( zero_zero @ real ) )
=> ( topolo3448309680560233919inuous @ C @ real @ ( topolo174197925503356063within @ C @ A3 @ S )
@ ^ [X2: C] : ( powr @ real @ ( F2 @ X2 ) @ ( G @ X2 ) ) ) ) ) ) ) ).
% continuous_at_within_powr
thf(fact_5902_continuous__within__ln,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [X: A,S: set @ A,F2: A > real] :
( ( topolo3448309680560233919inuous @ A @ real @ ( topolo174197925503356063within @ A @ X @ S ) @ F2 )
=> ( ( ( F2 @ X )
!= ( zero_zero @ real ) )
=> ( topolo3448309680560233919inuous @ A @ real @ ( topolo174197925503356063within @ A @ X @ S )
@ ^ [X2: A] : ( ln_ln @ real @ ( F2 @ X2 ) ) ) ) ) ) ).
% continuous_within_ln
thf(fact_5903_isCont__arsinh,axiom,
! [X: real] : ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) @ ( arsinh @ real ) ) ).
% isCont_arsinh
thf(fact_5904_DERIV__D,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D6: A,X: A] :
( ( has_field_derivative @ A @ F2 @ D6 @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) )
=> ( filterlim @ A @ A
@ ^ [H2: A] : ( divide_divide @ A @ ( minus_minus @ A @ ( F2 @ ( plus_plus @ A @ X @ H2 ) ) @ ( F2 @ X ) ) @ H2 )
@ ( topolo7230453075368039082e_nhds @ A @ D6 )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% DERIV_D
thf(fact_5905_DERIV__def,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D6: A,X: A] :
( ( has_field_derivative @ A @ F2 @ D6 @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) )
= ( filterlim @ A @ A
@ ^ [H2: A] : ( divide_divide @ A @ ( minus_minus @ A @ ( F2 @ ( plus_plus @ A @ X @ H2 ) ) @ ( F2 @ X ) ) @ H2 )
@ ( topolo7230453075368039082e_nhds @ A @ D6 )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% DERIV_def
thf(fact_5906_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_5907_lemma__termdiff4,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [K: real,F2: A > B,K6: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K )
=> ( ! [H3: A] :
( ( H3
!= ( zero_zero @ A ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ H3 ) @ K )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( F2 @ H3 ) ) @ ( times_times @ real @ K6 @ ( real_V7770717601297561774m_norm @ A @ H3 ) ) ) ) )
=> ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) @ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% lemma_termdiff4
thf(fact_5908_field__has__derivative__at,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D6: A,X: A] :
( ( has_derivative @ A @ A @ F2 @ ( times_times @ A @ D6 ) @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) )
= ( filterlim @ A @ A
@ ^ [H2: A] : ( divide_divide @ A @ ( minus_minus @ A @ ( F2 @ ( plus_plus @ A @ X @ H2 ) ) @ ( F2 @ X ) ) @ H2 )
@ ( topolo7230453075368039082e_nhds @ A @ D6 )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% field_has_derivative_at
thf(fact_5909_isCont__ln,axiom,
! [X: real] :
( ( X
!= ( zero_zero @ real ) )
=> ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) @ ( ln_ln @ real ) ) ) ).
% isCont_ln
thf(fact_5910_isCont__divide,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_V3459762299906320749_field @ B ) )
=> ! [A3: A,F2: A > B,G: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) @ F2 )
=> ( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) @ G )
=> ( ( ( G @ A3 )
!= ( zero_zero @ B ) )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) )
@ ^ [X2: A] : ( divide_divide @ B @ ( F2 @ X2 ) @ ( G @ X2 ) ) ) ) ) ) ) ).
% isCont_divide
thf(fact_5911_isCont__sgn,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [A3: A,F2: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) @ F2 )
=> ( ( ( F2 @ A3 )
!= ( zero_zero @ B ) )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) )
@ ^ [X2: A] : ( sgn_sgn @ B @ ( F2 @ X2 ) ) ) ) ) ) ).
% isCont_sgn
thf(fact_5912_filterlim__at__to__0,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: A > B,F5: filter @ B,A3: A] :
( ( filterlim @ A @ B @ F2 @ F5 @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) )
= ( filterlim @ A @ B
@ ^ [X2: A] : ( F2 @ ( plus_plus @ A @ X2 @ A3 ) )
@ F5
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% filterlim_at_to_0
thf(fact_5913_continuous__within__tan,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A,S: set @ A,F2: A > A] :
( ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ X @ S ) @ F2 )
=> ( ( ( cos @ A @ ( F2 @ X ) )
!= ( zero_zero @ A ) )
=> ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ X @ S )
@ ^ [X2: A] : ( tan @ A @ ( F2 @ X2 ) ) ) ) ) ) ).
% continuous_within_tan
thf(fact_5914_isCont__powr,axiom,
! [C: $tType] :
( ( topological_t2_space @ C )
=> ! [A3: C,F2: C > real,G: C > real] :
( ( topolo3448309680560233919inuous @ C @ real @ ( topolo174197925503356063within @ C @ A3 @ ( top_top @ ( set @ C ) ) ) @ F2 )
=> ( ( topolo3448309680560233919inuous @ C @ real @ ( topolo174197925503356063within @ C @ A3 @ ( top_top @ ( set @ C ) ) ) @ G )
=> ( ( ( F2 @ A3 )
!= ( zero_zero @ real ) )
=> ( topolo3448309680560233919inuous @ C @ real @ ( topolo174197925503356063within @ C @ A3 @ ( top_top @ ( set @ C ) ) )
@ ^ [X2: C] : ( powr @ real @ ( F2 @ X2 ) @ ( G @ X2 ) ) ) ) ) ) ) ).
% isCont_powr
thf(fact_5915_isCont__ln_H,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [X: A,F2: A > real] :
( ( topolo3448309680560233919inuous @ A @ real @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) @ F2 )
=> ( ( ( F2 @ X )
!= ( zero_zero @ real ) )
=> ( topolo3448309680560233919inuous @ A @ real @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) )
@ ^ [X2: A] : ( ln_ln @ real @ ( F2 @ X2 ) ) ) ) ) ) ).
% isCont_ln'
thf(fact_5916_continuous__within__cot,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A,S: set @ A,F2: A > A] :
( ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ X @ S ) @ F2 )
=> ( ( ( sin @ A @ ( F2 @ X ) )
!= ( zero_zero @ A ) )
=> ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ X @ S )
@ ^ [X2: A] : ( cot @ A @ ( F2 @ X2 ) ) ) ) ) ) ).
% continuous_within_cot
thf(fact_5917_continuous__at__within__tanh,axiom,
! [A: $tType,C: $tType] :
( ( ( topological_t2_space @ C )
& ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: C,A4: set @ C,F2: C > A] :
( ( topolo3448309680560233919inuous @ C @ A @ ( topolo174197925503356063within @ C @ X @ A4 ) @ F2 )
=> ( ( ( cosh @ A @ ( F2 @ X ) )
!= ( zero_zero @ A ) )
=> ( topolo3448309680560233919inuous @ C @ A @ ( topolo174197925503356063within @ C @ X @ A4 )
@ ^ [X2: C] : ( tanh @ A @ ( F2 @ X2 ) ) ) ) ) ) ).
% continuous_at_within_tanh
thf(fact_5918_CARAT__DERIV,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,L: A,X: A] :
( ( has_field_derivative @ A @ F2 @ L @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) )
= ( ? [G2: A > A] :
( ! [Z6: A] :
( ( minus_minus @ A @ ( F2 @ Z6 ) @ ( F2 @ 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_5919_isCont__has__Ub,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [A3: real,B2: real,F2: real > A] :
( ( ord_less_eq @ real @ A3 @ B2 )
=> ( ! [X3: real] :
( ( ( ord_less_eq @ real @ A3 @ X3 )
& ( ord_less_eq @ real @ X3 @ B2 ) )
=> ( topolo3448309680560233919inuous @ real @ A @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) @ F2 ) )
=> ? [M9: A] :
( ! [X4: real] :
( ( ( ord_less_eq @ real @ A3 @ X4 )
& ( ord_less_eq @ real @ X4 @ B2 ) )
=> ( ord_less_eq @ A @ ( F2 @ X4 ) @ M9 ) )
& ! [N9: A] :
( ( ord_less @ A @ N9 @ M9 )
=> ? [X3: real] :
( ( ord_less_eq @ real @ A3 @ X3 )
& ( ord_less_eq @ real @ X3 @ B2 )
& ( ord_less @ A @ N9 @ ( F2 @ X3 ) ) ) ) ) ) ) ) ).
% isCont_has_Ub
thf(fact_5920_isCont__tan,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A] :
( ( ( cos @ A @ X )
!= ( zero_zero @ A ) )
=> ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) @ ( tan @ A ) ) ) ) ).
% isCont_tan
thf(fact_5921_filterlim__shift__iff,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: A > B,D2: A,F5: filter @ B,A3: A] :
( ( filterlim @ A @ B @ ( comp @ A @ B @ A @ F2 @ ( plus_plus @ A @ D2 ) ) @ F5 @ ( topolo174197925503356063within @ A @ ( minus_minus @ A @ A3 @ D2 ) @ ( top_top @ ( set @ A ) ) ) )
= ( filterlim @ A @ B @ F2 @ F5 @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% filterlim_shift_iff
thf(fact_5922_filterlim__shift,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: A > B,F5: filter @ B,A3: A,D2: A] :
( ( filterlim @ A @ B @ F2 @ F5 @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) )
=> ( filterlim @ A @ B @ ( comp @ A @ B @ A @ F2 @ ( plus_plus @ A @ D2 ) ) @ F5 @ ( topolo174197925503356063within @ A @ ( minus_minus @ A @ A3 @ D2 ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% filterlim_shift
thf(fact_5923_isCont__cot,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A] :
( ( ( sin @ A @ X )
!= ( zero_zero @ A ) )
=> ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) @ ( cot @ A ) ) ) ) ).
% isCont_cot
thf(fact_5924_isCont__tanh,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A] :
( ( ( cosh @ A @ X )
!= ( zero_zero @ A ) )
=> ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) @ ( tanh @ A ) ) ) ) ).
% isCont_tanh
thf(fact_5925_powser__limit__0__strong,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [S: real,A3: nat > A,F2: A > A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ S )
=> ( ! [X3: A] :
( ( X3
!= ( zero_zero @ A ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X3 ) @ S )
=> ( sums @ A
@ ^ [N5: nat] : ( times_times @ A @ ( A3 @ N5 ) @ ( power_power @ A @ X3 @ N5 ) )
@ ( F2 @ X3 ) ) ) )
=> ( filterlim @ A @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ ( A3 @ ( zero_zero @ nat ) ) ) @ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% powser_limit_0_strong
thf(fact_5926_powser__limit__0,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [S: real,A3: nat > A,F2: A > A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ S )
=> ( ! [X3: A] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X3 ) @ S )
=> ( sums @ A
@ ^ [N5: nat] : ( times_times @ A @ ( A3 @ N5 ) @ ( power_power @ A @ X3 @ N5 ) )
@ ( F2 @ X3 ) ) )
=> ( filterlim @ A @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ ( A3 @ ( zero_zero @ nat ) ) ) @ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% powser_limit_0
thf(fact_5927_bij__int__encode,axiom,
bij_betw @ int @ nat @ nat_int_encode @ ( top_top @ ( set @ int ) ) @ ( top_top @ ( set @ nat ) ) ).
% bij_int_encode
thf(fact_5928_lemma__termdiff5,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_Vector_banach @ B ) )
=> ! [K: real,F2: nat > real,G: A > nat > B] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K )
=> ( ( summable @ real @ F2 )
=> ( ! [H3: A,N2: nat] :
( ( H3
!= ( zero_zero @ A ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ H3 ) @ K )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( G @ H3 @ N2 ) ) @ ( times_times @ real @ ( F2 @ N2 ) @ ( real_V7770717601297561774m_norm @ A @ H3 ) ) ) ) )
=> ( filterlim @ A @ B
@ ^ [H2: A] : ( suminf @ B @ ( G @ H2 ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ).
% lemma_termdiff5
thf(fact_5929_isCont__tan_H,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [A3: A,F2: A > A] :
( ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) @ F2 )
=> ( ( ( cos @ A @ ( F2 @ A3 ) )
!= ( zero_zero @ A ) )
=> ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) )
@ ^ [X2: A] : ( tan @ A @ ( F2 @ X2 ) ) ) ) ) ) ).
% isCont_tan'
thf(fact_5930_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_5931_continuous__at__within__log,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [A3: A,S: set @ A,F2: A > real,G: A > real] :
( ( topolo3448309680560233919inuous @ A @ real @ ( topolo174197925503356063within @ A @ A3 @ S ) @ F2 )
=> ( ( topolo3448309680560233919inuous @ A @ real @ ( topolo174197925503356063within @ A @ A3 @ S ) @ G )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( F2 @ A3 ) )
=> ( ( ( F2 @ A3 )
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( G @ A3 ) )
=> ( topolo3448309680560233919inuous @ A @ real @ ( topolo174197925503356063within @ A @ A3 @ S )
@ ^ [X2: A] : ( log2 @ ( F2 @ X2 ) @ ( G @ X2 ) ) ) ) ) ) ) ) ) ).
% continuous_at_within_log
thf(fact_5932_LIM__cos__div__sin,axiom,
( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( cos @ real @ X2 ) @ ( sin @ real @ X2 ) )
@ ( 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_5933_isCont__cot_H,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [A3: A,F2: A > A] :
( ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) @ F2 )
=> ( ( ( sin @ A @ ( F2 @ A3 ) )
!= ( zero_zero @ A ) )
=> ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) )
@ ^ [X2: A] : ( cot @ A @ ( F2 @ X2 ) ) ) ) ) ) ).
% isCont_cot'
thf(fact_5934_isCont__polynom,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [A3: A,C2: nat > A,N: nat] :
( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) )
@ ^ [W3: A] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( times_times @ A @ ( C2 @ I2 ) @ ( power_power @ A @ W3 @ I2 ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% isCont_polynom
thf(fact_5935_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_5936_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_5937_isCont__powser__converges__everywhere,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [C2: nat > A,X: A] :
( ! [Y3: A] :
( summable @ A
@ ^ [N5: nat] : ( times_times @ A @ ( C2 @ N5 ) @ ( power_power @ A @ Y3 @ N5 ) ) )
=> ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) )
@ ^ [X2: A] :
( suminf @ A
@ ^ [N5: nat] : ( times_times @ A @ ( C2 @ N5 ) @ ( power_power @ A @ X2 @ N5 ) ) ) ) ) ) ).
% isCont_powser_converges_everywhere
thf(fact_5938_isCont__log,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [A3: A,F2: A > real,G: A > real] :
( ( topolo3448309680560233919inuous @ A @ real @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) @ F2 )
=> ( ( topolo3448309680560233919inuous @ A @ real @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) @ G )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( F2 @ A3 ) )
=> ( ( ( F2 @ A3 )
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( G @ A3 ) )
=> ( topolo3448309680560233919inuous @ A @ real @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) )
@ ^ [X2: A] : ( log2 @ ( F2 @ X2 ) @ ( G @ X2 ) ) ) ) ) ) ) ) ) ).
% isCont_log
thf(fact_5939_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_5940_isCont__inverse__function,axiom,
! [D2: real,X: real,G: real > real,F2: real > real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D2 )
=> ( ! [Z4: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ ( minus_minus @ real @ Z4 @ X ) ) @ D2 )
=> ( ( G @ ( F2 @ Z4 ) )
= Z4 ) )
=> ( ! [Z4: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ ( minus_minus @ real @ Z4 @ X ) ) @ D2 )
=> ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ Z4 @ ( top_top @ ( set @ real ) ) ) @ F2 ) )
=> ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ ( F2 @ X ) @ ( top_top @ ( set @ real ) ) ) @ G ) ) ) ) ).
% isCont_inverse_function
thf(fact_5941_GMVT_H,axiom,
! [A3: real,B2: real,F2: real > real,G: real > real,G3: real > real,F6: real > real] :
( ( ord_less @ real @ A3 @ B2 )
=> ( ! [Z4: real] :
( ( ord_less_eq @ real @ A3 @ Z4 )
=> ( ( ord_less_eq @ real @ Z4 @ B2 )
=> ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ Z4 @ ( top_top @ ( set @ real ) ) ) @ F2 ) ) )
=> ( ! [Z4: real] :
( ( ord_less_eq @ real @ A3 @ Z4 )
=> ( ( ord_less_eq @ real @ Z4 @ B2 )
=> ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ Z4 @ ( top_top @ ( set @ real ) ) ) @ G ) ) )
=> ( ! [Z4: real] :
( ( ord_less @ real @ A3 @ Z4 )
=> ( ( ord_less @ real @ Z4 @ B2 )
=> ( has_field_derivative @ real @ G @ ( G3 @ Z4 ) @ ( topolo174197925503356063within @ real @ Z4 @ ( top_top @ ( set @ real ) ) ) ) ) )
=> ( ! [Z4: real] :
( ( ord_less @ real @ A3 @ Z4 )
=> ( ( ord_less @ real @ Z4 @ B2 )
=> ( has_field_derivative @ real @ F2 @ ( F6 @ Z4 ) @ ( topolo174197925503356063within @ real @ Z4 @ ( top_top @ ( set @ real ) ) ) ) ) )
=> ? [C3: real] :
( ( ord_less @ real @ A3 @ C3 )
& ( ord_less @ real @ C3 @ B2 )
& ( ( times_times @ real @ ( minus_minus @ real @ ( F2 @ B2 ) @ ( F2 @ A3 ) ) @ ( G3 @ C3 ) )
= ( times_times @ real @ ( minus_minus @ real @ ( G @ B2 ) @ ( G @ A3 ) ) @ ( F6 @ C3 ) ) ) ) ) ) ) ) ) ).
% GMVT'
thf(fact_5942_isCont__powser_H,axiom,
! [Aa: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_Vector_banach @ Aa )
& ( real_V3459762299906320749_field @ Aa ) )
=> ! [A3: A,F2: A > Aa,C2: nat > Aa,K6: Aa] :
( ( topolo3448309680560233919inuous @ A @ Aa @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) @ F2 )
=> ( ( summable @ Aa
@ ^ [N5: nat] : ( times_times @ Aa @ ( C2 @ N5 ) @ ( power_power @ Aa @ K6 @ N5 ) ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ Aa @ ( F2 @ A3 ) ) @ ( real_V7770717601297561774m_norm @ Aa @ K6 ) )
=> ( topolo3448309680560233919inuous @ A @ Aa @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) )
@ ^ [X2: A] :
( suminf @ Aa
@ ^ [N5: nat] : ( times_times @ Aa @ ( C2 @ N5 ) @ ( power_power @ Aa @ ( F2 @ X2 ) @ N5 ) ) ) ) ) ) ) ) ).
% isCont_powser'
thf(fact_5943_summable__Leibniz_I2_J,axiom,
! [A3: nat > real] :
( ( filterlim @ nat @ real @ A3 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) )
=> ( ( topological_monoseq @ real @ A3 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( A3 @ ( zero_zero @ nat ) ) )
=> ! [N4: nat] :
( member @ real
@ ( suminf @ real
@ ^ [I2: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I2 ) @ ( A3 @ I2 ) ) )
@ ( set_or1337092689740270186AtMost @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I2: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I2 ) @ ( A3 @ I2 ) )
@ ( set_ord_lessThan @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N4 ) ) )
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I2: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I2 ) @ ( A3 @ I2 ) )
@ ( set_ord_lessThan @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N4 ) @ ( one_one @ nat ) ) ) ) ) ) ) ) ) ).
% summable_Leibniz(2)
thf(fact_5944_summable__Leibniz_I3_J,axiom,
! [A3: nat > real] :
( ( filterlim @ nat @ real @ A3 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) )
=> ( ( topological_monoseq @ real @ A3 )
=> ( ( ord_less @ real @ ( A3 @ ( zero_zero @ nat ) ) @ ( zero_zero @ real ) )
=> ! [N4: nat] :
( member @ real
@ ( suminf @ real
@ ^ [I2: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I2 ) @ ( A3 @ I2 ) ) )
@ ( set_or1337092689740270186AtMost @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I2: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I2 ) @ ( A3 @ I2 ) )
@ ( set_ord_lessThan @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N4 ) @ ( one_one @ nat ) ) ) )
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I2: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I2 ) @ ( A3 @ I2 ) )
@ ( set_ord_lessThan @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N4 ) ) ) ) ) ) ) ) ).
% summable_Leibniz(3)
thf(fact_5945_summable__Leibniz_H_I4_J,axiom,
! [A3: nat > real,N: nat] :
( ( filterlim @ nat @ real @ A3 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) )
=> ( ! [N2: nat] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( A3 @ N2 ) )
=> ( ! [N2: nat] : ( ord_less_eq @ real @ ( A3 @ ( suc @ N2 ) ) @ ( A3 @ N2 ) )
=> ( ord_less_eq @ real
@ ( suminf @ real
@ ^ [I2: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I2 ) @ ( A3 @ I2 ) ) )
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I2: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I2 ) @ ( A3 @ I2 ) )
@ ( set_ord_lessThan @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% summable_Leibniz'(4)
thf(fact_5946_tendsto__zero__mult__left__iff,axiom,
! [A: $tType] :
( ( ( field @ A )
& ( topolo4211221413907600880p_mult @ A ) )
=> ! [C2: A,A3: nat > A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( filterlim @ nat @ A
@ ^ [N5: nat] : ( times_times @ A @ C2 @ ( A3 @ N5 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ ( at_top @ nat ) )
= ( filterlim @ nat @ A @ A3 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ ( at_top @ nat ) ) ) ) ) ).
% tendsto_zero_mult_left_iff
thf(fact_5947_tendsto__zero__mult__right__iff,axiom,
! [A: $tType] :
( ( ( field @ A )
& ( topolo4211221413907600880p_mult @ A ) )
=> ! [C2: A,A3: nat > A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( filterlim @ nat @ A
@ ^ [N5: nat] : ( times_times @ A @ ( A3 @ N5 ) @ C2 )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ ( at_top @ nat ) )
= ( filterlim @ nat @ A @ A3 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ ( at_top @ nat ) ) ) ) ) ).
% tendsto_zero_mult_right_iff
thf(fact_5948_tendsto__zero__divide__iff,axiom,
! [A: $tType] :
( ( ( field @ A )
& ( topolo4211221413907600880p_mult @ A ) )
=> ! [C2: A,A3: nat > A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( filterlim @ nat @ A
@ ^ [N5: nat] : ( divide_divide @ A @ ( A3 @ N5 ) @ C2 )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ ( at_top @ nat ) )
= ( filterlim @ nat @ A @ A3 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ ( at_top @ nat ) ) ) ) ) ).
% tendsto_zero_divide_iff
thf(fact_5949_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] :
( ! [N4: nat] : ( ord_less @ A @ ( U3 @ N4 ) @ X )
& ( filterlim @ nat @ A @ U3 @ ( topolo7230453075368039082e_nhds @ A @ X ) @ ( at_top @ nat ) ) ) ) ) ).
% approx_from_below_dense_linorder
thf(fact_5950_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] :
( ! [N4: nat] : ( ord_less @ A @ X @ ( U3 @ N4 ) )
& ( filterlim @ nat @ A @ U3 @ ( topolo7230453075368039082e_nhds @ A @ X ) @ ( at_top @ nat ) ) ) ) ) ).
% approx_from_above_dense_linorder
thf(fact_5951_LIMSEQ__Suc,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F2: nat > A,L: A] :
( ( filterlim @ nat @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ ( at_top @ nat ) )
=> ( filterlim @ nat @ A
@ ^ [N5: nat] : ( F2 @ ( suc @ N5 ) )
@ ( topolo7230453075368039082e_nhds @ A @ L )
@ ( at_top @ nat ) ) ) ) ).
% LIMSEQ_Suc
thf(fact_5952_LIMSEQ__imp__Suc,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F2: nat > A,L: A] :
( ( filterlim @ nat @ A
@ ^ [N5: nat] : ( F2 @ ( suc @ N5 ) )
@ ( topolo7230453075368039082e_nhds @ A @ L )
@ ( at_top @ nat ) )
=> ( filterlim @ nat @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ ( at_top @ nat ) ) ) ) ).
% LIMSEQ_imp_Suc
thf(fact_5953_seq__offset__neg,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F2: nat > A,L: A,K: nat] :
( ( filterlim @ nat @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ ( at_top @ nat ) )
=> ( filterlim @ nat @ A
@ ^ [I2: nat] : ( F2 @ ( minus_minus @ nat @ I2 @ K ) )
@ ( topolo7230453075368039082e_nhds @ A @ L )
@ ( at_top @ nat ) ) ) ) ).
% seq_offset_neg
thf(fact_5954_summable__LIMSEQ__zero,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A] :
( ( summable @ A @ F2 )
=> ( filterlim @ nat @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ ( at_top @ nat ) ) ) ) ).
% summable_LIMSEQ_zero
thf(fact_5955_mult__nat__right__at__top,axiom,
! [C2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ C2 )
=> ( filterlim @ nat @ nat
@ ^ [X2: nat] : ( times_times @ nat @ X2 @ C2 )
@ ( at_top @ nat )
@ ( at_top @ nat ) ) ) ).
% mult_nat_right_at_top
thf(fact_5956_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_5957_LIMSEQ__root,axiom,
( filterlim @ nat @ real
@ ^ [N5: nat] : ( root @ N5 @ ( semiring_1_of_nat @ real @ N5 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( one_one @ real ) )
@ ( at_top @ nat ) ) ).
% LIMSEQ_root
thf(fact_5958_lim__const__over__n,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [A3: A] :
( filterlim @ nat @ A
@ ^ [N5: nat] : ( divide_divide @ A @ A3 @ ( semiring_1_of_nat @ A @ N5 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ ( at_top @ nat ) ) ) ).
% lim_const_over_n
thf(fact_5959_lim__inverse__n,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ( filterlim @ nat @ A
@ ^ [N5: nat] : ( inverse_inverse @ A @ ( semiring_1_of_nat @ A @ N5 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ ( at_top @ nat ) ) ) ).
% lim_inverse_n
thf(fact_5960_LIMSEQ__linear,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [X8: nat > A,X: A,L: nat] :
( ( filterlim @ nat @ A @ X8 @ ( topolo7230453075368039082e_nhds @ A @ X ) @ ( at_top @ nat ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ L )
=> ( filterlim @ nat @ A
@ ^ [N5: nat] : ( X8 @ ( times_times @ nat @ N5 @ L ) )
@ ( topolo7230453075368039082e_nhds @ A @ X )
@ ( at_top @ nat ) ) ) ) ) ).
% LIMSEQ_linear
thf(fact_5961_telescope__summable,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A,C2: A] :
( ( filterlim @ nat @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ C2 ) @ ( at_top @ nat ) )
=> ( summable @ A
@ ^ [N5: nat] : ( minus_minus @ A @ ( F2 @ ( suc @ N5 ) ) @ ( F2 @ N5 ) ) ) ) ) ).
% telescope_summable
thf(fact_5962_telescope__summable_H,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A,C2: A] :
( ( filterlim @ nat @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ C2 ) @ ( at_top @ nat ) )
=> ( summable @ A
@ ^ [N5: nat] : ( minus_minus @ A @ ( F2 @ N5 ) @ ( F2 @ ( suc @ N5 ) ) ) ) ) ) ).
% telescope_summable'
thf(fact_5963_nested__sequence__unique,axiom,
! [F2: nat > real,G: nat > real] :
( ! [N2: nat] : ( ord_less_eq @ real @ ( F2 @ N2 ) @ ( F2 @ ( suc @ N2 ) ) )
=> ( ! [N2: nat] : ( ord_less_eq @ real @ ( G @ ( suc @ N2 ) ) @ ( G @ N2 ) )
=> ( ! [N2: nat] : ( ord_less_eq @ real @ ( F2 @ N2 ) @ ( G @ N2 ) )
=> ( ( filterlim @ nat @ real
@ ^ [N5: nat] : ( minus_minus @ real @ ( F2 @ N5 ) @ ( G @ N5 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( at_top @ nat ) )
=> ? [L3: real] :
( ! [N4: nat] : ( ord_less_eq @ real @ ( F2 @ N4 ) @ L3 )
& ( filterlim @ nat @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ L3 ) @ ( at_top @ nat ) )
& ! [N4: nat] : ( ord_less_eq @ real @ L3 @ ( G @ N4 ) )
& ( filterlim @ nat @ real @ G @ ( topolo7230453075368039082e_nhds @ real @ L3 ) @ ( at_top @ nat ) ) ) ) ) ) ) ).
% nested_sequence_unique
thf(fact_5964_lim__inverse__n_H,axiom,
( filterlim @ nat @ real
@ ^ [N5: nat] : ( divide_divide @ real @ ( one_one @ real ) @ ( semiring_1_of_nat @ real @ N5 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( at_top @ nat ) ) ).
% lim_inverse_n'
thf(fact_5965_LIMSEQ__root__const,axiom,
! [C2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ C2 )
=> ( filterlim @ nat @ real
@ ^ [N5: nat] : ( root @ N5 @ C2 )
@ ( topolo7230453075368039082e_nhds @ real @ ( one_one @ real ) )
@ ( at_top @ nat ) ) ) ).
% LIMSEQ_root_const
thf(fact_5966_LIMSEQ__inverse__real__of__nat,axiom,
( filterlim @ nat @ real
@ ^ [N5: nat] : ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ ( suc @ N5 ) ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( at_top @ nat ) ) ).
% LIMSEQ_inverse_real_of_nat
thf(fact_5967_LIMSEQ__inverse__real__of__nat__add,axiom,
! [R4: real] :
( filterlim @ nat @ real
@ ^ [N5: nat] : ( plus_plus @ real @ R4 @ ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ ( suc @ N5 ) ) ) )
@ ( topolo7230453075368039082e_nhds @ real @ R4 )
@ ( at_top @ nat ) ) ).
% LIMSEQ_inverse_real_of_nat_add
thf(fact_5968_increasing__LIMSEQ,axiom,
! [F2: nat > real,L: real] :
( ! [N2: nat] : ( ord_less_eq @ real @ ( F2 @ N2 ) @ ( F2 @ ( suc @ N2 ) ) )
=> ( ! [N2: nat] : ( ord_less_eq @ real @ ( F2 @ N2 ) @ L )
=> ( ! [E: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E )
=> ? [N4: nat] : ( ord_less_eq @ real @ L @ ( plus_plus @ real @ ( F2 @ N4 ) @ E ) ) )
=> ( filterlim @ nat @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ L ) @ ( at_top @ nat ) ) ) ) ) ).
% increasing_LIMSEQ
thf(fact_5969_lim__1__over__n,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ( filterlim @ nat @ A
@ ^ [N5: nat] : ( divide_divide @ A @ ( one_one @ A ) @ ( semiring_1_of_nat @ A @ N5 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ ( at_top @ nat ) ) ) ).
% lim_1_over_n
thf(fact_5970_LIMSEQ__n__over__Suc__n,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ( filterlim @ nat @ A
@ ^ [N5: nat] : ( divide_divide @ A @ ( semiring_1_of_nat @ A @ N5 ) @ ( semiring_1_of_nat @ A @ ( suc @ N5 ) ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( one_one @ A ) )
@ ( at_top @ nat ) ) ) ).
% LIMSEQ_n_over_Suc_n
thf(fact_5971_LIMSEQ__Suc__n__over__n,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ( filterlim @ nat @ A
@ ^ [N5: nat] : ( divide_divide @ A @ ( semiring_1_of_nat @ A @ ( suc @ N5 ) ) @ ( semiring_1_of_nat @ A @ N5 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( one_one @ A ) )
@ ( at_top @ nat ) ) ) ).
% LIMSEQ_Suc_n_over_n
thf(fact_5972_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_5973_telescope__sums_H,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A,C2: A] :
( ( filterlim @ nat @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ C2 ) @ ( at_top @ nat ) )
=> ( sums @ A
@ ^ [N5: nat] : ( minus_minus @ A @ ( F2 @ N5 ) @ ( F2 @ ( suc @ N5 ) ) )
@ ( minus_minus @ A @ ( F2 @ ( zero_zero @ nat ) ) @ C2 ) ) ) ) ).
% telescope_sums'
thf(fact_5974_telescope__sums,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A,C2: A] :
( ( filterlim @ nat @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ C2 ) @ ( at_top @ nat ) )
=> ( sums @ A
@ ^ [N5: nat] : ( minus_minus @ A @ ( F2 @ ( suc @ N5 ) ) @ ( F2 @ N5 ) )
@ ( minus_minus @ A @ C2 @ ( F2 @ ( zero_zero @ nat ) ) ) ) ) ) ).
% telescope_sums
thf(fact_5975_LIMSEQ__divide__realpow__zero,axiom,
! [X: real,A3: real] :
( ( ord_less @ real @ ( one_one @ real ) @ X )
=> ( filterlim @ nat @ real
@ ^ [N5: nat] : ( divide_divide @ real @ A3 @ ( power_power @ real @ X @ N5 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( at_top @ nat ) ) ) ).
% LIMSEQ_divide_realpow_zero
thf(fact_5976_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_5977_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_5978_LIMSEQ__inverse__realpow__zero,axiom,
! [X: real] :
( ( ord_less @ real @ ( one_one @ real ) @ X )
=> ( filterlim @ nat @ real
@ ^ [N5: nat] : ( inverse_inverse @ real @ ( power_power @ real @ X @ N5 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( at_top @ nat ) ) ) ).
% LIMSEQ_inverse_realpow_zero
thf(fact_5979_sums__def_H,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ( ( sums @ A )
= ( ^ [F4: nat > A,S5: A] :
( filterlim @ nat @ A
@ ^ [N5: nat] : ( groups7311177749621191930dd_sum @ nat @ A @ F4 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N5 ) )
@ ( topolo7230453075368039082e_nhds @ A @ S5 )
@ ( at_top @ nat ) ) ) ) ) ).
% sums_def'
thf(fact_5980_root__test__convergence,axiom,
! [A: $tType] :
( ( real_Vector_banach @ A )
=> ! [F2: nat > A,X: real] :
( ( filterlim @ nat @ real
@ ^ [N5: nat] : ( root @ N5 @ ( real_V7770717601297561774m_norm @ A @ ( F2 @ N5 ) ) )
@ ( topolo7230453075368039082e_nhds @ real @ X )
@ ( at_top @ nat ) )
=> ( ( ord_less @ real @ X @ ( one_one @ real ) )
=> ( summable @ A @ F2 ) ) ) ) ).
% root_test_convergence
thf(fact_5981_LIMSEQ__inverse__real__of__nat__add__minus,axiom,
! [R4: real] :
( filterlim @ nat @ real
@ ^ [N5: nat] : ( plus_plus @ real @ R4 @ ( uminus_uminus @ real @ ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ ( suc @ N5 ) ) ) ) )
@ ( topolo7230453075368039082e_nhds @ real @ R4 )
@ ( at_top @ nat ) ) ).
% LIMSEQ_inverse_real_of_nat_add_minus
thf(fact_5982_LIMSEQ__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X8: nat > A,L4: A] :
( ( filterlim @ nat @ A @ X8 @ ( topolo7230453075368039082e_nhds @ A @ L4 ) @ ( at_top @ nat ) )
= ( ! [R: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R )
=> ? [No: nat] :
! [N5: nat] :
( ( ord_less_eq @ nat @ No @ N5 )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( X8 @ N5 ) @ L4 ) ) @ R ) ) ) ) ) ) ).
% LIMSEQ_iff
thf(fact_5983_LIMSEQ__I,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X8: nat > A,L4: A] :
( ! [R3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R3 )
=> ? [No2: nat] :
! [N2: nat] :
( ( ord_less_eq @ nat @ No2 @ N2 )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( X8 @ N2 ) @ L4 ) ) @ R3 ) ) )
=> ( filterlim @ nat @ A @ X8 @ ( topolo7230453075368039082e_nhds @ A @ L4 ) @ ( at_top @ nat ) ) ) ) ).
% LIMSEQ_I
thf(fact_5984_LIMSEQ__D,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X8: nat > A,L4: A,R4: real] :
( ( filterlim @ nat @ A @ X8 @ ( topolo7230453075368039082e_nhds @ A @ L4 ) @ ( at_top @ nat ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ R4 )
=> ? [No3: nat] :
! [N4: nat] :
( ( ord_less_eq @ nat @ No3 @ N4 )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( X8 @ N4 ) @ L4 ) ) @ R4 ) ) ) ) ) ).
% LIMSEQ_D
thf(fact_5985_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_5986_tendsto__exp__limit__sequentially,axiom,
! [X: real] :
( filterlim @ nat @ real
@ ^ [N5: nat] : ( power_power @ real @ ( plus_plus @ real @ ( one_one @ real ) @ ( divide_divide @ real @ X @ ( semiring_1_of_nat @ real @ N5 ) ) ) @ N5 )
@ ( topolo7230453075368039082e_nhds @ real @ ( exp @ real @ X ) )
@ ( at_top @ nat ) ) ).
% tendsto_exp_limit_sequentially
thf(fact_5987_tendsto__at__iff__sequentially,axiom,
! [C: $tType,A: $tType] :
( ( ( topolo3112930676232923870pology @ A )
& ( topolo4958980785337419405_space @ C ) )
=> ! [F2: A > C,A3: C,X: A,S: set @ A] :
( ( filterlim @ A @ C @ F2 @ ( topolo7230453075368039082e_nhds @ C @ A3 ) @ ( topolo174197925503356063within @ A @ X @ S ) )
= ( ! [X7: nat > A] :
( ! [I2: nat] : ( member @ A @ ( X7 @ I2 ) @ ( minus_minus @ ( set @ A ) @ S @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( ( filterlim @ nat @ A @ X7 @ ( topolo7230453075368039082e_nhds @ A @ X ) @ ( at_top @ nat ) )
=> ( filterlim @ nat @ C @ ( comp @ A @ C @ nat @ F2 @ X7 ) @ ( topolo7230453075368039082e_nhds @ C @ A3 ) @ ( at_top @ nat ) ) ) ) ) ) ) ).
% tendsto_at_iff_sequentially
thf(fact_5988_tendsto__power__zero,axiom,
! [A: $tType,B: $tType] :
( ( real_V2822296259951069270ebra_1 @ A )
=> ! [F2: B > nat,F5: filter @ B,X: A] :
( ( filterlim @ B @ nat @ F2 @ ( at_top @ nat ) @ F5 )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ ( one_one @ real ) )
=> ( filterlim @ B @ A
@ ^ [Y5: B] : ( power_power @ A @ X @ ( F2 @ Y5 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ F5 ) ) ) ) ).
% tendsto_power_zero
thf(fact_5989_LIMSEQ__inverse__real__of__nat__add__minus__mult,axiom,
! [R4: real] :
( filterlim @ nat @ real
@ ^ [N5: nat] : ( times_times @ real @ R4 @ ( plus_plus @ real @ ( one_one @ real ) @ ( uminus_uminus @ real @ ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ ( suc @ N5 ) ) ) ) ) )
@ ( topolo7230453075368039082e_nhds @ real @ R4 )
@ ( at_top @ nat ) ) ).
% LIMSEQ_inverse_real_of_nat_add_minus_mult
thf(fact_5990_LIMSEQ__norm__0,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A] :
( ! [N2: nat] : ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( F2 @ N2 ) ) @ ( divide_divide @ real @ ( one_one @ real ) @ ( semiring_1_of_nat @ real @ ( suc @ N2 ) ) ) )
=> ( filterlim @ nat @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ ( at_top @ nat ) ) ) ) ).
% LIMSEQ_norm_0
thf(fact_5991_summable__Leibniz_I1_J,axiom,
! [A3: nat > real] :
( ( filterlim @ nat @ real @ A3 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) )
=> ( ( topological_monoseq @ real @ A3 )
=> ( summable @ real
@ ^ [N5: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ N5 ) @ ( A3 @ N5 ) ) ) ) ) ).
% summable_Leibniz(1)
thf(fact_5992_field__derivative__lim__unique,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,Df: A,Z2: A,S: nat > A,A3: A] :
( ( has_field_derivative @ A @ F2 @ Df @ ( topolo174197925503356063within @ A @ Z2 @ ( top_top @ ( set @ A ) ) ) )
=> ( ( filterlim @ nat @ A @ S @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ ( at_top @ nat ) )
=> ( ! [N2: nat] :
( ( S @ N2 )
!= ( zero_zero @ A ) )
=> ( ( filterlim @ nat @ A
@ ^ [N5: nat] : ( divide_divide @ A @ ( minus_minus @ A @ ( F2 @ ( plus_plus @ A @ Z2 @ ( S @ N5 ) ) ) @ ( F2 @ Z2 ) ) @ ( S @ N5 ) )
@ ( topolo7230453075368039082e_nhds @ A @ A3 )
@ ( at_top @ nat ) )
=> ( Df = A3 ) ) ) ) ) ) ).
% field_derivative_lim_unique
thf(fact_5993_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
@ ^ [N5: nat] : ( times_times @ A @ ( semiring_1_of_nat @ A @ N5 ) @ ( power_power @ A @ X @ N5 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ ( at_top @ nat ) ) ) ) ).
% powser_times_n_limit_0
thf(fact_5994_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
@ ^ [N5: nat] : ( divide_divide @ A @ ( semiring_1_of_nat @ A @ N5 ) @ ( power_power @ A @ X @ N5 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ ( at_top @ nat ) ) ) ) ).
% lim_n_over_pown
thf(fact_5995_summable,axiom,
! [A3: nat > real] :
( ( filterlim @ nat @ real @ A3 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) )
=> ( ! [N2: nat] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( A3 @ N2 ) )
=> ( ! [N2: nat] : ( ord_less_eq @ real @ ( A3 @ ( suc @ N2 ) ) @ ( A3 @ N2 ) )
=> ( summable @ real
@ ^ [N5: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ N5 ) @ ( A3 @ N5 ) ) ) ) ) ) ).
% summable
thf(fact_5996_cos__diff__limit__1,axiom,
! [Theta: nat > real,Theta2: real] :
( ( filterlim @ nat @ real
@ ^ [J3: nat] : ( cos @ real @ ( minus_minus @ real @ ( Theta @ J3 ) @ Theta2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( one_one @ real ) )
@ ( at_top @ nat ) )
=> ~ ! [K2: nat > int] :
~ ( filterlim @ nat @ real
@ ^ [J3: nat] : ( minus_minus @ real @ ( Theta @ J3 ) @ ( times_times @ real @ ( ring_1_of_int @ real @ ( K2 @ J3 ) ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) ) )
@ ( topolo7230453075368039082e_nhds @ real @ Theta2 )
@ ( at_top @ nat ) ) ) ).
% cos_diff_limit_1
thf(fact_5997_cos__limit__1,axiom,
! [Theta: nat > real] :
( ( filterlim @ nat @ real
@ ^ [J3: nat] : ( cos @ real @ ( Theta @ J3 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( one_one @ real ) )
@ ( at_top @ nat ) )
=> ? [K2: nat > int] :
( filterlim @ nat @ real
@ ^ [J3: nat] : ( minus_minus @ real @ ( Theta @ J3 ) @ ( times_times @ real @ ( ring_1_of_int @ real @ ( K2 @ J3 ) ) @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( at_top @ nat ) ) ) ).
% cos_limit_1
thf(fact_5998_summable__Leibniz_I4_J,axiom,
! [A3: nat > real] :
( ( filterlim @ nat @ real @ A3 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) )
=> ( ( topological_monoseq @ real @ A3 )
=> ( filterlim @ nat @ real
@ ^ [N5: nat] :
( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I2: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I2 ) @ ( A3 @ I2 ) )
@ ( set_ord_lessThan @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N5 ) ) )
@ ( topolo7230453075368039082e_nhds @ real
@ ( suminf @ real
@ ^ [I2: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I2 ) @ ( A3 @ I2 ) ) ) )
@ ( at_top @ nat ) ) ) ) ).
% summable_Leibniz(4)
thf(fact_5999_zeroseq__arctan__series,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ X ) @ ( one_one @ real ) )
=> ( filterlim @ nat @ real
@ ^ [N5: nat] : ( times_times @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( semiring_1_of_nat @ real @ ( plus_plus @ nat @ ( times_times @ nat @ N5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) @ ( power_power @ real @ X @ ( plus_plus @ nat @ ( times_times @ nat @ N5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( at_top @ nat ) ) ) ).
% zeroseq_arctan_series
thf(fact_6000_summable__Leibniz_H_I3_J,axiom,
! [A3: nat > real] :
( ( filterlim @ nat @ real @ A3 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) )
=> ( ! [N2: nat] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( A3 @ N2 ) )
=> ( ! [N2: nat] : ( ord_less_eq @ real @ ( A3 @ ( suc @ N2 ) ) @ ( A3 @ N2 ) )
=> ( filterlim @ nat @ real
@ ^ [N5: nat] :
( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I2: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I2 ) @ ( A3 @ I2 ) )
@ ( set_ord_lessThan @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N5 ) ) )
@ ( topolo7230453075368039082e_nhds @ real
@ ( suminf @ real
@ ^ [I2: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I2 ) @ ( A3 @ I2 ) ) ) )
@ ( at_top @ nat ) ) ) ) ) ).
% summable_Leibniz'(3)
thf(fact_6001_summable__Leibniz_H_I2_J,axiom,
! [A3: nat > real,N: nat] :
( ( filterlim @ nat @ real @ A3 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) )
=> ( ! [N2: nat] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( A3 @ N2 ) )
=> ( ! [N2: nat] : ( ord_less_eq @ real @ ( A3 @ ( suc @ N2 ) ) @ ( A3 @ N2 ) )
=> ( ord_less_eq @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I2: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I2 ) @ ( A3 @ I2 ) )
@ ( set_ord_lessThan @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) )
@ ( suminf @ real
@ ^ [I2: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I2 ) @ ( A3 @ I2 ) ) ) ) ) ) ) ).
% summable_Leibniz'(2)
thf(fact_6002_sums__alternating__upper__lower,axiom,
! [A3: nat > real] :
( ! [N2: nat] : ( ord_less_eq @ real @ ( A3 @ ( suc @ N2 ) ) @ ( A3 @ N2 ) )
=> ( ! [N2: nat] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( A3 @ N2 ) )
=> ( ( filterlim @ nat @ real @ A3 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) )
=> ? [L3: real] :
( ! [N4: nat] :
( ord_less_eq @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I2: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I2 ) @ ( A3 @ I2 ) )
@ ( set_ord_lessThan @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N4 ) ) )
@ L3 )
& ( filterlim @ nat @ real
@ ^ [N5: nat] :
( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I2: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I2 ) @ ( A3 @ I2 ) )
@ ( set_ord_lessThan @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N5 ) ) )
@ ( topolo7230453075368039082e_nhds @ real @ L3 )
@ ( at_top @ nat ) )
& ! [N4: nat] :
( ord_less_eq @ real @ L3
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I2: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I2 ) @ ( A3 @ I2 ) )
@ ( set_ord_lessThan @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N4 ) @ ( one_one @ nat ) ) ) ) )
& ( filterlim @ nat @ real
@ ^ [N5: nat] :
( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I2: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I2 ) @ ( A3 @ I2 ) )
@ ( set_ord_lessThan @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N5 ) @ ( one_one @ nat ) ) ) )
@ ( topolo7230453075368039082e_nhds @ real @ L3 )
@ ( at_top @ nat ) ) ) ) ) ) ).
% sums_alternating_upper_lower
thf(fact_6003_summable__Leibniz_I5_J,axiom,
! [A3: nat > real] :
( ( filterlim @ nat @ real @ A3 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) )
=> ( ( topological_monoseq @ real @ A3 )
=> ( filterlim @ nat @ real
@ ^ [N5: nat] :
( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I2: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I2 ) @ ( A3 @ I2 ) )
@ ( set_ord_lessThan @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N5 ) @ ( one_one @ nat ) ) ) )
@ ( topolo7230453075368039082e_nhds @ real
@ ( suminf @ real
@ ^ [I2: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I2 ) @ ( A3 @ I2 ) ) ) )
@ ( at_top @ nat ) ) ) ) ).
% summable_Leibniz(5)
thf(fact_6004_summable__Leibniz_H_I5_J,axiom,
! [A3: nat > real] :
( ( filterlim @ nat @ real @ A3 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) )
=> ( ! [N2: nat] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( A3 @ N2 ) )
=> ( ! [N2: nat] : ( ord_less_eq @ real @ ( A3 @ ( suc @ N2 ) ) @ ( A3 @ N2 ) )
=> ( filterlim @ nat @ real
@ ^ [N5: nat] :
( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I2: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I2 ) @ ( A3 @ I2 ) )
@ ( set_ord_lessThan @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N5 ) @ ( one_one @ nat ) ) ) )
@ ( topolo7230453075368039082e_nhds @ real
@ ( suminf @ real
@ ^ [I2: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I2 ) @ ( A3 @ I2 ) ) ) )
@ ( at_top @ nat ) ) ) ) ) ).
% summable_Leibniz'(5)
thf(fact_6005_filterlim__sequentially__Suc,axiom,
! [A: $tType,F2: nat > A,F5: filter @ A] :
( ( filterlim @ nat @ A
@ ^ [X2: nat] : ( F2 @ ( suc @ X2 ) )
@ F5
@ ( at_top @ nat ) )
= ( filterlim @ nat @ A @ F2 @ F5 @ ( at_top @ nat ) ) ) ).
% filterlim_sequentially_Suc
thf(fact_6006_summable__bounded__partials,axiom,
! [A: $tType] :
( ( ( real_V8037385150606011577_space @ A )
& ( real_V822414075346904944vector @ A ) )
=> ! [F2: nat > A,G: nat > real] :
( ( eventually @ nat
@ ^ [X02: nat] :
! [A5: nat] :
( ( ord_less_eq @ nat @ X02 @ A5 )
=> ! [B3: nat] :
( ( ord_less @ nat @ A5 @ B3 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F2 @ ( set_or3652927894154168847AtMost @ nat @ A5 @ B3 ) ) ) @ ( G @ A5 ) ) ) )
@ ( at_top @ nat ) )
=> ( ( filterlim @ nat @ real @ G @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) )
=> ( summable @ A @ F2 ) ) ) ) ).
% summable_bounded_partials
thf(fact_6007_has__derivative__at2,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,F6: A > B,X: A] :
( ( has_derivative @ A @ B @ F2 @ F6 @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) )
= ( ( real_V3181309239436604168linear @ A @ B @ F6 )
& ( filterlim @ A @ B
@ ^ [Y5: A] : ( real_V8093663219630862766scaleR @ B @ ( divide_divide @ real @ ( one_one @ real ) @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ Y5 @ X ) ) ) @ ( minus_minus @ B @ ( F2 @ Y5 ) @ ( plus_plus @ B @ ( F2 @ X ) @ ( F6 @ ( minus_minus @ A @ Y5 @ X ) ) ) ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% has_derivative_at2
thf(fact_6008_eventually__sequentially__Suc,axiom,
! [P: nat > $o] :
( ( eventually @ nat
@ ^ [I2: nat] : ( P @ ( suc @ I2 ) )
@ ( at_top @ nat ) )
= ( eventually @ nat @ P @ ( at_top @ nat ) ) ) ).
% eventually_sequentially_Suc
thf(fact_6009_filterlim__at__top__dense,axiom,
! [A: $tType,B: $tType] :
( ( unboun7993243217541854897norder @ B )
=> ! [F2: A > B,F5: filter @ A] :
( ( filterlim @ A @ B @ F2 @ ( at_top @ B ) @ F5 )
= ( ! [Z8: B] :
( eventually @ A
@ ^ [X2: A] : ( ord_less @ B @ Z8 @ ( F2 @ X2 ) )
@ F5 ) ) ) ) ).
% filterlim_at_top_dense
thf(fact_6010_eventually__at__top__dense,axiom,
! [A: $tType] :
( ( ( linorder @ A )
& ( no_top @ A ) )
=> ! [P: A > $o] :
( ( eventually @ A @ P @ ( at_top @ A ) )
= ( ? [N7: A] :
! [N5: A] :
( ( ord_less @ A @ N7 @ N5 )
=> ( P @ N5 ) ) ) ) ) ).
% eventually_at_top_dense
thf(fact_6011_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_6012_bounded__linear__zero,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ( real_V3181309239436604168linear @ A @ B
@ ^ [X2: A] : ( zero_zero @ B ) ) ) ).
% bounded_linear_zero
thf(fact_6013_bounded__linear__minus,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B] :
( ( real_V3181309239436604168linear @ A @ B @ F2 )
=> ( real_V3181309239436604168linear @ A @ B
@ ^ [X2: A] : ( uminus_uminus @ B @ ( F2 @ X2 ) ) ) ) ) ).
% bounded_linear_minus
thf(fact_6014_bounded__linear__divide,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [Y2: A] :
( real_V3181309239436604168linear @ A @ A
@ ^ [X2: A] : ( divide_divide @ A @ X2 @ Y2 ) ) ) ).
% bounded_linear_divide
thf(fact_6015_bounded__linear__sub,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,G: A > B] :
( ( real_V3181309239436604168linear @ A @ B @ F2 )
=> ( ( real_V3181309239436604168linear @ A @ B @ G )
=> ( real_V3181309239436604168linear @ A @ B
@ ^ [X2: A] : ( minus_minus @ B @ ( F2 @ X2 ) @ ( G @ X2 ) ) ) ) ) ) ).
% bounded_linear_sub
thf(fact_6016_eventually__nhds__top,axiom,
! [A: $tType] :
( ( ( order_top @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [B2: A,P: A > $o] :
( ( ord_less @ A @ B2 @ ( top_top @ A ) )
=> ( ( eventually @ A @ P @ ( topolo7230453075368039082e_nhds @ A @ ( top_top @ A ) ) )
= ( ? [B3: A] :
( ( ord_less @ A @ B3 @ ( top_top @ A ) )
& ! [Z6: A] :
( ( ord_less @ A @ B3 @ Z6 )
=> ( P @ Z6 ) ) ) ) ) ) ) ).
% eventually_nhds_top
thf(fact_6017_eventually__at__left__field,axiom,
! [A: $tType] :
( ( ( linordered_field @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [P: A > $o,X: A] :
( ( eventually @ A @ P @ ( topolo174197925503356063within @ A @ X @ ( set_ord_lessThan @ A @ X ) ) )
= ( ? [B3: A] :
( ( ord_less @ A @ B3 @ X )
& ! [Y5: A] :
( ( ord_less @ A @ B3 @ Y5 )
=> ( ( ord_less @ A @ Y5 @ X )
=> ( P @ Y5 ) ) ) ) ) ) ) ).
% eventually_at_left_field
thf(fact_6018_eventually__at__left,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [Y2: A,X: A,P: A > $o] :
( ( ord_less @ A @ Y2 @ X )
=> ( ( eventually @ A @ P @ ( topolo174197925503356063within @ A @ X @ ( set_ord_lessThan @ A @ X ) ) )
= ( ? [B3: A] :
( ( ord_less @ A @ B3 @ X )
& ! [Y5: A] :
( ( ord_less @ A @ B3 @ Y5 )
=> ( ( ord_less @ A @ Y5 @ X )
=> ( P @ Y5 ) ) ) ) ) ) ) ) ).
% eventually_at_left
thf(fact_6019_filterlim__at__top__gt,axiom,
! [A: $tType,B: $tType] :
( ( unboun7993243217541854897norder @ B )
=> ! [F2: A > B,F5: filter @ A,C2: B] :
( ( filterlim @ A @ B @ F2 @ ( at_top @ B ) @ F5 )
= ( ! [Z8: B] :
( ( ord_less @ B @ C2 @ Z8 )
=> ( eventually @ A
@ ^ [X2: A] : ( ord_less_eq @ B @ Z8 @ ( F2 @ X2 ) )
@ F5 ) ) ) ) ) ).
% filterlim_at_top_gt
thf(fact_6020_order__tendsto__iff,axiom,
! [B: $tType,A: $tType] :
( ( topolo2564578578187576103pology @ A )
=> ! [F2: B > A,X: A,F5: filter @ B] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ X ) @ F5 )
= ( ! [L2: A] :
( ( ord_less @ A @ L2 @ X )
=> ( eventually @ B
@ ^ [X2: B] : ( ord_less @ A @ L2 @ ( F2 @ X2 ) )
@ F5 ) )
& ! [U2: A] :
( ( ord_less @ A @ X @ U2 )
=> ( eventually @ B
@ ^ [X2: B] : ( ord_less @ A @ ( F2 @ X2 ) @ U2 )
@ F5 ) ) ) ) ) ).
% order_tendsto_iff
thf(fact_6021_order__tendstoI,axiom,
! [A: $tType,B: $tType] :
( ( topolo2564578578187576103pology @ A )
=> ! [Y2: A,F2: B > A,F5: filter @ B] :
( ! [A6: A] :
( ( ord_less @ A @ A6 @ Y2 )
=> ( eventually @ B
@ ^ [X2: B] : ( ord_less @ A @ A6 @ ( F2 @ X2 ) )
@ F5 ) )
=> ( ! [A6: A] :
( ( ord_less @ A @ Y2 @ A6 )
=> ( eventually @ B
@ ^ [X2: B] : ( ord_less @ A @ ( F2 @ X2 ) @ A6 )
@ F5 ) )
=> ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ Y2 ) @ F5 ) ) ) ) ).
% order_tendstoI
thf(fact_6022_order__tendstoD_I1_J,axiom,
! [A: $tType,B: $tType] :
( ( topolo2564578578187576103pology @ A )
=> ! [F2: B > A,Y2: A,F5: filter @ B,A3: A] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ Y2 ) @ F5 )
=> ( ( ord_less @ A @ A3 @ Y2 )
=> ( eventually @ B
@ ^ [X2: B] : ( ord_less @ A @ A3 @ ( F2 @ X2 ) )
@ F5 ) ) ) ) ).
% order_tendstoD(1)
thf(fact_6023_order__tendstoD_I2_J,axiom,
! [A: $tType,B: $tType] :
( ( topolo2564578578187576103pology @ A )
=> ! [F2: B > A,Y2: A,F5: filter @ B,A3: A] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ Y2 ) @ F5 )
=> ( ( ord_less @ A @ Y2 @ A3 )
=> ( eventually @ B
@ ^ [X2: B] : ( ord_less @ A @ ( F2 @ X2 ) @ A3 )
@ F5 ) ) ) ) ).
% order_tendstoD(2)
thf(fact_6024_bounded__linear_Otendsto__zero,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,G: C > A,F5: filter @ C] :
( ( real_V3181309239436604168linear @ A @ B @ F2 )
=> ( ( filterlim @ C @ A @ G @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ F5 )
=> ( filterlim @ C @ B
@ ^ [X2: C] : ( F2 @ ( G @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ F5 ) ) ) ) ).
% bounded_linear.tendsto_zero
thf(fact_6025_eventually__at__leftI,axiom,
! [A: $tType] :
( ( topolo2564578578187576103pology @ A )
=> ! [A3: A,B2: A,P: A > $o] :
( ! [X3: A] :
( ( member @ A @ X3 @ ( set_or5935395276787703475ssThan @ A @ A3 @ B2 ) )
=> ( P @ X3 ) )
=> ( ( ord_less @ A @ A3 @ B2 )
=> ( eventually @ A @ P @ ( topolo174197925503356063within @ A @ B2 @ ( set_ord_lessThan @ A @ B2 ) ) ) ) ) ) ).
% eventually_at_leftI
thf(fact_6026_eventually__at__to__0,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [P: A > $o,A3: A] :
( ( eventually @ A @ P @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) )
= ( eventually @ A
@ ^ [X2: A] : ( P @ ( plus_plus @ A @ X2 @ A3 ) )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% eventually_at_to_0
thf(fact_6027_decreasing__tendsto,axiom,
! [A: $tType,B: $tType] :
( ( topolo2564578578187576103pology @ A )
=> ! [L: A,F2: B > A,F5: filter @ B] :
( ( eventually @ B
@ ^ [N5: B] : ( ord_less_eq @ A @ L @ ( F2 @ N5 ) )
@ F5 )
=> ( ! [X3: A] :
( ( ord_less @ A @ L @ X3 )
=> ( eventually @ B
@ ^ [N5: B] : ( ord_less @ A @ ( F2 @ N5 ) @ X3 )
@ F5 ) )
=> ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F5 ) ) ) ) ).
% decreasing_tendsto
thf(fact_6028_increasing__tendsto,axiom,
! [A: $tType,B: $tType] :
( ( topolo2564578578187576103pology @ A )
=> ! [F2: B > A,L: A,F5: filter @ B] :
( ( eventually @ B
@ ^ [N5: B] : ( ord_less_eq @ A @ ( F2 @ N5 ) @ L )
@ F5 )
=> ( ! [X3: A] :
( ( ord_less @ A @ X3 @ L )
=> ( eventually @ B
@ ^ [N5: B] : ( ord_less @ A @ X3 @ ( F2 @ N5 ) )
@ F5 ) )
=> ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F5 ) ) ) ) ).
% increasing_tendsto
thf(fact_6029_continuous__arcosh__strong,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [F5: filter @ A,F2: A > real] :
( ( topolo3448309680560233919inuous @ A @ real @ F5 @ F2 )
=> ( ( eventually @ A
@ ^ [X2: A] : ( ord_less_eq @ real @ ( one_one @ real ) @ ( F2 @ X2 ) )
@ F5 )
=> ( topolo3448309680560233919inuous @ A @ real @ F5
@ ^ [X2: A] : ( arcosh @ real @ ( F2 @ X2 ) ) ) ) ) ) ).
% continuous_arcosh_strong
thf(fact_6030_tendsto__imp__filterlim__at__left,axiom,
! [B: $tType,A: $tType] :
( ( topolo2564578578187576103pology @ B )
=> ! [F2: A > B,L4: B,F5: filter @ A] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L4 ) @ F5 )
=> ( ( eventually @ A
@ ^ [X2: A] : ( ord_less @ B @ ( F2 @ X2 ) @ L4 )
@ F5 )
=> ( filterlim @ A @ B @ F2 @ ( topolo174197925503356063within @ B @ L4 @ ( set_ord_lessThan @ B @ L4 ) ) @ F5 ) ) ) ) ).
% tendsto_imp_filterlim_at_left
thf(fact_6031_tendsto__arcosh__strong,axiom,
! [B: $tType,F2: B > real,A3: real,F5: filter @ B] :
( ( filterlim @ B @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ A3 ) @ F5 )
=> ( ( ord_less_eq @ real @ ( one_one @ real ) @ A3 )
=> ( ( eventually @ B
@ ^ [X2: B] : ( ord_less_eq @ real @ ( one_one @ real ) @ ( F2 @ X2 ) )
@ F5 )
=> ( filterlim @ B @ real
@ ^ [X2: B] : ( arcosh @ real @ ( F2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( arcosh @ real @ A3 ) )
@ F5 ) ) ) ) ).
% tendsto_arcosh_strong
thf(fact_6032_filterlim__at__top__at__left,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo1944317154257567458pology @ A )
& ( linorder @ B ) )
=> ! [Q: A > $o,F2: A > B,P: B > $o,G: B > A,A3: A] :
( ! [X3: A,Y3: A] :
( ( Q @ X3 )
=> ( ( Q @ Y3 )
=> ( ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ord_less_eq @ B @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) ) ) )
=> ( ! [X3: B] :
( ( P @ X3 )
=> ( ( F2 @ ( G @ X3 ) )
= X3 ) )
=> ( ! [X3: B] :
( ( P @ X3 )
=> ( Q @ ( G @ X3 ) ) )
=> ( ( eventually @ A @ Q @ ( topolo174197925503356063within @ A @ A3 @ ( set_ord_lessThan @ A @ A3 ) ) )
=> ( ! [B4: A] :
( ( Q @ B4 )
=> ( ord_less @ A @ B4 @ A3 ) )
=> ( ( eventually @ B @ P @ ( at_top @ B ) )
=> ( filterlim @ A @ B @ F2 @ ( at_top @ B ) @ ( topolo174197925503356063within @ A @ A3 @ ( set_ord_lessThan @ A @ A3 ) ) ) ) ) ) ) ) ) ) ).
% filterlim_at_top_at_left
thf(fact_6033_tendsto__0__le,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,F5: filter @ A,G: A > C,K6: real] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) @ F5 )
=> ( ( eventually @ A
@ ^ [X2: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ C @ ( G @ X2 ) ) @ ( times_times @ real @ ( real_V7770717601297561774m_norm @ B @ ( F2 @ X2 ) ) @ K6 ) )
@ F5 )
=> ( filterlim @ A @ C @ G @ ( topolo7230453075368039082e_nhds @ C @ ( zero_zero @ C ) ) @ F5 ) ) ) ) ).
% tendsto_0_le
thf(fact_6034_filterlim__at__withinI,axiom,
! [A: $tType,B: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F2: B > A,C2: A,F5: filter @ B,A4: set @ A] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ C2 ) @ F5 )
=> ( ( eventually @ B
@ ^ [X2: B] : ( member @ A @ ( F2 @ X2 ) @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ C2 @ ( bot_bot @ ( set @ A ) ) ) ) )
@ F5 )
=> ( filterlim @ B @ A @ F2 @ ( topolo174197925503356063within @ A @ C2 @ A4 ) @ F5 ) ) ) ) ).
% filterlim_at_withinI
thf(fact_6035_tendsto__powr_H,axiom,
! [A: $tType,F2: A > real,A3: real,F5: filter @ A,G: A > real,B2: real] :
( ( filterlim @ A @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ A3 ) @ F5 )
=> ( ( filterlim @ A @ real @ G @ ( topolo7230453075368039082e_nhds @ real @ B2 ) @ F5 )
=> ( ( ( A3
!= ( zero_zero @ real ) )
| ( ( ord_less @ real @ ( zero_zero @ real ) @ B2 )
& ( eventually @ A
@ ^ [X2: A] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( F2 @ X2 ) )
@ F5 ) ) )
=> ( filterlim @ A @ real
@ ^ [X2: A] : ( powr @ real @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( powr @ real @ A3 @ B2 ) )
@ F5 ) ) ) ) ).
% tendsto_powr'
thf(fact_6036_tendsto__powr2,axiom,
! [A: $tType,F2: A > real,A3: real,F5: filter @ A,G: A > real,B2: real] :
( ( filterlim @ A @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ A3 ) @ F5 )
=> ( ( filterlim @ A @ real @ G @ ( topolo7230453075368039082e_nhds @ real @ B2 ) @ F5 )
=> ( ( eventually @ A
@ ^ [X2: A] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( F2 @ X2 ) )
@ F5 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ B2 )
=> ( filterlim @ A @ real
@ ^ [X2: A] : ( powr @ real @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( powr @ real @ A3 @ B2 ) )
@ F5 ) ) ) ) ) ).
% tendsto_powr2
thf(fact_6037_tendsto__zero__powrI,axiom,
! [A: $tType,F2: A > real,F5: filter @ A,G: A > real,B2: real] :
( ( filterlim @ A @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ F5 )
=> ( ( filterlim @ A @ real @ G @ ( topolo7230453075368039082e_nhds @ real @ B2 ) @ F5 )
=> ( ( eventually @ A
@ ^ [X2: A] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( F2 @ X2 ) )
@ F5 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ B2 )
=> ( filterlim @ A @ real
@ ^ [X2: A] : ( powr @ real @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ F5 ) ) ) ) ) ).
% tendsto_zero_powrI
thf(fact_6038_eventually__floor__less,axiom,
! [B: $tType,A: $tType] :
( ( ( archim2362893244070406136eiling @ B )
& ( topolo2564578578187576103pology @ B ) )
=> ! [F2: A > B,L: B,F5: filter @ A] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ F5 )
=> ( ~ ( member @ B @ L @ ( ring_1_Ints @ B ) )
=> ( eventually @ A
@ ^ [X2: A] : ( ord_less @ B @ ( ring_1_of_int @ B @ ( archim6421214686448440834_floor @ B @ L ) ) @ ( F2 @ X2 ) )
@ F5 ) ) ) ) ).
% eventually_floor_less
thf(fact_6039_eventually__less__ceiling,axiom,
! [B: $tType,A: $tType] :
( ( ( archim2362893244070406136eiling @ B )
& ( topolo2564578578187576103pology @ B ) )
=> ! [F2: A > B,L: B,F5: filter @ A] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ F5 )
=> ( ~ ( member @ B @ L @ ( ring_1_Ints @ B ) )
=> ( eventually @ A
@ ^ [X2: A] : ( ord_less @ B @ ( F2 @ X2 ) @ ( ring_1_of_int @ B @ ( archimedean_ceiling @ B @ L ) ) )
@ F5 ) ) ) ) ).
% eventually_less_ceiling
thf(fact_6040_filterlim__Suc,axiom,
filterlim @ nat @ nat @ suc @ ( at_top @ nat ) @ ( at_top @ nat ) ).
% filterlim_Suc
thf(fact_6041_has__derivative__iff__norm,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,F6: A > B,X: A,S: set @ A] :
( ( has_derivative @ A @ B @ F2 @ F6 @ ( topolo174197925503356063within @ A @ X @ S ) )
= ( ( real_V3181309239436604168linear @ A @ B @ F6 )
& ( filterlim @ A @ real
@ ^ [Y5: A] : ( divide_divide @ real @ ( real_V7770717601297561774m_norm @ B @ ( minus_minus @ B @ ( minus_minus @ B @ ( F2 @ Y5 ) @ ( F2 @ X ) ) @ ( F6 @ ( minus_minus @ A @ Y5 @ X ) ) ) ) @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ Y5 @ X ) ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ).
% has_derivative_iff_norm
thf(fact_6042_has__derivative__at__within,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,F6: A > B,X: A,S: set @ A] :
( ( has_derivative @ A @ B @ F2 @ F6 @ ( topolo174197925503356063within @ A @ X @ S ) )
= ( ( real_V3181309239436604168linear @ A @ B @ F6 )
& ( filterlim @ A @ B
@ ^ [Y5: A] : ( real_V8093663219630862766scaleR @ B @ ( inverse_inverse @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ Y5 @ X ) ) ) @ ( minus_minus @ B @ ( minus_minus @ B @ ( F2 @ Y5 ) @ ( F2 @ X ) ) @ ( F6 @ ( minus_minus @ A @ Y5 @ X ) ) ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ).
% has_derivative_at_within
thf(fact_6043_has__derivativeI,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F6: A > B,X: A,F2: A > B,S: set @ A] :
( ( real_V3181309239436604168linear @ A @ B @ F6 )
=> ( ( filterlim @ A @ B
@ ^ [Y5: A] : ( real_V8093663219630862766scaleR @ B @ ( inverse_inverse @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ Y5 @ X ) ) ) @ ( minus_minus @ B @ ( minus_minus @ B @ ( F2 @ Y5 ) @ ( F2 @ X ) ) @ ( F6 @ ( minus_minus @ A @ Y5 @ X ) ) ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( has_derivative @ A @ B @ F2 @ F6 @ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ).
% has_derivativeI
thf(fact_6044_has__derivative__iff__Ex,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,F6: A > B,X: A] :
( ( has_derivative @ A @ B @ F2 @ F6 @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) )
= ( ( real_V3181309239436604168linear @ A @ B @ F6 )
& ? [E3: A > B] :
( ! [H2: A] :
( ( F2 @ ( plus_plus @ A @ X @ H2 ) )
= ( plus_plus @ B @ ( plus_plus @ B @ ( F2 @ X ) @ ( F6 @ H2 ) ) @ ( E3 @ H2 ) ) )
& ( filterlim @ A @ real
@ ^ [H2: A] : ( divide_divide @ real @ ( real_V7770717601297561774m_norm @ B @ ( E3 @ H2 ) ) @ ( real_V7770717601297561774m_norm @ A @ H2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ).
% has_derivative_iff_Ex
thf(fact_6045_has__derivative__within,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,F6: A > B,X: A,S: set @ A] :
( ( has_derivative @ A @ B @ F2 @ F6 @ ( topolo174197925503356063within @ A @ X @ S ) )
= ( ( real_V3181309239436604168linear @ A @ B @ F6 )
& ( filterlim @ A @ B
@ ^ [Y5: A] : ( real_V8093663219630862766scaleR @ B @ ( divide_divide @ real @ ( one_one @ real ) @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ Y5 @ X ) ) ) @ ( minus_minus @ B @ ( F2 @ Y5 ) @ ( plus_plus @ B @ ( F2 @ X ) @ ( F6 @ ( minus_minus @ A @ Y5 @ X ) ) ) ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ).
% has_derivative_within
thf(fact_6046_has__derivative__at,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,D6: A > B,X: A] :
( ( has_derivative @ A @ B @ F2 @ D6 @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) )
= ( ( real_V3181309239436604168linear @ A @ B @ D6 )
& ( filterlim @ A @ real
@ ^ [H2: A] : ( divide_divide @ real @ ( real_V7770717601297561774m_norm @ B @ ( minus_minus @ B @ ( minus_minus @ B @ ( F2 @ ( plus_plus @ A @ X @ H2 ) ) @ ( F2 @ X ) ) @ ( D6 @ H2 ) ) ) @ ( real_V7770717601297561774m_norm @ A @ H2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% has_derivative_at
thf(fact_6047_polyfun__extremal,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [C2: nat > A,K: nat,N: nat,B7: real] :
( ( ( C2 @ K )
!= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ nat @ ( one_one @ nat ) @ K )
=> ( ( ord_less_eq @ nat @ K @ N )
=> ( eventually @ A
@ ^ [Z6: A] :
( ord_less_eq @ real @ B7
@ ( real_V7770717601297561774m_norm @ A
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I2: nat] : ( times_times @ A @ ( C2 @ I2 ) @ ( power_power @ A @ Z6 @ I2 ) )
@ ( set_ord_atMost @ nat @ N ) ) ) )
@ ( at_infinity @ A ) ) ) ) ) ) ).
% polyfun_extremal
thf(fact_6048_has__derivative__def,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ( ( has_derivative @ A @ B )
= ( ^ [F4: A > B,F7: A > B,F8: filter @ A] :
( ( real_V3181309239436604168linear @ A @ B @ F7 )
& ( filterlim @ A @ B
@ ^ [Y5: A] :
( real_V8093663219630862766scaleR @ B
@ ( inverse_inverse @ real
@ ( real_V7770717601297561774m_norm @ A
@ ( minus_minus @ A @ Y5
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F8
@ ^ [X2: A] : X2 ) ) ) )
@ ( minus_minus @ B
@ ( minus_minus @ B @ ( F4 @ Y5 )
@ ( F4
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F8
@ ^ [X2: A] : X2 ) ) )
@ ( F7
@ ( minus_minus @ A @ Y5
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F8
@ ^ [X2: A] : X2 ) ) ) ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ F8 ) ) ) ) ) ).
% has_derivative_def
thf(fact_6049_cosh__real__at__top,axiom,
filterlim @ real @ real @ ( cosh @ real ) @ ( at_top @ real ) @ ( at_top @ real ) ).
% cosh_real_at_top
thf(fact_6050_sinh__real__at__top,axiom,
filterlim @ real @ real @ ( sinh @ real ) @ ( at_top @ real ) @ ( at_top @ real ) ).
% sinh_real_at_top
thf(fact_6051_arcosh__real__at__top,axiom,
filterlim @ real @ real @ ( arcosh @ real ) @ ( at_top @ real ) @ ( at_top @ real ) ).
% arcosh_real_at_top
thf(fact_6052_arsinh__real__at__top,axiom,
filterlim @ real @ real @ ( arsinh @ real ) @ ( at_top @ real ) @ ( at_top @ real ) ).
% arsinh_real_at_top
thf(fact_6053_lhopital__at__top__at__top,axiom,
! [F2: real > real,A3: real,G: real > real,F6: real > real,G3: real > real] :
( ( filterlim @ real @ real @ F2 @ ( at_top @ real ) @ ( topolo174197925503356063within @ real @ A3 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( filterlim @ real @ real @ G @ ( at_top @ real ) @ ( topolo174197925503356063within @ real @ A3 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] : ( has_field_derivative @ real @ F2 @ ( F6 @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ A3 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] : ( has_field_derivative @ real @ G @ ( G3 @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ A3 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( F6 @ X2 ) @ ( G3 @ X2 ) )
@ ( at_top @ real )
@ ( topolo174197925503356063within @ real @ A3 @ ( top_top @ ( set @ real ) ) ) )
=> ( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( at_top @ real )
@ ( topolo174197925503356063within @ real @ A3 @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ) ) ).
% lhopital_at_top_at_top
thf(fact_6054_lhopital__left__at__top__at__top,axiom,
! [F2: real > real,A3: real,G: real > real,F6: real > real,G3: real > real] :
( ( filterlim @ real @ real @ F2 @ ( at_top @ real ) @ ( topolo174197925503356063within @ real @ A3 @ ( set_ord_lessThan @ real @ A3 ) ) )
=> ( ( filterlim @ real @ real @ G @ ( at_top @ real ) @ ( topolo174197925503356063within @ real @ A3 @ ( set_ord_lessThan @ real @ A3 ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] : ( has_field_derivative @ real @ F2 @ ( F6 @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ A3 @ ( set_ord_lessThan @ real @ A3 ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] : ( has_field_derivative @ real @ G @ ( G3 @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ A3 @ ( set_ord_lessThan @ real @ A3 ) ) )
=> ( ( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( F6 @ X2 ) @ ( G3 @ X2 ) )
@ ( at_top @ real )
@ ( topolo174197925503356063within @ real @ A3 @ ( set_ord_lessThan @ real @ A3 ) ) )
=> ( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( at_top @ real )
@ ( topolo174197925503356063within @ real @ A3 @ ( set_ord_lessThan @ real @ A3 ) ) ) ) ) ) ) ) ).
% lhopital_left_at_top_at_top
thf(fact_6055_ln__at__top,axiom,
filterlim @ real @ real @ ( ln_ln @ real ) @ ( at_top @ real ) @ ( at_top @ real ) ).
% ln_at_top
thf(fact_6056_exp__at__top,axiom,
filterlim @ real @ real @ ( exp @ real ) @ ( at_top @ real ) @ ( at_top @ real ) ).
% exp_at_top
thf(fact_6057_filterlim__int__sequentially,axiom,
filterlim @ nat @ int @ ( semiring_1_of_nat @ int ) @ ( at_top @ int ) @ ( at_top @ nat ) ).
% filterlim_int_sequentially
thf(fact_6058_lhospital__at__top__at__top,axiom,
! [G: real > real,G3: real > real,F2: real > real,F6: real > real,X: real] :
( ( filterlim @ real @ real @ G @ ( at_top @ real ) @ ( at_top @ real ) )
=> ( ( eventually @ real
@ ^ [X2: real] :
( ( G3 @ X2 )
!= ( zero_zero @ real ) )
@ ( at_top @ real ) )
=> ( ( eventually @ real
@ ^ [X2: real] : ( has_field_derivative @ real @ F2 @ ( F6 @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
@ ( at_top @ real ) )
=> ( ( eventually @ real
@ ^ [X2: real] : ( has_field_derivative @ real @ G @ ( G3 @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
@ ( at_top @ real ) )
=> ( ( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( F6 @ X2 ) @ ( G3 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ X )
@ ( at_top @ real ) )
=> ( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ X )
@ ( at_top @ real ) ) ) ) ) ) ) ).
% lhospital_at_top_at_top
thf(fact_6059_lhopital__at__top,axiom,
! [G: real > real,X: real,G3: real > real,F2: real > real,F6: real > real,Y2: real] :
( ( filterlim @ real @ real @ G @ ( at_top @ real ) @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] :
( ( G3 @ X2 )
!= ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] : ( has_field_derivative @ real @ F2 @ ( F6 @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] : ( has_field_derivative @ real @ G @ ( G3 @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( ( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( F6 @ X2 ) @ ( G3 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ Y2 )
@ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ Y2 )
@ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ) ) ).
% lhopital_at_top
thf(fact_6060_lhopital__left__at__top,axiom,
! [G: real > real,X: real,G3: real > real,F2: real > real,F6: real > real,Y2: real] :
( ( filterlim @ real @ real @ G @ ( at_top @ real ) @ ( topolo174197925503356063within @ real @ X @ ( set_ord_lessThan @ real @ X ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] :
( ( G3 @ X2 )
!= ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ real @ X @ ( set_ord_lessThan @ real @ X ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] : ( has_field_derivative @ real @ F2 @ ( F6 @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ X @ ( set_ord_lessThan @ real @ X ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] : ( has_field_derivative @ real @ G @ ( G3 @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ X @ ( set_ord_lessThan @ real @ X ) ) )
=> ( ( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( F6 @ X2 ) @ ( G3 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ Y2 )
@ ( topolo174197925503356063within @ real @ X @ ( set_ord_lessThan @ real @ X ) ) )
=> ( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ Y2 )
@ ( topolo174197925503356063within @ real @ X @ ( set_ord_lessThan @ real @ X ) ) ) ) ) ) ) ) ).
% lhopital_left_at_top
thf(fact_6061_tendsto__of__nat,axiom,
! [A: $tType] :
( ( real_V2822296259951069270ebra_1 @ A )
=> ( filterlim @ nat @ A @ ( semiring_1_of_nat @ A ) @ ( at_infinity @ A ) @ ( at_top @ nat ) ) ) ).
% tendsto_of_nat
thf(fact_6062_filterlim__pow__at__top,axiom,
! [A: $tType,N: nat,F2: A > real,F5: filter @ A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( filterlim @ A @ real @ F2 @ ( at_top @ real ) @ F5 )
=> ( filterlim @ A @ real
@ ^ [X2: A] : ( power_power @ real @ ( F2 @ X2 ) @ N )
@ ( at_top @ real )
@ F5 ) ) ) ).
% filterlim_pow_at_top
thf(fact_6063_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_6064_continuous__divide,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_V3459762299906320749_field @ B ) )
=> ! [F5: filter @ A,F2: A > B,G: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ F5 @ F2 )
=> ( ( topolo3448309680560233919inuous @ A @ B @ F5 @ G )
=> ( ( ( G
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F5
@ ^ [X2: A] : X2 ) )
!= ( zero_zero @ B ) )
=> ( topolo3448309680560233919inuous @ A @ B @ F5
@ ^ [X2: A] : ( divide_divide @ B @ ( F2 @ X2 ) @ ( G @ X2 ) ) ) ) ) ) ) ).
% continuous_divide
thf(fact_6065_real__tendsto__divide__at__top,axiom,
! [A: $tType,F2: A > real,C2: real,F5: filter @ A,G: A > real] :
( ( filterlim @ A @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ C2 ) @ F5 )
=> ( ( filterlim @ A @ real @ G @ ( at_top @ real ) @ F5 )
=> ( filterlim @ A @ real
@ ^ [X2: A] : ( divide_divide @ real @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ F5 ) ) ) ).
% real_tendsto_divide_at_top
thf(fact_6066_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_6067_continuous__inverse,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_V8999393235501362500lgebra @ B ) )
=> ! [F5: filter @ A,F2: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ F5 @ F2 )
=> ( ( ( F2
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F5
@ ^ [X2: A] : X2 ) )
!= ( zero_zero @ B ) )
=> ( topolo3448309680560233919inuous @ A @ B @ F5
@ ^ [X2: A] : ( inverse_inverse @ B @ ( F2 @ X2 ) ) ) ) ) ) ).
% continuous_inverse
thf(fact_6068_continuous__sgn,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F5: filter @ A,F2: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ F5 @ F2 )
=> ( ( ( F2
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F5
@ ^ [X2: A] : X2 ) )
!= ( zero_zero @ B ) )
=> ( topolo3448309680560233919inuous @ A @ B @ F5
@ ^ [X2: A] : ( sgn_sgn @ B @ ( F2 @ X2 ) ) ) ) ) ) ).
% continuous_sgn
thf(fact_6069_filterlim__int__of__nat__at__topD,axiom,
! [A: $tType,F2: int > A,F5: filter @ A] :
( ( filterlim @ nat @ A
@ ^ [X2: nat] : ( F2 @ ( semiring_1_of_nat @ int @ X2 ) )
@ F5
@ ( at_top @ nat ) )
=> ( filterlim @ int @ A @ F2 @ F5 @ ( at_top @ int ) ) ) ).
% filterlim_int_of_nat_at_topD
thf(fact_6070_continuous__powr,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [F5: filter @ A,F2: A > real,G: A > real] :
( ( topolo3448309680560233919inuous @ A @ real @ F5 @ F2 )
=> ( ( topolo3448309680560233919inuous @ A @ real @ F5 @ G )
=> ( ( ( F2
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F5
@ ^ [X2: A] : X2 ) )
!= ( zero_zero @ real ) )
=> ( topolo3448309680560233919inuous @ A @ real @ F5
@ ^ [X2: A] : ( powr @ real @ ( F2 @ X2 ) @ ( G @ X2 ) ) ) ) ) ) ) ).
% continuous_powr
thf(fact_6071_continuous__ln,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [F5: filter @ A,F2: A > real] :
( ( topolo3448309680560233919inuous @ A @ real @ F5 @ F2 )
=> ( ( ( F2
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F5
@ ^ [X2: A] : X2 ) )
!= ( zero_zero @ real ) )
=> ( topolo3448309680560233919inuous @ A @ real @ F5
@ ^ [X2: A] : ( ln_ln @ real @ ( F2 @ X2 ) ) ) ) ) ) ).
% continuous_ln
thf(fact_6072_tendsto__inverse__0,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ( filterlim @ A @ A @ ( inverse_inverse @ A ) @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ ( at_infinity @ A ) ) ) ).
% tendsto_inverse_0
thf(fact_6073_tendsto__neg__powr,axiom,
! [A: $tType,S: real,F2: A > real,F5: filter @ A] :
( ( ord_less @ real @ S @ ( zero_zero @ real ) )
=> ( ( filterlim @ A @ real @ F2 @ ( at_top @ real ) @ F5 )
=> ( filterlim @ A @ real
@ ^ [X2: A] : ( powr @ real @ ( F2 @ X2 ) @ S )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ F5 ) ) ) ).
% tendsto_neg_powr
thf(fact_6074_tendsto__mult__filterlim__at__infinity,axiom,
! [A: $tType,B: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: B > A,C2: A,F5: filter @ B,G: B > A] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ C2 ) @ F5 )
=> ( ( C2
!= ( zero_zero @ A ) )
=> ( ( filterlim @ B @ A @ G @ ( at_infinity @ A ) @ F5 )
=> ( filterlim @ B @ A
@ ^ [X2: B] : ( times_times @ A @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( at_infinity @ A )
@ F5 ) ) ) ) ) ).
% tendsto_mult_filterlim_at_infinity
thf(fact_6075_ln__x__over__x__tendsto__0,axiom,
( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( ln_ln @ real @ X2 ) @ X2 )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( at_top @ real ) ) ).
% ln_x_over_x_tendsto_0
thf(fact_6076_tendsto__divide__0,axiom,
! [A: $tType,C: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [F2: C > A,C2: A,F5: filter @ C,G: C > A] :
( ( filterlim @ C @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ C2 ) @ F5 )
=> ( ( filterlim @ C @ A @ G @ ( at_infinity @ A ) @ F5 )
=> ( filterlim @ C @ A
@ ^ [X2: C] : ( divide_divide @ A @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ F5 ) ) ) ) ).
% tendsto_divide_0
thf(fact_6077_filterlim__power__at__infinity,axiom,
! [B: $tType,A: $tType] :
( ( real_V8999393235501362500lgebra @ B )
=> ! [F2: A > B,F5: filter @ A,N: nat] :
( ( filterlim @ A @ B @ F2 @ ( at_infinity @ B ) @ F5 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( filterlim @ A @ B
@ ^ [X2: A] : ( power_power @ B @ ( F2 @ X2 ) @ N )
@ ( at_infinity @ B )
@ F5 ) ) ) ) ).
% filterlim_power_at_infinity
thf(fact_6078_continuous__tan,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [F5: filter @ A,F2: A > A] :
( ( topolo3448309680560233919inuous @ A @ A @ F5 @ F2 )
=> ( ( ( cos @ A
@ ( F2
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F5
@ ^ [X2: A] : X2 ) ) )
!= ( zero_zero @ A ) )
=> ( topolo3448309680560233919inuous @ A @ A @ F5
@ ^ [X2: A] : ( tan @ A @ ( F2 @ X2 ) ) ) ) ) ) ).
% continuous_tan
thf(fact_6079_continuous__cot,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [F5: filter @ A,F2: A > A] :
( ( topolo3448309680560233919inuous @ A @ A @ F5 @ F2 )
=> ( ( ( sin @ A
@ ( F2
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F5
@ ^ [X2: A] : X2 ) ) )
!= ( zero_zero @ A ) )
=> ( topolo3448309680560233919inuous @ A @ A @ F5
@ ^ [X2: A] : ( cot @ A @ ( F2 @ X2 ) ) ) ) ) ) ).
% continuous_cot
thf(fact_6080_continuous__tanh,axiom,
! [A: $tType,C: $tType] :
( ( ( topological_t2_space @ C )
& ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [F5: filter @ C,F2: C > A] :
( ( topolo3448309680560233919inuous @ C @ A @ F5 @ F2 )
=> ( ( ( cosh @ A
@ ( F2
@ ( topolo3827282254853284352ce_Lim @ C @ C @ F5
@ ^ [X2: C] : X2 ) ) )
!= ( zero_zero @ A ) )
=> ( topolo3448309680560233919inuous @ C @ A @ F5
@ ^ [X2: C] : ( tanh @ A @ ( F2 @ X2 ) ) ) ) ) ) ).
% continuous_tanh
thf(fact_6081_continuous__arcosh,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [F5: filter @ A,F2: A > real] :
( ( topolo3448309680560233919inuous @ A @ real @ F5 @ F2 )
=> ( ( ord_less @ real @ ( one_one @ real )
@ ( F2
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F5
@ ^ [X2: A] : X2 ) ) )
=> ( topolo3448309680560233919inuous @ A @ real @ F5
@ ^ [X2: A] : ( arcosh @ real @ ( F2 @ X2 ) ) ) ) ) ) ).
% continuous_arcosh
thf(fact_6082_filterlim__inverse__at__infinity,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ( filterlim @ A @ A @ ( inverse_inverse @ A ) @ ( at_infinity @ A ) @ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ).
% filterlim_inverse_at_infinity
thf(fact_6083_LIM__at__top__divide,axiom,
! [A: $tType,F2: A > real,A3: real,F5: filter @ A,G: A > real] :
( ( filterlim @ A @ real @ F2 @ ( topolo7230453075368039082e_nhds @ real @ A3 ) @ F5 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ A3 )
=> ( ( filterlim @ A @ real @ G @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ F5 )
=> ( ( eventually @ A
@ ^ [X2: A] : ( ord_less @ real @ ( zero_zero @ real ) @ ( G @ X2 ) )
@ F5 )
=> ( filterlim @ A @ real
@ ^ [X2: A] : ( divide_divide @ real @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( at_top @ real )
@ F5 ) ) ) ) ) ).
% LIM_at_top_divide
thf(fact_6084_tendsto__power__div__exp__0,axiom,
! [K: nat] :
( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( power_power @ real @ X2 @ K ) @ ( exp @ real @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( at_top @ real ) ) ).
% tendsto_power_div_exp_0
thf(fact_6085_filterlim__inverse__at__iff,axiom,
! [B: $tType,A: $tType] :
( ( real_V8999393235501362500lgebra @ B )
=> ! [G: A > B,F5: filter @ A] :
( ( filterlim @ A @ B
@ ^ [X2: A] : ( inverse_inverse @ B @ ( G @ X2 ) )
@ ( topolo174197925503356063within @ B @ ( zero_zero @ B ) @ ( top_top @ ( set @ B ) ) )
@ F5 )
= ( filterlim @ A @ B @ G @ ( at_infinity @ B ) @ F5 ) ) ) ).
% filterlim_inverse_at_iff
thf(fact_6086_lhopital,axiom,
! [F2: real > real,X: real,G: real > real,G3: real > real,F6: real > real,F5: filter @ real] :
( ( filterlim @ real @ real @ F2 @ ( 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
@ ^ [X2: real] :
( ( G @ X2 )
!= ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] :
( ( G3 @ X2 )
!= ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] : ( has_field_derivative @ real @ F2 @ ( F6 @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] : ( has_field_derivative @ real @ G @ ( G3 @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( ( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( F6 @ X2 ) @ ( G3 @ X2 ) )
@ F5
@ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ F5
@ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ) ) ) ) ).
% lhopital
thf(fact_6087_lhopital__left,axiom,
! [F2: real > real,X: real,G: real > real,G3: real > real,F6: real > real,F5: filter @ real] :
( ( filterlim @ real @ real @ F2 @ ( 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
@ ^ [X2: real] :
( ( G @ X2 )
!= ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ real @ X @ ( set_ord_lessThan @ real @ X ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] :
( ( G3 @ X2 )
!= ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ real @ X @ ( set_ord_lessThan @ real @ X ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] : ( has_field_derivative @ real @ F2 @ ( F6 @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ X @ ( set_ord_lessThan @ real @ X ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] : ( has_field_derivative @ real @ G @ ( G3 @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ X @ ( set_ord_lessThan @ real @ X ) ) )
=> ( ( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( F6 @ X2 ) @ ( G3 @ X2 ) )
@ F5
@ ( topolo174197925503356063within @ real @ X @ ( set_ord_lessThan @ real @ X ) ) )
=> ( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ F5
@ ( topolo174197925503356063within @ real @ X @ ( set_ord_lessThan @ real @ X ) ) ) ) ) ) ) ) ) ) ).
% lhopital_left
thf(fact_6088_continuous__log,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [F5: filter @ A,F2: A > real,G: A > real] :
( ( topolo3448309680560233919inuous @ A @ real @ F5 @ F2 )
=> ( ( topolo3448309680560233919inuous @ A @ real @ F5 @ G )
=> ( ( ord_less @ real @ ( zero_zero @ real )
@ ( F2
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F5
@ ^ [X2: A] : X2 ) ) )
=> ( ( ( F2
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F5
@ ^ [X2: A] : X2 ) )
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real )
@ ( G
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F5
@ ^ [X2: A] : X2 ) ) )
=> ( topolo3448309680560233919inuous @ A @ real @ F5
@ ^ [X2: A] : ( log2 @ ( F2 @ X2 ) @ ( G @ X2 ) ) ) ) ) ) ) ) ) ).
% continuous_log
thf(fact_6089_tendsto__exp__limit__at__top,axiom,
! [X: real] :
( filterlim @ real @ real
@ ^ [Y5: real] : ( powr @ real @ ( plus_plus @ real @ ( one_one @ real ) @ ( divide_divide @ real @ X @ Y5 ) ) @ Y5 )
@ ( topolo7230453075368039082e_nhds @ real @ ( exp @ real @ X ) )
@ ( at_top @ real ) ) ).
% tendsto_exp_limit_at_top
thf(fact_6090_filterlim__divide__at__infinity,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,C2: A,F5: filter @ A,G: A > A] :
( ( filterlim @ A @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ C2 ) @ F5 )
=> ( ( filterlim @ A @ A @ G @ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) @ F5 )
=> ( ( C2
!= ( zero_zero @ A ) )
=> ( filterlim @ A @ A
@ ^ [X2: A] : ( divide_divide @ A @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( at_infinity @ A )
@ F5 ) ) ) ) ) ).
% filterlim_divide_at_infinity
thf(fact_6091_continuous__artanh,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [F5: filter @ A,F2: A > real] :
( ( topolo3448309680560233919inuous @ A @ real @ F5 @ F2 )
=> ( ( member @ real
@ ( F2
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F5
@ ^ [X2: A] : X2 ) )
@ ( set_or5935395276787703475ssThan @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ ( one_one @ real ) ) )
=> ( topolo3448309680560233919inuous @ A @ real @ F5
@ ^ [X2: A] : ( artanh @ real @ ( F2 @ X2 ) ) ) ) ) ) ).
% continuous_artanh
thf(fact_6092_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_6093_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_6094_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_6095_lim__at__infinity__0,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,L: A] :
( ( filterlim @ A @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ ( at_infinity @ A ) )
= ( filterlim @ A @ A @ ( comp @ A @ A @ A @ F2 @ ( inverse_inverse @ A ) ) @ ( topolo7230453075368039082e_nhds @ A @ L ) @ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% lim_at_infinity_0
thf(fact_6096_lim__zero__infinity,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,L: A] :
( ( filterlim @ A @ A
@ ^ [X2: A] : ( F2 @ ( divide_divide @ A @ ( one_one @ A ) @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ L )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) )
=> ( filterlim @ A @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ ( at_infinity @ A ) ) ) ) ).
% lim_zero_infinity
thf(fact_6097_has__derivative__at__within__iff__Ex,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [X: A,S3: set @ A,F2: A > B,F6: A > B] :
( ( member @ A @ X @ S3 )
=> ( ( topolo1002775350975398744n_open @ A @ S3 )
=> ( ( has_derivative @ A @ B @ F2 @ F6 @ ( topolo174197925503356063within @ A @ X @ S3 ) )
= ( ( real_V3181309239436604168linear @ A @ B @ F6 )
& ? [E3: A > B] :
( ! [H2: A] :
( ( member @ A @ ( plus_plus @ A @ X @ H2 ) @ S3 )
=> ( ( F2 @ ( plus_plus @ A @ X @ H2 ) )
= ( plus_plus @ B @ ( plus_plus @ B @ ( F2 @ X ) @ ( F6 @ H2 ) ) @ ( E3 @ H2 ) ) ) )
& ( filterlim @ A @ real
@ ^ [H2: A] : ( divide_divide @ real @ ( real_V7770717601297561774m_norm @ B @ ( E3 @ H2 ) ) @ ( real_V7770717601297561774m_norm @ A @ H2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ) ) ).
% has_derivative_at_within_iff_Ex
thf(fact_6098_has__derivativeI__sandwich,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [E2: real,F6: A > B,S: set @ A,X: A,F2: A > B,H5: A > real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E2 )
=> ( ( real_V3181309239436604168linear @ A @ B @ F6 )
=> ( ! [Y3: A] :
( ( member @ A @ Y3 @ S )
=> ( ( Y3 != X )
=> ( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ Y3 @ X ) @ E2 )
=> ( ord_less_eq @ real @ ( divide_divide @ real @ ( real_V7770717601297561774m_norm @ B @ ( minus_minus @ B @ ( minus_minus @ B @ ( F2 @ Y3 ) @ ( F2 @ X ) ) @ ( F6 @ ( minus_minus @ A @ Y3 @ X ) ) ) ) @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ Y3 @ X ) ) ) @ ( H5 @ Y3 ) ) ) ) )
=> ( ( filterlim @ A @ real @ H5 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( has_derivative @ A @ B @ F2 @ F6 @ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ) ) ).
% has_derivativeI_sandwich
thf(fact_6099_dist__0__norm,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X: A] :
( ( real_V557655796197034286t_dist @ A @ ( zero_zero @ A ) @ X )
= ( real_V7770717601297561774m_norm @ A @ X ) ) ) ).
% dist_0_norm
thf(fact_6100_dist__diff_I1_J,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [A3: A,B2: A] :
( ( real_V557655796197034286t_dist @ A @ A3 @ ( minus_minus @ A @ A3 @ B2 ) )
= ( real_V7770717601297561774m_norm @ A @ B2 ) ) ) ).
% dist_diff(1)
thf(fact_6101_dist__diff_I2_J,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [A3: A,B2: A] :
( ( real_V557655796197034286t_dist @ A @ ( minus_minus @ A @ A3 @ B2 ) @ A3 )
= ( real_V7770717601297561774m_norm @ A @ B2 ) ) ) ).
% dist_diff(2)
thf(fact_6102_dist__scaleR,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X: real,A3: A,Y2: real] :
( ( real_V557655796197034286t_dist @ A @ ( real_V8093663219630862766scaleR @ A @ X @ A3 ) @ ( real_V8093663219630862766scaleR @ A @ Y2 @ A3 ) )
= ( times_times @ real @ ( abs_abs @ real @ ( minus_minus @ real @ X @ Y2 ) ) @ ( real_V7770717601297561774m_norm @ A @ A3 ) ) ) ) ).
% dist_scaleR
thf(fact_6103_dist__norm,axiom,
! [A: $tType] :
( ( real_V6936659425649961206t_norm @ A )
=> ( ( real_V557655796197034286t_dist @ A )
= ( ^ [X2: A,Y5: A] : ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ X2 @ Y5 ) ) ) ) ) ).
% dist_norm
thf(fact_6104_norm__conv__dist,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ( ( real_V7770717601297561774m_norm @ A )
= ( ^ [X2: A] : ( real_V557655796197034286t_dist @ A @ X2 @ ( zero_zero @ A ) ) ) ) ) ).
% norm_conv_dist
thf(fact_6105_dist__real__def,axiom,
( ( real_V557655796197034286t_dist @ real )
= ( ^ [X2: real,Y5: real] : ( abs_abs @ real @ ( minus_minus @ real @ X2 @ Y5 ) ) ) ) ).
% dist_real_def
thf(fact_6106_dist__complex__def,axiom,
( ( real_V557655796197034286t_dist @ complex )
= ( ^ [X2: complex,Y5: complex] : ( real_V7770717601297561774m_norm @ complex @ ( minus_minus @ complex @ X2 @ Y5 ) ) ) ) ).
% dist_complex_def
thf(fact_6107_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 )
=> ? [B4: A] :
( ( ord_less @ A @ X @ B4 )
& ( ord_less_eq @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ X @ B4 ) @ S3 ) ) ) ) ) ) ).
% open_right
thf(fact_6108_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 )
=> ? [B4: A] :
( ( ord_less @ A @ B4 @ X )
& ( ord_less_eq @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ B4 @ X ) @ S3 ) ) ) ) ) ) ).
% open_left
thf(fact_6109_abs__dist__diff__le,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [A3: A,B2: A,C2: A] : ( ord_less_eq @ real @ ( abs_abs @ real @ ( minus_minus @ real @ ( real_V557655796197034286t_dist @ A @ A3 @ B2 ) @ ( real_V557655796197034286t_dist @ A @ B2 @ C2 ) ) ) @ ( real_V557655796197034286t_dist @ A @ A3 @ C2 ) ) ) ).
% abs_dist_diff_le
thf(fact_6110_open__superdiagonal,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ( topolo1002775350975398744n_open @ ( product_prod @ A @ A )
@ ( collect @ ( product_prod @ A @ A )
@ ^ [Uu3: product_prod @ A @ A] :
? [X2: A,Y5: A] :
( ( Uu3
= ( product_Pair @ A @ A @ X2 @ Y5 ) )
& ( ord_less @ A @ Y5 @ X2 ) ) ) ) ) ).
% open_superdiagonal
thf(fact_6111_open__subdiagonal,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ( topolo1002775350975398744n_open @ ( product_prod @ A @ A )
@ ( collect @ ( product_prod @ A @ A )
@ ^ [Uu3: product_prod @ A @ A] :
? [X2: A,Y5: A] :
( ( Uu3
= ( product_Pair @ A @ A @ X2 @ Y5 ) )
& ( ord_less @ A @ X2 @ Y5 ) ) ) ) ) ).
% open_subdiagonal
thf(fact_6112_dist__of__int,axiom,
! [A: $tType] :
( ( real_V2822296259951069270ebra_1 @ A )
=> ! [M2: int,N: int] :
( ( real_V557655796197034286t_dist @ A @ ( ring_1_of_int @ A @ M2 ) @ ( ring_1_of_int @ A @ N ) )
= ( ring_1_of_int @ real @ ( abs_abs @ int @ ( minus_minus @ int @ M2 @ N ) ) ) ) ) ).
% dist_of_int
thf(fact_6113_dist__triangle__half__l,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X15: A,Y2: A,E2: real,X23: A] :
( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X15 @ Y2 ) @ ( divide_divide @ real @ E2 @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X23 @ Y2 ) @ ( divide_divide @ real @ E2 @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X15 @ X23 ) @ E2 ) ) ) ) ).
% dist_triangle_half_l
thf(fact_6114_dist__triangle__half__r,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [Y2: A,X15: A,E2: real,X23: A] :
( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ Y2 @ X15 ) @ ( divide_divide @ real @ E2 @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ Y2 @ X23 ) @ ( divide_divide @ real @ E2 @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X15 @ X23 ) @ E2 ) ) ) ) ).
% dist_triangle_half_r
thf(fact_6115_dist__triangle__third,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X15: A,X23: A,E2: real,X32: A,X42: A] :
( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X15 @ X23 ) @ ( divide_divide @ real @ E2 @ ( numeral_numeral @ real @ ( bit1 @ one2 ) ) ) )
=> ( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X23 @ X32 ) @ ( divide_divide @ real @ E2 @ ( numeral_numeral @ real @ ( bit1 @ one2 ) ) ) )
=> ( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X32 @ X42 ) @ ( divide_divide @ real @ E2 @ ( numeral_numeral @ real @ ( bit1 @ one2 ) ) ) )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X15 @ X42 ) @ E2 ) ) ) ) ) ).
% dist_triangle_third
thf(fact_6116_at__within__nhd,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [X: A,S3: set @ A,T6: set @ A,U4: set @ A] :
( ( member @ A @ X @ S3 )
=> ( ( topolo1002775350975398744n_open @ A @ S3 )
=> ( ( ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ T6 @ S3 ) @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ U4 @ S3 ) @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( ( topolo174197925503356063within @ A @ X @ T6 )
= ( topolo174197925503356063within @ A @ X @ U4 ) ) ) ) ) ) ).
% at_within_nhd
thf(fact_6117_CauchyI_H,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X8: nat > A] :
( ! [E: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E )
=> ? [M8: nat] :
! [M3: nat] :
( ( ord_less_eq @ nat @ M8 @ M3 )
=> ! [N2: nat] :
( ( ord_less @ nat @ M3 @ N2 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( X8 @ M3 ) @ ( X8 @ N2 ) ) @ E ) ) ) )
=> ( topolo3814608138187158403Cauchy @ A @ X8 ) ) ) ).
% CauchyI'
thf(fact_6118_Cauchy__altdef,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ( ( topolo3814608138187158403Cauchy @ A )
= ( ^ [F4: nat > A] :
! [E3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E3 )
=> ? [M7: nat] :
! [M5: nat] :
( ( ord_less_eq @ nat @ M7 @ M5 )
=> ! [N5: nat] :
( ( ord_less @ nat @ M5 @ N5 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( F4 @ M5 ) @ ( F4 @ N5 ) ) @ E3 ) ) ) ) ) ) ) ).
% Cauchy_altdef
thf(fact_6119_dist__of__nat,axiom,
! [A: $tType] :
( ( real_V2822296259951069270ebra_1 @ A )
=> ! [M2: nat,N: nat] :
( ( real_V557655796197034286t_dist @ A @ ( semiring_1_of_nat @ A @ M2 ) @ ( semiring_1_of_nat @ A @ N ) )
= ( ring_1_of_int @ real @ ( abs_abs @ int @ ( minus_minus @ int @ ( semiring_1_of_nat @ int @ M2 ) @ ( semiring_1_of_nat @ int @ N ) ) ) ) ) ) ).
% dist_of_nat
thf(fact_6120_metric__Cauchy__iff2,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ( ( topolo3814608138187158403Cauchy @ A )
= ( ^ [X7: nat > A] :
! [J3: nat] :
? [M7: nat] :
! [M5: nat] :
( ( ord_less_eq @ nat @ M7 @ M5 )
=> ! [N5: nat] :
( ( ord_less_eq @ nat @ M7 @ N5 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( X7 @ M5 ) @ ( X7 @ N5 ) ) @ ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ ( suc @ J3 ) ) ) ) ) ) ) ) ) ).
% metric_Cauchy_iff2
thf(fact_6121_tendsto__offset__zero__iff,axiom,
! [C: $tType,D: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( topolo4958980785337419405_space @ D )
& ( zero @ C ) )
=> ! [A3: A,S3: set @ A,F2: A > D,L4: D] :
( ( nO_MATCH @ C @ A @ ( zero_zero @ C ) @ A3 )
=> ( ( member @ A @ A3 @ S3 )
=> ( ( topolo1002775350975398744n_open @ A @ S3 )
=> ( ( filterlim @ A @ D @ F2 @ ( topolo7230453075368039082e_nhds @ D @ L4 ) @ ( topolo174197925503356063within @ A @ A3 @ S3 ) )
= ( filterlim @ A @ D
@ ^ [H2: A] : ( F2 @ ( plus_plus @ A @ A3 @ H2 ) )
@ ( topolo7230453075368039082e_nhds @ D @ L4 )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% tendsto_offset_zero_iff
thf(fact_6122_LIMSEQ__iff__nz,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X8: nat > A,L4: A] :
( ( filterlim @ nat @ A @ X8 @ ( topolo7230453075368039082e_nhds @ A @ L4 ) @ ( at_top @ nat ) )
= ( ! [R: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R )
=> ? [No: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ No )
& ! [N5: nat] :
( ( ord_less_eq @ nat @ No @ N5 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( X8 @ N5 ) @ L4 ) @ R ) ) ) ) ) ) ) ).
% LIMSEQ_iff_nz
thf(fact_6123_filterlim__pow__at__bot__even,axiom,
! [N: nat,F2: real > real,F5: filter @ real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( filterlim @ real @ real @ F2 @ ( at_bot @ real ) @ F5 )
=> ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( filterlim @ real @ real
@ ^ [X2: real] : ( power_power @ real @ ( F2 @ X2 ) @ N )
@ ( at_top @ real )
@ F5 ) ) ) ) ).
% filterlim_pow_at_bot_even
thf(fact_6124_tendsto__exp__limit__at__right,axiom,
! [X: real] :
( filterlim @ real @ real
@ ^ [Y5: real] : ( powr @ real @ ( plus_plus @ real @ ( one_one @ real ) @ ( times_times @ real @ X @ Y5 ) ) @ ( divide_divide @ real @ ( one_one @ real ) @ Y5 ) )
@ ( 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_6125_greaterThan__iff,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [I: A,K: A] :
( ( member @ A @ I @ ( set_ord_greaterThan @ A @ K ) )
= ( ord_less @ A @ K @ I ) ) ) ).
% greaterThan_iff
thf(fact_6126_Compl__greaterThan,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [K: A] :
( ( uminus_uminus @ ( set @ A ) @ ( set_ord_greaterThan @ A @ K ) )
= ( set_ord_atMost @ A @ K ) ) ) ).
% Compl_greaterThan
thf(fact_6127_Compl__atMost,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [K: A] :
( ( uminus_uminus @ ( set @ A ) @ ( set_ord_atMost @ A @ K ) )
= ( set_ord_greaterThan @ A @ K ) ) ) ).
% Compl_atMost
thf(fact_6128_image__uminus__greaterThan,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [X: A] :
( ( image @ A @ A @ ( uminus_uminus @ A ) @ ( set_ord_greaterThan @ A @ X ) )
= ( set_ord_lessThan @ A @ ( uminus_uminus @ A @ X ) ) ) ) ).
% image_uminus_greaterThan
thf(fact_6129_image__uminus__lessThan,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [X: A] :
( ( image @ A @ A @ ( uminus_uminus @ A ) @ ( set_ord_lessThan @ A @ X ) )
= ( set_ord_greaterThan @ A @ ( uminus_uminus @ A @ X ) ) ) ) ).
% image_uminus_lessThan
thf(fact_6130_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_6131_greaterThan__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( set_ord_greaterThan @ A )
= ( ^ [L2: A] : ( collect @ A @ ( ord_less @ A @ L2 ) ) ) ) ) ).
% greaterThan_def
thf(fact_6132_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_6133_sinh__real__at__bot,axiom,
filterlim @ real @ real @ ( sinh @ real ) @ ( at_bot @ real ) @ ( at_bot @ real ) ).
% sinh_real_at_bot
thf(fact_6134_arsinh__real__at__bot,axiom,
filterlim @ real @ real @ ( arsinh @ real ) @ ( at_bot @ real ) @ ( at_bot @ real ) ).
% arsinh_real_at_bot
thf(fact_6135_filterlim__at__bot__at__right,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo1944317154257567458pology @ A )
& ( linorder @ B ) )
=> ! [Q: A > $o,F2: A > B,P: B > $o,G: B > A,A3: A] :
( ! [X3: A,Y3: A] :
( ( Q @ X3 )
=> ( ( Q @ Y3 )
=> ( ( ord_less_eq @ A @ X3 @ Y3 )
=> ( ord_less_eq @ B @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) ) ) )
=> ( ! [X3: B] :
( ( P @ X3 )
=> ( ( F2 @ ( G @ X3 ) )
= X3 ) )
=> ( ! [X3: B] :
( ( P @ X3 )
=> ( Q @ ( G @ X3 ) ) )
=> ( ( eventually @ A @ Q @ ( topolo174197925503356063within @ A @ A3 @ ( set_ord_greaterThan @ A @ A3 ) ) )
=> ( ! [B4: A] :
( ( Q @ B4 )
=> ( ord_less @ A @ A3 @ B4 ) )
=> ( ( eventually @ B @ P @ ( at_bot @ B ) )
=> ( filterlim @ A @ B @ F2 @ ( at_bot @ B ) @ ( topolo174197925503356063within @ A @ A3 @ ( set_ord_greaterThan @ A @ A3 ) ) ) ) ) ) ) ) ) ) ).
% filterlim_at_bot_at_right
thf(fact_6136_eventually__at__right,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [X: A,Y2: A,P: A > $o] :
( ( ord_less @ A @ X @ Y2 )
=> ( ( eventually @ A @ P @ ( topolo174197925503356063within @ A @ X @ ( set_ord_greaterThan @ A @ X ) ) )
= ( ? [B3: A] :
( ( ord_less @ A @ X @ B3 )
& ! [Y5: A] :
( ( ord_less @ A @ X @ Y5 )
=> ( ( ord_less @ A @ Y5 @ B3 )
=> ( P @ Y5 ) ) ) ) ) ) ) ) ).
% eventually_at_right
thf(fact_6137_eventually__at__right__field,axiom,
! [A: $tType] :
( ( ( linordered_field @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [P: A > $o,X: A] :
( ( eventually @ A @ P @ ( topolo174197925503356063within @ A @ X @ ( set_ord_greaterThan @ A @ X ) ) )
= ( ? [B3: A] :
( ( ord_less @ A @ X @ B3 )
& ! [Y5: A] :
( ( ord_less @ A @ X @ Y5 )
=> ( ( ord_less @ A @ Y5 @ B3 )
=> ( P @ Y5 ) ) ) ) ) ) ) ).
% eventually_at_right_field
thf(fact_6138_at__within__Icc__at__right,axiom,
! [A: $tType] :
( ( topolo2564578578187576103pology @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( topolo174197925503356063within @ A @ A3 @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
= ( topolo174197925503356063within @ A @ A3 @ ( set_ord_greaterThan @ A @ A3 ) ) ) ) ) ).
% at_within_Icc_at_right
thf(fact_6139_eventually__at__bot__dense,axiom,
! [A: $tType] :
( ( ( linorder @ A )
& ( no_bot @ A ) )
=> ! [P: A > $o] :
( ( eventually @ A @ P @ ( at_bot @ A ) )
= ( ? [N7: A] :
! [N5: A] :
( ( ord_less @ A @ N5 @ N7 )
=> ( P @ N5 ) ) ) ) ) ).
% eventually_at_bot_dense
thf(fact_6140_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_6141_eventually__gt__at__bot,axiom,
! [A: $tType] :
( ( unboun7993243217541854897norder @ A )
=> ! [C2: A] :
( eventually @ A
@ ^ [X2: A] : ( ord_less @ A @ X2 @ C2 )
@ ( at_bot @ A ) ) ) ).
% eventually_gt_at_bot
thf(fact_6142_cosh__real__at__bot,axiom,
filterlim @ real @ real @ ( cosh @ real ) @ ( at_top @ real ) @ ( at_bot @ real ) ).
% cosh_real_at_bot
thf(fact_6143_lhopital__right__at__top__at__bot,axiom,
! [F2: real > real,A3: real,G: real > real,F6: real > real,G3: real > real] :
( ( filterlim @ real @ real @ F2 @ ( at_top @ real ) @ ( topolo174197925503356063within @ real @ A3 @ ( set_ord_greaterThan @ real @ A3 ) ) )
=> ( ( filterlim @ real @ real @ G @ ( at_bot @ real ) @ ( topolo174197925503356063within @ real @ A3 @ ( set_ord_greaterThan @ real @ A3 ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] : ( has_field_derivative @ real @ F2 @ ( F6 @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ A3 @ ( set_ord_greaterThan @ real @ A3 ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] : ( has_field_derivative @ real @ G @ ( G3 @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ A3 @ ( set_ord_greaterThan @ real @ A3 ) ) )
=> ( ( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( F6 @ X2 ) @ ( G3 @ X2 ) )
@ ( at_bot @ real )
@ ( topolo174197925503356063within @ real @ A3 @ ( set_ord_greaterThan @ real @ A3 ) ) )
=> ( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( at_bot @ real )
@ ( topolo174197925503356063within @ real @ A3 @ ( set_ord_greaterThan @ real @ A3 ) ) ) ) ) ) ) ) ).
% lhopital_right_at_top_at_bot
thf(fact_6144_filterlim__uminus__at__top,axiom,
! [A: $tType,F2: A > real,F5: filter @ A] :
( ( filterlim @ A @ real @ F2 @ ( at_top @ real ) @ F5 )
= ( filterlim @ A @ real
@ ^ [X2: A] : ( uminus_uminus @ real @ ( F2 @ X2 ) )
@ ( at_bot @ real )
@ F5 ) ) ).
% filterlim_uminus_at_top
thf(fact_6145_filterlim__uminus__at__bot,axiom,
! [A: $tType,F2: A > real,F5: filter @ A] :
( ( filterlim @ A @ real @ F2 @ ( at_bot @ real ) @ F5 )
= ( filterlim @ A @ real
@ ^ [X2: A] : ( uminus_uminus @ real @ ( F2 @ X2 ) )
@ ( at_top @ real )
@ F5 ) ) ).
% filterlim_uminus_at_bot
thf(fact_6146_filterlim__at__top__mirror,axiom,
! [A: $tType,F2: real > A,F5: filter @ A] :
( ( filterlim @ real @ A @ F2 @ F5 @ ( at_top @ real ) )
= ( filterlim @ real @ A
@ ^ [X2: real] : ( F2 @ ( uminus_uminus @ real @ X2 ) )
@ F5
@ ( at_bot @ real ) ) ) ).
% filterlim_at_top_mirror
thf(fact_6147_filterlim__at__bot__mirror,axiom,
! [A: $tType,F2: real > A,F5: filter @ A] :
( ( filterlim @ real @ A @ F2 @ F5 @ ( at_bot @ real ) )
= ( filterlim @ real @ A
@ ^ [X2: real] : ( F2 @ ( uminus_uminus @ real @ X2 ) )
@ F5
@ ( at_top @ real ) ) ) ).
% filterlim_at_bot_mirror
thf(fact_6148_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_6149_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_6150_less__separate,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,Y2: A] :
( ( ord_less @ A @ X @ Y2 )
=> ? [A6: A,B4: A] :
( ( member @ A @ X @ ( set_ord_lessThan @ A @ A6 ) )
& ( member @ A @ Y2 @ ( set_ord_greaterThan @ A @ B4 ) )
& ( ( inf_inf @ ( set @ A ) @ ( set_ord_lessThan @ A @ A6 ) @ ( set_ord_greaterThan @ A @ B4 ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% less_separate
thf(fact_6151_eventually__at__rightI,axiom,
! [A: $tType] :
( ( topolo2564578578187576103pology @ A )
=> ! [A3: A,B2: A,P: A > $o] :
( ! [X3: A] :
( ( member @ A @ X3 @ ( set_or5935395276787703475ssThan @ A @ A3 @ B2 ) )
=> ( P @ X3 ) )
=> ( ( ord_less @ A @ A3 @ B2 )
=> ( eventually @ A @ P @ ( topolo174197925503356063within @ A @ A3 @ ( set_ord_greaterThan @ A @ A3 ) ) ) ) ) ) ).
% eventually_at_rightI
thf(fact_6152_filterlim__at__left__to__right,axiom,
! [A: $tType,F2: real > A,F5: filter @ A,A3: real] :
( ( filterlim @ real @ A @ F2 @ F5 @ ( topolo174197925503356063within @ real @ A3 @ ( set_ord_lessThan @ real @ A3 ) ) )
= ( filterlim @ real @ A
@ ^ [X2: real] : ( F2 @ ( uminus_uminus @ real @ X2 ) )
@ F5
@ ( topolo174197925503356063within @ real @ ( uminus_uminus @ real @ A3 ) @ ( set_ord_greaterThan @ real @ ( uminus_uminus @ real @ A3 ) ) ) ) ) ).
% filterlim_at_left_to_right
thf(fact_6153_eventually__at__left__to__right,axiom,
! [P: real > $o,A3: real] :
( ( eventually @ real @ P @ ( topolo174197925503356063within @ real @ A3 @ ( set_ord_lessThan @ real @ A3 ) ) )
= ( eventually @ real
@ ^ [X2: real] : ( P @ ( uminus_uminus @ real @ X2 ) )
@ ( topolo174197925503356063within @ real @ ( uminus_uminus @ real @ A3 ) @ ( set_ord_greaterThan @ real @ ( uminus_uminus @ real @ A3 ) ) ) ) ) ).
% eventually_at_left_to_right
thf(fact_6154_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_6155_exp__at__bot,axiom,
filterlim @ real @ real @ ( exp @ real ) @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_bot @ real ) ).
% exp_at_bot
thf(fact_6156_filterlim__times__pos,axiom,
! [A: $tType,B: $tType] :
( ( ( linordered_field @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F2: B > A,P4: A,F13: filter @ B,C2: A,L: A] :
( ( filterlim @ B @ A @ F2 @ ( topolo174197925503356063within @ A @ P4 @ ( set_ord_greaterThan @ A @ P4 ) ) @ F13 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( L
= ( times_times @ A @ C2 @ P4 ) )
=> ( filterlim @ B @ A
@ ^ [X2: B] : ( times_times @ A @ C2 @ ( F2 @ X2 ) )
@ ( topolo174197925503356063within @ A @ L @ ( set_ord_greaterThan @ A @ L ) )
@ F13 ) ) ) ) ) ).
% filterlim_times_pos
thf(fact_6157_filterlim__at__bot__dense,axiom,
! [A: $tType,B: $tType] :
( ( ( dense_linorder @ B )
& ( no_bot @ B ) )
=> ! [F2: A > B,F5: filter @ A] :
( ( filterlim @ A @ B @ F2 @ ( at_bot @ B ) @ F5 )
= ( ! [Z8: B] :
( eventually @ A
@ ^ [X2: A] : ( ord_less @ B @ ( F2 @ X2 ) @ Z8 )
@ F5 ) ) ) ) ).
% filterlim_at_bot_dense
thf(fact_6158_tendsto__imp__filterlim__at__right,axiom,
! [B: $tType,A: $tType] :
( ( topolo2564578578187576103pology @ B )
=> ! [F2: A > B,L4: B,F5: filter @ A] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ L4 ) @ F5 )
=> ( ( eventually @ A
@ ^ [X2: A] : ( ord_less @ B @ L4 @ ( F2 @ X2 ) )
@ F5 )
=> ( filterlim @ A @ B @ F2 @ ( topolo174197925503356063within @ B @ L4 @ ( set_ord_greaterThan @ B @ L4 ) ) @ F5 ) ) ) ) ).
% tendsto_imp_filterlim_at_right
thf(fact_6159_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_6160_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_6161_filterlim__at__bot__lt,axiom,
! [A: $tType,B: $tType] :
( ( unboun7993243217541854897norder @ B )
=> ! [F2: A > B,F5: filter @ A,C2: B] :
( ( filterlim @ A @ B @ F2 @ ( at_bot @ B ) @ F5 )
= ( ! [Z8: B] :
( ( ord_less @ B @ Z8 @ C2 )
=> ( eventually @ A
@ ^ [X2: A] : ( ord_less_eq @ B @ ( F2 @ X2 ) @ Z8 )
@ F5 ) ) ) ) ) ).
% filterlim_at_bot_lt
thf(fact_6162_lhopital__right__at__top__at__top,axiom,
! [F2: real > real,A3: real,G: real > real,F6: real > real,G3: real > real] :
( ( filterlim @ real @ real @ F2 @ ( at_top @ real ) @ ( topolo174197925503356063within @ real @ A3 @ ( set_ord_greaterThan @ real @ A3 ) ) )
=> ( ( filterlim @ real @ real @ G @ ( at_top @ real ) @ ( topolo174197925503356063within @ real @ A3 @ ( set_ord_greaterThan @ real @ A3 ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] : ( has_field_derivative @ real @ F2 @ ( F6 @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ A3 @ ( set_ord_greaterThan @ real @ A3 ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] : ( has_field_derivative @ real @ G @ ( G3 @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ A3 @ ( set_ord_greaterThan @ real @ A3 ) ) )
=> ( ( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( F6 @ X2 ) @ ( G3 @ X2 ) )
@ ( at_top @ real )
@ ( topolo174197925503356063within @ real @ A3 @ ( set_ord_greaterThan @ real @ A3 ) ) )
=> ( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( at_top @ real )
@ ( topolo174197925503356063within @ real @ A3 @ ( set_ord_greaterThan @ real @ A3 ) ) ) ) ) ) ) ) ).
% lhopital_right_at_top_at_top
thf(fact_6163_lhopital__right__0,axiom,
! [F0: real > real,G0: real > real,G3: real > real,F6: real > real,F5: 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
@ ^ [X2: real] :
( ( G0 @ X2 )
!= ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] :
( ( G3 @ X2 )
!= ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] : ( has_field_derivative @ real @ F0 @ ( F6 @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] : ( has_field_derivative @ real @ G0 @ ( G3 @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) )
=> ( ( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( F6 @ X2 ) @ ( G3 @ X2 ) )
@ F5
@ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) )
=> ( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( F0 @ X2 ) @ ( G0 @ X2 ) )
@ F5
@ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) ) ) ) ) ) ) ) ) ).
% lhopital_right_0
thf(fact_6164_lhopital__right,axiom,
! [F2: real > real,X: real,G: real > real,G3: real > real,F6: real > real,F5: filter @ real] :
( ( filterlim @ real @ real @ F2 @ ( 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
@ ^ [X2: real] :
( ( G @ X2 )
!= ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ real @ X @ ( set_ord_greaterThan @ real @ X ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] :
( ( G3 @ X2 )
!= ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ real @ X @ ( set_ord_greaterThan @ real @ X ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] : ( has_field_derivative @ real @ F2 @ ( F6 @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ X @ ( set_ord_greaterThan @ real @ X ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] : ( has_field_derivative @ real @ G @ ( G3 @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ X @ ( set_ord_greaterThan @ real @ X ) ) )
=> ( ( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( F6 @ X2 ) @ ( G3 @ X2 ) )
@ F5
@ ( topolo174197925503356063within @ real @ X @ ( set_ord_greaterThan @ real @ X ) ) )
=> ( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ F5
@ ( topolo174197925503356063within @ real @ X @ ( set_ord_greaterThan @ real @ X ) ) ) ) ) ) ) ) ) ) ).
% lhopital_right
thf(fact_6165_lhopital__at__top__at__bot,axiom,
! [F2: real > real,A3: real,G: real > real,F6: real > real,G3: real > real] :
( ( filterlim @ real @ real @ F2 @ ( at_top @ real ) @ ( topolo174197925503356063within @ real @ A3 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( filterlim @ real @ real @ G @ ( at_bot @ real ) @ ( topolo174197925503356063within @ real @ A3 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] : ( has_field_derivative @ real @ F2 @ ( F6 @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ A3 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] : ( has_field_derivative @ real @ G @ ( G3 @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ A3 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( F6 @ X2 ) @ ( G3 @ X2 ) )
@ ( at_bot @ real )
@ ( topolo174197925503356063within @ real @ A3 @ ( top_top @ ( set @ real ) ) ) )
=> ( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( at_bot @ real )
@ ( topolo174197925503356063within @ real @ A3 @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ) ) ).
% lhopital_at_top_at_bot
thf(fact_6166_filterlim__pow__at__bot__odd,axiom,
! [N: nat,F2: real > real,F5: filter @ real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( filterlim @ real @ real @ F2 @ ( at_bot @ real ) @ F5 )
=> ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( filterlim @ real @ real
@ ^ [X2: real] : ( power_power @ real @ ( F2 @ X2 ) @ N )
@ ( at_bot @ real )
@ F5 ) ) ) ) ).
% filterlim_pow_at_bot_odd
thf(fact_6167_lhopital__right__at__top,axiom,
! [G: real > real,X: real,G3: real > real,F2: real > real,F6: real > real,Y2: real] :
( ( filterlim @ real @ real @ G @ ( at_top @ real ) @ ( topolo174197925503356063within @ real @ X @ ( set_ord_greaterThan @ real @ X ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] :
( ( G3 @ X2 )
!= ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ real @ X @ ( set_ord_greaterThan @ real @ X ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] : ( has_field_derivative @ real @ F2 @ ( F6 @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ X @ ( set_ord_greaterThan @ real @ X ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] : ( has_field_derivative @ real @ G @ ( G3 @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ X @ ( set_ord_greaterThan @ real @ X ) ) )
=> ( ( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( F6 @ X2 ) @ ( G3 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ Y2 )
@ ( topolo174197925503356063within @ real @ X @ ( set_ord_greaterThan @ real @ X ) ) )
=> ( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ Y2 )
@ ( topolo174197925503356063within @ real @ X @ ( set_ord_greaterThan @ real @ X ) ) ) ) ) ) ) ) ).
% lhopital_right_at_top
thf(fact_6168_lhopital__right__0__at__top,axiom,
! [G: real > real,G3: real > real,F2: real > real,F6: 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
@ ^ [X2: real] :
( ( G3 @ X2 )
!= ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] : ( has_field_derivative @ real @ F2 @ ( F6 @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] : ( has_field_derivative @ real @ G @ ( G3 @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) )
=> ( ( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( F6 @ X2 ) @ ( G3 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ X )
@ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) )
=> ( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ X )
@ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) ) ) ) ) ) ) ).
% lhopital_right_0_at_top
thf(fact_6169_lhopital__left__at__top__at__bot,axiom,
! [F2: real > real,A3: real,G: real > real,F6: real > real,G3: real > real] :
( ( filterlim @ real @ real @ F2 @ ( at_top @ real ) @ ( topolo174197925503356063within @ real @ A3 @ ( set_ord_lessThan @ real @ A3 ) ) )
=> ( ( filterlim @ real @ real @ G @ ( at_bot @ real ) @ ( topolo174197925503356063within @ real @ A3 @ ( set_ord_lessThan @ real @ A3 ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] : ( has_field_derivative @ real @ F2 @ ( F6 @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ A3 @ ( set_ord_lessThan @ real @ A3 ) ) )
=> ( ( eventually @ real
@ ^ [X2: real] : ( has_field_derivative @ real @ G @ ( G3 @ X2 ) @ ( topolo174197925503356063within @ real @ X2 @ ( top_top @ ( set @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ A3 @ ( set_ord_lessThan @ real @ A3 ) ) )
=> ( ( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( F6 @ X2 ) @ ( G3 @ X2 ) )
@ ( at_bot @ real )
@ ( topolo174197925503356063within @ real @ A3 @ ( set_ord_lessThan @ real @ A3 ) ) )
=> ( filterlim @ real @ real
@ ^ [X2: real] : ( divide_divide @ real @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( at_bot @ real )
@ ( topolo174197925503356063within @ real @ A3 @ ( set_ord_lessThan @ real @ A3 ) ) ) ) ) ) ) ) ).
% lhopital_left_at_top_at_bot
thf(fact_6170_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_6171_sequentially__imp__eventually__at__right,axiom,
! [A: $tType] :
( ( ( topolo3112930676232923870pology @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [A3: A,B2: A,P: A > $o] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ! [F3: nat > A] :
( ! [N4: nat] : ( ord_less @ A @ A3 @ ( F3 @ N4 ) )
=> ( ! [N4: nat] : ( ord_less @ A @ ( F3 @ N4 ) @ B2 )
=> ( ( order_antimono @ nat @ A @ F3 )
=> ( ( filterlim @ nat @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ A3 ) @ ( at_top @ nat ) )
=> ( eventually @ nat
@ ^ [N5: nat] : ( P @ ( F3 @ N5 ) )
@ ( at_top @ nat ) ) ) ) ) )
=> ( eventually @ A @ P @ ( topolo174197925503356063within @ A @ A3 @ ( set_ord_greaterThan @ A @ A3 ) ) ) ) ) ) ).
% sequentially_imp_eventually_at_right
thf(fact_6172_GMVT,axiom,
! [A3: real,B2: real,F2: real > real,G: real > real] :
( ( ord_less @ real @ A3 @ B2 )
=> ( ! [X3: real] :
( ( ( ord_less_eq @ real @ A3 @ X3 )
& ( ord_less_eq @ real @ X3 @ B2 ) )
=> ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) @ F2 ) )
=> ( ! [X3: real] :
( ( ( ord_less @ real @ A3 @ X3 )
& ( ord_less @ real @ X3 @ B2 ) )
=> ( differentiable @ real @ real @ F2 @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) )
=> ( ! [X3: real] :
( ( ( ord_less_eq @ real @ A3 @ X3 )
& ( ord_less_eq @ real @ X3 @ B2 ) )
=> ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) @ G ) )
=> ( ! [X3: real] :
( ( ( ord_less @ real @ A3 @ X3 )
& ( ord_less @ real @ X3 @ B2 ) )
=> ( differentiable @ real @ real @ G @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) )
=> ? [G_c: real,F_c: real,C3: real] :
( ( has_field_derivative @ real @ G @ G_c @ ( topolo174197925503356063within @ real @ C3 @ ( top_top @ ( set @ real ) ) ) )
& ( has_field_derivative @ real @ F2 @ F_c @ ( topolo174197925503356063within @ real @ C3 @ ( top_top @ ( set @ real ) ) ) )
& ( ord_less @ real @ A3 @ C3 )
& ( ord_less @ real @ C3 @ B2 )
& ( ( times_times @ real @ ( minus_minus @ real @ ( F2 @ B2 ) @ ( F2 @ A3 ) ) @ G_c )
= ( times_times @ real @ ( minus_minus @ real @ ( G @ B2 ) @ ( G @ A3 ) ) @ F_c ) ) ) ) ) ) ) ) ).
% GMVT
thf(fact_6173_differentiable__cmult__left__iff,axiom,
! [A: $tType,B: $tType] :
( ( ( real_V822414075346904944vector @ B )
& ( real_V3459762299906320749_field @ A ) )
=> ! [C2: A,Q5: B > A,T2: B] :
( ( differentiable @ B @ A
@ ^ [T3: B] : ( times_times @ A @ C2 @ ( Q5 @ T3 ) )
@ ( topolo174197925503356063within @ B @ T2 @ ( top_top @ ( set @ B ) ) ) )
= ( ( C2
= ( zero_zero @ A ) )
| ( differentiable @ B @ A @ Q5 @ ( topolo174197925503356063within @ B @ T2 @ ( top_top @ ( set @ B ) ) ) ) ) ) ) ).
% differentiable_cmult_left_iff
thf(fact_6174_differentiable__cmult__right__iff,axiom,
! [A: $tType,B: $tType] :
( ( ( real_V822414075346904944vector @ B )
& ( real_V3459762299906320749_field @ A ) )
=> ! [Q5: B > A,C2: A,T2: B] :
( ( differentiable @ B @ A
@ ^ [T3: B] : ( times_times @ A @ ( Q5 @ T3 ) @ C2 )
@ ( topolo174197925503356063within @ B @ T2 @ ( top_top @ ( set @ B ) ) ) )
= ( ( C2
= ( zero_zero @ A ) )
| ( differentiable @ B @ A @ Q5 @ ( topolo174197925503356063within @ B @ T2 @ ( top_top @ ( set @ B ) ) ) ) ) ) ) ).
% differentiable_cmult_right_iff
thf(fact_6175_differentiable__minus,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,F5: filter @ A] :
( ( differentiable @ A @ B @ F2 @ F5 )
=> ( differentiable @ A @ B
@ ^ [X2: A] : ( uminus_uminus @ B @ ( F2 @ X2 ) )
@ F5 ) ) ) ).
% differentiable_minus
thf(fact_6176_differentiable__diff,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F2: A > B,F5: filter @ A,G: A > B] :
( ( differentiable @ A @ B @ F2 @ F5 )
=> ( ( differentiable @ A @ B @ G @ F5 )
=> ( differentiable @ A @ B
@ ^ [X2: A] : ( minus_minus @ B @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ F5 ) ) ) ) ).
% differentiable_diff
thf(fact_6177_decseq__Suc__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( order_antimono @ nat @ A )
= ( ^ [F4: nat > A] :
! [N5: nat] : ( ord_less_eq @ A @ ( F4 @ ( suc @ N5 ) ) @ ( F4 @ N5 ) ) ) ) ) ).
% decseq_Suc_iff
thf(fact_6178_decseq__SucI,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X8: nat > A] :
( ! [N2: nat] : ( ord_less_eq @ A @ ( X8 @ ( suc @ N2 ) ) @ ( X8 @ N2 ) )
=> ( order_antimono @ nat @ A @ X8 ) ) ) ).
% decseq_SucI
thf(fact_6179_decseq__SucD,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A4: nat > A,I: nat] :
( ( order_antimono @ nat @ A @ A4 )
=> ( ord_less_eq @ A @ ( A4 @ ( suc @ I ) ) @ ( A4 @ I ) ) ) ) ).
% decseq_SucD
thf(fact_6180_differentiable__divide,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V3459762299906320749_field @ B ) )
=> ! [F2: A > B,X: A,S: set @ A,G: A > B] :
( ( differentiable @ A @ B @ F2 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( ( differentiable @ A @ B @ G @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( ( ( G @ X )
!= ( zero_zero @ B ) )
=> ( differentiable @ A @ B
@ ^ [X2: A] : ( divide_divide @ B @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ) ).
% differentiable_divide
thf(fact_6181_differentiable__inverse,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V3459762299906320749_field @ B ) )
=> ! [F2: A > B,X: A,S: set @ A] :
( ( differentiable @ A @ B @ F2 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( ( ( F2 @ X )
!= ( zero_zero @ B ) )
=> ( differentiable @ A @ B
@ ^ [X2: A] : ( inverse_inverse @ B @ ( F2 @ X2 ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ).
% differentiable_inverse
thf(fact_6182_max__of__antimono,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( linorder @ B ) )
=> ! [F2: A > B,X: A,Y2: A] :
( ( order_antimono @ A @ B @ F2 )
=> ( ( ord_max @ B @ ( F2 @ X ) @ ( F2 @ Y2 ) )
= ( F2 @ ( ord_min @ A @ X @ Y2 ) ) ) ) ) ).
% max_of_antimono
thf(fact_6183_min__of__antimono,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( linorder @ B ) )
=> ! [F2: A > B,X: A,Y2: A] :
( ( order_antimono @ A @ B @ F2 )
=> ( ( ord_min @ B @ ( F2 @ X ) @ ( F2 @ Y2 ) )
= ( F2 @ ( ord_max @ A @ X @ Y2 ) ) ) ) ) ).
% min_of_antimono
thf(fact_6184_greaterThan__0,axiom,
( ( set_ord_greaterThan @ nat @ ( zero_zero @ nat ) )
= ( image @ nat @ nat @ suc @ ( top_top @ ( set @ nat ) ) ) ) ).
% greaterThan_0
thf(fact_6185_greaterThan__Suc,axiom,
! [K: nat] :
( ( set_ord_greaterThan @ nat @ ( suc @ K ) )
= ( minus_minus @ ( set @ nat ) @ ( set_ord_greaterThan @ nat @ K ) @ ( insert @ nat @ ( suc @ K ) @ ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% greaterThan_Suc
thf(fact_6186_tendsto__at__right__sequentially,axiom,
! [C: $tType,B: $tType] :
( ( ( topolo3112930676232923870pology @ B )
& ( topolo1944317154257567458pology @ B )
& ( topolo4958980785337419405_space @ C ) )
=> ! [A3: B,B2: B,X8: B > C,L4: C] :
( ( ord_less @ B @ A3 @ B2 )
=> ( ! [S7: nat > B] :
( ! [N4: nat] : ( ord_less @ B @ A3 @ ( S7 @ N4 ) )
=> ( ! [N4: nat] : ( ord_less @ B @ ( S7 @ N4 ) @ B2 )
=> ( ( order_antimono @ nat @ B @ S7 )
=> ( ( filterlim @ nat @ B @ S7 @ ( topolo7230453075368039082e_nhds @ B @ A3 ) @ ( at_top @ nat ) )
=> ( filterlim @ nat @ C
@ ^ [N5: nat] : ( X8 @ ( S7 @ N5 ) )
@ ( topolo7230453075368039082e_nhds @ C @ L4 )
@ ( at_top @ nat ) ) ) ) ) )
=> ( filterlim @ B @ C @ X8 @ ( topolo7230453075368039082e_nhds @ C @ L4 ) @ ( topolo174197925503356063within @ B @ A3 @ ( set_ord_greaterThan @ B @ A3 ) ) ) ) ) ) ).
% tendsto_at_right_sequentially
thf(fact_6187_Bfun__inverse,axiom,
! [A: $tType,B: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [F2: B > A,A3: A,F5: filter @ B] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ A3 ) @ F5 )
=> ( ( A3
!= ( zero_zero @ A ) )
=> ( bfun @ B @ A
@ ^ [X2: B] : ( inverse_inverse @ A @ ( F2 @ X2 ) )
@ F5 ) ) ) ) ).
% Bfun_inverse
thf(fact_6188_MVT,axiom,
! [A3: real,B2: real,F2: real > real] :
( ( ord_less @ real @ A3 @ B2 )
=> ( ( topolo81223032696312382ous_on @ real @ real @ ( set_or1337092689740270186AtMost @ real @ A3 @ B2 ) @ F2 )
=> ( ! [X3: real] :
( ( ord_less @ real @ A3 @ X3 )
=> ( ( ord_less @ real @ X3 @ B2 )
=> ( differentiable @ real @ real @ F2 @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) ) )
=> ? [L3: real,Z4: real] :
( ( ord_less @ real @ A3 @ Z4 )
& ( ord_less @ real @ Z4 @ B2 )
& ( has_field_derivative @ real @ F2 @ L3 @ ( topolo174197925503356063within @ real @ Z4 @ ( top_top @ ( set @ real ) ) ) )
& ( ( minus_minus @ real @ ( F2 @ B2 ) @ ( F2 @ A3 ) )
= ( times_times @ real @ ( minus_minus @ real @ B2 @ A3 ) @ L3 ) ) ) ) ) ) ).
% MVT
thf(fact_6189_continuous__on__ln,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [S: set @ A,F2: A > real] :
( ( topolo81223032696312382ous_on @ A @ real @ S @ F2 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ S )
=> ( ( F2 @ X3 )
!= ( zero_zero @ real ) ) )
=> ( topolo81223032696312382ous_on @ A @ real @ S
@ ^ [X2: A] : ( ln_ln @ real @ ( F2 @ X2 ) ) ) ) ) ) ).
% continuous_on_ln
thf(fact_6190_continuous__on__inverse,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ A )
& ( real_V8999393235501362500lgebra @ B ) )
=> ! [S: set @ A,F2: A > B] :
( ( topolo81223032696312382ous_on @ A @ B @ S @ F2 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ S )
=> ( ( F2 @ X3 )
!= ( zero_zero @ B ) ) )
=> ( topolo81223032696312382ous_on @ A @ B @ S
@ ^ [X2: A] : ( inverse_inverse @ B @ ( F2 @ X2 ) ) ) ) ) ) ).
% continuous_on_inverse
thf(fact_6191_continuous__on__divide,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ A )
& ( real_V3459762299906320749_field @ B ) )
=> ! [S: set @ A,F2: A > B,G: A > B] :
( ( topolo81223032696312382ous_on @ A @ B @ S @ F2 )
=> ( ( topolo81223032696312382ous_on @ A @ B @ S @ G )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ S )
=> ( ( G @ X3 )
!= ( zero_zero @ B ) ) )
=> ( topolo81223032696312382ous_on @ A @ B @ S
@ ^ [X2: A] : ( divide_divide @ B @ ( F2 @ X2 ) @ ( G @ X2 ) ) ) ) ) ) ) ).
% continuous_on_divide
thf(fact_6192_continuous__on__op__minus,axiom,
! [A: $tType] :
( ( topolo1633459387980952147up_add @ A )
=> ! [S: set @ A,X: A] : ( topolo81223032696312382ous_on @ A @ A @ S @ ( minus_minus @ A @ X ) ) ) ).
% continuous_on_op_minus
thf(fact_6193_continuous__on__diff,axiom,
! [B: $tType,D: $tType] :
( ( ( topolo4958980785337419405_space @ D )
& ( topolo1633459387980952147up_add @ B ) )
=> ! [S: set @ D,F2: D > B,G: D > B] :
( ( topolo81223032696312382ous_on @ D @ B @ S @ F2 )
=> ( ( topolo81223032696312382ous_on @ D @ B @ S @ G )
=> ( topolo81223032696312382ous_on @ D @ B @ S
@ ^ [X2: D] : ( minus_minus @ B @ ( F2 @ X2 ) @ ( G @ X2 ) ) ) ) ) ) ).
% continuous_on_diff
thf(fact_6194_continuous__on__arsinh_H,axiom,
! [A4: set @ real,F2: real > real] :
( ( topolo81223032696312382ous_on @ real @ real @ A4 @ F2 )
=> ( topolo81223032696312382ous_on @ real @ real @ A4
@ ^ [X2: real] : ( arsinh @ real @ ( F2 @ X2 ) ) ) ) ).
% continuous_on_arsinh'
thf(fact_6195_continuous__on__arctan,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [S: set @ A,F2: A > real] :
( ( topolo81223032696312382ous_on @ A @ real @ S @ F2 )
=> ( topolo81223032696312382ous_on @ A @ real @ S
@ ^ [X2: A] : ( arctan @ ( F2 @ X2 ) ) ) ) ) ).
% continuous_on_arctan
thf(fact_6196_continuous__on__cosh,axiom,
! [A: $tType,C: $tType] :
( ( ( topolo4958980785337419405_space @ C )
& ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [A4: set @ C,F2: C > A] :
( ( topolo81223032696312382ous_on @ C @ A @ A4 @ F2 )
=> ( topolo81223032696312382ous_on @ C @ A @ A4
@ ^ [X2: C] : ( cosh @ A @ ( F2 @ X2 ) ) ) ) ) ).
% continuous_on_cosh
thf(fact_6197_continuous__on__exp,axiom,
! [A: $tType,C: $tType] :
( ( ( topolo4958980785337419405_space @ C )
& ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [S: set @ C,F2: C > A] :
( ( topolo81223032696312382ous_on @ C @ A @ S @ F2 )
=> ( topolo81223032696312382ous_on @ C @ A @ S
@ ^ [X2: C] : ( exp @ A @ ( F2 @ X2 ) ) ) ) ) ).
% continuous_on_exp
thf(fact_6198_continuous__on__sin,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_Vector_banach @ B )
& ( real_V3459762299906320749_field @ B ) )
=> ! [S: set @ A,F2: A > B] :
( ( topolo81223032696312382ous_on @ A @ B @ S @ F2 )
=> ( topolo81223032696312382ous_on @ A @ B @ S
@ ^ [X2: A] : ( sin @ B @ ( F2 @ X2 ) ) ) ) ) ).
% continuous_on_sin
thf(fact_6199_continuous__on__cos,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_Vector_banach @ B )
& ( real_V3459762299906320749_field @ B ) )
=> ! [S: set @ A,F2: A > B] :
( ( topolo81223032696312382ous_on @ A @ B @ S @ F2 )
=> ( topolo81223032696312382ous_on @ A @ B @ S
@ ^ [X2: A] : ( cos @ B @ ( F2 @ X2 ) ) ) ) ) ).
% continuous_on_cos
thf(fact_6200_continuous__on__sinh,axiom,
! [A: $tType,C: $tType] :
( ( ( topolo4958980785337419405_space @ C )
& ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [A4: set @ C,F2: C > A] :
( ( topolo81223032696312382ous_on @ C @ A @ A4 @ F2 )
=> ( topolo81223032696312382ous_on @ C @ A @ A4
@ ^ [X2: C] : ( sinh @ A @ ( F2 @ X2 ) ) ) ) ) ).
% continuous_on_sinh
thf(fact_6201_continuous__on__pochhammer,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [A4: set @ A,N: nat] :
( topolo81223032696312382ous_on @ A @ A @ A4
@ ^ [Z6: A] : ( comm_s3205402744901411588hammer @ A @ Z6 @ N ) ) ) ).
% continuous_on_pochhammer
thf(fact_6202_continuous__on__pochhammer_H,axiom,
! [A: $tType,C: $tType] :
( ( ( topolo4958980785337419405_space @ C )
& ( real_V3459762299906320749_field @ A ) )
=> ! [S: set @ C,F2: C > A,N: nat] :
( ( topolo81223032696312382ous_on @ C @ A @ S @ F2 )
=> ( topolo81223032696312382ous_on @ C @ A @ S
@ ^ [X2: C] : ( comm_s3205402744901411588hammer @ A @ ( F2 @ X2 ) @ N ) ) ) ) ).
% continuous_on_pochhammer'
thf(fact_6203_continuous__on__arsinh,axiom,
! [A4: set @ real] : ( topolo81223032696312382ous_on @ real @ real @ A4 @ ( arsinh @ real ) ) ).
% continuous_on_arsinh
thf(fact_6204_continuous__on__minus,axiom,
! [B: $tType,C: $tType] :
( ( ( topolo4958980785337419405_space @ C )
& ( topolo1633459387980952147up_add @ B ) )
=> ! [S: set @ C,F2: C > B] :
( ( topolo81223032696312382ous_on @ C @ B @ S @ F2 )
=> ( topolo81223032696312382ous_on @ C @ B @ S
@ ^ [X2: C] : ( uminus_uminus @ B @ ( F2 @ X2 ) ) ) ) ) ).
% continuous_on_minus
thf(fact_6205_continuous__on__sgn,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [S: set @ A,F2: A > B] :
( ( topolo81223032696312382ous_on @ A @ B @ S @ F2 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ S )
=> ( ( F2 @ X3 )
!= ( zero_zero @ B ) ) )
=> ( topolo81223032696312382ous_on @ A @ B @ S
@ ^ [X2: A] : ( sgn_sgn @ B @ ( F2 @ X2 ) ) ) ) ) ) ).
% continuous_on_sgn
thf(fact_6206_continuous__on__sin__real,axiom,
! [A3: real,B2: real] : ( topolo81223032696312382ous_on @ real @ real @ ( set_or1337092689740270186AtMost @ real @ A3 @ B2 ) @ ( sin @ real ) ) ).
% continuous_on_sin_real
thf(fact_6207_continuous__on__cos__real,axiom,
! [A3: real,B2: real] : ( topolo81223032696312382ous_on @ real @ real @ ( set_or1337092689740270186AtMost @ real @ A3 @ B2 ) @ ( cos @ real ) ) ).
% continuous_on_cos_real
thf(fact_6208_continuous__on__powr,axiom,
! [C: $tType] :
( ( topolo4958980785337419405_space @ C )
=> ! [S: set @ C,F2: C > real,G: C > real] :
( ( topolo81223032696312382ous_on @ C @ real @ S @ F2 )
=> ( ( topolo81223032696312382ous_on @ C @ real @ S @ G )
=> ( ! [X3: C] :
( ( member @ C @ X3 @ S )
=> ( ( F2 @ X3 )
!= ( zero_zero @ real ) ) )
=> ( topolo81223032696312382ous_on @ C @ real @ S
@ ^ [X2: C] : ( powr @ real @ ( F2 @ X2 ) @ ( G @ X2 ) ) ) ) ) ) ) ).
% continuous_on_powr
thf(fact_6209_Bseq__minus__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X8: nat > A] :
( ( bfun @ nat @ A
@ ^ [N5: nat] : ( uminus_uminus @ A @ ( X8 @ N5 ) )
@ ( at_top @ nat ) )
= ( bfun @ nat @ A @ X8 @ ( at_top @ nat ) ) ) ) ).
% Bseq_minus_iff
thf(fact_6210_Bseq__Suc__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F2: nat > A] :
( ( bfun @ nat @ A
@ ^ [N5: nat] : ( F2 @ ( suc @ N5 ) )
@ ( at_top @ nat ) )
= ( bfun @ nat @ A @ F2 @ ( at_top @ nat ) ) ) ) ).
% Bseq_Suc_iff
thf(fact_6211_open__Collect__less,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ A )
& ( topolo1944317154257567458pology @ B ) )
=> ! [F2: A > B,G: A > B] :
( ( topolo81223032696312382ous_on @ A @ B @ ( top_top @ ( set @ A ) ) @ F2 )
=> ( ( topolo81223032696312382ous_on @ A @ B @ ( top_top @ ( set @ A ) ) @ G )
=> ( topolo1002775350975398744n_open @ A
@ ( collect @ A
@ ^ [X2: A] : ( ord_less @ B @ ( F2 @ X2 ) @ ( G @ X2 ) ) ) ) ) ) ) ).
% open_Collect_less
thf(fact_6212_continuous__on__tan,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [S: set @ A,F2: A > A] :
( ( topolo81223032696312382ous_on @ A @ A @ S @ F2 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ S )
=> ( ( cos @ A @ ( F2 @ X3 ) )
!= ( zero_zero @ A ) ) )
=> ( topolo81223032696312382ous_on @ A @ A @ S
@ ^ [X2: A] : ( tan @ A @ ( F2 @ X2 ) ) ) ) ) ) ).
% continuous_on_tan
thf(fact_6213_continuous__on__cot,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [S: set @ A,F2: A > A] :
( ( topolo81223032696312382ous_on @ A @ A @ S @ F2 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ S )
=> ( ( sin @ A @ ( F2 @ X3 ) )
!= ( zero_zero @ A ) ) )
=> ( topolo81223032696312382ous_on @ A @ A @ S
@ ^ [X2: A] : ( cot @ A @ ( F2 @ X2 ) ) ) ) ) ) ).
% continuous_on_cot
thf(fact_6214_continuous__on__tanh,axiom,
! [A: $tType,C: $tType] :
( ( ( topolo4958980785337419405_space @ C )
& ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [A4: set @ C,F2: C > A] :
( ( topolo81223032696312382ous_on @ C @ A @ A4 @ F2 )
=> ( ! [X3: C] :
( ( member @ C @ X3 @ A4 )
=> ( ( cosh @ A @ ( F2 @ X3 ) )
!= ( zero_zero @ A ) ) )
=> ( topolo81223032696312382ous_on @ C @ A @ A4
@ ^ [X2: C] : ( tanh @ A @ ( F2 @ X2 ) ) ) ) ) ) ).
% continuous_on_tanh
thf(fact_6215_continuous__on__arcosh_H,axiom,
! [A4: set @ real,F2: real > real] :
( ( topolo81223032696312382ous_on @ real @ real @ A4 @ F2 )
=> ( ! [X3: real] :
( ( member @ real @ X3 @ A4 )
=> ( ord_less_eq @ real @ ( one_one @ real ) @ ( F2 @ X3 ) ) )
=> ( topolo81223032696312382ous_on @ real @ real @ A4
@ ^ [X2: real] : ( arcosh @ real @ ( F2 @ X2 ) ) ) ) ) ).
% continuous_on_arcosh'
thf(fact_6216_Bseq__cmult__iff,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [C2: A,F2: nat > A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( bfun @ nat @ A
@ ^ [X2: nat] : ( times_times @ A @ C2 @ ( F2 @ X2 ) )
@ ( at_top @ nat ) )
= ( bfun @ nat @ A @ F2 @ ( at_top @ nat ) ) ) ) ) ).
% Bseq_cmult_iff
thf(fact_6217_continuous__on__powr_H,axiom,
! [C: $tType] :
( ( topolo4958980785337419405_space @ C )
=> ! [S: set @ C,F2: C > real,G: C > real] :
( ( topolo81223032696312382ous_on @ C @ real @ S @ F2 )
=> ( ( topolo81223032696312382ous_on @ C @ real @ S @ G )
=> ( ! [X3: C] :
( ( member @ C @ X3 @ S )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( F2 @ X3 ) )
& ( ( ( F2 @ X3 )
= ( zero_zero @ real ) )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( G @ X3 ) ) ) ) )
=> ( topolo81223032696312382ous_on @ C @ real @ S
@ ^ [X2: C] : ( powr @ real @ ( F2 @ X2 ) @ ( G @ X2 ) ) ) ) ) ) ) ).
% continuous_on_powr'
thf(fact_6218_continuous__on__log,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [S: set @ A,F2: A > real,G: A > real] :
( ( topolo81223032696312382ous_on @ A @ real @ S @ F2 )
=> ( ( topolo81223032696312382ous_on @ A @ real @ S @ G )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ S )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( F2 @ X3 ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ S )
=> ( ( F2 @ X3 )
!= ( one_one @ real ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ S )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( G @ X3 ) ) )
=> ( topolo81223032696312382ous_on @ A @ real @ S
@ ^ [X2: A] : ( log2 @ ( F2 @ X2 ) @ ( G @ X2 ) ) ) ) ) ) ) ) ) ).
% continuous_on_log
thf(fact_6219_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_6220_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_6221_continuous__on__arccos,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [S: set @ A,F2: A > real] :
( ( topolo81223032696312382ous_on @ A @ real @ S @ F2 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ S )
=> ( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ ( F2 @ X3 ) )
& ( ord_less_eq @ real @ ( F2 @ X3 ) @ ( one_one @ real ) ) ) )
=> ( topolo81223032696312382ous_on @ A @ real @ S
@ ^ [X2: A] : ( arccos @ ( F2 @ X2 ) ) ) ) ) ) ).
% continuous_on_arccos
thf(fact_6222_continuous__on__arcsin,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [S: set @ A,F2: A > real] :
( ( topolo81223032696312382ous_on @ A @ real @ S @ F2 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ S )
=> ( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ ( F2 @ X3 ) )
& ( ord_less_eq @ real @ ( F2 @ X3 ) @ ( one_one @ real ) ) ) )
=> ( topolo81223032696312382ous_on @ A @ real @ S
@ ^ [X2: A] : ( arcsin @ ( F2 @ X2 ) ) ) ) ) ) ).
% continuous_on_arcsin
thf(fact_6223_continuous__on__artanh,axiom,
! [A4: set @ real] :
( ( ord_less_eq @ ( set @ real ) @ A4 @ ( set_or5935395276787703475ssThan @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ ( one_one @ real ) ) )
=> ( topolo81223032696312382ous_on @ real @ real @ A4 @ ( artanh @ real ) ) ) ).
% continuous_on_artanh
thf(fact_6224_continuous__on__artanh_H,axiom,
! [A4: set @ real,F2: real > real] :
( ( topolo81223032696312382ous_on @ real @ real @ A4 @ F2 )
=> ( ! [X3: real] :
( ( member @ real @ X3 @ A4 )
=> ( member @ real @ ( F2 @ X3 ) @ ( set_or5935395276787703475ssThan @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ ( one_one @ real ) ) ) )
=> ( topolo81223032696312382ous_on @ real @ real @ A4
@ ^ [X2: real] : ( artanh @ real @ ( F2 @ X2 ) ) ) ) ) ).
% continuous_on_artanh'
thf(fact_6225_mvt,axiom,
! [A3: real,B2: real,F2: real > real,F6: real > real > real] :
( ( ord_less @ real @ A3 @ B2 )
=> ( ( topolo81223032696312382ous_on @ real @ real @ ( set_or1337092689740270186AtMost @ real @ A3 @ B2 ) @ F2 )
=> ( ! [X3: real] :
( ( ord_less @ real @ A3 @ X3 )
=> ( ( ord_less @ real @ X3 @ B2 )
=> ( has_derivative @ real @ real @ F2 @ ( F6 @ X3 ) @ ( topolo174197925503356063within @ real @ X3 @ ( top_top @ ( set @ real ) ) ) ) ) )
=> ~ ! [Xi: real] :
( ( ord_less @ real @ A3 @ Xi )
=> ( ( ord_less @ real @ Xi @ B2 )
=> ( ( minus_minus @ real @ ( F2 @ B2 ) @ ( F2 @ A3 ) )
!= ( F6 @ Xi @ ( minus_minus @ real @ B2 @ A3 ) ) ) ) ) ) ) ) ).
% mvt
thf(fact_6226_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 ) )
@ ^ [X2: A] : ( ring_1_of_int @ B @ ( archim6421214686448440834_floor @ A @ X2 ) ) ) ) ).
% continuous_on_of_int_floor
thf(fact_6227_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 ) )
@ ^ [X2: A] : ( ring_1_of_int @ B @ ( archimedean_ceiling @ A @ X2 ) ) ) ) ).
% continuous_on_of_int_ceiling
thf(fact_6228_Bseq__iff1a,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X8: nat > A] :
( ( bfun @ nat @ A @ X8 @ ( at_top @ nat ) )
= ( ? [N7: nat] :
! [N5: nat] : ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( X8 @ N5 ) ) @ ( semiring_1_of_nat @ real @ ( suc @ N7 ) ) ) ) ) ) ).
% Bseq_iff1a
thf(fact_6229_Bseq__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X8: nat > A] :
( ( bfun @ nat @ A @ X8 @ ( at_top @ nat ) )
= ( ? [N7: nat] :
! [N5: nat] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( X8 @ N5 ) ) @ ( semiring_1_of_nat @ real @ ( suc @ N7 ) ) ) ) ) ) ).
% Bseq_iff
thf(fact_6230_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_6231_continuous__on__Icc__at__leftD,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo1944317154257567458pology @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [A3: A,B2: A,F2: A > B] :
( ( topolo81223032696312382ous_on @ A @ B @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) @ F2 )
=> ( ( ord_less @ A @ A3 @ B2 )
=> ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ ( F2 @ B2 ) ) @ ( topolo174197925503356063within @ A @ B2 @ ( set_ord_lessThan @ A @ B2 ) ) ) ) ) ) ).
% continuous_on_Icc_at_leftD
thf(fact_6232_continuous__on__Icc__at__rightD,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo1944317154257567458pology @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [A3: A,B2: A,F2: A > B] :
( ( topolo81223032696312382ous_on @ A @ B @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) @ F2 )
=> ( ( ord_less @ A @ A3 @ B2 )
=> ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ ( F2 @ A3 ) ) @ ( topolo174197925503356063within @ A @ A3 @ ( set_ord_greaterThan @ A @ A3 ) ) ) ) ) ) ).
% continuous_on_Icc_at_rightD
thf(fact_6233_continuous__on__IccI,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo1944317154257567458pology @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [F2: A > B,A3: A,B2: A] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ ( F2 @ A3 ) ) @ ( topolo174197925503356063within @ A @ A3 @ ( set_ord_greaterThan @ A @ A3 ) ) )
=> ( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ ( F2 @ B2 ) ) @ ( topolo174197925503356063within @ A @ B2 @ ( set_ord_lessThan @ A @ B2 ) ) )
=> ( ! [X3: A] :
( ( ord_less @ A @ A3 @ X3 )
=> ( ( ord_less @ A @ X3 @ B2 )
=> ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ ( F2 @ X3 ) ) @ ( topolo174197925503356063within @ A @ X3 @ ( top_top @ ( set @ A ) ) ) ) ) )
=> ( ( ord_less @ A @ A3 @ B2 )
=> ( topolo81223032696312382ous_on @ A @ B @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) @ F2 ) ) ) ) ) ) ).
% continuous_on_IccI
thf(fact_6234_Bseq__iff2,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X8: nat > A] :
( ( bfun @ nat @ A @ X8 @ ( at_top @ nat ) )
= ( ? [K3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K3 )
& ? [X2: A] :
! [N5: nat] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ ( X8 @ N5 ) @ ( uminus_uminus @ A @ X2 ) ) ) @ K3 ) ) ) ) ) ).
% Bseq_iff2
thf(fact_6235_Bseq__iff3,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X8: nat > A] :
( ( bfun @ nat @ A @ X8 @ ( at_top @ nat ) )
= ( ? [K3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K3 )
& ? [N7: nat] :
! [N5: nat] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ ( X8 @ N5 ) @ ( uminus_uminus @ A @ ( X8 @ N7 ) ) ) ) @ K3 ) ) ) ) ) ).
% Bseq_iff3
thf(fact_6236_eventually__filtercomap__at__topological,axiom,
! [A: $tType,B: $tType] :
( ( topolo4958980785337419405_space @ B )
=> ! [P: A > $o,F2: A > B,A4: B,B7: set @ B] :
( ( eventually @ A @ P @ ( filtercomap @ A @ B @ F2 @ ( topolo174197925503356063within @ B @ A4 @ B7 ) ) )
= ( ? [S8: set @ B] :
( ( topolo1002775350975398744n_open @ B @ S8 )
& ( member @ B @ A4 @ S8 )
& ! [X2: A] :
( ( member @ B @ ( F2 @ X2 ) @ ( minus_minus @ ( set @ B ) @ ( inf_inf @ ( set @ B ) @ S8 @ B7 ) @ ( insert @ B @ A4 @ ( bot_bot @ ( set @ B ) ) ) ) )
=> ( P @ X2 ) ) ) ) ) ) ).
% eventually_filtercomap_at_topological
thf(fact_6237_sequentially__imp__eventually__at__left,axiom,
! [A: $tType] :
( ( ( topolo3112930676232923870pology @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [B2: A,A3: A,P: A > $o] :
( ( ord_less @ A @ B2 @ A3 )
=> ( ! [F3: nat > A] :
( ! [N4: nat] : ( ord_less @ A @ B2 @ ( F3 @ N4 ) )
=> ( ! [N4: nat] : ( ord_less @ A @ ( F3 @ N4 ) @ A3 )
=> ( ( order_mono @ nat @ A @ F3 )
=> ( ( filterlim @ nat @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ A3 ) @ ( at_top @ nat ) )
=> ( eventually @ nat
@ ^ [N5: nat] : ( P @ ( F3 @ N5 ) )
@ ( at_top @ nat ) ) ) ) ) )
=> ( eventually @ A @ P @ ( topolo174197925503356063within @ A @ A3 @ ( set_ord_lessThan @ A @ A3 ) ) ) ) ) ) ).
% sequentially_imp_eventually_at_left
thf(fact_6238_decseq__eq__incseq,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ( ( order_antimono @ nat @ A )
= ( ^ [X7: nat > A] :
( order_mono @ nat @ A
@ ^ [N5: nat] : ( uminus_uminus @ A @ ( X7 @ N5 ) ) ) ) ) ) ).
% decseq_eq_incseq
thf(fact_6239_min__of__mono,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( linorder @ B ) )
=> ! [F2: A > B,M2: A,N: A] :
( ( order_mono @ A @ B @ F2 )
=> ( ( ord_min @ B @ ( F2 @ M2 ) @ ( F2 @ N ) )
= ( F2 @ ( ord_min @ A @ M2 @ N ) ) ) ) ) ).
% min_of_mono
thf(fact_6240_mono__funpow,axiom,
! [A: $tType] :
( ( ( lattice @ A )
& ( order_bot @ A ) )
=> ! [Q: A > A] :
( ( order_mono @ A @ A @ Q )
=> ( order_mono @ nat @ A
@ ^ [I2: nat] : ( compow @ ( A > A ) @ I2 @ Q @ ( bot_bot @ A ) ) ) ) ) ).
% mono_funpow
thf(fact_6241_funpow__mono,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [F2: A > A,A4: A,B7: A,N: nat] :
( ( order_mono @ A @ A @ F2 )
=> ( ( ord_less_eq @ A @ A4 @ B7 )
=> ( ord_less_eq @ A @ ( compow @ ( A > A ) @ N @ F2 @ A4 ) @ ( compow @ ( A > A ) @ N @ F2 @ B7 ) ) ) ) ) ).
% funpow_mono
thf(fact_6242_mono__invE,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( order @ B ) )
=> ! [F2: A > B,X: A,Y2: A] :
( ( order_mono @ A @ B @ F2 )
=> ( ( ord_less @ B @ ( F2 @ X ) @ ( F2 @ Y2 ) )
=> ( ord_less_eq @ A @ X @ Y2 ) ) ) ) ).
% mono_invE
thf(fact_6243_incseq__Suc__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( order_mono @ nat @ A )
= ( ^ [F4: nat > A] :
! [N5: nat] : ( ord_less_eq @ A @ ( F4 @ N5 ) @ ( F4 @ ( suc @ N5 ) ) ) ) ) ) ).
% incseq_Suc_iff
thf(fact_6244_incseq__SucI,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X8: nat > A] :
( ! [N2: nat] : ( ord_less_eq @ A @ ( X8 @ N2 ) @ ( X8 @ ( suc @ N2 ) ) )
=> ( order_mono @ nat @ A @ X8 ) ) ) ).
% incseq_SucI
thf(fact_6245_incseq__SucD,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A4: nat > A,I: nat] :
( ( order_mono @ nat @ A @ A4 )
=> ( ord_less_eq @ A @ ( A4 @ I ) @ ( A4 @ ( suc @ I ) ) ) ) ) ).
% incseq_SucD
thf(fact_6246_mono__add,axiom,
! [A: $tType] :
( ( ordere6658533253407199908up_add @ A )
=> ! [A3: A] : ( order_mono @ A @ A @ ( plus_plus @ A @ A3 ) ) ) ).
% mono_add
thf(fact_6247_mono__Suc,axiom,
order_mono @ nat @ nat @ suc ).
% mono_Suc
thf(fact_6248_mono__strict__invE,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( order @ B ) )
=> ! [F2: A > B,X: A,Y2: A] :
( ( order_mono @ A @ B @ F2 )
=> ( ( ord_less @ B @ ( F2 @ X ) @ ( F2 @ Y2 ) )
=> ( ord_less @ A @ X @ Y2 ) ) ) ) ).
% mono_strict_invE
thf(fact_6249_max__of__mono,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( linorder @ B ) )
=> ! [F2: A > B,M2: A,N: A] :
( ( order_mono @ A @ B @ F2 )
=> ( ( ord_max @ B @ ( F2 @ M2 ) @ ( F2 @ N ) )
= ( F2 @ ( ord_max @ A @ M2 @ N ) ) ) ) ) ).
% max_of_mono
thf(fact_6250_mono__pow,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F2: A > A,N: nat] :
( ( order_mono @ A @ A @ F2 )
=> ( order_mono @ A @ A @ ( compow @ ( A > A ) @ N @ F2 ) ) ) ) ).
% mono_pow
thf(fact_6251_mono__inf,axiom,
! [B: $tType,A: $tType] :
( ( ( semilattice_inf @ A )
& ( semilattice_inf @ B ) )
=> ! [F2: A > B,A4: A,B7: A] :
( ( order_mono @ A @ B @ F2 )
=> ( ord_less_eq @ B @ ( F2 @ ( inf_inf @ A @ A4 @ B7 ) ) @ ( inf_inf @ B @ ( F2 @ A4 ) @ ( F2 @ B7 ) ) ) ) ) ).
% mono_inf
thf(fact_6252_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_6253_mono__mult,axiom,
! [A: $tType] :
( ( ordered_semiring @ A )
=> ! [A3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( order_mono @ A @ A @ ( times_times @ A @ A3 ) ) ) ) ).
% mono_mult
thf(fact_6254_mono__image__least,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [F2: A > B,M2: A,N: A,M6: B,N3: B] :
( ( order_mono @ A @ B @ F2 )
=> ( ( ( image @ A @ B @ F2 @ ( set_or7035219750837199246ssThan @ A @ M2 @ N ) )
= ( set_or7035219750837199246ssThan @ B @ M6 @ N3 ) )
=> ( ( ord_less @ A @ M2 @ N )
=> ( ( F2 @ M2 )
= M6 ) ) ) ) ) ).
% mono_image_least
thf(fact_6255_Kleene__iter__gpfp,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [F2: A > A,P4: A,K: nat] :
( ( order_mono @ A @ A @ F2 )
=> ( ( ord_less_eq @ A @ P4 @ ( F2 @ P4 ) )
=> ( ord_less_eq @ A @ P4 @ ( compow @ ( A > A ) @ K @ F2 @ ( top_top @ A ) ) ) ) ) ) ).
% Kleene_iter_gpfp
thf(fact_6256_Kleene__iter__lpfp,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [F2: A > A,P4: A,K: nat] :
( ( order_mono @ A @ A @ F2 )
=> ( ( ord_less_eq @ A @ ( F2 @ P4 ) @ P4 )
=> ( ord_less_eq @ A @ ( compow @ ( A > A ) @ K @ F2 @ ( bot_bot @ A ) ) @ P4 ) ) ) ) ).
% Kleene_iter_lpfp
thf(fact_6257_funpow__mono2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [F2: A > A,I: nat,J2: nat,X: A,Y2: A] :
( ( order_mono @ A @ A @ F2 )
=> ( ( ord_less_eq @ nat @ I @ J2 )
=> ( ( ord_less_eq @ A @ X @ Y2 )
=> ( ( ord_less_eq @ A @ X @ ( F2 @ X ) )
=> ( ord_less_eq @ A @ ( compow @ ( A > A ) @ I @ F2 @ X ) @ ( compow @ ( A > A ) @ J2 @ F2 @ Y2 ) ) ) ) ) ) ) ).
% funpow_mono2
thf(fact_6258_antimono__funpow,axiom,
! [A: $tType] :
( ( ( lattice @ A )
& ( order_top @ A ) )
=> ! [Q: A > A] :
( ( order_mono @ A @ A @ Q )
=> ( order_antimono @ nat @ A
@ ^ [I2: nat] : ( compow @ ( A > A ) @ I2 @ Q @ ( top_top @ A ) ) ) ) ) ).
% antimono_funpow
thf(fact_6259_eventually__filtercomap__at__top__dense,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( no_top @ A ) )
=> ! [P: B > $o,F2: B > A] :
( ( eventually @ B @ P @ ( filtercomap @ B @ A @ F2 @ ( at_top @ A ) ) )
= ( ? [N7: A] :
! [X2: B] :
( ( ord_less @ A @ N7 @ ( F2 @ X2 ) )
=> ( P @ X2 ) ) ) ) ) ).
% eventually_filtercomap_at_top_dense
thf(fact_6260_eventually__filtercomap__at__bot__dense,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( no_bot @ A ) )
=> ! [P: B > $o,F2: B > A] :
( ( eventually @ B @ P @ ( filtercomap @ B @ A @ F2 @ ( at_bot @ A ) ) )
= ( ? [N7: A] :
! [X2: B] :
( ( ord_less @ A @ ( F2 @ X2 ) @ N7 )
=> ( P @ X2 ) ) ) ) ) ).
% eventually_filtercomap_at_bot_dense
thf(fact_6261_funpow__increasing,axiom,
! [A: $tType] :
( ( ( lattice @ A )
& ( order_top @ A ) )
=> ! [M2: nat,N: nat,F2: A > A] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( order_mono @ A @ A @ F2 )
=> ( ord_less_eq @ A @ ( compow @ ( A > A ) @ N @ F2 @ ( top_top @ A ) ) @ ( compow @ ( A > A ) @ M2 @ F2 @ ( top_top @ A ) ) ) ) ) ) ).
% funpow_increasing
thf(fact_6262_funpow__decreasing,axiom,
! [A: $tType] :
( ( ( lattice @ A )
& ( order_bot @ A ) )
=> ! [M2: nat,N: nat,F2: A > A] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( order_mono @ A @ A @ F2 )
=> ( ord_less_eq @ A @ ( compow @ ( A > A ) @ M2 @ F2 @ ( bot_bot @ A ) ) @ ( compow @ ( A > A ) @ N @ F2 @ ( bot_bot @ A ) ) ) ) ) ) ).
% funpow_decreasing
thf(fact_6263_mono__ge2__power__minus__self,axiom,
! [K: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K )
=> ( order_mono @ nat @ nat
@ ^ [M5: nat] : ( minus_minus @ nat @ ( power_power @ nat @ K @ M5 ) @ M5 ) ) ) ).
% mono_ge2_power_minus_self
thf(fact_6264_finite__mono__remains__stable__implies__strict__prefix,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [F2: nat > A] :
( ( finite_finite @ A @ ( image @ nat @ A @ F2 @ ( top_top @ ( set @ nat ) ) ) )
=> ( ( order_mono @ nat @ A @ F2 )
=> ( ! [N2: nat] :
( ( ( F2 @ N2 )
= ( F2 @ ( suc @ N2 ) ) )
=> ( ( F2 @ ( suc @ N2 ) )
= ( F2 @ ( suc @ ( suc @ N2 ) ) ) ) )
=> ? [N10: nat] :
( ! [N4: nat] :
( ( ord_less_eq @ nat @ N4 @ N10 )
=> ! [M: nat] :
( ( ord_less_eq @ nat @ M @ N10 )
=> ( ( ord_less @ nat @ M @ N4 )
=> ( ord_less @ A @ ( F2 @ M ) @ ( F2 @ N4 ) ) ) ) )
& ! [N4: nat] :
( ( ord_less_eq @ nat @ N10 @ N4 )
=> ( ( F2 @ N10 )
= ( F2 @ N4 ) ) ) ) ) ) ) ) ).
% finite_mono_remains_stable_implies_strict_prefix
thf(fact_6265_tendsto__at__left__sequentially,axiom,
! [A: $tType,B: $tType] :
( ( ( topolo3112930676232923870pology @ B )
& ( topolo1944317154257567458pology @ B )
& ( topolo4958980785337419405_space @ A ) )
=> ! [B2: B,A3: B,X8: B > A,L4: A] :
( ( ord_less @ B @ B2 @ A3 )
=> ( ! [S7: nat > B] :
( ! [N4: nat] : ( ord_less @ B @ ( S7 @ N4 ) @ A3 )
=> ( ! [N4: nat] : ( ord_less @ B @ B2 @ ( S7 @ N4 ) )
=> ( ( order_mono @ nat @ B @ S7 )
=> ( ( filterlim @ nat @ B @ S7 @ ( topolo7230453075368039082e_nhds @ B @ A3 ) @ ( at_top @ nat ) )
=> ( filterlim @ nat @ A
@ ^ [N5: nat] : ( X8 @ ( S7 @ N5 ) )
@ ( topolo7230453075368039082e_nhds @ A @ L4 )
@ ( at_top @ nat ) ) ) ) ) )
=> ( filterlim @ B @ A @ X8 @ ( topolo7230453075368039082e_nhds @ A @ L4 ) @ ( topolo174197925503356063within @ B @ A3 @ ( set_ord_lessThan @ B @ A3 ) ) ) ) ) ) ).
% tendsto_at_left_sequentially
thf(fact_6266_remdups__adj__altdef,axiom,
! [A: $tType,Xs: list @ A,Ys2: list @ A] :
( ( ( remdups_adj @ A @ Xs )
= Ys2 )
= ( ? [F4: nat > nat] :
( ( order_mono @ nat @ nat @ F4 )
& ( ( image @ nat @ nat @ F4 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Xs ) ) )
= ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Ys2 ) ) )
& ! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ A @ Xs @ I2 )
= ( nth @ A @ Ys2 @ ( F4 @ I2 ) ) ) )
& ! [I2: nat] :
( ( ord_less @ nat @ ( plus_plus @ nat @ I2 @ ( one_one @ nat ) ) @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ( nth @ A @ Xs @ I2 )
= ( nth @ A @ Xs @ ( plus_plus @ nat @ I2 @ ( one_one @ nat ) ) ) )
= ( ( F4 @ I2 )
= ( F4 @ ( plus_plus @ nat @ I2 @ ( one_one @ nat ) ) ) ) ) ) ) ) ) ).
% remdups_adj_altdef
thf(fact_6267_finite__mono__strict__prefix__implies__finite__fixpoint,axiom,
! [A: $tType,F2: nat > ( set @ A ),S3: set @ A] :
( ! [I3: nat] : ( ord_less_eq @ ( set @ A ) @ ( F2 @ I3 ) @ S3 )
=> ( ( finite_finite @ A @ S3 )
=> ( ? [N9: nat] :
( ! [N2: nat] :
( ( ord_less_eq @ nat @ N2 @ N9 )
=> ! [M3: nat] :
( ( ord_less_eq @ nat @ M3 @ N9 )
=> ( ( ord_less @ nat @ M3 @ N2 )
=> ( ord_less @ ( set @ A ) @ ( F2 @ M3 ) @ ( F2 @ N2 ) ) ) ) )
& ! [N2: nat] :
( ( ord_less_eq @ nat @ N9 @ N2 )
=> ( ( F2 @ N9 )
= ( F2 @ N2 ) ) ) )
=> ( ( F2 @ ( finite_card @ A @ S3 ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ F2 @ ( top_top @ ( set @ nat ) ) ) ) ) ) ) ) ).
% finite_mono_strict_prefix_implies_finite_fixpoint
thf(fact_6268_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_6269_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_6270_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_6271_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_6272_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_6273_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_6274_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_6275_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_6276_finite__subset__Union,axiom,
! [A: $tType,A4: set @ A,B10: set @ ( set @ A )] :
( ( finite_finite @ A @ A4 )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ ( complete_Sup_Sup @ ( set @ A ) @ B10 ) )
=> ~ ! [F9: set @ ( set @ A )] :
( ( finite_finite @ ( set @ A ) @ F9 )
=> ( ( ord_less_eq @ ( set @ ( set @ A ) ) @ F9 @ B10 )
=> ~ ( ord_less_eq @ ( set @ A ) @ A4 @ ( complete_Sup_Sup @ ( set @ A ) @ F9 ) ) ) ) ) ) ).
% finite_subset_Union
thf(fact_6277_SUP__inf,axiom,
! [A: $tType,B: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [F2: B > A,B7: set @ B,A3: A] :
( ( inf_inf @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ B7 ) ) @ A3 )
= ( complete_Sup_Sup @ A
@ ( image @ B @ A
@ ^ [B3: B] : ( inf_inf @ A @ ( F2 @ B3 ) @ A3 )
@ B7 ) ) ) ) ).
% SUP_inf
thf(fact_6278_Sup__inf,axiom,
! [A: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [B7: set @ A,A3: A] :
( ( inf_inf @ A @ ( complete_Sup_Sup @ A @ B7 ) @ A3 )
= ( complete_Sup_Sup @ A
@ ( image @ A @ A
@ ^ [B3: A] : ( inf_inf @ A @ B3 @ A3 )
@ B7 ) ) ) ) ).
% Sup_inf
thf(fact_6279_inf__SUP,axiom,
! [A: $tType,B: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [A3: A,F2: B > A,B7: set @ B] :
( ( inf_inf @ A @ A3 @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ B7 ) ) )
= ( complete_Sup_Sup @ A
@ ( image @ B @ A
@ ^ [B3: B] : ( inf_inf @ A @ A3 @ ( F2 @ B3 ) )
@ B7 ) ) ) ) ).
% inf_SUP
thf(fact_6280_SUP__inf__distrib2,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [F2: B > A,A4: set @ B,G: C > A,B7: set @ C] :
( ( inf_inf @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A4 ) ) @ ( complete_Sup_Sup @ A @ ( image @ C @ A @ G @ B7 ) ) )
= ( complete_Sup_Sup @ A
@ ( image @ B @ A
@ ^ [A5: B] :
( complete_Sup_Sup @ A
@ ( image @ C @ A
@ ^ [B3: C] : ( inf_inf @ A @ ( F2 @ A5 ) @ ( G @ B3 ) )
@ B7 ) )
@ A4 ) ) ) ) ).
% SUP_inf_distrib2
thf(fact_6281_finite__imp__Sup__less,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [X8: set @ A,X: A,A3: A] :
( ( finite_finite @ A @ X8 )
=> ( ( member @ A @ X @ X8 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ X8 )
=> ( ord_less @ A @ X3 @ A3 ) )
=> ( ord_less @ A @ ( complete_Sup_Sup @ A @ X8 ) @ A3 ) ) ) ) ) ).
% finite_imp_Sup_less
thf(fact_6282_Sup__inf__eq__bot__iff,axiom,
! [A: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [B7: set @ A,A3: A] :
( ( ( inf_inf @ A @ ( complete_Sup_Sup @ A @ B7 ) @ A3 )
= ( bot_bot @ A ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ B7 )
=> ( ( inf_inf @ A @ X2 @ A3 )
= ( bot_bot @ A ) ) ) ) ) ) ).
% Sup_inf_eq_bot_iff
thf(fact_6283_less__cSupE,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [Y2: A,X8: set @ A] :
( ( ord_less @ A @ Y2 @ ( complete_Sup_Sup @ A @ X8 ) )
=> ( ( X8
!= ( bot_bot @ ( set @ A ) ) )
=> ~ ! [X3: A] :
( ( member @ A @ X3 @ X8 )
=> ~ ( ord_less @ A @ Y2 @ X3 ) ) ) ) ) ).
% less_cSupE
thf(fact_6284_less__cSupD,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [X8: set @ A,Z2: A] :
( ( X8
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less @ A @ Z2 @ ( complete_Sup_Sup @ A @ X8 ) )
=> ? [X3: A] :
( ( member @ A @ X3 @ X8 )
& ( ord_less @ A @ Z2 @ X3 ) ) ) ) ) ).
% less_cSupD
thf(fact_6285_inf__Sup,axiom,
! [A: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [A3: A,B7: set @ A] :
( ( inf_inf @ A @ A3 @ ( complete_Sup_Sup @ A @ B7 ) )
= ( complete_Sup_Sup @ A @ ( image @ A @ A @ ( inf_inf @ A @ A3 ) @ B7 ) ) ) ) ).
% inf_Sup
thf(fact_6286_Sup__notin__open,axiom,
! [A: $tType] :
( ( topolo8458572112393995274pology @ A )
=> ! [A4: set @ A,X: A] :
( ( topolo1002775350975398744n_open @ A @ A4 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ A4 )
=> ( ord_less @ A @ X3 @ X ) )
=> ~ ( member @ A @ ( complete_Sup_Sup @ A @ A4 ) @ A4 ) ) ) ) ).
% Sup_notin_open
thf(fact_6287_finite__Sup__less__iff,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [X8: set @ A,A3: A] :
( ( finite_finite @ A @ X8 )
=> ( ( X8
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less @ A @ ( complete_Sup_Sup @ A @ X8 ) @ A3 )
= ( ! [X2: A] :
( ( member @ A @ X2 @ X8 )
=> ( ord_less @ A @ X2 @ A3 ) ) ) ) ) ) ) ).
% finite_Sup_less_iff
thf(fact_6288_sum_OUnion__comp,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [B7: set @ ( set @ B ),G: B > A] :
( ! [X3: set @ B] :
( ( member @ ( set @ B ) @ X3 @ B7 )
=> ( finite_finite @ B @ X3 ) )
=> ( ! [A14: set @ B] :
( ( member @ ( set @ B ) @ A14 @ B7 )
=> ! [A24: set @ B] :
( ( member @ ( set @ B ) @ A24 @ B7 )
=> ( ( A14 != A24 )
=> ! [X3: B] :
( ( member @ B @ X3 @ A14 )
=> ( ( member @ B @ X3 @ A24 )
=> ( ( G @ X3 )
= ( zero_zero @ A ) ) ) ) ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( complete_Sup_Sup @ ( set @ B ) @ B7 ) )
= ( comp @ ( ( set @ B ) > A ) @ ( ( set @ ( set @ B ) ) > A ) @ ( B > A ) @ ( groups7311177749621191930dd_sum @ ( set @ B ) @ A ) @ ( groups7311177749621191930dd_sum @ B @ A ) @ G @ B7 ) ) ) ) ) ).
% sum.Union_comp
thf(fact_6289_prod_OUnion__comp,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [B7: set @ ( set @ B ),G: B > A] :
( ! [X3: set @ B] :
( ( member @ ( set @ B ) @ X3 @ B7 )
=> ( finite_finite @ B @ X3 ) )
=> ( ! [A14: set @ B] :
( ( member @ ( set @ B ) @ A14 @ B7 )
=> ! [A24: set @ B] :
( ( member @ ( set @ B ) @ A24 @ B7 )
=> ( ( A14 != A24 )
=> ! [X3: B] :
( ( member @ B @ X3 @ A14 )
=> ( ( member @ B @ X3 @ A24 )
=> ( ( G @ X3 )
= ( one_one @ A ) ) ) ) ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( complete_Sup_Sup @ ( set @ B ) @ B7 ) )
= ( comp @ ( ( set @ B ) > A ) @ ( ( set @ ( set @ B ) ) > A ) @ ( B > A ) @ ( groups7121269368397514597t_prod @ ( set @ B ) @ A ) @ ( groups7121269368397514597t_prod @ B @ A ) @ G @ B7 ) ) ) ) ) ).
% prod.Union_comp
thf(fact_6290_UN__UN__finite__eq,axiom,
! [A: $tType,A4: nat > ( set @ A )] :
( ( complete_Sup_Sup @ ( set @ A )
@ ( image @ nat @ ( set @ A )
@ ^ [N5: nat] : ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ A4 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N5 ) ) )
@ ( top_top @ ( set @ nat ) ) ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ A4 @ ( top_top @ ( set @ nat ) ) ) ) ) ).
% UN_UN_finite_eq
thf(fact_6291_cSup__asclose,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( linordered_idom @ A ) )
=> ! [S3: set @ A,L: A,E2: A] :
( ( S3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ S3 )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X3 @ L ) ) @ E2 ) )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ ( complete_Sup_Sup @ A @ S3 ) @ L ) ) @ E2 ) ) ) ) ).
% cSup_asclose
thf(fact_6292_remdups__adj__adjacent,axiom,
! [A: $tType,I: nat,Xs: list @ A] :
( ( ord_less @ nat @ ( suc @ I ) @ ( size_size @ ( list @ A ) @ ( remdups_adj @ A @ Xs ) ) )
=> ( ( nth @ A @ ( remdups_adj @ A @ Xs ) @ I )
!= ( nth @ A @ ( remdups_adj @ A @ Xs ) @ ( suc @ I ) ) ) ) ).
% remdups_adj_adjacent
thf(fact_6293_UN__finite__subset,axiom,
! [A: $tType,A4: nat > ( set @ A ),C5: set @ A] :
( ! [N2: nat] : ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ A4 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) @ C5 )
=> ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ A4 @ ( top_top @ ( set @ nat ) ) ) ) @ C5 ) ) ).
% UN_finite_subset
thf(fact_6294_UN__finite2__eq,axiom,
! [A: $tType,A4: nat > ( set @ A ),B7: nat > ( set @ A ),K: nat] :
( ! [N2: nat] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ A4 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ B7 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( plus_plus @ nat @ N2 @ K ) ) ) ) )
=> ( ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ A4 @ ( top_top @ ( set @ nat ) ) ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ B7 @ ( top_top @ ( set @ nat ) ) ) ) ) ) ).
% UN_finite2_eq
thf(fact_6295_UN__finite2__subset,axiom,
! [A: $tType,A4: nat > ( set @ A ),B7: nat > ( set @ A ),K: nat] :
( ! [N2: nat] : ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ A4 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ B7 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( plus_plus @ nat @ N2 @ K ) ) ) ) )
=> ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ A4 @ ( top_top @ ( set @ nat ) ) ) ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ B7 @ ( top_top @ ( set @ nat ) ) ) ) ) ) ).
% UN_finite2_subset
thf(fact_6296_SUP__eq__top__iff,axiom,
! [B: $tType,A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [F2: B > A,A4: set @ B] :
( ( ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A4 ) )
= ( top_top @ A ) )
= ( ! [X2: A] :
( ( ord_less @ A @ X2 @ ( top_top @ A ) )
=> ? [Y5: B] :
( ( member @ B @ Y5 @ A4 )
& ( ord_less @ A @ X2 @ ( F2 @ Y5 ) ) ) ) ) ) ) ).
% SUP_eq_top_iff
thf(fact_6297_Sup__eq__top__iff,axiom,
! [A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [A4: set @ A] :
( ( ( complete_Sup_Sup @ A @ A4 )
= ( top_top @ A ) )
= ( ! [X2: A] :
( ( ord_less @ A @ X2 @ ( top_top @ A ) )
=> ? [Y5: A] :
( ( member @ A @ Y5 @ A4 )
& ( ord_less @ A @ X2 @ Y5 ) ) ) ) ) ) ).
% Sup_eq_top_iff
thf(fact_6298_Sup__nat__empty,axiom,
( ( complete_Sup_Sup @ nat @ ( bot_bot @ ( set @ nat ) ) )
= ( zero_zero @ nat ) ) ).
% Sup_nat_empty
thf(fact_6299_less__Sup__iff,axiom,
! [A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [A3: A,S3: set @ A] :
( ( ord_less @ A @ A3 @ ( complete_Sup_Sup @ A @ S3 ) )
= ( ? [X2: A] :
( ( member @ A @ X2 @ S3 )
& ( ord_less @ A @ A3 @ X2 ) ) ) ) ) ).
% less_Sup_iff
thf(fact_6300_le__Sup__iff,axiom,
! [A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [X: A,A4: set @ A] :
( ( ord_less_eq @ A @ X @ ( complete_Sup_Sup @ A @ A4 ) )
= ( ! [Y5: A] :
( ( ord_less @ A @ Y5 @ X )
=> ? [X2: A] :
( ( member @ A @ X2 @ A4 )
& ( ord_less @ A @ Y5 @ X2 ) ) ) ) ) ) ).
% le_Sup_iff
thf(fact_6301_less__SUP__iff,axiom,
! [A: $tType,B: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [A3: A,F2: B > A,A4: set @ B] :
( ( ord_less @ A @ A3 @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A4 ) ) )
= ( ? [X2: B] :
( ( member @ B @ X2 @ A4 )
& ( ord_less @ A @ A3 @ ( F2 @ X2 ) ) ) ) ) ) ).
% less_SUP_iff
thf(fact_6302_SUP__lessD,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F2: B > A,A4: set @ B,Y2: A,I: B] :
( ( ord_less @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A4 ) ) @ Y2 )
=> ( ( member @ B @ I @ A4 )
=> ( ord_less @ A @ ( F2 @ I ) @ Y2 ) ) ) ) ).
% SUP_lessD
thf(fact_6303_UN__extend__simps_I6_J,axiom,
! [L5: $tType,K8: $tType,A4: K8 > ( set @ L5 ),C5: set @ K8,B7: set @ L5] :
( ( minus_minus @ ( set @ L5 ) @ ( complete_Sup_Sup @ ( set @ L5 ) @ ( image @ K8 @ ( set @ L5 ) @ A4 @ C5 ) ) @ B7 )
= ( complete_Sup_Sup @ ( set @ L5 )
@ ( image @ K8 @ ( set @ L5 )
@ ^ [X2: K8] : ( minus_minus @ ( set @ L5 ) @ ( A4 @ X2 ) @ B7 )
@ C5 ) ) ) ).
% UN_extend_simps(6)
thf(fact_6304_le__SUP__iff,axiom,
! [B: $tType,A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [X: A,F2: B > A,A4: set @ B] :
( ( ord_less_eq @ A @ X @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A4 ) ) )
= ( ! [Y5: A] :
( ( ord_less @ A @ Y5 @ X )
=> ? [X2: B] :
( ( member @ B @ X2 @ A4 )
& ( ord_less @ A @ Y5 @ ( F2 @ X2 ) ) ) ) ) ) ) ).
% le_SUP_iff
thf(fact_6305_card__UNION,axiom,
! [A: $tType,A4: set @ ( set @ A )] :
( ( finite_finite @ ( set @ A ) @ A4 )
=> ( ! [X3: set @ A] :
( ( member @ ( set @ A ) @ X3 @ A4 )
=> ( finite_finite @ A @ X3 ) )
=> ( ( finite_card @ A @ ( complete_Sup_Sup @ ( set @ A ) @ A4 ) )
= ( 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 @ A4 )
& ( I7
!= ( bot_bot @ ( set @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ).
% card_UNION
thf(fact_6306_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_6307_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_6308_Inf__eq__bot__iff,axiom,
! [A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [A4: set @ A] :
( ( ( complete_Inf_Inf @ A @ A4 )
= ( bot_bot @ A ) )
= ( ! [X2: A] :
( ( ord_less @ A @ ( bot_bot @ A ) @ X2 )
=> ? [Y5: A] :
( ( member @ A @ Y5 @ A4 )
& ( ord_less @ A @ Y5 @ X2 ) ) ) ) ) ) ).
% Inf_eq_bot_iff
thf(fact_6309_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_6310_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_6311_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_6312_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_6313_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_6314_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_6315_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_6316_INF__eq__bot__iff,axiom,
! [B: $tType,A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [F2: B > A,A4: set @ B] :
( ( ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ A4 ) )
= ( bot_bot @ A ) )
= ( ! [X2: A] :
( ( ord_less @ A @ ( bot_bot @ A ) @ X2 )
=> ? [Y5: B] :
( ( member @ B @ Y5 @ A4 )
& ( ord_less @ A @ ( F2 @ Y5 ) @ X2 ) ) ) ) ) ) ).
% INF_eq_bot_iff
thf(fact_6317_Compl__INT,axiom,
! [A: $tType,B: $tType,B7: B > ( set @ A ),A4: set @ B] :
( ( uminus_uminus @ ( set @ A ) @ ( complete_Inf_Inf @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B7 @ A4 ) ) )
= ( complete_Sup_Sup @ ( set @ A )
@ ( image @ B @ ( set @ A )
@ ^ [X2: B] : ( uminus_uminus @ ( set @ A ) @ ( B7 @ X2 ) )
@ A4 ) ) ) ).
% Compl_INT
thf(fact_6318_Compl__UN,axiom,
! [A: $tType,B: $tType,B7: B > ( set @ A ),A4: set @ B] :
( ( uminus_uminus @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ B7 @ A4 ) ) )
= ( complete_Inf_Inf @ ( set @ A )
@ ( image @ B @ ( set @ A )
@ ^ [X2: B] : ( uminus_uminus @ ( set @ A ) @ ( B7 @ X2 ) )
@ A4 ) ) ) ).
% Compl_UN
thf(fact_6319_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_6320_INT__simps_I3_J,axiom,
! [E5: $tType,F: $tType,C5: set @ E5,A4: E5 > ( set @ F ),B7: set @ F] :
( ( ( C5
= ( bot_bot @ ( set @ E5 ) ) )
=> ( ( complete_Inf_Inf @ ( set @ F )
@ ( image @ E5 @ ( set @ F )
@ ^ [X2: E5] : ( minus_minus @ ( set @ F ) @ ( A4 @ X2 ) @ B7 )
@ C5 ) )
= ( top_top @ ( set @ F ) ) ) )
& ( ( C5
!= ( bot_bot @ ( set @ E5 ) ) )
=> ( ( complete_Inf_Inf @ ( set @ F )
@ ( image @ E5 @ ( set @ F )
@ ^ [X2: E5] : ( minus_minus @ ( set @ F ) @ ( A4 @ X2 ) @ B7 )
@ C5 ) )
= ( minus_minus @ ( set @ F ) @ ( complete_Inf_Inf @ ( set @ F ) @ ( image @ E5 @ ( set @ F ) @ A4 @ C5 ) ) @ B7 ) ) ) ) ).
% INT_simps(3)
thf(fact_6321_INT__simps_I4_J,axiom,
! [G4: $tType,H6: $tType,C5: set @ H6,A4: set @ G4,B7: H6 > ( set @ G4 )] :
( ( ( C5
= ( bot_bot @ ( set @ H6 ) ) )
=> ( ( complete_Inf_Inf @ ( set @ G4 )
@ ( image @ H6 @ ( set @ G4 )
@ ^ [X2: H6] : ( minus_minus @ ( set @ G4 ) @ A4 @ ( B7 @ X2 ) )
@ C5 ) )
= ( top_top @ ( set @ G4 ) ) ) )
& ( ( C5
!= ( bot_bot @ ( set @ H6 ) ) )
=> ( ( complete_Inf_Inf @ ( set @ G4 )
@ ( image @ H6 @ ( set @ G4 )
@ ^ [X2: H6] : ( minus_minus @ ( set @ G4 ) @ A4 @ ( B7 @ X2 ) )
@ C5 ) )
= ( minus_minus @ ( set @ G4 ) @ A4 @ ( complete_Sup_Sup @ ( set @ G4 ) @ ( image @ H6 @ ( set @ G4 ) @ B7 @ C5 ) ) ) ) ) ) ).
% INT_simps(4)
thf(fact_6322_less__INF__D,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [Y2: A,F2: B > A,A4: set @ B,I: B] :
( ( ord_less @ A @ Y2 @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ A4 ) ) )
=> ( ( member @ B @ I @ A4 )
=> ( ord_less @ A @ Y2 @ ( F2 @ I ) ) ) ) ) ).
% less_INF_D
thf(fact_6323_INF__less__iff,axiom,
! [A: $tType,B: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [F2: B > A,A4: set @ B,A3: A] :
( ( ord_less @ A @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ A4 ) ) @ A3 )
= ( ? [X2: B] :
( ( member @ B @ X2 @ A4 )
& ( ord_less @ A @ ( F2 @ X2 ) @ A3 ) ) ) ) ) ).
% INF_less_iff
thf(fact_6324_Inf__less__iff,axiom,
! [A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [S3: set @ A,A3: A] :
( ( ord_less @ A @ ( complete_Inf_Inf @ A @ S3 ) @ A3 )
= ( ? [X2: A] :
( ( member @ A @ X2 @ S3 )
& ( ord_less @ A @ X2 @ A3 ) ) ) ) ) ).
% Inf_less_iff
thf(fact_6325_finite__imp__less__Inf,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [X8: set @ A,X: A,A3: A] :
( ( finite_finite @ A @ X8 )
=> ( ( member @ A @ X @ X8 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ X8 )
=> ( ord_less @ A @ A3 @ X3 ) )
=> ( ord_less @ A @ A3 @ ( complete_Inf_Inf @ A @ X8 ) ) ) ) ) ) ).
% finite_imp_less_Inf
thf(fact_6326_Inf__le__iff,axiom,
! [A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [A4: set @ A,X: A] :
( ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A4 ) @ X )
= ( ! [Y5: A] :
( ( ord_less @ A @ X @ Y5 )
=> ? [X2: A] :
( ( member @ A @ X2 @ A4 )
& ( ord_less @ A @ X2 @ Y5 ) ) ) ) ) ) ).
% Inf_le_iff
thf(fact_6327_cInf__lessD,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [X8: set @ A,Z2: A] :
( ( X8
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less @ A @ ( complete_Inf_Inf @ A @ X8 ) @ Z2 )
=> ? [X3: A] :
( ( member @ A @ X3 @ X8 )
& ( ord_less @ A @ X3 @ Z2 ) ) ) ) ) ).
% cInf_lessD
thf(fact_6328_Inf__notin__open,axiom,
! [A: $tType] :
( ( topolo8458572112393995274pology @ A )
=> ! [A4: set @ A,X: A] :
( ( topolo1002775350975398744n_open @ A @ A4 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ A4 )
=> ( ord_less @ A @ X @ X3 ) )
=> ~ ( member @ A @ ( complete_Inf_Inf @ A @ A4 ) @ A4 ) ) ) ) ).
% Inf_notin_open
thf(fact_6329_INF__le__iff,axiom,
! [B: $tType,A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [F2: B > A,A4: set @ B,X: A] :
( ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ A4 ) ) @ X )
= ( ! [Y5: A] :
( ( ord_less @ A @ X @ Y5 )
=> ? [X2: B] :
( ( member @ B @ X2 @ A4 )
& ( ord_less @ A @ ( F2 @ X2 ) @ Y5 ) ) ) ) ) ) ).
% INF_le_iff
thf(fact_6330_Sup__Inf,axiom,
! [A: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [A4: set @ ( set @ A )] :
( ( complete_Sup_Sup @ A @ ( image @ ( set @ A ) @ A @ ( complete_Inf_Inf @ A ) @ A4 ) )
= ( complete_Inf_Inf @ A
@ ( image @ ( set @ A ) @ A @ ( complete_Sup_Sup @ A )
@ ( collect @ ( set @ A )
@ ^ [Uu3: set @ A] :
? [F4: ( set @ A ) > A] :
( ( Uu3
= ( image @ ( set @ A ) @ A @ F4 @ A4 ) )
& ! [X2: set @ A] :
( ( member @ ( set @ A ) @ X2 @ A4 )
=> ( member @ A @ ( F4 @ X2 ) @ X2 ) ) ) ) ) ) ) ) ).
% Sup_Inf
thf(fact_6331_finite__less__Inf__iff,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [X8: set @ A,A3: A] :
( ( finite_finite @ A @ X8 )
=> ( ( X8
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less @ A @ A3 @ ( complete_Inf_Inf @ A @ X8 ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ X8 )
=> ( ord_less @ A @ A3 @ X2 ) ) ) ) ) ) ) ).
% finite_less_Inf_iff
thf(fact_6332_cInf__abs__ge,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( linordered_idom @ A ) )
=> ! [S3: set @ A,A3: A] :
( ( S3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ S3 )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ X3 ) @ A3 ) )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ ( complete_Inf_Inf @ A @ S3 ) ) @ A3 ) ) ) ) ).
% cInf_abs_ge
thf(fact_6333_uminus__Inf,axiom,
! [A: $tType] :
( ( comple489889107523837845lgebra @ A )
=> ! [A4: set @ A] :
( ( uminus_uminus @ A @ ( complete_Inf_Inf @ A @ A4 ) )
= ( complete_Sup_Sup @ A @ ( image @ A @ A @ ( uminus_uminus @ A ) @ A4 ) ) ) ) ).
% uminus_Inf
thf(fact_6334_uminus__Sup,axiom,
! [A: $tType] :
( ( comple489889107523837845lgebra @ A )
=> ! [A4: set @ A] :
( ( uminus_uminus @ A @ ( complete_Sup_Sup @ A @ A4 ) )
= ( complete_Inf_Inf @ A @ ( image @ A @ A @ ( uminus_uminus @ A ) @ A4 ) ) ) ) ).
% uminus_Sup
thf(fact_6335_uminus__INF,axiom,
! [A: $tType,B: $tType] :
( ( comple489889107523837845lgebra @ A )
=> ! [B7: B > A,A4: set @ B] :
( ( uminus_uminus @ A @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ B7 @ A4 ) ) )
= ( complete_Sup_Sup @ A
@ ( image @ B @ A
@ ^ [X2: B] : ( uminus_uminus @ A @ ( B7 @ X2 ) )
@ A4 ) ) ) ) ).
% uminus_INF
thf(fact_6336_uminus__SUP,axiom,
! [A: $tType,B: $tType] :
( ( comple489889107523837845lgebra @ A )
=> ! [B7: B > A,A4: set @ B] :
( ( uminus_uminus @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ B7 @ A4 ) ) )
= ( complete_Inf_Inf @ A
@ ( image @ B @ A
@ ^ [X2: B] : ( uminus_uminus @ A @ ( B7 @ X2 ) )
@ A4 ) ) ) ) ).
% uminus_SUP
thf(fact_6337_UN__extend__simps_I7_J,axiom,
! [M11: $tType,N11: $tType,A4: set @ M11,B7: N11 > ( set @ M11 ),C5: set @ N11] :
( ( minus_minus @ ( set @ M11 ) @ A4 @ ( complete_Inf_Inf @ ( set @ M11 ) @ ( image @ N11 @ ( set @ M11 ) @ B7 @ C5 ) ) )
= ( complete_Sup_Sup @ ( set @ M11 )
@ ( image @ N11 @ ( set @ M11 )
@ ^ [X2: N11] : ( minus_minus @ ( set @ M11 ) @ A4 @ ( B7 @ X2 ) )
@ C5 ) ) ) ).
% UN_extend_simps(7)
thf(fact_6338_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_6339_SUP__INF__set,axiom,
! [A: $tType,B: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [G: B > A,A4: set @ ( set @ B )] :
( ( complete_Sup_Sup @ A
@ ( image @ ( set @ B ) @ A
@ ^ [X2: set @ B] : ( complete_Inf_Inf @ A @ ( image @ B @ A @ G @ X2 ) )
@ A4 ) )
= ( complete_Inf_Inf @ A
@ ( image @ ( set @ B ) @ A
@ ^ [X2: set @ B] : ( complete_Sup_Sup @ A @ ( image @ B @ A @ G @ X2 ) )
@ ( collect @ ( set @ B )
@ ^ [Uu3: set @ B] :
? [F4: ( set @ B ) > B] :
( ( Uu3
= ( image @ ( set @ B ) @ B @ F4 @ A4 ) )
& ! [X2: set @ B] :
( ( member @ ( set @ B ) @ X2 @ A4 )
=> ( member @ B @ ( F4 @ X2 ) @ X2 ) ) ) ) ) ) ) ) ).
% SUP_INF_set
thf(fact_6340_INF__SUP__set,axiom,
! [A: $tType,B: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [G: B > A,A4: set @ ( set @ B )] :
( ( complete_Inf_Inf @ A
@ ( image @ ( set @ B ) @ A
@ ^ [B5: set @ B] : ( complete_Sup_Sup @ A @ ( image @ B @ A @ G @ B5 ) )
@ A4 ) )
= ( complete_Sup_Sup @ A
@ ( image @ ( set @ B ) @ A
@ ^ [B5: set @ B] : ( complete_Inf_Inf @ A @ ( image @ B @ A @ G @ B5 ) )
@ ( collect @ ( set @ B )
@ ^ [Uu3: set @ B] :
? [F4: ( set @ B ) > B] :
( ( Uu3
= ( image @ ( set @ B ) @ B @ F4 @ A4 ) )
& ! [X2: set @ B] :
( ( member @ ( set @ B ) @ X2 @ A4 )
=> ( member @ B @ ( F4 @ X2 ) @ X2 ) ) ) ) ) ) ) ) ).
% INF_SUP_set
thf(fact_6341_cInf__asclose,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( linordered_idom @ A ) )
=> ! [S3: set @ A,L: A,E2: A] :
( ( S3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ S3 )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X3 @ L ) ) @ E2 ) )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ ( complete_Inf_Inf @ A @ S3 ) @ L ) ) @ E2 ) ) ) ) ).
% cInf_asclose
thf(fact_6342_mono__bij__Inf,axiom,
! [B: $tType,A: $tType] :
( ( ( comple5582772986160207858norder @ A )
& ( comple5582772986160207858norder @ B ) )
=> ! [F2: A > B,A4: set @ A] :
( ( order_mono @ A @ B @ F2 )
=> ( ( bij_betw @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
=> ( ( F2 @ ( complete_Inf_Inf @ A @ A4 ) )
= ( complete_Inf_Inf @ B @ ( image @ A @ B @ F2 @ A4 ) ) ) ) ) ) ).
% mono_bij_Inf
thf(fact_6343_INT__extend__simps_I3_J,axiom,
! [F: $tType,E5: $tType,C5: set @ E5,A4: E5 > ( set @ F ),B7: set @ F] :
( ( ( C5
= ( bot_bot @ ( set @ E5 ) ) )
=> ( ( minus_minus @ ( set @ F ) @ ( complete_Inf_Inf @ ( set @ F ) @ ( image @ E5 @ ( set @ F ) @ A4 @ C5 ) ) @ B7 )
= ( minus_minus @ ( set @ F ) @ ( top_top @ ( set @ F ) ) @ B7 ) ) )
& ( ( C5
!= ( bot_bot @ ( set @ E5 ) ) )
=> ( ( minus_minus @ ( set @ F ) @ ( complete_Inf_Inf @ ( set @ F ) @ ( image @ E5 @ ( set @ F ) @ A4 @ C5 ) ) @ B7 )
= ( complete_Inf_Inf @ ( set @ F )
@ ( image @ E5 @ ( set @ F )
@ ^ [X2: E5] : ( minus_minus @ ( set @ F ) @ ( A4 @ X2 ) @ B7 )
@ C5 ) ) ) ) ) ).
% INT_extend_simps(3)
thf(fact_6344_INT__extend__simps_I4_J,axiom,
! [G4: $tType,H6: $tType,C5: set @ H6,A4: set @ G4,B7: H6 > ( set @ G4 )] :
( ( ( C5
= ( bot_bot @ ( set @ H6 ) ) )
=> ( ( minus_minus @ ( set @ G4 ) @ A4 @ ( complete_Sup_Sup @ ( set @ G4 ) @ ( image @ H6 @ ( set @ G4 ) @ B7 @ C5 ) ) )
= A4 ) )
& ( ( C5
!= ( bot_bot @ ( set @ H6 ) ) )
=> ( ( minus_minus @ ( set @ G4 ) @ A4 @ ( complete_Sup_Sup @ ( set @ G4 ) @ ( image @ H6 @ ( set @ G4 ) @ B7 @ C5 ) ) )
= ( complete_Inf_Inf @ ( set @ G4 )
@ ( image @ H6 @ ( set @ G4 )
@ ^ [X2: H6] : ( minus_minus @ ( set @ G4 ) @ A4 @ ( B7 @ X2 ) )
@ C5 ) ) ) ) ) ).
% INT_extend_simps(4)
thf(fact_6345_INF__SUP,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [P: C > B > A] :
( ( complete_Inf_Inf @ A
@ ( image @ B @ A
@ ^ [Y5: B] :
( complete_Sup_Sup @ A
@ ( image @ C @ A
@ ^ [X2: C] : ( P @ X2 @ Y5 )
@ ( top_top @ ( set @ C ) ) ) )
@ ( top_top @ ( set @ B ) ) ) )
= ( complete_Sup_Sup @ A
@ ( image @ ( B > C ) @ A
@ ^ [F4: B > C] :
( complete_Inf_Inf @ A
@ ( image @ B @ A
@ ^ [X2: B] : ( P @ ( F4 @ X2 ) @ X2 )
@ ( top_top @ ( set @ B ) ) ) )
@ ( top_top @ ( set @ ( B > C ) ) ) ) ) ) ) ).
% INF_SUP
thf(fact_6346_SUP__INF,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [P: C > B > A] :
( ( complete_Sup_Sup @ A
@ ( image @ B @ A
@ ^ [Y5: B] :
( complete_Inf_Inf @ A
@ ( image @ C @ A
@ ^ [X2: C] : ( P @ X2 @ Y5 )
@ ( top_top @ ( set @ C ) ) ) )
@ ( top_top @ ( set @ B ) ) ) )
= ( complete_Inf_Inf @ A
@ ( image @ ( B > C ) @ A
@ ^ [X2: B > C] :
( complete_Sup_Sup @ A
@ ( image @ B @ A
@ ^ [Y5: B] : ( P @ ( X2 @ Y5 ) @ Y5 )
@ ( top_top @ ( set @ B ) ) ) )
@ ( top_top @ ( set @ ( B > C ) ) ) ) ) ) ) ).
% SUP_INF
thf(fact_6347_INF__nat__binary,axiom,
! [A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A4: A,B7: A] :
( ( inf_inf @ A @ A4
@ ( complete_Inf_Inf @ A
@ ( image @ nat @ A
@ ^ [X2: nat] : B7
@ ( collect @ nat @ ( ord_less @ nat @ ( zero_zero @ nat ) ) ) ) ) )
= ( inf_inf @ A @ A4 @ B7 ) ) ) ).
% INF_nat_binary
thf(fact_6348_inj__sgn__power,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( inj_on @ real @ real
@ ^ [Y5: real] : ( times_times @ real @ ( sgn_sgn @ real @ Y5 ) @ ( power_power @ real @ ( abs_abs @ real @ Y5 ) @ N ) )
@ ( top_top @ ( set @ real ) ) ) ) ).
% inj_sgn_power
thf(fact_6349_inj__uminus,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A4: set @ A] : ( inj_on @ A @ A @ ( uminus_uminus @ A ) @ A4 ) ) ).
% inj_uminus
thf(fact_6350_inj__mult__left,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [A3: A] :
( ( inj_on @ A @ A @ ( times_times @ A @ A3 ) @ ( top_top @ ( set @ A ) ) )
= ( A3
!= ( zero_zero @ A ) ) ) ) ).
% inj_mult_left
thf(fact_6351_inj__divide__right,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A3: A] :
( ( inj_on @ A @ A
@ ^ [B3: A] : ( divide_divide @ A @ B3 @ A3 )
@ ( top_top @ ( set @ A ) ) )
= ( A3
!= ( zero_zero @ A ) ) ) ) ).
% inj_divide_right
thf(fact_6352_inj__on__insert,axiom,
! [B: $tType,A: $tType,F2: A > B,A3: A,A4: set @ A] :
( ( inj_on @ A @ B @ F2 @ ( insert @ A @ A3 @ A4 ) )
= ( ( inj_on @ A @ B @ F2 @ A4 )
& ~ ( member @ B @ ( F2 @ A3 ) @ ( image @ A @ B @ F2 @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% inj_on_insert
thf(fact_6353_subset__image__inj,axiom,
! [A: $tType,B: $tType,S3: set @ A,F2: B > A,T6: set @ B] :
( ( ord_less_eq @ ( set @ A ) @ S3 @ ( image @ B @ A @ F2 @ T6 ) )
= ( ? [U5: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ U5 @ T6 )
& ( inj_on @ B @ A @ F2 @ U5 )
& ( S3
= ( image @ B @ A @ F2 @ U5 ) ) ) ) ) ).
% subset_image_inj
thf(fact_6354_Inf__real__def,axiom,
( ( complete_Inf_Inf @ real )
= ( ^ [X7: set @ real] : ( uminus_uminus @ real @ ( complete_Sup_Sup @ real @ ( image @ real @ real @ ( uminus_uminus @ real ) @ X7 ) ) ) ) ) ).
% Inf_real_def
thf(fact_6355_inj__diff__right,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A3: A] :
( inj_on @ A @ A
@ ^ [B3: A] : ( minus_minus @ A @ B3 @ A3 )
@ ( top_top @ ( set @ A ) ) ) ) ).
% inj_diff_right
thf(fact_6356_option_Oinj__map,axiom,
! [B: $tType,A: $tType,F2: A > B] :
( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
=> ( inj_on @ ( option @ A ) @ ( option @ B ) @ ( map_option @ A @ B @ F2 ) @ ( top_top @ ( set @ ( option @ A ) ) ) ) ) ).
% option.inj_map
thf(fact_6357_linorder__injI,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ A )
=> ! [F2: A > B] :
( ! [X3: A,Y3: A] :
( ( ord_less @ A @ X3 @ Y3 )
=> ( ( F2 @ X3 )
!= ( F2 @ Y3 ) ) )
=> ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) ) ) ) ).
% linorder_injI
thf(fact_6358_inj__fn,axiom,
! [A: $tType,F2: A > A,N: nat] :
( ( inj_on @ A @ A @ F2 @ ( top_top @ ( set @ A ) ) )
=> ( inj_on @ A @ A @ ( compow @ ( A > A ) @ N @ F2 ) @ ( top_top @ ( set @ A ) ) ) ) ).
% inj_fn
thf(fact_6359_inj__on__mult,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [A3: A,A4: set @ A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( inj_on @ A @ A @ ( times_times @ A @ A3 ) @ A4 ) ) ) ).
% inj_on_mult
thf(fact_6360_inj__on__diff,axiom,
! [B: $tType,A: $tType,F2: A > B,A4: set @ A,B7: set @ A] :
( ( inj_on @ A @ B @ F2 @ A4 )
=> ( inj_on @ A @ B @ F2 @ ( minus_minus @ ( set @ A ) @ A4 @ B7 ) ) ) ).
% inj_on_diff
thf(fact_6361_linorder__inj__onI,axiom,
! [B: $tType,A: $tType] :
( ( order @ A )
=> ! [A4: set @ A,F2: A > B] :
( ! [X3: A,Y3: A] :
( ( ord_less @ A @ X3 @ Y3 )
=> ( ( member @ A @ X3 @ A4 )
=> ( ( member @ A @ Y3 @ A4 )
=> ( ( F2 @ X3 )
!= ( F2 @ Y3 ) ) ) ) )
=> ( ! [X3: A,Y3: A] :
( ( member @ A @ X3 @ A4 )
=> ( ( member @ A @ Y3 @ A4 )
=> ( ( ord_less_eq @ A @ X3 @ Y3 )
| ( ord_less_eq @ A @ Y3 @ X3 ) ) ) )
=> ( inj_on @ A @ B @ F2 @ A4 ) ) ) ) ).
% linorder_inj_onI
thf(fact_6362_Inf__int__def,axiom,
( ( complete_Inf_Inf @ int )
= ( ^ [X7: set @ int] : ( uminus_uminus @ int @ ( complete_Sup_Sup @ int @ ( image @ int @ int @ ( uminus_uminus @ int ) @ X7 ) ) ) ) ) ).
% Inf_int_def
thf(fact_6363_inj__on__iff__surj,axiom,
! [A: $tType,B: $tType,A4: set @ A,A15: set @ B] :
( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ? [F4: A > B] :
( ( inj_on @ A @ B @ F4 @ A4 )
& ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F4 @ A4 ) @ A15 ) ) )
= ( ? [G2: B > A] :
( ( image @ B @ A @ G2 @ A15 )
= A4 ) ) ) ) ).
% inj_on_iff_surj
thf(fact_6364_image__set__diff,axiom,
! [B: $tType,A: $tType,F2: A > B,A4: set @ A,B7: set @ A] :
( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
=> ( ( image @ A @ B @ F2 @ ( minus_minus @ ( set @ A ) @ A4 @ B7 ) )
= ( minus_minus @ ( set @ B ) @ ( image @ A @ B @ F2 @ A4 ) @ ( image @ A @ B @ F2 @ B7 ) ) ) ) ).
% image_set_diff
thf(fact_6365_inj__on__image__set__diff,axiom,
! [B: $tType,A: $tType,F2: A > B,C5: set @ A,A4: set @ A,B7: set @ A] :
( ( inj_on @ A @ B @ F2 @ C5 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A4 @ B7 ) @ C5 )
=> ( ( ord_less_eq @ ( set @ A ) @ B7 @ C5 )
=> ( ( image @ A @ B @ F2 @ ( minus_minus @ ( set @ A ) @ A4 @ B7 ) )
= ( minus_minus @ ( set @ B ) @ ( image @ A @ B @ F2 @ A4 ) @ ( image @ A @ B @ F2 @ B7 ) ) ) ) ) ) ).
% inj_on_image_set_diff
thf(fact_6366_pigeonhole,axiom,
! [A: $tType,B: $tType,F2: B > A,A4: set @ B] :
( ( ord_less @ nat @ ( finite_card @ A @ ( image @ B @ A @ F2 @ A4 ) ) @ ( finite_card @ B @ A4 ) )
=> ~ ( inj_on @ B @ A @ F2 @ A4 ) ) ).
% pigeonhole
thf(fact_6367_continuous__inj__imp__mono,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo8458572112393995274pology @ A )
& ( topolo1944317154257567458pology @ B ) )
=> ! [A3: A,X: A,B2: A,F2: A > B] :
( ( ord_less @ A @ A3 @ X )
=> ( ( ord_less @ A @ X @ B2 )
=> ( ( topolo81223032696312382ous_on @ A @ B @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) @ F2 )
=> ( ( inj_on @ A @ B @ F2 @ ( set_or1337092689740270186AtMost @ A @ A3 @ B2 ) )
=> ( ( ( ord_less @ B @ ( F2 @ A3 ) @ ( F2 @ X ) )
& ( ord_less @ B @ ( F2 @ X ) @ ( F2 @ B2 ) ) )
| ( ( ord_less @ B @ ( F2 @ B2 ) @ ( F2 @ X ) )
& ( ord_less @ B @ ( F2 @ X ) @ ( F2 @ A3 ) ) ) ) ) ) ) ) ) ).
% continuous_inj_imp_mono
thf(fact_6368_inj__image__Compl__subset,axiom,
! [B: $tType,A: $tType,F2: A > B,A4: set @ A] :
( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
=> ( ord_less_eq @ ( set @ B ) @ ( image @ A @ B @ F2 @ ( uminus_uminus @ ( set @ A ) @ A4 ) ) @ ( uminus_uminus @ ( set @ B ) @ ( image @ A @ B @ F2 @ A4 ) ) ) ) ).
% inj_image_Compl_subset
thf(fact_6369_log__inj,axiom,
! [B2: real] :
( ( ord_less @ real @ ( one_one @ real ) @ B2 )
=> ( inj_on @ real @ real @ ( log2 @ B2 ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) ) ).
% log_inj
thf(fact_6370_funpow__inj__finite,axiom,
! [A: $tType,P4: A > A,X: A] :
( ( inj_on @ A @ A @ P4 @ ( top_top @ ( set @ A ) ) )
=> ( ( finite_finite @ A
@ ( collect @ A
@ ^ [Y5: A] :
? [N5: nat] :
( Y5
= ( compow @ ( A > A ) @ N5 @ P4 @ X ) ) ) )
=> ~ ! [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
=> ( ( compow @ ( A > A ) @ N2 @ P4 @ X )
!= X ) ) ) ) ).
% funpow_inj_finite
thf(fact_6371_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_6372_surj__int__decode,axiom,
( ( image @ nat @ int @ nat_int_decode @ ( top_top @ ( set @ nat ) ) )
= ( top_top @ ( set @ int ) ) ) ).
% surj_int_decode
thf(fact_6373_int__encode__inverse,axiom,
! [X: int] :
( ( nat_int_decode @ ( nat_int_encode @ X ) )
= X ) ).
% int_encode_inverse
thf(fact_6374_int__decode__inverse,axiom,
! [N: nat] :
( ( nat_int_encode @ ( nat_int_decode @ N ) )
= N ) ).
% int_decode_inverse
thf(fact_6375_inj__of__nat,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ( inj_on @ nat @ A @ ( semiring_1_of_nat @ A ) @ ( top_top @ ( set @ nat ) ) ) ) ).
% inj_of_nat
thf(fact_6376_inj__Some,axiom,
! [A: $tType,A4: set @ A] : ( inj_on @ A @ ( option @ A ) @ ( some @ A ) @ A4 ) ).
% inj_Some
thf(fact_6377_inj__prod__encode,axiom,
! [A4: set @ ( product_prod @ nat @ nat )] : ( inj_on @ ( product_prod @ nat @ nat ) @ nat @ nat_prod_encode @ A4 ) ).
% inj_prod_encode
thf(fact_6378_inj__setminus,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [A4: set @ ( set @ A )] : ( inj_on @ ( set @ A ) @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) ) @ A4 ) ) ).
% inj_setminus
thf(fact_6379_inj__Suc,axiom,
! [N6: set @ nat] : ( inj_on @ nat @ nat @ suc @ N6 ) ).
% inj_Suc
thf(fact_6380_inj__int__decode,axiom,
! [A4: set @ nat] : ( inj_on @ nat @ int @ nat_int_decode @ A4 ) ).
% inj_int_decode
thf(fact_6381_inj__on__of__nat,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [N6: set @ nat] : ( inj_on @ nat @ A @ ( semiring_1_of_nat @ A ) @ N6 ) ) ).
% inj_on_of_nat
thf(fact_6382_inj__int__encode,axiom,
! [A4: set @ int] : ( inj_on @ int @ nat @ nat_int_encode @ A4 ) ).
% inj_int_encode
thf(fact_6383_inj__on__diff__nat,axiom,
! [N6: set @ nat,K: nat] :
( ! [N2: nat] :
( ( member @ nat @ N2 @ N6 )
=> ( ord_less_eq @ nat @ K @ N2 ) )
=> ( inj_on @ nat @ nat
@ ^ [N5: nat] : ( minus_minus @ nat @ N5 @ K )
@ N6 ) ) ).
% inj_on_diff_nat
thf(fact_6384_inj__on__set__encode,axiom,
inj_on @ ( set @ nat ) @ nat @ nat_set_encode @ ( collect @ ( set @ nat ) @ ( finite_finite @ nat ) ) ).
% inj_on_set_encode
thf(fact_6385_int__decode__eq,axiom,
! [X: nat,Y2: nat] :
( ( ( nat_int_decode @ X )
= ( nat_int_decode @ Y2 ) )
= ( X = Y2 ) ) ).
% int_decode_eq
thf(fact_6386_finite__imp__nat__seg__image__inj__on,axiom,
! [A: $tType,A4: set @ A] :
( ( finite_finite @ A @ A4 )
=> ? [N2: nat,F3: nat > A] :
( ( A4
= ( image @ nat @ A @ F3
@ ( collect @ nat
@ ^ [I2: nat] : ( ord_less @ nat @ I2 @ N2 ) ) ) )
& ( inj_on @ nat @ A @ F3
@ ( collect @ nat
@ ^ [I2: nat] : ( ord_less @ nat @ I2 @ N2 ) ) ) ) ) ).
% finite_imp_nat_seg_image_inj_on
thf(fact_6387_finite__imp__inj__to__nat__seg,axiom,
! [A: $tType,A4: set @ A] :
( ( finite_finite @ A @ A4 )
=> ? [F3: A > nat,N2: nat] :
( ( ( image @ A @ nat @ F3 @ A4 )
= ( collect @ nat
@ ^ [I2: nat] : ( ord_less @ nat @ I2 @ N2 ) ) )
& ( inj_on @ A @ nat @ F3 @ A4 ) ) ) ).
% finite_imp_inj_to_nat_seg
thf(fact_6388_inj__on__nth,axiom,
! [A: $tType,Xs: list @ A,I5: set @ nat] :
( ( distinct @ A @ Xs )
=> ( ! [X3: nat] :
( ( member @ nat @ X3 @ I5 )
=> ( ord_less @ nat @ X3 @ ( size_size @ ( list @ A ) @ Xs ) ) )
=> ( inj_on @ nat @ A @ ( nth @ A @ Xs ) @ I5 ) ) ) ).
% inj_on_nth
thf(fact_6389_infinite__countable__subset,axiom,
! [A: $tType,S3: set @ A] :
( ~ ( finite_finite @ A @ S3 )
=> ? [F3: nat > A] :
( ( inj_on @ nat @ A @ F3 @ ( top_top @ ( set @ nat ) ) )
& ( ord_less_eq @ ( set @ A ) @ ( image @ nat @ A @ F3 @ ( top_top @ ( set @ nat ) ) ) @ S3 ) ) ) ).
% infinite_countable_subset
thf(fact_6390_infinite__iff__countable__subset,axiom,
! [A: $tType,S3: set @ A] :
( ( ~ ( finite_finite @ A @ S3 ) )
= ( ? [F4: nat > A] :
( ( inj_on @ nat @ A @ F4 @ ( top_top @ ( set @ nat ) ) )
& ( ord_less_eq @ ( set @ A ) @ ( image @ nat @ A @ F4 @ ( top_top @ ( set @ nat ) ) ) @ S3 ) ) ) ) ).
% infinite_iff_countable_subset
thf(fact_6391_inj__on__funpow__least,axiom,
! [A: $tType,N: nat,F2: A > A,S: A] :
( ( ( compow @ ( A > A ) @ N @ F2 @ S )
= S )
=> ( ! [M3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M3 )
=> ( ( ord_less @ nat @ M3 @ N )
=> ( ( compow @ ( A > A ) @ M3 @ F2 @ S )
!= S ) ) )
=> ( inj_on @ nat @ A
@ ^ [K3: nat] : ( compow @ ( A > A ) @ K3 @ F2 @ S )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% inj_on_funpow_least
thf(fact_6392_bij__int__decode,axiom,
bij_betw @ nat @ int @ nat_int_decode @ ( top_top @ ( set @ nat ) ) @ ( top_top @ ( set @ int ) ) ).
% bij_int_decode
thf(fact_6393_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_6394_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_6395_at__within__def,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ( ( topolo174197925503356063within @ A )
= ( ^ [A5: A,S5: set @ A] : ( inf_inf @ ( filter @ A ) @ ( topolo7230453075368039082e_nhds @ A @ A5 ) @ ( principal @ A @ ( minus_minus @ ( set @ A ) @ S5 @ ( insert @ A @ A5 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% at_within_def
thf(fact_6396_at__within__eq,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ( ( topolo174197925503356063within @ A )
= ( ^ [X2: A,S5: set @ A] :
( complete_Inf_Inf @ ( filter @ A )
@ ( image @ ( set @ A ) @ ( filter @ A )
@ ^ [S8: set @ A] : ( principal @ A @ ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ S8 @ S5 ) @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) )
@ ( collect @ ( set @ A )
@ ^ [S8: set @ A] :
( ( topolo1002775350975398744n_open @ A @ S8 )
& ( member @ A @ X2 @ S8 ) ) ) ) ) ) ) ) ).
% at_within_eq
thf(fact_6397_has__derivative__power__int_H,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [X: A,N: int,S3: set @ A] :
( ( X
!= ( zero_zero @ A ) )
=> ( has_derivative @ A @ A
@ ^ [X2: A] : ( power_int @ A @ X2 @ N )
@ ^ [Y5: A] : ( times_times @ A @ Y5 @ ( times_times @ A @ ( ring_1_of_int @ A @ N ) @ ( power_int @ A @ X @ ( minus_minus @ int @ N @ ( one_one @ int ) ) ) ) )
@ ( topolo174197925503356063within @ A @ X @ S3 ) ) ) ) ).
% has_derivative_power_int'
thf(fact_6398_has__derivative__power__int,axiom,
! [A: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V3459762299906320749_field @ A ) )
=> ! [F2: C > A,X: C,F6: C > A,S3: set @ C,N: int] :
( ( ( F2 @ X )
!= ( zero_zero @ A ) )
=> ( ( has_derivative @ C @ A @ F2 @ F6 @ ( topolo174197925503356063within @ C @ X @ S3 ) )
=> ( has_derivative @ C @ A
@ ^ [X2: C] : ( power_int @ A @ ( F2 @ X2 ) @ N )
@ ^ [H2: C] : ( times_times @ A @ ( F6 @ H2 ) @ ( times_times @ A @ ( ring_1_of_int @ A @ N ) @ ( power_int @ A @ ( F2 @ X ) @ ( minus_minus @ int @ N @ ( one_one @ int ) ) ) ) )
@ ( topolo174197925503356063within @ C @ X @ S3 ) ) ) ) ) ).
% has_derivative_power_int
thf(fact_6399_power__int__1__left,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [N: int] :
( ( power_int @ A @ ( one_one @ A ) @ N )
= ( one_one @ A ) ) ) ).
% power_int_1_left
thf(fact_6400_power__int__1__right,axiom,
! [A: $tType] :
( ( ( inverse @ A )
& ( monoid_mult @ A ) )
=> ! [Y2: A] :
( ( power_int @ A @ Y2 @ ( one_one @ int ) )
= Y2 ) ) ).
% power_int_1_right
thf(fact_6401_power__int__sgn,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,N: int] :
( ( sgn_sgn @ A @ ( power_int @ A @ A3 @ N ) )
= ( power_int @ A @ ( sgn_sgn @ A @ A3 ) @ N ) ) ) ).
% power_int_sgn
thf(fact_6402_power__int__mult__distrib__numeral2,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [X: A,W: num,M2: int] :
( ( power_int @ A @ ( times_times @ A @ X @ ( numeral_numeral @ A @ W ) ) @ M2 )
= ( times_times @ A @ ( power_int @ A @ X @ M2 ) @ ( power_int @ A @ ( numeral_numeral @ A @ W ) @ M2 ) ) ) ) ).
% power_int_mult_distrib_numeral2
thf(fact_6403_power__int__mult__distrib__numeral1,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [W: num,Y2: A,M2: int] :
( ( power_int @ A @ ( times_times @ A @ ( numeral_numeral @ A @ W ) @ Y2 ) @ M2 )
= ( times_times @ A @ ( power_int @ A @ ( numeral_numeral @ A @ W ) @ M2 ) @ ( power_int @ A @ Y2 @ M2 ) ) ) ) ).
% power_int_mult_distrib_numeral1
thf(fact_6404_power__int__0__left,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [M2: int] :
( ( M2
!= ( zero_zero @ int ) )
=> ( ( power_int @ A @ ( zero_zero @ A ) @ M2 )
= ( zero_zero @ A ) ) ) ) ).
% power_int_0_left
thf(fact_6405_power__int__eq__0__iff,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X: A,N: int] :
( ( ( power_int @ A @ X @ N )
= ( zero_zero @ A ) )
= ( ( X
= ( zero_zero @ A ) )
& ( N
!= ( zero_zero @ int ) ) ) ) ) ).
% power_int_eq_0_iff
thf(fact_6406_power__int__0__right,axiom,
! [B: $tType] :
( ( ( inverse @ B )
& ( power @ B ) )
=> ! [X: B] :
( ( power_int @ B @ X @ ( zero_zero @ int ) )
= ( one_one @ B ) ) ) ).
% power_int_0_right
thf(fact_6407_abs__power__int__minus,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,N: int] :
( ( abs_abs @ A @ ( power_int @ A @ ( uminus_uminus @ A @ A3 ) @ N ) )
= ( abs_abs @ A @ ( power_int @ A @ A3 @ N ) ) ) ) ).
% abs_power_int_minus
thf(fact_6408_power__int__of__nat,axiom,
! [A: $tType] :
( ( ( inverse @ A )
& ( power @ A ) )
=> ! [X: A,N: nat] :
( ( power_int @ A @ X @ ( semiring_1_of_nat @ int @ N ) )
= ( power_power @ A @ X @ N ) ) ) ).
% power_int_of_nat
thf(fact_6409_power__int__mult__numeral,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X: A,M2: num,N: num] :
( ( power_int @ A @ ( power_int @ A @ X @ ( numeral_numeral @ int @ M2 ) ) @ ( numeral_numeral @ int @ N ) )
= ( power_int @ A @ X @ ( numeral_numeral @ int @ ( times_times @ num @ M2 @ N ) ) ) ) ) ).
% power_int_mult_numeral
thf(fact_6410_power__int__minus__one__mult__self_H,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [M2: int,B2: A] :
( ( times_times @ A @ ( power_int @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ M2 ) @ ( times_times @ A @ ( power_int @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ M2 ) @ B2 ) )
= B2 ) ) ).
% power_int_minus_one_mult_self'
thf(fact_6411_power__int__minus__one__mult__self,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [M2: int] :
( ( times_times @ A @ ( power_int @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ M2 ) @ ( power_int @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ M2 ) )
= ( one_one @ A ) ) ) ).
% power_int_minus_one_mult_self
thf(fact_6412_power__int__numeral,axiom,
! [A: $tType] :
( ( ( inverse @ A )
& ( power @ A ) )
=> ! [X: A,N: num] :
( ( power_int @ A @ X @ ( numeral_numeral @ int @ N ) )
= ( power_power @ A @ X @ ( numeral_numeral @ nat @ N ) ) ) ) ).
% power_int_numeral
thf(fact_6413_power__int__minus1__right,axiom,
! [A: $tType] :
( ( ( inverse @ A )
& ( monoid_mult @ A ) )
=> ! [Y2: A] :
( ( power_int @ A @ Y2 @ ( uminus_uminus @ int @ ( one_one @ int ) ) )
= ( inverse_inverse @ A @ Y2 ) ) ) ).
% power_int_minus1_right
thf(fact_6414_power__int__add__numeral,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X: A,M2: num,N: num] :
( ( times_times @ A @ ( power_int @ A @ X @ ( numeral_numeral @ int @ M2 ) ) @ ( power_int @ A @ X @ ( numeral_numeral @ int @ N ) ) )
= ( power_int @ A @ X @ ( numeral_numeral @ int @ ( plus_plus @ num @ M2 @ N ) ) ) ) ) ).
% power_int_add_numeral
thf(fact_6415_power__int__add__numeral2,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X: A,M2: num,N: num,B2: A] :
( ( times_times @ A @ ( power_int @ A @ X @ ( numeral_numeral @ int @ M2 ) ) @ ( times_times @ A @ ( power_int @ A @ X @ ( numeral_numeral @ int @ N ) ) @ B2 ) )
= ( times_times @ A @ ( power_int @ A @ X @ ( numeral_numeral @ int @ ( plus_plus @ num @ M2 @ N ) ) ) @ B2 ) ) ) ).
% power_int_add_numeral2
thf(fact_6416_power__int__mono__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A,N: int] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ N )
=> ( ( ord_less_eq @ A @ ( power_int @ A @ A3 @ N ) @ ( power_int @ A @ B2 @ N ) )
= ( ord_less_eq @ A @ A3 @ B2 ) ) ) ) ) ) ).
% power_int_mono_iff
thf(fact_6417_power__int__minus__left__odd,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [N: int,A3: A] :
( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N )
=> ( ( power_int @ A @ ( uminus_uminus @ A @ A3 ) @ N )
= ( uminus_uminus @ A @ ( power_int @ A @ A3 @ N ) ) ) ) ) ).
% power_int_minus_left_odd
thf(fact_6418_power__int__minus__left__even,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [N: int,A3: A] :
( ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N )
=> ( ( power_int @ A @ ( uminus_uminus @ A @ A3 ) @ N )
= ( power_int @ A @ A3 @ N ) ) ) ) ).
% power_int_minus_left_even
thf(fact_6419_power__int__minus,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X: A,N: int] :
( ( power_int @ A @ X @ ( uminus_uminus @ int @ N ) )
= ( inverse_inverse @ A @ ( power_int @ A @ X @ N ) ) ) ) ).
% power_int_minus
thf(fact_6420_zero__less__power__int,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,N: int] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( power_int @ A @ X @ N ) ) ) ) ).
% zero_less_power_int
thf(fact_6421_power__int__abs,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,N: int] :
( ( abs_abs @ A @ ( power_int @ A @ A3 @ N ) )
= ( power_int @ A @ ( abs_abs @ A @ A3 ) @ N ) ) ) ).
% power_int_abs
thf(fact_6422_power__int__inverse,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X: A,N: int] :
( ( power_int @ A @ ( inverse_inverse @ A @ X ) @ N )
= ( inverse_inverse @ A @ ( power_int @ A @ X @ N ) ) ) ) ).
% power_int_inverse
thf(fact_6423_power__int__mult__distrib,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [X: A,Y2: A,M2: int] :
( ( power_int @ A @ ( times_times @ A @ X @ Y2 ) @ M2 )
= ( times_times @ A @ ( power_int @ A @ X @ M2 ) @ ( power_int @ A @ Y2 @ M2 ) ) ) ) ).
% power_int_mult_distrib
thf(fact_6424_power__int__commutes,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X: A,N: int] :
( ( times_times @ A @ ( power_int @ A @ X @ N ) @ X )
= ( times_times @ A @ X @ ( power_int @ A @ X @ N ) ) ) ) ).
% power_int_commutes
thf(fact_6425_power__int__mult,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X: A,M2: int,N: int] :
( ( power_int @ A @ X @ ( times_times @ int @ M2 @ N ) )
= ( power_int @ A @ ( power_int @ A @ X @ M2 ) @ N ) ) ) ).
% power_int_mult
thf(fact_6426_power__int__divide__distrib,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [X: A,Y2: A,M2: int] :
( ( power_int @ A @ ( divide_divide @ A @ X @ Y2 ) @ M2 )
= ( divide_divide @ A @ ( power_int @ A @ X @ M2 ) @ ( power_int @ A @ Y2 @ M2 ) ) ) ) ).
% power_int_divide_distrib
thf(fact_6427_power__int__not__zero,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X: A,N: int] :
( ( ( X
!= ( zero_zero @ A ) )
| ( N
= ( zero_zero @ int ) ) )
=> ( ( power_int @ A @ X @ N )
!= ( zero_zero @ A ) ) ) ) ).
% power_int_not_zero
thf(fact_6428_power__int__one__over,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X: A,N: int] :
( ( power_int @ A @ ( divide_divide @ A @ ( one_one @ A ) @ X ) @ N )
= ( divide_divide @ A @ ( one_one @ A ) @ ( power_int @ A @ X @ N ) ) ) ) ).
% power_int_one_over
thf(fact_6429_continuous__on__power__int,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ A )
& ( real_V8999393235501362500lgebra @ B ) )
=> ! [S: set @ A,F2: A > B,N: int] :
( ( topolo81223032696312382ous_on @ A @ B @ S @ F2 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ S )
=> ( ( F2 @ X3 )
!= ( zero_zero @ B ) ) )
=> ( topolo81223032696312382ous_on @ A @ B @ S
@ ^ [X2: A] : ( power_int @ B @ ( F2 @ X2 ) @ N ) ) ) ) ) ).
% continuous_on_power_int
thf(fact_6430_zero__le__power__int,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,N: int] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_int @ A @ X @ N ) ) ) ) ).
% zero_le_power_int
thf(fact_6431_power__int__0__left__If,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [M2: int] :
( ( ( M2
= ( zero_zero @ int ) )
=> ( ( power_int @ A @ ( zero_zero @ A ) @ M2 )
= ( one_one @ A ) ) )
& ( ( M2
!= ( zero_zero @ int ) )
=> ( ( power_int @ A @ ( zero_zero @ A ) @ M2 )
= ( zero_zero @ A ) ) ) ) ) ).
% power_int_0_left_If
thf(fact_6432_power__int__increasing,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [N: int,N6: int,A3: A] :
( ( ord_less_eq @ int @ N @ N6 )
=> ( ( ord_less_eq @ A @ ( one_one @ A ) @ A3 )
=> ( ord_less_eq @ A @ ( power_int @ A @ A3 @ N ) @ ( power_int @ A @ A3 @ N6 ) ) ) ) ) ).
% power_int_increasing
thf(fact_6433_power__int__strict__increasing,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [N: int,N6: int,A3: A] :
( ( ord_less @ int @ N @ N6 )
=> ( ( ord_less @ A @ ( one_one @ A ) @ A3 )
=> ( ord_less @ A @ ( power_int @ A @ A3 @ N ) @ ( power_int @ A @ A3 @ N6 ) ) ) ) ) ).
% power_int_strict_increasing
thf(fact_6434_power__int__minus__one__minus,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [N: int] :
( ( power_int @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( uminus_uminus @ int @ N ) )
= ( power_int @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N ) ) ) ).
% power_int_minus_one_minus
thf(fact_6435_power__int__diff,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [X: A,M2: int,N: int] :
( ( ( X
!= ( zero_zero @ A ) )
| ( M2 != N ) )
=> ( ( power_int @ A @ X @ ( minus_minus @ int @ M2 @ N ) )
= ( divide_divide @ A @ ( power_int @ A @ X @ M2 ) @ ( power_int @ A @ X @ N ) ) ) ) ) ).
% power_int_diff
thf(fact_6436_power__int__minus__one__diff__commute,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A3: int,B2: int] :
( ( power_int @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( minus_minus @ int @ A3 @ B2 ) )
= ( power_int @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( minus_minus @ int @ B2 @ A3 ) ) ) ) ).
% power_int_minus_one_diff_commute
thf(fact_6437_tendsto__power__int,axiom,
! [A: $tType,B: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [F2: B > A,A3: A,F5: filter @ B,N: int] :
( ( filterlim @ B @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ A3 ) @ F5 )
=> ( ( A3
!= ( zero_zero @ A ) )
=> ( filterlim @ B @ A
@ ^ [X2: B] : ( power_int @ A @ ( F2 @ X2 ) @ N )
@ ( topolo7230453075368039082e_nhds @ A @ ( power_int @ A @ A3 @ N ) )
@ F5 ) ) ) ) ).
% tendsto_power_int
thf(fact_6438_continuous__at__within__power__int,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_V8999393235501362500lgebra @ B ) )
=> ! [A3: A,S: set @ A,F2: A > B,N: int] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A3 @ S ) @ F2 )
=> ( ( ( F2 @ A3 )
!= ( zero_zero @ B ) )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A3 @ S )
@ ^ [X2: A] : ( power_int @ B @ ( F2 @ X2 ) @ N ) ) ) ) ) ).
% continuous_at_within_power_int
thf(fact_6439_differentiable__power__int,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V3459762299906320749_field @ B ) )
=> ! [F2: A > B,X: A,S: set @ A,N: int] :
( ( differentiable @ A @ B @ F2 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( ( ( F2 @ X )
!= ( zero_zero @ B ) )
=> ( differentiable @ A @ B
@ ^ [X2: A] : ( power_int @ B @ ( F2 @ X2 ) @ N )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ).
% differentiable_power_int
thf(fact_6440_continuous__power__int,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_V8999393235501362500lgebra @ B ) )
=> ! [F5: filter @ A,F2: A > B,N: int] :
( ( topolo3448309680560233919inuous @ A @ B @ F5 @ F2 )
=> ( ( ( F2
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F5
@ ^ [X2: A] : X2 ) )
!= ( zero_zero @ B ) )
=> ( topolo3448309680560233919inuous @ A @ B @ F5
@ ^ [X2: A] : ( power_int @ B @ ( F2 @ X2 ) @ N ) ) ) ) ) ).
% continuous_power_int
thf(fact_6441_power__int__strict__decreasing,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [N: int,N6: int,A3: A] :
( ( ord_less @ int @ N @ N6 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ A @ A3 @ ( one_one @ A ) )
=> ( ord_less @ A @ ( power_int @ A @ A3 @ N6 ) @ ( power_int @ A @ A3 @ N ) ) ) ) ) ) ).
% power_int_strict_decreasing
thf(fact_6442_power__int__mono,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y2: A,N: int] :
( ( ord_less_eq @ A @ X @ Y2 )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ N )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ord_less_eq @ A @ ( power_int @ A @ X @ N ) @ ( power_int @ A @ Y2 @ N ) ) ) ) ) ) ).
% power_int_mono
thf(fact_6443_power__int__strict__antimono,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A,N: int] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ int @ N @ ( zero_zero @ int ) )
=> ( ord_less @ A @ ( power_int @ A @ B2 @ N ) @ ( power_int @ A @ A3 @ N ) ) ) ) ) ) ).
% power_int_strict_antimono
thf(fact_6444_one__le__power__int,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,N: int] :
( ( ord_less_eq @ A @ ( one_one @ A ) @ X )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ N )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ ( power_int @ A @ X @ N ) ) ) ) ) ).
% one_le_power_int
thf(fact_6445_one__less__power__int,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,N: int] :
( ( ord_less @ A @ ( one_one @ A ) @ A3 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ N )
=> ( ord_less @ A @ ( one_one @ A ) @ ( power_int @ A @ A3 @ N ) ) ) ) ) ).
% one_less_power_int
thf(fact_6446_power__int__add,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X: A,M2: int,N: int] :
( ( ( X
!= ( zero_zero @ A ) )
| ( ( plus_plus @ int @ M2 @ N )
!= ( zero_zero @ int ) ) )
=> ( ( power_int @ A @ X @ ( plus_plus @ int @ M2 @ N ) )
= ( times_times @ A @ ( power_int @ A @ X @ M2 ) @ ( power_int @ A @ X @ N ) ) ) ) ) ).
% power_int_add
thf(fact_6447_power__int__minus__left__distrib,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( division_ring @ A )
& ( one @ B )
& ( uminus @ B ) )
=> ! [X: C,A3: A,N: int] :
( ( nO_MATCH @ B @ C @ ( uminus_uminus @ B @ ( one_one @ B ) ) @ X )
=> ( ( power_int @ A @ ( uminus_uminus @ A @ A3 ) @ N )
= ( times_times @ A @ ( power_int @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N ) @ ( power_int @ A @ A3 @ N ) ) ) ) ) ).
% power_int_minus_left_distrib
thf(fact_6448_power__int__strict__mono,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A,N: int] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ N )
=> ( ord_less @ A @ ( power_int @ A @ A3 @ N ) @ ( power_int @ A @ B2 @ N ) ) ) ) ) ) ).
% power_int_strict_mono
thf(fact_6449_power__int__antimono,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A3: A,B2: A,N: int] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less @ int @ N @ ( zero_zero @ int ) )
=> ( ord_less_eq @ A @ ( power_int @ A @ B2 @ N ) @ ( power_int @ A @ A3 @ N ) ) ) ) ) ) ).
% power_int_antimono
thf(fact_6450_power__int__le__one,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,N: int] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ N )
=> ( ( ord_less_eq @ A @ X @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( power_int @ A @ X @ N ) @ ( one_one @ A ) ) ) ) ) ) ).
% power_int_le_one
thf(fact_6451_power__int__decreasing,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [N: int,N6: int,A3: A] :
( ( ord_less_eq @ int @ N @ N6 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A3 )
=> ( ( ord_less_eq @ A @ A3 @ ( one_one @ A ) )
=> ( ( ( A3
!= ( zero_zero @ A ) )
| ( N6
!= ( zero_zero @ int ) )
| ( N
= ( zero_zero @ int ) ) )
=> ( ord_less_eq @ A @ ( power_int @ A @ A3 @ N6 ) @ ( power_int @ A @ A3 @ N ) ) ) ) ) ) ) ).
% power_int_decreasing
thf(fact_6452_power__int__le__imp__le__exp,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,M2: int,N: int] :
( ( ord_less @ A @ ( one_one @ A ) @ X )
=> ( ( ord_less_eq @ A @ ( power_int @ A @ X @ M2 ) @ ( power_int @ A @ X @ N ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ N )
=> ( ord_less_eq @ int @ M2 @ N ) ) ) ) ) ).
% power_int_le_imp_le_exp
thf(fact_6453_power__int__le__imp__less__exp,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,M2: int,N: int] :
( ( ord_less @ A @ ( one_one @ A ) @ X )
=> ( ( ord_less @ A @ ( power_int @ A @ X @ M2 ) @ ( power_int @ A @ X @ N ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ N )
=> ( ord_less @ int @ M2 @ N ) ) ) ) ) ).
% power_int_le_imp_less_exp
thf(fact_6454_power__int__minus__left,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [N: int,A3: A] :
( ( ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N )
=> ( ( power_int @ A @ ( uminus_uminus @ A @ A3 ) @ N )
= ( power_int @ A @ A3 @ N ) ) )
& ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N )
=> ( ( power_int @ A @ ( uminus_uminus @ A @ A3 ) @ N )
= ( uminus_uminus @ A @ ( power_int @ A @ A3 @ N ) ) ) ) ) ) ).
% power_int_minus_left
thf(fact_6455_power__int__minus__mult,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [X: A,N: int] :
( ( ( X
!= ( zero_zero @ A ) )
| ( N
!= ( zero_zero @ int ) ) )
=> ( ( times_times @ A @ ( power_int @ A @ X @ ( minus_minus @ int @ N @ ( one_one @ int ) ) ) @ X )
= ( power_int @ A @ X @ N ) ) ) ) ).
% power_int_minus_mult
thf(fact_6456_power__int__add__1_H,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X: A,M2: int] :
( ( ( X
!= ( zero_zero @ A ) )
| ( M2
!= ( uminus_uminus @ int @ ( one_one @ int ) ) ) )
=> ( ( power_int @ A @ X @ ( plus_plus @ int @ M2 @ ( one_one @ int ) ) )
= ( times_times @ A @ X @ ( power_int @ A @ X @ M2 ) ) ) ) ) ).
% power_int_add_1'
thf(fact_6457_power__int__add__1,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [X: A,M2: int] :
( ( ( X
!= ( zero_zero @ A ) )
| ( M2
!= ( uminus_uminus @ int @ ( one_one @ int ) ) ) )
=> ( ( power_int @ A @ X @ ( plus_plus @ int @ M2 @ ( one_one @ int ) ) )
= ( times_times @ A @ ( power_int @ A @ X @ M2 ) @ X ) ) ) ) ).
% power_int_add_1
thf(fact_6458_power__int__def,axiom,
! [A: $tType] :
( ( ( inverse @ A )
& ( power @ A ) )
=> ( ( power_int @ A )
= ( ^ [X2: A,N5: int] : ( if @ A @ ( ord_less_eq @ int @ ( zero_zero @ int ) @ N5 ) @ ( power_power @ A @ X2 @ ( nat2 @ N5 ) ) @ ( power_power @ A @ ( inverse_inverse @ A @ X2 ) @ ( nat2 @ ( uminus_uminus @ int @ N5 ) ) ) ) ) ) ) ).
% power_int_def
thf(fact_6459_powr__real__of__int_H,axiom,
! [X: real,N: int] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ( X
!= ( zero_zero @ real ) )
| ( ord_less @ int @ ( zero_zero @ int ) @ N ) )
=> ( ( powr @ real @ X @ ( ring_1_of_int @ real @ N ) )
= ( power_int @ real @ X @ N ) ) ) ) ).
% powr_real_of_int'
thf(fact_6460_DERIV__power__int,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F2: A > A,D2: A,X: A,S: set @ A,N: int] :
( ( has_field_derivative @ A @ F2 @ D2 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( ( ( F2 @ X )
!= ( zero_zero @ A ) )
=> ( has_field_derivative @ A
@ ^ [X2: A] : ( power_int @ A @ ( F2 @ X2 ) @ N )
@ ( times_times @ A @ ( times_times @ A @ ( ring_1_of_int @ A @ N ) @ ( power_int @ A @ ( F2 @ X ) @ ( minus_minus @ int @ N @ ( one_one @ int ) ) ) ) @ D2 )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ).
% DERIV_power_int
thf(fact_6461_power__int__numeral__neg__numeral,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [M2: num,N: num] :
( ( power_int @ A @ ( numeral_numeral @ A @ M2 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
= ( inverse_inverse @ A @ ( numeral_numeral @ A @ ( pow @ M2 @ N ) ) ) ) ) ).
% power_int_numeral_neg_numeral
thf(fact_6462_inverse__real_Otransfer,axiom,
( bNF_rel_fun @ ( nat > rat ) @ real @ ( nat > rat ) @ real @ pcr_real @ pcr_real
@ ^ [X7: nat > rat] :
( if @ ( nat > rat ) @ ( vanishes @ X7 )
@ ^ [N5: nat] : ( zero_zero @ rat )
@ ^ [N5: nat] : ( inverse_inverse @ rat @ ( X7 @ N5 ) ) )
@ ( inverse_inverse @ real ) ) ).
% inverse_real.transfer
thf(fact_6463_pow_Osimps_I1_J,axiom,
! [X: num] :
( ( pow @ X @ one2 )
= X ) ).
% pow.simps(1)
thf(fact_6464_zero__real_Otransfer,axiom,
( pcr_real
@ ^ [N5: nat] : ( zero_zero @ rat )
@ ( zero_zero @ real ) ) ).
% zero_real.transfer
thf(fact_6465_one__real_Otransfer,axiom,
( pcr_real
@ ^ [N5: nat] : ( one_one @ rat )
@ ( one_one @ real ) ) ).
% one_real.transfer
thf(fact_6466_uminus__real_Otransfer,axiom,
( bNF_rel_fun @ ( nat > rat ) @ real @ ( nat > rat ) @ real @ pcr_real @ pcr_real
@ ^ [X7: nat > rat,N5: nat] : ( uminus_uminus @ rat @ ( X7 @ N5 ) )
@ ( uminus_uminus @ real ) ) ).
% uminus_real.transfer
thf(fact_6467_plus__real_Otransfer,axiom,
( bNF_rel_fun @ ( nat > rat ) @ real @ ( ( nat > rat ) > nat > rat ) @ ( real > real ) @ pcr_real @ ( bNF_rel_fun @ ( nat > rat ) @ real @ ( nat > rat ) @ real @ pcr_real @ pcr_real )
@ ^ [X7: nat > rat,Y8: nat > rat,N5: nat] : ( plus_plus @ rat @ ( X7 @ N5 ) @ ( Y8 @ N5 ) )
@ ( plus_plus @ real ) ) ).
% plus_real.transfer
thf(fact_6468_times__real_Otransfer,axiom,
( bNF_rel_fun @ ( nat > rat ) @ real @ ( ( nat > rat ) > nat > rat ) @ ( real > real ) @ pcr_real @ ( bNF_rel_fun @ ( nat > rat ) @ real @ ( nat > rat ) @ real @ pcr_real @ pcr_real )
@ ^ [X7: nat > rat,Y8: nat > rat,N5: nat] : ( times_times @ rat @ ( X7 @ N5 ) @ ( Y8 @ N5 ) )
@ ( times_times @ real ) ) ).
% times_real.transfer
thf(fact_6469_Real_Opositive_Otransfer,axiom,
( bNF_rel_fun @ ( nat > rat ) @ real @ $o @ $o @ pcr_real
@ ^ [Y4: $o,Z: $o] : Y4 = Z
@ ^ [X7: nat > rat] :
? [R: rat] :
( ( ord_less @ rat @ ( zero_zero @ rat ) @ R )
& ? [K3: nat] :
! [N5: nat] :
( ( ord_less_eq @ nat @ K3 @ N5 )
=> ( ord_less @ rat @ R @ ( X7 @ N5 ) ) ) )
@ positive2 ) ).
% Real.positive.transfer
thf(fact_6470_pow_Osimps_I3_J,axiom,
! [X: num,Y2: num] :
( ( pow @ X @ ( bit1 @ Y2 ) )
= ( times_times @ num @ ( sqr @ ( pow @ X @ Y2 ) ) @ X ) ) ).
% pow.simps(3)
thf(fact_6471_sqr__conv__mult,axiom,
( sqr
= ( ^ [X2: num] : ( times_times @ num @ X2 @ X2 ) ) ) ).
% sqr_conv_mult
thf(fact_6472_Real_Opositive__mult,axiom,
! [X: real,Y2: real] :
( ( positive2 @ X )
=> ( ( positive2 @ Y2 )
=> ( positive2 @ ( times_times @ real @ X @ Y2 ) ) ) ) ).
% Real.positive_mult
thf(fact_6473_Real_Opositive__zero,axiom,
~ ( positive2 @ ( zero_zero @ real ) ) ).
% Real.positive_zero
thf(fact_6474_Real_Opositive__add,axiom,
! [X: real,Y2: real] :
( ( positive2 @ X )
=> ( ( positive2 @ Y2 )
=> ( positive2 @ ( plus_plus @ real @ X @ Y2 ) ) ) ) ).
% Real.positive_add
thf(fact_6475_sqr_Osimps_I1_J,axiom,
( ( sqr @ one2 )
= one2 ) ).
% sqr.simps(1)
thf(fact_6476_sqr_Osimps_I2_J,axiom,
! [N: num] :
( ( sqr @ ( bit0 @ N ) )
= ( bit0 @ ( bit0 @ ( sqr @ N ) ) ) ) ).
% sqr.simps(2)
thf(fact_6477_Real_Opositive__minus,axiom,
! [X: real] :
( ~ ( positive2 @ X )
=> ( ( X
!= ( zero_zero @ real ) )
=> ( positive2 @ ( uminus_uminus @ real @ X ) ) ) ) ).
% Real.positive_minus
thf(fact_6478_less__real__def,axiom,
( ( ord_less @ real )
= ( ^ [X2: real,Y5: real] : ( positive2 @ ( minus_minus @ real @ Y5 @ X2 ) ) ) ) ).
% less_real_def
thf(fact_6479_pow_Osimps_I2_J,axiom,
! [X: num,Y2: num] :
( ( pow @ X @ ( bit0 @ Y2 ) )
= ( sqr @ ( pow @ X @ Y2 ) ) ) ).
% pow.simps(2)
thf(fact_6480_sqr_Osimps_I3_J,axiom,
! [N: num] :
( ( sqr @ ( bit1 @ N ) )
= ( bit1 @ ( bit0 @ ( plus_plus @ num @ ( sqr @ N ) @ N ) ) ) ) ).
% sqr.simps(3)
thf(fact_6481_Real_Opositive_Orep__eq,axiom,
( positive2
= ( ^ [X2: real] :
? [R: rat] :
( ( ord_less @ rat @ ( zero_zero @ rat ) @ R )
& ? [K3: nat] :
! [N5: nat] :
( ( ord_less_eq @ nat @ K3 @ N5 )
=> ( ord_less @ rat @ R @ ( rep_real @ X2 @ N5 ) ) ) ) ) ) ).
% Real.positive.rep_eq
thf(fact_6482_Real_Opositive_Orsp,axiom,
( bNF_rel_fun @ ( nat > rat ) @ ( nat > rat ) @ $o @ $o @ realrel
@ ^ [Y4: $o,Z: $o] : Y4 = Z
@ ^ [X7: nat > rat] :
? [R: rat] :
( ( ord_less @ rat @ ( zero_zero @ rat ) @ R )
& ? [K3: nat] :
! [N5: nat] :
( ( ord_less_eq @ nat @ K3 @ N5 )
=> ( ord_less @ rat @ R @ ( X7 @ N5 ) ) ) )
@ ^ [X7: nat > rat] :
? [R: rat] :
( ( ord_less @ rat @ ( zero_zero @ rat ) @ R )
& ? [K3: nat] :
! [N5: nat] :
( ( ord_less_eq @ nat @ K3 @ N5 )
=> ( ord_less @ rat @ R @ ( X7 @ N5 ) ) ) ) ) ).
% Real.positive.rsp
thf(fact_6483_times__real_Orsp,axiom,
( bNF_rel_fun @ ( nat > rat ) @ ( nat > rat ) @ ( ( nat > rat ) > nat > rat ) @ ( ( nat > rat ) > nat > rat ) @ realrel @ ( bNF_rel_fun @ ( nat > rat ) @ ( nat > rat ) @ ( nat > rat ) @ ( nat > rat ) @ realrel @ realrel )
@ ^ [X7: nat > rat,Y8: nat > rat,N5: nat] : ( times_times @ rat @ ( X7 @ N5 ) @ ( Y8 @ N5 ) )
@ ^ [X7: nat > rat,Y8: nat > rat,N5: nat] : ( times_times @ rat @ ( X7 @ N5 ) @ ( Y8 @ N5 ) ) ) ).
% times_real.rsp
thf(fact_6484_uminus__real_Orsp,axiom,
( bNF_rel_fun @ ( nat > rat ) @ ( nat > rat ) @ ( nat > rat ) @ ( nat > rat ) @ realrel @ realrel
@ ^ [X7: nat > rat,N5: nat] : ( uminus_uminus @ rat @ ( X7 @ N5 ) )
@ ^ [X7: nat > rat,N5: nat] : ( uminus_uminus @ rat @ ( X7 @ N5 ) ) ) ).
% uminus_real.rsp
thf(fact_6485_plus__real_Orsp,axiom,
( bNF_rel_fun @ ( nat > rat ) @ ( nat > rat ) @ ( ( nat > rat ) > nat > rat ) @ ( ( nat > rat ) > nat > rat ) @ realrel @ ( bNF_rel_fun @ ( nat > rat ) @ ( nat > rat ) @ ( nat > rat ) @ ( nat > rat ) @ realrel @ realrel )
@ ^ [X7: nat > rat,Y8: nat > rat,N5: nat] : ( plus_plus @ rat @ ( X7 @ N5 ) @ ( Y8 @ N5 ) )
@ ^ [X7: nat > rat,Y8: nat > rat,N5: nat] : ( plus_plus @ rat @ ( X7 @ N5 ) @ ( Y8 @ N5 ) ) ) ).
% plus_real.rsp
thf(fact_6486_one__real_Orsp,axiom,
( realrel
@ ^ [N5: nat] : ( one_one @ rat )
@ ^ [N5: nat] : ( one_one @ rat ) ) ).
% one_real.rsp
thf(fact_6487_zero__real_Orsp,axiom,
( realrel
@ ^ [N5: nat] : ( zero_zero @ rat )
@ ^ [N5: nat] : ( zero_zero @ rat ) ) ).
% zero_real.rsp
thf(fact_6488_real_Orel__eq__transfer,axiom,
( bNF_rel_fun @ ( nat > rat ) @ real @ ( ( nat > rat ) > $o ) @ ( real > $o ) @ pcr_real
@ ( bNF_rel_fun @ ( nat > rat ) @ real @ $o @ $o @ pcr_real
@ ^ [Y4: $o,Z: $o] : Y4 = Z )
@ realrel
@ ^ [Y4: real,Z: real] : Y4 = Z ) ).
% real.rel_eq_transfer
thf(fact_6489_inverse__real_Orsp,axiom,
( bNF_rel_fun @ ( nat > rat ) @ ( nat > rat ) @ ( nat > rat ) @ ( nat > rat ) @ realrel @ realrel
@ ^ [X7: nat > rat] :
( if @ ( nat > rat ) @ ( vanishes @ X7 )
@ ^ [N5: nat] : ( zero_zero @ rat )
@ ^ [N5: nat] : ( inverse_inverse @ rat @ ( X7 @ N5 ) ) )
@ ^ [X7: nat > rat] :
( if @ ( nat > rat ) @ ( vanishes @ X7 )
@ ^ [N5: nat] : ( zero_zero @ rat )
@ ^ [N5: nat] : ( inverse_inverse @ rat @ ( X7 @ N5 ) ) ) ) ).
% inverse_real.rsp
thf(fact_6490_Real_Opositive__def,axiom,
( positive2
= ( map_fun @ real @ ( nat > rat ) @ $o @ $o @ rep_real @ ( id @ $o )
@ ^ [X7: nat > rat] :
? [R: rat] :
( ( ord_less @ rat @ ( zero_zero @ rat ) @ R )
& ? [K3: nat] :
! [N5: nat] :
( ( ord_less_eq @ nat @ K3 @ N5 )
=> ( ord_less @ rat @ R @ ( X7 @ N5 ) ) ) ) ) ) ).
% Real.positive_def
thf(fact_6491_UN__le__eq__Un0,axiom,
! [A: $tType,M10: nat > ( set @ A ),N: nat] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ M10 @ ( set_ord_atMost @ nat @ N ) ) )
= ( sup_sup @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ M10 @ ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ N ) ) ) @ ( M10 @ ( zero_zero @ nat ) ) ) ) ).
% UN_le_eq_Un0
thf(fact_6492_sup_Oidem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: A] :
( ( sup_sup @ A @ A3 @ A3 )
= A3 ) ) ).
% sup.idem
thf(fact_6493_sup__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A] :
( ( sup_sup @ A @ X @ X )
= X ) ) ).
% sup_idem
thf(fact_6494_sup_Oleft__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: A,B2: A] :
( ( sup_sup @ A @ A3 @ ( sup_sup @ A @ A3 @ B2 ) )
= ( sup_sup @ A @ A3 @ B2 ) ) ) ).
% sup.left_idem
thf(fact_6495_sup__left__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A,Y2: A] :
( ( sup_sup @ A @ X @ ( sup_sup @ A @ X @ Y2 ) )
= ( sup_sup @ A @ X @ Y2 ) ) ) ).
% sup_left_idem
thf(fact_6496_sup_Oright__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: A,B2: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ A3 @ B2 ) @ B2 )
= ( sup_sup @ A @ A3 @ B2 ) ) ) ).
% sup.right_idem
thf(fact_6497_sup__apply,axiom,
! [B: $tType,A: $tType] :
( ( semilattice_sup @ B )
=> ( ( sup_sup @ ( A > B ) )
= ( ^ [F4: A > B,G2: A > B,X2: A] : ( sup_sup @ B @ ( F4 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ).
% sup_apply
thf(fact_6498_sup_Obounded__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B2: A,C2: A,A3: A] :
( ( ord_less_eq @ A @ ( sup_sup @ A @ B2 @ C2 ) @ A3 )
= ( ( ord_less_eq @ A @ B2 @ A3 )
& ( ord_less_eq @ A @ C2 @ A3 ) ) ) ) ).
% sup.bounded_iff
thf(fact_6499_le__sup__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A,Y2: A,Z2: A] :
( ( ord_less_eq @ A @ ( sup_sup @ A @ X @ Y2 ) @ Z2 )
= ( ( ord_less_eq @ A @ X @ Z2 )
& ( ord_less_eq @ A @ Y2 @ Z2 ) ) ) ) ).
% le_sup_iff
thf(fact_6500_sup__top__right,axiom,
! [A: $tType] :
( ( bounded_lattice_top @ A )
=> ! [X: A] :
( ( sup_sup @ A @ X @ ( top_top @ A ) )
= ( top_top @ A ) ) ) ).
% sup_top_right
thf(fact_6501_sup__top__left,axiom,
! [A: $tType] :
( ( bounded_lattice_top @ A )
=> ! [X: A] :
( ( sup_sup @ A @ ( top_top @ A ) @ X )
= ( top_top @ A ) ) ) ).
% sup_top_left
thf(fact_6502_sup__bot__left,axiom,
! [A: $tType] :
( ( bounde4967611905675639751up_bot @ A )
=> ! [X: A] :
( ( sup_sup @ A @ ( bot_bot @ A ) @ X )
= X ) ) ).
% sup_bot_left
thf(fact_6503_sup__bot__right,axiom,
! [A: $tType] :
( ( bounde4967611905675639751up_bot @ A )
=> ! [X: A] :
( ( sup_sup @ A @ X @ ( bot_bot @ A ) )
= X ) ) ).
% sup_bot_right
thf(fact_6504_bot__eq__sup__iff,axiom,
! [A: $tType] :
( ( bounde4967611905675639751up_bot @ A )
=> ! [X: A,Y2: A] :
( ( ( bot_bot @ A )
= ( sup_sup @ A @ X @ Y2 ) )
= ( ( X
= ( bot_bot @ A ) )
& ( Y2
= ( bot_bot @ A ) ) ) ) ) ).
% bot_eq_sup_iff
thf(fact_6505_sup__eq__bot__iff,axiom,
! [A: $tType] :
( ( bounde4967611905675639751up_bot @ A )
=> ! [X: A,Y2: A] :
( ( ( sup_sup @ A @ X @ Y2 )
= ( bot_bot @ A ) )
= ( ( X
= ( bot_bot @ A ) )
& ( Y2
= ( bot_bot @ A ) ) ) ) ) ).
% sup_eq_bot_iff
thf(fact_6506_sup__bot_Oeq__neutr__iff,axiom,
! [A: $tType] :
( ( bounde4967611905675639751up_bot @ A )
=> ! [A3: A,B2: A] :
( ( ( sup_sup @ A @ A3 @ B2 )
= ( bot_bot @ A ) )
= ( ( A3
= ( bot_bot @ A ) )
& ( B2
= ( bot_bot @ A ) ) ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_6507_sup__bot_Oleft__neutral,axiom,
! [A: $tType] :
( ( bounde4967611905675639751up_bot @ A )
=> ! [A3: A] :
( ( sup_sup @ A @ ( bot_bot @ A ) @ A3 )
= A3 ) ) ).
% sup_bot.left_neutral
thf(fact_6508_sup__bot_Oneutr__eq__iff,axiom,
! [A: $tType] :
( ( bounde4967611905675639751up_bot @ A )
=> ! [A3: A,B2: A] :
( ( ( bot_bot @ A )
= ( sup_sup @ A @ A3 @ B2 ) )
= ( ( A3
= ( bot_bot @ A ) )
& ( B2
= ( bot_bot @ A ) ) ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_6509_sup__bot_Oright__neutral,axiom,
! [A: $tType] :
( ( bounde4967611905675639751up_bot @ A )
=> ! [A3: A] :
( ( sup_sup @ A @ A3 @ ( bot_bot @ A ) )
= A3 ) ) ).
% sup_bot.right_neutral
thf(fact_6510_sup__inf__absorb,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X: A,Y2: A] :
( ( sup_sup @ A @ X @ ( inf_inf @ A @ X @ Y2 ) )
= X ) ) ).
% sup_inf_absorb
thf(fact_6511_inf__sup__absorb,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X: A,Y2: A] :
( ( inf_inf @ A @ X @ ( sup_sup @ A @ X @ Y2 ) )
= X ) ) ).
% inf_sup_absorb
thf(fact_6512_Un__Diff__cancel2,axiom,
! [A: $tType,B7: set @ A,A4: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ B7 @ A4 ) @ A4 )
= ( sup_sup @ ( set @ A ) @ B7 @ A4 ) ) ).
% Un_Diff_cancel2
thf(fact_6513_Un__Diff__cancel,axiom,
! [A: $tType,A4: set @ A,B7: set @ A] :
( ( sup_sup @ ( set @ A ) @ A4 @ ( minus_minus @ ( set @ A ) @ B7 @ A4 ) )
= ( sup_sup @ ( set @ A ) @ A4 @ B7 ) ) ).
% Un_Diff_cancel
thf(fact_6514_sup__compl__top__left1,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y2: A] :
( ( sup_sup @ A @ ( uminus_uminus @ A @ X ) @ ( sup_sup @ A @ X @ Y2 ) )
= ( top_top @ A ) ) ) ).
% sup_compl_top_left1
thf(fact_6515_sup__compl__top__left2,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y2: A] :
( ( sup_sup @ A @ X @ ( sup_sup @ A @ ( uminus_uminus @ A @ X ) @ Y2 ) )
= ( top_top @ A ) ) ) ).
% sup_compl_top_left2
thf(fact_6516_boolean__algebra_Odisj__cancel__left,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A] :
( ( sup_sup @ A @ ( uminus_uminus @ A @ X ) @ X )
= ( top_top @ A ) ) ) ).
% boolean_algebra.disj_cancel_left
thf(fact_6517_boolean__algebra_Odisj__cancel__right,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A] :
( ( sup_sup @ A @ X @ ( uminus_uminus @ A @ X ) )
= ( top_top @ A ) ) ) ).
% boolean_algebra.disj_cancel_right
thf(fact_6518_boolean__algebra_Ode__Morgan__disj,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y2: A] :
( ( uminus_uminus @ A @ ( sup_sup @ A @ X @ Y2 ) )
= ( inf_inf @ A @ ( uminus_uminus @ A @ X ) @ ( uminus_uminus @ A @ Y2 ) ) ) ) ).
% boolean_algebra.de_Morgan_disj
thf(fact_6519_boolean__algebra_Ode__Morgan__conj,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y2: A] :
( ( uminus_uminus @ A @ ( inf_inf @ A @ X @ Y2 ) )
= ( sup_sup @ A @ ( uminus_uminus @ A @ X ) @ ( uminus_uminus @ A @ Y2 ) ) ) ) ).
% boolean_algebra.de_Morgan_conj
thf(fact_6520_Compl__Diff__eq,axiom,
! [A: $tType,A4: set @ A,B7: set @ A] :
( ( uminus_uminus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A4 @ B7 ) )
= ( sup_sup @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A4 ) @ B7 ) ) ).
% Compl_Diff_eq
thf(fact_6521_Inf__sup__eq__top__iff,axiom,
! [A: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [B7: set @ A,A3: A] :
( ( ( sup_sup @ A @ ( complete_Inf_Inf @ A @ B7 ) @ A3 )
= ( top_top @ A ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ B7 )
=> ( ( sup_sup @ A @ X2 @ A3 )
= ( top_top @ A ) ) ) ) ) ) ).
% Inf_sup_eq_top_iff
thf(fact_6522_distrib__sup__le,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X: A,Y2: A,Z2: A] : ( ord_less_eq @ A @ ( sup_sup @ A @ X @ ( inf_inf @ A @ Y2 @ Z2 ) ) @ ( inf_inf @ A @ ( sup_sup @ A @ X @ Y2 ) @ ( sup_sup @ A @ X @ Z2 ) ) ) ) ).
% distrib_sup_le
thf(fact_6523_distrib__inf__le,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X: A,Y2: A,Z2: A] : ( ord_less_eq @ A @ ( sup_sup @ A @ ( inf_inf @ A @ X @ Y2 ) @ ( inf_inf @ A @ X @ Z2 ) ) @ ( inf_inf @ A @ X @ ( sup_sup @ A @ Y2 @ Z2 ) ) ) ) ).
% distrib_inf_le
thf(fact_6524_sup__inf__distrib2,axiom,
! [A: $tType] :
( ( distrib_lattice @ A )
=> ! [Y2: A,Z2: A,X: A] :
( ( sup_sup @ A @ ( inf_inf @ A @ Y2 @ Z2 ) @ X )
= ( inf_inf @ A @ ( sup_sup @ A @ Y2 @ X ) @ ( sup_sup @ A @ Z2 @ X ) ) ) ) ).
% sup_inf_distrib2
thf(fact_6525_sup__inf__distrib1,axiom,
! [A: $tType] :
( ( distrib_lattice @ A )
=> ! [X: A,Y2: A,Z2: A] :
( ( sup_sup @ A @ X @ ( inf_inf @ A @ Y2 @ Z2 ) )
= ( inf_inf @ A @ ( sup_sup @ A @ X @ Y2 ) @ ( sup_sup @ A @ X @ Z2 ) ) ) ) ).
% sup_inf_distrib1
thf(fact_6526_inf__sup__distrib2,axiom,
! [A: $tType] :
( ( distrib_lattice @ A )
=> ! [Y2: A,Z2: A,X: A] :
( ( inf_inf @ A @ ( sup_sup @ A @ Y2 @ Z2 ) @ X )
= ( sup_sup @ A @ ( inf_inf @ A @ Y2 @ X ) @ ( inf_inf @ A @ Z2 @ X ) ) ) ) ).
% inf_sup_distrib2
thf(fact_6527_inf__sup__distrib1,axiom,
! [A: $tType] :
( ( distrib_lattice @ A )
=> ! [X: A,Y2: A,Z2: A] :
( ( inf_inf @ A @ X @ ( sup_sup @ A @ Y2 @ Z2 ) )
= ( sup_sup @ A @ ( inf_inf @ A @ X @ Y2 ) @ ( inf_inf @ A @ X @ Z2 ) ) ) ) ).
% inf_sup_distrib1
thf(fact_6528_distrib__imp2,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X: A,Y2: A,Z2: A] :
( ! [X3: A,Y3: A,Z4: A] :
( ( sup_sup @ A @ X3 @ ( inf_inf @ A @ Y3 @ Z4 ) )
= ( inf_inf @ A @ ( sup_sup @ A @ X3 @ Y3 ) @ ( sup_sup @ A @ X3 @ Z4 ) ) )
=> ( ( inf_inf @ A @ X @ ( sup_sup @ A @ Y2 @ Z2 ) )
= ( sup_sup @ A @ ( inf_inf @ A @ X @ Y2 ) @ ( inf_inf @ A @ X @ Z2 ) ) ) ) ) ).
% distrib_imp2
thf(fact_6529_distrib__imp1,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X: A,Y2: A,Z2: A] :
( ! [X3: A,Y3: A,Z4: A] :
( ( inf_inf @ A @ X3 @ ( sup_sup @ A @ Y3 @ Z4 ) )
= ( sup_sup @ A @ ( inf_inf @ A @ X3 @ Y3 ) @ ( inf_inf @ A @ X3 @ Z4 ) ) )
=> ( ( sup_sup @ A @ X @ ( inf_inf @ A @ Y2 @ Z2 ) )
= ( inf_inf @ A @ ( sup_sup @ A @ X @ Y2 ) @ ( sup_sup @ A @ X @ Z2 ) ) ) ) ) ).
% distrib_imp1
thf(fact_6530_Un__Diff__Int,axiom,
! [A: $tType,A4: set @ A,B7: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A4 @ B7 ) @ ( inf_inf @ ( set @ A ) @ A4 @ B7 ) )
= A4 ) ).
% Un_Diff_Int
thf(fact_6531_Int__Diff__Un,axiom,
! [A: $tType,A4: set @ A,B7: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A4 @ B7 ) @ ( minus_minus @ ( set @ A ) @ A4 @ B7 ) )
= A4 ) ).
% Int_Diff_Un
thf(fact_6532_Diff__Int,axiom,
! [A: $tType,A4: set @ A,B7: set @ A,C5: set @ A] :
( ( minus_minus @ ( set @ A ) @ A4 @ ( inf_inf @ ( set @ A ) @ B7 @ C5 ) )
= ( sup_sup @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A4 @ B7 ) @ ( minus_minus @ ( set @ A ) @ A4 @ C5 ) ) ) ).
% Diff_Int
thf(fact_6533_Diff__Un,axiom,
! [A: $tType,A4: set @ A,B7: set @ A,C5: set @ A] :
( ( minus_minus @ ( set @ A ) @ A4 @ ( sup_sup @ ( set @ A ) @ B7 @ C5 ) )
= ( inf_inf @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A4 @ B7 ) @ ( minus_minus @ ( set @ A ) @ A4 @ C5 ) ) ) ).
% Diff_Un
thf(fact_6534_Compl__Un,axiom,
! [A: $tType,A4: set @ A,B7: set @ A] :
( ( uminus_uminus @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A4 @ B7 ) )
= ( inf_inf @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A4 ) @ ( uminus_uminus @ ( set @ A ) @ B7 ) ) ) ).
% Compl_Un
thf(fact_6535_Compl__Int,axiom,
! [A: $tType,A4: set @ A,B7: set @ A] :
( ( uminus_uminus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A4 @ B7 ) )
= ( sup_sup @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A4 ) @ ( uminus_uminus @ ( set @ A ) @ B7 ) ) ) ).
% Compl_Int
thf(fact_6536_Compl__partition2,axiom,
! [A: $tType,A4: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A4 ) @ A4 )
= ( top_top @ ( set @ A ) ) ) ).
% Compl_partition2
thf(fact_6537_Compl__partition,axiom,
! [A: $tType,A4: set @ A] :
( ( sup_sup @ ( set @ A ) @ A4 @ ( uminus_uminus @ ( set @ A ) @ A4 ) )
= ( top_top @ ( set @ A ) ) ) ).
% Compl_partition
thf(fact_6538_sup__cancel__left2,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,A3: A,B2: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ ( uminus_uminus @ A @ X ) @ A3 ) @ ( sup_sup @ A @ X @ B2 ) )
= ( top_top @ A ) ) ) ).
% sup_cancel_left2
thf(fact_6539_sup__cancel__left1,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,A3: A,B2: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ X @ A3 ) @ ( sup_sup @ A @ ( uminus_uminus @ A @ X ) @ B2 ) )
= ( top_top @ A ) ) ) ).
% sup_cancel_left1
thf(fact_6540_Collect__imp__eq,axiom,
! [A: $tType,P: A > $o,Q: A > $o] :
( ( collect @ A
@ ^ [X2: A] :
( ( P @ X2 )
=> ( Q @ X2 ) ) )
= ( sup_sup @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ ( collect @ A @ P ) ) @ ( collect @ A @ Q ) ) ) ).
% Collect_imp_eq
thf(fact_6541_inf__sup__aci_I8_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X: A,Y2: A] :
( ( sup_sup @ A @ X @ ( sup_sup @ A @ X @ Y2 ) )
= ( sup_sup @ A @ X @ Y2 ) ) ) ).
% inf_sup_aci(8)
thf(fact_6542_inf__sup__aci_I7_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X: A,Y2: A,Z2: A] :
( ( sup_sup @ A @ X @ ( sup_sup @ A @ Y2 @ Z2 ) )
= ( sup_sup @ A @ Y2 @ ( sup_sup @ A @ X @ Z2 ) ) ) ) ).
% inf_sup_aci(7)
thf(fact_6543_inf__sup__aci_I6_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X: A,Y2: A,Z2: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ X @ Y2 ) @ Z2 )
= ( sup_sup @ A @ X @ ( sup_sup @ A @ Y2 @ Z2 ) ) ) ) ).
% inf_sup_aci(6)
thf(fact_6544_inf__sup__aci_I5_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ( ( sup_sup @ A )
= ( ^ [X2: A,Y5: A] : ( sup_sup @ A @ Y5 @ X2 ) ) ) ) ).
% inf_sup_aci(5)
thf(fact_6545_sup_Oassoc,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ A3 @ B2 ) @ C2 )
= ( sup_sup @ A @ A3 @ ( sup_sup @ A @ B2 @ C2 ) ) ) ) ).
% sup.assoc
thf(fact_6546_sup__assoc,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A,Y2: A,Z2: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ X @ Y2 ) @ Z2 )
= ( sup_sup @ A @ X @ ( sup_sup @ A @ Y2 @ Z2 ) ) ) ) ).
% sup_assoc
thf(fact_6547_sup_Ocommute,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( sup_sup @ A )
= ( ^ [A5: A,B3: A] : ( sup_sup @ A @ B3 @ A5 ) ) ) ) ).
% sup.commute
thf(fact_6548_sup__commute,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( sup_sup @ A )
= ( ^ [X2: A,Y5: A] : ( sup_sup @ A @ Y5 @ X2 ) ) ) ) ).
% sup_commute
thf(fact_6549_sup_Oleft__commute,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( sup_sup @ A @ B2 @ ( sup_sup @ A @ A3 @ C2 ) )
= ( sup_sup @ A @ A3 @ ( sup_sup @ A @ B2 @ C2 ) ) ) ) ).
% sup.left_commute
thf(fact_6550_sup__left__commute,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A,Y2: A,Z2: A] :
( ( sup_sup @ A @ X @ ( sup_sup @ A @ Y2 @ Z2 ) )
= ( sup_sup @ A @ Y2 @ ( sup_sup @ A @ X @ Z2 ) ) ) ) ).
% sup_left_commute
thf(fact_6551_sup__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( semilattice_sup @ B )
=> ( ( sup_sup @ ( A > B ) )
= ( ^ [F4: A > B,G2: A > B,X2: A] : ( sup_sup @ B @ ( F4 @ X2 ) @ ( G2 @ X2 ) ) ) ) ) ).
% sup_fun_def
thf(fact_6552_less__supI1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A,A3: A,B2: A] :
( ( ord_less @ A @ X @ A3 )
=> ( ord_less @ A @ X @ ( sup_sup @ A @ A3 @ B2 ) ) ) ) ).
% less_supI1
thf(fact_6553_less__supI2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A,B2: A,A3: A] :
( ( ord_less @ A @ X @ B2 )
=> ( ord_less @ A @ X @ ( sup_sup @ A @ A3 @ B2 ) ) ) ) ).
% less_supI2
thf(fact_6554_sup_Oabsorb3,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B2: A,A3: A] :
( ( ord_less @ A @ B2 @ A3 )
=> ( ( sup_sup @ A @ A3 @ B2 )
= A3 ) ) ) ).
% sup.absorb3
thf(fact_6555_sup_Oabsorb4,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: A,B2: A] :
( ( ord_less @ A @ A3 @ B2 )
=> ( ( sup_sup @ A @ A3 @ B2 )
= B2 ) ) ) ).
% sup.absorb4
thf(fact_6556_sup_Ostrict__boundedE,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B2: A,C2: A,A3: A] :
( ( ord_less @ A @ ( sup_sup @ A @ B2 @ C2 ) @ A3 )
=> ~ ( ( ord_less @ A @ B2 @ A3 )
=> ~ ( ord_less @ A @ C2 @ A3 ) ) ) ) ).
% sup.strict_boundedE
thf(fact_6557_sup_Ostrict__order__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( ord_less @ A )
= ( ^ [B3: A,A5: A] :
( ( A5
= ( sup_sup @ A @ A5 @ B3 ) )
& ( A5 != B3 ) ) ) ) ) ).
% sup.strict_order_iff
thf(fact_6558_sup_Ostrict__coboundedI1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( ord_less @ A @ C2 @ A3 )
=> ( ord_less @ A @ C2 @ ( sup_sup @ A @ A3 @ B2 ) ) ) ) ).
% sup.strict_coboundedI1
thf(fact_6559_sup_Ostrict__coboundedI2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [C2: A,B2: A,A3: A] :
( ( ord_less @ A @ C2 @ B2 )
=> ( ord_less @ A @ C2 @ ( sup_sup @ A @ A3 @ B2 ) ) ) ) ).
% sup.strict_coboundedI2
thf(fact_6560_sup__max,axiom,
! [A: $tType] :
( ( ( semilattice_sup @ A )
& ( linorder @ A ) )
=> ( ( sup_sup @ A )
= ( ord_max @ A ) ) ) ).
% sup_max
thf(fact_6561_Un__Diff,axiom,
! [A: $tType,A4: set @ A,B7: set @ A,C5: set @ A] :
( ( minus_minus @ ( set @ A ) @ ( sup_sup @ ( set @ A ) @ A4 @ B7 ) @ C5 )
= ( sup_sup @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A4 @ C5 ) @ ( minus_minus @ ( set @ A ) @ B7 @ C5 ) ) ) ).
% Un_Diff
thf(fact_6562_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_6563_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_6564_le__supE,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: A,B2: A,X: A] :
( ( ord_less_eq @ A @ ( sup_sup @ A @ A3 @ B2 ) @ X )
=> ~ ( ( ord_less_eq @ A @ A3 @ X )
=> ~ ( ord_less_eq @ A @ B2 @ X ) ) ) ) ).
% le_supE
thf(fact_6565_le__supI,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: A,X: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ X )
=> ( ( ord_less_eq @ A @ B2 @ X )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ A3 @ B2 ) @ X ) ) ) ) ).
% le_supI
thf(fact_6566_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_6567_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_6568_le__supI1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A,A3: A,B2: A] :
( ( ord_less_eq @ A @ X @ A3 )
=> ( ord_less_eq @ A @ X @ ( sup_sup @ A @ A3 @ B2 ) ) ) ) ).
% le_supI1
thf(fact_6569_le__supI2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A,B2: A,A3: A] :
( ( ord_less_eq @ A @ X @ B2 )
=> ( ord_less_eq @ A @ X @ ( sup_sup @ A @ A3 @ B2 ) ) ) ) ).
% le_supI2
thf(fact_6570_sup_Omono,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [C2: A,A3: A,D2: A,B2: A] :
( ( ord_less_eq @ A @ C2 @ A3 )
=> ( ( ord_less_eq @ A @ D2 @ B2 )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ C2 @ D2 ) @ ( sup_sup @ A @ A3 @ B2 ) ) ) ) ) ).
% sup.mono
thf(fact_6571_sup__mono,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: A,C2: A,B2: A,D2: A] :
( ( ord_less_eq @ A @ A3 @ C2 )
=> ( ( ord_less_eq @ A @ B2 @ D2 )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ A3 @ B2 ) @ ( sup_sup @ A @ C2 @ D2 ) ) ) ) ) ).
% sup_mono
thf(fact_6572_sup__least,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [Y2: A,X: A,Z2: A] :
( ( ord_less_eq @ A @ Y2 @ X )
=> ( ( ord_less_eq @ A @ Z2 @ X )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ Y2 @ Z2 ) @ X ) ) ) ) ).
% sup_least
thf(fact_6573_le__iff__sup,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [X2: A,Y5: A] :
( ( sup_sup @ A @ X2 @ Y5 )
= Y5 ) ) ) ) ).
% le_iff_sup
thf(fact_6574_sup_OorderE,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B2: A,A3: A] :
( ( ord_less_eq @ A @ B2 @ A3 )
=> ( A3
= ( sup_sup @ A @ A3 @ B2 ) ) ) ) ).
% sup.orderE
thf(fact_6575_sup_OorderI,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: A,B2: A] :
( ( A3
= ( sup_sup @ A @ A3 @ B2 ) )
=> ( ord_less_eq @ A @ B2 @ A3 ) ) ) ).
% sup.orderI
thf(fact_6576_sup__unique,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [F2: A > A > A,X: A,Y2: A] :
( ! [X3: A,Y3: A] : ( ord_less_eq @ A @ X3 @ ( F2 @ X3 @ Y3 ) )
=> ( ! [X3: A,Y3: A] : ( ord_less_eq @ A @ Y3 @ ( F2 @ X3 @ Y3 ) )
=> ( ! [X3: A,Y3: A,Z4: A] :
( ( ord_less_eq @ A @ Y3 @ X3 )
=> ( ( ord_less_eq @ A @ Z4 @ X3 )
=> ( ord_less_eq @ A @ ( F2 @ Y3 @ Z4 ) @ X3 ) ) )
=> ( ( sup_sup @ A @ X @ Y2 )
= ( F2 @ X @ Y2 ) ) ) ) ) ) ).
% sup_unique
thf(fact_6577_sup_Oabsorb1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B2: A,A3: A] :
( ( ord_less_eq @ A @ B2 @ A3 )
=> ( ( sup_sup @ A @ A3 @ B2 )
= A3 ) ) ) ).
% sup.absorb1
thf(fact_6578_sup_Oabsorb2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ A @ A3 @ B2 )
=> ( ( sup_sup @ A @ A3 @ B2 )
= B2 ) ) ) ).
% sup.absorb2
thf(fact_6579_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_6580_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_6581_sup_OboundedE,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B2: A,C2: A,A3: A] :
( ( ord_less_eq @ A @ ( sup_sup @ A @ B2 @ C2 ) @ A3 )
=> ~ ( ( ord_less_eq @ A @ B2 @ A3 )
=> ~ ( ord_less_eq @ A @ C2 @ A3 ) ) ) ) ).
% sup.boundedE
thf(fact_6582_sup_OboundedI,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B2: A,A3: A,C2: A] :
( ( ord_less_eq @ A @ B2 @ A3 )
=> ( ( ord_less_eq @ A @ C2 @ A3 )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ B2 @ C2 ) @ A3 ) ) ) ) ).
% sup.boundedI
thf(fact_6583_sup_Oorder__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B3: A,A5: A] :
( A5
= ( sup_sup @ A @ A5 @ B3 ) ) ) ) ) ).
% sup.order_iff
thf(fact_6584_sup_Ocobounded1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A3: A,B2: A] : ( ord_less_eq @ A @ A3 @ ( sup_sup @ A @ A3 @ B2 ) ) ) ).
% sup.cobounded1
thf(fact_6585_sup_Ocobounded2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B2: A,A3: A] : ( ord_less_eq @ A @ B2 @ ( sup_sup @ A @ A3 @ B2 ) ) ) ).
% sup.cobounded2
thf(fact_6586_sup_Oabsorb__iff1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B3: A,A5: A] :
( ( sup_sup @ A @ A5 @ B3 )
= A5 ) ) ) ) ).
% sup.absorb_iff1
thf(fact_6587_sup_Oabsorb__iff2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A5: A,B3: A] :
( ( sup_sup @ A @ A5 @ B3 )
= B3 ) ) ) ) ).
% sup.absorb_iff2
thf(fact_6588_sup_OcoboundedI1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [C2: A,A3: A,B2: A] :
( ( ord_less_eq @ A @ C2 @ A3 )
=> ( ord_less_eq @ A @ C2 @ ( sup_sup @ A @ A3 @ B2 ) ) ) ) ).
% sup.coboundedI1
thf(fact_6589_sup_OcoboundedI2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [C2: A,B2: A,A3: A] :
( ( ord_less_eq @ A @ C2 @ B2 )
=> ( ord_less_eq @ A @ C2 @ ( sup_sup @ A @ A3 @ B2 ) ) ) ) ).
% sup.coboundedI2
thf(fact_6590_Diff__partition,axiom,
! [A: $tType,A4: set @ A,B7: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A4 @ B7 )
=> ( ( sup_sup @ ( set @ A ) @ A4 @ ( minus_minus @ ( set @ A ) @ B7 @ A4 ) )
= B7 ) ) ).
% Diff_partition
thf(fact_6591_Diff__subset__conv,axiom,
! [A: $tType,A4: set @ A,B7: set @ A,C5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A4 @ B7 ) @ C5 )
= ( ord_less_eq @ ( set @ A ) @ A4 @ ( sup_sup @ ( set @ A ) @ B7 @ C5 ) ) ) ).
% Diff_subset_conv
thf(fact_6592_mono__sup,axiom,
! [B: $tType,A: $tType] :
( ( ( semilattice_sup @ A )
& ( semilattice_sup @ B ) )
=> ! [F2: A > B,A4: A,B7: A] :
( ( order_mono @ A @ B @ F2 )
=> ( ord_less_eq @ B @ ( sup_sup @ B @ ( F2 @ A4 ) @ ( F2 @ B7 ) ) @ ( F2 @ ( sup_sup @ A @ A4 @ B7 ) ) ) ) ) ).
% mono_sup
thf(fact_6593_sup__Inf,axiom,
! [A: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [A3: A,B7: set @ A] :
( ( sup_sup @ A @ A3 @ ( complete_Inf_Inf @ A @ B7 ) )
= ( complete_Inf_Inf @ A @ ( image @ A @ A @ ( sup_sup @ A @ A3 ) @ B7 ) ) ) ) ).
% sup_Inf
thf(fact_6594_INF__sup,axiom,
! [A: $tType,B: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [F2: B > A,B7: set @ B,A3: A] :
( ( sup_sup @ A @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ B7 ) ) @ A3 )
= ( complete_Inf_Inf @ A
@ ( image @ B @ A
@ ^ [B3: B] : ( sup_sup @ A @ ( F2 @ B3 ) @ A3 )
@ B7 ) ) ) ) ).
% INF_sup
thf(fact_6595_Inf__sup,axiom,
! [A: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [B7: set @ A,A3: A] :
( ( sup_sup @ A @ ( complete_Inf_Inf @ A @ B7 ) @ A3 )
= ( complete_Inf_Inf @ A
@ ( image @ A @ A
@ ^ [B3: A] : ( sup_sup @ A @ B3 @ A3 )
@ B7 ) ) ) ) ).
% Inf_sup
thf(fact_6596_sup__INF,axiom,
! [A: $tType,B: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [A3: A,F2: B > A,B7: set @ B] :
( ( sup_sup @ A @ A3 @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ B7 ) ) )
= ( complete_Inf_Inf @ A
@ ( image @ B @ A
@ ^ [B3: B] : ( sup_sup @ A @ A3 @ ( F2 @ B3 ) )
@ B7 ) ) ) ) ).
% sup_INF
thf(fact_6597_INF__sup__distrib2,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [F2: B > A,A4: set @ B,G: C > A,B7: set @ C] :
( ( sup_sup @ A @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ A4 ) ) @ ( complete_Inf_Inf @ A @ ( image @ C @ A @ G @ B7 ) ) )
= ( complete_Inf_Inf @ A
@ ( image @ B @ A
@ ^ [A5: B] :
( complete_Inf_Inf @ A
@ ( image @ C @ A
@ ^ [B3: C] : ( sup_sup @ A @ ( F2 @ A5 ) @ ( G @ B3 ) )
@ B7 ) )
@ A4 ) ) ) ) ).
% INF_sup_distrib2
thf(fact_6598_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_6599_shunt1,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y2: A,Z2: A] :
( ( ord_less_eq @ A @ ( inf_inf @ A @ X @ Y2 ) @ Z2 )
= ( ord_less_eq @ A @ X @ ( sup_sup @ A @ ( uminus_uminus @ A @ Y2 ) @ Z2 ) ) ) ) ).
% shunt1
thf(fact_6600_shunt2,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y2: A,Z2: A] :
( ( ord_less_eq @ A @ ( inf_inf @ A @ X @ ( uminus_uminus @ A @ Y2 ) ) @ Z2 )
= ( ord_less_eq @ A @ X @ ( sup_sup @ A @ Y2 @ Z2 ) ) ) ) ).
% shunt2
thf(fact_6601_sup__neg__inf,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [P4: A,Q5: A,R4: A] :
( ( ord_less_eq @ A @ P4 @ ( sup_sup @ A @ Q5 @ R4 ) )
= ( ord_less_eq @ A @ ( inf_inf @ A @ P4 @ ( uminus_uminus @ A @ Q5 ) ) @ R4 ) ) ) ).
% sup_neg_inf
thf(fact_6602_boolean__algebra__class_Oboolean__algebra_Ocompl__unique,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y2: A] :
( ( ( inf_inf @ A @ X @ Y2 )
= ( bot_bot @ A ) )
=> ( ( ( sup_sup @ A @ X @ Y2 )
= ( top_top @ A ) )
=> ( ( uminus_uminus @ A @ X )
= Y2 ) ) ) ) ).
% boolean_algebra_class.boolean_algebra.compl_unique
thf(fact_6603_ivl__disj__un__two__touch_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,M2: A,U: A] :
( ( ord_less_eq @ A @ L @ M2 )
=> ( ( ord_less @ A @ M2 @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ L @ M2 ) @ ( set_or7035219750837199246ssThan @ A @ M2 @ U ) )
= ( set_or7035219750837199246ssThan @ A @ L @ U ) ) ) ) ) ).
% ivl_disj_un_two_touch(2)
thf(fact_6604_infinite__imp__bij__betw2,axiom,
! [A: $tType,A4: set @ A,A3: A] :
( ~ ( finite_finite @ A @ A4 )
=> ? [H3: A > A] : ( bij_betw @ A @ A @ H3 @ A4 @ ( sup_sup @ ( set @ A ) @ A4 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% infinite_imp_bij_betw2
thf(fact_6605_ivl__disj__un__two__touch_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,M2: A,U: A] :
( ( ord_less @ A @ L @ M2 )
=> ( ( ord_less_eq @ A @ M2 @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ L @ M2 ) @ ( set_or1337092689740270186AtMost @ A @ M2 @ U ) )
= ( set_or3652927894154168847AtMost @ A @ L @ U ) ) ) ) ) ).
% ivl_disj_un_two_touch(3)
thf(fact_6606_ivl__disj__un__two_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,M2: A,U: A] :
( ( ord_less @ A @ L @ M2 )
=> ( ( ord_less_eq @ A @ M2 @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ L @ M2 ) @ ( set_or7035219750837199246ssThan @ A @ M2 @ U ) )
= ( set_or5935395276787703475ssThan @ A @ L @ U ) ) ) ) ) ).
% ivl_disj_un_two(1)
thf(fact_6607_ivl__disj__un__two_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,M2: A,U: A] :
( ( ord_less_eq @ A @ L @ M2 )
=> ( ( ord_less @ A @ M2 @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ L @ M2 ) @ ( set_or5935395276787703475ssThan @ A @ M2 @ U ) )
= ( set_or5935395276787703475ssThan @ A @ L @ U ) ) ) ) ) ).
% ivl_disj_un_two(2)
thf(fact_6608_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_6609_ivl__disj__un__two__touch_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,M2: A,U: A] :
( ( ord_less @ A @ L @ M2 )
=> ( ( ord_less @ A @ M2 @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ L @ M2 ) @ ( set_or7035219750837199246ssThan @ A @ M2 @ U ) )
= ( set_or5935395276787703475ssThan @ A @ L @ U ) ) ) ) ) ).
% ivl_disj_un_two_touch(1)
thf(fact_6610_SUP__nat__binary,axiom,
! [A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A4: A,B7: A] :
( ( sup_sup @ A @ A4
@ ( complete_Sup_Sup @ A
@ ( image @ nat @ A
@ ^ [X2: nat] : B7
@ ( collect @ nat @ ( ord_less @ nat @ ( zero_zero @ nat ) ) ) ) ) )
= ( sup_sup @ A @ A4 @ B7 ) ) ) ).
% SUP_nat_binary
thf(fact_6611_sup__bot_Osemilattice__neutr__order__axioms,axiom,
! [A: $tType] :
( ( bounde4967611905675639751up_bot @ A )
=> ( semila1105856199041335345_order @ A @ ( sup_sup @ A ) @ ( bot_bot @ A )
@ ^ [X2: A,Y5: A] : ( ord_less_eq @ A @ Y5 @ X2 )
@ ^ [X2: A,Y5: A] : ( ord_less @ A @ Y5 @ X2 ) ) ) ).
% sup_bot.semilattice_neutr_order_axioms
thf(fact_6612_sum_Ounion__inter__neutral,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A4: set @ B,B7: set @ B,G: B > A] :
( ( finite_finite @ B @ A4 )
=> ( ( finite_finite @ B @ B7 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( inf_inf @ ( set @ B ) @ A4 @ B7 ) )
=> ( ( G @ X3 )
= ( zero_zero @ A ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( sup_sup @ ( set @ B ) @ A4 @ B7 ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G @ A4 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ B7 ) ) ) ) ) ) ) ).
% sum.union_inter_neutral
thf(fact_6613_sum__Un,axiom,
! [A: $tType,B: $tType] :
( ( ab_group_add @ A )
=> ! [A4: set @ B,B7: set @ B,F2: B > A] :
( ( finite_finite @ B @ A4 )
=> ( ( finite_finite @ B @ B7 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ F2 @ ( sup_sup @ ( set @ B ) @ A4 @ B7 ) )
= ( minus_minus @ A @ ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ A4 ) @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ B7 ) ) @ ( groups7311177749621191930dd_sum @ B @ A @ F2 @ ( inf_inf @ ( set @ B ) @ A4 @ B7 ) ) ) ) ) ) ) ).
% sum_Un
thf(fact_6614_prod_Ounion__inter__neutral,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A4: set @ B,B7: set @ B,G: B > A] :
( ( finite_finite @ B @ A4 )
=> ( ( finite_finite @ B @ B7 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( inf_inf @ ( set @ B ) @ A4 @ B7 ) )
=> ( ( G @ X3 )
= ( one_one @ A ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( sup_sup @ ( set @ B ) @ A4 @ B7 ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G @ A4 ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ B7 ) ) ) ) ) ) ) ).
% prod.union_inter_neutral
thf(fact_6615_boolean__algebra_Oabstract__boolean__algebra__axioms,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ( boolea2506097494486148201lgebra @ A @ ( inf_inf @ A ) @ ( sup_sup @ A ) @ ( uminus_uminus @ A ) @ ( bot_bot @ A ) @ ( top_top @ A ) ) ) ).
% boolean_algebra.abstract_boolean_algebra_axioms
thf(fact_6616_sum__Un2,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ B )
=> ! [A4: set @ A,B7: set @ A,F2: A > B] :
( ( finite_finite @ A @ ( sup_sup @ ( set @ A ) @ A4 @ B7 ) )
=> ( ( groups7311177749621191930dd_sum @ A @ B @ F2 @ ( sup_sup @ ( set @ A ) @ A4 @ B7 ) )
= ( plus_plus @ B @ ( plus_plus @ B @ ( groups7311177749621191930dd_sum @ A @ B @ F2 @ ( minus_minus @ ( set @ A ) @ A4 @ B7 ) ) @ ( groups7311177749621191930dd_sum @ A @ B @ F2 @ ( minus_minus @ ( set @ A ) @ B7 @ A4 ) ) ) @ ( groups7311177749621191930dd_sum @ A @ B @ F2 @ ( inf_inf @ ( set @ A ) @ A4 @ B7 ) ) ) ) ) ) ).
% sum_Un2
thf(fact_6617_sum_Ounion__diff2,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A4: set @ B,B7: set @ B,G: B > A] :
( ( finite_finite @ B @ A4 )
=> ( ( finite_finite @ B @ B7 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G @ ( sup_sup @ ( set @ B ) @ A4 @ B7 ) )
= ( plus_plus @ A @ ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A4 @ B7 ) ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ B7 @ A4 ) ) ) @ ( groups7311177749621191930dd_sum @ B @ A @ G @ ( inf_inf @ ( set @ B ) @ A4 @ B7 ) ) ) ) ) ) ) ).
% sum.union_diff2
thf(fact_6618_prod_Ounion__diff2,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A4: set @ B,B7: set @ B,G: B > A] :
( ( finite_finite @ B @ A4 )
=> ( ( finite_finite @ B @ B7 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G @ ( sup_sup @ ( set @ B ) @ A4 @ B7 ) )
= ( times_times @ A @ ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ A4 @ B7 ) ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ ( minus_minus @ ( set @ B ) @ B7 @ A4 ) ) ) @ ( groups7121269368397514597t_prod @ B @ A @ G @ ( inf_inf @ ( set @ B ) @ A4 @ B7 ) ) ) ) ) ) ) ).
% prod.union_diff2
thf(fact_6619_inj__on__Un,axiom,
! [A: $tType,B: $tType,F2: A > B,A4: set @ A,B7: set @ A] :
( ( inj_on @ A @ B @ F2 @ ( sup_sup @ ( set @ A ) @ A4 @ B7 ) )
= ( ( inj_on @ A @ B @ F2 @ A4 )
& ( inj_on @ A @ B @ F2 @ B7 )
& ( ( inf_inf @ ( set @ B ) @ ( image @ A @ B @ F2 @ ( minus_minus @ ( set @ A ) @ A4 @ B7 ) ) @ ( image @ A @ B @ F2 @ ( minus_minus @ ( set @ A ) @ B7 @ A4 ) ) )
= ( bot_bot @ ( set @ B ) ) ) ) ) ).
% inj_on_Un
thf(fact_6620_ivl__disj__un__two_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,M2: A,U: A] :
( ( ord_less_eq @ A @ L @ M2 )
=> ( ( ord_less @ A @ M2 @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ L @ M2 ) @ ( set_or5935395276787703475ssThan @ A @ M2 @ U ) )
= ( set_or7035219750837199246ssThan @ A @ L @ U ) ) ) ) ) ).
% ivl_disj_un_two(4)
thf(fact_6621_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_6622_sum__Un__nat,axiom,
! [A: $tType,A4: set @ A,B7: set @ A,F2: A > nat] :
( ( finite_finite @ A @ A4 )
=> ( ( finite_finite @ A @ B7 )
=> ( ( groups7311177749621191930dd_sum @ A @ nat @ F2 @ ( sup_sup @ ( set @ A ) @ A4 @ B7 ) )
= ( minus_minus @ nat @ ( plus_plus @ nat @ ( groups7311177749621191930dd_sum @ A @ nat @ F2 @ A4 ) @ ( groups7311177749621191930dd_sum @ A @ nat @ F2 @ B7 ) ) @ ( groups7311177749621191930dd_sum @ A @ nat @ F2 @ ( inf_inf @ ( set @ A ) @ A4 @ B7 ) ) ) ) ) ) ).
% sum_Un_nat
thf(fact_6623_ivl__disj__un__two_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,M2: A,U: A] :
( ( ord_less @ A @ L @ M2 )
=> ( ( ord_less_eq @ A @ M2 @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ L @ M2 ) @ ( set_or1337092689740270186AtMost @ A @ M2 @ U ) )
= ( set_or3652927894154168847AtMost @ A @ L @ U ) ) ) ) ) ).
% ivl_disj_un_two(5)
thf(fact_6624_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_6625_prod__Un,axiom,
! [A: $tType,B: $tType] :
( ( field @ A )
=> ! [A4: set @ B,B7: set @ B,F2: B > A] :
( ( finite_finite @ B @ A4 )
=> ( ( finite_finite @ B @ B7 )
=> ( ! [X3: B] :
( ( member @ B @ X3 @ ( inf_inf @ ( set @ B ) @ A4 @ B7 ) )
=> ( ( F2 @ X3 )
!= ( zero_zero @ A ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ F2 @ ( sup_sup @ ( set @ B ) @ A4 @ B7 ) )
= ( divide_divide @ A @ ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ A4 ) @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ B7 ) ) @ ( groups7121269368397514597t_prod @ B @ A @ F2 @ ( inf_inf @ ( set @ B ) @ A4 @ B7 ) ) ) ) ) ) ) ) ).
% prod_Un
thf(fact_6626_UNION__fun__upd,axiom,
! [B: $tType,A: $tType,A4: B > ( set @ A ),I: B,B7: set @ A,J4: set @ B] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ ( fun_upd @ B @ ( set @ A ) @ A4 @ I @ B7 ) @ J4 ) )
= ( sup_sup @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ A4 @ ( minus_minus @ ( set @ B ) @ J4 @ ( insert @ B @ I @ ( bot_bot @ ( set @ B ) ) ) ) ) ) @ ( if @ ( set @ A ) @ ( member @ B @ I @ J4 ) @ B7 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% UNION_fun_upd
thf(fact_6627_times__int_Orsp,axiom,
( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) ) @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) ) @ intrel @ ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ intrel @ intrel )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X2: nat,Y5: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V5: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ X2 @ U2 ) @ ( times_times @ nat @ Y5 @ V5 ) ) @ ( plus_plus @ nat @ ( times_times @ nat @ X2 @ V5 ) @ ( times_times @ nat @ Y5 @ U2 ) ) ) ) )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X2: nat,Y5: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V5: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ X2 @ U2 ) @ ( times_times @ nat @ Y5 @ V5 ) ) @ ( plus_plus @ nat @ ( times_times @ nat @ X2 @ V5 ) @ ( times_times @ nat @ Y5 @ U2 ) ) ) ) ) ) ).
% times_int.rsp
thf(fact_6628_intrel__iff,axiom,
! [X: nat,Y2: nat,U: nat,V2: nat] :
( ( intrel @ ( product_Pair @ nat @ nat @ X @ Y2 ) @ ( product_Pair @ nat @ nat @ U @ V2 ) )
= ( ( plus_plus @ nat @ X @ V2 )
= ( plus_plus @ nat @ U @ Y2 ) ) ) ).
% intrel_iff
thf(fact_6629_sup__nat__def,axiom,
( ( sup_sup @ nat )
= ( ord_max @ nat ) ) ).
% sup_nat_def
thf(fact_6630_sup__int__def,axiom,
( ( sup_sup @ int )
= ( ord_max @ int ) ) ).
% sup_int_def
thf(fact_6631_zero__int_Orsp,axiom,
intrel @ ( product_Pair @ nat @ nat @ ( zero_zero @ nat ) @ ( zero_zero @ nat ) ) @ ( product_Pair @ nat @ nat @ ( zero_zero @ nat ) @ ( zero_zero @ nat ) ) ).
% zero_int.rsp
thf(fact_6632_int_Oabs__eq__iff,axiom,
! [X: product_prod @ nat @ nat,Y2: product_prod @ nat @ nat] :
( ( ( abs_Integ @ X )
= ( abs_Integ @ Y2 ) )
= ( intrel @ X @ Y2 ) ) ).
% int.abs_eq_iff
thf(fact_6633_uminus__int_Orsp,axiom,
( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ intrel @ intrel
@ ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [X2: nat,Y5: nat] : ( product_Pair @ nat @ nat @ Y5 @ X2 ) )
@ ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [X2: nat,Y5: nat] : ( product_Pair @ nat @ nat @ Y5 @ X2 ) ) ) ).
% uminus_int.rsp
thf(fact_6634_nat_Orsp,axiom,
( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ nat @ nat @ intrel
@ ^ [Y4: nat,Z: nat] : Y4 = Z
@ ( product_case_prod @ nat @ nat @ nat @ ( minus_minus @ nat ) )
@ ( product_case_prod @ nat @ nat @ nat @ ( minus_minus @ nat ) ) ) ).
% nat.rsp
thf(fact_6635_one__int_Orsp,axiom,
intrel @ ( product_Pair @ nat @ nat @ ( one_one @ nat ) @ ( zero_zero @ nat ) ) @ ( product_Pair @ nat @ nat @ ( one_one @ nat ) @ ( zero_zero @ nat ) ) ).
% one_int.rsp
thf(fact_6636_of__int_Orsp,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ A @ A @ intrel
@ ^ [Y4: A,Z: A] : Y4 = Z
@ ( product_case_prod @ nat @ nat @ A
@ ^ [I2: nat,J3: nat] : ( minus_minus @ A @ ( semiring_1_of_nat @ A @ I2 ) @ ( semiring_1_of_nat @ A @ J3 ) ) )
@ ( product_case_prod @ nat @ nat @ A
@ ^ [I2: nat,J3: nat] : ( minus_minus @ A @ ( semiring_1_of_nat @ A @ I2 ) @ ( semiring_1_of_nat @ A @ J3 ) ) ) ) ) ).
% of_int.rsp
thf(fact_6637_intrel__def,axiom,
( intrel
= ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [X2: nat,Y5: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U2: nat,V5: nat] :
( ( plus_plus @ nat @ X2 @ V5 )
= ( plus_plus @ nat @ U2 @ Y5 ) ) ) ) ) ).
% intrel_def
thf(fact_6638_fun__upd__image,axiom,
! [A: $tType,B: $tType,X: B,A4: set @ B,F2: B > A,Y2: A] :
( ( ( member @ B @ X @ A4 )
=> ( ( image @ B @ A @ ( fun_upd @ B @ A @ F2 @ X @ Y2 ) @ A4 )
= ( insert @ A @ Y2 @ ( image @ B @ A @ F2 @ ( minus_minus @ ( set @ B ) @ A4 @ ( insert @ B @ X @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) )
& ( ~ ( member @ B @ X @ A4 )
=> ( ( image @ B @ A @ ( fun_upd @ B @ A @ F2 @ X @ Y2 ) @ A4 )
= ( image @ B @ A @ F2 @ A4 ) ) ) ) ).
% fun_upd_image
thf(fact_6639_less__int_Orsp,axiom,
( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( ( product_prod @ nat @ nat ) > $o ) @ ( ( product_prod @ nat @ nat ) > $o ) @ intrel
@ ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ $o @ $o @ intrel
@ ^ [Y4: $o,Z: $o] : Y4 = Z )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [X2: nat,Y5: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U2: nat,V5: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ X2 @ V5 ) @ ( plus_plus @ nat @ U2 @ Y5 ) ) ) )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [X2: nat,Y5: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U2: nat,V5: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ X2 @ V5 ) @ ( plus_plus @ nat @ U2 @ Y5 ) ) ) ) ) ).
% less_int.rsp
thf(fact_6640_less__eq__int_Orsp,axiom,
( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( ( product_prod @ nat @ nat ) > $o ) @ ( ( product_prod @ nat @ nat ) > $o ) @ intrel
@ ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ $o @ $o @ intrel
@ ^ [Y4: $o,Z: $o] : Y4 = Z )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [X2: nat,Y5: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U2: nat,V5: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ X2 @ V5 ) @ ( plus_plus @ nat @ U2 @ Y5 ) ) ) )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [X2: nat,Y5: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U2: nat,V5: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ X2 @ V5 ) @ ( plus_plus @ nat @ U2 @ Y5 ) ) ) ) ) ).
% less_eq_int.rsp
thf(fact_6641_int_Orel__eq__transfer,axiom,
( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ int @ ( ( product_prod @ nat @ nat ) > $o ) @ ( int > $o ) @ pcr_int
@ ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ int @ $o @ $o @ pcr_int
@ ^ [Y4: $o,Z: $o] : Y4 = Z )
@ intrel
@ ^ [Y4: int,Z: int] : Y4 = Z ) ).
% int.rel_eq_transfer
thf(fact_6642_minus__int_Orsp,axiom,
( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) ) @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) ) @ intrel @ ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ intrel @ intrel )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X2: nat,Y5: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V5: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X2 @ V5 ) @ ( plus_plus @ nat @ Y5 @ U2 ) ) ) )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X2: nat,Y5: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V5: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X2 @ V5 ) @ ( plus_plus @ nat @ Y5 @ U2 ) ) ) ) ) ).
% minus_int.rsp
thf(fact_6643_plus__int_Orsp,axiom,
( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) ) @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) ) @ intrel @ ( bNF_rel_fun @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ intrel @ intrel )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X2: nat,Y5: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V5: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X2 @ U2 ) @ ( plus_plus @ nat @ Y5 @ V5 ) ) ) )
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X2: nat,Y5: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V5: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X2 @ U2 ) @ ( plus_plus @ nat @ Y5 @ V5 ) ) ) ) ) ).
% plus_int.rsp
thf(fact_6644_inverse__real_Oabs__eq,axiom,
! [X: nat > rat] :
( ( realrel @ X @ X )
=> ( ( inverse_inverse @ real @ ( real2 @ X ) )
= ( real2
@ ( if @ ( nat > rat ) @ ( vanishes @ X )
@ ^ [N5: nat] : ( zero_zero @ rat )
@ ^ [N5: nat] : ( inverse_inverse @ rat @ ( X @ N5 ) ) ) ) ) ) ).
% inverse_real.abs_eq
thf(fact_6645_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_6646_empty__upd__none,axiom,
! [B: $tType,A: $tType,X: A] :
( ( fun_upd @ A @ ( option @ B )
@ ^ [X2: A] : ( none @ B )
@ X
@ ( none @ B ) )
= ( ^ [X2: A] : ( none @ B ) ) ) ).
% empty_upd_none
thf(fact_6647_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_6648_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_6649_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_6650_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_6651_ran__map__upd,axiom,
! [A: $tType,B: $tType,M2: B > ( option @ A ),A3: B,B2: A] :
( ( ( M2 @ A3 )
= ( none @ A ) )
=> ( ( ran @ B @ A @ ( fun_upd @ B @ ( option @ A ) @ M2 @ A3 @ ( some @ A @ B2 ) ) )
= ( insert @ A @ B2 @ ( ran @ B @ A @ M2 ) ) ) ) ).
% ran_map_upd
thf(fact_6652_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_6653_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_6654_Cons__in__lex,axiom,
! [A: $tType,X: A,Xs: list @ A,Y2: A,Ys2: list @ A,R4: 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 @ Ys2 ) ) @ ( lex @ A @ R4 ) )
= ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y2 ) @ R4 )
& ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ A ) @ Ys2 ) ) )
| ( ( X = Y2 )
& ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys2 ) @ ( lex @ A @ R4 ) ) ) ) ) ).
% Cons_in_lex
thf(fact_6655_map__upd__nonempty,axiom,
! [B: $tType,A: $tType,T2: A > ( option @ B ),K: A,X: B] :
( ( fun_upd @ A @ ( option @ B ) @ T2 @ K @ ( some @ B @ X ) )
!= ( ^ [X2: A] : ( none @ B ) ) ) ).
% map_upd_nonempty
thf(fact_6656_real_Oabs__induct,axiom,
! [P: real > $o,X: real] :
( ! [Y3: nat > rat] :
( ( realrel @ Y3 @ Y3 )
=> ( P @ ( real2 @ Y3 ) ) )
=> ( P @ X ) ) ).
% real.abs_induct
thf(fact_6657_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_6658_list__update__code_I3_J,axiom,
! [A: $tType,X: A,Xs: list @ A,I: nat,Y2: A] :
( ( list_update @ A @ ( cons @ A @ X @ Xs ) @ ( suc @ I ) @ Y2 )
= ( cons @ A @ X @ ( list_update @ A @ Xs @ I @ Y2 ) ) ) ).
% list_update_code(3)
thf(fact_6659_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_6660_of__rat__Real,axiom,
( ( field_char_0_of_rat @ real )
= ( ^ [X2: rat] :
( real2
@ ^ [N5: nat] : X2 ) ) ) ).
% of_rat_Real
thf(fact_6661_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_6662_length__Suc__conv,axiom,
! [A: $tType,Xs: list @ A,N: nat] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( suc @ N ) )
= ( ? [Y5: A,Ys3: list @ A] :
( ( Xs
= ( cons @ A @ Y5 @ Ys3 ) )
& ( ( size_size @ ( list @ A ) @ Ys3 )
= N ) ) ) ) ).
% length_Suc_conv
thf(fact_6663_Suc__length__conv,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( ( suc @ N )
= ( size_size @ ( list @ A ) @ Xs ) )
= ( ? [Y5: A,Ys3: list @ A] :
( ( Xs
= ( cons @ A @ Y5 @ Ys3 ) )
& ( ( size_size @ ( list @ A ) @ Ys3 )
= N ) ) ) ) ).
% Suc_length_conv
thf(fact_6664_impossible__Cons,axiom,
! [A: $tType,Xs: list @ A,Ys2: list @ A,X: A] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ A ) @ Ys2 ) )
=> ( Xs
!= ( cons @ A @ X @ Ys2 ) ) ) ).
% impossible_Cons
thf(fact_6665_zero__real__def,axiom,
( ( zero_zero @ real )
= ( real2
@ ^ [N5: nat] : ( zero_zero @ rat ) ) ) ).
% zero_real_def
thf(fact_6666_one__real__def,axiom,
( ( one_one @ real )
= ( real2
@ ^ [N5: nat] : ( one_one @ rat ) ) ) ).
% one_real_def
thf(fact_6667_of__int__Real,axiom,
( ( ring_1_of_int @ real )
= ( ^ [X2: int] :
( real2
@ ^ [N5: nat] : ( ring_1_of_int @ rat @ X2 ) ) ) ) ).
% of_int_Real
thf(fact_6668_Suc__le__length__iff,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( ord_less_eq @ nat @ ( suc @ N ) @ ( size_size @ ( list @ A ) @ Xs ) )
= ( ? [X2: A,Ys3: list @ A] :
( ( Xs
= ( cons @ A @ X2 @ Ys3 ) )
& ( ord_less_eq @ nat @ N @ ( size_size @ ( list @ A ) @ Ys3 ) ) ) ) ) ).
% Suc_le_length_iff
thf(fact_6669_of__nat__Real,axiom,
( ( semiring_1_of_nat @ real )
= ( ^ [X2: nat] :
( real2
@ ^ [N5: nat] : ( semiring_1_of_nat @ rat @ X2 ) ) ) ) ).
% of_nat_Real
thf(fact_6670_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_6671_uminus__real_Oabs__eq,axiom,
! [X: nat > rat] :
( ( realrel @ X @ X )
=> ( ( uminus_uminus @ real @ ( real2 @ X ) )
= ( real2
@ ^ [N5: nat] : ( uminus_uminus @ rat @ ( X @ N5 ) ) ) ) ) ).
% uminus_real.abs_eq
thf(fact_6672_plus__real_Oabs__eq,axiom,
! [Xa: nat > rat,X: nat > rat] :
( ( realrel @ Xa @ Xa )
=> ( ( realrel @ X @ X )
=> ( ( plus_plus @ real @ ( real2 @ Xa ) @ ( real2 @ X ) )
= ( real2
@ ^ [N5: nat] : ( plus_plus @ rat @ ( Xa @ N5 ) @ ( X @ N5 ) ) ) ) ) ) ).
% plus_real.abs_eq
thf(fact_6673_times__real_Oabs__eq,axiom,
! [Xa: nat > rat,X: nat > rat] :
( ( realrel @ Xa @ Xa )
=> ( ( realrel @ X @ X )
=> ( ( times_times @ real @ ( real2 @ Xa ) @ ( real2 @ X ) )
= ( real2
@ ^ [N5: nat] : ( times_times @ rat @ ( Xa @ N5 ) @ ( X @ N5 ) ) ) ) ) ) ).
% times_real.abs_eq
thf(fact_6674_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_6675_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_6676_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_6677_sorted__list__of__set__greaterThanAtMost,axiom,
! [I: nat,J2: nat] :
( ( ord_less_eq @ nat @ ( suc @ I ) @ J2 )
=> ( ( linord4507533701916653071of_set @ nat @ ( set_or3652927894154168847AtMost @ nat @ I @ J2 ) )
= ( cons @ nat @ ( suc @ I ) @ ( linord4507533701916653071of_set @ nat @ ( set_or3652927894154168847AtMost @ nat @ ( suc @ I ) @ J2 ) ) ) ) ) ).
% sorted_list_of_set_greaterThanAtMost
thf(fact_6678_map__of__zip__upd,axiom,
! [A: $tType,B: $tType,Ys2: list @ B,Xs: list @ A,Zs: list @ B,X: A,Y2: B,Z2: B] :
( ( ( size_size @ ( list @ B ) @ Ys2 )
= ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ( size_size @ ( list @ B ) @ Zs )
= ( size_size @ ( list @ A ) @ Xs ) )
=> ( ~ ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ( ( fun_upd @ A @ ( option @ B ) @ ( map_of @ A @ B @ ( zip @ A @ B @ Xs @ Ys2 ) ) @ X @ ( some @ B @ Y2 ) )
= ( fun_upd @ A @ ( option @ B ) @ ( map_of @ A @ B @ ( zip @ A @ B @ Xs @ Zs ) ) @ X @ ( some @ B @ Z2 ) ) )
=> ( ( map_of @ A @ B @ ( zip @ A @ B @ Xs @ Ys2 ) )
= ( map_of @ A @ B @ ( zip @ A @ B @ Xs @ Zs ) ) ) ) ) ) ) ).
% map_of_zip_upd
thf(fact_6679_sorted__list__of__set__greaterThanLessThan,axiom,
! [I: nat,J2: nat] :
( ( ord_less @ nat @ ( suc @ I ) @ J2 )
=> ( ( linord4507533701916653071of_set @ nat @ ( set_or5935395276787703475ssThan @ nat @ I @ J2 ) )
= ( cons @ nat @ ( suc @ I ) @ ( linord4507533701916653071of_set @ nat @ ( set_or5935395276787703475ssThan @ nat @ ( suc @ I ) @ J2 ) ) ) ) ) ).
% sorted_list_of_set_greaterThanLessThan
thf(fact_6680_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_6681_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_6682_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_6683_Cons__lenlex__iff,axiom,
! [A: $tType,M2: A,Ms: list @ A,N: A,Ns: list @ A,R4: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ M2 @ Ms ) @ ( cons @ A @ N @ Ns ) ) @ ( lenlex @ A @ R4 ) )
= ( ( 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 @ M2 @ N ) @ R4 ) )
| ( ( M2 = N )
& ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ms @ Ns ) @ ( lenlex @ A @ R4 ) ) ) ) ) ).
% Cons_lenlex_iff
thf(fact_6684_Real_Opositive_Oabs__eq,axiom,
! [X: nat > rat] :
( ( realrel @ X @ X )
=> ( ( positive2 @ ( real2 @ X ) )
= ( ? [R: rat] :
( ( ord_less @ rat @ ( zero_zero @ rat ) @ R )
& ? [K3: nat] :
! [N5: nat] :
( ( ord_less_eq @ nat @ K3 @ N5 )
=> ( ord_less @ rat @ R @ ( X @ N5 ) ) ) ) ) ) ) ).
% Real.positive.abs_eq
thf(fact_6685_upto__aux__rec,axiom,
( upto_aux
= ( ^ [I2: int,J3: int,Js: list @ int] : ( if @ ( list @ int ) @ ( ord_less @ int @ J3 @ I2 ) @ Js @ ( upto_aux @ I2 @ ( minus_minus @ int @ J3 @ ( one_one @ int ) ) @ ( cons @ int @ J3 @ Js ) ) ) ) ) ).
% upto_aux_rec
thf(fact_6686_inverse__real__def,axiom,
( ( inverse_inverse @ real )
= ( map_fun @ real @ ( nat > rat ) @ ( nat > rat ) @ real @ rep_real @ real2
@ ^ [X7: nat > rat] :
( if @ ( nat > rat ) @ ( vanishes @ X7 )
@ ^ [N5: nat] : ( zero_zero @ rat )
@ ^ [N5: nat] : ( inverse_inverse @ rat @ ( X7 @ N5 ) ) ) ) ) ).
% inverse_real_def
thf(fact_6687_uminus__real__def,axiom,
( ( uminus_uminus @ real )
= ( map_fun @ real @ ( nat > rat ) @ ( nat > rat ) @ real @ rep_real @ real2
@ ^ [X7: nat > rat,N5: nat] : ( uminus_uminus @ rat @ ( X7 @ N5 ) ) ) ) ).
% uminus_real_def
thf(fact_6688_ran__map__upd__Some,axiom,
! [B: $tType,A: $tType,M2: B > ( option @ A ),X: B,Y2: A,Z2: A] :
( ( ( M2 @ X )
= ( some @ A @ Y2 ) )
=> ( ( inj_on @ B @ ( option @ A ) @ M2 @ ( dom @ B @ A @ M2 ) )
=> ( ~ ( member @ A @ Z2 @ ( ran @ B @ A @ M2 ) )
=> ( ( ran @ B @ A @ ( fun_upd @ B @ ( option @ A ) @ M2 @ X @ ( some @ A @ Z2 ) ) )
= ( sup_sup @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ ( ran @ B @ A @ M2 ) @ ( insert @ A @ Y2 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( insert @ A @ Z2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% ran_map_upd_Some
thf(fact_6689_le__Real,axiom,
! [X8: nat > rat,Y7: nat > rat] :
( ( cauchy @ X8 )
=> ( ( cauchy @ Y7 )
=> ( ( ord_less_eq @ real @ ( real2 @ X8 ) @ ( real2 @ Y7 ) )
= ( ! [R: rat] :
( ( ord_less @ rat @ ( zero_zero @ rat ) @ R )
=> ? [K3: nat] :
! [N5: nat] :
( ( ord_less_eq @ nat @ K3 @ N5 )
=> ( ord_less_eq @ rat @ ( X8 @ N5 ) @ ( plus_plus @ rat @ ( Y7 @ N5 ) @ R ) ) ) ) ) ) ) ) ).
% le_Real
thf(fact_6690_dom__eq__empty__conv,axiom,
! [B: $tType,A: $tType,F2: A > ( option @ B )] :
( ( ( dom @ A @ B @ F2 )
= ( bot_bot @ ( set @ A ) ) )
= ( F2
= ( ^ [X2: A] : ( none @ B ) ) ) ) ).
% dom_eq_empty_conv
thf(fact_6691_fun__upd__None__if__notin__dom,axiom,
! [B: $tType,A: $tType,K: A,M2: A > ( option @ B )] :
( ~ ( member @ A @ K @ ( dom @ A @ B @ M2 ) )
=> ( ( fun_upd @ A @ ( option @ B ) @ M2 @ K @ ( none @ B ) )
= M2 ) ) ).
% fun_upd_None_if_notin_dom
thf(fact_6692_dom__empty,axiom,
! [B: $tType,A: $tType] :
( ( dom @ A @ B
@ ^ [X2: A] : ( none @ B ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% dom_empty
thf(fact_6693_dom__map__of__zip,axiom,
! [B: $tType,A: $tType,Xs: list @ A,Ys2: list @ B] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys2 ) )
=> ( ( dom @ A @ B @ ( map_of @ A @ B @ ( zip @ A @ B @ Xs @ Ys2 ) ) )
= ( set2 @ A @ Xs ) ) ) ).
% dom_map_of_zip
thf(fact_6694_dom__fun__upd,axiom,
! [B: $tType,A: $tType,Y2: option @ B,F2: A > ( option @ B ),X: A] :
( ( ( Y2
= ( none @ B ) )
=> ( ( dom @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ F2 @ X @ Y2 ) )
= ( minus_minus @ ( set @ A ) @ ( dom @ A @ B @ F2 ) @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) )
& ( ( Y2
!= ( none @ B ) )
=> ( ( dom @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ F2 @ X @ Y2 ) )
= ( insert @ A @ X @ ( dom @ A @ B @ F2 ) ) ) ) ) ).
% dom_fun_upd
thf(fact_6695_realrel__refl,axiom,
! [X8: nat > rat] :
( ( cauchy @ X8 )
=> ( realrel @ X8 @ X8 ) ) ).
% realrel_refl
thf(fact_6696_cauchy__add,axiom,
! [X8: nat > rat,Y7: nat > rat] :
( ( cauchy @ X8 )
=> ( ( cauchy @ Y7 )
=> ( cauchy
@ ^ [N5: nat] : ( plus_plus @ rat @ ( X8 @ N5 ) @ ( Y7 @ N5 ) ) ) ) ) ).
% cauchy_add
thf(fact_6697_cauchy__const,axiom,
! [X: rat] :
( cauchy
@ ^ [N5: nat] : X ) ).
% cauchy_const
thf(fact_6698_cauchy__minus,axiom,
! [X8: nat > rat] :
( ( cauchy @ X8 )
=> ( cauchy
@ ^ [N5: nat] : ( uminus_uminus @ rat @ ( X8 @ N5 ) ) ) ) ).
% cauchy_minus
thf(fact_6699_cauchy__mult,axiom,
! [X8: nat > rat,Y7: nat > rat] :
( ( cauchy @ X8 )
=> ( ( cauchy @ Y7 )
=> ( cauchy
@ ^ [N5: nat] : ( times_times @ rat @ ( X8 @ N5 ) @ ( Y7 @ N5 ) ) ) ) ) ).
% cauchy_mult
thf(fact_6700_cauchy__diff,axiom,
! [X8: nat > rat,Y7: nat > rat] :
( ( cauchy @ X8 )
=> ( ( cauchy @ Y7 )
=> ( cauchy
@ ^ [N5: nat] : ( minus_minus @ rat @ ( X8 @ N5 ) @ ( Y7 @ N5 ) ) ) ) ) ).
% cauchy_diff
thf(fact_6701_domIff,axiom,
! [A: $tType,B: $tType,A3: A,M2: A > ( option @ B )] :
( ( member @ A @ A3 @ ( dom @ A @ B @ M2 ) )
= ( ( M2 @ A3 )
!= ( none @ B ) ) ) ).
% domIff
thf(fact_6702_dom__def,axiom,
! [B: $tType,A: $tType] :
( ( dom @ A @ B )
= ( ^ [M5: A > ( option @ B )] :
( collect @ A
@ ^ [A5: A] :
( ( M5 @ A5 )
!= ( none @ B ) ) ) ) ) ).
% dom_def
thf(fact_6703_Real__induct,axiom,
! [P: real > $o,X: real] :
( ! [X10: nat > rat] :
( ( cauchy @ X10 )
=> ( P @ ( real2 @ X10 ) ) )
=> ( P @ X ) ) ).
% Real_induct
thf(fact_6704_finite__map__freshness,axiom,
! [A: $tType,B: $tType,F2: A > ( option @ B )] :
( ( finite_finite @ A @ ( dom @ A @ B @ F2 ) )
=> ( ~ ( finite_finite @ A @ ( top_top @ ( set @ A ) ) )
=> ? [X3: A] :
( ( F2 @ X3 )
= ( none @ B ) ) ) ) ).
% finite_map_freshness
thf(fact_6705_cr__real__eq,axiom,
( pcr_real
= ( ^ [X2: nat > rat,Y5: real] :
( ( cauchy @ X2 )
& ( ( real2 @ X2 )
= Y5 ) ) ) ) ).
% cr_real_eq
thf(fact_6706_dom__minus,axiom,
! [A: $tType,B: $tType,F2: B > ( option @ A ),X: B,A4: set @ B] :
( ( ( F2 @ X )
= ( none @ A ) )
=> ( ( minus_minus @ ( set @ B ) @ ( dom @ B @ A @ F2 ) @ ( insert @ B @ X @ A4 ) )
= ( minus_minus @ ( set @ B ) @ ( dom @ B @ A @ F2 ) @ A4 ) ) ) ).
% dom_minus
thf(fact_6707_cauchy__inverse,axiom,
! [X8: nat > rat] :
( ( cauchy @ X8 )
=> ( ~ ( vanishes @ X8 )
=> ( cauchy
@ ^ [N5: nat] : ( inverse_inverse @ rat @ ( X8 @ N5 ) ) ) ) ) ).
% cauchy_inverse
thf(fact_6708_cauchy__imp__bounded,axiom,
! [X8: nat > rat] :
( ( cauchy @ X8 )
=> ? [B4: rat] :
( ( ord_less @ rat @ ( zero_zero @ rat ) @ B4 )
& ! [N4: nat] : ( ord_less @ rat @ ( abs_abs @ rat @ ( X8 @ N4 ) ) @ B4 ) ) ) ).
% cauchy_imp_bounded
thf(fact_6709_less__RealD,axiom,
! [Y7: nat > rat,X: real] :
( ( cauchy @ Y7 )
=> ( ( ord_less @ real @ X @ ( real2 @ Y7 ) )
=> ? [N2: nat] : ( ord_less @ real @ X @ ( field_char_0_of_rat @ real @ ( Y7 @ N2 ) ) ) ) ) ).
% less_RealD
thf(fact_6710_le__RealI,axiom,
! [Y7: nat > rat,X: real] :
( ( cauchy @ Y7 )
=> ( ! [N2: nat] : ( ord_less_eq @ real @ X @ ( field_char_0_of_rat @ real @ ( Y7 @ N2 ) ) )
=> ( ord_less_eq @ real @ X @ ( real2 @ Y7 ) ) ) ) ).
% le_RealI
thf(fact_6711_Real__leI,axiom,
! [X8: nat > rat,Y2: real] :
( ( cauchy @ X8 )
=> ( ! [N2: nat] : ( ord_less_eq @ real @ ( field_char_0_of_rat @ real @ ( X8 @ N2 ) ) @ Y2 )
=> ( ord_less_eq @ real @ ( real2 @ X8 ) @ Y2 ) ) ) ).
% Real_leI
thf(fact_6712_minus__Real,axiom,
! [X8: nat > rat] :
( ( cauchy @ X8 )
=> ( ( uminus_uminus @ real @ ( real2 @ X8 ) )
= ( real2
@ ^ [N5: nat] : ( uminus_uminus @ rat @ ( X8 @ N5 ) ) ) ) ) ).
% minus_Real
thf(fact_6713_add__Real,axiom,
! [X8: nat > rat,Y7: nat > rat] :
( ( cauchy @ X8 )
=> ( ( cauchy @ Y7 )
=> ( ( plus_plus @ real @ ( real2 @ X8 ) @ ( real2 @ Y7 ) )
= ( real2
@ ^ [N5: nat] : ( plus_plus @ rat @ ( X8 @ N5 ) @ ( Y7 @ N5 ) ) ) ) ) ) ).
% add_Real
thf(fact_6714_mult__Real,axiom,
! [X8: nat > rat,Y7: nat > rat] :
( ( cauchy @ X8 )
=> ( ( cauchy @ Y7 )
=> ( ( times_times @ real @ ( real2 @ X8 ) @ ( real2 @ Y7 ) )
= ( real2
@ ^ [N5: nat] : ( times_times @ rat @ ( X8 @ N5 ) @ ( Y7 @ N5 ) ) ) ) ) ) ).
% mult_Real
thf(fact_6715_diff__Real,axiom,
! [X8: nat > rat,Y7: nat > rat] :
( ( cauchy @ X8 )
=> ( ( cauchy @ Y7 )
=> ( ( minus_minus @ real @ ( real2 @ X8 ) @ ( real2 @ Y7 ) )
= ( real2
@ ^ [N5: nat] : ( minus_minus @ rat @ ( X8 @ N5 ) @ ( Y7 @ N5 ) ) ) ) ) ) ).
% diff_Real
thf(fact_6716_realrelI,axiom,
! [X8: nat > rat,Y7: nat > rat] :
( ( cauchy @ X8 )
=> ( ( cauchy @ Y7 )
=> ( ( vanishes
@ ^ [N5: nat] : ( minus_minus @ rat @ ( X8 @ N5 ) @ ( Y7 @ N5 ) ) )
=> ( realrel @ X8 @ Y7 ) ) ) ) ).
% realrelI
thf(fact_6717_eq__Real,axiom,
! [X8: nat > rat,Y7: nat > rat] :
( ( cauchy @ X8 )
=> ( ( cauchy @ Y7 )
=> ( ( ( real2 @ X8 )
= ( real2 @ Y7 ) )
= ( vanishes
@ ^ [N5: nat] : ( minus_minus @ rat @ ( X8 @ N5 ) @ ( Y7 @ N5 ) ) ) ) ) ) ).
% eq_Real
thf(fact_6718_vanishes__diff__inverse,axiom,
! [X8: nat > rat,Y7: nat > rat] :
( ( cauchy @ X8 )
=> ( ~ ( vanishes @ X8 )
=> ( ( cauchy @ Y7 )
=> ( ~ ( vanishes @ Y7 )
=> ( ( vanishes
@ ^ [N5: nat] : ( minus_minus @ rat @ ( X8 @ N5 ) @ ( Y7 @ N5 ) ) )
=> ( vanishes
@ ^ [N5: nat] : ( minus_minus @ rat @ ( inverse_inverse @ rat @ ( X8 @ N5 ) ) @ ( inverse_inverse @ rat @ ( Y7 @ N5 ) ) ) ) ) ) ) ) ) ).
% vanishes_diff_inverse
thf(fact_6719_finite__Map__induct,axiom,
! [B: $tType,A: $tType,M2: A > ( option @ B ),P: ( A > ( option @ B ) ) > $o] :
( ( finite_finite @ A @ ( dom @ A @ B @ M2 ) )
=> ( ( P
@ ^ [X2: A] : ( none @ B ) )
=> ( ! [K2: A,V4: B,M3: A > ( option @ B )] :
( ( finite_finite @ A @ ( dom @ A @ B @ M3 ) )
=> ( ~ ( member @ A @ K2 @ ( dom @ A @ B @ M3 ) )
=> ( ( P @ M3 )
=> ( P @ ( fun_upd @ A @ ( option @ B ) @ M3 @ K2 @ ( some @ B @ V4 ) ) ) ) ) )
=> ( P @ M2 ) ) ) ) ).
% finite_Map_induct
thf(fact_6720_realrel__def,axiom,
( realrel
= ( ^ [X7: nat > rat,Y8: nat > rat] :
( ( cauchy @ X7 )
& ( cauchy @ Y8 )
& ( vanishes
@ ^ [N5: nat] : ( minus_minus @ rat @ ( X7 @ N5 ) @ ( Y8 @ N5 ) ) ) ) ) ) ).
% realrel_def
thf(fact_6721_dom__eq__singleton__conv,axiom,
! [A: $tType,B: $tType,F2: A > ( option @ B ),X: A] :
( ( ( dom @ A @ B @ F2 )
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( ? [V5: B] :
( F2
= ( fun_upd @ A @ ( option @ B )
@ ^ [X2: A] : ( none @ B )
@ X
@ ( some @ B @ V5 ) ) ) ) ) ).
% dom_eq_singleton_conv
thf(fact_6722_cauchy__not__vanishes__cases,axiom,
! [X8: nat > rat] :
( ( cauchy @ X8 )
=> ( ~ ( vanishes @ X8 )
=> ? [B4: rat] :
( ( ord_less @ rat @ ( zero_zero @ rat ) @ B4 )
& ? [K2: nat] :
( ! [N4: nat] :
( ( ord_less_eq @ nat @ K2 @ N4 )
=> ( ord_less @ rat @ B4 @ ( uminus_uminus @ rat @ ( X8 @ N4 ) ) ) )
| ! [N4: nat] :
( ( ord_less_eq @ nat @ K2 @ N4 )
=> ( ord_less @ rat @ B4 @ ( X8 @ N4 ) ) ) ) ) ) ) ).
% cauchy_not_vanishes_cases
thf(fact_6723_positive__Real,axiom,
! [X8: nat > rat] :
( ( cauchy @ X8 )
=> ( ( positive2 @ ( real2 @ X8 ) )
= ( ? [R: rat] :
( ( ord_less @ rat @ ( zero_zero @ rat ) @ R )
& ? [K3: nat] :
! [N5: nat] :
( ( ord_less_eq @ nat @ K3 @ N5 )
=> ( ord_less @ rat @ R @ ( X8 @ N5 ) ) ) ) ) ) ) ).
% positive_Real
thf(fact_6724_cauchy__not__vanishes,axiom,
! [X8: nat > rat] :
( ( cauchy @ X8 )
=> ( ~ ( vanishes @ X8 )
=> ? [B4: rat] :
( ( ord_less @ rat @ ( zero_zero @ rat ) @ B4 )
& ? [K2: nat] :
! [N4: nat] :
( ( ord_less_eq @ nat @ K2 @ N4 )
=> ( ord_less @ rat @ B4 @ ( abs_abs @ rat @ ( X8 @ N4 ) ) ) ) ) ) ) ).
% cauchy_not_vanishes
thf(fact_6725_cauchy__def,axiom,
( cauchy
= ( ^ [X7: nat > rat] :
! [R: rat] :
( ( ord_less @ rat @ ( zero_zero @ rat ) @ R )
=> ? [K3: nat] :
! [M5: nat] :
( ( ord_less_eq @ nat @ K3 @ M5 )
=> ! [N5: nat] :
( ( ord_less_eq @ nat @ K3 @ N5 )
=> ( ord_less @ rat @ ( abs_abs @ rat @ ( minus_minus @ rat @ ( X7 @ M5 ) @ ( X7 @ N5 ) ) ) @ R ) ) ) ) ) ) ).
% cauchy_def
thf(fact_6726_cauchyI,axiom,
! [X8: nat > rat] :
( ! [R3: rat] :
( ( ord_less @ rat @ ( zero_zero @ rat ) @ R3 )
=> ? [K4: nat] :
! [M3: nat] :
( ( ord_less_eq @ nat @ K4 @ M3 )
=> ! [N2: nat] :
( ( ord_less_eq @ nat @ K4 @ N2 )
=> ( ord_less @ rat @ ( abs_abs @ rat @ ( minus_minus @ rat @ ( X8 @ M3 ) @ ( X8 @ N2 ) ) ) @ R3 ) ) ) )
=> ( cauchy @ X8 ) ) ).
% cauchyI
thf(fact_6727_cauchyD,axiom,
! [X8: nat > rat,R4: rat] :
( ( cauchy @ X8 )
=> ( ( ord_less @ rat @ ( zero_zero @ rat ) @ R4 )
=> ? [K2: nat] :
! [M: nat] :
( ( ord_less_eq @ nat @ K2 @ M )
=> ! [N4: nat] :
( ( ord_less_eq @ nat @ K2 @ N4 )
=> ( ord_less @ rat @ ( abs_abs @ rat @ ( minus_minus @ rat @ ( X8 @ M ) @ ( X8 @ N4 ) ) ) @ R4 ) ) ) ) ) ).
% cauchyD
thf(fact_6728_inverse__Real,axiom,
! [X8: nat > rat] :
( ( cauchy @ X8 )
=> ( ( ( vanishes @ X8 )
=> ( ( inverse_inverse @ real @ ( real2 @ X8 ) )
= ( zero_zero @ real ) ) )
& ( ~ ( vanishes @ X8 )
=> ( ( inverse_inverse @ real @ ( real2 @ X8 ) )
= ( real2
@ ^ [N5: nat] : ( inverse_inverse @ rat @ ( X8 @ N5 ) ) ) ) ) ) ) ).
% inverse_Real
thf(fact_6729_not__positive__Real,axiom,
! [X8: nat > rat] :
( ( cauchy @ X8 )
=> ( ( ~ ( positive2 @ ( real2 @ X8 ) ) )
= ( ! [R: rat] :
( ( ord_less @ rat @ ( zero_zero @ rat ) @ R )
=> ? [K3: nat] :
! [N5: nat] :
( ( ord_less_eq @ nat @ K3 @ N5 )
=> ( ord_less_eq @ rat @ ( X8 @ N5 ) @ R ) ) ) ) ) ) ).
% not_positive_Real
thf(fact_6730_dom__override__on,axiom,
! [B: $tType,A: $tType,F2: A > ( option @ B ),G: A > ( option @ B ),A4: set @ A] :
( ( dom @ A @ B @ ( override_on @ A @ ( option @ B ) @ F2 @ G @ A4 ) )
= ( sup_sup @ ( set @ A )
@ ( minus_minus @ ( set @ A ) @ ( dom @ A @ B @ F2 )
@ ( collect @ A
@ ^ [A5: A] : ( member @ A @ A5 @ ( minus_minus @ ( set @ A ) @ A4 @ ( dom @ A @ B @ G ) ) ) ) )
@ ( collect @ A
@ ^ [A5: A] : ( member @ A @ A5 @ ( inf_inf @ ( set @ A ) @ A4 @ ( dom @ A @ B @ G ) ) ) ) ) ) ).
% dom_override_on
thf(fact_6731_dom__map__upds,axiom,
! [B: $tType,A: $tType,M2: A > ( option @ B ),Xs: list @ A,Ys2: list @ B] :
( ( dom @ A @ B @ ( map_upds @ A @ B @ M2 @ Xs @ Ys2 ) )
= ( sup_sup @ ( set @ A ) @ ( set2 @ A @ ( take @ A @ ( size_size @ ( list @ B ) @ Ys2 ) @ Xs ) ) @ ( dom @ A @ B @ M2 ) ) ) ).
% dom_map_upds
thf(fact_6732_map__upds__list__update2__drop,axiom,
! [A: $tType,B: $tType,Xs: list @ A,I: nat,M2: A > ( option @ B ),Ys2: list @ B,Y2: B] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ I )
=> ( ( map_upds @ A @ B @ M2 @ Xs @ ( list_update @ B @ Ys2 @ I @ Y2 ) )
= ( map_upds @ A @ B @ M2 @ Xs @ Ys2 ) ) ) ).
% map_upds_list_update2_drop
thf(fact_6733_map__upd__upds__conv__if,axiom,
! [A: $tType,B: $tType,X: A,Ys2: list @ B,Xs: list @ A,F2: A > ( option @ B ),Y2: B] :
( ( ( member @ A @ X @ ( set2 @ A @ ( take @ A @ ( size_size @ ( list @ B ) @ Ys2 ) @ Xs ) ) )
=> ( ( map_upds @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ F2 @ X @ ( some @ B @ Y2 ) ) @ Xs @ Ys2 )
= ( map_upds @ A @ B @ F2 @ Xs @ Ys2 ) ) )
& ( ~ ( member @ A @ X @ ( set2 @ A @ ( take @ A @ ( size_size @ ( list @ B ) @ Ys2 ) @ Xs ) ) )
=> ( ( map_upds @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ F2 @ X @ ( some @ B @ Y2 ) ) @ Xs @ Ys2 )
= ( fun_upd @ A @ ( option @ B ) @ ( map_upds @ A @ B @ F2 @ Xs @ Ys2 ) @ X @ ( some @ B @ Y2 ) ) ) ) ) ).
% map_upd_upds_conv_if
thf(fact_6734_restrict__upd__same,axiom,
! [B: $tType,A: $tType,M2: A > ( option @ B ),X: A,Y2: B] :
( ( restrict_map @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ M2 @ X @ ( some @ B @ Y2 ) ) @ ( uminus_uminus @ ( set @ A ) @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( restrict_map @ A @ B @ M2 @ ( uminus_uminus @ ( set @ A ) @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% restrict_upd_same
thf(fact_6735_restrict__map__upds,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys2: list @ B,D6: set @ A,M2: A > ( option @ B )] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys2 ) )
=> ( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ D6 )
=> ( ( restrict_map @ A @ B @ ( map_upds @ A @ B @ M2 @ Xs @ Ys2 ) @ D6 )
= ( map_upds @ A @ B @ ( restrict_map @ A @ B @ M2 @ ( minus_minus @ ( set @ A ) @ D6 @ ( set2 @ A @ Xs ) ) ) @ Xs @ Ys2 ) ) ) ) ).
% restrict_map_upds
thf(fact_6736_restrict__out,axiom,
! [A: $tType,B: $tType,X: A,A4: set @ A,M2: A > ( option @ B )] :
( ~ ( member @ A @ X @ A4 )
=> ( ( restrict_map @ A @ B @ M2 @ A4 @ X )
= ( none @ B ) ) ) ).
% restrict_out
thf(fact_6737_restrict__map__empty,axiom,
! [B: $tType,A: $tType,D6: set @ A] :
( ( restrict_map @ A @ B
@ ^ [X2: A] : ( none @ B )
@ D6 )
= ( ^ [X2: A] : ( none @ B ) ) ) ).
% restrict_map_empty
thf(fact_6738_restrict__map__to__empty,axiom,
! [B: $tType,A: $tType,M2: A > ( option @ B )] :
( ( restrict_map @ A @ B @ M2 @ ( bot_bot @ ( set @ A ) ) )
= ( ^ [X2: A] : ( none @ B ) ) ) ).
% restrict_map_to_empty
thf(fact_6739_restrict__fun__upd,axiom,
! [B: $tType,A: $tType,X: A,D6: set @ A,M2: A > ( option @ B ),Y2: option @ B] :
( ( ( member @ A @ X @ D6 )
=> ( ( restrict_map @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ M2 @ X @ Y2 ) @ D6 )
= ( fun_upd @ A @ ( option @ B ) @ ( restrict_map @ A @ B @ M2 @ ( minus_minus @ ( set @ A ) @ D6 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) @ X @ Y2 ) ) )
& ( ~ ( member @ A @ X @ D6 )
=> ( ( restrict_map @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ M2 @ X @ Y2 ) @ D6 )
= ( restrict_map @ A @ B @ M2 @ D6 ) ) ) ) ).
% restrict_fun_upd
thf(fact_6740_fun__upd__restrict__conv,axiom,
! [A: $tType,B: $tType,X: A,D6: set @ A,M2: A > ( option @ B ),Y2: option @ B] :
( ( member @ A @ X @ D6 )
=> ( ( fun_upd @ A @ ( option @ B ) @ ( restrict_map @ A @ B @ M2 @ D6 ) @ X @ Y2 )
= ( fun_upd @ A @ ( option @ B ) @ ( restrict_map @ A @ B @ M2 @ ( minus_minus @ ( set @ A ) @ D6 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) @ X @ Y2 ) ) ) ).
% fun_upd_restrict_conv
thf(fact_6741_fun__upd__None__restrict,axiom,
! [B: $tType,A: $tType,X: A,D6: set @ A,M2: A > ( option @ B )] :
( ( ( member @ A @ X @ D6 )
=> ( ( fun_upd @ A @ ( option @ B ) @ ( restrict_map @ A @ B @ M2 @ D6 ) @ X @ ( none @ B ) )
= ( restrict_map @ A @ B @ M2 @ ( minus_minus @ ( set @ A ) @ D6 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) )
& ( ~ ( member @ A @ X @ D6 )
=> ( ( fun_upd @ A @ ( option @ B ) @ ( restrict_map @ A @ B @ M2 @ D6 ) @ X @ ( none @ B ) )
= ( restrict_map @ A @ B @ M2 @ D6 ) ) ) ) ).
% fun_upd_None_restrict
thf(fact_6742_restrict__map__def,axiom,
! [B: $tType,A: $tType] :
( ( restrict_map @ A @ B )
= ( ^ [M5: A > ( option @ B ),A7: set @ A,X2: A] : ( if @ ( option @ B ) @ ( member @ A @ X2 @ A7 ) @ ( M5 @ X2 ) @ ( none @ B ) ) ) ) ).
% restrict_map_def
thf(fact_6743_fun__upd__restrict,axiom,
! [A: $tType,B: $tType,M2: A > ( option @ B ),D6: set @ A,X: A,Y2: option @ B] :
( ( fun_upd @ A @ ( option @ B ) @ ( restrict_map @ A @ B @ M2 @ D6 ) @ X @ Y2 )
= ( fun_upd @ A @ ( option @ B ) @ ( restrict_map @ A @ B @ M2 @ ( minus_minus @ ( set @ A ) @ D6 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) @ X @ Y2 ) ) ).
% fun_upd_restrict
thf(fact_6744_restrict__complement__singleton__eq,axiom,
! [A: $tType,B: $tType,F2: A > ( option @ B ),X: A] :
( ( restrict_map @ A @ B @ F2 @ ( uminus_uminus @ ( set @ A ) @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( fun_upd @ A @ ( option @ B ) @ F2 @ X @ ( none @ B ) ) ) ).
% restrict_complement_singleton_eq
thf(fact_6745_times__real__def,axiom,
( ( times_times @ real )
= ( map_fun @ real @ ( nat > rat ) @ ( ( nat > rat ) > nat > rat ) @ ( real > real ) @ rep_real @ ( map_fun @ real @ ( nat > rat ) @ ( nat > rat ) @ real @ rep_real @ real2 )
@ ^ [X7: nat > rat,Y8: nat > rat,N5: nat] : ( times_times @ rat @ ( X7 @ N5 ) @ ( Y8 @ N5 ) ) ) ) ).
% times_real_def
thf(fact_6746_plus__real__def,axiom,
( ( plus_plus @ real )
= ( map_fun @ real @ ( nat > rat ) @ ( ( nat > rat ) > nat > rat ) @ ( real > real ) @ rep_real @ ( map_fun @ real @ ( nat > rat ) @ ( nat > rat ) @ real @ rep_real @ real2 )
@ ^ [X7: nat > rat,Y8: nat > rat,N5: nat] : ( plus_plus @ rat @ ( X7 @ N5 ) @ ( Y8 @ N5 ) ) ) ) ).
% plus_real_def
thf(fact_6747_list__encode_Osimps_I2_J,axiom,
! [X: nat,Xs: list @ nat] :
( ( nat_list_encode @ ( cons @ nat @ X @ Xs ) )
= ( suc @ ( nat_prod_encode @ ( product_Pair @ nat @ nat @ X @ ( nat_list_encode @ Xs ) ) ) ) ) ).
% list_encode.simps(2)
thf(fact_6748_cr__real__def,axiom,
( cr_real
= ( ^ [X2: nat > rat,Y5: real] :
( ( realrel @ X2 @ X2 )
& ( ( real2 @ X2 )
= Y5 ) ) ) ) ).
% cr_real_def
thf(fact_6749_list__encode__eq,axiom,
! [X: list @ nat,Y2: list @ nat] :
( ( ( nat_list_encode @ X )
= ( nat_list_encode @ Y2 ) )
= ( X = Y2 ) ) ).
% list_encode_eq
thf(fact_6750_bij__list__encode,axiom,
bij_betw @ ( list @ nat ) @ nat @ nat_list_encode @ ( top_top @ ( set @ ( list @ nat ) ) ) @ ( top_top @ ( set @ nat ) ) ).
% bij_list_encode
thf(fact_6751_inj__list__encode,axiom,
! [A4: set @ ( list @ nat )] : ( inj_on @ ( list @ nat ) @ nat @ nat_list_encode @ A4 ) ).
% inj_list_encode
thf(fact_6752_surj__list__encode,axiom,
( ( image @ ( list @ nat ) @ nat @ nat_list_encode @ ( top_top @ ( set @ ( list @ nat ) ) ) )
= ( top_top @ ( set @ nat ) ) ) ).
% surj_list_encode
thf(fact_6753_real_Opcr__cr__eq,axiom,
pcr_real = cr_real ).
% real.pcr_cr_eq
thf(fact_6754_list__encode_Oelims,axiom,
! [X: list @ nat,Y2: nat] :
( ( ( nat_list_encode @ X )
= Y2 )
=> ( ( ( X
= ( nil @ nat ) )
=> ( Y2
!= ( zero_zero @ nat ) ) )
=> ~ ! [X3: nat,Xs2: list @ nat] :
( ( X
= ( cons @ nat @ X3 @ Xs2 ) )
=> ( Y2
!= ( suc @ ( nat_prod_encode @ ( product_Pair @ nat @ nat @ X3 @ ( nat_list_encode @ Xs2 ) ) ) ) ) ) ) ) ).
% list_encode.elims
thf(fact_6755_lex__conv,axiom,
! [A: $tType] :
( ( lex @ A )
= ( ^ [R: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ ( list @ A ) @ ( list @ A ) )
@ ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ $o
@ ^ [Xs3: list @ A,Ys3: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs3 )
= ( size_size @ ( list @ A ) @ Ys3 ) )
& ? [Xys2: list @ A,X2: A,Y5: A,Xs4: list @ A,Ys4: list @ A] :
( ( Xs3
= ( append @ A @ Xys2 @ ( cons @ A @ X2 @ Xs4 ) ) )
& ( Ys3
= ( append @ A @ Xys2 @ ( cons @ A @ Y5 @ Ys4 ) ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y5 ) @ R ) ) ) ) ) ) ) ).
% lex_conv
thf(fact_6756_concat__inth,axiom,
! [A: $tType,Xs: list @ A,X: A,Ys2: list @ A] :
( ( nth @ A @ ( append @ A @ Xs @ ( append @ A @ ( cons @ A @ X @ ( nil @ A ) ) @ Ys2 ) ) @ ( size_size @ ( list @ A ) @ Xs ) )
= X ) ).
% concat_inth
thf(fact_6757_append__eq__append__conv,axiom,
! [A: $tType,Xs: list @ A,Ys2: list @ A,Us: list @ A,Vs: list @ A] :
( ( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ A ) @ Ys2 ) )
| ( ( size_size @ ( list @ A ) @ Us )
= ( size_size @ ( list @ A ) @ Vs ) ) )
=> ( ( ( append @ A @ Xs @ Us )
= ( append @ A @ Ys2 @ Vs ) )
= ( ( Xs = Ys2 )
& ( Us = Vs ) ) ) ) ).
% append_eq_append_conv
thf(fact_6758_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_6759_length__append,axiom,
! [A: $tType,Xs: list @ A,Ys2: list @ A] :
( ( size_size @ ( list @ A ) @ ( append @ A @ Xs @ Ys2 ) )
= ( plus_plus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ A ) @ Ys2 ) ) ) ).
% length_append
thf(fact_6760_sum__list_ONil,axiom,
! [A: $tType] :
( ( monoid_add @ A )
=> ( ( groups8242544230860333062m_list @ A @ ( nil @ A ) )
= ( zero_zero @ A ) ) ) ).
% sum_list.Nil
thf(fact_6761_take0,axiom,
! [A: $tType] :
( ( take @ A @ ( zero_zero @ nat ) )
= ( ^ [Xs3: list @ A] : ( nil @ A ) ) ) ).
% take0
thf(fact_6762_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_6763_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_6764_empty__replicate,axiom,
! [A: $tType,N: nat,X: A] :
( ( ( nil @ A )
= ( replicate @ A @ N @ X ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% empty_replicate
thf(fact_6765_replicate__empty,axiom,
! [A: $tType,N: nat,X: A] :
( ( ( replicate @ A @ N @ X )
= ( nil @ A ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% replicate_empty
thf(fact_6766_zip__append,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Us: list @ B,Ys2: list @ A,Vs: list @ B] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Us ) )
=> ( ( zip @ A @ B @ ( append @ A @ Xs @ Ys2 ) @ ( append @ B @ Us @ Vs ) )
= ( append @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Us ) @ ( zip @ A @ B @ Ys2 @ Vs ) ) ) ) ).
% zip_append
thf(fact_6767_horner__sum__simps_I1_J,axiom,
! [B: $tType,A: $tType] :
( ( comm_semiring_0 @ A )
=> ! [F2: B > A,A3: A] :
( ( groups4207007520872428315er_sum @ B @ A @ F2 @ A3 @ ( nil @ B ) )
= ( zero_zero @ A ) ) ) ).
% horner_sum_simps(1)
thf(fact_6768_fun__upds__append2__drop,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys2: list @ B,M2: A > ( option @ B ),Zs: list @ B] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys2 ) )
=> ( ( map_upds @ A @ B @ M2 @ Xs @ ( append @ B @ Ys2 @ Zs ) )
= ( map_upds @ A @ B @ M2 @ Xs @ Ys2 ) ) ) ).
% fun_upds_append2_drop
thf(fact_6769_fun__upds__append__drop,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys2: list @ B,M2: A > ( option @ B ),Zs: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys2 ) )
=> ( ( map_upds @ A @ B @ M2 @ ( append @ A @ Xs @ Zs ) @ Ys2 )
= ( map_upds @ A @ B @ M2 @ Xs @ Ys2 ) ) ) ).
% fun_upds_append_drop
thf(fact_6770_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_6771_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_6772_nth__append__length,axiom,
! [A: $tType,Xs: list @ A,X: A,Ys2: list @ A] :
( ( nth @ A @ ( append @ A @ Xs @ ( cons @ A @ X @ Ys2 ) ) @ ( size_size @ ( list @ A ) @ Xs ) )
= X ) ).
% nth_append_length
thf(fact_6773_nth__append__length__plus,axiom,
! [A: $tType,Xs: list @ A,Ys2: list @ A,N: nat] :
( ( nth @ A @ ( append @ A @ Xs @ Ys2 ) @ ( plus_plus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N ) )
= ( nth @ A @ Ys2 @ N ) ) ).
% nth_append_length_plus
thf(fact_6774_take__append,axiom,
! [A: $tType,N: nat,Xs: list @ A,Ys2: list @ A] :
( ( take @ A @ N @ ( append @ A @ Xs @ Ys2 ) )
= ( append @ A @ ( take @ A @ N @ Xs ) @ ( take @ A @ ( minus_minus @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) ) @ Ys2 ) ) ) ).
% take_append
thf(fact_6775_list__update__length,axiom,
! [A: $tType,Xs: list @ A,X: A,Ys2: list @ A,Y2: A] :
( ( list_update @ A @ ( append @ A @ Xs @ ( cons @ A @ X @ Ys2 ) ) @ ( size_size @ ( list @ A ) @ Xs ) @ Y2 )
= ( append @ A @ Xs @ ( cons @ A @ Y2 @ Ys2 ) ) ) ).
% list_update_length
thf(fact_6776_nths__singleton,axiom,
! [A: $tType,A4: set @ nat,X: A] :
( ( ( member @ nat @ ( zero_zero @ nat ) @ A4 )
=> ( ( nths @ A @ ( cons @ A @ X @ ( nil @ A ) ) @ A4 )
= ( cons @ A @ X @ ( nil @ A ) ) ) )
& ( ~ ( member @ nat @ ( zero_zero @ nat ) @ A4 )
=> ( ( nths @ A @ ( cons @ A @ X @ ( nil @ A ) ) @ A4 )
= ( nil @ A ) ) ) ) ).
% nths_singleton
thf(fact_6777_sorted__list__of__set__lessThan__Suc,axiom,
! [K: nat] :
( ( linord4507533701916653071of_set @ nat @ ( set_ord_lessThan @ nat @ ( suc @ K ) ) )
= ( append @ nat @ ( linord4507533701916653071of_set @ nat @ ( set_ord_lessThan @ nat @ K ) ) @ ( cons @ nat @ K @ ( nil @ nat ) ) ) ) ).
% sorted_list_of_set_lessThan_Suc
thf(fact_6778_sorted__list__of__set__atMost__Suc,axiom,
! [K: nat] :
( ( linord4507533701916653071of_set @ nat @ ( set_ord_atMost @ nat @ ( suc @ K ) ) )
= ( append @ nat @ ( linord4507533701916653071of_set @ nat @ ( set_ord_atMost @ nat @ K ) ) @ ( cons @ nat @ ( suc @ K ) @ ( nil @ nat ) ) ) ) ).
% sorted_list_of_set_atMost_Suc
thf(fact_6779_map__upds__append1,axiom,
! [B: $tType,A: $tType,Xs: list @ A,Ys2: list @ B,M2: A > ( option @ B ),X: A] :
( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ B ) @ Ys2 ) )
=> ( ( map_upds @ A @ B @ M2 @ ( append @ A @ Xs @ ( cons @ A @ X @ ( nil @ A ) ) ) @ Ys2 )
= ( fun_upd @ A @ ( option @ B ) @ ( map_upds @ A @ B @ M2 @ Xs @ Ys2 ) @ X @ ( some @ B @ ( nth @ B @ Ys2 @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ) ) ).
% map_upds_append1
thf(fact_6780_list__encode_Osimps_I1_J,axiom,
( ( nat_list_encode @ ( nil @ nat ) )
= ( zero_zero @ nat ) ) ).
% list_encode.simps(1)
thf(fact_6781_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_6782_take__0,axiom,
! [A: $tType,Xs: list @ A] :
( ( take @ A @ ( zero_zero @ nat ) @ Xs )
= ( nil @ A ) ) ).
% take_0
thf(fact_6783_replicate__0,axiom,
! [A: $tType,X: A] :
( ( replicate @ A @ ( zero_zero @ nat ) @ X )
= ( nil @ A ) ) ).
% replicate_0
thf(fact_6784_rotate__append,axiom,
! [A: $tType,L: list @ A,Q5: list @ A] :
( ( rotate @ A @ ( size_size @ ( list @ A ) @ L ) @ ( append @ A @ L @ Q5 ) )
= ( append @ A @ Q5 @ L ) ) ).
% rotate_append
thf(fact_6785_comm__append__is__replicate,axiom,
! [A: $tType,Xs: list @ A,Ys2: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( Ys2
!= ( nil @ A ) )
=> ( ( ( append @ A @ Xs @ Ys2 )
= ( append @ A @ Ys2 @ Xs ) )
=> ? [N2: nat,Zs2: list @ A] :
( ( ord_less @ nat @ ( one_one @ nat ) @ N2 )
& ( ( concat @ A @ ( replicate @ ( list @ A ) @ N2 @ Zs2 ) )
= ( append @ A @ Xs @ Ys2 ) ) ) ) ) ) ).
% comm_append_is_replicate
thf(fact_6786_enumerate__append__eq,axiom,
! [A: $tType,N: nat,Xs: list @ A,Ys2: list @ A] :
( ( enumerate @ A @ N @ ( append @ A @ Xs @ Ys2 ) )
= ( append @ ( product_prod @ nat @ A ) @ ( enumerate @ A @ N @ Xs ) @ ( enumerate @ A @ ( plus_plus @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) ) @ Ys2 ) ) ) ).
% enumerate_append_eq
thf(fact_6787_list_Osize_I3_J,axiom,
! [A: $tType] :
( ( size_size @ ( list @ A ) @ ( nil @ A ) )
= ( zero_zero @ nat ) ) ).
% list.size(3)
thf(fact_6788_find_Osimps_I1_J,axiom,
! [A: $tType,Uu: A > $o] :
( ( find @ A @ Uu @ ( nil @ A ) )
= ( none @ A ) ) ).
% find.simps(1)
thf(fact_6789_nths__Cons,axiom,
! [A: $tType,X: A,L: list @ A,A4: set @ nat] :
( ( nths @ A @ ( cons @ A @ X @ L ) @ A4 )
= ( append @ A @ ( if @ ( list @ A ) @ ( member @ nat @ ( zero_zero @ nat ) @ A4 ) @ ( cons @ A @ X @ ( nil @ A ) ) @ ( nil @ A ) )
@ ( nths @ A @ L
@ ( collect @ nat
@ ^ [J3: nat] : ( member @ nat @ ( suc @ J3 ) @ A4 ) ) ) ) ) ).
% nths_Cons
thf(fact_6790_list__encode_Ocases,axiom,
! [X: list @ nat] :
( ( X
!= ( nil @ nat ) )
=> ~ ! [X3: nat,Xs2: list @ nat] :
( X
!= ( cons @ nat @ X3 @ Xs2 ) ) ) ).
% list_encode.cases
thf(fact_6791_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_6792_same__length__different,axiom,
! [A: $tType,Xs: list @ A,Ys2: list @ A] :
( ( Xs != Ys2 )
=> ( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ A ) @ Ys2 ) )
=> ? [Pre: list @ A,X3: A,Xs5: list @ A,Y3: A,Ys5: list @ A] :
( ( X3 != Y3 )
& ( Xs
= ( append @ A @ Pre @ ( append @ A @ ( cons @ A @ X3 @ ( nil @ A ) ) @ Xs5 ) ) )
& ( Ys2
= ( append @ A @ Pre @ ( append @ A @ ( cons @ A @ Y3 @ ( nil @ A ) ) @ Ys5 ) ) ) ) ) ) ).
% same_length_different
thf(fact_6793_length__Suc__conv__rev,axiom,
! [A: $tType,Xs: list @ A,N: nat] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( suc @ N ) )
= ( ? [Y5: A,Ys3: list @ A] :
( ( Xs
= ( append @ A @ Ys3 @ ( cons @ A @ Y5 @ ( nil @ A ) ) ) )
& ( ( size_size @ ( list @ A ) @ Ys3 )
= N ) ) ) ) ).
% length_Suc_conv_rev
thf(fact_6794_list__induct4,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,Xs: list @ A,Ys2: list @ B,Zs: list @ C,Ws2: list @ D,P: ( list @ A ) > ( list @ B ) > ( list @ C ) > ( list @ D ) > $o] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys2 ) )
=> ( ( ( size_size @ ( list @ B ) @ Ys2 )
= ( size_size @ ( list @ C ) @ Zs ) )
=> ( ( ( size_size @ ( list @ C ) @ Zs )
= ( size_size @ ( list @ D ) @ Ws2 ) )
=> ( ( P @ ( nil @ A ) @ ( nil @ B ) @ ( nil @ C ) @ ( nil @ D ) )
=> ( ! [X3: A,Xs2: list @ A,Y3: B,Ys6: list @ B,Z4: C,Zs2: list @ C,W2: D,Ws: list @ D] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ B ) @ Ys6 ) )
=> ( ( ( size_size @ ( list @ B ) @ Ys6 )
= ( size_size @ ( list @ C ) @ Zs2 ) )
=> ( ( ( size_size @ ( list @ C ) @ Zs2 )
= ( size_size @ ( list @ D ) @ Ws ) )
=> ( ( P @ Xs2 @ Ys6 @ Zs2 @ Ws )
=> ( P @ ( cons @ A @ X3 @ Xs2 ) @ ( cons @ B @ Y3 @ Ys6 ) @ ( cons @ C @ Z4 @ Zs2 ) @ ( cons @ D @ W2 @ Ws ) ) ) ) ) )
=> ( P @ Xs @ Ys2 @ Zs @ Ws2 ) ) ) ) ) ) ).
% list_induct4
thf(fact_6795_list__induct3,axiom,
! [B: $tType,A: $tType,C: $tType,Xs: list @ A,Ys2: list @ B,Zs: list @ C,P: ( list @ A ) > ( list @ B ) > ( list @ C ) > $o] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys2 ) )
=> ( ( ( size_size @ ( list @ B ) @ Ys2 )
= ( size_size @ ( list @ C ) @ Zs ) )
=> ( ( P @ ( nil @ A ) @ ( nil @ B ) @ ( nil @ C ) )
=> ( ! [X3: A,Xs2: list @ A,Y3: B,Ys6: list @ B,Z4: C,Zs2: list @ C] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ B ) @ Ys6 ) )
=> ( ( ( size_size @ ( list @ B ) @ Ys6 )
= ( size_size @ ( list @ C ) @ Zs2 ) )
=> ( ( P @ Xs2 @ Ys6 @ Zs2 )
=> ( P @ ( cons @ A @ X3 @ Xs2 ) @ ( cons @ B @ Y3 @ Ys6 ) @ ( cons @ C @ Z4 @ Zs2 ) ) ) ) )
=> ( P @ Xs @ Ys2 @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_6796_list__induct2,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys2: list @ B,P: ( list @ A ) > ( list @ B ) > $o] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys2 ) )
=> ( ( P @ ( nil @ A ) @ ( nil @ B ) )
=> ( ! [X3: A,Xs2: list @ A,Y3: B,Ys6: list @ B] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ B ) @ Ys6 ) )
=> ( ( P @ Xs2 @ Ys6 )
=> ( P @ ( cons @ A @ X3 @ Xs2 ) @ ( cons @ B @ Y3 @ Ys6 ) ) ) )
=> ( P @ Xs @ Ys2 ) ) ) ) ).
% list_induct2
thf(fact_6797_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_6798_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_6799_take__Suc__conv__app__nth,axiom,
! [A: $tType,I: nat,Xs: list @ A] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( take @ A @ ( suc @ I ) @ Xs )
= ( append @ A @ ( take @ A @ I @ Xs ) @ ( cons @ A @ ( nth @ A @ Xs @ I ) @ ( nil @ A ) ) ) ) ) ).
% take_Suc_conv_app_nth
thf(fact_6800_list__update__append1,axiom,
! [A: $tType,I: nat,Xs: list @ A,Ys2: list @ A,X: A] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( list_update @ A @ ( append @ A @ Xs @ Ys2 ) @ I @ X )
= ( append @ A @ ( list_update @ A @ Xs @ I @ X ) @ Ys2 ) ) ) ).
% list_update_append1
thf(fact_6801_lexord__sufE,axiom,
! [A: $tType,Xs: list @ A,Zs: list @ A,Ys2: list @ A,Qs: list @ A,R4: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs @ Zs ) @ ( append @ A @ Ys2 @ Qs ) ) @ ( lexord @ A @ R4 ) )
=> ( ( Xs != Ys2 )
=> ( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ A ) @ Ys2 ) )
=> ( ( ( size_size @ ( list @ A ) @ Zs )
= ( size_size @ ( list @ A ) @ Qs ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys2 ) @ ( lexord @ A @ R4 ) ) ) ) ) ) ).
% lexord_sufE
thf(fact_6802_lex__append__rightI,axiom,
! [A: $tType,Xs: list @ A,Ys2: list @ A,R4: 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 @ Ys2 ) @ ( lex @ A @ R4 ) )
=> ( ( ( 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 @ Ys2 @ Vs ) ) @ ( lex @ A @ R4 ) ) ) ) ).
% lex_append_rightI
thf(fact_6803_lenlex__append1,axiom,
! [A: $tType,Us: list @ A,Xs: list @ A,R2: set @ ( product_prod @ A @ A ),Vs: list @ A,Ys2: list @ A] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Us @ Xs ) @ ( lenlex @ A @ R2 ) )
=> ( ( ( size_size @ ( list @ A ) @ Vs )
= ( size_size @ ( list @ A ) @ Ys2 ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Us @ Vs ) @ ( append @ A @ Xs @ Ys2 ) ) @ ( lenlex @ A @ R2 ) ) ) ) ).
% lenlex_append1
thf(fact_6804_nths__append,axiom,
! [A: $tType,L: list @ A,L6: list @ A,A4: set @ nat] :
( ( nths @ A @ ( append @ A @ L @ L6 ) @ A4 )
= ( append @ A @ ( nths @ A @ L @ A4 )
@ ( nths @ A @ L6
@ ( collect @ nat
@ ^ [J3: nat] : ( member @ nat @ ( plus_plus @ nat @ J3 @ ( size_size @ ( list @ A ) @ L ) ) @ A4 ) ) ) ) ) ).
% nths_append
thf(fact_6805_lexordp_Omono,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( order_mono @ ( ( list @ A ) > ( list @ A ) > $o ) @ ( ( list @ A ) > ( list @ A ) > $o )
@ ^ [P6: ( list @ A ) > ( list @ A ) > $o,X16: list @ A,X25: list @ A] :
( ? [Y5: A,Ys3: list @ A] :
( ( X16
= ( nil @ A ) )
& ( X25
= ( cons @ A @ Y5 @ Ys3 ) ) )
| ? [X2: A,Y5: A,Xs3: list @ A,Ys3: list @ A] :
( ( X16
= ( cons @ A @ X2 @ Xs3 ) )
& ( X25
= ( cons @ A @ Y5 @ Ys3 ) )
& ( ord_less @ A @ X2 @ Y5 ) )
| ? [X2: A,Y5: A,Xs3: list @ A,Ys3: list @ A] :
( ( X16
= ( cons @ A @ X2 @ Xs3 ) )
& ( X25
= ( cons @ A @ Y5 @ Ys3 ) )
& ~ ( ord_less @ A @ X2 @ Y5 )
& ~ ( ord_less @ A @ Y5 @ X2 )
& ( P6 @ Xs3 @ Ys3 ) ) ) ) ) ).
% lexordp.mono
thf(fact_6806_nth__append,axiom,
! [A: $tType,N: nat,Xs: list @ A,Ys2: list @ A] :
( ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ A @ ( append @ A @ Xs @ Ys2 ) @ N )
= ( nth @ A @ Xs @ N ) ) )
& ( ~ ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ A @ ( append @ A @ Xs @ Ys2 ) @ N )
= ( nth @ A @ Ys2 @ ( minus_minus @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ) ) ).
% nth_append
thf(fact_6807_list__update__append,axiom,
! [A: $tType,N: nat,Xs: list @ A,Ys2: list @ A,X: A] :
( ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( list_update @ A @ ( append @ A @ Xs @ Ys2 ) @ N @ X )
= ( append @ A @ ( list_update @ A @ Xs @ N @ X ) @ Ys2 ) ) )
& ( ~ ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( list_update @ A @ ( append @ A @ Xs @ Ys2 ) @ N @ X )
= ( append @ A @ Xs @ ( list_update @ A @ Ys2 @ ( minus_minus @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) ) @ X ) ) ) ) ) ).
% list_update_append
thf(fact_6808_lexord__sufI,axiom,
! [A: $tType,U: list @ A,W: list @ A,R4: set @ ( product_prod @ A @ A ),V2: list @ A,Z2: list @ A] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ U @ W ) @ ( lexord @ A @ R4 ) )
=> ( ( 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 @ Z2 ) ) @ ( lexord @ A @ R4 ) ) ) ) ).
% lexord_sufI
thf(fact_6809_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_6810_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_6811_horner__sum__append,axiom,
! [A: $tType,B: $tType] :
( ( comm_semiring_1 @ A )
=> ! [F2: B > A,A3: A,Xs: list @ B,Ys2: list @ B] :
( ( groups4207007520872428315er_sum @ B @ A @ F2 @ A3 @ ( append @ B @ Xs @ Ys2 ) )
= ( plus_plus @ A @ ( groups4207007520872428315er_sum @ B @ A @ F2 @ A3 @ Xs ) @ ( times_times @ A @ ( power_power @ A @ A3 @ ( size_size @ ( list @ B ) @ Xs ) ) @ ( groups4207007520872428315er_sum @ B @ A @ F2 @ A3 @ Ys2 ) ) ) ) ) ).
% horner_sum_append
thf(fact_6812_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_6813_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_6814_nth__repl,axiom,
! [A: $tType,M2: nat,Xs: list @ A,N: nat,X: A] :
( ( ord_less @ nat @ M2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( M2 != 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 ) ) ) @ M2 )
= ( nth @ A @ Xs @ M2 ) ) ) ) ) ).
% nth_repl
thf(fact_6815_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_6816_drop0,axiom,
! [A: $tType] :
( ( drop @ A @ ( zero_zero @ nat ) )
= ( ^ [X2: list @ A] : X2 ) ) ).
% drop0
thf(fact_6817_map__of__eq__empty__iff,axiom,
! [B: $tType,A: $tType,Xys: list @ ( product_prod @ A @ B )] :
( ( ( map_of @ A @ B @ Xys )
= ( ^ [X2: A] : ( none @ B ) ) )
= ( Xys
= ( nil @ ( product_prod @ A @ B ) ) ) ) ).
% map_of_eq_empty_iff
thf(fact_6818_empty__eq__map__of__iff,axiom,
! [B: $tType,A: $tType,Xys: list @ ( product_prod @ A @ B )] :
( ( ( ^ [X2: A] : ( none @ B ) )
= ( map_of @ A @ B @ Xys ) )
= ( Xys
= ( nil @ ( product_prod @ A @ B ) ) ) ) ).
% empty_eq_map_of_iff
thf(fact_6819_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_6820_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_6821_drop__update__cancel,axiom,
! [A: $tType,N: nat,M2: nat,Xs: list @ A,X: A] :
( ( ord_less @ nat @ N @ M2 )
=> ( ( drop @ A @ M2 @ ( list_update @ A @ Xs @ N @ X ) )
= ( drop @ A @ M2 @ Xs ) ) ) ).
% drop_update_cancel
thf(fact_6822_drop__replicate,axiom,
! [A: $tType,I: nat,K: nat,X: A] :
( ( drop @ A @ I @ ( replicate @ A @ K @ X ) )
= ( replicate @ A @ ( minus_minus @ nat @ K @ I ) @ X ) ) ).
% drop_replicate
thf(fact_6823_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_6824_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_6825_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_6826_drop__append,axiom,
! [A: $tType,N: nat,Xs: list @ A,Ys2: list @ A] :
( ( drop @ A @ N @ ( append @ A @ Xs @ Ys2 ) )
= ( append @ A @ ( drop @ A @ N @ Xs ) @ ( drop @ A @ ( minus_minus @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) ) @ Ys2 ) ) ) ).
% drop_append
thf(fact_6827_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_6828_nth__drop,axiom,
! [A: $tType,N: nat,Xs: list @ A,I: nat] :
( ( ord_less_eq @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ A @ ( drop @ A @ N @ Xs ) @ I )
= ( nth @ A @ Xs @ ( plus_plus @ nat @ N @ I ) ) ) ) ).
% nth_drop
thf(fact_6829_map__of__Cons__code_I1_J,axiom,
! [B: $tType,A: $tType,K: B] :
( ( map_of @ B @ A @ ( nil @ ( product_prod @ B @ A ) ) @ K )
= ( none @ A ) ) ).
% map_of_Cons_code(1)
thf(fact_6830_map__of_Osimps_I1_J,axiom,
! [B: $tType,A: $tType] :
( ( map_of @ A @ B @ ( nil @ ( product_prod @ A @ B ) ) )
= ( ^ [X2: A] : ( none @ B ) ) ) ).
% map_of.simps(1)
thf(fact_6831_drop__0,axiom,
! [A: $tType,Xs: list @ A] :
( ( drop @ A @ ( zero_zero @ nat ) @ Xs )
= Xs ) ).
% drop_0
thf(fact_6832_drop__take,axiom,
! [A: $tType,N: nat,M2: nat,Xs: list @ A] :
( ( drop @ A @ N @ ( take @ A @ M2 @ Xs ) )
= ( take @ A @ ( minus_minus @ nat @ M2 @ N ) @ ( drop @ A @ N @ Xs ) ) ) ).
% drop_take
thf(fact_6833_append__eq__conv__conj,axiom,
! [A: $tType,Xs: list @ A,Ys2: list @ A,Zs: list @ A] :
( ( ( append @ A @ Xs @ Ys2 )
= Zs )
= ( ( Xs
= ( take @ A @ ( size_size @ ( list @ A ) @ Xs ) @ Zs ) )
& ( Ys2
= ( drop @ A @ ( size_size @ ( list @ A ) @ Xs ) @ Zs ) ) ) ) ).
% append_eq_conv_conj
thf(fact_6834_drop__update__swap,axiom,
! [A: $tType,M2: nat,N: nat,Xs: list @ A,X: A] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( drop @ A @ M2 @ ( list_update @ A @ Xs @ N @ X ) )
= ( list_update @ A @ ( drop @ A @ M2 @ Xs ) @ ( minus_minus @ nat @ N @ M2 ) @ X ) ) ) ).
% drop_update_swap
thf(fact_6835_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_6836_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_6837_zip__append2,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys2: list @ B,Zs: list @ B] :
( ( zip @ A @ B @ Xs @ ( append @ B @ Ys2 @ Zs ) )
= ( append @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ ( take @ A @ ( size_size @ ( list @ B ) @ Ys2 ) @ Xs ) @ Ys2 ) @ ( zip @ A @ B @ ( drop @ A @ ( size_size @ ( list @ B ) @ Ys2 ) @ Xs ) @ Zs ) ) ) ).
% zip_append2
thf(fact_6838_zip__append1,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys2: list @ A,Zs: list @ B] :
( ( zip @ A @ B @ ( append @ A @ Xs @ Ys2 ) @ Zs )
= ( append @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ ( take @ B @ ( size_size @ ( list @ A ) @ Xs ) @ Zs ) ) @ ( zip @ A @ B @ Ys2 @ ( drop @ B @ ( size_size @ ( list @ A ) @ Xs ) @ Zs ) ) ) ) ).
% zip_append1
thf(fact_6839_Cons__nth__drop__Suc,axiom,
! [A: $tType,I: nat,Xs: list @ A] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( cons @ A @ ( nth @ A @ Xs @ I ) @ ( drop @ A @ ( suc @ I ) @ Xs ) )
= ( drop @ A @ I @ Xs ) ) ) ).
% Cons_nth_drop_Suc
thf(fact_6840_rotate__drop__take,axiom,
! [A: $tType] :
( ( rotate @ A )
= ( ^ [N5: nat,Xs3: list @ A] : ( append @ A @ ( drop @ A @ ( modulo_modulo @ nat @ N5 @ ( size_size @ ( list @ A ) @ Xs3 ) ) @ Xs3 ) @ ( take @ A @ ( modulo_modulo @ nat @ N5 @ ( size_size @ ( list @ A ) @ Xs3 ) ) @ Xs3 ) ) ) ) ).
% rotate_drop_take
thf(fact_6841_id__take__nth__drop,axiom,
! [A: $tType,I: nat,Xs: list @ A] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( Xs
= ( append @ A @ ( take @ A @ I @ Xs ) @ ( cons @ A @ ( nth @ A @ Xs @ I ) @ ( drop @ A @ ( suc @ I ) @ Xs ) ) ) ) ) ).
% id_take_nth_drop
thf(fact_6842_upd__conv__take__nth__drop,axiom,
! [A: $tType,I: nat,Xs: list @ A,A3: A] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( list_update @ A @ Xs @ I @ A3 )
= ( append @ A @ ( take @ A @ I @ Xs ) @ ( cons @ A @ A3 @ ( drop @ A @ ( suc @ I ) @ Xs ) ) ) ) ) ).
% upd_conv_take_nth_drop
thf(fact_6843_lexn__conv,axiom,
! [A: $tType] :
( ( lexn @ A )
= ( ^ [R: set @ ( product_prod @ A @ A ),N5: nat] :
( collect @ ( product_prod @ ( list @ A ) @ ( list @ A ) )
@ ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ $o
@ ^ [Xs3: list @ A,Ys3: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs3 )
= N5 )
& ( ( size_size @ ( list @ A ) @ Ys3 )
= N5 )
& ? [Xys2: list @ A,X2: A,Y5: A,Xs4: list @ A,Ys4: list @ A] :
( ( Xs3
= ( append @ A @ Xys2 @ ( cons @ A @ X2 @ Xs4 ) ) )
& ( Ys3
= ( append @ A @ Xys2 @ ( cons @ A @ Y5 @ Ys4 ) ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y5 ) @ R ) ) ) ) ) ) ) ).
% lexn_conv
thf(fact_6844_upto_Opsimps,axiom,
! [I: int,J2: int] :
( ( accp @ ( product_prod @ int @ int ) @ upto_rel @ ( product_Pair @ int @ int @ I @ J2 ) )
=> ( ( ( ord_less_eq @ int @ I @ J2 )
=> ( ( upto @ I @ J2 )
= ( cons @ int @ I @ ( upto @ ( plus_plus @ int @ I @ ( one_one @ int ) ) @ J2 ) ) ) )
& ( ~ ( ord_less_eq @ int @ I @ J2 )
=> ( ( upto @ I @ J2 )
= ( nil @ int ) ) ) ) ) ).
% upto.psimps
thf(fact_6845_upto__Nil,axiom,
! [I: int,J2: int] :
( ( ( upto @ I @ J2 )
= ( nil @ int ) )
= ( ord_less @ int @ J2 @ I ) ) ).
% upto_Nil
thf(fact_6846_upto__Nil2,axiom,
! [I: int,J2: int] :
( ( ( nil @ int )
= ( upto @ I @ J2 ) )
= ( ord_less @ int @ J2 @ I ) ) ).
% upto_Nil2
thf(fact_6847_upto__empty,axiom,
! [J2: int,I: int] :
( ( ord_less @ int @ J2 @ I )
=> ( ( upto @ I @ J2 )
= ( nil @ int ) ) ) ).
% upto_empty
thf(fact_6848_nth__upto,axiom,
! [I: int,K: nat,J2: int] :
( ( ord_less_eq @ int @ ( plus_plus @ int @ I @ ( semiring_1_of_nat @ int @ K ) ) @ J2 )
=> ( ( nth @ int @ ( upto @ I @ J2 ) @ K )
= ( plus_plus @ int @ I @ ( semiring_1_of_nat @ int @ K ) ) ) ) ).
% nth_upto
thf(fact_6849_length__upto,axiom,
! [I: int,J2: int] :
( ( size_size @ ( list @ int ) @ ( upto @ I @ J2 ) )
= ( nat2 @ ( plus_plus @ int @ ( minus_minus @ int @ J2 @ I ) @ ( one_one @ int ) ) ) ) ).
% length_upto
thf(fact_6850_upto__rec__numeral_I1_J,axiom,
! [M2: num,N: num] :
( ( ( ord_less_eq @ int @ ( numeral_numeral @ int @ M2 ) @ ( numeral_numeral @ int @ N ) )
=> ( ( upto @ ( numeral_numeral @ int @ M2 ) @ ( numeral_numeral @ int @ N ) )
= ( cons @ int @ ( numeral_numeral @ int @ M2 ) @ ( upto @ ( plus_plus @ int @ ( numeral_numeral @ int @ M2 ) @ ( one_one @ int ) ) @ ( numeral_numeral @ int @ N ) ) ) ) )
& ( ~ ( ord_less_eq @ int @ ( numeral_numeral @ int @ M2 ) @ ( numeral_numeral @ int @ N ) )
=> ( ( upto @ ( numeral_numeral @ int @ M2 ) @ ( numeral_numeral @ int @ N ) )
= ( nil @ int ) ) ) ) ).
% upto_rec_numeral(1)
thf(fact_6851_upto__rec__numeral_I4_J,axiom,
! [M2: num,N: num] :
( ( ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M2 ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
=> ( ( upto @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M2 ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
= ( cons @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M2 ) ) @ ( upto @ ( plus_plus @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M2 ) ) @ ( one_one @ int ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) ) ) ) )
& ( ~ ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M2 ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
=> ( ( upto @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M2 ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
= ( nil @ int ) ) ) ) ).
% upto_rec_numeral(4)
thf(fact_6852_upto__rec__numeral_I3_J,axiom,
! [M2: num,N: num] :
( ( ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M2 ) ) @ ( numeral_numeral @ int @ N ) )
=> ( ( upto @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M2 ) ) @ ( numeral_numeral @ int @ N ) )
= ( cons @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M2 ) ) @ ( upto @ ( plus_plus @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M2 ) ) @ ( one_one @ int ) ) @ ( numeral_numeral @ int @ N ) ) ) ) )
& ( ~ ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M2 ) ) @ ( numeral_numeral @ int @ N ) )
=> ( ( upto @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ M2 ) ) @ ( numeral_numeral @ int @ N ) )
= ( nil @ int ) ) ) ) ).
% upto_rec_numeral(3)
thf(fact_6853_upto__rec__numeral_I2_J,axiom,
! [M2: num,N: num] :
( ( ( ord_less_eq @ int @ ( numeral_numeral @ int @ M2 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
=> ( ( upto @ ( numeral_numeral @ int @ M2 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
= ( cons @ int @ ( numeral_numeral @ int @ M2 ) @ ( upto @ ( plus_plus @ int @ ( numeral_numeral @ int @ M2 ) @ ( one_one @ int ) ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) ) ) ) )
& ( ~ ( ord_less_eq @ int @ ( numeral_numeral @ int @ M2 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
=> ( ( upto @ ( numeral_numeral @ int @ M2 ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ N ) ) )
= ( nil @ int ) ) ) ) ).
% upto_rec_numeral(2)
thf(fact_6854_lexn_Osimps_I1_J,axiom,
! [A: $tType,R4: set @ ( product_prod @ A @ A )] :
( ( lexn @ A @ R4 @ ( zero_zero @ nat ) )
= ( bot_bot @ ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ) ) ).
% lexn.simps(1)
thf(fact_6855_upto__split2,axiom,
! [I: int,J2: int,K: int] :
( ( ord_less_eq @ int @ I @ J2 )
=> ( ( ord_less_eq @ int @ J2 @ K )
=> ( ( upto @ I @ K )
= ( append @ int @ ( upto @ I @ J2 ) @ ( upto @ ( plus_plus @ int @ J2 @ ( one_one @ int ) ) @ K ) ) ) ) ) ).
% upto_split2
thf(fact_6856_upto__split1,axiom,
! [I: int,J2: int,K: int] :
( ( ord_less_eq @ int @ I @ J2 )
=> ( ( ord_less_eq @ int @ J2 @ K )
=> ( ( upto @ I @ K )
= ( append @ int @ ( upto @ I @ ( minus_minus @ int @ J2 @ ( one_one @ int ) ) ) @ ( upto @ J2 @ K ) ) ) ) ) ).
% upto_split1
thf(fact_6857_atLeastLessThan__upto,axiom,
( ( set_or7035219750837199246ssThan @ int )
= ( ^ [I2: int,J3: int] : ( set2 @ int @ ( upto @ I2 @ ( minus_minus @ int @ J3 @ ( one_one @ int ) ) ) ) ) ) ).
% atLeastLessThan_upto
thf(fact_6858_greaterThanAtMost__upto,axiom,
( ( set_or3652927894154168847AtMost @ int )
= ( ^ [I2: int,J3: int] : ( set2 @ int @ ( upto @ ( plus_plus @ int @ I2 @ ( one_one @ int ) ) @ J3 ) ) ) ) ).
% greaterThanAtMost_upto
thf(fact_6859_lexn__length,axiom,
! [A: $tType,Xs: list @ A,Ys2: list @ A,R4: set @ ( product_prod @ A @ A ),N: nat] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys2 ) @ ( lexn @ A @ R4 @ N ) )
=> ( ( ( size_size @ ( list @ A ) @ Xs )
= N )
& ( ( size_size @ ( list @ A ) @ Ys2 )
= N ) ) ) ).
% lexn_length
thf(fact_6860_upto__rec1,axiom,
! [I: int,J2: int] :
( ( ord_less_eq @ int @ I @ J2 )
=> ( ( upto @ I @ J2 )
= ( cons @ int @ I @ ( upto @ ( plus_plus @ int @ I @ ( one_one @ int ) ) @ J2 ) ) ) ) ).
% upto_rec1
thf(fact_6861_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_6862_upto_Osimps,axiom,
( upto
= ( ^ [I2: int,J3: int] : ( if @ ( list @ int ) @ ( ord_less_eq @ int @ I2 @ J3 ) @ ( cons @ int @ I2 @ ( upto @ ( plus_plus @ int @ I2 @ ( one_one @ int ) ) @ J3 ) ) @ ( nil @ int ) ) ) ) ).
% upto.simps
thf(fact_6863_upto__rec2,axiom,
! [I: int,J2: int] :
( ( ord_less_eq @ int @ I @ J2 )
=> ( ( upto @ I @ J2 )
= ( append @ int @ ( upto @ I @ ( minus_minus @ int @ J2 @ ( one_one @ int ) ) ) @ ( cons @ int @ J2 @ ( nil @ int ) ) ) ) ) ).
% upto_rec2
thf(fact_6864_greaterThanLessThan__upto,axiom,
( ( set_or5935395276787703475ssThan @ int )
= ( ^ [I2: int,J3: int] : ( set2 @ int @ ( upto @ ( plus_plus @ int @ I2 @ ( one_one @ int ) ) @ ( minus_minus @ int @ J3 @ ( one_one @ int ) ) ) ) ) ) ).
% greaterThanLessThan_upto
thf(fact_6865_upto__split3,axiom,
! [I: int,J2: int,K: int] :
( ( ord_less_eq @ int @ I @ J2 )
=> ( ( ord_less_eq @ int @ J2 @ K )
=> ( ( upto @ I @ K )
= ( append @ int @ ( upto @ I @ ( minus_minus @ int @ J2 @ ( one_one @ int ) ) ) @ ( cons @ int @ J2 @ ( upto @ ( plus_plus @ int @ J2 @ ( one_one @ int ) ) @ K ) ) ) ) ) ) ).
% upto_split3
thf(fact_6866_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_6867_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_6868_graph__map__upd,axiom,
! [A: $tType,B: $tType,M2: A > ( option @ B ),K: A,V2: B] :
( ( graph @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ M2 @ K @ ( some @ B @ V2 ) ) )
= ( insert @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K @ V2 ) @ ( graph @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ M2 @ K @ ( none @ B ) ) ) ) ) ).
% graph_map_upd
thf(fact_6869_hd__replicate,axiom,
! [A: $tType,N: nat,X: A] :
( ( N
!= ( zero_zero @ nat ) )
=> ( ( hd @ A @ ( replicate @ A @ N @ X ) )
= X ) ) ).
% hd_replicate
thf(fact_6870_graph__empty,axiom,
! [B: $tType,A: $tType] :
( ( graph @ A @ B
@ ^ [X2: A] : ( none @ B ) )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% graph_empty
thf(fact_6871_hd__take,axiom,
! [A: $tType,J2: nat,Xs: list @ A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ J2 )
=> ( ( hd @ A @ ( take @ A @ J2 @ Xs ) )
= ( hd @ A @ Xs ) ) ) ).
% hd_take
thf(fact_6872_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_6873_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_6874_graph__fun__upd__None,axiom,
! [B: $tType,A: $tType,M2: A > ( option @ B ),K: A] :
( ( graph @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ M2 @ K @ ( none @ B ) ) )
= ( collect @ ( product_prod @ A @ B )
@ ^ [E3: product_prod @ A @ B] :
( ( member @ ( product_prod @ A @ B ) @ E3 @ ( graph @ A @ B @ M2 ) )
& ( ( product_fst @ A @ B @ E3 )
!= K ) ) ) ) ).
% graph_fun_upd_None
thf(fact_6875_hd__rotate__conv__nth,axiom,
! [A: $tType,Xs: list @ A,N: nat] :
( ( Xs
!= ( nil @ A ) )
=> ( ( hd @ A @ ( rotate @ A @ N @ Xs ) )
= ( nth @ A @ Xs @ ( modulo_modulo @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ) ).
% hd_rotate_conv_nth
thf(fact_6876_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_6877_Nitpick_Osize__list__simp_I1_J,axiom,
! [A: $tType] :
( ( size_list @ A )
= ( ^ [F4: A > nat,Xs3: list @ A] :
( if @ nat
@ ( Xs3
= ( nil @ A ) )
@ ( zero_zero @ nat )
@ ( suc @ ( plus_plus @ nat @ ( F4 @ ( hd @ A @ Xs3 ) ) @ ( size_list @ A @ F4 @ ( tl @ A @ Xs3 ) ) ) ) ) ) ) ).
% Nitpick.size_list_simp(1)
thf(fact_6878_list__encode_Opelims,axiom,
! [X: list @ nat,Y2: nat] :
( ( ( nat_list_encode @ X )
= Y2 )
=> ( ( accp @ ( list @ nat ) @ nat_list_encode_rel @ X )
=> ( ( ( X
= ( nil @ nat ) )
=> ( ( Y2
= ( zero_zero @ nat ) )
=> ~ ( accp @ ( list @ nat ) @ nat_list_encode_rel @ ( nil @ nat ) ) ) )
=> ~ ! [X3: nat,Xs2: list @ nat] :
( ( X
= ( cons @ nat @ X3 @ Xs2 ) )
=> ( ( Y2
= ( suc @ ( nat_prod_encode @ ( product_Pair @ nat @ nat @ X3 @ ( nat_list_encode @ Xs2 ) ) ) ) )
=> ~ ( accp @ ( list @ nat ) @ nat_list_encode_rel @ ( cons @ nat @ X3 @ Xs2 ) ) ) ) ) ) ) ).
% list_encode.pelims
thf(fact_6879_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_6880_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_6881_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_6882_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_6883_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_6884_Nitpick_Osize__list__simp_I2_J,axiom,
! [A: $tType] :
( ( size_size @ ( list @ A ) )
= ( ^ [Xs3: list @ A] :
( if @ nat
@ ( Xs3
= ( nil @ A ) )
@ ( zero_zero @ nat )
@ ( suc @ ( size_size @ ( list @ A ) @ ( tl @ A @ Xs3 ) ) ) ) ) ) ).
% Nitpick.size_list_simp(2)
thf(fact_6885_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_6886_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_6887_upt__rec__numeral,axiom,
! [M2: num,N: num] :
( ( ( ord_less @ nat @ ( numeral_numeral @ nat @ M2 ) @ ( numeral_numeral @ nat @ N ) )
=> ( ( upt @ ( numeral_numeral @ nat @ M2 ) @ ( numeral_numeral @ nat @ N ) )
= ( cons @ nat @ ( numeral_numeral @ nat @ M2 ) @ ( upt @ ( suc @ ( numeral_numeral @ nat @ M2 ) ) @ ( numeral_numeral @ nat @ N ) ) ) ) )
& ( ~ ( ord_less @ nat @ ( numeral_numeral @ nat @ M2 ) @ ( numeral_numeral @ nat @ N ) )
=> ( ( upt @ ( numeral_numeral @ nat @ M2 ) @ ( numeral_numeral @ nat @ N ) )
= ( nil @ nat ) ) ) ) ).
% upt_rec_numeral
thf(fact_6888_sorted__list__of__set__nonempty,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A] :
( ( finite_finite @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( linord4507533701916653071of_set @ A @ A4 )
= ( cons @ A @ ( lattic643756798350308766er_Min @ A @ A4 ) @ ( linord4507533701916653071of_set @ A @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ ( lattic643756798350308766er_Min @ A @ A4 ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ).
% sorted_list_of_set_nonempty
thf(fact_6889_tl__upt,axiom,
! [M2: nat,N: nat] :
( ( tl @ nat @ ( upt @ M2 @ N ) )
= ( upt @ ( suc @ M2 ) @ N ) ) ).
% tl_upt
thf(fact_6890_hd__upt,axiom,
! [I: nat,J2: nat] :
( ( ord_less @ nat @ I @ J2 )
=> ( ( hd @ nat @ ( upt @ I @ J2 ) )
= I ) ) ).
% hd_upt
thf(fact_6891_length__upt,axiom,
! [I: nat,J2: nat] :
( ( size_size @ ( list @ nat ) @ ( upt @ I @ J2 ) )
= ( minus_minus @ nat @ J2 @ I ) ) ).
% length_upt
thf(fact_6892_Min__gr__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less @ A @ X @ ( lattic643756798350308766er_Min @ A @ A4 ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ A4 )
=> ( ord_less @ A @ X @ X2 ) ) ) ) ) ) ) ).
% Min_gr_iff
thf(fact_6893_upt__eq__Nil__conv,axiom,
! [I: nat,J2: nat] :
( ( ( upt @ I @ J2 )
= ( nil @ nat ) )
= ( ( J2
= ( zero_zero @ nat ) )
| ( ord_less_eq @ nat @ J2 @ I ) ) ) ).
% upt_eq_Nil_conv
thf(fact_6894_nth__upt,axiom,
! [I: nat,K: nat,J2: nat] :
( ( ord_less @ nat @ ( plus_plus @ nat @ I @ K ) @ J2 )
=> ( ( nth @ nat @ ( upt @ I @ J2 ) @ K )
= ( plus_plus @ nat @ I @ K ) ) ) ).
% nth_upt
thf(fact_6895_upt__0,axiom,
! [I: nat] :
( ( upt @ I @ ( zero_zero @ nat ) )
= ( nil @ nat ) ) ).
% upt_0
thf(fact_6896_upt__conv__Cons__Cons,axiom,
! [M2: nat,N: nat,Ns: list @ nat,Q5: nat] :
( ( ( cons @ nat @ M2 @ ( cons @ nat @ N @ Ns ) )
= ( upt @ M2 @ Q5 ) )
= ( ( cons @ nat @ N @ Ns )
= ( upt @ ( suc @ M2 ) @ Q5 ) ) ) ).
% upt_conv_Cons_Cons
thf(fact_6897_greaterThanAtMost__upt,axiom,
( ( set_or3652927894154168847AtMost @ nat )
= ( ^ [N5: nat,M5: nat] : ( set2 @ nat @ ( upt @ ( suc @ N5 ) @ ( suc @ M5 ) ) ) ) ) ).
% greaterThanAtMost_upt
thf(fact_6898_atLeast__upt,axiom,
( ( set_ord_lessThan @ nat )
= ( ^ [N5: nat] : ( set2 @ nat @ ( upt @ ( zero_zero @ nat ) @ N5 ) ) ) ) ).
% atLeast_upt
thf(fact_6899_atLeastAtMost__upt,axiom,
( ( set_or1337092689740270186AtMost @ nat )
= ( ^ [N5: nat,M5: nat] : ( set2 @ nat @ ( upt @ N5 @ ( suc @ M5 ) ) ) ) ) ).
% atLeastAtMost_upt
thf(fact_6900_greaterThanLessThan__upt,axiom,
( ( set_or5935395276787703475ssThan @ nat )
= ( ^ [N5: nat,M5: nat] : ( set2 @ nat @ ( upt @ ( suc @ N5 ) @ M5 ) ) ) ) ).
% greaterThanLessThan_upt
thf(fact_6901_atMost__upto,axiom,
( ( set_ord_atMost @ nat )
= ( ^ [N5: nat] : ( set2 @ nat @ ( upt @ ( zero_zero @ nat ) @ ( suc @ N5 ) ) ) ) ) ).
% atMost_upto
thf(fact_6902_Min__less__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less @ A @ ( lattic643756798350308766er_Min @ A @ A4 ) @ X )
= ( ? [X2: A] :
( ( member @ A @ X2 @ A4 )
& ( ord_less @ A @ X2 @ X ) ) ) ) ) ) ) ).
% Min_less_iff
thf(fact_6903_upt__conv__Cons,axiom,
! [I: nat,J2: nat] :
( ( ord_less @ nat @ I @ J2 )
=> ( ( upt @ I @ J2 )
= ( cons @ nat @ I @ ( upt @ ( suc @ I ) @ J2 ) ) ) ) ).
% upt_conv_Cons
thf(fact_6904_enumerate__eq__zip,axiom,
! [A: $tType] :
( ( enumerate @ A )
= ( ^ [N5: nat,Xs3: list @ A] : ( zip @ nat @ A @ ( upt @ N5 @ ( plus_plus @ nat @ N5 @ ( size_size @ ( list @ A ) @ Xs3 ) ) ) @ Xs3 ) ) ) ).
% enumerate_eq_zip
thf(fact_6905_upt__eq__Cons__conv,axiom,
! [I: nat,J2: nat,X: nat,Xs: list @ nat] :
( ( ( upt @ I @ J2 )
= ( cons @ nat @ X @ Xs ) )
= ( ( ord_less @ nat @ I @ J2 )
& ( I = X )
& ( ( upt @ ( plus_plus @ nat @ I @ ( one_one @ nat ) ) @ J2 )
= Xs ) ) ) ).
% upt_eq_Cons_conv
thf(fact_6906_upt__rec,axiom,
( upt
= ( ^ [I2: nat,J3: nat] : ( if @ ( list @ nat ) @ ( ord_less @ nat @ I2 @ J3 ) @ ( cons @ nat @ I2 @ ( upt @ ( suc @ I2 ) @ J3 ) ) @ ( nil @ nat ) ) ) ) ).
% upt_rec
thf(fact_6907_Min_Oremove,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite @ A @ A4 )
=> ( ( member @ A @ X @ A4 )
=> ( ( ( ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798350308766er_Min @ A @ A4 )
= X ) )
& ( ( ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798350308766er_Min @ A @ A4 )
= ( ord_min @ A @ X @ ( lattic643756798350308766er_Min @ A @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ) ).
% Min.remove
thf(fact_6908_Min_Oinsert__remove,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite @ A @ A4 )
=> ( ( ( ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798350308766er_Min @ A @ ( insert @ A @ X @ A4 ) )
= X ) )
& ( ( ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798350308766er_Min @ A @ ( insert @ A @ X @ A4 ) )
= ( ord_min @ A @ X @ ( lattic643756798350308766er_Min @ A @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ).
% Min.insert_remove
thf(fact_6909_upt__Suc,axiom,
! [I: nat,J2: nat] :
( ( ( ord_less_eq @ nat @ I @ J2 )
=> ( ( upt @ I @ ( suc @ J2 ) )
= ( append @ nat @ ( upt @ I @ J2 ) @ ( cons @ nat @ J2 @ ( nil @ nat ) ) ) ) )
& ( ~ ( ord_less_eq @ nat @ I @ J2 )
=> ( ( upt @ I @ ( suc @ J2 ) )
= ( nil @ nat ) ) ) ) ).
% upt_Suc
thf(fact_6910_upt__Suc__append,axiom,
! [I: nat,J2: nat] :
( ( ord_less_eq @ nat @ I @ J2 )
=> ( ( upt @ I @ ( suc @ J2 ) )
= ( append @ nat @ ( upt @ I @ J2 ) @ ( cons @ nat @ J2 @ ( nil @ nat ) ) ) ) ) ).
% upt_Suc_append
thf(fact_6911_horner__sum__bit__eq__take__bit,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A3: A,N: nat] :
( ( groups4207007520872428315er_sum @ $o @ A @ ( zero_neq_one_of_bool @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( map @ nat @ $o @ ( bit_se5641148757651400278ts_bit @ A @ A3 ) @ ( upt @ ( zero_zero @ nat ) @ N ) ) )
= ( bit_se2584673776208193580ke_bit @ A @ N @ A3 ) ) ) ).
% horner_sum_bit_eq_take_bit
thf(fact_6912_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_6913_length__map,axiom,
! [A: $tType,B: $tType,F2: B > A,Xs: list @ B] :
( ( size_size @ ( list @ A ) @ ( map @ B @ A @ F2 @ Xs ) )
= ( size_size @ ( list @ B ) @ Xs ) ) ).
% length_map
thf(fact_6914_sum__list__0,axiom,
! [B: $tType,A: $tType] :
( ( monoid_add @ A )
=> ! [Xs: list @ B] :
( ( groups8242544230860333062m_list @ A
@ ( map @ B @ A
@ ^ [X2: B] : ( zero_zero @ A )
@ Xs ) )
= ( zero_zero @ A ) ) ) ).
% sum_list_0
thf(fact_6915_nth__map,axiom,
! [B: $tType,A: $tType,N: nat,Xs: list @ A,F2: A > B] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ B @ ( map @ A @ B @ F2 @ Xs ) @ N )
= ( F2 @ ( nth @ A @ Xs @ N ) ) ) ) ).
% nth_map
thf(fact_6916_map__fst__zip,axiom,
! [B: $tType,A: $tType,Xs: list @ A,Ys2: list @ B] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys2 ) )
=> ( ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys2 ) )
= Xs ) ) ).
% map_fst_zip
thf(fact_6917_map__snd__zip,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys2: list @ B] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys2 ) )
=> ( ( map @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys2 ) )
= Ys2 ) ) ).
% map_snd_zip
thf(fact_6918_map__eq__imp__length__eq,axiom,
! [A: $tType,B: $tType,C: $tType,F2: B > A,Xs: list @ B,G: C > A,Ys2: list @ C] :
( ( ( map @ B @ A @ F2 @ Xs )
= ( map @ C @ A @ G @ Ys2 ) )
=> ( ( size_size @ ( list @ B ) @ Xs )
= ( size_size @ ( list @ C ) @ Ys2 ) ) ) ).
% map_eq_imp_length_eq
thf(fact_6919_map__replicate__const,axiom,
! [B: $tType,A: $tType,K: A,Lst: list @ B] :
( ( map @ B @ A
@ ^ [X2: B] : K
@ Lst )
= ( replicate @ A @ ( size_size @ ( list @ B ) @ Lst ) @ K ) ) ).
% map_replicate_const
thf(fact_6920_sum__list__subtractf,axiom,
! [A: $tType,B: $tType] :
( ( ab_group_add @ A )
=> ! [F2: B > A,G: B > A,Xs: list @ B] :
( ( groups8242544230860333062m_list @ A
@ ( map @ B @ A
@ ^ [X2: B] : ( minus_minus @ A @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ Xs ) )
= ( minus_minus @ A @ ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ F2 @ Xs ) ) @ ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ G @ Xs ) ) ) ) ) ).
% sum_list_subtractf
thf(fact_6921_uminus__sum__list__map,axiom,
! [A: $tType,B: $tType] :
( ( ab_group_add @ A )
=> ! [F2: B > A,Xs: list @ B] :
( ( uminus_uminus @ A @ ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ F2 @ Xs ) ) )
= ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ ( comp @ A @ A @ B @ ( uminus_uminus @ A ) @ F2 ) @ Xs ) ) ) ) ).
% uminus_sum_list_map
thf(fact_6922_map__replicate__trivial,axiom,
! [A: $tType,X: A,I: nat] :
( ( map @ nat @ A
@ ^ [I2: nat] : X
@ ( upt @ ( zero_zero @ nat ) @ I ) )
= ( replicate @ A @ I @ X ) ) ).
% map_replicate_trivial
thf(fact_6923_sum__list__strict__mono,axiom,
! [B: $tType,A: $tType] :
( ( ( monoid_add @ B )
& ( strict9044650504122735259up_add @ B ) )
=> ! [Xs: list @ A,F2: A > B,G: A > B] :
( ( Xs
!= ( nil @ A ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( ord_less @ B @ ( F2 @ X3 ) @ ( G @ X3 ) ) )
=> ( ord_less @ B @ ( groups8242544230860333062m_list @ B @ ( map @ A @ B @ F2 @ Xs ) ) @ ( groups8242544230860333062m_list @ B @ ( map @ A @ B @ G @ Xs ) ) ) ) ) ) ).
% sum_list_strict_mono
thf(fact_6924_zip__eq__conv,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys2: list @ B,Zs: list @ ( product_prod @ A @ B )] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys2 ) )
=> ( ( ( zip @ A @ B @ Xs @ Ys2 )
= Zs )
= ( ( ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Zs )
= Xs )
& ( ( map @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ Zs )
= Ys2 ) ) ) ) ).
% zip_eq_conv
thf(fact_6925_sum__list__triv,axiom,
! [C: $tType,B: $tType] :
( ( semiring_1 @ B )
=> ! [R4: B,Xs: list @ C] :
( ( groups8242544230860333062m_list @ B
@ ( map @ C @ B
@ ^ [X2: C] : R4
@ Xs ) )
= ( times_times @ B @ ( semiring_1_of_nat @ B @ ( size_size @ ( list @ C ) @ Xs ) ) @ R4 ) ) ) ).
% sum_list_triv
thf(fact_6926_map__upt__Suc,axiom,
! [A: $tType,F2: nat > A,N: nat] :
( ( map @ nat @ A @ F2 @ ( upt @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
= ( cons @ A @ ( F2 @ ( zero_zero @ nat ) )
@ ( map @ nat @ A
@ ^ [I2: nat] : ( F2 @ ( suc @ I2 ) )
@ ( upt @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% map_upt_Suc
thf(fact_6927_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_6928_size__list__conv__sum__list,axiom,
! [B: $tType] :
( ( size_list @ B )
= ( ^ [F4: B > nat,Xs3: list @ B] : ( plus_plus @ nat @ ( groups8242544230860333062m_list @ nat @ ( map @ B @ nat @ F4 @ Xs3 ) ) @ ( size_size @ ( list @ B ) @ Xs3 ) ) ) ) ).
% size_list_conv_sum_list
thf(fact_6929_sum__list__Suc,axiom,
! [A: $tType,F2: A > nat,Xs: list @ A] :
( ( groups8242544230860333062m_list @ nat
@ ( map @ A @ nat
@ ^ [X2: A] : ( suc @ ( F2 @ X2 ) )
@ Xs ) )
= ( plus_plus @ nat @ ( groups8242544230860333062m_list @ nat @ ( map @ A @ nat @ F2 @ Xs ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ).
% sum_list_Suc
thf(fact_6930_nth__map__upt,axiom,
! [A: $tType,I: nat,N: nat,M2: nat,F2: nat > A] :
( ( ord_less @ nat @ I @ ( minus_minus @ nat @ N @ M2 ) )
=> ( ( nth @ A @ ( map @ nat @ A @ F2 @ ( upt @ M2 @ N ) ) @ I )
= ( F2 @ ( plus_plus @ nat @ M2 @ I ) ) ) ) ).
% nth_map_upt
thf(fact_6931_map__of__zip__map,axiom,
! [B: $tType,A: $tType,Xs: list @ A,F2: A > B] :
( ( map_of @ A @ B @ ( zip @ A @ B @ Xs @ ( map @ A @ B @ F2 @ Xs ) ) )
= ( ^ [X2: A] : ( if @ ( option @ B ) @ ( member @ A @ X2 @ ( set2 @ A @ Xs ) ) @ ( some @ B @ ( F2 @ X2 ) ) @ ( none @ B ) ) ) ) ).
% map_of_zip_map
thf(fact_6932_map__fst__zip__take,axiom,
! [B: $tType,A: $tType,Xs: list @ A,Ys2: list @ B] :
( ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys2 ) )
= ( take @ A @ ( ord_min @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ B ) @ Ys2 ) ) @ Xs ) ) ).
% map_fst_zip_take
thf(fact_6933_map__snd__zip__take,axiom,
! [B: $tType,A: $tType,Xs: list @ B,Ys2: list @ A] :
( ( map @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A ) @ ( zip @ B @ A @ Xs @ Ys2 ) )
= ( take @ A @ ( ord_min @ nat @ ( size_size @ ( list @ B ) @ Xs ) @ ( size_size @ ( list @ A ) @ Ys2 ) ) @ Ys2 ) ) ).
% map_snd_zip_take
thf(fact_6934_map__upt__eqI,axiom,
! [A: $tType,Xs: list @ A,N: nat,M2: nat,F2: nat > A] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( minus_minus @ nat @ N @ M2 ) )
=> ( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ A @ Xs @ I3 )
= ( F2 @ ( plus_plus @ nat @ M2 @ I3 ) ) ) )
=> ( ( map @ nat @ A @ F2 @ ( upt @ M2 @ N ) )
= Xs ) ) ) ).
% map_upt_eqI
thf(fact_6935_transpose__rectangle,axiom,
! [A: $tType,Xs: list @ ( list @ A ),N: nat] :
( ( ( Xs
= ( nil @ ( list @ A ) ) )
=> ( N
= ( zero_zero @ nat ) ) )
=> ( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ ( list @ A ) ) @ Xs ) )
=> ( ( size_size @ ( list @ A ) @ ( nth @ ( list @ A ) @ Xs @ I3 ) )
= N ) )
=> ( ( transpose @ A @ Xs )
= ( map @ nat @ ( list @ A )
@ ^ [I2: nat] :
( map @ nat @ A
@ ^ [J3: nat] : ( nth @ A @ ( nth @ ( list @ A ) @ Xs @ J3 ) @ I2 )
@ ( upt @ ( zero_zero @ nat ) @ ( size_size @ ( list @ ( list @ A ) ) @ Xs ) ) )
@ ( upt @ ( zero_zero @ nat ) @ N ) ) ) ) ) ).
% transpose_rectangle
thf(fact_6936_sorted__wrt__less__sum__mono__lowerbound,axiom,
! [B: $tType] :
( ( ordere6911136660526730532id_add @ B )
=> ! [F2: nat > B,Ns: list @ nat] :
( ! [X3: nat,Y3: nat] :
( ( ord_less_eq @ nat @ X3 @ Y3 )
=> ( ord_less_eq @ B @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( ( sorted_wrt @ nat @ ( ord_less @ nat ) @ Ns )
=> ( ord_less_eq @ B @ ( groups7311177749621191930dd_sum @ nat @ B @ F2 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ nat ) @ Ns ) ) ) @ ( groups8242544230860333062m_list @ B @ ( map @ nat @ B @ F2 @ Ns ) ) ) ) ) ) ).
% sorted_wrt_less_sum_mono_lowerbound
thf(fact_6937_map__Suc__upt,axiom,
! [M2: nat,N: nat] :
( ( map @ nat @ nat @ suc @ ( upt @ M2 @ N ) )
= ( upt @ ( suc @ M2 ) @ ( suc @ N ) ) ) ).
% map_Suc_upt
thf(fact_6938_strict__sorted__equal,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,Ys2: list @ A] :
( ( sorted_wrt @ A @ ( ord_less @ A ) @ Xs )
=> ( ( sorted_wrt @ A @ ( ord_less @ A ) @ Ys2 )
=> ( ( ( set2 @ A @ Ys2 )
= ( set2 @ A @ Xs ) )
=> ( Ys2 = Xs ) ) ) ) ) ).
% strict_sorted_equal
thf(fact_6939_strict__sorted__simps_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Ys2: list @ A] :
( ( sorted_wrt @ A @ ( ord_less @ A ) @ ( cons @ A @ X @ Ys2 ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ ( set2 @ A @ Ys2 ) )
=> ( ord_less @ A @ X @ X2 ) )
& ( sorted_wrt @ A @ ( ord_less @ A ) @ Ys2 ) ) ) ) ).
% strict_sorted_simps(2)
thf(fact_6940_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 )
@ ^ [Y5: A] : ( cons @ A @ Y5 @ Ys3 )
@ Xs )
@ ( n_lists @ A @ N @ Xs ) ) ) ) ).
% n_lists.simps(2)
thf(fact_6941_sorted__wrt01,axiom,
! [A: $tType,Xs: list @ A,P: A > A > $o] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) )
=> ( sorted_wrt @ A @ P @ Xs ) ) ).
% sorted_wrt01
thf(fact_6942_sorted__wrt__iff__nth__less,axiom,
! [A: $tType] :
( ( sorted_wrt @ A )
= ( ^ [P3: A > A > $o,Xs3: list @ A] :
! [I2: nat,J3: nat] :
( ( ord_less @ nat @ I2 @ J3 )
=> ( ( ord_less @ nat @ J3 @ ( size_size @ ( list @ A ) @ Xs3 ) )
=> ( P3 @ ( nth @ A @ Xs3 @ I2 ) @ ( nth @ A @ Xs3 @ J3 ) ) ) ) ) ) ).
% sorted_wrt_iff_nth_less
thf(fact_6943_sorted__wrt__nth__less,axiom,
! [A: $tType,P: A > A > $o,Xs: list @ A,I: nat,J2: nat] :
( ( sorted_wrt @ A @ P @ Xs )
=> ( ( ord_less @ nat @ I @ J2 )
=> ( ( ord_less @ nat @ J2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( P @ ( nth @ A @ Xs @ I ) @ ( nth @ A @ Xs @ J2 ) ) ) ) ) ).
% sorted_wrt_nth_less
thf(fact_6944_strict__sorted__simps_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( sorted_wrt @ A @ ( ord_less @ A ) @ ( nil @ A ) ) ) ).
% strict_sorted_simps(1)
thf(fact_6945_sorted__list__of__set_Ostrict__sorted__key__list__of__set,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A] : ( sorted_wrt @ A @ ( ord_less @ A ) @ ( linord4507533701916653071of_set @ A @ A4 ) ) ) ).
% sorted_list_of_set.strict_sorted_key_list_of_set
thf(fact_6946_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_6947_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_6948_sorted__wrt__upt,axiom,
! [M2: nat,N: nat] : ( sorted_wrt @ nat @ ( ord_less @ nat ) @ ( upt @ M2 @ N ) ) ).
% sorted_wrt_upt
thf(fact_6949_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_6950_map__add__upt,axiom,
! [N: nat,M2: nat] :
( ( map @ nat @ nat
@ ^ [I2: nat] : ( plus_plus @ nat @ I2 @ N )
@ ( upt @ ( zero_zero @ nat ) @ M2 ) )
= ( upt @ N @ ( plus_plus @ nat @ M2 @ N ) ) ) ).
% map_add_upt
thf(fact_6951_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_6952_sorted__iff__nth__mono__less,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
= ( ! [I2: nat,J3: nat] :
( ( ord_less @ nat @ I2 @ J3 )
=> ( ( ord_less @ nat @ J3 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ord_less_eq @ A @ ( nth @ A @ Xs @ I2 ) @ ( nth @ A @ Xs @ J3 ) ) ) ) ) ) ) ).
% sorted_iff_nth_mono_less
thf(fact_6953_map__decr__upt,axiom,
! [M2: nat,N: nat] :
( ( map @ nat @ nat
@ ^ [N5: nat] : ( minus_minus @ nat @ N5 @ ( suc @ ( zero_zero @ nat ) ) )
@ ( upt @ ( suc @ M2 ) @ ( suc @ N ) ) )
= ( upt @ M2 @ N ) ) ).
% map_decr_upt
thf(fact_6954_sorted__iff__nth__Suc,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
= ( ! [I2: nat] :
( ( ord_less @ nat @ ( suc @ I2 ) @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ord_less_eq @ A @ ( nth @ A @ Xs @ I2 ) @ ( nth @ A @ Xs @ ( suc @ I2 ) ) ) ) ) ) ) ).
% sorted_iff_nth_Suc
thf(fact_6955_sorted__list__of__set_Ofinite__set__strict__sorted,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A] :
( ( finite_finite @ A @ A4 )
=> ~ ! [L3: list @ A] :
( ( sorted_wrt @ A @ ( ord_less @ A ) @ L3 )
=> ( ( ( set2 @ A @ L3 )
= A4 )
=> ( ( size_size @ ( list @ A ) @ L3 )
!= ( finite_card @ A @ A4 ) ) ) ) ) ) ).
% sorted_list_of_set.finite_set_strict_sorted
thf(fact_6956_sorted__nth__mono,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,I: nat,J2: nat] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( ( ord_less_eq @ nat @ I @ J2 )
=> ( ( ord_less @ nat @ J2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ord_less_eq @ A @ ( nth @ A @ Xs @ I ) @ ( nth @ A @ Xs @ J2 ) ) ) ) ) ) ).
% sorted_nth_mono
thf(fact_6957_sorted__iff__nth__mono,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
= ( ! [I2: nat,J3: nat] :
( ( ord_less_eq @ nat @ I2 @ J3 )
=> ( ( ord_less @ nat @ J3 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ord_less_eq @ A @ ( nth @ A @ Xs @ I2 ) @ ( nth @ A @ Xs @ J3 ) ) ) ) ) ) ) ).
% sorted_iff_nth_mono
thf(fact_6958_sorted__wrt__less__idx,axiom,
! [Ns: list @ nat,I: nat] :
( ( sorted_wrt @ nat @ ( ord_less @ nat ) @ Ns )
=> ( ( ord_less @ nat @ I @ ( size_size @ ( list @ nat ) @ Ns ) )
=> ( ord_less_eq @ nat @ I @ ( nth @ nat @ Ns @ I ) ) ) ) ).
% sorted_wrt_less_idx
thf(fact_6959_sorted__list__of__set_Osorted__key__list__of__set__unique,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,L: list @ A] :
( ( finite_finite @ A @ A4 )
=> ( ( ( sorted_wrt @ A @ ( ord_less @ A ) @ L )
& ( ( set2 @ A @ L )
= A4 )
& ( ( size_size @ ( list @ A ) @ L )
= ( finite_card @ A @ A4 ) ) )
= ( ( linord4507533701916653071of_set @ A @ A4 )
= L ) ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_unique
thf(fact_6960_folding__insort__key_Ofinite__set__strict__sorted,axiom,
! [A: $tType,B: $tType,Less_eq: A > A > $o,Less: A > A > $o,S3: set @ B,F2: B > A,A4: set @ B] :
( ( folding_insort_key @ A @ B @ Less_eq @ Less @ S3 @ F2 )
=> ( ( ord_less_eq @ ( set @ B ) @ A4 @ S3 )
=> ( ( finite_finite @ B @ A4 )
=> ~ ! [L3: list @ B] :
( ( sorted_wrt @ A @ Less @ ( map @ B @ A @ F2 @ L3 ) )
=> ( ( ( set2 @ B @ L3 )
= A4 )
=> ( ( size_size @ ( list @ B ) @ L3 )
!= ( finite_card @ B @ A4 ) ) ) ) ) ) ) ).
% folding_insort_key.finite_set_strict_sorted
thf(fact_6961_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_6962_length__rev,axiom,
! [A: $tType,Xs: list @ A] :
( ( size_size @ ( list @ A ) @ ( rev @ A @ Xs ) )
= ( size_size @ ( list @ A ) @ Xs ) ) ).
% length_rev
thf(fact_6963_sorted__wrt__upto,axiom,
! [I: int,J2: int] : ( sorted_wrt @ int @ ( ord_less @ int ) @ ( upto @ I @ J2 ) ) ).
% sorted_wrt_upto
thf(fact_6964_zip__rev,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys2: list @ B] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys2 ) )
=> ( ( zip @ A @ B @ ( rev @ A @ Xs ) @ ( rev @ B @ Ys2 ) )
= ( rev @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys2 ) ) ) ) ).
% zip_rev
thf(fact_6965_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_6966_rev__take,axiom,
! [A: $tType,I: nat,Xs: list @ A] :
( ( rev @ A @ ( take @ A @ I @ Xs ) )
= ( drop @ A @ ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ I ) @ ( rev @ A @ Xs ) ) ) ).
% rev_take
thf(fact_6967_rev__drop,axiom,
! [A: $tType,I: nat,Xs: list @ A] :
( ( rev @ A @ ( drop @ A @ I @ Xs ) )
= ( take @ A @ ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ I ) @ ( rev @ A @ Xs ) ) ) ).
% rev_drop
thf(fact_6968_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_6969_rotate__rev,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( rotate @ A @ N @ ( rev @ A @ Xs ) )
= ( rev @ A @ ( rotate @ A @ ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( modulo_modulo @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) ) ) @ Xs ) ) ) ).
% rotate_rev
thf(fact_6970_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_6971_sorted__list__of__set_Ofolding__insort__key__axioms,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( folding_insort_key @ A @ A @ ( ord_less_eq @ A ) @ ( ord_less @ A ) @ ( top_top @ ( set @ A ) )
@ ^ [X2: A] : X2 ) ) ).
% sorted_list_of_set.folding_insort_key_axioms
thf(fact_6972_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_6973_rev__update,axiom,
! [A: $tType,K: nat,Xs: list @ A,Y2: A] :
( ( ord_less @ nat @ K @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( rev @ A @ ( list_update @ A @ Xs @ K @ Y2 ) )
= ( list_update @ A @ ( rev @ A @ Xs ) @ ( minus_minus @ nat @ ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ K ) @ ( one_one @ nat ) ) @ Y2 ) ) ) ).
% rev_update
thf(fact_6974_sorted__rev__iff__nth__Suc,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( rev @ A @ Xs ) )
= ( ! [I2: nat] :
( ( ord_less @ nat @ ( suc @ I2 ) @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ord_less_eq @ A @ ( nth @ A @ Xs @ ( suc @ I2 ) ) @ ( nth @ A @ Xs @ I2 ) ) ) ) ) ) ).
% sorted_rev_iff_nth_Suc
thf(fact_6975_sorted__rev__iff__nth__mono,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( rev @ A @ Xs ) )
= ( ! [I2: nat,J3: nat] :
( ( ord_less_eq @ nat @ I2 @ J3 )
=> ( ( ord_less @ nat @ J3 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ord_less_eq @ A @ ( nth @ A @ Xs @ J3 ) @ ( nth @ A @ Xs @ I2 ) ) ) ) ) ) ) ).
% sorted_rev_iff_nth_mono
thf(fact_6976_sorted__rev__nth__mono,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,I: nat,J2: nat] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( rev @ A @ Xs ) )
=> ( ( ord_less_eq @ nat @ I @ J2 )
=> ( ( ord_less @ nat @ J2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ord_less_eq @ A @ ( nth @ A @ Xs @ J2 ) @ ( nth @ A @ Xs @ I ) ) ) ) ) ) ).
% sorted_rev_nth_mono
thf(fact_6977_nth__nth__transpose__sorted,axiom,
! [A: $tType,Xs: list @ ( list @ A ),I: nat,J2: nat] :
( ( sorted_wrt @ nat @ ( ord_less_eq @ nat ) @ ( rev @ nat @ ( map @ ( list @ A ) @ nat @ ( size_size @ ( list @ A ) ) @ Xs ) ) )
=> ( ( ord_less @ nat @ I @ ( size_size @ ( list @ ( list @ A ) ) @ ( transpose @ A @ Xs ) ) )
=> ( ( ord_less @ nat @ J2
@ ( size_size @ ( list @ ( list @ A ) )
@ ( filter2 @ ( list @ A )
@ ^ [Ys3: list @ A] : ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Ys3 ) )
@ Xs ) ) )
=> ( ( nth @ A @ ( nth @ ( list @ A ) @ ( transpose @ A @ Xs ) @ I ) @ J2 )
= ( nth @ A @ ( nth @ ( list @ A ) @ Xs @ J2 ) @ I ) ) ) ) ) ).
% nth_nth_transpose_sorted
thf(fact_6978_transpose__column,axiom,
! [A: $tType,Xs: list @ ( list @ A ),I: nat] :
( ( sorted_wrt @ nat @ ( ord_less_eq @ nat ) @ ( rev @ nat @ ( map @ ( list @ A ) @ nat @ ( size_size @ ( list @ A ) ) @ Xs ) ) )
=> ( ( ord_less @ nat @ I @ ( size_size @ ( list @ ( list @ A ) ) @ Xs ) )
=> ( ( map @ ( list @ A ) @ A
@ ^ [Ys3: list @ A] : ( nth @ A @ Ys3 @ I )
@ ( filter2 @ ( list @ A )
@ ^ [Ys3: list @ A] : ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Ys3 ) )
@ ( transpose @ A @ Xs ) ) )
= ( nth @ ( list @ A ) @ Xs @ I ) ) ) ) ).
% transpose_column
thf(fact_6979_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_6980_length__filter__map,axiom,
! [A: $tType,B: $tType,P: A > $o,F2: B > A,Xs: list @ B] :
( ( size_size @ ( list @ A ) @ ( filter2 @ A @ P @ ( map @ B @ A @ F2 @ Xs ) ) )
= ( size_size @ ( list @ B ) @ ( filter2 @ B @ ( comp @ A @ $o @ B @ P @ F2 ) @ Xs ) ) ) ).
% length_filter_map
thf(fact_6981_length__filter__less,axiom,
! [A: $tType,X: A,Xs: list @ A,P: A > $o] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ~ ( P @ X )
=> ( ord_less @ nat @ ( size_size @ ( list @ A ) @ ( filter2 @ A @ P @ Xs ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ).
% length_filter_less
thf(fact_6982_replicate__length__filter,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( replicate @ A
@ ( size_size @ ( list @ A )
@ ( filter2 @ A
@ ( ^ [Y4: A,Z: A] : Y4 = Z
@ X )
@ Xs ) )
@ X )
= ( filter2 @ A
@ ( ^ [Y4: A,Z: A] : Y4 = Z
@ X )
@ Xs ) ) ).
% replicate_length_filter
thf(fact_6983_length__filter__le,axiom,
! [A: $tType,P: A > $o,Xs: list @ A] : ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ ( filter2 @ A @ P @ Xs ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) ).
% length_filter_le
thf(fact_6984_sum__length__filter__compl,axiom,
! [A: $tType,P: A > $o,Xs: list @ A] :
( ( plus_plus @ nat @ ( size_size @ ( list @ A ) @ ( filter2 @ A @ P @ Xs ) )
@ ( size_size @ ( list @ A )
@ ( filter2 @ A
@ ^ [X2: A] :
~ ( P @ X2 )
@ Xs ) ) )
= ( size_size @ ( list @ A ) @ Xs ) ) ).
% sum_length_filter_compl
thf(fact_6985_sum__list__map__filter_H,axiom,
! [A: $tType,B: $tType] :
( ( monoid_add @ A )
=> ! [F2: B > A,P: B > $o,Xs: list @ B] :
( ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ F2 @ ( filter2 @ B @ P @ Xs ) ) )
= ( groups8242544230860333062m_list @ A
@ ( map @ B @ A
@ ^ [X2: B] : ( if @ A @ ( P @ X2 ) @ ( F2 @ X2 ) @ ( zero_zero @ A ) )
@ Xs ) ) ) ) ).
% sum_list_map_filter'
thf(fact_6986_sum__list__map__filter,axiom,
! [A: $tType,B: $tType] :
( ( monoid_add @ A )
=> ! [Xs: list @ B,P: B > $o,F2: B > A] :
( ! [X3: B] :
( ( member @ B @ X3 @ ( set2 @ B @ Xs ) )
=> ( ~ ( P @ X3 )
=> ( ( F2 @ X3 )
= ( zero_zero @ A ) ) ) )
=> ( ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ F2 @ ( filter2 @ B @ P @ Xs ) ) )
= ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ F2 @ Xs ) ) ) ) ) ).
% sum_list_map_filter
thf(fact_6987_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
@ ^ [X2: A] : X2 != Y2
@ Xs ) ) ) ).
% set_minus_filter_out
thf(fact_6988_filter__eq__nths,axiom,
! [A: $tType] :
( ( filter2 @ A )
= ( ^ [P3: A > $o,Xs3: list @ A] :
( nths @ A @ Xs3
@ ( collect @ nat
@ ^ [I2: nat] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs3 ) )
& ( P3 @ ( nth @ A @ Xs3 @ I2 ) ) ) ) ) ) ) ).
% filter_eq_nths
thf(fact_6989_length__filter__conv__card,axiom,
! [A: $tType,P4: A > $o,Xs: list @ A] :
( ( size_size @ ( list @ A ) @ ( filter2 @ A @ P4 @ Xs ) )
= ( finite_card @ nat
@ ( collect @ nat
@ ^ [I2: nat] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs ) )
& ( P4 @ ( nth @ A @ Xs @ I2 ) ) ) ) ) ) ).
% length_filter_conv_card
thf(fact_6990_distinct__length__filter,axiom,
! [A: $tType,Xs: list @ A,P: A > $o] :
( ( distinct @ A @ Xs )
=> ( ( size_size @ ( list @ A ) @ ( filter2 @ A @ P @ Xs ) )
= ( finite_card @ A @ ( inf_inf @ ( set @ A ) @ ( collect @ A @ P ) @ ( set2 @ A @ Xs ) ) ) ) ) ).
% distinct_length_filter
thf(fact_6991_nth__transpose,axiom,
! [A: $tType,I: nat,Xs: list @ ( list @ A )] :
( ( ord_less @ nat @ I @ ( size_size @ ( list @ ( list @ A ) ) @ ( transpose @ A @ Xs ) ) )
=> ( ( nth @ ( list @ A ) @ ( transpose @ A @ Xs ) @ I )
= ( map @ ( list @ A ) @ A
@ ^ [Xs3: list @ A] : ( nth @ A @ Xs3 @ I )
@ ( filter2 @ ( list @ A )
@ ^ [Ys3: list @ A] : ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Ys3 ) )
@ Xs ) ) ) ) ).
% nth_transpose
thf(fact_6992_transpose__column__length,axiom,
! [A: $tType,Xs: list @ ( list @ A ),I: nat] :
( ( sorted_wrt @ nat @ ( ord_less_eq @ nat ) @ ( rev @ nat @ ( map @ ( list @ A ) @ nat @ ( size_size @ ( list @ A ) ) @ Xs ) ) )
=> ( ( ord_less @ nat @ I @ ( size_size @ ( list @ ( list @ A ) ) @ Xs ) )
=> ( ( size_size @ ( list @ ( list @ A ) )
@ ( filter2 @ ( list @ A )
@ ^ [Ys3: list @ A] : ( ord_less @ nat @ I @ ( size_size @ ( list @ A ) @ Ys3 ) )
@ ( transpose @ A @ Xs ) ) )
= ( size_size @ ( list @ A ) @ ( nth @ ( list @ A ) @ Xs @ I ) ) ) ) ) ).
% transpose_column_length
thf(fact_6993_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,F2: B > A,A4: set @ B,L: list @ B] :
( ( folding_insort_key @ A @ B @ Less_eq @ Less @ S3 @ F2 )
=> ( ( ord_less_eq @ ( set @ B ) @ A4 @ S3 )
=> ( ( finite_finite @ B @ A4 )
=> ( ( ( sorted_wrt @ A @ Less @ ( map @ B @ A @ F2 @ L ) )
& ( ( set2 @ B @ L )
= A4 )
& ( ( size_size @ ( list @ B ) @ L )
= ( finite_card @ B @ A4 ) ) )
= ( ( sorted8670434370408473282of_set @ A @ B @ Less_eq @ F2 @ A4 )
= L ) ) ) ) ) ).
% folding_insort_key.sorted_key_list_of_set_unique
thf(fact_6994_transpose__max__length,axiom,
! [A: $tType,Xs: list @ ( list @ A )] :
( ( foldr @ ( list @ A ) @ nat
@ ^ [Xs3: list @ A] : ( ord_max @ nat @ ( size_size @ ( list @ A ) @ Xs3 ) )
@ ( transpose @ A @ Xs )
@ ( zero_zero @ nat ) )
= ( size_size @ ( list @ ( list @ A ) )
@ ( filter2 @ ( list @ A )
@ ^ [X2: list @ A] :
( X2
!= ( nil @ A ) )
@ Xs ) ) ) ).
% transpose_max_length
thf(fact_6995_sum__list_Oeq__foldr,axiom,
! [A: $tType] :
( ( monoid_add @ A )
=> ( ( groups8242544230860333062m_list @ A )
= ( ^ [Xs3: list @ A] : ( foldr @ A @ A @ ( plus_plus @ A ) @ Xs3 @ ( zero_zero @ A ) ) ) ) ) ).
% sum_list.eq_foldr
thf(fact_6996_nths__shift__lemma__Suc,axiom,
! [A: $tType,P: nat > $o,Xs: list @ A,Is: list @ nat] :
( ( map @ ( product_prod @ A @ nat ) @ A @ ( product_fst @ A @ nat )
@ ( filter2 @ ( product_prod @ A @ nat )
@ ^ [P6: product_prod @ A @ nat] : ( P @ ( suc @ ( product_snd @ A @ nat @ P6 ) ) )
@ ( zip @ A @ nat @ Xs @ Is ) ) )
= ( map @ ( product_prod @ A @ nat ) @ A @ ( product_fst @ A @ nat )
@ ( filter2 @ ( product_prod @ A @ nat )
@ ^ [P6: product_prod @ A @ nat] : ( P @ ( product_snd @ A @ nat @ P6 ) )
@ ( zip @ A @ nat @ Xs @ ( map @ nat @ nat @ suc @ Is ) ) ) ) ) ).
% nths_shift_lemma_Suc
thf(fact_6997_horner__sum__foldr,axiom,
! [A: $tType,B: $tType] :
( ( comm_semiring_0 @ A )
=> ( ( groups4207007520872428315er_sum @ B @ A )
= ( ^ [F4: B > A,A5: A,Xs3: list @ B] :
( foldr @ B @ A
@ ^ [X2: B,B3: A] : ( plus_plus @ A @ ( F4 @ X2 ) @ ( times_times @ A @ A5 @ B3 ) )
@ Xs3
@ ( zero_zero @ A ) ) ) ) ) ).
% horner_sum_foldr
thf(fact_6998_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,F2: B > A,A4: set @ B] :
( ( folding_insort_key @ A @ B @ Less_eq @ Less @ S3 @ F2 )
=> ( ( ord_less_eq @ ( set @ B ) @ A4 @ S3 )
=> ( ( size_size @ ( list @ B ) @ ( sorted8670434370408473282of_set @ A @ B @ Less_eq @ F2 @ A4 ) )
= ( finite_card @ B @ A4 ) ) ) ) ).
% folding_insort_key.length_sorted_key_list_of_set
thf(fact_6999_nths__shift__lemma,axiom,
! [A: $tType,A4: set @ nat,Xs: list @ A,I: nat] :
( ( map @ ( product_prod @ A @ nat ) @ A @ ( product_fst @ A @ nat )
@ ( filter2 @ ( product_prod @ A @ nat )
@ ^ [P6: product_prod @ A @ nat] : ( member @ nat @ ( product_snd @ A @ nat @ P6 ) @ A4 )
@ ( zip @ A @ nat @ Xs @ ( upt @ I @ ( plus_plus @ nat @ I @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ) )
= ( map @ ( product_prod @ A @ nat ) @ A @ ( product_fst @ A @ nat )
@ ( filter2 @ ( product_prod @ A @ nat )
@ ^ [P6: product_prod @ A @ nat] : ( member @ nat @ ( plus_plus @ nat @ ( product_snd @ A @ nat @ P6 ) @ I ) @ A4 )
@ ( zip @ A @ nat @ Xs @ ( upt @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ) ) ).
% nths_shift_lemma
thf(fact_7000_nths__def,axiom,
! [A: $tType] :
( ( nths @ A )
= ( ^ [Xs3: list @ A,A7: set @ nat] :
( map @ ( product_prod @ A @ nat ) @ A @ ( product_fst @ A @ nat )
@ ( filter2 @ ( product_prod @ A @ nat )
@ ^ [P6: product_prod @ A @ nat] : ( member @ nat @ ( product_snd @ A @ nat @ P6 ) @ A7 )
@ ( zip @ A @ nat @ Xs3 @ ( upt @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Xs3 ) ) ) ) ) ) ) ).
% nths_def
thf(fact_7001_length__transpose,axiom,
! [A: $tType,Xs: list @ ( list @ A )] :
( ( size_size @ ( list @ ( list @ A ) ) @ ( transpose @ A @ Xs ) )
= ( foldr @ ( list @ A ) @ nat
@ ^ [Xs3: list @ A] : ( ord_max @ nat @ ( size_size @ ( list @ A ) @ Xs3 ) )
@ Xs
@ ( zero_zero @ nat ) ) ) ).
% length_transpose
thf(fact_7002_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_7003_transpose__aux__max,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Xss: list @ ( list @ B )] :
( ( ord_max @ nat @ ( suc @ ( size_size @ ( list @ A ) @ Xs ) )
@ ( foldr @ ( list @ B ) @ nat
@ ^ [Xs3: list @ B] : ( ord_max @ nat @ ( size_size @ ( list @ B ) @ Xs3 ) )
@ Xss
@ ( zero_zero @ nat ) ) )
= ( suc
@ ( ord_max @ nat @ ( size_size @ ( list @ A ) @ Xs )
@ ( foldr @ ( list @ B ) @ nat
@ ^ [X2: list @ B] : ( ord_max @ nat @ ( minus_minus @ nat @ ( size_size @ ( list @ B ) @ X2 ) @ ( suc @ ( zero_zero @ nat ) ) ) )
@ ( filter2 @ ( list @ B )
@ ^ [Ys3: list @ B] :
( Ys3
!= ( nil @ B ) )
@ Xss )
@ ( zero_zero @ nat ) ) ) ) ) ).
% transpose_aux_max
thf(fact_7004_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,F2: B > A,X: B,A4: set @ B] :
( ( folding_insort_key @ A @ B @ Less_eq @ Less @ S3 @ F2 )
=> ( ( ord_less_eq @ ( set @ B ) @ ( insert @ B @ X @ A4 ) @ S3 )
=> ( ( finite_finite @ B @ A4 )
=> ( ( sorted8670434370408473282of_set @ A @ B @ Less_eq @ F2 @ ( minus_minus @ ( set @ B ) @ A4 @ ( insert @ B @ X @ ( bot_bot @ ( set @ B ) ) ) ) )
= ( remove1 @ B @ X @ ( sorted8670434370408473282of_set @ A @ B @ Less_eq @ F2 @ A4 ) ) ) ) ) ) ).
% folding_insort_key.sorted_key_list_of_set_remove
thf(fact_7005_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,F2: B > A,X: B,A4: set @ B] :
( ( folding_insort_key @ A @ B @ Less_eq @ Less @ S3 @ F2 )
=> ( ( ord_less_eq @ ( set @ B ) @ ( insert @ B @ X @ A4 ) @ S3 )
=> ( ( finite_finite @ B @ A4 )
=> ( ( sorted8670434370408473282of_set @ A @ B @ Less_eq @ F2 @ ( insert @ B @ X @ A4 ) )
= ( insort_key @ A @ B @ Less_eq @ F2 @ X @ ( sorted8670434370408473282of_set @ A @ B @ Less_eq @ F2 @ ( minus_minus @ ( set @ B ) @ A4 @ ( insert @ B @ X @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ) ).
% folding_insort_key.sorted_key_list_of_set_insert_remove
thf(fact_7006_map__filter__map__filter,axiom,
! [A: $tType,B: $tType,F2: B > A,P: B > $o,Xs: list @ B] :
( ( map @ B @ A @ F2 @ ( filter2 @ B @ P @ Xs ) )
= ( map_filter @ B @ A
@ ^ [X2: B] : ( if @ ( option @ A ) @ ( P @ X2 ) @ ( some @ A @ ( F2 @ X2 ) ) @ ( none @ A ) )
@ Xs ) ) ).
% map_filter_map_filter
thf(fact_7007_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_7008_transpose__transpose,axiom,
! [A: $tType,Xs: list @ ( list @ A )] :
( ( sorted_wrt @ nat @ ( ord_less_eq @ nat ) @ ( rev @ nat @ ( map @ ( list @ A ) @ nat @ ( size_size @ ( list @ A ) ) @ Xs ) ) )
=> ( ( transpose @ A @ ( transpose @ A @ Xs ) )
= ( takeWhile @ ( list @ A )
@ ^ [X2: list @ A] :
( X2
!= ( nil @ A ) )
@ Xs ) ) ) ).
% transpose_transpose
thf(fact_7009_map__filter__def,axiom,
! [B: $tType,A: $tType] :
( ( map_filter @ A @ B )
= ( ^ [F4: A > ( option @ B ),Xs3: list @ A] :
( map @ A @ B @ ( comp @ ( option @ B ) @ B @ A @ ( the2 @ B ) @ F4 )
@ ( filter2 @ A
@ ^ [X2: A] :
( ( F4 @ X2 )
!= ( none @ B ) )
@ Xs3 ) ) ) ) ).
% map_filter_def
thf(fact_7010_option_Ocollapse,axiom,
! [A: $tType,Option: option @ A] :
( ( Option
!= ( none @ A ) )
=> ( ( some @ A @ ( the2 @ A @ Option ) )
= Option ) ) ).
% option.collapse
thf(fact_7011_comp__the__Some,axiom,
! [A: $tType] :
( ( comp @ ( option @ A ) @ A @ A @ ( the2 @ A ) @ ( some @ A ) )
= ( id @ A ) ) ).
% comp_the_Some
thf(fact_7012_takeWhile__eq__take,axiom,
! [A: $tType] :
( ( takeWhile @ A )
= ( ^ [P3: A > $o,Xs3: list @ A] : ( take @ A @ ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P3 @ Xs3 ) ) @ Xs3 ) ) ) ).
% takeWhile_eq_take
thf(fact_7013_length__takeWhile__le,axiom,
! [A: $tType,P: A > $o,Xs: list @ A] : ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P @ Xs ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) ).
% length_takeWhile_le
thf(fact_7014_option_Oexhaust__sel,axiom,
! [A: $tType,Option: option @ A] :
( ( Option
!= ( none @ A ) )
=> ( Option
= ( some @ A @ ( the2 @ A @ Option ) ) ) ) ).
% option.exhaust_sel
thf(fact_7015_option_Osel,axiom,
! [A: $tType,X23: A] :
( ( the2 @ A @ ( some @ A @ X23 ) )
= X23 ) ).
% option.sel
thf(fact_7016_option_Oexpand,axiom,
! [A: $tType,Option: option @ A,Option2: option @ A] :
( ( ( Option
= ( none @ A ) )
= ( Option2
= ( none @ A ) ) )
=> ( ( ( Option
!= ( none @ A ) )
=> ( ( Option2
!= ( none @ A ) )
=> ( ( the2 @ A @ Option )
= ( the2 @ A @ Option2 ) ) ) )
=> ( Option = Option2 ) ) ) ).
% option.expand
thf(fact_7017_option_Omap__sel,axiom,
! [B: $tType,A: $tType,A3: option @ A,F2: A > B] :
( ( A3
!= ( none @ A ) )
=> ( ( the2 @ B @ ( map_option @ A @ B @ F2 @ A3 ) )
= ( F2 @ ( the2 @ A @ A3 ) ) ) ) ).
% option.map_sel
thf(fact_7018_option_Ocase__eq__if,axiom,
! [A: $tType,B: $tType] :
( ( case_option @ B @ A )
= ( ^ [F12: B,F23: A > B,Option3: option @ A] :
( if @ B
@ ( Option3
= ( none @ A ) )
@ F12
@ ( F23 @ ( the2 @ A @ Option3 ) ) ) ) ) ).
% option.case_eq_if
thf(fact_7019_takeWhile__nth,axiom,
! [A: $tType,J2: nat,P: A > $o,Xs: list @ A] :
( ( ord_less @ nat @ J2 @ ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P @ Xs ) ) )
=> ( ( nth @ A @ ( takeWhile @ A @ P @ Xs ) @ J2 )
= ( nth @ A @ Xs @ J2 ) ) ) ).
% takeWhile_nth
thf(fact_7020_nth__length__takeWhile,axiom,
! [A: $tType,P: A > $o,Xs: list @ A] :
( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P @ Xs ) ) @ ( size_size @ ( list @ A ) @ Xs ) )
=> ~ ( P @ ( nth @ A @ Xs @ ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P @ Xs ) ) ) ) ) ).
% nth_length_takeWhile
thf(fact_7021_Option_Othese__def,axiom,
! [A: $tType] :
( ( these @ A )
= ( ^ [A7: set @ ( option @ A )] :
( image @ ( option @ A ) @ A @ ( the2 @ A )
@ ( collect @ ( option @ A )
@ ^ [X2: option @ A] :
( ( member @ ( option @ A ) @ X2 @ A7 )
& ( X2
!= ( none @ A ) ) ) ) ) ) ) ).
% Option.these_def
thf(fact_7022_Min_Oinfinite,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A] :
( ~ ( finite_finite @ A @ A4 )
=> ( ( lattic643756798350308766er_Min @ A @ A4 )
= ( the2 @ A @ ( none @ A ) ) ) ) ) ).
% Min.infinite
thf(fact_7023_option_Osplit__sel__asm,axiom,
! [B: $tType,A: $tType,P: B > $o,F1: B,F22: A > B,Option: option @ A] :
( ( P @ ( case_option @ B @ A @ F1 @ F22 @ Option ) )
= ( ~ ( ( ( Option
= ( none @ A ) )
& ~ ( P @ F1 ) )
| ( ( Option
= ( some @ A @ ( the2 @ A @ Option ) ) )
& ~ ( P @ ( F22 @ ( the2 @ A @ Option ) ) ) ) ) ) ) ).
% option.split_sel_asm
thf(fact_7024_option_Osplit__sel,axiom,
! [B: $tType,A: $tType,P: B > $o,F1: B,F22: A > B,Option: option @ A] :
( ( P @ ( case_option @ B @ A @ F1 @ F22 @ Option ) )
= ( ( ( Option
= ( none @ A ) )
=> ( P @ F1 ) )
& ( ( Option
= ( some @ A @ ( the2 @ A @ Option ) ) )
=> ( P @ ( F22 @ ( the2 @ A @ Option ) ) ) ) ) ) ).
% option.split_sel
thf(fact_7025_length__takeWhile__less__P__nth,axiom,
! [A: $tType,J2: nat,P: A > $o,Xs: list @ A] :
( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ J2 )
=> ( P @ ( nth @ A @ Xs @ I3 ) ) )
=> ( ( ord_less_eq @ nat @ J2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ord_less_eq @ nat @ J2 @ ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P @ Xs ) ) ) ) ) ).
% length_takeWhile_less_P_nth
thf(fact_7026_takeWhile__eq__take__P__nth,axiom,
! [A: $tType,N: nat,Xs: list @ A,P: A > $o] :
( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ N )
=> ( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( P @ ( nth @ A @ Xs @ I3 ) ) ) )
=> ( ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ~ ( P @ ( nth @ A @ Xs @ N ) ) )
=> ( ( takeWhile @ A @ P @ Xs )
= ( take @ A @ N @ Xs ) ) ) ) ).
% takeWhile_eq_take_P_nth
thf(fact_7027_filter__equals__takeWhile__sorted__rev,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F2: B > A,Xs: list @ B,T2: A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( rev @ A @ ( map @ B @ A @ F2 @ Xs ) ) )
=> ( ( filter2 @ B
@ ^ [X2: B] : ( ord_less @ A @ T2 @ ( F2 @ X2 ) )
@ Xs )
= ( takeWhile @ B
@ ^ [X2: B] : ( ord_less @ A @ T2 @ ( F2 @ X2 ) )
@ Xs ) ) ) ) ).
% filter_equals_takeWhile_sorted_rev
thf(fact_7028_Min_Oeq__fold_H,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( lattic643756798350308766er_Min @ A )
= ( ^ [A7: set @ A] :
( the2 @ A
@ ( finite_fold @ A @ ( option @ A )
@ ^ [X2: A,Y5: option @ A] : ( some @ A @ ( case_option @ A @ A @ X2 @ ( ord_min @ A @ X2 ) @ Y5 ) )
@ ( none @ A )
@ A7 ) ) ) ) ) ).
% Min.eq_fold'
thf(fact_7029_minus__Max__eq__Min,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [S3: set @ A] :
( ( finite_finite @ A @ S3 )
=> ( ( S3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( uminus_uminus @ A @ ( lattic643756798349783984er_Max @ A @ S3 ) )
= ( lattic643756798350308766er_Min @ A @ ( image @ A @ A @ ( uminus_uminus @ A ) @ S3 ) ) ) ) ) ) ).
% minus_Max_eq_Min
thf(fact_7030_Max__divisors__self__nat,axiom,
! [N: nat] :
( ( N
!= ( zero_zero @ nat ) )
=> ( ( lattic643756798349783984er_Max @ nat
@ ( collect @ nat
@ ^ [D4: nat] : ( dvd_dvd @ nat @ D4 @ N ) ) )
= N ) ) ).
% Max_divisors_self_nat
thf(fact_7031_Max__divisors__self__int,axiom,
! [N: int] :
( ( N
!= ( zero_zero @ int ) )
=> ( ( lattic643756798349783984er_Max @ int
@ ( collect @ int
@ ^ [D4: int] : ( dvd_dvd @ int @ D4 @ N ) ) )
= ( abs_abs @ int @ N ) ) ) ).
% Max_divisors_self_int
thf(fact_7032_Max__less__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less @ A @ ( lattic643756798349783984er_Max @ A @ A4 ) @ X )
= ( ! [X2: A] :
( ( member @ A @ X2 @ A4 )
=> ( ord_less @ A @ X2 @ X ) ) ) ) ) ) ) ).
% Max_less_iff
thf(fact_7033_minus__Min__eq__Max,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [S3: set @ A] :
( ( finite_finite @ A @ S3 )
=> ( ( S3
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( uminus_uminus @ A @ ( lattic643756798350308766er_Min @ A @ S3 ) )
= ( lattic643756798349783984er_Max @ A @ ( image @ A @ A @ ( uminus_uminus @ A ) @ S3 ) ) ) ) ) ) ).
% minus_Min_eq_Max
thf(fact_7034_Max_Oeq__fold_H,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( lattic643756798349783984er_Max @ A )
= ( ^ [A7: set @ A] :
( the2 @ A
@ ( finite_fold @ A @ ( option @ A )
@ ^ [X2: A,Y5: option @ A] : ( some @ A @ ( case_option @ A @ A @ X2 @ ( ord_max @ A @ X2 ) @ Y5 ) )
@ ( none @ A )
@ A7 ) ) ) ) ) ).
% Max.eq_fold'
thf(fact_7035_Max__gr__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite @ A @ A4 )
=> ( ( A4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less @ A @ X @ ( lattic643756798349783984er_Max @ A @ A4 ) )
= ( ? [X2: A] :
( ( member @ A @ X2 @ A4 )
& ( ord_less @ A @ X @ X2 ) ) ) ) ) ) ) ).
% Max_gr_iff
thf(fact_7036_Max_Oinfinite,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A] :
( ~ ( finite_finite @ A @ A4 )
=> ( ( lattic643756798349783984er_Max @ A @ A4 )
= ( the2 @ A @ ( none @ A ) ) ) ) ) ).
% Max.infinite
thf(fact_7037_gcd__is__Max__divisors__int,axiom,
! [N: int,M2: int] :
( ( N
!= ( zero_zero @ int ) )
=> ( ( gcd_gcd @ int @ M2 @ N )
= ( lattic643756798349783984er_Max @ int
@ ( collect @ int
@ ^ [D4: int] :
( ( dvd_dvd @ int @ D4 @ M2 )
& ( dvd_dvd @ int @ D4 @ N ) ) ) ) ) ) ).
% gcd_is_Max_divisors_int
thf(fact_7038_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_7039_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_7040_card__le__Suc__Max,axiom,
! [S3: set @ nat] :
( ( finite_finite @ nat @ S3 )
=> ( ord_less_eq @ nat @ ( finite_card @ nat @ S3 ) @ ( suc @ ( lattic643756798349783984er_Max @ nat @ S3 ) ) ) ) ).
% card_le_Suc_Max
thf(fact_7041_Sup__nat__def,axiom,
( ( complete_Sup_Sup @ nat )
= ( ^ [X7: set @ nat] :
( if @ nat
@ ( X7
= ( bot_bot @ ( set @ nat ) ) )
@ ( zero_zero @ nat )
@ ( lattic643756798349783984er_Max @ nat @ X7 ) ) ) ) ).
% Sup_nat_def
thf(fact_7042_divide__nat__def,axiom,
( ( divide_divide @ nat )
= ( ^ [M5: nat,N5: nat] :
( if @ nat
@ ( N5
= ( zero_zero @ nat ) )
@ ( zero_zero @ nat )
@ ( lattic643756798349783984er_Max @ nat
@ ( collect @ nat
@ ^ [K3: nat] : ( ord_less_eq @ nat @ ( times_times @ nat @ K3 @ N5 ) @ M5 ) ) ) ) ) ) ).
% divide_nat_def
thf(fact_7043_gcd__is__Max__divisors__nat,axiom,
! [N: nat,M2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( gcd_gcd @ nat @ M2 @ N )
= ( lattic643756798349783984er_Max @ nat
@ ( collect @ nat
@ ^ [D4: nat] :
( ( dvd_dvd @ nat @ D4 @ M2 )
& ( dvd_dvd @ nat @ D4 @ N ) ) ) ) ) ) ).
% gcd_is_Max_divisors_nat
thf(fact_7044_Max_Oremove,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite @ A @ A4 )
=> ( ( member @ A @ X @ A4 )
=> ( ( ( ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798349783984er_Max @ A @ A4 )
= X ) )
& ( ( ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798349783984er_Max @ A @ A4 )
= ( ord_max @ A @ X @ ( lattic643756798349783984er_Max @ A @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ) ).
% Max.remove
thf(fact_7045_Max_Oinsert__remove,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite @ A @ A4 )
=> ( ( ( ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798349783984er_Max @ A @ ( insert @ A @ X @ A4 ) )
= X ) )
& ( ( ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798349783984er_Max @ A @ ( insert @ A @ X @ A4 ) )
= ( ord_max @ A @ X @ ( lattic643756798349783984er_Max @ A @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ).
% Max.insert_remove
thf(fact_7046_Gcd__eq__Max,axiom,
! [M10: set @ nat] :
( ( finite_finite @ nat @ M10 )
=> ( ( M10
!= ( bot_bot @ ( set @ nat ) ) )
=> ( ~ ( member @ nat @ ( zero_zero @ nat ) @ M10 )
=> ( ( gcd_Gcd @ nat @ M10 )
= ( lattic643756798349783984er_Max @ nat
@ ( complete_Inf_Inf @ ( set @ nat )
@ ( image @ nat @ ( set @ nat )
@ ^ [M5: nat] :
( collect @ nat
@ ^ [D4: nat] : ( dvd_dvd @ nat @ D4 @ M5 ) )
@ M10 ) ) ) ) ) ) ) ).
% Gcd_eq_Max
thf(fact_7047_Sup__fin_Oeq__fold_H,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( lattic5882676163264333800up_fin @ A )
= ( ^ [A7: set @ A] :
( the2 @ A
@ ( finite_fold @ A @ ( option @ A )
@ ^ [X2: A,Y5: option @ A] : ( some @ A @ ( case_option @ A @ A @ X2 @ ( sup_sup @ A @ X2 ) @ Y5 ) )
@ ( none @ A )
@ A7 ) ) ) ) ) ).
% Sup_fin.eq_fold'
thf(fact_7048_Inf__fin_Oeq__fold_H,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( ( lattic7752659483105999362nf_fin @ A )
= ( ^ [A7: set @ A] :
( the2 @ A
@ ( finite_fold @ A @ ( option @ A )
@ ^ [X2: A,Y5: option @ A] : ( some @ A @ ( case_option @ A @ A @ X2 @ ( inf_inf @ A @ X2 ) @ Y5 ) )
@ ( none @ A )
@ A7 ) ) ) ) ) ).
% Inf_fin.eq_fold'
thf(fact_7049_card_Oeq__fold,axiom,
! [A: $tType] :
( ( finite_card @ A )
= ( finite_fold @ A @ nat
@ ^ [Uu3: A] : suc
@ ( zero_zero @ nat ) ) ) ).
% card.eq_fold
thf(fact_7050_Sup__fin_Oinfinite,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A4: set @ A] :
( ~ ( finite_finite @ A @ A4 )
=> ( ( lattic5882676163264333800up_fin @ A @ A4 )
= ( the2 @ A @ ( none @ A ) ) ) ) ) ).
% Sup_fin.infinite
thf(fact_7051_Inf__fin_Oinfinite,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A4: set @ A] :
( ~ ( finite_finite @ A @ A4 )
=> ( ( lattic7752659483105999362nf_fin @ A @ A4 )
= ( the2 @ A @ ( none @ A ) ) ) ) ) ).
% Inf_fin.infinite
thf(fact_7052_Inf__fin_Oinsert__remove,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite @ A @ A4 )
=> ( ( ( ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic7752659483105999362nf_fin @ A @ ( insert @ A @ X @ A4 ) )
= X ) )
& ( ( ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic7752659483105999362nf_fin @ A @ ( insert @ A @ X @ A4 ) )
= ( inf_inf @ A @ X @ ( lattic7752659483105999362nf_fin @ A @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ).
% Inf_fin.insert_remove
thf(fact_7053_Inf__fin_Oremove,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite @ A @ A4 )
=> ( ( member @ A @ X @ A4 )
=> ( ( ( ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic7752659483105999362nf_fin @ A @ A4 )
= X ) )
& ( ( ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic7752659483105999362nf_fin @ A @ A4 )
= ( inf_inf @ A @ X @ ( lattic7752659483105999362nf_fin @ A @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ) ).
% Inf_fin.remove
thf(fact_7054_Sup__fin_Oinsert__remove,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite @ A @ A4 )
=> ( ( ( ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic5882676163264333800up_fin @ A @ ( insert @ A @ X @ A4 ) )
= X ) )
& ( ( ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic5882676163264333800up_fin @ A @ ( insert @ A @ X @ A4 ) )
= ( sup_sup @ A @ X @ ( lattic5882676163264333800up_fin @ A @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ).
% Sup_fin.insert_remove
thf(fact_7055_Sup__fin_Oremove,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A4: set @ A,X: A] :
( ( finite_finite @ A @ A4 )
=> ( ( member @ A @ X @ A4 )
=> ( ( ( ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic5882676163264333800up_fin @ A @ A4 )
= X ) )
& ( ( ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic5882676163264333800up_fin @ A @ A4 )
= ( sup_sup @ A @ X @ ( lattic5882676163264333800up_fin @ A @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ) ).
% Sup_fin.remove
thf(fact_7056_comp__fun__commute__on_Ofold__rec,axiom,
! [B: $tType,A: $tType,S3: set @ A,F2: A > B > B,A4: set @ A,X: A,Z2: B] :
( ( finite4664212375090638736ute_on @ A @ B @ S3 @ F2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ S3 )
=> ( ( finite_finite @ A @ A4 )
=> ( ( member @ A @ X @ A4 )
=> ( ( finite_fold @ A @ B @ F2 @ Z2 @ A4 )
= ( F2 @ X @ ( finite_fold @ A @ B @ F2 @ Z2 @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ).
% comp_fun_commute_on.fold_rec
thf(fact_7057_comp__fun__commute__on_Ofold__insert__remove,axiom,
! [B: $tType,A: $tType,S3: set @ A,F2: A > B > B,X: A,A4: set @ A,Z2: B] :
( ( finite4664212375090638736ute_on @ A @ B @ S3 @ F2 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X @ A4 ) @ S3 )
=> ( ( finite_finite @ A @ A4 )
=> ( ( finite_fold @ A @ B @ F2 @ Z2 @ ( insert @ A @ X @ A4 ) )
= ( F2 @ X @ ( finite_fold @ A @ B @ F2 @ Z2 @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ).
% comp_fun_commute_on.fold_insert_remove
thf(fact_7058_comp__fun__commute__on_Ofold__graph__insertE__aux,axiom,
! [A: $tType,B: $tType,S3: set @ A,F2: A > B > B,A4: set @ A,Z2: B,Y2: B,A3: A] :
( ( finite4664212375090638736ute_on @ A @ B @ S3 @ F2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ S3 )
=> ( ( finite_fold_graph @ A @ B @ F2 @ Z2 @ A4 @ Y2 )
=> ( ( member @ A @ A3 @ A4 )
=> ? [Y9: B] :
( ( Y2
= ( F2 @ A3 @ Y9 ) )
& ( finite_fold_graph @ A @ B @ F2 @ Z2 @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) @ Y9 ) ) ) ) ) ) ).
% comp_fun_commute_on.fold_graph_insertE_aux
thf(fact_7059_butlast__take,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( ord_less_eq @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( butlast @ A @ ( take @ A @ N @ Xs ) )
= ( take @ A @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ Xs ) ) ) ).
% butlast_take
thf(fact_7060_length__butlast,axiom,
! [A: $tType,Xs: list @ A] :
( ( size_size @ ( list @ A ) @ ( butlast @ A @ Xs ) )
= ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) ) ) ).
% length_butlast
thf(fact_7061_nth__butlast,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ ( butlast @ A @ Xs ) ) )
=> ( ( nth @ A @ ( butlast @ A @ Xs ) @ N )
= ( nth @ A @ Xs @ N ) ) ) ).
% nth_butlast
thf(fact_7062_take__butlast,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( take @ A @ N @ ( butlast @ A @ Xs ) )
= ( take @ A @ N @ Xs ) ) ) ).
% take_butlast
thf(fact_7063_butlast__power,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( compow @ ( ( list @ A ) > ( list @ A ) ) @ N @ ( butlast @ A ) @ Xs )
= ( take @ A @ ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N ) @ Xs ) ) ).
% butlast_power
thf(fact_7064_butlast__conv__take,axiom,
! [A: $tType] :
( ( butlast @ A )
= ( ^ [Xs3: list @ A] : ( take @ A @ ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs3 ) @ ( one_one @ nat ) ) @ Xs3 ) ) ) ).
% butlast_conv_take
thf(fact_7065_butlast__list__update,axiom,
! [A: $tType,K: nat,Xs: list @ A,X: A] :
( ( ( K
= ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) ) )
=> ( ( butlast @ A @ ( list_update @ A @ Xs @ K @ X ) )
= ( butlast @ A @ Xs ) ) )
& ( ( K
!= ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) ) )
=> ( ( butlast @ A @ ( list_update @ A @ Xs @ K @ X ) )
= ( list_update @ A @ ( butlast @ A @ Xs ) @ K @ X ) ) ) ) ).
% butlast_list_update
thf(fact_7066_finite__enumerate__initial__segment,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [S3: set @ A,N: nat,S: A] :
( ( finite_finite @ A @ S3 )
=> ( ( ord_less @ nat @ N @ ( finite_card @ A @ ( inf_inf @ ( set @ A ) @ S3 @ ( set_ord_lessThan @ A @ S ) ) ) )
=> ( ( infini527867602293511546merate @ A @ ( inf_inf @ ( set @ A ) @ S3 @ ( set_ord_lessThan @ A @ S ) ) @ N )
= ( infini527867602293511546merate @ A @ S3 @ N ) ) ) ) ) ).
% finite_enumerate_initial_segment
thf(fact_7067_remove__def,axiom,
! [A: $tType] :
( ( remove @ A )
= ( ^ [X2: A,A7: set @ A] : ( minus_minus @ ( set @ A ) @ A7 @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% remove_def
thf(fact_7068_enumerate__mono__iff,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [S3: set @ A,M2: nat,N: nat] :
( ~ ( finite_finite @ A @ S3 )
=> ( ( ord_less @ A @ ( infini527867602293511546merate @ A @ S3 @ M2 ) @ ( infini527867602293511546merate @ A @ S3 @ N ) )
= ( ord_less @ nat @ M2 @ N ) ) ) ) ).
% enumerate_mono_iff
thf(fact_7069_finite__enumerate__mono__iff,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [S3: set @ A,M2: nat,N: nat] :
( ( finite_finite @ A @ S3 )
=> ( ( ord_less @ nat @ M2 @ ( finite_card @ A @ S3 ) )
=> ( ( ord_less @ nat @ N @ ( finite_card @ A @ S3 ) )
=> ( ( ord_less @ A @ ( infini527867602293511546merate @ A @ S3 @ M2 ) @ ( infini527867602293511546merate @ A @ S3 @ N ) )
= ( ord_less @ nat @ M2 @ N ) ) ) ) ) ) ).
% finite_enumerate_mono_iff
thf(fact_7070_enumerate__step,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [S3: set @ A,N: nat] :
( ~ ( finite_finite @ A @ S3 )
=> ( ord_less @ A @ ( infini527867602293511546merate @ A @ S3 @ N ) @ ( infini527867602293511546merate @ A @ S3 @ ( suc @ N ) ) ) ) ) ).
% enumerate_step
thf(fact_7071_enumerate__mono,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [M2: nat,N: nat,S3: set @ A] :
( ( ord_less @ nat @ M2 @ N )
=> ( ~ ( finite_finite @ A @ S3 )
=> ( ord_less @ A @ ( infini527867602293511546merate @ A @ S3 @ M2 ) @ ( infini527867602293511546merate @ A @ S3 @ N ) ) ) ) ) ).
% enumerate_mono
thf(fact_7072_finite__enumerate__in__set,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [S3: set @ A,N: nat] :
( ( finite_finite @ A @ S3 )
=> ( ( ord_less @ nat @ N @ ( finite_card @ A @ S3 ) )
=> ( member @ A @ ( infini527867602293511546merate @ A @ S3 @ N ) @ S3 ) ) ) ) ).
% finite_enumerate_in_set
thf(fact_7073_finite__enumerate__Ex,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [S3: set @ A,S: A] :
( ( finite_finite @ A @ S3 )
=> ( ( member @ A @ S @ S3 )
=> ? [N2: nat] :
( ( ord_less @ nat @ N2 @ ( finite_card @ A @ S3 ) )
& ( ( infini527867602293511546merate @ A @ S3 @ N2 )
= S ) ) ) ) ) ).
% finite_enumerate_Ex
thf(fact_7074_finite__enum__ext,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [X8: set @ A,Y7: set @ A] :
( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( finite_card @ A @ X8 ) )
=> ( ( infini527867602293511546merate @ A @ X8 @ I3 )
= ( infini527867602293511546merate @ A @ Y7 @ I3 ) ) )
=> ( ( finite_finite @ A @ X8 )
=> ( ( finite_finite @ A @ Y7 )
=> ( ( ( finite_card @ A @ X8 )
= ( finite_card @ A @ Y7 ) )
=> ( X8 = Y7 ) ) ) ) ) ) ).
% finite_enum_ext
thf(fact_7075_minus__fold__remove,axiom,
! [A: $tType,A4: set @ A,B7: set @ A] :
( ( finite_finite @ A @ A4 )
=> ( ( minus_minus @ ( set @ A ) @ B7 @ A4 )
= ( finite_fold @ A @ ( set @ A ) @ ( remove @ A ) @ B7 @ A4 ) ) ) ).
% minus_fold_remove
thf(fact_7076_finite__enumerate__mono,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [M2: nat,N: nat,S3: set @ A] :
( ( ord_less @ nat @ M2 @ N )
=> ( ( finite_finite @ A @ S3 )
=> ( ( ord_less @ nat @ N @ ( finite_card @ A @ S3 ) )
=> ( ord_less @ A @ ( infini527867602293511546merate @ A @ S3 @ M2 ) @ ( infini527867602293511546merate @ A @ S3 @ N ) ) ) ) ) ) ).
% finite_enumerate_mono
thf(fact_7077_finite__le__enumerate,axiom,
! [S3: set @ nat,N: nat] :
( ( finite_finite @ nat @ S3 )
=> ( ( ord_less @ nat @ N @ ( finite_card @ nat @ S3 ) )
=> ( ord_less_eq @ nat @ N @ ( infini527867602293511546merate @ nat @ S3 @ N ) ) ) ) ).
% finite_le_enumerate
thf(fact_7078_finite__enumerate__step,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [S3: set @ A,N: nat] :
( ( finite_finite @ A @ S3 )
=> ( ( ord_less @ nat @ ( suc @ N ) @ ( finite_card @ A @ S3 ) )
=> ( ord_less @ A @ ( infini527867602293511546merate @ A @ S3 @ N ) @ ( infini527867602293511546merate @ A @ S3 @ ( suc @ N ) ) ) ) ) ) ).
% finite_enumerate_step
thf(fact_7079_enumerate__Suc_H,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [S3: set @ A,N: nat] :
( ( infini527867602293511546merate @ A @ S3 @ ( suc @ N ) )
= ( infini527867602293511546merate @ A @ ( minus_minus @ ( set @ A ) @ S3 @ ( insert @ A @ ( infini527867602293511546merate @ A @ S3 @ ( zero_zero @ nat ) ) @ ( bot_bot @ ( set @ A ) ) ) ) @ N ) ) ) ).
% enumerate_Suc'
thf(fact_7080_finite__enum__subset,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [X8: set @ A,Y7: set @ A] :
( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( finite_card @ A @ X8 ) )
=> ( ( infini527867602293511546merate @ A @ X8 @ I3 )
= ( infini527867602293511546merate @ A @ Y7 @ I3 ) ) )
=> ( ( finite_finite @ A @ X8 )
=> ( ( finite_finite @ A @ Y7 )
=> ( ( ord_less_eq @ nat @ ( finite_card @ A @ X8 ) @ ( finite_card @ A @ Y7 ) )
=> ( ord_less_eq @ ( set @ A ) @ X8 @ Y7 ) ) ) ) ) ) ).
% finite_enum_subset
thf(fact_7081_finite__enumerate__Suc_H_H,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [S3: set @ A,N: nat] :
( ( finite_finite @ A @ S3 )
=> ( ( ord_less @ nat @ ( suc @ N ) @ ( finite_card @ A @ S3 ) )
=> ( ( infini527867602293511546merate @ A @ S3 @ ( suc @ N ) )
= ( ord_Least @ A
@ ^ [S5: A] :
( ( member @ A @ S5 @ S3 )
& ( ord_less @ A @ ( infini527867602293511546merate @ A @ S3 @ N ) @ S5 ) ) ) ) ) ) ) ).
% finite_enumerate_Suc''
thf(fact_7082_enumerate__Suc,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [S3: set @ A,N: nat] :
( ( infini527867602293511546merate @ A @ S3 @ ( suc @ N ) )
= ( infini527867602293511546merate @ A
@ ( minus_minus @ ( set @ A ) @ S3
@ ( insert @ A
@ ( ord_Least @ A
@ ^ [N5: A] : ( member @ A @ N5 @ S3 ) )
@ ( bot_bot @ ( set @ A ) ) ) )
@ N ) ) ) ).
% enumerate_Suc
thf(fact_7083_Least__eq__0,axiom,
! [P: nat > $o] :
( ( P @ ( zero_zero @ nat ) )
=> ( ( ord_Least @ nat @ P )
= ( zero_zero @ nat ) ) ) ).
% Least_eq_0
thf(fact_7084_not__less__Least,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [K: A,P: A > $o] :
( ( ord_less @ A @ K @ ( ord_Least @ A @ P ) )
=> ~ ( P @ K ) ) ) ).
% not_less_Least
thf(fact_7085_Least__Suc2,axiom,
! [P: nat > $o,N: nat,Q: nat > $o,M2: nat] :
( ( P @ N )
=> ( ( Q @ M2 )
=> ( ~ ( P @ ( zero_zero @ nat ) )
=> ( ! [K2: nat] :
( ( P @ ( suc @ K2 ) )
= ( Q @ K2 ) )
=> ( ( ord_Least @ nat @ P )
= ( suc @ ( ord_Least @ nat @ Q ) ) ) ) ) ) ) ).
% Least_Suc2
thf(fact_7086_Least__Suc,axiom,
! [P: nat > $o,N: nat] :
( ( P @ N )
=> ( ~ ( P @ ( zero_zero @ nat ) )
=> ( ( ord_Least @ nat @ P )
= ( suc
@ ( ord_Least @ nat
@ ^ [M5: nat] : ( P @ ( suc @ M5 ) ) ) ) ) ) ) ).
% Least_Suc
thf(fact_7087_enumerate__0,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [S3: set @ A] :
( ( infini527867602293511546merate @ A @ S3 @ ( zero_zero @ nat ) )
= ( ord_Least @ A
@ ^ [N5: A] : ( member @ A @ N5 @ S3 ) ) ) ) ).
% enumerate_0
thf(fact_7088_Sup__real__def,axiom,
( ( complete_Sup_Sup @ real )
= ( ^ [X7: set @ real] :
( ord_Least @ real
@ ^ [Z6: real] :
! [X2: real] :
( ( member @ real @ X2 @ X7 )
=> ( ord_less_eq @ real @ X2 @ Z6 ) ) ) ) ) ).
% Sup_real_def
thf(fact_7089_enumerate__Suc_H_H,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [S3: set @ A,N: nat] :
( ~ ( finite_finite @ A @ S3 )
=> ( ( infini527867602293511546merate @ A @ S3 @ ( suc @ N ) )
= ( ord_Least @ A
@ ^ [S5: A] :
( ( member @ A @ S5 @ S3 )
& ( ord_less @ A @ ( infini527867602293511546merate @ A @ S3 @ N ) @ S5 ) ) ) ) ) ) ).
% enumerate_Suc''
thf(fact_7090_Gcd__fin_Oeq__fold,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ( ( semiring_gcd_Gcd_fin @ A )
= ( ^ [A7: set @ A] : ( if @ A @ ( finite_finite @ A @ A7 ) @ ( finite_fold @ A @ A @ ( gcd_gcd @ A ) @ ( zero_zero @ A ) @ A7 ) @ ( one_one @ A ) ) ) ) ) ).
% Gcd_fin.eq_fold
thf(fact_7091_independentD,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [S: set @ A,T2: set @ A,U: A > real,V2: A] :
( ~ ( real_V358717886546972837endent @ A @ S )
=> ( ( finite_finite @ A @ T2 )
=> ( ( ord_less_eq @ ( set @ A ) @ T2 @ S )
=> ( ( ( groups7311177749621191930dd_sum @ A @ A
@ ^ [V5: A] : ( real_V8093663219630862766scaleR @ A @ ( U @ V5 ) @ V5 )
@ T2 )
= ( zero_zero @ A ) )
=> ( ( member @ A @ V2 @ T2 )
=> ( ( U @ V2 )
= ( zero_zero @ real ) ) ) ) ) ) ) ) ).
% independentD
thf(fact_7092_Gcd__fin_Oempty,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ( ( semiring_gcd_Gcd_fin @ A @ ( bot_bot @ ( set @ A ) ) )
= ( zero_zero @ A ) ) ) ).
% Gcd_fin.empty
thf(fact_7093_Gcd__fin_Oinfinite,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A4: set @ A] :
( ~ ( finite_finite @ A @ A4 )
=> ( ( semiring_gcd_Gcd_fin @ A @ A4 )
= ( one_one @ A ) ) ) ) ).
% Gcd_fin.infinite
thf(fact_7094_is__unit__Gcd__fin__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A4: set @ A] :
( ( dvd_dvd @ A @ ( semiring_gcd_Gcd_fin @ A @ A4 ) @ ( one_one @ A ) )
= ( ( semiring_gcd_Gcd_fin @ A @ A4 )
= ( one_one @ A ) ) ) ) ).
% is_unit_Gcd_fin_iff
thf(fact_7095_dependent__single,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [X: A] :
( ( real_V358717886546972837endent @ A @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( X
= ( zero_zero @ A ) ) ) ) ).
% dependent_single
thf(fact_7096_dependent__zero,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [A4: set @ A] :
( ( member @ A @ ( zero_zero @ A ) @ A4 )
=> ( real_V358717886546972837endent @ A @ A4 ) ) ) ).
% dependent_zero
thf(fact_7097_Gcd__fin_Oremove,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,A4: set @ A] :
( ( member @ A @ A3 @ A4 )
=> ( ( semiring_gcd_Gcd_fin @ A @ A4 )
= ( gcd_gcd @ A @ A3 @ ( semiring_gcd_Gcd_fin @ A @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% Gcd_fin.remove
thf(fact_7098_Gcd__fin_Oinsert__remove,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,A4: set @ A] :
( ( semiring_gcd_Gcd_fin @ A @ ( insert @ A @ A3 @ A4 ) )
= ( gcd_gcd @ A @ A3 @ ( semiring_gcd_Gcd_fin @ A @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% Gcd_fin.insert_remove
thf(fact_7099_independent__if__scalars__zero,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [A4: set @ A] :
( ( finite_finite @ A @ A4 )
=> ( ! [F3: A > real,X3: A] :
( ( ( groups7311177749621191930dd_sum @ A @ A
@ ^ [Y5: A] : ( real_V8093663219630862766scaleR @ A @ ( F3 @ Y5 ) @ Y5 )
@ A4 )
= ( zero_zero @ A ) )
=> ( ( member @ A @ X3 @ A4 )
=> ( ( F3 @ X3 )
= ( zero_zero @ real ) ) ) )
=> ~ ( real_V358717886546972837endent @ A @ A4 ) ) ) ) ).
% independent_if_scalars_zero
thf(fact_7100_dependent__finite,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [S3: set @ A] :
( ( finite_finite @ A @ S3 )
=> ( ( real_V358717886546972837endent @ A @ S3 )
= ( ? [U2: A > real] :
( ? [X2: A] :
( ( member @ A @ X2 @ S3 )
& ( ( U2 @ X2 )
!= ( zero_zero @ real ) ) )
& ( ( groups7311177749621191930dd_sum @ A @ A
@ ^ [V5: A] : ( real_V8093663219630862766scaleR @ A @ ( U2 @ V5 ) @ V5 )
@ S3 )
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% dependent_finite
thf(fact_7101_Gcd__fin__0__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A4: set @ A] :
( ( ( semiring_gcd_Gcd_fin @ A @ A4 )
= ( zero_zero @ A ) )
= ( ( ord_less_eq @ ( set @ A ) @ A4 @ ( insert @ A @ ( zero_zero @ A ) @ ( bot_bot @ ( set @ A ) ) ) )
& ( finite_finite @ A @ A4 ) ) ) ) ).
% Gcd_fin_0_iff
thf(fact_7102_independentD__alt,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [B7: set @ A,X8: A > real,X: A] :
( ~ ( real_V358717886546972837endent @ A @ B7 )
=> ( ( finite_finite @ A
@ ( collect @ A
@ ^ [X2: A] :
( ( X8 @ X2 )
!= ( zero_zero @ real ) ) ) )
=> ( ( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [X2: A] :
( ( X8 @ X2 )
!= ( zero_zero @ real ) ) )
@ B7 )
=> ( ( ( groups7311177749621191930dd_sum @ A @ A
@ ^ [X2: A] : ( real_V8093663219630862766scaleR @ A @ ( X8 @ X2 ) @ X2 )
@ ( collect @ A
@ ^ [X2: A] :
( ( X8 @ X2 )
!= ( zero_zero @ real ) ) ) )
= ( zero_zero @ A ) )
=> ( ( X8 @ X )
= ( zero_zero @ real ) ) ) ) ) ) ) ).
% independentD_alt
thf(fact_7103_independent__alt,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [B7: set @ A] :
( ( ~ ( real_V358717886546972837endent @ A @ B7 ) )
= ( ! [X7: A > real] :
( ( finite_finite @ A
@ ( collect @ A
@ ^ [X2: A] :
( ( X7 @ X2 )
!= ( zero_zero @ real ) ) ) )
=> ( ( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [X2: A] :
( ( X7 @ X2 )
!= ( zero_zero @ real ) ) )
@ B7 )
=> ( ( ( groups7311177749621191930dd_sum @ A @ A
@ ^ [X2: A] : ( real_V8093663219630862766scaleR @ A @ ( X7 @ X2 ) @ X2 )
@ ( collect @ A
@ ^ [X2: A] :
( ( X7 @ X2 )
!= ( zero_zero @ real ) ) ) )
= ( zero_zero @ A ) )
=> ! [X2: A] :
( ( X7 @ X2 )
= ( zero_zero @ real ) ) ) ) ) ) ) ) ).
% independent_alt
thf(fact_7104_dependent__alt,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ( ( real_V358717886546972837endent @ A )
= ( ^ [B5: set @ A] :
? [X7: A > real] :
( ( finite_finite @ A
@ ( collect @ A
@ ^ [X2: A] :
( ( X7 @ X2 )
!= ( zero_zero @ real ) ) ) )
& ( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [X2: A] :
( ( X7 @ X2 )
!= ( zero_zero @ real ) ) )
@ B5 )
& ( ( groups7311177749621191930dd_sum @ A @ A
@ ^ [X2: A] : ( real_V8093663219630862766scaleR @ A @ ( X7 @ X2 ) @ X2 )
@ ( collect @ A
@ ^ [X2: A] :
( ( X7 @ X2 )
!= ( zero_zero @ real ) ) ) )
= ( zero_zero @ A ) )
& ? [X2: A] :
( ( X7 @ X2 )
!= ( zero_zero @ real ) ) ) ) ) ) ).
% dependent_alt
thf(fact_7105_independent__explicit__finite__subsets,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [A4: set @ A] :
( ( ~ ( real_V358717886546972837endent @ A @ A4 ) )
= ( ! [S8: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ S8 @ A4 )
=> ( ( finite_finite @ A @ S8 )
=> ! [U2: A > real] :
( ( ( groups7311177749621191930dd_sum @ A @ A
@ ^ [V5: A] : ( real_V8093663219630862766scaleR @ A @ ( U2 @ V5 ) @ V5 )
@ S8 )
= ( zero_zero @ A ) )
=> ! [X2: A] :
( ( member @ A @ X2 @ S8 )
=> ( ( U2 @ X2 )
= ( zero_zero @ real ) ) ) ) ) ) ) ) ) ).
% independent_explicit_finite_subsets
thf(fact_7106_independent__explicit__module,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [S: set @ A] :
( ( ~ ( real_V358717886546972837endent @ A @ S ) )
= ( ! [T3: set @ A,U2: A > real,V5: A] :
( ( finite_finite @ A @ T3 )
=> ( ( ord_less_eq @ ( set @ A ) @ T3 @ S )
=> ( ( ( groups7311177749621191930dd_sum @ A @ A
@ ^ [W3: A] : ( real_V8093663219630862766scaleR @ A @ ( U2 @ W3 ) @ W3 )
@ T3 )
= ( zero_zero @ A ) )
=> ( ( member @ A @ V5 @ T3 )
=> ( ( U2 @ V5 )
= ( zero_zero @ real ) ) ) ) ) ) ) ) ) ).
% independent_explicit_module
thf(fact_7107_dependent__explicit,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ( ( real_V358717886546972837endent @ A )
= ( ^ [S5: set @ A] :
? [T3: set @ A] :
( ( finite_finite @ A @ T3 )
& ( ord_less_eq @ ( set @ A ) @ T3 @ S5 )
& ? [U2: A > real] :
( ( ( groups7311177749621191930dd_sum @ A @ A
@ ^ [V5: A] : ( real_V8093663219630862766scaleR @ A @ ( U2 @ V5 ) @ V5 )
@ T3 )
= ( zero_zero @ A ) )
& ? [X2: A] :
( ( member @ A @ X2 @ T3 )
& ( ( U2 @ X2 )
!= ( zero_zero @ real ) ) ) ) ) ) ) ) ).
% dependent_explicit
thf(fact_7108_possible__bit__def,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ( ( bit_se6407376104438227557le_bit @ A )
= ( ^ [Tyrep: itself @ A,N5: nat] :
( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N5 )
!= ( zero_zero @ A ) ) ) ) ) ).
% possible_bit_def
thf(fact_7109_continuous__on__arcosh,axiom,
! [A4: set @ real] :
( ( ord_less_eq @ ( set @ real ) @ A4 @ ( set_ord_atLeast @ real @ ( one_one @ real ) ) )
=> ( topolo81223032696312382ous_on @ real @ real @ A4 @ ( arcosh @ real ) ) ) ).
% continuous_on_arcosh
thf(fact_7110_Compl__lessThan,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [K: A] :
( ( uminus_uminus @ ( set @ A ) @ ( set_ord_lessThan @ A @ K ) )
= ( set_ord_atLeast @ A @ K ) ) ) ).
% Compl_lessThan
thf(fact_7111_Compl__atLeast,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [K: A] :
( ( uminus_uminus @ ( set @ A ) @ ( set_ord_atLeast @ A @ K ) )
= ( set_ord_lessThan @ A @ K ) ) ) ).
% Compl_atLeast
thf(fact_7112_image__minus__const__AtMost,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [C2: A,B2: A] :
( ( image @ A @ A @ ( minus_minus @ A @ C2 ) @ ( set_ord_atMost @ A @ B2 ) )
= ( set_ord_atLeast @ A @ ( minus_minus @ A @ C2 @ B2 ) ) ) ) ).
% image_minus_const_AtMost
thf(fact_7113_image__minus__const__atLeast,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [C2: A,A3: A] :
( ( image @ A @ A @ ( minus_minus @ A @ C2 ) @ ( set_ord_atLeast @ A @ A3 ) )
= ( set_ord_atMost @ A @ ( minus_minus @ A @ C2 @ A3 ) ) ) ) ).
% image_minus_const_atLeast
thf(fact_7114_image__uminus__atLeast,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [X: A] :
( ( image @ A @ A @ ( uminus_uminus @ A ) @ ( set_ord_atLeast @ A @ X ) )
= ( set_ord_atMost @ A @ ( uminus_uminus @ A @ X ) ) ) ) ).
% image_uminus_atLeast
thf(fact_7115_image__uminus__atMost,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [X: A] :
( ( image @ A @ A @ ( uminus_uminus @ A ) @ ( set_ord_atMost @ A @ X ) )
= ( set_ord_atLeast @ A @ ( uminus_uminus @ A @ X ) ) ) ) ).
% image_uminus_atMost
thf(fact_7116_possible__bit__0,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [Ty: itself @ A] : ( bit_se6407376104438227557le_bit @ A @ Ty @ ( zero_zero @ nat ) ) ) ).
% possible_bit_0
thf(fact_7117_Ici__subset__Ioi__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A3: A,B2: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_ord_atLeast @ A @ A3 ) @ ( set_ord_greaterThan @ A @ B2 ) )
= ( ord_less @ A @ B2 @ A3 ) ) ) ).
% Ici_subset_Ioi_iff
thf(fact_7118_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_7119_drop__bit__exp__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M2: nat,N: nat] :
( ( bit_se4197421643247451524op_bit @ A @ M2 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
= ( times_times @ A
@ ( zero_neq_one_of_bool @ A
@ ( ( ord_less_eq @ nat @ M2 @ N )
& ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N ) ) )
@ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ N @ M2 ) ) ) ) ) ).
% drop_bit_exp_eq
thf(fact_7120_bit__minus__2__iff,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ N )
= ( ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N )
& ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% bit_minus_2_iff
thf(fact_7121_atLeast__0,axiom,
( ( set_ord_atLeast @ nat @ ( zero_zero @ nat ) )
= ( top_top @ ( set @ nat ) ) ) ).
% atLeast_0
thf(fact_7122_CHAR__eq__0,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ( ( semiri4206861660011772517g_char @ A @ ( type2 @ A ) )
= ( zero_zero @ nat ) ) ) ).
% CHAR_eq_0
thf(fact_7123_of__nat__CHAR,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( ( semiring_1_of_nat @ A @ ( semiri4206861660011772517g_char @ A @ ( type2 @ A ) ) )
= ( zero_zero @ A ) ) ) ).
% of_nat_CHAR
thf(fact_7124_bit__minus__1__iff,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ N )
= ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N ) ) ) ).
% bit_minus_1_iff
thf(fact_7125_atLeast__Suc__greaterThan,axiom,
! [K: nat] :
( ( set_ord_atLeast @ nat @ ( suc @ K ) )
= ( set_ord_greaterThan @ nat @ K ) ) ).
% atLeast_Suc_greaterThan
thf(fact_7126_bit__mask__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M2: nat,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( bit_se2239418461657761734s_mask @ A @ M2 ) @ N )
= ( ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N )
& ( ord_less @ nat @ N @ M2 ) ) ) ) ).
% bit_mask_iff
thf(fact_7127_of__nat__eq__0__iff__char__dvd,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [N: nat] :
( ( ( semiring_1_of_nat @ A @ N )
= ( zero_zero @ A ) )
= ( dvd_dvd @ nat @ ( semiri4206861660011772517g_char @ A @ ( type2 @ A ) ) @ N ) ) ) ).
% of_nat_eq_0_iff_char_dvd
thf(fact_7128_CHAR__eqI,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [C2: nat] :
( ( ( semiring_1_of_nat @ A @ C2 )
= ( zero_zero @ A ) )
=> ( ! [X3: nat] :
( ( ( semiring_1_of_nat @ A @ X3 )
= ( zero_zero @ A ) )
=> ( dvd_dvd @ nat @ C2 @ X3 ) )
=> ( ( semiri4206861660011772517g_char @ A @ ( type2 @ A ) )
= C2 ) ) ) ) ).
% CHAR_eqI
thf(fact_7129_bit__of__nat__iff,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [M2: nat,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( semiring_1_of_nat @ A @ M2 ) @ N )
= ( ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N )
& ( bit_se5641148757651400278ts_bit @ nat @ M2 @ N ) ) ) ) ).
% bit_of_nat_iff
thf(fact_7130_bit__minus__iff,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [A3: A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( uminus_uminus @ A @ A3 ) @ N )
= ( ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N )
& ~ ( bit_se5641148757651400278ts_bit @ A @ ( minus_minus @ A @ A3 @ ( one_one @ A ) ) @ N ) ) ) ) ).
% bit_minus_iff
thf(fact_7131_CHAR__pos__iff,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( semiri4206861660011772517g_char @ A @ ( type2 @ A ) ) )
= ( ? [N5: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N5 )
& ( ( semiring_1_of_nat @ A @ N5 )
= ( zero_zero @ A ) ) ) ) ) ) ).
% CHAR_pos_iff
thf(fact_7132_CHAR__eq__posI,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [C2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ C2 )
=> ( ( ( semiring_1_of_nat @ A @ C2 )
= ( zero_zero @ A ) )
=> ( ! [X3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ X3 )
=> ( ( ord_less @ nat @ X3 @ C2 )
=> ( ( semiring_1_of_nat @ A @ X3 )
!= ( zero_zero @ A ) ) ) )
=> ( ( semiri4206861660011772517g_char @ A @ ( type2 @ A ) )
= C2 ) ) ) ) ) ).
% CHAR_eq_posI
thf(fact_7133_CHAR__eq0__iff,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( ( ( semiri4206861660011772517g_char @ A @ ( type2 @ A ) )
= ( zero_zero @ nat ) )
= ( ! [N5: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N5 )
=> ( ( semiring_1_of_nat @ A @ N5 )
!= ( zero_zero @ A ) ) ) ) ) ) ).
% CHAR_eq0_iff
thf(fact_7134_bit__push__bit__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M2: nat,A3: A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( bit_se4730199178511100633sh_bit @ A @ M2 @ A3 ) @ N )
= ( ( ord_less_eq @ nat @ M2 @ N )
& ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N )
& ( bit_se5641148757651400278ts_bit @ A @ A3 @ ( minus_minus @ nat @ N @ M2 ) ) ) ) ) ).
% bit_push_bit_iff
thf(fact_7135_atLeast__Suc,axiom,
! [K: nat] :
( ( set_ord_atLeast @ nat @ ( suc @ K ) )
= ( minus_minus @ ( set @ nat ) @ ( set_ord_atLeast @ nat @ K ) @ ( insert @ nat @ K @ ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% atLeast_Suc
thf(fact_7136_fold__possible__bit,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [N: nat] :
( ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N )
= ( zero_zero @ A ) )
= ( ~ ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N ) ) ) ) ).
% fold_possible_bit
thf(fact_7137_bit__2__iff,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N )
= ( ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ ( one_one @ nat ) )
& ( N
= ( one_one @ nat ) ) ) ) ) ).
% bit_2_iff
thf(fact_7138_bit__minus__exp__iff,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [M2: nat,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( uminus_uminus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M2 ) ) @ N )
= ( ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N )
& ( ord_less_eq @ nat @ M2 @ N ) ) ) ) ).
% bit_minus_exp_iff
thf(fact_7139_bit__mask__sub__iff,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [M2: nat,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M2 ) @ ( one_one @ A ) ) @ N )
= ( ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N )
& ( ord_less @ nat @ N @ M2 ) ) ) ) ).
% bit_mask_sub_iff
thf(fact_7140_bit__double__iff,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A3: A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A3 ) @ N )
= ( ( bit_se5641148757651400278ts_bit @ A @ A3 @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) )
& ( N
!= ( zero_zero @ nat ) )
& ( bit_se6407376104438227557le_bit @ A @ ( type2 @ A ) @ N ) ) ) ) ).
% bit_double_iff
thf(fact_7141_last__list__update,axiom,
! [A: $tType,Xs: list @ A,K: nat,X: A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( ( K
= ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) ) )
=> ( ( last @ A @ ( list_update @ A @ Xs @ K @ X ) )
= X ) )
& ( ( K
!= ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) ) )
=> ( ( last @ A @ ( list_update @ A @ Xs @ K @ X ) )
= ( last @ A @ Xs ) ) ) ) ) ).
% last_list_update
thf(fact_7142_minus__coset__filter,axiom,
! [A: $tType,A4: set @ A,Xs: list @ A] :
( ( minus_minus @ ( set @ A ) @ A4 @ ( coset @ A @ Xs ) )
= ( set2 @ A
@ ( filter2 @ A
@ ^ [X2: A] : ( member @ A @ X2 @ A4 )
@ Xs ) ) ) ).
% minus_coset_filter
thf(fact_7143_last__replicate,axiom,
! [A: $tType,N: nat,X: A] :
( ( N
!= ( zero_zero @ nat ) )
=> ( ( last @ A @ ( replicate @ A @ N @ X ) )
= X ) ) ).
% last_replicate
thf(fact_7144_last__upt,axiom,
! [I: nat,J2: nat] :
( ( ord_less @ nat @ I @ J2 )
=> ( ( last @ nat @ ( upt @ I @ J2 ) )
= ( minus_minus @ nat @ J2 @ ( one_one @ nat ) ) ) ) ).
% last_upt
thf(fact_7145_last__drop,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( last @ A @ ( drop @ A @ N @ Xs ) )
= ( last @ A @ Xs ) ) ) ).
% last_drop
thf(fact_7146_last__zip,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys2: list @ B] :
( ( Xs
!= ( nil @ A ) )
=> ( ( Ys2
!= ( nil @ B ) )
=> ( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys2 ) )
=> ( ( last @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys2 ) )
= ( product_Pair @ A @ B @ ( last @ A @ Xs ) @ ( last @ B @ Ys2 ) ) ) ) ) ) ).
% last_zip
thf(fact_7147_coset__def,axiom,
! [A: $tType] :
( ( coset @ A )
= ( ^ [Xs3: list @ A] : ( uminus_uminus @ ( set @ A ) @ ( set2 @ A @ Xs3 ) ) ) ) ).
% coset_def
thf(fact_7148_compl__coset,axiom,
! [A: $tType,Xs: list @ A] :
( ( uminus_uminus @ ( set @ A ) @ ( coset @ A @ Xs ) )
= ( set2 @ A @ Xs ) ) ).
% compl_coset
thf(fact_7149_last__conv__nth,axiom,
! [A: $tType,Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( last @ A @ Xs )
= ( nth @ A @ Xs @ ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) ) ) ) ) ).
% last_conv_nth
thf(fact_7150_flat__lub__def,axiom,
! [A: $tType] :
( ( partial_flat_lub @ A )
= ( ^ [B3: A,A7: set @ A] :
( if @ A @ ( ord_less_eq @ ( set @ A ) @ A7 @ ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) @ B3
@ ( the @ A
@ ^ [X2: A] : ( member @ A @ X2 @ ( minus_minus @ ( set @ A ) @ A7 @ ( insert @ A @ B3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% flat_lub_def
thf(fact_7151_in__measures_I2_J,axiom,
! [A: $tType,X: A,Y2: A,F2: A > nat,Fs: list @ ( A > nat )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y2 ) @ ( measures @ A @ ( cons @ ( A > nat ) @ F2 @ Fs ) ) )
= ( ( ord_less @ nat @ ( F2 @ X ) @ ( F2 @ Y2 ) )
| ( ( ( F2 @ X )
= ( F2 @ Y2 ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y2 ) @ ( measures @ A @ Fs ) ) ) ) ) ).
% in_measures(2)
thf(fact_7152_measures__less,axiom,
! [A: $tType,F2: A > nat,X: A,Y2: A,Fs: list @ ( A > nat )] :
( ( ord_less @ nat @ ( F2 @ X ) @ ( F2 @ Y2 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y2 ) @ ( measures @ A @ ( cons @ ( A > nat ) @ F2 @ Fs ) ) ) ) ).
% measures_less
thf(fact_7153_VEBT_Osimps_I7_J,axiom,
! [A: $tType,F1: ( option @ ( product_prod @ nat @ nat ) ) > nat > ( list @ ( product_prod @ vEBT_VEBT @ A ) ) > vEBT_VEBT > A > A,F22: $o > $o > A,X11: option @ ( product_prod @ nat @ nat ),X12: nat,X13: list @ vEBT_VEBT,X14: vEBT_VEBT] :
( ( vEBT_rec_VEBT @ A @ F1 @ F22 @ ( vEBT_Node @ X11 @ X12 @ X13 @ X14 ) )
= ( F1 @ X11 @ X12
@ ( map @ vEBT_VEBT @ ( product_prod @ vEBT_VEBT @ A )
@ ^ [VEBT: vEBT_VEBT] : ( product_Pair @ vEBT_VEBT @ A @ VEBT @ ( vEBT_rec_VEBT @ A @ F1 @ F22 @ VEBT ) )
@ X13 )
@ X14
@ ( vEBT_rec_VEBT @ A @ F1 @ F22 @ X14 ) ) ) ).
% VEBT.simps(7)
thf(fact_7154_and__not__num_Opelims,axiom,
! [X: num,Xa: num,Y2: option @ num] :
( ( ( bit_and_not_num @ X @ Xa )
= Y2 )
=> ( ( accp @ ( product_prod @ num @ num ) @ bit_and_not_num_rel @ ( product_Pair @ num @ num @ X @ Xa ) )
=> ( ( ( X = one2 )
=> ( ( Xa = one2 )
=> ( ( Y2
= ( none @ num ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_and_not_num_rel @ ( product_Pair @ num @ num @ one2 @ one2 ) ) ) ) )
=> ( ( ( X = one2 )
=> ! [N2: num] :
( ( Xa
= ( bit0 @ N2 ) )
=> ( ( Y2
= ( some @ num @ one2 ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_and_not_num_rel @ ( product_Pair @ num @ num @ one2 @ ( bit0 @ N2 ) ) ) ) ) )
=> ( ( ( X = one2 )
=> ! [N2: num] :
( ( Xa
= ( bit1 @ N2 ) )
=> ( ( Y2
= ( none @ num ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_and_not_num_rel @ ( product_Pair @ num @ num @ one2 @ ( bit1 @ N2 ) ) ) ) ) )
=> ( ! [M3: num] :
( ( X
= ( bit0 @ M3 ) )
=> ( ( Xa = one2 )
=> ( ( Y2
= ( some @ num @ ( bit0 @ M3 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_and_not_num_rel @ ( product_Pair @ num @ num @ ( bit0 @ M3 ) @ one2 ) ) ) ) )
=> ( ! [M3: num] :
( ( X
= ( bit0 @ M3 ) )
=> ! [N2: num] :
( ( Xa
= ( bit0 @ N2 ) )
=> ( ( Y2
= ( map_option @ num @ num @ bit0 @ ( bit_and_not_num @ M3 @ N2 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_and_not_num_rel @ ( product_Pair @ num @ num @ ( bit0 @ M3 ) @ ( bit0 @ N2 ) ) ) ) ) )
=> ( ! [M3: num] :
( ( X
= ( bit0 @ M3 ) )
=> ! [N2: num] :
( ( Xa
= ( bit1 @ N2 ) )
=> ( ( Y2
= ( map_option @ num @ num @ bit0 @ ( bit_and_not_num @ M3 @ N2 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_and_not_num_rel @ ( product_Pair @ num @ num @ ( bit0 @ M3 ) @ ( bit1 @ N2 ) ) ) ) ) )
=> ( ! [M3: num] :
( ( X
= ( bit1 @ M3 ) )
=> ( ( Xa = one2 )
=> ( ( Y2
= ( some @ num @ ( bit0 @ M3 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_and_not_num_rel @ ( product_Pair @ num @ num @ ( bit1 @ M3 ) @ one2 ) ) ) ) )
=> ( ! [M3: num] :
( ( X
= ( bit1 @ M3 ) )
=> ! [N2: num] :
( ( Xa
= ( bit0 @ N2 ) )
=> ( ( Y2
= ( case_option @ ( option @ num ) @ num @ ( some @ num @ one2 )
@ ^ [N8: num] : ( some @ num @ ( bit1 @ N8 ) )
@ ( bit_and_not_num @ M3 @ N2 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_and_not_num_rel @ ( product_Pair @ num @ num @ ( bit1 @ M3 ) @ ( bit0 @ N2 ) ) ) ) ) )
=> ~ ! [M3: num] :
( ( X
= ( bit1 @ M3 ) )
=> ! [N2: num] :
( ( Xa
= ( bit1 @ N2 ) )
=> ( ( Y2
= ( map_option @ num @ num @ bit0 @ ( bit_and_not_num @ M3 @ N2 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_and_not_num_rel @ ( product_Pair @ num @ num @ ( bit1 @ M3 ) @ ( bit1 @ N2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% and_not_num.pelims
thf(fact_7155_VEBT_Osimps_I8_J,axiom,
! [A: $tType,F1: ( option @ ( product_prod @ nat @ nat ) ) > nat > ( list @ ( product_prod @ vEBT_VEBT @ A ) ) > vEBT_VEBT > A > A,F22: $o > $o > A,X21: $o,X22: $o] :
( ( vEBT_rec_VEBT @ A @ F1 @ F22 @ ( vEBT_Leaf @ X21 @ X22 ) )
= ( F22 @ X21 @ X22 ) ) ).
% VEBT.simps(8)
thf(fact_7156_and__num_Opelims,axiom,
! [X: num,Xa: num,Y2: option @ num] :
( ( ( bit_un7362597486090784418nd_num @ X @ Xa )
= Y2 )
=> ( ( accp @ ( product_prod @ num @ num ) @ bit_un4731106466462545111um_rel @ ( product_Pair @ num @ num @ X @ Xa ) )
=> ( ( ( X = one2 )
=> ( ( Xa = one2 )
=> ( ( Y2
= ( some @ num @ one2 ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un4731106466462545111um_rel @ ( product_Pair @ num @ num @ one2 @ one2 ) ) ) ) )
=> ( ( ( X = one2 )
=> ! [N2: num] :
( ( Xa
= ( bit0 @ N2 ) )
=> ( ( Y2
= ( none @ num ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un4731106466462545111um_rel @ ( product_Pair @ num @ num @ one2 @ ( bit0 @ N2 ) ) ) ) ) )
=> ( ( ( X = one2 )
=> ! [N2: num] :
( ( Xa
= ( bit1 @ N2 ) )
=> ( ( Y2
= ( some @ num @ one2 ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un4731106466462545111um_rel @ ( product_Pair @ num @ num @ one2 @ ( bit1 @ N2 ) ) ) ) ) )
=> ( ! [M3: num] :
( ( X
= ( bit0 @ M3 ) )
=> ( ( Xa = one2 )
=> ( ( Y2
= ( none @ num ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un4731106466462545111um_rel @ ( product_Pair @ num @ num @ ( bit0 @ M3 ) @ one2 ) ) ) ) )
=> ( ! [M3: num] :
( ( X
= ( bit0 @ M3 ) )
=> ! [N2: num] :
( ( Xa
= ( bit0 @ N2 ) )
=> ( ( Y2
= ( map_option @ num @ num @ bit0 @ ( bit_un7362597486090784418nd_num @ M3 @ N2 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un4731106466462545111um_rel @ ( product_Pair @ num @ num @ ( bit0 @ M3 ) @ ( bit0 @ N2 ) ) ) ) ) )
=> ( ! [M3: num] :
( ( X
= ( bit0 @ M3 ) )
=> ! [N2: num] :
( ( Xa
= ( bit1 @ N2 ) )
=> ( ( Y2
= ( map_option @ num @ num @ bit0 @ ( bit_un7362597486090784418nd_num @ M3 @ N2 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un4731106466462545111um_rel @ ( product_Pair @ num @ num @ ( bit0 @ M3 ) @ ( bit1 @ N2 ) ) ) ) ) )
=> ( ! [M3: num] :
( ( X
= ( bit1 @ M3 ) )
=> ( ( Xa = one2 )
=> ( ( Y2
= ( some @ num @ one2 ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un4731106466462545111um_rel @ ( product_Pair @ num @ num @ ( bit1 @ M3 ) @ one2 ) ) ) ) )
=> ( ! [M3: num] :
( ( X
= ( bit1 @ M3 ) )
=> ! [N2: num] :
( ( Xa
= ( bit0 @ N2 ) )
=> ( ( Y2
= ( map_option @ num @ num @ bit0 @ ( bit_un7362597486090784418nd_num @ M3 @ N2 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un4731106466462545111um_rel @ ( product_Pair @ num @ num @ ( bit1 @ M3 ) @ ( bit0 @ N2 ) ) ) ) ) )
=> ~ ! [M3: num] :
( ( X
= ( bit1 @ M3 ) )
=> ! [N2: num] :
( ( Xa
= ( bit1 @ N2 ) )
=> ( ( Y2
= ( case_option @ ( option @ num ) @ num @ ( some @ num @ one2 )
@ ^ [N8: num] : ( some @ num @ ( bit1 @ N8 ) )
@ ( bit_un7362597486090784418nd_num @ M3 @ N2 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un4731106466462545111um_rel @ ( product_Pair @ num @ num @ ( bit1 @ M3 ) @ ( bit1 @ N2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% and_num.pelims
thf(fact_7157_xor__num_Opelims,axiom,
! [X: num,Xa: num,Y2: option @ num] :
( ( ( bit_un2480387367778600638or_num @ X @ Xa )
= Y2 )
=> ( ( accp @ ( product_prod @ num @ num ) @ bit_un2901131394128224187um_rel @ ( product_Pair @ num @ num @ X @ Xa ) )
=> ( ( ( X = one2 )
=> ( ( Xa = one2 )
=> ( ( Y2
= ( none @ num ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un2901131394128224187um_rel @ ( product_Pair @ num @ num @ one2 @ one2 ) ) ) ) )
=> ( ( ( X = one2 )
=> ! [N2: num] :
( ( Xa
= ( bit0 @ N2 ) )
=> ( ( Y2
= ( some @ num @ ( bit1 @ N2 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un2901131394128224187um_rel @ ( product_Pair @ num @ num @ one2 @ ( bit0 @ N2 ) ) ) ) ) )
=> ( ( ( X = one2 )
=> ! [N2: num] :
( ( Xa
= ( bit1 @ N2 ) )
=> ( ( Y2
= ( some @ num @ ( bit0 @ N2 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un2901131394128224187um_rel @ ( product_Pair @ num @ num @ one2 @ ( bit1 @ N2 ) ) ) ) ) )
=> ( ! [M3: num] :
( ( X
= ( bit0 @ M3 ) )
=> ( ( Xa = one2 )
=> ( ( Y2
= ( some @ num @ ( bit1 @ M3 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un2901131394128224187um_rel @ ( product_Pair @ num @ num @ ( bit0 @ M3 ) @ one2 ) ) ) ) )
=> ( ! [M3: num] :
( ( X
= ( bit0 @ M3 ) )
=> ! [N2: num] :
( ( Xa
= ( bit0 @ N2 ) )
=> ( ( Y2
= ( map_option @ num @ num @ bit0 @ ( bit_un2480387367778600638or_num @ M3 @ N2 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un2901131394128224187um_rel @ ( product_Pair @ num @ num @ ( bit0 @ M3 ) @ ( bit0 @ N2 ) ) ) ) ) )
=> ( ! [M3: num] :
( ( X
= ( bit0 @ M3 ) )
=> ! [N2: num] :
( ( Xa
= ( bit1 @ N2 ) )
=> ( ( Y2
= ( some @ num @ ( case_option @ num @ num @ one2 @ bit1 @ ( bit_un2480387367778600638or_num @ M3 @ N2 ) ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un2901131394128224187um_rel @ ( product_Pair @ num @ num @ ( bit0 @ M3 ) @ ( bit1 @ N2 ) ) ) ) ) )
=> ( ! [M3: num] :
( ( X
= ( bit1 @ M3 ) )
=> ( ( Xa = one2 )
=> ( ( Y2
= ( some @ num @ ( bit0 @ M3 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un2901131394128224187um_rel @ ( product_Pair @ num @ num @ ( bit1 @ M3 ) @ one2 ) ) ) ) )
=> ( ! [M3: num] :
( ( X
= ( bit1 @ M3 ) )
=> ! [N2: num] :
( ( Xa
= ( bit0 @ N2 ) )
=> ( ( Y2
= ( some @ num @ ( case_option @ num @ num @ one2 @ bit1 @ ( bit_un2480387367778600638or_num @ M3 @ N2 ) ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un2901131394128224187um_rel @ ( product_Pair @ num @ num @ ( bit1 @ M3 ) @ ( bit0 @ N2 ) ) ) ) ) )
=> ~ ! [M3: num] :
( ( X
= ( bit1 @ M3 ) )
=> ! [N2: num] :
( ( Xa
= ( bit1 @ N2 ) )
=> ( ( Y2
= ( map_option @ num @ num @ bit0 @ ( bit_un2480387367778600638or_num @ M3 @ N2 ) ) )
=> ~ ( accp @ ( product_prod @ num @ num ) @ bit_un2901131394128224187um_rel @ ( product_Pair @ num @ num @ ( bit1 @ M3 ) @ ( bit1 @ N2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% xor_num.pelims
thf(fact_7158_pair__lessI2,axiom,
! [A3: nat,B2: nat,S: nat,T2: nat] :
( ( ord_less_eq @ nat @ A3 @ B2 )
=> ( ( ord_less @ nat @ S @ T2 )
=> ( member @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ A3 @ S ) @ ( product_Pair @ nat @ nat @ B2 @ T2 ) ) @ fun_pair_less ) ) ) ).
% pair_lessI2
thf(fact_7159_horner__sum__transfer,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType] :
( ( ( comm_semiring_0 @ B )
& ( comm_semiring_0 @ A ) )
=> ! [A4: A > B > $o,B7: C > D > $o] :
( ( A4 @ ( zero_zero @ A ) @ ( zero_zero @ B ) )
=> ( ( bNF_rel_fun @ A @ B @ ( A > A ) @ ( B > B ) @ A4 @ ( bNF_rel_fun @ A @ B @ A @ B @ A4 @ A4 ) @ ( plus_plus @ A ) @ ( plus_plus @ B ) )
=> ( ( bNF_rel_fun @ A @ B @ ( A > A ) @ ( B > B ) @ A4 @ ( bNF_rel_fun @ A @ B @ A @ B @ A4 @ A4 ) @ ( times_times @ A ) @ ( times_times @ B ) )
=> ( bNF_rel_fun @ ( C > A ) @ ( D > B ) @ ( A > ( list @ C ) > A ) @ ( B > ( list @ D ) > B ) @ ( bNF_rel_fun @ C @ D @ A @ B @ B7 @ A4 ) @ ( bNF_rel_fun @ A @ B @ ( ( list @ C ) > A ) @ ( ( list @ D ) > B ) @ A4 @ ( bNF_rel_fun @ ( list @ C ) @ ( list @ D ) @ A @ B @ ( list_all2 @ C @ D @ B7 ) @ A4 ) ) @ ( groups4207007520872428315er_sum @ C @ A ) @ ( groups4207007520872428315er_sum @ D @ B ) ) ) ) ) ) ).
% horner_sum_transfer
thf(fact_7160_pair__less__iff1,axiom,
! [X: nat,Y2: nat,Z2: nat] :
( ( member @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ X @ Y2 ) @ ( product_Pair @ nat @ nat @ X @ Z2 ) ) @ fun_pair_less )
= ( ord_less @ nat @ Y2 @ Z2 ) ) ).
% pair_less_iff1
thf(fact_7161_list__all2__lengthD,axiom,
! [A: $tType,B: $tType,P: A > B > $o,Xs: list @ A,Ys2: list @ B] :
( ( list_all2 @ A @ B @ P @ Xs @ Ys2 )
=> ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys2 ) ) ) ).
% list_all2_lengthD
thf(fact_7162_length__transfer,axiom,
! [A: $tType,B: $tType,A4: A > B > $o] :
( bNF_rel_fun @ ( list @ A ) @ ( list @ B ) @ nat @ nat @ ( list_all2 @ A @ B @ A4 )
@ ^ [Y4: nat,Z: nat] : Y4 = Z
@ ( size_size @ ( list @ A ) )
@ ( size_size @ ( list @ B ) ) ) ).
% length_transfer
thf(fact_7163_list__all2__append,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys2: list @ B,P: A > B > $o,Us: list @ A,Vs: list @ B] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys2 ) )
=> ( ( list_all2 @ A @ B @ P @ ( append @ A @ Xs @ Us ) @ ( append @ B @ Ys2 @ Vs ) )
= ( ( list_all2 @ A @ B @ P @ Xs @ Ys2 )
& ( list_all2 @ A @ B @ P @ Us @ Vs ) ) ) ) ).
% list_all2_append
thf(fact_7164_list__all2__append1,axiom,
! [A: $tType,B: $tType,P: A > B > $o,Xs: list @ A,Ys2: list @ A,Zs: list @ B] :
( ( list_all2 @ A @ B @ P @ ( append @ A @ Xs @ Ys2 ) @ Zs )
= ( ? [Us2: list @ B,Vs2: list @ B] :
( ( Zs
= ( append @ B @ Us2 @ Vs2 ) )
& ( ( size_size @ ( list @ B ) @ Us2 )
= ( size_size @ ( list @ A ) @ Xs ) )
& ( ( size_size @ ( list @ B ) @ Vs2 )
= ( size_size @ ( list @ A ) @ Ys2 ) )
& ( list_all2 @ A @ B @ P @ Xs @ Us2 )
& ( list_all2 @ A @ B @ P @ Ys2 @ Vs2 ) ) ) ) ).
% list_all2_append1
thf(fact_7165_list__all2__append2,axiom,
! [B: $tType,A: $tType,P: A > B > $o,Xs: list @ A,Ys2: list @ B,Zs: list @ B] :
( ( list_all2 @ A @ B @ P @ Xs @ ( append @ B @ Ys2 @ Zs ) )
= ( ? [Us2: list @ A,Vs2: list @ A] :
( ( Xs
= ( append @ A @ Us2 @ Vs2 ) )
& ( ( size_size @ ( list @ A ) @ Us2 )
= ( size_size @ ( list @ B ) @ Ys2 ) )
& ( ( size_size @ ( list @ A ) @ Vs2 )
= ( size_size @ ( list @ B ) @ Zs ) )
& ( list_all2 @ A @ B @ P @ Us2 @ Ys2 )
& ( list_all2 @ A @ B @ P @ Vs2 @ Zs ) ) ) ) ).
% list_all2_append2
thf(fact_7166_list__all2__nthD,axiom,
! [A: $tType,B: $tType,P: A > B > $o,Xs: list @ A,Ys2: list @ B,P4: nat] :
( ( list_all2 @ A @ B @ P @ Xs @ Ys2 )
=> ( ( ord_less @ nat @ P4 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( P @ ( nth @ A @ Xs @ P4 ) @ ( nth @ B @ Ys2 @ P4 ) ) ) ) ).
% list_all2_nthD
thf(fact_7167_list__all2__nthD2,axiom,
! [A: $tType,B: $tType,P: A > B > $o,Xs: list @ A,Ys2: list @ B,P4: nat] :
( ( list_all2 @ A @ B @ P @ Xs @ Ys2 )
=> ( ( ord_less @ nat @ P4 @ ( size_size @ ( list @ B ) @ Ys2 ) )
=> ( P @ ( nth @ A @ Xs @ P4 ) @ ( nth @ B @ Ys2 @ P4 ) ) ) ) ).
% list_all2_nthD2
thf(fact_7168_list__all2__all__nthI,axiom,
! [A: $tType,B: $tType,A3: list @ A,B2: list @ B,P: A > B > $o] :
( ( ( size_size @ ( list @ A ) @ A3 )
= ( size_size @ ( list @ B ) @ B2 ) )
=> ( ! [N2: nat] :
( ( ord_less @ nat @ N2 @ ( size_size @ ( list @ A ) @ A3 ) )
=> ( P @ ( nth @ A @ A3 @ N2 ) @ ( nth @ B @ B2 @ N2 ) ) )
=> ( list_all2 @ A @ B @ P @ A3 @ B2 ) ) ) ).
% list_all2_all_nthI
thf(fact_7169_list__all2__conv__all__nth,axiom,
! [B: $tType,A: $tType] :
( ( list_all2 @ A @ B )
= ( ^ [P3: A > B > $o,Xs3: list @ A,Ys3: list @ B] :
( ( ( size_size @ ( list @ A ) @ Xs3 )
= ( size_size @ ( list @ B ) @ Ys3 ) )
& ! [I2: nat] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs3 ) )
=> ( P3 @ ( nth @ A @ Xs3 @ I2 ) @ ( nth @ B @ Ys3 @ I2 ) ) ) ) ) ) ).
% list_all2_conv_all_nth
thf(fact_7170_pair__lessI1,axiom,
! [A3: nat,B2: nat,S: nat,T2: nat] :
( ( ord_less @ nat @ A3 @ B2 )
=> ( member @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ A3 @ S ) @ ( product_Pair @ nat @ nat @ B2 @ T2 ) ) @ fun_pair_less ) ) ).
% pair_lessI1
thf(fact_7171_list__all2I,axiom,
! [A: $tType,B: $tType,A3: list @ A,B2: list @ B,P: A > B > $o] :
( ! [X3: product_prod @ A @ B] :
( ( member @ ( product_prod @ A @ B ) @ X3 @ ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ A3 @ B2 ) ) )
=> ( product_case_prod @ A @ B @ $o @ P @ X3 ) )
=> ( ( ( size_size @ ( list @ A ) @ A3 )
= ( size_size @ ( list @ B ) @ B2 ) )
=> ( list_all2 @ A @ B @ P @ A3 @ B2 ) ) ) ).
% list_all2I
thf(fact_7172_sum__list__transfer,axiom,
! [A: $tType,B: $tType] :
( ( ( monoid_add @ B )
& ( monoid_add @ A ) )
=> ! [A4: A > B > $o] :
( ( A4 @ ( zero_zero @ A ) @ ( zero_zero @ B ) )
=> ( ( bNF_rel_fun @ A @ B @ ( A > A ) @ ( B > B ) @ A4 @ ( bNF_rel_fun @ A @ B @ A @ B @ A4 @ A4 ) @ ( plus_plus @ A ) @ ( plus_plus @ B ) )
=> ( bNF_rel_fun @ ( list @ A ) @ ( list @ B ) @ A @ B @ ( list_all2 @ A @ B @ A4 ) @ A4 @ ( groups8242544230860333062m_list @ A ) @ ( groups8242544230860333062m_list @ B ) ) ) ) ) ).
% sum_list_transfer
thf(fact_7173_list__all2__iff,axiom,
! [B: $tType,A: $tType] :
( ( list_all2 @ A @ B )
= ( ^ [P3: A > B > $o,Xs3: list @ A,Ys3: list @ B] :
( ( ( size_size @ ( list @ A ) @ Xs3 )
= ( size_size @ ( list @ B ) @ Ys3 ) )
& ! [X2: product_prod @ A @ B] :
( ( member @ ( product_prod @ A @ B ) @ X2 @ ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs3 @ Ys3 ) ) )
=> ( product_case_prod @ A @ B @ $o @ P3 @ X2 ) ) ) ) ) ).
% list_all2_iff
thf(fact_7174_prod__list__transfer,axiom,
! [A: $tType,B: $tType] :
( ( ( monoid_mult @ B )
& ( monoid_mult @ A ) )
=> ! [A4: A > B > $o] :
( ( A4 @ ( one_one @ A ) @ ( one_one @ B ) )
=> ( ( bNF_rel_fun @ A @ B @ ( A > A ) @ ( B > B ) @ A4 @ ( bNF_rel_fun @ A @ B @ A @ B @ A4 @ A4 ) @ ( times_times @ A ) @ ( times_times @ B ) )
=> ( bNF_rel_fun @ ( list @ A ) @ ( list @ B ) @ A @ B @ ( list_all2 @ A @ B @ A4 ) @ A4 @ ( groups5270119922927024881d_list @ A ) @ ( groups5270119922927024881d_list @ B ) ) ) ) ) ).
% prod_list_transfer
thf(fact_7175_pair__leqI1,axiom,
! [A3: nat,B2: nat,S: nat,T2: nat] :
( ( ord_less @ nat @ A3 @ B2 )
=> ( member @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ A3 @ S ) @ ( product_Pair @ nat @ nat @ B2 @ T2 ) ) @ fun_pair_leq ) ) ).
% pair_leqI1
thf(fact_7176_prod__list_ONil,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ( ( groups5270119922927024881d_list @ A @ ( nil @ A ) )
= ( one_one @ A ) ) ) ).
% prod_list.Nil
thf(fact_7177_prod__list__zero__iff,axiom,
! [A: $tType] :
( ( ( semiring_1 @ A )
& ( semiri3467727345109120633visors @ A ) )
=> ! [Xs: list @ A] :
( ( ( groups5270119922927024881d_list @ A @ Xs )
= ( zero_zero @ A ) )
= ( member @ A @ ( zero_zero @ A ) @ ( set2 @ A @ Xs ) ) ) ) ).
% prod_list_zero_iff
thf(fact_7178_prod__list_Oeq__foldr,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ( ( groups5270119922927024881d_list @ A )
= ( ^ [Xs3: list @ A] : ( foldr @ A @ A @ ( times_times @ A ) @ Xs3 @ ( one_one @ A ) ) ) ) ) ).
% prod_list.eq_foldr
thf(fact_7179_bot_Oordering__top__axioms,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ( ordering_top @ A
@ ^ [X2: A,Y5: A] : ( ord_less_eq @ A @ Y5 @ X2 )
@ ^ [X2: A,Y5: A] : ( ord_less @ A @ Y5 @ X2 )
@ ( bot_bot @ A ) ) ) ).
% bot.ordering_top_axioms
thf(fact_7180_dropWhile__nth,axiom,
! [A: $tType,J2: nat,P: A > $o,Xs: list @ A] :
( ( ord_less @ nat @ J2 @ ( size_size @ ( list @ A ) @ ( dropWhile @ A @ P @ Xs ) ) )
=> ( ( nth @ A @ ( dropWhile @ A @ P @ Xs ) @ J2 )
= ( nth @ A @ Xs @ ( plus_plus @ nat @ J2 @ ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P @ Xs ) ) ) ) ) ) ).
% dropWhile_nth
thf(fact_7181_length__dropWhile__le,axiom,
! [A: $tType,P: A > $o,Xs: list @ A] : ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ ( dropWhile @ A @ P @ Xs ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) ).
% length_dropWhile_le
thf(fact_7182_dropWhile__eq__drop,axiom,
! [A: $tType] :
( ( dropWhile @ A )
= ( ^ [P3: A > $o,Xs3: list @ A] : ( drop @ A @ ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P3 @ Xs3 ) ) @ Xs3 ) ) ) ).
% dropWhile_eq_drop
thf(fact_7183_gcd__nat_Oordering__top__axioms,axiom,
( ordering_top @ nat @ ( dvd_dvd @ nat )
@ ^ [M5: nat,N5: nat] :
( ( dvd_dvd @ nat @ M5 @ N5 )
& ( M5 != N5 ) )
@ ( zero_zero @ nat ) ) ).
% gcd_nat.ordering_top_axioms
thf(fact_7184_top_Oordering__top__axioms,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ( ordering_top @ A @ ( ord_less_eq @ A ) @ ( ord_less @ A ) @ ( top_top @ A ) ) ) ).
% top.ordering_top_axioms
thf(fact_7185_bot__nat__0_Oordering__top__axioms,axiom,
( ordering_top @ nat
@ ^ [X2: nat,Y5: nat] : ( ord_less_eq @ nat @ Y5 @ X2 )
@ ^ [X2: nat,Y5: nat] : ( ord_less @ nat @ Y5 @ X2 )
@ ( zero_zero @ nat ) ) ).
% bot_nat_0.ordering_top_axioms
thf(fact_7186_find__dropWhile,axiom,
! [A: $tType] :
( ( find @ A )
= ( ^ [P3: A > $o,Xs3: list @ A] :
( case_list @ ( option @ A ) @ A @ ( none @ A )
@ ^ [X2: A,Xa4: list @ A] : ( some @ A @ X2 )
@ ( dropWhile @ A @ ( comp @ $o @ $o @ A @ (~) @ P3 ) @ Xs3 ) ) ) ) ).
% find_dropWhile
thf(fact_7187_set__encode__vimage__Suc,axiom,
! [A4: set @ nat] :
( ( nat_set_encode @ ( vimage @ nat @ nat @ suc @ A4 ) )
= ( divide_divide @ nat @ ( nat_set_encode @ A4 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% set_encode_vimage_Suc
thf(fact_7188_vimage__if,axiom,
! [B: $tType,A: $tType,C2: B,A4: set @ B,D2: B,B7: set @ A] :
( ( ( member @ B @ C2 @ A4 )
=> ( ( ( member @ B @ D2 @ A4 )
=> ( ( vimage @ A @ B
@ ^ [X2: A] : ( if @ B @ ( member @ A @ X2 @ B7 ) @ C2 @ D2 )
@ A4 )
= ( top_top @ ( set @ A ) ) ) )
& ( ~ ( member @ B @ D2 @ A4 )
=> ( ( vimage @ A @ B
@ ^ [X2: A] : ( if @ B @ ( member @ A @ X2 @ B7 ) @ C2 @ D2 )
@ A4 )
= B7 ) ) ) )
& ( ~ ( member @ B @ C2 @ A4 )
=> ( ( ( member @ B @ D2 @ A4 )
=> ( ( vimage @ A @ B
@ ^ [X2: A] : ( if @ B @ ( member @ A @ X2 @ B7 ) @ C2 @ D2 )
@ A4 )
= ( uminus_uminus @ ( set @ A ) @ B7 ) ) )
& ( ~ ( member @ B @ D2 @ A4 )
=> ( ( vimage @ A @ B
@ ^ [X2: A] : ( if @ B @ ( member @ A @ X2 @ B7 ) @ C2 @ D2 )
@ A4 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% vimage_if
thf(fact_7189_vimage__Compl,axiom,
! [A: $tType,B: $tType,F2: A > B,A4: set @ B] :
( ( vimage @ A @ B @ F2 @ ( uminus_uminus @ ( set @ B ) @ A4 ) )
= ( uminus_uminus @ ( set @ A ) @ ( vimage @ A @ B @ F2 @ A4 ) ) ) ).
% vimage_Compl
thf(fact_7190_vimage__Diff,axiom,
! [A: $tType,B: $tType,F2: A > B,A4: set @ B,B7: set @ B] :
( ( vimage @ A @ B @ F2 @ ( minus_minus @ ( set @ B ) @ A4 @ B7 ) )
= ( minus_minus @ ( set @ A ) @ ( vimage @ A @ B @ F2 @ A4 ) @ ( vimage @ A @ B @ F2 @ B7 ) ) ) ).
% vimage_Diff
thf(fact_7191_finite__vimage__Suc__iff,axiom,
! [F5: set @ nat] :
( ( finite_finite @ nat @ ( vimage @ nat @ nat @ suc @ F5 ) )
= ( finite_finite @ nat @ F5 ) ) ).
% finite_vimage_Suc_iff
thf(fact_7192_vimage__Suc__insert__Suc,axiom,
! [N: nat,A4: set @ nat] :
( ( vimage @ nat @ nat @ suc @ ( insert @ nat @ ( suc @ N ) @ A4 ) )
= ( insert @ nat @ N @ ( vimage @ nat @ nat @ suc @ A4 ) ) ) ).
% vimage_Suc_insert_Suc
thf(fact_7193_vimage__Suc__insert__0,axiom,
! [A4: set @ nat] :
( ( vimage @ nat @ nat @ suc @ ( insert @ nat @ ( zero_zero @ nat ) @ A4 ) )
= ( vimage @ nat @ nat @ suc @ A4 ) ) ).
% vimage_Suc_insert_0
thf(fact_7194_set__decode__div__2,axiom,
! [X: nat] :
( ( nat_set_decode @ ( divide_divide @ nat @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( vimage @ nat @ nat @ suc @ ( nat_set_decode @ X ) ) ) ).
% set_decode_div_2
thf(fact_7195_extract__def,axiom,
! [A: $tType] :
( ( extract @ A )
= ( ^ [P3: A > $o,Xs3: list @ A] :
( case_list @ ( option @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) ) @ A @ ( none @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) )
@ ^ [Y5: 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 @ (~) @ P3 ) @ Xs3 ) @ ( product_Pair @ A @ ( list @ A ) @ Y5 @ Ys3 ) ) )
@ ( dropWhile @ A @ ( comp @ $o @ $o @ A @ (~) @ P3 ) @ Xs3 ) ) ) ) ).
% extract_def
thf(fact_7196_euclidean__size__times__nonunit,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ! [A3: A,B2: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( B2
!= ( zero_zero @ A ) )
=> ( ~ ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ord_less @ nat @ ( euclid6346220572633701492n_size @ A @ B2 ) @ ( euclid6346220572633701492n_size @ A @ ( times_times @ A @ A3 @ B2 ) ) ) ) ) ) ) ).
% euclidean_size_times_nonunit
thf(fact_7197_euclidean__size__of__nat,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [N: nat] :
( ( euclid6346220572633701492n_size @ A @ ( semiring_1_of_nat @ A @ N ) )
= N ) ) ).
% euclidean_size_of_nat
thf(fact_7198_euclidean__size__eq__0__iff,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ! [B2: A] :
( ( ( euclid6346220572633701492n_size @ A @ B2 )
= ( zero_zero @ nat ) )
= ( B2
= ( zero_zero @ A ) ) ) ) ).
% euclidean_size_eq_0_iff
thf(fact_7199_size__0,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ( ( euclid6346220572633701492n_size @ A @ ( zero_zero @ A ) )
= ( zero_zero @ nat ) ) ) ).
% size_0
thf(fact_7200_euclidean__size__1,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ( ( euclid6346220572633701492n_size @ A @ ( one_one @ A ) )
= ( one_one @ nat ) ) ) ).
% euclidean_size_1
thf(fact_7201_euclidean__size__greater__0__iff,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ! [B2: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( euclid6346220572633701492n_size @ A @ B2 ) )
= ( B2
!= ( zero_zero @ A ) ) ) ) ).
% euclidean_size_greater_0_iff
thf(fact_7202_dvd__euclidean__size__eq__imp__dvd,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ! [A3: A,B2: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( ( euclid6346220572633701492n_size @ A @ A3 )
= ( euclid6346220572633701492n_size @ A @ B2 ) )
=> ( ( dvd_dvd @ A @ B2 @ A3 )
=> ( dvd_dvd @ A @ A3 @ B2 ) ) ) ) ) ).
% dvd_euclidean_size_eq_imp_dvd
thf(fact_7203_euclidean__size__unit,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ! [A3: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( euclid6346220572633701492n_size @ A @ A3 )
= ( euclid6346220572633701492n_size @ A @ ( one_one @ A ) ) ) ) ) ).
% euclidean_size_unit
thf(fact_7204_extract__Nil__code,axiom,
! [A: $tType,P: A > $o] :
( ( extract @ A @ P @ ( nil @ A ) )
= ( none @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) ) ) ).
% extract_Nil_code
thf(fact_7205_extract__None__iff,axiom,
! [A: $tType,P: A > $o,Xs: list @ A] :
( ( ( extract @ A @ P @ Xs )
= ( none @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) ) )
= ( ~ ? [X2: A] :
( ( member @ A @ X2 @ ( set2 @ A @ Xs ) )
& ( P @ X2 ) ) ) ) ).
% extract_None_iff
thf(fact_7206_unit__iff__euclidean__size,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ! [A3: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
= ( ( ( euclid6346220572633701492n_size @ A @ A3 )
= ( euclid6346220572633701492n_size @ A @ ( one_one @ A ) ) )
& ( A3
!= ( zero_zero @ A ) ) ) ) ) ).
% unit_iff_euclidean_size
thf(fact_7207_size__mult__mono,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ! [B2: A,A3: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ord_less_eq @ nat @ ( euclid6346220572633701492n_size @ A @ A3 ) @ ( euclid6346220572633701492n_size @ A @ ( times_times @ A @ A3 @ B2 ) ) ) ) ) ).
% size_mult_mono
thf(fact_7208_size__mult__mono_H,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ! [B2: A,A3: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ord_less_eq @ nat @ ( euclid6346220572633701492n_size @ A @ A3 ) @ ( euclid6346220572633701492n_size @ A @ ( times_times @ A @ B2 @ A3 ) ) ) ) ) ).
% size_mult_mono'
thf(fact_7209_euclidean__size__times__unit,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ! [A3: A,B2: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( euclid6346220572633701492n_size @ A @ ( times_times @ A @ A3 @ B2 ) )
= ( euclid6346220572633701492n_size @ A @ B2 ) ) ) ) ).
% euclidean_size_times_unit
thf(fact_7210_dvd__proper__imp__size__less,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ! [A3: A,B2: A] :
( ( dvd_dvd @ A @ A3 @ B2 )
=> ( ~ ( dvd_dvd @ A @ B2 @ A3 )
=> ( ( B2
!= ( zero_zero @ A ) )
=> ( ord_less @ nat @ ( euclid6346220572633701492n_size @ A @ A3 ) @ ( euclid6346220572633701492n_size @ A @ B2 ) ) ) ) ) ) ).
% dvd_proper_imp_size_less
thf(fact_7211_dvd__imp__size__le,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ! [A3: A,B2: A] :
( ( dvd_dvd @ A @ A3 @ B2 )
=> ( ( B2
!= ( zero_zero @ A ) )
=> ( ord_less_eq @ nat @ ( euclid6346220572633701492n_size @ A @ A3 ) @ ( euclid6346220572633701492n_size @ A @ B2 ) ) ) ) ) ).
% dvd_imp_size_le
thf(fact_7212_mod__size__less,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ! [B2: A,A3: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ord_less @ nat @ ( euclid6346220572633701492n_size @ A @ ( modulo_modulo @ A @ A3 @ B2 ) ) @ ( euclid6346220572633701492n_size @ A @ B2 ) ) ) ) ).
% mod_size_less
thf(fact_7213_divmod__cases,axiom,
! [A: $tType] :
( ( euclid3128863361964157862miring @ A )
=> ! [B2: A,A3: A] :
( ( ( B2
!= ( zero_zero @ A ) )
=> ( ( ( modulo_modulo @ A @ A3 @ B2 )
= ( zero_zero @ A ) )
=> ( A3
!= ( times_times @ A @ ( divide_divide @ A @ A3 @ B2 ) @ B2 ) ) ) )
=> ( ( ( B2
!= ( zero_zero @ A ) )
=> ! [Q3: A,R3: A] :
( ( ( euclid7384307370059645450egment @ A @ R3 )
= ( euclid7384307370059645450egment @ A @ B2 ) )
=> ( ( ord_less @ nat @ ( euclid6346220572633701492n_size @ A @ R3 ) @ ( euclid6346220572633701492n_size @ A @ B2 ) )
=> ( ( R3
!= ( zero_zero @ A ) )
=> ( ( ( divide_divide @ A @ A3 @ B2 )
= Q3 )
=> ( ( ( modulo_modulo @ A @ A3 @ B2 )
= R3 )
=> ( A3
!= ( plus_plus @ A @ ( times_times @ A @ Q3 @ B2 ) @ R3 ) ) ) ) ) ) ) )
=> ( B2
= ( zero_zero @ A ) ) ) ) ) ).
% divmod_cases
thf(fact_7214_mod__eqI,axiom,
! [A: $tType] :
( ( euclid3128863361964157862miring @ A )
=> ! [B2: A,R4: A,Q5: A,A3: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( ( euclid7384307370059645450egment @ A @ R4 )
= ( euclid7384307370059645450egment @ A @ B2 ) )
=> ( ( ord_less @ nat @ ( euclid6346220572633701492n_size @ A @ R4 ) @ ( euclid6346220572633701492n_size @ A @ B2 ) )
=> ( ( ( plus_plus @ A @ ( times_times @ A @ Q5 @ B2 ) @ R4 )
= A3 )
=> ( ( modulo_modulo @ A @ A3 @ B2 )
= R4 ) ) ) ) ) ) ).
% mod_eqI
thf(fact_7215_abs__division__segment,axiom,
! [K: int] :
( ( abs_abs @ int @ ( euclid7384307370059645450egment @ int @ K ) )
= ( one_one @ int ) ) ).
% abs_division_segment
thf(fact_7216_division__segment__1,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ( ( euclid7384307370059645450egment @ A @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% division_segment_1
thf(fact_7217_division__segment__numeral,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [K: num] :
( ( euclid7384307370059645450egment @ A @ ( numeral_numeral @ A @ K ) )
= ( one_one @ A ) ) ) ).
% division_segment_numeral
thf(fact_7218_division__segment__of__nat,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [N: nat] :
( ( euclid7384307370059645450egment @ A @ ( semiring_1_of_nat @ A @ N ) )
= ( one_one @ A ) ) ) ).
% division_segment_of_nat
thf(fact_7219_division__segment__euclidean__size,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [A3: A] :
( ( times_times @ A @ ( euclid7384307370059645450egment @ A @ A3 ) @ ( semiring_1_of_nat @ A @ ( euclid6346220572633701492n_size @ A @ A3 ) ) )
= A3 ) ) ).
% division_segment_euclidean_size
thf(fact_7220_division__segment__mult,axiom,
! [A: $tType] :
( ( euclid3128863361964157862miring @ A )
=> ! [A3: A,B2: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( B2
!= ( zero_zero @ A ) )
=> ( ( euclid7384307370059645450egment @ A @ ( times_times @ A @ A3 @ B2 ) )
= ( times_times @ A @ ( euclid7384307370059645450egment @ A @ A3 ) @ ( euclid7384307370059645450egment @ A @ B2 ) ) ) ) ) ) ).
% division_segment_mult
thf(fact_7221_division__segment__nat__def,axiom,
( ( euclid7384307370059645450egment @ nat )
= ( ^ [N5: nat] : ( one_one @ nat ) ) ) ).
% division_segment_nat_def
thf(fact_7222_division__segment__not__0,axiom,
! [A: $tType] :
( ( euclid3128863361964157862miring @ A )
=> ! [A3: A] :
( ( euclid7384307370059645450egment @ A @ A3 )
!= ( zero_zero @ A ) ) ) ).
% division_segment_not_0
thf(fact_7223_division__segment__eq__sgn,axiom,
! [K: int] :
( ( K
!= ( zero_zero @ int ) )
=> ( ( euclid7384307370059645450egment @ int @ K )
= ( sgn_sgn @ int @ K ) ) ) ).
% division_segment_eq_sgn
thf(fact_7224_is__unit__division__segment,axiom,
! [A: $tType] :
( ( euclid3128863361964157862miring @ A )
=> ! [A3: A] : ( dvd_dvd @ A @ ( euclid7384307370059645450egment @ A @ A3 ) @ ( one_one @ A ) ) ) ).
% is_unit_division_segment
thf(fact_7225_division__segment__mod,axiom,
! [A: $tType] :
( ( euclid3128863361964157862miring @ A )
=> ! [B2: A,A3: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ~ ( dvd_dvd @ A @ B2 @ A3 )
=> ( ( euclid7384307370059645450egment @ A @ ( modulo_modulo @ A @ A3 @ B2 ) )
= ( euclid7384307370059645450egment @ A @ B2 ) ) ) ) ) ).
% division_segment_mod
thf(fact_7226_of__nat__euclidean__size,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [A3: A] :
( ( semiring_1_of_nat @ A @ ( euclid6346220572633701492n_size @ A @ A3 ) )
= ( divide_divide @ A @ A3 @ ( euclid7384307370059645450egment @ A @ A3 ) ) ) ) ).
% of_nat_euclidean_size
thf(fact_7227_division__segment__int__def,axiom,
( ( euclid7384307370059645450egment @ int )
= ( ^ [K3: int] : ( if @ int @ ( ord_less_eq @ int @ ( zero_zero @ int ) @ K3 ) @ ( one_one @ int ) @ ( uminus_uminus @ int @ ( one_one @ int ) ) ) ) ) ).
% division_segment_int_def
thf(fact_7228_unique__euclidean__semiring__class_Odiv__eq__0__iff,axiom,
! [A: $tType] :
( ( euclid3128863361964157862miring @ A )
=> ! [A3: A,B2: A] :
( ( ( euclid7384307370059645450egment @ A @ A3 )
= ( euclid7384307370059645450egment @ A @ B2 ) )
=> ( ( ( divide_divide @ A @ A3 @ B2 )
= ( zero_zero @ A ) )
= ( ( ord_less @ nat @ ( euclid6346220572633701492n_size @ A @ A3 ) @ ( euclid6346220572633701492n_size @ A @ B2 ) )
| ( B2
= ( zero_zero @ A ) ) ) ) ) ) ).
% unique_euclidean_semiring_class.div_eq_0_iff
thf(fact_7229_div__bounded,axiom,
! [A: $tType] :
( ( euclid3128863361964157862miring @ A )
=> ! [B2: A,R4: A,Q5: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( ( euclid7384307370059645450egment @ A @ R4 )
= ( euclid7384307370059645450egment @ A @ B2 ) )
=> ( ( ord_less @ nat @ ( euclid6346220572633701492n_size @ A @ R4 ) @ ( euclid6346220572633701492n_size @ A @ B2 ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ Q5 @ B2 ) @ R4 ) @ B2 )
= Q5 ) ) ) ) ) ).
% div_bounded
thf(fact_7230_div__eqI,axiom,
! [A: $tType] :
( ( euclid3128863361964157862miring @ A )
=> ! [B2: A,R4: A,Q5: A,A3: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( ( euclid7384307370059645450egment @ A @ R4 )
= ( euclid7384307370059645450egment @ A @ B2 ) )
=> ( ( ord_less @ nat @ ( euclid6346220572633701492n_size @ A @ R4 ) @ ( euclid6346220572633701492n_size @ A @ B2 ) )
=> ( ( ( plus_plus @ A @ ( times_times @ A @ Q5 @ B2 ) @ R4 )
= A3 )
=> ( ( divide_divide @ A @ A3 @ B2 )
= Q5 ) ) ) ) ) ) ).
% div_eqI
thf(fact_7231_sorted__wrt__iff__nth__Suc__transp,axiom,
! [A: $tType,P: A > A > $o,Xs: list @ A] :
( ( transp @ A @ P )
=> ( ( sorted_wrt @ A @ P @ Xs )
= ( ! [I2: nat] :
( ( ord_less @ nat @ ( suc @ I2 ) @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( P @ ( nth @ A @ Xs @ I2 ) @ ( nth @ A @ Xs @ ( suc @ I2 ) ) ) ) ) ) ) ).
% sorted_wrt_iff_nth_Suc_transp
thf(fact_7232_mlex__eq,axiom,
! [A: $tType] :
( ( mlex_prod @ A )
= ( ^ [F4: A > nat,R6: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ A @ A )
@ ( product_case_prod @ A @ A @ $o
@ ^ [X2: A,Y5: A] :
( ( ord_less @ nat @ ( F4 @ X2 ) @ ( F4 @ Y5 ) )
| ( ( ord_less_eq @ nat @ ( F4 @ X2 ) @ ( F4 @ Y5 ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y5 ) @ R6 ) ) ) ) ) ) ) ).
% mlex_eq
thf(fact_7233_transp__realrel,axiom,
transp @ ( nat > rat ) @ realrel ).
% transp_realrel
thf(fact_7234_mlex__less,axiom,
! [A: $tType,F2: A > nat,X: A,Y2: A,R2: set @ ( product_prod @ A @ A )] :
( ( ord_less @ nat @ ( F2 @ X ) @ ( F2 @ Y2 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y2 ) @ ( mlex_prod @ A @ F2 @ R2 ) ) ) ).
% mlex_less
thf(fact_7235_mlex__iff,axiom,
! [A: $tType,X: A,Y2: A,F2: A > nat,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y2 ) @ ( mlex_prod @ A @ F2 @ R2 ) )
= ( ( ord_less @ nat @ ( F2 @ X ) @ ( F2 @ Y2 ) )
| ( ( ( F2 @ X )
= ( F2 @ Y2 ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y2 ) @ R2 ) ) ) ) ).
% mlex_iff
thf(fact_7236_in__measure,axiom,
! [A: $tType,X: A,Y2: A,F2: A > nat] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y2 ) @ ( measure @ A @ F2 ) )
= ( ord_less @ nat @ ( F2 @ X ) @ ( F2 @ Y2 ) ) ) ).
% in_measure
thf(fact_7237_transp__gr,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( transp @ A
@ ^ [X2: A,Y5: A] : ( ord_less @ A @ Y5 @ X2 ) ) ) ).
% transp_gr
thf(fact_7238_transp__less,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( transp @ A @ ( ord_less @ A ) ) ) ).
% transp_less
thf(fact_7239_pred__nat__def,axiom,
( pred_nat
= ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [M5: nat,N5: nat] :
( N5
= ( suc @ M5 ) ) ) ) ) ).
% pred_nat_def
thf(fact_7240_lexordp__conv__lexord,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_lexordp @ A )
= ( ^ [Xs3: list @ A,Ys3: list @ A] : ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs3 @ Ys3 ) @ ( lexord @ A @ ( collect @ ( product_prod @ A @ A ) @ ( product_case_prod @ A @ A @ $o @ ( ord_less @ A ) ) ) ) ) ) ) ) ).
% lexordp_conv_lexord
thf(fact_7241_lexordp__simps_I3_J,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X: A,Xs: list @ A,Y2: A,Ys2: list @ A] :
( ( ord_lexordp @ A @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y2 @ Ys2 ) )
= ( ( ord_less @ A @ X @ Y2 )
| ( ~ ( ord_less @ A @ Y2 @ X )
& ( ord_lexordp @ A @ Xs @ Ys2 ) ) ) ) ) ).
% lexordp_simps(3)
thf(fact_7242_lexordp__irreflexive,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [Xs: list @ A] :
( ! [X3: A] :
~ ( ord_less @ A @ X3 @ X3 )
=> ~ ( ord_lexordp @ A @ Xs @ Xs ) ) ) ).
% lexordp_irreflexive
thf(fact_7243_lexordp__append__leftD,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [Xs: list @ A,Us: list @ A,Vs: list @ A] :
( ( ord_lexordp @ A @ ( append @ A @ Xs @ Us ) @ ( append @ A @ Xs @ Vs ) )
=> ( ! [A6: A] :
~ ( ord_less @ A @ A6 @ A6 )
=> ( ord_lexordp @ A @ Us @ Vs ) ) ) ) ).
% lexordp_append_leftD
thf(fact_7244_lexordp_OCons__eq,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X: A,Y2: A,Xs: list @ A,Ys2: list @ A] :
( ~ ( ord_less @ A @ X @ Y2 )
=> ( ~ ( ord_less @ A @ Y2 @ X )
=> ( ( ord_lexordp @ A @ Xs @ Ys2 )
=> ( ord_lexordp @ A @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y2 @ Ys2 ) ) ) ) ) ) ).
% lexordp.Cons_eq
thf(fact_7245_lexordp_OCons,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X: A,Y2: A,Xs: list @ A,Ys2: list @ A] :
( ( ord_less @ A @ X @ Y2 )
=> ( ord_lexordp @ A @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y2 @ Ys2 ) ) ) ) ).
% lexordp.Cons
thf(fact_7246_lexordp__induct,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,Ys2: list @ A,P: ( list @ A ) > ( list @ A ) > $o] :
( ( ord_lexordp @ A @ Xs @ Ys2 )
=> ( ! [Y3: A,Ys6: list @ A] : ( P @ ( nil @ A ) @ ( cons @ A @ Y3 @ Ys6 ) )
=> ( ! [X3: A,Xs2: list @ A,Y3: A,Ys6: list @ A] :
( ( ord_less @ A @ X3 @ Y3 )
=> ( P @ ( cons @ A @ X3 @ Xs2 ) @ ( cons @ A @ Y3 @ Ys6 ) ) )
=> ( ! [X3: A,Xs2: list @ A,Ys6: list @ A] :
( ( ord_lexordp @ A @ Xs2 @ Ys6 )
=> ( ( P @ Xs2 @ Ys6 )
=> ( P @ ( cons @ A @ X3 @ Xs2 ) @ ( cons @ A @ X3 @ Ys6 ) ) ) )
=> ( P @ Xs @ Ys2 ) ) ) ) ) ) ).
% lexordp_induct
thf(fact_7247_lexordp__cases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,Ys2: list @ A] :
( ( ord_lexordp @ A @ Xs @ Ys2 )
=> ( ( ( Xs
= ( nil @ A ) )
=> ! [Y3: A,Ys5: list @ A] :
( Ys2
!= ( cons @ A @ Y3 @ Ys5 ) ) )
=> ( ! [X3: A] :
( ? [Xs5: list @ A] :
( Xs
= ( cons @ A @ X3 @ Xs5 ) )
=> ! [Y3: A] :
( ? [Ys5: list @ A] :
( Ys2
= ( cons @ A @ Y3 @ Ys5 ) )
=> ~ ( ord_less @ A @ X3 @ Y3 ) ) )
=> ~ ! [X3: A,Xs5: list @ A] :
( ( Xs
= ( cons @ A @ X3 @ Xs5 ) )
=> ! [Ys5: list @ A] :
( ( Ys2
= ( cons @ A @ X3 @ Ys5 ) )
=> ~ ( ord_lexordp @ A @ Xs5 @ Ys5 ) ) ) ) ) ) ) ).
% lexordp_cases
thf(fact_7248_lexordp_Osimps,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( ord_lexordp @ A )
= ( ^ [A12: list @ A,A23: list @ A] :
( ? [Y5: A,Ys3: list @ A] :
( ( A12
= ( nil @ A ) )
& ( A23
= ( cons @ A @ Y5 @ Ys3 ) ) )
| ? [X2: A,Y5: A,Xs3: list @ A,Ys3: list @ A] :
( ( A12
= ( cons @ A @ X2 @ Xs3 ) )
& ( A23
= ( cons @ A @ Y5 @ Ys3 ) )
& ( ord_less @ A @ X2 @ Y5 ) )
| ? [X2: A,Y5: A,Xs3: list @ A,Ys3: list @ A] :
( ( A12
= ( cons @ A @ X2 @ Xs3 ) )
& ( A23
= ( cons @ A @ Y5 @ Ys3 ) )
& ~ ( ord_less @ A @ X2 @ Y5 )
& ~ ( ord_less @ A @ Y5 @ X2 )
& ( ord_lexordp @ A @ Xs3 @ Ys3 ) ) ) ) ) ) ).
% lexordp.simps
thf(fact_7249_lexordp_Ocases,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A1: list @ A,A22: list @ A] :
( ( ord_lexordp @ A @ A1 @ A22 )
=> ( ( ( A1
= ( nil @ A ) )
=> ! [Y3: A,Ys6: list @ A] :
( A22
!= ( cons @ A @ Y3 @ Ys6 ) ) )
=> ( ! [X3: A] :
( ? [Xs2: list @ A] :
( A1
= ( cons @ A @ X3 @ Xs2 ) )
=> ! [Y3: A] :
( ? [Ys6: list @ A] :
( A22
= ( cons @ A @ Y3 @ Ys6 ) )
=> ~ ( ord_less @ A @ X3 @ Y3 ) ) )
=> ~ ! [X3: A,Y3: A,Xs2: list @ A] :
( ( A1
= ( cons @ A @ X3 @ Xs2 ) )
=> ! [Ys6: list @ A] :
( ( A22
= ( cons @ A @ Y3 @ Ys6 ) )
=> ( ~ ( ord_less @ A @ X3 @ Y3 )
=> ( ~ ( ord_less @ A @ Y3 @ X3 )
=> ~ ( ord_lexordp @ A @ Xs2 @ Ys6 ) ) ) ) ) ) ) ) ) ).
% lexordp.cases
thf(fact_7250_lexordp__append__left__rightI,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X: A,Y2: A,Us: list @ A,Xs: list @ A,Ys2: list @ A] :
( ( ord_less @ A @ X @ Y2 )
=> ( ord_lexordp @ A @ ( append @ A @ Us @ ( cons @ A @ X @ Xs ) ) @ ( append @ A @ Us @ ( cons @ A @ Y2 @ Ys2 ) ) ) ) ) ).
% lexordp_append_left_rightI
thf(fact_7251_lexordp__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_lexordp @ A )
= ( ^ [Xs3: list @ A,Ys3: list @ A] :
( ? [X2: A,Vs2: list @ A] :
( Ys3
= ( append @ A @ Xs3 @ ( cons @ A @ X2 @ Vs2 ) ) )
| ? [Us2: list @ A,A5: A,B3: A,Vs2: list @ A,Ws3: list @ A] :
( ( ord_less @ A @ A5 @ B3 )
& ( Xs3
= ( append @ A @ Us2 @ ( cons @ A @ A5 @ Vs2 ) ) )
& ( Ys3
= ( append @ A @ Us2 @ ( cons @ A @ B3 @ Ws3 ) ) ) ) ) ) ) ) ).
% lexordp_iff
thf(fact_7252_ord__class_Olexordp__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( ord_lexordp @ A )
= ( complete_lattice_lfp @ ( ( list @ A ) > ( list @ A ) > $o )
@ ^ [P6: ( list @ A ) > ( list @ A ) > $o,X16: list @ A,X25: list @ A] :
( ? [Y5: A,Ys3: list @ A] :
( ( X16
= ( nil @ A ) )
& ( X25
= ( cons @ A @ Y5 @ Ys3 ) ) )
| ? [X2: A,Y5: A,Xs3: list @ A,Ys3: list @ A] :
( ( X16
= ( cons @ A @ X2 @ Xs3 ) )
& ( X25
= ( cons @ A @ Y5 @ Ys3 ) )
& ( ord_less @ A @ X2 @ Y5 ) )
| ? [X2: A,Y5: A,Xs3: list @ A,Ys3: list @ A] :
( ( X16
= ( cons @ A @ X2 @ Xs3 ) )
& ( X25
= ( cons @ A @ Y5 @ Ys3 ) )
& ~ ( ord_less @ A @ X2 @ Y5 )
& ~ ( ord_less @ A @ Y5 @ X2 )
& ( P6 @ Xs3 @ Ys3 ) ) ) ) ) ) ).
% ord_class.lexordp_def
thf(fact_7253_less__enat__def,axiom,
( ( ord_less @ extended_enat )
= ( ^ [M5: extended_enat,N5: extended_enat] :
( extended_case_enat @ $o
@ ^ [M1: nat] : ( extended_case_enat @ $o @ ( ord_less @ nat @ M1 ) @ $true @ N5 )
@ $false
@ M5 ) ) ) ).
% less_enat_def
thf(fact_7254_lfp__funpow,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F2: A > A,N: nat] :
( ( order_mono @ A @ A @ F2 )
=> ( ( complete_lattice_lfp @ A @ ( compow @ ( A > A ) @ ( suc @ N ) @ F2 ) )
= ( complete_lattice_lfp @ A @ F2 ) ) ) ) ).
% lfp_funpow
thf(fact_7255_lfp__Kleene__iter,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F2: A > A,K: nat] :
( ( order_mono @ A @ A @ F2 )
=> ( ( ( compow @ ( A > A ) @ ( suc @ K ) @ F2 @ ( bot_bot @ A ) )
= ( compow @ ( A > A ) @ K @ F2 @ ( bot_bot @ A ) ) )
=> ( ( complete_lattice_lfp @ A @ F2 )
= ( compow @ ( A > A ) @ K @ F2 @ ( bot_bot @ A ) ) ) ) ) ) ).
% lfp_Kleene_iter
thf(fact_7256_Divides_Oadjust__div__def,axiom,
( adjust_div
= ( product_case_prod @ int @ int @ int
@ ^ [Q4: int,R: int] :
( plus_plus @ int @ Q4
@ ( zero_neq_one_of_bool @ int
@ ( R
!= ( zero_zero @ int ) ) ) ) ) ) ).
% Divides.adjust_div_def
thf(fact_7257_suminf__mono__reindex,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topological_t2_space @ A ) )
=> ! [G: nat > nat,F2: nat > A] :
( ( order_strict_mono @ nat @ nat @ G )
=> ( ! [N2: nat] :
( ~ ( member @ nat @ N2 @ ( image @ nat @ nat @ G @ ( top_top @ ( set @ nat ) ) ) )
=> ( ( F2 @ N2 )
= ( zero_zero @ A ) ) )
=> ( ( suminf @ A
@ ^ [N5: nat] : ( F2 @ ( G @ N5 ) ) )
= ( suminf @ A @ F2 ) ) ) ) ) ).
% suminf_mono_reindex
thf(fact_7258_strict__mono__Suc__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( order_strict_mono @ nat @ A )
= ( ^ [F4: nat > A] :
! [N5: nat] : ( ord_less @ A @ ( F4 @ N5 ) @ ( F4 @ ( suc @ N5 ) ) ) ) ) ) ).
% strict_mono_Suc_iff
thf(fact_7259_strict__monoD,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F2: A > B,X: A,Y2: A] :
( ( order_strict_mono @ A @ B @ F2 )
=> ( ( ord_less @ A @ X @ Y2 )
=> ( ord_less @ B @ ( F2 @ X ) @ ( F2 @ Y2 ) ) ) ) ) ).
% strict_monoD
thf(fact_7260_strict__monoI,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F2: A > B] :
( ! [X3: A,Y3: A] :
( ( ord_less @ A @ X3 @ Y3 )
=> ( ord_less @ B @ ( F2 @ X3 ) @ ( F2 @ Y3 ) ) )
=> ( order_strict_mono @ A @ B @ F2 ) ) ) ).
% strict_monoI
thf(fact_7261_strict__mono__def,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ( ( order_strict_mono @ A @ B )
= ( ^ [F4: A > B] :
! [X2: A,Y5: A] :
( ( ord_less @ A @ X2 @ Y5 )
=> ( ord_less @ B @ ( F4 @ X2 ) @ ( F4 @ Y5 ) ) ) ) ) ) ).
% strict_mono_def
thf(fact_7262_strict__mono__less,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( order @ B ) )
=> ! [F2: A > B,X: A,Y2: A] :
( ( order_strict_mono @ A @ B @ F2 )
=> ( ( ord_less @ B @ ( F2 @ X ) @ ( F2 @ Y2 ) )
= ( ord_less @ A @ X @ Y2 ) ) ) ) ).
% strict_mono_less
thf(fact_7263_strict__mono__imp__increasing,axiom,
! [F2: nat > nat,N: nat] :
( ( order_strict_mono @ nat @ nat @ F2 )
=> ( ord_less_eq @ nat @ N @ ( F2 @ N ) ) ) ).
% strict_mono_imp_increasing
thf(fact_7264_rat__floor__code,axiom,
( ( archim6421214686448440834_floor @ rat )
= ( ^ [P6: rat] : ( product_case_prod @ int @ int @ int @ ( divide_divide @ int ) @ ( quotient_of @ P6 ) ) ) ) ).
% rat_floor_code
thf(fact_7265_summable__mono__reindex,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [G: nat > nat,F2: nat > A] :
( ( order_strict_mono @ nat @ nat @ G )
=> ( ! [N2: nat] :
( ~ ( member @ nat @ N2 @ ( image @ nat @ nat @ G @ ( top_top @ ( set @ nat ) ) ) )
=> ( ( F2 @ N2 )
= ( zero_zero @ A ) ) )
=> ( ( summable @ A
@ ^ [N5: nat] : ( F2 @ ( G @ N5 ) ) )
= ( summable @ A @ F2 ) ) ) ) ) ).
% summable_mono_reindex
thf(fact_7266_sums__mono__reindex,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [G: nat > nat,F2: nat > A,C2: A] :
( ( order_strict_mono @ nat @ nat @ G )
=> ( ! [N2: nat] :
( ~ ( member @ nat @ N2 @ ( image @ nat @ nat @ G @ ( top_top @ ( set @ nat ) ) ) )
=> ( ( F2 @ N2 )
= ( zero_zero @ A ) ) )
=> ( ( sums @ A
@ ^ [N5: nat] : ( F2 @ ( G @ N5 ) )
@ C2 )
= ( sums @ A @ F2 @ C2 ) ) ) ) ) ).
% sums_mono_reindex
thf(fact_7267_prod__decode__triangle__add,axiom,
! [K: nat,M2: nat] :
( ( nat_prod_decode @ ( plus_plus @ nat @ ( nat_triangle @ K ) @ M2 ) )
= ( nat_prod_decode_aux @ K @ M2 ) ) ).
% prod_decode_triangle_add
thf(fact_7268_map__comp__def,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( map_comp @ B @ C @ A )
= ( ^ [F4: B > ( option @ C ),G2: A > ( option @ B ),K3: A] : ( case_option @ ( option @ C ) @ B @ ( none @ C ) @ F4 @ ( G2 @ K3 ) ) ) ) ).
% map_comp_def
thf(fact_7269_prod__decode__eq,axiom,
! [X: nat,Y2: nat] :
( ( ( nat_prod_decode @ X )
= ( nat_prod_decode @ Y2 ) )
= ( X = Y2 ) ) ).
% prod_decode_eq
thf(fact_7270_map__comp__simps_I1_J,axiom,
! [B: $tType,A: $tType,C: $tType,M22: B > ( option @ A ),K: B,M12: A > ( option @ C )] :
( ( ( M22 @ K )
= ( none @ A ) )
=> ( ( map_comp @ A @ C @ B @ M12 @ M22 @ K )
= ( none @ C ) ) ) ).
% map_comp_simps(1)
thf(fact_7271_prod__decode__inverse,axiom,
! [N: nat] :
( ( nat_prod_encode @ ( nat_prod_decode @ N ) )
= N ) ).
% prod_decode_inverse
thf(fact_7272_prod__encode__inverse,axiom,
! [X: product_prod @ nat @ nat] :
( ( nat_prod_decode @ ( nat_prod_encode @ X ) )
= X ) ).
% prod_encode_inverse
thf(fact_7273_map__comp__empty_I2_J,axiom,
! [B: $tType,D: $tType,C: $tType,M2: C > ( option @ B )] :
( ( map_comp @ B @ D @ C
@ ^ [X2: B] : ( none @ D )
@ M2 )
= ( ^ [X2: C] : ( none @ D ) ) ) ).
% map_comp_empty(2)
thf(fact_7274_map__comp__empty_I1_J,axiom,
! [C: $tType,B: $tType,A: $tType,M2: C > ( option @ B )] :
( ( map_comp @ C @ B @ A @ M2
@ ^ [X2: A] : ( none @ C ) )
= ( ^ [X2: A] : ( none @ B ) ) ) ).
% map_comp_empty(1)
thf(fact_7275_tanh__real__strict__mono,axiom,
order_strict_mono @ real @ real @ ( tanh @ real ) ).
% tanh_real_strict_mono
thf(fact_7276_sinh__real__strict__mono,axiom,
order_strict_mono @ real @ real @ ( sinh @ real ) ).
% sinh_real_strict_mono
thf(fact_7277_inj__prod__decode,axiom,
! [A4: set @ nat] : ( inj_on @ nat @ ( product_prod @ nat @ nat ) @ nat_prod_decode @ A4 ) ).
% inj_prod_decode
thf(fact_7278_prod__decode__def,axiom,
( nat_prod_decode
= ( nat_prod_decode_aux @ ( zero_zero @ nat ) ) ) ).
% prod_decode_def
thf(fact_7279_bij__prod__decode,axiom,
bij_betw @ nat @ ( product_prod @ nat @ nat ) @ nat_prod_decode @ ( top_top @ ( set @ nat ) ) @ ( top_top @ ( set @ ( product_prod @ nat @ nat ) ) ) ).
% bij_prod_decode
thf(fact_7280_surj__prod__decode,axiom,
( ( image @ nat @ ( product_prod @ nat @ nat ) @ nat_prod_decode @ ( top_top @ ( set @ nat ) ) )
= ( top_top @ ( set @ ( product_prod @ nat @ nat ) ) ) ) ).
% surj_prod_decode
thf(fact_7281_map__comp__None__iff,axiom,
! [A: $tType,C: $tType,B: $tType,M12: B > ( option @ A ),M22: C > ( option @ B ),K: C] :
( ( ( map_comp @ B @ A @ C @ M12 @ M22 @ K )
= ( none @ A ) )
= ( ( ( M22 @ K )
= ( none @ B ) )
| ? [K9: B] :
( ( ( M22 @ K )
= ( some @ B @ K9 ) )
& ( ( M12 @ K9 )
= ( none @ A ) ) ) ) ) ).
% map_comp_None_iff
thf(fact_7282_list__decode_Opinduct,axiom,
! [A0: nat,P: nat > $o] :
( ( accp @ nat @ nat_list_decode_rel @ A0 )
=> ( ( ( accp @ nat @ nat_list_decode_rel @ ( zero_zero @ nat ) )
=> ( P @ ( zero_zero @ nat ) ) )
=> ( ! [N2: nat] :
( ( accp @ nat @ nat_list_decode_rel @ ( suc @ N2 ) )
=> ( ! [X4: nat,Y: nat] :
( ( ( product_Pair @ nat @ nat @ X4 @ Y )
= ( nat_prod_decode @ N2 ) )
=> ( P @ Y ) )
=> ( P @ ( suc @ N2 ) ) ) )
=> ( P @ A0 ) ) ) ) ).
% list_decode.pinduct
thf(fact_7283_list__decode_Oelims,axiom,
! [X: nat,Y2: list @ nat] :
( ( ( nat_list_decode @ X )
= Y2 )
=> ( ( ( X
= ( zero_zero @ nat ) )
=> ( Y2
!= ( nil @ nat ) ) )
=> ~ ! [N2: nat] :
( ( X
= ( suc @ N2 ) )
=> ( Y2
!= ( product_case_prod @ nat @ nat @ ( list @ nat )
@ ^ [X2: nat,Y5: nat] : ( cons @ nat @ X2 @ ( nat_list_decode @ Y5 ) )
@ ( nat_prod_decode @ N2 ) ) ) ) ) ) ).
% list_decode.elims
thf(fact_7284_list__decode__inverse,axiom,
! [N: nat] :
( ( nat_list_encode @ ( nat_list_decode @ N ) )
= N ) ).
% list_decode_inverse
thf(fact_7285_list__encode__inverse,axiom,
! [X: list @ nat] :
( ( nat_list_decode @ ( nat_list_encode @ X ) )
= X ) ).
% list_encode_inverse
thf(fact_7286_list__decode_Opsimps_I1_J,axiom,
( ( accp @ nat @ nat_list_decode_rel @ ( zero_zero @ nat ) )
=> ( ( nat_list_decode @ ( zero_zero @ nat ) )
= ( nil @ nat ) ) ) ).
% list_decode.psimps(1)
thf(fact_7287_inj__list__decode,axiom,
! [A4: set @ nat] : ( inj_on @ nat @ ( list @ nat ) @ nat_list_decode @ A4 ) ).
% inj_list_decode
thf(fact_7288_list__decode__eq,axiom,
! [X: nat,Y2: nat] :
( ( ( nat_list_decode @ X )
= ( nat_list_decode @ Y2 ) )
= ( X = Y2 ) ) ).
% list_decode_eq
thf(fact_7289_list__decode_Osimps_I1_J,axiom,
( ( nat_list_decode @ ( zero_zero @ nat ) )
= ( nil @ nat ) ) ).
% list_decode.simps(1)
thf(fact_7290_list__decode_Opsimps_I2_J,axiom,
! [N: nat] :
( ( accp @ nat @ nat_list_decode_rel @ ( suc @ N ) )
=> ( ( nat_list_decode @ ( suc @ N ) )
= ( product_case_prod @ nat @ nat @ ( list @ nat )
@ ^ [X2: nat,Y5: nat] : ( cons @ nat @ X2 @ ( nat_list_decode @ Y5 ) )
@ ( nat_prod_decode @ N ) ) ) ) ).
% list_decode.psimps(2)
thf(fact_7291_bij__list__decode,axiom,
bij_betw @ nat @ ( list @ nat ) @ nat_list_decode @ ( top_top @ ( set @ nat ) ) @ ( top_top @ ( set @ ( list @ nat ) ) ) ).
% bij_list_decode
thf(fact_7292_surj__list__decode,axiom,
( ( image @ nat @ ( list @ nat ) @ nat_list_decode @ ( top_top @ ( set @ nat ) ) )
= ( top_top @ ( set @ ( list @ nat ) ) ) ) ).
% surj_list_decode
thf(fact_7293_list__decode_Osimps_I2_J,axiom,
! [N: nat] :
( ( nat_list_decode @ ( suc @ N ) )
= ( product_case_prod @ nat @ nat @ ( list @ nat )
@ ^ [X2: nat,Y5: nat] : ( cons @ nat @ X2 @ ( nat_list_decode @ Y5 ) )
@ ( nat_prod_decode @ N ) ) ) ).
% list_decode.simps(2)
thf(fact_7294_list__decode_Opelims,axiom,
! [X: nat,Y2: list @ nat] :
( ( ( nat_list_decode @ X )
= Y2 )
=> ( ( accp @ nat @ nat_list_decode_rel @ X )
=> ( ( ( X
= ( zero_zero @ nat ) )
=> ( ( Y2
= ( nil @ nat ) )
=> ~ ( accp @ nat @ nat_list_decode_rel @ ( zero_zero @ nat ) ) ) )
=> ~ ! [N2: nat] :
( ( X
= ( suc @ N2 ) )
=> ( ( Y2
= ( product_case_prod @ nat @ nat @ ( list @ nat )
@ ^ [X2: nat,Y5: nat] : ( cons @ nat @ X2 @ ( nat_list_decode @ Y5 ) )
@ ( nat_prod_decode @ N2 ) ) )
=> ~ ( accp @ nat @ nat_list_decode_rel @ ( suc @ N2 ) ) ) ) ) ) ) ).
% list_decode.pelims
thf(fact_7295_Max_Osemilattice__order__set__axioms,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( lattic4895041142388067077er_set @ A @ ( ord_max @ A )
@ ^ [X2: A,Y5: A] : ( ord_less_eq @ A @ Y5 @ X2 )
@ ^ [X2: A,Y5: A] : ( ord_less @ A @ Y5 @ X2 ) ) ) ).
% Max.semilattice_order_set_axioms
thf(fact_7296_Gcd__fin__def,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ( ( semiring_gcd_Gcd_fin @ A )
= ( bounde2362111253966948842tice_F @ A @ ( gcd_gcd @ A ) @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ) ).
% Gcd_fin_def
thf(fact_7297_Inf__fin_Osemilattice__order__set__axioms,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( lattic4895041142388067077er_set @ A @ ( inf_inf @ A ) @ ( ord_less_eq @ A ) @ ( ord_less @ A ) ) ) ).
% Inf_fin.semilattice_order_set_axioms
thf(fact_7298_Min_Osemilattice__order__set__axioms,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( lattic4895041142388067077er_set @ A @ ( ord_min @ A ) @ ( ord_less_eq @ A ) @ ( ord_less @ A ) ) ) ).
% Min.semilattice_order_set_axioms
thf(fact_7299_Sup__fin_Osemilattice__order__set__axioms,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( lattic4895041142388067077er_set @ A @ ( sup_sup @ A )
@ ^ [X2: A,Y5: A] : ( ord_less_eq @ A @ Y5 @ X2 )
@ ^ [X2: A,Y5: A] : ( ord_less @ A @ Y5 @ X2 ) ) ) ).
% Sup_fin.semilattice_order_set_axioms
thf(fact_7300_bounded__quasi__semilattice__set_Oremove,axiom,
! [A: $tType,F2: A > A > A,Top: A,Bot: A,Normalize: A > A,A3: A,A4: set @ A] :
( ( bounde6485984586167503788ce_set @ A @ F2 @ Top @ Bot @ Normalize )
=> ( ( member @ A @ A3 @ A4 )
=> ( ( bounde2362111253966948842tice_F @ A @ F2 @ Top @ Bot @ A4 )
= ( F2 @ A3 @ ( bounde2362111253966948842tice_F @ A @ F2 @ Top @ Bot @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% bounded_quasi_semilattice_set.remove
thf(fact_7301_bounded__quasi__semilattice__set_Oinsert__remove,axiom,
! [A: $tType,F2: A > A > A,Top: A,Bot: A,Normalize: A > A,A3: A,A4: set @ A] :
( ( bounde6485984586167503788ce_set @ A @ F2 @ Top @ Bot @ Normalize )
=> ( ( bounde2362111253966948842tice_F @ A @ F2 @ Top @ Bot @ ( insert @ A @ A3 @ A4 ) )
= ( F2 @ A3 @ ( bounde2362111253966948842tice_F @ A @ F2 @ Top @ Bot @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% bounded_quasi_semilattice_set.insert_remove
thf(fact_7302_gen__length__def,axiom,
! [A: $tType] :
( ( gen_length @ A )
= ( ^ [N5: nat,Xs3: list @ A] : ( plus_plus @ nat @ N5 @ ( size_size @ ( list @ A ) @ Xs3 ) ) ) ) ).
% gen_length_def
thf(fact_7303_compute__powr__real,axiom,
( powr_real
= ( ^ [B3: real,I2: real] :
( if @ real @ ( ord_less_eq @ real @ B3 @ ( zero_zero @ real ) )
@ ( abort @ real @ ( literal2 @ $false @ $false @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $false @ $true @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $true @ $true @ $false @ $true @ ( literal2 @ $false @ $true @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $true @ $false @ $false @ $true @ $true @ ( literal2 @ $true @ $false @ $false @ $false @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $false @ $false @ $true @ $false @ ( literal2 @ $true @ $true @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $false @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $false @ $false @ $true @ $false @ ( literal2 @ $false @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $true @ $true @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $false @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $false @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $true @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $true @ $false @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $false @ $false @ $true @ $false @ ( literal2 @ $false @ $true @ $false @ $false @ $false @ $true @ $true @ ( literal2 @ $true @ $false @ $false @ $false @ $false @ $true @ $true @ ( literal2 @ $true @ $true @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $true @ $false @ $false @ $true @ $true @ ( zero_zero @ literal ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
@ ^ [Uu3: product_unit] : ( powr_real @ B3 @ I2 ) )
@ ( if @ real
@ ( ( ring_1_of_int @ real @ ( archim6421214686448440834_floor @ real @ I2 ) )
= I2 )
@ ( if @ real @ ( ord_less_eq @ real @ ( zero_zero @ real ) @ I2 ) @ ( power_power @ real @ B3 @ ( nat2 @ ( archim6421214686448440834_floor @ real @ I2 ) ) ) @ ( divide_divide @ real @ ( one_one @ real ) @ ( power_power @ real @ B3 @ ( nat2 @ ( archim6421214686448440834_floor @ real @ ( uminus_uminus @ real @ I2 ) ) ) ) ) )
@ ( abort @ real @ ( literal2 @ $false @ $false @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $false @ $true @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $true @ $true @ $false @ $true @ ( literal2 @ $false @ $true @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $true @ $false @ $false @ $true @ $true @ ( literal2 @ $true @ $false @ $false @ $false @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $false @ $false @ $true @ $false @ ( literal2 @ $true @ $true @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $false @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $false @ $false @ $true @ $false @ ( literal2 @ $false @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $true @ $false @ $true @ $true @ $false @ $true @ $false @ ( literal2 @ $true @ $false @ $false @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $true @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $false @ $true @ $false @ $false @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $false @ $false @ $true @ $true @ ( literal2 @ $true @ $false @ $true @ $false @ $false @ $true @ $true @ ( literal2 @ $false @ $true @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $false @ $false @ $true @ $false @ ( literal2 @ $true @ $false @ $true @ $false @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $true @ $true @ $true @ $true @ ( literal2 @ $false @ $false @ $false @ $false @ $true @ $true @ $true @ ( literal2 @ $true @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $true @ $false @ $true @ $false @ $false @ $true @ $true @ ( literal2 @ $false @ $true @ $true @ $true @ $false @ $true @ $true @ ( literal2 @ $false @ $false @ $true @ $false @ $true @ $true @ $true @ ( zero_zero @ literal ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
@ ^ [Uu3: product_unit] : ( powr_real @ B3 @ I2 ) ) ) ) ) ) ).
% compute_powr_real
thf(fact_7304_powr__real__def,axiom,
( powr_real
= ( powr @ real ) ) ).
% powr_real_def
thf(fact_7305_gen__length__code_I2_J,axiom,
! [B: $tType,N: nat,X: B,Xs: list @ B] :
( ( gen_length @ B @ N @ ( cons @ B @ X @ Xs ) )
= ( gen_length @ B @ ( suc @ N ) @ Xs ) ) ).
% gen_length_code(2)
thf(fact_7306_length__code,axiom,
! [A: $tType] :
( ( size_size @ ( list @ A ) )
= ( gen_length @ A @ ( zero_zero @ nat ) ) ) ).
% length_code
thf(fact_7307_integer__of__nat_Orep__eq,axiom,
! [X: nat] :
( ( code_int_of_integer @ ( code_integer_of_nat @ X ) )
= ( semiring_1_of_nat @ int @ X ) ) ).
% integer_of_nat.rep_eq
thf(fact_7308_int__of__integer__integer__of__nat,axiom,
! [N: nat] :
( ( code_int_of_integer @ ( code_integer_of_nat @ N ) )
= ( semiring_1_of_nat @ int @ N ) ) ).
% int_of_integer_integer_of_nat
thf(fact_7309_integer__of__nat__0,axiom,
( ( code_integer_of_nat @ ( zero_zero @ nat ) )
= ( zero_zero @ code_integer ) ) ).
% integer_of_nat_0
thf(fact_7310_integer__of__nat_Oabs__eq,axiom,
( code_integer_of_nat
= ( ^ [X2: nat] : ( code_integer_of_int @ ( semiring_1_of_nat @ int @ X2 ) ) ) ) ).
% integer_of_nat.abs_eq
thf(fact_7311_integer__of__nat__1,axiom,
( ( code_integer_of_nat @ ( one_one @ nat ) )
= ( one_one @ code_integer ) ) ).
% integer_of_nat_1
thf(fact_7312_integer__of__nat__def,axiom,
( code_integer_of_nat
= ( map_fun @ nat @ nat @ int @ code_integer @ ( id @ nat ) @ code_integer_of_int @ ( semiring_1_of_nat @ int ) ) ) ).
% integer_of_nat_def
thf(fact_7313_pairs__le__eq__Sigma,axiom,
! [M2: nat] :
( ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [I2: nat,J3: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ I2 @ J3 ) @ M2 ) ) )
= ( product_Sigma @ nat @ nat @ ( set_ord_atMost @ nat @ M2 )
@ ^ [R: nat] : ( set_ord_atMost @ nat @ ( minus_minus @ nat @ M2 @ R ) ) ) ) ).
% pairs_le_eq_Sigma
thf(fact_7314_Sigma__interval__disjoint,axiom,
! [A: $tType,B: $tType] :
( ( order @ A )
=> ! [A4: set @ B,V2: B > A,W: A] :
( ( inf_inf @ ( set @ ( product_prod @ B @ A ) )
@ ( product_Sigma @ B @ A @ A4
@ ^ [I2: B] : ( set_ord_atMost @ A @ ( V2 @ I2 ) ) )
@ ( product_Sigma @ B @ A @ A4
@ ^ [I2: B] : ( set_or3652927894154168847AtMost @ A @ ( V2 @ I2 ) @ W ) ) )
= ( bot_bot @ ( set @ ( product_prod @ B @ A ) ) ) ) ) ).
% Sigma_interval_disjoint
thf(fact_7315_product__atMost__eq__Un,axiom,
! [A4: set @ nat,M2: nat] :
( ( product_Sigma @ nat @ nat @ A4
@ ^ [Uu3: nat] : ( set_ord_atMost @ nat @ M2 ) )
= ( sup_sup @ ( set @ ( product_prod @ nat @ nat ) )
@ ( product_Sigma @ nat @ nat @ A4
@ ^ [I2: nat] : ( set_ord_atMost @ nat @ ( minus_minus @ nat @ M2 @ I2 ) ) )
@ ( product_Sigma @ nat @ nat @ A4
@ ^ [I2: nat] : ( set_or3652927894154168847AtMost @ nat @ ( minus_minus @ nat @ M2 @ I2 ) @ M2 ) ) ) ) ).
% product_atMost_eq_Un
thf(fact_7316_Ex__inj__on__UNION__Sigma,axiom,
! [A: $tType,B: $tType,A4: B > ( set @ A ),I5: set @ B] :
? [F3: A > ( product_prod @ B @ A )] :
( ( inj_on @ A @ ( product_prod @ B @ A ) @ F3 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ A4 @ I5 ) ) )
& ( ord_less_eq @ ( set @ ( product_prod @ B @ A ) ) @ ( image @ A @ ( product_prod @ B @ A ) @ F3 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ A4 @ I5 ) ) ) @ ( product_Sigma @ B @ A @ I5 @ A4 ) ) ) ).
% Ex_inj_on_UNION_Sigma
thf(fact_7317_Compl__Times__UNIV1,axiom,
! [B: $tType,A: $tType,A4: set @ B] :
( ( uminus_uminus @ ( set @ ( product_prod @ A @ B ) )
@ ( product_Sigma @ A @ B @ ( top_top @ ( set @ A ) )
@ ^ [Uu3: A] : A4 ) )
= ( product_Sigma @ A @ B @ ( top_top @ ( set @ A ) )
@ ^ [Uu3: A] : ( uminus_uminus @ ( set @ B ) @ A4 ) ) ) ).
% Compl_Times_UNIV1
thf(fact_7318_Compl__Times__UNIV2,axiom,
! [B: $tType,A: $tType,A4: set @ A] :
( ( uminus_uminus @ ( set @ ( product_prod @ A @ B ) )
@ ( product_Sigma @ A @ B @ A4
@ ^ [Uu3: A] : ( top_top @ ( set @ B ) ) ) )
= ( product_Sigma @ A @ B @ ( uminus_uminus @ ( set @ A ) @ A4 )
@ ^ [Uu3: A] : ( top_top @ ( set @ B ) ) ) ) ).
% Compl_Times_UNIV2
thf(fact_7319_Sigma__Diff__distrib1,axiom,
! [B: $tType,A: $tType,I5: set @ A,J4: set @ A,C5: A > ( set @ B )] :
( ( product_Sigma @ A @ B @ ( minus_minus @ ( set @ A ) @ I5 @ J4 ) @ C5 )
= ( minus_minus @ ( set @ ( product_prod @ A @ B ) ) @ ( product_Sigma @ A @ B @ I5 @ C5 ) @ ( product_Sigma @ A @ B @ J4 @ C5 ) ) ) ).
% Sigma_Diff_distrib1
thf(fact_7320_Times__Diff__distrib1,axiom,
! [B: $tType,A: $tType,A4: set @ A,B7: set @ A,C5: set @ B] :
( ( product_Sigma @ A @ B @ ( minus_minus @ ( set @ A ) @ A4 @ B7 )
@ ^ [Uu3: A] : C5 )
= ( minus_minus @ ( set @ ( product_prod @ A @ B ) )
@ ( product_Sigma @ A @ B @ A4
@ ^ [Uu3: A] : C5 )
@ ( product_Sigma @ A @ B @ B7
@ ^ [Uu3: A] : C5 ) ) ) ).
% Times_Diff_distrib1
thf(fact_7321_Sigma__Diff__distrib2,axiom,
! [B: $tType,A: $tType,I5: set @ A,A4: A > ( set @ B ),B7: A > ( set @ B )] :
( ( product_Sigma @ A @ B @ I5
@ ^ [I2: A] : ( minus_minus @ ( set @ B ) @ ( A4 @ I2 ) @ ( B7 @ I2 ) ) )
= ( minus_minus @ ( set @ ( product_prod @ A @ B ) ) @ ( product_Sigma @ A @ B @ I5 @ A4 ) @ ( product_Sigma @ A @ B @ I5 @ B7 ) ) ) ).
% Sigma_Diff_distrib2
thf(fact_7322_lists__length__Suc__eq,axiom,
! [A: $tType,A4: set @ A,N: nat] :
( ( collect @ ( list @ A )
@ ^ [Xs3: list @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs3 ) @ A4 )
& ( ( size_size @ ( list @ A ) @ Xs3 )
= ( suc @ N ) ) ) )
= ( image @ ( product_prod @ ( list @ A ) @ A ) @ ( list @ A )
@ ( product_case_prod @ ( list @ A ) @ A @ ( list @ A )
@ ^ [Xs3: list @ A,N5: A] : ( cons @ A @ N5 @ Xs3 ) )
@ ( product_Sigma @ ( list @ A ) @ A
@ ( collect @ ( list @ A )
@ ^ [Xs3: list @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs3 ) @ A4 )
& ( ( size_size @ ( list @ A ) @ Xs3 )
= N ) ) )
@ ^ [Uu3: list @ A] : A4 ) ) ) ).
% lists_length_Suc_eq
thf(fact_7323_card__def,axiom,
! [B: $tType] :
( ( finite_card @ B )
= ( finite_folding_F @ B @ nat
@ ^ [Uu3: B] : suc
@ ( zero_zero @ nat ) ) ) ).
% card_def
thf(fact_7324_folding__on_Oremove,axiom,
! [B: $tType,A: $tType,S3: set @ A,F2: A > B > B,A4: set @ A,X: A,Z2: B] :
( ( finite_folding_on @ A @ B @ S3 @ F2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A4 @ S3 )
=> ( ( finite_finite @ A @ A4 )
=> ( ( member @ A @ X @ A4 )
=> ( ( finite_folding_F @ A @ B @ F2 @ Z2 @ A4 )
= ( F2 @ X @ ( finite_folding_F @ A @ B @ F2 @ Z2 @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ).
% folding_on.remove
thf(fact_7325_folding__on_Oinsert__remove,axiom,
! [B: $tType,A: $tType,S3: set @ A,F2: A > B > B,X: A,A4: set @ A,Z2: B] :
( ( finite_folding_on @ A @ B @ S3 @ F2 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X @ A4 ) @ S3 )
=> ( ( finite_finite @ A @ A4 )
=> ( ( finite_folding_F @ A @ B @ F2 @ Z2 @ ( insert @ A @ X @ A4 ) )
= ( F2 @ X @ ( finite_folding_F @ A @ B @ F2 @ Z2 @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ).
% folding_on.insert_remove
thf(fact_7326_card_Ofolding__on__axioms,axiom,
! [A: $tType] :
( finite_folding_on @ A @ nat @ ( top_top @ ( set @ A ) )
@ ^ [Uu3: A] : suc ) ).
% card.folding_on_axioms
thf(fact_7327_filtermap__ln__at__right,axiom,
( ( filtermap @ real @ real @ ( ln_ln @ real ) @ ( topolo174197925503356063within @ real @ ( zero_zero @ real ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) )
= ( at_bot @ real ) ) ).
% filtermap_ln_at_right
thf(fact_7328_Gcd__int__set__eq__fold,axiom,
! [Xs: list @ int] :
( ( gcd_Gcd @ int @ ( set2 @ int @ Xs ) )
= ( fold @ int @ int @ ( gcd_gcd @ int ) @ Xs @ ( zero_zero @ int ) ) ) ).
% Gcd_int_set_eq_fold
thf(fact_7329_filtermap__nhds__minus,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [A3: A] :
( ( filtermap @ A @ A @ ( uminus_uminus @ A ) @ ( topolo7230453075368039082e_nhds @ A @ A3 ) )
= ( topolo7230453075368039082e_nhds @ A @ ( uminus_uminus @ A @ A3 ) ) ) ) ).
% filtermap_nhds_minus
thf(fact_7330_filtermap__ln__at__top,axiom,
( ( filtermap @ real @ real @ ( ln_ln @ real ) @ ( at_top @ real ) )
= ( at_top @ real ) ) ).
% filtermap_ln_at_top
thf(fact_7331_filtermap__exp__at__top,axiom,
( ( filtermap @ real @ real @ ( exp @ real ) @ ( at_top @ real ) )
= ( at_top @ real ) ) ).
% filtermap_exp_at_top
thf(fact_7332_filtermap__nhds__times,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [C2: A,A3: A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( filtermap @ A @ A @ ( times_times @ A @ C2 ) @ ( topolo7230453075368039082e_nhds @ A @ A3 ) )
= ( topolo7230453075368039082e_nhds @ A @ ( times_times @ A @ C2 @ A3 ) ) ) ) ) ).
% filtermap_nhds_times
thf(fact_7333_filtermap__nhds__shift,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [D2: A,A3: A] :
( ( filtermap @ A @ A
@ ^ [X2: A] : ( minus_minus @ A @ X2 @ D2 )
@ ( topolo7230453075368039082e_nhds @ A @ A3 ) )
= ( topolo7230453075368039082e_nhds @ A @ ( minus_minus @ A @ A3 @ D2 ) ) ) ) ).
% filtermap_nhds_shift
thf(fact_7334_at__top__mirror,axiom,
! [A: $tType] :
( ( ( ordered_ab_group_add @ A )
& ( linorder @ A ) )
=> ( ( at_top @ A )
= ( filtermap @ A @ A @ ( uminus_uminus @ A ) @ ( at_bot @ A ) ) ) ) ).
% at_top_mirror
thf(fact_7335_at__bot__mirror,axiom,
! [A: $tType] :
( ( ( ordered_ab_group_add @ A )
& ( linorder @ A ) )
=> ( ( at_bot @ A )
= ( filtermap @ A @ A @ ( uminus_uminus @ A ) @ ( at_top @ A ) ) ) ) ).
% at_bot_mirror
thf(fact_7336_filtermap__at__shift,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [D2: A,A3: A] :
( ( filtermap @ A @ A
@ ^ [X2: A] : ( minus_minus @ A @ X2 @ D2 )
@ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) )
= ( topolo174197925503356063within @ A @ ( minus_minus @ A @ A3 @ D2 ) @ ( top_top @ ( set @ A ) ) ) ) ) ).
% filtermap_at_shift
thf(fact_7337_filtermap__at__minus,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [A3: A] :
( ( filtermap @ A @ A @ ( uminus_uminus @ A ) @ ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) ) )
= ( topolo174197925503356063within @ A @ ( uminus_uminus @ A @ A3 ) @ ( top_top @ ( set @ A ) ) ) ) ) ).
% filtermap_at_minus
thf(fact_7338_filtermap__at__right__shift,axiom,
! [D2: real,A3: real] :
( ( filtermap @ real @ real
@ ^ [X2: real] : ( minus_minus @ real @ X2 @ D2 )
@ ( topolo174197925503356063within @ real @ A3 @ ( set_ord_greaterThan @ real @ A3 ) ) )
= ( topolo174197925503356063within @ real @ ( minus_minus @ real @ A3 @ D2 ) @ ( set_ord_greaterThan @ real @ ( minus_minus @ real @ A3 @ D2 ) ) ) ) ).
% filtermap_at_right_shift
thf(fact_7339_at__to__0,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [A3: A] :
( ( topolo174197925503356063within @ A @ A3 @ ( top_top @ ( set @ A ) ) )
= ( filtermap @ A @ A
@ ^ [X2: A] : ( plus_plus @ A @ X2 @ A3 )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% at_to_0
thf(fact_7340_Gcd__set__eq__fold,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [Xs: list @ A] :
( ( gcd_Gcd @ A @ ( set2 @ A @ Xs ) )
= ( fold @ A @ A @ ( gcd_gcd @ A ) @ Xs @ ( zero_zero @ A ) ) ) ) ).
% Gcd_set_eq_fold
thf(fact_7341_Gcd__fin_Oset__eq__fold,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [Xs: list @ A] :
( ( semiring_gcd_Gcd_fin @ A @ ( set2 @ A @ Xs ) )
= ( fold @ A @ A @ ( gcd_gcd @ A ) @ Xs @ ( zero_zero @ A ) ) ) ) ).
% Gcd_fin.set_eq_fold
thf(fact_7342_at__right__minus,axiom,
! [A3: real] :
( ( topolo174197925503356063within @ real @ A3 @ ( set_ord_greaterThan @ real @ A3 ) )
= ( filtermap @ real @ real @ ( uminus_uminus @ real ) @ ( topolo174197925503356063within @ real @ ( uminus_uminus @ real @ A3 ) @ ( set_ord_lessThan @ real @ ( uminus_uminus @ real @ A3 ) ) ) ) ) ).
% at_right_minus
thf(fact_7343_at__left__minus,axiom,
! [A3: real] :
( ( topolo174197925503356063within @ real @ A3 @ ( set_ord_lessThan @ real @ A3 ) )
= ( filtermap @ real @ real @ ( uminus_uminus @ real ) @ ( topolo174197925503356063within @ real @ ( uminus_uminus @ real @ A3 ) @ ( set_ord_greaterThan @ real @ ( uminus_uminus @ real @ A3 ) ) ) ) ) ).
% at_left_minus
thf(fact_7344_filtermap__times__pos__at__right,axiom,
! [A: $tType] :
( ( ( linordered_field @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [C2: A,P4: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C2 )
=> ( ( filtermap @ A @ A @ ( times_times @ A @ C2 ) @ ( topolo174197925503356063within @ A @ P4 @ ( set_ord_greaterThan @ A @ P4 ) ) )
= ( topolo174197925503356063within @ A @ ( times_times @ A @ C2 @ P4 ) @ ( set_ord_greaterThan @ A @ ( times_times @ A @ C2 @ P4 ) ) ) ) ) ) ).
% filtermap_times_pos_at_right
thf(fact_7345_at__to__infinity,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ( ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) )
= ( filtermap @ A @ A @ ( inverse_inverse @ A ) @ ( at_infinity @ A ) ) ) ) ).
% at_to_infinity
thf(fact_7346_less__than__iff,axiom,
! [X: nat,Y2: nat] :
( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ X @ Y2 ) @ less_than )
= ( ord_less @ nat @ X @ Y2 ) ) ).
% less_than_iff
thf(fact_7347_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_7348_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_7349_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_7350_idiff__enat__enat,axiom,
! [A3: nat,B2: nat] :
( ( minus_minus @ extended_enat @ ( extended_enat2 @ A3 ) @ ( extended_enat2 @ B2 ) )
= ( extended_enat2 @ ( minus_minus @ nat @ A3 @ B2 ) ) ) ).
% idiff_enat_enat
thf(fact_7351_enat__ord__simps_I2_J,axiom,
! [M2: nat,N: nat] :
( ( ord_less @ extended_enat @ ( extended_enat2 @ M2 ) @ ( extended_enat2 @ N ) )
= ( ord_less @ nat @ M2 @ N ) ) ).
% enat_ord_simps(2)
thf(fact_7352_numeral__less__enat__iff,axiom,
! [M2: num,N: nat] :
( ( ord_less @ extended_enat @ ( numeral_numeral @ extended_enat @ M2 ) @ ( extended_enat2 @ N ) )
= ( ord_less @ nat @ ( numeral_numeral @ nat @ M2 ) @ N ) ) ).
% numeral_less_enat_iff
thf(fact_7353_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_7354_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_7355_zero__enat__def,axiom,
( ( zero_zero @ extended_enat )
= ( extended_enat2 @ ( zero_zero @ nat ) ) ) ).
% zero_enat_def
thf(fact_7356_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_7357_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_7358_one__enat__def,axiom,
( ( one_one @ extended_enat )
= ( extended_enat2 @ ( one_one @ nat ) ) ) ).
% one_enat_def
thf(fact_7359_VEBT__internal_Oelim__dead_Osimps_I1_J,axiom,
! [A3: $o,B2: $o,Uu: extended_enat] :
( ( vEBT_VEBT_elim_dead @ ( vEBT_Leaf @ A3 @ B2 ) @ Uu )
= ( vEBT_Leaf @ A3 @ B2 ) ) ).
% VEBT_internal.elim_dead.simps(1)
thf(fact_7360_less__enatE,axiom,
! [N: extended_enat,M2: nat] :
( ( ord_less @ extended_enat @ N @ ( extended_enat2 @ M2 ) )
=> ~ ! [K2: nat] :
( ( N
= ( extended_enat2 @ K2 ) )
=> ~ ( ord_less @ nat @ K2 @ M2 ) ) ) ).
% less_enatE
thf(fact_7361_Suc__ile__eq,axiom,
! [M2: nat,N: extended_enat] :
( ( ord_less_eq @ extended_enat @ ( extended_enat2 @ ( suc @ M2 ) ) @ N )
= ( ord_less @ extended_enat @ ( extended_enat2 @ M2 ) @ N ) ) ).
% Suc_ile_eq
thf(fact_7362_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_7363_minus__set__fold,axiom,
! [A: $tType,A4: set @ A,Xs: list @ A] :
( ( minus_minus @ ( set @ A ) @ A4 @ ( set2 @ A @ Xs ) )
= ( fold @ A @ ( set @ A ) @ ( remove @ A ) @ Xs @ A4 ) ) ).
% minus_set_fold
thf(fact_7364_Gcd__nat__set__eq__fold,axiom,
! [Xs: list @ nat] :
( ( gcd_Gcd @ nat @ ( set2 @ nat @ Xs ) )
= ( fold @ nat @ nat @ ( gcd_gcd @ nat ) @ Xs @ ( zero_zero @ nat ) ) ) ).
% Gcd_nat_set_eq_fold
thf(fact_7365_VEBT__internal_Oelim__dead_Oelims,axiom,
! [X: vEBT_VEBT,Xa: extended_enat,Y2: vEBT_VEBT] :
( ( ( vEBT_VEBT_elim_dead @ X @ Xa )
= Y2 )
=> ( ! [A6: $o,B4: $o] :
( ( X
= ( vEBT_Leaf @ A6 @ B4 ) )
=> ( Y2
!= ( vEBT_Leaf @ A6 @ B4 ) ) )
=> ( ! [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 ) )
=> ! [L3: nat] :
( ( Xa
= ( extended_enat2 @ L3 ) )
=> ( Y2
!= ( vEBT_Node @ Info2 @ Deg2
@ ( take @ vEBT_VEBT @ ( divide_divide @ nat @ L3 @ ( 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 @ L3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.elim_dead.elims
thf(fact_7366_elimcomplete,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 ) @ ( extend4730790105801354508finity @ extended_enat ) )
= ( vEBT_Node @ Info @ Deg @ TreeList @ Summary ) ) ) ).
% elimcomplete
thf(fact_7367_idiff__infinity,axiom,
! [N: extended_enat] :
( ( minus_minus @ extended_enat @ ( extend4730790105801354508finity @ extended_enat ) @ N )
= ( extend4730790105801354508finity @ extended_enat ) ) ).
% idiff_infinity
thf(fact_7368_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_7369_idiff__0,axiom,
! [N: extended_enat] :
( ( minus_minus @ extended_enat @ ( zero_zero @ extended_enat ) @ N )
= ( zero_zero @ extended_enat ) ) ).
% idiff_0
thf(fact_7370_idiff__0__right,axiom,
! [N: extended_enat] :
( ( minus_minus @ extended_enat @ N @ ( zero_zero @ extended_enat ) )
= N ) ).
% idiff_0_right
thf(fact_7371_idiff__self,axiom,
! [N: extended_enat] :
( ( N
!= ( extend4730790105801354508finity @ extended_enat ) )
=> ( ( minus_minus @ extended_enat @ N @ N )
= ( zero_zero @ extended_enat ) ) ) ).
% idiff_self
thf(fact_7372_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_7373_times__enat__simps_I4_J,axiom,
! [M2: nat] :
( ( ( M2
= ( zero_zero @ nat ) )
=> ( ( times_times @ extended_enat @ ( extended_enat2 @ M2 ) @ ( extend4730790105801354508finity @ extended_enat ) )
= ( zero_zero @ extended_enat ) ) )
& ( ( M2
!= ( zero_zero @ nat ) )
=> ( ( times_times @ extended_enat @ ( extended_enat2 @ M2 ) @ ( extend4730790105801354508finity @ extended_enat ) )
= ( extend4730790105801354508finity @ extended_enat ) ) ) ) ).
% times_enat_simps(4)
thf(fact_7374_idiff__infinity__right,axiom,
! [A3: nat] :
( ( minus_minus @ extended_enat @ ( extended_enat2 @ A3 ) @ ( extend4730790105801354508finity @ extended_enat ) )
= ( zero_zero @ extended_enat ) ) ).
% idiff_infinity_right
thf(fact_7375_infinity__ne__i1,axiom,
( ( extend4730790105801354508finity @ extended_enat )
!= ( one_one @ extended_enat ) ) ).
% infinity_ne_i1
thf(fact_7376_zero__one__enat__neq_I1_J,axiom,
( ( zero_zero @ extended_enat )
!= ( one_one @ extended_enat ) ) ).
% zero_one_enat_neq(1)
thf(fact_7377_add__diff__assoc__enat,axiom,
! [Z2: extended_enat,Y2: extended_enat,X: extended_enat] :
( ( ord_less_eq @ extended_enat @ Z2 @ Y2 )
=> ( ( plus_plus @ extended_enat @ X @ ( minus_minus @ extended_enat @ Y2 @ Z2 ) )
= ( minus_minus @ extended_enat @ ( plus_plus @ extended_enat @ X @ Y2 ) @ Z2 ) ) ) ).
% add_diff_assoc_enat
thf(fact_7378_VEBT__internal_Oelim__dead_Ocases,axiom,
! [X: product_prod @ vEBT_VEBT @ extended_enat] :
( ! [A6: $o,B4: $o,Uu2: extended_enat] :
( X
!= ( product_Pair @ vEBT_VEBT @ extended_enat @ ( vEBT_Leaf @ A6 @ B4 ) @ 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,L3: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ extended_enat @ ( vEBT_Node @ Info2 @ Deg2 @ TreeList2 @ Summary2 ) @ ( extended_enat2 @ L3 ) ) ) ) ) ).
% VEBT_internal.elim_dead.cases
thf(fact_7379_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_7380_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 ) )
=> ( ! [A6: $o,B4: $o] :
( ( X
= ( vEBT_Leaf @ A6 @ B4 ) )
=> ( ( Y2
= ( vEBT_Leaf @ A6 @ B4 ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ extended_enat ) @ vEBT_V312737461966249ad_rel @ ( product_Pair @ vEBT_VEBT @ extended_enat @ ( vEBT_Leaf @ A6 @ B4 ) @ 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 ) )
=> ! [L3: nat] :
( ( Xa
= ( extended_enat2 @ L3 ) )
=> ( ( Y2
= ( vEBT_Node @ Info2 @ Deg2
@ ( take @ vEBT_VEBT @ ( divide_divide @ nat @ L3 @ ( 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 @ L3 @ ( 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 @ L3 ) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.elim_dead.pelims
thf(fact_7381_diff__enat__def,axiom,
( ( minus_minus @ extended_enat )
= ( ^ [A5: extended_enat,B3: extended_enat] :
( extended_case_enat @ extended_enat
@ ^ [X2: nat] :
( extended_case_enat @ extended_enat
@ ^ [Y5: nat] : ( extended_enat2 @ ( minus_minus @ nat @ X2 @ Y5 ) )
@ ( zero_zero @ extended_enat )
@ B3 )
@ ( extend4730790105801354508finity @ extended_enat )
@ A5 ) ) ) ).
% diff_enat_def
thf(fact_7382_times__enat__def,axiom,
( ( times_times @ extended_enat )
= ( ^ [M5: extended_enat,N5: extended_enat] :
( extended_case_enat @ extended_enat
@ ^ [O: nat] :
( extended_case_enat @ extended_enat
@ ^ [P6: nat] : ( extended_enat2 @ ( times_times @ nat @ O @ P6 ) )
@ ( if @ extended_enat
@ ( O
= ( zero_zero @ nat ) )
@ ( zero_zero @ extended_enat )
@ ( extend4730790105801354508finity @ extended_enat ) )
@ N5 )
@ ( if @ extended_enat
@ ( N5
= ( zero_zero @ extended_enat ) )
@ ( zero_zero @ extended_enat )
@ ( extend4730790105801354508finity @ extended_enat ) )
@ M5 ) ) ) ).
% times_enat_def
thf(fact_7383_eSuc__def,axiom,
( extended_eSuc
= ( extended_case_enat @ extended_enat
@ ^ [N5: nat] : ( extended_enat2 @ ( suc @ N5 ) )
@ ( extend4730790105801354508finity @ extended_enat ) ) ) ).
% eSuc_def
thf(fact_7384_binomial__def,axiom,
( binomial
= ( ^ [N5: nat,K3: nat] :
( finite_card @ ( set @ nat )
@ ( collect @ ( set @ nat )
@ ^ [K5: set @ nat] :
( ( member @ ( set @ nat ) @ K5 @ ( pow2 @ nat @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N5 ) ) )
& ( ( finite_card @ nat @ K5 )
= K3 ) ) ) ) ) ) ).
% binomial_def
thf(fact_7385_eSuc__minus__eSuc,axiom,
! [N: extended_enat,M2: extended_enat] :
( ( minus_minus @ extended_enat @ ( extended_eSuc @ N ) @ ( extended_eSuc @ M2 ) )
= ( minus_minus @ extended_enat @ N @ M2 ) ) ).
% eSuc_minus_eSuc
thf(fact_7386_eSuc__minus__1,axiom,
! [N: extended_enat] :
( ( minus_minus @ extended_enat @ ( extended_eSuc @ N ) @ ( one_one @ extended_enat ) )
= N ) ).
% eSuc_minus_1
thf(fact_7387_enat__eSuc__iff,axiom,
! [Y2: nat,X: extended_enat] :
( ( ( extended_enat2 @ Y2 )
= ( extended_eSuc @ X ) )
= ( ? [N5: nat] :
( ( Y2
= ( suc @ N5 ) )
& ( ( extended_enat2 @ N5 )
= X ) ) ) ) ).
% enat_eSuc_iff
thf(fact_7388_eSuc__enat__iff,axiom,
! [X: extended_enat,Y2: nat] :
( ( ( extended_eSuc @ X )
= ( extended_enat2 @ Y2 ) )
= ( ? [N5: nat] :
( ( Y2
= ( suc @ N5 ) )
& ( X
= ( extended_enat2 @ N5 ) ) ) ) ) ).
% eSuc_enat_iff
thf(fact_7389_eSuc__enat,axiom,
! [N: nat] :
( ( extended_eSuc @ ( extended_enat2 @ N ) )
= ( extended_enat2 @ ( suc @ N ) ) ) ).
% eSuc_enat
thf(fact_7390_Pow__Compl,axiom,
! [A: $tType,A4: set @ A] :
( ( pow2 @ A @ ( uminus_uminus @ ( set @ A ) @ A4 ) )
= ( collect @ ( set @ A )
@ ^ [Uu3: set @ A] :
? [B5: set @ A] :
( ( Uu3
= ( uminus_uminus @ ( set @ A ) @ B5 ) )
& ( member @ ( set @ A ) @ A4 @ ( pow2 @ A @ B5 ) ) ) ) ) ).
% Pow_Compl
thf(fact_7391_one__eSuc,axiom,
( ( one_one @ extended_enat )
= ( extended_eSuc @ ( zero_zero @ extended_enat ) ) ) ).
% one_eSuc
thf(fact_7392_eSuc__plus__1,axiom,
( extended_eSuc
= ( ^ [N5: extended_enat] : ( plus_plus @ extended_enat @ N5 @ ( one_one @ extended_enat ) ) ) ) ).
% eSuc_plus_1
thf(fact_7393_plus__1__eSuc_I1_J,axiom,
! [Q5: extended_enat] :
( ( plus_plus @ extended_enat @ ( one_one @ extended_enat ) @ Q5 )
= ( extended_eSuc @ Q5 ) ) ).
% plus_1_eSuc(1)
thf(fact_7394_plus__1__eSuc_I2_J,axiom,
! [Q5: extended_enat] :
( ( plus_plus @ extended_enat @ Q5 @ ( one_one @ extended_enat ) )
= ( extended_eSuc @ Q5 ) ) ).
% plus_1_eSuc(2)
thf(fact_7395_Quotient__real,axiom,
quotient @ ( nat > rat ) @ real @ realrel @ real2 @ rep_real @ cr_real ).
% Quotient_real
thf(fact_7396_is__singleton__altdef,axiom,
! [A: $tType] :
( ( is_singleton @ A )
= ( ^ [A7: set @ A] :
( ( finite_card @ A @ A7 )
= ( one_one @ nat ) ) ) ) ).
% is_singleton_altdef
thf(fact_7397_extract__Cons__code,axiom,
! [A: $tType,P: A > $o,X: A,Xs: list @ A] :
( ( ( P @ X )
=> ( ( extract @ A @ P @ ( cons @ A @ X @ Xs ) )
= ( some @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) @ ( product_Pair @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) @ ( nil @ A ) @ ( product_Pair @ A @ ( list @ A ) @ X @ Xs ) ) ) ) )
& ( ~ ( P @ X )
=> ( ( extract @ A @ P @ ( cons @ A @ X @ Xs ) )
= ( case_option @ ( option @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) ) @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) @ ( none @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) )
@ ( product_case_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) @ ( option @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) )
@ ^ [Ys3: list @ A] :
( product_case_prod @ A @ ( list @ A ) @ ( option @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) )
@ ^ [Y5: 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 ) @ Y5 @ Zs3 ) ) ) ) )
@ ( extract @ A @ P @ Xs ) ) ) ) ) ).
% extract_Cons_code
thf(fact_7398_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_7399_char__of__nat,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [N: nat] :
( ( unique5772411509450598832har_of @ A @ ( semiring_1_of_nat @ A @ N ) )
= ( unique5772411509450598832har_of @ nat @ N ) ) ) ).
% char_of_nat
thf(fact_7400_inj__on__char__of__nat,axiom,
inj_on @ nat @ char @ ( unique5772411509450598832har_of @ nat ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% inj_on_char_of_nat
thf(fact_7401_char__of__def,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ( ( unique5772411509450598832har_of @ A )
= ( ^ [N5: A] :
( char2
@ ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N5 )
@ ( bit_se5641148757651400278ts_bit @ A @ N5 @ ( one_one @ nat ) )
@ ( bit_se5641148757651400278ts_bit @ A @ N5 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
@ ( bit_se5641148757651400278ts_bit @ A @ N5 @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) )
@ ( bit_se5641148757651400278ts_bit @ A @ N5 @ ( numeral_numeral @ nat @ ( bit0 @ ( bit0 @ one2 ) ) ) )
@ ( bit_se5641148757651400278ts_bit @ A @ N5 @ ( numeral_numeral @ nat @ ( bit1 @ ( bit0 @ one2 ) ) ) )
@ ( bit_se5641148757651400278ts_bit @ A @ N5 @ ( numeral_numeral @ nat @ ( bit0 @ ( bit1 @ one2 ) ) ) )
@ ( bit_se5641148757651400278ts_bit @ A @ N5 @ ( numeral_numeral @ nat @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ).
% char_of_def
thf(fact_7402_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_7403_char_Osize_I2_J,axiom,
! [X15: $o,X23: $o,X32: $o,X42: $o,X52: $o,X62: $o,X72: $o,X82: $o] :
( ( size_size @ char @ ( char2 @ X15 @ X23 @ X32 @ X42 @ X52 @ X62 @ X72 @ X82 ) )
= ( zero_zero @ nat ) ) ).
% char.size(2)
thf(fact_7404_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_7405_char_Osize__gen,axiom,
! [X15: $o,X23: $o,X32: $o,X42: $o,X52: $o,X62: $o,X72: $o,X82: $o] :
( ( size_char @ ( char2 @ X15 @ X23 @ X32 @ X42 @ X52 @ X62 @ X72 @ X82 ) )
= ( zero_zero @ nat ) ) ).
% char.size_gen
thf(fact_7406_prod__list__def,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ( ( groups5270119922927024881d_list @ A )
= ( groups_monoid_F @ A @ ( times_times @ A ) @ ( one_one @ A ) ) ) ) ).
% prod_list_def
thf(fact_7407_sum__list__def,axiom,
! [A: $tType] :
( ( monoid_add @ A )
=> ( ( groups8242544230860333062m_list @ A )
= ( groups_monoid_F @ A @ ( plus_plus @ A ) @ ( zero_zero @ A ) ) ) ) ).
% sum_list_def
thf(fact_7408_relpow__finite__bounded1,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),K: nat] :
( ( finite_finite @ ( product_prod @ A @ A ) @ R2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ K @ R2 )
@ ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ A ) )
@ ( image @ nat @ ( set @ ( product_prod @ A @ A ) )
@ ^ [N5: nat] : ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N5 @ R2 )
@ ( collect @ nat
@ ^ [N5: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N5 )
& ( ord_less_eq @ nat @ N5 @ ( finite_card @ ( product_prod @ A @ A ) @ R2 ) ) ) ) ) ) ) ) ) ).
% relpow_finite_bounded1
thf(fact_7409_cr__int__def,axiom,
( cr_int
= ( ^ [X2: product_prod @ nat @ nat] :
( ^ [Y4: int,Z: int] : Y4 = Z
@ ( abs_Integ @ X2 ) ) ) ) ).
% cr_int_def
thf(fact_7410_relpow__1,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( one_one @ nat ) @ R2 )
= R2 ) ).
% relpow_1
thf(fact_7411_finite__relpow,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),N: nat] :
( ( finite_finite @ ( product_prod @ A @ A ) @ R2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( finite_finite @ ( product_prod @ A @ A ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R2 ) ) ) ) ).
% finite_relpow
thf(fact_7412_relpow__Suc__I2,axiom,
! [A: $tType,X: A,Y2: A,R2: set @ ( product_prod @ A @ A ),Z2: A,N: nat] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y2 ) @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ Z2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R2 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( suc @ N ) @ R2 ) ) ) ) ).
% relpow_Suc_I2
thf(fact_7413_relpow__Suc__E2,axiom,
! [A: $tType,X: A,Z2: A,N: nat,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( suc @ N ) @ R2 ) )
=> ~ ! [Y3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y3 ) @ R2 )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R2 ) ) ) ) ).
% relpow_Suc_E2
thf(fact_7414_relpow__Suc__D2,axiom,
! [A: $tType,X: A,Z2: A,N: nat,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( suc @ N ) @ R2 ) )
=> ? [Y3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y3 ) @ R2 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R2 ) ) ) ) ).
% relpow_Suc_D2
thf(fact_7415_relpow__Suc__I,axiom,
! [A: $tType,X: A,Y2: A,N: nat,R2: set @ ( product_prod @ A @ A ),Z2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R2 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y2 @ Z2 ) @ R2 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( suc @ N ) @ R2 ) ) ) ) ).
% relpow_Suc_I
thf(fact_7416_relpow__Suc__E,axiom,
! [A: $tType,X: A,Z2: A,N: nat,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( suc @ N ) @ R2 ) )
=> ~ ! [Y3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y3 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R2 ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z2 ) @ R2 ) ) ) ).
% relpow_Suc_E
thf(fact_7417_relpow__0__E,axiom,
! [A: $tType,X: A,Y2: A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( zero_zero @ nat ) @ R2 ) )
=> ( X = Y2 ) ) ).
% relpow_0_E
thf(fact_7418_relpow__0__I,axiom,
! [A: $tType,X: A,R2: set @ ( product_prod @ A @ A )] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ X ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( zero_zero @ nat ) @ R2 ) ) ).
% relpow_0_I
thf(fact_7419_relpow__E2,axiom,
! [A: $tType,X: A,Z2: A,N: nat,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R2 ) )
=> ( ( ( N
= ( zero_zero @ nat ) )
=> ( X != Z2 ) )
=> ~ ! [Y3: A,M3: nat] :
( ( N
= ( suc @ M3 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y3 ) @ R2 )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ M3 @ R2 ) ) ) ) ) ) ).
% relpow_E2
thf(fact_7420_relpow__E,axiom,
! [A: $tType,X: A,Z2: A,N: nat,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R2 ) )
=> ( ( ( N
= ( zero_zero @ nat ) )
=> ( X != Z2 ) )
=> ~ ! [Y3: A,M3: nat] :
( ( N
= ( suc @ M3 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y3 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ M3 @ R2 ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z2 ) @ R2 ) ) ) ) ) ).
% relpow_E
thf(fact_7421_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_7422_int_Opcr__cr__eq,axiom,
pcr_int = cr_int ).
% int.pcr_cr_eq
thf(fact_7423_relpow__fun__conv,axiom,
! [A: $tType,A3: A,B2: A,N: nat,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R2 ) )
= ( ? [F4: nat > A] :
( ( ( F4 @ ( zero_zero @ nat ) )
= A3 )
& ( ( F4 @ N )
= B2 )
& ! [I2: nat] :
( ( ord_less @ nat @ I2 @ N )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( F4 @ I2 ) @ ( F4 @ ( suc @ I2 ) ) ) @ R2 ) ) ) ) ) ).
% relpow_fun_conv
thf(fact_7424_ntrancl__def,axiom,
! [A: $tType] :
( ( transitive_ntrancl @ A )
= ( ^ [N5: nat,R6: set @ ( product_prod @ A @ A )] :
( complete_Sup_Sup @ ( set @ ( product_prod @ A @ A ) )
@ ( image @ nat @ ( set @ ( product_prod @ A @ A ) )
@ ^ [I2: nat] : ( compow @ ( set @ ( product_prod @ A @ A ) ) @ I2 @ R6 )
@ ( collect @ nat
@ ^ [I2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ I2 )
& ( ord_less_eq @ nat @ I2 @ ( suc @ N5 ) ) ) ) ) ) ) ) ).
% ntrancl_def
thf(fact_7425_trancl__finite__eq__relpow,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( finite_finite @ ( product_prod @ A @ A ) @ R2 )
=> ( ( transitive_trancl @ A @ R2 )
= ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ A ) )
@ ( image @ nat @ ( set @ ( product_prod @ A @ A ) )
@ ^ [N5: nat] : ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N5 @ R2 )
@ ( collect @ nat
@ ^ [N5: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N5 )
& ( ord_less_eq @ nat @ N5 @ ( finite_card @ ( product_prod @ A @ A ) @ R2 ) ) ) ) ) ) ) ) ).
% trancl_finite_eq_relpow
thf(fact_7426_ntrancl__Zero,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( transitive_ntrancl @ A @ ( zero_zero @ nat ) @ R2 )
= R2 ) ).
% ntrancl_Zero
thf(fact_7427_finite__trancl__ntranl,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( finite_finite @ ( product_prod @ A @ A ) @ R2 )
=> ( ( transitive_trancl @ A @ R2 )
= ( transitive_ntrancl @ A @ ( minus_minus @ nat @ ( finite_card @ ( product_prod @ A @ A ) @ R2 ) @ ( one_one @ nat ) ) @ R2 ) ) ) ).
% finite_trancl_ntranl
thf(fact_7428_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_7429_trancl__power,axiom,
! [A: $tType,P4: product_prod @ A @ A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ P4 @ ( transitive_trancl @ A @ R2 ) )
= ( ? [N5: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N5 )
& ( member @ ( product_prod @ A @ A ) @ P4 @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N5 @ R2 ) ) ) ) ) ).
% trancl_power
thf(fact_7430_less__eq,axiom,
! [M2: nat,N: nat] :
( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ M2 @ N ) @ ( transitive_trancl @ nat @ pred_nat ) )
= ( ord_less @ nat @ M2 @ N ) ) ).
% less_eq
thf(fact_7431_Quotient__int,axiom,
quotient @ ( product_prod @ nat @ nat ) @ int @ intrel @ abs_Integ @ rep_Integ @ cr_int ).
% Quotient_int
thf(fact_7432_gfp__Kleene__iter,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F2: A > A,K: nat] :
( ( order_mono @ A @ A @ F2 )
=> ( ( ( compow @ ( A > A ) @ ( suc @ K ) @ F2 @ ( top_top @ A ) )
= ( compow @ ( A > A ) @ K @ F2 @ ( top_top @ A ) ) )
=> ( ( complete_lattice_gfp @ A @ F2 )
= ( compow @ ( A > A ) @ K @ F2 @ ( top_top @ A ) ) ) ) ) ) ).
% gfp_Kleene_iter
thf(fact_7433_gfp__funpow,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F2: A > A,N: nat] :
( ( order_mono @ A @ A @ F2 )
=> ( ( complete_lattice_gfp @ A @ ( compow @ ( A > A ) @ ( suc @ N ) @ F2 ) )
= ( complete_lattice_gfp @ A @ F2 ) ) ) ) ).
% gfp_funpow
thf(fact_7434_strict__sorted__equal__Uniq,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A4: set @ A] :
( uniq @ ( list @ A )
@ ^ [Xs3: list @ A] :
( ( sorted_wrt @ A @ ( ord_less @ A ) @ Xs3 )
& ( ( set2 @ A @ Xs3 )
= A4 ) ) ) ) ).
% strict_sorted_equal_Uniq
thf(fact_7435_list__ex__length,axiom,
! [A: $tType] :
( ( list_ex @ A )
= ( ^ [P3: A > $o,Xs3: list @ A] :
? [N5: nat] :
( ( ord_less @ nat @ N5 @ ( size_size @ ( list @ A ) @ Xs3 ) )
& ( P3 @ ( nth @ A @ Xs3 @ N5 ) ) ) ) ) ).
% list_ex_length
thf(fact_7436_span__breakdown,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [B2: A,S3: set @ A,A3: A] :
( ( member @ A @ B2 @ S3 )
=> ( ( member @ A @ A3 @ ( real_Vector_span @ A @ S3 ) )
=> ? [K2: real] : ( member @ A @ ( minus_minus @ A @ A3 @ ( real_V8093663219630862766scaleR @ A @ K2 @ B2 ) ) @ ( real_Vector_span @ A @ ( minus_minus @ ( set @ A ) @ S3 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% span_breakdown
thf(fact_7437_and_Osemilattice__neutr__axioms,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( semilattice_neutr @ A @ ( bit_se5824344872417868541ns_and @ A ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% and.semilattice_neutr_axioms
thf(fact_7438_span__insert__0,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [S3: set @ A] :
( ( real_Vector_span @ A @ ( insert @ A @ ( zero_zero @ A ) @ S3 ) )
= ( real_Vector_span @ A @ S3 ) ) ) ).
% span_insert_0
thf(fact_7439_span__empty,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ( ( real_Vector_span @ A @ ( bot_bot @ ( set @ A ) ) )
= ( insert @ A @ ( zero_zero @ A ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% span_empty
thf(fact_7440_span__delete__0,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [S3: set @ A] :
( ( real_Vector_span @ A @ ( minus_minus @ ( set @ A ) @ S3 @ ( insert @ A @ ( zero_zero @ A ) @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( real_Vector_span @ A @ S3 ) ) ) ).
% span_delete_0
thf(fact_7441_span__0,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [S3: set @ A] : ( member @ A @ ( zero_zero @ A ) @ ( real_Vector_span @ A @ S3 ) ) ) ).
% span_0
thf(fact_7442_span__breakdown__eq,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [X: A,A3: A,S3: set @ A] :
( ( member @ A @ X @ ( real_Vector_span @ A @ ( insert @ A @ A3 @ S3 ) ) )
= ( ? [K3: real] : ( member @ A @ ( minus_minus @ A @ X @ ( real_V8093663219630862766scaleR @ A @ K3 @ A3 ) ) @ ( real_Vector_span @ A @ S3 ) ) ) ) ) ).
% span_breakdown_eq
thf(fact_7443_in__span__delete,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [A3: A,S3: set @ A,B2: A] :
( ( member @ A @ A3 @ ( real_Vector_span @ A @ S3 ) )
=> ( ~ ( member @ A @ A3 @ ( real_Vector_span @ A @ ( minus_minus @ ( set @ A ) @ S3 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) )
=> ( member @ A @ B2 @ ( real_Vector_span @ A @ ( insert @ A @ A3 @ ( minus_minus @ ( set @ A ) @ S3 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ).
% in_span_delete
thf(fact_7444_span__diff,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [X: A,S3: set @ A,Y2: A] :
( ( member @ A @ X @ ( real_Vector_span @ A @ S3 ) )
=> ( ( member @ A @ Y2 @ ( real_Vector_span @ A @ S3 ) )
=> ( member @ A @ ( minus_minus @ A @ X @ Y2 ) @ ( real_Vector_span @ A @ S3 ) ) ) ) ) ).
% span_diff
thf(fact_7445_eq__span__insert__eq,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [X: A,Y2: A,S3: set @ A] :
( ( member @ A @ ( minus_minus @ A @ X @ Y2 ) @ ( real_Vector_span @ A @ S3 ) )
=> ( ( real_Vector_span @ A @ ( insert @ A @ X @ S3 ) )
= ( real_Vector_span @ A @ ( insert @ A @ Y2 @ S3 ) ) ) ) ) ).
% eq_span_insert_eq
thf(fact_7446_semilattice__neutr__order_Oaxioms_I1_J,axiom,
! [A: $tType,F2: A > A > A,Z2: A,Less_eq: A > A > $o,Less: A > A > $o] :
( ( semila1105856199041335345_order @ A @ F2 @ Z2 @ Less_eq @ Less )
=> ( semilattice_neutr @ A @ F2 @ Z2 ) ) ).
% semilattice_neutr_order.axioms(1)
thf(fact_7447_span__neg,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [X: A,S3: set @ A] :
( ( member @ A @ X @ ( real_Vector_span @ A @ S3 ) )
=> ( member @ A @ ( uminus_uminus @ A @ X ) @ ( real_Vector_span @ A @ S3 ) ) ) ) ).
% span_neg
thf(fact_7448_span__induct__alt,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [X: A,S3: set @ A,H: A > $o] :
( ( member @ A @ X @ ( real_Vector_span @ A @ S3 ) )
=> ( ( H @ ( zero_zero @ A ) )
=> ( ! [C3: real,X3: A,Y3: A] :
( ( member @ A @ X3 @ S3 )
=> ( ( H @ Y3 )
=> ( H @ ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ C3 @ X3 ) @ Y3 ) ) ) )
=> ( H @ X ) ) ) ) ) ).
% span_induct_alt
thf(fact_7449_inf__top_Osemilattice__neutr__axioms,axiom,
! [A: $tType] :
( ( bounde4346867609351753570nf_top @ A )
=> ( semilattice_neutr @ A @ ( inf_inf @ A ) @ ( top_top @ A ) ) ) ).
% inf_top.semilattice_neutr_axioms
thf(fact_7450_sup__bot_Osemilattice__neutr__axioms,axiom,
! [A: $tType] :
( ( bounde4967611905675639751up_bot @ A )
=> ( semilattice_neutr @ A @ ( sup_sup @ A ) @ ( bot_bot @ A ) ) ) ).
% sup_bot.semilattice_neutr_axioms
thf(fact_7451_gcd__nat_Osemilattice__neutr__axioms,axiom,
semilattice_neutr @ nat @ ( gcd_gcd @ nat ) @ ( zero_zero @ nat ) ).
% gcd_nat.semilattice_neutr_axioms
thf(fact_7452_span__insert,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [A3: A,S3: set @ A] :
( ( real_Vector_span @ A @ ( insert @ A @ A3 @ S3 ) )
= ( collect @ A
@ ^ [X2: A] :
? [K3: real] : ( member @ A @ ( minus_minus @ A @ X2 @ ( real_V8093663219630862766scaleR @ A @ K3 @ A3 ) ) @ ( real_Vector_span @ A @ S3 ) ) ) ) ) ).
% span_insert
thf(fact_7453_or_Osemilattice__neutr__axioms,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( semilattice_neutr @ A @ ( bit_se1065995026697491101ons_or @ A ) @ ( zero_zero @ A ) ) ) ).
% or.semilattice_neutr_axioms
thf(fact_7454_max__nat_Osemilattice__neutr__axioms,axiom,
semilattice_neutr @ nat @ ( ord_max @ nat ) @ ( zero_zero @ nat ) ).
% max_nat.semilattice_neutr_axioms
thf(fact_7455_dependent__def,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ( ( real_V358717886546972837endent @ A )
= ( ^ [P3: set @ A] :
? [X2: A] :
( ( member @ A @ X2 @ P3 )
& ( member @ A @ X2 @ ( real_Vector_span @ A @ ( minus_minus @ ( set @ A ) @ P3 @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ).
% dependent_def
thf(fact_7456_dim__def,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ( ( real_Vector_dim @ A )
= ( ^ [V6: set @ A] :
( if @ nat
@ ? [B3: set @ A] :
( ~ ( real_V358717886546972837endent @ A @ B3 )
& ( ( real_Vector_span @ A @ B3 )
= ( real_Vector_span @ A @ V6 ) ) )
@ ( finite_card @ A
@ ( fChoice @ ( set @ A )
@ ^ [B3: set @ A] :
( ~ ( real_V358717886546972837endent @ A @ B3 )
& ( ( real_Vector_span @ A @ B3 )
= ( real_Vector_span @ A @ V6 ) ) ) ) )
@ ( zero_zero @ nat ) ) ) ) ) ).
% dim_def
thf(fact_7457_linear__indep__image__lemma,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V4867850818363320053vector @ A )
& ( real_V4867850818363320053vector @ B ) )
=> ! [F2: A > B,B7: set @ A,X: A] :
( ( real_Vector_linear @ A @ B @ F2 )
=> ( ( finite_finite @ A @ B7 )
=> ( ~ ( real_V358717886546972837endent @ B @ ( image @ A @ B @ F2 @ B7 ) )
=> ( ( inj_on @ A @ B @ F2 @ B7 )
=> ( ( member @ A @ X @ ( real_Vector_span @ A @ B7 ) )
=> ( ( ( F2 @ X )
= ( zero_zero @ B ) )
=> ( X
= ( zero_zero @ A ) ) ) ) ) ) ) ) ) ).
% linear_indep_image_lemma
thf(fact_7458_linear__eq__0__on__span,axiom,
! [A: $tType,B: $tType] :
( ( ( real_V4867850818363320053vector @ B )
& ( real_V4867850818363320053vector @ A ) )
=> ! [F2: A > B,B2: set @ A,X: A] :
( ( real_Vector_linear @ A @ B @ F2 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ B2 )
=> ( ( F2 @ X3 )
= ( zero_zero @ B ) ) )
=> ( ( member @ A @ X @ ( real_Vector_span @ A @ B2 ) )
=> ( ( F2 @ X )
= ( zero_zero @ B ) ) ) ) ) ) ).
% linear_eq_0_on_span
thf(fact_7459_linear__injective__0,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V4867850818363320053vector @ A )
& ( real_V4867850818363320053vector @ B ) )
=> ! [F2: A > B] :
( ( real_Vector_linear @ A @ B @ F2 )
=> ( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
= ( ! [X2: A] :
( ( ( F2 @ X2 )
= ( zero_zero @ B ) )
=> ( X2
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% linear_injective_0
thf(fact_7460_linear__compose__neg,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V4867850818363320053vector @ A )
& ( real_V4867850818363320053vector @ B ) )
=> ! [F2: A > B] :
( ( real_Vector_linear @ A @ B @ F2 )
=> ( real_Vector_linear @ A @ B
@ ^ [X2: A] : ( uminus_uminus @ B @ ( F2 @ X2 ) ) ) ) ) ).
% linear_compose_neg
thf(fact_7461_linear__neg,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V4867850818363320053vector @ A )
& ( real_V4867850818363320053vector @ B ) )
=> ! [F2: A > B,X: A] :
( ( real_Vector_linear @ A @ B @ F2 )
=> ( ( F2 @ ( uminus_uminus @ A @ X ) )
= ( uminus_uminus @ B @ ( F2 @ X ) ) ) ) ) ).
% linear_neg
thf(fact_7462_module__hom__uminus,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ( real_Vector_linear @ A @ A @ ( uminus_uminus @ A ) ) ) ).
% module_hom_uminus
thf(fact_7463_linear__diff,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V4867850818363320053vector @ A )
& ( real_V4867850818363320053vector @ B ) )
=> ! [F2: A > B,X: A,Y2: A] :
( ( real_Vector_linear @ A @ B @ F2 )
=> ( ( F2 @ ( minus_minus @ A @ X @ Y2 ) )
= ( minus_minus @ B @ ( F2 @ X ) @ ( F2 @ Y2 ) ) ) ) ) ).
% linear_diff
thf(fact_7464_linear__compose__sub,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V4867850818363320053vector @ A )
& ( real_V4867850818363320053vector @ B ) )
=> ! [F2: A > B,G: A > B] :
( ( real_Vector_linear @ A @ B @ F2 )
=> ( ( real_Vector_linear @ A @ B @ G )
=> ( real_Vector_linear @ A @ B
@ ^ [X2: A] : ( minus_minus @ B @ ( F2 @ X2 ) @ ( G @ X2 ) ) ) ) ) ) ).
% linear_compose_sub
thf(fact_7465_linear__0,axiom,
! [A: $tType,B: $tType] :
( ( ( real_V4867850818363320053vector @ B )
& ( real_V4867850818363320053vector @ A ) )
=> ! [F2: A > B] :
( ( real_Vector_linear @ A @ B @ F2 )
=> ( ( F2 @ ( zero_zero @ A ) )
= ( zero_zero @ B ) ) ) ) ).
% linear_0
thf(fact_7466_module__hom__zero,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V4867850818363320053vector @ A )
& ( real_V4867850818363320053vector @ B ) )
=> ( real_Vector_linear @ A @ B
@ ^ [X2: A] : ( zero_zero @ B ) ) ) ).
% module_hom_zero
thf(fact_7467_representation__diff,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [Basis: set @ A,V2: A,U: A] :
( ~ ( real_V358717886546972837endent @ A @ Basis )
=> ( ( member @ A @ V2 @ ( real_Vector_span @ A @ Basis ) )
=> ( ( member @ A @ U @ ( real_Vector_span @ A @ Basis ) )
=> ( ( real_V7696804695334737415tation @ A @ Basis @ ( minus_minus @ A @ U @ V2 ) )
= ( ^ [B3: A] : ( minus_minus @ real @ ( real_V7696804695334737415tation @ A @ Basis @ U @ B3 ) @ ( real_V7696804695334737415tation @ A @ Basis @ V2 @ B3 ) ) ) ) ) ) ) ) ).
% representation_diff
thf(fact_7468_representation__neg,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [Basis: set @ A,V2: A] :
( ~ ( real_V358717886546972837endent @ A @ Basis )
=> ( ( member @ A @ V2 @ ( real_Vector_span @ A @ Basis ) )
=> ( ( real_V7696804695334737415tation @ A @ Basis @ ( uminus_uminus @ A @ V2 ) )
= ( ^ [B3: A] : ( uminus_uminus @ real @ ( real_V7696804695334737415tation @ A @ Basis @ V2 @ B3 ) ) ) ) ) ) ) ).
% representation_neg
thf(fact_7469_representation__zero,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [Basis: set @ A] :
( ( real_V7696804695334737415tation @ A @ Basis @ ( zero_zero @ A ) )
= ( ^ [B3: A] : ( zero_zero @ real ) ) ) ) ).
% representation_zero
thf(fact_7470_representation__basis,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [Basis: set @ A,B2: A] :
( ~ ( real_V358717886546972837endent @ A @ Basis )
=> ( ( member @ A @ B2 @ Basis )
=> ( ( real_V7696804695334737415tation @ A @ Basis @ B2 )
= ( ^ [V5: A] : ( if @ real @ ( V5 = B2 ) @ ( one_one @ real ) @ ( zero_zero @ real ) ) ) ) ) ) ) ).
% representation_basis
thf(fact_7471_finite__sequence__to__countable__set,axiom,
! [A: $tType,X8: set @ A] :
( ( countable_countable @ A @ X8 )
=> ~ ! [F10: nat > ( set @ A )] :
( ! [I4: nat] : ( ord_less_eq @ ( set @ A ) @ ( F10 @ I4 ) @ X8 )
=> ( ! [I4: nat] : ( ord_less_eq @ ( set @ A ) @ ( F10 @ I4 ) @ ( F10 @ ( suc @ I4 ) ) )
=> ( ! [I4: nat] : ( finite_finite @ A @ ( F10 @ I4 ) )
=> ( ( complete_Sup_Sup @ ( set @ A ) @ ( image @ nat @ ( set @ A ) @ F10 @ ( top_top @ ( set @ nat ) ) ) )
!= X8 ) ) ) ) ) ).
% finite_sequence_to_countable_set
thf(fact_7472_construct__def,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V4867850818363320053vector @ A )
& ( real_V4867850818363320053vector @ B ) )
=> ( ( real_V4425403222259421789struct @ A @ B )
= ( ^ [B5: set @ A,G2: A > B,V5: A] :
( groups7311177749621191930dd_sum @ A @ B
@ ^ [B3: A] : ( real_V8093663219630862766scaleR @ B @ ( real_V7696804695334737415tation @ A @ ( real_V4986007116245087402_basis @ A @ B5 ) @ V5 @ B3 ) @ ( if @ B @ ( member @ A @ B3 @ B5 ) @ ( G2 @ B3 ) @ ( zero_zero @ B ) ) )
@ ( collect @ A
@ ^ [B3: A] :
( ( real_V7696804695334737415tation @ A @ ( real_V4986007116245087402_basis @ A @ B5 ) @ V5 @ B3 )
!= ( zero_zero @ real ) ) ) ) ) ) ) ).
% construct_def
thf(fact_7473_countable__Diff,axiom,
! [A: $tType,A4: set @ A,B7: set @ A] :
( ( countable_countable @ A @ A4 )
=> ( countable_countable @ A @ ( minus_minus @ ( set @ A ) @ A4 @ B7 ) ) ) ).
% countable_Diff
thf(fact_7474_countable__Diff__eq,axiom,
! [A: $tType,A4: set @ A,X: A] :
( ( countable_countable @ A @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( countable_countable @ A @ A4 ) ) ).
% countable_Diff_eq
thf(fact_7475_less__ccSup__iff,axiom,
! [A: $tType] :
( ( ( counta3822494911875563373attice @ A )
& ( linorder @ A ) )
=> ! [S3: set @ A,A3: A] :
( ( countable_countable @ A @ S3 )
=> ( ( ord_less @ A @ A3 @ ( complete_Sup_Sup @ A @ S3 ) )
= ( ? [X2: A] :
( ( member @ A @ X2 @ S3 )
& ( ord_less @ A @ A3 @ X2 ) ) ) ) ) ) ).
% less_ccSup_iff
thf(fact_7476_ccInf__less__iff,axiom,
! [A: $tType] :
( ( ( counta3822494911875563373attice @ A )
& ( linorder @ A ) )
=> ! [S3: set @ A,A3: A] :
( ( countable_countable @ A @ S3 )
=> ( ( ord_less @ A @ ( complete_Inf_Inf @ A @ S3 ) @ A3 )
= ( ? [X2: A] :
( ( member @ A @ X2 @ S3 )
& ( ord_less @ A @ X2 @ A3 ) ) ) ) ) ) ).
% ccInf_less_iff
thf(fact_7477_uncountable__minus__countable,axiom,
! [A: $tType,A4: set @ A,B7: set @ A] :
( ~ ( countable_countable @ A @ A4 )
=> ( ( countable_countable @ A @ B7 )
=> ~ ( countable_countable @ A @ ( minus_minus @ ( set @ A ) @ A4 @ B7 ) ) ) ) ).
% uncountable_minus_countable
thf(fact_7478_less__ccSUP__iff,axiom,
! [A: $tType,B: $tType] :
( ( ( counta3822494911875563373attice @ A )
& ( linorder @ A ) )
=> ! [A4: set @ B,A3: A,F2: B > A] :
( ( countable_countable @ B @ A4 )
=> ( ( ord_less @ A @ A3 @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A4 ) ) )
= ( ? [X2: B] :
( ( member @ B @ X2 @ A4 )
& ( ord_less @ A @ A3 @ ( F2 @ X2 ) ) ) ) ) ) ) ).
% less_ccSUP_iff
thf(fact_7479_ccINF__less__iff,axiom,
! [A: $tType,B: $tType] :
( ( ( counta3822494911875563373attice @ A )
& ( linorder @ A ) )
=> ! [A4: set @ B,F2: B > A,A3: A] :
( ( countable_countable @ B @ A4 )
=> ( ( ord_less @ A @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ A4 ) ) @ A3 )
= ( ? [X2: B] :
( ( member @ B @ X2 @ A4 )
& ( ord_less @ A @ ( F2 @ X2 ) @ A3 ) ) ) ) ) ) ).
% ccINF_less_iff
thf(fact_7480_construct__outside,axiom,
! [A: $tType,B: $tType] :
( ( ( real_V4867850818363320053vector @ B )
& ( real_V4867850818363320053vector @ A ) )
=> ! [B7: set @ A,V2: A,F2: A > B] :
( ~ ( real_V358717886546972837endent @ A @ B7 )
=> ( ( member @ A @ V2 @ ( real_Vector_span @ A @ ( minus_minus @ ( set @ A ) @ ( real_V4986007116245087402_basis @ A @ B7 ) @ B7 ) ) )
=> ( ( real_V4425403222259421789struct @ A @ B @ B7 @ F2 @ V2 )
= ( zero_zero @ B ) ) ) ) ) ).
% construct_outside
thf(fact_7481_uminus__integer__def,axiom,
( ( uminus_uminus @ code_integer )
= ( map_fun @ code_integer @ int @ int @ code_integer @ code_int_of_integer @ code_integer_of_int @ ( uminus_uminus @ int ) ) ) ).
% uminus_integer_def
thf(fact_7482_natLess__def,axiom,
( bNF_Ca8459412986667044542atLess
= ( collect @ ( product_prod @ nat @ nat ) @ ( product_case_prod @ nat @ nat @ $o @ ( ord_less @ nat ) ) ) ) ).
% natLess_def
thf(fact_7483_Restr__natLeq,axiom,
! [N: nat] :
( ( inf_inf @ ( set @ ( product_prod @ nat @ nat ) ) @ bNF_Ca8665028551170535155natLeq
@ ( product_Sigma @ nat @ nat
@ ( collect @ nat
@ ^ [X2: nat] : ( ord_less @ nat @ X2 @ N ) )
@ ^ [Uu3: nat] :
( collect @ nat
@ ^ [X2: nat] : ( ord_less @ nat @ X2 @ N ) ) ) )
= ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [X2: nat,Y5: nat] :
( ( ord_less @ nat @ X2 @ N )
& ( ord_less @ nat @ Y5 @ N )
& ( ord_less_eq @ nat @ X2 @ Y5 ) ) ) ) ) ).
% Restr_natLeq
thf(fact_7484_divide__integer__def,axiom,
( ( divide_divide @ code_integer )
= ( map_fun @ code_integer @ int @ ( int > int ) @ ( code_integer > code_integer ) @ code_int_of_integer @ ( map_fun @ code_integer @ int @ int @ code_integer @ code_int_of_integer @ code_integer_of_int ) @ ( divide_divide @ int ) ) ) ).
% divide_integer_def
thf(fact_7485_minus__integer__def,axiom,
( ( minus_minus @ code_integer )
= ( map_fun @ code_integer @ int @ ( int > int ) @ ( code_integer > code_integer ) @ code_int_of_integer @ ( map_fun @ code_integer @ int @ int @ code_integer @ code_int_of_integer @ code_integer_of_int ) @ ( minus_minus @ int ) ) ) ).
% minus_integer_def
thf(fact_7486_Restr__natLeq2,axiom,
! [N: nat] :
( ( inf_inf @ ( set @ ( product_prod @ nat @ nat ) ) @ bNF_Ca8665028551170535155natLeq
@ ( product_Sigma @ nat @ nat @ ( order_underS @ nat @ bNF_Ca8665028551170535155natLeq @ N )
@ ^ [Uu3: nat] : ( order_underS @ nat @ bNF_Ca8665028551170535155natLeq @ N ) ) )
= ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [X2: nat,Y5: nat] :
( ( ord_less @ nat @ X2 @ N )
& ( ord_less @ nat @ Y5 @ N )
& ( ord_less_eq @ nat @ X2 @ Y5 ) ) ) ) ) ).
% Restr_natLeq2
thf(fact_7487_fun__cong__unused__0,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( zero @ B )
=> ! [F2: ( A > B ) > C,G: C] :
( ( F2
= ( ^ [X2: A > B] : G ) )
=> ( ( F2
@ ^ [X2: A] : ( zero_zero @ B ) )
= G ) ) ) ).
% fun_cong_unused_0
thf(fact_7488_natLeq__underS__less,axiom,
! [N: nat] :
( ( order_underS @ nat @ bNF_Ca8665028551170535155natLeq @ N )
= ( collect @ nat
@ ^ [X2: nat] : ( ord_less @ nat @ X2 @ N ) ) ) ).
% natLeq_underS_less
thf(fact_7489_add_Ogroup__axioms,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ( group @ A @ ( plus_plus @ A ) @ ( zero_zero @ A ) @ ( uminus_uminus @ A ) ) ) ).
% add.group_axioms
thf(fact_7490_sub_Oabs__eq,axiom,
( code_sub
= ( ^ [Xa4: num,X2: num] : ( code_integer_of_int @ ( minus_minus @ int @ ( numeral_numeral @ int @ Xa4 ) @ ( numeral_numeral @ int @ X2 ) ) ) ) ) ).
% sub.abs_eq
thf(fact_7491_group_Oinverse__distrib__swap,axiom,
! [A: $tType,F2: A > A > A,Z2: A,Inverse: A > A,A3: A,B2: A] :
( ( group @ A @ F2 @ Z2 @ Inverse )
=> ( ( Inverse @ ( F2 @ A3 @ B2 ) )
= ( F2 @ ( Inverse @ B2 ) @ ( Inverse @ A3 ) ) ) ) ).
% group.inverse_distrib_swap
thf(fact_7492_group_Ogroup__left__neutral,axiom,
! [A: $tType,F2: A > A > A,Z2: A,Inverse: A > A,A3: A] :
( ( group @ A @ F2 @ Z2 @ Inverse )
=> ( ( F2 @ Z2 @ A3 )
= A3 ) ) ).
% group.group_left_neutral
thf(fact_7493_group_Oinverse__neutral,axiom,
! [A: $tType,F2: A > A > A,Z2: A,Inverse: A > A] :
( ( group @ A @ F2 @ Z2 @ Inverse )
=> ( ( Inverse @ Z2 )
= Z2 ) ) ).
% group.inverse_neutral
thf(fact_7494_group_Oinverse__inverse,axiom,
! [A: $tType,F2: A > A > A,Z2: A,Inverse: A > A,A3: A] :
( ( group @ A @ F2 @ Z2 @ Inverse )
=> ( ( Inverse @ ( Inverse @ A3 ) )
= A3 ) ) ).
% group.inverse_inverse
thf(fact_7495_group_Oinverse__unique,axiom,
! [A: $tType,F2: A > A > A,Z2: A,Inverse: A > A,A3: A,B2: A] :
( ( group @ A @ F2 @ Z2 @ Inverse )
=> ( ( ( F2 @ A3 @ B2 )
= Z2 )
=> ( ( Inverse @ A3 )
= B2 ) ) ) ).
% group.inverse_unique
thf(fact_7496_group_Oright__inverse,axiom,
! [A: $tType,F2: A > A > A,Z2: A,Inverse: A > A,A3: A] :
( ( group @ A @ F2 @ Z2 @ Inverse )
=> ( ( F2 @ A3 @ ( Inverse @ A3 ) )
= Z2 ) ) ).
% group.right_inverse
thf(fact_7497_group_Oright__cancel,axiom,
! [A: $tType,F2: A > A > A,Z2: A,Inverse: A > A,B2: A,A3: A,C2: A] :
( ( group @ A @ F2 @ Z2 @ Inverse )
=> ( ( ( F2 @ B2 @ A3 )
= ( F2 @ C2 @ A3 ) )
= ( B2 = C2 ) ) ) ).
% group.right_cancel
thf(fact_7498_group_Oleft__inverse,axiom,
! [A: $tType,F2: A > A > A,Z2: A,Inverse: A > A,A3: A] :
( ( group @ A @ F2 @ Z2 @ Inverse )
=> ( ( F2 @ ( Inverse @ A3 ) @ A3 )
= Z2 ) ) ).
% group.left_inverse
thf(fact_7499_group_Oleft__cancel,axiom,
! [A: $tType,F2: A > A > A,Z2: A,Inverse: A > A,A3: A,B2: A,C2: A] :
( ( group @ A @ F2 @ Z2 @ Inverse )
=> ( ( ( F2 @ A3 @ B2 )
= ( F2 @ A3 @ C2 ) )
= ( B2 = C2 ) ) ) ).
% group.left_cancel
thf(fact_7500_sub_Orep__eq,axiom,
! [X: num,Xa: num] :
( ( code_int_of_integer @ ( code_sub @ X @ Xa ) )
= ( minus_minus @ int @ ( numeral_numeral @ int @ X ) @ ( numeral_numeral @ int @ Xa ) ) ) ).
% sub.rep_eq
thf(fact_7501_Code__Numeral_Osub__code_I9_J,axiom,
! [M2: num,N: num] :
( ( code_sub @ ( bit0 @ M2 ) @ ( bit1 @ N ) )
= ( minus_minus @ code_integer @ ( code_dup @ ( code_sub @ M2 @ N ) ) @ ( one_one @ code_integer ) ) ) ).
% Code_Numeral.sub_code(9)
thf(fact_7502_Code__Numeral_Osub__code_I8_J,axiom,
! [M2: num,N: num] :
( ( code_sub @ ( bit1 @ M2 ) @ ( bit0 @ N ) )
= ( plus_plus @ code_integer @ ( code_dup @ ( code_sub @ M2 @ N ) ) @ ( one_one @ code_integer ) ) ) ).
% Code_Numeral.sub_code(8)
thf(fact_7503_and_Omonoid__axioms,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( monoid @ A @ ( bit_se5824344872417868541ns_and @ A ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% and.monoid_axioms
thf(fact_7504_inv__o__cancel,axiom,
! [B: $tType,A: $tType,F2: A > B] :
( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
=> ( ( comp @ B @ A @ A @ ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F2 ) @ F2 )
= ( id @ A ) ) ) ).
% inv_o_cancel
thf(fact_7505_inv__into__f__f,axiom,
! [B: $tType,A: $tType,F2: A > B,A4: set @ A,X: A] :
( ( inj_on @ A @ B @ F2 @ A4 )
=> ( ( member @ A @ X @ A4 )
=> ( ( hilbert_inv_into @ A @ B @ A4 @ F2 @ ( F2 @ X ) )
= X ) ) ) ).
% inv_into_f_f
thf(fact_7506_inv__identity,axiom,
! [A: $tType] :
( ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) )
@ ^ [A5: A] : A5 )
= ( ^ [A5: A] : A5 ) ) ).
% inv_identity
thf(fact_7507_inv__id,axiom,
! [A: $tType] :
( ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ ( id @ A ) )
= ( id @ A ) ) ).
% inv_id
thf(fact_7508_inv__into__image__cancel,axiom,
! [B: $tType,A: $tType,F2: A > B,A4: set @ A,S3: set @ A] :
( ( inj_on @ A @ B @ F2 @ A4 )
=> ( ( ord_less_eq @ ( set @ A ) @ S3 @ A4 )
=> ( ( image @ B @ A @ ( hilbert_inv_into @ A @ B @ A4 @ F2 ) @ ( image @ A @ B @ F2 @ S3 ) )
= S3 ) ) ) ).
% inv_into_image_cancel
thf(fact_7509_o__inv__o__cancel,axiom,
! [B: $tType,C: $tType,A: $tType,F2: A > B,G: A > C] :
( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
=> ( ( comp @ B @ C @ A @ ( comp @ A @ C @ B @ G @ ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F2 ) ) @ F2 )
= G ) ) ).
% o_inv_o_cancel
thf(fact_7510_inv__fn,axiom,
! [A: $tType,F2: A > A,N: nat] :
( ( bij_betw @ A @ A @ F2 @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ A ) ) )
=> ( ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ ( compow @ ( A > A ) @ N @ F2 ) )
= ( compow @ ( A > A ) @ N @ ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ F2 ) ) ) ) ).
% inv_fn
thf(fact_7511_bij__betw__inv__into__subset,axiom,
! [B: $tType,A: $tType,F2: A > B,A4: set @ A,A15: set @ B,B7: set @ A,B11: set @ B] :
( ( bij_betw @ A @ B @ F2 @ A4 @ A15 )
=> ( ( ord_less_eq @ ( set @ A ) @ B7 @ A4 )
=> ( ( ( image @ A @ B @ F2 @ B7 )
= B11 )
=> ( bij_betw @ B @ A @ ( hilbert_inv_into @ A @ B @ A4 @ F2 ) @ B11 @ B7 ) ) ) ) ).
% bij_betw_inv_into_subset
thf(fact_7512_inj__on__inv__into,axiom,
! [B: $tType,A: $tType,B7: set @ A,F2: B > A,A4: set @ B] :
( ( ord_less_eq @ ( set @ A ) @ B7 @ ( image @ B @ A @ F2 @ A4 ) )
=> ( inj_on @ A @ B @ ( hilbert_inv_into @ B @ A @ A4 @ F2 ) @ B7 ) ) ).
% inj_on_inv_into
thf(fact_7513_image__inv__into__cancel,axiom,
! [B: $tType,A: $tType,F2: B > A,A4: set @ B,A15: set @ A,B11: set @ A] :
( ( ( image @ B @ A @ F2 @ A4 )
= A15 )
=> ( ( ord_less_eq @ ( set @ A ) @ B11 @ A15 )
=> ( ( image @ B @ A @ F2 @ ( image @ A @ B @ ( hilbert_inv_into @ B @ A @ A4 @ F2 ) @ B11 ) )
= B11 ) ) ) ).
% image_inv_into_cancel
thf(fact_7514_inj__transfer,axiom,
! [B: $tType,A: $tType,F2: A > B,P: A > $o,X: A] :
( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
=> ( ! [Y3: B] :
( ( member @ B @ Y3 @ ( image @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) ) )
=> ( P @ ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F2 @ Y3 ) ) )
=> ( P @ X ) ) ) ).
% inj_transfer
thf(fact_7515_image__inv__f__f,axiom,
! [B: $tType,A: $tType,F2: A > B,A4: set @ A] :
( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
=> ( ( image @ B @ A @ ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F2 ) @ ( image @ A @ B @ F2 @ A4 ) )
= A4 ) ) ).
% image_inv_f_f
thf(fact_7516_inj__imp__surj__inv,axiom,
! [B: $tType,A: $tType,F2: A > B] :
( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
=> ( ( image @ B @ A @ ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F2 ) @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) ) ) ).
% inj_imp_surj_inv
thf(fact_7517_surj__imp__inj__inv,axiom,
! [B: $tType,A: $tType,F2: B > A] :
( ( ( image @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
=> ( inj_on @ A @ B @ ( hilbert_inv_into @ B @ A @ ( top_top @ ( set @ B ) ) @ F2 ) @ ( top_top @ ( set @ A ) ) ) ) ).
% surj_imp_inj_inv
thf(fact_7518_surj__imp__inv__eq,axiom,
! [B: $tType,A: $tType,F2: B > A,G: A > B] :
( ( ( image @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
=> ( ! [X3: B] :
( ( G @ ( F2 @ X3 ) )
= X3 )
=> ( ( hilbert_inv_into @ B @ A @ ( top_top @ ( set @ B ) ) @ F2 )
= G ) ) ) ).
% surj_imp_inv_eq
thf(fact_7519_image__f__inv__f,axiom,
! [B: $tType,A: $tType,F2: B > A,A4: set @ A] :
( ( ( image @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
=> ( ( image @ B @ A @ F2 @ ( image @ A @ B @ ( hilbert_inv_into @ B @ A @ ( top_top @ ( set @ B ) ) @ F2 ) @ A4 ) )
= A4 ) ) ).
% image_f_inv_f
thf(fact_7520_surj__iff__all,axiom,
! [B: $tType,A: $tType,F2: B > A] :
( ( ( image @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
= ( ! [X2: A] :
( ( F2 @ ( hilbert_inv_into @ B @ A @ ( top_top @ ( set @ B ) ) @ F2 @ X2 ) )
= X2 ) ) ) ).
% surj_iff_all
thf(fact_7521_surj__f__inv__f,axiom,
! [B: $tType,A: $tType,F2: B > A,Y2: A] :
( ( ( image @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
=> ( ( F2 @ ( hilbert_inv_into @ B @ A @ ( top_top @ ( set @ B ) ) @ F2 @ Y2 ) )
= Y2 ) ) ).
% surj_f_inv_f
thf(fact_7522_inv__into__injective,axiom,
! [A: $tType,B: $tType,A4: set @ A,F2: A > B,X: B,Y2: B] :
( ( ( hilbert_inv_into @ A @ B @ A4 @ F2 @ X )
= ( hilbert_inv_into @ A @ B @ A4 @ F2 @ Y2 ) )
=> ( ( member @ B @ X @ ( image @ A @ B @ F2 @ A4 ) )
=> ( ( member @ B @ Y2 @ ( image @ A @ B @ F2 @ A4 ) )
=> ( X = Y2 ) ) ) ) ).
% inv_into_injective
thf(fact_7523_inv__into__into,axiom,
! [A: $tType,B: $tType,X: A,F2: B > A,A4: set @ B] :
( ( member @ A @ X @ ( image @ B @ A @ F2 @ A4 ) )
=> ( member @ B @ ( hilbert_inv_into @ B @ A @ A4 @ F2 @ X ) @ A4 ) ) ).
% inv_into_into
thf(fact_7524_f__inv__into__f,axiom,
! [B: $tType,A: $tType,Y2: A,F2: B > A,A4: set @ B] :
( ( member @ A @ Y2 @ ( image @ B @ A @ F2 @ A4 ) )
=> ( ( F2 @ ( hilbert_inv_into @ B @ A @ A4 @ F2 @ Y2 ) )
= Y2 ) ) ).
% f_inv_into_f
thf(fact_7525_inv__into__comp,axiom,
! [A: $tType,C: $tType,B: $tType,F2: A > B,G: C > A,A4: set @ C,X: B] :
( ( inj_on @ A @ B @ F2 @ ( image @ C @ A @ G @ A4 ) )
=> ( ( inj_on @ C @ A @ G @ A4 )
=> ( ( member @ B @ X @ ( image @ A @ B @ F2 @ ( image @ C @ A @ G @ A4 ) ) )
=> ( ( hilbert_inv_into @ C @ B @ A4 @ ( comp @ A @ B @ C @ F2 @ G ) @ X )
= ( comp @ A @ C @ B @ ( hilbert_inv_into @ C @ A @ A4 @ G ) @ ( hilbert_inv_into @ A @ B @ ( image @ C @ A @ G @ A4 ) @ F2 ) @ X ) ) ) ) ) ).
% inv_into_comp
thf(fact_7526_inj__imp__inv__eq,axiom,
! [A: $tType,B: $tType,F2: A > B,G: B > A] :
( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
=> ( ! [X3: B] :
( ( F2 @ ( G @ X3 ) )
= X3 )
=> ( ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F2 )
= G ) ) ) ).
% inj_imp_inv_eq
thf(fact_7527_inv__f__eq,axiom,
! [B: $tType,A: $tType,F2: A > B,X: A,Y2: B] :
( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
=> ( ( ( F2 @ X )
= Y2 )
=> ( ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F2 @ Y2 )
= X ) ) ) ).
% inv_f_eq
thf(fact_7528_inv__f__f,axiom,
! [B: $tType,A: $tType,F2: A > B,X: A] :
( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
=> ( ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F2 @ ( F2 @ X ) )
= X ) ) ).
% inv_f_f
thf(fact_7529_inv__into__f__eq,axiom,
! [B: $tType,A: $tType,F2: A > B,A4: set @ A,X: A,Y2: B] :
( ( inj_on @ A @ B @ F2 @ A4 )
=> ( ( member @ A @ X @ A4 )
=> ( ( ( F2 @ X )
= Y2 )
=> ( ( hilbert_inv_into @ A @ B @ A4 @ F2 @ Y2 )
= X ) ) ) ) ).
% inv_into_f_eq
thf(fact_7530_mult_Omonoid__axioms,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ( monoid @ A @ ( times_times @ A ) @ ( one_one @ A ) ) ) ).
% mult.monoid_axioms
thf(fact_7531_add_Omonoid__axioms,axiom,
! [A: $tType] :
( ( monoid_add @ A )
=> ( monoid @ A @ ( plus_plus @ A ) @ ( zero_zero @ A ) ) ) ).
% add.monoid_axioms
thf(fact_7532_monoid_Oright__neutral,axiom,
! [A: $tType,F2: A > A > A,Z2: A,A3: A] :
( ( monoid @ A @ F2 @ Z2 )
=> ( ( F2 @ A3 @ Z2 )
= A3 ) ) ).
% monoid.right_neutral
thf(fact_7533_monoid_Oleft__neutral,axiom,
! [A: $tType,F2: A > A > A,Z2: A,A3: A] :
( ( monoid @ A @ F2 @ Z2 )
=> ( ( F2 @ Z2 @ A3 )
= A3 ) ) ).
% monoid.left_neutral
thf(fact_7534_bij__betw__inv__into,axiom,
! [B: $tType,A: $tType,F2: A > B,A4: set @ A,B7: set @ B] :
( ( bij_betw @ A @ B @ F2 @ A4 @ B7 )
=> ( bij_betw @ B @ A @ ( hilbert_inv_into @ A @ B @ A4 @ F2 ) @ B7 @ A4 ) ) ).
% bij_betw_inv_into
thf(fact_7535_inv__into__inv__into__eq,axiom,
! [B: $tType,A: $tType,F2: A > B,A4: set @ A,A15: set @ B,A3: A] :
( ( bij_betw @ A @ B @ F2 @ A4 @ A15 )
=> ( ( member @ A @ A3 @ A4 )
=> ( ( hilbert_inv_into @ B @ A @ A15 @ ( hilbert_inv_into @ A @ B @ A4 @ F2 ) @ A3 )
= ( F2 @ A3 ) ) ) ) ).
% inv_into_inv_into_eq
thf(fact_7536_bij__betw__inv__into__left,axiom,
! [B: $tType,A: $tType,F2: A > B,A4: set @ A,A15: set @ B,A3: A] :
( ( bij_betw @ A @ B @ F2 @ A4 @ A15 )
=> ( ( member @ A @ A3 @ A4 )
=> ( ( hilbert_inv_into @ A @ B @ A4 @ F2 @ ( F2 @ A3 ) )
= A3 ) ) ) ).
% bij_betw_inv_into_left
thf(fact_7537_bij__betw__inv__into__right,axiom,
! [A: $tType,B: $tType,F2: A > B,A4: set @ A,A15: set @ B,A8: B] :
( ( bij_betw @ A @ B @ F2 @ A4 @ A15 )
=> ( ( member @ B @ A8 @ A15 )
=> ( ( F2 @ ( hilbert_inv_into @ A @ B @ A4 @ F2 @ A8 ) )
= A8 ) ) ) ).
% bij_betw_inv_into_right
thf(fact_7538_sup__bot_Omonoid__axioms,axiom,
! [A: $tType] :
( ( bounde4967611905675639751up_bot @ A )
=> ( monoid @ A @ ( sup_sup @ A ) @ ( bot_bot @ A ) ) ) ).
% sup_bot.monoid_axioms
thf(fact_7539_bij__imp__bij__inv,axiom,
! [B: $tType,A: $tType,F2: A > B] :
( ( bij_betw @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
=> ( bij_betw @ B @ A @ ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F2 ) @ ( top_top @ ( set @ B ) ) @ ( top_top @ ( set @ A ) ) ) ) ).
% bij_imp_bij_inv
thf(fact_7540_bij__inv__eq__iff,axiom,
! [A: $tType,B: $tType,P4: A > B,X: A,Y2: B] :
( ( bij_betw @ A @ B @ P4 @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
=> ( ( X
= ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ P4 @ Y2 ) )
= ( ( P4 @ X )
= Y2 ) ) ) ).
% bij_inv_eq_iff
thf(fact_7541_inv__inv__eq,axiom,
! [B: $tType,A: $tType,F2: A > B] :
( ( bij_betw @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
=> ( ( hilbert_inv_into @ B @ A @ ( top_top @ ( set @ B ) ) @ ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F2 ) )
= F2 ) ) ).
% inv_inv_eq
thf(fact_7542_mono__inv,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( linorder @ B ) )
=> ! [F2: A > B] :
( ( order_mono @ A @ B @ F2 )
=> ( ( bij_betw @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
=> ( order_mono @ B @ A @ ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F2 ) ) ) ) ) ).
% mono_inv
thf(fact_7543_o__inv__distrib,axiom,
! [C: $tType,B: $tType,A: $tType,F2: A > B,G: C > A] :
( ( bij_betw @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
=> ( ( bij_betw @ C @ A @ G @ ( top_top @ ( set @ C ) ) @ ( top_top @ ( set @ A ) ) )
=> ( ( hilbert_inv_into @ C @ B @ ( top_top @ ( set @ C ) ) @ ( comp @ A @ B @ C @ F2 @ G ) )
= ( comp @ A @ C @ B @ ( hilbert_inv_into @ C @ A @ ( top_top @ ( set @ C ) ) @ G ) @ ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F2 ) ) ) ) ) ).
% o_inv_distrib
thf(fact_7544_inv__equality,axiom,
! [A: $tType,B: $tType,G: B > A,F2: A > B] :
( ! [X3: A] :
( ( G @ ( F2 @ X3 ) )
= X3 )
=> ( ! [Y3: B] :
( ( F2 @ ( G @ Y3 ) )
= Y3 )
=> ( ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F2 )
= G ) ) ) ).
% inv_equality
thf(fact_7545_inv__unique__comp,axiom,
! [B: $tType,A: $tType,F2: B > A,G: A > B] :
( ( ( comp @ B @ A @ A @ F2 @ G )
= ( id @ A ) )
=> ( ( ( comp @ A @ B @ B @ G @ F2 )
= ( id @ B ) )
=> ( ( hilbert_inv_into @ B @ A @ ( top_top @ ( set @ B ) ) @ F2 )
= G ) ) ) ).
% inv_unique_comp
thf(fact_7546_gcd__nat_Omonoid__axioms,axiom,
monoid @ nat @ ( gcd_gcd @ nat ) @ ( zero_zero @ nat ) ).
% gcd_nat.monoid_axioms
thf(fact_7547_inv__def,axiom,
! [B: $tType,A: $tType,F2: B > A] :
( ( hilbert_inv_into @ B @ A @ ( top_top @ ( set @ B ) ) @ F2 )
= ( ^ [Y5: A] :
( fChoice @ B
@ ^ [X2: B] :
( ( F2 @ X2 )
= Y5 ) ) ) ) ).
% inv_def
thf(fact_7548_inv__into__def,axiom,
! [B: $tType,A: $tType] :
( ( hilbert_inv_into @ A @ B )
= ( ^ [A7: set @ A,F4: A > B,X2: B] :
( fChoice @ A
@ ^ [Y5: A] :
( ( member @ A @ Y5 @ A7 )
& ( ( F4 @ Y5 )
= X2 ) ) ) ) ) ).
% inv_into_def
thf(fact_7549_inv__into__def2,axiom,
! [B: $tType,A: $tType] :
( ( hilbert_inv_into @ A @ B )
= ( ^ [A7: set @ A,F4: A > B,X2: B] :
( fChoice @ A
@ ^ [Y5: A] :
( ( member @ A @ Y5 @ A7 )
& ( ( F4 @ Y5 )
= X2 ) ) ) ) ) ).
% inv_into_def2
thf(fact_7550_or_Omonoid__axioms,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( monoid @ A @ ( bit_se1065995026697491101ons_or @ A ) @ ( zero_zero @ A ) ) ) ).
% or.monoid_axioms
thf(fact_7551_inf__top_Omonoid__axioms,axiom,
! [A: $tType] :
( ( bounde4346867609351753570nf_top @ A )
=> ( monoid @ A @ ( inf_inf @ A ) @ ( top_top @ A ) ) ) ).
% inf_top.monoid_axioms
thf(fact_7552_xor_Omonoid__axioms,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( monoid @ A @ ( bit_se5824344971392196577ns_xor @ A ) @ ( zero_zero @ A ) ) ) ).
% xor.monoid_axioms
thf(fact_7553_bij__image__Collect__eq,axiom,
! [A: $tType,B: $tType,F2: A > B,P: A > $o] :
( ( bij_betw @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
=> ( ( image @ A @ B @ F2 @ ( collect @ A @ P ) )
= ( collect @ B
@ ^ [Y5: B] : ( P @ ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F2 @ Y5 ) ) ) ) ) ).
% bij_image_Collect_eq
thf(fact_7554_max__nat_Omonoid__axioms,axiom,
monoid @ nat @ ( ord_max @ nat ) @ ( zero_zero @ nat ) ).
% max_nat.monoid_axioms
thf(fact_7555_surj__iff,axiom,
! [B: $tType,A: $tType,F2: B > A] :
( ( ( image @ B @ A @ F2 @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
= ( ( comp @ B @ A @ A @ F2 @ ( hilbert_inv_into @ B @ A @ ( top_top @ ( set @ B ) ) @ F2 ) )
= ( id @ A ) ) ) ).
% surj_iff
thf(fact_7556_inj__imp__bij__betw__inv,axiom,
! [B: $tType,A: $tType,F2: A > B,M10: set @ A] :
( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
=> ( bij_betw @ B @ A @ ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F2 ) @ ( image @ A @ B @ F2 @ M10 ) @ M10 ) ) ).
% inj_imp_bij_betw_inv
thf(fact_7557_inj__iff,axiom,
! [B: $tType,A: $tType,F2: A > B] :
( ( inj_on @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) )
= ( ( comp @ B @ A @ A @ ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F2 ) @ F2 )
= ( id @ A ) ) ) ).
% inj_iff
thf(fact_7558_bij__vimage__eq__inv__image,axiom,
! [A: $tType,B: $tType,F2: A > B,A4: set @ B] :
( ( bij_betw @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
=> ( ( vimage @ A @ B @ F2 @ A4 )
= ( image @ B @ A @ ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F2 ) @ A4 ) ) ) ).
% bij_vimage_eq_inv_image
thf(fact_7559_inv__fn__o__fn__is__id,axiom,
! [A: $tType,F2: A > A,N: nat] :
( ( bij_betw @ A @ A @ F2 @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ A ) ) )
=> ( ( comp @ A @ A @ A @ ( compow @ ( A > A ) @ N @ ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ F2 ) ) @ ( compow @ ( A > A ) @ N @ F2 ) )
= ( ^ [X2: A] : X2 ) ) ) ).
% inv_fn_o_fn_is_id
thf(fact_7560_fn__o__inv__fn__is__id,axiom,
! [A: $tType,F2: A > A,N: nat] :
( ( bij_betw @ A @ A @ F2 @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ A ) ) )
=> ( ( comp @ A @ A @ A @ ( compow @ ( A > A ) @ N @ F2 ) @ ( compow @ ( A > A ) @ N @ ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ F2 ) ) )
= ( ^ [X2: A] : X2 ) ) ) ).
% fn_o_inv_fn_is_id
thf(fact_7561_strict__mono__inv__on__range,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( order @ B ) )
=> ! [F2: A > B] :
( ( order_strict_mono @ A @ B @ F2 )
=> ( strict_mono_on @ B @ A @ ( hilbert_inv_into @ A @ B @ ( top_top @ ( set @ A ) ) @ F2 ) @ ( image @ A @ B @ F2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% strict_mono_inv_on_range
thf(fact_7562_bijection_Oinv__comp__right,axiom,
! [A: $tType,F2: A > A] :
( ( hilbert_bijection @ A @ F2 )
=> ( ( comp @ A @ A @ A @ F2 @ ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ F2 ) )
= ( id @ A ) ) ) ).
% bijection.inv_comp_right
thf(fact_7563_bijection_Oinv__comp__left,axiom,
! [A: $tType,F2: A > A] :
( ( hilbert_bijection @ A @ F2 )
=> ( ( comp @ A @ A @ A @ ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ F2 ) @ F2 )
= ( id @ A ) ) ) ).
% bijection.inv_comp_left
thf(fact_7564_bijection_Oeq__invI,axiom,
! [A: $tType,F2: A > A,A3: A,B2: A] :
( ( hilbert_bijection @ A @ F2 )
=> ( ( ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ F2 @ A3 )
= ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ F2 @ B2 ) )
=> ( A3 = B2 ) ) ) ).
% bijection.eq_invI
thf(fact_7565_bijection_Oinv__left,axiom,
! [A: $tType,F2: A > A,A3: A] :
( ( hilbert_bijection @ A @ F2 )
=> ( ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ F2 @ ( F2 @ A3 ) )
= A3 ) ) ).
% bijection.inv_left
thf(fact_7566_bijection_Oinv__right,axiom,
! [A: $tType,F2: A > A,A3: A] :
( ( hilbert_bijection @ A @ F2 )
=> ( ( F2 @ ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ F2 @ A3 ) )
= A3 ) ) ).
% bijection.inv_right
thf(fact_7567_bijection_Oeq__inv__iff,axiom,
! [A: $tType,F2: A > A,A3: A,B2: A] :
( ( hilbert_bijection @ A @ F2 )
=> ( ( ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ F2 @ A3 )
= ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ F2 @ B2 ) )
= ( A3 = B2 ) ) ) ).
% bijection.eq_inv_iff
thf(fact_7568_bijection_Oinv__left__eq__iff,axiom,
! [A: $tType,F2: A > A,A3: A,B2: A] :
( ( hilbert_bijection @ A @ F2 )
=> ( ( ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ F2 @ A3 )
= B2 )
= ( ( F2 @ B2 )
= A3 ) ) ) ).
% bijection.inv_left_eq_iff
thf(fact_7569_bijection_Oinv__right__eq__iff,axiom,
! [A: $tType,F2: A > A,B2: A,A3: A] :
( ( hilbert_bijection @ A @ F2 )
=> ( ( B2
= ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ F2 @ A3 ) )
= ( ( F2 @ B2 )
= A3 ) ) ) ).
% bijection.inv_right_eq_iff
thf(fact_7570_bijection__def,axiom,
! [A: $tType] :
( ( hilbert_bijection @ A )
= ( ^ [F4: A > A] : ( bij_betw @ A @ A @ F4 @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ A ) ) ) ) ) ).
% bijection_def
thf(fact_7571_bijection_Ointro,axiom,
! [A: $tType,F2: A > A] :
( ( bij_betw @ A @ A @ F2 @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ A ) ) )
=> ( hilbert_bijection @ A @ F2 ) ) ).
% bijection.intro
thf(fact_7572_bijection_Obij,axiom,
! [A: $tType,F2: A > A] :
( ( hilbert_bijection @ A @ F2 )
=> ( bij_betw @ A @ A @ F2 @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ A ) ) ) ) ).
% bijection.bij
thf(fact_7573_bijection_OeqI,axiom,
! [A: $tType,F2: A > A,A3: A,B2: A] :
( ( hilbert_bijection @ A @ F2 )
=> ( ( ( F2 @ A3 )
= ( F2 @ B2 ) )
=> ( A3 = B2 ) ) ) ).
% bijection.eqI
thf(fact_7574_bijection_Oeq__iff,axiom,
! [A: $tType,F2: A > A,A3: A,B2: A] :
( ( hilbert_bijection @ A @ F2 )
=> ( ( ( F2 @ A3 )
= ( F2 @ B2 ) )
= ( A3 = B2 ) ) ) ).
% bijection.eq_iff
thf(fact_7575_bijection_Oinj,axiom,
! [A: $tType,F2: A > A] :
( ( hilbert_bijection @ A @ F2 )
=> ( inj_on @ A @ A @ F2 @ ( top_top @ ( set @ A ) ) ) ) ).
% bijection.inj
thf(fact_7576_bijection_Osurj,axiom,
! [A: $tType,F2: A > A] :
( ( hilbert_bijection @ A @ F2 )
=> ( ( image @ A @ A @ F2 @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) ) ) ).
% bijection.surj
thf(fact_7577_bijection_Osurj__inv,axiom,
! [A: $tType,F2: A > A] :
( ( hilbert_bijection @ A @ F2 )
=> ( ( image @ A @ A @ ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ F2 ) @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) ) ) ).
% bijection.surj_inv
thf(fact_7578_bijection_Oinj__inv,axiom,
! [A: $tType,F2: A > A] :
( ( hilbert_bijection @ A @ F2 )
=> ( inj_on @ A @ A @ ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ F2 ) @ ( top_top @ ( set @ A ) ) ) ) ).
% bijection.inj_inv
thf(fact_7579_bijection_Obij__inv,axiom,
! [A: $tType,F2: A > A] :
( ( hilbert_bijection @ A @ F2 )
=> ( bij_betw @ A @ A @ ( hilbert_inv_into @ A @ A @ ( top_top @ ( set @ A ) ) @ F2 ) @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ A ) ) ) ) ).
% bijection.bij_inv
thf(fact_7580_convergent__realpow,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ X @ ( one_one @ real ) )
=> ( topolo6863149650580417670ergent @ real @ ( power_power @ real @ X ) ) ) ) ).
% convergent_realpow
thf(fact_7581_less__cSUP__iff,axiom,
! [A: $tType,B: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [A4: set @ B,F2: B > A,A3: A] :
( ( A4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit941137186595557371_above @ A @ ( image @ B @ A @ F2 @ A4 ) )
=> ( ( ord_less @ A @ A3 @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A4 ) ) )
= ( ? [X2: B] :
( ( member @ B @ X2 @ A4 )
& ( ord_less @ A @ A3 @ ( F2 @ X2 ) ) ) ) ) ) ) ) ).
% less_cSUP_iff
thf(fact_7582_less__cSup__iff,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [X8: set @ A,Y2: A] :
( ( X8
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit941137186595557371_above @ A @ X8 )
=> ( ( ord_less @ A @ Y2 @ ( complete_Sup_Sup @ A @ X8 ) )
= ( ? [X2: A] :
( ( member @ A @ X2 @ X8 )
& ( ord_less @ A @ Y2 @ X2 ) ) ) ) ) ) ) ).
% less_cSup_iff
thf(fact_7583_convergent__mult__const__iff,axiom,
! [A: $tType] :
( ( ( field @ A )
& ( topolo4211221413907600880p_mult @ A ) )
=> ! [C2: A,F2: nat > A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( topolo6863149650580417670ergent @ A
@ ^ [N5: nat] : ( times_times @ A @ C2 @ ( F2 @ N5 ) ) )
= ( topolo6863149650580417670ergent @ A @ F2 ) ) ) ) ).
% convergent_mult_const_iff
thf(fact_7584_convergent__mult__const__right__iff,axiom,
! [A: $tType] :
( ( ( field @ A )
& ( topolo4211221413907600880p_mult @ A ) )
=> ! [C2: A,F2: nat > A] :
( ( C2
!= ( zero_zero @ A ) )
=> ( ( topolo6863149650580417670ergent @ A
@ ^ [N5: nat] : ( times_times @ A @ ( F2 @ N5 ) @ C2 ) )
= ( topolo6863149650580417670ergent @ A @ F2 ) ) ) ) ).
% convergent_mult_const_right_iff
thf(fact_7585_convergent__minus__iff,axiom,
! [A: $tType] :
( ( topolo1633459387980952147up_add @ A )
=> ( ( topolo6863149650580417670ergent @ A )
= ( ^ [X7: nat > A] :
( topolo6863149650580417670ergent @ A
@ ^ [N5: nat] : ( uminus_uminus @ A @ ( X7 @ N5 ) ) ) ) ) ) ).
% convergent_minus_iff
thf(fact_7586_convergent__Suc__iff,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F2: nat > A] :
( ( topolo6863149650580417670ergent @ A
@ ^ [N5: nat] : ( F2 @ ( suc @ N5 ) ) )
= ( topolo6863149650580417670ergent @ A @ F2 ) ) ) ).
% convergent_Suc_iff
thf(fact_7587_convergent__diff__const__right__iff,axiom,
! [A: $tType] :
( ( topolo1287966508704411220up_add @ A )
=> ! [F2: nat > A,C2: A] :
( ( topolo6863149650580417670ergent @ A
@ ^ [N5: nat] : ( minus_minus @ A @ ( F2 @ N5 ) @ C2 ) )
= ( topolo6863149650580417670ergent @ A @ F2 ) ) ) ).
% convergent_diff_const_right_iff
thf(fact_7588_convergent__diff,axiom,
! [A: $tType] :
( ( topolo1633459387980952147up_add @ A )
=> ! [X8: nat > A,Y7: nat > A] :
( ( topolo6863149650580417670ergent @ A @ X8 )
=> ( ( topolo6863149650580417670ergent @ A @ Y7 )
=> ( topolo6863149650580417670ergent @ A
@ ^ [N5: nat] : ( minus_minus @ A @ ( X8 @ N5 ) @ ( Y7 @ N5 ) ) ) ) ) ) ).
% convergent_diff
thf(fact_7589_cSUP__lessD,axiom,
! [B: $tType,A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [F2: B > A,A4: set @ B,Y2: A,I: B] :
( ( condit941137186595557371_above @ A @ ( image @ B @ A @ F2 @ A4 ) )
=> ( ( ord_less @ A @ ( complete_Sup_Sup @ A @ ( image @ B @ A @ F2 @ A4 ) ) @ Y2 )
=> ( ( member @ B @ I @ A4 )
=> ( ord_less @ A @ ( F2 @ I ) @ Y2 ) ) ) ) ) ).
% cSUP_lessD
thf(fact_7590_map__conv__bind__option,axiom,
! [A: $tType,B: $tType] :
( ( map_option @ B @ A )
= ( ^ [F4: B > A,X2: option @ B] : ( bind @ B @ A @ X2 @ ( comp @ A @ ( option @ A ) @ B @ ( some @ A ) @ F4 ) ) ) ) ).
% map_conv_bind_option
thf(fact_7591_cINF__less__iff,axiom,
! [A: $tType,B: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [A4: set @ B,F2: B > A,A3: A] :
( ( A4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit1013018076250108175_below @ A @ ( image @ B @ A @ F2 @ A4 ) )
=> ( ( ord_less @ A @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ A4 ) ) @ A3 )
= ( ? [X2: B] :
( ( member @ B @ X2 @ A4 )
& ( ord_less @ A @ ( F2 @ X2 ) @ A3 ) ) ) ) ) ) ) ).
% cINF_less_iff
thf(fact_7592_bind__assoc,axiom,
! [B: $tType,A: $tType,C: $tType,X: option @ C,F2: C > ( option @ B ),G: B > ( option @ A )] :
( ( bind @ B @ A @ ( bind @ C @ B @ X @ F2 ) @ G )
= ( bind @ C @ A @ X
@ ^ [Y5: C] : ( bind @ B @ A @ ( F2 @ Y5 ) @ G ) ) ) ).
% bind_assoc
thf(fact_7593_bind__runit,axiom,
! [A: $tType,X: option @ A] :
( ( bind @ A @ A @ X @ ( some @ A ) )
= X ) ).
% bind_runit
thf(fact_7594_bind__rzero,axiom,
! [B: $tType,A: $tType,X: option @ B] :
( ( bind @ B @ A @ X
@ ^ [X2: B] : ( none @ A ) )
= ( none @ A ) ) ).
% bind_rzero
thf(fact_7595_bdd__above__uminus,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [X8: set @ A] :
( ( condit941137186595557371_above @ A @ ( image @ A @ A @ ( uminus_uminus @ A ) @ X8 ) )
= ( condit1013018076250108175_below @ A @ X8 ) ) ) ).
% bdd_above_uminus
thf(fact_7596_bdd__below__uminus,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [X8: set @ A] :
( ( condit1013018076250108175_below @ A @ ( image @ A @ A @ ( uminus_uminus @ A ) @ X8 ) )
= ( condit941137186595557371_above @ A @ X8 ) ) ) ).
% bdd_below_uminus
thf(fact_7597_bind__map__option,axiom,
! [A: $tType,B: $tType,C: $tType,F2: C > B,X: option @ C,G: B > ( option @ A )] :
( ( bind @ B @ A @ ( map_option @ C @ B @ F2 @ X ) @ G )
= ( bind @ C @ A @ X @ ( comp @ B @ ( option @ A ) @ C @ G @ F2 ) ) ) ).
% bind_map_option
thf(fact_7598_cInf__less__iff,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [X8: set @ A,Y2: A] :
( ( X8
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit1013018076250108175_below @ A @ X8 )
=> ( ( ord_less @ A @ ( complete_Inf_Inf @ A @ X8 ) @ Y2 )
= ( ? [X2: A] :
( ( member @ A @ X2 @ X8 )
& ( ord_less @ A @ X2 @ Y2 ) ) ) ) ) ) ) ).
% cInf_less_iff
thf(fact_7599_bind__split__asm,axiom,
! [A: $tType,B: $tType,P: ( option @ A ) > $o,M2: option @ B,F2: B > ( option @ A )] :
( ( P @ ( bind @ B @ A @ M2 @ F2 ) )
= ( ~ ( ( ( M2
= ( none @ B ) )
& ~ ( P @ ( none @ A ) ) )
| ? [X2: B] :
( ( M2
= ( some @ B @ X2 ) )
& ~ ( P @ ( F2 @ X2 ) ) ) ) ) ) ).
% bind_split_asm
thf(fact_7600_bind__split,axiom,
! [A: $tType,B: $tType,P: ( option @ A ) > $o,M2: option @ B,F2: B > ( option @ A )] :
( ( P @ ( bind @ B @ A @ M2 @ F2 ) )
= ( ( ( M2
= ( none @ B ) )
=> ( P @ ( none @ A ) ) )
& ! [V5: B] :
( ( M2
= ( some @ B @ V5 ) )
=> ( P @ ( F2 @ V5 ) ) ) ) ) ).
% bind_split
thf(fact_7601_bind__eq__Some__conv,axiom,
! [A: $tType,B: $tType,F2: option @ B,G: B > ( option @ A ),X: A] :
( ( ( bind @ B @ A @ F2 @ G )
= ( some @ A @ X ) )
= ( ? [Y5: B] :
( ( F2
= ( some @ B @ Y5 ) )
& ( ( G @ Y5 )
= ( some @ A @ X ) ) ) ) ) ).
% bind_eq_Some_conv
thf(fact_7602_Option_Obind__cong,axiom,
! [B: $tType,A: $tType,X: option @ A,Y2: option @ A,F2: A > ( option @ B ),G: A > ( option @ B )] :
( ( X = Y2 )
=> ( ! [A6: A] :
( ( Y2
= ( some @ A @ A6 ) )
=> ( ( F2 @ A6 )
= ( G @ A6 ) ) )
=> ( ( bind @ A @ B @ X @ F2 )
= ( bind @ A @ B @ Y2 @ G ) ) ) ) ).
% Option.bind_cong
thf(fact_7603_bind_Obind__lunit,axiom,
! [B: $tType,A: $tType,X: A,F2: A > ( option @ B )] :
( ( bind @ A @ B @ ( some @ A @ X ) @ F2 )
= ( F2 @ X ) ) ).
% bind.bind_lunit
thf(fact_7604_bind_Obind__lzero,axiom,
! [A: $tType,B: $tType,F2: A > ( option @ B )] :
( ( bind @ A @ B @ ( none @ A ) @ F2 )
= ( none @ B ) ) ).
% bind.bind_lzero
thf(fact_7605_bind__eq__None__conv,axiom,
! [B: $tType,A: $tType,A3: option @ B,F2: B > ( option @ A )] :
( ( ( bind @ B @ A @ A3 @ F2 )
= ( none @ A ) )
= ( ( A3
= ( none @ B ) )
| ( ( F2 @ ( the2 @ B @ A3 ) )
= ( none @ A ) ) ) ) ).
% bind_eq_None_conv
thf(fact_7606_bind__option__cong__code,axiom,
! [B: $tType,A: $tType,X: option @ A,Y2: option @ A,F2: A > ( option @ B )] :
( ( X = Y2 )
=> ( ( bind @ A @ B @ X @ F2 )
= ( bind @ A @ B @ Y2 @ F2 ) ) ) ).
% bind_option_cong_code
thf(fact_7607_map__option__bind,axiom,
! [A: $tType,B: $tType,C: $tType,F2: B > A,X: option @ C,G: C > ( option @ B )] :
( ( map_option @ B @ A @ F2 @ ( bind @ C @ B @ X @ G ) )
= ( bind @ C @ A @ X @ ( comp @ ( option @ B ) @ ( option @ A ) @ C @ ( map_option @ B @ A @ F2 ) @ G ) ) ) ).
% map_option_bind
thf(fact_7608_less__cINF__D,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [F2: B > A,A4: set @ B,Y2: A,I: B] :
( ( condit1013018076250108175_below @ A @ ( image @ B @ A @ F2 @ A4 ) )
=> ( ( ord_less @ A @ Y2 @ ( complete_Inf_Inf @ A @ ( image @ B @ A @ F2 @ A4 ) ) )
=> ( ( member @ B @ I @ A4 )
=> ( ord_less @ A @ Y2 @ ( F2 @ I ) ) ) ) ) ) ).
% less_cINF_D
thf(fact_7609_set__bind__option,axiom,
! [A: $tType,B: $tType,X: option @ B,F2: B > ( option @ A )] :
( ( set_option @ A @ ( bind @ B @ A @ X @ F2 ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image @ B @ ( set @ A ) @ ( comp @ ( option @ A ) @ ( set @ A ) @ B @ ( set_option @ A ) @ F2 ) @ ( set_option @ B @ X ) ) ) ) ).
% set_bind_option
thf(fact_7610_combine__options__def,axiom,
! [A: $tType] :
( ( combine_options @ A )
= ( ^ [F4: A > A > A,X2: option @ A,Y5: option @ A] :
( case_option @ ( option @ A ) @ A @ Y5
@ ^ [Z6: A] :
( case_option @ ( option @ A ) @ A @ ( some @ A @ Z6 )
@ ^ [Aa2: A] : ( some @ A @ ( F4 @ Z6 @ Aa2 ) )
@ Y5 )
@ X2 ) ) ) ).
% combine_options_def
thf(fact_7611_elem__set,axiom,
! [A: $tType,X: A,Xo: option @ A] :
( ( member @ A @ X @ ( set_option @ A @ Xo ) )
= ( Xo
= ( some @ A @ X ) ) ) ).
% elem_set
thf(fact_7612_combine__options__simps_I3_J,axiom,
! [A: $tType,F2: A > A > A,A3: A,B2: A] :
( ( combine_options @ A @ F2 @ ( some @ A @ A3 ) @ ( some @ A @ B2 ) )
= ( some @ A @ ( F2 @ A3 @ B2 ) ) ) ).
% combine_options_simps(3)
thf(fact_7613_combine__options__simps_I2_J,axiom,
! [A: $tType,F2: A > A > A,X: option @ A] :
( ( combine_options @ A @ F2 @ X @ ( none @ A ) )
= X ) ).
% combine_options_simps(2)
thf(fact_7614_combine__options__simps_I1_J,axiom,
! [A: $tType,F2: A > A > A,Y2: option @ A] :
( ( combine_options @ A @ F2 @ ( none @ A ) @ Y2 )
= Y2 ) ).
% combine_options_simps(1)
thf(fact_7615_set__empty__eq,axiom,
! [A: $tType,Xo: option @ A] :
( ( ( set_option @ A @ Xo )
= ( bot_bot @ ( set @ A ) ) )
= ( Xo
= ( none @ A ) ) ) ).
% set_empty_eq
thf(fact_7616_bind__option__cong,axiom,
! [B: $tType,A: $tType,X: option @ A,Y2: option @ A,F2: A > ( option @ B ),G: A > ( option @ B )] :
( ( X = Y2 )
=> ( ! [Z4: A] :
( ( member @ A @ Z4 @ ( set_option @ A @ Y2 ) )
=> ( ( F2 @ Z4 )
= ( G @ Z4 ) ) )
=> ( ( bind @ A @ B @ X @ F2 )
= ( bind @ A @ B @ Y2 @ G ) ) ) ) ).
% bind_option_cong
thf(fact_7617_option_Oset__map,axiom,
! [B: $tType,A: $tType,F2: A > B,V2: option @ A] :
( ( set_option @ B @ ( map_option @ A @ B @ F2 @ V2 ) )
= ( image @ A @ B @ F2 @ ( set_option @ A @ V2 ) ) ) ).
% option.set_map
thf(fact_7618_map__option__idI,axiom,
! [A: $tType,X: option @ A,F2: A > A] :
( ! [Y3: A] :
( ( member @ A @ Y3 @ ( set_option @ A @ X ) )
=> ( ( F2 @ Y3 )
= Y3 ) )
=> ( ( map_option @ A @ A @ F2 @ X )
= X ) ) ).
% map_option_idI
thf(fact_7619_option_Oinj__map__strong,axiom,
! [B: $tType,A: $tType,X: option @ A,Xa: option @ A,F2: A > B,Fa: A > B] :
( ! [Z4: A,Za: A] :
( ( member @ A @ Z4 @ ( set_option @ A @ X ) )
=> ( ( member @ A @ Za @ ( set_option @ A @ Xa ) )
=> ( ( ( F2 @ Z4 )
= ( Fa @ Za ) )
=> ( Z4 = Za ) ) ) )
=> ( ( ( map_option @ A @ B @ F2 @ X )
= ( map_option @ A @ B @ Fa @ Xa ) )
=> ( X = Xa ) ) ) ).
% option.inj_map_strong
thf(fact_7620_option_Omap__cong0,axiom,
! [B: $tType,A: $tType,X: option @ A,F2: A > B,G: A > B] :
( ! [Z4: A] :
( ( member @ A @ Z4 @ ( set_option @ A @ X ) )
=> ( ( F2 @ Z4 )
= ( G @ Z4 ) ) )
=> ( ( map_option @ A @ B @ F2 @ X )
= ( map_option @ A @ B @ G @ X ) ) ) ).
% option.map_cong0
thf(fact_7621_option_Omap__cong,axiom,
! [B: $tType,A: $tType,X: option @ A,Ya: option @ A,F2: A > B,G: A > B] :
( ( X = Ya )
=> ( ! [Z4: A] :
( ( member @ A @ Z4 @ ( set_option @ A @ Ya ) )
=> ( ( F2 @ Z4 )
= ( G @ Z4 ) ) )
=> ( ( map_option @ A @ B @ F2 @ X )
= ( map_option @ A @ B @ G @ Ya ) ) ) ) ).
% option.map_cong
thf(fact_7622_combine__options__left__commute,axiom,
! [A: $tType,F2: A > A > A,Y2: option @ A,X: option @ A,Z2: option @ A] :
( ! [X3: A,Y3: A] :
( ( F2 @ X3 @ Y3 )
= ( F2 @ Y3 @ X3 ) )
=> ( ! [X3: A,Y3: A,Z4: A] :
( ( F2 @ ( F2 @ X3 @ Y3 ) @ Z4 )
= ( F2 @ X3 @ ( F2 @ Y3 @ Z4 ) ) )
=> ( ( combine_options @ A @ F2 @ Y2 @ ( combine_options @ A @ F2 @ X @ Z2 ) )
= ( combine_options @ A @ F2 @ X @ ( combine_options @ A @ F2 @ Y2 @ Z2 ) ) ) ) ) ).
% combine_options_left_commute
thf(fact_7623_combine__options__commute,axiom,
! [A: $tType,F2: A > A > A,X: option @ A,Y2: option @ A] :
( ! [X3: A,Y3: A] :
( ( F2 @ X3 @ Y3 )
= ( F2 @ Y3 @ X3 ) )
=> ( ( combine_options @ A @ F2 @ X @ Y2 )
= ( combine_options @ A @ F2 @ Y2 @ X ) ) ) ).
% combine_options_commute
thf(fact_7624_combine__options__assoc,axiom,
! [A: $tType,F2: A > A > A,X: option @ A,Y2: option @ A,Z2: option @ A] :
( ! [X3: A,Y3: A,Z4: A] :
( ( F2 @ ( F2 @ X3 @ Y3 ) @ Z4 )
= ( F2 @ X3 @ ( F2 @ Y3 @ Z4 ) ) )
=> ( ( combine_options @ A @ F2 @ ( combine_options @ A @ F2 @ X @ Y2 ) @ Z2 )
= ( combine_options @ A @ F2 @ X @ ( combine_options @ A @ F2 @ Y2 @ Z2 ) ) ) ) ).
% combine_options_assoc
thf(fact_7625_option_Osimps_I14_J,axiom,
! [A: $tType] :
( ( set_option @ A @ ( none @ A ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% option.simps(14)
thf(fact_7626_option_Oset__cases,axiom,
! [A: $tType,E2: A,A3: option @ A] :
( ( member @ A @ E2 @ ( set_option @ A @ A3 ) )
=> ( A3
= ( some @ A @ E2 ) ) ) ).
% option.set_cases
thf(fact_7627_option_Oset__intros,axiom,
! [A: $tType,X23: A] : ( member @ A @ X23 @ ( set_option @ A @ ( some @ A @ X23 ) ) ) ).
% option.set_intros
thf(fact_7628_ospec,axiom,
! [A: $tType,A4: option @ A,P: A > $o,X: A] :
( ! [X3: A] :
( ( member @ A @ X3 @ ( set_option @ A @ A4 ) )
=> ( P @ X3 ) )
=> ( ( A4
= ( some @ A @ X ) )
=> ( P @ X ) ) ) ).
% ospec
thf(fact_7629_option_Oset__sel,axiom,
! [A: $tType,A3: option @ A] :
( ( A3
!= ( none @ A ) )
=> ( member @ A @ ( the2 @ A @ A3 ) @ ( set_option @ A @ A3 ) ) ) ).
% option.set_sel
thf(fact_7630_option_Osimps_I15_J,axiom,
! [A: $tType,X23: A] :
( ( set_option @ A @ ( some @ A @ X23 ) )
= ( insert @ A @ X23 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% option.simps(15)
thf(fact_7631_option_Oin__rel,axiom,
! [B: $tType,A: $tType] :
( ( rel_option @ A @ B )
= ( ^ [R6: A > B > $o,A5: option @ A,B3: option @ B] :
? [Z6: option @ ( product_prod @ A @ B )] :
( ( member @ ( option @ ( product_prod @ A @ B ) ) @ Z6
@ ( collect @ ( option @ ( product_prod @ A @ B ) )
@ ^ [X2: option @ ( product_prod @ A @ B )] : ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( set_option @ ( product_prod @ A @ B ) @ X2 ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R6 ) ) ) ) )
& ( ( map_option @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Z6 )
= A5 )
& ( ( map_option @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ Z6 )
= B3 ) ) ) ) ).
% option.in_rel
thf(fact_7632_length__splice,axiom,
! [A: $tType,Xs: list @ A,Ys2: list @ A] :
( ( size_size @ ( list @ A ) @ ( splice @ A @ Xs @ Ys2 ) )
= ( plus_plus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ A ) @ Ys2 ) ) ) ).
% length_splice
thf(fact_7633_rel__option__None1,axiom,
! [A: $tType,B: $tType,P: A > B > $o,X: option @ B] :
( ( rel_option @ A @ B @ P @ ( none @ A ) @ X )
= ( X
= ( none @ B ) ) ) ).
% rel_option_None1
thf(fact_7634_rel__option__None2,axiom,
! [B: $tType,A: $tType,P: A > B > $o,X: option @ A] :
( ( rel_option @ A @ B @ P @ X @ ( none @ B ) )
= ( X
= ( none @ A ) ) ) ).
% rel_option_None2
thf(fact_7635_rel__option__reflI,axiom,
! [A: $tType,Y2: option @ A,P: A > A > $o] :
( ! [X3: A] :
( ( member @ A @ X3 @ ( set_option @ A @ Y2 ) )
=> ( P @ X3 @ X3 ) )
=> ( rel_option @ A @ A @ P @ Y2 @ Y2 ) ) ).
% rel_option_reflI
thf(fact_7636_option_Orel__refl__strong,axiom,
! [A: $tType,X: option @ A,Ra: A > A > $o] :
( ! [Z4: A] :
( ( member @ A @ Z4 @ ( set_option @ A @ X ) )
=> ( Ra @ Z4 @ Z4 ) )
=> ( rel_option @ A @ A @ Ra @ X @ X ) ) ).
% option.rel_refl_strong
thf(fact_7637_option_Orel__mono__strong,axiom,
! [A: $tType,B: $tType,R2: A > B > $o,X: option @ A,Y2: option @ B,Ra: A > B > $o] :
( ( rel_option @ A @ B @ R2 @ X @ Y2 )
=> ( ! [Z4: A,Yb: B] :
( ( member @ A @ Z4 @ ( set_option @ A @ X ) )
=> ( ( member @ B @ Yb @ ( set_option @ B @ Y2 ) )
=> ( ( R2 @ Z4 @ Yb )
=> ( Ra @ Z4 @ Yb ) ) ) )
=> ( rel_option @ A @ B @ Ra @ X @ Y2 ) ) ) ).
% option.rel_mono_strong
thf(fact_7638_option_Orel__cong,axiom,
! [A: $tType,B: $tType,X: option @ A,Ya: option @ A,Y2: option @ B,Xa: option @ B,R2: A > B > $o,Ra: A > B > $o] :
( ( X = Ya )
=> ( ( Y2 = Xa )
=> ( ! [Z4: A,Yb: B] :
( ( member @ A @ Z4 @ ( set_option @ A @ Ya ) )
=> ( ( member @ B @ Yb @ ( set_option @ B @ Xa ) )
=> ( ( R2 @ Z4 @ Yb )
= ( Ra @ Z4 @ Yb ) ) ) )
=> ( ( rel_option @ A @ B @ R2 @ X @ Y2 )
= ( rel_option @ A @ B @ Ra @ Ya @ Xa ) ) ) ) ) ).
% option.rel_cong
thf(fact_7639_rel__option__inf,axiom,
! [B: $tType,A: $tType,A4: A > B > $o,B7: A > B > $o] :
( ( inf_inf @ ( ( option @ A ) > ( option @ B ) > $o ) @ ( rel_option @ A @ B @ A4 ) @ ( rel_option @ A @ B @ B7 ) )
= ( rel_option @ A @ B @ ( inf_inf @ ( A > B > $o ) @ A4 @ B7 ) ) ) ).
% rel_option_inf
thf(fact_7640_rel__option__iff,axiom,
! [B: $tType,A: $tType] :
( ( rel_option @ A @ B )
= ( ^ [R6: A > B > $o,X2: option @ A,Y5: option @ B] :
( product_case_prod @ ( option @ A ) @ ( option @ B ) @ $o
@ ^ [A5: option @ A,B3: option @ B] :
( case_option @ $o @ A
@ ( case_option @ $o @ B @ $true
@ ^ [C4: B] : $false
@ B3 )
@ ^ [Z6: A] : ( case_option @ $o @ B @ $false @ ( R6 @ Z6 ) @ B3 )
@ A5 )
@ ( product_Pair @ ( option @ A ) @ ( option @ B ) @ X2 @ Y5 ) ) ) ) ).
% rel_option_iff
thf(fact_7641_option_Ocase__transfer,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,S3: C > D > $o,R2: A > B > $o] : ( bNF_rel_fun @ C @ D @ ( ( A > C ) > ( option @ A ) > C ) @ ( ( B > D ) > ( option @ B ) > D ) @ S3 @ ( bNF_rel_fun @ ( A > C ) @ ( B > D ) @ ( ( option @ A ) > C ) @ ( ( option @ B ) > D ) @ ( bNF_rel_fun @ A @ B @ C @ D @ R2 @ S3 ) @ ( bNF_rel_fun @ ( option @ A ) @ ( option @ B ) @ C @ D @ ( rel_option @ A @ B @ R2 ) @ S3 ) ) @ ( case_option @ C @ A ) @ ( case_option @ D @ B ) ) ).
% option.case_transfer
thf(fact_7642_option_Orel__mono,axiom,
! [B: $tType,A: $tType,R2: A > B > $o,Ra: A > B > $o] :
( ( ord_less_eq @ ( A > B > $o ) @ R2 @ Ra )
=> ( ord_less_eq @ ( ( option @ A ) > ( option @ B ) > $o ) @ ( rel_option @ A @ B @ R2 ) @ ( rel_option @ A @ B @ Ra ) ) ) ).
% option.rel_mono
thf(fact_7643_option_Odisc__transfer_I2_J,axiom,
! [A: $tType,B: $tType,R2: A > B > $o] :
( bNF_rel_fun @ ( option @ A ) @ ( option @ B ) @ $o @ $o @ ( rel_option @ A @ B @ R2 )
@ ^ [Y4: $o,Z: $o] : Y4 = Z
@ ^ [Option3: option @ A] :
( Option3
!= ( none @ A ) )
@ ^ [Option3: option @ B] :
( Option3
!= ( none @ B ) ) ) ).
% option.disc_transfer(2)
thf(fact_7644_option_Odisc__transfer_I1_J,axiom,
! [A: $tType,B: $tType,R2: A > B > $o] :
( bNF_rel_fun @ ( option @ A ) @ ( option @ B ) @ $o @ $o @ ( rel_option @ A @ B @ R2 )
@ ^ [Y4: $o,Z: $o] : Y4 = Z
@ ^ [Option3: option @ A] :
( Option3
= ( none @ A ) )
@ ^ [Option3: option @ B] :
( Option3
= ( none @ B ) ) ) ).
% option.disc_transfer(1)
thf(fact_7645_option_Octr__transfer_I2_J,axiom,
! [A: $tType,B: $tType,R2: A > B > $o] : ( bNF_rel_fun @ A @ B @ ( option @ A ) @ ( option @ B ) @ R2 @ ( rel_option @ A @ B @ R2 ) @ ( some @ A ) @ ( some @ B ) ) ).
% option.ctr_transfer(2)
thf(fact_7646_option_Orel__inject_I2_J,axiom,
! [A: $tType,B: $tType,R2: A > B > $o,X23: A,Y23: B] :
( ( rel_option @ A @ B @ R2 @ ( some @ A @ X23 ) @ ( some @ B @ Y23 ) )
= ( R2 @ X23 @ Y23 ) ) ).
% option.rel_inject(2)
thf(fact_7647_option_Orel__intros_I2_J,axiom,
! [A: $tType,B: $tType,R2: A > B > $o,X23: A,Y23: B] :
( ( R2 @ X23 @ Y23 )
=> ( rel_option @ A @ B @ R2 @ ( some @ A @ X23 ) @ ( some @ B @ Y23 ) ) ) ).
% option.rel_intros(2)
thf(fact_7648_option__rel__Some1,axiom,
! [A: $tType,B: $tType,A4: A > B > $o,X: A,Y2: option @ B] :
( ( rel_option @ A @ B @ A4 @ ( some @ A @ X ) @ Y2 )
= ( ? [Y6: B] :
( ( Y2
= ( some @ B @ Y6 ) )
& ( A4 @ X @ Y6 ) ) ) ) ).
% option_rel_Some1
thf(fact_7649_option__rel__Some2,axiom,
! [B: $tType,A: $tType,A4: A > B > $o,X: option @ A,Y2: B] :
( ( rel_option @ A @ B @ A4 @ X @ ( some @ B @ Y2 ) )
= ( ? [X9: A] :
( ( X
= ( some @ A @ X9 ) )
& ( A4 @ X9 @ Y2 ) ) ) ) ).
% option_rel_Some2
thf(fact_7650_option_Octr__transfer_I1_J,axiom,
! [A: $tType,B: $tType,R2: A > B > $o] : ( rel_option @ A @ B @ R2 @ ( none @ A ) @ ( none @ B ) ) ).
% option.ctr_transfer(1)
thf(fact_7651_option_Orel__induct,axiom,
! [A: $tType,B: $tType,R2: A > B > $o,X: option @ A,Y2: option @ B,Q: ( option @ A ) > ( option @ B ) > $o] :
( ( rel_option @ A @ B @ R2 @ X @ Y2 )
=> ( ( Q @ ( none @ A ) @ ( none @ B ) )
=> ( ! [A25: A,B22: B] :
( ( R2 @ A25 @ B22 )
=> ( Q @ ( some @ A @ A25 ) @ ( some @ B @ B22 ) ) )
=> ( Q @ X @ Y2 ) ) ) ) ).
% option.rel_induct
thf(fact_7652_option_Orel__cases,axiom,
! [A: $tType,B: $tType,R2: A > B > $o,A3: option @ A,B2: option @ B] :
( ( rel_option @ A @ B @ R2 @ A3 @ B2 )
=> ( ( ( A3
= ( none @ A ) )
=> ( B2
!= ( none @ B ) ) )
=> ~ ! [X3: A] :
( ( A3
= ( some @ A @ X3 ) )
=> ! [Y3: B] :
( ( B2
= ( some @ B @ Y3 ) )
=> ~ ( R2 @ X3 @ Y3 ) ) ) ) ) ).
% option.rel_cases
thf(fact_7653_option_Orel__distinct_I1_J,axiom,
! [A: $tType,B: $tType,R2: A > B > $o,Y23: B] :
~ ( rel_option @ A @ B @ R2 @ ( none @ A ) @ ( some @ B @ Y23 ) ) ).
% option.rel_distinct(1)
thf(fact_7654_option_Orel__distinct_I2_J,axiom,
! [A: $tType,B: $tType,R2: A > B > $o,Y23: A] :
~ ( rel_option @ A @ B @ R2 @ ( some @ A @ Y23 ) @ ( none @ B ) ) ).
% option.rel_distinct(2)
thf(fact_7655_option_Orel__sel,axiom,
! [B: $tType,A: $tType] :
( ( rel_option @ A @ B )
= ( ^ [R6: A > B > $o,A5: option @ A,B3: option @ B] :
( ( ( A5
= ( none @ A ) )
= ( B3
= ( none @ B ) ) )
& ( ( A5
!= ( none @ A ) )
=> ( ( B3
!= ( none @ B ) )
=> ( R6 @ ( the2 @ A @ A5 ) @ ( the2 @ B @ B3 ) ) ) ) ) ) ) ).
% option.rel_sel
thf(fact_7656_option_Orel__eq,axiom,
! [A: $tType] :
( ( rel_option @ A @ A
@ ^ [Y4: A,Z: A] : Y4 = Z )
= ( ^ [Y4: option @ A,Z: option @ A] : Y4 = Z ) ) ).
% option.rel_eq
thf(fact_7657_option_Orel__refl,axiom,
! [B: $tType,Ra: B > B > $o,X: option @ B] :
( ! [X3: B] : ( Ra @ X3 @ X3 )
=> ( rel_option @ B @ B @ Ra @ X @ X ) ) ).
% option.rel_refl
thf(fact_7658_option_Orel__map_I2_J,axiom,
! [A: $tType,C: $tType,B: $tType,Sa: A > C > $o,X: option @ A,G: B > C,Y2: option @ B] :
( ( rel_option @ A @ C @ Sa @ X @ ( map_option @ B @ C @ G @ Y2 ) )
= ( rel_option @ A @ B
@ ^ [X2: A,Y5: B] : ( Sa @ X2 @ ( G @ Y5 ) )
@ X
@ Y2 ) ) ).
% option.rel_map(2)
thf(fact_7659_option_Orel__map_I1_J,axiom,
! [A: $tType,C: $tType,B: $tType,Sb: C > B > $o,I: A > C,X: option @ A,Y2: option @ B] :
( ( rel_option @ C @ B @ Sb @ ( map_option @ A @ C @ I @ X ) @ Y2 )
= ( rel_option @ A @ B
@ ^ [X2: A] : ( Sb @ ( I @ X2 ) )
@ X
@ Y2 ) ) ).
% option.rel_map(1)
thf(fact_7660_option_Orec__transfer,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,S3: C > D > $o,R2: A > B > $o] : ( bNF_rel_fun @ C @ D @ ( ( A > C ) > ( option @ A ) > C ) @ ( ( B > D ) > ( option @ B ) > D ) @ S3 @ ( bNF_rel_fun @ ( A > C ) @ ( B > D ) @ ( ( option @ A ) > C ) @ ( ( option @ B ) > D ) @ ( bNF_rel_fun @ A @ B @ C @ D @ R2 @ S3 ) @ ( bNF_rel_fun @ ( option @ A ) @ ( option @ B ) @ C @ D @ ( rel_option @ A @ B @ R2 ) @ S3 ) ) @ ( rec_option @ C @ A ) @ ( rec_option @ D @ B ) ) ).
% option.rec_transfer
thf(fact_7661_option_Omap__transfer,axiom,
! [A: $tType,B: $tType,F: $tType,E5: $tType,Rb: A > E5 > $o,Sd: B > F > $o] : ( bNF_rel_fun @ ( A > B ) @ ( E5 > F ) @ ( ( option @ A ) > ( option @ B ) ) @ ( ( option @ E5 ) > ( option @ F ) ) @ ( bNF_rel_fun @ A @ E5 @ B @ F @ Rb @ Sd ) @ ( bNF_rel_fun @ ( option @ A ) @ ( option @ E5 ) @ ( option @ B ) @ ( option @ F ) @ ( rel_option @ A @ E5 @ Rb ) @ ( rel_option @ B @ F @ Sd ) ) @ ( map_option @ A @ B ) @ ( map_option @ E5 @ F ) ) ).
% option.map_transfer
thf(fact_7662_option_Orel__transfer,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,Sa: A > C > $o,Sc: B > D > $o] :
( bNF_rel_fun @ ( A > B > $o ) @ ( C > D > $o ) @ ( ( option @ A ) > ( option @ B ) > $o ) @ ( ( option @ C ) > ( option @ D ) > $o )
@ ( bNF_rel_fun @ A @ C @ ( B > $o ) @ ( D > $o ) @ Sa
@ ( bNF_rel_fun @ B @ D @ $o @ $o @ Sc
@ ^ [Y4: $o,Z: $o] : Y4 = Z ) )
@ ( bNF_rel_fun @ ( option @ A ) @ ( option @ C ) @ ( ( option @ B ) > $o ) @ ( ( option @ D ) > $o ) @ ( rel_option @ A @ C @ Sa )
@ ( bNF_rel_fun @ ( option @ B ) @ ( option @ D ) @ $o @ $o @ ( rel_option @ B @ D @ Sc )
@ ^ [Y4: $o,Z: $o] : Y4 = Z ) )
@ ( rel_option @ A @ B )
@ ( rel_option @ C @ D ) ) ).
% option.rel_transfer
thf(fact_7663_option_Orel__transp,axiom,
! [A: $tType,R2: A > A > $o] :
( ( transp @ A @ R2 )
=> ( transp @ ( option @ A ) @ ( rel_option @ A @ A @ R2 ) ) ) ).
% option.rel_transp
thf(fact_7664_option_OQuotient,axiom,
! [B: $tType,A: $tType,R2: A > A > $o,Abs: A > B,Rep: B > A,T6: A > B > $o] :
( ( quotient @ A @ B @ R2 @ Abs @ Rep @ T6 )
=> ( quotient @ ( option @ A ) @ ( option @ B ) @ ( rel_option @ A @ A @ R2 ) @ ( map_option @ A @ B @ Abs ) @ ( map_option @ B @ A @ Rep ) @ ( rel_option @ A @ B @ T6 ) ) ) ).
% option.Quotient
thf(fact_7665_option__bind__transfer,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,A4: A > B > $o,B7: C > D > $o] : ( bNF_rel_fun @ ( option @ A ) @ ( option @ B ) @ ( ( A > ( option @ C ) ) > ( option @ C ) ) @ ( ( B > ( option @ D ) ) > ( option @ D ) ) @ ( rel_option @ A @ B @ A4 ) @ ( bNF_rel_fun @ ( A > ( option @ C ) ) @ ( B > ( option @ D ) ) @ ( option @ C ) @ ( option @ D ) @ ( bNF_rel_fun @ A @ B @ ( option @ C ) @ ( option @ D ) @ A4 @ ( rel_option @ C @ D @ B7 ) ) @ ( rel_option @ C @ D @ B7 ) ) @ ( bind @ A @ C ) @ ( bind @ B @ D ) ) ).
% option_bind_transfer
thf(fact_7666_lenlex__def,axiom,
! [A: $tType] :
( ( lenlex @ A )
= ( ^ [R: set @ ( product_prod @ A @ A )] :
( inv_image @ ( product_prod @ nat @ ( list @ A ) ) @ ( list @ A ) @ ( lex_prod @ nat @ ( list @ A ) @ less_than @ ( lex @ A @ R ) )
@ ^ [Xs3: list @ A] : ( product_Pair @ nat @ ( list @ A ) @ ( size_size @ ( list @ A ) @ Xs3 ) @ Xs3 ) ) ) ) ).
% lenlex_def
thf(fact_7667_pairwise__alt,axiom,
! [A: $tType] :
( ( pairwise @ A )
= ( ^ [R6: A > A > $o,S8: set @ A] :
! [X2: A] :
( ( member @ A @ X2 @ S8 )
=> ! [Y5: A] :
( ( member @ A @ Y5 @ ( minus_minus @ ( set @ A ) @ S8 @ ( insert @ A @ X2 @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( R6 @ X2 @ Y5 ) ) ) ) ) ).
% pairwise_alt
thf(fact_7668_pred__option__parametric,axiom,
! [A: $tType,B: $tType,A4: A > B > $o] :
( bNF_rel_fun @ ( A > $o ) @ ( B > $o ) @ ( ( option @ A ) > $o ) @ ( ( option @ B ) > $o )
@ ( bNF_rel_fun @ A @ B @ $o @ $o @ A4
@ ^ [Y4: $o,Z: $o] : Y4 = Z )
@ ( bNF_rel_fun @ ( option @ A ) @ ( option @ B ) @ $o @ $o @ ( rel_option @ A @ B @ A4 )
@ ^ [Y4: $o,Z: $o] : Y4 = Z )
@ ( pred_option @ A )
@ ( pred_option @ B ) ) ).
% pred_option_parametric
thf(fact_7669_option_Opred__transfer,axiom,
! [A: $tType,B: $tType,R2: A > B > $o] :
( bNF_rel_fun @ ( A > $o ) @ ( B > $o ) @ ( ( option @ A ) > $o ) @ ( ( option @ B ) > $o )
@ ( bNF_rel_fun @ A @ B @ $o @ $o @ R2
@ ^ [Y4: $o,Z: $o] : Y4 = Z )
@ ( bNF_rel_fun @ ( option @ A ) @ ( option @ B ) @ $o @ $o @ ( rel_option @ A @ B @ R2 )
@ ^ [Y4: $o,Z: $o] : Y4 = Z )
@ ( pred_option @ A )
@ ( pred_option @ B ) ) ).
% option.pred_transfer
thf(fact_7670_option_Opred__inject_I2_J,axiom,
! [A: $tType,P: A > $o,A3: A] :
( ( pred_option @ A @ P @ ( some @ A @ A3 ) )
= ( P @ A3 ) ) ).
% option.pred_inject(2)
thf(fact_7671_option_Opred__mono,axiom,
! [A: $tType,P: A > $o,Pa: A > $o] :
( ( ord_less_eq @ ( A > $o ) @ P @ Pa )
=> ( ord_less_eq @ ( ( option @ A ) > $o ) @ ( pred_option @ A @ P ) @ ( pred_option @ A @ Pa ) ) ) ).
% option.pred_mono
thf(fact_7672_option_Opred__inject_I1_J,axiom,
! [A: $tType,P: A > $o] : ( pred_option @ A @ P @ ( none @ A ) ) ).
% option.pred_inject(1)
thf(fact_7673_option_Opred__True,axiom,
! [A: $tType] :
( ( pred_option @ A
@ ^ [Uu3: A] : $true )
= ( ^ [Uu3: option @ A] : $true ) ) ).
% option.pred_True
thf(fact_7674_option_Omap__cong__pred,axiom,
! [B: $tType,A: $tType,X: option @ A,Ya: option @ A,F2: A > B,G: A > B] :
( ( X = Ya )
=> ( ( pred_option @ A
@ ^ [Z6: A] :
( ( F2 @ Z6 )
= ( G @ Z6 ) )
@ Ya )
=> ( ( map_option @ A @ B @ F2 @ X )
= ( map_option @ A @ B @ G @ Ya ) ) ) ) ).
% option.map_cong_pred
thf(fact_7675_option_Opred__cong,axiom,
! [A: $tType,X: option @ A,Ya: option @ A,P: A > $o,Pa: A > $o] :
( ( X = Ya )
=> ( ! [Z4: A] :
( ( member @ A @ Z4 @ ( set_option @ A @ Ya ) )
=> ( ( P @ Z4 )
= ( Pa @ Z4 ) ) )
=> ( ( pred_option @ A @ P @ X )
= ( pred_option @ A @ Pa @ Ya ) ) ) ) ).
% option.pred_cong
thf(fact_7676_option_Opred__mono__strong,axiom,
! [A: $tType,P: A > $o,X: option @ A,Pa: A > $o] :
( ( pred_option @ A @ P @ X )
=> ( ! [Z4: A] :
( ( member @ A @ Z4 @ ( set_option @ A @ X ) )
=> ( ( P @ Z4 )
=> ( Pa @ Z4 ) ) )
=> ( pred_option @ A @ Pa @ X ) ) ) ).
% option.pred_mono_strong
thf(fact_7677_option_Opred__set,axiom,
! [A: $tType] :
( ( pred_option @ A )
= ( ^ [P3: A > $o,X2: option @ A] :
! [Y5: A] :
( ( member @ A @ Y5 @ ( set_option @ A @ X2 ) )
=> ( P3 @ Y5 ) ) ) ) ).
% option.pred_set
thf(fact_7678_option_Opred__map,axiom,
! [B: $tType,A: $tType,Q: B > $o,F2: A > B,X: option @ A] :
( ( pred_option @ B @ Q @ ( map_option @ A @ B @ F2 @ X ) )
= ( pred_option @ A @ ( comp @ B @ $o @ A @ Q @ F2 ) @ X ) ) ).
% option.pred_map
thf(fact_7679_map__le__imp__upd__le,axiom,
! [A: $tType,B: $tType,M12: A > ( option @ B ),M22: A > ( option @ B ),X: A,Y2: B] :
( ( map_le @ A @ B @ M12 @ M22 )
=> ( map_le @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ M12 @ X @ ( none @ B ) ) @ ( fun_upd @ A @ ( option @ B ) @ M22 @ X @ ( some @ B @ Y2 ) ) ) ) ).
% map_le_imp_upd_le
thf(fact_7680_lcm__altdef__int,axiom,
( ( gcd_lcm @ int )
= ( ^ [A5: int,B3: int] : ( divide_divide @ int @ ( times_times @ int @ ( abs_abs @ int @ A5 ) @ ( abs_abs @ int @ B3 ) ) @ ( gcd_gcd @ int @ A5 @ B3 ) ) ) ) ).
% lcm_altdef_int
thf(fact_7681_lcm_Obottom__right__bottom,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A] :
( ( gcd_lcm @ A @ A3 @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% lcm.bottom_right_bottom
thf(fact_7682_lcm_Obottom__left__bottom,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A] :
( ( gcd_lcm @ A @ ( zero_zero @ A ) @ A3 )
= ( zero_zero @ A ) ) ) ).
% lcm.bottom_left_bottom
thf(fact_7683_lcm__neg1,axiom,
! [A: $tType] :
( ( ring_gcd @ A )
=> ! [A3: A,B2: A] :
( ( gcd_lcm @ A @ ( uminus_uminus @ A @ A3 ) @ B2 )
= ( gcd_lcm @ A @ A3 @ B2 ) ) ) ).
% lcm_neg1
thf(fact_7684_lcm__neg2,axiom,
! [A: $tType] :
( ( ring_gcd @ A )
=> ! [A3: A,B2: A] :
( ( gcd_lcm @ A @ A3 @ ( uminus_uminus @ A @ B2 ) )
= ( gcd_lcm @ A @ A3 @ B2 ) ) ) ).
% lcm_neg2
thf(fact_7685_lcm__0__iff__int,axiom,
! [M2: int,N: int] :
( ( ( gcd_lcm @ int @ M2 @ N )
= ( zero_zero @ int ) )
= ( ( M2
= ( zero_zero @ int ) )
| ( N
= ( zero_zero @ int ) ) ) ) ).
% lcm_0_iff_int
thf(fact_7686_lcm__neg__numeral__2,axiom,
! [A: $tType] :
( ( ring_gcd @ A )
=> ! [A3: A,N: num] :
( ( gcd_lcm @ A @ A3 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) )
= ( gcd_lcm @ A @ A3 @ ( numeral_numeral @ A @ N ) ) ) ) ).
% lcm_neg_numeral_2
thf(fact_7687_lcm__neg__numeral__1,axiom,
! [A: $tType] :
( ( ring_gcd @ A )
=> ! [N: num,A3: A] :
( ( gcd_lcm @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ N ) ) @ A3 )
= ( gcd_lcm @ A @ ( numeral_numeral @ A @ N ) @ A3 ) ) ) ).
% lcm_neg_numeral_1
thf(fact_7688_lcm__eq__1__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A] :
( ( ( gcd_lcm @ A @ A3 @ B2 )
= ( one_one @ A ) )
= ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
& ( dvd_dvd @ A @ B2 @ ( one_one @ A ) ) ) ) ) ).
% lcm_eq_1_iff
thf(fact_7689_lcm__1__iff__int,axiom,
! [M2: int,N: int] :
( ( ( gcd_lcm @ int @ M2 @ N )
= ( one_one @ int ) )
= ( ( ( M2
= ( one_one @ int ) )
| ( M2
= ( uminus_uminus @ int @ ( one_one @ int ) ) ) )
& ( ( N
= ( one_one @ int ) )
| ( N
= ( uminus_uminus @ int @ ( one_one @ int ) ) ) ) ) ) ).
% lcm_1_iff_int
thf(fact_7690_lcm__mult__unit1,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( gcd_lcm @ A @ ( times_times @ A @ B2 @ A3 ) @ C2 )
= ( gcd_lcm @ A @ B2 @ C2 ) ) ) ) ).
% lcm_mult_unit1
thf(fact_7691_lcm__mult__unit2,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( gcd_lcm @ A @ B2 @ ( times_times @ A @ C2 @ A3 ) )
= ( gcd_lcm @ A @ B2 @ C2 ) ) ) ) ).
% lcm_mult_unit2
thf(fact_7692_lcm__neg1__int,axiom,
! [X: int,Y2: int] :
( ( gcd_lcm @ int @ ( uminus_uminus @ int @ X ) @ Y2 )
= ( gcd_lcm @ int @ X @ Y2 ) ) ).
% lcm_neg1_int
thf(fact_7693_lcm__neg2__int,axiom,
! [X: int,Y2: int] :
( ( gcd_lcm @ int @ X @ ( uminus_uminus @ int @ Y2 ) )
= ( gcd_lcm @ int @ X @ Y2 ) ) ).
% lcm_neg2_int
thf(fact_7694_map__le__empty,axiom,
! [B: $tType,A: $tType,G: A > ( option @ B )] :
( map_le @ A @ B
@ ^ [X2: A] : ( none @ B )
@ G ) ).
% map_le_empty
thf(fact_7695_upd__None__map__le,axiom,
! [B: $tType,A: $tType,F2: A > ( option @ B ),X: A] : ( map_le @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ F2 @ X @ ( none @ B ) ) @ F2 ) ).
% upd_None_map_le
thf(fact_7696_zero__eq__lcm__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A] :
( ( ( zero_zero @ A )
= ( gcd_lcm @ A @ A3 @ B2 ) )
= ( ( A3
= ( zero_zero @ A ) )
| ( B2
= ( zero_zero @ A ) ) ) ) ) ).
% zero_eq_lcm_iff
thf(fact_7697_lcm__eq__0__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A] :
( ( ( gcd_lcm @ A @ A3 @ B2 )
= ( zero_zero @ A ) )
= ( ( A3
= ( zero_zero @ A ) )
| ( B2
= ( zero_zero @ A ) ) ) ) ) ).
% lcm_eq_0_iff
thf(fact_7698_lcm__pos__int,axiom,
! [M2: int,N: int] :
( ( M2
!= ( zero_zero @ int ) )
=> ( ( N
!= ( zero_zero @ int ) )
=> ( ord_less @ int @ ( zero_zero @ int ) @ ( gcd_lcm @ int @ M2 @ N ) ) ) ) ).
% lcm_pos_int
thf(fact_7699_lcm__div__unit1,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( gcd_lcm @ A @ ( divide_divide @ A @ B2 @ A3 ) @ C2 )
= ( gcd_lcm @ A @ B2 @ C2 ) ) ) ) ).
% lcm_div_unit1
thf(fact_7700_lcm__div__unit2,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A,C2: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( gcd_lcm @ A @ B2 @ ( divide_divide @ A @ C2 @ A3 ) )
= ( gcd_lcm @ A @ B2 @ C2 ) ) ) ) ).
% lcm_div_unit2
thf(fact_7701_lcm__unique__int,axiom,
! [D2: int,A3: int,B2: int] :
( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ D2 )
& ( dvd_dvd @ int @ A3 @ D2 )
& ( dvd_dvd @ int @ B2 @ D2 )
& ! [E3: int] :
( ( ( dvd_dvd @ int @ A3 @ E3 )
& ( dvd_dvd @ int @ B2 @ E3 ) )
=> ( dvd_dvd @ int @ D2 @ E3 ) ) )
= ( D2
= ( gcd_lcm @ int @ A3 @ B2 ) ) ) ).
% lcm_unique_int
thf(fact_7702_lcm__ge__0__int,axiom,
! [X: int,Y2: int] : ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( gcd_lcm @ int @ X @ Y2 ) ) ).
% lcm_ge_0_int
thf(fact_7703_lcm__cases__int,axiom,
! [X: int,Y2: int,P: int > $o] :
( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y2 )
=> ( P @ ( gcd_lcm @ int @ X @ Y2 ) ) ) )
=> ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
=> ( ( ord_less_eq @ int @ Y2 @ ( zero_zero @ int ) )
=> ( P @ ( gcd_lcm @ int @ X @ ( uminus_uminus @ int @ Y2 ) ) ) ) )
=> ( ( ( ord_less_eq @ int @ X @ ( zero_zero @ int ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y2 )
=> ( P @ ( gcd_lcm @ int @ ( uminus_uminus @ int @ X ) @ Y2 ) ) ) )
=> ( ( ( ord_less_eq @ int @ X @ ( zero_zero @ int ) )
=> ( ( ord_less_eq @ int @ Y2 @ ( zero_zero @ int ) )
=> ( P @ ( gcd_lcm @ int @ ( uminus_uminus @ int @ X ) @ ( uminus_uminus @ int @ Y2 ) ) ) ) )
=> ( P @ ( gcd_lcm @ int @ X @ Y2 ) ) ) ) ) ) ).
% lcm_cases_int
thf(fact_7704_Lcm__int__set__eq__fold,axiom,
! [Xs: list @ int] :
( ( gcd_Lcm @ int @ ( set2 @ int @ Xs ) )
= ( fold @ int @ int @ ( gcd_lcm @ int ) @ Xs @ ( one_one @ int ) ) ) ).
% Lcm_int_set_eq_fold
thf(fact_7705_Lcm__fin_Oeq__fold,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ( ( semiring_gcd_Lcm_fin @ A )
= ( ^ [A7: set @ A] : ( if @ A @ ( finite_finite @ A @ A7 ) @ ( finite_fold @ A @ A @ ( gcd_lcm @ A ) @ ( one_one @ A ) @ A7 ) @ ( zero_zero @ A ) ) ) ) ) ).
% Lcm_fin.eq_fold
thf(fact_7706_lcm__0__iff__nat,axiom,
! [M2: nat,N: nat] :
( ( ( gcd_lcm @ nat @ M2 @ N )
= ( zero_zero @ nat ) )
= ( ( M2
= ( zero_zero @ nat ) )
| ( N
= ( zero_zero @ nat ) ) ) ) ).
% lcm_0_iff_nat
thf(fact_7707_lcm__1__iff__nat,axiom,
! [M2: nat,N: nat] :
( ( ( gcd_lcm @ nat @ M2 @ N )
= ( suc @ ( zero_zero @ nat ) ) )
= ( ( M2
= ( suc @ ( zero_zero @ nat ) ) )
& ( N
= ( suc @ ( zero_zero @ nat ) ) ) ) ) ).
% lcm_1_iff_nat
thf(fact_7708_lcm__int__int__eq,axiom,
! [M2: nat,N: nat] :
( ( gcd_lcm @ int @ ( semiring_1_of_nat @ int @ M2 ) @ ( semiring_1_of_nat @ int @ N ) )
= ( semiring_1_of_nat @ int @ ( gcd_lcm @ nat @ M2 @ N ) ) ) ).
% lcm_int_int_eq
thf(fact_7709_Lcm__UNIV,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ( ( gcd_Lcm @ A @ ( top_top @ ( set @ A ) ) )
= ( zero_zero @ A ) ) ) ).
% Lcm_UNIV
thf(fact_7710_Lcm__empty,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ( ( gcd_Lcm @ A @ ( bot_bot @ ( set @ A ) ) )
= ( one_one @ A ) ) ) ).
% Lcm_empty
thf(fact_7711_Lcm__1__iff,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A4: set @ A] :
( ( ( gcd_Lcm @ A @ A4 )
= ( one_one @ A ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ A4 )
=> ( dvd_dvd @ A @ X2 @ ( one_one @ A ) ) ) ) ) ) ).
% Lcm_1_iff
thf(fact_7712_Lcm__fin_Oinfinite,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A4: set @ A] :
( ~ ( finite_finite @ A @ A4 )
=> ( ( semiring_gcd_Lcm_fin @ A @ A4 )
= ( zero_zero @ A ) ) ) ) ).
% Lcm_fin.infinite
thf(fact_7713_Lcm__fin_Oempty,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ( ( semiring_gcd_Lcm_fin @ A @ ( bot_bot @ ( set @ A ) ) )
= ( one_one @ A ) ) ) ).
% Lcm_fin.empty
thf(fact_7714_is__unit__Lcm__fin__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A4: set @ A] :
( ( dvd_dvd @ A @ ( semiring_gcd_Lcm_fin @ A @ A4 ) @ ( one_one @ A ) )
= ( ( semiring_gcd_Lcm_fin @ A @ A4 )
= ( one_one @ A ) ) ) ) ).
% is_unit_Lcm_fin_iff
thf(fact_7715_lcm__nat__abs__right__eq,axiom,
! [N: nat,K: int] :
( ( gcd_lcm @ nat @ N @ ( nat2 @ ( abs_abs @ int @ K ) ) )
= ( nat2 @ ( gcd_lcm @ int @ ( semiring_1_of_nat @ int @ N ) @ K ) ) ) ).
% lcm_nat_abs_right_eq
thf(fact_7716_lcm__nat__abs__left__eq,axiom,
! [K: int,N: nat] :
( ( gcd_lcm @ nat @ ( nat2 @ ( abs_abs @ int @ K ) ) @ N )
= ( nat2 @ ( gcd_lcm @ int @ K @ ( semiring_1_of_nat @ int @ N ) ) ) ) ).
% lcm_nat_abs_left_eq
thf(fact_7717_lcm__nat__def,axiom,
( ( gcd_lcm @ nat )
= ( ^ [X2: nat,Y5: nat] : ( divide_divide @ nat @ ( times_times @ nat @ X2 @ Y5 ) @ ( gcd_gcd @ nat @ X2 @ Y5 ) ) ) ) ).
% lcm_nat_def
thf(fact_7718_lcm__pos__nat,axiom,
! [M2: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ ( gcd_lcm @ nat @ M2 @ N ) ) ) ) ).
% lcm_pos_nat
thf(fact_7719_Lcm__int__greater__eq__0,axiom,
! [K6: set @ int] : ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( gcd_Lcm @ int @ K6 ) ) ).
% Lcm_int_greater_eq_0
thf(fact_7720_Lcm__eq__0__I,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A4: set @ A] :
( ( member @ A @ ( zero_zero @ A ) @ A4 )
=> ( ( gcd_Lcm @ A @ A4 )
= ( zero_zero @ A ) ) ) ) ).
% Lcm_eq_0_I
thf(fact_7721_Lcm__no__multiple,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A4: set @ A] :
( ! [M3: A] :
( ( M3
!= ( zero_zero @ A ) )
=> ? [X4: A] :
( ( member @ A @ X4 @ A4 )
& ~ ( dvd_dvd @ A @ X4 @ M3 ) ) )
=> ( ( gcd_Lcm @ A @ A4 )
= ( zero_zero @ A ) ) ) ) ).
% Lcm_no_multiple
thf(fact_7722_Lcm__0__iff_H,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A4: set @ A] :
( ( ( gcd_Lcm @ A @ A4 )
= ( zero_zero @ A ) )
= ( ~ ? [L2: A] :
( ( L2
!= ( zero_zero @ A ) )
& ! [X2: A] :
( ( member @ A @ X2 @ A4 )
=> ( dvd_dvd @ A @ X2 @ L2 ) ) ) ) ) ) ).
% Lcm_0_iff'
thf(fact_7723_Lcm__fin__0__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A4: set @ A] :
( ( finite_finite @ A @ A4 )
=> ( ( ( semiring_gcd_Lcm_fin @ A @ A4 )
= ( zero_zero @ A ) )
= ( member @ A @ ( zero_zero @ A ) @ A4 ) ) ) ) ).
% Lcm_fin_0_iff
thf(fact_7724_Lcm__0__iff,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A4: set @ A] :
( ( finite_finite @ A @ A4 )
=> ( ( ( gcd_Lcm @ A @ A4 )
= ( zero_zero @ A ) )
= ( member @ A @ ( zero_zero @ A ) @ A4 ) ) ) ) ).
% Lcm_0_iff
thf(fact_7725_Lcm__fin__1__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A4: set @ A] :
( ( ( semiring_gcd_Lcm_fin @ A @ A4 )
= ( one_one @ A ) )
= ( ! [X2: A] :
( ( member @ A @ X2 @ A4 )
=> ( dvd_dvd @ A @ X2 @ ( one_one @ A ) ) )
& ( finite_finite @ A @ A4 ) ) ) ) ).
% Lcm_fin_1_iff
thf(fact_7726_lcm__code__integer,axiom,
( ( gcd_lcm @ code_integer )
= ( ^ [A5: code_integer,B3: code_integer] : ( divide_divide @ code_integer @ ( times_times @ code_integer @ ( abs_abs @ code_integer @ A5 ) @ ( abs_abs @ code_integer @ B3 ) ) @ ( gcd_gcd @ code_integer @ A5 @ B3 ) ) ) ) ).
% lcm_code_integer
thf(fact_7727_Lcm__no__units,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ( ( gcd_Lcm @ A )
= ( ^ [A7: set @ A] :
( gcd_Lcm @ A
@ ( minus_minus @ ( set @ A ) @ A7
@ ( collect @ A
@ ^ [A5: A] : ( dvd_dvd @ A @ A5 @ ( one_one @ A ) ) ) ) ) ) ) ) ).
% Lcm_no_units
thf(fact_7728_lcm__int__def,axiom,
( ( gcd_lcm @ int )
= ( ^ [X2: int,Y5: int] : ( semiring_1_of_nat @ int @ ( gcd_lcm @ nat @ ( nat2 @ ( abs_abs @ int @ X2 ) ) @ ( nat2 @ ( abs_abs @ int @ Y5 ) ) ) ) ) ) ).
% lcm_int_def
thf(fact_7729_Lcm__fin_Oinsert__remove,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,A4: set @ A] :
( ( semiring_gcd_Lcm_fin @ A @ ( insert @ A @ A3 @ A4 ) )
= ( gcd_lcm @ A @ A3 @ ( semiring_gcd_Lcm_fin @ A @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% Lcm_fin.insert_remove
thf(fact_7730_Lcm__fin_Oremove,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,A4: set @ A] :
( ( member @ A @ A3 @ A4 )
=> ( ( semiring_gcd_Lcm_fin @ A @ A4 )
= ( gcd_lcm @ A @ A3 @ ( semiring_gcd_Lcm_fin @ A @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ A3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% Lcm_fin.remove
thf(fact_7731_Lcm__set__eq__fold,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [Xs: list @ A] :
( ( gcd_Lcm @ A @ ( set2 @ A @ Xs ) )
= ( fold @ A @ A @ ( gcd_lcm @ A ) @ Xs @ ( one_one @ A ) ) ) ) ).
% Lcm_set_eq_fold
thf(fact_7732_Lcm__fin_Oset__eq__fold,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [Xs: list @ A] :
( ( semiring_gcd_Lcm_fin @ A @ ( set2 @ A @ Xs ) )
= ( fold @ A @ A @ ( gcd_lcm @ A ) @ Xs @ ( one_one @ A ) ) ) ) ).
% Lcm_fin.set_eq_fold
thf(fact_7733_Lcm__fin__def,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ( ( semiring_gcd_Lcm_fin @ A )
= ( bounde2362111253966948842tice_F @ A @ ( gcd_lcm @ A ) @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ) ).
% Lcm_fin_def
thf(fact_7734_Lcm__int__def,axiom,
( ( gcd_Lcm @ int )
= ( ^ [K5: set @ int] : ( semiring_1_of_nat @ int @ ( gcd_Lcm @ nat @ ( image @ int @ nat @ ( comp @ int @ nat @ int @ nat2 @ ( abs_abs @ int ) ) @ K5 ) ) ) ) ) ).
% Lcm_int_def
thf(fact_7735_Lcm__int__eq,axiom,
! [N6: set @ nat] :
( ( gcd_Lcm @ int @ ( image @ nat @ int @ ( semiring_1_of_nat @ int ) @ N6 ) )
= ( semiring_1_of_nat @ int @ ( gcd_Lcm @ nat @ N6 ) ) ) ).
% Lcm_int_eq
thf(fact_7736_Lcm__eq__0__I__nat,axiom,
! [A4: set @ nat] :
( ( member @ nat @ ( zero_zero @ nat ) @ A4 )
=> ( ( gcd_Lcm @ nat @ A4 )
= ( zero_zero @ nat ) ) ) ).
% Lcm_eq_0_I_nat
thf(fact_7737_Lcm__0__iff__nat,axiom,
! [A4: set @ nat] :
( ( finite_finite @ nat @ A4 )
=> ( ( ( gcd_Lcm @ nat @ A4 )
= ( zero_zero @ nat ) )
= ( member @ nat @ ( zero_zero @ nat ) @ A4 ) ) ) ).
% Lcm_0_iff_nat
thf(fact_7738_Lcm__nat__empty,axiom,
( ( gcd_Lcm @ nat @ ( bot_bot @ ( set @ nat ) ) )
= ( one_one @ nat ) ) ).
% Lcm_nat_empty
thf(fact_7739_Lcm__nat__infinite,axiom,
! [M10: set @ nat] :
( ~ ( finite_finite @ nat @ M10 )
=> ( ( gcd_Lcm @ nat @ M10 )
= ( zero_zero @ nat ) ) ) ).
% Lcm_nat_infinite
thf(fact_7740_Lcm__nat__set__eq__fold,axiom,
! [Xs: list @ nat] :
( ( gcd_Lcm @ nat @ ( set2 @ nat @ Xs ) )
= ( fold @ nat @ nat @ ( gcd_lcm @ nat ) @ Xs @ ( one_one @ nat ) ) ) ).
% Lcm_nat_set_eq_fold
thf(fact_7741_Lcm__eq__Max__nat,axiom,
! [M10: set @ nat] :
( ( finite_finite @ nat @ M10 )
=> ( ( M10
!= ( bot_bot @ ( set @ nat ) ) )
=> ( ~ ( member @ nat @ ( zero_zero @ nat ) @ M10 )
=> ( ! [M3: nat,N2: nat] :
( ( member @ nat @ M3 @ M10 )
=> ( ( member @ nat @ N2 @ M10 )
=> ( member @ nat @ ( gcd_lcm @ nat @ M3 @ N2 ) @ M10 ) ) )
=> ( ( gcd_Lcm @ nat @ M10 )
= ( lattic643756798349783984er_Max @ nat @ M10 ) ) ) ) ) ) ).
% Lcm_eq_Max_nat
thf(fact_7742_Lcm__nat__def,axiom,
( ( gcd_Lcm @ nat )
= ( ^ [M7: set @ nat] : ( if @ nat @ ( finite_finite @ nat @ M7 ) @ ( lattic5214292709420241887eutr_F @ nat @ ( gcd_lcm @ nat ) @ ( one_one @ nat ) @ M7 ) @ ( zero_zero @ nat ) ) ) ) ).
% Lcm_nat_def
thf(fact_7743_Lcm__coprime_H,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A4: set @ A] :
( ( ( finite_card @ A @ A4 )
!= ( zero_zero @ nat ) )
=> ( ! [A6: A,B4: A] :
( ( member @ A @ A6 @ A4 )
=> ( ( member @ A @ B4 @ A4 )
=> ( ( A6 != B4 )
=> ( algebr8660921524188924756oprime @ A @ A6 @ B4 ) ) ) )
=> ( ( gcd_Lcm @ A @ A4 )
= ( normal6383669964737779283malize @ A
@ ( groups7121269368397514597t_prod @ A @ A
@ ^ [X2: A] : X2
@ A4 ) ) ) ) ) ) ).
% Lcm_coprime'
thf(fact_7744_normalize__idem,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A3: A] :
( ( normal6383669964737779283malize @ A @ ( normal6383669964737779283malize @ A @ A3 ) )
= ( normal6383669964737779283malize @ A @ A3 ) ) ) ).
% normalize_idem
thf(fact_7745_normalize__0,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ( ( normal6383669964737779283malize @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% normalize_0
thf(fact_7746_normalize__eq__0__iff,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A3: A] :
( ( ( normal6383669964737779283malize @ A @ A3 )
= ( zero_zero @ A ) )
= ( A3
= ( zero_zero @ A ) ) ) ) ).
% normalize_eq_0_iff
thf(fact_7747_lcm_Onormalize__bottom,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ( ( normal6383669964737779283malize @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% lcm.normalize_bottom
thf(fact_7748_normalize__mult__normalize__left,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A3: A,B2: A] :
( ( normal6383669964737779283malize @ A @ ( times_times @ A @ ( normal6383669964737779283malize @ A @ A3 ) @ B2 ) )
= ( normal6383669964737779283malize @ A @ ( times_times @ A @ A3 @ B2 ) ) ) ) ).
% normalize_mult_normalize_left
thf(fact_7749_normalize__mult__normalize__right,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A3: A,B2: A] :
( ( normal6383669964737779283malize @ A @ ( times_times @ A @ A3 @ ( normal6383669964737779283malize @ A @ B2 ) ) )
= ( normal6383669964737779283malize @ A @ ( times_times @ A @ A3 @ B2 ) ) ) ) ).
% normalize_mult_normalize_right
thf(fact_7750_gcd_Onormalize__bottom,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ( ( normal6383669964737779283malize @ A @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% gcd.normalize_bottom
thf(fact_7751_normalize__1,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ( ( normal6383669964737779283malize @ A @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% normalize_1
thf(fact_7752_dvd__normalize__iff,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A3: A,B2: A] :
( ( dvd_dvd @ A @ A3 @ ( normal6383669964737779283malize @ A @ B2 ) )
= ( dvd_dvd @ A @ A3 @ B2 ) ) ) ).
% dvd_normalize_iff
thf(fact_7753_normalize__dvd__iff,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A3: A,B2: A] :
( ( dvd_dvd @ A @ ( normal6383669964737779283malize @ A @ A3 ) @ B2 )
= ( dvd_dvd @ A @ A3 @ B2 ) ) ) ).
% normalize_dvd_iff
thf(fact_7754_coprime__normalize__right__iff,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A3: A,B2: A] :
( ( algebr8660921524188924756oprime @ A @ A3 @ ( normal6383669964737779283malize @ A @ B2 ) )
= ( algebr8660921524188924756oprime @ A @ A3 @ B2 ) ) ) ).
% coprime_normalize_right_iff
thf(fact_7755_coprime__normalize__left__iff,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A3: A,B2: A] :
( ( algebr8660921524188924756oprime @ A @ ( normal6383669964737779283malize @ A @ A3 ) @ B2 )
= ( algebr8660921524188924756oprime @ A @ A3 @ B2 ) ) ) ).
% coprime_normalize_left_iff
thf(fact_7756_gcd_Otop__right__normalize,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A] :
( ( gcd_gcd @ A @ A3 @ ( zero_zero @ A ) )
= ( normal6383669964737779283malize @ A @ A3 ) ) ) ).
% gcd.top_right_normalize
thf(fact_7757_gcd_Otop__left__normalize,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A] :
( ( gcd_gcd @ A @ ( zero_zero @ A ) @ A3 )
= ( normal6383669964737779283malize @ A @ A3 ) ) ) ).
% gcd.top_left_normalize
thf(fact_7758_lcm_Otop__right__normalize,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A] :
( ( gcd_lcm @ A @ A3 @ ( one_one @ A ) )
= ( normal6383669964737779283malize @ A @ A3 ) ) ) ).
% lcm.top_right_normalize
thf(fact_7759_lcm_Otop__left__normalize,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A] :
( ( gcd_lcm @ A @ ( one_one @ A ) @ A3 )
= ( normal6383669964737779283malize @ A @ A3 ) ) ) ).
% lcm.top_left_normalize
thf(fact_7760_normalize__mult__unit__left,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A3: A,B2: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( normal6383669964737779283malize @ A @ ( times_times @ A @ A3 @ B2 ) )
= ( normal6383669964737779283malize @ A @ B2 ) ) ) ) ).
% normalize_mult_unit_left
thf(fact_7761_normalize__mult__unit__right,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [B2: A,A3: A] :
( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( normal6383669964737779283malize @ A @ ( times_times @ A @ A3 @ B2 ) )
= ( normal6383669964737779283malize @ A @ A3 ) ) ) ) ).
% normalize_mult_unit_right
thf(fact_7762_normalize__idem__imp__is__unit__iff,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A3: A] :
( ( ( normal6383669964737779283malize @ A @ A3 )
= A3 )
=> ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
= ( A3
= ( one_one @ A ) ) ) ) ) ).
% normalize_idem_imp_is_unit_iff
thf(fact_7763_is__unit__normalize,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A3: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( normal6383669964737779283malize @ A @ A3 )
= ( one_one @ A ) ) ) ) ).
% is_unit_normalize
thf(fact_7764_normalize__1__iff,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A3: A] :
( ( ( normal6383669964737779283malize @ A @ A3 )
= ( one_one @ A ) )
= ( dvd_dvd @ A @ A3 @ ( one_one @ A ) ) ) ) ).
% normalize_1_iff
thf(fact_7765_associated__unit,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A3: A,B2: A] :
( ( ( normal6383669964737779283malize @ A @ A3 )
= ( normal6383669964737779283malize @ A @ B2 ) )
=> ( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( dvd_dvd @ A @ B2 @ ( one_one @ A ) ) ) ) ) ).
% associated_unit
thf(fact_7766_associated__iff__dvd,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A3: A,B2: A] :
( ( ( normal6383669964737779283malize @ A @ A3 )
= ( normal6383669964737779283malize @ A @ B2 ) )
= ( ( dvd_dvd @ A @ A3 @ B2 )
& ( dvd_dvd @ A @ B2 @ A3 ) ) ) ) ).
% associated_iff_dvd
thf(fact_7767_associated__eqI,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A3: A,B2: A] :
( ( dvd_dvd @ A @ A3 @ B2 )
=> ( ( dvd_dvd @ A @ B2 @ A3 )
=> ( ( ( normal6383669964737779283malize @ A @ A3 )
= A3 )
=> ( ( ( normal6383669964737779283malize @ A @ B2 )
= B2 )
=> ( A3 = B2 ) ) ) ) ) ) ).
% associated_eqI
thf(fact_7768_associatedD2,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A3: A,B2: A] :
( ( ( normal6383669964737779283malize @ A @ A3 )
= ( normal6383669964737779283malize @ A @ B2 ) )
=> ( dvd_dvd @ A @ B2 @ A3 ) ) ) ).
% associatedD2
thf(fact_7769_associatedD1,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A3: A,B2: A] :
( ( ( normal6383669964737779283malize @ A @ A3 )
= ( normal6383669964737779283malize @ A @ B2 ) )
=> ( dvd_dvd @ A @ A3 @ B2 ) ) ) ).
% associatedD1
thf(fact_7770_associatedI,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A3: A,B2: A] :
( ( dvd_dvd @ A @ A3 @ B2 )
=> ( ( dvd_dvd @ A @ B2 @ A3 )
=> ( ( normal6383669964737779283malize @ A @ A3 )
= ( normal6383669964737779283malize @ A @ B2 ) ) ) ) ) ).
% associatedI
thf(fact_7771_dvd__normalize__div,axiom,
! [A: $tType] :
( ( normal6328177297339901930cative @ A )
=> ! [B2: A,A3: A] :
( ( dvd_dvd @ A @ B2 @ A3 )
=> ( ( normal6383669964737779283malize @ A @ ( divide_divide @ A @ A3 @ B2 ) )
= ( divide_divide @ A @ ( normal6383669964737779283malize @ A @ A3 ) @ ( normal6383669964737779283malize @ A @ B2 ) ) ) ) ) ).
% dvd_normalize_div
thf(fact_7772_normalize__mult,axiom,
! [A: $tType] :
( ( normal6328177297339901930cative @ A )
=> ! [A3: A,B2: A] :
( ( normal6383669964737779283malize @ A @ ( times_times @ A @ A3 @ B2 ) )
= ( times_times @ A @ ( normal6383669964737779283malize @ A @ A3 ) @ ( normal6383669964737779283malize @ A @ B2 ) ) ) ) ).
% normalize_mult
thf(fact_7773_lcm__gcd,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ( ( gcd_lcm @ A )
= ( ^ [A5: A,B3: A] : ( normal6383669964737779283malize @ A @ ( divide_divide @ A @ ( times_times @ A @ A5 @ B3 ) @ ( gcd_gcd @ A @ A5 @ B3 ) ) ) ) ) ) ).
% lcm_gcd
thf(fact_7774_Gcd__fin_Obounded__quasi__semilattice__set__axioms,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ( bounde6485984586167503788ce_set @ A @ ( gcd_gcd @ A ) @ ( zero_zero @ A ) @ ( one_one @ A ) @ ( normal6383669964737779283malize @ A ) ) ) ).
% Gcd_fin.bounded_quasi_semilattice_set_axioms
thf(fact_7775_Lcm__fin_Obounded__quasi__semilattice__set__axioms,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ( bounde6485984586167503788ce_set @ A @ ( gcd_lcm @ A ) @ ( one_one @ A ) @ ( zero_zero @ A ) @ ( normal6383669964737779283malize @ A ) ) ) ).
% Lcm_fin.bounded_quasi_semilattice_set_axioms
thf(fact_7776_gcd__lcm,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( B2
!= ( zero_zero @ A ) )
=> ( ( gcd_gcd @ A @ A3 @ B2 )
= ( normal6383669964737779283malize @ A @ ( divide_divide @ A @ ( times_times @ A @ A3 @ B2 ) @ ( gcd_lcm @ A @ A3 @ B2 ) ) ) ) ) ) ) ).
% gcd_lcm
thf(fact_7777_semilattice__neutr__set_Oremove,axiom,
! [A: $tType,F2: A > A > A,Z2: A,A4: set @ A,X: A] :
( ( lattic5652469242046573047tr_set @ A @ F2 @ Z2 )
=> ( ( finite_finite @ A @ A4 )
=> ( ( member @ A @ X @ A4 )
=> ( ( lattic5214292709420241887eutr_F @ A @ F2 @ Z2 @ A4 )
= ( F2 @ X @ ( lattic5214292709420241887eutr_F @ A @ F2 @ Z2 @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ).
% semilattice_neutr_set.remove
thf(fact_7778_semilattice__neutr__set_Oinsert__remove,axiom,
! [A: $tType,F2: A > A > A,Z2: A,A4: set @ A,X: A] :
( ( lattic5652469242046573047tr_set @ A @ F2 @ Z2 )
=> ( ( finite_finite @ A @ A4 )
=> ( ( lattic5214292709420241887eutr_F @ A @ F2 @ Z2 @ ( insert @ A @ X @ A4 ) )
= ( F2 @ X @ ( lattic5214292709420241887eutr_F @ A @ F2 @ Z2 @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% semilattice_neutr_set.insert_remove
thf(fact_7779_lcm_Obounded__quasi__semilattice__axioms,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ( bounde8507323023520639062attice @ A @ ( gcd_lcm @ A ) @ ( one_one @ A ) @ ( zero_zero @ A ) @ ( normal6383669964737779283malize @ A ) ) ) ).
% lcm.bounded_quasi_semilattice_axioms
thf(fact_7780_gcd_Obounded__quasi__semilattice__axioms,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ( bounde8507323023520639062attice @ A @ ( gcd_gcd @ A ) @ ( zero_zero @ A ) @ ( one_one @ A ) @ ( normal6383669964737779283malize @ A ) ) ) ).
% gcd.bounded_quasi_semilattice_axioms
thf(fact_7781_normalize__div,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A3: A] :
( ( divide_divide @ A @ ( normal6383669964737779283malize @ A @ A3 ) @ A3 )
= ( divide_divide @ A @ ( one_one @ A ) @ ( unit_f5069060285200089521factor @ A @ A3 ) ) ) ) ).
% normalize_div
thf(fact_7782_unit__factor__normalize,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( unit_f5069060285200089521factor @ A @ ( normal6383669964737779283malize @ A @ A3 ) )
= ( one_one @ A ) ) ) ) ).
% unit_factor_normalize
thf(fact_7783_unit__factor__simps_I1_J,axiom,
( ( unit_f5069060285200089521factor @ nat @ ( zero_zero @ nat ) )
= ( zero_zero @ nat ) ) ).
% unit_factor_simps(1)
thf(fact_7784_unit__factor__idem,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A3: A] :
( ( unit_f5069060285200089521factor @ A @ ( unit_f5069060285200089521factor @ A @ A3 ) )
= ( unit_f5069060285200089521factor @ A @ A3 ) ) ) ).
% unit_factor_idem
thf(fact_7785_unit__factor__simps_I2_J,axiom,
! [N: nat] :
( ( unit_f5069060285200089521factor @ nat @ ( suc @ N ) )
= ( one_one @ nat ) ) ).
% unit_factor_simps(2)
thf(fact_7786_unit__factor__eq__0__iff,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A3: A] :
( ( ( unit_f5069060285200089521factor @ A @ A3 )
= ( zero_zero @ A ) )
= ( A3
= ( zero_zero @ A ) ) ) ) ).
% unit_factor_eq_0_iff
thf(fact_7787_unit__factor__0,axiom,
! [A: $tType] :
( ( semido2269285787275462019factor @ A )
=> ( ( unit_f5069060285200089521factor @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% unit_factor_0
thf(fact_7788_unit__factor__1,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ( ( unit_f5069060285200089521factor @ A @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% unit_factor_1
thf(fact_7789_unit__factor__mult__normalize,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A3: A] :
( ( times_times @ A @ ( unit_f5069060285200089521factor @ A @ A3 ) @ ( normal6383669964737779283malize @ A @ A3 ) )
= A3 ) ) ).
% unit_factor_mult_normalize
thf(fact_7790_normalize__mult__unit__factor,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A3: A] :
( ( times_times @ A @ ( normal6383669964737779283malize @ A @ A3 ) @ ( unit_f5069060285200089521factor @ A @ A3 ) )
= A3 ) ) ).
% normalize_mult_unit_factor
thf(fact_7791_div__normalize,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A3: A] :
( ( divide_divide @ A @ A3 @ ( normal6383669964737779283malize @ A @ A3 ) )
= ( unit_f5069060285200089521factor @ A @ A3 ) ) ) ).
% div_normalize
thf(fact_7792_div__unit__factor,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A3: A] :
( ( divide_divide @ A @ A3 @ ( unit_f5069060285200089521factor @ A @ A3 ) )
= ( normal6383669964737779283malize @ A @ A3 ) ) ) ).
% div_unit_factor
thf(fact_7793_inv__unit__factor__eq__0__iff,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A3: A] :
( ( ( divide_divide @ A @ ( one_one @ A ) @ ( unit_f5069060285200089521factor @ A @ A3 ) )
= ( zero_zero @ A ) )
= ( A3
= ( zero_zero @ A ) ) ) ) ).
% inv_unit_factor_eq_0_iff
thf(fact_7794_mult__one__div__unit__factor,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A3: A,B2: A] :
( ( times_times @ A @ A3 @ ( divide_divide @ A @ ( one_one @ A ) @ ( unit_f5069060285200089521factor @ A @ B2 ) ) )
= ( divide_divide @ A @ A3 @ ( unit_f5069060285200089521factor @ A @ B2 ) ) ) ) ).
% mult_one_div_unit_factor
thf(fact_7795_unit__factor__mult__unit__left,axiom,
! [A: $tType] :
( ( semido2269285787275462019factor @ A )
=> ! [A3: A,B2: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( unit_f5069060285200089521factor @ A @ ( times_times @ A @ A3 @ B2 ) )
= ( times_times @ A @ A3 @ ( unit_f5069060285200089521factor @ A @ B2 ) ) ) ) ) ).
% unit_factor_mult_unit_left
thf(fact_7796_unit__factor__mult__unit__right,axiom,
! [A: $tType] :
( ( semido2269285787275462019factor @ A )
=> ! [A3: A,B2: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( unit_f5069060285200089521factor @ A @ ( times_times @ A @ B2 @ A3 ) )
= ( times_times @ A @ ( unit_f5069060285200089521factor @ A @ B2 ) @ A3 ) ) ) ) ).
% unit_factor_mult_unit_right
thf(fact_7797_unit__factor__lcm,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A] :
( ( ( ( A3
= ( zero_zero @ A ) )
| ( B2
= ( zero_zero @ A ) ) )
=> ( ( unit_f5069060285200089521factor @ A @ ( gcd_lcm @ A @ A3 @ B2 ) )
= ( zero_zero @ A ) ) )
& ( ~ ( ( A3
= ( zero_zero @ A ) )
| ( B2
= ( zero_zero @ A ) ) )
=> ( ( unit_f5069060285200089521factor @ A @ ( gcd_lcm @ A @ A3 @ B2 ) )
= ( one_one @ A ) ) ) ) ) ).
% unit_factor_lcm
thf(fact_7798_normalize__unit__factor,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( ( normal6383669964737779283malize @ A @ ( unit_f5069060285200089521factor @ A @ A3 ) )
= ( one_one @ A ) ) ) ) ).
% normalize_unit_factor
thf(fact_7799_normalize__unit__factor__eqI,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A3: A,B2: A] :
( ( ( normal6383669964737779283malize @ A @ A3 )
= ( normal6383669964737779283malize @ A @ B2 ) )
=> ( ( ( unit_f5069060285200089521factor @ A @ A3 )
= ( unit_f5069060285200089521factor @ A @ B2 ) )
=> ( A3 = B2 ) ) ) ) ).
% normalize_unit_factor_eqI
thf(fact_7800_unit__factor__is__unit,axiom,
! [A: $tType] :
( ( semido2269285787275462019factor @ A )
=> ! [A3: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( dvd_dvd @ A @ ( unit_f5069060285200089521factor @ A @ A3 ) @ ( one_one @ A ) ) ) ) ).
% unit_factor_is_unit
thf(fact_7801_unit__factor__mult,axiom,
! [A: $tType] :
( ( normal6328177297339901930cative @ A )
=> ! [A3: A,B2: A] :
( ( unit_f5069060285200089521factor @ A @ ( times_times @ A @ A3 @ B2 ) )
= ( times_times @ A @ ( unit_f5069060285200089521factor @ A @ A3 ) @ ( unit_f5069060285200089521factor @ A @ B2 ) ) ) ) ).
% unit_factor_mult
thf(fact_7802_unit__factor__self,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A3: A] : ( dvd_dvd @ A @ ( unit_f5069060285200089521factor @ A @ A3 ) @ A3 ) ) ).
% unit_factor_self
thf(fact_7803_dvd__unit__factor__div,axiom,
! [A: $tType] :
( ( normal6328177297339901930cative @ A )
=> ! [B2: A,A3: A] :
( ( dvd_dvd @ A @ B2 @ A3 )
=> ( ( unit_f5069060285200089521factor @ A @ ( divide_divide @ A @ A3 @ B2 ) )
= ( divide_divide @ A @ ( unit_f5069060285200089521factor @ A @ A3 ) @ ( unit_f5069060285200089521factor @ A @ B2 ) ) ) ) ) ).
% dvd_unit_factor_div
thf(fact_7804_unit__factor__nat__def,axiom,
( ( unit_f5069060285200089521factor @ nat )
= ( ^ [N5: nat] :
( if @ nat
@ ( N5
= ( zero_zero @ nat ) )
@ ( zero_zero @ nat )
@ ( one_one @ nat ) ) ) ) ).
% unit_factor_nat_def
thf(fact_7805_unit__factor__dvd,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A3: A,B2: A] :
( ( A3
!= ( zero_zero @ A ) )
=> ( dvd_dvd @ A @ ( unit_f5069060285200089521factor @ A @ A3 ) @ B2 ) ) ) ).
% unit_factor_dvd
thf(fact_7806_is__unit__unit__factor,axiom,
! [A: $tType] :
( ( semido2269285787275462019factor @ A )
=> ! [A3: A] :
( ( dvd_dvd @ A @ A3 @ ( one_one @ A ) )
=> ( ( unit_f5069060285200089521factor @ A @ A3 )
= A3 ) ) ) ).
% is_unit_unit_factor
thf(fact_7807_unit__factor__gcd,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A3: A,B2: A] :
( ( ( ( A3
= ( zero_zero @ A ) )
& ( B2
= ( zero_zero @ A ) ) )
=> ( ( unit_f5069060285200089521factor @ A @ ( gcd_gcd @ A @ A3 @ B2 ) )
= ( zero_zero @ A ) ) )
& ( ~ ( ( A3
= ( zero_zero @ A ) )
& ( B2
= ( zero_zero @ A ) ) )
=> ( ( unit_f5069060285200089521factor @ A @ ( gcd_gcd @ A @ A3 @ B2 ) )
= ( one_one @ A ) ) ) ) ) ).
% unit_factor_gcd
thf(fact_7808_coprime__crossproduct_H,axiom,
! [A: $tType] :
( ( semiri6843258321239162965malize @ A )
=> ! [B2: A,D2: A,A3: A,C2: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( ( unit_f5069060285200089521factor @ A @ B2 )
= ( unit_f5069060285200089521factor @ A @ D2 ) )
=> ( ( algebr8660921524188924756oprime @ A @ A3 @ B2 )
=> ( ( algebr8660921524188924756oprime @ A @ C2 @ D2 )
=> ( ( ( times_times @ A @ A3 @ D2 )
= ( times_times @ A @ B2 @ C2 ) )
= ( ( A3 = C2 )
& ( B2 = D2 ) ) ) ) ) ) ) ) ).
% coprime_crossproduct'
thf(fact_7809_unit__factor__Lcm,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A4: set @ A] :
( ( ( ( gcd_Lcm @ A @ A4 )
= ( zero_zero @ A ) )
=> ( ( unit_f5069060285200089521factor @ A @ ( gcd_Lcm @ A @ A4 ) )
= ( zero_zero @ A ) ) )
& ( ( ( gcd_Lcm @ A @ A4 )
!= ( zero_zero @ A ) )
=> ( ( unit_f5069060285200089521factor @ A @ ( gcd_Lcm @ A @ A4 ) )
= ( one_one @ A ) ) ) ) ) ).
% unit_factor_Lcm
thf(fact_7810_unit__factor__Gcd,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A4: set @ A] :
( ( ( ( gcd_Gcd @ A @ A4 )
= ( zero_zero @ A ) )
=> ( ( unit_f5069060285200089521factor @ A @ ( gcd_Gcd @ A @ A4 ) )
= ( zero_zero @ A ) ) )
& ( ( ( gcd_Gcd @ A @ A4 )
!= ( zero_zero @ A ) )
=> ( ( unit_f5069060285200089521factor @ A @ ( gcd_Gcd @ A @ A4 ) )
= ( one_one @ A ) ) ) ) ) ).
% unit_factor_Gcd
thf(fact_7811_normalize__idem__imp__unit__factor__eq,axiom,
! [A: $tType] :
( ( normal8620421768224518004emidom @ A )
=> ! [A3: A] :
( ( ( normal6383669964737779283malize @ A @ A3 )
= A3 )
=> ( ( unit_f5069060285200089521factor @ A @ A3 )
= ( zero_neq_one_of_bool @ A
@ ( A3
!= ( zero_zero @ A ) ) ) ) ) ) ).
% normalize_idem_imp_unit_factor_eq
thf(fact_7812_unit__factor__Lcm__fin,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A4: set @ A] :
( ( unit_f5069060285200089521factor @ A @ ( semiring_gcd_Lcm_fin @ A @ A4 ) )
= ( zero_neq_one_of_bool @ A
@ ( ( semiring_gcd_Lcm_fin @ A @ A4 )
!= ( zero_zero @ A ) ) ) ) ) ).
% unit_factor_Lcm_fin
thf(fact_7813_unit__factor__Gcd__fin,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A4: set @ A] :
( ( unit_f5069060285200089521factor @ A @ ( semiring_gcd_Gcd_fin @ A @ A4 ) )
= ( zero_neq_one_of_bool @ A
@ ( ( semiring_gcd_Gcd_fin @ A @ A4 )
!= ( zero_zero @ A ) ) ) ) ) ).
% unit_factor_Gcd_fin
thf(fact_7814_and_Ocomm__monoid__axioms,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( comm_monoid @ A @ ( bit_se5824344872417868541ns_and @ A ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ).
% and.comm_monoid_axioms
thf(fact_7815_card__Plus__conv__if,axiom,
! [B: $tType,A: $tType,A4: set @ A,B7: set @ B] :
( ( ( ( finite_finite @ A @ A4 )
& ( finite_finite @ B @ B7 ) )
=> ( ( finite_card @ ( sum_sum @ A @ B ) @ ( sum_Plus @ A @ B @ A4 @ B7 ) )
= ( plus_plus @ nat @ ( finite_card @ A @ A4 ) @ ( finite_card @ B @ B7 ) ) ) )
& ( ~ ( ( finite_finite @ A @ A4 )
& ( finite_finite @ B @ B7 ) )
=> ( ( finite_card @ ( sum_sum @ A @ B ) @ ( sum_Plus @ A @ B @ A4 @ B7 ) )
= ( zero_zero @ nat ) ) ) ) ).
% card_Plus_conv_if
thf(fact_7816_gcd__nat_Ocomm__monoid__axioms,axiom,
comm_monoid @ nat @ ( gcd_gcd @ nat ) @ ( zero_zero @ nat ) ).
% gcd_nat.comm_monoid_axioms
thf(fact_7817_inf__top_Ocomm__monoid__axioms,axiom,
! [A: $tType] :
( ( bounde4346867609351753570nf_top @ A )
=> ( comm_monoid @ A @ ( inf_inf @ A ) @ ( top_top @ A ) ) ) ).
% inf_top.comm_monoid_axioms
thf(fact_7818_semilattice__neutr_Oaxioms_I2_J,axiom,
! [A: $tType,F2: A > A > A,Z2: A] :
( ( semilattice_neutr @ A @ F2 @ Z2 )
=> ( comm_monoid @ A @ F2 @ Z2 ) ) ).
% semilattice_neutr.axioms(2)
thf(fact_7819_or_Ocomm__monoid__axioms,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( comm_monoid @ A @ ( bit_se1065995026697491101ons_or @ A ) @ ( zero_zero @ A ) ) ) ).
% or.comm_monoid_axioms
thf(fact_7820_xor_Ocomm__monoid__axioms,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( comm_monoid @ A @ ( bit_se5824344971392196577ns_xor @ A ) @ ( zero_zero @ A ) ) ) ).
% xor.comm_monoid_axioms
thf(fact_7821_max__nat_Ocomm__monoid__axioms,axiom,
comm_monoid @ nat @ ( ord_max @ nat ) @ ( zero_zero @ nat ) ).
% max_nat.comm_monoid_axioms
thf(fact_7822_comm__monoid_Ocomm__neutral,axiom,
! [A: $tType,F2: A > A > A,Z2: A,A3: A] :
( ( comm_monoid @ A @ F2 @ Z2 )
=> ( ( F2 @ A3 @ Z2 )
= A3 ) ) ).
% comm_monoid.comm_neutral
thf(fact_7823_sup__bot_Ocomm__monoid__axioms,axiom,
! [A: $tType] :
( ( bounde4967611905675639751up_bot @ A )
=> ( comm_monoid @ A @ ( sup_sup @ A ) @ ( bot_bot @ A ) ) ) ).
% sup_bot.comm_monoid_axioms
thf(fact_7824_add_Ocomm__monoid__axioms,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ( comm_monoid @ A @ ( plus_plus @ A ) @ ( zero_zero @ A ) ) ) ).
% add.comm_monoid_axioms
thf(fact_7825_mult_Ocomm__monoid__axioms,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ( comm_monoid @ A @ ( times_times @ A ) @ ( one_one @ A ) ) ) ).
% mult.comm_monoid_axioms
thf(fact_7826_lexn_Osimps_I2_J,axiom,
! [A: $tType,R4: set @ ( product_prod @ A @ A ),N: nat] :
( ( lexn @ A @ R4 @ ( suc @ N ) )
= ( inf_inf @ ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) @ ( image @ ( product_prod @ ( product_prod @ A @ ( list @ A ) ) @ ( product_prod @ A @ ( list @ A ) ) ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_map_prod @ ( product_prod @ A @ ( list @ A ) ) @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) @ ( list @ A ) @ ( product_case_prod @ A @ ( list @ A ) @ ( list @ A ) @ ( cons @ A ) ) @ ( product_case_prod @ A @ ( list @ A ) @ ( list @ A ) @ ( cons @ A ) ) ) @ ( lex_prod @ A @ ( list @ A ) @ R4 @ ( lexn @ A @ R4 @ N ) ) )
@ ( collect @ ( product_prod @ ( list @ A ) @ ( list @ A ) )
@ ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ $o
@ ^ [Xs3: list @ A,Ys3: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs3 )
= ( suc @ N ) )
& ( ( size_size @ ( list @ A ) @ Ys3 )
= ( suc @ N ) ) ) ) ) ) ) ).
% lexn.simps(2)
thf(fact_7827_is__num__normalize_I6_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [X: A,Y2: A] :
( ( neg_numeral_is_num @ A @ X )
=> ( ( neg_numeral_is_num @ A @ Y2 )
=> ( neg_numeral_is_num @ A @ ( plus_plus @ A @ X @ Y2 ) ) ) ) ) ).
% is_num_normalize(6)
thf(fact_7828_is__num__add__commute,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [X: A,Y2: A] :
( ( neg_numeral_is_num @ A @ X )
=> ( ( neg_numeral_is_num @ A @ Y2 )
=> ( ( plus_plus @ A @ X @ Y2 )
= ( plus_plus @ A @ Y2 @ X ) ) ) ) ) ).
% is_num_add_commute
thf(fact_7829_is__num__add__left__commute,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [X: A,Y2: A,Z2: A] :
( ( neg_numeral_is_num @ A @ X )
=> ( ( neg_numeral_is_num @ A @ Y2 )
=> ( ( plus_plus @ A @ X @ ( plus_plus @ A @ Y2 @ Z2 ) )
= ( plus_plus @ A @ Y2 @ ( plus_plus @ A @ X @ Z2 ) ) ) ) ) ) ).
% is_num_add_left_commute
thf(fact_7830_is__num__normalize_I5_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [X: A] :
( ( neg_numeral_is_num @ A @ X )
=> ( neg_numeral_is_num @ A @ ( uminus_uminus @ A @ X ) ) ) ) ).
% is_num_normalize(5)
thf(fact_7831_is__num__numeral,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [K: num] : ( neg_numeral_is_num @ A @ ( numeral_numeral @ A @ K ) ) ) ).
% is_num_numeral
thf(fact_7832_is__num__normalize_I4_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( neg_numeral_is_num @ A @ ( one_one @ A ) ) ) ).
% is_num_normalize(4)
thf(fact_7833_is__num_Ocases,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [A3: A] :
( ( neg_numeral_is_num @ A @ A3 )
=> ( ( A3
!= ( one_one @ A ) )
=> ( ! [X3: A] :
( ( A3
= ( uminus_uminus @ A @ X3 ) )
=> ~ ( neg_numeral_is_num @ A @ X3 ) )
=> ~ ! [X3: A,Y3: A] :
( ( A3
= ( plus_plus @ A @ X3 @ Y3 ) )
=> ( ( neg_numeral_is_num @ A @ X3 )
=> ~ ( neg_numeral_is_num @ A @ Y3 ) ) ) ) ) ) ) ).
% is_num.cases
thf(fact_7834_is__num_Osimps,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( neg_numeral_is_num @ A )
= ( ^ [A5: A] :
( ( A5
= ( one_one @ A ) )
| ? [X2: A] :
( ( A5
= ( uminus_uminus @ A @ X2 ) )
& ( neg_numeral_is_num @ A @ X2 ) )
| ? [X2: A,Y5: A] :
( ( A5
= ( plus_plus @ A @ X2 @ Y5 ) )
& ( neg_numeral_is_num @ A @ X2 )
& ( neg_numeral_is_num @ A @ Y5 ) ) ) ) ) ) ).
% is_num.simps
thf(fact_7835_times__num__def,axiom,
( ( times_times @ num )
= ( ^ [M5: num,N5: num] : ( num_of_nat @ ( times_times @ nat @ ( nat_of_num @ M5 ) @ ( nat_of_num @ N5 ) ) ) ) ) ).
% times_num_def
thf(fact_7836_arg__max__nat__lemma,axiom,
! [A: $tType,P: A > $o,K: A,F2: A > nat,B2: nat] :
( ( P @ K )
=> ( ! [Y3: A] :
( ( P @ Y3 )
=> ( ord_less @ nat @ ( F2 @ Y3 ) @ B2 ) )
=> ( ( P @ ( lattices_ord_arg_max @ A @ nat @ F2 @ P ) )
& ! [Y: A] :
( ( P @ Y )
=> ( ord_less_eq @ nat @ ( F2 @ Y ) @ ( F2 @ ( lattices_ord_arg_max @ A @ nat @ F2 @ P ) ) ) ) ) ) ) ).
% arg_max_nat_lemma
thf(fact_7837_less__eq__num__def,axiom,
( ( ord_less_eq @ num )
= ( ^ [M5: num,N5: num] : ( ord_less_eq @ nat @ ( nat_of_num @ M5 ) @ ( nat_of_num @ N5 ) ) ) ) ).
% less_eq_num_def
thf(fact_7838_nat__of__num__code_I1_J,axiom,
( ( nat_of_num @ one2 )
= ( one_one @ nat ) ) ).
% nat_of_num_code(1)
thf(fact_7839_nat__of__num__neq__0,axiom,
! [X: num] :
( ( nat_of_num @ X )
!= ( zero_zero @ nat ) ) ).
% nat_of_num_neq_0
thf(fact_7840_nat__of__num__pos,axiom,
! [X: num] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( nat_of_num @ X ) ) ).
% nat_of_num_pos
thf(fact_7841_arg__maxI,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [P: A > $o,X: A,F2: A > B,Q: A > $o] :
( ( P @ X )
=> ( ! [Y3: A] :
( ( P @ Y3 )
=> ~ ( ord_less @ B @ ( F2 @ X ) @ ( F2 @ Y3 ) ) )
=> ( ! [X3: A] :
( ( P @ X3 )
=> ( ! [Y: A] :
( ( P @ Y )
=> ~ ( ord_less @ B @ ( F2 @ X3 ) @ ( F2 @ Y ) ) )
=> ( Q @ X3 ) ) )
=> ( Q @ ( lattices_ord_arg_max @ A @ B @ F2 @ P ) ) ) ) ) ) ).
% arg_maxI
thf(fact_7842_nat__of__num__inc,axiom,
! [X: num] :
( ( nat_of_num @ ( inc @ X ) )
= ( suc @ ( nat_of_num @ X ) ) ) ).
% nat_of_num_inc
thf(fact_7843_arg__max__natI,axiom,
! [A: $tType,P: A > $o,K: A,F2: A > nat,B2: nat] :
( ( P @ K )
=> ( ! [Y3: A] :
( ( P @ Y3 )
=> ( ord_less @ nat @ ( F2 @ Y3 ) @ B2 ) )
=> ( P @ ( lattices_ord_arg_max @ A @ nat @ F2 @ P ) ) ) ) ).
% arg_max_natI
thf(fact_7844_num__eq__iff,axiom,
( ( ^ [Y4: num,Z: num] : Y4 = Z )
= ( ^ [X2: num,Y5: num] :
( ( nat_of_num @ X2 )
= ( nat_of_num @ Y5 ) ) ) ) ).
% num_eq_iff
thf(fact_7845_nat__of__num__numeral,axiom,
( nat_of_num
= ( numeral_numeral @ nat ) ) ).
% nat_of_num_numeral
thf(fact_7846_nat__of__num__inverse,axiom,
! [X: num] :
( ( num_of_nat @ ( nat_of_num @ X ) )
= X ) ).
% nat_of_num_inverse
thf(fact_7847_less__num__def,axiom,
( ( ord_less @ num )
= ( ^ [M5: num,N5: num] : ( ord_less @ nat @ ( nat_of_num @ M5 ) @ ( nat_of_num @ N5 ) ) ) ) ).
% less_num_def
thf(fact_7848_nat__of__num__code_I2_J,axiom,
! [N: num] :
( ( nat_of_num @ ( bit0 @ N ) )
= ( plus_plus @ nat @ ( nat_of_num @ N ) @ ( nat_of_num @ N ) ) ) ).
% nat_of_num_code(2)
thf(fact_7849_nat__of__num_Osimps_I2_J,axiom,
! [X: num] :
( ( nat_of_num @ ( bit0 @ X ) )
= ( plus_plus @ nat @ ( nat_of_num @ X ) @ ( nat_of_num @ X ) ) ) ).
% nat_of_num.simps(2)
thf(fact_7850_nat__of__num__add,axiom,
! [X: num,Y2: num] :
( ( nat_of_num @ ( plus_plus @ num @ X @ Y2 ) )
= ( plus_plus @ nat @ ( nat_of_num @ X ) @ ( nat_of_num @ Y2 ) ) ) ).
% nat_of_num_add
thf(fact_7851_nat__of__num__mult,axiom,
! [X: num,Y2: num] :
( ( nat_of_num @ ( times_times @ num @ X @ Y2 ) )
= ( times_times @ nat @ ( nat_of_num @ X ) @ ( nat_of_num @ Y2 ) ) ) ).
% nat_of_num_mult
thf(fact_7852_nat__of__num__sqr,axiom,
! [X: num] :
( ( nat_of_num @ ( sqr @ X ) )
= ( times_times @ nat @ ( nat_of_num @ X ) @ ( nat_of_num @ X ) ) ) ).
% nat_of_num_sqr
thf(fact_7853_nat__of__num_Osimps_I1_J,axiom,
( ( nat_of_num @ one2 )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% nat_of_num.simps(1)
thf(fact_7854_nat__of__num_Osimps_I3_J,axiom,
! [X: num] :
( ( nat_of_num @ ( bit1 @ X ) )
= ( suc @ ( plus_plus @ nat @ ( nat_of_num @ X ) @ ( nat_of_num @ X ) ) ) ) ).
% nat_of_num.simps(3)
thf(fact_7855_num__of__nat__inverse,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( nat_of_num @ ( num_of_nat @ N ) )
= N ) ) ).
% num_of_nat_inverse
thf(fact_7856_nat__of__num__code_I3_J,axiom,
! [N: num] :
( ( nat_of_num @ ( bit1 @ N ) )
= ( suc @ ( plus_plus @ nat @ ( nat_of_num @ N ) @ ( nat_of_num @ N ) ) ) ) ).
% nat_of_num_code(3)
thf(fact_7857_plus__num__def,axiom,
( ( plus_plus @ num )
= ( ^ [M5: num,N5: num] : ( num_of_nat @ ( plus_plus @ nat @ ( nat_of_num @ M5 ) @ ( nat_of_num @ N5 ) ) ) ) ) ).
% plus_num_def
thf(fact_7858_arg__max__nat__le,axiom,
! [A: $tType,P: A > $o,X: A,F2: A > nat,B2: nat] :
( ( P @ X )
=> ( ! [Y3: A] :
( ( P @ Y3 )
=> ( ord_less @ nat @ ( F2 @ Y3 ) @ B2 ) )
=> ( ord_less_eq @ nat @ ( F2 @ X ) @ ( F2 @ ( lattices_ord_arg_max @ A @ nat @ F2 @ P ) ) ) ) ) ).
% arg_max_nat_le
thf(fact_7859_real__floor__code,axiom,
! [X: rat] :
( ( archim6421214686448440834_floor @ real @ ( ratreal @ X ) )
= ( archim6421214686448440834_floor @ rat @ X ) ) ).
% real_floor_code
thf(fact_7860_prod_H__def,axiom,
! [C: $tType,A: $tType] :
( ( comm_monoid_mult @ A )
=> ( ( groups1962203154675924110t_prod @ C @ A )
= ( groups_comm_monoid_G @ A @ C @ ( times_times @ A ) @ ( one_one @ A ) ) ) ) ).
% prod'_def
thf(fact_7861_real__minus__code,axiom,
! [X: rat,Y2: rat] :
( ( minus_minus @ real @ ( ratreal @ X ) @ ( ratreal @ Y2 ) )
= ( ratreal @ ( minus_minus @ rat @ X @ Y2 ) ) ) ).
% real_minus_code
thf(fact_7862_real__inverse__code,axiom,
! [X: rat] :
( ( inverse_inverse @ real @ ( ratreal @ X ) )
= ( ratreal @ ( inverse_inverse @ rat @ X ) ) ) ).
% real_inverse_code
thf(fact_7863_real__plus__code,axiom,
! [X: rat,Y2: rat] :
( ( plus_plus @ real @ ( ratreal @ X ) @ ( ratreal @ Y2 ) )
= ( ratreal @ ( plus_plus @ rat @ X @ Y2 ) ) ) ).
% real_plus_code
thf(fact_7864_real__times__code,axiom,
! [X: rat,Y2: rat] :
( ( times_times @ real @ ( ratreal @ X ) @ ( ratreal @ Y2 ) )
= ( ratreal @ ( times_times @ rat @ X @ Y2 ) ) ) ).
% real_times_code
thf(fact_7865_one__real__code,axiom,
( ( one_one @ real )
= ( ratreal @ ( one_one @ rat ) ) ) ).
% one_real_code
thf(fact_7866_real__uminus__code,axiom,
! [X: rat] :
( ( uminus_uminus @ real @ ( ratreal @ X ) )
= ( ratreal @ ( uminus_uminus @ rat @ X ) ) ) ).
% real_uminus_code
thf(fact_7867_Ratreal__def,axiom,
( ratreal
= ( field_char_0_of_rat @ real ) ) ).
% Ratreal_def
thf(fact_7868_real__less__code,axiom,
! [X: rat,Y2: rat] :
( ( ord_less @ real @ ( ratreal @ X ) @ ( ratreal @ Y2 ) )
= ( ord_less @ rat @ X @ Y2 ) ) ).
% real_less_code
thf(fact_7869_zero__real__code,axiom,
( ( zero_zero @ real )
= ( ratreal @ ( zero_zero @ rat ) ) ) ).
% zero_real_code
thf(fact_7870_real__less__eq__code,axiom,
! [X: rat,Y2: rat] :
( ( ord_less_eq @ real @ ( ratreal @ X ) @ ( ratreal @ Y2 ) )
= ( ord_less_eq @ rat @ X @ Y2 ) ) ).
% real_less_eq_code
thf(fact_7871_real__divide__code,axiom,
! [X: rat,Y2: rat] :
( ( divide_divide @ real @ ( ratreal @ X ) @ ( ratreal @ Y2 ) )
= ( ratreal @ ( divide_divide @ rat @ X @ Y2 ) ) ) ).
% real_divide_code
thf(fact_7872_sum_H__def,axiom,
! [C: $tType,A: $tType] :
( ( comm_monoid_add @ A )
=> ( ( groups1027152243600224163dd_sum @ C @ A )
= ( groups_comm_monoid_G @ A @ C @ ( plus_plus @ A ) @ ( zero_zero @ A ) ) ) ) ).
% sum'_def
thf(fact_7873_fold__atLeastAtMost__nat_Opsimps,axiom,
! [A: $tType,F2: nat > A > A,A3: nat,B2: nat,Acc2: A] :
( ( accp @ ( product_prod @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) ) @ ( set_fo1817059534552279752at_rel @ A ) @ ( product_Pair @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) @ F2 @ ( product_Pair @ nat @ ( product_prod @ nat @ A ) @ A3 @ ( product_Pair @ nat @ A @ B2 @ Acc2 ) ) ) )
=> ( ( ( ord_less @ nat @ B2 @ A3 )
=> ( ( set_fo6178422350223883121st_nat @ A @ F2 @ A3 @ B2 @ Acc2 )
= Acc2 ) )
& ( ~ ( ord_less @ nat @ B2 @ A3 )
=> ( ( set_fo6178422350223883121st_nat @ A @ F2 @ A3 @ B2 @ Acc2 )
= ( set_fo6178422350223883121st_nat @ A @ F2 @ ( plus_plus @ nat @ A3 @ ( one_one @ nat ) ) @ B2 @ ( F2 @ A3 @ Acc2 ) ) ) ) ) ) ).
% fold_atLeastAtMost_nat.psimps
thf(fact_7874_fold__atLeastAtMost__nat_Opelims,axiom,
! [A: $tType,X: nat > A > A,Xa: nat,Xb3: nat,Xc: A,Y2: A] :
( ( ( set_fo6178422350223883121st_nat @ A @ X @ Xa @ Xb3 @ 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 @ Xb3 @ Xc ) ) ) )
=> ~ ( ( ( ( ord_less @ nat @ Xb3 @ Xa )
=> ( Y2 = Xc ) )
& ( ~ ( ord_less @ nat @ Xb3 @ Xa )
=> ( Y2
= ( set_fo6178422350223883121st_nat @ A @ X @ ( plus_plus @ nat @ Xa @ ( one_one @ nat ) ) @ Xb3 @ ( 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 @ Xb3 @ Xc ) ) ) ) ) ) ) ).
% fold_atLeastAtMost_nat.pelims
thf(fact_7875_fold__atLeastAtMost__nat_Opinduct,axiom,
! [A: $tType,A0: nat > A > A,A1: nat,A22: nat,A33: A,P: ( nat > A > A ) > nat > nat > A > $o] :
( ( accp @ ( product_prod @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) ) @ ( set_fo1817059534552279752at_rel @ A ) @ ( product_Pair @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) @ A0 @ ( product_Pair @ nat @ ( product_prod @ nat @ A ) @ A1 @ ( product_Pair @ nat @ A @ A22 @ A33 ) ) ) )
=> ( ! [F3: nat > A > A,A6: nat,B4: nat,Acc3: A] :
( ( accp @ ( product_prod @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) ) @ ( set_fo1817059534552279752at_rel @ A ) @ ( product_Pair @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) @ F3 @ ( product_Pair @ nat @ ( product_prod @ nat @ A ) @ A6 @ ( product_Pair @ nat @ A @ B4 @ Acc3 ) ) ) )
=> ( ( ~ ( ord_less @ nat @ B4 @ A6 )
=> ( P @ F3 @ ( plus_plus @ nat @ A6 @ ( one_one @ nat ) ) @ B4 @ ( F3 @ A6 @ Acc3 ) ) )
=> ( P @ F3 @ A6 @ B4 @ Acc3 ) ) )
=> ( P @ A0 @ A1 @ A22 @ A33 ) ) ) ).
% fold_atLeastAtMost_nat.pinduct
thf(fact_7876_prod_Ocomm__monoid__list__set__axioms,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ( groups4802862169904069756st_set @ A @ ( times_times @ A ) @ ( one_one @ A ) ) ) ).
% prod.comm_monoid_list_set_axioms
thf(fact_7877_sum_Ocomm__monoid__list__set__axioms,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ( groups4802862169904069756st_set @ A @ ( plus_plus @ A ) @ ( zero_zero @ A ) ) ) ).
% sum.comm_monoid_list_set_axioms
thf(fact_7878_natLeq__on__wo__rel,axiom,
! [N: nat] :
( bNF_Wellorder_wo_rel @ nat
@ ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [X2: nat,Y5: nat] :
( ( ord_less @ nat @ X2 @ N )
& ( ord_less @ nat @ Y5 @ N )
& ( ord_less_eq @ nat @ X2 @ Y5 ) ) ) ) ) ).
% natLeq_on_wo_rel
thf(fact_7879_max_Osemilattice__order__axioms,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( semilattice_order @ A @ ( ord_max @ A )
@ ^ [X2: A,Y5: A] : ( ord_less_eq @ A @ Y5 @ X2 )
@ ^ [X2: A,Y5: A] : ( ord_less @ A @ Y5 @ X2 ) ) ) ).
% max.semilattice_order_axioms
thf(fact_7880_semilattice__neutr__order__def,axiom,
! [A: $tType] :
( ( semila1105856199041335345_order @ A )
= ( ^ [F4: A > A > A,Z6: A,Less_eq2: A > A > $o,Less2: A > A > $o] :
( ( semilattice_neutr @ A @ F4 @ Z6 )
& ( semilattice_order @ A @ F4 @ Less_eq2 @ Less2 ) ) ) ) ).
% semilattice_neutr_order_def
thf(fact_7881_semilattice__neutr__order_Ointro,axiom,
! [A: $tType,F2: A > A > A,Z2: A,Less_eq: A > A > $o,Less: A > A > $o] :
( ( semilattice_neutr @ A @ F2 @ Z2 )
=> ( ( semilattice_order @ A @ F2 @ Less_eq @ Less )
=> ( semila1105856199041335345_order @ A @ F2 @ Z2 @ Less_eq @ Less ) ) ) ).
% semilattice_neutr_order.intro
thf(fact_7882_semilattice__order_Omono,axiom,
! [A: $tType,F2: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A3: A,C2: A,B2: A,D2: A] :
( ( semilattice_order @ A @ F2 @ Less_eq @ Less )
=> ( ( Less_eq @ A3 @ C2 )
=> ( ( Less_eq @ B2 @ D2 )
=> ( Less_eq @ ( F2 @ A3 @ B2 ) @ ( F2 @ C2 @ D2 ) ) ) ) ) ).
% semilattice_order.mono
thf(fact_7883_semilattice__order_OorderE,axiom,
! [A: $tType,F2: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B2: A] :
( ( semilattice_order @ A @ F2 @ Less_eq @ Less )
=> ( ( Less_eq @ A3 @ B2 )
=> ( A3
= ( F2 @ A3 @ B2 ) ) ) ) ).
% semilattice_order.orderE
thf(fact_7884_semilattice__order_OorderI,axiom,
! [A: $tType,F2: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B2: A] :
( ( semilattice_order @ A @ F2 @ Less_eq @ Less )
=> ( ( A3
= ( F2 @ A3 @ B2 ) )
=> ( Less_eq @ A3 @ B2 ) ) ) ).
% semilattice_order.orderI
thf(fact_7885_semilattice__order_Oabsorb1,axiom,
! [A: $tType,F2: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B2: A] :
( ( semilattice_order @ A @ F2 @ Less_eq @ Less )
=> ( ( Less_eq @ A3 @ B2 )
=> ( ( F2 @ A3 @ B2 )
= A3 ) ) ) ).
% semilattice_order.absorb1
thf(fact_7886_semilattice__order_Oabsorb2,axiom,
! [A: $tType,F2: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,B2: A,A3: A] :
( ( semilattice_order @ A @ F2 @ Less_eq @ Less )
=> ( ( Less_eq @ B2 @ A3 )
=> ( ( F2 @ A3 @ B2 )
= B2 ) ) ) ).
% semilattice_order.absorb2
thf(fact_7887_semilattice__order_Oabsorb3,axiom,
! [A: $tType,F2: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B2: A] :
( ( semilattice_order @ A @ F2 @ Less_eq @ Less )
=> ( ( Less @ A3 @ B2 )
=> ( ( F2 @ A3 @ B2 )
= A3 ) ) ) ).
% semilattice_order.absorb3
thf(fact_7888_semilattice__order_Oabsorb4,axiom,
! [A: $tType,F2: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,B2: A,A3: A] :
( ( semilattice_order @ A @ F2 @ Less_eq @ Less )
=> ( ( Less @ B2 @ A3 )
=> ( ( F2 @ A3 @ B2 )
= B2 ) ) ) ).
% semilattice_order.absorb4
thf(fact_7889_semilattice__order_OboundedE,axiom,
! [A: $tType,F2: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B2: A,C2: A] :
( ( semilattice_order @ A @ F2 @ Less_eq @ Less )
=> ( ( Less_eq @ A3 @ ( F2 @ B2 @ C2 ) )
=> ~ ( ( Less_eq @ A3 @ B2 )
=> ~ ( Less_eq @ A3 @ C2 ) ) ) ) ).
% semilattice_order.boundedE
thf(fact_7890_semilattice__order_OboundedI,axiom,
! [A: $tType,F2: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B2: A,C2: A] :
( ( semilattice_order @ A @ F2 @ Less_eq @ Less )
=> ( ( Less_eq @ A3 @ B2 )
=> ( ( Less_eq @ A3 @ C2 )
=> ( Less_eq @ A3 @ ( F2 @ B2 @ C2 ) ) ) ) ) ).
% semilattice_order.boundedI
thf(fact_7891_semilattice__order_Oorder__iff,axiom,
! [A: $tType,F2: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B2: A] :
( ( semilattice_order @ A @ F2 @ Less_eq @ Less )
=> ( ( Less_eq @ A3 @ B2 )
= ( A3
= ( F2 @ A3 @ B2 ) ) ) ) ).
% semilattice_order.order_iff
thf(fact_7892_semilattice__order_Ocobounded1,axiom,
! [A: $tType,F2: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B2: A] :
( ( semilattice_order @ A @ F2 @ Less_eq @ Less )
=> ( Less_eq @ ( F2 @ A3 @ B2 ) @ A3 ) ) ).
% semilattice_order.cobounded1
thf(fact_7893_semilattice__order_Ocobounded2,axiom,
! [A: $tType,F2: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B2: A] :
( ( semilattice_order @ A @ F2 @ Less_eq @ Less )
=> ( Less_eq @ ( F2 @ A3 @ B2 ) @ B2 ) ) ).
% semilattice_order.cobounded2
thf(fact_7894_semilattice__order_Oabsorb__iff1,axiom,
! [A: $tType,F2: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B2: A] :
( ( semilattice_order @ A @ F2 @ Less_eq @ Less )
=> ( ( Less_eq @ A3 @ B2 )
= ( ( F2 @ A3 @ B2 )
= A3 ) ) ) ).
% semilattice_order.absorb_iff1
thf(fact_7895_semilattice__order_Oabsorb__iff2,axiom,
! [A: $tType,F2: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,B2: A,A3: A] :
( ( semilattice_order @ A @ F2 @ Less_eq @ Less )
=> ( ( Less_eq @ B2 @ A3 )
= ( ( F2 @ A3 @ B2 )
= B2 ) ) ) ).
% semilattice_order.absorb_iff2
thf(fact_7896_semilattice__order_Obounded__iff,axiom,
! [A: $tType,F2: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B2: A,C2: A] :
( ( semilattice_order @ A @ F2 @ Less_eq @ Less )
=> ( ( Less_eq @ A3 @ ( F2 @ B2 @ C2 ) )
= ( ( Less_eq @ A3 @ B2 )
& ( Less_eq @ A3 @ C2 ) ) ) ) ).
% semilattice_order.bounded_iff
thf(fact_7897_semilattice__order_OcoboundedI1,axiom,
! [A: $tType,F2: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A3: A,C2: A,B2: A] :
( ( semilattice_order @ A @ F2 @ Less_eq @ Less )
=> ( ( Less_eq @ A3 @ C2 )
=> ( Less_eq @ ( F2 @ A3 @ B2 ) @ C2 ) ) ) ).
% semilattice_order.coboundedI1
thf(fact_7898_semilattice__order_OcoboundedI2,axiom,
! [A: $tType,F2: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,B2: A,C2: A,A3: A] :
( ( semilattice_order @ A @ F2 @ Less_eq @ Less )
=> ( ( Less_eq @ B2 @ C2 )
=> ( Less_eq @ ( F2 @ A3 @ B2 ) @ C2 ) ) ) ).
% semilattice_order.coboundedI2
thf(fact_7899_semilattice__order_Ostrict__boundedE,axiom,
! [A: $tType,F2: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B2: A,C2: A] :
( ( semilattice_order @ A @ F2 @ Less_eq @ Less )
=> ( ( Less @ A3 @ ( F2 @ B2 @ C2 ) )
=> ~ ( ( Less @ A3 @ B2 )
=> ~ ( Less @ A3 @ C2 ) ) ) ) ).
% semilattice_order.strict_boundedE
thf(fact_7900_semilattice__order_Ostrict__order__iff,axiom,
! [A: $tType,F2: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A3: A,B2: A] :
( ( semilattice_order @ A @ F2 @ Less_eq @ Less )
=> ( ( Less @ A3 @ B2 )
= ( ( A3
= ( F2 @ A3 @ B2 ) )
& ( A3 != B2 ) ) ) ) ).
% semilattice_order.strict_order_iff
thf(fact_7901_semilattice__order_Ostrict__coboundedI1,axiom,
! [A: $tType,F2: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,A3: A,C2: A,B2: A] :
( ( semilattice_order @ A @ F2 @ Less_eq @ Less )
=> ( ( Less @ A3 @ C2 )
=> ( Less @ ( F2 @ A3 @ B2 ) @ C2 ) ) ) ).
% semilattice_order.strict_coboundedI1
thf(fact_7902_semilattice__order_Ostrict__coboundedI2,axiom,
! [A: $tType,F2: A > A > A,Less_eq: A > A > $o,Less: A > A > $o,B2: A,C2: A,A3: A] :
( ( semilattice_order @ A @ F2 @ Less_eq @ Less )
=> ( ( Less @ B2 @ C2 )
=> ( Less @ ( F2 @ A3 @ B2 ) @ C2 ) ) ) ).
% semilattice_order.strict_coboundedI2
thf(fact_7903_semilattice__neutr__order_Oaxioms_I2_J,axiom,
! [A: $tType,F2: A > A > A,Z2: A,Less_eq: A > A > $o,Less: A > A > $o] :
( ( semila1105856199041335345_order @ A @ F2 @ Z2 @ Less_eq @ Less )
=> ( semilattice_order @ A @ F2 @ Less_eq @ Less ) ) ).
% semilattice_neutr_order.axioms(2)
thf(fact_7904_inf_Osemilattice__order__axioms,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( semilattice_order @ A @ ( inf_inf @ A ) @ ( ord_less_eq @ A ) @ ( ord_less @ A ) ) ) ).
% inf.semilattice_order_axioms
thf(fact_7905_min_Osemilattice__order__axioms,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( semilattice_order @ A @ ( ord_min @ A ) @ ( ord_less_eq @ A ) @ ( ord_less @ A ) ) ) ).
% min.semilattice_order_axioms
thf(fact_7906_sup_Osemilattice__order__axioms,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( semilattice_order @ A @ ( sup_sup @ A )
@ ^ [X2: A,Y5: A] : ( ord_less_eq @ A @ Y5 @ X2 )
@ ^ [X2: A,Y5: A] : ( ord_less @ A @ Y5 @ X2 ) ) ) ).
% sup.semilattice_order_axioms
thf(fact_7907_lexordp__eq_Osimps,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( ord_lexordp_eq @ A )
= ( ^ [A12: list @ A,A23: list @ A] :
( ? [Ys3: list @ A] :
( ( A12
= ( nil @ A ) )
& ( A23 = Ys3 ) )
| ? [X2: A,Y5: A,Xs3: list @ A,Ys3: list @ A] :
( ( A12
= ( cons @ A @ X2 @ Xs3 ) )
& ( A23
= ( cons @ A @ Y5 @ Ys3 ) )
& ( ord_less @ A @ X2 @ Y5 ) )
| ? [X2: A,Y5: A,Xs3: list @ A,Ys3: list @ A] :
( ( A12
= ( cons @ A @ X2 @ Xs3 ) )
& ( A23
= ( cons @ A @ Y5 @ Ys3 ) )
& ~ ( ord_less @ A @ X2 @ Y5 )
& ~ ( ord_less @ A @ Y5 @ X2 )
& ( ord_lexordp_eq @ A @ Xs3 @ Ys3 ) ) ) ) ) ) ).
% lexordp_eq.simps
thf(fact_7908_lexordp__eq__simps_I4_J,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X: A,Xs: list @ A,Y2: A,Ys2: list @ A] :
( ( ord_lexordp_eq @ A @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y2 @ Ys2 ) )
= ( ( ord_less @ A @ X @ Y2 )
| ( ~ ( ord_less @ A @ Y2 @ X )
& ( ord_lexordp_eq @ A @ Xs @ Ys2 ) ) ) ) ) ).
% lexordp_eq_simps(4)
thf(fact_7909_lexordp__eq_OCons__eq,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X: A,Y2: A,Xs: list @ A,Ys2: list @ A] :
( ~ ( ord_less @ A @ X @ Y2 )
=> ( ~ ( ord_less @ A @ Y2 @ X )
=> ( ( ord_lexordp_eq @ A @ Xs @ Ys2 )
=> ( ord_lexordp_eq @ A @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y2 @ Ys2 ) ) ) ) ) ) ).
% lexordp_eq.Cons_eq
thf(fact_7910_lexordp__eq_OCons,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X: A,Y2: A,Xs: list @ A,Ys2: list @ A] :
( ( ord_less @ A @ X @ Y2 )
=> ( ord_lexordp_eq @ A @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y2 @ Ys2 ) ) ) ) ).
% lexordp_eq.Cons
thf(fact_7911_lexordp__eq_Ocases,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A1: list @ A,A22: list @ A] :
( ( ord_lexordp_eq @ A @ A1 @ A22 )
=> ( ( A1
!= ( nil @ A ) )
=> ( ! [X3: A] :
( ? [Xs2: list @ A] :
( A1
= ( cons @ A @ X3 @ Xs2 ) )
=> ! [Y3: A] :
( ? [Ys6: list @ A] :
( A22
= ( cons @ A @ Y3 @ Ys6 ) )
=> ~ ( ord_less @ A @ X3 @ Y3 ) ) )
=> ~ ! [X3: A,Y3: A,Xs2: list @ A] :
( ( A1
= ( cons @ A @ X3 @ Xs2 ) )
=> ! [Ys6: list @ A] :
( ( A22
= ( cons @ A @ Y3 @ Ys6 ) )
=> ( ~ ( ord_less @ A @ X3 @ Y3 )
=> ( ~ ( ord_less @ A @ Y3 @ X3 )
=> ~ ( ord_lexordp_eq @ A @ Xs2 @ Ys6 ) ) ) ) ) ) ) ) ) ).
% lexordp_eq.cases
thf(fact_7912_those_Osimps_I2_J,axiom,
! [A: $tType,X: option @ A,Xs: list @ ( option @ A )] :
( ( those @ A @ ( cons @ ( option @ A ) @ X @ Xs ) )
= ( case_option @ ( option @ ( list @ A ) ) @ A @ ( none @ ( list @ A ) )
@ ^ [Y5: A] : ( map_option @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ Y5 ) @ ( those @ A @ Xs ) )
@ X ) ) ).
% those.simps(2)
thf(fact_7913_natLeq__natLess__Id,axiom,
( bNF_Ca8459412986667044542atLess
= ( minus_minus @ ( set @ ( product_prod @ nat @ nat ) ) @ bNF_Ca8665028551170535155natLeq @ ( id2 @ nat ) ) ) ).
% natLeq_natLess_Id
thf(fact_7914_relpow_Osimps_I1_J,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( zero_zero @ nat ) @ R2 )
= ( id2 @ A ) ) ).
% relpow.simps(1)
thf(fact_7915_wo__rel_Ocases__Total3,axiom,
! [A: $tType,R4: set @ ( product_prod @ A @ A ),A3: A,B2: A,Phi: A > A > $o] :
( ( bNF_Wellorder_wo_rel @ A @ R4 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ A3 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( field2 @ A @ R4 ) )
=> ( ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R4 @ ( id2 @ A ) ) )
| ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ A3 ) @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R4 @ ( id2 @ A ) ) ) )
=> ( Phi @ A3 @ B2 ) )
=> ( ( ( A3 = B2 )
=> ( Phi @ A3 @ B2 ) )
=> ( Phi @ A3 @ B2 ) ) ) ) ) ).
% wo_rel.cases_Total3
thf(fact_7916_ntrancl__Suc,axiom,
! [A: $tType,N: nat,R2: set @ ( product_prod @ A @ A )] :
( ( transitive_ntrancl @ A @ ( suc @ N ) @ R2 )
= ( relcomp @ A @ A @ A @ ( transitive_ntrancl @ A @ N @ R2 ) @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ ( id2 @ A ) @ R2 ) ) ) ).
% ntrancl_Suc
thf(fact_7917_relpow_Osimps_I2_J,axiom,
! [A: $tType,N: nat,R2: set @ ( product_prod @ A @ A )] :
( ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( suc @ N ) @ R2 )
= ( relcomp @ A @ A @ A @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R2 ) @ R2 ) ) ).
% relpow.simps(2)
thf(fact_7918_Field__natLeq__on,axiom,
! [N: nat] :
( ( field2 @ nat
@ ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [X2: nat,Y5: nat] :
( ( ord_less @ nat @ X2 @ N )
& ( ord_less @ nat @ Y5 @ N )
& ( ord_less_eq @ nat @ X2 @ Y5 ) ) ) ) )
= ( collect @ nat
@ ^ [X2: nat] : ( ord_less @ nat @ X2 @ N ) ) ) ).
% Field_natLeq_on
thf(fact_7919_bsqr__def,axiom,
! [A: $tType] :
( ( bNF_Wellorder_bsqr @ A )
= ( ^ [R: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ ( product_prod @ A @ A ) @ ( product_prod @ A @ A ) )
@ ( product_case_prod @ ( product_prod @ A @ A ) @ ( product_prod @ A @ A ) @ $o
@ ( product_case_prod @ A @ A @ ( ( product_prod @ A @ A ) > $o )
@ ^ [A12: A,A23: A] :
( product_case_prod @ A @ A @ $o
@ ^ [B1: A,B23: A] :
( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ A12 @ ( insert @ A @ A23 @ ( insert @ A @ B1 @ ( insert @ A @ B23 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) @ ( field2 @ A @ R ) )
& ( ( ( A12 = B1 )
& ( A23 = B23 ) )
| ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( bNF_We1388413361240627857o_max2 @ A @ R @ A12 @ A23 ) @ ( bNF_We1388413361240627857o_max2 @ A @ R @ B1 @ B23 ) ) @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R @ ( id2 @ A ) ) )
| ( ( ( bNF_We1388413361240627857o_max2 @ A @ R @ A12 @ A23 )
= ( bNF_We1388413361240627857o_max2 @ A @ R @ B1 @ B23 ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A12 @ B1 ) @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R @ ( id2 @ A ) ) ) )
| ( ( ( bNF_We1388413361240627857o_max2 @ A @ R @ A12 @ A23 )
= ( bNF_We1388413361240627857o_max2 @ A @ R @ B1 @ B23 ) )
& ( A12 = B1 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A23 @ B23 ) @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R @ ( id2 @ A ) ) ) ) ) ) ) ) ) ) ) ) ).
% bsqr_def
thf(fact_7920_Linear__order__in__diff__Id,axiom,
! [A: $tType,R4: set @ ( product_prod @ A @ A ),A3: A,B2: A] :
( ( order_679001287576687338der_on @ A @ ( field2 @ A @ R4 ) @ R4 )
=> ( ( member @ A @ A3 @ ( field2 @ A @ R4 ) )
=> ( ( member @ A @ B2 @ ( field2 @ A @ R4 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A3 @ B2 ) @ R4 )
= ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ A3 ) @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R4 @ ( id2 @ A ) ) ) ) ) ) ) ) ).
% Linear_order_in_diff_Id
thf(fact_7921_strict__linear__order__on__diff__Id,axiom,
! [A: $tType,A4: set @ A,R4: set @ ( product_prod @ A @ A )] :
( ( order_679001287576687338der_on @ A @ A4 @ R4 )
=> ( order_5396836661320670305der_on @ A @ A4 @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R4 @ ( id2 @ A ) ) ) ) ).
% strict_linear_order_on_diff_Id
thf(fact_7922_Linear__order__wf__diff__Id,axiom,
! [A: $tType,R4: set @ ( product_prod @ A @ A )] :
( ( order_679001287576687338der_on @ A @ ( field2 @ A @ R4 ) @ R4 )
=> ( ( wf @ A @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R4 @ ( id2 @ A ) ) )
= ( ! [A7: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A7 @ ( field2 @ A @ R4 ) )
=> ( ( A7
!= ( bot_bot @ ( set @ A ) ) )
=> ? [X2: A] :
( ( member @ A @ X2 @ A7 )
& ! [Y5: A] :
( ( member @ A @ Y5 @ A7 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X2 @ Y5 ) @ R4 ) ) ) ) ) ) ) ) ).
% Linear_order_wf_diff_Id
thf(fact_7923_wf,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ( wf @ A @ ( collect @ ( product_prod @ A @ A ) @ ( product_case_prod @ A @ A @ $o @ ( ord_less @ A ) ) ) ) ) ).
% wf
thf(fact_7924_wf__if__measure,axiom,
! [A: $tType,P: A > $o,F2: A > nat,G: A > A] :
( ! [X3: A] :
( ( P @ X3 )
=> ( ord_less @ nat @ ( F2 @ ( G @ X3 ) ) @ ( F2 @ X3 ) ) )
=> ( wf @ A
@ ( collect @ ( product_prod @ A @ A )
@ ( product_case_prod @ A @ A @ $o
@ ^ [Y5: A,X2: A] :
( ( P @ X2 )
& ( Y5
= ( G @ X2 ) ) ) ) ) ) ) ).
% wf_if_measure
thf(fact_7925_wf__less,axiom,
wf @ nat @ ( collect @ ( product_prod @ nat @ nat ) @ ( product_case_prod @ nat @ nat @ $o @ ( ord_less @ nat ) ) ) ).
% wf_less
thf(fact_7926_wf__bounded__measure,axiom,
! [A: $tType,R4: set @ ( product_prod @ A @ A ),Ub: A > nat,F2: A > nat] :
( ! [A6: A,B4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B4 @ A6 ) @ R4 )
=> ( ( ord_less_eq @ nat @ ( Ub @ B4 ) @ ( Ub @ A6 ) )
& ( ord_less_eq @ nat @ ( F2 @ B4 ) @ ( Ub @ A6 ) )
& ( ord_less @ nat @ ( F2 @ A6 ) @ ( F2 @ B4 ) ) ) )
=> ( wf @ A @ R4 ) ) ).
% wf_bounded_measure
thf(fact_7927_wf__iff__no__infinite__down__chain,axiom,
! [A: $tType] :
( ( wf @ A )
= ( ^ [R: set @ ( product_prod @ A @ A )] :
~ ? [F4: nat > A] :
! [I2: nat] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( F4 @ ( suc @ I2 ) ) @ ( F4 @ I2 ) ) @ R ) ) ) ).
% wf_iff_no_infinite_down_chain
thf(fact_7928_wf__no__infinite__down__chainE,axiom,
! [A: $tType,R4: set @ ( product_prod @ A @ A ),F2: nat > A] :
( ( wf @ A @ R4 )
=> ~ ! [K2: nat] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( F2 @ ( suc @ K2 ) ) @ ( F2 @ K2 ) ) @ R4 ) ) ).
% wf_no_infinite_down_chainE
thf(fact_7929_wf__int__ge__less__than,axiom,
! [D2: int] : ( wf @ int @ ( int_ge_less_than @ D2 ) ) ).
% wf_int_ge_less_than
thf(fact_7930_wf__int__ge__less__than2,axiom,
! [D2: int] : ( wf @ int @ ( int_ge_less_than2 @ D2 ) ) ).
% wf_int_ge_less_than2
thf(fact_7931_wo__rel_OWF,axiom,
! [A: $tType,R4: set @ ( product_prod @ A @ A )] :
( ( bNF_Wellorder_wo_rel @ A @ R4 )
=> ( wf @ A @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R4 @ ( id2 @ A ) ) ) ) ).
% wo_rel.WF
thf(fact_7932_partial__order__on__well__order__on,axiom,
! [A: $tType,R4: set @ ( product_prod @ A @ A ),A4: set @ A] :
( ( finite_finite @ ( product_prod @ A @ A ) @ R4 )
=> ( ( order_7125193373082350890der_on @ A @ A4 @ R4 )
=> ( wf @ A @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R4 @ ( id2 @ A ) ) ) ) ) ).
% partial_order_on_well_order_on
thf(fact_7933_dir__image__minus__Id,axiom,
! [B: $tType,A: $tType,F2: A > B,R4: set @ ( product_prod @ A @ A )] :
( ( inj_on @ A @ B @ F2 @ ( field2 @ A @ R4 ) )
=> ( ( minus_minus @ ( set @ ( product_prod @ B @ B ) ) @ ( bNF_We2720479622203943262_image @ A @ B @ R4 @ F2 ) @ ( id2 @ B ) )
= ( bNF_We2720479622203943262_image @ A @ B @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R4 @ ( id2 @ A ) ) @ F2 ) ) ) ).
% dir_image_minus_Id
thf(fact_7934_Total__Id__Field,axiom,
! [A: $tType,R4: set @ ( product_prod @ A @ A )] :
( ( total_on @ A @ ( field2 @ A @ R4 ) @ R4 )
=> ( ~ ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R4 @ ( id2 @ A ) )
=> ( ( field2 @ A @ R4 )
= ( field2 @ A @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R4 @ ( id2 @ A ) ) ) ) ) ) ).
% Total_Id_Field
thf(fact_7935_partial__order__on__acyclic,axiom,
! [A: $tType,A4: set @ A,R4: set @ ( product_prod @ A @ A )] :
( ( order_7125193373082350890der_on @ A @ A4 @ R4 )
=> ( transitive_acyclic @ A @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R4 @ ( id2 @ A ) ) ) ) ).
% partial_order_on_acyclic
thf(fact_7936_total__on__diff__Id,axiom,
! [A: $tType,A4: set @ A,R4: set @ ( product_prod @ A @ A )] :
( ( total_on @ A @ A4 @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R4 @ ( id2 @ A ) ) )
= ( total_on @ A @ A4 @ R4 ) ) ).
% total_on_diff_Id
thf(fact_7937_acyclicI__order,axiom,
! [A: $tType,B: $tType] :
( ( preorder @ A )
=> ! [R4: set @ ( product_prod @ B @ B ),F2: B > A] :
( ! [A6: B,B4: B] :
( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ A6 @ B4 ) @ R4 )
=> ( ord_less @ A @ ( F2 @ B4 ) @ ( F2 @ A6 ) ) )
=> ( transitive_acyclic @ B @ R4 ) ) ) ).
% acyclicI_order
thf(fact_7938_linear__order__on__acyclic,axiom,
! [A: $tType,A4: set @ A,R4: set @ ( product_prod @ A @ A )] :
( ( order_679001287576687338der_on @ A @ A4 @ R4 )
=> ( transitive_acyclic @ A @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R4 @ ( id2 @ A ) ) ) ) ).
% linear_order_on_acyclic
thf(fact_7939_natLeq__on__Well__order,axiom,
! [N: nat] :
( order_well_order_on @ nat
@ ( field2 @ nat
@ ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [X2: nat,Y5: nat] :
( ( ord_less @ nat @ X2 @ N )
& ( ord_less @ nat @ Y5 @ N )
& ( ord_less_eq @ nat @ X2 @ Y5 ) ) ) ) )
@ ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [X2: nat,Y5: nat] :
( ( ord_less @ nat @ X2 @ N )
& ( ord_less @ nat @ Y5 @ N )
& ( ord_less_eq @ nat @ X2 @ Y5 ) ) ) ) ) ).
% natLeq_on_Well_order
thf(fact_7940_well__order__on__def,axiom,
! [A: $tType] :
( ( order_well_order_on @ A )
= ( ^ [A7: set @ A,R: set @ ( product_prod @ A @ A )] :
( ( order_679001287576687338der_on @ A @ A7 @ R )
& ( wf @ A @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R @ ( id2 @ A ) ) ) ) ) ) ).
% well_order_on_def
thf(fact_7941_natLeq__on__well__order__on,axiom,
! [N: nat] :
( order_well_order_on @ nat
@ ( collect @ nat
@ ^ [X2: nat] : ( ord_less @ nat @ X2 @ N ) )
@ ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [X2: nat,Y5: nat] :
( ( ord_less @ nat @ X2 @ N )
& ( ord_less @ nat @ Y5 @ N )
& ( ord_less_eq @ nat @ X2 @ Y5 ) ) ) ) ) ).
% natLeq_on_well_order_on
thf(fact_7942_irrefl__diff__Id,axiom,
! [A: $tType,R4: set @ ( product_prod @ A @ A )] : ( irrefl @ A @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R4 @ ( id2 @ A ) ) ) ).
% irrefl_diff_Id
thf(fact_7943_Range__Diff__subset,axiom,
! [A: $tType,B: $tType,A4: set @ ( product_prod @ B @ A ),B7: set @ ( product_prod @ B @ A )] : ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ ( range2 @ B @ A @ A4 ) @ ( range2 @ B @ A @ B7 ) ) @ ( range2 @ B @ A @ ( minus_minus @ ( set @ ( product_prod @ B @ A ) ) @ A4 @ B7 ) ) ) ).
% Range_Diff_subset
thf(fact_7944_bdd__below_Opreordering__bdd__axioms,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( condit622319405099724424ng_bdd @ A
@ ^ [X2: A,Y5: A] : ( ord_less_eq @ A @ Y5 @ X2 )
@ ^ [X2: A,Y5: A] : ( ord_less @ A @ Y5 @ X2 ) ) ) ).
% bdd_below.preordering_bdd_axioms
thf(fact_7945_tendsto__Zfun__iff,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F2: A > B,A3: B,F5: filter @ A] :
( ( filterlim @ A @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ A3 ) @ F5 )
= ( zfun @ A @ B
@ ^ [X2: A] : ( minus_minus @ B @ ( F2 @ X2 ) @ A3 )
@ F5 ) ) ) ).
% tendsto_Zfun_iff
thf(fact_7946_bdd__above_Opreordering__bdd__axioms,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( condit622319405099724424ng_bdd @ A @ ( ord_less_eq @ A ) @ ( ord_less @ A ) ) ) ).
% bdd_above.preordering_bdd_axioms
thf(fact_7947_Zfun__zero,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F5: filter @ A] :
( zfun @ A @ B
@ ^ [X2: A] : ( zero_zero @ B )
@ F5 ) ) ).
% Zfun_zero
thf(fact_7948_Zfun__minus,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F2: A > B,F5: filter @ A] :
( ( zfun @ A @ B @ F2 @ F5 )
=> ( zfun @ A @ B
@ ^ [X2: A] : ( uminus_uminus @ B @ ( F2 @ X2 ) )
@ F5 ) ) ) ).
% Zfun_minus
thf(fact_7949_Zfun__diff,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F2: A > B,F5: filter @ A,G: A > B] :
( ( zfun @ A @ B @ F2 @ F5 )
=> ( ( zfun @ A @ B @ G @ F5 )
=> ( zfun @ A @ B
@ ^ [X2: A] : ( minus_minus @ B @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ F5 ) ) ) ) ).
% Zfun_diff
thf(fact_7950_natural__decr,axiom,
! [N: code_natural] :
( ( N
!= ( zero_zero @ code_natural ) )
=> ( ord_less @ nat @ ( minus_minus @ nat @ ( code_nat_of_natural @ N ) @ ( suc @ ( zero_zero @ nat ) ) ) @ ( code_nat_of_natural @ N ) ) ) ).
% natural_decr
thf(fact_7951_Domain__Diff__subset,axiom,
! [B: $tType,A: $tType,A4: set @ ( product_prod @ A @ B ),B7: set @ ( product_prod @ A @ B )] : ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ ( domain @ A @ B @ A4 ) @ ( domain @ A @ B @ B7 ) ) @ ( domain @ A @ B @ ( minus_minus @ ( set @ ( product_prod @ A @ B ) ) @ A4 @ B7 ) ) ) ).
% Domain_Diff_subset
thf(fact_7952_minus__natural_Orep__eq,axiom,
! [X: code_natural,Xa: code_natural] :
( ( code_nat_of_natural @ ( minus_minus @ code_natural @ X @ Xa ) )
= ( minus_minus @ nat @ ( code_nat_of_natural @ X ) @ ( code_nat_of_natural @ Xa ) ) ) ).
% minus_natural.rep_eq
thf(fact_7953_one__natural_Orep__eq,axiom,
( ( code_nat_of_natural @ ( one_one @ code_natural ) )
= ( one_one @ nat ) ) ).
% one_natural.rep_eq
thf(fact_7954_divide__natural_Orep__eq,axiom,
! [X: code_natural,Xa: code_natural] :
( ( code_nat_of_natural @ ( divide_divide @ code_natural @ X @ Xa ) )
= ( divide_divide @ nat @ ( code_nat_of_natural @ X ) @ ( code_nat_of_natural @ Xa ) ) ) ).
% divide_natural.rep_eq
thf(fact_7955_zero__natural_Orep__eq,axiom,
( ( code_nat_of_natural @ ( zero_zero @ code_natural ) )
= ( zero_zero @ nat ) ) ).
% zero_natural.rep_eq
thf(fact_7956_natural__zero__minus__one,axiom,
( ( minus_minus @ code_natural @ ( zero_zero @ code_natural ) @ ( one_one @ code_natural ) )
= ( zero_zero @ code_natural ) ) ).
% natural_zero_minus_one
thf(fact_7957_less__natural_Orep__eq,axiom,
( ( ord_less @ code_natural )
= ( ^ [X2: code_natural,Xa4: code_natural] : ( ord_less @ nat @ ( code_nat_of_natural @ X2 ) @ ( code_nat_of_natural @ Xa4 ) ) ) ) ).
% less_natural.rep_eq
thf(fact_7958_int__of__integer__of__natural,axiom,
! [N: code_natural] :
( ( code_int_of_integer @ ( code_i5400310926305786745atural @ N ) )
= ( semiring_1_of_nat @ int @ ( code_nat_of_natural @ N ) ) ) ).
% int_of_integer_of_natural
thf(fact_7959_integer__of__natural_Orep__eq,axiom,
! [X: code_natural] :
( ( code_int_of_integer @ ( code_i5400310926305786745atural @ X ) )
= ( semiring_1_of_nat @ int @ ( code_nat_of_natural @ X ) ) ) ).
% integer_of_natural.rep_eq
thf(fact_7960_log_Osimps,axiom,
( log
= ( ^ [B3: code_natural,I2: code_natural] :
( if @ code_natural
@ ( ( ord_less_eq @ code_natural @ B3 @ ( one_one @ code_natural ) )
| ( ord_less @ code_natural @ I2 @ B3 ) )
@ ( one_one @ code_natural )
@ ( plus_plus @ code_natural @ ( one_one @ code_natural ) @ ( log @ B3 @ ( divide_divide @ code_natural @ I2 @ B3 ) ) ) ) ) ) ).
% log.simps
thf(fact_7961_log_Oelims,axiom,
! [X: code_natural,Xa: code_natural,Y2: code_natural] :
( ( ( log @ X @ Xa )
= Y2 )
=> ( ( ( ( ord_less_eq @ code_natural @ X @ ( one_one @ code_natural ) )
| ( ord_less @ code_natural @ Xa @ X ) )
=> ( Y2
= ( one_one @ code_natural ) ) )
& ( ~ ( ( ord_less_eq @ code_natural @ X @ ( one_one @ code_natural ) )
| ( ord_less @ code_natural @ Xa @ X ) )
=> ( Y2
= ( plus_plus @ code_natural @ ( one_one @ code_natural ) @ ( log @ X @ ( divide_divide @ code_natural @ Xa @ X ) ) ) ) ) ) ) ).
% log.elims
thf(fact_7962_minus__shift__def,axiom,
( minus_shift
= ( ^ [R: code_natural,K3: code_natural,L2: code_natural] : ( if @ code_natural @ ( ord_less @ code_natural @ K3 @ L2 ) @ ( minus_minus @ code_natural @ ( plus_plus @ code_natural @ R @ K3 ) @ L2 ) @ ( minus_minus @ code_natural @ K3 @ L2 ) ) ) ) ).
% minus_shift_def
thf(fact_7963_next_Osimps,axiom,
! [V2: code_natural,W: code_natural] :
( ( next @ ( product_Pair @ code_natural @ code_natural @ V2 @ W ) )
= ( product_Pair @ code_natural @ ( product_prod @ code_natural @ code_natural ) @ ( plus_plus @ code_natural @ ( minus_shift @ ( numeral_numeral @ code_natural @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( minus_shift @ ( numeral_numeral @ code_natural @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( times_times @ code_natural @ ( modulo_modulo @ code_natural @ V2 @ ( numeral_numeral @ code_natural @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numeral_numeral @ code_natural @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( times_times @ code_natural @ ( divide_divide @ code_natural @ V2 @ ( numeral_numeral @ code_natural @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numeral_numeral @ code_natural @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( plus_plus @ code_natural @ ( minus_shift @ ( numeral_numeral @ code_natural @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( times_times @ code_natural @ ( modulo_modulo @ code_natural @ W @ ( numeral_numeral @ code_natural @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numeral_numeral @ code_natural @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( times_times @ code_natural @ ( divide_divide @ code_natural @ W @ ( numeral_numeral @ code_natural @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numeral_numeral @ code_natural @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( one_one @ code_natural ) ) ) @ ( one_one @ code_natural ) ) @ ( product_Pair @ code_natural @ code_natural @ ( minus_shift @ ( numeral_numeral @ code_natural @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( times_times @ code_natural @ ( modulo_modulo @ code_natural @ V2 @ ( numeral_numeral @ code_natural @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numeral_numeral @ code_natural @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( times_times @ code_natural @ ( divide_divide @ code_natural @ V2 @ ( numeral_numeral @ code_natural @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numeral_numeral @ code_natural @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( minus_shift @ ( numeral_numeral @ code_natural @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( times_times @ code_natural @ ( modulo_modulo @ code_natural @ W @ ( numeral_numeral @ code_natural @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numeral_numeral @ code_natural @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( times_times @ code_natural @ ( divide_divide @ code_natural @ W @ ( numeral_numeral @ code_natural @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ ( numeral_numeral @ code_natural @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% next.simps
thf(fact_7964_Random_Orange__def,axiom,
( range
= ( ^ [K3: code_natural] :
( product_scomp @ ( product_prod @ code_natural @ code_natural ) @ code_natural @ ( product_prod @ code_natural @ code_natural ) @ ( product_prod @ code_natural @ ( product_prod @ code_natural @ code_natural ) )
@ ( iterate @ code_natural @ ( product_prod @ code_natural @ code_natural ) @ ( log @ ( numeral_numeral @ code_natural @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) @ K3 )
@ ^ [L2: code_natural] :
( product_scomp @ ( product_prod @ code_natural @ code_natural ) @ code_natural @ ( product_prod @ code_natural @ code_natural ) @ ( product_prod @ code_natural @ ( product_prod @ code_natural @ code_natural ) ) @ next
@ ^ [V5: code_natural] : ( product_Pair @ code_natural @ ( product_prod @ code_natural @ code_natural ) @ ( plus_plus @ code_natural @ V5 @ ( times_times @ code_natural @ L2 @ ( numeral_numeral @ code_natural @ ( bit1 @ ( bit0 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ one2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
@ ( one_one @ code_natural ) )
@ ^ [V5: code_natural] : ( product_Pair @ code_natural @ ( product_prod @ code_natural @ code_natural ) @ ( modulo_modulo @ code_natural @ V5 @ K3 ) ) ) ) ) ).
% Random.range_def
thf(fact_7965_inc__shift__def,axiom,
( inc_shift
= ( ^ [V5: code_natural,K3: code_natural] : ( if @ code_natural @ ( V5 = K3 ) @ ( one_one @ code_natural ) @ ( plus_plus @ code_natural @ K3 @ ( one_one @ code_natural ) ) ) ) ) ).
% inc_shift_def
thf(fact_7966_iterate_Oelims,axiom,
! [A: $tType,B: $tType,X: code_natural,Xa: B > A > ( product_prod @ B @ A ),Xb3: B,Y2: A > ( product_prod @ B @ A )] :
( ( ( iterate @ B @ A @ X @ Xa @ Xb3 )
= Y2 )
=> ( ( ( X
= ( zero_zero @ code_natural ) )
=> ( Y2
= ( product_Pair @ B @ A @ Xb3 ) ) )
& ( ( X
!= ( zero_zero @ code_natural ) )
=> ( Y2
= ( product_scomp @ A @ B @ A @ ( product_prod @ B @ A ) @ ( Xa @ Xb3 ) @ ( iterate @ B @ A @ ( minus_minus @ code_natural @ X @ ( one_one @ code_natural ) ) @ Xa ) ) ) ) ) ) ).
% iterate.elims
thf(fact_7967_iterate_Osimps,axiom,
! [B: $tType,A: $tType] :
( ( iterate @ B @ A )
= ( ^ [K3: code_natural,F4: B > A > ( product_prod @ B @ A ),X2: B] :
( if @ ( A > ( product_prod @ B @ A ) )
@ ( K3
= ( zero_zero @ code_natural ) )
@ ( product_Pair @ B @ A @ X2 )
@ ( product_scomp @ A @ B @ A @ ( product_prod @ B @ A ) @ ( F4 @ X2 ) @ ( iterate @ B @ A @ ( minus_minus @ code_natural @ K3 @ ( one_one @ code_natural ) ) @ F4 ) ) ) ) ) ).
% iterate.simps
thf(fact_7968_log_Opelims,axiom,
! [X: code_natural,Xa: code_natural,Y2: code_natural] :
( ( ( log @ X @ Xa )
= Y2 )
=> ( ( accp @ ( product_prod @ code_natural @ code_natural ) @ log_rel @ ( product_Pair @ code_natural @ code_natural @ X @ Xa ) )
=> ~ ( ( ( ( ( ord_less_eq @ code_natural @ X @ ( one_one @ code_natural ) )
| ( ord_less @ code_natural @ Xa @ X ) )
=> ( Y2
= ( one_one @ code_natural ) ) )
& ( ~ ( ( ord_less_eq @ code_natural @ X @ ( one_one @ code_natural ) )
| ( ord_less @ code_natural @ Xa @ X ) )
=> ( Y2
= ( plus_plus @ code_natural @ ( one_one @ code_natural ) @ ( log @ X @ ( divide_divide @ code_natural @ Xa @ X ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ code_natural @ code_natural ) @ log_rel @ ( product_Pair @ code_natural @ code_natural @ X @ Xa ) ) ) ) ) ).
% log.pelims
thf(fact_7969_iterate_Opelims,axiom,
! [A: $tType,B: $tType,X: code_natural,Xa: B > A > ( product_prod @ B @ A ),Xb3: B,Y2: A > ( product_prod @ B @ A )] :
( ( ( iterate @ B @ A @ X @ Xa @ Xb3 )
= Y2 )
=> ( ( accp @ ( product_prod @ code_natural @ ( product_prod @ ( B > A > ( product_prod @ B @ A ) ) @ B ) ) @ ( iterate_rel @ B @ A ) @ ( product_Pair @ code_natural @ ( product_prod @ ( B > A > ( product_prod @ B @ A ) ) @ B ) @ X @ ( product_Pair @ ( B > A > ( product_prod @ B @ A ) ) @ B @ Xa @ Xb3 ) ) )
=> ~ ( ( ( ( X
= ( zero_zero @ code_natural ) )
=> ( Y2
= ( product_Pair @ B @ A @ Xb3 ) ) )
& ( ( X
!= ( zero_zero @ code_natural ) )
=> ( Y2
= ( product_scomp @ A @ B @ A @ ( product_prod @ B @ A ) @ ( Xa @ Xb3 ) @ ( iterate @ B @ A @ ( minus_minus @ code_natural @ X @ ( one_one @ code_natural ) ) @ Xa ) ) ) ) )
=> ~ ( accp @ ( product_prod @ code_natural @ ( product_prod @ ( B > A > ( product_prod @ B @ A ) ) @ B ) ) @ ( iterate_rel @ B @ A ) @ ( product_Pair @ code_natural @ ( product_prod @ ( B > A > ( product_prod @ B @ A ) ) @ B ) @ X @ ( product_Pair @ ( B > A > ( product_prod @ B @ A ) ) @ B @ Xa @ Xb3 ) ) ) ) ) ) ).
% iterate.pelims
thf(fact_7970_select__def,axiom,
! [A: $tType] :
( ( select @ A )
= ( ^ [Xs3: list @ A] :
( product_scomp @ ( product_prod @ code_natural @ code_natural ) @ code_natural @ ( product_prod @ code_natural @ code_natural ) @ ( product_prod @ A @ ( product_prod @ code_natural @ code_natural ) ) @ ( range @ ( code_natural_of_nat @ ( size_size @ ( list @ A ) @ Xs3 ) ) )
@ ^ [K3: code_natural] : ( product_Pair @ A @ ( product_prod @ code_natural @ code_natural ) @ ( nth @ A @ Xs3 @ ( code_nat_of_natural @ K3 ) ) ) ) ) ) ).
% select_def
thf(fact_7971_integer__of__natural__def,axiom,
( code_i5400310926305786745atural
= ( map_fun @ code_natural @ nat @ int @ code_integer @ code_nat_of_natural @ code_integer_of_int @ ( semiring_1_of_nat @ int ) ) ) ).
% integer_of_natural_def
thf(fact_7972_zero__natural__def,axiom,
( ( zero_zero @ code_natural )
= ( code_natural_of_nat @ ( zero_zero @ nat ) ) ) ).
% zero_natural_def
thf(fact_7973_minus__natural_Oabs__eq,axiom,
! [Xa: nat,X: nat] :
( ( minus_minus @ code_natural @ ( code_natural_of_nat @ Xa ) @ ( code_natural_of_nat @ X ) )
= ( code_natural_of_nat @ ( minus_minus @ nat @ Xa @ X ) ) ) ).
% minus_natural.abs_eq
thf(fact_7974_divide__natural_Oabs__eq,axiom,
! [Xa: nat,X: nat] :
( ( divide_divide @ code_natural @ ( code_natural_of_nat @ Xa ) @ ( code_natural_of_nat @ X ) )
= ( code_natural_of_nat @ ( divide_divide @ nat @ Xa @ X ) ) ) ).
% divide_natural.abs_eq
thf(fact_7975_less__natural_Oabs__eq,axiom,
! [Xa: nat,X: nat] :
( ( ord_less @ code_natural @ ( code_natural_of_nat @ Xa ) @ ( code_natural_of_nat @ X ) )
= ( ord_less @ nat @ Xa @ X ) ) ).
% less_natural.abs_eq
thf(fact_7976_one__natural__def,axiom,
( ( one_one @ code_natural )
= ( code_natural_of_nat @ ( one_one @ nat ) ) ) ).
% one_natural_def
thf(fact_7977_integer__of__natural_Oabs__eq,axiom,
! [X: nat] :
( ( code_i5400310926305786745atural @ ( code_natural_of_nat @ X ) )
= ( code_integer_of_int @ ( semiring_1_of_nat @ int @ X ) ) ) ).
% integer_of_natural.abs_eq
thf(fact_7978_pick__same,axiom,
! [A: $tType,L: nat,Xs: list @ A] :
( ( ord_less @ nat @ L @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( pick @ A @ ( map @ A @ ( product_prod @ code_natural @ A ) @ ( product_Pair @ code_natural @ A @ ( one_one @ code_natural ) ) @ Xs ) @ ( code_natural_of_nat @ L ) )
= ( nth @ A @ Xs @ L ) ) ) ).
% pick_same
thf(fact_7979_select__weight__select,axiom,
! [A: $tType,Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( select_weight @ A @ ( map @ A @ ( product_prod @ code_natural @ A ) @ ( product_Pair @ code_natural @ A @ ( one_one @ code_natural ) ) @ Xs ) )
= ( select @ A @ Xs ) ) ) ).
% select_weight_select
thf(fact_7980_pick_Osimps,axiom,
! [A: $tType,I: code_natural,X: product_prod @ code_natural @ A,Xs: list @ ( product_prod @ code_natural @ A )] :
( ( ( ord_less @ code_natural @ I @ ( product_fst @ code_natural @ A @ X ) )
=> ( ( pick @ A @ ( cons @ ( product_prod @ code_natural @ A ) @ X @ Xs ) @ I )
= ( product_snd @ code_natural @ A @ X ) ) )
& ( ~ ( ord_less @ code_natural @ I @ ( product_fst @ code_natural @ A @ X ) )
=> ( ( pick @ A @ ( cons @ ( product_prod @ code_natural @ A ) @ X @ Xs ) @ I )
= ( pick @ A @ Xs @ ( minus_minus @ code_natural @ I @ ( product_fst @ code_natural @ A @ X ) ) ) ) ) ) ).
% pick.simps
thf(fact_7981_Suc_Orep__eq,axiom,
! [X: code_natural] :
( ( code_nat_of_natural @ ( code_Suc @ X ) )
= ( suc @ ( code_nat_of_natural @ X ) ) ) ).
% Suc.rep_eq
thf(fact_7982_Suc__natural__minus__one,axiom,
! [N: code_natural] :
( ( minus_minus @ code_natural @ ( code_Suc @ N ) @ ( one_one @ code_natural ) )
= N ) ).
% Suc_natural_minus_one
thf(fact_7983_Suc_Oabs__eq,axiom,
! [X: nat] :
( ( code_Suc @ ( code_natural_of_nat @ X ) )
= ( code_natural_of_nat @ ( suc @ X ) ) ) ).
% Suc.abs_eq
thf(fact_7984_Code__Numeral_OSuc__def,axiom,
( code_Suc
= ( map_fun @ code_natural @ nat @ nat @ code_natural @ code_nat_of_natural @ code_natural_of_nat @ suc ) ) ).
% Code_Numeral.Suc_def
thf(fact_7985_semilattice__set_Oeq__fold_H,axiom,
! [A: $tType,F2: A > A > A,A4: set @ A] :
( ( lattic149705377957585745ce_set @ A @ F2 )
=> ( ( lattic1715443433743089157tice_F @ A @ F2 @ A4 )
= ( the2 @ A
@ ( finite_fold @ A @ ( option @ A )
@ ^ [X2: A,Y5: option @ A] : ( some @ A @ ( case_option @ A @ A @ X2 @ ( F2 @ X2 ) @ Y5 ) )
@ ( none @ A )
@ A4 ) ) ) ) ).
% semilattice_set.eq_fold'
thf(fact_7986_semilattice__set_Oinsert__remove,axiom,
! [A: $tType,F2: A > A > A,A4: set @ A,X: A] :
( ( lattic149705377957585745ce_set @ A @ F2 )
=> ( ( finite_finite @ A @ A4 )
=> ( ( ( ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic1715443433743089157tice_F @ A @ F2 @ ( insert @ A @ X @ A4 ) )
= X ) )
& ( ( ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic1715443433743089157tice_F @ A @ F2 @ ( insert @ A @ X @ A4 ) )
= ( F2 @ X @ ( lattic1715443433743089157tice_F @ A @ F2 @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ).
% semilattice_set.insert_remove
thf(fact_7987_semilattice__set_Oremove,axiom,
! [A: $tType,F2: A > A > A,A4: set @ A,X: A] :
( ( lattic149705377957585745ce_set @ A @ F2 )
=> ( ( finite_finite @ A @ A4 )
=> ( ( member @ A @ X @ A4 )
=> ( ( ( ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic1715443433743089157tice_F @ A @ F2 @ A4 )
= X ) )
& ( ( ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic1715443433743089157tice_F @ A @ F2 @ A4 )
= ( F2 @ X @ ( lattic1715443433743089157tice_F @ A @ F2 @ ( minus_minus @ ( set @ A ) @ A4 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ) ).
% semilattice_set.remove
thf(fact_7988_semilattice__set_Oinfinite,axiom,
! [A: $tType,F2: A > A > A,A4: set @ A] :
( ( lattic149705377957585745ce_set @ A @ F2 )
=> ( ~ ( finite_finite @ A @ A4 )
=> ( ( lattic1715443433743089157tice_F @ A @ F2 @ A4 )
= ( the2 @ A @ ( none @ A ) ) ) ) ) ).
% semilattice_set.infinite
thf(fact_7989_divide__natural__def,axiom,
( ( divide_divide @ code_natural )
= ( map_fun @ code_natural @ nat @ ( nat > nat ) @ ( code_natural > code_natural ) @ code_nat_of_natural @ ( map_fun @ code_natural @ nat @ nat @ code_natural @ code_nat_of_natural @ code_natural_of_nat ) @ ( divide_divide @ nat ) ) ) ).
% divide_natural_def
thf(fact_7990_minus__natural__def,axiom,
( ( minus_minus @ code_natural )
= ( map_fun @ code_natural @ nat @ ( nat > nat ) @ ( code_natural > code_natural ) @ code_nat_of_natural @ ( map_fun @ code_natural @ nat @ nat @ code_natural @ code_nat_of_natural @ code_natural_of_nat ) @ ( minus_minus @ nat ) ) ) ).
% minus_natural_def
thf(fact_7991_Quotient3__int,axiom,
quotient3 @ ( product_prod @ nat @ nat ) @ int @ intrel @ abs_Integ @ rep_Integ ).
% Quotient3_int
thf(fact_7992_map__option__o__case__sum,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,F2: D > C,G: A > ( option @ D ),H: B > ( option @ D )] :
( ( comp @ ( option @ D ) @ ( option @ C ) @ ( sum_sum @ A @ B ) @ ( map_option @ D @ C @ F2 ) @ ( sum_case_sum @ A @ ( option @ D ) @ B @ G @ H ) )
= ( sum_case_sum @ A @ ( option @ C ) @ B @ ( comp @ ( option @ D ) @ ( option @ C ) @ A @ ( map_option @ D @ C @ F2 ) @ G ) @ ( comp @ ( option @ D ) @ ( option @ C ) @ B @ ( map_option @ D @ C @ F2 ) @ H ) ) ) ).
% map_option_o_case_sum
thf(fact_7993_VEBT__internal_OminNull_Opelims_I1_J,axiom,
! [X: vEBT_VEBT,Y2: $o] :
( ( ( vEBT_VEBT_minNull @ X )
= Y2 )
=> ( ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ X )
=> ( ( ( X
= ( vEBT_Leaf @ $false @ $false ) )
=> ( Y2
=> ~ ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $false @ $false ) ) ) )
=> ( ! [Uv2: $o] :
( ( X
= ( vEBT_Leaf @ $true @ Uv2 ) )
=> ( ~ Y2
=> ~ ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $true @ Uv2 ) ) ) )
=> ( ! [Uu2: $o] :
( ( X
= ( vEBT_Leaf @ Uu2 @ $true ) )
=> ( ~ Y2
=> ~ ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ Uu2 @ $true ) ) ) )
=> ( ! [Uw2: nat,Ux2: list @ vEBT_VEBT,Uy: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uw2 @ Ux2 @ Uy ) )
=> ( Y2
=> ~ ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uw2 @ Ux2 @ Uy ) ) ) )
=> ~ ! [Uz2: product_prod @ nat @ nat,Va3: nat,Vb2: list @ vEBT_VEBT,Vc2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) )
=> ( ~ Y2
=> ~ ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.minNull.pelims(1)
thf(fact_7994_is__none__bind,axiom,
! [A: $tType,B: $tType,F2: option @ B,G: B > ( option @ A )] :
( ( is_none @ A @ ( bind @ B @ A @ F2 @ G ) )
= ( ( is_none @ B @ F2 )
| ( is_none @ A @ ( G @ ( the2 @ B @ F2 ) ) ) ) ) ).
% is_none_bind
thf(fact_7995_is__none__code_I2_J,axiom,
! [B: $tType,X: B] :
~ ( is_none @ B @ ( some @ B @ X ) ) ).
% is_none_code(2)
thf(fact_7996_is__none__code_I1_J,axiom,
! [A: $tType] : ( is_none @ A @ ( none @ A ) ) ).
% is_none_code(1)
thf(fact_7997_is__none__map__option,axiom,
! [A: $tType,B: $tType,F2: B > A,X: option @ B] :
( ( is_none @ A @ ( map_option @ B @ A @ F2 @ X ) )
= ( is_none @ B @ X ) ) ).
% is_none_map_option
thf(fact_7998_VEBT__internal_OminNull_Opelims_I2_J,axiom,
! [X: vEBT_VEBT] :
( ( vEBT_VEBT_minNull @ X )
=> ( ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ X )
=> ( ( ( X
= ( vEBT_Leaf @ $false @ $false ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $false @ $false ) ) )
=> ~ ! [Uw2: nat,Ux2: list @ vEBT_VEBT,Uy: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uw2 @ Ux2 @ Uy ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uw2 @ Ux2 @ Uy ) ) ) ) ) ) ).
% VEBT_internal.minNull.pelims(2)
thf(fact_7999_VEBT__internal_OminNull_Opelims_I3_J,axiom,
! [X: vEBT_VEBT] :
( ~ ( vEBT_VEBT_minNull @ X )
=> ( ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ X )
=> ( ! [Uv2: $o] :
( ( X
= ( vEBT_Leaf @ $true @ Uv2 ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $true @ Uv2 ) ) )
=> ( ! [Uu2: $o] :
( ( X
= ( vEBT_Leaf @ Uu2 @ $true ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ Uu2 @ $true ) ) )
=> ~ ! [Uz2: product_prod @ nat @ nat,Va3: nat,Vb2: list @ vEBT_VEBT,Vc2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) )
=> ~ ( accp @ vEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) ) ) ) ) ) ) ).
% VEBT_internal.minNull.pelims(3)
thf(fact_8000_is__none__simps_I2_J,axiom,
! [B: $tType,X: B] :
~ ( is_none @ B @ ( some @ B @ X ) ) ).
% is_none_simps(2)
thf(fact_8001_Option_Ois__none__def,axiom,
! [A: $tType] :
( ( is_none @ A )
= ( ^ [X2: option @ A] :
( X2
= ( none @ A ) ) ) ) ).
% Option.is_none_def
thf(fact_8002_is__none__simps_I1_J,axiom,
! [A: $tType] : ( is_none @ A @ ( none @ A ) ) ).
% is_none_simps(1)
thf(fact_8003_the__map__option,axiom,
! [B: $tType,A: $tType,X: option @ A,F2: A > B] :
( ~ ( is_none @ A @ X )
=> ( ( the2 @ B @ ( map_option @ A @ B @ F2 @ X ) )
= ( F2 @ ( the2 @ A @ X ) ) ) ) ).
% the_map_option
thf(fact_8004_rel__option__unfold,axiom,
! [B: $tType,A: $tType] :
( ( rel_option @ A @ B )
= ( ^ [R6: A > B > $o,X2: option @ A,Y5: option @ B] :
( ( ( is_none @ A @ X2 )
= ( is_none @ B @ Y5 ) )
& ( ~ ( is_none @ A @ X2 )
=> ( ~ ( is_none @ B @ Y5 )
=> ( R6 @ ( the2 @ A @ X2 ) @ ( the2 @ B @ Y5 ) ) ) ) ) ) ) ).
% rel_option_unfold
thf(fact_8005_rel__optionI,axiom,
! [A: $tType,B: $tType,X: option @ A,Y2: option @ B,P: A > B > $o] :
( ( ( is_none @ A @ X )
= ( is_none @ B @ Y2 ) )
=> ( ( ~ ( is_none @ A @ X )
=> ( ~ ( is_none @ B @ Y2 )
=> ( P @ ( the2 @ A @ X ) @ ( the2 @ B @ Y2 ) ) ) )
=> ( rel_option @ A @ B @ P @ X @ Y2 ) ) ) ).
% rel_optionI
thf(fact_8006_sub_Otransfer,axiom,
( bNF_rel_fun @ num @ num @ ( num > int ) @ ( num > code_integer )
@ ^ [Y4: num,Z: num] : Y4 = Z
@ ( bNF_rel_fun @ num @ num @ int @ code_integer
@ ^ [Y4: num,Z: num] : Y4 = Z
@ code_pcr_integer )
@ ^ [M5: num,N5: num] : ( minus_minus @ int @ ( numeral_numeral @ int @ M5 ) @ ( numeral_numeral @ int @ N5 ) )
@ code_sub ) ).
% sub.transfer
thf(fact_8007_bounded__bilinear_Otendsto__right__zero,axiom,
! [C: $tType,B: $tType,A: $tType,D: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B )
& ( real_V822414075346904944vector @ C ) )
=> ! [Prod: A > B > C,F2: D > B,F5: filter @ D,C2: A] :
( ( real_V2442710119149674383linear @ A @ B @ C @ Prod )
=> ( ( filterlim @ D @ B @ F2 @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) @ F5 )
=> ( filterlim @ D @ C
@ ^ [X2: D] : ( Prod @ C2 @ ( F2 @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ C @ ( zero_zero @ C ) )
@ F5 ) ) ) ) ).
% bounded_bilinear.tendsto_right_zero
thf(fact_8008_one__integer_Otransfer,axiom,
code_pcr_integer @ ( one_one @ int ) @ ( one_one @ code_integer ) ).
% one_integer.transfer
thf(fact_8009_less__integer_Otransfer,axiom,
( bNF_rel_fun @ int @ code_integer @ ( int > $o ) @ ( code_integer > $o ) @ code_pcr_integer
@ ( bNF_rel_fun @ int @ code_integer @ $o @ $o @ code_pcr_integer
@ ^ [Y4: $o,Z: $o] : Y4 = Z )
@ ( ord_less @ int )
@ ( ord_less @ code_integer ) ) ).
% less_integer.transfer
thf(fact_8010_divide__integer_Otransfer,axiom,
bNF_rel_fun @ int @ code_integer @ ( int > int ) @ ( code_integer > code_integer ) @ code_pcr_integer @ ( bNF_rel_fun @ int @ code_integer @ int @ code_integer @ code_pcr_integer @ code_pcr_integer ) @ ( divide_divide @ int ) @ ( divide_divide @ code_integer ) ).
% divide_integer.transfer
thf(fact_8011_minus__integer_Otransfer,axiom,
bNF_rel_fun @ int @ code_integer @ ( int > int ) @ ( code_integer > code_integer ) @ code_pcr_integer @ ( bNF_rel_fun @ int @ code_integer @ int @ code_integer @ code_pcr_integer @ code_pcr_integer ) @ ( minus_minus @ int ) @ ( minus_minus @ code_integer ) ).
% minus_integer.transfer
thf(fact_8012_bounded__bilinear_Odiff__right,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( ( real_V822414075346904944vector @ B )
& ( real_V822414075346904944vector @ C )
& ( real_V822414075346904944vector @ A ) )
=> ! [Prod: A > B > C,A3: A,B2: B,B6: B] :
( ( real_V2442710119149674383linear @ A @ B @ C @ Prod )
=> ( ( Prod @ A3 @ ( minus_minus @ B @ B2 @ B6 ) )
= ( minus_minus @ C @ ( Prod @ A3 @ B2 ) @ ( Prod @ A3 @ B6 ) ) ) ) ) ).
% bounded_bilinear.diff_right
thf(fact_8013_bounded__bilinear_Odiff__left,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( ( real_V822414075346904944vector @ B )
& ( real_V822414075346904944vector @ C )
& ( real_V822414075346904944vector @ A ) )
=> ! [Prod: A > B > C,A3: A,A8: A,B2: B] :
( ( real_V2442710119149674383linear @ A @ B @ C @ Prod )
=> ( ( Prod @ ( minus_minus @ A @ A3 @ A8 ) @ B2 )
= ( minus_minus @ C @ ( Prod @ A3 @ B2 ) @ ( Prod @ A8 @ B2 ) ) ) ) ) ).
% bounded_bilinear.diff_left
thf(fact_8014_uminus__integer_Otransfer,axiom,
bNF_rel_fun @ int @ code_integer @ int @ code_integer @ code_pcr_integer @ code_pcr_integer @ ( uminus_uminus @ int ) @ ( uminus_uminus @ code_integer ) ).
% uminus_integer.transfer
thf(fact_8015_integer__of__nat_Otransfer,axiom,
( bNF_rel_fun @ nat @ nat @ int @ code_integer
@ ^ [Y4: nat,Z: nat] : Y4 = Z
@ code_pcr_integer
@ ( semiring_1_of_nat @ int )
@ code_integer_of_nat ) ).
% integer_of_nat.transfer
thf(fact_8016_bounded__bilinear_Ominus__right,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( ( real_V822414075346904944vector @ B )
& ( real_V822414075346904944vector @ C )
& ( real_V822414075346904944vector @ A ) )
=> ! [Prod: A > B > C,A3: A,B2: B] :
( ( real_V2442710119149674383linear @ A @ B @ C @ Prod )
=> ( ( Prod @ A3 @ ( uminus_uminus @ B @ B2 ) )
= ( uminus_uminus @ C @ ( Prod @ A3 @ B2 ) ) ) ) ) ).
% bounded_bilinear.minus_right
thf(fact_8017_bounded__bilinear_Ominus__left,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( ( real_V822414075346904944vector @ B )
& ( real_V822414075346904944vector @ C )
& ( real_V822414075346904944vector @ A ) )
=> ! [Prod: A > B > C,A3: A,B2: B] :
( ( real_V2442710119149674383linear @ A @ B @ C @ Prod )
=> ( ( Prod @ ( uminus_uminus @ A @ A3 ) @ B2 )
= ( uminus_uminus @ C @ ( Prod @ A3 @ B2 ) ) ) ) ) ).
% bounded_bilinear.minus_left
thf(fact_8018_bounded__bilinear_Ozero__left,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [Prod: A > B > C,B2: B] :
( ( real_V2442710119149674383linear @ A @ B @ C @ Prod )
=> ( ( Prod @ ( zero_zero @ A ) @ B2 )
= ( zero_zero @ C ) ) ) ) ).
% bounded_bilinear.zero_left
thf(fact_8019_bounded__bilinear_Ozero__right,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [Prod: A > B > C,A3: A] :
( ( real_V2442710119149674383linear @ A @ B @ C @ Prod )
=> ( ( Prod @ A3 @ ( zero_zero @ B ) )
= ( zero_zero @ C ) ) ) ) ).
% bounded_bilinear.zero_right
thf(fact_8020_zero__integer_Otransfer,axiom,
code_pcr_integer @ ( zero_zero @ int ) @ ( zero_zero @ code_integer ) ).
% zero_integer.transfer
thf(fact_8021_bounded__bilinear_Oprod__diff__prod,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( ( real_V822414075346904944vector @ B )
& ( real_V822414075346904944vector @ C )
& ( real_V822414075346904944vector @ A ) )
=> ! [Prod: A > B > C,X: A,Y2: B,A3: A,B2: B] :
( ( real_V2442710119149674383linear @ A @ B @ C @ Prod )
=> ( ( minus_minus @ C @ ( Prod @ X @ Y2 ) @ ( Prod @ A3 @ B2 ) )
= ( plus_plus @ C @ ( plus_plus @ C @ ( Prod @ ( minus_minus @ A @ X @ A3 ) @ ( minus_minus @ B @ Y2 @ B2 ) ) @ ( Prod @ ( minus_minus @ A @ X @ A3 ) @ B2 ) ) @ ( Prod @ A3 @ ( minus_minus @ B @ Y2 @ B2 ) ) ) ) ) ) ).
% bounded_bilinear.prod_diff_prod
thf(fact_8022_bounded__bilinear_Otendsto__zero,axiom,
! [C: $tType,B: $tType,A: $tType,D: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B )
& ( real_V822414075346904944vector @ C ) )
=> ! [Prod: A > B > C,F2: D > A,F5: filter @ D,G: D > B] :
( ( real_V2442710119149674383linear @ A @ B @ C @ Prod )
=> ( ( filterlim @ D @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ F5 )
=> ( ( filterlim @ D @ B @ G @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) @ F5 )
=> ( filterlim @ D @ C
@ ^ [X2: D] : ( Prod @ ( F2 @ X2 ) @ ( G @ X2 ) )
@ ( topolo7230453075368039082e_nhds @ C @ ( zero_zero @ C ) )
@ F5 ) ) ) ) ) ).
% bounded_bilinear.tendsto_zero
thf(fact_8023_bounded__bilinear_Otendsto__left__zero,axiom,
! [C: $tType,B: $tType,A: $tType,D: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B )
& ( real_V822414075346904944vector @ C ) )
=> ! [Prod: A > B > C,F2: D > A,F5: filter @ D,C2: B] :
( ( real_V2442710119149674383linear @ A @ B @ C @ Prod )
=> ( ( filterlim @ D @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ F5 )
=> ( filterlim @ D @ C
@ ^ [X2: D] : ( Prod @ ( F2 @ X2 ) @ C2 )
@ ( topolo7230453075368039082e_nhds @ C @ ( zero_zero @ C ) )
@ F5 ) ) ) ) ).
% bounded_bilinear.tendsto_left_zero
thf(fact_8024_integer__of__natural_Otransfer,axiom,
bNF_rel_fun @ nat @ code_natural @ int @ code_integer @ code_pcr_natural @ code_pcr_integer @ ( semiring_1_of_nat @ int ) @ code_i5400310926305786745atural ).
% integer_of_natural.transfer
thf(fact_8025_list__all__length,axiom,
! [A: $tType] :
( ( list_all @ A )
= ( ^ [P3: A > $o,Xs3: list @ A] :
! [N5: nat] :
( ( ord_less @ nat @ N5 @ ( size_size @ ( list @ A ) @ Xs3 ) )
=> ( P3 @ ( nth @ A @ Xs3 @ N5 ) ) ) ) ) ).
% list_all_length
thf(fact_8026_Suc_Otransfer,axiom,
bNF_rel_fun @ nat @ code_natural @ nat @ code_natural @ code_pcr_natural @ code_pcr_natural @ suc @ code_Suc ).
% Suc.transfer
thf(fact_8027_divide__natural_Otransfer,axiom,
bNF_rel_fun @ nat @ code_natural @ ( nat > nat ) @ ( code_natural > code_natural ) @ code_pcr_natural @ ( bNF_rel_fun @ nat @ code_natural @ nat @ code_natural @ code_pcr_natural @ code_pcr_natural ) @ ( divide_divide @ nat ) @ ( divide_divide @ code_natural ) ).
% divide_natural.transfer
thf(fact_8028_less__natural_Otransfer,axiom,
( bNF_rel_fun @ nat @ code_natural @ ( nat > $o ) @ ( code_natural > $o ) @ code_pcr_natural
@ ( bNF_rel_fun @ nat @ code_natural @ $o @ $o @ code_pcr_natural
@ ^ [Y4: $o,Z: $o] : Y4 = Z )
@ ( ord_less @ nat )
@ ( ord_less @ code_natural ) ) ).
% less_natural.transfer
thf(fact_8029_minus__natural_Otransfer,axiom,
bNF_rel_fun @ nat @ code_natural @ ( nat > nat ) @ ( code_natural > code_natural ) @ code_pcr_natural @ ( bNF_rel_fun @ nat @ code_natural @ nat @ code_natural @ code_pcr_natural @ code_pcr_natural ) @ ( minus_minus @ nat ) @ ( minus_minus @ code_natural ) ).
% minus_natural.transfer
thf(fact_8030_zero__natural_Otransfer,axiom,
code_pcr_natural @ ( zero_zero @ nat ) @ ( zero_zero @ code_natural ) ).
% zero_natural.transfer
thf(fact_8031_one__natural_Otransfer,axiom,
code_pcr_natural @ ( one_one @ nat ) @ ( one_one @ code_natural ) ).
% one_natural.transfer
thf(fact_8032_linear__injective__on__subspace__0,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V4867850818363320053vector @ A )
& ( real_V4867850818363320053vector @ B ) )
=> ! [F2: A > B,S: set @ A] :
( ( real_Vector_linear @ A @ B @ F2 )
=> ( ( real_Vector_subspace @ A @ S )
=> ( ( inj_on @ A @ B @ F2 @ S )
= ( ! [X2: A] :
( ( member @ A @ X2 @ S )
=> ( ( ( F2 @ X2 )
= ( zero_zero @ B ) )
=> ( X2
= ( zero_zero @ A ) ) ) ) ) ) ) ) ) ).
% linear_injective_on_subspace_0
thf(fact_8033_of__int__code_I1_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [K: num] :
( ( ring_1_of_int @ A @ ( neg @ K ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ K ) ) ) ) ).
% of_int_code(1)
thf(fact_8034_less__int__code_I3_J,axiom,
! [L: num] :
~ ( ord_less @ int @ ( zero_zero @ int ) @ ( neg @ L ) ) ).
% less_int_code(3)
thf(fact_8035_less__int__code_I7_J,axiom,
! [K: num] : ( ord_less @ int @ ( neg @ K ) @ ( zero_zero @ int ) ) ).
% less_int_code(7)
thf(fact_8036_less__int__code_I9_J,axiom,
! [K: num,L: num] :
( ( ord_less @ int @ ( neg @ K ) @ ( neg @ L ) )
= ( ord_less @ num @ L @ K ) ) ).
% less_int_code(9)
thf(fact_8037_subspace__diff,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [S3: set @ A,X: A,Y2: A] :
( ( real_Vector_subspace @ A @ S3 )
=> ( ( member @ A @ X @ S3 )
=> ( ( member @ A @ Y2 @ S3 )
=> ( member @ A @ ( minus_minus @ A @ X @ Y2 ) @ S3 ) ) ) ) ) ).
% subspace_diff
thf(fact_8038_subspace__neg,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [S3: set @ A,X: A] :
( ( real_Vector_subspace @ A @ S3 )
=> ( ( member @ A @ X @ S3 )
=> ( member @ A @ ( uminus_uminus @ A @ X ) @ S3 ) ) ) ) ).
% subspace_neg
thf(fact_8039_subspace__single__0,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ( real_Vector_subspace @ A @ ( insert @ A @ ( zero_zero @ A ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% subspace_single_0
thf(fact_8040_linear__subspace__kernel,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V4867850818363320053vector @ A )
& ( real_V4867850818363320053vector @ B ) )
=> ! [F2: A > B] :
( ( real_Vector_linear @ A @ B @ F2 )
=> ( real_Vector_subspace @ A
@ ( collect @ A
@ ^ [X2: A] :
( ( F2 @ X2 )
= ( zero_zero @ B ) ) ) ) ) ) ).
% linear_subspace_kernel
thf(fact_8041_subspace__0,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [S3: set @ A] :
( ( real_Vector_subspace @ A @ S3 )
=> ( member @ A @ ( zero_zero @ A ) @ S3 ) ) ) ).
% subspace_0
thf(fact_8042_nat__code_I1_J,axiom,
! [K: num] :
( ( nat2 @ ( neg @ K ) )
= ( zero_zero @ nat ) ) ).
% nat_code(1)
thf(fact_8043_subspaceI,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [S3: set @ A] :
( ( member @ A @ ( zero_zero @ A ) @ S3 )
=> ( ! [X3: A,Y3: A] :
( ( member @ A @ X3 @ S3 )
=> ( ( member @ A @ Y3 @ S3 )
=> ( member @ A @ ( plus_plus @ A @ X3 @ Y3 ) @ S3 ) ) )
=> ( ! [C3: real,X3: A] :
( ( member @ A @ X3 @ S3 )
=> ( member @ A @ ( real_V8093663219630862766scaleR @ A @ C3 @ X3 ) @ S3 ) )
=> ( real_Vector_subspace @ A @ S3 ) ) ) ) ) ).
% subspaceI
thf(fact_8044_subspace__def,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ( ( real_Vector_subspace @ A )
= ( ^ [S8: set @ A] :
( ( member @ A @ ( zero_zero @ A ) @ S8 )
& ! [X2: A] :
( ( member @ A @ X2 @ S8 )
=> ! [Y5: A] :
( ( member @ A @ Y5 @ S8 )
=> ( member @ A @ ( plus_plus @ A @ X2 @ Y5 ) @ S8 ) ) )
& ! [C4: real,X2: A] :
( ( member @ A @ X2 @ S8 )
=> ( member @ A @ ( real_V8093663219630862766scaleR @ A @ C4 @ X2 ) @ S8 ) ) ) ) ) ) ).
% subspace_def
thf(fact_8045_less__eq__int__code_I7_J,axiom,
! [K: num] : ( ord_less_eq @ int @ ( neg @ K ) @ ( zero_zero @ int ) ) ).
% less_eq_int_code(7)
thf(fact_8046_less__eq__int__code_I3_J,axiom,
! [L: num] :
~ ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( neg @ L ) ) ).
% less_eq_int_code(3)
thf(fact_8047_less__eq__int__code_I9_J,axiom,
! [K: num,L: num] :
( ( ord_less_eq @ int @ ( neg @ K ) @ ( neg @ L ) )
= ( ord_less_eq @ num @ L @ K ) ) ).
% less_eq_int_code(9)
thf(fact_8048_plus__int__code_I6_J,axiom,
! [M2: num,N: num] :
( ( plus_plus @ int @ ( neg @ M2 ) @ ( neg @ N ) )
= ( neg @ ( plus_plus @ num @ M2 @ N ) ) ) ).
% plus_int_code(6)
thf(fact_8049_Int_Osub__code_I4_J,axiom,
! [N: num] :
( ( sub @ one2 @ ( bit0 @ N ) )
= ( neg @ ( bitM @ N ) ) ) ).
% Int.sub_code(4)
thf(fact_8050_Int_Osub__code_I5_J,axiom,
! [N: num] :
( ( sub @ one2 @ ( bit1 @ N ) )
= ( neg @ ( bit0 @ N ) ) ) ).
% Int.sub_code(5)
thf(fact_8051_minus__int__code_I6_J,axiom,
! [M2: num,N: num] :
( ( minus_minus @ int @ ( neg @ M2 ) @ ( neg @ N ) )
= ( sub @ N @ M2 ) ) ).
% minus_int_code(6)
thf(fact_8052_Int_Osub__def,axiom,
( sub
= ( ^ [M5: num,N5: num] : ( minus_minus @ int @ ( numeral_numeral @ int @ M5 ) @ ( numeral_numeral @ int @ N5 ) ) ) ) ).
% Int.sub_def
thf(fact_8053_Int_Osub__code_I1_J,axiom,
( ( sub @ one2 @ one2 )
= ( zero_zero @ int ) ) ).
% Int.sub_code(1)
thf(fact_8054_Int_Osub__code_I9_J,axiom,
! [M2: num,N: num] :
( ( sub @ ( bit0 @ M2 ) @ ( bit1 @ N ) )
= ( minus_minus @ int @ ( dup @ ( sub @ M2 @ N ) ) @ ( one_one @ int ) ) ) ).
% Int.sub_code(9)
thf(fact_8055_Int_Osub__code_I8_J,axiom,
! [M2: num,N: num] :
( ( sub @ ( bit1 @ M2 ) @ ( bit0 @ N ) )
= ( plus_plus @ int @ ( dup @ ( sub @ M2 @ N ) ) @ ( one_one @ int ) ) ) ).
% Int.sub_code(8)
thf(fact_8056_Int_Odup__code_I1_J,axiom,
( ( dup @ ( zero_zero @ int ) )
= ( zero_zero @ int ) ) ).
% Int.dup_code(1)
thf(fact_8057_Int_Odup__def,axiom,
( dup
= ( ^ [K3: int] : ( plus_plus @ int @ K3 @ K3 ) ) ) ).
% Int.dup_def
thf(fact_8058_Int_Odup__code_I3_J,axiom,
! [N: num] :
( ( dup @ ( neg @ N ) )
= ( neg @ ( bit0 @ N ) ) ) ).
% Int.dup_code(3)
thf(fact_8059_Int_Osub__code_I6_J,axiom,
! [M2: num,N: num] :
( ( sub @ ( bit0 @ M2 ) @ ( bit0 @ N ) )
= ( dup @ ( sub @ M2 @ N ) ) ) ).
% Int.sub_code(6)
thf(fact_8060_Int_Osub__code_I7_J,axiom,
! [M2: num,N: num] :
( ( sub @ ( bit1 @ M2 ) @ ( bit1 @ N ) )
= ( dup @ ( sub @ M2 @ N ) ) ) ).
% Int.sub_code(7)
thf(fact_8061_Int_Osub__code_I2_J,axiom,
! [M2: num] :
( ( sub @ ( bit0 @ M2 ) @ one2 )
= ( pos @ ( bitM @ M2 ) ) ) ).
% Int.sub_code(2)
thf(fact_8062_Int_Osub__code_I3_J,axiom,
! [M2: num] :
( ( sub @ ( bit1 @ M2 ) @ one2 )
= ( pos @ ( bit0 @ M2 ) ) ) ).
% Int.sub_code(3)
thf(fact_8063_Int_Odup__code_I2_J,axiom,
! [N: num] :
( ( dup @ ( pos @ N ) )
= ( pos @ ( bit0 @ N ) ) ) ).
% Int.dup_code(2)
thf(fact_8064_less__int__code_I6_J,axiom,
! [K: num,L: num] :
~ ( ord_less @ int @ ( pos @ K ) @ ( neg @ L ) ) ).
% less_int_code(6)
thf(fact_8065_less__int__code_I8_J,axiom,
! [K: num,L: num] : ( ord_less @ int @ ( neg @ K ) @ ( pos @ L ) ) ).
% less_int_code(8)
thf(fact_8066_uminus__int__code_I3_J,axiom,
! [M2: num] :
( ( uminus_uminus @ int @ ( neg @ M2 ) )
= ( pos @ M2 ) ) ).
% uminus_int_code(3)
thf(fact_8067_uminus__int__code_I2_J,axiom,
! [M2: num] :
( ( uminus_uminus @ int @ ( pos @ M2 ) )
= ( neg @ M2 ) ) ).
% uminus_int_code(2)
thf(fact_8068_Int_ONeg__def,axiom,
( neg
= ( ^ [N5: num] : ( uminus_uminus @ int @ ( pos @ N5 ) ) ) ) ).
% Int.Neg_def
thf(fact_8069_less__eq__int__code_I6_J,axiom,
! [K: num,L: num] :
~ ( ord_less_eq @ int @ ( pos @ K ) @ ( neg @ L ) ) ).
% less_eq_int_code(6)
thf(fact_8070_less__eq__int__code_I8_J,axiom,
! [K: num,L: num] : ( ord_less_eq @ int @ ( neg @ K ) @ ( pos @ L ) ) ).
% less_eq_int_code(8)
thf(fact_8071_times__int__code_I3_J,axiom,
! [M2: num,N: num] :
( ( times_times @ int @ ( pos @ M2 ) @ ( pos @ N ) )
= ( pos @ ( times_times @ num @ M2 @ N ) ) ) ).
% times_int_code(3)
thf(fact_8072_of__int__code_I3_J,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [K: num] :
( ( ring_1_of_int @ A @ ( pos @ K ) )
= ( numeral_numeral @ A @ K ) ) ) ).
% of_int_code(3)
thf(fact_8073_Int_OPos__def,axiom,
( pos
= ( numeral_numeral @ int ) ) ).
% Int.Pos_def
thf(fact_8074_plus__int__code_I3_J,axiom,
! [M2: num,N: num] :
( ( plus_plus @ int @ ( pos @ M2 ) @ ( pos @ N ) )
= ( pos @ ( plus_plus @ num @ M2 @ N ) ) ) ).
% plus_int_code(3)
thf(fact_8075_less__eq__int__code_I5_J,axiom,
! [K: num,L: num] :
( ( ord_less_eq @ int @ ( pos @ K ) @ ( pos @ L ) )
= ( ord_less_eq @ num @ K @ L ) ) ).
% less_eq_int_code(5)
thf(fact_8076_less__eq__int__code_I2_J,axiom,
! [L: num] : ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( pos @ L ) ) ).
% less_eq_int_code(2)
thf(fact_8077_less__eq__int__code_I4_J,axiom,
! [K: num] :
~ ( ord_less_eq @ int @ ( pos @ K ) @ ( zero_zero @ int ) ) ).
% less_eq_int_code(4)
thf(fact_8078_nat__code_I3_J,axiom,
! [K: num] :
( ( nat2 @ ( pos @ K ) )
= ( nat_of_num @ K ) ) ).
% nat_code(3)
thf(fact_8079_less__int__code_I5_J,axiom,
! [K: num,L: num] :
( ( ord_less @ int @ ( pos @ K ) @ ( pos @ L ) )
= ( ord_less @ num @ K @ L ) ) ).
% less_int_code(5)
thf(fact_8080_one__int__code,axiom,
( ( one_one @ int )
= ( pos @ one2 ) ) ).
% one_int_code
thf(fact_8081_less__int__code_I4_J,axiom,
! [K: num] :
~ ( ord_less @ int @ ( pos @ K ) @ ( zero_zero @ int ) ) ).
% less_int_code(4)
thf(fact_8082_less__int__code_I2_J,axiom,
! [L: num] : ( ord_less @ int @ ( zero_zero @ int ) @ ( pos @ L ) ) ).
% less_int_code(2)
thf(fact_8083_minus__int__code_I3_J,axiom,
! [M2: num,N: num] :
( ( minus_minus @ int @ ( pos @ M2 ) @ ( pos @ N ) )
= ( sub @ M2 @ N ) ) ).
% minus_int_code(3)
thf(fact_8084_minus__int__code_I4_J,axiom,
! [M2: num,N: num] :
( ( minus_minus @ int @ ( pos @ M2 ) @ ( neg @ N ) )
= ( pos @ ( plus_plus @ num @ M2 @ N ) ) ) ).
% minus_int_code(4)
thf(fact_8085_minus__int__code_I5_J,axiom,
! [M2: num,N: num] :
( ( minus_minus @ int @ ( neg @ M2 ) @ ( pos @ N ) )
= ( neg @ ( plus_plus @ num @ M2 @ N ) ) ) ).
% minus_int_code(5)
thf(fact_8086_times__int__code_I4_J,axiom,
! [M2: num,N: num] :
( ( times_times @ int @ ( pos @ M2 ) @ ( neg @ N ) )
= ( neg @ ( times_times @ num @ M2 @ N ) ) ) ).
% times_int_code(4)
thf(fact_8087_times__int__code_I5_J,axiom,
! [M2: num,N: num] :
( ( times_times @ int @ ( neg @ M2 ) @ ( pos @ N ) )
= ( neg @ ( times_times @ num @ M2 @ N ) ) ) ).
% times_int_code(5)
thf(fact_8088_times__int__code_I6_J,axiom,
! [M2: num,N: num] :
( ( times_times @ int @ ( neg @ M2 ) @ ( neg @ N ) )
= ( pos @ ( times_times @ num @ M2 @ N ) ) ) ).
% times_int_code(6)
thf(fact_8089_plus__int__code_I5_J,axiom,
! [M2: num,N: num] :
( ( plus_plus @ int @ ( neg @ M2 ) @ ( pos @ N ) )
= ( sub @ N @ M2 ) ) ).
% plus_int_code(5)
thf(fact_8090_plus__int__code_I4_J,axiom,
! [M2: num,N: num] :
( ( plus_plus @ int @ ( pos @ M2 ) @ ( neg @ N ) )
= ( sub @ M2 @ N ) ) ).
% plus_int_code(4)
thf(fact_8091_is__arg__max__def,axiom,
! [A: $tType,B: $tType] :
( ( ord @ A )
=> ( ( lattic501386751176901750rg_max @ B @ A )
= ( ^ [F4: B > A,P3: B > $o,X2: B] :
( ( P3 @ X2 )
& ~ ? [Y5: B] :
( ( P3 @ Y5 )
& ( ord_less @ A @ ( F4 @ X2 ) @ ( F4 @ Y5 ) ) ) ) ) ) ) ).
% is_arg_max_def
thf(fact_8092_prod__list_Ocomm__monoid__list__axioms,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ( groups1828464146339083142d_list @ A @ ( times_times @ A ) @ ( one_one @ A ) ) ) ).
% prod_list.comm_monoid_list_axioms
thf(fact_8093_sum__list_Ocomm__monoid__list__axioms,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ( groups1828464146339083142d_list @ A @ ( plus_plus @ A ) @ ( zero_zero @ A ) ) ) ).
% sum_list.comm_monoid_list_axioms
thf(fact_8094_VEBT_Orec__transfer,axiom,
! [A: $tType,B: $tType,S3: A > B > $o] :
( bNF_rel_fun @ ( ( option @ ( product_prod @ nat @ nat ) ) > nat > ( list @ ( product_prod @ vEBT_VEBT @ A ) ) > vEBT_VEBT > A > A ) @ ( ( option @ ( product_prod @ nat @ nat ) ) > nat > ( list @ ( product_prod @ vEBT_VEBT @ B ) ) > vEBT_VEBT > B > B ) @ ( ( $o > $o > A ) > vEBT_VEBT > A ) @ ( ( $o > $o > B ) > vEBT_VEBT > B )
@ ( bNF_rel_fun @ ( option @ ( product_prod @ nat @ nat ) ) @ ( option @ ( product_prod @ nat @ nat ) ) @ ( nat > ( list @ ( product_prod @ vEBT_VEBT @ A ) ) > vEBT_VEBT > A > A ) @ ( nat > ( list @ ( product_prod @ vEBT_VEBT @ B ) ) > vEBT_VEBT > B > B )
@ ^ [Y4: option @ ( product_prod @ nat @ nat ),Z: option @ ( product_prod @ nat @ nat )] : Y4 = Z
@ ( bNF_rel_fun @ nat @ nat @ ( ( list @ ( product_prod @ vEBT_VEBT @ A ) ) > vEBT_VEBT > A > A ) @ ( ( list @ ( product_prod @ vEBT_VEBT @ B ) ) > vEBT_VEBT > B > B )
@ ^ [Y4: nat,Z: nat] : Y4 = Z
@ ( bNF_rel_fun @ ( list @ ( product_prod @ vEBT_VEBT @ A ) ) @ ( list @ ( product_prod @ vEBT_VEBT @ B ) ) @ ( vEBT_VEBT > A > A ) @ ( vEBT_VEBT > B > B )
@ ( list_all2 @ ( product_prod @ vEBT_VEBT @ A ) @ ( product_prod @ vEBT_VEBT @ B )
@ ( basic_rel_prod @ vEBT_VEBT @ vEBT_VEBT @ A @ B
@ ^ [Y4: vEBT_VEBT,Z: vEBT_VEBT] : Y4 = Z
@ S3 ) )
@ ( bNF_rel_fun @ vEBT_VEBT @ vEBT_VEBT @ ( A > A ) @ ( B > B )
@ ^ [Y4: vEBT_VEBT,Z: vEBT_VEBT] : Y4 = Z
@ ( bNF_rel_fun @ A @ B @ A @ B @ S3 @ S3 ) ) ) ) )
@ ( bNF_rel_fun @ ( $o > $o > A ) @ ( $o > $o > B ) @ ( vEBT_VEBT > A ) @ ( vEBT_VEBT > B )
@ ( bNF_rel_fun @ $o @ $o @ ( $o > A ) @ ( $o > B )
@ ^ [Y4: $o,Z: $o] : Y4 = Z
@ ( bNF_rel_fun @ $o @ $o @ A @ B
@ ^ [Y4: $o,Z: $o] : Y4 = Z
@ S3 ) )
@ ( bNF_rel_fun @ vEBT_VEBT @ vEBT_VEBT @ A @ B
@ ^ [Y4: vEBT_VEBT,Z: vEBT_VEBT] : Y4 = Z
@ S3 ) )
@ ( vEBT_rec_VEBT @ A )
@ ( vEBT_rec_VEBT @ B ) ) ).
% VEBT.rec_transfer
thf(fact_8095_group_Oaxioms_I2_J,axiom,
! [A: $tType,F2: A > A > A,Z2: A,Inverse: A > A] :
( ( group @ A @ F2 @ Z2 @ Inverse )
=> ( group_axioms @ A @ F2 @ Z2 @ Inverse ) ) ).
% group.axioms(2)
thf(fact_8096_group__axioms_Ointro,axiom,
! [A: $tType,F2: A > A > A,Z2: A,Inverse: A > A] :
( ! [A6: A] :
( ( F2 @ Z2 @ A6 )
= A6 )
=> ( ! [A6: A] :
( ( F2 @ ( Inverse @ A6 ) @ A6 )
= Z2 )
=> ( group_axioms @ A @ F2 @ Z2 @ Inverse ) ) ) ).
% group_axioms.intro
thf(fact_8097_group__axioms__def,axiom,
! [A: $tType] :
( ( group_axioms @ A )
= ( ^ [F4: A > A > A,Z6: A,Inverse2: A > A] :
( ! [A5: A] :
( ( F4 @ Z6 @ A5 )
= A5 )
& ! [A5: A] :
( ( F4 @ ( Inverse2 @ A5 ) @ A5 )
= Z6 ) ) ) ) ).
% group_axioms_def
thf(fact_8098_VEBT_Ocase__transfer,axiom,
! [A: $tType,B: $tType,S3: A > B > $o] :
( bNF_rel_fun @ ( ( option @ ( product_prod @ nat @ nat ) ) > nat > ( list @ vEBT_VEBT ) > vEBT_VEBT > A ) @ ( ( option @ ( product_prod @ nat @ nat ) ) > nat > ( list @ vEBT_VEBT ) > vEBT_VEBT > B ) @ ( ( $o > $o > A ) > vEBT_VEBT > A ) @ ( ( $o > $o > B ) > vEBT_VEBT > B )
@ ( bNF_rel_fun @ ( option @ ( product_prod @ nat @ nat ) ) @ ( option @ ( product_prod @ nat @ nat ) ) @ ( nat > ( list @ vEBT_VEBT ) > vEBT_VEBT > A ) @ ( nat > ( list @ vEBT_VEBT ) > vEBT_VEBT > B )
@ ^ [Y4: option @ ( product_prod @ nat @ nat ),Z: option @ ( product_prod @ nat @ nat )] : Y4 = Z
@ ( bNF_rel_fun @ nat @ nat @ ( ( list @ vEBT_VEBT ) > vEBT_VEBT > A ) @ ( ( list @ vEBT_VEBT ) > vEBT_VEBT > B )
@ ^ [Y4: nat,Z: nat] : Y4 = Z
@ ( bNF_rel_fun @ ( list @ vEBT_VEBT ) @ ( list @ vEBT_VEBT ) @ ( vEBT_VEBT > A ) @ ( vEBT_VEBT > B )
@ ^ [Y4: list @ vEBT_VEBT,Z: list @ vEBT_VEBT] : Y4 = Z
@ ( bNF_rel_fun @ vEBT_VEBT @ vEBT_VEBT @ A @ B
@ ^ [Y4: vEBT_VEBT,Z: vEBT_VEBT] : Y4 = Z
@ S3 ) ) ) )
@ ( bNF_rel_fun @ ( $o > $o > A ) @ ( $o > $o > B ) @ ( vEBT_VEBT > A ) @ ( vEBT_VEBT > B )
@ ( bNF_rel_fun @ $o @ $o @ ( $o > A ) @ ( $o > B )
@ ^ [Y4: $o,Z: $o] : Y4 = Z
@ ( bNF_rel_fun @ $o @ $o @ A @ B
@ ^ [Y4: $o,Z: $o] : Y4 = Z
@ S3 ) )
@ ( bNF_rel_fun @ vEBT_VEBT @ vEBT_VEBT @ A @ B
@ ^ [Y4: vEBT_VEBT,Z: vEBT_VEBT] : Y4 = Z
@ S3 ) )
@ ( vEBT_case_VEBT @ A )
@ ( vEBT_case_VEBT @ B ) ) ).
% VEBT.case_transfer
thf(fact_8099_group__def,axiom,
! [A: $tType] :
( ( group @ A )
= ( ^ [F4: A > A > A,Z6: A,Inverse2: A > A] :
( ( semigroup @ A @ F4 )
& ( group_axioms @ A @ F4 @ Z6 @ Inverse2 ) ) ) ) ).
% group_def
thf(fact_8100_monoid_Oaxioms_I1_J,axiom,
! [A: $tType,F2: A > A > A,Z2: A] :
( ( monoid @ A @ F2 @ Z2 )
=> ( semigroup @ A @ F2 ) ) ).
% monoid.axioms(1)
thf(fact_8101_max_Osemigroup__axioms,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( semigroup @ A @ ( ord_max @ A ) ) ) ).
% max.semigroup_axioms
thf(fact_8102_VEBT_Ocase__distrib,axiom,
! [A: $tType,B: $tType,H: A > B,F1: ( option @ ( product_prod @ nat @ nat ) ) > nat > ( list @ vEBT_VEBT ) > vEBT_VEBT > A,F22: $o > $o > A,VEBT2: vEBT_VEBT] :
( ( H @ ( vEBT_case_VEBT @ A @ F1 @ F22 @ VEBT2 ) )
= ( vEBT_case_VEBT @ B
@ ^ [X16: option @ ( product_prod @ nat @ nat ),X25: nat,X34: list @ vEBT_VEBT,X43: vEBT_VEBT] : ( H @ ( F1 @ X16 @ X25 @ X34 @ X43 ) )
@ ^ [X16: $o,X25: $o] : ( H @ ( F22 @ X16 @ X25 ) )
@ VEBT2 ) ) ).
% VEBT.case_distrib
thf(fact_8103_semigroup__def,axiom,
! [A: $tType] :
( ( semigroup @ A )
= ( ^ [F4: A > A > A] :
! [A5: A,B3: A,C4: A] :
( ( F4 @ ( F4 @ A5 @ B3 ) @ C4 )
= ( F4 @ A5 @ ( F4 @ B3 @ C4 ) ) ) ) ) ).
% semigroup_def
thf(fact_8104_semigroup_Ointro,axiom,
! [A: $tType,F2: A > A > A] :
( ! [A6: A,B4: A,C3: A] :
( ( F2 @ ( F2 @ A6 @ B4 ) @ C3 )
= ( F2 @ A6 @ ( F2 @ B4 @ C3 ) ) )
=> ( semigroup @ A @ F2 ) ) ).
% semigroup.intro
thf(fact_8105_semigroup_Oassoc,axiom,
! [A: $tType,F2: A > A > A,A3: A,B2: A,C2: A] :
( ( semigroup @ A @ F2 )
=> ( ( F2 @ ( F2 @ A3 @ B2 ) @ C2 )
= ( F2 @ A3 @ ( F2 @ B2 @ C2 ) ) ) ) ).
% semigroup.assoc
thf(fact_8106_min_Osemigroup__axioms,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( semigroup @ A @ ( ord_min @ A ) ) ) ).
% min.semigroup_axioms
thf(fact_8107_VEBT_Osimps_I5_J,axiom,
! [A: $tType,F1: ( option @ ( product_prod @ nat @ nat ) ) > nat > ( list @ vEBT_VEBT ) > vEBT_VEBT > A,F22: $o > $o > A,X11: option @ ( product_prod @ nat @ nat ),X12: nat,X13: list @ vEBT_VEBT,X14: vEBT_VEBT] :
( ( vEBT_case_VEBT @ A @ F1 @ F22 @ ( vEBT_Node @ X11 @ X12 @ X13 @ X14 ) )
= ( F1 @ X11 @ X12 @ X13 @ X14 ) ) ).
% VEBT.simps(5)
thf(fact_8108_VEBT_Osimps_I6_J,axiom,
! [A: $tType,F1: ( option @ ( product_prod @ nat @ nat ) ) > nat > ( list @ vEBT_VEBT ) > vEBT_VEBT > A,F22: $o > $o > A,X21: $o,X22: $o] :
( ( vEBT_case_VEBT @ A @ F1 @ F22 @ ( vEBT_Leaf @ X21 @ X22 ) )
= ( F22 @ X21 @ X22 ) ) ).
% VEBT.simps(6)
thf(fact_8109_group_Oaxioms_I1_J,axiom,
! [A: $tType,F2: A > A > A,Z2: A,Inverse: A > A] :
( ( group @ A @ F2 @ Z2 @ Inverse )
=> ( semigroup @ A @ F2 ) ) ).
% group.axioms(1)
thf(fact_8110_inf_Osemigroup__axioms,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( semigroup @ A @ ( inf_inf @ A ) ) ) ).
% inf.semigroup_axioms
thf(fact_8111_sup_Osemigroup__axioms,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( semigroup @ A @ ( sup_sup @ A ) ) ) ).
% sup.semigroup_axioms
thf(fact_8112_add_Osemigroup__axioms,axiom,
! [A: $tType] :
( ( semigroup_add @ A )
=> ( semigroup @ A @ ( plus_plus @ A ) ) ) ).
% add.semigroup_axioms
thf(fact_8113_mult_Osemigroup__axioms,axiom,
! [A: $tType] :
( ( semigroup_mult @ A )
=> ( semigroup @ A @ ( times_times @ A ) ) ) ).
% mult.semigroup_axioms
thf(fact_8114_group_Ointro,axiom,
! [A: $tType,F2: A > A > A,Z2: A,Inverse: A > A] :
( ( semigroup @ A @ F2 )
=> ( ( group_axioms @ A @ F2 @ Z2 @ Inverse )
=> ( group @ A @ F2 @ Z2 @ Inverse ) ) ) ).
% group.intro
thf(fact_8115_monoid__def,axiom,
! [A: $tType] :
( ( monoid @ A )
= ( ^ [F4: A > A > A,Z6: A] :
( ( semigroup @ A @ F4 )
& ( monoid_axioms @ A @ F4 @ Z6 ) ) ) ) ).
% monoid_def
thf(fact_8116_monoid_Ointro,axiom,
! [A: $tType,F2: A > A > A,Z2: A] :
( ( semigroup @ A @ F2 )
=> ( ( monoid_axioms @ A @ F2 @ Z2 )
=> ( monoid @ A @ F2 @ Z2 ) ) ) ).
% monoid.intro
thf(fact_8117_monoid__axioms_Ointro,axiom,
! [A: $tType,F2: A > A > A,Z2: A] :
( ! [A6: A] :
( ( F2 @ Z2 @ A6 )
= A6 )
=> ( ! [A6: A] :
( ( F2 @ A6 @ Z2 )
= A6 )
=> ( monoid_axioms @ A @ F2 @ Z2 ) ) ) ).
% monoid_axioms.intro
thf(fact_8118_monoid__axioms__def,axiom,
! [A: $tType] :
( ( monoid_axioms @ A )
= ( ^ [F4: A > A > A,Z6: A] :
( ! [A5: A] :
( ( F4 @ Z6 @ A5 )
= A5 )
& ! [A5: A] :
( ( F4 @ A5 @ Z6 )
= A5 ) ) ) ) ).
% monoid_axioms_def
thf(fact_8119_monoid_Oaxioms_I2_J,axiom,
! [A: $tType,F2: A > A > A,Z2: A] :
( ( monoid @ A @ F2 @ Z2 )
=> ( monoid_axioms @ A @ F2 @ Z2 ) ) ).
% monoid.axioms(2)
thf(fact_8120_Quotient3__real,axiom,
quotient3 @ ( nat > rat ) @ real @ realrel @ real2 @ rep_real ).
% Quotient3_real
% Type constructors (884)
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,
! [A13: $tType] : ( bounded_lattice @ ( filter @ A13 ) ) ).
thf(tcon_HOL_Obool___Lattices_Obounded__lattice_3,axiom,
bounded_lattice @ $o ).
thf(tcon_Set_Oset___Lattices_Obounded__lattice_4,axiom,
! [A13: $tType] : ( bounded_lattice @ ( set @ A13 ) ) ).
thf(tcon_fun___Lattices_Obounded__lattice_5,axiom,
! [A13: $tType,A16: $tType] :
( ( bounded_lattice @ A16 )
=> ( bounded_lattice @ ( A13 > A16 ) ) ) ).
thf(tcon_fun___Conditionally__Complete__Lattices_Oconditionally__complete__lattice,axiom,
! [A13: $tType,A16: $tType] :
( ( comple6319245703460814977attice @ A16 )
=> ( condit1219197933456340205attice @ ( A13 > A16 ) ) ) ).
thf(tcon_fun___Countable__Complete__Lattices_Ocountable__complete__lattice,axiom,
! [A13: $tType,A16: $tType] :
( ( counta3822494911875563373attice @ A16 )
=> ( counta3822494911875563373attice @ ( A13 > A16 ) ) ) ).
thf(tcon_fun___Complete__Lattices_Ocomplete__distrib__lattice,axiom,
! [A13: $tType,A16: $tType] :
( ( comple592849572758109894attice @ A16 )
=> ( comple592849572758109894attice @ ( A13 > A16 ) ) ) ).
thf(tcon_fun___Complete__Lattices_Ocomplete__boolean__algebra,axiom,
! [A13: $tType,A16: $tType] :
( ( comple489889107523837845lgebra @ A16 )
=> ( comple489889107523837845lgebra @ ( A13 > A16 ) ) ) ).
thf(tcon_fun___Lattices_Obounded__semilattice__sup__bot,axiom,
! [A13: $tType,A16: $tType] :
( ( bounded_lattice @ A16 )
=> ( bounde4967611905675639751up_bot @ ( A13 > A16 ) ) ) ).
thf(tcon_fun___Lattices_Obounded__semilattice__inf__top,axiom,
! [A13: $tType,A16: $tType] :
( ( bounded_lattice @ A16 )
=> ( bounde4346867609351753570nf_top @ ( A13 > A16 ) ) ) ).
thf(tcon_fun___Complete__Lattices_Ocomplete__lattice,axiom,
! [A13: $tType,A16: $tType] :
( ( comple6319245703460814977attice @ A16 )
=> ( comple6319245703460814977attice @ ( A13 > A16 ) ) ) ).
thf(tcon_fun___Boolean__Algebras_Oboolean__algebra,axiom,
! [A13: $tType,A16: $tType] :
( ( boolea8198339166811842893lgebra @ A16 )
=> ( boolea8198339166811842893lgebra @ ( A13 > A16 ) ) ) ).
thf(tcon_fun___Lattices_Obounded__lattice__top,axiom,
! [A13: $tType,A16: $tType] :
( ( bounded_lattice @ A16 )
=> ( bounded_lattice_top @ ( A13 > A16 ) ) ) ).
thf(tcon_fun___Lattices_Obounded__lattice__bot,axiom,
! [A13: $tType,A16: $tType] :
( ( bounded_lattice @ A16 )
=> ( bounded_lattice_bot @ ( A13 > A16 ) ) ) ).
thf(tcon_fun___Lattices_Osemilattice__sup,axiom,
! [A13: $tType,A16: $tType] :
( ( semilattice_sup @ A16 )
=> ( semilattice_sup @ ( A13 > A16 ) ) ) ).
thf(tcon_fun___Lattices_Osemilattice__inf,axiom,
! [A13: $tType,A16: $tType] :
( ( semilattice_inf @ A16 )
=> ( semilattice_inf @ ( A13 > A16 ) ) ) ).
thf(tcon_fun___Lattices_Odistrib__lattice,axiom,
! [A13: $tType,A16: $tType] :
( ( distrib_lattice @ A16 )
=> ( distrib_lattice @ ( A13 > A16 ) ) ) ).
thf(tcon_fun___Orderings_Oorder__top,axiom,
! [A13: $tType,A16: $tType] :
( ( order_top @ A16 )
=> ( order_top @ ( A13 > A16 ) ) ) ).
thf(tcon_fun___Orderings_Oorder__bot,axiom,
! [A13: $tType,A16: $tType] :
( ( order_bot @ A16 )
=> ( order_bot @ ( A13 > A16 ) ) ) ).
thf(tcon_fun___Orderings_Opreorder,axiom,
! [A13: $tType,A16: $tType] :
( ( preorder @ A16 )
=> ( preorder @ ( A13 > A16 ) ) ) ).
thf(tcon_fun___Lattices_Olattice,axiom,
! [A13: $tType,A16: $tType] :
( ( lattice @ A16 )
=> ( lattice @ ( A13 > A16 ) ) ) ).
thf(tcon_fun___Orderings_Oorder,axiom,
! [A13: $tType,A16: $tType] :
( ( order @ A16 )
=> ( order @ ( A13 > A16 ) ) ) ).
thf(tcon_fun___Orderings_Oord,axiom,
! [A13: $tType,A16: $tType] :
( ( ord @ A16 )
=> ( ord @ ( A13 > A16 ) ) ) ).
thf(tcon_fun___Groups_Ouminus,axiom,
! [A13: $tType,A16: $tType] :
( ( uminus @ A16 )
=> ( uminus @ ( A13 > A16 ) ) ) ).
thf(tcon_fun___Groups_Ominus,axiom,
! [A13: $tType,A16: $tType] :
( ( minus @ A16 )
=> ( minus @ ( A13 > A16 ) ) ) ).
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___Rings_Onormalization__semidom__multiplicative,axiom,
normal6328177297339901930cative @ int ).
thf(tcon_Int_Oint___Divides_Ounique__euclidean__semiring__numeral,axiom,
unique1627219031080169319umeral @ int ).
thf(tcon_Int_Oint___Euclidean__Division_Oeuclidean__ring__cancel,axiom,
euclid8851590272496341667cancel @ int ).
thf(tcon_Int_Oint___Rings_Osemiring__no__zero__divisors__cancel,axiom,
semiri6575147826004484403cancel @ int ).
thf(tcon_Int_Oint___Groups_Ostrict__ordered__ab__semigroup__add,axiom,
strict9044650504122735259up_add @ int ).
thf(tcon_Int_Oint___Groups_Oordered__cancel__ab__semigroup__add,axiom,
ordere580206878836729694up_add @ int ).
thf(tcon_Int_Oint___Groups_Oordered__ab__semigroup__add__imp__le,axiom,
ordere2412721322843649153imp_le @ int ).
thf(tcon_Int_Oint___Bit__Operations_Osemiring__bit__operations,axiom,
bit_se359711467146920520ations @ int ).
thf(tcon_Int_Oint___Rings_Olinordered__comm__semiring__strict,axiom,
linord2810124833399127020strict @ int ).
thf(tcon_Int_Oint___Groups_Ostrict__ordered__comm__monoid__add,axiom,
strict7427464778891057005id_add @ int ).
thf(tcon_Int_Oint___Groups_Oordered__cancel__comm__monoid__add,axiom,
ordere8940638589300402666id_add @ int ).
thf(tcon_Int_Oint___Euclidean__Division_Oeuclidean__semiring,axiom,
euclid3725896446679973847miring @ int ).
thf(tcon_Int_Oint___Topological__Spaces_Otopological__space,axiom,
topolo4958980785337419405_space @ int ).
thf(tcon_Int_Oint___Topological__Spaces_Olinorder__topology,axiom,
topolo1944317154257567458pology @ 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_Osemidom__divide__unit__factor,axiom,
semido2269285787275462019factor @ int ).
thf(tcon_Int_Oint___Rings_Olinordered__semiring__strict,axiom,
linord8928482502909563296strict @ int ).
thf(tcon_Int_Oint___Rings_Osemiring__no__zero__divisors,axiom,
semiri3467727345109120633visors @ int ).
thf(tcon_Int_Oint___Groups_Oordered__ab__semigroup__add,axiom,
ordere6658533253407199908up_add @ int ).
thf(tcon_Int_Oint___Groups_Oordered__ab__group__add__abs,axiom,
ordere166539214618696060dd_abs @ int ).
thf(tcon_Int_Oint___GCD_Osemiring__gcd__mult__normalize,axiom,
semiri6843258321239162965malize @ int ).
thf(tcon_Int_Oint___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___Rings_Oordered__comm__semiring,axiom,
ordere2520102378445227354miring @ int ).
thf(tcon_Int_Oint___Rings_Onormalization__semidom,axiom,
normal8620421768224518004emidom @ int ).
thf(tcon_Int_Oint___Rings_Olinordered__semiring__1,axiom,
linord6961819062388156250ring_1 @ int ).
thf(tcon_Int_Oint___Groups_Oordered__ab__group__add,axiom,
ordered_ab_group_add @ int ).
thf(tcon_Int_Oint___Groups_Ocancel__semigroup__add,axiom,
cancel_semigroup_add @ int ).
thf(tcon_Int_Oint___Rings_Olinordered__semiring,axiom,
linordered_semiring @ int ).
thf(tcon_Int_Oint___Rings_Oordered__semiring__0,axiom,
ordered_semiring_0 @ int ).
thf(tcon_Int_Oint___Rings_Olinordered__semidom,axiom,
linordered_semidom @ int ).
thf(tcon_Int_Oint___Lattices_Osemilattice__sup_7,axiom,
semilattice_sup @ int ).
thf(tcon_Int_Oint___Lattices_Osemilattice__inf_8,axiom,
semilattice_inf @ int ).
thf(tcon_Int_Oint___Lattices_Odistrib__lattice_9,axiom,
distrib_lattice @ int ).
thf(tcon_Int_Oint___Groups_Oab__semigroup__mult,axiom,
ab_semigroup_mult @ int ).
thf(tcon_Int_Oint___Rings_Osemiring__1__cancel,axiom,
semiring_1_cancel @ int ).
thf(tcon_Int_Oint___Rings_Oalgebraic__semidom,axiom,
algebraic_semidom @ int ).
thf(tcon_Int_Oint___Groups_Ocomm__monoid__mult,axiom,
comm_monoid_mult @ int ).
thf(tcon_Int_Oint___Groups_Oab__semigroup__add,axiom,
ab_semigroup_add @ int ).
thf(tcon_Int_Oint___Rings_Oordered__semiring,axiom,
ordered_semiring @ int ).
thf(tcon_Int_Oint___Rings_Oordered__ring__abs,axiom,
ordered_ring_abs @ int ).
thf(tcon_Int_Oint___Parity_Osemiring__parity,axiom,
semiring_parity @ int ).
thf(tcon_Int_Oint___Groups_Ocomm__monoid__add,axiom,
comm_monoid_add @ int ).
thf(tcon_Int_Oint___Rings_Osemiring__modulo,axiom,
semiring_modulo @ int ).
thf(tcon_Int_Oint___Rings_Olinordered__ring,axiom,
linordered_ring @ int ).
thf(tcon_Int_Oint___Rings_Olinordered__idom,axiom,
linordered_idom @ int ).
thf(tcon_Int_Oint___Rings_Ocomm__semiring__1,axiom,
comm_semiring_1 @ int ).
thf(tcon_Int_Oint___Rings_Ocomm__semiring__0,axiom,
comm_semiring_0 @ int ).
thf(tcon_Int_Oint___Groups_Osemigroup__mult,axiom,
semigroup_mult @ int ).
thf(tcon_Int_Oint___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_10,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_11,axiom,
lattice @ int ).
thf(tcon_Int_Oint___Groups_Ogroup__add,axiom,
group_add @ int ).
thf(tcon_Int_Oint___GCD_Osemiring__gcd,axiom,
semiring_gcd @ int ).
thf(tcon_Int_Oint___GCD_Osemiring__Gcd,axiom,
semiring_Gcd @ int ).
thf(tcon_Int_Oint___Rings_Omult__zero,axiom,
mult_zero @ int ).
thf(tcon_Int_Oint___Rings_Ocomm__ring,axiom,
comm_ring @ int ).
thf(tcon_Int_Oint___Orderings_Oorder_12,axiom,
order @ int ).
thf(tcon_Int_Oint___Num_Oneg__numeral,axiom,
neg_numeral @ int ).
thf(tcon_Int_Oint___Nat_Oring__char__0,axiom,
ring_char_0 @ int ).
thf(tcon_Int_Oint___Rings_Osemiring,axiom,
semiring @ int ).
thf(tcon_Int_Oint___Rings_Osemidom,axiom,
semidom @ int ).
thf(tcon_Int_Oint___Orderings_Oord_13,axiom,
ord @ int ).
thf(tcon_Int_Oint___Groups_Ouminus_14,axiom,
uminus @ int ).
thf(tcon_Int_Oint___Rings_Oring__1,axiom,
ring_1 @ int ).
thf(tcon_Int_Oint___Rings_Oabs__if,axiom,
abs_if @ int ).
thf(tcon_Int_Oint___Groups_Ominus_15,axiom,
minus @ int ).
thf(tcon_Int_Oint___GCD_Oring__gcd,axiom,
ring_gcd @ int ).
thf(tcon_Int_Oint___Power_Opower,axiom,
power @ int ).
thf(tcon_Int_Oint___Num_Onumeral,axiom,
numeral @ int ).
thf(tcon_Int_Oint___Groups_Ozero,axiom,
zero @ int ).
thf(tcon_Int_Oint___Groups_Oplus,axiom,
plus @ int ).
thf(tcon_Int_Oint___Rings_Oring,axiom,
ring @ int ).
thf(tcon_Int_Oint___Rings_Oidom,axiom,
idom @ int ).
thf(tcon_Int_Oint___Groups_Oone,axiom,
one @ int ).
thf(tcon_Int_Oint___Rings_Odvd,axiom,
dvd @ int ).
thf(tcon_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___Rings_Onormalization__semidom__multiplicative_24,axiom,
normal6328177297339901930cative @ nat ).
thf(tcon_Nat_Onat___Divides_Ounique__euclidean__semiring__numeral_25,axiom,
unique1627219031080169319umeral @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors__cancel_26,axiom,
semiri6575147826004484403cancel @ nat ).
thf(tcon_Nat_Onat___Groups_Ostrict__ordered__ab__semigroup__add_27,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_28,axiom,
ordere580206878836729694up_add @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le_29,axiom,
ordere2412721322843649153imp_le @ nat ).
thf(tcon_Nat_Onat___Bit__Operations_Osemiring__bit__operations_30,axiom,
bit_se359711467146920520ations @ nat ).
thf(tcon_Nat_Onat___Rings_Olinordered__comm__semiring__strict_31,axiom,
linord2810124833399127020strict @ nat ).
thf(tcon_Nat_Onat___Groups_Ostrict__ordered__comm__monoid__add_32,axiom,
strict7427464778891057005id_add @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__cancel__comm__monoid__add_33,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_34,axiom,
euclid3725896446679973847miring @ nat ).
thf(tcon_Nat_Onat___Topological__Spaces_Otopological__space_35,axiom,
topolo4958980785337419405_space @ nat ).
thf(tcon_Nat_Onat___Topological__Spaces_Olinorder__topology_36,axiom,
topolo1944317154257567458pology @ 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_Osemidom__divide__unit__factor_43,axiom,
semido2269285787275462019factor @ nat ).
thf(tcon_Nat_Onat___Rings_Olinordered__semiring__strict_44,axiom,
linord8928482502909563296strict @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors_45,axiom,
semiri3467727345109120633visors @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add_46,axiom,
ordere6658533253407199908up_add @ nat ).
thf(tcon_Nat_Onat___GCD_Osemiring__gcd__mult__normalize_47,axiom,
semiri6843258321239162965malize @ nat ).
thf(tcon_Nat_Onat___Limits_Otopological__monoid__mult_48,axiom,
topolo1898628316856586783d_mult @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__comm__monoid__add_49,axiom,
ordere6911136660526730532id_add @ nat ).
thf(tcon_Nat_Onat___Groups_Ocancel__ab__semigroup__add_50,axiom,
cancel2418104881723323429up_add @ nat ).
thf(tcon_Nat_Onat___Limits_Otopological__monoid__add_51,axiom,
topolo6943815403480290642id_add @ nat ).
thf(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add_52,axiom,
cancel1802427076303600483id_add @ nat ).
thf(tcon_Nat_Onat___Rings_Ocomm__semiring__1__cancel_53,axiom,
comm_s4317794764714335236cancel @ nat ).
thf(tcon_Nat_Onat___Bit__Operations_Osemiring__bits_54,axiom,
bit_semiring_bits @ nat ).
thf(tcon_Nat_Onat___Topological__Spaces_Ot2__space_55,axiom,
topological_t2_space @ nat ).
thf(tcon_Nat_Onat___Rings_Oordered__comm__semiring_56,axiom,
ordere2520102378445227354miring @ nat ).
thf(tcon_Nat_Onat___Rings_Onormalization__semidom_57,axiom,
normal8620421768224518004emidom @ nat ).
thf(tcon_Nat_Onat___Groups_Ocancel__semigroup__add_58,axiom,
cancel_semigroup_add @ nat ).
thf(tcon_Nat_Onat___Rings_Olinordered__semiring_59,axiom,
linordered_semiring @ nat ).
thf(tcon_Nat_Onat___Rings_Oordered__semiring__0_60,axiom,
ordered_semiring_0 @ nat ).
thf(tcon_Nat_Onat___Rings_Olinordered__semidom_61,axiom,
linordered_semidom @ nat ).
thf(tcon_Nat_Onat___Lattices_Osemilattice__sup_62,axiom,
semilattice_sup @ nat ).
thf(tcon_Nat_Onat___Lattices_Osemilattice__inf_63,axiom,
semilattice_inf @ nat ).
thf(tcon_Nat_Onat___Lattices_Odistrib__lattice_64,axiom,
distrib_lattice @ nat ).
thf(tcon_Nat_Onat___Groups_Oab__semigroup__mult_65,axiom,
ab_semigroup_mult @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__1__cancel_66,axiom,
semiring_1_cancel @ nat ).
thf(tcon_Nat_Onat___Rings_Oalgebraic__semidom_67,axiom,
algebraic_semidom @ nat ).
thf(tcon_Nat_Onat___Groups_Ocomm__monoid__mult_68,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_69,axiom,
ab_semigroup_add @ nat ).
thf(tcon_Nat_Onat___Rings_Oordered__semiring_70,axiom,
ordered_semiring @ nat ).
thf(tcon_Nat_Onat___Parity_Osemiring__parity_71,axiom,
semiring_parity @ nat ).
thf(tcon_Nat_Onat___Groups_Ocomm__monoid__add_72,axiom,
comm_monoid_add @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__modulo_73,axiom,
semiring_modulo @ nat ).
thf(tcon_Nat_Onat___Rings_Ocomm__semiring__1_74,axiom,
comm_semiring_1 @ nat ).
thf(tcon_Nat_Onat___Rings_Ocomm__semiring__0_75,axiom,
comm_semiring_0 @ nat ).
thf(tcon_Nat_Onat___Groups_Osemigroup__mult_76,axiom,
semigroup_mult @ 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___GCD_Osemiring__Gcd_95,axiom,
semiring_Gcd @ nat ).
thf(tcon_Nat_Onat___Rings_Omult__zero_96,axiom,
mult_zero @ nat ).
thf(tcon_Nat_Onat___Orderings_Oorder_97,axiom,
order @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring_98,axiom,
semiring @ nat ).
thf(tcon_Nat_Onat___Rings_Osemidom_99,axiom,
semidom @ nat ).
thf(tcon_Nat_Onat___Orderings_Oord_100,axiom,
ord @ nat ).
thf(tcon_Nat_Onat___Groups_Ominus_101,axiom,
minus @ nat ).
thf(tcon_Nat_Onat___Power_Opower_102,axiom,
power @ nat ).
thf(tcon_Nat_Onat___Num_Onumeral_103,axiom,
numeral @ nat ).
thf(tcon_Nat_Onat___Groups_Ozero_104,axiom,
zero @ nat ).
thf(tcon_Nat_Onat___Groups_Oplus_105,axiom,
plus @ nat ).
thf(tcon_Nat_Onat___Groups_Oone_106,axiom,
one @ nat ).
thf(tcon_Nat_Onat___Rings_Odvd_107,axiom,
dvd @ nat ).
thf(tcon_Nat_Onat___Nat_Osize,axiom,
size @ nat ).
thf(tcon_Num_Onum___Orderings_Opreorder_108,axiom,
preorder @ num ).
thf(tcon_Num_Onum___Orderings_Olinorder_109,axiom,
linorder @ num ).
thf(tcon_Num_Onum___Orderings_Oorder_110,axiom,
order @ num ).
thf(tcon_Num_Onum___Orderings_Oord_111,axiom,
ord @ num ).
thf(tcon_Num_Onum___Groups_Oplus_112,axiom,
plus @ num ).
thf(tcon_Num_Onum___Nat_Osize_113,axiom,
size @ num ).
thf(tcon_Rat_Orat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_114,axiom,
semiri1453513574482234551roduct @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__monoid__add__imp__le_115,axiom,
ordere1937475149494474687imp_le @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring__no__zero__divisors__cancel_116,axiom,
semiri6575147826004484403cancel @ rat ).
thf(tcon_Rat_Orat___Groups_Ostrict__ordered__ab__semigroup__add_117,axiom,
strict9044650504122735259up_add @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__cancel__ab__semigroup__add_118,axiom,
ordere580206878836729694up_add @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__add__imp__le_119,axiom,
ordere2412721322843649153imp_le @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__comm__semiring__strict_120,axiom,
linord2810124833399127020strict @ rat ).
thf(tcon_Rat_Orat___Groups_Ostrict__ordered__comm__monoid__add_121,axiom,
strict7427464778891057005id_add @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__cancel__comm__monoid__add_122,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_123,axiom,
linord715952674999750819strict @ rat ).
thf(tcon_Rat_Orat___Orderings_Ounbounded__dense__linorder,axiom,
unboun7993243217541854897norder @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring__1__no__zero__divisors_124,axiom,
semiri2026040879449505780visors @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__nonzero__semiring_125,axiom,
linord181362715937106298miring @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__semiring__strict_126,axiom,
linord8928482502909563296strict @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring__no__zero__divisors_127,axiom,
semiri3467727345109120633visors @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__add_128,axiom,
ordere6658533253407199908up_add @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__ab__group__add__abs_129,axiom,
ordere166539214618696060dd_abs @ rat ).
thf(tcon_Rat_Orat___Archimedean__Field_Ofloor__ceiling,axiom,
archim2362893244070406136eiling @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__comm__monoid__add_130,axiom,
ordere6911136660526730532id_add @ rat ).
thf(tcon_Rat_Orat___Groups_Olinordered__ab__group__add_131,axiom,
linord5086331880401160121up_add @ rat ).
thf(tcon_Rat_Orat___Groups_Ocancel__ab__semigroup__add_132,axiom,
cancel2418104881723323429up_add @ rat ).
thf(tcon_Rat_Orat___Rings_Oring__1__no__zero__divisors_133,axiom,
ring_15535105094025558882visors @ rat ).
thf(tcon_Rat_Orat___Groups_Ocancel__comm__monoid__add_134,axiom,
cancel1802427076303600483id_add @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__ring__strict_135,axiom,
linord4710134922213307826strict @ rat ).
thf(tcon_Rat_Orat___Rings_Ocomm__semiring__1__cancel_136,axiom,
comm_s4317794764714335236cancel @ rat ).
thf(tcon_Rat_Orat___Rings_Oordered__comm__semiring_137,axiom,
ordere2520102378445227354miring @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__semiring__1_138,axiom,
linord6961819062388156250ring_1 @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__ab__group__add_139,axiom,
ordered_ab_group_add @ rat ).
thf(tcon_Rat_Orat___Groups_Ocancel__semigroup__add_140,axiom,
cancel_semigroup_add @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__semiring_141,axiom,
linordered_semiring @ rat ).
thf(tcon_Rat_Orat___Rings_Oordered__semiring__0_142,axiom,
ordered_semiring_0 @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__semidom_143,axiom,
linordered_semidom @ rat ).
thf(tcon_Rat_Orat___Orderings_Odense__linorder,axiom,
dense_linorder @ rat ).
thf(tcon_Rat_Orat___Lattices_Osemilattice__sup_144,axiom,
semilattice_sup @ rat ).
thf(tcon_Rat_Orat___Lattices_Osemilattice__inf_145,axiom,
semilattice_inf @ rat ).
thf(tcon_Rat_Orat___Lattices_Odistrib__lattice_146,axiom,
distrib_lattice @ rat ).
thf(tcon_Rat_Orat___Groups_Oab__semigroup__mult_147,axiom,
ab_semigroup_mult @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring__1__cancel_148,axiom,
semiring_1_cancel @ rat ).
thf(tcon_Rat_Orat___Groups_Ocomm__monoid__mult_149,axiom,
comm_monoid_mult @ rat ).
thf(tcon_Rat_Orat___Groups_Oab__semigroup__add_150,axiom,
ab_semigroup_add @ rat ).
thf(tcon_Rat_Orat___Fields_Olinordered__field,axiom,
linordered_field @ rat ).
thf(tcon_Rat_Orat___Rings_Oordered__semiring_151,axiom,
ordered_semiring @ rat ).
thf(tcon_Rat_Orat___Rings_Oordered__ring__abs_152,axiom,
ordered_ring_abs @ rat ).
thf(tcon_Rat_Orat___Groups_Ocomm__monoid__add_153,axiom,
comm_monoid_add @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__ring_154,axiom,
linordered_ring @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__idom_155,axiom,
linordered_idom @ rat ).
thf(tcon_Rat_Orat___Rings_Ocomm__semiring__1_156,axiom,
comm_semiring_1 @ rat ).
thf(tcon_Rat_Orat___Rings_Ocomm__semiring__0_157,axiom,
comm_semiring_0 @ rat ).
thf(tcon_Rat_Orat___Orderings_Odense__order,axiom,
dense_order @ rat ).
thf(tcon_Rat_Orat___Groups_Osemigroup__mult_158,axiom,
semigroup_mult @ rat ).
thf(tcon_Rat_Orat___Rings_Osemidom__divide_159,axiom,
semidom_divide @ rat ).
thf(tcon_Rat_Orat___Num_Osemiring__numeral_160,axiom,
semiring_numeral @ rat ).
thf(tcon_Rat_Orat___Groups_Osemigroup__add_161,axiom,
semigroup_add @ rat ).
thf(tcon_Rat_Orat___Fields_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_162,axiom,
zero_less_one @ rat ).
thf(tcon_Rat_Orat___Rings_Ocomm__semiring_163,axiom,
comm_semiring @ rat ).
thf(tcon_Rat_Orat___Nat_Osemiring__char__0_164,axiom,
semiring_char_0 @ rat ).
thf(tcon_Rat_Orat___Groups_Oab__group__add_165,axiom,
ab_group_add @ rat ).
thf(tcon_Rat_Orat___Fields_Ofield__char__0,axiom,
field_char_0 @ rat ).
thf(tcon_Rat_Orat___Rings_Ozero__neq__one_166,axiom,
zero_neq_one @ rat ).
thf(tcon_Rat_Orat___Rings_Oordered__ring_167,axiom,
ordered_ring @ rat ).
thf(tcon_Rat_Orat___Rings_Oidom__abs__sgn_168,axiom,
idom_abs_sgn @ rat ).
thf(tcon_Rat_Orat___Orderings_Opreorder_169,axiom,
preorder @ rat ).
thf(tcon_Rat_Orat___Orderings_Olinorder_170,axiom,
linorder @ rat ).
thf(tcon_Rat_Orat___Groups_Omonoid__mult_171,axiom,
monoid_mult @ rat ).
thf(tcon_Rat_Orat___Rings_Oidom__divide_172,axiom,
idom_divide @ rat ).
thf(tcon_Rat_Orat___Rings_Ocomm__ring__1_173,axiom,
comm_ring_1 @ rat ).
thf(tcon_Rat_Orat___Groups_Omonoid__add_174,axiom,
monoid_add @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring__1_175,axiom,
semiring_1 @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring__0_176,axiom,
semiring_0 @ rat ).
thf(tcon_Rat_Orat___Orderings_Ono__top_177,axiom,
no_top @ rat ).
thf(tcon_Rat_Orat___Orderings_Ono__bot_178,axiom,
no_bot @ rat ).
thf(tcon_Rat_Orat___Lattices_Olattice_179,axiom,
lattice @ rat ).
thf(tcon_Rat_Orat___Groups_Ogroup__add_180,axiom,
group_add @ rat ).
thf(tcon_Rat_Orat___Rings_Omult__zero_181,axiom,
mult_zero @ rat ).
thf(tcon_Rat_Orat___Rings_Ocomm__ring_182,axiom,
comm_ring @ rat ).
thf(tcon_Rat_Orat___Orderings_Oorder_183,axiom,
order @ rat ).
thf(tcon_Rat_Orat___Num_Oneg__numeral_184,axiom,
neg_numeral @ rat ).
thf(tcon_Rat_Orat___Nat_Oring__char__0_185,axiom,
ring_char_0 @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring_186,axiom,
semiring @ rat ).
thf(tcon_Rat_Orat___Fields_Oinverse,axiom,
inverse @ rat ).
thf(tcon_Rat_Orat___Rings_Osemidom_187,axiom,
semidom @ rat ).
thf(tcon_Rat_Orat___Orderings_Oord_188,axiom,
ord @ rat ).
thf(tcon_Rat_Orat___Groups_Ouminus_189,axiom,
uminus @ rat ).
thf(tcon_Rat_Orat___Rings_Oring__1_190,axiom,
ring_1 @ rat ).
thf(tcon_Rat_Orat___Rings_Oabs__if_191,axiom,
abs_if @ rat ).
thf(tcon_Rat_Orat___Groups_Ominus_192,axiom,
minus @ rat ).
thf(tcon_Rat_Orat___Fields_Ofield,axiom,
field @ rat ).
thf(tcon_Rat_Orat___Power_Opower_193,axiom,
power @ rat ).
thf(tcon_Rat_Orat___Num_Onumeral_194,axiom,
numeral @ rat ).
thf(tcon_Rat_Orat___Groups_Ozero_195,axiom,
zero @ rat ).
thf(tcon_Rat_Orat___Groups_Oplus_196,axiom,
plus @ rat ).
thf(tcon_Rat_Orat___Rings_Oring_197,axiom,
ring @ rat ).
thf(tcon_Rat_Orat___Rings_Oidom_198,axiom,
idom @ rat ).
thf(tcon_Rat_Orat___Groups_Oone_199,axiom,
one @ rat ).
thf(tcon_Rat_Orat___Rings_Odvd_200,axiom,
dvd @ rat ).
thf(tcon_Set_Oset___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_201,axiom,
! [A13: $tType] : ( condit1219197933456340205attice @ ( set @ A13 ) ) ).
thf(tcon_Set_Oset___Countable__Complete__Lattices_Ocountable__complete__lattice_202,axiom,
! [A13: $tType] : ( counta3822494911875563373attice @ ( set @ A13 ) ) ).
thf(tcon_Set_Oset___Complete__Lattices_Ocomplete__distrib__lattice_203,axiom,
! [A13: $tType] : ( comple592849572758109894attice @ ( set @ A13 ) ) ).
thf(tcon_Set_Oset___Complete__Lattices_Ocomplete__boolean__algebra_204,axiom,
! [A13: $tType] : ( comple489889107523837845lgebra @ ( set @ A13 ) ) ).
thf(tcon_Set_Oset___Lattices_Obounded__semilattice__sup__bot_205,axiom,
! [A13: $tType] : ( bounde4967611905675639751up_bot @ ( set @ A13 ) ) ).
thf(tcon_Set_Oset___Lattices_Obounded__semilattice__inf__top_206,axiom,
! [A13: $tType] : ( bounde4346867609351753570nf_top @ ( set @ A13 ) ) ).
thf(tcon_Set_Oset___Complete__Lattices_Ocomplete__lattice_207,axiom,
! [A13: $tType] : ( comple6319245703460814977attice @ ( set @ A13 ) ) ).
thf(tcon_Set_Oset___Boolean__Algebras_Oboolean__algebra_208,axiom,
! [A13: $tType] : ( boolea8198339166811842893lgebra @ ( set @ A13 ) ) ).
thf(tcon_Set_Oset___Lattices_Obounded__lattice__top_209,axiom,
! [A13: $tType] : ( bounded_lattice_top @ ( set @ A13 ) ) ).
thf(tcon_Set_Oset___Lattices_Obounded__lattice__bot_210,axiom,
! [A13: $tType] : ( bounded_lattice_bot @ ( set @ A13 ) ) ).
thf(tcon_Set_Oset___Lattices_Osemilattice__sup_211,axiom,
! [A13: $tType] : ( semilattice_sup @ ( set @ A13 ) ) ).
thf(tcon_Set_Oset___Lattices_Osemilattice__inf_212,axiom,
! [A13: $tType] : ( semilattice_inf @ ( set @ A13 ) ) ).
thf(tcon_Set_Oset___Lattices_Odistrib__lattice_213,axiom,
! [A13: $tType] : ( distrib_lattice @ ( set @ A13 ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder__top_214,axiom,
! [A13: $tType] : ( order_top @ ( set @ A13 ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder__bot_215,axiom,
! [A13: $tType] : ( order_bot @ ( set @ A13 ) ) ).
thf(tcon_Set_Oset___Orderings_Opreorder_216,axiom,
! [A13: $tType] : ( preorder @ ( set @ A13 ) ) ).
thf(tcon_Set_Oset___Lattices_Olattice_217,axiom,
! [A13: $tType] : ( lattice @ ( set @ A13 ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder_218,axiom,
! [A13: $tType] : ( order @ ( set @ A13 ) ) ).
thf(tcon_Set_Oset___Orderings_Oord_219,axiom,
! [A13: $tType] : ( ord @ ( set @ A13 ) ) ).
thf(tcon_Set_Oset___Groups_Ouminus_220,axiom,
! [A13: $tType] : ( uminus @ ( set @ A13 ) ) ).
thf(tcon_Set_Oset___Groups_Ominus_221,axiom,
! [A13: $tType] : ( minus @ ( set @ A13 ) ) ).
thf(tcon_HOL_Obool___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_222,axiom,
condit1219197933456340205attice @ $o ).
thf(tcon_HOL_Obool___Countable__Complete__Lattices_Ocountable__complete__lattice_223,axiom,
counta3822494911875563373attice @ $o ).
thf(tcon_HOL_Obool___Complete__Lattices_Ocomplete__distrib__lattice_224,axiom,
comple592849572758109894attice @ $o ).
thf(tcon_HOL_Obool___Complete__Lattices_Ocomplete__boolean__algebra_225,axiom,
comple489889107523837845lgebra @ $o ).
thf(tcon_HOL_Obool___Topological__Spaces_Otopological__space_226,axiom,
topolo4958980785337419405_space @ $o ).
thf(tcon_HOL_Obool___Topological__Spaces_Olinorder__topology_227,axiom,
topolo1944317154257567458pology @ $o ).
thf(tcon_HOL_Obool___Lattices_Obounded__semilattice__sup__bot_228,axiom,
bounde4967611905675639751up_bot @ $o ).
thf(tcon_HOL_Obool___Lattices_Obounded__semilattice__inf__top_229,axiom,
bounde4346867609351753570nf_top @ $o ).
thf(tcon_HOL_Obool___Complete__Lattices_Ocomplete__lattice_230,axiom,
comple6319245703460814977attice @ $o ).
thf(tcon_HOL_Obool___Topological__Spaces_Oorder__topology_231,axiom,
topolo2564578578187576103pology @ $o ).
thf(tcon_HOL_Obool___Boolean__Algebras_Oboolean__algebra_232,axiom,
boolea8198339166811842893lgebra @ $o ).
thf(tcon_HOL_Obool___Lattices_Obounded__lattice__top_233,axiom,
bounded_lattice_top @ $o ).
thf(tcon_HOL_Obool___Lattices_Obounded__lattice__bot_234,axiom,
bounded_lattice_bot @ $o ).
thf(tcon_HOL_Obool___Topological__Spaces_Ot2__space_235,axiom,
topological_t2_space @ $o ).
thf(tcon_HOL_Obool___Lattices_Osemilattice__sup_236,axiom,
semilattice_sup @ $o ).
thf(tcon_HOL_Obool___Lattices_Osemilattice__inf_237,axiom,
semilattice_inf @ $o ).
thf(tcon_HOL_Obool___Lattices_Odistrib__lattice_238,axiom,
distrib_lattice @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder__top_239,axiom,
order_top @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder__bot_240,axiom,
order_bot @ $o ).
thf(tcon_HOL_Obool___Orderings_Opreorder_241,axiom,
preorder @ $o ).
thf(tcon_HOL_Obool___Orderings_Olinorder_242,axiom,
linorder @ $o ).
thf(tcon_HOL_Obool___Lattices_Olattice_243,axiom,
lattice @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder_244,axiom,
order @ $o ).
thf(tcon_HOL_Obool___Orderings_Oord_245,axiom,
ord @ $o ).
thf(tcon_HOL_Obool___Groups_Ouminus_246,axiom,
uminus @ $o ).
thf(tcon_HOL_Obool___Groups_Ominus_247,axiom,
minus @ $o ).
thf(tcon_List_Olist___Nat_Osize_248,axiom,
! [A13: $tType] : ( size @ ( list @ A13 ) ) ).
thf(tcon_Real_Oreal___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_249,axiom,
condit6923001295902523014norder @ real ).
thf(tcon_Real_Oreal___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_250,axiom,
condit1219197933456340205attice @ real ).
thf(tcon_Real_Oreal___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_251,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_252,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_253,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_254,axiom,
strict9044650504122735259up_add @ real ).
thf(tcon_Real_Oreal___Groups_Oordered__cancel__ab__semigroup__add_255,axiom,
ordere580206878836729694up_add @ real ).
thf(tcon_Real_Oreal___Groups_Oordered__ab__semigroup__add__imp__le_256,axiom,
ordere2412721322843649153imp_le @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__comm__semiring__strict_257,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_258,axiom,
strict7427464778891057005id_add @ real ).
thf(tcon_Real_Oreal___Groups_Oordered__cancel__comm__monoid__add_259,axiom,
ordere8940638589300402666id_add @ real ).
thf(tcon_Real_Oreal___Topological__Spaces_Otopological__space_260,axiom,
topolo4958980785337419405_space @ real ).
thf(tcon_Real_Oreal___Topological__Spaces_Olinorder__topology_261,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_262,axiom,
archim462609752435547400_field @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__semiring__1__strict_263,axiom,
linord715952674999750819strict @ real ).
thf(tcon_Real_Oreal___Orderings_Ounbounded__dense__linorder_264,axiom,
unboun7993243217541854897norder @ real ).
thf(tcon_Real_Oreal___Limits_Otopological__comm__monoid__add_265,axiom,
topolo5987344860129210374id_add @ real ).
thf(tcon_Real_Oreal___Topological__Spaces_Oorder__topology_266,axiom,
topolo2564578578187576103pology @ real ).
thf(tcon_Real_Oreal___Rings_Osemiring__1__no__zero__divisors_267,axiom,
semiri2026040879449505780visors @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__nonzero__semiring_268,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_269,axiom,
topolo4211221413907600880p_mult @ real ).
thf(tcon_Real_Oreal___Topological__Spaces_Operfect__space,axiom,
topolo8386298272705272623_space @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__semiring__strict_270,axiom,
linord8928482502909563296strict @ real ).
thf(tcon_Real_Oreal___Rings_Osemiring__no__zero__divisors_271,axiom,
semiri3467727345109120633visors @ 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_272,axiom,
ordere6658533253407199908up_add @ real ).
thf(tcon_Real_Oreal___Groups_Oordered__ab__group__add__abs_273,axiom,
ordere166539214618696060dd_abs @ real ).
thf(tcon_Real_Oreal___Archimedean__Field_Ofloor__ceiling_274,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_275,axiom,
ordere6911136660526730532id_add @ real ).
thf(tcon_Real_Oreal___Groups_Olinordered__ab__group__add_276,axiom,
linord5086331880401160121up_add @ real ).
thf(tcon_Real_Oreal___Groups_Ocancel__ab__semigroup__add_277,axiom,
cancel2418104881723323429up_add @ real ).
thf(tcon_Real_Oreal___Rings_Oring__1__no__zero__divisors_278,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_279,axiom,
topolo6943815403480290642id_add @ real ).
thf(tcon_Real_Oreal___Groups_Ocancel__comm__monoid__add_280,axiom,
cancel1802427076303600483id_add @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__ring__strict_281,axiom,
linord4710134922213307826strict @ real ).
thf(tcon_Real_Oreal___Rings_Ocomm__semiring__1__cancel_282,axiom,
comm_s4317794764714335236cancel @ real ).
thf(tcon_Real_Oreal___Real__Vector__Spaces_Odist__norm,axiom,
real_V6936659425649961206t_norm @ real ).
thf(tcon_Real_Oreal___Limits_Otopological__group__add,axiom,
topolo1633459387980952147up_add @ real ).
thf(tcon_Real_Oreal___Topological__Spaces_Ot2__space_283,axiom,
topological_t2_space @ real ).
thf(tcon_Real_Oreal___Rings_Oordered__comm__semiring_284,axiom,
ordere2520102378445227354miring @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__semiring__1_285,axiom,
linord6961819062388156250ring_1 @ real ).
thf(tcon_Real_Oreal___Groups_Oordered__ab__group__add_286,axiom,
ordered_ab_group_add @ real ).
thf(tcon_Real_Oreal___Groups_Ocancel__semigroup__add_287,axiom,
cancel_semigroup_add @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__semiring_288,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_289,axiom,
ordered_semiring_0 @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__semidom_290,axiom,
linordered_semidom @ real ).
thf(tcon_Real_Oreal___Orderings_Odense__linorder_291,axiom,
dense_linorder @ real ).
thf(tcon_Real_Oreal___Lattices_Osemilattice__sup_292,axiom,
semilattice_sup @ real ).
thf(tcon_Real_Oreal___Lattices_Osemilattice__inf_293,axiom,
semilattice_inf @ real ).
thf(tcon_Real_Oreal___Lattices_Odistrib__lattice_294,axiom,
distrib_lattice @ real ).
thf(tcon_Real_Oreal___Groups_Oab__semigroup__mult_295,axiom,
ab_semigroup_mult @ real ).
thf(tcon_Real_Oreal___Rings_Osemiring__1__cancel_296,axiom,
semiring_1_cancel @ real ).
thf(tcon_Real_Oreal___Groups_Ocomm__monoid__mult_297,axiom,
comm_monoid_mult @ real ).
thf(tcon_Real_Oreal___Groups_Oab__semigroup__add_298,axiom,
ab_semigroup_add @ real ).
thf(tcon_Real_Oreal___Fields_Olinordered__field_299,axiom,
linordered_field @ real ).
thf(tcon_Real_Oreal___Rings_Oordered__semiring_300,axiom,
ordered_semiring @ real ).
thf(tcon_Real_Oreal___Rings_Oordered__ring__abs_301,axiom,
ordered_ring_abs @ real ).
thf(tcon_Real_Oreal___Groups_Ocomm__monoid__add_302,axiom,
comm_monoid_add @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__ring_303,axiom,
linordered_ring @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__idom_304,axiom,
linordered_idom @ real ).
thf(tcon_Real_Oreal___Rings_Ocomm__semiring__1_305,axiom,
comm_semiring_1 @ real ).
thf(tcon_Real_Oreal___Rings_Ocomm__semiring__0_306,axiom,
comm_semiring_0 @ real ).
thf(tcon_Real_Oreal___Orderings_Odense__order_307,axiom,
dense_order @ real ).
thf(tcon_Real_Oreal___Groups_Osemigroup__mult_308,axiom,
semigroup_mult @ real ).
thf(tcon_Real_Oreal___Rings_Osemidom__divide_309,axiom,
semidom_divide @ real ).
thf(tcon_Real_Oreal___Num_Osemiring__numeral_310,axiom,
semiring_numeral @ real ).
thf(tcon_Real_Oreal___Groups_Osemigroup__add_311,axiom,
semigroup_add @ real ).
thf(tcon_Real_Oreal___Fields_Ofield__abs__sgn_312,axiom,
field_abs_sgn @ real ).
thf(tcon_Real_Oreal___Fields_Odivision__ring_313,axiom,
division_ring @ real ).
thf(tcon_Real_Oreal___Rings_Ozero__less__one_314,axiom,
zero_less_one @ real ).
thf(tcon_Real_Oreal___Rings_Ocomm__semiring_315,axiom,
comm_semiring @ real ).
thf(tcon_Real_Oreal___Nat_Osemiring__char__0_316,axiom,
semiring_char_0 @ real ).
thf(tcon_Real_Oreal___Groups_Oab__group__add_317,axiom,
ab_group_add @ real ).
thf(tcon_Real_Oreal___Fields_Ofield__char__0_318,axiom,
field_char_0 @ real ).
thf(tcon_Real_Oreal___Rings_Ozero__neq__one_319,axiom,
zero_neq_one @ real ).
thf(tcon_Real_Oreal___Rings_Oordered__ring_320,axiom,
ordered_ring @ real ).
thf(tcon_Real_Oreal___Rings_Oidom__abs__sgn_321,axiom,
idom_abs_sgn @ real ).
thf(tcon_Real_Oreal___Orderings_Opreorder_322,axiom,
preorder @ real ).
thf(tcon_Real_Oreal___Orderings_Olinorder_323,axiom,
linorder @ real ).
thf(tcon_Real_Oreal___Groups_Omonoid__mult_324,axiom,
monoid_mult @ real ).
thf(tcon_Real_Oreal___Transcendental_Oln,axiom,
ln @ real ).
thf(tcon_Real_Oreal___Rings_Oidom__divide_325,axiom,
idom_divide @ real ).
thf(tcon_Real_Oreal___Rings_Ocomm__ring__1_326,axiom,
comm_ring_1 @ real ).
thf(tcon_Real_Oreal___Groups_Omonoid__add_327,axiom,
monoid_add @ real ).
thf(tcon_Real_Oreal___Rings_Osemiring__1_328,axiom,
semiring_1 @ real ).
thf(tcon_Real_Oreal___Rings_Osemiring__0_329,axiom,
semiring_0 @ real ).
thf(tcon_Real_Oreal___Orderings_Ono__top_330,axiom,
no_top @ real ).
thf(tcon_Real_Oreal___Orderings_Ono__bot_331,axiom,
no_bot @ real ).
thf(tcon_Real_Oreal___Lattices_Olattice_332,axiom,
lattice @ real ).
thf(tcon_Real_Oreal___Groups_Ogroup__add_333,axiom,
group_add @ real ).
thf(tcon_Real_Oreal___Rings_Omult__zero_334,axiom,
mult_zero @ real ).
thf(tcon_Real_Oreal___Rings_Ocomm__ring_335,axiom,
comm_ring @ real ).
thf(tcon_Real_Oreal___Orderings_Oorder_336,axiom,
order @ real ).
thf(tcon_Real_Oreal___Num_Oneg__numeral_337,axiom,
neg_numeral @ real ).
thf(tcon_Real_Oreal___Nat_Oring__char__0_338,axiom,
ring_char_0 @ real ).
thf(tcon_Real_Oreal___Rings_Osemiring_339,axiom,
semiring @ real ).
thf(tcon_Real_Oreal___Fields_Oinverse_340,axiom,
inverse @ real ).
thf(tcon_Real_Oreal___Rings_Osemidom_341,axiom,
semidom @ real ).
thf(tcon_Real_Oreal___Orderings_Oord_342,axiom,
ord @ real ).
thf(tcon_Real_Oreal___Groups_Ouminus_343,axiom,
uminus @ real ).
thf(tcon_Real_Oreal___Rings_Oring__1_344,axiom,
ring_1 @ real ).
thf(tcon_Real_Oreal___Rings_Oabs__if_345,axiom,
abs_if @ real ).
thf(tcon_Real_Oreal___Groups_Ominus_346,axiom,
minus @ real ).
thf(tcon_Real_Oreal___Fields_Ofield_347,axiom,
field @ real ).
thf(tcon_Real_Oreal___Power_Opower_348,axiom,
power @ real ).
thf(tcon_Real_Oreal___Num_Onumeral_349,axiom,
numeral @ real ).
thf(tcon_Real_Oreal___Groups_Ozero_350,axiom,
zero @ real ).
thf(tcon_Real_Oreal___Groups_Oplus_351,axiom,
plus @ real ).
thf(tcon_Real_Oreal___Rings_Oring_352,axiom,
ring @ real ).
thf(tcon_Real_Oreal___Rings_Oidom_353,axiom,
idom @ real ).
thf(tcon_Real_Oreal___Groups_Oone_354,axiom,
one @ real ).
thf(tcon_Real_Oreal___Rings_Odvd_355,axiom,
dvd @ real ).
thf(tcon_String_Ochar___Nat_Osize_356,axiom,
size @ char ).
thf(tcon_Sum__Type_Osum___Nat_Osize_357,axiom,
! [A13: $tType,A16: $tType] : ( size @ ( sum_sum @ A13 @ A16 ) ) ).
thf(tcon_Filter_Ofilter___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_358,axiom,
! [A13: $tType] : ( condit1219197933456340205attice @ ( filter @ A13 ) ) ).
thf(tcon_Filter_Ofilter___Countable__Complete__Lattices_Ocountable__complete__lattice_359,axiom,
! [A13: $tType] : ( counta3822494911875563373attice @ ( filter @ A13 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Obounded__semilattice__sup__bot_360,axiom,
! [A13: $tType] : ( bounde4967611905675639751up_bot @ ( filter @ A13 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Obounded__semilattice__inf__top_361,axiom,
! [A13: $tType] : ( bounde4346867609351753570nf_top @ ( filter @ A13 ) ) ).
thf(tcon_Filter_Ofilter___Complete__Lattices_Ocomplete__lattice_362,axiom,
! [A13: $tType] : ( comple6319245703460814977attice @ ( filter @ A13 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Obounded__lattice__top_363,axiom,
! [A13: $tType] : ( bounded_lattice_top @ ( filter @ A13 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Obounded__lattice__bot_364,axiom,
! [A13: $tType] : ( bounded_lattice_bot @ ( filter @ A13 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Osemilattice__sup_365,axiom,
! [A13: $tType] : ( semilattice_sup @ ( filter @ A13 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Osemilattice__inf_366,axiom,
! [A13: $tType] : ( semilattice_inf @ ( filter @ A13 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Odistrib__lattice_367,axiom,
! [A13: $tType] : ( distrib_lattice @ ( filter @ A13 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Oorder__top_368,axiom,
! [A13: $tType] : ( order_top @ ( filter @ A13 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Oorder__bot_369,axiom,
! [A13: $tType] : ( order_bot @ ( filter @ A13 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Opreorder_370,axiom,
! [A13: $tType] : ( preorder @ ( filter @ A13 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Olattice_371,axiom,
! [A13: $tType] : ( lattice @ ( filter @ A13 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Oorder_372,axiom,
! [A13: $tType] : ( order @ ( filter @ A13 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Oord_373,axiom,
! [A13: $tType] : ( ord @ ( filter @ A13 ) ) ).
thf(tcon_Option_Ooption___Nat_Osize_374,axiom,
! [A13: $tType] : ( size @ ( option @ A13 ) ) ).
thf(tcon_String_Oliteral___Groups_Osemigroup__add_375,axiom,
semigroup_add @ literal ).
thf(tcon_String_Oliteral___Orderings_Opreorder_376,axiom,
preorder @ literal ).
thf(tcon_String_Oliteral___Orderings_Olinorder_377,axiom,
linorder @ literal ).
thf(tcon_String_Oliteral___Groups_Omonoid__add_378,axiom,
monoid_add @ literal ).
thf(tcon_String_Oliteral___Orderings_Oorder_379,axiom,
order @ literal ).
thf(tcon_String_Oliteral___Orderings_Oord_380,axiom,
ord @ literal ).
thf(tcon_String_Oliteral___Groups_Ozero_381,axiom,
zero @ literal ).
thf(tcon_String_Oliteral___Groups_Oplus_382,axiom,
plus @ literal ).
thf(tcon_String_Oliteral___Nat_Osize_383,axiom,
size @ literal ).
thf(tcon_Complex_Ocomplex___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_384,axiom,
semiri1453513574482234551roduct @ complex ).
thf(tcon_Complex_Ocomplex___Topological__Spaces_Ofirst__countable__topology_385,axiom,
topolo3112930676232923870pology @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__div__algebra_386,axiom,
real_V8999393235501362500lgebra @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__algebra__1_387,axiom,
real_V2822296259951069270ebra_1 @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Osemiring__no__zero__divisors__cancel_388,axiom,
semiri6575147826004484403cancel @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__algebra_389,axiom,
real_V4412858255891104859lgebra @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__vector_390,axiom,
real_V822414075346904944vector @ complex ).
thf(tcon_Complex_Ocomplex___Topological__Spaces_Otopological__space_391,axiom,
topolo4958980785337419405_space @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__field_392,axiom,
real_V3459762299906320749_field @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__div__algebra_393,axiom,
real_V5047593784448816457lgebra @ complex ).
thf(tcon_Complex_Ocomplex___Limits_Otopological__comm__monoid__add_394,axiom,
topolo5987344860129210374id_add @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Osemiring__1__no__zero__divisors_395,axiom,
semiri2026040879449505780visors @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__algebra__1_396,axiom,
real_V2191834092415804123ebra_1 @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Ocomplete__space_397,axiom,
real_V8037385150606011577_space @ complex ).
thf(tcon_Complex_Ocomplex___Limits_Otopological__semigroup__mult_398,axiom,
topolo4211221413907600880p_mult @ complex ).
thf(tcon_Complex_Ocomplex___Topological__Spaces_Operfect__space_399,axiom,
topolo8386298272705272623_space @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Osemiring__no__zero__divisors_400,axiom,
semiri3467727345109120633visors @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Ometric__space_401,axiom,
real_V7819770556892013058_space @ complex ).
thf(tcon_Complex_Ocomplex___Limits_Otopological__ab__group__add_402,axiom,
topolo1287966508704411220up_add @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__vector_403,axiom,
real_V4867850818363320053vector @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Ocancel__ab__semigroup__add_404,axiom,
cancel2418104881723323429up_add @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Oring__1__no__zero__divisors_405,axiom,
ring_15535105094025558882visors @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__field_406,axiom,
real_V7773925162809079976_field @ complex ).
thf(tcon_Complex_Ocomplex___Limits_Otopological__monoid__add_407,axiom,
topolo6943815403480290642id_add @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Ocancel__comm__monoid__add_408,axiom,
cancel1802427076303600483id_add @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Ocomm__semiring__1__cancel_409,axiom,
comm_s4317794764714335236cancel @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Odist__norm_410,axiom,
real_V6936659425649961206t_norm @ complex ).
thf(tcon_Complex_Ocomplex___Limits_Otopological__group__add_411,axiom,
topolo1633459387980952147up_add @ complex ).
thf(tcon_Complex_Ocomplex___Topological__Spaces_Ot2__space_412,axiom,
topological_t2_space @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Ocancel__semigroup__add_413,axiom,
cancel_semigroup_add @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Obanach_414,axiom,
real_Vector_banach @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Oab__semigroup__mult_415,axiom,
ab_semigroup_mult @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Osemiring__1__cancel_416,axiom,
semiring_1_cancel @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Ocomm__monoid__mult_417,axiom,
comm_monoid_mult @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Oab__semigroup__add_418,axiom,
ab_semigroup_add @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Ocomm__monoid__add_419,axiom,
comm_monoid_add @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Ocomm__semiring__1_420,axiom,
comm_semiring_1 @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Ocomm__semiring__0_421,axiom,
comm_semiring_0 @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Osemigroup__mult_422,axiom,
semigroup_mult @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Osemidom__divide_423,axiom,
semidom_divide @ complex ).
thf(tcon_Complex_Ocomplex___Num_Osemiring__numeral_424,axiom,
semiring_numeral @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Osemigroup__add_425,axiom,
semigroup_add @ complex ).
thf(tcon_Complex_Ocomplex___Fields_Ofield__abs__sgn_426,axiom,
field_abs_sgn @ complex ).
thf(tcon_Complex_Ocomplex___Fields_Odivision__ring_427,axiom,
division_ring @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Ocomm__semiring_428,axiom,
comm_semiring @ complex ).
thf(tcon_Complex_Ocomplex___Nat_Osemiring__char__0_429,axiom,
semiring_char_0 @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Oab__group__add_430,axiom,
ab_group_add @ complex ).
thf(tcon_Complex_Ocomplex___Fields_Ofield__char__0_431,axiom,
field_char_0 @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Ozero__neq__one_432,axiom,
zero_neq_one @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Oidom__abs__sgn_433,axiom,
idom_abs_sgn @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Omonoid__mult_434,axiom,
monoid_mult @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Oidom__divide_435,axiom,
idom_divide @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Ocomm__ring__1_436,axiom,
comm_ring_1 @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Omonoid__add_437,axiom,
monoid_add @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Osemiring__1_438,axiom,
semiring_1 @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Osemiring__0_439,axiom,
semiring_0 @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Ogroup__add_440,axiom,
group_add @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Omult__zero_441,axiom,
mult_zero @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Ocomm__ring_442,axiom,
comm_ring @ complex ).
thf(tcon_Complex_Ocomplex___Num_Oneg__numeral_443,axiom,
neg_numeral @ complex ).
thf(tcon_Complex_Ocomplex___Nat_Oring__char__0_444,axiom,
ring_char_0 @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Osemiring_445,axiom,
semiring @ complex ).
thf(tcon_Complex_Ocomplex___Fields_Oinverse_446,axiom,
inverse @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Osemidom_447,axiom,
semidom @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Ouminus_448,axiom,
uminus @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Oring__1_449,axiom,
ring_1 @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Ominus_450,axiom,
minus @ complex ).
thf(tcon_Complex_Ocomplex___Fields_Ofield_451,axiom,
field @ complex ).
thf(tcon_Complex_Ocomplex___Power_Opower_452,axiom,
power @ complex ).
thf(tcon_Complex_Ocomplex___Num_Onumeral_453,axiom,
numeral @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Ozero_454,axiom,
zero @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Oplus_455,axiom,
plus @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Oring_456,axiom,
ring @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Oidom_457,axiom,
idom @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Oone_458,axiom,
one @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Odvd_459,axiom,
dvd @ complex ).
thf(tcon_Extended__Nat_Oenat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_460,axiom,
condit6923001295902523014norder @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_461,axiom,
condit1219197933456340205attice @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Countable__Complete__Lattices_Ocountable__complete__lattice_462,axiom,
counta3822494911875563373attice @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Complete__Lattices_Ocomplete__distrib__lattice_463,axiom,
comple592849572758109894attice @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Ostrict__ordered__ab__semigroup__add_464,axiom,
strict9044650504122735259up_add @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Ostrict__ordered__comm__monoid__add_465,axiom,
strict7427464778891057005id_add @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Ocanonically__ordered__monoid__add_466,axiom,
canoni5634975068530333245id_add @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Lattices_Obounded__semilattice__sup__bot_467,axiom,
bounde4967611905675639751up_bot @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Lattices_Obounded__semilattice__inf__top_468,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_469,axiom,
comple6319245703460814977attice @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Olinordered__nonzero__semiring_470,axiom,
linord181362715937106298miring @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Osemiring__no__zero__divisors_471,axiom,
semiri3467727345109120633visors @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Oordered__ab__semigroup__add_472,axiom,
ordere6658533253407199908up_add @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Oordered__comm__monoid__add_473,axiom,
ordere6911136660526730532id_add @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Lattices_Obounded__lattice__top_474,axiom,
bounded_lattice_top @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Lattices_Obounded__lattice__bot_475,axiom,
bounded_lattice_bot @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Oordered__comm__semiring_476,axiom,
ordere2520102378445227354miring @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Lattices_Osemilattice__sup_477,axiom,
semilattice_sup @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Lattices_Osemilattice__inf_478,axiom,
semilattice_inf @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Lattices_Odistrib__lattice_479,axiom,
distrib_lattice @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Oab__semigroup__mult_480,axiom,
ab_semigroup_mult @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Ocomm__monoid__mult_481,axiom,
comm_monoid_mult @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Oab__semigroup__add_482,axiom,
ab_semigroup_add @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Oordered__semiring_483,axiom,
ordered_semiring @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Ocomm__monoid__add_484,axiom,
comm_monoid_add @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Ocomm__semiring__1_485,axiom,
comm_semiring_1 @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Ocomm__semiring__0_486,axiom,
comm_semiring_0 @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Osemigroup__mult_487,axiom,
semigroup_mult @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Num_Osemiring__numeral_488,axiom,
semiring_numeral @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Osemigroup__add_489,axiom,
semigroup_add @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Ozero__less__one_490,axiom,
zero_less_one @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Ocomm__semiring_491,axiom,
comm_semiring @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Orderings_Owellorder_492,axiom,
wellorder @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Orderings_Oorder__top_493,axiom,
order_top @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Orderings_Oorder__bot_494,axiom,
order_bot @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Nat_Osemiring__char__0_495,axiom,
semiring_char_0 @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Ozero__neq__one_496,axiom,
zero_neq_one @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Orderings_Opreorder_497,axiom,
preorder @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Orderings_Olinorder_498,axiom,
linorder @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Omonoid__mult_499,axiom,
monoid_mult @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Omonoid__add_500,axiom,
monoid_add @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Osemiring__1_501,axiom,
semiring_1 @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Osemiring__0_502,axiom,
semiring_0 @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Lattices_Olattice_503,axiom,
lattice @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Omult__zero_504,axiom,
mult_zero @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Orderings_Oorder_505,axiom,
order @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Osemiring_506,axiom,
semiring @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Orderings_Oord_507,axiom,
ord @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Ominus_508,axiom,
minus @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Power_Opower_509,axiom,
power @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Num_Onumeral_510,axiom,
numeral @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Ozero_511,axiom,
zero @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Oplus_512,axiom,
plus @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Oone_513,axiom,
one @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Odvd_514,axiom,
dvd @ extended_enat ).
thf(tcon_Product__Type_Oprod___Topological__Spaces_Otopological__space_515,axiom,
! [A13: $tType,A16: $tType] :
( ( ( topolo4958980785337419405_space @ A13 )
& ( topolo4958980785337419405_space @ A16 ) )
=> ( topolo4958980785337419405_space @ ( product_prod @ A13 @ A16 ) ) ) ).
thf(tcon_Product__Type_Oprod___Topological__Spaces_Ot2__space_516,axiom,
! [A13: $tType,A16: $tType] :
( ( ( topological_t2_space @ A13 )
& ( topological_t2_space @ A16 ) )
=> ( topological_t2_space @ ( product_prod @ A13 @ A16 ) ) ) ).
thf(tcon_Product__Type_Oprod___Nat_Osize_517,axiom,
! [A13: $tType,A16: $tType] : ( size @ ( product_prod @ A13 @ A16 ) ) ).
thf(tcon_Product__Type_Ounit___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_518,axiom,
condit6923001295902523014norder @ product_unit ).
thf(tcon_Product__Type_Ounit___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_519,axiom,
condit1219197933456340205attice @ product_unit ).
thf(tcon_Product__Type_Ounit___Countable__Complete__Lattices_Ocountable__complete__lattice_520,axiom,
counta3822494911875563373attice @ product_unit ).
thf(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__distrib__lattice_521,axiom,
comple592849572758109894attice @ product_unit ).
thf(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__boolean__algebra_522,axiom,
comple489889107523837845lgebra @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Obounded__semilattice__sup__bot_523,axiom,
bounde4967611905675639751up_bot @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Obounded__semilattice__inf__top_524,axiom,
bounde4346867609351753570nf_top @ product_unit ).
thf(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__linorder_525,axiom,
comple5582772986160207858norder @ product_unit ).
thf(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__lattice_526,axiom,
comple6319245703460814977attice @ product_unit ).
thf(tcon_Product__Type_Ounit___Boolean__Algebras_Oboolean__algebra_527,axiom,
boolea8198339166811842893lgebra @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Obounded__lattice__top_528,axiom,
bounded_lattice_top @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Obounded__lattice__bot_529,axiom,
bounded_lattice_bot @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Osemilattice__sup_530,axiom,
semilattice_sup @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Osemilattice__inf_531,axiom,
semilattice_inf @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Odistrib__lattice_532,axiom,
distrib_lattice @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Owellorder_533,axiom,
wellorder @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Oorder__top_534,axiom,
order_top @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Oorder__bot_535,axiom,
order_bot @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Opreorder_536,axiom,
preorder @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Olinorder_537,axiom,
linorder @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Olattice_538,axiom,
lattice @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Oorder_539,axiom,
order @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Oord_540,axiom,
ord @ product_unit ).
thf(tcon_Product__Type_Ounit___Groups_Ouminus_541,axiom,
uminus @ product_unit ).
thf(tcon_Product__Type_Ounit___Groups_Ominus_542,axiom,
minus @ product_unit ).
thf(tcon_Code__Numeral_Ointeger___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_543,axiom,
bit_un5681908812861735899ations @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_544,axiom,
semiri1453513574482234551roduct @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__semiring__with__nat_545,axiom,
euclid5411537665997757685th_nat @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__ring__with__nat_546,axiom,
euclid8789492081693882211th_nat @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__monoid__add__imp__le_547,axiom,
ordere1937475149494474687imp_le @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Ounique__euclidean__semiring_548,axiom,
euclid3128863361964157862miring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__semiring__cancel_549,axiom,
euclid4440199948858584721cancel @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Divides_Ounique__euclidean__semiring__numeral_550,axiom,
unique1627219031080169319umeral @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__ring__cancel_551,axiom,
euclid8851590272496341667cancel @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__no__zero__divisors__cancel_552,axiom,
semiri6575147826004484403cancel @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ostrict__ordered__ab__semigroup__add_553,axiom,
strict9044650504122735259up_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__cancel__ab__semigroup__add_554,axiom,
ordere580206878836729694up_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__add__imp__le_555,axiom,
ordere2412721322843649153imp_le @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Bit__Operations_Osemiring__bit__operations_556,axiom,
bit_se359711467146920520ations @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__comm__semiring__strict_557,axiom,
linord2810124833399127020strict @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ostrict__ordered__comm__monoid__add_558,axiom,
strict7427464778891057005id_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__cancel__comm__monoid__add_559,axiom,
ordere8940638589300402666id_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__semiring_560,axiom,
euclid3725896446679973847miring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__1__strict_561,axiom,
linord715952674999750819strict @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Bit__Operations_Oring__bit__operations_562,axiom,
bit_ri3973907225187159222ations @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1__no__zero__divisors_563,axiom,
semiri2026040879449505780visors @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__nonzero__semiring_564,axiom,
linord181362715937106298miring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Euclidean__Division_Oeuclidean__ring_565,axiom,
euclid5891614535332579305n_ring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__strict_566,axiom,
linord8928482502909563296strict @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__no__zero__divisors_567,axiom,
semiri3467727345109120633visors @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__semigroup__add_568,axiom,
ordere6658533253407199908up_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__group__add__abs_569,axiom,
ordere166539214618696060dd_abs @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__comm__monoid__add_570,axiom,
ordere6911136660526730532id_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Olinordered__ab__group__add_571,axiom,
linord5086331880401160121up_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ocancel__ab__semigroup__add_572,axiom,
cancel2418104881723323429up_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oring__1__no__zero__divisors_573,axiom,
ring_15535105094025558882visors @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ocancel__comm__monoid__add_574,axiom,
cancel1802427076303600483id_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__ring__strict_575,axiom,
linord4710134922213307826strict @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__1__cancel_576,axiom,
comm_s4317794764714335236cancel @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Bit__Operations_Osemiring__bits_577,axiom,
bit_semiring_bits @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oordered__comm__semiring_578,axiom,
ordere2520102378445227354miring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring__1_579,axiom,
linord6961819062388156250ring_1 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oordered__ab__group__add_580,axiom,
ordered_ab_group_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ocancel__semigroup__add_581,axiom,
cancel_semigroup_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semiring_582,axiom,
linordered_semiring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oordered__semiring__0_583,axiom,
ordered_semiring_0 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__semidom_584,axiom,
linordered_semidom @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oab__semigroup__mult_585,axiom,
ab_semigroup_mult @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1__cancel_586,axiom,
semiring_1_cancel @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oalgebraic__semidom_587,axiom,
algebraic_semidom @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ocomm__monoid__mult_588,axiom,
comm_monoid_mult @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oab__semigroup__add_589,axiom,
ab_semigroup_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oordered__semiring_590,axiom,
ordered_semiring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oordered__ring__abs_591,axiom,
ordered_ring_abs @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Parity_Osemiring__parity_592,axiom,
semiring_parity @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ocomm__monoid__add_593,axiom,
comm_monoid_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__modulo_594,axiom,
semiring_modulo @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__ring_595,axiom,
linordered_ring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Olinordered__idom_596,axiom,
linordered_idom @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__1_597,axiom,
comm_semiring_1 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring__0_598,axiom,
comm_semiring_0 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Osemigroup__mult_599,axiom,
semigroup_mult @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemidom__modulo_600,axiom,
semidom_modulo @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemidom__divide_601,axiom,
semidom_divide @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Num_Osemiring__numeral_602,axiom,
semiring_numeral @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Osemigroup__add_603,axiom,
semigroup_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ozero__less__one_604,axiom,
zero_less_one @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__semiring_605,axiom,
comm_semiring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Nat_Osemiring__char__0_606,axiom,
semiring_char_0 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oab__group__add_607,axiom,
ab_group_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ozero__neq__one_608,axiom,
zero_neq_one @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oordered__ring_609,axiom,
ordered_ring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oidom__abs__sgn_610,axiom,
idom_abs_sgn @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Parity_Oring__parity_611,axiom,
ring_parity @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Orderings_Opreorder_612,axiom,
preorder @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Orderings_Olinorder_613,axiom,
linorder @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Omonoid__mult_614,axiom,
monoid_mult @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oidom__modulo_615,axiom,
idom_modulo @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oidom__divide_616,axiom,
idom_divide @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__ring__1_617,axiom,
comm_ring_1 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Omonoid__add_618,axiom,
monoid_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__1_619,axiom,
semiring_1 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring__0_620,axiom,
semiring_0 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ogroup__add_621,axiom,
group_add @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Omult__zero_622,axiom,
mult_zero @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Ocomm__ring_623,axiom,
comm_ring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Orderings_Oorder_624,axiom,
order @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Num_Oneg__numeral_625,axiom,
neg_numeral @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Nat_Oring__char__0_626,axiom,
ring_char_0 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemiring_627,axiom,
semiring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Osemidom_628,axiom,
semidom @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Orderings_Oord_629,axiom,
ord @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ouminus_630,axiom,
uminus @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oring__1_631,axiom,
ring_1 @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oabs__if_632,axiom,
abs_if @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ominus_633,axiom,
minus @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Power_Opower_634,axiom,
power @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Num_Onumeral_635,axiom,
numeral @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Ozero_636,axiom,
zero @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oplus_637,axiom,
plus @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oring_638,axiom,
ring @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Oidom_639,axiom,
idom @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Groups_Oone_640,axiom,
one @ code_integer ).
thf(tcon_Code__Numeral_Ointeger___Rings_Odvd_641,axiom,
dvd @ code_integer ).
thf(tcon_Code__Numeral_Onatural___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_642,axiom,
bit_un5681908812861735899ations @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Euclidean__Division_Ounique__euclidean__semiring__with__nat_643,axiom,
euclid5411537665997757685th_nat @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Oordered__ab__semigroup__monoid__add__imp__le_644,axiom,
ordere1937475149494474687imp_le @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Euclidean__Division_Ounique__euclidean__semiring_645,axiom,
euclid3128863361964157862miring @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Euclidean__Division_Oeuclidean__semiring__cancel_646,axiom,
euclid4440199948858584721cancel @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Osemiring__no__zero__divisors__cancel_647,axiom,
semiri6575147826004484403cancel @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Ostrict__ordered__ab__semigroup__add_648,axiom,
strict9044650504122735259up_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Oordered__cancel__ab__semigroup__add_649,axiom,
ordere580206878836729694up_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Oordered__ab__semigroup__add__imp__le_650,axiom,
ordere2412721322843649153imp_le @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Bit__Operations_Osemiring__bit__operations_651,axiom,
bit_se359711467146920520ations @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Olinordered__comm__semiring__strict_652,axiom,
linord2810124833399127020strict @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Ostrict__ordered__comm__monoid__add_653,axiom,
strict7427464778891057005id_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Oordered__cancel__comm__monoid__add_654,axiom,
ordere8940638589300402666id_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Euclidean__Division_Oeuclidean__semiring_655,axiom,
euclid3725896446679973847miring @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Osemiring__1__no__zero__divisors_656,axiom,
semiri2026040879449505780visors @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Olinordered__nonzero__semiring_657,axiom,
linord181362715937106298miring @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Olinordered__semiring__strict_658,axiom,
linord8928482502909563296strict @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Osemiring__no__zero__divisors_659,axiom,
semiri3467727345109120633visors @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Oordered__ab__semigroup__add_660,axiom,
ordere6658533253407199908up_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Oordered__comm__monoid__add_661,axiom,
ordere6911136660526730532id_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Ocancel__ab__semigroup__add_662,axiom,
cancel2418104881723323429up_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Ocancel__comm__monoid__add_663,axiom,
cancel1802427076303600483id_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Ocomm__semiring__1__cancel_664,axiom,
comm_s4317794764714335236cancel @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Bit__Operations_Osemiring__bits_665,axiom,
bit_semiring_bits @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Oordered__comm__semiring_666,axiom,
ordere2520102378445227354miring @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Ocancel__semigroup__add_667,axiom,
cancel_semigroup_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Olinordered__semiring_668,axiom,
linordered_semiring @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Oordered__semiring__0_669,axiom,
ordered_semiring_0 @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Olinordered__semidom_670,axiom,
linordered_semidom @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Oab__semigroup__mult_671,axiom,
ab_semigroup_mult @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Osemiring__1__cancel_672,axiom,
semiring_1_cancel @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Oalgebraic__semidom_673,axiom,
algebraic_semidom @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Ocomm__monoid__mult_674,axiom,
comm_monoid_mult @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Ocomm__monoid__diff_675,axiom,
comm_monoid_diff @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Oab__semigroup__add_676,axiom,
ab_semigroup_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Oordered__semiring_677,axiom,
ordered_semiring @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Parity_Osemiring__parity_678,axiom,
semiring_parity @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Ocomm__monoid__add_679,axiom,
comm_monoid_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Osemiring__modulo_680,axiom,
semiring_modulo @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Ocomm__semiring__1_681,axiom,
comm_semiring_1 @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Ocomm__semiring__0_682,axiom,
comm_semiring_0 @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Osemigroup__mult_683,axiom,
semigroup_mult @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Osemidom__modulo_684,axiom,
semidom_modulo @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Osemidom__divide_685,axiom,
semidom_divide @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Num_Osemiring__numeral_686,axiom,
semiring_numeral @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Osemigroup__add_687,axiom,
semigroup_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Ozero__less__one_688,axiom,
zero_less_one @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Ocomm__semiring_689,axiom,
comm_semiring @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Nat_Osemiring__char__0_690,axiom,
semiring_char_0 @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Ozero__neq__one_691,axiom,
zero_neq_one @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Orderings_Opreorder_692,axiom,
preorder @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Orderings_Olinorder_693,axiom,
linorder @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Omonoid__mult_694,axiom,
monoid_mult @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Omonoid__add_695,axiom,
monoid_add @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Osemiring__1_696,axiom,
semiring_1 @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Osemiring__0_697,axiom,
semiring_0 @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Omult__zero_698,axiom,
mult_zero @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Orderings_Oorder_699,axiom,
order @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Osemiring_700,axiom,
semiring @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Osemidom_701,axiom,
semidom @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Orderings_Oord_702,axiom,
ord @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Ominus_703,axiom,
minus @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Power_Opower_704,axiom,
power @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Num_Onumeral_705,axiom,
numeral @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Ozero_706,axiom,
zero @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Oplus_707,axiom,
plus @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Groups_Oone_708,axiom,
one @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Rings_Odvd_709,axiom,
dvd @ code_natural ).
thf(tcon_Code__Numeral_Onatural___Nat_Osize_710,axiom,
size @ code_natural ).
thf(tcon_VEBT__Definitions_OVEBT___Nat_Osize_711,axiom,
size @ vEBT_VEBT ).
% Helper facts (4)
thf(help_If_3_1_T,axiom,
! [P: $o] :
( ( P = $true )
| ( P = $false ) ) ).
thf(help_If_2_1_T,axiom,
! [A: $tType,X: A,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,P: A > $o] :
( ( P @ ( fChoice @ A @ P ) )
= ( ? [X7: A] : ( P @ X7 ) ) ) ).
% Conjectures (2)
thf(conj_0,hypothesis,
vEBT_invar_vebt @ t @ n ).
thf(conj_1,conjecture,
( ( vEBT_vebt_member @ t @ x )
= ( member @ nat @ x @ ( vEBT_set_vebt @ t ) ) ) ).
%------------------------------------------------------------------------------