TPTP Problem File: ITP229^2.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : ITP229^2 : TPTP v8.2.0. Released v8.0.0.
% Domain : Interactive Theorem Proving
% Problem : Sledgehammer problem VEBT_Insert 00617_038319
% 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 : 0066_VEBT_Insert_00617_038319 [Des22]
% Status : Theorem
% Rating : 1.00 v8.1.0
% Syntax : Number of formulae : 9587 (2772 unt; 651 typ; 0 def)
% Number of atoms : 29307 (9756 equ; 5 cnn)
% Maximal formula atoms : 71 ( 3 avg)
% Number of connectives : 178011 (2536 ~; 380 |;2461 &;158661 @)
% ( 0 <=>;13973 =>; 0 <=; 0 <~>)
% Maximal formula depth : 40 ( 8 avg)
% Number of types : 10 ( 9 usr)
% Number of type conns : 4542 (4542 >; 0 *; 0 +; 0 <<)
% Number of symbols : 646 ( 642 usr; 23 con; 0-9 aty)
% Number of variables : 32181 (2718 ^;27792 !;1026 ?;32181 :)
% ( 645 !>; 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 20:03:57.541
%------------------------------------------------------------------------------
% Could-be-implicit typings (16)
thf(ty_t_VEBT__Definitions_OVEBT,type,
vEBT_VEBT: $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_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_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 (635)
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_Groups_Ominus,type,
minus:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oabs__if,type,
abs_if:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oring__1,type,
ring_1:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ouminus,type,
uminus:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Obot,type,
bot:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oord,type,
ord:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Otop,type,
top:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemidom,type,
semidom:
!>[A: $tType] : $o ).
thf(sy_cl_Fields_Oinverse,type,
inverse:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemiring,type,
semiring:
!>[A: $tType] : $o ).
thf(sy_cl_Nat_Oring__char__0,type,
ring_char_0:
!>[A: $tType] : $o ).
thf(sy_cl_Num_Oneg__numeral,type,
neg_numeral:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Oorder,type,
order:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Ocomm__ring,type,
comm_ring:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Omult__zero,type,
mult_zero:
!>[A: $tType] : $o ).
thf(sy_cl_GCD_Osemiring__Gcd,type,
semiring_Gcd:
!>[A: $tType] : $o ).
thf(sy_cl_GCD_Osemiring__gcd,type,
semiring_gcd:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ogroup__add,type,
group_add:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Olattice,type,
lattice:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Ono__bot,type,
no_bot:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Ono__top,type,
no_top:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemiring__0,type,
semiring_0:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemiring__1,type,
semiring_1:
!>[A: $tType] : $o ).
thf(sy_cl_Finite__Set_Ofinite,type,
finite_finite:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Omonoid__add,type,
monoid_add:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Ocomm__ring__1,type,
comm_ring_1:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oidom__divide,type,
idom_divide:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oidom__modulo,type,
idom_modulo:
!>[A: $tType] : $o ).
thf(sy_cl_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_Countable_Ocountable,type,
countable:
!>[A: $tType] : $o ).
thf(sy_cl_Enum_Ofinite__lattice,type,
finite_lattice:
!>[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_Obounded__lattice,type,
bounded_lattice:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Odistrib__lattice,type,
distrib_lattice:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Osemilattice__inf,type,
semilattice_inf:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Osemilattice__sup,type,
semilattice_sup:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Odense__linorder,type,
dense_linorder:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__semidom,type,
linordered_semidom:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oordered__semiring__0,type,
ordered_semiring_0:
!>[A: $tType] : $o ).
thf(sy_cl_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_Complete__Partial__Order_Occpo,type,
comple9053668089753744459l_ccpo:
!>[A: $tType] : $o ).
thf(sy_cl_Enum_Ofinite__distrib__lattice,type,
finite8700451911770168679attice:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocancel__semigroup__add,type,
cancel_semigroup_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__ab__group__add,type,
ordered_ab_group_add:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__semiring__1,type,
linord6961819062388156250ring_1:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Oordered__comm__semiring,type,
ordere2520102378445227354miring:
!>[A: $tType] : $o ).
thf(sy_cl_Topological__Spaces_Ot1__space,type,
topological_t1_space:
!>[A: $tType] : $o ).
thf(sy_cl_Topological__Spaces_Ot2__space,type,
topological_t2_space:
!>[A: $tType] : $o ).
thf(sy_cl_Topological__Spaces_Ot3__space,type,
topological_t3_space:
!>[A: $tType] : $o ).
thf(sy_cl_Topological__Spaces_Ot4__space,type,
topological_t4_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_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_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_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_Topological__Spaces_Operfect__space,type,
topolo8386298272705272623_space:
!>[A: $tType] : $o ).
thf(sy_cl_Topological__Spaces_Ouniform__space,type,
topolo7287701948861334536_space:
!>[A: $tType] : $o ).
thf(sy_cl_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_Groups_Olinordered__ab__semigroup__add,type,
linord4140545234300271783up_add:
!>[A: $tType] : $o ).
thf(sy_cl_Limits_Otopological__comm__monoid__add,type,
topolo5987344860129210374id_add:
!>[A: $tType] : $o ).
thf(sy_cl_Orderings_Ounbounded__dense__linorder,type,
unboun7993243217541854897norder:
!>[A: $tType] : $o ).
thf(sy_cl_Real__Vector__Spaces_Ouniformity__dist,type,
real_V768167426530841204y_dist:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__semiring__1__strict,type,
linord715952674999750819strict:
!>[A: $tType] : $o ).
thf(sy_cl_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_Real__Vector__Spaces_Oreal__div__algebra,type,
real_V5047593784448816457lgebra:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Obounded__semilattice__inf__top,type,
bounde4346867609351753570nf_top:
!>[A: $tType] : $o ).
thf(sy_cl_Lattices_Obounded__semilattice__sup__bot,type,
bounde4967611905675639751up_bot:
!>[A: $tType] : $o ).
thf(sy_cl_Real__Vector__Spaces_Oreal__normed__field,type,
real_V3459762299906320749_field:
!>[A: $tType] : $o ).
thf(sy_cl_Topological__Spaces_Odiscrete__topology,type,
topolo8865339358273720382pology:
!>[A: $tType] : $o ).
thf(sy_cl_Topological__Spaces_Olinorder__topology,type,
topolo1944317154257567458pology:
!>[A: $tType] : $o ).
thf(sy_cl_Topological__Spaces_Otopological__space,type,
topolo4958980785337419405_space:
!>[A: $tType] : $o ).
thf(sy_cl_Euclidean__Division_Oeuclidean__semiring,type,
euclid3725896446679973847miring:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ocanonically__ordered__monoid__add,type,
canoni5634975068530333245id_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__cancel__comm__monoid__add,type,
ordere8940638589300402666id_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ostrict__ordered__comm__monoid__add,type,
strict7427464778891057005id_add:
!>[A: $tType] : $o ).
thf(sy_cl_Real__Vector__Spaces_Oreal__normed__vector,type,
real_V822414075346904944vector:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Olinordered__comm__semiring__strict,type,
linord2810124833399127020strict:
!>[A: $tType] : $o ).
thf(sy_cl_Bit__Operations_Osemiring__bit__operations,type,
bit_se359711467146920520ations:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__ab__semigroup__add__imp__le,type,
ordere2412721322843649153imp_le:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__cancel__ab__semigroup__add,type,
ordere580206878836729694up_add:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__cancel__comm__monoid__diff,type,
ordere1170586879665033532d_diff:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Ostrict__ordered__ab__semigroup__add,type,
strict9044650504122735259up_add:
!>[A: $tType] : $o ).
thf(sy_cl_Real__Vector__Spaces_Oordered__real__vector,type,
real_V5355595471888546746vector:
!>[A: $tType] : $o ).
thf(sy_cl_Real__Vector__Spaces_Oreal__normed__algebra,type,
real_V4412858255891104859lgebra:
!>[A: $tType] : $o ).
thf(sy_cl_Rings_Osemiring__no__zero__divisors__cancel,type,
semiri6575147826004484403cancel:
!>[A: $tType] : $o ).
thf(sy_cl_Euclidean__Division_Oeuclidean__ring__cancel,type,
euclid8851590272496341667cancel:
!>[A: $tType] : $o ).
thf(sy_cl_Real__Vector__Spaces_Oreal__normed__algebra__1,type,
real_V2822296259951069270ebra_1:
!>[A: $tType] : $o ).
thf(sy_cl_Divides_Ounique__euclidean__semiring__numeral,type,
unique1627219031080169319umeral:
!>[A: $tType] : $o ).
thf(sy_cl_Complete__Lattices_Ocomplete__distrib__lattice,type,
comple592849572758109894attice:
!>[A: $tType] : $o ).
thf(sy_cl_Real__Vector__Spaces_Oreal__normed__div__algebra,type,
real_V8999393235501362500lgebra:
!>[A: $tType] : $o ).
thf(sy_cl_Topological__Spaces_Ofirst__countable__topology,type,
topolo3112930676232923870pology:
!>[A: $tType] : $o ).
thf(sy_cl_Euclidean__Division_Oeuclidean__semiring__cancel,type,
euclid4440199948858584721cancel:
!>[A: $tType] : $o ).
thf(sy_cl_Euclidean__Division_Ounique__euclidean__semiring,type,
euclid3128863361964157862miring:
!>[A: $tType] : $o ).
thf(sy_cl_Topological__Spaces_Olinear__continuum__topology,type,
topolo8458572112393995274pology:
!>[A: $tType] : $o ).
thf(sy_cl_Groups_Oordered__ab__semigroup__monoid__add__imp__le,type,
ordere1937475149494474687imp_le:
!>[A: $tType] : $o ).
thf(sy_cl_Conditionally__Complete__Lattices_Olinear__continuum,type,
condit5016429287641298734tinuum:
!>[A: $tType] : $o ).
thf(sy_cl_Euclidean__Division_Ounique__euclidean__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_Countable__Complete__Lattices_Ocountable__complete__distrib__lattice,type,
counta4013691401010221786attice:
!>[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_Ocofinal,type,
bNF_Ca7293521722713021262ofinal:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_BNF__Cardinal__Order__Relation_OnatLeq,type,
bNF_Ca8665028551170535155natLeq: set @ ( product_prod @ nat @ nat ) ).
thf(sy_c_BNF__Cardinal__Order__Relation_OnatLess,type,
bNF_Ca8459412986667044542atLess: set @ ( product_prod @ nat @ nat ) ).
thf(sy_c_BNF__Cardinal__Order__Relation_OrelChain,type,
bNF_Ca3754400796208372196lChain:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( A > B ) > $o ) ).
thf(sy_c_BNF__Greatest__Fixpoint_Oimage2,type,
bNF_Greatest_image2:
!>[C: $tType,A: $tType,B: $tType] : ( ( set @ C ) > ( C > A ) > ( C > B ) > ( set @ ( product_prod @ A @ B ) ) ) ).
thf(sy_c_BNF__Wellorder__Constructions_OFunc,type,
bNF_Wellorder_Func:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( set @ B ) > ( set @ ( A > B ) ) ) ).
thf(sy_c_BNF__Wellorder__Constructions_OFunc__map,type,
bNF_We4925052301507509544nc_map:
!>[B: $tType,C: $tType,A: $tType,D: $tType] : ( ( set @ B ) > ( C > A ) > ( B > D ) > ( D > C ) > B > A ) ).
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__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_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_Oring__bit__operations__class_Onot,type,
bit_ri4277139882892585799ns_not:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Bit__Operations_Oring__bit__operations__class_Osigned__take__bit,type,
bit_ri4674362597316999326ke_bit:
!>[A: $tType] : ( nat > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oand,type,
bit_se5824344872417868541ns_and:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Odrop__bit,type,
bit_se4197421643247451524op_bit:
!>[A: $tType] : ( nat > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oflip__bit,type,
bit_se8732182000553998342ip_bit:
!>[A: $tType] : ( nat > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Omask,type,
bit_se2239418461657761734s_mask:
!>[A: $tType] : ( nat > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oor,type,
bit_se1065995026697491101ons_or:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Opush__bit,type,
bit_se4730199178511100633sh_bit:
!>[A: $tType] : ( nat > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oset__bit,type,
bit_se5668285175392031749et_bit:
!>[A: $tType] : ( nat > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Otake__bit,type,
bit_se2584673776208193580ke_bit:
!>[A: $tType] : ( nat > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Ounset__bit,type,
bit_se2638667681897837118et_bit:
!>[A: $tType] : ( nat > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oxor,type,
bit_se5824344971392196577ns_xor:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Bit__Operations_Osemiring__bits__class_Obit,type,
bit_se5641148757651400278ts_bit:
!>[A: $tType] : ( A > nat > $o ) ).
thf(sy_c_Bit__Operations_Otake__bit__num,type,
bit_take_bit_num: nat > num > ( option @ num ) ).
thf(sy_c_Boolean__Algebras_Oabstract__boolean__algebra,type,
boolea2506097494486148201lgebra:
!>[A: $tType] : ( ( A > A > A ) > ( A > A > A ) > ( A > A ) > A > A > $o ) ).
thf(sy_c_Boolean__Algebras_Oabstract__boolean__algebra__sym__diff,type,
boolea3799213064322606851m_diff:
!>[A: $tType] : ( ( A > A > A ) > ( A > A > A ) > ( A > A ) > A > A > ( A > A > A ) > $o ) ).
thf(sy_c_Complete__Lattices_OInf__class_OInf,type,
complete_Inf_Inf:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_Complete__Lattices_OSup__class_OSup,type,
complete_Sup_Sup:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_Complete__Partial__Order_Occpo__class_Oiteratesp,type,
comple7512665784863727008ratesp:
!>[A: $tType] : ( ( A > A ) > A > $o ) ).
thf(sy_c_Complete__Partial__Order_Ochain,type,
comple1602240252501008431_chain:
!>[A: $tType] : ( ( A > A > $o ) > ( set @ A ) > $o ) ).
thf(sy_c_Complex_OArg,type,
arg: complex > real ).
thf(sy_c_Complex_Ocis,type,
cis: real > complex ).
thf(sy_c_Complex_Ocomplex_OComplex,type,
complex2: 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_Countable__Set_Ocountable,type,
countable_countable:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Countable__Set_Ofrom__nat__into,type,
counta4804993851260445106t_into:
!>[A: $tType] : ( ( set @ A ) > nat > A ) ).
thf(sy_c_Countable__Set_Oto__nat__on,type,
countable_to_nat_on:
!>[A: $tType] : ( ( set @ A ) > A > nat ) ).
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_Odivmod__nat,type,
divmod_nat: nat > nat > ( product_prod @ nat @ nat ) ).
thf(sy_c_Divides_Oeucl__rel__int,type,
eucl_rel_int: int > int > ( product_prod @ int @ int ) > $o ).
thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivides__aux,type,
unique5940410009612947441es_aux:
!>[A: $tType] : ( ( product_prod @ A @ A ) > $o ) ).
thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod,type,
unique8689654367752047608divmod:
!>[A: $tType] : ( num > num > ( product_prod @ A @ A ) ) ).
thf(sy_c_Divides_Ounique__euclidean__semiring__numeral__class_Odivmod__step,type,
unique1321980374590559556d_step:
!>[A: $tType] : ( num > ( product_prod @ A @ A ) > ( product_prod @ A @ A ) ) ).
thf(sy_c_Equiv__Relations_Ocongruent,type,
equiv_congruent:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( A > B ) > $o ) ).
thf(sy_c_Equiv__Relations_Ocongruent2,type,
equiv_congruent2:
!>[A: $tType,B: $tType,C: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ B @ B ) ) > ( A > B > C ) > $o ) ).
thf(sy_c_Equiv__Relations_Oequiv,type,
equiv_equiv:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Equiv__Relations_Oproj,type,
equiv_proj:
!>[B: $tType,A: $tType] : ( ( set @ ( product_prod @ B @ A ) ) > B > ( set @ A ) ) ).
thf(sy_c_Equiv__Relations_Oquotient,type,
equiv_quotient:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > ( set @ ( set @ A ) ) ) ).
thf(sy_c_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_Ofinite__subsets__at__top,type,
finite5375528669736107172at_top:
!>[A: $tType] : ( ( set @ A ) > ( filter @ ( set @ A ) ) ) ).
thf(sy_c_Filter_Oprincipal,type,
principal:
!>[A: $tType] : ( ( set @ A ) > ( filter @ A ) ) ).
thf(sy_c_Filter_Oprod__filter,type,
prod_filter:
!>[A: $tType,B: $tType] : ( ( filter @ A ) > ( filter @ B ) > ( filter @ ( product_prod @ A @ B ) ) ) ).
thf(sy_c_Finite__Set_OFpow,type,
finite_Fpow:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( set @ A ) ) ) ).
thf(sy_c_Finite__Set_Ocard,type,
finite_card:
!>[B: $tType] : ( ( set @ B ) > nat ) ).
thf(sy_c_Finite__Set_Ocomp__fun__commute,type,
finite6289374366891150609ommute:
!>[A: $tType,B: $tType] : ( ( A > B > B ) > $o ) ).
thf(sy_c_Finite__Set_Ocomp__fun__commute__on,type,
finite4664212375090638736ute_on:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B > B ) > $o ) ).
thf(sy_c_Finite__Set_Ocomp__fun__idem__on,type,
finite673082921795544331dem_on:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B > B ) > $o ) ).
thf(sy_c_Finite__Set_Ocomp__fun__idem__on__axioms,type,
finite4980608107308702382axioms:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B > B ) > $o ) ).
thf(sy_c_Finite__Set_Ofinite,type,
finite_finite2:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Finite__Set_Ofold,type,
finite_fold:
!>[A: $tType,B: $tType] : ( ( A > B > B ) > B > ( set @ A ) > B ) ).
thf(sy_c_Finite__Set_Ofold__graph,type,
finite_fold_graph:
!>[A: $tType,B: $tType] : ( ( A > B > B ) > B > ( set @ A ) > B > $o ) ).
thf(sy_c_Finite__Set_Ofolding__idem__on,type,
finite1890593828518410140dem_on:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B > B ) > $o ) ).
thf(sy_c_Finite__Set_Ofolding__idem__on__axioms,type,
finite6916993218817215295axioms:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B > 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_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_Omax__strict,type,
fun_max_strict: set @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) ).
thf(sy_c_Fun__Def_Omax__weak,type,
fun_max_weak: set @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) ).
thf(sy_c_Fun__Def_Omin__strict,type,
fun_min_strict: set @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) ).
thf(sy_c_Fun__Def_Omin__weak,type,
fun_min_weak: set @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) ).
thf(sy_c_Fun__Def_Opair__leq,type,
fun_pair_leq: set @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) ).
thf(sy_c_Fun__Def_Opair__less,type,
fun_pair_less: set @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) ).
thf(sy_c_Fun__Def_Oreduction__pair,type,
fun_reduction_pair:
!>[A: $tType] : ( ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) > $o ) ).
thf(sy_c_Fun__Def_Orp__inv__image,type,
fun_rp_inv_image:
!>[A: $tType,B: $tType] : ( ( product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) ) > ( B > A ) > ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) ) ) ).
thf(sy_c_GCD_OGcd__class_OGcd,type,
gcd_Gcd:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_GCD_Obezw,type,
bezw: nat > nat > ( product_prod @ int @ int ) ).
thf(sy_c_GCD_Obezw__rel,type,
bezw_rel: ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) > $o ).
thf(sy_c_GCD_Ogcd__class_Ogcd,type,
gcd_gcd:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_GCD_Ogcd__nat__rel,type,
gcd_nat_rel: ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) > $o ).
thf(sy_c_GCD_Osemiring__1__class_Osemiring__char,type,
semiri4206861660011772517g_char:
!>[A: $tType] : ( ( itself @ A ) > nat ) ).
thf(sy_c_Groups_Oabs__class_Oabs,type,
abs_abs:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Groups_Ominus__class_Ominus,type,
minus_minus:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Groups_Oone__class_Oone,type,
one_one:
!>[A: $tType] : A ).
thf(sy_c_Groups_Oplus__class_Oplus,type,
plus_plus:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Groups_Osgn__class_Osgn,type,
sgn_sgn:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Groups_Otimes__class_Otimes,type,
times_times:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Groups_Ouminus__class_Ouminus,type,
uminus_uminus:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Groups_Ozero__class_Ozero,type,
zero_zero:
!>[A: $tType] : A ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum,type,
groups7311177749621191930dd_sum:
!>[B: $tType,A: $tType] : ( ( B > A ) > ( set @ B ) > A ) ).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_H,type,
groups1027152243600224163dd_sum:
!>[C: $tType,A: $tType] : ( ( C > A ) > ( set @ C ) > A ) ).
thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod,type,
groups7121269368397514597t_prod:
!>[B: $tType,A: $tType] : ( ( B > A ) > ( set @ B ) > A ) ).
thf(sy_c_Groups__Big_Ocomm__monoid__mult__class_Oprod_H,type,
groups1962203154675924110t_prod:
!>[C: $tType,A: $tType] : ( ( C > A ) > ( set @ C ) > A ) ).
thf(sy_c_Groups__List_Ocomm__semiring__0__class_Ohorner__sum,type,
groups4207007520872428315er_sum:
!>[B: $tType,A: $tType] : ( ( B > A ) > A > ( list @ B ) > A ) ).
thf(sy_c_Groups__List_Omonoid__add__class_Osum__list,type,
groups8242544230860333062m_list:
!>[A: $tType] : ( ( list @ A ) > A ) ).
thf(sy_c_Groups__List_Omonoid__mult__class_Oprod__list,type,
groups5270119922927024881d_list:
!>[A: $tType] : ( ( list @ A ) > A ) ).
thf(sy_c_HOL_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_Oundefined,type,
undefined:
!>[A: $tType] : A ).
thf(sy_c_If,type,
if:
!>[A: $tType] : ( $o > A > A > A ) ).
thf(sy_c_Inductive_Ocomplete__lattice__class_Olfp,type,
complete_lattice_lfp:
!>[A: $tType] : ( ( A > A ) > A ) ).
thf(sy_c_Infinite__Set_Owellorder__class_Oenumerate,type,
infini527867602293511546merate:
!>[A: $tType] : ( ( set @ A ) > nat > A ) ).
thf(sy_c_Int_OAbs__Integ,type,
abs_Integ: ( product_prod @ nat @ nat ) > int ).
thf(sy_c_Int_ORep__Integ,type,
rep_Integ: int > ( product_prod @ nat @ nat ) ).
thf(sy_c_Int_Oint__ge__less__than,type,
int_ge_less_than: int > ( set @ ( product_prod @ int @ int ) ) ).
thf(sy_c_Int_Oint__ge__less__than2,type,
int_ge_less_than2: int > ( set @ ( product_prod @ int @ int ) ) ).
thf(sy_c_Int_Onat,type,
nat2: int > nat ).
thf(sy_c_Int_Opower__int,type,
power_int:
!>[A: $tType] : ( A > int > A ) ).
thf(sy_c_Int_Oring__1__class_OInts,type,
ring_1_Ints:
!>[A: $tType] : ( set @ A ) ).
thf(sy_c_Int_Oring__1__class_Oof__int,type,
ring_1_of_int:
!>[A: $tType] : ( int > A ) ).
thf(sy_c_Lattices_Oinf__class_Oinf,type,
inf_inf:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Lattices_Osemilattice__neutr__order,type,
semila1105856199041335345_order:
!>[A: $tType] : ( ( A > A > A ) > A > ( A > A > $o ) > ( A > A > $o ) > $o ) ).
thf(sy_c_Lattices_Osup__class_Osup,type,
sup_sup:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Lattices__Big_Olinorder_OMax,type,
lattices_Max:
!>[A: $tType] : ( ( A > A > $o ) > ( set @ A ) > A ) ).
thf(sy_c_Lattices__Big_Olinorder_OMin,type,
lattices_Min:
!>[A: $tType] : ( ( A > A > $o ) > ( set @ A ) > A ) ).
thf(sy_c_Lattices__Big_Olinorder__class_OMax,type,
lattic643756798349783984er_Max:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_Lattices__Big_Olinorder__class_OMin,type,
lattic643756798350308766er_Min:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_Lattices__Big_Oord__class_Oarg__min__on,type,
lattic7623131987881927897min_on:
!>[B: $tType,A: $tType] : ( ( B > A ) > ( set @ B ) > B ) ).
thf(sy_c_Lattices__Big_Osemilattice__inf__class_OInf__fin,type,
lattic7752659483105999362nf_fin:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_Lattices__Big_Osemilattice__sup__class_OSup__fin,type,
lattic5882676163264333800up_fin:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_Limits_OBfun,type,
bfun:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( filter @ A ) > $o ) ).
thf(sy_c_Limits_Oat__infinity,type,
at_infinity:
!>[A: $tType] : ( filter @ A ) ).
thf(sy_c_List_OBleast,type,
bleast:
!>[A: $tType] : ( ( set @ A ) > ( A > $o ) > A ) ).
thf(sy_c_List_Oabort__Bleast,type,
abort_Bleast:
!>[A: $tType] : ( ( set @ A ) > ( A > $o ) > A ) ).
thf(sy_c_List_Oappend,type,
append:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Oarg__min__list,type,
arg_min_list:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( list @ A ) > A ) ).
thf(sy_c_List_Obind,type,
bind:
!>[A: $tType,B: $tType] : ( ( list @ A ) > ( A > ( list @ B ) ) > ( list @ B ) ) ).
thf(sy_c_List_Obutlast,type,
butlast:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Oconcat,type,
concat:
!>[A: $tType] : ( ( list @ ( list @ A ) ) > ( list @ A ) ) ).
thf(sy_c_List_Ocoset,type,
coset:
!>[A: $tType] : ( ( list @ A ) > ( set @ A ) ) ).
thf(sy_c_List_Ocount__list,type,
count_list:
!>[A: $tType] : ( ( list @ A ) > A > nat ) ).
thf(sy_c_List_Odistinct,type,
distinct:
!>[A: $tType] : ( ( list @ A ) > $o ) ).
thf(sy_c_List_Odrop,type,
drop:
!>[A: $tType] : ( nat > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_OdropWhile,type,
dropWhile:
!>[A: $tType] : ( ( A > $o ) > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Oenumerate,type,
enumerate:
!>[A: $tType] : ( nat > ( list @ A ) > ( list @ ( product_prod @ nat @ A ) ) ) ).
thf(sy_c_List_Oextract,type,
extract:
!>[A: $tType] : ( ( A > $o ) > ( list @ A ) > ( option @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) ) ) ).
thf(sy_c_List_Ofilter,type,
filter2:
!>[A: $tType] : ( ( A > $o ) > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Ofind,type,
find:
!>[A: $tType] : ( ( A > $o ) > ( list @ A ) > ( option @ A ) ) ).
thf(sy_c_List_Ofold,type,
fold:
!>[A: $tType,B: $tType] : ( ( A > B > B ) > ( list @ A ) > B > B ) ).
thf(sy_c_List_Ofolding__insort__key,type,
folding_insort_key:
!>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( A > A > $o ) > ( set @ B ) > ( B > A ) > $o ) ).
thf(sy_c_List_Ofoldl,type,
foldl:
!>[B: $tType,A: $tType] : ( ( B > A > B ) > B > ( list @ A ) > B ) ).
thf(sy_c_List_Ofoldr,type,
foldr:
!>[A: $tType,B: $tType] : ( ( A > B > B ) > ( list @ A ) > B > B ) ).
thf(sy_c_List_Olenlex,type,
lenlex:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ) ).
thf(sy_c_List_Olex,type,
lex:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ) ).
thf(sy_c_List_Olexn,type,
lexn:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > nat > ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ) ).
thf(sy_c_List_Olexord,type,
lexord:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ) ).
thf(sy_c_List_Olexordp,type,
lexordp:
!>[A: $tType] : ( ( A > A > $o ) > ( list @ A ) > ( list @ A ) > $o ) ).
thf(sy_c_List_Olinorder_Oinsort__key,type,
insort_key:
!>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( B > A ) > B > ( list @ B ) > ( list @ B ) ) ).
thf(sy_c_List_Olinorder_Osorted__key__list__of__set,type,
sorted8670434370408473282of_set:
!>[A: $tType,B: $tType] : ( ( A > A > $o ) > ( B > A ) > ( set @ B ) > ( list @ B ) ) ).
thf(sy_c_List_Olinorder__class_Oinsort__key,type,
linorder_insort_key:
!>[B: $tType,A: $tType] : ( ( B > A ) > B > ( list @ B ) > ( list @ B ) ) ).
thf(sy_c_List_Olinorder__class_Osort__key,type,
linorder_sort_key:
!>[B: $tType,A: $tType] : ( ( B > A ) > ( list @ B ) > ( list @ B ) ) ).
thf(sy_c_List_Olinorder__class_Osorted__key__list__of__set,type,
linord144544945434240204of_set:
!>[B: $tType,A: $tType] : ( ( B > A ) > ( set @ B ) > ( list @ B ) ) ).
thf(sy_c_List_Olinorder__class_Osorted__list__of__set,type,
linord4507533701916653071of_set:
!>[A: $tType] : ( ( set @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Olist_OCons,type,
cons:
!>[A: $tType] : ( A > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Olist_ONil,type,
nil:
!>[A: $tType] : ( list @ A ) ).
thf(sy_c_List_Olist_Ocase__list,type,
case_list:
!>[B: $tType,A: $tType] : ( B > ( A > ( list @ A ) > B ) > ( list @ A ) > B ) ).
thf(sy_c_List_Olist_Ohd,type,
hd:
!>[A: $tType] : ( ( list @ A ) > A ) ).
thf(sy_c_List_Olist_Olist__all2,type,
list_all2:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > ( list @ A ) > ( list @ B ) > $o ) ).
thf(sy_c_List_Olist_Omap,type,
map:
!>[A: $tType,Aa: $tType] : ( ( A > Aa ) > ( list @ A ) > ( list @ Aa ) ) ).
thf(sy_c_List_Olist_Oset,type,
set2:
!>[A: $tType] : ( ( list @ A ) > ( set @ A ) ) ).
thf(sy_c_List_Olist_Osize__list,type,
size_list:
!>[A: $tType] : ( ( A > nat ) > ( list @ A ) > nat ) ).
thf(sy_c_List_Olist_Otl,type,
tl:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Olist__update,type,
list_update:
!>[A: $tType] : ( ( list @ A ) > nat > A > ( list @ A ) ) ).
thf(sy_c_List_Olistrel,type,
listrel:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( set @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) ) ).
thf(sy_c_List_Olistrel1,type,
listrel1:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ) ).
thf(sy_c_List_Olistrel1p,type,
listrel1p:
!>[A: $tType] : ( ( A > A > $o ) > ( list @ A ) > ( list @ A ) > $o ) ).
thf(sy_c_List_Olistrelp,type,
listrelp:
!>[A: $tType,B: $tType] : ( ( A > B > $o ) > ( list @ A ) > ( list @ B ) > $o ) ).
thf(sy_c_List_Olists,type,
lists:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( list @ A ) ) ) ).
thf(sy_c_List_Olistset,type,
listset:
!>[A: $tType] : ( ( list @ ( set @ A ) ) > ( set @ ( list @ A ) ) ) ).
thf(sy_c_List_Omap__filter,type,
map_filter:
!>[A: $tType,B: $tType] : ( ( A > ( option @ B ) ) > ( list @ A ) > ( list @ B ) ) ).
thf(sy_c_List_Omap__project,type,
map_project:
!>[A: $tType,B: $tType] : ( ( A > ( option @ B ) ) > ( set @ A ) > ( set @ B ) ) ).
thf(sy_c_List_Omin__list,type,
min_list:
!>[A: $tType] : ( ( list @ A ) > A ) ).
thf(sy_c_List_Omin__list__rel,type,
min_list_rel:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) > $o ) ).
thf(sy_c_List_On__lists,type,
n_lists:
!>[A: $tType] : ( nat > ( list @ A ) > ( list @ ( list @ A ) ) ) ).
thf(sy_c_List_Onth,type,
nth:
!>[A: $tType] : ( ( list @ A ) > nat > A ) ).
thf(sy_c_List_Onths,type,
nths:
!>[A: $tType] : ( ( list @ A ) > ( set @ nat ) > ( list @ A ) ) ).
thf(sy_c_List_Oord_Olexordp,type,
lexordp2:
!>[A: $tType] : ( ( A > A > $o ) > ( list @ A ) > ( list @ A ) > $o ) ).
thf(sy_c_List_Oord__class_Olexordp,type,
ord_lexordp:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) > $o ) ).
thf(sy_c_List_Opartition,type,
partition:
!>[A: $tType] : ( ( A > $o ) > ( list @ A ) > ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ).
thf(sy_c_List_Oproduct,type,
product:
!>[A: $tType,B: $tType] : ( ( list @ A ) > ( list @ B ) > ( list @ ( product_prod @ A @ B ) ) ) ).
thf(sy_c_List_Oproduct__lists,type,
product_lists:
!>[A: $tType] : ( ( list @ ( list @ A ) ) > ( list @ ( list @ A ) ) ) ).
thf(sy_c_List_Oremdups,type,
remdups:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Oremdups__adj,type,
remdups_adj:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Oremdups__adj__rel,type,
remdups_adj_rel:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) > $o ) ).
thf(sy_c_List_Oremove1,type,
remove1:
!>[A: $tType] : ( A > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_OremoveAll,type,
removeAll:
!>[A: $tType] : ( A > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Oreplicate,type,
replicate:
!>[A: $tType] : ( nat > A > ( list @ A ) ) ).
thf(sy_c_List_Orev,type,
rev:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Orotate,type,
rotate:
!>[A: $tType] : ( nat > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Orotate1,type,
rotate1:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Oset__Cons,type,
set_Cons:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( list @ A ) ) > ( set @ ( list @ A ) ) ) ).
thf(sy_c_List_Oshuffles,type,
shuffles:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) > ( set @ ( list @ A ) ) ) ).
thf(sy_c_List_Oshuffles__rel,type,
shuffles_rel:
!>[A: $tType] : ( ( product_prod @ ( list @ A ) @ ( list @ A ) ) > ( product_prod @ ( list @ A ) @ ( list @ A ) ) > $o ) ).
thf(sy_c_List_Osorted__wrt,type,
sorted_wrt:
!>[A: $tType] : ( ( A > A > $o ) > ( list @ A ) > $o ) ).
thf(sy_c_List_Osplice,type,
splice:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Osplice__rel,type,
splice_rel:
!>[A: $tType] : ( ( product_prod @ ( list @ A ) @ ( list @ A ) ) > ( product_prod @ ( list @ A ) @ ( list @ A ) ) > $o ) ).
thf(sy_c_List_Osubseqs,type,
subseqs:
!>[A: $tType] : ( ( list @ A ) > ( list @ ( list @ A ) ) ) ).
thf(sy_c_List_Otake,type,
take:
!>[A: $tType] : ( nat > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_OtakeWhile,type,
takeWhile:
!>[A: $tType] : ( ( A > $o ) > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Otranspose,type,
transpose:
!>[A: $tType] : ( ( list @ ( list @ A ) ) > ( list @ ( list @ A ) ) ) ).
thf(sy_c_List_Otranspose__rel,type,
transpose_rel:
!>[A: $tType] : ( ( list @ ( list @ A ) ) > ( list @ ( list @ A ) ) > $o ) ).
thf(sy_c_List_Ounion,type,
union:
!>[A: $tType] : ( ( list @ A ) > ( list @ A ) > ( list @ A ) ) ).
thf(sy_c_List_Oupt,type,
upt: nat > nat > ( list @ nat ) ).
thf(sy_c_List_Oupto,type,
upto: int > int > ( list @ int ) ).
thf(sy_c_List_Oupto__aux,type,
upto_aux: int > int > ( list @ int ) > ( list @ int ) ).
thf(sy_c_List_Oupto__rel,type,
upto_rel: ( product_prod @ int @ int ) > ( product_prod @ int @ int ) > $o ).
thf(sy_c_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__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__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__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_Oinc,type,
inc: num > num ).
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_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_Osize__num,type,
size_num: num > nat ).
thf(sy_c_Num_Onum__of__nat,type,
num_of_nat: nat > num ).
thf(sy_c_Num_Onumeral__class_Onumeral,type,
numeral_numeral:
!>[A: $tType] : ( num > A ) ).
thf(sy_c_Num_Opow,type,
pow: num > num > num ).
thf(sy_c_Num_Opred__numeral,type,
pred_numeral: num > nat ).
thf(sy_c_Option_Ooption_ONone,type,
none:
!>[A: $tType] : ( option @ A ) ).
thf(sy_c_Option_Ooption_OSome,type,
some:
!>[A: $tType] : ( A > ( option @ A ) ) ).
thf(sy_c_Option_Ooption_Ocase__option,type,
case_option:
!>[B: $tType,A: $tType] : ( B > ( A > B ) > ( option @ A ) > B ) ).
thf(sy_c_Option_Ooption_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__Continuity_Ocountable__complete__lattice__class_Occlfp,type,
order_532582986084564980_cclfp:
!>[A: $tType] : ( ( A > A ) > A ) ).
thf(sy_c_Order__Continuity_Osup__continuous,type,
order_sup_continuous:
!>[A: $tType,B: $tType] : ( ( A > B ) > $o ) ).
thf(sy_c_Order__Relation_OAbove,type,
order_Above:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ A ) > ( set @ A ) ) ).
thf(sy_c_Order__Relation_OUnder,type,
order_Under:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ A ) > ( set @ A ) ) ).
thf(sy_c_Order__Relation_OUnderS,type,
order_UnderS:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ A ) > ( set @ A ) ) ).
thf(sy_c_Order__Relation_OaboveS,type,
order_aboveS:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > A > ( set @ A ) ) ).
thf(sy_c_Order__Relation_Olinear__order__on,type,
order_679001287576687338der_on:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Order__Relation_Opartial__order__on,type,
order_7125193373082350890der_on:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Order__Relation_Opreorder__on,type,
order_preorder_on:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Order__Relation_Ounder,type,
order_under:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > A > ( set @ A ) ) ).
thf(sy_c_Order__Relation_OunderS,type,
order_underS:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > A > ( set @ A ) ) ).
thf(sy_c_Orderings_Obot__class_Obot,type,
bot_bot:
!>[A: $tType] : A ).
thf(sy_c_Orderings_Oord_OLeast,type,
least:
!>[A: $tType] : ( ( A > A > $o ) > ( A > $o ) > A ) ).
thf(sy_c_Orderings_Oord_Omax,type,
max:
!>[A: $tType] : ( ( A > A > $o ) > A > A > A ) ).
thf(sy_c_Orderings_Oord_Omin,type,
min:
!>[A: $tType] : ( ( A > A > $o ) > A > A > A ) ).
thf(sy_c_Orderings_Oord__class_OLeast,type,
ord_Least:
!>[A: $tType] : ( ( A > $o ) > A ) ).
thf(sy_c_Orderings_Oord__class_Oless,type,
ord_less:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Orderings_Oord__class_Oless__eq,type,
ord_less_eq:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Orderings_Oord__class_Omax,type,
ord_max:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Orderings_Oord__class_Omin,type,
ord_min:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Orderings_Oorder__class_OGreatest,type,
order_Greatest:
!>[A: $tType] : ( ( A > $o ) > A ) ).
thf(sy_c_Orderings_Oorder__class_Oantimono,type,
order_antimono:
!>[A: $tType,B: $tType] : ( ( A > B ) > $o ) ).
thf(sy_c_Orderings_Oorder__class_Omono,type,
order_mono:
!>[A: $tType,B: $tType] : ( ( A > B ) > $o ) ).
thf(sy_c_Orderings_Oorder__class_Ostrict__mono,type,
order_strict_mono:
!>[A: $tType,B: $tType] : ( ( A > B ) > $o ) ).
thf(sy_c_Orderings_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_Oapfst,type,
product_apfst:
!>[A: $tType,C: $tType,B: $tType] : ( ( A > C ) > ( product_prod @ A @ B ) > ( product_prod @ C @ B ) ) ).
thf(sy_c_Product__Type_Oapsnd,type,
product_apsnd:
!>[B: $tType,C: $tType,A: $tType] : ( ( B > C ) > ( product_prod @ A @ B ) > ( product_prod @ A @ C ) ) ).
thf(sy_c_Product__Type_Omap__prod,type,
product_map_prod:
!>[A: $tType,C: $tType,B: $tType,D: $tType] : ( ( A > C ) > ( B > D ) > ( product_prod @ A @ B ) > ( product_prod @ C @ D ) ) ).
thf(sy_c_Product__Type_Oold_Oprod_Orec__prod,type,
product_rec_prod:
!>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T ) ).
thf(sy_c_Product__Type_Oold_Oprod_Orec__set__prod,type,
product_rec_set_prod:
!>[A: $tType,B: $tType,T: $tType] : ( ( A > B > T ) > ( product_prod @ A @ B ) > T > $o ) ).
thf(sy_c_Product__Type_Oprod_Ocase__prod,type,
product_case_prod:
!>[A: $tType,B: $tType,C: $tType] : ( ( A > B > C ) > ( product_prod @ A @ B ) > C ) ).
thf(sy_c_Product__Type_Oprod_Ofst,type,
product_fst:
!>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > A ) ).
thf(sy_c_Product__Type_Oprod_Osnd,type,
product_snd:
!>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > B ) ).
thf(sy_c_Product__Type_Oprod_Oswap,type,
product_swap:
!>[A: $tType,B: $tType] : ( ( product_prod @ A @ B ) > ( product_prod @ B @ A ) ) ).
thf(sy_c_Product__Type_Oproduct,type,
product_product:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( set @ B ) > ( set @ ( product_prod @ A @ B ) ) ) ).
thf(sy_c_Rat_OFract,type,
fract: 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_Onormalize,type,
normalize: ( product_prod @ int @ int ) > ( product_prod @ int @ int ) ).
thf(sy_c_Rat_Opositive,type,
positive: rat > $o ).
thf(sy_c_Real_OReal,type,
real2: ( nat > rat ) > real ).
thf(sy_c_Real_Ocauchy,type,
cauchy: ( nat > rat ) > $o ).
thf(sy_c_Real_Opositive,type,
positive2: real > $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_Obounded__linear,type,
real_V3181309239436604168linear:
!>[A: $tType,B: $tType] : ( ( A > B ) > $o ) ).
thf(sy_c_Real__Vector__Spaces_Odist__class_Odist,type,
real_V557655796197034286t_dist:
!>[A: $tType] : ( A > A > real ) ).
thf(sy_c_Real__Vector__Spaces_Onorm__class_Onorm,type,
real_V7770717601297561774m_norm:
!>[A: $tType] : ( A > real ) ).
thf(sy_c_Real__Vector__Spaces_Oof__real,type,
real_Vector_of_real:
!>[A: $tType] : ( real > A ) ).
thf(sy_c_Real__Vector__Spaces_OscaleR__class_OscaleR,type,
real_V8093663219630862766scaleR:
!>[A: $tType] : ( real > A > A ) ).
thf(sy_c_Relation_OField,type,
field2:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ A ) ) ).
thf(sy_c_Relation_OId,type,
id2:
!>[A: $tType] : ( set @ ( product_prod @ A @ A ) ) ).
thf(sy_c_Relation_OId__on,type,
id_on:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( product_prod @ A @ A ) ) ) ).
thf(sy_c_Relation_OImage,type,
image:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ B ) ) > ( set @ A ) > ( set @ B ) ) ).
thf(sy_c_Relation_Oantisym,type,
antisym:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Relation_Oinv__image,type,
inv_image:
!>[B: $tType,A: $tType] : ( ( set @ ( product_prod @ B @ B ) ) > ( A > B ) > ( set @ ( product_prod @ A @ A ) ) ) ).
thf(sy_c_Relation_Oirrefl,type,
irrefl:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Relation_Orefl__on,type,
refl_on:
!>[A: $tType] : ( ( set @ A ) > ( 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_Rings_Odivide__class_Odivide,type,
divide_divide:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Rings_Odvd__class_Odvd,type,
dvd_dvd:
!>[A: $tType] : ( A > A > $o ) ).
thf(sy_c_Rings_Omodulo__class_Omodulo,type,
modulo_modulo:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Rings_Ozero__neq__one__class_Oof__bool,type,
zero_neq_one_of_bool:
!>[A: $tType] : ( $o > A ) ).
thf(sy_c_Series_Osuminf,type,
suminf:
!>[A: $tType] : ( ( nat > A ) > A ) ).
thf(sy_c_Series_Osummable,type,
summable:
!>[A: $tType] : ( ( nat > A ) > $o ) ).
thf(sy_c_Series_Osums,type,
sums:
!>[A: $tType] : ( ( nat > A ) > A > $o ) ).
thf(sy_c_Set_OCollect,type,
collect:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) ) ).
thf(sy_c_Set_OPow,type,
pow2:
!>[A: $tType] : ( ( set @ A ) > ( set @ ( set @ A ) ) ) ).
thf(sy_c_Set_Ofilter,type,
filter3:
!>[A: $tType] : ( ( A > $o ) > ( set @ A ) > ( set @ A ) ) ).
thf(sy_c_Set_Oimage,type,
image2:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ A ) > ( set @ B ) ) ).
thf(sy_c_Set_Oinsert,type,
insert:
!>[A: $tType] : ( A > ( set @ A ) > ( set @ A ) ) ).
thf(sy_c_Set_Ois__singleton,type,
is_singleton:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Set_Oremove,type,
remove:
!>[A: $tType] : ( A > ( set @ A ) > ( set @ A ) ) ).
thf(sy_c_Set_Othe__elem,type,
the_elem:
!>[A: $tType] : ( ( set @ A ) > A ) ).
thf(sy_c_Set_Ovimage,type,
vimage:
!>[A: $tType,B: $tType] : ( ( A > B ) > ( set @ B ) > ( set @ A ) ) ).
thf(sy_c_Set__Interval_Ofold__atLeastAtMost__nat,type,
set_fo6178422350223883121st_nat:
!>[A: $tType] : ( ( nat > A > A ) > nat > nat > A > A ) ).
thf(sy_c_Set__Interval_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_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_Oclosed,type,
topolo7761053866217962861closed:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Topological__Spaces_Otopological__space__class_Ocompact,type,
topolo2193935891317330818ompact:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Topological__Spaces_Otopological__space__class_Onhds,type,
topolo7230453075368039082e_nhds:
!>[A: $tType] : ( A > ( filter @ A ) ) ).
thf(sy_c_Topological__Spaces_Ouniform__space__class_OCauchy,type,
topolo3814608138187158403Cauchy:
!>[A: $tType] : ( ( nat > A ) > $o ) ).
thf(sy_c_Topological__Spaces_Ouniform__space__class_Ocauchy__filter,type,
topolo6773858410816713723filter:
!>[A: $tType] : ( ( filter @ A ) > $o ) ).
thf(sy_c_Topological__Spaces_Ouniform__space__class_Ocomplete,type,
topolo2479028161051973599mplete:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Topological__Spaces_Ouniform__space__class_Ototally__bounded,type,
topolo6688025880775521714ounded:
!>[A: $tType] : ( ( set @ A ) > $o ) ).
thf(sy_c_Topological__Spaces_Ouniformity__class_Ouniformity,type,
topolo7806501430040627800ormity:
!>[A: $tType] : ( filter @ ( product_prod @ A @ A ) ) ).
thf(sy_c_Topological__Spaces_Ouniformly__continuous__on,type,
topolo6026614971017936543ous_on:
!>[A: $tType,B: $tType] : ( ( set @ A ) > ( A > B ) > $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,
log: real > real > real ).
thf(sy_c_Transcendental_Opi,type,
pi: real ).
thf(sy_c_Transcendental_Opowr,type,
powr:
!>[A: $tType] : ( A > A > A ) ).
thf(sy_c_Transcendental_Osin,type,
sin:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Transcendental_Osin__coeff,type,
sin_coeff: nat > real ).
thf(sy_c_Transcendental_Osinh,type,
sinh:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Transcendental_Otan,type,
tan:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Transcendental_Otanh,type,
tanh:
!>[A: $tType] : ( A > A ) ).
thf(sy_c_Transitive__Closure_Ontrancl,type,
transitive_ntrancl:
!>[A: $tType] : ( nat > ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ A @ A ) ) ) ).
thf(sy_c_Transitive__Closure_Ortrancl,type,
transitive_rtrancl:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ A @ A ) ) ) ).
thf(sy_c_Transitive__Closure_Otrancl,type,
transitive_trancl:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ A @ A ) ) ) ).
thf(sy_c_VEBT__Definitions_OVEBT_OLeaf,type,
vEBT_Leaf: $o > $o > vEBT_VEBT ).
thf(sy_c_VEBT__Definitions_OVEBT_ONode,type,
vEBT_Node: ( option @ ( product_prod @ nat @ nat ) ) > nat > ( list @ vEBT_VEBT ) > vEBT_VEBT > vEBT_VEBT ).
thf(sy_c_VEBT__Definitions_OVEBT_Osize__VEBT,type,
vEBT_size_VEBT: vEBT_VEBT > nat ).
thf(sy_c_VEBT__Definitions_OVEBT__internal_Oboth__member__options,type,
vEBT_V8194947554948674370ptions: vEBT_VEBT > nat > $o ).
thf(sy_c_VEBT__Definitions_OVEBT__internal_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__Insert_Ovebt__insert,type,
vEBT_vebt_insert: vEBT_VEBT > nat > vEBT_VEBT ).
thf(sy_c_VEBT__Insert_Ovebt__insert__rel,type,
vEBT_vebt_insert_rel: ( product_prod @ vEBT_VEBT @ nat ) > ( product_prod @ vEBT_VEBT @ 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_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_Ofinite__psubset,type,
finite_psubset:
!>[A: $tType] : ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ).
thf(sy_c_Wellfounded_Oless__than,type,
less_than: set @ ( product_prod @ nat @ nat ) ).
thf(sy_c_Wellfounded_Olex__prod,type,
lex_prod:
!>[A: $tType,B: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ B @ B ) ) > ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) ) ).
thf(sy_c_Wellfounded_Omax__ext,type,
max_ext:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ) ).
thf(sy_c_Wellfounded_Omax__extp,type,
max_extp:
!>[A: $tType] : ( ( A > A > $o ) > ( set @ A ) > ( set @ A ) > $o ) ).
thf(sy_c_Wellfounded_Omeasure,type,
measure:
!>[A: $tType] : ( ( A > nat ) > ( set @ ( product_prod @ A @ A ) ) ) ).
thf(sy_c_Wellfounded_Omin__ext,type,
min_ext:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ) ).
thf(sy_c_Wellfounded_Omlex__prod,type,
mlex_prod:
!>[A: $tType] : ( ( A > nat ) > ( set @ ( product_prod @ A @ A ) ) > ( set @ ( product_prod @ A @ A ) ) ) ).
thf(sy_c_Wellfounded_Owf,type,
wf:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > $o ) ).
thf(sy_c_Wfrec_Osame__fst,type,
same_fst:
!>[A: $tType,B: $tType] : ( ( A > $o ) > ( A > ( set @ ( product_prod @ B @ B ) ) ) > ( set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) ) ) ).
thf(sy_c_Zorn_OChains,type,
chains:
!>[A: $tType] : ( ( set @ ( product_prod @ A @ A ) ) > ( set @ ( set @ A ) ) ) ).
thf(sy_c_Zorn_Ochain__subset,type,
chain_subset:
!>[A: $tType] : ( ( set @ ( set @ A ) ) > $o ) ).
thf(sy_c_Zorn_Ochains,type,
chains2:
!>[A: $tType] : ( ( set @ ( set @ A ) ) > ( set @ ( set @ ( set @ A ) ) ) ) ).
thf(sy_c_Zorn_Opred__on_Ochain,type,
pred_chain:
!>[A: $tType] : ( ( set @ A ) > ( A > A > $o ) > ( set @ A ) > $o ) ).
thf(sy_c_fChoice,type,
fChoice:
!>[A: $tType] : ( ( A > $o ) > A ) ).
thf(sy_c_member,type,
member:
!>[A: $tType] : ( A > ( set @ A ) > $o ) ).
thf(sy_v_deg____,type,
deg: nat ).
thf(sy_v_m____,type,
m: nat ).
thf(sy_v_ma____,type,
ma: nat ).
thf(sy_v_mi____,type,
mi: nat ).
thf(sy_v_na____,type,
na: nat ).
thf(sy_v_summary____,type,
summary: vEBT_VEBT ).
thf(sy_v_treeList____,type,
treeList: list @ vEBT_VEBT ).
thf(sy_v_xa____,type,
xa: nat ).
thf(sy_v_ya____,type,
ya: nat ).
% Relevant facts (8185)
thf(fact_0_False,axiom,
~ ( ord_less @ nat @ mi @ ma ) ).
% False
thf(fact_1__092_060open_062x_A_060_Ami_092_060close_062,axiom,
ord_less @ nat @ xa @ mi ).
% \<open>x < mi\<close>
thf(fact_2__C4_Ohyps_C_I7_J,axiom,
ord_less_eq @ nat @ mi @ ma ).
% "4.hyps"(7)
thf(fact_3__C4_Oprems_C_I3_J,axiom,
vEBT_vebt_member @ ( vEBT_vebt_insert @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ mi @ ma ) ) @ deg @ treeList @ summary ) @ xa ) @ ya ).
% "4.prems"(3)
thf(fact_4_True,axiom,
( ( ya = xa )
| ( ya
= ( ord_max @ nat @ mi @ ma ) ) ) ).
% True
thf(fact_5__092_060open_062y_A_061_Amax_Ami_Ama_092_060close_062,axiom,
( ya
= ( ord_max @ nat @ mi @ ma ) ) ).
% \<open>y = max mi ma\<close>
thf(fact_6_min__Null__member,axiom,
! [T2: vEBT_VEBT,X: nat] :
( ( vEBT_VEBT_minNull @ T2 )
=> ~ ( vEBT_vebt_member @ T2 @ X ) ) ).
% min_Null_member
thf(fact_7_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_8_option_Oinject,axiom,
! [A: $tType,X2: A,Y2: A] :
( ( ( some @ A @ X2 )
= ( some @ A @ Y2 ) )
= ( X2 = Y2 ) ) ).
% option.inject
thf(fact_9_prod_Oinject,axiom,
! [A: $tType,B: $tType,X1: A,X2: B,Y1: A,Y2: B] :
( ( ( product_Pair @ A @ B @ X1 @ X2 )
= ( product_Pair @ A @ B @ Y1 @ Y2 ) )
= ( ( X1 = Y1 )
& ( X2 = Y2 ) ) ) ).
% prod.inject
thf(fact_10_old_Oprod_Oinject,axiom,
! [A: $tType,B: $tType,A2: A,B2: B,A3: A,B3: B] :
( ( ( product_Pair @ A @ B @ A2 @ B2 )
= ( product_Pair @ A @ B @ A3 @ B3 ) )
= ( ( A2 = A3 )
& ( B2 = B3 ) ) ) ).
% old.prod.inject
thf(fact_11__C4_Ohyps_C_I6_J,axiom,
( ( mi = ma )
=> ! [X3: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X3 @ ( set2 @ vEBT_VEBT @ treeList ) )
=> ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X_1 ) ) ) ).
% "4.hyps"(6)
thf(fact_12_prod__decode__aux_Ocases,axiom,
! [X: product_prod @ nat @ nat] :
~ ! [K: nat,M: nat] :
( X
!= ( product_Pair @ nat @ nat @ K @ M ) ) ).
% prod_decode_aux.cases
thf(fact_13__C4_Ohyps_C_I1_J,axiom,
vEBT_invar_vebt @ summary @ m ).
% "4.hyps"(1)
thf(fact_14_old_Oprod_Oexhaust,axiom,
! [A: $tType,B: $tType,Y: product_prod @ A @ B] :
~ ! [A4: A,B4: B] :
( Y
!= ( product_Pair @ A @ B @ A4 @ B4 ) ) ).
% old.prod.exhaust
thf(fact_15_surj__pair,axiom,
! [A: $tType,B: $tType,P: product_prod @ A @ B] :
? [X4: A,Y3: B] :
( P
= ( product_Pair @ A @ B @ X4 @ Y3 ) ) ).
% surj_pair
thf(fact_16_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_17_not__min__Null__member,axiom,
! [T2: vEBT_VEBT] :
( ~ ( vEBT_VEBT_minNull @ T2 )
=> ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ T2 @ X_12 ) ) ).
% not_min_Null_member
thf(fact_18_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_19_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_20_member__correct,axiom,
! [T2: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( vEBT_vebt_member @ T2 @ X )
= ( member @ nat @ X @ ( vEBT_set_vebt @ T2 ) ) ) ) ).
% member_correct
thf(fact_21_mi__eq__ma__no__ch,axiom,
! [Mi: nat,Ma: nat,Deg: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ Deg )
=> ( ( Mi = Ma )
=> ( ! [X3: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X3 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X_1 ) )
& ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_1 ) ) ) ) ).
% mi_eq_ma_no_ch
thf(fact_22_prod__induct7,axiom,
! [G: $tType,F: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) )] :
( ! [A4: A,B4: B,C2: C,D2: D,E2: E,F2: F,G2: G] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F @ G ) @ E2 @ ( product_Pair @ F @ G @ F2 @ G2 ) ) ) ) ) ) )
=> ( P2 @ X ) ) ).
% prod_induct7
thf(fact_23_prod__induct6,axiom,
! [F: $tType,E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) )] :
( ! [A4: A,B4: B,C2: C,D2: D,E2: E,F2: F] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ F ) @ D2 @ ( product_Pair @ E @ F @ E2 @ F2 ) ) ) ) ) )
=> ( P2 @ X ) ) ).
% prod_induct6
thf(fact_24_prod__induct5,axiom,
! [E: $tType,D: $tType,C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
( ! [A4: A,B4: B,C2: C,D2: D,E2: E] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C2 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) )
=> ( P2 @ X ) ) ).
% prod_induct5
thf(fact_25_prod__induct4,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
( ! [A4: A,B4: B,C2: C,D2: D] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B4 @ ( product_Pair @ C @ D @ C2 @ D2 ) ) ) )
=> ( P2 @ X ) ) ).
% prod_induct4
thf(fact_26_prod__induct3,axiom,
! [C: $tType,B: $tType,A: $tType,P2: ( product_prod @ A @ ( product_prod @ B @ C ) ) > $o,X: product_prod @ A @ ( product_prod @ B @ C )] :
( ! [A4: A,B4: B,C2: C] : ( P2 @ ( product_Pair @ A @ ( product_prod @ B @ C ) @ A4 @ ( product_Pair @ B @ C @ B4 @ C2 ) ) )
=> ( P2 @ X ) ) ).
% prod_induct3
thf(fact_27_prod__cases7,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F: $tType,G: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) )] :
~ ! [A4: A,B4: B,C2: C,D2: D,E2: E,F2: F,G2: G] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ ( product_prod @ F @ G ) ) @ D2 @ ( product_Pair @ E @ ( product_prod @ F @ G ) @ E2 @ ( product_Pair @ F @ G @ F2 @ G2 ) ) ) ) ) ) ) ).
% prod_cases7
thf(fact_28_prod__cases6,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,F: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) )] :
~ ! [A4: A,B4: B,C2: C,D2: D,E2: E,F2: F] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ ( product_prod @ E @ F ) ) @ C2 @ ( product_Pair @ D @ ( product_prod @ E @ F ) @ D2 @ ( product_Pair @ E @ F @ E2 @ F2 ) ) ) ) ) ) ).
% prod_cases6
thf(fact_29_prod__cases5,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,E: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) )] :
~ ! [A4: A,B4: B,C2: C,D2: D,E2: E] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ ( product_prod @ D @ E ) ) @ B4 @ ( product_Pair @ C @ ( product_prod @ D @ E ) @ C2 @ ( product_Pair @ D @ E @ D2 @ E2 ) ) ) ) ) ).
% prod_cases5
thf(fact_30_prod__cases4,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,Y: product_prod @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) )] :
~ ! [A4: A,B4: B,C2: C,D2: D] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ ( product_prod @ C @ D ) ) @ A4 @ ( product_Pair @ B @ ( product_prod @ C @ D ) @ B4 @ ( product_Pair @ C @ D @ C2 @ D2 ) ) ) ) ).
% prod_cases4
thf(fact_31_prod__cases3,axiom,
! [A: $tType,B: $tType,C: $tType,Y: product_prod @ A @ ( product_prod @ B @ C )] :
~ ! [A4: A,B4: B,C2: C] :
( Y
!= ( product_Pair @ A @ ( product_prod @ B @ C ) @ A4 @ ( product_Pair @ B @ C @ B4 @ C2 ) ) ) ).
% prod_cases3
thf(fact_32_Pair__inject,axiom,
! [A: $tType,B: $tType,A2: A,B2: B,A3: A,B3: B] :
( ( ( product_Pair @ A @ B @ A2 @ B2 )
= ( product_Pair @ A @ B @ A3 @ B3 ) )
=> ~ ( ( A2 = A3 )
=> ( B2 != B3 ) ) ) ).
% Pair_inject
thf(fact_33_prod__cases,axiom,
! [B: $tType,A: $tType,P2: ( product_prod @ A @ B ) > $o,P: product_prod @ A @ B] :
( ! [A4: A,B4: B] : ( P2 @ ( product_Pair @ A @ B @ A4 @ B4 ) )
=> ( P2 @ P ) ) ).
% prod_cases
thf(fact_34__C4_Ohyps_C_I3_J,axiom,
m = na ).
% "4.hyps"(3)
thf(fact_35_max_Oabsorb3,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,A2: A] :
( ( ord_less @ A @ B2 @ A2 )
=> ( ( ord_max @ A @ A2 @ B2 )
= A2 ) ) ) ).
% max.absorb3
thf(fact_36_max_Oabsorb4,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_max @ A @ A2 @ B2 )
= B2 ) ) ) ).
% max.absorb4
thf(fact_37_max__less__iff__conj,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A,Z: A] :
( ( ord_less @ A @ ( ord_max @ A @ X @ Y ) @ Z )
= ( ( ord_less @ A @ X @ Z )
& ( ord_less @ A @ Y @ Z ) ) ) ) ).
% max_less_iff_conj
thf(fact_38_max_Oabsorb1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,A2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_max @ A @ A2 @ B2 )
= A2 ) ) ) ).
% max.absorb1
thf(fact_39_max_Oabsorb2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_max @ A @ A2 @ B2 )
= B2 ) ) ) ).
% max.absorb2
thf(fact_40_max_Obounded__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,C3: A,A2: A] :
( ( ord_less_eq @ A @ ( ord_max @ A @ B2 @ C3 ) @ A2 )
= ( ( ord_less_eq @ A @ B2 @ A2 )
& ( ord_less_eq @ A @ C3 @ A2 ) ) ) ) ).
% max.bounded_iff
thf(fact_41_VEBT__internal_OminNull_Osimps_I5_J,axiom,
! [Uz: product_prod @ nat @ nat,Va: nat,Vb: list @ vEBT_VEBT,Vc: vEBT_VEBT] :
~ ( vEBT_VEBT_minNull @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ Uz ) @ Va @ Vb @ Vc ) ) ).
% VEBT_internal.minNull.simps(5)
thf(fact_42_old_Oprod_Orec,axiom,
! [A: $tType,T: $tType,B: $tType,F1: A > B > T,A2: A,B2: B] :
( ( product_rec_prod @ A @ B @ T @ F1 @ ( product_Pair @ A @ B @ A2 @ B2 ) )
= ( F1 @ A2 @ B2 ) ) ).
% old.prod.rec
thf(fact_43_set__vebt__set__vebt_H__valid,axiom,
! [T2: vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( vEBT_set_vebt @ T2 )
= ( vEBT_VEBT_set_vebt @ T2 ) ) ) ).
% set_vebt_set_vebt'_valid
thf(fact_44_mem__Collect__eq,axiom,
! [A: $tType,A2: A,P2: A > $o] :
( ( member @ A @ A2 @ ( collect @ A @ P2 ) )
= ( P2 @ A2 ) ) ).
% mem_Collect_eq
thf(fact_45_Collect__mem__eq,axiom,
! [A: $tType,A5: set @ A] :
( ( collect @ A
@ ^ [X5: A] : ( member @ A @ X5 @ A5 ) )
= A5 ) ).
% Collect_mem_eq
thf(fact_46_Collect__cong,axiom,
! [A: $tType,P2: A > $o,Q: A > $o] :
( ! [X4: A] :
( ( P2 @ X4 )
= ( Q @ X4 ) )
=> ( ( collect @ A @ P2 )
= ( collect @ A @ Q ) ) ) ).
% Collect_cong
thf(fact_47_ext,axiom,
! [B: $tType,A: $tType,F3: A > B,G3: A > B] :
( ! [X4: A] :
( ( F3 @ X4 )
= ( G3 @ X4 ) )
=> ( F3 = G3 ) ) ).
% ext
thf(fact_48_max_Oidem,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A] :
( ( ord_max @ A @ A2 @ A2 )
= A2 ) ) ).
% max.idem
thf(fact_49_max_Oleft__idem,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A] :
( ( ord_max @ A @ A2 @ ( ord_max @ A @ A2 @ B2 ) )
= ( ord_max @ A @ A2 @ B2 ) ) ) ).
% max.left_idem
thf(fact_50_max_Oright__idem,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A] :
( ( ord_max @ A @ ( ord_max @ A @ A2 @ B2 ) @ B2 )
= ( ord_max @ A @ A2 @ B2 ) ) ) ).
% max.right_idem
thf(fact_51__C4_Ohyps_C_I4_J,axiom,
( deg
= ( plus_plus @ nat @ na @ m ) ) ).
% "4.hyps"(4)
thf(fact_52_max_Oleft__commute,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,A2: A,C3: A] :
( ( ord_max @ A @ B2 @ ( ord_max @ A @ A2 @ C3 ) )
= ( ord_max @ A @ A2 @ ( ord_max @ A @ B2 @ C3 ) ) ) ) ).
% max.left_commute
thf(fact_53_max_Ocommute,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_max @ A )
= ( ^ [A6: A,B5: A] : ( ord_max @ A @ B5 @ A6 ) ) ) ) ).
% max.commute
thf(fact_54_max_Oassoc,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( ord_max @ A @ ( ord_max @ A @ A2 @ B2 ) @ C3 )
= ( ord_max @ A @ A2 @ ( ord_max @ A @ B2 @ C3 ) ) ) ) ).
% max.assoc
thf(fact_55_max_OcoboundedI2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C3: A,B2: A,A2: A] :
( ( ord_less_eq @ A @ C3 @ B2 )
=> ( ord_less_eq @ A @ C3 @ ( ord_max @ A @ A2 @ B2 ) ) ) ) ).
% max.coboundedI2
thf(fact_56_max_OcoboundedI1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C3: A,A2: A,B2: A] :
( ( ord_less_eq @ A @ C3 @ A2 )
=> ( ord_less_eq @ A @ C3 @ ( ord_max @ A @ A2 @ B2 ) ) ) ) ).
% max.coboundedI1
thf(fact_57_max_Oabsorb__iff2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A6: A,B5: A] :
( ( ord_max @ A @ A6 @ B5 )
= B5 ) ) ) ) ).
% max.absorb_iff2
thf(fact_58_max_Oabsorb__iff1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B5: A,A6: A] :
( ( ord_max @ A @ A6 @ B5 )
= A6 ) ) ) ) ).
% max.absorb_iff1
thf(fact_59_le__max__iff__disj,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Z: A,X: A,Y: A] :
( ( ord_less_eq @ A @ Z @ ( ord_max @ A @ X @ Y ) )
= ( ( ord_less_eq @ A @ Z @ X )
| ( ord_less_eq @ A @ Z @ Y ) ) ) ) ).
% le_max_iff_disj
thf(fact_60_max_Ocobounded2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,A2: A] : ( ord_less_eq @ A @ B2 @ ( ord_max @ A @ A2 @ B2 ) ) ) ).
% max.cobounded2
thf(fact_61_max_Ocobounded1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A] : ( ord_less_eq @ A @ A2 @ ( ord_max @ A @ A2 @ B2 ) ) ) ).
% max.cobounded1
thf(fact_62_max_Oorder__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B5: A,A6: A] :
( A6
= ( ord_max @ A @ A6 @ B5 ) ) ) ) ) ).
% max.order_iff
thf(fact_63_max_OboundedI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,A2: A,C3: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_less_eq @ A @ C3 @ A2 )
=> ( ord_less_eq @ A @ ( ord_max @ A @ B2 @ C3 ) @ A2 ) ) ) ) ).
% max.boundedI
thf(fact_64_max_OboundedE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,C3: A,A2: A] :
( ( ord_less_eq @ A @ ( ord_max @ A @ B2 @ C3 ) @ A2 )
=> ~ ( ( ord_less_eq @ A @ B2 @ A2 )
=> ~ ( ord_less_eq @ A @ C3 @ A2 ) ) ) ) ).
% max.boundedE
thf(fact_65_max_OorderI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A] :
( ( A2
= ( ord_max @ A @ A2 @ B2 ) )
=> ( ord_less_eq @ A @ B2 @ A2 ) ) ) ).
% max.orderI
thf(fact_66_max_OorderE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,A2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( A2
= ( ord_max @ A @ A2 @ B2 ) ) ) ) ).
% max.orderE
thf(fact_67_max_Omono,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C3: A,A2: A,D3: A,B2: A] :
( ( ord_less_eq @ A @ C3 @ A2 )
=> ( ( ord_less_eq @ A @ D3 @ B2 )
=> ( ord_less_eq @ A @ ( ord_max @ A @ C3 @ D3 ) @ ( ord_max @ A @ A2 @ B2 ) ) ) ) ) ).
% max.mono
thf(fact_68_max_Ostrict__coboundedI2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C3: A,B2: A,A2: A] :
( ( ord_less @ A @ C3 @ B2 )
=> ( ord_less @ A @ C3 @ ( ord_max @ A @ A2 @ B2 ) ) ) ) ).
% max.strict_coboundedI2
thf(fact_69_max_Ostrict__coboundedI1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C3: A,A2: A,B2: A] :
( ( ord_less @ A @ C3 @ A2 )
=> ( ord_less @ A @ C3 @ ( ord_max @ A @ A2 @ B2 ) ) ) ) ).
% max.strict_coboundedI1
thf(fact_70_max_Ostrict__order__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less @ A )
= ( ^ [B5: A,A6: A] :
( ( A6
= ( ord_max @ A @ A6 @ B5 ) )
& ( A6 != B5 ) ) ) ) ) ).
% max.strict_order_iff
thf(fact_71_max_Ostrict__boundedE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,C3: A,A2: A] :
( ( ord_less @ A @ ( ord_max @ A @ B2 @ C3 ) @ A2 )
=> ~ ( ( ord_less @ A @ B2 @ A2 )
=> ~ ( ord_less @ A @ C3 @ A2 ) ) ) ) ).
% max.strict_boundedE
thf(fact_72_less__max__iff__disj,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Z: A,X: A,Y: A] :
( ( ord_less @ A @ Z @ ( ord_max @ A @ X @ Y ) )
= ( ( ord_less @ A @ Z @ X )
| ( ord_less @ A @ Z @ Y ) ) ) ) ).
% less_max_iff_disj
thf(fact_73_valid__eq2,axiom,
! [T2: vEBT_VEBT,D3: nat] :
( ( vEBT_VEBT_valid @ T2 @ D3 )
=> ( vEBT_invar_vebt @ T2 @ D3 ) ) ).
% valid_eq2
thf(fact_74_valid__eq1,axiom,
! [T2: vEBT_VEBT,D3: nat] :
( ( vEBT_invar_vebt @ T2 @ D3 )
=> ( vEBT_VEBT_valid @ T2 @ D3 ) ) ).
% valid_eq1
thf(fact_75_valid__eq,axiom,
vEBT_VEBT_valid = vEBT_invar_vebt ).
% valid_eq
thf(fact_76_dual__order_Orefl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% dual_order.refl
thf(fact_77_order__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A] : ( ord_less_eq @ A @ X @ X ) ) ).
% order_refl
thf(fact_78_set__vebt__finite,axiom,
! [T2: vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( finite_finite2 @ nat @ ( vEBT_VEBT_set_vebt @ T2 ) ) ) ).
% set_vebt_finite
thf(fact_79_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_80_max__absorb2,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_max @ A @ X @ Y )
= Y ) ) ) ).
% max_absorb2
thf(fact_81_max__absorb1,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ Y @ X )
=> ( ( ord_max @ A @ X @ Y )
= X ) ) ) ).
% max_absorb1
thf(fact_82_max__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( ord_max @ A )
= ( ^ [A6: A,B5: A] : ( if @ A @ ( ord_less_eq @ A @ A6 @ B5 ) @ B5 @ A6 ) ) ) ) ).
% max_def
thf(fact_83_Lattices__Big_Oex__has__greatest__nat,axiom,
! [A: $tType,P2: A > $o,K2: A,F3: A > nat,B2: nat] :
( ( P2 @ K2 )
=> ( ! [Y3: A] :
( ( P2 @ Y3 )
=> ( ord_less @ nat @ ( F3 @ Y3 ) @ B2 ) )
=> ? [X4: A] :
( ( P2 @ X4 )
& ! [Y4: A] :
( ( P2 @ Y4 )
=> ( ord_less_eq @ nat @ ( F3 @ Y4 ) @ ( F3 @ X4 ) ) ) ) ) ) ).
% Lattices_Big.ex_has_greatest_nat
thf(fact_84_nat__descend__induct,axiom,
! [N: nat,P2: nat > $o,M2: nat] :
( ! [K: nat] :
( ( ord_less @ nat @ N @ K )
=> ( P2 @ K ) )
=> ( ! [K: nat] :
( ( ord_less_eq @ nat @ K @ N )
=> ( ! [I: nat] :
( ( ord_less @ nat @ K @ I )
=> ( P2 @ I ) )
=> ( P2 @ K ) ) )
=> ( P2 @ M2 ) ) ) ).
% nat_descend_induct
thf(fact_85_valid__0__not,axiom,
! [T2: vEBT_VEBT] :
~ ( vEBT_invar_vebt @ T2 @ ( zero_zero @ nat ) ) ).
% valid_0_not
thf(fact_86_valid__tree__deg__neq__0,axiom,
! [T2: vEBT_VEBT] :
~ ( vEBT_invar_vebt @ T2 @ ( zero_zero @ nat ) ) ).
% valid_tree_deg_neq_0
thf(fact_87_ex__has__greatest__nat__lemma,axiom,
! [A: $tType,P2: A > $o,K2: A,F3: A > nat,N: nat] :
( ( P2 @ K2 )
=> ( ! [X4: A] :
( ( P2 @ X4 )
=> ? [Y4: A] :
( ( P2 @ Y4 )
& ~ ( ord_less_eq @ nat @ ( F3 @ Y4 ) @ ( F3 @ X4 ) ) ) )
=> ? [Y3: A] :
( ( P2 @ Y3 )
& ~ ( ord_less @ nat @ ( F3 @ Y3 ) @ ( plus_plus @ nat @ ( F3 @ K2 ) @ N ) ) ) ) ) ).
% ex_has_greatest_nat_lemma
thf(fact_88_vebt__insert_Osimps_I2_J,axiom,
! [Info: option @ ( product_prod @ nat @ nat ),Ts: list @ vEBT_VEBT,S: vEBT_VEBT,X: nat] :
( ( vEBT_vebt_insert @ ( vEBT_Node @ Info @ ( zero_zero @ nat ) @ Ts @ S ) @ X )
= ( vEBT_Node @ Info @ ( zero_zero @ nat ) @ Ts @ S ) ) ).
% vebt_insert.simps(2)
thf(fact_89_nle__le,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A] :
( ( ~ ( ord_less_eq @ A @ A2 @ B2 ) )
= ( ( ord_less_eq @ A @ B2 @ A2 )
& ( B2 != A2 ) ) ) ) ).
% nle_le
thf(fact_90_le__cases3,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A,Z: A] :
( ( ( ord_less_eq @ A @ X @ Y )
=> ~ ( ord_less_eq @ A @ Y @ Z ) )
=> ( ( ( ord_less_eq @ A @ Y @ X )
=> ~ ( ord_less_eq @ A @ X @ Z ) )
=> ( ( ( ord_less_eq @ A @ X @ Z )
=> ~ ( ord_less_eq @ A @ Z @ Y ) )
=> ( ( ( ord_less_eq @ A @ Z @ Y )
=> ~ ( ord_less_eq @ A @ Y @ X ) )
=> ( ( ( ord_less_eq @ A @ Y @ Z )
=> ~ ( ord_less_eq @ A @ Z @ X ) )
=> ~ ( ( ord_less_eq @ A @ Z @ X )
=> ~ ( ord_less_eq @ A @ X @ Y ) ) ) ) ) ) ) ) ).
% le_cases3
thf(fact_91_order__class_Oorder__eq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y5: A,Z2: A] : Y5 = Z2 )
= ( ^ [X5: A,Y6: A] :
( ( ord_less_eq @ A @ X5 @ Y6 )
& ( ord_less_eq @ A @ Y6 @ X5 ) ) ) ) ) ).
% order_class.order_eq_iff
thf(fact_92_ord__eq__le__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( A2 = B2 )
=> ( ( ord_less_eq @ A @ B2 @ C3 )
=> ( ord_less_eq @ A @ A2 @ C3 ) ) ) ) ).
% ord_eq_le_trans
thf(fact_93_ord__le__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( B2 = C3 )
=> ( ord_less_eq @ A @ A2 @ C3 ) ) ) ) ).
% ord_le_eq_trans
thf(fact_94_order__antisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ Y @ X )
=> ( X = Y ) ) ) ) ).
% order_antisym
thf(fact_95_order_Otrans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ C3 )
=> ( ord_less_eq @ A @ A2 @ C3 ) ) ) ) ).
% order.trans
thf(fact_96_order__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A,Z: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ Y @ Z )
=> ( ord_less_eq @ A @ X @ Z ) ) ) ) ).
% order_trans
thf(fact_97_linorder__wlog,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P2: A > A > $o,A2: A,B2: A] :
( ! [A4: A,B4: A] :
( ( ord_less_eq @ A @ A4 @ B4 )
=> ( P2 @ A4 @ B4 ) )
=> ( ! [A4: A,B4: A] :
( ( P2 @ B4 @ A4 )
=> ( P2 @ A4 @ B4 ) )
=> ( P2 @ A2 @ B2 ) ) ) ) ).
% linorder_wlog
thf(fact_98_dual__order_Oeq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y5: A,Z2: A] : Y5 = Z2 )
= ( ^ [A6: A,B5: A] :
( ( ord_less_eq @ A @ B5 @ A6 )
& ( ord_less_eq @ A @ A6 @ B5 ) ) ) ) ) ).
% dual_order.eq_iff
thf(fact_99_dual__order_Oantisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B2: A,A2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_less_eq @ A @ A2 @ B2 )
=> ( A2 = B2 ) ) ) ) ).
% dual_order.antisym
thf(fact_100_dual__order_Otrans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [B2: A,A2: A,C3: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_less_eq @ A @ C3 @ B2 )
=> ( ord_less_eq @ A @ C3 @ A2 ) ) ) ) ).
% dual_order.trans
thf(fact_101_antisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ A2 )
=> ( A2 = B2 ) ) ) ) ).
% antisym
thf(fact_102_le__funD,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F3: A > B,G3: A > B,X: A] :
( ( ord_less_eq @ ( A > B ) @ F3 @ G3 )
=> ( ord_less_eq @ B @ ( F3 @ X ) @ ( G3 @ X ) ) ) ) ).
% le_funD
thf(fact_103_le__funE,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F3: A > B,G3: A > B,X: A] :
( ( ord_less_eq @ ( A > B ) @ F3 @ G3 )
=> ( ord_less_eq @ B @ ( F3 @ X ) @ ( G3 @ X ) ) ) ) ).
% le_funE
thf(fact_104_le__funI,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ! [F3: A > B,G3: A > B] :
( ! [X4: A] : ( ord_less_eq @ B @ ( F3 @ X4 ) @ ( G3 @ X4 ) )
=> ( ord_less_eq @ ( A > B ) @ F3 @ G3 ) ) ) ).
% le_funI
thf(fact_105_le__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ( ( ord_less_eq @ ( A > B ) )
= ( ^ [F4: A > B,G4: A > B] :
! [X5: A] : ( ord_less_eq @ B @ ( F4 @ X5 ) @ ( G4 @ X5 ) ) ) ) ) ).
% le_fun_def
thf(fact_106_Orderings_Oorder__eq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ^ [Y5: A,Z2: A] : Y5 = Z2 )
= ( ^ [A6: A,B5: A] :
( ( ord_less_eq @ A @ A6 @ B5 )
& ( ord_less_eq @ A @ B5 @ A6 ) ) ) ) ) ).
% Orderings.order_eq_iff
thf(fact_107_order__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A2: A,F3: B > A,B2: B,C3: B] :
( ( ord_less_eq @ A @ A2 @ ( F3 @ B2 ) )
=> ( ( ord_less_eq @ B @ B2 @ C3 )
=> ( ! [X4: B,Y3: B] :
( ( ord_less_eq @ B @ X4 @ Y3 )
=> ( ord_less_eq @ A @ ( F3 @ X4 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_eq @ A @ A2 @ ( F3 @ C3 ) ) ) ) ) ) ).
% order_subst1
thf(fact_108_order__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C )
& ( order @ A ) )
=> ! [A2: A,B2: A,F3: A > C,C3: C] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ C @ ( F3 @ B2 ) @ C3 )
=> ( ! [X4: A,Y3: A] :
( ( ord_less_eq @ A @ X4 @ Y3 )
=> ( ord_less_eq @ C @ ( F3 @ X4 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_eq @ C @ ( F3 @ A2 ) @ C3 ) ) ) ) ) ).
% order_subst2
thf(fact_109_order__eq__refl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A] :
( ( X = Y )
=> ( ord_less_eq @ A @ X @ Y ) ) ) ).
% order_eq_refl
thf(fact_110_linorder__linear,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
| ( ord_less_eq @ A @ Y @ X ) ) ) ).
% linorder_linear
thf(fact_111_ord__eq__le__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A2: A,F3: B > A,B2: B,C3: B] :
( ( A2
= ( F3 @ B2 ) )
=> ( ( ord_less_eq @ B @ B2 @ C3 )
=> ( ! [X4: B,Y3: B] :
( ( ord_less_eq @ B @ X4 @ Y3 )
=> ( ord_less_eq @ A @ ( F3 @ X4 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_eq @ A @ A2 @ ( F3 @ C3 ) ) ) ) ) ) ).
% ord_eq_le_subst
thf(fact_112_ord__le__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A2: A,B2: A,F3: A > B,C3: B] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ( F3 @ B2 )
= C3 )
=> ( ! [X4: A,Y3: A] :
( ( ord_less_eq @ A @ X4 @ Y3 )
=> ( ord_less_eq @ B @ ( F3 @ X4 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less_eq @ B @ ( F3 @ A2 ) @ C3 ) ) ) ) ) ).
% ord_le_eq_subst
thf(fact_113_linorder__le__cases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ~ ( ord_less_eq @ A @ X @ Y )
=> ( ord_less_eq @ A @ Y @ X ) ) ) ).
% linorder_le_cases
thf(fact_114_order__antisym__conv,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ Y @ X )
=> ( ( ord_less_eq @ A @ X @ Y )
= ( X = Y ) ) ) ) ).
% order_antisym_conv
thf(fact_115_lt__ex,axiom,
! [A: $tType] :
( ( no_bot @ A )
=> ! [X: A] :
? [Y3: A] : ( ord_less @ A @ Y3 @ X ) ) ).
% lt_ex
thf(fact_116_gt__ex,axiom,
! [A: $tType] :
( ( no_top @ A )
=> ! [X: A] :
? [X_12: A] : ( ord_less @ A @ X @ X_12 ) ) ).
% gt_ex
thf(fact_117_dense,axiom,
! [A: $tType] :
( ( dense_order @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ? [Z3: A] :
( ( ord_less @ A @ X @ Z3 )
& ( ord_less @ A @ Z3 @ Y ) ) ) ) ).
% dense
thf(fact_118_less__imp__neq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( X != Y ) ) ) ).
% less_imp_neq
thf(fact_119_order_Oasym,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ~ ( ord_less @ A @ B2 @ A2 ) ) ) ).
% order.asym
thf(fact_120_ord__eq__less__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( A2 = B2 )
=> ( ( ord_less @ A @ B2 @ C3 )
=> ( ord_less @ A @ A2 @ C3 ) ) ) ) ).
% ord_eq_less_trans
thf(fact_121_ord__less__eq__trans,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( B2 = C3 )
=> ( ord_less @ A @ A2 @ C3 ) ) ) ) ).
% ord_less_eq_trans
thf(fact_122_less__induct,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [P2: A > $o,A2: A] :
( ! [X4: A] :
( ! [Y4: A] :
( ( ord_less @ A @ Y4 @ X4 )
=> ( P2 @ Y4 ) )
=> ( P2 @ X4 ) )
=> ( P2 @ A2 ) ) ) ).
% less_induct
thf(fact_123_antisym__conv3,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Y: A,X: A] :
( ~ ( ord_less @ A @ Y @ X )
=> ( ( ~ ( ord_less @ A @ X @ Y ) )
= ( X = Y ) ) ) ) ).
% antisym_conv3
thf(fact_124_linorder__cases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ~ ( ord_less @ A @ X @ Y )
=> ( ( X != Y )
=> ( ord_less @ A @ Y @ X ) ) ) ) ).
% linorder_cases
thf(fact_125_dual__order_Oasym,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [B2: A,A2: A] :
( ( ord_less @ A @ B2 @ A2 )
=> ~ ( ord_less @ A @ A2 @ B2 ) ) ) ).
% dual_order.asym
thf(fact_126_dual__order_Oirrefl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A2: A] :
~ ( ord_less @ A @ A2 @ A2 ) ) ).
% dual_order.irrefl
thf(fact_127_exists__least__iff,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ( ( ^ [P3: A > $o] :
? [X6: A] : ( P3 @ X6 ) )
= ( ^ [P4: A > $o] :
? [N2: A] :
( ( P4 @ N2 )
& ! [M3: A] :
( ( ord_less @ A @ M3 @ N2 )
=> ~ ( P4 @ M3 ) ) ) ) ) ) ).
% exists_least_iff
thf(fact_128_linorder__less__wlog,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P2: A > A > $o,A2: A,B2: A] :
( ! [A4: A,B4: A] :
( ( ord_less @ A @ A4 @ B4 )
=> ( P2 @ A4 @ B4 ) )
=> ( ! [A4: A] : ( P2 @ A4 @ A4 )
=> ( ! [A4: A,B4: A] :
( ( P2 @ B4 @ A4 )
=> ( P2 @ A4 @ B4 ) )
=> ( P2 @ A2 @ B2 ) ) ) ) ) ).
% linorder_less_wlog
thf(fact_129_order_Ostrict__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ B2 @ C3 )
=> ( ord_less @ A @ A2 @ C3 ) ) ) ) ).
% order.strict_trans
thf(fact_130_not__less__iff__gr__or__eq,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( ~ ( ord_less @ A @ X @ Y ) )
= ( ( ord_less @ A @ Y @ X )
| ( X = Y ) ) ) ) ).
% not_less_iff_gr_or_eq
thf(fact_131_dual__order_Ostrict__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [B2: A,A2: A,C3: A] :
( ( ord_less @ A @ B2 @ A2 )
=> ( ( ord_less @ A @ C3 @ B2 )
=> ( ord_less @ A @ C3 @ A2 ) ) ) ) ).
% dual_order.strict_trans
thf(fact_132_order_Ostrict__implies__not__eq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( A2 != B2 ) ) ) ).
% order.strict_implies_not_eq
thf(fact_133_dual__order_Ostrict__implies__not__eq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B2: A,A2: A] :
( ( ord_less @ A @ B2 @ A2 )
=> ( A2 != B2 ) ) ) ).
% dual_order.strict_implies_not_eq
thf(fact_134_linorder__neqE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( X != Y )
=> ( ~ ( ord_less @ A @ X @ Y )
=> ( ord_less @ A @ Y @ X ) ) ) ) ).
% linorder_neqE
thf(fact_135_order__less__asym,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ~ ( ord_less @ A @ Y @ X ) ) ) ).
% order_less_asym
thf(fact_136_linorder__neq__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( X != Y )
= ( ( ord_less @ A @ X @ Y )
| ( ord_less @ A @ Y @ X ) ) ) ) ).
% linorder_neq_iff
thf(fact_137_order__less__asym_H,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ~ ( ord_less @ A @ B2 @ A2 ) ) ) ).
% order_less_asym'
thf(fact_138_order__less__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A,Z: A] :
( ( ord_less @ A @ X @ Y )
=> ( ( ord_less @ A @ Y @ Z )
=> ( ord_less @ A @ X @ Z ) ) ) ) ).
% order_less_trans
thf(fact_139_ord__eq__less__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A2: A,F3: B > A,B2: B,C3: B] :
( ( A2
= ( F3 @ B2 ) )
=> ( ( ord_less @ B @ B2 @ C3 )
=> ( ! [X4: B,Y3: B] :
( ( ord_less @ B @ X4 @ Y3 )
=> ( ord_less @ A @ ( F3 @ X4 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less @ A @ A2 @ ( F3 @ C3 ) ) ) ) ) ) ).
% ord_eq_less_subst
thf(fact_140_ord__less__eq__subst,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( ord @ A ) )
=> ! [A2: A,B2: A,F3: A > B,C3: B] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ( F3 @ B2 )
= C3 )
=> ( ! [X4: A,Y3: A] :
( ( ord_less @ A @ X4 @ Y3 )
=> ( ord_less @ B @ ( F3 @ X4 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less @ B @ ( F3 @ A2 ) @ C3 ) ) ) ) ) ).
% ord_less_eq_subst
thf(fact_141_order__less__irrefl,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A] :
~ ( ord_less @ A @ X @ X ) ) ).
% order_less_irrefl
thf(fact_142_order__less__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A2: A,F3: B > A,B2: B,C3: B] :
( ( ord_less @ A @ A2 @ ( F3 @ B2 ) )
=> ( ( ord_less @ B @ B2 @ C3 )
=> ( ! [X4: B,Y3: B] :
( ( ord_less @ B @ X4 @ Y3 )
=> ( ord_less @ A @ ( F3 @ X4 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less @ A @ A2 @ ( F3 @ C3 ) ) ) ) ) ) ).
% order_less_subst1
thf(fact_143_order__less__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C )
& ( order @ A ) )
=> ! [A2: A,B2: A,F3: A > C,C3: C] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ C @ ( F3 @ B2 ) @ C3 )
=> ( ! [X4: A,Y3: A] :
( ( ord_less @ A @ X4 @ Y3 )
=> ( ord_less @ C @ ( F3 @ X4 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less @ C @ ( F3 @ A2 ) @ C3 ) ) ) ) ) ).
% order_less_subst2
thf(fact_144_order__less__not__sym,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ~ ( ord_less @ A @ Y @ X ) ) ) ).
% order_less_not_sym
thf(fact_145_order__less__imp__triv,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A,P2: $o] :
( ( ord_less @ A @ X @ Y )
=> ( ( ord_less @ A @ Y @ X )
=> P2 ) ) ) ).
% order_less_imp_triv
thf(fact_146_linorder__less__linear,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
| ( X = Y )
| ( ord_less @ A @ Y @ X ) ) ) ).
% linorder_less_linear
thf(fact_147_order__less__imp__not__eq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( X != Y ) ) ) ).
% order_less_imp_not_eq
thf(fact_148_order__less__imp__not__eq2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( Y != X ) ) ) ).
% order_less_imp_not_eq2
thf(fact_149_order__less__imp__not__less,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ~ ( ord_less @ A @ Y @ X ) ) ) ).
% order_less_imp_not_less
thf(fact_150_ex__has__least__nat,axiom,
! [A: $tType,P2: A > $o,K2: A,M2: A > nat] :
( ( P2 @ K2 )
=> ? [X4: A] :
( ( P2 @ X4 )
& ! [Y4: A] :
( ( P2 @ Y4 )
=> ( ord_less_eq @ nat @ ( M2 @ X4 ) @ ( M2 @ Y4 ) ) ) ) ) ).
% ex_has_least_nat
thf(fact_151_vebt__member_Osimps_I3_J,axiom,
! [V: product_prod @ nat @ nat,Uy: list @ vEBT_VEBT,Uz: vEBT_VEBT,X: nat] :
~ ( vEBT_vebt_member @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V ) @ ( zero_zero @ nat ) @ Uy @ Uz ) @ X ) ).
% vebt_member.simps(3)
thf(fact_152_leD,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ Y @ X )
=> ~ ( ord_less @ A @ X @ Y ) ) ) ).
% leD
thf(fact_153_leI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ~ ( ord_less @ A @ X @ Y )
=> ( ord_less_eq @ A @ Y @ X ) ) ) ).
% leI
thf(fact_154_nless__le,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A,B2: A] :
( ( ~ ( ord_less @ A @ A2 @ B2 ) )
= ( ~ ( ord_less_eq @ A @ A2 @ B2 )
| ( A2 = B2 ) ) ) ) ).
% nless_le
thf(fact_155_antisym__conv1,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,Y: A] :
( ~ ( ord_less @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ X @ Y )
= ( X = Y ) ) ) ) ).
% antisym_conv1
thf(fact_156_antisym__conv2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ~ ( ord_less @ A @ X @ Y ) )
= ( X = Y ) ) ) ) ).
% antisym_conv2
thf(fact_157_dense__ge,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [Z: A,Y: A] :
( ! [X4: A] :
( ( ord_less @ A @ Z @ X4 )
=> ( ord_less_eq @ A @ Y @ X4 ) )
=> ( ord_less_eq @ A @ Y @ Z ) ) ) ).
% dense_ge
thf(fact_158_dense__le,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [Y: A,Z: A] :
( ! [X4: A] :
( ( ord_less @ A @ X4 @ Y )
=> ( ord_less_eq @ A @ X4 @ Z ) )
=> ( ord_less_eq @ A @ Y @ Z ) ) ) ).
% dense_le
thf(fact_159_less__le__not__le,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( ( ord_less @ A )
= ( ^ [X5: A,Y6: A] :
( ( ord_less_eq @ A @ X5 @ Y6 )
& ~ ( ord_less_eq @ A @ Y6 @ X5 ) ) ) ) ) ).
% less_le_not_le
thf(fact_160_not__le__imp__less,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Y: A,X: A] :
( ~ ( ord_less_eq @ A @ Y @ X )
=> ( ord_less @ A @ X @ Y ) ) ) ).
% not_le_imp_less
thf(fact_161_order_Oorder__iff__strict,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A6: A,B5: A] :
( ( ord_less @ A @ A6 @ B5 )
| ( A6 = B5 ) ) ) ) ) ).
% order.order_iff_strict
thf(fact_162_order_Ostrict__iff__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less @ A )
= ( ^ [A6: A,B5: A] :
( ( ord_less_eq @ A @ A6 @ B5 )
& ( A6 != B5 ) ) ) ) ) ).
% order.strict_iff_order
thf(fact_163_order_Ostrict__trans1,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ B2 @ C3 )
=> ( ord_less @ A @ A2 @ C3 ) ) ) ) ).
% order.strict_trans1
thf(fact_164_order_Ostrict__trans2,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ C3 )
=> ( ord_less @ A @ A2 @ C3 ) ) ) ) ).
% order.strict_trans2
thf(fact_165_order_Ostrict__iff__not,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( ( ord_less @ A )
= ( ^ [A6: A,B5: A] :
( ( ord_less_eq @ A @ A6 @ B5 )
& ~ ( ord_less_eq @ A @ B5 @ A6 ) ) ) ) ) ).
% order.strict_iff_not
thf(fact_166_dense__ge__bounded,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [Z: A,X: A,Y: A] :
( ( ord_less @ A @ Z @ X )
=> ( ! [W: A] :
( ( ord_less @ A @ Z @ W )
=> ( ( ord_less @ A @ W @ X )
=> ( ord_less_eq @ A @ Y @ W ) ) )
=> ( ord_less_eq @ A @ Y @ Z ) ) ) ) ).
% dense_ge_bounded
thf(fact_167_dense__le__bounded,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [X: A,Y: A,Z: A] :
( ( ord_less @ A @ X @ Y )
=> ( ! [W: A] :
( ( ord_less @ A @ X @ W )
=> ( ( ord_less @ A @ W @ Y )
=> ( ord_less_eq @ A @ W @ Z ) ) )
=> ( ord_less_eq @ A @ Y @ Z ) ) ) ) ).
% dense_le_bounded
thf(fact_168_dual__order_Oorder__iff__strict,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B5: A,A6: A] :
( ( ord_less @ A @ B5 @ A6 )
| ( A6 = B5 ) ) ) ) ) ).
% dual_order.order_iff_strict
thf(fact_169_dual__order_Ostrict__iff__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less @ A )
= ( ^ [B5: A,A6: A] :
( ( ord_less_eq @ A @ B5 @ A6 )
& ( A6 != B5 ) ) ) ) ) ).
% dual_order.strict_iff_order
thf(fact_170_dual__order_Ostrict__trans1,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [B2: A,A2: A,C3: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_less @ A @ C3 @ B2 )
=> ( ord_less @ A @ C3 @ A2 ) ) ) ) ).
% dual_order.strict_trans1
thf(fact_171_dual__order_Ostrict__trans2,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [B2: A,A2: A,C3: A] :
( ( ord_less @ A @ B2 @ A2 )
=> ( ( ord_less_eq @ A @ C3 @ B2 )
=> ( ord_less @ A @ C3 @ A2 ) ) ) ) ).
% dual_order.strict_trans2
thf(fact_172_dual__order_Ostrict__iff__not,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( ( ord_less @ A )
= ( ^ [B5: A,A6: A] :
( ( ord_less_eq @ A @ B5 @ A6 )
& ~ ( ord_less_eq @ A @ A6 @ B5 ) ) ) ) ) ).
% dual_order.strict_iff_not
thf(fact_173_order_Ostrict__implies__order,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% order.strict_implies_order
thf(fact_174_dual__order_Ostrict__implies__order,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [B2: A,A2: A] :
( ( ord_less @ A @ B2 @ A2 )
=> ( ord_less_eq @ A @ B2 @ A2 ) ) ) ).
% dual_order.strict_implies_order
thf(fact_175_order__le__less,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [X5: A,Y6: A] :
( ( ord_less @ A @ X5 @ Y6 )
| ( X5 = Y6 ) ) ) ) ) ).
% order_le_less
thf(fact_176_order__less__le,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( ord_less @ A )
= ( ^ [X5: A,Y6: A] :
( ( ord_less_eq @ A @ X5 @ Y6 )
& ( X5 != Y6 ) ) ) ) ) ).
% order_less_le
thf(fact_177_linorder__not__le,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( ~ ( ord_less_eq @ A @ X @ Y ) )
= ( ord_less @ A @ Y @ X ) ) ) ).
% linorder_not_le
thf(fact_178_linorder__not__less,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( ~ ( ord_less @ A @ X @ Y ) )
= ( ord_less_eq @ A @ Y @ X ) ) ) ).
% linorder_not_less
thf(fact_179_order__less__imp__le,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( ord_less_eq @ A @ X @ Y ) ) ) ).
% order_less_imp_le
thf(fact_180_order__le__neq__trans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( A2 != B2 )
=> ( ord_less @ A @ A2 @ B2 ) ) ) ) ).
% order_le_neq_trans
thf(fact_181_order__neq__le__trans,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A,B2: A] :
( ( A2 != B2 )
=> ( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ord_less @ A @ A2 @ B2 ) ) ) ) ).
% order_neq_le_trans
thf(fact_182_order__le__less__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A,Z: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less @ A @ Y @ Z )
=> ( ord_less @ A @ X @ Z ) ) ) ) ).
% order_le_less_trans
thf(fact_183_order__less__le__trans,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A,Z: A] :
( ( ord_less @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ Y @ Z )
=> ( ord_less @ A @ X @ Z ) ) ) ) ).
% order_less_le_trans
thf(fact_184_order__le__less__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A2: A,F3: B > A,B2: B,C3: B] :
( ( ord_less_eq @ A @ A2 @ ( F3 @ B2 ) )
=> ( ( ord_less @ B @ B2 @ C3 )
=> ( ! [X4: B,Y3: B] :
( ( ord_less @ B @ X4 @ Y3 )
=> ( ord_less @ A @ ( F3 @ X4 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less @ A @ A2 @ ( F3 @ C3 ) ) ) ) ) ) ).
% order_le_less_subst1
thf(fact_185_order__le__less__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C )
& ( order @ A ) )
=> ! [A2: A,B2: A,F3: A > C,C3: C] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less @ C @ ( F3 @ B2 ) @ C3 )
=> ( ! [X4: A,Y3: A] :
( ( ord_less_eq @ A @ X4 @ Y3 )
=> ( ord_less_eq @ C @ ( F3 @ X4 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less @ C @ ( F3 @ A2 ) @ C3 ) ) ) ) ) ).
% order_le_less_subst2
thf(fact_186_order__less__le__subst1,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [A2: A,F3: B > A,B2: B,C3: B] :
( ( ord_less @ A @ A2 @ ( F3 @ B2 ) )
=> ( ( ord_less_eq @ B @ B2 @ C3 )
=> ( ! [X4: B,Y3: B] :
( ( ord_less_eq @ B @ X4 @ Y3 )
=> ( ord_less_eq @ A @ ( F3 @ X4 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less @ A @ A2 @ ( F3 @ C3 ) ) ) ) ) ) ).
% order_less_le_subst1
thf(fact_187_order__less__le__subst2,axiom,
! [A: $tType,C: $tType] :
( ( ( order @ C )
& ( order @ A ) )
=> ! [A2: A,B2: A,F3: A > C,C3: C] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ C @ ( F3 @ B2 ) @ C3 )
=> ( ! [X4: A,Y3: A] :
( ( ord_less @ A @ X4 @ Y3 )
=> ( ord_less @ C @ ( F3 @ X4 ) @ ( F3 @ Y3 ) ) )
=> ( ord_less @ C @ ( F3 @ A2 ) @ C3 ) ) ) ) ) ).
% order_less_le_subst2
thf(fact_188_linorder__le__less__linear,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
| ( ord_less @ A @ Y @ X ) ) ) ).
% linorder_le_less_linear
thf(fact_189_order__le__imp__less__or__eq,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less @ A @ X @ Y )
| ( X = Y ) ) ) ) ).
% order_le_imp_less_or_eq
thf(fact_190_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_191_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_192_add__less__same__cancel1,axiom,
! [A: $tType] :
( ( ordere1937475149494474687imp_le @ A )
=> ! [B2: A,A2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ B2 @ A2 ) @ B2 )
= ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% add_less_same_cancel1
thf(fact_193_add__less__same__cancel2,axiom,
! [A: $tType] :
( ( ordere1937475149494474687imp_le @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ A2 @ B2 ) @ B2 )
= ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% add_less_same_cancel2
thf(fact_194_less__add__same__cancel1,axiom,
! [A: $tType] :
( ( ordere1937475149494474687imp_le @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ ( plus_plus @ A @ A2 @ B2 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) ) ) ).
% less_add_same_cancel1
thf(fact_195_less__add__same__cancel2,axiom,
! [A: $tType] :
( ( ordere1937475149494474687imp_le @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ ( plus_plus @ A @ B2 @ A2 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) ) ) ).
% less_add_same_cancel2
thf(fact_196_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ A2 @ A2 ) @ ( zero_zero @ A ) )
= ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% double_add_less_zero_iff_single_add_less_zero
thf(fact_197_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ A2 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% zero_less_double_add_iff_zero_less_single_add
thf(fact_198_add__le__same__cancel1,axiom,
! [A: $tType] :
( ( ordere1937475149494474687imp_le @ A )
=> ! [B2: A,A2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ B2 @ A2 ) @ B2 )
= ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% add_le_same_cancel1
thf(fact_199_add__le__same__cancel2,axiom,
! [A: $tType] :
( ( ordere1937475149494474687imp_le @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ B2 ) @ B2 )
= ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% add_le_same_cancel2
thf(fact_200_le__add__same__cancel1,axiom,
! [A: $tType] :
( ( ordere1937475149494474687imp_le @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ ( plus_plus @ A @ A2 @ B2 ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) ) ) ).
% le_add_same_cancel1
thf(fact_201_le__add__same__cancel2,axiom,
! [A: $tType] :
( ( ordere1937475149494474687imp_le @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ ( plus_plus @ A @ B2 @ A2 ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) ) ) ).
% le_add_same_cancel2
thf(fact_202_add__right__cancel,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [B2: A,A2: A,C3: A] :
( ( ( plus_plus @ A @ B2 @ A2 )
= ( plus_plus @ A @ C3 @ A2 ) )
= ( B2 = C3 ) ) ) ).
% add_right_cancel
thf(fact_203_add__left__cancel,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( ( plus_plus @ A @ A2 @ B2 )
= ( plus_plus @ A @ A2 @ C3 ) )
= ( B2 = C3 ) ) ) ).
% add_left_cancel
thf(fact_204_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_205_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_206_add__le__cancel__right,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [A2: A,C3: A,B2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C3 ) @ ( plus_plus @ A @ B2 @ C3 ) )
= ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% add_le_cancel_right
thf(fact_207_add__le__cancel__left,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [C3: A,A2: A,B2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ C3 @ A2 ) @ ( plus_plus @ A @ C3 @ B2 ) )
= ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% add_le_cancel_left
thf(fact_208_add__0,axiom,
! [A: $tType] :
( ( monoid_add @ A )
=> ! [A2: A] :
( ( plus_plus @ A @ ( zero_zero @ A ) @ A2 )
= A2 ) ) ).
% add_0
thf(fact_209_zero__eq__add__iff__both__eq__0,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [X: A,Y: A] :
( ( ( zero_zero @ A )
= ( plus_plus @ A @ X @ Y ) )
= ( ( X
= ( zero_zero @ A ) )
& ( Y
= ( zero_zero @ A ) ) ) ) ) ).
% zero_eq_add_iff_both_eq_0
thf(fact_210_add__eq__0__iff__both__eq__0,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [X: A,Y: A] :
( ( ( plus_plus @ A @ X @ Y )
= ( zero_zero @ A ) )
= ( ( X
= ( zero_zero @ A ) )
& ( Y
= ( zero_zero @ A ) ) ) ) ) ).
% add_eq_0_iff_both_eq_0
thf(fact_211_add__cancel__right__right,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add @ A )
=> ! [A2: A,B2: A] :
( ( A2
= ( plus_plus @ A @ A2 @ B2 ) )
= ( B2
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_right_right
thf(fact_212_add__cancel__right__left,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add @ A )
=> ! [A2: A,B2: A] :
( ( A2
= ( plus_plus @ A @ B2 @ A2 ) )
= ( B2
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_right_left
thf(fact_213_add__cancel__left__right,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add @ A )
=> ! [A2: A,B2: A] :
( ( ( plus_plus @ A @ A2 @ B2 )
= A2 )
= ( B2
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_left_right
thf(fact_214_add__cancel__left__left,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add @ A )
=> ! [B2: A,A2: A] :
( ( ( plus_plus @ A @ B2 @ A2 )
= A2 )
= ( B2
= ( zero_zero @ A ) ) ) ) ).
% add_cancel_left_left
thf(fact_215_double__zero__sym,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A2: A] :
( ( ( zero_zero @ A )
= ( plus_plus @ A @ A2 @ A2 ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% double_zero_sym
thf(fact_216_add_Oright__neutral,axiom,
! [A: $tType] :
( ( monoid_add @ A )
=> ! [A2: A] :
( ( plus_plus @ A @ A2 @ ( zero_zero @ A ) )
= A2 ) ) ).
% add.right_neutral
thf(fact_217_add__less__cancel__right,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [A2: A,C3: A,B2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ A2 @ C3 ) @ ( plus_plus @ A @ B2 @ C3 ) )
= ( ord_less @ A @ A2 @ B2 ) ) ) ).
% add_less_cancel_right
thf(fact_218_add__less__cancel__left,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [C3: A,A2: A,B2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ C3 @ A2 ) @ ( plus_plus @ A @ C3 @ B2 ) )
= ( ord_less @ A @ A2 @ B2 ) ) ) ).
% add_less_cancel_left
thf(fact_219_less__nat__zero__code,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% less_nat_zero_code
thf(fact_220_neq0__conv,axiom,
! [N: nat] :
( ( N
!= ( zero_zero @ nat ) )
= ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ).
% neq0_conv
thf(fact_221_bot__nat__0_Onot__eq__extremum,axiom,
! [A2: nat] :
( ( A2
!= ( zero_zero @ nat ) )
= ( ord_less @ nat @ ( zero_zero @ nat ) @ A2 ) ) ).
% bot_nat_0.not_eq_extremum
thf(fact_222_le0,axiom,
! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% le0
thf(fact_223_bot__nat__0_Oextremum,axiom,
! [A2: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ A2 ) ).
% bot_nat_0.extremum
thf(fact_224_Nat_Oadd__0__right,axiom,
! [M2: nat] :
( ( plus_plus @ nat @ M2 @ ( zero_zero @ nat ) )
= M2 ) ).
% Nat.add_0_right
thf(fact_225_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_226_nat__add__left__cancel__less,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( ord_less @ nat @ ( plus_plus @ nat @ K2 @ M2 ) @ ( plus_plus @ nat @ K2 @ N ) )
= ( ord_less @ nat @ M2 @ N ) ) ).
% nat_add_left_cancel_less
thf(fact_227_nat__add__left__cancel__le,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( ord_less_eq @ nat @ ( plus_plus @ nat @ K2 @ M2 ) @ ( plus_plus @ nat @ K2 @ N ) )
= ( ord_less_eq @ nat @ M2 @ N ) ) ).
% nat_add_left_cancel_le
thf(fact_228_max__0R,axiom,
! [N: nat] :
( ( ord_max @ nat @ N @ ( zero_zero @ nat ) )
= N ) ).
% max_0R
thf(fact_229_max__0L,axiom,
! [N: nat] :
( ( ord_max @ nat @ ( zero_zero @ nat ) @ N )
= N ) ).
% max_0L
thf(fact_230_max__nat_Oright__neutral,axiom,
! [A2: nat] :
( ( ord_max @ nat @ A2 @ ( zero_zero @ nat ) )
= A2 ) ).
% max_nat.right_neutral
thf(fact_231_max__nat_Oneutr__eq__iff,axiom,
! [A2: nat,B2: nat] :
( ( ( zero_zero @ nat )
= ( ord_max @ nat @ A2 @ B2 ) )
= ( ( A2
= ( zero_zero @ nat ) )
& ( B2
= ( zero_zero @ nat ) ) ) ) ).
% max_nat.neutr_eq_iff
thf(fact_232_max__nat_Oleft__neutral,axiom,
! [A2: nat] :
( ( ord_max @ nat @ ( zero_zero @ nat ) @ A2 )
= A2 ) ).
% max_nat.left_neutral
thf(fact_233_max__nat_Oeq__neutr__iff,axiom,
! [A2: nat,B2: nat] :
( ( ( ord_max @ nat @ A2 @ B2 )
= ( zero_zero @ nat ) )
= ( ( A2
= ( zero_zero @ nat ) )
& ( B2
= ( zero_zero @ nat ) ) ) ) ).
% max_nat.eq_neutr_iff
thf(fact_234_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ A2 ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% zero_le_double_add_iff_zero_le_single_add
thf(fact_235_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ A2 ) @ ( zero_zero @ A ) )
= ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% double_add_le_zero_iff_single_add_le_zero
thf(fact_236_less__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ( ( ord_less @ ( A > B ) )
= ( ^ [F4: A > B,G4: A > B] :
( ( ord_less_eq @ ( A > B ) @ F4 @ G4 )
& ~ ( ord_less_eq @ ( A > B ) @ G4 @ F4 ) ) ) ) ) ).
% less_fun_def
thf(fact_237_zero__reorient,axiom,
! [A: $tType] :
( ( zero @ A )
=> ! [X: A] :
( ( ( zero_zero @ A )
= X )
= ( X
= ( zero_zero @ A ) ) ) ) ).
% zero_reorient
thf(fact_238_measure__induct__rule,axiom,
! [B: $tType,A: $tType] :
( ( wellorder @ B )
=> ! [F3: A > B,P2: A > $o,A2: A] :
( ! [X4: A] :
( ! [Y4: A] :
( ( ord_less @ B @ ( F3 @ Y4 ) @ ( F3 @ X4 ) )
=> ( P2 @ Y4 ) )
=> ( P2 @ X4 ) )
=> ( P2 @ A2 ) ) ) ).
% measure_induct_rule
thf(fact_239_measure__induct,axiom,
! [B: $tType,A: $tType] :
( ( wellorder @ B )
=> ! [F3: A > B,P2: A > $o,A2: A] :
( ! [X4: A] :
( ! [Y4: A] :
( ( ord_less @ B @ ( F3 @ Y4 ) @ ( F3 @ X4 ) )
=> ( P2 @ Y4 ) )
=> ( P2 @ X4 ) )
=> ( P2 @ A2 ) ) ) ).
% measure_induct
thf(fact_240_add__right__imp__eq,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [B2: A,A2: A,C3: A] :
( ( ( plus_plus @ A @ B2 @ A2 )
= ( plus_plus @ A @ C3 @ A2 ) )
=> ( B2 = C3 ) ) ) ).
% add_right_imp_eq
thf(fact_241_add__left__imp__eq,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( ( plus_plus @ A @ A2 @ B2 )
= ( plus_plus @ A @ A2 @ C3 ) )
=> ( B2 = C3 ) ) ) ).
% add_left_imp_eq
thf(fact_242_add_Oleft__commute,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A )
=> ! [B2: A,A2: A,C3: A] :
( ( plus_plus @ A @ B2 @ ( plus_plus @ A @ A2 @ C3 ) )
= ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C3 ) ) ) ) ).
% add.left_commute
thf(fact_243_add_Ocommute,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A )
=> ( ( plus_plus @ A )
= ( ^ [A6: A,B5: A] : ( plus_plus @ A @ B5 @ A6 ) ) ) ) ).
% add.commute
thf(fact_244_add_Oright__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [B2: A,A2: A,C3: A] :
( ( ( plus_plus @ A @ B2 @ A2 )
= ( plus_plus @ A @ C3 @ A2 ) )
= ( B2 = C3 ) ) ) ).
% add.right_cancel
thf(fact_245_add_Oleft__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( ( plus_plus @ A @ A2 @ B2 )
= ( plus_plus @ A @ A2 @ C3 ) )
= ( B2 = C3 ) ) ) ).
% add.left_cancel
thf(fact_246_add_Oassoc,axiom,
! [A: $tType] :
( ( semigroup_add @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C3 )
= ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C3 ) ) ) ) ).
% add.assoc
thf(fact_247_group__cancel_Oadd2,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [B6: A,K2: A,B2: A,A2: A] :
( ( B6
= ( plus_plus @ A @ K2 @ B2 ) )
=> ( ( plus_plus @ A @ A2 @ B6 )
= ( plus_plus @ A @ K2 @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ) ).
% group_cancel.add2
thf(fact_248_group__cancel_Oadd1,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A5: A,K2: A,A2: A,B2: A] :
( ( A5
= ( plus_plus @ A @ K2 @ A2 ) )
=> ( ( plus_plus @ A @ A5 @ B2 )
= ( plus_plus @ A @ K2 @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ) ).
% group_cancel.add1
thf(fact_249_add__mono__thms__linordered__semiring_I4_J,axiom,
! [A: $tType] :
( ( ordere6658533253407199908up_add @ A )
=> ! [I2: A,J: A,K2: A,L: A] :
( ( ( I2 = J )
& ( K2 = L ) )
=> ( ( plus_plus @ A @ I2 @ K2 )
= ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_semiring(4)
thf(fact_250_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: $tType] :
( ( ab_semigroup_add @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C3 )
= ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C3 ) ) ) ) ).
% ab_semigroup_add_class.add_ac(1)
thf(fact_251_infinite__descent__measure,axiom,
! [A: $tType,P2: A > $o,V2: A > nat,X: A] :
( ! [X4: A] :
( ~ ( P2 @ X4 )
=> ? [Y4: A] :
( ( ord_less @ nat @ ( V2 @ Y4 ) @ ( V2 @ X4 ) )
& ~ ( P2 @ Y4 ) ) )
=> ( P2 @ X ) ) ).
% infinite_descent_measure
thf(fact_252_linorder__neqE__nat,axiom,
! [X: nat,Y: nat] :
( ( X != Y )
=> ( ~ ( ord_less @ nat @ X @ Y )
=> ( ord_less @ nat @ Y @ X ) ) ) ).
% linorder_neqE_nat
thf(fact_253_infinite__descent,axiom,
! [P2: nat > $o,N: nat] :
( ! [N3: nat] :
( ~ ( P2 @ N3 )
=> ? [M4: nat] :
( ( ord_less @ nat @ M4 @ N3 )
& ~ ( P2 @ M4 ) ) )
=> ( P2 @ N ) ) ).
% infinite_descent
thf(fact_254_nat__less__induct,axiom,
! [P2: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M4: nat] :
( ( ord_less @ nat @ M4 @ N3 )
=> ( P2 @ M4 ) )
=> ( P2 @ N3 ) )
=> ( P2 @ N ) ) ).
% nat_less_induct
thf(fact_255_less__irrefl__nat,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ N ) ).
% less_irrefl_nat
thf(fact_256_less__not__refl3,axiom,
! [S: nat,T2: nat] :
( ( ord_less @ nat @ S @ T2 )
=> ( S != T2 ) ) ).
% less_not_refl3
thf(fact_257_less__not__refl2,axiom,
! [N: nat,M2: nat] :
( ( ord_less @ nat @ N @ M2 )
=> ( M2 != N ) ) ).
% less_not_refl2
thf(fact_258_less__not__refl,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ N ) ).
% less_not_refl
thf(fact_259_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_260_Nat_Oex__has__greatest__nat,axiom,
! [P2: nat > $o,K2: nat,B2: nat] :
( ( P2 @ K2 )
=> ( ! [Y3: nat] :
( ( P2 @ Y3 )
=> ( ord_less_eq @ nat @ Y3 @ B2 ) )
=> ? [X4: nat] :
( ( P2 @ X4 )
& ! [Y4: nat] :
( ( P2 @ Y4 )
=> ( ord_less_eq @ nat @ Y4 @ X4 ) ) ) ) ) ).
% Nat.ex_has_greatest_nat
thf(fact_261_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_262_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_263_eq__imp__le,axiom,
! [M2: nat,N: nat] :
( ( M2 = N )
=> ( ord_less_eq @ nat @ M2 @ N ) ) ).
% eq_imp_le
thf(fact_264_le__trans,axiom,
! [I2: nat,J: nat,K2: nat] :
( ( ord_less_eq @ nat @ I2 @ J )
=> ( ( ord_less_eq @ nat @ J @ K2 )
=> ( ord_less_eq @ nat @ I2 @ K2 ) ) ) ).
% le_trans
thf(fact_265_le__refl,axiom,
! [N: nat] : ( ord_less_eq @ nat @ N @ N ) ).
% le_refl
thf(fact_266_zero__le,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [X: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ X ) ) ).
% zero_le
thf(fact_267_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_268_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_269_not__less__zero,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [N: A] :
~ ( ord_less @ A @ N @ ( zero_zero @ A ) ) ) ).
% not_less_zero
thf(fact_270_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_271_add__le__imp__le__right,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [A2: A,C3: A,B2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C3 ) @ ( plus_plus @ A @ B2 @ C3 ) )
=> ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% add_le_imp_le_right
thf(fact_272_add__le__imp__le__left,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [C3: A,A2: A,B2: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ C3 @ A2 ) @ ( plus_plus @ A @ C3 @ B2 ) )
=> ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% add_le_imp_le_left
thf(fact_273_le__iff__add,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A6: A,B5: A] :
? [C4: A] :
( B5
= ( plus_plus @ A @ A6 @ C4 ) ) ) ) ) ).
% le_iff_add
thf(fact_274_add__right__mono,axiom,
! [A: $tType] :
( ( ordere6658533253407199908up_add @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C3 ) @ ( plus_plus @ A @ B2 @ C3 ) ) ) ) ).
% add_right_mono
thf(fact_275_less__eqE,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ~ ! [C2: A] :
( B2
!= ( plus_plus @ A @ A2 @ C2 ) ) ) ) ).
% less_eqE
thf(fact_276_add__left__mono,axiom,
! [A: $tType] :
( ( ordere6658533253407199908up_add @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ C3 @ A2 ) @ ( plus_plus @ A @ C3 @ B2 ) ) ) ) ).
% add_left_mono
thf(fact_277_add__mono,axiom,
! [A: $tType] :
( ( ordere6658533253407199908up_add @ A )
=> ! [A2: A,B2: A,C3: A,D3: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C3 @ D3 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C3 ) @ ( plus_plus @ A @ B2 @ D3 ) ) ) ) ) ).
% add_mono
thf(fact_278_add__mono__thms__linordered__semiring_I1_J,axiom,
! [A: $tType] :
( ( ordere6658533253407199908up_add @ A )
=> ! [I2: A,J: A,K2: A,L: A] :
( ( ( ord_less_eq @ A @ I2 @ J )
& ( ord_less_eq @ A @ K2 @ L ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ I2 @ K2 ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_semiring(1)
thf(fact_279_add__mono__thms__linordered__semiring_I2_J,axiom,
! [A: $tType] :
( ( ordere6658533253407199908up_add @ A )
=> ! [I2: A,J: A,K2: A,L: A] :
( ( ( I2 = J )
& ( ord_less_eq @ A @ K2 @ L ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ I2 @ K2 ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_semiring(2)
thf(fact_280_add__mono__thms__linordered__semiring_I3_J,axiom,
! [A: $tType] :
( ( ordere6658533253407199908up_add @ A )
=> ! [I2: A,J: A,K2: A,L: A] :
( ( ( ord_less_eq @ A @ I2 @ J )
& ( K2 = L ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ I2 @ K2 ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_semiring(3)
thf(fact_281_add_Ogroup__left__neutral,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A] :
( ( plus_plus @ A @ ( zero_zero @ A ) @ A2 )
= A2 ) ) ).
% add.group_left_neutral
thf(fact_282_add_Ocomm__neutral,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A2: A] :
( ( plus_plus @ A @ A2 @ ( zero_zero @ A ) )
= A2 ) ) ).
% add.comm_neutral
thf(fact_283_comm__monoid__add__class_Oadd__0,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A2: A] :
( ( plus_plus @ A @ ( zero_zero @ A ) @ A2 )
= A2 ) ) ).
% comm_monoid_add_class.add_0
thf(fact_284_add__less__imp__less__right,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [A2: A,C3: A,B2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ A2 @ C3 ) @ ( plus_plus @ A @ B2 @ C3 ) )
=> ( ord_less @ A @ A2 @ B2 ) ) ) ).
% add_less_imp_less_right
thf(fact_285_add__less__imp__less__left,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [C3: A,A2: A,B2: A] :
( ( ord_less @ A @ ( plus_plus @ A @ C3 @ A2 ) @ ( plus_plus @ A @ C3 @ B2 ) )
=> ( ord_less @ A @ A2 @ B2 ) ) ) ).
% add_less_imp_less_left
thf(fact_286_add__strict__right__mono,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ord_less @ A @ ( plus_plus @ A @ A2 @ C3 ) @ ( plus_plus @ A @ B2 @ C3 ) ) ) ) ).
% add_strict_right_mono
thf(fact_287_add__strict__left__mono,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ord_less @ A @ ( plus_plus @ A @ C3 @ A2 ) @ ( plus_plus @ A @ C3 @ B2 ) ) ) ) ).
% add_strict_left_mono
thf(fact_288_add__strict__mono,axiom,
! [A: $tType] :
( ( strict9044650504122735259up_add @ A )
=> ! [A2: A,B2: A,C3: A,D3: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ C3 @ D3 )
=> ( ord_less @ A @ ( plus_plus @ A @ A2 @ C3 ) @ ( plus_plus @ A @ B2 @ D3 ) ) ) ) ) ).
% add_strict_mono
thf(fact_289_add__mono__thms__linordered__field_I1_J,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [I2: A,J: A,K2: A,L: A] :
( ( ( ord_less @ A @ I2 @ J )
& ( K2 = L ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I2 @ K2 ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_field(1)
thf(fact_290_add__mono__thms__linordered__field_I2_J,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [I2: A,J: A,K2: A,L: A] :
( ( ( I2 = J )
& ( ord_less @ A @ K2 @ L ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I2 @ K2 ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_field(2)
thf(fact_291_add__mono__thms__linordered__field_I5_J,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [I2: A,J: A,K2: A,L: A] :
( ( ( ord_less @ A @ I2 @ J )
& ( ord_less @ A @ K2 @ L ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I2 @ K2 ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_field(5)
thf(fact_292_infinite__descent0__measure,axiom,
! [A: $tType,V2: A > nat,P2: A > $o,X: A] :
( ! [X4: A] :
( ( ( V2 @ X4 )
= ( zero_zero @ nat ) )
=> ( P2 @ X4 ) )
=> ( ! [X4: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( V2 @ X4 ) )
=> ( ~ ( P2 @ X4 )
=> ? [Y4: A] :
( ( ord_less @ nat @ ( V2 @ Y4 ) @ ( V2 @ X4 ) )
& ~ ( P2 @ Y4 ) ) ) )
=> ( P2 @ X ) ) ) ).
% infinite_descent0_measure
thf(fact_293_infinite__descent0,axiom,
! [P2: nat > $o,N: nat] :
( ( P2 @ ( zero_zero @ nat ) )
=> ( ! [N3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( ~ ( P2 @ N3 )
=> ? [M4: nat] :
( ( ord_less @ nat @ M4 @ N3 )
& ~ ( P2 @ M4 ) ) ) )
=> ( P2 @ N ) ) ) ).
% infinite_descent0
thf(fact_294_gr__implies__not0,axiom,
! [M2: nat,N: nat] :
( ( ord_less @ nat @ M2 @ N )
=> ( N
!= ( zero_zero @ nat ) ) ) ).
% gr_implies_not0
thf(fact_295_less__zeroE,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% less_zeroE
thf(fact_296_not__less0,axiom,
! [N: nat] :
~ ( ord_less @ nat @ N @ ( zero_zero @ nat ) ) ).
% not_less0
thf(fact_297_not__gr0,axiom,
! [N: nat] :
( ( ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% not_gr0
thf(fact_298_gr0I,axiom,
! [N: nat] :
( ( N
!= ( zero_zero @ nat ) )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ).
% gr0I
thf(fact_299_bot__nat__0_Oextremum__strict,axiom,
! [A2: nat] :
~ ( ord_less @ nat @ A2 @ ( zero_zero @ nat ) ) ).
% bot_nat_0.extremum_strict
thf(fact_300_le__0__eq,axiom,
! [N: nat] :
( ( ord_less_eq @ nat @ N @ ( zero_zero @ nat ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% le_0_eq
thf(fact_301_bot__nat__0_Oextremum__uniqueI,axiom,
! [A2: nat] :
( ( ord_less_eq @ nat @ A2 @ ( zero_zero @ nat ) )
=> ( A2
= ( zero_zero @ nat ) ) ) ).
% bot_nat_0.extremum_uniqueI
thf(fact_302_bot__nat__0_Oextremum__unique,axiom,
! [A2: nat] :
( ( ord_less_eq @ nat @ A2 @ ( zero_zero @ nat ) )
= ( A2
= ( zero_zero @ nat ) ) ) ).
% bot_nat_0.extremum_unique
thf(fact_303_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ N ) ).
% less_eq_nat.simps(1)
thf(fact_304_less__mono__imp__le__mono,axiom,
! [F3: nat > nat,I2: nat,J: nat] :
( ! [I3: nat,J2: nat] :
( ( ord_less @ nat @ I3 @ J2 )
=> ( ord_less @ nat @ ( F3 @ I3 ) @ ( F3 @ J2 ) ) )
=> ( ( ord_less_eq @ nat @ I2 @ J )
=> ( ord_less_eq @ nat @ ( F3 @ I2 ) @ ( F3 @ J ) ) ) ) ).
% less_mono_imp_le_mono
thf(fact_305_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_306_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_307_le__eq__less__or__eq,axiom,
( ( ord_less_eq @ nat )
= ( ^ [M3: nat,N2: nat] :
( ( ord_less @ nat @ M3 @ N2 )
| ( M3 = N2 ) ) ) ) ).
% le_eq_less_or_eq
thf(fact_308_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_309_nat__less__le,axiom,
( ( ord_less @ nat )
= ( ^ [M3: nat,N2: nat] :
( ( ord_less_eq @ nat @ M3 @ N2 )
& ( M3 != N2 ) ) ) ) ).
% nat_less_le
thf(fact_310_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_311_plus__nat_Oadd__0,axiom,
! [N: nat] :
( ( plus_plus @ nat @ ( zero_zero @ nat ) @ N )
= N ) ).
% plus_nat.add_0
thf(fact_312_less__add__eq__less,axiom,
! [K2: nat,L: nat,M2: nat,N: nat] :
( ( ord_less @ nat @ K2 @ L )
=> ( ( ( plus_plus @ nat @ M2 @ L )
= ( plus_plus @ nat @ K2 @ N ) )
=> ( ord_less @ nat @ M2 @ N ) ) ) ).
% less_add_eq_less
thf(fact_313_trans__less__add2,axiom,
! [I2: nat,J: nat,M2: nat] :
( ( ord_less @ nat @ I2 @ J )
=> ( ord_less @ nat @ I2 @ ( plus_plus @ nat @ M2 @ J ) ) ) ).
% trans_less_add2
thf(fact_314_trans__less__add1,axiom,
! [I2: nat,J: nat,M2: nat] :
( ( ord_less @ nat @ I2 @ J )
=> ( ord_less @ nat @ I2 @ ( plus_plus @ nat @ J @ M2 ) ) ) ).
% trans_less_add1
thf(fact_315_add__less__mono1,axiom,
! [I2: nat,J: nat,K2: nat] :
( ( ord_less @ nat @ I2 @ J )
=> ( ord_less @ nat @ ( plus_plus @ nat @ I2 @ K2 ) @ ( plus_plus @ nat @ J @ K2 ) ) ) ).
% add_less_mono1
thf(fact_316_not__add__less2,axiom,
! [J: nat,I2: nat] :
~ ( ord_less @ nat @ ( plus_plus @ nat @ J @ I2 ) @ I2 ) ).
% not_add_less2
thf(fact_317_not__add__less1,axiom,
! [I2: nat,J: nat] :
~ ( ord_less @ nat @ ( plus_plus @ nat @ I2 @ J ) @ I2 ) ).
% not_add_less1
thf(fact_318_add__less__mono,axiom,
! [I2: nat,J: nat,K2: nat,L: nat] :
( ( ord_less @ nat @ I2 @ J )
=> ( ( ord_less @ nat @ K2 @ L )
=> ( ord_less @ nat @ ( plus_plus @ nat @ I2 @ K2 ) @ ( plus_plus @ nat @ J @ L ) ) ) ) ).
% add_less_mono
thf(fact_319_add__lessD1,axiom,
! [I2: nat,J: nat,K2: nat] :
( ( ord_less @ nat @ ( plus_plus @ nat @ I2 @ J ) @ K2 )
=> ( ord_less @ nat @ I2 @ K2 ) ) ).
% add_lessD1
thf(fact_320_nat__le__iff__add,axiom,
( ( ord_less_eq @ nat )
= ( ^ [M3: nat,N2: nat] :
? [K3: nat] :
( N2
= ( plus_plus @ nat @ M3 @ K3 ) ) ) ) ).
% nat_le_iff_add
thf(fact_321_trans__le__add2,axiom,
! [I2: nat,J: nat,M2: nat] :
( ( ord_less_eq @ nat @ I2 @ J )
=> ( ord_less_eq @ nat @ I2 @ ( plus_plus @ nat @ M2 @ J ) ) ) ).
% trans_le_add2
thf(fact_322_trans__le__add1,axiom,
! [I2: nat,J: nat,M2: nat] :
( ( ord_less_eq @ nat @ I2 @ J )
=> ( ord_less_eq @ nat @ I2 @ ( plus_plus @ nat @ J @ M2 ) ) ) ).
% trans_le_add1
thf(fact_323_add__le__mono1,axiom,
! [I2: nat,J: nat,K2: nat] :
( ( ord_less_eq @ nat @ I2 @ J )
=> ( ord_less_eq @ nat @ ( plus_plus @ nat @ I2 @ K2 ) @ ( plus_plus @ nat @ J @ K2 ) ) ) ).
% add_le_mono1
thf(fact_324_add__le__mono,axiom,
! [I2: nat,J: nat,K2: nat,L: nat] :
( ( ord_less_eq @ nat @ I2 @ J )
=> ( ( ord_less_eq @ nat @ K2 @ L )
=> ( ord_less_eq @ nat @ ( plus_plus @ nat @ I2 @ K2 ) @ ( plus_plus @ nat @ J @ L ) ) ) ) ).
% add_le_mono
thf(fact_325_le__Suc__ex,axiom,
! [K2: nat,L: nat] :
( ( ord_less_eq @ nat @ K2 @ L )
=> ? [N3: nat] :
( L
= ( plus_plus @ nat @ K2 @ N3 ) ) ) ).
% le_Suc_ex
thf(fact_326_add__leD2,axiom,
! [M2: nat,K2: nat,N: nat] :
( ( ord_less_eq @ nat @ ( plus_plus @ nat @ M2 @ K2 ) @ N )
=> ( ord_less_eq @ nat @ K2 @ N ) ) ).
% add_leD2
thf(fact_327_add__leD1,axiom,
! [M2: nat,K2: nat,N: nat] :
( ( ord_less_eq @ nat @ ( plus_plus @ nat @ M2 @ K2 ) @ N )
=> ( ord_less_eq @ nat @ M2 @ N ) ) ).
% add_leD1
thf(fact_328_le__add2,axiom,
! [N: nat,M2: nat] : ( ord_less_eq @ nat @ N @ ( plus_plus @ nat @ M2 @ N ) ) ).
% le_add2
thf(fact_329_le__add1,axiom,
! [N: nat,M2: nat] : ( ord_less_eq @ nat @ N @ ( plus_plus @ nat @ N @ M2 ) ) ).
% le_add1
thf(fact_330_add__leE,axiom,
! [M2: nat,K2: nat,N: nat] :
( ( ord_less_eq @ nat @ ( plus_plus @ nat @ M2 @ K2 ) @ N )
=> ~ ( ( ord_less_eq @ nat @ M2 @ N )
=> ~ ( ord_less_eq @ nat @ K2 @ N ) ) ) ).
% add_leE
thf(fact_331_max__add__distrib__right,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [X: A,Y: A,Z: A] :
( ( plus_plus @ A @ X @ ( ord_max @ A @ Y @ Z ) )
= ( ord_max @ A @ ( plus_plus @ A @ X @ Y ) @ ( plus_plus @ A @ X @ Z ) ) ) ) ).
% max_add_distrib_right
thf(fact_332_max__add__distrib__left,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [X: A,Y: A,Z: A] :
( ( plus_plus @ A @ ( ord_max @ A @ X @ Y ) @ Z )
= ( ord_max @ A @ ( plus_plus @ A @ X @ Z ) @ ( plus_plus @ A @ Y @ Z ) ) ) ) ).
% max_add_distrib_left
thf(fact_333_nat__add__max__right,axiom,
! [M2: nat,N: nat,Q2: nat] :
( ( plus_plus @ nat @ M2 @ ( ord_max @ nat @ N @ Q2 ) )
= ( ord_max @ nat @ ( plus_plus @ nat @ M2 @ N ) @ ( plus_plus @ nat @ M2 @ Q2 ) ) ) ).
% nat_add_max_right
thf(fact_334_nat__add__max__left,axiom,
! [M2: nat,N: nat,Q2: nat] :
( ( plus_plus @ nat @ ( ord_max @ nat @ M2 @ N ) @ Q2 )
= ( ord_max @ nat @ ( plus_plus @ nat @ M2 @ Q2 ) @ ( plus_plus @ nat @ N @ Q2 ) ) ) ).
% nat_add_max_left
thf(fact_335_add__nonpos__eq__0__iff,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ Y @ ( zero_zero @ A ) )
=> ( ( ( plus_plus @ A @ X @ Y )
= ( zero_zero @ A ) )
= ( ( X
= ( zero_zero @ A ) )
& ( Y
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% add_nonpos_eq_0_iff
thf(fact_336_add__nonneg__eq__0__iff,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
=> ( ( ( plus_plus @ A @ X @ Y )
= ( zero_zero @ A ) )
= ( ( X
= ( zero_zero @ A ) )
& ( Y
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% add_nonneg_eq_0_iff
thf(fact_337_add__nonpos__nonpos,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% add_nonpos_nonpos
thf(fact_338_add__nonneg__nonneg,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ) ).
% add_nonneg_nonneg
thf(fact_339_add__increasing2,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [C3: A,B2: A,A2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ord_less_eq @ A @ B2 @ ( plus_plus @ A @ A2 @ C3 ) ) ) ) ) ).
% add_increasing2
thf(fact_340_add__decreasing2,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [C3: A,A2: A,B2: A] :
( ( ord_less_eq @ A @ C3 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C3 ) @ B2 ) ) ) ) ).
% add_decreasing2
thf(fact_341_add__increasing,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ B2 @ C3 )
=> ( ord_less_eq @ A @ B2 @ ( plus_plus @ A @ A2 @ C3 ) ) ) ) ) ).
% add_increasing
thf(fact_342_add__decreasing,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A2: A,C3: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ C3 @ B2 )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ C3 ) @ B2 ) ) ) ) ).
% add_decreasing
thf(fact_343_add__less__le__mono,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [A2: A,B2: A,C3: A,D3: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C3 @ D3 )
=> ( ord_less @ A @ ( plus_plus @ A @ A2 @ C3 ) @ ( plus_plus @ A @ B2 @ D3 ) ) ) ) ) ).
% add_less_le_mono
thf(fact_344_add__le__less__mono,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [A2: A,B2: A,C3: A,D3: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ C3 @ D3 )
=> ( ord_less @ A @ ( plus_plus @ A @ A2 @ C3 ) @ ( plus_plus @ A @ B2 @ D3 ) ) ) ) ) ).
% add_le_less_mono
thf(fact_345_add__mono__thms__linordered__field_I3_J,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [I2: A,J: A,K2: A,L: A] :
( ( ( ord_less @ A @ I2 @ J )
& ( ord_less_eq @ A @ K2 @ L ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I2 @ K2 ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_field(3)
thf(fact_346_add__mono__thms__linordered__field_I4_J,axiom,
! [A: $tType] :
( ( ordere580206878836729694up_add @ A )
=> ! [I2: A,J: A,K2: A,L: A] :
( ( ( ord_less_eq @ A @ I2 @ J )
& ( ord_less @ A @ K2 @ L ) )
=> ( ord_less @ A @ ( plus_plus @ A @ I2 @ K2 ) @ ( plus_plus @ A @ J @ L ) ) ) ) ).
% add_mono_thms_linordered_field(4)
thf(fact_347_pos__add__strict,axiom,
! [A: $tType] :
( ( strict7427464778891057005id_add @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ B2 @ C3 )
=> ( ord_less @ A @ B2 @ ( plus_plus @ A @ A2 @ C3 ) ) ) ) ) ).
% pos_add_strict
thf(fact_348_canonically__ordered__monoid__add__class_OlessE,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ~ ! [C2: A] :
( ( B2
= ( plus_plus @ A @ A2 @ C2 ) )
=> ( C2
= ( zero_zero @ A ) ) ) ) ) ).
% canonically_ordered_monoid_add_class.lessE
thf(fact_349_add__pos__pos,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ) ).
% add_pos_pos
thf(fact_350_add__neg__neg,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( plus_plus @ A @ A2 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% add_neg_neg
thf(fact_351_ex__least__nat__le,axiom,
! [P2: nat > $o,N: nat] :
( ( P2 @ N )
=> ( ~ ( P2 @ ( zero_zero @ nat ) )
=> ? [K: nat] :
( ( ord_less_eq @ nat @ K @ N )
& ! [I: nat] :
( ( ord_less @ nat @ I @ K )
=> ~ ( P2 @ I ) )
& ( P2 @ K ) ) ) ) ).
% ex_least_nat_le
thf(fact_352_less__imp__add__positive,axiom,
! [I2: nat,J: nat] :
( ( ord_less @ nat @ I2 @ J )
=> ? [K: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K )
& ( ( plus_plus @ nat @ I2 @ K )
= J ) ) ) ).
% less_imp_add_positive
thf(fact_353_mono__nat__linear__lb,axiom,
! [F3: nat > nat,M2: nat,K2: nat] :
( ! [M: nat,N3: nat] :
( ( ord_less @ nat @ M @ N3 )
=> ( ord_less @ nat @ ( F3 @ M ) @ ( F3 @ N3 ) ) )
=> ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( F3 @ M2 ) @ K2 ) @ ( F3 @ ( plus_plus @ nat @ M2 @ K2 ) ) ) ) ).
% mono_nat_linear_lb
thf(fact_354_add__strict__increasing2,axiom,
! [A: $tType] :
( ( ordere8940638589300402666id_add @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ B2 @ C3 )
=> ( ord_less @ A @ B2 @ ( plus_plus @ A @ A2 @ C3 ) ) ) ) ) ).
% add_strict_increasing2
thf(fact_355_add__strict__increasing,axiom,
! [A: $tType] :
( ( ordere8940638589300402666id_add @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ B2 @ C3 )
=> ( ord_less @ A @ B2 @ ( plus_plus @ A @ A2 @ C3 ) ) ) ) ) ).
% add_strict_increasing
thf(fact_356_add__pos__nonneg,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ) ).
% add_pos_nonneg
thf(fact_357_add__nonpos__neg,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( plus_plus @ A @ A2 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% add_nonpos_neg
thf(fact_358_add__nonneg__pos,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ) ).
% add_nonneg_pos
thf(fact_359_add__neg__nonpos,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( plus_plus @ A @ A2 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% add_neg_nonpos
thf(fact_360_List_Ofinite__set,axiom,
! [A: $tType,Xs: list @ A] : ( finite_finite2 @ A @ ( set2 @ A @ Xs ) ) ).
% List.finite_set
thf(fact_361_double__eq__0__iff,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A2: A] :
( ( ( plus_plus @ A @ A2 @ A2 )
= ( zero_zero @ A ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% double_eq_0_iff
thf(fact_362_buildup__nothing__in__leaf,axiom,
! [N: nat,X: nat] :
~ ( vEBT_V5719532721284313246member @ ( vEBT_vebt_buildup @ N ) @ X ) ).
% buildup_nothing_in_leaf
thf(fact_363_field__le__epsilon,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y: A] :
( ! [E2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ E2 )
=> ( ord_less_eq @ A @ X @ ( plus_plus @ A @ Y @ E2 ) ) )
=> ( ord_less_eq @ A @ X @ Y ) ) ) ).
% field_le_epsilon
thf(fact_364_add__less__zeroD,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ ( plus_plus @ A @ X @ Y ) @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ X @ ( zero_zero @ A ) )
| ( ord_less @ A @ Y @ ( zero_zero @ A ) ) ) ) ) ).
% add_less_zeroD
thf(fact_365_buildup__gives__empty,axiom,
! [N: nat] :
( ( vEBT_VEBT_set_vebt @ ( vEBT_vebt_buildup @ N ) )
= ( bot_bot @ ( set @ nat ) ) ) ).
% buildup_gives_empty
thf(fact_366_finite__code,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ( ( finite_finite2 @ A )
= ( ^ [A7: set @ A] : $true ) ) ) ).
% finite_code
thf(fact_367_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_368_finite__nat__set__iff__bounded__le,axiom,
( ( finite_finite2 @ nat )
= ( ^ [N4: set @ nat] :
? [M3: nat] :
! [X5: nat] :
( ( member @ nat @ X5 @ N4 )
=> ( ord_less_eq @ nat @ X5 @ M3 ) ) ) ) ).
% finite_nat_set_iff_bounded_le
thf(fact_369_infinite__nat__iff__unbounded__le,axiom,
! [S2: set @ nat] :
( ( ~ ( finite_finite2 @ nat @ S2 ) )
= ( ! [M3: nat] :
? [N2: nat] :
( ( ord_less_eq @ nat @ M3 @ N2 )
& ( member @ nat @ N2 @ S2 ) ) ) ) ).
% infinite_nat_iff_unbounded_le
thf(fact_370_finite__nat__set__iff__bounded,axiom,
( ( finite_finite2 @ nat )
= ( ^ [N4: set @ nat] :
? [M3: nat] :
! [X5: nat] :
( ( member @ nat @ X5 @ N4 )
=> ( ord_less @ nat @ X5 @ M3 ) ) ) ) ).
% finite_nat_set_iff_bounded
thf(fact_371_both__member__options__def,axiom,
( vEBT_V8194947554948674370ptions
= ( ^ [T3: vEBT_VEBT,X5: nat] :
( ( vEBT_V5719532721284313246member @ T3 @ X5 )
| ( vEBT_VEBT_membermima @ T3 @ X5 ) ) ) ) ).
% both_member_options_def
thf(fact_372_bot__apply,axiom,
! [C: $tType,D: $tType] :
( ( bot @ C )
=> ( ( bot_bot @ ( D > C ) )
= ( ^ [X5: D] : ( bot_bot @ C ) ) ) ) ).
% bot_apply
thf(fact_373_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_374_max__bot,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [X: A] :
( ( ord_max @ A @ ( bot_bot @ A ) @ X )
= X ) ) ).
% max_bot
thf(fact_375_max__bot2,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [X: A] :
( ( ord_max @ A @ X @ ( bot_bot @ A ) )
= X ) ) ).
% max_bot2
thf(fact_376_bot__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( bot @ B )
=> ( ( bot_bot @ ( A > B ) )
= ( ^ [X5: A] : ( bot_bot @ B ) ) ) ) ).
% bot_fun_def
thf(fact_377_finite_OemptyI,axiom,
! [A: $tType] : ( finite_finite2 @ A @ ( bot_bot @ ( set @ A ) ) ) ).
% finite.emptyI
thf(fact_378_infinite__imp__nonempty,axiom,
! [A: $tType,S2: set @ A] :
( ~ ( finite_finite2 @ A @ S2 )
=> ( S2
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% infinite_imp_nonempty
thf(fact_379_finite__transitivity__chain,axiom,
! [A: $tType,A5: set @ A,R: A > A > $o] :
( ( finite_finite2 @ A @ A5 )
=> ( ! [X4: A] :
~ ( R @ X4 @ X4 )
=> ( ! [X4: A,Y3: A,Z3: A] :
( ( R @ X4 @ Y3 )
=> ( ( R @ Y3 @ Z3 )
=> ( R @ X4 @ Z3 ) ) )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ A5 )
=> ? [Y4: A] :
( ( member @ A @ Y4 @ A5 )
& ( R @ X4 @ Y4 ) ) )
=> ( A5
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% finite_transitivity_chain
thf(fact_380_bot_Oextremum,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [A2: A] : ( ord_less_eq @ A @ ( bot_bot @ A ) @ A2 ) ) ).
% bot.extremum
thf(fact_381_bot_Oextremum__unique,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ A2 @ ( bot_bot @ A ) )
= ( A2
= ( bot_bot @ A ) ) ) ) ).
% bot.extremum_unique
thf(fact_382_bot_Oextremum__uniqueI,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ A2 @ ( bot_bot @ A ) )
=> ( A2
= ( bot_bot @ A ) ) ) ) ).
% bot.extremum_uniqueI
thf(fact_383_bot_Oextremum__strict,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [A2: A] :
~ ( ord_less @ A @ A2 @ ( bot_bot @ A ) ) ) ).
% bot.extremum_strict
thf(fact_384_bot_Onot__eq__extremum,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [A2: A] :
( ( A2
!= ( bot_bot @ A ) )
= ( ord_less @ A @ ( bot_bot @ A ) @ A2 ) ) ) ).
% bot.not_eq_extremum
thf(fact_385_finite__has__maximal,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ? [X4: A] :
( ( member @ A @ X4 @ A5 )
& ! [Xa: A] :
( ( member @ A @ Xa @ A5 )
=> ( ( ord_less_eq @ A @ X4 @ Xa )
=> ( X4 = Xa ) ) ) ) ) ) ) ).
% finite_has_maximal
thf(fact_386_finite__has__minimal,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ? [X4: A] :
( ( member @ A @ X4 @ A5 )
& ! [Xa: A] :
( ( member @ A @ Xa @ A5 )
=> ( ( ord_less_eq @ A @ Xa @ X4 )
=> ( X4 = Xa ) ) ) ) ) ) ) ).
% finite_has_minimal
thf(fact_387_infinite__growing,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X7: set @ A] :
( ( X7
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ X7 )
=> ? [Xa: A] :
( ( member @ A @ Xa @ X7 )
& ( ord_less @ A @ X4 @ Xa ) ) )
=> ~ ( finite_finite2 @ A @ X7 ) ) ) ) ).
% infinite_growing
thf(fact_388_ex__min__if__finite,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [S2: set @ A] :
( ( finite_finite2 @ A @ S2 )
=> ( ( S2
!= ( bot_bot @ ( set @ A ) ) )
=> ? [X4: A] :
( ( member @ A @ X4 @ S2 )
& ~ ? [Xa: A] :
( ( member @ A @ Xa @ S2 )
& ( ord_less @ A @ Xa @ X4 ) ) ) ) ) ) ).
% ex_min_if_finite
thf(fact_389_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_390_linordered__field__no__lb,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X3: A] :
? [Y3: A] : ( ord_less @ A @ Y3 @ X3 ) ) ).
% linordered_field_no_lb
thf(fact_391_linordered__field__no__ub,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X3: A] :
? [X_12: A] : ( ord_less @ A @ X3 @ X_12 ) ) ).
% linordered_field_no_ub
thf(fact_392_linorder__neqE__linordered__idom,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,Y: A] :
( ( X != Y )
=> ( ~ ( ord_less @ A @ X @ Y )
=> ( ord_less @ A @ Y @ X ) ) ) ) ).
% linorder_neqE_linordered_idom
thf(fact_393_finite__subset,axiom,
! [A: $tType,A5: set @ A,B6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B6 )
=> ( ( finite_finite2 @ A @ B6 )
=> ( finite_finite2 @ A @ A5 ) ) ) ).
% finite_subset
thf(fact_394_infinite__super,axiom,
! [A: $tType,S2: set @ A,T4: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ S2 @ T4 )
=> ( ~ ( finite_finite2 @ A @ S2 )
=> ~ ( finite_finite2 @ A @ T4 ) ) ) ).
% infinite_super
thf(fact_395_rev__finite__subset,axiom,
! [A: $tType,B6: set @ A,A5: set @ A] :
( ( finite_finite2 @ A @ B6 )
=> ( ( ord_less_eq @ ( set @ A ) @ A5 @ B6 )
=> ( finite_finite2 @ A @ A5 ) ) ) ).
% rev_finite_subset
thf(fact_396_finite__psubset__induct,axiom,
! [A: $tType,A5: set @ A,P2: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ A5 )
=> ( ! [A8: set @ A] :
( ( finite_finite2 @ A @ A8 )
=> ( ! [B7: set @ A] :
( ( ord_less @ ( set @ A ) @ B7 @ A8 )
=> ( P2 @ B7 ) )
=> ( P2 @ A8 ) ) )
=> ( P2 @ A5 ) ) ) ).
% finite_psubset_induct
thf(fact_397_finite,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ! [A5: set @ A] : ( finite_finite2 @ A @ A5 ) ) ).
% finite
thf(fact_398_finite__set__choice,axiom,
! [B: $tType,A: $tType,A5: set @ A,P2: A > B > $o] :
( ( finite_finite2 @ A @ A5 )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ A5 )
=> ? [X_1: B] : ( P2 @ X4 @ X_1 ) )
=> ? [F2: A > B] :
! [X3: A] :
( ( member @ A @ X3 @ A5 )
=> ( P2 @ X3 @ ( F2 @ X3 ) ) ) ) ) ).
% finite_set_choice
thf(fact_399_bounded__Max__nat,axiom,
! [P2: nat > $o,X: nat,M5: nat] :
( ( P2 @ X )
=> ( ! [X4: nat] :
( ( P2 @ X4 )
=> ( ord_less_eq @ nat @ X4 @ M5 ) )
=> ~ ! [M: nat] :
( ( P2 @ M )
=> ~ ! [X3: nat] :
( ( P2 @ X3 )
=> ( ord_less_eq @ nat @ X3 @ M ) ) ) ) ) ).
% bounded_Max_nat
thf(fact_400_subset__code_I1_J,axiom,
! [A: $tType,Xs: list @ A,B6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ B6 )
= ( ! [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Xs ) )
=> ( member @ A @ X5 @ B6 ) ) ) ) ).
% subset_code(1)
thf(fact_401_fold__atLeastAtMost__nat_Ocases,axiom,
! [A: $tType,X: product_prod @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) )] :
~ ! [F2: nat > A > A,A4: nat,B4: nat,Acc: A] :
( X
!= ( product_Pair @ ( nat > A > A ) @ ( product_prod @ nat @ ( product_prod @ nat @ A ) ) @ F2 @ ( product_Pair @ nat @ ( product_prod @ nat @ A ) @ A4 @ ( product_Pair @ nat @ A @ B4 @ Acc ) ) ) ) ).
% fold_atLeastAtMost_nat.cases
thf(fact_402_VEBT__internal_Omembermima_Osimps_I3_J,axiom,
! [Mi: nat,Ma: nat,Va: list @ vEBT_VEBT,Vb: vEBT_VEBT,X: nat] :
( ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( zero_zero @ nat ) @ Va @ Vb ) @ X )
= ( ( X = Mi )
| ( X = Ma ) ) ) ).
% VEBT_internal.membermima.simps(3)
thf(fact_403_finite__has__maximal2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A5: set @ A,A2: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( member @ A @ A2 @ A5 )
=> ? [X4: A] :
( ( member @ A @ X4 @ A5 )
& ( ord_less_eq @ A @ A2 @ X4 )
& ! [Xa: A] :
( ( member @ A @ Xa @ A5 )
=> ( ( ord_less_eq @ A @ X4 @ Xa )
=> ( X4 = Xa ) ) ) ) ) ) ) ).
% finite_has_maximal2
thf(fact_404_finite__has__minimal2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A5: set @ A,A2: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( member @ A @ A2 @ A5 )
=> ? [X4: A] :
( ( member @ A @ X4 @ A5 )
& ( ord_less_eq @ A @ X4 @ A2 )
& ! [Xa: A] :
( ( member @ A @ Xa @ A5 )
=> ( ( ord_less_eq @ A @ Xa @ X4 )
=> ( X4 = Xa ) ) ) ) ) ) ) ).
% finite_has_minimal2
thf(fact_405_finite__list,axiom,
! [A: $tType,A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ? [Xs2: list @ A] :
( ( set2 @ A @ Xs2 )
= A5 ) ) ).
% finite_list
thf(fact_406_unbounded__k__infinite,axiom,
! [K2: nat,S2: set @ nat] :
( ! [M: nat] :
( ( ord_less @ nat @ K2 @ M )
=> ? [N5: nat] :
( ( ord_less @ nat @ M @ N5 )
& ( member @ nat @ N5 @ S2 ) ) )
=> ~ ( finite_finite2 @ nat @ S2 ) ) ).
% unbounded_k_infinite
thf(fact_407_bounded__nat__set__is__finite,axiom,
! [N6: set @ nat,N: nat] :
( ! [X4: nat] :
( ( member @ nat @ X4 @ N6 )
=> ( ord_less @ nat @ X4 @ N ) )
=> ( finite_finite2 @ nat @ N6 ) ) ).
% bounded_nat_set_is_finite
thf(fact_408_infinite__nat__iff__unbounded,axiom,
! [S2: set @ nat] :
( ( ~ ( finite_finite2 @ nat @ S2 ) )
= ( ! [M3: nat] :
? [N2: nat] :
( ( ord_less @ nat @ M3 @ N2 )
& ( member @ nat @ N2 @ S2 ) ) ) ) ).
% infinite_nat_iff_unbounded
thf(fact_409_empty__subsetI,axiom,
! [A: $tType,A5: set @ A] : ( ord_less_eq @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A5 ) ).
% empty_subsetI
thf(fact_410_subset__empty,axiom,
! [A: $tType,A5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ ( bot_bot @ ( set @ A ) ) )
= ( A5
= ( bot_bot @ ( set @ A ) ) ) ) ).
% subset_empty
thf(fact_411_empty__iff,axiom,
! [A: $tType,C3: A] :
~ ( member @ A @ C3 @ ( bot_bot @ ( set @ A ) ) ) ).
% empty_iff
thf(fact_412_all__not__in__conv,axiom,
! [A: $tType,A5: set @ A] :
( ( ! [X5: A] :
~ ( member @ A @ X5 @ A5 ) )
= ( A5
= ( bot_bot @ ( set @ A ) ) ) ) ).
% all_not_in_conv
thf(fact_413_Collect__empty__eq,axiom,
! [A: $tType,P2: A > $o] :
( ( ( collect @ A @ P2 )
= ( bot_bot @ ( set @ A ) ) )
= ( ! [X5: A] :
~ ( P2 @ X5 ) ) ) ).
% Collect_empty_eq
thf(fact_414_empty__Collect__eq,axiom,
! [A: $tType,P2: A > $o] :
( ( ( bot_bot @ ( set @ A ) )
= ( collect @ A @ P2 ) )
= ( ! [X5: A] :
~ ( P2 @ X5 ) ) ) ).
% empty_Collect_eq
thf(fact_415_arg__min__if__finite_I2_J,axiom,
! [B: $tType,A: $tType] :
( ( order @ B )
=> ! [S2: set @ A,F3: A > B] :
( ( finite_finite2 @ A @ S2 )
=> ( ( S2
!= ( bot_bot @ ( set @ A ) ) )
=> ~ ? [X3: A] :
( ( member @ A @ X3 @ S2 )
& ( ord_less @ B @ ( F3 @ X3 ) @ ( F3 @ ( lattic7623131987881927897min_on @ A @ B @ F3 @ S2 ) ) ) ) ) ) ) ).
% arg_min_if_finite(2)
thf(fact_416_arg__min__least,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ! [S2: set @ A,Y: A,F3: A > B] :
( ( finite_finite2 @ A @ S2 )
=> ( ( S2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( member @ A @ Y @ S2 )
=> ( ord_less_eq @ B @ ( F3 @ ( lattic7623131987881927897min_on @ A @ B @ F3 @ S2 ) ) @ ( F3 @ Y ) ) ) ) ) ) ).
% arg_min_least
thf(fact_417_Euclid__induct,axiom,
! [P2: nat > nat > $o,A2: nat,B2: nat] :
( ! [A4: nat,B4: nat] :
( ( P2 @ A4 @ B4 )
= ( P2 @ B4 @ A4 ) )
=> ( ! [A4: nat] : ( P2 @ A4 @ ( zero_zero @ nat ) )
=> ( ! [A4: nat,B4: nat] :
( ( P2 @ A4 @ B4 )
=> ( P2 @ A4 @ ( plus_plus @ nat @ A4 @ B4 ) ) )
=> ( P2 @ A2 @ B2 ) ) ) ) ).
% Euclid_induct
thf(fact_418_add__0__iff,axiom,
! [A: $tType] :
( ( semiri1453513574482234551roduct @ A )
=> ! [B2: A,A2: A] :
( ( B2
= ( plus_plus @ A @ B2 @ A2 ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% add_0_iff
thf(fact_419_verit__sum__simplify,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add @ A )
=> ! [A2: A] :
( ( plus_plus @ A @ A2 @ ( zero_zero @ A ) )
= A2 ) ) ).
% verit_sum_simplify
thf(fact_420_psubsetI,axiom,
! [A: $tType,A5: set @ A,B6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B6 )
=> ( ( A5 != B6 )
=> ( ord_less @ ( set @ A ) @ A5 @ B6 ) ) ) ).
% psubsetI
thf(fact_421_subset__iff__psubset__eq,axiom,
! [A: $tType] :
( ( ord_less_eq @ ( set @ A ) )
= ( ^ [A7: set @ A,B8: set @ A] :
( ( ord_less @ ( set @ A ) @ A7 @ B8 )
| ( A7 = B8 ) ) ) ) ).
% subset_iff_psubset_eq
thf(fact_422_subset__psubset__trans,axiom,
! [A: $tType,A5: set @ A,B6: set @ A,C5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B6 )
=> ( ( ord_less @ ( set @ A ) @ B6 @ C5 )
=> ( ord_less @ ( set @ A ) @ A5 @ C5 ) ) ) ).
% subset_psubset_trans
thf(fact_423_subset__not__subset__eq,axiom,
! [A: $tType] :
( ( ord_less @ ( set @ A ) )
= ( ^ [A7: set @ A,B8: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A7 @ B8 )
& ~ ( ord_less_eq @ ( set @ A ) @ B8 @ A7 ) ) ) ) ).
% subset_not_subset_eq
thf(fact_424_psubset__subset__trans,axiom,
! [A: $tType,A5: set @ A,B6: set @ A,C5: set @ A] :
( ( ord_less @ ( set @ A ) @ A5 @ B6 )
=> ( ( ord_less_eq @ ( set @ A ) @ B6 @ C5 )
=> ( ord_less @ ( set @ A ) @ A5 @ C5 ) ) ) ).
% psubset_subset_trans
thf(fact_425_psubset__imp__subset,axiom,
! [A: $tType,A5: set @ A,B6: set @ A] :
( ( ord_less @ ( set @ A ) @ A5 @ B6 )
=> ( ord_less_eq @ ( set @ A ) @ A5 @ B6 ) ) ).
% psubset_imp_subset
thf(fact_426_psubset__eq,axiom,
! [A: $tType] :
( ( ord_less @ ( set @ A ) )
= ( ^ [A7: set @ A,B8: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A7 @ B8 )
& ( A7 != B8 ) ) ) ) ).
% psubset_eq
thf(fact_427_psubsetE,axiom,
! [A: $tType,A5: set @ A,B6: set @ A] :
( ( ord_less @ ( set @ A ) @ A5 @ B6 )
=> ~ ( ( ord_less_eq @ ( set @ A ) @ A5 @ B6 )
=> ( ord_less_eq @ ( set @ A ) @ B6 @ A5 ) ) ) ).
% psubsetE
thf(fact_428_psubsetD,axiom,
! [A: $tType,A5: set @ A,B6: set @ A,C3: A] :
( ( ord_less @ ( set @ A ) @ A5 @ B6 )
=> ( ( member @ A @ C3 @ A5 )
=> ( member @ A @ C3 @ B6 ) ) ) ).
% psubsetD
thf(fact_429_psubset__trans,axiom,
! [A: $tType,A5: set @ A,B6: set @ A,C5: set @ A] :
( ( ord_less @ ( set @ A ) @ A5 @ B6 )
=> ( ( ord_less @ ( set @ A ) @ B6 @ C5 )
=> ( ord_less @ ( set @ A ) @ A5 @ C5 ) ) ) ).
% psubset_trans
thf(fact_430_bot__set__def,axiom,
! [A: $tType] :
( ( bot_bot @ ( set @ A ) )
= ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) ) ).
% bot_set_def
thf(fact_431_bot__nat__def,axiom,
( ( bot_bot @ nat )
= ( zero_zero @ nat ) ) ).
% bot_nat_def
thf(fact_432_verit__la__disequality,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A] :
( ( A2 = B2 )
| ~ ( ord_less_eq @ A @ A2 @ B2 )
| ~ ( ord_less_eq @ A @ B2 @ A2 ) ) ) ).
% verit_la_disequality
thf(fact_433_verit__comp__simplify1_I2_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A] : ( ord_less_eq @ A @ A2 @ A2 ) ) ).
% verit_comp_simplify1(2)
thf(fact_434_verit__comp__simplify1_I1_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A] :
~ ( ord_less @ A @ A2 @ A2 ) ) ).
% verit_comp_simplify1(1)
thf(fact_435_ex__in__conv,axiom,
! [A: $tType,A5: set @ A] :
( ( ? [X5: A] : ( member @ A @ X5 @ A5 ) )
= ( A5
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% ex_in_conv
thf(fact_436_equals0I,axiom,
! [A: $tType,A5: set @ A] :
( ! [Y3: A] :
~ ( member @ A @ Y3 @ A5 )
=> ( A5
= ( bot_bot @ ( set @ A ) ) ) ) ).
% equals0I
thf(fact_437_equals0D,axiom,
! [A: $tType,A5: set @ A,A2: A] :
( ( A5
= ( bot_bot @ ( set @ A ) ) )
=> ~ ( member @ A @ A2 @ A5 ) ) ).
% equals0D
thf(fact_438_emptyE,axiom,
! [A: $tType,A2: A] :
~ ( member @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ).
% emptyE
thf(fact_439_not__psubset__empty,axiom,
! [A: $tType,A5: set @ A] :
~ ( ord_less @ ( set @ A ) @ A5 @ ( bot_bot @ ( set @ A ) ) ) ).
% not_psubset_empty
thf(fact_440_arg__min__if__finite_I1_J,axiom,
! [B: $tType,A: $tType] :
( ( order @ B )
=> ! [S2: set @ A,F3: A > B] :
( ( finite_finite2 @ A @ S2 )
=> ( ( S2
!= ( bot_bot @ ( set @ A ) ) )
=> ( member @ A @ ( lattic7623131987881927897min_on @ A @ B @ F3 @ S2 ) @ S2 ) ) ) ) ).
% arg_min_if_finite(1)
thf(fact_441_verit__comp__simplify1_I3_J,axiom,
! [B: $tType] :
( ( linorder @ B )
=> ! [B3: B,A3: B] :
( ( ~ ( ord_less_eq @ B @ B3 @ A3 ) )
= ( ord_less @ B @ A3 @ B3 ) ) ) ).
% verit_comp_simplify1(3)
thf(fact_442_subset__emptyI,axiom,
! [A: $tType,A5: set @ A] :
( ! [X4: A] :
~ ( member @ A @ X4 @ A5 )
=> ( ord_less_eq @ ( set @ A ) @ A5 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% subset_emptyI
thf(fact_443_field__lbound__gt__zero,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [D1: A,D22: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ D1 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ D22 )
=> ? [E2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ E2 )
& ( ord_less @ A @ E2 @ D1 )
& ( ord_less @ A @ E2 @ D22 ) ) ) ) ) ).
% field_lbound_gt_zero
thf(fact_444_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_445_complete__interval,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [A2: A,B2: A,P2: A > $o] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( P2 @ A2 )
=> ( ~ ( P2 @ B2 )
=> ? [C2: A] :
( ( ord_less_eq @ A @ A2 @ C2 )
& ( ord_less_eq @ A @ C2 @ B2 )
& ! [X3: A] :
( ( ( ord_less_eq @ A @ A2 @ X3 )
& ( ord_less @ A @ X3 @ C2 ) )
=> ( P2 @ X3 ) )
& ! [D4: A] :
( ! [X4: A] :
( ( ( ord_less_eq @ A @ A2 @ X4 )
& ( ord_less @ A @ X4 @ D4 ) )
=> ( P2 @ X4 ) )
=> ( ord_less_eq @ A @ D4 @ C2 ) ) ) ) ) ) ) ).
% complete_interval
thf(fact_446_pinf_I6_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z3: A] :
! [X3: A] :
( ( ord_less @ A @ Z3 @ X3 )
=> ~ ( ord_less_eq @ A @ X3 @ T2 ) ) ) ).
% pinf(6)
thf(fact_447_pinf_I8_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z3: A] :
! [X3: A] :
( ( ord_less @ A @ Z3 @ X3 )
=> ( ord_less_eq @ A @ T2 @ X3 ) ) ) ).
% pinf(8)
thf(fact_448_minf_I6_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z3: A] :
! [X3: A] :
( ( ord_less @ A @ X3 @ Z3 )
=> ( ord_less_eq @ A @ X3 @ T2 ) ) ) ).
% minf(6)
thf(fact_449_minf_I8_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z3: A] :
! [X3: A] :
( ( ord_less @ A @ X3 @ Z3 )
=> ~ ( ord_less_eq @ A @ T2 @ X3 ) ) ) ).
% minf(8)
thf(fact_450_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_451_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,S3: vEBT_VEBT] :
( Tree
= ( vEBT_Node @ Info2 @ ( suc @ ( suc @ N ) ) @ TreeList2 @ S3 ) ) ) ).
% deg_SUcn_Node
thf(fact_452_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_453_even__odd__cases,axiom,
! [X: nat] :
( ! [N3: nat] :
( X
!= ( plus_plus @ nat @ N3 @ N3 ) )
=> ~ ! [N3: nat] :
( X
!= ( plus_plus @ nat @ N3 @ ( suc @ N3 ) ) ) ) ).
% even_odd_cases
thf(fact_454_old_Onat_Oinject,axiom,
! [Nat: nat,Nat2: nat] :
( ( ( suc @ Nat )
= ( suc @ Nat2 ) )
= ( Nat = Nat2 ) ) ).
% old.nat.inject
thf(fact_455_nat_Oinject,axiom,
! [X2: nat,Y2: nat] :
( ( ( suc @ X2 )
= ( suc @ Y2 ) )
= ( X2 = Y2 ) ) ).
% nat.inject
thf(fact_456_lessI,axiom,
! [N: nat] : ( ord_less @ nat @ N @ ( suc @ N ) ) ).
% lessI
thf(fact_457_Suc__mono,axiom,
! [M2: nat,N: nat] :
( ( ord_less @ nat @ M2 @ N )
=> ( ord_less @ nat @ ( suc @ M2 ) @ ( suc @ N ) ) ) ).
% Suc_mono
thf(fact_458_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_459_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_460_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_461_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_462_less__Suc0,axiom,
! [N: nat] :
( ( ord_less @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% less_Suc0
thf(fact_463_zero__less__Suc,axiom,
! [N: nat] : ( ord_less @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) ).
% zero_less_Suc
thf(fact_464_n__not__Suc__n,axiom,
! [N: nat] :
( N
!= ( suc @ N ) ) ).
% n_not_Suc_n
thf(fact_465_Suc__inject,axiom,
! [X: nat,Y: nat] :
( ( ( suc @ X )
= ( suc @ Y ) )
=> ( X = Y ) ) ).
% Suc_inject
thf(fact_466_vebt__buildup_Ocases,axiom,
! [X: nat] :
( ( X
!= ( zero_zero @ nat ) )
=> ( ( X
!= ( suc @ ( zero_zero @ nat ) ) )
=> ~ ! [Va2: nat] :
( X
!= ( suc @ ( suc @ Va2 ) ) ) ) ) ).
% vebt_buildup.cases
thf(fact_467_list__decode_Ocases,axiom,
! [X: nat] :
( ( X
!= ( zero_zero @ nat ) )
=> ~ ! [N3: nat] :
( X
!= ( suc @ N3 ) ) ) ).
% list_decode.cases
thf(fact_468_nat_Odistinct_I1_J,axiom,
! [X2: nat] :
( ( zero_zero @ nat )
!= ( suc @ X2 ) ) ).
% nat.distinct(1)
thf(fact_469_old_Onat_Odistinct_I2_J,axiom,
! [Nat3: nat] :
( ( suc @ Nat3 )
!= ( zero_zero @ nat ) ) ).
% old.nat.distinct(2)
thf(fact_470_old_Onat_Odistinct_I1_J,axiom,
! [Nat2: nat] :
( ( zero_zero @ nat )
!= ( suc @ Nat2 ) ) ).
% old.nat.distinct(1)
thf(fact_471_nat_OdiscI,axiom,
! [Nat: nat,X2: nat] :
( ( Nat
= ( suc @ X2 ) )
=> ( Nat
!= ( zero_zero @ nat ) ) ) ).
% nat.discI
thf(fact_472_old_Onat_Oexhaust,axiom,
! [Y: nat] :
( ( Y
!= ( zero_zero @ nat ) )
=> ~ ! [Nat4: nat] :
( Y
!= ( suc @ Nat4 ) ) ) ).
% old.nat.exhaust
thf(fact_473_nat__induct,axiom,
! [P2: nat > $o,N: nat] :
( ( P2 @ ( zero_zero @ nat ) )
=> ( ! [N3: nat] :
( ( P2 @ N3 )
=> ( P2 @ ( suc @ N3 ) ) )
=> ( P2 @ N ) ) ) ).
% nat_induct
thf(fact_474_diff__induct,axiom,
! [P2: nat > nat > $o,M2: nat,N: nat] :
( ! [X4: nat] : ( P2 @ X4 @ ( zero_zero @ nat ) )
=> ( ! [Y3: nat] : ( P2 @ ( zero_zero @ nat ) @ ( suc @ Y3 ) )
=> ( ! [X4: nat,Y3: nat] :
( ( P2 @ X4 @ Y3 )
=> ( P2 @ ( suc @ X4 ) @ ( suc @ Y3 ) ) )
=> ( P2 @ M2 @ N ) ) ) ) ).
% diff_induct
thf(fact_475_zero__induct,axiom,
! [P2: nat > $o,K2: nat] :
( ( P2 @ K2 )
=> ( ! [N3: nat] :
( ( P2 @ ( suc @ N3 ) )
=> ( P2 @ N3 ) )
=> ( P2 @ ( zero_zero @ nat ) ) ) ) ).
% zero_induct
thf(fact_476_Suc__neq__Zero,axiom,
! [M2: nat] :
( ( suc @ M2 )
!= ( zero_zero @ nat ) ) ).
% Suc_neq_Zero
thf(fact_477_Zero__neq__Suc,axiom,
! [M2: nat] :
( ( zero_zero @ nat )
!= ( suc @ M2 ) ) ).
% Zero_neq_Suc
thf(fact_478_Zero__not__Suc,axiom,
! [M2: nat] :
( ( zero_zero @ nat )
!= ( suc @ M2 ) ) ).
% Zero_not_Suc
thf(fact_479_not0__implies__Suc,axiom,
! [N: nat] :
( ( N
!= ( zero_zero @ nat ) )
=> ? [M: nat] :
( N
= ( suc @ M ) ) ) ).
% not0_implies_Suc
thf(fact_480_Nat_OlessE,axiom,
! [I2: nat,K2: nat] :
( ( ord_less @ nat @ I2 @ K2 )
=> ( ( K2
!= ( suc @ I2 ) )
=> ~ ! [J2: nat] :
( ( ord_less @ nat @ I2 @ J2 )
=> ( K2
!= ( suc @ J2 ) ) ) ) ) ).
% Nat.lessE
thf(fact_481_Suc__lessD,axiom,
! [M2: nat,N: nat] :
( ( ord_less @ nat @ ( suc @ M2 ) @ N )
=> ( ord_less @ nat @ M2 @ N ) ) ).
% Suc_lessD
thf(fact_482_Suc__lessE,axiom,
! [I2: nat,K2: nat] :
( ( ord_less @ nat @ ( suc @ I2 ) @ K2 )
=> ~ ! [J2: nat] :
( ( ord_less @ nat @ I2 @ J2 )
=> ( K2
!= ( suc @ J2 ) ) ) ) ).
% Suc_lessE
thf(fact_483_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_484_less__SucE,axiom,
! [M2: nat,N: nat] :
( ( ord_less @ nat @ M2 @ ( suc @ N ) )
=> ( ~ ( ord_less @ nat @ M2 @ N )
=> ( M2 = N ) ) ) ).
% less_SucE
thf(fact_485_less__SucI,axiom,
! [M2: nat,N: nat] :
( ( ord_less @ nat @ M2 @ N )
=> ( ord_less @ nat @ M2 @ ( suc @ N ) ) ) ).
% less_SucI
thf(fact_486_Ex__less__Suc,axiom,
! [N: nat,P2: nat > $o] :
( ( ? [I4: nat] :
( ( ord_less @ nat @ I4 @ ( suc @ N ) )
& ( P2 @ I4 ) ) )
= ( ( P2 @ N )
| ? [I4: nat] :
( ( ord_less @ nat @ I4 @ N )
& ( P2 @ I4 ) ) ) ) ).
% Ex_less_Suc
thf(fact_487_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_488_not__less__eq,axiom,
! [M2: nat,N: nat] :
( ( ~ ( ord_less @ nat @ M2 @ N ) )
= ( ord_less @ nat @ N @ ( suc @ M2 ) ) ) ).
% not_less_eq
thf(fact_489_All__less__Suc,axiom,
! [N: nat,P2: nat > $o] :
( ( ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( suc @ N ) )
=> ( P2 @ I4 ) ) )
= ( ( P2 @ N )
& ! [I4: nat] :
( ( ord_less @ nat @ I4 @ N )
=> ( P2 @ I4 ) ) ) ) ).
% All_less_Suc
thf(fact_490_Suc__less__eq2,axiom,
! [N: nat,M2: nat] :
( ( ord_less @ nat @ ( suc @ N ) @ M2 )
= ( ? [M6: nat] :
( ( M2
= ( suc @ M6 ) )
& ( ord_less @ nat @ N @ M6 ) ) ) ) ).
% Suc_less_eq2
thf(fact_491_less__antisym,axiom,
! [N: nat,M2: nat] :
( ~ ( ord_less @ nat @ N @ M2 )
=> ( ( ord_less @ nat @ N @ ( suc @ M2 ) )
=> ( M2 = N ) ) ) ).
% less_antisym
thf(fact_492_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_493_less__trans__Suc,axiom,
! [I2: nat,J: nat,K2: nat] :
( ( ord_less @ nat @ I2 @ J )
=> ( ( ord_less @ nat @ J @ K2 )
=> ( ord_less @ nat @ ( suc @ I2 ) @ K2 ) ) ) ).
% less_trans_Suc
thf(fact_494_less__Suc__induct,axiom,
! [I2: nat,J: nat,P2: nat > nat > $o] :
( ( ord_less @ nat @ I2 @ J )
=> ( ! [I3: nat] : ( P2 @ I3 @ ( suc @ I3 ) )
=> ( ! [I3: nat,J2: nat,K: nat] :
( ( ord_less @ nat @ I3 @ J2 )
=> ( ( ord_less @ nat @ J2 @ K )
=> ( ( P2 @ I3 @ J2 )
=> ( ( P2 @ J2 @ K )
=> ( P2 @ I3 @ K ) ) ) ) )
=> ( P2 @ I2 @ J ) ) ) ) ).
% less_Suc_induct
thf(fact_495_strict__inc__induct,axiom,
! [I2: nat,J: nat,P2: nat > $o] :
( ( ord_less @ nat @ I2 @ J )
=> ( ! [I3: nat] :
( ( J
= ( suc @ I3 ) )
=> ( P2 @ I3 ) )
=> ( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ J )
=> ( ( P2 @ ( suc @ I3 ) )
=> ( P2 @ I3 ) ) )
=> ( P2 @ I2 ) ) ) ) ).
% strict_inc_induct
thf(fact_496_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_497_Suc__leD,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq @ nat @ ( suc @ M2 ) @ N )
=> ( ord_less_eq @ nat @ M2 @ N ) ) ).
% Suc_leD
thf(fact_498_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_499_le__SucI,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ord_less_eq @ nat @ M2 @ ( suc @ N ) ) ) ).
% le_SucI
thf(fact_500_Suc__le__D,axiom,
! [N: nat,M7: nat] :
( ( ord_less_eq @ nat @ ( suc @ N ) @ M7 )
=> ? [M: nat] :
( M7
= ( suc @ M ) ) ) ).
% Suc_le_D
thf(fact_501_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_502_Suc__n__not__le__n,axiom,
! [N: nat] :
~ ( ord_less_eq @ nat @ ( suc @ N ) @ N ) ).
% Suc_n_not_le_n
thf(fact_503_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_504_full__nat__induct,axiom,
! [P2: nat > $o,N: nat] :
( ! [N3: nat] :
( ! [M4: nat] :
( ( ord_less_eq @ nat @ ( suc @ M4 ) @ N3 )
=> ( P2 @ M4 ) )
=> ( P2 @ N3 ) )
=> ( P2 @ N ) ) ).
% full_nat_induct
thf(fact_505_nat__induct__at__least,axiom,
! [M2: nat,N: nat,P2: nat > $o] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( P2 @ M2 )
=> ( ! [N3: nat] :
( ( ord_less_eq @ nat @ M2 @ N3 )
=> ( ( P2 @ N3 )
=> ( P2 @ ( suc @ N3 ) ) ) )
=> ( P2 @ N ) ) ) ) ).
% nat_induct_at_least
thf(fact_506_transitive__stepwise__le,axiom,
! [M2: nat,N: nat,R: nat > nat > $o] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ! [X4: nat] : ( R @ X4 @ X4 )
=> ( ! [X4: nat,Y3: nat,Z3: nat] :
( ( R @ X4 @ Y3 )
=> ( ( R @ Y3 @ Z3 )
=> ( R @ X4 @ Z3 ) ) )
=> ( ! [N3: nat] : ( R @ N3 @ ( suc @ N3 ) )
=> ( R @ M2 @ N ) ) ) ) ) ).
% transitive_stepwise_le
thf(fact_507_nat__arith_Osuc1,axiom,
! [A5: nat,K2: nat,A2: nat] :
( ( A5
= ( plus_plus @ nat @ K2 @ A2 ) )
=> ( ( suc @ A5 )
= ( plus_plus @ nat @ K2 @ ( suc @ A2 ) ) ) ) ).
% nat_arith.suc1
thf(fact_508_add__Suc,axiom,
! [M2: nat,N: nat] :
( ( plus_plus @ nat @ ( suc @ M2 ) @ N )
= ( suc @ ( plus_plus @ nat @ M2 @ N ) ) ) ).
% add_Suc
thf(fact_509_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_510_lift__Suc__mono__less,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [F3: nat > A,N: nat,N7: nat] :
( ! [N3: nat] : ( ord_less @ A @ ( F3 @ N3 ) @ ( F3 @ ( suc @ N3 ) ) )
=> ( ( ord_less @ nat @ N @ N7 )
=> ( ord_less @ A @ ( F3 @ N ) @ ( F3 @ N7 ) ) ) ) ) ).
% lift_Suc_mono_less
thf(fact_511_lift__Suc__mono__less__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [F3: nat > A,N: nat,M2: nat] :
( ! [N3: nat] : ( ord_less @ A @ ( F3 @ N3 ) @ ( F3 @ ( suc @ N3 ) ) )
=> ( ( ord_less @ A @ ( F3 @ N ) @ ( F3 @ M2 ) )
= ( ord_less @ nat @ N @ M2 ) ) ) ) ).
% lift_Suc_mono_less_iff
thf(fact_512_lift__Suc__mono__le,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [F3: nat > A,N: nat,N7: nat] :
( ! [N3: nat] : ( ord_less_eq @ A @ ( F3 @ N3 ) @ ( F3 @ ( suc @ N3 ) ) )
=> ( ( ord_less_eq @ nat @ N @ N7 )
=> ( ord_less_eq @ A @ ( F3 @ N ) @ ( F3 @ N7 ) ) ) ) ) ).
% lift_Suc_mono_le
thf(fact_513_lift__Suc__antimono__le,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [F3: nat > A,N: nat,N7: nat] :
( ! [N3: nat] : ( ord_less_eq @ A @ ( F3 @ ( suc @ N3 ) ) @ ( F3 @ N3 ) )
=> ( ( ord_less_eq @ nat @ N @ N7 )
=> ( ord_less_eq @ A @ ( F3 @ N7 ) @ ( F3 @ N ) ) ) ) ) ).
% lift_Suc_antimono_le
thf(fact_514_Ex__less__Suc2,axiom,
! [N: nat,P2: nat > $o] :
( ( ? [I4: nat] :
( ( ord_less @ nat @ I4 @ ( suc @ N ) )
& ( P2 @ I4 ) ) )
= ( ( P2 @ ( zero_zero @ nat ) )
| ? [I4: nat] :
( ( ord_less @ nat @ I4 @ N )
& ( P2 @ ( suc @ I4 ) ) ) ) ) ).
% Ex_less_Suc2
thf(fact_515_gr0__conv__Suc,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
= ( ? [M3: nat] :
( N
= ( suc @ M3 ) ) ) ) ).
% gr0_conv_Suc
thf(fact_516_All__less__Suc2,axiom,
! [N: nat,P2: nat > $o] :
( ( ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( suc @ N ) )
=> ( P2 @ I4 ) ) )
= ( ( P2 @ ( zero_zero @ nat ) )
& ! [I4: nat] :
( ( ord_less @ nat @ I4 @ N )
=> ( P2 @ ( suc @ I4 ) ) ) ) ) ).
% All_less_Suc2
thf(fact_517_gr0__implies__Suc,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ? [M: nat] :
( N
= ( suc @ M ) ) ) ).
% gr0_implies_Suc
thf(fact_518_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_519_Suc__leI,axiom,
! [M2: nat,N: nat] :
( ( ord_less @ nat @ M2 @ N )
=> ( ord_less_eq @ nat @ ( suc @ M2 ) @ N ) ) ).
% Suc_leI
thf(fact_520_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_521_dec__induct,axiom,
! [I2: nat,J: nat,P2: nat > $o] :
( ( ord_less_eq @ nat @ I2 @ J )
=> ( ( P2 @ I2 )
=> ( ! [N3: nat] :
( ( ord_less_eq @ nat @ I2 @ N3 )
=> ( ( ord_less @ nat @ N3 @ J )
=> ( ( P2 @ N3 )
=> ( P2 @ ( suc @ N3 ) ) ) ) )
=> ( P2 @ J ) ) ) ) ).
% dec_induct
thf(fact_522_inc__induct,axiom,
! [I2: nat,J: nat,P2: nat > $o] :
( ( ord_less_eq @ nat @ I2 @ J )
=> ( ( P2 @ J )
=> ( ! [N3: nat] :
( ( ord_less_eq @ nat @ I2 @ N3 )
=> ( ( ord_less @ nat @ N3 @ J )
=> ( ( P2 @ ( suc @ N3 ) )
=> ( P2 @ N3 ) ) ) )
=> ( P2 @ I2 ) ) ) ) ).
% inc_induct
thf(fact_523_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_524_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_525_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_526_less__eq__Suc__le,axiom,
( ( ord_less @ nat )
= ( ^ [N2: nat] : ( ord_less_eq @ nat @ ( suc @ N2 ) ) ) ) ).
% less_eq_Suc_le
thf(fact_527_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_528_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_529_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_530_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_531_less__add__Suc1,axiom,
! [I2: nat,M2: nat] : ( ord_less @ nat @ I2 @ ( suc @ ( plus_plus @ nat @ I2 @ M2 ) ) ) ).
% less_add_Suc1
thf(fact_532_less__add__Suc2,axiom,
! [I2: nat,M2: nat] : ( ord_less @ nat @ I2 @ ( suc @ ( plus_plus @ nat @ M2 @ I2 ) ) ) ).
% less_add_Suc2
thf(fact_533_less__iff__Suc__add,axiom,
( ( ord_less @ nat )
= ( ^ [M3: nat,N2: nat] :
? [K3: nat] :
( N2
= ( suc @ ( plus_plus @ nat @ M3 @ K3 ) ) ) ) ) ).
% less_iff_Suc_add
thf(fact_534_less__imp__Suc__add,axiom,
! [M2: nat,N: nat] :
( ( ord_less @ nat @ M2 @ N )
=> ? [K: nat] :
( N
= ( suc @ ( plus_plus @ nat @ M2 @ K ) ) ) ) ).
% less_imp_Suc_add
thf(fact_535_ex__least__nat__less,axiom,
! [P2: nat > $o,N: nat] :
( ( P2 @ N )
=> ( ~ ( P2 @ ( zero_zero @ nat ) )
=> ? [K: nat] :
( ( ord_less @ nat @ K @ N )
& ! [I: nat] :
( ( ord_less_eq @ nat @ I @ K )
=> ~ ( P2 @ I ) )
& ( P2 @ ( suc @ K ) ) ) ) ) ).
% ex_least_nat_less
thf(fact_536_minf_I11_J,axiom,
! [C: $tType,D: $tType] :
( ( ord @ C )
=> ! [F5: D] :
? [Z3: C] :
! [X3: C] :
( ( ord_less @ C @ X3 @ Z3 )
=> ( F5 = F5 ) ) ) ).
% minf(11)
thf(fact_537_minf_I7_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z3: A] :
! [X3: A] :
( ( ord_less @ A @ X3 @ Z3 )
=> ~ ( ord_less @ A @ T2 @ X3 ) ) ) ).
% minf(7)
thf(fact_538_minf_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z3: A] :
! [X3: A] :
( ( ord_less @ A @ X3 @ Z3 )
=> ( ord_less @ A @ X3 @ T2 ) ) ) ).
% minf(5)
thf(fact_539_minf_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z3: A] :
! [X3: A] :
( ( ord_less @ A @ X3 @ Z3 )
=> ( X3 != T2 ) ) ) ).
% minf(4)
thf(fact_540_minf_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z3: A] :
! [X3: A] :
( ( ord_less @ A @ X3 @ Z3 )
=> ( X3 != T2 ) ) ) ).
% minf(3)
thf(fact_541_minf_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P2: A > $o,P5: A > $o,Q: A > $o,Q4: A > $o] :
( ? [Z4: A] :
! [X4: A] :
( ( ord_less @ A @ X4 @ Z4 )
=> ( ( P2 @ X4 )
= ( P5 @ X4 ) ) )
=> ( ? [Z4: A] :
! [X4: A] :
( ( ord_less @ A @ X4 @ Z4 )
=> ( ( Q @ X4 )
= ( Q4 @ X4 ) ) )
=> ? [Z3: A] :
! [X3: A] :
( ( ord_less @ A @ X3 @ Z3 )
=> ( ( ( P2 @ X3 )
| ( Q @ X3 ) )
= ( ( P5 @ X3 )
| ( Q4 @ X3 ) ) ) ) ) ) ) ).
% minf(2)
thf(fact_542_minf_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P2: A > $o,P5: A > $o,Q: A > $o,Q4: A > $o] :
( ? [Z4: A] :
! [X4: A] :
( ( ord_less @ A @ X4 @ Z4 )
=> ( ( P2 @ X4 )
= ( P5 @ X4 ) ) )
=> ( ? [Z4: A] :
! [X4: A] :
( ( ord_less @ A @ X4 @ Z4 )
=> ( ( Q @ X4 )
= ( Q4 @ X4 ) ) )
=> ? [Z3: A] :
! [X3: A] :
( ( ord_less @ A @ X3 @ Z3 )
=> ( ( ( P2 @ X3 )
& ( Q @ X3 ) )
= ( ( P5 @ X3 )
& ( Q4 @ X3 ) ) ) ) ) ) ) ).
% minf(1)
thf(fact_543_pinf_I11_J,axiom,
! [C: $tType,D: $tType] :
( ( ord @ C )
=> ! [F5: D] :
? [Z3: C] :
! [X3: C] :
( ( ord_less @ C @ Z3 @ X3 )
=> ( F5 = F5 ) ) ) ).
% pinf(11)
thf(fact_544_pinf_I7_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z3: A] :
! [X3: A] :
( ( ord_less @ A @ Z3 @ X3 )
=> ( ord_less @ A @ T2 @ X3 ) ) ) ).
% pinf(7)
thf(fact_545_pinf_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z3: A] :
! [X3: A] :
( ( ord_less @ A @ Z3 @ X3 )
=> ~ ( ord_less @ A @ X3 @ T2 ) ) ) ).
% pinf(5)
thf(fact_546_pinf_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z3: A] :
! [X3: A] :
( ( ord_less @ A @ Z3 @ X3 )
=> ( X3 != T2 ) ) ) ).
% pinf(4)
thf(fact_547_pinf_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [T2: A] :
? [Z3: A] :
! [X3: A] :
( ( ord_less @ A @ Z3 @ X3 )
=> ( X3 != T2 ) ) ) ).
% pinf(3)
thf(fact_548_pinf_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P2: A > $o,P5: A > $o,Q: A > $o,Q4: A > $o] :
( ? [Z4: A] :
! [X4: A] :
( ( ord_less @ A @ Z4 @ X4 )
=> ( ( P2 @ X4 )
= ( P5 @ X4 ) ) )
=> ( ? [Z4: A] :
! [X4: A] :
( ( ord_less @ A @ Z4 @ X4 )
=> ( ( Q @ X4 )
= ( Q4 @ X4 ) ) )
=> ? [Z3: A] :
! [X3: A] :
( ( ord_less @ A @ Z3 @ X3 )
=> ( ( ( P2 @ X3 )
| ( Q @ X3 ) )
= ( ( P5 @ X3 )
| ( Q4 @ X3 ) ) ) ) ) ) ) ).
% pinf(2)
thf(fact_549_pinf_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P2: A > $o,P5: A > $o,Q: A > $o,Q4: A > $o] :
( ? [Z4: A] :
! [X4: A] :
( ( ord_less @ A @ Z4 @ X4 )
=> ( ( P2 @ X4 )
= ( P5 @ X4 ) ) )
=> ( ? [Z4: A] :
! [X4: A] :
( ( ord_less @ A @ Z4 @ X4 )
=> ( ( Q @ X4 )
= ( Q4 @ X4 ) ) )
=> ? [Z3: A] :
! [X3: A] :
( ( ord_less @ A @ Z3 @ X3 )
=> ( ( ( P2 @ X3 )
& ( Q @ X3 ) )
= ( ( P5 @ X3 )
& ( Q4 @ X3 ) ) ) ) ) ) ) ).
% pinf(1)
thf(fact_550_ex__gt__or__lt,axiom,
! [A: $tType] :
( ( condit5016429287641298734tinuum @ A )
=> ! [A2: A] :
? [B4: A] :
( ( ord_less @ A @ A2 @ B4 )
| ( ord_less @ A @ B4 @ A2 ) ) ) ).
% ex_gt_or_lt
thf(fact_551_is__num__normalize_I1_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C3 )
= ( plus_plus @ A @ A2 @ ( plus_plus @ A @ B2 @ C3 ) ) ) ) ).
% is_num_normalize(1)
thf(fact_552_ssubst__Pair__rhs,axiom,
! [B: $tType,A: $tType,R2: A,S: B,R: set @ ( product_prod @ A @ B ),S4: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R2 @ S ) @ R )
=> ( ( S4 = S )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ R2 @ S4 ) @ R ) ) ) ).
% ssubst_Pair_rhs
thf(fact_553_vebt__insert_Osimps_I3_J,axiom,
! [Info: option @ ( product_prod @ nat @ nat ),Ts: list @ vEBT_VEBT,S: vEBT_VEBT,X: nat] :
( ( vEBT_vebt_insert @ ( vEBT_Node @ Info @ ( suc @ ( zero_zero @ nat ) ) @ Ts @ S ) @ X )
= ( vEBT_Node @ Info @ ( suc @ ( zero_zero @ nat ) ) @ Ts @ S ) ) ).
% vebt_insert.simps(3)
thf(fact_554_vebt__member_Osimps_I4_J,axiom,
! [V: product_prod @ nat @ nat,Vb: list @ vEBT_VEBT,Vc: vEBT_VEBT,X: nat] :
~ ( vEBT_vebt_member @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb @ Vc ) @ X ) ).
% vebt_member.simps(4)
thf(fact_555_option_Osize__gen_I2_J,axiom,
! [A: $tType,X: A > nat,X2: A] :
( ( size_option @ A @ X @ ( some @ A @ X2 ) )
= ( plus_plus @ nat @ ( X @ X2 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% option.size_gen(2)
thf(fact_556_bot__empty__eq,axiom,
! [A: $tType] :
( ( bot_bot @ ( A > $o ) )
= ( ^ [X5: A] : ( member @ A @ X5 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% bot_empty_eq
thf(fact_557_Collect__empty__eq__bot,axiom,
! [A: $tType,P2: A > $o] :
( ( ( collect @ A @ P2 )
= ( bot_bot @ ( set @ A ) ) )
= ( P2
= ( bot_bot @ ( A > $o ) ) ) ) ).
% Collect_empty_eq_bot
thf(fact_558_option_Osize_I4_J,axiom,
! [A: $tType,X2: A] :
( ( size_size @ ( option @ A ) @ ( some @ A @ X2 ) )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% option.size(4)
thf(fact_559_triangle__Suc,axiom,
! [N: nat] :
( ( nat_triangle @ ( suc @ N ) )
= ( plus_plus @ nat @ ( nat_triangle @ N ) @ ( suc @ N ) ) ) ).
% triangle_Suc
thf(fact_560_dependent__nat__choice,axiom,
! [A: $tType,P2: nat > A > $o,Q: nat > A > A > $o] :
( ? [X_1: A] : ( P2 @ ( zero_zero @ nat ) @ X_1 )
=> ( ! [X4: A,N3: nat] :
( ( P2 @ N3 @ X4 )
=> ? [Y4: A] :
( ( P2 @ ( suc @ N3 ) @ Y4 )
& ( Q @ N3 @ X4 @ Y4 ) ) )
=> ? [F2: nat > A] :
! [N5: nat] :
( ( P2 @ N5 @ ( F2 @ N5 ) )
& ( Q @ N5 @ ( F2 @ N5 ) @ ( F2 @ ( suc @ N5 ) ) ) ) ) ) ).
% dependent_nat_choice
thf(fact_561_exists__least__lemma,axiom,
! [P2: nat > $o] :
( ~ ( P2 @ ( zero_zero @ nat ) )
=> ( ? [X_1: nat] : ( P2 @ X_1 )
=> ? [N3: nat] :
( ~ ( P2 @ N3 )
& ( P2 @ ( suc @ N3 ) ) ) ) ) ).
% exists_least_lemma
thf(fact_562__092_060open_062vebt__insert_A_INode_A_ISome_A_Imi_M_Ama_J_J_Adeg_AtreeList_Asummary_J_Ax_A_061_ANode_A_ISome_A_Ix_M_Amax_Ami_Ama_J_J_Adeg_A_ItreeList_091high_Ami_An_A_058_061_Avebt__insert_A_ItreeList_A_B_Ahigh_Ami_An_J_A_Ilow_Ami_An_J_093_J_A_Iif_AminNull_A_ItreeList_A_B_Ahigh_Ami_An_J_Athen_Avebt__insert_Asummary_A_Ihigh_Ami_An_J_Aelse_Asummary_J_092_060close_062,axiom,
( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ mi @ ma ) ) @ deg @ treeList @ summary ) @ xa )
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ xa @ ( ord_max @ nat @ mi @ ma ) ) ) @ deg @ ( list_update @ vEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ mi @ na ) @ ( vEBT_vebt_insert @ ( nth @ vEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ mi @ na ) ) @ ( vEBT_VEBT_low @ mi @ na ) ) ) @ ( if @ vEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth @ vEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ mi @ na ) ) ) @ ( vEBT_vebt_insert @ summary @ ( vEBT_VEBT_high @ mi @ na ) ) @ summary ) ) ) ).
% \<open>vebt_insert (Node (Some (mi, ma)) deg treeList summary) x = Node (Some (x, max mi ma)) deg (treeList[high mi n := vebt_insert (treeList ! high mi n) (low mi n)]) (if minNull (treeList ! high mi n) then vebt_insert summary (high mi n) else summary)\<close>
thf(fact_563_pair__lessI2,axiom,
! [A2: nat,B2: nat,S: nat,T2: nat] :
( ( ord_less_eq @ nat @ A2 @ 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 @ A2 @ S ) @ ( product_Pair @ nat @ nat @ B2 @ T2 ) ) @ fun_pair_less ) ) ) ).
% pair_lessI2
thf(fact_564_vebt__insert_Osimps_I4_J,axiom,
! [V: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
( ( vEBT_vebt_insert @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ V ) ) @ TreeList @ Summary ) @ X )
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ X @ X ) ) @ ( suc @ ( suc @ V ) ) @ TreeList @ Summary ) ) ).
% vebt_insert.simps(4)
thf(fact_565_pair__less__iff1,axiom,
! [X: nat,Y: nat,Z: nat] :
( ( member @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ X @ Y ) @ ( product_Pair @ nat @ nat @ X @ Z ) ) @ fun_pair_less )
= ( ord_less @ nat @ Y @ Z ) ) ).
% pair_less_iff1
thf(fact_566_bit__split__inv,axiom,
! [X: nat,D3: nat] :
( ( vEBT_VEBT_bit_concat @ ( vEBT_VEBT_high @ X @ D3 ) @ ( vEBT_VEBT_low @ X @ D3 ) @ D3 )
= X ) ).
% bit_split_inv
thf(fact_567_list__update__overwrite,axiom,
! [A: $tType,Xs: list @ A,I2: nat,X: A,Y: A] :
( ( list_update @ A @ ( list_update @ A @ Xs @ I2 @ X ) @ I2 @ Y )
= ( list_update @ A @ Xs @ I2 @ Y ) ) ).
% list_update_overwrite
thf(fact_568_not__None__eq,axiom,
! [A: $tType,X: option @ A] :
( ( X
!= ( none @ A ) )
= ( ? [Y6: A] :
( X
= ( some @ A @ Y6 ) ) ) ) ).
% not_None_eq
thf(fact_569_not__Some__eq,axiom,
! [A: $tType,X: option @ A] :
( ( ! [Y6: A] :
( X
!= ( some @ A @ Y6 ) ) )
= ( X
= ( none @ A ) ) ) ).
% not_Some_eq
thf(fact_570_list__update__id,axiom,
! [A: $tType,Xs: list @ A,I2: nat] :
( ( list_update @ A @ Xs @ I2 @ ( nth @ A @ Xs @ I2 ) )
= Xs ) ).
% list_update_id
thf(fact_571_nth__list__update__neq,axiom,
! [A: $tType,I2: nat,J: nat,Xs: list @ A,X: A] :
( ( I2 != J )
=> ( ( nth @ A @ ( list_update @ A @ Xs @ I2 @ X ) @ J )
= ( nth @ A @ Xs @ J ) ) ) ).
% nth_list_update_neq
thf(fact_572_triangle__0,axiom,
( ( nat_triangle @ ( zero_zero @ nat ) )
= ( zero_zero @ nat ) ) ).
% triangle_0
thf(fact_573_list__update__swap,axiom,
! [A: $tType,I2: nat,I5: nat,Xs: list @ A,X: A,X8: A] :
( ( I2 != I5 )
=> ( ( list_update @ A @ ( list_update @ A @ Xs @ I2 @ X ) @ I5 @ X8 )
= ( list_update @ A @ ( list_update @ A @ Xs @ I5 @ X8 ) @ I2 @ X ) ) ) ).
% list_update_swap
thf(fact_574_size__neq__size__imp__neq,axiom,
! [A: $tType] :
( ( size @ A )
=> ! [X: A,Y: A] :
( ( ( size_size @ A @ X )
!= ( size_size @ A @ Y ) )
=> ( X != Y ) ) ) ).
% size_neq_size_imp_neq
thf(fact_575_option_Odistinct_I1_J,axiom,
! [A: $tType,X2: A] :
( ( none @ A )
!= ( some @ A @ X2 ) ) ).
% option.distinct(1)
thf(fact_576_option_OdiscI,axiom,
! [A: $tType,Option: option @ A,X2: A] :
( ( Option
= ( some @ A @ X2 ) )
=> ( Option
!= ( none @ A ) ) ) ).
% option.discI
thf(fact_577_option_Oexhaust,axiom,
! [A: $tType,Y: option @ A] :
( ( Y
!= ( none @ A ) )
=> ~ ! [X22: A] :
( Y
!= ( some @ A @ X22 ) ) ) ).
% option.exhaust
thf(fact_578_split__option__ex,axiom,
! [A: $tType] :
( ( ^ [P3: ( option @ A ) > $o] :
? [X6: option @ A] : ( P3 @ X6 ) )
= ( ^ [P4: ( option @ A ) > $o] :
( ( P4 @ ( none @ A ) )
| ? [X5: A] : ( P4 @ ( some @ A @ X5 ) ) ) ) ) ).
% split_option_ex
thf(fact_579_split__option__all,axiom,
! [A: $tType] :
( ( ^ [P3: ( option @ A ) > $o] :
! [X6: option @ A] : ( P3 @ X6 ) )
= ( ^ [P4: ( option @ A ) > $o] :
( ( P4 @ ( none @ A ) )
& ! [X5: A] : ( P4 @ ( some @ A @ X5 ) ) ) ) ) ).
% split_option_all
thf(fact_580_combine__options__cases,axiom,
! [A: $tType,B: $tType,X: option @ A,P2: ( option @ A ) > ( option @ B ) > $o,Y: option @ B] :
( ( ( X
= ( none @ A ) )
=> ( P2 @ X @ Y ) )
=> ( ( ( Y
= ( none @ B ) )
=> ( P2 @ X @ Y ) )
=> ( ! [A4: A,B4: B] :
( ( X
= ( some @ A @ A4 ) )
=> ( ( Y
= ( some @ B @ B4 ) )
=> ( P2 @ X @ Y ) ) )
=> ( P2 @ X @ Y ) ) ) ) ).
% combine_options_cases
thf(fact_581_option_Osize_I3_J,axiom,
! [A: $tType] :
( ( size_size @ ( option @ A ) @ ( none @ A ) )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% option.size(3)
thf(fact_582_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_583_set__update__subsetI,axiom,
! [A: $tType,Xs: list @ A,A5: set @ A,X: A,I2: nat] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ A5 )
=> ( ( member @ A @ X @ A5 )
=> ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ ( list_update @ A @ Xs @ I2 @ X ) ) @ A5 ) ) ) ).
% set_update_subsetI
thf(fact_584_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_585_VEBT__internal_OminNull_Osimps_I4_J,axiom,
! [Uw: nat,Ux: list @ vEBT_VEBT,Uy: vEBT_VEBT] : ( vEBT_VEBT_minNull @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uw @ Ux @ Uy ) ) ).
% VEBT_internal.minNull.simps(4)
thf(fact_586_option_Osize__neq,axiom,
! [A: $tType,X: option @ A] :
( ( size_size @ ( option @ A ) @ X )
!= ( zero_zero @ nat ) ) ).
% option.size_neq
thf(fact_587_subrelI,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ A @ B )] :
( ! [X4: A,Y3: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y3 ) @ R2 )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y3 ) @ S ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R2 @ S ) ) ).
% subrelI
thf(fact_588_VEBT__internal_Omembermima_Osimps_I2_J,axiom,
! [Ux: list @ vEBT_VEBT,Uy: vEBT_VEBT,Uz: nat] :
~ ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( zero_zero @ nat ) @ Ux @ Uy ) @ Uz ) ).
% VEBT_internal.membermima.simps(2)
thf(fact_589_pair__lessI1,axiom,
! [A2: nat,B2: nat,S: nat,T2: nat] :
( ( ord_less @ nat @ A2 @ 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 @ A2 @ S ) @ ( product_Pair @ nat @ nat @ B2 @ T2 ) ) @ fun_pair_less ) ) ).
% pair_lessI1
thf(fact_590_in__children__def,axiom,
( vEBT_V5917875025757280293ildren
= ( ^ [N2: nat,TreeList3: list @ vEBT_VEBT,X5: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ X5 @ N2 ) ) @ ( vEBT_VEBT_low @ X5 @ N2 ) ) ) ) ).
% in_children_def
thf(fact_591_inthall,axiom,
! [A: $tType,Xs: list @ A,P2: A > $o,N: nat] :
( ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
=> ( P2 @ X4 ) )
=> ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( P2 @ ( nth @ A @ Xs @ N ) ) ) ) ).
% inthall
thf(fact_592_pair__leqI2,axiom,
! [A2: nat,B2: nat,S: nat,T2: nat] :
( ( ord_less_eq @ nat @ A2 @ B2 )
=> ( ( ord_less_eq @ nat @ S @ T2 )
=> ( member @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ A2 @ S ) @ ( product_Pair @ nat @ nat @ B2 @ T2 ) ) @ fun_pair_leq ) ) ) ).
% pair_leqI2
thf(fact_593_pair__leqI1,axiom,
! [A2: nat,B2: nat,S: nat,T2: nat] :
( ( ord_less @ nat @ A2 @ 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 @ A2 @ S ) @ ( product_Pair @ nat @ nat @ B2 @ T2 ) ) @ fun_pair_leq ) ) ).
% pair_leqI1
thf(fact_594__C4_Ohyps_C_I9_J,axiom,
( ( mi != ma )
=> ! [I: nat] :
( ( ord_less @ nat @ I @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ m ) )
=> ( ( ( ( vEBT_VEBT_high @ ma @ na )
= I )
=> ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ treeList @ I ) @ ( vEBT_VEBT_low @ ma @ na ) ) )
& ! [X3: nat] :
( ( ( ( vEBT_VEBT_high @ X3 @ na )
= I )
& ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ treeList @ I ) @ ( vEBT_VEBT_low @ X3 @ na ) ) )
=> ( ( ord_less @ nat @ mi @ X3 )
& ( ord_less_eq @ nat @ X3 @ ma ) ) ) ) ) ) ).
% "4.hyps"(9)
thf(fact_595_vebt__member_Ocases,axiom,
! [X: product_prod @ vEBT_VEBT @ nat] :
( ! [A4: $o,B4: $o,X4: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A4 @ B4 ) @ X4 ) )
=> ( ! [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT,X4: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) @ X4 ) )
=> ( ! [V3: product_prod @ nat @ nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT,X4: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Uy2 @ Uz2 ) @ X4 ) )
=> ( ! [V3: product_prod @ nat @ nat,Vb2: list @ vEBT_VEBT,Vc2: vEBT_VEBT,X4: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb2 @ Vc2 ) @ X4 ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT,X4: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary2 ) @ X4 ) ) ) ) ) ) ).
% vebt_member.cases
thf(fact_596_vebt__insert_Ocases,axiom,
! [X: product_prod @ vEBT_VEBT @ nat] :
( ! [A4: $o,B4: $o,X4: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A4 @ B4 ) @ X4 ) )
=> ( ! [Info2: option @ ( product_prod @ nat @ nat ),Ts2: list @ vEBT_VEBT,S3: vEBT_VEBT,X4: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Info2 @ ( zero_zero @ nat ) @ Ts2 @ S3 ) @ X4 ) )
=> ( ! [Info2: option @ ( product_prod @ nat @ nat ),Ts2: list @ vEBT_VEBT,S3: vEBT_VEBT,X4: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Info2 @ ( suc @ ( zero_zero @ nat ) ) @ Ts2 @ S3 ) @ X4 ) )
=> ( ! [V3: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT,X4: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ V3 ) ) @ TreeList2 @ Summary2 ) @ X4 ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT,X4: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary2 ) @ X4 ) ) ) ) ) ) ).
% vebt_insert.cases
thf(fact_597_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,Uy2: vEBT_VEBT,Uz2: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( zero_zero @ nat ) @ Ux2 @ Uy2 ) @ Uz2 ) )
=> ( ! [Mi2: nat,Ma2: nat,Va3: list @ vEBT_VEBT,Vb2: vEBT_VEBT,X4: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ Va3 @ Vb2 ) @ X4 ) )
=> ( ! [Mi2: nat,Ma2: nat,V3: nat,TreeList2: list @ vEBT_VEBT,Vc2: vEBT_VEBT,X4: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList2 @ Vc2 ) @ X4 ) )
=> ~ ! [V3: nat,TreeList2: list @ vEBT_VEBT,Vd: vEBT_VEBT,X4: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V3 ) @ TreeList2 @ Vd ) @ X4 ) ) ) ) ) ) ).
% VEBT_internal.membermima.cases
thf(fact_598_intind,axiom,
! [A: $tType,I2: nat,N: nat,P2: A > $o,X: A] :
( ( ord_less @ nat @ I2 @ N )
=> ( ( P2 @ X )
=> ( P2 @ ( nth @ A @ ( replicate @ A @ N @ X ) @ I2 ) ) ) ) ).
% intind
thf(fact_599_set__encode__empty,axiom,
( ( nat_set_encode @ ( bot_bot @ ( set @ nat ) ) )
= ( zero_zero @ nat ) ) ).
% set_encode_empty
thf(fact_600_Leaf__0__not,axiom,
! [A2: $o,B2: $o] :
~ ( vEBT_invar_vebt @ ( vEBT_Leaf @ A2 @ B2 ) @ ( zero_zero @ nat ) ) ).
% Leaf_0_not
thf(fact_601__C4_Ohyps_C_I2_J,axiom,
( ( size_size @ ( list @ vEBT_VEBT ) @ treeList )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ m ) ) ).
% "4.hyps"(2)
thf(fact_602__C4_Ohyps_C_I8_J,axiom,
ord_less @ nat @ ma @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ deg ) ).
% "4.hyps"(8)
thf(fact_603__C4_Oprems_C_I1_J,axiom,
ord_less @ nat @ xa @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ deg ) ).
% "4.prems"(1)
thf(fact_604__C4_Oprems_C_I2_J,axiom,
ord_less @ nat @ ya @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ deg ) ).
% "4.prems"(2)
thf(fact_605_VEBT_Oinject_I2_J,axiom,
! [X21: $o,X222: $o,Y21: $o,Y22: $o] :
( ( ( vEBT_Leaf @ X21 @ X222 )
= ( vEBT_Leaf @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X222 = Y22 ) ) ) ).
% VEBT.inject(2)
thf(fact_606__C4_OIH_C_I1_J,axiom,
! [X3: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X3 @ ( set2 @ vEBT_VEBT @ treeList ) )
=> ( ( vEBT_invar_vebt @ X3 @ na )
& ! [Xa: nat] :
( ( ord_less @ nat @ Xa @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ na ) )
=> ! [Xb: nat] :
( ( ord_less @ nat @ Xb @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ na ) )
=> ( ( vEBT_vebt_member @ ( vEBT_vebt_insert @ X3 @ Xa ) @ Xb )
=> ( ( vEBT_vebt_member @ X3 @ Xb )
| ( Xa = Xb ) ) ) ) ) ) ) ).
% "4.IH"(1)
thf(fact_607__C4_Ohyps_C_I5_J,axiom,
! [I: nat] :
( ( ord_less @ nat @ I @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ m ) )
=> ( ( ? [X9: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ treeList @ I ) @ X9 ) )
= ( vEBT_V8194947554948674370ptions @ summary @ I ) ) ) ).
% "4.hyps"(5)
thf(fact_608_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_609_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_610_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_611_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_612_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_613_add__numeral__left,axiom,
! [A: $tType] :
( ( numeral @ A )
=> ! [V: num,W2: num,Z: A] :
( ( plus_plus @ A @ ( numeral_numeral @ A @ V ) @ ( plus_plus @ A @ ( numeral_numeral @ A @ W2 ) @ Z ) )
= ( plus_plus @ A @ ( numeral_numeral @ A @ ( plus_plus @ num @ V @ W2 ) ) @ Z ) ) ) ).
% add_numeral_left
thf(fact_614_insert__simp__mima,axiom,
! [X: nat,Mi: nat,Ma: nat,Deg: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT] :
( ( ( X = Mi )
| ( X = Ma ) )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) ) ) ) ).
% insert_simp_mima
thf(fact_615_valid__insert__both__member__options__add,axiom,
! [T2: vEBT_VEBT,N: nat,X: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
=> ( vEBT_V8194947554948674370ptions @ ( vEBT_vebt_insert @ T2 @ X ) @ X ) ) ) ).
% valid_insert_both_member_options_add
thf(fact_616_valid__insert__both__member__options__pres,axiom,
! [T2: vEBT_VEBT,N: nat,X: nat,Y: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( ord_less @ nat @ Y @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( vEBT_V8194947554948674370ptions @ T2 @ X )
=> ( vEBT_V8194947554948674370ptions @ ( vEBT_vebt_insert @ T2 @ Y ) @ X ) ) ) ) ) ).
% valid_insert_both_member_options_pres
thf(fact_617_replicate__eq__replicate,axiom,
! [A: $tType,M2: nat,X: A,N: nat,Y: A] :
( ( ( replicate @ A @ M2 @ X )
= ( replicate @ A @ N @ Y ) )
= ( ( M2 = N )
& ( ( M2
!= ( zero_zero @ nat ) )
=> ( X = Y ) ) ) ) ).
% replicate_eq_replicate
thf(fact_618_length__replicate,axiom,
! [A: $tType,N: nat,X: A] :
( ( size_size @ ( list @ A ) @ ( replicate @ A @ N @ X ) )
= N ) ).
% length_replicate
thf(fact_619_length__list__update,axiom,
! [A: $tType,Xs: list @ A,I2: nat,X: A] :
( ( size_size @ ( list @ A ) @ ( list_update @ A @ Xs @ I2 @ X ) )
= ( size_size @ ( list @ A ) @ Xs ) ) ).
% length_list_update
thf(fact_620_mi__ma__2__deg,axiom,
! [Mi: nat,Ma: nat,Deg: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ N )
=> ( ( ord_less_eq @ nat @ Mi @ Ma )
& ( ord_less @ nat @ Ma @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg ) ) ) ) ).
% mi_ma_2_deg
thf(fact_621__C4_OIH_C_I2_J,axiom,
! [X: nat,Y: nat] :
( ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ m ) )
=> ( ( ord_less @ nat @ Y @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ m ) )
=> ( ( vEBT_vebt_member @ ( vEBT_vebt_insert @ summary @ X ) @ Y )
=> ( ( vEBT_vebt_member @ summary @ Y )
| ( X = Y ) ) ) ) ) ).
% "4.IH"(2)
thf(fact_622_max__number__of_I1_J,axiom,
! [A: $tType] :
( ( ( numeral @ A )
& ( ord @ A ) )
=> ! [U: num,V: num] :
( ( ( ord_less_eq @ A @ ( numeral_numeral @ A @ U ) @ ( numeral_numeral @ A @ V ) )
=> ( ( ord_max @ A @ ( numeral_numeral @ A @ U ) @ ( numeral_numeral @ A @ V ) )
= ( numeral_numeral @ A @ V ) ) )
& ( ~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ U ) @ ( numeral_numeral @ A @ V ) )
=> ( ( ord_max @ A @ ( numeral_numeral @ A @ U ) @ ( numeral_numeral @ A @ V ) )
= ( numeral_numeral @ A @ U ) ) ) ) ) ).
% max_number_of(1)
thf(fact_623_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_624_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_625_xyprop,axiom,
( ( ord_less @ nat @ ( vEBT_VEBT_high @ xa @ na ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ na ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ ya @ na ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ na ) ) ) ).
% xyprop
thf(fact_626__092_060open_062low_Ax_An_A_060_A2_A_094_An_A_092_060and_062_Alow_Ay_An_A_060_A2_A_094_An_092_060close_062,axiom,
( ( ord_less @ nat @ ( vEBT_VEBT_low @ xa @ na ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ na ) )
& ( ord_less @ nat @ ( vEBT_VEBT_low @ ya @ na ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ na ) ) ) ).
% \<open>low x n < 2 ^ n \<and> low y n < 2 ^ n\<close>
thf(fact_627_Ball__set__replicate,axiom,
! [A: $tType,N: nat,A2: A,P2: A > $o] :
( ( ! [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ ( replicate @ A @ N @ A2 ) ) )
=> ( P2 @ X5 ) ) )
= ( ( P2 @ A2 )
| ( N
= ( zero_zero @ nat ) ) ) ) ).
% Ball_set_replicate
thf(fact_628_Bex__set__replicate,axiom,
! [A: $tType,N: nat,A2: A,P2: A > $o] :
( ( ? [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ ( replicate @ A @ N @ A2 ) ) )
& ( P2 @ X5 ) ) )
= ( ( P2 @ A2 )
& ( N
!= ( zero_zero @ nat ) ) ) ) ).
% Bex_set_replicate
thf(fact_629_in__set__replicate,axiom,
! [A: $tType,X: A,N: nat,Y: A] :
( ( member @ A @ X @ ( set2 @ A @ ( replicate @ A @ N @ Y ) ) )
= ( ( X = Y )
& ( N
!= ( zero_zero @ nat ) ) ) ) ).
% in_set_replicate
thf(fact_630_list__update__beyond,axiom,
! [A: $tType,Xs: list @ A,I2: nat,X: A] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ I2 )
=> ( ( list_update @ A @ Xs @ I2 @ X )
= Xs ) ) ).
% list_update_beyond
thf(fact_631_nth__replicate,axiom,
! [A: $tType,I2: nat,N: nat,X: A] :
( ( ord_less @ nat @ I2 @ N )
=> ( ( nth @ A @ ( replicate @ A @ N @ X ) @ I2 )
= X ) ) ).
% nth_replicate
thf(fact_632__C00_C,axiom,
( ( deg
= ( plus_plus @ nat @ na @ m ) )
& ( ord_less_eq @ nat @ ( zero_zero @ nat ) @ na )
& ( na = m )
& ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ deg )
& ( ( size_size @ ( list @ vEBT_VEBT ) @ treeList )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ na ) ) ) ).
% "00"
thf(fact_633_nth__list__update__eq,axiom,
! [A: $tType,I2: nat,Xs: list @ A,X: A] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ A @ ( list_update @ A @ Xs @ I2 @ X ) @ I2 )
= X ) ) ).
% nth_list_update_eq
thf(fact_634_mimaxyprop,axiom,
( ~ ( ( xa = mi )
| ( xa = ma ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ xa @ na ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ na ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ mi @ na ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ na ) )
& ( ord_less @ nat @ ( vEBT_VEBT_low @ xa @ na ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ na ) )
& ( ord_less @ nat @ ( vEBT_VEBT_low @ mi @ na ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ na ) )
& ( ( size_size @ ( list @ vEBT_VEBT ) @ treeList )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ na ) ) ) ).
% mimaxyprop
thf(fact_635_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_636_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_637_set__swap,axiom,
! [A: $tType,I2: nat,Xs: list @ A,J: nat] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( set2 @ A @ ( list_update @ A @ ( list_update @ A @ Xs @ I2 @ ( nth @ A @ Xs @ J ) ) @ J @ ( nth @ A @ Xs @ I2 ) ) )
= ( set2 @ A @ Xs ) ) ) ) ).
% set_swap
thf(fact_638_Ex__list__of__length,axiom,
! [A: $tType,N: nat] :
? [Xs2: list @ A] :
( ( size_size @ ( list @ A ) @ Xs2 )
= N ) ).
% Ex_list_of_length
thf(fact_639_neq__if__length__neq,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs )
!= ( size_size @ ( list @ A ) @ Ys ) )
=> ( Xs != Ys ) ) ).
% neq_if_length_neq
thf(fact_640_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_641_replicate__length__same,axiom,
! [A: $tType,Xs: list @ A,X: A] :
( ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
=> ( X4 = X ) )
=> ( ( replicate @ A @ ( size_size @ ( list @ A ) @ Xs ) @ X )
= Xs ) ) ).
% replicate_length_same
thf(fact_642_less__by__empty,axiom,
! [A: $tType,A5: set @ ( product_prod @ A @ A ),B6: set @ ( product_prod @ A @ A )] :
( ( A5
= ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ A5 @ B6 ) ) ).
% less_by_empty
thf(fact_643_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_644_numeral__2__eq__2,axiom,
( ( numeral_numeral @ nat @ ( bit0 @ one2 ) )
= ( suc @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% numeral_2_eq_2
thf(fact_645_VEBT_Osize_I4_J,axiom,
! [X21: $o,X222: $o] :
( ( size_size @ vEBT_VEBT @ ( vEBT_Leaf @ X21 @ X222 ) )
= ( zero_zero @ nat ) ) ).
% VEBT.size(4)
thf(fact_646_num_Osize_I4_J,axiom,
( ( size_size @ num @ one2 )
= ( zero_zero @ nat ) ) ).
% num.size(4)
thf(fact_647_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_648_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_649_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_650_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_651_numeral__1__eq__Suc__0,axiom,
( ( numeral_numeral @ nat @ one2 )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% numeral_1_eq_Suc_0
thf(fact_652_Suc__nat__number__of__add,axiom,
! [V: num,N: nat] :
( ( suc @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ V ) @ N ) )
= ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( plus_plus @ num @ V @ one2 ) ) @ N ) ) ).
% Suc_nat_number_of_add
thf(fact_653_VEBT__internal_Ovalid_H_Ocases,axiom,
! [X: product_prod @ vEBT_VEBT @ nat] :
( ! [Uu2: $o,Uv2: $o,D2: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ D2 ) )
=> ~ ! [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_654_VEBT_Oexhaust,axiom,
! [Y: vEBT_VEBT] :
( ! [X112: option @ ( product_prod @ nat @ nat ),X122: nat,X132: list @ vEBT_VEBT,X142: vEBT_VEBT] :
( Y
!= ( vEBT_Node @ X112 @ X122 @ X132 @ X142 ) )
=> ~ ! [X212: $o,X223: $o] :
( Y
!= ( vEBT_Leaf @ X212 @ X223 ) ) ) ).
% VEBT.exhaust
thf(fact_655_VEBT_Odistinct_I1_J,axiom,
! [X11: option @ ( product_prod @ nat @ nat ),X12: nat,X13: list @ vEBT_VEBT,X14: vEBT_VEBT,X21: $o,X222: $o] :
( ( vEBT_Node @ X11 @ X12 @ X13 @ X14 )
!= ( vEBT_Leaf @ X21 @ X222 ) ) ).
% VEBT.distinct(1)
thf(fact_656_zero__neq__numeral,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [N: num] :
( ( zero_zero @ A )
!= ( numeral_numeral @ A @ N ) ) ) ).
% zero_neq_numeral
thf(fact_657_finite__maxlen,axiom,
! [A: $tType,M5: set @ ( list @ A )] :
( ( finite_finite2 @ ( list @ A ) @ M5 )
=> ? [N3: nat] :
! [X3: list @ A] :
( ( member @ ( list @ A ) @ X3 @ M5 )
=> ( ord_less @ nat @ ( size_size @ ( list @ A ) @ X3 ) @ N3 ) ) ) ).
% finite_maxlen
thf(fact_658_length__induct,axiom,
! [A: $tType,P2: ( list @ A ) > $o,Xs: list @ A] :
( ! [Xs2: list @ A] :
( ! [Ys2: list @ A] :
( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Ys2 ) @ ( size_size @ ( list @ A ) @ Xs2 ) )
=> ( P2 @ Ys2 ) )
=> ( P2 @ Xs2 ) )
=> ( P2 @ Xs ) ) ).
% length_induct
thf(fact_659_invar__vebt_Ointros_I2_J,axiom,
! [TreeList: list @ vEBT_VEBT,N: nat,Summary: vEBT_VEBT,M2: nat,Deg: nat] :
( ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ( vEBT_invar_vebt @ X4 @ 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_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_12 )
=> ( ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X_12 ) )
=> ( vEBT_invar_vebt @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg @ TreeList @ Summary ) @ Deg ) ) ) ) ) ) ) ) ).
% invar_vebt.intros(2)
thf(fact_660_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_661_VEBT__internal_OminNull_Osimps_I3_J,axiom,
! [Uu: $o] :
~ ( vEBT_VEBT_minNull @ ( vEBT_Leaf @ Uu @ $true ) ) ).
% VEBT_internal.minNull.simps(3)
thf(fact_662_VEBT__internal_OminNull_Osimps_I2_J,axiom,
! [Uv: $o] :
~ ( vEBT_VEBT_minNull @ ( vEBT_Leaf @ $true @ Uv ) ) ).
% VEBT_internal.minNull.simps(2)
thf(fact_663_VEBT__internal_OminNull_Osimps_I1_J,axiom,
vEBT_VEBT_minNull @ ( vEBT_Leaf @ $false @ $false ) ).
% VEBT_internal.minNull.simps(1)
thf(fact_664_invar__vebt_Ointros_I3_J,axiom,
! [TreeList: list @ vEBT_VEBT,N: nat,Summary: vEBT_VEBT,M2: nat,Deg: nat] :
( ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ( vEBT_invar_vebt @ X4 @ 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_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_12 )
=> ( ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X_12 ) )
=> ( vEBT_invar_vebt @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg @ TreeList @ Summary ) @ Deg ) ) ) ) ) ) ) ) ).
% invar_vebt.intros(3)
thf(fact_665_num_Osize_I5_J,axiom,
! [X2: num] :
( ( size_size @ num @ ( bit0 @ X2 ) )
= ( plus_plus @ nat @ ( size_size @ num @ X2 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% num.size(5)
thf(fact_666_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_667_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_668_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_669_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_670_list__eq__iff__nth__eq,axiom,
! [A: $tType] :
( ( ^ [Y5: list @ A,Z2: list @ A] : Y5 = Z2 )
= ( ^ [Xs3: list @ A,Ys3: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs3 )
= ( size_size @ ( list @ A ) @ Ys3 ) )
& ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ Xs3 ) )
=> ( ( nth @ A @ Xs3 @ I4 )
= ( nth @ A @ Ys3 @ I4 ) ) ) ) ) ) ).
% list_eq_iff_nth_eq
thf(fact_671_Skolem__list__nth,axiom,
! [A: $tType,K2: nat,P2: nat > A > $o] :
( ( ! [I4: nat] :
( ( ord_less @ nat @ I4 @ K2 )
=> ? [X9: A] : ( P2 @ I4 @ X9 ) ) )
= ( ? [Xs3: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs3 )
= K2 )
& ! [I4: nat] :
( ( ord_less @ nat @ I4 @ K2 )
=> ( P2 @ I4 @ ( nth @ A @ Xs3 @ I4 ) ) ) ) ) ) ).
% Skolem_list_nth
thf(fact_672_nth__equalityI,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ A ) @ Ys ) )
=> ( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ A @ Xs @ I3 )
= ( nth @ A @ Ys @ I3 ) ) )
=> ( Xs = Ys ) ) ) ).
% nth_equalityI
thf(fact_673_set__encode__eq,axiom,
! [A5: set @ nat,B6: set @ nat] :
( ( finite_finite2 @ nat @ A5 )
=> ( ( finite_finite2 @ nat @ B6 )
=> ( ( ( nat_set_encode @ A5 )
= ( nat_set_encode @ B6 ) )
= ( A5 = B6 ) ) ) ) ).
% set_encode_eq
thf(fact_674_vebt__buildup_Osimps_I1_J,axiom,
( ( vEBT_vebt_buildup @ ( zero_zero @ nat ) )
= ( vEBT_Leaf @ $false @ $false ) ) ).
% vebt_buildup.simps(1)
thf(fact_675_invar__vebt_Ointros_I4_J,axiom,
! [TreeList: list @ vEBT_VEBT,N: nat,Summary: vEBT_VEBT,M2: nat,Deg: nat,Mi: nat,Ma: nat] :
( ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ( vEBT_invar_vebt @ X4 @ 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 ) )
=> ( ( ? [X9: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ I3 ) @ X9 ) )
= ( vEBT_V8194947554948674370ptions @ Summary @ I3 ) ) )
=> ( ( ( Mi = Ma )
=> ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X_12 ) ) )
=> ( ( ord_less_eq @ nat @ Mi @ Ma )
=> ( ( ord_less @ nat @ Ma @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg ) )
=> ( ( ( Mi != Ma )
=> ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M2 ) )
=> ( ( ( ( vEBT_VEBT_high @ Ma @ N )
= I3 )
=> ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ I3 ) @ ( vEBT_VEBT_low @ Ma @ N ) ) )
& ! [X4: nat] :
( ( ( ( vEBT_VEBT_high @ X4 @ N )
= I3 )
& ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ I3 ) @ ( vEBT_VEBT_low @ X4 @ N ) ) )
=> ( ( ord_less @ nat @ Mi @ X4 )
& ( ord_less_eq @ nat @ X4 @ Ma ) ) ) ) ) )
=> ( vEBT_invar_vebt @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ Deg ) ) ) ) ) ) ) ) ) ) ) ).
% invar_vebt.intros(4)
thf(fact_676_invar__vebt_Ocases,axiom,
! [A1: vEBT_VEBT,A22: nat] :
( ( vEBT_invar_vebt @ A1 @ A22 )
=> ( ( ? [A4: $o,B4: $o] :
( A1
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ( A22
!= ( suc @ ( zero_zero @ nat ) ) ) )
=> ( ! [TreeList2: list @ vEBT_VEBT,N3: nat,Summary2: vEBT_VEBT,M: nat,Deg2: nat] :
( ( A1
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( ( A22 = Deg2 )
=> ( ! [X3: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X3 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ( vEBT_invar_vebt @ X3 @ N3 ) )
=> ( ( vEBT_invar_vebt @ Summary2 @ M )
=> ( ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) )
=> ( ( M = N3 )
=> ( ( Deg2
= ( plus_plus @ nat @ N3 @ M ) )
=> ( ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X_1 )
=> ~ ! [X3: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X3 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X_1 ) ) ) ) ) ) ) ) ) )
=> ( ! [TreeList2: list @ vEBT_VEBT,N3: nat,Summary2: vEBT_VEBT,M: nat,Deg2: nat] :
( ( A1
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( ( A22 = Deg2 )
=> ( ! [X3: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X3 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ( vEBT_invar_vebt @ X3 @ N3 ) )
=> ( ( vEBT_invar_vebt @ Summary2 @ M )
=> ( ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) )
=> ( ( M
= ( suc @ N3 ) )
=> ( ( Deg2
= ( plus_plus @ nat @ N3 @ M ) )
=> ( ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X_1 )
=> ~ ! [X3: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X3 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X_1 ) ) ) ) ) ) ) ) ) )
=> ( ! [TreeList2: list @ vEBT_VEBT,N3: nat,Summary2: vEBT_VEBT,M: nat,Deg2: nat,Mi2: nat,Ma2: nat] :
( ( A1
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( ( A22 = Deg2 )
=> ( ! [X3: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X3 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ( vEBT_invar_vebt @ X3 @ N3 ) )
=> ( ( vEBT_invar_vebt @ Summary2 @ M )
=> ( ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) )
=> ( ( M = N3 )
=> ( ( Deg2
= ( plus_plus @ nat @ N3 @ M ) )
=> ( ! [I: nat] :
( ( ord_less @ nat @ I @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) )
=> ( ( ? [X9: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I ) @ X9 ) )
= ( vEBT_V8194947554948674370ptions @ Summary2 @ I ) ) )
=> ( ( ( Mi2 = Ma2 )
=> ! [X3: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X3 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X_1 ) ) )
=> ( ( ord_less_eq @ nat @ Mi2 @ Ma2 )
=> ( ( ord_less @ nat @ Ma2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
=> ~ ( ( Mi2 != Ma2 )
=> ! [I: nat] :
( ( ord_less @ nat @ I @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) )
=> ( ( ( ( vEBT_VEBT_high @ Ma2 @ N3 )
= I )
=> ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I ) @ ( vEBT_VEBT_low @ Ma2 @ N3 ) ) )
& ! [X3: nat] :
( ( ( ( vEBT_VEBT_high @ X3 @ N3 )
= I )
& ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I ) @ ( vEBT_VEBT_low @ X3 @ N3 ) ) )
=> ( ( ord_less @ nat @ Mi2 @ X3 )
& ( ord_less_eq @ nat @ X3 @ Ma2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
=> ~ ! [TreeList2: list @ vEBT_VEBT,N3: nat,Summary2: vEBT_VEBT,M: nat,Deg2: nat,Mi2: nat,Ma2: nat] :
( ( A1
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( ( A22 = Deg2 )
=> ( ! [X3: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X3 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ( vEBT_invar_vebt @ X3 @ N3 ) )
=> ( ( vEBT_invar_vebt @ Summary2 @ M )
=> ( ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) )
=> ( ( M
= ( suc @ N3 ) )
=> ( ( Deg2
= ( plus_plus @ nat @ N3 @ M ) )
=> ( ! [I: nat] :
( ( ord_less @ nat @ I @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) )
=> ( ( ? [X9: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I ) @ X9 ) )
= ( vEBT_V8194947554948674370ptions @ Summary2 @ I ) ) )
=> ( ( ( Mi2 = Ma2 )
=> ! [X3: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X3 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X3 @ X_1 ) ) )
=> ( ( ord_less_eq @ nat @ Mi2 @ Ma2 )
=> ( ( ord_less @ nat @ Ma2 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
=> ~ ( ( Mi2 != Ma2 )
=> ! [I: nat] :
( ( ord_less @ nat @ I @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M ) )
=> ( ( ( ( vEBT_VEBT_high @ Ma2 @ N3 )
= I )
=> ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I ) @ ( vEBT_VEBT_low @ Ma2 @ N3 ) ) )
& ! [X3: nat] :
( ( ( ( vEBT_VEBT_high @ X3 @ N3 )
= I )
& ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I ) @ ( vEBT_VEBT_low @ X3 @ N3 ) ) )
=> ( ( ord_less @ nat @ Mi2 @ X3 )
& ( ord_less_eq @ nat @ X3 @ Ma2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% invar_vebt.cases
thf(fact_677_invar__vebt_Osimps,axiom,
( vEBT_invar_vebt
= ( ^ [A12: vEBT_VEBT,A23: nat] :
( ( ? [A6: $o,B5: $o] :
( A12
= ( vEBT_Leaf @ A6 @ B5 ) )
& ( A23
= ( suc @ ( zero_zero @ nat ) ) ) )
| ? [TreeList3: list @ vEBT_VEBT,N2: nat,Summary3: vEBT_VEBT] :
( ( A12
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ A23 @ TreeList3 @ Summary3 ) )
& ! [X5: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X5 @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
=> ( vEBT_invar_vebt @ X5 @ N2 ) )
& ( vEBT_invar_vebt @ Summary3 @ N2 )
& ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) )
& ( A23
= ( plus_plus @ nat @ N2 @ N2 ) )
& ~ ? [X9: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X9 )
& ! [X5: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X5 @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
=> ~ ? [X9: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X9 ) ) )
| ? [TreeList3: list @ vEBT_VEBT,N2: nat,Summary3: vEBT_VEBT] :
( ( A12
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ A23 @ TreeList3 @ Summary3 ) )
& ! [X5: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X5 @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
=> ( vEBT_invar_vebt @ X5 @ N2 ) )
& ( vEBT_invar_vebt @ Summary3 @ ( suc @ N2 ) )
& ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ N2 ) ) )
& ( A23
= ( plus_plus @ nat @ N2 @ ( suc @ N2 ) ) )
& ~ ? [X9: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X9 )
& ! [X5: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X5 @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
=> ~ ? [X9: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X9 ) ) )
| ? [TreeList3: list @ vEBT_VEBT,N2: 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 ) )
& ! [X5: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X5 @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
=> ( vEBT_invar_vebt @ X5 @ N2 ) )
& ( vEBT_invar_vebt @ Summary3 @ N2 )
& ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) )
& ( A23
= ( plus_plus @ nat @ N2 @ N2 ) )
& ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) )
=> ( ( ? [X9: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList3 @ I4 ) @ X9 ) )
= ( vEBT_V8194947554948674370ptions @ Summary3 @ I4 ) ) )
& ( ( Mi3 = Ma3 )
=> ! [X5: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X5 @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
=> ~ ? [X9: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X9 ) ) )
& ( ord_less_eq @ nat @ Mi3 @ Ma3 )
& ( ord_less @ nat @ Ma3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ A23 ) )
& ( ( Mi3 != Ma3 )
=> ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) )
=> ( ( ( ( vEBT_VEBT_high @ Ma3 @ N2 )
= I4 )
=> ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList3 @ I4 ) @ ( vEBT_VEBT_low @ Ma3 @ N2 ) ) )
& ! [X5: nat] :
( ( ( ( vEBT_VEBT_high @ X5 @ N2 )
= I4 )
& ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList3 @ I4 ) @ ( vEBT_VEBT_low @ X5 @ N2 ) ) )
=> ( ( ord_less @ nat @ Mi3 @ X5 )
& ( ord_less_eq @ nat @ X5 @ Ma3 ) ) ) ) ) ) )
| ? [TreeList3: list @ vEBT_VEBT,N2: 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 ) )
& ! [X5: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X5 @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
=> ( vEBT_invar_vebt @ X5 @ N2 ) )
& ( vEBT_invar_vebt @ Summary3 @ ( suc @ N2 ) )
& ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList3 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ N2 ) ) )
& ( A23
= ( plus_plus @ nat @ N2 @ ( suc @ N2 ) ) )
& ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ N2 ) ) )
=> ( ( ? [X9: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList3 @ I4 ) @ X9 ) )
= ( vEBT_V8194947554948674370ptions @ Summary3 @ I4 ) ) )
& ( ( Mi3 = Ma3 )
=> ! [X5: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X5 @ ( set2 @ vEBT_VEBT @ TreeList3 ) )
=> ~ ? [X9: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X9 ) ) )
& ( ord_less_eq @ nat @ Mi3 @ Ma3 )
& ( ord_less @ nat @ Ma3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ A23 ) )
& ( ( Mi3 != Ma3 )
=> ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ N2 ) ) )
=> ( ( ( ( vEBT_VEBT_high @ Ma3 @ N2 )
= I4 )
=> ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList3 @ I4 ) @ ( vEBT_VEBT_low @ Ma3 @ N2 ) ) )
& ! [X5: nat] :
( ( ( ( vEBT_VEBT_high @ X5 @ N2 )
= I4 )
& ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList3 @ I4 ) @ ( vEBT_VEBT_low @ X5 @ N2 ) ) )
=> ( ( ord_less @ nat @ Mi3 @ X5 )
& ( ord_less_eq @ nat @ X5 @ Ma3 ) ) ) ) ) ) ) ) ) ) ).
% invar_vebt.simps
thf(fact_678_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_679_invar__vebt_Ointros_I5_J,axiom,
! [TreeList: list @ vEBT_VEBT,N: nat,Summary: vEBT_VEBT,M2: nat,Deg: nat,Mi: nat,Ma: nat] :
( ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ( vEBT_invar_vebt @ X4 @ 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 ) )
=> ( ( ? [X9: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ I3 ) @ X9 ) )
= ( vEBT_V8194947554948674370ptions @ Summary @ I3 ) ) )
=> ( ( ( Mi = Ma )
=> ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X_12 ) ) )
=> ( ( ord_less_eq @ nat @ Mi @ Ma )
=> ( ( ord_less @ nat @ Ma @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg ) )
=> ( ( ( Mi != Ma )
=> ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M2 ) )
=> ( ( ( ( vEBT_VEBT_high @ Ma @ N )
= I3 )
=> ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ I3 ) @ ( vEBT_VEBT_low @ Ma @ N ) ) )
& ! [X4: nat] :
( ( ( ( vEBT_VEBT_high @ X4 @ N )
= I3 )
& ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ I3 ) @ ( vEBT_VEBT_low @ X4 @ N ) ) )
=> ( ( ord_less @ nat @ Mi @ X4 )
& ( ord_less_eq @ nat @ X4 @ Ma ) ) ) ) ) )
=> ( vEBT_invar_vebt @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ Deg ) ) ) ) ) ) ) ) ) ) ) ).
% invar_vebt.intros(5)
thf(fact_680_VEBT__internal_Onaive__member_Ocases,axiom,
! [X: product_prod @ vEBT_VEBT @ nat] :
( ! [A4: $o,B4: $o,X4: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A4 @ B4 ) @ X4 ) )
=> ( ! [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 ) )
=> ~ ! [Uy2: option @ ( product_prod @ nat @ nat ),V3: nat,TreeList2: list @ vEBT_VEBT,S3: vEBT_VEBT,X4: nat] :
( X
!= ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList2 @ S3 ) @ X4 ) ) ) ) ).
% VEBT_internal.naive_member.cases
thf(fact_681_invar__vebt_Ointros_I1_J,axiom,
! [A2: $o,B2: $o] : ( vEBT_invar_vebt @ ( vEBT_Leaf @ A2 @ B2 ) @ ( suc @ ( zero_zero @ nat ) ) ) ).
% invar_vebt.intros(1)
thf(fact_682_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_683_vebt__buildup_Osimps_I2_J,axiom,
( ( vEBT_vebt_buildup @ ( suc @ ( zero_zero @ nat ) ) )
= ( vEBT_Leaf @ $false @ $false ) ) ).
% vebt_buildup.simps(2)
thf(fact_684_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_685_list__ball__nth,axiom,
! [A: $tType,N: nat,Xs: list @ A,P2: A > $o] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
=> ( P2 @ X4 ) )
=> ( P2 @ ( nth @ A @ Xs @ N ) ) ) ) ).
% list_ball_nth
thf(fact_686_in__set__conv__nth,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
= ( ? [I4: nat] :
( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ Xs ) )
& ( ( nth @ A @ Xs @ I4 )
= X ) ) ) ) ).
% in_set_conv_nth
thf(fact_687_all__nth__imp__all__set,axiom,
! [A: $tType,Xs: list @ A,P2: A > $o,X: A] :
( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( P2 @ ( nth @ A @ Xs @ I3 ) ) )
=> ( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( P2 @ X ) ) ) ).
% all_nth_imp_all_set
thf(fact_688_all__set__conv__all__nth,axiom,
! [A: $tType,Xs: list @ A,P2: A > $o] :
( ( ! [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Xs ) )
=> ( P2 @ X5 ) ) )
= ( ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( P2 @ ( nth @ A @ Xs @ I4 ) ) ) ) ) ).
% all_set_conv_all_nth
thf(fact_689_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,Uy2: vEBT_VEBT] :
( X
!= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uw2 @ Ux2 @ Uy2 ) )
=> ~ ! [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_690_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_691_nth__list__update,axiom,
! [A: $tType,I2: nat,Xs: list @ A,J: nat,X: A] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ( I2 = J )
=> ( ( nth @ A @ ( list_update @ A @ Xs @ I2 @ X ) @ J )
= X ) )
& ( ( I2 != J )
=> ( ( nth @ A @ ( list_update @ A @ Xs @ I2 @ X ) @ J )
= ( nth @ A @ Xs @ J ) ) ) ) ) ).
% nth_list_update
thf(fact_692_list__update__same__conv,axiom,
! [A: $tType,I2: nat,Xs: list @ A,X: A] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ( list_update @ A @ Xs @ I2 @ X )
= Xs )
= ( ( nth @ A @ Xs @ I2 )
= X ) ) ) ).
% list_update_same_conv
thf(fact_693_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_694_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,Uy2: vEBT_VEBT] :
( X
!= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uw2 @ Ux2 @ Uy2 ) ) ) ) ).
% VEBT_internal.minNull.elims(2)
thf(fact_695_set__encode__inf,axiom,
! [A5: set @ nat] :
( ~ ( finite_finite2 @ nat @ A5 )
=> ( ( nat_set_encode @ A5 )
= ( zero_zero @ nat ) ) ) ).
% set_encode_inf
thf(fact_696_VEBT__internal_OminNull_Oelims_I1_J,axiom,
! [X: vEBT_VEBT,Y: $o] :
( ( ( vEBT_VEBT_minNull @ X )
= Y )
=> ( ( ( X
= ( vEBT_Leaf @ $false @ $false ) )
=> ~ Y )
=> ( ( ? [Uv2: $o] :
( X
= ( vEBT_Leaf @ $true @ Uv2 ) )
=> Y )
=> ( ( ? [Uu2: $o] :
( X
= ( vEBT_Leaf @ Uu2 @ $true ) )
=> Y )
=> ( ( ? [Uw2: nat,Ux2: list @ vEBT_VEBT,Uy2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uw2 @ Ux2 @ Uy2 ) )
=> ~ Y )
=> ~ ( ? [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 ) )
=> Y ) ) ) ) ) ) ).
% VEBT_internal.minNull.elims(1)
thf(fact_697_sum__power2__eq__zero__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,Y: A] :
( ( ( plus_plus @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( zero_zero @ A ) )
= ( ( X
= ( zero_zero @ A ) )
& ( Y
= ( zero_zero @ A ) ) ) ) ) ).
% sum_power2_eq_zero_iff
thf(fact_698_zero__less__power2,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( A2
!= ( zero_zero @ A ) ) ) ) ).
% zero_less_power2
thf(fact_699_power2__eq__iff__nonneg,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
=> ( ( ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( X = Y ) ) ) ) ) ).
% power2_eq_iff_nonneg
thf(fact_700_power2__less__eq__zero__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( zero_zero @ A ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% power2_less_eq_zero_iff
thf(fact_701_insert__simp__excp,axiom,
! [Mi: nat,Deg: nat,TreeList: list @ vEBT_VEBT,X: nat,Ma: nat,Summary: vEBT_VEBT] :
( ( ord_less @ nat @ ( vEBT_VEBT_high @ Mi @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
=> ( ( ord_less @ nat @ X @ Mi )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( X != Ma )
=> ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ X @ ( ord_max @ nat @ Mi @ Ma ) ) ) @ Deg @ ( list_update @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Mi @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_insert @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Mi @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Mi @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( if @ vEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ Mi @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary @ ( vEBT_VEBT_high @ Mi @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ Summary ) ) ) ) ) ) ) ).
% insert_simp_excp
thf(fact_702_insert__simp__norm,axiom,
! [X: nat,Deg: nat,TreeList: list @ vEBT_VEBT,Mi: nat,Ma: nat,Summary: vEBT_VEBT] :
( ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
=> ( ( ord_less @ nat @ Mi @ X )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
=> ( ( X != Ma )
=> ( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ ( ord_max @ nat @ X @ Ma ) ) ) @ Deg @ ( list_update @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_insert @ ( 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 ) ) ) ) ) ) @ ( if @ vEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ Summary ) ) ) ) ) ) ) ).
% insert_simp_norm
thf(fact_703_zero__eq__power2,axiom,
! [A: $tType] :
( ( semiri2026040879449505780visors @ A )
=> ! [A2: A] :
( ( ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( zero_zero @ A ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% zero_eq_power2
thf(fact_704_power__mono__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,B2: A,N: nat] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ A @ ( power_power @ A @ A2 @ N ) @ ( power_power @ A @ B2 @ N ) )
= ( ord_less_eq @ A @ A2 @ B2 ) ) ) ) ) ) ).
% power_mono_iff
thf(fact_705_power__eq__0__iff,axiom,
! [A: $tType] :
( ( semiri2026040879449505780visors @ A )
=> ! [A2: A,N: nat] :
( ( ( power_power @ A @ A2 @ N )
= ( zero_zero @ A ) )
= ( ( A2
= ( zero_zero @ A ) )
& ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% power_eq_0_iff
thf(fact_706_member__inv,axiom,
! [Mi: 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 @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg )
& ( ( X = Mi )
| ( X = Ma )
| ( ( ord_less @ nat @ X @ Ma )
& ( ord_less @ nat @ Mi @ 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_707_pow__sum,axiom,
! [A2: nat,B2: nat] :
( ( divide_divide @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( plus_plus @ nat @ A2 @ B2 ) ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ A2 ) )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ B2 ) ) ).
% pow_sum
thf(fact_708_high__def,axiom,
( vEBT_VEBT_high
= ( ^ [X5: nat,N2: nat] : ( divide_divide @ nat @ X5 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ).
% high_def
thf(fact_709_division__ring__divide__zero,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A] :
( ( divide_divide @ A @ A2 @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% division_ring_divide_zero
thf(fact_710_divide__cancel__right,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A2: A,C3: A,B2: A] :
( ( ( divide_divide @ A @ A2 @ C3 )
= ( divide_divide @ A @ B2 @ C3 ) )
= ( ( C3
= ( zero_zero @ A ) )
| ( A2 = B2 ) ) ) ) ).
% divide_cancel_right
thf(fact_711_divide__cancel__left,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [C3: A,A2: A,B2: A] :
( ( ( divide_divide @ A @ C3 @ A2 )
= ( divide_divide @ A @ C3 @ B2 ) )
= ( ( C3
= ( zero_zero @ A ) )
| ( A2 = B2 ) ) ) ) ).
% divide_cancel_left
thf(fact_712_div__by__0,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [A2: A] :
( ( divide_divide @ A @ A2 @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% div_by_0
thf(fact_713_divide__eq__0__iff,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A2: A,B2: A] :
( ( ( divide_divide @ A @ A2 @ B2 )
= ( zero_zero @ A ) )
= ( ( A2
= ( zero_zero @ A ) )
| ( B2
= ( zero_zero @ A ) ) ) ) ) ).
% divide_eq_0_iff
thf(fact_714_div__0,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [A2: A] :
( ( divide_divide @ A @ ( zero_zero @ A ) @ A2 )
= ( zero_zero @ A ) ) ) ).
% div_0
thf(fact_715_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_716_power__zero__numeral,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [K2: num] :
( ( power_power @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ nat @ K2 ) )
= ( zero_zero @ A ) ) ) ).
% power_zero_numeral
thf(fact_717_power__Suc0__right,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A2: A] :
( ( power_power @ A @ A2 @ ( suc @ ( zero_zero @ nat ) ) )
= A2 ) ) ).
% power_Suc0_right
thf(fact_718_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_719_power__Suc__0,axiom,
! [N: nat] :
( ( power_power @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% power_Suc_0
thf(fact_720_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_721_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_722_add__divide__distrib,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( divide_divide @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C3 )
= ( plus_plus @ A @ ( divide_divide @ A @ A2 @ C3 ) @ ( divide_divide @ A @ B2 @ C3 ) ) ) ) ).
% add_divide_distrib
thf(fact_723_divide__right__mono__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ C3 ) @ ( divide_divide @ A @ A2 @ C3 ) ) ) ) ) ).
% divide_right_mono_neg
thf(fact_724_divide__nonpos__nonpos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ Y @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X @ Y ) ) ) ) ) ).
% divide_nonpos_nonpos
thf(fact_725_divide__nonpos__nonneg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ X @ Y ) @ ( zero_zero @ A ) ) ) ) ) ).
% divide_nonpos_nonneg
thf(fact_726_divide__nonneg__nonpos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less_eq @ A @ Y @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ X @ Y ) @ ( zero_zero @ A ) ) ) ) ) ).
% divide_nonneg_nonpos
thf(fact_727_divide__nonneg__nonneg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X @ Y ) ) ) ) ) ).
% divide_nonneg_nonneg
thf(fact_728_zero__le__divide__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ A2 @ B2 ) )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) )
| ( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
& ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) ) ) ) ) ).
% zero_le_divide_iff
thf(fact_729_divide__right__mono,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ A2 @ C3 ) @ ( divide_divide @ A @ B2 @ C3 ) ) ) ) ) ).
% divide_right_mono
thf(fact_730_divide__le__0__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ A2 @ B2 ) @ ( zero_zero @ A ) )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
& ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) )
| ( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) ) ) ) ) ).
% divide_le_0_iff
thf(fact_731_divide__strict__right__mono__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,A2: A,C3: A] :
( ( ord_less @ A @ B2 @ A2 )
=> ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( divide_divide @ A @ A2 @ C3 ) @ ( divide_divide @ A @ B2 @ C3 ) ) ) ) ) ).
% divide_strict_right_mono_neg
thf(fact_732_divide__strict__right__mono,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ ( divide_divide @ A @ A2 @ C3 ) @ ( divide_divide @ A @ B2 @ C3 ) ) ) ) ) ).
% divide_strict_right_mono
thf(fact_733_zero__less__divide__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ A2 @ B2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
& ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) )
| ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
& ( ord_less @ A @ B2 @ ( zero_zero @ A ) ) ) ) ) ) ).
% zero_less_divide_iff
thf(fact_734_divide__less__cancel,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,C3: A,B2: A] :
( ( ord_less @ A @ ( divide_divide @ A @ A2 @ C3 ) @ ( divide_divide @ A @ B2 @ C3 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ A2 @ B2 ) )
& ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B2 @ A2 ) )
& ( C3
!= ( zero_zero @ A ) ) ) ) ) ).
% divide_less_cancel
thf(fact_735_divide__less__0__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( divide_divide @ A @ A2 @ B2 ) @ ( zero_zero @ A ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
& ( ord_less @ A @ B2 @ ( zero_zero @ A ) ) )
| ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
& ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) ) ) ) ) ).
% divide_less_0_iff
thf(fact_736_divide__pos__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ Y )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X @ Y ) ) ) ) ) ).
% divide_pos_pos
thf(fact_737_divide__pos__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less @ A @ Y @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( divide_divide @ A @ X @ Y ) @ ( zero_zero @ A ) ) ) ) ) ).
% divide_pos_neg
thf(fact_738_divide__neg__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ Y )
=> ( ord_less @ A @ ( divide_divide @ A @ X @ Y ) @ ( zero_zero @ A ) ) ) ) ) ).
% divide_neg_pos
thf(fact_739_divide__neg__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ Y @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X @ Y ) ) ) ) ) ).
% divide_neg_neg
thf(fact_740_divide__nonpos__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ Y )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ X @ Y ) @ ( zero_zero @ A ) ) ) ) ) ).
% divide_nonpos_pos
thf(fact_741_divide__nonpos__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ Y @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X @ Y ) ) ) ) ) ).
% divide_nonpos_neg
thf(fact_742_divide__nonneg__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ Y )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ X @ Y ) ) ) ) ) ).
% divide_nonneg_pos
thf(fact_743_divide__nonneg__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less @ A @ Y @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ X @ Y ) @ ( zero_zero @ A ) ) ) ) ) ).
% divide_nonneg_neg
thf(fact_744_divide__le__cancel,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,C3: A,B2: A] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ A2 @ C3 ) @ ( divide_divide @ A @ B2 @ C3 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ A2 @ B2 ) )
& ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B2 @ A2 ) ) ) ) ) ).
% divide_le_cancel
thf(fact_745_frac__less2,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y: A,W2: A,Z: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ W2 )
=> ( ( ord_less @ A @ W2 @ Z )
=> ( ord_less @ A @ ( divide_divide @ A @ X @ Z ) @ ( divide_divide @ A @ Y @ W2 ) ) ) ) ) ) ) ).
% frac_less2
thf(fact_746_frac__less,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y: A,W2: A,Z: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less @ A @ X @ Y )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ W2 )
=> ( ( ord_less_eq @ A @ W2 @ Z )
=> ( ord_less @ A @ ( divide_divide @ A @ X @ Z ) @ ( divide_divide @ A @ Y @ W2 ) ) ) ) ) ) ) ).
% frac_less
thf(fact_747_frac__le,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y: A,X: A,W2: A,Z: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
=> ( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ W2 )
=> ( ( ord_less_eq @ A @ W2 @ Z )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ X @ Z ) @ ( divide_divide @ A @ Y @ W2 ) ) ) ) ) ) ) ).
% frac_le
thf(fact_748_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_749_half__gt__zero,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% half_gt_zero
thf(fact_750_half__gt__zero__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% half_gt_zero_iff
thf(fact_751_field__less__half__sum,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( ord_less @ A @ X @ ( divide_divide @ A @ ( plus_plus @ A @ X @ Y ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% field_less_half_sum
thf(fact_752_power__not__zero,axiom,
! [A: $tType] :
( ( semiri2026040879449505780visors @ A )
=> ! [A2: A,N: nat] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( power_power @ A @ A2 @ N )
!= ( zero_zero @ A ) ) ) ) ).
% power_not_zero
thf(fact_753_zero__le__power,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,N: nat] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A2 @ N ) ) ) ) ).
% zero_le_power
thf(fact_754_power__mono,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,B2: A,N: nat] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ord_less_eq @ A @ ( power_power @ A @ A2 @ N ) @ ( power_power @ A @ B2 @ N ) ) ) ) ) ).
% power_mono
thf(fact_755_zero__less__power,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,N: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A2 @ N ) ) ) ) ).
% zero_less_power
thf(fact_756_nat__power__less__imp__less,axiom,
! [I2: nat,M2: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ I2 )
=> ( ( ord_less @ nat @ ( power_power @ nat @ I2 @ M2 ) @ ( power_power @ nat @ I2 @ N ) )
=> ( ord_less @ nat @ M2 @ N ) ) ) ).
% nat_power_less_imp_less
thf(fact_757_power__less__imp__less__base,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,N: nat,B2: A] :
( ( ord_less @ A @ ( power_power @ A @ A2 @ N ) @ ( power_power @ A @ B2 @ N ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less @ A @ A2 @ B2 ) ) ) ) ).
% power_less_imp_less_base
thf(fact_758_power__le__imp__le__base,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,N: nat,B2: A] :
( ( ord_less_eq @ A @ ( power_power @ A @ A2 @ ( suc @ N ) ) @ ( power_power @ A @ B2 @ ( suc @ N ) ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less_eq @ A @ A2 @ B2 ) ) ) ) ).
% power_le_imp_le_base
thf(fact_759_power__inject__base,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,N: nat,B2: A] :
( ( ( power_power @ A @ A2 @ ( suc @ N ) )
= ( power_power @ A @ B2 @ ( suc @ N ) ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( A2 = B2 ) ) ) ) ) ).
% power_inject_base
thf(fact_760_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_761_power__gt__expt,axiom,
! [N: nat,K2: nat] :
( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N )
=> ( ord_less @ nat @ K2 @ ( power_power @ nat @ N @ K2 ) ) ) ).
% power_gt_expt
thf(fact_762_nat__one__le__power,axiom,
! [I2: nat,N: nat] :
( ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ I2 )
=> ( ord_less_eq @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( power_power @ nat @ I2 @ N ) ) ) ).
% nat_one_le_power
thf(fact_763_power__eq__imp__eq__base,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,N: nat,B2: A] :
( ( ( power_power @ A @ A2 @ N )
= ( power_power @ A @ B2 @ N ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( A2 = B2 ) ) ) ) ) ) ).
% power_eq_imp_eq_base
thf(fact_764_power__eq__iff__eq__base,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N: nat,A2: A,B2: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ( power_power @ A @ A2 @ N )
= ( power_power @ A @ B2 @ N ) )
= ( A2 = B2 ) ) ) ) ) ) ).
% power_eq_iff_eq_base
thf(fact_765_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_766_less__exp,axiom,
! [N: nat] : ( ord_less @ nat @ N @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ).
% less_exp
thf(fact_767_power2__nat__le__imp__le,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq @ nat @ ( power_power @ nat @ M2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ N )
=> ( ord_less_eq @ nat @ M2 @ N ) ) ).
% power2_nat_le_imp_le
thf(fact_768_power2__nat__le__eq__le,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq @ nat @ ( power_power @ nat @ M2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( ord_less_eq @ nat @ M2 @ N ) ) ).
% power2_nat_le_eq_le
thf(fact_769_self__le__ge2__pow,axiom,
! [K2: nat,M2: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K2 )
=> ( ord_less_eq @ nat @ M2 @ ( power_power @ nat @ K2 @ M2 ) ) ) ).
% self_le_ge2_pow
thf(fact_770_zero__le__power2,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% zero_le_power2
thf(fact_771_power2__eq__imp__eq,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X: A,Y: A] :
( ( ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
=> ( X = Y ) ) ) ) ) ).
% power2_eq_imp_eq
thf(fact_772_power2__le__imp__le,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
=> ( ord_less_eq @ A @ X @ Y ) ) ) ) ).
% power2_le_imp_le
thf(fact_773_power__strict__mono,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,B2: A,N: nat] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ord_less @ A @ ( power_power @ A @ A2 @ N ) @ ( power_power @ A @ B2 @ N ) ) ) ) ) ) ).
% power_strict_mono
thf(fact_774_power2__less__0,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A] :
~ ( ord_less @ A @ ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( zero_zero @ A ) ) ) ).
% power2_less_0
thf(fact_775_power2__less__imp__less,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
=> ( ord_less @ A @ X @ Y ) ) ) ) ).
% power2_less_imp_less
thf(fact_776_sum__power2__le__zero__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( zero_zero @ A ) )
= ( ( X
= ( zero_zero @ A ) )
& ( Y
= ( zero_zero @ A ) ) ) ) ) ).
% sum_power2_le_zero_iff
thf(fact_777_sum__power2__ge__zero,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,Y: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% sum_power2_ge_zero
thf(fact_778_sum__power2__gt__zero__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
= ( ( X
!= ( zero_zero @ A ) )
| ( Y
!= ( zero_zero @ A ) ) ) ) ) ).
% sum_power2_gt_zero_iff
thf(fact_779_not__sum__power2__lt__zero,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,Y: A] :
~ ( ord_less @ A @ ( plus_plus @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( zero_zero @ A ) ) ) ).
% not_sum_power2_lt_zero
thf(fact_780_both__member__options__from__chilf__to__complete__tree,axiom,
! [X: nat,Deg: nat,TreeList: list @ vEBT_VEBT,Mi: nat,Ma: nat,Summary: vEBT_VEBT] :
( ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
=> ( ( ord_less_eq @ nat @ ( one_one @ nat ) @ Deg )
=> ( ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
=> ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X ) ) ) ) ).
% both_member_options_from_chilf_to_complete_tree
thf(fact_781_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_782_both__member__options__from__complete__tree__to__child,axiom,
! [Deg: nat,Mi: nat,Ma: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ Deg )
=> ( ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ Deg @ TreeList @ Summary ) @ X )
=> ( ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
| ( X = Mi )
| ( X = Ma ) ) ) ) ).
% both_member_options_from_complete_tree_to_child
thf(fact_783_semiring__norm_I76_J,axiom,
! [N: num] : ( ord_less @ num @ one2 @ ( bit0 @ N ) ) ).
% semiring_norm(76)
thf(fact_784_div__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less @ nat @ M2 @ N )
=> ( ( divide_divide @ nat @ M2 @ N )
= ( zero_zero @ nat ) ) ) ).
% div_less
thf(fact_785_div__by__Suc__0,axiom,
! [M2: nat] :
( ( divide_divide @ nat @ M2 @ ( suc @ ( zero_zero @ nat ) ) )
= M2 ) ).
% div_by_Suc_0
thf(fact_786_set__n__deg__not__0,axiom,
! [TreeList: list @ vEBT_VEBT,N: nat,M2: nat] :
( ! [X4: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X4 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ( vEBT_invar_vebt @ X4 @ 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_787_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_788_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_789_div__exp__eq,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A2: A,M2: nat,N: nat] :
( ( divide_divide @ A @ ( divide_divide @ A @ A2 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M2 ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
= ( divide_divide @ A @ A2 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( plus_plus @ nat @ M2 @ N ) ) ) ) ) ).
% div_exp_eq
thf(fact_790_deg1Leaf,axiom,
! [T2: vEBT_VEBT] :
( ( vEBT_invar_vebt @ T2 @ ( one_one @ nat ) )
= ( ? [A6: $o,B5: $o] :
( T2
= ( vEBT_Leaf @ A6 @ B5 ) ) ) ) ).
% deg1Leaf
thf(fact_791_deg__1__Leaf,axiom,
! [T2: vEBT_VEBT] :
( ( vEBT_invar_vebt @ T2 @ ( one_one @ nat ) )
=> ? [A4: $o,B4: $o] :
( T2
= ( vEBT_Leaf @ A4 @ B4 ) ) ) ).
% deg_1_Leaf
thf(fact_792_deg__1__Leafy,axiom,
! [T2: vEBT_VEBT,N: nat] :
( ( vEBT_invar_vebt @ T2 @ N )
=> ( ( N
= ( one_one @ nat ) )
=> ? [A4: $o,B4: $o] :
( T2
= ( vEBT_Leaf @ A4 @ B4 ) ) ) ) ).
% deg_1_Leafy
thf(fact_793_bits__div__0,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A2: A] :
( ( divide_divide @ A @ ( zero_zero @ A ) @ A2 )
= ( zero_zero @ A ) ) ) ).
% bits_div_0
thf(fact_794_bits__div__by__0,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A2: A] :
( ( divide_divide @ A @ A2 @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% bits_div_by_0
thf(fact_795_half__negative__int__iff,axiom,
! [K2: int] :
( ( ord_less @ int @ ( divide_divide @ int @ K2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( zero_zero @ int ) )
= ( ord_less @ int @ K2 @ ( zero_zero @ int ) ) ) ).
% half_negative_int_iff
thf(fact_796_div__by__1,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [A2: A] :
( ( divide_divide @ A @ A2 @ ( one_one @ A ) )
= A2 ) ) ).
% div_by_1
thf(fact_797_semiring__norm_I78_J,axiom,
! [M2: num,N: num] :
( ( ord_less @ num @ ( bit0 @ M2 ) @ ( bit0 @ N ) )
= ( ord_less @ num @ M2 @ N ) ) ).
% semiring_norm(78)
thf(fact_798_semiring__norm_I75_J,axiom,
! [M2: num] :
~ ( ord_less @ num @ M2 @ one2 ) ).
% semiring_norm(75)
thf(fact_799_divide__eq__1__iff,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A2: A,B2: A] :
( ( ( divide_divide @ A @ A2 @ B2 )
= ( one_one @ A ) )
= ( ( B2
!= ( zero_zero @ A ) )
& ( A2 = B2 ) ) ) ) ).
% divide_eq_1_iff
thf(fact_800_div__self,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [A2: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ A2 @ A2 )
= ( one_one @ A ) ) ) ) ).
% div_self
thf(fact_801_one__eq__divide__iff,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A2: A,B2: A] :
( ( ( one_one @ A )
= ( divide_divide @ A @ A2 @ B2 ) )
= ( ( B2
!= ( zero_zero @ A ) )
& ( A2 = B2 ) ) ) ) ).
% one_eq_divide_iff
thf(fact_802_divide__self,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ A2 @ A2 )
= ( one_one @ A ) ) ) ) ).
% divide_self
thf(fact_803_divide__self__if,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A] :
( ( ( A2
= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ A2 @ A2 )
= ( zero_zero @ A ) ) )
& ( ( A2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ A2 @ A2 )
= ( one_one @ A ) ) ) ) ) ).
% divide_self_if
thf(fact_804_divide__eq__eq__1,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,A2: A] :
( ( ( divide_divide @ A @ B2 @ A2 )
= ( one_one @ A ) )
= ( ( A2
!= ( zero_zero @ A ) )
& ( A2 = B2 ) ) ) ) ).
% divide_eq_eq_1
thf(fact_805_eq__divide__eq__1,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,A2: A] :
( ( ( one_one @ A )
= ( divide_divide @ A @ B2 @ A2 ) )
= ( ( A2
!= ( zero_zero @ A ) )
& ( A2 = B2 ) ) ) ) ).
% eq_divide_eq_1
thf(fact_806_one__divide__eq__0__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A] :
( ( ( divide_divide @ A @ ( one_one @ A ) @ A2 )
= ( zero_zero @ A ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% one_divide_eq_0_iff
thf(fact_807_zero__eq__1__divide__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A] :
( ( ( zero_zero @ A )
= ( divide_divide @ A @ ( one_one @ A ) @ A2 ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% zero_eq_1_divide_iff
thf(fact_808_power__inject__exp,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,M2: nat,N: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ A2 )
=> ( ( ( power_power @ A @ A2 @ M2 )
= ( power_power @ A @ A2 @ N ) )
= ( M2 = N ) ) ) ) ).
% power_inject_exp
thf(fact_809_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_810_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_811_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_812_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_813_less__one,axiom,
! [N: nat] :
( ( ord_less @ nat @ N @ ( one_one @ nat ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% less_one
thf(fact_814_divide__le__0__1__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ ( one_one @ A ) @ A2 ) @ ( zero_zero @ A ) )
= ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% divide_le_0_1_iff
thf(fact_815_zero__le__divide__1__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ ( one_one @ A ) @ A2 ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% zero_le_divide_1_iff
thf(fact_816_divide__less__0__1__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A] :
( ( ord_less @ A @ ( divide_divide @ A @ ( one_one @ A ) @ A2 ) @ ( zero_zero @ A ) )
= ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% divide_less_0_1_iff
thf(fact_817_divide__less__eq__1__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( divide_divide @ A @ B2 @ A2 ) @ ( one_one @ A ) )
= ( ord_less @ A @ A2 @ B2 ) ) ) ) ).
% divide_less_eq_1_neg
thf(fact_818_divide__less__eq__1__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ ( divide_divide @ A @ B2 @ A2 ) @ ( one_one @ A ) )
= ( ord_less @ A @ B2 @ A2 ) ) ) ) ).
% divide_less_eq_1_pos
thf(fact_819_less__divide__eq__1__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B2 @ A2 ) )
= ( ord_less @ A @ B2 @ A2 ) ) ) ) ).
% less_divide_eq_1_neg
thf(fact_820_less__divide__eq__1__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B2 @ A2 ) )
= ( ord_less @ A @ A2 @ B2 ) ) ) ) ).
% less_divide_eq_1_pos
thf(fact_821_zero__less__divide__1__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ ( one_one @ A ) @ A2 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% zero_less_divide_1_iff
thf(fact_822_power__strict__increasing__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [B2: A,X: nat,Y: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ B2 )
=> ( ( ord_less @ A @ ( power_power @ A @ B2 @ X ) @ ( power_power @ A @ B2 @ Y ) )
= ( ord_less @ nat @ X @ Y ) ) ) ) ).
% power_strict_increasing_iff
thf(fact_823_divide__le__eq__1__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ A2 ) @ ( one_one @ A ) )
= ( ord_less_eq @ A @ A2 @ B2 ) ) ) ) ).
% divide_le_eq_1_neg
thf(fact_824_divide__le__eq__1__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ A2 ) @ ( one_one @ A ) )
= ( ord_less_eq @ A @ B2 @ A2 ) ) ) ) ).
% divide_le_eq_1_pos
thf(fact_825_le__divide__eq__1__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B2 @ A2 ) )
= ( ord_less_eq @ A @ B2 @ A2 ) ) ) ) ).
% le_divide_eq_1_neg
thf(fact_826_le__divide__eq__1__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B2 @ A2 ) )
= ( ord_less_eq @ A @ A2 @ B2 ) ) ) ) ).
% le_divide_eq_1_pos
thf(fact_827_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_828_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_829_power__increasing__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [B2: A,X: nat,Y: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ B2 )
=> ( ( ord_less_eq @ A @ ( power_power @ A @ B2 @ X ) @ ( power_power @ A @ B2 @ Y ) )
= ( ord_less_eq @ nat @ X @ Y ) ) ) ) ).
% power_increasing_iff
thf(fact_830_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_831_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_832_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_833_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_834_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_835_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_836_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_837_one__reorient,axiom,
! [A: $tType] :
( ( one @ A )
=> ! [X: A] :
( ( ( one_one @ A )
= X )
= ( X
= ( one_one @ A ) ) ) ) ).
% one_reorient
thf(fact_838_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_839_zero__neq__one,axiom,
! [A: $tType] :
( ( zero_neq_one @ A )
=> ( ( zero_zero @ A )
!= ( one_one @ A ) ) ) ).
% zero_neq_one
thf(fact_840_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_841_div__add__self1,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [B2: A,A2: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ B2 @ A2 ) @ B2 )
= ( plus_plus @ A @ ( divide_divide @ A @ A2 @ B2 ) @ ( one_one @ A ) ) ) ) ) ).
% div_add_self1
thf(fact_842_div__add__self2,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [B2: A,A2: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ A2 @ B2 ) @ B2 )
= ( plus_plus @ A @ ( divide_divide @ A @ A2 @ B2 ) @ ( one_one @ A ) ) ) ) ) ).
% div_add_self2
thf(fact_843_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_844_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_845_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_846_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_847_zero__less__one,axiom,
! [A: $tType] :
( ( zero_less_one @ A )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( one_one @ A ) ) ) ).
% zero_less_one
thf(fact_848_not__one__less__zero,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ~ ( ord_less @ A @ ( one_one @ A ) @ ( zero_zero @ A ) ) ) ).
% not_one_less_zero
thf(fact_849_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_850_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_851_less__add__one,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A] : ( ord_less @ A @ A2 @ ( plus_plus @ A @ A2 @ ( one_one @ A ) ) ) ) ).
% less_add_one
thf(fact_852_add__mono1,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ord_less @ A @ ( plus_plus @ A @ A2 @ ( one_one @ A ) ) @ ( plus_plus @ A @ B2 @ ( one_one @ A ) ) ) ) ) ).
% add_mono1
thf(fact_853_right__inverse__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B2: A,A2: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( ( divide_divide @ A @ A2 @ B2 )
= ( one_one @ A ) )
= ( A2 = B2 ) ) ) ) ).
% right_inverse_eq
thf(fact_854_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_855_one__le__power,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,N: nat] :
( ( ord_less_eq @ A @ ( one_one @ A ) @ A2 )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ ( power_power @ A @ A2 @ N ) ) ) ) ).
% one_le_power
thf(fact_856_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_857_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_858_power__0,axiom,
! [A: $tType] :
( ( power @ A )
=> ! [A2: A] :
( ( power_power @ A @ A2 @ ( zero_zero @ nat ) )
= ( one_one @ A ) ) ) ).
% power_0
thf(fact_859_One__nat__def,axiom,
( ( one_one @ nat )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% One_nat_def
thf(fact_860_Suc__eq__plus1__left,axiom,
( suc
= ( plus_plus @ nat @ ( one_one @ nat ) ) ) ).
% Suc_eq_plus1_left
thf(fact_861_plus__1__eq__Suc,axiom,
( ( plus_plus @ nat @ ( one_one @ nat ) )
= suc ) ).
% plus_1_eq_Suc
thf(fact_862_Suc__eq__plus1,axiom,
( suc
= ( ^ [N2: nat] : ( plus_plus @ nat @ N2 @ ( one_one @ nat ) ) ) ) ).
% Suc_eq_plus1
thf(fact_863_VEBT__internal_Ovalid_H_Osimps_I1_J,axiom,
! [Uu: $o,Uv: $o,D3: nat] :
( ( vEBT_VEBT_valid @ ( vEBT_Leaf @ Uu @ Uv ) @ D3 )
= ( D3
= ( one_one @ nat ) ) ) ).
% VEBT_internal.valid'.simps(1)
thf(fact_864_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_865_power__le__one,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,N: nat] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ A2 @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( power_power @ A @ A2 @ N ) @ ( one_one @ A ) ) ) ) ) ).
% power_le_one
thf(fact_866_divide__less__eq__1,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,A2: A] :
( ( ord_less @ A @ ( divide_divide @ A @ B2 @ A2 ) @ ( one_one @ A ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
& ( ord_less @ A @ B2 @ A2 ) )
| ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
& ( ord_less @ A @ A2 @ B2 ) )
| ( A2
= ( zero_zero @ A ) ) ) ) ) ).
% divide_less_eq_1
thf(fact_867_less__divide__eq__1,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,A2: A] :
( ( ord_less @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B2 @ A2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
& ( ord_less @ A @ A2 @ B2 ) )
| ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
& ( ord_less @ A @ B2 @ A2 ) ) ) ) ) ).
% less_divide_eq_1
thf(fact_868_gt__half__sum,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ord_less @ A @ ( divide_divide @ A @ ( plus_plus @ A @ A2 @ B2 ) @ ( plus_plus @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) @ B2 ) ) ) ).
% gt_half_sum
thf(fact_869_less__half__sum,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ord_less @ A @ A2 @ ( divide_divide @ A @ ( plus_plus @ A @ A2 @ B2 ) @ ( plus_plus @ A @ ( one_one @ A ) @ ( one_one @ A ) ) ) ) ) ) ).
% less_half_sum
thf(fact_870_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_871_power__gt1,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,N: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ A2 )
=> ( ord_less @ A @ ( one_one @ A ) @ ( power_power @ A @ A2 @ ( suc @ N ) ) ) ) ) ).
% power_gt1
thf(fact_872_power__less__imp__less__exp,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,M2: nat,N: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ A2 )
=> ( ( ord_less @ A @ ( power_power @ A @ A2 @ M2 ) @ ( power_power @ A @ A2 @ N ) )
=> ( ord_less @ nat @ M2 @ N ) ) ) ) ).
% power_less_imp_less_exp
thf(fact_873_power__strict__increasing,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N: nat,N6: nat,A2: A] :
( ( ord_less @ nat @ N @ N6 )
=> ( ( ord_less @ A @ ( one_one @ A ) @ A2 )
=> ( ord_less @ A @ ( power_power @ A @ A2 @ N ) @ ( power_power @ A @ A2 @ N6 ) ) ) ) ) ).
% power_strict_increasing
thf(fact_874_power__increasing,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N: nat,N6: nat,A2: A] :
( ( ord_less_eq @ nat @ N @ N6 )
=> ( ( ord_less_eq @ A @ ( one_one @ A ) @ A2 )
=> ( ord_less_eq @ A @ ( power_power @ A @ A2 @ N ) @ ( power_power @ A @ A2 @ N6 ) ) ) ) ) ).
% power_increasing
thf(fact_875_nat__induct__non__zero,axiom,
! [N: nat,P2: nat > $o] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( P2 @ ( one_one @ nat ) )
=> ( ! [N3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( ( P2 @ N3 )
=> ( P2 @ ( suc @ N3 ) ) ) )
=> ( P2 @ N ) ) ) ) ).
% nat_induct_non_zero
thf(fact_876_vebt__member_Osimps_I1_J,axiom,
! [A2: $o,B2: $o,X: nat] :
( ( vEBT_vebt_member @ ( vEBT_Leaf @ A2 @ B2 ) @ X )
= ( ( ( X
= ( zero_zero @ nat ) )
=> A2 )
& ( ( X
!= ( zero_zero @ nat ) )
=> ( ( ( X
= ( one_one @ nat ) )
=> B2 )
& ( X
= ( one_one @ nat ) ) ) ) ) ) ).
% vebt_member.simps(1)
thf(fact_877_vebt__insert_Osimps_I1_J,axiom,
! [X: nat,A2: $o,B2: $o] :
( ( ( X
= ( zero_zero @ nat ) )
=> ( ( vEBT_vebt_insert @ ( vEBT_Leaf @ A2 @ B2 ) @ X )
= ( vEBT_Leaf @ $true @ B2 ) ) )
& ( ( X
!= ( zero_zero @ nat ) )
=> ( ( ( X
= ( one_one @ nat ) )
=> ( ( vEBT_vebt_insert @ ( vEBT_Leaf @ A2 @ B2 ) @ X )
= ( vEBT_Leaf @ A2 @ $true ) ) )
& ( ( X
!= ( one_one @ nat ) )
=> ( ( vEBT_vebt_insert @ ( vEBT_Leaf @ A2 @ B2 ) @ X )
= ( vEBT_Leaf @ A2 @ B2 ) ) ) ) ) ) ).
% vebt_insert.simps(1)
thf(fact_878_VEBT__internal_Onaive__member_Osimps_I1_J,axiom,
! [A2: $o,B2: $o,X: nat] :
( ( vEBT_V5719532721284313246member @ ( vEBT_Leaf @ A2 @ B2 ) @ X )
= ( ( ( X
= ( zero_zero @ nat ) )
=> A2 )
& ( ( X
!= ( zero_zero @ nat ) )
=> ( ( ( X
= ( one_one @ nat ) )
=> B2 )
& ( X
= ( one_one @ nat ) ) ) ) ) ) ).
% VEBT_internal.naive_member.simps(1)
thf(fact_879_divide__le__eq__1,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,A2: A] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ A2 ) @ ( one_one @ A ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
& ( ord_less_eq @ A @ B2 @ A2 ) )
| ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
& ( ord_less_eq @ A @ A2 @ B2 ) )
| ( A2
= ( zero_zero @ A ) ) ) ) ) ).
% divide_le_eq_1
thf(fact_880_le__divide__eq__1,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,A2: A] :
( ( ord_less_eq @ A @ ( one_one @ A ) @ ( divide_divide @ A @ B2 @ A2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
& ( ord_less_eq @ A @ A2 @ B2 ) )
| ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
& ( ord_less_eq @ A @ B2 @ A2 ) ) ) ) ) ).
% le_divide_eq_1
thf(fact_881_power__Suc__le__self,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,N: nat] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ A2 @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( power_power @ A @ A2 @ ( suc @ N ) ) @ A2 ) ) ) ) ).
% power_Suc_le_self
thf(fact_882_power__Suc__less__one,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,N: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ A2 @ ( one_one @ A ) )
=> ( ord_less @ A @ ( power_power @ A @ A2 @ ( suc @ N ) ) @ ( one_one @ A ) ) ) ) ) ).
% power_Suc_less_one
thf(fact_883_power__strict__decreasing,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N: nat,N6: nat,A2: A] :
( ( ord_less @ nat @ N @ N6 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ A2 @ ( one_one @ A ) )
=> ( ord_less @ A @ ( power_power @ A @ A2 @ N6 ) @ ( power_power @ A @ A2 @ N ) ) ) ) ) ) ).
% power_strict_decreasing
thf(fact_884_power__decreasing,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N: nat,N6: nat,A2: A] :
( ( ord_less_eq @ nat @ N @ N6 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ A2 @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( power_power @ A @ A2 @ N6 ) @ ( power_power @ A @ A2 @ N ) ) ) ) ) ) ).
% power_decreasing
thf(fact_885_power__le__imp__le__exp,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,M2: nat,N: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ ( power_power @ A @ A2 @ M2 ) @ ( power_power @ A @ A2 @ N ) )
=> ( ord_less_eq @ nat @ M2 @ N ) ) ) ) ).
% power_le_imp_le_exp
thf(fact_886_self__le__power,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,N: nat] :
( ( ord_less_eq @ A @ ( one_one @ A ) @ A2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ord_less_eq @ A @ A2 @ ( power_power @ A @ A2 @ N ) ) ) ) ) ).
% self_le_power
thf(fact_887_one__less__power,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,N: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ A2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ord_less @ A @ ( one_one @ A ) @ ( power_power @ A @ A2 @ N ) ) ) ) ) ).
% one_less_power
thf(fact_888_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_889_div__le__mono,axiom,
! [M2: nat,N: nat,K2: nat] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ord_less_eq @ nat @ ( divide_divide @ nat @ M2 @ K2 ) @ ( divide_divide @ nat @ N @ K2 ) ) ) ).
% div_le_mono
thf(fact_890_div__le__dividend,axiom,
! [M2: nat,N: nat] : ( ord_less_eq @ nat @ ( divide_divide @ nat @ M2 @ N ) @ M2 ) ).
% div_le_dividend
thf(fact_891_ex__power__ivl2,axiom,
! [B2: nat,K2: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ B2 )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K2 )
=> ? [N3: nat] :
( ( ord_less @ nat @ ( power_power @ nat @ B2 @ N3 ) @ K2 )
& ( ord_less_eq @ nat @ K2 @ ( power_power @ nat @ B2 @ ( plus_plus @ nat @ N3 @ ( one_one @ nat ) ) ) ) ) ) ) ).
% ex_power_ivl2
thf(fact_892_ex__power__ivl1,axiom,
! [B2: nat,K2: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ B2 )
=> ( ( ord_less_eq @ nat @ ( one_one @ nat ) @ K2 )
=> ? [N3: nat] :
( ( ord_less_eq @ nat @ ( power_power @ nat @ B2 @ N3 ) @ K2 )
& ( ord_less @ nat @ K2 @ ( power_power @ nat @ B2 @ ( plus_plus @ nat @ N3 @ ( one_one @ nat ) ) ) ) ) ) ) ).
% ex_power_ivl1
thf(fact_893_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_894_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_895_div__le__mono2,axiom,
! [M2: nat,N: nat,K2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M2 )
=> ( ( ord_less_eq @ nat @ M2 @ N )
=> ( ord_less_eq @ nat @ ( divide_divide @ nat @ K2 @ N ) @ ( divide_divide @ nat @ K2 @ M2 ) ) ) ) ).
% div_le_mono2
thf(fact_896_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_897_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_898_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_899_nat__induct2,axiom,
! [P2: nat > $o,N: nat] :
( ( P2 @ ( zero_zero @ nat ) )
=> ( ( P2 @ ( one_one @ nat ) )
=> ( ! [N3: nat] :
( ( P2 @ N3 )
=> ( P2 @ ( plus_plus @ nat @ N3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
=> ( P2 @ N ) ) ) ) ).
% nat_induct2
thf(fact_900_bit__concat__def,axiom,
( vEBT_VEBT_bit_concat
= ( ^ [H: nat,L2: nat,D5: nat] : ( plus_plus @ nat @ ( times_times @ nat @ H @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ D5 ) ) @ L2 ) ) ) ).
% bit_concat_def
thf(fact_901_low__inv,axiom,
! [X: nat,N: nat,Y: nat] :
( ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( vEBT_VEBT_low @ ( plus_plus @ nat @ ( times_times @ nat @ Y @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) @ X ) @ N )
= X ) ) ).
% low_inv
thf(fact_902_high__inv,axiom,
! [X: nat,N: nat,Y: nat] :
( ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( vEBT_VEBT_high @ ( plus_plus @ nat @ ( times_times @ nat @ Y @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) @ X ) @ N )
= Y ) ) ).
% high_inv
thf(fact_903_enat__ord__number_I1_J,axiom,
! [M2: num,N: num] :
( ( ord_less_eq @ extended_enat @ ( numeral_numeral @ extended_enat @ M2 ) @ ( numeral_numeral @ extended_enat @ N ) )
= ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ M2 ) @ ( numeral_numeral @ nat @ N ) ) ) ).
% enat_ord_number(1)
thf(fact_904_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_905_pos2,axiom,
ord_less @ nat @ ( zero_zero @ nat ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ).
% pos2
thf(fact_906_discrete,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ( ( ord_less @ A )
= ( ^ [A6: A] : ( ord_less_eq @ A @ ( plus_plus @ A @ A6 @ ( one_one @ A ) ) ) ) ) ) ).
% discrete
thf(fact_907_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ A2 @ B2 )
=> ( ( divide_divide @ A @ A2 @ B2 )
= ( zero_zero @ A ) ) ) ) ) ).
% unique_euclidean_semiring_numeral_class.div_less
thf(fact_908_div__positive,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [B2: A,A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ A2 @ B2 ) ) ) ) ) ).
% div_positive
thf(fact_909_zle__add1__eq__le,axiom,
! [W2: int,Z: int] :
( ( ord_less @ int @ W2 @ ( plus_plus @ int @ Z @ ( one_one @ int ) ) )
= ( ord_less_eq @ int @ W2 @ Z ) ) ).
% zle_add1_eq_le
thf(fact_910_i0__less,axiom,
! [N: extended_enat] :
( ( ord_less @ extended_enat @ ( zero_zero @ extended_enat ) @ N )
= ( N
!= ( zero_zero @ extended_enat ) ) ) ).
% i0_less
thf(fact_911_mult__cancel__right,axiom,
! [A: $tType] :
( ( semiri6575147826004484403cancel @ A )
=> ! [A2: A,C3: A,B2: A] :
( ( ( times_times @ A @ A2 @ C3 )
= ( times_times @ A @ B2 @ C3 ) )
= ( ( C3
= ( zero_zero @ A ) )
| ( A2 = B2 ) ) ) ) ).
% mult_cancel_right
thf(fact_912_mult__cancel__left,axiom,
! [A: $tType] :
( ( semiri6575147826004484403cancel @ A )
=> ! [C3: A,A2: A,B2: A] :
( ( ( times_times @ A @ C3 @ A2 )
= ( times_times @ A @ C3 @ B2 ) )
= ( ( C3
= ( zero_zero @ A ) )
| ( A2 = B2 ) ) ) ) ).
% mult_cancel_left
thf(fact_913_mult__eq__0__iff,axiom,
! [A: $tType] :
( ( semiri3467727345109120633visors @ A )
=> ! [A2: A,B2: A] :
( ( ( times_times @ A @ A2 @ B2 )
= ( zero_zero @ A ) )
= ( ( A2
= ( zero_zero @ A ) )
| ( B2
= ( zero_zero @ A ) ) ) ) ) ).
% mult_eq_0_iff
thf(fact_914_mult__zero__right,axiom,
! [A: $tType] :
( ( mult_zero @ A )
=> ! [A2: A] :
( ( times_times @ A @ A2 @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% mult_zero_right
thf(fact_915_mult__zero__left,axiom,
! [A: $tType] :
( ( mult_zero @ A )
=> ! [A2: A] :
( ( times_times @ A @ ( zero_zero @ A ) @ A2 )
= ( zero_zero @ A ) ) ) ).
% mult_zero_left
thf(fact_916_mult_Oright__neutral,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A2: A] :
( ( times_times @ A @ A2 @ ( one_one @ A ) )
= A2 ) ) ).
% mult.right_neutral
thf(fact_917_mult__1,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A2: A] :
( ( times_times @ A @ ( one_one @ A ) @ A2 )
= A2 ) ) ).
% mult_1
thf(fact_918_times__divide__eq__left,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [B2: A,C3: A,A2: A] :
( ( times_times @ A @ ( divide_divide @ A @ B2 @ C3 ) @ A2 )
= ( divide_divide @ A @ ( times_times @ A @ B2 @ A2 ) @ C3 ) ) ) ).
% times_divide_eq_left
thf(fact_919_divide__divide__eq__left,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( divide_divide @ A @ ( divide_divide @ A @ A2 @ B2 ) @ C3 )
= ( divide_divide @ A @ A2 @ ( times_times @ A @ B2 @ C3 ) ) ) ) ).
% divide_divide_eq_left
thf(fact_920_divide__divide__eq__right,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( divide_divide @ A @ A2 @ ( divide_divide @ A @ B2 @ C3 ) )
= ( divide_divide @ A @ ( times_times @ A @ A2 @ C3 ) @ B2 ) ) ) ).
% divide_divide_eq_right
thf(fact_921_times__divide__eq__right,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( times_times @ A @ A2 @ ( divide_divide @ A @ B2 @ C3 ) )
= ( divide_divide @ A @ ( times_times @ A @ A2 @ B2 ) @ C3 ) ) ) ).
% times_divide_eq_right
thf(fact_922_mult__cancel2,axiom,
! [M2: nat,K2: nat,N: nat] :
( ( ( times_times @ nat @ M2 @ K2 )
= ( times_times @ nat @ N @ K2 ) )
= ( ( M2 = N )
| ( K2
= ( zero_zero @ nat ) ) ) ) ).
% mult_cancel2
thf(fact_923_mult__cancel1,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( ( times_times @ nat @ K2 @ M2 )
= ( times_times @ nat @ K2 @ N ) )
= ( ( M2 = N )
| ( K2
= ( zero_zero @ nat ) ) ) ) ).
% mult_cancel1
thf(fact_924_mult__0__right,axiom,
! [M2: nat] :
( ( times_times @ nat @ M2 @ ( zero_zero @ nat ) )
= ( zero_zero @ nat ) ) ).
% mult_0_right
thf(fact_925_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_926_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_927_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_928_mult__cancel__right2,axiom,
! [A: $tType] :
( ( ring_15535105094025558882visors @ A )
=> ! [A2: A,C3: A] :
( ( ( times_times @ A @ A2 @ C3 )
= C3 )
= ( ( C3
= ( zero_zero @ A ) )
| ( A2
= ( one_one @ A ) ) ) ) ) ).
% mult_cancel_right2
thf(fact_929_mult__cancel__right1,axiom,
! [A: $tType] :
( ( ring_15535105094025558882visors @ A )
=> ! [C3: A,B2: A] :
( ( C3
= ( times_times @ A @ B2 @ C3 ) )
= ( ( C3
= ( zero_zero @ A ) )
| ( B2
= ( one_one @ A ) ) ) ) ) ).
% mult_cancel_right1
thf(fact_930_mult__cancel__left2,axiom,
! [A: $tType] :
( ( ring_15535105094025558882visors @ A )
=> ! [C3: A,A2: A] :
( ( ( times_times @ A @ C3 @ A2 )
= C3 )
= ( ( C3
= ( zero_zero @ A ) )
| ( A2
= ( one_one @ A ) ) ) ) ) ).
% mult_cancel_left2
thf(fact_931_mult__cancel__left1,axiom,
! [A: $tType] :
( ( ring_15535105094025558882visors @ A )
=> ! [C3: A,B2: A] :
( ( C3
= ( times_times @ A @ C3 @ B2 ) )
= ( ( C3
= ( zero_zero @ A ) )
| ( B2
= ( one_one @ A ) ) ) ) ) ).
% mult_cancel_left1
thf(fact_932_sum__squares__eq__zero__iff,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [X: A,Y: A] :
( ( ( plus_plus @ A @ ( times_times @ A @ X @ X ) @ ( times_times @ A @ Y @ Y ) )
= ( zero_zero @ A ) )
= ( ( X
= ( zero_zero @ A ) )
& ( Y
= ( zero_zero @ A ) ) ) ) ) ).
% sum_squares_eq_zero_iff
thf(fact_933_nonzero__mult__divide__mult__cancel__right2,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [C3: A,A2: A,B2: A] :
( ( C3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ A2 @ C3 ) @ ( times_times @ A @ C3 @ B2 ) )
= ( divide_divide @ A @ A2 @ B2 ) ) ) ) ).
% nonzero_mult_divide_mult_cancel_right2
thf(fact_934_nonzero__mult__div__cancel__right,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [B2: A,A2: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ A2 @ B2 ) @ B2 )
= A2 ) ) ) ).
% nonzero_mult_div_cancel_right
thf(fact_935_nonzero__mult__divide__mult__cancel__right,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [C3: A,A2: A,B2: A] :
( ( C3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ A2 @ C3 ) @ ( times_times @ A @ B2 @ C3 ) )
= ( divide_divide @ A @ A2 @ B2 ) ) ) ) ).
% nonzero_mult_divide_mult_cancel_right
thf(fact_936_nonzero__mult__divide__mult__cancel__left2,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [C3: A,A2: A,B2: A] :
( ( C3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ C3 @ A2 ) @ ( times_times @ A @ B2 @ C3 ) )
= ( divide_divide @ A @ A2 @ B2 ) ) ) ) ).
% nonzero_mult_divide_mult_cancel_left2
thf(fact_937_nonzero__mult__div__cancel__left,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [A2: A,B2: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ A2 @ B2 ) @ A2 )
= B2 ) ) ) ).
% nonzero_mult_div_cancel_left
thf(fact_938_nonzero__mult__divide__mult__cancel__left,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [C3: A,A2: A,B2: A] :
( ( C3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ C3 @ A2 ) @ ( times_times @ A @ C3 @ B2 ) )
= ( divide_divide @ A @ A2 @ B2 ) ) ) ) ).
% nonzero_mult_divide_mult_cancel_left
thf(fact_939_mult__divide__mult__cancel__left__if,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [C3: A,A2: A,B2: A] :
( ( ( C3
= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ C3 @ A2 ) @ ( times_times @ A @ C3 @ B2 ) )
= ( zero_zero @ A ) ) )
& ( ( C3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ C3 @ A2 ) @ ( times_times @ A @ C3 @ B2 ) )
= ( divide_divide @ A @ A2 @ B2 ) ) ) ) ) ).
% mult_divide_mult_cancel_left_if
thf(fact_940_div__mult__mult1__if,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [C3: A,A2: A,B2: A] :
( ( ( C3
= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ C3 @ A2 ) @ ( times_times @ A @ C3 @ B2 ) )
= ( zero_zero @ A ) ) )
& ( ( C3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ C3 @ A2 ) @ ( times_times @ A @ C3 @ B2 ) )
= ( divide_divide @ A @ A2 @ B2 ) ) ) ) ) ).
% div_mult_mult1_if
thf(fact_941_div__mult__mult2,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [C3: A,A2: A,B2: A] :
( ( C3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ A2 @ C3 ) @ ( times_times @ A @ B2 @ C3 ) )
= ( divide_divide @ A @ A2 @ B2 ) ) ) ) ).
% div_mult_mult2
thf(fact_942_div__mult__mult1,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [C3: A,A2: A,B2: A] :
( ( C3
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( times_times @ A @ C3 @ A2 ) @ ( times_times @ A @ C3 @ B2 ) )
= ( divide_divide @ A @ A2 @ B2 ) ) ) ) ).
% div_mult_mult1
thf(fact_943_distrib__right__numeral,axiom,
! [A: $tType] :
( ( ( numeral @ A )
& ( semiring @ A ) )
=> ! [A2: A,B2: A,V: num] :
( ( times_times @ A @ ( plus_plus @ A @ A2 @ B2 ) @ ( numeral_numeral @ A @ V ) )
= ( plus_plus @ A @ ( times_times @ A @ A2 @ ( numeral_numeral @ A @ V ) ) @ ( times_times @ A @ B2 @ ( numeral_numeral @ A @ V ) ) ) ) ) ).
% distrib_right_numeral
thf(fact_944_distrib__left__numeral,axiom,
! [A: $tType] :
( ( ( numeral @ A )
& ( semiring @ A ) )
=> ! [V: num,B2: A,C3: A] :
( ( times_times @ A @ ( numeral_numeral @ A @ V ) @ ( plus_plus @ A @ B2 @ C3 ) )
= ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ V ) @ B2 ) @ ( times_times @ A @ ( numeral_numeral @ A @ V ) @ C3 ) ) ) ) ).
% distrib_left_numeral
thf(fact_945_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_946_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_947_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_948_mult__less__cancel2,axiom,
! [M2: nat,K2: nat,N: nat] :
( ( ord_less @ nat @ ( times_times @ nat @ M2 @ K2 ) @ ( times_times @ nat @ N @ K2 ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K2 )
& ( ord_less @ nat @ M2 @ N ) ) ) ).
% mult_less_cancel2
thf(fact_949_nat__mult__less__cancel__disj,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( ord_less @ nat @ ( times_times @ nat @ K2 @ M2 ) @ ( times_times @ nat @ K2 @ N ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K2 )
& ( ord_less @ nat @ M2 @ N ) ) ) ).
% nat_mult_less_cancel_disj
thf(fact_950_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_951_nat__mult__div__cancel__disj,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( ( K2
= ( zero_zero @ nat ) )
=> ( ( divide_divide @ nat @ ( times_times @ nat @ K2 @ M2 ) @ ( times_times @ nat @ K2 @ N ) )
= ( zero_zero @ nat ) ) )
& ( ( K2
!= ( zero_zero @ nat ) )
=> ( ( divide_divide @ nat @ ( times_times @ nat @ K2 @ M2 ) @ ( times_times @ nat @ K2 @ N ) )
= ( divide_divide @ nat @ M2 @ N ) ) ) ) ).
% nat_mult_div_cancel_disj
thf(fact_952_div__neg__neg__trivial,axiom,
! [K2: int,L: int] :
( ( ord_less_eq @ int @ K2 @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ L @ K2 )
=> ( ( divide_divide @ int @ K2 @ L )
= ( zero_zero @ int ) ) ) ) ).
% div_neg_neg_trivial
thf(fact_953_div__pos__pos__trivial,axiom,
! [K2: int,L: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K2 )
=> ( ( ord_less @ int @ K2 @ L )
=> ( ( divide_divide @ int @ K2 @ L )
= ( zero_zero @ int ) ) ) ) ).
% div_pos_pos_trivial
thf(fact_954_divide__le__eq__numeral1_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,W2: num,A2: A] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ ( numeral_numeral @ A @ W2 ) ) @ A2 )
= ( ord_less_eq @ A @ B2 @ ( times_times @ A @ A2 @ ( numeral_numeral @ A @ W2 ) ) ) ) ) ).
% divide_le_eq_numeral1(1)
thf(fact_955_le__divide__eq__numeral1_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A,W2: num] :
( ( ord_less_eq @ A @ A2 @ ( divide_divide @ A @ B2 @ ( numeral_numeral @ A @ W2 ) ) )
= ( ord_less_eq @ A @ ( times_times @ A @ A2 @ ( numeral_numeral @ A @ W2 ) ) @ B2 ) ) ) ).
% le_divide_eq_numeral1(1)
thf(fact_956_divide__eq__eq__numeral1_I1_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B2: A,W2: num,A2: A] :
( ( ( divide_divide @ A @ B2 @ ( numeral_numeral @ A @ W2 ) )
= A2 )
= ( ( ( ( numeral_numeral @ A @ W2 )
!= ( zero_zero @ A ) )
=> ( B2
= ( times_times @ A @ A2 @ ( numeral_numeral @ A @ W2 ) ) ) )
& ( ( ( numeral_numeral @ A @ W2 )
= ( zero_zero @ A ) )
=> ( A2
= ( zero_zero @ A ) ) ) ) ) ) ).
% divide_eq_eq_numeral1(1)
thf(fact_957_eq__divide__eq__numeral1_I1_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A,B2: A,W2: num] :
( ( A2
= ( divide_divide @ A @ B2 @ ( numeral_numeral @ A @ W2 ) ) )
= ( ( ( ( numeral_numeral @ A @ W2 )
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ A2 @ ( numeral_numeral @ A @ W2 ) )
= B2 ) )
& ( ( ( numeral_numeral @ A @ W2 )
= ( zero_zero @ A ) )
=> ( A2
= ( zero_zero @ A ) ) ) ) ) ) ).
% eq_divide_eq_numeral1(1)
thf(fact_958_divide__less__eq__numeral1_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,W2: num,A2: A] :
( ( ord_less @ A @ ( divide_divide @ A @ B2 @ ( numeral_numeral @ A @ W2 ) ) @ A2 )
= ( ord_less @ A @ B2 @ ( times_times @ A @ A2 @ ( numeral_numeral @ A @ W2 ) ) ) ) ) ).
% divide_less_eq_numeral1(1)
thf(fact_959_less__divide__eq__numeral1_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A,W2: num] :
( ( ord_less @ A @ A2 @ ( divide_divide @ A @ B2 @ ( numeral_numeral @ A @ W2 ) ) )
= ( ord_less @ A @ ( times_times @ A @ A2 @ ( numeral_numeral @ A @ W2 ) ) @ B2 ) ) ) ).
% less_divide_eq_numeral1(1)
thf(fact_960_nonzero__divide__mult__cancel__left,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A2: A,B2: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ A2 @ ( times_times @ A @ A2 @ B2 ) )
= ( divide_divide @ A @ ( one_one @ A ) @ B2 ) ) ) ) ).
% nonzero_divide_mult_cancel_left
thf(fact_961_nonzero__divide__mult__cancel__right,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [B2: A,A2: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ B2 @ ( times_times @ A @ A2 @ B2 ) )
= ( divide_divide @ A @ ( one_one @ A ) @ A2 ) ) ) ) ).
% nonzero_divide_mult_cancel_right
thf(fact_962_div__mult__self4,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [B2: A,C3: A,A2: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ B2 @ C3 ) @ A2 ) @ B2 )
= ( plus_plus @ A @ C3 @ ( divide_divide @ A @ A2 @ B2 ) ) ) ) ) ).
% div_mult_self4
thf(fact_963_div__mult__self3,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [B2: A,C3: A,A2: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ C3 @ B2 ) @ A2 ) @ B2 )
= ( plus_plus @ A @ C3 @ ( divide_divide @ A @ A2 @ B2 ) ) ) ) ) ).
% div_mult_self3
thf(fact_964_div__mult__self2,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [B2: A,A2: A,C3: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ A2 @ ( times_times @ A @ B2 @ C3 ) ) @ B2 )
= ( plus_plus @ A @ C3 @ ( divide_divide @ A @ A2 @ B2 ) ) ) ) ) ).
% div_mult_self2
thf(fact_965_div__mult__self1,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [B2: A,A2: A,C3: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ A2 @ ( times_times @ A @ C3 @ B2 ) ) @ B2 )
= ( plus_plus @ A @ C3 @ ( divide_divide @ A @ A2 @ B2 ) ) ) ) ) ).
% div_mult_self1
thf(fact_966_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_967_mult__le__cancel2,axiom,
! [M2: nat,K2: nat,N: nat] :
( ( ord_less_eq @ nat @ ( times_times @ nat @ M2 @ K2 ) @ ( times_times @ nat @ N @ K2 ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K2 )
=> ( ord_less_eq @ nat @ M2 @ N ) ) ) ).
% mult_le_cancel2
thf(fact_968_nat__mult__le__cancel__disj,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( ord_less_eq @ nat @ ( times_times @ nat @ K2 @ M2 ) @ ( times_times @ nat @ K2 @ N ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K2 )
=> ( ord_less_eq @ nat @ M2 @ N ) ) ) ).
% nat_mult_le_cancel_disj
thf(fact_969_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_970_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_971_add1__zle__eq,axiom,
! [W2: int,Z: int] :
( ( ord_less_eq @ int @ ( plus_plus @ int @ W2 @ ( one_one @ int ) ) @ Z )
= ( ord_less @ int @ W2 @ Z ) ) ).
% add1_zle_eq
thf(fact_972_le__imp__0__less,axiom,
! [Z: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z )
=> ( ord_less @ int @ ( zero_zero @ int ) @ ( plus_plus @ int @ ( one_one @ int ) @ Z ) ) ) ).
% le_imp_0_less
thf(fact_973_zdiv__mono1,axiom,
! [A2: int,A3: int,B2: int] :
( ( ord_less_eq @ int @ A2 @ A3 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
=> ( ord_less_eq @ int @ ( divide_divide @ int @ A2 @ B2 ) @ ( divide_divide @ int @ A3 @ B2 ) ) ) ) ).
% zdiv_mono1
thf(fact_974_zdiv__mono2,axiom,
! [A2: int,B3: int,B2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A2 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B3 )
=> ( ( ord_less_eq @ int @ B3 @ B2 )
=> ( ord_less_eq @ int @ ( divide_divide @ int @ A2 @ B2 ) @ ( divide_divide @ int @ A2 @ B3 ) ) ) ) ) ).
% zdiv_mono2
thf(fact_975_zdiv__eq__0__iff,axiom,
! [I2: int,K2: int] :
( ( ( divide_divide @ int @ I2 @ K2 )
= ( zero_zero @ int ) )
= ( ( K2
= ( zero_zero @ int ) )
| ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ I2 )
& ( ord_less @ int @ I2 @ K2 ) )
| ( ( ord_less_eq @ int @ I2 @ ( zero_zero @ int ) )
& ( ord_less @ int @ K2 @ I2 ) ) ) ) ).
% zdiv_eq_0_iff
thf(fact_976_zdiv__mono1__neg,axiom,
! [A2: int,A3: int,B2: int] :
( ( ord_less_eq @ int @ A2 @ A3 )
=> ( ( ord_less @ int @ B2 @ ( zero_zero @ int ) )
=> ( ord_less_eq @ int @ ( divide_divide @ int @ A3 @ B2 ) @ ( divide_divide @ int @ A2 @ B2 ) ) ) ) ).
% zdiv_mono1_neg
thf(fact_977_zdiv__mono2__neg,axiom,
! [A2: int,B3: int,B2: int] :
( ( ord_less @ int @ A2 @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B3 )
=> ( ( ord_less_eq @ int @ B3 @ B2 )
=> ( ord_less_eq @ int @ ( divide_divide @ int @ A2 @ B3 ) @ ( divide_divide @ int @ A2 @ B2 ) ) ) ) ) ).
% zdiv_mono2_neg
thf(fact_978_zless__imp__add1__zle,axiom,
! [W2: int,Z: int] :
( ( ord_less @ int @ W2 @ Z )
=> ( ord_less_eq @ int @ ( plus_plus @ int @ W2 @ ( one_one @ int ) ) @ Z ) ) ).
% zless_imp_add1_zle
thf(fact_979_div__int__pos__iff,axiom,
! [K2: int,L: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( divide_divide @ int @ K2 @ L ) )
= ( ( K2
= ( zero_zero @ int ) )
| ( L
= ( zero_zero @ int ) )
| ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K2 )
& ( ord_less_eq @ int @ ( zero_zero @ int ) @ L ) )
| ( ( ord_less @ int @ K2 @ ( zero_zero @ int ) )
& ( ord_less @ int @ L @ ( zero_zero @ int ) ) ) ) ) ).
% div_int_pos_iff
thf(fact_980_div__positive__int,axiom,
! [L: int,K2: int] :
( ( ord_less_eq @ int @ L @ K2 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ L )
=> ( ord_less @ int @ ( zero_zero @ int ) @ ( divide_divide @ int @ K2 @ L ) ) ) ) ).
% div_positive_int
thf(fact_981_div__nonneg__neg__le0,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A2 )
=> ( ( ord_less @ int @ B2 @ ( zero_zero @ int ) )
=> ( ord_less_eq @ int @ ( divide_divide @ int @ A2 @ B2 ) @ ( zero_zero @ int ) ) ) ) ).
% div_nonneg_neg_le0
thf(fact_982_div__nonpos__pos__le0,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq @ int @ A2 @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
=> ( ord_less_eq @ int @ ( divide_divide @ int @ A2 @ B2 ) @ ( zero_zero @ int ) ) ) ) ).
% div_nonpos_pos_le0
thf(fact_983_pos__imp__zdiv__pos__iff,axiom,
! [K2: int,I2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ K2 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ ( divide_divide @ int @ I2 @ K2 ) )
= ( ord_less_eq @ int @ K2 @ I2 ) ) ) ).
% pos_imp_zdiv_pos_iff
thf(fact_984_int__one__le__iff__zero__less,axiom,
! [Z: int] :
( ( ord_less_eq @ int @ ( one_one @ int ) @ Z )
= ( ord_less @ int @ ( zero_zero @ int ) @ Z ) ) ).
% int_one_le_iff_zero_less
thf(fact_985_neg__imp__zdiv__nonneg__iff,axiom,
! [B2: int,A2: int] :
( ( ord_less @ int @ B2 @ ( zero_zero @ int ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( divide_divide @ int @ A2 @ B2 ) )
= ( ord_less_eq @ int @ A2 @ ( zero_zero @ int ) ) ) ) ).
% neg_imp_zdiv_nonneg_iff
thf(fact_986_pos__imp__zdiv__nonneg__iff,axiom,
! [B2: int,A2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( divide_divide @ int @ A2 @ B2 ) )
= ( ord_less_eq @ int @ ( zero_zero @ int ) @ A2 ) ) ) ).
% pos_imp_zdiv_nonneg_iff
thf(fact_987_nonneg1__imp__zdiv__pos__iff,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ A2 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ ( divide_divide @ int @ A2 @ B2 ) )
= ( ( ord_less_eq @ int @ B2 @ A2 )
& ( ord_less @ int @ ( zero_zero @ int ) @ B2 ) ) ) ) ).
% nonneg1_imp_zdiv_pos_iff
thf(fact_988_odd__nonzero,axiom,
! [Z: int] :
( ( plus_plus @ int @ ( plus_plus @ int @ ( one_one @ int ) @ Z ) @ Z )
!= ( zero_zero @ int ) ) ).
% odd_nonzero
thf(fact_989_pos__imp__zdiv__neg__iff,axiom,
! [B2: int,A2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
=> ( ( ord_less @ int @ ( divide_divide @ int @ A2 @ B2 ) @ ( zero_zero @ int ) )
= ( ord_less @ int @ A2 @ ( zero_zero @ int ) ) ) ) ).
% pos_imp_zdiv_neg_iff
thf(fact_990_neg__imp__zdiv__neg__iff,axiom,
! [B2: int,A2: int] :
( ( ord_less @ int @ B2 @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ ( divide_divide @ int @ A2 @ B2 ) @ ( zero_zero @ int ) )
= ( ord_less @ int @ ( zero_zero @ int ) @ A2 ) ) ) ).
% neg_imp_zdiv_neg_iff
thf(fact_991_int__div__less__self,axiom,
! [X: int,K2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ X )
=> ( ( ord_less @ int @ ( one_one @ int ) @ K2 )
=> ( ord_less @ int @ ( divide_divide @ int @ X @ K2 ) @ X ) ) ) ).
% int_div_less_self
thf(fact_992_div__neg__pos__less0,axiom,
! [A2: int,B2: int] :
( ( ord_less @ int @ A2 @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
=> ( ord_less @ int @ ( divide_divide @ int @ A2 @ B2 ) @ ( zero_zero @ int ) ) ) ) ).
% div_neg_pos_less0
thf(fact_993_odd__less__0__iff,axiom,
! [Z: int] :
( ( ord_less @ int @ ( plus_plus @ int @ ( plus_plus @ int @ ( one_one @ int ) @ Z ) @ Z ) @ ( zero_zero @ int ) )
= ( ord_less @ int @ Z @ ( zero_zero @ int ) ) ) ).
% odd_less_0_iff
thf(fact_994_zless__add1__eq,axiom,
! [W2: int,Z: int] :
( ( ord_less @ int @ W2 @ ( plus_plus @ int @ Z @ ( one_one @ int ) ) )
= ( ( ord_less @ int @ W2 @ Z )
| ( W2 = Z ) ) ) ).
% zless_add1_eq
thf(fact_995_int__gr__induct,axiom,
! [K2: int,I2: int,P2: int > $o] :
( ( ord_less @ int @ K2 @ I2 )
=> ( ( P2 @ ( plus_plus @ int @ K2 @ ( one_one @ int ) ) )
=> ( ! [I3: int] :
( ( ord_less @ int @ K2 @ I3 )
=> ( ( P2 @ I3 )
=> ( P2 @ ( plus_plus @ int @ I3 @ ( one_one @ int ) ) ) ) )
=> ( P2 @ I2 ) ) ) ) ).
% int_gr_induct
thf(fact_996_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: $tType] :
( ( ab_semigroup_mult @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( times_times @ A @ ( times_times @ A @ A2 @ B2 ) @ C3 )
= ( times_times @ A @ A2 @ ( times_times @ A @ B2 @ C3 ) ) ) ) ).
% ab_semigroup_mult_class.mult_ac(1)
thf(fact_997_mult_Oassoc,axiom,
! [A: $tType] :
( ( semigroup_mult @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( times_times @ A @ ( times_times @ A @ A2 @ B2 ) @ C3 )
= ( times_times @ A @ A2 @ ( times_times @ A @ B2 @ C3 ) ) ) ) ).
% mult.assoc
thf(fact_998_mult_Ocommute,axiom,
! [A: $tType] :
( ( ab_semigroup_mult @ A )
=> ( ( times_times @ A )
= ( ^ [A6: A,B5: A] : ( times_times @ A @ B5 @ A6 ) ) ) ) ).
% mult.commute
thf(fact_999_mult_Oleft__commute,axiom,
! [A: $tType] :
( ( ab_semigroup_mult @ A )
=> ! [B2: A,A2: A,C3: A] :
( ( times_times @ A @ B2 @ ( times_times @ A @ A2 @ C3 ) )
= ( times_times @ A @ A2 @ ( times_times @ A @ B2 @ C3 ) ) ) ) ).
% mult.left_commute
thf(fact_1000_less__int__code_I1_J,axiom,
~ ( ord_less @ int @ ( zero_zero @ int ) @ ( zero_zero @ int ) ) ).
% less_int_code(1)
thf(fact_1001_not__iless0,axiom,
! [N: extended_enat] :
~ ( ord_less @ extended_enat @ N @ ( zero_zero @ extended_enat ) ) ).
% not_iless0
thf(fact_1002_enat__less__induct,axiom,
! [P2: extended_enat > $o,N: extended_enat] :
( ! [N3: extended_enat] :
( ! [M4: extended_enat] :
( ( ord_less @ extended_enat @ M4 @ N3 )
=> ( P2 @ M4 ) )
=> ( P2 @ N3 ) )
=> ( P2 @ N ) ) ).
% enat_less_induct
thf(fact_1003_plus__int__code_I1_J,axiom,
! [K2: int] :
( ( plus_plus @ int @ K2 @ ( zero_zero @ int ) )
= K2 ) ).
% plus_int_code(1)
thf(fact_1004_plus__int__code_I2_J,axiom,
! [L: int] :
( ( plus_plus @ int @ ( zero_zero @ int ) @ L )
= L ) ).
% plus_int_code(2)
thf(fact_1005_less__eq__int__code_I1_J,axiom,
ord_less_eq @ int @ ( zero_zero @ int ) @ ( zero_zero @ int ) ).
% less_eq_int_code(1)
thf(fact_1006_mult__right__cancel,axiom,
! [A: $tType] :
( ( semiri6575147826004484403cancel @ A )
=> ! [C3: A,A2: A,B2: A] :
( ( C3
!= ( zero_zero @ A ) )
=> ( ( ( times_times @ A @ A2 @ C3 )
= ( times_times @ A @ B2 @ C3 ) )
= ( A2 = B2 ) ) ) ) ).
% mult_right_cancel
thf(fact_1007_mult__left__cancel,axiom,
! [A: $tType] :
( ( semiri6575147826004484403cancel @ A )
=> ! [C3: A,A2: A,B2: A] :
( ( C3
!= ( zero_zero @ A ) )
=> ( ( ( times_times @ A @ C3 @ A2 )
= ( times_times @ A @ C3 @ B2 ) )
= ( A2 = B2 ) ) ) ) ).
% mult_left_cancel
thf(fact_1008_no__zero__divisors,axiom,
! [A: $tType] :
( ( semiri3467727345109120633visors @ A )
=> ! [A2: A,B2: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( B2
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ A2 @ B2 )
!= ( zero_zero @ A ) ) ) ) ) ).
% no_zero_divisors
thf(fact_1009_divisors__zero,axiom,
! [A: $tType] :
( ( semiri3467727345109120633visors @ A )
=> ! [A2: A,B2: A] :
( ( ( times_times @ A @ A2 @ B2 )
= ( zero_zero @ A ) )
=> ( ( A2
= ( zero_zero @ A ) )
| ( B2
= ( zero_zero @ A ) ) ) ) ) ).
% divisors_zero
thf(fact_1010_mult__not__zero,axiom,
! [A: $tType] :
( ( mult_zero @ A )
=> ! [A2: A,B2: A] :
( ( ( times_times @ A @ A2 @ B2 )
!= ( zero_zero @ A ) )
=> ( ( A2
!= ( zero_zero @ A ) )
& ( B2
!= ( zero_zero @ A ) ) ) ) ) ).
% mult_not_zero
thf(fact_1011_comm__monoid__mult__class_Omult__1,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A2: A] :
( ( times_times @ A @ ( one_one @ A ) @ A2 )
= A2 ) ) ).
% comm_monoid_mult_class.mult_1
thf(fact_1012_mult_Ocomm__neutral,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A2: A] :
( ( times_times @ A @ A2 @ ( one_one @ A ) )
= A2 ) ) ).
% mult.comm_neutral
thf(fact_1013_crossproduct__noteq,axiom,
! [A: $tType] :
( ( semiri1453513574482234551roduct @ A )
=> ! [A2: A,B2: A,C3: A,D3: A] :
( ( ( A2 != B2 )
& ( C3 != D3 ) )
= ( ( plus_plus @ A @ ( times_times @ A @ A2 @ C3 ) @ ( times_times @ A @ B2 @ D3 ) )
!= ( plus_plus @ A @ ( times_times @ A @ A2 @ D3 ) @ ( times_times @ A @ B2 @ C3 ) ) ) ) ) ).
% crossproduct_noteq
thf(fact_1014_crossproduct__eq,axiom,
! [A: $tType] :
( ( semiri1453513574482234551roduct @ A )
=> ! [W2: A,Y: A,X: A,Z: A] :
( ( ( plus_plus @ A @ ( times_times @ A @ W2 @ Y ) @ ( times_times @ A @ X @ Z ) )
= ( plus_plus @ A @ ( times_times @ A @ W2 @ Z ) @ ( times_times @ A @ X @ Y ) ) )
= ( ( W2 = X )
| ( Y = Z ) ) ) ) ).
% crossproduct_eq
thf(fact_1015_combine__common__factor,axiom,
! [A: $tType] :
( ( semiring @ A )
=> ! [A2: A,E3: A,B2: A,C3: A] :
( ( plus_plus @ A @ ( times_times @ A @ A2 @ E3 ) @ ( plus_plus @ A @ ( times_times @ A @ B2 @ E3 ) @ C3 ) )
= ( plus_plus @ A @ ( times_times @ A @ ( plus_plus @ A @ A2 @ B2 ) @ E3 ) @ C3 ) ) ) ).
% combine_common_factor
thf(fact_1016_distrib__right,axiom,
! [A: $tType] :
( ( semiring @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( times_times @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C3 )
= ( plus_plus @ A @ ( times_times @ A @ A2 @ C3 ) @ ( times_times @ A @ B2 @ C3 ) ) ) ) ).
% distrib_right
thf(fact_1017_distrib__left,axiom,
! [A: $tType] :
( ( semiring @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( times_times @ A @ A2 @ ( plus_plus @ A @ B2 @ C3 ) )
= ( plus_plus @ A @ ( times_times @ A @ A2 @ B2 ) @ ( times_times @ A @ A2 @ C3 ) ) ) ) ).
% distrib_left
thf(fact_1018_comm__semiring__class_Odistrib,axiom,
! [A: $tType] :
( ( comm_semiring @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( times_times @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C3 )
= ( plus_plus @ A @ ( times_times @ A @ A2 @ C3 ) @ ( times_times @ A @ B2 @ C3 ) ) ) ) ).
% comm_semiring_class.distrib
thf(fact_1019_ring__class_Oring__distribs_I1_J,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( times_times @ A @ A2 @ ( plus_plus @ A @ B2 @ C3 ) )
= ( plus_plus @ A @ ( times_times @ A @ A2 @ B2 ) @ ( times_times @ A @ A2 @ C3 ) ) ) ) ).
% ring_class.ring_distribs(1)
thf(fact_1020_ring__class_Oring__distribs_I2_J,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( times_times @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C3 )
= ( plus_plus @ A @ ( times_times @ A @ A2 @ C3 ) @ ( times_times @ A @ B2 @ C3 ) ) ) ) ).
% ring_class.ring_distribs(2)
thf(fact_1021_times__divide__times__eq,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [X: A,Y: A,Z: A,W2: A] :
( ( times_times @ A @ ( divide_divide @ A @ X @ Y ) @ ( divide_divide @ A @ Z @ W2 ) )
= ( divide_divide @ A @ ( times_times @ A @ X @ Z ) @ ( times_times @ A @ Y @ W2 ) ) ) ) ).
% times_divide_times_eq
thf(fact_1022_divide__divide__times__eq,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [X: A,Y: A,Z: A,W2: A] :
( ( divide_divide @ A @ ( divide_divide @ A @ X @ Y ) @ ( divide_divide @ A @ Z @ W2 ) )
= ( divide_divide @ A @ ( times_times @ A @ X @ W2 ) @ ( times_times @ A @ Y @ Z ) ) ) ) ).
% divide_divide_times_eq
thf(fact_1023_divide__divide__eq__left_H,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( divide_divide @ A @ ( divide_divide @ A @ A2 @ B2 ) @ C3 )
= ( divide_divide @ A @ A2 @ ( times_times @ A @ C3 @ B2 ) ) ) ) ).
% divide_divide_eq_left'
thf(fact_1024_Suc__mult__cancel1,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( ( times_times @ nat @ ( suc @ K2 ) @ M2 )
= ( times_times @ nat @ ( suc @ K2 ) @ N ) )
= ( M2 = N ) ) ).
% Suc_mult_cancel1
thf(fact_1025_mult__0,axiom,
! [N: nat] :
( ( times_times @ nat @ ( zero_zero @ nat ) @ N )
= ( zero_zero @ nat ) ) ).
% mult_0
thf(fact_1026_nat__mult__eq__cancel__disj,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( ( times_times @ nat @ K2 @ M2 )
= ( times_times @ nat @ K2 @ N ) )
= ( ( K2
= ( zero_zero @ nat ) )
| ( M2 = N ) ) ) ).
% nat_mult_eq_cancel_disj
thf(fact_1027_mult__le__mono2,axiom,
! [I2: nat,J: nat,K2: nat] :
( ( ord_less_eq @ nat @ I2 @ J )
=> ( ord_less_eq @ nat @ ( times_times @ nat @ K2 @ I2 ) @ ( times_times @ nat @ K2 @ J ) ) ) ).
% mult_le_mono2
thf(fact_1028_mult__le__mono1,axiom,
! [I2: nat,J: nat,K2: nat] :
( ( ord_less_eq @ nat @ I2 @ J )
=> ( ord_less_eq @ nat @ ( times_times @ nat @ I2 @ K2 ) @ ( times_times @ nat @ J @ K2 ) ) ) ).
% mult_le_mono1
thf(fact_1029_mult__le__mono,axiom,
! [I2: nat,J: nat,K2: nat,L: nat] :
( ( ord_less_eq @ nat @ I2 @ J )
=> ( ( ord_less_eq @ nat @ K2 @ L )
=> ( ord_less_eq @ nat @ ( times_times @ nat @ I2 @ K2 ) @ ( times_times @ nat @ J @ L ) ) ) ) ).
% mult_le_mono
thf(fact_1030_le__square,axiom,
! [M2: nat] : ( ord_less_eq @ nat @ M2 @ ( times_times @ nat @ M2 @ M2 ) ) ).
% le_square
thf(fact_1031_le__cube,axiom,
! [M2: nat] : ( ord_less_eq @ nat @ M2 @ ( times_times @ nat @ M2 @ ( times_times @ nat @ M2 @ M2 ) ) ) ).
% le_cube
thf(fact_1032_add__mult__distrib2,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( times_times @ nat @ K2 @ ( plus_plus @ nat @ M2 @ N ) )
= ( plus_plus @ nat @ ( times_times @ nat @ K2 @ M2 ) @ ( times_times @ nat @ K2 @ N ) ) ) ).
% add_mult_distrib2
thf(fact_1033_add__mult__distrib,axiom,
! [M2: nat,N: nat,K2: nat] :
( ( times_times @ nat @ ( plus_plus @ nat @ M2 @ N ) @ K2 )
= ( plus_plus @ nat @ ( times_times @ nat @ M2 @ K2 ) @ ( times_times @ nat @ N @ K2 ) ) ) ).
% add_mult_distrib
thf(fact_1034_left__add__mult__distrib,axiom,
! [I2: nat,U: nat,J: nat,K2: nat] :
( ( plus_plus @ nat @ ( times_times @ nat @ I2 @ U ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ K2 ) )
= ( plus_plus @ nat @ ( times_times @ nat @ ( plus_plus @ nat @ I2 @ J ) @ U ) @ K2 ) ) ).
% left_add_mult_distrib
thf(fact_1035_nat__mult__1,axiom,
! [N: nat] :
( ( times_times @ nat @ ( one_one @ nat ) @ N )
= N ) ).
% nat_mult_1
thf(fact_1036_nat__mult__1__right,axiom,
! [N: nat] :
( ( times_times @ nat @ N @ ( one_one @ nat ) )
= N ) ).
% nat_mult_1_right
thf(fact_1037_nat__mult__max__right,axiom,
! [M2: nat,N: nat,Q2: nat] :
( ( times_times @ nat @ M2 @ ( ord_max @ nat @ N @ Q2 ) )
= ( ord_max @ nat @ ( times_times @ nat @ M2 @ N ) @ ( times_times @ nat @ M2 @ Q2 ) ) ) ).
% nat_mult_max_right
thf(fact_1038_nat__mult__max__left,axiom,
! [M2: nat,N: nat,Q2: nat] :
( ( times_times @ nat @ ( ord_max @ nat @ M2 @ N ) @ Q2 )
= ( ord_max @ nat @ ( times_times @ nat @ M2 @ Q2 ) @ ( times_times @ nat @ N @ Q2 ) ) ) ).
% nat_mult_max_left
thf(fact_1039_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [C3: A,A2: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( divide_divide @ A @ A2 @ ( times_times @ A @ B2 @ C3 ) )
= ( divide_divide @ A @ ( divide_divide @ A @ A2 @ B2 ) @ C3 ) ) ) ) ).
% unique_euclidean_semiring_numeral_class.div_mult2_eq
thf(fact_1040_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
! [A: $tType] :
( ( ordere2520102378445227354miring @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ ( times_times @ A @ C3 @ A2 ) @ ( times_times @ A @ C3 @ B2 ) ) ) ) ) ).
% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_1041_zero__le__mult__iff,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B2 ) )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) )
| ( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
& ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) ) ) ) ) ).
% zero_le_mult_iff
thf(fact_1042_mult__nonneg__nonpos2,axiom,
! [A: $tType] :
( ( ordered_semiring_0 @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ B2 @ A2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% mult_nonneg_nonpos2
thf(fact_1043_mult__nonpos__nonneg,axiom,
! [A: $tType] :
( ( ordered_semiring_0 @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ A2 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% mult_nonpos_nonneg
thf(fact_1044_mult__nonneg__nonpos,axiom,
! [A: $tType] :
( ( ordered_semiring_0 @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ A2 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% mult_nonneg_nonpos
thf(fact_1045_mult__nonneg__nonneg,axiom,
! [A: $tType] :
( ( ordered_semiring_0 @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B2 ) ) ) ) ) ).
% mult_nonneg_nonneg
thf(fact_1046_split__mult__neg__le,axiom,
! [A: $tType] :
( ( ordered_semiring_0 @ A )
=> ! [A2: A,B2: A] :
( ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
& ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) )
| ( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ A2 @ B2 ) @ ( zero_zero @ A ) ) ) ) ).
% split_mult_neg_le
thf(fact_1047_mult__le__0__iff,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( times_times @ A @ A2 @ B2 ) @ ( zero_zero @ A ) )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
& ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) )
| ( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) ) ) ) ) ).
% mult_le_0_iff
thf(fact_1048_mult__right__mono,axiom,
! [A: $tType] :
( ( ordered_semiring @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C3 ) @ ( times_times @ A @ B2 @ C3 ) ) ) ) ) ).
% mult_right_mono
thf(fact_1049_mult__right__mono__neg,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [B2: A,A2: A,C3: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_less_eq @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C3 ) @ ( times_times @ A @ B2 @ C3 ) ) ) ) ) ).
% mult_right_mono_neg
thf(fact_1050_mult__left__mono,axiom,
! [A: $tType] :
( ( ordered_semiring @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ ( times_times @ A @ C3 @ A2 ) @ ( times_times @ A @ C3 @ B2 ) ) ) ) ) ).
% mult_left_mono
thf(fact_1051_mult__nonpos__nonpos,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B2 ) ) ) ) ) ).
% mult_nonpos_nonpos
thf(fact_1052_mult__left__mono__neg,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [B2: A,A2: A,C3: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_less_eq @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ C3 @ A2 ) @ ( times_times @ A @ C3 @ B2 ) ) ) ) ) ).
% mult_left_mono_neg
thf(fact_1053_split__mult__pos__le,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [A2: A,B2: A] :
( ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) )
| ( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
& ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B2 ) ) ) ) ).
% split_mult_pos_le
thf(fact_1054_zero__le__square,axiom,
! [A: $tType] :
( ( linordered_ring @ A )
=> ! [A2: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ A2 ) ) ) ).
% zero_le_square
thf(fact_1055_mult__mono_H,axiom,
! [A: $tType] :
( ( ordered_semiring @ A )
=> ! [A2: A,B2: A,C3: A,D3: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C3 @ D3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C3 ) @ ( times_times @ A @ B2 @ D3 ) ) ) ) ) ) ) ).
% mult_mono'
thf(fact_1056_mult__mono,axiom,
! [A: $tType] :
( ( ordered_semiring @ A )
=> ! [A2: A,B2: A,C3: A,D3: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C3 @ D3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C3 ) @ ( times_times @ A @ B2 @ D3 ) ) ) ) ) ) ) ).
% mult_mono
thf(fact_1057_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
! [A: $tType] :
( ( linord2810124833399127020strict @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ ( times_times @ A @ C3 @ A2 ) @ ( times_times @ A @ C3 @ B2 ) ) ) ) ) ).
% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_1058_mult__less__cancel__right__disj,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [A2: A,C3: A,B2: A] :
( ( ord_less @ A @ ( times_times @ A @ A2 @ C3 ) @ ( times_times @ A @ B2 @ C3 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
& ( ord_less @ A @ A2 @ B2 ) )
| ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
& ( ord_less @ A @ B2 @ A2 ) ) ) ) ) ).
% mult_less_cancel_right_disj
thf(fact_1059_mult__strict__right__mono,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ ( times_times @ A @ A2 @ C3 ) @ ( times_times @ A @ B2 @ C3 ) ) ) ) ) ).
% mult_strict_right_mono
thf(fact_1060_mult__strict__right__mono__neg,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [B2: A,A2: A,C3: A] :
( ( ord_less @ A @ B2 @ A2 )
=> ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ A2 @ C3 ) @ ( times_times @ A @ B2 @ C3 ) ) ) ) ) ).
% mult_strict_right_mono_neg
thf(fact_1061_mult__less__cancel__left__disj,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [C3: A,A2: A,B2: A] :
( ( ord_less @ A @ ( times_times @ A @ C3 @ A2 ) @ ( times_times @ A @ C3 @ B2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
& ( ord_less @ A @ A2 @ B2 ) )
| ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
& ( ord_less @ A @ B2 @ A2 ) ) ) ) ) ).
% mult_less_cancel_left_disj
thf(fact_1062_mult__strict__left__mono,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ ( times_times @ A @ C3 @ A2 ) @ ( times_times @ A @ C3 @ B2 ) ) ) ) ) ).
% mult_strict_left_mono
thf(fact_1063_mult__strict__left__mono__neg,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [B2: A,A2: A,C3: A] :
( ( ord_less @ A @ B2 @ A2 )
=> ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ C3 @ A2 ) @ ( times_times @ A @ C3 @ B2 ) ) ) ) ) ).
% mult_strict_left_mono_neg
thf(fact_1064_mult__less__cancel__left__pos,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [C3: A,A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ord_less @ A @ ( times_times @ A @ C3 @ A2 ) @ ( times_times @ A @ C3 @ B2 ) )
= ( ord_less @ A @ A2 @ B2 ) ) ) ) ).
% mult_less_cancel_left_pos
thf(fact_1065_mult__less__cancel__left__neg,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [C3: A,A2: A,B2: A] :
( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( times_times @ A @ C3 @ A2 ) @ ( times_times @ A @ C3 @ B2 ) )
= ( ord_less @ A @ B2 @ A2 ) ) ) ) ).
% mult_less_cancel_left_neg
thf(fact_1066_zero__less__mult__pos2,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [B2: A,A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ B2 @ A2 ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) ) ) ) ).
% zero_less_mult_pos2
thf(fact_1067_zero__less__mult__pos,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B2 ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) ) ) ) ).
% zero_less_mult_pos
thf(fact_1068_zero__less__mult__iff,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
& ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) )
| ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
& ( ord_less @ A @ B2 @ ( zero_zero @ A ) ) ) ) ) ) ).
% zero_less_mult_iff
thf(fact_1069_mult__pos__neg2,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ B2 @ A2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% mult_pos_neg2
thf(fact_1070_mult__pos__pos,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B2 ) ) ) ) ) ).
% mult_pos_pos
thf(fact_1071_mult__pos__neg,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ A2 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% mult_pos_neg
thf(fact_1072_mult__neg__pos,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less @ A @ ( times_times @ A @ A2 @ B2 ) @ ( zero_zero @ A ) ) ) ) ) ).
% mult_neg_pos
thf(fact_1073_mult__less__0__iff,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( times_times @ A @ A2 @ B2 ) @ ( zero_zero @ A ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
& ( ord_less @ A @ B2 @ ( zero_zero @ A ) ) )
| ( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
& ( ord_less @ A @ ( zero_zero @ A ) @ B2 ) ) ) ) ) ).
% mult_less_0_iff
thf(fact_1074_not__square__less__zero,axiom,
! [A: $tType] :
( ( linordered_ring @ A )
=> ! [A2: A] :
~ ( ord_less @ A @ ( times_times @ A @ A2 @ A2 ) @ ( zero_zero @ A ) ) ) ).
% not_square_less_zero
thf(fact_1075_mult__neg__neg,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B2 ) ) ) ) ) ).
% mult_neg_neg
thf(fact_1076_add__scale__eq__noteq,axiom,
! [A: $tType] :
( ( semiri1453513574482234551roduct @ A )
=> ! [R2: A,A2: A,B2: A,C3: A,D3: A] :
( ( R2
!= ( zero_zero @ A ) )
=> ( ( ( A2 = B2 )
& ( C3 != D3 ) )
=> ( ( plus_plus @ A @ A2 @ ( times_times @ A @ R2 @ C3 ) )
!= ( plus_plus @ A @ B2 @ ( times_times @ A @ R2 @ D3 ) ) ) ) ) ) ).
% add_scale_eq_noteq
thf(fact_1077_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_1078_frac__eq__eq,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [Y: A,Z: A,X: A,W2: A] :
( ( Y
!= ( zero_zero @ A ) )
=> ( ( Z
!= ( zero_zero @ A ) )
=> ( ( ( divide_divide @ A @ X @ Y )
= ( divide_divide @ A @ W2 @ Z ) )
= ( ( times_times @ A @ X @ Z )
= ( times_times @ A @ W2 @ Y ) ) ) ) ) ) ).
% frac_eq_eq
thf(fact_1079_divide__eq__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B2: A,C3: A,A2: A] :
( ( ( divide_divide @ A @ B2 @ C3 )
= A2 )
= ( ( ( C3
!= ( zero_zero @ A ) )
=> ( B2
= ( times_times @ A @ A2 @ C3 ) ) )
& ( ( C3
= ( zero_zero @ A ) )
=> ( A2
= ( zero_zero @ A ) ) ) ) ) ) ).
% divide_eq_eq
thf(fact_1080_eq__divide__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( A2
= ( divide_divide @ A @ B2 @ C3 ) )
= ( ( ( C3
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ A2 @ C3 )
= B2 ) )
& ( ( C3
= ( zero_zero @ A ) )
=> ( A2
= ( zero_zero @ A ) ) ) ) ) ) ).
% eq_divide_eq
thf(fact_1081_divide__eq__imp,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [C3: A,B2: A,A2: A] :
( ( C3
!= ( zero_zero @ A ) )
=> ( ( B2
= ( times_times @ A @ A2 @ C3 ) )
=> ( ( divide_divide @ A @ B2 @ C3 )
= A2 ) ) ) ) ).
% divide_eq_imp
thf(fact_1082_eq__divide__imp,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [C3: A,A2: A,B2: A] :
( ( C3
!= ( zero_zero @ A ) )
=> ( ( ( times_times @ A @ A2 @ C3 )
= B2 )
=> ( A2
= ( divide_divide @ A @ B2 @ C3 ) ) ) ) ) ).
% eq_divide_imp
thf(fact_1083_nonzero__divide__eq__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [C3: A,B2: A,A2: A] :
( ( C3
!= ( zero_zero @ A ) )
=> ( ( ( divide_divide @ A @ B2 @ C3 )
= A2 )
= ( B2
= ( times_times @ A @ A2 @ C3 ) ) ) ) ) ).
% nonzero_divide_eq_eq
thf(fact_1084_nonzero__eq__divide__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [C3: A,A2: A,B2: A] :
( ( C3
!= ( zero_zero @ A ) )
=> ( ( A2
= ( divide_divide @ A @ B2 @ C3 ) )
= ( ( times_times @ A @ A2 @ C3 )
= B2 ) ) ) ) ).
% nonzero_eq_divide_eq
thf(fact_1085_Suc__mult__less__cancel1,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( ord_less @ nat @ ( times_times @ nat @ ( suc @ K2 ) @ M2 ) @ ( times_times @ nat @ ( suc @ K2 ) @ N ) )
= ( ord_less @ nat @ M2 @ N ) ) ).
% Suc_mult_less_cancel1
thf(fact_1086_power__add,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [A2: A,M2: nat,N: nat] :
( ( power_power @ A @ A2 @ ( plus_plus @ nat @ M2 @ N ) )
= ( times_times @ A @ ( power_power @ A @ A2 @ M2 ) @ ( power_power @ A @ A2 @ N ) ) ) ) ).
% power_add
thf(fact_1087_mult__less__mono2,axiom,
! [I2: nat,J: nat,K2: nat] :
( ( ord_less @ nat @ I2 @ J )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K2 )
=> ( ord_less @ nat @ ( times_times @ nat @ K2 @ I2 ) @ ( times_times @ nat @ K2 @ J ) ) ) ) ).
% mult_less_mono2
thf(fact_1088_mult__less__mono1,axiom,
! [I2: nat,J: nat,K2: nat] :
( ( ord_less @ nat @ I2 @ J )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K2 )
=> ( ord_less @ nat @ ( times_times @ nat @ I2 @ K2 ) @ ( times_times @ nat @ J @ K2 ) ) ) ) ).
% mult_less_mono1
thf(fact_1089_nat__mult__less__cancel1,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K2 )
=> ( ( ord_less @ nat @ ( times_times @ nat @ K2 @ M2 ) @ ( times_times @ nat @ K2 @ N ) )
= ( ord_less @ nat @ M2 @ N ) ) ) ).
% nat_mult_less_cancel1
thf(fact_1090_nat__mult__eq__cancel1,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K2 )
=> ( ( ( times_times @ nat @ K2 @ M2 )
= ( times_times @ nat @ K2 @ N ) )
= ( M2 = N ) ) ) ).
% nat_mult_eq_cancel1
thf(fact_1091_Suc__mult__le__cancel1,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( ord_less_eq @ nat @ ( times_times @ nat @ ( suc @ K2 ) @ M2 ) @ ( times_times @ nat @ ( suc @ K2 ) @ N ) )
= ( ord_less_eq @ nat @ M2 @ N ) ) ).
% Suc_mult_le_cancel1
thf(fact_1092_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_1093_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_1094_less__mult__imp__div__less,axiom,
! [M2: nat,I2: nat,N: nat] :
( ( ord_less @ nat @ M2 @ ( times_times @ nat @ I2 @ N ) )
=> ( ord_less @ nat @ ( divide_divide @ nat @ M2 @ N ) @ I2 ) ) ).
% less_mult_imp_div_less
thf(fact_1095_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_1096_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_1097_mult__less__le__imp__less,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A2: A,B2: A,C3: A,D3: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C3 @ D3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ ( times_times @ A @ A2 @ C3 ) @ ( times_times @ A @ B2 @ D3 ) ) ) ) ) ) ) ).
% mult_less_le_imp_less
thf(fact_1098_mult__le__less__imp__less,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A2: A,B2: A,C3: A,D3: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ C3 @ D3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ ( times_times @ A @ A2 @ C3 ) @ ( times_times @ A @ B2 @ D3 ) ) ) ) ) ) ) ).
% mult_le_less_imp_less
thf(fact_1099_mult__right__le__imp__le,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A2: A,C3: A,B2: A] :
( ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C3 ) @ ( times_times @ A @ B2 @ C3 ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ A2 @ B2 ) ) ) ) ).
% mult_right_le_imp_le
thf(fact_1100_mult__left__le__imp__le,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [C3: A,A2: A,B2: A] :
( ( ord_less_eq @ A @ ( times_times @ A @ C3 @ A2 ) @ ( times_times @ A @ C3 @ B2 ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ A2 @ B2 ) ) ) ) ).
% mult_left_le_imp_le
thf(fact_1101_mult__le__cancel__left__pos,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [C3: A,A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ord_less_eq @ A @ ( times_times @ A @ C3 @ A2 ) @ ( times_times @ A @ C3 @ B2 ) )
= ( ord_less_eq @ A @ A2 @ B2 ) ) ) ) ).
% mult_le_cancel_left_pos
thf(fact_1102_mult__le__cancel__left__neg,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [C3: A,A2: A,B2: A] :
( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( times_times @ A @ C3 @ A2 ) @ ( times_times @ A @ C3 @ B2 ) )
= ( ord_less_eq @ A @ B2 @ A2 ) ) ) ) ).
% mult_le_cancel_left_neg
thf(fact_1103_mult__less__cancel__right,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [A2: A,C3: A,B2: A] :
( ( ord_less @ A @ ( times_times @ A @ A2 @ C3 ) @ ( times_times @ A @ B2 @ C3 ) )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ A2 @ B2 ) )
& ( ( ord_less_eq @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B2 @ A2 ) ) ) ) ) ).
% mult_less_cancel_right
thf(fact_1104_mult__strict__mono_H,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A2: A,B2: A,C3: A,D3: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ C3 @ D3 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ ( times_times @ A @ A2 @ C3 ) @ ( times_times @ A @ B2 @ D3 ) ) ) ) ) ) ) ).
% mult_strict_mono'
thf(fact_1105_mult__right__less__imp__less,axiom,
! [A: $tType] :
( ( linordered_semiring @ A )
=> ! [A2: A,C3: A,B2: A] :
( ( ord_less @ A @ ( times_times @ A @ A2 @ C3 ) @ ( times_times @ A @ B2 @ C3 ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ A2 @ B2 ) ) ) ) ).
% mult_right_less_imp_less
thf(fact_1106_mult__less__cancel__left,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [C3: A,A2: A,B2: A] :
( ( ord_less @ A @ ( times_times @ A @ C3 @ A2 ) @ ( times_times @ A @ C3 @ B2 ) )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ A2 @ B2 ) )
& ( ( ord_less_eq @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B2 @ A2 ) ) ) ) ) ).
% mult_less_cancel_left
thf(fact_1107_mult__strict__mono,axiom,
! [A: $tType] :
( ( linord8928482502909563296strict @ A )
=> ! [A2: A,B2: A,C3: A,D3: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ C3 @ D3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ ( times_times @ A @ A2 @ C3 ) @ ( times_times @ A @ B2 @ D3 ) ) ) ) ) ) ) ).
% mult_strict_mono
thf(fact_1108_mult__left__less__imp__less,axiom,
! [A: $tType] :
( ( linordered_semiring @ A )
=> ! [C3: A,A2: A,B2: A] :
( ( ord_less @ A @ ( times_times @ A @ C3 @ A2 ) @ ( times_times @ A @ C3 @ B2 ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ A2 @ B2 ) ) ) ) ).
% mult_left_less_imp_less
thf(fact_1109_mult__le__cancel__right,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [A2: A,C3: A,B2: A] :
( ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C3 ) @ ( times_times @ A @ B2 @ C3 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ A2 @ B2 ) )
& ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B2 @ A2 ) ) ) ) ) ).
% mult_le_cancel_right
thf(fact_1110_mult__le__cancel__left,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [C3: A,A2: A,B2: A] :
( ( ord_less_eq @ A @ ( times_times @ A @ C3 @ A2 ) @ ( times_times @ A @ C3 @ B2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ A2 @ B2 ) )
& ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B2 @ A2 ) ) ) ) ) ).
% mult_le_cancel_left
thf(fact_1111_mult__left__le,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [C3: A,A2: A] :
( ( ord_less_eq @ A @ C3 @ ( one_one @ A ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C3 ) @ A2 ) ) ) ) ).
% mult_left_le
thf(fact_1112_mult__le__one,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ ( one_one @ A ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ A2 @ B2 ) @ ( one_one @ A ) ) ) ) ) ) ).
% mult_le_one
thf(fact_1113_mult__right__le__one__le,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
=> ( ( ord_less_eq @ A @ Y @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ X @ Y ) @ X ) ) ) ) ) ).
% mult_right_le_one_le
thf(fact_1114_mult__left__le__one__le,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
=> ( ( ord_less_eq @ A @ Y @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ Y @ X ) @ X ) ) ) ) ) ).
% mult_left_le_one_le
thf(fact_1115_sum__squares__le__zero__iff,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ ( times_times @ A @ X @ X ) @ ( times_times @ A @ Y @ Y ) ) @ ( zero_zero @ A ) )
= ( ( X
= ( zero_zero @ A ) )
& ( Y
= ( zero_zero @ A ) ) ) ) ) ).
% sum_squares_le_zero_iff
thf(fact_1116_sum__squares__ge__zero,axiom,
! [A: $tType] :
( ( linordered_ring @ A )
=> ! [X: A,Y: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ ( times_times @ A @ X @ X ) @ ( times_times @ A @ Y @ Y ) ) ) ) ).
% sum_squares_ge_zero
thf(fact_1117_sum__squares__gt__zero__iff,axiom,
! [A: $tType] :
( ( linord4710134922213307826strict @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( plus_plus @ A @ ( times_times @ A @ X @ X ) @ ( times_times @ A @ Y @ Y ) ) )
= ( ( X
!= ( zero_zero @ A ) )
| ( Y
!= ( zero_zero @ A ) ) ) ) ) ).
% sum_squares_gt_zero_iff
thf(fact_1118_not__sum__squares__lt__zero,axiom,
! [A: $tType] :
( ( linordered_ring @ A )
=> ! [X: A,Y: A] :
~ ( ord_less @ A @ ( plus_plus @ A @ ( times_times @ A @ X @ X ) @ ( times_times @ A @ Y @ Y ) ) @ ( zero_zero @ A ) ) ) ).
% not_sum_squares_lt_zero
thf(fact_1119_divide__strict__left__mono__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B2 ) )
=> ( ord_less @ A @ ( divide_divide @ A @ C3 @ A2 ) @ ( divide_divide @ A @ C3 @ B2 ) ) ) ) ) ) ).
% divide_strict_left_mono_neg
thf(fact_1120_divide__strict__left__mono,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,A2: A,C3: A] :
( ( ord_less @ A @ B2 @ A2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B2 ) )
=> ( ord_less @ A @ ( divide_divide @ A @ C3 @ A2 ) @ ( divide_divide @ A @ C3 @ B2 ) ) ) ) ) ) ).
% divide_strict_left_mono
thf(fact_1121_mult__imp__less__div__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y: A,Z: A,X: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Y )
=> ( ( ord_less @ A @ ( times_times @ A @ Z @ Y ) @ X )
=> ( ord_less @ A @ Z @ ( divide_divide @ A @ X @ Y ) ) ) ) ) ).
% mult_imp_less_div_pos
thf(fact_1122_mult__imp__div__pos__less,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y: A,X: A,Z: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Y )
=> ( ( ord_less @ A @ X @ ( times_times @ A @ Z @ Y ) )
=> ( ord_less @ A @ ( divide_divide @ A @ X @ Y ) @ Z ) ) ) ) ).
% mult_imp_div_pos_less
thf(fact_1123_pos__less__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C3: A,A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ord_less @ A @ A2 @ ( divide_divide @ A @ B2 @ C3 ) )
= ( ord_less @ A @ ( times_times @ A @ A2 @ C3 ) @ B2 ) ) ) ) ).
% pos_less_divide_eq
thf(fact_1124_pos__divide__less__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C3: A,B2: A,A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ord_less @ A @ ( divide_divide @ A @ B2 @ C3 ) @ A2 )
= ( ord_less @ A @ B2 @ ( times_times @ A @ A2 @ C3 ) ) ) ) ) ).
% pos_divide_less_eq
thf(fact_1125_neg__less__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C3: A,A2: A,B2: A] :
( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ A2 @ ( divide_divide @ A @ B2 @ C3 ) )
= ( ord_less @ A @ B2 @ ( times_times @ A @ A2 @ C3 ) ) ) ) ) ).
% neg_less_divide_eq
thf(fact_1126_neg__divide__less__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C3: A,B2: A,A2: A] :
( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( divide_divide @ A @ B2 @ C3 ) @ A2 )
= ( ord_less @ A @ ( times_times @ A @ A2 @ C3 ) @ B2 ) ) ) ) ).
% neg_divide_less_eq
thf(fact_1127_less__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( ord_less @ A @ A2 @ ( divide_divide @ A @ B2 @ C3 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ ( times_times @ A @ A2 @ C3 ) @ B2 ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B2 @ ( times_times @ A @ A2 @ C3 ) ) )
& ( ~ ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% less_divide_eq
thf(fact_1128_divide__less__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,C3: A,A2: A] :
( ( ord_less @ A @ ( divide_divide @ A @ B2 @ C3 ) @ A2 )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ B2 @ ( times_times @ A @ A2 @ C3 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ A2 @ C3 ) @ B2 ) )
& ( ~ ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ) ) ) ) ).
% divide_less_eq
thf(fact_1129_divide__eq__eq__numeral_I1_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B2: A,C3: A,W2: num] :
( ( ( divide_divide @ A @ B2 @ C3 )
= ( numeral_numeral @ A @ W2 ) )
= ( ( ( C3
!= ( zero_zero @ A ) )
=> ( B2
= ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C3 ) ) )
& ( ( C3
= ( zero_zero @ A ) )
=> ( ( numeral_numeral @ A @ W2 )
= ( zero_zero @ A ) ) ) ) ) ) ).
% divide_eq_eq_numeral(1)
thf(fact_1130_eq__divide__eq__numeral_I1_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [W2: num,B2: A,C3: A] :
( ( ( numeral_numeral @ A @ W2 )
= ( divide_divide @ A @ B2 @ C3 ) )
= ( ( ( C3
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C3 )
= B2 ) )
& ( ( C3
= ( zero_zero @ A ) )
=> ( ( numeral_numeral @ A @ W2 )
= ( zero_zero @ A ) ) ) ) ) ) ).
% eq_divide_eq_numeral(1)
thf(fact_1131_divide__add__eq__iff,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z: A,X: A,Y: A] :
( ( Z
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( divide_divide @ A @ X @ Z ) @ Y )
= ( divide_divide @ A @ ( plus_plus @ A @ X @ ( times_times @ A @ Y @ Z ) ) @ Z ) ) ) ) ).
% divide_add_eq_iff
thf(fact_1132_add__divide__eq__iff,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z: A,X: A,Y: A] :
( ( Z
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ X @ ( divide_divide @ A @ Y @ Z ) )
= ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ X @ Z ) @ Y ) @ Z ) ) ) ) ).
% add_divide_eq_iff
thf(fact_1133_add__num__frac,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [Y: A,Z: A,X: A] :
( ( Y
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ Z @ ( divide_divide @ A @ X @ Y ) )
= ( divide_divide @ A @ ( plus_plus @ A @ X @ ( times_times @ A @ Z @ Y ) ) @ Y ) ) ) ) ).
% add_num_frac
thf(fact_1134_add__frac__num,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [Y: A,X: A,Z: A] :
( ( Y
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( divide_divide @ A @ X @ Y ) @ Z )
= ( divide_divide @ A @ ( plus_plus @ A @ X @ ( times_times @ A @ Z @ Y ) ) @ Y ) ) ) ) ).
% add_frac_num
thf(fact_1135_add__frac__eq,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [Y: A,Z: A,X: A,W2: A] :
( ( Y
!= ( zero_zero @ A ) )
=> ( ( Z
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( divide_divide @ A @ X @ Y ) @ ( divide_divide @ A @ W2 @ Z ) )
= ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ X @ Z ) @ ( times_times @ A @ W2 @ Y ) ) @ ( times_times @ A @ Y @ Z ) ) ) ) ) ) ).
% add_frac_eq
thf(fact_1136_add__divide__eq__if__simps_I1_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z: A,A2: A,B2: A] :
( ( ( Z
= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ A2 @ ( divide_divide @ A @ B2 @ Z ) )
= A2 ) )
& ( ( Z
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ A2 @ ( divide_divide @ A @ B2 @ Z ) )
= ( divide_divide @ A @ ( plus_plus @ A @ ( times_times @ A @ A2 @ Z ) @ B2 ) @ Z ) ) ) ) ) ).
% add_divide_eq_if_simps(1)
thf(fact_1137_add__divide__eq__if__simps_I2_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z: A,A2: A,B2: A] :
( ( ( Z
= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( divide_divide @ A @ A2 @ Z ) @ B2 )
= B2 ) )
& ( ( Z
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( divide_divide @ A @ A2 @ Z ) @ B2 )
= ( divide_divide @ A @ ( plus_plus @ A @ A2 @ ( times_times @ A @ B2 @ Z ) ) @ Z ) ) ) ) ) ).
% add_divide_eq_if_simps(2)
thf(fact_1138_power__less__power__Suc,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,N: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ A2 )
=> ( ord_less @ A @ ( power_power @ A @ A2 @ N ) @ ( times_times @ A @ A2 @ ( power_power @ A @ A2 @ N ) ) ) ) ) ).
% power_less_power_Suc
thf(fact_1139_power__gt1__lemma,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,N: nat] :
( ( ord_less @ A @ ( one_one @ A ) @ A2 )
=> ( ord_less @ A @ ( one_one @ A ) @ ( times_times @ A @ A2 @ ( power_power @ A @ A2 @ N ) ) ) ) ) ).
% power_gt1_lemma
thf(fact_1140_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_1141_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_1142_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_1143_nat__mult__le__cancel1,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K2 )
=> ( ( ord_less_eq @ nat @ ( times_times @ nat @ K2 @ M2 ) @ ( times_times @ nat @ K2 @ N ) )
= ( ord_less_eq @ nat @ M2 @ N ) ) ) ).
% nat_mult_le_cancel1
thf(fact_1144_div__less__iff__less__mult,axiom,
! [Q2: nat,M2: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ Q2 )
=> ( ( ord_less @ nat @ ( divide_divide @ nat @ M2 @ Q2 ) @ N )
= ( ord_less @ nat @ M2 @ ( times_times @ nat @ N @ Q2 ) ) ) ) ).
% div_less_iff_less_mult
thf(fact_1145_nat__mult__div__cancel1,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K2 )
=> ( ( divide_divide @ nat @ ( times_times @ nat @ K2 @ M2 ) @ ( times_times @ nat @ K2 @ N ) )
= ( divide_divide @ nat @ M2 @ N ) ) ) ).
% nat_mult_div_cancel1
thf(fact_1146_realpow__pos__nth2,axiom,
! [A2: real,N: nat] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
=> ? [R3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R3 )
& ( ( power_power @ real @ R3 @ ( suc @ N ) )
= A2 ) ) ) ).
% realpow_pos_nth2
thf(fact_1147_field__le__mult__one__interval,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y: A] :
( ! [Z3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Z3 )
=> ( ( ord_less @ A @ Z3 @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ Z3 @ X ) @ Y ) ) )
=> ( ord_less_eq @ A @ X @ Y ) ) ) ).
% field_le_mult_one_interval
thf(fact_1148_mult__less__cancel__right2,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A,C3: A] :
( ( ord_less @ A @ ( times_times @ A @ A2 @ C3 ) @ C3 )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ A2 @ ( one_one @ A ) ) )
& ( ( ord_less_eq @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( one_one @ A ) @ A2 ) ) ) ) ) ).
% mult_less_cancel_right2
thf(fact_1149_mult__less__cancel__right1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [C3: A,B2: A] :
( ( ord_less @ A @ C3 @ ( times_times @ A @ B2 @ C3 ) )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ ( one_one @ A ) @ B2 ) )
& ( ( ord_less_eq @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B2 @ ( one_one @ A ) ) ) ) ) ) ).
% mult_less_cancel_right1
thf(fact_1150_mult__less__cancel__left2,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [C3: A,A2: A] :
( ( ord_less @ A @ ( times_times @ A @ C3 @ A2 ) @ C3 )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ A2 @ ( one_one @ A ) ) )
& ( ( ord_less_eq @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( one_one @ A ) @ A2 ) ) ) ) ) ).
% mult_less_cancel_left2
thf(fact_1151_mult__less__cancel__left1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [C3: A,B2: A] :
( ( ord_less @ A @ C3 @ ( times_times @ A @ C3 @ B2 ) )
= ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ ( one_one @ A ) @ B2 ) )
& ( ( ord_less_eq @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B2 @ ( one_one @ A ) ) ) ) ) ) ).
% mult_less_cancel_left1
thf(fact_1152_mult__le__cancel__right2,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A,C3: A] :
( ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C3 ) @ C3 )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ A2 @ ( one_one @ A ) ) )
& ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ A2 ) ) ) ) ) ).
% mult_le_cancel_right2
thf(fact_1153_mult__le__cancel__right1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [C3: A,B2: A] :
( ( ord_less_eq @ A @ C3 @ ( times_times @ A @ B2 @ C3 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ B2 ) )
& ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B2 @ ( one_one @ A ) ) ) ) ) ) ).
% mult_le_cancel_right1
thf(fact_1154_mult__le__cancel__left2,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [C3: A,A2: A] :
( ( ord_less_eq @ A @ ( times_times @ A @ C3 @ A2 ) @ C3 )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ A2 @ ( one_one @ A ) ) )
& ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ A2 ) ) ) ) ) ).
% mult_le_cancel_left2
thf(fact_1155_mult__le__cancel__left1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [C3: A,B2: A] :
( ( ord_less_eq @ A @ C3 @ ( times_times @ A @ C3 @ B2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ B2 ) )
& ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B2 @ ( one_one @ A ) ) ) ) ) ) ).
% mult_le_cancel_left1
thf(fact_1156_divide__left__mono__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C3 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B2 ) )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ C3 @ A2 ) @ ( divide_divide @ A @ C3 @ B2 ) ) ) ) ) ) ).
% divide_left_mono_neg
thf(fact_1157_mult__imp__le__div__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y: A,Z: A,X: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Y )
=> ( ( ord_less_eq @ A @ ( times_times @ A @ Z @ Y ) @ X )
=> ( ord_less_eq @ A @ Z @ ( divide_divide @ A @ X @ Y ) ) ) ) ) ).
% mult_imp_le_div_pos
thf(fact_1158_mult__imp__div__pos__le,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y: A,X: A,Z: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Y )
=> ( ( ord_less_eq @ A @ X @ ( times_times @ A @ Z @ Y ) )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ X @ Y ) @ Z ) ) ) ) ).
% mult_imp_div_pos_le
thf(fact_1159_pos__le__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C3: A,A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ord_less_eq @ A @ A2 @ ( divide_divide @ A @ B2 @ C3 ) )
= ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C3 ) @ B2 ) ) ) ) ).
% pos_le_divide_eq
thf(fact_1160_pos__divide__le__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C3: A,B2: A,A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ C3 ) @ A2 )
= ( ord_less_eq @ A @ B2 @ ( times_times @ A @ A2 @ C3 ) ) ) ) ) ).
% pos_divide_le_eq
thf(fact_1161_neg__le__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C3: A,A2: A,B2: A] :
( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ A2 @ ( divide_divide @ A @ B2 @ C3 ) )
= ( ord_less_eq @ A @ B2 @ ( times_times @ A @ A2 @ C3 ) ) ) ) ) ).
% neg_le_divide_eq
thf(fact_1162_neg__divide__le__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C3: A,B2: A,A2: A] :
( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ C3 ) @ A2 )
= ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C3 ) @ B2 ) ) ) ) ).
% neg_divide_le_eq
thf(fact_1163_divide__left__mono,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,A2: A,C3: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B2 ) )
=> ( ord_less_eq @ A @ ( divide_divide @ A @ C3 @ A2 ) @ ( divide_divide @ A @ C3 @ B2 ) ) ) ) ) ) ).
% divide_left_mono
thf(fact_1164_le__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( ord_less_eq @ A @ A2 @ ( divide_divide @ A @ B2 @ C3 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C3 ) @ B2 ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B2 @ ( times_times @ A @ A2 @ C3 ) ) )
& ( ~ ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% le_divide_eq
thf(fact_1165_divide__le__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,C3: A,A2: A] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ C3 ) @ A2 )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ B2 @ ( times_times @ A @ A2 @ C3 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C3 ) @ B2 ) )
& ( ~ ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ) ) ) ) ).
% divide_le_eq
thf(fact_1166_convex__bound__le,axiom,
! [A: $tType] :
( ( linord6961819062388156250ring_1 @ A )
=> ! [X: A,A2: A,Y: A,U: A,V: A] :
( ( ord_less_eq @ A @ X @ A2 )
=> ( ( ord_less_eq @ A @ Y @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ U )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ V )
=> ( ( ( plus_plus @ A @ U @ V )
= ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ ( times_times @ A @ U @ X ) @ ( times_times @ A @ V @ Y ) ) @ A2 ) ) ) ) ) ) ) ).
% convex_bound_le
thf(fact_1167_less__divide__eq__numeral_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [W2: num,B2: A,C3: A] :
( ( ord_less @ A @ ( numeral_numeral @ A @ W2 ) @ ( divide_divide @ A @ B2 @ C3 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C3 ) @ B2 ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B2 @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C3 ) ) )
& ( ~ ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( numeral_numeral @ A @ W2 ) @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% less_divide_eq_numeral(1)
thf(fact_1168_divide__less__eq__numeral_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,C3: A,W2: num] :
( ( ord_less @ A @ ( divide_divide @ A @ B2 @ C3 ) @ ( numeral_numeral @ A @ W2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ B2 @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C3 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C3 ) @ B2 ) )
& ( ~ ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ W2 ) ) ) ) ) ) ) ) ).
% divide_less_eq_numeral(1)
thf(fact_1169_power__Suc__less,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,N: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ A2 @ ( one_one @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ A2 @ ( power_power @ A @ A2 @ N ) ) @ ( power_power @ A @ A2 @ N ) ) ) ) ) ).
% power_Suc_less
thf(fact_1170_mult__2,axiom,
! [A: $tType] :
( ( semiring_numeral @ A )
=> ! [Z: A] :
( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Z )
= ( plus_plus @ A @ Z @ Z ) ) ) ).
% mult_2
thf(fact_1171_mult__2__right,axiom,
! [A: $tType] :
( ( semiring_numeral @ A )
=> ! [Z: A] :
( ( times_times @ A @ Z @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( plus_plus @ A @ Z @ Z ) ) ) ).
% mult_2_right
thf(fact_1172_left__add__twice,axiom,
! [A: $tType] :
( ( semiring_numeral @ A )
=> ! [A2: A,B2: A] :
( ( plus_plus @ A @ A2 @ ( plus_plus @ A @ A2 @ B2 ) )
= ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) @ B2 ) ) ) ).
% left_add_twice
thf(fact_1173_div__nat__eqI,axiom,
! [N: nat,Q2: nat,M2: nat] :
( ( ord_less_eq @ nat @ ( times_times @ nat @ N @ Q2 ) @ M2 )
=> ( ( ord_less @ nat @ M2 @ ( times_times @ nat @ N @ ( suc @ Q2 ) ) )
=> ( ( divide_divide @ nat @ M2 @ N )
= Q2 ) ) ) ).
% div_nat_eqI
thf(fact_1174_less__eq__div__iff__mult__less__eq,axiom,
! [Q2: nat,M2: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ Q2 )
=> ( ( ord_less_eq @ nat @ M2 @ ( divide_divide @ nat @ N @ Q2 ) )
= ( ord_less_eq @ nat @ ( times_times @ nat @ M2 @ Q2 ) @ N ) ) ) ).
% less_eq_div_iff_mult_less_eq
thf(fact_1175_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_1176_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_1177_split__div,axiom,
! [P2: nat > $o,M2: nat,N: nat] :
( ( P2 @ ( divide_divide @ nat @ M2 @ N ) )
= ( ( ( N
= ( zero_zero @ nat ) )
=> ( P2 @ ( zero_zero @ nat ) ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ! [I4: nat,J3: nat] :
( ( ord_less @ nat @ J3 @ N )
=> ( ( M2
= ( plus_plus @ nat @ ( times_times @ nat @ N @ I4 ) @ J3 ) )
=> ( P2 @ I4 ) ) ) ) ) ) ).
% split_div
thf(fact_1178_convex__bound__lt,axiom,
! [A: $tType] :
( ( linord715952674999750819strict @ A )
=> ! [X: A,A2: A,Y: A,U: A,V: A] :
( ( ord_less @ A @ X @ A2 )
=> ( ( ord_less @ A @ Y @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ U )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ V )
=> ( ( ( plus_plus @ A @ U @ V )
= ( one_one @ A ) )
=> ( ord_less @ A @ ( plus_plus @ A @ ( times_times @ A @ U @ X ) @ ( times_times @ A @ V @ Y ) ) @ A2 ) ) ) ) ) ) ) ).
% convex_bound_lt
thf(fact_1179_le__divide__eq__numeral_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [W2: num,B2: A,C3: A] :
( ( ord_less_eq @ A @ ( numeral_numeral @ A @ W2 ) @ ( divide_divide @ A @ B2 @ C3 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C3 ) @ B2 ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B2 @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C3 ) ) )
& ( ~ ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( numeral_numeral @ A @ W2 ) @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% le_divide_eq_numeral(1)
thf(fact_1180_divide__le__eq__numeral_I1_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,C3: A,W2: num] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ C3 ) @ ( numeral_numeral @ A @ W2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ B2 @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C3 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ C3 ) @ B2 ) )
& ( ~ ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ W2 ) ) ) ) ) ) ) ) ).
% divide_le_eq_numeral(1)
thf(fact_1181_sum__squares__bound,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y: A] : ( ord_less_eq @ A @ ( times_times @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X ) @ Y ) @ ( plus_plus @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% sum_squares_bound
thf(fact_1182_split__div_H,axiom,
! [P2: nat > $o,M2: nat,N: nat] :
( ( P2 @ ( divide_divide @ nat @ M2 @ N ) )
= ( ( ( N
= ( zero_zero @ nat ) )
& ( P2 @ ( zero_zero @ nat ) ) )
| ? [Q5: nat] :
( ( ord_less_eq @ nat @ ( times_times @ nat @ N @ Q5 ) @ M2 )
& ( ord_less @ nat @ M2 @ ( times_times @ nat @ N @ ( suc @ Q5 ) ) )
& ( P2 @ Q5 ) ) ) ) ).
% split_div'
thf(fact_1183_power2__sum,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [X: A,Y: A] :
( ( power_power @ A @ ( plus_plus @ A @ X @ Y ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( plus_plus @ A @ ( plus_plus @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( times_times @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X ) @ Y ) ) ) ) ).
% power2_sum
thf(fact_1184_zero__le__even__power_H,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A,N: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A2 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% zero_le_even_power'
thf(fact_1185_nat__bit__induct,axiom,
! [P2: nat > $o,N: nat] :
( ( P2 @ ( zero_zero @ nat ) )
=> ( ! [N3: nat] :
( ( P2 @ N3 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( P2 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) ) ) )
=> ( ! [N3: nat] :
( ( P2 @ N3 )
=> ( P2 @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N3 ) ) ) )
=> ( P2 @ N ) ) ) ) ).
% nat_bit_induct
thf(fact_1186_arith__geo__mean,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [U: A,X: A,Y: A] :
( ( ( power_power @ A @ U @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( times_times @ A @ X @ Y ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
=> ( ord_less_eq @ A @ U @ ( divide_divide @ A @ ( plus_plus @ A @ X @ Y ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% arith_geo_mean
thf(fact_1187_triangle__def,axiom,
( nat_triangle
= ( ^ [N2: nat] : ( divide_divide @ nat @ ( times_times @ nat @ N2 @ ( suc @ N2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% triangle_def
thf(fact_1188_realpow__pos__nth__unique,axiom,
! [N: nat,A2: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
=> ? [X4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X4 )
& ( ( power_power @ real @ X4 @ N )
= A2 )
& ! [Y4: real] :
( ( ( ord_less @ real @ ( zero_zero @ real ) @ Y4 )
& ( ( power_power @ real @ Y4 @ N )
= A2 ) )
=> ( Y4 = X4 ) ) ) ) ) ).
% realpow_pos_nth_unique
thf(fact_1189_realpow__pos__nth,axiom,
! [N: nat,A2: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
=> ? [R3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R3 )
& ( ( power_power @ real @ R3 @ N )
= A2 ) ) ) ) ).
% realpow_pos_nth
thf(fact_1190_odd__0__le__power__imp__0__le,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A,N: nat] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A2 @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% odd_0_le_power_imp_0_le
thf(fact_1191_odd__power__less__zero,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A,N: nat] :
( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( power_power @ A @ A2 @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) @ ( zero_zero @ A ) ) ) ) ).
% odd_power_less_zero
thf(fact_1192_set__bit__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A] :
( ( bit_se5668285175392031749et_bit @ A @ ( zero_zero @ nat ) @ A2 )
= ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% set_bit_0
thf(fact_1193_unset__bit__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A] :
( ( bit_se2638667681897837118et_bit @ A @ ( zero_zero @ nat ) @ A2 )
= ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% unset_bit_0
thf(fact_1194_mult__le__cancel__iff2,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z: A,X: A,Y: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Z )
=> ( ( ord_less_eq @ A @ ( times_times @ A @ Z @ X ) @ ( times_times @ A @ Z @ Y ) )
= ( ord_less_eq @ A @ X @ Y ) ) ) ) ).
% mult_le_cancel_iff2
thf(fact_1195_mult__le__cancel__iff1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z: A,X: A,Y: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Z )
=> ( ( ord_less_eq @ A @ ( times_times @ A @ X @ Z ) @ ( times_times @ A @ Y @ Z ) )
= ( ord_less_eq @ A @ X @ Y ) ) ) ) ).
% mult_le_cancel_iff1
thf(fact_1196_divides__aux__eq,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [Q2: A,R2: A] :
( ( unique5940410009612947441es_aux @ A @ ( product_Pair @ A @ A @ Q2 @ R2 ) )
= ( R2
= ( zero_zero @ A ) ) ) ) ).
% divides_aux_eq
thf(fact_1197_low__def,axiom,
( vEBT_VEBT_low
= ( ^ [X5: nat,N2: nat] : ( modulo_modulo @ nat @ X5 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ).
% low_def
thf(fact_1198_even__succ__div__exp,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A2: A,N: nat] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ ( one_one @ A ) @ A2 ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
= ( divide_divide @ A @ A2 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) ) ) ) ) ).
% even_succ_div_exp
thf(fact_1199_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_1200_vebt__insert_Oelims,axiom,
! [X: vEBT_VEBT,Xa2: nat,Y: vEBT_VEBT] :
( ( ( vEBT_vebt_insert @ X @ Xa2 )
= Y )
=> ( ! [A4: $o,B4: $o] :
( ( X
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ~ ( ( ( Xa2
= ( zero_zero @ nat ) )
=> ( Y
= ( vEBT_Leaf @ $true @ B4 ) ) )
& ( ( Xa2
!= ( zero_zero @ nat ) )
=> ( ( ( Xa2
= ( one_one @ nat ) )
=> ( Y
= ( vEBT_Leaf @ A4 @ $true ) ) )
& ( ( Xa2
!= ( one_one @ nat ) )
=> ( Y
= ( vEBT_Leaf @ A4 @ B4 ) ) ) ) ) ) )
=> ( ! [Info2: option @ ( product_prod @ nat @ nat ),Ts2: list @ vEBT_VEBT,S3: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Info2 @ ( zero_zero @ nat ) @ Ts2 @ S3 ) )
=> ( Y
!= ( vEBT_Node @ Info2 @ ( zero_zero @ nat ) @ Ts2 @ S3 ) ) )
=> ( ! [Info2: option @ ( product_prod @ nat @ nat ),Ts2: list @ vEBT_VEBT,S3: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Info2 @ ( suc @ ( zero_zero @ nat ) ) @ Ts2 @ S3 ) )
=> ( Y
!= ( vEBT_Node @ Info2 @ ( suc @ ( zero_zero @ nat ) ) @ Ts2 @ S3 ) ) )
=> ( ! [V3: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ V3 ) ) @ TreeList2 @ Summary2 ) )
=> ( Y
!= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Xa2 @ Xa2 ) ) @ ( suc @ ( suc @ V3 ) ) @ TreeList2 @ Summary2 ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary2 ) )
=> ( Y
!= ( if @ vEBT_VEBT
@ ( ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
& ~ ( ( Xa2 = Mi2 )
| ( Xa2 = Ma2 ) ) )
@ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ ( if @ nat @ ( ord_less @ nat @ Xa2 @ Mi2 ) @ Xa2 @ Mi2 ) @ ( ord_max @ nat @ ( if @ nat @ ( ord_less @ nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ Ma2 ) ) ) @ ( suc @ ( suc @ Va2 ) ) @ ( list_update @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_insert @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( ord_less @ nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( if @ vEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ Summary2 ) )
@ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary2 ) ) ) ) ) ) ) ) ) ).
% vebt_insert.elims
thf(fact_1201_set__vebt_H__def,axiom,
( vEBT_VEBT_set_vebt
= ( ^ [T3: vEBT_VEBT] : ( collect @ nat @ ( vEBT_vebt_member @ T3 ) ) ) ) ).
% set_vebt'_def
thf(fact_1202_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_1203_finite__Collect__disjI,axiom,
! [A: $tType,P2: A > $o,Q: A > $o] :
( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X5: A] :
( ( P2 @ X5 )
| ( Q @ X5 ) ) ) )
= ( ( finite_finite2 @ A @ ( collect @ A @ P2 ) )
& ( finite_finite2 @ A @ ( collect @ A @ Q ) ) ) ) ).
% finite_Collect_disjI
thf(fact_1204_finite__Collect__conjI,axiom,
! [A: $tType,P2: A > $o,Q: A > $o] :
( ( ( finite_finite2 @ A @ ( collect @ A @ P2 ) )
| ( finite_finite2 @ A @ ( collect @ A @ Q ) ) )
=> ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X5: A] :
( ( P2 @ X5 )
& ( Q @ X5 ) ) ) ) ) ).
% finite_Collect_conjI
thf(fact_1205_finite__interval__int1,axiom,
! [A2: int,B2: int] :
( finite_finite2 @ int
@ ( collect @ int
@ ^ [I4: int] :
( ( ord_less_eq @ int @ A2 @ I4 )
& ( ord_less_eq @ int @ I4 @ B2 ) ) ) ) ).
% finite_interval_int1
thf(fact_1206_finite__interval__int4,axiom,
! [A2: int,B2: int] :
( finite_finite2 @ int
@ ( collect @ int
@ ^ [I4: int] :
( ( ord_less @ int @ A2 @ I4 )
& ( ord_less @ int @ I4 @ B2 ) ) ) ) ).
% finite_interval_int4
thf(fact_1207_dvd__0__right,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A2: A] : ( dvd_dvd @ A @ A2 @ ( zero_zero @ A ) ) ) ).
% dvd_0_right
thf(fact_1208_dvd__0__left__iff,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A2: A] :
( ( dvd_dvd @ A @ ( zero_zero @ A ) @ A2 )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% dvd_0_left_iff
thf(fact_1209_dvd__add__triv__left__iff,axiom,
! [A: $tType] :
( ( comm_s4317794764714335236cancel @ A )
=> ! [A2: A,B2: A] :
( ( dvd_dvd @ A @ A2 @ ( plus_plus @ A @ A2 @ B2 ) )
= ( dvd_dvd @ A @ A2 @ B2 ) ) ) ).
% dvd_add_triv_left_iff
thf(fact_1210_dvd__add__triv__right__iff,axiom,
! [A: $tType] :
( ( comm_s4317794764714335236cancel @ A )
=> ! [A2: A,B2: A] :
( ( dvd_dvd @ A @ A2 @ ( plus_plus @ A @ B2 @ A2 ) )
= ( dvd_dvd @ A @ A2 @ B2 ) ) ) ).
% dvd_add_triv_right_iff
thf(fact_1211_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_1212_dvd__1__left,axiom,
! [K2: nat] : ( dvd_dvd @ nat @ ( suc @ ( zero_zero @ nat ) ) @ K2 ) ).
% dvd_1_left
thf(fact_1213_bits__mod__0,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A2: A] :
( ( modulo_modulo @ A @ ( zero_zero @ A ) @ A2 )
= ( zero_zero @ A ) ) ) ).
% bits_mod_0
thf(fact_1214_mod__0,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [A2: A] :
( ( modulo_modulo @ A @ ( zero_zero @ A ) @ A2 )
= ( zero_zero @ A ) ) ) ).
% mod_0
thf(fact_1215_mod__by__0,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [A2: A] :
( ( modulo_modulo @ A @ A2 @ ( zero_zero @ A ) )
= A2 ) ) ).
% mod_by_0
thf(fact_1216_mod__self,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [A2: A] :
( ( modulo_modulo @ A @ A2 @ A2 )
= ( zero_zero @ A ) ) ) ).
% mod_self
thf(fact_1217_div__dvd__div,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( dvd_dvd @ A @ A2 @ B2 )
=> ( ( dvd_dvd @ A @ A2 @ C3 )
=> ( ( dvd_dvd @ A @ ( divide_divide @ A @ B2 @ A2 ) @ ( divide_divide @ A @ C3 @ A2 ) )
= ( dvd_dvd @ A @ B2 @ C3 ) ) ) ) ) ).
% div_dvd_div
thf(fact_1218_not__real__square__gt__zero,axiom,
! [X: real] :
( ( ~ ( ord_less @ real @ ( zero_zero @ real ) @ ( times_times @ real @ X @ X ) ) )
= ( X
= ( zero_zero @ real ) ) ) ).
% not_real_square_gt_zero
thf(fact_1219_mod__add__self1,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [B2: A,A2: A] :
( ( modulo_modulo @ A @ ( plus_plus @ A @ B2 @ A2 ) @ B2 )
= ( modulo_modulo @ A @ A2 @ B2 ) ) ) ).
% mod_add_self1
thf(fact_1220_mod__add__self2,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A2: A,B2: A] :
( ( modulo_modulo @ A @ ( plus_plus @ A @ A2 @ B2 ) @ B2 )
= ( modulo_modulo @ A @ A2 @ B2 ) ) ) ).
% mod_add_self2
thf(fact_1221_nat__mult__dvd__cancel__disj,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( dvd_dvd @ nat @ ( times_times @ nat @ K2 @ M2 ) @ ( times_times @ nat @ K2 @ N ) )
= ( ( K2
= ( zero_zero @ nat ) )
| ( dvd_dvd @ nat @ M2 @ N ) ) ) ).
% nat_mult_dvd_cancel_disj
thf(fact_1222_unset__bit__negative__int__iff,axiom,
! [N: nat,K2: int] :
( ( ord_less @ int @ ( bit_se2638667681897837118et_bit @ int @ N @ K2 ) @ ( zero_zero @ int ) )
= ( ord_less @ int @ K2 @ ( zero_zero @ int ) ) ) ).
% unset_bit_negative_int_iff
thf(fact_1223_set__bit__negative__int__iff,axiom,
! [N: nat,K2: int] :
( ( ord_less @ int @ ( bit_se5668285175392031749et_bit @ int @ N @ K2 ) @ ( zero_zero @ int ) )
= ( ord_less @ int @ K2 @ ( zero_zero @ int ) ) ) ).
% set_bit_negative_int_iff
thf(fact_1224_mod__less,axiom,
! [M2: nat,N: nat] :
( ( ord_less @ nat @ M2 @ N )
=> ( ( modulo_modulo @ nat @ M2 @ N )
= M2 ) ) ).
% mod_less
thf(fact_1225_finite__Collect__subsets,axiom,
! [A: $tType,A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( finite_finite2 @ ( set @ A )
@ ( collect @ ( set @ A )
@ ^ [B8: set @ A] : ( ord_less_eq @ ( set @ A ) @ B8 @ A5 ) ) ) ) ).
% finite_Collect_subsets
thf(fact_1226_finite__interval__int3,axiom,
! [A2: int,B2: int] :
( finite_finite2 @ int
@ ( collect @ int
@ ^ [I4: int] :
( ( ord_less @ int @ A2 @ I4 )
& ( ord_less_eq @ int @ I4 @ B2 ) ) ) ) ).
% finite_interval_int3
thf(fact_1227_finite__interval__int2,axiom,
! [A2: int,B2: int] :
( finite_finite2 @ int
@ ( collect @ int
@ ^ [I4: int] :
( ( ord_less_eq @ int @ A2 @ I4 )
& ( ord_less @ int @ I4 @ B2 ) ) ) ) ).
% finite_interval_int2
thf(fact_1228_finite__Collect__less__nat,axiom,
! [K2: nat] :
( finite_finite2 @ nat
@ ( collect @ nat
@ ^ [N2: nat] : ( ord_less @ nat @ N2 @ K2 ) ) ) ).
% finite_Collect_less_nat
thf(fact_1229_finite__Collect__le__nat,axiom,
! [K2: nat] :
( finite_finite2 @ nat
@ ( collect @ nat
@ ^ [N2: nat] : ( ord_less_eq @ nat @ N2 @ K2 ) ) ) ).
% finite_Collect_le_nat
thf(fact_1230_set__decode__inverse,axiom,
! [N: nat] :
( ( nat_set_encode @ ( nat_set_decode @ N ) )
= N ) ).
% set_decode_inverse
thf(fact_1231_dvd__times__right__cancel__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ ( times_times @ A @ B2 @ A2 ) @ ( times_times @ A @ C3 @ A2 ) )
= ( dvd_dvd @ A @ B2 @ C3 ) ) ) ) ).
% dvd_times_right_cancel_iff
thf(fact_1232_dvd__times__left__cancel__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ ( times_times @ A @ A2 @ B2 ) @ ( times_times @ A @ A2 @ C3 ) )
= ( dvd_dvd @ A @ B2 @ C3 ) ) ) ) ).
% dvd_times_left_cancel_iff
thf(fact_1233_dvd__mult__cancel__right,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [A2: A,C3: A,B2: A] :
( ( dvd_dvd @ A @ ( times_times @ A @ A2 @ C3 ) @ ( times_times @ A @ B2 @ C3 ) )
= ( ( C3
= ( zero_zero @ A ) )
| ( dvd_dvd @ A @ A2 @ B2 ) ) ) ) ).
% dvd_mult_cancel_right
thf(fact_1234_dvd__mult__cancel__left,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [C3: A,A2: A,B2: A] :
( ( dvd_dvd @ A @ ( times_times @ A @ C3 @ A2 ) @ ( times_times @ A @ C3 @ B2 ) )
= ( ( C3
= ( zero_zero @ A ) )
| ( dvd_dvd @ A @ A2 @ B2 ) ) ) ) ).
% dvd_mult_cancel_left
thf(fact_1235_unit__prod,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A,B2: A] :
( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( dvd_dvd @ A @ ( times_times @ A @ A2 @ B2 ) @ ( one_one @ A ) ) ) ) ) ).
% unit_prod
thf(fact_1236_dvd__add__times__triv__left__iff,axiom,
! [A: $tType] :
( ( comm_s4317794764714335236cancel @ A )
=> ! [A2: A,C3: A,B2: A] :
( ( dvd_dvd @ A @ A2 @ ( plus_plus @ A @ ( times_times @ A @ C3 @ A2 ) @ B2 ) )
= ( dvd_dvd @ A @ A2 @ B2 ) ) ) ).
% dvd_add_times_triv_left_iff
thf(fact_1237_dvd__add__times__triv__right__iff,axiom,
! [A: $tType] :
( ( comm_s4317794764714335236cancel @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( dvd_dvd @ A @ A2 @ ( plus_plus @ A @ B2 @ ( times_times @ A @ C3 @ A2 ) ) )
= ( dvd_dvd @ A @ A2 @ B2 ) ) ) ).
% dvd_add_times_triv_right_iff
thf(fact_1238_mod__mult__self2__is__0,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A2: A,B2: A] :
( ( modulo_modulo @ A @ ( times_times @ A @ A2 @ B2 ) @ B2 )
= ( zero_zero @ A ) ) ) ).
% mod_mult_self2_is_0
thf(fact_1239_mod__mult__self1__is__0,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [B2: A,A2: A] :
( ( modulo_modulo @ A @ ( times_times @ A @ B2 @ A2 ) @ B2 )
= ( zero_zero @ A ) ) ) ).
% mod_mult_self1_is_0
thf(fact_1240_bits__mod__by__1,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A2: A] :
( ( modulo_modulo @ A @ A2 @ ( one_one @ A ) )
= ( zero_zero @ A ) ) ) ).
% bits_mod_by_1
thf(fact_1241_mod__by__1,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [A2: A] :
( ( modulo_modulo @ A @ A2 @ ( one_one @ A ) )
= ( zero_zero @ A ) ) ) ).
% mod_by_1
thf(fact_1242_dvd__mult__div__cancel,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A,B2: A] :
( ( dvd_dvd @ A @ A2 @ B2 )
=> ( ( times_times @ A @ A2 @ ( divide_divide @ A @ B2 @ A2 ) )
= B2 ) ) ) ).
% dvd_mult_div_cancel
thf(fact_1243_dvd__div__mult__self,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A,B2: A] :
( ( dvd_dvd @ A @ A2 @ B2 )
=> ( ( times_times @ A @ ( divide_divide @ A @ B2 @ A2 ) @ A2 )
= B2 ) ) ) ).
% dvd_div_mult_self
thf(fact_1244_unit__div__1__div__1,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A] :
( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) )
=> ( ( divide_divide @ A @ ( one_one @ A ) @ ( divide_divide @ A @ ( one_one @ A ) @ A2 ) )
= A2 ) ) ) ).
% unit_div_1_div_1
thf(fact_1245_unit__div__1__unit,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A] :
( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) )
=> ( dvd_dvd @ A @ ( divide_divide @ A @ ( one_one @ A ) @ A2 ) @ ( one_one @ A ) ) ) ) ).
% unit_div_1_unit
thf(fact_1246_unit__div,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A,B2: A] :
( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( dvd_dvd @ A @ ( divide_divide @ A @ A2 @ B2 ) @ ( one_one @ A ) ) ) ) ) ).
% unit_div
thf(fact_1247_div__add,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [C3: A,A2: A,B2: A] :
( ( dvd_dvd @ A @ C3 @ A2 )
=> ( ( dvd_dvd @ A @ C3 @ B2 )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C3 )
= ( plus_plus @ A @ ( divide_divide @ A @ A2 @ C3 ) @ ( divide_divide @ A @ B2 @ C3 ) ) ) ) ) ) ).
% div_add
thf(fact_1248_bits__mod__div__trivial,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A2: A,B2: A] :
( ( divide_divide @ A @ ( modulo_modulo @ A @ A2 @ B2 ) @ B2 )
= ( zero_zero @ A ) ) ) ).
% bits_mod_div_trivial
thf(fact_1249_mod__div__trivial,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A2: A,B2: A] :
( ( divide_divide @ A @ ( modulo_modulo @ A @ A2 @ B2 ) @ B2 )
= ( zero_zero @ A ) ) ) ).
% mod_div_trivial
thf(fact_1250_mod__mult__self1,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A2: A,C3: A,B2: A] :
( ( modulo_modulo @ A @ ( plus_plus @ A @ A2 @ ( times_times @ A @ C3 @ B2 ) ) @ B2 )
= ( modulo_modulo @ A @ A2 @ B2 ) ) ) ).
% mod_mult_self1
thf(fact_1251_mod__mult__self2,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( modulo_modulo @ A @ ( plus_plus @ A @ A2 @ ( times_times @ A @ B2 @ C3 ) ) @ B2 )
= ( modulo_modulo @ A @ A2 @ B2 ) ) ) ).
% mod_mult_self2
thf(fact_1252_mod__mult__self3,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [C3: A,B2: A,A2: A] :
( ( modulo_modulo @ A @ ( plus_plus @ A @ ( times_times @ A @ C3 @ B2 ) @ A2 ) @ B2 )
= ( modulo_modulo @ A @ A2 @ B2 ) ) ) ).
% mod_mult_self3
thf(fact_1253_mod__mult__self4,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [B2: A,C3: A,A2: A] :
( ( modulo_modulo @ A @ ( plus_plus @ A @ ( times_times @ A @ B2 @ C3 ) @ A2 ) @ B2 )
= ( modulo_modulo @ A @ A2 @ B2 ) ) ) ).
% mod_mult_self4
thf(fact_1254_dvd__imp__mod__0,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [A2: A,B2: A] :
( ( dvd_dvd @ A @ A2 @ B2 )
=> ( ( modulo_modulo @ A @ B2 @ A2 )
= ( zero_zero @ A ) ) ) ) ).
% dvd_imp_mod_0
thf(fact_1255_mod__by__Suc__0,axiom,
! [M2: nat] :
( ( modulo_modulo @ nat @ M2 @ ( suc @ ( zero_zero @ nat ) ) )
= ( zero_zero @ nat ) ) ).
% mod_by_Suc_0
thf(fact_1256_set__decode__zero,axiom,
( ( nat_set_decode @ ( zero_zero @ nat ) )
= ( bot_bot @ ( set @ nat ) ) ) ).
% set_decode_zero
thf(fact_1257_set__encode__inverse,axiom,
! [A5: set @ nat] :
( ( finite_finite2 @ nat @ A5 )
=> ( ( nat_set_decode @ ( nat_set_encode @ A5 ) )
= A5 ) ) ).
% set_encode_inverse
thf(fact_1258_unit__mult__div__div,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A,B2: A] :
( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) )
=> ( ( times_times @ A @ B2 @ ( divide_divide @ A @ ( one_one @ A ) @ A2 ) )
= ( divide_divide @ A @ B2 @ A2 ) ) ) ) ).
% unit_mult_div_div
thf(fact_1259_unit__div__mult__self,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A,B2: A] :
( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) )
=> ( ( times_times @ A @ ( divide_divide @ A @ B2 @ A2 ) @ A2 )
= B2 ) ) ) ).
% unit_div_mult_self
thf(fact_1260_pow__divides__pow__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [N: nat,A2: A,B2: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( dvd_dvd @ A @ ( power_power @ A @ A2 @ N ) @ ( power_power @ A @ B2 @ N ) )
= ( dvd_dvd @ A @ A2 @ B2 ) ) ) ) ).
% pow_divides_pow_iff
thf(fact_1261_Suc__mod__mult__self1,axiom,
! [M2: nat,K2: nat,N: nat] :
( ( modulo_modulo @ nat @ ( suc @ ( plus_plus @ nat @ M2 @ ( times_times @ nat @ K2 @ N ) ) ) @ N )
= ( modulo_modulo @ nat @ ( suc @ M2 ) @ N ) ) ).
% Suc_mod_mult_self1
thf(fact_1262_Suc__mod__mult__self2,axiom,
! [M2: nat,N: nat,K2: nat] :
( ( modulo_modulo @ nat @ ( suc @ ( plus_plus @ nat @ M2 @ ( times_times @ nat @ N @ K2 ) ) ) @ N )
= ( modulo_modulo @ nat @ ( suc @ M2 ) @ N ) ) ).
% Suc_mod_mult_self2
thf(fact_1263_Suc__mod__mult__self3,axiom,
! [K2: nat,N: nat,M2: nat] :
( ( modulo_modulo @ nat @ ( suc @ ( plus_plus @ nat @ ( times_times @ nat @ K2 @ N ) @ M2 ) ) @ N )
= ( modulo_modulo @ nat @ ( suc @ M2 ) @ N ) ) ).
% Suc_mod_mult_self3
thf(fact_1264_Suc__mod__mult__self4,axiom,
! [N: nat,K2: nat,M2: nat] :
( ( modulo_modulo @ nat @ ( suc @ ( plus_plus @ nat @ ( times_times @ nat @ N @ K2 ) @ M2 ) ) @ N )
= ( modulo_modulo @ nat @ ( suc @ M2 ) @ N ) ) ).
% Suc_mod_mult_self4
thf(fact_1265_odd__add,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A2: A,B2: A] :
( ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( plus_plus @ A @ A2 @ B2 ) ) )
= ( ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) )
!= ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) ) ) ) ).
% odd_add
thf(fact_1266_even__add,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A2: A,B2: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( plus_plus @ A @ A2 @ B2 ) )
= ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
= ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) ) ) ).
% even_add
thf(fact_1267_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_1268_zero__le__power__eq__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A,W2: num] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ W2 ) ) )
= ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W2 ) )
| ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W2 ) )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ) ) ).
% zero_le_power_eq_numeral
thf(fact_1269_power__less__zero__eq,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A,N: nat] :
( ( ord_less @ A @ ( power_power @ A @ A2 @ N ) @ ( zero_zero @ A ) )
= ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
& ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ) ).
% power_less_zero_eq
thf(fact_1270_power__less__zero__eq__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A,W2: num] :
( ( ord_less @ A @ ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ W2 ) ) @ ( zero_zero @ A ) )
= ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W2 ) )
& ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ) ).
% power_less_zero_eq_numeral
thf(fact_1271_even__plus__one__iff,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A2: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( plus_plus @ A @ A2 @ ( one_one @ A ) ) )
= ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) ) ) ) ).
% even_plus_one_iff
thf(fact_1272_not__mod__2__eq__0__eq__1,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A2: A] :
( ( ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
!= ( zero_zero @ A ) )
= ( ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( one_one @ A ) ) ) ) ).
% not_mod_2_eq_0_eq_1
thf(fact_1273_not__mod__2__eq__1__eq__0,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A2: A] :
( ( ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
!= ( one_one @ A ) )
= ( ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( zero_zero @ A ) ) ) ) ).
% not_mod_2_eq_1_eq_0
thf(fact_1274_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_1275_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_1276_zero__less__power__eq__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A,W2: num] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ W2 ) ) )
= ( ( ( numeral_numeral @ nat @ W2 )
= ( zero_zero @ nat ) )
| ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W2 ) )
& ( A2
!= ( zero_zero @ A ) ) )
| ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W2 ) )
& ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ) ) ).
% zero_less_power_eq_numeral
thf(fact_1277_even__succ__div__2,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A2: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ ( one_one @ A ) @ A2 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% even_succ_div_2
thf(fact_1278_odd__succ__div__two,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [A2: A] :
( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ A2 @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( plus_plus @ A @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) ) ) ) ) ).
% odd_succ_div_two
thf(fact_1279_even__succ__div__two,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [A2: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ A2 @ ( one_one @ A ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% even_succ_div_two
thf(fact_1280_even__power,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A2: A,N: nat] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( power_power @ A @ A2 @ N ) )
= ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
& ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% even_power
thf(fact_1281_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_1282_odd__two__times__div__two__succ,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [A2: A] :
( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
=> ( ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( one_one @ A ) )
= A2 ) ) ) ).
% odd_two_times_div_two_succ
thf(fact_1283_power__le__zero__eq__numeral,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A,W2: num] :
( ( ord_less_eq @ A @ ( power_power @ A @ A2 @ ( numeral_numeral @ nat @ W2 ) ) @ ( zero_zero @ A ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( numeral_numeral @ nat @ W2 ) )
& ( ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W2 ) )
& ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) )
| ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( numeral_numeral @ nat @ W2 ) )
& ( A2
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% power_le_zero_eq_numeral
thf(fact_1284_even__succ__mod__exp,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A2: A,N: nat] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( modulo_modulo @ A @ ( plus_plus @ A @ ( one_one @ A ) @ A2 ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
= ( plus_plus @ A @ ( one_one @ A ) @ ( modulo_modulo @ A @ A2 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) ) ) ) ) ) ).
% even_succ_mod_exp
thf(fact_1285_int__ge__induct,axiom,
! [K2: int,I2: int,P2: int > $o] :
( ( ord_less_eq @ int @ K2 @ I2 )
=> ( ( P2 @ K2 )
=> ( ! [I3: int] :
( ( ord_less_eq @ int @ K2 @ I3 )
=> ( ( P2 @ I3 )
=> ( P2 @ ( plus_plus @ int @ I3 @ ( one_one @ int ) ) ) ) )
=> ( P2 @ I2 ) ) ) ) ).
% int_ge_induct
thf(fact_1286_real__arch__pow,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ ( one_one @ real ) @ X )
=> ? [N3: nat] : ( ord_less @ real @ Y @ ( power_power @ real @ X @ N3 ) ) ) ).
% real_arch_pow
thf(fact_1287_real__arch__pow__inv,axiom,
! [Y: real,X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ Y )
=> ( ( ord_less @ real @ X @ ( one_one @ real ) )
=> ? [N3: nat] : ( ord_less @ real @ ( power_power @ real @ X @ N3 ) @ Y ) ) ) ).
% real_arch_pow_inv
thf(fact_1288_zdvd__mult__cancel,axiom,
! [K2: int,M2: int,N: int] :
( ( dvd_dvd @ int @ ( times_times @ int @ K2 @ M2 ) @ ( times_times @ int @ K2 @ N ) )
=> ( ( K2
!= ( zero_zero @ int ) )
=> ( dvd_dvd @ int @ M2 @ N ) ) ) ).
% zdvd_mult_cancel
thf(fact_1289_mod__0__imp__dvd,axiom,
! [A: $tType] :
( ( semiring_modulo @ A )
=> ! [A2: A,B2: A] :
( ( ( modulo_modulo @ A @ A2 @ B2 )
= ( zero_zero @ A ) )
=> ( dvd_dvd @ A @ B2 @ A2 ) ) ) ).
% mod_0_imp_dvd
thf(fact_1290_dvd__eq__mod__eq__0,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ( ( dvd_dvd @ A )
= ( ^ [A6: A,B5: A] :
( ( modulo_modulo @ A @ B5 @ A6 )
= ( zero_zero @ A ) ) ) ) ) ).
% dvd_eq_mod_eq_0
thf(fact_1291_mod__eq__0__iff__dvd,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [A2: A,B2: A] :
( ( ( modulo_modulo @ A @ A2 @ B2 )
= ( zero_zero @ A ) )
= ( dvd_dvd @ A @ B2 @ A2 ) ) ) ).
% mod_eq_0_iff_dvd
thf(fact_1292_finite__divisors__int,axiom,
! [I2: int] :
( ( I2
!= ( zero_zero @ int ) )
=> ( finite_finite2 @ int
@ ( collect @ int
@ ^ [D5: int] : ( dvd_dvd @ int @ D5 @ I2 ) ) ) ) ).
% finite_divisors_int
thf(fact_1293_strict__subset__divisors__dvd,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ ( set @ A )
@ ( collect @ A
@ ^ [C4: A] : ( dvd_dvd @ A @ C4 @ A2 ) )
@ ( collect @ A
@ ^ [C4: A] : ( dvd_dvd @ A @ C4 @ B2 ) ) )
= ( ( dvd_dvd @ A @ A2 @ B2 )
& ~ ( dvd_dvd @ A @ B2 @ A2 ) ) ) ) ).
% strict_subset_divisors_dvd
thf(fact_1294_zdvd__reduce,axiom,
! [K2: int,N: int,M2: int] :
( ( dvd_dvd @ int @ K2 @ ( plus_plus @ int @ N @ ( times_times @ int @ K2 @ M2 ) ) )
= ( dvd_dvd @ int @ K2 @ N ) ) ).
% zdvd_reduce
thf(fact_1295_zdvd__period,axiom,
! [A2: int,D3: int,X: int,T2: int,C3: int] :
( ( dvd_dvd @ int @ A2 @ D3 )
=> ( ( dvd_dvd @ int @ A2 @ ( plus_plus @ int @ X @ T2 ) )
= ( dvd_dvd @ int @ A2 @ ( plus_plus @ int @ ( plus_plus @ int @ X @ ( times_times @ int @ C3 @ D3 ) ) @ T2 ) ) ) ) ).
% zdvd_period
thf(fact_1296_int__distrib_I1_J,axiom,
! [Z1: int,Z22: int,W2: int] :
( ( times_times @ int @ ( plus_plus @ int @ Z1 @ Z22 ) @ W2 )
= ( plus_plus @ int @ ( times_times @ int @ Z1 @ W2 ) @ ( times_times @ int @ Z22 @ W2 ) ) ) ).
% int_distrib(1)
thf(fact_1297_int__distrib_I2_J,axiom,
! [W2: int,Z1: int,Z22: int] :
( ( times_times @ int @ W2 @ ( plus_plus @ int @ Z1 @ Z22 ) )
= ( plus_plus @ int @ ( times_times @ int @ W2 @ Z1 ) @ ( times_times @ int @ W2 @ Z22 ) ) ) ).
% int_distrib(2)
thf(fact_1298_less__set__def,axiom,
! [A: $tType] :
( ( ord_less @ ( set @ A ) )
= ( ^ [A7: set @ A,B8: set @ A] :
( ord_less @ ( A > $o )
@ ^ [X5: A] : ( member @ A @ X5 @ A7 )
@ ^ [X5: A] : ( member @ A @ X5 @ B8 ) ) ) ) ).
% less_set_def
thf(fact_1299_bot__enat__def,axiom,
( ( bot_bot @ extended_enat )
= ( zero_zero @ extended_enat ) ) ).
% bot_enat_def
thf(fact_1300_less__eq__real__def,axiom,
( ( ord_less_eq @ real )
= ( ^ [X5: real,Y6: real] :
( ( ord_less @ real @ X5 @ Y6 )
| ( X5 = Y6 ) ) ) ) ).
% less_eq_real_def
thf(fact_1301_subset__divisors__dvd,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [C4: A] : ( dvd_dvd @ A @ C4 @ A2 ) )
@ ( collect @ A
@ ^ [C4: A] : ( dvd_dvd @ A @ C4 @ B2 ) ) )
= ( dvd_dvd @ A @ A2 @ B2 ) ) ) ).
% subset_divisors_dvd
thf(fact_1302_empty__def,axiom,
! [A: $tType] :
( ( bot_bot @ ( set @ A ) )
= ( collect @ A
@ ^ [X5: A] : $false ) ) ).
% empty_def
thf(fact_1303_pigeonhole__infinite__rel,axiom,
! [B: $tType,A: $tType,A5: set @ A,B6: set @ B,R: A > B > $o] :
( ~ ( finite_finite2 @ A @ A5 )
=> ( ( finite_finite2 @ B @ B6 )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ A5 )
=> ? [Xa: B] :
( ( member @ B @ Xa @ B6 )
& ( R @ X4 @ Xa ) ) )
=> ? [X4: B] :
( ( member @ B @ X4 @ B6 )
& ~ ( finite_finite2 @ A
@ ( collect @ A
@ ^ [A6: A] :
( ( member @ A @ A6 @ A5 )
& ( R @ A6 @ X4 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite_rel
thf(fact_1304_not__finite__existsD,axiom,
! [A: $tType,P2: A > $o] :
( ~ ( finite_finite2 @ A @ ( collect @ A @ P2 ) )
=> ? [X_12: A] : ( P2 @ X_12 ) ) ).
% not_finite_existsD
thf(fact_1305_dvd__mod__imp__dvd,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [C3: A,A2: A,B2: A] :
( ( dvd_dvd @ A @ C3 @ ( modulo_modulo @ A @ A2 @ B2 ) )
=> ( ( dvd_dvd @ A @ C3 @ B2 )
=> ( dvd_dvd @ A @ C3 @ A2 ) ) ) ) ).
% dvd_mod_imp_dvd
thf(fact_1306_dvd__mod__iff,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [C3: A,B2: A,A2: A] :
( ( dvd_dvd @ A @ C3 @ B2 )
=> ( ( dvd_dvd @ A @ C3 @ ( modulo_modulo @ A @ A2 @ B2 ) )
= ( dvd_dvd @ A @ C3 @ A2 ) ) ) ) ).
% dvd_mod_iff
thf(fact_1307_dvd__trans,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( dvd_dvd @ A @ A2 @ B2 )
=> ( ( dvd_dvd @ A @ B2 @ C3 )
=> ( dvd_dvd @ A @ A2 @ C3 ) ) ) ) ).
% dvd_trans
thf(fact_1308_dvd__refl,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A2: A] : ( dvd_dvd @ A @ A2 @ A2 ) ) ).
% dvd_refl
thf(fact_1309_pred__equals__eq2,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S2: set @ ( product_prod @ A @ B )] :
( ( ( ^ [X5: A,Y6: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X5 @ Y6 ) @ R ) )
= ( ^ [X5: A,Y6: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X5 @ Y6 ) @ S2 ) ) )
= ( R = S2 ) ) ).
% pred_equals_eq2
thf(fact_1310_lambda__zero,axiom,
! [A: $tType] :
( ( mult_zero @ A )
=> ( ( ^ [H: A] : ( zero_zero @ A ) )
= ( times_times @ A @ ( zero_zero @ A ) ) ) ) ).
% lambda_zero
thf(fact_1311_lambda__one,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ( ( ^ [X5: A] : X5 )
= ( times_times @ A @ ( one_one @ A ) ) ) ) ).
% lambda_one
thf(fact_1312_max__def__raw,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( ord_max @ A )
= ( ^ [A6: A,B5: A] : ( if @ A @ ( ord_less_eq @ A @ A6 @ B5 ) @ B5 @ A6 ) ) ) ) ).
% max_def_raw
thf(fact_1313_finite__divisors__nat,axiom,
! [M2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M2 )
=> ( finite_finite2 @ nat
@ ( collect @ nat
@ ^ [D5: nat] : ( dvd_dvd @ nat @ D5 @ M2 ) ) ) ) ).
% finite_divisors_nat
thf(fact_1314_pred__subset__eq2,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S2: set @ ( product_prod @ A @ B )] :
( ( ord_less_eq @ ( A > B > $o )
@ ^ [X5: A,Y6: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X5 @ Y6 ) @ R )
@ ^ [X5: A,Y6: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X5 @ Y6 ) @ S2 ) )
= ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R @ S2 ) ) ).
% pred_subset_eq2
thf(fact_1315_unit__imp__mod__eq__0,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ! [B2: A,A2: A] :
( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( modulo_modulo @ A @ A2 @ B2 )
= ( zero_zero @ A ) ) ) ) ).
% unit_imp_mod_eq_0
thf(fact_1316_bot__empty__eq2,axiom,
! [B: $tType,A: $tType] :
( ( bot_bot @ ( A > B > $o ) )
= ( ^ [X5: A,Y6: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X5 @ Y6 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ) ).
% bot_empty_eq2
thf(fact_1317_finite__M__bounded__by__nat,axiom,
! [P2: nat > $o,I2: nat] :
( finite_finite2 @ nat
@ ( collect @ nat
@ ^ [K3: nat] :
( ( P2 @ K3 )
& ( ord_less @ nat @ K3 @ I2 ) ) ) ) ).
% finite_M_bounded_by_nat
thf(fact_1318_finite__less__ub,axiom,
! [F3: nat > nat,U: nat] :
( ! [N3: nat] : ( ord_less_eq @ nat @ N3 @ ( F3 @ N3 ) )
=> ( finite_finite2 @ nat
@ ( collect @ nat
@ ^ [N2: nat] : ( ord_less_eq @ nat @ ( F3 @ N2 ) @ U ) ) ) ) ).
% finite_less_ub
thf(fact_1319_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_1320_mod__add__eq,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A2: A,C3: A,B2: A] :
( ( modulo_modulo @ A @ ( plus_plus @ A @ ( modulo_modulo @ A @ A2 @ C3 ) @ ( modulo_modulo @ A @ B2 @ C3 ) ) @ C3 )
= ( modulo_modulo @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C3 ) ) ) ).
% mod_add_eq
thf(fact_1321_mod__add__cong,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A2: A,C3: A,A3: A,B2: A,B3: A] :
( ( ( modulo_modulo @ A @ A2 @ C3 )
= ( modulo_modulo @ A @ A3 @ C3 ) )
=> ( ( ( modulo_modulo @ A @ B2 @ C3 )
= ( modulo_modulo @ A @ B3 @ C3 ) )
=> ( ( modulo_modulo @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C3 )
= ( modulo_modulo @ A @ ( plus_plus @ A @ A3 @ B3 ) @ C3 ) ) ) ) ) ).
% mod_add_cong
thf(fact_1322_mod__add__left__eq,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A2: A,C3: A,B2: A] :
( ( modulo_modulo @ A @ ( plus_plus @ A @ ( modulo_modulo @ A @ A2 @ C3 ) @ B2 ) @ C3 )
= ( modulo_modulo @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C3 ) ) ) ).
% mod_add_left_eq
thf(fact_1323_mod__add__right__eq,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( modulo_modulo @ A @ ( plus_plus @ A @ A2 @ ( modulo_modulo @ A @ B2 @ C3 ) ) @ C3 )
= ( modulo_modulo @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C3 ) ) ) ).
% mod_add_right_eq
thf(fact_1324_dvd__field__iff,axiom,
! [A: $tType] :
( ( field @ A )
=> ( ( dvd_dvd @ A )
= ( ^ [A6: A,B5: A] :
( ( A6
= ( zero_zero @ A ) )
=> ( B5
= ( zero_zero @ A ) ) ) ) ) ) ).
% dvd_field_iff
thf(fact_1325_dvd__0__left,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A2: A] :
( ( dvd_dvd @ A @ ( zero_zero @ A ) @ A2 )
=> ( A2
= ( zero_zero @ A ) ) ) ) ).
% dvd_0_left
thf(fact_1326_dvd__triv__right,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A2: A,B2: A] : ( dvd_dvd @ A @ A2 @ ( times_times @ A @ B2 @ A2 ) ) ) ).
% dvd_triv_right
thf(fact_1327_dvd__mult__right,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( dvd_dvd @ A @ ( times_times @ A @ A2 @ B2 ) @ C3 )
=> ( dvd_dvd @ A @ B2 @ C3 ) ) ) ).
% dvd_mult_right
thf(fact_1328_mult__dvd__mono,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A2: A,B2: A,C3: A,D3: A] :
( ( dvd_dvd @ A @ A2 @ B2 )
=> ( ( dvd_dvd @ A @ C3 @ D3 )
=> ( dvd_dvd @ A @ ( times_times @ A @ A2 @ C3 ) @ ( times_times @ A @ B2 @ D3 ) ) ) ) ) ).
% mult_dvd_mono
thf(fact_1329_dvd__triv__left,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A2: A,B2: A] : ( dvd_dvd @ A @ A2 @ ( times_times @ A @ A2 @ B2 ) ) ) ).
% dvd_triv_left
thf(fact_1330_dvd__mult__left,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( dvd_dvd @ A @ ( times_times @ A @ A2 @ B2 ) @ C3 )
=> ( dvd_dvd @ A @ A2 @ C3 ) ) ) ).
% dvd_mult_left
thf(fact_1331_dvd__mult2,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( dvd_dvd @ A @ A2 @ B2 )
=> ( dvd_dvd @ A @ A2 @ ( times_times @ A @ B2 @ C3 ) ) ) ) ).
% dvd_mult2
thf(fact_1332_dvd__mult,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A2: A,C3: A,B2: A] :
( ( dvd_dvd @ A @ A2 @ C3 )
=> ( dvd_dvd @ A @ A2 @ ( times_times @ A @ B2 @ C3 ) ) ) ) ).
% dvd_mult
thf(fact_1333_dvd__def,axiom,
! [A: $tType] :
( ( dvd @ A )
=> ( ( dvd_dvd @ A )
= ( ^ [B5: A,A6: A] :
? [K3: A] :
( A6
= ( times_times @ A @ B5 @ K3 ) ) ) ) ) ).
% dvd_def
thf(fact_1334_dvdI,axiom,
! [A: $tType] :
( ( dvd @ A )
=> ! [A2: A,B2: A,K2: A] :
( ( A2
= ( times_times @ A @ B2 @ K2 ) )
=> ( dvd_dvd @ A @ B2 @ A2 ) ) ) ).
% dvdI
thf(fact_1335_dvdE,axiom,
! [A: $tType] :
( ( dvd @ A )
=> ! [B2: A,A2: A] :
( ( dvd_dvd @ A @ B2 @ A2 )
=> ~ ! [K: A] :
( A2
!= ( times_times @ A @ B2 @ K ) ) ) ) ).
% dvdE
thf(fact_1336_dvd__unit__imp__unit,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A,B2: A] :
( ( dvd_dvd @ A @ A2 @ B2 )
=> ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( dvd_dvd @ A @ A2 @ ( one_one @ A ) ) ) ) ) ).
% dvd_unit_imp_unit
thf(fact_1337_unit__imp__dvd,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,A2: A] :
( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( dvd_dvd @ A @ B2 @ A2 ) ) ) ).
% unit_imp_dvd
thf(fact_1338_one__dvd,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A2: A] : ( dvd_dvd @ A @ ( one_one @ A ) @ A2 ) ) ).
% one_dvd
thf(fact_1339_dvd__add,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( dvd_dvd @ A @ A2 @ B2 )
=> ( ( dvd_dvd @ A @ A2 @ C3 )
=> ( dvd_dvd @ A @ A2 @ ( plus_plus @ A @ B2 @ C3 ) ) ) ) ) ).
% dvd_add
thf(fact_1340_dvd__add__left__iff,axiom,
! [A: $tType] :
( ( comm_s4317794764714335236cancel @ A )
=> ! [A2: A,C3: A,B2: A] :
( ( dvd_dvd @ A @ A2 @ C3 )
=> ( ( dvd_dvd @ A @ A2 @ ( plus_plus @ A @ B2 @ C3 ) )
= ( dvd_dvd @ A @ A2 @ B2 ) ) ) ) ).
% dvd_add_left_iff
thf(fact_1341_dvd__add__right__iff,axiom,
! [A: $tType] :
( ( comm_s4317794764714335236cancel @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( dvd_dvd @ A @ A2 @ B2 )
=> ( ( dvd_dvd @ A @ A2 @ ( plus_plus @ A @ B2 @ C3 ) )
= ( dvd_dvd @ A @ A2 @ C3 ) ) ) ) ).
% dvd_add_right_iff
thf(fact_1342_div__div__div__same,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [D3: A,B2: A,A2: A] :
( ( dvd_dvd @ A @ D3 @ B2 )
=> ( ( dvd_dvd @ A @ B2 @ A2 )
=> ( ( divide_divide @ A @ ( divide_divide @ A @ A2 @ D3 ) @ ( divide_divide @ A @ B2 @ D3 ) )
= ( divide_divide @ A @ A2 @ B2 ) ) ) ) ) ).
% div_div_div_same
thf(fact_1343_dvd__div__eq__cancel,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [A2: A,C3: A,B2: A] :
( ( ( divide_divide @ A @ A2 @ C3 )
= ( divide_divide @ A @ B2 @ C3 ) )
=> ( ( dvd_dvd @ A @ C3 @ A2 )
=> ( ( dvd_dvd @ A @ C3 @ B2 )
=> ( A2 = B2 ) ) ) ) ) ).
% dvd_div_eq_cancel
thf(fact_1344_dvd__div__eq__iff,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [C3: A,A2: A,B2: A] :
( ( dvd_dvd @ A @ C3 @ A2 )
=> ( ( dvd_dvd @ A @ C3 @ B2 )
=> ( ( ( divide_divide @ A @ A2 @ C3 )
= ( divide_divide @ A @ B2 @ C3 ) )
= ( A2 = B2 ) ) ) ) ) ).
% dvd_div_eq_iff
thf(fact_1345_mod__less__eq__dividend,axiom,
! [M2: nat,N: nat] : ( ord_less_eq @ nat @ ( modulo_modulo @ nat @ M2 @ N ) @ M2 ) ).
% mod_less_eq_dividend
thf(fact_1346_gcd__nat_Oextremum,axiom,
! [A2: nat] : ( dvd_dvd @ nat @ A2 @ ( zero_zero @ nat ) ) ).
% gcd_nat.extremum
thf(fact_1347_gcd__nat_Oextremum__strict,axiom,
! [A2: nat] :
~ ( ( dvd_dvd @ nat @ ( zero_zero @ nat ) @ A2 )
& ( ( zero_zero @ nat )
!= A2 ) ) ).
% gcd_nat.extremum_strict
thf(fact_1348_gcd__nat_Oextremum__unique,axiom,
! [A2: nat] :
( ( dvd_dvd @ nat @ ( zero_zero @ nat ) @ A2 )
= ( A2
= ( zero_zero @ nat ) ) ) ).
% gcd_nat.extremum_unique
thf(fact_1349_gcd__nat_Onot__eq__extremum,axiom,
! [A2: nat] :
( ( A2
!= ( zero_zero @ nat ) )
= ( ( dvd_dvd @ nat @ A2 @ ( zero_zero @ nat ) )
& ( A2
!= ( zero_zero @ nat ) ) ) ) ).
% gcd_nat.not_eq_extremum
thf(fact_1350_gcd__nat_Oextremum__uniqueI,axiom,
! [A2: nat] :
( ( dvd_dvd @ nat @ ( zero_zero @ nat ) @ A2 )
=> ( A2
= ( zero_zero @ nat ) ) ) ).
% gcd_nat.extremum_uniqueI
thf(fact_1351_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_1352_set__decode__def,axiom,
( nat_set_decode
= ( ^ [X5: nat] :
( collect @ nat
@ ^ [N2: nat] :
~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( divide_divide @ nat @ X5 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) ) ) ) ).
% set_decode_def
thf(fact_1353_times__int__code_I1_J,axiom,
! [K2: int] :
( ( times_times @ int @ K2 @ ( zero_zero @ int ) )
= ( zero_zero @ int ) ) ).
% times_int_code(1)
thf(fact_1354_times__int__code_I2_J,axiom,
! [L: int] :
( ( times_times @ int @ ( zero_zero @ int ) @ L )
= ( zero_zero @ int ) ) ).
% times_int_code(2)
thf(fact_1355_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_1356_set__vebt__def,axiom,
( vEBT_set_vebt
= ( ^ [T3: vEBT_VEBT] : ( collect @ nat @ ( vEBT_V8194947554948674370ptions @ T3 ) ) ) ) ).
% set_vebt_def
thf(fact_1357_enat__0__less__mult__iff,axiom,
! [M2: extended_enat,N: extended_enat] :
( ( ord_less @ extended_enat @ ( zero_zero @ extended_enat ) @ ( times_times @ extended_enat @ M2 @ N ) )
= ( ( ord_less @ extended_enat @ ( zero_zero @ extended_enat ) @ M2 )
& ( ord_less @ extended_enat @ ( zero_zero @ extended_enat ) @ N ) ) ) ).
% enat_0_less_mult_iff
thf(fact_1358_even__iff__mod__2__eq__zero,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A2: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
= ( ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( zero_zero @ A ) ) ) ) ).
% even_iff_mod_2_eq_zero
thf(fact_1359_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_1360_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ord_less_eq @ A @ ( modulo_modulo @ A @ A2 @ B2 ) @ A2 ) ) ) ).
% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
thf(fact_1361_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [B2: A,A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less @ A @ ( modulo_modulo @ A @ A2 @ B2 ) @ B2 ) ) ) ).
% unique_euclidean_semiring_numeral_class.pos_mod_bound
thf(fact_1362_mod__eq__self__iff__div__eq__0,axiom,
! [A: $tType] :
( ( euclid3725896446679973847miring @ A )
=> ! [A2: A,B2: A] :
( ( ( modulo_modulo @ A @ A2 @ B2 )
= A2 )
= ( ( divide_divide @ A @ A2 @ B2 )
= ( zero_zero @ A ) ) ) ) ).
% mod_eq_self_iff_div_eq_0
thf(fact_1363_mod__eqE,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel @ A )
=> ! [A2: A,C3: A,B2: A] :
( ( ( modulo_modulo @ A @ A2 @ C3 )
= ( modulo_modulo @ A @ B2 @ C3 ) )
=> ~ ! [D2: A] :
( B2
!= ( plus_plus @ A @ A2 @ ( times_times @ A @ C3 @ D2 ) ) ) ) ) ).
% mod_eqE
thf(fact_1364_div__add1__eq,axiom,
! [A: $tType] :
( ( euclid3128863361964157862miring @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( divide_divide @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C3 )
= ( plus_plus @ A @ ( plus_plus @ A @ ( divide_divide @ A @ A2 @ C3 ) @ ( divide_divide @ A @ B2 @ C3 ) ) @ ( divide_divide @ A @ ( plus_plus @ A @ ( modulo_modulo @ A @ A2 @ C3 ) @ ( modulo_modulo @ A @ B2 @ C3 ) ) @ C3 ) ) ) ) ).
% div_add1_eq
thf(fact_1365_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_1366_minf_I10_J,axiom,
! [B: $tType] :
( ( ( plus @ B )
& ( linorder @ B )
& ( dvd @ B ) )
=> ! [D3: B,S: B] :
? [Z3: B] :
! [X3: B] :
( ( ord_less @ B @ X3 @ Z3 )
=> ( ( ~ ( dvd_dvd @ B @ D3 @ ( plus_plus @ B @ X3 @ S ) ) )
= ( ~ ( dvd_dvd @ B @ D3 @ ( plus_plus @ B @ X3 @ S ) ) ) ) ) ) ).
% minf(10)
thf(fact_1367_minf_I9_J,axiom,
! [B: $tType] :
( ( ( plus @ B )
& ( linorder @ B )
& ( dvd @ B ) )
=> ! [D3: B,S: B] :
? [Z3: B] :
! [X3: B] :
( ( ord_less @ B @ X3 @ Z3 )
=> ( ( dvd_dvd @ B @ D3 @ ( plus_plus @ B @ X3 @ S ) )
= ( dvd_dvd @ B @ D3 @ ( plus_plus @ B @ X3 @ S ) ) ) ) ) ).
% minf(9)
thf(fact_1368_pinf_I10_J,axiom,
! [B: $tType] :
( ( ( plus @ B )
& ( linorder @ B )
& ( dvd @ B ) )
=> ! [D3: B,S: B] :
? [Z3: B] :
! [X3: B] :
( ( ord_less @ B @ Z3 @ X3 )
=> ( ( ~ ( dvd_dvd @ B @ D3 @ ( plus_plus @ B @ X3 @ S ) ) )
= ( ~ ( dvd_dvd @ B @ D3 @ ( plus_plus @ B @ X3 @ S ) ) ) ) ) ) ).
% pinf(10)
thf(fact_1369_pinf_I9_J,axiom,
! [B: $tType] :
( ( ( plus @ B )
& ( linorder @ B )
& ( dvd @ B ) )
=> ! [D3: B,S: B] :
? [Z3: B] :
! [X3: B] :
( ( ord_less @ B @ Z3 @ X3 )
=> ( ( dvd_dvd @ B @ D3 @ ( plus_plus @ B @ X3 @ S ) )
= ( dvd_dvd @ B @ D3 @ ( plus_plus @ B @ X3 @ S ) ) ) ) ) ).
% pinf(9)
thf(fact_1370_dvd__div__eq__0__iff,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [B2: A,A2: A] :
( ( dvd_dvd @ A @ B2 @ A2 )
=> ( ( ( divide_divide @ A @ A2 @ B2 )
= ( zero_zero @ A ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ) ).
% dvd_div_eq_0_iff
thf(fact_1371_is__unit__mult__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A,B2: A] :
( ( dvd_dvd @ A @ ( times_times @ A @ A2 @ B2 ) @ ( one_one @ A ) )
= ( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) )
& ( dvd_dvd @ A @ B2 @ ( one_one @ A ) ) ) ) ) ).
% is_unit_mult_iff
thf(fact_1372_dvd__mult__unit__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,A2: A,C3: A] :
( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ A2 @ ( times_times @ A @ C3 @ B2 ) )
= ( dvd_dvd @ A @ A2 @ C3 ) ) ) ) ).
% dvd_mult_unit_iff
thf(fact_1373_mult__unit__dvd__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,A2: A,C3: A] :
( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ ( times_times @ A @ A2 @ B2 ) @ C3 )
= ( dvd_dvd @ A @ A2 @ C3 ) ) ) ) ).
% mult_unit_dvd_iff
thf(fact_1374_dvd__mult__unit__iff_H,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,A2: A,C3: A] :
( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ A2 @ ( times_times @ A @ B2 @ C3 ) )
= ( dvd_dvd @ A @ A2 @ C3 ) ) ) ) ).
% dvd_mult_unit_iff'
thf(fact_1375_mult__unit__dvd__iff_H,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ ( times_times @ A @ A2 @ B2 ) @ C3 )
= ( dvd_dvd @ A @ B2 @ C3 ) ) ) ) ).
% mult_unit_dvd_iff'
thf(fact_1376_unit__mult__left__cancel,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) )
=> ( ( ( times_times @ A @ A2 @ B2 )
= ( times_times @ A @ A2 @ C3 ) )
= ( B2 = C3 ) ) ) ) ).
% unit_mult_left_cancel
thf(fact_1377_unit__mult__right__cancel,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) )
=> ( ( ( times_times @ A @ B2 @ A2 )
= ( times_times @ A @ C3 @ A2 ) )
= ( B2 = C3 ) ) ) ) ).
% unit_mult_right_cancel
thf(fact_1378_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_1379_finite__set__decode,axiom,
! [N: nat] : ( finite_finite2 @ nat @ ( nat_set_decode @ N ) ) ).
% finite_set_decode
thf(fact_1380_mod__induct,axiom,
! [P2: nat > $o,N: nat,P: nat,M2: nat] :
( ( P2 @ N )
=> ( ( ord_less @ nat @ N @ P )
=> ( ( ord_less @ nat @ M2 @ P )
=> ( ! [N3: nat] :
( ( ord_less @ nat @ N3 @ P )
=> ( ( P2 @ N3 )
=> ( P2 @ ( modulo_modulo @ nat @ ( suc @ N3 ) @ P ) ) ) )
=> ( P2 @ M2 ) ) ) ) ) ).
% mod_induct
thf(fact_1381_div__mult__div__if__dvd,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,A2: A,D3: A,C3: A] :
( ( dvd_dvd @ A @ B2 @ A2 )
=> ( ( dvd_dvd @ A @ D3 @ C3 )
=> ( ( times_times @ A @ ( divide_divide @ A @ A2 @ B2 ) @ ( divide_divide @ A @ C3 @ D3 ) )
= ( divide_divide @ A @ ( times_times @ A @ A2 @ C3 ) @ ( times_times @ A @ B2 @ D3 ) ) ) ) ) ) ).
% div_mult_div_if_dvd
thf(fact_1382_dvd__mult__imp__div,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A,C3: A,B2: A] :
( ( dvd_dvd @ A @ ( times_times @ A @ A2 @ C3 ) @ B2 )
=> ( dvd_dvd @ A @ A2 @ ( divide_divide @ A @ B2 @ C3 ) ) ) ) ).
% dvd_mult_imp_div
thf(fact_1383_dvd__div__mult2__eq,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,C3: A,A2: A] :
( ( dvd_dvd @ A @ ( times_times @ A @ B2 @ C3 ) @ A2 )
=> ( ( divide_divide @ A @ A2 @ ( times_times @ A @ B2 @ C3 ) )
= ( divide_divide @ A @ ( divide_divide @ A @ A2 @ B2 ) @ C3 ) ) ) ) ).
% dvd_div_mult2_eq
thf(fact_1384_div__div__eq__right,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [C3: A,B2: A,A2: A] :
( ( dvd_dvd @ A @ C3 @ B2 )
=> ( ( dvd_dvd @ A @ B2 @ A2 )
=> ( ( divide_divide @ A @ A2 @ ( divide_divide @ A @ B2 @ C3 ) )
= ( times_times @ A @ ( divide_divide @ A @ A2 @ B2 ) @ C3 ) ) ) ) ) ).
% div_div_eq_right
thf(fact_1385_div__mult__swap,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [C3: A,B2: A,A2: A] :
( ( dvd_dvd @ A @ C3 @ B2 )
=> ( ( times_times @ A @ A2 @ ( divide_divide @ A @ B2 @ C3 ) )
= ( divide_divide @ A @ ( times_times @ A @ A2 @ B2 ) @ C3 ) ) ) ) ).
% div_mult_swap
thf(fact_1386_dvd__div__mult,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [C3: A,B2: A,A2: A] :
( ( dvd_dvd @ A @ C3 @ B2 )
=> ( ( times_times @ A @ ( divide_divide @ A @ B2 @ C3 ) @ A2 )
= ( divide_divide @ A @ ( times_times @ A @ B2 @ A2 ) @ C3 ) ) ) ) ).
% dvd_div_mult
thf(fact_1387_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_1388_gcd__nat__induct,axiom,
! [P2: nat > nat > $o,M2: nat,N: nat] :
( ! [M: nat] : ( P2 @ M @ ( zero_zero @ nat ) )
=> ( ! [M: nat,N3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( ( P2 @ N3 @ ( modulo_modulo @ nat @ M @ N3 ) )
=> ( P2 @ M @ N3 ) ) )
=> ( P2 @ M2 @ N ) ) ) ).
% gcd_nat_induct
thf(fact_1389_unit__div__cancel,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) )
=> ( ( ( divide_divide @ A @ B2 @ A2 )
= ( divide_divide @ A @ C3 @ A2 ) )
= ( B2 = C3 ) ) ) ) ).
% unit_div_cancel
thf(fact_1390_div__unit__dvd__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,A2: A,C3: A] :
( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ ( divide_divide @ A @ A2 @ B2 ) @ C3 )
= ( dvd_dvd @ A @ A2 @ C3 ) ) ) ) ).
% div_unit_dvd_iff
thf(fact_1391_dvd__div__unit__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,A2: A,C3: A] :
( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ A2 @ ( divide_divide @ A @ C3 @ B2 ) )
= ( dvd_dvd @ A @ A2 @ C3 ) ) ) ) ).
% dvd_div_unit_iff
thf(fact_1392_div__plus__div__distrib__dvd__left,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [C3: A,A2: A,B2: A] :
( ( dvd_dvd @ A @ C3 @ A2 )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C3 )
= ( plus_plus @ A @ ( divide_divide @ A @ A2 @ C3 ) @ ( divide_divide @ A @ B2 @ C3 ) ) ) ) ) ).
% div_plus_div_distrib_dvd_left
thf(fact_1393_div__plus__div__distrib__dvd__right,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [C3: A,B2: A,A2: A] :
( ( dvd_dvd @ A @ C3 @ B2 )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C3 )
= ( plus_plus @ A @ ( divide_divide @ A @ A2 @ C3 ) @ ( divide_divide @ A @ B2 @ C3 ) ) ) ) ) ).
% div_plus_div_distrib_dvd_right
thf(fact_1394_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_1395_mod__eq__0D,axiom,
! [M2: nat,D3: nat] :
( ( ( modulo_modulo @ nat @ M2 @ D3 )
= ( zero_zero @ nat ) )
=> ? [Q3: nat] :
( M2
= ( times_times @ nat @ D3 @ Q3 ) ) ) ).
% mod_eq_0D
thf(fact_1396_dvd__power__le,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [X: A,Y: A,N: nat,M2: nat] :
( ( dvd_dvd @ A @ X @ Y )
=> ( ( ord_less_eq @ nat @ N @ M2 )
=> ( dvd_dvd @ A @ ( power_power @ A @ X @ N ) @ ( power_power @ A @ Y @ M2 ) ) ) ) ) ).
% dvd_power_le
thf(fact_1397_power__le__dvd,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A2: A,N: nat,B2: A,M2: nat] :
( ( dvd_dvd @ A @ ( power_power @ A @ A2 @ N ) @ B2 )
=> ( ( ord_less_eq @ nat @ M2 @ N )
=> ( dvd_dvd @ A @ ( power_power @ A @ A2 @ M2 ) @ B2 ) ) ) ) ).
% power_le_dvd
thf(fact_1398_le__imp__power__dvd,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [M2: nat,N: nat,A2: A] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( dvd_dvd @ A @ ( power_power @ A @ A2 @ M2 ) @ ( power_power @ A @ A2 @ N ) ) ) ) ).
% le_imp_power_dvd
thf(fact_1399_nat__mod__eq__iff,axiom,
! [X: nat,N: nat,Y: nat] :
( ( ( modulo_modulo @ nat @ X @ N )
= ( modulo_modulo @ nat @ Y @ N ) )
= ( ? [Q1: nat,Q22: nat] :
( ( plus_plus @ nat @ X @ ( times_times @ nat @ N @ Q1 ) )
= ( plus_plus @ nat @ Y @ ( times_times @ nat @ N @ Q22 ) ) ) ) ) ).
% nat_mod_eq_iff
thf(fact_1400_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_1401_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_1402_zdvd__imp__le,axiom,
! [Z: int,N: int] :
( ( dvd_dvd @ int @ Z @ N )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ N )
=> ( ord_less_eq @ int @ Z @ N ) ) ) ).
% zdvd_imp_le
thf(fact_1403_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_1404_bezout__lemma__nat,axiom,
! [D3: nat,A2: nat,B2: nat,X: nat,Y: nat] :
( ( dvd_dvd @ nat @ D3 @ A2 )
=> ( ( dvd_dvd @ nat @ D3 @ B2 )
=> ( ( ( ( times_times @ nat @ A2 @ X )
= ( plus_plus @ nat @ ( times_times @ nat @ B2 @ Y ) @ D3 ) )
| ( ( times_times @ nat @ B2 @ X )
= ( plus_plus @ nat @ ( times_times @ nat @ A2 @ Y ) @ D3 ) ) )
=> ? [X4: nat,Y3: nat] :
( ( dvd_dvd @ nat @ D3 @ A2 )
& ( dvd_dvd @ nat @ D3 @ ( plus_plus @ nat @ A2 @ B2 ) )
& ( ( ( times_times @ nat @ A2 @ X4 )
= ( plus_plus @ nat @ ( times_times @ nat @ ( plus_plus @ nat @ A2 @ B2 ) @ Y3 ) @ D3 ) )
| ( ( times_times @ nat @ ( plus_plus @ nat @ A2 @ B2 ) @ X4 )
= ( plus_plus @ nat @ ( times_times @ nat @ A2 @ Y3 ) @ D3 ) ) ) ) ) ) ) ).
% bezout_lemma_nat
thf(fact_1405_bezout__add__nat,axiom,
! [A2: nat,B2: nat] :
? [D2: nat,X4: nat,Y3: nat] :
( ( dvd_dvd @ nat @ D2 @ A2 )
& ( dvd_dvd @ nat @ D2 @ B2 )
& ( ( ( times_times @ nat @ A2 @ X4 )
= ( plus_plus @ nat @ ( times_times @ nat @ B2 @ Y3 ) @ D2 ) )
| ( ( times_times @ nat @ B2 @ X4 )
= ( plus_plus @ nat @ ( times_times @ nat @ A2 @ Y3 ) @ D2 ) ) ) ) ).
% bezout_add_nat
thf(fact_1406_q__pos__lemma,axiom,
! [B3: int,Q6: int,R4: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( plus_plus @ int @ ( times_times @ int @ B3 @ Q6 ) @ R4 ) )
=> ( ( ord_less @ int @ R4 @ B3 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B3 )
=> ( ord_less_eq @ int @ ( zero_zero @ int ) @ Q6 ) ) ) ) ).
% q_pos_lemma
thf(fact_1407_zdiv__mono2__lemma,axiom,
! [B2: int,Q2: int,R2: int,B3: int,Q6: int,R4: int] :
( ( ( plus_plus @ int @ ( times_times @ int @ B2 @ Q2 ) @ R2 )
= ( plus_plus @ int @ ( times_times @ int @ B3 @ Q6 ) @ R4 ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( plus_plus @ int @ ( times_times @ int @ B3 @ Q6 ) @ R4 ) )
=> ( ( ord_less @ int @ R4 @ B3 )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ R2 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B3 )
=> ( ( ord_less_eq @ int @ B3 @ B2 )
=> ( ord_less_eq @ int @ Q2 @ Q6 ) ) ) ) ) ) ) ).
% zdiv_mono2_lemma
thf(fact_1408_zdiv__mono2__neg__lemma,axiom,
! [B2: int,Q2: int,R2: int,B3: int,Q6: int,R4: int] :
( ( ( plus_plus @ int @ ( times_times @ int @ B2 @ Q2 ) @ R2 )
= ( plus_plus @ int @ ( times_times @ int @ B3 @ Q6 ) @ R4 ) )
=> ( ( ord_less @ int @ ( plus_plus @ int @ ( times_times @ int @ B3 @ Q6 ) @ R4 ) @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ R2 @ B2 )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ R4 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ B3 )
=> ( ( ord_less_eq @ int @ B3 @ B2 )
=> ( ord_less_eq @ int @ Q6 @ Q2 ) ) ) ) ) ) ) ).
% zdiv_mono2_neg_lemma
thf(fact_1409_unique__quotient__lemma,axiom,
! [B2: int,Q6: int,R4: int,Q2: int,R2: int] :
( ( ord_less_eq @ int @ ( plus_plus @ int @ ( times_times @ int @ B2 @ Q6 ) @ R4 ) @ ( plus_plus @ int @ ( times_times @ int @ B2 @ Q2 ) @ R2 ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ R4 )
=> ( ( ord_less @ int @ R4 @ B2 )
=> ( ( ord_less @ int @ R2 @ B2 )
=> ( ord_less_eq @ int @ Q6 @ Q2 ) ) ) ) ) ).
% unique_quotient_lemma
thf(fact_1410_unique__quotient__lemma__neg,axiom,
! [B2: int,Q6: int,R4: int,Q2: int,R2: int] :
( ( ord_less_eq @ int @ ( plus_plus @ int @ ( times_times @ int @ B2 @ Q6 ) @ R4 ) @ ( plus_plus @ int @ ( times_times @ int @ B2 @ Q2 ) @ R2 ) )
=> ( ( ord_less_eq @ int @ R2 @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ B2 @ R2 )
=> ( ( ord_less @ int @ B2 @ R4 )
=> ( ord_less_eq @ int @ Q2 @ Q6 ) ) ) ) ) ).
% unique_quotient_lemma_neg
thf(fact_1411_incr__mult__lemma,axiom,
! [D3: int,P2: int > $o,K2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D3 )
=> ( ! [X4: int] :
( ( P2 @ X4 )
=> ( P2 @ ( plus_plus @ int @ X4 @ D3 ) ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K2 )
=> ! [X3: int] :
( ( P2 @ X3 )
=> ( P2 @ ( plus_plus @ int @ X3 @ ( times_times @ int @ K2 @ D3 ) ) ) ) ) ) ) ).
% incr_mult_lemma
thf(fact_1412_zmult__zless__mono2,axiom,
! [I2: int,J: int,K2: int] :
( ( ord_less @ int @ I2 @ J )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ K2 )
=> ( ord_less @ int @ ( times_times @ int @ K2 @ I2 ) @ ( times_times @ int @ K2 @ J ) ) ) ) ).
% zmult_zless_mono2
thf(fact_1413_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_1414_mod2__eq__if,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A2: A] :
( ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
=> ( ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( zero_zero @ A ) ) )
& ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
=> ( ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( one_one @ A ) ) ) ) ) ).
% mod2_eq_if
thf(fact_1415_parity__cases,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A2: A] :
( ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
=> ( ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
!= ( zero_zero @ A ) ) )
=> ~ ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
=> ( ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
!= ( one_one @ A ) ) ) ) ) ).
% parity_cases
thf(fact_1416_finite__lists__length__eq,axiom,
! [A: $tType,A5: set @ A,N: nat] :
( ( finite_finite2 @ A @ A5 )
=> ( finite_finite2 @ ( list @ A )
@ ( collect @ ( list @ A )
@ ^ [Xs3: list @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs3 ) @ A5 )
& ( ( size_size @ ( list @ A ) @ Xs3 )
= N ) ) ) ) ) ).
% finite_lists_length_eq
thf(fact_1417_even__set__bit__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M2: nat,A2: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5668285175392031749et_bit @ A @ M2 @ A2 ) )
= ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
& ( M2
!= ( zero_zero @ nat ) ) ) ) ) ).
% even_set_bit_iff
thf(fact_1418_even__unset__bit__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M2: nat,A2: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se2638667681897837118et_bit @ A @ M2 @ A2 ) )
= ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
| ( M2
= ( zero_zero @ nat ) ) ) ) ) ).
% even_unset_bit_iff
thf(fact_1419_vebt__buildup_Osimps_I3_J,axiom,
! [Va: nat] :
( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va ) ) )
=> ( ( vEBT_vebt_buildup @ ( suc @ ( suc @ Va ) ) )
= ( 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 ) ) )
=> ( ( vEBT_vebt_buildup @ ( suc @ ( suc @ Va ) ) )
= ( 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.simps(3)
thf(fact_1420_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ A2 @ B2 )
=> ( ( modulo_modulo @ A @ A2 @ B2 )
= A2 ) ) ) ) ).
% unique_euclidean_semiring_numeral_class.mod_less
thf(fact_1421_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [B2: A,A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( modulo_modulo @ A @ A2 @ B2 ) ) ) ) ).
% unique_euclidean_semiring_numeral_class.pos_mod_sign
thf(fact_1422_cong__exp__iff__simps_I2_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [N: num,Q2: num] :
( ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit0 @ N ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q2 ) ) )
= ( zero_zero @ A ) )
= ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ N ) @ ( numeral_numeral @ A @ Q2 ) )
= ( zero_zero @ A ) ) ) ) ).
% cong_exp_iff_simps(2)
thf(fact_1423_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_1424_finite__lists__length__le,axiom,
! [A: $tType,A5: set @ A,N: nat] :
( ( finite_finite2 @ A @ A5 )
=> ( finite_finite2 @ ( list @ A )
@ ( collect @ ( list @ A )
@ ^ [Xs3: list @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs3 ) @ A5 )
& ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs3 ) @ N ) ) ) ) ) ).
% finite_lists_length_le
thf(fact_1425_div__mult1__eq,axiom,
! [A: $tType] :
( ( euclid3128863361964157862miring @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( divide_divide @ A @ ( times_times @ A @ A2 @ B2 ) @ C3 )
= ( plus_plus @ A @ ( times_times @ A @ A2 @ ( divide_divide @ A @ B2 @ C3 ) ) @ ( divide_divide @ A @ ( times_times @ A @ A2 @ ( modulo_modulo @ A @ B2 @ C3 ) ) @ C3 ) ) ) ) ).
% div_mult1_eq
thf(fact_1426_mult__div__mod__eq,axiom,
! [A: $tType] :
( ( semiring_modulo @ A )
=> ! [B2: A,A2: A] :
( ( plus_plus @ A @ ( times_times @ A @ B2 @ ( divide_divide @ A @ A2 @ B2 ) ) @ ( modulo_modulo @ A @ A2 @ B2 ) )
= A2 ) ) ).
% mult_div_mod_eq
thf(fact_1427_mod__mult__div__eq,axiom,
! [A: $tType] :
( ( semiring_modulo @ A )
=> ! [A2: A,B2: A] :
( ( plus_plus @ A @ ( modulo_modulo @ A @ A2 @ B2 ) @ ( times_times @ A @ B2 @ ( divide_divide @ A @ A2 @ B2 ) ) )
= A2 ) ) ).
% mod_mult_div_eq
thf(fact_1428_mod__div__mult__eq,axiom,
! [A: $tType] :
( ( semiring_modulo @ A )
=> ! [A2: A,B2: A] :
( ( plus_plus @ A @ ( modulo_modulo @ A @ A2 @ B2 ) @ ( times_times @ A @ ( divide_divide @ A @ A2 @ B2 ) @ B2 ) )
= A2 ) ) ).
% mod_div_mult_eq
thf(fact_1429_div__mult__mod__eq,axiom,
! [A: $tType] :
( ( semiring_modulo @ A )
=> ! [A2: A,B2: A] :
( ( plus_plus @ A @ ( times_times @ A @ ( divide_divide @ A @ A2 @ B2 ) @ B2 ) @ ( modulo_modulo @ A @ A2 @ B2 ) )
= A2 ) ) ).
% div_mult_mod_eq
thf(fact_1430_mod__div__decomp,axiom,
! [A: $tType] :
( ( semiring_modulo @ A )
=> ! [A2: A,B2: A] :
( A2
= ( plus_plus @ A @ ( times_times @ A @ ( divide_divide @ A @ A2 @ B2 ) @ B2 ) @ ( modulo_modulo @ A @ A2 @ B2 ) ) ) ) ).
% mod_div_decomp
thf(fact_1431_cancel__div__mod__rules_I1_J,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ ( times_times @ A @ ( divide_divide @ A @ A2 @ B2 ) @ B2 ) @ ( modulo_modulo @ A @ A2 @ B2 ) ) @ C3 )
= ( plus_plus @ A @ A2 @ C3 ) ) ) ).
% cancel_div_mod_rules(1)
thf(fact_1432_cancel__div__mod__rules_I2_J,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [B2: A,A2: A,C3: A] :
( ( plus_plus @ A @ ( plus_plus @ A @ ( times_times @ A @ B2 @ ( divide_divide @ A @ A2 @ B2 ) ) @ ( modulo_modulo @ A @ A2 @ B2 ) ) @ C3 )
= ( plus_plus @ A @ A2 @ C3 ) ) ) ).
% cancel_div_mod_rules(2)
thf(fact_1433_unit__dvdE,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A,B2: A] :
( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) )
=> ~ ( ( A2
!= ( zero_zero @ A ) )
=> ! [C2: A] :
( B2
!= ( times_times @ A @ A2 @ C2 ) ) ) ) ) ).
% unit_dvdE
thf(fact_1434_unity__coeff__ex,axiom,
! [A: $tType] :
( ( ( dvd @ A )
& ( semiring_0 @ A ) )
=> ! [P2: A > $o,L: A] :
( ( ? [X5: A] : ( P2 @ ( times_times @ A @ L @ X5 ) ) )
= ( ? [X5: A] :
( ( dvd_dvd @ A @ L @ ( plus_plus @ A @ X5 @ ( zero_zero @ A ) ) )
& ( P2 @ X5 ) ) ) ) ) ).
% unity_coeff_ex
thf(fact_1435_dvd__div__div__eq__mult,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A,C3: A,B2: A,D3: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( C3
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ A2 @ B2 )
=> ( ( dvd_dvd @ A @ C3 @ D3 )
=> ( ( ( divide_divide @ A @ B2 @ A2 )
= ( divide_divide @ A @ D3 @ C3 ) )
= ( ( times_times @ A @ B2 @ C3 )
= ( times_times @ A @ A2 @ D3 ) ) ) ) ) ) ) ) ).
% dvd_div_div_eq_mult
thf(fact_1436_dvd__div__iff__mult,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [C3: A,B2: A,A2: A] :
( ( C3
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ C3 @ B2 )
=> ( ( dvd_dvd @ A @ A2 @ ( divide_divide @ A @ B2 @ C3 ) )
= ( dvd_dvd @ A @ ( times_times @ A @ A2 @ C3 ) @ B2 ) ) ) ) ) ).
% dvd_div_iff_mult
thf(fact_1437_div__dvd__iff__mult,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,A2: A,C3: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ B2 @ A2 )
=> ( ( dvd_dvd @ A @ ( divide_divide @ A @ A2 @ B2 ) @ C3 )
= ( dvd_dvd @ A @ A2 @ ( times_times @ A @ C3 @ B2 ) ) ) ) ) ) ).
% div_dvd_iff_mult
thf(fact_1438_dvd__div__eq__mult,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ A2 @ B2 )
=> ( ( ( divide_divide @ A @ B2 @ A2 )
= C3 )
= ( B2
= ( times_times @ A @ C3 @ A2 ) ) ) ) ) ) ).
% dvd_div_eq_mult
thf(fact_1439_unit__div__eq__0__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,A2: A] :
( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( ( divide_divide @ A @ A2 @ B2 )
= ( zero_zero @ A ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ) ).
% unit_div_eq_0_iff
thf(fact_1440_unit__eq__div1,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,A2: A,C3: A] :
( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( ( divide_divide @ A @ A2 @ B2 )
= C3 )
= ( A2
= ( times_times @ A @ C3 @ B2 ) ) ) ) ) ).
% unit_eq_div1
thf(fact_1441_unit__eq__div2,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,A2: A,C3: A] :
( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( A2
= ( divide_divide @ A @ C3 @ B2 ) )
= ( ( times_times @ A @ A2 @ B2 )
= C3 ) ) ) ) ).
% unit_eq_div2
thf(fact_1442_div__mult__unit2,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [C3: A,B2: A,A2: A] :
( ( dvd_dvd @ A @ C3 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ B2 @ A2 )
=> ( ( divide_divide @ A @ A2 @ ( times_times @ A @ B2 @ C3 ) )
= ( divide_divide @ A @ ( divide_divide @ A @ A2 @ B2 ) @ C3 ) ) ) ) ) ).
% div_mult_unit2
thf(fact_1443_unit__div__commute,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,A2: A,C3: A] :
( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( times_times @ A @ ( divide_divide @ A @ A2 @ B2 ) @ C3 )
= ( divide_divide @ A @ ( times_times @ A @ A2 @ C3 ) @ B2 ) ) ) ) ).
% unit_div_commute
thf(fact_1444_unit__div__mult__swap,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [C3: A,A2: A,B2: A] :
( ( dvd_dvd @ A @ C3 @ ( one_one @ A ) )
=> ( ( times_times @ A @ A2 @ ( divide_divide @ A @ B2 @ C3 ) )
= ( divide_divide @ A @ ( times_times @ A @ A2 @ B2 ) @ C3 ) ) ) ) ).
% unit_div_mult_swap
thf(fact_1445_is__unit__div__mult2__eq,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [B2: A,C3: A,A2: A] :
( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( dvd_dvd @ A @ C3 @ ( one_one @ A ) )
=> ( ( divide_divide @ A @ A2 @ ( times_times @ A @ B2 @ C3 ) )
= ( divide_divide @ A @ ( divide_divide @ A @ A2 @ B2 ) @ C3 ) ) ) ) ) ).
% is_unit_div_mult2_eq
thf(fact_1446_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_1447_is__unit__power__iff,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A,N: nat] :
( ( dvd_dvd @ A @ ( power_power @ A @ A2 @ N ) @ ( one_one @ A ) )
= ( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) )
| ( N
= ( zero_zero @ nat ) ) ) ) ) ).
% is_unit_power_iff
thf(fact_1448_div__less__mono,axiom,
! [A5: nat,B6: nat,N: nat] :
( ( ord_less @ nat @ A5 @ B6 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ( modulo_modulo @ nat @ A5 @ N )
= ( zero_zero @ nat ) )
=> ( ( ( modulo_modulo @ nat @ B6 @ N )
= ( zero_zero @ nat ) )
=> ( ord_less @ nat @ ( divide_divide @ nat @ A5 @ N ) @ ( divide_divide @ nat @ B6 @ N ) ) ) ) ) ) ).
% div_less_mono
thf(fact_1449_mod__eq__nat1E,axiom,
! [M2: nat,Q2: nat,N: nat] :
( ( ( modulo_modulo @ nat @ M2 @ Q2 )
= ( modulo_modulo @ nat @ N @ Q2 ) )
=> ( ( ord_less_eq @ nat @ N @ M2 )
=> ~ ! [S3: nat] :
( M2
!= ( plus_plus @ nat @ N @ ( times_times @ nat @ Q2 @ S3 ) ) ) ) ) ).
% mod_eq_nat1E
thf(fact_1450_mod__eq__nat2E,axiom,
! [M2: nat,Q2: nat,N: nat] :
( ( ( modulo_modulo @ nat @ M2 @ Q2 )
= ( modulo_modulo @ nat @ N @ Q2 ) )
=> ( ( ord_less_eq @ nat @ M2 @ N )
=> ~ ! [S3: nat] :
( N
!= ( plus_plus @ nat @ M2 @ ( times_times @ nat @ Q2 @ S3 ) ) ) ) ) ).
% mod_eq_nat2E
thf(fact_1451_nat__mod__eq__lemma,axiom,
! [X: nat,N: nat,Y: nat] :
( ( ( modulo_modulo @ nat @ X @ N )
= ( modulo_modulo @ nat @ Y @ N ) )
=> ( ( ord_less_eq @ nat @ Y @ X )
=> ? [Q3: nat] :
( X
= ( plus_plus @ nat @ Y @ ( times_times @ nat @ N @ Q3 ) ) ) ) ) ).
% nat_mod_eq_lemma
thf(fact_1452_dvd__imp__le,axiom,
! [K2: nat,N: nat] :
( ( dvd_dvd @ nat @ K2 @ N )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ord_less_eq @ nat @ K2 @ N ) ) ) ).
% dvd_imp_le
thf(fact_1453_mod__mult2__eq,axiom,
! [M2: nat,N: nat,Q2: nat] :
( ( modulo_modulo @ nat @ M2 @ ( times_times @ nat @ N @ Q2 ) )
= ( plus_plus @ nat @ ( times_times @ nat @ N @ ( modulo_modulo @ nat @ ( divide_divide @ nat @ M2 @ N ) @ Q2 ) ) @ ( modulo_modulo @ nat @ M2 @ N ) ) ) ).
% mod_mult2_eq
thf(fact_1454_div__mod__decomp,axiom,
! [A5: nat,N: nat] :
( A5
= ( plus_plus @ nat @ ( times_times @ nat @ ( divide_divide @ nat @ A5 @ N ) @ N ) @ ( modulo_modulo @ nat @ A5 @ N ) ) ) ).
% div_mod_decomp
thf(fact_1455_nat__mult__dvd__cancel1,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K2 )
=> ( ( dvd_dvd @ nat @ ( times_times @ nat @ K2 @ M2 ) @ ( times_times @ nat @ K2 @ N ) )
= ( dvd_dvd @ nat @ M2 @ N ) ) ) ).
% nat_mult_dvd_cancel1
thf(fact_1456_dvd__mult__cancel,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( dvd_dvd @ nat @ ( times_times @ nat @ K2 @ M2 ) @ ( times_times @ nat @ K2 @ N ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K2 )
=> ( dvd_dvd @ nat @ M2 @ N ) ) ) ).
% dvd_mult_cancel
thf(fact_1457_bezout__add__strong__nat,axiom,
! [A2: nat,B2: nat] :
( ( A2
!= ( zero_zero @ nat ) )
=> ? [D2: nat,X4: nat,Y3: nat] :
( ( dvd_dvd @ nat @ D2 @ A2 )
& ( dvd_dvd @ nat @ D2 @ B2 )
& ( ( times_times @ nat @ A2 @ X4 )
= ( plus_plus @ nat @ ( times_times @ nat @ B2 @ Y3 ) @ D2 ) ) ) ) ).
% bezout_add_strong_nat
thf(fact_1458_int__div__pos__eq,axiom,
! [A2: int,B2: int,Q2: int,R2: int] :
( ( A2
= ( plus_plus @ int @ ( times_times @ int @ B2 @ Q2 ) @ R2 ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ R2 )
=> ( ( ord_less @ int @ R2 @ B2 )
=> ( ( divide_divide @ int @ A2 @ B2 )
= Q2 ) ) ) ) ).
% int_div_pos_eq
thf(fact_1459_int__div__neg__eq,axiom,
! [A2: int,B2: int,Q2: int,R2: int] :
( ( A2
= ( plus_plus @ int @ ( times_times @ int @ B2 @ Q2 ) @ R2 ) )
=> ( ( ord_less_eq @ int @ R2 @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ B2 @ R2 )
=> ( ( divide_divide @ int @ A2 @ B2 )
= Q2 ) ) ) ) ).
% int_div_neg_eq
thf(fact_1460_split__zdiv,axiom,
! [P2: int > $o,N: int,K2: int] :
( ( P2 @ ( divide_divide @ int @ N @ K2 ) )
= ( ( ( K2
= ( zero_zero @ int ) )
=> ( P2 @ ( zero_zero @ int ) ) )
& ( ( ord_less @ int @ ( zero_zero @ int ) @ K2 )
=> ! [I4: int,J3: int] :
( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ J3 )
& ( ord_less @ int @ J3 @ K2 )
& ( N
= ( plus_plus @ int @ ( times_times @ int @ K2 @ I4 ) @ J3 ) ) )
=> ( P2 @ I4 ) ) )
& ( ( ord_less @ int @ K2 @ ( zero_zero @ int ) )
=> ! [I4: int,J3: int] :
( ( ( ord_less @ int @ K2 @ J3 )
& ( ord_less_eq @ int @ J3 @ ( zero_zero @ int ) )
& ( N
= ( plus_plus @ int @ ( times_times @ int @ K2 @ I4 ) @ J3 ) ) )
=> ( P2 @ I4 ) ) ) ) ) ).
% split_zdiv
thf(fact_1461_vebt__buildup_Oelims,axiom,
! [X: nat,Y: vEBT_VEBT] :
( ( ( vEBT_vebt_buildup @ X )
= Y )
=> ( ( ( X
= ( zero_zero @ nat ) )
=> ( Y
!= ( vEBT_Leaf @ $false @ $false ) ) )
=> ( ( ( X
= ( suc @ ( zero_zero @ nat ) ) )
=> ( Y
!= ( vEBT_Leaf @ $false @ $false ) ) )
=> ~ ! [Va2: nat] :
( ( X
= ( suc @ ( suc @ Va2 ) ) )
=> ~ ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va2 ) ) )
=> ( Y
= ( 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 ) ) )
=> ( Y
= ( 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.elims
thf(fact_1462_even__zero,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( zero_zero @ A ) ) ) ).
% even_zero
thf(fact_1463_is__unit__div__mult__cancel__right,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A,B2: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( divide_divide @ A @ A2 @ ( times_times @ A @ B2 @ A2 ) )
= ( divide_divide @ A @ ( one_one @ A ) @ B2 ) ) ) ) ) ).
% is_unit_div_mult_cancel_right
thf(fact_1464_is__unit__div__mult__cancel__left,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A,B2: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ B2 @ ( one_one @ A ) )
=> ( ( divide_divide @ A @ A2 @ ( times_times @ A @ A2 @ B2 ) )
= ( divide_divide @ A @ ( one_one @ A ) @ B2 ) ) ) ) ) ).
% is_unit_div_mult_cancel_left
thf(fact_1465_is__unitE,axiom,
! [A: $tType] :
( ( algebraic_semidom @ A )
=> ! [A2: A,C3: A] :
( ( dvd_dvd @ A @ A2 @ ( one_one @ A ) )
=> ~ ( ( A2
!= ( zero_zero @ A ) )
=> ! [B4: A] :
( ( B4
!= ( zero_zero @ A ) )
=> ( ( dvd_dvd @ A @ B4 @ ( one_one @ A ) )
=> ( ( ( divide_divide @ A @ ( one_one @ A ) @ A2 )
= B4 )
=> ( ( ( divide_divide @ A @ ( one_one @ A ) @ B4 )
= A2 )
=> ( ( ( times_times @ A @ A2 @ B4 )
= ( one_one @ A ) )
=> ( ( divide_divide @ A @ C3 @ A2 )
!= ( times_times @ A @ C3 @ B4 ) ) ) ) ) ) ) ) ) ) ).
% is_unitE
thf(fact_1466_odd__even__add,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A2: A,B2: A] :
( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
=> ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 )
=> ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ) ).
% odd_even_add
thf(fact_1467_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_1468_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_1469_split__mod,axiom,
! [P2: nat > $o,M2: nat,N: nat] :
( ( P2 @ ( modulo_modulo @ nat @ M2 @ N ) )
= ( ( ( N
= ( zero_zero @ nat ) )
=> ( P2 @ M2 ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ! [I4: nat,J3: nat] :
( ( ord_less @ nat @ J3 @ N )
=> ( ( M2
= ( plus_plus @ nat @ ( times_times @ nat @ N @ I4 ) @ J3 ) )
=> ( P2 @ J3 ) ) ) ) ) ) ).
% split_mod
thf(fact_1470_div2__even__ext__nat,axiom,
! [X: nat,Y: nat] :
( ( ( divide_divide @ nat @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( divide_divide @ nat @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
=> ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ X )
= ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Y ) )
=> ( X = Y ) ) ) ).
% div2_even_ext_nat
thf(fact_1471_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_1472_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_1473_power__dvd__imp__le,axiom,
! [I2: nat,M2: nat,N: nat] :
( ( dvd_dvd @ nat @ ( power_power @ nat @ I2 @ M2 ) @ ( power_power @ nat @ I2 @ N ) )
=> ( ( ord_less @ nat @ ( one_one @ nat ) @ I2 )
=> ( ord_less_eq @ nat @ M2 @ N ) ) ) ).
% power_dvd_imp_le
thf(fact_1474_unset__bit__Suc,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A2: A] :
( ( bit_se2638667681897837118et_bit @ A @ ( suc @ N ) @ A2 )
= ( plus_plus @ A @ ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se2638667681897837118et_bit @ A @ N @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% unset_bit_Suc
thf(fact_1475_set__bit__Suc,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A2: A] :
( ( bit_se5668285175392031749et_bit @ A @ ( suc @ N ) @ A2 )
= ( plus_plus @ A @ ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5668285175392031749et_bit @ A @ N @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% set_bit_Suc
thf(fact_1476_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [C3: A,A2: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( modulo_modulo @ A @ A2 @ ( times_times @ A @ B2 @ C3 ) )
= ( plus_plus @ A @ ( times_times @ A @ B2 @ ( modulo_modulo @ A @ ( divide_divide @ A @ A2 @ B2 ) @ C3 ) ) @ ( modulo_modulo @ A @ A2 @ B2 ) ) ) ) ) ).
% unique_euclidean_semiring_numeral_class.mod_mult2_eq
thf(fact_1477_power__mono__odd,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: nat,A2: A,B2: A] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ord_less_eq @ A @ ( power_power @ A @ A2 @ N ) @ ( power_power @ A @ B2 @ N ) ) ) ) ) ).
% power_mono_odd
thf(fact_1478_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_1479_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_1480_dvd__power__iff__le,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K2 )
=> ( ( dvd_dvd @ nat @ ( power_power @ nat @ K2 @ M2 ) @ ( power_power @ nat @ K2 @ N ) )
= ( ord_less_eq @ nat @ M2 @ N ) ) ) ).
% dvd_power_iff_le
thf(fact_1481_divmod__digit__0_I2_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [B2: A,A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less @ A @ ( modulo_modulo @ A @ A2 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) @ B2 )
=> ( ( modulo_modulo @ A @ A2 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) )
= ( modulo_modulo @ A @ A2 @ B2 ) ) ) ) ) ).
% divmod_digit_0(2)
thf(fact_1482_bits__stable__imp__add__self,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A2: A] :
( ( ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= A2 )
=> ( ( plus_plus @ A @ A2 @ ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
= ( zero_zero @ A ) ) ) ) ).
% bits_stable_imp_add_self
thf(fact_1483_VEBT__internal_Onaive__member_Osimps_I3_J,axiom,
! [Uy: option @ ( product_prod @ nat @ nat ),V: nat,TreeList: list @ vEBT_VEBT,S: vEBT_VEBT,X: nat] :
( ( vEBT_V5719532721284313246member @ ( vEBT_Node @ Uy @ ( suc @ V ) @ TreeList @ S ) @ X )
= ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ V ) @ ( 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 @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) ) ) ) ).
% VEBT_internal.naive_member.simps(3)
thf(fact_1484_div__exp__mod__exp__eq,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A2: A,N: nat,M2: nat] :
( ( modulo_modulo @ A @ ( divide_divide @ A @ A2 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M2 ) )
= ( divide_divide @ A @ ( modulo_modulo @ A @ A2 @ ( 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_1485_oddE,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [A2: A] :
( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
=> ~ ! [B4: A] :
( A2
!= ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B4 ) @ ( one_one @ A ) ) ) ) ) ).
% oddE
thf(fact_1486_zero__le__even__power,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: nat,A2: A] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A2 @ N ) ) ) ) ).
% zero_le_even_power
thf(fact_1487_zero__le__odd__power,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: nat,A2: A] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A2 @ N ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ) ).
% zero_le_odd_power
thf(fact_1488_zero__le__power__eq,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A,N: nat] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A2 @ N ) )
= ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
| ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ) ) ).
% zero_le_power_eq
thf(fact_1489_verit__le__mono__div,axiom,
! [A5: nat,B6: nat,N: nat] :
( ( ord_less @ nat @ A5 @ B6 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ord_less_eq @ nat
@ ( plus_plus @ nat @ ( divide_divide @ nat @ A5 @ N )
@ ( if @ nat
@ ( ( modulo_modulo @ nat @ B6 @ N )
= ( zero_zero @ nat ) )
@ ( one_one @ nat )
@ ( zero_zero @ nat ) ) )
@ ( divide_divide @ nat @ B6 @ N ) ) ) ) ).
% verit_le_mono_div
thf(fact_1490_VEBT__internal_Omembermima_Osimps_I5_J,axiom,
! [V: nat,TreeList: list @ vEBT_VEBT,Vd2: vEBT_VEBT,X: nat] :
( ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V ) @ TreeList @ Vd2 ) @ X )
= ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ V ) @ ( 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 @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) ) ) ) ).
% VEBT_internal.membermima.simps(5)
thf(fact_1491_even__set__encode__iff,axiom,
! [A5: set @ nat] :
( ( finite_finite2 @ nat @ A5 )
=> ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( nat_set_encode @ A5 ) )
= ( ~ ( member @ nat @ ( zero_zero @ nat ) @ A5 ) ) ) ) ).
% even_set_encode_iff
thf(fact_1492_divmod__digit__0_I1_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [B2: A,A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less @ A @ ( modulo_modulo @ A @ A2 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) @ B2 )
=> ( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A2 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) )
= ( divide_divide @ A @ A2 @ B2 ) ) ) ) ) ).
% divmod_digit_0(1)
thf(fact_1493_vebt__member_Osimps_I5_J,axiom,
! [Mi: nat,Ma: nat,Va: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
( ( vEBT_vebt_member @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X )
= ( ( X != Mi )
=> ( ( X != Ma )
=> ( ~ ( ord_less @ nat @ X @ Mi )
& ( ~ ( ord_less @ nat @ X @ Mi )
=> ( ~ ( ord_less @ nat @ Ma @ X )
& ( ~ ( ord_less @ nat @ Ma @ X )
=> ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( 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 @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) ) ) ) ) ) ) ) ) ) ).
% vebt_member.simps(5)
thf(fact_1494_VEBT__internal_Omembermima_Osimps_I4_J,axiom,
! [Mi: nat,Ma: nat,V: nat,TreeList: list @ vEBT_VEBT,Vc: vEBT_VEBT,X: nat] :
( ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ V ) @ TreeList @ Vc ) @ X )
= ( ( X = Mi )
| ( X = Ma )
| ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ V ) @ ( 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 @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ X @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ X @ ( divide_divide @ nat @ ( suc @ V ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) ) ) ) ) ).
% VEBT_internal.membermima.simps(4)
thf(fact_1495_zero__less__power__eq,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A,N: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( power_power @ A @ A2 @ N ) )
= ( ( N
= ( zero_zero @ nat ) )
| ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
& ( A2
!= ( zero_zero @ A ) ) )
| ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
& ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ) ) ).
% zero_less_power_eq
thf(fact_1496_VEBT__internal_Onaive__member_Oelims_I1_J,axiom,
! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
( ( ( vEBT_V5719532721284313246member @ X @ Xa2 )
= Y )
=> ( ! [A4: $o,B4: $o] :
( ( X
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ( Y
= ( ~ ( ( ( Xa2
= ( zero_zero @ nat ) )
=> A4 )
& ( ( Xa2
!= ( zero_zero @ nat ) )
=> ( ( ( Xa2
= ( one_one @ nat ) )
=> B4 )
& ( Xa2
= ( 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 ) )
=> Y )
=> ~ ! [Uy2: option @ ( product_prod @ nat @ nat ),V3: nat,TreeList2: list @ vEBT_VEBT] :
( ? [S3: vEBT_VEBT] :
( X
= ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList2 @ S3 ) )
=> ( Y
= ( ~ ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_V5719532721284313246member @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.naive_member.elims(1)
thf(fact_1497_VEBT__internal_Onaive__member_Oelims_I2_J,axiom,
! [X: vEBT_VEBT,Xa2: nat] :
( ( vEBT_V5719532721284313246member @ X @ Xa2 )
=> ( ! [A4: $o,B4: $o] :
( ( X
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ~ ( ( ( Xa2
= ( zero_zero @ nat ) )
=> A4 )
& ( ( Xa2
!= ( zero_zero @ nat ) )
=> ( ( ( Xa2
= ( one_one @ nat ) )
=> B4 )
& ( Xa2
= ( one_one @ nat ) ) ) ) ) )
=> ~ ! [Uy2: option @ ( product_prod @ nat @ nat ),V3: nat,TreeList2: list @ vEBT_VEBT] :
( ? [S3: vEBT_VEBT] :
( X
= ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList2 @ S3 ) )
=> ~ ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_V5719532721284313246member @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ).
% VEBT_internal.naive_member.elims(2)
thf(fact_1498_VEBT__internal_Onaive__member_Oelims_I3_J,axiom,
! [X: vEBT_VEBT,Xa2: nat] :
( ~ ( vEBT_V5719532721284313246member @ X @ Xa2 )
=> ( ! [A4: $o,B4: $o] :
( ( X
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ( ( ( Xa2
= ( zero_zero @ nat ) )
=> A4 )
& ( ( Xa2
!= ( zero_zero @ nat ) )
=> ( ( ( Xa2
= ( one_one @ nat ) )
=> B4 )
& ( Xa2
= ( 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 ) )
=> ~ ! [Uy2: option @ ( product_prod @ nat @ nat ),V3: nat,TreeList2: list @ vEBT_VEBT] :
( ? [S3: vEBT_VEBT] :
( X
= ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList2 @ S3 ) )
=> ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_V5719532721284313246member @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ).
% VEBT_internal.naive_member.elims(3)
thf(fact_1499_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_1500_VEBT__internal_Omembermima_Oelims_I2_J,axiom,
! [X: vEBT_VEBT,Xa2: nat] :
( ( vEBT_VEBT_membermima @ X @ Xa2 )
=> ( ! [Mi2: nat,Ma2: nat] :
( ? [Va3: list @ vEBT_VEBT,Vb2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ Va3 @ Vb2 ) )
=> ~ ( ( Xa2 = Mi2 )
| ( Xa2 = Ma2 ) ) )
=> ( ! [Mi2: nat,Ma2: nat,V3: nat,TreeList2: list @ vEBT_VEBT] :
( ? [Vc2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList2 @ Vc2 ) )
=> ~ ( ( Xa2 = Mi2 )
| ( Xa2 = Ma2 )
| ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) )
=> ~ ! [V3: nat,TreeList2: list @ vEBT_VEBT] :
( ? [Vd: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V3 ) @ TreeList2 @ Vd ) )
=> ~ ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ).
% VEBT_internal.membermima.elims(2)
thf(fact_1501_power__le__zero__eq,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A,N: nat] :
( ( ord_less_eq @ A @ ( power_power @ A @ A2 @ N ) @ ( zero_zero @ A ) )
= ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
& ( ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
& ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) )
| ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
& ( A2
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% power_le_zero_eq
thf(fact_1502_vebt__member_Oelims_I2_J,axiom,
! [X: vEBT_VEBT,Xa2: nat] :
( ( vEBT_vebt_member @ X @ Xa2 )
=> ( ! [A4: $o,B4: $o] :
( ( X
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ~ ( ( ( Xa2
= ( zero_zero @ nat ) )
=> A4 )
& ( ( Xa2
!= ( zero_zero @ nat ) )
=> ( ( ( Xa2
= ( one_one @ nat ) )
=> B4 )
& ( Xa2
= ( one_one @ nat ) ) ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list @ vEBT_VEBT] :
( ? [Summary2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary2 ) )
=> ~ ( ( Xa2 != Mi2 )
=> ( ( Xa2 != Ma2 )
=> ( ~ ( ord_less @ nat @ Xa2 @ Mi2 )
& ( ~ ( ord_less @ nat @ Xa2 @ Mi2 )
=> ( ~ ( ord_less @ nat @ Ma2 @ Xa2 )
& ( ~ ( ord_less @ nat @ Ma2 @ Xa2 )
=> ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_vebt_member @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_member.elims(2)
thf(fact_1503_VEBT__internal_Omembermima_Oelims_I1_J,axiom,
! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
( ( ( vEBT_VEBT_membermima @ X @ Xa2 )
= Y )
=> ( ( ? [Uu2: $o,Uv2: $o] :
( X
= ( vEBT_Leaf @ Uu2 @ Uv2 ) )
=> Y )
=> ( ( ? [Ux2: list @ vEBT_VEBT,Uy2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( zero_zero @ nat ) @ Ux2 @ Uy2 ) )
=> Y )
=> ( ! [Mi2: nat,Ma2: nat] :
( ? [Va3: list @ vEBT_VEBT,Vb2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ Va3 @ Vb2 ) )
=> ( Y
= ( ~ ( ( Xa2 = Mi2 )
| ( Xa2 = Ma2 ) ) ) ) )
=> ( ! [Mi2: nat,Ma2: nat,V3: nat,TreeList2: list @ vEBT_VEBT] :
( ? [Vc2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList2 @ Vc2 ) )
=> ( Y
= ( ~ ( ( Xa2 = Mi2 )
| ( Xa2 = Ma2 )
| ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) )
=> ~ ! [V3: nat,TreeList2: list @ vEBT_VEBT] :
( ? [Vd: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V3 ) @ TreeList2 @ Vd ) )
=> ( Y
= ( ~ ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.membermima.elims(1)
thf(fact_1504_VEBT__internal_Omembermima_Oelims_I3_J,axiom,
! [X: vEBT_VEBT,Xa2: nat] :
( ~ ( vEBT_VEBT_membermima @ X @ Xa2 )
=> ( ! [Uu2: $o,Uv2: $o] :
( X
!= ( vEBT_Leaf @ Uu2 @ Uv2 ) )
=> ( ! [Ux2: list @ vEBT_VEBT,Uy2: vEBT_VEBT] :
( X
!= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( zero_zero @ nat ) @ Ux2 @ Uy2 ) )
=> ( ! [Mi2: nat,Ma2: nat] :
( ? [Va3: list @ vEBT_VEBT,Vb2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ Va3 @ Vb2 ) )
=> ( ( Xa2 = Mi2 )
| ( Xa2 = Ma2 ) ) )
=> ( ! [Mi2: nat,Ma2: nat,V3: nat,TreeList2: list @ vEBT_VEBT] :
( ? [Vc2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList2 @ Vc2 ) )
=> ( ( Xa2 = Mi2 )
| ( Xa2 = Ma2 )
| ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) )
=> ~ ! [V3: nat,TreeList2: list @ vEBT_VEBT] :
( ? [Vd: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V3 ) @ TreeList2 @ Vd ) )
=> ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.membermima.elims(3)
thf(fact_1505_vebt__insert_Osimps_I5_J,axiom,
! [Mi: nat,Ma: nat,Va: nat,TreeList: list @ vEBT_VEBT,Summary: vEBT_VEBT,X: nat] :
( ( vEBT_vebt_insert @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) @ X )
= ( if @ vEBT_VEBT
@ ( ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList ) )
& ~ ( ( X = Mi )
| ( X = Ma ) ) )
@ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ ( if @ nat @ ( ord_less @ nat @ X @ Mi ) @ X @ Mi ) @ ( ord_max @ nat @ ( if @ nat @ ( ord_less @ nat @ X @ Mi ) @ Mi @ X ) @ Ma ) ) ) @ ( suc @ ( suc @ Va ) ) @ ( list_update @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_insert @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( ord_less @ nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( if @ vEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth @ vEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ X @ Mi ) @ Mi @ X ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ Summary ) )
@ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList @ Summary ) ) ) ).
% vebt_insert.simps(5)
thf(fact_1506_vebt__member_Oelims_I1_J,axiom,
! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
( ( ( vEBT_vebt_member @ X @ Xa2 )
= Y )
=> ( ! [A4: $o,B4: $o] :
( ( X
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ( Y
= ( ~ ( ( ( Xa2
= ( zero_zero @ nat ) )
=> A4 )
& ( ( Xa2
!= ( zero_zero @ nat ) )
=> ( ( ( Xa2
= ( one_one @ nat ) )
=> B4 )
& ( Xa2
= ( one_one @ nat ) ) ) ) ) ) ) )
=> ( ( ? [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) )
=> Y )
=> ( ( ? [V3: product_prod @ nat @ nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Uy2 @ Uz2 ) )
=> Y )
=> ( ( ? [V3: product_prod @ nat @ nat,Vb2: list @ vEBT_VEBT,Vc2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb2 @ Vc2 ) )
=> Y )
=> ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list @ vEBT_VEBT] :
( ? [Summary2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary2 ) )
=> ( Y
= ( ~ ( ( Xa2 != Mi2 )
=> ( ( Xa2 != Ma2 )
=> ( ~ ( ord_less @ nat @ Xa2 @ Mi2 )
& ( ~ ( ord_less @ nat @ Xa2 @ Mi2 )
=> ( ~ ( ord_less @ nat @ Ma2 @ Xa2 )
& ( ~ ( ord_less @ nat @ Ma2 @ Xa2 )
=> ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_vebt_member @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_member.elims(1)
thf(fact_1507_vebt__member_Oelims_I3_J,axiom,
! [X: vEBT_VEBT,Xa2: nat] :
( ~ ( vEBT_vebt_member @ X @ Xa2 )
=> ( ! [A4: $o,B4: $o] :
( ( X
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ( ( ( Xa2
= ( zero_zero @ nat ) )
=> A4 )
& ( ( Xa2
!= ( zero_zero @ nat ) )
=> ( ( ( Xa2
= ( one_one @ nat ) )
=> B4 )
& ( Xa2
= ( one_one @ nat ) ) ) ) ) )
=> ( ! [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] :
( X
!= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) )
=> ( ! [V3: product_prod @ nat @ nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( X
!= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Uy2 @ Uz2 ) )
=> ( ! [V3: product_prod @ nat @ nat,Vb2: list @ vEBT_VEBT,Vc2: vEBT_VEBT] :
( X
!= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb2 @ Vc2 ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list @ vEBT_VEBT] :
( ? [Summary2: vEBT_VEBT] :
( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary2 ) )
=> ( ( Xa2 != Mi2 )
=> ( ( Xa2 != Ma2 )
=> ( ~ ( ord_less @ nat @ Xa2 @ Mi2 )
& ( ~ ( ord_less @ nat @ Xa2 @ Mi2 )
=> ( ~ ( ord_less @ nat @ Ma2 @ Xa2 )
& ( ~ ( ord_less @ nat @ Ma2 @ Xa2 )
=> ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_vebt_member @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_member.elims(3)
thf(fact_1508_divmod__digit__1_I1_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ ( modulo_modulo @ A @ A2 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) )
=> ( ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A2 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) ) @ ( one_one @ A ) )
= ( divide_divide @ A @ A2 @ B2 ) ) ) ) ) ) ).
% divmod_digit_1(1)
thf(fact_1509_mult__less__iff1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z: A,X: A,Y: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Z )
=> ( ( ord_less @ A @ ( times_times @ A @ X @ Z ) @ ( times_times @ A @ Y @ Z ) )
= ( ord_less @ A @ X @ Y ) ) ) ) ).
% mult_less_iff1
thf(fact_1510_finite__nth__roots,axiom,
! [N: nat,C3: complex] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( finite_finite2 @ complex
@ ( collect @ complex
@ ^ [Z5: complex] :
( ( power_power @ complex @ Z5 @ N )
= C3 ) ) ) ) ).
% finite_nth_roots
thf(fact_1511_finite__roots__unity,axiom,
! [A: $tType] :
( ( ( real_V8999393235501362500lgebra @ A )
& ( idom @ A ) )
=> ! [N: nat] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ N )
=> ( finite_finite2 @ A
@ ( collect @ A
@ ^ [Z5: A] :
( ( power_power @ A @ Z5 @ N )
= ( one_one @ A ) ) ) ) ) ) ).
% finite_roots_unity
thf(fact_1512_vebt__insert_Opelims,axiom,
! [X: vEBT_VEBT,Xa2: nat,Y: vEBT_VEBT] :
( ( ( vEBT_vebt_insert @ X @ Xa2 )
= Y )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_insert_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa2 ) )
=> ( ! [A4: $o,B4: $o] :
( ( X
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ( ( ( ( Xa2
= ( zero_zero @ nat ) )
=> ( Y
= ( vEBT_Leaf @ $true @ B4 ) ) )
& ( ( Xa2
!= ( zero_zero @ nat ) )
=> ( ( ( Xa2
= ( one_one @ nat ) )
=> ( Y
= ( vEBT_Leaf @ A4 @ $true ) ) )
& ( ( Xa2
!= ( one_one @ nat ) )
=> ( Y
= ( vEBT_Leaf @ A4 @ B4 ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_insert_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A4 @ B4 ) @ Xa2 ) ) ) )
=> ( ! [Info2: option @ ( product_prod @ nat @ nat ),Ts2: list @ vEBT_VEBT,S3: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Info2 @ ( zero_zero @ nat ) @ Ts2 @ S3 ) )
=> ( ( Y
= ( vEBT_Node @ Info2 @ ( zero_zero @ nat ) @ Ts2 @ S3 ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_insert_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Info2 @ ( zero_zero @ nat ) @ Ts2 @ S3 ) @ Xa2 ) ) ) )
=> ( ! [Info2: option @ ( product_prod @ nat @ nat ),Ts2: list @ vEBT_VEBT,S3: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Info2 @ ( suc @ ( zero_zero @ nat ) ) @ Ts2 @ S3 ) )
=> ( ( Y
= ( vEBT_Node @ Info2 @ ( suc @ ( zero_zero @ nat ) ) @ Ts2 @ S3 ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_insert_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Info2 @ ( suc @ ( zero_zero @ nat ) ) @ Ts2 @ S3 ) @ Xa2 ) ) ) )
=> ( ! [V3: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ V3 ) ) @ TreeList2 @ Summary2 ) )
=> ( ( Y
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Xa2 @ Xa2 ) ) @ ( suc @ ( suc @ V3 ) ) @ TreeList2 @ Summary2 ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_insert_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ ( suc @ V3 ) ) @ TreeList2 @ Summary2 ) @ Xa2 ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary2 ) )
=> ( ( Y
= ( if @ vEBT_VEBT
@ ( ( ord_less @ nat @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
& ~ ( ( Xa2 = Mi2 )
| ( Xa2 = Ma2 ) ) )
@ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ ( if @ nat @ ( ord_less @ nat @ Xa2 @ Mi2 ) @ Xa2 @ Mi2 ) @ ( ord_max @ nat @ ( if @ nat @ ( ord_less @ nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ Ma2 ) ) ) @ ( suc @ ( suc @ Va2 ) ) @ ( list_update @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( vEBT_vebt_insert @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ ( if @ nat @ ( ord_less @ nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( if @ vEBT_VEBT @ ( vEBT_VEBT_minNull @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( vEBT_vebt_insert @ Summary2 @ ( vEBT_VEBT_high @ ( if @ nat @ ( ord_less @ nat @ Xa2 @ Mi2 ) @ Mi2 @ Xa2 ) @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ Summary2 ) )
@ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary2 ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_insert_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ).
% vebt_insert.pelims
thf(fact_1513_flip__bit__Suc,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A2: A] :
( ( bit_se8732182000553998342ip_bit @ A @ ( suc @ N ) @ A2 )
= ( plus_plus @ A @ ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se8732182000553998342ip_bit @ A @ N @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% flip_bit_Suc
thf(fact_1514_product__nth,axiom,
! [A: $tType,B: $tType,N: nat,Xs: list @ A,Ys: list @ B] :
( ( ord_less @ nat @ N @ ( times_times @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ B ) @ Ys ) ) )
=> ( ( nth @ ( product_prod @ A @ B ) @ ( product @ A @ B @ Xs @ Ys ) @ N )
= ( product_Pair @ A @ B @ ( nth @ A @ Xs @ ( divide_divide @ nat @ N @ ( size_size @ ( list @ B ) @ Ys ) ) ) @ ( nth @ B @ Ys @ ( modulo_modulo @ nat @ N @ ( size_size @ ( list @ B ) @ Ys ) ) ) ) ) ) ).
% product_nth
thf(fact_1515_prod_Ofinite__Collect__op,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [I6: set @ B,X: B > A,Y: B > A] :
( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [I4: B] :
( ( member @ B @ I4 @ I6 )
& ( ( X @ I4 )
!= ( one_one @ A ) ) ) ) )
=> ( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [I4: B] :
( ( member @ B @ I4 @ I6 )
& ( ( Y @ I4 )
!= ( one_one @ A ) ) ) ) )
=> ( finite_finite2 @ B
@ ( collect @ B
@ ^ [I4: B] :
( ( member @ B @ I4 @ I6 )
& ( ( times_times @ A @ ( X @ I4 ) @ ( Y @ I4 ) )
!= ( one_one @ A ) ) ) ) ) ) ) ) ).
% prod.finite_Collect_op
thf(fact_1516_sum_Ofinite__Collect__op,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [I6: set @ B,X: B > A,Y: B > A] :
( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [I4: B] :
( ( member @ B @ I4 @ I6 )
& ( ( X @ I4 )
!= ( zero_zero @ A ) ) ) ) )
=> ( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [I4: B] :
( ( member @ B @ I4 @ I6 )
& ( ( Y @ I4 )
!= ( zero_zero @ A ) ) ) ) )
=> ( finite_finite2 @ B
@ ( collect @ B
@ ^ [I4: B] :
( ( member @ B @ I4 @ I6 )
& ( ( plus_plus @ A @ ( X @ I4 ) @ ( Y @ I4 ) )
!= ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% sum.finite_Collect_op
thf(fact_1517_vebt__member_Opelims_I1_J,axiom,
! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
( ( ( vEBT_vebt_member @ X @ Xa2 )
= Y )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa2 ) )
=> ( ! [A4: $o,B4: $o] :
( ( X
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ( ( Y
= ( ( ( Xa2
= ( zero_zero @ nat ) )
=> A4 )
& ( ( Xa2
!= ( zero_zero @ nat ) )
=> ( ( ( Xa2
= ( one_one @ nat ) )
=> B4 )
& ( Xa2
= ( one_one @ nat ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A4 @ B4 ) @ Xa2 ) ) ) )
=> ( ! [Uu2: nat,Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ Uu2 @ Uv2 @ Uw2 ) )
=> ( ~ Y
=> ~ ( 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 ) @ Xa2 ) ) ) )
=> ( ! [V3: product_prod @ nat @ nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Uy2 @ Uz2 ) )
=> ( ~ Y
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Uy2 @ Uz2 ) @ Xa2 ) ) ) )
=> ( ! [V3: product_prod @ nat @ nat,Vb2: list @ vEBT_VEBT,Vc2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb2 @ Vc2 ) )
=> ( ~ Y
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb2 @ Vc2 ) @ Xa2 ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary2 ) )
=> ( ( Y
= ( ( Xa2 != Mi2 )
=> ( ( Xa2 != Ma2 )
=> ( ~ ( ord_less @ nat @ Xa2 @ Mi2 )
& ( ~ ( ord_less @ nat @ Xa2 @ Mi2 )
=> ( ~ ( ord_less @ nat @ Ma2 @ Xa2 )
& ( ~ ( ord_less @ nat @ Ma2 @ Xa2 )
=> ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_vebt_member @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( 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 @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ).
% vebt_member.pelims(1)
thf(fact_1518_vebt__member_Opelims_I3_J,axiom,
! [X: vEBT_VEBT,Xa2: nat] :
( ~ ( vEBT_vebt_member @ X @ Xa2 )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa2 ) )
=> ( ! [A4: $o,B4: $o] :
( ( X
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A4 @ B4 ) @ Xa2 ) )
=> ( ( ( Xa2
= ( zero_zero @ nat ) )
=> A4 )
& ( ( Xa2
!= ( zero_zero @ nat ) )
=> ( ( ( Xa2
= ( one_one @ nat ) )
=> B4 )
& ( Xa2
= ( 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 ) @ Xa2 ) ) )
=> ( ! [V3: product_prod @ nat @ nat,Uy2: list @ vEBT_VEBT,Uz2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Uy2 @ Uz2 ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( zero_zero @ nat ) @ Uy2 @ Uz2 ) @ Xa2 ) ) )
=> ( ! [V3: product_prod @ nat @ nat,Vb2: list @ vEBT_VEBT,Vc2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ V3 ) @ ( 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 ) @ V3 ) @ ( suc @ ( zero_zero @ nat ) ) @ Vb2 @ Vc2 ) @ Xa2 ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ 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 @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary2 ) @ Xa2 ) )
=> ( ( Xa2 != Mi2 )
=> ( ( Xa2 != Ma2 )
=> ( ~ ( ord_less @ nat @ Xa2 @ Mi2 )
& ( ~ ( ord_less @ nat @ Xa2 @ Mi2 )
=> ( ~ ( ord_less @ nat @ Ma2 @ Xa2 )
& ( ~ ( ord_less @ nat @ Ma2 @ Xa2 )
=> ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_vebt_member @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_member.pelims(3)
thf(fact_1519_flip__bit__negative__int__iff,axiom,
! [N: nat,K2: int] :
( ( ord_less @ int @ ( bit_se8732182000553998342ip_bit @ int @ N @ K2 ) @ ( zero_zero @ int ) )
= ( ord_less @ int @ K2 @ ( zero_zero @ int ) ) ) ).
% flip_bit_negative_int_iff
thf(fact_1520_mod__neg__neg__trivial,axiom,
! [K2: int,L: int] :
( ( ord_less_eq @ int @ K2 @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ L @ K2 )
=> ( ( modulo_modulo @ int @ K2 @ L )
= K2 ) ) ) ).
% mod_neg_neg_trivial
thf(fact_1521_mod__pos__pos__trivial,axiom,
! [K2: int,L: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K2 )
=> ( ( ord_less @ int @ K2 @ L )
=> ( ( modulo_modulo @ int @ K2 @ L )
= K2 ) ) ) ).
% mod_pos_pos_trivial
thf(fact_1522_length__product,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B] :
( ( size_size @ ( list @ ( product_prod @ A @ B ) ) @ ( product @ A @ B @ Xs @ Ys ) )
= ( times_times @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ B ) @ Ys ) ) ) ).
% length_product
thf(fact_1523_dvd__antisym,axiom,
! [M2: nat,N: nat] :
( ( dvd_dvd @ nat @ M2 @ N )
=> ( ( dvd_dvd @ nat @ N @ M2 )
=> ( M2 = N ) ) ) ).
% dvd_antisym
thf(fact_1524_bot2E,axiom,
! [A: $tType,B: $tType,X: A,Y: B] :
~ ( bot_bot @ ( A > B > $o ) @ X @ Y ) ).
% bot2E
thf(fact_1525_Euclidean__Division_Opos__mod__bound,axiom,
! [L: int,K2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ L )
=> ( ord_less @ int @ ( modulo_modulo @ int @ K2 @ L ) @ L ) ) ).
% Euclidean_Division.pos_mod_bound
thf(fact_1526_neg__mod__bound,axiom,
! [L: int,K2: int] :
( ( ord_less @ int @ L @ ( zero_zero @ int ) )
=> ( ord_less @ int @ L @ ( modulo_modulo @ int @ K2 @ L ) ) ) ).
% neg_mod_bound
thf(fact_1527_neg__mod__sign,axiom,
! [L: int,K2: int] :
( ( ord_less @ int @ L @ ( zero_zero @ int ) )
=> ( ord_less_eq @ int @ ( modulo_modulo @ int @ K2 @ L ) @ ( zero_zero @ int ) ) ) ).
% neg_mod_sign
thf(fact_1528_Euclidean__Division_Opos__mod__sign,axiom,
! [L: int,K2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ L )
=> ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( modulo_modulo @ int @ K2 @ L ) ) ) ).
% Euclidean_Division.pos_mod_sign
thf(fact_1529_zmod__trivial__iff,axiom,
! [I2: int,K2: int] :
( ( ( modulo_modulo @ int @ I2 @ K2 )
= I2 )
= ( ( K2
= ( zero_zero @ int ) )
| ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ I2 )
& ( ord_less @ int @ I2 @ K2 ) )
| ( ( ord_less_eq @ int @ I2 @ ( zero_zero @ int ) )
& ( ord_less @ int @ K2 @ I2 ) ) ) ) ).
% zmod_trivial_iff
thf(fact_1530_pos__mod__conj,axiom,
! [B2: int,A2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( modulo_modulo @ int @ A2 @ B2 ) )
& ( ord_less @ int @ ( modulo_modulo @ int @ A2 @ B2 ) @ B2 ) ) ) ).
% pos_mod_conj
thf(fact_1531_neg__mod__conj,axiom,
! [B2: int,A2: int] :
( ( ord_less @ int @ B2 @ ( zero_zero @ int ) )
=> ( ( ord_less_eq @ int @ ( modulo_modulo @ int @ A2 @ B2 ) @ ( zero_zero @ int ) )
& ( ord_less @ int @ B2 @ ( modulo_modulo @ int @ A2 @ B2 ) ) ) ) ).
% neg_mod_conj
thf(fact_1532_mod__int__pos__iff,axiom,
! [K2: int,L: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( modulo_modulo @ int @ K2 @ L ) )
= ( ( dvd_dvd @ int @ L @ K2 )
| ( ( L
= ( zero_zero @ int ) )
& ( ord_less_eq @ int @ ( zero_zero @ int ) @ K2 ) )
| ( ord_less @ int @ ( zero_zero @ int ) @ L ) ) ) ).
% mod_int_pos_iff
thf(fact_1533_zdiv__mono__strict,axiom,
! [A5: int,B6: int,N: int] :
( ( ord_less @ int @ A5 @ B6 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ N )
=> ( ( ( modulo_modulo @ int @ A5 @ N )
= ( zero_zero @ int ) )
=> ( ( ( modulo_modulo @ int @ B6 @ N )
= ( zero_zero @ int ) )
=> ( ord_less @ int @ ( divide_divide @ int @ A5 @ N ) @ ( divide_divide @ int @ B6 @ N ) ) ) ) ) ) ).
% zdiv_mono_strict
thf(fact_1534_mod__pos__neg__trivial,axiom,
! [K2: int,L: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ K2 )
=> ( ( ord_less_eq @ int @ ( plus_plus @ int @ K2 @ L ) @ ( zero_zero @ int ) )
=> ( ( modulo_modulo @ int @ K2 @ L )
= ( plus_plus @ int @ K2 @ L ) ) ) ) ).
% mod_pos_neg_trivial
thf(fact_1535_int__mod__pos__eq,axiom,
! [A2: int,B2: int,Q2: int,R2: int] :
( ( A2
= ( plus_plus @ int @ ( times_times @ int @ B2 @ Q2 ) @ R2 ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ R2 )
=> ( ( ord_less @ int @ R2 @ B2 )
=> ( ( modulo_modulo @ int @ A2 @ B2 )
= R2 ) ) ) ) ).
% int_mod_pos_eq
thf(fact_1536_int__mod__neg__eq,axiom,
! [A2: int,B2: int,Q2: int,R2: int] :
( ( A2
= ( plus_plus @ int @ ( times_times @ int @ B2 @ Q2 ) @ R2 ) )
=> ( ( ord_less_eq @ int @ R2 @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ B2 @ R2 )
=> ( ( modulo_modulo @ int @ A2 @ B2 )
= R2 ) ) ) ) ).
% int_mod_neg_eq
thf(fact_1537_split__zmod,axiom,
! [P2: int > $o,N: int,K2: int] :
( ( P2 @ ( modulo_modulo @ int @ N @ K2 ) )
= ( ( ( K2
= ( zero_zero @ int ) )
=> ( P2 @ N ) )
& ( ( ord_less @ int @ ( zero_zero @ int ) @ K2 )
=> ! [I4: int,J3: int] :
( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ J3 )
& ( ord_less @ int @ J3 @ K2 )
& ( N
= ( plus_plus @ int @ ( times_times @ int @ K2 @ I4 ) @ J3 ) ) )
=> ( P2 @ J3 ) ) )
& ( ( ord_less @ int @ K2 @ ( zero_zero @ int ) )
=> ! [I4: int,J3: int] :
( ( ( ord_less @ int @ K2 @ J3 )
& ( ord_less_eq @ int @ J3 @ ( zero_zero @ int ) )
& ( N
= ( plus_plus @ int @ ( times_times @ int @ K2 @ I4 ) @ J3 ) ) )
=> ( P2 @ J3 ) ) ) ) ) ).
% split_zmod
thf(fact_1538_split__neg__lemma,axiom,
! [K2: int,P2: int > int > $o,N: int] :
( ( ord_less @ int @ K2 @ ( zero_zero @ int ) )
=> ( ( P2 @ ( divide_divide @ int @ N @ K2 ) @ ( modulo_modulo @ int @ N @ K2 ) )
= ( ! [I4: int,J3: int] :
( ( ( ord_less @ int @ K2 @ J3 )
& ( ord_less_eq @ int @ J3 @ ( zero_zero @ int ) )
& ( N
= ( plus_plus @ int @ ( times_times @ int @ K2 @ I4 ) @ J3 ) ) )
=> ( P2 @ I4 @ J3 ) ) ) ) ) ).
% split_neg_lemma
thf(fact_1539_split__pos__lemma,axiom,
! [K2: int,P2: int > int > $o,N: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ K2 )
=> ( ( P2 @ ( divide_divide @ int @ N @ K2 ) @ ( modulo_modulo @ int @ N @ K2 ) )
= ( ! [I4: int,J3: int] :
( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ J3 )
& ( ord_less @ int @ J3 @ K2 )
& ( N
= ( plus_plus @ int @ ( times_times @ int @ K2 @ I4 ) @ J3 ) ) )
=> ( P2 @ I4 @ J3 ) ) ) ) ) ).
% split_pos_lemma
thf(fact_1540_verit__le__mono__div__int,axiom,
! [A5: int,B6: int,N: int] :
( ( ord_less @ int @ A5 @ B6 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ N )
=> ( ord_less_eq @ int
@ ( plus_plus @ int @ ( divide_divide @ int @ A5 @ N )
@ ( if @ int
@ ( ( modulo_modulo @ int @ B6 @ N )
= ( zero_zero @ int ) )
@ ( one_one @ int )
@ ( zero_zero @ int ) ) )
@ ( divide_divide @ int @ B6 @ N ) ) ) ) ).
% verit_le_mono_div_int
thf(fact_1541_even__flip__bit__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M2: nat,A2: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se8732182000553998342ip_bit @ A @ M2 @ A2 ) )
= ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
!= ( M2
= ( zero_zero @ nat ) ) ) ) ) ).
% even_flip_bit_iff
thf(fact_1542_vebt__member_Opelims_I2_J,axiom,
! [X: vEBT_VEBT,Xa2: nat] :
( ( vEBT_vebt_member @ X @ Xa2 )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa2 ) )
=> ( ! [A4: $o,B4: $o] :
( ( X
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_vebt_member_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A4 @ B4 ) @ Xa2 ) )
=> ~ ( ( ( Xa2
= ( zero_zero @ nat ) )
=> A4 )
& ( ( Xa2
!= ( zero_zero @ nat ) )
=> ( ( ( Xa2
= ( one_one @ nat ) )
=> B4 )
& ( Xa2
= ( one_one @ nat ) ) ) ) ) ) )
=> ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ 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 @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList2 @ Summary2 ) @ Xa2 ) )
=> ~ ( ( Xa2 != Mi2 )
=> ( ( Xa2 != Ma2 )
=> ( ~ ( ord_less @ nat @ Xa2 @ Mi2 )
& ( ~ ( ord_less @ nat @ Xa2 @ Mi2 )
=> ( ~ ( ord_less @ nat @ Ma2 @ Xa2 )
& ( ~ ( ord_less @ nat @ Ma2 @ Xa2 )
=> ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_vebt_member @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% vebt_member.pelims(2)
thf(fact_1543_VEBT__internal_Onaive__member_Opelims_I3_J,axiom,
! [X: vEBT_VEBT,Xa2: nat] :
( ~ ( vEBT_V5719532721284313246member @ X @ Xa2 )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa2 ) )
=> ( ! [A4: $o,B4: $o] :
( ( X
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A4 @ B4 ) @ Xa2 ) )
=> ( ( ( Xa2
= ( zero_zero @ nat ) )
=> A4 )
& ( ( Xa2
!= ( zero_zero @ nat ) )
=> ( ( ( Xa2
= ( one_one @ nat ) )
=> B4 )
& ( Xa2
= ( 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 ) @ Xa2 ) ) )
=> ~ ! [Uy2: option @ ( product_prod @ nat @ nat ),V3: nat,TreeList2: list @ vEBT_VEBT,S3: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList2 @ S3 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList2 @ S3 ) @ Xa2 ) )
=> ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_V5719532721284313246member @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.naive_member.pelims(3)
thf(fact_1544_VEBT__internal_Onaive__member_Opelims_I2_J,axiom,
! [X: vEBT_VEBT,Xa2: nat] :
( ( vEBT_V5719532721284313246member @ X @ Xa2 )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa2 ) )
=> ( ! [A4: $o,B4: $o] :
( ( X
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A4 @ B4 ) @ Xa2 ) )
=> ~ ( ( ( Xa2
= ( zero_zero @ nat ) )
=> A4 )
& ( ( Xa2
!= ( zero_zero @ nat ) )
=> ( ( ( Xa2
= ( one_one @ nat ) )
=> B4 )
& ( Xa2
= ( one_one @ nat ) ) ) ) ) ) )
=> ~ ! [Uy2: option @ ( product_prod @ nat @ nat ),V3: nat,TreeList2: list @ vEBT_VEBT,S3: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList2 @ S3 ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList2 @ S3 ) @ Xa2 ) )
=> ~ ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_V5719532721284313246member @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ) ).
% VEBT_internal.naive_member.pelims(2)
thf(fact_1545_VEBT__internal_Onaive__member_Opelims_I1_J,axiom,
! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
( ( ( vEBT_V5719532721284313246member @ X @ Xa2 )
= Y )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa2 ) )
=> ( ! [A4: $o,B4: $o] :
( ( X
= ( vEBT_Leaf @ A4 @ B4 ) )
=> ( ( Y
= ( ( ( Xa2
= ( zero_zero @ nat ) )
=> A4 )
& ( ( Xa2
!= ( zero_zero @ nat ) )
=> ( ( ( Xa2
= ( one_one @ nat ) )
=> B4 )
& ( Xa2
= ( one_one @ nat ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ A4 @ B4 ) @ Xa2 ) ) ) )
=> ( ! [Uu2: option @ ( product_prod @ nat @ nat ),Uv2: list @ vEBT_VEBT,Uw2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Uu2 @ ( zero_zero @ nat ) @ Uv2 @ Uw2 ) )
=> ( ~ Y
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Uu2 @ ( zero_zero @ nat ) @ Uv2 @ Uw2 ) @ Xa2 ) ) ) )
=> ~ ! [Uy2: option @ ( product_prod @ nat @ nat ),V3: nat,TreeList2: list @ vEBT_VEBT,S3: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList2 @ S3 ) )
=> ( ( Y
= ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_V5719532721284313246member @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V5765760719290551771er_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Uy2 @ ( suc @ V3 ) @ TreeList2 @ S3 ) @ Xa2 ) ) ) ) ) ) ) ) ).
% VEBT_internal.naive_member.pelims(1)
thf(fact_1546_VEBT__internal_Omembermima_Opelims_I3_J,axiom,
! [X: vEBT_VEBT,Xa2: nat] :
( ~ ( vEBT_VEBT_membermima @ X @ Xa2 )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa2 ) )
=> ( ! [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 ) @ Xa2 ) ) )
=> ( ! [Ux2: list @ vEBT_VEBT,Uy2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( zero_zero @ nat ) @ Ux2 @ Uy2 ) )
=> ~ ( 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 @ Uy2 ) @ Xa2 ) ) )
=> ( ! [Mi2: nat,Ma2: nat,Va3: list @ vEBT_VEBT,Vb2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ 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 @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ Va3 @ Vb2 ) @ Xa2 ) )
=> ( ( Xa2 = Mi2 )
| ( Xa2 = Ma2 ) ) ) )
=> ( ! [Mi2: nat,Ma2: nat,V3: nat,TreeList2: list @ vEBT_VEBT,Vc2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ V3 ) @ 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 @ Mi2 @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList2 @ Vc2 ) @ Xa2 ) )
=> ( ( Xa2 = Mi2 )
| ( Xa2 = Ma2 )
| ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) )
=> ~ ! [V3: nat,TreeList2: list @ vEBT_VEBT,Vd: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V3 ) @ TreeList2 @ Vd ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V3 ) @ TreeList2 @ Vd ) @ Xa2 ) )
=> ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.membermima.pelims(3)
thf(fact_1547_VEBT__internal_Omembermima_Opelims_I1_J,axiom,
! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
( ( ( vEBT_VEBT_membermima @ X @ Xa2 )
= Y )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa2 ) )
=> ( ! [Uu2: $o,Uv2: $o] :
( ( X
= ( vEBT_Leaf @ Uu2 @ Uv2 ) )
=> ( ~ Y
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa2 ) ) ) )
=> ( ! [Ux2: list @ vEBT_VEBT,Uy2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( zero_zero @ nat ) @ Ux2 @ Uy2 ) )
=> ( ~ Y
=> ~ ( 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 @ Uy2 ) @ Xa2 ) ) ) )
=> ( ! [Mi2: nat,Ma2: nat,Va3: list @ vEBT_VEBT,Vb2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ Va3 @ Vb2 ) )
=> ( ( Y
= ( ( Xa2 = Mi2 )
| ( Xa2 = 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 @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ Va3 @ Vb2 ) @ Xa2 ) ) ) )
=> ( ! [Mi2: nat,Ma2: nat,V3: nat,TreeList2: list @ vEBT_VEBT,Vc2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList2 @ Vc2 ) )
=> ( ( Y
= ( ( Xa2 = Mi2 )
| ( Xa2 = Ma2 )
| ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList2 @ Vc2 ) @ Xa2 ) ) ) )
=> ~ ! [V3: nat,TreeList2: list @ vEBT_VEBT,Vd: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V3 ) @ TreeList2 @ Vd ) )
=> ( ( Y
= ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V3 ) @ TreeList2 @ Vd ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.membermima.pelims(1)
thf(fact_1548_VEBT__internal_Omembermima_Opelims_I2_J,axiom,
! [X: vEBT_VEBT,Xa2: nat] :
( ( vEBT_VEBT_membermima @ X @ Xa2 )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa2 ) )
=> ( ! [Mi2: nat,Ma2: nat,Va3: list @ vEBT_VEBT,Vb2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ 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 @ Mi2 @ Ma2 ) ) @ ( zero_zero @ nat ) @ Va3 @ Vb2 ) @ Xa2 ) )
=> ~ ( ( Xa2 = Mi2 )
| ( Xa2 = Ma2 ) ) ) )
=> ( ! [Mi2: nat,Ma2: nat,V3: nat,TreeList2: list @ vEBT_VEBT,Vc2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ Mi2 @ Ma2 ) ) @ ( suc @ V3 ) @ 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 @ Mi2 @ Ma2 ) ) @ ( suc @ V3 ) @ TreeList2 @ Vc2 ) @ Xa2 ) )
=> ~ ( ( Xa2 = Mi2 )
| ( Xa2 = Ma2 )
| ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) )
=> ~ ! [V3: nat,TreeList2: list @ vEBT_VEBT,Vd: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V3 ) @ TreeList2 @ Vd ) )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_V4351362008482014158ma_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ ( none @ ( product_prod @ nat @ nat ) ) @ ( suc @ V3 ) @ TreeList2 @ Vd ) @ Xa2 ) )
=> ~ ( ( ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) )
=> ( vEBT_VEBT_membermima @ ( nth @ vEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( ord_less @ nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide @ nat @ ( suc @ V3 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 ) ) ) ) ) ) ) ) ) ).
% VEBT_internal.membermima.pelims(2)
thf(fact_1549_arcosh__1,axiom,
! [A: $tType] :
( ( ln @ A )
=> ( ( arcosh @ A @ ( one_one @ A ) )
= ( zero_zero @ A ) ) ) ).
% arcosh_1
thf(fact_1550_artanh__0,axiom,
! [A: $tType] :
( ( ( real_V3459762299906320749_field @ A )
& ( ln @ A ) )
=> ( ( artanh @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% artanh_0
thf(fact_1551_arsinh__0,axiom,
! [A: $tType] :
( ( ln @ A )
=> ( ( arsinh @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% arsinh_0
thf(fact_1552_vebt__buildup_Opelims,axiom,
! [X: nat,Y: vEBT_VEBT] :
( ( ( vEBT_vebt_buildup @ X )
= Y )
=> ( ( accp @ nat @ vEBT_v4011308405150292612up_rel @ X )
=> ( ( ( X
= ( zero_zero @ nat ) )
=> ( ( Y
= ( vEBT_Leaf @ $false @ $false ) )
=> ~ ( accp @ nat @ vEBT_v4011308405150292612up_rel @ ( zero_zero @ nat ) ) ) )
=> ( ( ( X
= ( suc @ ( zero_zero @ nat ) ) )
=> ( ( Y
= ( vEBT_Leaf @ $false @ $false ) )
=> ~ ( accp @ nat @ vEBT_v4011308405150292612up_rel @ ( suc @ ( zero_zero @ nat ) ) ) ) )
=> ~ ! [Va2: nat] :
( ( X
= ( suc @ ( suc @ Va2 ) ) )
=> ( ( ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( suc @ ( suc @ Va2 ) ) )
=> ( Y
= ( 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 ) ) )
=> ( Y
= ( 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 ) ) ) ) ) ) ) ) )
=> ~ ( accp @ nat @ vEBT_v4011308405150292612up_rel @ ( suc @ ( suc @ Va2 ) ) ) ) ) ) ) ) ) ).
% vebt_buildup.pelims
thf(fact_1553_flip__bit__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A] :
( ( bit_se8732182000553998342ip_bit @ A @ ( zero_zero @ nat ) @ A2 )
= ( plus_plus @ A @ ( zero_neq_one_of_bool @ A @ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% flip_bit_0
thf(fact_1554_signed__take__bit__Suc,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: nat,A2: A] :
( ( bit_ri4674362597316999326ke_bit @ A @ ( suc @ N ) @ A2 )
= ( plus_plus @ A @ ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_ri4674362597316999326ke_bit @ A @ N @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% signed_take_bit_Suc
thf(fact_1555_even__mult__exp__div__exp__iff,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A2: A,M2: nat,N: nat] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( divide_divide @ A @ ( times_times @ A @ A2 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ 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 @ A2 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ N @ M2 ) ) ) ) ) ) ) ) ).
% even_mult_exp_div_exp_iff
thf(fact_1556_num_Osize__gen_I2_J,axiom,
! [X2: num] :
( ( size_num @ ( bit0 @ X2 ) )
= ( plus_plus @ nat @ ( size_num @ X2 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% num.size_gen(2)
thf(fact_1557_minus__apply,axiom,
! [B: $tType,A: $tType] :
( ( minus @ B )
=> ( ( minus_minus @ ( A > B ) )
= ( ^ [A7: A > B,B8: A > B,X5: A] : ( minus_minus @ B @ ( A7 @ X5 ) @ ( B8 @ X5 ) ) ) ) ) ).
% minus_apply
thf(fact_1558_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add @ A )
=> ! [A2: A] :
( ( minus_minus @ A @ A2 @ A2 )
= ( zero_zero @ A ) ) ) ).
% cancel_comm_monoid_add_class.diff_cancel
thf(fact_1559_diff__zero,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add @ A )
=> ! [A2: A] :
( ( minus_minus @ A @ A2 @ ( zero_zero @ A ) )
= A2 ) ) ).
% diff_zero
thf(fact_1560_zero__diff,axiom,
! [A: $tType] :
( ( comm_monoid_diff @ A )
=> ! [A2: A] :
( ( minus_minus @ A @ ( zero_zero @ A ) @ A2 )
= ( zero_zero @ A ) ) ) ).
% zero_diff
thf(fact_1561_diff__0__right,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A] :
( ( minus_minus @ A @ A2 @ ( zero_zero @ A ) )
= A2 ) ) ).
% diff_0_right
thf(fact_1562_diff__self,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A] :
( ( minus_minus @ A @ A2 @ A2 )
= ( zero_zero @ A ) ) ) ).
% diff_self
thf(fact_1563_add__diff__cancel__right_H,axiom,
! [A: $tType] :
( ( cancel2418104881723323429up_add @ A )
=> ! [A2: A,B2: A] :
( ( minus_minus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ B2 )
= A2 ) ) ).
% add_diff_cancel_right'
thf(fact_1564_add__diff__cancel__right,axiom,
! [A: $tType] :
( ( cancel2418104881723323429up_add @ A )
=> ! [A2: A,C3: A,B2: A] :
( ( minus_minus @ A @ ( plus_plus @ A @ A2 @ C3 ) @ ( plus_plus @ A @ B2 @ C3 ) )
= ( minus_minus @ A @ A2 @ B2 ) ) ) ).
% add_diff_cancel_right
thf(fact_1565_add__diff__cancel__left_H,axiom,
! [A: $tType] :
( ( cancel2418104881723323429up_add @ A )
=> ! [A2: A,B2: A] :
( ( minus_minus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ A2 )
= B2 ) ) ).
% add_diff_cancel_left'
thf(fact_1566_add__diff__cancel__left,axiom,
! [A: $tType] :
( ( cancel2418104881723323429up_add @ A )
=> ! [C3: A,A2: A,B2: A] :
( ( minus_minus @ A @ ( plus_plus @ A @ C3 @ A2 ) @ ( plus_plus @ A @ C3 @ B2 ) )
= ( minus_minus @ A @ A2 @ B2 ) ) ) ).
% add_diff_cancel_left
thf(fact_1567_diff__add__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A] :
( ( plus_plus @ A @ ( minus_minus @ A @ A2 @ B2 ) @ B2 )
= A2 ) ) ).
% diff_add_cancel
thf(fact_1568_add__diff__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A] :
( ( minus_minus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ B2 )
= A2 ) ) ).
% add_diff_cancel
thf(fact_1569_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_1570_Suc__diff__diff,axiom,
! [M2: nat,N: nat,K2: nat] :
( ( minus_minus @ nat @ ( minus_minus @ nat @ ( suc @ M2 ) @ N ) @ ( suc @ K2 ) )
= ( minus_minus @ nat @ ( minus_minus @ nat @ M2 @ N ) @ K2 ) ) ).
% Suc_diff_diff
thf(fact_1571_diff__self__eq__0,axiom,
! [M2: nat] :
( ( minus_minus @ nat @ M2 @ M2 )
= ( zero_zero @ nat ) ) ).
% diff_self_eq_0
thf(fact_1572_diff__0__eq__0,axiom,
! [N: nat] :
( ( minus_minus @ nat @ ( zero_zero @ nat ) @ N )
= ( zero_zero @ nat ) ) ).
% diff_0_eq_0
thf(fact_1573_diff__diff__cancel,axiom,
! [I2: nat,N: nat] :
( ( ord_less_eq @ nat @ I2 @ N )
=> ( ( minus_minus @ nat @ N @ ( minus_minus @ nat @ N @ I2 ) )
= I2 ) ) ).
% diff_diff_cancel
thf(fact_1574_diff__diff__left,axiom,
! [I2: nat,J: nat,K2: nat] :
( ( minus_minus @ nat @ ( minus_minus @ nat @ I2 @ J ) @ K2 )
= ( minus_minus @ nat @ I2 @ ( plus_plus @ nat @ J @ K2 ) ) ) ).
% diff_diff_left
thf(fact_1575_of__bool__less__eq__iff,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [P2: $o,Q: $o] :
( ( ord_less_eq @ A @ ( zero_neq_one_of_bool @ A @ P2 ) @ ( zero_neq_one_of_bool @ A @ Q ) )
= ( P2
=> Q ) ) ) ).
% of_bool_less_eq_iff
thf(fact_1576_of__bool__eq__0__iff,axiom,
! [A: $tType] :
( ( zero_neq_one @ A )
=> ! [P2: $o] :
( ( ( zero_neq_one_of_bool @ A @ P2 )
= ( zero_zero @ A ) )
= ~ P2 ) ) ).
% of_bool_eq_0_iff
thf(fact_1577_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_1578_of__bool__less__iff,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [P2: $o,Q: $o] :
( ( ord_less @ A @ ( zero_neq_one_of_bool @ A @ P2 ) @ ( zero_neq_one_of_bool @ A @ Q ) )
= ( ~ P2
& Q ) ) ) ).
% of_bool_less_iff
thf(fact_1579_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_1580_of__bool__eq__1__iff,axiom,
! [A: $tType] :
( ( zero_neq_one @ A )
=> ! [P2: $o] :
( ( ( zero_neq_one_of_bool @ A @ P2 )
= ( one_one @ A ) )
= P2 ) ) ).
% of_bool_eq_1_iff
thf(fact_1581_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_1582_of__bool__or__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [P2: $o,Q: $o] :
( ( zero_neq_one_of_bool @ A
@ ( P2
| Q ) )
= ( ord_max @ A @ ( zero_neq_one_of_bool @ A @ P2 ) @ ( zero_neq_one_of_bool @ A @ Q ) ) ) ) ).
% of_bool_or_iff
thf(fact_1583_diff__ge__0__iff__ge,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( minus_minus @ A @ A2 @ B2 ) )
= ( ord_less_eq @ A @ B2 @ A2 ) ) ) ).
% diff_ge_0_iff_ge
thf(fact_1584_diff__gt__0__iff__gt,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( minus_minus @ A @ A2 @ B2 ) )
= ( ord_less @ A @ B2 @ A2 ) ) ) ).
% diff_gt_0_iff_gt
thf(fact_1585_le__add__diff__inverse2,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [B2: A,A2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( plus_plus @ A @ ( minus_minus @ A @ A2 @ B2 ) @ B2 )
= A2 ) ) ) ).
% le_add_diff_inverse2
thf(fact_1586_le__add__diff__inverse,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [B2: A,A2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( plus_plus @ A @ B2 @ ( minus_minus @ A @ A2 @ B2 ) )
= A2 ) ) ) ).
% le_add_diff_inverse
thf(fact_1587_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_1588_diff__add__zero,axiom,
! [A: $tType] :
( ( comm_monoid_diff @ A )
=> ! [A2: A,B2: A] :
( ( minus_minus @ A @ A2 @ ( plus_plus @ A @ A2 @ B2 ) )
= ( zero_zero @ A ) ) ) ).
% diff_add_zero
thf(fact_1589_div__diff,axiom,
! [A: $tType] :
( ( idom_modulo @ A )
=> ! [C3: A,A2: A,B2: A] :
( ( dvd_dvd @ A @ C3 @ A2 )
=> ( ( dvd_dvd @ A @ C3 @ B2 )
=> ( ( divide_divide @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C3 )
= ( minus_minus @ A @ ( divide_divide @ A @ A2 @ C3 ) @ ( divide_divide @ A @ B2 @ C3 ) ) ) ) ) ) ).
% div_diff
thf(fact_1590_zero__less__of__bool__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [P2: $o] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( zero_neq_one_of_bool @ A @ P2 ) )
= P2 ) ) ).
% zero_less_of_bool_iff
thf(fact_1591_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_1592_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_1593_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_1594_of__bool__less__one__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [P2: $o] :
( ( ord_less @ A @ ( zero_neq_one_of_bool @ A @ P2 ) @ ( one_one @ A ) )
= ~ P2 ) ) ).
% of_bool_less_one_iff
thf(fact_1595_of__bool__not__iff,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [P2: $o] :
( ( zero_neq_one_of_bool @ A @ ~ P2 )
= ( minus_minus @ A @ ( one_one @ A ) @ ( zero_neq_one_of_bool @ A @ P2 ) ) ) ) ).
% of_bool_not_iff
thf(fact_1596_Nat_Odiff__diff__right,axiom,
! [K2: nat,J: nat,I2: nat] :
( ( ord_less_eq @ nat @ K2 @ J )
=> ( ( minus_minus @ nat @ I2 @ ( minus_minus @ nat @ J @ K2 ) )
= ( minus_minus @ nat @ ( plus_plus @ nat @ I2 @ K2 ) @ J ) ) ) ).
% Nat.diff_diff_right
thf(fact_1597_Nat_Oadd__diff__assoc2,axiom,
! [K2: nat,J: nat,I2: nat] :
( ( ord_less_eq @ nat @ K2 @ J )
=> ( ( plus_plus @ nat @ ( minus_minus @ nat @ J @ K2 ) @ I2 )
= ( minus_minus @ nat @ ( plus_plus @ nat @ J @ I2 ) @ K2 ) ) ) ).
% Nat.add_diff_assoc2
thf(fact_1598_Nat_Oadd__diff__assoc,axiom,
! [K2: nat,J: nat,I2: nat] :
( ( ord_less_eq @ nat @ K2 @ J )
=> ( ( plus_plus @ nat @ I2 @ ( minus_minus @ nat @ J @ K2 ) )
= ( minus_minus @ nat @ ( plus_plus @ nat @ I2 @ J ) @ K2 ) ) ) ).
% Nat.add_diff_assoc
thf(fact_1599_diff__Suc__1,axiom,
! [N: nat] :
( ( minus_minus @ nat @ ( suc @ N ) @ ( one_one @ nat ) )
= N ) ).
% diff_Suc_1
thf(fact_1600_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_1601_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_1602_diff__Suc__diff__eq2,axiom,
! [K2: nat,J: nat,I2: nat] :
( ( ord_less_eq @ nat @ K2 @ J )
=> ( ( minus_minus @ nat @ ( suc @ ( minus_minus @ nat @ J @ K2 ) ) @ I2 )
= ( minus_minus @ nat @ ( suc @ J ) @ ( plus_plus @ nat @ K2 @ I2 ) ) ) ) ).
% diff_Suc_diff_eq2
thf(fact_1603_diff__Suc__diff__eq1,axiom,
! [K2: nat,J: nat,I2: nat] :
( ( ord_less_eq @ nat @ K2 @ J )
=> ( ( minus_minus @ nat @ I2 @ ( suc @ ( minus_minus @ nat @ J @ K2 ) ) )
= ( minus_minus @ nat @ ( plus_plus @ nat @ I2 @ K2 ) @ ( suc @ J ) ) ) ) ).
% diff_Suc_diff_eq1
thf(fact_1604_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_1605_even__diff,axiom,
! [A: $tType] :
( ( ring_parity @ A )
=> ! [A2: A,B2: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( minus_minus @ A @ A2 @ B2 ) )
= ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ).
% even_diff
thf(fact_1606_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_1607_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_1608_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_1609_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_1610_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_1611_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_1612_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_1613_of__bool__eq__iff,axiom,
! [A: $tType] :
( ( zero_neq_one @ A )
=> ! [P: $o,Q2: $o] :
( ( ( zero_neq_one_of_bool @ A @ P )
= ( zero_neq_one_of_bool @ A @ Q2 ) )
= ( P = Q2 ) ) ) ).
% of_bool_eq_iff
thf(fact_1614_fun__diff__def,axiom,
! [B: $tType,A: $tType] :
( ( minus @ B )
=> ( ( minus_minus @ ( A > B ) )
= ( ^ [A7: A > B,B8: A > B,X5: A] : ( minus_minus @ B @ ( A7 @ X5 ) @ ( B8 @ X5 ) ) ) ) ) ).
% fun_diff_def
thf(fact_1615_diff__eq__diff__eq,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A,C3: A,D3: A] :
( ( ( minus_minus @ A @ A2 @ B2 )
= ( minus_minus @ A @ C3 @ D3 ) )
=> ( ( A2 = B2 )
= ( C3 = D3 ) ) ) ) ).
% diff_eq_diff_eq
thf(fact_1616_diff__right__commute,axiom,
! [A: $tType] :
( ( cancel2418104881723323429up_add @ A )
=> ! [A2: A,C3: A,B2: A] :
( ( minus_minus @ A @ ( minus_minus @ A @ A2 @ C3 ) @ B2 )
= ( minus_minus @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C3 ) ) ) ).
% diff_right_commute
thf(fact_1617_diff__commute,axiom,
! [I2: nat,J: nat,K2: nat] :
( ( minus_minus @ nat @ ( minus_minus @ nat @ I2 @ J ) @ K2 )
= ( minus_minus @ nat @ ( minus_minus @ nat @ I2 @ K2 ) @ J ) ) ).
% diff_commute
thf(fact_1618_of__bool__conj,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [P2: $o,Q: $o] :
( ( zero_neq_one_of_bool @ A
@ ( P2
& Q ) )
= ( times_times @ A @ ( zero_neq_one_of_bool @ A @ P2 ) @ ( zero_neq_one_of_bool @ A @ Q ) ) ) ) ).
% of_bool_conj
thf(fact_1619_diff__eq__diff__less__eq,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B2: A,C3: A,D3: A] :
( ( ( minus_minus @ A @ A2 @ B2 )
= ( minus_minus @ A @ C3 @ D3 ) )
=> ( ( ord_less_eq @ A @ A2 @ B2 )
= ( ord_less_eq @ A @ C3 @ D3 ) ) ) ) ).
% diff_eq_diff_less_eq
thf(fact_1620_diff__right__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ A2 @ C3 ) @ ( minus_minus @ A @ B2 @ C3 ) ) ) ) ).
% diff_right_mono
thf(fact_1621_diff__left__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [B2: A,A2: A,C3: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ C3 @ A2 ) @ ( minus_minus @ A @ C3 @ B2 ) ) ) ) ).
% diff_left_mono
thf(fact_1622_diff__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B2: A,D3: A,C3: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ D3 @ C3 )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ A2 @ C3 ) @ ( minus_minus @ A @ B2 @ D3 ) ) ) ) ) ).
% diff_mono
thf(fact_1623_eq__iff__diff__eq__0,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ( ( ^ [Y5: A,Z2: A] : Y5 = Z2 )
= ( ^ [A6: A,B5: A] :
( ( minus_minus @ A @ A6 @ B5 )
= ( zero_zero @ A ) ) ) ) ) ).
% eq_iff_diff_eq_0
thf(fact_1624_diff__strict__right__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ord_less @ A @ ( minus_minus @ A @ A2 @ C3 ) @ ( minus_minus @ A @ B2 @ C3 ) ) ) ) ).
% diff_strict_right_mono
thf(fact_1625_diff__strict__left__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [B2: A,A2: A,C3: A] :
( ( ord_less @ A @ B2 @ A2 )
=> ( ord_less @ A @ ( minus_minus @ A @ C3 @ A2 ) @ ( minus_minus @ A @ C3 @ B2 ) ) ) ) ).
% diff_strict_left_mono
thf(fact_1626_diff__eq__diff__less,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B2: A,C3: A,D3: A] :
( ( ( minus_minus @ A @ A2 @ B2 )
= ( minus_minus @ A @ C3 @ D3 ) )
=> ( ( ord_less @ A @ A2 @ B2 )
= ( ord_less @ A @ C3 @ D3 ) ) ) ) ).
% diff_eq_diff_less
thf(fact_1627_diff__strict__mono,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B2: A,D3: A,C3: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ D3 @ C3 )
=> ( ord_less @ A @ ( minus_minus @ A @ A2 @ C3 ) @ ( minus_minus @ A @ B2 @ D3 ) ) ) ) ) ).
% diff_strict_mono
thf(fact_1628_left__diff__distrib,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( times_times @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C3 )
= ( minus_minus @ A @ ( times_times @ A @ A2 @ C3 ) @ ( times_times @ A @ B2 @ C3 ) ) ) ) ).
% left_diff_distrib
thf(fact_1629_right__diff__distrib,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( times_times @ A @ A2 @ ( minus_minus @ A @ B2 @ C3 ) )
= ( minus_minus @ A @ ( times_times @ A @ A2 @ B2 ) @ ( times_times @ A @ A2 @ C3 ) ) ) ) ).
% right_diff_distrib
thf(fact_1630_left__diff__distrib_H,axiom,
! [A: $tType] :
( ( comm_s4317794764714335236cancel @ A )
=> ! [B2: A,C3: A,A2: A] :
( ( times_times @ A @ ( minus_minus @ A @ B2 @ C3 ) @ A2 )
= ( minus_minus @ A @ ( times_times @ A @ B2 @ A2 ) @ ( times_times @ A @ C3 @ A2 ) ) ) ) ).
% left_diff_distrib'
thf(fact_1631_right__diff__distrib_H,axiom,
! [A: $tType] :
( ( comm_s4317794764714335236cancel @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( times_times @ A @ A2 @ ( minus_minus @ A @ B2 @ C3 ) )
= ( minus_minus @ A @ ( times_times @ A @ A2 @ B2 ) @ ( times_times @ A @ A2 @ C3 ) ) ) ) ).
% right_diff_distrib'
thf(fact_1632_diff__diff__eq,axiom,
! [A: $tType] :
( ( cancel2418104881723323429up_add @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( minus_minus @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C3 )
= ( minus_minus @ A @ A2 @ ( plus_plus @ A @ B2 @ C3 ) ) ) ) ).
% diff_diff_eq
thf(fact_1633_add__implies__diff,axiom,
! [A: $tType] :
( ( cancel1802427076303600483id_add @ A )
=> ! [C3: A,B2: A,A2: A] :
( ( ( plus_plus @ A @ C3 @ B2 )
= A2 )
=> ( C3
= ( minus_minus @ A @ A2 @ B2 ) ) ) ) ).
% add_implies_diff
thf(fact_1634_diff__add__eq__diff__diff__swap,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( minus_minus @ A @ A2 @ ( plus_plus @ A @ B2 @ C3 ) )
= ( minus_minus @ A @ ( minus_minus @ A @ A2 @ C3 ) @ B2 ) ) ) ).
% diff_add_eq_diff_diff_swap
thf(fact_1635_diff__add__eq,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( plus_plus @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C3 )
= ( minus_minus @ A @ ( plus_plus @ A @ A2 @ C3 ) @ B2 ) ) ) ).
% diff_add_eq
thf(fact_1636_diff__diff__eq2,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( minus_minus @ A @ A2 @ ( minus_minus @ A @ B2 @ C3 ) )
= ( minus_minus @ A @ ( plus_plus @ A @ A2 @ C3 ) @ B2 ) ) ) ).
% diff_diff_eq2
thf(fact_1637_add__diff__eq,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( plus_plus @ A @ A2 @ ( minus_minus @ A @ B2 @ C3 ) )
= ( minus_minus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C3 ) ) ) ).
% add_diff_eq
thf(fact_1638_eq__diff__eq,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,C3: A,B2: A] :
( ( A2
= ( minus_minus @ A @ C3 @ B2 ) )
= ( ( plus_plus @ A @ A2 @ B2 )
= C3 ) ) ) ).
% eq_diff_eq
thf(fact_1639_diff__eq__eq,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( ( minus_minus @ A @ A2 @ B2 )
= C3 )
= ( A2
= ( plus_plus @ A @ C3 @ B2 ) ) ) ) ).
% diff_eq_eq
thf(fact_1640_group__cancel_Osub1,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A5: A,K2: A,A2: A,B2: A] :
( ( A5
= ( plus_plus @ A @ K2 @ A2 ) )
=> ( ( minus_minus @ A @ A5 @ B2 )
= ( plus_plus @ A @ K2 @ ( minus_minus @ A @ A2 @ B2 ) ) ) ) ) ).
% group_cancel.sub1
thf(fact_1641_add__diff__add,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A2: A,C3: A,B2: A,D3: A] :
( ( minus_minus @ A @ ( plus_plus @ A @ A2 @ C3 ) @ ( plus_plus @ A @ B2 @ D3 ) )
= ( plus_plus @ A @ ( minus_minus @ A @ A2 @ B2 ) @ ( minus_minus @ A @ C3 @ D3 ) ) ) ) ).
% add_diff_add
thf(fact_1642_diff__divide__distrib,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( divide_divide @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C3 )
= ( minus_minus @ A @ ( divide_divide @ A @ A2 @ C3 ) @ ( divide_divide @ A @ B2 @ C3 ) ) ) ) ).
% diff_divide_distrib
thf(fact_1643_dvd__diff,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [X: A,Y: A,Z: A] :
( ( dvd_dvd @ A @ X @ Y )
=> ( ( dvd_dvd @ A @ X @ Z )
=> ( dvd_dvd @ A @ X @ ( minus_minus @ A @ Y @ Z ) ) ) ) ) ).
% dvd_diff
thf(fact_1644_zero__induct__lemma,axiom,
! [P2: nat > $o,K2: nat,I2: nat] :
( ( P2 @ K2 )
=> ( ! [N3: nat] :
( ( P2 @ ( suc @ N3 ) )
=> ( P2 @ N3 ) )
=> ( P2 @ ( minus_minus @ nat @ K2 @ I2 ) ) ) ) ).
% zero_induct_lemma
thf(fact_1645_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_1646_minus__nat_Odiff__0,axiom,
! [M2: nat] :
( ( minus_minus @ nat @ M2 @ ( zero_zero @ nat ) )
= M2 ) ).
% minus_nat.diff_0
thf(fact_1647_less__imp__diff__less,axiom,
! [J: nat,K2: nat,N: nat] :
( ( ord_less @ nat @ J @ K2 )
=> ( ord_less @ nat @ ( minus_minus @ nat @ J @ N ) @ K2 ) ) ).
% less_imp_diff_less
thf(fact_1648_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_1649_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_1650_le__diff__iff_H,axiom,
! [A2: nat,C3: nat,B2: nat] :
( ( ord_less_eq @ nat @ A2 @ C3 )
=> ( ( ord_less_eq @ nat @ B2 @ C3 )
=> ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ C3 @ A2 ) @ ( minus_minus @ nat @ C3 @ B2 ) )
= ( ord_less_eq @ nat @ B2 @ A2 ) ) ) ) ).
% le_diff_iff'
thf(fact_1651_diff__le__self,axiom,
! [M2: nat,N: nat] : ( ord_less_eq @ nat @ ( minus_minus @ nat @ M2 @ N ) @ M2 ) ).
% diff_le_self
thf(fact_1652_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_1653_Nat_Odiff__diff__eq,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( ord_less_eq @ nat @ K2 @ M2 )
=> ( ( ord_less_eq @ nat @ K2 @ N )
=> ( ( minus_minus @ nat @ ( minus_minus @ nat @ M2 @ K2 ) @ ( minus_minus @ nat @ N @ K2 ) )
= ( minus_minus @ nat @ M2 @ N ) ) ) ) ).
% Nat.diff_diff_eq
thf(fact_1654_le__diff__iff,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( ord_less_eq @ nat @ K2 @ M2 )
=> ( ( ord_less_eq @ nat @ K2 @ N )
=> ( ( ord_less_eq @ nat @ ( minus_minus @ nat @ M2 @ K2 ) @ ( minus_minus @ nat @ N @ K2 ) )
= ( ord_less_eq @ nat @ M2 @ N ) ) ) ) ).
% le_diff_iff
thf(fact_1655_eq__diff__iff,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( ord_less_eq @ nat @ K2 @ M2 )
=> ( ( ord_less_eq @ nat @ K2 @ N )
=> ( ( ( minus_minus @ nat @ M2 @ K2 )
= ( minus_minus @ nat @ N @ K2 ) )
= ( M2 = N ) ) ) ) ).
% eq_diff_iff
thf(fact_1656_diff__add__inverse2,axiom,
! [M2: nat,N: nat] :
( ( minus_minus @ nat @ ( plus_plus @ nat @ M2 @ N ) @ N )
= M2 ) ).
% diff_add_inverse2
thf(fact_1657_diff__add__inverse,axiom,
! [N: nat,M2: nat] :
( ( minus_minus @ nat @ ( plus_plus @ nat @ N @ M2 ) @ N )
= M2 ) ).
% diff_add_inverse
thf(fact_1658_diff__cancel2,axiom,
! [M2: nat,K2: nat,N: nat] :
( ( minus_minus @ nat @ ( plus_plus @ nat @ M2 @ K2 ) @ ( plus_plus @ nat @ N @ K2 ) )
= ( minus_minus @ nat @ M2 @ N ) ) ).
% diff_cancel2
thf(fact_1659_Nat_Odiff__cancel,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( minus_minus @ nat @ ( plus_plus @ nat @ K2 @ M2 ) @ ( plus_plus @ nat @ K2 @ N ) )
= ( minus_minus @ nat @ M2 @ N ) ) ).
% Nat.diff_cancel
thf(fact_1660_diff__mult__distrib,axiom,
! [M2: nat,N: nat,K2: nat] :
( ( times_times @ nat @ ( minus_minus @ nat @ M2 @ N ) @ K2 )
= ( minus_minus @ nat @ ( times_times @ nat @ M2 @ K2 ) @ ( times_times @ nat @ N @ K2 ) ) ) ).
% diff_mult_distrib
thf(fact_1661_diff__mult__distrib2,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( times_times @ nat @ K2 @ ( minus_minus @ nat @ M2 @ N ) )
= ( minus_minus @ nat @ ( times_times @ nat @ K2 @ M2 ) @ ( times_times @ nat @ K2 @ N ) ) ) ).
% diff_mult_distrib2
thf(fact_1662_max__diff__distrib__left,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [X: A,Y: A,Z: A] :
( ( minus_minus @ A @ ( ord_max @ A @ X @ Y ) @ Z )
= ( ord_max @ A @ ( minus_minus @ A @ X @ Z ) @ ( minus_minus @ A @ Y @ Z ) ) ) ) ).
% max_diff_distrib_left
thf(fact_1663_dvd__diff__nat,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( dvd_dvd @ nat @ K2 @ M2 )
=> ( ( dvd_dvd @ nat @ K2 @ N )
=> ( dvd_dvd @ nat @ K2 @ ( minus_minus @ nat @ M2 @ N ) ) ) ) ).
% dvd_diff_nat
thf(fact_1664_zero__less__eq__of__bool,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [P2: $o] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( zero_neq_one_of_bool @ A @ P2 ) ) ) ).
% zero_less_eq_of_bool
thf(fact_1665_of__bool__less__eq__one,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [P2: $o] : ( ord_less_eq @ A @ ( zero_neq_one_of_bool @ A @ P2 ) @ ( one_one @ A ) ) ) ).
% of_bool_less_eq_one
thf(fact_1666_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_1667_split__of__bool,axiom,
! [A: $tType] :
( ( zero_neq_one @ A )
=> ! [P2: A > $o,P: $o] :
( ( P2 @ ( zero_neq_one_of_bool @ A @ P ) )
= ( ( P
=> ( P2 @ ( one_one @ A ) ) )
& ( ~ P
=> ( P2 @ ( zero_zero @ A ) ) ) ) ) ) ).
% split_of_bool
thf(fact_1668_split__of__bool__asm,axiom,
! [A: $tType] :
( ( zero_neq_one @ A )
=> ! [P2: A > $o,P: $o] :
( ( P2 @ ( zero_neq_one_of_bool @ A @ P ) )
= ( ~ ( ( P
& ~ ( P2 @ ( one_one @ A ) ) )
| ( ~ P
& ~ ( P2 @ ( zero_zero @ A ) ) ) ) ) ) ) ).
% split_of_bool_asm
thf(fact_1669_le__iff__diff__le__0,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A6: A,B5: A] : ( ord_less_eq @ A @ ( minus_minus @ A @ A6 @ B5 ) @ ( zero_zero @ A ) ) ) ) ) ).
% le_iff_diff_le_0
thf(fact_1670_less__iff__diff__less__0,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ( ( ord_less @ A )
= ( ^ [A6: A,B5: A] : ( ord_less @ A @ ( minus_minus @ A @ A6 @ B5 ) @ ( zero_zero @ A ) ) ) ) ) ).
% less_iff_diff_less_0
thf(fact_1671_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ( minus_minus @ A @ B2 @ A2 )
= C3 )
= ( B2
= ( plus_plus @ A @ C3 @ A2 ) ) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_1672_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( plus_plus @ A @ A2 @ ( minus_minus @ A @ B2 @ A2 ) )
= B2 ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
thf(fact_1673_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( minus_minus @ A @ C3 @ ( minus_minus @ A @ B2 @ A2 ) )
= ( minus_minus @ A @ ( plus_plus @ A @ C3 @ A2 ) @ B2 ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_1674_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( minus_minus @ A @ ( plus_plus @ A @ B2 @ C3 ) @ A2 )
= ( plus_plus @ A @ ( minus_minus @ A @ B2 @ A2 ) @ C3 ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_1675_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( plus_plus @ A @ ( minus_minus @ A @ B2 @ A2 ) @ C3 )
= ( minus_minus @ A @ ( plus_plus @ A @ B2 @ C3 ) @ A2 ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_1676_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( minus_minus @ A @ ( plus_plus @ A @ C3 @ B2 ) @ A2 )
= ( plus_plus @ A @ C3 @ ( minus_minus @ A @ B2 @ A2 ) ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_1677_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( plus_plus @ A @ C3 @ ( minus_minus @ A @ B2 @ A2 ) )
= ( minus_minus @ A @ ( plus_plus @ A @ C3 @ B2 ) @ A2 ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_1678_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C3 @ ( minus_minus @ A @ B2 @ A2 ) )
= ( ord_less_eq @ A @ ( plus_plus @ A @ C3 @ A2 ) @ B2 ) ) ) ) ).
% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_1679_le__add__diff,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ord_less_eq @ A @ C3 @ ( minus_minus @ A @ ( plus_plus @ A @ B2 @ C3 ) @ A2 ) ) ) ) ).
% le_add_diff
thf(fact_1680_diff__add,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( plus_plus @ A @ ( minus_minus @ A @ B2 @ A2 ) @ A2 )
= B2 ) ) ) ).
% diff_add
thf(fact_1681_le__diff__eq,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,C3: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ ( minus_minus @ A @ C3 @ B2 ) )
= ( ord_less_eq @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C3 ) ) ) ).
% le_diff_eq
thf(fact_1682_diff__le__eq,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( ord_less_eq @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C3 )
= ( ord_less_eq @ A @ A2 @ ( plus_plus @ A @ C3 @ B2 ) ) ) ) ).
% diff_le_eq
thf(fact_1683_add__le__add__imp__diff__le,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [I2: A,K2: A,N: A,J: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ I2 @ K2 ) @ N )
=> ( ( ord_less_eq @ A @ N @ ( plus_plus @ A @ J @ K2 ) )
=> ( ( ord_less_eq @ A @ ( plus_plus @ A @ I2 @ K2 ) @ N )
=> ( ( ord_less_eq @ A @ N @ ( plus_plus @ A @ J @ K2 ) )
=> ( ord_less_eq @ A @ ( minus_minus @ A @ N @ K2 ) @ J ) ) ) ) ) ) ).
% add_le_add_imp_diff_le
thf(fact_1684_add__le__imp__le__diff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [I2: A,K2: A,N: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ I2 @ K2 ) @ N )
=> ( ord_less_eq @ A @ I2 @ ( minus_minus @ A @ N @ K2 ) ) ) ) ).
% add_le_imp_le_diff
thf(fact_1685_less__diff__eq,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,C3: A,B2: A] :
( ( ord_less @ A @ A2 @ ( minus_minus @ A @ C3 @ B2 ) )
= ( ord_less @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C3 ) ) ) ).
% less_diff_eq
thf(fact_1686_diff__less__eq,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( ord_less @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C3 )
= ( ord_less @ A @ A2 @ ( plus_plus @ A @ C3 @ B2 ) ) ) ) ).
% diff_less_eq
thf(fact_1687_linordered__semidom__class_Oadd__diff__inverse,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [A2: A,B2: A] :
( ~ ( ord_less @ A @ A2 @ B2 )
=> ( ( plus_plus @ A @ B2 @ ( minus_minus @ A @ A2 @ B2 ) )
= A2 ) ) ) ).
% linordered_semidom_class.add_diff_inverse
thf(fact_1688_square__diff__square__factored,axiom,
! [A: $tType] :
( ( comm_ring @ A )
=> ! [X: A,Y: A] :
( ( minus_minus @ A @ ( times_times @ A @ X @ X ) @ ( times_times @ A @ Y @ Y ) )
= ( times_times @ A @ ( plus_plus @ A @ X @ Y ) @ ( minus_minus @ A @ X @ Y ) ) ) ) ).
% square_diff_square_factored
thf(fact_1689_eq__add__iff2,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A2: A,E3: A,C3: A,B2: A,D3: A] :
( ( ( plus_plus @ A @ ( times_times @ A @ A2 @ E3 ) @ C3 )
= ( plus_plus @ A @ ( times_times @ A @ B2 @ E3 ) @ D3 ) )
= ( C3
= ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ B2 @ A2 ) @ E3 ) @ D3 ) ) ) ) ).
% eq_add_iff2
thf(fact_1690_eq__add__iff1,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A2: A,E3: A,C3: A,B2: A,D3: A] :
( ( ( plus_plus @ A @ ( times_times @ A @ A2 @ E3 ) @ C3 )
= ( plus_plus @ A @ ( times_times @ A @ B2 @ E3 ) @ D3 ) )
= ( ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ A2 @ B2 ) @ E3 ) @ C3 )
= D3 ) ) ) ).
% eq_add_iff1
thf(fact_1691_mult__diff__mult,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [X: A,Y: A,A2: A,B2: A] :
( ( minus_minus @ A @ ( times_times @ A @ X @ Y ) @ ( times_times @ A @ A2 @ B2 ) )
= ( plus_plus @ A @ ( times_times @ A @ X @ ( minus_minus @ A @ Y @ B2 ) ) @ ( times_times @ A @ ( minus_minus @ A @ X @ A2 ) @ B2 ) ) ) ) ).
% mult_diff_mult
thf(fact_1692_dvd__minus__mod,axiom,
! [A: $tType] :
( ( semidom_modulo @ A )
=> ! [B2: A,A2: A] : ( dvd_dvd @ A @ B2 @ ( minus_minus @ A @ A2 @ ( modulo_modulo @ A @ A2 @ B2 ) ) ) ) ).
% dvd_minus_mod
thf(fact_1693_diff__less__Suc,axiom,
! [M2: nat,N: nat] : ( ord_less @ nat @ ( minus_minus @ nat @ M2 @ N ) @ ( suc @ M2 ) ) ).
% diff_less_Suc
thf(fact_1694_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_1695_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_1696_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_1697_diff__less__mono,axiom,
! [A2: nat,B2: nat,C3: nat] :
( ( ord_less @ nat @ A2 @ B2 )
=> ( ( ord_less_eq @ nat @ C3 @ A2 )
=> ( ord_less @ nat @ ( minus_minus @ nat @ A2 @ C3 ) @ ( minus_minus @ nat @ B2 @ C3 ) ) ) ) ).
% diff_less_mono
thf(fact_1698_less__diff__iff,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( ord_less_eq @ nat @ K2 @ M2 )
=> ( ( ord_less_eq @ nat @ K2 @ N )
=> ( ( ord_less @ nat @ ( minus_minus @ nat @ M2 @ K2 ) @ ( minus_minus @ nat @ N @ K2 ) )
= ( ord_less @ nat @ M2 @ N ) ) ) ) ).
% less_diff_iff
thf(fact_1699_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_1700_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_1701_less__diff__conv,axiom,
! [I2: nat,J: nat,K2: nat] :
( ( ord_less @ nat @ I2 @ ( minus_minus @ nat @ J @ K2 ) )
= ( ord_less @ nat @ ( plus_plus @ nat @ I2 @ K2 ) @ J ) ) ).
% less_diff_conv
thf(fact_1702_Nat_Ole__imp__diff__is__add,axiom,
! [I2: nat,J: nat,K2: nat] :
( ( ord_less_eq @ nat @ I2 @ J )
=> ( ( ( minus_minus @ nat @ J @ I2 )
= K2 )
= ( J
= ( plus_plus @ nat @ K2 @ I2 ) ) ) ) ).
% Nat.le_imp_diff_is_add
thf(fact_1703_Nat_Odiff__add__assoc2,axiom,
! [K2: nat,J: nat,I2: nat] :
( ( ord_less_eq @ nat @ K2 @ J )
=> ( ( minus_minus @ nat @ ( plus_plus @ nat @ J @ I2 ) @ K2 )
= ( plus_plus @ nat @ ( minus_minus @ nat @ J @ K2 ) @ I2 ) ) ) ).
% Nat.diff_add_assoc2
thf(fact_1704_Nat_Odiff__add__assoc,axiom,
! [K2: nat,J: nat,I2: nat] :
( ( ord_less_eq @ nat @ K2 @ J )
=> ( ( minus_minus @ nat @ ( plus_plus @ nat @ I2 @ J ) @ K2 )
= ( plus_plus @ nat @ I2 @ ( minus_minus @ nat @ J @ K2 ) ) ) ) ).
% Nat.diff_add_assoc
thf(fact_1705_Nat_Ole__diff__conv2,axiom,
! [K2: nat,J: nat,I2: nat] :
( ( ord_less_eq @ nat @ K2 @ J )
=> ( ( ord_less_eq @ nat @ I2 @ ( minus_minus @ nat @ J @ K2 ) )
= ( ord_less_eq @ nat @ ( plus_plus @ nat @ I2 @ K2 ) @ J ) ) ) ).
% Nat.le_diff_conv2
thf(fact_1706_le__diff__conv,axiom,
! [J: nat,K2: nat,I2: nat] :
( ( ord_less_eq @ nat @ ( minus_minus @ nat @ J @ K2 ) @ I2 )
= ( ord_less_eq @ nat @ J @ ( plus_plus @ nat @ I2 @ K2 ) ) ) ).
% le_diff_conv
thf(fact_1707_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_1708_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_1709_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_1710_dvd__diffD1,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( dvd_dvd @ nat @ K2 @ ( minus_minus @ nat @ M2 @ N ) )
=> ( ( dvd_dvd @ nat @ K2 @ M2 )
=> ( ( ord_less_eq @ nat @ N @ M2 )
=> ( dvd_dvd @ nat @ K2 @ N ) ) ) ) ).
% dvd_diffD1
thf(fact_1711_dvd__diffD,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( dvd_dvd @ nat @ K2 @ ( minus_minus @ nat @ M2 @ N ) )
=> ( ( dvd_dvd @ nat @ K2 @ N )
=> ( ( ord_less_eq @ nat @ N @ M2 )
=> ( dvd_dvd @ nat @ K2 @ M2 ) ) ) ) ).
% dvd_diffD
thf(fact_1712_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_1713_mod__if,axiom,
( ( modulo_modulo @ nat )
= ( ^ [M3: nat,N2: nat] : ( if @ nat @ ( ord_less @ nat @ M3 @ N2 ) @ M3 @ ( modulo_modulo @ nat @ ( minus_minus @ nat @ M3 @ N2 ) @ N2 ) ) ) ) ).
% mod_if
thf(fact_1714_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_1715_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_1716_ordered__ring__class_Ole__add__iff2,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [A2: A,E3: A,C3: A,B2: A,D3: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ ( times_times @ A @ A2 @ E3 ) @ C3 ) @ ( plus_plus @ A @ ( times_times @ A @ B2 @ E3 ) @ D3 ) )
= ( ord_less_eq @ A @ C3 @ ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ B2 @ A2 ) @ E3 ) @ D3 ) ) ) ) ).
% ordered_ring_class.le_add_iff2
thf(fact_1717_ordered__ring__class_Ole__add__iff1,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [A2: A,E3: A,C3: A,B2: A,D3: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ ( times_times @ A @ A2 @ E3 ) @ C3 ) @ ( plus_plus @ A @ ( times_times @ A @ B2 @ E3 ) @ D3 ) )
= ( ord_less_eq @ A @ ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ A2 @ B2 ) @ E3 ) @ C3 ) @ D3 ) ) ) ).
% ordered_ring_class.le_add_iff1
thf(fact_1718_less__add__iff2,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [A2: A,E3: A,C3: A,B2: A,D3: A] :
( ( ord_less @ A @ ( plus_plus @ A @ ( times_times @ A @ A2 @ E3 ) @ C3 ) @ ( plus_plus @ A @ ( times_times @ A @ B2 @ E3 ) @ D3 ) )
= ( ord_less @ A @ C3 @ ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ B2 @ A2 ) @ E3 ) @ D3 ) ) ) ) ).
% less_add_iff2
thf(fact_1719_less__add__iff1,axiom,
! [A: $tType] :
( ( ordered_ring @ A )
=> ! [A2: A,E3: A,C3: A,B2: A,D3: A] :
( ( ord_less @ A @ ( plus_plus @ A @ ( times_times @ A @ A2 @ E3 ) @ C3 ) @ ( plus_plus @ A @ ( times_times @ A @ B2 @ E3 ) @ D3 ) )
= ( ord_less @ A @ ( plus_plus @ A @ ( times_times @ A @ ( minus_minus @ A @ A2 @ B2 ) @ E3 ) @ C3 ) @ D3 ) ) ) ).
% less_add_iff1
thf(fact_1720_add__divide__eq__if__simps_I4_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z: A,A2: A,B2: A] :
( ( ( Z
= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ A2 @ ( divide_divide @ A @ B2 @ Z ) )
= A2 ) )
& ( ( Z
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ A2 @ ( divide_divide @ A @ B2 @ Z ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ A2 @ Z ) @ B2 ) @ Z ) ) ) ) ) ).
% add_divide_eq_if_simps(4)
thf(fact_1721_diff__frac__eq,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [Y: A,Z: A,X: A,W2: A] :
( ( Y
!= ( zero_zero @ A ) )
=> ( ( Z
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( divide_divide @ A @ X @ Y ) @ ( divide_divide @ A @ W2 @ Z ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ X @ Z ) @ ( times_times @ A @ W2 @ Y ) ) @ ( times_times @ A @ Y @ Z ) ) ) ) ) ) ).
% diff_frac_eq
thf(fact_1722_diff__divide__eq__iff,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z: A,X: A,Y: A] :
( ( Z
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ X @ ( divide_divide @ A @ Y @ Z ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ X @ Z ) @ Y ) @ Z ) ) ) ) ).
% diff_divide_eq_iff
thf(fact_1723_divide__diff__eq__iff,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z: A,X: A,Y: A] :
( ( Z
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( divide_divide @ A @ X @ Z ) @ Y )
= ( divide_divide @ A @ ( minus_minus @ A @ X @ ( times_times @ A @ Y @ Z ) ) @ Z ) ) ) ) ).
% divide_diff_eq_iff
thf(fact_1724_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_1725_inf__period_I3_J,axiom,
! [A: $tType] :
( ( ( comm_ring @ A )
& ( dvd @ A ) )
=> ! [D3: A,D6: A,T2: A] :
( ( dvd_dvd @ A @ D3 @ D6 )
=> ! [X3: A,K4: A] :
( ( dvd_dvd @ A @ D3 @ ( plus_plus @ A @ X3 @ T2 ) )
= ( dvd_dvd @ A @ D3 @ ( plus_plus @ A @ ( minus_minus @ A @ X3 @ ( times_times @ A @ K4 @ D6 ) ) @ T2 ) ) ) ) ) ).
% inf_period(3)
thf(fact_1726_inf__period_I4_J,axiom,
! [A: $tType] :
( ( ( comm_ring @ A )
& ( dvd @ A ) )
=> ! [D3: A,D6: A,T2: A] :
( ( dvd_dvd @ A @ D3 @ D6 )
=> ! [X3: A,K4: A] :
( ( ~ ( dvd_dvd @ A @ D3 @ ( plus_plus @ A @ X3 @ T2 ) ) )
= ( ~ ( dvd_dvd @ A @ D3 @ ( plus_plus @ A @ ( minus_minus @ A @ X3 @ ( times_times @ A @ K4 @ D6 ) ) @ T2 ) ) ) ) ) ) ).
% inf_period(4)
thf(fact_1727_minus__div__mult__eq__mod,axiom,
! [A: $tType] :
( ( semiring_modulo @ A )
=> ! [A2: A,B2: A] :
( ( minus_minus @ A @ A2 @ ( times_times @ A @ ( divide_divide @ A @ A2 @ B2 ) @ B2 ) )
= ( modulo_modulo @ A @ A2 @ B2 ) ) ) ).
% minus_div_mult_eq_mod
thf(fact_1728_minus__mod__eq__div__mult,axiom,
! [A: $tType] :
( ( semiring_modulo @ A )
=> ! [A2: A,B2: A] :
( ( minus_minus @ A @ A2 @ ( modulo_modulo @ A @ A2 @ B2 ) )
= ( times_times @ A @ ( divide_divide @ A @ A2 @ B2 ) @ B2 ) ) ) ).
% minus_mod_eq_div_mult
thf(fact_1729_minus__mod__eq__mult__div,axiom,
! [A: $tType] :
( ( semiring_modulo @ A )
=> ! [A2: A,B2: A] :
( ( minus_minus @ A @ A2 @ ( modulo_modulo @ A @ A2 @ B2 ) )
= ( times_times @ A @ B2 @ ( divide_divide @ A @ A2 @ B2 ) ) ) ) ).
% minus_mod_eq_mult_div
thf(fact_1730_minus__mult__div__eq__mod,axiom,
! [A: $tType] :
( ( semiring_modulo @ A )
=> ! [A2: A,B2: A] :
( ( minus_minus @ A @ A2 @ ( times_times @ A @ B2 @ ( divide_divide @ A @ A2 @ B2 ) ) )
= ( modulo_modulo @ A @ A2 @ B2 ) ) ) ).
% minus_mult_div_eq_mod
thf(fact_1731_diff__Suc__less,axiom,
! [N: nat,I2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ord_less @ nat @ ( minus_minus @ nat @ N @ ( suc @ I2 ) ) @ N ) ) ).
% diff_Suc_less
thf(fact_1732_nat__diff__split__asm,axiom,
! [P2: nat > $o,A2: nat,B2: nat] :
( ( P2 @ ( minus_minus @ nat @ A2 @ B2 ) )
= ( ~ ( ( ( ord_less @ nat @ A2 @ B2 )
& ~ ( P2 @ ( zero_zero @ nat ) ) )
| ? [D5: nat] :
( ( A2
= ( plus_plus @ nat @ B2 @ D5 ) )
& ~ ( P2 @ D5 ) ) ) ) ) ).
% nat_diff_split_asm
thf(fact_1733_nat__diff__split,axiom,
! [P2: nat > $o,A2: nat,B2: nat] :
( ( P2 @ ( minus_minus @ nat @ A2 @ B2 ) )
= ( ( ( ord_less @ nat @ A2 @ B2 )
=> ( P2 @ ( zero_zero @ nat ) ) )
& ! [D5: nat] :
( ( A2
= ( plus_plus @ nat @ B2 @ D5 ) )
=> ( P2 @ D5 ) ) ) ) ).
% nat_diff_split
thf(fact_1734_less__diff__conv2,axiom,
! [K2: nat,J: nat,I2: nat] :
( ( ord_less_eq @ nat @ K2 @ J )
=> ( ( ord_less @ nat @ ( minus_minus @ nat @ J @ K2 ) @ I2 )
= ( ord_less @ nat @ J @ ( plus_plus @ nat @ I2 @ K2 ) ) ) ) ).
% less_diff_conv2
thf(fact_1735_nat__diff__add__eq2,axiom,
! [I2: nat,J: nat,U: nat,M2: nat,N: nat] :
( ( ord_less_eq @ nat @ I2 @ J )
=> ( ( minus_minus @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I2 @ U ) @ M2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N ) )
= ( minus_minus @ nat @ M2 @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ J @ I2 ) @ U ) @ N ) ) ) ) ).
% nat_diff_add_eq2
thf(fact_1736_nat__diff__add__eq1,axiom,
! [J: nat,I2: nat,U: nat,M2: nat,N: nat] :
( ( ord_less_eq @ nat @ J @ I2 )
=> ( ( minus_minus @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I2 @ U ) @ M2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N ) )
= ( minus_minus @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ I2 @ J ) @ U ) @ M2 ) @ N ) ) ) ).
% nat_diff_add_eq1
thf(fact_1737_nat__le__add__iff2,axiom,
! [I2: nat,J: nat,U: nat,M2: nat,N: nat] :
( ( ord_less_eq @ nat @ I2 @ J )
=> ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I2 @ U ) @ M2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N ) )
= ( ord_less_eq @ nat @ M2 @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ J @ I2 ) @ U ) @ N ) ) ) ) ).
% nat_le_add_iff2
thf(fact_1738_nat__le__add__iff1,axiom,
! [J: nat,I2: nat,U: nat,M2: nat,N: nat] :
( ( ord_less_eq @ nat @ J @ I2 )
=> ( ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I2 @ U ) @ M2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N ) )
= ( ord_less_eq @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ I2 @ J ) @ U ) @ M2 ) @ N ) ) ) ).
% nat_le_add_iff1
thf(fact_1739_nat__eq__add__iff2,axiom,
! [I2: nat,J: nat,U: nat,M2: nat,N: nat] :
( ( ord_less_eq @ nat @ I2 @ J )
=> ( ( ( plus_plus @ nat @ ( times_times @ nat @ I2 @ U ) @ M2 )
= ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N ) )
= ( M2
= ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ J @ I2 ) @ U ) @ N ) ) ) ) ).
% nat_eq_add_iff2
thf(fact_1740_nat__eq__add__iff1,axiom,
! [J: nat,I2: nat,U: nat,M2: nat,N: nat] :
( ( ord_less_eq @ nat @ J @ I2 )
=> ( ( ( plus_plus @ nat @ ( times_times @ nat @ I2 @ U ) @ M2 )
= ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N ) )
= ( ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ I2 @ J ) @ U ) @ M2 )
= N ) ) ) ).
% nat_eq_add_iff1
thf(fact_1741_mod__eq__dvd__iff__nat,axiom,
! [N: nat,M2: nat,Q2: nat] :
( ( ord_less_eq @ nat @ N @ M2 )
=> ( ( ( modulo_modulo @ nat @ M2 @ Q2 )
= ( modulo_modulo @ nat @ N @ Q2 ) )
= ( dvd_dvd @ nat @ Q2 @ ( minus_minus @ nat @ M2 @ N ) ) ) ) ).
% mod_eq_dvd_iff_nat
thf(fact_1742_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_1743_frac__le__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y: A,Z: A,X: A,W2: A] :
( ( Y
!= ( zero_zero @ A ) )
=> ( ( Z
!= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( divide_divide @ A @ X @ Y ) @ ( divide_divide @ A @ W2 @ Z ) )
= ( ord_less_eq @ A @ ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ X @ Z ) @ ( times_times @ A @ W2 @ Y ) ) @ ( times_times @ A @ Y @ Z ) ) @ ( zero_zero @ A ) ) ) ) ) ) ).
% frac_le_eq
thf(fact_1744_frac__less__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y: A,Z: A,X: A,W2: A] :
( ( Y
!= ( zero_zero @ A ) )
=> ( ( Z
!= ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( divide_divide @ A @ X @ Y ) @ ( divide_divide @ A @ W2 @ Z ) )
= ( ord_less @ A @ ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ X @ Z ) @ ( times_times @ A @ W2 @ Y ) ) @ ( times_times @ A @ Y @ Z ) ) @ ( zero_zero @ A ) ) ) ) ) ) ).
% frac_less_eq
thf(fact_1745_signed__take__bit__int__less__exp,axiom,
! [N: nat,K2: int] : ( ord_less @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N @ K2 ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ).
% signed_take_bit_int_less_exp
thf(fact_1746_power__diff,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [A2: A,N: nat,M2: nat] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ nat @ N @ M2 )
=> ( ( power_power @ A @ A2 @ ( minus_minus @ nat @ M2 @ N ) )
= ( divide_divide @ A @ ( power_power @ A @ A2 @ M2 ) @ ( power_power @ A @ A2 @ N ) ) ) ) ) ) ).
% power_diff
thf(fact_1747_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_1748_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_1749_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_1750_div__if,axiom,
( ( divide_divide @ nat )
= ( ^ [M3: nat,N2: nat] :
( if @ nat
@ ( ( ord_less @ nat @ M3 @ N2 )
| ( N2
= ( zero_zero @ nat ) ) )
@ ( zero_zero @ nat )
@ ( suc @ ( divide_divide @ nat @ ( minus_minus @ nat @ M3 @ N2 ) @ N2 ) ) ) ) ) ).
% div_if
thf(fact_1751_add__eq__if,axiom,
( ( plus_plus @ nat )
= ( ^ [M3: nat,N2: nat] :
( if @ nat
@ ( M3
= ( zero_zero @ nat ) )
@ N2
@ ( suc @ ( plus_plus @ nat @ ( minus_minus @ nat @ M3 @ ( one_one @ nat ) ) @ N2 ) ) ) ) ) ).
% add_eq_if
thf(fact_1752_nat__less__add__iff2,axiom,
! [I2: nat,J: nat,U: nat,M2: nat,N: nat] :
( ( ord_less_eq @ nat @ I2 @ J )
=> ( ( ord_less @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I2 @ U ) @ M2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N ) )
= ( ord_less @ nat @ M2 @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ J @ I2 ) @ U ) @ N ) ) ) ) ).
% nat_less_add_iff2
thf(fact_1753_nat__less__add__iff1,axiom,
! [J: nat,I2: nat,U: nat,M2: nat,N: nat] :
( ( ord_less_eq @ nat @ J @ I2 )
=> ( ( ord_less @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ I2 @ U ) @ M2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ J @ U ) @ N ) )
= ( ord_less @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( minus_minus @ nat @ I2 @ J ) @ U ) @ M2 ) @ N ) ) ) ).
% nat_less_add_iff1
thf(fact_1754_mult__eq__if,axiom,
( ( times_times @ nat )
= ( ^ [M3: nat,N2: nat] :
( if @ nat
@ ( M3
= ( zero_zero @ nat ) )
@ ( zero_zero @ nat )
@ ( plus_plus @ nat @ N2 @ ( times_times @ nat @ ( minus_minus @ nat @ M3 @ ( one_one @ nat ) ) @ N2 ) ) ) ) ) ).
% mult_eq_if
thf(fact_1755_dvd__minus__add,axiom,
! [Q2: nat,N: nat,R2: nat,M2: nat] :
( ( ord_less_eq @ nat @ Q2 @ N )
=> ( ( ord_less_eq @ nat @ Q2 @ ( times_times @ nat @ R2 @ M2 ) )
=> ( ( dvd_dvd @ nat @ M2 @ ( minus_minus @ nat @ N @ Q2 ) )
= ( dvd_dvd @ nat @ M2 @ ( plus_plus @ nat @ N @ ( minus_minus @ nat @ ( times_times @ nat @ R2 @ M2 ) @ Q2 ) ) ) ) ) ) ).
% dvd_minus_add
thf(fact_1756_mod__nat__eqI,axiom,
! [R2: nat,N: nat,M2: nat] :
( ( ord_less @ nat @ R2 @ N )
=> ( ( ord_less_eq @ nat @ R2 @ M2 )
=> ( ( dvd_dvd @ nat @ N @ ( minus_minus @ nat @ M2 @ R2 ) )
=> ( ( modulo_modulo @ nat @ M2 @ N )
= R2 ) ) ) ) ).
% mod_nat_eqI
thf(fact_1757_scaling__mono,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [U: A,V: A,R2: A,S: A] :
( ( ord_less_eq @ A @ U @ V )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ R2 )
=> ( ( ord_less_eq @ A @ R2 @ S )
=> ( ord_less_eq @ A @ ( plus_plus @ A @ U @ ( divide_divide @ A @ ( times_times @ A @ R2 @ ( minus_minus @ A @ V @ U ) ) @ S ) ) @ V ) ) ) ) ) ).
% scaling_mono
thf(fact_1758_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_1759_signed__take__bit__int__greater__eq__self__iff,axiom,
! [K2: int,N: nat] :
( ( ord_less_eq @ int @ K2 @ ( bit_ri4674362597316999326ke_bit @ int @ N @ K2 ) )
= ( ord_less @ int @ K2 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ) ).
% signed_take_bit_int_greater_eq_self_iff
thf(fact_1760_signed__take__bit__int__less__self__iff,axiom,
! [N: nat,K2: int] :
( ( ord_less @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N @ K2 ) @ K2 )
= ( ord_less_eq @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) @ K2 ) ) ).
% signed_take_bit_int_less_self_iff
thf(fact_1761_power__diff__power__eq,axiom,
! [A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A2: A,N: nat,M2: nat] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( ( ord_less_eq @ nat @ N @ M2 )
=> ( ( divide_divide @ A @ ( power_power @ A @ A2 @ M2 ) @ ( power_power @ A @ A2 @ N ) )
= ( power_power @ A @ A2 @ ( minus_minus @ nat @ M2 @ N ) ) ) )
& ( ~ ( ord_less_eq @ nat @ N @ M2 )
=> ( ( divide_divide @ A @ ( power_power @ A @ A2 @ M2 ) @ ( power_power @ A @ A2 @ N ) )
= ( divide_divide @ A @ ( one_one @ A ) @ ( power_power @ A @ A2 @ ( minus_minus @ nat @ N @ M2 ) ) ) ) ) ) ) ) ).
% power_diff_power_eq
thf(fact_1762_power__eq__if,axiom,
! [A: $tType] :
( ( power @ A )
=> ( ( power_power @ A )
= ( ^ [P6: A,M3: nat] :
( if @ A
@ ( M3
= ( zero_zero @ nat ) )
@ ( one_one @ A )
@ ( times_times @ A @ P6 @ ( power_power @ A @ P6 @ ( minus_minus @ nat @ M3 @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% power_eq_if
thf(fact_1763_power__minus__mult,axiom,
! [A: $tType] :
( ( monoid_mult @ A )
=> ! [N: nat,A2: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( times_times @ A @ ( power_power @ A @ A2 @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) @ A2 )
= ( power_power @ A @ A2 @ N ) ) ) ) ).
% power_minus_mult
thf(fact_1764_diff__le__diff__pow,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K2 )
=> ( ord_less_eq @ nat @ ( minus_minus @ nat @ M2 @ N ) @ ( minus_minus @ nat @ ( power_power @ nat @ K2 @ M2 ) @ ( power_power @ nat @ K2 @ N ) ) ) ) ).
% diff_le_diff_pow
thf(fact_1765_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_1766_num_Osize__gen_I1_J,axiom,
( ( size_num @ one2 )
= ( zero_zero @ nat ) ) ).
% num.size_gen(1)
thf(fact_1767_bits__induct,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [P2: A > $o,A2: A] :
( ! [A4: A] :
( ( ( divide_divide @ A @ A4 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= A4 )
=> ( P2 @ A4 ) )
=> ( ! [A4: A,B4: $o] :
( ( P2 @ A4 )
=> ( ( ( divide_divide @ A @ ( plus_plus @ A @ ( zero_neq_one_of_bool @ A @ B4 ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A4 ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= A4 )
=> ( P2 @ ( plus_plus @ A @ ( zero_neq_one_of_bool @ A @ B4 ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A4 ) ) ) ) )
=> ( P2 @ A2 ) ) ) ) ).
% bits_induct
thf(fact_1768_int__power__div__base,axiom,
! [M2: nat,K2: int] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M2 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ K2 )
=> ( ( divide_divide @ int @ ( power_power @ int @ K2 @ M2 ) @ K2 )
= ( power_power @ int @ K2 @ ( minus_minus @ nat @ M2 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ).
% int_power_div_base
thf(fact_1769_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_1770_power2__diff,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [X: A,Y: A] :
( ( power_power @ A @ ( minus_minus @ A @ X @ Y ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( minus_minus @ A @ ( plus_plus @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( times_times @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X ) @ Y ) ) ) ) ).
% power2_diff
thf(fact_1771_mult__exp__mod__exp__eq,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [M2: nat,N: nat,A2: A] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( modulo_modulo @ A @ ( times_times @ A @ A2 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M2 ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
= ( times_times @ A @ ( modulo_modulo @ A @ A2 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ N @ M2 ) ) ) @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ M2 ) ) ) ) ) ).
% mult_exp_mod_exp_eq
thf(fact_1772_divmod__digit__1_I2_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less_eq @ A @ B2 @ ( modulo_modulo @ A @ A2 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) )
=> ( ( minus_minus @ A @ ( modulo_modulo @ A @ A2 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) @ B2 )
= ( modulo_modulo @ A @ A2 @ B2 ) ) ) ) ) ) ).
% divmod_digit_1(2)
thf(fact_1773_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_1774_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_1775_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_1776_divmod__step__eq,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [L: num,R2: A,Q2: A] :
( ( ( ord_less_eq @ A @ ( numeral_numeral @ A @ L ) @ R2 )
=> ( ( unique1321980374590559556d_step @ A @ L @ ( product_Pair @ A @ A @ Q2 @ R2 ) )
= ( product_Pair @ A @ A @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Q2 ) @ ( one_one @ A ) ) @ ( minus_minus @ A @ R2 @ ( numeral_numeral @ A @ L ) ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ L ) @ R2 )
=> ( ( unique1321980374590559556d_step @ A @ L @ ( product_Pair @ A @ A @ Q2 @ R2 ) )
= ( product_Pair @ A @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Q2 ) @ R2 ) ) ) ) ) ).
% divmod_step_eq
thf(fact_1777_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_1778_artanh__def,axiom,
! [A: $tType] :
( ( ( real_V3459762299906320749_field @ A )
& ( ln @ A ) )
=> ( ( artanh @ A )
= ( ^ [X5: A] : ( divide_divide @ A @ ( ln_ln @ A @ ( divide_divide @ A @ ( plus_plus @ A @ ( one_one @ A ) @ X5 ) @ ( minus_minus @ A @ ( one_one @ A ) @ X5 ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% artanh_def
thf(fact_1779_Suc__if__eq,axiom,
! [A: $tType,F3: nat > A,H2: nat > A,G3: A,N: nat] :
( ! [N3: nat] :
( ( F3 @ ( suc @ N3 ) )
= ( H2 @ N3 ) )
=> ( ( ( F3 @ ( zero_zero @ nat ) )
= G3 )
=> ( ( ( N
= ( zero_zero @ nat ) )
=> ( ( F3 @ N )
= G3 ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ( ( F3 @ N )
= ( H2 @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% Suc_if_eq
thf(fact_1780_signed__take__bit__rec,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( ( bit_ri4674362597316999326ke_bit @ A )
= ( ^ [N2: nat,A6: A] :
( if @ A
@ ( N2
= ( zero_zero @ nat ) )
@ ( uminus_uminus @ A @ ( modulo_modulo @ A @ A6 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
@ ( plus_plus @ A @ ( modulo_modulo @ A @ A6 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_ri4674362597316999326ke_bit @ A @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) @ ( divide_divide @ A @ A6 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% signed_take_bit_rec
thf(fact_1781_diff__shunt__var,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y: A] :
( ( ( minus_minus @ A @ X @ Y )
= ( bot_bot @ A ) )
= ( ord_less_eq @ A @ X @ Y ) ) ) ).
% diff_shunt_var
thf(fact_1782_take__bit__rec,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se2584673776208193580ke_bit @ A )
= ( ^ [N2: nat,A6: A] :
( if @ A
@ ( N2
= ( zero_zero @ nat ) )
@ ( zero_zero @ A )
@ ( plus_plus @ A @ ( times_times @ A @ ( bit_se2584673776208193580ke_bit @ A @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) @ ( divide_divide @ A @ A6 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ A @ A6 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% take_bit_rec
thf(fact_1783_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_1784_neg__equal__iff__equal,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A] :
( ( ( uminus_uminus @ A @ A2 )
= ( uminus_uminus @ A @ B2 ) )
= ( A2 = B2 ) ) ) ).
% neg_equal_iff_equal
thf(fact_1785_add_Oinverse__inverse,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A] :
( ( uminus_uminus @ A @ ( uminus_uminus @ A @ A2 ) )
= A2 ) ) ).
% add.inverse_inverse
thf(fact_1786_uminus__apply,axiom,
! [B: $tType,A: $tType] :
( ( uminus @ B )
=> ( ( uminus_uminus @ ( A > B ) )
= ( ^ [A7: A > B,X5: A] : ( uminus_uminus @ B @ ( A7 @ X5 ) ) ) ) ) ).
% uminus_apply
thf(fact_1787_Diff__empty,axiom,
! [A: $tType,A5: set @ A] :
( ( minus_minus @ ( set @ A ) @ A5 @ ( bot_bot @ ( set @ A ) ) )
= A5 ) ).
% Diff_empty
thf(fact_1788_empty__Diff,axiom,
! [A: $tType,A5: set @ A] :
( ( minus_minus @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ A5 )
= ( bot_bot @ ( set @ A ) ) ) ).
% empty_Diff
thf(fact_1789_Diff__cancel,axiom,
! [A: $tType,A5: set @ A] :
( ( minus_minus @ ( set @ A ) @ A5 @ A5 )
= ( bot_bot @ ( set @ A ) ) ) ).
% Diff_cancel
thf(fact_1790_finite__Diff,axiom,
! [A: $tType,A5: set @ A,B6: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ A5 @ B6 ) ) ) ).
% finite_Diff
thf(fact_1791_finite__Diff2,axiom,
! [A: $tType,B6: set @ A,A5: set @ A] :
( ( finite_finite2 @ A @ B6 )
=> ( ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ A5 @ B6 ) )
= ( finite_finite2 @ A @ A5 ) ) ) ).
% finite_Diff2
thf(fact_1792_compl__le__compl__iff,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ X ) @ ( uminus_uminus @ A @ Y ) )
= ( ord_less_eq @ A @ Y @ X ) ) ) ).
% compl_le_compl_iff
thf(fact_1793_neg__le__iff__le,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [B2: A,A2: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A2 ) )
= ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% neg_le_iff_le
thf(fact_1794_add_Oinverse__neutral,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ( ( uminus_uminus @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% add.inverse_neutral
thf(fact_1795_neg__0__equal__iff__equal,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A] :
( ( ( zero_zero @ A )
= ( uminus_uminus @ A @ A2 ) )
= ( ( zero_zero @ A )
= A2 ) ) ) ).
% neg_0_equal_iff_equal
thf(fact_1796_neg__equal__0__iff__equal,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A] :
( ( ( uminus_uminus @ A @ A2 )
= ( zero_zero @ A ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% neg_equal_0_iff_equal
thf(fact_1797_equal__neg__zero,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A2: A] :
( ( A2
= ( uminus_uminus @ A @ A2 ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% equal_neg_zero
thf(fact_1798_neg__equal__zero,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A2: A] :
( ( ( uminus_uminus @ A @ A2 )
= A2 )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% neg_equal_zero
thf(fact_1799_compl__less__compl__iff,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ X ) @ ( uminus_uminus @ A @ Y ) )
= ( ord_less @ A @ Y @ X ) ) ) ).
% compl_less_compl_iff
thf(fact_1800_neg__less__iff__less,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [B2: A,A2: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A2 ) )
= ( ord_less @ A @ A2 @ B2 ) ) ) ).
% neg_less_iff_less
thf(fact_1801_mult__minus__right,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A2: A,B2: A] :
( ( times_times @ A @ A2 @ ( uminus_uminus @ A @ B2 ) )
= ( uminus_uminus @ A @ ( times_times @ A @ A2 @ B2 ) ) ) ) ).
% mult_minus_right
thf(fact_1802_minus__mult__minus,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A2: A,B2: A] :
( ( times_times @ A @ ( uminus_uminus @ A @ A2 ) @ ( uminus_uminus @ A @ B2 ) )
= ( times_times @ A @ A2 @ B2 ) ) ) ).
% minus_mult_minus
thf(fact_1803_mult__minus__left,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A2: A,B2: A] :
( ( times_times @ A @ ( uminus_uminus @ A @ A2 ) @ B2 )
= ( uminus_uminus @ A @ ( times_times @ A @ A2 @ B2 ) ) ) ) ).
% mult_minus_left
thf(fact_1804_add__minus__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A] :
( ( plus_plus @ A @ A2 @ ( plus_plus @ A @ ( uminus_uminus @ A @ A2 ) @ B2 ) )
= B2 ) ) ).
% add_minus_cancel
thf(fact_1805_minus__add__cancel,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ A2 ) @ ( plus_plus @ A @ A2 @ B2 ) )
= B2 ) ) ).
% minus_add_cancel
thf(fact_1806_minus__add__distrib,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A2: A,B2: A] :
( ( uminus_uminus @ A @ ( plus_plus @ A @ A2 @ B2 ) )
= ( plus_plus @ A @ ( uminus_uminus @ A @ A2 ) @ ( uminus_uminus @ A @ B2 ) ) ) ) ).
% minus_add_distrib
thf(fact_1807_minus__diff__eq,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A] :
( ( uminus_uminus @ A @ ( minus_minus @ A @ A2 @ B2 ) )
= ( minus_minus @ A @ B2 @ A2 ) ) ) ).
% minus_diff_eq
thf(fact_1808_minus__dvd__iff,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [X: A,Y: A] :
( ( dvd_dvd @ A @ ( uminus_uminus @ A @ X ) @ Y )
= ( dvd_dvd @ A @ X @ Y ) ) ) ).
% minus_dvd_iff
thf(fact_1809_dvd__minus__iff,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [X: A,Y: A] :
( ( dvd_dvd @ A @ X @ ( uminus_uminus @ A @ Y ) )
= ( dvd_dvd @ A @ X @ Y ) ) ) ).
% dvd_minus_iff
thf(fact_1810_Diff__eq__empty__iff,axiom,
! [A: $tType,A5: set @ A,B6: set @ A] :
( ( ( minus_minus @ ( set @ A ) @ A5 @ B6 )
= ( bot_bot @ ( set @ A ) ) )
= ( ord_less_eq @ ( set @ A ) @ A5 @ B6 ) ) ).
% Diff_eq_empty_iff
thf(fact_1811_Icc__eq__Icc,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,H2: A,L3: A,H3: A] :
( ( ( set_or1337092689740270186AtMost @ A @ L @ H2 )
= ( set_or1337092689740270186AtMost @ A @ L3 @ H3 ) )
= ( ( ( L = L3 )
& ( H2 = H3 ) )
| ( ~ ( ord_less_eq @ A @ L @ H2 )
& ~ ( ord_less_eq @ A @ L3 @ H3 ) ) ) ) ) ).
% Icc_eq_Icc
thf(fact_1812_atLeastAtMost__iff,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [I2: A,L: A,U: A] :
( ( member @ A @ I2 @ ( set_or1337092689740270186AtMost @ A @ L @ U ) )
= ( ( ord_less_eq @ A @ L @ I2 )
& ( ord_less_eq @ A @ I2 @ U ) ) ) ) ).
% atLeastAtMost_iff
thf(fact_1813_ln__inj__iff,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ Y )
=> ( ( ( ln_ln @ real @ X )
= ( ln_ln @ real @ Y ) )
= ( X = Y ) ) ) ) ).
% ln_inj_iff
thf(fact_1814_ln__less__cancel__iff,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ Y )
=> ( ( ord_less @ real @ ( ln_ln @ real @ X ) @ ( ln_ln @ real @ Y ) )
= ( ord_less @ real @ X @ Y ) ) ) ) ).
% ln_less_cancel_iff
thf(fact_1815_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_1816_finite__atLeastAtMost,axiom,
! [L: nat,U: nat] : ( finite_finite2 @ nat @ ( set_or1337092689740270186AtMost @ nat @ L @ U ) ) ).
% finite_atLeastAtMost
thf(fact_1817_semiring__norm_I80_J,axiom,
! [M2: num,N: num] :
( ( ord_less @ num @ ( bit1 @ M2 ) @ ( bit1 @ N ) )
= ( ord_less @ num @ M2 @ N ) ) ).
% semiring_norm(80)
thf(fact_1818_neg__0__le__iff__le,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ A2 ) )
= ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% neg_0_le_iff_le
thf(fact_1819_neg__le__0__iff__le,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A2 ) @ ( zero_zero @ A ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% neg_le_0_iff_le
thf(fact_1820_less__eq__neg__nonpos,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ A2 @ ( uminus_uminus @ A @ A2 ) )
= ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% less_eq_neg_nonpos
thf(fact_1821_neg__less__eq__nonneg,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A2 ) @ A2 )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% neg_less_eq_nonneg
thf(fact_1822_less__neg__neg,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A2: A] :
( ( ord_less @ A @ A2 @ ( uminus_uminus @ A @ A2 ) )
= ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% less_neg_neg
thf(fact_1823_neg__less__pos,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [A2: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ A2 ) @ A2 )
= ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% neg_less_pos
thf(fact_1824_neg__0__less__iff__less,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ A2 ) )
= ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% neg_0_less_iff_less
thf(fact_1825_neg__less__0__iff__less,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ A2 ) @ ( zero_zero @ A ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% neg_less_0_iff_less
thf(fact_1826_add_Oright__inverse,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A] :
( ( plus_plus @ A @ A2 @ ( uminus_uminus @ A @ A2 ) )
= ( zero_zero @ A ) ) ) ).
% add.right_inverse
thf(fact_1827_ab__left__minus,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ A2 ) @ A2 )
= ( zero_zero @ A ) ) ) ).
% ab_left_minus
thf(fact_1828_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_1829_diff__0,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A] :
( ( minus_minus @ A @ ( zero_zero @ A ) @ A2 )
= ( uminus_uminus @ A @ A2 ) ) ) ).
% diff_0
thf(fact_1830_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_1831_uminus__add__conv__diff,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A2: A,B2: A] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ A2 ) @ B2 )
= ( minus_minus @ A @ B2 @ A2 ) ) ) ).
% uminus_add_conv_diff
thf(fact_1832_diff__minus__eq__add,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A] :
( ( minus_minus @ A @ A2 @ ( uminus_uminus @ A @ B2 ) )
= ( plus_plus @ A @ A2 @ B2 ) ) ) ).
% diff_minus_eq_add
thf(fact_1833_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_1834_atLeastatMost__empty__iff2,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A2: A,B2: A] :
( ( ( bot_bot @ ( set @ A ) )
= ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
= ( ~ ( ord_less_eq @ A @ A2 @ B2 ) ) ) ) ).
% atLeastatMost_empty_iff2
thf(fact_1835_atLeastatMost__empty__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A2: A,B2: A] :
( ( ( set_or1337092689740270186AtMost @ A @ A2 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
= ( ~ ( ord_less_eq @ A @ A2 @ B2 ) ) ) ) ).
% atLeastatMost_empty_iff
thf(fact_1836_atLeastatMost__subset__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A2: A,B2: A,C3: A,D3: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) @ ( set_or1337092689740270186AtMost @ A @ C3 @ D3 ) )
= ( ~ ( ord_less_eq @ A @ A2 @ B2 )
| ( ( ord_less_eq @ A @ C3 @ A2 )
& ( ord_less_eq @ A @ B2 @ D3 ) ) ) ) ) ).
% atLeastatMost_subset_iff
thf(fact_1837_atLeastatMost__empty,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B2: A,A2: A] :
( ( ord_less @ A @ B2 @ A2 )
=> ( ( set_or1337092689740270186AtMost @ A @ A2 @ B2 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% atLeastatMost_empty
thf(fact_1838_infinite__Icc__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A2: A,B2: A] :
( ( ~ ( finite_finite2 @ A @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) ) )
= ( ord_less @ A @ A2 @ B2 ) ) ) ).
% infinite_Icc_iff
thf(fact_1839_ln__le__cancel__iff,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ Y )
=> ( ( ord_less_eq @ real @ ( ln_ln @ real @ X ) @ ( ln_ln @ real @ Y ) )
= ( ord_less_eq @ real @ X @ Y ) ) ) ) ).
% ln_le_cancel_iff
thf(fact_1840_take__bit__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A] :
( ( bit_se2584673776208193580ke_bit @ A @ ( zero_zero @ nat ) @ A2 )
= ( zero_zero @ A ) ) ) ).
% take_bit_0
thf(fact_1841_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_1842_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_1843_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_1844_ln__one,axiom,
! [A: $tType] :
( ( ln @ A )
=> ( ( ln_ln @ A @ ( one_one @ A ) )
= ( zero_zero @ A ) ) ) ).
% ln_one
thf(fact_1845_zle__diff1__eq,axiom,
! [W2: int,Z: int] :
( ( ord_less_eq @ int @ W2 @ ( minus_minus @ int @ Z @ ( one_one @ int ) ) )
= ( ord_less @ int @ W2 @ Z ) ) ).
% zle_diff1_eq
thf(fact_1846_semiring__norm_I81_J,axiom,
! [M2: num,N: num] :
( ( ord_less @ num @ ( bit1 @ M2 ) @ ( bit0 @ N ) )
= ( ord_less @ num @ M2 @ N ) ) ).
% semiring_norm(81)
thf(fact_1847_semiring__norm_I77_J,axiom,
! [N: num] : ( ord_less @ num @ one2 @ ( bit1 @ N ) ) ).
% semiring_norm(77)
thf(fact_1848_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_1849_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_1850_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_1851_mod__minus1__right,axiom,
! [A: $tType] :
( ( euclid8851590272496341667cancel @ A )
=> ! [A2: A] :
( ( modulo_modulo @ A @ A2 @ ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( zero_zero @ A ) ) ) ).
% mod_minus1_right
thf(fact_1852_max__number__of_I4_J,axiom,
! [A: $tType] :
( ( ( uminus @ A )
& ( numeral @ A )
& ( ord @ A ) )
=> ! [U: num,V: num] :
( ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) )
=> ( ( ord_max @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) )
=> ( ( ord_max @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) ) ) ) ) ).
% max_number_of(4)
thf(fact_1853_max__number__of_I3_J,axiom,
! [A: $tType] :
( ( ( uminus @ A )
& ( numeral @ A )
& ( ord @ A ) )
=> ! [U: num,V: num] :
( ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( numeral_numeral @ A @ V ) )
=> ( ( ord_max @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( numeral_numeral @ A @ V ) )
= ( numeral_numeral @ A @ V ) ) )
& ( ~ ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( numeral_numeral @ A @ V ) )
=> ( ( ord_max @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( numeral_numeral @ A @ V ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) ) ) ) ) ).
% max_number_of(3)
thf(fact_1854_max__number__of_I2_J,axiom,
! [A: $tType] :
( ( ( uminus @ A )
& ( numeral @ A )
& ( ord @ A ) )
=> ! [U: num,V: num] :
( ( ( ord_less_eq @ A @ ( numeral_numeral @ A @ U ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) )
=> ( ( ord_max @ A @ ( numeral_numeral @ A @ U ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ U ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) )
=> ( ( ord_max @ A @ ( numeral_numeral @ A @ U ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) )
= ( numeral_numeral @ A @ U ) ) ) ) ) ).
% max_number_of(2)
thf(fact_1855_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_1856_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_1857_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_1858_semiring__norm_I168_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [V: num,W2: num,Y: A] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) @ ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ Y ) )
= ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ ( plus_plus @ num @ V @ W2 ) ) ) @ Y ) ) ) ).
% semiring_norm(168)
thf(fact_1859_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_1860_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_1861_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_1862_semiring__norm_I74_J,axiom,
! [M2: num,N: num] :
( ( ord_less_eq @ num @ ( bit1 @ M2 ) @ ( bit0 @ N ) )
= ( ord_less @ num @ M2 @ N ) ) ).
% semiring_norm(74)
thf(fact_1863_semiring__norm_I79_J,axiom,
! [M2: num,N: num] :
( ( ord_less @ num @ ( bit0 @ M2 ) @ ( bit1 @ N ) )
= ( ord_less_eq @ num @ M2 @ N ) ) ).
% semiring_norm(79)
thf(fact_1864_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_1865_le__divide__eq__numeral1_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A,W2: num] :
( ( ord_less_eq @ A @ A2 @ ( divide_divide @ A @ B2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) )
= ( ord_less_eq @ A @ B2 @ ( times_times @ A @ A2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) ) ) ) ).
% le_divide_eq_numeral1(2)
thf(fact_1866_divide__le__eq__numeral1_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,W2: num,A2: A] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) @ A2 )
= ( ord_less_eq @ A @ ( times_times @ A @ A2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) @ B2 ) ) ) ).
% divide_le_eq_numeral1(2)
thf(fact_1867_eq__divide__eq__numeral1_I2_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A,B2: A,W2: num] :
( ( A2
= ( divide_divide @ A @ B2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) )
= ( ( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) )
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ A2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) )
= B2 ) )
& ( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) )
= ( zero_zero @ A ) )
=> ( A2
= ( zero_zero @ A ) ) ) ) ) ) ).
% eq_divide_eq_numeral1(2)
thf(fact_1868_divide__eq__eq__numeral1_I2_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B2: A,W2: num,A2: A] :
( ( ( divide_divide @ A @ B2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) )
= A2 )
= ( ( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) )
!= ( zero_zero @ A ) )
=> ( B2
= ( times_times @ A @ A2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) ) )
& ( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) )
= ( zero_zero @ A ) )
=> ( A2
= ( zero_zero @ A ) ) ) ) ) ) ).
% divide_eq_eq_numeral1(2)
thf(fact_1869_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_1870_less__divide__eq__numeral1_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A,W2: num] :
( ( ord_less @ A @ A2 @ ( divide_divide @ A @ B2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) )
= ( ord_less @ A @ B2 @ ( times_times @ A @ A2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) ) ) ) ).
% less_divide_eq_numeral1(2)
thf(fact_1871_divide__less__eq__numeral1_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,W2: num,A2: A] :
( ( ord_less @ A @ ( divide_divide @ A @ B2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) @ A2 )
= ( ord_less @ A @ ( times_times @ A @ A2 @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) @ B2 ) ) ) ).
% divide_less_eq_numeral1(2)
thf(fact_1872_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_1873_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_1874_even__take__bit__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A2: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se2584673776208193580ke_bit @ A @ N @ A2 ) )
= ( ( N
= ( zero_zero @ nat ) )
| ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) ) ) ) ).
% even_take_bit_eq
thf(fact_1875_Suc__div__eq__add3__div__numeral,axiom,
! [M2: nat,V: num] :
( ( divide_divide @ nat @ ( suc @ ( suc @ ( suc @ M2 ) ) ) @ ( numeral_numeral @ nat @ V ) )
= ( divide_divide @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) @ M2 ) @ ( numeral_numeral @ nat @ V ) ) ) ).
% Suc_div_eq_add3_div_numeral
thf(fact_1876_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_1877_Suc__mod__eq__add3__mod__numeral,axiom,
! [M2: nat,V: num] :
( ( modulo_modulo @ nat @ ( suc @ ( suc @ ( suc @ M2 ) ) ) @ ( numeral_numeral @ nat @ V ) )
= ( modulo_modulo @ nat @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) @ M2 ) @ ( numeral_numeral @ nat @ V ) ) ) ).
% Suc_mod_eq_add3_mod_numeral
thf(fact_1878_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_1879_take__bit__Suc__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A] :
( ( bit_se2584673776208193580ke_bit @ A @ ( suc @ ( zero_zero @ nat ) ) @ A2 )
= ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% take_bit_Suc_0
thf(fact_1880_signed__take__bit__0,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [A2: A] :
( ( bit_ri4674362597316999326ke_bit @ A @ ( zero_zero @ nat ) @ A2 )
= ( uminus_uminus @ A @ ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% signed_take_bit_0
thf(fact_1881_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_1882_take__bit__of__2,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [N: nat] :
( ( bit_se2584673776208193580ke_bit @ A @ N @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= ( times_times @ A @ ( zero_neq_one_of_bool @ A @ ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% take_bit_of_2
thf(fact_1883_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_1884_compl__le__swap2,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ Y ) @ X )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ X ) @ Y ) ) ) ).
% compl_le_swap2
thf(fact_1885_compl__le__swap1,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ Y @ ( uminus_uminus @ A @ X ) )
=> ( ord_less_eq @ A @ X @ ( uminus_uminus @ A @ Y ) ) ) ) ).
% compl_le_swap1
thf(fact_1886_compl__mono,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ Y ) @ ( uminus_uminus @ A @ X ) ) ) ) ).
% compl_mono
thf(fact_1887_compl__less__swap2,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [Y: A,X: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ Y ) @ X )
=> ( ord_less @ A @ ( uminus_uminus @ A @ X ) @ Y ) ) ) ).
% compl_less_swap2
thf(fact_1888_compl__less__swap1,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [Y: A,X: A] :
( ( ord_less @ A @ Y @ ( uminus_uminus @ A @ X ) )
=> ( ord_less @ A @ X @ ( uminus_uminus @ A @ Y ) ) ) ) ).
% compl_less_swap1
thf(fact_1889_minus__equation__iff,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A] :
( ( ( uminus_uminus @ A @ A2 )
= B2 )
= ( ( uminus_uminus @ A @ B2 )
= A2 ) ) ) ).
% minus_equation_iff
thf(fact_1890_equation__minus__iff,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A] :
( ( A2
= ( uminus_uminus @ A @ B2 ) )
= ( B2
= ( uminus_uminus @ A @ A2 ) ) ) ) ).
% equation_minus_iff
thf(fact_1891_fun__Compl__def,axiom,
! [B: $tType,A: $tType] :
( ( uminus @ B )
=> ( ( uminus_uminus @ ( A > B ) )
= ( ^ [A7: A > B,X5: A] : ( uminus_uminus @ B @ ( A7 @ X5 ) ) ) ) ) ).
% fun_Compl_def
thf(fact_1892_ln__div,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ Y )
=> ( ( ln_ln @ real @ ( divide_divide @ real @ X @ Y ) )
= ( minus_minus @ real @ ( ln_ln @ real @ X ) @ ( ln_ln @ real @ Y ) ) ) ) ) ).
% ln_div
thf(fact_1893_take__bit__add,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [N: nat,A2: A,B2: A] :
( ( bit_se2584673776208193580ke_bit @ A @ N @ ( plus_plus @ A @ ( bit_se2584673776208193580ke_bit @ A @ N @ A2 ) @ ( bit_se2584673776208193580ke_bit @ A @ N @ B2 ) ) )
= ( bit_se2584673776208193580ke_bit @ A @ N @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ).
% take_bit_add
thf(fact_1894_take__bit__tightened,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A2: A,B2: A,M2: nat] :
( ( ( bit_se2584673776208193580ke_bit @ A @ N @ A2 )
= ( bit_se2584673776208193580ke_bit @ A @ N @ B2 ) )
=> ( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( bit_se2584673776208193580ke_bit @ A @ M2 @ A2 )
= ( bit_se2584673776208193580ke_bit @ A @ M2 @ B2 ) ) ) ) ) ).
% take_bit_tightened
thf(fact_1895_take__bit__nat__less__eq__self,axiom,
! [N: nat,M2: nat] : ( ord_less_eq @ nat @ ( bit_se2584673776208193580ke_bit @ nat @ N @ M2 ) @ M2 ) ).
% take_bit_nat_less_eq_self
thf(fact_1896_take__bit__tightened__less__eq__nat,axiom,
! [M2: nat,N: nat,Q2: nat] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ord_less_eq @ nat @ ( bit_se2584673776208193580ke_bit @ nat @ M2 @ Q2 ) @ ( bit_se2584673776208193580ke_bit @ nat @ N @ Q2 ) ) ) ).
% take_bit_tightened_less_eq_nat
thf(fact_1897_le__minus__iff,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ ( uminus_uminus @ A @ B2 ) )
= ( ord_less_eq @ A @ B2 @ ( uminus_uminus @ A @ A2 ) ) ) ) ).
% le_minus_iff
thf(fact_1898_minus__le__iff,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A2 ) @ B2 )
= ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ A2 ) ) ) ).
% minus_le_iff
thf(fact_1899_le__imp__neg__le,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A2 ) ) ) ) ).
% le_imp_neg_le
thf(fact_1900_less__minus__iff,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ ( uminus_uminus @ A @ B2 ) )
= ( ord_less @ A @ B2 @ ( uminus_uminus @ A @ A2 ) ) ) ) ).
% less_minus_iff
thf(fact_1901_minus__less__iff,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ A2 ) @ B2 )
= ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ A2 ) ) ) ).
% minus_less_iff
thf(fact_1902_verit__negate__coefficient_I2_J,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A2 ) ) ) ) ).
% verit_negate_coefficient(2)
thf(fact_1903_minus__mult__commute,axiom,
! [A: $tType] :
( ( ring @ A )
=> ! [A2: A,B2: A] :
( ( times_times @ A @ ( uminus_uminus @ A @ A2 ) @ B2 )
= ( times_times @ A @ A2 @ ( uminus_uminus @ A @ B2 ) ) ) ) ).
% minus_mult_commute
thf(fact_1904_square__eq__iff,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [A2: A,B2: A] :
( ( ( times_times @ A @ A2 @ A2 )
= ( times_times @ A @ B2 @ B2 ) )
= ( ( A2 = B2 )
| ( A2
= ( uminus_uminus @ A @ B2 ) ) ) ) ) ).
% square_eq_iff
thf(fact_1905_is__num__normalize_I8_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [A2: A,B2: A] :
( ( uminus_uminus @ A @ ( plus_plus @ A @ A2 @ B2 ) )
= ( plus_plus @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A2 ) ) ) ) ).
% is_num_normalize(8)
thf(fact_1906_group__cancel_Oneg1,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A5: A,K2: A,A2: A] :
( ( A5
= ( plus_plus @ A @ K2 @ A2 ) )
=> ( ( uminus_uminus @ A @ A5 )
= ( plus_plus @ A @ ( uminus_uminus @ A @ K2 ) @ ( uminus_uminus @ A @ A2 ) ) ) ) ) ).
% group_cancel.neg1
thf(fact_1907_add_Oinverse__distrib__swap,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A] :
( ( uminus_uminus @ A @ ( plus_plus @ A @ A2 @ B2 ) )
= ( plus_plus @ A @ ( uminus_uminus @ A @ B2 ) @ ( uminus_uminus @ A @ A2 ) ) ) ) ).
% add.inverse_distrib_swap
thf(fact_1908_minus__diff__commute,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [B2: A,A2: A] :
( ( minus_minus @ A @ ( uminus_uminus @ A @ B2 ) @ A2 )
= ( minus_minus @ A @ ( uminus_uminus @ A @ A2 ) @ B2 ) ) ) ).
% minus_diff_commute
thf(fact_1909_minus__divide__left,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A,B2: A] :
( ( uminus_uminus @ A @ ( divide_divide @ A @ A2 @ B2 ) )
= ( divide_divide @ A @ ( uminus_uminus @ A @ A2 ) @ B2 ) ) ) ).
% minus_divide_left
thf(fact_1910_minus__divide__divide,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A2: A,B2: A] :
( ( divide_divide @ A @ ( uminus_uminus @ A @ A2 ) @ ( uminus_uminus @ A @ B2 ) )
= ( divide_divide @ A @ A2 @ B2 ) ) ) ).
% minus_divide_divide
thf(fact_1911_minus__divide__right,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A2: A,B2: A] :
( ( uminus_uminus @ A @ ( divide_divide @ A @ A2 @ B2 ) )
= ( divide_divide @ A @ A2 @ ( uminus_uminus @ A @ B2 ) ) ) ) ).
% minus_divide_right
thf(fact_1912_Diff__infinite__finite,axiom,
! [A: $tType,T4: set @ A,S2: set @ A] :
( ( finite_finite2 @ A @ T4 )
=> ( ~ ( finite_finite2 @ A @ S2 )
=> ~ ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ S2 @ T4 ) ) ) ) ).
% Diff_infinite_finite
thf(fact_1913_minus__int__code_I1_J,axiom,
! [K2: int] :
( ( minus_minus @ int @ K2 @ ( zero_zero @ int ) )
= K2 ) ).
% minus_int_code(1)
thf(fact_1914_uminus__int__code_I1_J,axiom,
( ( uminus_uminus @ int @ ( zero_zero @ int ) )
= ( zero_zero @ int ) ) ).
% uminus_int_code(1)
thf(fact_1915_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_1916_int__distrib_I3_J,axiom,
! [Z1: int,Z22: int,W2: int] :
( ( times_times @ int @ ( minus_minus @ int @ Z1 @ Z22 ) @ W2 )
= ( minus_minus @ int @ ( times_times @ int @ Z1 @ W2 ) @ ( times_times @ int @ Z22 @ W2 ) ) ) ).
% int_distrib(3)
thf(fact_1917_int__distrib_I4_J,axiom,
! [W2: int,Z1: int,Z22: int] :
( ( times_times @ int @ W2 @ ( minus_minus @ int @ Z1 @ Z22 ) )
= ( minus_minus @ int @ ( times_times @ int @ W2 @ Z1 ) @ ( times_times @ int @ W2 @ Z22 ) ) ) ).
% int_distrib(4)
thf(fact_1918_psubset__imp__ex__mem,axiom,
! [A: $tType,A5: set @ A,B6: set @ A] :
( ( ord_less @ ( set @ A ) @ A5 @ B6 )
=> ? [B4: A] : ( member @ A @ B4 @ ( minus_minus @ ( set @ A ) @ B6 @ A5 ) ) ) ).
% psubset_imp_ex_mem
thf(fact_1919_zdvd__zdiffD,axiom,
! [K2: int,M2: int,N: int] :
( ( dvd_dvd @ int @ K2 @ ( minus_minus @ int @ M2 @ N ) )
=> ( ( dvd_dvd @ int @ K2 @ N )
=> ( dvd_dvd @ int @ K2 @ M2 ) ) ) ).
% zdvd_zdiffD
thf(fact_1920_ln__diff__le,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ Y )
=> ( ord_less_eq @ real @ ( minus_minus @ real @ ( ln_ln @ real @ X ) @ ( ln_ln @ real @ Y ) ) @ ( divide_divide @ real @ ( minus_minus @ real @ X @ Y ) @ Y ) ) ) ) ).
% ln_diff_le
thf(fact_1921_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_1922_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_1923_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_1924_take__bit__minus__small__eq,axiom,
! [K2: int,N: nat] :
( ( ord_less @ int @ ( zero_zero @ int ) @ K2 )
=> ( ( ord_less_eq @ int @ K2 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( bit_se2584673776208193580ke_bit @ int @ N @ ( uminus_uminus @ int @ K2 ) )
= ( minus_minus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) @ K2 ) ) ) ) ).
% take_bit_minus_small_eq
thf(fact_1925_infinite__Icc,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ~ ( finite_finite2 @ A @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) ) ) ) ).
% infinite_Icc
thf(fact_1926_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_1927_take__bit__tightened__less__eq__int,axiom,
! [M2: nat,N: nat,K2: int] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ord_less_eq @ int @ ( bit_se2584673776208193580ke_bit @ int @ M2 @ K2 ) @ ( bit_se2584673776208193580ke_bit @ int @ N @ K2 ) ) ) ).
% take_bit_tightened_less_eq_int
thf(fact_1928_not__take__bit__negative,axiom,
! [N: nat,K2: int] :
~ ( ord_less @ int @ ( bit_se2584673776208193580ke_bit @ int @ N @ K2 ) @ ( zero_zero @ int ) ) ).
% not_take_bit_negative
thf(fact_1929_take__bit__int__greater__self__iff,axiom,
! [K2: int,N: nat] :
( ( ord_less @ int @ K2 @ ( bit_se2584673776208193580ke_bit @ int @ N @ K2 ) )
= ( ord_less @ int @ K2 @ ( zero_zero @ int ) ) ) ).
% take_bit_int_greater_self_iff
thf(fact_1930_signed__take__bit__take__bit,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [M2: nat,N: nat,A2: A] :
( ( bit_ri4674362597316999326ke_bit @ A @ M2 @ ( bit_se2584673776208193580ke_bit @ A @ N @ A2 ) )
= ( if @ ( A > A ) @ ( ord_less_eq @ nat @ N @ M2 ) @ ( bit_se2584673776208193580ke_bit @ A @ N ) @ ( bit_ri4674362597316999326ke_bit @ A @ M2 ) @ A2 ) ) ) ).
% signed_take_bit_take_bit
thf(fact_1931_ex__nat__less,axiom,
! [N: nat,P2: nat > $o] :
( ( ? [M3: nat] :
( ( ord_less_eq @ nat @ M3 @ N )
& ( P2 @ M3 ) ) )
= ( ? [X5: nat] :
( ( member @ nat @ X5 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
& ( P2 @ X5 ) ) ) ) ).
% ex_nat_less
thf(fact_1932_all__nat__less,axiom,
! [N: nat,P2: nat > $o] :
( ( ! [M3: nat] :
( ( ord_less_eq @ nat @ M3 @ N )
=> ( P2 @ M3 ) ) )
= ( ! [X5: nat] :
( ( member @ nat @ X5 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
=> ( P2 @ X5 ) ) ) ) ).
% all_nat_less
thf(fact_1933_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_1934_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_1935_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_1936_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_1937_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_1938_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_1939_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_1940_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_1941_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_1942_add__eq__0__iff,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A] :
( ( ( plus_plus @ A @ A2 @ B2 )
= ( zero_zero @ A ) )
= ( B2
= ( uminus_uminus @ A @ A2 ) ) ) ) ).
% add_eq_0_iff
thf(fact_1943_ab__group__add__class_Oab__left__minus,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A2: A] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ A2 ) @ A2 )
= ( zero_zero @ A ) ) ) ).
% ab_group_add_class.ab_left_minus
thf(fact_1944_add_Oinverse__unique,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A] :
( ( ( plus_plus @ A @ A2 @ B2 )
= ( zero_zero @ A ) )
=> ( ( uminus_uminus @ A @ A2 )
= B2 ) ) ) ).
% add.inverse_unique
thf(fact_1945_eq__neg__iff__add__eq__0,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A] :
( ( A2
= ( uminus_uminus @ A @ B2 ) )
= ( ( plus_plus @ A @ A2 @ B2 )
= ( zero_zero @ A ) ) ) ) ).
% eq_neg_iff_add_eq_0
thf(fact_1946_neg__eq__iff__add__eq__0,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A,B2: A] :
( ( ( uminus_uminus @ A @ A2 )
= B2 )
= ( ( plus_plus @ A @ A2 @ B2 )
= ( zero_zero @ A ) ) ) ) ).
% neg_eq_iff_add_eq_0
thf(fact_1947_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_1948_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_1949_nonzero__minus__divide__divide,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B2: A,A2: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( uminus_uminus @ A @ A2 ) @ ( uminus_uminus @ A @ B2 ) )
= ( divide_divide @ A @ A2 @ B2 ) ) ) ) ).
% nonzero_minus_divide_divide
thf(fact_1950_nonzero__minus__divide__right,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B2: A,A2: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( uminus_uminus @ A @ ( divide_divide @ A @ A2 @ B2 ) )
= ( divide_divide @ A @ A2 @ ( uminus_uminus @ A @ B2 ) ) ) ) ) ).
% nonzero_minus_divide_right
thf(fact_1951_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_1952_group__cancel_Osub2,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [B6: A,K2: A,B2: A,A2: A] :
( ( B6
= ( plus_plus @ A @ K2 @ B2 ) )
=> ( ( minus_minus @ A @ A2 @ B6 )
= ( plus_plus @ A @ ( uminus_uminus @ A @ K2 ) @ ( minus_minus @ A @ A2 @ B2 ) ) ) ) ) ).
% group_cancel.sub2
thf(fact_1953_diff__conv__add__uminus,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ( ( minus_minus @ A )
= ( ^ [A6: A,B5: A] : ( plus_plus @ A @ A6 @ ( uminus_uminus @ A @ B5 ) ) ) ) ) ).
% diff_conv_add_uminus
thf(fact_1954_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ( ( minus_minus @ A )
= ( ^ [A6: A,B5: A] : ( plus_plus @ A @ A6 @ ( uminus_uminus @ A @ B5 ) ) ) ) ) ).
% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_1955_take__bit__unset__bit__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,M2: nat,A2: A] :
( ( ( ord_less_eq @ nat @ N @ M2 )
=> ( ( bit_se2584673776208193580ke_bit @ A @ N @ ( bit_se2638667681897837118et_bit @ A @ M2 @ A2 ) )
= ( bit_se2584673776208193580ke_bit @ A @ N @ A2 ) ) )
& ( ~ ( ord_less_eq @ nat @ N @ M2 )
=> ( ( bit_se2584673776208193580ke_bit @ A @ N @ ( bit_se2638667681897837118et_bit @ A @ M2 @ A2 ) )
= ( bit_se2638667681897837118et_bit @ A @ M2 @ ( bit_se2584673776208193580ke_bit @ A @ N @ A2 ) ) ) ) ) ) ).
% take_bit_unset_bit_eq
thf(fact_1956_take__bit__set__bit__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,M2: nat,A2: A] :
( ( ( ord_less_eq @ nat @ N @ M2 )
=> ( ( bit_se2584673776208193580ke_bit @ A @ N @ ( bit_se5668285175392031749et_bit @ A @ M2 @ A2 ) )
= ( bit_se2584673776208193580ke_bit @ A @ N @ A2 ) ) )
& ( ~ ( ord_less_eq @ nat @ N @ M2 )
=> ( ( bit_se2584673776208193580ke_bit @ A @ N @ ( bit_se5668285175392031749et_bit @ A @ M2 @ A2 ) )
= ( bit_se5668285175392031749et_bit @ A @ M2 @ ( bit_se2584673776208193580ke_bit @ A @ N @ A2 ) ) ) ) ) ) ).
% take_bit_set_bit_eq
thf(fact_1957_take__bit__flip__bit__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,M2: nat,A2: A] :
( ( ( ord_less_eq @ nat @ N @ M2 )
=> ( ( bit_se2584673776208193580ke_bit @ A @ N @ ( bit_se8732182000553998342ip_bit @ A @ M2 @ A2 ) )
= ( bit_se2584673776208193580ke_bit @ A @ N @ A2 ) ) )
& ( ~ ( ord_less_eq @ nat @ N @ M2 )
=> ( ( bit_se2584673776208193580ke_bit @ A @ N @ ( bit_se8732182000553998342ip_bit @ A @ M2 @ A2 ) )
= ( bit_se8732182000553998342ip_bit @ A @ M2 @ ( bit_se2584673776208193580ke_bit @ A @ N @ A2 ) ) ) ) ) ) ).
% take_bit_flip_bit_eq
thf(fact_1958_dvd__div__neg,axiom,
! [A: $tType] :
( ( idom_divide @ A )
=> ! [B2: A,A2: A] :
( ( dvd_dvd @ A @ B2 @ A2 )
=> ( ( divide_divide @ A @ A2 @ ( uminus_uminus @ A @ B2 ) )
= ( uminus_uminus @ A @ ( divide_divide @ A @ A2 @ B2 ) ) ) ) ) ).
% dvd_div_neg
thf(fact_1959_dvd__neg__div,axiom,
! [A: $tType] :
( ( idom_divide @ A )
=> ! [B2: A,A2: A] :
( ( dvd_dvd @ A @ B2 @ A2 )
=> ( ( divide_divide @ A @ ( uminus_uminus @ A @ A2 ) @ B2 )
= ( uminus_uminus @ A @ ( divide_divide @ A @ A2 @ B2 ) ) ) ) ) ).
% dvd_neg_div
thf(fact_1960_subset__Compl__self__eq,axiom,
! [A: $tType,A5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ ( uminus_uminus @ ( set @ A ) @ A5 ) )
= ( A5
= ( bot_bot @ ( set @ A ) ) ) ) ).
% subset_Compl_self_eq
thf(fact_1961_real__add__less__0__iff,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ ( plus_plus @ real @ X @ Y ) @ ( zero_zero @ real ) )
= ( ord_less @ real @ Y @ ( uminus_uminus @ real @ X ) ) ) ).
% real_add_less_0_iff
thf(fact_1962_real__0__less__add__iff,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( plus_plus @ real @ X @ Y ) )
= ( ord_less @ real @ ( uminus_uminus @ real @ X ) @ Y ) ) ).
% real_0_less_add_iff
thf(fact_1963_int__le__induct,axiom,
! [I2: int,K2: int,P2: int > $o] :
( ( ord_less_eq @ int @ I2 @ K2 )
=> ( ( P2 @ K2 )
=> ( ! [I3: int] :
( ( ord_less_eq @ int @ I3 @ K2 )
=> ( ( P2 @ I3 )
=> ( P2 @ ( minus_minus @ int @ I3 @ ( one_one @ int ) ) ) ) )
=> ( P2 @ I2 ) ) ) ) ).
% int_le_induct
thf(fact_1964_int__less__induct,axiom,
! [I2: int,K2: int,P2: int > $o] :
( ( ord_less @ int @ I2 @ K2 )
=> ( ( P2 @ ( minus_minus @ int @ K2 @ ( one_one @ int ) ) )
=> ( ! [I3: int] :
( ( ord_less @ int @ I3 @ K2 )
=> ( ( P2 @ I3 )
=> ( P2 @ ( minus_minus @ int @ I3 @ ( one_one @ int ) ) ) ) )
=> ( P2 @ I2 ) ) ) ) ).
% int_less_induct
thf(fact_1965_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_1966_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_1967_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_1968_take__bit__int__less__eq,axiom,
! [N: nat,K2: int] :
( ( ord_less_eq @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) @ K2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ord_less_eq @ int @ ( bit_se2584673776208193580ke_bit @ int @ N @ K2 ) @ ( minus_minus @ int @ K2 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ) ) ) ).
% take_bit_int_less_eq
thf(fact_1969_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_1970_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_1971_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_1972_ln__mult,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ Y )
=> ( ( ln_ln @ real @ ( times_times @ real @ X @ Y ) )
= ( plus_plus @ real @ ( ln_ln @ real @ X ) @ ( ln_ln @ real @ Y ) ) ) ) ) ).
% ln_mult
thf(fact_1973_atLeastatMost__psubset__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A2: A,B2: A,C3: A,D3: A] :
( ( ord_less @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) @ ( set_or1337092689740270186AtMost @ A @ C3 @ D3 ) )
= ( ( ~ ( ord_less_eq @ A @ A2 @ B2 )
| ( ( ord_less_eq @ A @ C3 @ A2 )
& ( ord_less_eq @ A @ B2 @ D3 )
& ( ( ord_less @ A @ C3 @ A2 )
| ( ord_less @ A @ B2 @ D3 ) ) ) )
& ( ord_less_eq @ A @ C3 @ D3 ) ) ) ) ).
% atLeastatMost_psubset_iff
thf(fact_1974_take__bit__signed__take__bit,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [M2: nat,N: nat,A2: A] :
( ( ord_less_eq @ nat @ M2 @ ( suc @ N ) )
=> ( ( bit_se2584673776208193580ke_bit @ A @ M2 @ ( bit_ri4674362597316999326ke_bit @ A @ N @ A2 ) )
= ( bit_se2584673776208193580ke_bit @ A @ M2 @ A2 ) ) ) ) ).
% take_bit_signed_take_bit
thf(fact_1975_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_1976_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_1977_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_1978_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_1979_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_1980_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_1981_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_1982_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_1983_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_1984_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_1985_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_1986_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_1987_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_1988_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_1989_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_1990_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_1991_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_1992_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_1993_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_1994_nonzero__neg__divide__eq__eq2,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B2: A,C3: A,A2: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( C3
= ( uminus_uminus @ A @ ( divide_divide @ A @ A2 @ B2 ) ) )
= ( ( times_times @ A @ C3 @ B2 )
= ( uminus_uminus @ A @ A2 ) ) ) ) ) ).
% nonzero_neg_divide_eq_eq2
thf(fact_1995_nonzero__neg__divide__eq__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B2: A,A2: A,C3: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( ( uminus_uminus @ A @ ( divide_divide @ A @ A2 @ B2 ) )
= C3 )
= ( ( uminus_uminus @ A @ A2 )
= ( times_times @ A @ C3 @ B2 ) ) ) ) ) ).
% nonzero_neg_divide_eq_eq
thf(fact_1996_minus__divide__eq__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B2: A,C3: A,A2: A] :
( ( ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C3 ) )
= A2 )
= ( ( ( C3
!= ( zero_zero @ A ) )
=> ( ( uminus_uminus @ A @ B2 )
= ( times_times @ A @ A2 @ C3 ) ) )
& ( ( C3
= ( zero_zero @ A ) )
=> ( A2
= ( zero_zero @ A ) ) ) ) ) ) ).
% minus_divide_eq_eq
thf(fact_1997_eq__minus__divide__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( A2
= ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C3 ) ) )
= ( ( ( C3
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ A2 @ C3 )
= ( uminus_uminus @ A @ B2 ) ) )
& ( ( C3
= ( zero_zero @ A ) )
=> ( A2
= ( zero_zero @ A ) ) ) ) ) ) ).
% eq_minus_divide_eq
thf(fact_1998_divide__eq__minus__1__iff,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A2: A,B2: A] :
( ( ( divide_divide @ A @ A2 @ B2 )
= ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( ( B2
!= ( zero_zero @ A ) )
& ( A2
= ( uminus_uminus @ A @ B2 ) ) ) ) ) ).
% divide_eq_minus_1_iff
thf(fact_1999_take__bit__Suc__bit1,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [N: nat,K2: num] :
( ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N ) @ ( numeral_numeral @ A @ ( bit1 @ K2 ) ) )
= ( plus_plus @ A @ ( times_times @ A @ ( bit_se2584673776208193580ke_bit @ A @ N @ ( numeral_numeral @ A @ K2 ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) ) ) ) ).
% take_bit_Suc_bit1
thf(fact_2000_plusinfinity,axiom,
! [D3: int,P5: int > $o,P2: int > $o] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D3 )
=> ( ! [X4: int,K: int] :
( ( P5 @ X4 )
= ( P5 @ ( minus_minus @ int @ X4 @ ( times_times @ int @ K @ D3 ) ) ) )
=> ( ? [Z4: int] :
! [X4: int] :
( ( ord_less @ int @ Z4 @ X4 )
=> ( ( P2 @ X4 )
= ( P5 @ X4 ) ) )
=> ( ? [X_1: int] : ( P5 @ X_1 )
=> ? [X_12: int] : ( P2 @ X_12 ) ) ) ) ) ).
% plusinfinity
thf(fact_2001_minusinfinity,axiom,
! [D3: int,P1: int > $o,P2: int > $o] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D3 )
=> ( ! [X4: int,K: int] :
( ( P1 @ X4 )
= ( P1 @ ( minus_minus @ int @ X4 @ ( times_times @ int @ K @ D3 ) ) ) )
=> ( ? [Z4: int] :
! [X4: int] :
( ( ord_less @ int @ X4 @ Z4 )
=> ( ( P2 @ X4 )
= ( P1 @ X4 ) ) )
=> ( ? [X_1: int] : ( P1 @ X_1 )
=> ? [X_12: int] : ( P2 @ X_12 ) ) ) ) ) ).
% minusinfinity
thf(fact_2002_int__induct,axiom,
! [P2: int > $o,K2: int,I2: int] :
( ( P2 @ K2 )
=> ( ! [I3: int] :
( ( ord_less_eq @ int @ K2 @ I3 )
=> ( ( P2 @ I3 )
=> ( P2 @ ( plus_plus @ int @ I3 @ ( one_one @ int ) ) ) ) )
=> ( ! [I3: int] :
( ( ord_less_eq @ int @ I3 @ K2 )
=> ( ( P2 @ I3 )
=> ( P2 @ ( minus_minus @ int @ I3 @ ( one_one @ int ) ) ) ) )
=> ( P2 @ I2 ) ) ) ) ).
% int_induct
thf(fact_2003_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_2004_subset__eq__atLeast0__atMost__finite,axiom,
! [N6: set @ nat,N: nat] :
( ( ord_less_eq @ ( set @ nat ) @ N6 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
=> ( finite_finite2 @ nat @ N6 ) ) ).
% subset_eq_atLeast0_atMost_finite
thf(fact_2005_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_2006_take__bit__int__less__exp,axiom,
! [N: nat,K2: int] : ( ord_less @ int @ ( bit_se2584673776208193580ke_bit @ int @ N @ K2 ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ).
% take_bit_int_less_exp
thf(fact_2007_less__minus__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( ord_less @ A @ A2 @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C3 ) ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ ( times_times @ A @ A2 @ C3 ) @ ( uminus_uminus @ A @ B2 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ ( times_times @ A @ A2 @ C3 ) ) )
& ( ~ ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% less_minus_divide_eq
thf(fact_2008_minus__divide__less__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,C3: A,A2: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C3 ) ) @ A2 )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ ( times_times @ A @ A2 @ C3 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ A2 @ C3 ) @ ( uminus_uminus @ A @ B2 ) ) )
& ( ~ ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ) ) ) ) ).
% minus_divide_less_eq
thf(fact_2009_neg__less__minus__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C3: A,A2: A,B2: A] :
( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ A2 @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C3 ) ) )
= ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ ( times_times @ A @ A2 @ C3 ) ) ) ) ) ).
% neg_less_minus_divide_eq
thf(fact_2010_neg__minus__divide__less__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C3: A,B2: A,A2: A] :
( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C3 ) ) @ A2 )
= ( ord_less @ A @ ( times_times @ A @ A2 @ C3 ) @ ( uminus_uminus @ A @ B2 ) ) ) ) ) ).
% neg_minus_divide_less_eq
thf(fact_2011_pos__less__minus__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C3: A,A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ord_less @ A @ A2 @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C3 ) ) )
= ( ord_less @ A @ ( times_times @ A @ A2 @ C3 ) @ ( uminus_uminus @ A @ B2 ) ) ) ) ) ).
% pos_less_minus_divide_eq
thf(fact_2012_pos__minus__divide__less__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C3: A,B2: A,A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ord_less @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C3 ) ) @ A2 )
= ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ ( times_times @ A @ A2 @ C3 ) ) ) ) ) ).
% pos_minus_divide_less_eq
thf(fact_2013_eq__divide__eq__numeral_I2_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [W2: num,B2: A,C3: A] :
( ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) )
= ( divide_divide @ A @ B2 @ C3 ) )
= ( ( ( C3
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ C3 )
= B2 ) )
& ( ( C3
= ( zero_zero @ A ) )
=> ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) )
= ( zero_zero @ A ) ) ) ) ) ) ).
% eq_divide_eq_numeral(2)
thf(fact_2014_divide__eq__eq__numeral_I2_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [B2: A,C3: A,W2: num] :
( ( ( divide_divide @ A @ B2 @ C3 )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) )
= ( ( ( C3
!= ( zero_zero @ A ) )
=> ( B2
= ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ C3 ) ) )
& ( ( C3
= ( zero_zero @ A ) )
=> ( ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) )
= ( zero_zero @ A ) ) ) ) ) ) ).
% divide_eq_eq_numeral(2)
thf(fact_2015_cong__exp__iff__simps_I3_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [N: num,Q2: num] :
( ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit1 @ N ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q2 ) ) )
!= ( zero_zero @ A ) ) ) ).
% cong_exp_iff_simps(3)
thf(fact_2016_minus__divide__add__eq__iff,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z: A,X: A,Y: A] :
( ( Z
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ X @ Z ) ) @ Y )
= ( divide_divide @ A @ ( plus_plus @ A @ ( uminus_uminus @ A @ X ) @ ( times_times @ A @ Y @ Z ) ) @ Z ) ) ) ) ).
% minus_divide_add_eq_iff
thf(fact_2017_add__divide__eq__if__simps_I3_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z: A,A2: A,B2: A] :
( ( ( Z
= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ A2 @ Z ) ) @ B2 )
= B2 ) )
& ( ( Z
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ A2 @ Z ) ) @ B2 )
= ( divide_divide @ A @ ( plus_plus @ A @ ( uminus_uminus @ A @ A2 ) @ ( times_times @ A @ B2 @ Z ) ) @ Z ) ) ) ) ) ).
% add_divide_eq_if_simps(3)
thf(fact_2018_minus__divide__diff__eq__iff,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z: A,X: A,Y: A] :
( ( Z
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ X @ Z ) ) @ Y )
= ( divide_divide @ A @ ( minus_minus @ A @ ( uminus_uminus @ A @ X ) @ ( times_times @ A @ Y @ Z ) ) @ Z ) ) ) ) ).
% minus_divide_diff_eq_iff
thf(fact_2019_add__divide__eq__if__simps_I5_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z: A,A2: A,B2: A] :
( ( ( Z
= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( divide_divide @ A @ A2 @ Z ) @ B2 )
= ( uminus_uminus @ A @ B2 ) ) )
& ( ( Z
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( divide_divide @ A @ A2 @ Z ) @ B2 )
= ( divide_divide @ A @ ( minus_minus @ A @ A2 @ ( times_times @ A @ B2 @ Z ) ) @ Z ) ) ) ) ) ).
% add_divide_eq_if_simps(5)
thf(fact_2020_add__divide__eq__if__simps_I6_J,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Z: A,A2: A,B2: A] :
( ( ( Z
= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ A2 @ Z ) ) @ B2 )
= ( uminus_uminus @ A @ B2 ) ) )
& ( ( Z
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ A2 @ Z ) ) @ B2 )
= ( divide_divide @ A @ ( minus_minus @ A @ ( uminus_uminus @ A @ A2 ) @ ( times_times @ A @ B2 @ Z ) ) @ Z ) ) ) ) ) ).
% add_divide_eq_if_simps(6)
thf(fact_2021_numeral__3__eq__3,axiom,
( ( numeral_numeral @ nat @ ( bit1 @ one2 ) )
= ( suc @ ( suc @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ).
% numeral_3_eq_3
thf(fact_2022_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_2023_signed__take__bit__int__greater__self__iff,axiom,
! [K2: int,N: nat] :
( ( ord_less @ int @ K2 @ ( bit_ri4674362597316999326ke_bit @ int @ N @ K2 ) )
= ( ord_less @ int @ K2 @ ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% signed_take_bit_int_greater_self_iff
thf(fact_2024_decr__mult__lemma,axiom,
! [D3: int,P2: int > $o,K2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D3 )
=> ( ! [X4: int] :
( ( P2 @ X4 )
=> ( P2 @ ( minus_minus @ int @ X4 @ D3 ) ) )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K2 )
=> ! [X3: int] :
( ( P2 @ X3 )
=> ( P2 @ ( minus_minus @ int @ X3 @ ( times_times @ int @ K2 @ D3 ) ) ) ) ) ) ) ).
% decr_mult_lemma
thf(fact_2025_mod__pos__geq,axiom,
! [L: int,K2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ L )
=> ( ( ord_less_eq @ int @ L @ K2 )
=> ( ( modulo_modulo @ int @ K2 @ L )
= ( modulo_modulo @ int @ ( minus_minus @ int @ K2 @ L ) @ L ) ) ) ) ).
% mod_pos_geq
thf(fact_2026_verit__less__mono__div__int2,axiom,
! [A5: int,B6: int,N: int] :
( ( ord_less_eq @ int @ A5 @ B6 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ ( uminus_uminus @ int @ N ) )
=> ( ord_less_eq @ int @ ( divide_divide @ int @ B6 @ N ) @ ( divide_divide @ int @ A5 @ N ) ) ) ) ).
% verit_less_mono_div_int2
thf(fact_2027_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_2028_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_2029_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_2030_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_2031_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_2032_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_2033_take__bit__int__less__self__iff,axiom,
! [N: nat,K2: int] :
( ( ord_less @ int @ ( bit_se2584673776208193580ke_bit @ int @ N @ K2 ) @ K2 )
= ( ord_less_eq @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) @ K2 ) ) ).
% take_bit_int_less_self_iff
thf(fact_2034_take__bit__int__greater__eq__self__iff,axiom,
! [K2: int,N: nat] :
( ( ord_less_eq @ int @ K2 @ ( bit_se2584673776208193580ke_bit @ int @ N @ K2 ) )
= ( ord_less @ int @ K2 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ) ).
% take_bit_int_greater_eq_self_iff
thf(fact_2035_le__minus__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( ord_less_eq @ A @ A2 @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C3 ) ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C3 ) @ ( uminus_uminus @ A @ B2 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( times_times @ A @ A2 @ C3 ) ) )
& ( ~ ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% le_minus_divide_eq
thf(fact_2036_minus__divide__le__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,C3: A,A2: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C3 ) ) @ A2 )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( times_times @ A @ A2 @ C3 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C3 ) @ ( uminus_uminus @ A @ B2 ) ) )
& ( ~ ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ) ) ) ) ).
% minus_divide_le_eq
thf(fact_2037_neg__le__minus__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C3: A,A2: A,B2: A] :
( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ A2 @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C3 ) ) )
= ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( times_times @ A @ A2 @ C3 ) ) ) ) ) ).
% neg_le_minus_divide_eq
thf(fact_2038_neg__minus__divide__le__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C3: A,B2: A,A2: A] :
( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C3 ) ) @ A2 )
= ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C3 ) @ ( uminus_uminus @ A @ B2 ) ) ) ) ) ).
% neg_minus_divide_le_eq
thf(fact_2039_pos__le__minus__divide__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C3: A,A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ord_less_eq @ A @ A2 @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C3 ) ) )
= ( ord_less_eq @ A @ ( times_times @ A @ A2 @ C3 ) @ ( uminus_uminus @ A @ B2 ) ) ) ) ) ).
% pos_le_minus_divide_eq
thf(fact_2040_pos__minus__divide__le__eq,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C3: A,B2: A,A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( divide_divide @ A @ B2 @ C3 ) ) @ A2 )
= ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( times_times @ A @ A2 @ C3 ) ) ) ) ) ).
% pos_minus_divide_le_eq
thf(fact_2041_less__divide__eq__numeral_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [W2: num,B2: A,C3: A] :
( ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ ( divide_divide @ A @ B2 @ C3 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ C3 ) @ B2 ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B2 @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ C3 ) ) )
& ( ~ ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% less_divide_eq_numeral(2)
thf(fact_2042_divide__less__eq__numeral_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,C3: A,W2: num] :
( ( ord_less @ A @ ( divide_divide @ A @ B2 @ C3 ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less @ A @ B2 @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ C3 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ C3 ) @ B2 ) )
& ( ~ ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) ) ) ) ) ) ) ).
% divide_less_eq_numeral(2)
thf(fact_2043_cong__exp__iff__simps_I11_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M2: num,Q2: num] :
( ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit1 @ M2 ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q2 ) ) )
= ( modulo_modulo @ A @ ( numeral_numeral @ A @ one2 ) @ ( numeral_numeral @ A @ ( bit0 @ Q2 ) ) ) )
= ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ M2 ) @ ( numeral_numeral @ A @ Q2 ) )
= ( zero_zero @ A ) ) ) ) ).
% cong_exp_iff_simps(11)
thf(fact_2044_cong__exp__iff__simps_I7_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [Q2: num,N: num] :
( ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ one2 ) @ ( numeral_numeral @ A @ ( bit0 @ Q2 ) ) )
= ( modulo_modulo @ A @ ( numeral_numeral @ A @ ( bit1 @ N ) ) @ ( numeral_numeral @ A @ ( bit0 @ Q2 ) ) ) )
= ( ( modulo_modulo @ A @ ( numeral_numeral @ A @ N ) @ ( numeral_numeral @ A @ Q2 ) )
= ( zero_zero @ A ) ) ) ) ).
% cong_exp_iff_simps(7)
thf(fact_2045_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_2046_neg__one__power__add__eq__neg__one__power__diff,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [K2: nat,N: nat] :
( ( ord_less_eq @ nat @ K2 @ N )
=> ( ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( plus_plus @ nat @ N @ K2 ) )
= ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ ( minus_minus @ nat @ N @ K2 ) ) ) ) ) ).
% neg_one_power_add_eq_neg_one_power_diff
thf(fact_2047_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_2048_signed__take__bit__int__eq__self,axiom,
! [N: nat,K2: int] :
( ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) @ K2 )
=> ( ( ord_less @ int @ K2 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( bit_ri4674362597316999326ke_bit @ int @ N @ K2 )
= K2 ) ) ) ).
% signed_take_bit_int_eq_self
thf(fact_2049_signed__take__bit__int__eq__self__iff,axiom,
! [N: nat,K2: int] :
( ( ( bit_ri4674362597316999326ke_bit @ int @ N @ K2 )
= K2 )
= ( ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) @ K2 )
& ( ord_less @ int @ K2 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% signed_take_bit_int_eq_self_iff
thf(fact_2050_take__bit__eq__0__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A2: A] :
( ( ( bit_se2584673776208193580ke_bit @ A @ N @ A2 )
= ( zero_zero @ A ) )
= ( dvd_dvd @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) @ A2 ) ) ) ).
% take_bit_eq_0_iff
thf(fact_2051_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_2052_take__bit__int__eq__self__iff,axiom,
! [N: nat,K2: int] :
( ( ( bit_se2584673776208193580ke_bit @ int @ N @ K2 )
= K2 )
= ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K2 )
& ( ord_less @ int @ K2 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% take_bit_int_eq_self_iff
thf(fact_2053_take__bit__int__eq__self,axiom,
! [K2: int,N: nat] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K2 )
=> ( ( ord_less @ int @ K2 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) )
=> ( ( bit_se2584673776208193580ke_bit @ int @ N @ K2 )
= K2 ) ) ) ).
% take_bit_int_eq_self
thf(fact_2054_divide__le__eq__numeral_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,C3: A,W2: num] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ B2 @ C3 ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ B2 @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ C3 ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ C3 ) @ B2 ) )
& ( ~ ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) ) ) ) ) ) ) ) ).
% divide_le_eq_numeral(2)
thf(fact_2055_le__divide__eq__numeral_I2_J,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [W2: num,B2: A,C3: A] :
( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ ( divide_divide @ A @ B2 @ C3 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ C3 ) @ B2 ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B2 @ ( times_times @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ C3 ) ) )
& ( ~ ( ord_less @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% le_divide_eq_numeral(2)
thf(fact_2056_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_2057_div__pos__geq,axiom,
! [L: int,K2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ L )
=> ( ( ord_less_eq @ int @ L @ K2 )
=> ( ( divide_divide @ int @ K2 @ L )
= ( plus_plus @ int @ ( divide_divide @ int @ ( minus_minus @ int @ K2 @ L ) @ L ) @ ( one_one @ int ) ) ) ) ) ).
% div_pos_geq
thf(fact_2058_div__pos__neg__trivial,axiom,
! [K2: int,L: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ K2 )
=> ( ( ord_less_eq @ int @ ( plus_plus @ int @ K2 @ L ) @ ( zero_zero @ int ) )
=> ( ( divide_divide @ int @ K2 @ L )
= ( uminus_uminus @ int @ ( one_one @ int ) ) ) ) ) ).
% div_pos_neg_trivial
thf(fact_2059_take__bit__int__greater__eq,axiom,
! [K2: int,N: nat] :
( ( ord_less @ int @ K2 @ ( zero_zero @ int ) )
=> ( ord_less_eq @ int @ ( plus_plus @ int @ K2 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) @ ( bit_se2584673776208193580ke_bit @ int @ N @ K2 ) ) ) ).
% take_bit_int_greater_eq
thf(fact_2060_take__bit__Suc,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A2: A] :
( ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N ) @ A2 )
= ( plus_plus @ A @ ( times_times @ A @ ( bit_se2584673776208193580ke_bit @ A @ N @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% take_bit_Suc
thf(fact_2061_signed__take__bit__int__greater__eq,axiom,
! [K2: int,N: nat] :
( ( ord_less @ int @ K2 @ ( uminus_uminus @ int @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) )
=> ( ord_less_eq @ int @ ( plus_plus @ int @ K2 @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( suc @ N ) ) ) @ ( bit_ri4674362597316999326ke_bit @ int @ N @ K2 ) ) ) ).
% signed_take_bit_int_greater_eq
thf(fact_2062_stable__imp__take__bit__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A,N: nat] :
( ( ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
= A2 )
=> ( ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
=> ( ( bit_se2584673776208193580ke_bit @ A @ N @ A2 )
= ( zero_zero @ A ) ) )
& ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
=> ( ( bit_se2584673776208193580ke_bit @ A @ N @ A2 )
= ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ A ) ) ) ) ) ) ) ).
% stable_imp_take_bit_eq
thf(fact_2063_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_2064_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_2065_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_2066_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_2067_Bolzano,axiom,
! [A2: real,B2: real,P2: real > real > $o] :
( ( ord_less_eq @ real @ A2 @ B2 )
=> ( ! [A4: real,B4: real,C2: real] :
( ( P2 @ A4 @ B4 )
=> ( ( P2 @ B4 @ C2 )
=> ( ( ord_less_eq @ real @ A4 @ B4 )
=> ( ( ord_less_eq @ real @ B4 @ C2 )
=> ( P2 @ A4 @ C2 ) ) ) ) )
=> ( ! [X4: real] :
( ( ord_less_eq @ real @ A2 @ X4 )
=> ( ( ord_less_eq @ real @ X4 @ B2 )
=> ? [D4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D4 )
& ! [A4: real,B4: real] :
( ( ( ord_less_eq @ real @ A4 @ X4 )
& ( ord_less_eq @ real @ X4 @ B4 )
& ( ord_less @ real @ ( minus_minus @ real @ B4 @ A4 ) @ D4 ) )
=> ( P2 @ A4 @ B4 ) ) ) ) )
=> ( P2 @ A2 @ B2 ) ) ) ) ).
% Bolzano
thf(fact_2068_divmod__step__def,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ( ( unique1321980374590559556d_step @ A )
= ( ^ [L2: num] :
( product_case_prod @ A @ A @ ( product_prod @ A @ A )
@ ^ [Q5: A,R5: A] : ( if @ ( product_prod @ A @ A ) @ ( ord_less_eq @ A @ ( numeral_numeral @ A @ L2 ) @ R5 ) @ ( product_Pair @ A @ A @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Q5 ) @ ( one_one @ A ) ) @ ( minus_minus @ A @ R5 @ ( numeral_numeral @ A @ L2 ) ) ) @ ( product_Pair @ A @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Q5 ) @ R5 ) ) ) ) ) ) ).
% divmod_step_def
thf(fact_2069_finite__atLeastAtMost__int,axiom,
! [L: int,U: int] : ( finite_finite2 @ int @ ( set_or1337092689740270186AtMost @ int @ L @ U ) ) ).
% finite_atLeastAtMost_int
thf(fact_2070_case__prod__conv,axiom,
! [B: $tType,A: $tType,C: $tType,F3: B > C > A,A2: B,B2: C] :
( ( product_case_prod @ B @ C @ A @ F3 @ ( product_Pair @ B @ C @ A2 @ B2 ) )
= ( F3 @ A2 @ B2 ) ) ).
% case_prod_conv
thf(fact_2071_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_2072_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_2073_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_2074_divmod__algorithm__code_I5_J,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ! [M2: num,N: num] :
( ( unique8689654367752047608divmod @ A @ ( bit0 @ M2 ) @ ( bit0 @ N ) )
= ( product_case_prod @ A @ A @ ( product_prod @ A @ A )
@ ^ [Q5: A,R5: A] : ( product_Pair @ A @ A @ Q5 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ R5 ) )
@ ( unique8689654367752047608divmod @ A @ M2 @ N ) ) ) ) ).
% divmod_algorithm_code(5)
thf(fact_2075_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 )
@ ^ [Q5: A,R5: A] : ( product_Pair @ A @ A @ Q5 @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ R5 ) @ ( one_one @ A ) ) )
@ ( unique8689654367752047608divmod @ A @ M2 @ N ) ) ) ) ).
% divmod_algorithm_code(6)
thf(fact_2076_prod_Ocase__distrib,axiom,
! [C: $tType,D: $tType,B: $tType,A: $tType,H2: C > D,F3: A > B > C,Prod: product_prod @ A @ B] :
( ( H2 @ ( product_case_prod @ A @ B @ C @ F3 @ Prod ) )
= ( product_case_prod @ A @ B @ D
@ ^ [X15: A,X23: B] : ( H2 @ ( F3 @ X15 @ X23 ) )
@ Prod ) ) ).
% prod.case_distrib
thf(fact_2077_old_Oprod_Ocase,axiom,
! [A: $tType,C: $tType,B: $tType,F3: A > B > C,X1: A,X2: B] :
( ( product_case_prod @ A @ B @ C @ F3 @ ( product_Pair @ A @ B @ X1 @ X2 ) )
= ( F3 @ X1 @ X2 ) ) ).
% old.prod.case
thf(fact_2078_split__cong,axiom,
! [C: $tType,B: $tType,A: $tType,Q2: product_prod @ A @ B,F3: A > B > C,G3: A > B > C,P: product_prod @ A @ B] :
( ! [X4: A,Y3: B] :
( ( ( product_Pair @ A @ B @ X4 @ Y3 )
= Q2 )
=> ( ( F3 @ X4 @ Y3 )
= ( G3 @ X4 @ Y3 ) ) )
=> ( ( P = Q2 )
=> ( ( product_case_prod @ A @ B @ C @ F3 @ P )
= ( product_case_prod @ A @ B @ C @ G3 @ Q2 ) ) ) ) ).
% split_cong
thf(fact_2079_case__prodE2,axiom,
! [B: $tType,A: $tType,C: $tType,Q: A > $o,P2: B > C > A,Z: product_prod @ B @ C] :
( ( Q @ ( product_case_prod @ B @ C @ A @ P2 @ Z ) )
=> ~ ! [X4: B,Y3: C] :
( ( Z
= ( product_Pair @ B @ C @ X4 @ Y3 ) )
=> ~ ( Q @ ( P2 @ X4 @ Y3 ) ) ) ) ).
% case_prodE2
thf(fact_2080_case__prod__eta,axiom,
! [C: $tType,B: $tType,A: $tType,F3: ( product_prod @ A @ B ) > C] :
( ( product_case_prod @ A @ B @ C
@ ^ [X5: A,Y6: B] : ( F3 @ ( product_Pair @ A @ B @ X5 @ Y6 ) ) )
= F3 ) ).
% case_prod_eta
thf(fact_2081_cond__case__prod__eta,axiom,
! [C: $tType,B: $tType,A: $tType,F3: A > B > C,G3: ( product_prod @ A @ B ) > C] :
( ! [X4: A,Y3: B] :
( ( F3 @ X4 @ Y3 )
= ( G3 @ ( product_Pair @ A @ B @ X4 @ Y3 ) ) )
=> ( ( product_case_prod @ A @ B @ C @ F3 )
= G3 ) ) ).
% cond_case_prod_eta
thf(fact_2082_periodic__finite__ex,axiom,
! [D3: int,P2: int > $o] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D3 )
=> ( ! [X4: int,K: int] :
( ( P2 @ X4 )
= ( P2 @ ( minus_minus @ int @ X4 @ ( times_times @ int @ K @ D3 ) ) ) )
=> ( ( ? [X9: int] : ( P2 @ X9 ) )
= ( ? [X5: int] :
( ( member @ int @ X5 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D3 ) )
& ( P2 @ X5 ) ) ) ) ) ) ).
% periodic_finite_ex
thf(fact_2083_aset_I7_J,axiom,
! [D6: int,A5: set @ int,T2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D6 )
=> ! [X3: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D6 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ A5 )
=> ( X3
!= ( minus_minus @ int @ Xb2 @ Xa3 ) ) ) )
=> ( ( ord_less @ int @ T2 @ X3 )
=> ( ord_less @ int @ T2 @ ( plus_plus @ int @ X3 @ D6 ) ) ) ) ) ).
% aset(7)
thf(fact_2084_aset_I5_J,axiom,
! [D6: int,T2: int,A5: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D6 )
=> ( ( member @ int @ T2 @ A5 )
=> ! [X3: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D6 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ A5 )
=> ( X3
!= ( minus_minus @ int @ Xb2 @ Xa3 ) ) ) )
=> ( ( ord_less @ int @ X3 @ T2 )
=> ( ord_less @ int @ ( plus_plus @ int @ X3 @ D6 ) @ T2 ) ) ) ) ) ).
% aset(5)
thf(fact_2085_aset_I4_J,axiom,
! [D6: int,T2: int,A5: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D6 )
=> ( ( member @ int @ T2 @ A5 )
=> ! [X3: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D6 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ A5 )
=> ( X3
!= ( minus_minus @ int @ Xb2 @ Xa3 ) ) ) )
=> ( ( X3 != T2 )
=> ( ( plus_plus @ int @ X3 @ D6 )
!= T2 ) ) ) ) ) ).
% aset(4)
thf(fact_2086_aset_I3_J,axiom,
! [D6: int,T2: int,A5: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D6 )
=> ( ( member @ int @ ( plus_plus @ int @ T2 @ ( one_one @ int ) ) @ A5 )
=> ! [X3: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D6 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ A5 )
=> ( X3
!= ( minus_minus @ int @ Xb2 @ Xa3 ) ) ) )
=> ( ( X3 = T2 )
=> ( ( plus_plus @ int @ X3 @ D6 )
= T2 ) ) ) ) ) ).
% aset(3)
thf(fact_2087_bset_I7_J,axiom,
! [D6: int,T2: int,B6: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D6 )
=> ( ( member @ int @ T2 @ B6 )
=> ! [X3: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D6 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ B6 )
=> ( X3
!= ( plus_plus @ int @ Xb2 @ Xa3 ) ) ) )
=> ( ( ord_less @ int @ T2 @ X3 )
=> ( ord_less @ int @ T2 @ ( minus_minus @ int @ X3 @ D6 ) ) ) ) ) ) ).
% bset(7)
thf(fact_2088_bset_I5_J,axiom,
! [D6: int,B6: set @ int,T2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D6 )
=> ! [X3: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D6 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ B6 )
=> ( X3
!= ( plus_plus @ int @ Xb2 @ Xa3 ) ) ) )
=> ( ( ord_less @ int @ X3 @ T2 )
=> ( ord_less @ int @ ( minus_minus @ int @ X3 @ D6 ) @ T2 ) ) ) ) ).
% bset(5)
thf(fact_2089_bset_I4_J,axiom,
! [D6: int,T2: int,B6: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D6 )
=> ( ( member @ int @ T2 @ B6 )
=> ! [X3: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D6 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ B6 )
=> ( X3
!= ( plus_plus @ int @ Xb2 @ Xa3 ) ) ) )
=> ( ( X3 != T2 )
=> ( ( minus_minus @ int @ X3 @ D6 )
!= T2 ) ) ) ) ) ).
% bset(4)
thf(fact_2090_bset_I3_J,axiom,
! [D6: int,T2: int,B6: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D6 )
=> ( ( member @ int @ ( minus_minus @ int @ T2 @ ( one_one @ int ) ) @ B6 )
=> ! [X3: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D6 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ B6 )
=> ( X3
!= ( plus_plus @ int @ Xb2 @ Xa3 ) ) ) )
=> ( ( X3 = T2 )
=> ( ( minus_minus @ int @ X3 @ D6 )
= T2 ) ) ) ) ) ).
% bset(3)
thf(fact_2091_bset_I6_J,axiom,
! [D6: int,B6: set @ int,T2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D6 )
=> ! [X3: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D6 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ B6 )
=> ( X3
!= ( plus_plus @ int @ Xb2 @ Xa3 ) ) ) )
=> ( ( ord_less_eq @ int @ X3 @ T2 )
=> ( ord_less_eq @ int @ ( minus_minus @ int @ X3 @ D6 ) @ T2 ) ) ) ) ).
% bset(6)
thf(fact_2092_bset_I8_J,axiom,
! [D6: int,T2: int,B6: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D6 )
=> ( ( member @ int @ ( minus_minus @ int @ T2 @ ( one_one @ int ) ) @ B6 )
=> ! [X3: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D6 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ B6 )
=> ( X3
!= ( plus_plus @ int @ Xb2 @ Xa3 ) ) ) )
=> ( ( ord_less_eq @ int @ T2 @ X3 )
=> ( ord_less_eq @ int @ T2 @ ( minus_minus @ int @ X3 @ D6 ) ) ) ) ) ) ).
% bset(8)
thf(fact_2093_aset_I6_J,axiom,
! [D6: int,T2: int,A5: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D6 )
=> ( ( member @ int @ ( plus_plus @ int @ T2 @ ( one_one @ int ) ) @ A5 )
=> ! [X3: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D6 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ A5 )
=> ( X3
!= ( minus_minus @ int @ Xb2 @ Xa3 ) ) ) )
=> ( ( ord_less_eq @ int @ X3 @ T2 )
=> ( ord_less_eq @ int @ ( plus_plus @ int @ X3 @ D6 ) @ T2 ) ) ) ) ) ).
% aset(6)
thf(fact_2094_aset_I8_J,axiom,
! [D6: int,A5: set @ int,T2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D6 )
=> ! [X3: int] :
( ! [Xa3: int] :
( ( member @ int @ Xa3 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D6 ) )
=> ! [Xb2: int] :
( ( member @ int @ Xb2 @ A5 )
=> ( X3
!= ( minus_minus @ int @ Xb2 @ Xa3 ) ) ) )
=> ( ( ord_less_eq @ int @ T2 @ X3 )
=> ( ord_less_eq @ int @ T2 @ ( plus_plus @ int @ X3 @ D6 ) ) ) ) ) ).
% aset(8)
thf(fact_2095_cpmi,axiom,
! [D6: int,P2: int > $o,P5: int > $o,B6: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D6 )
=> ( ? [Z4: int] :
! [X4: int] :
( ( ord_less @ int @ X4 @ Z4 )
=> ( ( P2 @ X4 )
= ( P5 @ X4 ) ) )
=> ( ! [X4: int] :
( ! [Xa: int] :
( ( member @ int @ Xa @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D6 ) )
=> ! [Xb: int] :
( ( member @ int @ Xb @ B6 )
=> ( X4
!= ( plus_plus @ int @ Xb @ Xa ) ) ) )
=> ( ( P2 @ X4 )
=> ( P2 @ ( minus_minus @ int @ X4 @ D6 ) ) ) )
=> ( ! [X4: int,K: int] :
( ( P5 @ X4 )
= ( P5 @ ( minus_minus @ int @ X4 @ ( times_times @ int @ K @ D6 ) ) ) )
=> ( ( ? [X9: int] : ( P2 @ X9 ) )
= ( ? [X5: int] :
( ( member @ int @ X5 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D6 ) )
& ( P5 @ X5 ) )
| ? [X5: int] :
( ( member @ int @ X5 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D6 ) )
& ? [Y6: int] :
( ( member @ int @ Y6 @ B6 )
& ( P2 @ ( plus_plus @ int @ Y6 @ X5 ) ) ) ) ) ) ) ) ) ) ).
% cpmi
thf(fact_2096_cppi,axiom,
! [D6: int,P2: int > $o,P5: int > $o,A5: set @ int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D6 )
=> ( ? [Z4: int] :
! [X4: int] :
( ( ord_less @ int @ Z4 @ X4 )
=> ( ( P2 @ X4 )
= ( P5 @ X4 ) ) )
=> ( ! [X4: int] :
( ! [Xa: int] :
( ( member @ int @ Xa @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D6 ) )
=> ! [Xb: int] :
( ( member @ int @ Xb @ A5 )
=> ( X4
!= ( minus_minus @ int @ Xb @ Xa ) ) ) )
=> ( ( P2 @ X4 )
=> ( P2 @ ( plus_plus @ int @ X4 @ D6 ) ) ) )
=> ( ! [X4: int,K: int] :
( ( P5 @ X4 )
= ( P5 @ ( minus_minus @ int @ X4 @ ( times_times @ int @ K @ D6 ) ) ) )
=> ( ( ? [X9: int] : ( P2 @ X9 ) )
= ( ? [X5: int] :
( ( member @ int @ X5 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D6 ) )
& ( P5 @ X5 ) )
| ? [X5: int] :
( ( member @ int @ X5 @ ( set_or1337092689740270186AtMost @ int @ ( one_one @ int ) @ D6 ) )
& ? [Y6: int] :
( ( member @ int @ Y6 @ A5 )
& ( P2 @ ( minus_minus @ int @ Y6 @ X5 ) ) ) ) ) ) ) ) ) ) ).
% cppi
thf(fact_2097_divmod__def,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ( ( unique8689654367752047608divmod @ A )
= ( ^ [M3: num,N2: num] : ( product_Pair @ A @ A @ ( divide_divide @ A @ ( numeral_numeral @ A @ M3 ) @ ( numeral_numeral @ A @ N2 ) ) @ ( modulo_modulo @ A @ ( numeral_numeral @ A @ M3 ) @ ( numeral_numeral @ A @ N2 ) ) ) ) ) ) ).
% divmod_def
thf(fact_2098_divmod_H__nat__def,axiom,
( ( unique8689654367752047608divmod @ nat )
= ( ^ [M3: num,N2: num] : ( product_Pair @ nat @ nat @ ( divide_divide @ nat @ ( numeral_numeral @ nat @ M3 ) @ ( numeral_numeral @ nat @ N2 ) ) @ ( modulo_modulo @ nat @ ( numeral_numeral @ nat @ M3 ) @ ( numeral_numeral @ nat @ N2 ) ) ) ) ) ).
% divmod'_nat_def
thf(fact_2099_divmod__step__nat__def,axiom,
( ( unique1321980374590559556d_step @ nat )
= ( ^ [L2: num] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [Q5: nat,R5: nat] : ( if @ ( product_prod @ nat @ nat ) @ ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ L2 ) @ R5 ) @ ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Q5 ) @ ( one_one @ nat ) ) @ ( minus_minus @ nat @ R5 @ ( numeral_numeral @ nat @ L2 ) ) ) @ ( product_Pair @ nat @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Q5 ) @ R5 ) ) ) ) ) ).
% divmod_step_nat_def
thf(fact_2100_divmod__divmod__step,axiom,
! [A: $tType] :
( ( unique1627219031080169319umeral @ A )
=> ( ( unique8689654367752047608divmod @ A )
= ( ^ [M3: num,N2: num] : ( if @ ( product_prod @ A @ A ) @ ( ord_less @ num @ M3 @ N2 ) @ ( product_Pair @ A @ A @ ( zero_zero @ A ) @ ( numeral_numeral @ A @ M3 ) ) @ ( unique1321980374590559556d_step @ A @ N2 @ ( unique8689654367752047608divmod @ A @ M3 @ ( bit0 @ N2 ) ) ) ) ) ) ) ).
% divmod_divmod_step
thf(fact_2101_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_2102_divmod__nat__if,axiom,
( divmod_nat
= ( ^ [M3: nat,N2: nat] :
( if @ ( product_prod @ nat @ nat )
@ ( ( N2
= ( zero_zero @ nat ) )
| ( ord_less @ nat @ M3 @ N2 ) )
@ ( product_Pair @ nat @ nat @ ( zero_zero @ nat ) @ M3 )
@ ( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [Q5: nat] : ( product_Pair @ nat @ nat @ ( suc @ Q5 ) )
@ ( divmod_nat @ ( minus_minus @ nat @ M3 @ N2 ) @ N2 ) ) ) ) ) ).
% divmod_nat_if
thf(fact_2103_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_2104_take__bit__numeral__bit1,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [L: num,K2: num] :
( ( bit_se2584673776208193580ke_bit @ A @ ( numeral_numeral @ nat @ L ) @ ( numeral_numeral @ A @ ( bit1 @ K2 ) ) )
= ( plus_plus @ A @ ( times_times @ A @ ( bit_se2584673776208193580ke_bit @ A @ ( pred_numeral @ L ) @ ( numeral_numeral @ A @ K2 ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( one_one @ A ) ) ) ) ).
% take_bit_numeral_bit1
thf(fact_2105_sqrt__sum__squares__half__less,axiom,
! [X: real,U: real,Y: real] :
( ( ord_less @ real @ X @ ( divide_divide @ real @ U @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less @ real @ Y @ ( divide_divide @ real @ U @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y )
=> ( ord_less @ real @ ( sqrt @ ( plus_plus @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ U ) ) ) ) ) ).
% sqrt_sum_squares_half_less
thf(fact_2106_of__int__less__neg__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: int,X: num,N: nat] :
( ( ord_less @ A @ ( ring_1_of_int @ A @ A2 ) @ ( power_power @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ X ) ) @ N ) )
= ( ord_less @ int @ A2 @ ( power_power @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ X ) ) @ N ) ) ) ) ).
% of_int_less_neg_numeral_power_cancel_iff
thf(fact_2107_of__int__eq__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [W2: int,Z: int] :
( ( ( ring_1_of_int @ A @ W2 )
= ( ring_1_of_int @ A @ Z ) )
= ( W2 = Z ) ) ) ).
% of_int_eq_iff
thf(fact_2108_case__prodI2,axiom,
! [B: $tType,A: $tType,P: product_prod @ A @ B,C3: A > B > $o] :
( ! [A4: A,B4: B] :
( ( P
= ( product_Pair @ A @ B @ A4 @ B4 ) )
=> ( C3 @ A4 @ B4 ) )
=> ( product_case_prod @ A @ B @ $o @ C3 @ P ) ) ).
% case_prodI2
thf(fact_2109_case__prodI,axiom,
! [A: $tType,B: $tType,F3: A > B > $o,A2: A,B2: B] :
( ( F3 @ A2 @ B2 )
=> ( product_case_prod @ A @ B @ $o @ F3 @ ( product_Pair @ A @ B @ A2 @ B2 ) ) ) ).
% case_prodI
thf(fact_2110_mem__case__prodI2,axiom,
! [C: $tType,B: $tType,A: $tType,P: product_prod @ A @ B,Z: C,C3: A > B > ( set @ C )] :
( ! [A4: A,B4: B] :
( ( P
= ( product_Pair @ A @ B @ A4 @ B4 ) )
=> ( member @ C @ Z @ ( C3 @ A4 @ B4 ) ) )
=> ( member @ C @ Z @ ( product_case_prod @ A @ B @ ( set @ C ) @ C3 @ P ) ) ) ).
% mem_case_prodI2
thf(fact_2111_mem__case__prodI,axiom,
! [A: $tType,B: $tType,C: $tType,Z: A,C3: B > C > ( set @ A ),A2: B,B2: C] :
( ( member @ A @ Z @ ( C3 @ A2 @ B2 ) )
=> ( member @ A @ Z @ ( product_case_prod @ B @ C @ ( set @ A ) @ C3 @ ( product_Pair @ B @ C @ A2 @ B2 ) ) ) ) ).
% mem_case_prodI
thf(fact_2112_case__prodI2_H,axiom,
! [A: $tType,B: $tType,C: $tType,P: product_prod @ A @ B,C3: A > B > C > $o,X: C] :
( ! [A4: A,B4: B] :
( ( ( product_Pair @ A @ B @ A4 @ B4 )
= P )
=> ( C3 @ A4 @ B4 @ X ) )
=> ( product_case_prod @ A @ B @ ( C > $o ) @ C3 @ P @ X ) ) ).
% case_prodI2'
thf(fact_2113_real__sqrt__less__iff,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ ( sqrt @ X ) @ ( sqrt @ Y ) )
= ( ord_less @ real @ X @ Y ) ) ).
% real_sqrt_less_iff
thf(fact_2114_of__int__of__bool,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [P2: $o] :
( ( ring_1_of_int @ A @ ( zero_neq_one_of_bool @ int @ P2 ) )
= ( zero_neq_one_of_bool @ A @ P2 ) ) ) ).
% of_int_of_bool
thf(fact_2115_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_2116_tanh__real__less__iff,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ ( tanh @ real @ X ) @ ( tanh @ real @ Y ) )
= ( ord_less @ real @ X @ Y ) ) ).
% tanh_real_less_iff
thf(fact_2117_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_2118_of__int__0__eq__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [Z: int] :
( ( ( zero_zero @ A )
= ( ring_1_of_int @ A @ Z ) )
= ( Z
= ( zero_zero @ int ) ) ) ) ).
% of_int_0_eq_iff
thf(fact_2119_of__int__eq__0__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [Z: int] :
( ( ( ring_1_of_int @ A @ Z )
= ( zero_zero @ A ) )
= ( Z
= ( zero_zero @ int ) ) ) ) ).
% of_int_eq_0_iff
thf(fact_2120_of__int__le__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [W2: int,Z: int] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ W2 ) @ ( ring_1_of_int @ A @ Z ) )
= ( ord_less_eq @ int @ W2 @ Z ) ) ) ).
% of_int_le_iff
thf(fact_2121_of__int__eq__numeral__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [Z: int,N: num] :
( ( ( ring_1_of_int @ A @ Z )
= ( numeral_numeral @ A @ N ) )
= ( Z
= ( numeral_numeral @ int @ N ) ) ) ) ).
% of_int_eq_numeral_iff
thf(fact_2122_of__int__numeral,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [K2: num] :
( ( ring_1_of_int @ A @ ( numeral_numeral @ int @ K2 ) )
= ( numeral_numeral @ A @ K2 ) ) ) ).
% of_int_numeral
thf(fact_2123_of__int__less__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [W2: int,Z: int] :
( ( ord_less @ A @ ( ring_1_of_int @ A @ W2 ) @ ( ring_1_of_int @ A @ Z ) )
= ( ord_less @ int @ W2 @ Z ) ) ) ).
% of_int_less_iff
thf(fact_2124_of__int__eq__1__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [Z: int] :
( ( ( ring_1_of_int @ A @ Z )
= ( one_one @ A ) )
= ( Z
= ( one_one @ int ) ) ) ) ).
% of_int_eq_1_iff
thf(fact_2125_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_2126_of__int__mult,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [W2: int,Z: int] :
( ( ring_1_of_int @ A @ ( times_times @ int @ W2 @ Z ) )
= ( times_times @ A @ ( ring_1_of_int @ A @ W2 ) @ ( ring_1_of_int @ A @ Z ) ) ) ) ).
% of_int_mult
thf(fact_2127_of__int__add,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [W2: int,Z: int] :
( ( ring_1_of_int @ A @ ( plus_plus @ int @ W2 @ Z ) )
= ( plus_plus @ A @ ( ring_1_of_int @ A @ W2 ) @ ( ring_1_of_int @ A @ Z ) ) ) ) ).
% of_int_add
thf(fact_2128_of__int__minus,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [Z: int] :
( ( ring_1_of_int @ A @ ( uminus_uminus @ int @ Z ) )
= ( uminus_uminus @ A @ ( ring_1_of_int @ A @ Z ) ) ) ) ).
% of_int_minus
thf(fact_2129_real__sqrt__gt__0__iff,axiom,
! [Y: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( sqrt @ Y ) )
= ( ord_less @ real @ ( zero_zero @ real ) @ Y ) ) ).
% real_sqrt_gt_0_iff
thf(fact_2130_real__sqrt__lt__0__iff,axiom,
! [X: real] :
( ( ord_less @ real @ ( sqrt @ X ) @ ( zero_zero @ real ) )
= ( ord_less @ real @ X @ ( zero_zero @ real ) ) ) ).
% real_sqrt_lt_0_iff
thf(fact_2131_of__int__diff,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [W2: int,Z: int] :
( ( ring_1_of_int @ A @ ( minus_minus @ int @ W2 @ Z ) )
= ( minus_minus @ A @ ( ring_1_of_int @ A @ W2 ) @ ( ring_1_of_int @ A @ Z ) ) ) ) ).
% of_int_diff
thf(fact_2132_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_2133_real__sqrt__gt__1__iff,axiom,
! [Y: real] :
( ( ord_less @ real @ ( one_one @ real ) @ ( sqrt @ Y ) )
= ( ord_less @ real @ ( one_one @ real ) @ Y ) ) ).
% real_sqrt_gt_1_iff
thf(fact_2134_pred__numeral__simps_I1_J,axiom,
( ( pred_numeral @ one2 )
= ( zero_zero @ nat ) ) ).
% pred_numeral_simps(1)
thf(fact_2135_of__int__power__eq__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [X: int,B2: int,W2: nat] :
( ( ( ring_1_of_int @ A @ X )
= ( power_power @ A @ ( ring_1_of_int @ A @ B2 ) @ W2 ) )
= ( X
= ( power_power @ int @ B2 @ W2 ) ) ) ) ).
% of_int_power_eq_of_int_cancel_iff
thf(fact_2136_of__int__eq__of__int__power__cancel__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [B2: int,W2: nat,X: int] :
( ( ( power_power @ A @ ( ring_1_of_int @ A @ B2 ) @ W2 )
= ( ring_1_of_int @ A @ X ) )
= ( ( power_power @ int @ B2 @ W2 )
= X ) ) ) ).
% of_int_eq_of_int_power_cancel_iff
thf(fact_2137_of__int__power,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [Z: int,N: nat] :
( ( ring_1_of_int @ A @ ( power_power @ int @ Z @ N ) )
= ( power_power @ A @ ( ring_1_of_int @ A @ Z ) @ N ) ) ) ).
% of_int_power
thf(fact_2138_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_2139_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_2140_less__Suc__numeral,axiom,
! [N: nat,K2: num] :
( ( ord_less @ nat @ ( suc @ N ) @ ( numeral_numeral @ nat @ K2 ) )
= ( ord_less @ nat @ N @ ( pred_numeral @ K2 ) ) ) ).
% less_Suc_numeral
thf(fact_2141_less__numeral__Suc,axiom,
! [K2: num,N: nat] :
( ( ord_less @ nat @ ( numeral_numeral @ nat @ K2 ) @ ( suc @ N ) )
= ( ord_less @ nat @ ( pred_numeral @ K2 ) @ N ) ) ).
% less_numeral_Suc
thf(fact_2142_le__Suc__numeral,axiom,
! [N: nat,K2: num] :
( ( ord_less_eq @ nat @ ( suc @ N ) @ ( numeral_numeral @ nat @ K2 ) )
= ( ord_less_eq @ nat @ N @ ( pred_numeral @ K2 ) ) ) ).
% le_Suc_numeral
thf(fact_2143_le__numeral__Suc,axiom,
! [K2: num,N: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ K2 ) @ ( suc @ N ) )
= ( ord_less_eq @ nat @ ( pred_numeral @ K2 ) @ N ) ) ).
% le_numeral_Suc
thf(fact_2144_max__Suc__numeral,axiom,
! [N: nat,K2: num] :
( ( ord_max @ nat @ ( suc @ N ) @ ( numeral_numeral @ nat @ K2 ) )
= ( suc @ ( ord_max @ nat @ N @ ( pred_numeral @ K2 ) ) ) ) ).
% max_Suc_numeral
thf(fact_2145_max__numeral__Suc,axiom,
! [K2: num,N: nat] :
( ( ord_max @ nat @ ( numeral_numeral @ nat @ K2 ) @ ( suc @ N ) )
= ( suc @ ( ord_max @ nat @ ( pred_numeral @ K2 ) @ N ) ) ) ).
% max_numeral_Suc
thf(fact_2146_of__int__0__le__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z: int] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( ring_1_of_int @ A @ Z ) )
= ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z ) ) ) ).
% of_int_0_le_iff
thf(fact_2147_of__int__le__0__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z: int] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Z ) @ ( zero_zero @ A ) )
= ( ord_less_eq @ int @ Z @ ( zero_zero @ int ) ) ) ) ).
% of_int_le_0_iff
thf(fact_2148_of__int__0__less__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z: int] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( ring_1_of_int @ A @ Z ) )
= ( ord_less @ int @ ( zero_zero @ int ) @ Z ) ) ) ).
% of_int_0_less_iff
thf(fact_2149_of__int__less__0__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z: int] :
( ( ord_less @ A @ ( ring_1_of_int @ A @ Z ) @ ( zero_zero @ A ) )
= ( ord_less @ int @ Z @ ( zero_zero @ int ) ) ) ) ).
% of_int_less_0_iff
thf(fact_2150_of__int__numeral__le__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: num,Z: int] :
( ( ord_less_eq @ A @ ( numeral_numeral @ A @ N ) @ ( ring_1_of_int @ A @ Z ) )
= ( ord_less_eq @ int @ ( numeral_numeral @ int @ N ) @ Z ) ) ) ).
% of_int_numeral_le_iff
thf(fact_2151_of__int__le__numeral__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z: int,N: num] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Z ) @ ( numeral_numeral @ A @ N ) )
= ( ord_less_eq @ int @ Z @ ( numeral_numeral @ int @ N ) ) ) ) ).
% of_int_le_numeral_iff
thf(fact_2152_of__int__less__numeral__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z: int,N: num] :
( ( ord_less @ A @ ( ring_1_of_int @ A @ Z ) @ ( numeral_numeral @ A @ N ) )
= ( ord_less @ int @ Z @ ( numeral_numeral @ int @ N ) ) ) ) ).
% of_int_less_numeral_iff
thf(fact_2153_of__int__numeral__less__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: num,Z: int] :
( ( ord_less @ A @ ( numeral_numeral @ A @ N ) @ ( ring_1_of_int @ A @ Z ) )
= ( ord_less @ int @ ( numeral_numeral @ int @ N ) @ Z ) ) ) ).
% of_int_numeral_less_iff
thf(fact_2154_of__int__le__1__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z: int] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Z ) @ ( one_one @ A ) )
= ( ord_less_eq @ int @ Z @ ( one_one @ int ) ) ) ) ).
% of_int_le_1_iff
thf(fact_2155_of__int__1__le__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z: int] :
( ( ord_less_eq @ A @ ( one_one @ A ) @ ( ring_1_of_int @ A @ Z ) )
= ( ord_less_eq @ int @ ( one_one @ int ) @ Z ) ) ) ).
% of_int_1_le_iff
thf(fact_2156_of__int__less__1__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z: int] :
( ( ord_less @ A @ ( ring_1_of_int @ A @ Z ) @ ( one_one @ A ) )
= ( ord_less @ int @ Z @ ( one_one @ int ) ) ) ) ).
% of_int_less_1_iff
thf(fact_2157_of__int__1__less__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z: int] :
( ( ord_less @ A @ ( one_one @ A ) @ ( ring_1_of_int @ A @ Z ) )
= ( ord_less @ int @ ( one_one @ int ) @ Z ) ) ) ).
% of_int_1_less_iff
thf(fact_2158_of__int__power__le__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: int,B2: int,W2: nat] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ X ) @ ( power_power @ A @ ( ring_1_of_int @ A @ B2 ) @ W2 ) )
= ( ord_less_eq @ int @ X @ ( power_power @ int @ B2 @ W2 ) ) ) ) ).
% of_int_power_le_of_int_cancel_iff
thf(fact_2159_of__int__le__of__int__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [B2: int,W2: nat,X: int] :
( ( ord_less_eq @ A @ ( power_power @ A @ ( ring_1_of_int @ A @ B2 ) @ W2 ) @ ( ring_1_of_int @ A @ X ) )
= ( ord_less_eq @ int @ ( power_power @ int @ B2 @ W2 ) @ X ) ) ) ).
% of_int_le_of_int_power_cancel_iff
thf(fact_2160_of__int__eq__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [Y: int,X: num,N: nat] :
( ( ( ring_1_of_int @ A @ Y )
= ( power_power @ A @ ( numeral_numeral @ A @ X ) @ N ) )
= ( Y
= ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N ) ) ) ) ).
% of_int_eq_numeral_power_cancel_iff
thf(fact_2161_numeral__power__eq__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [X: num,N: nat,Y: int] :
( ( ( power_power @ A @ ( numeral_numeral @ A @ X ) @ N )
= ( ring_1_of_int @ A @ Y ) )
= ( ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N )
= Y ) ) ) ).
% numeral_power_eq_of_int_cancel_iff
thf(fact_2162_of__int__less__of__int__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [B2: int,W2: nat,X: int] :
( ( ord_less @ A @ ( power_power @ A @ ( ring_1_of_int @ A @ B2 ) @ W2 ) @ ( ring_1_of_int @ A @ X ) )
= ( ord_less @ int @ ( power_power @ int @ B2 @ W2 ) @ X ) ) ) ).
% of_int_less_of_int_power_cancel_iff
thf(fact_2163_of__int__power__less__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: int,B2: int,W2: nat] :
( ( ord_less @ A @ ( ring_1_of_int @ A @ X ) @ ( power_power @ A @ ( ring_1_of_int @ A @ B2 ) @ W2 ) )
= ( ord_less @ int @ X @ ( power_power @ int @ B2 @ W2 ) ) ) ) ).
% of_int_power_less_of_int_cancel_iff
thf(fact_2164_of__int__le__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: int,X: num,N: nat] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ A2 ) @ ( power_power @ A @ ( numeral_numeral @ A @ X ) @ N ) )
= ( ord_less_eq @ int @ A2 @ ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N ) ) ) ) ).
% of_int_le_numeral_power_cancel_iff
thf(fact_2165_numeral__power__le__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: num,N: nat,A2: int] :
( ( ord_less_eq @ A @ ( power_power @ A @ ( numeral_numeral @ A @ X ) @ N ) @ ( ring_1_of_int @ A @ A2 ) )
= ( ord_less_eq @ int @ ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N ) @ A2 ) ) ) ).
% numeral_power_le_of_int_cancel_iff
thf(fact_2166_numeral__power__less__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: num,N: nat,A2: int] :
( ( ord_less @ A @ ( power_power @ A @ ( numeral_numeral @ A @ X ) @ N ) @ ( ring_1_of_int @ A @ A2 ) )
= ( ord_less @ int @ ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N ) @ A2 ) ) ) ).
% numeral_power_less_of_int_cancel_iff
thf(fact_2167_of__int__less__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: int,X: num,N: nat] :
( ( ord_less @ A @ ( ring_1_of_int @ A @ A2 ) @ ( power_power @ A @ ( numeral_numeral @ A @ X ) @ N ) )
= ( ord_less @ int @ A2 @ ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N ) ) ) ) ).
% of_int_less_numeral_power_cancel_iff
thf(fact_2168_of__int__eq__neg__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [Y: int,X: num,N: nat] :
( ( ( ring_1_of_int @ A @ Y )
= ( power_power @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ X ) ) @ N ) )
= ( Y
= ( power_power @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ X ) ) @ N ) ) ) ) ).
% of_int_eq_neg_numeral_power_cancel_iff
thf(fact_2169_neg__numeral__power__eq__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [X: num,N: nat,Y: int] :
( ( ( power_power @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ X ) ) @ N )
= ( ring_1_of_int @ A @ Y ) )
= ( ( power_power @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ X ) ) @ N )
= Y ) ) ) ).
% neg_numeral_power_eq_of_int_cancel_iff
thf(fact_2170_of__int__le__neg__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: int,X: num,N: nat] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ A2 ) @ ( power_power @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ X ) ) @ N ) )
= ( ord_less_eq @ int @ A2 @ ( power_power @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ X ) ) @ N ) ) ) ) ).
% of_int_le_neg_numeral_power_cancel_iff
thf(fact_2171_neg__numeral__power__le__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: num,N: nat,A2: int] :
( ( ord_less_eq @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ X ) ) @ N ) @ ( ring_1_of_int @ A @ A2 ) )
= ( ord_less_eq @ int @ ( power_power @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ X ) ) @ N ) @ A2 ) ) ) ).
% neg_numeral_power_le_of_int_cancel_iff
thf(fact_2172_neg__numeral__power__less__of__int__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: num,N: nat,A2: int] :
( ( ord_less @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ X ) ) @ N ) @ ( ring_1_of_int @ A @ A2 ) )
= ( ord_less @ int @ ( power_power @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ X ) ) @ N ) @ A2 ) ) ) ).
% neg_numeral_power_less_of_int_cancel_iff
thf(fact_2173_ex__le__of__int,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X: A] :
? [Z3: int] : ( ord_less_eq @ A @ X @ ( ring_1_of_int @ A @ Z3 ) ) ) ).
% ex_le_of_int
thf(fact_2174_real__sqrt__less__mono,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ X @ Y )
=> ( ord_less @ real @ ( sqrt @ X ) @ ( sqrt @ Y ) ) ) ).
% real_sqrt_less_mono
thf(fact_2175_ex__less__of__int,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X: A] :
? [Z3: int] : ( ord_less @ A @ X @ ( ring_1_of_int @ A @ Z3 ) ) ) ).
% ex_less_of_int
thf(fact_2176_ex__of__int__less,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X: A] :
? [Z3: int] : ( ord_less @ A @ ( ring_1_of_int @ A @ Z3 ) @ X ) ) ).
% ex_of_int_less
thf(fact_2177_mult__of__int__commute,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [X: int,Y: A] :
( ( times_times @ A @ ( ring_1_of_int @ A @ X ) @ Y )
= ( times_times @ A @ Y @ ( ring_1_of_int @ A @ X ) ) ) ) ).
% mult_of_int_commute
thf(fact_2178_mem__case__prodE,axiom,
! [B: $tType,A: $tType,C: $tType,Z: A,C3: B > C > ( set @ A ),P: product_prod @ B @ C] :
( ( member @ A @ Z @ ( product_case_prod @ B @ C @ ( set @ A ) @ C3 @ P ) )
=> ~ ! [X4: B,Y3: C] :
( ( P
= ( product_Pair @ B @ C @ X4 @ Y3 ) )
=> ~ ( member @ A @ Z @ ( C3 @ X4 @ Y3 ) ) ) ) ).
% mem_case_prodE
thf(fact_2179_of__int__max,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: int,Y: int] :
( ( ring_1_of_int @ A @ ( ord_max @ int @ X @ Y ) )
= ( ord_max @ A @ ( ring_1_of_int @ A @ X ) @ ( ring_1_of_int @ A @ Y ) ) ) ) ).
% of_int_max
thf(fact_2180_case__prodE,axiom,
! [A: $tType,B: $tType,C3: A > B > $o,P: product_prod @ A @ B] :
( ( product_case_prod @ A @ B @ $o @ C3 @ P )
=> ~ ! [X4: A,Y3: B] :
( ( P
= ( product_Pair @ A @ B @ X4 @ Y3 ) )
=> ~ ( C3 @ X4 @ Y3 ) ) ) ).
% case_prodE
thf(fact_2181_case__prodD,axiom,
! [A: $tType,B: $tType,F3: A > B > $o,A2: A,B2: B] :
( ( product_case_prod @ A @ B @ $o @ F3 @ ( product_Pair @ A @ B @ A2 @ B2 ) )
=> ( F3 @ A2 @ B2 ) ) ).
% case_prodD
thf(fact_2182_case__prodE_H,axiom,
! [B: $tType,A: $tType,C: $tType,C3: A > B > C > $o,P: product_prod @ A @ B,Z: C] :
( ( product_case_prod @ A @ B @ ( C > $o ) @ C3 @ P @ Z )
=> ~ ! [X4: A,Y3: B] :
( ( P
= ( product_Pair @ A @ B @ X4 @ Y3 ) )
=> ~ ( C3 @ X4 @ Y3 @ Z ) ) ) ).
% case_prodE'
thf(fact_2183_case__prodD_H,axiom,
! [B: $tType,A: $tType,C: $tType,R: A > B > C > $o,A2: A,B2: B,C3: C] :
( ( product_case_prod @ A @ B @ ( C > $o ) @ R @ ( product_Pair @ A @ B @ A2 @ B2 ) @ C3 )
=> ( R @ A2 @ B2 @ C3 ) ) ).
% case_prodD'
thf(fact_2184_real__sqrt__gt__zero,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( sqrt @ X ) ) ) ).
% real_sqrt_gt_zero
thf(fact_2185_Collect__case__prod__mono,axiom,
! [B: $tType,A: $tType,A5: A > B > $o,B6: A > B > $o] :
( ( ord_less_eq @ ( A > B > $o ) @ A5 @ B6 )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ A5 ) ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ B6 ) ) ) ) ).
% Collect_case_prod_mono
thf(fact_2186_tanh__real__lt__1,axiom,
! [X: real] : ( ord_less @ real @ ( tanh @ real @ X ) @ ( one_one @ real ) ) ).
% tanh_real_lt_1
thf(fact_2187_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_2188_sqrt2__less__2,axiom,
ord_less @ real @ ( sqrt @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ).
% sqrt2_less_2
thf(fact_2189_of__int__nonneg,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( ring_1_of_int @ A @ Z ) ) ) ) ).
% of_int_nonneg
thf(fact_2190_of__int__pos,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Z: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ Z )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( ring_1_of_int @ A @ Z ) ) ) ) ).
% of_int_pos
thf(fact_2191_floor__exists1,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X: A] :
? [X4: int] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ X4 ) @ X )
& ( ord_less @ A @ X @ ( ring_1_of_int @ A @ ( plus_plus @ int @ X4 @ ( one_one @ int ) ) ) )
& ! [Y4: int] :
( ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Y4 ) @ X )
& ( ord_less @ A @ X @ ( ring_1_of_int @ A @ ( plus_plus @ int @ Y4 @ ( one_one @ int ) ) ) ) )
=> ( Y4 = X4 ) ) ) ) ).
% floor_exists1
thf(fact_2192_floor__exists,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X: A] :
? [Z3: int] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Z3 ) @ X )
& ( ord_less @ A @ X @ ( ring_1_of_int @ A @ ( plus_plus @ int @ Z3 @ ( one_one @ int ) ) ) ) ) ) ).
% floor_exists
thf(fact_2193_of__int__neg__numeral,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [K2: num] :
( ( ring_1_of_int @ A @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K2 ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ K2 ) ) ) ) ).
% of_int_neg_numeral
thf(fact_2194_int__le__real__less,axiom,
( ( ord_less_eq @ int )
= ( ^ [N2: int,M3: int] : ( ord_less @ real @ ( ring_1_of_int @ real @ N2 ) @ ( plus_plus @ real @ ( ring_1_of_int @ real @ M3 ) @ ( one_one @ real ) ) ) ) ) ).
% int_le_real_less
thf(fact_2195_int__less__real__le,axiom,
( ( ord_less @ int )
= ( ^ [N2: int,M3: int] : ( ord_less_eq @ real @ ( plus_plus @ real @ ( ring_1_of_int @ real @ N2 ) @ ( one_one @ real ) ) @ ( ring_1_of_int @ real @ M3 ) ) ) ) ).
% int_less_real_le
thf(fact_2196_real__less__rsqrt,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ Y )
=> ( ord_less @ real @ X @ ( sqrt @ Y ) ) ) ).
% real_less_rsqrt
thf(fact_2197_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_2198_real__less__lsqrt,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y )
=> ( ( ord_less @ real @ X @ ( power_power @ real @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
=> ( ord_less @ real @ ( sqrt @ X ) @ Y ) ) ) ) ).
% real_less_lsqrt
thf(fact_2199_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_2200_divmod__nat__def,axiom,
( divmod_nat
= ( ^ [M3: nat,N2: nat] : ( product_Pair @ nat @ nat @ ( divide_divide @ nat @ M3 @ N2 ) @ ( modulo_modulo @ nat @ M3 @ N2 ) ) ) ) ).
% divmod_nat_def
thf(fact_2201_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_2202_round__unique,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Y: int] :
( ( ord_less @ A @ ( minus_minus @ A @ X @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( ring_1_of_int @ A @ Y ) )
=> ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Y ) @ ( plus_plus @ A @ X @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) )
=> ( ( archimedean_round @ A @ X )
= Y ) ) ) ) ).
% round_unique
thf(fact_2203_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_2204_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_2205_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_2206_int__ge__less__than__def,axiom,
( int_ge_less_than
= ( ^ [D5: int] :
( collect @ ( product_prod @ int @ int )
@ ( product_case_prod @ int @ int @ $o
@ ^ [Z6: int,Z5: int] :
( ( ord_less_eq @ int @ D5 @ Z6 )
& ( ord_less @ int @ Z6 @ Z5 ) ) ) ) ) ) ).
% int_ge_less_than_def
thf(fact_2207_int__ge__less__than2__def,axiom,
( int_ge_less_than2
= ( ^ [D5: int] :
( collect @ ( product_prod @ int @ int )
@ ( product_case_prod @ int @ int @ $o
@ ^ [Z6: int,Z5: int] :
( ( ord_less_eq @ int @ D5 @ Z5 )
& ( ord_less @ int @ Z6 @ Z5 ) ) ) ) ) ) ).
% int_ge_less_than2_def
thf(fact_2208_Sum__Icc__int,axiom,
! [M2: int,N: int] :
( ( ord_less_eq @ int @ M2 @ N )
=> ( ( groups7311177749621191930dd_sum @ int @ int
@ ^ [X5: int] : X5
@ ( 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_2209_split__part,axiom,
! [B: $tType,A: $tType,P2: $o,Q: A > B > $o] :
( ( product_case_prod @ A @ B @ $o
@ ^ [A6: A,B5: B] :
( P2
& ( Q @ A6 @ B5 ) ) )
= ( ^ [Ab: product_prod @ A @ B] :
( P2
& ( product_case_prod @ A @ B @ $o @ Q @ Ab ) ) ) ) ).
% split_part
thf(fact_2210_sum_Oneutral__const,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A5: set @ B] :
( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [Uu3: B] : ( zero_zero @ A )
@ A5 )
= ( zero_zero @ A ) ) ) ).
% sum.neutral_const
thf(fact_2211_sum_Oempty,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G3: B > A] :
( ( groups7311177749621191930dd_sum @ B @ A @ G3 @ ( bot_bot @ ( set @ B ) ) )
= ( zero_zero @ A ) ) ) ).
% sum.empty
thf(fact_2212_sum__eq__0__iff,axiom,
! [A: $tType,B: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [F5: set @ B,F3: B > A] :
( ( finite_finite2 @ B @ F5 )
=> ( ( ( groups7311177749621191930dd_sum @ B @ A @ F3 @ F5 )
= ( zero_zero @ A ) )
= ( ! [X5: B] :
( ( member @ B @ X5 @ F5 )
=> ( ( F3 @ X5 )
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% sum_eq_0_iff
thf(fact_2213_sum_Oinfinite,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A5: set @ B,G3: B > A] :
( ~ ( finite_finite2 @ B @ A5 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G3 @ A5 )
= ( zero_zero @ A ) ) ) ) ).
% sum.infinite
thf(fact_2214_round__0,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archimedean_round @ A @ ( zero_zero @ A ) )
= ( zero_zero @ int ) ) ) ).
% round_0
thf(fact_2215_sum_Odelta,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ A )
=> ! [S2: set @ B,A2: B,B2: B > A] :
( ( finite_finite2 @ B @ S2 )
=> ( ( ( member @ B @ A2 @ S2 )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [K3: B] : ( if @ A @ ( K3 = A2 ) @ ( B2 @ K3 ) @ ( zero_zero @ A ) )
@ S2 )
= ( B2 @ A2 ) ) )
& ( ~ ( member @ B @ A2 @ S2 )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [K3: B] : ( if @ A @ ( K3 = A2 ) @ ( B2 @ K3 ) @ ( zero_zero @ A ) )
@ S2 )
= ( zero_zero @ A ) ) ) ) ) ) ).
% sum.delta
thf(fact_2216_sum_Odelta_H,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ A )
=> ! [S2: set @ B,A2: B,B2: B > A] :
( ( finite_finite2 @ B @ S2 )
=> ( ( ( member @ B @ A2 @ S2 )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [K3: B] : ( if @ A @ ( A2 = K3 ) @ ( B2 @ K3 ) @ ( zero_zero @ A ) )
@ S2 )
= ( B2 @ A2 ) ) )
& ( ~ ( member @ B @ A2 @ S2 )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [K3: B] : ( if @ A @ ( A2 = K3 ) @ ( B2 @ K3 ) @ ( zero_zero @ A ) )
@ S2 )
= ( zero_zero @ A ) ) ) ) ) ) ).
% sum.delta'
thf(fact_2217_of__int__sum,axiom,
! [A: $tType,B: $tType] :
( ( ring_1 @ A )
=> ! [F3: B > int,A5: set @ B] :
( ( ring_1_of_int @ A @ ( groups7311177749621191930dd_sum @ B @ int @ F3 @ A5 ) )
= ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X5: B] : ( ring_1_of_int @ A @ ( F3 @ X5 ) )
@ A5 ) ) ) ).
% of_int_sum
thf(fact_2218_prod_Odisc__eq__case,axiom,
! [B: $tType,A: $tType,Prod: product_prod @ A @ B] :
( product_case_prod @ A @ B @ $o
@ ^ [Uu3: A,Uv3: B] : $true
@ Prod ) ).
% prod.disc_eq_case
thf(fact_2219_sum_Oneutral,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A5: set @ B,G3: B > A] :
( ! [X4: B] :
( ( member @ B @ X4 @ A5 )
=> ( ( G3 @ X4 )
= ( zero_zero @ A ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G3 @ A5 )
= ( zero_zero @ A ) ) ) ) ).
% sum.neutral
thf(fact_2220_sum_Onot__neutral__contains__not__neutral,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G3: B > A,A5: set @ B] :
( ( ( groups7311177749621191930dd_sum @ B @ A @ G3 @ A5 )
!= ( zero_zero @ A ) )
=> ~ ! [A4: B] :
( ( member @ B @ A4 @ A5 )
=> ( ( G3 @ A4 )
= ( zero_zero @ A ) ) ) ) ) ).
% sum.not_neutral_contains_not_neutral
thf(fact_2221_sum__mono,axiom,
! [A: $tType,B: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [K5: set @ B,F3: B > A,G3: B > A] :
( ! [I3: B] :
( ( member @ B @ I3 @ K5 )
=> ( ord_less_eq @ A @ ( F3 @ I3 ) @ ( G3 @ I3 ) ) )
=> ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ K5 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G3 @ K5 ) ) ) ) ).
% sum_mono
thf(fact_2222_sum_Odistrib,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [G3: B > A,H2: B > A,A5: set @ B] :
( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X5: B] : ( plus_plus @ A @ ( G3 @ X5 ) @ ( H2 @ X5 ) )
@ A5 )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G3 @ A5 ) @ ( groups7311177749621191930dd_sum @ B @ A @ H2 @ A5 ) ) ) ) ).
% sum.distrib
thf(fact_2223_sum_Oswap__restrict,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( comm_monoid_add @ A )
=> ! [A5: set @ B,B6: set @ C,G3: B > C > A,R: B > C > $o] :
( ( finite_finite2 @ B @ A5 )
=> ( ( finite_finite2 @ C @ B6 )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X5: B] :
( groups7311177749621191930dd_sum @ C @ A @ ( G3 @ X5 )
@ ( collect @ C
@ ^ [Y6: C] :
( ( member @ C @ Y6 @ B6 )
& ( R @ X5 @ Y6 ) ) ) )
@ A5 )
= ( groups7311177749621191930dd_sum @ C @ A
@ ^ [Y6: C] :
( groups7311177749621191930dd_sum @ B @ A
@ ^ [X5: B] : ( G3 @ X5 @ Y6 )
@ ( collect @ B
@ ^ [X5: B] :
( ( member @ B @ X5 @ A5 )
& ( R @ X5 @ Y6 ) ) ) )
@ B6 ) ) ) ) ) ).
% sum.swap_restrict
thf(fact_2224_sum__nonpos,axiom,
! [B: $tType,A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A5: set @ B,F3: B > A] :
( ! [X4: B] :
( ( member @ B @ X4 @ A5 )
=> ( ord_less_eq @ A @ ( F3 @ X4 ) @ ( zero_zero @ A ) ) )
=> ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ A5 ) @ ( zero_zero @ A ) ) ) ) ).
% sum_nonpos
thf(fact_2225_sum__nonneg,axiom,
! [A: $tType,B: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A5: set @ B,F3: B > A] :
( ! [X4: B] :
( ( member @ B @ X4 @ A5 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F3 @ X4 ) ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ A5 ) ) ) ) ).
% sum_nonneg
thf(fact_2226_sum__mono__inv,axiom,
! [A: $tType,I7: $tType] :
( ( ordere8940638589300402666id_add @ A )
=> ! [F3: I7 > A,I6: set @ I7,G3: I7 > A,I2: I7] :
( ( ( groups7311177749621191930dd_sum @ I7 @ A @ F3 @ I6 )
= ( groups7311177749621191930dd_sum @ I7 @ A @ G3 @ I6 ) )
=> ( ! [I3: I7] :
( ( member @ I7 @ I3 @ I6 )
=> ( ord_less_eq @ A @ ( F3 @ I3 ) @ ( G3 @ I3 ) ) )
=> ( ( member @ I7 @ I2 @ I6 )
=> ( ( finite_finite2 @ I7 @ I6 )
=> ( ( F3 @ I2 )
= ( G3 @ I2 ) ) ) ) ) ) ) ).
% sum_mono_inv
thf(fact_2227_sum_Ointer__filter,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A5: set @ B,G3: B > A,P2: B > $o] :
( ( finite_finite2 @ B @ A5 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G3
@ ( collect @ B
@ ^ [X5: B] :
( ( member @ B @ X5 @ A5 )
& ( P2 @ X5 ) ) ) )
= ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X5: B] : ( if @ A @ ( P2 @ X5 ) @ ( G3 @ X5 ) @ ( zero_zero @ A ) )
@ A5 ) ) ) ) ).
% sum.inter_filter
thf(fact_2228_sum__nonneg__eq__0__iff,axiom,
! [A: $tType,B: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [A5: set @ B,F3: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ A5 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F3 @ X4 ) ) )
=> ( ( ( groups7311177749621191930dd_sum @ B @ A @ F3 @ A5 )
= ( zero_zero @ A ) )
= ( ! [X5: B] :
( ( member @ B @ X5 @ A5 )
=> ( ( F3 @ X5 )
= ( zero_zero @ A ) ) ) ) ) ) ) ) ).
% sum_nonneg_eq_0_iff
thf(fact_2229_sum__le__included,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [S: set @ B,T2: set @ C,G3: C > A,I2: C > B,F3: B > A] :
( ( finite_finite2 @ B @ S )
=> ( ( finite_finite2 @ C @ T2 )
=> ( ! [X4: C] :
( ( member @ C @ X4 @ T2 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( G3 @ X4 ) ) )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ S )
=> ? [Xa: C] :
( ( member @ C @ Xa @ T2 )
& ( ( I2 @ Xa )
= X4 )
& ( ord_less_eq @ A @ ( F3 @ X4 ) @ ( G3 @ Xa ) ) ) )
=> ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ S ) @ ( groups7311177749621191930dd_sum @ C @ A @ G3 @ T2 ) ) ) ) ) ) ) ).
% sum_le_included
thf(fact_2230_sum__strict__mono__ex1,axiom,
! [A: $tType,I7: $tType] :
( ( ordere8940638589300402666id_add @ A )
=> ! [A5: set @ I7,F3: I7 > A,G3: I7 > A] :
( ( finite_finite2 @ I7 @ A5 )
=> ( ! [X4: I7] :
( ( member @ I7 @ X4 @ A5 )
=> ( ord_less_eq @ A @ ( F3 @ X4 ) @ ( G3 @ X4 ) ) )
=> ( ? [X3: I7] :
( ( member @ I7 @ X3 @ A5 )
& ( ord_less @ A @ ( F3 @ X3 ) @ ( G3 @ X3 ) ) )
=> ( ord_less @ A @ ( groups7311177749621191930dd_sum @ I7 @ A @ F3 @ A5 ) @ ( groups7311177749621191930dd_sum @ I7 @ A @ G3 @ A5 ) ) ) ) ) ) ).
% sum_strict_mono_ex1
thf(fact_2231_sum_Orelated,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [R: A > A > $o,S2: set @ B,H2: B > A,G3: B > A] :
( ( R @ ( zero_zero @ A ) @ ( zero_zero @ A ) )
=> ( ! [X16: A,Y15: A,X22: A,Y23: A] :
( ( ( R @ X16 @ X22 )
& ( R @ Y15 @ Y23 ) )
=> ( R @ ( plus_plus @ A @ X16 @ Y15 ) @ ( plus_plus @ A @ X22 @ Y23 ) ) )
=> ( ( finite_finite2 @ B @ S2 )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ S2 )
=> ( R @ ( H2 @ X4 ) @ ( G3 @ X4 ) ) )
=> ( R @ ( groups7311177749621191930dd_sum @ B @ A @ H2 @ S2 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G3 @ S2 ) ) ) ) ) ) ) ).
% sum.related
thf(fact_2232_sum__strict__mono,axiom,
! [A: $tType,B: $tType] :
( ( strict7427464778891057005id_add @ A )
=> ! [A5: set @ B,F3: B > A,G3: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ A5 )
=> ( ord_less @ A @ ( F3 @ X4 ) @ ( G3 @ X4 ) ) )
=> ( ord_less @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ A5 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G3 @ A5 ) ) ) ) ) ) ).
% sum_strict_mono
thf(fact_2233_sum_Oreindex__bij__witness__not__neutral,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comm_monoid_add @ A )
=> ! [S5: set @ B,T5: set @ C,S2: set @ B,I2: C > B,J: B > C,T4: set @ C,G3: B > A,H2: C > A] :
( ( finite_finite2 @ B @ S5 )
=> ( ( finite_finite2 @ C @ T5 )
=> ( ! [A4: B] :
( ( member @ B @ A4 @ ( minus_minus @ ( set @ B ) @ S2 @ S5 ) )
=> ( ( I2 @ ( J @ A4 ) )
= A4 ) )
=> ( ! [A4: B] :
( ( member @ B @ A4 @ ( minus_minus @ ( set @ B ) @ S2 @ S5 ) )
=> ( member @ C @ ( J @ A4 ) @ ( minus_minus @ ( set @ C ) @ T4 @ T5 ) ) )
=> ( ! [B4: C] :
( ( member @ C @ B4 @ ( minus_minus @ ( set @ C ) @ T4 @ T5 ) )
=> ( ( J @ ( I2 @ B4 ) )
= B4 ) )
=> ( ! [B4: C] :
( ( member @ C @ B4 @ ( minus_minus @ ( set @ C ) @ T4 @ T5 ) )
=> ( member @ B @ ( I2 @ B4 ) @ ( minus_minus @ ( set @ B ) @ S2 @ S5 ) ) )
=> ( ! [A4: B] :
( ( member @ B @ A4 @ S5 )
=> ( ( G3 @ A4 )
= ( zero_zero @ A ) ) )
=> ( ! [B4: C] :
( ( member @ C @ B4 @ T5 )
=> ( ( H2 @ B4 )
= ( zero_zero @ A ) ) )
=> ( ! [A4: B] :
( ( member @ B @ A4 @ S2 )
=> ( ( H2 @ ( J @ A4 ) )
= ( G3 @ A4 ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G3 @ S2 )
= ( groups7311177749621191930dd_sum @ C @ A @ H2 @ T4 ) ) ) ) ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_witness_not_neutral
thf(fact_2234_round__mono,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ord_less_eq @ int @ ( archimedean_round @ A @ X ) @ ( archimedean_round @ A @ Y ) ) ) ) ).
% round_mono
thf(fact_2235_sum__nonneg__leq__bound,axiom,
! [B: $tType,A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [S: set @ B,F3: B > A,B6: A,I2: B] :
( ( finite_finite2 @ B @ S )
=> ( ! [I3: B] :
( ( member @ B @ I3 @ S )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F3 @ I3 ) ) )
=> ( ( ( groups7311177749621191930dd_sum @ B @ A @ F3 @ S )
= B6 )
=> ( ( member @ B @ I2 @ S )
=> ( ord_less_eq @ A @ ( F3 @ I2 ) @ B6 ) ) ) ) ) ) ).
% sum_nonneg_leq_bound
thf(fact_2236_sum__nonneg__0,axiom,
! [B: $tType,A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [S: set @ B,F3: B > A,I2: B] :
( ( finite_finite2 @ B @ S )
=> ( ! [I3: B] :
( ( member @ B @ I3 @ S )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F3 @ I3 ) ) )
=> ( ( ( groups7311177749621191930dd_sum @ B @ A @ F3 @ S )
= ( zero_zero @ A ) )
=> ( ( member @ B @ I2 @ S )
=> ( ( F3 @ I2 )
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% sum_nonneg_0
thf(fact_2237_sum_Osetdiff__irrelevant,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A5: set @ B,G3: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G3
@ ( minus_minus @ ( set @ B ) @ A5
@ ( collect @ B
@ ^ [X5: B] :
( ( G3 @ X5 )
= ( zero_zero @ A ) ) ) ) )
= ( groups7311177749621191930dd_sum @ B @ A @ G3 @ A5 ) ) ) ) ).
% sum.setdiff_irrelevant
thf(fact_2238_sum__pos2,axiom,
! [A: $tType,B: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [I6: set @ B,I2: B,F3: B > A] :
( ( finite_finite2 @ B @ I6 )
=> ( ( member @ B @ I2 @ I6 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ ( F3 @ I2 ) )
=> ( ! [I3: B] :
( ( member @ B @ I3 @ I6 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F3 @ I3 ) ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ I6 ) ) ) ) ) ) ) ).
% sum_pos2
thf(fact_2239_sum__pos,axiom,
! [A: $tType,B: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [I6: set @ B,F3: B > A] :
( ( finite_finite2 @ B @ I6 )
=> ( ( I6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [I3: B] :
( ( member @ B @ I3 @ I6 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( F3 @ I3 ) ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ I6 ) ) ) ) ) ) ).
% sum_pos
thf(fact_2240_sum_Omono__neutral__cong__right,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [T4: set @ B,S2: set @ B,G3: B > A,H2: B > A] :
( ( finite_finite2 @ B @ T4 )
=> ( ( ord_less_eq @ ( set @ B ) @ S2 @ T4 )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ ( minus_minus @ ( set @ B ) @ T4 @ S2 ) )
=> ( ( G3 @ X4 )
= ( zero_zero @ A ) ) )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ S2 )
=> ( ( G3 @ X4 )
= ( H2 @ X4 ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G3 @ T4 )
= ( groups7311177749621191930dd_sum @ B @ A @ H2 @ S2 ) ) ) ) ) ) ) ).
% sum.mono_neutral_cong_right
thf(fact_2241_sum_Omono__neutral__cong__left,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [T4: set @ B,S2: set @ B,H2: B > A,G3: B > A] :
( ( finite_finite2 @ B @ T4 )
=> ( ( ord_less_eq @ ( set @ B ) @ S2 @ T4 )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ ( minus_minus @ ( set @ B ) @ T4 @ S2 ) )
=> ( ( H2 @ X4 )
= ( zero_zero @ A ) ) )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ S2 )
=> ( ( G3 @ X4 )
= ( H2 @ X4 ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G3 @ S2 )
= ( groups7311177749621191930dd_sum @ B @ A @ H2 @ T4 ) ) ) ) ) ) ) ).
% sum.mono_neutral_cong_left
thf(fact_2242_sum_Omono__neutral__right,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [T4: set @ B,S2: set @ B,G3: B > A] :
( ( finite_finite2 @ B @ T4 )
=> ( ( ord_less_eq @ ( set @ B ) @ S2 @ T4 )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ ( minus_minus @ ( set @ B ) @ T4 @ S2 ) )
=> ( ( G3 @ X4 )
= ( zero_zero @ A ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G3 @ T4 )
= ( groups7311177749621191930dd_sum @ B @ A @ G3 @ S2 ) ) ) ) ) ) ).
% sum.mono_neutral_right
thf(fact_2243_sum_Omono__neutral__left,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [T4: set @ B,S2: set @ B,G3: B > A] :
( ( finite_finite2 @ B @ T4 )
=> ( ( ord_less_eq @ ( set @ B ) @ S2 @ T4 )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ ( minus_minus @ ( set @ B ) @ T4 @ S2 ) )
=> ( ( G3 @ X4 )
= ( zero_zero @ A ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G3 @ S2 )
= ( groups7311177749621191930dd_sum @ B @ A @ G3 @ T4 ) ) ) ) ) ) ).
% sum.mono_neutral_left
thf(fact_2244_sum_Osame__carrierI,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [C5: set @ B,A5: set @ B,B6: set @ B,G3: B > A,H2: B > A] :
( ( finite_finite2 @ B @ C5 )
=> ( ( ord_less_eq @ ( set @ B ) @ A5 @ C5 )
=> ( ( ord_less_eq @ ( set @ B ) @ B6 @ C5 )
=> ( ! [A4: B] :
( ( member @ B @ A4 @ ( minus_minus @ ( set @ B ) @ C5 @ A5 ) )
=> ( ( G3 @ A4 )
= ( zero_zero @ A ) ) )
=> ( ! [B4: B] :
( ( member @ B @ B4 @ ( minus_minus @ ( set @ B ) @ C5 @ B6 ) )
=> ( ( H2 @ B4 )
= ( zero_zero @ A ) ) )
=> ( ( ( groups7311177749621191930dd_sum @ B @ A @ G3 @ C5 )
= ( groups7311177749621191930dd_sum @ B @ A @ H2 @ C5 ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G3 @ A5 )
= ( groups7311177749621191930dd_sum @ B @ A @ H2 @ B6 ) ) ) ) ) ) ) ) ) ).
% sum.same_carrierI
thf(fact_2245_sum_Osame__carrier,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [C5: set @ B,A5: set @ B,B6: set @ B,G3: B > A,H2: B > A] :
( ( finite_finite2 @ B @ C5 )
=> ( ( ord_less_eq @ ( set @ B ) @ A5 @ C5 )
=> ( ( ord_less_eq @ ( set @ B ) @ B6 @ C5 )
=> ( ! [A4: B] :
( ( member @ B @ A4 @ ( minus_minus @ ( set @ B ) @ C5 @ A5 ) )
=> ( ( G3 @ A4 )
= ( zero_zero @ A ) ) )
=> ( ! [B4: B] :
( ( member @ B @ B4 @ ( minus_minus @ ( set @ B ) @ C5 @ B6 ) )
=> ( ( H2 @ B4 )
= ( zero_zero @ A ) ) )
=> ( ( ( groups7311177749621191930dd_sum @ B @ A @ G3 @ A5 )
= ( groups7311177749621191930dd_sum @ B @ A @ H2 @ B6 ) )
= ( ( groups7311177749621191930dd_sum @ B @ A @ G3 @ C5 )
= ( groups7311177749621191930dd_sum @ B @ A @ H2 @ C5 ) ) ) ) ) ) ) ) ) ).
% sum.same_carrier
thf(fact_2246_sum_Osubset__diff,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [B6: set @ B,A5: set @ B,G3: B > A] :
( ( ord_less_eq @ ( set @ B ) @ B6 @ A5 )
=> ( ( finite_finite2 @ B @ A5 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G3 @ A5 )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G3 @ ( minus_minus @ ( set @ B ) @ A5 @ B6 ) ) @ ( groups7311177749621191930dd_sum @ B @ A @ G3 @ B6 ) ) ) ) ) ) ).
% sum.subset_diff
thf(fact_2247_sum__diff,axiom,
! [A: $tType,B: $tType] :
( ( ab_group_add @ A )
=> ! [A5: set @ B,B6: set @ B,F3: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( ord_less_eq @ ( set @ B ) @ B6 @ A5 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ F3 @ ( minus_minus @ ( set @ B ) @ A5 @ B6 ) )
= ( minus_minus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ A5 ) @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ B6 ) ) ) ) ) ) ).
% sum_diff
thf(fact_2248_sum__mono2,axiom,
! [A: $tType,B: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [B6: set @ B,A5: set @ B,F3: B > A] :
( ( finite_finite2 @ B @ B6 )
=> ( ( ord_less_eq @ ( set @ B ) @ A5 @ B6 )
=> ( ! [B4: B] :
( ( member @ B @ B4 @ ( minus_minus @ ( set @ B ) @ B6 @ A5 ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F3 @ B4 ) ) )
=> ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ A5 ) @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ B6 ) ) ) ) ) ) ).
% sum_mono2
thf(fact_2249_sum__strict__mono2,axiom,
! [B: $tType,A: $tType] :
( ( ordere8940638589300402666id_add @ B )
=> ! [B6: set @ A,A5: set @ A,B2: A,F3: A > B] :
( ( finite_finite2 @ A @ B6 )
=> ( ( ord_less_eq @ ( set @ A ) @ A5 @ B6 )
=> ( ( member @ A @ B2 @ ( minus_minus @ ( set @ A ) @ B6 @ A5 ) )
=> ( ( ord_less @ B @ ( zero_zero @ B ) @ ( F3 @ B2 ) )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ B6 )
=> ( ord_less_eq @ B @ ( zero_zero @ B ) @ ( F3 @ X4 ) ) )
=> ( ord_less @ B @ ( groups7311177749621191930dd_sum @ A @ B @ F3 @ A5 ) @ ( groups7311177749621191930dd_sum @ A @ B @ F3 @ B6 ) ) ) ) ) ) ) ) ).
% sum_strict_mono2
thf(fact_2250_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_2251_set__encode__insert,axiom,
! [A5: set @ nat,N: nat] :
( ( finite_finite2 @ nat @ A5 )
=> ( ~ ( member @ nat @ N @ A5 )
=> ( ( nat_set_encode @ ( insert @ nat @ N @ A5 ) )
= ( plus_plus @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ ( nat_set_encode @ A5 ) ) ) ) ) ).
% set_encode_insert
thf(fact_2252_Sum__Icc__nat,axiom,
! [M2: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [X5: nat] : X5
@ ( 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_2253_neg__numeral__le__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V: num,X: A] :
( ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V ) ) @ ( archimedean_ceiling @ A @ X ) )
= ( ord_less @ A @ ( minus_minus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) @ ( one_one @ A ) ) @ X ) ) ) ).
% neg_numeral_le_ceiling
thf(fact_2254_ceiling__less__neg__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,V: num] :
( ( ord_less @ int @ ( archimedean_ceiling @ A @ X ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V ) ) )
= ( ord_less_eq @ A @ X @ ( minus_minus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) @ ( one_one @ A ) ) ) ) ) ).
% ceiling_less_neg_numeral
thf(fact_2255_arith__series__nat,axiom,
! [A2: nat,D3: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [I4: nat] : ( plus_plus @ nat @ A2 @ ( times_times @ nat @ I4 @ D3 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
= ( divide_divide @ nat @ ( times_times @ nat @ ( suc @ N ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ A2 ) @ ( times_times @ nat @ N @ D3 ) ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% arith_series_nat
thf(fact_2256_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_2257_singletonI,axiom,
! [A: $tType,A2: A] : ( member @ A @ A2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% singletonI
thf(fact_2258_finite__insert,axiom,
! [A: $tType,A2: A,A5: set @ A] :
( ( finite_finite2 @ A @ ( insert @ A @ A2 @ A5 ) )
= ( finite_finite2 @ A @ A5 ) ) ).
% finite_insert
thf(fact_2259_singleton__conv,axiom,
! [A: $tType,A2: A] :
( ( collect @ A
@ ^ [X5: A] : X5 = A2 )
= ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% singleton_conv
thf(fact_2260_singleton__conv2,axiom,
! [A: $tType,A2: A] :
( ( collect @ A
@ ( ^ [Y5: A,Z2: A] : Y5 = Z2
@ A2 ) )
= ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ).
% singleton_conv2
thf(fact_2261_singleton__insert__inj__eq_H,axiom,
! [A: $tType,A2: A,A5: set @ A,B2: A] :
( ( ( insert @ A @ A2 @ A5 )
= ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) )
= ( ( A2 = B2 )
& ( ord_less_eq @ ( set @ A ) @ A5 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% singleton_insert_inj_eq'
thf(fact_2262_singleton__insert__inj__eq,axiom,
! [A: $tType,B2: A,A2: A,A5: set @ A] :
( ( ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) )
= ( insert @ A @ A2 @ A5 ) )
= ( ( A2 = B2 )
& ( ord_less_eq @ ( set @ A ) @ A5 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% singleton_insert_inj_eq
thf(fact_2263_atLeastAtMost__singleton,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A] :
( ( set_or1337092689740270186AtMost @ A @ A2 @ A2 )
= ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% atLeastAtMost_singleton
thf(fact_2264_atLeastAtMost__singleton__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( ( set_or1337092689740270186AtMost @ A @ A2 @ B2 )
= ( insert @ A @ C3 @ ( bot_bot @ ( set @ A ) ) ) )
= ( ( A2 = B2 )
& ( B2 = C3 ) ) ) ) ).
% atLeastAtMost_singleton_iff
thf(fact_2265_insert__Diff__single,axiom,
! [A: $tType,A2: A,A5: set @ A] :
( ( insert @ A @ A2 @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( insert @ A @ A2 @ A5 ) ) ).
% insert_Diff_single
thf(fact_2266_finite__Diff__insert,axiom,
! [A: $tType,A5: set @ A,A2: A,B6: set @ A] :
( ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ A2 @ B6 ) ) )
= ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ A5 @ B6 ) ) ) ).
% finite_Diff_insert
thf(fact_2267_ceiling__zero,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archimedean_ceiling @ A @ ( zero_zero @ A ) )
= ( zero_zero @ int ) ) ) ).
% ceiling_zero
thf(fact_2268_mask__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se2239418461657761734s_mask @ A @ ( zero_zero @ nat ) )
= ( zero_zero @ A ) ) ) ).
% mask_0
thf(fact_2269_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_2270_sum_Oinsert,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A5: set @ B,X: B,G3: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ~ ( member @ B @ X @ A5 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G3 @ ( insert @ B @ X @ A5 ) )
= ( plus_plus @ A @ ( G3 @ X ) @ ( groups7311177749621191930dd_sum @ B @ A @ G3 @ A5 ) ) ) ) ) ) ).
% sum.insert
thf(fact_2271_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_2272_subset__Compl__singleton,axiom,
! [A: $tType,A5: set @ A,B2: A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ ( uminus_uminus @ ( set @ A ) @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( ~ ( member @ A @ B2 @ A5 ) ) ) ).
% subset_Compl_singleton
thf(fact_2273_ceiling__add__of__int,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Z: int] :
( ( archimedean_ceiling @ A @ ( plus_plus @ A @ X @ ( ring_1_of_int @ A @ Z ) ) )
= ( plus_plus @ int @ ( archimedean_ceiling @ A @ X ) @ Z ) ) ) ).
% ceiling_add_of_int
thf(fact_2274_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_2275_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_2276_ceiling__le__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,V: num] :
( ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ X ) @ ( numeral_numeral @ int @ V ) )
= ( ord_less_eq @ A @ X @ ( numeral_numeral @ A @ V ) ) ) ) ).
% ceiling_le_numeral
thf(fact_2277_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_2278_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_2279_numeral__less__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V: num,X: A] :
( ( ord_less @ int @ ( numeral_numeral @ int @ V ) @ ( archimedean_ceiling @ A @ X ) )
= ( ord_less @ A @ ( numeral_numeral @ A @ V ) @ X ) ) ) ).
% numeral_less_ceiling
thf(fact_2280_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_2281_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_2282_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_2283_ceiling__add__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,V: num] :
( ( archimedean_ceiling @ A @ ( plus_plus @ A @ X @ ( numeral_numeral @ A @ V ) ) )
= ( plus_plus @ int @ ( archimedean_ceiling @ A @ X ) @ ( numeral_numeral @ int @ V ) ) ) ) ).
% ceiling_add_numeral
thf(fact_2284_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_2285_sum_Ocl__ivl__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [N: nat,M2: nat,G3: nat > A] :
( ( ( ord_less @ nat @ ( suc @ N ) @ M2 )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ M2 @ ( suc @ N ) ) )
= ( zero_zero @ A ) ) )
& ( ~ ( ord_less @ nat @ ( suc @ N ) @ M2 )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ M2 @ ( suc @ N ) ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) @ ( G3 @ ( suc @ N ) ) ) ) ) ) ) ).
% sum.cl_ivl_Suc
thf(fact_2286_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_2287_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_2288_sum__zero__power,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A5: set @ nat,C3: nat > A] :
( ( ( ( finite_finite2 @ nat @ A5 )
& ( member @ nat @ ( zero_zero @ nat ) @ A5 ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( C3 @ I4 ) @ ( power_power @ A @ ( zero_zero @ A ) @ I4 ) )
@ A5 )
= ( C3 @ ( zero_zero @ nat ) ) ) )
& ( ~ ( ( finite_finite2 @ nat @ A5 )
& ( member @ nat @ ( zero_zero @ nat ) @ A5 ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( C3 @ I4 ) @ ( power_power @ A @ ( zero_zero @ A ) @ I4 ) )
@ A5 )
= ( zero_zero @ A ) ) ) ) ) ).
% sum_zero_power
thf(fact_2289_ceiling__less__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,V: num] :
( ( ord_less @ int @ ( archimedean_ceiling @ A @ X ) @ ( numeral_numeral @ int @ V ) )
= ( ord_less_eq @ A @ X @ ( minus_minus @ A @ ( numeral_numeral @ A @ V ) @ ( one_one @ A ) ) ) ) ) ).
% ceiling_less_numeral
thf(fact_2290_numeral__le__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V: num,X: A] :
( ( ord_less_eq @ int @ ( numeral_numeral @ int @ V ) @ ( archimedean_ceiling @ A @ X ) )
= ( ord_less @ A @ ( minus_minus @ A @ ( numeral_numeral @ A @ V ) @ ( one_one @ A ) ) @ X ) ) ) ).
% numeral_le_ceiling
thf(fact_2291_ceiling__le__neg__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,V: num] :
( ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ X ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V ) ) )
= ( ord_less_eq @ A @ X @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) ) ) ) ).
% ceiling_le_neg_numeral
thf(fact_2292_neg__numeral__less__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V: num,X: A] :
( ( ord_less @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V ) ) @ ( archimedean_ceiling @ A @ X ) )
= ( ord_less @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) @ X ) ) ) ).
% neg_numeral_less_ceiling
thf(fact_2293_sum__zero__power_H,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A5: set @ nat,C3: nat > A,D3: nat > A] :
( ( ( ( finite_finite2 @ nat @ A5 )
& ( member @ nat @ ( zero_zero @ nat ) @ A5 ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( divide_divide @ A @ ( times_times @ A @ ( C3 @ I4 ) @ ( power_power @ A @ ( zero_zero @ A ) @ I4 ) ) @ ( D3 @ I4 ) )
@ A5 )
= ( divide_divide @ A @ ( C3 @ ( zero_zero @ nat ) ) @ ( D3 @ ( zero_zero @ nat ) ) ) ) )
& ( ~ ( ( finite_finite2 @ nat @ A5 )
& ( member @ nat @ ( zero_zero @ nat ) @ A5 ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( divide_divide @ A @ ( times_times @ A @ ( C3 @ I4 ) @ ( power_power @ A @ ( zero_zero @ A ) @ I4 ) ) @ ( D3 @ I4 ) )
@ A5 )
= ( zero_zero @ A ) ) ) ) ) ).
% sum_zero_power'
thf(fact_2294_singletonD,axiom,
! [A: $tType,B2: A,A2: A] :
( ( member @ A @ B2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( B2 = A2 ) ) ).
% singletonD
thf(fact_2295_singleton__iff,axiom,
! [A: $tType,B2: A,A2: A] :
( ( member @ A @ B2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
= ( B2 = A2 ) ) ).
% singleton_iff
thf(fact_2296_doubleton__eq__iff,axiom,
! [A: $tType,A2: A,B2: A,C3: A,D3: A] :
( ( ( insert @ A @ A2 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) )
= ( insert @ A @ C3 @ ( insert @ A @ D3 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( ( ( A2 = C3 )
& ( B2 = D3 ) )
| ( ( A2 = D3 )
& ( B2 = C3 ) ) ) ) ).
% doubleton_eq_iff
thf(fact_2297_insert__not__empty,axiom,
! [A: $tType,A2: A,A5: set @ A] :
( ( insert @ A @ A2 @ A5 )
!= ( bot_bot @ ( set @ A ) ) ) ).
% insert_not_empty
thf(fact_2298_singleton__inject,axiom,
! [A: $tType,A2: A,B2: A] :
( ( ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) )
= ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( A2 = B2 ) ) ).
% singleton_inject
thf(fact_2299_finite_OinsertI,axiom,
! [A: $tType,A5: set @ A,A2: A] :
( ( finite_finite2 @ A @ A5 )
=> ( finite_finite2 @ A @ ( insert @ A @ A2 @ A5 ) ) ) ).
% finite.insertI
thf(fact_2300_less__eq__mask,axiom,
! [N: nat] : ( ord_less_eq @ nat @ N @ ( bit_se2239418461657761734s_mask @ nat @ N ) ) ).
% less_eq_mask
thf(fact_2301_sum__diff1__nat,axiom,
! [A: $tType,A2: A,A5: set @ A,F3: A > nat] :
( ( ( member @ A @ A2 @ A5 )
=> ( ( groups7311177749621191930dd_sum @ A @ nat @ F3 @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( minus_minus @ nat @ ( groups7311177749621191930dd_sum @ A @ nat @ F3 @ A5 ) @ ( F3 @ A2 ) ) ) )
& ( ~ ( member @ A @ A2 @ A5 )
=> ( ( groups7311177749621191930dd_sum @ A @ nat @ F3 @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( groups7311177749621191930dd_sum @ A @ nat @ F3 @ A5 ) ) ) ) ).
% sum_diff1_nat
thf(fact_2302_Collect__conv__if,axiom,
! [A: $tType,P2: A > $o,A2: A] :
( ( ( P2 @ A2 )
=> ( ( collect @ A
@ ^ [X5: A] :
( ( X5 = A2 )
& ( P2 @ X5 ) ) )
= ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) )
& ( ~ ( P2 @ A2 )
=> ( ( collect @ A
@ ^ [X5: A] :
( ( X5 = A2 )
& ( P2 @ X5 ) ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Collect_conv_if
thf(fact_2303_Collect__conv__if2,axiom,
! [A: $tType,P2: A > $o,A2: A] :
( ( ( P2 @ A2 )
=> ( ( collect @ A
@ ^ [X5: A] :
( ( A2 = X5 )
& ( P2 @ X5 ) ) )
= ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) )
& ( ~ ( P2 @ A2 )
=> ( ( collect @ A
@ ^ [X5: A] :
( ( A2 = X5 )
& ( P2 @ X5 ) ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Collect_conv_if2
thf(fact_2304_sum__cong__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A5: set @ nat,F3: nat > A,G3: nat > A] :
( ~ ( member @ nat @ ( zero_zero @ nat ) @ A5 )
=> ( ! [X4: nat] :
( ( member @ nat @ ( suc @ X4 ) @ A5 )
=> ( ( F3 @ ( suc @ X4 ) )
= ( G3 @ ( suc @ X4 ) ) ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ A5 )
= ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ A5 ) ) ) ) ) ).
% sum_cong_Suc
thf(fact_2305_ceiling__mono,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ Y @ X )
=> ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ Y ) @ ( archimedean_ceiling @ A @ X ) ) ) ) ).
% ceiling_mono
thf(fact_2306_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_2307_ceiling__less__cancel,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ int @ ( archimedean_ceiling @ A @ X ) @ ( archimedean_ceiling @ A @ Y ) )
=> ( ord_less @ A @ X @ Y ) ) ) ).
% ceiling_less_cancel
thf(fact_2308_infinite__finite__induct,axiom,
! [A: $tType,P2: ( set @ A ) > $o,A5: set @ A] :
( ! [A8: set @ A] :
( ~ ( finite_finite2 @ A @ A8 )
=> ( P2 @ A8 ) )
=> ( ( P2 @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [X4: A,F6: set @ A] :
( ( finite_finite2 @ A @ F6 )
=> ( ~ ( member @ A @ X4 @ F6 )
=> ( ( P2 @ F6 )
=> ( P2 @ ( insert @ A @ X4 @ F6 ) ) ) ) )
=> ( P2 @ A5 ) ) ) ) ).
% infinite_finite_induct
thf(fact_2309_finite__ne__induct,axiom,
! [A: $tType,F5: set @ A,P2: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ F5 )
=> ( ( F5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X4: A] : ( P2 @ ( insert @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( ! [X4: A,F6: set @ A] :
( ( finite_finite2 @ A @ F6 )
=> ( ( F6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ~ ( member @ A @ X4 @ F6 )
=> ( ( P2 @ F6 )
=> ( P2 @ ( insert @ A @ X4 @ F6 ) ) ) ) ) )
=> ( P2 @ F5 ) ) ) ) ) ).
% finite_ne_induct
thf(fact_2310_finite__induct,axiom,
! [A: $tType,F5: set @ A,P2: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ F5 )
=> ( ( P2 @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [X4: A,F6: set @ A] :
( ( finite_finite2 @ A @ F6 )
=> ( ~ ( member @ A @ X4 @ F6 )
=> ( ( P2 @ F6 )
=> ( P2 @ ( insert @ A @ X4 @ F6 ) ) ) ) )
=> ( P2 @ F5 ) ) ) ) ).
% finite_induct
thf(fact_2311_finite_Osimps,axiom,
! [A: $tType] :
( ( finite_finite2 @ A )
= ( ^ [A6: set @ A] :
( ( A6
= ( bot_bot @ ( set @ A ) ) )
| ? [A7: set @ A,B5: A] :
( ( A6
= ( insert @ A @ B5 @ A7 ) )
& ( finite_finite2 @ A @ A7 ) ) ) ) ) ).
% finite.simps
thf(fact_2312_finite_Ocases,axiom,
! [A: $tType,A2: set @ A] :
( ( finite_finite2 @ A @ A2 )
=> ( ( A2
!= ( bot_bot @ ( set @ A ) ) )
=> ~ ! [A8: set @ A] :
( ? [A4: A] :
( A2
= ( insert @ A @ A4 @ A8 ) )
=> ~ ( finite_finite2 @ A @ A8 ) ) ) ) ).
% finite.cases
thf(fact_2313_subset__singleton__iff,axiom,
! [A: $tType,X7: set @ A,A2: A] :
( ( ord_less_eq @ ( set @ A ) @ X7 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
= ( ( X7
= ( bot_bot @ ( set @ A ) ) )
| ( X7
= ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% subset_singleton_iff
thf(fact_2314_subset__singletonD,axiom,
! [A: $tType,A5: set @ A,X: A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
=> ( ( A5
= ( bot_bot @ ( set @ A ) ) )
| ( A5
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% subset_singletonD
thf(fact_2315_atLeastAtMost__singleton_H,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A2: A,B2: A] :
( ( A2 = B2 )
=> ( ( set_or1337092689740270186AtMost @ A @ A2 @ B2 )
= ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% atLeastAtMost_singleton'
thf(fact_2316_Diff__insert__absorb,axiom,
! [A: $tType,X: A,A5: set @ A] :
( ~ ( member @ A @ X @ A5 )
=> ( ( minus_minus @ ( set @ A ) @ ( insert @ A @ X @ A5 ) @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= A5 ) ) ).
% Diff_insert_absorb
thf(fact_2317_Diff__insert2,axiom,
! [A: $tType,A5: set @ A,A2: A,B6: set @ A] :
( ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ A2 @ B6 ) )
= ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) @ B6 ) ) ).
% Diff_insert2
thf(fact_2318_insert__Diff,axiom,
! [A: $tType,A2: A,A5: set @ A] :
( ( member @ A @ A2 @ A5 )
=> ( ( insert @ A @ A2 @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= A5 ) ) ).
% insert_Diff
thf(fact_2319_Diff__insert,axiom,
! [A: $tType,A5: set @ A,A2: A,B6: set @ A] :
( ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ A2 @ B6 ) )
= ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A5 @ B6 ) @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Diff_insert
thf(fact_2320_sum__subtractf__nat,axiom,
! [A: $tType,A5: set @ A,G3: A > nat,F3: A > nat] :
( ! [X4: A] :
( ( member @ A @ X4 @ A5 )
=> ( ord_less_eq @ nat @ ( G3 @ X4 ) @ ( F3 @ X4 ) ) )
=> ( ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [X5: A] : ( minus_minus @ nat @ ( F3 @ X5 ) @ ( G3 @ X5 ) )
@ A5 )
= ( minus_minus @ nat @ ( groups7311177749621191930dd_sum @ A @ nat @ F3 @ A5 ) @ ( groups7311177749621191930dd_sum @ A @ nat @ G3 @ A5 ) ) ) ) ).
% sum_subtractf_nat
thf(fact_2321_sum_Oshift__bounds__cl__Suc__ivl,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G3: nat > A,M2: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M2 ) @ ( suc @ N ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( G3 @ ( suc @ I4 ) )
@ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ).
% sum.shift_bounds_cl_Suc_ivl
thf(fact_2322_not__mask__negative__int,axiom,
! [N: nat] :
~ ( ord_less @ int @ ( bit_se2239418461657761734s_mask @ int @ N ) @ ( zero_zero @ int ) ) ).
% not_mask_negative_int
thf(fact_2323_sum_Oshift__bounds__cl__nat__ivl,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G3: nat > A,M2: nat,K2: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ M2 @ K2 ) @ ( plus_plus @ nat @ N @ K2 ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( G3 @ ( plus_plus @ nat @ I4 @ K2 ) )
@ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ).
% sum.shift_bounds_cl_nat_ivl
thf(fact_2324_sum__eq__Suc0__iff,axiom,
! [A: $tType,A5: set @ A,F3: A > nat] :
( ( finite_finite2 @ A @ A5 )
=> ( ( ( groups7311177749621191930dd_sum @ A @ nat @ F3 @ A5 )
= ( suc @ ( zero_zero @ nat ) ) )
= ( ? [X5: A] :
( ( member @ A @ X5 @ A5 )
& ( ( F3 @ X5 )
= ( suc @ ( zero_zero @ nat ) ) )
& ! [Y6: A] :
( ( member @ A @ Y6 @ A5 )
=> ( ( X5 != Y6 )
=> ( ( F3 @ Y6 )
= ( zero_zero @ nat ) ) ) ) ) ) ) ) ).
% sum_eq_Suc0_iff
thf(fact_2325_sum__SucD,axiom,
! [A: $tType,F3: A > nat,A5: set @ A,N: nat] :
( ( ( groups7311177749621191930dd_sum @ A @ nat @ F3 @ A5 )
= ( suc @ N ) )
=> ? [X4: A] :
( ( member @ A @ X4 @ A5 )
& ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F3 @ X4 ) ) ) ) ).
% sum_SucD
thf(fact_2326_sum__eq__1__iff,axiom,
! [A: $tType,A5: set @ A,F3: A > nat] :
( ( finite_finite2 @ A @ A5 )
=> ( ( ( groups7311177749621191930dd_sum @ A @ nat @ F3 @ A5 )
= ( one_one @ nat ) )
= ( ? [X5: A] :
( ( member @ A @ X5 @ A5 )
& ( ( F3 @ X5 )
= ( one_one @ nat ) )
& ! [Y6: A] :
( ( member @ A @ Y6 @ A5 )
=> ( ( X5 != Y6 )
=> ( ( F3 @ Y6 )
= ( zero_zero @ nat ) ) ) ) ) ) ) ) ).
% sum_eq_1_iff
thf(fact_2327_ceiling__le,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,A2: int] :
( ( ord_less_eq @ A @ X @ ( ring_1_of_int @ A @ A2 ) )
=> ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ X ) @ A2 ) ) ) ).
% ceiling_le
thf(fact_2328_ceiling__le__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Z: int] :
( ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ X ) @ Z )
= ( ord_less_eq @ A @ X @ ( ring_1_of_int @ A @ Z ) ) ) ) ).
% ceiling_le_iff
thf(fact_2329_less__ceiling__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z: int,X: A] :
( ( ord_less @ int @ Z @ ( archimedean_ceiling @ A @ X ) )
= ( ord_less @ A @ ( ring_1_of_int @ A @ Z ) @ X ) ) ) ).
% less_ceiling_iff
thf(fact_2330_ceiling__add__le,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Y: A] : ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ ( plus_plus @ A @ X @ Y ) ) @ ( plus_plus @ int @ ( archimedean_ceiling @ A @ X ) @ ( archimedean_ceiling @ A @ Y ) ) ) ) ).
% ceiling_add_le
thf(fact_2331_finite__ranking__induct,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [S2: set @ B,P2: ( set @ B ) > $o,F3: B > A] :
( ( finite_finite2 @ B @ S2 )
=> ( ( P2 @ ( bot_bot @ ( set @ B ) ) )
=> ( ! [X4: B,S6: set @ B] :
( ( finite_finite2 @ B @ S6 )
=> ( ! [Y4: B] :
( ( member @ B @ Y4 @ S6 )
=> ( ord_less_eq @ A @ ( F3 @ Y4 ) @ ( F3 @ X4 ) ) )
=> ( ( P2 @ S6 )
=> ( P2 @ ( insert @ B @ X4 @ S6 ) ) ) ) )
=> ( P2 @ S2 ) ) ) ) ) ).
% finite_ranking_induct
thf(fact_2332_finite__linorder__max__induct,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,P2: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ A5 )
=> ( ( P2 @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [B4: A,A8: set @ A] :
( ( finite_finite2 @ A @ A8 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ A8 )
=> ( ord_less @ A @ X3 @ B4 ) )
=> ( ( P2 @ A8 )
=> ( P2 @ ( insert @ A @ B4 @ A8 ) ) ) ) )
=> ( P2 @ A5 ) ) ) ) ) ).
% finite_linorder_max_induct
thf(fact_2333_finite__linorder__min__induct,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,P2: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ A5 )
=> ( ( P2 @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [B4: A,A8: set @ A] :
( ( finite_finite2 @ A @ A8 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ A8 )
=> ( ord_less @ A @ B4 @ X3 ) )
=> ( ( P2 @ A8 )
=> ( P2 @ ( insert @ A @ B4 @ A8 ) ) ) ) )
=> ( P2 @ A5 ) ) ) ) ) ).
% finite_linorder_min_induct
thf(fact_2334_sum_Oinsert__if,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A5: set @ B,X: B,G3: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( ( member @ B @ X @ A5 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G3 @ ( insert @ B @ X @ A5 ) )
= ( groups7311177749621191930dd_sum @ B @ A @ G3 @ A5 ) ) )
& ( ~ ( member @ B @ X @ A5 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G3 @ ( insert @ B @ X @ A5 ) )
= ( plus_plus @ A @ ( G3 @ X ) @ ( groups7311177749621191930dd_sum @ B @ A @ G3 @ A5 ) ) ) ) ) ) ) ).
% sum.insert_if
thf(fact_2335_finite__subset__induct,axiom,
! [A: $tType,F5: set @ A,A5: set @ A,P2: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ F5 )
=> ( ( ord_less_eq @ ( set @ A ) @ F5 @ A5 )
=> ( ( P2 @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [A4: A,F6: set @ A] :
( ( finite_finite2 @ A @ F6 )
=> ( ( member @ A @ A4 @ A5 )
=> ( ~ ( member @ A @ A4 @ F6 )
=> ( ( P2 @ F6 )
=> ( P2 @ ( insert @ A @ A4 @ F6 ) ) ) ) ) )
=> ( P2 @ F5 ) ) ) ) ) ).
% finite_subset_induct
thf(fact_2336_finite__subset__induct_H,axiom,
! [A: $tType,F5: set @ A,A5: set @ A,P2: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ F5 )
=> ( ( ord_less_eq @ ( set @ A ) @ F5 @ A5 )
=> ( ( P2 @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [A4: A,F6: set @ A] :
( ( finite_finite2 @ A @ F6 )
=> ( ( member @ A @ A4 @ A5 )
=> ( ( ord_less_eq @ ( set @ A ) @ F6 @ A5 )
=> ( ~ ( member @ A @ A4 @ F6 )
=> ( ( P2 @ F6 )
=> ( P2 @ ( insert @ A @ A4 @ F6 ) ) ) ) ) ) )
=> ( P2 @ F5 ) ) ) ) ) ).
% finite_subset_induct'
thf(fact_2337_finite__empty__induct,axiom,
! [A: $tType,A5: set @ A,P2: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ A5 )
=> ( ( P2 @ A5 )
=> ( ! [A4: A,A8: set @ A] :
( ( finite_finite2 @ A @ A8 )
=> ( ( member @ A @ A4 @ A8 )
=> ( ( P2 @ A8 )
=> ( P2 @ ( minus_minus @ ( set @ A ) @ A8 @ ( insert @ A @ A4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) )
=> ( P2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% finite_empty_induct
thf(fact_2338_infinite__coinduct,axiom,
! [A: $tType,X7: ( set @ A ) > $o,A5: set @ A] :
( ( X7 @ A5 )
=> ( ! [A8: set @ A] :
( ( X7 @ A8 )
=> ? [X3: A] :
( ( member @ A @ X3 @ A8 )
& ( ( X7 @ ( minus_minus @ ( set @ A ) @ A8 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) )
| ~ ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ A8 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) )
=> ~ ( finite_finite2 @ A @ A5 ) ) ) ).
% infinite_coinduct
thf(fact_2339_infinite__remove,axiom,
! [A: $tType,S2: set @ A,A2: A] :
( ~ ( finite_finite2 @ A @ S2 )
=> ~ ( finite_finite2 @ A @ ( minus_minus @ ( set @ A ) @ S2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% infinite_remove
thf(fact_2340_subset__insert__iff,axiom,
! [A: $tType,A5: set @ A,X: A,B6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ ( insert @ A @ X @ B6 ) )
= ( ( ( member @ A @ X @ A5 )
=> ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ B6 ) )
& ( ~ ( member @ A @ X @ A5 )
=> ( ord_less_eq @ ( set @ A ) @ A5 @ B6 ) ) ) ) ).
% subset_insert_iff
thf(fact_2341_Diff__single__insert,axiom,
! [A: $tType,A5: set @ A,X: A,B6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ B6 )
=> ( ord_less_eq @ ( set @ A ) @ A5 @ ( insert @ A @ X @ B6 ) ) ) ).
% Diff_single_insert
thf(fact_2342_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_2343_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_2344_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_2345_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_2346_set__update__subset__insert,axiom,
! [A: $tType,Xs: list @ A,I2: nat,X: A] : ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ ( list_update @ A @ Xs @ I2 @ X ) ) @ ( insert @ A @ X @ ( set2 @ A @ Xs ) ) ) ).
% set_update_subset_insert
thf(fact_2347_Compl__insert,axiom,
! [A: $tType,X: A,A5: set @ A] :
( ( uminus_uminus @ ( set @ A ) @ ( insert @ A @ X @ A5 ) )
= ( minus_minus @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A5 ) @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Compl_insert
thf(fact_2348_sum__power__add,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [X: A,M2: nat,I6: set @ nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( power_power @ A @ X @ ( plus_plus @ nat @ M2 @ I4 ) )
@ I6 )
= ( times_times @ A @ ( power_power @ A @ X @ M2 ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ X ) @ I6 ) ) ) ) ).
% sum_power_add
thf(fact_2349_sum_OatLeastAtMost__rev,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G3: nat > A,N: nat,M2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ N @ M2 ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( G3 @ ( minus_minus @ nat @ ( plus_plus @ nat @ M2 @ N ) @ I4 ) )
@ ( set_or1337092689740270186AtMost @ nat @ N @ M2 ) ) ) ) ).
% sum.atLeastAtMost_rev
thf(fact_2350_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_2351_sum__nth__roots,axiom,
! [N: nat,C3: complex] :
( ( ord_less @ nat @ ( one_one @ nat ) @ N )
=> ( ( groups7311177749621191930dd_sum @ complex @ complex
@ ^ [X5: complex] : X5
@ ( collect @ complex
@ ^ [Z5: complex] :
( ( power_power @ complex @ Z5 @ N )
= C3 ) ) )
= ( zero_zero @ complex ) ) ) ).
% sum_nth_roots
thf(fact_2352_sum__roots__unity,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( one_one @ nat ) @ N )
=> ( ( groups7311177749621191930dd_sum @ complex @ complex
@ ^ [X5: complex] : X5
@ ( collect @ complex
@ ^ [Z5: complex] :
( ( power_power @ complex @ Z5 @ N )
= ( one_one @ complex ) ) ) )
= ( zero_zero @ complex ) ) ) ).
% sum_roots_unity
thf(fact_2353_of__int__ceiling__le__add__one,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [R2: A] : ( ord_less_eq @ A @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ R2 ) ) @ ( plus_plus @ A @ R2 @ ( one_one @ A ) ) ) ) ).
% of_int_ceiling_le_add_one
thf(fact_2354_of__int__ceiling__diff__one__le,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [R2: A] : ( ord_less_eq @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ R2 ) ) @ ( one_one @ A ) ) @ R2 ) ) ).
% of_int_ceiling_diff_one_le
thf(fact_2355_sum__diff__nat,axiom,
! [A: $tType,B6: set @ A,A5: set @ A,F3: A > nat] :
( ( finite_finite2 @ A @ B6 )
=> ( ( ord_less_eq @ ( set @ A ) @ B6 @ A5 )
=> ( ( groups7311177749621191930dd_sum @ A @ nat @ F3 @ ( minus_minus @ ( set @ A ) @ A5 @ B6 ) )
= ( minus_minus @ nat @ ( groups7311177749621191930dd_sum @ A @ nat @ F3 @ A5 ) @ ( groups7311177749621191930dd_sum @ A @ nat @ F3 @ B6 ) ) ) ) ) ).
% sum_diff_nat
thf(fact_2356_sum__shift__lb__Suc0__0,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [F3: nat > A,K2: nat] :
( ( ( F3 @ ( zero_zero @ nat ) )
= ( zero_zero @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ ( zero_zero @ nat ) ) @ K2 ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ K2 ) ) ) ) ) ).
% sum_shift_lb_Suc0_0
thf(fact_2357_sum_OatLeast0__atMost__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G3: nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) @ ( G3 @ ( suc @ N ) ) ) ) ) ).
% sum.atLeast0_atMost_Suc
thf(fact_2358_sum_Onat__ivl__Suc_H,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [M2: nat,N: nat,G3: nat > A] :
( ( ord_less_eq @ nat @ M2 @ ( suc @ N ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ M2 @ ( suc @ N ) ) )
= ( plus_plus @ A @ ( G3 @ ( suc @ N ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ) ) ).
% sum.nat_ivl_Suc'
thf(fact_2359_sum_OatLeast__Suc__atMost,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [M2: nat,N: nat,G3: nat > A] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
= ( plus_plus @ A @ ( G3 @ M2 ) @ ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M2 ) @ N ) ) ) ) ) ) ).
% sum.atLeast_Suc_atMost
thf(fact_2360_remove__induct,axiom,
! [A: $tType,P2: ( set @ A ) > $o,B6: set @ A] :
( ( P2 @ ( bot_bot @ ( set @ A ) ) )
=> ( ( ~ ( finite_finite2 @ A @ B6 )
=> ( P2 @ B6 ) )
=> ( ! [A8: set @ A] :
( ( finite_finite2 @ A @ A8 )
=> ( ( A8
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ A8 @ B6 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ A8 )
=> ( P2 @ ( minus_minus @ ( set @ A ) @ A8 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) )
=> ( P2 @ A8 ) ) ) ) )
=> ( P2 @ B6 ) ) ) ) ).
% remove_induct
thf(fact_2361_finite__remove__induct,axiom,
! [A: $tType,B6: set @ A,P2: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ B6 )
=> ( ( P2 @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [A8: set @ A] :
( ( finite_finite2 @ A @ A8 )
=> ( ( A8
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ A8 @ B6 )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ A8 )
=> ( P2 @ ( minus_minus @ ( set @ A ) @ A8 @ ( insert @ A @ X3 @ ( bot_bot @ ( set @ A ) ) ) ) ) )
=> ( P2 @ A8 ) ) ) ) )
=> ( P2 @ B6 ) ) ) ) ).
% finite_remove_induct
thf(fact_2362_finite__induct__select,axiom,
! [A: $tType,S2: set @ A,P2: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ S2 )
=> ( ( P2 @ ( bot_bot @ ( set @ A ) ) )
=> ( ! [T6: set @ A] :
( ( ord_less @ ( set @ A ) @ T6 @ S2 )
=> ( ( P2 @ T6 )
=> ? [X3: A] :
( ( member @ A @ X3 @ ( minus_minus @ ( set @ A ) @ S2 @ T6 ) )
& ( P2 @ ( insert @ A @ X3 @ T6 ) ) ) ) )
=> ( P2 @ S2 ) ) ) ) ).
% finite_induct_select
thf(fact_2363_psubset__insert__iff,axiom,
! [A: $tType,A5: set @ A,X: A,B6: set @ A] :
( ( ord_less @ ( set @ A ) @ A5 @ ( insert @ A @ X @ B6 ) )
= ( ( ( member @ A @ X @ B6 )
=> ( ord_less @ ( set @ A ) @ A5 @ B6 ) )
& ( ~ ( member @ A @ X @ B6 )
=> ( ( ( member @ A @ X @ A5 )
=> ( ord_less @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ B6 ) )
& ( ~ ( member @ A @ X @ A5 )
=> ( ord_less_eq @ ( set @ A ) @ A5 @ B6 ) ) ) ) ) ) ).
% psubset_insert_iff
thf(fact_2364_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_2365_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_2366_sum_OSuc__reindex__ivl,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [M2: nat,N: nat,G3: nat > A] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) @ ( G3 @ ( suc @ N ) ) )
= ( plus_plus @ A @ ( G3 @ M2 )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( G3 @ ( suc @ I4 ) )
@ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ) ) ).
% sum.Suc_reindex_ivl
thf(fact_2367_sum__Suc__diff,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [M2: nat,N: nat,F3: nat > A] :
( ( ord_less_eq @ nat @ M2 @ ( suc @ N ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( minus_minus @ A @ ( F3 @ ( suc @ I4 ) ) @ ( F3 @ I4 ) )
@ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
= ( minus_minus @ A @ ( F3 @ ( suc @ N ) ) @ ( F3 @ M2 ) ) ) ) ) ).
% sum_Suc_diff
thf(fact_2368_ceiling__split,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [P2: int > $o,T2: A] :
( ( P2 @ ( archimedean_ceiling @ A @ T2 ) )
= ( ! [I4: int] :
( ( ( ord_less @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ I4 ) @ ( one_one @ A ) ) @ T2 )
& ( ord_less_eq @ A @ T2 @ ( ring_1_of_int @ A @ I4 ) ) )
=> ( P2 @ I4 ) ) ) ) ) ).
% ceiling_split
thf(fact_2369_ceiling__eq__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,A2: int] :
( ( ( archimedean_ceiling @ A @ X )
= A2 )
= ( ( ord_less @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ A2 ) @ ( one_one @ A ) ) @ X )
& ( ord_less_eq @ A @ X @ ( ring_1_of_int @ A @ A2 ) ) ) ) ) ).
% ceiling_eq_iff
thf(fact_2370_ceiling__unique,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z: int,X: A] :
( ( ord_less @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ Z ) @ ( one_one @ A ) ) @ X )
=> ( ( ord_less_eq @ A @ X @ ( ring_1_of_int @ A @ Z ) )
=> ( ( archimedean_ceiling @ A @ X )
= Z ) ) ) ) ).
% ceiling_unique
thf(fact_2371_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_2372_mult__ceiling__le,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less_eq @ int @ ( archimedean_ceiling @ A @ ( times_times @ A @ A2 @ B2 ) ) @ ( times_times @ int @ ( archimedean_ceiling @ A @ A2 ) @ ( archimedean_ceiling @ A @ B2 ) ) ) ) ) ) ).
% mult_ceiling_le
thf(fact_2373_ceiling__less__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Z: int] :
( ( ord_less @ int @ ( archimedean_ceiling @ A @ X ) @ Z )
= ( ord_less_eq @ A @ X @ ( minus_minus @ A @ ( ring_1_of_int @ A @ Z ) @ ( one_one @ A ) ) ) ) ) ).
% ceiling_less_iff
thf(fact_2374_le__ceiling__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z: int,X: A] :
( ( ord_less_eq @ int @ Z @ ( archimedean_ceiling @ A @ X ) )
= ( ord_less @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ Z ) @ ( one_one @ A ) ) @ X ) ) ) ).
% le_ceiling_iff
thf(fact_2375_sum_Oub__add__nat,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [M2: nat,N: nat,G3: nat > A,P: nat] :
( ( ord_less_eq @ nat @ M2 @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ M2 @ ( plus_plus @ nat @ N @ P ) ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) @ ( plus_plus @ nat @ N @ P ) ) ) ) ) ) ) ).
% sum.ub_add_nat
thf(fact_2376_sum_Oremove,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A5: set @ B,X: B,G3: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( member @ B @ X @ A5 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G3 @ A5 )
= ( plus_plus @ A @ ( G3 @ X ) @ ( groups7311177749621191930dd_sum @ B @ A @ G3 @ ( minus_minus @ ( set @ B ) @ A5 @ ( insert @ B @ X @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ) ).
% sum.remove
thf(fact_2377_sum_Oinsert__remove,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A5: set @ B,G3: B > A,X: B] :
( ( finite_finite2 @ B @ A5 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G3 @ ( insert @ B @ X @ A5 ) )
= ( plus_plus @ A @ ( G3 @ X ) @ ( groups7311177749621191930dd_sum @ B @ A @ G3 @ ( minus_minus @ ( set @ B ) @ A5 @ ( insert @ B @ X @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ).
% sum.insert_remove
thf(fact_2378_sum__diff1,axiom,
! [A: $tType,B: $tType] :
( ( ab_group_add @ A )
=> ! [A5: set @ B,A2: B,F3: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( ( member @ B @ A2 @ A5 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ F3 @ ( minus_minus @ ( set @ B ) @ A5 @ ( insert @ B @ A2 @ ( bot_bot @ ( set @ B ) ) ) ) )
= ( minus_minus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ A5 ) @ ( F3 @ A2 ) ) ) )
& ( ~ ( member @ B @ A2 @ A5 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ F3 @ ( minus_minus @ ( set @ B ) @ A5 @ ( insert @ B @ A2 @ ( bot_bot @ ( set @ B ) ) ) ) )
= ( groups7311177749621191930dd_sum @ B @ A @ F3 @ A5 ) ) ) ) ) ) ).
% sum_diff1
thf(fact_2379_sum__count__set,axiom,
! [A: $tType,Xs: list @ A,X7: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ X7 )
=> ( ( finite_finite2 @ A @ X7 )
=> ( ( groups7311177749621191930dd_sum @ A @ nat @ ( count_list @ A @ Xs ) @ X7 )
= ( size_size @ ( list @ A ) @ Xs ) ) ) ) ).
% sum_count_set
thf(fact_2380_set__encode__def,axiom,
( nat_set_encode
= ( groups7311177749621191930dd_sum @ nat @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% set_encode_def
thf(fact_2381_sum_Odelta__remove,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [S2: set @ B,A2: B,B2: B > A,C3: B > A] :
( ( finite_finite2 @ B @ S2 )
=> ( ( ( member @ B @ A2 @ S2 )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [K3: B] : ( if @ A @ ( K3 = A2 ) @ ( B2 @ K3 ) @ ( C3 @ K3 ) )
@ S2 )
= ( plus_plus @ A @ ( B2 @ A2 ) @ ( groups7311177749621191930dd_sum @ B @ A @ C3 @ ( minus_minus @ ( set @ B ) @ S2 @ ( insert @ B @ A2 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) )
& ( ~ ( member @ B @ A2 @ S2 )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [K3: B] : ( if @ A @ ( K3 = A2 ) @ ( B2 @ K3 ) @ ( C3 @ K3 ) )
@ S2 )
= ( groups7311177749621191930dd_sum @ B @ A @ C3 @ ( minus_minus @ ( set @ B ) @ S2 @ ( insert @ B @ A2 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ) ).
% sum.delta_remove
thf(fact_2382_ceiling__divide__upper,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Q2: A,P: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Q2 )
=> ( ord_less_eq @ A @ P @ ( times_times @ A @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ ( divide_divide @ A @ P @ Q2 ) ) ) @ Q2 ) ) ) ) ).
% ceiling_divide_upper
thf(fact_2383_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_2384_member__le__sum,axiom,
! [B: $tType,C: $tType] :
( ( ( ordere6911136660526730532id_add @ B )
& ( semiring_1 @ B ) )
=> ! [I2: C,A5: set @ C,F3: C > B] :
( ( member @ C @ I2 @ A5 )
=> ( ! [X4: C] :
( ( member @ C @ X4 @ ( minus_minus @ ( set @ C ) @ A5 @ ( insert @ C @ I2 @ ( bot_bot @ ( set @ C ) ) ) ) )
=> ( ord_less_eq @ B @ ( zero_zero @ B ) @ ( F3 @ X4 ) ) )
=> ( ( finite_finite2 @ C @ A5 )
=> ( ord_less_eq @ B @ ( F3 @ I2 ) @ ( groups7311177749621191930dd_sum @ C @ B @ F3 @ A5 ) ) ) ) ) ) ).
% member_le_sum
thf(fact_2385_sum__natinterval__diff,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [M2: nat,N: nat,F3: nat > A] :
( ( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( minus_minus @ A @ ( F3 @ K3 ) @ ( F3 @ ( plus_plus @ nat @ K3 @ ( one_one @ nat ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
= ( minus_minus @ A @ ( F3 @ M2 ) @ ( F3 @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) ) ) )
& ( ~ ( ord_less_eq @ nat @ M2 @ N )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( minus_minus @ A @ ( F3 @ K3 ) @ ( F3 @ ( plus_plus @ nat @ K3 @ ( one_one @ nat ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
= ( zero_zero @ A ) ) ) ) ) ).
% sum_natinterval_diff
thf(fact_2386_sum__telescope_H_H,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [M2: nat,N: nat,F3: nat > A] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( minus_minus @ A @ ( F3 @ K3 ) @ ( F3 @ ( minus_minus @ nat @ K3 @ ( one_one @ nat ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M2 ) @ N ) )
= ( minus_minus @ A @ ( F3 @ N ) @ ( F3 @ M2 ) ) ) ) ) ).
% sum_telescope''
thf(fact_2387_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_2388_set__decode__plus__power__2,axiom,
! [N: nat,Z: nat] :
( ~ ( member @ nat @ N @ ( nat_set_decode @ Z ) )
=> ( ( nat_set_decode @ ( plus_plus @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ Z ) )
= ( insert @ nat @ N @ ( nat_set_decode @ Z ) ) ) ) ).
% set_decode_plus_power_2
thf(fact_2389_ceiling__divide__lower,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Q2: A,P: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Q2 )
=> ( ord_less @ A @ ( times_times @ A @ ( minus_minus @ A @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ A @ ( divide_divide @ A @ P @ Q2 ) ) ) @ ( one_one @ A ) ) @ Q2 ) @ P ) ) ) ).
% ceiling_divide_lower
thf(fact_2390_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_2391_mask__eq__sum__exp,axiom,
! [A: $tType] :
( ( semiring_parity @ A )
=> ! [N: nat] :
( ( minus_minus @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ A ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) )
@ ( collect @ nat
@ ^ [Q5: nat] : ( ord_less @ nat @ Q5 @ N ) ) ) ) ) ).
% mask_eq_sum_exp
thf(fact_2392_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_2393_sum_Oin__pairs,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G3: nat > A,M2: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( 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
@ ^ [I4: nat] : ( plus_plus @ A @ ( G3 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I4 ) ) @ ( G3 @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I4 ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ).
% sum.in_pairs
thf(fact_2394_mask__eq__sum__exp__nat,axiom,
! [N: nat] :
( ( minus_minus @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ ( suc @ ( zero_zero @ nat ) ) )
= ( groups7311177749621191930dd_sum @ nat @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
@ ( collect @ nat
@ ^ [Q5: nat] : ( ord_less @ nat @ Q5 @ N ) ) ) ) ).
% mask_eq_sum_exp_nat
thf(fact_2395_gauss__sum__nat,axiom,
! [N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [X5: nat] : X5
@ ( 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_2396_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_2397_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_2398_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_2399_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_2400_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_2401_arith__series,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [A2: A,D3: A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( plus_plus @ A @ A2 @ ( times_times @ A @ ( semiring_1_of_nat @ A @ I4 ) @ D3 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
= ( divide_divide @ A @ ( times_times @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N ) @ ( one_one @ A ) ) @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ D3 ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% arith_series
thf(fact_2402_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_2403_int__eq__iff__numeral,axiom,
! [M2: nat,V: num] :
( ( ( semiring_1_of_nat @ int @ M2 )
= ( numeral_numeral @ int @ V ) )
= ( M2
= ( numeral_numeral @ nat @ V ) ) ) ).
% int_eq_iff_numeral
thf(fact_2404_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_2405_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_2406_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_2407_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_2408_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_2409_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_2410_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_2411_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_2412_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_2413_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_2414_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_2415_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_2416_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_2417_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_2418_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_2419_of__nat__of__bool,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [P2: $o] :
( ( semiring_1_of_nat @ A @ ( zero_neq_one_of_bool @ nat @ P2 ) )
= ( zero_neq_one_of_bool @ A @ P2 ) ) ) ).
% of_nat_of_bool
thf(fact_2420_of__nat__sum,axiom,
! [A: $tType,B: $tType] :
( ( semiring_1 @ A )
=> ! [F3: B > nat,A5: set @ B] :
( ( semiring_1_of_nat @ A @ ( groups7311177749621191930dd_sum @ B @ nat @ F3 @ A5 ) )
= ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X5: B] : ( semiring_1_of_nat @ A @ ( F3 @ X5 ) )
@ A5 ) ) ) ).
% of_nat_sum
thf(fact_2421_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_2422_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_2423_numeral__less__real__of__nat__iff,axiom,
! [W2: num,N: nat] :
( ( ord_less @ real @ ( numeral_numeral @ real @ W2 ) @ ( semiring_1_of_nat @ real @ N ) )
= ( ord_less @ nat @ ( numeral_numeral @ nat @ W2 ) @ N ) ) ).
% numeral_less_real_of_nat_iff
thf(fact_2424_real__of__nat__less__numeral__iff,axiom,
! [N: nat,W2: num] :
( ( ord_less @ real @ ( semiring_1_of_nat @ real @ N ) @ ( numeral_numeral @ real @ W2 ) )
= ( ord_less @ nat @ N @ ( numeral_numeral @ nat @ W2 ) ) ) ).
% real_of_nat_less_numeral_iff
thf(fact_2425_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_2426_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_2427_of__nat__power__less__of__nat__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X: nat,B2: nat,W2: nat] :
( ( ord_less @ A @ ( semiring_1_of_nat @ A @ X ) @ ( power_power @ A @ ( semiring_1_of_nat @ A @ B2 ) @ W2 ) )
= ( ord_less @ nat @ X @ ( power_power @ nat @ B2 @ W2 ) ) ) ) ).
% of_nat_power_less_of_nat_cancel_iff
thf(fact_2428_of__nat__less__of__nat__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [B2: nat,W2: nat,X: nat] :
( ( ord_less @ A @ ( power_power @ A @ ( semiring_1_of_nat @ A @ B2 ) @ W2 ) @ ( semiring_1_of_nat @ A @ X ) )
= ( ord_less @ nat @ ( power_power @ nat @ B2 @ W2 ) @ X ) ) ) ).
% of_nat_less_of_nat_power_cancel_iff
thf(fact_2429_of__nat__power__le__of__nat__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X: nat,B2: nat,W2: nat] :
( ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ X ) @ ( power_power @ A @ ( semiring_1_of_nat @ A @ B2 ) @ W2 ) )
= ( ord_less_eq @ nat @ X @ ( power_power @ nat @ B2 @ W2 ) ) ) ) ).
% of_nat_power_le_of_nat_cancel_iff
thf(fact_2430_of__nat__le__of__nat__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [B2: nat,W2: nat,X: nat] :
( ( ord_less_eq @ A @ ( power_power @ A @ ( semiring_1_of_nat @ A @ B2 ) @ W2 ) @ ( semiring_1_of_nat @ A @ X ) )
= ( ord_less_eq @ nat @ ( power_power @ nat @ B2 @ W2 ) @ X ) ) ) ).
% of_nat_le_of_nat_power_cancel_iff
thf(fact_2431_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_2432_of__nat__less__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X: nat,I2: num,N: nat] :
( ( ord_less @ A @ ( semiring_1_of_nat @ A @ X ) @ ( power_power @ A @ ( numeral_numeral @ A @ I2 ) @ N ) )
= ( ord_less @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ I2 ) @ N ) ) ) ) ).
% of_nat_less_numeral_power_cancel_iff
thf(fact_2433_numeral__power__less__of__nat__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [I2: num,N: nat,X: nat] :
( ( ord_less @ A @ ( power_power @ A @ ( numeral_numeral @ A @ I2 ) @ N ) @ ( semiring_1_of_nat @ A @ X ) )
= ( ord_less @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ I2 ) @ N ) @ X ) ) ) ).
% numeral_power_less_of_nat_cancel_iff
thf(fact_2434_of__nat__le__numeral__power__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [X: nat,I2: num,N: nat] :
( ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ X ) @ ( power_power @ A @ ( numeral_numeral @ A @ I2 ) @ N ) )
= ( ord_less_eq @ nat @ X @ ( power_power @ nat @ ( numeral_numeral @ nat @ I2 ) @ N ) ) ) ) ).
% of_nat_le_numeral_power_cancel_iff
thf(fact_2435_numeral__power__le__of__nat__cancel__iff,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [I2: num,N: nat,X: nat] :
( ( ord_less_eq @ A @ ( power_power @ A @ ( numeral_numeral @ A @ I2 ) @ N ) @ ( semiring_1_of_nat @ A @ X ) )
= ( ord_less_eq @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ I2 ) @ N ) @ X ) ) ) ).
% numeral_power_le_of_nat_cancel_iff
thf(fact_2436_real__arch__simple,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X: A] :
? [N3: nat] : ( ord_less_eq @ A @ X @ ( semiring_1_of_nat @ A @ N3 ) ) ) ).
% real_arch_simple
thf(fact_2437_reals__Archimedean2,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X: A] :
? [N3: nat] : ( ord_less @ A @ X @ ( semiring_1_of_nat @ A @ N3 ) ) ) ).
% reals_Archimedean2
thf(fact_2438_mult__of__nat__commute,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [X: nat,Y: A] :
( ( times_times @ A @ ( semiring_1_of_nat @ A @ X ) @ Y )
= ( times_times @ A @ Y @ ( semiring_1_of_nat @ A @ X ) ) ) ) ).
% mult_of_nat_commute
thf(fact_2439_int__cases2,axiom,
! [Z: int] :
( ! [N3: nat] :
( Z
!= ( semiring_1_of_nat @ int @ N3 ) )
=> ~ ! [N3: nat] :
( Z
!= ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N3 ) ) ) ) ).
% int_cases2
thf(fact_2440_int__diff__cases,axiom,
! [Z: int] :
~ ! [M: nat,N3: nat] :
( Z
!= ( minus_minus @ int @ ( semiring_1_of_nat @ int @ M ) @ ( semiring_1_of_nat @ int @ N3 ) ) ) ).
% int_diff_cases
thf(fact_2441_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_2442_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_2443_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_2444_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_2445_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_2446_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_2447_of__nat__mono,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [I2: nat,J: nat] :
( ( ord_less_eq @ nat @ I2 @ J )
=> ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ I2 ) @ ( semiring_1_of_nat @ A @ J ) ) ) ) ).
% of_nat_mono
thf(fact_2448_int__ops_I1_J,axiom,
( ( semiring_1_of_nat @ int @ ( zero_zero @ nat ) )
= ( zero_zero @ int ) ) ).
% int_ops(1)
thf(fact_2449_nat__int__comparison_I2_J,axiom,
( ( ord_less @ nat )
= ( ^ [A6: nat,B5: nat] : ( ord_less @ int @ ( semiring_1_of_nat @ int @ A6 ) @ ( semiring_1_of_nat @ int @ B5 ) ) ) ) ).
% nat_int_comparison(2)
thf(fact_2450_int__of__nat__induct,axiom,
! [P2: int > $o,Z: int] :
( ! [N3: nat] : ( P2 @ ( semiring_1_of_nat @ int @ N3 ) )
=> ( ! [N3: nat] : ( P2 @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ ( suc @ N3 ) ) ) )
=> ( P2 @ Z ) ) ) ).
% int_of_nat_induct
thf(fact_2451_int__cases,axiom,
! [Z: int] :
( ! [N3: nat] :
( Z
!= ( semiring_1_of_nat @ int @ N3 ) )
=> ~ ! [N3: nat] :
( Z
!= ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ ( suc @ N3 ) ) ) ) ) ).
% int_cases
thf(fact_2452_nat__int__comparison_I3_J,axiom,
( ( ord_less_eq @ nat )
= ( ^ [A6: nat,B5: nat] : ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ A6 ) @ ( semiring_1_of_nat @ int @ B5 ) ) ) ) ).
% nat_int_comparison(3)
thf(fact_2453_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_2454_zero__le__imp__eq__int,axiom,
! [K2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K2 )
=> ? [N3: nat] :
( K2
= ( semiring_1_of_nat @ int @ N3 ) ) ) ).
% zero_le_imp_eq_int
thf(fact_2455_nonneg__int__cases,axiom,
! [K2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K2 )
=> ~ ! [N3: nat] :
( K2
!= ( semiring_1_of_nat @ int @ N3 ) ) ) ).
% nonneg_int_cases
thf(fact_2456_int__ops_I5_J,axiom,
! [A2: nat,B2: nat] :
( ( semiring_1_of_nat @ int @ ( plus_plus @ nat @ A2 @ B2 ) )
= ( plus_plus @ int @ ( semiring_1_of_nat @ int @ A2 ) @ ( semiring_1_of_nat @ int @ B2 ) ) ) ).
% int_ops(5)
thf(fact_2457_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_2458_zadd__int__left,axiom,
! [M2: nat,N: nat,Z: int] :
( ( plus_plus @ int @ ( semiring_1_of_nat @ int @ M2 ) @ ( plus_plus @ int @ ( semiring_1_of_nat @ int @ N ) @ Z ) )
= ( plus_plus @ int @ ( semiring_1_of_nat @ int @ ( plus_plus @ nat @ M2 @ N ) ) @ Z ) ) ).
% zadd_int_left
thf(fact_2459_zle__iff__zadd,axiom,
( ( ord_less_eq @ int )
= ( ^ [W3: int,Z5: int] :
? [N2: nat] :
( Z5
= ( plus_plus @ int @ W3 @ ( semiring_1_of_nat @ int @ N2 ) ) ) ) ) ).
% zle_iff_zadd
thf(fact_2460_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_2461_of__nat__max,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [X: nat,Y: nat] :
( ( semiring_1_of_nat @ A @ ( ord_max @ nat @ X @ Y ) )
= ( ord_max @ A @ ( semiring_1_of_nat @ A @ X ) @ ( semiring_1_of_nat @ A @ Y ) ) ) ) ).
% of_nat_max
thf(fact_2462_nat__less__as__int,axiom,
( ( ord_less @ nat )
= ( ^ [A6: nat,B5: nat] : ( ord_less @ int @ ( semiring_1_of_nat @ int @ A6 ) @ ( semiring_1_of_nat @ int @ B5 ) ) ) ) ).
% nat_less_as_int
thf(fact_2463_nat__leq__as__int,axiom,
( ( ord_less_eq @ nat )
= ( ^ [A6: nat,B5: nat] : ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ A6 ) @ ( semiring_1_of_nat @ int @ B5 ) ) ) ) ).
% nat_leq_as_int
thf(fact_2464_ex__less__of__nat__mult,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X )
=> ? [N3: nat] : ( ord_less @ A @ Y @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N3 ) @ X ) ) ) ) ).
% ex_less_of_nat_mult
thf(fact_2465_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_2466_reals__Archimedean3,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ! [Y4: real] :
? [N3: nat] : ( ord_less @ real @ Y4 @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N3 ) @ X ) ) ) ).
% reals_Archimedean3
thf(fact_2467_int__cases4,axiom,
! [M2: int] :
( ! [N3: nat] :
( M2
!= ( semiring_1_of_nat @ int @ N3 ) )
=> ~ ! [N3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( M2
!= ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N3 ) ) ) ) ) ).
% int_cases4
thf(fact_2468_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_2469_zless__iff__Suc__zadd,axiom,
( ( ord_less @ int )
= ( ^ [W3: int,Z5: int] :
? [N2: nat] :
( Z5
= ( plus_plus @ int @ W3 @ ( semiring_1_of_nat @ int @ ( suc @ N2 ) ) ) ) ) ) ).
% zless_iff_Suc_zadd
thf(fact_2470_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_2471_nonpos__int__cases,axiom,
! [K2: int] :
( ( ord_less_eq @ int @ K2 @ ( zero_zero @ int ) )
=> ~ ! [N3: nat] :
( K2
!= ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N3 ) ) ) ) ).
% nonpos_int_cases
thf(fact_2472_int__sum,axiom,
! [B: $tType,F3: B > nat,A5: set @ B] :
( ( semiring_1_of_nat @ int @ ( groups7311177749621191930dd_sum @ B @ nat @ F3 @ A5 ) )
= ( groups7311177749621191930dd_sum @ B @ int
@ ^ [X5: B] : ( semiring_1_of_nat @ int @ ( F3 @ X5 ) )
@ A5 ) ) ).
% int_sum
thf(fact_2473_mod__mult2__eq_H,axiom,
! [A: $tType] :
( ( euclid5411537665997757685th_nat @ A )
=> ! [A2: A,M2: nat,N: nat] :
( ( modulo_modulo @ A @ A2 @ ( 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 @ A2 @ ( semiring_1_of_nat @ A @ M2 ) ) @ ( semiring_1_of_nat @ A @ N ) ) ) @ ( modulo_modulo @ A @ A2 @ ( semiring_1_of_nat @ A @ M2 ) ) ) ) ) ).
% mod_mult2_eq'
thf(fact_2474_zero__less__imp__eq__int,axiom,
! [K2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ K2 )
=> ? [N3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
& ( K2
= ( semiring_1_of_nat @ int @ N3 ) ) ) ) ).
% zero_less_imp_eq_int
thf(fact_2475_pos__int__cases,axiom,
! [K2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ K2 )
=> ~ ! [N3: nat] :
( ( K2
= ( semiring_1_of_nat @ int @ N3 ) )
=> ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 ) ) ) ).
% pos_int_cases
thf(fact_2476_int__cases3,axiom,
! [K2: int] :
( ( K2
!= ( zero_zero @ int ) )
=> ( ! [N3: nat] :
( ( K2
= ( semiring_1_of_nat @ int @ N3 ) )
=> ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 ) )
=> ~ ! [N3: nat] :
( ( K2
= ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N3 ) ) )
=> ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 ) ) ) ) ).
% int_cases3
thf(fact_2477_nat__less__real__le,axiom,
( ( ord_less @ nat )
= ( ^ [N2: nat,M3: nat] : ( ord_less_eq @ real @ ( plus_plus @ real @ ( semiring_1_of_nat @ real @ N2 ) @ ( one_one @ real ) ) @ ( semiring_1_of_nat @ real @ M3 ) ) ) ) ).
% nat_less_real_le
thf(fact_2478_nat__le__real__less,axiom,
( ( ord_less_eq @ nat )
= ( ^ [N2: nat,M3: nat] : ( ord_less @ real @ ( semiring_1_of_nat @ real @ N2 ) @ ( plus_plus @ real @ ( semiring_1_of_nat @ real @ M3 ) @ ( one_one @ real ) ) ) ) ) ).
% nat_le_real_less
thf(fact_2479_zmult__zless__mono2__lemma,axiom,
! [I2: int,J: int,K2: nat] :
( ( ord_less @ int @ I2 @ J )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K2 )
=> ( ord_less @ int @ ( times_times @ int @ ( semiring_1_of_nat @ int @ K2 ) @ I2 ) @ ( times_times @ int @ ( semiring_1_of_nat @ int @ K2 ) @ J ) ) ) ) ).
% zmult_zless_mono2_lemma
thf(fact_2480_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_2481_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_2482_negD,axiom,
! [X: int] :
( ( ord_less @ int @ X @ ( zero_zero @ int ) )
=> ? [N3: nat] :
( X
= ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ ( suc @ N3 ) ) ) ) ) ).
% negD
thf(fact_2483_int__ops_I6_J,axiom,
! [A2: nat,B2: nat] :
( ( ( ord_less @ int @ ( semiring_1_of_nat @ int @ A2 ) @ ( semiring_1_of_nat @ int @ B2 ) )
=> ( ( semiring_1_of_nat @ int @ ( minus_minus @ nat @ A2 @ B2 ) )
= ( zero_zero @ int ) ) )
& ( ~ ( ord_less @ int @ ( semiring_1_of_nat @ int @ A2 ) @ ( semiring_1_of_nat @ int @ B2 ) )
=> ( ( semiring_1_of_nat @ int @ ( minus_minus @ nat @ A2 @ B2 ) )
= ( minus_minus @ int @ ( semiring_1_of_nat @ int @ A2 ) @ ( semiring_1_of_nat @ int @ B2 ) ) ) ) ) ).
% int_ops(6)
thf(fact_2484_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_2485_simp__from__to,axiom,
( ( set_or1337092689740270186AtMost @ int )
= ( ^ [I4: int,J3: int] : ( if @ ( set @ int ) @ ( ord_less @ int @ J3 @ I4 ) @ ( bot_bot @ ( set @ int ) ) @ ( insert @ int @ I4 @ ( set_or1337092689740270186AtMost @ int @ ( plus_plus @ int @ I4 @ ( one_one @ int ) ) @ J3 ) ) ) ) ) ).
% simp_from_to
thf(fact_2486_nat__approx__posE,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [E3: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ E3 )
=> ~ ! [N3: nat] :
~ ( ord_less @ A @ ( divide_divide @ A @ ( one_one @ A ) @ ( semiring_1_of_nat @ A @ ( suc @ N3 ) ) ) @ E3 ) ) ) ).
% nat_approx_posE
thf(fact_2487_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_2488_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_2489_real__archimedian__rdiv__eq__0,axiom,
! [X: real,C3: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ C3 )
=> ( ! [M: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
=> ( ord_less_eq @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ M ) @ X ) @ C3 ) )
=> ( X
= ( zero_zero @ real ) ) ) ) ) ).
% real_archimedian_rdiv_eq_0
thf(fact_2490_neg__int__cases,axiom,
! [K2: int] :
( ( ord_less @ int @ K2 @ ( zero_zero @ int ) )
=> ~ ! [N3: nat] :
( ( K2
= ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N3 ) ) )
=> ~ ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 ) ) ) ).
% neg_int_cases
thf(fact_2491_zdiff__int__split,axiom,
! [P2: int > $o,X: nat,Y: nat] :
( ( P2 @ ( semiring_1_of_nat @ int @ ( minus_minus @ nat @ X @ Y ) ) )
= ( ( ( ord_less_eq @ nat @ Y @ X )
=> ( P2 @ ( minus_minus @ int @ ( semiring_1_of_nat @ int @ X ) @ ( semiring_1_of_nat @ int @ Y ) ) ) )
& ( ( ord_less @ nat @ X @ Y )
=> ( P2 @ ( zero_zero @ int ) ) ) ) ) ).
% zdiff_int_split
thf(fact_2492_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_2493_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_2494_double__arith__series,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A2: A,D3: A,N: nat] :
( ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( plus_plus @ A @ A2 @ ( times_times @ A @ ( semiring_1_of_nat @ A @ I4 ) @ D3 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) )
= ( times_times @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N ) @ ( one_one @ A ) ) @ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ D3 ) ) ) ) ) ).
% double_arith_series
thf(fact_2495_of__nat__code__if,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( ( semiring_1_of_nat @ A )
= ( ^ [N2: nat] :
( if @ A
@ ( N2
= ( zero_zero @ nat ) )
@ ( zero_zero @ A )
@ ( product_case_prod @ nat @ nat @ A
@ ^ [M3: nat,Q5: nat] :
( if @ A
@ ( Q5
= ( zero_zero @ nat ) )
@ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ A @ M3 ) )
@ ( plus_plus @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ A @ M3 ) ) @ ( one_one @ A ) ) )
@ ( divmod_nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% of_nat_code_if
thf(fact_2496_lemma__termdiff3,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [H2: A,Z: A,K5: real,N: nat] :
( ( H2
!= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ Z ) @ K5 )
=> ( ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ Z @ H2 ) ) @ K5 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( divide_divide @ A @ ( minus_minus @ A @ ( power_power @ A @ ( plus_plus @ A @ Z @ H2 ) @ N ) @ ( power_power @ A @ Z @ N ) ) @ H2 ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ ( power_power @ A @ Z @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) @ ( times_times @ real @ ( times_times @ real @ ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ ( semiring_1_of_nat @ real @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) ) ) @ ( power_power @ real @ K5 @ ( minus_minus @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( real_V7770717601297561774m_norm @ A @ H2 ) ) ) ) ) ) ) ).
% lemma_termdiff3
thf(fact_2497_ln__series,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ X @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) )
=> ( ( ln_ln @ real @ X )
= ( suminf @ real
@ ^ [N2: nat] : ( times_times @ real @ ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ N2 ) @ ( divide_divide @ real @ ( one_one @ real ) @ ( semiring_1_of_nat @ real @ ( plus_plus @ nat @ N2 @ ( one_one @ nat ) ) ) ) ) @ ( power_power @ real @ ( minus_minus @ real @ X @ ( one_one @ real ) ) @ ( suc @ N2 ) ) ) ) ) ) ) ).
% ln_series
thf(fact_2498_lemma__termdiff2,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [H2: A,Z: A,N: nat] :
( ( H2
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( divide_divide @ A @ ( minus_minus @ A @ ( power_power @ A @ ( plus_plus @ A @ Z @ H2 ) @ N ) @ ( power_power @ A @ Z @ N ) ) @ H2 ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ ( power_power @ A @ Z @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) ) ) )
= ( times_times @ A @ H2
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [P6: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [Q5: nat] : ( times_times @ A @ ( power_power @ A @ ( plus_plus @ A @ Z @ H2 ) @ Q5 ) @ ( power_power @ A @ Z @ ( minus_minus @ nat @ ( minus_minus @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ Q5 ) ) )
@ ( set_ord_lessThan @ nat @ ( minus_minus @ nat @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) @ P6 ) ) )
@ ( set_ord_lessThan @ nat @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ) ) ).
% lemma_termdiff2
thf(fact_2499_pochhammer__double,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [Z: A,N: nat] :
( ( comm_s3205402744901411588hammer @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ Z ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) )
= ( times_times @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) @ ( comm_s3205402744901411588hammer @ A @ Z @ N ) ) @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ Z @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ N ) ) ) ) ).
% pochhammer_double
thf(fact_2500_of__nat__code,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( ( semiring_1_of_nat @ A )
= ( ^ [N2: nat] :
( semiri8178284476397505188at_aux @ A
@ ^ [I4: A] : ( plus_plus @ A @ I4 @ ( one_one @ A ) )
@ N2
@ ( zero_zero @ A ) ) ) ) ) ).
% of_nat_code
thf(fact_2501_lessThan__eq__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( ( set_ord_lessThan @ A @ X )
= ( set_ord_lessThan @ A @ Y ) )
= ( X = Y ) ) ) ).
% lessThan_eq_iff
thf(fact_2502_lessThan__iff,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [I2: A,K2: A] :
( ( member @ A @ I2 @ ( set_ord_lessThan @ A @ K2 ) )
= ( ord_less @ A @ I2 @ K2 ) ) ) ).
% lessThan_iff
thf(fact_2503_finite__lessThan,axiom,
! [K2: nat] : ( finite_finite2 @ nat @ ( set_ord_lessThan @ nat @ K2 ) ) ).
% finite_lessThan
thf(fact_2504_lessThan__subset__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_ord_lessThan @ A @ X ) @ ( set_ord_lessThan @ A @ Y ) )
= ( ord_less_eq @ A @ X @ Y ) ) ) ).
% lessThan_subset_iff
thf(fact_2505_pochhammer__0,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A2: A] :
( ( comm_s3205402744901411588hammer @ A @ A2 @ ( zero_zero @ nat ) )
= ( one_one @ A ) ) ) ).
% pochhammer_0
thf(fact_2506_pochhammer__Suc0,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A2: A] :
( ( comm_s3205402744901411588hammer @ A @ A2 @ ( suc @ ( zero_zero @ nat ) ) )
= A2 ) ) ).
% pochhammer_Suc0
thf(fact_2507_lessThan__0,axiom,
( ( set_ord_lessThan @ nat @ ( zero_zero @ nat ) )
= ( bot_bot @ ( set @ nat ) ) ) ).
% lessThan_0
thf(fact_2508_single__Diff__lessThan,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [K2: A] :
( ( minus_minus @ ( set @ A ) @ ( insert @ A @ K2 @ ( bot_bot @ ( set @ A ) ) ) @ ( set_ord_lessThan @ A @ K2 ) )
= ( insert @ A @ K2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% single_Diff_lessThan
thf(fact_2509_sum_OlessThan__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G3: nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_ord_lessThan @ nat @ ( suc @ N ) ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_ord_lessThan @ nat @ N ) ) @ ( G3 @ N ) ) ) ) ).
% sum.lessThan_Suc
thf(fact_2510_powser__zero,axiom,
! [A: $tType] :
( ( real_V2822296259951069270ebra_1 @ A )
=> ! [F3: nat > A] :
( ( suminf @ A
@ ^ [N2: nat] : ( times_times @ A @ ( F3 @ N2 ) @ ( power_power @ A @ ( zero_zero @ A ) @ N2 ) ) )
= ( F3 @ ( zero_zero @ nat ) ) ) ) ).
% powser_zero
thf(fact_2511_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_2512_lessThan__non__empty,axiom,
! [A: $tType] :
( ( no_bot @ A )
=> ! [X: A] :
( ( set_ord_lessThan @ A @ X )
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% lessThan_non_empty
thf(fact_2513_infinite__Iio,axiom,
! [A: $tType] :
( ( ( linorder @ A )
& ( no_bot @ A ) )
=> ! [A2: A] :
~ ( finite_finite2 @ A @ ( set_ord_lessThan @ A @ A2 ) ) ) ).
% infinite_Iio
thf(fact_2514_lessThan__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( set_ord_lessThan @ A )
= ( ^ [U2: A] :
( collect @ A
@ ^ [X5: A] : ( ord_less @ A @ X5 @ U2 ) ) ) ) ) ).
% lessThan_def
thf(fact_2515_Iio__eq__empty__iff,axiom,
! [A: $tType] :
( ( ( linorder @ A )
& ( order_bot @ A ) )
=> ! [N: A] :
( ( ( set_ord_lessThan @ A @ N )
= ( bot_bot @ ( set @ A ) ) )
= ( N
= ( bot_bot @ A ) ) ) ) ).
% Iio_eq_empty_iff
thf(fact_2516_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_2517_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_2518_pochhammer__eq__0__mono,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A2: A,N: nat,M2: nat] :
( ( ( comm_s3205402744901411588hammer @ A @ A2 @ N )
= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ nat @ N @ M2 )
=> ( ( comm_s3205402744901411588hammer @ A @ A2 @ M2 )
= ( zero_zero @ A ) ) ) ) ) ).
% pochhammer_eq_0_mono
thf(fact_2519_pochhammer__neq__0__mono,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A2: A,M2: nat,N: nat] :
( ( ( comm_s3205402744901411588hammer @ A @ A2 @ M2 )
!= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ nat @ N @ M2 )
=> ( ( comm_s3205402744901411588hammer @ A @ A2 @ N )
!= ( zero_zero @ A ) ) ) ) ) ).
% pochhammer_neq_0_mono
thf(fact_2520_lessThan__Suc,axiom,
! [K2: nat] :
( ( set_ord_lessThan @ nat @ ( suc @ K2 ) )
= ( insert @ nat @ K2 @ ( set_ord_lessThan @ nat @ K2 ) ) ) ).
% lessThan_Suc
thf(fact_2521_lessThan__empty__iff,axiom,
! [N: nat] :
( ( ( set_ord_lessThan @ nat @ N )
= ( bot_bot @ ( set @ nat ) ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% lessThan_empty_iff
thf(fact_2522_finite__nat__bounded,axiom,
! [S2: set @ nat] :
( ( finite_finite2 @ nat @ S2 )
=> ? [K: nat] : ( ord_less_eq @ ( set @ nat ) @ S2 @ ( set_ord_lessThan @ nat @ K ) ) ) ).
% finite_nat_bounded
thf(fact_2523_finite__nat__iff__bounded,axiom,
( ( finite_finite2 @ nat )
= ( ^ [S7: set @ nat] :
? [K3: nat] : ( ord_less_eq @ ( set @ nat ) @ S7 @ ( set_ord_lessThan @ nat @ K3 ) ) ) ) ).
% finite_nat_iff_bounded
thf(fact_2524_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_2525_of__nat__aux_Osimps_I2_J,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [Inc: A > A,N: nat,I2: A] :
( ( semiri8178284476397505188at_aux @ A @ Inc @ ( suc @ N ) @ I2 )
= ( semiri8178284476397505188at_aux @ A @ Inc @ N @ ( Inc @ I2 ) ) ) ) ).
% of_nat_aux.simps(2)
thf(fact_2526_of__nat__aux_Osimps_I1_J,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [Inc: A > A,I2: A] :
( ( semiri8178284476397505188at_aux @ A @ Inc @ ( zero_zero @ nat ) @ I2 )
= I2 ) ) ).
% of_nat_aux.simps(1)
thf(fact_2527_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_2528_lessThan__nat__numeral,axiom,
! [K2: num] :
( ( set_ord_lessThan @ nat @ ( numeral_numeral @ nat @ K2 ) )
= ( insert @ nat @ ( pred_numeral @ K2 ) @ ( set_ord_lessThan @ nat @ ( pred_numeral @ K2 ) ) ) ) ).
% lessThan_nat_numeral
thf(fact_2529_sum_Onat__diff__reindex,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G3: nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( G3 @ ( minus_minus @ nat @ N @ ( suc @ I4 ) ) )
@ ( set_ord_lessThan @ nat @ N ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% sum.nat_diff_reindex
thf(fact_2530_sum__diff__distrib,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [Q: A > nat,P2: A > nat,N: A] :
( ! [X4: A] : ( ord_less_eq @ nat @ ( Q @ X4 ) @ ( P2 @ X4 ) )
=> ( ( minus_minus @ nat @ ( groups7311177749621191930dd_sum @ A @ nat @ P2 @ ( set_ord_lessThan @ A @ N ) ) @ ( groups7311177749621191930dd_sum @ A @ nat @ Q @ ( set_ord_lessThan @ A @ N ) ) )
= ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [X5: A] : ( minus_minus @ nat @ ( P2 @ X5 ) @ ( Q @ X5 ) )
@ ( set_ord_lessThan @ A @ N ) ) ) ) ) ).
% sum_diff_distrib
thf(fact_2531_lemma__NBseq__def,axiom,
! [A: $tType,B: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [X7: A > B] :
( ( ? [K6: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K6 )
& ! [N2: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( X7 @ N2 ) ) @ K6 ) ) )
= ( ? [N4: nat] :
! [N2: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( X7 @ N2 ) ) @ ( semiring_1_of_nat @ real @ ( suc @ N4 ) ) ) ) ) ) ).
% lemma_NBseq_def
thf(fact_2532_lemma__NBseq__def2,axiom,
! [A: $tType,B: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [X7: A > B] :
( ( ? [K6: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K6 )
& ! [N2: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( X7 @ N2 ) ) @ K6 ) ) )
= ( ? [N4: nat] :
! [N2: A] : ( ord_less @ real @ ( real_V7770717601297561774m_norm @ B @ ( X7 @ N2 ) ) @ ( semiring_1_of_nat @ real @ ( suc @ N4 ) ) ) ) ) ) ).
% lemma_NBseq_def2
thf(fact_2533_pochhammer__rec,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A2: A,N: nat] :
( ( comm_s3205402744901411588hammer @ A @ A2 @ ( suc @ N ) )
= ( times_times @ A @ A2 @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ A2 @ ( one_one @ A ) ) @ N ) ) ) ) ).
% pochhammer_rec
thf(fact_2534_pochhammer__Suc,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A2: A,N: nat] :
( ( comm_s3205402744901411588hammer @ A @ A2 @ ( suc @ N ) )
= ( times_times @ A @ ( comm_s3205402744901411588hammer @ A @ A2 @ N ) @ ( plus_plus @ A @ A2 @ ( semiring_1_of_nat @ A @ N ) ) ) ) ) ).
% pochhammer_Suc
thf(fact_2535_pochhammer__rec_H,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [Z: A,N: nat] :
( ( comm_s3205402744901411588hammer @ A @ Z @ ( suc @ N ) )
= ( times_times @ A @ ( plus_plus @ A @ Z @ ( semiring_1_of_nat @ A @ N ) ) @ ( comm_s3205402744901411588hammer @ A @ Z @ N ) ) ) ) ).
% pochhammer_rec'
thf(fact_2536_pochhammer__of__nat__eq__0__lemma,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [N: nat,K2: nat] :
( ( ord_less @ nat @ N @ K2 )
=> ( ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ ( semiring_1_of_nat @ A @ N ) ) @ K2 )
= ( zero_zero @ A ) ) ) ) ).
% pochhammer_of_nat_eq_0_lemma
thf(fact_2537_pochhammer__of__nat__eq__0__iff,axiom,
! [A: $tType] :
( ( ( ring_char_0 @ A )
& ( idom @ A ) )
=> ! [N: nat,K2: nat] :
( ( ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ ( semiring_1_of_nat @ A @ N ) ) @ K2 )
= ( zero_zero @ A ) )
= ( ord_less @ nat @ N @ K2 ) ) ) ).
% pochhammer_of_nat_eq_0_iff
thf(fact_2538_pochhammer__eq__0__iff,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A2: A,N: nat] :
( ( ( comm_s3205402744901411588hammer @ A @ A2 @ N )
= ( zero_zero @ A ) )
= ( ? [K3: nat] :
( ( ord_less @ nat @ K3 @ N )
& ( A2
= ( uminus_uminus @ A @ ( semiring_1_of_nat @ A @ K3 ) ) ) ) ) ) ) ).
% pochhammer_eq_0_iff
thf(fact_2539_pochhammer__of__nat__eq__0__lemma_H,axiom,
! [A: $tType] :
( ( ( ring_char_0 @ A )
& ( idom @ A ) )
=> ! [K2: nat,N: nat] :
( ( ord_less_eq @ nat @ K2 @ N )
=> ( ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ ( semiring_1_of_nat @ A @ N ) ) @ K2 )
!= ( zero_zero @ A ) ) ) ) ).
% pochhammer_of_nat_eq_0_lemma'
thf(fact_2540_pochhammer__product_H,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [Z: A,N: nat,M2: nat] :
( ( comm_s3205402744901411588hammer @ A @ Z @ ( plus_plus @ nat @ N @ M2 ) )
= ( times_times @ A @ ( comm_s3205402744901411588hammer @ A @ Z @ N ) @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ Z @ ( semiring_1_of_nat @ A @ N ) ) @ M2 ) ) ) ) ).
% pochhammer_product'
thf(fact_2541_sum_OlessThan__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G3: nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_ord_lessThan @ nat @ ( suc @ N ) ) )
= ( plus_plus @ A @ ( G3 @ ( zero_zero @ nat ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( G3 @ ( suc @ I4 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% sum.lessThan_Suc_shift
thf(fact_2542_sum__lessThan__telescope,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [F3: nat > A,M2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [N2: nat] : ( minus_minus @ A @ ( F3 @ ( suc @ N2 ) ) @ ( F3 @ N2 ) )
@ ( set_ord_lessThan @ nat @ M2 ) )
= ( minus_minus @ A @ ( F3 @ M2 ) @ ( F3 @ ( zero_zero @ nat ) ) ) ) ) ).
% sum_lessThan_telescope
thf(fact_2543_sum__lessThan__telescope_H,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [F3: nat > A,M2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [N2: nat] : ( minus_minus @ A @ ( F3 @ N2 ) @ ( F3 @ ( suc @ N2 ) ) )
@ ( set_ord_lessThan @ nat @ M2 ) )
= ( minus_minus @ A @ ( F3 @ ( zero_zero @ nat ) ) @ ( F3 @ M2 ) ) ) ) ).
% sum_lessThan_telescope'
thf(fact_2544_sum_OatLeast1__atMost__eq,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G3: nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( G3 @ ( suc @ K3 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% sum.atLeast1_atMost_eq
thf(fact_2545_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_2546_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_2547_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_2548_pochhammer__product,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [M2: nat,N: nat,Z: A] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( comm_s3205402744901411588hammer @ A @ Z @ N )
= ( times_times @ A @ ( comm_s3205402744901411588hammer @ A @ Z @ M2 ) @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ Z @ ( semiring_1_of_nat @ A @ M2 ) ) @ ( minus_minus @ nat @ N @ M2 ) ) ) ) ) ) ).
% pochhammer_product
thf(fact_2549_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_2550_lemma__termdiff1,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [Z: A,H2: A,M2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [P6: nat] : ( minus_minus @ A @ ( times_times @ A @ ( power_power @ A @ ( plus_plus @ A @ Z @ H2 ) @ ( minus_minus @ nat @ M2 @ P6 ) ) @ ( power_power @ A @ Z @ P6 ) ) @ ( power_power @ A @ Z @ M2 ) )
@ ( set_ord_lessThan @ nat @ M2 ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [P6: nat] : ( times_times @ A @ ( power_power @ A @ Z @ P6 ) @ ( minus_minus @ A @ ( power_power @ A @ ( plus_plus @ A @ Z @ H2 ) @ ( minus_minus @ nat @ M2 @ P6 ) ) @ ( power_power @ A @ Z @ ( minus_minus @ nat @ M2 @ P6 ) ) ) )
@ ( set_ord_lessThan @ nat @ M2 ) ) ) ) ).
% lemma_termdiff1
thf(fact_2551_power__diff__sumr2,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [X: A,N: nat,Y: A] :
( ( minus_minus @ A @ ( power_power @ A @ X @ N ) @ ( power_power @ A @ Y @ N ) )
= ( times_times @ A @ ( minus_minus @ A @ X @ Y )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( power_power @ A @ Y @ ( minus_minus @ nat @ N @ ( suc @ I4 ) ) ) @ ( power_power @ A @ X @ I4 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% power_diff_sumr2
thf(fact_2552_diff__power__eq__sum,axiom,
! [A: $tType] :
( ( ( monoid_mult @ A )
& ( comm_ring @ A ) )
=> ! [X: A,N: nat,Y: A] :
( ( minus_minus @ A @ ( power_power @ A @ X @ ( suc @ N ) ) @ ( power_power @ A @ Y @ ( suc @ N ) ) )
= ( times_times @ A @ ( minus_minus @ A @ X @ Y )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [P6: nat] : ( times_times @ A @ ( power_power @ A @ X @ P6 ) @ ( power_power @ A @ Y @ ( minus_minus @ nat @ N @ P6 ) ) )
@ ( set_ord_lessThan @ nat @ ( suc @ N ) ) ) ) ) ) ).
% diff_power_eq_sum
thf(fact_2553_pochhammer__absorb__comp,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [R2: A,K2: nat] :
( ( times_times @ A @ ( minus_minus @ A @ R2 @ ( semiring_1_of_nat @ A @ K2 ) ) @ ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ R2 ) @ K2 ) )
= ( times_times @ A @ R2 @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ ( uminus_uminus @ A @ R2 ) @ ( one_one @ A ) ) @ K2 ) ) ) ) ).
% pochhammer_absorb_comp
thf(fact_2554_real__sum__nat__ivl__bounded2,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N: nat,F3: nat > A,K5: A,K2: nat] :
( ! [P7: nat] :
( ( ord_less @ nat @ P7 @ N )
=> ( ord_less_eq @ A @ ( F3 @ P7 ) @ K5 ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ K5 )
=> ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_ord_lessThan @ nat @ ( minus_minus @ nat @ N @ K2 ) ) ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ K5 ) ) ) ) ) ).
% real_sum_nat_ivl_bounded2
thf(fact_2555_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
@ ^ [I4: nat] : ( power_power @ A @ X @ ( minus_minus @ nat @ N @ ( suc @ I4 ) ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% one_diff_power_eq'
thf(fact_2556_pochhammer__minus_H,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [B2: A,K2: nat] :
( ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ ( minus_minus @ A @ B2 @ ( semiring_1_of_nat @ A @ K2 ) ) @ ( one_one @ A ) ) @ K2 )
= ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ K2 ) @ ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ B2 ) @ K2 ) ) ) ) ).
% pochhammer_minus'
thf(fact_2557_pochhammer__minus,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [B2: A,K2: nat] :
( ( comm_s3205402744901411588hammer @ A @ ( uminus_uminus @ A @ B2 ) @ K2 )
= ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ K2 ) @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ ( minus_minus @ A @ B2 @ ( semiring_1_of_nat @ A @ K2 ) ) @ ( one_one @ A ) ) @ K2 ) ) ) ) ).
% pochhammer_minus
thf(fact_2558_sum__split__even__odd,axiom,
! [F3: nat > real,G3: nat > real,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I4: nat] : ( if @ real @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I4 ) @ ( F3 @ I4 ) @ ( G3 @ I4 ) )
@ ( set_ord_lessThan @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I4: nat] : ( F3 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I4 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I4: nat] : ( G3 @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I4 ) @ ( one_one @ nat ) ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% sum_split_even_odd
thf(fact_2559_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_2560_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_2561_suminf__geometric,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [C3: A] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ C3 ) @ ( one_one @ real ) )
=> ( ( suminf @ A @ ( power_power @ A @ C3 ) )
= ( divide_divide @ A @ ( one_one @ A ) @ ( minus_minus @ A @ ( one_one @ A ) @ C3 ) ) ) ) ) ).
% suminf_geometric
thf(fact_2562_norm__zero,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ( ( real_V7770717601297561774m_norm @ A @ ( zero_zero @ A ) )
= ( zero_zero @ real ) ) ) ).
% norm_zero
thf(fact_2563_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_2564_suminf__zero,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topological_t2_space @ A ) )
=> ( ( suminf @ A
@ ^ [N2: nat] : ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% suminf_zero
thf(fact_2565_norm__not__less__zero,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X: A] :
~ ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ ( zero_zero @ real ) ) ) ).
% norm_not_less_zero
thf(fact_2566_norm__uminus__minus,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X: A,Y: A] :
( ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( uminus_uminus @ A @ X ) @ Y ) )
= ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ X @ Y ) ) ) ) ).
% norm_uminus_minus
thf(fact_2567_nonzero__norm__divide,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [B2: A,A2: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( real_V7770717601297561774m_norm @ A @ ( divide_divide @ A @ A2 @ B2 ) )
= ( divide_divide @ real @ ( real_V7770717601297561774m_norm @ A @ A2 ) @ ( real_V7770717601297561774m_norm @ A @ B2 ) ) ) ) ) ).
% nonzero_norm_divide
thf(fact_2568_power__eq__imp__eq__norm,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [W2: A,N: nat,Z: A] :
( ( ( power_power @ A @ W2 @ N )
= ( power_power @ A @ Z @ N ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( real_V7770717601297561774m_norm @ A @ W2 )
= ( real_V7770717601297561774m_norm @ A @ Z ) ) ) ) ) ).
% power_eq_imp_eq_norm
thf(fact_2569_norm__mult__less,axiom,
! [A: $tType] :
( ( real_V4412858255891104859lgebra @ A )
=> ! [X: A,R2: real,Y: A,S: real] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ R2 )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ Y ) @ S )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( times_times @ A @ X @ Y ) ) @ ( times_times @ real @ R2 @ S ) ) ) ) ) ).
% norm_mult_less
thf(fact_2570_norm__triangle__lt,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X: A,Y: A,E3: real] :
( ( ord_less @ real @ ( plus_plus @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ ( real_V7770717601297561774m_norm @ A @ Y ) ) @ E3 )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ X @ Y ) ) @ E3 ) ) ) ).
% norm_triangle_lt
thf(fact_2571_norm__add__less,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X: A,R2: real,Y: A,S: real] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ R2 )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ Y ) @ S )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ X @ Y ) ) @ ( plus_plus @ real @ R2 @ S ) ) ) ) ) ).
% norm_add_less
thf(fact_2572_norm__add__leD,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [A2: A,B2: A,C3: real] :
( ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ A2 @ B2 ) ) @ C3 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ B2 ) @ ( plus_plus @ real @ ( real_V7770717601297561774m_norm @ A @ A2 ) @ C3 ) ) ) ) ).
% norm_add_leD
thf(fact_2573_norm__triangle__le,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X: A,Y: A,E3: real] :
( ( ord_less_eq @ real @ ( plus_plus @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ ( real_V7770717601297561774m_norm @ A @ Y ) ) @ E3 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ X @ Y ) ) @ E3 ) ) ) ).
% norm_triangle_le
thf(fact_2574_norm__triangle__ineq,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X: A,Y: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ X @ Y ) ) @ ( plus_plus @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ ( real_V7770717601297561774m_norm @ A @ Y ) ) ) ) ).
% norm_triangle_ineq
thf(fact_2575_norm__triangle__mono,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [A2: A,R2: real,B2: A,S: real] :
( ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ A2 ) @ R2 )
=> ( ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ B2 ) @ S )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ A2 @ B2 ) ) @ ( plus_plus @ real @ R2 @ S ) ) ) ) ) ).
% norm_triangle_mono
thf(fact_2576_norm__diff__triangle__less,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X: A,Y: A,E1: real,Z: A,E22: real] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ X @ Y ) ) @ E1 )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ Y @ Z ) ) @ E22 )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ X @ Z ) ) @ ( plus_plus @ real @ E1 @ E22 ) ) ) ) ) ).
% norm_diff_triangle_less
thf(fact_2577_norm__diff__ineq,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [A2: A,B2: A] : ( ord_less_eq @ real @ ( minus_minus @ real @ ( real_V7770717601297561774m_norm @ A @ A2 ) @ ( real_V7770717601297561774m_norm @ A @ B2 ) ) @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ).
% norm_diff_ineq
thf(fact_2578_suminf__finite,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topological_t2_space @ A ) )
=> ! [N6: set @ nat,F3: nat > A] :
( ( finite_finite2 @ nat @ N6 )
=> ( ! [N3: nat] :
( ~ ( member @ nat @ N3 @ N6 )
=> ( ( F3 @ N3 )
= ( zero_zero @ A ) ) )
=> ( ( suminf @ A @ F3 )
= ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ N6 ) ) ) ) ) ).
% suminf_finite
thf(fact_2579_power__eq__1__iff,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [W2: A,N: nat] :
( ( ( power_power @ A @ W2 @ N )
= ( one_one @ A ) )
=> ( ( ( real_V7770717601297561774m_norm @ A @ W2 )
= ( one_one @ real ) )
| ( N
= ( zero_zero @ nat ) ) ) ) ) ).
% power_eq_1_iff
thf(fact_2580_norm__diff__triangle__ineq,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [A2: A,B2: A,C3: A,D3: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ ( plus_plus @ A @ C3 @ D3 ) ) ) @ ( plus_plus @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ A2 @ C3 ) ) @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ B2 @ D3 ) ) ) ) ) ).
% norm_diff_triangle_ineq
thf(fact_2581_pochhammer__times__pochhammer__half,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [Z: A,N: nat] :
( ( times_times @ A @ ( comm_s3205402744901411588hammer @ A @ Z @ ( suc @ N ) ) @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ Z @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) @ ( suc @ N ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [K3: nat] : ( plus_plus @ A @ Z @ ( divide_divide @ A @ ( semiring_1_of_nat @ A @ K3 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ nat ) ) ) ) ) ) ).
% pochhammer_times_pochhammer_half
thf(fact_2582_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_2583_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_2584_and__int_Oelims,axiom,
! [X: int,Xa2: int,Y: int] :
( ( ( bit_se5824344872417868541ns_and @ int @ X @ Xa2 )
= Y )
=> ( ( ( ( member @ int @ X @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
& ( member @ int @ Xa2 @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) )
=> ( Y
= ( uminus_uminus @ int
@ ( zero_neq_one_of_bool @ int
@ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ X )
& ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Xa2 ) ) ) ) ) )
& ( ~ ( ( member @ int @ X @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
& ( member @ int @ Xa2 @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) )
=> ( Y
= ( plus_plus @ int
@ ( zero_neq_one_of_bool @ int
@ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ X )
& ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Xa2 ) ) )
@ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( divide_divide @ int @ X @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ Xa2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% and_int.elims
thf(fact_2585_pochhammer__code,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ( ( comm_s3205402744901411588hammer @ A )
= ( ^ [A6: A,N2: nat] :
( if @ A
@ ( N2
= ( zero_zero @ nat ) )
@ ( one_one @ A )
@ ( set_fo6178422350223883121st_nat @ A
@ ^ [O: nat] : ( times_times @ A @ ( plus_plus @ A @ A6 @ ( semiring_1_of_nat @ A @ O ) ) )
@ ( zero_zero @ nat )
@ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) )
@ ( one_one @ A ) ) ) ) ) ) ).
% pochhammer_code
thf(fact_2586_ceiling__log__nat__eq__powr__iff,axiom,
! [B2: nat,K2: nat,N: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ B2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K2 )
=> ( ( ( archimedean_ceiling @ real @ ( log @ ( semiring_1_of_nat @ real @ B2 ) @ ( semiring_1_of_nat @ real @ K2 ) ) )
= ( plus_plus @ int @ ( semiring_1_of_nat @ int @ N ) @ ( one_one @ int ) ) )
= ( ( ord_less @ nat @ ( power_power @ nat @ B2 @ N ) @ K2 )
& ( ord_less_eq @ nat @ K2 @ ( power_power @ nat @ B2 @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% ceiling_log_nat_eq_powr_iff
thf(fact_2587_and__zero__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A] :
( ( bit_se5824344872417868541ns_and @ A @ A2 @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% and_zero_eq
thf(fact_2588_zero__and__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A] :
( ( bit_se5824344872417868541ns_and @ A @ ( zero_zero @ A ) @ A2 )
= ( zero_zero @ A ) ) ) ).
% zero_and_eq
thf(fact_2589_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_2590_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_2591_of__nat__prod,axiom,
! [A: $tType,B: $tType] :
( ( comm_semiring_1 @ A )
=> ! [F3: B > nat,A5: set @ B] :
( ( semiring_1_of_nat @ A @ ( groups7121269368397514597t_prod @ B @ nat @ F3 @ A5 ) )
= ( groups7121269368397514597t_prod @ B @ A
@ ^ [X5: B] : ( semiring_1_of_nat @ A @ ( F3 @ X5 ) )
@ A5 ) ) ) ).
% of_nat_prod
thf(fact_2592_of__int__prod,axiom,
! [A: $tType,B: $tType] :
( ( comm_ring_1 @ A )
=> ! [F3: B > int,A5: set @ B] :
( ( ring_1_of_int @ A @ ( groups7121269368397514597t_prod @ B @ int @ F3 @ A5 ) )
= ( groups7121269368397514597t_prod @ B @ A
@ ^ [X5: B] : ( ring_1_of_int @ A @ ( F3 @ X5 ) )
@ A5 ) ) ) ).
% of_int_prod
thf(fact_2593_prod__zero__iff,axiom,
! [A: $tType,B: $tType] :
( ( semidom @ A )
=> ! [A5: set @ B,F3: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( ( groups7121269368397514597t_prod @ B @ A @ F3 @ A5 )
= ( zero_zero @ A ) )
= ( ? [X5: B] :
( ( member @ B @ X5 @ A5 )
& ( ( F3 @ X5 )
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% prod_zero_iff
thf(fact_2594_prod_Oempty,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G3: B > A] :
( ( groups7121269368397514597t_prod @ B @ A @ G3 @ ( bot_bot @ ( set @ B ) ) )
= ( one_one @ A ) ) ) ).
% prod.empty
thf(fact_2595_prod_Oinfinite,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A5: set @ B,G3: B > A] :
( ~ ( finite_finite2 @ B @ A5 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G3 @ A5 )
= ( one_one @ A ) ) ) ) ).
% prod.infinite
thf(fact_2596_dvd__prod__eqI,axiom,
! [A: $tType,B: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A5: set @ B,A2: B,B2: A,F3: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( member @ B @ A2 @ A5 )
=> ( ( B2
= ( F3 @ A2 ) )
=> ( dvd_dvd @ A @ B2 @ ( groups7121269368397514597t_prod @ B @ A @ F3 @ A5 ) ) ) ) ) ) ).
% dvd_prod_eqI
thf(fact_2597_dvd__prodI,axiom,
! [A: $tType,B: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A5: set @ B,A2: B,F3: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( member @ B @ A2 @ A5 )
=> ( dvd_dvd @ A @ ( F3 @ A2 ) @ ( groups7121269368397514597t_prod @ B @ A @ F3 @ A5 ) ) ) ) ) ).
% dvd_prodI
thf(fact_2598_and__negative__int__iff,axiom,
! [K2: int,L: int] :
( ( ord_less @ int @ ( bit_se5824344872417868541ns_and @ int @ K2 @ L ) @ ( zero_zero @ int ) )
= ( ( ord_less @ int @ K2 @ ( zero_zero @ int ) )
& ( ord_less @ int @ L @ ( zero_zero @ int ) ) ) ) ).
% and_negative_int_iff
thf(fact_2599_prod_Odelta_H,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S2: set @ B,A2: B,B2: B > A] :
( ( finite_finite2 @ B @ S2 )
=> ( ( ( member @ B @ A2 @ S2 )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K3: B] : ( if @ A @ ( A2 = K3 ) @ ( B2 @ K3 ) @ ( one_one @ A ) )
@ S2 )
= ( B2 @ A2 ) ) )
& ( ~ ( member @ B @ A2 @ S2 )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K3: B] : ( if @ A @ ( A2 = K3 ) @ ( B2 @ K3 ) @ ( one_one @ A ) )
@ S2 )
= ( one_one @ A ) ) ) ) ) ) ).
% prod.delta'
thf(fact_2600_prod_Odelta,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S2: set @ B,A2: B,B2: B > A] :
( ( finite_finite2 @ B @ S2 )
=> ( ( ( member @ B @ A2 @ S2 )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K3: B] : ( if @ A @ ( K3 = A2 ) @ ( B2 @ K3 ) @ ( one_one @ A ) )
@ S2 )
= ( B2 @ A2 ) ) )
& ( ~ ( member @ B @ A2 @ S2 )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K3: B] : ( if @ A @ ( K3 = A2 ) @ ( B2 @ K3 ) @ ( one_one @ A ) )
@ S2 )
= ( one_one @ A ) ) ) ) ) ) ).
% prod.delta
thf(fact_2601_prod_Oinsert,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A5: set @ B,X: B,G3: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ~ ( member @ B @ X @ A5 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G3 @ ( insert @ B @ X @ A5 ) )
= ( times_times @ A @ ( G3 @ X ) @ ( groups7121269368397514597t_prod @ B @ A @ G3 @ A5 ) ) ) ) ) ) ).
% prod.insert
thf(fact_2602_prod_OlessThan__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G3: nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_ord_lessThan @ nat @ ( suc @ N ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_ord_lessThan @ nat @ N ) ) @ ( G3 @ N ) ) ) ) ).
% prod.lessThan_Suc
thf(fact_2603_zero__less__log__cancel__iff,axiom,
! [A2: real,X: real] :
( ( ord_less @ real @ ( one_one @ real ) @ A2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( log @ A2 @ X ) )
= ( ord_less @ real @ ( one_one @ real ) @ X ) ) ) ) ).
% zero_less_log_cancel_iff
thf(fact_2604_log__less__zero__cancel__iff,axiom,
! [A2: real,X: real] :
( ( ord_less @ real @ ( one_one @ real ) @ A2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( log @ A2 @ X ) @ ( zero_zero @ real ) )
= ( ord_less @ real @ X @ ( one_one @ real ) ) ) ) ) ).
% log_less_zero_cancel_iff
thf(fact_2605_one__less__log__cancel__iff,axiom,
! [A2: real,X: real] :
( ( ord_less @ real @ ( one_one @ real ) @ A2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( one_one @ real ) @ ( log @ A2 @ X ) )
= ( ord_less @ real @ A2 @ X ) ) ) ) ).
% one_less_log_cancel_iff
thf(fact_2606_log__less__one__cancel__iff,axiom,
! [A2: real,X: real] :
( ( ord_less @ real @ ( one_one @ real ) @ A2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( log @ A2 @ X ) @ ( one_one @ real ) )
= ( ord_less @ real @ X @ A2 ) ) ) ) ).
% log_less_one_cancel_iff
thf(fact_2607_log__less__cancel__iff,axiom,
! [A2: real,X: real,Y: real] :
( ( ord_less @ real @ ( one_one @ real ) @ A2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ Y )
=> ( ( ord_less @ real @ ( log @ A2 @ X ) @ ( log @ A2 @ Y ) )
= ( ord_less @ real @ X @ Y ) ) ) ) ) ).
% log_less_cancel_iff
thf(fact_2608_log__eq__one,axiom,
! [A2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
=> ( ( A2
!= ( one_one @ real ) )
=> ( ( log @ A2 @ A2 )
= ( one_one @ real ) ) ) ) ).
% log_eq_one
thf(fact_2609_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_2610_and__numerals_I1_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [Y: num] :
( ( bit_se5824344872417868541ns_and @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ Y ) ) )
= ( zero_zero @ A ) ) ) ).
% and_numerals(1)
thf(fact_2611_log__le__cancel__iff,axiom,
! [A2: real,X: real,Y: real] :
( ( ord_less @ real @ ( one_one @ real ) @ A2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ Y )
=> ( ( ord_less_eq @ real @ ( log @ A2 @ X ) @ ( log @ A2 @ Y ) )
= ( ord_less_eq @ real @ X @ Y ) ) ) ) ) ).
% log_le_cancel_iff
thf(fact_2612_log__le__one__cancel__iff,axiom,
! [A2: real,X: real] :
( ( ord_less @ real @ ( one_one @ real ) @ A2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( log @ A2 @ X ) @ ( one_one @ real ) )
= ( ord_less_eq @ real @ X @ A2 ) ) ) ) ).
% log_le_one_cancel_iff
thf(fact_2613_one__le__log__cancel__iff,axiom,
! [A2: real,X: real] :
( ( ord_less @ real @ ( one_one @ real ) @ A2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( one_one @ real ) @ ( log @ A2 @ X ) )
= ( ord_less_eq @ real @ A2 @ X ) ) ) ) ).
% one_le_log_cancel_iff
thf(fact_2614_log__le__zero__cancel__iff,axiom,
! [A2: real,X: real] :
( ( ord_less @ real @ ( one_one @ real ) @ A2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( log @ A2 @ X ) @ ( zero_zero @ real ) )
= ( ord_less_eq @ real @ X @ ( one_one @ real ) ) ) ) ) ).
% log_le_zero_cancel_iff
thf(fact_2615_zero__le__log__cancel__iff,axiom,
! [A2: real,X: real] :
( ( ord_less @ real @ ( one_one @ real ) @ A2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( log @ A2 @ X ) )
= ( ord_less_eq @ real @ ( one_one @ real ) @ X ) ) ) ) ).
% zero_le_log_cancel_iff
thf(fact_2616_prod_Ocl__ivl__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [N: nat,M2: nat,G3: nat > A] :
( ( ( ord_less @ nat @ ( suc @ N ) @ M2 )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ M2 @ ( suc @ N ) ) )
= ( one_one @ A ) ) )
& ( ~ ( ord_less @ nat @ ( suc @ N ) @ M2 )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ M2 @ ( suc @ N ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) @ ( G3 @ ( suc @ N ) ) ) ) ) ) ) ).
% prod.cl_ivl_Suc
thf(fact_2617_log__pow__cancel,axiom,
! [A2: real,B2: nat] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
=> ( ( A2
!= ( one_one @ real ) )
=> ( ( log @ A2 @ ( power_power @ real @ A2 @ B2 ) )
= ( semiring_1_of_nat @ real @ B2 ) ) ) ) ).
% log_pow_cancel
thf(fact_2618_and__numerals_I7_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X: num,Y: num] :
( ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ ( bit1 @ X ) ) @ ( numeral_numeral @ A @ ( bit1 @ Y ) ) )
= ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ A @ ( numeral_numeral @ A @ X ) @ ( numeral_numeral @ A @ Y ) ) ) ) ) ) ).
% and_numerals(7)
thf(fact_2619_prod_Oswap__restrict,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A5: set @ B,B6: set @ C,G3: B > C > A,R: B > C > $o] :
( ( finite_finite2 @ B @ A5 )
=> ( ( finite_finite2 @ C @ B6 )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [X5: B] :
( groups7121269368397514597t_prod @ C @ A @ ( G3 @ X5 )
@ ( collect @ C
@ ^ [Y6: C] :
( ( member @ C @ Y6 @ B6 )
& ( R @ X5 @ Y6 ) ) ) )
@ A5 )
= ( groups7121269368397514597t_prod @ C @ A
@ ^ [Y6: C] :
( groups7121269368397514597t_prod @ B @ A
@ ^ [X5: B] : ( G3 @ X5 @ Y6 )
@ ( collect @ B
@ ^ [X5: B] :
( ( member @ B @ X5 @ A5 )
& ( R @ X5 @ Y6 ) ) ) )
@ B6 ) ) ) ) ) ).
% prod.swap_restrict
thf(fact_2620_prod__mono,axiom,
! [A: $tType,B: $tType] :
( ( linordered_semidom @ A )
=> ! [A5: set @ B,F3: B > A,G3: B > A] :
( ! [I3: B] :
( ( member @ B @ I3 @ A5 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F3 @ I3 ) )
& ( ord_less_eq @ A @ ( F3 @ I3 ) @ ( G3 @ I3 ) ) ) )
=> ( ord_less_eq @ A @ ( groups7121269368397514597t_prod @ B @ A @ F3 @ A5 ) @ ( groups7121269368397514597t_prod @ B @ A @ G3 @ A5 ) ) ) ) ).
% prod_mono
thf(fact_2621_prod__nonneg,axiom,
! [A: $tType,B: $tType] :
( ( linordered_semidom @ A )
=> ! [A5: set @ B,F3: B > A] :
( ! [X4: B] :
( ( member @ B @ X4 @ A5 )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F3 @ X4 ) ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( groups7121269368397514597t_prod @ B @ A @ F3 @ A5 ) ) ) ) ).
% prod_nonneg
thf(fact_2622_prod__pos,axiom,
! [A: $tType,B: $tType] :
( ( linordered_semidom @ A )
=> ! [A5: set @ B,F3: B > A] :
( ! [X4: B] :
( ( member @ B @ X4 @ A5 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( F3 @ X4 ) ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( groups7121269368397514597t_prod @ B @ A @ F3 @ A5 ) ) ) ) ).
% prod_pos
thf(fact_2623_prod__ge__1,axiom,
! [A: $tType,B: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [A5: set @ B,F3: B > A] :
( ! [X4: B] :
( ( member @ B @ X4 @ A5 )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ ( F3 @ X4 ) ) )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ ( groups7121269368397514597t_prod @ B @ A @ F3 @ A5 ) ) ) ) ).
% prod_ge_1
thf(fact_2624_prod__zero,axiom,
! [B: $tType,A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A5: set @ B,F3: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ? [X3: B] :
( ( member @ B @ X3 @ A5 )
& ( ( F3 @ X3 )
= ( zero_zero @ A ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ F3 @ A5 )
= ( zero_zero @ A ) ) ) ) ) ).
% prod_zero
thf(fact_2625_prod__atLeastAtMost__code,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [F3: nat > A,A2: nat,B2: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ F3 @ ( set_or1337092689740270186AtMost @ nat @ A2 @ B2 ) )
= ( set_fo6178422350223883121st_nat @ A
@ ^ [A6: nat] : ( times_times @ A @ ( F3 @ A6 ) )
@ A2
@ B2
@ ( one_one @ A ) ) ) ) ).
% prod_atLeastAtMost_code
thf(fact_2626_pi__gt__zero,axiom,
ord_less @ real @ ( zero_zero @ real ) @ pi ).
% pi_gt_zero
thf(fact_2627_pi__not__less__zero,axiom,
~ ( ord_less @ real @ pi @ ( zero_zero @ real ) ) ).
% pi_not_less_zero
thf(fact_2628_prod_Ointer__filter,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A5: set @ B,G3: B > A,P2: B > $o] :
( ( finite_finite2 @ B @ A5 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G3
@ ( collect @ B
@ ^ [X5: B] :
( ( member @ B @ X5 @ A5 )
& ( P2 @ X5 ) ) ) )
= ( groups7121269368397514597t_prod @ B @ A
@ ^ [X5: B] : ( if @ A @ ( P2 @ X5 ) @ ( G3 @ X5 ) @ ( one_one @ A ) )
@ A5 ) ) ) ) ).
% prod.inter_filter
thf(fact_2629_prod_Oshift__bounds__cl__Suc__ivl,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G3: nat > A,M2: nat,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M2 ) @ ( suc @ N ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( G3 @ ( suc @ I4 ) )
@ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ).
% prod.shift_bounds_cl_Suc_ivl
thf(fact_2630_prod_Oshift__bounds__cl__nat__ivl,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G3: nat > A,M2: nat,K2: nat,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ M2 @ K2 ) @ ( plus_plus @ nat @ N @ K2 ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( G3 @ ( plus_plus @ nat @ I4 @ K2 ) )
@ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ).
% prod.shift_bounds_cl_nat_ivl
thf(fact_2631_prod__le__1,axiom,
! [B: $tType,A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [A5: set @ B,F3: B > A] :
( ! [X4: B] :
( ( member @ B @ X4 @ A5 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F3 @ X4 ) )
& ( ord_less_eq @ A @ ( F3 @ X4 ) @ ( one_one @ A ) ) ) )
=> ( ord_less_eq @ A @ ( groups7121269368397514597t_prod @ B @ A @ F3 @ A5 ) @ ( one_one @ A ) ) ) ) ).
% prod_le_1
thf(fact_2632_prod_Orelated,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [R: A > A > $o,S2: set @ B,H2: B > A,G3: B > A] :
( ( R @ ( one_one @ A ) @ ( one_one @ A ) )
=> ( ! [X16: A,Y15: A,X22: A,Y23: A] :
( ( ( R @ X16 @ X22 )
& ( R @ Y15 @ Y23 ) )
=> ( R @ ( times_times @ A @ X16 @ Y15 ) @ ( times_times @ A @ X22 @ Y23 ) ) )
=> ( ( finite_finite2 @ B @ S2 )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ S2 )
=> ( R @ ( H2 @ X4 ) @ ( G3 @ X4 ) ) )
=> ( R @ ( groups7121269368397514597t_prod @ B @ A @ H2 @ S2 ) @ ( groups7121269368397514597t_prod @ B @ A @ G3 @ S2 ) ) ) ) ) ) ) ).
% prod.related
thf(fact_2633_prod_Oinsert__if,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A5: set @ B,X: B,G3: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( ( member @ B @ X @ A5 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G3 @ ( insert @ B @ X @ A5 ) )
= ( groups7121269368397514597t_prod @ B @ A @ G3 @ A5 ) ) )
& ( ~ ( member @ B @ X @ A5 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G3 @ ( insert @ B @ X @ A5 ) )
= ( times_times @ A @ ( G3 @ X ) @ ( groups7121269368397514597t_prod @ B @ A @ G3 @ A5 ) ) ) ) ) ) ) ).
% prod.insert_if
thf(fact_2634_prod__dvd__prod__subset2,axiom,
! [A: $tType,B: $tType] :
( ( comm_semiring_1 @ A )
=> ! [B6: set @ B,A5: set @ B,F3: B > A,G3: B > A] :
( ( finite_finite2 @ B @ B6 )
=> ( ( ord_less_eq @ ( set @ B ) @ A5 @ B6 )
=> ( ! [A4: B] :
( ( member @ B @ A4 @ A5 )
=> ( dvd_dvd @ A @ ( F3 @ A4 ) @ ( G3 @ A4 ) ) )
=> ( dvd_dvd @ A @ ( groups7121269368397514597t_prod @ B @ A @ F3 @ A5 ) @ ( groups7121269368397514597t_prod @ B @ A @ G3 @ B6 ) ) ) ) ) ) ).
% prod_dvd_prod_subset2
thf(fact_2635_prod__dvd__prod__subset,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [B6: set @ B,A5: set @ B,F3: B > A] :
( ( finite_finite2 @ B @ B6 )
=> ( ( ord_less_eq @ ( set @ B ) @ A5 @ B6 )
=> ( dvd_dvd @ A @ ( groups7121269368397514597t_prod @ B @ A @ F3 @ A5 ) @ ( groups7121269368397514597t_prod @ B @ A @ F3 @ B6 ) ) ) ) ) ).
% prod_dvd_prod_subset
thf(fact_2636_prod_Oreindex__bij__witness__not__neutral,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S5: set @ B,T5: set @ C,S2: set @ B,I2: C > B,J: B > C,T4: set @ C,G3: B > A,H2: C > A] :
( ( finite_finite2 @ B @ S5 )
=> ( ( finite_finite2 @ C @ T5 )
=> ( ! [A4: B] :
( ( member @ B @ A4 @ ( minus_minus @ ( set @ B ) @ S2 @ S5 ) )
=> ( ( I2 @ ( J @ A4 ) )
= A4 ) )
=> ( ! [A4: B] :
( ( member @ B @ A4 @ ( minus_minus @ ( set @ B ) @ S2 @ S5 ) )
=> ( member @ C @ ( J @ A4 ) @ ( minus_minus @ ( set @ C ) @ T4 @ T5 ) ) )
=> ( ! [B4: C] :
( ( member @ C @ B4 @ ( minus_minus @ ( set @ C ) @ T4 @ T5 ) )
=> ( ( J @ ( I2 @ B4 ) )
= B4 ) )
=> ( ! [B4: C] :
( ( member @ C @ B4 @ ( minus_minus @ ( set @ C ) @ T4 @ T5 ) )
=> ( member @ B @ ( I2 @ B4 ) @ ( minus_minus @ ( set @ B ) @ S2 @ S5 ) ) )
=> ( ! [A4: B] :
( ( member @ B @ A4 @ S5 )
=> ( ( G3 @ A4 )
= ( one_one @ A ) ) )
=> ( ! [B4: C] :
( ( member @ C @ B4 @ T5 )
=> ( ( H2 @ B4 )
= ( one_one @ A ) ) )
=> ( ! [A4: B] :
( ( member @ B @ A4 @ S2 )
=> ( ( H2 @ ( J @ A4 ) )
= ( G3 @ A4 ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G3 @ S2 )
= ( groups7121269368397514597t_prod @ C @ A @ H2 @ T4 ) ) ) ) ) ) ) ) ) ) ) ) ).
% prod.reindex_bij_witness_not_neutral
thf(fact_2637_AND__upper2_H_H,axiom,
! [Y: int,Z: int,X: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y )
=> ( ( ord_less @ int @ Y @ Z )
=> ( ord_less @ int @ ( bit_se5824344872417868541ns_and @ int @ X @ Y ) @ Z ) ) ) ).
% AND_upper2''
thf(fact_2638_AND__upper1_H_H,axiom,
! [Y: int,Z: int,Ya: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y )
=> ( ( ord_less @ int @ Y @ Z )
=> ( ord_less @ int @ ( bit_se5824344872417868541ns_and @ int @ Y @ Ya ) @ Z ) ) ) ).
% AND_upper1''
thf(fact_2639_and__less__eq,axiom,
! [L: int,K2: int] :
( ( ord_less @ int @ L @ ( zero_zero @ int ) )
=> ( ord_less_eq @ int @ ( bit_se5824344872417868541ns_and @ int @ K2 @ L ) @ K2 ) ) ).
% and_less_eq
thf(fact_2640_prod_Osetdiff__irrelevant,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A5: set @ B,G3: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G3
@ ( minus_minus @ ( set @ B ) @ A5
@ ( collect @ B
@ ^ [X5: B] :
( ( G3 @ X5 )
= ( one_one @ A ) ) ) ) )
= ( groups7121269368397514597t_prod @ B @ A @ G3 @ A5 ) ) ) ) ).
% prod.setdiff_irrelevant
thf(fact_2641_prod_Onat__diff__reindex,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G3: nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( G3 @ ( minus_minus @ nat @ N @ ( suc @ I4 ) ) )
@ ( set_ord_lessThan @ nat @ N ) )
= ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% prod.nat_diff_reindex
thf(fact_2642_prod_OatLeastAtMost__rev,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G3: nat > A,N: nat,M2: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ N @ M2 ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( G3 @ ( minus_minus @ nat @ ( plus_plus @ nat @ M2 @ N ) @ I4 ) )
@ ( set_or1337092689740270186AtMost @ nat @ N @ M2 ) ) ) ) ).
% prod.atLeastAtMost_rev
thf(fact_2643_less__1__prod2,axiom,
! [B: $tType,A: $tType] :
( ( linordered_idom @ B )
=> ! [I6: set @ A,I2: A,F3: A > B] :
( ( finite_finite2 @ A @ I6 )
=> ( ( member @ A @ I2 @ I6 )
=> ( ( ord_less @ B @ ( one_one @ B ) @ ( F3 @ I2 ) )
=> ( ! [I3: A] :
( ( member @ A @ I3 @ I6 )
=> ( ord_less_eq @ B @ ( one_one @ B ) @ ( F3 @ I3 ) ) )
=> ( ord_less @ B @ ( one_one @ B ) @ ( groups7121269368397514597t_prod @ A @ B @ F3 @ I6 ) ) ) ) ) ) ) ).
% less_1_prod2
thf(fact_2644_less__1__prod,axiom,
! [B: $tType,A: $tType] :
( ( linordered_idom @ B )
=> ! [I6: set @ A,F3: A > B] :
( ( finite_finite2 @ A @ I6 )
=> ( ( I6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [I3: A] :
( ( member @ A @ I3 @ I6 )
=> ( ord_less @ B @ ( one_one @ B ) @ ( F3 @ I3 ) ) )
=> ( ord_less @ B @ ( one_one @ B ) @ ( groups7121269368397514597t_prod @ A @ B @ F3 @ I6 ) ) ) ) ) ) ).
% less_1_prod
thf(fact_2645_prod_Osubset__diff,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [B6: set @ B,A5: set @ B,G3: B > A] :
( ( ord_less_eq @ ( set @ B ) @ B6 @ A5 )
=> ( ( finite_finite2 @ B @ A5 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G3 @ A5 )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G3 @ ( minus_minus @ ( set @ B ) @ A5 @ B6 ) ) @ ( groups7121269368397514597t_prod @ B @ A @ G3 @ B6 ) ) ) ) ) ) ).
% prod.subset_diff
thf(fact_2646_prod_Osame__carrier,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [C5: set @ B,A5: set @ B,B6: set @ B,G3: B > A,H2: B > A] :
( ( finite_finite2 @ B @ C5 )
=> ( ( ord_less_eq @ ( set @ B ) @ A5 @ C5 )
=> ( ( ord_less_eq @ ( set @ B ) @ B6 @ C5 )
=> ( ! [A4: B] :
( ( member @ B @ A4 @ ( minus_minus @ ( set @ B ) @ C5 @ A5 ) )
=> ( ( G3 @ A4 )
= ( one_one @ A ) ) )
=> ( ! [B4: B] :
( ( member @ B @ B4 @ ( minus_minus @ ( set @ B ) @ C5 @ B6 ) )
=> ( ( H2 @ B4 )
= ( one_one @ A ) ) )
=> ( ( ( groups7121269368397514597t_prod @ B @ A @ G3 @ A5 )
= ( groups7121269368397514597t_prod @ B @ A @ H2 @ B6 ) )
= ( ( groups7121269368397514597t_prod @ B @ A @ G3 @ C5 )
= ( groups7121269368397514597t_prod @ B @ A @ H2 @ C5 ) ) ) ) ) ) ) ) ) ).
% prod.same_carrier
thf(fact_2647_prod_Osame__carrierI,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [C5: set @ B,A5: set @ B,B6: set @ B,G3: B > A,H2: B > A] :
( ( finite_finite2 @ B @ C5 )
=> ( ( ord_less_eq @ ( set @ B ) @ A5 @ C5 )
=> ( ( ord_less_eq @ ( set @ B ) @ B6 @ C5 )
=> ( ! [A4: B] :
( ( member @ B @ A4 @ ( minus_minus @ ( set @ B ) @ C5 @ A5 ) )
=> ( ( G3 @ A4 )
= ( one_one @ A ) ) )
=> ( ! [B4: B] :
( ( member @ B @ B4 @ ( minus_minus @ ( set @ B ) @ C5 @ B6 ) )
=> ( ( H2 @ B4 )
= ( one_one @ A ) ) )
=> ( ( ( groups7121269368397514597t_prod @ B @ A @ G3 @ C5 )
= ( groups7121269368397514597t_prod @ B @ A @ H2 @ C5 ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G3 @ A5 )
= ( groups7121269368397514597t_prod @ B @ A @ H2 @ B6 ) ) ) ) ) ) ) ) ) ).
% prod.same_carrierI
thf(fact_2648_prod_Omono__neutral__left,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [T4: set @ B,S2: set @ B,G3: B > A] :
( ( finite_finite2 @ B @ T4 )
=> ( ( ord_less_eq @ ( set @ B ) @ S2 @ T4 )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ ( minus_minus @ ( set @ B ) @ T4 @ S2 ) )
=> ( ( G3 @ X4 )
= ( one_one @ A ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G3 @ S2 )
= ( groups7121269368397514597t_prod @ B @ A @ G3 @ T4 ) ) ) ) ) ) ).
% prod.mono_neutral_left
thf(fact_2649_prod_Omono__neutral__right,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [T4: set @ B,S2: set @ B,G3: B > A] :
( ( finite_finite2 @ B @ T4 )
=> ( ( ord_less_eq @ ( set @ B ) @ S2 @ T4 )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ ( minus_minus @ ( set @ B ) @ T4 @ S2 ) )
=> ( ( G3 @ X4 )
= ( one_one @ A ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G3 @ T4 )
= ( groups7121269368397514597t_prod @ B @ A @ G3 @ S2 ) ) ) ) ) ) ).
% prod.mono_neutral_right
thf(fact_2650_prod_Omono__neutral__cong__left,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [T4: set @ B,S2: set @ B,H2: B > A,G3: B > A] :
( ( finite_finite2 @ B @ T4 )
=> ( ( ord_less_eq @ ( set @ B ) @ S2 @ T4 )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ ( minus_minus @ ( set @ B ) @ T4 @ S2 ) )
=> ( ( H2 @ X4 )
= ( one_one @ A ) ) )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ S2 )
=> ( ( G3 @ X4 )
= ( H2 @ X4 ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G3 @ S2 )
= ( groups7121269368397514597t_prod @ B @ A @ H2 @ T4 ) ) ) ) ) ) ) ).
% prod.mono_neutral_cong_left
thf(fact_2651_prod_Omono__neutral__cong__right,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [T4: set @ B,S2: set @ B,G3: B > A,H2: B > A] :
( ( finite_finite2 @ B @ T4 )
=> ( ( ord_less_eq @ ( set @ B ) @ S2 @ T4 )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ ( minus_minus @ ( set @ B ) @ T4 @ S2 ) )
=> ( ( G3 @ X4 )
= ( one_one @ A ) ) )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ S2 )
=> ( ( G3 @ X4 )
= ( H2 @ X4 ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G3 @ T4 )
= ( groups7121269368397514597t_prod @ B @ A @ H2 @ S2 ) ) ) ) ) ) ) ).
% prod.mono_neutral_cong_right
thf(fact_2652_prod_OatLeast0__atMost__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G3: nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) @ ( G3 @ ( suc @ N ) ) ) ) ) ).
% prod.atLeast0_atMost_Suc
thf(fact_2653_prod_OatLeast__Suc__atMost,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M2: nat,N: nat,G3: nat > A] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
= ( times_times @ A @ ( G3 @ M2 ) @ ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M2 ) @ N ) ) ) ) ) ) ).
% prod.atLeast_Suc_atMost
thf(fact_2654_prod_Onat__ivl__Suc_H,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M2: nat,N: nat,G3: nat > A] :
( ( ord_less_eq @ nat @ M2 @ ( suc @ N ) )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ M2 @ ( suc @ N ) ) )
= ( times_times @ A @ ( G3 @ ( suc @ N ) ) @ ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ) ) ).
% prod.nat_ivl_Suc'
thf(fact_2655_log__base__change,axiom,
! [A2: real,B2: real,X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
=> ( ( A2
!= ( one_one @ real ) )
=> ( ( log @ B2 @ X )
= ( divide_divide @ real @ ( log @ A2 @ X ) @ ( log @ A2 @ B2 ) ) ) ) ) ).
% log_base_change
thf(fact_2656_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 )
= ( log @ B2 @ ( semiring_1_of_nat @ real @ M2 ) ) ) ) ) ).
% log_of_power_eq
thf(fact_2657_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 ) @ ( log @ B2 @ M2 ) ) ) ) ).
% less_log_of_power
thf(fact_2658_pi__less__4,axiom,
ord_less @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ one2 ) ) ) ).
% pi_less_4
thf(fact_2659_prod_OlessThan__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G3: nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_ord_lessThan @ nat @ ( suc @ N ) ) )
= ( times_times @ A @ ( G3 @ ( zero_zero @ nat ) )
@ ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( G3 @ ( suc @ I4 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% prod.lessThan_Suc_shift
thf(fact_2660_prod_OSuc__reindex__ivl,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M2: nat,N: nat,G3: nat > A] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) @ ( G3 @ ( suc @ N ) ) )
= ( times_times @ A @ ( G3 @ M2 )
@ ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( G3 @ ( suc @ I4 ) )
@ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ) ) ).
% prod.Suc_reindex_ivl
thf(fact_2661_prod_OatLeast1__atMost__eq,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G3: nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ ( zero_zero @ nat ) ) @ N ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [K3: nat] : ( G3 @ ( suc @ K3 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% prod.atLeast1_atMost_eq
thf(fact_2662_prod__mono__strict,axiom,
! [A: $tType,B: $tType] :
( ( linordered_semidom @ A )
=> ! [A5: set @ B,F3: B > A,G3: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ! [I3: B] :
( ( member @ B @ I3 @ A5 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F3 @ I3 ) )
& ( ord_less @ A @ ( F3 @ I3 ) @ ( G3 @ I3 ) ) ) )
=> ( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ord_less @ A @ ( groups7121269368397514597t_prod @ B @ A @ F3 @ A5 ) @ ( groups7121269368397514597t_prod @ B @ A @ G3 @ A5 ) ) ) ) ) ) ).
% prod_mono_strict
thf(fact_2663_even__prod__iff,axiom,
! [A: $tType,B: $tType] :
( ( semiring_parity @ A )
=> ! [A5: set @ B,F3: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( groups7121269368397514597t_prod @ B @ A @ F3 @ A5 ) )
= ( ? [X5: B] :
( ( member @ B @ X5 @ A5 )
& ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( F3 @ X5 ) ) ) ) ) ) ) ).
% even_prod_iff
thf(fact_2664_prod_Oinsert__remove,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A5: set @ B,G3: B > A,X: B] :
( ( finite_finite2 @ B @ A5 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G3 @ ( insert @ B @ X @ A5 ) )
= ( times_times @ A @ ( G3 @ X ) @ ( groups7121269368397514597t_prod @ B @ A @ G3 @ ( minus_minus @ ( set @ B ) @ A5 @ ( insert @ B @ X @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ).
% prod.insert_remove
thf(fact_2665_prod_Oremove,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A5: set @ B,X: B,G3: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( member @ B @ X @ A5 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G3 @ A5 )
= ( times_times @ A @ ( G3 @ X ) @ ( groups7121269368397514597t_prod @ B @ A @ G3 @ ( minus_minus @ ( set @ B ) @ A5 @ ( insert @ B @ X @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ) ).
% prod.remove
thf(fact_2666_prod_Oub__add__nat,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M2: nat,N: nat,G3: nat > A,P: nat] :
( ( ord_less_eq @ nat @ M2 @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ M2 @ ( plus_plus @ nat @ N @ P ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) @ ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) @ ( plus_plus @ nat @ N @ P ) ) ) ) ) ) ) ).
% prod.ub_add_nat
thf(fact_2667_log__mult,axiom,
! [A2: real,X: real,Y: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
=> ( ( A2
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ Y )
=> ( ( log @ A2 @ ( times_times @ real @ X @ Y ) )
= ( plus_plus @ real @ ( log @ A2 @ X ) @ ( log @ A2 @ Y ) ) ) ) ) ) ) ).
% log_mult
thf(fact_2668_log__divide,axiom,
! [A2: real,X: real,Y: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
=> ( ( A2
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ Y )
=> ( ( log @ A2 @ ( divide_divide @ real @ X @ Y ) )
= ( minus_minus @ real @ ( log @ A2 @ X ) @ ( log @ A2 @ Y ) ) ) ) ) ) ) ).
% log_divide
thf(fact_2669_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 ) @ ( log @ B2 @ M2 ) ) ) ) ).
% le_log_of_power
thf(fact_2670_log__base__pow,axiom,
! [A2: real,N: nat,X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
=> ( ( log @ ( power_power @ real @ A2 @ N ) @ X )
= ( divide_divide @ real @ ( log @ A2 @ X ) @ ( semiring_1_of_nat @ real @ N ) ) ) ) ).
% log_base_pow
thf(fact_2671_log__nat__power,axiom,
! [X: real,B2: real,N: nat] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( log @ B2 @ ( power_power @ real @ X @ N ) )
= ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ ( log @ B2 @ X ) ) ) ) ).
% log_nat_power
thf(fact_2672_prod_Odelta__remove,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S2: set @ B,A2: B,B2: B > A,C3: B > A] :
( ( finite_finite2 @ B @ S2 )
=> ( ( ( member @ B @ A2 @ S2 )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K3: B] : ( if @ A @ ( K3 = A2 ) @ ( B2 @ K3 ) @ ( C3 @ K3 ) )
@ S2 )
= ( times_times @ A @ ( B2 @ A2 ) @ ( groups7121269368397514597t_prod @ B @ A @ C3 @ ( minus_minus @ ( set @ B ) @ S2 @ ( insert @ B @ A2 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) )
& ( ~ ( member @ B @ A2 @ S2 )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K3: B] : ( if @ A @ ( K3 = A2 ) @ ( B2 @ K3 ) @ ( C3 @ K3 ) )
@ S2 )
= ( groups7121269368397514597t_prod @ B @ A @ C3 @ ( minus_minus @ ( set @ B ) @ S2 @ ( insert @ B @ A2 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ) ).
% prod.delta_remove
thf(fact_2673_fold__atLeastAtMost__nat_Oelims,axiom,
! [A: $tType,X: nat > A > A,Xa2: nat,Xb3: nat,Xc: A,Y: A] :
( ( ( set_fo6178422350223883121st_nat @ A @ X @ Xa2 @ Xb3 @ Xc )
= Y )
=> ( ( ( ord_less @ nat @ Xb3 @ Xa2 )
=> ( Y = Xc ) )
& ( ~ ( ord_less @ nat @ Xb3 @ Xa2 )
=> ( Y
= ( set_fo6178422350223883121st_nat @ A @ X @ ( plus_plus @ nat @ Xa2 @ ( one_one @ nat ) ) @ Xb3 @ ( X @ Xa2 @ Xc ) ) ) ) ) ) ).
% fold_atLeastAtMost_nat.elims
thf(fact_2674_fold__atLeastAtMost__nat_Osimps,axiom,
! [A: $tType] :
( ( set_fo6178422350223883121st_nat @ A )
= ( ^ [F4: nat > A > A,A6: nat,B5: nat,Acc2: A] : ( if @ A @ ( ord_less @ nat @ B5 @ A6 ) @ Acc2 @ ( set_fo6178422350223883121st_nat @ A @ F4 @ ( plus_plus @ nat @ A6 @ ( one_one @ nat ) ) @ B5 @ ( F4 @ A6 @ Acc2 ) ) ) ) ) ).
% fold_atLeastAtMost_nat.simps
thf(fact_2675_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_2676_prod__mono2,axiom,
! [B: $tType,A: $tType] :
( ( linordered_idom @ B )
=> ! [B6: set @ A,A5: set @ A,F3: A > B] :
( ( finite_finite2 @ A @ B6 )
=> ( ( ord_less_eq @ ( set @ A ) @ A5 @ B6 )
=> ( ! [B4: A] :
( ( member @ A @ B4 @ ( minus_minus @ ( set @ A ) @ B6 @ A5 ) )
=> ( ord_less_eq @ B @ ( one_one @ B ) @ ( F3 @ B4 ) ) )
=> ( ! [A4: A] :
( ( member @ A @ A4 @ A5 )
=> ( ord_less_eq @ B @ ( zero_zero @ B ) @ ( F3 @ A4 ) ) )
=> ( ord_less_eq @ B @ ( groups7121269368397514597t_prod @ A @ B @ F3 @ A5 ) @ ( groups7121269368397514597t_prod @ A @ B @ F3 @ B6 ) ) ) ) ) ) ) ).
% prod_mono2
thf(fact_2677_prod__diff1,axiom,
! [A: $tType,B: $tType] :
( ( semidom_divide @ A )
=> ! [A5: set @ B,F3: B > A,A2: B] :
( ( finite_finite2 @ B @ A5 )
=> ( ( ( F3 @ A2 )
!= ( zero_zero @ A ) )
=> ( ( ( member @ B @ A2 @ A5 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ F3 @ ( minus_minus @ ( set @ B ) @ A5 @ ( insert @ B @ A2 @ ( bot_bot @ ( set @ B ) ) ) ) )
= ( divide_divide @ A @ ( groups7121269368397514597t_prod @ B @ A @ F3 @ A5 ) @ ( F3 @ A2 ) ) ) )
& ( ~ ( member @ B @ A2 @ A5 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ F3 @ ( minus_minus @ ( set @ B ) @ A5 @ ( insert @ B @ A2 @ ( bot_bot @ ( set @ B ) ) ) ) )
= ( groups7121269368397514597t_prod @ B @ A @ F3 @ A5 ) ) ) ) ) ) ) ).
% prod_diff1
thf(fact_2678_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 @ ( log @ B2 @ ( semiring_1_of_nat @ real @ M2 ) ) @ ( semiring_1_of_nat @ real @ N ) ) ) ) ) ).
% log_of_power_less
thf(fact_2679_log__eq__div__ln__mult__log,axiom,
! [A2: real,B2: real,X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
=> ( ( A2
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ B2 )
=> ( ( B2
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( log @ A2 @ X )
= ( times_times @ real @ ( divide_divide @ real @ ( ln_ln @ real @ B2 ) @ ( ln_ln @ real @ A2 ) ) @ ( log @ B2 @ X ) ) ) ) ) ) ) ) ).
% log_eq_div_ln_mult_log
thf(fact_2680_pochhammer__Suc__prod,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A2: A,N: nat] :
( ( comm_s3205402744901411588hammer @ A @ A2 @ ( suc @ N ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( plus_plus @ A @ A2 @ ( semiring_1_of_nat @ A @ I4 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% pochhammer_Suc_prod
thf(fact_2681_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_2682_pochhammer__prod__rev,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ( ( comm_s3205402744901411588hammer @ A )
= ( ^ [A6: A,N2: nat] :
( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( plus_plus @ A @ A6 @ ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ N2 @ I4 ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ N2 ) ) ) ) ) ).
% pochhammer_prod_rev
thf(fact_2683_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_2684_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 @ ( log @ B2 @ ( semiring_1_of_nat @ real @ M2 ) ) @ ( semiring_1_of_nat @ real @ N ) ) ) ) ) ).
% log_of_power_le
thf(fact_2685_prod_Oin__pairs,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G3: nat > A,M2: nat,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( 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
@ ^ [I4: nat] : ( times_times @ A @ ( G3 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I4 ) ) @ ( G3 @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I4 ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ).
% prod.in_pairs
thf(fact_2686_sum__atLeastAtMost__code,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [F3: nat > A,A2: nat,B2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_or1337092689740270186AtMost @ nat @ A2 @ B2 ) )
= ( set_fo6178422350223883121st_nat @ A
@ ^ [A6: nat] : ( plus_plus @ A @ ( F3 @ A6 ) )
@ A2
@ B2
@ ( zero_zero @ A ) ) ) ) ).
% sum_atLeastAtMost_code
thf(fact_2687_pochhammer__Suc__prod__rev,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A2: A,N: nat] :
( ( comm_s3205402744901411588hammer @ A @ A2 @ ( suc @ N ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( plus_plus @ A @ A2 @ ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ N @ I4 ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% pochhammer_Suc_prod_rev
thf(fact_2688_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_2689_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 ) @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ M2 ) ) ) ) ).
% less_log2_of_power
thf(fact_2690_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 ) @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ M2 ) ) ) ) ).
% le_log2_of_power
thf(fact_2691_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 @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ M2 ) ) @ ( semiring_1_of_nat @ real @ N ) ) ) ) ).
% log2_of_power_less
thf(fact_2692_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 @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ M2 ) ) @ ( semiring_1_of_nat @ real @ N ) ) ) ) ).
% log2_of_power_le
thf(fact_2693_ceiling__log__nat__eq__if,axiom,
! [B2: nat,N: nat,K2: nat] :
( ( ord_less @ nat @ ( power_power @ nat @ B2 @ N ) @ K2 )
=> ( ( ord_less_eq @ nat @ K2 @ ( power_power @ nat @ B2 @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ B2 )
=> ( ( archimedean_ceiling @ real @ ( log @ ( semiring_1_of_nat @ real @ B2 ) @ ( semiring_1_of_nat @ real @ K2 ) ) )
= ( plus_plus @ int @ ( semiring_1_of_nat @ int @ N ) @ ( one_one @ int ) ) ) ) ) ) ).
% ceiling_log_nat_eq_if
thf(fact_2694_ceiling__log2__div2,axiom,
! [N: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( archimedean_ceiling @ real @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ N ) ) )
= ( plus_plus @ int @ ( archimedean_ceiling @ real @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ ( plus_plus @ nat @ ( divide_divide @ nat @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) ) @ ( one_one @ int ) ) ) ) ).
% ceiling_log2_div2
thf(fact_2695_ceiling__log__eq__powr__iff,axiom,
! [X: real,B2: real,K2: nat] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( one_one @ real ) @ B2 )
=> ( ( ( archimedean_ceiling @ real @ ( log @ B2 @ X ) )
= ( plus_plus @ int @ ( semiring_1_of_nat @ int @ K2 ) @ ( one_one @ int ) ) )
= ( ( ord_less @ real @ ( powr @ real @ B2 @ ( semiring_1_of_nat @ real @ K2 ) ) @ X )
& ( ord_less_eq @ real @ X @ ( powr @ real @ B2 @ ( semiring_1_of_nat @ real @ ( plus_plus @ nat @ K2 @ ( one_one @ nat ) ) ) ) ) ) ) ) ) ).
% ceiling_log_eq_powr_iff
thf(fact_2696_geometric__deriv__sums,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [Z: A] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ Z ) @ ( one_one @ real ) )
=> ( sums @ A
@ ^ [N2: nat] : ( times_times @ A @ ( semiring_1_of_nat @ A @ ( suc @ N2 ) ) @ ( power_power @ A @ Z @ N2 ) )
@ ( divide_divide @ A @ ( one_one @ A ) @ ( power_power @ A @ ( minus_minus @ A @ ( one_one @ A ) @ Z ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% geometric_deriv_sums
thf(fact_2697_floor__log__nat__eq__powr__iff,axiom,
! [B2: nat,K2: nat,N: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ B2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K2 )
=> ( ( ( archim6421214686448440834_floor @ real @ ( log @ ( semiring_1_of_nat @ real @ B2 ) @ ( semiring_1_of_nat @ real @ K2 ) ) )
= ( semiring_1_of_nat @ int @ N ) )
= ( ( ord_less_eq @ nat @ ( power_power @ nat @ B2 @ N ) @ K2 )
& ( ord_less @ nat @ K2 @ ( power_power @ nat @ B2 @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% floor_log_nat_eq_powr_iff
thf(fact_2698_and__int_Opelims,axiom,
! [X: int,Xa2: int,Y: int] :
( ( ( bit_se5824344872417868541ns_and @ int @ X @ Xa2 )
= Y )
=> ( ( accp @ ( product_prod @ int @ int ) @ bit_and_int_rel @ ( product_Pair @ int @ int @ X @ Xa2 ) )
=> ~ ( ( ( ( ( member @ int @ X @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
& ( member @ int @ Xa2 @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) )
=> ( Y
= ( uminus_uminus @ int
@ ( zero_neq_one_of_bool @ int
@ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ X )
& ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Xa2 ) ) ) ) ) )
& ( ~ ( ( member @ int @ X @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
& ( member @ int @ Xa2 @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) )
=> ( Y
= ( plus_plus @ int
@ ( zero_neq_one_of_bool @ int
@ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ X )
& ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ Xa2 ) ) )
@ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( divide_divide @ int @ X @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ Xa2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ int @ int ) @ bit_and_int_rel @ ( product_Pair @ int @ int @ X @ Xa2 ) ) ) ) ) ).
% and_int.pelims
thf(fact_2699_and__int_Opsimps,axiom,
! [K2: int,L: int] :
( ( accp @ ( product_prod @ int @ int ) @ bit_and_int_rel @ ( product_Pair @ int @ int @ K2 @ L ) )
=> ( ( ( ( member @ int @ K2 @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
& ( member @ int @ L @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) )
=> ( ( bit_se5824344872417868541ns_and @ int @ K2 @ L )
= ( uminus_uminus @ int
@ ( zero_neq_one_of_bool @ int
@ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K2 )
& ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ L ) ) ) ) ) )
& ( ~ ( ( member @ int @ K2 @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
& ( member @ int @ L @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) )
=> ( ( bit_se5824344872417868541ns_and @ int @ K2 @ L )
= ( plus_plus @ int
@ ( zero_neq_one_of_bool @ int
@ ( ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ K2 )
& ~ ( dvd_dvd @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ L ) ) )
@ ( times_times @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ int @ ( divide_divide @ int @ K2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% and_int.psimps
thf(fact_2700_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_2701_powr__eq__0__iff,axiom,
! [A: $tType] :
( ( ln @ A )
=> ! [W2: A,Z: A] :
( ( ( powr @ A @ W2 @ Z )
= ( zero_zero @ A ) )
= ( W2
= ( zero_zero @ A ) ) ) ) ).
% powr_eq_0_iff
thf(fact_2702_powr__0,axiom,
! [A: $tType] :
( ( ln @ A )
=> ! [Z: A] :
( ( powr @ A @ ( zero_zero @ A ) @ Z )
= ( zero_zero @ A ) ) ) ).
% powr_0
thf(fact_2703_sums__zero,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ( sums @ A
@ ^ [N2: nat] : ( zero_zero @ A )
@ ( zero_zero @ A ) ) ) ).
% sums_zero
thf(fact_2704_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_2705_floor__zero,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archim6421214686448440834_floor @ A @ ( zero_zero @ A ) )
= ( zero_zero @ int ) ) ) ).
% floor_zero
thf(fact_2706_powr__gt__zero,axiom,
! [X: real,A2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( powr @ real @ X @ A2 ) )
= ( X
!= ( zero_zero @ real ) ) ) ).
% powr_gt_zero
thf(fact_2707_powr__less__cancel__iff,axiom,
! [X: real,A2: real,B2: real] :
( ( ord_less @ real @ ( one_one @ real ) @ X )
=> ( ( ord_less @ real @ ( powr @ real @ X @ A2 ) @ ( powr @ real @ X @ B2 ) )
= ( ord_less @ real @ A2 @ B2 ) ) ) ).
% powr_less_cancel_iff
thf(fact_2708_prod__eq__1__iff,axiom,
! [A: $tType,A5: set @ A,F3: A > nat] :
( ( finite_finite2 @ A @ A5 )
=> ( ( ( groups7121269368397514597t_prod @ A @ nat @ F3 @ A5 )
= ( one_one @ nat ) )
= ( ! [X5: A] :
( ( member @ A @ X5 @ A5 )
=> ( ( F3 @ X5 )
= ( one_one @ nat ) ) ) ) ) ) ).
% prod_eq_1_iff
thf(fact_2709_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_2710_and__nat__numerals_I1_J,axiom,
! [Y: num] :
( ( bit_se5824344872417868541ns_and @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( numeral_numeral @ nat @ ( bit0 @ Y ) ) )
= ( zero_zero @ nat ) ) ).
% and_nat_numerals(1)
thf(fact_2711_powr__eq__one__iff,axiom,
! [A2: real,X: real] :
( ( ord_less @ real @ ( one_one @ real ) @ A2 )
=> ( ( ( powr @ real @ A2 @ X )
= ( one_one @ real ) )
= ( X
= ( zero_zero @ real ) ) ) ) ).
% powr_eq_one_iff
thf(fact_2712_powr__le__cancel__iff,axiom,
! [X: real,A2: real,B2: real] :
( ( ord_less @ real @ ( one_one @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( powr @ real @ X @ A2 ) @ ( powr @ real @ X @ B2 ) )
= ( ord_less_eq @ real @ A2 @ B2 ) ) ) ).
% powr_le_cancel_iff
thf(fact_2713_prod__pos__nat__iff,axiom,
! [A: $tType,A5: set @ A,F3: A > nat] :
( ( finite_finite2 @ A @ A5 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( groups7121269368397514597t_prod @ A @ nat @ F3 @ A5 ) )
= ( ! [X5: A] :
( ( member @ A @ X5 @ A5 )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ ( F3 @ X5 ) ) ) ) ) ) ).
% prod_pos_nat_iff
thf(fact_2714_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_2715_and__nat__numerals_I2_J,axiom,
! [Y: num] :
( ( bit_se5824344872417868541ns_and @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( numeral_numeral @ nat @ ( bit1 @ Y ) ) )
= ( one_one @ nat ) ) ).
% and_nat_numerals(2)
thf(fact_2716_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_2717_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_2718_numeral__le__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V: num,X: A] :
( ( ord_less_eq @ int @ ( numeral_numeral @ int @ V ) @ ( archim6421214686448440834_floor @ A @ X ) )
= ( ord_less_eq @ A @ ( numeral_numeral @ A @ V ) @ X ) ) ) ).
% numeral_le_floor
thf(fact_2719_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_2720_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_2721_floor__less__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,V: num] :
( ( ord_less @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( numeral_numeral @ int @ V ) )
= ( ord_less @ A @ X @ ( numeral_numeral @ A @ V ) ) ) ) ).
% floor_less_numeral
thf(fact_2722_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_2723_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_2724_log__powr__cancel,axiom,
! [A2: real,Y: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
=> ( ( A2
!= ( one_one @ real ) )
=> ( ( log @ A2 @ ( powr @ real @ A2 @ Y ) )
= Y ) ) ) ).
% log_powr_cancel
thf(fact_2725_powr__log__cancel,axiom,
! [A2: real,X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
=> ( ( A2
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( powr @ real @ A2 @ ( log @ A2 @ X ) )
= X ) ) ) ) ).
% powr_log_cancel
thf(fact_2726_powser__sums__zero__iff,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [A2: nat > A,X: A] :
( ( sums @ A
@ ^ [N2: nat] : ( times_times @ A @ ( A2 @ N2 ) @ ( power_power @ A @ ( zero_zero @ A ) @ N2 ) )
@ X )
= ( ( A2 @ ( zero_zero @ nat ) )
= X ) ) ) ).
% powser_sums_zero_iff
thf(fact_2727_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_2728_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_2729_numeral__less__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V: num,X: A] :
( ( ord_less @ int @ ( numeral_numeral @ int @ V ) @ ( archim6421214686448440834_floor @ A @ X ) )
= ( ord_less_eq @ A @ ( plus_plus @ A @ ( numeral_numeral @ A @ V ) @ ( one_one @ A ) ) @ X ) ) ) ).
% numeral_less_floor
thf(fact_2730_floor__le__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,V: num] :
( ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( numeral_numeral @ int @ V ) )
= ( ord_less @ A @ X @ ( plus_plus @ A @ ( numeral_numeral @ A @ V ) @ ( one_one @ A ) ) ) ) ) ).
% floor_le_numeral
thf(fact_2731_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_2732_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_2733_neg__numeral__le__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V: num,X: A] :
( ( ord_less_eq @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V ) ) @ ( archim6421214686448440834_floor @ A @ X ) )
= ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) @ X ) ) ) ).
% neg_numeral_le_floor
thf(fact_2734_floor__less__neg__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,V: num] :
( ( ord_less @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V ) ) )
= ( ord_less @ A @ X @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) ) ) ) ).
% floor_less_neg_numeral
thf(fact_2735_neg__numeral__less__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [V: num,X: A] :
( ( ord_less @ int @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V ) ) @ ( archim6421214686448440834_floor @ A @ X ) )
= ( ord_less_eq @ A @ ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) @ ( one_one @ A ) ) @ X ) ) ) ).
% neg_numeral_less_floor
thf(fact_2736_floor__le__neg__numeral,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,V: num] :
( ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ V ) ) )
= ( ord_less @ A @ X @ ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) @ ( one_one @ A ) ) ) ) ) ).
% floor_le_neg_numeral
thf(fact_2737_int__prod,axiom,
! [B: $tType,F3: B > nat,A5: set @ B] :
( ( semiring_1_of_nat @ int @ ( groups7121269368397514597t_prod @ B @ nat @ F3 @ A5 ) )
= ( groups7121269368397514597t_prod @ B @ int
@ ^ [X5: B] : ( semiring_1_of_nat @ int @ ( F3 @ X5 ) )
@ A5 ) ) ).
% int_prod
thf(fact_2738_sums__0,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [F3: nat > A] :
( ! [N3: nat] :
( ( F3 @ N3 )
= ( zero_zero @ A ) )
=> ( sums @ A @ F3 @ ( zero_zero @ A ) ) ) ) ).
% sums_0
thf(fact_2739_sums__le,axiom,
! [A: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F3: nat > A,G3: nat > A,S: A,T2: A] :
( ! [N3: nat] : ( ord_less_eq @ A @ ( F3 @ N3 ) @ ( G3 @ N3 ) )
=> ( ( sums @ A @ F3 @ S )
=> ( ( sums @ A @ G3 @ T2 )
=> ( ord_less_eq @ A @ S @ T2 ) ) ) ) ) ).
% sums_le
thf(fact_2740_sums__single,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [I2: nat,F3: nat > A] :
( sums @ A
@ ^ [R5: nat] : ( if @ A @ ( R5 = I2 ) @ ( F3 @ R5 ) @ ( zero_zero @ A ) )
@ ( F3 @ I2 ) ) ) ).
% sums_single
thf(fact_2741_sums__add,axiom,
! [A: $tType] :
( ( ( topolo5987344860129210374id_add @ A )
& ( topological_t2_space @ A ) )
=> ! [F3: nat > A,A2: A,G3: nat > A,B2: A] :
( ( sums @ A @ F3 @ A2 )
=> ( ( sums @ A @ G3 @ B2 )
=> ( sums @ A
@ ^ [N2: nat] : ( plus_plus @ A @ ( F3 @ N2 ) @ ( G3 @ N2 ) )
@ ( plus_plus @ A @ A2 @ B2 ) ) ) ) ) ).
% sums_add
thf(fact_2742_floor__mono,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( archim6421214686448440834_floor @ A @ Y ) ) ) ) ).
% floor_mono
thf(fact_2743_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_2744_floor__less__cancel,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( archim6421214686448440834_floor @ A @ Y ) )
=> ( ord_less @ A @ X @ Y ) ) ) ).
% floor_less_cancel
thf(fact_2745_powr__less__mono2__neg,axiom,
! [A2: real,X: real,Y: real] :
( ( ord_less @ real @ A2 @ ( zero_zero @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ X @ Y )
=> ( ord_less @ real @ ( powr @ real @ Y @ A2 ) @ ( powr @ real @ X @ A2 ) ) ) ) ) ).
% powr_less_mono2_neg
thf(fact_2746_powr__non__neg,axiom,
! [A2: real,X: real] :
~ ( ord_less @ real @ ( powr @ real @ A2 @ X ) @ ( zero_zero @ real ) ) ).
% powr_non_neg
thf(fact_2747_powr__less__cancel,axiom,
! [X: real,A2: real,B2: real] :
( ( ord_less @ real @ ( powr @ real @ X @ A2 ) @ ( powr @ real @ X @ B2 ) )
=> ( ( ord_less @ real @ ( one_one @ real ) @ X )
=> ( ord_less @ real @ A2 @ B2 ) ) ) ).
% powr_less_cancel
thf(fact_2748_powr__less__mono,axiom,
! [A2: real,B2: real,X: real] :
( ( ord_less @ real @ A2 @ B2 )
=> ( ( ord_less @ real @ ( one_one @ real ) @ X )
=> ( ord_less @ real @ ( powr @ real @ X @ A2 ) @ ( powr @ real @ X @ B2 ) ) ) ) ).
% powr_less_mono
thf(fact_2749_sums__mult__iff,axiom,
! [A: $tType] :
( ( ( field @ A )
& ( real_V4412858255891104859lgebra @ A ) )
=> ! [C3: A,F3: nat > A,D3: A] :
( ( C3
!= ( zero_zero @ A ) )
=> ( ( sums @ A
@ ^ [N2: nat] : ( times_times @ A @ C3 @ ( F3 @ N2 ) )
@ ( times_times @ A @ C3 @ D3 ) )
= ( sums @ A @ F3 @ D3 ) ) ) ) ).
% sums_mult_iff
thf(fact_2750_sums__mult2__iff,axiom,
! [A: $tType] :
( ( ( field @ A )
& ( real_V4412858255891104859lgebra @ A ) )
=> ! [C3: A,F3: nat > A,D3: A] :
( ( C3
!= ( zero_zero @ A ) )
=> ( ( sums @ A
@ ^ [N2: nat] : ( times_times @ A @ ( F3 @ N2 ) @ C3 )
@ ( times_times @ A @ D3 @ C3 ) )
= ( sums @ A @ F3 @ D3 ) ) ) ) ).
% sums_mult2_iff
thf(fact_2751_le__floor__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z: int,X: A] :
( ( ord_less_eq @ int @ Z @ ( archim6421214686448440834_floor @ A @ X ) )
= ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Z ) @ X ) ) ) ).
% le_floor_iff
thf(fact_2752_floor__less__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Z: int] :
( ( ord_less @ int @ ( archim6421214686448440834_floor @ A @ X ) @ Z )
= ( ord_less @ A @ X @ ( ring_1_of_int @ A @ Z ) ) ) ) ).
% floor_less_iff
thf(fact_2753_powr__mono2_H,axiom,
! [A2: real,X: real,Y: real] :
( ( ord_less_eq @ real @ A2 @ ( zero_zero @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ X @ Y )
=> ( ord_less_eq @ real @ ( powr @ real @ Y @ A2 ) @ ( powr @ real @ X @ A2 ) ) ) ) ) ).
% powr_mono2'
thf(fact_2754_powr__less__mono2,axiom,
! [A2: real,X: real,Y: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ X @ Y )
=> ( ord_less @ real @ ( powr @ real @ X @ A2 ) @ ( powr @ real @ Y @ A2 ) ) ) ) ) ).
% powr_less_mono2
thf(fact_2755_le__floor__add,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Y: A] : ( ord_less_eq @ int @ ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( archim6421214686448440834_floor @ A @ Y ) ) @ ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X @ Y ) ) ) ) ).
% le_floor_add
thf(fact_2756_powr__inj,axiom,
! [A2: real,X: real,Y: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
=> ( ( A2
!= ( one_one @ real ) )
=> ( ( ( powr @ real @ A2 @ X )
= ( powr @ real @ A2 @ Y ) )
= ( X = Y ) ) ) ) ).
% powr_inj
thf(fact_2757_gr__one__powr,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ ( one_one @ real ) @ X )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ Y )
=> ( ord_less @ real @ ( one_one @ real ) @ ( powr @ real @ X @ Y ) ) ) ) ).
% gr_one_powr
thf(fact_2758_int__add__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z: int,X: A] :
( ( plus_plus @ int @ Z @ ( archim6421214686448440834_floor @ A @ X ) )
= ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ ( ring_1_of_int @ A @ Z ) @ X ) ) ) ) ).
% int_add_floor
thf(fact_2759_floor__add__int,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Z: int] :
( ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X ) @ Z )
= ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X @ ( ring_1_of_int @ A @ Z ) ) ) ) ) ).
% floor_add_int
thf(fact_2760_sums__mult__D,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [C3: A,F3: nat > A,A2: A] :
( ( sums @ A
@ ^ [N2: nat] : ( times_times @ A @ C3 @ ( F3 @ N2 ) )
@ A2 )
=> ( ( C3
!= ( zero_zero @ A ) )
=> ( sums @ A @ F3 @ ( divide_divide @ A @ A2 @ C3 ) ) ) ) ) ).
% sums_mult_D
thf(fact_2761_sums__Suc__imp,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: nat > A,S: A] :
( ( ( F3 @ ( zero_zero @ nat ) )
= ( zero_zero @ A ) )
=> ( ( sums @ A
@ ^ [N2: nat] : ( F3 @ ( suc @ N2 ) )
@ S )
=> ( sums @ A @ F3 @ S ) ) ) ) ).
% sums_Suc_imp
thf(fact_2762_sums__Suc__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: nat > A,S: A] :
( ( sums @ A
@ ^ [N2: nat] : ( F3 @ ( suc @ N2 ) )
@ S )
= ( sums @ A @ F3 @ ( plus_plus @ A @ S @ ( F3 @ ( zero_zero @ nat ) ) ) ) ) ) ).
% sums_Suc_iff
thf(fact_2763_sums__Suc,axiom,
! [A: $tType] :
( ( ( topolo5987344860129210374id_add @ A )
& ( topological_t2_space @ A ) )
=> ! [F3: nat > A,L: A] :
( ( sums @ A
@ ^ [N2: nat] : ( F3 @ ( suc @ N2 ) )
@ L )
=> ( sums @ A @ F3 @ ( plus_plus @ A @ L @ ( F3 @ ( zero_zero @ nat ) ) ) ) ) ) ).
% sums_Suc
thf(fact_2764_sums__zero__iff__shift,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [N: nat,F3: nat > A,S: A] :
( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ N )
=> ( ( F3 @ I3 )
= ( zero_zero @ A ) ) )
=> ( ( sums @ A
@ ^ [I4: nat] : ( F3 @ ( plus_plus @ nat @ I4 @ N ) )
@ S )
= ( sums @ A @ F3 @ S ) ) ) ) ).
% sums_zero_iff_shift
thf(fact_2765_powr__add,axiom,
! [A: $tType] :
( ( ( real_V3459762299906320749_field @ A )
& ( ln @ A ) )
=> ! [X: A,A2: A,B2: A] :
( ( powr @ A @ X @ ( plus_plus @ A @ A2 @ B2 ) )
= ( times_times @ A @ ( powr @ A @ X @ A2 ) @ ( powr @ A @ X @ B2 ) ) ) ) ).
% powr_add
thf(fact_2766_sums__finite,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [N6: set @ nat,F3: nat > A] :
( ( finite_finite2 @ nat @ N6 )
=> ( ! [N3: nat] :
( ~ ( member @ nat @ N3 @ N6 )
=> ( ( F3 @ N3 )
= ( zero_zero @ A ) ) )
=> ( sums @ A @ F3 @ ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ N6 ) ) ) ) ) ).
% sums_finite
thf(fact_2767_sums__If__finite,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [P2: nat > $o,F3: nat > A] :
( ( finite_finite2 @ nat @ ( collect @ nat @ P2 ) )
=> ( sums @ A
@ ^ [R5: nat] : ( if @ A @ ( P2 @ R5 ) @ ( F3 @ R5 ) @ ( zero_zero @ A ) )
@ ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( collect @ nat @ P2 ) ) ) ) ) ).
% sums_If_finite
thf(fact_2768_sums__If__finite__set,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [A5: set @ nat,F3: nat > A] :
( ( finite_finite2 @ nat @ A5 )
=> ( sums @ A
@ ^ [R5: nat] : ( if @ A @ ( member @ nat @ R5 @ A5 ) @ ( F3 @ R5 ) @ ( zero_zero @ A ) )
@ ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ A5 ) ) ) ) ).
% sums_If_finite_set
thf(fact_2769_prod__int__eq,axiom,
! [I2: nat,J: nat] :
( ( groups7121269368397514597t_prod @ nat @ int @ ( semiring_1_of_nat @ int ) @ ( set_or1337092689740270186AtMost @ nat @ I2 @ J ) )
= ( groups7121269368397514597t_prod @ int @ int
@ ^ [X5: int] : X5
@ ( set_or1337092689740270186AtMost @ int @ ( semiring_1_of_nat @ int @ I2 ) @ ( semiring_1_of_nat @ int @ J ) ) ) ) ).
% prod_int_eq
thf(fact_2770_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_2771_floor__log__eq__powr__iff,axiom,
! [X: real,B2: real,K2: int] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( one_one @ real ) @ B2 )
=> ( ( ( archim6421214686448440834_floor @ real @ ( log @ B2 @ X ) )
= K2 )
= ( ( ord_less_eq @ real @ ( powr @ real @ B2 @ ( ring_1_of_int @ real @ K2 ) ) @ X )
& ( ord_less @ real @ X @ ( powr @ real @ B2 @ ( ring_1_of_int @ real @ ( plus_plus @ int @ K2 @ ( one_one @ int ) ) ) ) ) ) ) ) ) ).
% floor_log_eq_powr_iff
thf(fact_2772_powser__sums__if,axiom,
! [A: $tType] :
( ( ( ring_1 @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [M2: nat,Z: A] :
( sums @ A
@ ^ [N2: nat] : ( times_times @ A @ ( if @ A @ ( N2 = M2 ) @ ( one_one @ A ) @ ( zero_zero @ A ) ) @ ( power_power @ A @ Z @ N2 ) )
@ ( power_power @ A @ Z @ M2 ) ) ) ).
% powser_sums_if
thf(fact_2773_powser__sums__zero,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [A2: nat > A] :
( sums @ A
@ ^ [N2: nat] : ( times_times @ A @ ( A2 @ N2 ) @ ( power_power @ A @ ( zero_zero @ A ) @ N2 ) )
@ ( A2 @ ( zero_zero @ nat ) ) ) ) ).
% powser_sums_zero
thf(fact_2774_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_2775_powr__less__iff,axiom,
! [B2: real,X: real,Y: real] :
( ( ord_less @ real @ ( one_one @ real ) @ B2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( powr @ real @ B2 @ Y ) @ X )
= ( ord_less @ real @ Y @ ( log @ B2 @ X ) ) ) ) ) ).
% powr_less_iff
thf(fact_2776_less__powr__iff,axiom,
! [B2: real,X: real,Y: real] :
( ( ord_less @ real @ ( one_one @ real ) @ B2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ X @ ( powr @ real @ B2 @ Y ) )
= ( ord_less @ real @ ( log @ B2 @ X ) @ Y ) ) ) ) ).
% less_powr_iff
thf(fact_2777_log__less__iff,axiom,
! [B2: real,X: real,Y: real] :
( ( ord_less @ real @ ( one_one @ real ) @ B2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ ( log @ B2 @ X ) @ Y )
= ( ord_less @ real @ X @ ( powr @ real @ B2 @ Y ) ) ) ) ) ).
% log_less_iff
thf(fact_2778_less__log__iff,axiom,
! [B2: real,X: real,Y: real] :
( ( ord_less @ real @ ( one_one @ real ) @ B2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ Y @ ( log @ B2 @ X ) )
= ( ord_less @ real @ ( powr @ real @ B2 @ Y ) @ X ) ) ) ) ).
% less_log_iff
thf(fact_2779_sums__iff__shift,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: nat > A,N: nat,S: A] :
( ( sums @ A
@ ^ [I4: nat] : ( F3 @ ( plus_plus @ nat @ I4 @ N ) )
@ S )
= ( sums @ A @ F3 @ ( plus_plus @ A @ S @ ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ) ).
% sums_iff_shift
thf(fact_2780_sums__split__initial__segment,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: nat > A,S: A,N: nat] :
( ( sums @ A @ F3 @ S )
=> ( sums @ A
@ ^ [I4: nat] : ( F3 @ ( plus_plus @ nat @ I4 @ N ) )
@ ( minus_minus @ A @ S @ ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ) ).
% sums_split_initial_segment
thf(fact_2781_sums__iff__shift_H,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: nat > A,N: nat,S: A] :
( ( sums @ A
@ ^ [I4: nat] : ( F3 @ ( plus_plus @ nat @ I4 @ N ) )
@ ( minus_minus @ A @ S @ ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_ord_lessThan @ nat @ N ) ) ) )
= ( sums @ A @ F3 @ S ) ) ) ).
% sums_iff_shift'
thf(fact_2782_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_2783_real__of__int__floor__add__one__gt,axiom,
! [R2: real] : ( ord_less @ real @ R2 @ ( plus_plus @ real @ ( ring_1_of_int @ real @ ( archim6421214686448440834_floor @ real @ R2 ) ) @ ( one_one @ real ) ) ) ).
% real_of_int_floor_add_one_gt
thf(fact_2784_sums__If__finite__set_H,axiom,
! [A: $tType] :
( ( ( topolo1287966508704411220up_add @ A )
& ( topological_t2_space @ A ) )
=> ! [G3: nat > A,S2: A,A5: set @ nat,S5: A,F3: nat > A] :
( ( sums @ A @ G3 @ S2 )
=> ( ( finite_finite2 @ nat @ A5 )
=> ( ( S5
= ( plus_plus @ A @ S2
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [N2: nat] : ( minus_minus @ A @ ( F3 @ N2 ) @ ( G3 @ N2 ) )
@ A5 ) ) )
=> ( sums @ A
@ ^ [N2: nat] : ( if @ A @ ( member @ nat @ N2 @ A5 ) @ ( F3 @ N2 ) @ ( G3 @ N2 ) )
@ S5 ) ) ) ) ) ).
% sums_If_finite_set'
thf(fact_2785_real__of__int__floor__gt__diff__one,axiom,
! [R2: real] : ( ord_less @ real @ ( minus_minus @ real @ R2 @ ( one_one @ real ) ) @ ( ring_1_of_int @ real @ ( archim6421214686448440834_floor @ real @ R2 ) ) ) ).
% real_of_int_floor_gt_diff_one
thf(fact_2786_prod__int__plus__eq,axiom,
! [I2: nat,J: nat] :
( ( groups7121269368397514597t_prod @ nat @ int @ ( semiring_1_of_nat @ int ) @ ( set_or1337092689740270186AtMost @ nat @ I2 @ ( plus_plus @ nat @ I2 @ J ) ) )
= ( groups7121269368397514597t_prod @ int @ int
@ ^ [X5: int] : X5
@ ( set_or1337092689740270186AtMost @ int @ ( semiring_1_of_nat @ int @ I2 ) @ ( semiring_1_of_nat @ int @ ( plus_plus @ nat @ I2 @ J ) ) ) ) ) ).
% prod_int_plus_eq
thf(fact_2787_floor__unique,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z: int,X: A] :
( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ Z ) @ X )
=> ( ( ord_less @ A @ X @ ( plus_plus @ A @ ( ring_1_of_int @ A @ Z ) @ ( one_one @ A ) ) )
=> ( ( archim6421214686448440834_floor @ A @ X )
= Z ) ) ) ) ).
% floor_unique
thf(fact_2788_floor__eq__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,A2: int] :
( ( ( archim6421214686448440834_floor @ A @ X )
= A2 )
= ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ A2 ) @ X )
& ( ord_less @ A @ X @ ( plus_plus @ A @ ( ring_1_of_int @ A @ A2 ) @ ( one_one @ A ) ) ) ) ) ) ).
% floor_eq_iff
thf(fact_2789_floor__split,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [P2: int > $o,T2: A] :
( ( P2 @ ( archim6421214686448440834_floor @ A @ T2 ) )
= ( ! [I4: int] :
( ( ( ord_less_eq @ A @ ( ring_1_of_int @ A @ I4 ) @ T2 )
& ( ord_less @ A @ T2 @ ( plus_plus @ A @ ( ring_1_of_int @ A @ I4 ) @ ( one_one @ A ) ) ) )
=> ( P2 @ I4 ) ) ) ) ) ).
% floor_split
thf(fact_2790_le__mult__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ord_less_eq @ int @ ( times_times @ int @ ( archim6421214686448440834_floor @ A @ A2 ) @ ( archim6421214686448440834_floor @ A @ B2 ) ) @ ( archim6421214686448440834_floor @ A @ ( times_times @ A @ A2 @ B2 ) ) ) ) ) ) ).
% le_mult_floor
thf(fact_2791_less__floor__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z: int,X: A] :
( ( ord_less @ int @ Z @ ( archim6421214686448440834_floor @ A @ X ) )
= ( ord_less_eq @ A @ ( plus_plus @ A @ ( ring_1_of_int @ A @ Z ) @ ( one_one @ A ) ) @ X ) ) ) ).
% less_floor_iff
thf(fact_2792_floor__le__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Z: int] :
( ( ord_less_eq @ int @ ( archim6421214686448440834_floor @ A @ X ) @ Z )
= ( ord_less @ A @ X @ ( plus_plus @ A @ ( ring_1_of_int @ A @ Z ) @ ( one_one @ A ) ) ) ) ) ).
% floor_le_iff
thf(fact_2793_binomial__maximum_H,axiom,
! [N: nat,K2: nat] : ( ord_less_eq @ nat @ ( binomial @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ K2 ) @ ( binomial @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ N ) ) ).
% binomial_maximum'
thf(fact_2794_binomial__mono,axiom,
! [K2: nat,K7: nat,N: nat] :
( ( ord_less_eq @ nat @ K2 @ K7 )
=> ( ( ord_less_eq @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K7 ) @ N )
=> ( ord_less_eq @ nat @ ( binomial @ N @ K2 ) @ ( binomial @ N @ K7 ) ) ) ) ).
% binomial_mono
thf(fact_2795_binomial__antimono,axiom,
! [K2: nat,K7: nat,N: nat] :
( ( ord_less_eq @ nat @ K2 @ K7 )
=> ( ( ord_less_eq @ nat @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ K2 )
=> ( ( ord_less_eq @ nat @ K7 @ N )
=> ( ord_less_eq @ nat @ ( binomial @ N @ K7 ) @ ( binomial @ N @ K2 ) ) ) ) ) ).
% binomial_antimono
thf(fact_2796_binomial__maximum,axiom,
! [N: nat,K2: nat] : ( ord_less_eq @ nat @ ( binomial @ N @ K2 ) @ ( binomial @ N @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ).
% binomial_maximum
thf(fact_2797_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_2798_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_2799_le__log__iff,axiom,
! [B2: real,X: real,Y: real] :
( ( ord_less @ real @ ( one_one @ real ) @ B2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ Y @ ( log @ B2 @ X ) )
= ( ord_less_eq @ real @ ( powr @ real @ B2 @ Y ) @ X ) ) ) ) ).
% le_log_iff
thf(fact_2800_log__le__iff,axiom,
! [B2: real,X: real,Y: real] :
( ( ord_less @ real @ ( one_one @ real ) @ B2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( log @ B2 @ X ) @ Y )
= ( ord_less_eq @ real @ X @ ( powr @ real @ B2 @ Y ) ) ) ) ) ).
% log_le_iff
thf(fact_2801_le__powr__iff,axiom,
! [B2: real,X: real,Y: real] :
( ( ord_less @ real @ ( one_one @ real ) @ B2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ X @ ( powr @ real @ B2 @ Y ) )
= ( ord_less_eq @ real @ ( log @ B2 @ X ) @ Y ) ) ) ) ).
% le_powr_iff
thf(fact_2802_powr__le__iff,axiom,
! [B2: real,X: real,Y: real] :
( ( ord_less @ real @ ( one_one @ real ) @ B2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( powr @ real @ B2 @ Y ) @ X )
= ( ord_less_eq @ real @ Y @ ( log @ B2 @ X ) ) ) ) ) ).
% powr_le_iff
thf(fact_2803_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_2804_ln__prod,axiom,
! [A: $tType,I6: set @ A,F3: A > real] :
( ( finite_finite2 @ A @ I6 )
=> ( ! [I3: A] :
( ( member @ A @ I3 @ I6 )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( F3 @ I3 ) ) )
=> ( ( ln_ln @ real @ ( groups7121269368397514597t_prod @ A @ real @ F3 @ I6 ) )
= ( groups7311177749621191930dd_sum @ A @ real
@ ^ [X5: A] : ( ln_ln @ real @ ( F3 @ X5 ) )
@ I6 ) ) ) ) ).
% ln_prod
thf(fact_2805_floor__divide__lower,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Q2: A,P: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Q2 )
=> ( ord_less_eq @ A @ ( times_times @ A @ ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ A @ ( divide_divide @ A @ P @ Q2 ) ) ) @ Q2 ) @ P ) ) ) ).
% floor_divide_lower
thf(fact_2806_binomial__less__binomial__Suc,axiom,
! [K2: nat,N: nat] :
( ( ord_less @ nat @ K2 @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
=> ( ord_less @ nat @ ( binomial @ N @ K2 ) @ ( binomial @ N @ ( suc @ K2 ) ) ) ) ).
% binomial_less_binomial_Suc
thf(fact_2807_binomial__strict__mono,axiom,
! [K2: nat,K7: nat,N: nat] :
( ( ord_less @ nat @ K2 @ K7 )
=> ( ( ord_less_eq @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K7 ) @ N )
=> ( ord_less @ nat @ ( binomial @ N @ K2 ) @ ( binomial @ N @ K7 ) ) ) ) ).
% binomial_strict_mono
thf(fact_2808_binomial__strict__antimono,axiom,
! [K2: nat,K7: nat,N: nat] :
( ( ord_less @ nat @ K2 @ K7 )
=> ( ( ord_less_eq @ nat @ N @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K2 ) )
=> ( ( ord_less_eq @ nat @ K7 @ N )
=> ( ord_less @ nat @ ( binomial @ N @ K7 ) @ ( binomial @ N @ K2 ) ) ) ) ) ).
% binomial_strict_antimono
thf(fact_2809_ln__powr__bound,axiom,
! [X: real,A2: real] :
( ( ord_less_eq @ real @ ( one_one @ real ) @ X )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
=> ( ord_less_eq @ real @ ( ln_ln @ real @ X ) @ ( divide_divide @ real @ ( powr @ real @ X @ A2 ) @ A2 ) ) ) ) ).
% ln_powr_bound
thf(fact_2810_ln__powr__bound2,axiom,
! [X: real,A2: real] :
( ( ord_less @ real @ ( one_one @ real ) @ X )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
=> ( ord_less_eq @ real @ ( powr @ real @ ( ln_ln @ real @ X ) @ A2 ) @ ( times_times @ real @ ( powr @ real @ A2 @ A2 ) @ X ) ) ) ) ).
% ln_powr_bound2
thf(fact_2811_log__add__eq__powr,axiom,
! [B2: real,X: real,Y: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ B2 )
=> ( ( B2
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( plus_plus @ real @ ( log @ B2 @ X ) @ Y )
= ( log @ B2 @ ( times_times @ real @ X @ ( powr @ real @ B2 @ Y ) ) ) ) ) ) ) ).
% log_add_eq_powr
thf(fact_2812_add__log__eq__powr,axiom,
! [B2: real,X: real,Y: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ B2 )
=> ( ( B2
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( plus_plus @ real @ Y @ ( log @ B2 @ X ) )
= ( log @ B2 @ ( times_times @ real @ ( powr @ real @ B2 @ Y ) @ X ) ) ) ) ) ) ).
% add_log_eq_powr
thf(fact_2813_minus__log__eq__powr,axiom,
! [B2: real,X: real,Y: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ B2 )
=> ( ( B2
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( minus_minus @ real @ Y @ ( log @ B2 @ X ) )
= ( log @ B2 @ ( divide_divide @ real @ ( powr @ real @ B2 @ Y ) @ X ) ) ) ) ) ) ).
% minus_log_eq_powr
thf(fact_2814_floor__divide__upper,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Q2: A,P: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Q2 )
=> ( ord_less @ A @ P @ ( times_times @ A @ ( plus_plus @ A @ ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ A @ ( divide_divide @ A @ P @ Q2 ) ) ) @ ( one_one @ A ) ) @ Q2 ) ) ) ) ).
% floor_divide_upper
thf(fact_2815_geometric__sums,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [C3: A] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ C3 ) @ ( one_one @ real ) )
=> ( sums @ A @ ( power_power @ A @ C3 ) @ ( divide_divide @ A @ ( one_one @ A ) @ ( minus_minus @ A @ ( one_one @ A ) @ C3 ) ) ) ) ) ).
% geometric_sums
thf(fact_2816_round__def,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archimedean_round @ A )
= ( ^ [X5: A] : ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X5 @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% round_def
thf(fact_2817_log__minus__eq__powr,axiom,
! [B2: real,X: real,Y: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ B2 )
=> ( ( B2
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( minus_minus @ real @ ( log @ B2 @ X ) @ Y )
= ( log @ B2 @ ( times_times @ real @ X @ ( powr @ real @ B2 @ ( uminus_uminus @ real @ Y ) ) ) ) ) ) ) ) ).
% log_minus_eq_powr
thf(fact_2818_and__int_Opinduct,axiom,
! [A0: int,A1: int,P2: int > int > $o] :
( ( accp @ ( product_prod @ int @ int ) @ bit_and_int_rel @ ( product_Pair @ int @ int @ A0 @ A1 ) )
=> ( ! [K: int,L4: int] :
( ( accp @ ( product_prod @ int @ int ) @ bit_and_int_rel @ ( product_Pair @ int @ int @ K @ L4 ) )
=> ( ( ~ ( ( member @ int @ K @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) )
& ( member @ int @ L4 @ ( insert @ int @ ( zero_zero @ int ) @ ( insert @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( bot_bot @ ( set @ int ) ) ) ) ) )
=> ( P2 @ ( divide_divide @ int @ K @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ int @ L4 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) ) ) )
=> ( P2 @ K @ L4 ) ) )
=> ( P2 @ A0 @ A1 ) ) ) ).
% and_int.pinduct
thf(fact_2819_and__nat__unfold,axiom,
( ( bit_se5824344872417868541ns_and @ nat )
= ( ^ [M3: nat,N2: nat] :
( if @ nat
@ ( ( M3
= ( zero_zero @ nat ) )
| ( N2
= ( zero_zero @ nat ) ) )
@ ( zero_zero @ nat )
@ ( plus_plus @ nat @ ( times_times @ nat @ ( modulo_modulo @ nat @ M3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ nat @ ( divide_divide @ nat @ M3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ).
% and_nat_unfold
thf(fact_2820_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_2821_and__nat__rec,axiom,
( ( bit_se5824344872417868541ns_and @ nat )
= ( ^ [M3: nat,N2: nat] :
( plus_plus @ nat
@ ( zero_neq_one_of_bool @ nat
@ ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M3 )
& ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) )
@ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( bit_se5824344872417868541ns_and @ nat @ ( divide_divide @ nat @ M3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% and_nat_rec
thf(fact_2822_floor__log2__div2,axiom,
! [N: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( archim6421214686448440834_floor @ real @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ N ) ) )
= ( plus_plus @ int @ ( archim6421214686448440834_floor @ real @ ( log @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ ( semiring_1_of_nat @ real @ ( divide_divide @ nat @ N @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) @ ( one_one @ int ) ) ) ) ).
% floor_log2_div2
thf(fact_2823_floor__log__nat__eq__if,axiom,
! [B2: nat,N: nat,K2: nat] :
( ( ord_less_eq @ nat @ ( power_power @ nat @ B2 @ N ) @ K2 )
=> ( ( ord_less @ nat @ K2 @ ( power_power @ nat @ B2 @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) ) )
=> ( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ B2 )
=> ( ( archim6421214686448440834_floor @ real @ ( log @ ( semiring_1_of_nat @ real @ B2 ) @ ( semiring_1_of_nat @ real @ K2 ) ) )
= ( semiring_1_of_nat @ int @ N ) ) ) ) ) ).
% floor_log_nat_eq_if
thf(fact_2824_zero__less__binomial__iff,axiom,
! [N: nat,K2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( binomial @ N @ K2 ) )
= ( ord_less_eq @ nat @ K2 @ N ) ) ).
% zero_less_binomial_iff
thf(fact_2825_binomial__n__0,axiom,
! [N: nat] :
( ( binomial @ N @ ( zero_zero @ nat ) )
= ( one_one @ nat ) ) ).
% binomial_n_0
thf(fact_2826_binomial__Suc__Suc,axiom,
! [N: nat,K2: nat] :
( ( binomial @ ( suc @ N ) @ ( suc @ K2 ) )
= ( plus_plus @ nat @ ( binomial @ N @ K2 ) @ ( binomial @ N @ ( suc @ K2 ) ) ) ) ).
% binomial_Suc_Suc
thf(fact_2827_binomial__eq__0__iff,axiom,
! [N: nat,K2: nat] :
( ( ( binomial @ N @ K2 )
= ( zero_zero @ nat ) )
= ( ord_less @ nat @ N @ K2 ) ) ).
% binomial_eq_0_iff
thf(fact_2828_binomial__0__Suc,axiom,
! [K2: nat] :
( ( binomial @ ( zero_zero @ nat ) @ ( suc @ K2 ) )
= ( zero_zero @ nat ) ) ).
% binomial_0_Suc
thf(fact_2829_binomial__1,axiom,
! [N: nat] :
( ( binomial @ N @ ( suc @ ( zero_zero @ nat ) ) )
= N ) ).
% binomial_1
thf(fact_2830_binomial__eq__0,axiom,
! [N: nat,K2: nat] :
( ( ord_less @ nat @ N @ K2 )
=> ( ( binomial @ N @ K2 )
= ( zero_zero @ nat ) ) ) ).
% binomial_eq_0
thf(fact_2831_binomial__symmetric,axiom,
! [K2: nat,N: nat] :
( ( ord_less_eq @ nat @ K2 @ N )
=> ( ( binomial @ N @ K2 )
= ( binomial @ N @ ( minus_minus @ nat @ N @ K2 ) ) ) ) ).
% binomial_symmetric
thf(fact_2832_choose__mult__lemma,axiom,
! [M2: nat,R2: nat,K2: nat] :
( ( times_times @ nat @ ( binomial @ ( plus_plus @ nat @ ( plus_plus @ nat @ M2 @ R2 ) @ K2 ) @ ( plus_plus @ nat @ M2 @ K2 ) ) @ ( binomial @ ( plus_plus @ nat @ M2 @ K2 ) @ K2 ) )
= ( times_times @ nat @ ( binomial @ ( plus_plus @ nat @ ( plus_plus @ nat @ M2 @ R2 ) @ K2 ) @ K2 ) @ ( binomial @ ( plus_plus @ nat @ M2 @ R2 ) @ M2 ) ) ) ).
% choose_mult_lemma
thf(fact_2833_binomial__le__pow,axiom,
! [R2: nat,N: nat] :
( ( ord_less_eq @ nat @ R2 @ N )
=> ( ord_less_eq @ nat @ ( binomial @ N @ R2 ) @ ( power_power @ nat @ N @ R2 ) ) ) ).
% binomial_le_pow
thf(fact_2834_zero__less__binomial,axiom,
! [K2: nat,N: nat] :
( ( ord_less_eq @ nat @ K2 @ N )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ ( binomial @ N @ K2 ) ) ) ).
% zero_less_binomial
thf(fact_2835_Suc__times__binomial__add,axiom,
! [A2: nat,B2: nat] :
( ( times_times @ nat @ ( suc @ A2 ) @ ( binomial @ ( suc @ ( plus_plus @ nat @ A2 @ B2 ) ) @ ( suc @ A2 ) ) )
= ( times_times @ nat @ ( suc @ B2 ) @ ( binomial @ ( suc @ ( plus_plus @ nat @ A2 @ B2 ) ) @ A2 ) ) ) ).
% Suc_times_binomial_add
thf(fact_2836_choose__mult,axiom,
! [K2: nat,M2: nat,N: nat] :
( ( ord_less_eq @ nat @ K2 @ M2 )
=> ( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( times_times @ nat @ ( binomial @ N @ M2 ) @ ( binomial @ M2 @ K2 ) )
= ( times_times @ nat @ ( binomial @ N @ K2 ) @ ( binomial @ ( minus_minus @ nat @ N @ K2 ) @ ( minus_minus @ nat @ M2 @ K2 ) ) ) ) ) ) ).
% choose_mult
thf(fact_2837_binomial__ge__n__over__k__pow__k,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [K2: nat,N: nat] :
( ( ord_less_eq @ nat @ K2 @ N )
=> ( ord_less_eq @ A @ ( power_power @ A @ ( divide_divide @ A @ ( semiring_1_of_nat @ A @ N ) @ ( semiring_1_of_nat @ A @ K2 ) ) @ K2 ) @ ( semiring_1_of_nat @ A @ ( binomial @ N @ K2 ) ) ) ) ) ).
% binomial_ge_n_over_k_pow_k
thf(fact_2838_binomial__le__pow2,axiom,
! [N: nat,K2: nat] : ( ord_less_eq @ nat @ ( binomial @ N @ K2 ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ).
% binomial_le_pow2
thf(fact_2839_choose__reduce__nat,axiom,
! [N: nat,K2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K2 )
=> ( ( binomial @ N @ K2 )
= ( plus_plus @ nat @ ( binomial @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ ( minus_minus @ nat @ K2 @ ( one_one @ nat ) ) ) @ ( binomial @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ K2 ) ) ) ) ) ).
% choose_reduce_nat
thf(fact_2840_times__binomial__minus1__eq,axiom,
! [K2: nat,N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K2 )
=> ( ( times_times @ nat @ K2 @ ( binomial @ N @ K2 ) )
= ( times_times @ nat @ N @ ( binomial @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ ( minus_minus @ nat @ K2 @ ( one_one @ nat ) ) ) ) ) ) ).
% times_binomial_minus1_eq
thf(fact_2841_binomial__addition__formula,axiom,
! [N: nat,K2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( binomial @ N @ ( suc @ K2 ) )
= ( plus_plus @ nat @ ( binomial @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ ( suc @ K2 ) ) @ ( binomial @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ K2 ) ) ) ) ).
% binomial_addition_formula
thf(fact_2842_upto_Opinduct,axiom,
! [A0: int,A1: int,P2: int > int > $o] :
( ( accp @ ( product_prod @ int @ int ) @ upto_rel @ ( product_Pair @ int @ int @ A0 @ A1 ) )
=> ( ! [I3: int,J2: int] :
( ( accp @ ( product_prod @ int @ int ) @ upto_rel @ ( product_Pair @ int @ int @ I3 @ J2 ) )
=> ( ( ( ord_less_eq @ int @ I3 @ J2 )
=> ( P2 @ ( plus_plus @ int @ I3 @ ( one_one @ int ) ) @ J2 ) )
=> ( P2 @ I3 @ J2 ) ) )
=> ( P2 @ A0 @ A1 ) ) ) ).
% upto.pinduct
thf(fact_2843_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
@ ^ [I4: nat] :
( if @ A
@ ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I4 )
@ ( semiring_1_of_nat @ A @ ( binomial @ N @ I4 ) )
@ ( zero_zero @ A ) )
@ ( set_ord_atMost @ nat @ N ) ) )
= ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% choose_odd_sum
thf(fact_2844_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
@ ^ [I4: nat] : ( if @ A @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I4 ) @ ( semiring_1_of_nat @ A @ ( binomial @ N @ I4 ) ) @ ( zero_zero @ A ) )
@ ( set_ord_atMost @ nat @ N ) ) )
= ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) ) ) ) ).
% choose_even_sum
thf(fact_2845_arcosh__def,axiom,
! [A: $tType] :
( ( ln @ A )
=> ( ( arcosh @ A )
= ( ^ [X5: A] : ( ln_ln @ A @ ( plus_plus @ A @ X5 @ ( powr @ A @ ( minus_minus @ A @ ( power_power @ A @ X5 @ ( 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_2846_round__altdef,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ( ( archimedean_round @ A )
= ( ^ [X5: A] : ( if @ int @ ( ord_less_eq @ A @ ( divide_divide @ A @ ( one_one @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( archimedean_frac @ A @ X5 ) ) @ ( archimedean_ceiling @ A @ X5 ) @ ( archim6421214686448440834_floor @ A @ X5 ) ) ) ) ) ).
% round_altdef
thf(fact_2847_atMost__eq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,Y: A] :
( ( ( set_ord_atMost @ A @ X )
= ( set_ord_atMost @ A @ Y ) )
= ( X = Y ) ) ) ).
% atMost_eq_iff
thf(fact_2848_atMost__iff,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [I2: A,K2: A] :
( ( member @ A @ I2 @ ( set_ord_atMost @ A @ K2 ) )
= ( ord_less_eq @ A @ I2 @ K2 ) ) ) ).
% atMost_iff
thf(fact_2849_finite__atMost,axiom,
! [K2: nat] : ( finite_finite2 @ nat @ ( set_ord_atMost @ nat @ K2 ) ) ).
% finite_atMost
thf(fact_2850_atMost__subset__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_ord_atMost @ A @ X ) @ ( set_ord_atMost @ A @ Y ) )
= ( ord_less_eq @ A @ X @ Y ) ) ) ).
% atMost_subset_iff
thf(fact_2851_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_2852_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_2853_of__real__add,axiom,
! [A: $tType] :
( ( real_V2191834092415804123ebra_1 @ A )
=> ! [X: real,Y: real] :
( ( real_Vector_of_real @ A @ ( plus_plus @ real @ X @ Y ) )
= ( plus_plus @ A @ ( real_Vector_of_real @ A @ X ) @ ( real_Vector_of_real @ A @ Y ) ) ) ) ).
% of_real_add
thf(fact_2854_frac__of__int,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z: int] :
( ( archimedean_frac @ A @ ( ring_1_of_int @ A @ Z ) )
= ( zero_zero @ A ) ) ) ).
% frac_of_int
thf(fact_2855_Icc__subset__Iic__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [L: A,H2: A,H3: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ L @ H2 ) @ ( set_ord_atMost @ A @ H3 ) )
= ( ~ ( ord_less_eq @ A @ L @ H2 )
| ( ord_less_eq @ A @ H2 @ H3 ) ) ) ) ).
% Icc_subset_Iic_iff
thf(fact_2856_sum_OatMost__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G3: nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_ord_atMost @ nat @ ( suc @ N ) ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_ord_atMost @ nat @ N ) ) @ ( G3 @ ( suc @ N ) ) ) ) ) ).
% sum.atMost_Suc
thf(fact_2857_prod_OatMost__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G3: nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_ord_atMost @ nat @ ( suc @ N ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_ord_atMost @ nat @ N ) ) @ ( G3 @ ( suc @ N ) ) ) ) ) ).
% prod.atMost_Suc
thf(fact_2858_atMost__0,axiom,
( ( set_ord_atMost @ nat @ ( zero_zero @ nat ) )
= ( insert @ nat @ ( zero_zero @ nat ) @ ( bot_bot @ ( set @ nat ) ) ) ) ).
% atMost_0
thf(fact_2859_not__empty__eq__Iic__eq__empty,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [H2: A] :
( ( bot_bot @ ( set @ A ) )
!= ( set_ord_atMost @ A @ H2 ) ) ) ).
% not_empty_eq_Iic_eq_empty
thf(fact_2860_infinite__Iic,axiom,
! [A: $tType] :
( ( ( linorder @ A )
& ( no_bot @ A ) )
=> ! [A2: A] :
~ ( finite_finite2 @ A @ ( set_ord_atMost @ A @ A2 ) ) ) ).
% infinite_Iic
thf(fact_2861_not__Iic__eq__Icc,axiom,
! [A: $tType] :
( ( no_bot @ A )
=> ! [H3: A,L: A,H2: A] :
( ( set_ord_atMost @ A @ H3 )
!= ( set_or1337092689740270186AtMost @ A @ L @ H2 ) ) ) ).
% not_Iic_eq_Icc
thf(fact_2862_atMost__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( set_ord_atMost @ A )
= ( ^ [U2: A] :
( collect @ A
@ ^ [X5: A] : ( ord_less_eq @ A @ X5 @ U2 ) ) ) ) ) ).
% atMost_def
thf(fact_2863_atMost__atLeast0,axiom,
( ( set_ord_atMost @ nat )
= ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) ) ) ).
% atMost_atLeast0
thf(fact_2864_lessThan__Suc__atMost,axiom,
! [K2: nat] :
( ( set_ord_lessThan @ nat @ ( suc @ K2 ) )
= ( set_ord_atMost @ nat @ K2 ) ) ).
% lessThan_Suc_atMost
thf(fact_2865_atMost__Suc,axiom,
! [K2: nat] :
( ( set_ord_atMost @ nat @ ( suc @ K2 ) )
= ( insert @ nat @ ( suc @ K2 ) @ ( set_ord_atMost @ nat @ K2 ) ) ) ).
% atMost_Suc
thf(fact_2866_not__Iic__le__Icc,axiom,
! [A: $tType] :
( ( no_bot @ A )
=> ! [H2: A,L3: A,H3: A] :
~ ( ord_less_eq @ ( set @ A ) @ ( set_ord_atMost @ A @ H2 ) @ ( set_or1337092689740270186AtMost @ A @ L3 @ H3 ) ) ) ).
% not_Iic_le_Icc
thf(fact_2867_finite__nat__iff__bounded__le,axiom,
( ( finite_finite2 @ nat )
= ( ^ [S7: set @ nat] :
? [K3: nat] : ( ord_less_eq @ ( set @ nat ) @ S7 @ ( set_ord_atMost @ nat @ K3 ) ) ) ) ).
% finite_nat_iff_bounded_le
thf(fact_2868_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_2869_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_2870_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_2871_atMost__nat__numeral,axiom,
! [K2: num] :
( ( set_ord_atMost @ nat @ ( numeral_numeral @ nat @ K2 ) )
= ( insert @ nat @ ( numeral_numeral @ nat @ K2 ) @ ( set_ord_atMost @ nat @ ( pred_numeral @ K2 ) ) ) ) ).
% atMost_nat_numeral
thf(fact_2872_Iic__subset__Iio__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_ord_atMost @ A @ A2 ) @ ( set_ord_lessThan @ A @ B2 ) )
= ( ord_less @ A @ A2 @ B2 ) ) ) ).
% Iic_subset_Iio_iff
thf(fact_2873_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_2874_sum_OatMost__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G3: nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_ord_atMost @ nat @ ( suc @ N ) ) )
= ( plus_plus @ A @ ( G3 @ ( zero_zero @ nat ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( G3 @ ( suc @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ) ).
% sum.atMost_Suc_shift
thf(fact_2875_sum__telescope,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [F3: nat > A,I2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( minus_minus @ A @ ( F3 @ I4 ) @ ( F3 @ ( suc @ I4 ) ) )
@ ( set_ord_atMost @ nat @ I2 ) )
= ( minus_minus @ A @ ( F3 @ ( zero_zero @ nat ) ) @ ( F3 @ ( suc @ I2 ) ) ) ) ) ).
% sum_telescope
thf(fact_2876_polyfun__eq__coeffs,axiom,
! [A: $tType] :
( ( ( real_V8999393235501362500lgebra @ A )
& ( idom @ A ) )
=> ! [C3: nat > A,N: nat,D3: nat > A] :
( ( ! [X5: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( C3 @ I4 ) @ ( power_power @ A @ X5 @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( D3 @ I4 ) @ ( power_power @ A @ X5 @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) ) ) )
= ( ! [I4: nat] :
( ( ord_less_eq @ nat @ I4 @ N )
=> ( ( C3 @ I4 )
= ( D3 @ I4 ) ) ) ) ) ) ).
% polyfun_eq_coeffs
thf(fact_2877_prod_OatMost__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G3: nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_ord_atMost @ nat @ ( suc @ N ) ) )
= ( times_times @ A @ ( G3 @ ( zero_zero @ nat ) )
@ ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( G3 @ ( suc @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ) ).
% prod.atMost_Suc_shift
thf(fact_2878_sum_Onested__swap_H,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A2: nat > nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( groups7311177749621191930dd_sum @ nat @ A @ ( A2 @ I4 ) @ ( set_ord_lessThan @ nat @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [J3: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( A2 @ I4 @ J3 )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ J3 ) @ N ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% sum.nested_swap'
thf(fact_2879_prod_Onested__swap_H,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A2: nat > nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( groups7121269368397514597t_prod @ nat @ A @ ( A2 @ I4 ) @ ( set_ord_lessThan @ nat @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [J3: nat] :
( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( A2 @ I4 @ J3 )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ J3 ) @ N ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% prod.nested_swap'
thf(fact_2880_sum__choose__lower,axiom,
! [R2: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [K3: nat] : ( binomial @ ( plus_plus @ nat @ R2 @ K3 ) @ K3 )
@ ( set_ord_atMost @ nat @ N ) )
= ( binomial @ ( suc @ ( plus_plus @ nat @ R2 @ N ) ) @ N ) ) ).
% sum_choose_lower
thf(fact_2881_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_2882_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_2883_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_2884_polyfun__eq__0,axiom,
! [A: $tType] :
( ( ( real_V8999393235501362500lgebra @ A )
& ( idom @ A ) )
=> ! [C3: nat > A,N: nat] :
( ( ! [X5: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( C3 @ I4 ) @ ( power_power @ A @ X5 @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( zero_zero @ A ) ) )
= ( ! [I4: nat] :
( ( ord_less_eq @ nat @ I4 @ N )
=> ( ( C3 @ I4 )
= ( zero_zero @ A ) ) ) ) ) ) ).
% polyfun_eq_0
thf(fact_2885_zero__polynom__imp__zero__coeffs,axiom,
! [A: $tType] :
( ( ( ab_semigroup_mult @ A )
& ( real_V8999393235501362500lgebra @ A ) )
=> ! [C3: nat > A,N: nat,K2: nat] :
( ! [W: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( C3 @ I4 ) @ ( power_power @ A @ W @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ nat @ K2 @ N )
=> ( ( C3 @ K2 )
= ( zero_zero @ A ) ) ) ) ) ).
% zero_polynom_imp_zero_coeffs
thf(fact_2886_frac__add,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Y: A] :
( ( ( ord_less @ A @ ( plus_plus @ A @ ( archimedean_frac @ A @ X ) @ ( archimedean_frac @ A @ Y ) ) @ ( one_one @ A ) )
=> ( ( archimedean_frac @ A @ ( plus_plus @ A @ X @ Y ) )
= ( plus_plus @ A @ ( archimedean_frac @ A @ X ) @ ( archimedean_frac @ A @ Y ) ) ) )
& ( ~ ( ord_less @ A @ ( plus_plus @ A @ ( archimedean_frac @ A @ X ) @ ( archimedean_frac @ A @ Y ) ) @ ( one_one @ A ) )
=> ( ( archimedean_frac @ A @ ( plus_plus @ A @ X @ Y ) )
= ( minus_minus @ A @ ( plus_plus @ A @ ( archimedean_frac @ A @ X ) @ ( archimedean_frac @ A @ Y ) ) @ ( one_one @ A ) ) ) ) ) ) ).
% frac_add
thf(fact_2887_sum_OatMost__shift,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G3: nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_ord_atMost @ nat @ N ) )
= ( plus_plus @ A @ ( G3 @ ( zero_zero @ nat ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( G3 @ ( suc @ I4 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% sum.atMost_shift
thf(fact_2888_sum__up__index__split,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [F3: nat > A,M2: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_ord_atMost @ nat @ ( plus_plus @ nat @ M2 @ N ) ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_ord_atMost @ nat @ M2 ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M2 ) @ ( plus_plus @ nat @ M2 @ N ) ) ) ) ) ) ).
% sum_up_index_split
thf(fact_2889_prod_OatMost__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G3: nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_ord_atMost @ nat @ N ) )
= ( times_times @ A @ ( G3 @ ( zero_zero @ nat ) )
@ ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( G3 @ ( suc @ I4 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ).
% prod.atMost_shift
thf(fact_2890_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_2891_sum_Otriangle__reindex__eq,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G3: nat > nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ ( product_prod @ nat @ nat ) @ A @ ( product_case_prod @ nat @ nat @ A @ G3 )
@ ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [I4: nat,J3: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ I4 @ J3 ) @ N ) ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( G3 @ I4 @ ( minus_minus @ nat @ K3 @ I4 ) )
@ ( set_ord_atMost @ nat @ K3 ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% sum.triangle_reindex_eq
thf(fact_2892_prod_Otriangle__reindex__eq,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G3: nat > nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ ( product_prod @ nat @ nat ) @ A @ ( product_case_prod @ nat @ nat @ A @ G3 )
@ ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [I4: nat,J3: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ I4 @ J3 ) @ N ) ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [K3: nat] :
( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( G3 @ I4 @ ( minus_minus @ nat @ K3 @ I4 ) )
@ ( set_ord_atMost @ nat @ K3 ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% prod.triangle_reindex_eq
thf(fact_2893_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_2894_vandermonde,axiom,
! [M2: nat,N: nat,R2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [K3: nat] : ( times_times @ nat @ ( binomial @ M2 @ K3 ) @ ( binomial @ N @ ( minus_minus @ nat @ R2 @ K3 ) ) )
@ ( set_ord_atMost @ nat @ R2 ) )
= ( binomial @ ( plus_plus @ nat @ M2 @ N ) @ R2 ) ) ).
% vandermonde
thf(fact_2895_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_2896_polyfun__finite__roots,axiom,
! [A: $tType] :
( ( ( real_V8999393235501362500lgebra @ A )
& ( idom @ A ) )
=> ! [C3: nat > A,N: nat] :
( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X5: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( C3 @ I4 ) @ ( power_power @ A @ X5 @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( zero_zero @ A ) ) ) )
= ( ? [I4: nat] :
( ( ord_less_eq @ nat @ I4 @ N )
& ( ( C3 @ I4 )
!= ( zero_zero @ A ) ) ) ) ) ) ).
% polyfun_finite_roots
thf(fact_2897_polyfun__roots__finite,axiom,
! [A: $tType] :
( ( ( real_V8999393235501362500lgebra @ A )
& ( idom @ A ) )
=> ! [C3: nat > A,K2: nat,N: nat] :
( ( ( C3 @ K2 )
!= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ nat @ K2 @ N )
=> ( finite_finite2 @ A
@ ( collect @ A
@ ^ [Z5: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( C3 @ I4 ) @ ( power_power @ A @ Z5 @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% polyfun_roots_finite
thf(fact_2898_polyfun__linear__factor__root,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [C3: nat > A,A2: A,N: nat] :
( ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( C3 @ I4 ) @ ( power_power @ A @ A2 @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( zero_zero @ A ) )
=> ~ ! [B4: nat > A] :
~ ! [Z4: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( C3 @ I4 ) @ ( power_power @ A @ Z4 @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( times_times @ A @ ( minus_minus @ A @ Z4 @ A2 )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( B4 @ I4 ) @ ( power_power @ A @ Z4 @ I4 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ) ).
% polyfun_linear_factor_root
thf(fact_2899_polyfun__linear__factor,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [C3: nat > A,N: nat,A2: A] :
? [B4: nat > A] :
! [Z4: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( C3 @ I4 ) @ ( power_power @ A @ Z4 @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( plus_plus @ A
@ ( times_times @ A @ ( minus_minus @ A @ Z4 @ A2 )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( B4 @ I4 ) @ ( power_power @ A @ Z4 @ I4 ) )
@ ( set_ord_lessThan @ nat @ N ) ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( C3 @ I4 ) @ ( power_power @ A @ A2 @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ) ).
% polyfun_linear_factor
thf(fact_2900_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_2901_sum_Otriangle__reindex,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G3: nat > nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ ( product_prod @ nat @ nat ) @ A @ ( product_case_prod @ nat @ nat @ A @ G3 )
@ ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [I4: nat,J3: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ I4 @ J3 ) @ N ) ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( G3 @ I4 @ ( minus_minus @ nat @ K3 @ I4 ) )
@ ( set_ord_atMost @ nat @ K3 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% sum.triangle_reindex
thf(fact_2902_prod_Otriangle__reindex,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G3: nat > nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ ( product_prod @ nat @ nat ) @ A @ ( product_case_prod @ nat @ nat @ A @ G3 )
@ ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [I4: nat,J3: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ I4 @ J3 ) @ N ) ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [K3: nat] :
( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( G3 @ I4 @ ( minus_minus @ nat @ K3 @ I4 ) )
@ ( set_ord_atMost @ nat @ K3 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% prod.triangle_reindex
thf(fact_2903_binomial,axiom,
! [A2: nat,B2: nat,N: nat] :
( ( power_power @ nat @ ( plus_plus @ nat @ A2 @ B2 ) @ N )
= ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [K3: nat] : ( times_times @ nat @ ( times_times @ nat @ ( semiring_1_of_nat @ nat @ ( binomial @ N @ K3 ) ) @ ( power_power @ nat @ A2 @ K3 ) ) @ ( power_power @ nat @ B2 @ ( minus_minus @ nat @ N @ K3 ) ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ).
% binomial
thf(fact_2904_sum_Oin__pairs__0,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G3: nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_ord_atMost @ nat @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( plus_plus @ A @ ( G3 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I4 ) ) @ ( G3 @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I4 ) ) ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% sum.in_pairs_0
thf(fact_2905_polynomial__product,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [M2: nat,A2: nat > A,N: nat,B2: nat > A,X: A] :
( ! [I3: nat] :
( ( ord_less @ nat @ M2 @ I3 )
=> ( ( A2 @ I3 )
= ( zero_zero @ A ) ) )
=> ( ! [J2: nat] :
( ( ord_less @ nat @ N @ J2 )
=> ( ( B2 @ J2 )
= ( zero_zero @ A ) ) )
=> ( ( times_times @ A
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( A2 @ I4 ) @ ( power_power @ A @ X @ I4 ) )
@ ( 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
@ ^ [R5: nat] :
( times_times @ A
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( times_times @ A @ ( A2 @ K3 ) @ ( B2 @ ( minus_minus @ nat @ R5 @ K3 ) ) )
@ ( set_ord_atMost @ nat @ R5 ) )
@ ( power_power @ A @ X @ R5 ) )
@ ( set_ord_atMost @ nat @ ( plus_plus @ nat @ M2 @ N ) ) ) ) ) ) ) ).
% polynomial_product
thf(fact_2906_prod_Oin__pairs__0,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G3: nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_ord_atMost @ nat @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( G3 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I4 ) ) @ ( G3 @ ( suc @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ I4 ) ) ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% prod.in_pairs_0
thf(fact_2907_polyfun__eq__const,axiom,
! [A: $tType] :
( ( ( real_V8999393235501362500lgebra @ A )
& ( idom @ A ) )
=> ! [C3: nat > A,N: nat,K2: A] :
( ( ! [X5: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( C3 @ I4 ) @ ( power_power @ A @ X5 @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) )
= K2 ) )
= ( ( ( C3 @ ( zero_zero @ nat ) )
= K2 )
& ! [X5: nat] :
( ( member @ nat @ X5 @ ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ N ) )
=> ( ( C3 @ X5 )
= ( zero_zero @ A ) ) ) ) ) ) ).
% polyfun_eq_const
thf(fact_2908_binomial__ring,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ! [A2: A,B2: A,N: nat] :
( ( power_power @ A @ ( plus_plus @ A @ A2 @ B2 ) @ N )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( times_times @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( binomial @ N @ K3 ) ) @ ( power_power @ A @ A2 @ K3 ) ) @ ( power_power @ A @ B2 @ ( minus_minus @ nat @ N @ K3 ) ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% binomial_ring
thf(fact_2909_pochhammer__binomial__sum,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [A2: A,B2: A,N: nat] :
( ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ A2 @ B2 ) @ N )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( times_times @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( binomial @ N @ K3 ) ) @ ( comm_s3205402744901411588hammer @ A @ A2 @ K3 ) ) @ ( comm_s3205402744901411588hammer @ A @ B2 @ ( minus_minus @ nat @ N @ K3 ) ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% pochhammer_binomial_sum
thf(fact_2910_polynomial__product__nat,axiom,
! [M2: nat,A2: nat > nat,N: nat,B2: nat > nat,X: nat] :
( ! [I3: nat] :
( ( ord_less @ nat @ M2 @ I3 )
=> ( ( A2 @ I3 )
= ( zero_zero @ nat ) ) )
=> ( ! [J2: nat] :
( ( ord_less @ nat @ N @ J2 )
=> ( ( B2 @ J2 )
= ( zero_zero @ nat ) ) )
=> ( ( times_times @ nat
@ ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [I4: nat] : ( times_times @ nat @ ( A2 @ I4 ) @ ( power_power @ nat @ X @ I4 ) )
@ ( 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
@ ^ [R5: nat] :
( times_times @ nat
@ ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [K3: nat] : ( times_times @ nat @ ( A2 @ K3 ) @ ( B2 @ ( minus_minus @ nat @ R5 @ K3 ) ) )
@ ( set_ord_atMost @ nat @ R5 ) )
@ ( power_power @ nat @ X @ R5 ) )
@ ( set_ord_atMost @ nat @ ( plus_plus @ nat @ M2 @ N ) ) ) ) ) ) ).
% polynomial_product_nat
thf(fact_2911_floor__add,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Y: A] :
( ( ( ord_less @ A @ ( plus_plus @ A @ ( archimedean_frac @ A @ X ) @ ( archimedean_frac @ A @ Y ) ) @ ( one_one @ A ) )
=> ( ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X @ Y ) )
= ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( archim6421214686448440834_floor @ A @ Y ) ) ) )
& ( ~ ( ord_less @ A @ ( plus_plus @ A @ ( archimedean_frac @ A @ X ) @ ( archimedean_frac @ A @ Y ) ) @ ( one_one @ A ) )
=> ( ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X @ Y ) )
= ( plus_plus @ int @ ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( archim6421214686448440834_floor @ A @ Y ) ) @ ( one_one @ int ) ) ) ) ) ) ).
% floor_add
thf(fact_2912_sum_Ozero__middle,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [P: nat,K2: nat,G3: nat > A,H2: nat > A] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ P )
=> ( ( ord_less_eq @ nat @ K2 @ P )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [J3: nat] : ( if @ A @ ( ord_less @ nat @ J3 @ K2 ) @ ( G3 @ J3 ) @ ( if @ A @ ( J3 = K2 ) @ ( zero_zero @ A ) @ ( H2 @ ( minus_minus @ nat @ J3 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) )
@ ( set_ord_atMost @ nat @ P ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [J3: nat] : ( if @ A @ ( ord_less @ nat @ J3 @ K2 ) @ ( G3 @ J3 ) @ ( H2 @ J3 ) )
@ ( set_ord_atMost @ nat @ ( minus_minus @ nat @ P @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ) ) ).
% sum.zero_middle
thf(fact_2913_prod_Ozero__middle,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [P: nat,K2: nat,G3: nat > A,H2: nat > A] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ P )
=> ( ( ord_less_eq @ nat @ K2 @ P )
=> ( ( groups7121269368397514597t_prod @ nat @ A
@ ^ [J3: nat] : ( if @ A @ ( ord_less @ nat @ J3 @ K2 ) @ ( G3 @ J3 ) @ ( if @ A @ ( J3 = K2 ) @ ( one_one @ A ) @ ( H2 @ ( minus_minus @ nat @ J3 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) )
@ ( set_ord_atMost @ nat @ P ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [J3: nat] : ( if @ A @ ( ord_less @ nat @ J3 @ K2 ) @ ( G3 @ J3 ) @ ( H2 @ J3 ) )
@ ( set_ord_atMost @ nat @ ( minus_minus @ nat @ P @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) ) ) ).
% prod.zero_middle
thf(fact_2914_root__polyfun,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [N: nat,Z: A,A2: A] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ N )
=> ( ( ( power_power @ A @ Z @ N )
= A2 )
= ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] :
( times_times @ A
@ ( if @ A
@ ( I4
= ( zero_zero @ nat ) )
@ ( uminus_uminus @ A @ A2 )
@ ( if @ A @ ( I4 = N ) @ ( one_one @ A ) @ ( zero_zero @ A ) ) )
@ ( power_power @ A @ Z @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( zero_zero @ A ) ) ) ) ) ).
% root_polyfun
thf(fact_2915_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_2916_choose__alternating__linear__sum,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [N: nat] :
( ( N
!= ( one_one @ nat ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ I4 ) @ ( semiring_1_of_nat @ A @ I4 ) ) @ ( semiring_1_of_nat @ A @ ( binomial @ N @ I4 ) ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( zero_zero @ A ) ) ) ) ).
% choose_alternating_linear_sum
thf(fact_2917_polyfun__diff__alt,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [N: nat,A2: nat > A,X: A,Y: A] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ N )
=> ( ( minus_minus @ A
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( A2 @ I4 ) @ ( power_power @ A @ X @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( A2 @ I4 ) @ ( power_power @ A @ Y @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) ) )
= ( times_times @ A @ ( minus_minus @ A @ X @ Y )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [J3: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( times_times @ A @ ( times_times @ A @ ( A2 @ ( plus_plus @ nat @ ( plus_plus @ nat @ J3 @ K3 ) @ ( one_one @ nat ) ) ) @ ( power_power @ A @ Y @ 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_2918_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_2919_choose__alternating__sum,axiom,
! [A: $tType] :
( ( comm_ring_1 @ A )
=> ! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ I4 ) @ ( semiring_1_of_nat @ A @ ( binomial @ N @ I4 ) ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( zero_zero @ A ) ) ) ) ).
% choose_alternating_sum
thf(fact_2920_polyfun__extremal__lemma,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [E3: real,C3: nat > A,N: nat] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E3 )
=> ? [M8: real] :
! [Z4: A] :
( ( ord_less_eq @ real @ M8 @ ( real_V7770717601297561774m_norm @ A @ Z4 ) )
=> ( ord_less_eq @ real
@ ( real_V7770717601297561774m_norm @ A
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( C3 @ I4 ) @ ( power_power @ A @ Z4 @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) ) )
@ ( times_times @ real @ E3 @ ( power_power @ real @ ( real_V7770717601297561774m_norm @ A @ Z4 ) @ ( suc @ N ) ) ) ) ) ) ) ).
% polyfun_extremal_lemma
thf(fact_2921_polyfun__diff,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [N: nat,A2: nat > A,X: A,Y: A] :
( ( ord_less_eq @ nat @ ( one_one @ nat ) @ N )
=> ( ( minus_minus @ A
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( A2 @ I4 ) @ ( power_power @ A @ X @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( A2 @ I4 ) @ ( power_power @ A @ Y @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) ) )
= ( times_times @ A @ ( minus_minus @ A @ X @ Y )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [J3: nat] :
( times_times @ A
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( A2 @ I4 ) @ ( power_power @ A @ Y @ ( minus_minus @ nat @ ( minus_minus @ nat @ I4 @ J3 ) @ ( one_one @ nat ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ J3 ) @ N ) )
@ ( power_power @ A @ X @ J3 ) )
@ ( set_ord_lessThan @ nat @ N ) ) ) ) ) ) ).
% polyfun_diff
thf(fact_2922_arsinh__def,axiom,
! [A: $tType] :
( ( ln @ A )
=> ( ( arsinh @ A )
= ( ^ [X5: A] : ( ln_ln @ A @ ( plus_plus @ A @ X5 @ ( powr @ A @ ( plus_plus @ A @ ( power_power @ A @ X5 @ ( 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_2923_gbinomial__partial__row__sum,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A2: A,M2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( times_times @ A @ ( gbinomial @ A @ A2 @ K3 ) @ ( minus_minus @ A @ ( divide_divide @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( semiring_1_of_nat @ A @ K3 ) ) )
@ ( set_ord_atMost @ nat @ 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 @ A2 @ ( plus_plus @ nat @ M2 @ ( one_one @ nat ) ) ) ) ) ) ).
% gbinomial_partial_row_sum
thf(fact_2924_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_2925_gchoose__row__sum__weighted,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [R2: A,M2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( times_times @ A @ ( gbinomial @ A @ R2 @ K3 ) @ ( minus_minus @ A @ ( divide_divide @ A @ R2 @ ( 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 @ R2 @ ( suc @ M2 ) ) ) ) ) ).
% gchoose_row_sum_weighted
thf(fact_2926_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_2927_log__base__10__eq1,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( log @ ( numeral_numeral @ real @ ( bit0 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) @ X )
= ( times_times @ real @ ( divide_divide @ real @ ( ln_ln @ real @ ( exp @ real @ ( one_one @ real ) ) ) @ ( ln_ln @ real @ ( numeral_numeral @ real @ ( bit0 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) ) ) @ ( ln_ln @ real @ X ) ) ) ) ).
% log_base_10_eq1
thf(fact_2928_of__nat__id,axiom,
( ( semiring_1_of_nat @ nat )
= ( ^ [N2: nat] : N2 ) ) ).
% of_nat_id
thf(fact_2929_exp__less__cancel__iff,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ ( exp @ real @ X ) @ ( exp @ real @ Y ) )
= ( ord_less @ real @ X @ Y ) ) ).
% exp_less_cancel_iff
thf(fact_2930_exp__less__mono,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ X @ Y )
=> ( ord_less @ real @ ( exp @ real @ X ) @ ( exp @ real @ Y ) ) ) ).
% exp_less_mono
thf(fact_2931_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_2932_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_2933_gbinomial__0_I2_J,axiom,
! [B: $tType] :
( ( ( semiring_char_0 @ B )
& ( semidom_divide @ B ) )
=> ! [K2: nat] :
( ( gbinomial @ B @ ( zero_zero @ B ) @ ( suc @ K2 ) )
= ( zero_zero @ B ) ) ) ).
% gbinomial_0(2)
thf(fact_2934_gbinomial__0_I1_J,axiom,
! [A: $tType] :
( ( ( semiring_char_0 @ A )
& ( semidom_divide @ A ) )
=> ! [A2: A] :
( ( gbinomial @ A @ A2 @ ( zero_zero @ nat ) )
= ( one_one @ A ) ) ) ).
% gbinomial_0(1)
thf(fact_2935_gbinomial__Suc0,axiom,
! [A: $tType] :
( ( ( semiring_char_0 @ A )
& ( semidom_divide @ A ) )
=> ! [A2: A] :
( ( gbinomial @ A @ A2 @ ( suc @ ( zero_zero @ nat ) ) )
= A2 ) ) ).
% gbinomial_Suc0
thf(fact_2936_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_2937_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_2938_exp__ln,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( exp @ real @ ( ln_ln @ real @ X ) )
= X ) ) ).
% exp_ln
thf(fact_2939_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_2940_exp__less__cancel,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ ( exp @ real @ X ) @ ( exp @ real @ Y ) )
=> ( ord_less @ real @ X @ Y ) ) ).
% exp_less_cancel
thf(fact_2941_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_2942_exp__total,axiom,
! [Y: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ Y )
=> ? [X4: real] :
( ( exp @ real @ X4 )
= Y ) ) ).
% exp_total
thf(fact_2943_exp__gt__zero,axiom,
! [X: real] : ( ord_less @ real @ ( zero_zero @ real ) @ ( exp @ real @ X ) ) ).
% exp_gt_zero
thf(fact_2944_not__exp__less__zero,axiom,
! [X: real] :
~ ( ord_less @ real @ ( exp @ real @ X ) @ ( zero_zero @ real ) ) ).
% not_exp_less_zero
thf(fact_2945_exp__add__commuting,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X: A,Y: A] :
( ( ( times_times @ A @ X @ Y )
= ( times_times @ A @ Y @ X ) )
=> ( ( exp @ A @ ( plus_plus @ A @ X @ Y ) )
= ( times_times @ A @ ( exp @ A @ X ) @ ( exp @ A @ Y ) ) ) ) ) ).
% exp_add_commuting
thf(fact_2946_mult__exp__exp,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A,Y: A] :
( ( times_times @ A @ ( exp @ A @ X ) @ ( exp @ A @ Y ) )
= ( exp @ A @ ( plus_plus @ A @ X @ Y ) ) ) ) ).
% mult_exp_exp
thf(fact_2947_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_2948_gbinomial__Suc__Suc,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A2: A,K2: nat] :
( ( gbinomial @ A @ ( plus_plus @ A @ A2 @ ( one_one @ A ) ) @ ( suc @ K2 ) )
= ( plus_plus @ A @ ( gbinomial @ A @ A2 @ K2 ) @ ( gbinomial @ A @ A2 @ ( suc @ K2 ) ) ) ) ) ).
% gbinomial_Suc_Suc
thf(fact_2949_gbinomial__of__nat__symmetric,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K2: nat,N: nat] :
( ( ord_less_eq @ nat @ K2 @ N )
=> ( ( gbinomial @ A @ ( semiring_1_of_nat @ A @ N ) @ K2 )
= ( gbinomial @ A @ ( semiring_1_of_nat @ A @ N ) @ ( minus_minus @ nat @ N @ K2 ) ) ) ) ) ).
% gbinomial_of_nat_symmetric
thf(fact_2950_exp__sum,axiom,
! [B: $tType,A: $tType] :
( ( ( comm_monoid_mult @ B )
& ( real_Vector_banach @ B )
& ( real_V2822296259951069270ebra_1 @ B ) )
=> ! [I6: set @ A,F3: A > B] :
( ( finite_finite2 @ A @ I6 )
=> ( ( exp @ B @ ( groups7311177749621191930dd_sum @ A @ B @ F3 @ I6 ) )
= ( groups7121269368397514597t_prod @ A @ B
@ ^ [X5: A] : ( exp @ B @ ( F3 @ X5 ) )
@ I6 ) ) ) ) ).
% exp_sum
thf(fact_2951_ln__ge__iff,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ Y @ ( ln_ln @ real @ X ) )
= ( ord_less_eq @ real @ ( exp @ real @ Y ) @ X ) ) ) ).
% ln_ge_iff
thf(fact_2952_gbinomial__addition__formula,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A2: A,K2: nat] :
( ( gbinomial @ A @ A2 @ ( suc @ K2 ) )
= ( plus_plus @ A @ ( gbinomial @ A @ ( minus_minus @ A @ A2 @ ( one_one @ A ) ) @ ( suc @ K2 ) ) @ ( gbinomial @ A @ ( minus_minus @ A @ A2 @ ( one_one @ A ) ) @ K2 ) ) ) ) ).
% gbinomial_addition_formula
thf(fact_2953_gbinomial__ge__n__over__k__pow__k,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [K2: nat,A2: A] :
( ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ K2 ) @ A2 )
=> ( ord_less_eq @ A @ ( power_power @ A @ ( divide_divide @ A @ A2 @ ( semiring_1_of_nat @ A @ K2 ) ) @ K2 ) @ ( gbinomial @ A @ A2 @ K2 ) ) ) ) ).
% gbinomial_ge_n_over_k_pow_k
thf(fact_2954_gbinomial__mult__1,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A2: A,K2: nat] :
( ( times_times @ A @ A2 @ ( gbinomial @ A @ A2 @ K2 ) )
= ( plus_plus @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ K2 ) @ ( gbinomial @ A @ A2 @ K2 ) ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( suc @ K2 ) ) @ ( gbinomial @ A @ A2 @ ( suc @ K2 ) ) ) ) ) ) ).
% gbinomial_mult_1
thf(fact_2955_gbinomial__mult__1_H,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A2: A,K2: nat] :
( ( times_times @ A @ ( gbinomial @ A @ A2 @ K2 ) @ A2 )
= ( plus_plus @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ K2 ) @ ( gbinomial @ A @ A2 @ K2 ) ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( suc @ K2 ) ) @ ( gbinomial @ A @ A2 @ ( suc @ K2 ) ) ) ) ) ) ).
% gbinomial_mult_1'
thf(fact_2956_powr__def,axiom,
! [A: $tType] :
( ( ln @ A )
=> ( ( powr @ A )
= ( ^ [X5: A,A6: A] :
( if @ A
@ ( X5
= ( zero_zero @ A ) )
@ ( zero_zero @ A )
@ ( exp @ A @ ( times_times @ A @ A6 @ ( ln_ln @ A @ X5 ) ) ) ) ) ) ) ).
% powr_def
thf(fact_2957_Suc__times__gbinomial,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K2: nat,A2: A] :
( ( times_times @ A @ ( semiring_1_of_nat @ A @ ( suc @ K2 ) ) @ ( gbinomial @ A @ ( plus_plus @ A @ A2 @ ( one_one @ A ) ) @ ( suc @ K2 ) ) )
= ( times_times @ A @ ( plus_plus @ A @ A2 @ ( one_one @ A ) ) @ ( gbinomial @ A @ A2 @ K2 ) ) ) ) ).
% Suc_times_gbinomial
thf(fact_2958_gbinomial__trinomial__revision,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K2: nat,M2: nat,A2: A] :
( ( ord_less_eq @ nat @ K2 @ M2 )
=> ( ( times_times @ A @ ( gbinomial @ A @ A2 @ M2 ) @ ( gbinomial @ A @ ( semiring_1_of_nat @ A @ M2 ) @ K2 ) )
= ( times_times @ A @ ( gbinomial @ A @ A2 @ K2 ) @ ( gbinomial @ A @ ( minus_minus @ A @ A2 @ ( semiring_1_of_nat @ A @ K2 ) ) @ ( minus_minus @ nat @ M2 @ K2 ) ) ) ) ) ) ).
% gbinomial_trinomial_revision
thf(fact_2959_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_2960_tanh__altdef,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ( ( tanh @ A )
= ( ^ [X5: A] : ( divide_divide @ A @ ( minus_minus @ A @ ( exp @ A @ X5 ) @ ( exp @ A @ ( uminus_uminus @ A @ X5 ) ) ) @ ( plus_plus @ A @ ( exp @ A @ X5 ) @ ( exp @ A @ ( uminus_uminus @ A @ X5 ) ) ) ) ) ) ) ).
% tanh_altdef
thf(fact_2961_gbinomial__parallel__sum,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A2: A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( gbinomial @ A @ ( plus_plus @ A @ A2 @ ( semiring_1_of_nat @ A @ K3 ) ) @ K3 )
@ ( set_ord_atMost @ nat @ N ) )
= ( gbinomial @ A @ ( plus_plus @ A @ ( plus_plus @ A @ A2 @ ( semiring_1_of_nat @ A @ N ) ) @ ( one_one @ A ) ) @ N ) ) ) ).
% gbinomial_parallel_sum
thf(fact_2962_gbinomial__rec,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A2: A,K2: nat] :
( ( gbinomial @ A @ ( plus_plus @ A @ A2 @ ( one_one @ A ) ) @ ( suc @ K2 ) )
= ( times_times @ A @ ( gbinomial @ A @ A2 @ K2 ) @ ( divide_divide @ A @ ( plus_plus @ A @ A2 @ ( one_one @ A ) ) @ ( semiring_1_of_nat @ A @ ( suc @ K2 ) ) ) ) ) ) ).
% gbinomial_rec
thf(fact_2963_gbinomial__factors,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A2: A,K2: nat] :
( ( gbinomial @ A @ ( plus_plus @ A @ A2 @ ( one_one @ A ) ) @ ( suc @ K2 ) )
= ( times_times @ A @ ( divide_divide @ A @ ( plus_plus @ A @ A2 @ ( one_one @ A ) ) @ ( semiring_1_of_nat @ A @ ( suc @ K2 ) ) ) @ ( gbinomial @ A @ A2 @ K2 ) ) ) ) ).
% gbinomial_factors
thf(fact_2964_gbinomial__minus,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A2: A,K2: nat] :
( ( gbinomial @ A @ ( uminus_uminus @ A @ A2 ) @ K2 )
= ( times_times @ A @ ( power_power @ A @ ( uminus_uminus @ A @ ( one_one @ A ) ) @ K2 ) @ ( gbinomial @ A @ ( minus_minus @ A @ ( plus_plus @ A @ A2 @ ( semiring_1_of_nat @ A @ K2 ) ) @ ( one_one @ A ) ) @ K2 ) ) ) ) ).
% gbinomial_minus
thf(fact_2965_gbinomial__reduce__nat,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K2: nat,A2: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K2 )
=> ( ( gbinomial @ A @ A2 @ K2 )
= ( plus_plus @ A @ ( gbinomial @ A @ ( minus_minus @ A @ A2 @ ( one_one @ A ) ) @ ( minus_minus @ nat @ K2 @ ( one_one @ nat ) ) ) @ ( gbinomial @ A @ ( minus_minus @ A @ A2 @ ( one_one @ A ) ) @ K2 ) ) ) ) ) ).
% gbinomial_reduce_nat
thf(fact_2966_gbinomial__partial__sum__poly,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [M2: nat,A2: A,X: A,Y: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( times_times @ A @ ( times_times @ A @ ( gbinomial @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ M2 ) @ A2 ) @ K3 ) @ ( power_power @ A @ X @ K3 ) ) @ ( power_power @ A @ Y @ ( 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 @ A2 ) @ K3 ) @ ( power_power @ A @ ( uminus_uminus @ A @ X ) @ K3 ) ) @ ( power_power @ A @ ( plus_plus @ A @ X @ Y ) @ ( minus_minus @ nat @ M2 @ K3 ) ) )
@ ( set_ord_atMost @ nat @ M2 ) ) ) ) ).
% gbinomial_partial_sum_poly
thf(fact_2967_gbinomial__sum__up__index,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K2: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [J3: nat] : ( gbinomial @ A @ ( semiring_1_of_nat @ A @ J3 ) @ K2 )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
= ( gbinomial @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ N ) @ ( one_one @ A ) ) @ ( plus_plus @ nat @ K2 @ ( one_one @ nat ) ) ) ) ) ).
% gbinomial_sum_up_index
thf(fact_2968_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_2969_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_2970_gbinomial__absorption_H,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K2: nat,A2: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ K2 )
=> ( ( gbinomial @ A @ A2 @ K2 )
= ( times_times @ A @ ( divide_divide @ A @ A2 @ ( semiring_1_of_nat @ A @ K2 ) ) @ ( gbinomial @ A @ ( minus_minus @ A @ A2 @ ( one_one @ A ) ) @ ( minus_minus @ nat @ K2 @ ( one_one @ nat ) ) ) ) ) ) ) ).
% gbinomial_absorption'
thf(fact_2971_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_2972_gbinomial__partial__sum__poly__xpos,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [M2: nat,A2: A,X: A,Y: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( times_times @ A @ ( times_times @ A @ ( gbinomial @ A @ ( plus_plus @ A @ ( semiring_1_of_nat @ A @ M2 ) @ A2 ) @ K3 ) @ ( power_power @ A @ X @ K3 ) ) @ ( power_power @ A @ Y @ ( 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 ) @ A2 ) @ ( one_one @ A ) ) @ K3 ) @ ( power_power @ A @ X @ K3 ) ) @ ( power_power @ A @ ( plus_plus @ A @ X @ Y ) @ ( minus_minus @ nat @ M2 @ K3 ) ) )
@ ( set_ord_atMost @ nat @ M2 ) ) ) ) ).
% gbinomial_partial_sum_poly_xpos
thf(fact_2973_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_2974_log__base__10__eq2,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( log @ ( numeral_numeral @ real @ ( bit0 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) @ X )
= ( times_times @ real @ ( log @ ( numeral_numeral @ real @ ( bit0 @ ( bit1 @ ( bit0 @ one2 ) ) ) ) @ ( exp @ real @ ( one_one @ real ) ) ) @ ( ln_ln @ real @ X ) ) ) ) ).
% log_base_10_eq2
thf(fact_2975_gbinomial__code,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( ( gbinomial @ A )
= ( ^ [A6: A,K3: nat] :
( if @ A
@ ( K3
= ( zero_zero @ nat ) )
@ ( one_one @ A )
@ ( divide_divide @ A
@ ( set_fo6178422350223883121st_nat @ A
@ ^ [L2: nat] : ( times_times @ A @ ( minus_minus @ A @ A6 @ ( 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_2976_sum__pos__lt__pair,axiom,
! [F3: nat > real,K2: nat] :
( ( summable @ real @ F3 )
=> ( ! [D2: nat] : ( ord_less @ real @ ( zero_zero @ real ) @ ( plus_plus @ real @ ( F3 @ ( plus_plus @ nat @ K2 @ ( times_times @ nat @ ( suc @ ( suc @ ( zero_zero @ nat ) ) ) @ D2 ) ) ) @ ( F3 @ ( plus_plus @ nat @ K2 @ ( plus_plus @ nat @ ( times_times @ nat @ ( suc @ ( suc @ ( zero_zero @ nat ) ) ) @ D2 ) @ ( one_one @ nat ) ) ) ) ) )
=> ( ord_less @ real @ ( groups7311177749621191930dd_sum @ nat @ real @ F3 @ ( set_ord_lessThan @ nat @ K2 ) ) @ ( suminf @ real @ F3 ) ) ) ) ).
% sum_pos_lt_pair
thf(fact_2977_binomial__code,axiom,
( binomial
= ( ^ [N2: nat,K3: nat] : ( if @ nat @ ( ord_less @ nat @ N2 @ K3 ) @ ( zero_zero @ nat ) @ ( if @ nat @ ( ord_less @ nat @ N2 @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K3 ) ) @ ( binomial @ N2 @ ( minus_minus @ nat @ N2 @ K3 ) ) @ ( divide_divide @ nat @ ( set_fo6178422350223883121st_nat @ nat @ ( times_times @ nat ) @ ( plus_plus @ nat @ ( minus_minus @ nat @ N2 @ K3 ) @ ( one_one @ nat ) ) @ N2 @ ( one_one @ nat ) ) @ ( semiring_char_0_fact @ nat @ K3 ) ) ) ) ) ) ).
% binomial_code
thf(fact_2978_modulo__int__unfold,axiom,
! [L: int,K2: int,N: nat,M2: nat] :
( ( ( ( ( sgn_sgn @ int @ L )
= ( zero_zero @ int ) )
| ( ( sgn_sgn @ int @ K2 )
= ( zero_zero @ int ) )
| ( N
= ( zero_zero @ nat ) ) )
=> ( ( modulo_modulo @ int @ ( times_times @ int @ ( sgn_sgn @ int @ K2 ) @ ( semiring_1_of_nat @ int @ M2 ) ) @ ( times_times @ int @ ( sgn_sgn @ int @ L ) @ ( semiring_1_of_nat @ int @ N ) ) )
= ( times_times @ int @ ( sgn_sgn @ int @ K2 ) @ ( semiring_1_of_nat @ int @ M2 ) ) ) )
& ( ~ ( ( ( sgn_sgn @ int @ L )
= ( zero_zero @ int ) )
| ( ( sgn_sgn @ int @ K2 )
= ( zero_zero @ int ) )
| ( N
= ( zero_zero @ nat ) ) )
=> ( ( ( ( sgn_sgn @ int @ K2 )
= ( sgn_sgn @ int @ L ) )
=> ( ( modulo_modulo @ int @ ( times_times @ int @ ( sgn_sgn @ int @ K2 ) @ ( semiring_1_of_nat @ int @ 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 @ K2 )
!= ( sgn_sgn @ int @ L ) )
=> ( ( modulo_modulo @ int @ ( times_times @ int @ ( sgn_sgn @ int @ K2 ) @ ( 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_2979_powr__int,axiom,
! [X: real,I2: int] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ I2 )
=> ( ( powr @ real @ X @ ( ring_1_of_int @ real @ I2 ) )
= ( power_power @ real @ X @ ( nat2 @ I2 ) ) ) )
& ( ~ ( ord_less_eq @ int @ ( zero_zero @ int ) @ I2 )
=> ( ( powr @ real @ X @ ( ring_1_of_int @ real @ I2 ) )
= ( divide_divide @ real @ ( one_one @ real ) @ ( power_power @ real @ X @ ( nat2 @ ( uminus_uminus @ int @ I2 ) ) ) ) ) ) ) ) ).
% powr_int
thf(fact_2980_sgn__sgn,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ! [A2: A] :
( ( sgn_sgn @ A @ ( sgn_sgn @ A @ A2 ) )
= ( sgn_sgn @ A @ A2 ) ) ) ).
% sgn_sgn
thf(fact_2981_sgn__zero,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ( ( sgn_sgn @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% sgn_zero
thf(fact_2982_sgn__0,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ( ( sgn_sgn @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% sgn_0
thf(fact_2983_sgn__1,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ( ( sgn_sgn @ A @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% sgn_1
thf(fact_2984_sgn__divide,axiom,
! [A: $tType] :
( ( field_abs_sgn @ A )
=> ! [A2: A,B2: A] :
( ( sgn_sgn @ A @ ( divide_divide @ A @ A2 @ B2 ) )
= ( divide_divide @ A @ ( sgn_sgn @ A @ A2 ) @ ( sgn_sgn @ A @ B2 ) ) ) ) ).
% sgn_divide
thf(fact_2985_idom__abs__sgn__class_Osgn__minus,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ! [A2: A] :
( ( sgn_sgn @ A @ ( uminus_uminus @ A @ A2 ) )
= ( uminus_uminus @ A @ ( sgn_sgn @ A @ A2 ) ) ) ) ).
% idom_abs_sgn_class.sgn_minus
thf(fact_2986_nat__int,axiom,
! [N: nat] :
( ( nat2 @ ( semiring_1_of_nat @ int @ N ) )
= N ) ).
% nat_int
thf(fact_2987_summable__zero,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ( summable @ A
@ ^ [N2: nat] : ( zero_zero @ A ) ) ) ).
% summable_zero
thf(fact_2988_summable__single,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [I2: nat,F3: nat > A] :
( summable @ A
@ ^ [R5: nat] : ( if @ A @ ( R5 = I2 ) @ ( F3 @ R5 ) @ ( zero_zero @ A ) ) ) ) ).
% summable_single
thf(fact_2989_summable__iff__shift,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: nat > A,K2: nat] :
( ( summable @ A
@ ^ [N2: nat] : ( F3 @ ( plus_plus @ nat @ N2 @ K2 ) ) )
= ( summable @ A @ F3 ) ) ) ).
% summable_iff_shift
thf(fact_2990_sgn__greater,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( sgn_sgn @ A @ A2 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% sgn_greater
thf(fact_2991_sgn__less,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A] :
( ( ord_less @ A @ ( sgn_sgn @ A @ A2 ) @ ( zero_zero @ A ) )
= ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% sgn_less
thf(fact_2992_divide__sgn,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( divide_divide @ A @ A2 @ ( sgn_sgn @ A @ B2 ) )
= ( times_times @ A @ A2 @ ( sgn_sgn @ A @ B2 ) ) ) ) ).
% divide_sgn
thf(fact_2993_fact__0,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ( ( semiring_char_0_fact @ A @ ( zero_zero @ nat ) )
= ( one_one @ A ) ) ) ).
% fact_0
thf(fact_2994_nat__numeral,axiom,
! [K2: num] :
( ( nat2 @ ( numeral_numeral @ int @ K2 ) )
= ( numeral_numeral @ nat @ K2 ) ) ).
% nat_numeral
thf(fact_2995_summable__cmult__iff,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [C3: A,F3: nat > A] :
( ( summable @ A
@ ^ [N2: nat] : ( times_times @ A @ C3 @ ( F3 @ N2 ) ) )
= ( ( C3
= ( zero_zero @ A ) )
| ( summable @ A @ F3 ) ) ) ) ).
% summable_cmult_iff
thf(fact_2996_summable__divide__iff,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: nat > A,C3: A] :
( ( summable @ A
@ ^ [N2: nat] : ( divide_divide @ A @ ( F3 @ N2 ) @ C3 ) )
= ( ( C3
= ( zero_zero @ A ) )
| ( summable @ A @ F3 ) ) ) ) ).
% summable_divide_iff
thf(fact_2997_summable__If__finite__set,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [A5: set @ nat,F3: nat > A] :
( ( finite_finite2 @ nat @ A5 )
=> ( summable @ A
@ ^ [R5: nat] : ( if @ A @ ( member @ nat @ R5 @ A5 ) @ ( F3 @ R5 ) @ ( zero_zero @ A ) ) ) ) ) ).
% summable_If_finite_set
thf(fact_2998_summable__If__finite,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [P2: nat > $o,F3: nat > A] :
( ( finite_finite2 @ nat @ ( collect @ nat @ P2 ) )
=> ( summable @ A
@ ^ [R5: nat] : ( if @ A @ ( P2 @ R5 ) @ ( F3 @ R5 ) @ ( zero_zero @ A ) ) ) ) ) ).
% summable_If_finite
thf(fact_2999_nat__of__bool,axiom,
! [P2: $o] :
( ( nat2 @ ( zero_neq_one_of_bool @ int @ P2 ) )
= ( zero_neq_one_of_bool @ nat @ P2 ) ) ).
% nat_of_bool
thf(fact_3000_sgn__pos,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( sgn_sgn @ A @ A2 )
= ( one_one @ A ) ) ) ) ).
% sgn_pos
thf(fact_3001_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_3002_nat__1,axiom,
( ( nat2 @ ( one_one @ int ) )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% nat_1
thf(fact_3003_sgn__mult__self__eq,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A] :
( ( times_times @ A @ ( sgn_sgn @ A @ A2 ) @ ( sgn_sgn @ A @ A2 ) )
= ( zero_neq_one_of_bool @ A
@ ( A2
!= ( zero_zero @ A ) ) ) ) ) ).
% sgn_mult_self_eq
thf(fact_3004_nat__le__0,axiom,
! [Z: int] :
( ( ord_less_eq @ int @ Z @ ( zero_zero @ int ) )
=> ( ( nat2 @ Z )
= ( zero_zero @ nat ) ) ) ).
% nat_le_0
thf(fact_3005_nat__0__iff,axiom,
! [I2: int] :
( ( ( nat2 @ I2 )
= ( zero_zero @ nat ) )
= ( ord_less_eq @ int @ I2 @ ( zero_zero @ int ) ) ) ).
% nat_0_iff
thf(fact_3006_zless__nat__conj,axiom,
! [W2: int,Z: int] :
( ( ord_less @ nat @ ( nat2 @ W2 ) @ ( nat2 @ Z ) )
= ( ( ord_less @ int @ ( zero_zero @ int ) @ Z )
& ( ord_less @ int @ W2 @ Z ) ) ) ).
% zless_nat_conj
thf(fact_3007_nat__neg__numeral,axiom,
! [K2: num] :
( ( nat2 @ ( uminus_uminus @ int @ ( numeral_numeral @ int @ K2 ) ) )
= ( zero_zero @ nat ) ) ).
% nat_neg_numeral
thf(fact_3008_nat__zminus__int,axiom,
! [N: nat] :
( ( nat2 @ ( uminus_uminus @ int @ ( semiring_1_of_nat @ int @ N ) ) )
= ( zero_zero @ nat ) ) ).
% nat_zminus_int
thf(fact_3009_int__nat__eq,axiom,
! [Z: int] :
( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z )
=> ( ( semiring_1_of_nat @ int @ ( nat2 @ Z ) )
= Z ) )
& ( ~ ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z )
=> ( ( semiring_1_of_nat @ int @ ( nat2 @ Z ) )
= ( zero_zero @ int ) ) ) ) ).
% int_nat_eq
thf(fact_3010_sgn__mult__dvd__iff,axiom,
! [R2: int,L: int,K2: int] :
( ( dvd_dvd @ int @ ( times_times @ int @ ( sgn_sgn @ int @ R2 ) @ L ) @ K2 )
= ( ( dvd_dvd @ int @ L @ K2 )
& ( ( R2
= ( zero_zero @ int ) )
=> ( K2
= ( zero_zero @ int ) ) ) ) ) ).
% sgn_mult_dvd_iff
thf(fact_3011_mult__sgn__dvd__iff,axiom,
! [L: int,R2: int,K2: int] :
( ( dvd_dvd @ int @ ( times_times @ int @ L @ ( sgn_sgn @ int @ R2 ) ) @ K2 )
= ( ( dvd_dvd @ int @ L @ K2 )
& ( ( R2
= ( zero_zero @ int ) )
=> ( K2
= ( zero_zero @ int ) ) ) ) ) ).
% mult_sgn_dvd_iff
thf(fact_3012_dvd__sgn__mult__iff,axiom,
! [L: int,R2: int,K2: int] :
( ( dvd_dvd @ int @ L @ ( times_times @ int @ ( sgn_sgn @ int @ R2 ) @ K2 ) )
= ( ( dvd_dvd @ int @ L @ K2 )
| ( R2
= ( zero_zero @ int ) ) ) ) ).
% dvd_sgn_mult_iff
thf(fact_3013_dvd__mult__sgn__iff,axiom,
! [L: int,K2: int,R2: int] :
( ( dvd_dvd @ int @ L @ ( times_times @ int @ K2 @ ( sgn_sgn @ int @ R2 ) ) )
= ( ( dvd_dvd @ int @ L @ K2 )
| ( R2
= ( zero_zero @ int ) ) ) ) ).
% dvd_mult_sgn_iff
thf(fact_3014_sgn__neg,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A] :
( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( sgn_sgn @ A @ A2 )
= ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ).
% sgn_neg
thf(fact_3015_zero__less__nat__eq,axiom,
! [Z: int] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( nat2 @ Z ) )
= ( ord_less @ int @ ( zero_zero @ int ) @ Z ) ) ).
% zero_less_nat_eq
thf(fact_3016_of__nat__nat,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [Z: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z )
=> ( ( semiring_1_of_nat @ A @ ( nat2 @ Z ) )
= ( ring_1_of_int @ A @ Z ) ) ) ) ).
% of_nat_nat
thf(fact_3017_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_3018_diff__nat__numeral,axiom,
! [V: num,V4: num] :
( ( minus_minus @ nat @ ( numeral_numeral @ nat @ V ) @ ( numeral_numeral @ nat @ V4 ) )
= ( nat2 @ ( minus_minus @ int @ ( numeral_numeral @ int @ V ) @ ( numeral_numeral @ int @ V4 ) ) ) ) ).
% diff_nat_numeral
thf(fact_3019_nat__eq__numeral__power__cancel__iff,axiom,
! [Y: int,X: num,N: nat] :
( ( ( nat2 @ Y )
= ( power_power @ nat @ ( numeral_numeral @ nat @ X ) @ N ) )
= ( Y
= ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N ) ) ) ).
% nat_eq_numeral_power_cancel_iff
thf(fact_3020_numeral__power__eq__nat__cancel__iff,axiom,
! [X: num,N: nat,Y: int] :
( ( ( power_power @ nat @ ( numeral_numeral @ nat @ X ) @ N )
= ( nat2 @ Y ) )
= ( ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N )
= Y ) ) ).
% numeral_power_eq_nat_cancel_iff
thf(fact_3021_summable__geometric__iff,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [C3: A] :
( ( summable @ A @ ( power_power @ A @ C3 ) )
= ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ C3 ) @ ( one_one @ real ) ) ) ) ).
% summable_geometric_iff
thf(fact_3022_nat__ceiling__le__eq,axiom,
! [X: real,A2: nat] :
( ( ord_less_eq @ nat @ ( nat2 @ ( archimedean_ceiling @ real @ X ) ) @ A2 )
= ( ord_less_eq @ real @ X @ ( semiring_1_of_nat @ real @ A2 ) ) ) ).
% nat_ceiling_le_eq
thf(fact_3023_one__less__nat__eq,axiom,
! [Z: int] :
( ( ord_less @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( nat2 @ Z ) )
= ( ord_less @ int @ ( one_one @ int ) @ Z ) ) ).
% one_less_nat_eq
thf(fact_3024_nat__numeral__diff__1,axiom,
! [V: num] :
( ( minus_minus @ nat @ ( numeral_numeral @ nat @ V ) @ ( one_one @ nat ) )
= ( nat2 @ ( minus_minus @ int @ ( numeral_numeral @ int @ V ) @ ( one_one @ int ) ) ) ) ).
% nat_numeral_diff_1
thf(fact_3025_numeral__power__less__nat__cancel__iff,axiom,
! [X: num,N: nat,A2: int] :
( ( ord_less @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ X ) @ N ) @ ( nat2 @ A2 ) )
= ( ord_less @ int @ ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N ) @ A2 ) ) ).
% numeral_power_less_nat_cancel_iff
thf(fact_3026_nat__less__numeral__power__cancel__iff,axiom,
! [A2: int,X: num,N: nat] :
( ( ord_less @ nat @ ( nat2 @ A2 ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ X ) @ N ) )
= ( ord_less @ int @ A2 @ ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N ) ) ) ).
% nat_less_numeral_power_cancel_iff
thf(fact_3027_nat__le__numeral__power__cancel__iff,axiom,
! [A2: int,X: num,N: nat] :
( ( ord_less_eq @ nat @ ( nat2 @ A2 ) @ ( power_power @ nat @ ( numeral_numeral @ nat @ X ) @ N ) )
= ( ord_less_eq @ int @ A2 @ ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N ) ) ) ).
% nat_le_numeral_power_cancel_iff
thf(fact_3028_numeral__power__le__nat__cancel__iff,axiom,
! [X: num,N: nat,A2: int] :
( ( ord_less_eq @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ X ) @ N ) @ ( nat2 @ A2 ) )
= ( ord_less_eq @ int @ ( power_power @ int @ ( numeral_numeral @ int @ X ) @ N ) @ A2 ) ) ).
% numeral_power_le_nat_cancel_iff
thf(fact_3029_sgn__0__0,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A] :
( ( ( sgn_sgn @ A @ A2 )
= ( zero_zero @ A ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% sgn_0_0
thf(fact_3030_sgn__eq__0__iff,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ! [A2: A] :
( ( ( sgn_sgn @ A @ A2 )
= ( zero_zero @ A ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% sgn_eq_0_iff
thf(fact_3031_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_3032_sgn__mult,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ! [A2: A,B2: A] :
( ( sgn_sgn @ A @ ( times_times @ A @ A2 @ B2 ) )
= ( times_times @ A @ ( sgn_sgn @ A @ A2 ) @ ( sgn_sgn @ A @ B2 ) ) ) ) ).
% sgn_mult
thf(fact_3033_same__sgn__sgn__add,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [B2: A,A2: A] :
( ( ( sgn_sgn @ A @ B2 )
= ( sgn_sgn @ A @ A2 ) )
=> ( ( sgn_sgn @ A @ ( plus_plus @ A @ A2 @ B2 ) )
= ( sgn_sgn @ A @ A2 ) ) ) ) ).
% same_sgn_sgn_add
thf(fact_3034_fact__ge__self,axiom,
! [N: nat] : ( ord_less_eq @ nat @ N @ ( semiring_char_0_fact @ nat @ N ) ) ).
% fact_ge_self
thf(fact_3035_fact__mono__nat,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ord_less_eq @ nat @ ( semiring_char_0_fact @ nat @ M2 ) @ ( semiring_char_0_fact @ nat @ N ) ) ) ).
% fact_mono_nat
thf(fact_3036_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_3037_summable__const__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [C3: A] :
( ( summable @ A
@ ^ [Uu3: nat] : C3 )
= ( C3
= ( zero_zero @ A ) ) ) ) ).
% summable_const_iff
thf(fact_3038_summable__comparison__test,axiom,
! [A: $tType] :
( ( real_Vector_banach @ A )
=> ! [F3: nat > A,G3: nat > real] :
( ? [N8: nat] :
! [N3: nat] :
( ( ord_less_eq @ nat @ N8 @ N3 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( F3 @ N3 ) ) @ ( G3 @ N3 ) ) )
=> ( ( summable @ real @ G3 )
=> ( summable @ A @ F3 ) ) ) ) ).
% summable_comparison_test
thf(fact_3039_summable__comparison__test_H,axiom,
! [A: $tType] :
( ( real_Vector_banach @ A )
=> ! [G3: nat > real,N6: nat,F3: nat > A] :
( ( summable @ real @ G3 )
=> ( ! [N3: nat] :
( ( ord_less_eq @ nat @ N6 @ N3 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( F3 @ N3 ) ) @ ( G3 @ N3 ) ) )
=> ( summable @ A @ F3 ) ) ) ) ).
% summable_comparison_test'
thf(fact_3040_summable__add,axiom,
! [A: $tType] :
( ( ( topolo5987344860129210374id_add @ A )
& ( topological_t2_space @ A ) )
=> ! [F3: nat > A,G3: nat > A] :
( ( summable @ A @ F3 )
=> ( ( summable @ A @ G3 )
=> ( summable @ A
@ ^ [N2: nat] : ( plus_plus @ A @ ( F3 @ N2 ) @ ( G3 @ N2 ) ) ) ) ) ) ).
% summable_add
thf(fact_3041_summable__ignore__initial__segment,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: nat > A,K2: nat] :
( ( summable @ A @ F3 )
=> ( summable @ A
@ ^ [N2: nat] : ( F3 @ ( plus_plus @ nat @ N2 @ K2 ) ) ) ) ) ).
% summable_ignore_initial_segment
thf(fact_3042_powser__insidea,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [F3: nat > A,X: A,Z: A] :
( ( summable @ A
@ ^ [N2: nat] : ( times_times @ A @ ( F3 @ N2 ) @ ( power_power @ A @ X @ N2 ) ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ Z ) @ ( real_V7770717601297561774m_norm @ A @ X ) )
=> ( summable @ real
@ ^ [N2: nat] : ( real_V7770717601297561774m_norm @ A @ ( times_times @ A @ ( F3 @ N2 ) @ ( power_power @ A @ Z @ N2 ) ) ) ) ) ) ) ).
% powser_insidea
thf(fact_3043_suminf__le,axiom,
! [A: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F3: nat > A,G3: nat > A] :
( ! [N3: nat] : ( ord_less_eq @ A @ ( F3 @ N3 ) @ ( G3 @ N3 ) )
=> ( ( summable @ A @ F3 )
=> ( ( summable @ A @ G3 )
=> ( ord_less_eq @ A @ ( suminf @ A @ F3 ) @ ( suminf @ A @ G3 ) ) ) ) ) ) ).
% suminf_le
thf(fact_3044_summable__finite,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [N6: set @ nat,F3: nat > A] :
( ( finite_finite2 @ nat @ N6 )
=> ( ! [N3: nat] :
( ~ ( member @ nat @ N3 @ N6 )
=> ( ( F3 @ N3 )
= ( zero_zero @ A ) ) )
=> ( summable @ A @ F3 ) ) ) ) ).
% summable_finite
thf(fact_3045_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_3046_sgn__not__eq__imp,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [B2: A,A2: A] :
( ( ( sgn_sgn @ A @ B2 )
!= ( sgn_sgn @ A @ A2 ) )
=> ( ( ( sgn_sgn @ A @ A2 )
!= ( zero_zero @ A ) )
=> ( ( ( sgn_sgn @ A @ B2 )
!= ( zero_zero @ A ) )
=> ( ( sgn_sgn @ A @ A2 )
= ( uminus_uminus @ A @ ( sgn_sgn @ A @ B2 ) ) ) ) ) ) ) ).
% sgn_not_eq_imp
thf(fact_3047_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_3048_nat__zero__as__int,axiom,
( ( zero_zero @ nat )
= ( nat2 @ ( zero_zero @ int ) ) ) ).
% nat_zero_as_int
thf(fact_3049_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_3050_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_3051_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_3052_nat__mono,axiom,
! [X: int,Y: int] :
( ( ord_less_eq @ int @ X @ Y )
=> ( ord_less_eq @ nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ).
% nat_mono
thf(fact_3053_int__sgnE,axiom,
! [K2: int] :
~ ! [N3: nat,L4: int] :
( K2
!= ( times_times @ int @ ( sgn_sgn @ int @ L4 ) @ ( semiring_1_of_nat @ int @ N3 ) ) ) ).
% int_sgnE
thf(fact_3054_eq__nat__nat__iff,axiom,
! [Z: int,Z7: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z7 )
=> ( ( ( nat2 @ Z )
= ( nat2 @ Z7 ) )
= ( Z = Z7 ) ) ) ) ).
% eq_nat_nat_iff
thf(fact_3055_all__nat,axiom,
( ( ^ [P3: nat > $o] :
! [X6: nat] : ( P3 @ X6 ) )
= ( ^ [P4: nat > $o] :
! [X5: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X5 )
=> ( P4 @ ( nat2 @ X5 ) ) ) ) ) ).
% all_nat
thf(fact_3056_ex__nat,axiom,
( ( ^ [P3: nat > $o] :
? [X6: nat] : ( P3 @ X6 ) )
= ( ^ [P4: nat > $o] :
? [X5: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X5 )
& ( P4 @ ( nat2 @ X5 ) ) ) ) ) ).
% ex_nat
thf(fact_3057_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_3058_fact__mono,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [M2: nat,N: nat] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ord_less_eq @ A @ ( semiring_char_0_fact @ A @ M2 ) @ ( semiring_char_0_fact @ A @ N ) ) ) ) ).
% fact_mono
thf(fact_3059_summable__mult__D,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [C3: A,F3: nat > A] :
( ( summable @ A
@ ^ [N2: nat] : ( times_times @ A @ C3 @ ( F3 @ N2 ) ) )
=> ( ( C3
!= ( zero_zero @ A ) )
=> ( summable @ A @ F3 ) ) ) ) ).
% summable_mult_D
thf(fact_3060_fact__dvd,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [N: nat,M2: nat] :
( ( ord_less_eq @ nat @ N @ M2 )
=> ( dvd_dvd @ A @ ( semiring_char_0_fact @ A @ N ) @ ( semiring_char_0_fact @ A @ M2 ) ) ) ) ).
% fact_dvd
thf(fact_3061_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_3062_suminf__add,axiom,
! [A: $tType] :
( ( ( topolo5987344860129210374id_add @ A )
& ( topological_t2_space @ A ) )
=> ! [F3: nat > A,G3: nat > A] :
( ( summable @ A @ F3 )
=> ( ( summable @ A @ G3 )
=> ( ( plus_plus @ A @ ( suminf @ A @ F3 ) @ ( suminf @ A @ G3 ) )
= ( suminf @ A
@ ^ [N2: nat] : ( plus_plus @ A @ ( F3 @ N2 ) @ ( G3 @ N2 ) ) ) ) ) ) ) ).
% suminf_add
thf(fact_3063_suminf__eq__zero__iff,axiom,
! [A: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F3: nat > A] :
( ( summable @ A @ F3 )
=> ( ! [N3: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F3 @ N3 ) )
=> ( ( ( suminf @ A @ F3 )
= ( zero_zero @ A ) )
= ( ! [N2: nat] :
( ( F3 @ N2 )
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% suminf_eq_zero_iff
thf(fact_3064_suminf__nonneg,axiom,
! [A: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F3: nat > A] :
( ( summable @ A @ F3 )
=> ( ! [N3: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F3 @ N3 ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( suminf @ A @ F3 ) ) ) ) ) ).
% suminf_nonneg
thf(fact_3065_suminf__pos,axiom,
! [A: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F3: nat > A] :
( ( summable @ A @ F3 )
=> ( ! [N3: nat] : ( ord_less @ A @ ( zero_zero @ A ) @ ( F3 @ N3 ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( suminf @ A @ F3 ) ) ) ) ) ).
% suminf_pos
thf(fact_3066_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_3067_sgn__1__pos,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A] :
( ( ( sgn_sgn @ A @ A2 )
= ( one_one @ A ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% sgn_1_pos
thf(fact_3068_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_3069_summable__zero__power_H,axiom,
! [A: $tType] :
( ( ( ring_1 @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [F3: nat > A] :
( summable @ A
@ ^ [N2: nat] : ( times_times @ A @ ( F3 @ N2 ) @ ( power_power @ A @ ( zero_zero @ A ) @ N2 ) ) ) ) ).
% summable_zero_power'
thf(fact_3070_summable__0__powser,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [F3: nat > A] :
( summable @ A
@ ^ [N2: nat] : ( times_times @ A @ ( F3 @ N2 ) @ ( power_power @ A @ ( zero_zero @ A ) @ N2 ) ) ) ) ).
% summable_0_powser
thf(fact_3071_nat__mono__iff,axiom,
! [Z: int,W2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ Z )
=> ( ( ord_less @ nat @ ( nat2 @ W2 ) @ ( nat2 @ Z ) )
= ( ord_less @ int @ W2 @ Z ) ) ) ).
% nat_mono_iff
thf(fact_3072_of__nat__ceiling,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [R2: A] : ( ord_less_eq @ A @ R2 @ ( semiring_1_of_nat @ A @ ( nat2 @ ( archimedean_ceiling @ A @ R2 ) ) ) ) ) ).
% of_nat_ceiling
thf(fact_3073_zless__nat__eq__int__zless,axiom,
! [M2: nat,Z: int] :
( ( ord_less @ nat @ M2 @ ( nat2 @ Z ) )
= ( ord_less @ int @ ( semiring_1_of_nat @ int @ M2 ) @ Z ) ) ).
% zless_nat_eq_int_zless
thf(fact_3074_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_3075_nat__0__le,axiom,
! [Z: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z )
=> ( ( semiring_1_of_nat @ int @ ( nat2 @ Z ) )
= Z ) ) ).
% nat_0_le
thf(fact_3076_int__eq__iff,axiom,
! [M2: nat,Z: int] :
( ( ( semiring_1_of_nat @ int @ M2 )
= Z )
= ( ( M2
= ( nat2 @ Z ) )
& ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z ) ) ) ).
% int_eq_iff
thf(fact_3077_summable__powser__ignore__initial__segment,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [F3: nat > A,M2: nat,Z: A] :
( ( summable @ A
@ ^ [N2: nat] : ( times_times @ A @ ( F3 @ ( plus_plus @ nat @ N2 @ M2 ) ) @ ( power_power @ A @ Z @ N2 ) ) )
= ( summable @ A
@ ^ [N2: nat] : ( times_times @ A @ ( F3 @ N2 ) @ ( power_power @ A @ Z @ N2 ) ) ) ) ) ).
% summable_powser_ignore_initial_segment
thf(fact_3078_nat__int__add,axiom,
! [A2: nat,B2: nat] :
( ( nat2 @ ( plus_plus @ int @ ( semiring_1_of_nat @ int @ A2 ) @ ( semiring_1_of_nat @ int @ B2 ) ) )
= ( plus_plus @ nat @ A2 @ B2 ) ) ).
% nat_int_add
thf(fact_3079_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_3080_fact__fact__dvd__fact,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [K2: nat,N: nat] : ( dvd_dvd @ A @ ( times_times @ A @ ( semiring_char_0_fact @ A @ K2 ) @ ( semiring_char_0_fact @ A @ N ) ) @ ( semiring_char_0_fact @ A @ ( plus_plus @ nat @ K2 @ N ) ) ) ) ).
% fact_fact_dvd_fact
thf(fact_3081_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_3082_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_3083_summable__norm__comparison__test,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: nat > A,G3: nat > real] :
( ? [N8: nat] :
! [N3: nat] :
( ( ord_less_eq @ nat @ N8 @ N3 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( F3 @ N3 ) ) @ ( G3 @ N3 ) ) )
=> ( ( summable @ real @ G3 )
=> ( summable @ real
@ ^ [N2: nat] : ( real_V7770717601297561774m_norm @ A @ ( F3 @ N2 ) ) ) ) ) ) ).
% summable_norm_comparison_test
thf(fact_3084_nat__plus__as__int,axiom,
( ( plus_plus @ nat )
= ( ^ [A6: nat,B5: nat] : ( nat2 @ ( plus_plus @ int @ ( semiring_1_of_nat @ int @ A6 ) @ ( semiring_1_of_nat @ int @ B5 ) ) ) ) ) ).
% nat_plus_as_int
thf(fact_3085_suminf__pos2,axiom,
! [A: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F3: nat > A,I2: nat] :
( ( summable @ A @ F3 )
=> ( ! [N3: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F3 @ N3 ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ ( F3 @ I2 ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( suminf @ A @ F3 ) ) ) ) ) ) ).
% suminf_pos2
thf(fact_3086_suminf__pos__iff,axiom,
! [A: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F3: nat > A] :
( ( summable @ A @ F3 )
=> ( ! [N3: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F3 @ N3 ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ ( suminf @ A @ F3 ) )
= ( ? [I4: nat] : ( ord_less @ A @ ( zero_zero @ A ) @ ( F3 @ I4 ) ) ) ) ) ) ) ).
% suminf_pos_iff
thf(fact_3087_suminf__le__const,axiom,
! [A: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F3: nat > A,X: A] :
( ( summable @ A @ F3 )
=> ( ! [N3: nat] : ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_ord_lessThan @ nat @ N3 ) ) @ X )
=> ( ord_less_eq @ A @ ( suminf @ A @ F3 ) @ X ) ) ) ) ).
% suminf_le_const
thf(fact_3088_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_3089_sgn__1__neg,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A] :
( ( ( sgn_sgn @ A @ A2 )
= ( uminus_uminus @ A @ ( one_one @ A ) ) )
= ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% sgn_1_neg
thf(fact_3090_sgn__if,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( ( sgn_sgn @ A )
= ( ^ [X5: A] :
( if @ A
@ ( X5
= ( zero_zero @ A ) )
@ ( zero_zero @ A )
@ ( if @ A @ ( ord_less @ A @ ( zero_zero @ A ) @ X5 ) @ ( one_one @ A ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) ) ) ) ).
% sgn_if
thf(fact_3091_of__nat__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [R2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ R2 )
=> ( ord_less_eq @ A @ ( semiring_1_of_nat @ A @ ( nat2 @ ( archim6421214686448440834_floor @ A @ R2 ) ) ) @ R2 ) ) ) ).
% of_nat_floor
thf(fact_3092_zsgn__def,axiom,
( ( sgn_sgn @ int )
= ( ^ [I4: int] :
( if @ int
@ ( I4
= ( zero_zero @ int ) )
@ ( zero_zero @ int )
@ ( if @ int @ ( ord_less @ int @ ( zero_zero @ int ) @ I4 ) @ ( one_one @ int ) @ ( uminus_uminus @ int @ ( one_one @ int ) ) ) ) ) ) ).
% zsgn_def
thf(fact_3093_fact__div__fact__le__pow,axiom,
! [R2: nat,N: nat] :
( ( ord_less_eq @ nat @ R2 @ N )
=> ( ord_less_eq @ nat @ ( divide_divide @ nat @ ( semiring_char_0_fact @ nat @ N ) @ ( semiring_char_0_fact @ nat @ ( minus_minus @ nat @ N @ R2 ) ) ) @ ( power_power @ nat @ N @ R2 ) ) ) ).
% fact_div_fact_le_pow
thf(fact_3094_powser__inside,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V8999393235501362500lgebra @ A ) )
=> ! [F3: nat > A,X: A,Z: A] :
( ( summable @ A
@ ^ [N2: nat] : ( times_times @ A @ ( F3 @ N2 ) @ ( power_power @ A @ X @ N2 ) ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ Z ) @ ( real_V7770717601297561774m_norm @ A @ X ) )
=> ( summable @ A
@ ^ [N2: nat] : ( times_times @ A @ ( F3 @ N2 ) @ ( power_power @ A @ Z @ N2 ) ) ) ) ) ) ).
% powser_inside
thf(fact_3095_binomial__fact__lemma,axiom,
! [K2: nat,N: nat] :
( ( ord_less_eq @ nat @ K2 @ N )
=> ( ( times_times @ nat @ ( times_times @ nat @ ( semiring_char_0_fact @ nat @ K2 ) @ ( semiring_char_0_fact @ nat @ ( minus_minus @ nat @ N @ K2 ) ) ) @ ( binomial @ N @ K2 ) )
= ( semiring_char_0_fact @ nat @ N ) ) ) ).
% binomial_fact_lemma
thf(fact_3096_nat__less__eq__zless,axiom,
! [W2: int,Z: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ W2 )
=> ( ( ord_less @ nat @ ( nat2 @ W2 ) @ ( nat2 @ Z ) )
= ( ord_less @ int @ W2 @ Z ) ) ) ).
% nat_less_eq_zless
thf(fact_3097_summableI__nonneg__bounded,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( ordere6911136660526730532id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F3: nat > A,X: A] :
( ! [N3: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F3 @ N3 ) )
=> ( ! [N3: nat] : ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_ord_lessThan @ nat @ N3 ) ) @ X )
=> ( summable @ A @ F3 ) ) ) ) ).
% summableI_nonneg_bounded
thf(fact_3098_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_3099_nat__le__eq__zle,axiom,
! [W2: int,Z: int] :
( ( ( ord_less @ int @ ( zero_zero @ int ) @ W2 )
| ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z ) )
=> ( ( ord_less_eq @ nat @ ( nat2 @ W2 ) @ ( nat2 @ Z ) )
= ( ord_less_eq @ int @ W2 @ Z ) ) ) ).
% nat_le_eq_zle
thf(fact_3100_bounded__imp__summable,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( linord2810124833399127020strict @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [A2: nat > A,B6: A] :
( ! [N3: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( A2 @ N3 ) )
=> ( ! [N3: nat] : ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ A2 @ ( set_ord_atMost @ nat @ N3 ) ) @ B6 )
=> ( summable @ A @ A2 ) ) ) ) ).
% bounded_imp_summable
thf(fact_3101_nat__eq__iff,axiom,
! [W2: int,M2: nat] :
( ( ( nat2 @ W2 )
= M2 )
= ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ W2 )
=> ( W2
= ( semiring_1_of_nat @ int @ M2 ) ) )
& ( ~ ( ord_less_eq @ int @ ( zero_zero @ int ) @ W2 )
=> ( M2
= ( zero_zero @ nat ) ) ) ) ) ).
% nat_eq_iff
thf(fact_3102_nat__eq__iff2,axiom,
! [M2: nat,W2: int] :
( ( M2
= ( nat2 @ W2 ) )
= ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ W2 )
=> ( W2
= ( semiring_1_of_nat @ int @ M2 ) ) )
& ( ~ ( ord_less_eq @ int @ ( zero_zero @ int ) @ W2 )
=> ( M2
= ( zero_zero @ nat ) ) ) ) ) ).
% nat_eq_iff2
thf(fact_3103_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_3104_summable__geometric,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [C3: A] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ C3 ) @ ( one_one @ real ) )
=> ( summable @ A @ ( power_power @ A @ C3 ) ) ) ) ).
% summable_geometric
thf(fact_3105_split__nat,axiom,
! [P2: nat > $o,I2: int] :
( ( P2 @ ( nat2 @ I2 ) )
= ( ! [N2: nat] :
( ( I2
= ( semiring_1_of_nat @ int @ N2 ) )
=> ( P2 @ N2 ) )
& ( ( ord_less @ int @ I2 @ ( zero_zero @ int ) )
=> ( P2 @ ( zero_zero @ nat ) ) ) ) ) ).
% split_nat
thf(fact_3106_le__mult__nat__floor,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [A2: A,B2: A] : ( ord_less_eq @ nat @ ( times_times @ nat @ ( nat2 @ ( archim6421214686448440834_floor @ A @ A2 ) ) @ ( nat2 @ ( archim6421214686448440834_floor @ A @ B2 ) ) ) @ ( nat2 @ ( archim6421214686448440834_floor @ A @ ( times_times @ A @ A2 @ B2 ) ) ) ) ) ).
% le_mult_nat_floor
thf(fact_3107_le__nat__iff,axiom,
! [K2: int,N: nat] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ K2 )
=> ( ( ord_less_eq @ nat @ N @ ( nat2 @ K2 ) )
= ( ord_less_eq @ int @ ( semiring_1_of_nat @ int @ N ) @ K2 ) ) ) ).
% le_nat_iff
thf(fact_3108_suminf__split__head,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: nat > A] :
( ( summable @ A @ F3 )
=> ( ( suminf @ A
@ ^ [N2: nat] : ( F3 @ ( suc @ N2 ) ) )
= ( minus_minus @ A @ ( suminf @ A @ F3 ) @ ( F3 @ ( zero_zero @ nat ) ) ) ) ) ) ).
% suminf_split_head
thf(fact_3109_nat__add__distrib,axiom,
! [Z: int,Z7: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z7 )
=> ( ( nat2 @ ( plus_plus @ int @ Z @ Z7 ) )
= ( plus_plus @ nat @ ( nat2 @ Z ) @ ( nat2 @ Z7 ) ) ) ) ) ).
% nat_add_distrib
thf(fact_3110_nat__mult__distrib,axiom,
! [Z: int,Z7: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z )
=> ( ( nat2 @ ( times_times @ int @ Z @ Z7 ) )
= ( times_times @ nat @ ( nat2 @ Z ) @ ( nat2 @ Z7 ) ) ) ) ).
% nat_mult_distrib
thf(fact_3111_nat__diff__distrib,axiom,
! [Z7: int,Z: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z7 )
=> ( ( ord_less_eq @ int @ Z7 @ Z )
=> ( ( nat2 @ ( minus_minus @ int @ Z @ Z7 ) )
= ( minus_minus @ nat @ ( nat2 @ Z ) @ ( nat2 @ Z7 ) ) ) ) ) ).
% nat_diff_distrib
thf(fact_3112_nat__diff__distrib_H,axiom,
! [X: int,Y: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ X )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Y )
=> ( ( nat2 @ ( minus_minus @ int @ X @ Y ) )
= ( minus_minus @ nat @ ( nat2 @ X ) @ ( nat2 @ Y ) ) ) ) ) ).
% nat_diff_distrib'
thf(fact_3113_nat__power__eq,axiom,
! [Z: int,N: nat] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z )
=> ( ( nat2 @ ( power_power @ int @ Z @ N ) )
= ( power_power @ nat @ ( nat2 @ Z ) @ N ) ) ) ).
% nat_power_eq
thf(fact_3114_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_3115_choose__dvd,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [K2: nat,N: nat] :
( ( ord_less_eq @ nat @ K2 @ N )
=> ( dvd_dvd @ A @ ( times_times @ A @ ( semiring_char_0_fact @ A @ K2 ) @ ( semiring_char_0_fact @ A @ ( minus_minus @ nat @ N @ K2 ) ) ) @ ( semiring_char_0_fact @ A @ N ) ) ) ) ).
% choose_dvd
thf(fact_3116_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_3117_le__nat__floor,axiom,
! [X: nat,A2: real] :
( ( ord_less_eq @ real @ ( semiring_1_of_nat @ real @ X ) @ A2 )
=> ( ord_less_eq @ nat @ X @ ( nat2 @ ( archim6421214686448440834_floor @ real @ A2 ) ) ) ) ).
% le_nat_floor
thf(fact_3118_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
@ ^ [X5: nat] : X5
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ N ) @ M2 ) ) ) ) ) ).
% fact_eq_fact_times
thf(fact_3119_sum__le__suminf,axiom,
! [A: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F3: nat > A,I6: set @ nat] :
( ( summable @ A @ F3 )
=> ( ( finite_finite2 @ nat @ I6 )
=> ( ! [N3: nat] :
( ( member @ nat @ N3 @ ( uminus_uminus @ ( set @ nat ) @ I6 ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F3 @ N3 ) ) )
=> ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ I6 ) @ ( suminf @ A @ F3 ) ) ) ) ) ) ).
% sum_le_suminf
thf(fact_3120_nat__2,axiom,
( ( nat2 @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) )
= ( suc @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% nat_2
thf(fact_3121_binomial__altdef__nat,axiom,
! [K2: nat,N: nat] :
( ( ord_less_eq @ nat @ K2 @ N )
=> ( ( binomial @ N @ K2 )
= ( divide_divide @ nat @ ( semiring_char_0_fact @ nat @ N ) @ ( times_times @ nat @ ( semiring_char_0_fact @ nat @ K2 ) @ ( semiring_char_0_fact @ nat @ ( minus_minus @ nat @ N @ K2 ) ) ) ) ) ) ).
% binomial_altdef_nat
thf(fact_3122_Suc__nat__eq__nat__zadd1,axiom,
! [Z: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z )
=> ( ( suc @ ( nat2 @ Z ) )
= ( nat2 @ ( plus_plus @ int @ ( one_one @ int ) @ Z ) ) ) ) ).
% Suc_nat_eq_nat_zadd1
thf(fact_3123_nat__less__iff,axiom,
! [W2: int,M2: nat] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ W2 )
=> ( ( ord_less @ nat @ ( nat2 @ W2 ) @ M2 )
= ( ord_less @ int @ W2 @ ( semiring_1_of_nat @ int @ M2 ) ) ) ) ).
% nat_less_iff
thf(fact_3124_nat__mult__distrib__neg,axiom,
! [Z: int,Z7: int] :
( ( ord_less_eq @ int @ Z @ ( zero_zero @ int ) )
=> ( ( nat2 @ ( times_times @ int @ Z @ Z7 ) )
= ( times_times @ nat @ ( nat2 @ ( uminus_uminus @ int @ Z ) ) @ ( nat2 @ ( uminus_uminus @ int @ Z7 ) ) ) ) ) ).
% nat_mult_distrib_neg
thf(fact_3125_suminf__split__initial__segment,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: nat > A,K2: nat] :
( ( summable @ A @ F3 )
=> ( ( suminf @ A @ F3 )
= ( plus_plus @ A
@ ( suminf @ A
@ ^ [N2: nat] : ( F3 @ ( plus_plus @ nat @ N2 @ K2 ) ) )
@ ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_ord_lessThan @ nat @ K2 ) ) ) ) ) ) ).
% suminf_split_initial_segment
thf(fact_3126_suminf__minus__initial__segment,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: nat > A,K2: nat] :
( ( summable @ A @ F3 )
=> ( ( suminf @ A
@ ^ [N2: nat] : ( F3 @ ( plus_plus @ nat @ N2 @ K2 ) ) )
= ( minus_minus @ A @ ( suminf @ A @ F3 ) @ ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_ord_lessThan @ nat @ K2 ) ) ) ) ) ) ).
% suminf_minus_initial_segment
thf(fact_3127_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_3128_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
@ ^ [X5: nat] : X5
@ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ N @ ( one_one @ nat ) ) @ M2 ) ) ) ) ).
% fact_div_fact
thf(fact_3129_diff__nat__eq__if,axiom,
! [Z7: int,Z: int] :
( ( ( ord_less @ int @ Z7 @ ( zero_zero @ int ) )
=> ( ( minus_minus @ nat @ ( nat2 @ Z ) @ ( nat2 @ Z7 ) )
= ( nat2 @ Z ) ) )
& ( ~ ( ord_less @ int @ Z7 @ ( zero_zero @ int ) )
=> ( ( minus_minus @ nat @ ( nat2 @ Z ) @ ( nat2 @ Z7 ) )
= ( if @ nat @ ( ord_less @ int @ ( minus_minus @ int @ Z @ Z7 ) @ ( zero_zero @ int ) ) @ ( zero_zero @ nat ) @ ( nat2 @ ( minus_minus @ int @ Z @ Z7 ) ) ) ) ) ) ).
% diff_nat_eq_if
thf(fact_3130_sum__less__suminf,axiom,
! [A: $tType] :
( ( ( ordere8940638589300402666id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F3: nat > A,N: nat] :
( ( summable @ A @ F3 )
=> ( ! [M: nat] :
( ( ord_less_eq @ nat @ N @ M )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( F3 @ M ) ) )
=> ( ord_less @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_ord_lessThan @ nat @ N ) ) @ ( suminf @ A @ F3 ) ) ) ) ) ).
% sum_less_suminf
thf(fact_3131_powser__split__head_I1_J,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V8999393235501362500lgebra @ A ) )
=> ! [F3: nat > A,Z: A] :
( ( summable @ A
@ ^ [N2: nat] : ( times_times @ A @ ( F3 @ N2 ) @ ( power_power @ A @ Z @ N2 ) ) )
=> ( ( suminf @ A
@ ^ [N2: nat] : ( times_times @ A @ ( F3 @ N2 ) @ ( power_power @ A @ Z @ N2 ) ) )
= ( plus_plus @ A @ ( F3 @ ( zero_zero @ nat ) )
@ ( times_times @ A
@ ( suminf @ A
@ ^ [N2: nat] : ( times_times @ A @ ( F3 @ ( suc @ N2 ) ) @ ( power_power @ A @ Z @ N2 ) ) )
@ Z ) ) ) ) ) ).
% powser_split_head(1)
thf(fact_3132_powser__split__head_I2_J,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V8999393235501362500lgebra @ A ) )
=> ! [F3: nat > A,Z: A] :
( ( summable @ A
@ ^ [N2: nat] : ( times_times @ A @ ( F3 @ N2 ) @ ( power_power @ A @ Z @ N2 ) ) )
=> ( ( times_times @ A
@ ( suminf @ A
@ ^ [N2: nat] : ( times_times @ A @ ( F3 @ ( suc @ N2 ) ) @ ( power_power @ A @ Z @ N2 ) ) )
@ Z )
= ( minus_minus @ A
@ ( suminf @ A
@ ^ [N2: nat] : ( times_times @ A @ ( F3 @ N2 ) @ ( power_power @ A @ Z @ N2 ) ) )
@ ( F3 @ ( zero_zero @ nat ) ) ) ) ) ) ).
% powser_split_head(2)
thf(fact_3133_summable__partial__sum__bound,axiom,
! [A: $tType] :
( ( real_Vector_banach @ A )
=> ! [F3: nat > A,E3: real] :
( ( summable @ A @ F3 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ E3 )
=> ~ ! [N9: nat] :
~ ! [M4: nat] :
( ( ord_less_eq @ nat @ N9 @ M4 )
=> ! [N5: nat] : ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_or1337092689740270186AtMost @ nat @ M4 @ N5 ) ) ) @ E3 ) ) ) ) ) ).
% summable_partial_sum_bound
thf(fact_3134_suminf__exist__split,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [R2: real,F3: nat > A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R2 )
=> ( ( summable @ A @ F3 )
=> ? [N9: nat] :
! [N5: nat] :
( ( ord_less_eq @ nat @ N9 @ N5 )
=> ( ord_less @ real
@ ( real_V7770717601297561774m_norm @ A
@ ( suminf @ A
@ ^ [I4: nat] : ( F3 @ ( plus_plus @ nat @ I4 @ N5 ) ) ) )
@ R2 ) ) ) ) ) ).
% suminf_exist_split
thf(fact_3135_summable__power__series,axiom,
! [F3: nat > real,Z: real] :
( ! [I3: nat] : ( ord_less_eq @ real @ ( F3 @ I3 ) @ ( one_one @ real ) )
=> ( ! [I3: nat] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( F3 @ I3 ) )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Z )
=> ( ( ord_less @ real @ Z @ ( one_one @ real ) )
=> ( summable @ real
@ ^ [I4: nat] : ( times_times @ real @ ( F3 @ I4 ) @ ( power_power @ real @ Z @ I4 ) ) ) ) ) ) ) ).
% summable_power_series
thf(fact_3136_Abel__lemma,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [R2: real,R0: real,A2: nat > A,M5: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ R2 )
=> ( ( ord_less @ real @ R2 @ R0 )
=> ( ! [N3: nat] : ( ord_less_eq @ real @ ( times_times @ real @ ( real_V7770717601297561774m_norm @ A @ ( A2 @ N3 ) ) @ ( power_power @ real @ R0 @ N3 ) ) @ M5 )
=> ( summable @ real
@ ^ [N2: nat] : ( times_times @ real @ ( real_V7770717601297561774m_norm @ A @ ( A2 @ N2 ) ) @ ( power_power @ real @ R2 @ N2 ) ) ) ) ) ) ) ).
% Abel_lemma
thf(fact_3137_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_3138_nat__dvd__iff,axiom,
! [Z: int,M2: nat] :
( ( dvd_dvd @ nat @ ( nat2 @ Z ) @ M2 )
= ( ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z )
=> ( dvd_dvd @ int @ Z @ ( semiring_1_of_nat @ int @ M2 ) ) )
& ( ~ ( ord_less_eq @ int @ ( zero_zero @ int ) @ Z )
=> ( M2
= ( zero_zero @ nat ) ) ) ) ) ).
% nat_dvd_iff
thf(fact_3139_fact__num__eq__if,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ( ( semiring_char_0_fact @ A )
= ( ^ [M3: nat] :
( if @ A
@ ( M3
= ( zero_zero @ nat ) )
@ ( one_one @ A )
@ ( times_times @ A @ ( semiring_1_of_nat @ A @ M3 ) @ ( semiring_char_0_fact @ A @ ( minus_minus @ nat @ M3 @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% fact_num_eq_if
thf(fact_3140_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_3141_fact__binomial,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K2: nat,N: nat] :
( ( ord_less_eq @ nat @ K2 @ N )
=> ( ( times_times @ A @ ( semiring_char_0_fact @ A @ K2 ) @ ( semiring_1_of_nat @ A @ ( binomial @ N @ K2 ) ) )
= ( divide_divide @ A @ ( semiring_char_0_fact @ A @ N ) @ ( semiring_char_0_fact @ A @ ( minus_minus @ nat @ N @ K2 ) ) ) ) ) ) ).
% fact_binomial
thf(fact_3142_binomial__fact,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K2: nat,N: nat] :
( ( ord_less_eq @ nat @ K2 @ N )
=> ( ( semiring_1_of_nat @ A @ ( binomial @ N @ K2 ) )
= ( divide_divide @ A @ ( semiring_char_0_fact @ A @ N ) @ ( times_times @ A @ ( semiring_char_0_fact @ A @ K2 ) @ ( semiring_char_0_fact @ A @ ( minus_minus @ nat @ N @ K2 ) ) ) ) ) ) ) ).
% binomial_fact
thf(fact_3143_summable__ratio__test,axiom,
! [A: $tType] :
( ( real_Vector_banach @ A )
=> ! [C3: real,N6: nat,F3: nat > A] :
( ( ord_less @ real @ C3 @ ( one_one @ real ) )
=> ( ! [N3: nat] :
( ( ord_less_eq @ nat @ N6 @ N3 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( F3 @ ( suc @ N3 ) ) ) @ ( times_times @ real @ C3 @ ( real_V7770717601297561774m_norm @ A @ ( F3 @ N3 ) ) ) ) )
=> ( summable @ A @ F3 ) ) ) ) ).
% summable_ratio_test
thf(fact_3144_sum__less__suminf2,axiom,
! [A: $tType] :
( ( ( ordere8940638589300402666id_add @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F3: nat > A,N: nat,I2: nat] :
( ( summable @ A @ F3 )
=> ( ! [M: nat] :
( ( ord_less_eq @ nat @ N @ M )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F3 @ M ) ) )
=> ( ( ord_less_eq @ nat @ N @ I2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ ( F3 @ I2 ) )
=> ( ord_less @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_ord_lessThan @ nat @ N ) ) @ ( suminf @ A @ F3 ) ) ) ) ) ) ) ).
% sum_less_suminf2
thf(fact_3145_gbinomial__pochhammer_H,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( ( gbinomial @ A )
= ( ^ [A6: A,K3: nat] : ( divide_divide @ A @ ( comm_s3205402744901411588hammer @ A @ ( plus_plus @ A @ ( minus_minus @ A @ A6 @ ( semiring_1_of_nat @ A @ K3 ) ) @ ( one_one @ A ) ) @ K3 ) @ ( semiring_char_0_fact @ A @ K3 ) ) ) ) ) ).
% gbinomial_pochhammer'
thf(fact_3146_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
@ ^ [M3: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M3 @ ( zero_zero @ A ) ) @ ( semiring_char_0_fact @ real @ M3 ) ) @ ( power_power @ real @ X @ M3 ) )
@ ( set_ord_lessThan @ nat @ N ) )
= ( Diff @ ( zero_zero @ nat ) @ ( zero_zero @ A ) ) ) ) ) ) ).
% Maclaurin_zero
thf(fact_3147_Maclaurin__lemma,axiom,
! [H2: real,F3: real > real,J: nat > real,N: nat] :
( ( ord_less @ real @ ( zero_zero @ real ) @ H2 )
=> ? [B9: real] :
( ( F3 @ H2 )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M3: nat] : ( times_times @ real @ ( divide_divide @ real @ ( J @ M3 ) @ ( semiring_char_0_fact @ real @ M3 ) ) @ ( power_power @ real @ H2 @ M3 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ B9 @ ( divide_divide @ real @ ( power_power @ real @ H2 @ N ) @ ( semiring_char_0_fact @ real @ N ) ) ) ) ) ) ).
% Maclaurin_lemma
thf(fact_3148_gbinomial__Suc,axiom,
! [A: $tType] :
( ( ( semiring_char_0 @ A )
& ( semidom_divide @ A ) )
=> ! [A2: A,K2: nat] :
( ( gbinomial @ A @ A2 @ ( suc @ K2 ) )
= ( divide_divide @ A
@ ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( minus_minus @ A @ A2 @ ( semiring_1_of_nat @ A @ I4 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ K2 ) )
@ ( semiring_char_0_fact @ A @ ( suc @ K2 ) ) ) ) ) ).
% gbinomial_Suc
thf(fact_3149_divide__int__unfold,axiom,
! [L: int,K2: int,N: nat,M2: nat] :
( ( ( ( ( sgn_sgn @ int @ L )
= ( zero_zero @ int ) )
| ( ( sgn_sgn @ int @ K2 )
= ( zero_zero @ int ) )
| ( N
= ( zero_zero @ nat ) ) )
=> ( ( divide_divide @ int @ ( times_times @ int @ ( sgn_sgn @ int @ K2 ) @ ( semiring_1_of_nat @ int @ 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 @ K2 )
= ( zero_zero @ int ) )
| ( N
= ( zero_zero @ nat ) ) )
=> ( ( ( ( sgn_sgn @ int @ K2 )
= ( sgn_sgn @ int @ L ) )
=> ( ( divide_divide @ int @ ( times_times @ int @ ( sgn_sgn @ int @ K2 ) @ ( semiring_1_of_nat @ int @ 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 @ K2 )
!= ( sgn_sgn @ int @ L ) )
=> ( ( divide_divide @ int @ ( times_times @ int @ ( sgn_sgn @ int @ K2 ) @ ( 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_3150_sin__coeff__def,axiom,
( sin_coeff
= ( ^ [N2: nat] : ( if @ real @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) @ ( zero_zero @ real ) @ ( divide_divide @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ ( divide_divide @ nat @ ( minus_minus @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( semiring_char_0_fact @ real @ N2 ) ) ) ) ) ).
% sin_coeff_def
thf(fact_3151_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_3152_diffs__equiv,axiom,
! [A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( ring_1 @ A ) )
=> ! [C3: nat > A,X: A] :
( ( summable @ A
@ ^ [N2: nat] : ( times_times @ A @ ( diffs @ A @ C3 @ N2 ) @ ( power_power @ A @ X @ N2 ) ) )
=> ( sums @ A
@ ^ [N2: nat] : ( times_times @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( C3 @ N2 ) ) @ ( power_power @ A @ X @ ( minus_minus @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) ) ) )
@ ( suminf @ A
@ ^ [N2: nat] : ( times_times @ A @ ( diffs @ A @ C3 @ N2 ) @ ( power_power @ A @ X @ N2 ) ) ) ) ) ) ).
% diffs_equiv
thf(fact_3153_Maclaurin__exp__lt,axiom,
! [X: real,N: nat] :
( ( X
!= ( zero_zero @ real ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ? [T7: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( abs_abs @ real @ T7 ) )
& ( ord_less @ real @ ( abs_abs @ real @ T7 ) @ ( abs_abs @ real @ X ) )
& ( ( exp @ real @ X )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M3: nat] : ( divide_divide @ real @ ( power_power @ real @ X @ M3 ) @ ( semiring_char_0_fact @ real @ M3 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( exp @ real @ T7 ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ X @ N ) ) ) ) ) ) ) ).
% Maclaurin_exp_lt
thf(fact_3154_sin__paired,axiom,
! [X: real] :
( sums @ real
@ ^ [N2: nat] : ( times_times @ real @ ( divide_divide @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ N2 ) @ ( semiring_char_0_fact @ real @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) @ ( one_one @ nat ) ) ) ) @ ( power_power @ real @ X @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) @ ( one_one @ nat ) ) ) )
@ ( sin @ real @ X ) ) ).
% sin_paired
thf(fact_3155_abs__idempotent,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A2: A] :
( ( abs_abs @ A @ ( abs_abs @ A @ A2 ) )
= ( abs_abs @ A @ A2 ) ) ) ).
% abs_idempotent
thf(fact_3156_abs__abs,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ! [A2: A] :
( ( abs_abs @ A @ ( abs_abs @ A @ A2 ) )
= ( abs_abs @ A @ A2 ) ) ) ).
% abs_abs
thf(fact_3157_abs__0,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ( ( abs_abs @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% abs_0
thf(fact_3158_abs__0__eq,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A2: A] :
( ( ( zero_zero @ A )
= ( abs_abs @ A @ A2 ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% abs_0_eq
thf(fact_3159_abs__eq__0,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A2: A] :
( ( ( abs_abs @ A @ A2 )
= ( zero_zero @ A ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% abs_eq_0
thf(fact_3160_abs__zero,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ( ( abs_abs @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% abs_zero
thf(fact_3161_abs__mult__self__eq,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A] :
( ( times_times @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ A2 ) )
= ( times_times @ A @ A2 @ A2 ) ) ) ).
% abs_mult_self_eq
thf(fact_3162_abs__1,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ( ( abs_abs @ A @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% abs_1
thf(fact_3163_abs__add__abs,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A2: A,B2: A] :
( ( abs_abs @ A @ ( plus_plus @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B2 ) ) )
= ( plus_plus @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B2 ) ) ) ) ).
% abs_add_abs
thf(fact_3164_abs__divide,axiom,
! [A: $tType] :
( ( field_abs_sgn @ A )
=> ! [A2: A,B2: A] :
( ( abs_abs @ A @ ( divide_divide @ A @ A2 @ B2 ) )
= ( divide_divide @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B2 ) ) ) ) ).
% abs_divide
thf(fact_3165_abs__minus__cancel,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A2: A] :
( ( abs_abs @ A @ ( uminus_uminus @ A @ A2 ) )
= ( abs_abs @ A @ A2 ) ) ) ).
% abs_minus_cancel
thf(fact_3166_abs__minus,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ! [A2: A] :
( ( abs_abs @ A @ ( uminus_uminus @ A @ A2 ) )
= ( abs_abs @ A @ A2 ) ) ) ).
% abs_minus
thf(fact_3167_dvd__abs__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M2: A,K2: A] :
( ( dvd_dvd @ A @ M2 @ ( abs_abs @ A @ K2 ) )
= ( dvd_dvd @ A @ M2 @ K2 ) ) ) ).
% dvd_abs_iff
thf(fact_3168_abs__dvd__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [M2: A,K2: A] :
( ( dvd_dvd @ A @ ( abs_abs @ A @ M2 ) @ K2 )
= ( dvd_dvd @ A @ M2 @ K2 ) ) ) ).
% abs_dvd_iff
thf(fact_3169_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_3170_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_3171_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_3172_abs__bool__eq,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [P2: $o] :
( ( abs_abs @ A @ ( zero_neq_one_of_bool @ A @ P2 ) )
= ( zero_neq_one_of_bool @ A @ P2 ) ) ) ).
% abs_bool_eq
thf(fact_3173_abs__of__nonneg,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( abs_abs @ A @ A2 )
= A2 ) ) ) ).
% abs_of_nonneg
thf(fact_3174_abs__le__self__iff,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ ( abs_abs @ A @ A2 ) @ A2 )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% abs_le_self_iff
thf(fact_3175_abs__le__zero__iff,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ ( abs_abs @ A @ A2 ) @ ( zero_zero @ A ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% abs_le_zero_iff
thf(fact_3176_zero__less__abs__iff,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( abs_abs @ A @ A2 ) )
= ( A2
!= ( zero_zero @ A ) ) ) ) ).
% zero_less_abs_iff
thf(fact_3177_sin__coeff__0,axiom,
( ( sin_coeff @ ( zero_zero @ nat ) )
= ( zero_zero @ real ) ) ).
% sin_coeff_0
thf(fact_3178_sum__abs,axiom,
! [B: $tType,A: $tType] :
( ( ordere166539214618696060dd_abs @ B )
=> ! [F3: A > B,A5: set @ A] :
( ord_less_eq @ B @ ( abs_abs @ B @ ( groups7311177749621191930dd_sum @ A @ B @ F3 @ A5 ) )
@ ( groups7311177749621191930dd_sum @ A @ B
@ ^ [I4: A] : ( abs_abs @ B @ ( F3 @ I4 ) )
@ A5 ) ) ) ).
% sum_abs
thf(fact_3179_divide__le__0__abs__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( divide_divide @ A @ A2 @ ( abs_abs @ A @ B2 ) ) @ ( zero_zero @ A ) )
= ( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
| ( B2
= ( zero_zero @ A ) ) ) ) ) ).
% divide_le_0_abs_iff
thf(fact_3180_zero__le__divide__abs__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( divide_divide @ A @ A2 @ ( abs_abs @ A @ B2 ) ) )
= ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
| ( B2
= ( zero_zero @ A ) ) ) ) ) ).
% zero_le_divide_abs_iff
thf(fact_3181_abs__of__nonpos,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( abs_abs @ A @ A2 )
= ( uminus_uminus @ A @ A2 ) ) ) ) ).
% abs_of_nonpos
thf(fact_3182_abs__sgn__eq__1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( abs_abs @ A @ ( sgn_sgn @ A @ A2 ) )
= ( one_one @ A ) ) ) ) ).
% abs_sgn_eq_1
thf(fact_3183_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_3184_idom__abs__sgn__class_Oabs__sgn,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ! [A2: A] :
( ( sgn_sgn @ A @ ( abs_abs @ A @ A2 ) )
= ( zero_neq_one_of_bool @ A
@ ( A2
!= ( zero_zero @ A ) ) ) ) ) ).
% idom_abs_sgn_class.abs_sgn
thf(fact_3185_sgn__abs,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ! [A2: A] :
( ( abs_abs @ A @ ( sgn_sgn @ A @ A2 ) )
= ( zero_neq_one_of_bool @ A
@ ( A2
!= ( zero_zero @ A ) ) ) ) ) ).
% sgn_abs
thf(fact_3186_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_3187_sum__abs__ge__zero,axiom,
! [B: $tType,A: $tType] :
( ( ordere166539214618696060dd_abs @ B )
=> ! [F3: A > B,A5: set @ A] :
( ord_less_eq @ B @ ( zero_zero @ B )
@ ( groups7311177749621191930dd_sum @ A @ B
@ ^ [I4: A] : ( abs_abs @ B @ ( F3 @ I4 ) )
@ A5 ) ) ) ).
% sum_abs_ge_zero
thf(fact_3188_zero__less__power__abs__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A,N: nat] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( power_power @ A @ ( abs_abs @ A @ A2 ) @ N ) )
= ( ( A2
!= ( zero_zero @ A ) )
| ( N
= ( zero_zero @ nat ) ) ) ) ) ).
% zero_less_power_abs_iff
thf(fact_3189_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_3190_norm__of__real__addn,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [X: real,B2: num] :
( ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ ( real_Vector_of_real @ A @ X ) @ ( numeral_numeral @ A @ B2 ) ) )
= ( abs_abs @ real @ ( plus_plus @ real @ X @ ( numeral_numeral @ real @ B2 ) ) ) ) ) ).
% norm_of_real_addn
thf(fact_3191_abs__eq__iff,axiom,
! [A: $tType] :
( ( linordered_ring @ A )
=> ! [X: A,Y: A] :
( ( ( abs_abs @ A @ X )
= ( abs_abs @ A @ Y ) )
= ( ( X = Y )
| ( X
= ( uminus_uminus @ A @ Y ) ) ) ) ) ).
% abs_eq_iff
thf(fact_3192_abs__one,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( ( abs_abs @ A @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% abs_one
thf(fact_3193_abs__mult,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ! [A2: A,B2: A] :
( ( abs_abs @ A @ ( times_times @ A @ A2 @ B2 ) )
= ( times_times @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B2 ) ) ) ) ).
% abs_mult
thf(fact_3194_abs__eq__0__iff,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ! [A2: A] :
( ( ( abs_abs @ A @ A2 )
= ( zero_zero @ A ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% abs_eq_0_iff
thf(fact_3195_abs__ge__self,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A2: A] : ( ord_less_eq @ A @ A2 @ ( abs_abs @ A @ A2 ) ) ) ).
% abs_ge_self
thf(fact_3196_abs__le__D1,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( abs_abs @ A @ A2 ) @ B2 )
=> ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% abs_le_D1
thf(fact_3197_dvd__if__abs__eq,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [L: A,K2: A] :
( ( ( abs_abs @ A @ L )
= ( abs_abs @ A @ K2 ) )
=> ( dvd_dvd @ A @ L @ K2 ) ) ) ).
% dvd_if_abs_eq
thf(fact_3198_abs__minus__commute,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A2: A,B2: A] :
( ( abs_abs @ A @ ( minus_minus @ A @ A2 @ B2 ) )
= ( abs_abs @ A @ ( minus_minus @ A @ B2 @ A2 ) ) ) ) ).
% abs_minus_commute
thf(fact_3199_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_3200_sgn__power__injE,axiom,
! [A2: real,N: nat,X: real,B2: real] :
( ( ( times_times @ real @ ( sgn_sgn @ real @ A2 ) @ ( power_power @ real @ ( abs_abs @ real @ A2 ) @ N ) )
= X )
=> ( ( X
= ( times_times @ real @ ( sgn_sgn @ real @ B2 ) @ ( power_power @ real @ ( abs_abs @ real @ B2 ) @ N ) ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( A2 = B2 ) ) ) ) ).
% sgn_power_injE
thf(fact_3201_abs__ge__zero,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A2: A] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( abs_abs @ A @ A2 ) ) ) ).
% abs_ge_zero
thf(fact_3202_abs__of__pos,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( abs_abs @ A @ A2 )
= A2 ) ) ) ).
% abs_of_pos
thf(fact_3203_abs__not__less__zero,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A2: A] :
~ ( ord_less @ A @ ( abs_abs @ A @ A2 ) @ ( zero_zero @ A ) ) ) ).
% abs_not_less_zero
thf(fact_3204_abs__triangle__ineq,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A2: A,B2: A] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( plus_plus @ A @ A2 @ B2 ) ) @ ( plus_plus @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B2 ) ) ) ) ).
% abs_triangle_ineq
thf(fact_3205_abs__mult__less,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A,C3: A,B2: A,D3: A] :
( ( ord_less @ A @ ( abs_abs @ A @ A2 ) @ C3 )
=> ( ( ord_less @ A @ ( abs_abs @ A @ B2 ) @ D3 )
=> ( ord_less @ A @ ( times_times @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B2 ) ) @ ( times_times @ A @ C3 @ D3 ) ) ) ) ) ).
% abs_mult_less
thf(fact_3206_abs__triangle__ineq2__sym,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A2: A,B2: A] : ( ord_less_eq @ A @ ( minus_minus @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B2 ) ) @ ( abs_abs @ A @ ( minus_minus @ A @ B2 @ A2 ) ) ) ) ).
% abs_triangle_ineq2_sym
thf(fact_3207_abs__triangle__ineq3,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A2: A,B2: A] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B2 ) ) ) @ ( abs_abs @ A @ ( minus_minus @ A @ A2 @ B2 ) ) ) ) ).
% abs_triangle_ineq3
thf(fact_3208_abs__triangle__ineq2,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A2: A,B2: A] : ( ord_less_eq @ A @ ( minus_minus @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B2 ) ) @ ( abs_abs @ A @ ( minus_minus @ A @ A2 @ B2 ) ) ) ) ).
% abs_triangle_ineq2
thf(fact_3209_nonzero__abs__divide,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [B2: A,A2: A] :
( ( B2
!= ( zero_zero @ A ) )
=> ( ( abs_abs @ A @ ( divide_divide @ A @ A2 @ B2 ) )
= ( divide_divide @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B2 ) ) ) ) ) ).
% nonzero_abs_divide
thf(fact_3210_abs__ge__minus__self,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A2: A] : ( ord_less_eq @ A @ ( uminus_uminus @ A @ A2 ) @ ( abs_abs @ A @ A2 ) ) ) ).
% abs_ge_minus_self
thf(fact_3211_abs__le__iff,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( abs_abs @ A @ A2 ) @ B2 )
= ( ( ord_less_eq @ A @ A2 @ B2 )
& ( ord_less_eq @ A @ ( uminus_uminus @ A @ A2 ) @ B2 ) ) ) ) ).
% abs_le_iff
thf(fact_3212_abs__le__D2,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( abs_abs @ A @ A2 ) @ B2 )
=> ( ord_less_eq @ A @ ( uminus_uminus @ A @ A2 ) @ B2 ) ) ) ).
% abs_le_D2
thf(fact_3213_abs__leI,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ A2 ) @ B2 )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ A2 ) @ B2 ) ) ) ) ).
% abs_leI
thf(fact_3214_abs__less__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( abs_abs @ A @ A2 ) @ B2 )
= ( ( ord_less @ A @ A2 @ B2 )
& ( ord_less @ A @ ( uminus_uminus @ A @ A2 ) @ B2 ) ) ) ) ).
% abs_less_iff
thf(fact_3215_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_3216_abs__mult__sgn,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ! [A2: A] :
( ( times_times @ A @ ( abs_abs @ A @ A2 ) @ ( sgn_sgn @ A @ A2 ) )
= A2 ) ) ).
% abs_mult_sgn
thf(fact_3217_sgn__mult__abs,axiom,
! [A: $tType] :
( ( idom_abs_sgn @ A )
=> ! [A2: A] :
( ( times_times @ A @ ( sgn_sgn @ A @ A2 ) @ ( abs_abs @ A @ A2 ) )
= A2 ) ) ).
% sgn_mult_abs
thf(fact_3218_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_3219_same__sgn__abs__add,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [B2: A,A2: A] :
( ( ( sgn_sgn @ A @ B2 )
= ( sgn_sgn @ A @ A2 ) )
=> ( ( abs_abs @ A @ ( plus_plus @ A @ A2 @ B2 ) )
= ( plus_plus @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B2 ) ) ) ) ) ).
% same_sgn_abs_add
thf(fact_3220_dense__eq0__I,axiom,
! [A: $tType] :
( ( ( ordere166539214618696060dd_abs @ A )
& ( dense_linorder @ A ) )
=> ! [X: A] :
( ! [E2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ E2 )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ X ) @ E2 ) )
=> ( X
= ( zero_zero @ A ) ) ) ) ).
% dense_eq0_I
thf(fact_3221_abs__mult__pos,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( times_times @ A @ ( abs_abs @ A @ Y ) @ X )
= ( abs_abs @ A @ ( times_times @ A @ Y @ X ) ) ) ) ) ).
% abs_mult_pos
thf(fact_3222_abs__eq__mult,axiom,
! [A: $tType] :
( ( ordered_ring_abs @ A )
=> ! [A2: A,B2: A] :
( ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
| ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) )
& ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
| ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) ) )
=> ( ( abs_abs @ A @ ( times_times @ A @ A2 @ B2 ) )
= ( times_times @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B2 ) ) ) ) ) ).
% abs_eq_mult
thf(fact_3223_abs__eq__iff_H,axiom,
! [A: $tType] :
( ( linordered_ring @ A )
=> ! [A2: A,B2: A] :
( ( ( abs_abs @ A @ A2 )
= B2 )
= ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
& ( ( A2 = B2 )
| ( A2
= ( uminus_uminus @ A @ B2 ) ) ) ) ) ) ).
% abs_eq_iff'
thf(fact_3224_eq__abs__iff_H,axiom,
! [A: $tType] :
( ( linordered_ring @ A )
=> ! [A2: A,B2: A] :
( ( A2
= ( abs_abs @ A @ B2 ) )
= ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
& ( ( B2 = A2 )
| ( B2
= ( uminus_uminus @ A @ A2 ) ) ) ) ) ) ).
% eq_abs_iff'
thf(fact_3225_abs__minus__le__zero,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A2: A] : ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( abs_abs @ A @ A2 ) ) @ ( zero_zero @ A ) ) ) ).
% abs_minus_le_zero
thf(fact_3226_zero__le__power__abs,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A,N: nat] : ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( power_power @ A @ ( abs_abs @ A @ A2 ) @ N ) ) ) ).
% zero_le_power_abs
thf(fact_3227_abs__div__pos,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [Y: A,X: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ Y )
=> ( ( divide_divide @ A @ ( abs_abs @ A @ X ) @ Y )
= ( abs_abs @ A @ ( divide_divide @ A @ X @ Y ) ) ) ) ) ).
% abs_div_pos
thf(fact_3228_abs__if,axiom,
! [A: $tType] :
( ( abs_if @ A )
=> ( ( abs_abs @ A )
= ( ^ [A6: A] : ( if @ A @ ( ord_less @ A @ A6 @ ( zero_zero @ A ) ) @ ( uminus_uminus @ A @ A6 ) @ A6 ) ) ) ) ).
% abs_if
thf(fact_3229_abs__of__neg,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A2: A] :
( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( abs_abs @ A @ A2 )
= ( uminus_uminus @ A @ A2 ) ) ) ) ).
% abs_of_neg
thf(fact_3230_abs__if__raw,axiom,
! [A: $tType] :
( ( abs_if @ A )
=> ( ( abs_abs @ A )
= ( ^ [A6: A] : ( if @ A @ ( ord_less @ A @ A6 @ ( zero_zero @ A ) ) @ ( uminus_uminus @ A @ A6 ) @ A6 ) ) ) ) ).
% abs_if_raw
thf(fact_3231_abs__diff__le__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,A2: A,R2: A] :
( ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X @ A2 ) ) @ R2 )
= ( ( ord_less_eq @ A @ ( minus_minus @ A @ A2 @ R2 ) @ X )
& ( ord_less_eq @ A @ X @ ( plus_plus @ A @ A2 @ R2 ) ) ) ) ) ).
% abs_diff_le_iff
thf(fact_3232_abs__triangle__ineq4,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A2: A,B2: A] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ A2 @ B2 ) ) @ ( plus_plus @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B2 ) ) ) ) ).
% abs_triangle_ineq4
thf(fact_3233_abs__diff__triangle__ineq,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [A2: A,B2: A,C3: A,D3: A] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ ( plus_plus @ A @ A2 @ B2 ) @ ( plus_plus @ A @ C3 @ D3 ) ) ) @ ( plus_plus @ A @ ( abs_abs @ A @ ( minus_minus @ A @ A2 @ C3 ) ) @ ( abs_abs @ A @ ( minus_minus @ A @ B2 @ D3 ) ) ) ) ) ).
% abs_diff_triangle_ineq
thf(fact_3234_abs__diff__less__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,A2: A,R2: A] :
( ( ord_less @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X @ A2 ) ) @ R2 )
= ( ( ord_less @ A @ ( minus_minus @ A @ A2 @ R2 ) @ X )
& ( ord_less @ A @ X @ ( plus_plus @ A @ A2 @ R2 ) ) ) ) ) ).
% abs_diff_less_iff
thf(fact_3235_abs__sgn__eq,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A] :
( ( ( A2
= ( zero_zero @ A ) )
=> ( ( abs_abs @ A @ ( sgn_sgn @ A @ A2 ) )
= ( zero_zero @ A ) ) )
& ( ( A2
!= ( zero_zero @ A ) )
=> ( ( abs_abs @ A @ ( sgn_sgn @ A @ A2 ) )
= ( one_one @ A ) ) ) ) ) ).
% abs_sgn_eq
thf(fact_3236_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_3237_abs__real__def,axiom,
( ( abs_abs @ real )
= ( ^ [A6: real] : ( if @ real @ ( ord_less @ real @ A6 @ ( zero_zero @ real ) ) @ ( uminus_uminus @ real @ A6 ) @ A6 ) ) ) ).
% abs_real_def
thf(fact_3238_summable__rabs__comparison__test,axiom,
! [F3: nat > real,G3: nat > real] :
( ? [N8: nat] :
! [N3: nat] :
( ( ord_less_eq @ nat @ N8 @ N3 )
=> ( ord_less_eq @ real @ ( abs_abs @ real @ ( F3 @ N3 ) ) @ ( G3 @ N3 ) ) )
=> ( ( summable @ real @ G3 )
=> ( summable @ real
@ ^ [N2: nat] : ( abs_abs @ real @ ( F3 @ N2 ) ) ) ) ) ).
% summable_rabs_comparison_test
thf(fact_3239_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_3240_sgn__real__def,axiom,
( ( sgn_sgn @ real )
= ( ^ [A6: real] :
( if @ real
@ ( A6
= ( zero_zero @ real ) )
@ ( zero_zero @ real )
@ ( if @ real @ ( ord_less @ real @ ( zero_zero @ real ) @ A6 ) @ ( one_one @ real ) @ ( uminus_uminus @ real @ ( one_one @ real ) ) ) ) ) ) ).
% sgn_real_def
thf(fact_3241_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_3242_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_3243_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_3244_round__diff__minimal,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [Z: A,M2: int] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ Z @ ( ring_1_of_int @ A @ ( archimedean_round @ A @ Z ) ) ) ) @ ( abs_abs @ A @ ( minus_minus @ A @ Z @ ( ring_1_of_int @ A @ M2 ) ) ) ) ) ).
% round_diff_minimal
thf(fact_3245_abs__le__square__iff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ ( abs_abs @ A @ X ) @ ( abs_abs @ A @ Y ) )
= ( ord_less_eq @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% abs_le_square_iff
thf(fact_3246_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_3247_power2__le__iff__abs__le,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ Y )
=> ( ( ord_less_eq @ A @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ A @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( ord_less_eq @ A @ ( abs_abs @ A @ X ) @ Y ) ) ) ) ).
% power2_le_iff_abs_le
thf(fact_3248_abs__sqrt__wlog,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [P2: A > A > $o,X: A] :
( ! [X4: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X4 )
=> ( P2 @ X4 @ ( power_power @ A @ X4 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
=> ( P2 @ ( abs_abs @ A @ X ) @ ( power_power @ A @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ).
% abs_sqrt_wlog
thf(fact_3249_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_3250_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_3251_power__mono__even,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: nat,A2: A,B2: A] :
( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( ( ord_less_eq @ A @ ( abs_abs @ A @ A2 ) @ ( abs_abs @ A @ B2 ) )
=> ( ord_less_eq @ A @ ( power_power @ A @ A2 @ N ) @ ( power_power @ A @ B2 @ N ) ) ) ) ) ).
% power_mono_even
thf(fact_3252_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_3253_convex__sum__bound__le,axiom,
! [A: $tType,B: $tType] :
( ( linordered_idom @ B )
=> ! [I6: set @ A,X: A > B,A2: A > B,B2: B,Delta: B] :
( ! [I3: A] :
( ( member @ A @ I3 @ I6 )
=> ( ord_less_eq @ B @ ( zero_zero @ B ) @ ( X @ I3 ) ) )
=> ( ( ( groups7311177749621191930dd_sum @ A @ B @ X @ I6 )
= ( one_one @ B ) )
=> ( ! [I3: A] :
( ( member @ A @ I3 @ I6 )
=> ( ord_less_eq @ B @ ( abs_abs @ B @ ( minus_minus @ B @ ( A2 @ I3 ) @ B2 ) ) @ Delta ) )
=> ( ord_less_eq @ B
@ ( abs_abs @ B
@ ( minus_minus @ B
@ ( groups7311177749621191930dd_sum @ A @ B
@ ^ [I4: A] : ( times_times @ B @ ( A2 @ I4 ) @ ( X @ I4 ) )
@ I6 )
@ B2 ) )
@ Delta ) ) ) ) ) ).
% convex_sum_bound_le
thf(fact_3254_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_3255_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_3256_termdiff__converges,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A,K5: real,C3: nat > A] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ K5 )
=> ( ! [X4: A] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X4 ) @ K5 )
=> ( summable @ A
@ ^ [N2: nat] : ( times_times @ A @ ( C3 @ N2 ) @ ( power_power @ A @ X4 @ N2 ) ) ) )
=> ( summable @ A
@ ^ [N2: nat] : ( times_times @ A @ ( diffs @ A @ C3 @ N2 ) @ ( power_power @ A @ X @ N2 ) ) ) ) ) ) ).
% termdiff_converges
thf(fact_3257_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_3258_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_3259_sin__monotone__2pi,axiom,
! [Y: real,X: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ Y )
=> ( ( ord_less @ real @ Y @ X )
=> ( ( ord_less_eq @ real @ X @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less @ real @ ( sin @ real @ Y ) @ ( sin @ real @ X ) ) ) ) ) ).
% sin_monotone_2pi
thf(fact_3260_sin__mono__less__eq,axiom,
! [X: real,Y: 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 ) ) ) ) @ Y )
=> ( ( ord_less_eq @ real @ Y @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less @ real @ ( sin @ real @ X ) @ ( sin @ real @ Y ) )
= ( ord_less @ real @ X @ Y ) ) ) ) ) ) ).
% sin_mono_less_eq
thf(fact_3261_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_3262_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_3263_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_3264_real__sqrt__sum__squares__less,axiom,
! [X: real,U: real,Y: real] :
( ( ord_less @ real @ ( abs_abs @ real @ X ) @ ( divide_divide @ real @ U @ ( sqrt @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) )
=> ( ( ord_less @ real @ ( abs_abs @ real @ Y ) @ ( 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 @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ U ) ) ) ).
% real_sqrt_sum_squares_less
thf(fact_3265_Maclaurin__sin__expansion4,axiom,
! [X: real,N: nat] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ? [T7: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ T7 )
& ( ord_less_eq @ real @ T7 @ X )
& ( ( sin @ real @ X )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M3: nat] : ( times_times @ real @ ( sin_coeff @ M3 ) @ ( power_power @ real @ X @ M3 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( sin @ real @ ( plus_plus @ real @ T7 @ ( 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_3266_Maclaurin__sin__expansion3,axiom,
! [N: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ? [T7: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ T7 )
& ( ord_less @ real @ T7 @ X )
& ( ( sin @ real @ X )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M3: nat] : ( times_times @ real @ ( sin_coeff @ M3 ) @ ( power_power @ real @ X @ M3 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( sin @ real @ ( plus_plus @ real @ T7 @ ( 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_3267_sin__x__sin__y,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A,Y: A] :
( sums @ A
@ ^ [P6: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [N2: nat] :
( if @ A
@ ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ P6 )
& ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) )
@ ( times_times @ A @ ( real_V8093663219630862766scaleR @ A @ ( uminus_uminus @ real @ ( divide_divide @ real @ ( ring_1_of_int @ real @ ( times_times @ int @ ( power_power @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( divide_divide @ nat @ P6 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( semiring_1_of_nat @ int @ ( binomial @ P6 @ N2 ) ) ) ) @ ( semiring_char_0_fact @ real @ P6 ) ) ) @ ( power_power @ A @ X @ N2 ) ) @ ( power_power @ A @ Y @ ( minus_minus @ nat @ P6 @ N2 ) ) )
@ ( zero_zero @ A ) )
@ ( set_ord_atMost @ nat @ P6 ) )
@ ( times_times @ A @ ( sin @ A @ X ) @ ( sin @ A @ Y ) ) ) ) ).
% sin_x_sin_y
thf(fact_3268_lemma__interval,axiom,
! [A2: real,X: real,B2: real] :
( ( ord_less @ real @ A2 @ X )
=> ( ( ord_less @ real @ X @ B2 )
=> ? [D2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D2 )
& ! [Y4: real] :
( ( ord_less @ real @ ( abs_abs @ real @ ( minus_minus @ real @ X @ Y4 ) ) @ D2 )
=> ( ( ord_less_eq @ real @ A2 @ Y4 )
& ( ord_less_eq @ real @ Y4 @ B2 ) ) ) ) ) ) ).
% lemma_interval
thf(fact_3269_monoseq__arctan__series,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ X ) @ ( one_one @ real ) )
=> ( topological_monoseq @ real
@ ^ [N2: nat] : ( times_times @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( semiring_1_of_nat @ real @ ( plus_plus @ nat @ ( times_times @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) @ ( power_power @ real @ X @ ( plus_plus @ nat @ ( times_times @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) ) ) ).
% monoseq_arctan_series
thf(fact_3270_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_3271_lemma__interval__lt,axiom,
! [A2: real,X: real,B2: real] :
( ( ord_less @ real @ A2 @ X )
=> ( ( ord_less @ real @ X @ B2 )
=> ? [D2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D2 )
& ! [Y4: real] :
( ( ord_less @ real @ ( abs_abs @ real @ ( minus_minus @ real @ X @ Y4 ) ) @ D2 )
=> ( ( ord_less @ real @ A2 @ Y4 )
& ( ord_less @ real @ Y4 @ B2 ) ) ) ) ) ) ).
% lemma_interval_lt
thf(fact_3272_scaleR__cancel__right,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [A2: real,X: A,B2: real] :
( ( ( real_V8093663219630862766scaleR @ A @ A2 @ X )
= ( real_V8093663219630862766scaleR @ A @ B2 @ X ) )
= ( ( A2 = B2 )
| ( X
= ( zero_zero @ A ) ) ) ) ) ).
% scaleR_cancel_right
thf(fact_3273_scaleR__zero__right,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [A2: real] :
( ( real_V8093663219630862766scaleR @ A @ A2 @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% scaleR_zero_right
thf(fact_3274_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_3275_scaleR__eq__0__iff,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [A2: real,X: A] :
( ( ( real_V8093663219630862766scaleR @ A @ A2 @ X )
= ( zero_zero @ A ) )
= ( ( A2
= ( zero_zero @ real ) )
| ( X
= ( zero_zero @ A ) ) ) ) ) ).
% scaleR_eq_0_iff
thf(fact_3276_scaleR__eq__iff,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [B2: A,U: real,A2: A] :
( ( ( plus_plus @ A @ B2 @ ( real_V8093663219630862766scaleR @ A @ U @ A2 ) )
= ( plus_plus @ A @ A2 @ ( real_V8093663219630862766scaleR @ A @ U @ B2 ) ) )
= ( ( A2 = B2 )
| ( U
= ( one_one @ real ) ) ) ) ) ).
% scaleR_eq_iff
thf(fact_3277_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_3278_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_3279_zdvd1__eq,axiom,
! [X: int] :
( ( dvd_dvd @ int @ X @ ( one_one @ int ) )
= ( ( abs_abs @ int @ X )
= ( one_one @ int ) ) ) ).
% zdvd1_eq
thf(fact_3280_scaleR__collapse,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [U: real,A2: A] :
( ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ ( minus_minus @ real @ ( one_one @ real ) @ U ) @ A2 ) @ ( real_V8093663219630862766scaleR @ A @ U @ A2 ) )
= A2 ) ) ).
% scaleR_collapse
thf(fact_3281_zabs__less__one__iff,axiom,
! [Z: int] :
( ( ord_less @ int @ ( abs_abs @ int @ Z ) @ ( one_one @ int ) )
= ( Z
= ( zero_zero @ int ) ) ) ).
% zabs_less_one_iff
thf(fact_3282_dvd__nat__abs__iff,axiom,
! [N: nat,K2: int] :
( ( dvd_dvd @ nat @ N @ ( nat2 @ ( abs_abs @ int @ K2 ) ) )
= ( dvd_dvd @ int @ ( semiring_1_of_nat @ int @ N ) @ K2 ) ) ).
% dvd_nat_abs_iff
thf(fact_3283_nat__abs__dvd__iff,axiom,
! [K2: int,N: nat] :
( ( dvd_dvd @ nat @ ( nat2 @ ( abs_abs @ int @ K2 ) ) @ N )
= ( dvd_dvd @ int @ K2 @ ( semiring_1_of_nat @ int @ N ) ) ) ).
% nat_abs_dvd_iff
thf(fact_3284_scaleR__half__double,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [A2: A] :
( ( real_V8093663219630862766scaleR @ A @ ( divide_divide @ real @ ( one_one @ real ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( plus_plus @ A @ A2 @ A2 ) )
= A2 ) ) ).
% scaleR_half_double
thf(fact_3285_scaleR__right__imp__eq,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [X: A,A2: real,B2: real] :
( ( X
!= ( zero_zero @ A ) )
=> ( ( ( real_V8093663219630862766scaleR @ A @ A2 @ X )
= ( real_V8093663219630862766scaleR @ A @ B2 @ X ) )
=> ( A2 = B2 ) ) ) ) ).
% scaleR_right_imp_eq
thf(fact_3286_scaleR__right__distrib,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [A2: real,X: A,Y: A] :
( ( real_V8093663219630862766scaleR @ A @ A2 @ ( plus_plus @ A @ X @ Y ) )
= ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ X ) @ ( real_V8093663219630862766scaleR @ A @ A2 @ Y ) ) ) ) ).
% scaleR_right_distrib
thf(fact_3287_zdvd__antisym__abs,axiom,
! [A2: int,B2: int] :
( ( dvd_dvd @ int @ A2 @ B2 )
=> ( ( dvd_dvd @ int @ B2 @ A2 )
=> ( ( abs_abs @ int @ A2 )
= ( abs_abs @ int @ B2 ) ) ) ) ).
% zdvd_antisym_abs
thf(fact_3288_arctan__less__iff,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ ( arctan @ X ) @ ( arctan @ Y ) )
= ( ord_less @ real @ X @ Y ) ) ).
% arctan_less_iff
thf(fact_3289_arctan__monotone,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ X @ Y )
=> ( ord_less @ real @ ( arctan @ X ) @ ( arctan @ Y ) ) ) ).
% arctan_monotone
thf(fact_3290_scaleR__left__distrib,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [A2: real,B2: real,X: A] :
( ( real_V8093663219630862766scaleR @ A @ ( plus_plus @ real @ A2 @ B2 ) @ X )
= ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ X ) @ ( real_V8093663219630862766scaleR @ A @ B2 @ X ) ) ) ) ).
% scaleR_left_distrib
thf(fact_3291_scaleR__left_Oadd,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [X: real,Y: real,Xa2: A] :
( ( real_V8093663219630862766scaleR @ A @ ( plus_plus @ real @ X @ Y ) @ Xa2 )
= ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ X @ Xa2 ) @ ( real_V8093663219630862766scaleR @ A @ Y @ Xa2 ) ) ) ) ).
% scaleR_left.add
thf(fact_3292_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_3293_infinite__int__iff__unbounded__le,axiom,
! [S2: set @ int] :
( ( ~ ( finite_finite2 @ int @ S2 ) )
= ( ! [M3: int] :
? [N2: int] :
( ( ord_less_eq @ int @ M3 @ ( abs_abs @ int @ N2 ) )
& ( member @ int @ N2 @ S2 ) ) ) ) ).
% infinite_int_iff_unbounded_le
thf(fact_3294_infinite__int__iff__unbounded,axiom,
! [S2: set @ int] :
( ( ~ ( finite_finite2 @ int @ S2 ) )
= ( ! [M3: int] :
? [N2: int] :
( ( ord_less @ int @ M3 @ ( abs_abs @ int @ N2 ) )
& ( member @ int @ N2 @ S2 ) ) ) ) ).
% infinite_int_iff_unbounded
thf(fact_3295_scaleR__right__mono__neg,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [B2: real,A2: real,C3: A] :
( ( ord_less_eq @ real @ B2 @ A2 )
=> ( ( ord_less_eq @ A @ C3 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ C3 ) @ ( real_V8093663219630862766scaleR @ A @ B2 @ C3 ) ) ) ) ) ).
% scaleR_right_mono_neg
thf(fact_3296_scaleR__right__mono,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A2: real,B2: real,X: A] :
( ( ord_less_eq @ real @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ X ) @ ( real_V8093663219630862766scaleR @ A @ B2 @ X ) ) ) ) ) ).
% scaleR_right_mono
thf(fact_3297_scaleR__le__cancel__left,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C3: real,A2: A,B2: A] :
( ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ C3 @ A2 ) @ ( real_V8093663219630862766scaleR @ A @ C3 @ B2 ) )
= ( ( ( ord_less @ real @ ( zero_zero @ real ) @ C3 )
=> ( ord_less_eq @ A @ A2 @ B2 ) )
& ( ( ord_less @ real @ C3 @ ( zero_zero @ real ) )
=> ( ord_less_eq @ A @ B2 @ A2 ) ) ) ) ) ).
% scaleR_le_cancel_left
thf(fact_3298_scaleR__le__cancel__left__neg,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C3: real,A2: A,B2: A] :
( ( ord_less @ real @ C3 @ ( zero_zero @ real ) )
=> ( ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ C3 @ A2 ) @ ( real_V8093663219630862766scaleR @ A @ C3 @ B2 ) )
= ( ord_less_eq @ A @ B2 @ A2 ) ) ) ) ).
% scaleR_le_cancel_left_neg
thf(fact_3299_scaleR__le__cancel__left__pos,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C3: real,A2: A,B2: A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ C3 )
=> ( ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ C3 @ A2 ) @ ( real_V8093663219630862766scaleR @ A @ C3 @ B2 ) )
= ( ord_less_eq @ A @ A2 @ B2 ) ) ) ) ).
% scaleR_le_cancel_left_pos
thf(fact_3300_scaleR__left__mono,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [X: A,Y: A,A2: real] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ A2 )
=> ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ X ) @ ( real_V8093663219630862766scaleR @ A @ A2 @ Y ) ) ) ) ) ).
% scaleR_left_mono
thf(fact_3301_scaleR__left__mono__neg,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [B2: A,A2: A,C3: real] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_less_eq @ real @ C3 @ ( zero_zero @ real ) )
=> ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ C3 @ A2 ) @ ( real_V8093663219630862766scaleR @ A @ C3 @ B2 ) ) ) ) ) ).
% scaleR_left_mono_neg
thf(fact_3302_eq__vector__fraction__iff,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [X: A,U: real,V: real,A2: A] :
( ( X
= ( real_V8093663219630862766scaleR @ A @ ( divide_divide @ real @ U @ V ) @ A2 ) )
= ( ( ( V
= ( zero_zero @ real ) )
=> ( X
= ( zero_zero @ A ) ) )
& ( ( V
!= ( zero_zero @ real ) )
=> ( ( real_V8093663219630862766scaleR @ A @ V @ X )
= ( real_V8093663219630862766scaleR @ A @ U @ A2 ) ) ) ) ) ) ).
% eq_vector_fraction_iff
thf(fact_3303_vector__fraction__eq__iff,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [U: real,V: real,A2: A,X: A] :
( ( ( real_V8093663219630862766scaleR @ A @ ( divide_divide @ real @ U @ V ) @ A2 )
= X )
= ( ( ( V
= ( zero_zero @ real ) )
=> ( X
= ( zero_zero @ A ) ) )
& ( ( V
!= ( zero_zero @ real ) )
=> ( ( real_V8093663219630862766scaleR @ A @ U @ A2 )
= ( real_V8093663219630862766scaleR @ A @ V @ X ) ) ) ) ) ) ).
% vector_fraction_eq_iff
thf(fact_3304_Real__Vector__Spaces_Ole__add__iff1,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A2: real,E3: A,C3: A,B2: real,D3: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ E3 ) @ C3 ) @ ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ B2 @ E3 ) @ D3 ) )
= ( ord_less_eq @ A @ ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ ( minus_minus @ real @ A2 @ B2 ) @ E3 ) @ C3 ) @ D3 ) ) ) ).
% Real_Vector_Spaces.le_add_iff1
thf(fact_3305_Real__Vector__Spaces_Ole__add__iff2,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A2: real,E3: A,C3: A,B2: real,D3: A] :
( ( ord_less_eq @ A @ ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ E3 ) @ C3 ) @ ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ B2 @ E3 ) @ D3 ) )
= ( ord_less_eq @ A @ C3 @ ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ ( minus_minus @ real @ B2 @ A2 ) @ E3 ) @ D3 ) ) ) ) ).
% Real_Vector_Spaces.le_add_iff2
thf(fact_3306_zabs__def,axiom,
( ( abs_abs @ int )
= ( ^ [I4: int] : ( if @ int @ ( ord_less @ int @ I4 @ ( zero_zero @ int ) ) @ ( uminus_uminus @ int @ I4 ) @ I4 ) ) ) ).
% zabs_def
thf(fact_3307_dvd__imp__le__int,axiom,
! [I2: int,D3: int] :
( ( I2
!= ( zero_zero @ int ) )
=> ( ( dvd_dvd @ int @ D3 @ I2 )
=> ( ord_less_eq @ int @ ( abs_abs @ int @ D3 ) @ ( abs_abs @ int @ I2 ) ) ) ) ).
% dvd_imp_le_int
thf(fact_3308_nat__abs__mult__distrib,axiom,
! [W2: int,Z: int] :
( ( nat2 @ ( abs_abs @ int @ ( times_times @ int @ W2 @ Z ) ) )
= ( times_times @ nat @ ( nat2 @ ( abs_abs @ int @ W2 ) ) @ ( nat2 @ ( abs_abs @ int @ Z ) ) ) ) ).
% nat_abs_mult_distrib
thf(fact_3309_abs__mod__less,axiom,
! [L: int,K2: int] :
( ( L
!= ( zero_zero @ int ) )
=> ( ord_less @ int @ ( abs_abs @ int @ ( modulo_modulo @ int @ K2 @ L ) ) @ ( abs_abs @ int @ L ) ) ) ).
% abs_mod_less
thf(fact_3310_zero__le__scaleR__iff,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A2: real,B2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( real_V8093663219630862766scaleR @ A @ A2 @ B2 ) )
= ( ( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) )
| ( ( ord_less @ real @ A2 @ ( zero_zero @ real ) )
& ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) )
| ( A2
= ( zero_zero @ real ) ) ) ) ) ).
% zero_le_scaleR_iff
thf(fact_3311_scaleR__le__0__iff,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A2: real,B2: A] :
( ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ B2 ) @ ( zero_zero @ A ) )
= ( ( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
& ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) )
| ( ( ord_less @ real @ A2 @ ( zero_zero @ real ) )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) )
| ( A2
= ( zero_zero @ real ) ) ) ) ) ).
% scaleR_le_0_iff
thf(fact_3312_scaleR__nonpos__nonpos,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A2: real,B2: A] :
( ( ord_less_eq @ real @ A2 @ ( zero_zero @ real ) )
=> ( ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( real_V8093663219630862766scaleR @ A @ A2 @ B2 ) ) ) ) ) ).
% scaleR_nonpos_nonpos
thf(fact_3313_scaleR__nonpos__nonneg,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A2: real,X: A] :
( ( ord_less_eq @ real @ A2 @ ( zero_zero @ real ) )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ X ) @ ( zero_zero @ A ) ) ) ) ) ).
% scaleR_nonpos_nonneg
thf(fact_3314_scaleR__nonneg__nonpos,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A2: real,X: A] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ A2 )
=> ( ( ord_less_eq @ A @ X @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ X ) @ ( zero_zero @ A ) ) ) ) ) ).
% scaleR_nonneg_nonpos
thf(fact_3315_scaleR__nonneg__nonneg,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A2: real,X: A] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( real_V8093663219630862766scaleR @ A @ A2 @ X ) ) ) ) ) ).
% scaleR_nonneg_nonneg
thf(fact_3316_split__scaleR__pos__le,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A2: real,B2: A] :
( ( ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ A2 )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 ) )
| ( ( ord_less_eq @ real @ A2 @ ( zero_zero @ real ) )
& ( ord_less_eq @ A @ B2 @ ( zero_zero @ A ) ) ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( real_V8093663219630862766scaleR @ A @ A2 @ B2 ) ) ) ) ).
% split_scaleR_pos_le
thf(fact_3317_split__scaleR__neg__le,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A2: real,X: A] :
( ( ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ A2 )
& ( ord_less_eq @ A @ X @ ( zero_zero @ A ) ) )
| ( ( ord_less_eq @ real @ A2 @ ( zero_zero @ real ) )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ X ) ) )
=> ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ X ) @ ( zero_zero @ A ) ) ) ) ).
% split_scaleR_neg_le
thf(fact_3318_scaleR__mono_H,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A2: real,B2: real,C3: A,D3: A] :
( ( ord_less_eq @ real @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C3 @ D3 )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ C3 )
=> ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ C3 ) @ ( real_V8093663219630862766scaleR @ A @ B2 @ D3 ) ) ) ) ) ) ) ).
% scaleR_mono'
thf(fact_3319_scaleR__mono,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [A2: real,B2: real,X: A,Y: A] :
( ( ord_less_eq @ real @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ B2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ X ) @ ( real_V8093663219630862766scaleR @ A @ B2 @ Y ) ) ) ) ) ) ) ).
% scaleR_mono
thf(fact_3320_scaleR__left__le__one__le,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [X: A,A2: real] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ X )
=> ( ( ord_less_eq @ real @ A2 @ ( one_one @ real ) )
=> ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ X ) @ X ) ) ) ) ).
% scaleR_left_le_one_le
thf(fact_3321_scaleR__2,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [X: A] :
( ( real_V8093663219630862766scaleR @ A @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ X )
= ( plus_plus @ A @ X @ X ) ) ) ).
% scaleR_2
thf(fact_3322_nat__abs__triangle__ineq,axiom,
! [K2: int,L: int] : ( ord_less_eq @ nat @ ( nat2 @ ( abs_abs @ int @ ( plus_plus @ int @ K2 @ L ) ) ) @ ( plus_plus @ nat @ ( nat2 @ ( abs_abs @ int @ K2 ) ) @ ( nat2 @ ( abs_abs @ int @ L ) ) ) ) ).
% nat_abs_triangle_ineq
thf(fact_3323_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_3324_nat__abs__int__diff,axiom,
! [A2: nat,B2: nat] :
( ( ( ord_less_eq @ nat @ A2 @ B2 )
=> ( ( nat2 @ ( abs_abs @ int @ ( minus_minus @ int @ ( semiring_1_of_nat @ int @ A2 ) @ ( semiring_1_of_nat @ int @ B2 ) ) ) )
= ( minus_minus @ nat @ B2 @ A2 ) ) )
& ( ~ ( ord_less_eq @ nat @ A2 @ B2 )
=> ( ( nat2 @ ( abs_abs @ int @ ( minus_minus @ int @ ( semiring_1_of_nat @ int @ A2 ) @ ( semiring_1_of_nat @ int @ B2 ) ) ) )
= ( minus_minus @ nat @ A2 @ B2 ) ) ) ) ).
% nat_abs_int_diff
thf(fact_3325_nat__intermed__int__val,axiom,
! [M2: nat,N: nat,F3: nat > int,K2: int] :
( ! [I3: nat] :
( ( ( ord_less_eq @ nat @ M2 @ I3 )
& ( ord_less @ nat @ I3 @ N ) )
=> ( ord_less_eq @ int @ ( abs_abs @ int @ ( minus_minus @ int @ ( F3 @ ( suc @ I3 ) ) @ ( F3 @ I3 ) ) ) @ ( one_one @ int ) ) )
=> ( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( ord_less_eq @ int @ ( F3 @ M2 ) @ K2 )
=> ( ( ord_less_eq @ int @ K2 @ ( F3 @ N ) )
=> ? [I3: nat] :
( ( ord_less_eq @ nat @ M2 @ I3 )
& ( ord_less_eq @ nat @ I3 @ N )
& ( ( F3 @ I3 )
= K2 ) ) ) ) ) ) ).
% nat_intermed_int_val
thf(fact_3326_decr__lemma,axiom,
! [D3: int,X: int,Z: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D3 )
=> ( ord_less @ int @ ( minus_minus @ int @ X @ ( times_times @ int @ ( plus_plus @ int @ ( abs_abs @ int @ ( minus_minus @ int @ X @ Z ) ) @ ( one_one @ int ) ) @ D3 ) ) @ Z ) ) ).
% decr_lemma
thf(fact_3327_incr__lemma,axiom,
! [D3: int,Z: int,X: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ D3 )
=> ( ord_less @ int @ Z @ ( plus_plus @ int @ X @ ( times_times @ int @ ( plus_plus @ int @ ( abs_abs @ int @ ( minus_minus @ int @ X @ Z ) ) @ ( one_one @ int ) ) @ D3 ) ) ) ) ).
% incr_lemma
thf(fact_3328_arctan__ubound,axiom,
! [Y: real] : ( ord_less @ real @ ( arctan @ Y ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ).
% arctan_ubound
thf(fact_3329_nat__ivt__aux,axiom,
! [N: nat,F3: nat > int,K2: int] :
( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ N )
=> ( ord_less_eq @ int @ ( abs_abs @ int @ ( minus_minus @ int @ ( F3 @ ( suc @ I3 ) ) @ ( F3 @ I3 ) ) ) @ ( one_one @ int ) ) )
=> ( ( ord_less_eq @ int @ ( F3 @ ( zero_zero @ nat ) ) @ K2 )
=> ( ( ord_less_eq @ int @ K2 @ ( F3 @ N ) )
=> ? [I3: nat] :
( ( ord_less_eq @ nat @ I3 @ N )
& ( ( F3 @ I3 )
= K2 ) ) ) ) ) ).
% nat_ivt_aux
thf(fact_3330_arctan__bounded,axiom,
! [Y: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ ( arctan @ Y ) )
& ( ord_less @ real @ ( arctan @ Y ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ).
% arctan_bounded
thf(fact_3331_arctan__lbound,axiom,
! [Y: real] : ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ ( arctan @ Y ) ) ).
% arctan_lbound
thf(fact_3332_nat0__intermed__int__val,axiom,
! [N: nat,F3: nat > int,K2: int] :
( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ N )
=> ( ord_less_eq @ int @ ( abs_abs @ int @ ( minus_minus @ int @ ( F3 @ ( plus_plus @ nat @ I3 @ ( one_one @ nat ) ) ) @ ( F3 @ I3 ) ) ) @ ( one_one @ int ) ) )
=> ( ( ord_less_eq @ int @ ( F3 @ ( zero_zero @ nat ) ) @ K2 )
=> ( ( ord_less_eq @ int @ K2 @ ( F3 @ N ) )
=> ? [I3: nat] :
( ( ord_less_eq @ nat @ I3 @ N )
& ( ( F3 @ I3 )
= K2 ) ) ) ) ) ).
% nat0_intermed_int_val
thf(fact_3333_arctan__add,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ X ) @ ( one_one @ real ) )
=> ( ( ord_less @ real @ ( abs_abs @ real @ Y ) @ ( one_one @ real ) )
=> ( ( plus_plus @ real @ ( arctan @ X ) @ ( arctan @ Y ) )
= ( arctan @ ( divide_divide @ real @ ( plus_plus @ real @ X @ Y ) @ ( minus_minus @ real @ ( one_one @ real ) @ ( times_times @ real @ X @ Y ) ) ) ) ) ) ) ).
% arctan_add
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_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_3336_sums__cos__x__plus__y,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A,Y: A] :
( sums @ A
@ ^ [P6: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [N2: 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 @ N2 ) ) ) ) @ ( semiring_char_0_fact @ real @ P6 ) ) @ ( power_power @ A @ X @ N2 ) ) @ ( power_power @ A @ Y @ ( minus_minus @ nat @ P6 @ N2 ) ) ) @ ( zero_zero @ A ) )
@ ( set_ord_atMost @ nat @ P6 ) )
@ ( cos @ A @ ( plus_plus @ A @ X @ Y ) ) ) ) ).
% sums_cos_x_plus_y
thf(fact_3337_cos__x__cos__y,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A,Y: A] :
( sums @ A
@ ^ [P6: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [N2: nat] :
( if @ A
@ ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ P6 )
& ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) )
@ ( times_times @ A @ ( real_V8093663219630862766scaleR @ A @ ( divide_divide @ real @ ( ring_1_of_int @ real @ ( times_times @ int @ ( power_power @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( divide_divide @ nat @ P6 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( semiring_1_of_nat @ int @ ( binomial @ P6 @ N2 ) ) ) ) @ ( semiring_char_0_fact @ real @ P6 ) ) @ ( power_power @ A @ X @ N2 ) ) @ ( power_power @ A @ Y @ ( minus_minus @ nat @ P6 @ N2 ) ) )
@ ( zero_zero @ A ) )
@ ( set_ord_atMost @ nat @ P6 ) )
@ ( times_times @ A @ ( cos @ A @ X ) @ ( cos @ A @ Y ) ) ) ) ).
% cos_x_cos_y
thf(fact_3338_sincos__total__2pi,axiom,
! [X: real,Y: real] :
( ( ( plus_plus @ real @ ( power_power @ real @ X @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( power_power @ real @ Y @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) )
= ( one_one @ real ) )
=> ~ ! [T7: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ T7 )
=> ( ( ord_less @ real @ T7 @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) )
=> ( ( X
= ( cos @ real @ T7 ) )
=> ( Y
!= ( sin @ real @ T7 ) ) ) ) ) ) ).
% sincos_total_2pi
thf(fact_3339_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_3340_monoseq__def,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( topological_monoseq @ A )
= ( ^ [X9: nat > A] :
( ! [M3: nat,N2: nat] :
( ( ord_less_eq @ nat @ M3 @ N2 )
=> ( ord_less_eq @ A @ ( X9 @ M3 ) @ ( X9 @ N2 ) ) )
| ! [M3: nat,N2: nat] :
( ( ord_less_eq @ nat @ M3 @ N2 )
=> ( ord_less_eq @ A @ ( X9 @ N2 ) @ ( X9 @ M3 ) ) ) ) ) ) ) ).
% monoseq_def
thf(fact_3341_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_3342_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_3343_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_3344_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_3345_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_3346_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_3347_add__tan__eq,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A,Y: A] :
( ( ( cos @ A @ X )
!= ( zero_zero @ A ) )
=> ( ( ( cos @ A @ Y )
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( tan @ A @ X ) @ ( tan @ A @ Y ) )
= ( divide_divide @ A @ ( sin @ A @ ( plus_plus @ A @ X @ Y ) ) @ ( times_times @ A @ ( cos @ A @ X ) @ ( cos @ A @ Y ) ) ) ) ) ) ) ).
% add_tan_eq
thf(fact_3348_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_3349_sin__add,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A,Y: A] :
( ( sin @ A @ ( plus_plus @ A @ X @ Y ) )
= ( plus_plus @ A @ ( times_times @ A @ ( sin @ A @ X ) @ ( cos @ A @ Y ) ) @ ( times_times @ A @ ( cos @ A @ X ) @ ( sin @ A @ Y ) ) ) ) ) ).
% sin_add
thf(fact_3350_lemma__tan__add1,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A,Y: A] :
( ( ( cos @ A @ X )
!= ( zero_zero @ A ) )
=> ( ( ( cos @ A @ Y )
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( tan @ A @ X ) @ ( tan @ A @ Y ) ) )
= ( divide_divide @ A @ ( cos @ A @ ( plus_plus @ A @ X @ Y ) ) @ ( times_times @ A @ ( cos @ A @ X ) @ ( cos @ A @ Y ) ) ) ) ) ) ) ).
% lemma_tan_add1
thf(fact_3351_tan__diff,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A,Y: A] :
( ( ( cos @ A @ X )
!= ( zero_zero @ A ) )
=> ( ( ( cos @ A @ Y )
!= ( zero_zero @ A ) )
=> ( ( ( cos @ A @ ( minus_minus @ A @ X @ Y ) )
!= ( zero_zero @ A ) )
=> ( ( tan @ A @ ( minus_minus @ A @ X @ Y ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( tan @ A @ X ) @ ( tan @ A @ Y ) ) @ ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( tan @ A @ X ) @ ( tan @ A @ Y ) ) ) ) ) ) ) ) ) ).
% tan_diff
thf(fact_3352_tan__add,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A,Y: A] :
( ( ( cos @ A @ X )
!= ( zero_zero @ A ) )
=> ( ( ( cos @ A @ Y )
!= ( zero_zero @ A ) )
=> ( ( ( cos @ A @ ( plus_plus @ A @ X @ Y ) )
!= ( zero_zero @ A ) )
=> ( ( tan @ A @ ( plus_plus @ A @ X @ Y ) )
= ( divide_divide @ A @ ( plus_plus @ A @ ( tan @ A @ X ) @ ( tan @ A @ Y ) ) @ ( minus_minus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( tan @ A @ X ) @ ( tan @ A @ Y ) ) ) ) ) ) ) ) ) ).
% tan_add
thf(fact_3353_cos__add,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A,Y: A] :
( ( cos @ A @ ( plus_plus @ A @ X @ Y ) )
= ( minus_minus @ A @ ( times_times @ A @ ( cos @ A @ X ) @ ( cos @ A @ Y ) ) @ ( times_times @ A @ ( sin @ A @ X ) @ ( sin @ A @ Y ) ) ) ) ) ).
% cos_add
thf(fact_3354_cos__diff,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A,Y: A] :
( ( cos @ A @ ( minus_minus @ A @ X @ Y ) )
= ( plus_plus @ A @ ( times_times @ A @ ( cos @ A @ X ) @ ( cos @ A @ Y ) ) @ ( times_times @ A @ ( sin @ A @ X ) @ ( sin @ A @ Y ) ) ) ) ) ).
% cos_diff
thf(fact_3355_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_3356_cos__monotone__0__pi,axiom,
! [Y: real,X: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y )
=> ( ( ord_less @ real @ Y @ X )
=> ( ( ord_less_eq @ real @ X @ pi )
=> ( ord_less @ real @ ( cos @ real @ X ) @ ( cos @ real @ Y ) ) ) ) ) ).
% cos_monotone_0_pi
thf(fact_3357_cos__mono__less__eq,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ X @ pi )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y )
=> ( ( ord_less_eq @ real @ Y @ pi )
=> ( ( ord_less @ real @ ( cos @ real @ X ) @ ( cos @ real @ Y ) )
= ( ord_less @ real @ Y @ X ) ) ) ) ) ) ).
% cos_mono_less_eq
thf(fact_3358_tan__half,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ( ( tan @ A )
= ( ^ [X5: A] : ( divide_divide @ A @ ( sin @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X5 ) ) @ ( plus_plus @ A @ ( cos @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ X5 ) ) @ ( one_one @ A ) ) ) ) ) ) ).
% tan_half
thf(fact_3359_cos__two__less__zero,axiom,
ord_less @ real @ ( cos @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( zero_zero @ real ) ).
% cos_two_less_zero
thf(fact_3360_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_3361_cos__monotone__minus__pi__0,axiom,
! [Y: real,X: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ pi ) @ Y )
=> ( ( ord_less @ real @ Y @ X )
=> ( ( ord_less_eq @ real @ X @ ( zero_zero @ real ) )
=> ( ord_less @ real @ ( cos @ real @ Y ) @ ( cos @ real @ X ) ) ) ) ) ).
% cos_monotone_minus_pi_0
thf(fact_3362_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_3363_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_3364_cos__plus__cos,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [W2: A,Z: A] :
( ( plus_plus @ A @ ( cos @ A @ W2 ) @ ( cos @ A @ Z ) )
= ( times_times @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( cos @ A @ ( divide_divide @ A @ ( plus_plus @ A @ W2 @ Z ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) @ ( cos @ A @ ( divide_divide @ A @ ( minus_minus @ A @ W2 @ Z ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% cos_plus_cos
thf(fact_3365_cos__times__cos,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [W2: A,Z: A] :
( ( times_times @ A @ ( cos @ A @ W2 ) @ ( cos @ A @ Z ) )
= ( divide_divide @ A @ ( plus_plus @ A @ ( cos @ A @ ( minus_minus @ A @ W2 @ Z ) ) @ ( cos @ A @ ( plus_plus @ A @ W2 @ Z ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% cos_times_cos
thf(fact_3366_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_3367_lemma__tan__total,axiom,
! [Y: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ Y )
=> ? [X4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X4 )
& ( ord_less @ real @ X4 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
& ( ord_less @ real @ Y @ ( tan @ real @ X4 ) ) ) ) ).
% lemma_tan_total
thf(fact_3368_tan__total,axiom,
! [Y: real] :
? [X4: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X4 )
& ( ord_less @ real @ X4 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
& ( ( tan @ real @ X4 )
= Y )
& ! [Y4: real] :
( ( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ Y4 )
& ( ord_less @ real @ Y4 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
& ( ( tan @ real @ Y4 )
= Y ) )
=> ( Y4 = X4 ) ) ) ).
% tan_total
thf(fact_3369_tan__monotone,axiom,
! [Y: real,X: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ Y )
=> ( ( ord_less @ real @ Y @ X )
=> ( ( ord_less @ real @ X @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less @ real @ ( tan @ real @ Y ) @ ( tan @ real @ X ) ) ) ) ) ).
% tan_monotone
thf(fact_3370_tan__monotone_H,axiom,
! [Y: real,X: 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 ) ) ) )
=> ( ( 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 @ Y @ X )
= ( ord_less @ real @ ( tan @ real @ Y ) @ ( tan @ real @ X ) ) ) ) ) ) ) ).
% tan_monotone'
thf(fact_3371_tan__mono__lt__eq,axiom,
! [X: real,Y: 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 ) ) ) ) @ Y )
=> ( ( ord_less @ real @ Y @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less @ real @ ( tan @ real @ X ) @ ( tan @ real @ Y ) )
= ( ord_less @ real @ X @ Y ) ) ) ) ) ) ).
% tan_mono_lt_eq
thf(fact_3372_lemma__tan__total1,axiom,
! [Y: real] :
? [X4: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X4 )
& ( ord_less @ real @ X4 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
& ( ( tan @ real @ X4 )
= Y ) ) ).
% lemma_tan_total1
thf(fact_3373_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_3374_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_3375_cos__diff__cos,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [W2: A,Z: A] :
( ( minus_minus @ A @ ( cos @ A @ W2 ) @ ( cos @ A @ Z ) )
= ( times_times @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( sin @ A @ ( divide_divide @ A @ ( plus_plus @ A @ W2 @ Z ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) @ ( sin @ A @ ( divide_divide @ A @ ( minus_minus @ A @ Z @ W2 ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% cos_diff_cos
thf(fact_3376_sin__diff__sin,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [W2: A,Z: A] :
( ( minus_minus @ A @ ( sin @ A @ W2 ) @ ( sin @ A @ Z ) )
= ( times_times @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( sin @ A @ ( divide_divide @ A @ ( minus_minus @ A @ W2 @ Z ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) @ ( cos @ A @ ( divide_divide @ A @ ( plus_plus @ A @ W2 @ Z ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% sin_diff_sin
thf(fact_3377_sin__plus__sin,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [W2: A,Z: A] :
( ( plus_plus @ A @ ( sin @ A @ W2 ) @ ( sin @ A @ Z ) )
= ( times_times @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( sin @ A @ ( divide_divide @ A @ ( plus_plus @ A @ W2 @ Z ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) @ ( cos @ A @ ( divide_divide @ A @ ( minus_minus @ A @ W2 @ Z ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% sin_plus_sin
thf(fact_3378_cos__times__sin,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [W2: A,Z: A] :
( ( times_times @ A @ ( cos @ A @ W2 ) @ ( sin @ A @ Z ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( sin @ A @ ( plus_plus @ A @ W2 @ Z ) ) @ ( sin @ A @ ( minus_minus @ A @ W2 @ Z ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% cos_times_sin
thf(fact_3379_sin__times__cos,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [W2: A,Z: A] :
( ( times_times @ A @ ( sin @ A @ W2 ) @ ( cos @ A @ Z ) )
= ( divide_divide @ A @ ( plus_plus @ A @ ( sin @ A @ ( plus_plus @ A @ W2 @ Z ) ) @ ( sin @ A @ ( minus_minus @ A @ W2 @ Z ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% sin_times_cos
thf(fact_3380_sin__times__sin,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [W2: A,Z: A] :
( ( times_times @ A @ ( sin @ A @ W2 ) @ ( sin @ A @ Z ) )
= ( divide_divide @ A @ ( minus_minus @ A @ ( cos @ A @ ( minus_minus @ A @ W2 @ Z ) ) @ ( cos @ A @ ( plus_plus @ A @ W2 @ Z ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ).
% sin_times_sin
thf(fact_3381_tan__total__pos,axiom,
! [Y: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y )
=> ? [X4: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X4 )
& ( ord_less @ real @ X4 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
& ( ( tan @ real @ X4 )
= Y ) ) ) ).
% tan_total_pos
thf(fact_3382_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_3383_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_3384_tan__mono__le__eq,axiom,
! [X: real,Y: 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 ) ) ) ) @ Y )
=> ( ( ord_less @ real @ Y @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less_eq @ real @ ( tan @ real @ X ) @ ( tan @ real @ Y ) )
= ( ord_less_eq @ real @ X @ Y ) ) ) ) ) ) ).
% tan_mono_le_eq
thf(fact_3385_tan__mono__le,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X )
=> ( ( ord_less_eq @ real @ X @ Y )
=> ( ( ord_less @ real @ Y @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less_eq @ real @ ( tan @ real @ X ) @ ( tan @ real @ Y ) ) ) ) ) ).
% tan_mono_le
thf(fact_3386_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_3387_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_3388_arctan,axiom,
! [Y: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ ( arctan @ Y ) )
& ( ord_less @ real @ ( arctan @ Y ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
& ( ( tan @ real @ ( arctan @ Y ) )
= Y ) ) ).
% arctan
thf(fact_3389_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_3390_arctan__unique,axiom,
! [X: real,Y: 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 )
= Y )
=> ( ( arctan @ Y )
= X ) ) ) ) ).
% arctan_unique
thf(fact_3391_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_3392_tan__total__pi4,axiom,
! [X: real] :
( ( ord_less @ real @ ( abs_abs @ real @ X ) @ ( one_one @ real ) )
=> ? [Z3: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ one2 ) ) ) ) ) @ Z3 )
& ( ord_less @ real @ Z3 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ ( bit0 @ one2 ) ) ) ) )
& ( ( tan @ real @ Z3 )
= X ) ) ) ).
% tan_total_pi4
thf(fact_3393_mono__SucI1,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X7: nat > A] :
( ! [N3: nat] : ( ord_less_eq @ A @ ( X7 @ N3 ) @ ( X7 @ ( suc @ N3 ) ) )
=> ( topological_monoseq @ A @ X7 ) ) ) ).
% mono_SucI1
thf(fact_3394_mono__SucI2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X7: nat > A] :
( ! [N3: nat] : ( ord_less_eq @ A @ ( X7 @ ( suc @ N3 ) ) @ ( X7 @ N3 ) )
=> ( topological_monoseq @ A @ X7 ) ) ) ).
% mono_SucI2
thf(fact_3395_monoseq__Suc,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( topological_monoseq @ A )
= ( ^ [X9: nat > A] :
( ! [N2: nat] : ( ord_less_eq @ A @ ( X9 @ N2 ) @ ( X9 @ ( suc @ N2 ) ) )
| ! [N2: nat] : ( ord_less_eq @ A @ ( X9 @ ( suc @ N2 ) ) @ ( X9 @ N2 ) ) ) ) ) ) ).
% monoseq_Suc
thf(fact_3396_monoI1,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X7: nat > A] :
( ! [M: nat,N3: nat] :
( ( ord_less_eq @ nat @ M @ N3 )
=> ( ord_less_eq @ A @ ( X7 @ M ) @ ( X7 @ N3 ) ) )
=> ( topological_monoseq @ A @ X7 ) ) ) ).
% monoI1
thf(fact_3397_monoI2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X7: nat > A] :
( ! [M: nat,N3: nat] :
( ( ord_less_eq @ nat @ M @ N3 )
=> ( ord_less_eq @ A @ ( X7 @ N3 ) @ ( X7 @ M ) ) )
=> ( topological_monoseq @ A @ X7 ) ) ) ).
% monoI2
thf(fact_3398_Maclaurin__cos__expansion2,axiom,
! [X: real,N: nat] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ? [T7: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ T7 )
& ( ord_less @ real @ T7 @ X )
& ( ( cos @ real @ X )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M3: nat] : ( times_times @ real @ ( cos_coeff @ M3 ) @ ( power_power @ real @ X @ M3 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( cos @ real @ ( plus_plus @ real @ T7 @ ( 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_3399_Maclaurin__minus__cos__expansion,axiom,
! [N: nat,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ X @ ( zero_zero @ real ) )
=> ? [T7: real] :
( ( ord_less @ real @ X @ T7 )
& ( ord_less @ real @ T7 @ ( zero_zero @ real ) )
& ( ( cos @ real @ X )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M3: nat] : ( times_times @ real @ ( cos_coeff @ M3 ) @ ( power_power @ real @ X @ M3 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( cos @ real @ ( plus_plus @ real @ T7 @ ( 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_3400_exp__first__two__terms,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ( ( exp @ A )
= ( ^ [X5: A] :
( plus_plus @ A @ ( plus_plus @ A @ ( one_one @ A ) @ X5 )
@ ( suminf @ A
@ ^ [N2: nat] : ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ ( plus_plus @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) @ ( power_power @ A @ X5 @ ( plus_plus @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% exp_first_two_terms
thf(fact_3401_complex__unimodular__polar,axiom,
! [Z: complex] :
( ( ( real_V7770717601297561774m_norm @ complex @ Z )
= ( one_one @ real ) )
=> ~ ! [T7: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ T7 )
=> ( ( ord_less @ real @ T7 @ ( times_times @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) @ pi ) )
=> ( Z
!= ( complex2 @ ( cos @ real @ T7 ) @ ( sin @ real @ T7 ) ) ) ) ) ) ).
% complex_unimodular_polar
thf(fact_3402_eucl__rel__int_Ocases,axiom,
! [A1: int,A22: int,A32: product_prod @ int @ int] :
( ( eucl_rel_int @ A1 @ A22 @ A32 )
=> ( ( ( A22
= ( zero_zero @ int ) )
=> ( A32
!= ( product_Pair @ int @ int @ ( zero_zero @ int ) @ A1 ) ) )
=> ( ! [Q3: int] :
( ( A32
= ( product_Pair @ int @ int @ Q3 @ ( zero_zero @ int ) ) )
=> ( ( A22
!= ( zero_zero @ int ) )
=> ( A1
!= ( times_times @ int @ Q3 @ A22 ) ) ) )
=> ~ ! [R3: int,Q3: int] :
( ( A32
= ( 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_3403_inverse__eq__iff__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A,B2: A] :
( ( ( inverse_inverse @ A @ A2 )
= ( inverse_inverse @ A @ B2 ) )
= ( A2 = B2 ) ) ) ).
% inverse_eq_iff_eq
thf(fact_3404_inverse__inverse__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A] :
( ( inverse_inverse @ A @ ( inverse_inverse @ A @ A2 ) )
= A2 ) ) ).
% inverse_inverse_eq
thf(fact_3405_inverse__nonzero__iff__nonzero,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A] :
( ( ( inverse_inverse @ A @ A2 )
= ( zero_zero @ A ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% inverse_nonzero_iff_nonzero
thf(fact_3406_inverse__zero,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ( ( inverse_inverse @ A @ ( zero_zero @ A ) )
= ( zero_zero @ A ) ) ) ).
% inverse_zero
thf(fact_3407_inverse__mult__distrib,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A2: A,B2: A] :
( ( inverse_inverse @ A @ ( times_times @ A @ A2 @ B2 ) )
= ( times_times @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) ) ) ) ).
% inverse_mult_distrib
thf(fact_3408_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_3409_inverse__1,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ( ( inverse_inverse @ A @ ( one_one @ A ) )
= ( one_one @ A ) ) ) ).
% inverse_1
thf(fact_3410_inverse__divide,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A2: A,B2: A] :
( ( inverse_inverse @ A @ ( divide_divide @ A @ A2 @ B2 ) )
= ( divide_divide @ A @ B2 @ A2 ) ) ) ).
% inverse_divide
thf(fact_3411_inverse__minus__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A] :
( ( inverse_inverse @ A @ ( uminus_uminus @ A @ A2 ) )
= ( uminus_uminus @ A @ ( inverse_inverse @ A @ A2 ) ) ) ) ).
% inverse_minus_eq
thf(fact_3412_abs__inverse,axiom,
! [A: $tType] :
( ( field_abs_sgn @ A )
=> ! [A2: A] :
( ( abs_abs @ A @ ( inverse_inverse @ A @ A2 ) )
= ( inverse_inverse @ A @ ( abs_abs @ A @ A2 ) ) ) ) ).
% abs_inverse
thf(fact_3413_sgn__inverse,axiom,
! [A: $tType] :
( ( field_abs_sgn @ A )
=> ! [A2: A] :
( ( sgn_sgn @ A @ ( inverse_inverse @ A @ A2 ) )
= ( inverse_inverse @ A @ ( sgn_sgn @ A @ A2 ) ) ) ) ).
% sgn_inverse
thf(fact_3414_inverse__sgn,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A] :
( ( inverse_inverse @ A @ ( sgn_sgn @ A @ A2 ) )
= ( sgn_sgn @ A @ A2 ) ) ) ).
% inverse_sgn
thf(fact_3415_inverse__nonpositive__iff__nonpositive,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ ( inverse_inverse @ A @ A2 ) @ ( zero_zero @ A ) )
= ( ord_less_eq @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% inverse_nonpositive_iff_nonpositive
thf(fact_3416_inverse__nonnegative__iff__nonnegative,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( inverse_inverse @ A @ A2 ) )
= ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% inverse_nonnegative_iff_nonnegative
thf(fact_3417_inverse__positive__iff__positive,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( inverse_inverse @ A @ A2 ) )
= ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ).
% inverse_positive_iff_positive
thf(fact_3418_inverse__negative__iff__negative,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A] :
( ( ord_less @ A @ ( inverse_inverse @ A @ A2 ) @ ( zero_zero @ A ) )
= ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ).
% inverse_negative_iff_negative
thf(fact_3419_inverse__less__iff__less__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) )
= ( ord_less @ A @ B2 @ A2 ) ) ) ) ) ).
% inverse_less_iff_less_neg
thf(fact_3420_inverse__less__iff__less,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) )
= ( ord_less @ A @ B2 @ A2 ) ) ) ) ) ).
% inverse_less_iff_less
thf(fact_3421_cos__coeff__0,axiom,
( ( cos_coeff @ ( zero_zero @ nat ) )
= ( one_one @ real ) ) ).
% cos_coeff_0
thf(fact_3422_inverse__le__iff__le__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
=> ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
=> ( ( ord_less_eq @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) )
= ( ord_less_eq @ A @ B2 @ A2 ) ) ) ) ) ).
% inverse_le_iff_le_neg
thf(fact_3423_inverse__le__iff__le,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less_eq @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) )
= ( ord_less_eq @ A @ B2 @ A2 ) ) ) ) ) ).
% inverse_le_iff_le
thf(fact_3424_left__inverse,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ ( inverse_inverse @ A @ A2 ) @ A2 )
= ( one_one @ A ) ) ) ) ).
% left_inverse
thf(fact_3425_right__inverse,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ A2 @ ( inverse_inverse @ A @ A2 ) )
= ( one_one @ A ) ) ) ) ).
% right_inverse
thf(fact_3426_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_3427_inverse__zero__imp__zero,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A] :
( ( ( inverse_inverse @ A @ A2 )
= ( zero_zero @ A ) )
=> ( A2
= ( zero_zero @ A ) ) ) ) ).
% inverse_zero_imp_zero
thf(fact_3428_nonzero__inverse__eq__imp__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A,B2: A] :
( ( ( inverse_inverse @ A @ A2 )
= ( inverse_inverse @ A @ B2 ) )
=> ( ( A2
!= ( zero_zero @ A ) )
=> ( ( B2
!= ( zero_zero @ A ) )
=> ( A2 = B2 ) ) ) ) ) ).
% nonzero_inverse_eq_imp_eq
thf(fact_3429_nonzero__inverse__inverse__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( inverse_inverse @ A @ ( inverse_inverse @ A @ A2 ) )
= A2 ) ) ) ).
% nonzero_inverse_inverse_eq
thf(fact_3430_nonzero__imp__inverse__nonzero,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( inverse_inverse @ A @ A2 )
!= ( zero_zero @ A ) ) ) ) ).
% nonzero_imp_inverse_nonzero
thf(fact_3431_mult__commute__imp__mult__inverse__commute,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Y: A,X: A] :
( ( ( times_times @ A @ Y @ X )
= ( times_times @ A @ X @ Y ) )
=> ( ( times_times @ A @ ( inverse_inverse @ A @ Y ) @ X )
= ( times_times @ A @ X @ ( inverse_inverse @ A @ Y ) ) ) ) ) ).
% mult_commute_imp_mult_inverse_commute
thf(fact_3432_inverse__eq__imp__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A,B2: A] :
( ( ( inverse_inverse @ A @ A2 )
= ( inverse_inverse @ A @ B2 ) )
=> ( A2 = B2 ) ) ) ).
% inverse_eq_imp_eq
thf(fact_3433_nonzero__norm__inverse,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [A2: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( real_V7770717601297561774m_norm @ A @ ( inverse_inverse @ A @ A2 ) )
= ( inverse_inverse @ real @ ( real_V7770717601297561774m_norm @ A @ A2 ) ) ) ) ) ).
% nonzero_norm_inverse
thf(fact_3434_nonzero__inverse__scaleR__distrib,axiom,
! [A: $tType] :
( ( real_V5047593784448816457lgebra @ A )
=> ! [A2: real,X: A] :
( ( A2
!= ( zero_zero @ real ) )
=> ( ( X
!= ( zero_zero @ A ) )
=> ( ( inverse_inverse @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ X ) )
= ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ A2 ) @ ( inverse_inverse @ A @ X ) ) ) ) ) ) ).
% nonzero_inverse_scaleR_distrib
thf(fact_3435_norm__inverse__le__norm,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [R2: real,X: A] :
( ( ord_less_eq @ real @ R2 @ ( real_V7770717601297561774m_norm @ A @ X ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ R2 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( inverse_inverse @ A @ X ) ) @ ( inverse_inverse @ real @ R2 ) ) ) ) ) ).
% norm_inverse_le_norm
thf(fact_3436_positive__imp__inverse__positive,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ord_less @ A @ ( zero_zero @ A ) @ ( inverse_inverse @ A @ A2 ) ) ) ) ).
% positive_imp_inverse_positive
thf(fact_3437_negative__imp__inverse__negative,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A] :
( ( ord_less @ A @ A2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( inverse_inverse @ A @ A2 ) @ ( zero_zero @ A ) ) ) ) ).
% negative_imp_inverse_negative
thf(fact_3438_inverse__positive__imp__positive,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ ( inverse_inverse @ A @ A2 ) )
=> ( ( A2
!= ( zero_zero @ A ) )
=> ( ord_less @ A @ ( zero_zero @ A ) @ A2 ) ) ) ) ).
% inverse_positive_imp_positive
thf(fact_3439_inverse__negative__imp__negative,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A] :
( ( ord_less @ A @ ( inverse_inverse @ A @ A2 ) @ ( zero_zero @ A ) )
=> ( ( A2
!= ( zero_zero @ A ) )
=> ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ) ).
% inverse_negative_imp_negative
thf(fact_3440_less__imp__inverse__less__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ ( inverse_inverse @ A @ B2 ) @ ( inverse_inverse @ A @ A2 ) ) ) ) ) ).
% less_imp_inverse_less_neg
thf(fact_3441_inverse__less__imp__less__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) )
=> ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less @ A @ B2 @ A2 ) ) ) ) ).
% inverse_less_imp_less_neg
thf(fact_3442_less__imp__inverse__less,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ord_less @ A @ ( inverse_inverse @ A @ B2 ) @ ( inverse_inverse @ A @ A2 ) ) ) ) ) ).
% less_imp_inverse_less
thf(fact_3443_inverse__less__imp__less,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ord_less @ A @ B2 @ A2 ) ) ) ) ).
% inverse_less_imp_less
thf(fact_3444_nonzero__inverse__mult__distrib,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A,B2: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( B2
!= ( zero_zero @ A ) )
=> ( ( inverse_inverse @ A @ ( times_times @ A @ A2 @ B2 ) )
= ( times_times @ A @ ( inverse_inverse @ A @ B2 ) @ ( inverse_inverse @ A @ A2 ) ) ) ) ) ) ).
% nonzero_inverse_mult_distrib
thf(fact_3445_nonzero__inverse__minus__eq,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( inverse_inverse @ A @ ( uminus_uminus @ A @ A2 ) )
= ( uminus_uminus @ A @ ( inverse_inverse @ A @ A2 ) ) ) ) ) ).
% nonzero_inverse_minus_eq
thf(fact_3446_inverse__unique,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A,B2: A] :
( ( ( times_times @ A @ A2 @ B2 )
= ( one_one @ A ) )
=> ( ( inverse_inverse @ A @ A2 )
= B2 ) ) ) ).
% inverse_unique
thf(fact_3447_field__class_Ofield__divide__inverse,axiom,
! [A: $tType] :
( ( field @ A )
=> ( ( divide_divide @ A )
= ( ^ [A6: A,B5: A] : ( times_times @ A @ A6 @ ( inverse_inverse @ A @ B5 ) ) ) ) ) ).
% field_class.field_divide_inverse
thf(fact_3448_divide__inverse,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ( ( divide_divide @ A )
= ( ^ [A6: A,B5: A] : ( times_times @ A @ A6 @ ( inverse_inverse @ A @ B5 ) ) ) ) ) ).
% divide_inverse
thf(fact_3449_divide__inverse__commute,axiom,
! [A: $tType] :
( ( field @ A )
=> ( ( divide_divide @ A )
= ( ^ [A6: A,B5: A] : ( times_times @ A @ ( inverse_inverse @ A @ B5 ) @ A6 ) ) ) ) ).
% divide_inverse_commute
thf(fact_3450_inverse__eq__divide,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ( ( inverse_inverse @ A )
= ( divide_divide @ A @ ( one_one @ A ) ) ) ) ).
% inverse_eq_divide
thf(fact_3451_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_3452_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_3453_mult__inverse__of__nat__commute,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Xa2: nat,X: A] :
( ( times_times @ A @ ( inverse_inverse @ A @ ( semiring_1_of_nat @ A @ Xa2 ) ) @ X )
= ( times_times @ A @ X @ ( inverse_inverse @ A @ ( semiring_1_of_nat @ A @ Xa2 ) ) ) ) ) ).
% mult_inverse_of_nat_commute
thf(fact_3454_nonzero__abs__inverse,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( abs_abs @ A @ ( inverse_inverse @ A @ A2 ) )
= ( inverse_inverse @ A @ ( abs_abs @ A @ A2 ) ) ) ) ) ).
% nonzero_abs_inverse
thf(fact_3455_mult__inverse__of__int__commute,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [Xa2: int,X: A] :
( ( times_times @ A @ ( inverse_inverse @ A @ ( ring_1_of_int @ A @ Xa2 ) ) @ X )
= ( times_times @ A @ X @ ( inverse_inverse @ A @ ( ring_1_of_int @ A @ Xa2 ) ) ) ) ) ).
% mult_inverse_of_int_commute
thf(fact_3456_le__imp__inverse__le__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ ( inverse_inverse @ A @ B2 ) @ ( inverse_inverse @ A @ A2 ) ) ) ) ) ).
% le_imp_inverse_le_neg
thf(fact_3457_inverse__le__imp__le__neg,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) )
=> ( ( ord_less @ A @ B2 @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ B2 @ A2 ) ) ) ) ).
% inverse_le_imp_le_neg
thf(fact_3458_le__imp__inverse__le,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ord_less_eq @ A @ ( inverse_inverse @ A @ B2 ) @ ( inverse_inverse @ A @ A2 ) ) ) ) ) ).
% le_imp_inverse_le
thf(fact_3459_inverse__le__imp__le,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ord_less_eq @ A @ B2 @ A2 ) ) ) ) ).
% inverse_le_imp_le
thf(fact_3460_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_3461_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_3462_one__less__inverse,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ A2 @ ( one_one @ A ) )
=> ( ord_less @ A @ ( one_one @ A ) @ ( inverse_inverse @ A @ A2 ) ) ) ) ) ).
% one_less_inverse
thf(fact_3463_field__class_Ofield__inverse,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A2: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( times_times @ A @ ( inverse_inverse @ A @ A2 ) @ A2 )
= ( one_one @ A ) ) ) ) ).
% field_class.field_inverse
thf(fact_3464_division__ring__inverse__add,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A,B2: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( B2
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) )
= ( times_times @ A @ ( times_times @ A @ ( inverse_inverse @ A @ A2 ) @ ( plus_plus @ A @ A2 @ B2 ) ) @ ( inverse_inverse @ A @ B2 ) ) ) ) ) ) ).
% division_ring_inverse_add
thf(fact_3465_inverse__add,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A2: A,B2: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( B2
!= ( zero_zero @ A ) )
=> ( ( plus_plus @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) )
= ( times_times @ A @ ( times_times @ A @ ( plus_plus @ A @ A2 @ B2 ) @ ( inverse_inverse @ A @ A2 ) ) @ ( inverse_inverse @ A @ B2 ) ) ) ) ) ) ).
% inverse_add
thf(fact_3466_division__ring__inverse__diff,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A,B2: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( B2
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) )
= ( times_times @ A @ ( times_times @ A @ ( inverse_inverse @ A @ A2 ) @ ( minus_minus @ A @ B2 @ A2 ) ) @ ( inverse_inverse @ A @ B2 ) ) ) ) ) ) ).
% division_ring_inverse_diff
thf(fact_3467_nonzero__inverse__eq__divide,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( inverse_inverse @ A @ A2 )
= ( divide_divide @ A @ ( one_one @ A ) @ A2 ) ) ) ) ).
% nonzero_inverse_eq_divide
thf(fact_3468_inverse__less__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B2 ) )
=> ( ord_less @ A @ B2 @ A2 ) )
& ( ( ord_less_eq @ A @ ( times_times @ A @ A2 @ B2 ) @ ( zero_zero @ A ) )
=> ( ord_less @ A @ A2 @ B2 ) ) ) ) ) ).
% inverse_less_iff
thf(fact_3469_inverse__le__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) )
= ( ( ( ord_less @ A @ ( zero_zero @ A ) @ ( times_times @ A @ A2 @ B2 ) )
=> ( ord_less_eq @ A @ B2 @ A2 ) )
& ( ( ord_less_eq @ A @ ( times_times @ A @ A2 @ B2 ) @ ( zero_zero @ A ) )
=> ( ord_less_eq @ A @ A2 @ B2 ) ) ) ) ) ).
% inverse_le_iff
thf(fact_3470_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_3471_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_3472_one__le__inverse,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ A2 @ ( one_one @ A ) )
=> ( ord_less_eq @ A @ ( one_one @ A ) @ ( inverse_inverse @ A @ A2 ) ) ) ) ) ).
% one_le_inverse
thf(fact_3473_inverse__diff__inverse,axiom,
! [A: $tType] :
( ( division_ring @ A )
=> ! [A2: A,B2: A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( ( B2
!= ( zero_zero @ A ) )
=> ( ( minus_minus @ A @ ( inverse_inverse @ A @ A2 ) @ ( inverse_inverse @ A @ B2 ) )
= ( uminus_uminus @ A @ ( times_times @ A @ ( times_times @ A @ ( inverse_inverse @ A @ A2 ) @ ( minus_minus @ A @ A2 @ B2 ) ) @ ( inverse_inverse @ A @ B2 ) ) ) ) ) ) ) ).
% inverse_diff_inverse
thf(fact_3474_reals__Archimedean,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X )
=> ? [N3: nat] : ( ord_less @ A @ ( inverse_inverse @ A @ ( semiring_1_of_nat @ A @ ( suc @ N3 ) ) ) @ X ) ) ) ).
% reals_Archimedean
thf(fact_3475_real__vector__affinity__eq,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [M2: real,X: A,C3: A,Y: A] :
( ( M2
!= ( zero_zero @ real ) )
=> ( ( ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ M2 @ X ) @ C3 )
= Y )
= ( X
= ( minus_minus @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ M2 ) @ Y ) @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ M2 ) @ C3 ) ) ) ) ) ) ).
% real_vector_affinity_eq
thf(fact_3476_real__vector__eq__affinity,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [M2: real,Y: A,X: A,C3: A] :
( ( M2
!= ( zero_zero @ real ) )
=> ( ( Y
= ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ M2 @ X ) @ C3 ) )
= ( ( minus_minus @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ M2 ) @ Y ) @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ M2 ) @ C3 ) )
= X ) ) ) ) ).
% real_vector_eq_affinity
thf(fact_3477_pos__divideR__le__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C3: real,B2: A,A2: A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ C3 )
=> ( ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C3 ) @ B2 ) @ A2 )
= ( ord_less_eq @ A @ B2 @ ( real_V8093663219630862766scaleR @ A @ C3 @ A2 ) ) ) ) ) ).
% pos_divideR_le_eq
thf(fact_3478_pos__le__divideR__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C3: real,A2: A,B2: A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ C3 )
=> ( ( ord_less_eq @ A @ A2 @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C3 ) @ B2 ) )
= ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ C3 @ A2 ) @ B2 ) ) ) ) ).
% pos_le_divideR_eq
thf(fact_3479_neg__divideR__le__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C3: real,B2: A,A2: A] :
( ( ord_less @ real @ C3 @ ( zero_zero @ real ) )
=> ( ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C3 ) @ B2 ) @ A2 )
= ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ C3 @ A2 ) @ B2 ) ) ) ) ).
% neg_divideR_le_eq
thf(fact_3480_neg__le__divideR__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C3: real,A2: A,B2: A] :
( ( ord_less @ real @ C3 @ ( zero_zero @ real ) )
=> ( ( ord_less_eq @ A @ A2 @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C3 ) @ B2 ) )
= ( ord_less_eq @ A @ B2 @ ( real_V8093663219630862766scaleR @ A @ C3 @ A2 ) ) ) ) ) ).
% neg_le_divideR_eq
thf(fact_3481_neg__less__divideR__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C3: real,A2: A,B2: A] :
( ( ord_less @ real @ C3 @ ( zero_zero @ real ) )
=> ( ( ord_less @ A @ A2 @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C3 ) @ B2 ) )
= ( ord_less @ A @ B2 @ ( real_V8093663219630862766scaleR @ A @ C3 @ A2 ) ) ) ) ) ).
% neg_less_divideR_eq
thf(fact_3482_neg__divideR__less__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C3: real,B2: A,A2: A] :
( ( ord_less @ real @ C3 @ ( zero_zero @ real ) )
=> ( ( ord_less @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C3 ) @ B2 ) @ A2 )
= ( ord_less @ A @ ( real_V8093663219630862766scaleR @ A @ C3 @ A2 ) @ B2 ) ) ) ) ).
% neg_divideR_less_eq
thf(fact_3483_pos__less__divideR__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C3: real,A2: A,B2: A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ C3 )
=> ( ( ord_less @ A @ A2 @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C3 ) @ B2 ) )
= ( ord_less @ A @ ( real_V8093663219630862766scaleR @ A @ C3 @ A2 ) @ B2 ) ) ) ) ).
% pos_less_divideR_eq
thf(fact_3484_pos__divideR__less__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C3: real,B2: A,A2: A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ C3 )
=> ( ( ord_less @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C3 ) @ B2 ) @ A2 )
= ( ord_less @ A @ B2 @ ( real_V8093663219630862766scaleR @ A @ C3 @ A2 ) ) ) ) ) ).
% pos_divideR_less_eq
thf(fact_3485_forall__pos__mono__1,axiom,
! [P2: real > $o,E3: real] :
( ! [D2: real,E2: real] :
( ( ord_less @ real @ D2 @ E2 )
=> ( ( P2 @ D2 )
=> ( P2 @ E2 ) ) )
=> ( ! [N3: nat] : ( P2 @ ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ ( suc @ N3 ) ) ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ E3 )
=> ( P2 @ E3 ) ) ) ) ).
% forall_pos_mono_1
thf(fact_3486_forall__pos__mono,axiom,
! [P2: real > $o,E3: real] :
( ! [D2: real,E2: real] :
( ( ord_less @ real @ D2 @ E2 )
=> ( ( P2 @ D2 )
=> ( P2 @ E2 ) ) )
=> ( ! [N3: nat] :
( ( N3
!= ( zero_zero @ nat ) )
=> ( P2 @ ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ N3 ) ) ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ E3 )
=> ( P2 @ E3 ) ) ) ) ).
% forall_pos_mono
thf(fact_3487_real__arch__inverse,axiom,
! [E3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E3 )
= ( ? [N2: nat] :
( ( N2
!= ( zero_zero @ nat ) )
& ( ord_less @ real @ ( zero_zero @ real ) @ ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ N2 ) ) )
& ( ord_less @ real @ ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ N2 ) ) @ E3 ) ) ) ) ).
% real_arch_inverse
thf(fact_3488_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_3489_ex__inverse__of__nat__less,axiom,
! [A: $tType] :
( ( archim462609752435547400_field @ A )
=> ! [X: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ X )
=> ? [N3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
& ( ord_less @ A @ ( inverse_inverse @ A @ ( semiring_1_of_nat @ A @ N3 ) ) @ X ) ) ) ) ).
% ex_inverse_of_nat_less
thf(fact_3490_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_3491_neg__minus__divideR__le__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C3: real,B2: A,A2: A] :
( ( ord_less @ real @ C3 @ ( zero_zero @ real ) )
=> ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C3 ) @ B2 ) ) @ A2 )
= ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ C3 @ A2 ) @ ( uminus_uminus @ A @ B2 ) ) ) ) ) ).
% neg_minus_divideR_le_eq
thf(fact_3492_neg__le__minus__divideR__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C3: real,A2: A,B2: A] :
( ( ord_less @ real @ C3 @ ( zero_zero @ real ) )
=> ( ( ord_less_eq @ A @ A2 @ ( uminus_uminus @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C3 ) @ B2 ) ) )
= ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( real_V8093663219630862766scaleR @ A @ C3 @ A2 ) ) ) ) ) ).
% neg_le_minus_divideR_eq
thf(fact_3493_pos__minus__divideR__le__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C3: real,B2: A,A2: A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ C3 )
=> ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C3 ) @ B2 ) ) @ A2 )
= ( ord_less_eq @ A @ ( uminus_uminus @ A @ B2 ) @ ( real_V8093663219630862766scaleR @ A @ C3 @ A2 ) ) ) ) ) ).
% pos_minus_divideR_le_eq
thf(fact_3494_pos__le__minus__divideR__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C3: real,A2: A,B2: A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ C3 )
=> ( ( ord_less_eq @ A @ A2 @ ( uminus_uminus @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C3 ) @ B2 ) ) )
= ( ord_less_eq @ A @ ( real_V8093663219630862766scaleR @ A @ C3 @ A2 ) @ ( uminus_uminus @ A @ B2 ) ) ) ) ) ).
% pos_le_minus_divideR_eq
thf(fact_3495_pos__less__minus__divideR__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C3: real,A2: A,B2: A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ C3 )
=> ( ( ord_less @ A @ A2 @ ( uminus_uminus @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C3 ) @ B2 ) ) )
= ( ord_less @ A @ ( real_V8093663219630862766scaleR @ A @ C3 @ A2 ) @ ( uminus_uminus @ A @ B2 ) ) ) ) ) ).
% pos_less_minus_divideR_eq
thf(fact_3496_pos__minus__divideR__less__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C3: real,B2: A,A2: A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ C3 )
=> ( ( ord_less @ A @ ( uminus_uminus @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C3 ) @ B2 ) ) @ A2 )
= ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ ( real_V8093663219630862766scaleR @ A @ C3 @ A2 ) ) ) ) ) ).
% pos_minus_divideR_less_eq
thf(fact_3497_neg__less__minus__divideR__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C3: real,A2: A,B2: A] :
( ( ord_less @ real @ C3 @ ( zero_zero @ real ) )
=> ( ( ord_less @ A @ A2 @ ( uminus_uminus @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C3 ) @ B2 ) ) )
= ( ord_less @ A @ ( uminus_uminus @ A @ B2 ) @ ( real_V8093663219630862766scaleR @ A @ C3 @ A2 ) ) ) ) ) ).
% neg_less_minus_divideR_eq
thf(fact_3498_neg__minus__divideR__less__eq,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C3: real,B2: A,A2: A] :
( ( ord_less @ real @ C3 @ ( zero_zero @ real ) )
=> ( ( ord_less @ A @ ( uminus_uminus @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ C3 ) @ B2 ) ) @ A2 )
= ( ord_less @ A @ ( real_V8093663219630862766scaleR @ A @ C3 @ A2 ) @ ( uminus_uminus @ A @ B2 ) ) ) ) ) ).
% neg_minus_divideR_less_eq
thf(fact_3499_log__inverse,axiom,
! [A2: real,X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
=> ( ( A2
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( log @ A2 @ ( inverse_inverse @ real @ X ) )
= ( uminus_uminus @ real @ ( log @ A2 @ X ) ) ) ) ) ) ).
% log_inverse
thf(fact_3500_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_3501_real__inv__sqrt__pow2,axiom,
! [X: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( ( power_power @ real @ ( inverse_inverse @ real @ ( sqrt @ X ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( inverse_inverse @ real @ X ) ) ) ).
% real_inv_sqrt_pow2
thf(fact_3502_exp__series__add__commuting,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X: A,Y: A,N: nat] :
( ( ( times_times @ A @ X @ Y )
= ( times_times @ A @ Y @ X ) )
=> ( ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ A @ ( plus_plus @ A @ X @ Y ) @ N ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ I4 ) ) @ ( power_power @ A @ X @ I4 ) ) @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ ( minus_minus @ nat @ N @ I4 ) ) ) @ ( power_power @ A @ Y @ ( minus_minus @ nat @ N @ I4 ) ) ) )
@ ( set_ord_atMost @ nat @ N ) ) ) ) ) ).
% exp_series_add_commuting
thf(fact_3503_exp__first__term,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ( ( exp @ A )
= ( ^ [X5: A] :
( plus_plus @ A @ ( one_one @ A )
@ ( suminf @ A
@ ^ [N2: nat] : ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ ( suc @ N2 ) ) ) @ ( power_power @ A @ X5 @ ( suc @ N2 ) ) ) ) ) ) ) ) ).
% exp_first_term
thf(fact_3504_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_3505_eucl__rel__int__iff,axiom,
! [K2: int,L: int,Q2: int,R2: int] :
( ( eucl_rel_int @ K2 @ L @ ( product_Pair @ int @ int @ Q2 @ R2 ) )
= ( ( K2
= ( plus_plus @ int @ ( times_times @ int @ L @ Q2 ) @ R2 ) )
& ( ( ord_less @ int @ ( zero_zero @ int ) @ L )
=> ( ( ord_less_eq @ int @ ( zero_zero @ int ) @ R2 )
& ( ord_less @ int @ R2 @ L ) ) )
& ( ~ ( ord_less @ int @ ( zero_zero @ int ) @ L )
=> ( ( ( ord_less @ int @ L @ ( zero_zero @ int ) )
=> ( ( ord_less @ int @ L @ R2 )
& ( ord_less_eq @ int @ R2 @ ( zero_zero @ int ) ) ) )
& ( ~ ( ord_less @ int @ L @ ( zero_zero @ int ) )
=> ( Q2
= ( zero_zero @ int ) ) ) ) ) ) ) ).
% eucl_rel_int_iff
thf(fact_3506_eucl__rel__int__remainderI,axiom,
! [R2: int,L: int,K2: int,Q2: int] :
( ( ( sgn_sgn @ int @ R2 )
= ( sgn_sgn @ int @ L ) )
=> ( ( ord_less @ int @ ( abs_abs @ int @ R2 ) @ ( abs_abs @ int @ L ) )
=> ( ( K2
= ( plus_plus @ int @ ( times_times @ int @ Q2 @ L ) @ R2 ) )
=> ( eucl_rel_int @ K2 @ L @ ( product_Pair @ int @ int @ Q2 @ R2 ) ) ) ) ) ).
% eucl_rel_int_remainderI
thf(fact_3507_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_3508_exp__first__terms,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [K2: nat] :
( ( exp @ A )
= ( ^ [X5: A] :
( plus_plus @ A
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [N2: nat] : ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ N2 ) ) @ ( power_power @ A @ X5 @ N2 ) )
@ ( set_ord_lessThan @ nat @ K2 ) )
@ ( suminf @ A
@ ^ [N2: nat] : ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ ( plus_plus @ nat @ N2 @ K2 ) ) ) @ ( power_power @ A @ X5 @ ( plus_plus @ nat @ N2 @ K2 ) ) ) ) ) ) ) ) ).
% exp_first_terms
thf(fact_3509_eucl__rel__int_Osimps,axiom,
( eucl_rel_int
= ( ^ [A12: int,A23: int,A33: product_prod @ int @ int] :
( ? [K3: int] :
( ( A12 = K3 )
& ( A23
= ( zero_zero @ int ) )
& ( A33
= ( product_Pair @ int @ int @ ( zero_zero @ int ) @ K3 ) ) )
| ? [L2: int,K3: int,Q5: int] :
( ( A12 = K3 )
& ( A23 = L2 )
& ( A33
= ( product_Pair @ int @ int @ Q5 @ ( zero_zero @ int ) ) )
& ( L2
!= ( zero_zero @ int ) )
& ( K3
= ( times_times @ int @ Q5 @ L2 ) ) )
| ? [R5: int,L2: int,K3: int,Q5: int] :
( ( A12 = K3 )
& ( A23 = L2 )
& ( A33
= ( product_Pair @ int @ int @ Q5 @ R5 ) )
& ( ( sgn_sgn @ int @ R5 )
= ( sgn_sgn @ int @ L2 ) )
& ( ord_less @ int @ ( abs_abs @ int @ R5 ) @ ( abs_abs @ int @ L2 ) )
& ( K3
= ( plus_plus @ int @ ( times_times @ int @ Q5 @ L2 ) @ R5 ) ) ) ) ) ) ).
% eucl_rel_int.simps
thf(fact_3510_sinh__converges,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X: A] :
( sums @ A
@ ^ [N2: nat] : ( if @ A @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) @ ( zero_zero @ A ) @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ N2 ) ) @ ( power_power @ A @ X @ N2 ) ) )
@ ( sinh @ A @ X ) ) ) ).
% sinh_converges
thf(fact_3511_cosh__converges,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ! [X: A] :
( sums @ A
@ ^ [N2: nat] : ( if @ A @ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) @ ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( semiring_char_0_fact @ real @ N2 ) ) @ ( power_power @ A @ X @ N2 ) ) @ ( zero_zero @ A ) )
@ ( cosh @ A @ X ) ) ) ).
% cosh_converges
thf(fact_3512_arctan__def,axiom,
( arctan
= ( ^ [Y6: real] :
( the @ real
@ ^ [X5: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ X5 )
& ( ord_less @ real @ X5 @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
& ( ( tan @ real @ X5 )
= Y6 ) ) ) ) ) ).
% arctan_def
thf(fact_3513_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_3514_arcsin__lt__bounded,axiom,
! [Y: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y )
=> ( ( ord_less @ real @ Y @ ( one_one @ real ) )
=> ( ( ord_less @ real @ ( uminus_uminus @ real @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) @ ( arcsin @ Y ) )
& ( ord_less @ real @ ( arcsin @ Y ) @ ( divide_divide @ real @ pi @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) ) ) ) ) ).
% arcsin_lt_bounded
thf(fact_3515_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_3516_sinh__real__less__iff,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ ( sinh @ real @ X ) @ ( sinh @ real @ Y ) )
= ( ord_less @ real @ X @ Y ) ) ).
% sinh_real_less_iff
thf(fact_3517_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_3518_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_3519_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_3520_sinh__less__cosh__real,axiom,
! [X: real] : ( ord_less @ real @ ( sinh @ real @ X ) @ ( cosh @ real @ X ) ) ).
% sinh_less_cosh_real
thf(fact_3521_cosh__add,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A,Y: A] :
( ( cosh @ A @ ( plus_plus @ A @ X @ Y ) )
= ( plus_plus @ A @ ( times_times @ A @ ( cosh @ A @ X ) @ ( cosh @ A @ Y ) ) @ ( times_times @ A @ ( sinh @ A @ X ) @ ( sinh @ A @ Y ) ) ) ) ) ).
% cosh_add
thf(fact_3522_sinh__add,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A,Y: A] :
( ( sinh @ A @ ( plus_plus @ A @ X @ Y ) )
= ( plus_plus @ A @ ( times_times @ A @ ( sinh @ A @ X ) @ ( cosh @ A @ Y ) ) @ ( times_times @ A @ ( cosh @ A @ X ) @ ( sinh @ A @ Y ) ) ) ) ) ).
% sinh_add
thf(fact_3523_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_3524_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_3525_cosh__real__pos,axiom,
! [X: real] : ( ord_less @ real @ ( zero_zero @ real ) @ ( cosh @ real @ X ) ) ).
% cosh_real_pos
thf(fact_3526_cosh__real__nonpos__less__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ X @ ( zero_zero @ real ) )
=> ( ( ord_less_eq @ real @ Y @ ( zero_zero @ real ) )
=> ( ( ord_less @ real @ ( cosh @ real @ X ) @ ( cosh @ real @ Y ) )
= ( ord_less @ real @ Y @ X ) ) ) ) ).
% cosh_real_nonpos_less_iff
thf(fact_3527_cosh__real__nonneg__less__iff,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y )
=> ( ( ord_less @ real @ ( cosh @ real @ X ) @ ( cosh @ real @ Y ) )
= ( ord_less @ real @ X @ Y ) ) ) ) ).
% cosh_real_nonneg_less_iff
thf(fact_3528_cosh__real__strict__mono,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ X )
=> ( ( ord_less @ real @ X @ Y )
=> ( ord_less @ real @ ( cosh @ real @ X ) @ ( cosh @ real @ Y ) ) ) ) ).
% cosh_real_strict_mono
thf(fact_3529_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_3530_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_3531_arcsin__less__arcsin,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X )
=> ( ( ord_less @ real @ X @ Y )
=> ( ( ord_less_eq @ real @ Y @ ( one_one @ real ) )
=> ( ord_less @ real @ ( arcsin @ X ) @ ( arcsin @ Y ) ) ) ) ) ).
% arcsin_less_arcsin
thf(fact_3532_arcsin__less__mono,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ X ) @ ( one_one @ real ) )
=> ( ( ord_less_eq @ real @ ( abs_abs @ real @ Y ) @ ( one_one @ real ) )
=> ( ( ord_less @ real @ ( arcsin @ X ) @ ( arcsin @ Y ) )
= ( ord_less @ real @ X @ Y ) ) ) ) ).
% arcsin_less_mono
thf(fact_3533_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_3534_tanh__add,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A,Y: A] :
( ( ( cosh @ A @ X )
!= ( zero_zero @ A ) )
=> ( ( ( cosh @ A @ Y )
!= ( zero_zero @ A ) )
=> ( ( tanh @ A @ ( plus_plus @ A @ X @ Y ) )
= ( divide_divide @ A @ ( plus_plus @ A @ ( tanh @ A @ X ) @ ( tanh @ A @ Y ) ) @ ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( tanh @ A @ X ) @ ( tanh @ A @ Y ) ) ) ) ) ) ) ) ).
% tanh_add
thf(fact_3535_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_3536_cosh__field__def,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ( ( cosh @ A )
= ( ^ [Z5: A] : ( divide_divide @ A @ ( plus_plus @ A @ ( exp @ A @ Z5 ) @ ( exp @ A @ ( uminus_uminus @ A @ Z5 ) ) ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ).
% cosh_field_def
thf(fact_3537_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_3538_cosh__def,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V2822296259951069270ebra_1 @ A ) )
=> ( ( cosh @ A )
= ( ^ [X5: A] : ( real_V8093663219630862766scaleR @ A @ ( inverse_inverse @ real @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) @ ( plus_plus @ A @ ( exp @ A @ X5 ) @ ( exp @ A @ ( uminus_uminus @ A @ X5 ) ) ) ) ) ) ) ).
% cosh_def
thf(fact_3539_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_3540_old_Orec__prod__def,axiom,
! [T: $tType,B: $tType,A: $tType] :
( ( product_rec_prod @ A @ B @ T )
= ( ^ [F12: A > B > T,X5: product_prod @ A @ B] : ( the @ T @ ( product_rec_set_prod @ A @ B @ T @ F12 @ X5 ) ) ) ) ).
% old.rec_prod_def
thf(fact_3541_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_3542_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_3543_the__elem__def,axiom,
! [A: $tType] :
( ( the_elem @ A )
= ( ^ [X9: set @ A] :
( the @ A
@ ^ [X5: A] :
( X9
= ( insert @ A @ X5 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% the_elem_def
thf(fact_3544_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_3545_xor_Oright__neutral,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A] :
( ( bit_se5824344971392196577ns_xor @ A @ A2 @ ( zero_zero @ A ) )
= A2 ) ) ).
% xor.right_neutral
thf(fact_3546_xor_Oleft__neutral,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A] :
( ( bit_se5824344971392196577ns_xor @ A @ ( zero_zero @ A ) @ A2 )
= A2 ) ) ).
% xor.left_neutral
thf(fact_3547_xor__self__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A] :
( ( bit_se5824344971392196577ns_xor @ A @ A2 @ A2 )
= ( zero_zero @ A ) ) ) ).
% xor_self_eq
thf(fact_3548_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_3549_The__split__eq,axiom,
! [A: $tType,B: $tType,X: A,Y: B] :
( ( the @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [X10: A,Y7: B] :
( ( X = X10 )
& ( Y = Y7 ) ) ) )
= ( product_Pair @ A @ B @ X @ Y ) ) ).
% The_split_eq
thf(fact_3550_the__elem__eq,axiom,
! [A: $tType,X: A] :
( ( the_elem @ A @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= X ) ).
% the_elem_eq
thf(fact_3551_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_3552_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_3553_xor__nat__numerals_I2_J,axiom,
! [Y: num] :
( ( bit_se5824344971392196577ns_xor @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( numeral_numeral @ nat @ ( bit1 @ Y ) ) )
= ( numeral_numeral @ nat @ ( bit0 @ Y ) ) ) ).
% xor_nat_numerals(2)
thf(fact_3554_xor__nat__numerals_I1_J,axiom,
! [Y: num] :
( ( bit_se5824344971392196577ns_xor @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( numeral_numeral @ nat @ ( bit0 @ Y ) ) )
= ( numeral_numeral @ nat @ ( bit1 @ Y ) ) ) ).
% xor_nat_numerals(1)
thf(fact_3555_xor__numerals_I4_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X: num,Y: num] :
( ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ ( bit0 @ X ) ) @ ( numeral_numeral @ A @ ( bit1 @ Y ) ) )
= ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ X ) @ ( numeral_numeral @ A @ Y ) ) ) ) ) ) ).
% xor_numerals(4)
thf(fact_3556_xor__numerals_I6_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X: num,Y: num] :
( ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ ( bit1 @ X ) ) @ ( numeral_numeral @ A @ ( bit0 @ Y ) ) )
= ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se5824344971392196577ns_xor @ A @ ( numeral_numeral @ A @ X ) @ ( numeral_numeral @ A @ Y ) ) ) ) ) ) ).
% xor_numerals(6)
thf(fact_3557_floor__real__def,axiom,
( ( archim6421214686448440834_floor @ real )
= ( ^ [X5: real] :
( the @ int
@ ^ [Z5: int] :
( ( ord_less_eq @ real @ ( ring_1_of_int @ real @ Z5 ) @ X5 )
& ( ord_less @ real @ X5 @ ( ring_1_of_int @ real @ ( plus_plus @ int @ Z5 @ ( one_one @ int ) ) ) ) ) ) ) ) ).
% floor_real_def
thf(fact_3558_xor__nat__unfold,axiom,
( ( bit_se5824344971392196577ns_xor @ nat )
= ( ^ [M3: nat,N2: nat] :
( if @ nat
@ ( M3
= ( zero_zero @ nat ) )
@ N2
@ ( if @ nat
@ ( N2
= ( zero_zero @ nat ) )
@ M3
@ ( plus_plus @ nat @ ( modulo_modulo @ nat @ ( plus_plus @ nat @ ( modulo_modulo @ nat @ M3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( bit_se5824344971392196577ns_xor @ nat @ ( divide_divide @ nat @ M3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% xor_nat_unfold
thf(fact_3559_xor__nat__rec,axiom,
( ( bit_se5824344971392196577ns_xor @ nat )
= ( ^ [M3: nat,N2: nat] :
( plus_plus @ nat
@ ( zero_neq_one_of_bool @ nat
@ ( ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M3 ) )
!= ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) ) )
@ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( bit_se5824344971392196577ns_xor @ nat @ ( divide_divide @ nat @ M3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% xor_nat_rec
thf(fact_3560_xor__one__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A] :
( ( bit_se5824344971392196577ns_xor @ A @ A2 @ ( one_one @ A ) )
= ( minus_minus @ A @ ( plus_plus @ A @ A2 @ ( zero_neq_one_of_bool @ A @ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) ) )
@ ( zero_neq_one_of_bool @ A
@ ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) ) ) ) ) ).
% xor_one_eq
thf(fact_3561_one__xor__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A] :
( ( bit_se5824344971392196577ns_xor @ A @ ( one_one @ A ) @ A2 )
= ( minus_minus @ A @ ( plus_plus @ A @ A2 @ ( zero_neq_one_of_bool @ A @ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) ) )
@ ( zero_neq_one_of_bool @ A
@ ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) ) ) ) ) ).
% one_xor_eq
thf(fact_3562_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_3563_case__prod__Pair__iden,axiom,
! [B: $tType,A: $tType,P: product_prod @ A @ B] :
( ( product_case_prod @ A @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B ) @ P )
= P ) ).
% case_prod_Pair_iden
thf(fact_3564_floor__rat__def,axiom,
( ( archim6421214686448440834_floor @ rat )
= ( ^ [X5: rat] :
( the @ int
@ ^ [Z5: int] :
( ( ord_less_eq @ rat @ ( ring_1_of_int @ rat @ Z5 ) @ X5 )
& ( ord_less @ rat @ X5 @ ( ring_1_of_int @ rat @ ( plus_plus @ int @ Z5 @ ( one_one @ int ) ) ) ) ) ) ) ) ).
% floor_rat_def
thf(fact_3565_sum__diff1_H__aux,axiom,
! [B: $tType,A: $tType] :
( ( ab_group_add @ B )
=> ! [F5: set @ A,I6: set @ A,F3: A > B,I2: A] :
( ( finite_finite2 @ A @ F5 )
=> ( ( ord_less_eq @ ( set @ A )
@ ( collect @ A
@ ^ [I4: A] :
( ( member @ A @ I4 @ I6 )
& ( ( F3 @ I4 )
!= ( zero_zero @ B ) ) ) )
@ F5 )
=> ( ( ( member @ A @ I2 @ I6 )
=> ( ( groups1027152243600224163dd_sum @ A @ B @ F3 @ ( minus_minus @ ( set @ A ) @ I6 @ ( insert @ A @ I2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( minus_minus @ B @ ( groups1027152243600224163dd_sum @ A @ B @ F3 @ I6 ) @ ( F3 @ I2 ) ) ) )
& ( ~ ( member @ A @ I2 @ I6 )
=> ( ( groups1027152243600224163dd_sum @ A @ B @ F3 @ ( minus_minus @ ( set @ A ) @ I6 @ ( insert @ A @ I2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( groups1027152243600224163dd_sum @ A @ B @ F3 @ I6 ) ) ) ) ) ) ) ).
% sum_diff1'_aux
thf(fact_3566_or__nat__unfold,axiom,
( ( bit_se1065995026697491101ons_or @ nat )
= ( ^ [M3: nat,N2: nat] :
( if @ nat
@ ( M3
= ( zero_zero @ nat ) )
@ N2
@ ( if @ nat
@ ( N2
= ( zero_zero @ nat ) )
@ M3
@ ( plus_plus @ nat @ ( ord_max @ nat @ ( modulo_modulo @ nat @ M3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( modulo_modulo @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ nat @ ( divide_divide @ nat @ M3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ) ) ).
% or_nat_unfold
thf(fact_3567_or_Oright__neutral,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A] :
( ( bit_se1065995026697491101ons_or @ A @ A2 @ ( zero_zero @ A ) )
= A2 ) ) ).
% or.right_neutral
thf(fact_3568_or_Oleft__neutral,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A] :
( ( bit_se1065995026697491101ons_or @ A @ ( zero_zero @ A ) @ A2 )
= A2 ) ) ).
% or.left_neutral
thf(fact_3569_xor__negative__int__iff,axiom,
! [K2: int,L: int] :
( ( ord_less @ int @ ( bit_se5824344971392196577ns_xor @ int @ K2 @ L ) @ ( zero_zero @ int ) )
= ( ( ord_less @ int @ K2 @ ( zero_zero @ int ) )
!= ( ord_less @ int @ L @ ( zero_zero @ int ) ) ) ) ).
% xor_negative_int_iff
thf(fact_3570_sum_Oempty_H,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ A )
=> ! [P: B > A] :
( ( groups1027152243600224163dd_sum @ B @ A @ P @ ( bot_bot @ ( set @ B ) ) )
= ( zero_zero @ A ) ) ) ).
% sum.empty'
thf(fact_3571_sum_Oeq__sum,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [I6: set @ B,P: B > A] :
( ( finite_finite2 @ B @ I6 )
=> ( ( groups1027152243600224163dd_sum @ B @ A @ P @ I6 )
= ( groups7311177749621191930dd_sum @ B @ A @ P @ I6 ) ) ) ) ).
% sum.eq_sum
thf(fact_3572_or__nat__numerals_I2_J,axiom,
! [Y: num] :
( ( bit_se1065995026697491101ons_or @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( numeral_numeral @ nat @ ( bit1 @ Y ) ) )
= ( numeral_numeral @ nat @ ( bit1 @ Y ) ) ) ).
% or_nat_numerals(2)
thf(fact_3573_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_3574_sum_Oinsert_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [I6: set @ B,P: B > A,I2: B] :
( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X5: B] :
( ( member @ B @ X5 @ I6 )
& ( ( P @ X5 )
!= ( zero_zero @ A ) ) ) ) )
=> ( ( ( member @ B @ I2 @ I6 )
=> ( ( groups1027152243600224163dd_sum @ B @ A @ P @ ( insert @ B @ I2 @ I6 ) )
= ( groups1027152243600224163dd_sum @ B @ A @ P @ I6 ) ) )
& ( ~ ( member @ B @ I2 @ I6 )
=> ( ( groups1027152243600224163dd_sum @ B @ A @ P @ ( insert @ B @ I2 @ I6 ) )
= ( plus_plus @ A @ ( P @ I2 ) @ ( groups1027152243600224163dd_sum @ B @ A @ P @ I6 ) ) ) ) ) ) ) ).
% sum.insert'
thf(fact_3575_or__nat__numerals_I1_J,axiom,
! [Y: num] :
( ( bit_se1065995026697491101ons_or @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( numeral_numeral @ nat @ ( bit0 @ Y ) ) )
= ( numeral_numeral @ nat @ ( bit1 @ Y ) ) ) ).
% or_nat_numerals(1)
thf(fact_3576_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_3577_or__numerals_I7_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X: num,Y: num] :
( ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ ( bit1 @ X ) ) @ ( numeral_numeral @ A @ ( bit1 @ Y ) ) )
= ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ X ) @ ( numeral_numeral @ A @ Y ) ) ) ) ) ) ).
% or_numerals(7)
thf(fact_3578_or__numerals_I6_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X: num,Y: num] :
( ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ ( bit1 @ X ) ) @ ( numeral_numeral @ A @ ( bit0 @ Y ) ) )
= ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ X ) @ ( numeral_numeral @ A @ Y ) ) ) ) ) ) ).
% or_numerals(6)
thf(fact_3579_or__numerals_I4_J,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [X: num,Y: num] :
( ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ ( bit0 @ X ) ) @ ( numeral_numeral @ A @ ( bit1 @ Y ) ) )
= ( plus_plus @ A @ ( one_one @ A ) @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ A @ ( numeral_numeral @ A @ X ) @ ( numeral_numeral @ A @ Y ) ) ) ) ) ) ).
% or_numerals(4)
thf(fact_3580_sgn__rat__def,axiom,
( ( sgn_sgn @ rat )
= ( ^ [A6: rat] :
( if @ rat
@ ( A6
= ( zero_zero @ rat ) )
@ ( zero_zero @ rat )
@ ( if @ rat @ ( ord_less @ rat @ ( zero_zero @ rat ) @ A6 ) @ ( one_one @ rat ) @ ( uminus_uminus @ rat @ ( one_one @ rat ) ) ) ) ) ) ).
% sgn_rat_def
thf(fact_3581_abs__rat__def,axiom,
( ( abs_abs @ rat )
= ( ^ [A6: rat] : ( if @ rat @ ( ord_less @ rat @ A6 @ ( zero_zero @ rat ) ) @ ( uminus_uminus @ rat @ A6 ) @ A6 ) ) ) ).
% abs_rat_def
thf(fact_3582_obtain__pos__sum,axiom,
! [R2: rat] :
( ( ord_less @ rat @ ( zero_zero @ rat ) @ R2 )
=> ~ ! [S3: rat] :
( ( ord_less @ rat @ ( zero_zero @ rat ) @ S3 )
=> ! [T7: rat] :
( ( ord_less @ rat @ ( zero_zero @ rat ) @ T7 )
=> ( R2
!= ( plus_plus @ rat @ S3 @ T7 ) ) ) ) ) ).
% obtain_pos_sum
thf(fact_3583_less__eq__rat__def,axiom,
( ( ord_less_eq @ rat )
= ( ^ [X5: rat,Y6: rat] :
( ( ord_less @ rat @ X5 @ Y6 )
| ( X5 = Y6 ) ) ) ) ).
% less_eq_rat_def
thf(fact_3584_or__eq__0__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A,B2: A] :
( ( ( bit_se1065995026697491101ons_or @ A @ A2 @ B2 )
= ( zero_zero @ A ) )
= ( ( A2
= ( zero_zero @ A ) )
& ( B2
= ( zero_zero @ A ) ) ) ) ) ).
% or_eq_0_iff
thf(fact_3585_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_3586_sum_Onon__neutral_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [G3: B > A,I6: set @ B] :
( ( groups1027152243600224163dd_sum @ B @ A @ G3
@ ( collect @ B
@ ^ [X5: B] :
( ( member @ B @ X5 @ I6 )
& ( ( G3 @ X5 )
!= ( zero_zero @ A ) ) ) ) )
= ( groups1027152243600224163dd_sum @ B @ A @ G3 @ I6 ) ) ) ).
% sum.non_neutral'
thf(fact_3587_sum_Odistrib__triv_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [I6: set @ B,G3: B > A,H2: B > A] :
( ( finite_finite2 @ B @ I6 )
=> ( ( groups1027152243600224163dd_sum @ B @ A
@ ^ [I4: B] : ( plus_plus @ A @ ( G3 @ I4 ) @ ( H2 @ I4 ) )
@ I6 )
= ( plus_plus @ A @ ( groups1027152243600224163dd_sum @ B @ A @ G3 @ I6 ) @ ( groups1027152243600224163dd_sum @ B @ A @ H2 @ I6 ) ) ) ) ) ).
% sum.distrib_triv'
thf(fact_3588_sum_Omono__neutral__left_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [S2: set @ B,T4: set @ B,G3: B > A] :
( ( ord_less_eq @ ( set @ B ) @ S2 @ T4 )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ ( minus_minus @ ( set @ B ) @ T4 @ S2 ) )
=> ( ( G3 @ X4 )
= ( zero_zero @ A ) ) )
=> ( ( groups1027152243600224163dd_sum @ B @ A @ G3 @ S2 )
= ( groups1027152243600224163dd_sum @ B @ A @ G3 @ T4 ) ) ) ) ) ).
% sum.mono_neutral_left'
thf(fact_3589_sum_Omono__neutral__right_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [S2: set @ B,T4: set @ B,G3: B > A] :
( ( ord_less_eq @ ( set @ B ) @ S2 @ T4 )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ ( minus_minus @ ( set @ B ) @ T4 @ S2 ) )
=> ( ( G3 @ X4 )
= ( zero_zero @ A ) ) )
=> ( ( groups1027152243600224163dd_sum @ B @ A @ G3 @ T4 )
= ( groups1027152243600224163dd_sum @ B @ A @ G3 @ S2 ) ) ) ) ) ).
% sum.mono_neutral_right'
thf(fact_3590_sum_Omono__neutral__cong__left_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [S2: set @ B,T4: set @ B,H2: B > A,G3: B > A] :
( ( ord_less_eq @ ( set @ B ) @ S2 @ T4 )
=> ( ! [I3: B] :
( ( member @ B @ I3 @ ( minus_minus @ ( set @ B ) @ T4 @ S2 ) )
=> ( ( H2 @ I3 )
= ( zero_zero @ A ) ) )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ S2 )
=> ( ( G3 @ X4 )
= ( H2 @ X4 ) ) )
=> ( ( groups1027152243600224163dd_sum @ B @ A @ G3 @ S2 )
= ( groups1027152243600224163dd_sum @ B @ A @ H2 @ T4 ) ) ) ) ) ) ).
% sum.mono_neutral_cong_left'
thf(fact_3591_sum_Omono__neutral__cong__right_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [S2: set @ B,T4: set @ B,G3: B > A,H2: B > A] :
( ( ord_less_eq @ ( set @ B ) @ S2 @ T4 )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ ( minus_minus @ ( set @ B ) @ T4 @ S2 ) )
=> ( ( G3 @ X4 )
= ( zero_zero @ A ) ) )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ S2 )
=> ( ( G3 @ X4 )
= ( H2 @ X4 ) ) )
=> ( ( groups1027152243600224163dd_sum @ B @ A @ G3 @ T4 )
= ( groups1027152243600224163dd_sum @ B @ A @ H2 @ S2 ) ) ) ) ) ) ).
% sum.mono_neutral_cong_right'
thf(fact_3592_bit_Ocomplement__unique,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [A2: A,X: A,Y: A] :
( ( ( bit_se5824344872417868541ns_and @ A @ A2 @ X )
= ( zero_zero @ A ) )
=> ( ( ( bit_se1065995026697491101ons_or @ A @ A2 @ X )
= ( uminus_uminus @ A @ ( one_one @ A ) ) )
=> ( ( ( bit_se5824344872417868541ns_and @ A @ A2 @ Y )
= ( zero_zero @ A ) )
=> ( ( ( bit_se1065995026697491101ons_or @ A @ A2 @ Y )
= ( uminus_uminus @ A @ ( one_one @ A ) ) )
=> ( X = Y ) ) ) ) ) ) ).
% bit.complement_unique
thf(fact_3593_sum_Odistrib_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [I6: set @ B,G3: B > A,H2: B > A] :
( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X5: B] :
( ( member @ B @ X5 @ I6 )
& ( ( G3 @ X5 )
!= ( zero_zero @ A ) ) ) ) )
=> ( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X5: B] :
( ( member @ B @ X5 @ I6 )
& ( ( H2 @ X5 )
!= ( zero_zero @ A ) ) ) ) )
=> ( ( groups1027152243600224163dd_sum @ B @ A
@ ^ [I4: B] : ( plus_plus @ A @ ( G3 @ I4 ) @ ( H2 @ I4 ) )
@ I6 )
= ( plus_plus @ A @ ( groups1027152243600224163dd_sum @ B @ A @ G3 @ I6 ) @ ( groups1027152243600224163dd_sum @ B @ A @ H2 @ I6 ) ) ) ) ) ) ).
% sum.distrib'
thf(fact_3594_sum_OG__def,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ( ( groups1027152243600224163dd_sum @ B @ A )
= ( ^ [P6: B > A,I8: set @ B] :
( if @ A
@ ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X5: B] :
( ( member @ B @ X5 @ I8 )
& ( ( P6 @ X5 )
!= ( zero_zero @ A ) ) ) ) )
@ ( groups7311177749621191930dd_sum @ B @ A @ P6
@ ( collect @ B
@ ^ [X5: B] :
( ( member @ B @ X5 @ I8 )
& ( ( P6 @ X5 )
!= ( zero_zero @ A ) ) ) ) )
@ ( zero_zero @ A ) ) ) ) ) ).
% sum.G_def
thf(fact_3595_fun__cong__unused__0,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( zero @ B )
=> ! [F3: ( A > B ) > C,G3: C] :
( ( F3
= ( ^ [X5: A > B] : G3 ) )
=> ( ( F3
@ ^ [X5: A] : ( zero_zero @ B ) )
= G3 ) ) ) ).
% fun_cong_unused_0
thf(fact_3596_dbl__dec__def,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ( ( neg_numeral_dbl_dec @ A )
= ( ^ [X5: A] : ( minus_minus @ A @ ( plus_plus @ A @ X5 @ X5 ) @ ( one_one @ A ) ) ) ) ) ).
% dbl_dec_def
thf(fact_3597_one__or__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A] :
( ( bit_se1065995026697491101ons_or @ A @ ( one_one @ A ) @ A2 )
= ( plus_plus @ A @ A2 @ ( zero_neq_one_of_bool @ A @ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) ) ) ) ) ).
% one_or_eq
thf(fact_3598_or__one__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A] :
( ( bit_se1065995026697491101ons_or @ A @ A2 @ ( one_one @ A ) )
= ( plus_plus @ A @ A2 @ ( zero_neq_one_of_bool @ A @ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) ) ) ) ) ).
% or_one_eq
thf(fact_3599_XOR__upper,axiom,
! [X: int,N: nat,Y: 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 @ Y @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) )
=> ( ord_less @ int @ ( bit_se5824344971392196577ns_xor @ int @ X @ Y ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ) ) ) ).
% XOR_upper
thf(fact_3600_sum__diff1_H,axiom,
! [B: $tType,A: $tType] :
( ( ab_group_add @ B )
=> ! [I6: set @ A,F3: A > B,I2: A] :
( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [I4: A] :
( ( member @ A @ I4 @ I6 )
& ( ( F3 @ I4 )
!= ( zero_zero @ B ) ) ) ) )
=> ( ( ( member @ A @ I2 @ I6 )
=> ( ( groups1027152243600224163dd_sum @ A @ B @ F3 @ ( minus_minus @ ( set @ A ) @ I6 @ ( insert @ A @ I2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( minus_minus @ B @ ( groups1027152243600224163dd_sum @ A @ B @ F3 @ I6 ) @ ( F3 @ I2 ) ) ) )
& ( ~ ( member @ A @ I2 @ I6 )
=> ( ( groups1027152243600224163dd_sum @ A @ B @ F3 @ ( minus_minus @ ( set @ A ) @ I6 @ ( insert @ A @ I2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( groups1027152243600224163dd_sum @ A @ B @ F3 @ I6 ) ) ) ) ) ) ).
% sum_diff1'
thf(fact_3601_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_3602_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_3603_or__nat__rec,axiom,
( ( bit_se1065995026697491101ons_or @ nat )
= ( ^ [M3: nat,N2: nat] :
( plus_plus @ nat
@ ( zero_neq_one_of_bool @ nat
@ ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ M3 )
| ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) )
@ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( bit_se1065995026697491101ons_or @ nat @ ( divide_divide @ nat @ M3 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( divide_divide @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% or_nat_rec
thf(fact_3604_normalize__negative,axiom,
! [Q2: int,P: int] :
( ( ord_less @ int @ Q2 @ ( zero_zero @ int ) )
=> ( ( normalize @ ( product_Pair @ int @ int @ P @ Q2 ) )
= ( normalize @ ( product_Pair @ int @ int @ ( uminus_uminus @ int @ P ) @ ( uminus_uminus @ int @ Q2 ) ) ) ) ) ).
% normalize_negative
thf(fact_3605_Sum__Ico__nat,axiom,
! [M2: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [X5: nat] : X5
@ ( 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_3606_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_3607_Cauchy__iff2,axiom,
( ( topolo3814608138187158403Cauchy @ real )
= ( ^ [X9: nat > real] :
! [J3: nat] :
? [M9: nat] :
! [M3: nat] :
( ( ord_less_eq @ nat @ M9 @ M3 )
=> ! [N2: nat] :
( ( ord_less_eq @ nat @ M9 @ N2 )
=> ( ord_less @ real @ ( abs_abs @ real @ ( minus_minus @ real @ ( X9 @ M3 ) @ ( X9 @ N2 ) ) ) @ ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ ( suc @ J3 ) ) ) ) ) ) ) ) ).
% Cauchy_iff2
thf(fact_3608_sum__power2,axiom,
! [K2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K2 ) )
= ( minus_minus @ nat @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K2 ) @ ( one_one @ nat ) ) ) ).
% sum_power2
thf(fact_3609_or__negative__int__iff,axiom,
! [K2: int,L: int] :
( ( ord_less @ int @ ( bit_se1065995026697491101ons_or @ int @ K2 @ L ) @ ( zero_zero @ int ) )
= ( ( ord_less @ int @ K2 @ ( zero_zero @ int ) )
| ( ord_less @ int @ L @ ( zero_zero @ int ) ) ) ) ).
% or_negative_int_iff
thf(fact_3610_finite__atLeastLessThan,axiom,
! [L: nat,U: nat] : ( finite_finite2 @ nat @ ( set_or7035219750837199246ssThan @ nat @ L @ U ) ) ).
% finite_atLeastLessThan
thf(fact_3611_atLeastLessThan__iff,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [I2: A,L: A,U: A] :
( ( member @ A @ I2 @ ( set_or7035219750837199246ssThan @ A @ L @ U ) )
= ( ( ord_less_eq @ A @ L @ I2 )
& ( ord_less @ A @ I2 @ U ) ) ) ) ).
% atLeastLessThan_iff
thf(fact_3612_atLeastLessThan__empty,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [B2: A,A2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( set_or7035219750837199246ssThan @ A @ A2 @ B2 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% atLeastLessThan_empty
thf(fact_3613_ivl__subset,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [I2: A,J: A,M2: A,N: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ I2 @ J ) @ ( set_or7035219750837199246ssThan @ A @ M2 @ N ) )
= ( ( ord_less_eq @ A @ J @ I2 )
| ( ( ord_less_eq @ A @ M2 @ I2 )
& ( ord_less_eq @ A @ J @ N ) ) ) ) ) ).
% ivl_subset
thf(fact_3614_atLeastLessThan__empty__iff2,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A2: A,B2: A] :
( ( ( bot_bot @ ( set @ A ) )
= ( set_or7035219750837199246ssThan @ A @ A2 @ B2 ) )
= ( ~ ( ord_less @ A @ A2 @ B2 ) ) ) ) ).
% atLeastLessThan_empty_iff2
thf(fact_3615_atLeastLessThan__empty__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A2: A,B2: A] :
( ( ( set_or7035219750837199246ssThan @ A @ A2 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
= ( ~ ( ord_less @ A @ A2 @ B2 ) ) ) ) ).
% atLeastLessThan_empty_iff
thf(fact_3616_infinite__Ico__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A2: A,B2: A] :
( ( ~ ( finite_finite2 @ A @ ( set_or7035219750837199246ssThan @ A @ A2 @ B2 ) ) )
= ( ord_less @ A @ A2 @ B2 ) ) ) ).
% infinite_Ico_iff
thf(fact_3617_ivl__diff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [I2: A,N: A,M2: A] :
( ( ord_less_eq @ A @ I2 @ N )
=> ( ( minus_minus @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ I2 @ M2 ) @ ( set_or7035219750837199246ssThan @ A @ I2 @ N ) )
= ( set_or7035219750837199246ssThan @ A @ N @ M2 ) ) ) ) ).
% ivl_diff
thf(fact_3618_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_3619_atLeastLessThan__singleton,axiom,
! [M2: nat] :
( ( set_or7035219750837199246ssThan @ nat @ M2 @ ( suc @ M2 ) )
= ( insert @ nat @ M2 @ ( bot_bot @ ( set @ nat ) ) ) ) ).
% atLeastLessThan_singleton
thf(fact_3620_sum_Oop__ivl__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [N: nat,M2: nat,G3: nat > A] :
( ( ( ord_less @ nat @ N @ M2 )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ M2 @ ( suc @ N ) ) )
= ( zero_zero @ A ) ) )
& ( ~ ( ord_less @ nat @ N @ M2 )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ M2 @ ( suc @ N ) ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) @ ( G3 @ N ) ) ) ) ) ) ).
% sum.op_ivl_Suc
thf(fact_3621_prod_Oop__ivl__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [N: nat,M2: nat,G3: nat > A] :
( ( ( ord_less @ nat @ N @ M2 )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ M2 @ ( suc @ N ) ) )
= ( one_one @ A ) ) )
& ( ~ ( ord_less @ nat @ N @ M2 )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ M2 @ ( suc @ N ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) @ ( G3 @ N ) ) ) ) ) ) ).
% prod.op_ivl_Suc
thf(fact_3622_atLeastLessThan__eq__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A,C3: A,D3: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ C3 @ D3 )
=> ( ( ( set_or7035219750837199246ssThan @ A @ A2 @ B2 )
= ( set_or7035219750837199246ssThan @ A @ C3 @ D3 ) )
= ( ( A2 = C3 )
& ( B2 = D3 ) ) ) ) ) ) ).
% atLeastLessThan_eq_iff
thf(fact_3623_atLeastLessThan__inj_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A,C3: A,D3: A] :
( ( ( set_or7035219750837199246ssThan @ A @ A2 @ B2 )
= ( set_or7035219750837199246ssThan @ A @ C3 @ D3 ) )
=> ( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ C3 @ D3 )
=> ( A2 = C3 ) ) ) ) ) ).
% atLeastLessThan_inj(1)
thf(fact_3624_atLeastLessThan__inj_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A,C3: A,D3: A] :
( ( ( set_or7035219750837199246ssThan @ A @ A2 @ B2 )
= ( set_or7035219750837199246ssThan @ A @ C3 @ D3 ) )
=> ( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ C3 @ D3 )
=> ( B2 = D3 ) ) ) ) ) ).
% atLeastLessThan_inj(2)
thf(fact_3625_atLeastLessThan__subset__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A,C3: A,D3: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ A2 @ B2 ) @ ( set_or7035219750837199246ssThan @ A @ C3 @ D3 ) )
=> ( ( ord_less_eq @ A @ B2 @ A2 )
| ( ( ord_less_eq @ A @ C3 @ A2 )
& ( ord_less_eq @ A @ B2 @ D3 ) ) ) ) ) ).
% atLeastLessThan_subset_iff
thf(fact_3626_infinite__Ico,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ~ ( finite_finite2 @ A @ ( set_or7035219750837199246ssThan @ A @ A2 @ B2 ) ) ) ) ).
% infinite_Ico
thf(fact_3627_all__nat__less__eq,axiom,
! [N: nat,P2: nat > $o] :
( ( ! [M3: nat] :
( ( ord_less @ nat @ M3 @ N )
=> ( P2 @ M3 ) ) )
= ( ! [X5: nat] :
( ( member @ nat @ X5 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) )
=> ( P2 @ X5 ) ) ) ) ).
% all_nat_less_eq
thf(fact_3628_ex__nat__less__eq,axiom,
! [N: nat,P2: nat > $o] :
( ( ? [M3: nat] :
( ( ord_less @ nat @ M3 @ N )
& ( P2 @ M3 ) ) )
= ( ? [X5: nat] :
( ( member @ nat @ X5 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) )
& ( P2 @ X5 ) ) ) ) ).
% ex_nat_less_eq
thf(fact_3629_atLeastLessThanSuc__atLeastAtMost,axiom,
! [L: nat,U: nat] :
( ( set_or7035219750837199246ssThan @ nat @ L @ ( suc @ U ) )
= ( set_or1337092689740270186AtMost @ nat @ L @ U ) ) ).
% atLeastLessThanSuc_atLeastAtMost
thf(fact_3630_lessThan__atLeast0,axiom,
( ( set_ord_lessThan @ nat )
= ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) ) ) ).
% lessThan_atLeast0
thf(fact_3631_atLeastLessThan0,axiom,
! [M2: nat] :
( ( set_or7035219750837199246ssThan @ nat @ M2 @ ( zero_zero @ nat ) )
= ( bot_bot @ ( set @ nat ) ) ) ).
% atLeastLessThan0
thf(fact_3632_sum_Oshift__bounds__Suc__ivl,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G3: nat > A,M2: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M2 ) @ ( suc @ N ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( G3 @ ( suc @ I4 ) )
@ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) ) ) ).
% sum.shift_bounds_Suc_ivl
thf(fact_3633_sum_Oshift__bounds__nat__ivl,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G3: nat > A,M2: nat,K2: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ ( plus_plus @ nat @ M2 @ K2 ) @ ( plus_plus @ nat @ N @ K2 ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( G3 @ ( plus_plus @ nat @ I4 @ K2 ) )
@ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) ) ) ).
% sum.shift_bounds_nat_ivl
thf(fact_3634_prod_Oshift__bounds__Suc__ivl,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G3: nat > A,M2: nat,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M2 ) @ ( suc @ N ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( G3 @ ( suc @ I4 ) )
@ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) ) ) ).
% prod.shift_bounds_Suc_ivl
thf(fact_3635_prod_Oshift__bounds__nat__ivl,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G3: nat > A,M2: nat,K2: nat,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ ( plus_plus @ nat @ M2 @ K2 ) @ ( plus_plus @ nat @ N @ K2 ) ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( G3 @ ( plus_plus @ nat @ I4 @ K2 ) )
@ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) ) ) ).
% prod.shift_bounds_nat_ivl
thf(fact_3636_sum_Oivl__cong,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( comm_monoid_add @ A ) )
=> ! [A2: B,C3: B,B2: B,D3: B,G3: B > A,H2: B > A] :
( ( A2 = C3 )
=> ( ( B2 = D3 )
=> ( ! [X4: B] :
( ( ord_less_eq @ B @ C3 @ X4 )
=> ( ( ord_less @ B @ X4 @ D3 )
=> ( ( G3 @ X4 )
= ( H2 @ X4 ) ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G3 @ ( set_or7035219750837199246ssThan @ B @ A2 @ B2 ) )
= ( groups7311177749621191930dd_sum @ B @ A @ H2 @ ( set_or7035219750837199246ssThan @ B @ C3 @ D3 ) ) ) ) ) ) ) ).
% sum.ivl_cong
thf(fact_3637_prod_Oivl__cong,axiom,
! [A: $tType,B: $tType] :
( ( ( ord @ B )
& ( comm_monoid_mult @ A ) )
=> ! [A2: B,C3: B,B2: B,D3: B,G3: B > A,H2: B > A] :
( ( A2 = C3 )
=> ( ( B2 = D3 )
=> ( ! [X4: B] :
( ( ord_less_eq @ B @ C3 @ X4 )
=> ( ( ord_less @ B @ X4 @ D3 )
=> ( ( G3 @ X4 )
= ( H2 @ X4 ) ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G3 @ ( set_or7035219750837199246ssThan @ B @ A2 @ B2 ) )
= ( groups7121269368397514597t_prod @ B @ A @ H2 @ ( set_or7035219750837199246ssThan @ B @ C3 @ D3 ) ) ) ) ) ) ) ).
% prod.ivl_cong
thf(fact_3638_sum_OatLeastLessThan__concat,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [M2: nat,N: nat,P: nat,G3: nat > A] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( ord_less_eq @ nat @ N @ P )
=> ( ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ N @ P ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ M2 @ P ) ) ) ) ) ) ).
% sum.atLeastLessThan_concat
thf(fact_3639_sum__diff__nat__ivl,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [M2: nat,N: nat,P: nat,F3: nat > A] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( ord_less_eq @ nat @ N @ P )
=> ( ( minus_minus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_or7035219750837199246ssThan @ nat @ M2 @ P ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_or7035219750837199246ssThan @ nat @ N @ P ) ) ) ) ) ) ).
% sum_diff_nat_ivl
thf(fact_3640_size__list__estimation,axiom,
! [A: $tType,X: A,Xs: list @ A,Y: nat,F3: A > nat] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ( ord_less @ nat @ Y @ ( F3 @ X ) )
=> ( ord_less @ nat @ Y @ ( size_list @ A @ F3 @ Xs ) ) ) ) ).
% size_list_estimation
thf(fact_3641_size__list__estimation_H,axiom,
! [A: $tType,X: A,Xs: list @ A,Y: nat,F3: A > nat] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ( ord_less_eq @ nat @ Y @ ( F3 @ X ) )
=> ( ord_less_eq @ nat @ Y @ ( size_list @ A @ F3 @ Xs ) ) ) ) ).
% size_list_estimation'
thf(fact_3642_size__list__pointwise,axiom,
! [A: $tType,Xs: list @ A,F3: A > nat,G3: A > nat] :
( ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
=> ( ord_less_eq @ nat @ ( F3 @ X4 ) @ ( G3 @ X4 ) ) )
=> ( ord_less_eq @ nat @ ( size_list @ A @ F3 @ Xs ) @ ( size_list @ A @ G3 @ Xs ) ) ) ).
% size_list_pointwise
thf(fact_3643_prod_OatLeastLessThan__concat,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M2: nat,N: nat,P: nat,G3: nat > A] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( ord_less_eq @ nat @ N @ P )
=> ( ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) @ ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ N @ P ) ) )
= ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ M2 @ P ) ) ) ) ) ) ).
% prod.atLeastLessThan_concat
thf(fact_3644_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_3645_subset__eq__atLeast0__lessThan__finite,axiom,
! [N6: set @ nat,N: nat] :
( ( ord_less_eq @ ( set @ nat ) @ N6 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) )
=> ( finite_finite2 @ nat @ N6 ) ) ).
% subset_eq_atLeast0_lessThan_finite
thf(fact_3646_atLeastLessThan__subseteq__atLeastAtMost__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A2: A,B2: A,C3: A,D3: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ A2 @ B2 ) @ ( set_or1337092689740270186AtMost @ A @ C3 @ D3 ) )
= ( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C3 @ A2 )
& ( ord_less_eq @ A @ B2 @ D3 ) ) ) ) ) ).
% atLeastLessThan_subseteq_atLeastAtMost_iff
thf(fact_3647_atLeastAtMost__subseteq__atLeastLessThan__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A2: A,B2: A,C3: A,D3: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) @ ( set_or7035219750837199246ssThan @ A @ C3 @ D3 ) )
= ( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C3 @ A2 )
& ( ord_less @ A @ B2 @ D3 ) ) ) ) ) ).
% atLeastAtMost_subseteq_atLeastLessThan_iff
thf(fact_3648_sum__shift__lb__Suc0__0__upt,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [F3: nat > A,K2: nat] :
( ( ( F3 @ ( zero_zero @ nat ) )
= ( zero_zero @ A ) )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ ( zero_zero @ nat ) ) @ K2 ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K2 ) ) ) ) ) ).
% sum_shift_lb_Suc0_0_upt
thf(fact_3649_sum_OatLeast0__lessThan__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G3: nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) @ ( G3 @ N ) ) ) ) ).
% sum.atLeast0_lessThan_Suc
thf(fact_3650_sum_OatLeast__Suc__lessThan,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [M2: nat,N: nat,G3: nat > A] :
( ( ord_less @ nat @ M2 @ N )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) )
= ( plus_plus @ A @ ( G3 @ M2 ) @ ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M2 ) @ N ) ) ) ) ) ) ).
% sum.atLeast_Suc_lessThan
thf(fact_3651_sum_OatLeastLessThan__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A2: nat,B2: nat,G3: nat > A] :
( ( ord_less_eq @ nat @ A2 @ B2 )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ A2 @ ( suc @ B2 ) ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ A2 @ B2 ) ) @ ( G3 @ B2 ) ) ) ) ) ).
% sum.atLeastLessThan_Suc
thf(fact_3652_atLeastLessThan__eq__atLeastAtMost__diff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( set_or7035219750837199246ssThan @ A )
= ( ^ [A6: A,B5: A] : ( minus_minus @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A6 @ B5 ) @ ( insert @ A @ B5 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% atLeastLessThan_eq_atLeastAtMost_diff
thf(fact_3653_prod_OatLeast0__lessThan__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G3: nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) @ ( G3 @ N ) ) ) ) ).
% prod.atLeast0_lessThan_Suc
thf(fact_3654_prod_OatLeast__Suc__lessThan,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M2: nat,N: nat,G3: nat > A] :
( ( ord_less @ nat @ M2 @ N )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) )
= ( times_times @ A @ ( G3 @ M2 ) @ ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M2 ) @ N ) ) ) ) ) ) ).
% prod.atLeast_Suc_lessThan
thf(fact_3655_prod_OatLeastLessThan__Suc,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A2: nat,B2: nat,G3: nat > A] :
( ( ord_less_eq @ nat @ A2 @ B2 )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ A2 @ ( suc @ B2 ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ A2 @ B2 ) ) @ ( G3 @ B2 ) ) ) ) ) ).
% prod.atLeastLessThan_Suc
thf(fact_3656_sum_Olast__plus,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [M2: nat,N: nat,G3: nat > A] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
= ( plus_plus @ A @ ( G3 @ N ) @ ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) ) ) ) ) ).
% sum.last_plus
thf(fact_3657_prod_Olast__plus,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M2: nat,N: nat,G3: nat > A] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
= ( times_times @ A @ ( G3 @ N ) @ ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) ) ) ) ) ).
% prod.last_plus
thf(fact_3658_sum__Suc__diff_H,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [M2: nat,N: nat,F3: nat > A] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( minus_minus @ A @ ( F3 @ ( suc @ I4 ) ) @ ( F3 @ I4 ) )
@ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) )
= ( minus_minus @ A @ ( F3 @ N ) @ ( F3 @ M2 ) ) ) ) ) ).
% sum_Suc_diff'
thf(fact_3659_sum_OatLeastLessThan__rev,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G3: nat > A,N: nat,M2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ N @ M2 ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( G3 @ ( minus_minus @ nat @ ( plus_plus @ nat @ M2 @ N ) @ ( suc @ I4 ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ N @ M2 ) ) ) ) ).
% sum.atLeastLessThan_rev
thf(fact_3660_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_3661_sum_Onested__swap,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [A2: nat > nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( groups7311177749621191930dd_sum @ nat @ A @ ( A2 @ I4 ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ I4 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [J3: nat] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( A2 @ I4 @ J3 )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ J3 ) @ N ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% sum.nested_swap
thf(fact_3662_prod_OatLeastLessThan__rev,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G3: nat > A,N: nat,M2: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ N @ M2 ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( G3 @ ( minus_minus @ nat @ ( plus_plus @ nat @ M2 @ N ) @ ( suc @ I4 ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ N @ M2 ) ) ) ) ).
% prod.atLeastLessThan_rev
thf(fact_3663_prod_Onested__swap,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A2: nat > nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( groups7121269368397514597t_prod @ nat @ A @ ( A2 @ I4 ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ I4 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [J3: nat] :
( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( A2 @ I4 @ J3 )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ J3 ) @ N ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% prod.nested_swap
thf(fact_3664_sum_Onat__group,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G3: nat > A,K2: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [M3: nat] : ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ ( times_times @ nat @ M3 @ K2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ M3 @ K2 ) @ K2 ) ) )
@ ( set_ord_lessThan @ nat @ N ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_ord_lessThan @ nat @ ( times_times @ nat @ N @ K2 ) ) ) ) ) ).
% sum.nat_group
thf(fact_3665_prod_Onat__group,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G3: nat > A,K2: nat,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A
@ ^ [M3: nat] : ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ ( times_times @ nat @ M3 @ K2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ M3 @ K2 ) @ K2 ) ) )
@ ( set_ord_lessThan @ nat @ N ) )
= ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_ord_lessThan @ nat @ ( times_times @ nat @ N @ K2 ) ) ) ) ) ).
% prod.nat_group
thf(fact_3666_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_3667_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_3668_sum_Ohead__if,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [N: nat,M2: nat,G3: nat > A] :
( ( ( ord_less @ nat @ N @ M2 )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
= ( zero_zero @ A ) ) )
& ( ~ ( ord_less @ nat @ N @ M2 )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) @ ( G3 @ N ) ) ) ) ) ) ).
% sum.head_if
thf(fact_3669_prod_Ohead__if,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [N: nat,M2: nat,G3: nat > A] :
( ( ( ord_less @ nat @ N @ M2 )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
= ( one_one @ A ) ) )
& ( ~ ( ord_less @ nat @ N @ M2 )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) @ ( G3 @ N ) ) ) ) ) ) ).
% prod.head_if
thf(fact_3670_normalize__denom__pos,axiom,
! [R2: product_prod @ int @ int,P: int,Q2: int] :
( ( ( normalize @ R2 )
= ( product_Pair @ int @ int @ P @ Q2 ) )
=> ( ord_less @ int @ ( zero_zero @ int ) @ Q2 ) ) ).
% normalize_denom_pos
thf(fact_3671_fact__prod__Suc,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ( ( semiring_char_0_fact @ A )
= ( ^ [N2: nat] : ( semiring_1_of_nat @ A @ ( groups7121269368397514597t_prod @ nat @ nat @ suc @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ) ) ) ).
% fact_prod_Suc
thf(fact_3672_sum_OatLeastLessThan__rev__at__least__Suc__atMost,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G3: nat > A,N: nat,M2: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ N @ M2 ) )
= ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( G3 @ ( minus_minus @ nat @ ( plus_plus @ nat @ M2 @ N ) @ I4 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ N ) @ M2 ) ) ) ) ).
% sum.atLeastLessThan_rev_at_least_Suc_atMost
thf(fact_3673_prod_OatLeastLessThan__rev__at__least__Suc__atMost,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G3: nat > A,N: nat,M2: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ N @ M2 ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( G3 @ ( minus_minus @ nat @ ( plus_plus @ nat @ M2 @ N ) @ I4 ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ N ) @ M2 ) ) ) ) ).
% prod.atLeastLessThan_rev_at_least_Suc_atMost
thf(fact_3674_pochhammer__prod,axiom,
! [A: $tType] :
( ( comm_semiring_1 @ A )
=> ( ( comm_s3205402744901411588hammer @ A )
= ( ^ [A6: A,N2: nat] :
( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( plus_plus @ A @ A6 @ ( semiring_1_of_nat @ A @ I4 ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ) ) ).
% pochhammer_prod
thf(fact_3675_atLeastLessThan__nat__numeral,axiom,
! [M2: nat,K2: num] :
( ( ( ord_less_eq @ nat @ M2 @ ( pred_numeral @ K2 ) )
=> ( ( set_or7035219750837199246ssThan @ nat @ M2 @ ( numeral_numeral @ nat @ K2 ) )
= ( insert @ nat @ ( pred_numeral @ K2 ) @ ( set_or7035219750837199246ssThan @ nat @ M2 @ ( pred_numeral @ K2 ) ) ) ) )
& ( ~ ( ord_less_eq @ nat @ M2 @ ( pred_numeral @ K2 ) )
=> ( ( set_or7035219750837199246ssThan @ nat @ M2 @ ( numeral_numeral @ nat @ K2 ) )
= ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% atLeastLessThan_nat_numeral
thf(fact_3676_fact__prod__rev,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ( ( semiring_char_0_fact @ A )
= ( ^ [N2: nat] : ( semiring_1_of_nat @ A @ ( groups7121269368397514597t_prod @ nat @ nat @ ( minus_minus @ nat @ N2 ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ) ) ) ).
% fact_prod_rev
thf(fact_3677_summable__Cauchy,axiom,
! [A: $tType] :
( ( real_Vector_banach @ A )
=> ( ( summable @ A )
= ( ^ [F4: nat > A] :
! [E4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E4 )
=> ? [N4: nat] :
! [M3: nat] :
( ( ord_less_eq @ nat @ N4 @ M3 )
=> ! [N2: nat] : ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F4 @ ( set_or7035219750837199246ssThan @ nat @ M3 @ N2 ) ) ) @ E4 ) ) ) ) ) ) ).
% summable_Cauchy
thf(fact_3678_CauchyD,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X7: nat > A,E3: real] :
( ( topolo3814608138187158403Cauchy @ A @ X7 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ E3 )
=> ? [M8: nat] :
! [M4: nat] :
( ( ord_less_eq @ nat @ M8 @ M4 )
=> ! [N5: nat] :
( ( ord_less_eq @ nat @ M8 @ N5 )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( X7 @ M4 ) @ ( X7 @ N5 ) ) ) @ E3 ) ) ) ) ) ) ).
% CauchyD
thf(fact_3679_CauchyI,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X7: nat > A] :
( ! [E2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E2 )
=> ? [M10: nat] :
! [M: nat] :
( ( ord_less_eq @ nat @ M10 @ M )
=> ! [N3: nat] :
( ( ord_less_eq @ nat @ M10 @ N3 )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( X7 @ M ) @ ( X7 @ N3 ) ) ) @ E2 ) ) ) )
=> ( topolo3814608138187158403Cauchy @ A @ X7 ) ) ) ).
% CauchyI
thf(fact_3680_Cauchy__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ( ( topolo3814608138187158403Cauchy @ A )
= ( ^ [X9: nat > A] :
! [E4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E4 )
=> ? [M9: nat] :
! [M3: nat] :
( ( ord_less_eq @ nat @ M9 @ M3 )
=> ! [N2: nat] :
( ( ord_less_eq @ nat @ M9 @ N2 )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( X9 @ M3 ) @ ( X9 @ N2 ) ) ) @ E4 ) ) ) ) ) ) ) ).
% Cauchy_iff
thf(fact_3681_sums__group,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [F3: nat > A,S: A,K2: nat] :
( ( sums @ A @ F3 @ S )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K2 )
=> ( sums @ A
@ ^ [N2: nat] : ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_or7035219750837199246ssThan @ nat @ ( times_times @ nat @ N2 @ K2 ) @ ( plus_plus @ nat @ ( times_times @ nat @ N2 @ K2 ) @ K2 ) ) )
@ S ) ) ) ) ).
% sums_group
thf(fact_3682_OR__upper,axiom,
! [X: int,N: nat,Y: 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 @ Y @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) )
=> ( ord_less @ int @ ( bit_se1065995026697491101ons_or @ int @ X @ Y ) @ ( power_power @ int @ ( numeral_numeral @ int @ ( bit0 @ one2 ) ) @ N ) ) ) ) ) ).
% OR_upper
thf(fact_3683_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_3684_fact__split,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [K2: nat,N: nat] :
( ( ord_less_eq @ nat @ K2 @ N )
=> ( ( semiring_char_0_fact @ A @ N )
= ( times_times @ A @ ( semiring_1_of_nat @ A @ ( groups7121269368397514597t_prod @ nat @ nat @ suc @ ( set_or7035219750837199246ssThan @ nat @ ( minus_minus @ nat @ N @ K2 ) @ N ) ) ) @ ( semiring_char_0_fact @ A @ ( minus_minus @ nat @ N @ K2 ) ) ) ) ) ) ).
% fact_split
thf(fact_3685_binomial__altdef__of__nat,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K2: nat,N: nat] :
( ( ord_less_eq @ nat @ K2 @ N )
=> ( ( semiring_1_of_nat @ A @ ( binomial @ N @ K2 ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( divide_divide @ A @ ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ N @ I4 ) ) @ ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ K2 @ I4 ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K2 ) ) ) ) ) ).
% binomial_altdef_of_nat
thf(fact_3686_gbinomial__altdef__of__nat,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( ( gbinomial @ A )
= ( ^ [A6: A,K3: nat] :
( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( divide_divide @ A @ ( minus_minus @ A @ A6 @ ( semiring_1_of_nat @ A @ I4 ) ) @ ( semiring_1_of_nat @ A @ ( minus_minus @ nat @ K3 @ I4 ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K3 ) ) ) ) ) ).
% gbinomial_altdef_of_nat
thf(fact_3687_gbinomial__mult__fact_H,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A2: A,K2: nat] :
( ( times_times @ A @ ( gbinomial @ A @ A2 @ K2 ) @ ( semiring_char_0_fact @ A @ K2 ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( minus_minus @ A @ A2 @ ( semiring_1_of_nat @ A @ I4 ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K2 ) ) ) ) ).
% gbinomial_mult_fact'
thf(fact_3688_gbinomial__mult__fact,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [K2: nat,A2: A] :
( ( times_times @ A @ ( semiring_char_0_fact @ A @ K2 ) @ ( gbinomial @ A @ A2 @ K2 ) )
= ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( minus_minus @ A @ A2 @ ( semiring_1_of_nat @ A @ I4 ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K2 ) ) ) ) ).
% gbinomial_mult_fact
thf(fact_3689_gbinomial__prod__rev,axiom,
! [A: $tType] :
( ( ( semiring_char_0 @ A )
& ( semidom_divide @ A ) )
=> ( ( gbinomial @ A )
= ( ^ [A6: A,K3: nat] :
( divide_divide @ A
@ ( groups7121269368397514597t_prod @ nat @ A
@ ^ [I4: nat] : ( minus_minus @ A @ A6 @ ( semiring_1_of_nat @ A @ I4 ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ K3 ) )
@ ( semiring_char_0_fact @ A @ K3 ) ) ) ) ) ).
% gbinomial_prod_rev
thf(fact_3690_horner__sum__eq__sum,axiom,
! [A: $tType,B: $tType] :
( ( comm_semiring_1 @ A )
=> ( ( groups4207007520872428315er_sum @ B @ A )
= ( ^ [F4: B > A,A6: A,Xs3: list @ B] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [N2: nat] : ( times_times @ A @ ( F4 @ ( nth @ B @ Xs3 @ N2 ) ) @ ( power_power @ A @ A6 @ N2 ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ B ) @ Xs3 ) ) ) ) ) ) ).
% horner_sum_eq_sum
thf(fact_3691_Chebyshev__sum__upper,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [N: nat,A2: nat > A,B2: nat > A] :
( ! [I3: nat,J2: nat] :
( ( ord_less_eq @ nat @ I3 @ J2 )
=> ( ( ord_less @ nat @ J2 @ N )
=> ( ord_less_eq @ A @ ( A2 @ I3 ) @ ( A2 @ J2 ) ) ) )
=> ( ! [I3: nat,J2: nat] :
( ( ord_less_eq @ nat @ I3 @ J2 )
=> ( ( ord_less @ nat @ J2 @ N )
=> ( ord_less_eq @ A @ ( B2 @ J2 ) @ ( B2 @ I3 ) ) ) )
=> ( ord_less_eq @ A
@ ( times_times @ A @ ( semiring_1_of_nat @ A @ N )
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( times_times @ A @ ( A2 @ K3 ) @ ( B2 @ K3 ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) )
@ ( times_times @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ A2 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ B2 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ) ) ) ).
% Chebyshev_sum_upper
thf(fact_3692_Chebyshev__sum__upper__nat,axiom,
! [N: nat,A2: nat > nat,B2: nat > nat] :
( ! [I3: nat,J2: nat] :
( ( ord_less_eq @ nat @ I3 @ J2 )
=> ( ( ord_less @ nat @ J2 @ N )
=> ( ord_less_eq @ nat @ ( A2 @ I3 ) @ ( A2 @ J2 ) ) ) )
=> ( ! [I3: nat,J2: nat] :
( ( ord_less_eq @ nat @ I3 @ J2 )
=> ( ( ord_less @ nat @ J2 @ N )
=> ( ord_less_eq @ nat @ ( B2 @ J2 ) @ ( B2 @ I3 ) ) ) )
=> ( ord_less_eq @ nat
@ ( times_times @ nat @ N
@ ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [I4: nat] : ( times_times @ nat @ ( A2 @ I4 ) @ ( B2 @ I4 ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) )
@ ( times_times @ nat @ ( groups7311177749621191930dd_sum @ nat @ nat @ A2 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) @ ( groups7311177749621191930dd_sum @ nat @ nat @ B2 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ) ) ).
% Chebyshev_sum_upper_nat
thf(fact_3693_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_3694_finite__atLeastLessThan__int,axiom,
! [L: int,U: int] : ( finite_finite2 @ int @ ( set_or7035219750837199246ssThan @ int @ L @ U ) ) ).
% finite_atLeastLessThan_int
thf(fact_3695_finite__atLeastZeroLessThan__int,axiom,
! [U: int] : ( finite_finite2 @ int @ ( set_or7035219750837199246ssThan @ int @ ( zero_zero @ int ) @ U ) ) ).
% finite_atLeastZeroLessThan_int
thf(fact_3696_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_3697_VEBT_Osize__gen_I2_J,axiom,
! [X21: $o,X222: $o] :
( ( vEBT_size_VEBT @ ( vEBT_Leaf @ X21 @ X222 ) )
= ( zero_zero @ nat ) ) ).
% VEBT.size_gen(2)
thf(fact_3698_is__singleton__the__elem,axiom,
! [A: $tType] :
( ( is_singleton @ A )
= ( ^ [A7: set @ A] :
( A7
= ( insert @ A @ ( the_elem @ A @ A7 ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% is_singleton_the_elem
thf(fact_3699_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_3700_even__sum__iff,axiom,
! [A: $tType,B: $tType] :
( ( semiring_parity @ A )
=> ! [A5: set @ B,F3: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ A5 ) )
= ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) )
@ ( finite_card @ B
@ ( collect @ B
@ ^ [A6: B] :
( ( member @ B @ A6 @ A5 )
& ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( F3 @ A6 ) ) ) ) ) ) ) ) ) ).
% even_sum_iff
thf(fact_3701_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_3702_card__lessThan,axiom,
! [U: nat] :
( ( finite_card @ nat @ ( set_ord_lessThan @ nat @ U ) )
= U ) ).
% card_lessThan
thf(fact_3703_card__Collect__less__nat,axiom,
! [N: nat] :
( ( finite_card @ nat
@ ( collect @ nat
@ ^ [I4: nat] : ( ord_less @ nat @ I4 @ N ) ) )
= N ) ).
% card_Collect_less_nat
thf(fact_3704_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_3705_card__atMost,axiom,
! [U: nat] :
( ( finite_card @ nat @ ( set_ord_atMost @ nat @ U ) )
= ( suc @ U ) ) ).
% card_atMost
thf(fact_3706_card__atLeastLessThan,axiom,
! [L: nat,U: nat] :
( ( finite_card @ nat @ ( set_or7035219750837199246ssThan @ nat @ L @ U ) )
= ( minus_minus @ nat @ U @ L ) ) ).
% card_atLeastLessThan
thf(fact_3707_card__Collect__le__nat,axiom,
! [N: nat] :
( ( finite_card @ nat
@ ( collect @ nat
@ ^ [I4: nat] : ( ord_less_eq @ nat @ I4 @ N ) ) )
= ( suc @ N ) ) ).
% card_Collect_le_nat
thf(fact_3708_card_Oempty,axiom,
! [A: $tType] :
( ( finite_card @ A @ ( bot_bot @ ( set @ A ) ) )
= ( zero_zero @ nat ) ) ).
% card.empty
thf(fact_3709_card_Oinfinite,axiom,
! [A: $tType,A5: set @ A] :
( ~ ( finite_finite2 @ A @ A5 )
=> ( ( finite_card @ A @ A5 )
= ( zero_zero @ nat ) ) ) ).
% card.infinite
thf(fact_3710_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_3711_signed__take__bit__negative__iff,axiom,
! [N: nat,K2: int] :
( ( ord_less @ int @ ( bit_ri4674362597316999326ke_bit @ int @ N @ K2 ) @ ( zero_zero @ int ) )
= ( bit_se5641148757651400278ts_bit @ int @ K2 @ N ) ) ).
% signed_take_bit_negative_iff
thf(fact_3712_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_3713_is__singletonI,axiom,
! [A: $tType,X: A] : ( is_singleton @ A @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ).
% is_singletonI
thf(fact_3714_card__0__eq,axiom,
! [A: $tType,A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( ( finite_card @ A @ A5 )
= ( zero_zero @ nat ) )
= ( A5
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% card_0_eq
thf(fact_3715_card__insert__disjoint,axiom,
! [A: $tType,A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ~ ( member @ A @ X @ A5 )
=> ( ( finite_card @ A @ ( insert @ A @ X @ A5 ) )
= ( suc @ ( finite_card @ A @ A5 ) ) ) ) ) ).
% card_insert_disjoint
thf(fact_3716_card__Diff__insert,axiom,
! [A: $tType,A2: A,A5: set @ A,B6: set @ A] :
( ( member @ A @ A2 @ A5 )
=> ( ~ ( member @ A @ A2 @ B6 )
=> ( ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ A2 @ B6 ) ) )
= ( minus_minus @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A5 @ B6 ) ) @ ( one_one @ nat ) ) ) ) ) ).
% card_Diff_insert
thf(fact_3717_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_3718_bit__0,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A2: A] :
( ( bit_se5641148757651400278ts_bit @ A @ A2 @ ( zero_zero @ nat ) )
= ( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) ) ) ) ).
% bit_0
thf(fact_3719_bit__mod__2__iff,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A2: A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( modulo_modulo @ A @ A2 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ N )
= ( ( N
= ( zero_zero @ nat ) )
& ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) ) ) ) ).
% bit_mod_2_iff
thf(fact_3720_subseqs__refl,axiom,
! [A: $tType,Xs: list @ A] : ( member @ ( list @ A ) @ Xs @ ( set2 @ ( list @ A ) @ ( subseqs @ A @ Xs ) ) ) ).
% subseqs_refl
thf(fact_3721_bit__disjunctive__add__iff,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A2: A,B2: A,N: nat] :
( ! [N3: nat] :
( ~ ( bit_se5641148757651400278ts_bit @ A @ A2 @ N3 )
| ~ ( bit_se5641148757651400278ts_bit @ A @ B2 @ N3 ) )
=> ( ( bit_se5641148757651400278ts_bit @ A @ ( plus_plus @ A @ A2 @ B2 ) @ N )
= ( ( bit_se5641148757651400278ts_bit @ A @ A2 @ N )
| ( bit_se5641148757651400278ts_bit @ A @ B2 @ N ) ) ) ) ) ).
% bit_disjunctive_add_iff
thf(fact_3722_is__singleton__altdef,axiom,
! [A: $tType] :
( ( is_singleton @ A )
= ( ^ [A7: set @ A] :
( ( finite_card @ A @ A7 )
= ( one_one @ nat ) ) ) ) ).
% is_singleton_altdef
thf(fact_3723_n__subsets,axiom,
! [A: $tType,A5: set @ A,K2: nat] :
( ( finite_finite2 @ A @ A5 )
=> ( ( finite_card @ ( set @ A )
@ ( collect @ ( set @ A )
@ ^ [B8: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ B8 @ A5 )
& ( ( finite_card @ A @ B8 )
= K2 ) ) ) )
= ( binomial @ ( finite_card @ A @ A5 ) @ K2 ) ) ) ).
% n_subsets
thf(fact_3724_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_3725_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_3726_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_3727_disjunctive__add,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A,B2: A] :
( ! [N3: nat] :
( ~ ( bit_se5641148757651400278ts_bit @ A @ A2 @ N3 )
| ~ ( bit_se5641148757651400278ts_bit @ A @ B2 @ N3 ) )
=> ( ( plus_plus @ A @ A2 @ B2 )
= ( bit_se1065995026697491101ons_or @ A @ A2 @ B2 ) ) ) ) ).
% disjunctive_add
thf(fact_3728_bit__take__bit__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M2: nat,A2: A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( bit_se2584673776208193580ke_bit @ A @ M2 @ A2 ) @ N )
= ( ( ord_less @ nat @ N @ M2 )
& ( bit_se5641148757651400278ts_bit @ A @ A2 @ N ) ) ) ) ).
% bit_take_bit_iff
thf(fact_3729_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_3730_infinite__arbitrarily__large,axiom,
! [A: $tType,A5: set @ A,N: nat] :
( ~ ( finite_finite2 @ A @ A5 )
=> ? [B9: set @ A] :
( ( finite_finite2 @ A @ B9 )
& ( ( finite_card @ A @ B9 )
= N )
& ( ord_less_eq @ ( set @ A ) @ B9 @ A5 ) ) ) ).
% infinite_arbitrarily_large
thf(fact_3731_card__subset__eq,axiom,
! [A: $tType,B6: set @ A,A5: set @ A] :
( ( finite_finite2 @ A @ B6 )
=> ( ( ord_less_eq @ ( set @ A ) @ A5 @ B6 )
=> ( ( ( finite_card @ A @ A5 )
= ( finite_card @ A @ B6 ) )
=> ( A5 = B6 ) ) ) ) ).
% card_subset_eq
thf(fact_3732_card__le__if__inj__on__rel,axiom,
! [B: $tType,A: $tType,B6: set @ A,A5: set @ B,R2: B > A > $o] :
( ( finite_finite2 @ A @ B6 )
=> ( ! [A4: B] :
( ( member @ B @ A4 @ A5 )
=> ? [B10: A] :
( ( member @ A @ B10 @ B6 )
& ( R2 @ A4 @ B10 ) ) )
=> ( ! [A13: B,A24: B,B4: A] :
( ( member @ B @ A13 @ A5 )
=> ( ( member @ B @ A24 @ A5 )
=> ( ( member @ A @ B4 @ B6 )
=> ( ( R2 @ A13 @ B4 )
=> ( ( R2 @ A24 @ B4 )
=> ( A13 = A24 ) ) ) ) ) )
=> ( ord_less_eq @ nat @ ( finite_card @ B @ A5 ) @ ( finite_card @ A @ B6 ) ) ) ) ) ).
% card_le_if_inj_on_rel
thf(fact_3733_card__insert__le,axiom,
! [A: $tType,A5: set @ A,X: A] : ( ord_less_eq @ nat @ ( finite_card @ A @ A5 ) @ ( finite_card @ A @ ( insert @ A @ X @ A5 ) ) ) ).
% card_insert_le
thf(fact_3734_card__atLeastZeroLessThan__int,axiom,
! [U: int] :
( ( finite_card @ int @ ( set_or7035219750837199246ssThan @ int @ ( zero_zero @ int ) @ U ) )
= ( nat2 @ U ) ) ).
% card_atLeastZeroLessThan_int
thf(fact_3735_sum__multicount__gen,axiom,
! [A: $tType,B: $tType,S: set @ A,T2: set @ B,R: A > B > $o,K2: B > nat] :
( ( finite_finite2 @ A @ S )
=> ( ( finite_finite2 @ B @ T2 )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ T2 )
=> ( ( finite_card @ A
@ ( collect @ A
@ ^ [I4: A] :
( ( member @ A @ I4 @ S )
& ( R @ I4 @ X4 ) ) ) )
= ( K2 @ X4 ) ) )
=> ( ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [I4: A] :
( finite_card @ B
@ ( collect @ B
@ ^ [J3: B] :
( ( member @ B @ J3 @ T2 )
& ( R @ I4 @ J3 ) ) ) )
@ S )
= ( groups7311177749621191930dd_sum @ B @ nat @ K2 @ T2 ) ) ) ) ) ).
% sum_multicount_gen
thf(fact_3736_card__lists__length__eq,axiom,
! [A: $tType,A5: set @ A,N: nat] :
( ( finite_finite2 @ A @ A5 )
=> ( ( finite_card @ ( list @ A )
@ ( collect @ ( list @ A )
@ ^ [Xs3: list @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs3 ) @ A5 )
& ( ( size_size @ ( list @ A ) @ Xs3 )
= N ) ) ) )
= ( power_power @ nat @ ( finite_card @ A @ A5 ) @ N ) ) ) ).
% card_lists_length_eq
thf(fact_3737_is__singletonI_H,axiom,
! [A: $tType,A5: set @ A] :
( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X4: A,Y3: A] :
( ( member @ A @ X4 @ A5 )
=> ( ( member @ A @ Y3 @ A5 )
=> ( X4 = Y3 ) ) )
=> ( is_singleton @ A @ A5 ) ) ) ).
% is_singletonI'
thf(fact_3738_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_3739_card__2__iff_H,axiom,
! [A: $tType,S2: set @ A] :
( ( ( finite_card @ A @ S2 )
= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( ? [X5: A] :
( ( member @ A @ X5 @ S2 )
& ? [Y6: A] :
( ( member @ A @ Y6 @ S2 )
& ( X5 != Y6 )
& ! [Z5: A] :
( ( member @ A @ Z5 @ S2 )
=> ( ( Z5 = X5 )
| ( Z5 = Y6 ) ) ) ) ) ) ) ).
% card_2_iff'
thf(fact_3740_card__eq__0__iff,axiom,
! [A: $tType,A5: set @ A] :
( ( ( finite_card @ A @ A5 )
= ( zero_zero @ nat ) )
= ( ( A5
= ( bot_bot @ ( set @ A ) ) )
| ~ ( finite_finite2 @ A @ A5 ) ) ) ).
% card_eq_0_iff
thf(fact_3741_card__ge__0__finite,axiom,
! [A: $tType,A5: set @ A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( finite_card @ A @ A5 ) )
=> ( finite_finite2 @ A @ A5 ) ) ).
% card_ge_0_finite
thf(fact_3742_card__Suc__eq__finite,axiom,
! [A: $tType,A5: set @ A,K2: nat] :
( ( ( finite_card @ A @ A5 )
= ( suc @ K2 ) )
= ( ? [B5: A,B8: set @ A] :
( ( A5
= ( insert @ A @ B5 @ B8 ) )
& ~ ( member @ A @ B5 @ B8 )
& ( ( finite_card @ A @ B8 )
= K2 )
& ( finite_finite2 @ A @ B8 ) ) ) ) ).
% card_Suc_eq_finite
thf(fact_3743_card__insert__if,axiom,
! [A: $tType,A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( ( member @ A @ X @ A5 )
=> ( ( finite_card @ A @ ( insert @ A @ X @ A5 ) )
= ( finite_card @ A @ A5 ) ) )
& ( ~ ( member @ A @ X @ A5 )
=> ( ( finite_card @ A @ ( insert @ A @ X @ A5 ) )
= ( suc @ ( finite_card @ A @ A5 ) ) ) ) ) ) ).
% card_insert_if
thf(fact_3744_finite__if__finite__subsets__card__bdd,axiom,
! [A: $tType,F5: set @ A,C5: nat] :
( ! [G5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ G5 @ F5 )
=> ( ( finite_finite2 @ A @ G5 )
=> ( ord_less_eq @ nat @ ( finite_card @ A @ G5 ) @ C5 ) ) )
=> ( ( finite_finite2 @ A @ F5 )
& ( ord_less_eq @ nat @ ( finite_card @ A @ F5 ) @ C5 ) ) ) ).
% finite_if_finite_subsets_card_bdd
thf(fact_3745_obtain__subset__with__card__n,axiom,
! [A: $tType,N: nat,S2: set @ A] :
( ( ord_less_eq @ nat @ N @ ( finite_card @ A @ S2 ) )
=> ~ ! [T6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ T6 @ S2 )
=> ( ( ( finite_card @ A @ T6 )
= N )
=> ~ ( finite_finite2 @ A @ T6 ) ) ) ) ).
% obtain_subset_with_card_n
thf(fact_3746_card__seteq,axiom,
! [A: $tType,B6: set @ A,A5: set @ A] :
( ( finite_finite2 @ A @ B6 )
=> ( ( ord_less_eq @ ( set @ A ) @ A5 @ B6 )
=> ( ( ord_less_eq @ nat @ ( finite_card @ A @ B6 ) @ ( finite_card @ A @ A5 ) )
=> ( A5 = B6 ) ) ) ) ).
% card_seteq
thf(fact_3747_card__mono,axiom,
! [A: $tType,B6: set @ A,A5: set @ A] :
( ( finite_finite2 @ A @ B6 )
=> ( ( ord_less_eq @ ( set @ A ) @ A5 @ B6 )
=> ( ord_less_eq @ nat @ ( finite_card @ A @ A5 ) @ ( finite_card @ A @ B6 ) ) ) ) ).
% card_mono
thf(fact_3748_card__less__sym__Diff,axiom,
! [A: $tType,A5: set @ A,B6: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( finite_finite2 @ A @ B6 )
=> ( ( ord_less @ nat @ ( finite_card @ A @ A5 ) @ ( finite_card @ A @ B6 ) )
=> ( ord_less @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A5 @ B6 ) ) @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ B6 @ A5 ) ) ) ) ) ) ).
% card_less_sym_Diff
thf(fact_3749_card__le__sym__Diff,axiom,
! [A: $tType,A5: set @ A,B6: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( finite_finite2 @ A @ B6 )
=> ( ( ord_less_eq @ nat @ ( finite_card @ A @ A5 ) @ ( finite_card @ A @ B6 ) )
=> ( ord_less_eq @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A5 @ B6 ) ) @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ B6 @ A5 ) ) ) ) ) ) ).
% card_le_sym_Diff
thf(fact_3750_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_3751_card__1__singletonE,axiom,
! [A: $tType,A5: set @ A] :
( ( ( finite_card @ A @ A5 )
= ( one_one @ nat ) )
=> ~ ! [X4: A] :
( A5
!= ( insert @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% card_1_singletonE
thf(fact_3752_psubset__card__mono,axiom,
! [A: $tType,B6: set @ A,A5: set @ A] :
( ( finite_finite2 @ A @ B6 )
=> ( ( ord_less @ ( set @ A ) @ A5 @ B6 )
=> ( ord_less @ nat @ ( finite_card @ A @ A5 ) @ ( finite_card @ A @ B6 ) ) ) ) ).
% psubset_card_mono
thf(fact_3753_card__less__Suc2,axiom,
! [M5: set @ nat,I2: nat] :
( ~ ( member @ nat @ ( zero_zero @ nat ) @ M5 )
=> ( ( finite_card @ nat
@ ( collect @ nat
@ ^ [K3: nat] :
( ( member @ nat @ ( suc @ K3 ) @ M5 )
& ( ord_less @ nat @ K3 @ I2 ) ) ) )
= ( finite_card @ nat
@ ( collect @ nat
@ ^ [K3: nat] :
( ( member @ nat @ K3 @ M5 )
& ( ord_less @ nat @ K3 @ ( suc @ I2 ) ) ) ) ) ) ) ).
% card_less_Suc2
thf(fact_3754_card__less__Suc,axiom,
! [M5: set @ nat,I2: nat] :
( ( member @ nat @ ( zero_zero @ nat ) @ M5 )
=> ( ( suc
@ ( finite_card @ nat
@ ( collect @ nat
@ ^ [K3: nat] :
( ( member @ nat @ ( suc @ K3 ) @ M5 )
& ( ord_less @ nat @ K3 @ I2 ) ) ) ) )
= ( finite_card @ nat
@ ( collect @ nat
@ ^ [K3: nat] :
( ( member @ nat @ K3 @ M5 )
& ( ord_less @ nat @ K3 @ ( suc @ I2 ) ) ) ) ) ) ) ).
% card_less_Suc
thf(fact_3755_card__less,axiom,
! [M5: set @ nat,I2: nat] :
( ( member @ nat @ ( zero_zero @ nat ) @ M5 )
=> ( ( finite_card @ nat
@ ( collect @ nat
@ ^ [K3: nat] :
( ( member @ nat @ K3 @ M5 )
& ( ord_less @ nat @ K3 @ ( suc @ I2 ) ) ) ) )
!= ( zero_zero @ nat ) ) ) ).
% card_less
thf(fact_3756_sum__Suc,axiom,
! [A: $tType,F3: A > nat,A5: set @ A] :
( ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [X5: A] : ( suc @ ( F3 @ X5 ) )
@ A5 )
= ( plus_plus @ nat @ ( groups7311177749621191930dd_sum @ A @ nat @ F3 @ A5 ) @ ( finite_card @ A @ A5 ) ) ) ).
% sum_Suc
thf(fact_3757_subset__card__intvl__is__intvl,axiom,
! [A5: set @ nat,K2: nat] :
( ( ord_less_eq @ ( set @ nat ) @ A5 @ ( set_or7035219750837199246ssThan @ nat @ K2 @ ( plus_plus @ nat @ K2 @ ( finite_card @ nat @ A5 ) ) ) )
=> ( A5
= ( set_or7035219750837199246ssThan @ nat @ K2 @ ( plus_plus @ nat @ K2 @ ( finite_card @ nat @ A5 ) ) ) ) ) ).
% subset_card_intvl_is_intvl
thf(fact_3758_sum__multicount,axiom,
! [A: $tType,B: $tType,S2: set @ A,T4: set @ B,R: A > B > $o,K2: nat] :
( ( finite_finite2 @ A @ S2 )
=> ( ( finite_finite2 @ B @ T4 )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ T4 )
=> ( ( finite_card @ A
@ ( collect @ A
@ ^ [I4: A] :
( ( member @ A @ I4 @ S2 )
& ( R @ I4 @ X4 ) ) ) )
= K2 ) )
=> ( ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [I4: A] :
( finite_card @ B
@ ( collect @ B
@ ^ [J3: B] :
( ( member @ B @ J3 @ T4 )
& ( R @ I4 @ J3 ) ) ) )
@ S2 )
= ( times_times @ nat @ K2 @ ( finite_card @ B @ T4 ) ) ) ) ) ) ).
% sum_multicount
thf(fact_3759_sum__bounded__above,axiom,
! [B: $tType,A: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( semiring_1 @ A ) )
=> ! [A5: set @ B,F3: B > A,K5: A] :
( ! [I3: B] :
( ( member @ B @ I3 @ A5 )
=> ( ord_less_eq @ A @ ( F3 @ I3 ) @ K5 ) )
=> ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ A5 ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( finite_card @ B @ A5 ) ) @ K5 ) ) ) ) ).
% sum_bounded_above
thf(fact_3760_sum__bounded__below,axiom,
! [A: $tType,B: $tType] :
( ( ( ordere6911136660526730532id_add @ A )
& ( semiring_1 @ A ) )
=> ! [A5: set @ B,K5: A,F3: B > A] :
( ! [I3: B] :
( ( member @ B @ I3 @ A5 )
=> ( ord_less_eq @ A @ K5 @ ( F3 @ I3 ) ) )
=> ( ord_less_eq @ A @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( finite_card @ B @ A5 ) ) @ K5 ) @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ A5 ) ) ) ) ).
% sum_bounded_below
thf(fact_3761_card__gt__0__iff,axiom,
! [A: $tType,A5: set @ A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( finite_card @ A @ A5 ) )
= ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
& ( finite_finite2 @ A @ A5 ) ) ) ).
% card_gt_0_iff
thf(fact_3762_card__1__singleton__iff,axiom,
! [A: $tType,A5: set @ A] :
( ( ( finite_card @ A @ A5 )
= ( suc @ ( zero_zero @ nat ) ) )
= ( ? [X5: A] :
( A5
= ( insert @ A @ X5 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% card_1_singleton_iff
thf(fact_3763_card__eq__SucD,axiom,
! [A: $tType,A5: set @ A,K2: nat] :
( ( ( finite_card @ A @ A5 )
= ( suc @ K2 ) )
=> ? [B4: A,B9: set @ A] :
( ( A5
= ( insert @ A @ B4 @ B9 ) )
& ~ ( member @ A @ B4 @ B9 )
& ( ( finite_card @ A @ B9 )
= K2 )
& ( ( K2
= ( zero_zero @ nat ) )
=> ( B9
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% card_eq_SucD
thf(fact_3764_card__Suc__eq,axiom,
! [A: $tType,A5: set @ A,K2: nat] :
( ( ( finite_card @ A @ A5 )
= ( suc @ K2 ) )
= ( ? [B5: A,B8: set @ A] :
( ( A5
= ( insert @ A @ B5 @ B8 ) )
& ~ ( member @ A @ B5 @ B8 )
& ( ( finite_card @ A @ B8 )
= K2 )
& ( ( K2
= ( zero_zero @ nat ) )
=> ( B8
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% card_Suc_eq
thf(fact_3765_card__le__Suc0__iff__eq,axiom,
! [A: $tType,A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( ord_less_eq @ nat @ ( finite_card @ A @ A5 ) @ ( suc @ ( zero_zero @ nat ) ) )
= ( ! [X5: A] :
( ( member @ A @ X5 @ A5 )
=> ! [Y6: A] :
( ( member @ A @ Y6 @ A5 )
=> ( X5 = Y6 ) ) ) ) ) ) ).
% card_le_Suc0_iff_eq
thf(fact_3766_card__le__Suc__iff,axiom,
! [A: $tType,N: nat,A5: set @ A] :
( ( ord_less_eq @ nat @ ( suc @ N ) @ ( finite_card @ A @ A5 ) )
= ( ? [A6: A,B8: set @ A] :
( ( A5
= ( insert @ A @ A6 @ B8 ) )
& ~ ( member @ A @ A6 @ B8 )
& ( ord_less_eq @ nat @ N @ ( finite_card @ A @ B8 ) )
& ( finite_finite2 @ A @ B8 ) ) ) ) ).
% card_le_Suc_iff
thf(fact_3767_card__Diff1__le,axiom,
! [A: $tType,A5: set @ A,X: A] : ( ord_less_eq @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) @ ( finite_card @ A @ A5 ) ) ).
% card_Diff1_le
thf(fact_3768_card__Diff__subset,axiom,
! [A: $tType,B6: set @ A,A5: set @ A] :
( ( finite_finite2 @ A @ B6 )
=> ( ( ord_less_eq @ ( set @ A ) @ B6 @ A5 )
=> ( ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A5 @ B6 ) )
= ( minus_minus @ nat @ ( finite_card @ A @ A5 ) @ ( finite_card @ A @ B6 ) ) ) ) ) ).
% card_Diff_subset
thf(fact_3769_bit__imp__take__bit__positive,axiom,
! [N: nat,M2: nat,K2: int] :
( ( ord_less @ nat @ N @ M2 )
=> ( ( bit_se5641148757651400278ts_bit @ int @ K2 @ N )
=> ( ord_less @ int @ ( zero_zero @ int ) @ ( bit_se2584673776208193580ke_bit @ int @ M2 @ K2 ) ) ) ) ).
% bit_imp_take_bit_positive
thf(fact_3770_card__psubset,axiom,
! [A: $tType,B6: set @ A,A5: set @ A] :
( ( finite_finite2 @ A @ B6 )
=> ( ( ord_less_eq @ ( set @ A ) @ A5 @ B6 )
=> ( ( ord_less @ nat @ ( finite_card @ A @ A5 ) @ ( finite_card @ A @ B6 ) )
=> ( ord_less @ ( set @ A ) @ A5 @ B6 ) ) ) ) ).
% card_psubset
thf(fact_3771_diff__card__le__card__Diff,axiom,
! [A: $tType,B6: set @ A,A5: set @ A] :
( ( finite_finite2 @ A @ B6 )
=> ( ord_less_eq @ nat @ ( minus_minus @ nat @ ( finite_card @ A @ A5 ) @ ( finite_card @ A @ B6 ) ) @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A5 @ B6 ) ) ) ) ).
% diff_card_le_card_Diff
thf(fact_3772_card__lists__length__le,axiom,
! [A: $tType,A5: set @ A,N: nat] :
( ( finite_finite2 @ A @ A5 )
=> ( ( finite_card @ ( list @ A )
@ ( collect @ ( list @ A )
@ ^ [Xs3: list @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs3 ) @ A5 )
& ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs3 ) @ N ) ) ) )
= ( groups7311177749621191930dd_sum @ nat @ nat @ ( power_power @ nat @ ( finite_card @ A @ A5 ) ) @ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% card_lists_length_le
thf(fact_3773_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
@ ^ [Z5: A] :
( ( power_power @ A @ Z5 @ N )
= ( one_one @ A ) ) ) )
@ N ) ) ) ).
% card_roots_unity
thf(fact_3774_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_3775_exp__eq__0__imp__not__bit,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [N: nat,A2: A] :
( ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N )
= ( zero_zero @ A ) )
=> ~ ( bit_se5641148757651400278ts_bit @ A @ A2 @ N ) ) ) ).
% exp_eq_0_imp_not_bit
thf(fact_3776_card__sum__le__nat__sum,axiom,
! [S2: set @ nat] :
( ord_less_eq @ nat
@ ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [X5: nat] : X5
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( finite_card @ nat @ S2 ) ) )
@ ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [X5: nat] : X5
@ S2 ) ) ).
% card_sum_le_nat_sum
thf(fact_3777_card__nth__roots,axiom,
! [C3: complex,N: nat] :
( ( C3
!= ( zero_zero @ complex ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( finite_card @ complex
@ ( collect @ complex
@ ^ [Z5: complex] :
( ( power_power @ complex @ Z5 @ N )
= C3 ) ) )
= N ) ) ) ).
% card_nth_roots
thf(fact_3778_card__roots__unity__eq,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( finite_card @ complex
@ ( collect @ complex
@ ^ [Z5: complex] :
( ( power_power @ complex @ Z5 @ N )
= ( one_one @ complex ) ) ) )
= N ) ) ).
% card_roots_unity_eq
thf(fact_3779_int__bit__bound,axiom,
! [K2: int] :
~ ! [N3: nat] :
( ! [M4: nat] :
( ( ord_less_eq @ nat @ N3 @ M4 )
=> ( ( bit_se5641148757651400278ts_bit @ int @ K2 @ M4 )
= ( bit_se5641148757651400278ts_bit @ int @ K2 @ N3 ) ) )
=> ~ ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( ( bit_se5641148757651400278ts_bit @ int @ K2 @ ( minus_minus @ nat @ N3 @ ( one_one @ nat ) ) )
= ( ~ ( bit_se5641148757651400278ts_bit @ int @ K2 @ N3 ) ) ) ) ) ).
% int_bit_bound
thf(fact_3780_is__singleton__def,axiom,
! [A: $tType] :
( ( is_singleton @ A )
= ( ^ [A7: set @ A] :
? [X5: A] :
( A7
= ( insert @ A @ X5 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% is_singleton_def
thf(fact_3781_is__singletonE,axiom,
! [A: $tType,A5: set @ A] :
( ( is_singleton @ A @ A5 )
=> ~ ! [X4: A] :
( A5
!= ( insert @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% is_singletonE
thf(fact_3782_card__2__iff,axiom,
! [A: $tType,S2: set @ A] :
( ( ( finite_card @ A @ S2 )
= ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) )
= ( ? [X5: A,Y6: A] :
( ( S2
= ( insert @ A @ X5 @ ( insert @ A @ Y6 @ ( bot_bot @ ( set @ A ) ) ) ) )
& ( X5 != Y6 ) ) ) ) ).
% card_2_iff
thf(fact_3783_card__3__iff,axiom,
! [A: $tType,S2: set @ A] :
( ( ( finite_card @ A @ S2 )
= ( numeral_numeral @ nat @ ( bit1 @ one2 ) ) )
= ( ? [X5: A,Y6: A,Z5: A] :
( ( S2
= ( insert @ A @ X5 @ ( insert @ A @ Y6 @ ( insert @ A @ Z5 @ ( bot_bot @ ( set @ A ) ) ) ) ) )
& ( X5 != Y6 )
& ( Y6 != Z5 )
& ( X5 != Z5 ) ) ) ) ).
% card_3_iff
thf(fact_3784_odd__card__imp__not__empty,axiom,
! [A: $tType,A5: set @ A] :
( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( finite_card @ A @ A5 ) )
=> ( A5
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% odd_card_imp_not_empty
thf(fact_3785_card_Oremove,axiom,
! [A: $tType,A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( member @ A @ X @ A5 )
=> ( ( finite_card @ A @ A5 )
= ( suc @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% card.remove
thf(fact_3786_card_Oinsert__remove,axiom,
! [A: $tType,A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( finite_card @ A @ ( insert @ A @ X @ A5 ) )
= ( suc @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% card.insert_remove
thf(fact_3787_card__Suc__Diff1,axiom,
! [A: $tType,A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( member @ A @ X @ A5 )
=> ( ( suc @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) )
= ( finite_card @ A @ A5 ) ) ) ) ).
% card_Suc_Diff1
thf(fact_3788_card__Diff1__less,axiom,
! [A: $tType,A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( member @ A @ X @ A5 )
=> ( ord_less @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) @ ( finite_card @ A @ A5 ) ) ) ) ).
% card_Diff1_less
thf(fact_3789_card__Diff2__less,axiom,
! [A: $tType,A5: set @ A,X: A,Y: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( member @ A @ X @ A5 )
=> ( ( member @ A @ Y @ A5 )
=> ( ord_less @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ ( insert @ A @ Y @ ( bot_bot @ ( set @ A ) ) ) ) ) @ ( finite_card @ A @ A5 ) ) ) ) ) ).
% card_Diff2_less
thf(fact_3790_card__Diff1__less__iff,axiom,
! [A: $tType,A5: set @ A,X: A] :
( ( ord_less @ nat @ ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) @ ( finite_card @ A @ A5 ) )
= ( ( finite_finite2 @ A @ A5 )
& ( member @ A @ X @ A5 ) ) ) ).
% card_Diff1_less_iff
thf(fact_3791_card__Diff__singleton__if,axiom,
! [A: $tType,X: A,A5: set @ A] :
( ( ( member @ A @ X @ A5 )
=> ( ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( minus_minus @ nat @ ( finite_card @ A @ A5 ) @ ( one_one @ nat ) ) ) )
& ( ~ ( member @ A @ X @ A5 )
=> ( ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( finite_card @ A @ A5 ) ) ) ) ).
% card_Diff_singleton_if
thf(fact_3792_card__Diff__singleton,axiom,
! [A: $tType,X: A,A5: set @ A] :
( ( member @ A @ X @ A5 )
=> ( ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( minus_minus @ nat @ ( finite_card @ A @ A5 ) @ ( one_one @ nat ) ) ) ) ).
% card_Diff_singleton
thf(fact_3793_and__exp__eq__0__iff__not__bit,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A,N: nat] :
( ( ( bit_se5824344872417868541ns_and @ A @ A2 @ ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N ) )
= ( zero_zero @ A ) )
= ( ~ ( bit_se5641148757651400278ts_bit @ A @ A2 @ N ) ) ) ) ).
% and_exp_eq_0_iff_not_bit
thf(fact_3794_prod__le__power,axiom,
! [B: $tType,A: $tType] :
( ( linordered_semidom @ A )
=> ! [A5: set @ B,F3: B > A,N: A,K2: nat] :
( ! [I3: B] :
( ( member @ B @ I3 @ A5 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( F3 @ I3 ) )
& ( ord_less_eq @ A @ ( F3 @ I3 ) @ N ) ) )
=> ( ( ord_less_eq @ nat @ ( finite_card @ B @ A5 ) @ K2 )
=> ( ( ord_less_eq @ A @ ( one_one @ A ) @ N )
=> ( ord_less_eq @ A @ ( groups7121269368397514597t_prod @ B @ A @ F3 @ A5 ) @ ( power_power @ A @ N @ K2 ) ) ) ) ) ) ).
% prod_le_power
thf(fact_3795_sum__bounded__above__strict,axiom,
! [B: $tType,A: $tType] :
( ( ( ordere8940638589300402666id_add @ A )
& ( semiring_1 @ A ) )
=> ! [A5: set @ B,F3: B > A,K5: A] :
( ! [I3: B] :
( ( member @ B @ I3 @ A5 )
=> ( ord_less @ A @ ( F3 @ I3 ) @ K5 ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ ( finite_card @ B @ A5 ) )
=> ( ord_less @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ A5 ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ ( finite_card @ B @ A5 ) ) @ K5 ) ) ) ) ) ).
% sum_bounded_above_strict
thf(fact_3796_sum__bounded__above__divide,axiom,
! [B: $tType,A: $tType] :
( ( linordered_field @ A )
=> ! [A5: set @ B,F3: B > A,K5: A] :
( ! [I3: B] :
( ( member @ B @ I3 @ A5 )
=> ( ord_less_eq @ A @ ( F3 @ I3 ) @ ( divide_divide @ A @ K5 @ ( semiring_1_of_nat @ A @ ( finite_card @ B @ A5 ) ) ) ) )
=> ( ( finite_finite2 @ B @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ord_less_eq @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ A5 ) @ K5 ) ) ) ) ) ).
% sum_bounded_above_divide
thf(fact_3797_card__insert__le__m1,axiom,
! [A: $tType,N: nat,Y: set @ A,X: A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ nat @ ( finite_card @ A @ Y ) @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) )
=> ( ord_less_eq @ nat @ ( finite_card @ A @ ( insert @ A @ X @ Y ) ) @ N ) ) ) ).
% card_insert_le_m1
thf(fact_3798_polyfun__roots__card,axiom,
! [A: $tType] :
( ( ( real_V8999393235501362500lgebra @ A )
& ( idom @ A ) )
=> ! [C3: nat > A,K2: nat,N: nat] :
( ( ( C3 @ K2 )
!= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ nat @ K2 @ N )
=> ( ord_less_eq @ nat
@ ( finite_card @ A
@ ( collect @ A
@ ^ [Z5: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( C3 @ I4 ) @ ( power_power @ A @ Z5 @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( zero_zero @ A ) ) ) )
@ N ) ) ) ) ).
% polyfun_roots_card
thf(fact_3799_prod__gen__delta,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S2: set @ B,A2: B,B2: B > A,C3: A] :
( ( finite_finite2 @ B @ S2 )
=> ( ( ( member @ B @ A2 @ S2 )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K3: B] : ( if @ A @ ( K3 = A2 ) @ ( B2 @ K3 ) @ C3 )
@ S2 )
= ( times_times @ A @ ( B2 @ A2 ) @ ( power_power @ A @ C3 @ ( minus_minus @ nat @ ( finite_card @ B @ S2 ) @ ( one_one @ nat ) ) ) ) ) )
& ( ~ ( member @ B @ A2 @ S2 )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [K3: B] : ( if @ A @ ( K3 = A2 ) @ ( B2 @ K3 ) @ C3 )
@ S2 )
= ( power_power @ A @ C3 @ ( finite_card @ B @ S2 ) ) ) ) ) ) ) ).
% prod_gen_delta
thf(fact_3800_even__bit__succ__iff,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A2: A,N: nat] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
=> ( ( bit_se5641148757651400278ts_bit @ A @ ( plus_plus @ A @ ( one_one @ A ) @ A2 ) @ N )
= ( ( bit_se5641148757651400278ts_bit @ A @ A2 @ N )
| ( N
= ( zero_zero @ nat ) ) ) ) ) ) ).
% even_bit_succ_iff
thf(fact_3801_odd__bit__iff__bit__pred,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ! [A2: A,N: nat] :
( ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 )
=> ( ( bit_se5641148757651400278ts_bit @ A @ A2 @ N )
= ( ( bit_se5641148757651400278ts_bit @ A @ ( minus_minus @ A @ A2 @ ( one_one @ A ) ) @ N )
| ( N
= ( zero_zero @ nat ) ) ) ) ) ) ).
% odd_bit_iff_bit_pred
thf(fact_3802_bit__sum__mult__2__cases,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A,B2: A,N: nat] :
( ! [J2: nat] :
~ ( bit_se5641148757651400278ts_bit @ A @ A2 @ ( suc @ J2 ) )
=> ( ( bit_se5641148757651400278ts_bit @ A @ ( plus_plus @ A @ A2 @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) ) @ N )
= ( ( ( N
= ( zero_zero @ nat ) )
=> ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) )
& ( ( N
!= ( zero_zero @ nat ) )
=> ( bit_se5641148757651400278ts_bit @ A @ ( times_times @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ B2 ) @ N ) ) ) ) ) ) ).
% bit_sum_mult_2_cases
thf(fact_3803_polyfun__rootbound,axiom,
! [A: $tType] :
( ( ( real_V8999393235501362500lgebra @ A )
& ( idom @ A ) )
=> ! [C3: nat > A,K2: nat,N: nat] :
( ( ( C3 @ K2 )
!= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ nat @ K2 @ N )
=> ( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [Z5: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( C3 @ I4 ) @ ( power_power @ A @ Z5 @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( zero_zero @ A ) ) ) )
& ( ord_less_eq @ nat
@ ( finite_card @ A
@ ( collect @ A
@ ^ [Z5: A] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( C3 @ I4 ) @ ( power_power @ A @ Z5 @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) )
= ( zero_zero @ A ) ) ) )
@ N ) ) ) ) ) ).
% polyfun_rootbound
thf(fact_3804_bit__rec,axiom,
! [A: $tType] :
( ( bit_semiring_bits @ A )
=> ( ( bit_se5641148757651400278ts_bit @ A )
= ( ^ [A6: A,N2: nat] :
( ( ( N2
= ( zero_zero @ nat ) )
=> ~ ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A6 ) )
& ( ( N2
!= ( zero_zero @ nat ) )
=> ( bit_se5641148757651400278ts_bit @ A @ ( divide_divide @ A @ A6 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% bit_rec
thf(fact_3805_card__lists__distinct__length__eq,axiom,
! [A: $tType,A5: set @ A,K2: nat] :
( ( finite_finite2 @ A @ A5 )
=> ( ( ord_less_eq @ nat @ K2 @ ( finite_card @ A @ A5 ) )
=> ( ( finite_card @ ( list @ A )
@ ( collect @ ( list @ A )
@ ^ [Xs3: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs3 )
= K2 )
& ( distinct @ A @ Xs3 )
& ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs3 ) @ A5 ) ) ) )
= ( groups7121269368397514597t_prod @ nat @ nat
@ ^ [X5: nat] : X5
@ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ ( minus_minus @ nat @ ( finite_card @ A @ A5 ) @ K2 ) @ ( one_one @ nat ) ) @ ( finite_card @ A @ A5 ) ) ) ) ) ) ).
% card_lists_distinct_length_eq
thf(fact_3806_length__mul__elem,axiom,
! [A: $tType,Xs: list @ ( list @ A ),N: nat] :
( ! [X4: list @ A] :
( ( member @ ( list @ A ) @ X4 @ ( set2 @ ( list @ A ) @ Xs ) )
=> ( ( size_size @ ( list @ A ) @ X4 )
= N ) )
=> ( ( size_size @ ( list @ A ) @ ( concat @ A @ Xs ) )
= ( times_times @ nat @ ( size_size @ ( list @ ( list @ A ) ) @ Xs ) @ N ) ) ) ).
% length_mul_elem
thf(fact_3807_card__lists__distinct__length__eq_H,axiom,
! [A: $tType,K2: nat,A5: set @ A] :
( ( ord_less @ nat @ K2 @ ( finite_card @ A @ A5 ) )
=> ( ( finite_card @ ( list @ A )
@ ( collect @ ( list @ A )
@ ^ [Xs3: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs3 )
= K2 )
& ( distinct @ A @ Xs3 )
& ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs3 ) @ A5 ) ) ) )
= ( groups7121269368397514597t_prod @ nat @ nat
@ ^ [X5: nat] : X5
@ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ ( minus_minus @ nat @ ( finite_card @ A @ A5 ) @ K2 ) @ ( one_one @ nat ) ) @ ( finite_card @ A @ A5 ) ) ) ) ) ).
% card_lists_distinct_length_eq'
thf(fact_3808_case__nat__add__eq__if,axiom,
! [A: $tType,A2: A,F3: nat > A,V: num,N: nat] :
( ( case_nat @ A @ A2 @ F3 @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ V ) @ N ) )
= ( F3 @ ( plus_plus @ nat @ ( pred_numeral @ V ) @ N ) ) ) ).
% case_nat_add_eq_if
thf(fact_3809_distinct__swap,axiom,
! [A: $tType,I2: nat,Xs: list @ A,J: nat] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( distinct @ A @ ( list_update @ A @ ( list_update @ A @ Xs @ I2 @ ( nth @ A @ Xs @ J ) ) @ J @ ( nth @ A @ Xs @ I2 ) ) )
= ( distinct @ A @ Xs ) ) ) ) ).
% distinct_swap
thf(fact_3810_finite__lists__distinct__length__eq,axiom,
! [A: $tType,A5: set @ A,N: nat] :
( ( finite_finite2 @ A @ A5 )
=> ( finite_finite2 @ ( list @ A )
@ ( collect @ ( list @ A )
@ ^ [Xs3: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs3 )
= N )
& ( distinct @ A @ Xs3 )
& ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs3 ) @ A5 ) ) ) ) ) ).
% finite_lists_distinct_length_eq
thf(fact_3811_nat_Ocase__distrib,axiom,
! [B: $tType,A: $tType,H2: A > B,F1: A,F22: nat > A,Nat: nat] :
( ( H2 @ ( case_nat @ A @ F1 @ F22 @ Nat ) )
= ( case_nat @ B @ ( H2 @ F1 )
@ ^ [X5: nat] : ( H2 @ ( F22 @ X5 ) )
@ Nat ) ) ).
% nat.case_distrib
thf(fact_3812_distinct__product,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B] :
( ( distinct @ A @ Xs )
=> ( ( distinct @ B @ Ys )
=> ( distinct @ ( product_prod @ A @ B ) @ ( product @ A @ B @ Xs @ Ys ) ) ) ) ).
% distinct_product
thf(fact_3813_sorted__list__of__set_Odistinct__if__distinct__map,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] :
( ( distinct @ A @ Xs )
=> ( distinct @ A @ Xs ) ) ) ).
% sorted_list_of_set.distinct_if_distinct_map
thf(fact_3814_subseqs__distinctD,axiom,
! [A: $tType,Ys: list @ A,Xs: list @ A] :
( ( member @ ( list @ A ) @ Ys @ ( set2 @ ( list @ A ) @ ( subseqs @ A @ Xs ) ) )
=> ( ( distinct @ A @ Xs )
=> ( distinct @ A @ Ys ) ) ) ).
% subseqs_distinctD
thf(fact_3815_old_Onat_Osimps_I5_J,axiom,
! [A: $tType,F1: A,F22: nat > A,X2: nat] :
( ( case_nat @ A @ F1 @ F22 @ ( suc @ X2 ) )
= ( F22 @ X2 ) ) ).
% old.nat.simps(5)
thf(fact_3816_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_3817_finite__distinct__list,axiom,
! [A: $tType,A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ? [Xs2: list @ A] :
( ( ( set2 @ A @ Xs2 )
= A5 )
& ( distinct @ A @ Xs2 ) ) ) ).
% finite_distinct_list
thf(fact_3818_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_3819_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_3820_distinct__conv__nth,axiom,
! [A: $tType] :
( ( distinct @ A )
= ( ^ [Xs3: list @ A] :
! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ Xs3 ) )
=> ! [J3: nat] :
( ( ord_less @ nat @ J3 @ ( size_size @ ( list @ A ) @ Xs3 ) )
=> ( ( I4 != J3 )
=> ( ( nth @ A @ Xs3 @ I4 )
!= ( nth @ A @ Xs3 @ J3 ) ) ) ) ) ) ) ).
% distinct_conv_nth
thf(fact_3821_nth__eq__iff__index__eq,axiom,
! [A: $tType,Xs: list @ A,I2: nat,J: nat] :
( ( distinct @ A @ Xs )
=> ( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ( nth @ A @ Xs @ I2 )
= ( nth @ A @ Xs @ J ) )
= ( I2 = J ) ) ) ) ) ).
% nth_eq_iff_index_eq
thf(fact_3822_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_3823_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_3824_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_3825_max__Suc2,axiom,
! [M2: nat,N: nat] :
( ( ord_max @ nat @ M2 @ ( suc @ N ) )
= ( case_nat @ nat @ ( suc @ N )
@ ^ [M6: nat] : ( suc @ ( ord_max @ nat @ M6 @ N ) )
@ M2 ) ) ).
% max_Suc2
thf(fact_3826_max__Suc1,axiom,
! [N: nat,M2: nat] :
( ( ord_max @ nat @ ( suc @ N ) @ M2 )
= ( case_nat @ nat @ ( suc @ N )
@ ^ [M6: nat] : ( suc @ ( ord_max @ nat @ N @ M6 ) )
@ M2 ) ) ).
% max_Suc1
thf(fact_3827_distinct__Ex1,axiom,
! [A: $tType,Xs: list @ A,X: A] :
( ( distinct @ A @ Xs )
=> ( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ? [X4: nat] :
( ( ord_less @ nat @ X4 @ ( size_size @ ( list @ A ) @ Xs ) )
& ( ( nth @ A @ Xs @ X4 )
= X )
& ! [Y4: nat] :
( ( ( ord_less @ nat @ Y4 @ ( size_size @ ( list @ A ) @ Xs ) )
& ( ( nth @ A @ Xs @ Y4 )
= X ) )
=> ( Y4 = X4 ) ) ) ) ) ).
% distinct_Ex1
thf(fact_3828_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_3829_Nitpick_Ocase__nat__unfold,axiom,
! [A: $tType] :
( ( case_nat @ A )
= ( ^ [X5: A,F4: nat > A,N2: nat] :
( if @ A
@ ( N2
= ( zero_zero @ nat ) )
@ X5
@ ( F4 @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) ) ) ) ) ).
% Nitpick.case_nat_unfold
thf(fact_3830_distinct__list__update,axiom,
! [A: $tType,Xs: list @ A,A2: A,I2: nat] :
( ( distinct @ A @ Xs )
=> ( ~ ( member @ A @ A2 @ ( minus_minus @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( insert @ A @ ( nth @ A @ Xs @ I2 ) @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( distinct @ A @ ( list_update @ A @ Xs @ I2 @ A2 ) ) ) ) ).
% distinct_list_update
thf(fact_3831_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_3832_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_3833_distinct__union,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( distinct @ A @ ( union @ A @ Xs @ Ys ) )
= ( distinct @ A @ Ys ) ) ).
% distinct_union
thf(fact_3834_nat_Osplit__sels_I1_J,axiom,
! [A: $tType,P2: A > $o,F1: A,F22: nat > A,Nat: nat] :
( ( P2 @ ( case_nat @ A @ F1 @ F22 @ Nat ) )
= ( ( ( Nat
= ( zero_zero @ nat ) )
=> ( P2 @ F1 ) )
& ( ( Nat
= ( suc @ ( pred @ Nat ) ) )
=> ( P2 @ ( F22 @ ( pred @ Nat ) ) ) ) ) ) ).
% nat.split_sels(1)
thf(fact_3835_nat_Osplit__sels_I2_J,axiom,
! [A: $tType,P2: A > $o,F1: A,F22: nat > A,Nat: nat] :
( ( P2 @ ( case_nat @ A @ F1 @ F22 @ Nat ) )
= ( ~ ( ( ( Nat
= ( zero_zero @ nat ) )
& ~ ( P2 @ F1 ) )
| ( ( Nat
= ( suc @ ( pred @ Nat ) ) )
& ~ ( P2 @ ( F22 @ ( pred @ Nat ) ) ) ) ) ) ) ).
% nat.split_sels(2)
thf(fact_3836_pred__def,axiom,
( pred
= ( case_nat @ nat @ ( zero_zero @ nat )
@ ^ [X23: nat] : X23 ) ) ).
% pred_def
thf(fact_3837_length__n__lists__elem,axiom,
! [A: $tType,Ys: list @ A,N: nat,Xs: list @ A] :
( ( member @ ( list @ A ) @ Ys @ ( set2 @ ( list @ A ) @ ( n_lists @ A @ N @ Xs ) ) )
=> ( ( size_size @ ( list @ A ) @ Ys )
= N ) ) ).
% length_n_lists_elem
thf(fact_3838_card__Pow,axiom,
! [A: $tType,A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( finite_card @ ( set @ A ) @ ( pow2 @ A @ A5 ) )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( finite_card @ A @ A5 ) ) ) ) ).
% card_Pow
thf(fact_3839_rec__nat__add__eq__if,axiom,
! [A: $tType,A2: A,F3: nat > A > A,V: num,N: nat] :
( ( rec_nat @ A @ A2 @ F3 @ ( plus_plus @ nat @ ( numeral_numeral @ nat @ V ) @ N ) )
= ( F3 @ ( plus_plus @ nat @ ( pred_numeral @ V ) @ N ) @ ( rec_nat @ A @ A2 @ F3 @ ( plus_plus @ nat @ ( pred_numeral @ V ) @ N ) ) ) ) ).
% rec_nat_add_eq_if
thf(fact_3840_signed__take__bit__code,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( ( bit_ri4674362597316999326ke_bit @ A )
= ( ^ [N2: nat,A6: A] : ( if @ A @ ( bit_se5641148757651400278ts_bit @ A @ ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N2 ) @ A6 ) @ N2 ) @ ( plus_plus @ A @ ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N2 ) @ A6 ) @ ( bit_se4730199178511100633sh_bit @ A @ ( suc @ N2 ) @ ( uminus_uminus @ A @ ( one_one @ A ) ) ) ) @ ( bit_se2584673776208193580ke_bit @ A @ ( suc @ N2 ) @ A6 ) ) ) ) ) ).
% signed_take_bit_code
thf(fact_3841_wmin__insertI,axiom,
! [X: product_prod @ nat @ nat,XS: set @ ( product_prod @ nat @ nat ),Y: product_prod @ nat @ nat,YS: set @ ( product_prod @ nat @ nat )] :
( ( member @ ( product_prod @ nat @ nat ) @ X @ XS )
=> ( ( member @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ X @ Y ) @ fun_pair_leq )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ XS @ YS ) @ fun_min_weak )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ XS @ ( insert @ ( product_prod @ nat @ nat ) @ Y @ YS ) ) @ fun_min_weak ) ) ) ) ).
% wmin_insertI
thf(fact_3842_push__bit__negative__int__iff,axiom,
! [N: nat,K2: int] :
( ( ord_less @ int @ ( bit_se4730199178511100633sh_bit @ int @ N @ K2 ) @ ( zero_zero @ int ) )
= ( ord_less @ int @ K2 @ ( zero_zero @ int ) ) ) ).
% push_bit_negative_int_iff
thf(fact_3843_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_3844_push__bit__eq__0__iff,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ! [N: nat,A2: A] :
( ( ( bit_se4730199178511100633sh_bit @ A @ N @ A2 )
= ( zero_zero @ A ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ).
% push_bit_eq_0_iff
thf(fact_3845_push__bit__push__bit,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M2: nat,N: nat,A2: A] :
( ( bit_se4730199178511100633sh_bit @ A @ M2 @ ( bit_se4730199178511100633sh_bit @ A @ N @ A2 ) )
= ( bit_se4730199178511100633sh_bit @ A @ ( plus_plus @ nat @ M2 @ N ) @ A2 ) ) ) ).
% push_bit_push_bit
thf(fact_3846_Pow__empty,axiom,
! [A: $tType] :
( ( pow2 @ A @ ( bot_bot @ ( set @ A ) ) )
= ( insert @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ ( bot_bot @ ( set @ ( set @ A ) ) ) ) ) ).
% Pow_empty
thf(fact_3847_Pow__singleton__iff,axiom,
! [A: $tType,X7: set @ A,Y8: set @ A] :
( ( ( pow2 @ A @ X7 )
= ( insert @ ( set @ A ) @ Y8 @ ( bot_bot @ ( set @ ( set @ A ) ) ) ) )
= ( ( X7
= ( bot_bot @ ( set @ A ) ) )
& ( Y8
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Pow_singleton_iff
thf(fact_3848_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_3849_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_3850_finite__Pow__iff,axiom,
! [A: $tType,A5: set @ A] :
( ( finite_finite2 @ ( set @ A ) @ ( pow2 @ A @ A5 ) )
= ( finite_finite2 @ A @ A5 ) ) ).
% finite_Pow_iff
thf(fact_3851_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_3852_even__push__bit__iff,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A2: A] :
( ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( bit_se4730199178511100633sh_bit @ A @ N @ A2 ) )
= ( ( N
!= ( zero_zero @ nat ) )
| ( dvd_dvd @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ A2 ) ) ) ) ).
% even_push_bit_iff
thf(fact_3853_push__bit__add,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A2: A,B2: A] :
( ( bit_se4730199178511100633sh_bit @ A @ N @ ( plus_plus @ A @ A2 @ B2 ) )
= ( plus_plus @ A @ ( bit_se4730199178511100633sh_bit @ A @ N @ A2 ) @ ( bit_se4730199178511100633sh_bit @ A @ N @ B2 ) ) ) ) ).
% push_bit_add
thf(fact_3854_Pow__bottom,axiom,
! [A: $tType,B6: set @ A] : ( member @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ ( pow2 @ A @ B6 ) ) ).
% Pow_bottom
thf(fact_3855_Pow__not__empty,axiom,
! [A: $tType,A5: set @ A] :
( ( pow2 @ A @ A5 )
!= ( bot_bot @ ( set @ ( set @ A ) ) ) ) ).
% Pow_not_empty
thf(fact_3856_wmin__emptyI,axiom,
! [X7: set @ ( product_prod @ nat @ nat )] : ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ X7 @ ( bot_bot @ ( set @ ( product_prod @ nat @ nat ) ) ) ) @ fun_min_weak ) ).
% wmin_emptyI
thf(fact_3857_push__bit__take__bit,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M2: nat,N: nat,A2: A] :
( ( bit_se4730199178511100633sh_bit @ A @ M2 @ ( bit_se2584673776208193580ke_bit @ A @ N @ A2 ) )
= ( bit_se2584673776208193580ke_bit @ A @ ( plus_plus @ nat @ M2 @ N ) @ ( bit_se4730199178511100633sh_bit @ A @ M2 @ A2 ) ) ) ) ).
% push_bit_take_bit
thf(fact_3858_bit__push__bit__iff__int,axiom,
! [M2: nat,K2: int,N: nat] :
( ( bit_se5641148757651400278ts_bit @ int @ ( bit_se4730199178511100633sh_bit @ int @ M2 @ K2 ) @ N )
= ( ( ord_less_eq @ nat @ M2 @ N )
& ( bit_se5641148757651400278ts_bit @ int @ K2 @ ( minus_minus @ nat @ N @ M2 ) ) ) ) ).
% bit_push_bit_iff_int
thf(fact_3859_bit__push__bit__iff__nat,axiom,
! [M2: nat,Q2: nat,N: nat] :
( ( bit_se5641148757651400278ts_bit @ nat @ ( bit_se4730199178511100633sh_bit @ nat @ M2 @ Q2 ) @ N )
= ( ( ord_less_eq @ nat @ M2 @ N )
& ( bit_se5641148757651400278ts_bit @ nat @ Q2 @ ( minus_minus @ nat @ N @ M2 ) ) ) ) ).
% bit_push_bit_iff_nat
thf(fact_3860_bit__iff__and__push__bit__not__eq__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se5641148757651400278ts_bit @ A )
= ( ^ [A6: A,N2: nat] :
( ( bit_se5824344872417868541ns_and @ A @ A6 @ ( bit_se4730199178511100633sh_bit @ A @ N2 @ ( one_one @ A ) ) )
!= ( zero_zero @ A ) ) ) ) ) ).
% bit_iff_and_push_bit_not_eq_0
thf(fact_3861_binomial__def,axiom,
( binomial
= ( ^ [N2: nat,K3: nat] :
( finite_card @ ( set @ nat )
@ ( collect @ ( set @ nat )
@ ^ [K6: set @ nat] :
( ( member @ ( set @ nat ) @ K6 @ ( pow2 @ nat @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) ) )
& ( ( finite_card @ nat @ K6 )
= K3 ) ) ) ) ) ) ).
% binomial_def
thf(fact_3862_take__bit__sum,axiom,
! [A: $tType] :
( ( bit_un5681908812861735899ations @ A )
=> ( ( bit_se2584673776208193580ke_bit @ A )
= ( ^ [N2: nat,A6: A] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [K3: nat] : ( bit_se4730199178511100633sh_bit @ A @ K3 @ ( zero_neq_one_of_bool @ A @ ( bit_se5641148757651400278ts_bit @ A @ A6 @ K3 ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) ) ) ) ) ).
% take_bit_sum
thf(fact_3863_old_Orec__nat__def,axiom,
! [T: $tType] :
( ( rec_nat @ T )
= ( ^ [F12: T,F23: nat > T > T,X5: nat] : ( the @ T @ ( rec_set_nat @ T @ F12 @ F23 @ X5 ) ) ) ) ).
% old.rec_nat_def
thf(fact_3864_wmax__insertI,axiom,
! [Y: product_prod @ nat @ nat,YS: set @ ( product_prod @ nat @ nat ),X: product_prod @ nat @ nat,XS: set @ ( product_prod @ nat @ nat )] :
( ( member @ ( product_prod @ nat @ nat ) @ Y @ YS )
=> ( ( member @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ X @ Y ) @ fun_pair_leq )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ XS @ YS ) @ fun_max_weak )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ ( insert @ ( product_prod @ nat @ nat ) @ X @ XS ) @ YS ) @ fun_max_weak ) ) ) ) ).
% wmax_insertI
thf(fact_3865_smax__insertI,axiom,
! [Y: product_prod @ nat @ nat,Y8: set @ ( product_prod @ nat @ nat ),X: product_prod @ nat @ nat,X7: set @ ( product_prod @ nat @ nat )] :
( ( member @ ( product_prod @ nat @ nat ) @ Y @ Y8 )
=> ( ( member @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ X @ Y ) @ fun_pair_less )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ X7 @ Y8 ) @ fun_max_strict )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ ( insert @ ( product_prod @ nat @ nat ) @ X @ X7 ) @ Y8 ) @ fun_max_strict ) ) ) ) ).
% smax_insertI
thf(fact_3866_smin__insertI,axiom,
! [X: product_prod @ nat @ nat,XS: set @ ( product_prod @ nat @ nat ),Y: product_prod @ nat @ nat,YS: set @ ( product_prod @ nat @ nat )] :
( ( member @ ( product_prod @ nat @ nat ) @ X @ XS )
=> ( ( member @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) @ X @ Y ) @ fun_pair_less )
=> ( ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ XS @ YS ) @ fun_min_strict )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ XS @ ( insert @ ( product_prod @ nat @ nat ) @ Y @ YS ) ) @ fun_min_strict ) ) ) ) ).
% smin_insertI
thf(fact_3867_wmax__emptyI,axiom,
! [X7: set @ ( product_prod @ nat @ nat )] :
( ( finite_finite2 @ ( product_prod @ nat @ nat ) @ X7 )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ ( bot_bot @ ( set @ ( product_prod @ nat @ nat ) ) ) @ X7 ) @ fun_max_weak ) ) ).
% wmax_emptyI
thf(fact_3868_smin__emptyI,axiom,
! [X7: set @ ( product_prod @ nat @ nat )] :
( ( X7
!= ( bot_bot @ ( set @ ( product_prod @ nat @ nat ) ) ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ X7 @ ( bot_bot @ ( set @ ( product_prod @ nat @ nat ) ) ) ) @ fun_min_strict ) ) ).
% smin_emptyI
thf(fact_3869_smax__emptyI,axiom,
! [Y8: set @ ( product_prod @ nat @ nat )] :
( ( finite_finite2 @ ( product_prod @ nat @ nat ) @ Y8 )
=> ( ( Y8
!= ( bot_bot @ ( set @ ( product_prod @ nat @ nat ) ) ) )
=> ( member @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ ( bot_bot @ ( set @ ( product_prod @ nat @ nat ) ) ) @ Y8 ) @ fun_max_strict ) ) ) ).
% smax_emptyI
thf(fact_3870_rec__nat__0__imp,axiom,
! [A: $tType,F3: nat > A,F1: A,F22: nat > A > A] :
( ( F3
= ( rec_nat @ A @ F1 @ F22 ) )
=> ( ( F3 @ ( zero_zero @ nat ) )
= F1 ) ) ).
% rec_nat_0_imp
thf(fact_3871_bezw__0,axiom,
! [X: nat] :
( ( bezw @ X @ ( zero_zero @ nat ) )
= ( product_Pair @ int @ int @ ( one_one @ int ) @ ( zero_zero @ int ) ) ) ).
% bezw_0
thf(fact_3872_prod__decode__aux_Osimps,axiom,
( nat_prod_decode_aux
= ( ^ [K3: nat,M3: nat] : ( if @ ( product_prod @ nat @ nat ) @ ( ord_less_eq @ nat @ M3 @ K3 ) @ ( product_Pair @ nat @ nat @ M3 @ ( minus_minus @ nat @ K3 @ M3 ) ) @ ( nat_prod_decode_aux @ ( suc @ K3 ) @ ( minus_minus @ nat @ M3 @ ( suc @ K3 ) ) ) ) ) ) ).
% prod_decode_aux.simps
thf(fact_3873_prod__decode__aux_Oelims,axiom,
! [X: nat,Xa2: nat,Y: product_prod @ nat @ nat] :
( ( ( nat_prod_decode_aux @ X @ Xa2 )
= Y )
=> ( ( ( ord_less_eq @ nat @ Xa2 @ X )
=> ( Y
= ( product_Pair @ nat @ nat @ Xa2 @ ( minus_minus @ nat @ X @ Xa2 ) ) ) )
& ( ~ ( ord_less_eq @ nat @ Xa2 @ X )
=> ( Y
= ( nat_prod_decode_aux @ ( suc @ X ) @ ( minus_minus @ nat @ Xa2 @ ( suc @ X ) ) ) ) ) ) ) ).
% prod_decode_aux.elims
thf(fact_3874_Suc__0__div__numeral,axiom,
! [K2: num] :
( ( divide_divide @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( numeral_numeral @ nat @ K2 ) )
= ( product_fst @ nat @ nat @ ( unique8689654367752047608divmod @ nat @ one2 @ K2 ) ) ) ).
% Suc_0_div_numeral
thf(fact_3875_Suc__0__mod__numeral,axiom,
! [K2: num] :
( ( modulo_modulo @ nat @ ( suc @ ( zero_zero @ nat ) ) @ ( numeral_numeral @ nat @ K2 ) )
= ( product_snd @ nat @ nat @ ( unique8689654367752047608divmod @ nat @ one2 @ K2 ) ) ) ).
% Suc_0_mod_numeral
thf(fact_3876_finite__enumerate,axiom,
! [S2: set @ nat] :
( ( finite_finite2 @ nat @ S2 )
=> ? [R3: nat > nat] :
( ( strict_mono_on @ nat @ nat @ R3 @ ( set_ord_lessThan @ nat @ ( finite_card @ nat @ S2 ) ) )
& ! [N5: nat] :
( ( ord_less @ nat @ N5 @ ( finite_card @ nat @ S2 ) )
=> ( member @ nat @ ( R3 @ N5 ) @ S2 ) ) ) ) ).
% finite_enumerate
thf(fact_3877_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_3878_set__rotate1,axiom,
! [A: $tType,Xs: list @ A] :
( ( set2 @ A @ ( rotate1 @ A @ Xs ) )
= ( set2 @ A @ Xs ) ) ).
% set_rotate1
thf(fact_3879_length__rotate1,axiom,
! [A: $tType,Xs: list @ A] :
( ( size_size @ ( list @ A ) @ ( rotate1 @ A @ Xs ) )
= ( size_size @ ( list @ A ) @ Xs ) ) ).
% length_rotate1
thf(fact_3880_distinct1__rotate,axiom,
! [A: $tType,Xs: list @ A] :
( ( distinct @ A @ ( rotate1 @ A @ Xs ) )
= ( distinct @ A @ Xs ) ) ).
% distinct1_rotate
thf(fact_3881_prod_Ocollapse,axiom,
! [B: $tType,A: $tType,Prod: product_prod @ A @ B] :
( ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) )
= Prod ) ).
% prod.collapse
thf(fact_3882_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_3883_BNF__Greatest__Fixpoint_Osubst__Pair,axiom,
! [B: $tType,A: $tType,P2: A > B > $o,X: A,Y: B,A2: product_prod @ A @ B] :
( ( P2 @ X @ Y )
=> ( ( A2
= ( product_Pair @ A @ B @ X @ Y ) )
=> ( P2 @ ( product_fst @ A @ B @ A2 ) @ ( product_snd @ A @ B @ A2 ) ) ) ) ).
% BNF_Greatest_Fixpoint.subst_Pair
thf(fact_3884_surjective__pairing,axiom,
! [B: $tType,A: $tType,T2: product_prod @ A @ B] :
( T2
= ( product_Pair @ A @ B @ ( product_fst @ A @ B @ T2 ) @ ( product_snd @ A @ B @ T2 ) ) ) ).
% surjective_pairing
thf(fact_3885_prod_Oexhaust__sel,axiom,
! [B: $tType,A: $tType,Prod: product_prod @ A @ B] :
( Prod
= ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) ) ).
% prod.exhaust_sel
thf(fact_3886_fst__eqD,axiom,
! [B: $tType,A: $tType,X: A,Y: B,A2: A] :
( ( ( product_fst @ A @ B @ ( product_Pair @ A @ B @ X @ Y ) )
= A2 )
=> ( X = A2 ) ) ).
% fst_eqD
thf(fact_3887_fst__conv,axiom,
! [B: $tType,A: $tType,X1: A,X2: B] :
( ( product_fst @ A @ B @ ( product_Pair @ A @ B @ X1 @ X2 ) )
= X1 ) ).
% fst_conv
thf(fact_3888_snd__eqD,axiom,
! [B: $tType,A: $tType,X: B,Y: A,A2: A] :
( ( ( product_snd @ B @ A @ ( product_Pair @ B @ A @ X @ Y ) )
= A2 )
=> ( Y = A2 ) ) ).
% snd_eqD
thf(fact_3889_snd__conv,axiom,
! [Aa: $tType,A: $tType,X1: Aa,X2: A] :
( ( product_snd @ Aa @ A @ ( product_Pair @ Aa @ A @ X1 @ X2 ) )
= X2 ) ).
% snd_conv
thf(fact_3890_prod_Osplit__sel__asm,axiom,
! [C: $tType,B: $tType,A: $tType,P2: C > $o,F3: A > B > C,Prod: product_prod @ A @ B] :
( ( P2 @ ( product_case_prod @ A @ B @ C @ F3 @ Prod ) )
= ( ~ ( ( Prod
= ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) )
& ~ ( P2 @ ( F3 @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) ) ) ) ) ).
% prod.split_sel_asm
thf(fact_3891_prod_Osplit__sel,axiom,
! [C: $tType,B: $tType,A: $tType,P2: C > $o,F3: A > B > C,Prod: product_prod @ A @ B] :
( ( P2 @ ( product_case_prod @ A @ B @ C @ F3 @ Prod ) )
= ( ( Prod
= ( product_Pair @ A @ B @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) )
=> ( P2 @ ( F3 @ ( product_fst @ A @ B @ Prod ) @ ( product_snd @ A @ B @ Prod ) ) ) ) ) ).
% prod.split_sel
thf(fact_3892_Product__Type_OCollect__case__prodD,axiom,
! [B: $tType,A: $tType,X: product_prod @ A @ B,A5: A > B > $o] :
( ( member @ ( product_prod @ A @ B ) @ X @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ A5 ) ) )
=> ( A5 @ ( product_fst @ A @ B @ X ) @ ( product_snd @ A @ B @ X ) ) ) ).
% Product_Type.Collect_case_prodD
thf(fact_3893_case__prod__beta,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( product_case_prod @ B @ C @ A )
= ( ^ [F4: B > C > A,P6: product_prod @ B @ C] : ( F4 @ ( product_fst @ B @ C @ P6 ) @ ( product_snd @ B @ C @ P6 ) ) ) ) ).
% case_prod_beta
thf(fact_3894_split__beta,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( product_case_prod @ A @ B @ C )
= ( ^ [F4: A > B > C,Prod2: product_prod @ A @ B] : ( F4 @ ( product_fst @ A @ B @ Prod2 ) @ ( product_snd @ A @ B @ Prod2 ) ) ) ) ).
% split_beta
thf(fact_3895_prod_Oexpand,axiom,
! [B: $tType,A: $tType,Prod: product_prod @ A @ B,Prod3: product_prod @ A @ B] :
( ( ( ( product_fst @ A @ B @ Prod )
= ( product_fst @ A @ B @ Prod3 ) )
& ( ( product_snd @ A @ B @ Prod )
= ( product_snd @ A @ B @ Prod3 ) ) )
=> ( Prod = Prod3 ) ) ).
% prod.expand
thf(fact_3896_prod__eqI,axiom,
! [B: $tType,A: $tType,P: product_prod @ A @ B,Q2: product_prod @ A @ B] :
( ( ( product_fst @ A @ B @ P )
= ( product_fst @ A @ B @ Q2 ) )
=> ( ( ( product_snd @ A @ B @ P )
= ( product_snd @ A @ B @ Q2 ) )
=> ( P = Q2 ) ) ) ).
% prod_eqI
thf(fact_3897_prod__eq__iff,axiom,
! [B: $tType,A: $tType] :
( ( ^ [Y5: product_prod @ A @ B,Z2: product_prod @ A @ B] : Y5 = Z2 )
= ( ^ [S8: product_prod @ A @ B,T3: product_prod @ A @ B] :
( ( ( product_fst @ A @ B @ S8 )
= ( product_fst @ A @ B @ T3 ) )
& ( ( product_snd @ A @ B @ S8 )
= ( product_snd @ A @ B @ T3 ) ) ) ) ) ).
% prod_eq_iff
thf(fact_3898_case__prod__unfold,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( product_case_prod @ A @ B @ C )
= ( ^ [C4: A > B > C,P6: product_prod @ A @ B] : ( C4 @ ( product_fst @ A @ B @ P6 ) @ ( product_snd @ A @ B @ P6 ) ) ) ) ).
% case_prod_unfold
thf(fact_3899_case__prod__beta_H,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( product_case_prod @ A @ B @ C )
= ( ^ [F4: A > B > C,X5: product_prod @ A @ B] : ( F4 @ ( product_fst @ A @ B @ X5 ) @ ( product_snd @ A @ B @ X5 ) ) ) ) ).
% case_prod_beta'
thf(fact_3900_split__comp__eq,axiom,
! [A: $tType,C: $tType,B: $tType,D: $tType,F3: A > B > C,G3: D > A] :
( ( ^ [U2: product_prod @ D @ B] : ( F3 @ ( G3 @ ( product_fst @ D @ B @ U2 ) ) @ ( product_snd @ D @ B @ U2 ) ) )
= ( product_case_prod @ D @ B @ C
@ ^ [X5: D] : ( F3 @ ( G3 @ X5 ) ) ) ) ).
% split_comp_eq
thf(fact_3901_The__case__prod,axiom,
! [B: $tType,A: $tType,P2: A > B > $o] :
( ( the @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ P2 ) )
= ( the @ ( product_prod @ A @ B )
@ ^ [Xy: product_prod @ A @ B] : ( P2 @ ( product_fst @ A @ B @ Xy ) @ ( product_snd @ A @ B @ Xy ) ) ) ) ).
% The_case_prod
thf(fact_3902_snd__def,axiom,
! [B: $tType,A: $tType] :
( ( product_snd @ A @ B )
= ( product_case_prod @ A @ B @ B
@ ^ [X15: A,X23: B] : X23 ) ) ).
% snd_def
thf(fact_3903_fst__def,axiom,
! [B: $tType,A: $tType] :
( ( product_fst @ A @ B )
= ( product_case_prod @ A @ B @ A
@ ^ [X15: A,X23: B] : X15 ) ) ).
% fst_def
thf(fact_3904_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_3905_size__prod__simp,axiom,
! [B: $tType,A: $tType] :
( ( basic_BNF_size_prod @ A @ B )
= ( ^ [F4: A > nat,G4: B > nat,P6: product_prod @ A @ B] : ( plus_plus @ nat @ ( plus_plus @ nat @ ( F4 @ ( product_fst @ A @ B @ P6 ) ) @ ( G4 @ ( product_snd @ A @ B @ P6 ) ) ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ).
% size_prod_simp
thf(fact_3906_in__set__enumerate__eq,axiom,
! [A: $tType,P: product_prod @ nat @ A,N: nat,Xs: list @ A] :
( ( member @ ( product_prod @ nat @ A ) @ P @ ( set2 @ ( product_prod @ nat @ A ) @ ( enumerate @ A @ N @ Xs ) ) )
= ( ( ord_less_eq @ nat @ N @ ( product_fst @ nat @ A @ P ) )
& ( ord_less @ nat @ ( product_fst @ nat @ A @ P ) @ ( plus_plus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N ) )
& ( ( nth @ A @ Xs @ ( minus_minus @ nat @ ( product_fst @ nat @ A @ P ) @ N ) )
= ( product_snd @ nat @ A @ P ) ) ) ) ).
% in_set_enumerate_eq
thf(fact_3907_exI__realizer,axiom,
! [B: $tType,A: $tType,P2: A > B > $o,Y: A,X: B] :
( ( P2 @ Y @ X )
=> ( P2 @ ( product_snd @ B @ A @ ( product_Pair @ B @ A @ X @ Y ) ) @ ( product_fst @ B @ A @ ( product_Pair @ B @ A @ X @ Y ) ) ) ) ).
% exI_realizer
thf(fact_3908_conjI__realizer,axiom,
! [A: $tType,B: $tType,P2: A > $o,P: A,Q: B > $o,Q2: B] :
( ( P2 @ P )
=> ( ( Q @ Q2 )
=> ( ( P2 @ ( product_fst @ A @ B @ ( product_Pair @ A @ B @ P @ Q2 ) ) )
& ( Q @ ( product_snd @ A @ B @ ( product_Pair @ A @ B @ P @ Q2 ) ) ) ) ) ) ).
% conjI_realizer
thf(fact_3909_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_3910_distinct__enumerate,axiom,
! [A: $tType,N: nat,Xs: list @ A] : ( distinct @ ( product_prod @ nat @ A ) @ ( enumerate @ A @ N @ Xs ) ) ).
% distinct_enumerate
thf(fact_3911_bezw__non__0,axiom,
! [Y: nat,X: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ Y )
=> ( ( bezw @ X @ Y )
= ( product_Pair @ int @ int @ ( product_snd @ int @ int @ ( bezw @ Y @ ( modulo_modulo @ nat @ X @ Y ) ) ) @ ( minus_minus @ int @ ( product_fst @ int @ int @ ( bezw @ Y @ ( modulo_modulo @ nat @ X @ Y ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ ( bezw @ Y @ ( modulo_modulo @ nat @ X @ Y ) ) ) @ ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ X @ Y ) ) ) ) ) ) ) ).
% bezw_non_0
thf(fact_3912_bezw_Oelims,axiom,
! [X: nat,Xa2: nat,Y: product_prod @ int @ int] :
( ( ( bezw @ X @ Xa2 )
= Y )
=> ( ( ( Xa2
= ( zero_zero @ nat ) )
=> ( Y
= ( product_Pair @ int @ int @ ( one_one @ int ) @ ( zero_zero @ int ) ) ) )
& ( ( Xa2
!= ( zero_zero @ nat ) )
=> ( Y
= ( product_Pair @ int @ int @ ( product_snd @ int @ int @ ( bezw @ Xa2 @ ( modulo_modulo @ nat @ X @ Xa2 ) ) ) @ ( minus_minus @ int @ ( product_fst @ int @ int @ ( bezw @ Xa2 @ ( modulo_modulo @ nat @ X @ Xa2 ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ ( bezw @ Xa2 @ ( modulo_modulo @ nat @ X @ Xa2 ) ) ) @ ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ X @ Xa2 ) ) ) ) ) ) ) ) ) ).
% bezw.elims
thf(fact_3913_bezw_Osimps,axiom,
( bezw
= ( ^ [X5: nat,Y6: nat] :
( if @ ( product_prod @ int @ int )
@ ( Y6
= ( zero_zero @ nat ) )
@ ( product_Pair @ int @ int @ ( one_one @ int ) @ ( zero_zero @ int ) )
@ ( product_Pair @ int @ int @ ( product_snd @ int @ int @ ( bezw @ Y6 @ ( modulo_modulo @ nat @ X5 @ Y6 ) ) ) @ ( minus_minus @ int @ ( product_fst @ int @ int @ ( bezw @ Y6 @ ( modulo_modulo @ nat @ X5 @ Y6 ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ ( bezw @ Y6 @ ( modulo_modulo @ nat @ X5 @ Y6 ) ) ) @ ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ X5 @ Y6 ) ) ) ) ) ) ) ) ).
% bezw.simps
thf(fact_3914_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_3915_bezw_Opelims,axiom,
! [X: nat,Xa2: nat,Y: product_prod @ int @ int] :
( ( ( bezw @ X @ Xa2 )
= Y )
=> ( ( accp @ ( product_prod @ nat @ nat ) @ bezw_rel @ ( product_Pair @ nat @ nat @ X @ Xa2 ) )
=> ~ ( ( ( ( Xa2
= ( zero_zero @ nat ) )
=> ( Y
= ( product_Pair @ int @ int @ ( one_one @ int ) @ ( zero_zero @ int ) ) ) )
& ( ( Xa2
!= ( zero_zero @ nat ) )
=> ( Y
= ( product_Pair @ int @ int @ ( product_snd @ int @ int @ ( bezw @ Xa2 @ ( modulo_modulo @ nat @ X @ Xa2 ) ) ) @ ( minus_minus @ int @ ( product_fst @ int @ int @ ( bezw @ Xa2 @ ( modulo_modulo @ nat @ X @ Xa2 ) ) ) @ ( times_times @ int @ ( product_snd @ int @ int @ ( bezw @ Xa2 @ ( modulo_modulo @ nat @ X @ Xa2 ) ) ) @ ( semiring_1_of_nat @ int @ ( divide_divide @ nat @ X @ Xa2 ) ) ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ nat @ nat ) @ bezw_rel @ ( product_Pair @ nat @ nat @ X @ Xa2 ) ) ) ) ) ).
% bezw.pelims
thf(fact_3916_strict__mono__onD,axiom,
! [B: $tType,A: $tType] :
( ( ( ord @ A )
& ( ord @ B ) )
=> ! [F3: A > B,A5: set @ A,R2: A,S: A] :
( ( strict_mono_on @ A @ B @ F3 @ A5 )
=> ( ( member @ A @ R2 @ A5 )
=> ( ( member @ A @ S @ A5 )
=> ( ( ord_less @ A @ R2 @ S )
=> ( ord_less @ B @ ( F3 @ R2 ) @ ( F3 @ S ) ) ) ) ) ) ) ).
% strict_mono_onD
thf(fact_3917_strict__mono__onI,axiom,
! [B: $tType,A: $tType] :
( ( ( ord @ A )
& ( ord @ B ) )
=> ! [A5: set @ A,F3: A > B] :
( ! [R3: A,S3: A] :
( ( member @ A @ R3 @ A5 )
=> ( ( member @ A @ S3 @ A5 )
=> ( ( ord_less @ A @ R3 @ S3 )
=> ( ord_less @ B @ ( F3 @ R3 ) @ ( F3 @ S3 ) ) ) ) )
=> ( strict_mono_on @ A @ B @ F3 @ A5 ) ) ) ).
% strict_mono_onI
thf(fact_3918_strict__mono__on__def,axiom,
! [B: $tType,A: $tType] :
( ( ( ord @ A )
& ( ord @ B ) )
=> ( ( strict_mono_on @ A @ B )
= ( ^ [F4: A > B,A7: set @ A] :
! [R5: A,S8: A] :
( ( ( member @ A @ R5 @ A7 )
& ( member @ A @ S8 @ A7 )
& ( ord_less @ A @ R5 @ S8 ) )
=> ( ord_less @ B @ ( F4 @ R5 ) @ ( F4 @ S8 ) ) ) ) ) ) ).
% strict_mono_on_def
thf(fact_3919_strict__mono__on__leD,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( preorder @ B ) )
=> ! [F3: A > B,A5: set @ A,X: A,Y: A] :
( ( strict_mono_on @ A @ B @ F3 @ A5 )
=> ( ( member @ A @ X @ A5 )
=> ( ( member @ A @ Y @ A5 )
=> ( ( ord_less_eq @ A @ X @ Y )
=> ( ord_less_eq @ B @ ( F3 @ X ) @ ( F3 @ Y ) ) ) ) ) ) ) ).
% strict_mono_on_leD
thf(fact_3920_prod__decode__aux_Opelims,axiom,
! [X: nat,Xa2: nat,Y: product_prod @ nat @ nat] :
( ( ( nat_prod_decode_aux @ X @ Xa2 )
= Y )
=> ( ( accp @ ( product_prod @ nat @ nat ) @ nat_pr5047031295181774490ux_rel @ ( product_Pair @ nat @ nat @ X @ Xa2 ) )
=> ~ ( ( ( ( ord_less_eq @ nat @ Xa2 @ X )
=> ( Y
= ( product_Pair @ nat @ nat @ Xa2 @ ( minus_minus @ nat @ X @ Xa2 ) ) ) )
& ( ~ ( ord_less_eq @ nat @ Xa2 @ X )
=> ( Y
= ( nat_prod_decode_aux @ ( suc @ X ) @ ( minus_minus @ nat @ Xa2 @ ( suc @ X ) ) ) ) ) )
=> ~ ( accp @ ( product_prod @ nat @ nat ) @ nat_pr5047031295181774490ux_rel @ ( product_Pair @ nat @ nat @ X @ Xa2 ) ) ) ) ) ).
% prod_decode_aux.pelims
thf(fact_3921_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_3922_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_3923_drop__bit__rec,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se4197421643247451524op_bit @ A )
= ( ^ [N2: nat,A6: A] :
( if @ A
@ ( N2
= ( zero_zero @ nat ) )
@ A6
@ ( bit_se4197421643247451524op_bit @ A @ ( minus_minus @ nat @ N2 @ ( one_one @ nat ) ) @ ( divide_divide @ A @ A6 @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) ) ) ) ) ) ) ).
% drop_bit_rec
thf(fact_3924_gcd__eq__0__iff,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [A2: A,B2: A] :
( ( ( gcd_gcd @ A @ A2 @ B2 )
= ( zero_zero @ A ) )
= ( ( A2
= ( zero_zero @ A ) )
& ( B2
= ( zero_zero @ A ) ) ) ) ) ).
% gcd_eq_0_iff
thf(fact_3925_gcd__add1,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [M2: A,N: A] :
( ( gcd_gcd @ A @ ( plus_plus @ A @ M2 @ N ) @ N )
= ( gcd_gcd @ A @ M2 @ N ) ) ) ).
% gcd_add1
thf(fact_3926_gcd__add2,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [M2: A,N: A] :
( ( gcd_gcd @ A @ M2 @ ( plus_plus @ A @ M2 @ N ) )
= ( gcd_gcd @ A @ M2 @ N ) ) ) ).
% gcd_add2
thf(fact_3927_drop__bit__negative__int__iff,axiom,
! [N: nat,K2: int] :
( ( ord_less @ int @ ( bit_se4197421643247451524op_bit @ int @ N @ K2 ) @ ( zero_zero @ int ) )
= ( ord_less @ int @ K2 @ ( zero_zero @ int ) ) ) ).
% drop_bit_negative_int_iff
thf(fact_3928_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_3929_drop__bit__drop__bit,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M2: nat,N: nat,A2: A] :
( ( bit_se4197421643247451524op_bit @ A @ M2 @ ( bit_se4197421643247451524op_bit @ A @ N @ A2 ) )
= ( bit_se4197421643247451524op_bit @ A @ ( plus_plus @ nat @ M2 @ N ) @ A2 ) ) ) ).
% drop_bit_drop_bit
thf(fact_3930_in__set__remove1,axiom,
! [A: $tType,A2: A,B2: A,Xs: list @ A] :
( ( A2 != B2 )
=> ( ( member @ A @ A2 @ ( set2 @ A @ ( remove1 @ A @ B2 @ Xs ) ) )
= ( member @ A @ A2 @ ( set2 @ A @ Xs ) ) ) ) ).
% in_set_remove1
thf(fact_3931_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_3932_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_3933_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_3934_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_3935_remove1__commute,axiom,
! [A: $tType,X: A,Y: A,Zs: list @ A] :
( ( remove1 @ A @ X @ ( remove1 @ A @ Y @ Zs ) )
= ( remove1 @ A @ Y @ ( remove1 @ A @ X @ Zs ) ) ) ).
% remove1_commute
thf(fact_3936_remove1__idem,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ~ ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ( remove1 @ A @ X @ Xs )
= Xs ) ) ).
% remove1_idem
thf(fact_3937_notin__set__remove1,axiom,
! [A: $tType,X: A,Xs: list @ A,Y: A] :
( ~ ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ~ ( member @ A @ X @ ( set2 @ A @ ( remove1 @ A @ Y @ Xs ) ) ) ) ).
% notin_set_remove1
thf(fact_3938_distinct__remove1,axiom,
! [A: $tType,Xs: list @ A,X: A] :
( ( distinct @ A @ Xs )
=> ( distinct @ A @ ( remove1 @ A @ X @ Xs ) ) ) ).
% distinct_remove1
thf(fact_3939_gcd__add__mult,axiom,
! [A: $tType] :
( ( semiring_gcd @ A )
=> ! [M2: A,K2: A,N: A] :
( ( gcd_gcd @ A @ M2 @ ( plus_plus @ A @ ( times_times @ A @ K2 @ M2 ) @ N ) )
= ( gcd_gcd @ A @ M2 @ N ) ) ) ).
% gcd_add_mult
thf(fact_3940_take__bit__eq__self__iff__drop__bit__eq__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A2: A] :
( ( ( bit_se2584673776208193580ke_bit @ A @ N @ A2 )
= A2 )
= ( ( bit_se4197421643247451524op_bit @ A @ N @ A2 )
= ( zero_zero @ A ) ) ) ) ).
% take_bit_eq_self_iff_drop_bit_eq_0
thf(fact_3941_take__bit__drop__bit,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [M2: nat,N: nat,A2: A] :
( ( bit_se2584673776208193580ke_bit @ A @ M2 @ ( bit_se4197421643247451524op_bit @ A @ N @ A2 ) )
= ( bit_se4197421643247451524op_bit @ A @ N @ ( bit_se2584673776208193580ke_bit @ A @ ( plus_plus @ nat @ M2 @ N ) @ A2 ) ) ) ) ).
% take_bit_drop_bit
thf(fact_3942_set__remove1__subset,axiom,
! [A: $tType,X: A,Xs: list @ A] : ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ ( remove1 @ A @ X @ Xs ) ) @ ( set2 @ A @ Xs ) ) ).
% set_remove1_subset
thf(fact_3943_gcd__le1__int,axiom,
! [A2: int,B2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ A2 )
=> ( ord_less_eq @ int @ ( gcd_gcd @ int @ A2 @ B2 ) @ A2 ) ) ).
% gcd_le1_int
thf(fact_3944_gcd__le2__int,axiom,
! [B2: int,A2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
=> ( ord_less_eq @ int @ ( gcd_gcd @ int @ A2 @ B2 ) @ B2 ) ) ).
% gcd_le2_int
thf(fact_3945_gcd__non__0__int,axiom,
! [Y: int,X: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ Y )
=> ( ( gcd_gcd @ int @ X @ Y )
= ( gcd_gcd @ int @ Y @ ( modulo_modulo @ int @ X @ Y ) ) ) ) ).
% gcd_non_0_int
thf(fact_3946_bits__ident,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A2: A] :
( ( plus_plus @ A @ ( bit_se4730199178511100633sh_bit @ A @ N @ ( bit_se4197421643247451524op_bit @ A @ N @ A2 ) ) @ ( bit_se2584673776208193580ke_bit @ A @ N @ A2 ) )
= A2 ) ) ).
% bits_ident
thf(fact_3947_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_3948_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_3949_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_3950_div__add__self2__no__field,axiom,
! [B: $tType,A: $tType] :
( ( ( euclid4440199948858584721cancel @ A )
& ( field @ B ) )
=> ! [X: B,B2: A,A2: A] :
( ( nO_MATCH @ B @ A @ X @ B2 )
=> ( ( B2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ A2 @ B2 ) @ B2 )
= ( plus_plus @ A @ ( divide_divide @ A @ A2 @ B2 ) @ ( one_one @ A ) ) ) ) ) ) ).
% div_add_self2_no_field
thf(fact_3951_div__add__self1__no__field,axiom,
! [B: $tType,A: $tType] :
( ( ( euclid4440199948858584721cancel @ A )
& ( field @ B ) )
=> ! [X: B,B2: A,A2: A] :
( ( nO_MATCH @ B @ A @ X @ B2 )
=> ( ( B2
!= ( zero_zero @ A ) )
=> ( ( divide_divide @ A @ ( plus_plus @ A @ B2 @ A2 ) @ B2 )
= ( plus_plus @ A @ ( divide_divide @ A @ A2 @ B2 ) @ ( one_one @ A ) ) ) ) ) ) ).
% div_add_self1_no_field
thf(fact_3952_gcd__0__left__nat,axiom,
! [X: nat] :
( ( gcd_gcd @ nat @ ( zero_zero @ nat ) @ X )
= X ) ).
% gcd_0_left_nat
thf(fact_3953_gcd__0__nat,axiom,
! [X: nat] :
( ( gcd_gcd @ nat @ X @ ( zero_zero @ nat ) )
= X ) ).
% gcd_0_nat
thf(fact_3954_gcd__nat_Oright__neutral,axiom,
! [A2: nat] :
( ( gcd_gcd @ nat @ A2 @ ( zero_zero @ nat ) )
= A2 ) ).
% gcd_nat.right_neutral
thf(fact_3955_gcd__nat_Oneutr__eq__iff,axiom,
! [A2: nat,B2: nat] :
( ( ( zero_zero @ nat )
= ( gcd_gcd @ nat @ A2 @ B2 ) )
= ( ( A2
= ( zero_zero @ nat ) )
& ( B2
= ( zero_zero @ nat ) ) ) ) ).
% gcd_nat.neutr_eq_iff
thf(fact_3956_gcd__nat_Oleft__neutral,axiom,
! [A2: nat] :
( ( gcd_gcd @ nat @ ( zero_zero @ nat ) @ A2 )
= A2 ) ).
% gcd_nat.left_neutral
thf(fact_3957_gcd__nat_Oeq__neutr__iff,axiom,
! [A2: nat,B2: nat] :
( ( ( gcd_gcd @ nat @ A2 @ B2 )
= ( zero_zero @ nat ) )
= ( ( A2
= ( zero_zero @ nat ) )
& ( B2
= ( zero_zero @ nat ) ) ) ) ).
% gcd_nat.eq_neutr_iff
thf(fact_3958_gcd__Suc__0,axiom,
! [M2: nat] :
( ( gcd_gcd @ nat @ M2 @ ( suc @ ( zero_zero @ nat ) ) )
= ( suc @ ( zero_zero @ nat ) ) ) ).
% gcd_Suc_0
thf(fact_3959_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_3960_greaterThanLessThan__iff,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [I2: A,L: A,U: A] :
( ( member @ A @ I2 @ ( set_or5935395276787703475ssThan @ A @ L @ U ) )
= ( ( ord_less @ A @ L @ I2 )
& ( ord_less @ A @ I2 @ U ) ) ) ) ).
% greaterThanLessThan_iff
thf(fact_3961_finite__greaterThanLessThan__int,axiom,
! [L: int,U: int] : ( finite_finite2 @ int @ ( set_or5935395276787703475ssThan @ int @ L @ U ) ) ).
% finite_greaterThanLessThan_int
thf(fact_3962_greaterThanLessThan__empty,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,K2: A] :
( ( ord_less_eq @ A @ L @ K2 )
=> ( ( set_or5935395276787703475ssThan @ A @ K2 @ L )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% greaterThanLessThan_empty
thf(fact_3963_greaterThanLessThan__empty__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A2: A,B2: A] :
( ( ( set_or5935395276787703475ssThan @ A @ A2 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
= ( ord_less_eq @ A @ B2 @ A2 ) ) ) ).
% greaterThanLessThan_empty_iff
thf(fact_3964_greaterThanLessThan__empty__iff2,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A2: A,B2: A] :
( ( ( bot_bot @ ( set @ A ) )
= ( set_or5935395276787703475ssThan @ A @ A2 @ B2 ) )
= ( ord_less_eq @ A @ B2 @ A2 ) ) ) ).
% greaterThanLessThan_empty_iff2
thf(fact_3965_infinite__Ioo__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A2: A,B2: A] :
( ( ~ ( finite_finite2 @ A @ ( set_or5935395276787703475ssThan @ A @ A2 @ B2 ) ) )
= ( ord_less @ A @ A2 @ B2 ) ) ) ).
% infinite_Ioo_iff
thf(fact_3966_real__root__Suc__0,axiom,
! [X: real] :
( ( root @ ( suc @ ( zero_zero @ nat ) ) @ X )
= X ) ).
% real_root_Suc_0
thf(fact_3967_root__0,axiom,
! [X: real] :
( ( root @ ( zero_zero @ nat ) @ X )
= ( zero_zero @ real ) ) ).
% root_0
thf(fact_3968_real__root__eq__iff,axiom,
! [N: nat,X: real,Y: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ( root @ N @ X )
= ( root @ N @ Y ) )
= ( X = Y ) ) ) ).
% real_root_eq_iff
thf(fact_3969_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_3970_real__root__less__iff,axiom,
! [N: nat,X: real,Y: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( root @ N @ X ) @ ( root @ N @ Y ) )
= ( ord_less @ real @ X @ Y ) ) ) ).
% real_root_less_iff
thf(fact_3971_real__root__le__iff,axiom,
! [N: nat,X: real,Y: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ real @ ( root @ N @ X ) @ ( root @ N @ Y ) )
= ( ord_less_eq @ real @ X @ Y ) ) ) ).
% real_root_le_iff
thf(fact_3972_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_3973_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_3974_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_3975_real__root__gt__0__iff,axiom,
! [N: nat,Y: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( root @ N @ Y ) )
= ( ord_less @ real @ ( zero_zero @ real ) @ Y ) ) ) ).
% real_root_gt_0_iff
thf(fact_3976_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_3977_real__root__ge__0__iff,axiom,
! [N: nat,Y: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( root @ N @ Y ) )
= ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y ) ) ) ).
% real_root_ge_0_iff
thf(fact_3978_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_3979_real__root__gt__1__iff,axiom,
! [N: nat,Y: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( one_one @ real ) @ ( root @ N @ Y ) )
= ( ord_less @ real @ ( one_one @ real ) @ Y ) ) ) ).
% real_root_gt_1_iff
thf(fact_3980_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_3981_real__root__ge__1__iff,axiom,
! [N: nat,Y: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ real @ ( one_one @ real ) @ ( root @ N @ Y ) )
= ( ord_less_eq @ real @ ( one_one @ real ) @ Y ) ) ) ).
% real_root_ge_1_iff
thf(fact_3982_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_3983_gcd__le2__nat,axiom,
! [B2: nat,A2: nat] :
( ( B2
!= ( zero_zero @ nat ) )
=> ( ord_less_eq @ nat @ ( gcd_gcd @ nat @ A2 @ B2 ) @ B2 ) ) ).
% gcd_le2_nat
thf(fact_3984_gcd__le1__nat,axiom,
! [A2: nat,B2: nat] :
( ( A2
!= ( zero_zero @ nat ) )
=> ( ord_less_eq @ nat @ ( gcd_gcd @ nat @ A2 @ B2 ) @ A2 ) ) ).
% gcd_le1_nat
thf(fact_3985_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_3986_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_3987_gcd__non__0__nat,axiom,
! [Y: nat,X: nat] :
( ( Y
!= ( zero_zero @ nat ) )
=> ( ( gcd_gcd @ nat @ X @ Y )
= ( gcd_gcd @ nat @ Y @ ( modulo_modulo @ nat @ X @ Y ) ) ) ) ).
% gcd_non_0_nat
thf(fact_3988_gcd__nat_Osimps,axiom,
( ( gcd_gcd @ nat )
= ( ^ [X5: nat,Y6: nat] :
( if @ nat
@ ( Y6
= ( zero_zero @ nat ) )
@ X5
@ ( gcd_gcd @ nat @ Y6 @ ( modulo_modulo @ nat @ X5 @ Y6 ) ) ) ) ) ).
% gcd_nat.simps
thf(fact_3989_gcd__nat_Oelims,axiom,
! [X: nat,Xa2: nat,Y: nat] :
( ( ( gcd_gcd @ nat @ X @ Xa2 )
= Y )
=> ( ( ( Xa2
= ( zero_zero @ nat ) )
=> ( Y = X ) )
& ( ( Xa2
!= ( zero_zero @ nat ) )
=> ( Y
= ( gcd_gcd @ nat @ Xa2 @ ( modulo_modulo @ nat @ X @ Xa2 ) ) ) ) ) ) ).
% gcd_nat.elims
thf(fact_3990_infinite__Ioo,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ~ ( finite_finite2 @ A @ ( set_or5935395276787703475ssThan @ A @ A2 @ B2 ) ) ) ) ).
% infinite_Ioo
thf(fact_3991_real__root__less__mono,axiom,
! [N: nat,X: real,Y: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ X @ Y )
=> ( ord_less @ real @ ( root @ N @ X ) @ ( root @ N @ Y ) ) ) ) ).
% real_root_less_mono
thf(fact_3992_real__root__le__mono,axiom,
! [N: nat,X: real,Y: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ real @ X @ Y )
=> ( ord_less_eq @ real @ ( root @ N @ X ) @ ( root @ N @ Y ) ) ) ) ).
% real_root_le_mono
thf(fact_3993_real__root__power,axiom,
! [N: nat,X: real,K2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( root @ N @ ( power_power @ real @ X @ K2 ) )
= ( power_power @ real @ ( root @ N @ X ) @ K2 ) ) ) ).
% real_root_power
thf(fact_3994_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_3995_bezout__nat,axiom,
! [A2: nat,B2: nat] :
( ( A2
!= ( zero_zero @ nat ) )
=> ? [X4: nat,Y3: nat] :
( ( times_times @ nat @ A2 @ X4 )
= ( plus_plus @ nat @ ( times_times @ nat @ B2 @ Y3 ) @ ( gcd_gcd @ nat @ A2 @ B2 ) ) ) ) ).
% bezout_nat
thf(fact_3996_bezout__gcd__nat_H,axiom,
! [B2: nat,A2: nat] :
? [X4: nat,Y3: nat] :
( ( ( ord_less_eq @ nat @ ( times_times @ nat @ B2 @ Y3 ) @ ( times_times @ nat @ A2 @ X4 ) )
& ( ( minus_minus @ nat @ ( times_times @ nat @ A2 @ X4 ) @ ( times_times @ nat @ B2 @ Y3 ) )
= ( gcd_gcd @ nat @ A2 @ B2 ) ) )
| ( ( ord_less_eq @ nat @ ( times_times @ nat @ A2 @ Y3 ) @ ( times_times @ nat @ B2 @ X4 ) )
& ( ( minus_minus @ nat @ ( times_times @ nat @ B2 @ X4 ) @ ( times_times @ nat @ A2 @ Y3 ) )
= ( gcd_gcd @ nat @ A2 @ B2 ) ) ) ) ).
% bezout_gcd_nat'
thf(fact_3997_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_3998_greaterThanLessThan__subseteq__greaterThanLessThan,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A2: A,B2: A,C3: A,D3: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ A2 @ B2 ) @ ( set_or5935395276787703475ssThan @ A @ C3 @ D3 ) )
= ( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C3 @ A2 )
& ( ord_less_eq @ A @ B2 @ D3 ) ) ) ) ) ).
% greaterThanLessThan_subseteq_greaterThanLessThan
thf(fact_3999_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_4000_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_4001_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_4002_root__abs__power,axiom,
! [N: nat,Y: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( abs_abs @ real @ ( root @ N @ ( power_power @ real @ Y @ N ) ) )
= ( abs_abs @ real @ Y ) ) ) ).
% root_abs_power
thf(fact_4003_scale__right__distrib__NO__MATCH,axiom,
! [A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [X: A,Y: A,A2: real] :
( ( nO_MATCH @ A @ real @ ( divide_divide @ A @ X @ Y ) @ A2 )
=> ( ( real_V8093663219630862766scaleR @ A @ A2 @ ( plus_plus @ A @ X @ Y ) )
= ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ X ) @ ( real_V8093663219630862766scaleR @ A @ A2 @ Y ) ) ) ) ) ).
% scale_right_distrib_NO_MATCH
thf(fact_4004_greaterThanLessThan__subseteq__atLeastAtMost__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A2: A,B2: A,C3: A,D3: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ A2 @ B2 ) @ ( set_or1337092689740270186AtMost @ A @ C3 @ D3 ) )
= ( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C3 @ A2 )
& ( ord_less_eq @ A @ B2 @ D3 ) ) ) ) ) ).
% greaterThanLessThan_subseteq_atLeastAtMost_iff
thf(fact_4005_greaterThanLessThan__subseteq__atLeastLessThan__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A2: A,B2: A,C3: A,D3: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ A2 @ B2 ) @ ( set_or7035219750837199246ssThan @ A @ C3 @ D3 ) )
= ( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C3 @ A2 )
& ( ord_less_eq @ A @ B2 @ D3 ) ) ) ) ) ).
% greaterThanLessThan_subseteq_atLeastLessThan_iff
thf(fact_4006_distrib__left__NO__MATCH,axiom,
! [B: $tType,A: $tType] :
( ( semiring @ A )
=> ! [X: B,Y: B,A2: A,B2: A,C3: A] :
( ( nO_MATCH @ B @ A @ ( divide_divide @ B @ X @ Y ) @ A2 )
=> ( ( times_times @ A @ A2 @ ( plus_plus @ A @ B2 @ C3 ) )
= ( plus_plus @ A @ ( times_times @ A @ A2 @ B2 ) @ ( times_times @ A @ A2 @ C3 ) ) ) ) ) ).
% distrib_left_NO_MATCH
thf(fact_4007_distrib__right__NO__MATCH,axiom,
! [B: $tType,A: $tType] :
( ( semiring @ A )
=> ! [X: B,Y: B,C3: A,A2: A,B2: A] :
( ( nO_MATCH @ B @ A @ ( divide_divide @ B @ X @ Y ) @ C3 )
=> ( ( times_times @ A @ ( plus_plus @ A @ A2 @ B2 ) @ C3 )
= ( plus_plus @ A @ ( times_times @ A @ A2 @ C3 ) @ ( times_times @ A @ B2 @ C3 ) ) ) ) ) ).
% distrib_right_NO_MATCH
thf(fact_4008_left__diff__distrib__NO__MATCH,axiom,
! [B: $tType,A: $tType] :
( ( ring @ A )
=> ! [X: B,Y: B,C3: A,A2: A,B2: A] :
( ( nO_MATCH @ B @ A @ ( divide_divide @ B @ X @ Y ) @ C3 )
=> ( ( times_times @ A @ ( minus_minus @ A @ A2 @ B2 ) @ C3 )
= ( minus_minus @ A @ ( times_times @ A @ A2 @ C3 ) @ ( times_times @ A @ B2 @ C3 ) ) ) ) ) ).
% left_diff_distrib_NO_MATCH
thf(fact_4009_right__diff__distrib__NO__MATCH,axiom,
! [B: $tType,A: $tType] :
( ( ring @ A )
=> ! [X: B,Y: B,A2: A,B2: A,C3: A] :
( ( nO_MATCH @ B @ A @ ( divide_divide @ B @ X @ Y ) @ A2 )
=> ( ( times_times @ A @ A2 @ ( minus_minus @ A @ B2 @ C3 ) )
= ( minus_minus @ A @ ( times_times @ A @ A2 @ B2 ) @ ( times_times @ A @ A2 @ C3 ) ) ) ) ) ).
% right_diff_distrib_NO_MATCH
thf(fact_4010_atLeastAtMost__diff__ends,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A] :
( ( minus_minus @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) @ ( insert @ A @ A2 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( set_or5935395276787703475ssThan @ A @ A2 @ B2 ) ) ) ).
% atLeastAtMost_diff_ends
thf(fact_4011_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_4012_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_4013_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_4014_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_4015_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_4016_real__root__pos__unique,axiom,
! [N: nat,Y: real,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ Y )
=> ( ( ( power_power @ real @ Y @ N )
= X )
=> ( ( root @ N @ X )
= Y ) ) ) ) ).
% real_root_pos_unique
thf(fact_4017_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_4018_scale__left__distrib__NO__MATCH,axiom,
! [C: $tType,A: $tType] :
( ( real_V4867850818363320053vector @ A )
=> ! [X: A,Y: A,C3: C,A2: real,B2: real] :
( ( nO_MATCH @ A @ C @ ( divide_divide @ A @ X @ Y ) @ C3 )
=> ( ( real_V8093663219630862766scaleR @ A @ ( plus_plus @ real @ A2 @ B2 ) @ X )
= ( plus_plus @ A @ ( real_V8093663219630862766scaleR @ A @ A2 @ X ) @ ( real_V8093663219630862766scaleR @ A @ B2 @ X ) ) ) ) ) ).
% scale_left_distrib_NO_MATCH
thf(fact_4019_root__sgn__power,axiom,
! [N: nat,Y: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( root @ N @ ( times_times @ real @ ( sgn_sgn @ real @ Y ) @ ( power_power @ real @ ( abs_abs @ real @ Y ) @ N ) ) )
= Y ) ) ).
% root_sgn_power
thf(fact_4020_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_4021_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_4022_log__root,axiom,
! [N: nat,A2: real,B2: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
=> ( ( log @ B2 @ ( root @ N @ A2 ) )
= ( divide_divide @ real @ ( log @ B2 @ A2 ) @ ( semiring_1_of_nat @ real @ N ) ) ) ) ) ).
% log_root
thf(fact_4023_log__base__root,axiom,
! [N: nat,B2: real,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ B2 )
=> ( ( log @ ( root @ N @ B2 ) @ X )
= ( times_times @ real @ ( semiring_1_of_nat @ real @ N ) @ ( log @ B2 @ X ) ) ) ) ) ).
% log_base_root
thf(fact_4024_split__root,axiom,
! [P2: real > $o,N: nat,X: real] :
( ( P2 @ ( root @ N @ X ) )
= ( ( ( N
= ( zero_zero @ nat ) )
=> ( P2 @ ( zero_zero @ real ) ) )
& ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ! [Y6: real] :
( ( ( times_times @ real @ ( sgn_sgn @ real @ Y6 ) @ ( power_power @ real @ ( abs_abs @ real @ Y6 ) @ N ) )
= X )
=> ( P2 @ Y6 ) ) ) ) ) ).
% split_root
thf(fact_4025_gcd__nat_Opelims,axiom,
! [X: nat,Xa2: nat,Y: nat] :
( ( ( gcd_gcd @ nat @ X @ Xa2 )
= Y )
=> ( ( accp @ ( product_prod @ nat @ nat ) @ gcd_nat_rel @ ( product_Pair @ nat @ nat @ X @ Xa2 ) )
=> ~ ( ( ( ( Xa2
= ( zero_zero @ nat ) )
=> ( Y = X ) )
& ( ( Xa2
!= ( zero_zero @ nat ) )
=> ( Y
= ( gcd_gcd @ nat @ Xa2 @ ( modulo_modulo @ nat @ X @ Xa2 ) ) ) ) )
=> ~ ( accp @ ( product_prod @ nat @ nat ) @ gcd_nat_rel @ ( product_Pair @ nat @ nat @ X @ Xa2 ) ) ) ) ) ).
% gcd_nat.pelims
thf(fact_4026_horner__sum__eq__sum__funpow,axiom,
! [A: $tType,B: $tType] :
( ( comm_semiring_0 @ A )
=> ( ( groups4207007520872428315er_sum @ B @ A )
= ( ^ [F4: B > A,A6: A,Xs3: list @ B] :
( groups7311177749621191930dd_sum @ nat @ A
@ ^ [N2: nat] : ( compow @ ( A > A ) @ N2 @ ( times_times @ A @ A6 ) @ ( F4 @ ( nth @ B @ Xs3 @ N2 ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ B ) @ Xs3 ) ) ) ) ) ) ).
% horner_sum_eq_sum_funpow
thf(fact_4027_card__UNION,axiom,
! [A: $tType,A5: set @ ( set @ A )] :
( ( finite_finite2 @ ( set @ A ) @ A5 )
=> ( ! [X4: set @ A] :
( ( member @ ( set @ A ) @ X4 @ A5 )
=> ( finite_finite2 @ A @ X4 ) )
=> ( ( finite_card @ A @ ( complete_Sup_Sup @ ( set @ A ) @ A5 ) )
= ( nat2
@ ( groups7311177749621191930dd_sum @ ( set @ ( set @ A ) ) @ int
@ ^ [I8: set @ ( set @ A )] : ( times_times @ int @ ( power_power @ int @ ( uminus_uminus @ int @ ( one_one @ int ) ) @ ( plus_plus @ nat @ ( finite_card @ ( set @ A ) @ I8 ) @ ( one_one @ nat ) ) ) @ ( semiring_1_of_nat @ int @ ( finite_card @ A @ ( complete_Inf_Inf @ ( set @ A ) @ I8 ) ) ) )
@ ( collect @ ( set @ ( set @ A ) )
@ ^ [I8: set @ ( set @ A )] :
( ( ord_less_eq @ ( set @ ( set @ A ) ) @ I8 @ A5 )
& ( I8
!= ( bot_bot @ ( set @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ).
% card_UNION
thf(fact_4028_max__nat_Osemilattice__neutr__order__axioms,axiom,
( semila1105856199041335345_order @ nat @ ( ord_max @ nat ) @ ( zero_zero @ nat )
@ ^ [X5: nat,Y6: nat] : ( ord_less_eq @ nat @ Y6 @ X5 )
@ ^ [X5: nat,Y6: nat] : ( ord_less @ nat @ Y6 @ X5 ) ) ).
% max_nat.semilattice_neutr_order_axioms
thf(fact_4029_finite__greaterThanLessThan,axiom,
! [L: nat,U: nat] : ( finite_finite2 @ nat @ ( set_or5935395276787703475ssThan @ nat @ L @ U ) ) ).
% finite_greaterThanLessThan
thf(fact_4030_Suc__funpow,axiom,
! [N: nat] :
( ( compow @ ( nat > nat ) @ N @ suc )
= ( plus_plus @ nat @ N ) ) ).
% Suc_funpow
thf(fact_4031_Sup__lessThan,axiom,
! [A: $tType] :
( ( ( comple6319245703460814977attice @ A )
& ( dense_linorder @ A ) )
=> ! [Y: A] :
( ( complete_Sup_Sup @ A @ ( set_ord_lessThan @ A @ Y ) )
= Y ) ) ).
% Sup_lessThan
thf(fact_4032_finite__Inter,axiom,
! [A: $tType,M5: set @ ( set @ A )] :
( ? [X3: set @ A] :
( ( member @ ( set @ A ) @ X3 @ M5 )
& ( finite_finite2 @ A @ X3 ) )
=> ( finite_finite2 @ A @ ( complete_Inf_Inf @ ( set @ A ) @ M5 ) ) ) ).
% finite_Inter
thf(fact_4033_Sup__atMost,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [Y: A] :
( ( complete_Sup_Sup @ A @ ( set_ord_atMost @ A @ Y ) )
= Y ) ) ).
% Sup_atMost
thf(fact_4034_funpow__0,axiom,
! [A: $tType,F3: A > A,X: A] :
( ( compow @ ( A > A ) @ ( zero_zero @ nat ) @ F3 @ X )
= X ) ).
% funpow_0
thf(fact_4035_cSup__atLeastAtMost,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ Y @ X )
=> ( ( complete_Sup_Sup @ A @ ( set_or1337092689740270186AtMost @ A @ Y @ X ) )
= X ) ) ) ).
% cSup_atLeastAtMost
thf(fact_4036_Sup__atLeastAtMost,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( complete_Sup_Sup @ A @ ( set_or1337092689740270186AtMost @ A @ X @ Y ) )
= Y ) ) ) ).
% Sup_atLeastAtMost
thf(fact_4037_cSup__singleton,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X: A] :
( ( complete_Sup_Sup @ A @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= X ) ) ).
% cSup_singleton
thf(fact_4038_cInf__atLeastAtMost,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ Y @ X )
=> ( ( complete_Inf_Inf @ A @ ( set_or1337092689740270186AtMost @ A @ Y @ X ) )
= Y ) ) ) ).
% cInf_atLeastAtMost
thf(fact_4039_Inf__atLeastAtMost,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( complete_Inf_Inf @ A @ ( set_or1337092689740270186AtMost @ A @ X @ Y ) )
= X ) ) ) ).
% Inf_atLeastAtMost
thf(fact_4040_cInf__singleton,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X: A] :
( ( complete_Inf_Inf @ A @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= X ) ) ).
% cInf_singleton
thf(fact_4041_Sup__atLeastLessThan,axiom,
! [A: $tType] :
( ( ( comple6319245703460814977attice @ A )
& ( dense_linorder @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( ( complete_Sup_Sup @ A @ ( set_or7035219750837199246ssThan @ A @ X @ Y ) )
= Y ) ) ) ).
% Sup_atLeastLessThan
thf(fact_4042_cSup__atLeastLessThan,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( dense_linorder @ A ) )
=> ! [Y: A,X: A] :
( ( ord_less @ A @ Y @ X )
=> ( ( complete_Sup_Sup @ A @ ( set_or7035219750837199246ssThan @ A @ Y @ X ) )
= X ) ) ) ).
% cSup_atLeastLessThan
thf(fact_4043_Inf__atLeastLessThan,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( ( complete_Inf_Inf @ A @ ( set_or7035219750837199246ssThan @ A @ X @ Y ) )
= X ) ) ) ).
% Inf_atLeastLessThan
thf(fact_4044_cInf__atLeastLessThan,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [Y: A,X: A] :
( ( ord_less @ A @ Y @ X )
=> ( ( complete_Inf_Inf @ A @ ( set_or7035219750837199246ssThan @ A @ Y @ X ) )
= Y ) ) ) ).
% cInf_atLeastLessThan
thf(fact_4045_Inf__atMost,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X: A] :
( ( complete_Inf_Inf @ A @ ( set_ord_atMost @ A @ X ) )
= ( bot_bot @ A ) ) ) ).
% Inf_atMost
thf(fact_4046_cSup__greaterThanLessThan,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( dense_linorder @ A ) )
=> ! [Y: A,X: A] :
( ( ord_less @ A @ Y @ X )
=> ( ( complete_Sup_Sup @ A @ ( set_or5935395276787703475ssThan @ A @ Y @ X ) )
= X ) ) ) ).
% cSup_greaterThanLessThan
thf(fact_4047_Sup__greaterThanLessThan,axiom,
! [A: $tType] :
( ( ( comple6319245703460814977attice @ A )
& ( dense_linorder @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( ( complete_Sup_Sup @ A @ ( set_or5935395276787703475ssThan @ A @ X @ Y ) )
= Y ) ) ) ).
% Sup_greaterThanLessThan
thf(fact_4048_Inf__greaterThanLessThan,axiom,
! [A: $tType] :
( ( ( comple6319245703460814977attice @ A )
& ( dense_linorder @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( ( complete_Inf_Inf @ A @ ( set_or5935395276787703475ssThan @ A @ X @ Y ) )
= X ) ) ) ).
% Inf_greaterThanLessThan
thf(fact_4049_cInf__greaterThanLessThan,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( dense_linorder @ A ) )
=> ! [Y: A,X: A] :
( ( ord_less @ A @ Y @ X )
=> ( ( complete_Inf_Inf @ A @ ( set_or5935395276787703475ssThan @ A @ Y @ X ) )
= Y ) ) ) ).
% cInf_greaterThanLessThan
thf(fact_4050_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_4051_finite__Union,axiom,
! [A: $tType,A5: set @ ( set @ A )] :
( ( finite_finite2 @ ( set @ A ) @ A5 )
=> ( ! [M8: set @ A] :
( ( member @ ( set @ A ) @ M8 @ A5 )
=> ( finite_finite2 @ A @ M8 ) )
=> ( finite_finite2 @ A @ ( complete_Sup_Sup @ ( set @ A ) @ A5 ) ) ) ) ).
% finite_Union
thf(fact_4052_funpow__mult,axiom,
! [A: $tType,N: nat,M2: nat,F3: A > A] :
( ( compow @ ( A > A ) @ N @ ( compow @ ( A > A ) @ M2 @ F3 ) )
= ( compow @ ( A > A ) @ ( times_times @ nat @ M2 @ N ) @ F3 ) ) ).
% funpow_mult
thf(fact_4053_funpow__swap1,axiom,
! [A: $tType,F3: A > A,N: nat,X: A] :
( ( F3 @ ( compow @ ( A > A ) @ N @ F3 @ X ) )
= ( compow @ ( A > A ) @ N @ F3 @ ( F3 @ X ) ) ) ).
% funpow_swap1
thf(fact_4054_semilattice__neutr__order_Oneutr__eq__iff,axiom,
! [A: $tType,F3: A > A > A,Z: A,Less_eq: A > A > $o,Less: A > A > $o,A2: A,B2: A] :
( ( semila1105856199041335345_order @ A @ F3 @ Z @ Less_eq @ Less )
=> ( ( Z
= ( F3 @ A2 @ B2 ) )
= ( ( A2 = Z )
& ( B2 = Z ) ) ) ) ).
% semilattice_neutr_order.neutr_eq_iff
thf(fact_4055_semilattice__neutr__order_Oeq__neutr__iff,axiom,
! [A: $tType,F3: A > A > A,Z: A,Less_eq: A > A > $o,Less: A > A > $o,A2: A,B2: A] :
( ( semila1105856199041335345_order @ A @ F3 @ Z @ Less_eq @ Less )
=> ( ( ( F3 @ A2 @ B2 )
= Z )
= ( ( A2 = Z )
& ( B2 = Z ) ) ) ) ).
% semilattice_neutr_order.eq_neutr_iff
thf(fact_4056_cSup__eq__maximum,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [Z: A,X7: set @ A] :
( ( member @ A @ Z @ X7 )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ X7 )
=> ( ord_less_eq @ A @ X4 @ Z ) )
=> ( ( complete_Sup_Sup @ A @ X7 )
= Z ) ) ) ) ).
% cSup_eq_maximum
thf(fact_4057_cSup__eq,axiom,
! [A: $tType] :
( ( ( condit1219197933456340205attice @ A )
& ( no_bot @ A ) )
=> ! [X7: set @ A,A2: A] :
( ! [X4: A] :
( ( member @ A @ X4 @ X7 )
=> ( ord_less_eq @ A @ X4 @ A2 ) )
=> ( ! [Y3: A] :
( ! [X3: A] :
( ( member @ A @ X3 @ X7 )
=> ( ord_less_eq @ A @ X3 @ Y3 ) )
=> ( ord_less_eq @ A @ A2 @ Y3 ) )
=> ( ( complete_Sup_Sup @ A @ X7 )
= A2 ) ) ) ) ).
% cSup_eq
thf(fact_4058_cInf__eq__minimum,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [Z: A,X7: set @ A] :
( ( member @ A @ Z @ X7 )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ X7 )
=> ( ord_less_eq @ A @ Z @ X4 ) )
=> ( ( complete_Inf_Inf @ A @ X7 )
= Z ) ) ) ) ).
% cInf_eq_minimum
thf(fact_4059_cInf__eq,axiom,
! [A: $tType] :
( ( ( condit1219197933456340205attice @ A )
& ( no_top @ A ) )
=> ! [X7: set @ A,A2: A] :
( ! [X4: A] :
( ( member @ A @ X4 @ X7 )
=> ( ord_less_eq @ A @ A2 @ X4 ) )
=> ( ! [Y3: A] :
( ! [X3: A] :
( ( member @ A @ X3 @ X7 )
=> ( ord_less_eq @ A @ Y3 @ X3 ) )
=> ( ord_less_eq @ A @ Y3 @ A2 ) )
=> ( ( complete_Inf_Inf @ A @ X7 )
= A2 ) ) ) ) ).
% cInf_eq
thf(fact_4060_cInf__eq__non__empty,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X7: set @ A,A2: A] :
( ( X7
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ X7 )
=> ( ord_less_eq @ A @ A2 @ X4 ) )
=> ( ! [Y3: A] :
( ! [X3: A] :
( ( member @ A @ X3 @ X7 )
=> ( ord_less_eq @ A @ Y3 @ X3 ) )
=> ( ord_less_eq @ A @ Y3 @ A2 ) )
=> ( ( complete_Inf_Inf @ A @ X7 )
= A2 ) ) ) ) ) ).
% cInf_eq_non_empty
thf(fact_4061_cInf__greatest,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X7: set @ A,Z: A] :
( ( X7
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ X7 )
=> ( ord_less_eq @ A @ Z @ X4 ) )
=> ( ord_less_eq @ A @ Z @ ( complete_Inf_Inf @ A @ X7 ) ) ) ) ) ).
% cInf_greatest
thf(fact_4062_cInf__le__finite,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X7: set @ A,X: A] :
( ( finite_finite2 @ A @ X7 )
=> ( ( member @ A @ X @ X7 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ X7 ) @ X ) ) ) ) ).
% cInf_le_finite
thf(fact_4063_cInf__lessD,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [X7: set @ A,Z: A] :
( ( X7
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less @ A @ ( complete_Inf_Inf @ A @ X7 ) @ Z )
=> ? [X4: A] :
( ( member @ A @ X4 @ X7 )
& ( ord_less @ A @ X4 @ Z ) ) ) ) ) ).
% cInf_lessD
thf(fact_4064_finite__imp__less__Inf,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [X7: set @ A,X: A,A2: A] :
( ( finite_finite2 @ A @ X7 )
=> ( ( member @ A @ X @ X7 )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ X7 )
=> ( ord_less @ A @ A2 @ X4 ) )
=> ( ord_less @ A @ A2 @ ( complete_Inf_Inf @ A @ X7 ) ) ) ) ) ) ).
% finite_imp_less_Inf
thf(fact_4065_cSup__eq__non__empty,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X7: set @ A,A2: A] :
( ( X7
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ X7 )
=> ( ord_less_eq @ A @ X4 @ A2 ) )
=> ( ! [Y3: A] :
( ! [X3: A] :
( ( member @ A @ X3 @ X7 )
=> ( ord_less_eq @ A @ X3 @ Y3 ) )
=> ( ord_less_eq @ A @ A2 @ Y3 ) )
=> ( ( complete_Sup_Sup @ A @ X7 )
= A2 ) ) ) ) ) ).
% cSup_eq_non_empty
thf(fact_4066_cSup__least,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X7: set @ A,Z: A] :
( ( X7
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ X7 )
=> ( ord_less_eq @ A @ X4 @ Z ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ X7 ) @ Z ) ) ) ) ).
% cSup_least
thf(fact_4067_le__cSup__finite,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X7: set @ A,X: A] :
( ( finite_finite2 @ A @ X7 )
=> ( ( member @ A @ X @ X7 )
=> ( ord_less_eq @ A @ X @ ( complete_Sup_Sup @ A @ X7 ) ) ) ) ) ).
% le_cSup_finite
thf(fact_4068_less__cSupD,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [X7: set @ A,Z: A] :
( ( X7
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less @ A @ Z @ ( complete_Sup_Sup @ A @ X7 ) )
=> ? [X4: A] :
( ( member @ A @ X4 @ X7 )
& ( ord_less @ A @ Z @ X4 ) ) ) ) ) ).
% less_cSupD
thf(fact_4069_less__cSupE,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [Y: A,X7: set @ A] :
( ( ord_less @ A @ Y @ ( complete_Sup_Sup @ A @ X7 ) )
=> ( ( X7
!= ( bot_bot @ ( set @ A ) ) )
=> ~ ! [X4: A] :
( ( member @ A @ X4 @ X7 )
=> ~ ( ord_less @ A @ Y @ X4 ) ) ) ) ) ).
% less_cSupE
thf(fact_4070_finite__imp__Sup__less,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [X7: set @ A,X: A,A2: A] :
( ( finite_finite2 @ A @ X7 )
=> ( ( member @ A @ X @ X7 )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ X7 )
=> ( ord_less @ A @ X4 @ A2 ) )
=> ( ord_less @ A @ ( complete_Sup_Sup @ A @ X7 ) @ A2 ) ) ) ) ) ).
% finite_imp_Sup_less
thf(fact_4071_card__Union__le__sum__card,axiom,
! [A: $tType,U3: set @ ( set @ A )] : ( ord_less_eq @ nat @ ( finite_card @ A @ ( complete_Sup_Sup @ ( set @ A ) @ U3 ) ) @ ( groups7311177749621191930dd_sum @ ( set @ A ) @ nat @ ( finite_card @ A ) @ U3 ) ) ).
% card_Union_le_sum_card
thf(fact_4072_atLeastSucLessThan__greaterThanLessThan,axiom,
! [L: nat,U: nat] :
( ( set_or7035219750837199246ssThan @ nat @ ( suc @ L ) @ U )
= ( set_or5935395276787703475ssThan @ nat @ L @ U ) ) ).
% atLeastSucLessThan_greaterThanLessThan
thf(fact_4073_finite__UnionD,axiom,
! [A: $tType,A5: set @ ( set @ A )] :
( ( finite_finite2 @ A @ ( complete_Sup_Sup @ ( set @ A ) @ A5 ) )
=> ( finite_finite2 @ ( set @ A ) @ A5 ) ) ).
% finite_UnionD
thf(fact_4074_finite__less__Inf__iff,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [X7: set @ A,A2: A] :
( ( finite_finite2 @ A @ X7 )
=> ( ( X7
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less @ A @ A2 @ ( complete_Inf_Inf @ A @ X7 ) )
= ( ! [X5: A] :
( ( member @ A @ X5 @ X7 )
=> ( ord_less @ A @ A2 @ X5 ) ) ) ) ) ) ) ).
% finite_less_Inf_iff
thf(fact_4075_finite__Sup__less__iff,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [X7: set @ A,A2: A] :
( ( finite_finite2 @ A @ X7 )
=> ( ( X7
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less @ A @ ( complete_Sup_Sup @ A @ X7 ) @ A2 )
= ( ! [X5: A] :
( ( member @ A @ X5 @ X7 )
=> ( ord_less @ A @ X5 @ A2 ) ) ) ) ) ) ) ).
% finite_Sup_less_iff
thf(fact_4076_cInf__abs__ge,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( linordered_idom @ A ) )
=> ! [S2: set @ A,A2: A] :
( ( S2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ S2 )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ X4 ) @ A2 ) )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ ( complete_Inf_Inf @ A @ S2 ) ) @ A2 ) ) ) ) ).
% cInf_abs_ge
thf(fact_4077_cSup__abs__le,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( linordered_idom @ A ) )
=> ! [S2: set @ A,A2: A] :
( ( S2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ S2 )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ X4 ) @ A2 ) )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ ( complete_Sup_Sup @ A @ S2 ) ) @ A2 ) ) ) ) ).
% cSup_abs_le
thf(fact_4078_card__Union__le__sum__card__weak,axiom,
! [A: $tType,U3: set @ ( set @ A )] :
( ! [X4: set @ A] :
( ( member @ ( set @ A ) @ X4 @ U3 )
=> ( finite_finite2 @ A @ X4 ) )
=> ( ord_less_eq @ nat @ ( finite_card @ A @ ( complete_Sup_Sup @ ( set @ A ) @ U3 ) ) @ ( groups7311177749621191930dd_sum @ ( set @ A ) @ nat @ ( finite_card @ A ) @ U3 ) ) ) ).
% card_Union_le_sum_card_weak
thf(fact_4079_finite__subset__Union,axiom,
! [A: $tType,A5: set @ A,B11: set @ ( set @ A )] :
( ( finite_finite2 @ A @ A5 )
=> ( ( ord_less_eq @ ( set @ A ) @ A5 @ ( complete_Sup_Sup @ ( set @ A ) @ B11 ) )
=> ~ ! [F7: set @ ( set @ A )] :
( ( finite_finite2 @ ( set @ A ) @ F7 )
=> ( ( ord_less_eq @ ( set @ ( set @ A ) ) @ F7 @ B11 )
=> ~ ( ord_less_eq @ ( set @ A ) @ A5 @ ( complete_Sup_Sup @ ( set @ A ) @ F7 ) ) ) ) ) ) ).
% finite_subset_Union
thf(fact_4080_of__nat__def,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( ( semiring_1_of_nat @ A )
= ( ^ [N2: nat] : ( compow @ ( A > A ) @ N2 @ ( plus_plus @ A @ ( one_one @ A ) ) @ ( zero_zero @ A ) ) ) ) ) ).
% of_nat_def
thf(fact_4081_numeral__add__unfold__funpow,axiom,
! [A: $tType] :
( ( semiring_numeral @ A )
=> ! [K2: num,A2: A] :
( ( plus_plus @ A @ ( numeral_numeral @ A @ K2 ) @ A2 )
= ( compow @ ( A > A ) @ ( numeral_numeral @ nat @ K2 ) @ ( plus_plus @ A @ ( one_one @ A ) ) @ A2 ) ) ) ).
% numeral_add_unfold_funpow
thf(fact_4082_cInf__asclose,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( linordered_idom @ A ) )
=> ! [S2: set @ A,L: A,E3: A] :
( ( S2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ S2 )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X4 @ L ) ) @ E3 ) )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ ( complete_Inf_Inf @ A @ S2 ) @ L ) ) @ E3 ) ) ) ) ).
% cInf_asclose
thf(fact_4083_cSup__asclose,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( linordered_idom @ A ) )
=> ! [S2: set @ A,L: A,E3: A] :
( ( S2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ S2 )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X4 @ L ) ) @ E3 ) )
=> ( ord_less_eq @ A @ ( abs_abs @ A @ ( minus_minus @ A @ ( complete_Sup_Sup @ A @ S2 ) @ L ) ) @ E3 ) ) ) ) ).
% cSup_asclose
thf(fact_4084_gcd__nat_Osemilattice__neutr__order__axioms,axiom,
( semila1105856199041335345_order @ nat @ ( gcd_gcd @ nat ) @ ( zero_zero @ nat ) @ ( dvd_dvd @ nat )
@ ^ [M3: nat,N2: nat] :
( ( dvd_dvd @ nat @ M3 @ N2 )
& ( M3 != N2 ) ) ) ).
% gcd_nat.semilattice_neutr_order_axioms
thf(fact_4085_Sup__insert__finite,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [S2: set @ A,X: A] :
( ( finite_finite2 @ A @ S2 )
=> ( ( ( S2
= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Sup_Sup @ A @ ( insert @ A @ X @ S2 ) )
= X ) )
& ( ( S2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Sup_Sup @ A @ ( insert @ A @ X @ S2 ) )
= ( ord_max @ A @ X @ ( complete_Sup_Sup @ A @ S2 ) ) ) ) ) ) ) ).
% Sup_insert_finite
thf(fact_4086_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_4087_ccpo__Sup__singleton,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ! [X: A] :
( ( complete_Sup_Sup @ A @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= X ) ) ).
% ccpo_Sup_singleton
thf(fact_4088_Sup__empty,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ( ( complete_Sup_Sup @ A @ ( bot_bot @ ( set @ A ) ) )
= ( bot_bot @ A ) ) ) ).
% Sup_empty
thf(fact_4089_ccSup__empty,axiom,
! [A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ( ( complete_Sup_Sup @ A @ ( bot_bot @ ( set @ A ) ) )
= ( bot_bot @ A ) ) ) ).
% ccSup_empty
thf(fact_4090_Inf__eq__bot__iff,axiom,
! [A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [A5: set @ A] :
( ( ( complete_Inf_Inf @ A @ A5 )
= ( bot_bot @ A ) )
= ( ! [X5: A] :
( ( ord_less @ A @ ( bot_bot @ A ) @ X5 )
=> ? [Y6: A] :
( ( member @ A @ Y6 @ A5 )
& ( ord_less @ A @ Y6 @ X5 ) ) ) ) ) ) ).
% Inf_eq_bot_iff
thf(fact_4091_Sup__bot__conv_I1_J,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ A] :
( ( ( complete_Sup_Sup @ A @ A5 )
= ( bot_bot @ A ) )
= ( ! [X5: A] :
( ( member @ A @ X5 @ A5 )
=> ( X5
= ( bot_bot @ A ) ) ) ) ) ) ).
% Sup_bot_conv(1)
thf(fact_4092_Sup__bot__conv_I2_J,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ A] :
( ( ( bot_bot @ A )
= ( complete_Sup_Sup @ A @ A5 ) )
= ( ! [X5: A] :
( ( member @ A @ X5 @ A5 )
=> ( X5
= ( bot_bot @ A ) ) ) ) ) ) ).
% Sup_bot_conv(2)
thf(fact_4093_Sup__nat__empty,axiom,
( ( complete_Sup_Sup @ nat @ ( bot_bot @ ( set @ nat ) ) )
= ( zero_zero @ nat ) ) ).
% Sup_nat_empty
thf(fact_4094_Inf__nat__def1,axiom,
! [K5: set @ nat] :
( ( K5
!= ( bot_bot @ ( set @ nat ) ) )
=> ( member @ nat @ ( complete_Inf_Inf @ nat @ K5 ) @ K5 ) ) ).
% Inf_nat_def1
thf(fact_4095_Sup__eqI,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ A,X: A] :
( ! [Y3: A] :
( ( member @ A @ Y3 @ A5 )
=> ( ord_less_eq @ A @ Y3 @ X ) )
=> ( ! [Y3: A] :
( ! [Z4: A] :
( ( member @ A @ Z4 @ A5 )
=> ( ord_less_eq @ A @ Z4 @ Y3 ) )
=> ( ord_less_eq @ A @ X @ Y3 ) )
=> ( ( complete_Sup_Sup @ A @ A5 )
= X ) ) ) ) ).
% Sup_eqI
thf(fact_4096_Sup__mono,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ A,B6: set @ A] :
( ! [A4: A] :
( ( member @ A @ A4 @ A5 )
=> ? [X3: A] :
( ( member @ A @ X3 @ B6 )
& ( ord_less_eq @ A @ A4 @ X3 ) ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A5 ) @ ( complete_Sup_Sup @ A @ B6 ) ) ) ) ).
% Sup_mono
thf(fact_4097_Sup__least,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ A,Z: A] :
( ! [X4: A] :
( ( member @ A @ X4 @ A5 )
=> ( ord_less_eq @ A @ X4 @ Z ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A5 ) @ Z ) ) ) ).
% Sup_least
thf(fact_4098_Sup__upper,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X: A,A5: set @ A] :
( ( member @ A @ X @ A5 )
=> ( ord_less_eq @ A @ X @ ( complete_Sup_Sup @ A @ A5 ) ) ) ) ).
% Sup_upper
thf(fact_4099_Sup__le__iff,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ A,B2: A] :
( ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A5 ) @ B2 )
= ( ! [X5: A] :
( ( member @ A @ X5 @ A5 )
=> ( ord_less_eq @ A @ X5 @ B2 ) ) ) ) ) ).
% Sup_le_iff
thf(fact_4100_Sup__upper2,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [U: A,A5: set @ A,V: A] :
( ( member @ A @ U @ A5 )
=> ( ( ord_less_eq @ A @ V @ U )
=> ( ord_less_eq @ A @ V @ ( complete_Sup_Sup @ A @ A5 ) ) ) ) ) ).
% Sup_upper2
thf(fact_4101_less__Sup__iff,axiom,
! [A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [A2: A,S2: set @ A] :
( ( ord_less @ A @ A2 @ ( complete_Sup_Sup @ A @ S2 ) )
= ( ? [X5: A] :
( ( member @ A @ X5 @ S2 )
& ( ord_less @ A @ A2 @ X5 ) ) ) ) ) ).
% less_Sup_iff
thf(fact_4102_Inf__eqI,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ A,X: A] :
( ! [I3: A] :
( ( member @ A @ I3 @ A5 )
=> ( ord_less_eq @ A @ X @ I3 ) )
=> ( ! [Y3: A] :
( ! [I: A] :
( ( member @ A @ I @ A5 )
=> ( ord_less_eq @ A @ Y3 @ I ) )
=> ( ord_less_eq @ A @ Y3 @ X ) )
=> ( ( complete_Inf_Inf @ A @ A5 )
= X ) ) ) ) ).
% Inf_eqI
thf(fact_4103_Inf__mono,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [B6: set @ A,A5: set @ A] :
( ! [B4: A] :
( ( member @ A @ B4 @ B6 )
=> ? [X3: A] :
( ( member @ A @ X3 @ A5 )
& ( ord_less_eq @ A @ X3 @ B4 ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A5 ) @ ( complete_Inf_Inf @ A @ B6 ) ) ) ) ).
% Inf_mono
thf(fact_4104_Inf__lower,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X: A,A5: set @ A] :
( ( member @ A @ X @ A5 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A5 ) @ X ) ) ) ).
% Inf_lower
thf(fact_4105_Inf__lower2,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [U: A,A5: set @ A,V: A] :
( ( member @ A @ U @ A5 )
=> ( ( ord_less_eq @ A @ U @ V )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A5 ) @ V ) ) ) ) ).
% Inf_lower2
thf(fact_4106_le__Inf__iff,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [B2: A,A5: set @ A] :
( ( ord_less_eq @ A @ B2 @ ( complete_Inf_Inf @ A @ A5 ) )
= ( ! [X5: A] :
( ( member @ A @ X5 @ A5 )
=> ( ord_less_eq @ A @ B2 @ X5 ) ) ) ) ) ).
% le_Inf_iff
thf(fact_4107_Inf__greatest,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ A,Z: A] :
( ! [X4: A] :
( ( member @ A @ X4 @ A5 )
=> ( ord_less_eq @ A @ Z @ X4 ) )
=> ( ord_less_eq @ A @ Z @ ( complete_Inf_Inf @ A @ A5 ) ) ) ) ).
% Inf_greatest
thf(fact_4108_Inf__less__iff,axiom,
! [A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [S2: set @ A,A2: A] :
( ( ord_less @ A @ ( complete_Inf_Inf @ A @ S2 ) @ A2 )
= ( ? [X5: A] :
( ( member @ A @ X5 @ S2 )
& ( ord_less @ A @ X5 @ A2 ) ) ) ) ) ).
% Inf_less_iff
thf(fact_4109_Union__empty__conv,axiom,
! [A: $tType,A5: set @ ( set @ A )] :
( ( ( complete_Sup_Sup @ ( set @ A ) @ A5 )
= ( bot_bot @ ( set @ A ) ) )
= ( ! [X5: set @ A] :
( ( member @ ( set @ A ) @ X5 @ A5 )
=> ( X5
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% Union_empty_conv
thf(fact_4110_empty__Union__conv,axiom,
! [A: $tType,A5: set @ ( set @ A )] :
( ( ( bot_bot @ ( set @ A ) )
= ( complete_Sup_Sup @ ( set @ A ) @ A5 ) )
= ( ! [X5: set @ A] :
( ( member @ ( set @ A ) @ X5 @ A5 )
=> ( X5
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% empty_Union_conv
thf(fact_4111_le__Sup__iff,axiom,
! [A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [X: A,A5: set @ A] :
( ( ord_less_eq @ A @ X @ ( complete_Sup_Sup @ A @ A5 ) )
= ( ! [Y6: A] :
( ( ord_less @ A @ Y6 @ X )
=> ? [X5: A] :
( ( member @ A @ X5 @ A5 )
& ( ord_less @ A @ Y6 @ X5 ) ) ) ) ) ) ).
% le_Sup_iff
thf(fact_4112_Inf__le__iff,axiom,
! [A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [A5: set @ A,X: A] :
( ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A5 ) @ X )
= ( ! [Y6: A] :
( ( ord_less @ A @ X @ Y6 )
=> ? [X5: A] :
( ( member @ A @ X5 @ A5 )
& ( ord_less @ A @ X5 @ Y6 ) ) ) ) ) ) ).
% Inf_le_iff
thf(fact_4113_less__eq__Sup,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ A,U: A] :
( ! [V3: A] :
( ( member @ A @ V3 @ A5 )
=> ( ord_less_eq @ A @ U @ V3 ) )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ord_less_eq @ A @ U @ ( complete_Sup_Sup @ A @ A5 ) ) ) ) ) ).
% less_eq_Sup
thf(fact_4114_Sup__subset__mono,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ A,B6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B6 )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A5 ) @ ( complete_Sup_Sup @ A @ B6 ) ) ) ) ).
% Sup_subset_mono
thf(fact_4115_Inf__less__eq,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ A,U: A] :
( ! [V3: A] :
( ( member @ A @ V3 @ A5 )
=> ( ord_less_eq @ A @ V3 @ U ) )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A5 ) @ U ) ) ) ) ).
% Inf_less_eq
thf(fact_4116_Inf__superset__mono,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [B6: set @ A,A5: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ B6 @ A5 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A5 ) @ ( complete_Inf_Inf @ A @ B6 ) ) ) ) ).
% Inf_superset_mono
thf(fact_4117_Union__empty,axiom,
! [A: $tType] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( bot_bot @ ( set @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Union_empty
thf(fact_4118_Inter__subset,axiom,
! [A: $tType,A5: set @ ( set @ A ),B6: set @ A] :
( ! [X17: set @ A] :
( ( member @ ( set @ A ) @ X17 @ A5 )
=> ( ord_less_eq @ ( set @ A ) @ X17 @ B6 ) )
=> ( ( A5
!= ( bot_bot @ ( set @ ( set @ A ) ) ) )
=> ( ord_less_eq @ ( set @ A ) @ ( complete_Inf_Inf @ ( set @ A ) @ A5 ) @ B6 ) ) ) ).
% Inter_subset
thf(fact_4119_Inf__le__Sup,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ A] :
( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A5 ) @ ( complete_Sup_Sup @ A @ A5 ) ) ) ) ).
% Inf_le_Sup
thf(fact_4120_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_4121_relpowp__fun__conv,axiom,
! [A: $tType] :
( ( compow @ ( A > A > $o ) )
= ( ^ [N2: nat,P4: A > A > $o,X5: A,Y6: A] :
? [F4: nat > A] :
( ( ( F4 @ ( zero_zero @ nat ) )
= X5 )
& ( ( F4 @ N2 )
= Y6 )
& ! [I4: nat] :
( ( ord_less @ nat @ I4 @ N2 )
=> ( P4 @ ( F4 @ I4 ) @ ( F4 @ ( suc @ I4 ) ) ) ) ) ) ) ).
% relpowp_fun_conv
thf(fact_4122_Nat_Ofunpow__code__def,axiom,
! [A: $tType] :
( ( funpow @ A )
= ( compow @ ( A > A ) ) ) ).
% Nat.funpow_code_def
thf(fact_4123_card__partition,axiom,
! [A: $tType,C5: set @ ( set @ A ),K2: nat] :
( ( finite_finite2 @ ( set @ A ) @ C5 )
=> ( ( finite_finite2 @ A @ ( complete_Sup_Sup @ ( set @ A ) @ C5 ) )
=> ( ! [C2: set @ A] :
( ( member @ ( set @ A ) @ C2 @ C5 )
=> ( ( finite_card @ A @ C2 )
= K2 ) )
=> ( ! [C1: set @ A,C22: set @ A] :
( ( member @ ( set @ A ) @ C1 @ C5 )
=> ( ( member @ ( set @ A ) @ C22 @ C5 )
=> ( ( C1 != C22 )
=> ( ( inf_inf @ ( set @ A ) @ C1 @ C22 )
= ( bot_bot @ ( set @ A ) ) ) ) ) )
=> ( ( times_times @ nat @ K2 @ ( finite_card @ ( set @ A ) @ C5 ) )
= ( finite_card @ A @ ( complete_Sup_Sup @ ( set @ A ) @ C5 ) ) ) ) ) ) ) ).
% card_partition
thf(fact_4124_inf__apply,axiom,
! [B: $tType,A: $tType] :
( ( semilattice_inf @ B )
=> ( ( inf_inf @ ( A > B ) )
= ( ^ [F4: A > B,G4: A > B,X5: A] : ( inf_inf @ B @ ( F4 @ X5 ) @ ( G4 @ X5 ) ) ) ) ) ).
% inf_apply
thf(fact_4125_inf__right__idem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X: A,Y: A] :
( ( inf_inf @ A @ ( inf_inf @ A @ X @ Y ) @ Y )
= ( inf_inf @ A @ X @ Y ) ) ) ).
% inf_right_idem
thf(fact_4126_inf_Oright__idem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A2: A,B2: A] :
( ( inf_inf @ A @ ( inf_inf @ A @ A2 @ B2 ) @ B2 )
= ( inf_inf @ A @ A2 @ B2 ) ) ) ).
% inf.right_idem
thf(fact_4127_inf__left__idem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X: A,Y: A] :
( ( inf_inf @ A @ X @ ( inf_inf @ A @ X @ Y ) )
= ( inf_inf @ A @ X @ Y ) ) ) ).
% inf_left_idem
thf(fact_4128_inf_Oleft__idem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A2: A,B2: A] :
( ( inf_inf @ A @ A2 @ ( inf_inf @ A @ A2 @ B2 ) )
= ( inf_inf @ A @ A2 @ B2 ) ) ) ).
% inf.left_idem
thf(fact_4129_inf__idem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X: A] :
( ( inf_inf @ A @ X @ X )
= X ) ) ).
% inf_idem
thf(fact_4130_inf_Oidem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A2: A] :
( ( inf_inf @ A @ A2 @ A2 )
= A2 ) ) ).
% inf.idem
thf(fact_4131_le__inf__iff,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X: A,Y: A,Z: A] :
( ( ord_less_eq @ A @ X @ ( inf_inf @ A @ Y @ Z ) )
= ( ( ord_less_eq @ A @ X @ Y )
& ( ord_less_eq @ A @ X @ Z ) ) ) ) ).
% le_inf_iff
thf(fact_4132_inf_Obounded__iff,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( ord_less_eq @ A @ A2 @ ( inf_inf @ A @ B2 @ C3 ) )
= ( ( ord_less_eq @ A @ A2 @ B2 )
& ( ord_less_eq @ A @ A2 @ C3 ) ) ) ) ).
% inf.bounded_iff
thf(fact_4133_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_4134_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_4135_boolean__algebra_Oconj__zero__left,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A] :
( ( inf_inf @ A @ ( bot_bot @ A ) @ X )
= ( bot_bot @ A ) ) ) ).
% boolean_algebra.conj_zero_left
thf(fact_4136_boolean__algebra_Oconj__zero__right,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A] :
( ( inf_inf @ A @ X @ ( bot_bot @ A ) )
= ( bot_bot @ A ) ) ) ).
% boolean_algebra.conj_zero_right
thf(fact_4137_finite__Int,axiom,
! [A: $tType,F5: set @ A,G6: set @ A] :
( ( ( finite_finite2 @ A @ F5 )
| ( finite_finite2 @ A @ G6 ) )
=> ( finite_finite2 @ A @ ( inf_inf @ ( set @ A ) @ F5 @ G6 ) ) ) ).
% finite_Int
thf(fact_4138_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_4139_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_4140_inf__compl__bot__right,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y: A] :
( ( inf_inf @ A @ X @ ( inf_inf @ A @ Y @ ( uminus_uminus @ A @ X ) ) )
= ( bot_bot @ A ) ) ) ).
% inf_compl_bot_right
thf(fact_4141_inf__compl__bot__left2,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y: A] :
( ( inf_inf @ A @ X @ ( inf_inf @ A @ ( uminus_uminus @ A @ X ) @ Y ) )
= ( bot_bot @ A ) ) ) ).
% inf_compl_bot_left2
thf(fact_4142_inf__compl__bot__left1,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y: A] :
( ( inf_inf @ A @ ( uminus_uminus @ A @ X ) @ ( inf_inf @ A @ X @ Y ) )
= ( bot_bot @ A ) ) ) ).
% inf_compl_bot_left1
thf(fact_4143_insert__disjoint_I1_J,axiom,
! [A: $tType,A2: A,A5: set @ A,B6: set @ A] :
( ( ( inf_inf @ ( set @ A ) @ ( insert @ A @ A2 @ A5 ) @ B6 )
= ( bot_bot @ ( set @ A ) ) )
= ( ~ ( member @ A @ A2 @ B6 )
& ( ( inf_inf @ ( set @ A ) @ A5 @ B6 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% insert_disjoint(1)
thf(fact_4144_insert__disjoint_I2_J,axiom,
! [A: $tType,A2: A,A5: set @ A,B6: set @ A] :
( ( ( bot_bot @ ( set @ A ) )
= ( inf_inf @ ( set @ A ) @ ( insert @ A @ A2 @ A5 ) @ B6 ) )
= ( ~ ( member @ A @ A2 @ B6 )
& ( ( bot_bot @ ( set @ A ) )
= ( inf_inf @ ( set @ A ) @ A5 @ B6 ) ) ) ) ).
% insert_disjoint(2)
thf(fact_4145_disjoint__insert_I1_J,axiom,
! [A: $tType,B6: set @ A,A2: A,A5: set @ A] :
( ( ( inf_inf @ ( set @ A ) @ B6 @ ( insert @ A @ A2 @ A5 ) )
= ( bot_bot @ ( set @ A ) ) )
= ( ~ ( member @ A @ A2 @ B6 )
& ( ( inf_inf @ ( set @ A ) @ B6 @ A5 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% disjoint_insert(1)
thf(fact_4146_disjoint__insert_I2_J,axiom,
! [A: $tType,A5: set @ A,B2: A,B6: set @ A] :
( ( ( bot_bot @ ( set @ A ) )
= ( inf_inf @ ( set @ A ) @ A5 @ ( insert @ A @ B2 @ B6 ) ) )
= ( ~ ( member @ A @ B2 @ A5 )
& ( ( bot_bot @ ( set @ A ) )
= ( inf_inf @ ( set @ A ) @ A5 @ B6 ) ) ) ) ).
% disjoint_insert(2)
thf(fact_4147_Diff__disjoint,axiom,
! [A: $tType,A5: set @ A,B6: set @ A] :
( ( inf_inf @ ( set @ A ) @ A5 @ ( minus_minus @ ( set @ A ) @ B6 @ A5 ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Diff_disjoint
thf(fact_4148_Compl__disjoint2,axiom,
! [A: $tType,A5: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( uminus_uminus @ ( set @ A ) @ A5 ) @ A5 )
= ( bot_bot @ ( set @ A ) ) ) ).
% Compl_disjoint2
thf(fact_4149_Compl__disjoint,axiom,
! [A: $tType,A5: set @ A] :
( ( inf_inf @ ( set @ A ) @ A5 @ ( uminus_uminus @ ( set @ A ) @ A5 ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Compl_disjoint
thf(fact_4150_sum__mult__of__bool__eq,axiom,
! [A: $tType,B: $tType] :
( ( semiring_1 @ A )
=> ! [A5: set @ B,F3: B > A,P2: B > $o] :
( ( finite_finite2 @ B @ A5 )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X5: B] : ( times_times @ A @ ( F3 @ X5 ) @ ( zero_neq_one_of_bool @ A @ ( P2 @ X5 ) ) )
@ A5 )
= ( groups7311177749621191930dd_sum @ B @ A @ F3 @ ( inf_inf @ ( set @ B ) @ A5 @ ( collect @ B @ P2 ) ) ) ) ) ) ).
% sum_mult_of_bool_eq
thf(fact_4151_sum__of__bool__mult__eq,axiom,
! [A: $tType,B: $tType] :
( ( semiring_1 @ A )
=> ! [A5: set @ B,P2: B > $o,F3: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X5: B] : ( times_times @ A @ ( zero_neq_one_of_bool @ A @ ( P2 @ X5 ) ) @ ( F3 @ X5 ) )
@ A5 )
= ( groups7311177749621191930dd_sum @ B @ A @ F3 @ ( inf_inf @ ( set @ B ) @ A5 @ ( collect @ B @ P2 ) ) ) ) ) ) ).
% sum_of_bool_mult_eq
thf(fact_4152_sum__of__bool__eq,axiom,
! [A: $tType,B: $tType] :
( ( semiring_1 @ A )
=> ! [A5: set @ B,P2: B > $o] :
( ( finite_finite2 @ B @ A5 )
=> ( ( finite_finite2 @ B @ A5 )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X5: B] : ( zero_neq_one_of_bool @ A @ ( P2 @ X5 ) )
@ A5 )
= ( semiring_1_of_nat @ A @ ( finite_card @ B @ ( inf_inf @ ( set @ B ) @ A5 @ ( collect @ B @ P2 ) ) ) ) ) ) ) ) ).
% sum_of_bool_eq
thf(fact_4153_Sup__inter__less__eq,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ A,B6: set @ A] : ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( inf_inf @ ( set @ A ) @ A5 @ B6 ) ) @ ( inf_inf @ A @ ( complete_Sup_Sup @ A @ A5 ) @ ( complete_Sup_Sup @ A @ B6 ) ) ) ) ).
% Sup_inter_less_eq
thf(fact_4154_disjoint__iff__not__equal,axiom,
! [A: $tType,A5: set @ A,B6: set @ A] :
( ( ( inf_inf @ ( set @ A ) @ A5 @ B6 )
= ( bot_bot @ ( set @ A ) ) )
= ( ! [X5: A] :
( ( member @ A @ X5 @ A5 )
=> ! [Y6: A] :
( ( member @ A @ Y6 @ B6 )
=> ( X5 != Y6 ) ) ) ) ) ).
% disjoint_iff_not_equal
thf(fact_4155_Int__empty__right,axiom,
! [A: $tType,A5: set @ A] :
( ( inf_inf @ ( set @ A ) @ A5 @ ( bot_bot @ ( set @ A ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Int_empty_right
thf(fact_4156_Int__empty__left,axiom,
! [A: $tType,B6: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ B6 )
= ( bot_bot @ ( set @ A ) ) ) ).
% Int_empty_left
thf(fact_4157_disjoint__iff,axiom,
! [A: $tType,A5: set @ A,B6: set @ A] :
( ( ( inf_inf @ ( set @ A ) @ A5 @ B6 )
= ( bot_bot @ ( set @ A ) ) )
= ( ! [X5: A] :
( ( member @ A @ X5 @ A5 )
=> ~ ( member @ A @ X5 @ B6 ) ) ) ) ).
% disjoint_iff
thf(fact_4158_Int__emptyI,axiom,
! [A: $tType,A5: set @ A,B6: set @ A] :
( ! [X4: A] :
( ( member @ A @ X4 @ A5 )
=> ~ ( member @ A @ X4 @ B6 ) )
=> ( ( inf_inf @ ( set @ A ) @ A5 @ B6 )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% Int_emptyI
thf(fact_4159_inf__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( semilattice_inf @ B )
=> ( ( inf_inf @ ( A > B ) )
= ( ^ [F4: A > B,G4: A > B,X5: A] : ( inf_inf @ B @ ( F4 @ X5 ) @ ( G4 @ X5 ) ) ) ) ) ).
% inf_fun_def
thf(fact_4160_inf__left__commute,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X: A,Y: A,Z: A] :
( ( inf_inf @ A @ X @ ( inf_inf @ A @ Y @ Z ) )
= ( inf_inf @ A @ Y @ ( inf_inf @ A @ X @ Z ) ) ) ) ).
% inf_left_commute
thf(fact_4161_inf_Oleft__commute,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [B2: A,A2: A,C3: A] :
( ( inf_inf @ A @ B2 @ ( inf_inf @ A @ A2 @ C3 ) )
= ( inf_inf @ A @ A2 @ ( inf_inf @ A @ B2 @ C3 ) ) ) ) ).
% inf.left_commute
thf(fact_4162_inf__commute,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( ( inf_inf @ A )
= ( ^ [X5: A,Y6: A] : ( inf_inf @ A @ Y6 @ X5 ) ) ) ) ).
% inf_commute
thf(fact_4163_inf_Ocommute,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( ( inf_inf @ A )
= ( ^ [A6: A,B5: A] : ( inf_inf @ A @ B5 @ A6 ) ) ) ) ).
% inf.commute
thf(fact_4164_inf__assoc,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X: A,Y: A,Z: A] :
( ( inf_inf @ A @ ( inf_inf @ A @ X @ Y ) @ Z )
= ( inf_inf @ A @ X @ ( inf_inf @ A @ Y @ Z ) ) ) ) ).
% inf_assoc
thf(fact_4165_inf_Oassoc,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( inf_inf @ A @ ( inf_inf @ A @ A2 @ B2 ) @ C3 )
= ( inf_inf @ A @ A2 @ ( inf_inf @ A @ B2 @ C3 ) ) ) ) ).
% inf.assoc
thf(fact_4166_inf__sup__aci_I1_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ( ( inf_inf @ A )
= ( ^ [X5: A,Y6: A] : ( inf_inf @ A @ Y6 @ X5 ) ) ) ) ).
% inf_sup_aci(1)
thf(fact_4167_inf__sup__aci_I2_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X: A,Y: A,Z: A] :
( ( inf_inf @ A @ ( inf_inf @ A @ X @ Y ) @ Z )
= ( inf_inf @ A @ X @ ( inf_inf @ A @ Y @ Z ) ) ) ) ).
% inf_sup_aci(2)
thf(fact_4168_inf__sup__aci_I3_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X: A,Y: A,Z: A] :
( ( inf_inf @ A @ X @ ( inf_inf @ A @ Y @ Z ) )
= ( inf_inf @ A @ Y @ ( inf_inf @ A @ X @ Z ) ) ) ) ).
% inf_sup_aci(3)
thf(fact_4169_inf__sup__aci_I4_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X: A,Y: A] :
( ( inf_inf @ A @ X @ ( inf_inf @ A @ X @ Y ) )
= ( inf_inf @ A @ X @ Y ) ) ) ).
% inf_sup_aci(4)
thf(fact_4170_less__infI1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A2: A,X: A,B2: A] :
( ( ord_less @ A @ A2 @ X )
=> ( ord_less @ A @ ( inf_inf @ A @ A2 @ B2 ) @ X ) ) ) ).
% less_infI1
thf(fact_4171_less__infI2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [B2: A,X: A,A2: A] :
( ( ord_less @ A @ B2 @ X )
=> ( ord_less @ A @ ( inf_inf @ A @ A2 @ B2 ) @ X ) ) ) ).
% less_infI2
thf(fact_4172_inf_Oabsorb3,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( inf_inf @ A @ A2 @ B2 )
= A2 ) ) ) ).
% inf.absorb3
thf(fact_4173_inf_Oabsorb4,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [B2: A,A2: A] :
( ( ord_less @ A @ B2 @ A2 )
=> ( ( inf_inf @ A @ A2 @ B2 )
= B2 ) ) ) ).
% inf.absorb4
thf(fact_4174_inf_Ostrict__boundedE,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( ord_less @ A @ A2 @ ( inf_inf @ A @ B2 @ C3 ) )
=> ~ ( ( ord_less @ A @ A2 @ B2 )
=> ~ ( ord_less @ A @ A2 @ C3 ) ) ) ) ).
% inf.strict_boundedE
thf(fact_4175_inf_Ostrict__order__iff,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( ( ord_less @ A )
= ( ^ [A6: A,B5: A] :
( ( A6
= ( inf_inf @ A @ A6 @ B5 ) )
& ( A6 != B5 ) ) ) ) ) ).
% inf.strict_order_iff
thf(fact_4176_inf_Ostrict__coboundedI1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A2: A,C3: A,B2: A] :
( ( ord_less @ A @ A2 @ C3 )
=> ( ord_less @ A @ ( inf_inf @ A @ A2 @ B2 ) @ C3 ) ) ) ).
% inf.strict_coboundedI1
thf(fact_4177_inf_Ostrict__coboundedI2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [B2: A,C3: A,A2: A] :
( ( ord_less @ A @ B2 @ C3 )
=> ( ord_less @ A @ ( inf_inf @ A @ A2 @ B2 ) @ C3 ) ) ) ).
% inf.strict_coboundedI2
thf(fact_4178_inf_OcoboundedI2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [B2: A,C3: A,A2: A] :
( ( ord_less_eq @ A @ B2 @ C3 )
=> ( ord_less_eq @ A @ ( inf_inf @ A @ A2 @ B2 ) @ C3 ) ) ) ).
% inf.coboundedI2
thf(fact_4179_inf_OcoboundedI1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A2: A,C3: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ C3 )
=> ( ord_less_eq @ A @ ( inf_inf @ A @ A2 @ B2 ) @ C3 ) ) ) ).
% inf.coboundedI1
thf(fact_4180_inf_Oabsorb__iff2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B5: A,A6: A] :
( ( inf_inf @ A @ A6 @ B5 )
= B5 ) ) ) ) ).
% inf.absorb_iff2
thf(fact_4181_inf_Oabsorb__iff1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A6: A,B5: A] :
( ( inf_inf @ A @ A6 @ B5 )
= A6 ) ) ) ) ).
% inf.absorb_iff1
thf(fact_4182_inf_Ocobounded2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A2: A,B2: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ A2 @ B2 ) @ B2 ) ) ).
% inf.cobounded2
thf(fact_4183_inf_Ocobounded1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A2: A,B2: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ A2 @ B2 ) @ A2 ) ) ).
% inf.cobounded1
thf(fact_4184_inf_Oorder__iff,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A6: A,B5: A] :
( A6
= ( inf_inf @ A @ A6 @ B5 ) ) ) ) ) ).
% inf.order_iff
thf(fact_4185_inf__greatest,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X: A,Y: A,Z: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ X @ Z )
=> ( ord_less_eq @ A @ X @ ( inf_inf @ A @ Y @ Z ) ) ) ) ) ).
% inf_greatest
thf(fact_4186_inf_OboundedI,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ A2 @ C3 )
=> ( ord_less_eq @ A @ A2 @ ( inf_inf @ A @ B2 @ C3 ) ) ) ) ) ).
% inf.boundedI
thf(fact_4187_inf_OboundedE,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( ord_less_eq @ A @ A2 @ ( inf_inf @ A @ B2 @ C3 ) )
=> ~ ( ( ord_less_eq @ A @ A2 @ B2 )
=> ~ ( ord_less_eq @ A @ A2 @ C3 ) ) ) ) ).
% inf.boundedE
thf(fact_4188_inf__absorb2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ Y @ X )
=> ( ( inf_inf @ A @ X @ Y )
= Y ) ) ) ).
% inf_absorb2
thf(fact_4189_inf__absorb1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( inf_inf @ A @ X @ Y )
= X ) ) ) ).
% inf_absorb1
thf(fact_4190_inf_Oabsorb2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [B2: A,A2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( inf_inf @ A @ A2 @ B2 )
= B2 ) ) ) ).
% inf.absorb2
thf(fact_4191_inf_Oabsorb1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( inf_inf @ A @ A2 @ B2 )
= A2 ) ) ) ).
% inf.absorb1
thf(fact_4192_le__iff__inf,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [X5: A,Y6: A] :
( ( inf_inf @ A @ X5 @ Y6 )
= X5 ) ) ) ) ).
% le_iff_inf
thf(fact_4193_inf__unique,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [F3: A > A > A,X: A,Y: A] :
( ! [X4: A,Y3: A] : ( ord_less_eq @ A @ ( F3 @ X4 @ Y3 ) @ X4 )
=> ( ! [X4: A,Y3: A] : ( ord_less_eq @ A @ ( F3 @ X4 @ Y3 ) @ Y3 )
=> ( ! [X4: A,Y3: A,Z3: A] :
( ( ord_less_eq @ A @ X4 @ Y3 )
=> ( ( ord_less_eq @ A @ X4 @ Z3 )
=> ( ord_less_eq @ A @ X4 @ ( F3 @ Y3 @ Z3 ) ) ) )
=> ( ( inf_inf @ A @ X @ Y )
= ( F3 @ X @ Y ) ) ) ) ) ) ).
% inf_unique
thf(fact_4194_inf_OorderI,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A2: A,B2: A] :
( ( A2
= ( inf_inf @ A @ A2 @ B2 ) )
=> ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% inf.orderI
thf(fact_4195_inf_OorderE,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( A2
= ( inf_inf @ A @ A2 @ B2 ) ) ) ) ).
% inf.orderE
thf(fact_4196_le__infI2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [B2: A,X: A,A2: A] :
( ( ord_less_eq @ A @ B2 @ X )
=> ( ord_less_eq @ A @ ( inf_inf @ A @ A2 @ B2 ) @ X ) ) ) ).
% le_infI2
thf(fact_4197_le__infI1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A2: A,X: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ X )
=> ( ord_less_eq @ A @ ( inf_inf @ A @ A2 @ B2 ) @ X ) ) ) ).
% le_infI1
thf(fact_4198_inf__mono,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A2: A,C3: A,B2: A,D3: A] :
( ( ord_less_eq @ A @ A2 @ C3 )
=> ( ( ord_less_eq @ A @ B2 @ D3 )
=> ( ord_less_eq @ A @ ( inf_inf @ A @ A2 @ B2 ) @ ( inf_inf @ A @ C3 @ D3 ) ) ) ) ) ).
% inf_mono
thf(fact_4199_le__infI,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X: A,A2: A,B2: A] :
( ( ord_less_eq @ A @ X @ A2 )
=> ( ( ord_less_eq @ A @ X @ B2 )
=> ( ord_less_eq @ A @ X @ ( inf_inf @ A @ A2 @ B2 ) ) ) ) ) ).
% le_infI
thf(fact_4200_le__infE,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X: A,A2: A,B2: A] :
( ( ord_less_eq @ A @ X @ ( inf_inf @ A @ A2 @ B2 ) )
=> ~ ( ( ord_less_eq @ A @ X @ A2 )
=> ~ ( ord_less_eq @ A @ X @ B2 ) ) ) ) ).
% le_infE
thf(fact_4201_inf__le2,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X: A,Y: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ X @ Y ) @ Y ) ) ).
% inf_le2
thf(fact_4202_inf__le1,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X: A,Y: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ X @ Y ) @ X ) ) ).
% inf_le1
thf(fact_4203_inf__sup__ord_I1_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X: A,Y: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ X @ Y ) @ X ) ) ).
% inf_sup_ord(1)
thf(fact_4204_inf__sup__ord_I2_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X: A,Y: A] : ( ord_less_eq @ A @ ( inf_inf @ A @ X @ Y ) @ Y ) ) ).
% inf_sup_ord(2)
thf(fact_4205_inf__cancel__left2,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,A2: A,B2: A] :
( ( inf_inf @ A @ ( inf_inf @ A @ ( uminus_uminus @ A @ X ) @ A2 ) @ ( inf_inf @ A @ X @ B2 ) )
= ( bot_bot @ A ) ) ) ).
% inf_cancel_left2
thf(fact_4206_inf__cancel__left1,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,A2: A,B2: A] :
( ( inf_inf @ A @ ( inf_inf @ A @ X @ A2 ) @ ( inf_inf @ A @ ( uminus_uminus @ A @ X ) @ B2 ) )
= ( bot_bot @ A ) ) ) ).
% inf_cancel_left1
thf(fact_4207_Sup__inf__eq__bot__iff,axiom,
! [A: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [B6: set @ A,A2: A] :
( ( ( inf_inf @ A @ ( complete_Sup_Sup @ A @ B6 ) @ A2 )
= ( bot_bot @ A ) )
= ( ! [X5: A] :
( ( member @ A @ X5 @ B6 )
=> ( ( inf_inf @ A @ X5 @ A2 )
= ( bot_bot @ A ) ) ) ) ) ) ).
% Sup_inf_eq_bot_iff
thf(fact_4208_Int__Diff__disjoint,axiom,
! [A: $tType,A5: set @ A,B6: set @ A] :
( ( inf_inf @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A5 @ B6 ) @ ( minus_minus @ ( set @ A ) @ A5 @ B6 ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Int_Diff_disjoint
thf(fact_4209_Diff__triv,axiom,
! [A: $tType,A5: set @ A,B6: set @ A] :
( ( ( inf_inf @ ( set @ A ) @ A5 @ B6 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( minus_minus @ ( set @ A ) @ A5 @ B6 )
= A5 ) ) ).
% Diff_triv
thf(fact_4210_Union__disjoint,axiom,
! [A: $tType,C5: set @ ( set @ A ),A5: set @ A] :
( ( ( inf_inf @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ C5 ) @ A5 )
= ( bot_bot @ ( set @ A ) ) )
= ( ! [X5: set @ A] :
( ( member @ ( set @ A ) @ X5 @ C5 )
=> ( ( inf_inf @ ( set @ A ) @ X5 @ A5 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% Union_disjoint
thf(fact_4211_ivl__disj__int__two_I3_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,M2: A,U: A] :
( ( inf_inf @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ L @ M2 ) @ ( set_or7035219750837199246ssThan @ A @ M2 @ U ) )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% ivl_disj_int_two(3)
thf(fact_4212_inf__shunt,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y: A] :
( ( ( inf_inf @ A @ X @ Y )
= ( bot_bot @ A ) )
= ( ord_less_eq @ A @ X @ ( uminus_uminus @ A @ Y ) ) ) ) ).
% inf_shunt
thf(fact_4213_finite__Inf__in,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X4: A,Y3: A] :
( ( member @ A @ X4 @ A5 )
=> ( ( member @ A @ Y3 @ A5 )
=> ( member @ A @ ( inf_inf @ A @ X4 @ Y3 ) @ A5 ) ) )
=> ( member @ A @ ( complete_Inf_Inf @ A @ A5 ) @ A5 ) ) ) ) ) ).
% finite_Inf_in
thf(fact_4214_ivl__disj__int__two_I7_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,M2: A,U: A] :
( ( inf_inf @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ L @ M2 ) @ ( set_or1337092689740270186AtMost @ A @ M2 @ U ) )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% ivl_disj_int_two(7)
thf(fact_4215_ivl__disj__int__one_I4_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,U: A] :
( ( inf_inf @ ( set @ A ) @ ( set_ord_lessThan @ A @ L ) @ ( set_or1337092689740270186AtMost @ A @ L @ U ) )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% ivl_disj_int_one(4)
thf(fact_4216_ivl__disj__int__one_I2_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,U: A] :
( ( inf_inf @ ( set @ A ) @ ( set_ord_lessThan @ A @ L ) @ ( set_or7035219750837199246ssThan @ A @ L @ U ) )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% ivl_disj_int_one(2)
thf(fact_4217_disjoint__eq__subset__Compl,axiom,
! [A: $tType,A5: set @ A,B6: set @ A] :
( ( ( inf_inf @ ( set @ A ) @ A5 @ B6 )
= ( bot_bot @ ( set @ A ) ) )
= ( ord_less_eq @ ( set @ A ) @ A5 @ ( uminus_uminus @ ( set @ A ) @ B6 ) ) ) ).
% disjoint_eq_subset_Compl
thf(fact_4218_ivl__disj__int__two_I5_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,M2: A,U: A] :
( ( inf_inf @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ L @ M2 ) @ ( set_or1337092689740270186AtMost @ A @ M2 @ U ) )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% ivl_disj_int_two(5)
thf(fact_4219_ivl__disj__int__two_I4_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,M2: A,U: A] :
( ( inf_inf @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ L @ M2 ) @ ( set_or5935395276787703475ssThan @ A @ M2 @ U ) )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% ivl_disj_int_two(4)
thf(fact_4220_ivl__disj__int__two_I1_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,M2: A,U: A] :
( ( inf_inf @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ L @ M2 ) @ ( set_or7035219750837199246ssThan @ A @ M2 @ U ) )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% ivl_disj_int_two(1)
thf(fact_4221_ivl__disj__int__one_I1_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,U: A] :
( ( inf_inf @ ( set @ A ) @ ( set_ord_atMost @ A @ L ) @ ( set_or5935395276787703475ssThan @ A @ L @ U ) )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% ivl_disj_int_one(1)
thf(fact_4222_relpowp__0__I,axiom,
! [A: $tType,P2: A > A > $o,X: A] : ( compow @ ( A > A > $o ) @ ( zero_zero @ nat ) @ P2 @ X @ X ) ).
% relpowp_0_I
thf(fact_4223_relpowp__0__E,axiom,
! [A: $tType,P2: A > A > $o,X: A,Y: A] :
( ( compow @ ( A > A > $o ) @ ( zero_zero @ nat ) @ P2 @ X @ Y )
=> ( X = Y ) ) ).
% relpowp_0_E
thf(fact_4224_relpowp_Osimps_I1_J,axiom,
! [A: $tType,R: A > A > $o] :
( ( compow @ ( A > A > $o ) @ ( zero_zero @ nat ) @ R )
= ( ^ [Y5: A,Z2: A] : Y5 = Z2 ) ) ).
% relpowp.simps(1)
thf(fact_4225_insert__partition,axiom,
! [A: $tType,X: set @ A,F5: set @ ( set @ A )] :
( ~ ( member @ ( set @ A ) @ X @ F5 )
=> ( ! [X4: set @ A] :
( ( member @ ( set @ A ) @ X4 @ ( insert @ ( set @ A ) @ X @ F5 ) )
=> ! [Xa3: set @ A] :
( ( member @ ( set @ A ) @ Xa3 @ ( insert @ ( set @ A ) @ X @ F5 ) )
=> ( ( X4 != Xa3 )
=> ( ( inf_inf @ ( set @ A ) @ X4 @ Xa3 )
= ( bot_bot @ ( set @ A ) ) ) ) ) )
=> ( ( inf_inf @ ( set @ A ) @ X @ ( complete_Sup_Sup @ ( set @ A ) @ F5 ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% insert_partition
thf(fact_4226_sum_Ointer__restrict,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A5: set @ B,G3: B > A,B6: set @ B] :
( ( finite_finite2 @ B @ A5 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G3 @ ( inf_inf @ ( set @ B ) @ A5 @ B6 ) )
= ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X5: B] : ( if @ A @ ( member @ B @ X5 @ B6 ) @ ( G3 @ X5 ) @ ( zero_zero @ A ) )
@ A5 ) ) ) ) ).
% sum.inter_restrict
thf(fact_4227_prod_Ointer__restrict,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A5: set @ B,G3: B > A,B6: set @ B] :
( ( finite_finite2 @ B @ A5 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G3 @ ( inf_inf @ ( set @ B ) @ A5 @ B6 ) )
= ( groups7121269368397514597t_prod @ B @ A
@ ^ [X5: B] : ( if @ A @ ( member @ B @ X5 @ B6 ) @ ( G3 @ X5 ) @ ( one_one @ A ) )
@ A5 ) ) ) ) ).
% prod.inter_restrict
thf(fact_4228_sum_Omono__neutral__cong,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [T4: set @ B,S2: set @ B,H2: B > A,G3: B > A] :
( ( finite_finite2 @ B @ T4 )
=> ( ( finite_finite2 @ B @ S2 )
=> ( ! [I3: B] :
( ( member @ B @ I3 @ ( minus_minus @ ( set @ B ) @ T4 @ S2 ) )
=> ( ( H2 @ I3 )
= ( zero_zero @ A ) ) )
=> ( ! [I3: B] :
( ( member @ B @ I3 @ ( minus_minus @ ( set @ B ) @ S2 @ T4 ) )
=> ( ( G3 @ I3 )
= ( zero_zero @ A ) ) )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ ( inf_inf @ ( set @ B ) @ S2 @ T4 ) )
=> ( ( G3 @ X4 )
= ( H2 @ X4 ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G3 @ S2 )
= ( groups7311177749621191930dd_sum @ B @ A @ H2 @ T4 ) ) ) ) ) ) ) ) ).
% sum.mono_neutral_cong
thf(fact_4229_Iio__Int__singleton,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,K2: A] :
( ( ( ord_less @ A @ X @ K2 )
=> ( ( inf_inf @ ( set @ A ) @ ( set_ord_lessThan @ A @ K2 ) @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
& ( ~ ( ord_less @ A @ X @ K2 )
=> ( ( inf_inf @ ( set @ A ) @ ( set_ord_lessThan @ A @ K2 ) @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% Iio_Int_singleton
thf(fact_4230_sum_OInt__Diff,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A5: set @ B,G3: B > A,B6: set @ B] :
( ( finite_finite2 @ B @ A5 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G3 @ A5 )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G3 @ ( inf_inf @ ( set @ B ) @ A5 @ B6 ) ) @ ( groups7311177749621191930dd_sum @ B @ A @ G3 @ ( minus_minus @ ( set @ B ) @ A5 @ B6 ) ) ) ) ) ) ).
% sum.Int_Diff
thf(fact_4231_prod_OInt__Diff,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A5: set @ B,G3: B > A,B6: set @ B] :
( ( finite_finite2 @ B @ A5 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G3 @ A5 )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G3 @ ( inf_inf @ ( set @ B ) @ A5 @ B6 ) ) @ ( groups7121269368397514597t_prod @ B @ A @ G3 @ ( minus_minus @ ( set @ B ) @ A5 @ B6 ) ) ) ) ) ) ).
% prod.Int_Diff
thf(fact_4232_prod_Omono__neutral__cong,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [T4: set @ B,S2: set @ B,H2: B > A,G3: B > A] :
( ( finite_finite2 @ B @ T4 )
=> ( ( finite_finite2 @ B @ S2 )
=> ( ! [I3: B] :
( ( member @ B @ I3 @ ( minus_minus @ ( set @ B ) @ T4 @ S2 ) )
=> ( ( H2 @ I3 )
= ( one_one @ A ) ) )
=> ( ! [I3: B] :
( ( member @ B @ I3 @ ( minus_minus @ ( set @ B ) @ S2 @ T4 ) )
=> ( ( G3 @ I3 )
= ( one_one @ A ) ) )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ ( inf_inf @ ( set @ B ) @ S2 @ T4 ) )
=> ( ( G3 @ X4 )
= ( H2 @ X4 ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G3 @ S2 )
= ( groups7121269368397514597t_prod @ B @ A @ H2 @ T4 ) ) ) ) ) ) ) ) ).
% prod.mono_neutral_cong
thf(fact_4233_card__Diff__subset__Int,axiom,
! [A: $tType,A5: set @ A,B6: set @ A] :
( ( finite_finite2 @ A @ ( inf_inf @ ( set @ A ) @ A5 @ B6 ) )
=> ( ( finite_card @ A @ ( minus_minus @ ( set @ A ) @ A5 @ B6 ) )
= ( minus_minus @ nat @ ( finite_card @ A @ A5 ) @ ( finite_card @ A @ ( inf_inf @ ( set @ A ) @ A5 @ B6 ) ) ) ) ) ).
% card_Diff_subset_Int
thf(fact_4234_sum_OIf__cases,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A5: set @ B,P2: B > $o,H2: B > A,G3: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X5: B] : ( if @ A @ ( P2 @ X5 ) @ ( H2 @ X5 ) @ ( G3 @ X5 ) )
@ A5 )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ H2 @ ( inf_inf @ ( set @ B ) @ A5 @ ( collect @ B @ P2 ) ) ) @ ( groups7311177749621191930dd_sum @ B @ A @ G3 @ ( inf_inf @ ( set @ B ) @ A5 @ ( uminus_uminus @ ( set @ B ) @ ( collect @ B @ P2 ) ) ) ) ) ) ) ) ).
% sum.If_cases
thf(fact_4235_prod_OIf__cases,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A5: set @ B,P2: B > $o,H2: B > A,G3: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [X5: B] : ( if @ A @ ( P2 @ X5 ) @ ( H2 @ X5 ) @ ( G3 @ X5 ) )
@ A5 )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ H2 @ ( inf_inf @ ( set @ B ) @ A5 @ ( collect @ B @ P2 ) ) ) @ ( groups7121269368397514597t_prod @ B @ A @ G3 @ ( inf_inf @ ( set @ B ) @ A5 @ ( uminus_uminus @ ( set @ B ) @ ( collect @ B @ P2 ) ) ) ) ) ) ) ) ).
% prod.If_cases
thf(fact_4236_dvd__partition,axiom,
! [A: $tType,C5: set @ ( set @ A ),K2: nat] :
( ( finite_finite2 @ A @ ( complete_Sup_Sup @ ( set @ A ) @ C5 ) )
=> ( ! [X4: set @ A] :
( ( member @ ( set @ A ) @ X4 @ C5 )
=> ( dvd_dvd @ nat @ K2 @ ( finite_card @ A @ X4 ) ) )
=> ( ! [X4: set @ A] :
( ( member @ ( set @ A ) @ X4 @ C5 )
=> ! [Xa3: set @ A] :
( ( member @ ( set @ A ) @ Xa3 @ C5 )
=> ( ( X4 != Xa3 )
=> ( ( inf_inf @ ( set @ A ) @ X4 @ Xa3 )
= ( bot_bot @ ( set @ A ) ) ) ) ) )
=> ( dvd_dvd @ nat @ K2 @ ( finite_card @ A @ ( complete_Sup_Sup @ ( set @ A ) @ C5 ) ) ) ) ) ) ).
% dvd_partition
thf(fact_4237_sum__div__partition,axiom,
! [B: $tType,A: $tType] :
( ( euclid4440199948858584721cancel @ A )
=> ! [A5: set @ B,F3: B > A,B2: A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( divide_divide @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ A5 ) @ B2 )
= ( plus_plus @ A
@ ( groups7311177749621191930dd_sum @ B @ A
@ ^ [A6: B] : ( divide_divide @ A @ ( F3 @ A6 ) @ B2 )
@ ( inf_inf @ ( set @ B ) @ A5
@ ( collect @ B
@ ^ [A6: B] : ( dvd_dvd @ A @ B2 @ ( F3 @ A6 ) ) ) ) )
@ ( divide_divide @ A
@ ( groups7311177749621191930dd_sum @ B @ A @ F3
@ ( inf_inf @ ( set @ B ) @ A5
@ ( collect @ B
@ ^ [A6: B] :
~ ( dvd_dvd @ A @ B2 @ ( F3 @ A6 ) ) ) ) )
@ B2 ) ) ) ) ) ).
% sum_div_partition
thf(fact_4238_distinct__concat,axiom,
! [A: $tType,Xs: list @ ( list @ A )] :
( ( distinct @ ( list @ A ) @ Xs )
=> ( ! [Ys4: list @ A] :
( ( member @ ( list @ A ) @ Ys4 @ ( set2 @ ( list @ A ) @ Xs ) )
=> ( distinct @ A @ Ys4 ) )
=> ( ! [Ys4: list @ A,Zs2: list @ A] :
( ( member @ ( list @ A ) @ Ys4 @ ( set2 @ ( list @ A ) @ Xs ) )
=> ( ( member @ ( list @ A ) @ Zs2 @ ( set2 @ ( list @ A ) @ Xs ) )
=> ( ( Ys4 != Zs2 )
=> ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ Ys4 ) @ ( set2 @ A @ Zs2 ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) )
=> ( distinct @ A @ ( concat @ A @ Xs ) ) ) ) ) ).
% distinct_concat
thf(fact_4239_relpowp__E,axiom,
! [A: $tType,N: nat,P2: A > A > $o,X: A,Z: A] :
( ( compow @ ( A > A > $o ) @ N @ P2 @ X @ Z )
=> ( ( ( N
= ( zero_zero @ nat ) )
=> ( X != Z ) )
=> ~ ! [Y3: A,M: nat] :
( ( N
= ( suc @ M ) )
=> ( ( compow @ ( A > A > $o ) @ M @ P2 @ X @ Y3 )
=> ~ ( P2 @ Y3 @ Z ) ) ) ) ) ).
% relpowp_E
thf(fact_4240_relpowp__E2,axiom,
! [A: $tType,N: nat,P2: A > A > $o,X: A,Z: A] :
( ( compow @ ( A > A > $o ) @ N @ P2 @ X @ Z )
=> ( ( ( N
= ( zero_zero @ nat ) )
=> ( X != Z ) )
=> ~ ! [Y3: A,M: nat] :
( ( N
= ( suc @ M ) )
=> ( ( P2 @ X @ Y3 )
=> ~ ( compow @ ( A > A > $o ) @ M @ P2 @ Y3 @ Z ) ) ) ) ) ).
% relpowp_E2
thf(fact_4241_card__disjoint__shuffles,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ Ys ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_card @ ( list @ A ) @ ( shuffles @ A @ Xs @ Ys ) )
= ( binomial @ ( plus_plus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ A ) @ Ys ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ).
% card_disjoint_shuffles
thf(fact_4242_totally__bounded__Union,axiom,
! [A: $tType] :
( ( topolo7287701948861334536_space @ A )
=> ! [M5: set @ ( set @ A )] :
( ( finite_finite2 @ ( set @ A ) @ M5 )
=> ( ! [S6: set @ A] :
( ( member @ ( set @ A ) @ S6 @ M5 )
=> ( topolo6688025880775521714ounded @ A @ S6 ) )
=> ( topolo6688025880775521714ounded @ A @ ( complete_Sup_Sup @ ( set @ A ) @ M5 ) ) ) ) ) ).
% totally_bounded_Union
thf(fact_4243_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_4244_times__int_Oabs__eq,axiom,
! [Xa2: product_prod @ nat @ nat,X: product_prod @ nat @ nat] :
( ( times_times @ int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X ) )
= ( abs_Integ
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X5: nat,Y6: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V5: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ X5 @ U2 ) @ ( times_times @ nat @ Y6 @ V5 ) ) @ ( plus_plus @ nat @ ( times_times @ nat @ X5 @ V5 ) @ ( times_times @ nat @ Y6 @ U2 ) ) ) )
@ Xa2
@ X ) ) ) ).
% times_int.abs_eq
thf(fact_4245_removeAll__id,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ~ ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ( removeAll @ A @ X @ Xs )
= Xs ) ) ).
% removeAll_id
thf(fact_4246_totally__bounded__empty,axiom,
! [A: $tType] :
( ( topolo7287701948861334536_space @ A )
=> ( topolo6688025880775521714ounded @ A @ ( bot_bot @ ( set @ A ) ) ) ) ).
% totally_bounded_empty
thf(fact_4247_finite__shuffles,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] : ( finite_finite2 @ ( list @ A ) @ ( shuffles @ A @ Xs @ Ys ) ) ).
% finite_shuffles
thf(fact_4248_inf__Int__eq2,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S2: set @ ( product_prod @ A @ B )] :
( ( inf_inf @ ( A > B > $o )
@ ^ [X5: A,Y6: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X5 @ Y6 ) @ R )
@ ^ [X5: A,Y6: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X5 @ Y6 ) @ S2 ) )
= ( ^ [X5: A,Y6: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X5 @ Y6 ) @ ( inf_inf @ ( set @ ( product_prod @ A @ B ) ) @ R @ S2 ) ) ) ) ).
% inf_Int_eq2
thf(fact_4249_int_Oabs__induct,axiom,
! [P2: int > $o,X: int] :
( ! [Y3: product_prod @ nat @ nat] : ( P2 @ ( abs_Integ @ Y3 ) )
=> ( P2 @ X ) ) ).
% int.abs_induct
thf(fact_4250_shuffles__commutes,axiom,
! [A: $tType] :
( ( shuffles @ A )
= ( ^ [Xs3: list @ A,Ys3: list @ A] : ( shuffles @ A @ Ys3 @ Xs3 ) ) ) ).
% shuffles_commutes
thf(fact_4251_eq__Abs__Integ,axiom,
! [Z: int] :
~ ! [X4: nat,Y3: nat] :
( Z
!= ( abs_Integ @ ( product_Pair @ nat @ nat @ X4 @ Y3 ) ) ) ).
% eq_Abs_Integ
thf(fact_4252_distinct__removeAll,axiom,
! [A: $tType,Xs: list @ A,X: A] :
( ( distinct @ A @ Xs )
=> ( distinct @ A @ ( removeAll @ A @ X @ Xs ) ) ) ).
% distinct_removeAll
thf(fact_4253_length__shuffles,axiom,
! [A: $tType,Zs: list @ A,Xs: list @ A,Ys: list @ A] :
( ( member @ ( list @ A ) @ Zs @ ( shuffles @ A @ Xs @ Ys ) )
=> ( ( size_size @ ( list @ A ) @ Zs )
= ( plus_plus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ A ) @ Ys ) ) ) ) ).
% length_shuffles
thf(fact_4254_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_4255_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_4256_distinct__remove1__removeAll,axiom,
! [A: $tType,Xs: list @ A,X: A] :
( ( distinct @ A @ Xs )
=> ( ( remove1 @ A @ X @ Xs )
= ( removeAll @ A @ X @ Xs ) ) ) ).
% distinct_remove1_removeAll
thf(fact_4257_zero__int__def,axiom,
( ( zero_zero @ int )
= ( abs_Integ @ ( product_Pair @ nat @ nat @ ( zero_zero @ nat ) @ ( zero_zero @ nat ) ) ) ) ).
% zero_int_def
thf(fact_4258_int__def,axiom,
( ( semiring_1_of_nat @ int )
= ( ^ [N2: nat] : ( abs_Integ @ ( product_Pair @ nat @ nat @ N2 @ ( zero_zero @ nat ) ) ) ) ) ).
% int_def
thf(fact_4259_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_4260_distinct__disjoint__shuffles,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Zs: list @ A] :
( ( distinct @ A @ Xs )
=> ( ( distinct @ A @ Ys )
=> ( ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ Ys ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( member @ ( list @ A ) @ Zs @ ( shuffles @ A @ Xs @ Ys ) )
=> ( distinct @ A @ Zs ) ) ) ) ) ).
% distinct_disjoint_shuffles
thf(fact_4261_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 )
@ ^ [X5: nat,Y6: nat] : ( product_Pair @ nat @ nat @ Y6 @ X5 )
@ X ) ) ) ).
% uminus_int.abs_eq
thf(fact_4262_one__int__def,axiom,
( ( one_one @ int )
= ( abs_Integ @ ( product_Pair @ nat @ nat @ ( one_one @ nat ) @ ( zero_zero @ nat ) ) ) ) ).
% one_int_def
thf(fact_4263_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
@ ^ [I4: nat,J3: nat] : ( minus_minus @ A @ ( semiring_1_of_nat @ A @ I4 ) @ ( semiring_1_of_nat @ A @ J3 ) )
@ X ) ) ) ).
% of_int.abs_eq
thf(fact_4264_less__int_Oabs__eq,axiom,
! [Xa2: product_prod @ nat @ nat,X: product_prod @ nat @ nat] :
( ( ord_less @ int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X ) )
= ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [X5: nat,Y6: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U2: nat,V5: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ X5 @ V5 ) @ ( plus_plus @ nat @ U2 @ Y6 ) ) )
@ Xa2
@ X ) ) ).
% less_int.abs_eq
thf(fact_4265_less__eq__int_Oabs__eq,axiom,
! [Xa2: product_prod @ nat @ nat,X: product_prod @ nat @ nat] :
( ( ord_less_eq @ int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X ) )
= ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [X5: nat,Y6: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U2: nat,V5: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ X5 @ V5 ) @ ( plus_plus @ nat @ U2 @ Y6 ) ) )
@ Xa2
@ X ) ) ).
% less_eq_int.abs_eq
thf(fact_4266_plus__int_Oabs__eq,axiom,
! [Xa2: product_prod @ nat @ nat,X: product_prod @ nat @ nat] :
( ( plus_plus @ int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X ) )
= ( abs_Integ
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X5: nat,Y6: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V5: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X5 @ U2 ) @ ( plus_plus @ nat @ Y6 @ V5 ) ) )
@ Xa2
@ X ) ) ) ).
% plus_int.abs_eq
thf(fact_4267_minus__int_Oabs__eq,axiom,
! [Xa2: product_prod @ nat @ nat,X: product_prod @ nat @ nat] :
( ( minus_minus @ int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X ) )
= ( abs_Integ
@ ( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > ( product_prod @ nat @ nat ) )
@ ^ [X5: nat,Y6: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V5: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X5 @ V5 ) @ ( plus_plus @ nat @ Y6 @ U2 ) ) )
@ Xa2
@ X ) ) ) ).
% minus_int.abs_eq
thf(fact_4268_distinct__concat__iff,axiom,
! [A: $tType,Xs: list @ ( list @ A )] :
( ( distinct @ A @ ( concat @ A @ Xs ) )
= ( ( distinct @ ( list @ A ) @ ( removeAll @ ( list @ A ) @ ( nil @ A ) @ Xs ) )
& ! [Ys3: list @ A] :
( ( member @ ( list @ A ) @ Ys3 @ ( set2 @ ( list @ A ) @ Xs ) )
=> ( distinct @ A @ Ys3 ) )
& ! [Ys3: list @ A,Zs3: list @ A] :
( ( ( member @ ( list @ A ) @ Ys3 @ ( set2 @ ( list @ A ) @ Xs ) )
& ( member @ ( list @ A ) @ Zs3 @ ( set2 @ ( list @ A ) @ Xs ) )
& ( Ys3 != Zs3 ) )
=> ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ Ys3 ) @ ( set2 @ A @ Zs3 ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% distinct_concat_iff
thf(fact_4269_distinct__product__lists,axiom,
! [A: $tType,Xss: list @ ( list @ A )] :
( ! [X4: list @ A] :
( ( member @ ( list @ A ) @ X4 @ ( set2 @ ( list @ A ) @ Xss ) )
=> ( distinct @ A @ X4 ) )
=> ( distinct @ ( list @ A ) @ ( product_lists @ A @ Xss ) ) ) ).
% distinct_product_lists
thf(fact_4270_max__rpair__set,axiom,
fun_reduction_pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) ) @ ( set @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) ) @ fun_max_strict @ fun_max_weak ) ).
% max_rpair_set
thf(fact_4271_min__rpair__set,axiom,
fun_reduction_pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( product_Pair @ ( set @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) ) @ ( set @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) ) @ fun_min_strict @ fun_min_weak ) ).
% min_rpair_set
thf(fact_4272_list__update__nonempty,axiom,
! [A: $tType,Xs: list @ A,K2: nat,X: A] :
( ( ( list_update @ A @ Xs @ K2 @ X )
= ( nil @ A ) )
= ( Xs
= ( nil @ A ) ) ) ).
% list_update_nonempty
thf(fact_4273_concat__replicate__trivial,axiom,
! [A: $tType,I2: nat] :
( ( concat @ A @ ( replicate @ ( list @ A ) @ I2 @ ( nil @ A ) ) )
= ( nil @ A ) ) ).
% concat_replicate_trivial
thf(fact_4274_Nil__in__shuffles,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( member @ ( list @ A ) @ ( nil @ A ) @ ( shuffles @ A @ Xs @ Ys ) )
= ( ( Xs
= ( nil @ A ) )
& ( Ys
= ( nil @ A ) ) ) ) ).
% Nil_in_shuffles
thf(fact_4275_enumerate__simps_I1_J,axiom,
! [A: $tType,N: nat] :
( ( enumerate @ A @ N @ ( nil @ A ) )
= ( nil @ ( product_prod @ nat @ A ) ) ) ).
% enumerate_simps(1)
thf(fact_4276_rotate1__is__Nil__conv,axiom,
! [A: $tType,Xs: list @ A] :
( ( ( rotate1 @ A @ Xs )
= ( nil @ A ) )
= ( Xs
= ( nil @ A ) ) ) ).
% rotate1_is_Nil_conv
thf(fact_4277_set__empty2,axiom,
! [A: $tType,Xs: list @ A] :
( ( ( bot_bot @ ( set @ A ) )
= ( set2 @ A @ Xs ) )
= ( Xs
= ( nil @ A ) ) ) ).
% set_empty2
thf(fact_4278_set__empty,axiom,
! [A: $tType,Xs: list @ A] :
( ( ( set2 @ A @ Xs )
= ( bot_bot @ ( set @ A ) ) )
= ( Xs
= ( nil @ A ) ) ) ).
% set_empty
thf(fact_4279_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_4280_empty__replicate,axiom,
! [A: $tType,N: nat,X: A] :
( ( ( nil @ A )
= ( replicate @ A @ N @ X ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% empty_replicate
thf(fact_4281_replicate__empty,axiom,
! [A: $tType,N: nat,X: A] :
( ( ( replicate @ A @ N @ X )
= ( nil @ A ) )
= ( N
= ( zero_zero @ nat ) ) ) ).
% replicate_empty
thf(fact_4282_horner__sum__simps_I1_J,axiom,
! [B: $tType,A: $tType] :
( ( comm_semiring_0 @ A )
=> ! [F3: B > A,A2: A] :
( ( groups4207007520872428315er_sum @ B @ A @ F3 @ A2 @ ( nil @ B ) )
= ( zero_zero @ A ) ) ) ).
% horner_sum_simps(1)
thf(fact_4283_Nil__eq__concat__conv,axiom,
! [A: $tType,Xss: list @ ( list @ A )] :
( ( ( nil @ A )
= ( concat @ A @ Xss ) )
= ( ! [X5: list @ A] :
( ( member @ ( list @ A ) @ X5 @ ( set2 @ ( list @ A ) @ Xss ) )
=> ( X5
= ( nil @ A ) ) ) ) ) ).
% Nil_eq_concat_conv
thf(fact_4284_concat__eq__Nil__conv,axiom,
! [A: $tType,Xss: list @ ( list @ A )] :
( ( ( concat @ A @ Xss )
= ( nil @ A ) )
= ( ! [X5: list @ A] :
( ( member @ ( list @ A ) @ X5 @ ( set2 @ ( list @ A ) @ Xss ) )
=> ( X5
= ( nil @ A ) ) ) ) ) ).
% concat_eq_Nil_conv
thf(fact_4285_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_4286_removeAll_Osimps_I1_J,axiom,
! [A: $tType,X: A] :
( ( removeAll @ A @ X @ ( nil @ A ) )
= ( nil @ A ) ) ).
% removeAll.simps(1)
thf(fact_4287_Nil__in__shufflesI,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( Xs
= ( nil @ A ) )
=> ( ( Ys
= ( nil @ A ) )
=> ( member @ ( list @ A ) @ ( nil @ A ) @ ( shuffles @ A @ Xs @ Ys ) ) ) ) ).
% Nil_in_shufflesI
thf(fact_4288_shuffles_Osimps_I2_J,axiom,
! [A: $tType,Xs: list @ A] :
( ( shuffles @ A @ Xs @ ( nil @ A ) )
= ( insert @ ( list @ A ) @ Xs @ ( bot_bot @ ( set @ ( list @ A ) ) ) ) ) ).
% shuffles.simps(2)
thf(fact_4289_shuffles_Osimps_I1_J,axiom,
! [A: $tType,Ys: list @ A] :
( ( shuffles @ A @ ( nil @ A ) @ Ys )
= ( insert @ ( list @ A ) @ Ys @ ( bot_bot @ ( set @ ( list @ A ) ) ) ) ) ).
% shuffles.simps(1)
thf(fact_4290_distinct_Osimps_I1_J,axiom,
! [A: $tType] : ( distinct @ A @ ( nil @ A ) ) ).
% distinct.simps(1)
thf(fact_4291_rotate1_Osimps_I1_J,axiom,
! [A: $tType] :
( ( rotate1 @ A @ ( nil @ A ) )
= ( nil @ A ) ) ).
% rotate1.simps(1)
thf(fact_4292_remove1_Osimps_I1_J,axiom,
! [A: $tType,X: A] :
( ( remove1 @ A @ X @ ( nil @ A ) )
= ( nil @ A ) ) ).
% remove1.simps(1)
thf(fact_4293_product_Osimps_I1_J,axiom,
! [B: $tType,A: $tType,Uu: list @ B] :
( ( product @ A @ B @ ( nil @ A ) @ Uu )
= ( nil @ ( product_prod @ A @ B ) ) ) ).
% product.simps(1)
thf(fact_4294_list__update__code_I1_J,axiom,
! [A: $tType,I2: nat,Y: A] :
( ( list_update @ A @ ( nil @ A ) @ I2 @ Y )
= ( nil @ A ) ) ).
% list_update_code(1)
thf(fact_4295_list__update_Osimps_I1_J,axiom,
! [A: $tType,I2: nat,V: A] :
( ( list_update @ A @ ( nil @ A ) @ I2 @ V )
= ( nil @ A ) ) ).
% list_update.simps(1)
thf(fact_4296_concat_Osimps_I1_J,axiom,
! [A: $tType] :
( ( concat @ A @ ( nil @ ( list @ A ) ) )
= ( nil @ A ) ) ).
% concat.simps(1)
thf(fact_4297_empty__set,axiom,
! [A: $tType] :
( ( bot_bot @ ( set @ A ) )
= ( set2 @ A @ ( nil @ A ) ) ) ).
% empty_set
thf(fact_4298_list_Osize_I3_J,axiom,
! [A: $tType] :
( ( size_size @ ( list @ A ) @ ( nil @ A ) )
= ( zero_zero @ nat ) ) ).
% list.size(3)
thf(fact_4299_replicate__0,axiom,
! [A: $tType,X: A] :
( ( replicate @ A @ ( zero_zero @ nat ) @ X )
= ( nil @ A ) ) ).
% replicate_0
thf(fact_4300_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_4301_count__list_Osimps_I1_J,axiom,
! [A: $tType,Y: A] :
( ( count_list @ A @ ( nil @ A ) @ Y )
= ( zero_zero @ nat ) ) ).
% count_list.simps(1)
thf(fact_4302_Pow__set_I1_J,axiom,
! [A: $tType] :
( ( pow2 @ A @ ( set2 @ A @ ( nil @ A ) ) )
= ( insert @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ ( bot_bot @ ( set @ ( set @ A ) ) ) ) ) ).
% Pow_set(1)
thf(fact_4303_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_4304_less__eq__int_Orep__eq,axiom,
( ( ord_less_eq @ int )
= ( ^ [X5: int,Xa4: int] :
( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [Y6: nat,Z5: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U2: nat,V5: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ Y6 @ V5 ) @ ( plus_plus @ nat @ U2 @ Z5 ) ) )
@ ( rep_Integ @ X5 )
@ ( rep_Integ @ Xa4 ) ) ) ) ).
% less_eq_int.rep_eq
thf(fact_4305_less__int_Orep__eq,axiom,
( ( ord_less @ int )
= ( ^ [X5: int,Xa4: int] :
( product_case_prod @ nat @ nat @ ( ( product_prod @ nat @ nat ) > $o )
@ ^ [Y6: nat,Z5: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U2: nat,V5: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ Y6 @ V5 ) @ ( plus_plus @ nat @ U2 @ Z5 ) ) )
@ ( rep_Integ @ X5 )
@ ( rep_Integ @ Xa4 ) ) ) ) ).
% less_int.rep_eq
thf(fact_4306_prod__encode__def,axiom,
( nat_prod_encode
= ( product_case_prod @ nat @ nat @ nat
@ ^ [M3: nat,N2: nat] : ( plus_plus @ nat @ ( nat_triangle @ ( plus_plus @ nat @ M3 @ N2 ) ) @ M3 ) ) ) ).
% prod_encode_def
thf(fact_4307_Gcd__remove0__nat,axiom,
! [M5: set @ nat] :
( ( finite_finite2 @ nat @ M5 )
=> ( ( gcd_Gcd @ nat @ M5 )
= ( gcd_Gcd @ nat @ ( minus_minus @ ( set @ nat ) @ M5 @ ( insert @ nat @ ( zero_zero @ nat ) @ ( bot_bot @ ( set @ nat ) ) ) ) ) ) ) ).
% Gcd_remove0_nat
thf(fact_4308_prod__encode__eq,axiom,
! [X: product_prod @ nat @ nat,Y: product_prod @ nat @ nat] :
( ( ( nat_prod_encode @ X )
= ( nat_prod_encode @ Y ) )
= ( X = Y ) ) ).
% prod_encode_eq
thf(fact_4309_Gcd__empty,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ( ( gcd_Gcd @ A @ ( bot_bot @ ( set @ A ) ) )
= ( zero_zero @ A ) ) ) ).
% Gcd_empty
thf(fact_4310_Gcd__2,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A2: A,B2: A] :
( ( gcd_Gcd @ A @ ( insert @ A @ A2 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( gcd_gcd @ A @ A2 @ B2 ) ) ) ).
% Gcd_2
thf(fact_4311_Gcd__0__iff,axiom,
! [A: $tType] :
( ( semiring_Gcd @ A )
=> ! [A5: set @ A] :
( ( ( gcd_Gcd @ A @ A5 )
= ( zero_zero @ A ) )
= ( ord_less_eq @ ( set @ A ) @ A5 @ ( insert @ A @ ( zero_zero @ A ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% Gcd_0_iff
thf(fact_4312_Gcd__in,axiom,
! [A5: set @ nat] :
( ! [A4: nat,B4: nat] :
( ( member @ nat @ A4 @ A5 )
=> ( ( member @ nat @ B4 @ A5 )
=> ( member @ nat @ ( gcd_gcd @ nat @ A4 @ B4 ) @ A5 ) ) )
=> ( ( A5
!= ( bot_bot @ ( set @ nat ) ) )
=> ( member @ nat @ ( gcd_Gcd @ nat @ A5 ) @ A5 ) ) ) ).
% Gcd_in
thf(fact_4313_le__prod__encode__2,axiom,
! [B2: nat,A2: nat] : ( ord_less_eq @ nat @ B2 @ ( nat_prod_encode @ ( product_Pair @ nat @ nat @ A2 @ B2 ) ) ) ).
% le_prod_encode_2
thf(fact_4314_le__prod__encode__1,axiom,
! [A2: nat,B2: nat] : ( ord_less_eq @ nat @ A2 @ ( nat_prod_encode @ ( product_Pair @ nat @ nat @ A2 @ B2 ) ) ) ).
% le_prod_encode_1
thf(fact_4315_nat_Orep__eq,axiom,
( nat2
= ( ^ [X5: int] : ( product_case_prod @ nat @ nat @ nat @ ( minus_minus @ nat ) @ ( rep_Integ @ X5 ) ) ) ) ).
% nat.rep_eq
thf(fact_4316_of__int_Orep__eq,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( ( ring_1_of_int @ A )
= ( ^ [X5: int] :
( product_case_prod @ nat @ nat @ A
@ ^ [I4: nat,J3: nat] : ( minus_minus @ A @ ( semiring_1_of_nat @ A @ I4 ) @ ( semiring_1_of_nat @ A @ J3 ) )
@ ( rep_Integ @ X5 ) ) ) ) ) ).
% of_int.rep_eq
thf(fact_4317_prod__encode__prod__decode__aux,axiom,
! [K2: nat,M2: nat] :
( ( nat_prod_encode @ ( nat_prod_decode_aux @ K2 @ M2 ) )
= ( plus_plus @ nat @ ( nat_triangle @ K2 ) @ M2 ) ) ).
% prod_encode_prod_decode_aux
thf(fact_4318_listset_Osimps_I1_J,axiom,
! [A: $tType] :
( ( listset @ A @ ( nil @ ( set @ A ) ) )
= ( insert @ ( list @ A ) @ ( nil @ A ) @ ( bot_bot @ ( set @ ( list @ A ) ) ) ) ) ).
% listset.simps(1)
thf(fact_4319_semiring__char__def,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( ( semiri4206861660011772517g_char @ A )
= ( ^ [Uu4: itself @ A] :
( gcd_Gcd @ nat
@ ( collect @ nat
@ ^ [N2: nat] :
( ( semiring_1_of_nat @ A @ N2 )
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% semiring_char_def
thf(fact_4320_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 )
@ ^ [X5: nat,Y6: nat] : ( product_Pair @ nat @ nat @ Y6 @ X5 ) ) ) ) ).
% uminus_int_def
thf(fact_4321_eq__snd__iff,axiom,
! [A: $tType,B: $tType,B2: A,P: product_prod @ B @ A] :
( ( B2
= ( product_snd @ B @ A @ P ) )
= ( ? [A6: B] :
( P
= ( product_Pair @ B @ A @ A6 @ B2 ) ) ) ) ).
% eq_snd_iff
thf(fact_4322_sum_Osize__neq,axiom,
! [A: $tType,B: $tType,X: sum_sum @ A @ B] :
( ( size_size @ ( sum_sum @ A @ B ) @ X )
!= ( zero_zero @ nat ) ) ).
% sum.size_neq
thf(fact_4323_prod_Osize__neq,axiom,
! [A: $tType,B: $tType,X: product_prod @ A @ B] :
( ( size_size @ ( product_prod @ A @ B ) @ X )
!= ( zero_zero @ nat ) ) ).
% prod.size_neq
thf(fact_4324_eq__fst__iff,axiom,
! [A: $tType,B: $tType,A2: A,P: product_prod @ A @ B] :
( ( A2
= ( product_fst @ A @ B @ P ) )
= ( ? [B5: B] :
( P
= ( product_Pair @ A @ B @ A2 @ B5 ) ) ) ) ).
% eq_fst_iff
thf(fact_4325_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 ) )
@ ^ [X5: nat,Y6: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V5: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ X5 @ U2 ) @ ( times_times @ nat @ Y6 @ V5 ) ) @ ( plus_plus @ nat @ ( times_times @ nat @ X5 @ V5 ) @ ( times_times @ nat @ Y6 @ U2 ) ) ) ) ) ) ) ).
% times_int_def
thf(fact_4326_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 ) )
@ ^ [X5: nat,Y6: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V5: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X5 @ V5 ) @ ( plus_plus @ nat @ Y6 @ U2 ) ) ) ) ) ) ).
% minus_int_def
thf(fact_4327_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 ) )
@ ^ [X5: nat,Y6: nat] :
( product_case_prod @ nat @ nat @ ( product_prod @ nat @ nat )
@ ^ [U2: nat,V5: nat] : ( product_Pair @ nat @ nat @ ( plus_plus @ nat @ X5 @ U2 ) @ ( plus_plus @ nat @ Y6 @ V5 ) ) ) ) ) ) ).
% plus_int_def
thf(fact_4328_sndI,axiom,
! [A: $tType,B: $tType,X: product_prod @ A @ B,Y: A,Z: B] :
( ( X
= ( product_Pair @ A @ B @ Y @ Z ) )
=> ( ( product_snd @ A @ B @ X )
= Z ) ) ).
% sndI
thf(fact_4329_fstI,axiom,
! [B: $tType,A: $tType,X: product_prod @ A @ B,Y: A,Z: B] :
( ( X
= ( product_Pair @ A @ B @ Y @ Z ) )
=> ( ( product_fst @ A @ B @ X )
= Y ) ) ).
% fstI
thf(fact_4330_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_4331_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_4332_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_4333_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_4334_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_4335_semiring__norm_I167_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [V: num,W2: num,Y: A] :
( ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) @ ( plus_plus @ A @ ( numeral_numeral @ A @ W2 ) @ Y ) )
= ( plus_plus @ A @ ( neg_numeral_sub @ A @ W2 @ V ) @ Y ) ) ) ).
% semiring_norm(167)
thf(fact_4336_semiring__norm_I166_J,axiom,
! [A: $tType] :
( ( neg_numeral @ A )
=> ! [V: num,W2: num,Y: A] :
( ( plus_plus @ A @ ( numeral_numeral @ A @ V ) @ ( plus_plus @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ W2 ) ) @ Y ) )
= ( plus_plus @ A @ ( neg_numeral_sub @ A @ V @ W2 ) @ Y ) ) ) ).
% semiring_norm(166)
thf(fact_4337_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_4338_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_4339_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_4340_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_4341_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_4342_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_4343_prod_Oinsert_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [I6: set @ B,P: B > A,I2: B] :
( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X5: B] :
( ( member @ B @ X5 @ I6 )
& ( ( P @ X5 )
!= ( one_one @ A ) ) ) ) )
=> ( ( ( member @ B @ I2 @ I6 )
=> ( ( groups1962203154675924110t_prod @ B @ A @ P @ ( insert @ B @ I2 @ I6 ) )
= ( groups1962203154675924110t_prod @ B @ A @ P @ I6 ) ) )
& ( ~ ( member @ B @ I2 @ I6 )
=> ( ( groups1962203154675924110t_prod @ B @ A @ P @ ( insert @ B @ I2 @ I6 ) )
= ( times_times @ A @ ( P @ I2 ) @ ( groups1962203154675924110t_prod @ B @ A @ P @ I6 ) ) ) ) ) ) ) ).
% prod.insert'
thf(fact_4344_sorted__list__of__set_Osorted__key__list__of__set__remove,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( linord4507533701916653071of_set @ A @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( remove1 @ A @ X @ ( linord4507533701916653071of_set @ A @ A5 ) ) ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_remove
thf(fact_4345_image__minus__const__atLeastLessThan__nat,axiom,
! [C3: nat,Y: nat,X: nat] :
( ( ( ord_less @ nat @ C3 @ Y )
=> ( ( image2 @ nat @ nat
@ ^ [I4: nat] : ( minus_minus @ nat @ I4 @ C3 )
@ ( set_or7035219750837199246ssThan @ nat @ X @ Y ) )
= ( set_or7035219750837199246ssThan @ nat @ ( minus_minus @ nat @ X @ C3 ) @ ( minus_minus @ nat @ Y @ C3 ) ) ) )
& ( ~ ( ord_less @ nat @ C3 @ Y )
=> ( ( ( ord_less @ nat @ X @ Y )
=> ( ( image2 @ nat @ nat
@ ^ [I4: nat] : ( minus_minus @ nat @ I4 @ C3 )
@ ( set_or7035219750837199246ssThan @ nat @ X @ Y ) )
= ( insert @ nat @ ( zero_zero @ nat ) @ ( bot_bot @ ( set @ nat ) ) ) ) )
& ( ~ ( ord_less @ nat @ X @ Y )
=> ( ( image2 @ nat @ nat
@ ^ [I4: nat] : ( minus_minus @ nat @ I4 @ C3 )
@ ( set_or7035219750837199246ssThan @ nat @ X @ Y ) )
= ( bot_bot @ ( set @ nat ) ) ) ) ) ) ) ).
% image_minus_const_atLeastLessThan_nat
thf(fact_4346_rat__floor__lemma,axiom,
! [A2: int,B2: int] :
( ( ord_less_eq @ rat @ ( ring_1_of_int @ rat @ ( divide_divide @ int @ A2 @ B2 ) ) @ ( fract @ A2 @ B2 ) )
& ( ord_less @ rat @ ( fract @ A2 @ B2 ) @ ( ring_1_of_int @ rat @ ( plus_plus @ int @ ( divide_divide @ int @ A2 @ B2 ) @ ( one_one @ int ) ) ) ) ) ).
% rat_floor_lemma
thf(fact_4347_image__is__empty,axiom,
! [A: $tType,B: $tType,F3: B > A,A5: set @ B] :
( ( ( image2 @ B @ A @ F3 @ A5 )
= ( bot_bot @ ( set @ A ) ) )
= ( A5
= ( bot_bot @ ( set @ B ) ) ) ) ).
% image_is_empty
thf(fact_4348_empty__is__image,axiom,
! [A: $tType,B: $tType,F3: B > A,A5: set @ B] :
( ( ( bot_bot @ ( set @ A ) )
= ( image2 @ B @ A @ F3 @ A5 ) )
= ( A5
= ( bot_bot @ ( set @ B ) ) ) ) ).
% empty_is_image
thf(fact_4349_image__empty,axiom,
! [B: $tType,A: $tType,F3: B > A] :
( ( image2 @ B @ A @ F3 @ ( bot_bot @ ( set @ B ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% image_empty
thf(fact_4350_finite__imageI,axiom,
! [B: $tType,A: $tType,F5: set @ A,H2: A > B] :
( ( finite_finite2 @ A @ F5 )
=> ( finite_finite2 @ B @ ( image2 @ A @ B @ H2 @ F5 ) ) ) ).
% finite_imageI
thf(fact_4351_image__add__0,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [S2: set @ A] :
( ( image2 @ A @ A @ ( plus_plus @ A @ ( zero_zero @ A ) ) @ S2 )
= S2 ) ) ).
% image_add_0
thf(fact_4352_image__add__atLeastAtMost,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [K2: A,I2: A,J: A] :
( ( image2 @ A @ A @ ( plus_plus @ A @ K2 ) @ ( set_or1337092689740270186AtMost @ A @ I2 @ J ) )
= ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ I2 @ K2 ) @ ( plus_plus @ A @ J @ K2 ) ) ) ) ).
% image_add_atLeastAtMost
thf(fact_4353_image__diff__atLeastAtMost,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [D3: A,A2: A,B2: A] :
( ( image2 @ A @ A @ ( minus_minus @ A @ D3 ) @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ D3 @ B2 ) @ ( minus_minus @ A @ D3 @ A2 ) ) ) ) ).
% image_diff_atLeastAtMost
thf(fact_4354_image__uminus__atLeastAtMost,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [X: A,Y: A] :
( ( image2 @ A @ A @ ( uminus_uminus @ A ) @ ( set_or1337092689740270186AtMost @ A @ X @ Y ) )
= ( set_or1337092689740270186AtMost @ A @ ( uminus_uminus @ A @ Y ) @ ( uminus_uminus @ A @ X ) ) ) ) ).
% image_uminus_atLeastAtMost
thf(fact_4355_image__add__atLeastLessThan,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [K2: A,I2: A,J: A] :
( ( image2 @ A @ A @ ( plus_plus @ A @ K2 ) @ ( set_or7035219750837199246ssThan @ A @ I2 @ J ) )
= ( set_or7035219750837199246ssThan @ A @ ( plus_plus @ A @ I2 @ K2 ) @ ( plus_plus @ A @ J @ K2 ) ) ) ) ).
% image_add_atLeastLessThan
thf(fact_4356_image__add__atMost,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [C3: A,A2: A] :
( ( image2 @ A @ A @ ( plus_plus @ A @ C3 ) @ ( set_ord_atMost @ A @ A2 ) )
= ( set_ord_atMost @ A @ ( plus_plus @ A @ C3 @ A2 ) ) ) ) ).
% image_add_atMost
thf(fact_4357_image__uminus__greaterThanLessThan,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [X: A,Y: A] :
( ( image2 @ A @ A @ ( uminus_uminus @ A ) @ ( set_or5935395276787703475ssThan @ A @ X @ Y ) )
= ( set_or5935395276787703475ssThan @ A @ ( uminus_uminus @ A @ Y ) @ ( uminus_uminus @ A @ X ) ) ) ) ).
% image_uminus_greaterThanLessThan
thf(fact_4358_sorted__list__of__set_Osorted__key__list__of__set__empty,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( linord4507533701916653071of_set @ A @ ( bot_bot @ ( set @ A ) ) )
= ( nil @ A ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_empty
thf(fact_4359_sorted__list__of__set_Ofold__insort__key_Oinfinite,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A] :
( ~ ( finite_finite2 @ A @ A5 )
=> ( ( linord4507533701916653071of_set @ A @ A5 )
= ( nil @ A ) ) ) ) ).
% sorted_list_of_set.fold_insort_key.infinite
thf(fact_4360_sorted__list__of__set_Oset__sorted__key__list__of__set,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( set2 @ A @ ( linord4507533701916653071of_set @ A @ A5 ) )
= A5 ) ) ) ).
% sorted_list_of_set.set_sorted_key_list_of_set
thf(fact_4361_prod_Oempty_H,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [P: B > A] :
( ( groups1962203154675924110t_prod @ B @ A @ P @ ( bot_bot @ ( set @ B ) ) )
= ( one_one @ A ) ) ) ).
% prod.empty'
thf(fact_4362_sorted__list__of__set_Olength__sorted__key__list__of__set,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A] :
( ( size_size @ ( list @ A ) @ ( linord4507533701916653071of_set @ A @ A5 ) )
= ( finite_card @ A @ A5 ) ) ) ).
% sorted_list_of_set.length_sorted_key_list_of_set
thf(fact_4363_prod_Oeq__sum,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [I6: set @ B,P: B > A] :
( ( finite_finite2 @ B @ I6 )
=> ( ( groups1962203154675924110t_prod @ B @ A @ P @ I6 )
= ( groups7121269368397514597t_prod @ B @ A @ P @ I6 ) ) ) ) ).
% prod.eq_sum
thf(fact_4364_image__Suc__atLeastAtMost,axiom,
! [I2: nat,J: nat] :
( ( image2 @ nat @ nat @ suc @ ( set_or1337092689740270186AtMost @ nat @ I2 @ J ) )
= ( set_or1337092689740270186AtMost @ nat @ ( suc @ I2 ) @ ( suc @ J ) ) ) ).
% image_Suc_atLeastAtMost
thf(fact_4365_image__Suc__atLeastLessThan,axiom,
! [I2: nat,J: nat] :
( ( image2 @ nat @ nat @ suc @ ( set_or7035219750837199246ssThan @ nat @ I2 @ J ) )
= ( set_or7035219750837199246ssThan @ nat @ ( suc @ I2 ) @ ( suc @ J ) ) ) ).
% image_Suc_atLeastLessThan
thf(fact_4366_SUP__bot__conv_I2_J,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [B6: B > A,A5: set @ B] :
( ( ( bot_bot @ A )
= ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ B6 @ A5 ) ) )
= ( ! [X5: B] :
( ( member @ B @ X5 @ A5 )
=> ( ( B6 @ X5 )
= ( bot_bot @ A ) ) ) ) ) ) ).
% SUP_bot_conv(2)
thf(fact_4367_SUP__bot__conv_I1_J,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [B6: B > A,A5: set @ B] :
( ( ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ B6 @ A5 ) )
= ( bot_bot @ A ) )
= ( ! [X5: B] :
( ( member @ B @ X5 @ A5 )
=> ( ( B6 @ X5 )
= ( bot_bot @ A ) ) ) ) ) ) ).
% SUP_bot_conv(1)
thf(fact_4368_SUP__bot,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ B] :
( ( complete_Sup_Sup @ A
@ ( image2 @ B @ A
@ ^ [X5: B] : ( bot_bot @ A )
@ A5 ) )
= ( bot_bot @ A ) ) ) ).
% SUP_bot
thf(fact_4369_ccSUP__bot,axiom,
! [B: $tType,A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A5: set @ B] :
( ( complete_Sup_Sup @ A
@ ( image2 @ B @ A
@ ^ [X5: B] : ( bot_bot @ A )
@ A5 ) )
= ( bot_bot @ A ) ) ) ).
% ccSUP_bot
thf(fact_4370_SUP__const,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ B,F3: A] :
( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ A
@ ( image2 @ B @ A
@ ^ [I4: B] : F3
@ A5 ) )
= F3 ) ) ) ).
% SUP_const
thf(fact_4371_ccSUP__const,axiom,
! [B: $tType,A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A5: set @ B,F3: A] :
( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ A
@ ( image2 @ B @ A
@ ^ [I4: B] : F3
@ A5 ) )
= F3 ) ) ) ).
% ccSUP_const
thf(fact_4372_cSUP__const,axiom,
! [B: $tType,A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A5: set @ B,C3: A] :
( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ A
@ ( image2 @ B @ A
@ ^ [X5: B] : C3
@ A5 ) )
= C3 ) ) ) ).
% cSUP_const
thf(fact_4373_image__add__atLeastAtMost_H,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [K2: A,I2: A,J: A] :
( ( image2 @ A @ A
@ ^ [N2: A] : ( plus_plus @ A @ N2 @ K2 )
@ ( set_or1337092689740270186AtMost @ A @ I2 @ J ) )
= ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ I2 @ K2 ) @ ( plus_plus @ A @ J @ K2 ) ) ) ) ).
% image_add_atLeastAtMost'
thf(fact_4374_INF__const,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ B,F3: A] :
( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Inf_Inf @ A
@ ( image2 @ B @ A
@ ^ [I4: B] : F3
@ A5 ) )
= F3 ) ) ) ).
% INF_const
thf(fact_4375_ccINF__const,axiom,
! [B: $tType,A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A5: set @ B,F3: A] :
( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Inf_Inf @ A
@ ( image2 @ B @ A
@ ^ [I4: B] : F3
@ A5 ) )
= F3 ) ) ) ).
% ccINF_const
thf(fact_4376_cINF__const,axiom,
! [B: $tType,A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A5: set @ B,C3: A] :
( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Inf_Inf @ A
@ ( image2 @ B @ A
@ ^ [X5: B] : C3
@ A5 ) )
= C3 ) ) ) ).
% cINF_const
thf(fact_4377_image__minus__const__atLeastAtMost_H,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [D3: A,A2: A,B2: A] :
( ( image2 @ A @ A
@ ^ [T3: A] : ( minus_minus @ A @ T3 @ D3 )
@ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ A2 @ D3 ) @ ( minus_minus @ A @ B2 @ D3 ) ) ) ) ).
% image_minus_const_atLeastAtMost'
thf(fact_4378_image__add__atLeastLessThan_H,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [K2: A,I2: A,J: A] :
( ( image2 @ A @ A
@ ^ [N2: A] : ( plus_plus @ A @ N2 @ K2 )
@ ( set_or7035219750837199246ssThan @ A @ I2 @ J ) )
= ( set_or7035219750837199246ssThan @ A @ ( plus_plus @ A @ I2 @ K2 ) @ ( plus_plus @ A @ J @ K2 ) ) ) ) ).
% image_add_atLeastLessThan'
thf(fact_4379_sorted__list__of__set_Osorted__key__list__of__set__eq__Nil__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( ( linord4507533701916653071of_set @ A @ A5 )
= ( nil @ A ) )
= ( A5
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_eq_Nil_iff
thf(fact_4380_INF__eq__bot__iff,axiom,
! [B: $tType,A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [F3: B > A,A5: set @ B] :
( ( ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ A5 ) )
= ( bot_bot @ A ) )
= ( ! [X5: A] :
( ( ord_less @ A @ ( bot_bot @ A ) @ X5 )
=> ? [Y6: B] :
( ( member @ B @ Y6 @ A5 )
& ( ord_less @ A @ ( F3 @ Y6 ) @ X5 ) ) ) ) ) ) ).
% INF_eq_bot_iff
thf(fact_4381_ccSUP__empty,axiom,
! [B: $tType,A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [F3: B > A] :
( ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ ( bot_bot @ ( set @ B ) ) ) )
= ( bot_bot @ A ) ) ) ).
% ccSUP_empty
thf(fact_4382_image__mult__atLeastAtMost,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [D3: A,A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ D3 )
=> ( ( image2 @ A @ A @ ( times_times @ A @ D3 ) @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( times_times @ A @ D3 @ A2 ) @ ( times_times @ A @ D3 @ B2 ) ) ) ) ) ).
% image_mult_atLeastAtMost
thf(fact_4383_image__divide__atLeastAtMost,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [D3: A,A2: A,B2: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ D3 )
=> ( ( image2 @ A @ A
@ ^ [C4: A] : ( divide_divide @ A @ C4 @ D3 )
@ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( divide_divide @ A @ A2 @ D3 ) @ ( divide_divide @ A @ B2 @ D3 ) ) ) ) ) ).
% image_divide_atLeastAtMost
thf(fact_4384_less__rat,axiom,
! [B2: int,D3: int,A2: int,C3: int] :
( ( B2
!= ( zero_zero @ int ) )
=> ( ( D3
!= ( zero_zero @ int ) )
=> ( ( ord_less @ rat @ ( fract @ A2 @ B2 ) @ ( fract @ C3 @ D3 ) )
= ( ord_less @ int @ ( times_times @ int @ ( times_times @ int @ A2 @ D3 ) @ ( times_times @ int @ B2 @ D3 ) ) @ ( times_times @ int @ ( times_times @ int @ C3 @ B2 ) @ ( times_times @ int @ B2 @ D3 ) ) ) ) ) ) ).
% less_rat
thf(fact_4385_all__subset__image,axiom,
! [A: $tType,B: $tType,F3: B > A,A5: set @ B,P2: ( set @ A ) > $o] :
( ( ! [B8: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ B8 @ ( image2 @ B @ A @ F3 @ A5 ) )
=> ( P2 @ B8 ) ) )
= ( ! [B8: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ B8 @ A5 )
=> ( P2 @ ( image2 @ B @ A @ F3 @ B8 ) ) ) ) ) ).
% all_subset_image
thf(fact_4386_pigeonhole__infinite,axiom,
! [B: $tType,A: $tType,A5: set @ A,F3: A > B] :
( ~ ( finite_finite2 @ A @ A5 )
=> ( ( finite_finite2 @ B @ ( image2 @ A @ B @ F3 @ A5 ) )
=> ? [X4: A] :
( ( member @ A @ X4 @ A5 )
& ~ ( finite_finite2 @ A
@ ( collect @ A
@ ^ [A6: A] :
( ( member @ A @ A6 @ A5 )
& ( ( F3 @ A6 )
= ( F3 @ X4 ) ) ) ) ) ) ) ) ).
% pigeonhole_infinite
thf(fact_4387_sorted__list__of__set_Osorted__key__list__of__set__inject,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,B6: set @ A] :
( ( ( linord4507533701916653071of_set @ A @ A5 )
= ( linord4507533701916653071of_set @ A @ B6 ) )
=> ( ( finite_finite2 @ A @ A5 )
=> ( ( finite_finite2 @ A @ B6 )
=> ( A5 = B6 ) ) ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_inject
thf(fact_4388_UNION__singleton__eq__range,axiom,
! [A: $tType,B: $tType,F3: B > A,A5: set @ B] :
( ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [X5: B] : ( insert @ A @ ( F3 @ X5 ) @ ( bot_bot @ ( set @ A ) ) )
@ A5 ) )
= ( image2 @ B @ A @ F3 @ A5 ) ) ).
% UNION_singleton_eq_range
thf(fact_4389_sorted__list__of__set_Odistinct__sorted__key__list__of__set,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A] : ( distinct @ A @ ( linord4507533701916653071of_set @ A @ A5 ) ) ) ).
% sorted_list_of_set.distinct_sorted_key_list_of_set
thf(fact_4390_SUP__eq,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ B,B6: set @ C,F3: B > A,G3: C > A] :
( ! [I3: B] :
( ( member @ B @ I3 @ A5 )
=> ? [X3: C] :
( ( member @ C @ X3 @ B6 )
& ( ord_less_eq @ A @ ( F3 @ I3 ) @ ( G3 @ X3 ) ) ) )
=> ( ! [J2: C] :
( ( member @ C @ J2 @ B6 )
=> ? [X3: B] :
( ( member @ B @ X3 @ A5 )
& ( ord_less_eq @ A @ ( G3 @ J2 ) @ ( F3 @ X3 ) ) ) )
=> ( ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A5 ) )
= ( complete_Sup_Sup @ A @ ( image2 @ C @ A @ G3 @ B6 ) ) ) ) ) ) ).
% SUP_eq
thf(fact_4391_INF__eq,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ B,B6: set @ C,G3: C > A,F3: B > A] :
( ! [I3: B] :
( ( member @ B @ I3 @ A5 )
=> ? [X3: C] :
( ( member @ C @ X3 @ B6 )
& ( ord_less_eq @ A @ ( G3 @ X3 ) @ ( F3 @ I3 ) ) ) )
=> ( ! [J2: C] :
( ( member @ C @ J2 @ B6 )
=> ? [X3: B] :
( ( member @ B @ X3 @ A5 )
& ( ord_less_eq @ A @ ( F3 @ X3 ) @ ( G3 @ J2 ) ) ) )
=> ( ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ A5 ) )
= ( complete_Inf_Inf @ A @ ( image2 @ C @ A @ G3 @ B6 ) ) ) ) ) ) ).
% INF_eq
thf(fact_4392_zero__notin__Suc__image,axiom,
! [A5: set @ nat] :
~ ( member @ nat @ ( zero_zero @ nat ) @ ( image2 @ nat @ nat @ suc @ A5 ) ) ).
% zero_notin_Suc_image
thf(fact_4393_all__finite__subset__image,axiom,
! [A: $tType,B: $tType,F3: B > A,A5: set @ B,P2: ( set @ A ) > $o] :
( ( ! [B8: set @ A] :
( ( ( finite_finite2 @ A @ B8 )
& ( ord_less_eq @ ( set @ A ) @ B8 @ ( image2 @ B @ A @ F3 @ A5 ) ) )
=> ( P2 @ B8 ) ) )
= ( ! [B8: set @ B] :
( ( ( finite_finite2 @ B @ B8 )
& ( ord_less_eq @ ( set @ B ) @ B8 @ A5 ) )
=> ( P2 @ ( image2 @ B @ A @ F3 @ B8 ) ) ) ) ) ).
% all_finite_subset_image
thf(fact_4394_ex__finite__subset__image,axiom,
! [A: $tType,B: $tType,F3: B > A,A5: set @ B,P2: ( set @ A ) > $o] :
( ( ? [B8: set @ A] :
( ( finite_finite2 @ A @ B8 )
& ( ord_less_eq @ ( set @ A ) @ B8 @ ( image2 @ B @ A @ F3 @ A5 ) )
& ( P2 @ B8 ) ) )
= ( ? [B8: set @ B] :
( ( finite_finite2 @ B @ B8 )
& ( ord_less_eq @ ( set @ B ) @ B8 @ A5 )
& ( P2 @ ( image2 @ B @ A @ F3 @ B8 ) ) ) ) ) ).
% ex_finite_subset_image
thf(fact_4395_finite__subset__image,axiom,
! [A: $tType,B: $tType,B6: set @ A,F3: B > A,A5: set @ B] :
( ( finite_finite2 @ A @ B6 )
=> ( ( ord_less_eq @ ( set @ A ) @ B6 @ ( image2 @ B @ A @ F3 @ A5 ) )
=> ? [C6: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ C6 @ A5 )
& ( finite_finite2 @ B @ C6 )
& ( B6
= ( image2 @ B @ A @ F3 @ C6 ) ) ) ) ) ).
% finite_subset_image
thf(fact_4396_finite__surj,axiom,
! [A: $tType,B: $tType,A5: set @ A,B6: set @ B,F3: A > B] :
( ( finite_finite2 @ A @ A5 )
=> ( ( ord_less_eq @ ( set @ B ) @ B6 @ ( image2 @ A @ B @ F3 @ A5 ) )
=> ( finite_finite2 @ B @ B6 ) ) ) ).
% finite_surj
thf(fact_4397_translation__Int,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A2: A,S: set @ A,T2: set @ A] :
( ( image2 @ A @ A @ ( plus_plus @ A @ A2 ) @ ( inf_inf @ ( set @ A ) @ S @ T2 ) )
= ( inf_inf @ ( set @ A ) @ ( image2 @ A @ A @ ( plus_plus @ A @ A2 ) @ S ) @ ( image2 @ A @ A @ ( plus_plus @ A @ A2 ) @ T2 ) ) ) ) ).
% translation_Int
thf(fact_4398_SUP__eq__const,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I6: set @ B,F3: B > A,X: A] :
( ( I6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [I3: B] :
( ( member @ B @ I3 @ I6 )
=> ( ( F3 @ I3 )
= X ) )
=> ( ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ I6 ) )
= X ) ) ) ) ).
% SUP_eq_const
thf(fact_4399_translation__diff,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A2: A,S: set @ A,T2: set @ A] :
( ( image2 @ A @ A @ ( plus_plus @ A @ A2 ) @ ( minus_minus @ ( set @ A ) @ S @ T2 ) )
= ( minus_minus @ ( set @ A ) @ ( image2 @ A @ A @ ( plus_plus @ A @ A2 ) @ S ) @ ( image2 @ A @ A @ ( plus_plus @ A @ A2 ) @ T2 ) ) ) ) ).
% translation_diff
thf(fact_4400_INF__eq__const,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I6: set @ B,F3: B > A,X: A] :
( ( I6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [I3: B] :
( ( member @ B @ I3 @ I6 )
=> ( ( F3 @ I3 )
= X ) )
=> ( ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ I6 ) )
= X ) ) ) ) ).
% INF_eq_const
thf(fact_4401_finite__image__absD,axiom,
! [A: $tType] :
( ( linordered_ring @ A )
=> ! [S2: set @ A] :
( ( finite_finite2 @ A @ ( image2 @ A @ A @ ( abs_abs @ A ) @ S2 ) )
=> ( finite_finite2 @ A @ S2 ) ) ) ).
% finite_image_absD
thf(fact_4402_Rat__induct__pos,axiom,
! [P2: rat > $o,Q2: rat] :
( ! [A4: int,B4: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ B4 )
=> ( P2 @ ( fract @ A4 @ B4 ) ) )
=> ( P2 @ Q2 ) ) ).
% Rat_induct_pos
thf(fact_4403_translation__Compl,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A2: A,T2: set @ A] :
( ( image2 @ A @ A @ ( plus_plus @ A @ A2 ) @ ( uminus_uminus @ ( set @ A ) @ T2 ) )
= ( uminus_uminus @ ( set @ A ) @ ( image2 @ A @ A @ ( plus_plus @ A @ A2 ) @ T2 ) ) ) ) ).
% translation_Compl
thf(fact_4404_SUP__eqI,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ B,F3: B > A,X: A] :
( ! [I3: B] :
( ( member @ B @ I3 @ A5 )
=> ( ord_less_eq @ A @ ( F3 @ I3 ) @ X ) )
=> ( ! [Y3: A] :
( ! [I: B] :
( ( member @ B @ I @ A5 )
=> ( ord_less_eq @ A @ ( F3 @ I ) @ Y3 ) )
=> ( ord_less_eq @ A @ X @ Y3 ) )
=> ( ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A5 ) )
= X ) ) ) ) ).
% SUP_eqI
thf(fact_4405_SUP__mono,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ B,B6: set @ C,F3: B > A,G3: C > A] :
( ! [N3: B] :
( ( member @ B @ N3 @ A5 )
=> ? [X3: C] :
( ( member @ C @ X3 @ B6 )
& ( ord_less_eq @ A @ ( F3 @ N3 ) @ ( G3 @ X3 ) ) ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A5 ) ) @ ( complete_Sup_Sup @ A @ ( image2 @ C @ A @ G3 @ B6 ) ) ) ) ) ).
% SUP_mono
thf(fact_4406_SUP__least,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ B,F3: B > A,U: A] :
( ! [I3: B] :
( ( member @ B @ I3 @ A5 )
=> ( ord_less_eq @ A @ ( F3 @ I3 ) @ U ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A5 ) ) @ U ) ) ) ).
% SUP_least
thf(fact_4407_SUP__mono_H,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F3: B > A,G3: B > A,A5: set @ B] :
( ! [X4: B] : ( ord_less_eq @ A @ ( F3 @ X4 ) @ ( G3 @ X4 ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A5 ) ) @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ G3 @ A5 ) ) ) ) ) ).
% SUP_mono'
thf(fact_4408_SUP__upper,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I2: B,A5: set @ B,F3: B > A] :
( ( member @ B @ I2 @ A5 )
=> ( ord_less_eq @ A @ ( F3 @ I2 ) @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A5 ) ) ) ) ) ).
% SUP_upper
thf(fact_4409_SUP__le__iff,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F3: B > A,A5: set @ B,U: A] :
( ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A5 ) ) @ U )
= ( ! [X5: B] :
( ( member @ B @ X5 @ A5 )
=> ( ord_less_eq @ A @ ( F3 @ X5 ) @ U ) ) ) ) ) ).
% SUP_le_iff
thf(fact_4410_SUP__upper2,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I2: B,A5: set @ B,U: A,F3: B > A] :
( ( member @ B @ I2 @ A5 )
=> ( ( ord_less_eq @ A @ U @ ( F3 @ I2 ) )
=> ( ord_less_eq @ A @ U @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A5 ) ) ) ) ) ) ).
% SUP_upper2
thf(fact_4411_less__SUP__iff,axiom,
! [A: $tType,B: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [A2: A,F3: B > A,A5: set @ B] :
( ( ord_less @ A @ A2 @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A5 ) ) )
= ( ? [X5: B] :
( ( member @ B @ X5 @ A5 )
& ( ord_less @ A @ A2 @ ( F3 @ X5 ) ) ) ) ) ) ).
% less_SUP_iff
thf(fact_4412_SUP__lessD,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F3: B > A,A5: set @ B,Y: A,I2: B] :
( ( ord_less @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A5 ) ) @ Y )
=> ( ( member @ B @ I2 @ A5 )
=> ( ord_less @ A @ ( F3 @ I2 ) @ Y ) ) ) ) ).
% SUP_lessD
thf(fact_4413_INF__eqI,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ B,X: A,F3: B > A] :
( ! [I3: B] :
( ( member @ B @ I3 @ A5 )
=> ( ord_less_eq @ A @ X @ ( F3 @ I3 ) ) )
=> ( ! [Y3: A] :
( ! [I: B] :
( ( member @ B @ I @ A5 )
=> ( ord_less_eq @ A @ Y3 @ ( F3 @ I ) ) )
=> ( ord_less_eq @ A @ Y3 @ X ) )
=> ( ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ A5 ) )
= X ) ) ) ) ).
% INF_eqI
thf(fact_4414_INF__mono,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [B6: set @ B,A5: set @ C,F3: C > A,G3: B > A] :
( ! [M: B] :
( ( member @ B @ M @ B6 )
=> ? [X3: C] :
( ( member @ C @ X3 @ A5 )
& ( ord_less_eq @ A @ ( F3 @ X3 ) @ ( G3 @ M ) ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image2 @ C @ A @ F3 @ A5 ) ) @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ G3 @ B6 ) ) ) ) ) ).
% INF_mono
thf(fact_4415_INF__lower,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I2: B,A5: set @ B,F3: B > A] :
( ( member @ B @ I2 @ A5 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ A5 ) ) @ ( F3 @ I2 ) ) ) ) ).
% INF_lower
thf(fact_4416_INF__mono_H,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F3: B > A,G3: B > A,A5: set @ B] :
( ! [X4: B] : ( ord_less_eq @ A @ ( F3 @ X4 ) @ ( G3 @ X4 ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ A5 ) ) @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ G3 @ A5 ) ) ) ) ) ).
% INF_mono'
thf(fact_4417_INF__lower2,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I2: B,A5: set @ B,F3: B > A,U: A] :
( ( member @ B @ I2 @ A5 )
=> ( ( ord_less_eq @ A @ ( F3 @ I2 ) @ U )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ A5 ) ) @ U ) ) ) ) ).
% INF_lower2
thf(fact_4418_le__INF__iff,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [U: A,F3: B > A,A5: set @ B] :
( ( ord_less_eq @ A @ U @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ A5 ) ) )
= ( ! [X5: B] :
( ( member @ B @ X5 @ A5 )
=> ( ord_less_eq @ A @ U @ ( F3 @ X5 ) ) ) ) ) ) ).
% le_INF_iff
thf(fact_4419_INF__greatest,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ B,U: A,F3: B > A] :
( ! [I3: B] :
( ( member @ B @ I3 @ A5 )
=> ( ord_less_eq @ A @ U @ ( F3 @ I3 ) ) )
=> ( ord_less_eq @ A @ U @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ A5 ) ) ) ) ) ).
% INF_greatest
thf(fact_4420_INF__less__iff,axiom,
! [A: $tType,B: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [F3: B > A,A5: set @ B,A2: A] :
( ( ord_less @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ A5 ) ) @ A2 )
= ( ? [X5: B] :
( ( member @ B @ X5 @ A5 )
& ( ord_less @ A @ ( F3 @ X5 ) @ A2 ) ) ) ) ) ).
% INF_less_iff
thf(fact_4421_less__INF__D,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [Y: A,F3: B > A,A5: set @ B,I2: B] :
( ( ord_less @ A @ Y @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ A5 ) ) )
=> ( ( member @ B @ I2 @ A5 )
=> ( ord_less @ A @ Y @ ( F3 @ I2 ) ) ) ) ) ).
% less_INF_D
thf(fact_4422_nat__seg__image__imp__finite,axiom,
! [A: $tType,A5: set @ A,F3: nat > A,N: nat] :
( ( A5
= ( image2 @ nat @ A @ F3
@ ( collect @ nat
@ ^ [I4: nat] : ( ord_less @ nat @ I4 @ N ) ) ) )
=> ( finite_finite2 @ A @ A5 ) ) ).
% nat_seg_image_imp_finite
thf(fact_4423_finite__conv__nat__seg__image,axiom,
! [A: $tType] :
( ( finite_finite2 @ A )
= ( ^ [A7: set @ A] :
? [N2: nat,F4: nat > A] :
( A7
= ( image2 @ nat @ A @ F4
@ ( collect @ nat
@ ^ [I4: nat] : ( ord_less @ nat @ I4 @ N2 ) ) ) ) ) ) ).
% finite_conv_nat_seg_image
thf(fact_4424_image__constant,axiom,
! [A: $tType,B: $tType,X: A,A5: set @ A,C3: B] :
( ( member @ A @ X @ A5 )
=> ( ( image2 @ A @ B
@ ^ [X5: A] : C3
@ A5 )
= ( insert @ B @ C3 @ ( bot_bot @ ( set @ B ) ) ) ) ) ).
% image_constant
thf(fact_4425_image__constant__conv,axiom,
! [B: $tType,A: $tType,A5: set @ B,C3: A] :
( ( ( A5
= ( bot_bot @ ( set @ B ) ) )
=> ( ( image2 @ B @ A
@ ^ [X5: B] : C3
@ A5 )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( image2 @ B @ A
@ ^ [X5: B] : C3
@ A5 )
= ( insert @ A @ C3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% image_constant_conv
thf(fact_4426_sum_Oimage__gen,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [S2: set @ B,H2: B > A,G3: B > C] :
( ( finite_finite2 @ B @ S2 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ H2 @ S2 )
= ( groups7311177749621191930dd_sum @ C @ A
@ ^ [Y6: C] :
( groups7311177749621191930dd_sum @ B @ A @ H2
@ ( collect @ B
@ ^ [X5: B] :
( ( member @ B @ X5 @ S2 )
& ( ( G3 @ X5 )
= Y6 ) ) ) )
@ ( image2 @ B @ C @ G3 @ S2 ) ) ) ) ) ).
% sum.image_gen
thf(fact_4427_prod_Oimage__gen,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S2: set @ B,H2: B > A,G3: B > C] :
( ( finite_finite2 @ B @ S2 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ H2 @ S2 )
= ( groups7121269368397514597t_prod @ C @ A
@ ^ [Y6: C] :
( groups7121269368397514597t_prod @ B @ A @ H2
@ ( collect @ B
@ ^ [X5: B] :
( ( member @ B @ X5 @ S2 )
& ( ( G3 @ X5 )
= Y6 ) ) ) )
@ ( image2 @ B @ C @ G3 @ S2 ) ) ) ) ) ).
% prod.image_gen
thf(fact_4428_the__elem__image__unique,axiom,
! [B: $tType,A: $tType,A5: set @ A,F3: A > B,X: A] :
( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [Y3: A] :
( ( member @ A @ Y3 @ A5 )
=> ( ( F3 @ Y3 )
= ( F3 @ X ) ) )
=> ( ( the_elem @ B @ ( image2 @ A @ B @ F3 @ A5 ) )
= ( F3 @ X ) ) ) ) ).
% the_elem_image_unique
thf(fact_4429_le__SUP__iff,axiom,
! [B: $tType,A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [X: A,F3: B > A,A5: set @ B] :
( ( ord_less_eq @ A @ X @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A5 ) ) )
= ( ! [Y6: A] :
( ( ord_less @ A @ Y6 @ X )
=> ? [X5: B] :
( ( member @ B @ X5 @ A5 )
& ( ord_less @ A @ Y6 @ ( F3 @ X5 ) ) ) ) ) ) ) ).
% le_SUP_iff
thf(fact_4430_INF__le__iff,axiom,
! [B: $tType,A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [F3: B > A,A5: set @ B,X: A] :
( ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ A5 ) ) @ X )
= ( ! [Y6: A] :
( ( ord_less @ A @ X @ Y6 )
=> ? [X5: B] :
( ( member @ B @ X5 @ A5 )
& ( ord_less @ A @ ( F3 @ X5 ) @ Y6 ) ) ) ) ) ) ).
% INF_le_iff
thf(fact_4431_SUP__eq__iff,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I6: set @ B,C3: A,F3: B > A] :
( ( I6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [I3: B] :
( ( member @ B @ I3 @ I6 )
=> ( ord_less_eq @ A @ C3 @ ( F3 @ I3 ) ) )
=> ( ( ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ I6 ) )
= C3 )
= ( ! [X5: B] :
( ( member @ B @ X5 @ I6 )
=> ( ( F3 @ X5 )
= C3 ) ) ) ) ) ) ) ).
% SUP_eq_iff
thf(fact_4432_cSUP__least,axiom,
! [B: $tType,A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A5: set @ B,F3: B > A,M5: A] :
( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ A5 )
=> ( ord_less_eq @ A @ ( F3 @ X4 ) @ M5 ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A5 ) ) @ M5 ) ) ) ) ).
% cSUP_least
thf(fact_4433_INF__eq__iff,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I6: set @ B,F3: B > A,C3: A] :
( ( I6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [I3: B] :
( ( member @ B @ I3 @ I6 )
=> ( ord_less_eq @ A @ ( F3 @ I3 ) @ C3 ) )
=> ( ( ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ I6 ) )
= C3 )
= ( ! [X5: B] :
( ( member @ B @ X5 @ I6 )
=> ( ( F3 @ X5 )
= C3 ) ) ) ) ) ) ) ).
% INF_eq_iff
thf(fact_4434_cINF__greatest,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A5: set @ B,M2: A,F3: B > A] :
( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ A5 )
=> ( ord_less_eq @ A @ M2 @ ( F3 @ X4 ) ) )
=> ( ord_less_eq @ A @ M2 @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ A5 ) ) ) ) ) ) ).
% cINF_greatest
thf(fact_4435_card__image__le,axiom,
! [B: $tType,A: $tType,A5: set @ A,F3: A > B] :
( ( finite_finite2 @ A @ A5 )
=> ( ord_less_eq @ nat @ ( finite_card @ B @ ( image2 @ A @ B @ F3 @ A5 ) ) @ ( finite_card @ A @ A5 ) ) ) ).
% card_image_le
thf(fact_4436_prod_Odistrib__triv_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [I6: set @ B,G3: B > A,H2: B > A] :
( ( finite_finite2 @ B @ I6 )
=> ( ( groups1962203154675924110t_prod @ B @ A
@ ^ [I4: B] : ( times_times @ A @ ( G3 @ I4 ) @ ( H2 @ I4 ) )
@ I6 )
= ( times_times @ A @ ( groups1962203154675924110t_prod @ B @ A @ G3 @ I6 ) @ ( groups1962203154675924110t_prod @ B @ A @ H2 @ I6 ) ) ) ) ) ).
% prod.distrib_triv'
thf(fact_4437_SUP__subset__mono,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ B,B6: set @ B,F3: B > A,G3: B > A] :
( ( ord_less_eq @ ( set @ B ) @ A5 @ B6 )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ A5 )
=> ( ord_less_eq @ A @ ( F3 @ X4 ) @ ( G3 @ X4 ) ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A5 ) ) @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ G3 @ B6 ) ) ) ) ) ) ).
% SUP_subset_mono
thf(fact_4438_INF__superset__mono,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [B6: set @ B,A5: set @ B,F3: B > A,G3: B > A] :
( ( ord_less_eq @ ( set @ B ) @ B6 @ A5 )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ B6 )
=> ( ord_less_eq @ A @ ( F3 @ X4 ) @ ( G3 @ X4 ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ A5 ) ) @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ G3 @ B6 ) ) ) ) ) ) ).
% INF_superset_mono
thf(fact_4439_SUP__constant,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ B,C3: A] :
( ( ( A5
= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ A
@ ( image2 @ B @ A
@ ^ [Y6: B] : C3
@ A5 ) )
= ( bot_bot @ A ) ) )
& ( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ A
@ ( image2 @ B @ A
@ ^ [Y6: B] : C3
@ A5 ) )
= C3 ) ) ) ) ).
% SUP_constant
thf(fact_4440_SUP__empty,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F3: B > A] :
( ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ ( bot_bot @ ( set @ B ) ) ) )
= ( bot_bot @ A ) ) ) ).
% SUP_empty
thf(fact_4441_sum_Ogroup,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [S2: set @ B,T4: set @ C,G3: B > C,H2: B > A] :
( ( finite_finite2 @ B @ S2 )
=> ( ( finite_finite2 @ C @ T4 )
=> ( ( ord_less_eq @ ( set @ C ) @ ( image2 @ B @ C @ G3 @ S2 ) @ T4 )
=> ( ( groups7311177749621191930dd_sum @ C @ A
@ ^ [Y6: C] :
( groups7311177749621191930dd_sum @ B @ A @ H2
@ ( collect @ B
@ ^ [X5: B] :
( ( member @ B @ X5 @ S2 )
& ( ( G3 @ X5 )
= Y6 ) ) ) )
@ T4 )
= ( groups7311177749621191930dd_sum @ B @ A @ H2 @ S2 ) ) ) ) ) ) ).
% sum.group
thf(fact_4442_INF__inf__const1,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I6: set @ B,X: A,F3: B > A] :
( ( I6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Inf_Inf @ A
@ ( image2 @ B @ A
@ ^ [I4: B] : ( inf_inf @ A @ X @ ( F3 @ I4 ) )
@ I6 ) )
= ( inf_inf @ A @ X @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ I6 ) ) ) ) ) ) ).
% INF_inf_const1
thf(fact_4443_INF__inf__const2,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [I6: set @ B,F3: B > A,X: A] :
( ( I6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Inf_Inf @ A
@ ( image2 @ B @ A
@ ^ [I4: B] : ( inf_inf @ A @ ( F3 @ I4 ) @ X )
@ I6 ) )
= ( inf_inf @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ I6 ) ) @ X ) ) ) ) ).
% INF_inf_const2
thf(fact_4444_prod_Ogroup,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S2: set @ B,T4: set @ C,G3: B > C,H2: B > A] :
( ( finite_finite2 @ B @ S2 )
=> ( ( finite_finite2 @ C @ T4 )
=> ( ( ord_less_eq @ ( set @ C ) @ ( image2 @ B @ C @ G3 @ S2 ) @ T4 )
=> ( ( groups7121269368397514597t_prod @ C @ A
@ ^ [Y6: C] :
( groups7121269368397514597t_prod @ B @ A @ H2
@ ( collect @ B
@ ^ [X5: B] :
( ( member @ B @ X5 @ S2 )
& ( ( G3 @ X5 )
= Y6 ) ) ) )
@ T4 )
= ( groups7121269368397514597t_prod @ B @ A @ H2 @ S2 ) ) ) ) ) ) ).
% prod.group
thf(fact_4445_INF__le__SUP,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ B,F3: B > A] :
( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ A5 ) ) @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A5 ) ) ) ) ) ).
% INF_le_SUP
thf(fact_4446_surj__card__le,axiom,
! [B: $tType,A: $tType,A5: set @ A,B6: set @ B,F3: A > B] :
( ( finite_finite2 @ A @ A5 )
=> ( ( ord_less_eq @ ( set @ B ) @ B6 @ ( image2 @ A @ B @ F3 @ A5 ) )
=> ( ord_less_eq @ nat @ ( finite_card @ B @ B6 ) @ ( finite_card @ A @ A5 ) ) ) ) ).
% surj_card_le
thf(fact_4447_scaleR__image__atLeastAtMost,axiom,
! [A: $tType] :
( ( real_V5355595471888546746vector @ A )
=> ! [C3: real,X: A,Y: A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ C3 )
=> ( ( image2 @ A @ A @ ( real_V8093663219630862766scaleR @ A @ C3 ) @ ( set_or1337092689740270186AtMost @ A @ X @ Y ) )
= ( set_or1337092689740270186AtMost @ A @ ( real_V8093663219630862766scaleR @ A @ C3 @ X ) @ ( real_V8093663219630862766scaleR @ A @ C3 @ Y ) ) ) ) ) ).
% scaleR_image_atLeastAtMost
thf(fact_4448_image__Suc__lessThan,axiom,
! [N: nat] :
( ( image2 @ nat @ nat @ suc @ ( set_ord_lessThan @ nat @ N ) )
= ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ N ) ) ).
% image_Suc_lessThan
thf(fact_4449_image__Suc__atMost,axiom,
! [N: nat] :
( ( image2 @ nat @ nat @ suc @ ( set_ord_atMost @ nat @ N ) )
= ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ ( suc @ N ) ) ) ).
% image_Suc_atMost
thf(fact_4450_atLeast0__atMost__Suc__eq__insert__0,axiom,
! [N: nat] :
( ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) )
= ( insert @ nat @ ( zero_zero @ nat ) @ ( image2 @ nat @ nat @ suc @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% atLeast0_atMost_Suc_eq_insert_0
thf(fact_4451_atLeast0__lessThan__Suc__eq__insert__0,axiom,
! [N: nat] :
( ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) )
= ( insert @ nat @ ( zero_zero @ nat ) @ ( image2 @ nat @ nat @ suc @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% atLeast0_lessThan_Suc_eq_insert_0
thf(fact_4452_lessThan__Suc__eq__insert__0,axiom,
! [N: nat] :
( ( set_ord_lessThan @ nat @ ( suc @ N ) )
= ( insert @ nat @ ( zero_zero @ nat ) @ ( image2 @ nat @ nat @ suc @ ( set_ord_lessThan @ nat @ N ) ) ) ) ).
% lessThan_Suc_eq_insert_0
thf(fact_4453_atMost__Suc__eq__insert__0,axiom,
! [N: nat] :
( ( set_ord_atMost @ nat @ ( suc @ N ) )
= ( insert @ nat @ ( zero_zero @ nat ) @ ( image2 @ nat @ nat @ suc @ ( set_ord_atMost @ nat @ N ) ) ) ) ).
% atMost_Suc_eq_insert_0
thf(fact_4454_Fract__less__zero__iff,axiom,
! [B2: int,A2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
=> ( ( ord_less @ rat @ ( fract @ A2 @ B2 ) @ ( zero_zero @ rat ) )
= ( ord_less @ int @ A2 @ ( zero_zero @ int ) ) ) ) ).
% Fract_less_zero_iff
thf(fact_4455_zero__less__Fract__iff,axiom,
! [B2: int,A2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
=> ( ( ord_less @ rat @ ( zero_zero @ rat ) @ ( fract @ A2 @ B2 ) )
= ( ord_less @ int @ ( zero_zero @ int ) @ A2 ) ) ) ).
% zero_less_Fract_iff
thf(fact_4456_prod_Odistrib_H,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [I6: set @ B,G3: B > A,H2: B > A] :
( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X5: B] :
( ( member @ B @ X5 @ I6 )
& ( ( G3 @ X5 )
!= ( one_one @ A ) ) ) ) )
=> ( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X5: B] :
( ( member @ B @ X5 @ I6 )
& ( ( H2 @ X5 )
!= ( one_one @ A ) ) ) ) )
=> ( ( groups1962203154675924110t_prod @ B @ A
@ ^ [I4: B] : ( times_times @ A @ ( G3 @ I4 ) @ ( H2 @ I4 ) )
@ I6 )
= ( times_times @ A @ ( groups1962203154675924110t_prod @ B @ A @ G3 @ I6 ) @ ( groups1962203154675924110t_prod @ B @ A @ H2 @ I6 ) ) ) ) ) ) ).
% prod.distrib'
thf(fact_4457_one__less__Fract__iff,axiom,
! [B2: int,A2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
=> ( ( ord_less @ rat @ ( one_one @ rat ) @ ( fract @ A2 @ B2 ) )
= ( ord_less @ int @ B2 @ A2 ) ) ) ).
% one_less_Fract_iff
thf(fact_4458_Fract__less__one__iff,axiom,
! [B2: int,A2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
=> ( ( ord_less @ rat @ ( fract @ A2 @ B2 ) @ ( one_one @ rat ) )
= ( ord_less @ int @ A2 @ B2 ) ) ) ).
% Fract_less_one_iff
thf(fact_4459_prod_OG__def,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ( ( groups1962203154675924110t_prod @ B @ A )
= ( ^ [P6: B > A,I8: set @ B] :
( if @ A
@ ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X5: B] :
( ( member @ B @ X5 @ I8 )
& ( ( P6 @ X5 )
!= ( one_one @ A ) ) ) ) )
@ ( groups7121269368397514597t_prod @ B @ A @ P6
@ ( collect @ B
@ ^ [X5: B] :
( ( member @ B @ X5 @ I8 )
& ( ( P6 @ X5 )
!= ( one_one @ A ) ) ) ) )
@ ( one_one @ A ) ) ) ) ) ).
% prod.G_def
thf(fact_4460_zero__le__Fract__iff,axiom,
! [B2: int,A2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
=> ( ( ord_less_eq @ rat @ ( zero_zero @ rat ) @ ( fract @ A2 @ B2 ) )
= ( ord_less_eq @ int @ ( zero_zero @ int ) @ A2 ) ) ) ).
% zero_le_Fract_iff
thf(fact_4461_Fract__le__zero__iff,axiom,
! [B2: int,A2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
=> ( ( ord_less_eq @ rat @ ( fract @ A2 @ B2 ) @ ( zero_zero @ rat ) )
= ( ord_less_eq @ int @ A2 @ ( zero_zero @ int ) ) ) ) ).
% Fract_le_zero_iff
thf(fact_4462_Fract__le__one__iff,axiom,
! [B2: int,A2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
=> ( ( ord_less_eq @ rat @ ( fract @ A2 @ B2 ) @ ( one_one @ rat ) )
= ( ord_less_eq @ int @ A2 @ B2 ) ) ) ).
% Fract_le_one_iff
thf(fact_4463_one__le__Fract__iff,axiom,
! [B2: int,A2: int] :
( ( ord_less @ int @ ( zero_zero @ int ) @ B2 )
=> ( ( ord_less_eq @ rat @ ( one_one @ rat ) @ ( fract @ A2 @ B2 ) )
= ( ord_less_eq @ int @ B2 @ A2 ) ) ) ).
% one_le_Fract_iff
thf(fact_4464_image__mult__atLeastAtMost__if,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [C3: A,X: A,Y: A] :
( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( image2 @ A @ A @ ( times_times @ A @ C3 ) @ ( set_or1337092689740270186AtMost @ A @ X @ Y ) )
= ( set_or1337092689740270186AtMost @ A @ ( times_times @ A @ C3 @ X ) @ ( times_times @ A @ C3 @ Y ) ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( ( ord_less_eq @ A @ X @ Y )
=> ( ( image2 @ A @ A @ ( times_times @ A @ C3 ) @ ( set_or1337092689740270186AtMost @ A @ X @ Y ) )
= ( set_or1337092689740270186AtMost @ A @ ( times_times @ A @ C3 @ Y ) @ ( times_times @ A @ C3 @ X ) ) ) )
& ( ~ ( ord_less_eq @ A @ X @ Y )
=> ( ( image2 @ A @ A @ ( times_times @ A @ C3 ) @ ( set_or1337092689740270186AtMost @ A @ X @ Y ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% image_mult_atLeastAtMost_if
thf(fact_4465_image__mult__atLeastAtMost__if_H,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y: A,C3: A] :
( ( ( ord_less_eq @ A @ X @ Y )
=> ( ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( image2 @ A @ A
@ ^ [X5: A] : ( times_times @ A @ X5 @ C3 )
@ ( set_or1337092689740270186AtMost @ A @ X @ Y ) )
= ( set_or1337092689740270186AtMost @ A @ ( times_times @ A @ X @ C3 ) @ ( times_times @ A @ Y @ C3 ) ) ) )
& ( ~ ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( image2 @ A @ A
@ ^ [X5: A] : ( times_times @ A @ X5 @ C3 )
@ ( set_or1337092689740270186AtMost @ A @ X @ Y ) )
= ( set_or1337092689740270186AtMost @ A @ ( times_times @ A @ Y @ C3 ) @ ( times_times @ A @ X @ C3 ) ) ) ) ) )
& ( ~ ( ord_less_eq @ A @ X @ Y )
=> ( ( image2 @ A @ A
@ ^ [X5: A] : ( times_times @ A @ X5 @ C3 )
@ ( set_or1337092689740270186AtMost @ A @ X @ Y ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% image_mult_atLeastAtMost_if'
thf(fact_4466_image__affinity__atLeastAtMost,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A,M2: A,C3: A] :
( ( ( ( set_or1337092689740270186AtMost @ A @ A2 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( image2 @ A @ A
@ ^ [X5: A] : ( plus_plus @ A @ ( times_times @ A @ M2 @ X5 ) @ C3 )
@ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( ( set_or1337092689740270186AtMost @ A @ A2 @ B2 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ M2 )
=> ( ( image2 @ A @ A
@ ^ [X5: A] : ( plus_plus @ A @ ( times_times @ A @ M2 @ X5 ) @ C3 )
@ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ ( times_times @ A @ M2 @ A2 ) @ C3 ) @ ( plus_plus @ A @ ( times_times @ A @ M2 @ B2 ) @ C3 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ M2 )
=> ( ( image2 @ A @ A
@ ^ [X5: A] : ( plus_plus @ A @ ( times_times @ A @ M2 @ X5 ) @ C3 )
@ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ ( times_times @ A @ M2 @ B2 ) @ C3 ) @ ( plus_plus @ A @ ( times_times @ A @ M2 @ A2 ) @ C3 ) ) ) ) ) ) ) ) ).
% image_affinity_atLeastAtMost
thf(fact_4467_image__affinity__atLeastAtMost__diff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A,M2: A,C3: A] :
( ( ( ( set_or1337092689740270186AtMost @ A @ A2 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( image2 @ A @ A
@ ^ [X5: A] : ( minus_minus @ A @ ( times_times @ A @ M2 @ X5 ) @ C3 )
@ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( ( set_or1337092689740270186AtMost @ A @ A2 @ B2 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ M2 )
=> ( ( image2 @ A @ A
@ ^ [X5: A] : ( minus_minus @ A @ ( times_times @ A @ M2 @ X5 ) @ C3 )
@ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ ( times_times @ A @ M2 @ A2 ) @ C3 ) @ ( minus_minus @ A @ ( times_times @ A @ M2 @ B2 ) @ C3 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ M2 )
=> ( ( image2 @ A @ A
@ ^ [X5: A] : ( minus_minus @ A @ ( times_times @ A @ M2 @ X5 ) @ C3 )
@ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ ( times_times @ A @ M2 @ B2 ) @ C3 ) @ ( minus_minus @ A @ ( times_times @ A @ M2 @ A2 ) @ C3 ) ) ) ) ) ) ) ) ).
% image_affinity_atLeastAtMost_diff
thf(fact_4468_image__affinity__atLeastAtMost__div,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A,M2: A,C3: A] :
( ( ( ( set_or1337092689740270186AtMost @ A @ A2 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( image2 @ A @ A
@ ^ [X5: A] : ( plus_plus @ A @ ( divide_divide @ A @ X5 @ M2 ) @ C3 )
@ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( ( set_or1337092689740270186AtMost @ A @ A2 @ B2 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ M2 )
=> ( ( image2 @ A @ A
@ ^ [X5: A] : ( plus_plus @ A @ ( divide_divide @ A @ X5 @ M2 ) @ C3 )
@ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ ( divide_divide @ A @ A2 @ M2 ) @ C3 ) @ ( plus_plus @ A @ ( divide_divide @ A @ B2 @ M2 ) @ C3 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ M2 )
=> ( ( image2 @ A @ A
@ ^ [X5: A] : ( plus_plus @ A @ ( divide_divide @ A @ X5 @ M2 ) @ C3 )
@ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( plus_plus @ A @ ( divide_divide @ A @ B2 @ M2 ) @ C3 ) @ ( plus_plus @ A @ ( divide_divide @ A @ A2 @ M2 ) @ C3 ) ) ) ) ) ) ) ) ).
% image_affinity_atLeastAtMost_div
thf(fact_4469_image__affinity__atLeastAtMost__div__diff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A,M2: A,C3: A] :
( ( ( ( set_or1337092689740270186AtMost @ A @ A2 @ B2 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( image2 @ A @ A
@ ^ [X5: A] : ( minus_minus @ A @ ( divide_divide @ A @ X5 @ M2 ) @ C3 )
@ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( ( set_or1337092689740270186AtMost @ A @ A2 @ B2 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ M2 )
=> ( ( image2 @ A @ A
@ ^ [X5: A] : ( minus_minus @ A @ ( divide_divide @ A @ X5 @ M2 ) @ C3 )
@ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ ( divide_divide @ A @ A2 @ M2 ) @ C3 ) @ ( minus_minus @ A @ ( divide_divide @ A @ B2 @ M2 ) @ C3 ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ M2 )
=> ( ( image2 @ A @ A
@ ^ [X5: A] : ( minus_minus @ A @ ( divide_divide @ A @ X5 @ M2 ) @ C3 )
@ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
= ( set_or1337092689740270186AtMost @ A @ ( minus_minus @ A @ ( divide_divide @ A @ B2 @ M2 ) @ C3 ) @ ( minus_minus @ A @ ( divide_divide @ A @ A2 @ M2 ) @ C3 ) ) ) ) ) ) ) ) ).
% image_affinity_atLeastAtMost_div_diff
thf(fact_4470_sum__fun__comp,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( semiring_1 @ C )
=> ! [S2: set @ A,R: set @ B,G3: A > B,F3: B > C] :
( ( finite_finite2 @ A @ S2 )
=> ( ( finite_finite2 @ B @ R )
=> ( ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ G3 @ S2 ) @ R )
=> ( ( groups7311177749621191930dd_sum @ A @ C
@ ^ [X5: A] : ( F3 @ ( G3 @ X5 ) )
@ S2 )
= ( groups7311177749621191930dd_sum @ B @ C
@ ^ [Y6: B] :
( times_times @ C
@ ( semiring_1_of_nat @ C
@ ( finite_card @ A
@ ( collect @ A
@ ^ [X5: A] :
( ( member @ A @ X5 @ S2 )
& ( ( G3 @ X5 )
= Y6 ) ) ) ) )
@ ( F3 @ Y6 ) )
@ R ) ) ) ) ) ) ).
% sum_fun_comp
thf(fact_4471_nth__sorted__list__of__set__greaterThanLessThan,axiom,
! [N: nat,J: nat,I2: nat] :
( ( ord_less @ nat @ N @ ( minus_minus @ nat @ J @ ( suc @ I2 ) ) )
=> ( ( nth @ nat @ ( linord4507533701916653071of_set @ nat @ ( set_or5935395276787703475ssThan @ nat @ I2 @ J ) ) @ N )
= ( suc @ ( plus_plus @ nat @ I2 @ N ) ) ) ) ).
% nth_sorted_list_of_set_greaterThanLessThan
thf(fact_4472_INF__nat__binary,axiom,
! [A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A5: A,B6: A] :
( ( inf_inf @ A @ A5
@ ( complete_Inf_Inf @ A
@ ( image2 @ nat @ A
@ ^ [X5: nat] : B6
@ ( collect @ nat @ ( ord_less @ nat @ ( zero_zero @ nat ) ) ) ) ) )
= ( inf_inf @ A @ A5 @ B6 ) ) ) ).
% INF_nat_binary
thf(fact_4473_sorted__list__of__set_Osorted__key__list__of__set__insert__remove,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( linord4507533701916653071of_set @ A @ ( insert @ A @ X @ A5 ) )
= ( linorder_insort_key @ A @ A
@ ^ [X5: A] : X5
@ X
@ ( linord4507533701916653071of_set @ A @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_insert_remove
thf(fact_4474_sorted__list__of__set__def,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( linord4507533701916653071of_set @ A )
= ( linord144544945434240204of_set @ A @ A
@ ^ [X5: A] : X5 ) ) ) ).
% sorted_list_of_set_def
thf(fact_4475_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
@ ^ [I4: nat,J3: nat] : ( minus_minus @ A @ ( semiring_1_of_nat @ A @ I4 ) @ ( semiring_1_of_nat @ A @ J3 ) ) ) ) ) ) ).
% ring_1_class.of_int_def
thf(fact_4476_of__nat__eq__id,axiom,
( ( semiring_1_of_nat @ nat )
= ( id @ nat ) ) ).
% of_nat_eq_id
thf(fact_4477_case__prod__Pair,axiom,
! [B: $tType,A: $tType] :
( ( product_case_prod @ A @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B ) )
= ( id @ ( product_prod @ A @ B ) ) ) ).
% case_prod_Pair
thf(fact_4478_id__funpow,axiom,
! [A: $tType,N: nat] :
( ( compow @ ( A > A ) @ N @ ( id @ A ) )
= ( id @ A ) ) ).
% id_funpow
thf(fact_4479_remove1__insort__key,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [X: B,F3: B > A,Xs: list @ B] :
( ( remove1 @ B @ X @ ( linorder_insort_key @ B @ A @ F3 @ X @ Xs ) )
= Xs ) ) ).
% remove1_insort_key
thf(fact_4480_pair__imageI,axiom,
! [C: $tType,B: $tType,A: $tType,A2: A,B2: B,A5: set @ ( product_prod @ A @ B ),F3: A > B > C] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ A5 )
=> ( member @ C @ ( F3 @ A2 @ B2 ) @ ( image2 @ ( product_prod @ A @ B ) @ C @ ( product_case_prod @ A @ B @ C @ F3 ) @ A5 ) ) ) ).
% pair_imageI
thf(fact_4481_finite__UN,axiom,
! [B: $tType,A: $tType,A5: set @ A,B6: A > ( set @ B )] :
( ( finite_finite2 @ A @ A5 )
=> ( ( finite_finite2 @ B @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ B6 @ A5 ) ) )
= ( ! [X5: A] :
( ( member @ A @ X5 @ A5 )
=> ( finite_finite2 @ B @ ( B6 @ X5 ) ) ) ) ) ) ).
% finite_UN
thf(fact_4482_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_4483_drop__bit__0,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ( ( bit_se4197421643247451524op_bit @ A @ ( zero_zero @ nat ) )
= ( id @ A ) ) ) ).
% drop_bit_0
thf(fact_4484_length__insort,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F3: B > A,X: B,Xs: list @ B] :
( ( size_size @ ( list @ B ) @ ( linorder_insort_key @ B @ A @ F3 @ X @ Xs ) )
= ( suc @ ( size_size @ ( list @ B ) @ Xs ) ) ) ) ).
% length_insort
thf(fact_4485_UN__constant,axiom,
! [B: $tType,A: $tType,A5: set @ B,C3: set @ A] :
( ( ( A5
= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [Y6: B] : C3
@ A5 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [Y6: B] : C3
@ A5 ) )
= C3 ) ) ) ).
% UN_constant
thf(fact_4486_finite__UN__I,axiom,
! [B: $tType,A: $tType,A5: set @ A,B6: A > ( set @ B )] :
( ( finite_finite2 @ A @ A5 )
=> ( ! [A4: A] :
( ( member @ A @ A4 @ A5 )
=> ( finite_finite2 @ B @ ( B6 @ A4 ) ) )
=> ( finite_finite2 @ B @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ B6 @ A5 ) ) ) ) ) ).
% finite_UN_I
thf(fact_4487_finite__INT,axiom,
! [B: $tType,A: $tType,I6: set @ A,A5: A > ( set @ B )] :
( ? [X3: A] :
( ( member @ A @ X3 @ I6 )
& ( finite_finite2 @ B @ ( A5 @ X3 ) ) )
=> ( finite_finite2 @ B @ ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ A5 @ I6 ) ) ) ) ).
% finite_INT
thf(fact_4488_UN__simps_I1_J,axiom,
! [A: $tType,B: $tType,C5: set @ B,A2: A,B6: B > ( set @ A )] :
( ( ( C5
= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [X5: B] : ( insert @ A @ A2 @ ( B6 @ X5 ) )
@ C5 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( C5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [X5: B] : ( insert @ A @ A2 @ ( B6 @ X5 ) )
@ C5 ) )
= ( insert @ A @ A2 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B6 @ C5 ) ) ) ) ) ) ).
% UN_simps(1)
thf(fact_4489_UN__singleton,axiom,
! [A: $tType,A5: set @ A] :
( ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ A @ ( set @ A )
@ ^ [X5: A] : ( insert @ A @ X5 @ ( bot_bot @ ( set @ A ) ) )
@ A5 ) )
= A5 ) ).
% UN_singleton
thf(fact_4490_sorted__list__of__set_Osorted__key__list__of__set__insert,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ~ ( member @ A @ X @ A5 )
=> ( ( linord4507533701916653071of_set @ A @ ( insert @ A @ X @ A5 ) )
= ( linorder_insort_key @ A @ A
@ ^ [X5: A] : X5
@ X
@ ( linord4507533701916653071of_set @ A @ A5 ) ) ) ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_insert
thf(fact_4491_set__concat,axiom,
! [A: $tType,Xs: list @ ( list @ A )] :
( ( set2 @ A @ ( concat @ A @ Xs ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ ( list @ A ) @ ( set @ A ) @ ( set2 @ A ) @ ( set2 @ ( list @ A ) @ Xs ) ) ) ) ).
% set_concat
thf(fact_4492_Inf__INT__eq2,axiom,
! [B: $tType,A: $tType] :
( ( complete_Inf_Inf @ ( A > B > $o ) )
= ( ^ [S7: set @ ( A > B > $o ),X5: A,Y6: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X5 @ Y6 ) @ ( complete_Inf_Inf @ ( set @ ( product_prod @ A @ B ) ) @ ( image2 @ ( ( product_prod @ A @ B ) > $o ) @ ( set @ ( product_prod @ A @ B ) ) @ ( collect @ ( product_prod @ A @ B ) ) @ ( image2 @ ( A > B > $o ) @ ( ( product_prod @ A @ B ) > $o ) @ ( product_case_prod @ A @ B @ $o ) @ S7 ) ) ) ) ) ) ).
% Inf_INT_eq2
thf(fact_4493_Sup__SUP__eq2,axiom,
! [B: $tType,A: $tType] :
( ( complete_Sup_Sup @ ( A > B > $o ) )
= ( ^ [S7: set @ ( A > B > $o ),X5: A,Y6: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X5 @ Y6 ) @ ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) ) @ ( image2 @ ( ( product_prod @ A @ B ) > $o ) @ ( set @ ( product_prod @ A @ B ) ) @ ( collect @ ( product_prod @ A @ B ) ) @ ( image2 @ ( A > B > $o ) @ ( ( product_prod @ A @ B ) > $o ) @ ( product_case_prod @ A @ B @ $o ) @ S7 ) ) ) ) ) ) ).
% Sup_SUP_eq2
thf(fact_4494_SUP__UN__eq2,axiom,
! [B: $tType,C: $tType,A: $tType,R2: C > ( set @ ( product_prod @ A @ B ) ),S2: set @ C] :
( ( complete_Sup_Sup @ ( A > B > $o )
@ ( image2 @ C @ ( A > B > $o )
@ ^ [I4: C,X5: A,Y6: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X5 @ Y6 ) @ ( R2 @ I4 ) )
@ S2 ) )
= ( ^ [X5: A,Y6: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X5 @ Y6 ) @ ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) ) @ ( image2 @ C @ ( set @ ( product_prod @ A @ B ) ) @ R2 @ S2 ) ) ) ) ) ).
% SUP_UN_eq2
thf(fact_4495_INF__INT__eq2,axiom,
! [B: $tType,C: $tType,A: $tType,R2: C > ( set @ ( product_prod @ A @ B ) ),S2: set @ C] :
( ( complete_Inf_Inf @ ( A > B > $o )
@ ( image2 @ C @ ( A > B > $o )
@ ^ [I4: C,X5: A,Y6: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X5 @ Y6 ) @ ( R2 @ I4 ) )
@ S2 ) )
= ( ^ [X5: A,Y6: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X5 @ Y6 ) @ ( complete_Inf_Inf @ ( set @ ( product_prod @ A @ B ) ) @ ( image2 @ C @ ( set @ ( product_prod @ A @ B ) ) @ R2 @ S2 ) ) ) ) ) ).
% INF_INT_eq2
thf(fact_4496_INF__Int__eq2,axiom,
! [B: $tType,A: $tType,S2: set @ ( set @ ( product_prod @ A @ B ) )] :
( ( complete_Inf_Inf @ ( A > B > $o )
@ ( image2 @ ( set @ ( product_prod @ A @ B ) ) @ ( A > B > $o )
@ ^ [I4: set @ ( product_prod @ A @ B ),X5: A,Y6: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X5 @ Y6 ) @ I4 )
@ S2 ) )
= ( ^ [X5: A,Y6: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X5 @ Y6 ) @ ( complete_Inf_Inf @ ( set @ ( product_prod @ A @ B ) ) @ S2 ) ) ) ) ).
% INF_Int_eq2
thf(fact_4497_SUP__Sup__eq2,axiom,
! [B: $tType,A: $tType,S2: set @ ( set @ ( product_prod @ A @ B ) )] :
( ( complete_Sup_Sup @ ( A > B > $o )
@ ( image2 @ ( set @ ( product_prod @ A @ B ) ) @ ( A > B > $o )
@ ^ [I4: set @ ( product_prod @ A @ B ),X5: A,Y6: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X5 @ Y6 ) @ I4 )
@ S2 ) )
= ( ^ [X5: A,Y6: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X5 @ Y6 ) @ ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) ) @ S2 ) ) ) ) ).
% SUP_Sup_eq2
thf(fact_4498_insort__left__comm,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A,Xs: list @ A] :
( ( linorder_insort_key @ A @ A
@ ^ [X5: A] : X5
@ X
@ ( linorder_insort_key @ A @ A
@ ^ [X5: A] : X5
@ Y
@ Xs ) )
= ( linorder_insort_key @ A @ A
@ ^ [X5: A] : X5
@ Y
@ ( linorder_insort_key @ A @ A
@ ^ [X5: A] : X5
@ X
@ Xs ) ) ) ) ).
% insort_left_comm
thf(fact_4499_insort__key__left__comm,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F3: B > A,X: B,Y: B,Xs: list @ B] :
( ( ( F3 @ X )
!= ( F3 @ Y ) )
=> ( ( linorder_insort_key @ B @ A @ F3 @ Y @ ( linorder_insort_key @ B @ A @ F3 @ X @ Xs ) )
= ( linorder_insort_key @ B @ A @ F3 @ X @ ( linorder_insort_key @ B @ A @ F3 @ Y @ Xs ) ) ) ) ) ).
% insort_key_left_comm
thf(fact_4500_insort__not__Nil,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F3: B > A,A2: B,Xs: list @ B] :
( ( linorder_insort_key @ B @ A @ F3 @ A2 @ Xs )
!= ( nil @ B ) ) ) ).
% insort_not_Nil
thf(fact_4501_funpow__simps__right_I1_J,axiom,
! [A: $tType,F3: A > A] :
( ( compow @ ( A > A ) @ ( zero_zero @ nat ) @ F3 )
= ( id @ A ) ) ).
% funpow_simps_right(1)
thf(fact_4502_None__notin__image__Some,axiom,
! [A: $tType,A5: set @ A] :
~ ( member @ ( option @ A ) @ ( none @ A ) @ ( image2 @ A @ ( option @ A ) @ ( some @ A ) @ A5 ) ) ).
% None_notin_image_Some
thf(fact_4503_set__insort__key,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F3: B > A,X: B,Xs: list @ B] :
( ( set2 @ B @ ( linorder_insort_key @ B @ A @ F3 @ X @ Xs ) )
= ( insert @ B @ X @ ( set2 @ B @ Xs ) ) ) ) ).
% set_insort_key
thf(fact_4504_distinct__insort,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F3: B > A,X: B,Xs: list @ B] :
( ( distinct @ B @ ( linorder_insort_key @ B @ A @ F3 @ X @ Xs ) )
= ( ~ ( member @ B @ X @ ( set2 @ B @ Xs ) )
& ( distinct @ B @ Xs ) ) ) ) ).
% distinct_insort
thf(fact_4505_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 )
@ ^ [X5: nat,Y6: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U2: nat,V5: nat] : ( ord_less @ nat @ ( plus_plus @ nat @ X5 @ V5 ) @ ( plus_plus @ nat @ U2 @ Y6 ) ) ) ) ) ) ).
% less_int_def
thf(fact_4506_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 )
@ ^ [X5: nat,Y6: nat] :
( product_case_prod @ nat @ nat @ $o
@ ^ [U2: nat,V5: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ X5 @ V5 ) @ ( plus_plus @ nat @ U2 @ Y6 ) ) ) ) ) ) ).
% less_eq_int_def
thf(fact_4507_INF__filter__not__bot,axiom,
! [I7: $tType,A: $tType,B6: set @ I7,F5: I7 > ( filter @ A )] :
( ! [X17: set @ I7] :
( ( ord_less_eq @ ( set @ I7 ) @ X17 @ B6 )
=> ( ( finite_finite2 @ I7 @ X17 )
=> ( ( complete_Inf_Inf @ ( filter @ A ) @ ( image2 @ I7 @ ( filter @ A ) @ F5 @ X17 ) )
!= ( bot_bot @ ( filter @ A ) ) ) ) )
=> ( ( complete_Inf_Inf @ ( filter @ A ) @ ( image2 @ I7 @ ( filter @ A ) @ F5 @ B6 ) )
!= ( bot_bot @ ( filter @ A ) ) ) ) ).
% INF_filter_not_bot
thf(fact_4508_finite__int__iff__bounded__le,axiom,
( ( finite_finite2 @ int )
= ( ^ [S7: set @ int] :
? [K3: int] : ( ord_less_eq @ ( set @ int ) @ ( image2 @ int @ int @ ( abs_abs @ int ) @ S7 ) @ ( set_ord_atMost @ int @ K3 ) ) ) ) ).
% finite_int_iff_bounded_le
thf(fact_4509_finite__int__iff__bounded,axiom,
( ( finite_finite2 @ int )
= ( ^ [S7: set @ int] :
? [K3: int] : ( ord_less_eq @ ( set @ int ) @ ( image2 @ int @ int @ ( abs_abs @ int ) @ S7 ) @ ( set_ord_lessThan @ int @ K3 ) ) ) ) ).
% finite_int_iff_bounded
thf(fact_4510_UN__empty2,axiom,
! [B: $tType,A: $tType,A5: set @ B] :
( ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [X5: B] : ( bot_bot @ ( set @ A ) )
@ A5 ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% UN_empty2
thf(fact_4511_UN__empty,axiom,
! [B: $tType,A: $tType,B6: B > ( set @ A )] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B6 @ ( bot_bot @ ( set @ B ) ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% UN_empty
thf(fact_4512_UNION__empty__conv_I1_J,axiom,
! [A: $tType,B: $tType,B6: B > ( set @ A ),A5: set @ B] :
( ( ( bot_bot @ ( set @ A ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B6 @ A5 ) ) )
= ( ! [X5: B] :
( ( member @ B @ X5 @ A5 )
=> ( ( B6 @ X5 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% UNION_empty_conv(1)
thf(fact_4513_UNION__empty__conv_I2_J,axiom,
! [A: $tType,B: $tType,B6: B > ( set @ A ),A5: set @ B] :
( ( ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B6 @ A5 ) )
= ( bot_bot @ ( set @ A ) ) )
= ( ! [X5: B] :
( ( member @ B @ X5 @ A5 )
=> ( ( B6 @ X5 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% UNION_empty_conv(2)
thf(fact_4514_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_4515_in__image__insert__iff,axiom,
! [A: $tType,B6: set @ ( set @ A ),X: A,A5: set @ A] :
( ! [C6: set @ A] :
( ( member @ ( set @ A ) @ C6 @ B6 )
=> ~ ( member @ A @ X @ C6 ) )
=> ( ( member @ ( set @ A ) @ A5 @ ( image2 @ ( set @ A ) @ ( set @ A ) @ ( insert @ A @ X ) @ B6 ) )
= ( ( member @ A @ X @ A5 )
& ( member @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ B6 ) ) ) ) ).
% in_image_insert_iff
thf(fact_4516_image__int__atLeastAtMost,axiom,
! [A2: nat,B2: nat] :
( ( image2 @ nat @ int @ ( semiring_1_of_nat @ int ) @ ( set_or1337092689740270186AtMost @ nat @ A2 @ B2 ) )
= ( set_or1337092689740270186AtMost @ int @ ( semiring_1_of_nat @ int @ A2 ) @ ( semiring_1_of_nat @ int @ B2 ) ) ) ).
% image_int_atLeastAtMost
thf(fact_4517_UN__extend__simps_I1_J,axiom,
! [A: $tType,B: $tType,C5: set @ B,A2: A,B6: B > ( set @ A )] :
( ( ( C5
= ( bot_bot @ ( set @ B ) ) )
=> ( ( insert @ A @ A2 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B6 @ C5 ) ) )
= ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) )
& ( ( C5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( insert @ A @ A2 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B6 @ C5 ) ) )
= ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [X5: B] : ( insert @ A @ A2 @ ( B6 @ X5 ) )
@ C5 ) ) ) ) ) ).
% UN_extend_simps(1)
thf(fact_4518_image__int__atLeastLessThan,axiom,
! [A2: nat,B2: nat] :
( ( image2 @ nat @ int @ ( semiring_1_of_nat @ int ) @ ( set_or7035219750837199246ssThan @ nat @ A2 @ B2 ) )
= ( set_or7035219750837199246ssThan @ int @ ( semiring_1_of_nat @ int @ A2 ) @ ( semiring_1_of_nat @ int @ B2 ) ) ) ).
% image_int_atLeastLessThan
thf(fact_4519_INT__extend__simps_I1_J,axiom,
! [B: $tType,A: $tType,C5: set @ A,A5: A > ( set @ B ),B6: set @ B] :
( ( ( C5
= ( bot_bot @ ( set @ A ) ) )
=> ( ( inf_inf @ ( set @ B ) @ ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ A5 @ C5 ) ) @ B6 )
= B6 ) )
& ( ( C5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( inf_inf @ ( set @ B ) @ ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ A5 @ C5 ) ) @ B6 )
= ( complete_Inf_Inf @ ( set @ B )
@ ( image2 @ A @ ( set @ B )
@ ^ [X5: A] : ( inf_inf @ ( set @ B ) @ ( A5 @ X5 ) @ B6 )
@ C5 ) ) ) ) ) ).
% INT_extend_simps(1)
thf(fact_4520_INT__extend__simps_I2_J,axiom,
! [C: $tType,D: $tType,C5: set @ D,A5: set @ C,B6: D > ( set @ C )] :
( ( ( C5
= ( bot_bot @ ( set @ D ) ) )
=> ( ( inf_inf @ ( set @ C ) @ A5 @ ( complete_Inf_Inf @ ( set @ C ) @ ( image2 @ D @ ( set @ C ) @ B6 @ C5 ) ) )
= A5 ) )
& ( ( C5
!= ( bot_bot @ ( set @ D ) ) )
=> ( ( inf_inf @ ( set @ C ) @ A5 @ ( complete_Inf_Inf @ ( set @ C ) @ ( image2 @ D @ ( set @ C ) @ B6 @ C5 ) ) )
= ( complete_Inf_Inf @ ( set @ C )
@ ( image2 @ D @ ( set @ C )
@ ^ [X5: D] : ( inf_inf @ ( set @ C ) @ A5 @ ( B6 @ X5 ) )
@ C5 ) ) ) ) ) ).
% INT_extend_simps(2)
thf(fact_4521_Int__Inter__eq_I1_J,axiom,
! [A: $tType,B11: set @ ( set @ A ),A5: set @ A] :
( ( ( B11
= ( bot_bot @ ( set @ ( set @ A ) ) ) )
=> ( ( inf_inf @ ( set @ A ) @ A5 @ ( complete_Inf_Inf @ ( set @ A ) @ B11 ) )
= A5 ) )
& ( ( B11
!= ( bot_bot @ ( set @ ( set @ A ) ) ) )
=> ( ( inf_inf @ ( set @ A ) @ A5 @ ( complete_Inf_Inf @ ( set @ A ) @ B11 ) )
= ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ ( set @ A ) @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ A5 ) @ B11 ) ) ) ) ) ).
% Int_Inter_eq(1)
thf(fact_4522_Int__Inter__eq_I2_J,axiom,
! [A: $tType,B11: set @ ( set @ A ),A5: set @ A] :
( ( ( B11
= ( bot_bot @ ( set @ ( set @ A ) ) ) )
=> ( ( inf_inf @ ( set @ A ) @ ( complete_Inf_Inf @ ( set @ A ) @ B11 ) @ A5 )
= A5 ) )
& ( ( B11
!= ( bot_bot @ ( set @ ( set @ A ) ) ) )
=> ( ( inf_inf @ ( set @ A ) @ ( complete_Inf_Inf @ ( set @ A ) @ B11 ) @ A5 )
= ( complete_Inf_Inf @ ( set @ A )
@ ( image2 @ ( set @ A ) @ ( set @ A )
@ ^ [B8: set @ A] : ( inf_inf @ ( set @ A ) @ B8 @ A5 )
@ B11 ) ) ) ) ) ).
% Int_Inter_eq(2)
thf(fact_4523_Collect__split__mono__strong,axiom,
! [B: $tType,A: $tType,X7: set @ A,A5: set @ ( product_prod @ A @ B ),Y8: set @ B,P2: A > B > $o,Q: A > B > $o] :
( ( X7
= ( image2 @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ A5 ) )
=> ( ( Y8
= ( image2 @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ A5 ) )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ X7 )
=> ! [Xa3: B] :
( ( member @ B @ Xa3 @ Y8 )
=> ( ( P2 @ X4 @ Xa3 )
=> ( Q @ X4 @ Xa3 ) ) ) )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ A5 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ P2 ) ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ A5 @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ Q ) ) ) ) ) ) ) ).
% Collect_split_mono_strong
thf(fact_4524_INT__extend__simps_I4_J,axiom,
! [G: $tType,H4: $tType,C5: set @ H4,A5: set @ G,B6: H4 > ( set @ G )] :
( ( ( C5
= ( bot_bot @ ( set @ H4 ) ) )
=> ( ( minus_minus @ ( set @ G ) @ A5 @ ( complete_Sup_Sup @ ( set @ G ) @ ( image2 @ H4 @ ( set @ G ) @ B6 @ C5 ) ) )
= A5 ) )
& ( ( C5
!= ( bot_bot @ ( set @ H4 ) ) )
=> ( ( minus_minus @ ( set @ G ) @ A5 @ ( complete_Sup_Sup @ ( set @ G ) @ ( image2 @ H4 @ ( set @ G ) @ B6 @ C5 ) ) )
= ( complete_Inf_Inf @ ( set @ G )
@ ( image2 @ H4 @ ( set @ G )
@ ^ [X5: H4] : ( minus_minus @ ( set @ G ) @ A5 @ ( B6 @ X5 ) )
@ C5 ) ) ) ) ) ).
% INT_extend_simps(4)
thf(fact_4525_UN__le__add__shift__strict,axiom,
! [A: $tType,M5: nat > ( set @ A ),K2: nat,N: nat] :
( ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ nat @ ( set @ A )
@ ^ [I4: nat] : ( M5 @ ( plus_plus @ nat @ I4 @ K2 ) )
@ ( set_ord_lessThan @ nat @ N ) ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ M5 @ ( set_or7035219750837199246ssThan @ nat @ K2 @ ( plus_plus @ nat @ N @ K2 ) ) ) ) ) ).
% UN_le_add_shift_strict
thf(fact_4526_UN__le__add__shift,axiom,
! [A: $tType,M5: nat > ( set @ A ),K2: nat,N: nat] :
( ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ nat @ ( set @ A )
@ ^ [I4: nat] : ( M5 @ ( plus_plus @ nat @ I4 @ K2 ) )
@ ( set_ord_atMost @ nat @ N ) ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ M5 @ ( set_or1337092689740270186AtMost @ nat @ K2 @ ( plus_plus @ nat @ N @ K2 ) ) ) ) ) ).
% UN_le_add_shift
thf(fact_4527_subset__subseqs,axiom,
! [A: $tType,X7: set @ A,Xs: list @ A] :
( ( ord_less_eq @ ( set @ A ) @ X7 @ ( set2 @ A @ Xs ) )
=> ( member @ ( set @ A ) @ X7 @ ( image2 @ ( list @ A ) @ ( set @ A ) @ ( set2 @ A ) @ ( set2 @ ( list @ A ) @ ( subseqs @ A @ Xs ) ) ) ) ) ).
% subset_subseqs
thf(fact_4528_subseqs__powset,axiom,
! [A: $tType,Xs: list @ A] :
( ( image2 @ ( list @ A ) @ ( set @ A ) @ ( set2 @ A ) @ ( set2 @ ( list @ A ) @ ( subseqs @ A @ Xs ) ) )
= ( pow2 @ A @ ( set2 @ A @ Xs ) ) ) ).
% subseqs_powset
thf(fact_4529_image__add__int__atLeastLessThan,axiom,
! [L: int,U: int] :
( ( image2 @ int @ int
@ ^ [X5: int] : ( plus_plus @ int @ X5 @ L )
@ ( set_or7035219750837199246ssThan @ int @ ( zero_zero @ int ) @ ( minus_minus @ int @ U @ L ) ) )
= ( set_or7035219750837199246ssThan @ int @ L @ U ) ) ).
% image_add_int_atLeastLessThan
thf(fact_4530_sum_OUNION__disjoint,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [I6: set @ B,A5: B > ( set @ C ),G3: C > A] :
( ( finite_finite2 @ B @ I6 )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ I6 )
=> ( finite_finite2 @ C @ ( A5 @ X4 ) ) )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ I6 )
=> ! [Xa3: B] :
( ( member @ B @ Xa3 @ I6 )
=> ( ( X4 != Xa3 )
=> ( ( inf_inf @ ( set @ C ) @ ( A5 @ X4 ) @ ( A5 @ Xa3 ) )
= ( bot_bot @ ( set @ C ) ) ) ) ) )
=> ( ( groups7311177749621191930dd_sum @ C @ A @ G3 @ ( complete_Sup_Sup @ ( set @ C ) @ ( image2 @ B @ ( set @ C ) @ A5 @ I6 ) ) )
= ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X5: B] : ( groups7311177749621191930dd_sum @ C @ A @ G3 @ ( A5 @ X5 ) )
@ I6 ) ) ) ) ) ) ).
% sum.UNION_disjoint
thf(fact_4531_prod_OUNION__disjoint,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [I6: set @ B,A5: B > ( set @ C ),G3: C > A] :
( ( finite_finite2 @ B @ I6 )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ I6 )
=> ( finite_finite2 @ C @ ( A5 @ X4 ) ) )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ I6 )
=> ! [Xa3: B] :
( ( member @ B @ Xa3 @ I6 )
=> ( ( X4 != Xa3 )
=> ( ( inf_inf @ ( set @ C ) @ ( A5 @ X4 ) @ ( A5 @ Xa3 ) )
= ( bot_bot @ ( set @ C ) ) ) ) ) )
=> ( ( groups7121269368397514597t_prod @ C @ A @ G3 @ ( complete_Sup_Sup @ ( set @ C ) @ ( image2 @ B @ ( set @ C ) @ A5 @ I6 ) ) )
= ( groups7121269368397514597t_prod @ B @ A
@ ^ [X5: B] : ( groups7121269368397514597t_prod @ C @ A @ G3 @ ( A5 @ X5 ) )
@ I6 ) ) ) ) ) ) ).
% prod.UNION_disjoint
thf(fact_4532_card__UN__le,axiom,
! [B: $tType,A: $tType,I6: set @ A,A5: A > ( set @ B )] :
( ( finite_finite2 @ A @ I6 )
=> ( ord_less_eq @ nat @ ( finite_card @ B @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ A5 @ I6 ) ) )
@ ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [I4: A] : ( finite_card @ B @ ( A5 @ I4 ) )
@ I6 ) ) ) ).
% card_UN_le
thf(fact_4533_sorted__list__of__set_Ofold__insort__key_Oremove,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( member @ A @ X @ A5 )
=> ( ( linord4507533701916653071of_set @ A @ A5 )
= ( linorder_insort_key @ A @ A
@ ^ [X5: A] : X5
@ X
@ ( linord4507533701916653071of_set @ A @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ).
% sorted_list_of_set.fold_insort_key.remove
thf(fact_4534_image__atLeastZeroLessThan__int,axiom,
! [U: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ U )
=> ( ( set_or7035219750837199246ssThan @ int @ ( zero_zero @ int ) @ U )
= ( image2 @ nat @ int @ ( semiring_1_of_nat @ int ) @ ( set_ord_lessThan @ nat @ ( nat2 @ U ) ) ) ) ) ).
% image_atLeastZeroLessThan_int
thf(fact_4535_card__UN__disjoint,axiom,
! [B: $tType,A: $tType,I6: set @ A,A5: A > ( set @ B )] :
( ( finite_finite2 @ A @ I6 )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ I6 )
=> ( finite_finite2 @ B @ ( A5 @ X4 ) ) )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ I6 )
=> ! [Xa3: A] :
( ( member @ A @ Xa3 @ I6 )
=> ( ( X4 != Xa3 )
=> ( ( inf_inf @ ( set @ B ) @ ( A5 @ X4 ) @ ( A5 @ Xa3 ) )
= ( bot_bot @ ( set @ B ) ) ) ) ) )
=> ( ( finite_card @ B @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ A5 @ I6 ) ) )
= ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [I4: A] : ( finite_card @ B @ ( A5 @ I4 ) )
@ I6 ) ) ) ) ) ).
% card_UN_disjoint
thf(fact_4536_finite__mono__strict__prefix__implies__finite__fixpoint,axiom,
! [A: $tType,F3: nat > ( set @ A ),S2: set @ A] :
( ! [I3: nat] : ( ord_less_eq @ ( set @ A ) @ ( F3 @ I3 ) @ S2 )
=> ( ( finite_finite2 @ A @ S2 )
=> ( ? [N8: nat] :
( ! [N3: nat] :
( ( ord_less_eq @ nat @ N3 @ N8 )
=> ! [M: nat] :
( ( ord_less_eq @ nat @ M @ N8 )
=> ( ( ord_less @ nat @ M @ N3 )
=> ( ord_less @ ( set @ A ) @ ( F3 @ M ) @ ( F3 @ N3 ) ) ) ) )
& ! [N3: nat] :
( ( ord_less_eq @ nat @ N8 @ N3 )
=> ( ( F3 @ N8 )
= ( F3 @ N3 ) ) ) )
=> ( ( F3 @ ( finite_card @ A @ S2 ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ F3 @ ( top_top @ ( set @ nat ) ) ) ) ) ) ) ) ).
% finite_mono_strict_prefix_implies_finite_fixpoint
thf(fact_4537_length__remdups__concat,axiom,
! [A: $tType,Xss: list @ ( list @ A )] :
( ( size_size @ ( list @ A ) @ ( remdups @ A @ ( concat @ A @ Xss ) ) )
= ( finite_card @ A @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ ( list @ A ) @ ( set @ A ) @ ( set2 @ A ) @ ( set2 @ ( list @ A ) @ Xss ) ) ) ) ) ).
% length_remdups_concat
thf(fact_4538_INT__simps_I4_J,axiom,
! [G: $tType,H4: $tType,C5: set @ H4,A5: set @ G,B6: H4 > ( set @ G )] :
( ( ( C5
= ( bot_bot @ ( set @ H4 ) ) )
=> ( ( complete_Inf_Inf @ ( set @ G )
@ ( image2 @ H4 @ ( set @ G )
@ ^ [X5: H4] : ( minus_minus @ ( set @ G ) @ A5 @ ( B6 @ X5 ) )
@ C5 ) )
= ( top_top @ ( set @ G ) ) ) )
& ( ( C5
!= ( bot_bot @ ( set @ H4 ) ) )
=> ( ( complete_Inf_Inf @ ( set @ G )
@ ( image2 @ H4 @ ( set @ G )
@ ^ [X5: H4] : ( minus_minus @ ( set @ G ) @ A5 @ ( B6 @ X5 ) )
@ C5 ) )
= ( minus_minus @ ( set @ G ) @ A5 @ ( complete_Sup_Sup @ ( set @ G ) @ ( image2 @ H4 @ ( set @ G ) @ B6 @ C5 ) ) ) ) ) ) ).
% INT_simps(4)
thf(fact_4539_totally__bounded__metric,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ( ( topolo6688025880775521714ounded @ A )
= ( ^ [S7: set @ A] :
! [E4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E4 )
=> ? [K3: set @ A] :
( ( finite_finite2 @ A @ K3 )
& ( ord_less_eq @ ( set @ A ) @ S7
@ ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ A @ ( set @ A )
@ ^ [X5: A] :
( collect @ A
@ ^ [Y6: A] : ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X5 @ Y6 ) @ E4 ) )
@ K3 ) ) ) ) ) ) ) ) ).
% totally_bounded_metric
thf(fact_4540_top__apply,axiom,
! [C: $tType,D: $tType] :
( ( top @ C )
=> ( ( top_top @ ( D > C ) )
= ( ^ [X5: D] : ( top_top @ C ) ) ) ) ).
% top_apply
thf(fact_4541_atMost__UNIV__triv,axiom,
! [A: $tType] :
( ( set_ord_atMost @ ( set @ A ) @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ ( set @ A ) ) ) ) ).
% atMost_UNIV_triv
thf(fact_4542_card__UNIV__unit,axiom,
( ( finite_card @ product_unit @ ( top_top @ ( set @ product_unit ) ) )
= ( one_one @ nat ) ) ).
% card_UNIV_unit
thf(fact_4543_finite__option__UNIV,axiom,
! [A: $tType] :
( ( finite_finite2 @ ( option @ A ) @ ( top_top @ ( set @ ( option @ A ) ) ) )
= ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) ) ).
% finite_option_UNIV
thf(fact_4544_finite__Plus__UNIV__iff,axiom,
! [A: $tType,B: $tType] :
( ( finite_finite2 @ ( sum_sum @ A @ B ) @ ( top_top @ ( set @ ( sum_sum @ A @ B ) ) ) )
= ( ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) )
& ( finite_finite2 @ B @ ( top_top @ ( set @ B ) ) ) ) ) ).
% finite_Plus_UNIV_iff
thf(fact_4545_inf__top_Oright__neutral,axiom,
! [A: $tType] :
( ( bounde4346867609351753570nf_top @ A )
=> ! [A2: A] :
( ( inf_inf @ A @ A2 @ ( top_top @ A ) )
= A2 ) ) ).
% inf_top.right_neutral
thf(fact_4546_inf__top_Oneutr__eq__iff,axiom,
! [A: $tType] :
( ( bounde4346867609351753570nf_top @ A )
=> ! [A2: A,B2: A] :
( ( ( top_top @ A )
= ( inf_inf @ A @ A2 @ B2 ) )
= ( ( A2
= ( top_top @ A ) )
& ( B2
= ( top_top @ A ) ) ) ) ) ).
% inf_top.neutr_eq_iff
thf(fact_4547_inf__top_Oleft__neutral,axiom,
! [A: $tType] :
( ( bounde4346867609351753570nf_top @ A )
=> ! [A2: A] :
( ( inf_inf @ A @ ( top_top @ A ) @ A2 )
= A2 ) ) ).
% inf_top.left_neutral
thf(fact_4548_inf__top_Oeq__neutr__iff,axiom,
! [A: $tType] :
( ( bounde4346867609351753570nf_top @ A )
=> ! [A2: A,B2: A] :
( ( ( inf_inf @ A @ A2 @ B2 )
= ( top_top @ A ) )
= ( ( A2
= ( top_top @ A ) )
& ( B2
= ( top_top @ A ) ) ) ) ) ).
% inf_top.eq_neutr_iff
thf(fact_4549_top__eq__inf__iff,axiom,
! [A: $tType] :
( ( bounde4346867609351753570nf_top @ A )
=> ! [X: A,Y: A] :
( ( ( top_top @ A )
= ( inf_inf @ A @ X @ Y ) )
= ( ( X
= ( top_top @ A ) )
& ( Y
= ( top_top @ A ) ) ) ) ) ).
% top_eq_inf_iff
thf(fact_4550_inf__eq__top__iff,axiom,
! [A: $tType] :
( ( bounde4346867609351753570nf_top @ A )
=> ! [X: A,Y: A] :
( ( ( inf_inf @ A @ X @ Y )
= ( top_top @ A ) )
= ( ( X
= ( top_top @ A ) )
& ( Y
= ( top_top @ A ) ) ) ) ) ).
% inf_eq_top_iff
thf(fact_4551_inf__top__right,axiom,
! [A: $tType] :
( ( bounde4346867609351753570nf_top @ A )
=> ! [X: A] :
( ( inf_inf @ A @ X @ ( top_top @ A ) )
= X ) ) ).
% inf_top_right
thf(fact_4552_inf__top__left,axiom,
! [A: $tType] :
( ( bounde4346867609351753570nf_top @ A )
=> ! [X: A] :
( ( inf_inf @ A @ ( top_top @ A ) @ X )
= X ) ) ).
% inf_top_left
thf(fact_4553_range__mult,axiom,
! [A2: real] :
( ( ( A2
= ( zero_zero @ real ) )
=> ( ( image2 @ real @ real @ ( times_times @ real @ A2 ) @ ( top_top @ ( set @ real ) ) )
= ( insert @ real @ ( zero_zero @ real ) @ ( bot_bot @ ( set @ real ) ) ) ) )
& ( ( A2
!= ( zero_zero @ real ) )
=> ( ( image2 @ real @ real @ ( times_times @ real @ A2 ) @ ( top_top @ ( set @ real ) ) )
= ( top_top @ ( set @ real ) ) ) ) ) ).
% range_mult
thf(fact_4554_max__top,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [X: A] :
( ( ord_max @ A @ ( top_top @ A ) @ X )
= ( top_top @ A ) ) ) ).
% max_top
thf(fact_4555_max__top2,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [X: A] :
( ( ord_max @ A @ X @ ( top_top @ A ) )
= ( top_top @ A ) ) ) ).
% max_top2
thf(fact_4556_dist__add__cancel2,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [B2: A,A2: A,C3: A] :
( ( real_V557655796197034286t_dist @ A @ ( plus_plus @ A @ B2 @ A2 ) @ ( plus_plus @ A @ C3 @ A2 ) )
= ( real_V557655796197034286t_dist @ A @ B2 @ C3 ) ) ) ).
% dist_add_cancel2
thf(fact_4557_dist__add__cancel,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( real_V557655796197034286t_dist @ A @ ( plus_plus @ A @ A2 @ B2 ) @ ( plus_plus @ A @ A2 @ C3 ) )
= ( real_V557655796197034286t_dist @ A @ B2 @ C3 ) ) ) ).
% dist_add_cancel
thf(fact_4558_remdups__eq__nil__iff,axiom,
! [A: $tType,X: list @ A] :
( ( ( remdups @ A @ X )
= ( nil @ A ) )
= ( X
= ( nil @ A ) ) ) ).
% remdups_eq_nil_iff
thf(fact_4559_remdups__eq__nil__right__iff,axiom,
! [A: $tType,X: list @ A] :
( ( ( nil @ A )
= ( remdups @ A @ X ) )
= ( X
= ( nil @ A ) ) ) ).
% remdups_eq_nil_right_iff
thf(fact_4560_set__remdups,axiom,
! [A: $tType,Xs: list @ A] :
( ( set2 @ A @ ( remdups @ A @ Xs ) )
= ( set2 @ A @ Xs ) ) ).
% set_remdups
thf(fact_4561_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_4562_distinct__remdups,axiom,
! [A: $tType,Xs: list @ A] : ( distinct @ A @ ( remdups @ A @ Xs ) ) ).
% distinct_remdups
thf(fact_4563_remdups__id__iff__distinct,axiom,
! [A: $tType,Xs: list @ A] :
( ( ( remdups @ A @ Xs )
= Xs )
= ( distinct @ A @ Xs ) ) ).
% remdups_id_iff_distinct
thf(fact_4564_Collect__const,axiom,
! [A: $tType,P2: $o] :
( ( P2
=> ( ( collect @ A
@ ^ [S8: A] : P2 )
= ( top_top @ ( set @ A ) ) ) )
& ( ~ P2
=> ( ( collect @ A
@ ^ [S8: A] : P2 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Collect_const
thf(fact_4565_finite__Collect__not,axiom,
! [A: $tType,P2: A > $o] :
( ( finite_finite2 @ A @ ( collect @ A @ P2 ) )
=> ( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X5: A] :
~ ( P2 @ X5 ) ) )
= ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) ) ) ).
% finite_Collect_not
thf(fact_4566_Collect__const__case__prod,axiom,
! [B: $tType,A: $tType,P2: $o] :
( ( P2
=> ( ( collect @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [A6: A,B5: B] : P2 ) )
= ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) )
& ( ~ P2
=> ( ( collect @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [A6: A,B5: B] : P2 ) )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ) ).
% Collect_const_case_prod
thf(fact_4567_range__add,axiom,
! [A: $tType] :
( ( group_add @ A )
=> ! [A2: A] :
( ( image2 @ A @ A @ ( plus_plus @ A @ A2 ) @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) ) ) ).
% range_add
thf(fact_4568_surj__plus,axiom,
! [A: $tType] :
( ( ab_group_add @ A )
=> ! [A2: A] :
( ( image2 @ A @ A @ ( plus_plus @ A @ A2 ) @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) ) ) ).
% surj_plus
thf(fact_4569_Sup__eq__top__iff,axiom,
! [A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [A5: set @ A] :
( ( ( complete_Sup_Sup @ A @ A5 )
= ( top_top @ A ) )
= ( ! [X5: A] :
( ( ord_less @ A @ X5 @ ( top_top @ A ) )
=> ? [Y6: A] :
( ( member @ A @ Y6 @ A5 )
& ( ord_less @ A @ X5 @ Y6 ) ) ) ) ) ) ).
% Sup_eq_top_iff
thf(fact_4570_boolean__algebra_Ocompl__one,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ( ( uminus_uminus @ A @ ( top_top @ A ) )
= ( bot_bot @ A ) ) ) ).
% boolean_algebra.compl_one
thf(fact_4571_boolean__algebra_Ocompl__zero,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ( ( uminus_uminus @ A @ ( bot_bot @ A ) )
= ( top_top @ A ) ) ) ).
% boolean_algebra.compl_zero
thf(fact_4572_Inf__UNIV,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ( ( complete_Inf_Inf @ A @ ( top_top @ ( set @ A ) ) )
= ( bot_bot @ A ) ) ) ).
% Inf_UNIV
thf(fact_4573_ccInf__empty,axiom,
! [A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ( ( complete_Inf_Inf @ A @ ( bot_bot @ ( set @ A ) ) )
= ( top_top @ A ) ) ) ).
% ccInf_empty
thf(fact_4574_Inf__empty,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ( ( complete_Inf_Inf @ A @ ( bot_bot @ ( set @ A ) ) )
= ( top_top @ A ) ) ) ).
% Inf_empty
thf(fact_4575_Diff__UNIV,axiom,
! [A: $tType,A5: set @ A] :
( ( minus_minus @ ( set @ A ) @ A5 @ ( top_top @ ( set @ A ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Diff_UNIV
thf(fact_4576_surj__fn,axiom,
! [A: $tType,F3: A > A,N: nat] :
( ( ( image2 @ A @ A @ F3 @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) )
=> ( ( image2 @ A @ A @ ( compow @ ( A > A ) @ N @ F3 ) @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) ) ) ).
% surj_fn
thf(fact_4577_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_4578_zero__less__dist__iff,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( real_V557655796197034286t_dist @ A @ X @ Y ) )
= ( X != Y ) ) ) ).
% zero_less_dist_iff
thf(fact_4579_finite__compl,axiom,
! [A: $tType,A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( finite_finite2 @ A @ ( uminus_uminus @ ( set @ A ) @ A5 ) )
= ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) ) ) ).
% finite_compl
thf(fact_4580_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_4581_SUP__eq__top__iff,axiom,
! [B: $tType,A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [F3: B > A,A5: set @ B] :
( ( ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A5 ) )
= ( top_top @ A ) )
= ( ! [X5: A] :
( ( ord_less @ A @ X5 @ ( top_top @ A ) )
=> ? [Y6: B] :
( ( member @ B @ Y6 @ A5 )
& ( ord_less @ A @ X5 @ ( F3 @ Y6 ) ) ) ) ) ) ) ).
% SUP_eq_top_iff
thf(fact_4582_range__constant,axiom,
! [B: $tType,A: $tType,X: A] :
( ( image2 @ B @ A
@ ^ [Uu3: B] : X
@ ( top_top @ ( set @ B ) ) )
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ).
% range_constant
thf(fact_4583_ccINF__empty,axiom,
! [B: $tType,A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [F3: B > A] :
( ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ ( bot_bot @ ( set @ B ) ) ) )
= ( top_top @ A ) ) ) ).
% ccINF_empty
thf(fact_4584_INT__constant,axiom,
! [B: $tType,A: $tType,A5: set @ B,C3: set @ A] :
( ( ( A5
= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Inf_Inf @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [Y6: B] : C3
@ A5 ) )
= ( top_top @ ( set @ A ) ) ) )
& ( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Inf_Inf @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [Y6: B] : C3
@ A5 ) )
= C3 ) ) ) ).
% INT_constant
thf(fact_4585_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_4586_INT__simps_I2_J,axiom,
! [C: $tType,D: $tType,C5: set @ D,A5: set @ C,B6: D > ( set @ C )] :
( ( ( C5
= ( bot_bot @ ( set @ D ) ) )
=> ( ( complete_Inf_Inf @ ( set @ C )
@ ( image2 @ D @ ( set @ C )
@ ^ [X5: D] : ( inf_inf @ ( set @ C ) @ A5 @ ( B6 @ X5 ) )
@ C5 ) )
= ( top_top @ ( set @ C ) ) ) )
& ( ( C5
!= ( bot_bot @ ( set @ D ) ) )
=> ( ( complete_Inf_Inf @ ( set @ C )
@ ( image2 @ D @ ( set @ C )
@ ^ [X5: D] : ( inf_inf @ ( set @ C ) @ A5 @ ( B6 @ X5 ) )
@ C5 ) )
= ( inf_inf @ ( set @ C ) @ A5 @ ( complete_Inf_Inf @ ( set @ C ) @ ( image2 @ D @ ( set @ C ) @ B6 @ C5 ) ) ) ) ) ) ).
% INT_simps(2)
thf(fact_4587_INT__simps_I1_J,axiom,
! [A: $tType,B: $tType,C5: set @ A,A5: A > ( set @ B ),B6: set @ B] :
( ( ( C5
= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Inf_Inf @ ( set @ B )
@ ( image2 @ A @ ( set @ B )
@ ^ [X5: A] : ( inf_inf @ ( set @ B ) @ ( A5 @ X5 ) @ B6 )
@ C5 ) )
= ( top_top @ ( set @ B ) ) ) )
& ( ( C5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Inf_Inf @ ( set @ B )
@ ( image2 @ A @ ( set @ B )
@ ^ [X5: A] : ( inf_inf @ ( set @ B ) @ ( A5 @ X5 ) @ B6 )
@ C5 ) )
= ( inf_inf @ ( set @ B ) @ ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ A5 @ C5 ) ) @ B6 ) ) ) ) ).
% INT_simps(1)
thf(fact_4588_INT__simps_I3_J,axiom,
! [E: $tType,F: $tType,C5: set @ E,A5: E > ( set @ F ),B6: set @ F] :
( ( ( C5
= ( bot_bot @ ( set @ E ) ) )
=> ( ( complete_Inf_Inf @ ( set @ F )
@ ( image2 @ E @ ( set @ F )
@ ^ [X5: E] : ( minus_minus @ ( set @ F ) @ ( A5 @ X5 ) @ B6 )
@ C5 ) )
= ( top_top @ ( set @ F ) ) ) )
& ( ( C5
!= ( bot_bot @ ( set @ E ) ) )
=> ( ( complete_Inf_Inf @ ( set @ F )
@ ( image2 @ E @ ( set @ F )
@ ^ [X5: E] : ( minus_minus @ ( set @ F ) @ ( A5 @ X5 ) @ B6 )
@ C5 ) )
= ( minus_minus @ ( set @ F ) @ ( complete_Inf_Inf @ ( set @ F ) @ ( image2 @ E @ ( set @ F ) @ A5 @ C5 ) ) @ B6 ) ) ) ) ).
% INT_simps(3)
thf(fact_4589_UNIV__option__conv,axiom,
! [A: $tType] :
( ( top_top @ ( set @ ( option @ A ) ) )
= ( insert @ ( option @ A ) @ ( none @ A ) @ ( image2 @ A @ ( option @ A ) @ ( some @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ).
% UNIV_option_conv
thf(fact_4590_surj__prod__encode,axiom,
( ( image2 @ ( product_prod @ nat @ nat ) @ nat @ nat_prod_encode @ ( top_top @ ( set @ ( product_prod @ nat @ nat ) ) ) )
= ( top_top @ ( set @ nat ) ) ) ).
% surj_prod_encode
thf(fact_4591_INF__filter__bot__base,axiom,
! [B: $tType,A: $tType,I6: set @ A,F5: A > ( filter @ B )] :
( ! [I3: A] :
( ( member @ A @ I3 @ I6 )
=> ! [J2: A] :
( ( member @ A @ J2 @ I6 )
=> ? [X3: A] :
( ( member @ A @ X3 @ I6 )
& ( ord_less_eq @ ( filter @ B ) @ ( F5 @ X3 ) @ ( inf_inf @ ( filter @ B ) @ ( F5 @ I3 ) @ ( F5 @ J2 ) ) ) ) ) )
=> ( ( ( complete_Inf_Inf @ ( filter @ B ) @ ( image2 @ A @ ( filter @ B ) @ F5 @ I6 ) )
= ( bot_bot @ ( filter @ B ) ) )
= ( ? [X5: A] :
( ( member @ A @ X5 @ I6 )
& ( ( F5 @ X5 )
= ( bot_bot @ ( filter @ B ) ) ) ) ) ) ) ).
% INF_filter_bot_base
thf(fact_4592_remdups_Osimps_I1_J,axiom,
! [A: $tType] :
( ( remdups @ A @ ( nil @ A ) )
= ( nil @ A ) ) ).
% remdups.simps(1)
thf(fact_4593_top__greatest,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [A2: A] : ( ord_less_eq @ A @ A2 @ ( top_top @ A ) ) ) ).
% top_greatest
thf(fact_4594_top_Oextremum__unique,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ ( top_top @ A ) @ A2 )
= ( A2
= ( top_top @ A ) ) ) ) ).
% top.extremum_unique
thf(fact_4595_top_Oextremum__uniqueI,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ ( top_top @ A ) @ A2 )
=> ( A2
= ( top_top @ A ) ) ) ) ).
% top.extremum_uniqueI
thf(fact_4596_infinite__UNIV__nat,axiom,
~ ( finite_finite2 @ nat @ ( top_top @ ( set @ nat ) ) ) ).
% infinite_UNIV_nat
thf(fact_4597_nat__not__finite,axiom,
~ ( finite_finite2 @ nat @ ( top_top @ ( set @ nat ) ) ) ).
% nat_not_finite
thf(fact_4598_infinite__UNIV__char__0,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ~ ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) ) ).
% infinite_UNIV_char_0
thf(fact_4599_ex__new__if__finite,axiom,
! [A: $tType,A5: set @ A] :
( ~ ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ A5 )
=> ? [A4: A] :
~ ( member @ A @ A4 @ A5 ) ) ) ).
% ex_new_if_finite
thf(fact_4600_finite__UNIV,axiom,
! [A: $tType] :
( ( finite_finite @ A )
=> ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) ) ).
% finite_UNIV
thf(fact_4601_Finite__Set_Ofinite__set,axiom,
! [A: $tType] :
( ( finite_finite2 @ ( set @ A ) @ ( top_top @ ( set @ ( set @ A ) ) ) )
= ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) ) ).
% Finite_Set.finite_set
thf(fact_4602_finite__Prod__UNIV,axiom,
! [B: $tType,A: $tType] :
( ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) )
=> ( ( finite_finite2 @ B @ ( top_top @ ( set @ B ) ) )
=> ( finite_finite2 @ ( product_prod @ A @ B ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ) ).
% finite_Prod_UNIV
thf(fact_4603_finite__prod,axiom,
! [A: $tType,B: $tType] :
( ( finite_finite2 @ ( product_prod @ A @ B ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) )
= ( ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) )
& ( finite_finite2 @ B @ ( top_top @ ( set @ B ) ) ) ) ) ).
% finite_prod
thf(fact_4604_finite__fun__UNIVD2,axiom,
! [A: $tType,B: $tType] :
( ( finite_finite2 @ ( A > B ) @ ( top_top @ ( set @ ( A > B ) ) ) )
=> ( finite_finite2 @ B @ ( top_top @ ( set @ B ) ) ) ) ).
% finite_fun_UNIVD2
thf(fact_4605_top_Onot__eq__extremum,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [A2: A] :
( ( A2
!= ( top_top @ A ) )
= ( ord_less @ A @ A2 @ ( top_top @ A ) ) ) ) ).
% top.not_eq_extremum
thf(fact_4606_top_Oextremum__strict,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [A2: A] :
~ ( ord_less @ A @ ( top_top @ A ) @ A2 ) ) ).
% top.extremum_strict
thf(fact_4607_dist__commute__lessI,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [Y: A,X: A,E3: real] :
( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ Y @ X ) @ E3 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X @ Y ) @ E3 ) ) ) ).
% dist_commute_lessI
thf(fact_4608_not__UNIV__eq__Icc,axiom,
! [A: $tType] :
( ( no_top @ A )
=> ! [L3: A,H3: A] :
( ( top_top @ ( set @ A ) )
!= ( set_or1337092689740270186AtMost @ A @ L3 @ H3 ) ) ) ).
% not_UNIV_eq_Icc
thf(fact_4609_atMost__eq__UNIV__iff,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [X: A] :
( ( ( set_ord_atMost @ A @ X )
= ( top_top @ ( set @ A ) ) )
= ( X
= ( top_top @ A ) ) ) ) ).
% atMost_eq_UNIV_iff
thf(fact_4610_remdups__remdups,axiom,
! [A: $tType,Xs: list @ A] :
( ( remdups @ A @ ( remdups @ A @ Xs ) )
= ( remdups @ A @ Xs ) ) ).
% remdups_remdups
thf(fact_4611_not__UNIV__eq__Iic,axiom,
! [A: $tType] :
( ( no_top @ A )
=> ! [H3: A] :
( ( top_top @ ( set @ A ) )
!= ( set_ord_atMost @ A @ H3 ) ) ) ).
% not_UNIV_eq_Iic
thf(fact_4612_distinct__remdups__id,axiom,
! [A: $tType,Xs: list @ A] :
( ( distinct @ A @ Xs )
=> ( ( remdups @ A @ Xs )
= Xs ) ) ).
% distinct_remdups_id
thf(fact_4613_empty__not__UNIV,axiom,
! [A: $tType] :
( ( bot_bot @ ( set @ A ) )
!= ( top_top @ ( set @ A ) ) ) ).
% empty_not_UNIV
thf(fact_4614_atLeastAtMost__eq__UNIV__iff,axiom,
! [A: $tType] :
( ( bounded_lattice @ A )
=> ! [X: A,Y: A] :
( ( ( set_or1337092689740270186AtMost @ A @ X @ Y )
= ( top_top @ ( set @ A ) ) )
= ( ( X
= ( bot_bot @ A ) )
& ( Y
= ( top_top @ A ) ) ) ) ) ).
% atLeastAtMost_eq_UNIV_iff
thf(fact_4615_Inf__filter__not__bot,axiom,
! [A: $tType,B6: set @ ( filter @ A )] :
( ! [X17: set @ ( filter @ A )] :
( ( ord_less_eq @ ( set @ ( filter @ A ) ) @ X17 @ B6 )
=> ( ( finite_finite2 @ ( filter @ A ) @ X17 )
=> ( ( complete_Inf_Inf @ ( filter @ A ) @ X17 )
!= ( bot_bot @ ( filter @ A ) ) ) ) )
=> ( ( complete_Inf_Inf @ ( filter @ A ) @ B6 )
!= ( bot_bot @ ( filter @ A ) ) ) ) ).
% Inf_filter_not_bot
thf(fact_4616_infinite__UNIV__int,axiom,
~ ( finite_finite2 @ int @ ( top_top @ ( set @ int ) ) ) ).
% infinite_UNIV_int
thf(fact_4617_UN__lessThan__UNIV,axiom,
( ( complete_Sup_Sup @ ( set @ nat ) @ ( image2 @ nat @ ( set @ nat ) @ ( set_ord_lessThan @ nat ) @ ( top_top @ ( set @ nat ) ) ) )
= ( top_top @ ( set @ nat ) ) ) ).
% UN_lessThan_UNIV
thf(fact_4618_UN__atMost__UNIV,axiom,
( ( complete_Sup_Sup @ ( set @ nat ) @ ( image2 @ nat @ ( set @ nat ) @ ( set_ord_atMost @ nat ) @ ( top_top @ ( set @ nat ) ) ) )
= ( top_top @ ( set @ nat ) ) ) ).
% UN_atMost_UNIV
thf(fact_4619_infinite__UNIV__listI,axiom,
! [A: $tType] :
~ ( finite_finite2 @ ( list @ A ) @ ( top_top @ ( set @ ( list @ A ) ) ) ) ).
% infinite_UNIV_listI
thf(fact_4620_norm__conv__dist,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ( ( real_V7770717601297561774m_norm @ A )
= ( ^ [X5: A] : ( real_V557655796197034286t_dist @ A @ X5 @ ( zero_zero @ A ) ) ) ) ) ).
% norm_conv_dist
thf(fact_4621_dist__not__less__zero,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X: A,Y: A] :
~ ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X @ Y ) @ ( zero_zero @ real ) ) ) ).
% dist_not_less_zero
thf(fact_4622_dist__pos__lt,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X: A,Y: A] :
( ( X != Y )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( real_V557655796197034286t_dist @ A @ X @ Y ) ) ) ) ).
% dist_pos_lt
thf(fact_4623_dist__triangle__less__add,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X1: A,Y: A,E1: real,X2: A,E22: real] :
( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X1 @ Y ) @ E1 )
=> ( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X2 @ Y ) @ E22 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X1 @ X2 ) @ ( plus_plus @ real @ E1 @ E22 ) ) ) ) ) ).
% dist_triangle_less_add
thf(fact_4624_dist__triangle__lt,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X: A,Z: A,Y: A,E3: real] :
( ( ord_less @ real @ ( plus_plus @ real @ ( real_V557655796197034286t_dist @ A @ X @ Z ) @ ( real_V557655796197034286t_dist @ A @ Y @ Z ) ) @ E3 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X @ Y ) @ E3 ) ) ) ).
% dist_triangle_lt
thf(fact_4625_finite__fun__UNIVD1,axiom,
! [B: $tType,A: $tType] :
( ( finite_finite2 @ ( A > B ) @ ( top_top @ ( set @ ( A > B ) ) ) )
=> ( ( ( finite_card @ B @ ( top_top @ ( set @ B ) ) )
!= ( suc @ ( zero_zero @ nat ) ) )
=> ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) ) ) ).
% finite_fun_UNIVD1
thf(fact_4626_perfect__space__class_OUNIV__not__singleton,axiom,
! [A: $tType] :
( ( topolo8386298272705272623_space @ A )
=> ! [X: A] :
( ( top_top @ ( set @ A ) )
!= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% perfect_space_class.UNIV_not_singleton
thf(fact_4627_not__UNIV__le__Icc,axiom,
! [A: $tType] :
( ( no_top @ A )
=> ! [L: A,H2: A] :
~ ( ord_less_eq @ ( set @ A ) @ ( top_top @ ( set @ A ) ) @ ( set_or1337092689740270186AtMost @ A @ L @ H2 ) ) ) ).
% not_UNIV_le_Icc
thf(fact_4628_card__eq__UNIV__imp__eq__UNIV,axiom,
! [A: $tType,A5: set @ A] :
( ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) )
=> ( ( ( finite_card @ A @ A5 )
= ( finite_card @ A @ ( top_top @ ( set @ A ) ) ) )
=> ( A5
= ( top_top @ ( set @ A ) ) ) ) ) ).
% card_eq_UNIV_imp_eq_UNIV
thf(fact_4629_finite__range__Some,axiom,
! [A: $tType] :
( ( finite_finite2 @ ( option @ A ) @ ( image2 @ A @ ( option @ A ) @ ( some @ A ) @ ( top_top @ ( set @ A ) ) ) )
= ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) ) ) ).
% finite_range_Some
thf(fact_4630_not__UNIV__le__Iic,axiom,
! [A: $tType] :
( ( no_top @ A )
=> ! [H2: A] :
~ ( ord_less_eq @ ( set @ A ) @ ( top_top @ ( set @ A ) ) @ ( set_ord_atMost @ A @ H2 ) ) ) ).
% not_UNIV_le_Iic
thf(fact_4631_Inter__UNIV,axiom,
! [A: $tType] :
( ( complete_Inf_Inf @ ( set @ A ) @ ( top_top @ ( set @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Inter_UNIV
thf(fact_4632_notin__range__Some,axiom,
! [A: $tType,X: option @ A] :
( ( ~ ( member @ ( option @ A ) @ X @ ( image2 @ A @ ( option @ A ) @ ( some @ A ) @ ( top_top @ ( set @ A ) ) ) ) )
= ( X
= ( none @ A ) ) ) ).
% notin_range_Some
thf(fact_4633_Compl__UNIV__eq,axiom,
! [A: $tType] :
( ( uminus_uminus @ ( set @ A ) @ ( top_top @ ( set @ A ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Compl_UNIV_eq
thf(fact_4634_Compl__empty__eq,axiom,
! [A: $tType] :
( ( uminus_uminus @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) ) ).
% Compl_empty_eq
thf(fact_4635_int__in__range__abs,axiom,
! [N: nat] : ( member @ int @ ( semiring_1_of_nat @ int @ N ) @ ( image2 @ int @ int @ ( abs_abs @ int ) @ ( top_top @ ( set @ int ) ) ) ) ).
% int_in_range_abs
thf(fact_4636_Inter__empty,axiom,
! [A: $tType] :
( ( complete_Inf_Inf @ ( set @ A ) @ ( bot_bot @ ( set @ ( set @ A ) ) ) )
= ( top_top @ ( set @ A ) ) ) ).
% Inter_empty
thf(fact_4637_finite__range__imageI,axiom,
! [C: $tType,A: $tType,B: $tType,G3: B > A,F3: A > C] :
( ( finite_finite2 @ A @ ( image2 @ B @ A @ G3 @ ( top_top @ ( set @ B ) ) ) )
=> ( finite_finite2 @ C
@ ( image2 @ B @ C
@ ^ [X5: B] : ( F3 @ ( G3 @ X5 ) )
@ ( top_top @ ( set @ B ) ) ) ) ) ).
% finite_range_imageI
thf(fact_4638_remove1__remdups,axiom,
! [A: $tType,Xs: list @ A,X: A] :
( ( distinct @ A @ Xs )
=> ( ( remove1 @ A @ X @ ( remdups @ A @ Xs ) )
= ( remdups @ A @ ( remove1 @ A @ X @ Xs ) ) ) ) ).
% remove1_remdups
thf(fact_4639_range__eq__singletonD,axiom,
! [B: $tType,A: $tType,F3: B > A,A2: A,X: B] :
( ( ( image2 @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) )
= ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
=> ( ( F3 @ X )
= A2 ) ) ).
% range_eq_singletonD
thf(fact_4640_INF__empty,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F3: B > A] :
( ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ ( bot_bot @ ( set @ B ) ) ) )
= ( top_top @ A ) ) ) ).
% INF_empty
thf(fact_4641_INF__constant,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ B,C3: A] :
( ( ( A5
= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Inf_Inf @ A
@ ( image2 @ B @ A
@ ^ [Y6: B] : C3
@ A5 ) )
= ( top_top @ A ) ) )
& ( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( complete_Inf_Inf @ A
@ ( image2 @ B @ A
@ ^ [Y6: B] : C3
@ A5 ) )
= C3 ) ) ) ) ).
% INF_constant
thf(fact_4642_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_4643_INT__empty,axiom,
! [B: $tType,A: $tType,B6: B > ( set @ A )] :
( ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B6 @ ( bot_bot @ ( set @ B ) ) ) )
= ( top_top @ ( set @ A ) ) ) ).
% INT_empty
thf(fact_4644_UN__UN__finite__eq,axiom,
! [A: $tType,A5: nat > ( set @ A )] :
( ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ nat @ ( set @ A )
@ ^ [N2: nat] : ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ A5 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N2 ) ) )
@ ( top_top @ ( set @ nat ) ) ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ A5 @ ( top_top @ ( set @ nat ) ) ) ) ) ).
% UN_UN_finite_eq
thf(fact_4645_Cauchy__def,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ( ( topolo3814608138187158403Cauchy @ A )
= ( ^ [X9: nat > A] :
! [E4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E4 )
=> ? [M9: nat] :
! [M3: nat] :
( ( ord_less_eq @ nat @ M9 @ M3 )
=> ! [N2: nat] :
( ( ord_less_eq @ nat @ M9 @ N2 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( X9 @ M3 ) @ ( X9 @ N2 ) ) @ E4 ) ) ) ) ) ) ) ).
% Cauchy_def
thf(fact_4646_Cauchy__altdef2,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ( ( topolo3814608138187158403Cauchy @ A )
= ( ^ [S8: nat > A] :
! [E4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E4 )
=> ? [N4: nat] :
! [N2: nat] :
( ( ord_less_eq @ nat @ N4 @ N2 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( S8 @ N2 ) @ ( S8 @ N4 ) ) @ E4 ) ) ) ) ) ) ).
% Cauchy_altdef2
thf(fact_4647_metric__CauchyD,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X7: nat > A,E3: real] :
( ( topolo3814608138187158403Cauchy @ A @ X7 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ E3 )
=> ? [M8: nat] :
! [M4: nat] :
( ( ord_less_eq @ nat @ M8 @ M4 )
=> ! [N5: nat] :
( ( ord_less_eq @ nat @ M8 @ N5 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( X7 @ M4 ) @ ( X7 @ N5 ) ) @ E3 ) ) ) ) ) ) ).
% metric_CauchyD
thf(fact_4648_metric__CauchyI,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X7: nat > A] :
( ! [E2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E2 )
=> ? [M10: nat] :
! [M: nat] :
( ( ord_less_eq @ nat @ M10 @ M )
=> ! [N3: nat] :
( ( ord_less_eq @ nat @ M10 @ N3 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( X7 @ M ) @ ( X7 @ N3 ) ) @ E2 ) ) ) )
=> ( topolo3814608138187158403Cauchy @ A @ X7 ) ) ) ).
% metric_CauchyI
thf(fact_4649_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_4650_finite__UNIV__card__ge__0,axiom,
! [A: $tType] :
( ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ ( finite_card @ A @ ( top_top @ ( set @ A ) ) ) ) ) ).
% finite_UNIV_card_ge_0
thf(fact_4651_dist__triangle__half__l,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X1: A,Y: A,E3: real,X2: A] :
( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X1 @ Y ) @ ( divide_divide @ real @ E3 @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X2 @ Y ) @ ( divide_divide @ real @ E3 @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X1 @ X2 ) @ E3 ) ) ) ) ).
% dist_triangle_half_l
thf(fact_4652_dist__triangle__half__r,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [Y: A,X1: A,E3: real,X2: A] :
( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ Y @ X1 ) @ ( divide_divide @ real @ E3 @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ Y @ X2 ) @ ( divide_divide @ real @ E3 @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X1 @ X2 ) @ E3 ) ) ) ) ).
% dist_triangle_half_r
thf(fact_4653_dist__triangle__third,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X1: A,X2: A,E3: real,X32: A,X42: A] :
( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X1 @ X2 ) @ ( divide_divide @ real @ E3 @ ( numeral_numeral @ real @ ( bit1 @ one2 ) ) ) )
=> ( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X2 @ X32 ) @ ( divide_divide @ real @ E3 @ ( numeral_numeral @ real @ ( bit1 @ one2 ) ) ) )
=> ( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X32 @ X42 ) @ ( divide_divide @ real @ E3 @ ( numeral_numeral @ real @ ( bit1 @ one2 ) ) ) )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X1 @ X42 ) @ E3 ) ) ) ) ) ).
% dist_triangle_third
thf(fact_4654_CauchyI_H,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X7: nat > A] :
( ! [E2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E2 )
=> ? [M10: nat] :
! [M: nat] :
( ( ord_less_eq @ nat @ M10 @ M )
=> ! [N3: nat] :
( ( ord_less @ nat @ M @ N3 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( X7 @ M ) @ ( X7 @ N3 ) ) @ E2 ) ) ) )
=> ( topolo3814608138187158403Cauchy @ A @ X7 ) ) ) ).
% CauchyI'
thf(fact_4655_Cauchy__altdef,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ( ( topolo3814608138187158403Cauchy @ A )
= ( ^ [F4: nat > A] :
! [E4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E4 )
=> ? [M9: nat] :
! [M3: nat] :
( ( ord_less_eq @ nat @ M9 @ M3 )
=> ! [N2: nat] :
( ( ord_less @ nat @ M3 @ N2 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( F4 @ M3 ) @ ( F4 @ N2 ) ) @ E4 ) ) ) ) ) ) ) ).
% Cauchy_altdef
thf(fact_4656_UN__finite__subset,axiom,
! [A: $tType,A5: nat > ( set @ A ),C5: set @ A] :
( ! [N3: nat] : ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ A5 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N3 ) ) ) @ C5 )
=> ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ A5 @ ( top_top @ ( set @ nat ) ) ) ) @ C5 ) ) ).
% UN_finite_subset
thf(fact_4657_UNIV__nat__eq,axiom,
( ( top_top @ ( set @ nat ) )
= ( insert @ nat @ ( zero_zero @ nat ) @ ( image2 @ nat @ nat @ suc @ ( top_top @ ( set @ nat ) ) ) ) ) ).
% UNIV_nat_eq
thf(fact_4658_UN__finite2__eq,axiom,
! [A: $tType,A5: nat > ( set @ A ),B6: nat > ( set @ A ),K2: nat] :
( ! [N3: nat] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ A5 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N3 ) ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ B6 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( plus_plus @ nat @ N3 @ K2 ) ) ) ) )
=> ( ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ A5 @ ( top_top @ ( set @ nat ) ) ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ B6 @ ( top_top @ ( set @ nat ) ) ) ) ) ) ).
% UN_finite2_eq
thf(fact_4659_INT__extend__simps_I3_J,axiom,
! [F: $tType,E: $tType,C5: set @ E,A5: E > ( set @ F ),B6: set @ F] :
( ( ( C5
= ( bot_bot @ ( set @ E ) ) )
=> ( ( minus_minus @ ( set @ F ) @ ( complete_Inf_Inf @ ( set @ F ) @ ( image2 @ E @ ( set @ F ) @ A5 @ C5 ) ) @ B6 )
= ( minus_minus @ ( set @ F ) @ ( top_top @ ( set @ F ) ) @ B6 ) ) )
& ( ( C5
!= ( bot_bot @ ( set @ E ) ) )
=> ( ( minus_minus @ ( set @ F ) @ ( complete_Inf_Inf @ ( set @ F ) @ ( image2 @ E @ ( set @ F ) @ A5 @ C5 ) ) @ B6 )
= ( complete_Inf_Inf @ ( set @ F )
@ ( image2 @ E @ ( set @ F )
@ ^ [X5: E] : ( minus_minus @ ( set @ F ) @ ( A5 @ X5 ) @ B6 )
@ C5 ) ) ) ) ) ).
% INT_extend_simps(3)
thf(fact_4660_card__range__greater__zero,axiom,
! [A: $tType,B: $tType,F3: B > A] :
( ( finite_finite2 @ A @ ( image2 @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) ) )
=> ( ord_less @ nat @ ( zero_zero @ nat ) @ ( finite_card @ A @ ( image2 @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) ) ) ) ) ).
% card_range_greater_zero
thf(fact_4661_metric__Cauchy__iff2,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ( ( topolo3814608138187158403Cauchy @ A )
= ( ^ [X9: nat > A] :
! [J3: nat] :
? [M9: nat] :
! [M3: nat] :
( ( ord_less_eq @ nat @ M9 @ M3 )
=> ! [N2: nat] :
( ( ord_less_eq @ nat @ M9 @ N2 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( X9 @ M3 ) @ ( X9 @ N2 ) ) @ ( inverse_inverse @ real @ ( semiring_1_of_nat @ real @ ( suc @ J3 ) ) ) ) ) ) ) ) ) ).
% metric_Cauchy_iff2
thf(fact_4662_UN__finite2__subset,axiom,
! [A: $tType,A5: nat > ( set @ A ),B6: nat > ( set @ A ),K2: nat] :
( ! [N3: nat] : ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ A5 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N3 ) ) ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ B6 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( plus_plus @ nat @ N3 @ K2 ) ) ) ) )
=> ( ord_less_eq @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ A5 @ ( top_top @ ( set @ nat ) ) ) ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ B6 @ ( top_top @ ( set @ nat ) ) ) ) ) ) ).
% UN_finite2_subset
thf(fact_4663_range__mod,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( image2 @ nat @ nat
@ ^ [M3: nat] : ( modulo_modulo @ nat @ M3 @ N )
@ ( top_top @ ( set @ nat ) ) )
= ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ).
% range_mod
thf(fact_4664_Sup__finite__empty,axiom,
! [A: $tType] :
( ( finite_lattice @ A )
=> ( ( complete_Sup_Sup @ A @ ( bot_bot @ ( set @ A ) ) )
= ( complete_Inf_Inf @ A @ ( top_top @ ( set @ A ) ) ) ) ) ).
% Sup_finite_empty
thf(fact_4665_Inf__finite__empty,axiom,
! [A: $tType] :
( ( finite_lattice @ A )
=> ( ( complete_Inf_Inf @ A @ ( bot_bot @ ( set @ A ) ) )
= ( complete_Sup_Sup @ A @ ( top_top @ ( set @ A ) ) ) ) ) ).
% Inf_finite_empty
thf(fact_4666_cclfp__def,axiom,
! [A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ( ( order_532582986084564980_cclfp @ A )
= ( ^ [F4: A > A] :
( complete_Sup_Sup @ A
@ ( image2 @ nat @ A
@ ^ [I4: nat] : ( compow @ ( A > A ) @ I4 @ F4 @ ( bot_bot @ A ) )
@ ( top_top @ ( set @ nat ) ) ) ) ) ) ) ).
% cclfp_def
thf(fact_4667_root__def,axiom,
( root
= ( ^ [N2: nat,X5: real] :
( if @ real
@ ( N2
= ( zero_zero @ nat ) )
@ ( zero_zero @ real )
@ ( the_inv_into @ real @ real @ ( top_top @ ( set @ real ) )
@ ^ [Y6: real] : ( times_times @ real @ ( sgn_sgn @ real @ Y6 ) @ ( power_power @ real @ ( abs_abs @ real @ Y6 ) @ N2 ) )
@ X5 ) ) ) ) ).
% root_def
thf(fact_4668_top__empty__eq2,axiom,
! [B: $tType,A: $tType] :
( ( top_top @ ( A > B > $o ) )
= ( ^ [X5: A,Y6: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X5 @ Y6 ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) ) ) ).
% top_empty_eq2
thf(fact_4669_less__filter__def,axiom,
! [A: $tType] :
( ( ord_less @ ( filter @ A ) )
= ( ^ [F8: filter @ A,F9: filter @ A] :
( ( ord_less_eq @ ( filter @ A ) @ F8 @ F9 )
& ~ ( ord_less_eq @ ( filter @ A ) @ F9 @ F8 ) ) ) ) ).
% less_filter_def
thf(fact_4670_bot__finite__def,axiom,
! [A: $tType] :
( ( finite_lattice @ A )
=> ( ( bot_bot @ A )
= ( complete_Inf_Inf @ A @ ( top_top @ ( set @ A ) ) ) ) ) ).
% bot_finite_def
thf(fact_4671_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
@ ^ [X5: A,Y6: A] :
( ( ord_less @ nat @ ( F4 @ X5 ) @ ( F4 @ Y6 ) )
| ( ( ord_less_eq @ nat @ ( F4 @ X5 ) @ ( F4 @ Y6 ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X5 @ Y6 ) @ R6 ) ) ) ) ) ) ) ).
% mlex_eq
thf(fact_4672_these__insert__Some,axiom,
! [A: $tType,X: A,A5: set @ ( option @ A )] :
( ( these @ A @ ( insert @ ( option @ A ) @ ( some @ A @ X ) @ A5 ) )
= ( insert @ A @ X @ ( these @ A @ A5 ) ) ) ).
% these_insert_Some
thf(fact_4673_Id__on__def,axiom,
! [A: $tType] :
( ( id_on @ A )
= ( ^ [A7: set @ A] :
( complete_Sup_Sup @ ( set @ ( product_prod @ A @ A ) )
@ ( image2 @ A @ ( set @ ( product_prod @ A @ A ) )
@ ^ [X5: A] : ( insert @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X5 @ X5 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
@ A7 ) ) ) ) ).
% Id_on_def
thf(fact_4674_top2I,axiom,
! [A: $tType,B: $tType,X: A,Y: B] : ( top_top @ ( A > B > $o ) @ X @ Y ) ).
% top2I
thf(fact_4675_Id__onI,axiom,
! [A: $tType,A2: A,A5: set @ A] :
( ( member @ A @ A2 @ A5 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ A2 ) @ ( id_on @ A @ A5 ) ) ) ).
% Id_onI
thf(fact_4676_these__empty,axiom,
! [A: $tType] :
( ( these @ A @ ( bot_bot @ ( set @ ( option @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% these_empty
thf(fact_4677_Id__on__empty,axiom,
! [A: $tType] :
( ( id_on @ A @ ( bot_bot @ ( set @ A ) ) )
= ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ).
% Id_on_empty
thf(fact_4678_these__image__Some__eq,axiom,
! [A: $tType,A5: set @ A] :
( ( these @ A @ ( image2 @ A @ ( option @ A ) @ ( some @ A ) @ A5 ) )
= A5 ) ).
% these_image_Some_eq
thf(fact_4679_these__insert__None,axiom,
! [A: $tType,A5: set @ ( option @ A )] :
( ( these @ A @ ( insert @ ( option @ A ) @ ( none @ A ) @ A5 ) )
= ( these @ A @ A5 ) ) ).
% these_insert_None
thf(fact_4680_Id__on__iff,axiom,
! [A: $tType,X: A,Y: A,A5: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( id_on @ A @ A5 ) )
= ( ( X = Y )
& ( member @ A @ X @ A5 ) ) ) ).
% Id_on_iff
thf(fact_4681_Id__on__eqI,axiom,
! [A: $tType,A2: A,B2: A,A5: set @ A] :
( ( A2 = B2 )
=> ( ( member @ A @ A2 @ A5 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ ( id_on @ A @ A5 ) ) ) ) ).
% Id_on_eqI
thf(fact_4682_Id__onE,axiom,
! [A: $tType,C3: product_prod @ A @ A,A5: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ C3 @ ( id_on @ A @ A5 ) )
=> ~ ! [X4: A] :
( ( member @ A @ X4 @ A5 )
=> ( C3
!= ( product_Pair @ A @ A @ X4 @ X4 ) ) ) ) ).
% Id_onE
thf(fact_4683_in__these__eq,axiom,
! [A: $tType,X: A,A5: set @ ( option @ A )] :
( ( member @ A @ X @ ( these @ A @ A5 ) )
= ( member @ ( option @ A ) @ ( some @ A @ X ) @ A5 ) ) ).
% in_these_eq
thf(fact_4684_these__not__empty__eq,axiom,
! [A: $tType,B6: set @ ( option @ A )] :
( ( ( these @ A @ B6 )
!= ( bot_bot @ ( set @ A ) ) )
= ( ( B6
!= ( bot_bot @ ( set @ ( option @ A ) ) ) )
& ( B6
!= ( insert @ ( option @ A ) @ ( none @ A ) @ ( bot_bot @ ( set @ ( option @ A ) ) ) ) ) ) ) ).
% these_not_empty_eq
thf(fact_4685_these__empty__eq,axiom,
! [A: $tType,B6: set @ ( option @ A )] :
( ( ( these @ A @ B6 )
= ( bot_bot @ ( set @ A ) ) )
= ( ( B6
= ( bot_bot @ ( set @ ( option @ A ) ) ) )
| ( B6
= ( insert @ ( option @ A ) @ ( none @ A ) @ ( bot_bot @ ( set @ ( option @ A ) ) ) ) ) ) ) ).
% these_empty_eq
thf(fact_4686_Some__image__these__eq,axiom,
! [A: $tType,A5: set @ ( option @ A )] :
( ( image2 @ A @ ( option @ A ) @ ( some @ A ) @ ( these @ A @ A5 ) )
= ( collect @ ( option @ A )
@ ^ [X5: option @ A] :
( ( member @ ( option @ A ) @ X5 @ A5 )
& ( X5
!= ( none @ A ) ) ) ) ) ).
% Some_image_these_eq
thf(fact_4687_mlex__leq,axiom,
! [A: $tType,F3: A > nat,X: A,Y: A,R: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ nat @ ( F3 @ X ) @ ( F3 @ Y ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( mlex_prod @ A @ F3 @ R ) ) ) ) ).
% mlex_leq
thf(fact_4688_mlex__iff,axiom,
! [A: $tType,X: A,Y: A,F3: A > nat,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( mlex_prod @ A @ F3 @ R ) )
= ( ( ord_less @ nat @ ( F3 @ X ) @ ( F3 @ Y ) )
| ( ( ( F3 @ X )
= ( F3 @ Y ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R ) ) ) ) ).
% mlex_iff
thf(fact_4689_mlex__less,axiom,
! [A: $tType,F3: A > nat,X: A,Y: A,R: set @ ( product_prod @ A @ A )] :
( ( ord_less @ nat @ ( F3 @ X ) @ ( F3 @ Y ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( mlex_prod @ A @ F3 @ R ) ) ) ).
% mlex_less
thf(fact_4690_in__finite__psubset,axiom,
! [A: $tType,A5: set @ A,B6: set @ A] :
( ( member @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ A5 @ B6 ) @ ( finite_psubset @ A ) )
= ( ( ord_less @ ( set @ A ) @ A5 @ B6 )
& ( finite_finite2 @ A @ B6 ) ) ) ).
% in_finite_psubset
thf(fact_4691_in__measure,axiom,
! [A: $tType,X: A,Y: A,F3: A > nat] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( measure @ A @ F3 ) )
= ( ord_less @ nat @ ( F3 @ X ) @ ( F3 @ Y ) ) ) ).
% in_measure
thf(fact_4692_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_4693_at__within__empty,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [A2: A] :
( ( topolo174197925503356063within @ A @ A2 @ ( bot_bot @ ( set @ A ) ) )
= ( bot_bot @ ( filter @ A ) ) ) ) ).
% at_within_empty
thf(fact_4694_at__discrete,axiom,
! [A: $tType] :
( ( topolo8865339358273720382pology @ A )
=> ( ( topolo174197925503356063within @ A )
= ( ^ [X5: A,S7: set @ A] : ( bot_bot @ ( filter @ A ) ) ) ) ) ).
% at_discrete
thf(fact_4695_DERIV__at__within__shift,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,Y: A,Z: A,X: A,S2: set @ A] :
( ( has_field_derivative @ A @ F3 @ Y @ ( topolo174197925503356063within @ A @ ( plus_plus @ A @ Z @ X ) @ ( image2 @ A @ A @ ( plus_plus @ A @ Z ) @ S2 ) ) )
= ( has_field_derivative @ A
@ ^ [X5: A] : ( F3 @ ( plus_plus @ A @ Z @ X5 ) )
@ Y
@ ( topolo174197925503356063within @ A @ X @ S2 ) ) ) ) ).
% DERIV_at_within_shift
thf(fact_4696_DERIV__mult,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,Da: A,X: A,S: set @ A,G3: A > A,Db: A] :
( ( has_field_derivative @ A @ F3 @ Da @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( ( has_field_derivative @ A @ G3 @ Db @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( has_field_derivative @ A
@ ^ [X5: A] : ( times_times @ A @ ( F3 @ X5 ) @ ( G3 @ X5 ) )
@ ( plus_plus @ A @ ( times_times @ A @ Da @ ( G3 @ X ) ) @ ( times_times @ A @ Db @ ( F3 @ X ) ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ).
% DERIV_mult
thf(fact_4697_DERIV__mult_H,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,D6: A,X: A,S: set @ A,G3: A > A,E5: A] :
( ( has_field_derivative @ A @ F3 @ D6 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( ( has_field_derivative @ A @ G3 @ E5 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( has_field_derivative @ A
@ ^ [X5: A] : ( times_times @ A @ ( F3 @ X5 ) @ ( G3 @ X5 ) )
@ ( plus_plus @ A @ ( times_times @ A @ ( F3 @ X ) @ E5 ) @ ( times_times @ A @ D6 @ ( G3 @ X ) ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ).
% DERIV_mult'
thf(fact_4698_DERIV__add,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,D6: A,X: A,S: set @ A,G3: A > A,E5: A] :
( ( has_field_derivative @ A @ F3 @ D6 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( ( has_field_derivative @ A @ G3 @ E5 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( has_field_derivative @ A
@ ^ [X5: A] : ( plus_plus @ A @ ( F3 @ X5 ) @ ( G3 @ X5 ) )
@ ( plus_plus @ A @ D6 @ E5 )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ).
% DERIV_add
thf(fact_4699_DERIV__const,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [K2: A,F5: filter @ A] :
( has_field_derivative @ A
@ ^ [X5: A] : K2
@ ( zero_zero @ A )
@ F5 ) ) ).
% DERIV_const
thf(fact_4700_field__differentiable__add,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,F10: A,F5: filter @ A,G3: A > A,G7: A] :
( ( has_field_derivative @ A @ F3 @ F10 @ F5 )
=> ( ( has_field_derivative @ A @ G3 @ G7 @ F5 )
=> ( has_field_derivative @ A
@ ^ [Z5: A] : ( plus_plus @ A @ ( F3 @ Z5 ) @ ( G3 @ Z5 ) )
@ ( plus_plus @ A @ F10 @ G7 )
@ F5 ) ) ) ) ).
% field_differentiable_add
thf(fact_4701_has__field__derivative__transform__within,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,F10: A,A2: A,S2: set @ A,D3: real,G3: A > A] :
( ( has_field_derivative @ A @ F3 @ F10 @ ( topolo174197925503356063within @ A @ A2 @ S2 ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ D3 )
=> ( ( member @ A @ A2 @ S2 )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ S2 )
=> ( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X4 @ A2 ) @ D3 )
=> ( ( F3 @ X4 )
= ( G3 @ X4 ) ) ) )
=> ( has_field_derivative @ A @ G3 @ F10 @ ( topolo174197925503356063within @ A @ A2 @ S2 ) ) ) ) ) ) ) ).
% has_field_derivative_transform_within
thf(fact_4702_has__real__derivative__neg__dec__left,axiom,
! [F3: real > real,L: real,X: real,S2: set @ real] :
( ( has_field_derivative @ real @ F3 @ L @ ( topolo174197925503356063within @ real @ X @ S2 ) )
=> ( ( ord_less @ real @ L @ ( zero_zero @ real ) )
=> ? [D2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D2 )
& ! [H5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ H5 )
=> ( ( member @ real @ ( minus_minus @ real @ X @ H5 ) @ S2 )
=> ( ( ord_less @ real @ H5 @ D2 )
=> ( ord_less @ real @ ( F3 @ X ) @ ( F3 @ ( minus_minus @ real @ X @ H5 ) ) ) ) ) ) ) ) ) ).
% has_real_derivative_neg_dec_left
thf(fact_4703_has__real__derivative__pos__inc__left,axiom,
! [F3: real > real,L: real,X: real,S2: set @ real] :
( ( has_field_derivative @ real @ F3 @ L @ ( topolo174197925503356063within @ real @ X @ S2 ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ L )
=> ? [D2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D2 )
& ! [H5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ H5 )
=> ( ( member @ real @ ( minus_minus @ real @ X @ H5 ) @ S2 )
=> ( ( ord_less @ real @ H5 @ D2 )
=> ( ord_less @ real @ ( F3 @ ( minus_minus @ real @ X @ H5 ) ) @ ( F3 @ X ) ) ) ) ) ) ) ) ).
% has_real_derivative_pos_inc_left
thf(fact_4704_has__real__derivative__pos__inc__right,axiom,
! [F3: real > real,L: real,X: real,S2: set @ real] :
( ( has_field_derivative @ real @ F3 @ L @ ( topolo174197925503356063within @ real @ X @ S2 ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ L )
=> ? [D2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D2 )
& ! [H5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ H5 )
=> ( ( member @ real @ ( plus_plus @ real @ X @ H5 ) @ S2 )
=> ( ( ord_less @ real @ H5 @ D2 )
=> ( ord_less @ real @ ( F3 @ X ) @ ( F3 @ ( plus_plus @ real @ X @ H5 ) ) ) ) ) ) ) ) ) ).
% has_real_derivative_pos_inc_right
thf(fact_4705_has__real__derivative__neg__dec__right,axiom,
! [F3: real > real,L: real,X: real,S2: set @ real] :
( ( has_field_derivative @ real @ F3 @ L @ ( topolo174197925503356063within @ real @ X @ S2 ) )
=> ( ( ord_less @ real @ L @ ( zero_zero @ real ) )
=> ? [D2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D2 )
& ! [H5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ H5 )
=> ( ( member @ real @ ( plus_plus @ real @ X @ H5 ) @ S2 )
=> ( ( ord_less @ real @ H5 @ D2 )
=> ( ord_less @ real @ ( F3 @ ( plus_plus @ real @ X @ H5 ) ) @ ( F3 @ X ) ) ) ) ) ) ) ) ).
% has_real_derivative_neg_dec_right
thf(fact_4706_DERIV__isconst3,axiom,
! [A2: real,B2: real,X: real,Y: real,F3: real > real] :
( ( ord_less @ real @ A2 @ B2 )
=> ( ( member @ real @ X @ ( set_or5935395276787703475ssThan @ real @ A2 @ B2 ) )
=> ( ( member @ real @ Y @ ( set_or5935395276787703475ssThan @ real @ A2 @ B2 ) )
=> ( ! [X4: real] :
( ( member @ real @ X4 @ ( set_or5935395276787703475ssThan @ real @ A2 @ B2 ) )
=> ( has_field_derivative @ real @ F3 @ ( zero_zero @ real ) @ ( topolo174197925503356063within @ real @ X4 @ ( top_top @ ( set @ real ) ) ) ) )
=> ( ( F3 @ X )
= ( F3 @ Y ) ) ) ) ) ) ).
% DERIV_isconst3
thf(fact_4707_DERIV__shift,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,Y: A,X: A,Z: A] :
( ( has_field_derivative @ A @ F3 @ Y @ ( topolo174197925503356063within @ A @ ( plus_plus @ A @ X @ Z ) @ ( top_top @ ( set @ A ) ) ) )
= ( has_field_derivative @ A
@ ^ [X5: A] : ( F3 @ ( plus_plus @ A @ X5 @ Z ) )
@ Y
@ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% DERIV_shift
thf(fact_4708_DERIV__neg__imp__decreasing,axiom,
! [A2: real,B2: real,F3: real > real] :
( ( ord_less @ real @ A2 @ B2 )
=> ( ! [X4: real] :
( ( ord_less_eq @ real @ A2 @ X4 )
=> ( ( ord_less_eq @ real @ X4 @ B2 )
=> ? [Y4: real] :
( ( has_field_derivative @ real @ F3 @ Y4 @ ( topolo174197925503356063within @ real @ X4 @ ( top_top @ ( set @ real ) ) ) )
& ( ord_less @ real @ Y4 @ ( zero_zero @ real ) ) ) ) )
=> ( ord_less @ real @ ( F3 @ B2 ) @ ( F3 @ A2 ) ) ) ) ).
% DERIV_neg_imp_decreasing
thf(fact_4709_DERIV__pos__imp__increasing,axiom,
! [A2: real,B2: real,F3: real > real] :
( ( ord_less @ real @ A2 @ B2 )
=> ( ! [X4: real] :
( ( ord_less_eq @ real @ A2 @ X4 )
=> ( ( ord_less_eq @ real @ X4 @ B2 )
=> ? [Y4: real] :
( ( has_field_derivative @ real @ F3 @ Y4 @ ( topolo174197925503356063within @ real @ X4 @ ( top_top @ ( set @ real ) ) ) )
& ( ord_less @ real @ ( zero_zero @ real ) @ Y4 ) ) ) )
=> ( ord_less @ real @ ( F3 @ A2 ) @ ( F3 @ B2 ) ) ) ) ).
% DERIV_pos_imp_increasing
thf(fact_4710_DERIV__pos__inc__right,axiom,
! [F3: real > real,L: real,X: real] :
( ( has_field_derivative @ real @ F3 @ L @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ L )
=> ? [D2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D2 )
& ! [H5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ H5 )
=> ( ( ord_less @ real @ H5 @ D2 )
=> ( ord_less @ real @ ( F3 @ X ) @ ( F3 @ ( plus_plus @ real @ X @ H5 ) ) ) ) ) ) ) ) ).
% DERIV_pos_inc_right
thf(fact_4711_DERIV__neg__dec__right,axiom,
! [F3: real > real,L: real,X: real] :
( ( has_field_derivative @ real @ F3 @ L @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( ( ord_less @ real @ L @ ( zero_zero @ real ) )
=> ? [D2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D2 )
& ! [H5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ H5 )
=> ( ( ord_less @ real @ H5 @ D2 )
=> ( ord_less @ real @ ( F3 @ ( plus_plus @ real @ X @ H5 ) ) @ ( F3 @ X ) ) ) ) ) ) ) ).
% DERIV_neg_dec_right
thf(fact_4712_DERIV__neg__dec__left,axiom,
! [F3: real > real,L: real,X: real] :
( ( has_field_derivative @ real @ F3 @ L @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( ( ord_less @ real @ L @ ( zero_zero @ real ) )
=> ? [D2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D2 )
& ! [H5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ H5 )
=> ( ( ord_less @ real @ H5 @ D2 )
=> ( ord_less @ real @ ( F3 @ X ) @ ( F3 @ ( minus_minus @ real @ X @ H5 ) ) ) ) ) ) ) ) ).
% DERIV_neg_dec_left
thf(fact_4713_DERIV__pos__inc__left,axiom,
! [F3: real > real,L: real,X: real] :
( ( has_field_derivative @ real @ F3 @ L @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ L )
=> ? [D2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D2 )
& ! [H5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ H5 )
=> ( ( ord_less @ real @ H5 @ D2 )
=> ( ord_less @ real @ ( F3 @ ( minus_minus @ real @ X @ H5 ) ) @ ( F3 @ X ) ) ) ) ) ) ) ).
% DERIV_pos_inc_left
thf(fact_4714_DERIV__divide,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,D6: A,X: A,S: set @ A,G3: A > A,E5: A] :
( ( has_field_derivative @ A @ F3 @ D6 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( ( has_field_derivative @ A @ G3 @ E5 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( ( ( G3 @ X )
!= ( zero_zero @ A ) )
=> ( has_field_derivative @ A
@ ^ [X5: A] : ( divide_divide @ A @ ( F3 @ X5 ) @ ( G3 @ X5 ) )
@ ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ D6 @ ( G3 @ X ) ) @ ( times_times @ A @ ( F3 @ X ) @ E5 ) ) @ ( times_times @ A @ ( G3 @ X ) @ ( G3 @ X ) ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ) ).
% DERIV_divide
thf(fact_4715_DERIV__inverse_H,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,D6: A,X: A,S: set @ A] :
( ( has_field_derivative @ A @ F3 @ D6 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( ( ( F3 @ X )
!= ( zero_zero @ A ) )
=> ( has_field_derivative @ A
@ ^ [X5: A] : ( inverse_inverse @ A @ ( F3 @ X5 ) )
@ ( uminus_uminus @ A @ ( times_times @ A @ ( times_times @ A @ ( inverse_inverse @ A @ ( F3 @ X ) ) @ D6 ) @ ( inverse_inverse @ A @ ( F3 @ X ) ) ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ).
% DERIV_inverse'
thf(fact_4716_at__neq__bot,axiom,
! [A: $tType] :
( ( topolo8386298272705272623_space @ A )
=> ! [A2: A] :
( ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) )
!= ( bot_bot @ ( filter @ A ) ) ) ) ).
% at_neq_bot
thf(fact_4717_trivial__limit__at__left__real,axiom,
! [A: $tType] :
( ( ( dense_order @ A )
& ( no_bot @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [X: A] :
( ( topolo174197925503356063within @ A @ X @ ( set_ord_lessThan @ A @ X ) )
!= ( bot_bot @ ( filter @ A ) ) ) ) ).
% trivial_limit_at_left_real
thf(fact_4718_MVT2,axiom,
! [A2: real,B2: real,F3: real > real,F10: real > real] :
( ( ord_less @ real @ A2 @ B2 )
=> ( ! [X4: real] :
( ( ord_less_eq @ real @ A2 @ X4 )
=> ( ( ord_less_eq @ real @ X4 @ B2 )
=> ( has_field_derivative @ real @ F3 @ ( F10 @ X4 ) @ ( topolo174197925503356063within @ real @ X4 @ ( top_top @ ( set @ real ) ) ) ) ) )
=> ? [Z3: real] :
( ( ord_less @ real @ A2 @ Z3 )
& ( ord_less @ real @ Z3 @ B2 )
& ( ( minus_minus @ real @ ( F3 @ B2 ) @ ( F3 @ A2 ) )
= ( times_times @ real @ ( minus_minus @ real @ B2 @ A2 ) @ ( F10 @ Z3 ) ) ) ) ) ) ).
% MVT2
thf(fact_4719_DERIV__local__const,axiom,
! [F3: real > real,L: real,X: real,D3: real] :
( ( has_field_derivative @ real @ F3 @ L @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ D3 )
=> ( ! [Y3: real] :
( ( ord_less @ real @ ( abs_abs @ real @ ( minus_minus @ real @ X @ Y3 ) ) @ D3 )
=> ( ( F3 @ X )
= ( F3 @ Y3 ) ) )
=> ( L
= ( zero_zero @ real ) ) ) ) ) ).
% DERIV_local_const
thf(fact_4720_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_4721_DERIV__cos__add,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [K2: A,Xa2: A] :
( has_field_derivative @ A
@ ^ [X5: A] : ( cos @ A @ ( plus_plus @ A @ X5 @ K2 ) )
@ ( uminus_uminus @ A @ ( sin @ A @ ( plus_plus @ A @ Xa2 @ K2 ) ) )
@ ( topolo174197925503356063within @ A @ Xa2 @ ( top_top @ ( set @ A ) ) ) ) ) ).
% DERIV_cos_add
thf(fact_4722_DERIV__power__Suc,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,D6: A,X: A,S: set @ A,N: nat] :
( ( has_field_derivative @ A @ F3 @ D6 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( has_field_derivative @ A
@ ^ [X5: A] : ( power_power @ A @ ( F3 @ X5 ) @ ( suc @ N ) )
@ ( times_times @ A @ ( plus_plus @ A @ ( one_one @ A ) @ ( semiring_1_of_nat @ A @ N ) ) @ ( times_times @ A @ D6 @ ( power_power @ A @ ( F3 @ X ) @ N ) ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ).
% DERIV_power_Suc
thf(fact_4723_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_4724_DERIV__power,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,D6: A,X: A,S: set @ A,N: nat] :
( ( has_field_derivative @ A @ F3 @ D6 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( has_field_derivative @ A
@ ^ [X5: A] : ( power_power @ A @ ( F3 @ X5 ) @ N )
@ ( times_times @ A @ ( semiring_1_of_nat @ A @ N ) @ ( times_times @ A @ D6 @ ( power_power @ A @ ( F3 @ X ) @ ( minus_minus @ nat @ N @ ( suc @ ( zero_zero @ nat ) ) ) ) ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ).
% DERIV_power
thf(fact_4725_DERIV__local__max,axiom,
! [F3: real > real,L: real,X: real,D3: real] :
( ( has_field_derivative @ real @ F3 @ L @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ D3 )
=> ( ! [Y3: real] :
( ( ord_less @ real @ ( abs_abs @ real @ ( minus_minus @ real @ X @ Y3 ) ) @ D3 )
=> ( ord_less_eq @ real @ ( F3 @ Y3 ) @ ( F3 @ X ) ) )
=> ( L
= ( zero_zero @ real ) ) ) ) ) ).
% DERIV_local_max
thf(fact_4726_DERIV__local__min,axiom,
! [F3: real > real,L: real,X: real,D3: real] :
( ( has_field_derivative @ real @ F3 @ L @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ D3 )
=> ( ! [Y3: real] :
( ( ord_less @ real @ ( abs_abs @ real @ ( minus_minus @ real @ X @ Y3 ) ) @ D3 )
=> ( ord_less_eq @ real @ ( F3 @ X ) @ ( F3 @ Y3 ) ) )
=> ( L
= ( zero_zero @ real ) ) ) ) ) ).
% DERIV_local_min
thf(fact_4727_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_4728_DERIV__pow,axiom,
! [N: nat,X: real,S: set @ real] :
( has_field_derivative @ real
@ ^ [X5: real] : ( power_power @ real @ X5 @ 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_4729_at__within__Icc__at,axiom,
! [A: $tType] :
( ( topolo2564578578187576103pology @ A )
=> ! [A2: A,X: A,B2: A] :
( ( ord_less @ A @ A2 @ X )
=> ( ( ord_less @ A @ X @ B2 )
=> ( ( topolo174197925503356063within @ A @ X @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
= ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% at_within_Icc_at
thf(fact_4730_at__within__Icc__at__left,axiom,
! [A: $tType] :
( ( topolo2564578578187576103pology @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( topolo174197925503356063within @ A @ B2 @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
= ( topolo174197925503356063within @ A @ B2 @ ( set_ord_lessThan @ A @ B2 ) ) ) ) ) ).
% at_within_Icc_at_left
thf(fact_4731_trivial__limit__at__left__bot,axiom,
! [A: $tType] :
( ( ( order_bot @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ( ( topolo174197925503356063within @ A @ ( bot_bot @ A ) @ ( set_ord_lessThan @ A @ ( bot_bot @ A ) ) )
= ( bot_bot @ ( filter @ A ) ) ) ) ).
% trivial_limit_at_left_bot
thf(fact_4732_DERIV__quotient,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,D3: A,X: A,S: set @ A,G3: A > A,E3: A] :
( ( has_field_derivative @ A @ F3 @ D3 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( ( has_field_derivative @ A @ G3 @ E3 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( ( ( G3 @ X )
!= ( zero_zero @ A ) )
=> ( has_field_derivative @ A
@ ^ [Y6: A] : ( divide_divide @ A @ ( F3 @ Y6 ) @ ( G3 @ Y6 ) )
@ ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ D3 @ ( G3 @ X ) ) @ ( times_times @ A @ E3 @ ( F3 @ X ) ) ) @ ( power_power @ A @ ( G3 @ X ) @ ( suc @ ( suc @ ( zero_zero @ nat ) ) ) ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ) ).
% DERIV_quotient
thf(fact_4733_DERIV__inverse__fun,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,D3: A,X: A,S: set @ A] :
( ( has_field_derivative @ A @ F3 @ D3 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( ( ( F3 @ X )
!= ( zero_zero @ A ) )
=> ( has_field_derivative @ A
@ ^ [X5: A] : ( inverse_inverse @ A @ ( F3 @ X5 ) )
@ ( uminus_uminus @ A @ ( times_times @ A @ D3 @ ( inverse_inverse @ A @ ( power_power @ A @ ( F3 @ X ) @ ( suc @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ).
% DERIV_inverse_fun
thf(fact_4734_termdiffs__sums__strong,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [K5: real,C3: nat > A,F3: A > A,F10: A,Z: A] :
( ! [Z3: A] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ Z3 ) @ K5 )
=> ( sums @ A
@ ^ [N2: nat] : ( times_times @ A @ ( C3 @ N2 ) @ ( power_power @ A @ Z3 @ N2 ) )
@ ( F3 @ Z3 ) ) )
=> ( ( has_field_derivative @ A @ F3 @ F10 @ ( topolo174197925503356063within @ A @ Z @ ( top_top @ ( set @ A ) ) ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ Z ) @ K5 )
=> ( sums @ A
@ ^ [N2: nat] : ( times_times @ A @ ( diffs @ A @ C3 @ N2 ) @ ( power_power @ A @ Z @ N2 ) )
@ F10 ) ) ) ) ) ).
% termdiffs_sums_strong
thf(fact_4735_has__real__derivative__powr,axiom,
! [Z: real,R2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ Z )
=> ( has_field_derivative @ real
@ ^ [Z5: real] : ( powr @ real @ Z5 @ R2 )
@ ( times_times @ real @ R2 @ ( powr @ real @ Z @ ( minus_minus @ real @ R2 @ ( one_one @ real ) ) ) )
@ ( topolo174197925503356063within @ real @ Z @ ( top_top @ ( set @ real ) ) ) ) ) ).
% has_real_derivative_powr
thf(fact_4736_termdiffs__strong_H,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [K5: real,C3: nat > A,Z: A] :
( ! [Z3: A] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ Z3 ) @ K5 )
=> ( summable @ A
@ ^ [N2: nat] : ( times_times @ A @ ( C3 @ N2 ) @ ( power_power @ A @ Z3 @ N2 ) ) ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ Z ) @ K5 )
=> ( has_field_derivative @ A
@ ^ [Z5: A] :
( suminf @ A
@ ^ [N2: nat] : ( times_times @ A @ ( C3 @ N2 ) @ ( power_power @ A @ Z5 @ N2 ) ) )
@ ( suminf @ A
@ ^ [N2: nat] : ( times_times @ A @ ( diffs @ A @ C3 @ N2 ) @ ( power_power @ A @ Z @ N2 ) ) )
@ ( topolo174197925503356063within @ A @ Z @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% termdiffs_strong'
thf(fact_4737_termdiffs__strong,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [C3: nat > A,K5: A,X: A] :
( ( summable @ A
@ ^ [N2: nat] : ( times_times @ A @ ( C3 @ N2 ) @ ( power_power @ A @ K5 @ N2 ) ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ ( real_V7770717601297561774m_norm @ A @ K5 ) )
=> ( has_field_derivative @ A
@ ^ [X5: A] :
( suminf @ A
@ ^ [N2: nat] : ( times_times @ A @ ( C3 @ N2 ) @ ( power_power @ A @ X5 @ N2 ) ) )
@ ( suminf @ A
@ ^ [N2: nat] : ( times_times @ A @ ( diffs @ A @ C3 @ N2 ) @ ( power_power @ A @ X @ N2 ) ) )
@ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% termdiffs_strong
thf(fact_4738_termdiffs,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [C3: nat > A,K5: A,X: A] :
( ( summable @ A
@ ^ [N2: nat] : ( times_times @ A @ ( C3 @ N2 ) @ ( power_power @ A @ K5 @ N2 ) ) )
=> ( ( summable @ A
@ ^ [N2: nat] : ( times_times @ A @ ( diffs @ A @ C3 @ N2 ) @ ( power_power @ A @ K5 @ N2 ) ) )
=> ( ( summable @ A
@ ^ [N2: nat] : ( times_times @ A @ ( diffs @ A @ ( diffs @ A @ C3 ) @ N2 ) @ ( power_power @ A @ K5 @ N2 ) ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ ( real_V7770717601297561774m_norm @ A @ K5 ) )
=> ( has_field_derivative @ A
@ ^ [X5: A] :
( suminf @ A
@ ^ [N2: nat] : ( times_times @ A @ ( C3 @ N2 ) @ ( power_power @ A @ X5 @ N2 ) ) )
@ ( suminf @ A
@ ^ [N2: nat] : ( times_times @ A @ ( diffs @ A @ C3 @ N2 ) @ ( power_power @ A @ X @ N2 ) ) )
@ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% termdiffs
thf(fact_4739_DERIV__log,axiom,
! [X: real,B2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ X )
=> ( has_field_derivative @ real @ ( log @ B2 ) @ ( divide_divide @ real @ ( one_one @ real ) @ ( times_times @ real @ ( ln_ln @ real @ B2 ) @ X ) ) @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) ) ) ).
% DERIV_log
thf(fact_4740_DERIV__fun__powr,axiom,
! [G3: real > real,M2: real,X: real,R2: real] :
( ( has_field_derivative @ real @ G3 @ M2 @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( G3 @ X ) )
=> ( has_field_derivative @ real
@ ^ [X5: real] : ( powr @ real @ ( G3 @ X5 ) @ R2 )
@ ( times_times @ real @ ( times_times @ real @ R2 @ ( powr @ real @ ( G3 @ X ) @ ( minus_minus @ real @ R2 @ ( semiring_1_of_nat @ real @ ( one_one @ nat ) ) ) ) ) @ M2 )
@ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) ) ) ) ).
% DERIV_fun_powr
thf(fact_4741_DERIV__powr,axiom,
! [G3: real > real,M2: real,X: real,F3: real > real,R2: real] :
( ( has_field_derivative @ real @ G3 @ M2 @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( G3 @ X ) )
=> ( ( has_field_derivative @ real @ F3 @ R2 @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) )
=> ( has_field_derivative @ real
@ ^ [X5: real] : ( powr @ real @ ( G3 @ X5 ) @ ( F3 @ X5 ) )
@ ( times_times @ real @ ( powr @ real @ ( G3 @ X ) @ ( F3 @ X ) ) @ ( plus_plus @ real @ ( times_times @ real @ R2 @ ( ln_ln @ real @ ( G3 @ X ) ) ) @ ( divide_divide @ real @ ( times_times @ real @ M2 @ ( F3 @ X ) ) @ ( G3 @ X ) ) ) )
@ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ).
% DERIV_powr
thf(fact_4742_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_4743_arcosh__real__has__field__derivative,axiom,
! [X: real,A5: 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 @ A5 ) ) ) ).
% arcosh_real_has_field_derivative
thf(fact_4744_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_4745_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_4746_has__field__derivative__tanh,axiom,
! [A9: $tType] :
( ( ( real_Vector_banach @ A9 )
& ( real_V3459762299906320749_field @ A9 ) )
=> ! [G3: A9 > A9,X: A9,Db: A9,S: set @ A9] :
( ( ( cosh @ A9 @ ( G3 @ X ) )
!= ( zero_zero @ A9 ) )
=> ( ( has_field_derivative @ A9 @ G3 @ Db @ ( topolo174197925503356063within @ A9 @ X @ S ) )
=> ( has_field_derivative @ A9
@ ^ [X5: A9] : ( tanh @ A9 @ ( G3 @ X5 ) )
@ ( times_times @ A9 @ ( minus_minus @ A9 @ ( one_one @ A9 ) @ ( power_power @ A9 @ ( tanh @ A9 @ ( G3 @ X ) ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) @ Db )
@ ( topolo174197925503356063within @ A9 @ X @ S ) ) ) ) ) ).
% has_field_derivative_tanh
thf(fact_4747_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_4748_artanh__real__has__field__derivative,axiom,
! [X: real,A5: 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 @ A5 ) ) ) ).
% artanh_real_has_field_derivative
thf(fact_4749_DERIV__power__series_H,axiom,
! [R: real,F3: nat > real,X0: real] :
( ! [X4: real] :
( ( member @ real @ X4 @ ( set_or5935395276787703475ssThan @ real @ ( uminus_uminus @ real @ R ) @ R ) )
=> ( summable @ real
@ ^ [N2: nat] : ( times_times @ real @ ( times_times @ real @ ( F3 @ N2 ) @ ( semiring_1_of_nat @ real @ ( suc @ N2 ) ) ) @ ( power_power @ real @ X4 @ N2 ) ) ) )
=> ( ( member @ real @ X0 @ ( set_or5935395276787703475ssThan @ real @ ( uminus_uminus @ real @ R ) @ R ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ R )
=> ( has_field_derivative @ real
@ ^ [X5: real] :
( suminf @ real
@ ^ [N2: nat] : ( times_times @ real @ ( F3 @ N2 ) @ ( power_power @ real @ X5 @ ( suc @ N2 ) ) ) )
@ ( suminf @ real
@ ^ [N2: nat] : ( times_times @ real @ ( times_times @ real @ ( F3 @ N2 ) @ ( semiring_1_of_nat @ real @ ( suc @ N2 ) ) ) @ ( power_power @ real @ X0 @ N2 ) ) )
@ ( topolo174197925503356063within @ real @ X0 @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ).
% DERIV_power_series'
thf(fact_4750_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_4751_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_4752_Maclaurin__all__le,axiom,
! [Diff: nat > real > real,F3: real > real,X: real,N: nat] :
( ( ( Diff @ ( zero_zero @ nat ) )
= F3 )
=> ( ! [M: nat,X4: real] : ( has_field_derivative @ real @ ( Diff @ M ) @ ( Diff @ ( suc @ M ) @ X4 ) @ ( topolo174197925503356063within @ real @ X4 @ ( top_top @ ( set @ real ) ) ) )
=> ? [T7: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ T7 ) @ ( abs_abs @ real @ X ) )
& ( ( F3 @ X )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M3: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M3 @ ( zero_zero @ real ) ) @ ( semiring_char_0_fact @ real @ M3 ) ) @ ( power_power @ real @ X @ M3 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N @ T7 ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ X @ N ) ) ) ) ) ) ) ).
% Maclaurin_all_le
thf(fact_4753_Maclaurin__all__le__objl,axiom,
! [Diff: nat > real > real,F3: real > real,X: real,N: nat] :
( ( ( ( Diff @ ( zero_zero @ nat ) )
= F3 )
& ! [M: nat,X4: real] : ( has_field_derivative @ real @ ( Diff @ M ) @ ( Diff @ ( suc @ M ) @ X4 ) @ ( topolo174197925503356063within @ real @ X4 @ ( top_top @ ( set @ real ) ) ) ) )
=> ? [T7: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ T7 ) @ ( abs_abs @ real @ X ) )
& ( ( F3 @ X )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M3: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M3 @ ( zero_zero @ real ) ) @ ( semiring_char_0_fact @ real @ M3 ) ) @ ( power_power @ real @ X @ M3 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N @ T7 ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ X @ N ) ) ) ) ) ) ).
% Maclaurin_all_le_objl
thf(fact_4754_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_4755_Maclaurin,axiom,
! [H2: real,N: nat,Diff: nat > real > real,F3: real > real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ H2 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ( Diff @ ( zero_zero @ nat ) )
= F3 )
=> ( ! [M: nat,T7: real] :
( ( ( ord_less @ nat @ M @ N )
& ( ord_less_eq @ real @ ( zero_zero @ real ) @ T7 )
& ( ord_less_eq @ real @ T7 @ H2 ) )
=> ( has_field_derivative @ real @ ( Diff @ M ) @ ( Diff @ ( suc @ M ) @ T7 ) @ ( topolo174197925503356063within @ real @ T7 @ ( top_top @ ( set @ real ) ) ) ) )
=> ? [T7: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ T7 )
& ( ord_less @ real @ T7 @ H2 )
& ( ( F3 @ H2 )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M3: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M3 @ ( zero_zero @ real ) ) @ ( semiring_char_0_fact @ real @ M3 ) ) @ ( power_power @ real @ H2 @ M3 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N @ T7 ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ H2 @ N ) ) ) ) ) ) ) ) ) ).
% Maclaurin
thf(fact_4756_Maclaurin2,axiom,
! [H2: real,Diff: nat > real > real,F3: real > real,N: nat] :
( ( ord_less @ real @ ( zero_zero @ real ) @ H2 )
=> ( ( ( Diff @ ( zero_zero @ nat ) )
= F3 )
=> ( ! [M: nat,T7: real] :
( ( ( ord_less @ nat @ M @ N )
& ( ord_less_eq @ real @ ( zero_zero @ real ) @ T7 )
& ( ord_less_eq @ real @ T7 @ H2 ) )
=> ( has_field_derivative @ real @ ( Diff @ M ) @ ( Diff @ ( suc @ M ) @ T7 ) @ ( topolo174197925503356063within @ real @ T7 @ ( top_top @ ( set @ real ) ) ) ) )
=> ? [T7: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ T7 )
& ( ord_less_eq @ real @ T7 @ H2 )
& ( ( F3 @ H2 )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M3: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M3 @ ( zero_zero @ real ) ) @ ( semiring_char_0_fact @ real @ M3 ) ) @ ( power_power @ real @ H2 @ M3 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N @ T7 ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ H2 @ N ) ) ) ) ) ) ) ) ).
% Maclaurin2
thf(fact_4757_Maclaurin__minus,axiom,
! [H2: real,N: nat,Diff: nat > real > real,F3: real > real] :
( ( ord_less @ real @ H2 @ ( zero_zero @ real ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ( Diff @ ( zero_zero @ nat ) )
= F3 )
=> ( ! [M: nat,T7: real] :
( ( ( ord_less @ nat @ M @ N )
& ( ord_less_eq @ real @ H2 @ T7 )
& ( ord_less_eq @ real @ T7 @ ( zero_zero @ real ) ) )
=> ( has_field_derivative @ real @ ( Diff @ M ) @ ( Diff @ ( suc @ M ) @ T7 ) @ ( topolo174197925503356063within @ real @ T7 @ ( top_top @ ( set @ real ) ) ) ) )
=> ? [T7: real] :
( ( ord_less @ real @ H2 @ T7 )
& ( ord_less @ real @ T7 @ ( zero_zero @ real ) )
& ( ( F3 @ H2 )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M3: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M3 @ ( zero_zero @ real ) ) @ ( semiring_char_0_fact @ real @ M3 ) ) @ ( power_power @ real @ H2 @ M3 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N @ T7 ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ H2 @ N ) ) ) ) ) ) ) ) ) ).
% Maclaurin_minus
thf(fact_4758_Maclaurin__all__lt,axiom,
! [Diff: nat > real > real,F3: real > real,N: nat,X: real] :
( ( ( Diff @ ( zero_zero @ nat ) )
= F3 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( X
!= ( zero_zero @ real ) )
=> ( ! [M: nat,X4: real] : ( has_field_derivative @ real @ ( Diff @ M ) @ ( Diff @ ( suc @ M ) @ X4 ) @ ( topolo174197925503356063within @ real @ X4 @ ( top_top @ ( set @ real ) ) ) )
=> ? [T7: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( abs_abs @ real @ T7 ) )
& ( ord_less @ real @ ( abs_abs @ real @ T7 ) @ ( abs_abs @ real @ X ) )
& ( ( F3 @ X )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M3: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M3 @ ( zero_zero @ real ) ) @ ( semiring_char_0_fact @ real @ M3 ) ) @ ( power_power @ real @ X @ M3 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N @ T7 ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ X @ N ) ) ) ) ) ) ) ) ) ).
% Maclaurin_all_lt
thf(fact_4759_Maclaurin__bi__le,axiom,
! [Diff: nat > real > real,F3: real > real,N: nat,X: real] :
( ( ( Diff @ ( zero_zero @ nat ) )
= F3 )
=> ( ! [M: nat,T7: real] :
( ( ( ord_less @ nat @ M @ N )
& ( ord_less_eq @ real @ ( abs_abs @ real @ T7 ) @ ( abs_abs @ real @ X ) ) )
=> ( has_field_derivative @ real @ ( Diff @ M ) @ ( Diff @ ( suc @ M ) @ T7 ) @ ( topolo174197925503356063within @ real @ T7 @ ( top_top @ ( set @ real ) ) ) ) )
=> ? [T7: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ T7 ) @ ( abs_abs @ real @ X ) )
& ( ( F3 @ X )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M3: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M3 @ ( zero_zero @ real ) ) @ ( semiring_char_0_fact @ real @ M3 ) ) @ ( power_power @ real @ X @ M3 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N @ T7 ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ X @ N ) ) ) ) ) ) ) ).
% Maclaurin_bi_le
thf(fact_4760_Taylor__down,axiom,
! [N: nat,Diff: nat > real > real,F3: real > real,A2: real,B2: real,C3: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ( Diff @ ( zero_zero @ nat ) )
= F3 )
=> ( ! [M: nat,T7: real] :
( ( ( ord_less @ nat @ M @ N )
& ( ord_less_eq @ real @ A2 @ T7 )
& ( ord_less_eq @ real @ T7 @ B2 ) )
=> ( has_field_derivative @ real @ ( Diff @ M ) @ ( Diff @ ( suc @ M ) @ T7 ) @ ( topolo174197925503356063within @ real @ T7 @ ( top_top @ ( set @ real ) ) ) ) )
=> ( ( ord_less @ real @ A2 @ C3 )
=> ( ( ord_less_eq @ real @ C3 @ B2 )
=> ? [T7: real] :
( ( ord_less @ real @ A2 @ T7 )
& ( ord_less @ real @ T7 @ C3 )
& ( ( F3 @ A2 )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M3: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M3 @ C3 ) @ ( semiring_char_0_fact @ real @ M3 ) ) @ ( power_power @ real @ ( minus_minus @ real @ A2 @ C3 ) @ M3 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N @ T7 ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ ( minus_minus @ real @ A2 @ C3 ) @ N ) ) ) ) ) ) ) ) ) ) ).
% Taylor_down
thf(fact_4761_Taylor__up,axiom,
! [N: nat,Diff: nat > real > real,F3: real > real,A2: real,B2: real,C3: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ( Diff @ ( zero_zero @ nat ) )
= F3 )
=> ( ! [M: nat,T7: real] :
( ( ( ord_less @ nat @ M @ N )
& ( ord_less_eq @ real @ A2 @ T7 )
& ( ord_less_eq @ real @ T7 @ B2 ) )
=> ( has_field_derivative @ real @ ( Diff @ M ) @ ( Diff @ ( suc @ M ) @ T7 ) @ ( topolo174197925503356063within @ real @ T7 @ ( top_top @ ( set @ real ) ) ) ) )
=> ( ( ord_less_eq @ real @ A2 @ C3 )
=> ( ( ord_less @ real @ C3 @ B2 )
=> ? [T7: real] :
( ( ord_less @ real @ C3 @ T7 )
& ( ord_less @ real @ T7 @ B2 )
& ( ( F3 @ B2 )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M3: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M3 @ C3 ) @ ( semiring_char_0_fact @ real @ M3 ) ) @ ( power_power @ real @ ( minus_minus @ real @ B2 @ C3 ) @ M3 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N @ T7 ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ ( minus_minus @ real @ B2 @ C3 ) @ N ) ) ) ) ) ) ) ) ) ) ).
% Taylor_up
thf(fact_4762_Taylor,axiom,
! [N: nat,Diff: nat > real > real,F3: real > real,A2: real,B2: real,C3: real,X: real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( ( Diff @ ( zero_zero @ nat ) )
= F3 )
=> ( ! [M: nat,T7: real] :
( ( ( ord_less @ nat @ M @ N )
& ( ord_less_eq @ real @ A2 @ T7 )
& ( ord_less_eq @ real @ T7 @ B2 ) )
=> ( has_field_derivative @ real @ ( Diff @ M ) @ ( Diff @ ( suc @ M ) @ T7 ) @ ( topolo174197925503356063within @ real @ T7 @ ( top_top @ ( set @ real ) ) ) ) )
=> ( ( ord_less_eq @ real @ A2 @ C3 )
=> ( ( ord_less_eq @ real @ C3 @ B2 )
=> ( ( ord_less_eq @ real @ A2 @ X )
=> ( ( ord_less_eq @ real @ X @ B2 )
=> ( ( X != C3 )
=> ? [T7: real] :
( ( ( ord_less @ real @ X @ C3 )
=> ( ( ord_less @ real @ X @ T7 )
& ( ord_less @ real @ T7 @ C3 ) ) )
& ( ~ ( ord_less @ real @ X @ C3 )
=> ( ( ord_less @ real @ C3 @ T7 )
& ( ord_less @ real @ T7 @ X ) ) )
& ( ( F3 @ X )
= ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [M3: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ M3 @ C3 ) @ ( semiring_char_0_fact @ real @ M3 ) ) @ ( power_power @ real @ ( minus_minus @ real @ X @ C3 ) @ M3 ) )
@ ( set_ord_lessThan @ nat @ N ) )
@ ( times_times @ real @ ( divide_divide @ real @ ( Diff @ N @ T7 ) @ ( semiring_char_0_fact @ real @ N ) ) @ ( power_power @ real @ ( minus_minus @ real @ X @ C3 ) @ N ) ) ) ) ) ) ) ) ) ) ) ) ) ).
% Taylor
thf(fact_4763_finite__psubset__def,axiom,
! [A: $tType] :
( ( finite_psubset @ A )
= ( collect @ ( product_prod @ ( set @ A ) @ ( set @ A ) )
@ ( product_case_prod @ ( set @ A ) @ ( set @ A ) @ $o
@ ^ [A7: set @ A,B8: set @ A] :
( ( ord_less @ ( set @ A ) @ A7 @ B8 )
& ( finite_finite2 @ A @ B8 ) ) ) ) ) ).
% finite_psubset_def
thf(fact_4764_Maclaurin__lemma2,axiom,
! [N: nat,H2: real,Diff: nat > real > real,K2: nat,B6: real] :
( ! [M: nat,T7: real] :
( ( ( ord_less @ nat @ M @ N )
& ( ord_less_eq @ real @ ( zero_zero @ real ) @ T7 )
& ( ord_less_eq @ real @ T7 @ H2 ) )
=> ( has_field_derivative @ real @ ( Diff @ M ) @ ( Diff @ ( suc @ M ) @ T7 ) @ ( topolo174197925503356063within @ real @ T7 @ ( top_top @ ( set @ real ) ) ) ) )
=> ( ( N
= ( suc @ K2 ) )
=> ! [M4: nat,T8: real] :
( ( ( ord_less @ nat @ M4 @ N )
& ( ord_less_eq @ real @ ( zero_zero @ real ) @ T8 )
& ( ord_less_eq @ real @ T8 @ H2 ) )
=> ( has_field_derivative @ real
@ ^ [U2: real] :
( minus_minus @ real @ ( Diff @ M4 @ U2 )
@ ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [P6: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ ( plus_plus @ nat @ M4 @ P6 ) @ ( zero_zero @ real ) ) @ ( semiring_char_0_fact @ real @ P6 ) ) @ ( power_power @ real @ U2 @ P6 ) )
@ ( set_ord_lessThan @ nat @ ( minus_minus @ nat @ N @ M4 ) ) )
@ ( times_times @ real @ B6 @ ( divide_divide @ real @ ( power_power @ real @ U2 @ ( minus_minus @ nat @ N @ M4 ) ) @ ( semiring_char_0_fact @ real @ ( minus_minus @ nat @ N @ M4 ) ) ) ) ) )
@ ( minus_minus @ real @ ( Diff @ ( suc @ M4 ) @ T8 )
@ ( plus_plus @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [P6: nat] : ( times_times @ real @ ( divide_divide @ real @ ( Diff @ ( plus_plus @ nat @ ( suc @ M4 ) @ P6 ) @ ( zero_zero @ real ) ) @ ( semiring_char_0_fact @ real @ P6 ) ) @ ( power_power @ real @ T8 @ P6 ) )
@ ( set_ord_lessThan @ nat @ ( minus_minus @ nat @ N @ ( suc @ M4 ) ) ) )
@ ( times_times @ real @ B6 @ ( divide_divide @ real @ ( power_power @ real @ T8 @ ( minus_minus @ nat @ N @ ( suc @ M4 ) ) ) @ ( semiring_char_0_fact @ real @ ( minus_minus @ nat @ N @ ( suc @ M4 ) ) ) ) ) ) )
@ ( topolo174197925503356063within @ real @ T8 @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ).
% Maclaurin_lemma2
thf(fact_4765_DERIV__arctan__series,axiom,
! [X: real] :
( ( ord_less @ real @ ( abs_abs @ real @ X ) @ ( one_one @ real ) )
=> ( has_field_derivative @ real
@ ^ [X10: 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 @ X10 @ ( 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_4766_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_4767_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_4768_has__derivative__arcsin,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [G3: A > real,X: A,G7: A > real,S: set @ A] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ ( G3 @ X ) )
=> ( ( ord_less @ real @ ( G3 @ X ) @ ( one_one @ real ) )
=> ( ( has_derivative @ A @ real @ G3 @ G7 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( has_derivative @ A @ real
@ ^ [X5: A] : ( arcsin @ ( G3 @ X5 ) )
@ ^ [X5: A] : ( times_times @ real @ ( G7 @ X5 ) @ ( inverse_inverse @ real @ ( sqrt @ ( minus_minus @ real @ ( one_one @ real ) @ ( power_power @ real @ ( G3 @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ) ).
% has_derivative_arcsin
thf(fact_4769_has__derivative__real__sqrt,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [G3: A > real,X: A,G7: A > real,S: set @ A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( G3 @ X ) )
=> ( ( has_derivative @ A @ real @ G3 @ G7 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( has_derivative @ A @ real
@ ^ [X5: A] : ( sqrt @ ( G3 @ X5 ) )
@ ^ [X5: A] : ( times_times @ real @ ( G7 @ X5 ) @ ( divide_divide @ real @ ( inverse_inverse @ real @ ( sqrt @ ( G3 @ X ) ) ) @ ( numeral_numeral @ real @ ( bit0 @ one2 ) ) ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ).
% has_derivative_real_sqrt
thf(fact_4770_has__derivative__scaleR,axiom,
! [C: $tType,D: $tType] :
( ( ( real_V822414075346904944vector @ D )
& ( real_V822414075346904944vector @ C ) )
=> ! [F3: D > real,F10: D > real,X: D,S: set @ D,G3: D > C,G7: D > C] :
( ( has_derivative @ D @ real @ F3 @ F10 @ ( topolo174197925503356063within @ D @ X @ S ) )
=> ( ( has_derivative @ D @ C @ G3 @ G7 @ ( topolo174197925503356063within @ D @ X @ S ) )
=> ( has_derivative @ D @ C
@ ^ [X5: D] : ( real_V8093663219630862766scaleR @ C @ ( F3 @ X5 ) @ ( G3 @ X5 ) )
@ ^ [H: D] : ( plus_plus @ C @ ( real_V8093663219630862766scaleR @ C @ ( F3 @ X ) @ ( G7 @ H ) ) @ ( real_V8093663219630862766scaleR @ C @ ( F10 @ H ) @ ( G3 @ X ) ) )
@ ( topolo174197925503356063within @ D @ X @ S ) ) ) ) ) ).
% has_derivative_scaleR
thf(fact_4771_has__derivative__const,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [C3: B,F5: filter @ A] :
( has_derivative @ A @ B
@ ^ [X5: A] : C3
@ ^ [X5: A] : ( zero_zero @ B )
@ F5 ) ) ).
% has_derivative_const
thf(fact_4772_has__derivative__add,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F3: A > B,F10: A > B,F5: filter @ A,G3: A > B,G7: A > B] :
( ( has_derivative @ A @ B @ F3 @ F10 @ F5 )
=> ( ( has_derivative @ A @ B @ G3 @ G7 @ F5 )
=> ( has_derivative @ A @ B
@ ^ [X5: A] : ( plus_plus @ B @ ( F3 @ X5 ) @ ( G3 @ X5 ) )
@ ^ [X5: A] : ( plus_plus @ B @ ( F10 @ X5 ) @ ( G7 @ X5 ) )
@ F5 ) ) ) ) ).
% has_derivative_add
thf(fact_4773_has__derivative__zero__unique,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F5: A > B,X: A] :
( ( has_derivative @ A @ B
@ ^ [X5: A] : ( zero_zero @ B )
@ F5
@ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) )
=> ( F5
= ( ^ [H: A] : ( zero_zero @ B ) ) ) ) ) ).
% has_derivative_zero_unique
thf(fact_4774_has__derivative__mult,axiom,
! [A: $tType,D: $tType] :
( ( ( real_V822414075346904944vector @ D )
& ( real_V4412858255891104859lgebra @ A ) )
=> ! [F3: D > A,F10: D > A,X: D,S: set @ D,G3: D > A,G7: D > A] :
( ( has_derivative @ D @ A @ F3 @ F10 @ ( topolo174197925503356063within @ D @ X @ S ) )
=> ( ( has_derivative @ D @ A @ G3 @ G7 @ ( topolo174197925503356063within @ D @ X @ S ) )
=> ( has_derivative @ D @ A
@ ^ [X5: D] : ( times_times @ A @ ( F3 @ X5 ) @ ( G3 @ X5 ) )
@ ^ [H: D] : ( plus_plus @ A @ ( times_times @ A @ ( F3 @ X ) @ ( G7 @ H ) ) @ ( times_times @ A @ ( F10 @ H ) @ ( G3 @ X ) ) )
@ ( topolo174197925503356063within @ D @ X @ S ) ) ) ) ) ).
% has_derivative_mult
thf(fact_4775_has__derivative__transform__within,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F3: A > B,F10: A > B,X: A,S: set @ A,D3: real,G3: A > B] :
( ( has_derivative @ A @ B @ F3 @ F10 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ D3 )
=> ( ( member @ A @ X @ S )
=> ( ! [X18: A] :
( ( member @ A @ X18 @ S )
=> ( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X18 @ X ) @ D3 )
=> ( ( F3 @ X18 )
= ( G3 @ X18 ) ) ) )
=> ( has_derivative @ A @ B @ G3 @ F10 @ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ) ) ).
% has_derivative_transform_within
thf(fact_4776_has__derivative__divide_H,axiom,
! [A: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V3459762299906320749_field @ A ) )
=> ! [F3: C > A,F10: C > A,X: C,S2: set @ C,G3: C > A,G7: C > A] :
( ( has_derivative @ C @ A @ F3 @ F10 @ ( topolo174197925503356063within @ C @ X @ S2 ) )
=> ( ( has_derivative @ C @ A @ G3 @ G7 @ ( topolo174197925503356063within @ C @ X @ S2 ) )
=> ( ( ( G3 @ X )
!= ( zero_zero @ A ) )
=> ( has_derivative @ C @ A
@ ^ [X5: C] : ( divide_divide @ A @ ( F3 @ X5 ) @ ( G3 @ X5 ) )
@ ^ [H: C] : ( divide_divide @ A @ ( minus_minus @ A @ ( times_times @ A @ ( F10 @ H ) @ ( G3 @ X ) ) @ ( times_times @ A @ ( F3 @ X ) @ ( G7 @ H ) ) ) @ ( times_times @ A @ ( G3 @ X ) @ ( G3 @ X ) ) )
@ ( topolo174197925503356063within @ C @ X @ S2 ) ) ) ) ) ) ).
% has_derivative_divide'
thf(fact_4777_arccos__less__arccos,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ X )
=> ( ( ord_less @ real @ X @ Y )
=> ( ( ord_less_eq @ real @ Y @ ( one_one @ real ) )
=> ( ord_less @ real @ ( arccos @ Y ) @ ( arccos @ X ) ) ) ) ) ).
% arccos_less_arccos
thf(fact_4778_arccos__less__mono,axiom,
! [X: real,Y: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ X ) @ ( one_one @ real ) )
=> ( ( ord_less_eq @ real @ ( abs_abs @ real @ Y ) @ ( one_one @ real ) )
=> ( ( ord_less @ real @ ( arccos @ X ) @ ( arccos @ Y ) )
= ( ord_less @ real @ Y @ X ) ) ) ) ).
% arccos_less_mono
thf(fact_4779_has__derivative__inverse,axiom,
! [A: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V8999393235501362500lgebra @ A ) )
=> ! [F3: C > A,X: C,F10: C > A,S2: set @ C] :
( ( ( F3 @ X )
!= ( zero_zero @ A ) )
=> ( ( has_derivative @ C @ A @ F3 @ F10 @ ( topolo174197925503356063within @ C @ X @ S2 ) )
=> ( has_derivative @ C @ A
@ ^ [X5: C] : ( inverse_inverse @ A @ ( F3 @ X5 ) )
@ ^ [H: C] : ( uminus_uminus @ A @ ( times_times @ A @ ( times_times @ A @ ( inverse_inverse @ A @ ( F3 @ X ) ) @ ( F10 @ H ) ) @ ( inverse_inverse @ A @ ( F3 @ X ) ) ) )
@ ( topolo174197925503356063within @ C @ X @ S2 ) ) ) ) ) ).
% has_derivative_inverse
thf(fact_4780_has__derivative__inverse_H,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [X: A,S2: set @ A] :
( ( X
!= ( zero_zero @ A ) )
=> ( has_derivative @ A @ A @ ( inverse_inverse @ A )
@ ^ [H: A] : ( uminus_uminus @ A @ ( times_times @ A @ ( times_times @ A @ ( inverse_inverse @ A @ X ) @ H ) @ ( inverse_inverse @ A @ X ) ) )
@ ( topolo174197925503356063within @ A @ X @ S2 ) ) ) ) ).
% has_derivative_inverse'
thf(fact_4781_arccos__lt__bounded,axiom,
! [Y: real] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ Y )
=> ( ( ord_less @ real @ Y @ ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( arccos @ Y ) )
& ( ord_less @ real @ ( arccos @ Y ) @ pi ) ) ) ) ).
% arccos_lt_bounded
thf(fact_4782_has__derivative__ln,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [G3: A > real,X: A,G7: A > real,S: set @ A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ ( G3 @ X ) )
=> ( ( has_derivative @ A @ real @ G3 @ G7 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( has_derivative @ A @ real
@ ^ [X5: A] : ( ln_ln @ real @ ( G3 @ X5 ) )
@ ^ [X5: A] : ( times_times @ real @ ( G7 @ X5 ) @ ( inverse_inverse @ real @ ( G3 @ X ) ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ).
% has_derivative_ln
thf(fact_4783_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_4784_has__derivative__arccos,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [G3: A > real,X: A,G7: A > real,S: set @ A] :
( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ ( G3 @ X ) )
=> ( ( ord_less @ real @ ( G3 @ X ) @ ( one_one @ real ) )
=> ( ( has_derivative @ A @ real @ G3 @ G7 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( has_derivative @ A @ real
@ ^ [X5: A] : ( arccos @ ( G3 @ X5 ) )
@ ^ [X5: A] : ( times_times @ real @ ( G7 @ X5 ) @ ( inverse_inverse @ real @ ( uminus_uminus @ real @ ( sqrt @ ( minus_minus @ real @ ( one_one @ real ) @ ( power_power @ real @ ( G3 @ X ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) ) ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ) ).
% has_derivative_arccos
thf(fact_4785_has__derivative__divide,axiom,
! [A: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V8999393235501362500lgebra @ A ) )
=> ! [F3: C > A,F10: C > A,X: C,S2: set @ C,G3: C > A,G7: C > A] :
( ( has_derivative @ C @ A @ F3 @ F10 @ ( topolo174197925503356063within @ C @ X @ S2 ) )
=> ( ( has_derivative @ C @ A @ G3 @ G7 @ ( topolo174197925503356063within @ C @ X @ S2 ) )
=> ( ( ( G3 @ X )
!= ( zero_zero @ A ) )
=> ( has_derivative @ C @ A
@ ^ [X5: C] : ( divide_divide @ A @ ( F3 @ X5 ) @ ( G3 @ X5 ) )
@ ^ [H: C] : ( plus_plus @ A @ ( times_times @ A @ ( uminus_uminus @ A @ ( F3 @ X ) ) @ ( times_times @ A @ ( times_times @ A @ ( inverse_inverse @ A @ ( G3 @ X ) ) @ ( G7 @ H ) ) @ ( inverse_inverse @ A @ ( G3 @ X ) ) ) ) @ ( divide_divide @ A @ ( F10 @ H ) @ ( G3 @ X ) ) )
@ ( topolo174197925503356063within @ C @ X @ S2 ) ) ) ) ) ) ).
% has_derivative_divide
thf(fact_4786_has__derivative__prod,axiom,
! [B: $tType,I7: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V3459762299906320749_field @ B ) )
=> ! [I6: set @ I7,F3: I7 > A > B,F10: I7 > A > B,X: A,S2: set @ A] :
( ! [I3: I7] :
( ( member @ I7 @ I3 @ I6 )
=> ( has_derivative @ A @ B @ ( F3 @ I3 ) @ ( F10 @ I3 ) @ ( topolo174197925503356063within @ A @ X @ S2 ) ) )
=> ( has_derivative @ A @ B
@ ^ [X5: A] :
( groups7121269368397514597t_prod @ I7 @ B
@ ^ [I4: I7] : ( F3 @ I4 @ X5 )
@ I6 )
@ ^ [Y6: A] :
( groups7311177749621191930dd_sum @ I7 @ B
@ ^ [I4: I7] :
( times_times @ B @ ( F10 @ I4 @ Y6 )
@ ( groups7121269368397514597t_prod @ I7 @ B
@ ^ [J3: I7] : ( F3 @ J3 @ X )
@ ( minus_minus @ ( set @ I7 ) @ I6 @ ( insert @ I7 @ I4 @ ( bot_bot @ ( set @ I7 ) ) ) ) ) )
@ I6 )
@ ( topolo174197925503356063within @ A @ X @ S2 ) ) ) ) ).
% has_derivative_prod
thf(fact_4787_has__derivative__powr,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [G3: A > real,G7: A > real,X: A,X7: set @ A,F3: A > real,F10: A > real] :
( ( has_derivative @ A @ real @ G3 @ G7 @ ( topolo174197925503356063within @ A @ X @ X7 ) )
=> ( ( has_derivative @ A @ real @ F3 @ F10 @ ( topolo174197925503356063within @ A @ X @ X7 ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( G3 @ X ) )
=> ( ( member @ A @ X @ X7 )
=> ( has_derivative @ A @ real
@ ^ [X5: A] : ( powr @ real @ ( G3 @ X5 ) @ ( F3 @ X5 ) )
@ ^ [H: A] : ( times_times @ real @ ( powr @ real @ ( G3 @ X ) @ ( F3 @ X ) ) @ ( plus_plus @ real @ ( times_times @ real @ ( F10 @ H ) @ ( ln_ln @ real @ ( G3 @ X ) ) ) @ ( divide_divide @ real @ ( times_times @ real @ ( G7 @ H ) @ ( F3 @ X ) ) @ ( G3 @ X ) ) ) )
@ ( topolo174197925503356063within @ A @ X @ X7 ) ) ) ) ) ) ) ).
% has_derivative_powr
thf(fact_4788_termdiffs__aux,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [C3: nat > A,K5: A,X: A] :
( ( summable @ A
@ ^ [N2: nat] : ( times_times @ A @ ( diffs @ A @ ( diffs @ A @ C3 ) @ N2 ) @ ( power_power @ A @ K5 @ N2 ) ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ ( real_V7770717601297561774m_norm @ A @ K5 ) )
=> ( filterlim @ A @ A
@ ^ [H: A] :
( suminf @ A
@ ^ [N2: nat] : ( times_times @ A @ ( C3 @ N2 ) @ ( minus_minus @ A @ ( divide_divide @ A @ ( minus_minus @ A @ ( power_power @ A @ ( plus_plus @ A @ X @ H ) @ N2 ) @ ( power_power @ A @ X @ N2 ) ) @ H ) @ ( times_times @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( power_power @ A @ X @ ( minus_minus @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) ) ) ) ) ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% termdiffs_aux
thf(fact_4789_lex__prod__def,axiom,
! [B: $tType,A: $tType] :
( ( lex_prod @ A @ B )
= ( ^ [Ra: set @ ( product_prod @ A @ A ),Rb: set @ ( product_prod @ B @ B )] :
( collect @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) )
@ ( product_case_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ $o
@ ( product_case_prod @ A @ B @ ( ( product_prod @ A @ B ) > $o )
@ ^ [A6: A,B5: B] :
( product_case_prod @ A @ B @ $o
@ ^ [A10: A,B12: B] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A6 @ A10 ) @ Ra )
| ( ( A6 = A10 )
& ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ B5 @ B12 ) @ Rb ) ) ) ) ) ) ) ) ) ).
% lex_prod_def
thf(fact_4790_Gcd__eq__Max,axiom,
! [M5: set @ nat] :
( ( finite_finite2 @ nat @ M5 )
=> ( ( M5
!= ( bot_bot @ ( set @ nat ) ) )
=> ( ~ ( member @ nat @ ( zero_zero @ nat ) @ M5 )
=> ( ( gcd_Gcd @ nat @ M5 )
= ( lattic643756798349783984er_Max @ nat
@ ( complete_Inf_Inf @ ( set @ nat )
@ ( image2 @ nat @ ( set @ nat )
@ ^ [M3: nat] :
( collect @ nat
@ ^ [D5: nat] : ( dvd_dvd @ nat @ D5 @ M3 ) )
@ M5 ) ) ) ) ) ) ) ).
% Gcd_eq_Max
thf(fact_4791_Max__singleton,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A] :
( ( lattic643756798349783984er_Max @ A @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= X ) ) ).
% Max_singleton
thf(fact_4792_in__lex__prod,axiom,
! [A: $tType,B: $tType,A2: A,B2: B,A3: A,B3: B,R2: set @ ( product_prod @ A @ A ),S: set @ ( product_prod @ B @ B )] :
( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ ( product_Pair @ A @ B @ A3 @ B3 ) ) @ ( lex_prod @ A @ B @ R2 @ S ) )
= ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ A3 ) @ R2 )
| ( ( A2 = A3 )
& ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ B2 @ B3 ) @ S ) ) ) ) ).
% in_lex_prod
thf(fact_4793_Max__divisors__self__nat,axiom,
! [N: nat] :
( ( N
!= ( zero_zero @ nat ) )
=> ( ( lattic643756798349783984er_Max @ nat
@ ( collect @ nat
@ ^ [D5: nat] : ( dvd_dvd @ nat @ D5 @ N ) ) )
= N ) ) ).
% Max_divisors_self_nat
thf(fact_4794_Max_Obounded__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ ( lattic643756798349783984er_Max @ A @ A5 ) @ X )
= ( ! [X5: A] :
( ( member @ A @ X5 @ A5 )
=> ( ord_less_eq @ A @ X5 @ X ) ) ) ) ) ) ) ).
% Max.bounded_iff
thf(fact_4795_Max__less__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less @ A @ ( lattic643756798349783984er_Max @ A @ A5 ) @ X )
= ( ! [X5: A] :
( ( member @ A @ X5 @ A5 )
=> ( ord_less @ A @ X5 @ X ) ) ) ) ) ) ) ).
% Max_less_iff
thf(fact_4796_tendsto__mult__right__iff,axiom,
! [A: $tType,B: $tType] :
( ( ( field @ A )
& ( topolo4211221413907600880p_mult @ A ) )
=> ! [C3: A,F3: B > A,L: A,F5: filter @ B] :
( ( C3
!= ( zero_zero @ A ) )
=> ( ( filterlim @ B @ A
@ ^ [X5: B] : ( times_times @ A @ ( F3 @ X5 ) @ C3 )
@ ( topolo7230453075368039082e_nhds @ A @ ( times_times @ A @ L @ C3 ) )
@ F5 )
= ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F5 ) ) ) ) ).
% tendsto_mult_right_iff
thf(fact_4797_tendsto__mult__left__iff,axiom,
! [A: $tType,B: $tType] :
( ( ( field @ A )
& ( topolo4211221413907600880p_mult @ A ) )
=> ! [C3: A,F3: B > A,L: A,F5: filter @ B] :
( ( C3
!= ( zero_zero @ A ) )
=> ( ( filterlim @ B @ A
@ ^ [X5: B] : ( times_times @ A @ C3 @ ( F3 @ X5 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( times_times @ A @ C3 @ L ) )
@ F5 )
= ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F5 ) ) ) ) ).
% tendsto_mult_left_iff
thf(fact_4798_power__tendsto__0__iff,axiom,
! [A: $tType,N: nat,F3: A > real,F5: filter @ A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( filterlim @ A @ real
@ ^ [X5: A] : ( power_power @ real @ ( F3 @ X5 ) @ N )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ F5 )
= ( filterlim @ A @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ F5 ) ) ) ).
% power_tendsto_0_iff
thf(fact_4799_Max__const,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ B,C3: A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( lattic643756798349783984er_Max @ A
@ ( image2 @ B @ A
@ ^ [Uu3: B] : C3
@ A5 ) )
= C3 ) ) ) ) ).
% Max_const
thf(fact_4800_Max__insert,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798349783984er_Max @ A @ ( insert @ A @ X @ A5 ) )
= ( ord_max @ A @ X @ ( lattic643756798349783984er_Max @ A @ A5 ) ) ) ) ) ) ).
% Max_insert
thf(fact_4801_LIM__not__zero,axiom,
! [Aa: $tType,A: $tType] :
( ( ( topolo8386298272705272623_space @ A )
& ( zero @ Aa )
& ( topological_t2_space @ Aa ) )
=> ! [K2: Aa,A2: A] :
( ( K2
!= ( zero_zero @ Aa ) )
=> ~ ( filterlim @ A @ Aa
@ ^ [X5: A] : K2
@ ( topolo7230453075368039082e_nhds @ Aa @ ( zero_zero @ Aa ) )
@ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% LIM_not_zero
thf(fact_4802_tendsto__unique,axiom,
! [B: $tType,A: $tType] :
( ( topological_t2_space @ A )
=> ! [F5: filter @ B,F3: B > A,A2: A,B2: A] :
( ( F5
!= ( bot_bot @ ( filter @ B ) ) )
=> ( ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ F5 )
=> ( ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ B2 ) @ F5 )
=> ( A2 = B2 ) ) ) ) ) ).
% tendsto_unique
thf(fact_4803_tendsto__bot,axiom,
! [A: $tType,B: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F3: B > A,A2: A] : ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ ( bot_bot @ ( filter @ B ) ) ) ) ).
% tendsto_bot
thf(fact_4804_nhds__neq__bot,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [A2: A] :
( ( topolo7230453075368039082e_nhds @ A @ A2 )
!= ( bot_bot @ ( filter @ A ) ) ) ) ).
% nhds_neq_bot
thf(fact_4805_tendsto__const__iff,axiom,
! [B: $tType,A: $tType] :
( ( topological_t2_space @ A )
=> ! [F5: filter @ B,A2: A,B2: A] :
( ( F5
!= ( bot_bot @ ( filter @ B ) ) )
=> ( ( filterlim @ B @ A
@ ^ [X5: B] : A2
@ ( topolo7230453075368039082e_nhds @ A @ B2 )
@ F5 )
= ( A2 = B2 ) ) ) ) ).
% tendsto_const_iff
thf(fact_4806_tendsto__cot,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [F3: A > A,A2: A,F5: filter @ A] :
( ( filterlim @ A @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ F5 )
=> ( ( ( sin @ A @ A2 )
!= ( zero_zero @ A ) )
=> ( filterlim @ A @ A
@ ^ [X5: A] : ( cot @ A @ ( F3 @ X5 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( cot @ A @ A2 ) )
@ F5 ) ) ) ) ).
% tendsto_cot
thf(fact_4807_tendsto__tanh,axiom,
! [A: $tType,C: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [F3: C > A,A2: A,F5: filter @ C] :
( ( filterlim @ C @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ F5 )
=> ( ( ( cosh @ A @ A2 )
!= ( zero_zero @ A ) )
=> ( filterlim @ C @ A
@ ^ [X5: C] : ( tanh @ A @ ( F3 @ X5 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( tanh @ A @ A2 ) )
@ F5 ) ) ) ) ).
% tendsto_tanh
thf(fact_4808_tendsto__inverse,axiom,
! [A: $tType,B: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [F3: B > A,A2: A,F5: filter @ B] :
( ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ F5 )
=> ( ( A2
!= ( zero_zero @ A ) )
=> ( filterlim @ B @ A
@ ^ [X5: B] : ( inverse_inverse @ A @ ( F3 @ X5 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( inverse_inverse @ A @ A2 ) )
@ F5 ) ) ) ) ).
% tendsto_inverse
thf(fact_4809_Lim__transform__eq,axiom,
! [A: $tType,B: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: B > A,G3: B > A,F5: filter @ B,A2: A] :
( ( filterlim @ B @ A
@ ^ [X5: B] : ( minus_minus @ A @ ( F3 @ X5 ) @ ( G3 @ X5 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ F5 )
=> ( ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ F5 )
= ( filterlim @ B @ A @ G3 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ F5 ) ) ) ) ).
% Lim_transform_eq
thf(fact_4810_LIM__zero__cancel,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F3: A > B,L: B,F5: filter @ A] :
( ( filterlim @ A @ B
@ ^ [X5: A] : ( minus_minus @ B @ ( F3 @ X5 ) @ L )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ F5 )
=> ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ F5 ) ) ) ).
% LIM_zero_cancel
thf(fact_4811_Lim__transform2,axiom,
! [A: $tType,B: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: B > A,A2: A,F5: filter @ B,G3: B > A] :
( ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ F5 )
=> ( ( filterlim @ B @ A
@ ^ [X5: B] : ( minus_minus @ A @ ( F3 @ X5 ) @ ( G3 @ X5 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ F5 )
=> ( filterlim @ B @ A @ G3 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ F5 ) ) ) ) ).
% Lim_transform2
thf(fact_4812_Lim__transform,axiom,
! [A: $tType,B: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [G3: B > A,A2: A,F5: filter @ B,F3: B > A] :
( ( filterlim @ B @ A @ G3 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ F5 )
=> ( ( filterlim @ B @ A
@ ^ [X5: B] : ( minus_minus @ A @ ( F3 @ X5 ) @ ( G3 @ X5 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ F5 )
=> ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ F5 ) ) ) ) ).
% Lim_transform
thf(fact_4813_LIM__zero__iff,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F3: A > B,L: B,F5: filter @ A] :
( ( filterlim @ A @ B
@ ^ [X5: A] : ( minus_minus @ B @ ( F3 @ X5 ) @ L )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ F5 )
= ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ F5 ) ) ) ).
% LIM_zero_iff
thf(fact_4814_LIM__zero,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F3: A > B,L: B,F5: filter @ A] :
( ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ F5 )
=> ( filterlim @ A @ B
@ ^ [X5: A] : ( minus_minus @ B @ ( F3 @ X5 ) @ L )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ F5 ) ) ) ).
% LIM_zero
thf(fact_4815_tendsto__divide,axiom,
! [A: $tType,B: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: B > A,A2: A,F5: filter @ B,G3: B > A,B2: A] :
( ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ F5 )
=> ( ( filterlim @ B @ A @ G3 @ ( topolo7230453075368039082e_nhds @ A @ B2 ) @ F5 )
=> ( ( B2
!= ( zero_zero @ A ) )
=> ( filterlim @ B @ A
@ ^ [X5: B] : ( divide_divide @ A @ ( F3 @ X5 ) @ ( G3 @ X5 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( divide_divide @ A @ A2 @ B2 ) )
@ F5 ) ) ) ) ) ).
% tendsto_divide
thf(fact_4816_tendsto__divide__zero,axiom,
! [A: $tType,B: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: B > A,F5: filter @ B,C3: A] :
( ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ F5 )
=> ( filterlim @ B @ A
@ ^ [X5: B] : ( divide_divide @ A @ ( F3 @ X5 ) @ C3 )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ F5 ) ) ) ).
% tendsto_divide_zero
thf(fact_4817_tendsto__add,axiom,
! [A: $tType,B: $tType] :
( ( topolo6943815403480290642id_add @ A )
=> ! [F3: B > A,A2: A,F5: filter @ B,G3: B > A,B2: A] :
( ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ F5 )
=> ( ( filterlim @ B @ A @ G3 @ ( topolo7230453075368039082e_nhds @ A @ B2 ) @ F5 )
=> ( filterlim @ B @ A
@ ^ [X5: B] : ( plus_plus @ A @ ( F3 @ X5 ) @ ( G3 @ X5 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( plus_plus @ A @ A2 @ B2 ) )
@ F5 ) ) ) ) ).
% tendsto_add
thf(fact_4818_tendsto__add__const__iff,axiom,
! [A: $tType,B: $tType] :
( ( topolo1633459387980952147up_add @ A )
=> ! [C3: A,F3: B > A,D3: A,F5: filter @ B] :
( ( filterlim @ B @ A
@ ^ [X5: B] : ( plus_plus @ A @ C3 @ ( F3 @ X5 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( plus_plus @ A @ C3 @ D3 ) )
@ F5 )
= ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ D3 ) @ F5 ) ) ) ).
% tendsto_add_const_iff
thf(fact_4819_tendsto__add__zero,axiom,
! [B: $tType,D: $tType] :
( ( topolo6943815403480290642id_add @ B )
=> ! [F3: D > B,F5: filter @ D,G3: D > B] :
( ( filterlim @ D @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) @ F5 )
=> ( ( filterlim @ D @ B @ G3 @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) @ F5 )
=> ( filterlim @ D @ B
@ ^ [X5: D] : ( plus_plus @ B @ ( F3 @ X5 ) @ ( G3 @ X5 ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ F5 ) ) ) ) ).
% tendsto_add_zero
thf(fact_4820_tendsto__mult__zero,axiom,
! [A: $tType,D: $tType] :
( ( real_V4412858255891104859lgebra @ A )
=> ! [F3: D > A,F5: filter @ D,G3: D > A] :
( ( filterlim @ D @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ F5 )
=> ( ( filterlim @ D @ A @ G3 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ F5 )
=> ( filterlim @ D @ A
@ ^ [X5: D] : ( times_times @ A @ ( F3 @ X5 ) @ ( G3 @ X5 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ F5 ) ) ) ) ).
% tendsto_mult_zero
thf(fact_4821_tendsto__mult__left__zero,axiom,
! [A: $tType,D: $tType] :
( ( real_V4412858255891104859lgebra @ A )
=> ! [F3: D > A,F5: filter @ D,C3: A] :
( ( filterlim @ D @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ F5 )
=> ( filterlim @ D @ A
@ ^ [X5: D] : ( times_times @ A @ ( F3 @ X5 ) @ C3 )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ F5 ) ) ) ).
% tendsto_mult_left_zero
thf(fact_4822_tendsto__mult__right__zero,axiom,
! [A: $tType,D: $tType] :
( ( real_V4412858255891104859lgebra @ A )
=> ! [F3: D > A,F5: filter @ D,C3: A] :
( ( filterlim @ D @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ F5 )
=> ( filterlim @ D @ A
@ ^ [X5: D] : ( times_times @ A @ C3 @ ( F3 @ X5 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ F5 ) ) ) ).
% tendsto_mult_right_zero
thf(fact_4823_tendsto__sgn,axiom,
! [A: $tType,B: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: B > A,L: A,F5: filter @ B] :
( ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F5 )
=> ( ( L
!= ( zero_zero @ A ) )
=> ( filterlim @ B @ A
@ ^ [X5: B] : ( sgn_sgn @ A @ ( F3 @ X5 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( sgn_sgn @ A @ L ) )
@ F5 ) ) ) ) ).
% tendsto_sgn
thf(fact_4824_tendsto__norm__zero__cancel,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F3: A > B,F5: filter @ A] :
( ( filterlim @ A @ real
@ ^ [X5: A] : ( real_V7770717601297561774m_norm @ B @ ( F3 @ X5 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ F5 )
=> ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) @ F5 ) ) ) ).
% tendsto_norm_zero_cancel
thf(fact_4825_tendsto__norm__zero__iff,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F3: A > B,F5: filter @ A] :
( ( filterlim @ A @ real
@ ^ [X5: A] : ( real_V7770717601297561774m_norm @ B @ ( F3 @ X5 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ F5 )
= ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) @ F5 ) ) ) ).
% tendsto_norm_zero_iff
thf(fact_4826_tendsto__norm__zero,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F3: A > B,F5: filter @ A] :
( ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) @ F5 )
=> ( filterlim @ A @ real
@ ^ [X5: A] : ( real_V7770717601297561774m_norm @ B @ ( F3 @ X5 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ F5 ) ) ) ).
% tendsto_norm_zero
thf(fact_4827_tendsto__max,axiom,
! [A: $tType,B: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [X7: B > A,X: A,Net: filter @ B,Y8: B > A,Y: A] :
( ( filterlim @ B @ A @ X7 @ ( topolo7230453075368039082e_nhds @ A @ X ) @ Net )
=> ( ( filterlim @ B @ A @ Y8 @ ( topolo7230453075368039082e_nhds @ A @ Y ) @ Net )
=> ( filterlim @ B @ A
@ ^ [X5: B] : ( ord_max @ A @ ( X7 @ X5 ) @ ( Y8 @ X5 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( ord_max @ A @ X @ Y ) )
@ Net ) ) ) ) ).
% tendsto_max
thf(fact_4828_tendsto__null__sum,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( topolo5987344860129210374id_add @ C )
=> ! [I6: set @ B,F3: A > B > C,F5: filter @ A] :
( ! [I3: B] :
( ( member @ B @ I3 @ I6 )
=> ( filterlim @ A @ C
@ ^ [X5: A] : ( F3 @ X5 @ I3 )
@ ( topolo7230453075368039082e_nhds @ C @ ( zero_zero @ C ) )
@ F5 ) )
=> ( filterlim @ A @ C
@ ^ [I4: A] : ( groups7311177749621191930dd_sum @ B @ C @ ( F3 @ I4 ) @ I6 )
@ ( topolo7230453075368039082e_nhds @ C @ ( zero_zero @ C ) )
@ F5 ) ) ) ).
% tendsto_null_sum
thf(fact_4829_tendsto__tan,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [F3: A > A,A2: A,F5: filter @ A] :
( ( filterlim @ A @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ F5 )
=> ( ( ( cos @ A @ A2 )
!= ( zero_zero @ A ) )
=> ( filterlim @ A @ A
@ ^ [X5: A] : ( tan @ A @ ( F3 @ X5 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( tan @ A @ A2 ) )
@ F5 ) ) ) ) ).
% tendsto_tan
thf(fact_4830_tendsto__Pair,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ B )
& ( topolo4958980785337419405_space @ C ) )
=> ! [F3: A > B,A2: B,F5: filter @ A,G3: A > C,B2: C] :
( ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ A2 ) @ F5 )
=> ( ( filterlim @ A @ C @ G3 @ ( topolo7230453075368039082e_nhds @ C @ B2 ) @ F5 )
=> ( filterlim @ A @ ( product_prod @ B @ C )
@ ^ [X5: A] : ( product_Pair @ B @ C @ ( F3 @ X5 ) @ ( G3 @ X5 ) )
@ ( topolo7230453075368039082e_nhds @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ A2 @ B2 ) )
@ F5 ) ) ) ) ).
% tendsto_Pair
thf(fact_4831_tendsto__arcosh,axiom,
! [B: $tType,F3: B > real,A2: real,F5: filter @ B] :
( ( filterlim @ B @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ A2 ) @ F5 )
=> ( ( ord_less @ real @ ( one_one @ real ) @ A2 )
=> ( filterlim @ B @ real
@ ^ [X5: B] : ( arcosh @ real @ ( F3 @ X5 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( arcosh @ real @ A2 ) )
@ F5 ) ) ) ).
% tendsto_arcosh
thf(fact_4832_Lim__transform__within,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V7819770556892013058_space @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [F3: A > B,L: B,X: A,S2: set @ A,D3: real,G3: A > B] :
( ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ ( topolo174197925503356063within @ A @ X @ S2 ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ D3 )
=> ( ! [X18: A] :
( ( member @ A @ X18 @ S2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( real_V557655796197034286t_dist @ A @ X18 @ X ) )
=> ( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X18 @ X ) @ D3 )
=> ( ( F3 @ X18 )
= ( G3 @ X18 ) ) ) ) )
=> ( filterlim @ A @ B @ G3 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ ( topolo174197925503356063within @ A @ X @ S2 ) ) ) ) ) ) ).
% Lim_transform_within
thf(fact_4833_LIM__offset__zero__cancel,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [F3: A > B,A2: A,L5: B] :
( ( filterlim @ A @ B
@ ^ [H: A] : ( F3 @ ( plus_plus @ A @ A2 @ H ) )
@ ( topolo7230453075368039082e_nhds @ B @ L5 )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) )
=> ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ L5 ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% LIM_offset_zero_cancel
thf(fact_4834_LIM__offset__zero,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [F3: A > B,L5: B,A2: A] :
( ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ L5 ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) )
=> ( filterlim @ A @ B
@ ^ [H: A] : ( F3 @ ( plus_plus @ A @ A2 @ H ) )
@ ( topolo7230453075368039082e_nhds @ B @ L5 )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% LIM_offset_zero
thf(fact_4835_LIM__isCont__iff,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [F3: A > B,A2: A] :
( ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ ( F3 @ A2 ) ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) )
= ( filterlim @ A @ B
@ ^ [H: A] : ( F3 @ ( plus_plus @ A @ A2 @ H ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( F3 @ A2 ) )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% LIM_isCont_iff
thf(fact_4836_tendsto__null__power,axiom,
! [B: $tType,A: $tType] :
( ( real_V2822296259951069270ebra_1 @ B )
=> ! [F3: A > B,F5: filter @ A,N: nat] :
( ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) @ F5 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( filterlim @ A @ B
@ ^ [X5: A] : ( power_power @ B @ ( F3 @ X5 ) @ N )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ F5 ) ) ) ) ).
% tendsto_null_power
thf(fact_4837_LIM__offset,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [F3: A > B,L5: B,A2: A,K2: A] :
( ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ L5 ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) )
=> ( filterlim @ A @ B
@ ^ [X5: A] : ( F3 @ ( plus_plus @ A @ X5 @ K2 ) )
@ ( topolo7230453075368039082e_nhds @ B @ L5 )
@ ( topolo174197925503356063within @ A @ ( minus_minus @ A @ A2 @ K2 ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% LIM_offset
thf(fact_4838_tendsto__log,axiom,
! [A: $tType,F3: A > real,A2: real,F5: filter @ A,G3: A > real,B2: real] :
( ( filterlim @ A @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ A2 ) @ F5 )
=> ( ( filterlim @ A @ real @ G3 @ ( topolo7230453075368039082e_nhds @ real @ B2 ) @ F5 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
=> ( ( A2
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ B2 )
=> ( filterlim @ A @ real
@ ^ [X5: A] : ( log @ ( F3 @ X5 ) @ ( G3 @ X5 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( log @ A2 @ B2 ) )
@ F5 ) ) ) ) ) ) ).
% tendsto_log
thf(fact_4839_Max_OcoboundedI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,A2: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( member @ A @ A2 @ A5 )
=> ( ord_less_eq @ A @ A2 @ ( lattic643756798349783984er_Max @ A @ A5 ) ) ) ) ) ).
% Max.coboundedI
thf(fact_4840_Max__eq__if,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,B6: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( finite_finite2 @ A @ B6 )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ A5 )
=> ? [Xa: A] :
( ( member @ A @ Xa @ B6 )
& ( ord_less_eq @ A @ X4 @ Xa ) ) )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ B6 )
=> ? [Xa: A] :
( ( member @ A @ Xa @ A5 )
& ( ord_less_eq @ A @ X4 @ Xa ) ) )
=> ( ( lattic643756798349783984er_Max @ A @ A5 )
= ( lattic643756798349783984er_Max @ A @ B6 ) ) ) ) ) ) ) ).
% Max_eq_if
thf(fact_4841_Max__eqI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ! [Y3: A] :
( ( member @ A @ Y3 @ A5 )
=> ( ord_less_eq @ A @ Y3 @ X ) )
=> ( ( member @ A @ X @ A5 )
=> ( ( lattic643756798349783984er_Max @ A @ A5 )
= X ) ) ) ) ) ).
% Max_eqI
thf(fact_4842_Max__ge,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( member @ A @ X @ A5 )
=> ( ord_less_eq @ A @ X @ ( lattic643756798349783984er_Max @ A @ A5 ) ) ) ) ) ).
% Max_ge
thf(fact_4843_Max__in,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( member @ A @ ( lattic643756798349783984er_Max @ A @ A5 ) @ A5 ) ) ) ) ).
% Max_in
thf(fact_4844_tendsto__artanh,axiom,
! [A: $tType,F3: A > real,A2: real,F5: filter @ A] :
( ( filterlim @ A @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ A2 ) @ F5 )
=> ( ( ord_less @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ A2 )
=> ( ( ord_less @ real @ A2 @ ( one_one @ real ) )
=> ( filterlim @ A @ real
@ ^ [X5: A] : ( artanh @ real @ ( F3 @ X5 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( artanh @ real @ A2 ) )
@ F5 ) ) ) ) ).
% tendsto_artanh
thf(fact_4845_Max_Oin__idem,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( member @ A @ X @ A5 )
=> ( ( ord_max @ A @ X @ ( lattic643756798349783984er_Max @ A @ A5 ) )
= ( lattic643756798349783984er_Max @ A @ A5 ) ) ) ) ) ).
% Max.in_idem
thf(fact_4846_LIM__def,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V7819770556892013058_space @ A )
& ( real_V7819770556892013058_space @ B ) )
=> ! [F3: A > B,L5: B,A2: A] :
( ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ L5 ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) )
= ( ! [R5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R5 )
=> ? [S8: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ S8 )
& ! [X5: A] :
( ( ( X5 != A2 )
& ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X5 @ A2 ) @ S8 ) )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ B @ ( F3 @ X5 ) @ L5 ) @ R5 ) ) ) ) ) ) ) ).
% LIM_def
thf(fact_4847_metric__LIM__D,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V7819770556892013058_space @ A )
& ( real_V7819770556892013058_space @ B ) )
=> ! [F3: A > B,L5: B,A2: A,R2: real] :
( ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ L5 ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ R2 )
=> ? [S3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ S3 )
& ! [X3: A] :
( ( ( X3 != A2 )
& ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X3 @ A2 ) @ S3 ) )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ B @ ( F3 @ X3 ) @ L5 ) @ R2 ) ) ) ) ) ) ).
% metric_LIM_D
thf(fact_4848_metric__LIM__I,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V7819770556892013058_space @ A )
& ( real_V7819770556892013058_space @ B ) )
=> ! [A2: A,F3: A > B,L5: B] :
( ! [R3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R3 )
=> ? [S9: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ S9 )
& ! [X4: A] :
( ( ( X4 != A2 )
& ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X4 @ A2 ) @ S9 ) )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ B @ ( F3 @ X4 ) @ L5 ) @ R3 ) ) ) )
=> ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ L5 ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% metric_LIM_I
thf(fact_4849_metric__LIM__equal2,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V7819770556892013058_space @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [G3: A > B,L: B,A2: A,R: real,F3: A > B] :
( ( filterlim @ A @ B @ G3 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ R )
=> ( ! [X4: A] :
( ( X4 != A2 )
=> ( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X4 @ A2 ) @ R )
=> ( ( F3 @ X4 )
= ( G3 @ X4 ) ) ) )
=> ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ).
% metric_LIM_equal2
thf(fact_4850_metric__LIM__compose2,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( real_V7819770556892013058_space @ A )
& ( topolo4958980785337419405_space @ B )
& ( topolo4958980785337419405_space @ C ) )
=> ! [F3: A > B,B2: B,A2: A,G3: B > C,C3: C] :
( ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ B2 ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) )
=> ( ( filterlim @ B @ C @ G3 @ ( topolo7230453075368039082e_nhds @ C @ C3 ) @ ( topolo174197925503356063within @ B @ B2 @ ( top_top @ ( set @ B ) ) ) )
=> ( ? [D4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D4 )
& ! [X4: A] :
( ( ( X4 != A2 )
& ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X4 @ A2 ) @ D4 ) )
=> ( ( F3 @ X4 )
!= B2 ) ) )
=> ( filterlim @ A @ C
@ ^ [X5: A] : ( G3 @ ( F3 @ X5 ) )
@ ( topolo7230453075368039082e_nhds @ C @ C3 )
@ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ).
% metric_LIM_compose2
thf(fact_4851_LIM__offset__zero__iff,axiom,
! [C: $tType,D: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( topolo4958980785337419405_space @ D )
& ( zero @ C ) )
=> ! [A2: A,F3: A > D,L5: D] :
( ( nO_MATCH @ C @ A @ ( zero_zero @ C ) @ A2 )
=> ( ( filterlim @ A @ D @ F3 @ ( topolo7230453075368039082e_nhds @ D @ L5 ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) )
= ( filterlim @ A @ D
@ ^ [H: A] : ( F3 @ ( plus_plus @ A @ A2 @ H ) )
@ ( topolo7230453075368039082e_nhds @ D @ L5 )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% LIM_offset_zero_iff
thf(fact_4852_LIM__equal2,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [R: real,A2: A,F3: A > B,G3: A > B,L: B] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R )
=> ( ! [X4: A] :
( ( X4 != A2 )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ X4 @ A2 ) ) @ R )
=> ( ( F3 @ X4 )
= ( G3 @ X4 ) ) ) )
=> ( ( filterlim @ A @ B @ G3 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) )
=> ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ).
% LIM_equal2
thf(fact_4853_LIM__eq,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F3: A > B,L5: B,A2: A] :
( ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ L5 ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) )
= ( ! [R5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R5 )
=> ? [S8: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ S8 )
& ! [X5: A] :
( ( ( X5 != A2 )
& ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ X5 @ A2 ) ) @ S8 ) )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ B @ ( minus_minus @ B @ ( F3 @ X5 ) @ L5 ) ) @ R5 ) ) ) ) ) ) ) ).
% LIM_eq
thf(fact_4854_LIM__I,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [A2: A,F3: A > B,L5: B] :
( ! [R3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R3 )
=> ? [S9: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ S9 )
& ! [X4: A] :
( ( ( X4 != A2 )
& ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ X4 @ A2 ) ) @ S9 ) )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ B @ ( minus_minus @ B @ ( F3 @ X4 ) @ L5 ) ) @ R3 ) ) ) )
=> ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ L5 ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% LIM_I
thf(fact_4855_LIM__D,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F3: A > B,L5: B,A2: A,R2: real] :
( ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ L5 ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ R2 )
=> ? [S3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ S3 )
& ! [X3: A] :
( ( ( X3 != A2 )
& ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ X3 @ A2 ) ) @ S3 ) )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ B @ ( minus_minus @ B @ ( F3 @ X3 ) @ L5 ) ) @ R2 ) ) ) ) ) ) ).
% LIM_D
thf(fact_4856_DERIV__LIM__iff,axiom,
! [A: $tType] :
( ( ( inverse @ A )
& ( real_V822414075346904944vector @ A ) )
=> ! [F3: A > A,A2: A,D6: A] :
( ( filterlim @ A @ A
@ ^ [H: A] : ( divide_divide @ A @ ( minus_minus @ A @ ( F3 @ ( plus_plus @ A @ A2 @ H ) ) @ ( F3 @ A2 ) ) @ H )
@ ( topolo7230453075368039082e_nhds @ A @ D6 )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) )
= ( filterlim @ A @ A
@ ^ [X5: A] : ( divide_divide @ A @ ( minus_minus @ A @ ( F3 @ X5 ) @ ( F3 @ A2 ) ) @ ( minus_minus @ A @ X5 @ A2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ D6 )
@ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% DERIV_LIM_iff
thf(fact_4857_LIM__fun__less__zero,axiom,
! [F3: real > real,L: real,C3: real] :
( ( filterlim @ real @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ L ) @ ( topolo174197925503356063within @ real @ C3 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( ord_less @ real @ L @ ( zero_zero @ real ) )
=> ? [R3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R3 )
& ! [X3: real] :
( ( ( X3 != C3 )
& ( ord_less @ real @ ( abs_abs @ real @ ( minus_minus @ real @ C3 @ X3 ) ) @ R3 ) )
=> ( ord_less @ real @ ( F3 @ X3 ) @ ( zero_zero @ real ) ) ) ) ) ) ).
% LIM_fun_less_zero
thf(fact_4858_LIM__fun__not__zero,axiom,
! [F3: real > real,L: real,C3: real] :
( ( filterlim @ real @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ L ) @ ( topolo174197925503356063within @ real @ C3 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( L
!= ( zero_zero @ real ) )
=> ? [R3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R3 )
& ! [X3: real] :
( ( ( X3 != C3 )
& ( ord_less @ real @ ( abs_abs @ real @ ( minus_minus @ real @ C3 @ X3 ) ) @ R3 ) )
=> ( ( F3 @ X3 )
!= ( zero_zero @ real ) ) ) ) ) ) ).
% LIM_fun_not_zero
thf(fact_4859_LIM__fun__gt__zero,axiom,
! [F3: real > real,L: real,C3: real] :
( ( filterlim @ real @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ L ) @ ( topolo174197925503356063within @ real @ C3 @ ( top_top @ ( set @ real ) ) ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ L )
=> ? [R3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R3 )
& ! [X3: real] :
( ( ( X3 != C3 )
& ( ord_less @ real @ ( abs_abs @ real @ ( minus_minus @ real @ C3 @ X3 ) ) @ R3 ) )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( F3 @ X3 ) ) ) ) ) ) ).
% LIM_fun_gt_zero
thf(fact_4860_LIM__compose2,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( topolo4958980785337419405_space @ B )
& ( topolo4958980785337419405_space @ C ) )
=> ! [F3: A > B,B2: B,A2: A,G3: B > C,C3: C] :
( ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ B2 ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) )
=> ( ( filterlim @ B @ C @ G3 @ ( topolo7230453075368039082e_nhds @ C @ C3 ) @ ( topolo174197925503356063within @ B @ B2 @ ( top_top @ ( set @ B ) ) ) )
=> ( ? [D4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D4 )
& ! [X4: A] :
( ( ( X4 != A2 )
& ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ X4 @ A2 ) ) @ D4 ) )
=> ( ( F3 @ X4 )
!= B2 ) ) )
=> ( filterlim @ A @ C
@ ^ [X5: A] : ( G3 @ ( F3 @ X5 ) )
@ ( topolo7230453075368039082e_nhds @ C @ C3 )
@ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ).
% LIM_compose2
thf(fact_4861_Max_OboundedI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [A4: A] :
( ( member @ A @ A4 @ A5 )
=> ( ord_less_eq @ A @ A4 @ X ) )
=> ( ord_less_eq @ A @ ( lattic643756798349783984er_Max @ A @ A5 ) @ X ) ) ) ) ) ).
% Max.boundedI
thf(fact_4862_Max_OboundedE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ ( lattic643756798349783984er_Max @ A @ A5 ) @ X )
=> ! [A11: A] :
( ( member @ A @ A11 @ A5 )
=> ( ord_less_eq @ A @ A11 @ X ) ) ) ) ) ) ).
% Max.boundedE
thf(fact_4863_eq__Max__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,M2: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( M2
= ( lattic643756798349783984er_Max @ A @ A5 ) )
= ( ( member @ A @ M2 @ A5 )
& ! [X5: A] :
( ( member @ A @ X5 @ A5 )
=> ( ord_less_eq @ A @ X5 @ M2 ) ) ) ) ) ) ) ).
% eq_Max_iff
thf(fact_4864_Max__ge__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ X @ ( lattic643756798349783984er_Max @ A @ A5 ) )
= ( ? [X5: A] :
( ( member @ A @ X5 @ A5 )
& ( ord_less_eq @ A @ X @ X5 ) ) ) ) ) ) ) ).
% Max_ge_iff
thf(fact_4865_Max__eq__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,M2: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( lattic643756798349783984er_Max @ A @ A5 )
= M2 )
= ( ( member @ A @ M2 @ A5 )
& ! [X5: A] :
( ( member @ A @ X5 @ A5 )
=> ( ord_less_eq @ A @ X5 @ M2 ) ) ) ) ) ) ) ).
% Max_eq_iff
thf(fact_4866_Max__gr__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less @ A @ X @ ( lattic643756798349783984er_Max @ A @ A5 ) )
= ( ? [X5: A] :
( ( member @ A @ X5 @ A5 )
& ( ord_less @ A @ X @ X5 ) ) ) ) ) ) ) ).
% Max_gr_iff
thf(fact_4867_Max__insert2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,A2: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ! [B4: A] :
( ( member @ A @ B4 @ A5 )
=> ( ord_less_eq @ A @ B4 @ A2 ) )
=> ( ( lattic643756798349783984er_Max @ A @ ( insert @ A @ A2 @ A5 ) )
= A2 ) ) ) ) ).
% Max_insert2
thf(fact_4868_Max__Sup,axiom,
! [A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798349783984er_Max @ A @ A5 )
= ( complete_Sup_Sup @ A @ A5 ) ) ) ) ) ).
% Max_Sup
thf(fact_4869_cSup__eq__Max,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [X7: set @ A] :
( ( finite_finite2 @ A @ X7 )
=> ( ( X7
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Sup_Sup @ A @ X7 )
= ( lattic643756798349783984er_Max @ A @ X7 ) ) ) ) ) ).
% cSup_eq_Max
thf(fact_4870_DERIV__D,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,D6: A,X: A] :
( ( has_field_derivative @ A @ F3 @ D6 @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) )
=> ( filterlim @ A @ A
@ ^ [H: A] : ( divide_divide @ A @ ( minus_minus @ A @ ( F3 @ ( plus_plus @ A @ X @ H ) ) @ ( F3 @ X ) ) @ H )
@ ( topolo7230453075368039082e_nhds @ A @ D6 )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% DERIV_D
thf(fact_4871_DERIV__def,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,D6: A,X: A] :
( ( has_field_derivative @ A @ F3 @ D6 @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) )
= ( filterlim @ A @ A
@ ^ [H: A] : ( divide_divide @ A @ ( minus_minus @ A @ ( F3 @ ( plus_plus @ A @ X @ H ) ) @ ( F3 @ X ) ) @ H )
@ ( topolo7230453075368039082e_nhds @ A @ D6 )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% DERIV_def
thf(fact_4872_lim__exp__minus__1,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ( filterlim @ A @ A
@ ^ [Z5: A] : ( divide_divide @ A @ ( minus_minus @ A @ ( exp @ A @ Z5 ) @ ( one_one @ A ) ) @ Z5 )
@ ( topolo7230453075368039082e_nhds @ A @ ( one_one @ A ) )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ).
% lim_exp_minus_1
thf(fact_4873_lemma__termdiff4,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [K2: real,F3: A > B,K5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K2 )
=> ( ! [H6: A] :
( ( H6
!= ( zero_zero @ A ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ H6 ) @ K2 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( F3 @ H6 ) ) @ ( times_times @ real @ K5 @ ( real_V7770717601297561774m_norm @ A @ H6 ) ) ) ) )
=> ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) @ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% lemma_termdiff4
thf(fact_4874_field__has__derivative__at,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,D6: A,X: A] :
( ( has_derivative @ A @ A @ F3 @ ( times_times @ A @ D6 ) @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) )
= ( filterlim @ A @ A
@ ^ [H: A] : ( divide_divide @ A @ ( minus_minus @ A @ ( F3 @ ( plus_plus @ A @ X @ H ) ) @ ( F3 @ X ) ) @ H )
@ ( topolo7230453075368039082e_nhds @ A @ D6 )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% field_has_derivative_at
thf(fact_4875_filterlim__at__to__0,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: A > B,F5: filter @ B,A2: A] :
( ( filterlim @ A @ B @ F3 @ F5 @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) )
= ( filterlim @ A @ B
@ ^ [X5: A] : ( F3 @ ( plus_plus @ A @ X5 @ A2 ) )
@ F5
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% filterlim_at_to_0
thf(fact_4876_Max_Osubset__imp,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,B6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B6 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B6 )
=> ( ord_less_eq @ A @ ( lattic643756798349783984er_Max @ A @ A5 ) @ ( lattic643756798349783984er_Max @ A @ B6 ) ) ) ) ) ) ).
% Max.subset_imp
thf(fact_4877_Max__mono,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [M5: set @ A,N6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ M5 @ N6 )
=> ( ( M5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ N6 )
=> ( ord_less_eq @ A @ ( lattic643756798349783984er_Max @ A @ M5 ) @ ( lattic643756798349783984er_Max @ A @ N6 ) ) ) ) ) ) ).
% Max_mono
thf(fact_4878_hom__Max__commute,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [H2: A > A,N6: set @ A] :
( ! [X4: A,Y3: A] :
( ( H2 @ ( ord_max @ A @ X4 @ Y3 ) )
= ( ord_max @ A @ ( H2 @ X4 ) @ ( H2 @ Y3 ) ) )
=> ( ( finite_finite2 @ A @ N6 )
=> ( ( N6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( H2 @ ( lattic643756798349783984er_Max @ A @ N6 ) )
= ( lattic643756798349783984er_Max @ A @ ( image2 @ A @ A @ H2 @ N6 ) ) ) ) ) ) ) ).
% hom_Max_commute
thf(fact_4879_Max_Osubset,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,B6: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( B6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ B6 @ A5 )
=> ( ( ord_max @ A @ ( lattic643756798349783984er_Max @ A @ B6 ) @ ( lattic643756798349783984er_Max @ A @ A5 ) )
= ( lattic643756798349783984er_Max @ A @ A5 ) ) ) ) ) ) ).
% Max.subset
thf(fact_4880_Max_Oinsert__not__elem,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ~ ( member @ A @ X @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798349783984er_Max @ A @ ( insert @ A @ X @ A5 ) )
= ( ord_max @ A @ X @ ( lattic643756798349783984er_Max @ A @ A5 ) ) ) ) ) ) ) ).
% Max.insert_not_elem
thf(fact_4881_Max_Oclosed,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X4: A,Y3: A] : ( member @ A @ ( ord_max @ A @ X4 @ Y3 ) @ ( insert @ A @ X4 @ ( insert @ A @ Y3 @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( member @ A @ ( lattic643756798349783984er_Max @ A @ A5 ) @ A5 ) ) ) ) ) ).
% Max.closed
thf(fact_4882_card__le__Suc__Max,axiom,
! [S2: set @ nat] :
( ( finite_finite2 @ nat @ S2 )
=> ( ord_less_eq @ nat @ ( finite_card @ nat @ S2 ) @ ( suc @ ( lattic643756798349783984er_Max @ nat @ S2 ) ) ) ) ).
% card_le_Suc_Max
thf(fact_4883_Sup__nat__def,axiom,
( ( complete_Sup_Sup @ nat )
= ( ^ [X9: set @ nat] :
( if @ nat
@ ( X9
= ( bot_bot @ ( set @ nat ) ) )
@ ( zero_zero @ nat )
@ ( lattic643756798349783984er_Max @ nat @ X9 ) ) ) ) ).
% Sup_nat_def
thf(fact_4884_divide__nat__def,axiom,
( ( divide_divide @ nat )
= ( ^ [M3: nat,N2: nat] :
( if @ nat
@ ( N2
= ( zero_zero @ nat ) )
@ ( zero_zero @ nat )
@ ( lattic643756798349783984er_Max @ nat
@ ( collect @ nat
@ ^ [K3: nat] : ( ord_less_eq @ nat @ ( times_times @ nat @ K3 @ N2 ) @ M3 ) ) ) ) ) ) ).
% divide_nat_def
thf(fact_4885_Max__add__commute,axiom,
! [B: $tType,A: $tType] :
( ( linord4140545234300271783up_add @ A )
=> ! [S2: set @ B,F3: B > A,K2: A] :
( ( finite_finite2 @ B @ S2 )
=> ( ( S2
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( lattic643756798349783984er_Max @ A
@ ( image2 @ B @ A
@ ^ [X5: B] : ( plus_plus @ A @ ( F3 @ X5 ) @ K2 )
@ S2 ) )
= ( plus_plus @ A @ ( lattic643756798349783984er_Max @ A @ ( image2 @ B @ A @ F3 @ S2 ) ) @ K2 ) ) ) ) ) ).
% Max_add_commute
thf(fact_4886_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
@ ^ [D5: nat] :
( ( dvd_dvd @ nat @ D5 @ M2 )
& ( dvd_dvd @ nat @ D5 @ N ) ) ) ) ) ) ).
% gcd_is_Max_divisors_nat
thf(fact_4887_powser__limit__0,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [S: real,A2: nat > A,F3: A > A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ S )
=> ( ! [X4: A] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X4 ) @ S )
=> ( sums @ A
@ ^ [N2: nat] : ( times_times @ A @ ( A2 @ N2 ) @ ( power_power @ A @ X4 @ N2 ) )
@ ( F3 @ X4 ) ) )
=> ( filterlim @ A @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ ( A2 @ ( zero_zero @ nat ) ) ) @ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% powser_limit_0
thf(fact_4888_powser__limit__0__strong,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [S: real,A2: nat > A,F3: A > A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ S )
=> ( ! [X4: A] :
( ( X4
!= ( zero_zero @ A ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X4 ) @ S )
=> ( sums @ A
@ ^ [N2: nat] : ( times_times @ A @ ( A2 @ N2 ) @ ( power_power @ A @ X4 @ N2 ) )
@ ( F3 @ X4 ) ) ) )
=> ( filterlim @ A @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ ( A2 @ ( zero_zero @ nat ) ) ) @ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% powser_limit_0_strong
thf(fact_4889_filterlim__transform__within,axiom,
! [B: $tType,A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [G3: A > B,G6: filter @ B,X: A,S2: set @ A,F5: filter @ B,D3: real,F3: A > B] :
( ( filterlim @ A @ B @ G3 @ G6 @ ( topolo174197925503356063within @ A @ X @ S2 ) )
=> ( ( ord_less_eq @ ( filter @ B ) @ G6 @ F5 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ D3 )
=> ( ! [X18: A] :
( ( member @ A @ X18 @ S2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( real_V557655796197034286t_dist @ A @ X18 @ X ) )
=> ( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X18 @ X ) @ D3 )
=> ( ( F3 @ X18 )
= ( G3 @ X18 ) ) ) ) )
=> ( filterlim @ A @ B @ F3 @ F5 @ ( topolo174197925503356063within @ A @ X @ S2 ) ) ) ) ) ) ) ).
% filterlim_transform_within
thf(fact_4890_lemma__termdiff5,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_Vector_banach @ B ) )
=> ! [K2: real,F3: nat > real,G3: A > nat > B] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K2 )
=> ( ( summable @ real @ F3 )
=> ( ! [H6: A,N3: nat] :
( ( H6
!= ( zero_zero @ A ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ H6 ) @ K2 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( G3 @ H6 @ N3 ) ) @ ( times_times @ real @ ( F3 @ N3 ) @ ( real_V7770717601297561774m_norm @ A @ H6 ) ) ) ) )
=> ( filterlim @ A @ B
@ ^ [H: A] : ( suminf @ B @ ( G3 @ H ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ).
% lemma_termdiff5
thf(fact_4891_Max_Oinsert__remove,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( ( ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798349783984er_Max @ A @ ( insert @ A @ X @ A5 ) )
= X ) )
& ( ( ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798349783984er_Max @ A @ ( insert @ A @ X @ A5 ) )
= ( ord_max @ A @ X @ ( lattic643756798349783984er_Max @ A @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ).
% Max.insert_remove
thf(fact_4892_Max_Oremove,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( member @ A @ X @ A5 )
=> ( ( ( ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798349783984er_Max @ A @ A5 )
= X ) )
& ( ( ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798349783984er_Max @ A @ A5 )
= ( ord_max @ A @ X @ ( lattic643756798349783984er_Max @ A @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ) ).
% Max.remove
thf(fact_4893_sum__le__card__Max,axiom,
! [A: $tType,A5: set @ A,F3: A > nat] :
( ( finite_finite2 @ A @ A5 )
=> ( ord_less_eq @ nat @ ( groups7311177749621191930dd_sum @ A @ nat @ F3 @ A5 ) @ ( times_times @ nat @ ( finite_card @ A @ A5 ) @ ( lattic643756798349783984er_Max @ nat @ ( image2 @ A @ nat @ F3 @ A5 ) ) ) ) ) ).
% sum_le_card_Max
thf(fact_4894_summable__Leibniz_I2_J,axiom,
! [A2: nat > real] :
( ( filterlim @ nat @ real @ A2 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) )
=> ( ( topological_monoseq @ real @ A2 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( A2 @ ( zero_zero @ nat ) ) )
=> ! [N5: nat] :
( member @ real
@ ( suminf @ real
@ ^ [I4: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I4 ) @ ( A2 @ I4 ) ) )
@ ( set_or1337092689740270186AtMost @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I4: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I4 ) @ ( A2 @ I4 ) )
@ ( set_ord_lessThan @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N5 ) ) )
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I4: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I4 ) @ ( A2 @ I4 ) )
@ ( set_ord_lessThan @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N5 ) @ ( one_one @ nat ) ) ) ) ) ) ) ) ) ).
% summable_Leibniz(2)
thf(fact_4895_summable__Leibniz_I3_J,axiom,
! [A2: nat > real] :
( ( filterlim @ nat @ real @ A2 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) )
=> ( ( topological_monoseq @ real @ A2 )
=> ( ( ord_less @ real @ ( A2 @ ( zero_zero @ nat ) ) @ ( zero_zero @ real ) )
=> ! [N5: nat] :
( member @ real
@ ( suminf @ real
@ ^ [I4: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I4 ) @ ( A2 @ I4 ) ) )
@ ( set_or1337092689740270186AtMost @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I4: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I4 ) @ ( A2 @ I4 ) )
@ ( set_ord_lessThan @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N5 ) @ ( one_one @ nat ) ) ) )
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I4: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I4 ) @ ( A2 @ I4 ) )
@ ( set_ord_lessThan @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N5 ) ) ) ) ) ) ) ) ).
% summable_Leibniz(3)
thf(fact_4896_summable__Leibniz_H_I4_J,axiom,
! [A2: nat > real,N: nat] :
( ( filterlim @ nat @ real @ A2 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) )
=> ( ! [N3: nat] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( A2 @ N3 ) )
=> ( ! [N3: nat] : ( ord_less_eq @ real @ ( A2 @ ( suc @ N3 ) ) @ ( A2 @ N3 ) )
=> ( ord_less_eq @ real
@ ( suminf @ real
@ ^ [I4: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I4 ) @ ( A2 @ I4 ) ) )
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I4: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I4 ) @ ( A2 @ I4 ) )
@ ( set_ord_lessThan @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N ) @ ( one_one @ nat ) ) ) ) ) ) ) ) ).
% summable_Leibniz'(4)
thf(fact_4897_trivial__limit__sequentially,axiom,
( ( at_top @ nat )
!= ( bot_bot @ ( filter @ nat ) ) ) ).
% trivial_limit_sequentially
thf(fact_4898_tendsto__zero__mult__right__iff,axiom,
! [A: $tType] :
( ( ( field @ A )
& ( topolo4211221413907600880p_mult @ A ) )
=> ! [C3: A,A2: nat > A] :
( ( C3
!= ( zero_zero @ A ) )
=> ( ( filterlim @ nat @ A
@ ^ [N2: nat] : ( times_times @ A @ ( A2 @ N2 ) @ C3 )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ ( at_top @ nat ) )
= ( filterlim @ nat @ A @ A2 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ ( at_top @ nat ) ) ) ) ) ).
% tendsto_zero_mult_right_iff
thf(fact_4899_tendsto__zero__mult__left__iff,axiom,
! [A: $tType] :
( ( ( field @ A )
& ( topolo4211221413907600880p_mult @ A ) )
=> ! [C3: A,A2: nat > A] :
( ( C3
!= ( zero_zero @ A ) )
=> ( ( filterlim @ nat @ A
@ ^ [N2: nat] : ( times_times @ A @ C3 @ ( A2 @ N2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ ( at_top @ nat ) )
= ( filterlim @ nat @ A @ A2 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ ( at_top @ nat ) ) ) ) ) ).
% tendsto_zero_mult_left_iff
thf(fact_4900_tendsto__zero__divide__iff,axiom,
! [A: $tType] :
( ( ( field @ A )
& ( topolo4211221413907600880p_mult @ A ) )
=> ! [C3: A,A2: nat > A] :
( ( C3
!= ( zero_zero @ A ) )
=> ( ( filterlim @ nat @ A
@ ^ [N2: nat] : ( divide_divide @ A @ ( A2 @ N2 ) @ C3 )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ ( at_top @ nat ) )
= ( filterlim @ nat @ A @ A2 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ ( at_top @ nat ) ) ) ) ) ).
% tendsto_zero_divide_iff
thf(fact_4901_trivial__limit__at__top__linorder,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( at_top @ A )
!= ( bot_bot @ ( filter @ A ) ) ) ) ).
% trivial_limit_at_top_linorder
thf(fact_4902_approx__from__below__dense__linorder,axiom,
! [A: $tType] :
( ( ( dense_linorder @ A )
& ( topolo3112930676232923870pology @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [Y: A,X: A] :
( ( ord_less @ A @ Y @ X )
=> ? [U4: nat > A] :
( ! [N5: nat] : ( ord_less @ A @ ( U4 @ N5 ) @ X )
& ( filterlim @ nat @ A @ U4 @ ( topolo7230453075368039082e_nhds @ A @ X ) @ ( at_top @ nat ) ) ) ) ) ).
% approx_from_below_dense_linorder
thf(fact_4903_approx__from__above__dense__linorder,axiom,
! [A: $tType] :
( ( ( dense_linorder @ A )
& ( topolo3112930676232923870pology @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ? [U4: nat > A] :
( ! [N5: nat] : ( ord_less @ A @ X @ ( U4 @ N5 ) )
& ( filterlim @ nat @ A @ U4 @ ( topolo7230453075368039082e_nhds @ A @ X ) @ ( at_top @ nat ) ) ) ) ) ).
% approx_from_above_dense_linorder
thf(fact_4904_LIMSEQ__offset,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F3: nat > A,K2: nat,A2: A] :
( ( filterlim @ nat @ A
@ ^ [N2: nat] : ( F3 @ ( plus_plus @ nat @ N2 @ K2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ A2 )
@ ( at_top @ nat ) )
=> ( filterlim @ nat @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ ( at_top @ nat ) ) ) ) ).
% LIMSEQ_offset
thf(fact_4905_LIMSEQ__ignore__initial__segment,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F3: nat > A,A2: A,K2: nat] :
( ( filterlim @ nat @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ ( at_top @ nat ) )
=> ( filterlim @ nat @ A
@ ^ [N2: nat] : ( F3 @ ( plus_plus @ nat @ N2 @ K2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ A2 )
@ ( at_top @ nat ) ) ) ) ).
% LIMSEQ_ignore_initial_segment
thf(fact_4906_LIMSEQ__le__const2,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [X7: nat > A,X: A,A2: A] :
( ( filterlim @ nat @ A @ X7 @ ( topolo7230453075368039082e_nhds @ A @ X ) @ ( at_top @ nat ) )
=> ( ? [N8: nat] :
! [N3: nat] :
( ( ord_less_eq @ nat @ N8 @ N3 )
=> ( ord_less_eq @ A @ ( X7 @ N3 ) @ A2 ) )
=> ( ord_less_eq @ A @ X @ A2 ) ) ) ) ).
% LIMSEQ_le_const2
thf(fact_4907_LIMSEQ__le__const,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [X7: nat > A,X: A,A2: A] :
( ( filterlim @ nat @ A @ X7 @ ( topolo7230453075368039082e_nhds @ A @ X ) @ ( at_top @ nat ) )
=> ( ? [N8: nat] :
! [N3: nat] :
( ( ord_less_eq @ nat @ N8 @ N3 )
=> ( ord_less_eq @ A @ A2 @ ( X7 @ N3 ) ) )
=> ( ord_less_eq @ A @ A2 @ X ) ) ) ) ).
% LIMSEQ_le_const
thf(fact_4908_Lim__bounded2,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [F3: nat > A,L: A,N6: nat,C5: A] :
( ( filterlim @ nat @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ ( at_top @ nat ) )
=> ( ! [N3: nat] :
( ( ord_less_eq @ nat @ N6 @ N3 )
=> ( ord_less_eq @ A @ C5 @ ( F3 @ N3 ) ) )
=> ( ord_less_eq @ A @ C5 @ L ) ) ) ) ).
% Lim_bounded2
thf(fact_4909_Lim__bounded,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [F3: nat > A,L: A,M5: nat,C5: A] :
( ( filterlim @ nat @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ ( at_top @ nat ) )
=> ( ! [N3: nat] :
( ( ord_less_eq @ nat @ M5 @ N3 )
=> ( ord_less_eq @ A @ ( F3 @ N3 ) @ C5 ) )
=> ( ord_less_eq @ A @ L @ C5 ) ) ) ) ).
% Lim_bounded
thf(fact_4910_LIMSEQ__le,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [X7: nat > A,X: A,Y8: nat > A,Y: A] :
( ( filterlim @ nat @ A @ X7 @ ( topolo7230453075368039082e_nhds @ A @ X ) @ ( at_top @ nat ) )
=> ( ( filterlim @ nat @ A @ Y8 @ ( topolo7230453075368039082e_nhds @ A @ Y ) @ ( at_top @ nat ) )
=> ( ? [N8: nat] :
! [N3: nat] :
( ( ord_less_eq @ nat @ N8 @ N3 )
=> ( ord_less_eq @ A @ ( X7 @ N3 ) @ ( Y8 @ N3 ) ) )
=> ( ord_less_eq @ A @ X @ Y ) ) ) ) ) ).
% LIMSEQ_le
thf(fact_4911_lim__mono,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [N6: nat,X7: nat > A,Y8: nat > A,X: A,Y: A] :
( ! [N3: nat] :
( ( ord_less_eq @ nat @ N6 @ N3 )
=> ( ord_less_eq @ A @ ( X7 @ N3 ) @ ( Y8 @ N3 ) ) )
=> ( ( filterlim @ nat @ A @ X7 @ ( topolo7230453075368039082e_nhds @ A @ X ) @ ( at_top @ nat ) )
=> ( ( filterlim @ nat @ A @ Y8 @ ( topolo7230453075368039082e_nhds @ A @ Y ) @ ( at_top @ nat ) )
=> ( ord_less_eq @ A @ X @ Y ) ) ) ) ) ).
% lim_mono
thf(fact_4912_Sup__lim,axiom,
! [A: $tType] :
( ( ( comple5582772986160207858norder @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [B2: nat > A,S: set @ A,A2: A] :
( ! [N3: nat] : ( member @ A @ ( B2 @ N3 ) @ S )
=> ( ( filterlim @ nat @ A @ B2 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ ( at_top @ nat ) )
=> ( ord_less_eq @ A @ A2 @ ( complete_Sup_Sup @ A @ S ) ) ) ) ) ).
% Sup_lim
thf(fact_4913_Inf__lim,axiom,
! [A: $tType] :
( ( ( comple5582772986160207858norder @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [B2: nat > A,S: set @ A,A2: A] :
( ! [N3: nat] : ( member @ A @ ( B2 @ N3 ) @ S )
=> ( ( filterlim @ nat @ A @ B2 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ ( at_top @ nat ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ S ) @ A2 ) ) ) ) ).
% Inf_lim
thf(fact_4914_Inf__as__limit,axiom,
! [A: $tType] :
( ( ( comple5582772986160207858norder @ A )
& ( topolo3112930676232923870pology @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [A5: set @ A] :
( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ? [U4: nat > A] :
( ! [N5: nat] : ( member @ A @ ( U4 @ N5 ) @ A5 )
& ( filterlim @ nat @ A @ U4 @ ( topolo7230453075368039082e_nhds @ A @ ( complete_Inf_Inf @ A @ A5 ) ) @ ( at_top @ nat ) ) ) ) ) ).
% Inf_as_limit
thf(fact_4915_summable__LIMSEQ__zero,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: nat > A] :
( ( summable @ A @ F3 )
=> ( filterlim @ nat @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ ( at_top @ nat ) ) ) ) ).
% summable_LIMSEQ_zero
thf(fact_4916_mult__nat__left__at__top,axiom,
! [C3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ C3 )
=> ( filterlim @ nat @ nat @ ( times_times @ nat @ C3 ) @ ( at_top @ nat ) @ ( at_top @ nat ) ) ) ).
% mult_nat_left_at_top
thf(fact_4917_mult__nat__right__at__top,axiom,
! [C3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ C3 )
=> ( filterlim @ nat @ nat
@ ^ [X5: nat] : ( times_times @ nat @ X5 @ C3 )
@ ( at_top @ nat )
@ ( at_top @ nat ) ) ) ).
% mult_nat_right_at_top
thf(fact_4918_monoseq__le,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [A2: nat > A,X: A] :
( ( topological_monoseq @ A @ A2 )
=> ( ( filterlim @ nat @ A @ A2 @ ( topolo7230453075368039082e_nhds @ A @ X ) @ ( at_top @ nat ) )
=> ( ( ! [N5: nat] : ( ord_less_eq @ A @ ( A2 @ N5 ) @ X )
& ! [M4: nat,N5: nat] :
( ( ord_less_eq @ nat @ M4 @ N5 )
=> ( ord_less_eq @ A @ ( A2 @ M4 ) @ ( A2 @ N5 ) ) ) )
| ( ! [N5: nat] : ( ord_less_eq @ A @ X @ ( A2 @ N5 ) )
& ! [M4: nat,N5: nat] :
( ( ord_less_eq @ nat @ M4 @ N5 )
=> ( ord_less_eq @ A @ ( A2 @ N5 ) @ ( A2 @ M4 ) ) ) ) ) ) ) ) ).
% monoseq_le
thf(fact_4919_lim__const__over__n,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [A2: A] :
( filterlim @ nat @ A
@ ^ [N2: nat] : ( divide_divide @ A @ A2 @ ( semiring_1_of_nat @ A @ N2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ ( at_top @ nat ) ) ) ).
% lim_const_over_n
thf(fact_4920_lim__inverse__n,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ( filterlim @ nat @ A
@ ^ [N2: nat] : ( inverse_inverse @ A @ ( semiring_1_of_nat @ A @ N2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ ( at_top @ nat ) ) ) ).
% lim_inverse_n
thf(fact_4921_LIMSEQ__linear,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [X7: nat > A,X: A,L: nat] :
( ( filterlim @ nat @ A @ X7 @ ( topolo7230453075368039082e_nhds @ A @ X ) @ ( at_top @ nat ) )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ L )
=> ( filterlim @ nat @ A
@ ^ [N2: nat] : ( X7 @ ( times_times @ nat @ N2 @ L ) )
@ ( topolo7230453075368039082e_nhds @ A @ X )
@ ( at_top @ nat ) ) ) ) ) ).
% LIMSEQ_linear
thf(fact_4922_LIMSEQ__inverse__zero,axiom,
! [X7: nat > real] :
( ! [R3: real] :
? [N8: nat] :
! [N3: nat] :
( ( ord_less_eq @ nat @ N8 @ N3 )
=> ( ord_less @ real @ R3 @ ( X7 @ N3 ) ) )
=> ( filterlim @ nat @ real
@ ^ [N2: nat] : ( inverse_inverse @ real @ ( X7 @ N2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( at_top @ nat ) ) ) ).
% LIMSEQ_inverse_zero
thf(fact_4923_LIMSEQ__root__const,axiom,
! [C3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ C3 )
=> ( filterlim @ nat @ real
@ ^ [N2: nat] : ( root @ N2 @ C3 )
@ ( topolo7230453075368039082e_nhds @ real @ ( one_one @ real ) )
@ ( at_top @ nat ) ) ) ).
% LIMSEQ_root_const
thf(fact_4924_increasing__LIMSEQ,axiom,
! [F3: nat > real,L: real] :
( ! [N3: nat] : ( ord_less_eq @ real @ ( F3 @ N3 ) @ ( F3 @ ( suc @ N3 ) ) )
=> ( ! [N3: nat] : ( ord_less_eq @ real @ ( F3 @ N3 ) @ L )
=> ( ! [E2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E2 )
=> ? [N5: nat] : ( ord_less_eq @ real @ L @ ( plus_plus @ real @ ( F3 @ N5 ) @ E2 ) ) )
=> ( filterlim @ nat @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ L ) @ ( at_top @ nat ) ) ) ) ) ).
% increasing_LIMSEQ
thf(fact_4925_lim__sequentially,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X7: nat > A,L5: A] :
( ( filterlim @ nat @ A @ X7 @ ( topolo7230453075368039082e_nhds @ A @ L5 ) @ ( at_top @ nat ) )
= ( ! [R5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R5 )
=> ? [No: nat] :
! [N2: nat] :
( ( ord_less_eq @ nat @ No @ N2 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( X7 @ N2 ) @ L5 ) @ R5 ) ) ) ) ) ) ).
% lim_sequentially
thf(fact_4926_metric__LIMSEQ__I,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X7: nat > A,L5: A] :
( ! [R3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R3 )
=> ? [No2: nat] :
! [N3: nat] :
( ( ord_less_eq @ nat @ No2 @ N3 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( X7 @ N3 ) @ L5 ) @ R3 ) ) )
=> ( filterlim @ nat @ A @ X7 @ ( topolo7230453075368039082e_nhds @ A @ L5 ) @ ( at_top @ nat ) ) ) ) ).
% metric_LIMSEQ_I
thf(fact_4927_metric__LIMSEQ__D,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X7: nat > A,L5: A,R2: real] :
( ( filterlim @ nat @ A @ X7 @ ( topolo7230453075368039082e_nhds @ A @ L5 ) @ ( at_top @ nat ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ R2 )
=> ? [No3: nat] :
! [N5: nat] :
( ( ord_less_eq @ nat @ No3 @ N5 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( X7 @ N5 ) @ L5 ) @ R2 ) ) ) ) ) ).
% metric_LIMSEQ_D
thf(fact_4928_lim__1__over__n,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ( filterlim @ nat @ A
@ ^ [N2: nat] : ( divide_divide @ A @ ( one_one @ A ) @ ( semiring_1_of_nat @ A @ N2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ ( at_top @ nat ) ) ) ).
% lim_1_over_n
thf(fact_4929_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_4930_telescope__sums,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: nat > A,C3: A] :
( ( filterlim @ nat @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ C3 ) @ ( at_top @ nat ) )
=> ( sums @ A
@ ^ [N2: nat] : ( minus_minus @ A @ ( F3 @ ( suc @ N2 ) ) @ ( F3 @ N2 ) )
@ ( minus_minus @ A @ C3 @ ( F3 @ ( zero_zero @ nat ) ) ) ) ) ) ).
% telescope_sums
thf(fact_4931_telescope__sums_H,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: nat > A,C3: A] :
( ( filterlim @ nat @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ C3 ) @ ( at_top @ nat ) )
=> ( sums @ A
@ ^ [N2: nat] : ( minus_minus @ A @ ( F3 @ N2 ) @ ( F3 @ ( suc @ N2 ) ) )
@ ( minus_minus @ A @ ( F3 @ ( zero_zero @ nat ) ) @ C3 ) ) ) ) ).
% telescope_sums'
thf(fact_4932_LIMSEQ__divide__realpow__zero,axiom,
! [X: real,A2: real] :
( ( ord_less @ real @ ( one_one @ real ) @ X )
=> ( filterlim @ nat @ real
@ ^ [N2: nat] : ( divide_divide @ real @ A2 @ ( power_power @ real @ X @ N2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( at_top @ nat ) ) ) ).
% LIMSEQ_divide_realpow_zero
thf(fact_4933_LIMSEQ__abs__realpow__zero,axiom,
! [C3: real] :
( ( ord_less @ real @ ( abs_abs @ real @ C3 ) @ ( one_one @ real ) )
=> ( filterlim @ nat @ real @ ( power_power @ real @ ( abs_abs @ real @ C3 ) ) @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) ) ) ).
% LIMSEQ_abs_realpow_zero
thf(fact_4934_LIMSEQ__abs__realpow__zero2,axiom,
! [C3: real] :
( ( ord_less @ real @ ( abs_abs @ real @ C3 ) @ ( one_one @ real ) )
=> ( filterlim @ nat @ real @ ( power_power @ real @ C3 ) @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) ) ) ).
% LIMSEQ_abs_realpow_zero2
thf(fact_4935_LIMSEQ__inverse__realpow__zero,axiom,
! [X: real] :
( ( ord_less @ real @ ( one_one @ real ) @ X )
=> ( filterlim @ nat @ real
@ ^ [N2: nat] : ( inverse_inverse @ real @ ( power_power @ real @ X @ N2 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( at_top @ nat ) ) ) ).
% LIMSEQ_inverse_realpow_zero
thf(fact_4936_sums__def_H,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ( ( sums @ A )
= ( ^ [F4: nat > A,S8: A] :
( filterlim @ nat @ A
@ ^ [N2: nat] : ( groups7311177749621191930dd_sum @ nat @ A @ F4 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ S8 )
@ ( at_top @ nat ) ) ) ) ) ).
% sums_def'
thf(fact_4937_root__test__convergence,axiom,
! [A: $tType] :
( ( real_Vector_banach @ A )
=> ! [F3: nat > A,X: real] :
( ( filterlim @ nat @ real
@ ^ [N2: nat] : ( root @ N2 @ ( real_V7770717601297561774m_norm @ A @ ( F3 @ N2 ) ) )
@ ( topolo7230453075368039082e_nhds @ real @ X )
@ ( at_top @ nat ) )
=> ( ( ord_less @ real @ X @ ( one_one @ real ) )
=> ( summable @ A @ F3 ) ) ) ) ).
% root_test_convergence
thf(fact_4938_LIMSEQ__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X7: nat > A,L5: A] :
( ( filterlim @ nat @ A @ X7 @ ( topolo7230453075368039082e_nhds @ A @ L5 ) @ ( at_top @ nat ) )
= ( ! [R5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R5 )
=> ? [No: nat] :
! [N2: nat] :
( ( ord_less_eq @ nat @ No @ N2 )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( X7 @ N2 ) @ L5 ) ) @ R5 ) ) ) ) ) ) ).
% LIMSEQ_iff
thf(fact_4939_LIMSEQ__I,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X7: nat > A,L5: A] :
( ! [R3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R3 )
=> ? [No2: nat] :
! [N3: nat] :
( ( ord_less_eq @ nat @ No2 @ N3 )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( X7 @ N3 ) @ L5 ) ) @ R3 ) ) )
=> ( filterlim @ nat @ A @ X7 @ ( topolo7230453075368039082e_nhds @ A @ L5 ) @ ( at_top @ nat ) ) ) ) ).
% LIMSEQ_I
thf(fact_4940_LIMSEQ__D,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X7: nat > A,L5: A,R2: real] :
( ( filterlim @ nat @ A @ X7 @ ( topolo7230453075368039082e_nhds @ A @ L5 ) @ ( at_top @ nat ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ R2 )
=> ? [No3: nat] :
! [N5: nat] :
( ( ord_less_eq @ nat @ No3 @ N5 )
=> ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ ( X7 @ N5 ) @ L5 ) ) @ R2 ) ) ) ) ) ).
% LIMSEQ_D
thf(fact_4941_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_4942_LIMSEQ__iff__nz,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X7: nat > A,L5: A] :
( ( filterlim @ nat @ A @ X7 @ ( topolo7230453075368039082e_nhds @ A @ L5 ) @ ( at_top @ nat ) )
= ( ! [R5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R5 )
=> ? [No: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ No )
& ! [N2: nat] :
( ( ord_less_eq @ nat @ No @ N2 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( X7 @ N2 ) @ L5 ) @ R5 ) ) ) ) ) ) ) ).
% LIMSEQ_iff_nz
thf(fact_4943_tendsto__power__zero,axiom,
! [A: $tType,B: $tType] :
( ( real_V2822296259951069270ebra_1 @ A )
=> ! [F3: B > nat,F5: filter @ B,X: A] :
( ( filterlim @ B @ nat @ F3 @ ( at_top @ nat ) @ F5 )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ ( one_one @ real ) )
=> ( filterlim @ B @ A
@ ^ [Y6: B] : ( power_power @ A @ X @ ( F3 @ Y6 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ F5 ) ) ) ) ).
% tendsto_power_zero
thf(fact_4944_LIMSEQ__norm__0,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: nat > A] :
( ! [N3: nat] : ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( F3 @ N3 ) ) @ ( divide_divide @ real @ ( one_one @ real ) @ ( semiring_1_of_nat @ real @ ( suc @ N3 ) ) ) )
=> ( filterlim @ nat @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ ( at_top @ nat ) ) ) ) ).
% LIMSEQ_norm_0
thf(fact_4945_field__derivative__lim__unique,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,Df: A,Z: A,S: nat > A,A2: A] :
( ( has_field_derivative @ A @ F3 @ Df @ ( topolo174197925503356063within @ A @ Z @ ( top_top @ ( set @ A ) ) ) )
=> ( ( filterlim @ nat @ A @ S @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ ( at_top @ nat ) )
=> ( ! [N3: nat] :
( ( S @ N3 )
!= ( zero_zero @ A ) )
=> ( ( filterlim @ nat @ A
@ ^ [N2: nat] : ( divide_divide @ A @ ( minus_minus @ A @ ( F3 @ ( plus_plus @ A @ Z @ ( S @ N2 ) ) ) @ ( F3 @ Z ) ) @ ( S @ N2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ A2 )
@ ( at_top @ nat ) )
=> ( Df = A2 ) ) ) ) ) ) ).
% field_derivative_lim_unique
thf(fact_4946_powser__times__n__limit__0,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V8999393235501362500lgebra @ A ) )
=> ! [X: A] :
( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ ( one_one @ real ) )
=> ( filterlim @ nat @ A
@ ^ [N2: nat] : ( times_times @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( power_power @ A @ X @ N2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ ( at_top @ nat ) ) ) ) ).
% powser_times_n_limit_0
thf(fact_4947_lim__n__over__pown,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A] :
( ( ord_less @ real @ ( one_one @ real ) @ ( real_V7770717601297561774m_norm @ A @ X ) )
=> ( filterlim @ nat @ A
@ ^ [N2: nat] : ( divide_divide @ A @ ( semiring_1_of_nat @ A @ N2 ) @ ( power_power @ A @ X @ N2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ ( at_top @ nat ) ) ) ) ).
% lim_n_over_pown
thf(fact_4948_zeroseq__arctan__series,axiom,
! [X: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ X ) @ ( one_one @ real ) )
=> ( filterlim @ nat @ real
@ ^ [N2: nat] : ( times_times @ real @ ( divide_divide @ real @ ( one_one @ real ) @ ( semiring_1_of_nat @ real @ ( plus_plus @ nat @ ( times_times @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) ) @ ( power_power @ real @ X @ ( plus_plus @ nat @ ( times_times @ nat @ N2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ ( one_one @ nat ) ) ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( at_top @ nat ) ) ) ).
% zeroseq_arctan_series
thf(fact_4949_sums__alternating__upper__lower,axiom,
! [A2: nat > real] :
( ! [N3: nat] : ( ord_less_eq @ real @ ( A2 @ ( suc @ N3 ) ) @ ( A2 @ N3 ) )
=> ( ! [N3: nat] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( A2 @ N3 ) )
=> ( ( filterlim @ nat @ real @ A2 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) )
=> ? [L4: real] :
( ! [N5: nat] :
( ord_less_eq @ real
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I4: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I4 ) @ ( A2 @ I4 ) )
@ ( set_ord_lessThan @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N5 ) ) )
@ L4 )
& ( filterlim @ nat @ real
@ ^ [N2: nat] :
( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I4: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I4 ) @ ( A2 @ I4 ) )
@ ( set_ord_lessThan @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) ) )
@ ( topolo7230453075368039082e_nhds @ real @ L4 )
@ ( at_top @ nat ) )
& ! [N5: nat] :
( ord_less_eq @ real @ L4
@ ( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I4: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I4 ) @ ( A2 @ I4 ) )
@ ( set_ord_lessThan @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N5 ) @ ( one_one @ nat ) ) ) ) )
& ( filterlim @ nat @ real
@ ^ [N2: nat] :
( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I4: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I4 ) @ ( A2 @ I4 ) )
@ ( set_ord_lessThan @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) @ ( one_one @ nat ) ) ) )
@ ( topolo7230453075368039082e_nhds @ real @ L4 )
@ ( at_top @ nat ) ) ) ) ) ) ).
% sums_alternating_upper_lower
thf(fact_4950_summable__Leibniz_I5_J,axiom,
! [A2: nat > real] :
( ( filterlim @ nat @ real @ A2 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) )
=> ( ( topological_monoseq @ real @ A2 )
=> ( filterlim @ nat @ real
@ ^ [N2: nat] :
( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I4: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I4 ) @ ( A2 @ I4 ) )
@ ( set_ord_lessThan @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) @ ( one_one @ nat ) ) ) )
@ ( topolo7230453075368039082e_nhds @ real
@ ( suminf @ real
@ ^ [I4: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I4 ) @ ( A2 @ I4 ) ) ) )
@ ( at_top @ nat ) ) ) ) ).
% summable_Leibniz(5)
thf(fact_4951_summable__Leibniz_H_I5_J,axiom,
! [A2: nat > real] :
( ( filterlim @ nat @ real @ A2 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) )
=> ( ! [N3: nat] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( A2 @ N3 ) )
=> ( ! [N3: nat] : ( ord_less_eq @ real @ ( A2 @ ( suc @ N3 ) ) @ ( A2 @ N3 ) )
=> ( filterlim @ nat @ real
@ ^ [N2: nat] :
( groups7311177749621191930dd_sum @ nat @ real
@ ^ [I4: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I4 ) @ ( A2 @ I4 ) )
@ ( set_ord_lessThan @ nat @ ( plus_plus @ nat @ ( times_times @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N2 ) @ ( one_one @ nat ) ) ) )
@ ( topolo7230453075368039082e_nhds @ real
@ ( suminf @ real
@ ^ [I4: nat] : ( times_times @ real @ ( power_power @ real @ ( uminus_uminus @ real @ ( one_one @ real ) ) @ I4 ) @ ( A2 @ I4 ) ) ) )
@ ( at_top @ nat ) ) ) ) ) ).
% summable_Leibniz'(5)
thf(fact_4952_has__derivative__at2,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F3: A > B,F10: A > B,X: A] :
( ( has_derivative @ A @ B @ F3 @ F10 @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) )
= ( ( real_V3181309239436604168linear @ A @ B @ F10 )
& ( filterlim @ A @ B
@ ^ [Y6: A] : ( real_V8093663219630862766scaleR @ B @ ( divide_divide @ real @ ( one_one @ real ) @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ Y6 @ X ) ) ) @ ( minus_minus @ B @ ( F3 @ Y6 ) @ ( plus_plus @ B @ ( F3 @ X ) @ ( F10 @ ( minus_minus @ A @ Y6 @ X ) ) ) ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% has_derivative_at2
thf(fact_4953_has__derivative__at,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F3: A > B,D6: A > B,X: A] :
( ( has_derivative @ A @ B @ F3 @ D6 @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) )
= ( ( real_V3181309239436604168linear @ A @ B @ D6 )
& ( filterlim @ A @ real
@ ^ [H: A] : ( divide_divide @ real @ ( real_V7770717601297561774m_norm @ B @ ( minus_minus @ B @ ( minus_minus @ B @ ( F3 @ ( plus_plus @ A @ X @ H ) ) @ ( F3 @ X ) ) @ ( D6 @ H ) ) ) @ ( real_V7770717601297561774m_norm @ A @ H ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ).
% has_derivative_at
thf(fact_4954_has__derivative__within,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F3: A > B,F10: A > B,X: A,S: set @ A] :
( ( has_derivative @ A @ B @ F3 @ F10 @ ( topolo174197925503356063within @ A @ X @ S ) )
= ( ( real_V3181309239436604168linear @ A @ B @ F10 )
& ( filterlim @ A @ B
@ ^ [Y6: A] : ( real_V8093663219630862766scaleR @ B @ ( divide_divide @ real @ ( one_one @ real ) @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ Y6 @ X ) ) ) @ ( minus_minus @ B @ ( F3 @ Y6 ) @ ( plus_plus @ B @ ( F3 @ X ) @ ( F10 @ ( minus_minus @ A @ Y6 @ X ) ) ) ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ).
% has_derivative_within
thf(fact_4955_bounded__linear__add,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F3: A > B,G3: A > B] :
( ( real_V3181309239436604168linear @ A @ B @ F3 )
=> ( ( real_V3181309239436604168linear @ A @ B @ G3 )
=> ( real_V3181309239436604168linear @ A @ B
@ ^ [X5: A] : ( plus_plus @ B @ ( F3 @ X5 ) @ ( G3 @ X5 ) ) ) ) ) ) ).
% bounded_linear_add
thf(fact_4956_bounded__linear__zero,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ( real_V3181309239436604168linear @ A @ B
@ ^ [X5: A] : ( zero_zero @ B ) ) ) ).
% bounded_linear_zero
thf(fact_4957_bounded__linear_Otendsto__zero,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F3: A > B,G3: C > A,F5: filter @ C] :
( ( real_V3181309239436604168linear @ A @ B @ F3 )
=> ( ( filterlim @ C @ A @ G3 @ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) ) @ F5 )
=> ( filterlim @ C @ B
@ ^ [X5: C] : ( F3 @ ( G3 @ X5 ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ F5 ) ) ) ) ).
% bounded_linear.tendsto_zero
thf(fact_4958_has__derivative__within__singleton__iff,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F3: A > B,G3: A > B,X: A] :
( ( has_derivative @ A @ B @ F3 @ G3 @ ( topolo174197925503356063within @ A @ X @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( real_V3181309239436604168linear @ A @ B @ G3 ) ) ) ).
% has_derivative_within_singleton_iff
thf(fact_4959_filterlim__pow__at__top,axiom,
! [A: $tType,N: nat,F3: A > real,F5: filter @ A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( filterlim @ A @ real @ F3 @ ( at_top @ real ) @ F5 )
=> ( filterlim @ A @ real
@ ^ [X5: A] : ( power_power @ real @ ( F3 @ X5 ) @ N )
@ ( at_top @ real )
@ F5 ) ) ) ).
% filterlim_pow_at_top
thf(fact_4960_bounded__linear_Opos__bounded,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F3: A > B] :
( ( real_V3181309239436604168linear @ A @ B @ F3 )
=> ? [K8: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K8 )
& ! [X3: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( F3 @ X3 ) ) @ ( times_times @ real @ ( real_V7770717601297561774m_norm @ A @ X3 ) @ K8 ) ) ) ) ) ).
% bounded_linear.pos_bounded
thf(fact_4961_bounded__linear__intro,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F3: A > B,K5: real] :
( ! [X4: A,Y3: A] :
( ( F3 @ ( plus_plus @ A @ X4 @ Y3 ) )
= ( plus_plus @ B @ ( F3 @ X4 ) @ ( F3 @ Y3 ) ) )
=> ( ! [R3: real,X4: A] :
( ( F3 @ ( real_V8093663219630862766scaleR @ A @ R3 @ X4 ) )
= ( real_V8093663219630862766scaleR @ B @ R3 @ ( F3 @ X4 ) ) )
=> ( ! [X4: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( F3 @ X4 ) ) @ ( times_times @ real @ ( real_V7770717601297561774m_norm @ A @ X4 ) @ K5 ) )
=> ( real_V3181309239436604168linear @ A @ B @ F3 ) ) ) ) ) ).
% bounded_linear_intro
thf(fact_4962_filterlim__tendsto__pos__mult__at__top,axiom,
! [A: $tType,F3: A > real,C3: real,F5: filter @ A,G3: A > real] :
( ( filterlim @ A @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ C3 ) @ F5 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ C3 )
=> ( ( filterlim @ A @ real @ G3 @ ( at_top @ real ) @ F5 )
=> ( filterlim @ A @ real
@ ^ [X5: A] : ( times_times @ real @ ( F3 @ X5 ) @ ( G3 @ X5 ) )
@ ( at_top @ real )
@ F5 ) ) ) ) ).
% filterlim_tendsto_pos_mult_at_top
thf(fact_4963_filterlim__at__top__mult__tendsto__pos,axiom,
! [A: $tType,F3: A > real,C3: real,F5: filter @ A,G3: A > real] :
( ( filterlim @ A @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ C3 ) @ F5 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ C3 )
=> ( ( filterlim @ A @ real @ G3 @ ( at_top @ real ) @ F5 )
=> ( filterlim @ A @ real
@ ^ [X5: A] : ( times_times @ real @ ( G3 @ X5 ) @ ( F3 @ X5 ) )
@ ( at_top @ real )
@ F5 ) ) ) ) ).
% filterlim_at_top_mult_tendsto_pos
thf(fact_4964_tendsto__neg__powr,axiom,
! [A: $tType,S: real,F3: A > real,F5: filter @ A] :
( ( ord_less @ real @ S @ ( zero_zero @ real ) )
=> ( ( filterlim @ A @ real @ F3 @ ( at_top @ real ) @ F5 )
=> ( filterlim @ A @ real
@ ^ [X5: A] : ( powr @ real @ ( F3 @ X5 ) @ S )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ F5 ) ) ) ).
% tendsto_neg_powr
thf(fact_4965_DERIV__neg__imp__decreasing__at__top,axiom,
! [B2: real,F3: real > real,Flim: real] :
( ! [X4: real] :
( ( ord_less_eq @ real @ B2 @ X4 )
=> ? [Y4: real] :
( ( has_field_derivative @ real @ F3 @ Y4 @ ( topolo174197925503356063within @ real @ X4 @ ( top_top @ ( set @ real ) ) ) )
& ( ord_less @ real @ Y4 @ ( zero_zero @ real ) ) ) )
=> ( ( filterlim @ real @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ Flim ) @ ( at_top @ real ) )
=> ( ord_less @ real @ Flim @ ( F3 @ B2 ) ) ) ) ).
% DERIV_neg_imp_decreasing_at_top
thf(fact_4966_has__derivative__at__within,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F3: A > B,F10: A > B,X: A,S: set @ A] :
( ( has_derivative @ A @ B @ F3 @ F10 @ ( topolo174197925503356063within @ A @ X @ S ) )
= ( ( real_V3181309239436604168linear @ A @ B @ F10 )
& ( filterlim @ A @ B
@ ^ [Y6: A] : ( real_V8093663219630862766scaleR @ B @ ( inverse_inverse @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ Y6 @ X ) ) ) @ ( minus_minus @ B @ ( minus_minus @ B @ ( F3 @ Y6 ) @ ( F3 @ X ) ) @ ( F10 @ ( minus_minus @ A @ Y6 @ X ) ) ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ).
% has_derivative_at_within
thf(fact_4967_has__derivativeI,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F10: A > B,X: A,F3: A > B,S: set @ A] :
( ( real_V3181309239436604168linear @ A @ B @ F10 )
=> ( ( filterlim @ A @ B
@ ^ [Y6: A] : ( real_V8093663219630862766scaleR @ B @ ( inverse_inverse @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ Y6 @ X ) ) ) @ ( minus_minus @ B @ ( minus_minus @ B @ ( F3 @ Y6 ) @ ( F3 @ X ) ) @ ( F10 @ ( minus_minus @ A @ Y6 @ X ) ) ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( has_derivative @ A @ B @ F3 @ F10 @ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ).
% has_derivativeI
thf(fact_4968_has__derivative__iff__Ex,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F3: A > B,F10: A > B,X: A] :
( ( has_derivative @ A @ B @ F3 @ F10 @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) )
= ( ( real_V3181309239436604168linear @ A @ B @ F10 )
& ? [E4: A > B] :
( ! [H: A] :
( ( F3 @ ( plus_plus @ A @ X @ H ) )
= ( plus_plus @ B @ ( plus_plus @ B @ ( F3 @ X ) @ ( F10 @ H ) ) @ ( E4 @ H ) ) )
& ( filterlim @ A @ real
@ ^ [H: A] : ( divide_divide @ real @ ( real_V7770717601297561774m_norm @ B @ ( E4 @ H ) ) @ ( real_V7770717601297561774m_norm @ A @ H ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ).
% has_derivative_iff_Ex
thf(fact_4969_has__derivativeI__sandwich,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [E3: real,F10: A > B,S: set @ A,X: A,F3: A > B,H7: A > real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E3 )
=> ( ( real_V3181309239436604168linear @ A @ B @ F10 )
=> ( ! [Y3: A] :
( ( member @ A @ Y3 @ S )
=> ( ( Y3 != X )
=> ( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ Y3 @ X ) @ E3 )
=> ( ord_less_eq @ real @ ( divide_divide @ real @ ( real_V7770717601297561774m_norm @ B @ ( minus_minus @ B @ ( minus_minus @ B @ ( F3 @ Y3 ) @ ( F3 @ X ) ) @ ( F10 @ ( minus_minus @ A @ Y3 @ X ) ) ) ) @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ Y3 @ X ) ) ) @ ( H7 @ Y3 ) ) ) ) )
=> ( ( filterlim @ A @ real @ H7 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( has_derivative @ A @ B @ F3 @ F10 @ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ) ) ).
% has_derivativeI_sandwich
thf(fact_4970_has__derivative__def,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ( ( has_derivative @ A @ B )
= ( ^ [F4: A > B,F11: A > B,F8: filter @ A] :
( ( real_V3181309239436604168linear @ A @ B @ F11 )
& ( filterlim @ A @ B
@ ^ [Y6: A] :
( real_V8093663219630862766scaleR @ B
@ ( inverse_inverse @ real
@ ( real_V7770717601297561774m_norm @ A
@ ( minus_minus @ A @ Y6
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F8
@ ^ [X5: A] : X5 ) ) ) )
@ ( minus_minus @ B
@ ( minus_minus @ B @ ( F4 @ Y6 )
@ ( F4
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F8
@ ^ [X5: A] : X5 ) ) )
@ ( F11
@ ( minus_minus @ A @ Y6
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F8
@ ^ [X5: A] : X5 ) ) ) ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) )
@ F8 ) ) ) ) ) ).
% has_derivative_def
thf(fact_4971_has__derivative__at__within__iff__Ex,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [X: A,S2: set @ A,F3: A > B,F10: A > B] :
( ( member @ A @ X @ S2 )
=> ( ( topolo1002775350975398744n_open @ A @ S2 )
=> ( ( has_derivative @ A @ B @ F3 @ F10 @ ( topolo174197925503356063within @ A @ X @ S2 ) )
= ( ( real_V3181309239436604168linear @ A @ B @ F10 )
& ? [E4: A > B] :
( ! [H: A] :
( ( member @ A @ ( plus_plus @ A @ X @ H ) @ S2 )
=> ( ( F3 @ ( plus_plus @ A @ X @ H ) )
= ( plus_plus @ B @ ( plus_plus @ B @ ( F3 @ X ) @ ( F10 @ H ) ) @ ( E4 @ H ) ) ) )
& ( filterlim @ A @ real
@ ^ [H: A] : ( divide_divide @ real @ ( real_V7770717601297561774m_norm @ B @ ( E4 @ H ) ) @ ( real_V7770717601297561774m_norm @ A @ H ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ) ) ).
% has_derivative_at_within_iff_Ex
thf(fact_4972_lim__zero__infinity,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,L: A] :
( ( filterlim @ A @ A
@ ^ [X5: A] : ( F3 @ ( divide_divide @ A @ ( one_one @ A ) @ X5 ) )
@ ( topolo7230453075368039082e_nhds @ A @ L )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) )
=> ( filterlim @ A @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ ( at_infinity @ A ) ) ) ) ).
% lim_zero_infinity
thf(fact_4973_open__empty,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ( topolo1002775350975398744n_open @ A @ ( bot_bot @ ( set @ A ) ) ) ) ).
% open_empty
thf(fact_4974_open__Inter,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [S2: set @ ( set @ A )] :
( ( finite_finite2 @ ( set @ A ) @ S2 )
=> ( ! [X4: set @ A] :
( ( member @ ( set @ A ) @ X4 @ S2 )
=> ( topolo1002775350975398744n_open @ A @ X4 ) )
=> ( topolo1002775350975398744n_open @ A @ ( complete_Inf_Inf @ ( set @ A ) @ S2 ) ) ) ) ) ).
% open_Inter
thf(fact_4975_open__INT,axiom,
! [A: $tType,B: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [A5: set @ B,B6: B > ( set @ A )] :
( ( finite_finite2 @ B @ A5 )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ A5 )
=> ( topolo1002775350975398744n_open @ A @ ( B6 @ X4 ) ) )
=> ( topolo1002775350975398744n_open @ A @ ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B6 @ A5 ) ) ) ) ) ) ).
% open_INT
thf(fact_4976_Sup__notin__open,axiom,
! [A: $tType] :
( ( topolo8458572112393995274pology @ A )
=> ! [A5: set @ A,X: A] :
( ( topolo1002775350975398744n_open @ A @ A5 )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ A5 )
=> ( ord_less @ A @ X4 @ X ) )
=> ~ ( member @ A @ ( complete_Sup_Sup @ A @ A5 ) @ A5 ) ) ) ) ).
% Sup_notin_open
thf(fact_4977_Inf__notin__open,axiom,
! [A: $tType] :
( ( topolo8458572112393995274pology @ A )
=> ! [A5: set @ A,X: A] :
( ( topolo1002775350975398744n_open @ A @ A5 )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ A5 )
=> ( ord_less @ A @ X @ X4 ) )
=> ~ ( member @ A @ ( complete_Inf_Inf @ A @ A5 ) @ A5 ) ) ) ) ).
% Inf_notin_open
thf(fact_4978_not__open__singleton,axiom,
! [A: $tType] :
( ( topolo8386298272705272623_space @ A )
=> ! [X: A] :
~ ( topolo1002775350975398744n_open @ A @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% not_open_singleton
thf(fact_4979_separation__t2,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [X: A,Y: A] :
( ( X != Y )
= ( ? [U5: set @ A,V6: set @ A] :
( ( topolo1002775350975398744n_open @ A @ U5 )
& ( topolo1002775350975398744n_open @ A @ V6 )
& ( member @ A @ X @ U5 )
& ( member @ A @ Y @ V6 )
& ( ( inf_inf @ ( set @ A ) @ U5 @ V6 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% separation_t2
thf(fact_4980_hausdorff,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [X: A,Y: A] :
( ( X != Y )
=> ? [U6: set @ A,V7: set @ A] :
( ( topolo1002775350975398744n_open @ A @ U6 )
& ( topolo1002775350975398744n_open @ A @ V7 )
& ( member @ A @ X @ U6 )
& ( member @ A @ Y @ V7 )
& ( ( inf_inf @ ( set @ A ) @ U6 @ V7 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% hausdorff
thf(fact_4981_open__ball,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X: A,D3: real] :
( topolo1002775350975398744n_open @ A
@ ( collect @ A
@ ^ [Y6: A] : ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X @ Y6 ) @ D3 ) ) ) ) ).
% open_ball
thf(fact_4982_open__right,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [S2: set @ A,X: A,Y: A] :
( ( topolo1002775350975398744n_open @ A @ S2 )
=> ( ( member @ A @ X @ S2 )
=> ( ( ord_less @ A @ X @ Y )
=> ? [B4: A] :
( ( ord_less @ A @ X @ B4 )
& ( ord_less_eq @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ X @ B4 ) @ S2 ) ) ) ) ) ) ).
% open_right
thf(fact_4983_open__dist,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ( ( topolo1002775350975398744n_open @ A )
= ( ^ [S7: set @ A] :
! [X5: A] :
( ( member @ A @ X5 @ S7 )
=> ? [E4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E4 )
& ! [Y6: A] :
( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ Y6 @ X5 ) @ E4 )
=> ( member @ A @ Y6 @ S7 ) ) ) ) ) ) ) ).
% open_dist
thf(fact_4984_Lim__ident__at,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [X: A,S: set @ A] :
( ( ( topolo174197925503356063within @ A @ X @ S )
!= ( bot_bot @ ( filter @ A ) ) )
=> ( ( topolo3827282254853284352ce_Lim @ A @ A @ ( topolo174197925503356063within @ A @ X @ S )
@ ^ [X5: A] : X5 )
= X ) ) ) ).
% Lim_ident_at
thf(fact_4985_lim__explicit,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F3: nat > A,F0: A] :
( ( filterlim @ nat @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ F0 ) @ ( at_top @ nat ) )
= ( ! [S7: set @ A] :
( ( topolo1002775350975398744n_open @ A @ S7 )
=> ( ( member @ A @ F0 @ S7 )
=> ? [N4: nat] :
! [N2: nat] :
( ( ord_less_eq @ nat @ N4 @ N2 )
=> ( member @ A @ ( F3 @ N2 ) @ S7 ) ) ) ) ) ) ) ).
% lim_explicit
thf(fact_4986_not__tendsto__and__filterlim__at__infinity,axiom,
! [A: $tType,B: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F5: filter @ B,F3: B > A,C3: A] :
( ( F5
!= ( bot_bot @ ( filter @ B ) ) )
=> ( ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ C3 ) @ F5 )
=> ~ ( filterlim @ B @ A @ F3 @ ( at_infinity @ A ) @ F5 ) ) ) ) ).
% not_tendsto_and_filterlim_at_infinity
thf(fact_4987_tendsto__Lim,axiom,
! [A: $tType,B: $tType] :
( ( topological_t2_space @ B )
=> ! [Net: filter @ A,F3: A > B,L: B] :
( ( Net
!= ( bot_bot @ ( filter @ A ) ) )
=> ( ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ Net )
=> ( ( topolo3827282254853284352ce_Lim @ A @ B @ Net @ F3 )
= L ) ) ) ) ).
% tendsto_Lim
thf(fact_4988_tendsto__add__filterlim__at__infinity_H,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F3: A > B,F5: filter @ A,G3: A > B,C3: B] :
( ( filterlim @ A @ B @ F3 @ ( at_infinity @ B ) @ F5 )
=> ( ( filterlim @ A @ B @ G3 @ ( topolo7230453075368039082e_nhds @ B @ C3 ) @ F5 )
=> ( filterlim @ A @ B
@ ^ [X5: A] : ( plus_plus @ B @ ( F3 @ X5 ) @ ( G3 @ X5 ) )
@ ( at_infinity @ B )
@ F5 ) ) ) ) ).
% tendsto_add_filterlim_at_infinity'
thf(fact_4989_tendsto__add__filterlim__at__infinity,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F3: A > B,C3: B,F5: filter @ A,G3: A > B] :
( ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ C3 ) @ F5 )
=> ( ( filterlim @ A @ B @ G3 @ ( at_infinity @ B ) @ F5 )
=> ( filterlim @ A @ B
@ ^ [X5: A] : ( plus_plus @ B @ ( F3 @ X5 ) @ ( G3 @ X5 ) )
@ ( at_infinity @ B )
@ F5 ) ) ) ) ).
% tendsto_add_filterlim_at_infinity
thf(fact_4990_at__within__nhd,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [X: A,S2: set @ A,T4: set @ A,U3: set @ A] :
( ( member @ A @ X @ S2 )
=> ( ( topolo1002775350975398744n_open @ A @ S2 )
=> ( ( ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ T4 @ S2 ) @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ U3 @ S2 ) @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( ( topolo174197925503356063within @ A @ X @ T4 )
= ( topolo174197925503356063within @ A @ X @ U3 ) ) ) ) ) ) ).
% at_within_nhd
thf(fact_4991_at__eq__bot__iff,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [A2: A] :
( ( ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) )
= ( bot_bot @ ( filter @ A ) ) )
= ( topolo1002775350975398744n_open @ A @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% at_eq_bot_iff
thf(fact_4992_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_4993_tendsto__mult__filterlim__at__infinity,axiom,
! [A: $tType,B: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: B > A,C3: A,F5: filter @ B,G3: B > A] :
( ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ C3 ) @ F5 )
=> ( ( C3
!= ( zero_zero @ A ) )
=> ( ( filterlim @ B @ A @ G3 @ ( at_infinity @ A ) @ F5 )
=> ( filterlim @ B @ A
@ ^ [X5: B] : ( times_times @ A @ ( F3 @ X5 ) @ ( G3 @ X5 ) )
@ ( at_infinity @ A )
@ F5 ) ) ) ) ) ).
% tendsto_mult_filterlim_at_infinity
thf(fact_4994_tendsto__divide__0,axiom,
! [A: $tType,C: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [F3: C > A,C3: A,F5: filter @ C,G3: C > A] :
( ( filterlim @ C @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ C3 ) @ F5 )
=> ( ( filterlim @ C @ A @ G3 @ ( at_infinity @ A ) @ F5 )
=> ( filterlim @ C @ A
@ ^ [X5: C] : ( divide_divide @ A @ ( F3 @ X5 ) @ ( G3 @ X5 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( zero_zero @ A ) )
@ F5 ) ) ) ) ).
% tendsto_divide_0
thf(fact_4995_filterlim__power__at__infinity,axiom,
! [B: $tType,A: $tType] :
( ( real_V8999393235501362500lgebra @ B )
=> ! [F3: A > B,F5: filter @ A,N: nat] :
( ( filterlim @ A @ B @ F3 @ ( at_infinity @ B ) @ F5 )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( filterlim @ A @ B
@ ^ [X5: A] : ( power_power @ B @ ( F3 @ X5 ) @ N )
@ ( at_infinity @ B )
@ F5 ) ) ) ) ).
% filterlim_power_at_infinity
thf(fact_4996_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_4997_filterlim__inverse__at__iff,axiom,
! [B: $tType,A: $tType] :
( ( real_V8999393235501362500lgebra @ B )
=> ! [G3: A > B,F5: filter @ A] :
( ( filterlim @ A @ B
@ ^ [X5: A] : ( inverse_inverse @ B @ ( G3 @ X5 ) )
@ ( topolo174197925503356063within @ B @ ( zero_zero @ B ) @ ( top_top @ ( set @ B ) ) )
@ F5 )
= ( filterlim @ A @ B @ G3 @ ( at_infinity @ B ) @ F5 ) ) ) ).
% filterlim_inverse_at_iff
thf(fact_4998_tendsto__offset__zero__iff,axiom,
! [C: $tType,D: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( topolo4958980785337419405_space @ D )
& ( zero @ C ) )
=> ! [A2: A,S2: set @ A,F3: A > D,L5: D] :
( ( nO_MATCH @ C @ A @ ( zero_zero @ C ) @ A2 )
=> ( ( member @ A @ A2 @ S2 )
=> ( ( topolo1002775350975398744n_open @ A @ S2 )
=> ( ( filterlim @ A @ D @ F3 @ ( topolo7230453075368039082e_nhds @ D @ L5 ) @ ( topolo174197925503356063within @ A @ A2 @ S2 ) )
= ( filterlim @ A @ D
@ ^ [H: A] : ( F3 @ ( plus_plus @ A @ A2 @ H ) )
@ ( topolo7230453075368039082e_nhds @ D @ L5 )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% tendsto_offset_zero_iff
thf(fact_4999_filterlim__divide__at__infinity,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,C3: A,F5: filter @ A,G3: A > A] :
( ( filterlim @ A @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ C3 ) @ F5 )
=> ( ( filterlim @ A @ A @ G3 @ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) @ F5 )
=> ( ( C3
!= ( zero_zero @ A ) )
=> ( filterlim @ A @ A
@ ^ [X5: A] : ( divide_divide @ A @ ( F3 @ X5 ) @ ( G3 @ X5 ) )
@ ( at_infinity @ A )
@ F5 ) ) ) ) ) ).
% filterlim_divide_at_infinity
thf(fact_5000_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_5001_filterlim__pow__at__bot__even,axiom,
! [N: nat,F3: real > real,F5: filter @ real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( filterlim @ real @ real @ F3 @ ( at_bot @ real ) @ F5 )
=> ( ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( filterlim @ real @ real
@ ^ [X5: real] : ( power_power @ real @ ( F3 @ X5 ) @ N )
@ ( at_top @ real )
@ F5 ) ) ) ) ).
% filterlim_pow_at_bot_even
thf(fact_5002_polyfun__extremal,axiom,
! [A: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [C3: nat > A,K2: nat,N: nat,B6: real] :
( ( ( C3 @ K2 )
!= ( zero_zero @ A ) )
=> ( ( ord_less_eq @ nat @ ( one_one @ nat ) @ K2 )
=> ( ( ord_less_eq @ nat @ K2 @ N )
=> ( eventually @ A
@ ^ [Z5: A] :
( ord_less_eq @ real @ B6
@ ( real_V7770717601297561774m_norm @ A
@ ( groups7311177749621191930dd_sum @ nat @ A
@ ^ [I4: nat] : ( times_times @ A @ ( C3 @ I4 ) @ ( power_power @ A @ Z5 @ I4 ) )
@ ( set_ord_atMost @ nat @ N ) ) ) )
@ ( at_infinity @ A ) ) ) ) ) ) ).
% polyfun_extremal
thf(fact_5003_filterlim__pow__at__bot__odd,axiom,
! [N: nat,F3: real > real,F5: filter @ real] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( ( filterlim @ real @ real @ F3 @ ( at_bot @ real ) @ F5 )
=> ( ~ ( dvd_dvd @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ N )
=> ( filterlim @ real @ real
@ ^ [X5: real] : ( power_power @ real @ ( F3 @ X5 ) @ N )
@ ( at_bot @ real )
@ F5 ) ) ) ) ).
% filterlim_pow_at_bot_odd
thf(fact_5004_eventually__sequentially__seg,axiom,
! [P2: nat > $o,K2: nat] :
( ( eventually @ nat
@ ^ [N2: nat] : ( P2 @ ( plus_plus @ nat @ N2 @ K2 ) )
@ ( at_top @ nat ) )
= ( eventually @ nat @ P2 @ ( at_top @ nat ) ) ) ).
% eventually_sequentially_seg
thf(fact_5005_eventually__const,axiom,
! [A: $tType,F5: filter @ A,P2: $o] :
( ( F5
!= ( bot_bot @ ( filter @ A ) ) )
=> ( ( eventually @ A
@ ^ [X5: A] : P2
@ F5 )
= P2 ) ) ).
% eventually_const
thf(fact_5006_filterlim__at__infinity__imp__filterlim__at__bot,axiom,
! [A: $tType,F3: A > real,F5: filter @ A] :
( ( filterlim @ A @ real @ F3 @ ( at_infinity @ real ) @ F5 )
=> ( ( eventually @ A
@ ^ [X5: A] : ( ord_less @ real @ ( F3 @ X5 ) @ ( zero_zero @ real ) )
@ F5 )
=> ( filterlim @ A @ real @ F3 @ ( at_bot @ real ) @ F5 ) ) ) ).
% filterlim_at_infinity_imp_filterlim_at_bot
thf(fact_5007_eventually__le__at__bot,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C3: A] :
( eventually @ A
@ ^ [X5: A] : ( ord_less_eq @ A @ X5 @ C3 )
@ ( at_bot @ A ) ) ) ).
% eventually_le_at_bot
thf(fact_5008_eventually__at__bot__linorder,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P2: A > $o] :
( ( eventually @ A @ P2 @ ( at_bot @ A ) )
= ( ? [N4: A] :
! [N2: A] :
( ( ord_less_eq @ A @ N2 @ N4 )
=> ( P2 @ N2 ) ) ) ) ) ).
% eventually_at_bot_linorder
thf(fact_5009_eventually__gt__at__bot,axiom,
! [A: $tType] :
( ( unboun7993243217541854897norder @ A )
=> ! [C3: A] :
( eventually @ A
@ ^ [X5: A] : ( ord_less @ A @ X5 @ C3 )
@ ( at_bot @ A ) ) ) ).
% eventually_gt_at_bot
thf(fact_5010_eventually__at__bot__dense,axiom,
! [A: $tType] :
( ( ( linorder @ A )
& ( no_bot @ A ) )
=> ! [P2: A > $o] :
( ( eventually @ A @ P2 @ ( at_bot @ A ) )
= ( ? [N4: A] :
! [N2: A] :
( ( ord_less @ A @ N2 @ N4 )
=> ( P2 @ N2 ) ) ) ) ) ).
% eventually_at_bot_dense
thf(fact_5011_eventually__happens_H,axiom,
! [A: $tType,F5: filter @ A,P2: A > $o] :
( ( F5
!= ( bot_bot @ ( filter @ A ) ) )
=> ( ( eventually @ A @ P2 @ F5 )
=> ? [X_12: A] : ( P2 @ X_12 ) ) ) ).
% eventually_happens'
thf(fact_5012_eventually__happens,axiom,
! [A: $tType,P2: A > $o,Net: filter @ A] :
( ( eventually @ A @ P2 @ Net )
=> ( ( Net
= ( bot_bot @ ( filter @ A ) ) )
| ? [X_12: A] : ( P2 @ X_12 ) ) ) ).
% eventually_happens
thf(fact_5013_eventually__bot,axiom,
! [A: $tType,P2: A > $o] : ( eventually @ A @ P2 @ ( bot_bot @ ( filter @ A ) ) ) ).
% eventually_bot
thf(fact_5014_trivial__limit__def,axiom,
! [A: $tType,F5: filter @ A] :
( ( F5
= ( bot_bot @ ( filter @ A ) ) )
= ( eventually @ A
@ ^ [X5: A] : $false
@ F5 ) ) ).
% trivial_limit_def
thf(fact_5015_eventually__const__iff,axiom,
! [A: $tType,P2: $o,F5: filter @ A] :
( ( eventually @ A
@ ^ [X5: A] : P2
@ F5 )
= ( P2
| ( F5
= ( bot_bot @ ( filter @ A ) ) ) ) ) ).
% eventually_const_iff
thf(fact_5016_False__imp__not__eventually,axiom,
! [A: $tType,P2: A > $o,Net: filter @ A] :
( ! [X4: A] :
~ ( P2 @ X4 )
=> ( ( Net
!= ( bot_bot @ ( filter @ A ) ) )
=> ~ ( eventually @ A @ P2 @ Net ) ) ) ).
% False_imp_not_eventually
thf(fact_5017_filterlim__at__bot__dense,axiom,
! [A: $tType,B: $tType] :
( ( ( dense_linorder @ B )
& ( no_bot @ B ) )
=> ! [F3: A > B,F5: filter @ A] :
( ( filterlim @ A @ B @ F3 @ ( at_bot @ B ) @ F5 )
= ( ! [Z8: B] :
( eventually @ A
@ ^ [X5: A] : ( ord_less @ B @ ( F3 @ X5 ) @ Z8 )
@ F5 ) ) ) ) ).
% filterlim_at_bot_dense
thf(fact_5018_filterlim__at__bot__le,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ B )
=> ! [F3: A > B,F5: filter @ A,C3: B] :
( ( filterlim @ A @ B @ F3 @ ( at_bot @ B ) @ F5 )
= ( ! [Z8: B] :
( ( ord_less_eq @ B @ Z8 @ C3 )
=> ( eventually @ A
@ ^ [X5: A] : ( ord_less_eq @ B @ ( F3 @ X5 ) @ Z8 )
@ F5 ) ) ) ) ) ).
% filterlim_at_bot_le
thf(fact_5019_filterlim__at__bot,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ B )
=> ! [F3: A > B,F5: filter @ A] :
( ( filterlim @ A @ B @ F3 @ ( at_bot @ B ) @ F5 )
= ( ! [Z8: B] :
( eventually @ A
@ ^ [X5: A] : ( ord_less_eq @ B @ ( F3 @ X5 ) @ Z8 )
@ F5 ) ) ) ) ).
% filterlim_at_bot
thf(fact_5020_filterlim__at__bot__lt,axiom,
! [A: $tType,B: $tType] :
( ( unboun7993243217541854897norder @ B )
=> ! [F3: A > B,F5: filter @ A,C3: B] :
( ( filterlim @ A @ B @ F3 @ ( at_bot @ B ) @ F5 )
= ( ! [Z8: B] :
( ( ord_less @ B @ Z8 @ C3 )
=> ( eventually @ A
@ ^ [X5: A] : ( ord_less_eq @ B @ ( F3 @ X5 ) @ Z8 )
@ F5 ) ) ) ) ) ).
% filterlim_at_bot_lt
thf(fact_5021_eventually__at__top__linorderI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C3: A,P2: A > $o] :
( ! [X4: A] :
( ( ord_less_eq @ A @ C3 @ X4 )
=> ( P2 @ X4 ) )
=> ( eventually @ A @ P2 @ ( at_top @ A ) ) ) ) ).
% eventually_at_top_linorderI
thf(fact_5022_eventually__at__top__linorder,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P2: A > $o] :
( ( eventually @ A @ P2 @ ( at_top @ A ) )
= ( ? [N4: A] :
! [N2: A] :
( ( ord_less_eq @ A @ N4 @ N2 )
=> ( P2 @ N2 ) ) ) ) ) ).
% eventually_at_top_linorder
thf(fact_5023_eventually__at__top__dense,axiom,
! [A: $tType] :
( ( ( linorder @ A )
& ( no_top @ A ) )
=> ! [P2: A > $o] :
( ( eventually @ A @ P2 @ ( at_top @ A ) )
= ( ? [N4: A] :
! [N2: A] :
( ( ord_less @ A @ N4 @ N2 )
=> ( P2 @ N2 ) ) ) ) ) ).
% eventually_at_top_dense
thf(fact_5024_eventually__sequentiallyI,axiom,
! [C3: nat,P2: nat > $o] :
( ! [X4: nat] :
( ( ord_less_eq @ nat @ C3 @ X4 )
=> ( P2 @ X4 ) )
=> ( eventually @ nat @ P2 @ ( at_top @ nat ) ) ) ).
% eventually_sequentiallyI
thf(fact_5025_eventually__sequentially,axiom,
! [P2: nat > $o] :
( ( eventually @ nat @ P2 @ ( at_top @ nat ) )
= ( ? [N4: nat] :
! [N2: nat] :
( ( ord_less_eq @ nat @ N4 @ N2 )
=> ( P2 @ N2 ) ) ) ) ).
% eventually_sequentially
thf(fact_5026_trivial__limit__at__bot__linorder,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( at_bot @ A )
!= ( bot_bot @ ( filter @ A ) ) ) ) ).
% trivial_limit_at_bot_linorder
thf(fact_5027_eventually__ge__at__top,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [C3: A] : ( eventually @ A @ ( ord_less_eq @ A @ C3 ) @ ( at_top @ A ) ) ) ).
% eventually_ge_at_top
thf(fact_5028_eventually__gt__at__top,axiom,
! [A: $tType] :
( ( ( linorder @ A )
& ( no_top @ A ) )
=> ! [C3: A] : ( eventually @ A @ ( ord_less @ A @ C3 ) @ ( at_top @ A ) ) ) ).
% eventually_gt_at_top
thf(fact_5029_le__sequentially,axiom,
! [F5: filter @ nat] :
( ( ord_less_eq @ ( filter @ nat ) @ F5 @ ( at_top @ nat ) )
= ( ! [N4: nat] : ( eventually @ nat @ ( ord_less_eq @ nat @ N4 ) @ F5 ) ) ) ).
% le_sequentially
thf(fact_5030_sequentially__offset,axiom,
! [P2: nat > $o,K2: nat] :
( ( eventually @ nat @ P2 @ ( at_top @ nat ) )
=> ( eventually @ nat
@ ^ [I4: nat] : ( P2 @ ( plus_plus @ nat @ I4 @ K2 ) )
@ ( at_top @ nat ) ) ) ).
% sequentially_offset
thf(fact_5031_filterlim__inverse__at__bot,axiom,
! [A: $tType,F3: A > real,F5: filter @ A] :
( ( filterlim @ A @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ F5 )
=> ( ( eventually @ A
@ ^ [X5: A] : ( ord_less @ real @ ( F3 @ X5 ) @ ( zero_zero @ real ) )
@ F5 )
=> ( filterlim @ A @ real
@ ^ [X5: A] : ( inverse_inverse @ real @ ( F3 @ X5 ) )
@ ( at_bot @ real )
@ F5 ) ) ) ).
% filterlim_inverse_at_bot
thf(fact_5032_eventually__nhds__top,axiom,
! [A: $tType] :
( ( ( order_top @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [B2: A,P2: A > $o] :
( ( ord_less @ A @ B2 @ ( top_top @ A ) )
=> ( ( eventually @ A @ P2 @ ( topolo7230453075368039082e_nhds @ A @ ( top_top @ A ) ) )
= ( ? [B5: A] :
( ( ord_less @ A @ B5 @ ( top_top @ A ) )
& ! [Z5: A] :
( ( ord_less @ A @ B5 @ Z5 )
=> ( P2 @ Z5 ) ) ) ) ) ) ) ).
% eventually_nhds_top
thf(fact_5033_filterlim__at__top__at__top,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( linorder @ B ) )
=> ! [Q: A > $o,F3: A > B,P2: B > $o,G3: B > A] :
( ! [X4: A,Y3: A] :
( ( Q @ X4 )
=> ( ( Q @ Y3 )
=> ( ( ord_less_eq @ A @ X4 @ Y3 )
=> ( ord_less_eq @ B @ ( F3 @ X4 ) @ ( F3 @ Y3 ) ) ) ) )
=> ( ! [X4: B] :
( ( P2 @ X4 )
=> ( ( F3 @ ( G3 @ X4 ) )
= X4 ) )
=> ( ! [X4: B] :
( ( P2 @ X4 )
=> ( Q @ ( G3 @ X4 ) ) )
=> ( ( eventually @ A @ Q @ ( at_top @ A ) )
=> ( ( eventually @ B @ P2 @ ( at_top @ B ) )
=> ( filterlim @ A @ B @ F3 @ ( at_top @ B ) @ ( at_top @ A ) ) ) ) ) ) ) ) ).
% filterlim_at_top_at_top
thf(fact_5034_eventually__at__left__field,axiom,
! [A: $tType] :
( ( ( linordered_field @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [P2: A > $o,X: A] :
( ( eventually @ A @ P2 @ ( topolo174197925503356063within @ A @ X @ ( set_ord_lessThan @ A @ X ) ) )
= ( ? [B5: A] :
( ( ord_less @ A @ B5 @ X )
& ! [Y6: A] :
( ( ord_less @ A @ B5 @ Y6 )
=> ( ( ord_less @ A @ Y6 @ X )
=> ( P2 @ Y6 ) ) ) ) ) ) ) ).
% eventually_at_left_field
thf(fact_5035_eventually__at__left,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [Y: A,X: A,P2: A > $o] :
( ( ord_less @ A @ Y @ X )
=> ( ( eventually @ A @ P2 @ ( topolo174197925503356063within @ A @ X @ ( set_ord_lessThan @ A @ X ) ) )
= ( ? [B5: A] :
( ( ord_less @ A @ B5 @ X )
& ! [Y6: A] :
( ( ord_less @ A @ B5 @ Y6 )
=> ( ( ord_less @ A @ Y6 @ X )
=> ( P2 @ Y6 ) ) ) ) ) ) ) ) ).
% eventually_at_left
thf(fact_5036_tendsto__sandwich,axiom,
! [A: $tType,B: $tType] :
( ( topolo2564578578187576103pology @ A )
=> ! [F3: B > A,G3: B > A,Net: filter @ B,H2: B > A,C3: A] :
( ( eventually @ B
@ ^ [N2: B] : ( ord_less_eq @ A @ ( F3 @ N2 ) @ ( G3 @ N2 ) )
@ Net )
=> ( ( eventually @ B
@ ^ [N2: B] : ( ord_less_eq @ A @ ( G3 @ N2 ) @ ( H2 @ N2 ) )
@ Net )
=> ( ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ C3 ) @ Net )
=> ( ( filterlim @ B @ A @ H2 @ ( topolo7230453075368039082e_nhds @ A @ C3 ) @ Net )
=> ( filterlim @ B @ A @ G3 @ ( topolo7230453075368039082e_nhds @ A @ C3 ) @ Net ) ) ) ) ) ) ).
% tendsto_sandwich
thf(fact_5037_order__tendsto__iff,axiom,
! [B: $tType,A: $tType] :
( ( topolo2564578578187576103pology @ A )
=> ! [F3: B > A,X: A,F5: filter @ B] :
( ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ X ) @ F5 )
= ( ! [L2: A] :
( ( ord_less @ A @ L2 @ X )
=> ( eventually @ B
@ ^ [X5: B] : ( ord_less @ A @ L2 @ ( F3 @ X5 ) )
@ F5 ) )
& ! [U2: A] :
( ( ord_less @ A @ X @ U2 )
=> ( eventually @ B
@ ^ [X5: B] : ( ord_less @ A @ ( F3 @ X5 ) @ U2 )
@ F5 ) ) ) ) ) ).
% order_tendsto_iff
thf(fact_5038_order__tendstoI,axiom,
! [A: $tType,B: $tType] :
( ( topolo2564578578187576103pology @ A )
=> ! [Y: A,F3: B > A,F5: filter @ B] :
( ! [A4: A] :
( ( ord_less @ A @ A4 @ Y )
=> ( eventually @ B
@ ^ [X5: B] : ( ord_less @ A @ A4 @ ( F3 @ X5 ) )
@ F5 ) )
=> ( ! [A4: A] :
( ( ord_less @ A @ Y @ A4 )
=> ( eventually @ B
@ ^ [X5: B] : ( ord_less @ A @ ( F3 @ X5 ) @ A4 )
@ F5 ) )
=> ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ Y ) @ F5 ) ) ) ) ).
% order_tendstoI
thf(fact_5039_order__tendstoD_I1_J,axiom,
! [A: $tType,B: $tType] :
( ( topolo2564578578187576103pology @ A )
=> ! [F3: B > A,Y: A,F5: filter @ B,A2: A] :
( ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ Y ) @ F5 )
=> ( ( ord_less @ A @ A2 @ Y )
=> ( eventually @ B
@ ^ [X5: B] : ( ord_less @ A @ A2 @ ( F3 @ X5 ) )
@ F5 ) ) ) ) ).
% order_tendstoD(1)
thf(fact_5040_order__tendstoD_I2_J,axiom,
! [A: $tType,B: $tType] :
( ( topolo2564578578187576103pology @ A )
=> ! [F3: B > A,Y: A,F5: filter @ B,A2: A] :
( ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ Y ) @ F5 )
=> ( ( ord_less @ A @ Y @ A2 )
=> ( eventually @ B
@ ^ [X5: B] : ( ord_less @ A @ ( F3 @ X5 ) @ A2 )
@ F5 ) ) ) ) ).
% order_tendstoD(2)
thf(fact_5041_filterlim__at__top__mono,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F3: B > A,F5: filter @ B,G3: B > A] :
( ( filterlim @ B @ A @ F3 @ ( at_top @ A ) @ F5 )
=> ( ( eventually @ B
@ ^ [X5: B] : ( ord_less_eq @ A @ ( F3 @ X5 ) @ ( G3 @ X5 ) )
@ F5 )
=> ( filterlim @ B @ A @ G3 @ ( at_top @ A ) @ F5 ) ) ) ) ).
% filterlim_at_top_mono
thf(fact_5042_filterlim__at__top__ge,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ B )
=> ! [F3: A > B,F5: filter @ A,C3: B] :
( ( filterlim @ A @ B @ F3 @ ( at_top @ B ) @ F5 )
= ( ! [Z8: B] :
( ( ord_less_eq @ B @ C3 @ Z8 )
=> ( eventually @ A
@ ^ [X5: A] : ( ord_less_eq @ B @ Z8 @ ( F3 @ X5 ) )
@ F5 ) ) ) ) ) ).
% filterlim_at_top_ge
thf(fact_5043_filterlim__at__top,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ B )
=> ! [F3: A > B,F5: filter @ A] :
( ( filterlim @ A @ B @ F3 @ ( at_top @ B ) @ F5 )
= ( ! [Z8: B] :
( eventually @ A
@ ^ [X5: A] : ( ord_less_eq @ B @ Z8 @ ( F3 @ X5 ) )
@ F5 ) ) ) ) ).
% filterlim_at_top
thf(fact_5044_filterlim__at__top__dense,axiom,
! [A: $tType,B: $tType] :
( ( unboun7993243217541854897norder @ B )
=> ! [F3: A > B,F5: filter @ A] :
( ( filterlim @ A @ B @ F3 @ ( at_top @ B ) @ F5 )
= ( ! [Z8: B] :
( eventually @ A
@ ^ [X5: A] : ( ord_less @ B @ Z8 @ ( F3 @ X5 ) )
@ F5 ) ) ) ) ).
% filterlim_at_top_dense
thf(fact_5045_eventually__Inf__base,axiom,
! [A: $tType,B6: set @ ( filter @ A ),P2: A > $o] :
( ( B6
!= ( bot_bot @ ( set @ ( filter @ A ) ) ) )
=> ( ! [F6: filter @ A] :
( ( member @ ( filter @ A ) @ F6 @ B6 )
=> ! [G5: filter @ A] :
( ( member @ ( filter @ A ) @ G5 @ B6 )
=> ? [X3: filter @ A] :
( ( member @ ( filter @ A ) @ X3 @ B6 )
& ( ord_less_eq @ ( filter @ A ) @ X3 @ ( inf_inf @ ( filter @ A ) @ F6 @ G5 ) ) ) ) )
=> ( ( eventually @ A @ P2 @ ( complete_Inf_Inf @ ( filter @ A ) @ B6 ) )
= ( ? [X5: filter @ A] :
( ( member @ ( filter @ A ) @ X5 @ B6 )
& ( eventually @ A @ P2 @ X5 ) ) ) ) ) ) ).
% eventually_Inf_base
thf(fact_5046_eventually__INF__finite,axiom,
! [B: $tType,A: $tType,A5: set @ A,P2: B > $o,F5: A > ( filter @ B )] :
( ( finite_finite2 @ A @ A5 )
=> ( ( eventually @ B @ P2 @ ( complete_Inf_Inf @ ( filter @ B ) @ ( image2 @ A @ ( filter @ B ) @ F5 @ A5 ) ) )
= ( ? [Q7: A > B > $o] :
( ! [X5: A] :
( ( member @ A @ X5 @ A5 )
=> ( eventually @ B @ ( Q7 @ X5 ) @ ( F5 @ X5 ) ) )
& ! [Y6: B] :
( ! [X5: A] :
( ( member @ A @ X5 @ A5 )
=> ( Q7 @ X5 @ Y6 ) )
=> ( P2 @ Y6 ) ) ) ) ) ) ).
% eventually_INF_finite
thf(fact_5047_eventually__at__left__real,axiom,
! [B2: real,A2: real] :
( ( ord_less @ real @ B2 @ A2 )
=> ( eventually @ real
@ ^ [X5: real] : ( member @ real @ X5 @ ( set_or5935395276787703475ssThan @ real @ B2 @ A2 ) )
@ ( topolo174197925503356063within @ real @ A2 @ ( set_ord_lessThan @ real @ A2 ) ) ) ) ).
% eventually_at_left_real
thf(fact_5048_eventually__at,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [P2: A > $o,A2: A,S2: set @ A] :
( ( eventually @ A @ P2 @ ( topolo174197925503356063within @ A @ A2 @ S2 ) )
= ( ? [D5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D5 )
& ! [X5: A] :
( ( member @ A @ X5 @ S2 )
=> ( ( ( X5 != A2 )
& ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X5 @ A2 ) @ D5 ) )
=> ( P2 @ X5 ) ) ) ) ) ) ) ).
% eventually_at
thf(fact_5049_eventually__nhds__metric,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [P2: A > $o,A2: A] :
( ( eventually @ A @ P2 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) )
= ( ? [D5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D5 )
& ! [X5: A] :
( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X5 @ A2 ) @ D5 )
=> ( P2 @ X5 ) ) ) ) ) ) ).
% eventually_nhds_metric
thf(fact_5050_eventually__at__leftI,axiom,
! [A: $tType] :
( ( topolo2564578578187576103pology @ A )
=> ! [A2: A,B2: A,P2: A > $o] :
( ! [X4: A] :
( ( member @ A @ X4 @ ( set_or5935395276787703475ssThan @ A @ A2 @ B2 ) )
=> ( P2 @ X4 ) )
=> ( ( ord_less @ A @ A2 @ B2 )
=> ( eventually @ A @ P2 @ ( topolo174197925503356063within @ A @ B2 @ ( set_ord_lessThan @ A @ B2 ) ) ) ) ) ) ).
% eventually_at_leftI
thf(fact_5051_eventually__at__to__0,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [P2: A > $o,A2: A] :
( ( eventually @ A @ P2 @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) )
= ( eventually @ A
@ ^ [X5: A] : ( P2 @ ( plus_plus @ A @ X5 @ A2 ) )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% eventually_at_to_0
thf(fact_5052_decreasing__tendsto,axiom,
! [A: $tType,B: $tType] :
( ( topolo2564578578187576103pology @ A )
=> ! [L: A,F3: B > A,F5: filter @ B] :
( ( eventually @ B
@ ^ [N2: B] : ( ord_less_eq @ A @ L @ ( F3 @ N2 ) )
@ F5 )
=> ( ! [X4: A] :
( ( ord_less @ A @ L @ X4 )
=> ( eventually @ B
@ ^ [N2: B] : ( ord_less @ A @ ( F3 @ N2 ) @ X4 )
@ F5 ) )
=> ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F5 ) ) ) ) ).
% decreasing_tendsto
thf(fact_5053_increasing__tendsto,axiom,
! [A: $tType,B: $tType] :
( ( topolo2564578578187576103pology @ A )
=> ! [F3: B > A,L: A,F5: filter @ B] :
( ( eventually @ B
@ ^ [N2: B] : ( ord_less_eq @ A @ ( F3 @ N2 ) @ L )
@ F5 )
=> ( ! [X4: A] :
( ( ord_less @ A @ X4 @ L )
=> ( eventually @ B
@ ^ [N2: B] : ( ord_less @ A @ X4 @ ( F3 @ N2 ) )
@ F5 ) )
=> ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F5 ) ) ) ) ).
% increasing_tendsto
thf(fact_5054_filterlim__at__top__gt,axiom,
! [A: $tType,B: $tType] :
( ( unboun7993243217541854897norder @ B )
=> ! [F3: A > B,F5: filter @ A,C3: B] :
( ( filterlim @ A @ B @ F3 @ ( at_top @ B ) @ F5 )
= ( ! [Z8: B] :
( ( ord_less @ B @ C3 @ Z8 )
=> ( eventually @ A
@ ^ [X5: A] : ( ord_less_eq @ B @ Z8 @ ( F3 @ X5 ) )
@ F5 ) ) ) ) ) ).
% filterlim_at_top_gt
thf(fact_5055_tendsto__le,axiom,
! [B: $tType,A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [F5: filter @ B,F3: B > A,X: A,G3: B > A,Y: A] :
( ( F5
!= ( bot_bot @ ( filter @ B ) ) )
=> ( ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ X ) @ F5 )
=> ( ( filterlim @ B @ A @ G3 @ ( topolo7230453075368039082e_nhds @ A @ Y ) @ F5 )
=> ( ( eventually @ B
@ ^ [X5: B] : ( ord_less_eq @ A @ ( G3 @ X5 ) @ ( F3 @ X5 ) )
@ F5 )
=> ( ord_less_eq @ A @ Y @ X ) ) ) ) ) ) ).
% tendsto_le
thf(fact_5056_tendsto__lowerbound,axiom,
! [B: $tType,A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [F3: B > A,X: A,F5: filter @ B,A2: A] :
( ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ X ) @ F5 )
=> ( ( eventually @ B
@ ^ [I4: B] : ( ord_less_eq @ A @ A2 @ ( F3 @ I4 ) )
@ F5 )
=> ( ( F5
!= ( bot_bot @ ( filter @ B ) ) )
=> ( ord_less_eq @ A @ A2 @ X ) ) ) ) ) ).
% tendsto_lowerbound
thf(fact_5057_tendsto__upperbound,axiom,
! [B: $tType,A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [F3: B > A,X: A,F5: filter @ B,A2: A] :
( ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ X ) @ F5 )
=> ( ( eventually @ B
@ ^ [I4: B] : ( ord_less_eq @ A @ ( F3 @ I4 ) @ A2 )
@ F5 )
=> ( ( F5
!= ( bot_bot @ ( filter @ B ) ) )
=> ( ord_less_eq @ A @ X @ A2 ) ) ) ) ) ).
% tendsto_upperbound
thf(fact_5058_filterlim__at__infinity__imp__filterlim__at__top,axiom,
! [A: $tType,F3: A > real,F5: filter @ A] :
( ( filterlim @ A @ real @ F3 @ ( at_infinity @ real ) @ F5 )
=> ( ( eventually @ A
@ ^ [X5: A] : ( ord_less @ real @ ( zero_zero @ real ) @ ( F3 @ X5 ) )
@ F5 )
=> ( filterlim @ A @ real @ F3 @ ( at_top @ real ) @ F5 ) ) ) ).
% filterlim_at_infinity_imp_filterlim_at_top
thf(fact_5059_eventually__INF,axiom,
! [A: $tType,B: $tType,P2: A > $o,F5: B > ( filter @ A ),B6: set @ B] :
( ( eventually @ A @ P2 @ ( complete_Inf_Inf @ ( filter @ A ) @ ( image2 @ B @ ( filter @ A ) @ F5 @ B6 ) ) )
= ( ? [X9: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ X9 @ B6 )
& ( finite_finite2 @ B @ X9 )
& ( eventually @ A @ P2 @ ( complete_Inf_Inf @ ( filter @ A ) @ ( image2 @ B @ ( filter @ A ) @ F5 @ X9 ) ) ) ) ) ) ).
% eventually_INF
thf(fact_5060_eventually__at__le,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [P2: A > $o,A2: A,S2: set @ A] :
( ( eventually @ A @ P2 @ ( topolo174197925503356063within @ A @ A2 @ S2 ) )
= ( ? [D5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D5 )
& ! [X5: A] :
( ( member @ A @ X5 @ S2 )
=> ( ( ( X5 != A2 )
& ( ord_less_eq @ real @ ( real_V557655796197034286t_dist @ A @ X5 @ A2 ) @ D5 ) )
=> ( P2 @ X5 ) ) ) ) ) ) ) ).
% eventually_at_le
thf(fact_5061_eventually__at__infinity__pos,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [P: A > $o] :
( ( eventually @ A @ P @ ( at_infinity @ A ) )
= ( ? [B5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ B5 )
& ! [X5: A] :
( ( ord_less_eq @ real @ B5 @ ( real_V7770717601297561774m_norm @ A @ X5 ) )
=> ( P @ X5 ) ) ) ) ) ) ).
% eventually_at_infinity_pos
thf(fact_5062_tendsto__imp__filterlim__at__left,axiom,
! [B: $tType,A: $tType] :
( ( topolo2564578578187576103pology @ B )
=> ! [F3: A > B,L5: B,F5: filter @ A] :
( ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ L5 ) @ F5 )
=> ( ( eventually @ A
@ ^ [X5: A] : ( ord_less @ B @ ( F3 @ X5 ) @ L5 )
@ F5 )
=> ( filterlim @ A @ B @ F3 @ ( topolo174197925503356063within @ B @ L5 @ ( set_ord_lessThan @ B @ L5 ) ) @ F5 ) ) ) ) ).
% tendsto_imp_filterlim_at_left
thf(fact_5063_tendstoD,axiom,
! [A: $tType,B: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [F3: B > A,L: A,F5: filter @ B,E3: real] :
( ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F5 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ E3 )
=> ( eventually @ B
@ ^ [X5: B] : ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( F3 @ X5 ) @ L ) @ E3 )
@ F5 ) ) ) ) ).
% tendstoD
thf(fact_5064_tendstoI,axiom,
! [A: $tType,B: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [F3: B > A,L: A,F5: filter @ B] :
( ! [E2: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E2 )
=> ( eventually @ B
@ ^ [X5: B] : ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( F3 @ X5 ) @ L ) @ E2 )
@ F5 ) )
=> ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F5 ) ) ) ).
% tendstoI
thf(fact_5065_tendsto__iff,axiom,
! [B: $tType,A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [F3: B > A,L: A,F5: filter @ B] :
( ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F5 )
= ( ! [E4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E4 )
=> ( eventually @ B
@ ^ [X5: B] : ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ ( F3 @ X5 ) @ L ) @ E4 )
@ F5 ) ) ) ) ) ).
% tendsto_iff
thf(fact_5066_eventually__Inf,axiom,
! [A: $tType,P2: A > $o,B6: set @ ( filter @ A )] :
( ( eventually @ A @ P2 @ ( complete_Inf_Inf @ ( filter @ A ) @ B6 ) )
= ( ? [X9: set @ ( filter @ A )] :
( ( ord_less_eq @ ( set @ ( filter @ A ) ) @ X9 @ B6 )
& ( finite_finite2 @ ( filter @ A ) @ X9 )
& ( eventually @ A @ P2 @ ( complete_Inf_Inf @ ( filter @ A ) @ X9 ) ) ) ) ) ).
% eventually_Inf
thf(fact_5067_filterlim__at__top__at__left,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo1944317154257567458pology @ A )
& ( linorder @ B ) )
=> ! [Q: A > $o,F3: A > B,P2: B > $o,G3: B > A,A2: A] :
( ! [X4: A,Y3: A] :
( ( Q @ X4 )
=> ( ( Q @ Y3 )
=> ( ( ord_less_eq @ A @ X4 @ Y3 )
=> ( ord_less_eq @ B @ ( F3 @ X4 ) @ ( F3 @ Y3 ) ) ) ) )
=> ( ! [X4: B] :
( ( P2 @ X4 )
=> ( ( F3 @ ( G3 @ X4 ) )
= X4 ) )
=> ( ! [X4: B] :
( ( P2 @ X4 )
=> ( Q @ ( G3 @ X4 ) ) )
=> ( ( eventually @ A @ Q @ ( topolo174197925503356063within @ A @ A2 @ ( set_ord_lessThan @ A @ A2 ) ) )
=> ( ! [B4: A] :
( ( Q @ B4 )
=> ( ord_less @ A @ B4 @ A2 ) )
=> ( ( eventually @ B @ P2 @ ( at_top @ B ) )
=> ( filterlim @ A @ B @ F3 @ ( at_top @ B ) @ ( topolo174197925503356063within @ A @ A2 @ ( set_ord_lessThan @ A @ A2 ) ) ) ) ) ) ) ) ) ) ).
% filterlim_at_top_at_left
thf(fact_5068_eventually__INF__base,axiom,
! [B: $tType,A: $tType,B6: set @ A,F5: A > ( filter @ B ),P2: B > $o] :
( ( B6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [A4: A] :
( ( member @ A @ A4 @ B6 )
=> ! [B4: A] :
( ( member @ A @ B4 @ B6 )
=> ? [X3: A] :
( ( member @ A @ X3 @ B6 )
& ( ord_less_eq @ ( filter @ B ) @ ( F5 @ X3 ) @ ( inf_inf @ ( filter @ B ) @ ( F5 @ A4 ) @ ( F5 @ B4 ) ) ) ) ) )
=> ( ( eventually @ B @ P2 @ ( complete_Inf_Inf @ ( filter @ B ) @ ( image2 @ A @ ( filter @ B ) @ F5 @ B6 ) ) )
= ( ? [X5: A] :
( ( member @ A @ X5 @ B6 )
& ( eventually @ B @ P2 @ ( F5 @ X5 ) ) ) ) ) ) ) ).
% eventually_INF_base
thf(fact_5069_tendsto__0__le,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V822414075346904944vector @ B ) )
=> ! [F3: A > B,F5: filter @ A,G3: A > C,K5: real] :
( ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ ( zero_zero @ B ) ) @ F5 )
=> ( ( eventually @ A
@ ^ [X5: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ C @ ( G3 @ X5 ) ) @ ( times_times @ real @ ( real_V7770717601297561774m_norm @ B @ ( F3 @ X5 ) ) @ K5 ) )
@ F5 )
=> ( filterlim @ A @ C @ G3 @ ( topolo7230453075368039082e_nhds @ C @ ( zero_zero @ C ) ) @ F5 ) ) ) ) ).
% tendsto_0_le
thf(fact_5070_filterlim__at__withinI,axiom,
! [A: $tType,B: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F3: B > A,C3: A,F5: filter @ B,A5: set @ A] :
( ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ C3 ) @ F5 )
=> ( ( eventually @ B
@ ^ [X5: B] : ( member @ A @ ( F3 @ X5 ) @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ C3 @ ( bot_bot @ ( set @ A ) ) ) ) )
@ F5 )
=> ( filterlim @ B @ A @ F3 @ ( topolo174197925503356063within @ A @ C3 @ A5 ) @ F5 ) ) ) ) ).
% filterlim_at_withinI
thf(fact_5071_filterlim__tendsto__pos__mult__at__bot,axiom,
! [A: $tType,F3: A > real,C3: real,F5: filter @ A,G3: A > real] :
( ( filterlim @ A @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ C3 ) @ F5 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ C3 )
=> ( ( filterlim @ A @ real @ G3 @ ( at_bot @ real ) @ F5 )
=> ( filterlim @ A @ real
@ ^ [X5: A] : ( times_times @ real @ ( F3 @ X5 ) @ ( G3 @ X5 ) )
@ ( at_bot @ real )
@ F5 ) ) ) ) ).
% filterlim_tendsto_pos_mult_at_bot
thf(fact_5072_filterlim__at__infinity,axiom,
! [C: $tType,A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [C3: real,F3: C > A,F5: filter @ C] :
( ( ord_less_eq @ real @ ( zero_zero @ real ) @ C3 )
=> ( ( filterlim @ C @ A @ F3 @ ( at_infinity @ A ) @ F5 )
= ( ! [R5: real] :
( ( ord_less @ real @ C3 @ R5 )
=> ( eventually @ C
@ ^ [X5: C] : ( ord_less_eq @ real @ R5 @ ( real_V7770717601297561774m_norm @ A @ ( F3 @ X5 ) ) )
@ F5 ) ) ) ) ) ) ).
% filterlim_at_infinity
thf(fact_5073_tendsto__zero__powrI,axiom,
! [A: $tType,F3: A > real,F5: filter @ A,G3: A > real,B2: real] :
( ( filterlim @ A @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ F5 )
=> ( ( filterlim @ A @ real @ G3 @ ( topolo7230453075368039082e_nhds @ real @ B2 ) @ F5 )
=> ( ( eventually @ A
@ ^ [X5: A] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( F3 @ X5 ) )
@ F5 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ B2 )
=> ( filterlim @ A @ real
@ ^ [X5: A] : ( powr @ real @ ( F3 @ X5 ) @ ( G3 @ X5 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) )
@ F5 ) ) ) ) ) ).
% tendsto_zero_powrI
thf(fact_5074_tendsto__powr2,axiom,
! [A: $tType,F3: A > real,A2: real,F5: filter @ A,G3: A > real,B2: real] :
( ( filterlim @ A @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ A2 ) @ F5 )
=> ( ( filterlim @ A @ real @ G3 @ ( topolo7230453075368039082e_nhds @ real @ B2 ) @ F5 )
=> ( ( eventually @ A
@ ^ [X5: A] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( F3 @ X5 ) )
@ F5 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ B2 )
=> ( filterlim @ A @ real
@ ^ [X5: A] : ( powr @ real @ ( F3 @ X5 ) @ ( G3 @ X5 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( powr @ real @ A2 @ B2 ) )
@ F5 ) ) ) ) ) ).
% tendsto_powr2
thf(fact_5075_tendsto__powr_H,axiom,
! [A: $tType,F3: A > real,A2: real,F5: filter @ A,G3: A > real,B2: real] :
( ( filterlim @ A @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ A2 ) @ F5 )
=> ( ( filterlim @ A @ real @ G3 @ ( topolo7230453075368039082e_nhds @ real @ B2 ) @ F5 )
=> ( ( ( A2
!= ( zero_zero @ real ) )
| ( ( ord_less @ real @ ( zero_zero @ real ) @ B2 )
& ( eventually @ A
@ ^ [X5: A] : ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( F3 @ X5 ) )
@ F5 ) ) )
=> ( filterlim @ A @ real
@ ^ [X5: A] : ( powr @ real @ ( F3 @ X5 ) @ ( G3 @ X5 ) )
@ ( topolo7230453075368039082e_nhds @ real @ ( powr @ real @ A2 @ B2 ) )
@ F5 ) ) ) ) ).
% tendsto_powr'
thf(fact_5076_LIM__at__top__divide,axiom,
! [A: $tType,F3: A > real,A2: real,F5: filter @ A,G3: A > real] :
( ( filterlim @ A @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ A2 ) @ F5 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ A2 )
=> ( ( filterlim @ A @ real @ G3 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ F5 )
=> ( ( eventually @ A
@ ^ [X5: A] : ( ord_less @ real @ ( zero_zero @ real ) @ ( G3 @ X5 ) )
@ F5 )
=> ( filterlim @ A @ real
@ ^ [X5: A] : ( divide_divide @ real @ ( F3 @ X5 ) @ ( G3 @ X5 ) )
@ ( at_top @ real )
@ F5 ) ) ) ) ) ).
% LIM_at_top_divide
thf(fact_5077_filterlim__inverse__at__top,axiom,
! [A: $tType,F3: A > real,F5: filter @ A] :
( ( filterlim @ A @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ F5 )
=> ( ( eventually @ A
@ ^ [X5: A] : ( ord_less @ real @ ( zero_zero @ real ) @ ( F3 @ X5 ) )
@ F5 )
=> ( filterlim @ A @ real
@ ^ [X5: A] : ( inverse_inverse @ real @ ( F3 @ X5 ) )
@ ( at_top @ real )
@ F5 ) ) ) ).
% filterlim_inverse_at_top
thf(fact_5078_filterlim__inverse__at__top__iff,axiom,
! [A: $tType,F3: A > real,F5: filter @ A] :
( ( eventually @ A
@ ^ [X5: A] : ( ord_less @ real @ ( zero_zero @ real ) @ ( F3 @ X5 ) )
@ F5 )
=> ( ( filterlim @ A @ real
@ ^ [X5: A] : ( inverse_inverse @ real @ ( F3 @ X5 ) )
@ ( at_top @ real )
@ F5 )
= ( filterlim @ A @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ F5 ) ) ) ).
% filterlim_inverse_at_top_iff
thf(fact_5079_filterlim__tendsto__neg__mult__at__bot,axiom,
! [A: $tType,F3: A > real,C3: real,F5: filter @ A,G3: A > real] :
( ( filterlim @ A @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ C3 ) @ F5 )
=> ( ( ord_less @ real @ C3 @ ( zero_zero @ real ) )
=> ( ( filterlim @ A @ real @ G3 @ ( at_top @ real ) @ F5 )
=> ( filterlim @ A @ real
@ ^ [X5: A] : ( times_times @ real @ ( F3 @ X5 ) @ ( G3 @ X5 ) )
@ ( at_bot @ real )
@ F5 ) ) ) ) ).
% filterlim_tendsto_neg_mult_at_bot
thf(fact_5080_DERIV__pos__imp__increasing__at__bot,axiom,
! [B2: real,F3: real > real,Flim: real] :
( ! [X4: real] :
( ( ord_less_eq @ real @ X4 @ B2 )
=> ? [Y4: real] :
( ( has_field_derivative @ real @ F3 @ Y4 @ ( topolo174197925503356063within @ real @ X4 @ ( top_top @ ( set @ real ) ) ) )
& ( ord_less @ real @ ( zero_zero @ real ) @ Y4 ) ) )
=> ( ( filterlim @ real @ real @ F3 @ ( topolo7230453075368039082e_nhds @ real @ Flim ) @ ( at_bot @ real ) )
=> ( ord_less @ real @ Flim @ ( F3 @ B2 ) ) ) ) ).
% DERIV_pos_imp_increasing_at_bot
thf(fact_5081_summable__Cauchy_H,axiom,
! [A: $tType] :
( ( real_Vector_banach @ A )
=> ! [F3: nat > A,G3: nat > real] :
( ( eventually @ nat
@ ^ [M3: nat] :
! [N2: nat] :
( ( ord_less_eq @ nat @ M3 @ N2 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_or7035219750837199246ssThan @ nat @ M3 @ N2 ) ) ) @ ( G3 @ M3 ) ) )
@ ( at_top @ nat ) )
=> ( ( filterlim @ nat @ real @ G3 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) )
=> ( summable @ A @ F3 ) ) ) ) ).
% summable_Cauchy'
thf(fact_5082_Bfun__metric__def,axiom,
! [B: $tType,A: $tType] :
( ( real_V7819770556892013058_space @ B )
=> ( ( bfun @ A @ B )
= ( ^ [F4: A > B,F8: filter @ A] :
? [Y6: B,K6: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K6 )
& ( eventually @ A
@ ^ [X5: A] : ( ord_less_eq @ real @ ( real_V557655796197034286t_dist @ B @ ( F4 @ X5 ) @ Y6 ) @ K6 )
@ F8 ) ) ) ) ) ).
% Bfun_metric_def
thf(fact_5083_Bfun__def,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ( ( bfun @ A @ B )
= ( ^ [F4: A > B,F8: filter @ A] :
? [K6: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K6 )
& ( eventually @ A
@ ^ [X5: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( F4 @ X5 ) ) @ K6 )
@ F8 ) ) ) ) ) ).
% Bfun_def
thf(fact_5084_Bseq__add__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: nat > A,C3: A] :
( ( bfun @ nat @ A
@ ^ [X5: nat] : ( plus_plus @ A @ ( F3 @ X5 ) @ C3 )
@ ( at_top @ nat ) )
= ( bfun @ nat @ A @ F3 @ ( at_top @ nat ) ) ) ) ).
% Bseq_add_iff
thf(fact_5085_Bseq__add,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: nat > A,C3: A] :
( ( bfun @ nat @ A @ F3 @ ( at_top @ nat ) )
=> ( bfun @ nat @ A
@ ^ [X5: nat] : ( plus_plus @ A @ ( F3 @ X5 ) @ C3 )
@ ( at_top @ nat ) ) ) ) ).
% Bseq_add
thf(fact_5086_Bseq__ignore__initial__segment,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X7: nat > A,K2: nat] :
( ( bfun @ nat @ A @ X7 @ ( at_top @ nat ) )
=> ( bfun @ nat @ A
@ ^ [N2: nat] : ( X7 @ ( plus_plus @ nat @ N2 @ K2 ) )
@ ( at_top @ nat ) ) ) ) ).
% Bseq_ignore_initial_segment
thf(fact_5087_Bseq__offset,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ! [X7: nat > A,K2: nat] :
( ( bfun @ nat @ A
@ ^ [N2: nat] : ( X7 @ ( plus_plus @ nat @ N2 @ K2 ) )
@ ( at_top @ nat ) )
=> ( bfun @ nat @ A @ X7 @ ( at_top @ nat ) ) ) ) ).
% Bseq_offset
thf(fact_5088_eventually__all__ge__at__top,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P2: A > $o] :
( ( eventually @ A @ P2 @ ( at_top @ A ) )
=> ( eventually @ A
@ ^ [X5: A] :
! [Y6: A] :
( ( ord_less_eq @ A @ X5 @ Y6 )
=> ( P2 @ Y6 ) )
@ ( at_top @ A ) ) ) ) ).
% eventually_all_ge_at_top
thf(fact_5089_finite__set__of__finite__funs,axiom,
! [B: $tType,A: $tType,A5: set @ A,B6: set @ B,D3: B] :
( ( finite_finite2 @ A @ A5 )
=> ( ( finite_finite2 @ B @ B6 )
=> ( finite_finite2 @ ( A > B )
@ ( collect @ ( A > B )
@ ^ [F4: A > B] :
! [X5: A] :
( ( ( member @ A @ X5 @ A5 )
=> ( member @ B @ ( F4 @ X5 ) @ B6 ) )
& ( ~ ( member @ A @ X5 @ A5 )
=> ( ( F4 @ X5 )
= D3 ) ) ) ) ) ) ) ).
% finite_set_of_finite_funs
thf(fact_5090_Bseq__cmult__iff,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [C3: A,F3: nat > A] :
( ( C3
!= ( zero_zero @ A ) )
=> ( ( bfun @ nat @ A
@ ^ [X5: nat] : ( times_times @ A @ C3 @ ( F3 @ X5 ) )
@ ( at_top @ nat ) )
= ( bfun @ nat @ A @ F3 @ ( at_top @ nat ) ) ) ) ) ).
% Bseq_cmult_iff
thf(fact_5091_BseqD,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X7: nat > A] :
( ( bfun @ nat @ A @ X7 @ ( at_top @ nat ) )
=> ? [K8: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K8 )
& ! [N5: nat] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( X7 @ N5 ) ) @ K8 ) ) ) ) ).
% BseqD
thf(fact_5092_BseqE,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X7: nat > A] :
( ( bfun @ nat @ A @ X7 @ ( at_top @ nat ) )
=> ~ ! [K8: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K8 )
=> ~ ! [N5: nat] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( X7 @ N5 ) ) @ K8 ) ) ) ) ).
% BseqE
thf(fact_5093_BseqI,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [K5: real,X7: nat > A] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K5 )
=> ( ! [N3: nat] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( X7 @ N3 ) ) @ K5 )
=> ( bfun @ nat @ A @ X7 @ ( at_top @ nat ) ) ) ) ) ).
% BseqI
thf(fact_5094_Bseq__def,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X7: nat > A] :
( ( bfun @ nat @ A @ X7 @ ( at_top @ nat ) )
= ( ? [K6: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K6 )
& ! [N2: nat] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( X7 @ N2 ) ) @ K6 ) ) ) ) ) ).
% Bseq_def
thf(fact_5095_Bseq__iff1a,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X7: nat > A] :
( ( bfun @ nat @ A @ X7 @ ( at_top @ nat ) )
= ( ? [N4: nat] :
! [N2: nat] : ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( X7 @ N2 ) ) @ ( semiring_1_of_nat @ real @ ( suc @ N4 ) ) ) ) ) ) ).
% Bseq_iff1a
thf(fact_5096_Bseq__iff2,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X7: nat > A] :
( ( bfun @ nat @ A @ X7 @ ( at_top @ nat ) )
= ( ? [K3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K3 )
& ? [X5: A] :
! [N2: nat] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ ( X7 @ N2 ) @ ( uminus_uminus @ A @ X5 ) ) ) @ K3 ) ) ) ) ) ).
% Bseq_iff2
thf(fact_5097_Bseq__iff3,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [X7: nat > A] :
( ( bfun @ nat @ A @ X7 @ ( at_top @ nat ) )
= ( ? [K3: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ K3 )
& ? [N4: nat] :
! [N2: nat] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( plus_plus @ A @ ( X7 @ N2 ) @ ( uminus_uminus @ A @ ( X7 @ N4 ) ) ) ) @ K3 ) ) ) ) ) ).
% Bseq_iff3
thf(fact_5098_Bfun__inverse,axiom,
! [A: $tType,B: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [F3: B > A,A2: A,F5: filter @ B] :
( ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ F5 )
=> ( ( A2
!= ( zero_zero @ A ) )
=> ( bfun @ B @ A
@ ^ [X5: B] : ( inverse_inverse @ A @ ( F3 @ X5 ) )
@ F5 ) ) ) ) ).
% Bfun_inverse
thf(fact_5099_BfunE,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ B )
=> ! [F3: A > B,F5: filter @ A] :
( ( bfun @ A @ B @ F3 @ F5 )
=> ~ ! [B9: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ B9 )
=> ~ ( eventually @ A
@ ^ [X5: A] : ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ B @ ( F3 @ X5 ) ) @ B9 )
@ F5 ) ) ) ) ).
% BfunE
thf(fact_5100_summable__bounded__partials,axiom,
! [A: $tType] :
( ( ( real_V8037385150606011577_space @ A )
& ( real_V822414075346904944vector @ A ) )
=> ! [F3: nat > A,G3: nat > real] :
( ( eventually @ nat
@ ^ [X02: nat] :
! [A6: nat] :
( ( ord_less_eq @ nat @ X02 @ A6 )
=> ! [B5: nat] :
( ( ord_less @ nat @ A6 @ B5 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set_or3652927894154168847AtMost @ nat @ A6 @ B5 ) ) ) @ ( G3 @ A6 ) ) ) )
@ ( at_top @ nat ) )
=> ( ( filterlim @ nat @ real @ G3 @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ ( at_top @ nat ) )
=> ( summable @ A @ F3 ) ) ) ) ).
% summable_bounded_partials
thf(fact_5101_Greatest__def,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( order_Greatest @ A )
= ( ^ [P4: A > $o] :
( the @ A
@ ^ [X5: A] :
( ( P4 @ X5 )
& ! [Y6: A] :
( ( P4 @ Y6 )
=> ( ord_less_eq @ A @ Y6 @ X5 ) ) ) ) ) ) ) ).
% Greatest_def
thf(fact_5102_cauchy__filter__metric,axiom,
! [A: $tType] :
( ( ( real_V768167426530841204y_dist @ A )
& ( topolo7287701948861334536_space @ A ) )
=> ( ( topolo6773858410816713723filter @ A )
= ( ^ [F8: filter @ A] :
! [E4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E4 )
=> ? [P4: A > $o] :
( ( eventually @ A @ P4 @ F8 )
& ! [X5: A,Y6: A] :
( ( ( P4 @ X5 )
& ( P4 @ Y6 ) )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X5 @ Y6 ) @ E4 ) ) ) ) ) ) ) ).
% cauchy_filter_metric
thf(fact_5103_finite__greaterThanAtMost,axiom,
! [L: nat,U: nat] : ( finite_finite2 @ nat @ ( set_or3652927894154168847AtMost @ nat @ L @ U ) ) ).
% finite_greaterThanAtMost
thf(fact_5104_greaterThanAtMost__iff,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [I2: A,L: A,U: A] :
( ( member @ A @ I2 @ ( set_or3652927894154168847AtMost @ A @ L @ U ) )
= ( ( ord_less @ A @ L @ I2 )
& ( ord_less_eq @ A @ I2 @ U ) ) ) ) ).
% greaterThanAtMost_iff
thf(fact_5105_greaterThanAtMost__empty,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,K2: A] :
( ( ord_less_eq @ A @ L @ K2 )
=> ( ( set_or3652927894154168847AtMost @ A @ K2 @ L )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% greaterThanAtMost_empty
thf(fact_5106_greaterThanAtMost__empty__iff2,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [K2: A,L: A] :
( ( ( bot_bot @ ( set @ A ) )
= ( set_or3652927894154168847AtMost @ A @ K2 @ L ) )
= ( ~ ( ord_less @ A @ K2 @ L ) ) ) ) ).
% greaterThanAtMost_empty_iff2
thf(fact_5107_greaterThanAtMost__empty__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [K2: A,L: A] :
( ( ( set_or3652927894154168847AtMost @ A @ K2 @ L )
= ( bot_bot @ ( set @ A ) ) )
= ( ~ ( ord_less @ A @ K2 @ L ) ) ) ) ).
% greaterThanAtMost_empty_iff
thf(fact_5108_infinite__Ioc__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A2: A,B2: A] :
( ( ~ ( finite_finite2 @ A @ ( set_or3652927894154168847AtMost @ A @ A2 @ B2 ) ) )
= ( ord_less @ A @ A2 @ B2 ) ) ) ).
% infinite_Ioc_iff
thf(fact_5109_image__add__greaterThanAtMost,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [C3: A,A2: A,B2: A] :
( ( image2 @ A @ A @ ( plus_plus @ A @ C3 ) @ ( set_or3652927894154168847AtMost @ A @ A2 @ B2 ) )
= ( set_or3652927894154168847AtMost @ A @ ( plus_plus @ A @ C3 @ A2 ) @ ( plus_plus @ A @ C3 @ B2 ) ) ) ) ).
% image_add_greaterThanAtMost
thf(fact_5110_cSup__greaterThanAtMost,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [Y: A,X: A] :
( ( ord_less @ A @ Y @ X )
=> ( ( complete_Sup_Sup @ A @ ( set_or3652927894154168847AtMost @ A @ Y @ X ) )
= X ) ) ) ).
% cSup_greaterThanAtMost
thf(fact_5111_Sup__greaterThanAtMost,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( ( complete_Sup_Sup @ A @ ( set_or3652927894154168847AtMost @ A @ X @ Y ) )
= Y ) ) ) ).
% Sup_greaterThanAtMost
thf(fact_5112_cInf__greaterThanAtMost,axiom,
! [A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( dense_linorder @ A ) )
=> ! [Y: A,X: A] :
( ( ord_less @ A @ Y @ X )
=> ( ( complete_Inf_Inf @ A @ ( set_or3652927894154168847AtMost @ A @ Y @ X ) )
= Y ) ) ) ).
% cInf_greaterThanAtMost
thf(fact_5113_Inf__greaterThanAtMost,axiom,
! [A: $tType] :
( ( ( comple6319245703460814977attice @ A )
& ( dense_linorder @ A ) )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( ( complete_Inf_Inf @ A @ ( set_or3652927894154168847AtMost @ A @ X @ Y ) )
= X ) ) ) ).
% Inf_greaterThanAtMost
thf(fact_5114_card__greaterThanAtMost,axiom,
! [L: nat,U: nat] :
( ( finite_card @ nat @ ( set_or3652927894154168847AtMost @ nat @ L @ U ) )
= ( minus_minus @ nat @ U @ L ) ) ).
% card_greaterThanAtMost
thf(fact_5115_image__minus__const__greaterThanAtMost,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [C3: A,A2: A,B2: A] :
( ( image2 @ A @ A @ ( minus_minus @ A @ C3 ) @ ( set_or3652927894154168847AtMost @ A @ A2 @ B2 ) )
= ( set_or7035219750837199246ssThan @ A @ ( minus_minus @ A @ C3 @ B2 ) @ ( minus_minus @ A @ C3 @ A2 ) ) ) ) ).
% image_minus_const_greaterThanAtMost
thf(fact_5116_image__diff__atLeastLessThan,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [C3: A,A2: A,B2: A] :
( ( image2 @ A @ A @ ( minus_minus @ A @ C3 ) @ ( set_or7035219750837199246ssThan @ A @ A2 @ B2 ) )
= ( set_or3652927894154168847AtMost @ A @ ( minus_minus @ A @ C3 @ B2 ) @ ( minus_minus @ A @ C3 @ A2 ) ) ) ) ).
% image_diff_atLeastLessThan
thf(fact_5117_image__uminus__atLeastLessThan,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [X: A,Y: A] :
( ( image2 @ A @ A @ ( uminus_uminus @ A ) @ ( set_or7035219750837199246ssThan @ A @ X @ Y ) )
= ( set_or3652927894154168847AtMost @ A @ ( uminus_uminus @ A @ Y ) @ ( uminus_uminus @ A @ X ) ) ) ) ).
% image_uminus_atLeastLessThan
thf(fact_5118_image__uminus__greaterThanAtMost,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [X: A,Y: A] :
( ( image2 @ A @ A @ ( uminus_uminus @ A ) @ ( set_or3652927894154168847AtMost @ A @ X @ Y ) )
= ( set_or7035219750837199246ssThan @ A @ ( uminus_uminus @ A @ Y ) @ ( uminus_uminus @ A @ X ) ) ) ) ).
% image_uminus_greaterThanAtMost
thf(fact_5119_Ioc__inj,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A,C3: A,D3: A] :
( ( ( set_or3652927894154168847AtMost @ A @ A2 @ B2 )
= ( set_or3652927894154168847AtMost @ A @ C3 @ D3 ) )
= ( ( ( ord_less_eq @ A @ B2 @ A2 )
& ( ord_less_eq @ A @ D3 @ C3 ) )
| ( ( A2 = C3 )
& ( B2 = D3 ) ) ) ) ) ).
% Ioc_inj
thf(fact_5120_atLeastSucAtMost__greaterThanAtMost,axiom,
! [L: nat,U: nat] :
( ( set_or1337092689740270186AtMost @ nat @ ( suc @ L ) @ U )
= ( set_or3652927894154168847AtMost @ nat @ L @ U ) ) ).
% atLeastSucAtMost_greaterThanAtMost
thf(fact_5121_Ioc__subset__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A,C3: A,D3: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ A2 @ B2 ) @ ( set_or3652927894154168847AtMost @ A @ C3 @ D3 ) )
= ( ( ord_less_eq @ A @ B2 @ A2 )
| ( ( ord_less_eq @ A @ C3 @ A2 )
& ( ord_less_eq @ A @ B2 @ D3 ) ) ) ) ) ).
% Ioc_subset_iff
thf(fact_5122_infinite__Ioc,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ~ ( finite_finite2 @ A @ ( set_or3652927894154168847AtMost @ A @ A2 @ B2 ) ) ) ) ).
% infinite_Ioc
thf(fact_5123_ivl__disj__int__two_I6_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,M2: A,U: A] :
( ( inf_inf @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ L @ M2 ) @ ( set_or3652927894154168847AtMost @ A @ M2 @ U ) )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% ivl_disj_int_two(6)
thf(fact_5124_GreatestI__ex__nat,axiom,
! [P2: nat > $o,B2: nat] :
( ? [X_1: nat] : ( P2 @ X_1 )
=> ( ! [Y3: nat] :
( ( P2 @ Y3 )
=> ( ord_less_eq @ nat @ Y3 @ B2 ) )
=> ( P2 @ ( order_Greatest @ nat @ P2 ) ) ) ) ).
% GreatestI_ex_nat
thf(fact_5125_Greatest__le__nat,axiom,
! [P2: nat > $o,K2: nat,B2: nat] :
( ( P2 @ K2 )
=> ( ! [Y3: nat] :
( ( P2 @ Y3 )
=> ( ord_less_eq @ nat @ Y3 @ B2 ) )
=> ( ord_less_eq @ nat @ K2 @ ( order_Greatest @ nat @ P2 ) ) ) ) ).
% Greatest_le_nat
thf(fact_5126_GreatestI__nat,axiom,
! [P2: nat > $o,K2: nat,B2: nat] :
( ( P2 @ K2 )
=> ( ! [Y3: nat] :
( ( P2 @ Y3 )
=> ( ord_less_eq @ nat @ Y3 @ B2 ) )
=> ( P2 @ ( order_Greatest @ nat @ P2 ) ) ) ) ).
% GreatestI_nat
thf(fact_5127_Ioc__disjoint,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A,C3: A,D3: A] :
( ( ( inf_inf @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ A2 @ B2 ) @ ( set_or3652927894154168847AtMost @ A @ C3 @ D3 ) )
= ( bot_bot @ ( set @ A ) ) )
= ( ( ord_less_eq @ A @ B2 @ A2 )
| ( ord_less_eq @ A @ D3 @ C3 )
| ( ord_less_eq @ A @ B2 @ C3 )
| ( ord_less_eq @ A @ D3 @ A2 ) ) ) ) ).
% Ioc_disjoint
thf(fact_5128_open__left,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [S2: set @ A,X: A,Y: A] :
( ( topolo1002775350975398744n_open @ A @ S2 )
=> ( ( member @ A @ X @ S2 )
=> ( ( ord_less @ A @ Y @ X )
=> ? [B4: A] :
( ( ord_less @ A @ B4 @ X )
& ( ord_less_eq @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ B4 @ X ) @ S2 ) ) ) ) ) ) ).
% open_left
thf(fact_5129_ivl__disj__int__two_I8_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,M2: A,U: A] :
( ( inf_inf @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ L @ M2 ) @ ( set_or3652927894154168847AtMost @ A @ M2 @ U ) )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% ivl_disj_int_two(8)
thf(fact_5130_ivl__disj__int__one_I3_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,U: A] :
( ( inf_inf @ ( set @ A ) @ ( set_ord_atMost @ A @ L ) @ ( set_or3652927894154168847AtMost @ A @ L @ U ) )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% ivl_disj_int_one(3)
thf(fact_5131_ivl__disj__int__two_I2_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,M2: A,U: A] :
( ( inf_inf @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ L @ M2 ) @ ( set_or5935395276787703475ssThan @ A @ M2 @ U ) )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% ivl_disj_int_two(2)
thf(fact_5132_Greatest__equality,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [P2: A > $o,X: A] :
( ( P2 @ X )
=> ( ! [Y3: A] :
( ( P2 @ Y3 )
=> ( ord_less_eq @ A @ Y3 @ X ) )
=> ( ( order_Greatest @ A @ P2 )
= X ) ) ) ) ).
% Greatest_equality
thf(fact_5133_GreatestI2__order,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [P2: A > $o,X: A,Q: A > $o] :
( ( P2 @ X )
=> ( ! [Y3: A] :
( ( P2 @ Y3 )
=> ( ord_less_eq @ A @ Y3 @ X ) )
=> ( ! [X4: A] :
( ( P2 @ X4 )
=> ( ! [Y4: A] :
( ( P2 @ Y4 )
=> ( ord_less_eq @ A @ Y4 @ X4 ) )
=> ( Q @ X4 ) ) )
=> ( Q @ ( order_Greatest @ A @ P2 ) ) ) ) ) ) ).
% GreatestI2_order
thf(fact_5134_sum_Ohead,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [M2: nat,N: nat,G3: nat > A] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
= ( plus_plus @ A @ ( G3 @ M2 ) @ ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or3652927894154168847AtMost @ nat @ M2 @ N ) ) ) ) ) ) ).
% sum.head
thf(fact_5135_prod_Ohead,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M2: nat,N: nat,G3: nat > A] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
= ( times_times @ A @ ( G3 @ M2 ) @ ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or3652927894154168847AtMost @ nat @ M2 @ N ) ) ) ) ) ) ).
% prod.head
thf(fact_5136_greaterThanAtMost__subseteq__atLeastAtMost__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A2: A,B2: A,C3: A,D3: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ A2 @ B2 ) @ ( set_or1337092689740270186AtMost @ A @ C3 @ D3 ) )
= ( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C3 @ A2 )
& ( ord_less_eq @ A @ B2 @ D3 ) ) ) ) ) ).
% greaterThanAtMost_subseteq_atLeastAtMost_iff
thf(fact_5137_greaterThanAtMost__subseteq__atLeastLessThan__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A2: A,B2: A,C3: A,D3: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ A2 @ B2 ) @ ( set_or7035219750837199246ssThan @ A @ C3 @ D3 ) )
= ( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C3 @ A2 )
& ( ord_less @ A @ B2 @ D3 ) ) ) ) ) ).
% greaterThanAtMost_subseteq_atLeastLessThan_iff
thf(fact_5138_greaterThanLessThan__subseteq__greaterThanAtMost__iff,axiom,
! [A: $tType] :
( ( dense_linorder @ A )
=> ! [A2: A,B2: A,C3: A,D3: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ A2 @ B2 ) @ ( set_or3652927894154168847AtMost @ A @ C3 @ D3 ) )
= ( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ C3 @ A2 )
& ( ord_less_eq @ A @ B2 @ D3 ) ) ) ) ) ).
% greaterThanLessThan_subseteq_greaterThanAtMost_iff
thf(fact_5139_greaterThanAtMost__eq__atLeastAtMost__diff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( set_or3652927894154168847AtMost @ A )
= ( ^ [A6: A,B5: A] : ( minus_minus @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A6 @ B5 ) @ ( insert @ A @ A6 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% greaterThanAtMost_eq_atLeastAtMost_diff
thf(fact_5140_nth__sorted__list__of__set__greaterThanAtMost,axiom,
! [N: nat,J: nat,I2: nat] :
( ( ord_less @ nat @ N @ ( minus_minus @ nat @ J @ I2 ) )
=> ( ( nth @ nat @ ( linord4507533701916653071of_set @ nat @ ( set_or3652927894154168847AtMost @ nat @ I2 @ J ) ) @ N )
= ( suc @ ( plus_plus @ nat @ I2 @ N ) ) ) ) ).
% nth_sorted_list_of_set_greaterThanAtMost
thf(fact_5141_same__fst__def,axiom,
! [B: $tType,A: $tType] :
( ( same_fst @ A @ B )
= ( ^ [P4: A > $o,R6: A > ( set @ ( product_prod @ B @ B ) )] :
( collect @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) )
@ ( product_case_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ $o
@ ( product_case_prod @ A @ B @ ( ( product_prod @ A @ B ) > $o )
@ ^ [X10: A,Y7: B] :
( product_case_prod @ A @ B @ $o
@ ^ [X5: A,Y6: B] :
( ( X10 = X5 )
& ( P4 @ X5 )
& ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ Y7 @ Y6 ) @ ( R6 @ X5 ) ) ) ) ) ) ) ) ) ).
% same_fst_def
thf(fact_5142_ord_OLeast__def,axiom,
! [A: $tType] :
( ( least @ A )
= ( ^ [Less_eq2: A > A > $o,P4: A > $o] :
( the @ A
@ ^ [X5: A] :
( ( P4 @ X5 )
& ! [Y6: A] :
( ( P4 @ Y6 )
=> ( Less_eq2 @ X5 @ Y6 ) ) ) ) ) ) ).
% ord.Least_def
thf(fact_5143_eventually__filtercomap__at__topological,axiom,
! [A: $tType,B: $tType] :
( ( topolo4958980785337419405_space @ B )
=> ! [P2: A > $o,F3: A > B,A5: B,B6: set @ B] :
( ( eventually @ A @ P2 @ ( filtercomap @ A @ B @ F3 @ ( topolo174197925503356063within @ B @ A5 @ B6 ) ) )
= ( ? [S7: set @ B] :
( ( topolo1002775350975398744n_open @ B @ S7 )
& ( member @ B @ A5 @ S7 )
& ! [X5: A] :
( ( member @ B @ ( F3 @ X5 ) @ ( minus_minus @ ( set @ B ) @ ( inf_inf @ ( set @ B ) @ S7 @ B6 ) @ ( insert @ B @ A5 @ ( bot_bot @ ( set @ B ) ) ) ) )
=> ( P2 @ X5 ) ) ) ) ) ) ).
% eventually_filtercomap_at_topological
thf(fact_5144_finite__greaterThanAtMost__int,axiom,
! [L: int,U: int] : ( finite_finite2 @ int @ ( set_or3652927894154168847AtMost @ int @ L @ U ) ) ).
% finite_greaterThanAtMost_int
thf(fact_5145_filtercomap__bot,axiom,
! [B: $tType,A: $tType,F3: A > B] :
( ( filtercomap @ A @ B @ F3 @ ( bot_bot @ ( filter @ B ) ) )
= ( bot_bot @ ( filter @ A ) ) ) ).
% filtercomap_bot
thf(fact_5146_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_5147_same__fstI,axiom,
! [B: $tType,A: $tType,P2: A > $o,X: A,Y9: B,Y: B,R: A > ( set @ ( product_prod @ B @ B ) )] :
( ( P2 @ X )
=> ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ Y9 @ Y ) @ ( R @ X ) )
=> ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y9 ) @ ( product_Pair @ A @ B @ X @ Y ) ) @ ( same_fst @ A @ B @ P2 @ R ) ) ) ) ).
% same_fstI
thf(fact_5148_ord_OLeast_Ocong,axiom,
! [A: $tType] :
( ( least @ A )
= ( least @ A ) ) ).
% ord.Least.cong
thf(fact_5149_filtercomap__neq__bot,axiom,
! [A: $tType,B: $tType,F5: filter @ A,F3: B > A] :
( ! [P8: A > $o] :
( ( eventually @ A @ P8 @ F5 )
=> ? [X3: B] : ( P8 @ ( F3 @ X3 ) ) )
=> ( ( filtercomap @ B @ A @ F3 @ F5 )
!= ( bot_bot @ ( filter @ B ) ) ) ) ).
% filtercomap_neq_bot
thf(fact_5150_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_5151_eventually__filtercomap__at__top__linorder,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ A )
=> ! [P2: B > $o,F3: B > A] :
( ( eventually @ B @ P2 @ ( filtercomap @ B @ A @ F3 @ ( at_top @ A ) ) )
= ( ? [N4: A] :
! [X5: B] :
( ( ord_less_eq @ A @ N4 @ ( F3 @ X5 ) )
=> ( P2 @ X5 ) ) ) ) ) ).
% eventually_filtercomap_at_top_linorder
thf(fact_5152_eventually__filtercomap__at__top__dense,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( no_top @ A ) )
=> ! [P2: B > $o,F3: B > A] :
( ( eventually @ B @ P2 @ ( filtercomap @ B @ A @ F3 @ ( at_top @ A ) ) )
= ( ? [N4: A] :
! [X5: B] :
( ( ord_less @ A @ N4 @ ( F3 @ X5 ) )
=> ( P2 @ X5 ) ) ) ) ) ).
% eventually_filtercomap_at_top_dense
thf(fact_5153_filtercomap__neq__bot__surj,axiom,
! [A: $tType,B: $tType,F5: filter @ A,F3: B > A] :
( ( F5
!= ( bot_bot @ ( filter @ A ) ) )
=> ( ( ( image2 @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
=> ( ( filtercomap @ B @ A @ F3 @ F5 )
!= ( bot_bot @ ( filter @ B ) ) ) ) ) ).
% filtercomap_neq_bot_surj
thf(fact_5154_eventually__filtercomap__at__bot__linorder,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ A )
=> ! [P2: B > $o,F3: B > A] :
( ( eventually @ B @ P2 @ ( filtercomap @ B @ A @ F3 @ ( at_bot @ A ) ) )
= ( ? [N4: A] :
! [X5: B] :
( ( ord_less_eq @ A @ ( F3 @ X5 ) @ N4 )
=> ( P2 @ X5 ) ) ) ) ) ).
% eventually_filtercomap_at_bot_linorder
thf(fact_5155_eventually__filtercomap__at__bot__dense,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( no_bot @ A ) )
=> ! [P2: B > $o,F3: B > A] :
( ( eventually @ B @ P2 @ ( filtercomap @ B @ A @ F3 @ ( at_bot @ A ) ) )
= ( ? [N4: A] :
! [X5: B] :
( ( ord_less @ A @ ( F3 @ X5 ) @ N4 )
=> ( P2 @ X5 ) ) ) ) ) ).
% eventually_filtercomap_at_bot_dense
thf(fact_5156_dual__Min,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( lattices_Min @ A
@ ^ [X5: A,Y6: A] : ( ord_less_eq @ A @ Y6 @ X5 ) )
= ( lattic643756798349783984er_Max @ A ) ) ) ).
% dual_Min
thf(fact_5157_at__within__eq,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ( ( topolo174197925503356063within @ A )
= ( ^ [X5: A,S8: set @ A] :
( complete_Inf_Inf @ ( filter @ A )
@ ( image2 @ ( set @ A ) @ ( filter @ A )
@ ^ [S7: set @ A] : ( principal @ A @ ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ S7 @ S8 ) @ ( insert @ A @ X5 @ ( bot_bot @ ( set @ A ) ) ) ) )
@ ( collect @ ( set @ A )
@ ^ [S7: set @ A] :
( ( topolo1002775350975398744n_open @ A @ S7 )
& ( member @ A @ X5 @ S7 ) ) ) ) ) ) ) ) ).
% at_within_eq
thf(fact_5158_isCont__powser,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [C3: nat > A,K5: A,X: A] :
( ( summable @ A
@ ^ [N2: nat] : ( times_times @ A @ ( C3 @ N2 ) @ ( power_power @ A @ K5 @ N2 ) ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ X ) @ ( real_V7770717601297561774m_norm @ A @ K5 ) )
=> ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) )
@ ^ [X5: A] :
( suminf @ A
@ ^ [N2: nat] : ( times_times @ A @ ( C3 @ N2 ) @ ( power_power @ A @ X5 @ N2 ) ) ) ) ) ) ) ).
% isCont_powser
thf(fact_5159_continuous__bot,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [F3: A > B] : ( topolo3448309680560233919inuous @ A @ B @ ( bot_bot @ ( filter @ A ) ) @ F3 ) ) ).
% continuous_bot
thf(fact_5160_continuous__trivial__limit,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [Net: filter @ A,F3: A > B] :
( ( Net
= ( bot_bot @ ( filter @ A ) ) )
=> ( topolo3448309680560233919inuous @ A @ B @ Net @ F3 ) ) ) ).
% continuous_trivial_limit
thf(fact_5161_continuous__Pair,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( topolo4958980785337419405_space @ B )
& ( topolo4958980785337419405_space @ C ) )
=> ! [F5: filter @ A,F3: A > B,G3: A > C] :
( ( topolo3448309680560233919inuous @ A @ B @ F5 @ F3 )
=> ( ( topolo3448309680560233919inuous @ A @ C @ F5 @ G3 )
=> ( topolo3448309680560233919inuous @ A @ ( product_prod @ B @ C ) @ F5
@ ^ [X5: A] : ( product_Pair @ B @ C @ ( F3 @ X5 ) @ ( G3 @ X5 ) ) ) ) ) ) ).
% continuous_Pair
thf(fact_5162_isCont__Pair,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( topolo4958980785337419405_space @ B )
& ( topolo4958980785337419405_space @ C ) )
=> ! [A2: A,F3: A > B,G3: A > C] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) @ F3 )
=> ( ( topolo3448309680560233919inuous @ A @ C @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) @ G3 )
=> ( topolo3448309680560233919inuous @ A @ ( product_prod @ B @ C ) @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) )
@ ^ [X5: A] : ( product_Pair @ B @ C @ ( F3 @ X5 ) @ ( G3 @ X5 ) ) ) ) ) ) ).
% isCont_Pair
thf(fact_5163_continuous__max,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( topolo1944317154257567458pology @ B ) )
=> ! [F5: filter @ A,F3: A > B,G3: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ F5 @ F3 )
=> ( ( topolo3448309680560233919inuous @ A @ B @ F5 @ G3 )
=> ( topolo3448309680560233919inuous @ A @ B @ F5
@ ^ [X5: A] : ( ord_max @ B @ ( F3 @ X5 ) @ ( G3 @ X5 ) ) ) ) ) ) ).
% continuous_max
thf(fact_5164_linorder_OMin_Ocong,axiom,
! [A: $tType] :
( ( lattices_Min @ A )
= ( lattices_Min @ A ) ) ).
% linorder.Min.cong
thf(fact_5165_continuous__add,axiom,
! [B: $tType,D: $tType] :
( ( ( topological_t2_space @ D )
& ( topolo6943815403480290642id_add @ B ) )
=> ! [F5: filter @ D,F3: D > B,G3: D > B] :
( ( topolo3448309680560233919inuous @ D @ B @ F5 @ F3 )
=> ( ( topolo3448309680560233919inuous @ D @ B @ F5 @ G3 )
=> ( topolo3448309680560233919inuous @ D @ B @ F5
@ ^ [X5: D] : ( plus_plus @ B @ ( F3 @ X5 ) @ ( G3 @ X5 ) ) ) ) ) ) ).
% continuous_add
thf(fact_5166_principal__eq__bot__iff,axiom,
! [A: $tType,X7: set @ A] :
( ( ( principal @ A @ X7 )
= ( bot_bot @ ( filter @ A ) ) )
= ( X7
= ( bot_bot @ ( set @ A ) ) ) ) ).
% principal_eq_bot_iff
thf(fact_5167_bot__eq__principal__empty,axiom,
! [A: $tType] :
( ( bot_bot @ ( filter @ A ) )
= ( principal @ A @ ( bot_bot @ ( set @ A ) ) ) ) ).
% bot_eq_principal_empty
thf(fact_5168_IVT,axiom,
! [A: $tType,B: $tType] :
( ( ( topolo1944317154257567458pology @ B )
& ( topolo8458572112393995274pology @ A ) )
=> ! [F3: A > B,A2: A,Y: B,B2: A] :
( ( ord_less_eq @ B @ ( F3 @ A2 ) @ Y )
=> ( ( ord_less_eq @ B @ Y @ ( F3 @ B2 ) )
=> ( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ! [X4: A] :
( ( ( ord_less_eq @ A @ A2 @ X4 )
& ( ord_less_eq @ A @ X4 @ B2 ) )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ X4 @ ( top_top @ ( set @ A ) ) ) @ F3 ) )
=> ? [X4: A] :
( ( ord_less_eq @ A @ A2 @ X4 )
& ( ord_less_eq @ A @ X4 @ B2 )
& ( ( F3 @ X4 )
= Y ) ) ) ) ) ) ) ).
% IVT
thf(fact_5169_IVT2,axiom,
! [A: $tType,B: $tType] :
( ( ( topolo1944317154257567458pology @ B )
& ( topolo8458572112393995274pology @ A ) )
=> ! [F3: A > B,B2: A,Y: B,A2: A] :
( ( ord_less_eq @ B @ ( F3 @ B2 ) @ Y )
=> ( ( ord_less_eq @ B @ Y @ ( F3 @ A2 ) )
=> ( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ! [X4: A] :
( ( ( ord_less_eq @ A @ A2 @ X4 )
& ( ord_less_eq @ A @ X4 @ B2 ) )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ X4 @ ( top_top @ ( set @ A ) ) ) @ F3 ) )
=> ? [X4: A] :
( ( ord_less_eq @ A @ A2 @ X4 )
& ( ord_less_eq @ A @ X4 @ B2 )
& ( ( F3 @ X4 )
= Y ) ) ) ) ) ) ) ).
% IVT2
thf(fact_5170_continuous__at__within__divide,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_V3459762299906320749_field @ B ) )
=> ! [A2: A,S: set @ A,F3: A > B,G3: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ S ) @ F3 )
=> ( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ S ) @ G3 )
=> ( ( ( G3 @ A2 )
!= ( zero_zero @ B ) )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ S )
@ ^ [X5: A] : ( divide_divide @ B @ ( F3 @ X5 ) @ ( G3 @ X5 ) ) ) ) ) ) ) ).
% continuous_at_within_divide
thf(fact_5171_isCont__add,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( topolo6943815403480290642id_add @ B ) )
=> ! [A2: A,F3: A > B,G3: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) @ F3 )
=> ( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) @ G3 )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) )
@ ^ [X5: A] : ( plus_plus @ B @ ( F3 @ X5 ) @ ( G3 @ X5 ) ) ) ) ) ) ).
% isCont_add
thf(fact_5172_continuous__at__within__inverse,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_V8999393235501362500lgebra @ B ) )
=> ! [A2: A,S: set @ A,F3: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ S ) @ F3 )
=> ( ( ( F3 @ A2 )
!= ( zero_zero @ B ) )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ S )
@ ^ [X5: A] : ( inverse_inverse @ B @ ( F3 @ X5 ) ) ) ) ) ) ).
% continuous_at_within_inverse
thf(fact_5173_nhds__discrete,axiom,
! [A: $tType] :
( ( topolo8865339358273720382pology @ A )
=> ( ( topolo7230453075368039082e_nhds @ A )
= ( ^ [X5: A] : ( principal @ A @ ( insert @ A @ X5 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% nhds_discrete
thf(fact_5174_continuous__at__within__sgn,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [A2: A,S: set @ A,F3: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ S ) @ F3 )
=> ( ( ( F3 @ A2 )
!= ( zero_zero @ B ) )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ S )
@ ^ [X5: A] : ( sgn_sgn @ B @ ( F3 @ X5 ) ) ) ) ) ) ).
% continuous_at_within_sgn
thf(fact_5175_continuous__divide,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_V3459762299906320749_field @ B ) )
=> ! [F5: filter @ A,F3: A > B,G3: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ F5 @ F3 )
=> ( ( topolo3448309680560233919inuous @ A @ B @ F5 @ G3 )
=> ( ( ( G3
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F5
@ ^ [X5: A] : X5 ) )
!= ( zero_zero @ B ) )
=> ( topolo3448309680560233919inuous @ A @ B @ F5
@ ^ [X5: A] : ( divide_divide @ B @ ( F3 @ X5 ) @ ( G3 @ X5 ) ) ) ) ) ) ) ).
% continuous_divide
thf(fact_5176_continuous__inverse,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_V8999393235501362500lgebra @ B ) )
=> ! [F5: filter @ A,F3: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ F5 @ F3 )
=> ( ( ( F3
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F5
@ ^ [X5: A] : X5 ) )
!= ( zero_zero @ B ) )
=> ( topolo3448309680560233919inuous @ A @ B @ F5
@ ^ [X5: A] : ( inverse_inverse @ B @ ( F3 @ X5 ) ) ) ) ) ) ).
% continuous_inverse
thf(fact_5177_continuous__sgn,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F5: filter @ A,F3: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ F5 @ F3 )
=> ( ( ( F3
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F5
@ ^ [X5: A] : X5 ) )
!= ( zero_zero @ B ) )
=> ( topolo3448309680560233919inuous @ A @ B @ F5
@ ^ [X5: A] : ( sgn_sgn @ B @ ( F3 @ X5 ) ) ) ) ) ) ).
% continuous_sgn
thf(fact_5178_isCont__eq__Lb,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [A2: real,B2: real,F3: real > A] :
( ( ord_less_eq @ real @ A2 @ B2 )
=> ( ! [X4: real] :
( ( ( ord_less_eq @ real @ A2 @ X4 )
& ( ord_less_eq @ real @ X4 @ B2 ) )
=> ( topolo3448309680560233919inuous @ real @ A @ ( topolo174197925503356063within @ real @ X4 @ ( top_top @ ( set @ real ) ) ) @ F3 ) )
=> ? [M8: A] :
( ! [X3: real] :
( ( ( ord_less_eq @ real @ A2 @ X3 )
& ( ord_less_eq @ real @ X3 @ B2 ) )
=> ( ord_less_eq @ A @ M8 @ ( F3 @ X3 ) ) )
& ? [X4: real] :
( ( ord_less_eq @ real @ A2 @ X4 )
& ( ord_less_eq @ real @ X4 @ B2 )
& ( ( F3 @ X4 )
= M8 ) ) ) ) ) ) ).
% isCont_eq_Lb
thf(fact_5179_isCont__eq__Ub,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [A2: real,B2: real,F3: real > A] :
( ( ord_less_eq @ real @ A2 @ B2 )
=> ( ! [X4: real] :
( ( ( ord_less_eq @ real @ A2 @ X4 )
& ( ord_less_eq @ real @ X4 @ B2 ) )
=> ( topolo3448309680560233919inuous @ real @ A @ ( topolo174197925503356063within @ real @ X4 @ ( top_top @ ( set @ real ) ) ) @ F3 ) )
=> ? [M8: A] :
( ! [X3: real] :
( ( ( ord_less_eq @ real @ A2 @ X3 )
& ( ord_less_eq @ real @ X3 @ B2 ) )
=> ( ord_less_eq @ A @ ( F3 @ X3 ) @ M8 ) )
& ? [X4: real] :
( ( ord_less_eq @ real @ A2 @ X4 )
& ( ord_less_eq @ real @ X4 @ B2 )
& ( ( F3 @ X4 )
= M8 ) ) ) ) ) ) ).
% isCont_eq_Ub
thf(fact_5180_isCont__bounded,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [A2: real,B2: real,F3: real > A] :
( ( ord_less_eq @ real @ A2 @ B2 )
=> ( ! [X4: real] :
( ( ( ord_less_eq @ real @ A2 @ X4 )
& ( ord_less_eq @ real @ X4 @ B2 ) )
=> ( topolo3448309680560233919inuous @ real @ A @ ( topolo174197925503356063within @ real @ X4 @ ( top_top @ ( set @ real ) ) ) @ F3 ) )
=> ? [M8: A] :
! [X3: real] :
( ( ( ord_less_eq @ real @ A2 @ X3 )
& ( ord_less_eq @ real @ X3 @ B2 ) )
=> ( ord_less_eq @ A @ ( F3 @ X3 ) @ M8 ) ) ) ) ) ).
% isCont_bounded
thf(fact_5181_isCont__inverse__function2,axiom,
! [A2: real,X: real,B2: real,G3: real > real,F3: real > real] :
( ( ord_less @ real @ A2 @ X )
=> ( ( ord_less @ real @ X @ B2 )
=> ( ! [Z3: real] :
( ( ord_less_eq @ real @ A2 @ Z3 )
=> ( ( ord_less_eq @ real @ Z3 @ B2 )
=> ( ( G3 @ ( F3 @ Z3 ) )
= Z3 ) ) )
=> ( ! [Z3: real] :
( ( ord_less_eq @ real @ A2 @ Z3 )
=> ( ( ord_less_eq @ real @ Z3 @ B2 )
=> ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ Z3 @ ( top_top @ ( set @ real ) ) ) @ F3 ) ) )
=> ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ ( F3 @ X ) @ ( top_top @ ( set @ real ) ) ) @ G3 ) ) ) ) ) ).
% isCont_inverse_function2
thf(fact_5182_isCont__divide,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_V3459762299906320749_field @ B ) )
=> ! [A2: A,F3: A > B,G3: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) @ F3 )
=> ( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) @ G3 )
=> ( ( ( G3 @ A2 )
!= ( zero_zero @ B ) )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) )
@ ^ [X5: A] : ( divide_divide @ B @ ( F3 @ X5 ) @ ( G3 @ X5 ) ) ) ) ) ) ) ).
% isCont_divide
thf(fact_5183_isCont__sgn,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [A2: A,F3: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) @ F3 )
=> ( ( ( F3 @ A2 )
!= ( zero_zero @ B ) )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) )
@ ^ [X5: A] : ( sgn_sgn @ B @ ( F3 @ X5 ) ) ) ) ) ) ).
% isCont_sgn
thf(fact_5184_tendsto__principal__singleton,axiom,
! [A: $tType,B: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [F3: B > A,X: B] : ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ ( F3 @ X ) ) @ ( principal @ B @ ( insert @ B @ X @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ).
% tendsto_principal_singleton
thf(fact_5185_continuous__within__tan,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A,S: set @ A,F3: A > A] :
( ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ X @ S ) @ F3 )
=> ( ( ( cos @ A @ ( F3 @ X ) )
!= ( zero_zero @ A ) )
=> ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ X @ S )
@ ^ [X5: A] : ( tan @ A @ ( F3 @ X5 ) ) ) ) ) ) ).
% continuous_within_tan
thf(fact_5186_nhds__discrete__open,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [X: A] :
( ( topolo1002775350975398744n_open @ A @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
=> ( ( topolo7230453075368039082e_nhds @ A @ X )
= ( principal @ A @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% nhds_discrete_open
thf(fact_5187_continuous__within__cot,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: A,S: set @ A,F3: A > A] :
( ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ X @ S ) @ F3 )
=> ( ( ( sin @ A @ ( F3 @ X ) )
!= ( zero_zero @ A ) )
=> ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ X @ S )
@ ^ [X5: A] : ( cot @ A @ ( F3 @ X5 ) ) ) ) ) ) ).
% continuous_within_cot
thf(fact_5188_continuous__at__within__tanh,axiom,
! [A: $tType,C: $tType] :
( ( ( topological_t2_space @ C )
& ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [X: C,A5: set @ C,F3: C > A] :
( ( topolo3448309680560233919inuous @ C @ A @ ( topolo174197925503356063within @ C @ X @ A5 ) @ F3 )
=> ( ( ( cosh @ A @ ( F3 @ X ) )
!= ( zero_zero @ A ) )
=> ( topolo3448309680560233919inuous @ C @ A @ ( topolo174197925503356063within @ C @ X @ A5 )
@ ^ [X5: C] : ( tanh @ A @ ( F3 @ X5 ) ) ) ) ) ) ).
% continuous_at_within_tanh
thf(fact_5189_continuous__tan,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [F5: filter @ A,F3: A > A] :
( ( topolo3448309680560233919inuous @ A @ A @ F5 @ F3 )
=> ( ( ( cos @ A
@ ( F3
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F5
@ ^ [X5: A] : X5 ) ) )
!= ( zero_zero @ A ) )
=> ( topolo3448309680560233919inuous @ A @ A @ F5
@ ^ [X5: A] : ( tan @ A @ ( F3 @ X5 ) ) ) ) ) ) ).
% continuous_tan
thf(fact_5190_continuous__cot,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [F5: filter @ A,F3: A > A] :
( ( topolo3448309680560233919inuous @ A @ A @ F5 @ F3 )
=> ( ( ( sin @ A
@ ( F3
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F5
@ ^ [X5: A] : X5 ) ) )
!= ( zero_zero @ A ) )
=> ( topolo3448309680560233919inuous @ A @ A @ F5
@ ^ [X5: A] : ( cot @ A @ ( F3 @ X5 ) ) ) ) ) ) ).
% continuous_cot
thf(fact_5191_continuous__tanh,axiom,
! [A: $tType,C: $tType] :
( ( ( topological_t2_space @ C )
& ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [F5: filter @ C,F3: C > A] :
( ( topolo3448309680560233919inuous @ C @ A @ F5 @ F3 )
=> ( ( ( cosh @ A
@ ( F3
@ ( topolo3827282254853284352ce_Lim @ C @ C @ F5
@ ^ [X5: C] : X5 ) ) )
!= ( zero_zero @ A ) )
=> ( topolo3448309680560233919inuous @ C @ A @ F5
@ ^ [X5: C] : ( tanh @ A @ ( F3 @ X5 ) ) ) ) ) ) ).
% continuous_tanh
thf(fact_5192_isCont__has__Ub,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [A2: real,B2: real,F3: real > A] :
( ( ord_less_eq @ real @ A2 @ B2 )
=> ( ! [X4: real] :
( ( ( ord_less_eq @ real @ A2 @ X4 )
& ( ord_less_eq @ real @ X4 @ B2 ) )
=> ( topolo3448309680560233919inuous @ real @ A @ ( topolo174197925503356063within @ real @ X4 @ ( top_top @ ( set @ real ) ) ) @ F3 ) )
=> ? [M8: A] :
( ! [X3: real] :
( ( ( ord_less_eq @ real @ A2 @ X3 )
& ( ord_less_eq @ real @ X3 @ B2 ) )
=> ( ord_less_eq @ A @ ( F3 @ X3 ) @ M8 ) )
& ! [N8: A] :
( ( ord_less @ A @ N8 @ M8 )
=> ? [X4: real] :
( ( ord_less_eq @ real @ A2 @ X4 )
& ( ord_less_eq @ real @ X4 @ B2 )
& ( ord_less @ A @ N8 @ ( F3 @ X4 ) ) ) ) ) ) ) ) ).
% isCont_has_Ub
thf(fact_5193_continuous__arcosh,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [F5: filter @ A,F3: A > real] :
( ( topolo3448309680560233919inuous @ A @ real @ F5 @ F3 )
=> ( ( ord_less @ real @ ( one_one @ real )
@ ( F3
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F5
@ ^ [X5: A] : X5 ) ) )
=> ( topolo3448309680560233919inuous @ A @ real @ F5
@ ^ [X5: A] : ( arcosh @ real @ ( F3 @ X5 ) ) ) ) ) ) ).
% continuous_arcosh
thf(fact_5194_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_5195_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_5196_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_5197_isCont__tan_H,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [A2: A,F3: A > A] :
( ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) @ F3 )
=> ( ( ( cos @ A @ ( F3 @ A2 ) )
!= ( zero_zero @ A ) )
=> ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) )
@ ^ [X5: A] : ( tan @ A @ ( F3 @ X5 ) ) ) ) ) ) ).
% isCont_tan'
thf(fact_5198_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_5199_continuous__at__within__log,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [A2: A,S: set @ A,F3: A > real,G3: A > real] :
( ( topolo3448309680560233919inuous @ A @ real @ ( topolo174197925503356063within @ A @ A2 @ S ) @ F3 )
=> ( ( topolo3448309680560233919inuous @ A @ real @ ( topolo174197925503356063within @ A @ A2 @ S ) @ G3 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( F3 @ A2 ) )
=> ( ( ( F3 @ A2 )
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( G3 @ A2 ) )
=> ( topolo3448309680560233919inuous @ A @ real @ ( topolo174197925503356063within @ A @ A2 @ S )
@ ^ [X5: A] : ( log @ ( F3 @ X5 ) @ ( G3 @ X5 ) ) ) ) ) ) ) ) ) ).
% continuous_at_within_log
thf(fact_5200_isCont__cot_H,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [A2: A,F3: A > A] :
( ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) @ F3 )
=> ( ( ( sin @ A @ ( F3 @ A2 ) )
!= ( zero_zero @ A ) )
=> ( topolo3448309680560233919inuous @ A @ A @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) )
@ ^ [X5: A] : ( cot @ A @ ( F3 @ X5 ) ) ) ) ) ) ).
% isCont_cot'
thf(fact_5201_continuous__log,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [F5: filter @ A,F3: A > real,G3: A > real] :
( ( topolo3448309680560233919inuous @ A @ real @ F5 @ F3 )
=> ( ( topolo3448309680560233919inuous @ A @ real @ F5 @ G3 )
=> ( ( ord_less @ real @ ( zero_zero @ real )
@ ( F3
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F5
@ ^ [X5: A] : X5 ) ) )
=> ( ( ( F3
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F5
@ ^ [X5: A] : X5 ) )
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real )
@ ( G3
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F5
@ ^ [X5: A] : X5 ) ) )
=> ( topolo3448309680560233919inuous @ A @ real @ F5
@ ^ [X5: A] : ( log @ ( F3 @ X5 ) @ ( G3 @ X5 ) ) ) ) ) ) ) ) ) ).
% continuous_log
thf(fact_5202_isCont__iff,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [X: A,F3: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ X @ ( top_top @ ( set @ A ) ) ) @ F3 )
= ( filterlim @ A @ B
@ ^ [H: A] : ( F3 @ ( plus_plus @ A @ X @ H ) )
@ ( topolo7230453075368039082e_nhds @ B @ ( F3 @ X ) )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% isCont_iff
thf(fact_5203_filterlim__base__iff,axiom,
! [A: $tType,C: $tType,B: $tType,D: $tType,I6: set @ A,F5: A > ( set @ B ),F3: B > C,G6: D > ( set @ C ),J4: set @ D] :
( ( I6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [I3: A] :
( ( member @ A @ I3 @ I6 )
=> ! [J2: A] :
( ( member @ A @ J2 @ I6 )
=> ( ( ord_less_eq @ ( set @ B ) @ ( F5 @ I3 ) @ ( F5 @ J2 ) )
| ( ord_less_eq @ ( set @ B ) @ ( F5 @ J2 ) @ ( F5 @ I3 ) ) ) ) )
=> ( ( filterlim @ B @ C @ F3
@ ( complete_Inf_Inf @ ( filter @ C )
@ ( image2 @ D @ ( filter @ C )
@ ^ [J3: D] : ( principal @ C @ ( G6 @ J3 ) )
@ J4 ) )
@ ( complete_Inf_Inf @ ( filter @ B )
@ ( image2 @ A @ ( filter @ B )
@ ^ [I4: A] : ( principal @ B @ ( F5 @ I4 ) )
@ I6 ) ) )
= ( ! [X5: D] :
( ( member @ D @ X5 @ J4 )
=> ? [Y6: A] :
( ( member @ A @ Y6 @ I6 )
& ! [Z5: B] :
( ( member @ B @ Z5 @ ( F5 @ Y6 ) )
=> ( member @ C @ ( F3 @ Z5 ) @ ( G6 @ X5 ) ) ) ) ) ) ) ) ) ).
% filterlim_base_iff
thf(fact_5204_DERIV__inverse__function,axiom,
! [F3: real > real,D6: real,G3: real > real,X: real,A2: real,B2: real] :
( ( has_field_derivative @ real @ F3 @ D6 @ ( topolo174197925503356063within @ real @ ( G3 @ X ) @ ( top_top @ ( set @ real ) ) ) )
=> ( ( D6
!= ( zero_zero @ real ) )
=> ( ( ord_less @ real @ A2 @ X )
=> ( ( ord_less @ real @ X @ B2 )
=> ( ! [Y3: real] :
( ( ord_less @ real @ A2 @ Y3 )
=> ( ( ord_less @ real @ Y3 @ B2 )
=> ( ( F3 @ ( G3 @ Y3 ) )
= Y3 ) ) )
=> ( ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) @ G3 )
=> ( has_field_derivative @ real @ G3 @ ( inverse_inverse @ real @ D6 ) @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ) ) ) ).
% DERIV_inverse_function
thf(fact_5205_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_5206_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_5207_INF__principal__finite,axiom,
! [B: $tType,A: $tType,X7: set @ A,F3: A > ( set @ B )] :
( ( finite_finite2 @ A @ X7 )
=> ( ( complete_Inf_Inf @ ( filter @ B )
@ ( image2 @ A @ ( filter @ B )
@ ^ [X5: A] : ( principal @ B @ ( F3 @ X5 ) )
@ X7 ) )
= ( principal @ B @ ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ F3 @ X7 ) ) ) ) ) ).
% INF_principal_finite
thf(fact_5208_LIM__less__bound,axiom,
! [B2: real,X: real,F3: real > real] :
( ( ord_less @ real @ B2 @ X )
=> ( ! [X4: real] :
( ( member @ real @ X4 @ ( set_or5935395276787703475ssThan @ real @ B2 @ X ) )
=> ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( F3 @ X4 ) ) )
=> ( ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ X @ ( top_top @ ( set @ real ) ) ) @ F3 )
=> ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( F3 @ X ) ) ) ) ) ).
% LIM_less_bound
thf(fact_5209_isCont__log,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ! [A2: A,F3: A > real,G3: A > real] :
( ( topolo3448309680560233919inuous @ A @ real @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) @ F3 )
=> ( ( topolo3448309680560233919inuous @ A @ real @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) @ G3 )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( F3 @ A2 ) )
=> ( ( ( F3 @ A2 )
!= ( one_one @ real ) )
=> ( ( ord_less @ real @ ( zero_zero @ real ) @ ( G3 @ A2 ) )
=> ( topolo3448309680560233919inuous @ A @ real @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) )
@ ^ [X5: A] : ( log @ ( F3 @ X5 ) @ ( G3 @ X5 ) ) ) ) ) ) ) ) ) ).
% isCont_log
thf(fact_5210_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_5211_at__within__def,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ( ( topolo174197925503356063within @ A )
= ( ^ [A6: A,S8: set @ A] : ( inf_inf @ ( filter @ A ) @ ( topolo7230453075368039082e_nhds @ A @ A6 ) @ ( principal @ A @ ( minus_minus @ ( set @ A ) @ S8 @ ( insert @ A @ A6 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% at_within_def
thf(fact_5212_isCont__inverse__function,axiom,
! [D3: real,X: real,G3: real > real,F3: real > real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D3 )
=> ( ! [Z3: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ ( minus_minus @ real @ Z3 @ X ) ) @ D3 )
=> ( ( G3 @ ( F3 @ Z3 ) )
= Z3 ) )
=> ( ! [Z3: real] :
( ( ord_less_eq @ real @ ( abs_abs @ real @ ( minus_minus @ real @ Z3 @ X ) ) @ D3 )
=> ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ Z3 @ ( top_top @ ( set @ real ) ) ) @ F3 ) )
=> ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ ( F3 @ X ) @ ( top_top @ ( set @ real ) ) ) @ G3 ) ) ) ) ).
% isCont_inverse_function
thf(fact_5213_GMVT_H,axiom,
! [A2: real,B2: real,F3: real > real,G3: real > real,G7: real > real,F10: real > real] :
( ( ord_less @ real @ A2 @ B2 )
=> ( ! [Z3: real] :
( ( ord_less_eq @ real @ A2 @ Z3 )
=> ( ( ord_less_eq @ real @ Z3 @ B2 )
=> ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ Z3 @ ( top_top @ ( set @ real ) ) ) @ F3 ) ) )
=> ( ! [Z3: real] :
( ( ord_less_eq @ real @ A2 @ Z3 )
=> ( ( ord_less_eq @ real @ Z3 @ B2 )
=> ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ Z3 @ ( top_top @ ( set @ real ) ) ) @ G3 ) ) )
=> ( ! [Z3: real] :
( ( ord_less @ real @ A2 @ Z3 )
=> ( ( ord_less @ real @ Z3 @ B2 )
=> ( has_field_derivative @ real @ G3 @ ( G7 @ Z3 ) @ ( topolo174197925503356063within @ real @ Z3 @ ( top_top @ ( set @ real ) ) ) ) ) )
=> ( ! [Z3: real] :
( ( ord_less @ real @ A2 @ Z3 )
=> ( ( ord_less @ real @ Z3 @ B2 )
=> ( has_field_derivative @ real @ F3 @ ( F10 @ Z3 ) @ ( topolo174197925503356063within @ real @ Z3 @ ( top_top @ ( set @ real ) ) ) ) ) )
=> ? [C2: real] :
( ( ord_less @ real @ A2 @ C2 )
& ( ord_less @ real @ C2 @ B2 )
& ( ( times_times @ real @ ( minus_minus @ real @ ( F3 @ B2 ) @ ( F3 @ A2 ) ) @ ( G7 @ C2 ) )
= ( times_times @ real @ ( minus_minus @ real @ ( G3 @ B2 ) @ ( G3 @ A2 ) ) @ ( F10 @ C2 ) ) ) ) ) ) ) ) ) ).
% GMVT'
thf(fact_5214_metric__isCont__LIM__compose2,axiom,
! [D: $tType,C: $tType,A: $tType] :
( ( ( real_V7819770556892013058_space @ A )
& ( topolo4958980785337419405_space @ C )
& ( topolo4958980785337419405_space @ D ) )
=> ! [A2: A,F3: A > C,G3: C > D,L: D] :
( ( topolo3448309680560233919inuous @ A @ C @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) @ F3 )
=> ( ( filterlim @ C @ D @ G3 @ ( topolo7230453075368039082e_nhds @ D @ L ) @ ( topolo174197925503356063within @ C @ ( F3 @ A2 ) @ ( top_top @ ( set @ C ) ) ) )
=> ( ? [D4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D4 )
& ! [X4: A] :
( ( ( X4 != A2 )
& ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X4 @ A2 ) @ D4 ) )
=> ( ( F3 @ X4 )
!= ( F3 @ A2 ) ) ) )
=> ( filterlim @ A @ D
@ ^ [X5: A] : ( G3 @ ( F3 @ X5 ) )
@ ( topolo7230453075368039082e_nhds @ D @ L )
@ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ).
% metric_isCont_LIM_compose2
thf(fact_5215_at__left__eq,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [Y: A,X: A] :
( ( ord_less @ A @ Y @ X )
=> ( ( topolo174197925503356063within @ A @ X @ ( set_ord_lessThan @ A @ X ) )
= ( complete_Inf_Inf @ ( filter @ A )
@ ( image2 @ A @ ( filter @ A )
@ ^ [A6: A] : ( principal @ A @ ( set_or5935395276787703475ssThan @ A @ A6 @ X ) )
@ ( set_ord_lessThan @ A @ X ) ) ) ) ) ) ).
% at_left_eq
thf(fact_5216_isCont__LIM__compose2,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( topolo4958980785337419405_space @ B )
& ( topolo4958980785337419405_space @ C ) )
=> ! [A2: A,F3: A > B,G3: B > C,L: C] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) @ F3 )
=> ( ( filterlim @ B @ C @ G3 @ ( topolo7230453075368039082e_nhds @ C @ L ) @ ( topolo174197925503356063within @ B @ ( F3 @ A2 ) @ ( top_top @ ( set @ B ) ) ) )
=> ( ? [D4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D4 )
& ! [X4: A] :
( ( ( X4 != A2 )
& ( ord_less @ real @ ( real_V7770717601297561774m_norm @ A @ ( minus_minus @ A @ X4 @ A2 ) ) @ D4 ) )
=> ( ( F3 @ X4 )
!= ( F3 @ A2 ) ) ) )
=> ( filterlim @ A @ C
@ ^ [X5: A] : ( G3 @ ( F3 @ X5 ) )
@ ( topolo7230453075368039082e_nhds @ C @ L )
@ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ) ) ).
% isCont_LIM_compose2
thf(fact_5217_isCont__powser_H,axiom,
! [Aa: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_Vector_banach @ Aa )
& ( real_V3459762299906320749_field @ Aa ) )
=> ! [A2: A,F3: A > Aa,C3: nat > Aa,K5: Aa] :
( ( topolo3448309680560233919inuous @ A @ Aa @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) @ F3 )
=> ( ( summable @ Aa
@ ^ [N2: nat] : ( times_times @ Aa @ ( C3 @ N2 ) @ ( power_power @ Aa @ K5 @ N2 ) ) )
=> ( ( ord_less @ real @ ( real_V7770717601297561774m_norm @ Aa @ ( F3 @ A2 ) ) @ ( real_V7770717601297561774m_norm @ Aa @ K5 ) )
=> ( topolo3448309680560233919inuous @ A @ Aa @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) )
@ ^ [X5: A] :
( suminf @ Aa
@ ^ [N2: nat] : ( times_times @ Aa @ ( C3 @ N2 ) @ ( power_power @ Aa @ ( F3 @ X5 ) @ N2 ) ) ) ) ) ) ) ) ).
% isCont_powser'
thf(fact_5218_complete__uniform,axiom,
! [A: $tType] :
( ( topolo7287701948861334536_space @ A )
=> ( ( topolo2479028161051973599mplete @ A )
= ( ^ [S7: set @ A] :
! [F8: filter @ A] :
( ( ord_less_eq @ ( filter @ A ) @ F8 @ ( principal @ A @ S7 ) )
=> ( ( F8
!= ( bot_bot @ ( filter @ A ) ) )
=> ( ( topolo6773858410816713723filter @ A @ F8 )
=> ? [X5: A] :
( ( member @ A @ X5 @ S7 )
& ( ord_less_eq @ ( filter @ A ) @ F8 @ ( topolo7230453075368039082e_nhds @ A @ X5 ) ) ) ) ) ) ) ) ) ).
% complete_uniform
thf(fact_5219_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 )
@ ( image2 @ A @ ( filter @ A )
@ ^ [A6: A] : ( principal @ A @ ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ ( set_ord_lessThan @ A @ A6 ) @ S ) @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
@ ( set_ord_greaterThan @ A @ X ) ) )
@ ( complete_Inf_Inf @ ( filter @ A )
@ ( image2 @ A @ ( filter @ A )
@ ^ [A6: A] : ( principal @ A @ ( minus_minus @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ ( set_ord_greaterThan @ A @ A6 ) @ S ) @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
@ ( set_ord_lessThan @ A @ X ) ) ) ) ) ) ) ).
% at_within_order
thf(fact_5220_image__Fpow__mono,axiom,
! [B: $tType,A: $tType,F3: B > A,A5: set @ B,B6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( image2 @ B @ A @ F3 @ A5 ) @ B6 )
=> ( ord_less_eq @ ( set @ ( set @ A ) ) @ ( image2 @ ( set @ B ) @ ( set @ A ) @ ( image2 @ B @ A @ F3 ) @ ( finite_Fpow @ B @ A5 ) ) @ ( finite_Fpow @ A @ B6 ) ) ) ).
% image_Fpow_mono
thf(fact_5221_greaterThan__eq__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( ( set_ord_greaterThan @ A @ X )
= ( set_ord_greaterThan @ A @ Y ) )
= ( X = Y ) ) ) ).
% greaterThan_eq_iff
thf(fact_5222_greaterThan__iff,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [I2: A,K2: A] :
( ( member @ A @ I2 @ ( set_ord_greaterThan @ A @ K2 ) )
= ( ord_less @ A @ K2 @ I2 ) ) ) ).
% greaterThan_iff
thf(fact_5223_Inf__greaterThan,axiom,
! [A: $tType] :
( ( ( comple6319245703460814977attice @ A )
& ( dense_linorder @ A ) )
=> ! [X: A] :
( ( complete_Inf_Inf @ A @ ( set_ord_greaterThan @ A @ X ) )
= X ) ) ).
% Inf_greaterThan
thf(fact_5224_greaterThan__subset__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_ord_greaterThan @ A @ X ) @ ( set_ord_greaterThan @ A @ Y ) )
= ( ord_less_eq @ A @ Y @ X ) ) ) ).
% greaterThan_subset_iff
thf(fact_5225_Compl__greaterThan,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [K2: A] :
( ( uminus_uminus @ ( set @ A ) @ ( set_ord_greaterThan @ A @ K2 ) )
= ( set_ord_atMost @ A @ K2 ) ) ) ).
% Compl_greaterThan
thf(fact_5226_Compl__atMost,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [K2: A] :
( ( uminus_uminus @ ( set @ A ) @ ( set_ord_atMost @ A @ K2 ) )
= ( set_ord_greaterThan @ A @ K2 ) ) ) ).
% Compl_atMost
thf(fact_5227_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_5228_image__uminus__lessThan,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [X: A] :
( ( image2 @ A @ A @ ( uminus_uminus @ A ) @ ( set_ord_lessThan @ A @ X ) )
= ( set_ord_greaterThan @ A @ ( uminus_uminus @ A @ X ) ) ) ) ).
% image_uminus_lessThan
thf(fact_5229_image__uminus__greaterThan,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [X: A] :
( ( image2 @ A @ A @ ( uminus_uminus @ A ) @ ( set_ord_greaterThan @ A @ X ) )
= ( set_ord_lessThan @ A @ ( uminus_uminus @ A @ X ) ) ) ) ).
% image_uminus_greaterThan
thf(fact_5230_greaterThan__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( set_ord_greaterThan @ A )
= ( ^ [L2: A] : ( collect @ A @ ( ord_less @ A @ L2 ) ) ) ) ) ).
% greaterThan_def
thf(fact_5231_infinite__Ioi,axiom,
! [A: $tType] :
( ( ( linorder @ A )
& ( no_top @ A ) )
=> ! [A2: A] :
~ ( finite_finite2 @ A @ ( set_ord_greaterThan @ A @ A2 ) ) ) ).
% infinite_Ioi
thf(fact_5232_greaterThan__non__empty,axiom,
! [A: $tType] :
( ( no_top @ A )
=> ! [X: A] :
( ( set_ord_greaterThan @ A @ X )
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% greaterThan_non_empty
thf(fact_5233_empty__in__Fpow,axiom,
! [A: $tType,A5: set @ A] : ( member @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ ( finite_Fpow @ A @ A5 ) ) ).
% empty_in_Fpow
thf(fact_5234_lessThan__Int__lessThan,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A] :
( ( inf_inf @ ( set @ A ) @ ( set_ord_greaterThan @ A @ A2 ) @ ( set_ord_greaterThan @ A @ B2 ) )
= ( set_ord_greaterThan @ A @ ( ord_max @ A @ A2 @ B2 ) ) ) ) ).
% lessThan_Int_lessThan
thf(fact_5235_trivial__limit__at__right__real,axiom,
! [A: $tType] :
( ( ( dense_order @ A )
& ( no_top @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [X: A] :
( ( topolo174197925503356063within @ A @ X @ ( set_ord_greaterThan @ A @ X ) )
!= ( bot_bot @ ( filter @ A ) ) ) ) ).
% trivial_limit_at_right_real
thf(fact_5236_Fpow__not__empty,axiom,
! [A: $tType,A5: set @ A] :
( ( finite_Fpow @ A @ A5 )
!= ( bot_bot @ ( set @ ( set @ A ) ) ) ) ).
% Fpow_not_empty
thf(fact_5237_eventually__at__right__field,axiom,
! [A: $tType] :
( ( ( linordered_field @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [P2: A > $o,X: A] :
( ( eventually @ A @ P2 @ ( topolo174197925503356063within @ A @ X @ ( set_ord_greaterThan @ A @ X ) ) )
= ( ? [B5: A] :
( ( ord_less @ A @ X @ B5 )
& ! [Y6: A] :
( ( ord_less @ A @ X @ Y6 )
=> ( ( ord_less @ A @ Y6 @ B5 )
=> ( P2 @ Y6 ) ) ) ) ) ) ) ).
% eventually_at_right_field
thf(fact_5238_eventually__at__right,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [X: A,Y: A,P2: A > $o] :
( ( ord_less @ A @ X @ Y )
=> ( ( eventually @ A @ P2 @ ( topolo174197925503356063within @ A @ X @ ( set_ord_greaterThan @ A @ X ) ) )
= ( ? [B5: A] :
( ( ord_less @ A @ X @ B5 )
& ! [Y6: A] :
( ( ord_less @ A @ X @ Y6 )
=> ( ( ord_less @ A @ Y6 @ B5 )
=> ( P2 @ Y6 ) ) ) ) ) ) ) ) ).
% eventually_at_right
thf(fact_5239_at__within__Icc__at__right,axiom,
! [A: $tType] :
( ( topolo2564578578187576103pology @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( topolo174197925503356063within @ A @ A2 @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
= ( topolo174197925503356063within @ A @ A2 @ ( set_ord_greaterThan @ A @ A2 ) ) ) ) ) ).
% at_within_Icc_at_right
thf(fact_5240_ivl__disj__int__one_I7_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,U: A] :
( ( inf_inf @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ L @ U ) @ ( set_ord_greaterThan @ A @ U ) )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% ivl_disj_int_one(7)
thf(fact_5241_trivial__limit__at__right__top,axiom,
! [A: $tType] :
( ( ( order_top @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ( ( topolo174197925503356063within @ A @ ( top_top @ A ) @ ( set_ord_greaterThan @ A @ ( top_top @ A ) ) )
= ( bot_bot @ ( filter @ A ) ) ) ) ).
% trivial_limit_at_right_top
thf(fact_5242_ivl__disj__int__one_I5_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,U: A] :
( ( inf_inf @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ L @ U ) @ ( set_ord_greaterThan @ A @ U ) )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% ivl_disj_int_one(5)
thf(fact_5243_greaterThanLessThan__eq,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( set_or5935395276787703475ssThan @ A )
= ( ^ [A6: A,B5: A] : ( inf_inf @ ( set @ A ) @ ( set_ord_greaterThan @ A @ A6 ) @ ( set_ord_lessThan @ A @ B5 ) ) ) ) ) ).
% greaterThanLessThan_eq
thf(fact_5244_greaterThanLessThan__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( set_or5935395276787703475ssThan @ A )
= ( ^ [L2: A,U2: A] : ( inf_inf @ ( set @ A ) @ ( set_ord_greaterThan @ A @ L2 ) @ ( set_ord_lessThan @ A @ U2 ) ) ) ) ) ).
% greaterThanLessThan_def
thf(fact_5245_greaterThanAtMost__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( set_or3652927894154168847AtMost @ A )
= ( ^ [L2: A,U2: A] : ( inf_inf @ ( set @ A ) @ ( set_ord_greaterThan @ A @ L2 ) @ ( set_ord_atMost @ A @ U2 ) ) ) ) ) ).
% greaterThanAtMost_def
thf(fact_5246_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_5247_Fpow__mono,axiom,
! [A: $tType,A5: set @ A,B6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B6 )
=> ( ord_less_eq @ ( set @ ( set @ A ) ) @ ( finite_Fpow @ A @ A5 ) @ ( finite_Fpow @ A @ B6 ) ) ) ).
% Fpow_mono
thf(fact_5248_less__separate,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ? [A4: A,B4: A] :
( ( member @ A @ X @ ( set_ord_lessThan @ A @ A4 ) )
& ( member @ A @ Y @ ( set_ord_greaterThan @ A @ B4 ) )
& ( ( inf_inf @ ( set @ A ) @ ( set_ord_lessThan @ A @ A4 ) @ ( set_ord_greaterThan @ A @ B4 ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% less_separate
thf(fact_5249_eventually__at__rightI,axiom,
! [A: $tType] :
( ( topolo2564578578187576103pology @ A )
=> ! [A2: A,B2: A,P2: A > $o] :
( ! [X4: A] :
( ( member @ A @ X4 @ ( set_or5935395276787703475ssThan @ A @ A2 @ B2 ) )
=> ( P2 @ X4 ) )
=> ( ( ord_less @ A @ A2 @ B2 )
=> ( eventually @ A @ P2 @ ( topolo174197925503356063within @ A @ A2 @ ( set_ord_greaterThan @ A @ A2 ) ) ) ) ) ) ).
% eventually_at_rightI
thf(fact_5250_Fpow__subset__Pow,axiom,
! [A: $tType,A5: set @ A] : ( ord_less_eq @ ( set @ ( set @ A ) ) @ ( finite_Fpow @ A @ A5 ) @ ( pow2 @ A @ A5 ) ) ).
% Fpow_subset_Pow
thf(fact_5251_greaterThan__0,axiom,
( ( set_ord_greaterThan @ nat @ ( zero_zero @ nat ) )
= ( image2 @ nat @ nat @ suc @ ( top_top @ ( set @ nat ) ) ) ) ).
% greaterThan_0
thf(fact_5252_Fpow__def,axiom,
! [A: $tType] :
( ( finite_Fpow @ A )
= ( ^ [A7: set @ A] :
( collect @ ( set @ A )
@ ^ [X9: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ X9 @ A7 )
& ( finite_finite2 @ A @ X9 ) ) ) ) ) ).
% Fpow_def
thf(fact_5253_Fpow__Pow__finite,axiom,
! [A: $tType] :
( ( finite_Fpow @ A )
= ( ^ [A7: set @ A] : ( inf_inf @ ( set @ ( set @ A ) ) @ ( pow2 @ A @ A7 ) @ ( collect @ ( set @ A ) @ ( finite_finite2 @ A ) ) ) ) ) ).
% Fpow_Pow_finite
thf(fact_5254_eventually__at__right__real,axiom,
! [A2: real,B2: real] :
( ( ord_less @ real @ A2 @ B2 )
=> ( eventually @ real
@ ^ [X5: real] : ( member @ real @ X5 @ ( set_or5935395276787703475ssThan @ real @ A2 @ B2 ) )
@ ( topolo174197925503356063within @ real @ A2 @ ( set_ord_greaterThan @ real @ A2 ) ) ) ) ).
% eventually_at_right_real
thf(fact_5255_greaterThan__Suc,axiom,
! [K2: nat] :
( ( set_ord_greaterThan @ nat @ ( suc @ K2 ) )
= ( minus_minus @ ( set @ nat ) @ ( set_ord_greaterThan @ nat @ K2 ) @ ( insert @ nat @ ( suc @ K2 ) @ ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% greaterThan_Suc
thf(fact_5256_filterlim__times__pos,axiom,
! [A: $tType,B: $tType] :
( ( ( linordered_field @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [F3: B > A,P: A,F13: filter @ B,C3: A,L: A] :
( ( filterlim @ B @ A @ F3 @ ( topolo174197925503356063within @ A @ P @ ( set_ord_greaterThan @ A @ P ) ) @ F13 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( L
= ( times_times @ A @ C3 @ P ) )
=> ( filterlim @ B @ A
@ ^ [X5: B] : ( times_times @ A @ C3 @ ( F3 @ X5 ) )
@ ( topolo174197925503356063within @ A @ L @ ( set_ord_greaterThan @ A @ L ) )
@ F13 ) ) ) ) ) ).
% filterlim_times_pos
thf(fact_5257_tendsto__imp__filterlim__at__right,axiom,
! [B: $tType,A: $tType] :
( ( topolo2564578578187576103pology @ B )
=> ! [F3: A > B,L5: B,F5: filter @ A] :
( ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ L5 ) @ F5 )
=> ( ( eventually @ A
@ ^ [X5: A] : ( ord_less @ B @ L5 @ ( F3 @ X5 ) )
@ F5 )
=> ( filterlim @ A @ B @ F3 @ ( topolo174197925503356063within @ B @ L5 @ ( set_ord_greaterThan @ B @ L5 ) ) @ F5 ) ) ) ) ).
% tendsto_imp_filterlim_at_right
thf(fact_5258_filterlim__at__bot__at__right,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo1944317154257567458pology @ A )
& ( linorder @ B ) )
=> ! [Q: A > $o,F3: A > B,P2: B > $o,G3: B > A,A2: A] :
( ! [X4: A,Y3: A] :
( ( Q @ X4 )
=> ( ( Q @ Y3 )
=> ( ( ord_less_eq @ A @ X4 @ Y3 )
=> ( ord_less_eq @ B @ ( F3 @ X4 ) @ ( F3 @ Y3 ) ) ) ) )
=> ( ! [X4: B] :
( ( P2 @ X4 )
=> ( ( F3 @ ( G3 @ X4 ) )
= X4 ) )
=> ( ! [X4: B] :
( ( P2 @ X4 )
=> ( Q @ ( G3 @ X4 ) ) )
=> ( ( eventually @ A @ Q @ ( topolo174197925503356063within @ A @ A2 @ ( set_ord_greaterThan @ A @ A2 ) ) )
=> ( ! [B4: A] :
( ( Q @ B4 )
=> ( ord_less @ A @ A2 @ B4 ) )
=> ( ( eventually @ B @ P2 @ ( at_bot @ B ) )
=> ( filterlim @ A @ B @ F3 @ ( at_bot @ B ) @ ( topolo174197925503356063within @ A @ A2 @ ( set_ord_greaterThan @ A @ A2 ) ) ) ) ) ) ) ) ) ) ).
% filterlim_at_bot_at_right
thf(fact_5259_INT__greaterThan__UNIV,axiom,
( ( complete_Inf_Inf @ ( set @ nat ) @ ( image2 @ nat @ ( set @ nat ) @ ( set_ord_greaterThan @ nat ) @ ( top_top @ ( set @ nat ) ) ) )
= ( bot_bot @ ( set @ nat ) ) ) ).
% INT_greaterThan_UNIV
thf(fact_5260_nhds__metric,axiom,
! [A: $tType] :
( ( real_V7819770556892013058_space @ A )
=> ( ( topolo7230453075368039082e_nhds @ A )
= ( ^ [X5: A] :
( complete_Inf_Inf @ ( filter @ A )
@ ( image2 @ real @ ( filter @ A )
@ ^ [E4: real] :
( principal @ A
@ ( collect @ A
@ ^ [Y6: A] : ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ Y6 @ X5 ) @ E4 ) ) )
@ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) ) ) ) ) ).
% nhds_metric
thf(fact_5261_at__right__eq,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [X: A,Y: A] :
( ( ord_less @ A @ X @ Y )
=> ( ( topolo174197925503356063within @ A @ X @ ( set_ord_greaterThan @ A @ X ) )
= ( complete_Inf_Inf @ ( filter @ A )
@ ( image2 @ A @ ( filter @ A )
@ ^ [A6: A] : ( principal @ A @ ( set_or5935395276787703475ssThan @ A @ X @ A6 ) )
@ ( set_ord_greaterThan @ A @ X ) ) ) ) ) ) ).
% at_right_eq
thf(fact_5262_isCont__If__ge,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo1944317154257567458pology @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [A2: A,G3: A > B,F3: A > B] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ ( set_ord_lessThan @ A @ A2 ) ) @ G3 )
=> ( ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ ( G3 @ A2 ) ) @ ( topolo174197925503356063within @ A @ A2 @ ( set_ord_greaterThan @ A @ A2 ) ) )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) )
@ ^ [X5: A] : ( if @ B @ ( ord_less_eq @ A @ X5 @ A2 ) @ ( G3 @ X5 ) @ ( F3 @ X5 ) ) ) ) ) ) ).
% isCont_If_ge
thf(fact_5263_interval__cases,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [S2: set @ A] :
( ! [A4: A,B4: A,X4: A] :
( ( member @ A @ A4 @ S2 )
=> ( ( member @ A @ B4 @ S2 )
=> ( ( ord_less_eq @ A @ A4 @ X4 )
=> ( ( ord_less_eq @ A @ X4 @ B4 )
=> ( member @ A @ X4 @ S2 ) ) ) ) )
=> ? [A4: A,B4: A] :
( ( S2
= ( bot_bot @ ( set @ A ) ) )
| ( S2
= ( top_top @ ( set @ A ) ) )
| ( S2
= ( set_ord_lessThan @ A @ B4 ) )
| ( S2
= ( set_ord_atMost @ A @ B4 ) )
| ( S2
= ( set_ord_greaterThan @ A @ A4 ) )
| ( S2
= ( set_ord_atLeast @ A @ A4 ) )
| ( S2
= ( set_or5935395276787703475ssThan @ A @ A4 @ B4 ) )
| ( S2
= ( set_or3652927894154168847AtMost @ A @ A4 @ B4 ) )
| ( S2
= ( set_or7035219750837199246ssThan @ A @ A4 @ B4 ) )
| ( S2
= ( set_or1337092689740270186AtMost @ A @ A4 @ B4 ) ) ) ) ) ).
% interval_cases
thf(fact_5264_sequentially__imp__eventually__at__right,axiom,
! [A: $tType] :
( ( ( topolo3112930676232923870pology @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [A2: A,B2: A,P2: A > $o] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ! [F2: nat > A] :
( ! [N5: nat] : ( ord_less @ A @ A2 @ ( F2 @ N5 ) )
=> ( ! [N5: nat] : ( ord_less @ A @ ( F2 @ N5 ) @ B2 )
=> ( ( order_antimono @ nat @ A @ F2 )
=> ( ( filterlim @ nat @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ ( at_top @ nat ) )
=> ( eventually @ nat
@ ^ [N2: nat] : ( P2 @ ( F2 @ N2 ) )
@ ( at_top @ nat ) ) ) ) ) )
=> ( eventually @ A @ P2 @ ( topolo174197925503356063within @ A @ A2 @ ( set_ord_greaterThan @ A @ A2 ) ) ) ) ) ) ).
% sequentially_imp_eventually_at_right
thf(fact_5265_GMVT,axiom,
! [A2: real,B2: real,F3: real > real,G3: real > real] :
( ( ord_less @ real @ A2 @ B2 )
=> ( ! [X4: real] :
( ( ( ord_less_eq @ real @ A2 @ X4 )
& ( ord_less_eq @ real @ X4 @ B2 ) )
=> ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ X4 @ ( top_top @ ( set @ real ) ) ) @ F3 ) )
=> ( ! [X4: real] :
( ( ( ord_less @ real @ A2 @ X4 )
& ( ord_less @ real @ X4 @ B2 ) )
=> ( differentiable @ real @ real @ F3 @ ( topolo174197925503356063within @ real @ X4 @ ( top_top @ ( set @ real ) ) ) ) )
=> ( ! [X4: real] :
( ( ( ord_less_eq @ real @ A2 @ X4 )
& ( ord_less_eq @ real @ X4 @ B2 ) )
=> ( topolo3448309680560233919inuous @ real @ real @ ( topolo174197925503356063within @ real @ X4 @ ( top_top @ ( set @ real ) ) ) @ G3 ) )
=> ( ! [X4: real] :
( ( ( ord_less @ real @ A2 @ X4 )
& ( ord_less @ real @ X4 @ B2 ) )
=> ( differentiable @ real @ real @ G3 @ ( topolo174197925503356063within @ real @ X4 @ ( top_top @ ( set @ real ) ) ) ) )
=> ? [G_c: real,F_c: real,C2: real] :
( ( has_field_derivative @ real @ G3 @ G_c @ ( topolo174197925503356063within @ real @ C2 @ ( top_top @ ( set @ real ) ) ) )
& ( has_field_derivative @ real @ F3 @ F_c @ ( topolo174197925503356063within @ real @ C2 @ ( top_top @ ( set @ real ) ) ) )
& ( ord_less @ real @ A2 @ C2 )
& ( ord_less @ real @ C2 @ B2 )
& ( ( times_times @ real @ ( minus_minus @ real @ ( F3 @ B2 ) @ ( F3 @ A2 ) ) @ G_c )
= ( times_times @ real @ ( minus_minus @ real @ ( G3 @ B2 ) @ ( G3 @ A2 ) ) @ F_c ) ) ) ) ) ) ) ) ).
% GMVT
thf(fact_5266_atLeast__eq__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X: A,Y: A] :
( ( ( set_ord_atLeast @ A @ X )
= ( set_ord_atLeast @ A @ Y ) )
= ( X = Y ) ) ) ).
% atLeast_eq_iff
thf(fact_5267_atLeast__0,axiom,
( ( set_ord_atLeast @ nat @ ( zero_zero @ nat ) )
= ( top_top @ ( set @ nat ) ) ) ).
% atLeast_0
thf(fact_5268_atLeast__iff,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [I2: A,K2: A] :
( ( member @ A @ I2 @ ( set_ord_atLeast @ A @ K2 ) )
= ( ord_less_eq @ A @ K2 @ I2 ) ) ) ).
% atLeast_iff
thf(fact_5269_Inf__atLeast,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X: A] :
( ( complete_Inf_Inf @ A @ ( set_ord_atLeast @ A @ X ) )
= X ) ) ).
% Inf_atLeast
thf(fact_5270_atLeast__empty__triv,axiom,
! [A: $tType] :
( ( set_ord_atLeast @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) )
= ( top_top @ ( set @ ( set @ A ) ) ) ) ).
% atLeast_empty_triv
thf(fact_5271_atLeast__subset__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_ord_atLeast @ A @ X ) @ ( set_ord_atLeast @ A @ Y ) )
= ( ord_less_eq @ A @ Y @ X ) ) ) ).
% atLeast_subset_iff
thf(fact_5272_image__add__atLeast,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [K2: A,I2: A] :
( ( image2 @ A @ A @ ( plus_plus @ A @ K2 ) @ ( set_ord_atLeast @ A @ I2 ) )
= ( set_ord_atLeast @ A @ ( plus_plus @ A @ K2 @ I2 ) ) ) ) ).
% image_add_atLeast
thf(fact_5273_Sup__atLeast,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [X: A] :
( ( complete_Sup_Sup @ A @ ( set_ord_atLeast @ A @ X ) )
= ( top_top @ A ) ) ) ).
% Sup_atLeast
thf(fact_5274_Compl__atLeast,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [K2: A] :
( ( uminus_uminus @ ( set @ A ) @ ( set_ord_atLeast @ A @ K2 ) )
= ( set_ord_lessThan @ A @ K2 ) ) ) ).
% Compl_atLeast
thf(fact_5275_Compl__lessThan,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [K2: A] :
( ( uminus_uminus @ ( set @ A ) @ ( set_ord_lessThan @ A @ K2 ) )
= ( set_ord_atLeast @ A @ K2 ) ) ) ).
% Compl_lessThan
thf(fact_5276_Icc__subset__Ici__iff,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [L: A,H2: A,L3: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ L @ H2 ) @ ( set_ord_atLeast @ A @ L3 ) )
= ( ~ ( ord_less_eq @ A @ L @ H2 )
| ( ord_less_eq @ A @ L3 @ L ) ) ) ) ).
% Icc_subset_Ici_iff
thf(fact_5277_image__minus__const__AtMost,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [C3: A,B2: A] :
( ( image2 @ A @ A @ ( minus_minus @ A @ C3 ) @ ( set_ord_atMost @ A @ B2 ) )
= ( set_ord_atLeast @ A @ ( minus_minus @ A @ C3 @ B2 ) ) ) ) ).
% image_minus_const_AtMost
thf(fact_5278_image__minus__const__atLeast,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [C3: A,A2: A] :
( ( image2 @ A @ A @ ( minus_minus @ A @ C3 ) @ ( set_ord_atLeast @ A @ A2 ) )
= ( set_ord_atMost @ A @ ( minus_minus @ A @ C3 @ A2 ) ) ) ) ).
% image_minus_const_atLeast
thf(fact_5279_image__uminus__atMost,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [X: A] :
( ( image2 @ A @ A @ ( uminus_uminus @ A ) @ ( set_ord_atMost @ A @ X ) )
= ( set_ord_atLeast @ A @ ( uminus_uminus @ A @ X ) ) ) ) ).
% image_uminus_atMost
thf(fact_5280_image__uminus__atLeast,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [X: A] :
( ( image2 @ A @ A @ ( uminus_uminus @ A ) @ ( set_ord_atLeast @ A @ X ) )
= ( set_ord_atMost @ A @ ( uminus_uminus @ A @ X ) ) ) ) ).
% image_uminus_atLeast
thf(fact_5281_Int__atLeastAtMostR2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,C3: A,D3: A] :
( ( inf_inf @ ( set @ A ) @ ( set_ord_atLeast @ A @ A2 ) @ ( set_or1337092689740270186AtMost @ A @ C3 @ D3 ) )
= ( set_or1337092689740270186AtMost @ A @ ( ord_max @ A @ A2 @ C3 ) @ D3 ) ) ) ).
% Int_atLeastAtMostR2
thf(fact_5282_Int__atLeastAtMostL2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( inf_inf @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) @ ( set_ord_atLeast @ A @ C3 ) )
= ( set_or1337092689740270186AtMost @ A @ ( ord_max @ A @ A2 @ C3 ) @ B2 ) ) ) ).
% Int_atLeastAtMostL2
thf(fact_5283_differentiable__cmult__left__iff,axiom,
! [A: $tType,B: $tType] :
( ( ( real_V822414075346904944vector @ B )
& ( real_V3459762299906320749_field @ A ) )
=> ! [C3: A,Q2: B > A,T2: B] :
( ( differentiable @ B @ A
@ ^ [T3: B] : ( times_times @ A @ C3 @ ( Q2 @ T3 ) )
@ ( topolo174197925503356063within @ B @ T2 @ ( top_top @ ( set @ B ) ) ) )
= ( ( C3
= ( zero_zero @ A ) )
| ( differentiable @ B @ A @ Q2 @ ( topolo174197925503356063within @ B @ T2 @ ( top_top @ ( set @ B ) ) ) ) ) ) ) ).
% differentiable_cmult_left_iff
thf(fact_5284_differentiable__cmult__right__iff,axiom,
! [A: $tType,B: $tType] :
( ( ( real_V822414075346904944vector @ B )
& ( real_V3459762299906320749_field @ A ) )
=> ! [Q2: B > A,C3: A,T2: B] :
( ( differentiable @ B @ A
@ ^ [T3: B] : ( times_times @ A @ ( Q2 @ T3 ) @ C3 )
@ ( topolo174197925503356063within @ B @ T2 @ ( top_top @ ( set @ B ) ) ) )
= ( ( C3
= ( zero_zero @ A ) )
| ( differentiable @ B @ A @ Q2 @ ( topolo174197925503356063within @ B @ T2 @ ( top_top @ ( set @ B ) ) ) ) ) ) ) ).
% differentiable_cmult_right_iff
thf(fact_5285_atLeast__Suc__greaterThan,axiom,
! [K2: nat] :
( ( set_ord_atLeast @ nat @ ( suc @ K2 ) )
= ( set_ord_greaterThan @ nat @ K2 ) ) ).
% atLeast_Suc_greaterThan
thf(fact_5286_not__empty__eq__Ici__eq__empty,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [L: A] :
( ( bot_bot @ ( set @ A ) )
!= ( set_ord_atLeast @ A @ L ) ) ) ).
% not_empty_eq_Ici_eq_empty
thf(fact_5287_atLeast__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( set_ord_atLeast @ A )
= ( ^ [L2: A] : ( collect @ A @ ( ord_less_eq @ A @ L2 ) ) ) ) ) ).
% atLeast_def
thf(fact_5288_not__Ici__eq__Icc,axiom,
! [A: $tType] :
( ( no_top @ A )
=> ! [L3: A,L: A,H2: A] :
( ( set_ord_atLeast @ A @ L3 )
!= ( set_or1337092689740270186AtMost @ A @ L @ H2 ) ) ) ).
% not_Ici_eq_Icc
thf(fact_5289_not__Iic__eq__Ici,axiom,
! [A: $tType] :
( ( no_top @ A )
=> ! [H2: A,L3: A] :
( ( set_ord_atMost @ A @ H2 )
!= ( set_ord_atLeast @ A @ L3 ) ) ) ).
% not_Iic_eq_Ici
thf(fact_5290_infinite__Ici,axiom,
! [A: $tType] :
( ( ( linorder @ A )
& ( no_top @ A ) )
=> ! [A2: A] :
~ ( finite_finite2 @ A @ ( set_ord_atLeast @ A @ A2 ) ) ) ).
% infinite_Ici
thf(fact_5291_differentiable__add,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [F3: A > B,F5: filter @ A,G3: A > B] :
( ( differentiable @ A @ B @ F3 @ F5 )
=> ( ( differentiable @ A @ B @ G3 @ F5 )
=> ( differentiable @ A @ B
@ ^ [X5: A] : ( plus_plus @ B @ ( F3 @ X5 ) @ ( G3 @ X5 ) )
@ F5 ) ) ) ) ).
% differentiable_add
thf(fact_5292_not__UNIV__eq__Ici,axiom,
! [A: $tType] :
( ( no_bot @ A )
=> ! [L3: A] :
( ( top_top @ ( set @ A ) )
!= ( set_ord_atLeast @ A @ L3 ) ) ) ).
% not_UNIV_eq_Ici
thf(fact_5293_antimono__def,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ( ( order_antimono @ A @ B )
= ( ^ [F4: A > B] :
! [X5: A,Y6: A] :
( ( ord_less_eq @ A @ X5 @ Y6 )
=> ( ord_less_eq @ B @ ( F4 @ Y6 ) @ ( F4 @ X5 ) ) ) ) ) ) ).
% antimono_def
thf(fact_5294_antimonoI,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F3: A > B] :
( ! [X4: A,Y3: A] :
( ( ord_less_eq @ A @ X4 @ Y3 )
=> ( ord_less_eq @ B @ ( F3 @ Y3 ) @ ( F3 @ X4 ) ) )
=> ( order_antimono @ A @ B @ F3 ) ) ) ).
% antimonoI
thf(fact_5295_antimonoE,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F3: A > B,X: A,Y: A] :
( ( order_antimono @ A @ B @ F3 )
=> ( ( ord_less_eq @ A @ X @ Y )
=> ( ord_less_eq @ B @ ( F3 @ Y ) @ ( F3 @ X ) ) ) ) ) ).
% antimonoE
thf(fact_5296_antimonoD,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F3: A > B,X: A,Y: A] :
( ( order_antimono @ A @ B @ F3 )
=> ( ( ord_less_eq @ A @ X @ Y )
=> ( ord_less_eq @ B @ ( F3 @ Y ) @ ( F3 @ X ) ) ) ) ) ).
% antimonoD
thf(fact_5297_atLeast__eq__UNIV__iff,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [X: A] :
( ( ( set_ord_atLeast @ A @ X )
= ( top_top @ ( set @ A ) ) )
= ( X
= ( bot_bot @ A ) ) ) ) ).
% atLeast_eq_UNIV_iff
thf(fact_5298_not__UNIV__le__Ici,axiom,
! [A: $tType] :
( ( no_bot @ A )
=> ! [L: A] :
~ ( ord_less_eq @ ( set @ A ) @ ( top_top @ ( set @ A ) ) @ ( set_ord_atLeast @ A @ L ) ) ) ).
% not_UNIV_le_Ici
thf(fact_5299_not__Ici__le__Icc,axiom,
! [A: $tType] :
( ( no_top @ A )
=> ! [L: A,L3: A,H3: A] :
~ ( ord_less_eq @ ( set @ A ) @ ( set_ord_atLeast @ A @ L ) @ ( set_or1337092689740270186AtMost @ A @ L3 @ H3 ) ) ) ).
% not_Ici_le_Icc
thf(fact_5300_not__Ici__le__Iic,axiom,
! [A: $tType] :
( ( no_top @ A )
=> ! [L: A,H3: A] :
~ ( ord_less_eq @ ( set @ A ) @ ( set_ord_atLeast @ A @ L ) @ ( set_ord_atMost @ A @ H3 ) ) ) ).
% not_Ici_le_Iic
thf(fact_5301_not__Iic__le__Ici,axiom,
! [A: $tType] :
( ( no_bot @ A )
=> ! [H2: A,L3: A] :
~ ( ord_less_eq @ ( set @ A ) @ ( set_ord_atMost @ A @ H2 ) @ ( set_ord_atLeast @ A @ L3 ) ) ) ).
% not_Iic_le_Ici
thf(fact_5302_Ioi__le__Ico,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A2: A] : ( ord_less_eq @ ( set @ A ) @ ( set_ord_greaterThan @ A @ A2 ) @ ( set_ord_atLeast @ A @ A2 ) ) ) ).
% Ioi_le_Ico
thf(fact_5303_decseq__Suc__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( order_antimono @ nat @ A )
= ( ^ [F4: nat > A] :
! [N2: nat] : ( ord_less_eq @ A @ ( F4 @ ( suc @ N2 ) ) @ ( F4 @ N2 ) ) ) ) ) ).
% decseq_Suc_iff
thf(fact_5304_decseq__SucI,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X7: nat > A] :
( ! [N3: nat] : ( ord_less_eq @ A @ ( X7 @ ( suc @ N3 ) ) @ ( X7 @ N3 ) )
=> ( order_antimono @ nat @ A @ X7 ) ) ) ).
% decseq_SucI
thf(fact_5305_decseq__SucD,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A5: nat > A,I2: nat] :
( ( order_antimono @ nat @ A @ A5 )
=> ( ord_less_eq @ A @ ( A5 @ ( suc @ I2 ) ) @ ( A5 @ I2 ) ) ) ) ).
% decseq_SucD
thf(fact_5306_decseq__def,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( order_antimono @ nat @ A )
= ( ^ [X9: nat > A] :
! [M3: nat,N2: nat] :
( ( ord_less_eq @ nat @ M3 @ N2 )
=> ( ord_less_eq @ A @ ( X9 @ N2 ) @ ( X9 @ M3 ) ) ) ) ) ) ).
% decseq_def
thf(fact_5307_decseqD,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [F3: nat > A,I2: nat,J: nat] :
( ( order_antimono @ nat @ A @ F3 )
=> ( ( ord_less_eq @ nat @ I2 @ J )
=> ( ord_less_eq @ A @ ( F3 @ J ) @ ( F3 @ I2 ) ) ) ) ) ).
% decseqD
thf(fact_5308_differentiable__sum,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( ( real_V822414075346904944vector @ B )
& ( real_V822414075346904944vector @ C ) )
=> ! [S: set @ A,F3: A > B > C,Net: filter @ B] :
( ( finite_finite2 @ A @ S )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ S )
=> ( differentiable @ B @ C @ ( F3 @ X4 ) @ Net ) )
=> ( differentiable @ B @ C
@ ^ [X5: B] :
( groups7311177749621191930dd_sum @ A @ C
@ ^ [A6: A] : ( F3 @ A6 @ X5 )
@ S )
@ Net ) ) ) ) ).
% differentiable_sum
thf(fact_5309_Ici__subset__Ioi__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ ( set @ A ) @ ( set_ord_atLeast @ A @ A2 ) @ ( set_ord_greaterThan @ A @ B2 ) )
= ( ord_less @ A @ B2 @ A2 ) ) ) ).
% Ici_subset_Ioi_iff
thf(fact_5310_ivl__disj__int__one_I8_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,U: A] :
( ( inf_inf @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ L @ U ) @ ( set_ord_atLeast @ A @ U ) )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% ivl_disj_int_one(8)
thf(fact_5311_atLeastLessThan__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( set_or7035219750837199246ssThan @ A )
= ( ^ [L2: A,U2: A] : ( inf_inf @ ( set @ A ) @ ( set_ord_atLeast @ A @ L2 ) @ ( set_ord_lessThan @ A @ U2 ) ) ) ) ) ).
% atLeastLessThan_def
thf(fact_5312_atLeastAtMost__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( set_or1337092689740270186AtMost @ A )
= ( ^ [L2: A,U2: A] : ( inf_inf @ ( set @ A ) @ ( set_ord_atLeast @ A @ L2 ) @ ( set_ord_atMost @ A @ U2 ) ) ) ) ) ).
% atLeastAtMost_def
thf(fact_5313_ivl__disj__int__one_I6_J,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [L: A,U: A] :
( ( inf_inf @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ L @ U ) @ ( set_ord_atLeast @ A @ U ) )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% ivl_disj_int_one(6)
thf(fact_5314_differentiable__divide,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V3459762299906320749_field @ B ) )
=> ! [F3: A > B,X: A,S: set @ A,G3: A > B] :
( ( differentiable @ A @ B @ F3 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( ( differentiable @ A @ B @ G3 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( ( ( G3 @ X )
!= ( zero_zero @ B ) )
=> ( differentiable @ A @ B
@ ^ [X5: A] : ( divide_divide @ B @ ( F3 @ X5 ) @ ( G3 @ X5 ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ) ).
% differentiable_divide
thf(fact_5315_differentiable__inverse,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V3459762299906320749_field @ B ) )
=> ! [F3: A > B,X: A,S: set @ A] :
( ( differentiable @ A @ B @ F3 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( ( ( F3 @ X )
!= ( zero_zero @ B ) )
=> ( differentiable @ A @ B
@ ^ [X5: A] : ( inverse_inverse @ B @ ( F3 @ X5 ) )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ).
% differentiable_inverse
thf(fact_5316_atMost__Int__atLeast,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [N: A] :
( ( inf_inf @ ( set @ A ) @ ( set_ord_atMost @ A @ N ) @ ( set_ord_atLeast @ A @ N ) )
= ( insert @ A @ N @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% atMost_Int_atLeast
thf(fact_5317_decseq__ge,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [X7: nat > A,L5: A,N: nat] :
( ( order_antimono @ nat @ A @ X7 )
=> ( ( filterlim @ nat @ A @ X7 @ ( topolo7230453075368039082e_nhds @ A @ L5 ) @ ( at_top @ nat ) )
=> ( ord_less_eq @ A @ L5 @ ( X7 @ N ) ) ) ) ) ).
% decseq_ge
thf(fact_5318_UN__atLeast__UNIV,axiom,
( ( complete_Sup_Sup @ ( set @ nat ) @ ( image2 @ nat @ ( set @ nat ) @ ( set_ord_atLeast @ nat ) @ ( top_top @ ( set @ nat ) ) ) )
= ( top_top @ ( set @ nat ) ) ) ).
% UN_atLeast_UNIV
thf(fact_5319_atLeast__Suc,axiom,
! [K2: nat] :
( ( set_ord_atLeast @ nat @ ( suc @ K2 ) )
= ( minus_minus @ ( set @ nat ) @ ( set_ord_atLeast @ nat @ K2 ) @ ( insert @ nat @ K2 @ ( bot_bot @ ( set @ nat ) ) ) ) ) ).
% atLeast_Suc
thf(fact_5320_tendsto__at__right__sequentially,axiom,
! [C: $tType,B: $tType] :
( ( ( topolo3112930676232923870pology @ B )
& ( topolo1944317154257567458pology @ B )
& ( topolo4958980785337419405_space @ C ) )
=> ! [A2: B,B2: B,X7: B > C,L5: C] :
( ( ord_less @ B @ A2 @ B2 )
=> ( ! [S6: nat > B] :
( ! [N5: nat] : ( ord_less @ B @ A2 @ ( S6 @ N5 ) )
=> ( ! [N5: nat] : ( ord_less @ B @ ( S6 @ N5 ) @ B2 )
=> ( ( order_antimono @ nat @ B @ S6 )
=> ( ( filterlim @ nat @ B @ S6 @ ( topolo7230453075368039082e_nhds @ B @ A2 ) @ ( at_top @ nat ) )
=> ( filterlim @ nat @ C
@ ^ [N2: nat] : ( X7 @ ( S6 @ N2 ) )
@ ( topolo7230453075368039082e_nhds @ C @ L5 )
@ ( at_top @ nat ) ) ) ) ) )
=> ( filterlim @ B @ C @ X7 @ ( topolo7230453075368039082e_nhds @ C @ L5 ) @ ( topolo174197925503356063within @ B @ A2 @ ( set_ord_greaterThan @ B @ A2 ) ) ) ) ) ) ).
% tendsto_at_right_sequentially
thf(fact_5321_continuous__at__Sup__antimono,axiom,
! [B: $tType,A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( topolo1944317154257567458pology @ A )
& ( condit6923001295902523014norder @ B )
& ( topolo1944317154257567458pology @ B ) )
=> ! [F3: A > B,S2: set @ A] :
( ( order_antimono @ A @ B @ F3 )
=> ( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ ( complete_Sup_Sup @ A @ S2 ) @ ( set_ord_lessThan @ A @ ( complete_Sup_Sup @ A @ S2 ) ) ) @ F3 )
=> ( ( S2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit941137186595557371_above @ A @ S2 )
=> ( ( F3 @ ( complete_Sup_Sup @ A @ S2 ) )
= ( complete_Inf_Inf @ B @ ( image2 @ A @ B @ F3 @ S2 ) ) ) ) ) ) ) ) ).
% continuous_at_Sup_antimono
thf(fact_5322_continuous__at__Inf__antimono,axiom,
! [B: $tType,A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( topolo1944317154257567458pology @ A )
& ( condit6923001295902523014norder @ B )
& ( topolo1944317154257567458pology @ B ) )
=> ! [F3: A > B,S2: set @ A] :
( ( order_antimono @ A @ B @ F3 )
=> ( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ ( complete_Inf_Inf @ A @ S2 ) @ ( set_ord_greaterThan @ A @ ( complete_Inf_Inf @ A @ S2 ) ) ) @ F3 )
=> ( ( S2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit1013018076250108175_below @ A @ S2 )
=> ( ( F3 @ ( complete_Inf_Inf @ A @ S2 ) )
= ( complete_Sup_Sup @ B @ ( image2 @ A @ B @ F3 @ S2 ) ) ) ) ) ) ) ) ).
% continuous_at_Inf_antimono
thf(fact_5323_MVT,axiom,
! [A2: real,B2: real,F3: real > real] :
( ( ord_less @ real @ A2 @ B2 )
=> ( ( topolo81223032696312382ous_on @ real @ real @ ( set_or1337092689740270186AtMost @ real @ A2 @ B2 ) @ F3 )
=> ( ! [X4: real] :
( ( ord_less @ real @ A2 @ X4 )
=> ( ( ord_less @ real @ X4 @ B2 )
=> ( differentiable @ real @ real @ F3 @ ( topolo174197925503356063within @ real @ X4 @ ( top_top @ ( set @ real ) ) ) ) ) )
=> ? [L4: real,Z3: real] :
( ( ord_less @ real @ A2 @ Z3 )
& ( ord_less @ real @ Z3 @ B2 )
& ( has_field_derivative @ real @ F3 @ L4 @ ( topolo174197925503356063within @ real @ Z3 @ ( top_top @ ( set @ real ) ) ) )
& ( ( minus_minus @ real @ ( F3 @ B2 ) @ ( F3 @ A2 ) )
= ( times_times @ real @ ( minus_minus @ real @ B2 @ A2 ) @ L4 ) ) ) ) ) ) ).
% MVT
thf(fact_5324_bdd__below_OI,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A5: set @ A,M5: A] :
( ! [X4: A] :
( ( member @ A @ X4 @ A5 )
=> ( ord_less_eq @ A @ M5 @ X4 ) )
=> ( condit1013018076250108175_below @ A @ A5 ) ) ) ).
% bdd_below.I
thf(fact_5325_bdd__belowI,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A5: set @ A,M2: A] :
( ! [X4: A] :
( ( member @ A @ X4 @ A5 )
=> ( ord_less_eq @ A @ M2 @ X4 ) )
=> ( condit1013018076250108175_below @ A @ A5 ) ) ) ).
% bdd_belowI
thf(fact_5326_bdd__above_OI,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A5: set @ A,M5: A] :
( ! [X4: A] :
( ( member @ A @ X4 @ A5 )
=> ( ord_less_eq @ A @ X4 @ M5 ) )
=> ( condit941137186595557371_above @ A @ A5 ) ) ) ).
% bdd_above.I
thf(fact_5327_bdd__below__empty,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( condit1013018076250108175_below @ A @ ( bot_bot @ ( set @ A ) ) ) ) ).
% bdd_below_empty
thf(fact_5328_bdd__above__empty,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( condit941137186595557371_above @ A @ ( bot_bot @ ( set @ A ) ) ) ) ).
% bdd_above_empty
thf(fact_5329_bdd__below__UN,axiom,
! [A: $tType,B: $tType] :
( ( lattice @ A )
=> ! [I6: set @ B,A5: B > ( set @ A )] :
( ( finite_finite2 @ B @ I6 )
=> ( ( condit1013018076250108175_below @ A @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ A5 @ I6 ) ) )
= ( ! [X5: B] :
( ( member @ B @ X5 @ I6 )
=> ( condit1013018076250108175_below @ A @ ( A5 @ X5 ) ) ) ) ) ) ) ).
% bdd_below_UN
thf(fact_5330_bdd__above__UN,axiom,
! [A: $tType,B: $tType] :
( ( lattice @ A )
=> ! [I6: set @ B,A5: B > ( set @ A )] :
( ( finite_finite2 @ B @ I6 )
=> ( ( condit941137186595557371_above @ A @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ A5 @ I6 ) ) )
= ( ! [X5: B] :
( ( member @ B @ X5 @ I6 )
=> ( condit941137186595557371_above @ A @ ( A5 @ X5 ) ) ) ) ) ) ) ).
% bdd_above_UN
thf(fact_5331_cInf__le__cSup,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A5: set @ A] :
( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit941137186595557371_above @ A @ A5 )
=> ( ( condit1013018076250108175_below @ A @ A5 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A5 ) @ ( complete_Sup_Sup @ A @ A5 ) ) ) ) ) ) ).
% cInf_le_cSup
thf(fact_5332_continuous__on__add,axiom,
! [B: $tType,D: $tType] :
( ( ( topolo4958980785337419405_space @ D )
& ( topolo6943815403480290642id_add @ B ) )
=> ! [S: set @ D,F3: D > B,G3: D > B] :
( ( topolo81223032696312382ous_on @ D @ B @ S @ F3 )
=> ( ( topolo81223032696312382ous_on @ D @ B @ S @ G3 )
=> ( topolo81223032696312382ous_on @ D @ B @ S
@ ^ [X5: D] : ( plus_plus @ B @ ( F3 @ X5 ) @ ( G3 @ X5 ) ) ) ) ) ) ).
% continuous_on_add
thf(fact_5333_continuous__on__max,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ A )
& ( topolo1944317154257567458pology @ B ) )
=> ! [A5: set @ A,F3: A > B,G3: A > B] :
( ( topolo81223032696312382ous_on @ A @ B @ A5 @ F3 )
=> ( ( topolo81223032696312382ous_on @ A @ B @ A5 @ G3 )
=> ( topolo81223032696312382ous_on @ A @ B @ A5
@ ^ [X5: A] : ( ord_max @ B @ ( F3 @ X5 ) @ ( G3 @ X5 ) ) ) ) ) ) ).
% continuous_on_max
thf(fact_5334_bdd__above__finite,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( condit941137186595557371_above @ A @ A5 ) ) ) ).
% bdd_above_finite
thf(fact_5335_bdd__below__finite,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( condit1013018076250108175_below @ A @ A5 ) ) ) ).
% bdd_below_finite
thf(fact_5336_bdd__above__nat,axiom,
( ( condit941137186595557371_above @ nat )
= ( finite_finite2 @ nat ) ) ).
% bdd_above_nat
thf(fact_5337_continuous__on__Pair,axiom,
! [C: $tType,B: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ A )
& ( topolo4958980785337419405_space @ B )
& ( topolo4958980785337419405_space @ C ) )
=> ! [S: set @ A,F3: A > B,G3: A > C] :
( ( topolo81223032696312382ous_on @ A @ B @ S @ F3 )
=> ( ( topolo81223032696312382ous_on @ A @ C @ S @ G3 )
=> ( topolo81223032696312382ous_on @ A @ ( product_prod @ B @ C ) @ S
@ ^ [X5: A] : ( product_Pair @ B @ C @ ( F3 @ X5 ) @ ( G3 @ X5 ) ) ) ) ) ) ).
% continuous_on_Pair
thf(fact_5338_continuous__on__sing,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [X: A,F3: A > B] : ( topolo81223032696312382ous_on @ A @ B @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) @ F3 ) ) ).
% continuous_on_sing
thf(fact_5339_IVT2_H,axiom,
! [A: $tType,B: $tType] :
( ( ( topolo1944317154257567458pology @ B )
& ( topolo8458572112393995274pology @ A ) )
=> ! [F3: A > B,B2: A,Y: B,A2: A] :
( ( ord_less_eq @ B @ ( F3 @ B2 ) @ Y )
=> ( ( ord_less_eq @ B @ Y @ ( F3 @ A2 ) )
=> ( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( topolo81223032696312382ous_on @ A @ B @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) @ F3 )
=> ? [X4: A] :
( ( ord_less_eq @ A @ A2 @ X4 )
& ( ord_less_eq @ A @ X4 @ B2 )
& ( ( F3 @ X4 )
= Y ) ) ) ) ) ) ) ).
% IVT2'
thf(fact_5340_IVT_H,axiom,
! [A: $tType,B: $tType] :
( ( ( topolo1944317154257567458pology @ B )
& ( topolo8458572112393995274pology @ A ) )
=> ! [F3: A > B,A2: A,Y: B,B2: A] :
( ( ord_less_eq @ B @ ( F3 @ A2 ) @ Y )
=> ( ( ord_less_eq @ B @ Y @ ( F3 @ B2 ) )
=> ( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( topolo81223032696312382ous_on @ A @ B @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) @ F3 )
=> ? [X4: A] :
( ( ord_less_eq @ A @ A2 @ X4 )
& ( ord_less_eq @ A @ X4 @ B2 )
& ( ( F3 @ X4 )
= Y ) ) ) ) ) ) ) ).
% IVT'
thf(fact_5341_bdd__below_OE,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A5: set @ A] :
( ( condit1013018076250108175_below @ A @ A5 )
=> ~ ! [M8: A] :
~ ! [X3: A] :
( ( member @ A @ X3 @ A5 )
=> ( ord_less_eq @ A @ M8 @ X3 ) ) ) ) ).
% bdd_below.E
thf(fact_5342_bdd__below_Ounfold,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( ( condit1013018076250108175_below @ A )
= ( ^ [A7: set @ A] :
? [M9: A] :
! [X5: A] :
( ( member @ A @ X5 @ A7 )
=> ( ord_less_eq @ A @ M9 @ X5 ) ) ) ) ) ).
% bdd_below.unfold
thf(fact_5343_bdd__above_OE,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ! [A5: set @ A] :
( ( condit941137186595557371_above @ A @ A5 )
=> ~ ! [M8: A] :
~ ! [X3: A] :
( ( member @ A @ X3 @ A5 )
=> ( ord_less_eq @ A @ X3 @ M8 ) ) ) ) ).
% bdd_above.E
thf(fact_5344_bdd__above_Ounfold,axiom,
! [A: $tType] :
( ( preorder @ A )
=> ( ( condit941137186595557371_above @ A )
= ( ^ [A7: set @ A] :
? [M9: A] :
! [X5: A] :
( ( member @ A @ X5 @ A7 )
=> ( ord_less_eq @ A @ X5 @ M9 ) ) ) ) ) ).
% bdd_above.unfold
thf(fact_5345_continuous__on__empty,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [F3: A > B] : ( topolo81223032696312382ous_on @ A @ B @ ( bot_bot @ ( set @ A ) ) @ F3 ) ) ).
% continuous_on_empty
thf(fact_5346_bdd__belowI2,axiom,
! [A: $tType,B: $tType] :
( ( preorder @ A )
=> ! [A5: set @ B,M2: A,F3: B > A] :
( ! [X4: B] :
( ( member @ B @ X4 @ A5 )
=> ( ord_less_eq @ A @ M2 @ ( F3 @ X4 ) ) )
=> ( condit1013018076250108175_below @ A @ ( image2 @ B @ A @ F3 @ A5 ) ) ) ) ).
% bdd_belowI2
thf(fact_5347_bdd__below_OI2,axiom,
! [A: $tType,B: $tType] :
( ( preorder @ A )
=> ! [A5: set @ B,M5: A,F3: B > A] :
( ! [X4: B] :
( ( member @ B @ X4 @ A5 )
=> ( ord_less_eq @ A @ M5 @ ( F3 @ X4 ) ) )
=> ( condit1013018076250108175_below @ A @ ( image2 @ B @ A @ F3 @ A5 ) ) ) ) ).
% bdd_below.I2
thf(fact_5348_bdd__above_OI2,axiom,
! [A: $tType,B: $tType] :
( ( preorder @ A )
=> ! [A5: set @ B,F3: B > A,M5: A] :
( ! [X4: B] :
( ( member @ B @ X4 @ A5 )
=> ( ord_less_eq @ A @ ( F3 @ X4 ) @ M5 ) )
=> ( condit941137186595557371_above @ A @ ( image2 @ B @ A @ F3 @ A5 ) ) ) ) ).
% bdd_above.I2
thf(fact_5349_cInf__lower2,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X: A,X7: set @ A,Y: A] :
( ( member @ A @ X @ X7 )
=> ( ( ord_less_eq @ A @ X @ Y )
=> ( ( condit1013018076250108175_below @ A @ X7 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ X7 ) @ Y ) ) ) ) ) ).
% cInf_lower2
thf(fact_5350_cInf__lower,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X: A,X7: set @ A] :
( ( member @ A @ X @ X7 )
=> ( ( condit1013018076250108175_below @ A @ X7 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ X7 ) @ X ) ) ) ) ).
% cInf_lower
thf(fact_5351_cSup__upper2,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X: A,X7: set @ A,Y: A] :
( ( member @ A @ X @ X7 )
=> ( ( ord_less_eq @ A @ Y @ X )
=> ( ( condit941137186595557371_above @ A @ X7 )
=> ( ord_less_eq @ A @ Y @ ( complete_Sup_Sup @ A @ X7 ) ) ) ) ) ) ).
% cSup_upper2
thf(fact_5352_cSup__upper,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X: A,X7: set @ A] :
( ( member @ A @ X @ X7 )
=> ( ( condit941137186595557371_above @ A @ X7 )
=> ( ord_less_eq @ A @ X @ ( complete_Sup_Sup @ A @ X7 ) ) ) ) ) ).
% cSup_upper
thf(fact_5353_continuous__on__divide,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ A )
& ( real_V3459762299906320749_field @ B ) )
=> ! [S: set @ A,F3: A > B,G3: A > B] :
( ( topolo81223032696312382ous_on @ A @ B @ S @ F3 )
=> ( ( topolo81223032696312382ous_on @ A @ B @ S @ G3 )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ S )
=> ( ( G3 @ X4 )
!= ( zero_zero @ B ) ) )
=> ( topolo81223032696312382ous_on @ A @ B @ S
@ ^ [X5: A] : ( divide_divide @ B @ ( F3 @ X5 ) @ ( G3 @ X5 ) ) ) ) ) ) ) ).
% continuous_on_divide
thf(fact_5354_continuous__on__inverse,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ A )
& ( real_V8999393235501362500lgebra @ B ) )
=> ! [S: set @ A,F3: A > B] :
( ( topolo81223032696312382ous_on @ A @ B @ S @ F3 )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ S )
=> ( ( F3 @ X4 )
!= ( zero_zero @ B ) ) )
=> ( topolo81223032696312382ous_on @ A @ B @ S
@ ^ [X5: A] : ( inverse_inverse @ B @ ( F3 @ X5 ) ) ) ) ) ) ).
% continuous_on_inverse
thf(fact_5355_continuous__on__sgn,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ A )
& ( real_V822414075346904944vector @ B ) )
=> ! [S: set @ A,F3: A > B] :
( ( topolo81223032696312382ous_on @ A @ B @ S @ F3 )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ S )
=> ( ( F3 @ X4 )
!= ( zero_zero @ B ) ) )
=> ( topolo81223032696312382ous_on @ A @ B @ S
@ ^ [X5: A] : ( sgn_sgn @ B @ ( F3 @ X5 ) ) ) ) ) ) ).
% continuous_on_sgn
thf(fact_5356_continuous__onI__mono,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo1944317154257567458pology @ A )
& ( dense_order @ B )
& ( topolo1944317154257567458pology @ B ) )
=> ! [F3: A > B,A5: set @ A] :
( ( topolo1002775350975398744n_open @ B @ ( image2 @ A @ B @ F3 @ A5 ) )
=> ( ! [X4: A,Y3: A] :
( ( member @ A @ X4 @ A5 )
=> ( ( member @ A @ Y3 @ A5 )
=> ( ( ord_less_eq @ A @ X4 @ Y3 )
=> ( ord_less_eq @ B @ ( F3 @ X4 ) @ ( F3 @ Y3 ) ) ) ) )
=> ( topolo81223032696312382ous_on @ A @ B @ A5 @ F3 ) ) ) ) ).
% continuous_onI_mono
thf(fact_5357_cINF__lower,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [F3: B > A,A5: set @ B,X: B] :
( ( condit1013018076250108175_below @ A @ ( image2 @ B @ A @ F3 @ A5 ) )
=> ( ( member @ B @ X @ A5 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ A5 ) ) @ ( F3 @ X ) ) ) ) ) ).
% cINF_lower
thf(fact_5358_cINF__lower2,axiom,
! [B: $tType,A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [F3: B > A,A5: set @ B,X: B,U: A] :
( ( condit1013018076250108175_below @ A @ ( image2 @ B @ A @ F3 @ A5 ) )
=> ( ( member @ B @ X @ A5 )
=> ( ( ord_less_eq @ A @ ( F3 @ X ) @ U )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ A5 ) ) @ U ) ) ) ) ) ).
% cINF_lower2
thf(fact_5359_cInf__mono,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [B6: set @ A,A5: set @ A] :
( ( B6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit1013018076250108175_below @ A @ A5 )
=> ( ! [B4: A] :
( ( member @ A @ B4 @ B6 )
=> ? [X3: A] :
( ( member @ A @ X3 @ A5 )
& ( ord_less_eq @ A @ X3 @ B4 ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A5 ) @ ( complete_Inf_Inf @ A @ B6 ) ) ) ) ) ) ).
% cInf_mono
thf(fact_5360_le__cInf__iff,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [S2: set @ A,A2: A] :
( ( S2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit1013018076250108175_below @ A @ S2 )
=> ( ( ord_less_eq @ A @ A2 @ ( complete_Inf_Inf @ A @ S2 ) )
= ( ! [X5: A] :
( ( member @ A @ X5 @ S2 )
=> ( ord_less_eq @ A @ A2 @ X5 ) ) ) ) ) ) ) ).
% le_cInf_iff
thf(fact_5361_cSUP__upper,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X: B,A5: set @ B,F3: B > A] :
( ( member @ B @ X @ A5 )
=> ( ( condit941137186595557371_above @ A @ ( image2 @ B @ A @ F3 @ A5 ) )
=> ( ord_less_eq @ A @ ( F3 @ X ) @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A5 ) ) ) ) ) ) ).
% cSUP_upper
thf(fact_5362_cSUP__upper2,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [F3: B > A,A5: set @ B,X: B,U: A] :
( ( condit941137186595557371_above @ A @ ( image2 @ B @ A @ F3 @ A5 ) )
=> ( ( member @ B @ X @ A5 )
=> ( ( ord_less_eq @ A @ U @ ( F3 @ X ) )
=> ( ord_less_eq @ A @ U @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A5 ) ) ) ) ) ) ) ).
% cSUP_upper2
thf(fact_5363_cInf__less__iff,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [X7: set @ A,Y: A] :
( ( X7
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit1013018076250108175_below @ A @ X7 )
=> ( ( ord_less @ A @ ( complete_Inf_Inf @ A @ X7 ) @ Y )
= ( ? [X5: A] :
( ( member @ A @ X5 @ X7 )
& ( ord_less @ A @ X5 @ Y ) ) ) ) ) ) ) ).
% cInf_less_iff
thf(fact_5364_cSup__mono,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [B6: set @ A,A5: set @ A] :
( ( B6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit941137186595557371_above @ A @ A5 )
=> ( ! [B4: A] :
( ( member @ A @ B4 @ B6 )
=> ? [X3: A] :
( ( member @ A @ X3 @ A5 )
& ( ord_less_eq @ A @ B4 @ X3 ) ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ B6 ) @ ( complete_Sup_Sup @ A @ A5 ) ) ) ) ) ) ).
% cSup_mono
thf(fact_5365_cSup__le__iff,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [S2: set @ A,A2: A] :
( ( S2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit941137186595557371_above @ A @ S2 )
=> ( ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ S2 ) @ A2 )
= ( ! [X5: A] :
( ( member @ A @ X5 @ S2 )
=> ( ord_less_eq @ A @ X5 @ A2 ) ) ) ) ) ) ) ).
% cSup_le_iff
thf(fact_5366_less__cSup__iff,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [X7: set @ A,Y: A] :
( ( X7
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit941137186595557371_above @ A @ X7 )
=> ( ( ord_less @ A @ Y @ ( complete_Sup_Sup @ A @ X7 ) )
= ( ? [X5: A] :
( ( member @ A @ X5 @ X7 )
& ( ord_less @ A @ Y @ X5 ) ) ) ) ) ) ) ).
% less_cSup_iff
thf(fact_5367_open__Collect__less,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ A )
& ( topolo1944317154257567458pology @ B ) )
=> ! [F3: A > B,G3: A > B] :
( ( topolo81223032696312382ous_on @ A @ B @ ( top_top @ ( set @ A ) ) @ F3 )
=> ( ( topolo81223032696312382ous_on @ A @ B @ ( top_top @ ( set @ A ) ) @ G3 )
=> ( topolo1002775350975398744n_open @ A
@ ( collect @ A
@ ^ [X5: A] : ( ord_less @ B @ ( F3 @ X5 ) @ ( G3 @ X5 ) ) ) ) ) ) ) ).
% open_Collect_less
thf(fact_5368_continuous__on__tan,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [S: set @ A,F3: A > A] :
( ( topolo81223032696312382ous_on @ A @ A @ S @ F3 )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ S )
=> ( ( cos @ A @ ( F3 @ X4 ) )
!= ( zero_zero @ A ) ) )
=> ( topolo81223032696312382ous_on @ A @ A @ S
@ ^ [X5: A] : ( tan @ A @ ( F3 @ X5 ) ) ) ) ) ) ).
% continuous_on_tan
thf(fact_5369_open__Collect__less__Int,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [S: set @ A,F3: A > real,G3: A > real] :
( ( topolo81223032696312382ous_on @ A @ real @ S @ F3 )
=> ( ( topolo81223032696312382ous_on @ A @ real @ S @ G3 )
=> ? [A8: set @ A] :
( ( topolo1002775350975398744n_open @ A @ A8 )
& ( ( inf_inf @ ( set @ A ) @ A8 @ S )
= ( collect @ A
@ ^ [X5: A] :
( ( member @ A @ X5 @ S )
& ( ord_less @ real @ ( F3 @ X5 ) @ ( G3 @ X5 ) ) ) ) ) ) ) ) ) ).
% open_Collect_less_Int
thf(fact_5370_continuous__on__cot,axiom,
! [A: $tType] :
( ( ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [S: set @ A,F3: A > A] :
( ( topolo81223032696312382ous_on @ A @ A @ S @ F3 )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ S )
=> ( ( sin @ A @ ( F3 @ X4 ) )
!= ( zero_zero @ A ) ) )
=> ( topolo81223032696312382ous_on @ A @ A @ S
@ ^ [X5: A] : ( cot @ A @ ( F3 @ X5 ) ) ) ) ) ) ).
% continuous_on_cot
thf(fact_5371_continuous__on__tanh,axiom,
! [A: $tType,C: $tType] :
( ( ( topolo4958980785337419405_space @ C )
& ( real_Vector_banach @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [A5: set @ C,F3: C > A] :
( ( topolo81223032696312382ous_on @ C @ A @ A5 @ F3 )
=> ( ! [X4: C] :
( ( member @ C @ X4 @ A5 )
=> ( ( cosh @ A @ ( F3 @ X4 ) )
!= ( zero_zero @ A ) ) )
=> ( topolo81223032696312382ous_on @ C @ A @ A5
@ ^ [X5: C] : ( tanh @ A @ ( F3 @ X5 ) ) ) ) ) ) ).
% continuous_on_tanh
thf(fact_5372_less__cINF__D,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [F3: B > A,A5: set @ B,Y: A,I2: B] :
( ( condit1013018076250108175_below @ A @ ( image2 @ B @ A @ F3 @ A5 ) )
=> ( ( ord_less @ A @ Y @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ A5 ) ) )
=> ( ( member @ B @ I2 @ A5 )
=> ( ord_less @ A @ Y @ ( F3 @ I2 ) ) ) ) ) ) ).
% less_cINF_D
thf(fact_5373_cSUP__lessD,axiom,
! [B: $tType,A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [F3: B > A,A5: set @ B,Y: A,I2: B] :
( ( condit941137186595557371_above @ A @ ( image2 @ B @ A @ F3 @ A5 ) )
=> ( ( ord_less @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A5 ) ) @ Y )
=> ( ( member @ B @ I2 @ A5 )
=> ( ord_less @ A @ ( F3 @ I2 ) @ Y ) ) ) ) ) ).
% cSUP_lessD
thf(fact_5374_cINF__mono,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [B6: set @ B,F3: C > A,A5: set @ C,G3: B > A] :
( ( B6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit1013018076250108175_below @ A @ ( image2 @ C @ A @ F3 @ A5 ) )
=> ( ! [M: B] :
( ( member @ B @ M @ B6 )
=> ? [X3: C] :
( ( member @ C @ X3 @ A5 )
& ( ord_less_eq @ A @ ( F3 @ X3 ) @ ( G3 @ M ) ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image2 @ C @ A @ F3 @ A5 ) ) @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ G3 @ B6 ) ) ) ) ) ) ) ).
% cINF_mono
thf(fact_5375_le__cINF__iff,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A5: set @ B,F3: B > A,U: A] :
( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit1013018076250108175_below @ A @ ( image2 @ B @ A @ F3 @ A5 ) )
=> ( ( ord_less_eq @ A @ U @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ A5 ) ) )
= ( ! [X5: B] :
( ( member @ B @ X5 @ A5 )
=> ( ord_less_eq @ A @ U @ ( F3 @ X5 ) ) ) ) ) ) ) ) ).
% le_cINF_iff
thf(fact_5376_cInf__superset__mono,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A5: set @ A,B6: set @ A] :
( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit1013018076250108175_below @ A @ B6 )
=> ( ( ord_less_eq @ ( set @ A ) @ A5 @ B6 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ B6 ) @ ( complete_Inf_Inf @ A @ A5 ) ) ) ) ) ) ).
% cInf_superset_mono
thf(fact_5377_cSUP__mono,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A5: set @ B,G3: C > A,B6: set @ C,F3: B > A] :
( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit941137186595557371_above @ A @ ( image2 @ C @ A @ G3 @ B6 ) )
=> ( ! [N3: B] :
( ( member @ B @ N3 @ A5 )
=> ? [X3: C] :
( ( member @ C @ X3 @ B6 )
& ( ord_less_eq @ A @ ( F3 @ N3 ) @ ( G3 @ X3 ) ) ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A5 ) ) @ ( complete_Sup_Sup @ A @ ( image2 @ C @ A @ G3 @ B6 ) ) ) ) ) ) ) ).
% cSUP_mono
thf(fact_5378_cSUP__le__iff,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A5: set @ B,F3: B > A,U: A] :
( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit941137186595557371_above @ A @ ( image2 @ B @ A @ F3 @ A5 ) )
=> ( ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A5 ) ) @ U )
= ( ! [X5: B] :
( ( member @ B @ X5 @ A5 )
=> ( ord_less_eq @ A @ ( F3 @ X5 ) @ U ) ) ) ) ) ) ) ).
% cSUP_le_iff
thf(fact_5379_cSup__subset__mono,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A5: set @ A,B6: set @ A] :
( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit941137186595557371_above @ A @ B6 )
=> ( ( ord_less_eq @ ( set @ A ) @ A5 @ B6 )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A5 ) @ ( complete_Sup_Sup @ A @ B6 ) ) ) ) ) ) ).
% cSup_subset_mono
thf(fact_5380_cInf__insert__If,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X7: set @ A,A2: A] :
( ( condit1013018076250108175_below @ A @ X7 )
=> ( ( ( X7
= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Inf_Inf @ A @ ( insert @ A @ A2 @ X7 ) )
= A2 ) )
& ( ( X7
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Inf_Inf @ A @ ( insert @ A @ A2 @ X7 ) )
= ( inf_inf @ A @ A2 @ ( complete_Inf_Inf @ A @ X7 ) ) ) ) ) ) ) ).
% cInf_insert_If
thf(fact_5381_cInf__insert,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X7: set @ A,A2: A] :
( ( X7
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit1013018076250108175_below @ A @ X7 )
=> ( ( complete_Inf_Inf @ A @ ( insert @ A @ A2 @ X7 ) )
= ( inf_inf @ A @ A2 @ ( complete_Inf_Inf @ A @ X7 ) ) ) ) ) ) ).
% cInf_insert
thf(fact_5382_open__Collect__positive,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [S: set @ A,F3: A > real] :
( ( topolo81223032696312382ous_on @ A @ real @ S @ F3 )
=> ? [A8: set @ A] :
( ( topolo1002775350975398744n_open @ A @ A8 )
& ( ( inf_inf @ ( set @ A ) @ A8 @ S )
= ( collect @ A
@ ^ [X5: A] :
( ( member @ A @ X5 @ S )
& ( ord_less @ real @ ( zero_zero @ real ) @ ( F3 @ X5 ) ) ) ) ) ) ) ) ).
% open_Collect_positive
thf(fact_5383_continuous__on__powr_H,axiom,
! [C: $tType] :
( ( topolo4958980785337419405_space @ C )
=> ! [S: set @ C,F3: C > real,G3: C > real] :
( ( topolo81223032696312382ous_on @ C @ real @ S @ F3 )
=> ( ( topolo81223032696312382ous_on @ C @ real @ S @ G3 )
=> ( ! [X4: C] :
( ( member @ C @ X4 @ S )
=> ( ( ord_less_eq @ real @ ( zero_zero @ real ) @ ( F3 @ X4 ) )
& ( ( ( F3 @ X4 )
= ( zero_zero @ real ) )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( G3 @ X4 ) ) ) ) )
=> ( topolo81223032696312382ous_on @ C @ real @ S
@ ^ [X5: C] : ( powr @ real @ ( F3 @ X5 ) @ ( G3 @ X5 ) ) ) ) ) ) ) ).
% continuous_on_powr'
thf(fact_5384_continuous__on__log,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [S: set @ A,F3: A > real,G3: A > real] :
( ( topolo81223032696312382ous_on @ A @ real @ S @ F3 )
=> ( ( topolo81223032696312382ous_on @ A @ real @ S @ G3 )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ S )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( F3 @ X4 ) ) )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ S )
=> ( ( F3 @ X4 )
!= ( one_one @ real ) ) )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ S )
=> ( ord_less @ real @ ( zero_zero @ real ) @ ( G3 @ X4 ) ) )
=> ( topolo81223032696312382ous_on @ A @ real @ S
@ ^ [X5: A] : ( log @ ( F3 @ X5 ) @ ( G3 @ X5 ) ) ) ) ) ) ) ) ) ).
% continuous_on_log
thf(fact_5385_cINF__less__iff,axiom,
! [A: $tType,B: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [A5: set @ B,F3: B > A,A2: A] :
( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit1013018076250108175_below @ A @ ( image2 @ B @ A @ F3 @ A5 ) )
=> ( ( ord_less @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ A5 ) ) @ A2 )
= ( ? [X5: B] :
( ( member @ B @ X5 @ A5 )
& ( ord_less @ A @ ( F3 @ X5 ) @ A2 ) ) ) ) ) ) ) ).
% cINF_less_iff
thf(fact_5386_less__cSUP__iff,axiom,
! [A: $tType,B: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [A5: set @ B,F3: B > A,A2: A] :
( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit941137186595557371_above @ A @ ( image2 @ B @ A @ F3 @ A5 ) )
=> ( ( ord_less @ A @ A2 @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A5 ) ) )
= ( ? [X5: B] :
( ( member @ B @ X5 @ A5 )
& ( ord_less @ A @ A2 @ ( F3 @ X5 ) ) ) ) ) ) ) ) ).
% less_cSUP_iff
thf(fact_5387_cINF__inf__distrib,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A5: set @ B,F3: B > A,G3: B > A] :
( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit1013018076250108175_below @ A @ ( image2 @ B @ A @ F3 @ A5 ) )
=> ( ( condit1013018076250108175_below @ A @ ( image2 @ B @ A @ G3 @ A5 ) )
=> ( ( inf_inf @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ A5 ) ) @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ G3 @ A5 ) ) )
= ( complete_Inf_Inf @ A
@ ( image2 @ B @ A
@ ^ [A6: B] : ( inf_inf @ A @ ( F3 @ A6 ) @ ( G3 @ A6 ) )
@ A5 ) ) ) ) ) ) ) ).
% cINF_inf_distrib
thf(fact_5388_DERIV__atLeastAtMost__imp__continuous__on,axiom,
! [A: $tType] :
( ( ( ord @ A )
& ( real_V3459762299906320749_field @ A ) )
=> ! [A2: A,B2: A,F3: A > A] :
( ! [X4: A] :
( ( ord_less_eq @ A @ A2 @ X4 )
=> ( ( ord_less_eq @ A @ X4 @ B2 )
=> ? [Y4: A] : ( has_field_derivative @ A @ F3 @ Y4 @ ( topolo174197925503356063within @ A @ X4 @ ( top_top @ ( set @ A ) ) ) ) ) )
=> ( topolo81223032696312382ous_on @ A @ A @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) @ F3 ) ) ) ).
% DERIV_atLeastAtMost_imp_continuous_on
thf(fact_5389_cINF__superset__mono,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A5: set @ B,G3: B > A,B6: set @ B,F3: B > A] :
( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit1013018076250108175_below @ A @ ( image2 @ B @ A @ G3 @ B6 ) )
=> ( ( ord_less_eq @ ( set @ B ) @ A5 @ B6 )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ B6 )
=> ( ord_less_eq @ A @ ( G3 @ X4 ) @ ( F3 @ X4 ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ G3 @ B6 ) ) @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ A5 ) ) ) ) ) ) ) ) ).
% cINF_superset_mono
thf(fact_5390_Rolle__deriv,axiom,
! [A2: real,B2: real,F3: real > real,F10: real > real > real] :
( ( ord_less @ real @ A2 @ B2 )
=> ( ( ( F3 @ A2 )
= ( F3 @ B2 ) )
=> ( ( topolo81223032696312382ous_on @ real @ real @ ( set_or1337092689740270186AtMost @ real @ A2 @ B2 ) @ F3 )
=> ( ! [X4: real] :
( ( ord_less @ real @ A2 @ X4 )
=> ( ( ord_less @ real @ X4 @ B2 )
=> ( has_derivative @ real @ real @ F3 @ ( F10 @ X4 ) @ ( topolo174197925503356063within @ real @ X4 @ ( top_top @ ( set @ real ) ) ) ) ) )
=> ? [Z3: real] :
( ( ord_less @ real @ A2 @ Z3 )
& ( ord_less @ real @ Z3 @ B2 )
& ( ( F10 @ Z3 )
= ( ^ [V5: real] : ( zero_zero @ real ) ) ) ) ) ) ) ) ).
% Rolle_deriv
thf(fact_5391_cSUP__subset__mono,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A5: set @ B,G3: B > A,B6: set @ B,F3: B > A] :
( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit941137186595557371_above @ A @ ( image2 @ B @ A @ G3 @ B6 ) )
=> ( ( ord_less_eq @ ( set @ B ) @ A5 @ B6 )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ A5 )
=> ( ord_less_eq @ A @ ( F3 @ X4 ) @ ( G3 @ X4 ) ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A5 ) ) @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ G3 @ B6 ) ) ) ) ) ) ) ) ).
% cSUP_subset_mono
thf(fact_5392_less__eq__cInf__inter,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A5: set @ A,B6: set @ A] :
( ( condit1013018076250108175_below @ A @ A5 )
=> ( ( condit1013018076250108175_below @ A @ B6 )
=> ( ( ( inf_inf @ ( set @ A ) @ A5 @ B6 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ord_less_eq @ A @ ( inf_inf @ A @ ( complete_Inf_Inf @ A @ A5 ) @ ( complete_Inf_Inf @ A @ B6 ) ) @ ( complete_Inf_Inf @ A @ ( inf_inf @ ( set @ A ) @ A5 @ B6 ) ) ) ) ) ) ) ).
% less_eq_cInf_inter
thf(fact_5393_cINF__insert,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A5: set @ B,F3: B > A,A2: B] :
( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit1013018076250108175_below @ A @ ( image2 @ B @ A @ F3 @ A5 ) )
=> ( ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ ( insert @ B @ A2 @ A5 ) ) )
= ( inf_inf @ A @ ( F3 @ A2 ) @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ A5 ) ) ) ) ) ) ) ).
% cINF_insert
thf(fact_5394_mvt,axiom,
! [A2: real,B2: real,F3: real > real,F10: real > real > real] :
( ( ord_less @ real @ A2 @ B2 )
=> ( ( topolo81223032696312382ous_on @ real @ real @ ( set_or1337092689740270186AtMost @ real @ A2 @ B2 ) @ F3 )
=> ( ! [X4: real] :
( ( ord_less @ real @ A2 @ X4 )
=> ( ( ord_less @ real @ X4 @ B2 )
=> ( has_derivative @ real @ real @ F3 @ ( F10 @ X4 ) @ ( topolo174197925503356063within @ real @ X4 @ ( top_top @ ( set @ real ) ) ) ) ) )
=> ~ ! [Xi: real] :
( ( ord_less @ real @ A2 @ Xi )
=> ( ( ord_less @ real @ Xi @ B2 )
=> ( ( minus_minus @ real @ ( F3 @ B2 ) @ ( F3 @ A2 ) )
!= ( F10 @ Xi @ ( minus_minus @ real @ B2 @ A2 ) ) ) ) ) ) ) ) ).
% mvt
thf(fact_5395_cINF__UNION,axiom,
! [B: $tType,D: $tType,C: $tType] :
( ( condit1219197933456340205attice @ B )
=> ! [A5: set @ C,B6: C > ( set @ D ),F3: D > B] :
( ( A5
!= ( bot_bot @ ( set @ C ) ) )
=> ( ! [X4: C] :
( ( member @ C @ X4 @ A5 )
=> ( ( B6 @ X4 )
!= ( bot_bot @ ( set @ D ) ) ) )
=> ( ( condit1013018076250108175_below @ B
@ ( complete_Sup_Sup @ ( set @ B )
@ ( image2 @ C @ ( set @ B )
@ ^ [X5: C] : ( image2 @ D @ B @ F3 @ ( B6 @ X5 ) )
@ A5 ) ) )
=> ( ( complete_Inf_Inf @ B @ ( image2 @ D @ B @ F3 @ ( complete_Sup_Sup @ ( set @ D ) @ ( image2 @ C @ ( set @ D ) @ B6 @ A5 ) ) ) )
= ( complete_Inf_Inf @ B
@ ( image2 @ C @ B
@ ^ [X5: C] : ( complete_Inf_Inf @ B @ ( image2 @ D @ B @ F3 @ ( B6 @ X5 ) ) )
@ A5 ) ) ) ) ) ) ) ).
% cINF_UNION
thf(fact_5396_cSUP__UNION,axiom,
! [B: $tType,D: $tType,C: $tType] :
( ( condit1219197933456340205attice @ B )
=> ! [A5: set @ C,B6: C > ( set @ D ),F3: D > B] :
( ( A5
!= ( bot_bot @ ( set @ C ) ) )
=> ( ! [X4: C] :
( ( member @ C @ X4 @ A5 )
=> ( ( B6 @ X4 )
!= ( bot_bot @ ( set @ D ) ) ) )
=> ( ( condit941137186595557371_above @ B
@ ( complete_Sup_Sup @ ( set @ B )
@ ( image2 @ C @ ( set @ B )
@ ^ [X5: C] : ( image2 @ D @ B @ F3 @ ( B6 @ X5 ) )
@ A5 ) ) )
=> ( ( complete_Sup_Sup @ B @ ( image2 @ D @ B @ F3 @ ( complete_Sup_Sup @ ( set @ D ) @ ( image2 @ C @ ( set @ D ) @ B6 @ A5 ) ) ) )
= ( complete_Sup_Sup @ B
@ ( image2 @ C @ B
@ ^ [X5: C] : ( complete_Sup_Sup @ B @ ( image2 @ D @ B @ F3 @ ( B6 @ X5 ) ) )
@ A5 ) ) ) ) ) ) ) ).
% cSUP_UNION
thf(fact_5397_continuous__on__Icc__at__leftD,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo1944317154257567458pology @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [A2: A,B2: A,F3: A > B] :
( ( topolo81223032696312382ous_on @ A @ B @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) @ F3 )
=> ( ( ord_less @ A @ A2 @ B2 )
=> ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ ( F3 @ B2 ) ) @ ( topolo174197925503356063within @ A @ B2 @ ( set_ord_lessThan @ A @ B2 ) ) ) ) ) ) ).
% continuous_on_Icc_at_leftD
thf(fact_5398_continuous__on__Icc__at__rightD,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo1944317154257567458pology @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [A2: A,B2: A,F3: A > B] :
( ( topolo81223032696312382ous_on @ A @ B @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) @ F3 )
=> ( ( ord_less @ A @ A2 @ B2 )
=> ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ ( F3 @ A2 ) ) @ ( topolo174197925503356063within @ A @ A2 @ ( set_ord_greaterThan @ A @ A2 ) ) ) ) ) ) ).
% continuous_on_Icc_at_rightD
thf(fact_5399_DERIV__isconst__end,axiom,
! [A2: real,B2: real,F3: real > real] :
( ( ord_less @ real @ A2 @ B2 )
=> ( ( topolo81223032696312382ous_on @ real @ real @ ( set_or1337092689740270186AtMost @ real @ A2 @ B2 ) @ F3 )
=> ( ! [X4: real] :
( ( ord_less @ real @ A2 @ X4 )
=> ( ( ord_less @ real @ X4 @ B2 )
=> ( has_field_derivative @ real @ F3 @ ( zero_zero @ real ) @ ( topolo174197925503356063within @ real @ X4 @ ( top_top @ ( set @ real ) ) ) ) ) )
=> ( ( F3 @ B2 )
= ( F3 @ A2 ) ) ) ) ) ).
% DERIV_isconst_end
thf(fact_5400_DERIV__neg__imp__decreasing__open,axiom,
! [A2: real,B2: real,F3: real > real] :
( ( ord_less @ real @ A2 @ B2 )
=> ( ! [X4: real] :
( ( ord_less @ real @ A2 @ X4 )
=> ( ( ord_less @ real @ X4 @ B2 )
=> ? [Y4: real] :
( ( has_field_derivative @ real @ F3 @ Y4 @ ( topolo174197925503356063within @ real @ X4 @ ( top_top @ ( set @ real ) ) ) )
& ( ord_less @ real @ Y4 @ ( zero_zero @ real ) ) ) ) )
=> ( ( topolo81223032696312382ous_on @ real @ real @ ( set_or1337092689740270186AtMost @ real @ A2 @ B2 ) @ F3 )
=> ( ord_less @ real @ ( F3 @ B2 ) @ ( F3 @ A2 ) ) ) ) ) ).
% DERIV_neg_imp_decreasing_open
thf(fact_5401_DERIV__pos__imp__increasing__open,axiom,
! [A2: real,B2: real,F3: real > real] :
( ( ord_less @ real @ A2 @ B2 )
=> ( ! [X4: real] :
( ( ord_less @ real @ A2 @ X4 )
=> ( ( ord_less @ real @ X4 @ B2 )
=> ? [Y4: real] :
( ( has_field_derivative @ real @ F3 @ Y4 @ ( topolo174197925503356063within @ real @ X4 @ ( top_top @ ( set @ real ) ) ) )
& ( ord_less @ real @ ( zero_zero @ real ) @ Y4 ) ) ) )
=> ( ( topolo81223032696312382ous_on @ real @ real @ ( set_or1337092689740270186AtMost @ real @ A2 @ B2 ) @ F3 )
=> ( ord_less @ real @ ( F3 @ A2 ) @ ( F3 @ B2 ) ) ) ) ) ).
% DERIV_pos_imp_increasing_open
thf(fact_5402_DERIV__isconst2,axiom,
! [A2: real,B2: real,F3: real > real,X: real] :
( ( ord_less @ real @ A2 @ B2 )
=> ( ( topolo81223032696312382ous_on @ real @ real @ ( set_or1337092689740270186AtMost @ real @ A2 @ B2 ) @ F3 )
=> ( ! [X4: real] :
( ( ord_less @ real @ A2 @ X4 )
=> ( ( ord_less @ real @ X4 @ B2 )
=> ( has_field_derivative @ real @ F3 @ ( zero_zero @ real ) @ ( topolo174197925503356063within @ real @ X4 @ ( top_top @ ( set @ real ) ) ) ) ) )
=> ( ( ord_less_eq @ real @ A2 @ X )
=> ( ( ord_less_eq @ real @ X @ B2 )
=> ( ( F3 @ X )
= ( F3 @ A2 ) ) ) ) ) ) ) ).
% DERIV_isconst2
thf(fact_5403_continuous__on__IccI,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo1944317154257567458pology @ A )
& ( topolo4958980785337419405_space @ B ) )
=> ! [F3: A > B,A2: A,B2: A] :
( ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ ( F3 @ A2 ) ) @ ( topolo174197925503356063within @ A @ A2 @ ( set_ord_greaterThan @ A @ A2 ) ) )
=> ( ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ ( F3 @ B2 ) ) @ ( topolo174197925503356063within @ A @ B2 @ ( set_ord_lessThan @ A @ B2 ) ) )
=> ( ! [X4: A] :
( ( ord_less @ A @ A2 @ X4 )
=> ( ( ord_less @ A @ X4 @ B2 )
=> ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ ( F3 @ X4 ) ) @ ( topolo174197925503356063within @ A @ X4 @ ( top_top @ ( set @ A ) ) ) ) ) )
=> ( ( ord_less @ A @ A2 @ B2 )
=> ( topolo81223032696312382ous_on @ A @ B @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) @ F3 ) ) ) ) ) ) ).
% continuous_on_IccI
thf(fact_5404_Rolle,axiom,
! [A2: real,B2: real,F3: real > real] :
( ( ord_less @ real @ A2 @ B2 )
=> ( ( ( F3 @ A2 )
= ( F3 @ B2 ) )
=> ( ( topolo81223032696312382ous_on @ real @ real @ ( set_or1337092689740270186AtMost @ real @ A2 @ B2 ) @ F3 )
=> ( ! [X4: real] :
( ( ord_less @ real @ A2 @ X4 )
=> ( ( ord_less @ real @ X4 @ B2 )
=> ( differentiable @ real @ real @ F3 @ ( topolo174197925503356063within @ real @ X4 @ ( top_top @ ( set @ real ) ) ) ) ) )
=> ? [Z3: real] :
( ( ord_less @ real @ A2 @ Z3 )
& ( ord_less @ real @ Z3 @ B2 )
& ( has_field_derivative @ real @ F3 @ ( zero_zero @ real ) @ ( topolo174197925503356063within @ real @ Z3 @ ( top_top @ ( set @ real ) ) ) ) ) ) ) ) ) ).
% Rolle
thf(fact_5405_uniformity__dist,axiom,
! [A: $tType] :
( ( real_V768167426530841204y_dist @ A )
=> ( ( topolo7806501430040627800ormity @ A )
= ( complete_Inf_Inf @ ( filter @ ( product_prod @ A @ A ) )
@ ( image2 @ real @ ( filter @ ( product_prod @ A @ A ) )
@ ^ [E4: real] :
( principal @ ( product_prod @ A @ A )
@ ( collect @ ( product_prod @ A @ A )
@ ( product_case_prod @ A @ A @ $o
@ ^ [X5: A,Y6: A] : ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X5 @ Y6 ) @ E4 ) ) ) )
@ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) ) ) ) ).
% uniformity_dist
thf(fact_5406_compactE__image,axiom,
! [A: $tType,B: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [S2: set @ A,C5: set @ B,F3: B > ( set @ A )] :
( ( topolo2193935891317330818ompact @ A @ S2 )
=> ( ! [T6: B] :
( ( member @ B @ T6 @ C5 )
=> ( topolo1002775350975398744n_open @ A @ ( F3 @ T6 ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ S2 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ F3 @ C5 ) ) )
=> ~ ! [C7: set @ B] :
( ( ord_less_eq @ ( set @ B ) @ C7 @ C5 )
=> ( ( finite_finite2 @ B @ C7 )
=> ~ ( ord_less_eq @ ( set @ A ) @ S2 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ F3 @ C7 ) ) ) ) ) ) ) ) ) ).
% compactE_image
thf(fact_5407_compactE,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [S2: set @ A,T9: set @ ( set @ A )] :
( ( topolo2193935891317330818ompact @ A @ S2 )
=> ( ( ord_less_eq @ ( set @ A ) @ S2 @ ( complete_Sup_Sup @ ( set @ A ) @ T9 ) )
=> ( ! [B9: set @ A] :
( ( member @ ( set @ A ) @ B9 @ T9 )
=> ( topolo1002775350975398744n_open @ A @ B9 ) )
=> ~ ! [T10: set @ ( set @ A )] :
( ( ord_less_eq @ ( set @ ( set @ A ) ) @ T10 @ T9 )
=> ( ( finite_finite2 @ ( set @ A ) @ T10 )
=> ~ ( ord_less_eq @ ( set @ A ) @ S2 @ ( complete_Sup_Sup @ ( set @ A ) @ T10 ) ) ) ) ) ) ) ) ).
% compactE
thf(fact_5408_uniformity__refl,axiom,
! [A: $tType] :
( ( topolo7287701948861334536_space @ A )
=> ! [E5: ( product_prod @ A @ A ) > $o,X: A] :
( ( eventually @ ( product_prod @ A @ A ) @ E5 @ ( topolo7806501430040627800ormity @ A ) )
=> ( E5 @ ( product_Pair @ A @ A @ X @ X ) ) ) ) ).
% uniformity_refl
thf(fact_5409_uniformity__trans,axiom,
! [A: $tType] :
( ( topolo7287701948861334536_space @ A )
=> ! [E5: ( product_prod @ A @ A ) > $o] :
( ( eventually @ ( product_prod @ A @ A ) @ E5 @ ( topolo7806501430040627800ormity @ A ) )
=> ? [D7: ( product_prod @ A @ A ) > $o] :
( ( eventually @ ( product_prod @ A @ A ) @ D7 @ ( topolo7806501430040627800ormity @ A ) )
& ! [X3: A,Y4: A,Z4: A] :
( ( D7 @ ( product_Pair @ A @ A @ X3 @ Y4 ) )
=> ( ( D7 @ ( product_Pair @ A @ A @ Y4 @ Z4 ) )
=> ( E5 @ ( product_Pair @ A @ A @ X3 @ Z4 ) ) ) ) ) ) ) ).
% uniformity_trans
thf(fact_5410_uniformity__transE,axiom,
! [A: $tType] :
( ( topolo7287701948861334536_space @ A )
=> ! [E5: ( product_prod @ A @ A ) > $o] :
( ( eventually @ ( product_prod @ A @ A ) @ E5 @ ( topolo7806501430040627800ormity @ A ) )
=> ~ ! [D7: ( product_prod @ A @ A ) > $o] :
( ( eventually @ ( product_prod @ A @ A ) @ D7 @ ( topolo7806501430040627800ormity @ A ) )
=> ~ ! [X3: A,Y4: A] :
( ( D7 @ ( product_Pair @ A @ A @ X3 @ Y4 ) )
=> ! [Z4: A] :
( ( D7 @ ( product_Pair @ A @ A @ Y4 @ Z4 ) )
=> ( E5 @ ( product_Pair @ A @ A @ X3 @ Z4 ) ) ) ) ) ) ) ).
% uniformity_transE
thf(fact_5411_uniformity__bot,axiom,
! [A: $tType] :
( ( topolo7287701948861334536_space @ A )
=> ( ( topolo7806501430040627800ormity @ A )
!= ( bot_bot @ ( filter @ ( product_prod @ A @ A ) ) ) ) ) ).
% uniformity_bot
thf(fact_5412_compact__empty,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ( topolo2193935891317330818ompact @ A @ ( bot_bot @ ( set @ A ) ) ) ) ).
% compact_empty
thf(fact_5413_compact__attains__inf,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [S2: set @ A] :
( ( topolo2193935891317330818ompact @ A @ S2 )
=> ( ( S2
!= ( bot_bot @ ( set @ A ) ) )
=> ? [X4: A] :
( ( member @ A @ X4 @ S2 )
& ! [Xa: A] :
( ( member @ A @ Xa @ S2 )
=> ( ord_less_eq @ A @ X4 @ Xa ) ) ) ) ) ) ).
% compact_attains_inf
thf(fact_5414_compact__attains__sup,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [S2: set @ A] :
( ( topolo2193935891317330818ompact @ A @ S2 )
=> ( ( S2
!= ( bot_bot @ ( set @ A ) ) )
=> ? [X4: A] :
( ( member @ A @ X4 @ S2 )
& ! [Xa: A] :
( ( member @ A @ Xa @ S2 )
=> ( ord_less_eq @ A @ Xa @ X4 ) ) ) ) ) ) ).
% compact_attains_sup
thf(fact_5415_uniformity__sym,axiom,
! [A: $tType] :
( ( topolo7287701948861334536_space @ A )
=> ! [E5: ( product_prod @ A @ A ) > $o] :
( ( eventually @ ( product_prod @ A @ A ) @ E5 @ ( topolo7806501430040627800ormity @ A ) )
=> ( eventually @ ( product_prod @ A @ A )
@ ( product_case_prod @ A @ A @ $o
@ ^ [X5: A,Y6: A] : ( E5 @ ( product_Pair @ A @ A @ Y6 @ X5 ) ) )
@ ( topolo7806501430040627800ormity @ A ) ) ) ) ).
% uniformity_sym
thf(fact_5416_continuous__attains__sup,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ A )
& ( topolo1944317154257567458pology @ B ) )
=> ! [S: set @ A,F3: A > B] :
( ( topolo2193935891317330818ompact @ A @ S )
=> ( ( S
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( topolo81223032696312382ous_on @ A @ B @ S @ F3 )
=> ? [X4: A] :
( ( member @ A @ X4 @ S )
& ! [Xa: A] :
( ( member @ A @ Xa @ S )
=> ( ord_less_eq @ B @ ( F3 @ Xa ) @ ( F3 @ X4 ) ) ) ) ) ) ) ) ).
% continuous_attains_sup
thf(fact_5417_continuous__attains__inf,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ A )
& ( topolo1944317154257567458pology @ B ) )
=> ! [S: set @ A,F3: A > B] :
( ( topolo2193935891317330818ompact @ A @ S )
=> ( ( S
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( topolo81223032696312382ous_on @ A @ B @ S @ F3 )
=> ? [X4: A] :
( ( member @ A @ X4 @ S )
& ! [Xa: A] :
( ( member @ A @ Xa @ S )
=> ( ord_less_eq @ B @ ( F3 @ X4 ) @ ( F3 @ Xa ) ) ) ) ) ) ) ) ).
% continuous_attains_inf
thf(fact_5418_Cauchy__uniform__iff,axiom,
! [A: $tType] :
( ( topolo7287701948861334536_space @ A )
=> ( ( topolo3814608138187158403Cauchy @ A )
= ( ^ [X9: nat > A] :
! [P4: ( product_prod @ A @ A ) > $o] :
( ( eventually @ ( product_prod @ A @ A ) @ P4 @ ( topolo7806501430040627800ormity @ A ) )
=> ? [N4: nat] :
! [N2: nat] :
( ( ord_less_eq @ nat @ N4 @ N2 )
=> ! [M3: nat] :
( ( ord_less_eq @ nat @ N4 @ M3 )
=> ( P4 @ ( product_Pair @ A @ A @ ( X9 @ N2 ) @ ( X9 @ M3 ) ) ) ) ) ) ) ) ) ).
% Cauchy_uniform_iff
thf(fact_5419_uniformity__complex__def,axiom,
( ( topolo7806501430040627800ormity @ complex )
= ( complete_Inf_Inf @ ( filter @ ( product_prod @ complex @ complex ) )
@ ( image2 @ real @ ( filter @ ( product_prod @ complex @ complex ) )
@ ^ [E4: real] :
( principal @ ( product_prod @ complex @ complex )
@ ( collect @ ( product_prod @ complex @ complex )
@ ( product_case_prod @ complex @ complex @ $o
@ ^ [X5: complex,Y6: complex] : ( ord_less @ real @ ( real_V557655796197034286t_dist @ complex @ X5 @ Y6 ) @ E4 ) ) ) )
@ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) ) ) ).
% uniformity_complex_def
thf(fact_5420_uniformity__real__def,axiom,
( ( topolo7806501430040627800ormity @ real )
= ( complete_Inf_Inf @ ( filter @ ( product_prod @ real @ real ) )
@ ( image2 @ real @ ( filter @ ( product_prod @ real @ real ) )
@ ^ [E4: real] :
( principal @ ( product_prod @ real @ real )
@ ( collect @ ( product_prod @ real @ real )
@ ( product_case_prod @ real @ real @ $o
@ ^ [X5: real,Y6: real] : ( ord_less @ real @ ( real_V557655796197034286t_dist @ real @ X5 @ Y6 ) @ E4 ) ) ) )
@ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) ) ) ).
% uniformity_real_def
thf(fact_5421_totally__bounded__def,axiom,
! [A: $tType] :
( ( topolo7287701948861334536_space @ A )
=> ( ( topolo6688025880775521714ounded @ A )
= ( ^ [S7: set @ A] :
! [E6: ( product_prod @ A @ A ) > $o] :
( ( eventually @ ( product_prod @ A @ A ) @ E6 @ ( topolo7806501430040627800ormity @ A ) )
=> ? [X9: set @ A] :
( ( finite_finite2 @ A @ X9 )
& ! [X5: A] :
( ( member @ A @ X5 @ S7 )
=> ? [Y6: A] :
( ( member @ A @ Y6 @ X9 )
& ( E6 @ ( product_Pair @ A @ A @ Y6 @ X5 ) ) ) ) ) ) ) ) ) ).
% totally_bounded_def
thf(fact_5422_eventually__uniformity__metric,axiom,
! [A: $tType] :
( ( real_V768167426530841204y_dist @ A )
=> ! [P2: ( product_prod @ A @ A ) > $o] :
( ( eventually @ ( product_prod @ A @ A ) @ P2 @ ( topolo7806501430040627800ormity @ A ) )
= ( ? [E4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E4 )
& ! [X5: A,Y6: A] :
( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X5 @ Y6 ) @ E4 )
=> ( P2 @ ( product_Pair @ A @ A @ X5 @ Y6 ) ) ) ) ) ) ) ).
% eventually_uniformity_metric
thf(fact_5423_tendsto__iff__uniformity,axiom,
! [A: $tType,B: $tType] :
( ( topolo7287701948861334536_space @ B )
=> ! [F3: A > B,L: B,F5: filter @ A] :
( ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ F5 )
= ( ! [E6: ( product_prod @ B @ B ) > $o] :
( ( eventually @ ( product_prod @ B @ B ) @ E6 @ ( topolo7806501430040627800ormity @ B ) )
=> ( eventually @ A
@ ^ [X5: A] : ( E6 @ ( product_Pair @ B @ B @ ( F3 @ X5 ) @ L ) )
@ F5 ) ) ) ) ) ).
% tendsto_iff_uniformity
thf(fact_5424_compact__eq__Heine__Borel,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ( ( topolo2193935891317330818ompact @ A )
= ( ^ [S7: set @ A] :
! [C8: set @ ( set @ A )] :
( ( ! [X5: set @ A] :
( ( member @ ( set @ A ) @ X5 @ C8 )
=> ( topolo1002775350975398744n_open @ A @ X5 ) )
& ( ord_less_eq @ ( set @ A ) @ S7 @ ( complete_Sup_Sup @ ( set @ A ) @ C8 ) ) )
=> ? [D8: set @ ( set @ A )] :
( ( ord_less_eq @ ( set @ ( set @ A ) ) @ D8 @ C8 )
& ( finite_finite2 @ ( set @ A ) @ D8 )
& ( ord_less_eq @ ( set @ A ) @ S7 @ ( complete_Sup_Sup @ ( set @ A ) @ D8 ) ) ) ) ) ) ) ).
% compact_eq_Heine_Borel
thf(fact_5425_compactI,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [S: set @ A] :
( ! [C6: set @ ( set @ A )] :
( ! [X3: set @ A] :
( ( member @ ( set @ A ) @ X3 @ C6 )
=> ( topolo1002775350975398744n_open @ A @ X3 ) )
=> ( ( ord_less_eq @ ( set @ A ) @ S @ ( complete_Sup_Sup @ ( set @ A ) @ C6 ) )
=> ? [C9: set @ ( set @ A )] :
( ( ord_less_eq @ ( set @ ( set @ A ) ) @ C9 @ C6 )
& ( finite_finite2 @ ( set @ A ) @ C9 )
& ( ord_less_eq @ ( set @ A ) @ S @ ( complete_Sup_Sup @ ( set @ A ) @ C9 ) ) ) ) )
=> ( topolo2193935891317330818ompact @ A @ S ) ) ) ).
% compactI
thf(fact_5426_prod__filter__INF,axiom,
! [C: $tType,D: $tType,B: $tType,A: $tType,I6: set @ A,J4: set @ B,A5: A > ( filter @ C ),B6: B > ( filter @ D )] :
( ( I6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( J4
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( prod_filter @ C @ D @ ( complete_Inf_Inf @ ( filter @ C ) @ ( image2 @ A @ ( filter @ C ) @ A5 @ I6 ) ) @ ( complete_Inf_Inf @ ( filter @ D ) @ ( image2 @ B @ ( filter @ D ) @ B6 @ J4 ) ) )
= ( complete_Inf_Inf @ ( filter @ ( product_prod @ C @ D ) )
@ ( image2 @ A @ ( filter @ ( product_prod @ C @ D ) )
@ ^ [I4: A] :
( complete_Inf_Inf @ ( filter @ ( product_prod @ C @ D ) )
@ ( image2 @ B @ ( filter @ ( product_prod @ C @ D ) )
@ ^ [J3: B] : ( prod_filter @ C @ D @ ( A5 @ I4 ) @ ( B6 @ J3 ) )
@ J4 ) )
@ I6 ) ) ) ) ) ).
% prod_filter_INF
thf(fact_5427_prod__filter__INF1,axiom,
! [B: $tType,C: $tType,A: $tType,I6: set @ A,A5: A > ( filter @ B ),B6: filter @ C] :
( ( I6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( prod_filter @ B @ C @ ( complete_Inf_Inf @ ( filter @ B ) @ ( image2 @ A @ ( filter @ B ) @ A5 @ I6 ) ) @ B6 )
= ( complete_Inf_Inf @ ( filter @ ( product_prod @ B @ C ) )
@ ( image2 @ A @ ( filter @ ( product_prod @ B @ C ) )
@ ^ [I4: A] : ( prod_filter @ B @ C @ ( A5 @ I4 ) @ B6 )
@ I6 ) ) ) ) ).
% prod_filter_INF1
thf(fact_5428_prod__filter__INF2,axiom,
! [B: $tType,C: $tType,A: $tType,J4: set @ A,A5: filter @ B,B6: A > ( filter @ C )] :
( ( J4
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( prod_filter @ B @ C @ A5 @ ( complete_Inf_Inf @ ( filter @ C ) @ ( image2 @ A @ ( filter @ C ) @ B6 @ J4 ) ) )
= ( complete_Inf_Inf @ ( filter @ ( product_prod @ B @ C ) )
@ ( image2 @ A @ ( filter @ ( product_prod @ B @ C ) )
@ ^ [I4: A] : ( prod_filter @ B @ C @ A5 @ ( B6 @ I4 ) )
@ J4 ) ) ) ) ).
% prod_filter_INF2
thf(fact_5429_prod__filter__eq__bot,axiom,
! [A: $tType,B: $tType,A5: filter @ A,B6: filter @ B] :
( ( ( prod_filter @ A @ B @ A5 @ B6 )
= ( bot_bot @ ( filter @ ( product_prod @ A @ B ) ) ) )
= ( ( A5
= ( bot_bot @ ( filter @ A ) ) )
| ( B6
= ( bot_bot @ ( filter @ B ) ) ) ) ) ).
% prod_filter_eq_bot
thf(fact_5430_eventually__prod__same,axiom,
! [A: $tType,P2: ( product_prod @ A @ A ) > $o,F5: filter @ A] :
( ( eventually @ ( product_prod @ A @ A ) @ P2 @ ( prod_filter @ A @ A @ F5 @ F5 ) )
= ( ? [Q7: A > $o] :
( ( eventually @ A @ Q7 @ F5 )
& ! [X5: A,Y6: A] :
( ( Q7 @ X5 )
=> ( ( Q7 @ Y6 )
=> ( P2 @ ( product_Pair @ A @ A @ X5 @ Y6 ) ) ) ) ) ) ) ).
% eventually_prod_same
thf(fact_5431_eventually__prod__filter,axiom,
! [B: $tType,A: $tType,P2: ( product_prod @ A @ B ) > $o,F5: filter @ A,G6: filter @ B] :
( ( eventually @ ( product_prod @ A @ B ) @ P2 @ ( prod_filter @ A @ B @ F5 @ G6 ) )
= ( ? [Pf: A > $o,Pg: B > $o] :
( ( eventually @ A @ Pf @ F5 )
& ( eventually @ B @ Pg @ G6 )
& ! [X5: A,Y6: B] :
( ( Pf @ X5 )
=> ( ( Pg @ Y6 )
=> ( P2 @ ( product_Pair @ A @ B @ X5 @ Y6 ) ) ) ) ) ) ) ).
% eventually_prod_filter
thf(fact_5432_prod__filter__mono__iff,axiom,
! [A: $tType,B: $tType,A5: filter @ A,B6: filter @ B,C5: filter @ A,D6: filter @ B] :
( ( A5
!= ( bot_bot @ ( filter @ A ) ) )
=> ( ( B6
!= ( bot_bot @ ( filter @ B ) ) )
=> ( ( ord_less_eq @ ( filter @ ( product_prod @ A @ B ) ) @ ( prod_filter @ A @ B @ A5 @ B6 ) @ ( prod_filter @ A @ B @ C5 @ D6 ) )
= ( ( ord_less_eq @ ( filter @ A ) @ A5 @ C5 )
& ( ord_less_eq @ ( filter @ B ) @ B6 @ D6 ) ) ) ) ) ).
% prod_filter_mono_iff
thf(fact_5433_eventually__prod__sequentially,axiom,
! [P2: ( product_prod @ nat @ nat ) > $o] :
( ( eventually @ ( product_prod @ nat @ nat ) @ P2 @ ( prod_filter @ nat @ nat @ ( at_top @ nat ) @ ( at_top @ nat ) ) )
= ( ? [N4: nat] :
! [M3: nat] :
( ( ord_less_eq @ nat @ N4 @ M3 )
=> ! [N2: nat] :
( ( ord_less_eq @ nat @ N4 @ N2 )
=> ( P2 @ ( product_Pair @ nat @ nat @ N2 @ M3 ) ) ) ) ) ) ).
% eventually_prod_sequentially
thf(fact_5434_eventually__prod2,axiom,
! [A: $tType,B: $tType,A5: filter @ A,P2: B > $o,B6: filter @ B] :
( ( A5
!= ( bot_bot @ ( filter @ A ) ) )
=> ( ( eventually @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [X5: A] : P2 )
@ ( prod_filter @ A @ B @ A5 @ B6 ) )
= ( eventually @ B @ P2 @ B6 ) ) ) ).
% eventually_prod2
thf(fact_5435_eventually__prod1,axiom,
! [A: $tType,B: $tType,B6: filter @ A,P2: B > $o,A5: filter @ B] :
( ( B6
!= ( bot_bot @ ( filter @ A ) ) )
=> ( ( eventually @ ( product_prod @ B @ A )
@ ( product_case_prod @ B @ A @ $o
@ ^ [X5: B,Y6: A] : ( P2 @ X5 ) )
@ ( prod_filter @ B @ A @ A5 @ B6 ) )
= ( eventually @ B @ P2 @ A5 ) ) ) ).
% eventually_prod1
thf(fact_5436_nhds__prod,axiom,
! [A: $tType,B: $tType] :
( ( ( topolo4958980785337419405_space @ B )
& ( topolo4958980785337419405_space @ A ) )
=> ! [A2: A,B2: B] :
( ( topolo7230453075368039082e_nhds @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) )
= ( prod_filter @ A @ B @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ ( topolo7230453075368039082e_nhds @ B @ B2 ) ) ) ) ).
% nhds_prod
thf(fact_5437_filterlim__Pair,axiom,
! [C: $tType,B: $tType,A: $tType,F3: A > B,G6: filter @ B,F5: filter @ A,G3: A > C,H7: filter @ C] :
( ( filterlim @ A @ B @ F3 @ G6 @ F5 )
=> ( ( filterlim @ A @ C @ G3 @ H7 @ F5 )
=> ( filterlim @ A @ ( product_prod @ B @ C )
@ ^ [X5: A] : ( product_Pair @ B @ C @ ( F3 @ X5 ) @ ( G3 @ X5 ) )
@ ( prod_filter @ B @ C @ G6 @ H7 )
@ F5 ) ) ) ).
% filterlim_Pair
thf(fact_5438_tendsto__add__Pair,axiom,
! [A: $tType] :
( ( topolo6943815403480290642id_add @ A )
=> ! [A2: A,B2: A] :
( filterlim @ ( product_prod @ A @ A ) @ A
@ ^ [X5: product_prod @ A @ A] : ( plus_plus @ A @ ( product_fst @ A @ A @ X5 ) @ ( product_snd @ A @ A @ X5 ) )
@ ( topolo7230453075368039082e_nhds @ A @ ( plus_plus @ A @ A2 @ B2 ) )
@ ( prod_filter @ A @ A @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ ( topolo7230453075368039082e_nhds @ A @ B2 ) ) ) ) ).
% tendsto_add_Pair
thf(fact_5439_uniformity__trans_H,axiom,
! [A: $tType] :
( ( topolo7287701948861334536_space @ A )
=> ! [E5: ( product_prod @ A @ A ) > $o] :
( ( eventually @ ( product_prod @ A @ A ) @ E5 @ ( topolo7806501430040627800ormity @ A ) )
=> ( eventually @ ( product_prod @ ( product_prod @ A @ A ) @ ( product_prod @ A @ A ) )
@ ( product_case_prod @ ( product_prod @ A @ A ) @ ( product_prod @ A @ A ) @ $o
@ ( product_case_prod @ A @ A @ ( ( product_prod @ A @ A ) > $o )
@ ^ [X5: A,Y6: A] :
( product_case_prod @ A @ A @ $o
@ ^ [Y7: A,Z5: A] :
( ( Y6 = Y7 )
=> ( E5 @ ( product_Pair @ A @ A @ X5 @ Z5 ) ) ) ) ) )
@ ( prod_filter @ ( product_prod @ A @ A ) @ ( product_prod @ A @ A ) @ ( topolo7806501430040627800ormity @ A ) @ ( topolo7806501430040627800ormity @ A ) ) ) ) ) ).
% uniformity_trans'
thf(fact_5440_tendsto__at__iff__sequentially,axiom,
! [C: $tType,A: $tType] :
( ( ( topolo3112930676232923870pology @ A )
& ( topolo4958980785337419405_space @ C ) )
=> ! [F3: A > C,A2: C,X: A,S: set @ A] :
( ( filterlim @ A @ C @ F3 @ ( topolo7230453075368039082e_nhds @ C @ A2 ) @ ( topolo174197925503356063within @ A @ X @ S ) )
= ( ! [X9: nat > A] :
( ! [I4: nat] : ( member @ A @ ( X9 @ I4 ) @ ( minus_minus @ ( set @ A ) @ S @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( ( filterlim @ nat @ A @ X9 @ ( topolo7230453075368039082e_nhds @ A @ X ) @ ( at_top @ nat ) )
=> ( filterlim @ nat @ C @ ( comp @ A @ C @ nat @ F3 @ X9 ) @ ( topolo7230453075368039082e_nhds @ C @ A2 ) @ ( at_top @ nat ) ) ) ) ) ) ) ).
% tendsto_at_iff_sequentially
thf(fact_5441_sequentially__imp__eventually__at__left,axiom,
! [A: $tType] :
( ( ( topolo3112930676232923870pology @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [B2: A,A2: A,P2: A > $o] :
( ( ord_less @ A @ B2 @ A2 )
=> ( ! [F2: nat > A] :
( ! [N5: nat] : ( ord_less @ A @ B2 @ ( F2 @ N5 ) )
=> ( ! [N5: nat] : ( ord_less @ A @ ( F2 @ N5 ) @ A2 )
=> ( ( order_mono @ nat @ A @ F2 )
=> ( ( filterlim @ nat @ A @ F2 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ ( at_top @ nat ) )
=> ( eventually @ nat
@ ^ [N2: nat] : ( P2 @ ( F2 @ N2 ) )
@ ( at_top @ nat ) ) ) ) ) )
=> ( eventually @ A @ P2 @ ( topolo174197925503356063within @ A @ A2 @ ( set_ord_lessThan @ A @ A2 ) ) ) ) ) ) ).
% sequentially_imp_eventually_at_left
thf(fact_5442_relpow__finite__bounded1,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),K2: nat] :
( ( finite_finite2 @ ( product_prod @ A @ A ) @ R )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ K2 )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ K2 @ R )
@ ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ A ) )
@ ( image2 @ nat @ ( set @ ( product_prod @ A @ A ) )
@ ^ [N2: nat] : ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N2 @ R )
@ ( collect @ nat
@ ^ [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
& ( ord_less_eq @ nat @ N2 @ ( finite_card @ ( product_prod @ A @ A ) @ R ) ) ) ) ) ) ) ) ) ).
% relpow_finite_bounded1
thf(fact_5443_finite__relpow,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),N: nat] :
( ( finite_finite2 @ ( product_prod @ A @ A ) @ R )
=> ( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( finite_finite2 @ ( product_prod @ A @ A ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R ) ) ) ) ).
% finite_relpow
thf(fact_5444_mono__funpow,axiom,
! [A: $tType] :
( ( ( lattice @ A )
& ( order_bot @ A ) )
=> ! [Q: A > A] :
( ( order_mono @ A @ A @ Q )
=> ( order_mono @ nat @ A
@ ^ [I4: nat] : ( compow @ ( A > A ) @ I4 @ Q @ ( bot_bot @ A ) ) ) ) ) ).
% mono_funpow
thf(fact_5445_funpow__mono,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [F3: A > A,A5: A,B6: A,N: nat] :
( ( order_mono @ A @ A @ F3 )
=> ( ( ord_less_eq @ A @ A5 @ B6 )
=> ( ord_less_eq @ A @ ( compow @ ( A > A ) @ N @ F3 @ A5 ) @ ( compow @ ( A > A ) @ N @ F3 @ B6 ) ) ) ) ) ).
% funpow_mono
thf(fact_5446_mono__def,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ( ( order_mono @ A @ B )
= ( ^ [F4: A > B] :
! [X5: A,Y6: A] :
( ( ord_less_eq @ A @ X5 @ Y6 )
=> ( ord_less_eq @ B @ ( F4 @ X5 ) @ ( F4 @ Y6 ) ) ) ) ) ) ).
% mono_def
thf(fact_5447_monoI,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F3: A > B] :
( ! [X4: A,Y3: A] :
( ( ord_less_eq @ A @ X4 @ Y3 )
=> ( ord_less_eq @ B @ ( F3 @ X4 ) @ ( F3 @ Y3 ) ) )
=> ( order_mono @ A @ B @ F3 ) ) ) ).
% monoI
thf(fact_5448_monoE,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F3: A > B,X: A,Y: A] :
( ( order_mono @ A @ B @ F3 )
=> ( ( ord_less_eq @ A @ X @ Y )
=> ( ord_less_eq @ B @ ( F3 @ X ) @ ( F3 @ Y ) ) ) ) ) ).
% monoE
thf(fact_5449_monoD,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F3: A > B,X: A,Y: A] :
( ( order_mono @ A @ B @ F3 )
=> ( ( ord_less_eq @ A @ X @ Y )
=> ( ord_less_eq @ B @ ( F3 @ X ) @ ( F3 @ Y ) ) ) ) ) ).
% monoD
thf(fact_5450_incseqD,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [F3: nat > A,I2: nat,J: nat] :
( ( order_mono @ nat @ A @ F3 )
=> ( ( ord_less_eq @ nat @ I2 @ J )
=> ( ord_less_eq @ A @ ( F3 @ I2 ) @ ( F3 @ J ) ) ) ) ) ).
% incseqD
thf(fact_5451_incseq__def,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( order_mono @ nat @ A )
= ( ^ [X9: nat > A] :
! [M3: nat,N2: nat] :
( ( ord_less_eq @ nat @ M3 @ N2 )
=> ( ord_less_eq @ A @ ( X9 @ M3 ) @ ( X9 @ N2 ) ) ) ) ) ) ).
% incseq_def
thf(fact_5452_incseq__SucD,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [A5: nat > A,I2: nat] :
( ( order_mono @ nat @ A @ A5 )
=> ( ord_less_eq @ A @ ( A5 @ I2 ) @ ( A5 @ ( suc @ I2 ) ) ) ) ) ).
% incseq_SucD
thf(fact_5453_incseq__SucI,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [X7: nat > A] :
( ! [N3: nat] : ( ord_less_eq @ A @ ( X7 @ N3 ) @ ( X7 @ ( suc @ N3 ) ) )
=> ( order_mono @ nat @ A @ X7 ) ) ) ).
% incseq_SucI
thf(fact_5454_incseq__Suc__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( order_mono @ nat @ A )
= ( ^ [F4: nat > A] :
! [N2: nat] : ( ord_less_eq @ A @ ( F4 @ N2 ) @ ( F4 @ ( suc @ N2 ) ) ) ) ) ) ).
% incseq_Suc_iff
thf(fact_5455_mono__invE,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( order @ B ) )
=> ! [F3: A > B,X: A,Y: A] :
( ( order_mono @ A @ B @ F3 )
=> ( ( ord_less @ B @ ( F3 @ X ) @ ( F3 @ Y ) )
=> ( ord_less_eq @ A @ X @ Y ) ) ) ) ).
% mono_invE
thf(fact_5456_relpow__Suc__D2_H,axiom,
! [A: $tType,N: nat,R: set @ ( product_prod @ A @ A ),X3: A,Y4: A,Z4: A] :
( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Y4 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Z4 ) @ R ) )
=> ? [W: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ W ) @ R )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ W @ Z4 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R ) ) ) ) ).
% relpow_Suc_D2'
thf(fact_5457_comp__funpow,axiom,
! [B: $tType,A: $tType,N: nat,F3: A > A] :
( ( compow @ ( ( B > A ) > B > A ) @ N @ ( comp @ A @ A @ B @ F3 ) )
= ( comp @ A @ A @ B @ ( compow @ ( A > A ) @ N @ F3 ) ) ) ).
% comp_funpow
thf(fact_5458_mono__pow,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F3: A > A,N: nat] :
( ( order_mono @ A @ A @ F3 )
=> ( order_mono @ A @ A @ ( compow @ ( A > A ) @ N @ F3 ) ) ) ) ).
% mono_pow
thf(fact_5459_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_5460_mono__Suc,axiom,
order_mono @ nat @ nat @ suc ).
% mono_Suc
thf(fact_5461_comp__cong,axiom,
! [C: $tType,B: $tType,D: $tType,A: $tType,E: $tType,F3: B > A,G3: C > B,X: C,F10: D > A,G7: E > D,X8: E] :
( ( ( F3 @ ( G3 @ X ) )
= ( F10 @ ( G7 @ X8 ) ) )
=> ( ( comp @ B @ A @ C @ F3 @ G3 @ X )
= ( comp @ D @ A @ E @ F10 @ G7 @ X8 ) ) ) ).
% comp_cong
thf(fact_5462_max__of__mono,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( linorder @ B ) )
=> ! [F3: A > B,M2: A,N: A] :
( ( order_mono @ A @ B @ F3 )
=> ( ( ord_max @ B @ ( F3 @ M2 ) @ ( F3 @ N ) )
= ( F3 @ ( ord_max @ A @ M2 @ N ) ) ) ) ) ).
% max_of_mono
thf(fact_5463_mono__add,axiom,
! [A: $tType] :
( ( ordere6658533253407199908up_add @ A )
=> ! [A2: A] : ( order_mono @ A @ A @ ( plus_plus @ A @ A2 ) ) ) ).
% mono_add
thf(fact_5464_mono__strict__invE,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( order @ B ) )
=> ! [F3: A > B,X: A,Y: A] :
( ( order_mono @ A @ B @ F3 )
=> ( ( ord_less @ B @ ( F3 @ X ) @ ( F3 @ Y ) )
=> ( ord_less @ A @ X @ Y ) ) ) ) ).
% mono_strict_invE
thf(fact_5465_mono__inf,axiom,
! [B: $tType,A: $tType] :
( ( ( semilattice_inf @ A )
& ( semilattice_inf @ B ) )
=> ! [F3: A > B,A5: A,B6: A] :
( ( order_mono @ A @ B @ F3 )
=> ( ord_less_eq @ B @ ( F3 @ ( inf_inf @ A @ A5 @ B6 ) ) @ ( inf_inf @ B @ ( F3 @ A5 ) @ ( F3 @ B6 ) ) ) ) ) ).
% mono_inf
thf(fact_5466_funpow__Suc__right,axiom,
! [A: $tType,N: nat,F3: A > A] :
( ( compow @ ( A > A ) @ ( suc @ N ) @ F3 )
= ( comp @ A @ A @ A @ ( compow @ ( A > A ) @ N @ F3 ) @ F3 ) ) ).
% funpow_Suc_right
thf(fact_5467_funpow_Osimps_I2_J,axiom,
! [A: $tType,N: nat,F3: A > A] :
( ( compow @ ( A > A ) @ ( suc @ N ) @ F3 )
= ( comp @ A @ A @ A @ F3 @ ( compow @ ( A > A ) @ N @ F3 ) ) ) ).
% funpow.simps(2)
thf(fact_5468_funpow__add,axiom,
! [A: $tType,M2: nat,N: nat,F3: A > A] :
( ( compow @ ( A > A ) @ ( plus_plus @ nat @ M2 @ N ) @ F3 )
= ( comp @ A @ A @ A @ ( compow @ ( A > A ) @ M2 @ F3 ) @ ( compow @ ( A > A ) @ N @ F3 ) ) ) ).
% funpow_add
thf(fact_5469_relpow__Suc__E,axiom,
! [A: $tType,X: A,Z: A,N: nat,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( suc @ N ) @ R ) )
=> ~ ! [Y3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y3 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z ) @ R ) ) ) ).
% relpow_Suc_E
thf(fact_5470_relpow__Suc__I,axiom,
! [A: $tType,X: A,Y: A,N: nat,R: set @ ( product_prod @ A @ A ),Z: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z ) @ R )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( suc @ N ) @ R ) ) ) ) ).
% relpow_Suc_I
thf(fact_5471_relpow__Suc__D2,axiom,
! [A: $tType,X: A,Z: A,N: nat,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( suc @ N ) @ R ) )
=> ? [Y3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y3 ) @ R )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R ) ) ) ) ).
% relpow_Suc_D2
thf(fact_5472_relpow__Suc__E2,axiom,
! [A: $tType,X: A,Z: A,N: nat,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( suc @ N ) @ R ) )
=> ~ ! [Y3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y3 ) @ R )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R ) ) ) ) ).
% relpow_Suc_E2
thf(fact_5473_relpow__Suc__I2,axiom,
! [A: $tType,X: A,Y: A,R: set @ ( product_prod @ A @ A ),Z: A,N: nat] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( suc @ N ) @ R ) ) ) ) ).
% relpow_Suc_I2
thf(fact_5474_relpow__0__E,axiom,
! [A: $tType,X: A,Y: A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( zero_zero @ nat ) @ R ) )
=> ( X = Y ) ) ).
% relpow_0_E
thf(fact_5475_relpow__0__I,axiom,
! [A: $tType,X: A,R: 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 ) @ R ) ) ).
% relpow_0_I
thf(fact_5476_cclfp__lowerbound,axiom,
! [A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [F3: A > A,A5: A] :
( ( order_mono @ A @ A @ F3 )
=> ( ( ord_less_eq @ A @ ( F3 @ A5 ) @ A5 )
=> ( ord_less_eq @ A @ ( order_532582986084564980_cclfp @ A @ F3 ) @ A5 ) ) ) ) ).
% cclfp_lowerbound
thf(fact_5477_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_5478_mono__mult,axiom,
! [A: $tType] :
( ( ordered_semiring @ A )
=> ! [A2: A] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( order_mono @ A @ A @ ( times_times @ A @ A2 ) ) ) ) ).
% mono_mult
thf(fact_5479_mono__image__least,axiom,
! [A: $tType,B: $tType] :
( ( ( order @ B )
& ( order @ A ) )
=> ! [F3: A > B,M2: A,N: A,M7: B,N7: B] :
( ( order_mono @ A @ B @ F3 )
=> ( ( ( image2 @ A @ B @ F3 @ ( set_or7035219750837199246ssThan @ A @ M2 @ N ) )
= ( set_or7035219750837199246ssThan @ B @ M7 @ N7 ) )
=> ( ( ord_less @ A @ M2 @ N )
=> ( ( F3 @ M2 )
= M7 ) ) ) ) ) ).
% mono_image_least
thf(fact_5480_sum__comp__morphism,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( ( comm_monoid_add @ B )
& ( comm_monoid_add @ A ) )
=> ! [H2: B > A,G3: C > B,A5: set @ C] :
( ( ( H2 @ ( zero_zero @ B ) )
= ( zero_zero @ A ) )
=> ( ! [X4: B,Y3: B] :
( ( H2 @ ( plus_plus @ B @ X4 @ Y3 ) )
= ( plus_plus @ A @ ( H2 @ X4 ) @ ( H2 @ Y3 ) ) )
=> ( ( groups7311177749621191930dd_sum @ C @ A @ ( comp @ B @ A @ C @ H2 @ G3 ) @ A5 )
= ( H2 @ ( groups7311177749621191930dd_sum @ C @ B @ G3 @ A5 ) ) ) ) ) ) ).
% sum_comp_morphism
thf(fact_5481_Kleene__iter__gpfp,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [F3: A > A,P: A,K2: nat] :
( ( order_mono @ A @ A @ F3 )
=> ( ( ord_less_eq @ A @ P @ ( F3 @ P ) )
=> ( ord_less_eq @ A @ P @ ( compow @ ( A > A ) @ K2 @ F3 @ ( top_top @ A ) ) ) ) ) ) ).
% Kleene_iter_gpfp
thf(fact_5482_Kleene__iter__lpfp,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [F3: A > A,P: A,K2: nat] :
( ( order_mono @ A @ A @ F3 )
=> ( ( ord_less_eq @ A @ ( F3 @ P ) @ P )
=> ( ord_less_eq @ A @ ( compow @ ( A > A ) @ K2 @ F3 @ ( bot_bot @ A ) ) @ P ) ) ) ) ).
% Kleene_iter_lpfp
thf(fact_5483_funpow__mono2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [F3: A > A,I2: nat,J: nat,X: A,Y: A] :
( ( order_mono @ A @ A @ F3 )
=> ( ( ord_less_eq @ nat @ I2 @ J )
=> ( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_less_eq @ A @ X @ ( F3 @ X ) )
=> ( ord_less_eq @ A @ ( compow @ ( A > A ) @ I2 @ F3 @ X ) @ ( compow @ ( A > A ) @ J @ F3 @ Y ) ) ) ) ) ) ) ).
% funpow_mono2
thf(fact_5484_relpowp__relpow__eq,axiom,
! [A: $tType,N: nat,R: set @ ( product_prod @ A @ A )] :
( ( compow @ ( A > A > $o ) @ N
@ ^ [X5: A,Y6: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X5 @ Y6 ) @ R ) )
= ( ^ [X5: A,Y6: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X5 @ Y6 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R ) ) ) ) ).
% relpowp_relpow_eq
thf(fact_5485_sum_OatLeast__Suc__atMost__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G3: nat > A,M2: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M2 ) @ ( suc @ N ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ nat @ A @ nat @ G3 @ suc ) @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ).
% sum.atLeast_Suc_atMost_Suc_shift
thf(fact_5486_sum_OatLeast__Suc__lessThan__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G3: nat > A,M2: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M2 ) @ ( suc @ N ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ nat @ A @ nat @ G3 @ suc ) @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) ) ) ).
% sum.atLeast_Suc_lessThan_Suc_shift
thf(fact_5487_sum_OatLeastAtMost__shift__bounds,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G3: nat > A,M2: nat,K2: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ M2 @ K2 ) @ ( plus_plus @ nat @ N @ K2 ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ nat @ A @ nat @ G3 @ ( plus_plus @ nat @ K2 ) ) @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ).
% sum.atLeastAtMost_shift_bounds
thf(fact_5488_sum_OatLeastLessThan__shift__bounds,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G3: nat > A,M2: nat,K2: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ ( plus_plus @ nat @ M2 @ K2 ) @ ( plus_plus @ nat @ N @ K2 ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ nat @ A @ nat @ G3 @ ( plus_plus @ nat @ K2 ) ) @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) ) ) ).
% sum.atLeastLessThan_shift_bounds
thf(fact_5489_prod_OatLeast__Suc__atMost__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G3: nat > A,M2: nat,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M2 ) @ ( suc @ N ) ) )
= ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G3 @ suc ) @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ).
% prod.atLeast_Suc_atMost_Suc_shift
thf(fact_5490_prod_OatLeast__Suc__lessThan__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G3: nat > A,M2: nat,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M2 ) @ ( suc @ N ) ) )
= ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G3 @ suc ) @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) ) ) ).
% prod.atLeast_Suc_lessThan_Suc_shift
thf(fact_5491_prod_OatLeastAtMost__shift__bounds,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G3: nat > A,M2: nat,K2: nat,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ ( plus_plus @ nat @ M2 @ K2 ) @ ( plus_plus @ nat @ N @ K2 ) ) )
= ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G3 @ ( plus_plus @ nat @ K2 ) ) @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ).
% prod.atLeastAtMost_shift_bounds
thf(fact_5492_prod_OatLeastLessThan__shift__bounds,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G3: nat > A,M2: nat,K2: nat,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ ( plus_plus @ nat @ M2 @ K2 ) @ ( plus_plus @ nat @ N @ K2 ) ) )
= ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G3 @ ( plus_plus @ nat @ K2 ) ) @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) ) ) ).
% prod.atLeastLessThan_shift_bounds
thf(fact_5493_bit__drop__bit__eq,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [N: nat,A2: A] :
( ( bit_se5641148757651400278ts_bit @ A @ ( bit_se4197421643247451524op_bit @ A @ N @ A2 ) )
= ( comp @ nat @ $o @ nat @ ( bit_se5641148757651400278ts_bit @ A @ A2 ) @ ( plus_plus @ nat @ N ) ) ) ) ).
% bit_drop_bit_eq
thf(fact_5494_mono__Sup,axiom,
! [B: $tType,A: $tType] :
( ( ( comple6319245703460814977attice @ A )
& ( comple6319245703460814977attice @ B ) )
=> ! [F3: A > B,A5: set @ A] :
( ( order_mono @ A @ B @ F3 )
=> ( ord_less_eq @ B @ ( complete_Sup_Sup @ B @ ( image2 @ A @ B @ F3 @ A5 ) ) @ ( F3 @ ( complete_Sup_Sup @ A @ A5 ) ) ) ) ) ).
% mono_Sup
thf(fact_5495_mono__SUP,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( ( comple6319245703460814977attice @ A )
& ( comple6319245703460814977attice @ B ) )
=> ! [F3: A > B,A5: C > A,I6: set @ C] :
( ( order_mono @ A @ B @ F3 )
=> ( ord_less_eq @ B
@ ( complete_Sup_Sup @ B
@ ( image2 @ C @ B
@ ^ [X5: C] : ( F3 @ ( A5 @ X5 ) )
@ I6 ) )
@ ( F3 @ ( complete_Sup_Sup @ A @ ( image2 @ C @ A @ A5 @ I6 ) ) ) ) ) ) ).
% mono_SUP
thf(fact_5496_mono__Inf,axiom,
! [B: $tType,A: $tType] :
( ( ( comple6319245703460814977attice @ A )
& ( comple6319245703460814977attice @ B ) )
=> ! [F3: A > B,A5: set @ A] :
( ( order_mono @ A @ B @ F3 )
=> ( ord_less_eq @ B @ ( F3 @ ( complete_Inf_Inf @ A @ A5 ) ) @ ( complete_Inf_Inf @ B @ ( image2 @ A @ B @ F3 @ A5 ) ) ) ) ) ).
% mono_Inf
thf(fact_5497_mono__INF,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( ( comple6319245703460814977attice @ A )
& ( comple6319245703460814977attice @ B ) )
=> ! [F3: A > B,A5: C > A,I6: set @ C] :
( ( order_mono @ A @ B @ F3 )
=> ( ord_less_eq @ B @ ( F3 @ ( complete_Inf_Inf @ A @ ( image2 @ C @ A @ A5 @ I6 ) ) )
@ ( complete_Inf_Inf @ B
@ ( image2 @ C @ B
@ ^ [X5: C] : ( F3 @ ( A5 @ X5 ) )
@ I6 ) ) ) ) ) ).
% mono_INF
thf(fact_5498_relpow__E2,axiom,
! [A: $tType,X: A,Z: A,N: nat,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R ) )
=> ( ( ( N
= ( zero_zero @ nat ) )
=> ( X != Z ) )
=> ~ ! [Y3: A,M: nat] :
( ( N
= ( suc @ M ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y3 ) @ R )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ M @ R ) ) ) ) ) ) ).
% relpow_E2
thf(fact_5499_relpow__E,axiom,
! [A: $tType,X: A,Z: A,N: nat,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R ) )
=> ( ( ( N
= ( zero_zero @ nat ) )
=> ( X != Z ) )
=> ~ ! [Y3: A,M: nat] :
( ( N
= ( suc @ M ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y3 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ M @ R ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z ) @ R ) ) ) ) ) ).
% relpow_E
thf(fact_5500_antimono__funpow,axiom,
! [A: $tType] :
( ( ( lattice @ A )
& ( order_top @ A ) )
=> ! [Q: A > A] :
( ( order_mono @ A @ A @ Q )
=> ( order_antimono @ nat @ A
@ ^ [I4: nat] : ( compow @ ( A > A ) @ I4 @ Q @ ( top_top @ A ) ) ) ) ) ).
% antimono_funpow
thf(fact_5501_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_5502_incseq__le,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ! [X7: nat > A,L5: A,N: nat] :
( ( order_mono @ nat @ A @ X7 )
=> ( ( filterlim @ nat @ A @ X7 @ ( topolo7230453075368039082e_nhds @ A @ L5 ) @ ( at_top @ nat ) )
=> ( ord_less_eq @ A @ ( X7 @ N ) @ L5 ) ) ) ) ).
% incseq_le
thf(fact_5503_sum_Oreindex__nontrivial,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A5: set @ B,H2: B > C,G3: C > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ! [X4: B,Y3: B] :
( ( member @ B @ X4 @ A5 )
=> ( ( member @ B @ Y3 @ A5 )
=> ( ( X4 != Y3 )
=> ( ( ( H2 @ X4 )
= ( H2 @ Y3 ) )
=> ( ( G3 @ ( H2 @ X4 ) )
= ( zero_zero @ A ) ) ) ) ) )
=> ( ( groups7311177749621191930dd_sum @ C @ A @ G3 @ ( image2 @ B @ C @ H2 @ A5 ) )
= ( groups7311177749621191930dd_sum @ B @ A @ ( comp @ C @ A @ B @ G3 @ H2 ) @ A5 ) ) ) ) ) ).
% sum.reindex_nontrivial
thf(fact_5504_funpow__increasing,axiom,
! [A: $tType] :
( ( ( lattice @ A )
& ( order_top @ A ) )
=> ! [M2: nat,N: nat,F3: A > A] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( order_mono @ A @ A @ F3 )
=> ( ord_less_eq @ A @ ( compow @ ( A > A ) @ N @ F3 @ ( top_top @ A ) ) @ ( compow @ ( A > A ) @ M2 @ F3 @ ( top_top @ A ) ) ) ) ) ) ).
% funpow_increasing
thf(fact_5505_funpow__decreasing,axiom,
! [A: $tType] :
( ( ( lattice @ A )
& ( order_bot @ A ) )
=> ! [M2: nat,N: nat,F3: A > A] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( order_mono @ A @ A @ F3 )
=> ( ord_less_eq @ A @ ( compow @ ( A > A ) @ M2 @ F3 @ ( bot_bot @ A ) ) @ ( compow @ ( A > A ) @ N @ F3 @ ( bot_bot @ A ) ) ) ) ) ) ).
% funpow_decreasing
thf(fact_5506_prod_Oreindex__nontrivial,axiom,
! [C: $tType,A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A5: set @ B,H2: B > C,G3: C > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ! [X4: B,Y3: B] :
( ( member @ B @ X4 @ A5 )
=> ( ( member @ B @ Y3 @ A5 )
=> ( ( X4 != Y3 )
=> ( ( ( H2 @ X4 )
= ( H2 @ Y3 ) )
=> ( ( G3 @ ( H2 @ X4 ) )
= ( one_one @ A ) ) ) ) ) )
=> ( ( groups7121269368397514597t_prod @ C @ A @ G3 @ ( image2 @ B @ C @ H2 @ A5 ) )
= ( groups7121269368397514597t_prod @ B @ A @ ( comp @ C @ A @ B @ G3 @ H2 ) @ A5 ) ) ) ) ) ).
% prod.reindex_nontrivial
thf(fact_5507_DERIV__at__within__shift__lemma,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,Y: A,Z: A,X: A,S2: set @ A] :
( ( has_field_derivative @ A @ F3 @ Y @ ( topolo174197925503356063within @ A @ ( plus_plus @ A @ Z @ X ) @ ( image2 @ A @ A @ ( plus_plus @ A @ Z ) @ S2 ) ) )
=> ( has_field_derivative @ A @ ( comp @ A @ A @ A @ F3 @ ( plus_plus @ A @ Z ) ) @ Y @ ( topolo174197925503356063within @ A @ X @ S2 ) ) ) ) ).
% DERIV_at_within_shift_lemma
thf(fact_5508_mono__Max__commute,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( linorder @ B ) )
=> ! [F3: A > B,A5: set @ A] :
( ( order_mono @ A @ B @ F3 )
=> ( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( F3 @ ( lattic643756798349783984er_Max @ A @ A5 ) )
= ( lattic643756798349783984er_Max @ B @ ( image2 @ A @ B @ F3 @ A5 ) ) ) ) ) ) ) ).
% mono_Max_commute
thf(fact_5509_relpow__fun__conv,axiom,
! [A: $tType,A2: A,B2: A,N: nat,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R ) )
= ( ? [F4: nat > A] :
( ( ( F4 @ ( zero_zero @ nat ) )
= A2 )
& ( ( F4 @ N )
= B2 )
& ! [I4: nat] :
( ( ord_less @ nat @ I4 @ N )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( F4 @ I4 ) @ ( F4 @ ( suc @ I4 ) ) ) @ R ) ) ) ) ) ).
% relpow_fun_conv
thf(fact_5510_sum__image__le,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( ordere6911136660526730532id_add @ B )
=> ! [I6: set @ C,G3: A > B,F3: C > A] :
( ( finite_finite2 @ C @ I6 )
=> ( ! [I3: C] :
( ( member @ C @ I3 @ I6 )
=> ( ord_less_eq @ B @ ( zero_zero @ B ) @ ( G3 @ ( F3 @ I3 ) ) ) )
=> ( ord_less_eq @ B @ ( groups7311177749621191930dd_sum @ A @ B @ G3 @ ( image2 @ C @ A @ F3 @ I6 ) ) @ ( groups7311177749621191930dd_sum @ C @ B @ ( comp @ A @ B @ C @ G3 @ F3 ) @ I6 ) ) ) ) ) ).
% sum_image_le
thf(fact_5511_filterlim__shift__iff,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: A > B,D3: A,F5: filter @ B,A2: A] :
( ( filterlim @ A @ B @ ( comp @ A @ B @ A @ F3 @ ( plus_plus @ A @ D3 ) ) @ F5 @ ( topolo174197925503356063within @ A @ ( minus_minus @ A @ A2 @ D3 ) @ ( top_top @ ( set @ A ) ) ) )
= ( filterlim @ A @ B @ F3 @ F5 @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% filterlim_shift_iff
thf(fact_5512_filterlim__shift,axiom,
! [B: $tType,A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: A > B,F5: filter @ B,A2: A,D3: A] :
( ( filterlim @ A @ B @ F3 @ F5 @ ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) ) )
=> ( filterlim @ A @ B @ ( comp @ A @ B @ A @ F3 @ ( plus_plus @ A @ D3 ) ) @ F5 @ ( topolo174197925503356063within @ A @ ( minus_minus @ A @ A2 @ D3 ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% filterlim_shift
thf(fact_5513_sum_OatLeast0__atMost__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G3: nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
= ( plus_plus @ A @ ( G3 @ ( zero_zero @ nat ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ nat @ A @ nat @ G3 @ suc ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ) ).
% sum.atLeast0_atMost_Suc_shift
thf(fact_5514_sum_OatLeast0__lessThan__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G3: nat > A,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
= ( plus_plus @ A @ ( G3 @ ( zero_zero @ nat ) ) @ ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ nat @ A @ nat @ G3 @ suc ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ) ).
% sum.atLeast0_lessThan_Suc_shift
thf(fact_5515_prod_OatLeast0__atMost__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G3: nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
= ( times_times @ A @ ( G3 @ ( zero_zero @ nat ) ) @ ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G3 @ suc ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ) ).
% prod.atLeast0_atMost_Suc_shift
thf(fact_5516_prod_OatLeast0__lessThan__Suc__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G3: nat > A,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
= ( times_times @ A @ ( G3 @ ( zero_zero @ nat ) ) @ ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G3 @ suc ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ) ).
% prod.atLeast0_lessThan_Suc_shift
thf(fact_5517_sum_OatLeastLessThan__shift__0,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G3: nat > A,M2: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ nat @ A @ nat @ G3 @ ( plus_plus @ nat @ M2 ) ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ N @ M2 ) ) ) ) ) ).
% sum.atLeastLessThan_shift_0
thf(fact_5518_prod_OatLeastLessThan__shift__0,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G3: nat > A,M2: nat,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) )
= ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G3 @ ( plus_plus @ nat @ M2 ) ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ N @ M2 ) ) ) ) ) ).
% prod.atLeastLessThan_shift_0
thf(fact_5519_mono__cSUP,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( ( condit1219197933456340205attice @ A )
& ( condit1219197933456340205attice @ B ) )
=> ! [F3: A > B,A5: C > A,I6: set @ C] :
( ( order_mono @ A @ B @ F3 )
=> ( ( condit941137186595557371_above @ A @ ( image2 @ C @ A @ A5 @ I6 ) )
=> ( ( I6
!= ( bot_bot @ ( set @ C ) ) )
=> ( ord_less_eq @ B
@ ( complete_Sup_Sup @ B
@ ( image2 @ C @ B
@ ^ [X5: C] : ( F3 @ ( A5 @ X5 ) )
@ I6 ) )
@ ( F3 @ ( complete_Sup_Sup @ A @ ( image2 @ C @ A @ A5 @ I6 ) ) ) ) ) ) ) ) ).
% mono_cSUP
thf(fact_5520_mono__cSup,axiom,
! [B: $tType,A: $tType] :
( ( ( condit1219197933456340205attice @ A )
& ( condit1219197933456340205attice @ B ) )
=> ! [F3: A > B,A5: set @ A] :
( ( order_mono @ A @ B @ F3 )
=> ( ( condit941137186595557371_above @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ord_less_eq @ B @ ( complete_Sup_Sup @ B @ ( image2 @ A @ B @ F3 @ A5 ) ) @ ( F3 @ ( complete_Sup_Sup @ A @ A5 ) ) ) ) ) ) ) ).
% mono_cSup
thf(fact_5521_mono__cINF,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( ( condit1219197933456340205attice @ A )
& ( condit1219197933456340205attice @ B ) )
=> ! [F3: A > B,A5: C > A,I6: set @ C] :
( ( order_mono @ A @ B @ F3 )
=> ( ( condit1013018076250108175_below @ A @ ( image2 @ C @ A @ A5 @ I6 ) )
=> ( ( I6
!= ( bot_bot @ ( set @ C ) ) )
=> ( ord_less_eq @ B @ ( F3 @ ( complete_Inf_Inf @ A @ ( image2 @ C @ A @ A5 @ I6 ) ) )
@ ( complete_Inf_Inf @ B
@ ( image2 @ C @ B
@ ^ [X5: C] : ( F3 @ ( A5 @ X5 ) )
@ I6 ) ) ) ) ) ) ) ).
% mono_cINF
thf(fact_5522_mono__cInf,axiom,
! [B: $tType,A: $tType] :
( ( ( condit1219197933456340205attice @ A )
& ( condit1219197933456340205attice @ B ) )
=> ! [F3: A > B,A5: set @ A] :
( ( order_mono @ A @ B @ F3 )
=> ( ( condit1013018076250108175_below @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ord_less_eq @ B @ ( F3 @ ( complete_Inf_Inf @ A @ A5 ) ) @ ( complete_Inf_Inf @ B @ ( image2 @ A @ B @ F3 @ A5 ) ) ) ) ) ) ) ).
% mono_cInf
thf(fact_5523_sum_OatLeast__atMost__pred__shift,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G3: nat > A,M2: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ( comp @ nat @ A @ nat @ G3
@ ^ [N2: nat] : ( minus_minus @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M2 ) @ ( suc @ N ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ).
% sum.atLeast_atMost_pred_shift
thf(fact_5524_sum_OatLeast__lessThan__pred__shift,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G3: nat > A,M2: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A
@ ( comp @ nat @ A @ nat @ G3
@ ^ [N2: nat] : ( minus_minus @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M2 ) @ ( suc @ N ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) ) ) ).
% sum.atLeast_lessThan_pred_shift
thf(fact_5525_prod_OatLeast__atMost__pred__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G3: nat > A,M2: nat,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A
@ ( comp @ nat @ A @ nat @ G3
@ ^ [N2: nat] : ( minus_minus @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) ) )
@ ( set_or1337092689740270186AtMost @ nat @ ( suc @ M2 ) @ ( suc @ N ) ) )
= ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ).
% prod.atLeast_atMost_pred_shift
thf(fact_5526_prod_OatLeast__lessThan__pred__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G3: nat > A,M2: nat,N: nat] :
( ( groups7121269368397514597t_prod @ nat @ A
@ ( comp @ nat @ A @ nat @ G3
@ ^ [N2: nat] : ( minus_minus @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) ) )
@ ( set_or7035219750837199246ssThan @ nat @ ( suc @ M2 ) @ ( suc @ N ) ) )
= ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) ) ) ).
% prod.atLeast_lessThan_pred_shift
thf(fact_5527_relpow__finite__bounded,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),K2: nat] :
( ( finite_finite2 @ ( product_prod @ A @ A ) @ R )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ K2 @ R )
@ ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ A ) )
@ ( image2 @ nat @ ( set @ ( product_prod @ A @ A ) )
@ ^ [N2: nat] : ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N2 @ R )
@ ( collect @ nat
@ ^ [N2: nat] : ( ord_less_eq @ nat @ N2 @ ( finite_card @ ( product_prod @ A @ A ) @ R ) ) ) ) ) ) ) ).
% relpow_finite_bounded
thf(fact_5528_sum_OatLeast__int__atMost__int__shift,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G3: int > A,M2: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ int @ A @ G3 @ ( set_or1337092689740270186AtMost @ int @ ( semiring_1_of_nat @ int @ M2 ) @ ( semiring_1_of_nat @ int @ N ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ int @ A @ nat @ G3 @ ( semiring_1_of_nat @ int ) ) @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ).
% sum.atLeast_int_atMost_int_shift
thf(fact_5529_mono__ge2__power__minus__self,axiom,
! [K2: nat] :
( ( ord_less_eq @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ K2 )
=> ( order_mono @ nat @ nat
@ ^ [M3: nat] : ( minus_minus @ nat @ ( power_power @ nat @ K2 @ M3 ) @ M3 ) ) ) ).
% mono_ge2_power_minus_self
thf(fact_5530_prod_OatLeast__int__atMost__int__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G3: int > A,M2: nat,N: nat] :
( ( groups7121269368397514597t_prod @ int @ A @ G3 @ ( set_or1337092689740270186AtMost @ int @ ( semiring_1_of_nat @ int @ M2 ) @ ( semiring_1_of_nat @ int @ N ) ) )
= ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ int @ A @ nat @ G3 @ ( semiring_1_of_nat @ int ) ) @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ).
% prod.atLeast_int_atMost_int_shift
thf(fact_5531_sum_OatLeast__int__lessThan__int__shift,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [G3: int > A,M2: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ int @ A @ G3 @ ( set_or7035219750837199246ssThan @ int @ ( semiring_1_of_nat @ int @ M2 ) @ ( semiring_1_of_nat @ int @ N ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ int @ A @ nat @ G3 @ ( semiring_1_of_nat @ int ) ) @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) ) ) ).
% sum.atLeast_int_lessThan_int_shift
thf(fact_5532_sum_OatLeastAtMost__shift__0,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [M2: nat,N: nat,G3: nat > A] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( groups7311177749621191930dd_sum @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ nat @ A @ nat @ G3 @ ( plus_plus @ nat @ M2 ) ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ N @ M2 ) ) ) ) ) ) ).
% sum.atLeastAtMost_shift_0
thf(fact_5533_prod_OatLeastAtMost__shift__0,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [M2: nat,N: nat,G3: nat > A] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( groups7121269368397514597t_prod @ nat @ A @ G3 @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) )
= ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ nat @ A @ nat @ G3 @ ( plus_plus @ nat @ M2 ) ) @ ( set_or1337092689740270186AtMost @ nat @ ( zero_zero @ nat ) @ ( minus_minus @ nat @ N @ M2 ) ) ) ) ) ) ).
% prod.atLeastAtMost_shift_0
thf(fact_5534_prod_OatLeast__int__lessThan__int__shift,axiom,
! [A: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G3: int > A,M2: nat,N: nat] :
( ( groups7121269368397514597t_prod @ int @ A @ G3 @ ( set_or7035219750837199246ssThan @ int @ ( semiring_1_of_nat @ int @ M2 ) @ ( semiring_1_of_nat @ int @ N ) ) )
= ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ int @ A @ nat @ G3 @ ( semiring_1_of_nat @ int ) ) @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) ) ) ).
% prod.atLeast_int_lessThan_int_shift
thf(fact_5535_finite__mono__remains__stable__implies__strict__prefix,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [F3: nat > A] :
( ( finite_finite2 @ A @ ( image2 @ nat @ A @ F3 @ ( top_top @ ( set @ nat ) ) ) )
=> ( ( order_mono @ nat @ A @ F3 )
=> ( ! [N3: nat] :
( ( ( F3 @ N3 )
= ( F3 @ ( suc @ N3 ) ) )
=> ( ( F3 @ ( suc @ N3 ) )
= ( F3 @ ( suc @ ( suc @ N3 ) ) ) ) )
=> ? [N9: nat] :
( ! [N5: nat] :
( ( ord_less_eq @ nat @ N5 @ N9 )
=> ! [M4: nat] :
( ( ord_less_eq @ nat @ M4 @ N9 )
=> ( ( ord_less @ nat @ M4 @ N5 )
=> ( ord_less @ A @ ( F3 @ M4 ) @ ( F3 @ N5 ) ) ) ) )
& ! [N5: nat] :
( ( ord_less_eq @ nat @ N9 @ N5 )
=> ( ( F3 @ N9 )
= ( F3 @ N5 ) ) ) ) ) ) ) ) ).
% finite_mono_remains_stable_implies_strict_prefix
thf(fact_5536_tendsto__at__left__sequentially,axiom,
! [A: $tType,B: $tType] :
( ( ( topolo3112930676232923870pology @ B )
& ( topolo1944317154257567458pology @ B )
& ( topolo4958980785337419405_space @ A ) )
=> ! [B2: B,A2: B,X7: B > A,L5: A] :
( ( ord_less @ B @ B2 @ A2 )
=> ( ! [S6: nat > B] :
( ! [N5: nat] : ( ord_less @ B @ ( S6 @ N5 ) @ A2 )
=> ( ! [N5: nat] : ( ord_less @ B @ B2 @ ( S6 @ N5 ) )
=> ( ( order_mono @ nat @ B @ S6 )
=> ( ( filterlim @ nat @ B @ S6 @ ( topolo7230453075368039082e_nhds @ B @ A2 ) @ ( at_top @ nat ) )
=> ( filterlim @ nat @ A
@ ^ [N2: nat] : ( X7 @ ( S6 @ N2 ) )
@ ( topolo7230453075368039082e_nhds @ A @ L5 )
@ ( at_top @ nat ) ) ) ) ) )
=> ( filterlim @ B @ A @ X7 @ ( topolo7230453075368039082e_nhds @ A @ L5 ) @ ( topolo174197925503356063within @ B @ A2 @ ( set_ord_lessThan @ B @ A2 ) ) ) ) ) ) ).
% tendsto_at_left_sequentially
thf(fact_5537_lim__at__infinity__0,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,L: A] :
( ( filterlim @ A @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ ( at_infinity @ A ) )
= ( filterlim @ A @ A @ ( comp @ A @ A @ A @ F3 @ ( inverse_inverse @ A ) ) @ ( topolo7230453075368039082e_nhds @ A @ L ) @ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% lim_at_infinity_0
thf(fact_5538_continuous__at__Sup__mono,axiom,
! [B: $tType,A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( topolo1944317154257567458pology @ A )
& ( condit6923001295902523014norder @ B )
& ( topolo1944317154257567458pology @ B ) )
=> ! [F3: A > B,S2: set @ A] :
( ( order_mono @ A @ B @ F3 )
=> ( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ ( complete_Sup_Sup @ A @ S2 ) @ ( set_ord_lessThan @ A @ ( complete_Sup_Sup @ A @ S2 ) ) ) @ F3 )
=> ( ( S2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit941137186595557371_above @ A @ S2 )
=> ( ( F3 @ ( complete_Sup_Sup @ A @ S2 ) )
= ( complete_Sup_Sup @ B @ ( image2 @ A @ B @ F3 @ S2 ) ) ) ) ) ) ) ) ).
% continuous_at_Sup_mono
thf(fact_5539_continuous__at__Inf__mono,axiom,
! [B: $tType,A: $tType] :
( ( ( condit6923001295902523014norder @ A )
& ( topolo1944317154257567458pology @ A )
& ( condit6923001295902523014norder @ B )
& ( topolo1944317154257567458pology @ B ) )
=> ! [F3: A > B,S2: set @ A] :
( ( order_mono @ A @ B @ F3 )
=> ( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ ( complete_Inf_Inf @ A @ S2 ) @ ( set_ord_greaterThan @ A @ ( complete_Inf_Inf @ A @ S2 ) ) ) @ F3 )
=> ( ( S2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit1013018076250108175_below @ A @ S2 )
=> ( ( F3 @ ( complete_Inf_Inf @ A @ S2 ) )
= ( complete_Inf_Inf @ B @ ( image2 @ A @ B @ F3 @ S2 ) ) ) ) ) ) ) ) ).
% continuous_at_Inf_mono
thf(fact_5540_ntrancl__def,axiom,
! [A: $tType] :
( ( transitive_ntrancl @ A )
= ( ^ [N2: nat,R6: set @ ( product_prod @ A @ A )] :
( complete_Sup_Sup @ ( set @ ( product_prod @ A @ A ) )
@ ( image2 @ nat @ ( set @ ( product_prod @ A @ A ) )
@ ^ [I4: nat] : ( compow @ ( set @ ( product_prod @ A @ A ) ) @ I4 @ R6 )
@ ( collect @ nat
@ ^ [I4: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ I4 )
& ( ord_less_eq @ nat @ I4 @ ( suc @ N2 ) ) ) ) ) ) ) ) ).
% ntrancl_def
thf(fact_5541_trancl__finite__eq__relpow,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( finite_finite2 @ ( product_prod @ A @ A ) @ R )
=> ( ( transitive_trancl @ A @ R )
= ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ A ) )
@ ( image2 @ nat @ ( set @ ( product_prod @ A @ A ) )
@ ^ [N2: nat] : ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N2 @ R )
@ ( collect @ nat
@ ^ [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
& ( ord_less_eq @ nat @ N2 @ ( finite_card @ ( product_prod @ A @ A ) @ R ) ) ) ) ) ) ) ) ).
% trancl_finite_eq_relpow
thf(fact_5542_remdups__adj__altdef,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( remdups_adj @ A @ Xs )
= Ys )
= ( ? [F4: nat > nat] :
( ( order_mono @ nat @ nat @ F4 )
& ( ( image2 @ nat @ nat @ F4 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Xs ) ) )
= ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Ys ) ) )
& ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ A @ Xs @ I4 )
= ( nth @ A @ Ys @ ( F4 @ I4 ) ) ) )
& ! [I4: nat] :
( ( ord_less @ nat @ ( plus_plus @ nat @ I4 @ ( one_one @ nat ) ) @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ( nth @ A @ Xs @ I4 )
= ( nth @ A @ Xs @ ( plus_plus @ nat @ I4 @ ( one_one @ nat ) ) ) )
= ( ( F4 @ I4 )
= ( F4 @ ( plus_plus @ nat @ I4 @ ( one_one @ nat ) ) ) ) ) ) ) ) ) ).
% remdups_adj_altdef
thf(fact_5543_remdups__adj__Nil__iff,axiom,
! [A: $tType,Xs: list @ A] :
( ( ( remdups_adj @ A @ Xs )
= ( nil @ A ) )
= ( Xs
= ( nil @ A ) ) ) ).
% remdups_adj_Nil_iff
thf(fact_5544_remdups__adj__set,axiom,
! [A: $tType,Xs: list @ A] :
( ( set2 @ A @ ( remdups_adj @ A @ Xs ) )
= ( set2 @ A @ Xs ) ) ).
% remdups_adj_set
thf(fact_5545_trancl__empty,axiom,
! [A: $tType] :
( ( transitive_trancl @ A @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
= ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ).
% trancl_empty
thf(fact_5546_finite__trancl,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( finite_finite2 @ ( product_prod @ A @ A ) @ ( transitive_trancl @ A @ R2 ) )
= ( finite_finite2 @ ( product_prod @ A @ A ) @ R2 ) ) ).
% finite_trancl
thf(fact_5547_ntrancl__Zero,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( transitive_ntrancl @ A @ ( zero_zero @ nat ) @ R )
= R ) ).
% ntrancl_Zero
thf(fact_5548_snd__diag__snd,axiom,
! [B: $tType,A: $tType] :
( ( comp @ ( product_prod @ B @ B ) @ B @ ( product_prod @ A @ B ) @ ( product_snd @ B @ B )
@ ( comp @ B @ ( product_prod @ B @ B ) @ ( product_prod @ A @ B )
@ ^ [X5: B] : ( product_Pair @ B @ B @ X5 @ X5 )
@ ( product_snd @ A @ B ) ) )
= ( product_snd @ A @ B ) ) ).
% snd_diag_snd
thf(fact_5549_fst__diag__fst,axiom,
! [B: $tType,A: $tType] :
( ( comp @ ( product_prod @ A @ A ) @ A @ ( product_prod @ A @ B ) @ ( product_fst @ A @ A )
@ ( comp @ A @ ( product_prod @ A @ A ) @ ( product_prod @ A @ B )
@ ^ [X5: A] : ( product_Pair @ A @ A @ X5 @ X5 )
@ ( product_fst @ A @ B ) ) )
= ( product_fst @ A @ B ) ) ).
% fst_diag_fst
thf(fact_5550_fst__diag__snd,axiom,
! [B: $tType,A: $tType] :
( ( comp @ ( product_prod @ B @ B ) @ B @ ( product_prod @ A @ B ) @ ( product_fst @ B @ B )
@ ( comp @ B @ ( product_prod @ B @ B ) @ ( product_prod @ A @ B )
@ ^ [X5: B] : ( product_Pair @ B @ B @ X5 @ X5 )
@ ( product_snd @ A @ B ) ) )
= ( product_snd @ A @ B ) ) ).
% fst_diag_snd
thf(fact_5551_snd__diag__fst,axiom,
! [B: $tType,A: $tType] :
( ( comp @ ( product_prod @ A @ A ) @ A @ ( product_prod @ A @ B ) @ ( product_snd @ A @ A )
@ ( comp @ A @ ( product_prod @ A @ A ) @ ( product_prod @ A @ B )
@ ^ [X5: A] : ( product_Pair @ A @ A @ X5 @ X5 )
@ ( product_fst @ A @ B ) ) )
= ( product_fst @ A @ B ) ) ).
% snd_diag_fst
thf(fact_5552_sorted__list__of__set_Ofold__insort__key_Ocomp__fun__commute__on,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Y: A,X: A] :
( ( comp @ ( list @ A ) @ ( list @ A ) @ ( list @ A )
@ ( linorder_insort_key @ A @ A
@ ^ [X5: A] : X5
@ Y )
@ ( linorder_insort_key @ A @ A
@ ^ [X5: A] : X5
@ X ) )
= ( comp @ ( list @ A ) @ ( list @ A ) @ ( list @ A )
@ ( linorder_insort_key @ A @ A
@ ^ [X5: A] : X5
@ X )
@ ( linorder_insort_key @ A @ A
@ ^ [X5: A] : X5
@ Y ) ) ) ) ).
% sorted_list_of_set.fold_insort_key.comp_fun_commute_on
thf(fact_5553_remdups__adj__distinct,axiom,
! [A: $tType,Xs: list @ A] :
( ( distinct @ A @ Xs )
=> ( ( remdups_adj @ A @ Xs )
= Xs ) ) ).
% remdups_adj_distinct
thf(fact_5554_remdups__adj_Osimps_I1_J,axiom,
! [A: $tType] :
( ( remdups_adj @ A @ ( nil @ A ) )
= ( nil @ A ) ) ).
% remdups_adj.simps(1)
thf(fact_5555_trancl_Ocases,axiom,
! [A: $tType,A1: A,A22: A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A1 @ A22 ) @ ( transitive_trancl @ A @ R2 ) )
=> ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A1 @ A22 ) @ R2 )
=> ~ ! [B4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A1 @ B4 ) @ ( transitive_trancl @ A @ R2 ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B4 @ A22 ) @ R2 ) ) ) ) ).
% trancl.cases
thf(fact_5556_trancl_Osimps,axiom,
! [A: $tType,A1: A,A22: A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A1 @ A22 ) @ ( transitive_trancl @ A @ R2 ) )
= ( ? [A6: A,B5: A] :
( ( A1 = A6 )
& ( A22 = B5 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A6 @ B5 ) @ R2 ) )
| ? [A6: A,B5: A,C4: A] :
( ( A1 = A6 )
& ( A22 = C4 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A6 @ B5 ) @ ( transitive_trancl @ A @ R2 ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B5 @ C4 ) @ R2 ) ) ) ) ).
% trancl.simps
thf(fact_5557_trancl_Or__into__trancl,axiom,
! [A: $tType,A2: A,B2: A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ R2 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ ( transitive_trancl @ A @ R2 ) ) ) ).
% trancl.r_into_trancl
thf(fact_5558_tranclE,axiom,
! [A: $tType,A2: A,B2: A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ ( transitive_trancl @ A @ R2 ) )
=> ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ R2 )
=> ~ ! [C2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ C2 ) @ ( transitive_trancl @ A @ R2 ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ C2 @ B2 ) @ R2 ) ) ) ) ).
% tranclE
thf(fact_5559_trancl__trans,axiom,
! [A: $tType,X: A,Y: A,R2: set @ ( product_prod @ A @ A ),Z: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( transitive_trancl @ A @ R2 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z ) @ ( transitive_trancl @ A @ R2 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z ) @ ( transitive_trancl @ A @ R2 ) ) ) ) ).
% trancl_trans
thf(fact_5560_trancl__induct,axiom,
! [A: $tType,A2: A,B2: A,R2: set @ ( product_prod @ A @ A ),P2: A > $o] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ ( transitive_trancl @ A @ R2 ) )
=> ( ! [Y3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ Y3 ) @ R2 )
=> ( P2 @ Y3 ) )
=> ( ! [Y3: A,Z3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ Y3 ) @ ( transitive_trancl @ A @ R2 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z3 ) @ R2 )
=> ( ( P2 @ Y3 )
=> ( P2 @ Z3 ) ) ) )
=> ( P2 @ B2 ) ) ) ) ).
% trancl_induct
thf(fact_5561_r__r__into__trancl,axiom,
! [A: $tType,A2: A,B2: A,R: set @ ( product_prod @ A @ A ),C3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ R )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ C3 ) @ R )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ C3 ) @ ( transitive_trancl @ A @ R ) ) ) ) ).
% r_r_into_trancl
thf(fact_5562_converse__tranclE,axiom,
! [A: $tType,X: A,Z: A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z ) @ ( transitive_trancl @ A @ R2 ) )
=> ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z ) @ 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 @ Z ) @ ( transitive_trancl @ A @ R2 ) ) ) ) ) ).
% converse_tranclE
thf(fact_5563_irrefl__trancl__rD,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),X: A,Y: A] :
( ! [X4: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ X4 ) @ ( transitive_trancl @ A @ R2 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R2 )
=> ( X != Y ) ) ) ).
% irrefl_trancl_rD
thf(fact_5564_Transitive__Closure_Otrancl__into__trancl,axiom,
! [A: $tType,A2: A,B2: A,R2: set @ ( product_prod @ A @ A ),C3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ ( transitive_trancl @ A @ R2 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ C3 ) @ R2 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ C3 ) @ ( transitive_trancl @ A @ R2 ) ) ) ) ).
% Transitive_Closure.trancl_into_trancl
thf(fact_5565_trancl__into__trancl2,axiom,
! [A: $tType,A2: A,B2: A,R2: set @ ( product_prod @ A @ A ),C3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ C3 ) @ ( transitive_trancl @ A @ R2 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ C3 ) @ ( transitive_trancl @ A @ R2 ) ) ) ) ).
% trancl_into_trancl2
thf(fact_5566_trancl__trans__induct,axiom,
! [A: $tType,X: A,Y: A,R2: set @ ( product_prod @ A @ A ),P2: A > A > $o] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( transitive_trancl @ A @ R2 ) )
=> ( ! [X4: A,Y3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y3 ) @ R2 )
=> ( P2 @ X4 @ Y3 ) )
=> ( ! [X4: A,Y3: A,Z3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y3 ) @ ( transitive_trancl @ A @ R2 ) )
=> ( ( P2 @ X4 @ Y3 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z3 ) @ ( transitive_trancl @ A @ R2 ) )
=> ( ( P2 @ Y3 @ Z3 )
=> ( P2 @ X4 @ Z3 ) ) ) ) )
=> ( P2 @ X @ Y ) ) ) ) ).
% trancl_trans_induct
thf(fact_5567_converse__trancl__induct,axiom,
! [A: $tType,A2: A,B2: A,R2: set @ ( product_prod @ A @ A ),P2: A > $o] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ ( transitive_trancl @ A @ R2 ) )
=> ( ! [Y3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ B2 ) @ R2 )
=> ( P2 @ Y3 ) )
=> ( ! [Y3: A,Z3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z3 ) @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z3 @ B2 ) @ ( transitive_trancl @ A @ R2 ) )
=> ( ( P2 @ Z3 )
=> ( P2 @ Y3 ) ) ) )
=> ( P2 @ A2 ) ) ) ) ).
% converse_trancl_induct
thf(fact_5568_trancl__induct2,axiom,
! [A: $tType,B: $tType,Ax: A,Ay: B,Bx: A,By: B,R2: set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ),P2: A > B > $o] :
( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Ax @ Ay ) @ ( product_Pair @ A @ B @ Bx @ By ) ) @ ( transitive_trancl @ ( product_prod @ A @ B ) @ R2 ) )
=> ( ! [A4: A,B4: B] :
( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Ax @ Ay ) @ ( product_Pair @ A @ B @ A4 @ B4 ) ) @ R2 )
=> ( P2 @ A4 @ B4 ) )
=> ( ! [A4: A,B4: B,Aa2: A,Ba: B] :
( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Ax @ Ay ) @ ( product_Pair @ A @ B @ A4 @ B4 ) ) @ ( transitive_trancl @ ( product_prod @ A @ B ) @ R2 ) )
=> ( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B4 ) @ ( product_Pair @ A @ B @ Aa2 @ Ba ) ) @ R2 )
=> ( ( P2 @ A4 @ B4 )
=> ( P2 @ Aa2 @ Ba ) ) ) )
=> ( P2 @ Bx @ By ) ) ) ) ).
% trancl_induct2
thf(fact_5569_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_5570_finite__trancl__ntranl,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( finite_finite2 @ ( product_prod @ A @ A ) @ R )
=> ( ( transitive_trancl @ A @ R )
= ( transitive_ntrancl @ A @ ( minus_minus @ nat @ ( finite_card @ ( product_prod @ A @ A ) @ R ) @ ( one_one @ nat ) ) @ R ) ) ) ).
% finite_trancl_ntranl
thf(fact_5571_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_5572_trancl__power,axiom,
! [A: $tType,P: product_prod @ A @ A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ P @ ( transitive_trancl @ A @ R ) )
= ( ? [N2: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N2 )
& ( member @ ( product_prod @ A @ A ) @ P @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N2 @ R ) ) ) ) ) ).
% trancl_power
thf(fact_5573_fst__diag__id,axiom,
! [A: $tType,Z: A] :
( ( comp @ ( product_prod @ A @ A ) @ A @ A @ ( product_fst @ A @ A )
@ ^ [X5: A] : ( product_Pair @ A @ A @ X5 @ X5 )
@ Z )
= ( id @ A @ Z ) ) ).
% fst_diag_id
thf(fact_5574_snd__diag__id,axiom,
! [A: $tType,Z: A] :
( ( comp @ ( product_prod @ A @ A ) @ A @ A @ ( product_snd @ A @ A )
@ ^ [X5: A] : ( product_Pair @ A @ A @ X5 @ X5 )
@ Z )
= ( id @ A @ Z ) ) ).
% snd_diag_id
thf(fact_5575_case__prod__comp,axiom,
! [D: $tType,A: $tType,C: $tType,B: $tType,F3: D > C > A,G3: B > D,X: product_prod @ B @ C] :
( ( product_case_prod @ B @ C @ A @ ( comp @ D @ ( C > A ) @ B @ F3 @ G3 ) @ X )
= ( F3 @ ( G3 @ ( product_fst @ B @ C @ X ) ) @ ( product_snd @ B @ C @ X ) ) ) ).
% case_prod_comp
thf(fact_5576_prod_OUnion__comp,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [B6: set @ ( set @ B ),G3: B > A] :
( ! [X4: set @ B] :
( ( member @ ( set @ B ) @ X4 @ B6 )
=> ( finite_finite2 @ B @ X4 ) )
=> ( ! [A14: set @ B] :
( ( member @ ( set @ B ) @ A14 @ B6 )
=> ! [A25: set @ B] :
( ( member @ ( set @ B ) @ A25 @ B6 )
=> ( ( A14 != A25 )
=> ! [X4: B] :
( ( member @ B @ X4 @ A14 )
=> ( ( member @ B @ X4 @ A25 )
=> ( ( G3 @ X4 )
= ( one_one @ A ) ) ) ) ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G3 @ ( complete_Sup_Sup @ ( set @ B ) @ B6 ) )
= ( comp @ ( ( set @ B ) > A ) @ ( ( set @ ( set @ B ) ) > A ) @ ( B > A ) @ ( groups7121269368397514597t_prod @ ( set @ B ) @ A ) @ ( groups7121269368397514597t_prod @ B @ A ) @ G3 @ B6 ) ) ) ) ) ).
% prod.Union_comp
thf(fact_5577_remdups__adj__adjacent,axiom,
! [A: $tType,I2: nat,Xs: list @ A] :
( ( ord_less @ nat @ ( suc @ I2 ) @ ( size_size @ ( list @ A ) @ ( remdups_adj @ A @ Xs ) ) )
=> ( ( nth @ A @ ( remdups_adj @ A @ Xs ) @ I2 )
!= ( nth @ A @ ( remdups_adj @ A @ Xs ) @ ( suc @ I2 ) ) ) ) ).
% remdups_adj_adjacent
thf(fact_5578_infinite__int__iff__infinite__nat__abs,axiom,
! [S2: set @ int] :
( ( ~ ( finite_finite2 @ int @ S2 ) )
= ( ~ ( finite_finite2 @ nat @ ( image2 @ int @ nat @ ( comp @ int @ nat @ int @ nat2 @ ( abs_abs @ int ) ) @ S2 ) ) ) ) ).
% infinite_int_iff_infinite_nat_abs
thf(fact_5579_fst__snd__flip,axiom,
! [B: $tType,A: $tType] :
( ( product_fst @ A @ B )
= ( comp @ ( product_prod @ B @ A ) @ A @ ( product_prod @ A @ B ) @ ( product_snd @ B @ A )
@ ( product_case_prod @ A @ B @ ( product_prod @ B @ A )
@ ^ [X5: A,Y6: B] : ( product_Pair @ B @ A @ Y6 @ X5 ) ) ) ) ).
% fst_snd_flip
thf(fact_5580_snd__fst__flip,axiom,
! [A: $tType,B: $tType] :
( ( product_snd @ B @ A )
= ( comp @ ( product_prod @ A @ B ) @ A @ ( product_prod @ B @ A ) @ ( product_fst @ A @ B )
@ ( product_case_prod @ B @ A @ ( product_prod @ A @ B )
@ ^ [X5: B,Y6: A] : ( product_Pair @ A @ B @ Y6 @ X5 ) ) ) ) ).
% snd_fst_flip
thf(fact_5581_prod_OUnion__disjoint,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [C5: set @ ( set @ B ),G3: B > A] :
( ! [X4: set @ B] :
( ( member @ ( set @ B ) @ X4 @ C5 )
=> ( finite_finite2 @ B @ X4 ) )
=> ( ! [X4: set @ B] :
( ( member @ ( set @ B ) @ X4 @ C5 )
=> ! [Xa3: set @ B] :
( ( member @ ( set @ B ) @ Xa3 @ C5 )
=> ( ( X4 != Xa3 )
=> ( ( inf_inf @ ( set @ B ) @ X4 @ Xa3 )
= ( bot_bot @ ( set @ B ) ) ) ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G3 @ ( complete_Sup_Sup @ ( set @ B ) @ C5 ) )
= ( comp @ ( ( set @ B ) > A ) @ ( ( set @ ( set @ B ) ) > A ) @ ( B > A ) @ ( groups7121269368397514597t_prod @ ( set @ B ) @ A ) @ ( groups7121269368397514597t_prod @ B @ A ) @ G3 @ C5 ) ) ) ) ) ).
% prod.Union_disjoint
thf(fact_5582_sum_OUnion__comp,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [B6: set @ ( set @ B ),G3: B > A] :
( ! [X4: set @ B] :
( ( member @ ( set @ B ) @ X4 @ B6 )
=> ( finite_finite2 @ B @ X4 ) )
=> ( ! [A14: set @ B] :
( ( member @ ( set @ B ) @ A14 @ B6 )
=> ! [A25: set @ B] :
( ( member @ ( set @ B ) @ A25 @ B6 )
=> ( ( A14 != A25 )
=> ! [X4: B] :
( ( member @ B @ X4 @ A14 )
=> ( ( member @ B @ X4 @ A25 )
=> ( ( G3 @ X4 )
= ( zero_zero @ A ) ) ) ) ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G3 @ ( complete_Sup_Sup @ ( set @ B ) @ B6 ) )
= ( comp @ ( ( set @ B ) > A ) @ ( ( set @ ( set @ B ) ) > A ) @ ( B > A ) @ ( groups7311177749621191930dd_sum @ ( set @ B ) @ A ) @ ( groups7311177749621191930dd_sum @ B @ A ) @ G3 @ B6 ) ) ) ) ) ).
% sum.Union_comp
thf(fact_5583_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_5584_sum_OUnion__disjoint,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [C5: set @ ( set @ B ),G3: B > A] :
( ! [X4: set @ B] :
( ( member @ ( set @ B ) @ X4 @ C5 )
=> ( finite_finite2 @ B @ X4 ) )
=> ( ! [X4: set @ B] :
( ( member @ ( set @ B ) @ X4 @ C5 )
=> ! [Xa3: set @ B] :
( ( member @ ( set @ B ) @ Xa3 @ C5 )
=> ( ( X4 != Xa3 )
=> ( ( inf_inf @ ( set @ B ) @ X4 @ Xa3 )
= ( bot_bot @ ( set @ B ) ) ) ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G3 @ ( complete_Sup_Sup @ ( set @ B ) @ C5 ) )
= ( comp @ ( ( set @ B ) > A ) @ ( ( set @ ( set @ B ) ) > A ) @ ( B > A ) @ ( groups7311177749621191930dd_sum @ ( set @ B ) @ A ) @ ( groups7311177749621191930dd_sum @ B @ A ) @ G3 @ C5 ) ) ) ) ) ).
% sum.Union_disjoint
thf(fact_5585_filterlim__at__top__iff__inverse__0,axiom,
! [A: $tType,F3: A > real,F5: filter @ A] :
( ( eventually @ A
@ ^ [X5: A] : ( ord_less @ real @ ( zero_zero @ real ) @ ( F3 @ X5 ) )
@ F5 )
=> ( ( filterlim @ A @ real @ F3 @ ( at_top @ real ) @ F5 )
= ( filterlim @ A @ real @ ( comp @ real @ real @ A @ ( inverse_inverse @ real ) @ F3 ) @ ( topolo7230453075368039082e_nhds @ real @ ( zero_zero @ real ) ) @ F5 ) ) ) ).
% filterlim_at_top_iff_inverse_0
thf(fact_5586_empty__natural,axiom,
! [C: $tType,B: $tType,D: $tType,A: $tType,F3: A > C,G3: D > B] :
( ( comp @ C @ ( set @ B ) @ A
@ ^ [Uu3: C] : ( bot_bot @ ( set @ B ) )
@ F3 )
= ( comp @ ( set @ D ) @ ( set @ B ) @ A @ ( image2 @ D @ B @ G3 )
@ ^ [Uu3: A] : ( bot_bot @ ( set @ D ) ) ) ) ).
% empty_natural
thf(fact_5587_prod__filter__assoc,axiom,
! [A: $tType,B: $tType,C: $tType,F5: filter @ A,G6: filter @ B,H7: filter @ C] :
( ( prod_filter @ ( product_prod @ A @ B ) @ C @ ( prod_filter @ A @ B @ F5 @ G6 ) @ H7 )
= ( filtermap @ ( product_prod @ A @ ( product_prod @ B @ C ) ) @ ( product_prod @ ( product_prod @ A @ B ) @ C )
@ ( product_case_prod @ A @ ( product_prod @ B @ C ) @ ( product_prod @ ( product_prod @ A @ B ) @ C )
@ ^ [X5: A] :
( product_case_prod @ B @ C @ ( product_prod @ ( product_prod @ A @ B ) @ C )
@ ^ [Y6: B] : ( product_Pair @ ( product_prod @ A @ B ) @ C @ ( product_Pair @ A @ B @ X5 @ Y6 ) ) ) )
@ ( prod_filter @ A @ ( product_prod @ B @ C ) @ F5 @ ( prod_filter @ B @ C @ G6 @ H7 ) ) ) ) ).
% prod_filter_assoc
thf(fact_5588_range__abs__Nats,axiom,
( ( image2 @ int @ int @ ( abs_abs @ int ) @ ( top_top @ ( set @ int ) ) )
= ( semiring_1_Nats @ int ) ) ).
% range_abs_Nats
thf(fact_5589_filtermap__bot,axiom,
! [B: $tType,A: $tType,F3: B > A] :
( ( filtermap @ B @ A @ F3 @ ( bot_bot @ ( filter @ B ) ) )
= ( bot_bot @ ( filter @ A ) ) ) ).
% filtermap_bot
thf(fact_5590_filtermap__bot__iff,axiom,
! [A: $tType,B: $tType,F3: B > A,F5: filter @ B] :
( ( ( filtermap @ B @ A @ F3 @ F5 )
= ( bot_bot @ ( filter @ A ) ) )
= ( F5
= ( bot_bot @ ( filter @ B ) ) ) ) ).
% filtermap_bot_iff
thf(fact_5591_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_5592_Nats__induct,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [X: A,P2: A > $o] :
( ( member @ A @ X @ ( semiring_1_Nats @ A ) )
=> ( ! [N3: nat] : ( P2 @ ( semiring_1_of_nat @ A @ N3 ) )
=> ( P2 @ X ) ) ) ) ).
% Nats_induct
thf(fact_5593_Nats__cases,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [X: A] :
( ( member @ A @ X @ ( semiring_1_Nats @ A ) )
=> ~ ! [N3: nat] :
( X
!= ( semiring_1_of_nat @ A @ N3 ) ) ) ) ).
% Nats_cases
thf(fact_5594_Nats__0,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( member @ A @ ( zero_zero @ A ) @ ( semiring_1_Nats @ A ) ) ) ).
% Nats_0
thf(fact_5595_Nats__numeral,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [W2: num] : ( member @ A @ ( numeral_numeral @ A @ W2 ) @ ( semiring_1_Nats @ A ) ) ) ).
% Nats_numeral
thf(fact_5596_Nats__mult,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [A2: A,B2: A] :
( ( member @ A @ A2 @ ( semiring_1_Nats @ A ) )
=> ( ( member @ A @ B2 @ ( semiring_1_Nats @ A ) )
=> ( member @ A @ ( times_times @ A @ A2 @ B2 ) @ ( semiring_1_Nats @ A ) ) ) ) ) ).
% Nats_mult
thf(fact_5597_Nats__1,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( member @ A @ ( one_one @ A ) @ ( semiring_1_Nats @ A ) ) ) ).
% Nats_1
thf(fact_5598_Nats__add,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ! [A2: A,B2: A] :
( ( member @ A @ A2 @ ( semiring_1_Nats @ A ) )
=> ( ( member @ A @ B2 @ ( semiring_1_Nats @ A ) )
=> ( member @ A @ ( plus_plus @ A @ A2 @ B2 ) @ ( semiring_1_Nats @ A ) ) ) ) ) ).
% Nats_add
thf(fact_5599_filtermap__Pair,axiom,
! [A: $tType,B: $tType,C: $tType,F3: C > A,G3: C > B,F5: filter @ C] :
( ord_less_eq @ ( filter @ ( product_prod @ A @ B ) )
@ ( filtermap @ C @ ( product_prod @ A @ B )
@ ^ [X5: C] : ( product_Pair @ A @ B @ ( F3 @ X5 ) @ ( G3 @ X5 ) )
@ F5 )
@ ( prod_filter @ A @ B @ ( filtermap @ C @ A @ F3 @ F5 ) @ ( filtermap @ C @ B @ G3 @ F5 ) ) ) ).
% filtermap_Pair
thf(fact_5600_filtermap__nhds__times,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [C3: A,A2: A] :
( ( C3
!= ( zero_zero @ A ) )
=> ( ( filtermap @ A @ A @ ( times_times @ A @ C3 ) @ ( topolo7230453075368039082e_nhds @ A @ A2 ) )
= ( topolo7230453075368039082e_nhds @ A @ ( times_times @ A @ C3 @ A2 ) ) ) ) ) ).
% filtermap_nhds_times
thf(fact_5601_Nats__diff,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A,B2: A] :
( ( member @ A @ A2 @ ( semiring_1_Nats @ A ) )
=> ( ( member @ A @ B2 @ ( semiring_1_Nats @ A ) )
=> ( ( ord_less_eq @ A @ B2 @ A2 )
=> ( member @ A @ ( minus_minus @ A @ A2 @ B2 ) @ ( semiring_1_Nats @ A ) ) ) ) ) ) ).
% Nats_diff
thf(fact_5602_at__to__0,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [A2: A] :
( ( topolo174197925503356063within @ A @ A2 @ ( top_top @ ( set @ A ) ) )
= ( filtermap @ A @ A
@ ^ [X5: A] : ( plus_plus @ A @ X5 @ A2 )
@ ( topolo174197925503356063within @ A @ ( zero_zero @ A ) @ ( top_top @ ( set @ A ) ) ) ) ) ) ).
% at_to_0
thf(fact_5603_Nats__def,axiom,
! [A: $tType] :
( ( semiring_1 @ A )
=> ( ( semiring_1_Nats @ A )
= ( image2 @ nat @ A @ ( semiring_1_of_nat @ A ) @ ( top_top @ ( set @ nat ) ) ) ) ) ).
% Nats_def
thf(fact_5604_filtermap__times__pos__at__right,axiom,
! [A: $tType] :
( ( ( linordered_field @ A )
& ( topolo1944317154257567458pology @ A ) )
=> ! [C3: A,P: A] :
( ( ord_less @ A @ ( zero_zero @ A ) @ C3 )
=> ( ( filtermap @ A @ A @ ( times_times @ A @ C3 ) @ ( topolo174197925503356063within @ A @ P @ ( set_ord_greaterThan @ A @ P ) ) )
= ( topolo174197925503356063within @ A @ ( times_times @ A @ C3 @ P ) @ ( set_ord_greaterThan @ A @ ( times_times @ A @ C3 @ P ) ) ) ) ) ) ).
% filtermap_times_pos_at_right
thf(fact_5605_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_5606_prod__filter__principal__singleton,axiom,
! [A: $tType,B: $tType,X: A,F5: filter @ B] :
( ( prod_filter @ A @ B @ ( principal @ A @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ F5 )
= ( filtermap @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X ) @ F5 ) ) ).
% prod_filter_principal_singleton
thf(fact_5607_prod__filter__principal__singleton2,axiom,
! [B: $tType,A: $tType,F5: filter @ A,X: B] :
( ( prod_filter @ A @ B @ F5 @ ( principal @ B @ ( insert @ B @ X @ ( bot_bot @ ( set @ B ) ) ) ) )
= ( filtermap @ A @ ( product_prod @ A @ B )
@ ^ [A6: A] : ( product_Pair @ A @ B @ A6 @ X )
@ F5 ) ) ).
% prod_filter_principal_singleton2
thf(fact_5608_cauchy__filter__metric__filtermap,axiom,
! [A: $tType,B: $tType] :
( ( ( real_V768167426530841204y_dist @ B )
& ( topolo7287701948861334536_space @ B ) )
=> ! [F3: A > B,F5: filter @ A] :
( ( topolo6773858410816713723filter @ B @ ( filtermap @ A @ B @ F3 @ F5 ) )
= ( ! [E4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E4 )
=> ? [P4: A > $o] :
( ( eventually @ A @ P4 @ F5 )
& ! [X5: A,Y6: A] :
( ( ( P4 @ X5 )
& ( P4 @ Y6 ) )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ B @ ( F3 @ X5 ) @ ( F3 @ Y6 ) ) @ E4 ) ) ) ) ) ) ) ).
% cauchy_filter_metric_filtermap
thf(fact_5609_surj__int__encode,axiom,
( ( image2 @ int @ nat @ nat_int_encode @ ( top_top @ ( set @ int ) ) )
= ( top_top @ ( set @ nat ) ) ) ).
% surj_int_encode
thf(fact_5610_compact__imp__fip__image,axiom,
! [B: $tType,A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [S: set @ A,I6: set @ B,F3: B > ( set @ A )] :
( ( topolo2193935891317330818ompact @ A @ S )
=> ( ! [I3: B] :
( ( member @ B @ I3 @ I6 )
=> ( topolo7761053866217962861closed @ A @ ( F3 @ I3 ) ) )
=> ( ! [I9: set @ B] :
( ( finite_finite2 @ B @ I9 )
=> ( ( ord_less_eq @ ( set @ B ) @ I9 @ I6 )
=> ( ( inf_inf @ ( set @ A ) @ S @ ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ F3 @ I9 ) ) )
!= ( bot_bot @ ( set @ A ) ) ) ) )
=> ( ( inf_inf @ ( set @ A ) @ S @ ( complete_Inf_Inf @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ F3 @ I6 ) ) )
!= ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% compact_imp_fip_image
thf(fact_5611_inj__sgn__power,axiom,
! [N: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
=> ( inj_on @ real @ real
@ ^ [Y6: real] : ( times_times @ real @ ( sgn_sgn @ real @ Y6 ) @ ( power_power @ real @ ( abs_abs @ real @ Y6 ) @ N ) )
@ ( top_top @ ( set @ real ) ) ) ) ).
% inj_sgn_power
thf(fact_5612_inj__on__empty,axiom,
! [B: $tType,A: $tType,F3: A > B] : ( inj_on @ A @ B @ F3 @ ( bot_bot @ ( set @ A ) ) ) ).
% inj_on_empty
thf(fact_5613_closed__empty,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ( topolo7761053866217962861closed @ A @ ( bot_bot @ ( set @ A ) ) ) ) ).
% closed_empty
thf(fact_5614_closed__singleton,axiom,
! [A: $tType] :
( ( topological_t1_space @ A )
=> ! [A2: A] : ( topolo7761053866217962861closed @ A @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% closed_singleton
thf(fact_5615_closed__Union,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [S2: set @ ( set @ A )] :
( ( finite_finite2 @ ( set @ A ) @ S2 )
=> ( ! [X4: set @ A] :
( ( member @ ( set @ A ) @ X4 @ S2 )
=> ( topolo7761053866217962861closed @ A @ X4 ) )
=> ( topolo7761053866217962861closed @ A @ ( complete_Sup_Sup @ ( set @ A ) @ S2 ) ) ) ) ) ).
% closed_Union
thf(fact_5616_inj__mult__left,axiom,
! [A: $tType] :
( ( idom @ A )
=> ! [A2: A] :
( ( inj_on @ A @ A @ ( times_times @ A @ A2 ) @ ( top_top @ ( set @ A ) ) )
= ( A2
!= ( zero_zero @ A ) ) ) ) ).
% inj_mult_left
thf(fact_5617_inj__divide__right,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [A2: A] :
( ( inj_on @ A @ A
@ ^ [B5: A] : ( divide_divide @ A @ B5 @ A2 )
@ ( top_top @ ( set @ A ) ) )
= ( A2
!= ( zero_zero @ A ) ) ) ) ).
% inj_divide_right
thf(fact_5618_closed__UN,axiom,
! [A: $tType,B: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [A5: set @ B,B6: B > ( set @ A )] :
( ( finite_finite2 @ B @ A5 )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ A5 )
=> ( topolo7761053866217962861closed @ A @ ( B6 @ X4 ) ) )
=> ( topolo7761053866217962861closed @ A @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B6 @ A5 ) ) ) ) ) ) ).
% closed_UN
thf(fact_5619_inj__on__insert,axiom,
! [B: $tType,A: $tType,F3: A > B,A2: A,A5: set @ A] :
( ( inj_on @ A @ B @ F3 @ ( insert @ A @ A2 @ A5 ) )
= ( ( inj_on @ A @ B @ F3 @ A5 )
& ~ ( member @ B @ ( F3 @ A2 ) @ ( image2 @ A @ B @ F3 @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% inj_on_insert
thf(fact_5620_inj__on__strict__subset,axiom,
! [B: $tType,A: $tType,F3: A > B,B6: set @ A,A5: set @ A] :
( ( inj_on @ A @ B @ F3 @ B6 )
=> ( ( ord_less @ ( set @ A ) @ A5 @ B6 )
=> ( ord_less @ ( set @ B ) @ ( image2 @ A @ B @ F3 @ A5 ) @ ( image2 @ A @ B @ F3 @ B6 ) ) ) ) ).
% inj_on_strict_subset
thf(fact_5621_inj__on__image__Fpow,axiom,
! [B: $tType,A: $tType,F3: A > B,A5: set @ A] :
( ( inj_on @ A @ B @ F3 @ A5 )
=> ( inj_on @ ( set @ A ) @ ( set @ B ) @ ( image2 @ A @ B @ F3 ) @ ( finite_Fpow @ A @ A5 ) ) ) ).
% inj_on_image_Fpow
thf(fact_5622_finite__image__iff,axiom,
! [B: $tType,A: $tType,F3: A > B,A5: set @ A] :
( ( inj_on @ A @ B @ F3 @ A5 )
=> ( ( finite_finite2 @ B @ ( image2 @ A @ B @ F3 @ A5 ) )
= ( finite_finite2 @ A @ A5 ) ) ) ).
% finite_image_iff
thf(fact_5623_finite__imageD,axiom,
! [A: $tType,B: $tType,F3: B > A,A5: set @ B] :
( ( finite_finite2 @ A @ ( image2 @ B @ A @ F3 @ A5 ) )
=> ( ( inj_on @ B @ A @ F3 @ A5 )
=> ( finite_finite2 @ B @ A5 ) ) ) ).
% finite_imageD
thf(fact_5624_card__image,axiom,
! [B: $tType,A: $tType,F3: A > B,A5: set @ A] :
( ( inj_on @ A @ B @ F3 @ A5 )
=> ( ( finite_card @ B @ ( image2 @ A @ B @ F3 @ A5 ) )
= ( finite_card @ A @ A5 ) ) ) ).
% card_image
thf(fact_5625_linorder__inj__onI,axiom,
! [B: $tType,A: $tType] :
( ( order @ A )
=> ! [A5: set @ A,F3: A > B] :
( ! [X4: A,Y3: A] :
( ( ord_less @ A @ X4 @ Y3 )
=> ( ( member @ A @ X4 @ A5 )
=> ( ( member @ A @ Y3 @ A5 )
=> ( ( F3 @ X4 )
!= ( F3 @ Y3 ) ) ) ) )
=> ( ! [X4: A,Y3: A] :
( ( member @ A @ X4 @ A5 )
=> ( ( member @ A @ Y3 @ A5 )
=> ( ( ord_less_eq @ A @ X4 @ Y3 )
| ( ord_less_eq @ A @ Y3 @ X4 ) ) ) )
=> ( inj_on @ A @ B @ F3 @ A5 ) ) ) ) ).
% linorder_inj_onI
thf(fact_5626_inj__add__left,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [A2: A] : ( inj_on @ A @ A @ ( plus_plus @ A @ A2 ) @ ( top_top @ ( set @ A ) ) ) ) ).
% inj_add_left
thf(fact_5627_sorted__list__of__set_Oinj__on,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( inj_on @ A @ A
@ ^ [X5: A] : X5
@ ( top_top @ ( set @ A ) ) ) ) ).
% sorted_list_of_set.inj_on
thf(fact_5628_inj__on__add,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [A2: A,A5: set @ A] : ( inj_on @ A @ A @ ( plus_plus @ A @ A2 ) @ A5 ) ) ).
% inj_on_add
thf(fact_5629_inj__on__add_H,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [A2: A,A5: set @ A] :
( inj_on @ A @ A
@ ^ [B5: A] : ( plus_plus @ A @ B5 @ A2 )
@ A5 ) ) ).
% inj_on_add'
thf(fact_5630_int__encode__eq,axiom,
! [X: int,Y: int] :
( ( ( nat_int_encode @ X )
= ( nat_int_encode @ Y ) )
= ( X = Y ) ) ).
% int_encode_eq
thf(fact_5631_finite__inverse__image__gen,axiom,
! [A: $tType,B: $tType,A5: set @ A,F3: B > A,D6: set @ B] :
( ( finite_finite2 @ A @ A5 )
=> ( ( inj_on @ B @ A @ F3 @ D6 )
=> ( finite_finite2 @ B
@ ( collect @ B
@ ^ [J3: B] :
( ( member @ B @ J3 @ D6 )
& ( member @ A @ ( F3 @ J3 ) @ A5 ) ) ) ) ) ) ).
% finite_inverse_image_gen
thf(fact_5632_finite__imp__closed,axiom,
! [A: $tType] :
( ( topological_t1_space @ A )
=> ! [S2: set @ A] :
( ( finite_finite2 @ A @ S2 )
=> ( topolo7761053866217962861closed @ A @ S2 ) ) ) ).
% finite_imp_closed
thf(fact_5633_inj__on__mult,axiom,
! [A: $tType] :
( ( semidom_divide @ A )
=> ! [A2: A,A5: set @ A] :
( ( A2
!= ( zero_zero @ A ) )
=> ( inj_on @ A @ A @ ( times_times @ A @ A2 ) @ A5 ) ) ) ).
% inj_on_mult
thf(fact_5634_linorder__injI,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ A )
=> ! [F3: A > B] :
( ! [X4: A,Y3: A] :
( ( ord_less @ A @ X4 @ Y3 )
=> ( ( F3 @ X4 )
!= ( F3 @ Y3 ) ) )
=> ( inj_on @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) ) ) ) ).
% linorder_injI
thf(fact_5635_inj__fn,axiom,
! [A: $tType,F3: A > A,N: nat] :
( ( inj_on @ A @ A @ F3 @ ( top_top @ ( set @ A ) ) )
=> ( inj_on @ A @ A @ ( compow @ ( A > A ) @ N @ F3 ) @ ( top_top @ ( set @ A ) ) ) ) ).
% inj_fn
thf(fact_5636_inj__on__Inter,axiom,
! [B: $tType,A: $tType,S2: set @ ( set @ A ),F3: A > B] :
( ( S2
!= ( bot_bot @ ( set @ ( set @ A ) ) ) )
=> ( ! [A8: set @ A] :
( ( member @ ( set @ A ) @ A8 @ S2 )
=> ( inj_on @ A @ B @ F3 @ A8 ) )
=> ( inj_on @ A @ B @ F3 @ ( complete_Inf_Inf @ ( set @ A ) @ S2 ) ) ) ) ).
% inj_on_Inter
thf(fact_5637_filtermap__sequentually__ne__bot,axiom,
! [A: $tType,F3: nat > A] :
( ( filtermap @ nat @ A @ F3 @ ( at_top @ nat ) )
!= ( bot_bot @ ( filter @ A ) ) ) ).
% filtermap_sequentually_ne_bot
thf(fact_5638_finite__inverse__image,axiom,
! [A: $tType,B: $tType,A5: set @ A,F3: B > A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( inj_on @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) )
=> ( finite_finite2 @ B
@ ( collect @ B
@ ^ [J3: B] : ( member @ A @ ( F3 @ J3 ) @ A5 ) ) ) ) ) ).
% finite_inverse_image
thf(fact_5639_finite__UNIV__surj__inj,axiom,
! [A: $tType,F3: A > A] :
( ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) )
=> ( ( ( image2 @ A @ A @ F3 @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) )
=> ( inj_on @ A @ A @ F3 @ ( top_top @ ( set @ A ) ) ) ) ) ).
% finite_UNIV_surj_inj
thf(fact_5640_finite__UNIV__inj__surj,axiom,
! [A: $tType,F3: A > A] :
( ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) )
=> ( ( inj_on @ A @ A @ F3 @ ( top_top @ ( set @ A ) ) )
=> ( ( image2 @ A @ A @ F3 @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ A ) ) ) ) ) ).
% finite_UNIV_inj_surj
thf(fact_5641_inj__on__iff__surj,axiom,
! [A: $tType,B: $tType,A5: set @ A,A15: set @ B] :
( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ? [F4: A > B] :
( ( inj_on @ A @ B @ F4 @ A5 )
& ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F4 @ A5 ) @ A15 ) ) )
= ( ? [G4: B > A] :
( ( image2 @ B @ A @ G4 @ A15 )
= A5 ) ) ) ) ).
% inj_on_iff_surj
thf(fact_5642_finite__surj__inj,axiom,
! [A: $tType,A5: set @ A,F3: A > A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( ord_less_eq @ ( set @ A ) @ A5 @ ( image2 @ A @ A @ F3 @ A5 ) )
=> ( inj_on @ A @ A @ F3 @ A5 ) ) ) ).
% finite_surj_inj
thf(fact_5643_inj__on__finite,axiom,
! [B: $tType,A: $tType,F3: A > B,A5: set @ A,B6: set @ B] :
( ( inj_on @ A @ B @ F3 @ A5 )
=> ( ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F3 @ A5 ) @ B6 )
=> ( ( finite_finite2 @ B @ B6 )
=> ( finite_finite2 @ A @ A5 ) ) ) ) ).
% inj_on_finite
thf(fact_5644_endo__inj__surj,axiom,
! [A: $tType,A5: set @ A,F3: A > A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image2 @ A @ A @ F3 @ A5 ) @ A5 )
=> ( ( inj_on @ A @ A @ F3 @ A5 )
=> ( ( image2 @ A @ A @ F3 @ A5 )
= A5 ) ) ) ) ).
% endo_inj_surj
thf(fact_5645_eq__card__imp__inj__on,axiom,
! [B: $tType,A: $tType,A5: set @ A,F3: A > B] :
( ( finite_finite2 @ A @ A5 )
=> ( ( ( finite_card @ B @ ( image2 @ A @ B @ F3 @ A5 ) )
= ( finite_card @ A @ A5 ) )
=> ( inj_on @ A @ B @ F3 @ A5 ) ) ) ).
% eq_card_imp_inj_on
thf(fact_5646_inj__on__iff__eq__card,axiom,
! [B: $tType,A: $tType,A5: set @ A,F3: A > B] :
( ( finite_finite2 @ A @ A5 )
=> ( ( inj_on @ A @ B @ F3 @ A5 )
= ( ( finite_card @ B @ ( image2 @ A @ B @ F3 @ A5 ) )
= ( finite_card @ A @ A5 ) ) ) ) ).
% inj_on_iff_eq_card
thf(fact_5647_pigeonhole,axiom,
! [A: $tType,B: $tType,F3: B > A,A5: set @ B] :
( ( ord_less @ nat @ ( finite_card @ A @ ( image2 @ B @ A @ F3 @ A5 ) ) @ ( finite_card @ B @ A5 ) )
=> ~ ( inj_on @ B @ A @ F3 @ A5 ) ) ).
% pigeonhole
thf(fact_5648_continuous__inj__imp__mono,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo8458572112393995274pology @ A )
& ( topolo1944317154257567458pology @ B ) )
=> ! [A2: A,X: A,B2: A,F3: A > B] :
( ( ord_less @ A @ A2 @ X )
=> ( ( ord_less @ A @ X @ B2 )
=> ( ( topolo81223032696312382ous_on @ A @ B @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) @ F3 )
=> ( ( inj_on @ A @ B @ F3 @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) )
=> ( ( ( ord_less @ B @ ( F3 @ A2 ) @ ( F3 @ X ) )
& ( ord_less @ B @ ( F3 @ X ) @ ( F3 @ B2 ) ) )
| ( ( ord_less @ B @ ( F3 @ B2 ) @ ( F3 @ X ) )
& ( ord_less @ B @ ( F3 @ X ) @ ( F3 @ A2 ) ) ) ) ) ) ) ) ) ).
% continuous_inj_imp_mono
thf(fact_5649_inj__on__INTER,axiom,
! [C: $tType,B: $tType,A: $tType,I6: set @ A,F3: B > C,A5: A > ( set @ B )] :
( ( I6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [I3: A] :
( ( member @ A @ I3 @ I6 )
=> ( inj_on @ B @ C @ F3 @ ( A5 @ I3 ) ) )
=> ( inj_on @ B @ C @ F3 @ ( complete_Inf_Inf @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ A5 @ I6 ) ) ) ) ) ).
% inj_on_INTER
thf(fact_5650_closed__Collect__le,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ A )
& ( topolo1944317154257567458pology @ B ) )
=> ! [F3: A > B,G3: A > B] :
( ( topolo81223032696312382ous_on @ A @ B @ ( top_top @ ( set @ A ) ) @ F3 )
=> ( ( topolo81223032696312382ous_on @ A @ B @ ( top_top @ ( set @ A ) ) @ G3 )
=> ( topolo7761053866217962861closed @ A
@ ( collect @ A
@ ^ [X5: A] : ( ord_less_eq @ B @ ( F3 @ X5 ) @ ( G3 @ X5 ) ) ) ) ) ) ) ).
% closed_Collect_le
thf(fact_5651_surjective__iff__injective__gen,axiom,
! [B: $tType,A: $tType,S2: set @ A,T4: set @ B,F3: A > B] :
( ( finite_finite2 @ A @ S2 )
=> ( ( finite_finite2 @ B @ T4 )
=> ( ( ( finite_card @ A @ S2 )
= ( finite_card @ B @ T4 ) )
=> ( ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F3 @ S2 ) @ T4 )
=> ( ( ! [X5: B] :
( ( member @ B @ X5 @ T4 )
=> ? [Y6: A] :
( ( member @ A @ Y6 @ S2 )
& ( ( F3 @ Y6 )
= X5 ) ) ) )
= ( inj_on @ A @ B @ F3 @ S2 ) ) ) ) ) ) ).
% surjective_iff_injective_gen
thf(fact_5652_card__bij__eq,axiom,
! [A: $tType,B: $tType,F3: A > B,A5: set @ A,B6: set @ B,G3: B > A] :
( ( inj_on @ A @ B @ F3 @ A5 )
=> ( ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F3 @ A5 ) @ B6 )
=> ( ( inj_on @ B @ A @ G3 @ B6 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image2 @ B @ A @ G3 @ B6 ) @ A5 )
=> ( ( finite_finite2 @ A @ A5 )
=> ( ( finite_finite2 @ B @ B6 )
=> ( ( finite_card @ A @ A5 )
= ( finite_card @ B @ B6 ) ) ) ) ) ) ) ) ).
% card_bij_eq
thf(fact_5653_t4__space,axiom,
! [A: $tType] :
( ( topological_t4_space @ A )
=> ! [S2: set @ A,T4: set @ A] :
( ( topolo7761053866217962861closed @ A @ S2 )
=> ( ( topolo7761053866217962861closed @ A @ T4 )
=> ( ( ( inf_inf @ ( set @ A ) @ S2 @ T4 )
= ( bot_bot @ ( set @ A ) ) )
=> ? [U6: set @ A,V7: set @ A] :
( ( topolo1002775350975398744n_open @ A @ U6 )
& ( topolo1002775350975398744n_open @ A @ V7 )
& ( ord_less_eq @ ( set @ A ) @ S2 @ U6 )
& ( ord_less_eq @ ( set @ A ) @ T4 @ V7 )
& ( ( inf_inf @ ( set @ A ) @ U6 @ V7 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% t4_space
thf(fact_5654_t3__space,axiom,
! [A: $tType] :
( ( topological_t3_space @ A )
=> ! [S2: set @ A,Y: A] :
( ( topolo7761053866217962861closed @ A @ S2 )
=> ( ~ ( member @ A @ Y @ S2 )
=> ? [U6: set @ A,V7: set @ A] :
( ( topolo1002775350975398744n_open @ A @ U6 )
& ( topolo1002775350975398744n_open @ A @ V7 )
& ( member @ A @ Y @ U6 )
& ( ord_less_eq @ ( set @ A ) @ S2 @ V7 )
& ( ( inf_inf @ ( set @ A ) @ U6 @ V7 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% t3_space
thf(fact_5655_continuous__on__closed__Union,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( ( topolo4958980785337419405_space @ B )
& ( topolo4958980785337419405_space @ C ) )
=> ! [I6: set @ A,U3: A > ( set @ B ),F3: B > C] :
( ( finite_finite2 @ A @ I6 )
=> ( ! [I3: A] :
( ( member @ A @ I3 @ I6 )
=> ( topolo7761053866217962861closed @ B @ ( U3 @ I3 ) ) )
=> ( ! [I3: A] :
( ( member @ A @ I3 @ I6 )
=> ( topolo81223032696312382ous_on @ B @ C @ ( U3 @ I3 ) @ F3 ) )
=> ( topolo81223032696312382ous_on @ B @ C @ ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ U3 @ I6 ) ) @ F3 ) ) ) ) ) ).
% continuous_on_closed_Union
thf(fact_5656_Lim__in__closed__set,axiom,
! [B: $tType,A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [S2: set @ A,F3: B > A,F5: filter @ B,L: A] :
( ( topolo7761053866217962861closed @ A @ S2 )
=> ( ( eventually @ B
@ ^ [X5: B] : ( member @ A @ ( F3 @ X5 ) @ S2 )
@ F5 )
=> ( ( F5
!= ( bot_bot @ ( filter @ B ) ) )
=> ( ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ L ) @ F5 )
=> ( member @ A @ L @ S2 ) ) ) ) ) ) ).
% Lim_in_closed_set
thf(fact_5657_card__le__inj,axiom,
! [B: $tType,A: $tType,A5: set @ A,B6: set @ B] :
( ( finite_finite2 @ A @ A5 )
=> ( ( finite_finite2 @ B @ B6 )
=> ( ( ord_less_eq @ nat @ ( finite_card @ A @ A5 ) @ ( finite_card @ B @ B6 ) )
=> ? [F2: A > B] :
( ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F2 @ A5 ) @ B6 )
& ( inj_on @ A @ B @ F2 @ A5 ) ) ) ) ) ).
% card_le_inj
thf(fact_5658_card__inj__on__le,axiom,
! [A: $tType,B: $tType,F3: A > B,A5: set @ A,B6: set @ B] :
( ( inj_on @ A @ B @ F3 @ A5 )
=> ( ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F3 @ A5 ) @ B6 )
=> ( ( finite_finite2 @ B @ B6 )
=> ( ord_less_eq @ nat @ ( finite_card @ A @ A5 ) @ ( finite_card @ B @ B6 ) ) ) ) ) ).
% card_inj_on_le
thf(fact_5659_inj__on__iff__card__le,axiom,
! [A: $tType,B: $tType,A5: set @ A,B6: set @ B] :
( ( finite_finite2 @ A @ A5 )
=> ( ( finite_finite2 @ B @ B6 )
=> ( ( ? [F4: A > B] :
( ( inj_on @ A @ B @ F4 @ A5 )
& ( ord_less_eq @ ( set @ B ) @ ( image2 @ A @ B @ F4 @ A5 ) @ B6 ) ) )
= ( ord_less_eq @ nat @ ( finite_card @ A @ A5 ) @ ( finite_card @ B @ B6 ) ) ) ) ) ).
% inj_on_iff_card_le
thf(fact_5660_log__inj,axiom,
! [B2: real] :
( ( ord_less @ real @ ( one_one @ real ) @ B2 )
=> ( inj_on @ real @ real @ ( log @ B2 ) @ ( set_ord_greaterThan @ real @ ( zero_zero @ real ) ) ) ) ).
% log_inj
thf(fact_5661_compact__fip,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ( ( topolo2193935891317330818ompact @ A )
= ( ^ [U5: set @ A] :
! [A7: set @ ( set @ A )] :
( ! [X5: set @ A] :
( ( member @ ( set @ A ) @ X5 @ A7 )
=> ( topolo7761053866217962861closed @ A @ X5 ) )
=> ( ! [B8: set @ ( set @ A )] :
( ( ord_less_eq @ ( set @ ( set @ A ) ) @ B8 @ A7 )
=> ( ( finite_finite2 @ ( set @ A ) @ B8 )
=> ( ( inf_inf @ ( set @ A ) @ U5 @ ( complete_Inf_Inf @ ( set @ A ) @ B8 ) )
!= ( bot_bot @ ( set @ A ) ) ) ) )
=> ( ( inf_inf @ ( set @ A ) @ U5 @ ( complete_Inf_Inf @ ( set @ A ) @ A7 ) )
!= ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% compact_fip
thf(fact_5662_compact__imp__fip,axiom,
! [A: $tType] :
( ( topolo4958980785337419405_space @ A )
=> ! [S2: set @ A,F5: set @ ( set @ A )] :
( ( topolo2193935891317330818ompact @ A @ S2 )
=> ( ! [T6: set @ A] :
( ( member @ ( set @ A ) @ T6 @ F5 )
=> ( topolo7761053866217962861closed @ A @ T6 ) )
=> ( ! [F14: set @ ( set @ A )] :
( ( finite_finite2 @ ( set @ A ) @ F14 )
=> ( ( ord_less_eq @ ( set @ ( set @ A ) ) @ F14 @ F5 )
=> ( ( inf_inf @ ( set @ A ) @ S2 @ ( complete_Inf_Inf @ ( set @ A ) @ F14 ) )
!= ( bot_bot @ ( set @ A ) ) ) ) )
=> ( ( inf_inf @ ( set @ A ) @ S2 @ ( complete_Inf_Inf @ ( set @ A ) @ F5 ) )
!= ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% compact_imp_fip
thf(fact_5663_rtrancl__finite__eq__relpow,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( finite_finite2 @ ( product_prod @ A @ A ) @ R )
=> ( ( transitive_rtrancl @ A @ R )
= ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ A ) )
@ ( image2 @ nat @ ( set @ ( product_prod @ A @ A ) )
@ ^ [N2: nat] : ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N2 @ R )
@ ( collect @ nat
@ ^ [N2: nat] : ( ord_less_eq @ nat @ N2 @ ( finite_card @ ( product_prod @ A @ A ) @ R ) ) ) ) ) ) ) ).
% rtrancl_finite_eq_relpow
thf(fact_5664_uniformly__continuous__onD,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo7287701948861334536_space @ A )
& ( topolo7287701948861334536_space @ B ) )
=> ! [S: set @ A,F3: A > B,E5: ( product_prod @ B @ B ) > $o] :
( ( topolo6026614971017936543ous_on @ A @ B @ S @ F3 )
=> ( ( eventually @ ( product_prod @ B @ B ) @ E5 @ ( topolo7806501430040627800ormity @ B ) )
=> ( eventually @ ( product_prod @ A @ A )
@ ( product_case_prod @ A @ A @ $o
@ ^ [X5: A,Y6: A] :
( ( member @ A @ X5 @ S )
=> ( ( member @ A @ Y6 @ S )
=> ( E5 @ ( product_Pair @ B @ B @ ( F3 @ X5 ) @ ( F3 @ Y6 ) ) ) ) ) )
@ ( topolo7806501430040627800ormity @ A ) ) ) ) ) ).
% uniformly_continuous_onD
thf(fact_5665_measure__function__int,axiom,
fun_is_measure @ int @ ( comp @ int @ nat @ int @ nat2 @ ( abs_abs @ int ) ) ).
% measure_function_int
thf(fact_5666_inj__on__convol__ident,axiom,
! [B: $tType,A: $tType,F3: A > B,X7: set @ A] :
( inj_on @ A @ ( product_prod @ A @ B )
@ ^ [X5: A] : ( product_Pair @ A @ B @ X5 @ ( F3 @ X5 ) )
@ X7 ) ).
% inj_on_convol_ident
thf(fact_5667_inj__prod__encode,axiom,
! [A5: set @ ( product_prod @ nat @ nat )] : ( inj_on @ ( product_prod @ nat @ nat ) @ nat @ nat_prod_encode @ A5 ) ).
% inj_prod_encode
thf(fact_5668_inj__int__encode,axiom,
! [A5: set @ int] : ( inj_on @ int @ nat @ nat_int_encode @ A5 ) ).
% inj_int_encode
thf(fact_5669_inj__Suc,axiom,
! [N6: set @ nat] : ( inj_on @ nat @ nat @ suc @ N6 ) ).
% inj_Suc
thf(fact_5670_inj__Some,axiom,
! [A: $tType,A5: set @ A] : ( inj_on @ A @ ( option @ A ) @ ( some @ A ) @ A5 ) ).
% inj_Some
thf(fact_5671_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_5672_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_5673_measure__fst,axiom,
! [B: $tType,A: $tType,F3: A > nat] :
( ( fun_is_measure @ A @ F3 )
=> ( fun_is_measure @ ( product_prod @ A @ B )
@ ^ [P6: product_prod @ A @ B] : ( F3 @ ( product_fst @ A @ B @ P6 ) ) ) ) ).
% measure_fst
thf(fact_5674_is__measure_Osimps,axiom,
! [A: $tType] :
( ( fun_is_measure @ A )
= ( ^ [A6: A > nat] :
? [X9: A > nat] :
( ^ [Y5: A > nat,Z2: A > nat] : Y5 = Z2
@ A6
@ X9 ) ) ) ).
% is_measure.simps
thf(fact_5675_is__measure__trivial,axiom,
! [A: $tType,F3: A > nat] : ( fun_is_measure @ A @ F3 ) ).
% is_measure_trivial
thf(fact_5676_measure__snd,axiom,
! [A: $tType,B: $tType,F3: A > nat] :
( ( fun_is_measure @ A @ F3 )
=> ( fun_is_measure @ ( product_prod @ B @ A )
@ ^ [P6: product_prod @ B @ A] : ( F3 @ ( product_snd @ B @ A @ P6 ) ) ) ) ).
% measure_snd
thf(fact_5677_rtrancl_Ocases,axiom,
! [A: $tType,A1: A,A22: A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A1 @ A22 ) @ ( transitive_rtrancl @ A @ R2 ) )
=> ( ( A22 != A1 )
=> ~ ! [B4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A1 @ B4 ) @ ( transitive_rtrancl @ A @ R2 ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B4 @ A22 ) @ R2 ) ) ) ) ).
% rtrancl.cases
thf(fact_5678_rtrancl_Osimps,axiom,
! [A: $tType,A1: A,A22: A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A1 @ A22 ) @ ( transitive_rtrancl @ A @ R2 ) )
= ( ? [A6: A] :
( ( A1 = A6 )
& ( A22 = A6 ) )
| ? [A6: A,B5: A,C4: A] :
( ( A1 = A6 )
& ( A22 = C4 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A6 @ B5 ) @ ( transitive_rtrancl @ A @ R2 ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B5 @ C4 ) @ R2 ) ) ) ) ).
% rtrancl.simps
thf(fact_5679_rtrancl_Ortrancl__refl,axiom,
! [A: $tType,A2: A,R2: set @ ( product_prod @ A @ A )] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ A2 ) @ ( transitive_rtrancl @ A @ R2 ) ) ).
% rtrancl.rtrancl_refl
thf(fact_5680_rtrancl_Ortrancl__into__rtrancl,axiom,
! [A: $tType,A2: A,B2: A,R2: set @ ( product_prod @ A @ A ),C3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ ( transitive_rtrancl @ A @ R2 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ C3 ) @ R2 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ C3 ) @ ( transitive_rtrancl @ A @ R2 ) ) ) ) ).
% rtrancl.rtrancl_into_rtrancl
thf(fact_5681_rtranclE,axiom,
! [A: $tType,A2: A,B2: A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ ( transitive_rtrancl @ A @ R2 ) )
=> ( ( A2 != B2 )
=> ~ ! [Y3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ Y3 ) @ ( transitive_rtrancl @ A @ R2 ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ B2 ) @ R2 ) ) ) ) ).
% rtranclE
thf(fact_5682_rtrancl__trans,axiom,
! [A: $tType,X: A,Y: A,R2: set @ ( product_prod @ A @ A ),Z: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( transitive_rtrancl @ A @ R2 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z ) @ ( transitive_rtrancl @ A @ R2 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z ) @ ( transitive_rtrancl @ A @ R2 ) ) ) ) ).
% rtrancl_trans
thf(fact_5683_rtrancl__induct,axiom,
! [A: $tType,A2: A,B2: A,R2: set @ ( product_prod @ A @ A ),P2: A > $o] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ ( transitive_rtrancl @ A @ R2 ) )
=> ( ( P2 @ A2 )
=> ( ! [Y3: A,Z3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ Y3 ) @ ( transitive_rtrancl @ A @ R2 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z3 ) @ R2 )
=> ( ( P2 @ Y3 )
=> ( P2 @ Z3 ) ) ) )
=> ( P2 @ B2 ) ) ) ) ).
% rtrancl_induct
thf(fact_5684_converse__rtranclE,axiom,
! [A: $tType,X: A,Z: A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z ) @ ( transitive_rtrancl @ A @ R2 ) )
=> ( ( X != Z )
=> ~ ! [Y3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y3 ) @ R2 )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z ) @ ( transitive_rtrancl @ A @ R2 ) ) ) ) ) ).
% converse_rtranclE
thf(fact_5685_converse__rtrancl__induct,axiom,
! [A: $tType,A2: A,B2: A,R2: set @ ( product_prod @ A @ A ),P2: A > $o] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ ( transitive_rtrancl @ A @ R2 ) )
=> ( ( P2 @ B2 )
=> ( ! [Y3: A,Z3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z3 ) @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z3 @ B2 ) @ ( transitive_rtrancl @ A @ R2 ) )
=> ( ( P2 @ Z3 )
=> ( P2 @ Y3 ) ) ) )
=> ( P2 @ A2 ) ) ) ) ).
% converse_rtrancl_induct
thf(fact_5686_converse__rtrancl__into__rtrancl,axiom,
! [A: $tType,A2: A,B2: A,R2: set @ ( product_prod @ A @ A ),C3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ C3 ) @ ( transitive_rtrancl @ A @ R2 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ C3 ) @ ( transitive_rtrancl @ A @ R2 ) ) ) ) ).
% converse_rtrancl_into_rtrancl
thf(fact_5687_tranclD,axiom,
! [A: $tType,X: A,Y: A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( transitive_trancl @ A @ R ) )
=> ? [Z3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z3 ) @ R )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z3 @ Y ) @ ( transitive_rtrancl @ A @ R ) ) ) ) ).
% tranclD
thf(fact_5688_rtranclD,axiom,
! [A: $tType,A2: A,B2: A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ ( transitive_rtrancl @ A @ R ) )
=> ( ( A2 = B2 )
| ( ( A2 != B2 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ ( transitive_trancl @ A @ R ) ) ) ) ) ).
% rtranclD
thf(fact_5689_tranclD2,axiom,
! [A: $tType,X: A,Y: A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( transitive_trancl @ A @ R ) )
=> ? [Z3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z3 ) @ ( transitive_rtrancl @ A @ R ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z3 @ Y ) @ R ) ) ) ).
% tranclD2
thf(fact_5690_trancl__into__rtrancl,axiom,
! [A: $tType,A2: A,B2: A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ ( transitive_trancl @ A @ R2 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ ( transitive_rtrancl @ A @ R2 ) ) ) ).
% trancl_into_rtrancl
thf(fact_5691_rtrancl__eq__or__trancl,axiom,
! [A: $tType,X: A,Y: A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( transitive_rtrancl @ A @ R ) )
= ( ( X = Y )
| ( ( X != Y )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( transitive_trancl @ A @ R ) ) ) ) ) ).
% rtrancl_eq_or_trancl
thf(fact_5692_rtrancl__into__trancl1,axiom,
! [A: $tType,A2: A,B2: A,R2: set @ ( product_prod @ A @ A ),C3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ ( transitive_rtrancl @ A @ R2 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ C3 ) @ R2 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ C3 ) @ ( transitive_trancl @ A @ R2 ) ) ) ) ).
% rtrancl_into_trancl1
thf(fact_5693_rtrancl__into__trancl2,axiom,
! [A: $tType,A2: A,B2: A,R2: set @ ( product_prod @ A @ A ),C3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ C3 ) @ ( transitive_rtrancl @ A @ R2 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ C3 ) @ ( transitive_trancl @ A @ R2 ) ) ) ) ).
% rtrancl_into_trancl2
thf(fact_5694_rtrancl__trancl__trancl,axiom,
! [A: $tType,X: A,Y: A,R2: set @ ( product_prod @ A @ A ),Z: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( transitive_rtrancl @ A @ R2 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z ) @ ( transitive_trancl @ A @ R2 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Z ) @ ( transitive_trancl @ A @ R2 ) ) ) ) ).
% rtrancl_trancl_trancl
thf(fact_5695_trancl__rtrancl__trancl,axiom,
! [A: $tType,A2: A,B2: A,R2: set @ ( product_prod @ A @ A ),C3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ ( transitive_trancl @ A @ R2 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ C3 ) @ ( transitive_rtrancl @ A @ R2 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ C3 ) @ ( transitive_trancl @ A @ R2 ) ) ) ) ).
% trancl_rtrancl_trancl
thf(fact_5696_converse__rtrancl__induct2,axiom,
! [A: $tType,B: $tType,Ax: A,Ay: B,Bx: A,By: B,R2: set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ),P2: A > B > $o] :
( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Ax @ Ay ) @ ( product_Pair @ A @ B @ Bx @ By ) ) @ ( transitive_rtrancl @ ( product_prod @ A @ B ) @ R2 ) )
=> ( ( P2 @ Bx @ By )
=> ( ! [A4: A,B4: B,Aa2: A,Ba: B] :
( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B4 ) @ ( product_Pair @ A @ B @ Aa2 @ Ba ) ) @ R2 )
=> ( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Aa2 @ Ba ) @ ( product_Pair @ A @ B @ Bx @ By ) ) @ ( transitive_rtrancl @ ( product_prod @ A @ B ) @ R2 ) )
=> ( ( P2 @ Aa2 @ Ba )
=> ( P2 @ A4 @ B4 ) ) ) )
=> ( P2 @ Ax @ Ay ) ) ) ) ).
% converse_rtrancl_induct2
thf(fact_5697_converse__rtranclE2,axiom,
! [B: $tType,A: $tType,Xa2: A,Xb3: B,Za: A,Zb: B,R2: set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) )] :
( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Xa2 @ Xb3 ) @ ( product_Pair @ A @ B @ Za @ Zb ) ) @ ( transitive_rtrancl @ ( product_prod @ A @ B ) @ R2 ) )
=> ( ( ( product_Pair @ A @ B @ Xa2 @ Xb3 )
!= ( product_Pair @ A @ B @ Za @ Zb ) )
=> ~ ! [A4: A,B4: B] :
( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Xa2 @ Xb3 ) @ ( product_Pair @ A @ B @ A4 @ B4 ) ) @ R2 )
=> ~ ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B4 ) @ ( product_Pair @ A @ B @ Za @ Zb ) ) @ ( transitive_rtrancl @ ( product_prod @ A @ B ) @ R2 ) ) ) ) ) ).
% converse_rtranclE2
thf(fact_5698_rtrancl__induct2,axiom,
! [A: $tType,B: $tType,Ax: A,Ay: B,Bx: A,By: B,R2: set @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ),P2: A > B > $o] :
( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Ax @ Ay ) @ ( product_Pair @ A @ B @ Bx @ By ) ) @ ( transitive_rtrancl @ ( product_prod @ A @ B ) @ R2 ) )
=> ( ( P2 @ Ax @ Ay )
=> ( ! [A4: A,B4: B,Aa2: A,Ba: B] :
( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Ax @ Ay ) @ ( product_Pair @ A @ B @ A4 @ B4 ) ) @ ( transitive_rtrancl @ ( product_prod @ A @ B ) @ R2 ) )
=> ( ( member @ ( product_prod @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) ) @ ( product_Pair @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B4 ) @ ( product_Pair @ A @ B @ Aa2 @ Ba ) ) @ R2 )
=> ( ( P2 @ A4 @ B4 )
=> ( P2 @ Aa2 @ Ba ) ) ) )
=> ( P2 @ Bx @ By ) ) ) ) ).
% rtrancl_induct2
thf(fact_5699_inj__singleton,axiom,
! [A: $tType,A5: set @ A] :
( inj_on @ A @ ( set @ A )
@ ^ [X5: A] : ( insert @ A @ X5 @ ( bot_bot @ ( set @ A ) ) )
@ A5 ) ).
% inj_singleton
thf(fact_5700_inj__on__diff__nat,axiom,
! [N6: set @ nat,K2: nat] :
( ! [N3: nat] :
( ( member @ nat @ N3 @ N6 )
=> ( ord_less_eq @ nat @ K2 @ N3 ) )
=> ( inj_on @ nat @ nat
@ ^ [N2: nat] : ( minus_minus @ nat @ N2 @ K2 )
@ N6 ) ) ).
% inj_on_diff_nat
thf(fact_5701_swap__inj__on,axiom,
! [B: $tType,A: $tType,A5: set @ ( product_prod @ A @ B )] :
( inj_on @ ( product_prod @ A @ B ) @ ( product_prod @ B @ A )
@ ( product_case_prod @ A @ B @ ( product_prod @ B @ A )
@ ^ [I4: A,J3: B] : ( product_Pair @ B @ A @ J3 @ I4 ) )
@ A5 ) ).
% swap_inj_on
thf(fact_5702_inj__on__set__encode,axiom,
inj_on @ ( set @ nat ) @ nat @ nat_set_encode @ ( collect @ ( set @ nat ) @ ( finite_finite2 @ nat ) ) ).
% inj_on_set_encode
thf(fact_5703_measure__size,axiom,
! [A: $tType] :
( ( size @ A )
=> ( fun_is_measure @ A @ ( size_size @ A ) ) ) ).
% measure_size
thf(fact_5704_inj__graph,axiom,
! [B: $tType,A: $tType] :
( inj_on @ ( A > B ) @ ( set @ ( product_prod @ A @ B ) )
@ ^ [F4: A > B] :
( collect @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [X5: A,Y6: B] :
( Y6
= ( F4 @ X5 ) ) ) )
@ ( top_top @ ( set @ ( A > B ) ) ) ) ).
% inj_graph
thf(fact_5705_range__inj__infinite,axiom,
! [A: $tType,F3: nat > A] :
( ( inj_on @ nat @ A @ F3 @ ( top_top @ ( set @ nat ) ) )
=> ~ ( finite_finite2 @ A @ ( image2 @ nat @ A @ F3 @ ( top_top @ ( set @ nat ) ) ) ) ) ).
% range_inj_infinite
thf(fact_5706_finite__imp__nat__seg__image__inj__on,axiom,
! [A: $tType,A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ? [N3: nat,F2: nat > A] :
( ( A5
= ( image2 @ nat @ A @ F2
@ ( collect @ nat
@ ^ [I4: nat] : ( ord_less @ nat @ I4 @ N3 ) ) ) )
& ( inj_on @ nat @ A @ F2
@ ( collect @ nat
@ ^ [I4: nat] : ( ord_less @ nat @ I4 @ N3 ) ) ) ) ) ).
% finite_imp_nat_seg_image_inj_on
thf(fact_5707_finite__imp__inj__to__nat__seg,axiom,
! [A: $tType,A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ? [F2: A > nat,N3: nat] :
( ( ( image2 @ A @ nat @ F2 @ A5 )
= ( collect @ nat
@ ^ [I4: nat] : ( ord_less @ nat @ I4 @ N3 ) ) )
& ( inj_on @ A @ nat @ F2 @ A5 ) ) ) ).
% finite_imp_inj_to_nat_seg
thf(fact_5708_inj__on__nth,axiom,
! [A: $tType,Xs: list @ A,I6: set @ nat] :
( ( distinct @ A @ Xs )
=> ( ! [X4: nat] :
( ( member @ nat @ X4 @ I6 )
=> ( ord_less @ nat @ X4 @ ( size_size @ ( list @ A ) @ Xs ) ) )
=> ( inj_on @ nat @ A @ ( nth @ A @ Xs ) @ I6 ) ) ) ).
% inj_on_nth
thf(fact_5709_infinite__countable__subset,axiom,
! [A: $tType,S2: set @ A] :
( ~ ( finite_finite2 @ A @ S2 )
=> ? [F2: nat > A] :
( ( inj_on @ nat @ A @ F2 @ ( top_top @ ( set @ nat ) ) )
& ( ord_less_eq @ ( set @ A ) @ ( image2 @ nat @ A @ F2 @ ( top_top @ ( set @ nat ) ) ) @ S2 ) ) ) ).
% infinite_countable_subset
thf(fact_5710_infinite__iff__countable__subset,axiom,
! [A: $tType,S2: set @ A] :
( ( ~ ( finite_finite2 @ A @ S2 ) )
= ( ? [F4: nat > A] :
( ( inj_on @ nat @ A @ F4 @ ( top_top @ ( set @ nat ) ) )
& ( ord_less_eq @ ( set @ A ) @ ( image2 @ nat @ A @ F4 @ ( top_top @ ( set @ nat ) ) ) @ S2 ) ) ) ) ).
% infinite_iff_countable_subset
thf(fact_5711_inj__on__funpow__least,axiom,
! [A: $tType,N: nat,F3: A > A,S: A] :
( ( ( compow @ ( A > A ) @ N @ F3 @ S )
= S )
=> ( ! [M: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ M )
=> ( ( ord_less @ nat @ M @ N )
=> ( ( compow @ ( A > A ) @ M @ F3 @ S )
!= S ) ) )
=> ( inj_on @ nat @ A
@ ^ [K3: nat] : ( compow @ ( A > A ) @ K3 @ F3 @ S )
@ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% inj_on_funpow_least
thf(fact_5712_uniformly__continuous__on__def,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V7819770556892013058_space @ A )
& ( real_V7819770556892013058_space @ B ) )
=> ( ( topolo6026614971017936543ous_on @ A @ B )
= ( ^ [S8: set @ A,F4: A > B] :
! [E4: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ E4 )
=> ? [D5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ D5 )
& ! [X5: A] :
( ( member @ A @ X5 @ S8 )
=> ! [Y6: A] :
( ( member @ A @ Y6 @ S8 )
=> ( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ Y6 @ X5 ) @ D5 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ B @ ( F4 @ Y6 ) @ ( F4 @ X5 ) ) @ E4 ) ) ) ) ) ) ) ) ) ).
% uniformly_continuous_on_def
thf(fact_5713_isUCont__def,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V7819770556892013058_space @ A )
& ( real_V7819770556892013058_space @ B ) )
=> ! [F3: A > B] :
( ( topolo6026614971017936543ous_on @ A @ B @ ( top_top @ ( set @ A ) ) @ F3 )
= ( ! [R5: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ R5 )
=> ? [S8: real] :
( ( ord_less @ real @ ( zero_zero @ real ) @ S8 )
& ! [X5: A,Y6: A] :
( ( ord_less @ real @ ( real_V557655796197034286t_dist @ A @ X5 @ Y6 ) @ S8 )
=> ( ord_less @ real @ ( real_V557655796197034286t_dist @ B @ ( F3 @ X5 ) @ ( F3 @ Y6 ) ) @ R5 ) ) ) ) ) ) ) ).
% isUCont_def
thf(fact_5714_surj__int__decode,axiom,
( ( image2 @ nat @ int @ nat_int_decode @ ( top_top @ ( set @ nat ) ) )
= ( top_top @ ( set @ int ) ) ) ).
% surj_int_decode
thf(fact_5715_set__list__bind,axiom,
! [A: $tType,B: $tType,Xs: list @ B,F3: B > ( list @ A )] :
( ( set2 @ A @ ( bind @ B @ A @ Xs @ F3 ) )
= ( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [X5: B] : ( set2 @ A @ ( F3 @ X5 ) )
@ ( set2 @ B @ Xs ) ) ) ) ).
% set_list_bind
thf(fact_5716_has__derivative__power__int_H,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [X: A,N: int,S2: set @ A] :
( ( X
!= ( zero_zero @ A ) )
=> ( has_derivative @ A @ A
@ ^ [X5: A] : ( power_int @ A @ X5 @ N )
@ ^ [Y6: A] : ( times_times @ A @ Y6 @ ( times_times @ A @ ( ring_1_of_int @ A @ N ) @ ( power_int @ A @ X @ ( minus_minus @ int @ N @ ( one_one @ int ) ) ) ) )
@ ( topolo174197925503356063within @ A @ X @ S2 ) ) ) ) ).
% has_derivative_power_int'
thf(fact_5717_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_5718_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_5719_power__int__1__right,axiom,
! [A: $tType] :
( ( ( inverse @ A )
& ( monoid_mult @ A ) )
=> ! [Y: A] :
( ( power_int @ A @ Y @ ( one_one @ int ) )
= Y ) ) ).
% power_int_1_right
thf(fact_5720_power__int__sgn,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,N: int] :
( ( sgn_sgn @ A @ ( power_int @ A @ A2 @ N ) )
= ( power_int @ A @ ( sgn_sgn @ A @ A2 ) @ N ) ) ) ).
% power_int_sgn
thf(fact_5721_bind__simps_I1_J,axiom,
! [B: $tType,A: $tType,F3: B > ( list @ A )] :
( ( bind @ B @ A @ ( nil @ B ) @ F3 )
= ( nil @ A ) ) ).
% bind_simps(1)
thf(fact_5722_int__encode__inverse,axiom,
! [X: int] :
( ( nat_int_decode @ ( nat_int_encode @ X ) )
= X ) ).
% int_encode_inverse
thf(fact_5723_int__decode__inverse,axiom,
! [N: nat] :
( ( nat_int_encode @ ( nat_int_decode @ N ) )
= N ) ).
% int_decode_inverse
thf(fact_5724_power__int__mult__distrib__numeral1,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [W2: num,Y: A,M2: int] :
( ( power_int @ A @ ( times_times @ A @ ( numeral_numeral @ A @ W2 ) @ Y ) @ M2 )
= ( times_times @ A @ ( power_int @ A @ ( numeral_numeral @ A @ W2 ) @ M2 ) @ ( power_int @ A @ Y @ M2 ) ) ) ) ).
% power_int_mult_distrib_numeral1
thf(fact_5725_power__int__mult__distrib__numeral2,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [X: A,W2: num,M2: int] :
( ( power_int @ A @ ( times_times @ A @ X @ ( numeral_numeral @ A @ W2 ) ) @ M2 )
= ( times_times @ A @ ( power_int @ A @ X @ M2 ) @ ( power_int @ A @ ( numeral_numeral @ A @ W2 ) @ M2 ) ) ) ) ).
% power_int_mult_distrib_numeral2
thf(fact_5726_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_5727_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_5728_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_5729_abs__power__int__minus,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,N: int] :
( ( abs_abs @ A @ ( power_int @ A @ ( uminus_uminus @ A @ A2 ) @ N ) )
= ( abs_abs @ A @ ( power_int @ A @ A2 @ N ) ) ) ) ).
% abs_power_int_minus
thf(fact_5730_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_5731_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_5732_power__int__minus1__right,axiom,
! [A: $tType] :
( ( ( inverse @ A )
& ( monoid_mult @ A ) )
=> ! [Y: A] :
( ( power_int @ A @ Y @ ( uminus_uminus @ int @ ( one_one @ int ) ) )
= ( inverse_inverse @ A @ Y ) ) ) ).
% power_int_minus1_right
thf(fact_5733_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_5734_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_5735_power__int__mono__iff,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A,N: int] :
( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ B2 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ N )
=> ( ( ord_less_eq @ A @ ( power_int @ A @ A2 @ N ) @ ( power_int @ A @ B2 @ N ) )
= ( ord_less_eq @ A @ A2 @ B2 ) ) ) ) ) ) ).
% power_int_mono_iff
thf(fact_5736_inj__int__decode,axiom,
! [A5: set @ nat] : ( inj_on @ nat @ int @ nat_int_decode @ A5 ) ).
% inj_int_decode
thf(fact_5737_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_5738_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_5739_power__int__abs,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,N: int] :
( ( abs_abs @ A @ ( power_int @ A @ A2 @ N ) )
= ( power_int @ A @ ( abs_abs @ A @ A2 ) @ N ) ) ) ).
% power_int_abs
thf(fact_5740_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_5741_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_5742_power__int__mult__distrib,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [X: A,Y: A,M2: int] :
( ( power_int @ A @ ( times_times @ A @ X @ Y ) @ M2 )
= ( times_times @ A @ ( power_int @ A @ X @ M2 ) @ ( power_int @ A @ Y @ M2 ) ) ) ) ).
% power_int_mult_distrib
thf(fact_5743_int__decode__eq,axiom,
! [X: nat,Y: nat] :
( ( ( nat_int_decode @ X )
= ( nat_int_decode @ Y ) )
= ( X = Y ) ) ).
% int_decode_eq
thf(fact_5744_power__int__divide__distrib,axiom,
! [A: $tType] :
( ( field @ A )
=> ! [X: A,Y: A,M2: int] :
( ( power_int @ A @ ( divide_divide @ A @ X @ Y ) @ M2 )
= ( divide_divide @ A @ ( power_int @ A @ X @ M2 ) @ ( power_int @ A @ Y @ M2 ) ) ) ) ).
% power_int_divide_distrib
thf(fact_5745_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_5746_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_5747_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_5748_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_5749_continuous__on__power__int,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo4958980785337419405_space @ A )
& ( real_V8999393235501362500lgebra @ B ) )
=> ! [S: set @ A,F3: A > B,N: int] :
( ( topolo81223032696312382ous_on @ A @ B @ S @ F3 )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ S )
=> ( ( F3 @ X4 )
!= ( zero_zero @ B ) ) )
=> ( topolo81223032696312382ous_on @ A @ B @ S
@ ^ [X5: A] : ( power_int @ B @ ( F3 @ X5 ) @ N ) ) ) ) ) ).
% continuous_on_power_int
thf(fact_5750_list__bind__cong,axiom,
! [B: $tType,A: $tType,Xs: list @ A,Ys: list @ A,F3: A > ( list @ B ),G3: A > ( list @ B )] :
( ( Xs = Ys )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
=> ( ( F3 @ X4 )
= ( G3 @ X4 ) ) )
=> ( ( bind @ A @ B @ Xs @ F3 )
= ( bind @ A @ B @ Ys @ G3 ) ) ) ) ).
% list_bind_cong
thf(fact_5751_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_5752_power__int__increasing,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [N: int,N6: int,A2: A] :
( ( ord_less_eq @ int @ N @ N6 )
=> ( ( ord_less_eq @ A @ ( one_one @ A ) @ A2 )
=> ( ord_less_eq @ A @ ( power_int @ A @ A2 @ N ) @ ( power_int @ A @ A2 @ N6 ) ) ) ) ) ).
% power_int_increasing
thf(fact_5753_power__int__strict__increasing,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [N: int,N6: int,A2: A] :
( ( ord_less @ int @ N @ N6 )
=> ( ( ord_less @ A @ ( one_one @ A ) @ A2 )
=> ( ord_less @ A @ ( power_int @ A @ A2 @ N ) @ ( power_int @ A @ A2 @ N6 ) ) ) ) ) ).
% power_int_strict_increasing
thf(fact_5754_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_5755_tendsto__power__int,axiom,
! [A: $tType,B: $tType] :
( ( real_V8999393235501362500lgebra @ A )
=> ! [F3: B > A,A2: A,F5: filter @ B,N: int] :
( ( filterlim @ B @ A @ F3 @ ( topolo7230453075368039082e_nhds @ A @ A2 ) @ F5 )
=> ( ( A2
!= ( zero_zero @ A ) )
=> ( filterlim @ B @ A
@ ^ [X5: B] : ( power_int @ A @ ( F3 @ X5 ) @ N )
@ ( topolo7230453075368039082e_nhds @ A @ ( power_int @ A @ A2 @ N ) )
@ F5 ) ) ) ) ).
% tendsto_power_int
thf(fact_5756_continuous__at__within__power__int,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_V8999393235501362500lgebra @ B ) )
=> ! [A2: A,S: set @ A,F3: A > B,N: int] :
( ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ S ) @ F3 )
=> ( ( ( F3 @ A2 )
!= ( zero_zero @ B ) )
=> ( topolo3448309680560233919inuous @ A @ B @ ( topolo174197925503356063within @ A @ A2 @ S )
@ ^ [X5: A] : ( power_int @ B @ ( F3 @ X5 ) @ N ) ) ) ) ) ).
% continuous_at_within_power_int
thf(fact_5757_differentiable__power__int,axiom,
! [B: $tType,A: $tType] :
( ( ( real_V822414075346904944vector @ A )
& ( real_V3459762299906320749_field @ B ) )
=> ! [F3: A > B,X: A,S: set @ A,N: int] :
( ( differentiable @ A @ B @ F3 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( ( ( F3 @ X )
!= ( zero_zero @ B ) )
=> ( differentiable @ A @ B
@ ^ [X5: A] : ( power_int @ B @ ( F3 @ X5 ) @ N )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ).
% differentiable_power_int
thf(fact_5758_continuous__power__int,axiom,
! [B: $tType,A: $tType] :
( ( ( topological_t2_space @ A )
& ( real_V8999393235501362500lgebra @ B ) )
=> ! [F5: filter @ A,F3: A > B,N: int] :
( ( topolo3448309680560233919inuous @ A @ B @ F5 @ F3 )
=> ( ( ( F3
@ ( topolo3827282254853284352ce_Lim @ A @ A @ F5
@ ^ [X5: A] : X5 ) )
!= ( zero_zero @ B ) )
=> ( topolo3448309680560233919inuous @ A @ B @ F5
@ ^ [X5: A] : ( power_int @ B @ ( F3 @ X5 ) @ N ) ) ) ) ) ).
% continuous_power_int
thf(fact_5759_power__int__strict__decreasing,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [N: int,N6: int,A2: A] :
( ( ord_less @ int @ N @ N6 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ A @ A2 @ ( one_one @ A ) )
=> ( ord_less @ A @ ( power_int @ A @ A2 @ N6 ) @ ( power_int @ A @ A2 @ N ) ) ) ) ) ) ).
% power_int_strict_decreasing
thf(fact_5760_power__int__mono,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [X: A,Y: A,N: int] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( 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 @ Y @ N ) ) ) ) ) ) ).
% power_int_mono
thf(fact_5761_power__int__strict__antimono,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A,N: int] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ int @ N @ ( zero_zero @ int ) )
=> ( ord_less @ A @ ( power_int @ A @ B2 @ N ) @ ( power_int @ A @ A2 @ N ) ) ) ) ) ) ).
% power_int_strict_antimono
thf(fact_5762_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_5763_one__less__power__int,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,N: int] :
( ( ord_less @ A @ ( one_one @ A ) @ A2 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ N )
=> ( ord_less @ A @ ( one_one @ A ) @ ( power_int @ A @ A2 @ N ) ) ) ) ) ).
% one_less_power_int
thf(fact_5764_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_5765_power__int__antimono,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A,N: int] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ int @ N @ ( zero_zero @ int ) )
=> ( ord_less_eq @ A @ ( power_int @ A @ B2 @ N ) @ ( power_int @ A @ A2 @ N ) ) ) ) ) ) ).
% power_int_antimono
thf(fact_5766_power__int__strict__mono,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [A2: A,B2: A,N: int] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less @ int @ ( zero_zero @ int ) @ N )
=> ( ord_less @ A @ ( power_int @ A @ A2 @ N ) @ ( power_int @ A @ B2 @ N ) ) ) ) ) ) ).
% power_int_strict_mono
thf(fact_5767_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_5768_power__int__decreasing,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [N: int,N6: int,A2: A] :
( ( ord_less_eq @ int @ N @ N6 )
=> ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
=> ( ( ord_less_eq @ A @ A2 @ ( one_one @ A ) )
=> ( ( ( A2
!= ( zero_zero @ A ) )
| ( N6
!= ( zero_zero @ int ) )
| ( N
= ( zero_zero @ int ) ) )
=> ( ord_less_eq @ A @ ( power_int @ A @ A2 @ N6 ) @ ( power_int @ A @ A2 @ N ) ) ) ) ) ) ) ).
% power_int_decreasing
thf(fact_5769_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_5770_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_5771_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_5772_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_5773_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_5774_power__int__def,axiom,
! [A: $tType] :
( ( ( inverse @ A )
& ( power @ A ) )
=> ( ( power_int @ A )
= ( ^ [X5: A,N2: int] : ( if @ A @ ( ord_less_eq @ int @ ( zero_zero @ int ) @ N2 ) @ ( power_power @ A @ X5 @ ( nat2 @ N2 ) ) @ ( power_power @ A @ ( inverse_inverse @ A @ X5 ) @ ( nat2 @ ( uminus_uminus @ int @ N2 ) ) ) ) ) ) ) ).
% power_int_def
thf(fact_5775_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_5776_DERIV__power__int,axiom,
! [A: $tType] :
( ( real_V3459762299906320749_field @ A )
=> ! [F3: A > A,D3: A,X: A,S: set @ A,N: int] :
( ( has_field_derivative @ A @ F3 @ D3 @ ( topolo174197925503356063within @ A @ X @ S ) )
=> ( ( ( F3 @ X )
!= ( zero_zero @ A ) )
=> ( has_field_derivative @ A
@ ^ [X5: A] : ( power_int @ A @ ( F3 @ X5 ) @ N )
@ ( times_times @ A @ ( times_times @ A @ ( ring_1_of_int @ A @ N ) @ ( power_int @ A @ ( F3 @ X ) @ ( minus_minus @ int @ N @ ( one_one @ int ) ) ) ) @ D3 )
@ ( topolo174197925503356063within @ A @ X @ S ) ) ) ) ) ).
% DERIV_power_int
thf(fact_5777_has__derivative__power__int,axiom,
! [A: $tType,C: $tType] :
( ( ( real_V822414075346904944vector @ C )
& ( real_V3459762299906320749_field @ A ) )
=> ! [F3: C > A,X: C,F10: C > A,S2: set @ C,N: int] :
( ( ( F3 @ X )
!= ( zero_zero @ A ) )
=> ( ( has_derivative @ C @ A @ F3 @ F10 @ ( topolo174197925503356063within @ C @ X @ S2 ) )
=> ( has_derivative @ C @ A
@ ^ [X5: C] : ( power_int @ A @ ( F3 @ X5 ) @ N )
@ ^ [H: C] : ( times_times @ A @ ( F10 @ H ) @ ( times_times @ A @ ( ring_1_of_int @ A @ N ) @ ( power_int @ A @ ( F3 @ X ) @ ( minus_minus @ int @ N @ ( one_one @ int ) ) ) ) )
@ ( topolo174197925503356063within @ C @ X @ S2 ) ) ) ) ) ).
% has_derivative_power_int
thf(fact_5778_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_5779_inj__apsnd,axiom,
! [A: $tType,C: $tType,B: $tType,F3: B > C] :
( ( inj_on @ ( product_prod @ A @ B ) @ ( product_prod @ A @ C ) @ ( product_apsnd @ B @ C @ A @ F3 ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) )
= ( inj_on @ B @ C @ F3 @ ( top_top @ ( set @ B ) ) ) ) ).
% inj_apsnd
thf(fact_5780_inj__apfst,axiom,
! [B: $tType,C: $tType,A: $tType,F3: A > C] :
( ( inj_on @ ( product_prod @ A @ B ) @ ( product_prod @ C @ B ) @ ( product_apfst @ A @ C @ B @ F3 ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) )
= ( inj_on @ A @ C @ F3 @ ( top_top @ ( set @ A ) ) ) ) ).
% inj_apfst
thf(fact_5781_apfst__conv,axiom,
! [C: $tType,A: $tType,B: $tType,F3: C > A,X: C,Y: B] :
( ( product_apfst @ C @ A @ B @ F3 @ ( product_Pair @ C @ B @ X @ Y ) )
= ( product_Pair @ A @ B @ ( F3 @ X ) @ Y ) ) ).
% apfst_conv
thf(fact_5782_apsnd__conv,axiom,
! [A: $tType,B: $tType,C: $tType,F3: C > B,X: A,Y: C] :
( ( product_apsnd @ C @ B @ A @ F3 @ ( product_Pair @ A @ C @ X @ Y ) )
= ( product_Pair @ A @ B @ X @ ( F3 @ Y ) ) ) ).
% apsnd_conv
thf(fact_5783_apfst__eq__conv,axiom,
! [A: $tType,B: $tType,C: $tType,F3: C > A,X: product_prod @ C @ B,G3: C > A] :
( ( ( product_apfst @ C @ A @ B @ F3 @ X )
= ( product_apfst @ C @ A @ B @ G3 @ X ) )
= ( ( F3 @ ( product_fst @ C @ B @ X ) )
= ( G3 @ ( product_fst @ C @ B @ X ) ) ) ) ).
% apfst_eq_conv
thf(fact_5784_fst__apfst,axiom,
! [A: $tType,B: $tType,C: $tType,F3: C > A,X: product_prod @ C @ B] :
( ( product_fst @ A @ B @ ( product_apfst @ C @ A @ B @ F3 @ X ) )
= ( F3 @ ( product_fst @ C @ B @ X ) ) ) ).
% fst_apfst
thf(fact_5785_snd__apfst,axiom,
! [B: $tType,A: $tType,C: $tType,F3: C > B,X: product_prod @ C @ A] :
( ( product_snd @ B @ A @ ( product_apfst @ C @ B @ A @ F3 @ X ) )
= ( product_snd @ C @ A @ X ) ) ).
% snd_apfst
thf(fact_5786_fst__apsnd,axiom,
! [B: $tType,C: $tType,A: $tType,F3: C > B,X: product_prod @ A @ C] :
( ( product_fst @ A @ B @ ( product_apsnd @ C @ B @ A @ F3 @ X ) )
= ( product_fst @ A @ C @ X ) ) ).
% fst_apsnd
thf(fact_5787_apsnd__eq__conv,axiom,
! [B: $tType,C: $tType,A: $tType,F3: C > B,X: product_prod @ A @ C,G3: C > B] :
( ( ( product_apsnd @ C @ B @ A @ F3 @ X )
= ( product_apsnd @ C @ B @ A @ G3 @ X ) )
= ( ( F3 @ ( product_snd @ A @ C @ X ) )
= ( G3 @ ( product_snd @ A @ C @ X ) ) ) ) ).
% apsnd_eq_conv
thf(fact_5788_snd__apsnd,axiom,
! [A: $tType,C: $tType,B: $tType,F3: C > A,X: product_prod @ B @ C] :
( ( product_snd @ B @ A @ ( product_apsnd @ C @ A @ B @ F3 @ X ) )
= ( F3 @ ( product_snd @ B @ C @ X ) ) ) ).
% snd_apsnd
thf(fact_5789_snd__comp__apfst,axiom,
! [C: $tType,B: $tType,A: $tType,F3: A > C] :
( ( comp @ ( product_prod @ C @ B ) @ B @ ( product_prod @ A @ B ) @ ( product_snd @ C @ B ) @ ( product_apfst @ A @ C @ B @ F3 ) )
= ( product_snd @ A @ B ) ) ).
% snd_comp_apfst
thf(fact_5790_fst__comp__apsnd,axiom,
! [C: $tType,B: $tType,A: $tType,F3: B > C] :
( ( comp @ ( product_prod @ A @ C ) @ A @ ( product_prod @ A @ B ) @ ( product_fst @ A @ C ) @ ( product_apsnd @ B @ C @ A @ F3 ) )
= ( product_fst @ A @ B ) ) ).
% fst_comp_apsnd
thf(fact_5791_apfst__id,axiom,
! [B: $tType,A: $tType] :
( ( product_apfst @ A @ A @ B @ ( id @ A ) )
= ( id @ ( product_prod @ A @ B ) ) ) ).
% apfst_id
thf(fact_5792_apsnd__id,axiom,
! [B: $tType,A: $tType] :
( ( product_apsnd @ B @ B @ A @ ( id @ B ) )
= ( id @ ( product_prod @ A @ B ) ) ) ).
% apsnd_id
thf(fact_5793_fst__comp__apfst,axiom,
! [C: $tType,B: $tType,A: $tType,F3: A > C] :
( ( comp @ ( product_prod @ C @ B ) @ C @ ( product_prod @ A @ B ) @ ( product_fst @ C @ B ) @ ( product_apfst @ A @ C @ B @ F3 ) )
= ( comp @ A @ C @ ( product_prod @ A @ B ) @ F3 @ ( product_fst @ A @ B ) ) ) ).
% fst_comp_apfst
thf(fact_5794_snd__comp__apsnd,axiom,
! [C: $tType,B: $tType,A: $tType,F3: B > C] :
( ( comp @ ( product_prod @ A @ C ) @ C @ ( product_prod @ A @ B ) @ ( product_snd @ A @ C ) @ ( product_apsnd @ B @ C @ A @ F3 ) )
= ( comp @ B @ C @ ( product_prod @ A @ B ) @ F3 @ ( product_snd @ A @ B ) ) ) ).
% snd_comp_apsnd
thf(fact_5795_apfst__apsnd,axiom,
! [A: $tType,B: $tType,D: $tType,C: $tType,F3: C > A,G3: D > B,X: product_prod @ C @ D] :
( ( product_apfst @ C @ A @ B @ F3 @ ( product_apsnd @ D @ B @ C @ G3 @ X ) )
= ( product_Pair @ A @ B @ ( F3 @ ( product_fst @ C @ D @ X ) ) @ ( G3 @ ( product_snd @ C @ D @ X ) ) ) ) ).
% apfst_apsnd
thf(fact_5796_apsnd__apfst,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,F3: C > B,G3: D > A,X: product_prod @ D @ C] :
( ( product_apsnd @ C @ B @ A @ F3 @ ( product_apfst @ D @ A @ C @ G3 @ X ) )
= ( product_Pair @ A @ B @ ( G3 @ ( product_fst @ D @ C @ X ) ) @ ( F3 @ ( product_snd @ D @ C @ X ) ) ) ) ).
% apsnd_apfst
thf(fact_5797_apsnd__apfst__commute,axiom,
! [A: $tType,B: $tType,C: $tType,D: $tType,F3: C > B,G3: D > A,P: product_prod @ D @ C] :
( ( product_apsnd @ C @ B @ A @ F3 @ ( product_apfst @ D @ A @ C @ G3 @ P ) )
= ( product_apfst @ D @ A @ B @ G3 @ ( product_apsnd @ C @ B @ D @ F3 @ P ) ) ) ).
% apsnd_apfst_commute
thf(fact_5798_apsnd__compose,axiom,
! [C: $tType,B: $tType,D: $tType,A: $tType,F3: C > B,G3: D > C,X: product_prod @ A @ D] :
( ( product_apsnd @ C @ B @ A @ F3 @ ( product_apsnd @ D @ C @ A @ G3 @ X ) )
= ( product_apsnd @ D @ B @ A @ ( comp @ C @ B @ D @ F3 @ G3 ) @ X ) ) ).
% apsnd_compose
thf(fact_5799_apfst__compose,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,F3: C > A,G3: D > C,X: product_prod @ D @ B] :
( ( product_apfst @ C @ A @ B @ F3 @ ( product_apfst @ D @ C @ B @ G3 @ X ) )
= ( product_apfst @ D @ A @ B @ ( comp @ C @ A @ D @ F3 @ G3 ) @ X ) ) ).
% apfst_compose
thf(fact_5800_apfst__convE,axiom,
! [C: $tType,A: $tType,B: $tType,Q2: product_prod @ A @ B,F3: C > A,P: product_prod @ C @ B] :
( ( Q2
= ( product_apfst @ C @ A @ B @ F3 @ P ) )
=> ~ ! [X4: C,Y3: B] :
( ( P
= ( product_Pair @ C @ B @ X4 @ Y3 ) )
=> ( Q2
!= ( product_Pair @ A @ B @ ( F3 @ X4 ) @ Y3 ) ) ) ) ).
% apfst_convE
thf(fact_5801_eventually__less__ceiling,axiom,
! [B: $tType,A: $tType] :
( ( ( archim2362893244070406136eiling @ B )
& ( topolo2564578578187576103pology @ B ) )
=> ! [F3: A > B,L: B,F5: filter @ A] :
( ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ F5 )
=> ( ~ ( member @ B @ L @ ( ring_1_Ints @ B ) )
=> ( eventually @ A
@ ^ [X5: A] : ( ord_less @ B @ ( F3 @ X5 ) @ ( ring_1_of_int @ B @ ( archimedean_ceiling @ B @ L ) ) )
@ F5 ) ) ) ) ).
% eventually_less_ceiling
thf(fact_5802_eventually__floor__less,axiom,
! [B: $tType,A: $tType] :
( ( ( archim2362893244070406136eiling @ B )
& ( topolo2564578578187576103pology @ B ) )
=> ! [F3: A > B,L: B,F5: filter @ A] :
( ( filterlim @ A @ B @ F3 @ ( topolo7230453075368039082e_nhds @ B @ L ) @ F5 )
=> ( ~ ( member @ B @ L @ ( ring_1_Ints @ B ) )
=> ( eventually @ A
@ ^ [X5: A] : ( ord_less @ B @ ( ring_1_of_int @ B @ ( archim6421214686448440834_floor @ B @ L ) ) @ ( F3 @ X5 ) )
@ F5 ) ) ) ) ).
% eventually_floor_less
thf(fact_5803_Ints__sum,axiom,
! [A: $tType,B: $tType] :
( ( ring_1 @ B )
=> ! [A5: set @ A,F3: A > B] :
( ! [X4: A] :
( ( member @ A @ X4 @ A5 )
=> ( member @ B @ ( F3 @ X4 ) @ ( ring_1_Ints @ B ) ) )
=> ( member @ B @ ( groups7311177749621191930dd_sum @ A @ B @ F3 @ A5 ) @ ( ring_1_Ints @ B ) ) ) ) ).
% Ints_sum
thf(fact_5804_Ints__prod,axiom,
! [A: $tType,B: $tType] :
( ( ( comm_monoid_mult @ B )
& ( ring_1 @ B ) )
=> ! [A5: set @ A,F3: A > B] :
( ! [X4: A] :
( ( member @ A @ X4 @ A5 )
=> ( member @ B @ ( F3 @ X4 ) @ ( ring_1_Ints @ B ) ) )
=> ( member @ B @ ( groups7121269368397514597t_prod @ A @ B @ F3 @ A5 ) @ ( ring_1_Ints @ B ) ) ) ) ).
% Ints_prod
thf(fact_5805_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_5806_floor__add2,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,Y: A] :
( ( ( member @ A @ X @ ( ring_1_Ints @ A ) )
| ( member @ A @ Y @ ( ring_1_Ints @ A ) ) )
=> ( ( archim6421214686448440834_floor @ A @ ( plus_plus @ A @ X @ Y ) )
= ( plus_plus @ int @ ( archim6421214686448440834_floor @ A @ X ) @ ( archim6421214686448440834_floor @ A @ Y ) ) ) ) ) ).
% floor_add2
thf(fact_5807_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_5808_Ints__mult,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [A2: A,B2: A] :
( ( member @ A @ A2 @ ( ring_1_Ints @ A ) )
=> ( ( member @ A @ B2 @ ( ring_1_Ints @ A ) )
=> ( member @ A @ ( times_times @ A @ A2 @ B2 ) @ ( ring_1_Ints @ A ) ) ) ) ) ).
% Ints_mult
thf(fact_5809_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_5810_Ints__minus,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [A2: A] :
( ( member @ A @ A2 @ ( ring_1_Ints @ A ) )
=> ( member @ A @ ( uminus_uminus @ A @ A2 ) @ ( ring_1_Ints @ A ) ) ) ) ).
% Ints_minus
thf(fact_5811_Ints__add,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [A2: A,B2: A] :
( ( member @ A @ A2 @ ( ring_1_Ints @ A ) )
=> ( ( member @ A @ B2 @ ( ring_1_Ints @ A ) )
=> ( member @ A @ ( plus_plus @ A @ A2 @ B2 ) @ ( ring_1_Ints @ A ) ) ) ) ) ).
% Ints_add
thf(fact_5812_Ints__1,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( member @ A @ ( one_one @ A ) @ ( ring_1_Ints @ A ) ) ) ).
% Ints_1
thf(fact_5813_Ints__0,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( member @ A @ ( zero_zero @ A ) @ ( ring_1_Ints @ A ) ) ) ).
% Ints_0
thf(fact_5814_Ints__abs,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A] :
( ( member @ A @ A2 @ ( ring_1_Ints @ A ) )
=> ( member @ A @ ( abs_abs @ A @ A2 ) @ ( ring_1_Ints @ A ) ) ) ) ).
% Ints_abs
thf(fact_5815_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_5816_Ints__numeral,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [N: num] : ( member @ A @ ( numeral_numeral @ A @ N ) @ ( ring_1_Ints @ A ) ) ) ).
% Ints_numeral
thf(fact_5817_Ints__diff,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [A2: A,B2: A] :
( ( member @ A @ A2 @ ( ring_1_Ints @ A ) )
=> ( ( member @ A @ B2 @ ( ring_1_Ints @ A ) )
=> ( member @ A @ ( minus_minus @ A @ A2 @ B2 ) @ ( ring_1_Ints @ A ) ) ) ) ) ).
% Ints_diff
thf(fact_5818_Ints__power,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [A2: A,N: nat] :
( ( member @ A @ A2 @ ( ring_1_Ints @ A ) )
=> ( member @ A @ ( power_power @ A @ A2 @ N ) @ ( ring_1_Ints @ A ) ) ) ) ).
% Ints_power
thf(fact_5819_Ints__of__int,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [Z: int] : ( member @ A @ ( ring_1_of_int @ A @ Z ) @ ( ring_1_Ints @ A ) ) ) ).
% Ints_of_int
thf(fact_5820_Ints__induct,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [Q2: A,P2: A > $o] :
( ( member @ A @ Q2 @ ( ring_1_Ints @ A ) )
=> ( ! [Z3: int] : ( P2 @ ( ring_1_of_int @ A @ Z3 ) )
=> ( P2 @ Q2 ) ) ) ) ).
% Ints_induct
thf(fact_5821_Ints__cases,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ! [Q2: A] :
( ( member @ A @ Q2 @ ( ring_1_Ints @ A ) )
=> ~ ! [Z3: int] :
( Q2
!= ( ring_1_of_int @ A @ Z3 ) ) ) ) ).
% Ints_cases
thf(fact_5822_Ints__double__eq__0__iff,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [A2: A] :
( ( member @ A @ A2 @ ( ring_1_Ints @ A ) )
=> ( ( ( plus_plus @ A @ A2 @ A2 )
= ( zero_zero @ A ) )
= ( A2
= ( zero_zero @ A ) ) ) ) ) ).
% Ints_double_eq_0_iff
thf(fact_5823_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_5824_finite__int__segment,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [A2: A,B2: A] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [X5: A] :
( ( member @ A @ X5 @ ( ring_1_Ints @ A ) )
& ( ord_less_eq @ A @ A2 @ X5 )
& ( ord_less_eq @ A @ X5 @ B2 ) ) ) ) ) ).
% finite_int_segment
thf(fact_5825_Ints__odd__nonzero,axiom,
! [A: $tType] :
( ( ring_char_0 @ A )
=> ! [A2: A] :
( ( member @ A @ A2 @ ( ring_1_Ints @ A ) )
=> ( ( plus_plus @ A @ ( plus_plus @ A @ ( one_one @ A ) @ A2 ) @ A2 )
!= ( zero_zero @ A ) ) ) ) ).
% Ints_odd_nonzero
thf(fact_5826_of__int__divide__in__Ints,axiom,
! [A: $tType] :
( ( idom_divide @ A )
=> ! [B2: int,A2: int] :
( ( dvd_dvd @ int @ B2 @ A2 )
=> ( member @ A @ ( divide_divide @ A @ ( ring_1_of_int @ A @ A2 ) @ ( ring_1_of_int @ A @ B2 ) ) @ ( ring_1_Ints @ A ) ) ) ) ).
% of_int_divide_in_Ints
thf(fact_5827_finite__abs__int__segment,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [A2: A] :
( finite_finite2 @ A
@ ( collect @ A
@ ^ [K3: A] :
( ( member @ A @ K3 @ ( ring_1_Ints @ A ) )
& ( ord_less_eq @ A @ ( abs_abs @ A @ K3 ) @ A2 ) ) ) ) ) ).
% finite_abs_int_segment
thf(fact_5828_Nats__altdef2,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ( ( semiring_1_Nats @ A )
= ( collect @ A
@ ^ [N2: A] :
( ( member @ A @ N2 @ ( ring_1_Ints @ A ) )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ N2 ) ) ) ) ) ).
% Nats_altdef2
thf(fact_5829_Ints__def,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( ( ring_1_Ints @ A )
= ( image2 @ int @ A @ ( ring_1_of_int @ A ) @ ( top_top @ ( set @ int ) ) ) ) ) ).
% Ints_def
thf(fact_5830_Ints__odd__less__0,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [A2: A] :
( ( member @ A @ A2 @ ( ring_1_Ints @ A ) )
=> ( ( ord_less @ A @ ( plus_plus @ A @ ( plus_plus @ A @ ( one_one @ A ) @ A2 ) @ A2 ) @ ( zero_zero @ A ) )
= ( ord_less @ A @ A2 @ ( zero_zero @ A ) ) ) ) ) ).
% Ints_odd_less_0
thf(fact_5831_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_5832_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_5833_Ints__eq__abs__less1,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: A,Y: A] :
( ( member @ A @ X @ ( ring_1_Ints @ A ) )
=> ( ( member @ A @ Y @ ( ring_1_Ints @ A ) )
=> ( ( X = Y )
= ( ord_less @ A @ ( abs_abs @ A @ ( minus_minus @ A @ X @ Y ) ) @ ( one_one @ A ) ) ) ) ) ) ).
% Ints_eq_abs_less1
thf(fact_5834_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_5835_le__mult__floor__Ints,axiom,
! [A: $tType,B: $tType] :
( ( ( archim2362893244070406136eiling @ B )
& ( linordered_idom @ A ) )
=> ! [A2: B,B2: B] :
( ( ord_less_eq @ B @ ( zero_zero @ B ) @ A2 )
=> ( ( member @ B @ A2 @ ( ring_1_Ints @ B ) )
=> ( ord_less_eq @ A @ ( ring_1_of_int @ A @ ( times_times @ int @ ( archim6421214686448440834_floor @ B @ A2 ) @ ( archim6421214686448440834_floor @ B @ B2 ) ) ) @ ( ring_1_of_int @ A @ ( archim6421214686448440834_floor @ B @ ( times_times @ B @ A2 @ B2 ) ) ) ) ) ) ) ).
% le_mult_floor_Ints
thf(fact_5836_frac__unique__iff,axiom,
! [A: $tType] :
( ( archim2362893244070406136eiling @ A )
=> ! [X: A,A2: A] :
( ( ( archimedean_frac @ A @ X )
= A2 )
= ( ( member @ A @ ( minus_minus @ A @ X @ A2 ) @ ( ring_1_Ints @ A ) )
& ( ord_less_eq @ A @ ( zero_zero @ A ) @ A2 )
& ( ord_less @ A @ A2 @ ( one_one @ A ) ) ) ) ) ).
% frac_unique_iff
thf(fact_5837_mult__ceiling__le__Ints,axiom,
! [A: $tType,B: $tType] :
( ( ( archim2362893244070406136eiling @ B )
& ( linordered_idom @ A ) )
=> ! [A2: B,B2: B] :
( ( ord_less_eq @ B @ ( zero_zero @ B ) @ A2 )
=> ( ( member @ B @ A2 @ ( ring_1_Ints @ B ) )
=> ( ord_less_eq @ A @ ( ring_1_of_int @ A @ ( archimedean_ceiling @ B @ ( times_times @ B @ A2 @ B2 ) ) ) @ ( ring_1_of_int @ A @ ( times_times @ int @ ( archimedean_ceiling @ B @ A2 ) @ ( archimedean_ceiling @ B @ B2 ) ) ) ) ) ) ) ).
% mult_ceiling_le_Ints
thf(fact_5838_VEBT__internal_Ovalid_H_Oelims_I1_J,axiom,
! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
( ( ( vEBT_VEBT_valid @ X @ Xa2 )
= Y )
=> ( ( ? [Uu2: $o,Uv2: $o] :
( X
= ( vEBT_Leaf @ Uu2 @ Uv2 ) )
=> ( Y
= ( Xa2
!= ( 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 ) )
=> ( Y
= ( ~ ( ( Deg2 = Xa2 )
& ! [X5: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X5 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ( vEBT_VEBT_valid @ X5 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
& ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
& ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( case_option @ $o @ ( product_prod @ nat @ nat )
@ ( ~ ? [X9: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X9 )
& ! [X5: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X5 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X9: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X9 ) ) )
@ ( 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 ) )
& ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
=> ( ( ? [X9: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I4 ) @ X9 ) )
= ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
& ( ( Mi3 = Ma3 )
=> ! [X5: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X5 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X9: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X9 ) ) )
& ( ( Mi3 != Ma3 )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList2 @ Ma3 )
& ! [X5: nat] :
( ( ord_less @ nat @ X5 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList2 @ X5 )
=> ( ( ord_less @ nat @ Mi3 @ X5 )
& ( ord_less_eq @ nat @ X5 @ Ma3 ) ) ) ) ) ) ) )
@ Mima ) ) ) ) ) ) ) ).
% VEBT_internal.valid'.elims(1)
thf(fact_5839_VEBT__internal_Ovalid_H_Oelims_I2_J,axiom,
! [X: vEBT_VEBT,Xa2: nat] :
( ( vEBT_VEBT_valid @ X @ Xa2 )
=> ( ( ? [Uu2: $o,Uv2: $o] :
( X
= ( vEBT_Leaf @ Uu2 @ Uv2 ) )
=> ( Xa2
!= ( 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 = Xa2 )
& ! [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 )
@ ( ~ ? [X9: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X9 )
& ! [X5: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X5 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X9: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X9 ) ) )
@ ( 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 ) )
& ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
=> ( ( ? [X9: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I4 ) @ X9 ) )
= ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
& ( ( Mi3 = Ma3 )
=> ! [X5: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X5 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X9: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X9 ) ) )
& ( ( Mi3 != Ma3 )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList2 @ Ma3 )
& ! [X5: nat] :
( ( ord_less @ nat @ X5 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList2 @ X5 )
=> ( ( ord_less @ nat @ Mi3 @ X5 )
& ( ord_less_eq @ nat @ X5 @ Ma3 ) ) ) ) ) ) ) )
@ Mima ) ) ) ) ) ).
% VEBT_internal.valid'.elims(2)
thf(fact_5840_VEBT__internal_Ovalid_H_Oelims_I3_J,axiom,
! [X: vEBT_VEBT,Xa2: nat] :
( ~ ( vEBT_VEBT_valid @ X @ Xa2 )
=> ( ( ? [Uu2: $o,Uv2: $o] :
( X
= ( vEBT_Leaf @ Uu2 @ Uv2 ) )
=> ( Xa2
= ( 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 = Xa2 )
& ! [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 )
@ ( ~ ? [X9: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X9 )
& ! [X5: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X5 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X9: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X9 ) ) )
@ ( 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 ) )
& ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
=> ( ( ? [X9: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I4 ) @ X9 ) )
= ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
& ( ( Mi3 = Ma3 )
=> ! [X5: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X5 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X9: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X9 ) ) )
& ( ( Mi3 != Ma3 )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList2 @ Ma3 )
& ! [X5: nat] :
( ( ord_less @ nat @ X5 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList2 @ X5 )
=> ( ( ord_less @ nat @ Mi3 @ X5 )
& ( ord_less_eq @ nat @ X5 @ Ma3 ) ) ) ) ) ) ) )
@ Mima ) ) ) ) ) ).
% VEBT_internal.valid'.elims(3)
thf(fact_5841_ball__empty,axiom,
! [A: $tType,P2: A > $o,X3: A] :
( ( member @ A @ X3 @ ( bot_bot @ ( set @ A ) ) )
=> ( P2 @ X3 ) ) ).
% ball_empty
thf(fact_5842_finite__Collect__bounded__ex,axiom,
! [B: $tType,A: $tType,P2: A > $o,Q: B > A > $o] :
( ( finite_finite2 @ A @ ( collect @ A @ P2 ) )
=> ( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X5: B] :
? [Y6: A] :
( ( P2 @ Y6 )
& ( Q @ X5 @ Y6 ) ) ) )
= ( ! [Y6: A] :
( ( P2 @ Y6 )
=> ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X5: B] : ( Q @ X5 @ Y6 ) ) ) ) ) ) ) ).
% finite_Collect_bounded_ex
thf(fact_5843_closed__diagonal,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ( topolo7761053866217962861closed @ ( product_prod @ A @ A )
@ ( collect @ ( product_prod @ A @ A )
@ ^ [Y6: product_prod @ A @ A] :
? [X5: A] :
( Y6
= ( product_Pair @ A @ A @ X5 @ X5 ) ) ) ) ) ).
% closed_diagonal
thf(fact_5844_finite_Omono,axiom,
! [A: $tType] :
( order_mono @ ( ( set @ A ) > $o ) @ ( ( set @ A ) > $o )
@ ^ [P6: ( set @ A ) > $o,X5: set @ A] :
( ( X5
= ( bot_bot @ ( set @ A ) ) )
| ? [A7: set @ A,A6: A] :
( ( X5
= ( insert @ A @ A6 @ A7 ) )
& ( P6 @ A7 ) ) ) ) ).
% finite.mono
thf(fact_5845_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_5846_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_5847_finite__image__set,axiom,
! [A: $tType,B: $tType,P2: A > $o,F3: A > B] :
( ( finite_finite2 @ A @ ( collect @ A @ P2 ) )
=> ( finite_finite2 @ B
@ ( collect @ B
@ ^ [Uu3: B] :
? [X5: A] :
( ( Uu3
= ( F3 @ X5 ) )
& ( P2 @ X5 ) ) ) ) ) ).
% finite_image_set
thf(fact_5848_finite__image__set2,axiom,
! [A: $tType,B: $tType,C: $tType,P2: A > $o,Q: B > $o,F3: A > B > C] :
( ( finite_finite2 @ A @ ( collect @ A @ P2 ) )
=> ( ( finite_finite2 @ B @ ( collect @ B @ Q ) )
=> ( finite_finite2 @ C
@ ( collect @ C
@ ^ [Uu3: C] :
? [X5: A,Y6: B] :
( ( Uu3
= ( F3 @ X5 @ Y6 ) )
& ( P2 @ X5 )
& ( Q @ Y6 ) ) ) ) ) ) ).
% finite_image_set2
thf(fact_5849_option_Osimps_I5_J,axiom,
! [B: $tType,A: $tType,F1: B,F22: A > B,X2: A] :
( ( case_option @ B @ A @ F1 @ F22 @ ( some @ A @ X2 ) )
= ( F22 @ X2 ) ) ).
% option.simps(5)
thf(fact_5850_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_5851_option_Ocase__distrib,axiom,
! [C: $tType,B: $tType,A: $tType,H2: B > C,F1: B,F22: A > B,Option: option @ A] :
( ( H2 @ ( case_option @ B @ A @ F1 @ F22 @ Option ) )
= ( case_option @ C @ A @ ( H2 @ F1 )
@ ^ [X5: A] : ( H2 @ ( F22 @ X5 ) )
@ Option ) ) ).
% option.case_distrib
thf(fact_5852_open__diagonal__complement,axiom,
! [A: $tType] :
( ( topological_t2_space @ A )
=> ( topolo1002775350975398744n_open @ ( product_prod @ A @ A )
@ ( collect @ ( product_prod @ A @ A )
@ ^ [Uu3: product_prod @ A @ A] :
? [X5: A,Y6: A] :
( ( Uu3
= ( product_Pair @ A @ A @ X5 @ Y6 ) )
& ( X5 != Y6 ) ) ) ) ) ).
% open_diagonal_complement
thf(fact_5853_closed__subdiagonal,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ( topolo7761053866217962861closed @ ( product_prod @ A @ A )
@ ( collect @ ( product_prod @ A @ A )
@ ^ [Uu3: product_prod @ A @ A] :
? [X5: A,Y6: A] :
( ( Uu3
= ( product_Pair @ A @ A @ X5 @ Y6 ) )
& ( ord_less_eq @ A @ X5 @ Y6 ) ) ) ) ) ).
% closed_subdiagonal
thf(fact_5854_closed__superdiagonal,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ( topolo7761053866217962861closed @ ( product_prod @ A @ A )
@ ( collect @ ( product_prod @ A @ A )
@ ^ [Uu3: product_prod @ A @ A] :
? [X5: A,Y6: A] :
( ( Uu3
= ( product_Pair @ A @ A @ X5 @ Y6 ) )
& ( ord_less_eq @ A @ Y6 @ X5 ) ) ) ) ) ).
% closed_superdiagonal
thf(fact_5855_open__superdiagonal,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ( topolo1002775350975398744n_open @ ( product_prod @ A @ A )
@ ( collect @ ( product_prod @ A @ A )
@ ^ [Uu3: product_prod @ A @ A] :
? [X5: A,Y6: A] :
( ( Uu3
= ( product_Pair @ A @ A @ X5 @ Y6 ) )
& ( ord_less @ A @ Y6 @ X5 ) ) ) ) ) ).
% open_superdiagonal
thf(fact_5856_open__subdiagonal,axiom,
! [A: $tType] :
( ( topolo1944317154257567458pology @ A )
=> ( topolo1002775350975398744n_open @ ( product_prod @ A @ A )
@ ( collect @ ( product_prod @ A @ A )
@ ^ [Uu3: product_prod @ A @ A] :
? [X5: A,Y6: A] :
( ( Uu3
= ( product_Pair @ A @ A @ X5 @ Y6 ) )
& ( ord_less @ A @ X5 @ Y6 ) ) ) ) ) ).
% open_subdiagonal
thf(fact_5857_eventually__ball__finite,axiom,
! [A: $tType,B: $tType,A5: set @ A,P2: B > A > $o,Net: filter @ B] :
( ( finite_finite2 @ A @ A5 )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ A5 )
=> ( eventually @ B
@ ^ [Y6: B] : ( P2 @ Y6 @ X4 )
@ Net ) )
=> ( eventually @ B
@ ^ [X5: B] :
! [Y6: A] :
( ( member @ A @ Y6 @ A5 )
=> ( P2 @ X5 @ Y6 ) )
@ Net ) ) ) ).
% eventually_ball_finite
thf(fact_5858_eventually__ball__finite__distrib,axiom,
! [B: $tType,A: $tType,A5: set @ A,P2: B > A > $o,Net: filter @ B] :
( ( finite_finite2 @ A @ A5 )
=> ( ( eventually @ B
@ ^ [X5: B] :
! [Y6: A] :
( ( member @ A @ Y6 @ A5 )
=> ( P2 @ X5 @ Y6 ) )
@ Net )
= ( ! [X5: A] :
( ( member @ A @ X5 @ A5 )
=> ( eventually @ B
@ ^ [Y6: B] : ( P2 @ Y6 @ X5 )
@ Net ) ) ) ) ) ).
% eventually_ball_finite_distrib
thf(fact_5859_case__optionE,axiom,
! [A: $tType,P2: $o,Q: A > $o,X: option @ A] :
( ( case_option @ $o @ A @ P2 @ Q @ X )
=> ( ( ( X
= ( none @ A ) )
=> ~ P2 )
=> ~ ! [Y3: A] :
( ( X
= ( some @ A @ Y3 ) )
=> ~ ( Q @ Y3 ) ) ) ) ).
% case_optionE
thf(fact_5860_Inf__eq__Sup,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ( ( complete_Inf_Inf @ A )
= ( ^ [A7: set @ A] :
( complete_Sup_Sup @ A
@ ( collect @ A
@ ^ [B5: A] :
! [X5: A] :
( ( member @ A @ X5 @ A7 )
=> ( ord_less_eq @ A @ B5 @ X5 ) ) ) ) ) ) ) ).
% Inf_eq_Sup
thf(fact_5861_Sup__eq__Inf,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ( ( complete_Sup_Sup @ A )
= ( ^ [A7: set @ A] :
( complete_Inf_Inf @ A
@ ( collect @ A
@ ^ [B5: A] :
! [X5: A] :
( ( member @ A @ X5 @ A7 )
=> ( ord_less_eq @ A @ X5 @ B5 ) ) ) ) ) ) ) ).
% Sup_eq_Inf
thf(fact_5862_set__conv__nth,axiom,
! [A: $tType] :
( ( set2 @ A )
= ( ^ [Xs3: list @ A] :
( collect @ A
@ ^ [Uu3: A] :
? [I4: nat] :
( ( Uu3
= ( nth @ A @ Xs3 @ I4 ) )
& ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ Xs3 ) ) ) ) ) ) ).
% set_conv_nth
thf(fact_5863_cInf__cSup,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [S2: set @ A] :
( ( S2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit1013018076250108175_below @ A @ S2 )
=> ( ( complete_Inf_Inf @ A @ S2 )
= ( complete_Sup_Sup @ A
@ ( collect @ A
@ ^ [X5: A] :
! [Y6: A] :
( ( member @ A @ Y6 @ S2 )
=> ( ord_less_eq @ A @ X5 @ Y6 ) ) ) ) ) ) ) ) ).
% cInf_cSup
thf(fact_5864_cSup__cInf,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [S2: set @ A] :
( ( S2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit941137186595557371_above @ A @ S2 )
=> ( ( complete_Sup_Sup @ A @ S2 )
= ( complete_Inf_Inf @ A
@ ( collect @ A
@ ^ [X5: A] :
! [Y6: A] :
( ( member @ A @ Y6 @ S2 )
=> ( ord_less_eq @ A @ Y6 @ X5 ) ) ) ) ) ) ) ) ).
% cSup_cInf
thf(fact_5865_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 )
& ! [X5: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X5 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ( vEBT_VEBT_valid @ X5 @ ( 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 )
@ ( ~ ? [X9: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X9 )
& ! [X5: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X5 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ~ ? [X9: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X9 ) ) )
@ ( 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 ) )
& ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
=> ( ( ? [X9: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList @ I4 ) @ X9 ) )
= ( vEBT_V8194947554948674370ptions @ Summary @ I4 ) ) )
& ( ( Mi3 = Ma3 )
=> ! [X5: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X5 @ ( set2 @ vEBT_VEBT @ TreeList ) )
=> ~ ? [X9: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X9 ) ) )
& ( ( Mi3 != Ma3 )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList @ Ma3 )
& ! [X5: nat] :
( ( ord_less @ nat @ X5 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg ) )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList @ X5 )
=> ( ( ord_less @ nat @ Mi3 @ X5 )
& ( ord_less_eq @ nat @ X5 @ Ma3 ) ) ) ) ) ) ) )
@ Mima2 ) ) ) ).
% VEBT_internal.valid'.simps(2)
thf(fact_5866_funpow__inj__finite,axiom,
! [A: $tType,P: A > A,X: A] :
( ( inj_on @ A @ A @ P @ ( top_top @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [Y6: A] :
? [N2: nat] :
( Y6
= ( compow @ ( A > A ) @ N2 @ P @ X ) ) ) )
=> ~ ! [N3: nat] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ N3 )
=> ( ( compow @ ( A > A ) @ N3 @ P @ X )
!= X ) ) ) ) ).
% funpow_inj_finite
thf(fact_5867_VEBT__internal_Ovalid_H_Opelims_I3_J,axiom,
! [X: vEBT_VEBT,Xa2: nat] :
( ~ ( vEBT_VEBT_valid @ X @ Xa2 )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_VEBT_valid_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa2 ) )
=> ( ! [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 ) @ Xa2 ) )
=> ( Xa2
= ( 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 ) @ Xa2 ) )
=> ( ( Deg2 = Xa2 )
& ! [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 )
@ ( ~ ? [X9: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X9 )
& ! [X5: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X5 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X9: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X9 ) ) )
@ ( 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 ) )
& ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
=> ( ( ? [X9: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I4 ) @ X9 ) )
= ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
& ( ( Mi3 = Ma3 )
=> ! [X5: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X5 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X9: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X9 ) ) )
& ( ( Mi3 != Ma3 )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList2 @ Ma3 )
& ! [X5: nat] :
( ( ord_less @ nat @ X5 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList2 @ X5 )
=> ( ( ord_less @ nat @ Mi3 @ X5 )
& ( ord_less_eq @ nat @ X5 @ Ma3 ) ) ) ) ) ) ) )
@ Mima ) ) ) ) ) ) ) ).
% VEBT_internal.valid'.pelims(3)
thf(fact_5868_VEBT__internal_Ovalid_H_Opelims_I2_J,axiom,
! [X: vEBT_VEBT,Xa2: nat] :
( ( vEBT_VEBT_valid @ X @ Xa2 )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_VEBT_valid_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa2 ) )
=> ( ! [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 ) @ Xa2 ) )
=> ( Xa2
!= ( 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 ) @ Xa2 ) )
=> ~ ( ( Deg2 = Xa2 )
& ! [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 )
@ ( ~ ? [X9: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X9 )
& ! [X5: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X5 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X9: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X9 ) ) )
@ ( 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 ) )
& ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
=> ( ( ? [X9: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I4 ) @ X9 ) )
= ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
& ( ( Mi3 = Ma3 )
=> ! [X5: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X5 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X9: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X9 ) ) )
& ( ( Mi3 != Ma3 )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList2 @ Ma3 )
& ! [X5: nat] :
( ( ord_less @ nat @ X5 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList2 @ X5 )
=> ( ( ord_less @ nat @ Mi3 @ X5 )
& ( ord_less_eq @ nat @ X5 @ Ma3 ) ) ) ) ) ) ) )
@ Mima ) ) ) ) ) ) ) ).
% VEBT_internal.valid'.pelims(2)
thf(fact_5869_VEBT__internal_Ovalid_H_Opelims_I1_J,axiom,
! [X: vEBT_VEBT,Xa2: nat,Y: $o] :
( ( ( vEBT_VEBT_valid @ X @ Xa2 )
= Y )
=> ( ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_VEBT_valid_rel @ ( product_Pair @ vEBT_VEBT @ nat @ X @ Xa2 ) )
=> ( ! [Uu2: $o,Uv2: $o] :
( ( X
= ( vEBT_Leaf @ Uu2 @ Uv2 ) )
=> ( ( Y
= ( Xa2
= ( one_one @ nat ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_VEBT_valid_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa2 ) ) ) )
=> ~ ! [Mima: option @ ( product_prod @ nat @ nat ),Deg2: nat,TreeList2: list @ vEBT_VEBT,Summary2: vEBT_VEBT] :
( ( X
= ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary2 ) )
=> ( ( Y
= ( ( Deg2 = Xa2 )
& ! [X5: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X5 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ( vEBT_VEBT_valid @ X5 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
& ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) )
& ( ( size_size @ ( list @ vEBT_VEBT ) @ TreeList2 )
= ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
& ( case_option @ $o @ ( product_prod @ nat @ nat )
@ ( ~ ? [X9: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X9 )
& ! [X5: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X5 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X9: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X9 ) ) )
@ ( 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 ) )
& ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ ( minus_minus @ nat @ Deg2 @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ) )
=> ( ( ? [X9: nat] : ( vEBT_V8194947554948674370ptions @ ( nth @ vEBT_VEBT @ TreeList2 @ I4 ) @ X9 ) )
= ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
& ( ( Mi3 = Ma3 )
=> ! [X5: vEBT_VEBT] :
( ( member @ vEBT_VEBT @ X5 @ ( set2 @ vEBT_VEBT @ TreeList2 ) )
=> ~ ? [X9: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X9 ) ) )
& ( ( Mi3 != Ma3 )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList2 @ Ma3 )
& ! [X5: nat] :
( ( ord_less @ nat @ X5 @ ( power_power @ nat @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) @ Deg2 ) )
=> ( ( vEBT_V5917875025757280293ildren @ ( divide_divide @ nat @ Deg2 @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) @ TreeList2 @ X5 )
=> ( ( ord_less @ nat @ Mi3 @ X5 )
& ( ord_less_eq @ nat @ X5 @ Ma3 ) ) ) ) ) ) ) )
@ Mima ) ) )
=> ~ ( accp @ ( product_prod @ vEBT_VEBT @ nat ) @ vEBT_VEBT_valid_rel @ ( product_Pair @ vEBT_VEBT @ nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList2 @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ).
% VEBT_internal.valid'.pelims(1)
thf(fact_5870_finite__Inf__Sup,axiom,
! [A: $tType] :
( ( finite8700451911770168679attice @ A )
=> ! [A5: set @ ( set @ A )] :
( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image2 @ ( set @ A ) @ A @ ( complete_Sup_Sup @ A ) @ A5 ) )
@ ( complete_Sup_Sup @ A
@ ( image2 @ ( set @ A ) @ A @ ( complete_Inf_Inf @ A )
@ ( collect @ ( set @ A )
@ ^ [Uu3: set @ A] :
? [F4: ( set @ A ) > A] :
( ( Uu3
= ( image2 @ ( set @ A ) @ A @ F4 @ A5 ) )
& ! [X5: set @ A] :
( ( member @ ( set @ A ) @ X5 @ A5 )
=> ( member @ A @ ( F4 @ X5 ) @ X5 ) ) ) ) ) ) ) ) ).
% finite_Inf_Sup
thf(fact_5871_Inf__Sup__le,axiom,
! [A: $tType] :
( ( comple592849572758109894attice @ A )
=> ! [A5: set @ ( set @ A )] :
( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image2 @ ( set @ A ) @ A @ ( complete_Sup_Sup @ A ) @ A5 ) )
@ ( complete_Sup_Sup @ A
@ ( image2 @ ( set @ A ) @ A @ ( complete_Inf_Inf @ A )
@ ( collect @ ( set @ A )
@ ^ [Uu3: set @ A] :
? [F4: ( set @ A ) > A] :
( ( Uu3
= ( image2 @ ( set @ A ) @ A @ F4 @ A5 ) )
& ! [X5: set @ A] :
( ( member @ ( set @ A ) @ X5 @ A5 )
=> ( member @ A @ ( F4 @ X5 ) @ X5 ) ) ) ) ) ) ) ) ).
% Inf_Sup_le
thf(fact_5872_Sup__Inf__le,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ ( set @ A )] :
( ord_less_eq @ A
@ ( complete_Sup_Sup @ A
@ ( image2 @ ( set @ A ) @ A @ ( complete_Inf_Inf @ A )
@ ( collect @ ( set @ A )
@ ^ [Uu3: set @ A] :
? [F4: ( set @ A ) > A] :
( ( Uu3
= ( image2 @ ( set @ A ) @ A @ F4 @ A5 ) )
& ! [X5: set @ A] :
( ( member @ ( set @ A ) @ X5 @ A5 )
=> ( member @ A @ ( F4 @ X5 ) @ X5 ) ) ) ) ) )
@ ( complete_Inf_Inf @ A @ ( image2 @ ( set @ A ) @ A @ ( complete_Sup_Sup @ A ) @ A5 ) ) ) ) ).
% Sup_Inf_le
thf(fact_5873_mono__compose,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ C ) )
=> ! [Q: A > B > C,F3: D > B] :
( ( order_mono @ A @ ( B > C ) @ Q )
=> ( order_mono @ A @ ( D > C )
@ ^ [I4: A,X5: D] : ( Q @ I4 @ ( F3 @ X5 ) ) ) ) ) ).
% mono_compose
thf(fact_5874_Union__maximal__sets,axiom,
! [A: $tType,F15: set @ ( set @ A )] :
( ( finite_finite2 @ ( set @ A ) @ F15 )
=> ( ( complete_Sup_Sup @ ( set @ A )
@ ( collect @ ( set @ A )
@ ^ [T11: set @ A] :
( ( member @ ( set @ A ) @ T11 @ F15 )
& ! [X5: set @ A] :
( ( member @ ( set @ A ) @ X5 @ F15 )
=> ~ ( ord_less @ ( set @ A ) @ T11 @ X5 ) ) ) ) )
= ( complete_Sup_Sup @ ( set @ A ) @ F15 ) ) ) ).
% Union_maximal_sets
thf(fact_5875_Nats__altdef1,axiom,
! [A: $tType] :
( ( ring_1 @ A )
=> ( ( semiring_1_Nats @ A )
= ( collect @ A
@ ^ [Uu3: A] :
? [N2: int] :
( ( Uu3
= ( ring_1_of_int @ A @ N2 ) )
& ( ord_less_eq @ int @ ( zero_zero @ int ) @ N2 ) ) ) ) ) ).
% Nats_altdef1
thf(fact_5876_iteratesp_Omono,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ! [F3: A > A] :
( order_mono @ ( A > $o ) @ ( A > $o )
@ ^ [P6: A > $o,X5: A] :
( ? [Y6: A] :
( ( X5
= ( F3 @ Y6 ) )
& ( P6 @ Y6 ) )
| ? [M9: set @ A] :
( ( X5
= ( complete_Sup_Sup @ A @ M9 ) )
& ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ M9 )
& ! [Y6: A] :
( ( member @ A @ Y6 @ M9 )
=> ( P6 @ Y6 ) ) ) ) ) ) ).
% iteratesp.mono
thf(fact_5877_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_5878_image2__def,axiom,
! [A: $tType,B: $tType,C: $tType] :
( ( bNF_Greatest_image2 @ C @ A @ B )
= ( ^ [A7: set @ C,F4: C > A,G4: C > B] :
( collect @ ( product_prod @ A @ B )
@ ^ [Uu3: product_prod @ A @ B] :
? [A6: C] :
( ( Uu3
= ( product_Pair @ A @ B @ ( F4 @ A6 ) @ ( G4 @ A6 ) ) )
& ( member @ C @ A6 @ A7 ) ) ) ) ) ).
% image2_def
thf(fact_5879_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_5880_chain__empty,axiom,
! [A: $tType,Ord: A > A > $o] : ( comple1602240252501008431_chain @ A @ Ord @ ( bot_bot @ ( set @ A ) ) ) ).
% chain_empty
thf(fact_5881_ccpo__Sup__least,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ! [A5: set @ A,Z: A] :
( ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ A5 )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ A5 )
=> ( ord_less_eq @ A @ X4 @ Z ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A5 ) @ Z ) ) ) ) ).
% ccpo_Sup_least
thf(fact_5882_ccpo__Sup__upper,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ! [A5: set @ A,X: A] :
( ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ A5 )
=> ( ( member @ A @ X @ A5 )
=> ( ord_less_eq @ A @ X @ ( complete_Sup_Sup @ A @ A5 ) ) ) ) ) ).
% ccpo_Sup_upper
thf(fact_5883_chain__singleton,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ! [X: A] : ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% chain_singleton
thf(fact_5884_image2__eqI,axiom,
! [A: $tType,C: $tType,B: $tType,B2: A,F3: B > A,X: B,C3: C,G3: B > C,A5: set @ B] :
( ( B2
= ( F3 @ X ) )
=> ( ( C3
= ( G3 @ X ) )
=> ( ( member @ B @ X @ A5 )
=> ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ B2 @ C3 ) @ ( bNF_Greatest_image2 @ B @ A @ C @ A5 @ F3 @ G3 ) ) ) ) ) ).
% image2_eqI
thf(fact_5885_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_5886_in__chain__finite,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ! [A5: set @ A] :
( ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ A5 )
=> ( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( member @ A @ ( complete_Sup_Sup @ A @ A5 ) @ A5 ) ) ) ) ) ).
% in_chain_finite
thf(fact_5887_Rats__eq__int__div__nat,axiom,
( ( field_char_0_Rats @ real )
= ( collect @ real
@ ^ [Uu3: real] :
? [I4: int,N2: nat] :
( ( Uu3
= ( divide_divide @ real @ ( ring_1_of_int @ real @ I4 ) @ ( semiring_1_of_nat @ real @ N2 ) ) )
& ( N2
!= ( zero_zero @ nat ) ) ) ) ) ).
% Rats_eq_int_div_nat
thf(fact_5888_UN__le__eq__Un0,axiom,
! [A: $tType,M5: nat > ( set @ A ),N: nat] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ M5 @ ( set_ord_atMost @ nat @ N ) ) )
= ( sup_sup @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ M5 @ ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ N ) ) ) @ ( M5 @ ( zero_zero @ nat ) ) ) ) ).
% UN_le_eq_Un0
thf(fact_5889_sup__apply,axiom,
! [B: $tType,A: $tType] :
( ( semilattice_sup @ B )
=> ( ( sup_sup @ ( A > B ) )
= ( ^ [F4: A > B,G4: A > B,X5: A] : ( sup_sup @ B @ ( F4 @ X5 ) @ ( G4 @ X5 ) ) ) ) ) ).
% sup_apply
thf(fact_5890_sup_Oright__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A2: A,B2: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ A2 @ B2 ) @ B2 )
= ( sup_sup @ A @ A2 @ B2 ) ) ) ).
% sup.right_idem
thf(fact_5891_sup__left__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A,Y: A] :
( ( sup_sup @ A @ X @ ( sup_sup @ A @ X @ Y ) )
= ( sup_sup @ A @ X @ Y ) ) ) ).
% sup_left_idem
thf(fact_5892_sup_Oleft__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A2: A,B2: A] :
( ( sup_sup @ A @ A2 @ ( sup_sup @ A @ A2 @ B2 ) )
= ( sup_sup @ A @ A2 @ B2 ) ) ) ).
% sup.left_idem
thf(fact_5893_sup__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A] :
( ( sup_sup @ A @ X @ X )
= X ) ) ).
% sup_idem
thf(fact_5894_sup_Oidem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A2: A] :
( ( sup_sup @ A @ A2 @ A2 )
= A2 ) ) ).
% sup.idem
thf(fact_5895_le__sup__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A,Y: A,Z: A] :
( ( ord_less_eq @ A @ ( sup_sup @ A @ X @ Y ) @ Z )
= ( ( ord_less_eq @ A @ X @ Z )
& ( ord_less_eq @ A @ Y @ Z ) ) ) ) ).
% le_sup_iff
thf(fact_5896_sup_Obounded__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B2: A,C3: A,A2: A] :
( ( ord_less_eq @ A @ ( sup_sup @ A @ B2 @ C3 ) @ A2 )
= ( ( ord_less_eq @ A @ B2 @ A2 )
& ( ord_less_eq @ A @ C3 @ A2 ) ) ) ) ).
% sup.bounded_iff
thf(fact_5897_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_5898_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_5899_sup__bot_Oright__neutral,axiom,
! [A: $tType] :
( ( bounde4967611905675639751up_bot @ A )
=> ! [A2: A] :
( ( sup_sup @ A @ A2 @ ( bot_bot @ A ) )
= A2 ) ) ).
% sup_bot.right_neutral
thf(fact_5900_sup__bot_Oneutr__eq__iff,axiom,
! [A: $tType] :
( ( bounde4967611905675639751up_bot @ A )
=> ! [A2: A,B2: A] :
( ( ( bot_bot @ A )
= ( sup_sup @ A @ A2 @ B2 ) )
= ( ( A2
= ( bot_bot @ A ) )
& ( B2
= ( bot_bot @ A ) ) ) ) ) ).
% sup_bot.neutr_eq_iff
thf(fact_5901_sup__bot_Oleft__neutral,axiom,
! [A: $tType] :
( ( bounde4967611905675639751up_bot @ A )
=> ! [A2: A] :
( ( sup_sup @ A @ ( bot_bot @ A ) @ A2 )
= A2 ) ) ).
% sup_bot.left_neutral
thf(fact_5902_sup__bot_Oeq__neutr__iff,axiom,
! [A: $tType] :
( ( bounde4967611905675639751up_bot @ A )
=> ! [A2: A,B2: A] :
( ( ( sup_sup @ A @ A2 @ B2 )
= ( bot_bot @ A ) )
= ( ( A2
= ( bot_bot @ A ) )
& ( B2
= ( bot_bot @ A ) ) ) ) ) ).
% sup_bot.eq_neutr_iff
thf(fact_5903_sup__eq__bot__iff,axiom,
! [A: $tType] :
( ( bounde4967611905675639751up_bot @ A )
=> ! [X: A,Y: A] :
( ( ( sup_sup @ A @ X @ Y )
= ( bot_bot @ A ) )
= ( ( X
= ( bot_bot @ A ) )
& ( Y
= ( bot_bot @ A ) ) ) ) ) ).
% sup_eq_bot_iff
thf(fact_5904_bot__eq__sup__iff,axiom,
! [A: $tType] :
( ( bounde4967611905675639751up_bot @ A )
=> ! [X: A,Y: A] :
( ( ( bot_bot @ A )
= ( sup_sup @ A @ X @ Y ) )
= ( ( X
= ( bot_bot @ A ) )
& ( Y
= ( bot_bot @ A ) ) ) ) ) ).
% bot_eq_sup_iff
thf(fact_5905_sup__bot__right,axiom,
! [A: $tType] :
( ( bounde4967611905675639751up_bot @ A )
=> ! [X: A] :
( ( sup_sup @ A @ X @ ( bot_bot @ A ) )
= X ) ) ).
% sup_bot_right
thf(fact_5906_sup__bot__left,axiom,
! [A: $tType] :
( ( bounde4967611905675639751up_bot @ A )
=> ! [X: A] :
( ( sup_sup @ A @ ( bot_bot @ A ) @ X )
= X ) ) ).
% sup_bot_left
thf(fact_5907_inf__sup__absorb,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X: A,Y: A] :
( ( inf_inf @ A @ X @ ( sup_sup @ A @ X @ Y ) )
= X ) ) ).
% inf_sup_absorb
thf(fact_5908_sup__inf__absorb,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X: A,Y: A] :
( ( sup_sup @ A @ X @ ( inf_inf @ A @ X @ Y ) )
= X ) ) ).
% sup_inf_absorb
thf(fact_5909_Un__empty,axiom,
! [A: $tType,A5: set @ A,B6: set @ A] :
( ( ( sup_sup @ ( set @ A ) @ A5 @ B6 )
= ( bot_bot @ ( set @ A ) ) )
= ( ( A5
= ( bot_bot @ ( set @ A ) ) )
& ( B6
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Un_empty
thf(fact_5910_finite__Un,axiom,
! [A: $tType,F5: set @ A,G6: set @ A] :
( ( finite_finite2 @ A @ ( sup_sup @ ( set @ A ) @ F5 @ G6 ) )
= ( ( finite_finite2 @ A @ F5 )
& ( finite_finite2 @ A @ G6 ) ) ) ).
% finite_Un
thf(fact_5911_set__union,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( set2 @ A @ ( union @ A @ Xs @ Ys ) )
= ( sup_sup @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ Ys ) ) ) ).
% set_union
thf(fact_5912_UN__simps_I3_J,axiom,
! [E: $tType,F: $tType,C5: set @ F,A5: set @ E,B6: F > ( set @ E )] :
( ( ( C5
= ( bot_bot @ ( set @ F ) ) )
=> ( ( complete_Sup_Sup @ ( set @ E )
@ ( image2 @ F @ ( set @ E )
@ ^ [X5: F] : ( sup_sup @ ( set @ E ) @ A5 @ ( B6 @ X5 ) )
@ C5 ) )
= ( bot_bot @ ( set @ E ) ) ) )
& ( ( C5
!= ( bot_bot @ ( set @ F ) ) )
=> ( ( complete_Sup_Sup @ ( set @ E )
@ ( image2 @ F @ ( set @ E )
@ ^ [X5: F] : ( sup_sup @ ( set @ E ) @ A5 @ ( B6 @ X5 ) )
@ C5 ) )
= ( sup_sup @ ( set @ E ) @ A5 @ ( complete_Sup_Sup @ ( set @ E ) @ ( image2 @ F @ ( set @ E ) @ B6 @ C5 ) ) ) ) ) ) ).
% UN_simps(3)
thf(fact_5913_UN__simps_I2_J,axiom,
! [C: $tType,D: $tType,C5: set @ C,A5: C > ( set @ D ),B6: set @ D] :
( ( ( C5
= ( bot_bot @ ( set @ C ) ) )
=> ( ( complete_Sup_Sup @ ( set @ D )
@ ( image2 @ C @ ( set @ D )
@ ^ [X5: C] : ( sup_sup @ ( set @ D ) @ ( A5 @ X5 ) @ B6 )
@ C5 ) )
= ( bot_bot @ ( set @ D ) ) ) )
& ( ( C5
!= ( bot_bot @ ( set @ C ) ) )
=> ( ( complete_Sup_Sup @ ( set @ D )
@ ( image2 @ C @ ( set @ D )
@ ^ [X5: C] : ( sup_sup @ ( set @ D ) @ ( A5 @ X5 ) @ B6 )
@ C5 ) )
= ( sup_sup @ ( set @ D ) @ ( complete_Sup_Sup @ ( set @ D ) @ ( image2 @ C @ ( set @ D ) @ A5 @ C5 ) ) @ B6 ) ) ) ) ).
% UN_simps(2)
thf(fact_5914_mono__sup,axiom,
! [B: $tType,A: $tType] :
( ( ( semilattice_sup @ A )
& ( semilattice_sup @ B ) )
=> ! [F3: A > B,A5: A,B6: A] :
( ( order_mono @ A @ B @ F3 )
=> ( ord_less_eq @ B @ ( sup_sup @ B @ ( F3 @ A5 ) @ ( F3 @ B6 ) ) @ ( F3 @ ( sup_sup @ A @ A5 @ B6 ) ) ) ) ) ).
% mono_sup
thf(fact_5915_cSUP__union,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A5: set @ B,F3: B > A,B6: set @ B] :
( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit941137186595557371_above @ A @ ( image2 @ B @ A @ F3 @ A5 ) )
=> ( ( B6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit941137186595557371_above @ A @ ( image2 @ B @ A @ F3 @ B6 ) )
=> ( ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ ( sup_sup @ ( set @ B ) @ A5 @ B6 ) ) )
= ( sup_sup @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A5 ) ) @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ B6 ) ) ) ) ) ) ) ) ) ).
% cSUP_union
thf(fact_5916_rtrancl__Un__separator__converseE,axiom,
! [A: $tType,A2: A,B2: A,P2: set @ ( product_prod @ A @ A ),Q: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ ( transitive_rtrancl @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ P2 @ Q ) ) )
=> ( ! [X4: A,Y3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ B2 ) @ ( transitive_rtrancl @ A @ P2 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ X4 ) @ Q )
=> ( Y3 = X4 ) ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ ( transitive_rtrancl @ A @ P2 ) ) ) ) ).
% rtrancl_Un_separator_converseE
thf(fact_5917_rtrancl__Un__separatorE,axiom,
! [A: $tType,A2: A,B2: A,P2: set @ ( product_prod @ A @ A ),Q: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ ( transitive_rtrancl @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ P2 @ Q ) ) )
=> ( ! [X4: A,Y3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ X4 ) @ ( transitive_rtrancl @ A @ P2 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y3 ) @ Q )
=> ( X4 = Y3 ) ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ ( transitive_rtrancl @ A @ P2 ) ) ) ) ).
% rtrancl_Un_separatorE
thf(fact_5918_ivl__disj__un__two__touch_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,M2: A,U: A] :
( ( ord_less_eq @ A @ L @ M2 )
=> ( ( ord_less_eq @ A @ M2 @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ L @ M2 ) @ ( set_or1337092689740270186AtMost @ A @ M2 @ U ) )
= ( set_or1337092689740270186AtMost @ A @ L @ U ) ) ) ) ) ).
% ivl_disj_un_two_touch(4)
thf(fact_5919_sup_OcoboundedI2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [C3: A,B2: A,A2: A] :
( ( ord_less_eq @ A @ C3 @ B2 )
=> ( ord_less_eq @ A @ C3 @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ).
% sup.coboundedI2
thf(fact_5920_sup_OcoboundedI1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [C3: A,A2: A,B2: A] :
( ( ord_less_eq @ A @ C3 @ A2 )
=> ( ord_less_eq @ A @ C3 @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ).
% sup.coboundedI1
thf(fact_5921_sup_Oabsorb__iff2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A6: A,B5: A] :
( ( sup_sup @ A @ A6 @ B5 )
= B5 ) ) ) ) ).
% sup.absorb_iff2
thf(fact_5922_sup_Oabsorb__iff1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B5: A,A6: A] :
( ( sup_sup @ A @ A6 @ B5 )
= A6 ) ) ) ) ).
% sup.absorb_iff1
thf(fact_5923_sup_Ocobounded2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B2: A,A2: A] : ( ord_less_eq @ A @ B2 @ ( sup_sup @ A @ A2 @ B2 ) ) ) ).
% sup.cobounded2
thf(fact_5924_sup_Ocobounded1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A2: A,B2: A] : ( ord_less_eq @ A @ A2 @ ( sup_sup @ A @ A2 @ B2 ) ) ) ).
% sup.cobounded1
thf(fact_5925_sup_Oorder__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B5: A,A6: A] :
( A6
= ( sup_sup @ A @ A6 @ B5 ) ) ) ) ) ).
% sup.order_iff
thf(fact_5926_sup_OboundedI,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B2: A,A2: A,C3: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_less_eq @ A @ C3 @ A2 )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ B2 @ C3 ) @ A2 ) ) ) ) ).
% sup.boundedI
thf(fact_5927_sup_OboundedE,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B2: A,C3: A,A2: A] :
( ( ord_less_eq @ A @ ( sup_sup @ A @ B2 @ C3 ) @ A2 )
=> ~ ( ( ord_less_eq @ A @ B2 @ A2 )
=> ~ ( ord_less_eq @ A @ C3 @ A2 ) ) ) ) ).
% sup.boundedE
thf(fact_5928_sup__absorb2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( sup_sup @ A @ X @ Y )
= Y ) ) ) ).
% sup_absorb2
thf(fact_5929_sup__absorb1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ Y @ X )
=> ( ( sup_sup @ A @ X @ Y )
= X ) ) ) ).
% sup_absorb1
thf(fact_5930_sup_Oabsorb2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( sup_sup @ A @ A2 @ B2 )
= B2 ) ) ) ).
% sup.absorb2
thf(fact_5931_sup_Oabsorb1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B2: A,A2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( sup_sup @ A @ A2 @ B2 )
= A2 ) ) ) ).
% sup.absorb1
thf(fact_5932_sup__unique,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [F3: A > A > A,X: A,Y: A] :
( ! [X4: A,Y3: A] : ( ord_less_eq @ A @ X4 @ ( F3 @ X4 @ Y3 ) )
=> ( ! [X4: A,Y3: A] : ( ord_less_eq @ A @ Y3 @ ( F3 @ X4 @ Y3 ) )
=> ( ! [X4: A,Y3: A,Z3: A] :
( ( ord_less_eq @ A @ Y3 @ X4 )
=> ( ( ord_less_eq @ A @ Z3 @ X4 )
=> ( ord_less_eq @ A @ ( F3 @ Y3 @ Z3 ) @ X4 ) ) )
=> ( ( sup_sup @ A @ X @ Y )
= ( F3 @ X @ Y ) ) ) ) ) ) ).
% sup_unique
thf(fact_5933_sup_OorderI,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A2: A,B2: A] :
( ( A2
= ( sup_sup @ A @ A2 @ B2 ) )
=> ( ord_less_eq @ A @ B2 @ A2 ) ) ) ).
% sup.orderI
thf(fact_5934_sup_OorderE,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B2: A,A2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( A2
= ( sup_sup @ A @ A2 @ B2 ) ) ) ) ).
% sup.orderE
thf(fact_5935_le__iff__sup,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [X5: A,Y6: A] :
( ( sup_sup @ A @ X5 @ Y6 )
= Y6 ) ) ) ) ).
% le_iff_sup
thf(fact_5936_sup__least,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [Y: A,X: A,Z: A] :
( ( ord_less_eq @ A @ Y @ X )
=> ( ( ord_less_eq @ A @ Z @ X )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ Y @ Z ) @ X ) ) ) ) ).
% sup_least
thf(fact_5937_sup__mono,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A2: A,C3: A,B2: A,D3: A] :
( ( ord_less_eq @ A @ A2 @ C3 )
=> ( ( ord_less_eq @ A @ B2 @ D3 )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ A2 @ B2 ) @ ( sup_sup @ A @ C3 @ D3 ) ) ) ) ) ).
% sup_mono
thf(fact_5938_sup_Omono,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [C3: A,A2: A,D3: A,B2: A] :
( ( ord_less_eq @ A @ C3 @ A2 )
=> ( ( ord_less_eq @ A @ D3 @ B2 )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ C3 @ D3 ) @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ) ).
% sup.mono
thf(fact_5939_le__supI2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A,B2: A,A2: A] :
( ( ord_less_eq @ A @ X @ B2 )
=> ( ord_less_eq @ A @ X @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ).
% le_supI2
thf(fact_5940_le__supI1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A,A2: A,B2: A] :
( ( ord_less_eq @ A @ X @ A2 )
=> ( ord_less_eq @ A @ X @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ).
% le_supI1
thf(fact_5941_sup__ge2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [Y: A,X: A] : ( ord_less_eq @ A @ Y @ ( sup_sup @ A @ X @ Y ) ) ) ).
% sup_ge2
thf(fact_5942_sup__ge1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A,Y: A] : ( ord_less_eq @ A @ X @ ( sup_sup @ A @ X @ Y ) ) ) ).
% sup_ge1
thf(fact_5943_le__supI,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A2: A,X: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ X )
=> ( ( ord_less_eq @ A @ B2 @ X )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ A2 @ B2 ) @ X ) ) ) ) ).
% le_supI
thf(fact_5944_le__supE,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A2: A,B2: A,X: A] :
( ( ord_less_eq @ A @ ( sup_sup @ A @ A2 @ B2 ) @ X )
=> ~ ( ( ord_less_eq @ A @ A2 @ X )
=> ~ ( ord_less_eq @ A @ B2 @ X ) ) ) ) ).
% le_supE
thf(fact_5945_inf__sup__ord_I3_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X: A,Y: A] : ( ord_less_eq @ A @ X @ ( sup_sup @ A @ X @ Y ) ) ) ).
% inf_sup_ord(3)
thf(fact_5946_inf__sup__ord_I4_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [Y: A,X: A] : ( ord_less_eq @ A @ Y @ ( sup_sup @ A @ X @ Y ) ) ) ).
% inf_sup_ord(4)
thf(fact_5947_distrib__inf__le,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X: A,Y: A,Z: A] : ( ord_less_eq @ A @ ( sup_sup @ A @ ( inf_inf @ A @ X @ Y ) @ ( inf_inf @ A @ X @ Z ) ) @ ( inf_inf @ A @ X @ ( sup_sup @ A @ Y @ Z ) ) ) ) ).
% distrib_inf_le
thf(fact_5948_distrib__sup__le,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X: A,Y: A,Z: A] : ( ord_less_eq @ A @ ( sup_sup @ A @ X @ ( inf_inf @ A @ Y @ Z ) ) @ ( inf_inf @ A @ ( sup_sup @ A @ X @ Y ) @ ( sup_sup @ A @ X @ Z ) ) ) ) ).
% distrib_sup_le
thf(fact_5949_distrib__imp1,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X: A,Y: A,Z: A] :
( ! [X4: A,Y3: A,Z3: A] :
( ( inf_inf @ A @ X4 @ ( sup_sup @ A @ Y3 @ Z3 ) )
= ( sup_sup @ A @ ( inf_inf @ A @ X4 @ Y3 ) @ ( inf_inf @ A @ X4 @ Z3 ) ) )
=> ( ( sup_sup @ A @ X @ ( inf_inf @ A @ Y @ Z ) )
= ( inf_inf @ A @ ( sup_sup @ A @ X @ Y ) @ ( sup_sup @ A @ X @ Z ) ) ) ) ) ).
% distrib_imp1
thf(fact_5950_distrib__imp2,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X: A,Y: A,Z: A] :
( ! [X4: A,Y3: A,Z3: A] :
( ( sup_sup @ A @ X4 @ ( inf_inf @ A @ Y3 @ Z3 ) )
= ( inf_inf @ A @ ( sup_sup @ A @ X4 @ Y3 ) @ ( sup_sup @ A @ X4 @ Z3 ) ) )
=> ( ( inf_inf @ A @ X @ ( sup_sup @ A @ Y @ Z ) )
= ( sup_sup @ A @ ( inf_inf @ A @ X @ Y ) @ ( inf_inf @ A @ X @ Z ) ) ) ) ) ).
% distrib_imp2
thf(fact_5951_inf__sup__distrib1,axiom,
! [A: $tType] :
( ( distrib_lattice @ A )
=> ! [X: A,Y: A,Z: A] :
( ( inf_inf @ A @ X @ ( sup_sup @ A @ Y @ Z ) )
= ( sup_sup @ A @ ( inf_inf @ A @ X @ Y ) @ ( inf_inf @ A @ X @ Z ) ) ) ) ).
% inf_sup_distrib1
thf(fact_5952_inf__sup__distrib2,axiom,
! [A: $tType] :
( ( distrib_lattice @ A )
=> ! [Y: A,Z: A,X: A] :
( ( inf_inf @ A @ ( sup_sup @ A @ Y @ Z ) @ X )
= ( sup_sup @ A @ ( inf_inf @ A @ Y @ X ) @ ( inf_inf @ A @ Z @ X ) ) ) ) ).
% inf_sup_distrib2
thf(fact_5953_sup__inf__distrib1,axiom,
! [A: $tType] :
( ( distrib_lattice @ A )
=> ! [X: A,Y: A,Z: A] :
( ( sup_sup @ A @ X @ ( inf_inf @ A @ Y @ Z ) )
= ( inf_inf @ A @ ( sup_sup @ A @ X @ Y ) @ ( sup_sup @ A @ X @ Z ) ) ) ) ).
% sup_inf_distrib1
thf(fact_5954_sup__inf__distrib2,axiom,
! [A: $tType] :
( ( distrib_lattice @ A )
=> ! [Y: A,Z: A,X: A] :
( ( sup_sup @ A @ ( inf_inf @ A @ Y @ Z ) @ X )
= ( inf_inf @ A @ ( sup_sup @ A @ Y @ X ) @ ( sup_sup @ A @ Z @ X ) ) ) ) ).
% sup_inf_distrib2
thf(fact_5955_sup_Ostrict__coboundedI2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [C3: A,B2: A,A2: A] :
( ( ord_less @ A @ C3 @ B2 )
=> ( ord_less @ A @ C3 @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ).
% sup.strict_coboundedI2
thf(fact_5956_sup_Ostrict__coboundedI1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [C3: A,A2: A,B2: A] :
( ( ord_less @ A @ C3 @ A2 )
=> ( ord_less @ A @ C3 @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ).
% sup.strict_coboundedI1
thf(fact_5957_sup_Ostrict__order__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( ord_less @ A )
= ( ^ [B5: A,A6: A] :
( ( A6
= ( sup_sup @ A @ A6 @ B5 ) )
& ( A6 != B5 ) ) ) ) ) ).
% sup.strict_order_iff
thf(fact_5958_sup_Ostrict__boundedE,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B2: A,C3: A,A2: A] :
( ( ord_less @ A @ ( sup_sup @ A @ B2 @ C3 ) @ A2 )
=> ~ ( ( ord_less @ A @ B2 @ A2 )
=> ~ ( ord_less @ A @ C3 @ A2 ) ) ) ) ).
% sup.strict_boundedE
thf(fact_5959_sup_Oabsorb4,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( sup_sup @ A @ A2 @ B2 )
= B2 ) ) ) ).
% sup.absorb4
thf(fact_5960_sup_Oabsorb3,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B2: A,A2: A] :
( ( ord_less @ A @ B2 @ A2 )
=> ( ( sup_sup @ A @ A2 @ B2 )
= A2 ) ) ) ).
% sup.absorb3
thf(fact_5961_less__supI2,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A,B2: A,A2: A] :
( ( ord_less @ A @ X @ B2 )
=> ( ord_less @ A @ X @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ).
% less_supI2
thf(fact_5962_less__supI1,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A,A2: A,B2: A] :
( ( ord_less @ A @ X @ A2 )
=> ( ord_less @ A @ X @ ( sup_sup @ A @ A2 @ B2 ) ) ) ) ).
% less_supI1
thf(fact_5963_Rats__dense__in__real,axiom,
! [X: real,Y: real] :
( ( ord_less @ real @ X @ Y )
=> ? [X4: real] :
( ( member @ real @ X4 @ ( field_char_0_Rats @ real ) )
& ( ord_less @ real @ X @ X4 )
& ( ord_less @ real @ X4 @ Y ) ) ) ).
% Rats_dense_in_real
thf(fact_5964_Rats__no__bot__less,axiom,
! [X: real] :
? [X4: real] :
( ( member @ real @ X4 @ ( field_char_0_Rats @ real ) )
& ( ord_less @ real @ X4 @ X ) ) ).
% Rats_no_bot_less
thf(fact_5965_complete__linorder__sup__max,axiom,
! [A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ( ( sup_sup @ A )
= ( ord_max @ A ) ) ) ).
% complete_linorder_sup_max
thf(fact_5966_Rats__0,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ( member @ A @ ( zero_zero @ A ) @ ( field_char_0_Rats @ A ) ) ) ).
% Rats_0
thf(fact_5967_sup__fun__def,axiom,
! [B: $tType,A: $tType] :
( ( semilattice_sup @ B )
=> ( ( sup_sup @ ( A > B ) )
= ( ^ [F4: A > B,G4: A > B,X5: A] : ( sup_sup @ B @ ( F4 @ X5 ) @ ( G4 @ X5 ) ) ) ) ) ).
% sup_fun_def
thf(fact_5968_sup__left__commute,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A,Y: A,Z: A] :
( ( sup_sup @ A @ X @ ( sup_sup @ A @ Y @ Z ) )
= ( sup_sup @ A @ Y @ ( sup_sup @ A @ X @ Z ) ) ) ) ).
% sup_left_commute
thf(fact_5969_sup_Oleft__commute,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [B2: A,A2: A,C3: A] :
( ( sup_sup @ A @ B2 @ ( sup_sup @ A @ A2 @ C3 ) )
= ( sup_sup @ A @ A2 @ ( sup_sup @ A @ B2 @ C3 ) ) ) ) ).
% sup.left_commute
thf(fact_5970_sup__commute,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( sup_sup @ A )
= ( ^ [X5: A,Y6: A] : ( sup_sup @ A @ Y6 @ X5 ) ) ) ) ).
% sup_commute
thf(fact_5971_sup_Ocommute,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( sup_sup @ A )
= ( ^ [A6: A,B5: A] : ( sup_sup @ A @ B5 @ A6 ) ) ) ) ).
% sup.commute
thf(fact_5972_sup__assoc,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A,Y: A,Z: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ X @ Y ) @ Z )
= ( sup_sup @ A @ X @ ( sup_sup @ A @ Y @ Z ) ) ) ) ).
% sup_assoc
thf(fact_5973_sup_Oassoc,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ A2 @ B2 ) @ C3 )
= ( sup_sup @ A @ A2 @ ( sup_sup @ A @ B2 @ C3 ) ) ) ) ).
% sup.assoc
thf(fact_5974_inf__sup__aci_I5_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ( ( sup_sup @ A )
= ( ^ [X5: A,Y6: A] : ( sup_sup @ A @ Y6 @ X5 ) ) ) ) ).
% inf_sup_aci(5)
thf(fact_5975_inf__sup__aci_I6_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X: A,Y: A,Z: A] :
( ( sup_sup @ A @ ( sup_sup @ A @ X @ Y ) @ Z )
= ( sup_sup @ A @ X @ ( sup_sup @ A @ Y @ Z ) ) ) ) ).
% inf_sup_aci(6)
thf(fact_5976_inf__sup__aci_I7_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X: A,Y: A,Z: A] :
( ( sup_sup @ A @ X @ ( sup_sup @ A @ Y @ Z ) )
= ( sup_sup @ A @ Y @ ( sup_sup @ A @ X @ Z ) ) ) ) ).
% inf_sup_aci(7)
thf(fact_5977_inf__sup__aci_I8_J,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [X: A,Y: A] :
( ( sup_sup @ A @ X @ ( sup_sup @ A @ X @ Y ) )
= ( sup_sup @ A @ X @ Y ) ) ) ).
% inf_sup_aci(8)
thf(fact_5978_sup__max,axiom,
! [A: $tType] :
( ( ( semilattice_sup @ A )
& ( linorder @ A ) )
=> ( ( sup_sup @ A )
= ( ord_max @ A ) ) ) ).
% sup_max
thf(fact_5979_Rats__add,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ! [A2: A,B2: A] :
( ( member @ A @ A2 @ ( field_char_0_Rats @ A ) )
=> ( ( member @ A @ B2 @ ( field_char_0_Rats @ A ) )
=> ( member @ A @ ( plus_plus @ A @ A2 @ B2 ) @ ( field_char_0_Rats @ A ) ) ) ) ) ).
% Rats_add
thf(fact_5980_infinite__Un,axiom,
! [A: $tType,S2: set @ A,T4: set @ A] :
( ( ~ ( finite_finite2 @ A @ ( sup_sup @ ( set @ A ) @ S2 @ T4 ) ) )
= ( ~ ( finite_finite2 @ A @ S2 )
| ~ ( finite_finite2 @ A @ T4 ) ) ) ).
% infinite_Un
thf(fact_5981_Un__infinite,axiom,
! [A: $tType,S2: set @ A,T4: set @ A] :
( ~ ( finite_finite2 @ A @ S2 )
=> ~ ( finite_finite2 @ A @ ( sup_sup @ ( set @ A ) @ S2 @ T4 ) ) ) ).
% Un_infinite
thf(fact_5982_finite__UnI,axiom,
! [A: $tType,F5: set @ A,G6: set @ A] :
( ( finite_finite2 @ A @ F5 )
=> ( ( finite_finite2 @ A @ G6 )
=> ( finite_finite2 @ A @ ( sup_sup @ ( set @ A ) @ F5 @ G6 ) ) ) ) ).
% finite_UnI
thf(fact_5983_Rats__infinite,axiom,
! [A: $tType] :
( ( field_char_0 @ A )
=> ~ ( finite_finite2 @ A @ ( field_char_0_Rats @ A ) ) ) ).
% Rats_infinite
thf(fact_5984_Un__empty__left,axiom,
! [A: $tType,B6: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ B6 )
= B6 ) ).
% Un_empty_left
thf(fact_5985_Un__empty__right,axiom,
! [A: $tType,A5: set @ A] :
( ( sup_sup @ ( set @ A ) @ A5 @ ( bot_bot @ ( set @ A ) ) )
= A5 ) ).
% Un_empty_right
thf(fact_5986_boolean__algebra_Odisj__zero__right,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A] :
( ( sup_sup @ A @ X @ ( bot_bot @ A ) )
= X ) ) ).
% boolean_algebra.disj_zero_right
thf(fact_5987_cSup__union__distrib,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A5: set @ A,B6: set @ A] :
( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit941137186595557371_above @ A @ A5 )
=> ( ( B6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit941137186595557371_above @ A @ B6 )
=> ( ( complete_Sup_Sup @ A @ ( sup_sup @ ( set @ A ) @ A5 @ B6 ) )
= ( sup_sup @ A @ ( complete_Sup_Sup @ A @ A5 ) @ ( complete_Sup_Sup @ A @ B6 ) ) ) ) ) ) ) ) ).
% cSup_union_distrib
thf(fact_5988_ivl__disj__un__two_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,M2: A,U: A] :
( ( ord_less_eq @ A @ L @ M2 )
=> ( ( ord_less_eq @ A @ M2 @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ L @ M2 ) @ ( set_or7035219750837199246ssThan @ A @ M2 @ U ) )
= ( set_or7035219750837199246ssThan @ A @ L @ U ) ) ) ) ) ).
% ivl_disj_un_two(3)
thf(fact_5989_singleton__Un__iff,axiom,
! [A: $tType,X: A,A5: set @ A,B6: set @ A] :
( ( ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) )
= ( sup_sup @ ( set @ A ) @ A5 @ B6 ) )
= ( ( ( A5
= ( bot_bot @ ( set @ A ) ) )
& ( B6
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
| ( ( A5
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
& ( B6
= ( bot_bot @ ( set @ A ) ) ) )
| ( ( A5
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
& ( B6
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% singleton_Un_iff
thf(fact_5990_Un__singleton__iff,axiom,
! [A: $tType,A5: set @ A,B6: set @ A,X: A] :
( ( ( sup_sup @ ( set @ A ) @ A5 @ B6 )
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( ( ( A5
= ( bot_bot @ ( set @ A ) ) )
& ( B6
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
| ( ( A5
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
& ( B6
= ( bot_bot @ ( set @ A ) ) ) )
| ( ( A5
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
& ( B6
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% Un_singleton_iff
thf(fact_5991_insert__is__Un,axiom,
! [A: $tType] :
( ( insert @ A )
= ( ^ [A6: A] : ( sup_sup @ ( set @ A ) @ ( insert @ A @ A6 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% insert_is_Un
thf(fact_5992_ivl__disj__un__two_I6_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,M2: A,U: A] :
( ( ord_less_eq @ A @ L @ M2 )
=> ( ( ord_less_eq @ A @ M2 @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ L @ M2 ) @ ( set_or3652927894154168847AtMost @ A @ M2 @ U ) )
= ( set_or3652927894154168847AtMost @ A @ L @ U ) ) ) ) ) ).
% ivl_disj_un_two(6)
thf(fact_5993_set__shuffles,axiom,
! [A: $tType,Zs: list @ A,Xs: list @ A,Ys: list @ A] :
( ( member @ ( list @ A ) @ Zs @ ( shuffles @ A @ Xs @ Ys ) )
=> ( ( set2 @ A @ Zs )
= ( sup_sup @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ Ys ) ) ) ) ).
% set_shuffles
thf(fact_5994_sup__shunt,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y: A] :
( ( ( sup_sup @ A @ X @ Y )
= ( top_top @ A ) )
= ( ord_less_eq @ A @ ( uminus_uminus @ A @ X ) @ Y ) ) ) ).
% sup_shunt
thf(fact_5995_shunt1,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y: A,Z: A] :
( ( ord_less_eq @ A @ ( inf_inf @ A @ X @ Y ) @ Z )
= ( ord_less_eq @ A @ X @ ( sup_sup @ A @ ( uminus_uminus @ A @ Y ) @ Z ) ) ) ) ).
% shunt1
thf(fact_5996_shunt2,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y: A,Z: A] :
( ( ord_less_eq @ A @ ( inf_inf @ A @ X @ ( uminus_uminus @ A @ Y ) ) @ Z )
= ( ord_less_eq @ A @ X @ ( sup_sup @ A @ Y @ Z ) ) ) ) ).
% shunt2
thf(fact_5997_sup__neg__inf,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [P: A,Q2: A,R2: A] :
( ( ord_less_eq @ A @ P @ ( sup_sup @ A @ Q2 @ R2 ) )
= ( ord_less_eq @ A @ ( inf_inf @ A @ P @ ( uminus_uminus @ A @ Q2 ) ) @ R2 ) ) ) ).
% sup_neg_inf
thf(fact_5998_boolean__algebra_Ocomplement__unique,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [A2: A,X: A,Y: A] :
( ( ( inf_inf @ A @ A2 @ X )
= ( bot_bot @ A ) )
=> ( ( ( sup_sup @ A @ A2 @ X )
= ( top_top @ A ) )
=> ( ( ( inf_inf @ A @ A2 @ Y )
= ( bot_bot @ A ) )
=> ( ( ( sup_sup @ A @ A2 @ Y )
= ( top_top @ A ) )
=> ( X = Y ) ) ) ) ) ) ).
% boolean_algebra.complement_unique
thf(fact_5999_finite__Sup__in,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X4: A,Y3: A] :
( ( member @ A @ X4 @ A5 )
=> ( ( member @ A @ Y3 @ A5 )
=> ( member @ A @ ( sup_sup @ A @ X4 @ Y3 ) @ A5 ) ) )
=> ( member @ A @ ( complete_Sup_Sup @ A @ A5 ) @ A5 ) ) ) ) ) ).
% finite_Sup_in
thf(fact_6000_less__eq__Inf__inter,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ A,B6: set @ A] : ( ord_less_eq @ A @ ( sup_sup @ A @ ( complete_Inf_Inf @ A @ A5 ) @ ( complete_Inf_Inf @ A @ B6 ) ) @ ( complete_Inf_Inf @ A @ ( inf_inf @ ( set @ A ) @ A5 @ B6 ) ) ) ) ).
% less_eq_Inf_inter
thf(fact_6001_ivl__disj__un__two_I7_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,M2: A,U: A] :
( ( ord_less_eq @ A @ L @ M2 )
=> ( ( ord_less_eq @ A @ M2 @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ L @ M2 ) @ ( set_or1337092689740270186AtMost @ A @ M2 @ U ) )
= ( set_or1337092689740270186AtMost @ A @ L @ U ) ) ) ) ) ).
% ivl_disj_un_two(7)
thf(fact_6002_ivl__disj__un__one_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,U: A] :
( ( ord_less_eq @ A @ L @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_ord_lessThan @ A @ L ) @ ( set_or7035219750837199246ssThan @ A @ L @ U ) )
= ( set_ord_lessThan @ A @ U ) ) ) ) ).
% ivl_disj_un_one(2)
thf(fact_6003_Union__image__empty,axiom,
! [B: $tType,A: $tType,A5: set @ A,F3: B > ( set @ A )] :
( ( sup_sup @ ( set @ A ) @ A5 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ F3 @ ( bot_bot @ ( set @ B ) ) ) ) )
= A5 ) ).
% Union_image_empty
thf(fact_6004_card__Un__le,axiom,
! [A: $tType,A5: set @ A,B6: set @ A] : ( ord_less_eq @ nat @ ( finite_card @ A @ ( sup_sup @ ( set @ A ) @ A5 @ B6 ) ) @ ( plus_plus @ nat @ ( finite_card @ A @ A5 ) @ ( finite_card @ A @ B6 ) ) ) ).
% card_Un_le
thf(fact_6005_atLeastLessThan__add__Un,axiom,
! [I2: nat,J: nat,K2: nat] :
( ( ord_less_eq @ nat @ I2 @ J )
=> ( ( set_or7035219750837199246ssThan @ nat @ I2 @ ( plus_plus @ nat @ J @ K2 ) )
= ( sup_sup @ ( set @ nat ) @ ( set_or7035219750837199246ssThan @ nat @ I2 @ J ) @ ( set_or7035219750837199246ssThan @ nat @ J @ ( plus_plus @ nat @ J @ K2 ) ) ) ) ) ).
% atLeastLessThan_add_Un
thf(fact_6006_ivl__disj__un__two_I8_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,M2: A,U: A] :
( ( ord_less_eq @ A @ L @ M2 )
=> ( ( ord_less_eq @ A @ M2 @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ L @ M2 ) @ ( set_or3652927894154168847AtMost @ A @ M2 @ U ) )
= ( set_or1337092689740270186AtMost @ A @ L @ U ) ) ) ) ) ).
% ivl_disj_un_two(8)
thf(fact_6007_ivl__disj__un__one_I8_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,U: A] :
( ( ord_less_eq @ A @ L @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ L @ U ) @ ( set_ord_atLeast @ A @ U ) )
= ( set_ord_atLeast @ A @ L ) ) ) ) ).
% ivl_disj_un_one(8)
thf(fact_6008_ivl__disj__un__one_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,U: A] :
( ( ord_less_eq @ A @ L @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_ord_atMost @ A @ L ) @ ( set_or3652927894154168847AtMost @ A @ L @ U ) )
= ( set_ord_atMost @ A @ U ) ) ) ) ).
% ivl_disj_un_one(3)
thf(fact_6009_ivl__disj__un__one_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,U: A] :
( ( ord_less_eq @ A @ L @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ L @ U ) @ ( set_ord_greaterThan @ A @ U ) )
= ( set_ord_greaterThan @ A @ L ) ) ) ) ).
% ivl_disj_un_one(5)
thf(fact_6010_UN__extend__simps_I3_J,axiom,
! [E: $tType,F: $tType,C5: set @ F,A5: set @ E,B6: F > ( set @ E )] :
( ( ( C5
= ( bot_bot @ ( set @ F ) ) )
=> ( ( sup_sup @ ( set @ E ) @ A5 @ ( complete_Sup_Sup @ ( set @ E ) @ ( image2 @ F @ ( set @ E ) @ B6 @ C5 ) ) )
= A5 ) )
& ( ( C5
!= ( bot_bot @ ( set @ F ) ) )
=> ( ( sup_sup @ ( set @ E ) @ A5 @ ( complete_Sup_Sup @ ( set @ E ) @ ( image2 @ F @ ( set @ E ) @ B6 @ C5 ) ) )
= ( complete_Sup_Sup @ ( set @ E )
@ ( image2 @ F @ ( set @ E )
@ ^ [X5: F] : ( sup_sup @ ( set @ E ) @ A5 @ ( B6 @ X5 ) )
@ C5 ) ) ) ) ) ).
% UN_extend_simps(3)
thf(fact_6011_UN__extend__simps_I2_J,axiom,
! [D: $tType,C: $tType,C5: set @ C,A5: C > ( set @ D ),B6: set @ D] :
( ( ( C5
= ( bot_bot @ ( set @ C ) ) )
=> ( ( sup_sup @ ( set @ D ) @ ( complete_Sup_Sup @ ( set @ D ) @ ( image2 @ C @ ( set @ D ) @ A5 @ C5 ) ) @ B6 )
= B6 ) )
& ( ( C5
!= ( bot_bot @ ( set @ C ) ) )
=> ( ( sup_sup @ ( set @ D ) @ ( complete_Sup_Sup @ ( set @ D ) @ ( image2 @ C @ ( set @ D ) @ A5 @ C5 ) ) @ B6 )
= ( complete_Sup_Sup @ ( set @ D )
@ ( image2 @ C @ ( set @ D )
@ ^ [X5: C] : ( sup_sup @ ( set @ D ) @ ( A5 @ X5 ) @ B6 )
@ C5 ) ) ) ) ) ).
% UN_extend_simps(2)
thf(fact_6012_boolean__algebra__class_Oboolean__algebra_Ocompl__unique,axiom,
! [A: $tType] :
( ( boolea8198339166811842893lgebra @ A )
=> ! [X: A,Y: A] :
( ( ( inf_inf @ A @ X @ Y )
= ( bot_bot @ A ) )
=> ( ( ( sup_sup @ A @ X @ Y )
= ( top_top @ A ) )
=> ( ( uminus_uminus @ A @ X )
= Y ) ) ) ) ).
% boolean_algebra_class.boolean_algebra.compl_unique
thf(fact_6013_cSup__insert__If,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X7: set @ A,A2: A] :
( ( condit941137186595557371_above @ A @ X7 )
=> ( ( ( X7
= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Sup_Sup @ A @ ( insert @ A @ A2 @ X7 ) )
= A2 ) )
& ( ( X7
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Sup_Sup @ A @ ( insert @ A @ A2 @ X7 ) )
= ( sup_sup @ A @ A2 @ ( complete_Sup_Sup @ A @ X7 ) ) ) ) ) ) ) ).
% cSup_insert_If
thf(fact_6014_cSup__insert,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X7: set @ A,A2: A] :
( ( X7
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit941137186595557371_above @ A @ X7 )
=> ( ( complete_Sup_Sup @ A @ ( insert @ A @ A2 @ X7 ) )
= ( sup_sup @ A @ A2 @ ( complete_Sup_Sup @ A @ X7 ) ) ) ) ) ) ).
% cSup_insert
thf(fact_6015_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_6016_sum_Ounion__inter,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A5: set @ B,B6: set @ B,G3: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( finite_finite2 @ B @ B6 )
=> ( ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G3 @ ( sup_sup @ ( set @ B ) @ A5 @ B6 ) ) @ ( groups7311177749621191930dd_sum @ B @ A @ G3 @ ( inf_inf @ ( set @ B ) @ A5 @ B6 ) ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G3 @ A5 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G3 @ B6 ) ) ) ) ) ) ).
% sum.union_inter
thf(fact_6017_prod_Ounion__inter,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A5: set @ B,B6: set @ B,G3: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( finite_finite2 @ B @ B6 )
=> ( ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G3 @ ( sup_sup @ ( set @ B ) @ A5 @ B6 ) ) @ ( groups7121269368397514597t_prod @ B @ A @ G3 @ ( inf_inf @ ( set @ B ) @ A5 @ B6 ) ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G3 @ A5 ) @ ( groups7121269368397514597t_prod @ B @ A @ G3 @ B6 ) ) ) ) ) ) ).
% prod.union_inter
thf(fact_6018_card__Un__Int,axiom,
! [A: $tType,A5: set @ A,B6: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( finite_finite2 @ A @ B6 )
=> ( ( plus_plus @ nat @ ( finite_card @ A @ A5 ) @ ( finite_card @ A @ B6 ) )
= ( plus_plus @ nat @ ( finite_card @ A @ ( sup_sup @ ( set @ A ) @ A5 @ B6 ) ) @ ( finite_card @ A @ ( inf_inf @ ( set @ A ) @ A5 @ B6 ) ) ) ) ) ) ).
% card_Un_Int
thf(fact_6019_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_6020_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_6021_ivl__disj__un__one_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,U: A] :
( ( ord_less_eq @ A @ L @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_ord_lessThan @ A @ L ) @ ( set_or1337092689740270186AtMost @ A @ L @ U ) )
= ( set_ord_atMost @ A @ U ) ) ) ) ).
% ivl_disj_un_one(4)
thf(fact_6022_ivl__disj__un__singleton_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [U: A] :
( ( sup_sup @ ( set @ A ) @ ( set_ord_lessThan @ A @ U ) @ ( insert @ A @ U @ ( bot_bot @ ( set @ A ) ) ) )
= ( set_ord_atMost @ A @ U ) ) ) ).
% ivl_disj_un_singleton(2)
thf(fact_6023_cInf__union__distrib,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A5: set @ A,B6: set @ A] :
( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit1013018076250108175_below @ A @ A5 )
=> ( ( B6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( condit1013018076250108175_below @ A @ B6 )
=> ( ( complete_Inf_Inf @ A @ ( sup_sup @ ( set @ A ) @ A5 @ B6 ) )
= ( inf_inf @ A @ ( complete_Inf_Inf @ A @ A5 ) @ ( complete_Inf_Inf @ A @ B6 ) ) ) ) ) ) ) ) ).
% cInf_union_distrib
thf(fact_6024_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_6025_Max_Ounion,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,B6: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B6 )
=> ( ( B6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798349783984er_Max @ A @ ( sup_sup @ ( set @ A ) @ A5 @ B6 ) )
= ( ord_max @ A @ ( lattic643756798349783984er_Max @ A @ A5 ) @ ( lattic643756798349783984er_Max @ A @ B6 ) ) ) ) ) ) ) ) ).
% Max.union
thf(fact_6026_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_6027_ivl__disj__un__one_I7_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,U: A] :
( ( ord_less_eq @ A @ L @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ L @ U ) @ ( set_ord_greaterThan @ A @ U ) )
= ( set_ord_atLeast @ A @ L ) ) ) ) ).
% ivl_disj_un_one(7)
thf(fact_6028_ivl__disj__un__singleton_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A] :
( ( sup_sup @ ( set @ A ) @ ( insert @ A @ L @ ( bot_bot @ ( set @ A ) ) ) @ ( set_ord_greaterThan @ A @ L ) )
= ( set_ord_atLeast @ A @ L ) ) ) ).
% ivl_disj_un_singleton(1)
thf(fact_6029_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_6030_SUP__nat__binary,axiom,
! [A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A5: A,B6: A] :
( ( sup_sup @ A @ A5
@ ( complete_Sup_Sup @ A
@ ( image2 @ nat @ A
@ ^ [X5: nat] : B6
@ ( collect @ nat @ ( ord_less @ nat @ ( zero_zero @ nat ) ) ) ) ) )
= ( sup_sup @ A @ A5 @ B6 ) ) ) ).
% SUP_nat_binary
thf(fact_6031_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_6032_conditionally__complete__lattice__class_OSUP__sup__distrib,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A5: set @ B,F3: B > A,G3: B > A] :
( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit941137186595557371_above @ A @ ( image2 @ B @ A @ F3 @ A5 ) )
=> ( ( condit941137186595557371_above @ A @ ( image2 @ B @ A @ G3 @ A5 ) )
=> ( ( sup_sup @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A5 ) ) @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ G3 @ A5 ) ) )
= ( complete_Sup_Sup @ A
@ ( image2 @ B @ A
@ ^ [A6: B] : ( sup_sup @ A @ ( F3 @ A6 ) @ ( G3 @ A6 ) )
@ A5 ) ) ) ) ) ) ) ).
% conditionally_complete_lattice_class.SUP_sup_distrib
thf(fact_6033_inj__on__disjoint__Un,axiom,
! [B: $tType,A: $tType,F3: A > B,A5: set @ A,G3: A > B,B6: set @ A] :
( ( inj_on @ A @ B @ F3 @ A5 )
=> ( ( inj_on @ A @ B @ G3 @ B6 )
=> ( ( ( inf_inf @ ( set @ B ) @ ( image2 @ A @ B @ F3 @ A5 ) @ ( image2 @ A @ B @ G3 @ B6 ) )
= ( bot_bot @ ( set @ B ) ) )
=> ( inj_on @ A @ B
@ ^ [X5: A] : ( if @ B @ ( member @ A @ X5 @ A5 ) @ ( F3 @ X5 ) @ ( G3 @ X5 ) )
@ ( sup_sup @ ( set @ A ) @ A5 @ B6 ) ) ) ) ) ).
% inj_on_disjoint_Un
thf(fact_6034_sup__bot_Osemilattice__neutr__order__axioms,axiom,
! [A: $tType] :
( ( bounde4967611905675639751up_bot @ A )
=> ( semila1105856199041335345_order @ A @ ( sup_sup @ A ) @ ( bot_bot @ A )
@ ^ [X5: A,Y6: A] : ( ord_less_eq @ A @ Y6 @ X5 )
@ ^ [X5: A,Y6: A] : ( ord_less @ A @ Y6 @ X5 ) ) ) ).
% sup_bot.semilattice_neutr_order_axioms
thf(fact_6035_rtrancl__insert,axiom,
! [A: $tType,A2: A,B2: A,R2: set @ ( product_prod @ A @ A )] :
( ( transitive_rtrancl @ A @ ( insert @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ R2 ) )
= ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ ( transitive_rtrancl @ A @ R2 )
@ ( collect @ ( product_prod @ A @ A )
@ ( product_case_prod @ A @ A @ $o
@ ^ [X5: A,Y6: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X5 @ A2 ) @ ( transitive_rtrancl @ A @ R2 ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ Y6 ) @ ( transitive_rtrancl @ A @ R2 ) ) ) ) ) ) ) ).
% rtrancl_insert
thf(fact_6036_trancl__insert2,axiom,
! [A: $tType,A2: A,B2: A,R2: set @ ( product_prod @ A @ A )] :
( ( transitive_trancl @ A @ ( insert @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ R2 ) )
= ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ ( transitive_trancl @ A @ R2 )
@ ( collect @ ( product_prod @ A @ A )
@ ( product_case_prod @ A @ A @ $o
@ ^ [X5: A,Y6: A] :
( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X5 @ A2 ) @ ( transitive_trancl @ A @ R2 ) )
| ( X5 = A2 ) )
& ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ Y6 ) @ ( transitive_trancl @ A @ R2 ) )
| ( Y6 = B2 ) ) ) ) ) ) ) ).
% trancl_insert2
thf(fact_6037_cSup__inter__less__eq,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A5: set @ A,B6: set @ A] :
( ( condit941137186595557371_above @ A @ A5 )
=> ( ( condit941137186595557371_above @ A @ B6 )
=> ( ( ( inf_inf @ ( set @ A ) @ A5 @ B6 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( inf_inf @ ( set @ A ) @ A5 @ B6 ) ) @ ( sup_sup @ A @ ( complete_Sup_Sup @ A @ A5 ) @ ( complete_Sup_Sup @ A @ B6 ) ) ) ) ) ) ) ).
% cSup_inter_less_eq
thf(fact_6038_cSUP__insert,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A5: set @ B,F3: B > A,A2: B] :
( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit941137186595557371_above @ A @ ( image2 @ B @ A @ F3 @ A5 ) )
=> ( ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ ( insert @ B @ A2 @ A5 ) ) )
= ( sup_sup @ A @ ( F3 @ A2 ) @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A5 ) ) ) ) ) ) ) ).
% cSUP_insert
thf(fact_6039_sum_Ounion__inter__neutral,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A5: set @ B,B6: set @ B,G3: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( finite_finite2 @ B @ B6 )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ ( inf_inf @ ( set @ B ) @ A5 @ B6 ) )
=> ( ( G3 @ X4 )
= ( zero_zero @ A ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G3 @ ( sup_sup @ ( set @ B ) @ A5 @ B6 ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G3 @ A5 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G3 @ B6 ) ) ) ) ) ) ) ).
% sum.union_inter_neutral
thf(fact_6040_sum__Un,axiom,
! [A: $tType,B: $tType] :
( ( ab_group_add @ A )
=> ! [A5: set @ B,B6: set @ B,F3: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( finite_finite2 @ B @ B6 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ F3 @ ( sup_sup @ ( set @ B ) @ A5 @ B6 ) )
= ( minus_minus @ A @ ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ A5 ) @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ B6 ) ) @ ( groups7311177749621191930dd_sum @ B @ A @ F3 @ ( inf_inf @ ( set @ B ) @ A5 @ B6 ) ) ) ) ) ) ) ).
% sum_Un
thf(fact_6041_sum_Ounion__disjoint,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A5: set @ B,B6: set @ B,G3: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( finite_finite2 @ B @ B6 )
=> ( ( ( inf_inf @ ( set @ B ) @ A5 @ B6 )
= ( bot_bot @ ( set @ B ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G3 @ ( sup_sup @ ( set @ B ) @ A5 @ B6 ) )
= ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G3 @ A5 ) @ ( groups7311177749621191930dd_sum @ B @ A @ G3 @ B6 ) ) ) ) ) ) ) ).
% sum.union_disjoint
thf(fact_6042_prod_Ounion__inter__neutral,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A5: set @ B,B6: set @ B,G3: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( finite_finite2 @ B @ B6 )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ ( inf_inf @ ( set @ B ) @ A5 @ B6 ) )
=> ( ( G3 @ X4 )
= ( one_one @ A ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G3 @ ( sup_sup @ ( set @ B ) @ A5 @ B6 ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G3 @ A5 ) @ ( groups7121269368397514597t_prod @ B @ A @ G3 @ B6 ) ) ) ) ) ) ) ).
% prod.union_inter_neutral
thf(fact_6043_prod_Ounion__disjoint,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A5: set @ B,B6: set @ B,G3: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( finite_finite2 @ B @ B6 )
=> ( ( ( inf_inf @ ( set @ B ) @ A5 @ B6 )
= ( bot_bot @ ( set @ B ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G3 @ ( sup_sup @ ( set @ B ) @ A5 @ B6 ) )
= ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G3 @ A5 ) @ ( groups7121269368397514597t_prod @ B @ A @ G3 @ B6 ) ) ) ) ) ) ) ).
% prod.union_disjoint
thf(fact_6044_ivl__disj__un__singleton_I6_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,U: A] :
( ( ord_less_eq @ A @ L @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ L @ U ) @ ( insert @ A @ U @ ( bot_bot @ ( set @ A ) ) ) )
= ( set_or1337092689740270186AtMost @ A @ L @ U ) ) ) ) ).
% ivl_disj_un_singleton(6)
thf(fact_6045_sum_Ounion__diff2,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A5: set @ B,B6: set @ B,G3: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( finite_finite2 @ B @ B6 )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G3 @ ( sup_sup @ ( set @ B ) @ A5 @ B6 ) )
= ( plus_plus @ A @ ( plus_plus @ A @ ( groups7311177749621191930dd_sum @ B @ A @ G3 @ ( minus_minus @ ( set @ B ) @ A5 @ B6 ) ) @ ( groups7311177749621191930dd_sum @ B @ A @ G3 @ ( minus_minus @ ( set @ B ) @ B6 @ A5 ) ) ) @ ( groups7311177749621191930dd_sum @ B @ A @ G3 @ ( inf_inf @ ( set @ B ) @ A5 @ B6 ) ) ) ) ) ) ) ).
% sum.union_diff2
thf(fact_6046_sum__Un2,axiom,
! [B: $tType,A: $tType] :
( ( comm_monoid_add @ B )
=> ! [A5: set @ A,B6: set @ A,F3: A > B] :
( ( finite_finite2 @ A @ ( sup_sup @ ( set @ A ) @ A5 @ B6 ) )
=> ( ( groups7311177749621191930dd_sum @ A @ B @ F3 @ ( sup_sup @ ( set @ A ) @ A5 @ B6 ) )
= ( plus_plus @ B @ ( plus_plus @ B @ ( groups7311177749621191930dd_sum @ A @ B @ F3 @ ( minus_minus @ ( set @ A ) @ A5 @ B6 ) ) @ ( groups7311177749621191930dd_sum @ A @ B @ F3 @ ( minus_minus @ ( set @ A ) @ B6 @ A5 ) ) ) @ ( groups7311177749621191930dd_sum @ A @ B @ F3 @ ( inf_inf @ ( set @ A ) @ A5 @ B6 ) ) ) ) ) ) ).
% sum_Un2
thf(fact_6047_prod_Ounion__diff2,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A5: set @ B,B6: set @ B,G3: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( finite_finite2 @ B @ B6 )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G3 @ ( sup_sup @ ( set @ B ) @ A5 @ B6 ) )
= ( times_times @ A @ ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ G3 @ ( minus_minus @ ( set @ B ) @ A5 @ B6 ) ) @ ( groups7121269368397514597t_prod @ B @ A @ G3 @ ( minus_minus @ ( set @ B ) @ B6 @ A5 ) ) ) @ ( groups7121269368397514597t_prod @ B @ A @ G3 @ ( inf_inf @ ( set @ B ) @ A5 @ B6 ) ) ) ) ) ) ) ).
% prod.union_diff2
thf(fact_6048_card__Un__disjoint,axiom,
! [A: $tType,A5: set @ A,B6: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( finite_finite2 @ A @ B6 )
=> ( ( ( inf_inf @ ( set @ A ) @ A5 @ B6 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_card @ A @ ( sup_sup @ ( set @ A ) @ A5 @ B6 ) )
= ( plus_plus @ nat @ ( finite_card @ A @ A5 ) @ ( finite_card @ A @ B6 ) ) ) ) ) ) ).
% card_Un_disjoint
thf(fact_6049_inj__on__Un,axiom,
! [A: $tType,B: $tType,F3: A > B,A5: set @ A,B6: set @ A] :
( ( inj_on @ A @ B @ F3 @ ( sup_sup @ ( set @ A ) @ A5 @ B6 ) )
= ( ( inj_on @ A @ B @ F3 @ A5 )
& ( inj_on @ A @ B @ F3 @ B6 )
& ( ( inf_inf @ ( set @ B ) @ ( image2 @ A @ B @ F3 @ ( minus_minus @ ( set @ A ) @ A5 @ B6 ) ) @ ( image2 @ A @ B @ F3 @ ( minus_minus @ ( set @ A ) @ B6 @ A5 ) ) )
= ( bot_bot @ ( set @ B ) ) ) ) ) ).
% inj_on_Un
thf(fact_6050_ivl__disj__un__singleton_I5_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [L: A,U: A] :
( ( ord_less_eq @ A @ L @ U )
=> ( ( sup_sup @ ( set @ A ) @ ( insert @ A @ L @ ( bot_bot @ ( set @ A ) ) ) @ ( set_or3652927894154168847AtMost @ A @ L @ U ) )
= ( set_or1337092689740270186AtMost @ A @ L @ U ) ) ) ) ).
% ivl_disj_un_singleton(5)
thf(fact_6051_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_6052_cINF__union,axiom,
! [A: $tType,B: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [A5: set @ B,F3: B > A,B6: set @ B] :
( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit1013018076250108175_below @ A @ ( image2 @ B @ A @ F3 @ A5 ) )
=> ( ( B6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( condit1013018076250108175_below @ A @ ( image2 @ B @ A @ F3 @ B6 ) )
=> ( ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ ( sup_sup @ ( set @ B ) @ A5 @ B6 ) ) )
= ( inf_inf @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ A5 ) ) @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ B6 ) ) ) ) ) ) ) ) ) ).
% cINF_union
thf(fact_6053_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_6054_sum__Un__nat,axiom,
! [A: $tType,A5: set @ A,B6: set @ A,F3: A > nat] :
( ( finite_finite2 @ A @ A5 )
=> ( ( finite_finite2 @ A @ B6 )
=> ( ( groups7311177749621191930dd_sum @ A @ nat @ F3 @ ( sup_sup @ ( set @ A ) @ A5 @ B6 ) )
= ( minus_minus @ nat @ ( plus_plus @ nat @ ( groups7311177749621191930dd_sum @ A @ nat @ F3 @ A5 ) @ ( groups7311177749621191930dd_sum @ A @ nat @ F3 @ B6 ) ) @ ( groups7311177749621191930dd_sum @ A @ nat @ F3 @ ( inf_inf @ ( set @ A ) @ A5 @ B6 ) ) ) ) ) ) ).
% sum_Un_nat
thf(fact_6055_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_6056_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_6057_trancl__insert,axiom,
! [A: $tType,Y: A,X: A,R2: set @ ( product_prod @ A @ A )] :
( ( transitive_trancl @ A @ ( insert @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ X ) @ R2 ) )
= ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ ( transitive_trancl @ A @ R2 )
@ ( collect @ ( product_prod @ A @ A )
@ ( product_case_prod @ A @ A @ $o
@ ^ [A6: A,B5: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A6 @ Y ) @ ( transitive_rtrancl @ A @ R2 ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ B5 ) @ ( transitive_rtrancl @ A @ R2 ) ) ) ) ) ) ) ).
% trancl_insert
thf(fact_6058_prod__Un,axiom,
! [A: $tType,B: $tType] :
( ( field @ A )
=> ! [A5: set @ B,B6: set @ B,F3: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( finite_finite2 @ B @ B6 )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ ( inf_inf @ ( set @ B ) @ A5 @ B6 ) )
=> ( ( F3 @ X4 )
!= ( zero_zero @ A ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ F3 @ ( sup_sup @ ( set @ B ) @ A5 @ B6 ) )
= ( divide_divide @ A @ ( times_times @ A @ ( groups7121269368397514597t_prod @ B @ A @ F3 @ A5 ) @ ( groups7121269368397514597t_prod @ B @ A @ F3 @ B6 ) ) @ ( groups7121269368397514597t_prod @ B @ A @ F3 @ ( inf_inf @ ( set @ B ) @ A5 @ B6 ) ) ) ) ) ) ) ) ).
% prod_Un
thf(fact_6059_If__the__inv__into__in__Func,axiom,
! [B: $tType,A: $tType,G3: A > B,C5: set @ A,B6: set @ A,X: A] :
( ( inj_on @ A @ B @ G3 @ C5 )
=> ( ( ord_less_eq @ ( set @ A ) @ C5 @ ( sup_sup @ ( set @ A ) @ B6 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( member @ ( B > A )
@ ^ [I4: B] : ( if @ A @ ( member @ B @ I4 @ ( image2 @ A @ B @ G3 @ C5 ) ) @ ( the_inv_into @ A @ B @ C5 @ G3 @ I4 ) @ X )
@ ( bNF_Wellorder_Func @ B @ A @ ( top_top @ ( set @ B ) ) @ ( sup_sup @ ( set @ A ) @ B6 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% If_the_inv_into_in_Func
thf(fact_6060_min__weak__def,axiom,
( fun_min_weak
= ( sup_sup @ ( set @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) ) @ ( min_ext @ ( product_prod @ nat @ nat ) @ fun_pair_leq ) @ ( insert @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ ( bot_bot @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( bot_bot @ ( set @ ( product_prod @ nat @ nat ) ) ) ) @ ( bot_bot @ ( set @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) ) ) ) ) ) ).
% min_weak_def
thf(fact_6061_flat__lub__def,axiom,
! [A: $tType] :
( ( partial_flat_lub @ A )
= ( ^ [B5: A,A7: set @ A] :
( if @ A @ ( ord_less_eq @ ( set @ A ) @ A7 @ ( insert @ A @ B5 @ ( bot_bot @ ( set @ A ) ) ) ) @ B5
@ ( the @ A
@ ^ [X5: A] : ( member @ A @ X5 @ ( minus_minus @ ( set @ A ) @ A7 @ ( insert @ A @ B5 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ).
% flat_lub_def
thf(fact_6062_sup__Un__eq2,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S2: set @ ( product_prod @ A @ B )] :
( ( sup_sup @ ( A > B > $o )
@ ^ [X5: A,Y6: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X5 @ Y6 ) @ R )
@ ^ [X5: A,Y6: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X5 @ Y6 ) @ S2 ) )
= ( ^ [X5: A,Y6: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X5 @ Y6 ) @ ( sup_sup @ ( set @ ( product_prod @ A @ B ) ) @ R @ S2 ) ) ) ) ).
% sup_Un_eq2
thf(fact_6063_sup__nat__def,axiom,
( ( sup_sup @ nat )
= ( ord_max @ nat ) ) ).
% sup_nat_def
thf(fact_6064_sup__int__def,axiom,
( ( sup_sup @ int )
= ( ord_max @ int ) ) ).
% sup_int_def
thf(fact_6065_min__strict__def,axiom,
( fun_min_strict
= ( min_ext @ ( product_prod @ nat @ nat ) @ fun_pair_less ) ) ).
% min_strict_def
thf(fact_6066_max__weak__def,axiom,
( fun_max_weak
= ( sup_sup @ ( set @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) ) @ ( max_ext @ ( product_prod @ nat @ nat ) @ fun_pair_leq ) @ ( insert @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( product_Pair @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) @ ( bot_bot @ ( set @ ( product_prod @ nat @ nat ) ) ) @ ( bot_bot @ ( set @ ( product_prod @ nat @ nat ) ) ) ) @ ( bot_bot @ ( set @ ( product_prod @ ( set @ ( product_prod @ nat @ nat ) ) @ ( set @ ( product_prod @ nat @ nat ) ) ) ) ) ) ) ) ).
% max_weak_def
thf(fact_6067_Func__map__surj,axiom,
! [C: $tType,A: $tType,D: $tType,B: $tType,F1: B > A,A16: set @ B,B1: set @ A,F22: C > D,B22: set @ C,A26: set @ D] :
( ( ( image2 @ B @ A @ F1 @ A16 )
= B1 )
=> ( ( inj_on @ C @ D @ F22 @ B22 )
=> ( ( ord_less_eq @ ( set @ D ) @ ( image2 @ C @ D @ F22 @ B22 ) @ A26 )
=> ( ( ( B22
= ( bot_bot @ ( set @ C ) ) )
=> ( A26
= ( bot_bot @ ( set @ D ) ) ) )
=> ( ( bNF_Wellorder_Func @ C @ A @ B22 @ B1 )
= ( image2 @ ( D > B ) @ ( C > A ) @ ( bNF_We4925052301507509544nc_map @ C @ B @ A @ D @ B22 @ F1 @ F22 ) @ ( bNF_Wellorder_Func @ D @ B @ A26 @ A16 ) ) ) ) ) ) ) ).
% Func_map_surj
thf(fact_6068_max__strict__def,axiom,
( fun_max_strict
= ( max_ext @ ( product_prod @ nat @ nat ) @ fun_pair_less ) ) ).
% max_strict_def
thf(fact_6069_max__ext_Ocases,axiom,
! [A: $tType,A1: set @ A,A22: set @ A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ A1 @ A22 ) @ ( max_ext @ A @ R ) )
=> ~ ( ( finite_finite2 @ A @ A1 )
=> ( ( finite_finite2 @ A @ A22 )
=> ( ( A22
!= ( bot_bot @ ( set @ A ) ) )
=> ~ ! [X3: A] :
( ( member @ A @ X3 @ A1 )
=> ? [Xa3: A] :
( ( member @ A @ Xa3 @ A22 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X3 @ Xa3 ) @ R ) ) ) ) ) ) ) ).
% max_ext.cases
thf(fact_6070_max__ext_Osimps,axiom,
! [A: $tType,A1: set @ A,A22: set @ A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ A1 @ A22 ) @ ( max_ext @ A @ R ) )
= ( ( finite_finite2 @ A @ A1 )
& ( finite_finite2 @ A @ A22 )
& ( A22
!= ( bot_bot @ ( set @ A ) ) )
& ! [X5: A] :
( ( member @ A @ X5 @ A1 )
=> ? [Y6: A] :
( ( member @ A @ Y6 @ A22 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X5 @ Y6 ) @ R ) ) ) ) ) ).
% max_ext.simps
thf(fact_6071_max__ext_Omax__extI,axiom,
! [A: $tType,X7: set @ A,Y8: set @ A,R: set @ ( product_prod @ A @ A )] :
( ( finite_finite2 @ A @ X7 )
=> ( ( finite_finite2 @ A @ Y8 )
=> ( ( Y8
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ X7 )
=> ? [Xa: A] :
( ( member @ A @ Xa @ Y8 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Xa ) @ R ) ) )
=> ( member @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ X7 @ Y8 ) @ ( max_ext @ A @ R ) ) ) ) ) ) ).
% max_ext.max_extI
thf(fact_6072_Func__non__emp,axiom,
! [A: $tType,B: $tType,B6: set @ A,A5: set @ B] :
( ( B6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( bNF_Wellorder_Func @ B @ A @ A5 @ B6 )
!= ( bot_bot @ ( set @ ( B > A ) ) ) ) ) ).
% Func_non_emp
thf(fact_6073_Func__is__emp,axiom,
! [A: $tType,B: $tType,A5: set @ A,B6: set @ B] :
( ( ( bNF_Wellorder_Func @ A @ B @ A5 @ B6 )
= ( bot_bot @ ( set @ ( A > B ) ) ) )
= ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
& ( B6
= ( bot_bot @ ( set @ B ) ) ) ) ) ).
% Func_is_emp
thf(fact_6074_max__ext__def,axiom,
! [A: $tType] :
( ( max_ext @ A )
= ( ^ [R6: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ ( set @ A ) @ ( set @ A ) )
@ ( product_case_prod @ ( set @ A ) @ ( set @ A ) @ $o
@ ( max_extp @ A
@ ^ [X5: A,Y6: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X5 @ Y6 ) @ R6 ) ) ) ) ) ) ).
% max_ext_def
thf(fact_6075_Pow__fold,axiom,
! [A: $tType,A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( pow2 @ A @ A5 )
= ( finite_fold @ A @ ( set @ ( set @ A ) )
@ ^ [X5: A,A7: set @ ( set @ A )] : ( sup_sup @ ( set @ ( set @ A ) ) @ A7 @ ( image2 @ ( set @ A ) @ ( set @ A ) @ ( insert @ A @ X5 ) @ A7 ) )
@ ( insert @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ ( bot_bot @ ( set @ ( set @ A ) ) ) )
@ A5 ) ) ) ).
% Pow_fold
thf(fact_6076_fold__empty,axiom,
! [B: $tType,A: $tType,F3: B > A > A,Z: A] :
( ( finite_fold @ B @ A @ F3 @ Z @ ( bot_bot @ ( set @ B ) ) )
= Z ) ).
% fold_empty
thf(fact_6077_fold__infinite,axiom,
! [A: $tType,B: $tType,A5: set @ A,F3: A > B > B,Z: B] :
( ~ ( finite_finite2 @ A @ A5 )
=> ( ( finite_fold @ A @ B @ F3 @ Z @ A5 )
= Z ) ) ).
% fold_infinite
thf(fact_6078_fold__closed__eq,axiom,
! [B: $tType,A: $tType,A5: set @ A,B6: set @ B,F3: A > B > B,G3: A > B > B,Z: B] :
( ! [A4: A,B4: B] :
( ( member @ A @ A4 @ A5 )
=> ( ( member @ B @ B4 @ B6 )
=> ( ( F3 @ A4 @ B4 )
= ( G3 @ A4 @ B4 ) ) ) )
=> ( ! [A4: A,B4: B] :
( ( member @ A @ A4 @ A5 )
=> ( ( member @ B @ B4 @ B6 )
=> ( member @ B @ ( G3 @ A4 @ B4 ) @ B6 ) ) )
=> ( ( member @ B @ Z @ B6 )
=> ( ( finite_fold @ A @ B @ F3 @ Z @ A5 )
= ( finite_fold @ A @ B @ G3 @ Z @ A5 ) ) ) ) ) ).
% fold_closed_eq
thf(fact_6079_union__fold__insert,axiom,
! [A: $tType,A5: set @ A,B6: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( sup_sup @ ( set @ A ) @ A5 @ B6 )
= ( finite_fold @ A @ ( set @ A ) @ ( insert @ A ) @ B6 @ A5 ) ) ) ).
% union_fold_insert
thf(fact_6080_sup__Sup__fold__sup,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ A,B6: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( sup_sup @ A @ ( complete_Sup_Sup @ A @ A5 ) @ B6 )
= ( finite_fold @ A @ A @ ( sup_sup @ A ) @ B6 @ A5 ) ) ) ) ).
% sup_Sup_fold_sup
thf(fact_6081_inf__Inf__fold__inf,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ A,B6: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( inf_inf @ A @ ( complete_Inf_Inf @ A @ A5 ) @ B6 )
= ( finite_fold @ A @ A @ ( inf_inf @ A ) @ B6 @ A5 ) ) ) ) ).
% inf_Inf_fold_inf
thf(fact_6082_fold__image,axiom,
! [C: $tType,B: $tType,A: $tType,G3: A > B,A5: set @ A,F3: B > C > C,Z: C] :
( ( inj_on @ A @ B @ G3 @ A5 )
=> ( ( finite_fold @ B @ C @ F3 @ Z @ ( image2 @ A @ B @ G3 @ A5 ) )
= ( finite_fold @ A @ C @ ( comp @ B @ ( C > C ) @ A @ F3 @ G3 ) @ Z @ A5 ) ) ) ).
% fold_image
thf(fact_6083_Sup__fold__sup,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( complete_Sup_Sup @ A @ A5 )
= ( finite_fold @ A @ A @ ( sup_sup @ A ) @ ( bot_bot @ A ) @ A5 ) ) ) ) ).
% Sup_fold_sup
thf(fact_6084_Inf__fold__inf,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( complete_Inf_Inf @ A @ A5 )
= ( finite_fold @ A @ A @ ( inf_inf @ A ) @ ( top_top @ A ) @ A5 ) ) ) ) ).
% Inf_fold_inf
thf(fact_6085_sum_Oeq__fold,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ( ( groups7311177749621191930dd_sum @ B @ A )
= ( ^ [G4: B > A] : ( finite_fold @ B @ A @ ( comp @ A @ ( A > A ) @ B @ ( plus_plus @ A ) @ G4 ) @ ( zero_zero @ A ) ) ) ) ) ).
% sum.eq_fold
thf(fact_6086_Max_Oeq__fold,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( lattic643756798349783984er_Max @ A @ ( insert @ A @ X @ A5 ) )
= ( finite_fold @ A @ A @ ( ord_max @ A ) @ X @ A5 ) ) ) ) ).
% Max.eq_fold
thf(fact_6087_image__fold__insert,axiom,
! [B: $tType,A: $tType,A5: set @ A,F3: A > B] :
( ( finite_finite2 @ A @ A5 )
=> ( ( image2 @ A @ B @ F3 @ A5 )
= ( finite_fold @ A @ ( set @ B )
@ ^ [K3: A] : ( insert @ B @ ( F3 @ K3 ) )
@ ( bot_bot @ ( set @ B ) )
@ A5 ) ) ) ).
% image_fold_insert
thf(fact_6088_sup__SUP__fold__sup,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ B,B6: A,F3: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( sup_sup @ A @ B6 @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A5 ) ) )
= ( finite_fold @ B @ A @ ( comp @ A @ ( A > A ) @ B @ ( sup_sup @ A ) @ F3 ) @ B6 @ A5 ) ) ) ) ).
% sup_SUP_fold_sup
thf(fact_6089_inf__INF__fold__inf,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ B,B6: A,F3: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( inf_inf @ A @ B6 @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ A5 ) ) )
= ( finite_fold @ B @ A @ ( comp @ A @ ( A > A ) @ B @ ( inf_inf @ A ) @ F3 ) @ B6 @ A5 ) ) ) ) ).
% inf_INF_fold_inf
thf(fact_6090_SUP__fold__sup,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ B,F3: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A5 ) )
= ( finite_fold @ B @ A @ ( comp @ A @ ( A > A ) @ B @ ( sup_sup @ A ) @ F3 ) @ ( bot_bot @ A ) @ A5 ) ) ) ) ).
% SUP_fold_sup
thf(fact_6091_INF__fold__inf,axiom,
! [A: $tType,B: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ B,F3: B > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ A5 ) )
= ( finite_fold @ B @ A @ ( comp @ A @ ( A > A ) @ B @ ( inf_inf @ A ) @ F3 ) @ ( top_top @ A ) @ A5 ) ) ) ) ).
% INF_fold_inf
thf(fact_6092_max__extp__max__ext__eq,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( max_extp @ A
@ ^ [X5: A,Y6: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X5 @ Y6 ) @ R ) )
= ( ^ [X5: set @ A,Y6: set @ A] : ( member @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ X5 @ Y6 ) @ ( max_ext @ A @ R ) ) ) ) ).
% max_extp_max_ext_eq
thf(fact_6093_Set__filter__fold,axiom,
! [A: $tType,A5: set @ A,P2: A > $o] :
( ( finite_finite2 @ A @ A5 )
=> ( ( filter3 @ A @ P2 @ A5 )
= ( finite_fold @ A @ ( set @ A )
@ ^ [X5: A,A17: set @ A] : ( if @ ( set @ A ) @ ( P2 @ X5 ) @ ( insert @ A @ X5 @ A17 ) @ A17 )
@ ( bot_bot @ ( set @ A ) )
@ A5 ) ) ) ).
% Set_filter_fold
thf(fact_6094_fold__union__pair,axiom,
! [B: $tType,A: $tType,B6: set @ A,X: B,A5: set @ ( product_prod @ B @ A )] :
( ( finite_finite2 @ A @ B6 )
=> ( ( sup_sup @ ( set @ ( product_prod @ B @ A ) )
@ ( complete_Sup_Sup @ ( set @ ( product_prod @ B @ A ) )
@ ( image2 @ A @ ( set @ ( product_prod @ B @ A ) )
@ ^ [Y6: A] : ( insert @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X @ Y6 ) @ ( bot_bot @ ( set @ ( product_prod @ B @ A ) ) ) )
@ B6 ) )
@ A5 )
= ( finite_fold @ A @ ( set @ ( product_prod @ B @ A ) )
@ ^ [Y6: A] : ( insert @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X @ Y6 ) )
@ A5
@ B6 ) ) ) ).
% fold_union_pair
thf(fact_6095_finite__filter,axiom,
! [A: $tType,S2: set @ A,P2: A > $o] :
( ( finite_finite2 @ A @ S2 )
=> ( finite_finite2 @ A @ ( filter3 @ A @ P2 @ S2 ) ) ) ).
% finite_filter
thf(fact_6096_Ball__fold,axiom,
! [A: $tType,A5: set @ A,P2: A > $o] :
( ( finite_finite2 @ A @ A5 )
=> ( ( ! [X5: A] :
( ( member @ A @ X5 @ A5 )
=> ( P2 @ X5 ) ) )
= ( finite_fold @ A @ $o
@ ^ [K3: A,S8: $o] :
( S8
& ( P2 @ K3 ) )
@ $true
@ A5 ) ) ) ).
% Ball_fold
thf(fact_6097_card_Oeq__fold,axiom,
! [A: $tType] :
( ( finite_card @ A )
= ( finite_fold @ A @ nat
@ ^ [Uu3: A] : suc
@ ( zero_zero @ nat ) ) ) ).
% card.eq_fold
thf(fact_6098_sorted__list__of__set_Ofold__insort__key_Oeq__fold,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( linord4507533701916653071of_set @ A )
= ( finite_fold @ A @ ( list @ A )
@ ( linorder_insort_key @ A @ A
@ ^ [X5: A] : X5 )
@ ( nil @ A ) ) ) ) ).
% sorted_list_of_set.fold_insort_key.eq_fold
thf(fact_6099_inter__Set__filter,axiom,
! [A: $tType,B6: set @ A,A5: set @ A] :
( ( finite_finite2 @ A @ B6 )
=> ( ( inf_inf @ ( set @ A ) @ A5 @ B6 )
= ( filter3 @ A
@ ^ [X5: A] : ( member @ A @ X5 @ A5 )
@ B6 ) ) ) ).
% inter_Set_filter
thf(fact_6100_Id__on__fold,axiom,
! [A: $tType,A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( id_on @ A @ A5 )
= ( finite_fold @ A @ ( set @ ( product_prod @ A @ A ) )
@ ^ [X5: A] : ( insert @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X5 @ X5 ) )
@ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) )
@ A5 ) ) ) ).
% Id_on_fold
thf(fact_6101_take__bit__num__def,axiom,
( bit_take_bit_num
= ( ^ [N2: nat,M3: num] :
( if @ ( option @ num )
@ ( ( bit_se2584673776208193580ke_bit @ nat @ N2 @ ( numeral_numeral @ nat @ M3 ) )
= ( zero_zero @ nat ) )
@ ( none @ num )
@ ( some @ num @ ( num_of_nat @ ( bit_se2584673776208193580ke_bit @ nat @ N2 @ ( numeral_numeral @ nat @ M3 ) ) ) ) ) ) ) ).
% take_bit_num_def
thf(fact_6102_insert__relcomp__union__fold,axiom,
! [C: $tType,B: $tType,A: $tType,S2: set @ ( product_prod @ A @ B ),X: product_prod @ C @ A,X7: set @ ( product_prod @ C @ B )] :
( ( finite_finite2 @ ( product_prod @ A @ B ) @ S2 )
=> ( ( sup_sup @ ( set @ ( product_prod @ C @ B ) ) @ ( relcomp @ C @ A @ B @ ( insert @ ( product_prod @ C @ A ) @ X @ ( bot_bot @ ( set @ ( product_prod @ C @ A ) ) ) ) @ S2 ) @ X7 )
= ( finite_fold @ ( product_prod @ A @ B ) @ ( set @ ( product_prod @ C @ B ) )
@ ( product_case_prod @ A @ B @ ( ( set @ ( product_prod @ C @ B ) ) > ( set @ ( product_prod @ C @ B ) ) )
@ ^ [W3: A,Z5: B,A17: set @ ( product_prod @ C @ B )] :
( if @ ( set @ ( product_prod @ C @ B ) )
@ ( ( product_snd @ C @ A @ X )
= W3 )
@ ( insert @ ( product_prod @ C @ B ) @ ( product_Pair @ C @ B @ ( product_fst @ C @ A @ X ) @ Z5 ) @ A17 )
@ A17 ) )
@ X7
@ S2 ) ) ) ).
% insert_relcomp_union_fold
thf(fact_6103_relcomp__empty2,axiom,
! [C: $tType,B: $tType,A: $tType,R: set @ ( product_prod @ A @ C )] :
( ( relcomp @ A @ C @ B @ R @ ( bot_bot @ ( set @ ( product_prod @ C @ B ) ) ) )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% relcomp_empty2
thf(fact_6104_relcomp__empty1,axiom,
! [C: $tType,B: $tType,A: $tType,R: set @ ( product_prod @ C @ B )] :
( ( relcomp @ A @ C @ B @ ( bot_bot @ ( set @ ( product_prod @ A @ C ) ) ) @ R )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% relcomp_empty1
thf(fact_6105_finite__relcomp,axiom,
! [A: $tType,C: $tType,B: $tType,R: set @ ( product_prod @ A @ B ),S2: set @ ( product_prod @ B @ C )] :
( ( finite_finite2 @ ( product_prod @ A @ B ) @ R )
=> ( ( finite_finite2 @ ( product_prod @ B @ C ) @ S2 )
=> ( finite_finite2 @ ( product_prod @ A @ C ) @ ( relcomp @ A @ B @ C @ R @ S2 ) ) ) ) ).
% finite_relcomp
thf(fact_6106_relpow__add,axiom,
! [A: $tType,M2: nat,N: nat,R: set @ ( product_prod @ A @ A )] :
( ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( plus_plus @ nat @ M2 @ N ) @ R )
= ( relcomp @ A @ A @ A @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ M2 @ R ) @ ( compow @ ( set @ ( product_prod @ A @ A ) ) @ N @ R ) ) ) ).
% relpow_add
thf(fact_6107_relcomp_Ocases,axiom,
! [A: $tType,C: $tType,B: $tType,A1: A,A22: C,R2: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ B @ C )] :
( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ A1 @ A22 ) @ ( relcomp @ A @ B @ C @ R2 @ S ) )
=> ~ ! [B4: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A1 @ B4 ) @ R2 )
=> ~ ( member @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ B4 @ A22 ) @ S ) ) ) ).
% relcomp.cases
thf(fact_6108_relcomp_Osimps,axiom,
! [B: $tType,C: $tType,A: $tType,A1: A,A22: C,R2: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ B @ C )] :
( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ A1 @ A22 ) @ ( relcomp @ A @ B @ C @ R2 @ S ) )
= ( ? [A6: A,B5: B,C4: C] :
( ( A1 = A6 )
& ( A22 = C4 )
& ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A6 @ B5 ) @ R2 )
& ( member @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ B5 @ C4 ) @ S ) ) ) ) ).
% relcomp.simps
thf(fact_6109_relcomp_OrelcompI,axiom,
! [A: $tType,C: $tType,B: $tType,A2: A,B2: B,R2: set @ ( product_prod @ A @ B ),C3: C,S: set @ ( product_prod @ B @ C )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ R2 )
=> ( ( member @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ B2 @ C3 ) @ S )
=> ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ A2 @ C3 ) @ ( relcomp @ A @ B @ C @ R2 @ S ) ) ) ) ).
% relcomp.relcompI
thf(fact_6110_relcompE,axiom,
! [A: $tType,B: $tType,C: $tType,Xz: product_prod @ A @ B,R2: set @ ( product_prod @ A @ C ),S: set @ ( product_prod @ C @ B )] :
( ( member @ ( product_prod @ A @ B ) @ Xz @ ( relcomp @ A @ C @ B @ R2 @ S ) )
=> ~ ! [X4: A,Y3: C,Z3: B] :
( ( Xz
= ( product_Pair @ A @ B @ X4 @ Z3 ) )
=> ( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ X4 @ Y3 ) @ R2 )
=> ~ ( member @ ( product_prod @ C @ B ) @ ( product_Pair @ C @ B @ Y3 @ Z3 ) @ S ) ) ) ) ).
% relcompE
thf(fact_6111_relcompEpair,axiom,
! [A: $tType,B: $tType,C: $tType,A2: A,C3: B,R2: set @ ( product_prod @ A @ C ),S: set @ ( product_prod @ C @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ C3 ) @ ( relcomp @ A @ C @ B @ R2 @ S ) )
=> ~ ! [B4: C] :
( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ A2 @ B4 ) @ R2 )
=> ~ ( member @ ( product_prod @ C @ B ) @ ( product_Pair @ C @ B @ B4 @ C3 ) @ S ) ) ) ).
% relcompEpair
thf(fact_6112_union__comp__emptyR,axiom,
! [A: $tType,A5: set @ ( product_prod @ A @ A ),B6: set @ ( product_prod @ A @ A ),C5: set @ ( product_prod @ A @ A )] :
( ( ( relcomp @ A @ A @ A @ A5 @ B6 )
= ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
=> ( ( ( relcomp @ A @ A @ A @ A5 @ C5 )
= ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
=> ( ( relcomp @ A @ A @ A @ A5 @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ B6 @ C5 ) )
= ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ) ).
% union_comp_emptyR
thf(fact_6113_union__comp__emptyL,axiom,
! [A: $tType,A5: set @ ( product_prod @ A @ A ),C5: set @ ( product_prod @ A @ A ),B6: set @ ( product_prod @ A @ A )] :
( ( ( relcomp @ A @ A @ A @ A5 @ C5 )
= ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
=> ( ( ( relcomp @ A @ A @ A @ B6 @ C5 )
= ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
=> ( ( relcomp @ A @ A @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ A5 @ B6 ) @ C5 )
= ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ) ).
% union_comp_emptyL
thf(fact_6114_num__of__nat_Osimps_I1_J,axiom,
( ( num_of_nat @ ( zero_zero @ nat ) )
= one2 ) ).
% num_of_nat.simps(1)
thf(fact_6115_relcomp__unfold,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( relcomp @ A @ C @ B )
= ( ^ [R5: set @ ( product_prod @ A @ C ),S8: set @ ( product_prod @ C @ B )] :
( collect @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [X5: A,Z5: B] :
? [Y6: C] :
( ( member @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ X5 @ Y6 ) @ R5 )
& ( member @ ( product_prod @ C @ B ) @ ( product_Pair @ C @ B @ Y6 @ Z5 ) @ S8 ) ) ) ) ) ) ).
% relcomp_unfold
thf(fact_6116_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_6117_max__ext__compat,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),S2: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( relcomp @ A @ A @ A @ R @ S2 ) @ R )
=> ( ord_less_eq @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( relcomp @ ( set @ A ) @ ( set @ A ) @ ( set @ A ) @ ( max_ext @ A @ R ) @ ( sup_sup @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( max_ext @ A @ S2 ) @ ( insert @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ ( bot_bot @ ( set @ A ) ) ) @ ( bot_bot @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ) ) ) ) @ ( max_ext @ A @ R ) ) ) ).
% max_ext_compat
thf(fact_6118_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_6119_min__ext__compat,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),S2: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( relcomp @ A @ A @ A @ R @ S2 ) @ R )
=> ( ord_less_eq @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( relcomp @ ( set @ A ) @ ( set @ A ) @ ( set @ A ) @ ( min_ext @ A @ R ) @ ( sup_sup @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) @ ( min_ext @ A @ S2 ) @ ( insert @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) @ ( product_Pair @ ( set @ A ) @ ( set @ A ) @ ( bot_bot @ ( set @ A ) ) @ ( bot_bot @ ( set @ A ) ) ) @ ( bot_bot @ ( set @ ( product_prod @ ( set @ A ) @ ( set @ A ) ) ) ) ) ) ) @ ( min_ext @ A @ R ) ) ) ).
% min_ext_compat
thf(fact_6120_relcomp__fold,axiom,
! [C: $tType,B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S2: set @ ( product_prod @ B @ C )] :
( ( finite_finite2 @ ( product_prod @ A @ B ) @ R )
=> ( ( finite_finite2 @ ( product_prod @ B @ C ) @ S2 )
=> ( ( relcomp @ A @ B @ C @ R @ S2 )
= ( finite_fold @ ( product_prod @ A @ B ) @ ( set @ ( product_prod @ A @ C ) )
@ ( product_case_prod @ A @ B @ ( ( set @ ( product_prod @ A @ C ) ) > ( set @ ( product_prod @ A @ C ) ) )
@ ^ [X5: A,Y6: B,A7: set @ ( product_prod @ A @ C )] :
( finite_fold @ ( product_prod @ B @ C ) @ ( set @ ( product_prod @ A @ C ) )
@ ( product_case_prod @ B @ C @ ( ( set @ ( product_prod @ A @ C ) ) > ( set @ ( product_prod @ A @ C ) ) )
@ ^ [W3: B,Z5: C,A17: set @ ( product_prod @ A @ C )] : ( if @ ( set @ ( product_prod @ A @ C ) ) @ ( Y6 = W3 ) @ ( insert @ ( product_prod @ A @ C ) @ ( product_Pair @ A @ C @ X5 @ Z5 ) @ A17 ) @ A17 ) )
@ A7
@ S2 ) )
@ ( bot_bot @ ( set @ ( product_prod @ A @ C ) ) )
@ R ) ) ) ) ).
% relcomp_fold
thf(fact_6121_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_6122_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_6123_insert__relcomp__fold,axiom,
! [C: $tType,B: $tType,A: $tType,S2: set @ ( product_prod @ A @ B ),X: product_prod @ C @ A,R: set @ ( product_prod @ C @ A )] :
( ( finite_finite2 @ ( product_prod @ A @ B ) @ S2 )
=> ( ( relcomp @ C @ A @ B @ ( insert @ ( product_prod @ C @ A ) @ X @ R ) @ S2 )
= ( finite_fold @ ( product_prod @ A @ B ) @ ( set @ ( product_prod @ C @ B ) )
@ ( product_case_prod @ A @ B @ ( ( set @ ( product_prod @ C @ B ) ) > ( set @ ( product_prod @ C @ B ) ) )
@ ^ [W3: A,Z5: B,A17: set @ ( product_prod @ C @ B )] :
( if @ ( set @ ( product_prod @ C @ B ) )
@ ( ( product_snd @ C @ A @ X )
= W3 )
@ ( insert @ ( product_prod @ C @ B ) @ ( product_Pair @ C @ B @ ( product_fst @ C @ A @ X ) @ Z5 ) @ A17 )
@ A17 ) )
@ ( relcomp @ C @ A @ B @ R @ S2 )
@ S2 ) ) ) ).
% insert_relcomp_fold
thf(fact_6124_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_6125_comp__fun__commute__relcomp__fold,axiom,
! [A: $tType,B: $tType,C: $tType,S2: set @ ( product_prod @ A @ B )] :
( ( finite_finite2 @ ( product_prod @ A @ B ) @ S2 )
=> ( finite6289374366891150609ommute @ ( product_prod @ C @ A ) @ ( set @ ( product_prod @ C @ B ) )
@ ( product_case_prod @ C @ A @ ( ( set @ ( product_prod @ C @ B ) ) > ( set @ ( product_prod @ C @ B ) ) )
@ ^ [X5: C,Y6: A,A7: set @ ( product_prod @ C @ B )] :
( finite_fold @ ( product_prod @ A @ B ) @ ( set @ ( product_prod @ C @ B ) )
@ ( product_case_prod @ A @ B @ ( ( set @ ( product_prod @ C @ B ) ) > ( set @ ( product_prod @ C @ B ) ) )
@ ^ [W3: A,Z5: B,A17: set @ ( product_prod @ C @ B )] : ( if @ ( set @ ( product_prod @ C @ B ) ) @ ( Y6 = W3 ) @ ( insert @ ( product_prod @ C @ B ) @ ( product_Pair @ C @ B @ X5 @ Z5 ) @ A17 ) @ A17 ) )
@ A7
@ S2 ) ) ) ) ).
% comp_fun_commute_relcomp_fold
thf(fact_6126_comp__fun__commute__product__fold,axiom,
! [A: $tType,B: $tType,B6: set @ A] :
( ( finite_finite2 @ A @ B6 )
=> ( finite6289374366891150609ommute @ B @ ( set @ ( product_prod @ B @ A ) )
@ ^ [X5: B,Z5: set @ ( product_prod @ B @ A )] :
( finite_fold @ A @ ( set @ ( product_prod @ B @ A ) )
@ ^ [Y6: A] : ( insert @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X5 @ Y6 ) )
@ Z5
@ B6 ) ) ) ).
% comp_fun_commute_product_fold
thf(fact_6127_comp__fun__commute__const,axiom,
! [B: $tType,A: $tType,F3: B > B] :
( finite6289374366891150609ommute @ A @ B
@ ^ [Uu3: A] : F3 ) ).
% comp_fun_commute_const
thf(fact_6128_comp__fun__commute__def,axiom,
! [B: $tType,A: $tType] :
( ( finite6289374366891150609ommute @ A @ B )
= ( ^ [F4: A > B > B] :
! [Y6: A,X5: A] :
( ( comp @ B @ B @ B @ ( F4 @ Y6 ) @ ( F4 @ X5 ) )
= ( comp @ B @ B @ B @ ( F4 @ X5 ) @ ( F4 @ Y6 ) ) ) ) ) ).
% comp_fun_commute_def
thf(fact_6129_comp__fun__commute_Ocomp__fun__commute,axiom,
! [B: $tType,A: $tType,F3: A > B > B,Y: A,X: A] :
( ( finite6289374366891150609ommute @ A @ B @ F3 )
=> ( ( comp @ B @ B @ B @ ( F3 @ Y ) @ ( F3 @ X ) )
= ( comp @ B @ B @ B @ ( F3 @ X ) @ ( F3 @ Y ) ) ) ) ).
% comp_fun_commute.comp_fun_commute
thf(fact_6130_comp__fun__commute_Ointro,axiom,
! [B: $tType,A: $tType,F3: A > B > B] :
( ! [Y3: A,X4: A] :
( ( comp @ B @ B @ B @ ( F3 @ Y3 ) @ ( F3 @ X4 ) )
= ( comp @ B @ B @ B @ ( F3 @ X4 ) @ ( F3 @ Y3 ) ) )
=> ( finite6289374366891150609ommute @ A @ B @ F3 ) ) ).
% comp_fun_commute.intro
thf(fact_6131_comp__fun__commute_Ocomp__comp__fun__commute,axiom,
! [B: $tType,A: $tType,C: $tType,F3: A > B > B,G3: C > A] :
( ( finite6289374366891150609ommute @ A @ B @ F3 )
=> ( finite6289374366891150609ommute @ C @ B @ ( comp @ A @ ( B > B ) @ C @ F3 @ G3 ) ) ) ).
% comp_fun_commute.comp_comp_fun_commute
thf(fact_6132_comp__fun__commute__filter__fold,axiom,
! [A: $tType,P2: A > $o] :
( finite6289374366891150609ommute @ A @ ( set @ A )
@ ^ [X5: A,A17: set @ A] : ( if @ ( set @ A ) @ ( P2 @ X5 ) @ ( insert @ A @ X5 @ A17 ) @ A17 ) ) ).
% comp_fun_commute_filter_fold
thf(fact_6133_comp__fun__commute_Ocomp__fun__commute__funpow,axiom,
! [B: $tType,A: $tType,F3: A > B > B,G3: A > nat] :
( ( finite6289374366891150609ommute @ A @ B @ F3 )
=> ( finite6289374366891150609ommute @ A @ B
@ ^ [X5: A] : ( compow @ ( B > B ) @ ( G3 @ X5 ) @ ( F3 @ X5 ) ) ) ) ).
% comp_fun_commute.comp_fun_commute_funpow
thf(fact_6134_UNION__fun__upd,axiom,
! [B: $tType,A: $tType,A5: B > ( set @ A ),I2: B,B6: set @ A,J4: set @ B] :
( ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ ( fun_upd @ B @ ( set @ A ) @ A5 @ I2 @ B6 ) @ J4 ) )
= ( sup_sup @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ A5 @ ( minus_minus @ ( set @ B ) @ J4 @ ( insert @ B @ I2 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) @ ( if @ ( set @ A ) @ ( member @ B @ I2 @ J4 ) @ B6 @ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% UNION_fun_upd
thf(fact_6135_comp__fun__commute__on_Ofold__set__union__disj,axiom,
! [B: $tType,A: $tType,S2: set @ A,F3: A > B > B,A5: set @ A,B6: set @ A,Z: B] :
( ( finite4664212375090638736ute_on @ A @ B @ S2 @ F3 )
=> ( ( ord_less_eq @ ( set @ A ) @ A5 @ S2 )
=> ( ( ord_less_eq @ ( set @ A ) @ B6 @ S2 )
=> ( ( finite_finite2 @ A @ A5 )
=> ( ( finite_finite2 @ A @ B6 )
=> ( ( ( inf_inf @ ( set @ A ) @ A5 @ B6 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_fold @ A @ B @ F3 @ Z @ ( sup_sup @ ( set @ A ) @ A5 @ B6 ) )
= ( finite_fold @ A @ B @ F3 @ ( finite_fold @ A @ B @ F3 @ Z @ A5 ) @ B6 ) ) ) ) ) ) ) ) ).
% comp_fun_commute_on.fold_set_union_disj
thf(fact_6136_comp__fun__commute__on_Ointro,axiom,
! [B: $tType,A: $tType,S2: set @ A,F3: A > B > B] :
( ! [X4: A,Y3: A] :
( ( member @ A @ X4 @ S2 )
=> ( ( member @ A @ Y3 @ S2 )
=> ( ( comp @ B @ B @ B @ ( F3 @ Y3 ) @ ( F3 @ X4 ) )
= ( comp @ B @ B @ B @ ( F3 @ X4 ) @ ( F3 @ Y3 ) ) ) ) )
=> ( finite4664212375090638736ute_on @ A @ B @ S2 @ F3 ) ) ).
% comp_fun_commute_on.intro
thf(fact_6137_comp__fun__commute__on_Ocommute__left__comp,axiom,
! [A: $tType,B: $tType,C: $tType,S2: set @ A,F3: A > B > B,X: A,Y: A,G3: C > B] :
( ( finite4664212375090638736ute_on @ A @ B @ S2 @ F3 )
=> ( ( member @ A @ X @ S2 )
=> ( ( member @ A @ Y @ S2 )
=> ( ( comp @ B @ B @ C @ ( F3 @ Y ) @ ( comp @ B @ B @ C @ ( F3 @ X ) @ G3 ) )
= ( comp @ B @ B @ C @ ( F3 @ X ) @ ( comp @ B @ B @ C @ ( F3 @ Y ) @ G3 ) ) ) ) ) ) ).
% comp_fun_commute_on.commute_left_comp
thf(fact_6138_comp__fun__commute__on_Ocomp__fun__commute__on,axiom,
! [B: $tType,A: $tType,S2: set @ A,F3: A > B > B,X: A,Y: A] :
( ( finite4664212375090638736ute_on @ A @ B @ S2 @ F3 )
=> ( ( member @ A @ X @ S2 )
=> ( ( member @ A @ Y @ S2 )
=> ( ( comp @ B @ B @ B @ ( F3 @ Y ) @ ( F3 @ X ) )
= ( comp @ B @ B @ B @ ( F3 @ X ) @ ( F3 @ Y ) ) ) ) ) ) ).
% comp_fun_commute_on.comp_fun_commute_on
thf(fact_6139_comp__fun__commute__on__def,axiom,
! [B: $tType,A: $tType] :
( ( finite4664212375090638736ute_on @ A @ B )
= ( ^ [S7: set @ A,F4: A > B > B] :
! [X5: A,Y6: A] :
( ( member @ A @ X5 @ S7 )
=> ( ( member @ A @ Y6 @ S7 )
=> ( ( comp @ B @ B @ B @ ( F4 @ Y6 ) @ ( F4 @ X5 ) )
= ( comp @ B @ B @ B @ ( F4 @ X5 ) @ ( F4 @ Y6 ) ) ) ) ) ) ) ).
% comp_fun_commute_on_def
thf(fact_6140_comp__fun__commute__on_Ocomp__fun__commute__on__funpow,axiom,
! [B: $tType,A: $tType,S2: set @ A,F3: A > B > B,G3: A > nat] :
( ( finite4664212375090638736ute_on @ A @ B @ S2 @ F3 )
=> ( finite4664212375090638736ute_on @ A @ B @ S2
@ ^ [X5: A] : ( compow @ ( B > B ) @ ( G3 @ X5 ) @ ( F3 @ X5 ) ) ) ) ).
% comp_fun_commute_on.comp_fun_commute_on_funpow
thf(fact_6141_comp__fun__commute__on_Ofun__left__comm,axiom,
! [A: $tType,B: $tType,S2: set @ A,F3: A > B > B,X: A,Y: A,Z: B] :
( ( finite4664212375090638736ute_on @ A @ B @ S2 @ F3 )
=> ( ( member @ A @ X @ S2 )
=> ( ( member @ A @ Y @ S2 )
=> ( ( F3 @ Y @ ( F3 @ X @ Z ) )
= ( F3 @ X @ ( F3 @ Y @ Z ) ) ) ) ) ) ).
% comp_fun_commute_on.fun_left_comm
thf(fact_6142_finite__update__induct,axiom,
! [B: $tType,A: $tType,F3: A > B,C3: B,P2: ( A > B ) > $o] :
( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [A6: A] :
( ( F3 @ A6 )
!= C3 ) ) )
=> ( ( P2
@ ^ [A6: A] : C3 )
=> ( ! [A4: A,B4: B,F2: A > B] :
( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [C4: A] :
( ( F2 @ C4 )
!= C3 ) ) )
=> ( ( ( F2 @ A4 )
= C3 )
=> ( ( B4 != C3 )
=> ( ( P2 @ F2 )
=> ( P2 @ ( fun_upd @ A @ B @ F2 @ A4 @ B4 ) ) ) ) ) )
=> ( P2 @ F3 ) ) ) ) ).
% finite_update_induct
thf(fact_6143_comp__fun__commute__insort,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( finite6289374366891150609ommute @ A @ ( list @ A )
@ ( linorder_insort_key @ A @ A
@ ^ [X5: A] : X5 ) ) ) ).
% comp_fun_commute_insort
thf(fact_6144_comp__fun__commute__def_H,axiom,
! [B: $tType,A: $tType] :
( ( finite6289374366891150609ommute @ A @ B )
= ( finite4664212375090638736ute_on @ A @ B @ ( top_top @ ( set @ A ) ) ) ) ).
% comp_fun_commute_def'
thf(fact_6145_Finite__Set_Ofold__cong,axiom,
! [B: $tType,A: $tType,S2: set @ A,F3: A > B > B,G3: A > B > B,A5: set @ A,S: B,T2: B,B6: set @ A] :
( ( finite4664212375090638736ute_on @ A @ B @ S2 @ F3 )
=> ( ( finite4664212375090638736ute_on @ A @ B @ S2 @ G3 )
=> ( ( ord_less_eq @ ( set @ A ) @ A5 @ S2 )
=> ( ( finite_finite2 @ A @ A5 )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ A5 )
=> ( ( F3 @ X4 )
= ( G3 @ X4 ) ) )
=> ( ( S = T2 )
=> ( ( A5 = B6 )
=> ( ( finite_fold @ A @ B @ F3 @ S @ A5 )
= ( finite_fold @ A @ B @ G3 @ T2 @ B6 ) ) ) ) ) ) ) ) ) ).
% Finite_Set.fold_cong
thf(fact_6146_fold__atLeastAtMost__nat,axiom,
! [A: $tType,F3: nat > A > A,A2: nat,B2: nat,Acc3: A] :
( ( finite6289374366891150609ommute @ nat @ A @ F3 )
=> ( ( set_fo6178422350223883121st_nat @ A @ F3 @ A2 @ B2 @ Acc3 )
= ( finite_fold @ nat @ A @ F3 @ Acc3 @ ( set_or1337092689740270186AtMost @ nat @ A2 @ B2 ) ) ) ) ).
% fold_atLeastAtMost_nat
thf(fact_6147_fun__upd__image,axiom,
! [A: $tType,B: $tType,X: B,A5: set @ B,F3: B > A,Y: A] :
( ( ( member @ B @ X @ A5 )
=> ( ( image2 @ B @ A @ ( fun_upd @ B @ A @ F3 @ X @ Y ) @ A5 )
= ( insert @ A @ Y @ ( image2 @ B @ A @ F3 @ ( minus_minus @ ( set @ B ) @ A5 @ ( insert @ B @ X @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) )
& ( ~ ( member @ B @ X @ A5 )
=> ( ( image2 @ B @ A @ ( fun_upd @ B @ A @ F3 @ X @ Y ) @ A5 )
= ( image2 @ B @ A @ F3 @ A5 ) ) ) ) ).
% fun_upd_image
thf(fact_6148_comp__fun__commute__on_Ocomp__comp__fun__commute__on,axiom,
! [B: $tType,A: $tType,C: $tType,S2: set @ A,F3: A > B > B,G3: C > A,R: set @ C] :
( ( finite4664212375090638736ute_on @ A @ B @ S2 @ F3 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image2 @ C @ A @ G3 @ ( top_top @ ( set @ C ) ) ) @ S2 )
=> ( finite4664212375090638736ute_on @ C @ B @ R @ ( comp @ A @ ( B > B ) @ C @ F3 @ G3 ) ) ) ) ).
% comp_fun_commute_on.comp_comp_fun_commute_on
thf(fact_6149_comp__fun__commute__on_Ofold__fun__left__comm,axiom,
! [B: $tType,A: $tType,S2: set @ A,F3: A > B > B,X: A,A5: set @ A,Z: B] :
( ( finite4664212375090638736ute_on @ A @ B @ S2 @ F3 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X @ A5 ) @ S2 )
=> ( ( finite_finite2 @ A @ A5 )
=> ( ( F3 @ X @ ( finite_fold @ A @ B @ F3 @ Z @ A5 ) )
= ( finite_fold @ A @ B @ F3 @ ( F3 @ X @ Z ) @ A5 ) ) ) ) ) ).
% comp_fun_commute_on.fold_fun_left_comm
thf(fact_6150_comp__fun__commute__on_Ofold__insert2,axiom,
! [B: $tType,A: $tType,S2: set @ A,F3: A > B > B,X: A,A5: set @ A,Z: B] :
( ( finite4664212375090638736ute_on @ A @ B @ S2 @ F3 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X @ A5 ) @ S2 )
=> ( ( finite_finite2 @ A @ A5 )
=> ( ~ ( member @ A @ X @ A5 )
=> ( ( finite_fold @ A @ B @ F3 @ Z @ ( insert @ A @ X @ A5 ) )
= ( finite_fold @ A @ B @ F3 @ ( F3 @ X @ Z ) @ A5 ) ) ) ) ) ) ).
% comp_fun_commute_on.fold_insert2
thf(fact_6151_comp__fun__commute__on_Ofold__insert,axiom,
! [B: $tType,A: $tType,S2: set @ A,F3: A > B > B,X: A,A5: set @ A,Z: B] :
( ( finite4664212375090638736ute_on @ A @ B @ S2 @ F3 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X @ A5 ) @ S2 )
=> ( ( finite_finite2 @ A @ A5 )
=> ( ~ ( member @ A @ X @ A5 )
=> ( ( finite_fold @ A @ B @ F3 @ Z @ ( insert @ A @ X @ A5 ) )
= ( F3 @ X @ ( finite_fold @ A @ B @ F3 @ Z @ A5 ) ) ) ) ) ) ) ).
% comp_fun_commute_on.fold_insert
thf(fact_6152_comp__fun__commute__Pow__fold,axiom,
! [A: $tType] :
( finite6289374366891150609ommute @ A @ ( set @ ( set @ A ) )
@ ^ [X5: A,A7: set @ ( set @ A )] : ( sup_sup @ ( set @ ( set @ A ) ) @ A7 @ ( image2 @ ( set @ A ) @ ( set @ A ) @ ( insert @ A @ X5 ) @ A7 ) ) ) ).
% comp_fun_commute_Pow_fold
thf(fact_6153_comp__fun__commute__on_Ofold__rec,axiom,
! [B: $tType,A: $tType,S2: set @ A,F3: A > B > B,A5: set @ A,X: A,Z: B] :
( ( finite4664212375090638736ute_on @ A @ B @ S2 @ F3 )
=> ( ( ord_less_eq @ ( set @ A ) @ A5 @ S2 )
=> ( ( finite_finite2 @ A @ A5 )
=> ( ( member @ A @ X @ A5 )
=> ( ( finite_fold @ A @ B @ F3 @ Z @ A5 )
= ( F3 @ X @ ( finite_fold @ A @ B @ F3 @ Z @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ).
% comp_fun_commute_on.fold_rec
thf(fact_6154_comp__fun__commute__on_Ofold__insert__remove,axiom,
! [B: $tType,A: $tType,S2: set @ A,F3: A > B > B,X: A,A5: set @ A,Z: B] :
( ( finite4664212375090638736ute_on @ A @ B @ S2 @ F3 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X @ A5 ) @ S2 )
=> ( ( finite_finite2 @ A @ A5 )
=> ( ( finite_fold @ A @ B @ F3 @ Z @ ( insert @ A @ X @ A5 ) )
= ( F3 @ X @ ( finite_fold @ A @ B @ F3 @ Z @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ).
% comp_fun_commute_on.fold_insert_remove
thf(fact_6155_set__nths,axiom,
! [A: $tType,Xs: list @ A,I6: set @ nat] :
( ( set2 @ A @ ( nths @ A @ Xs @ I6 ) )
= ( collect @ A
@ ^ [Uu3: A] :
? [I4: nat] :
( ( Uu3
= ( nth @ A @ Xs @ I4 ) )
& ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ Xs ) )
& ( member @ nat @ I4 @ I6 ) ) ) ) ).
% set_nths
thf(fact_6156_minus__fold__remove,axiom,
! [A: $tType,A5: set @ A,B6: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( minus_minus @ ( set @ A ) @ B6 @ A5 )
= ( finite_fold @ A @ ( set @ A ) @ ( remove @ A ) @ B6 @ A5 ) ) ) ).
% minus_fold_remove
thf(fact_6157_nths__nil,axiom,
! [A: $tType,A5: set @ nat] :
( ( nths @ A @ ( nil @ A ) @ A5 )
= ( nil @ A ) ) ).
% nths_nil
thf(fact_6158_nths__empty,axiom,
! [A: $tType,Xs: list @ A] :
( ( nths @ A @ Xs @ ( bot_bot @ ( set @ nat ) ) )
= ( nil @ A ) ) ).
% nths_empty
thf(fact_6159_in__set__nthsD,axiom,
! [A: $tType,X: A,Xs: list @ A,I6: set @ nat] :
( ( member @ A @ X @ ( set2 @ A @ ( nths @ A @ Xs @ I6 ) ) )
=> ( member @ A @ X @ ( set2 @ A @ Xs ) ) ) ).
% in_set_nthsD
thf(fact_6160_notin__set__nthsI,axiom,
! [A: $tType,X: A,Xs: list @ A,I6: set @ nat] :
( ~ ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ~ ( member @ A @ X @ ( set2 @ A @ ( nths @ A @ Xs @ I6 ) ) ) ) ).
% notin_set_nthsI
thf(fact_6161_distinct__nthsI,axiom,
! [A: $tType,Xs: list @ A,I6: set @ nat] :
( ( distinct @ A @ Xs )
=> ( distinct @ A @ ( nths @ A @ Xs @ I6 ) ) ) ).
% distinct_nthsI
thf(fact_6162_set__nths__subset,axiom,
! [A: $tType,Xs: list @ A,I6: set @ nat] : ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ ( nths @ A @ Xs @ I6 ) ) @ ( set2 @ A @ Xs ) ) ).
% set_nths_subset
thf(fact_6163_nths__all,axiom,
! [A: $tType,Xs: list @ A,I6: set @ nat] :
( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( member @ nat @ I3 @ I6 ) )
=> ( ( nths @ A @ Xs @ I6 )
= Xs ) ) ).
% nths_all
thf(fact_6164_length__nths,axiom,
! [A: $tType,Xs: list @ A,I6: set @ nat] :
( ( size_size @ ( list @ A ) @ ( nths @ A @ Xs @ I6 ) )
= ( finite_card @ nat
@ ( collect @ nat
@ ^ [I4: nat] :
( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ Xs ) )
& ( member @ nat @ I4 @ I6 ) ) ) ) ) ).
% length_nths
thf(fact_6165_remove__code_I1_J,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( remove @ A @ X @ ( set2 @ A @ Xs ) )
= ( set2 @ A @ ( removeAll @ A @ X @ Xs ) ) ) ).
% remove_code(1)
thf(fact_6166_remove__def,axiom,
! [A: $tType] :
( ( remove @ A )
= ( ^ [X5: A,A7: set @ A] : ( minus_minus @ ( set @ A ) @ A7 @ ( insert @ A @ X5 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% remove_def
thf(fact_6167_image__map__upd,axiom,
! [B: $tType,A: $tType,X: A,A5: set @ A,M2: A > ( option @ B ),Y: B] :
( ~ ( member @ A @ X @ A5 )
=> ( ( image2 @ A @ ( option @ B ) @ ( fun_upd @ A @ ( option @ B ) @ M2 @ X @ ( some @ B @ Y ) ) @ A5 )
= ( image2 @ A @ ( option @ B ) @ M2 @ A5 ) ) ) ).
% image_map_upd
thf(fact_6168_finite__range__updI,axiom,
! [A: $tType,B: $tType,F3: B > ( option @ A ),A2: B,B2: A] :
( ( finite_finite2 @ ( option @ A ) @ ( image2 @ B @ ( option @ A ) @ F3 @ ( top_top @ ( set @ B ) ) ) )
=> ( finite_finite2 @ ( option @ A ) @ ( image2 @ B @ ( option @ A ) @ ( fun_upd @ B @ ( option @ A ) @ F3 @ A2 @ ( some @ A @ B2 ) ) @ ( top_top @ ( set @ B ) ) ) ) ) ).
% finite_range_updI
thf(fact_6169_map__upd__Some__unfold,axiom,
! [B: $tType,A: $tType,M2: B > ( option @ A ),A2: B,B2: A,X: B,Y: A] :
( ( ( fun_upd @ B @ ( option @ A ) @ M2 @ A2 @ ( some @ A @ B2 ) @ X )
= ( some @ A @ Y ) )
= ( ( ( X = A2 )
& ( B2 = Y ) )
| ( ( X != A2 )
& ( ( M2 @ X )
= ( some @ A @ Y ) ) ) ) ) ).
% map_upd_Some_unfold
thf(fact_6170_map__upd__triv,axiom,
! [A: $tType,B: $tType,T2: B > ( option @ A ),K2: B,X: A] :
( ( ( T2 @ K2 )
= ( some @ A @ X ) )
=> ( ( fun_upd @ B @ ( option @ A ) @ T2 @ K2 @ ( some @ A @ X ) )
= T2 ) ) ).
% map_upd_triv
thf(fact_6171_map__upd__eqD1,axiom,
! [A: $tType,B: $tType,M2: A > ( option @ B ),A2: A,X: B,N: A > ( option @ B ),Y: B] :
( ( ( fun_upd @ A @ ( option @ B ) @ M2 @ A2 @ ( some @ B @ X ) )
= ( fun_upd @ A @ ( option @ B ) @ N @ A2 @ ( some @ B @ Y ) ) )
=> ( X = Y ) ) ).
% map_upd_eqD1
thf(fact_6172_map__upd__nonempty,axiom,
! [B: $tType,A: $tType,T2: A > ( option @ B ),K2: A,X: B] :
( ( fun_upd @ A @ ( option @ B ) @ T2 @ K2 @ ( some @ B @ X ) )
!= ( ^ [X5: A] : ( none @ B ) ) ) ).
% map_upd_nonempty
thf(fact_6173_graph__map__upd,axiom,
! [A: $tType,B: $tType,M2: A > ( option @ B ),K2: A,V: B] :
( ( graph @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ M2 @ K2 @ ( some @ B @ V ) ) )
= ( insert @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K2 @ V ) @ ( graph @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ M2 @ K2 @ ( none @ B ) ) ) ) ) ).
% graph_map_upd
thf(fact_6174_restrict__upd__same,axiom,
! [B: $tType,A: $tType,M2: A > ( option @ B ),X: A,Y: B] :
( ( restrict_map @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ M2 @ X @ ( some @ B @ Y ) ) @ ( 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_6175_restrict__map__to__empty,axiom,
! [B: $tType,A: $tType,M2: A > ( option @ B )] :
( ( restrict_map @ A @ B @ M2 @ ( bot_bot @ ( set @ A ) ) )
= ( ^ [X5: A] : ( none @ B ) ) ) ).
% restrict_map_to_empty
thf(fact_6176_graph__empty,axiom,
! [B: $tType,A: $tType] :
( ( graph @ A @ B
@ ^ [X5: A] : ( none @ B ) )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% graph_empty
thf(fact_6177_restrict__fun__upd,axiom,
! [B: $tType,A: $tType,X: A,D6: set @ A,M2: A > ( option @ B ),Y: option @ B] :
( ( ( member @ A @ X @ D6 )
=> ( ( restrict_map @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ M2 @ X @ Y ) @ D6 )
= ( fun_upd @ A @ ( option @ B ) @ ( restrict_map @ A @ B @ M2 @ ( minus_minus @ ( set @ A ) @ D6 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) @ X @ Y ) ) )
& ( ~ ( member @ A @ X @ D6 )
=> ( ( restrict_map @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ M2 @ X @ Y ) @ D6 )
= ( restrict_map @ A @ B @ M2 @ D6 ) ) ) ) ).
% restrict_fun_upd
thf(fact_6178_fun__upd__restrict__conv,axiom,
! [A: $tType,B: $tType,X: A,D6: set @ A,M2: A > ( option @ B ),Y: option @ B] :
( ( member @ A @ X @ D6 )
=> ( ( fun_upd @ A @ ( option @ B ) @ ( restrict_map @ A @ B @ M2 @ D6 ) @ X @ Y )
= ( fun_upd @ A @ ( option @ B ) @ ( restrict_map @ A @ B @ M2 @ ( minus_minus @ ( set @ A ) @ D6 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) @ X @ Y ) ) ) ).
% fun_upd_restrict_conv
thf(fact_6179_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_6180_graph__restrictD_I1_J,axiom,
! [B: $tType,A: $tType,K2: A,V: B,M2: A > ( option @ B ),A5: set @ A] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K2 @ V ) @ ( graph @ A @ B @ ( restrict_map @ A @ B @ M2 @ A5 ) ) )
=> ( member @ A @ K2 @ A5 ) ) ).
% graph_restrictD(1)
thf(fact_6181_graph__restrictD_I2_J,axiom,
! [A: $tType,B: $tType,K2: A,V: B,M2: A > ( option @ B ),A5: set @ A] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K2 @ V ) @ ( graph @ A @ B @ ( restrict_map @ A @ B @ M2 @ A5 ) ) )
=> ( ( M2 @ K2 )
= ( some @ B @ V ) ) ) ).
% graph_restrictD(2)
thf(fact_6182_in__graphD,axiom,
! [A: $tType,B: $tType,K2: A,V: B,M2: A > ( option @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K2 @ V ) @ ( graph @ A @ B @ M2 ) )
=> ( ( M2 @ K2 )
= ( some @ B @ V ) ) ) ).
% in_graphD
thf(fact_6183_in__graphI,axiom,
! [A: $tType,B: $tType,M2: B > ( option @ A ),K2: B,V: A] :
( ( ( M2 @ K2 )
= ( some @ A @ V ) )
=> ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ K2 @ V ) @ ( graph @ B @ A @ M2 ) ) ) ).
% in_graphI
thf(fact_6184_graph__def,axiom,
! [B: $tType,A: $tType] :
( ( graph @ A @ B )
= ( ^ [M3: A > ( option @ B )] :
( collect @ ( product_prod @ A @ B )
@ ^ [Uu3: product_prod @ A @ B] :
? [A6: A,B5: B] :
( ( Uu3
= ( product_Pair @ A @ B @ A6 @ B5 ) )
& ( ( M3 @ A6 )
= ( some @ B @ B5 ) ) ) ) ) ) ).
% graph_def
thf(fact_6185_fun__upd__restrict,axiom,
! [A: $tType,B: $tType,M2: A > ( option @ B ),D6: set @ A,X: A,Y: option @ B] :
( ( fun_upd @ A @ ( option @ B ) @ ( restrict_map @ A @ B @ M2 @ D6 ) @ X @ Y )
= ( fun_upd @ A @ ( option @ B ) @ ( restrict_map @ A @ B @ M2 @ ( minus_minus @ ( set @ A ) @ D6 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) @ X @ Y ) ) ).
% fun_upd_restrict
thf(fact_6186_restrict__complement__singleton__eq,axiom,
! [A: $tType,B: $tType,F3: A > ( option @ B ),X: A] :
( ( restrict_map @ A @ B @ F3 @ ( uminus_uminus @ ( set @ A ) @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( fun_upd @ A @ ( option @ B ) @ F3 @ X @ ( none @ B ) ) ) ).
% restrict_complement_singleton_eq
thf(fact_6187_restrict__map__upds,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B,D6: set @ A,M2: A > ( option @ B )] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ D6 )
=> ( ( restrict_map @ A @ B @ ( map_upds @ A @ B @ M2 @ Xs @ Ys ) @ D6 )
= ( map_upds @ A @ B @ ( restrict_map @ A @ B @ M2 @ ( minus_minus @ ( set @ A ) @ D6 @ ( set2 @ A @ Xs ) ) ) @ Xs @ Ys ) ) ) ) ).
% restrict_map_upds
thf(fact_6188_ran__map__upd,axiom,
! [A: $tType,B: $tType,M2: B > ( option @ A ),A2: B,B2: A] :
( ( ( M2 @ A2 )
= ( none @ A ) )
=> ( ( ran @ B @ A @ ( fun_upd @ B @ ( option @ A ) @ M2 @ A2 @ ( some @ A @ B2 ) ) )
= ( insert @ A @ B2 @ ( ran @ B @ A @ M2 ) ) ) ) ).
% ran_map_upd
thf(fact_6189_map__upds__apply__nontin,axiom,
! [B: $tType,A: $tType,X: A,Xs: list @ A,F3: A > ( option @ B ),Ys: list @ B] :
( ~ ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ( map_upds @ A @ B @ F3 @ Xs @ Ys @ X )
= ( F3 @ X ) ) ) ).
% map_upds_apply_nontin
thf(fact_6190_ran__empty,axiom,
! [B: $tType,A: $tType] :
( ( ran @ B @ A
@ ^ [X5: B] : ( none @ A ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% ran_empty
thf(fact_6191_map__upds__list__update2__drop,axiom,
! [A: $tType,B: $tType,Xs: list @ A,I2: nat,M2: A > ( option @ B ),Ys: list @ B,Y: B] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ I2 )
=> ( ( map_upds @ A @ B @ M2 @ Xs @ ( list_update @ B @ Ys @ I2 @ Y ) )
= ( map_upds @ A @ B @ M2 @ Xs @ Ys ) ) ) ).
% map_upds_list_update2_drop
thf(fact_6192_map__upds__twist,axiom,
! [A: $tType,B: $tType,A2: A,As: list @ A,M2: A > ( option @ B ),B2: B,Bs: list @ B] :
( ~ ( member @ A @ A2 @ ( set2 @ A @ As ) )
=> ( ( map_upds @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ M2 @ A2 @ ( some @ B @ B2 ) ) @ As @ Bs )
= ( fun_upd @ A @ ( option @ B ) @ ( map_upds @ A @ B @ M2 @ As @ Bs ) @ A2 @ ( some @ B @ B2 ) ) ) ) ).
% map_upds_twist
thf(fact_6193_ranI,axiom,
! [A: $tType,B: $tType,M2: B > ( option @ A ),A2: B,B2: A] :
( ( ( M2 @ A2 )
= ( some @ A @ B2 ) )
=> ( member @ A @ B2 @ ( ran @ B @ A @ M2 ) ) ) ).
% ranI
thf(fact_6194_ran__restrictD,axiom,
! [B: $tType,A: $tType,Y: A,M2: B > ( option @ A ),A5: set @ B] :
( ( member @ A @ Y @ ( ran @ B @ A @ ( restrict_map @ B @ A @ M2 @ A5 ) ) )
=> ? [X4: B] :
( ( member @ B @ X4 @ A5 )
& ( ( M2 @ X4 )
= ( some @ A @ Y ) ) ) ) ).
% ran_restrictD
thf(fact_6195_ran__def,axiom,
! [B: $tType,A: $tType] :
( ( ran @ A @ B )
= ( ^ [M3: A > ( option @ B )] :
( collect @ B
@ ^ [B5: B] :
? [A6: A] :
( ( M3 @ A6 )
= ( some @ B @ B5 ) ) ) ) ) ).
% ran_def
thf(fact_6196_ran__map__upd__Some,axiom,
! [B: $tType,A: $tType,M2: B > ( option @ A ),X: B,Y: A,Z: A] :
( ( ( M2 @ X )
= ( some @ A @ Y ) )
=> ( ( inj_on @ B @ ( option @ A ) @ M2 @ ( dom @ B @ A @ M2 ) )
=> ( ~ ( member @ A @ Z @ ( ran @ B @ A @ M2 ) )
=> ( ( ran @ B @ A @ ( fun_upd @ B @ ( option @ A ) @ M2 @ X @ ( some @ A @ Z ) ) )
= ( sup_sup @ ( set @ A ) @ ( minus_minus @ ( set @ A ) @ ( ran @ B @ A @ M2 ) @ ( insert @ A @ Y @ ( bot_bot @ ( set @ A ) ) ) ) @ ( insert @ A @ Z @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% ran_map_upd_Some
thf(fact_6197_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_6198_dom__eq__empty__conv,axiom,
! [B: $tType,A: $tType,F3: A > ( option @ B )] :
( ( ( dom @ A @ B @ F3 )
= ( bot_bot @ ( set @ A ) ) )
= ( F3
= ( ^ [X5: A] : ( none @ B ) ) ) ) ).
% dom_eq_empty_conv
thf(fact_6199_dom__const,axiom,
! [B: $tType,A: $tType,F3: A > B] :
( ( dom @ A @ B
@ ^ [X5: A] : ( some @ B @ ( F3 @ X5 ) ) )
= ( top_top @ ( set @ A ) ) ) ).
% dom_const
thf(fact_6200_dom__empty,axiom,
! [B: $tType,A: $tType] :
( ( dom @ A @ B
@ ^ [X5: A] : ( none @ B ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% dom_empty
thf(fact_6201_finite__graph__iff__finite__dom,axiom,
! [B: $tType,A: $tType,M2: A > ( option @ B )] :
( ( finite_finite2 @ ( product_prod @ A @ B ) @ ( graph @ A @ B @ M2 ) )
= ( finite_finite2 @ A @ ( dom @ A @ B @ M2 ) ) ) ).
% finite_graph_iff_finite_dom
thf(fact_6202_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_6203_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_6204_dom__fun__upd,axiom,
! [B: $tType,A: $tType,Y: option @ B,F3: A > ( option @ B ),X: A] :
( ( ( Y
= ( none @ B ) )
=> ( ( dom @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ F3 @ X @ Y ) )
= ( minus_minus @ ( set @ A ) @ ( dom @ A @ B @ F3 ) @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) )
& ( ( Y
!= ( none @ B ) )
=> ( ( dom @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ F3 @ X @ Y ) )
= ( insert @ A @ X @ ( dom @ A @ B @ F3 ) ) ) ) ) ).
% dom_fun_upd
thf(fact_6205_finite__ran,axiom,
! [B: $tType,A: $tType,P: A > ( option @ B )] :
( ( finite_finite2 @ A @ ( dom @ A @ B @ P ) )
=> ( finite_finite2 @ B @ ( ran @ A @ B @ P ) ) ) ).
% finite_ran
thf(fact_6206_insert__dom,axiom,
! [A: $tType,B: $tType,F3: B > ( option @ A ),X: B,Y: A] :
( ( ( F3 @ X )
= ( some @ A @ Y ) )
=> ( ( insert @ B @ X @ ( dom @ B @ A @ F3 ) )
= ( dom @ B @ A @ F3 ) ) ) ).
% insert_dom
thf(fact_6207_domD,axiom,
! [A: $tType,B: $tType,A2: A,M2: A > ( option @ B )] :
( ( member @ A @ A2 @ ( dom @ A @ B @ M2 ) )
=> ? [B4: B] :
( ( M2 @ A2 )
= ( some @ B @ B4 ) ) ) ).
% domD
thf(fact_6208_domI,axiom,
! [A: $tType,B: $tType,M2: B > ( option @ A ),A2: B,B2: A] :
( ( ( M2 @ A2 )
= ( some @ A @ B2 ) )
=> ( member @ B @ A2 @ ( dom @ B @ A @ M2 ) ) ) ).
% domI
thf(fact_6209_finite__map__freshness,axiom,
! [A: $tType,B: $tType,F3: A > ( option @ B )] :
( ( finite_finite2 @ A @ ( dom @ A @ B @ F3 ) )
=> ( ~ ( finite_finite2 @ A @ ( top_top @ ( set @ A ) ) )
=> ? [X4: A] :
( ( F3 @ X4 )
= ( none @ B ) ) ) ) ).
% finite_map_freshness
thf(fact_6210_finite__set__of__finite__maps,axiom,
! [B: $tType,A: $tType,A5: set @ A,B6: set @ B] :
( ( finite_finite2 @ A @ A5 )
=> ( ( finite_finite2 @ B @ B6 )
=> ( finite_finite2 @ ( A > ( option @ B ) )
@ ( collect @ ( A > ( option @ B ) )
@ ^ [M3: A > ( option @ B )] :
( ( ( dom @ A @ B @ M3 )
= A5 )
& ( ord_less_eq @ ( set @ B ) @ ( ran @ A @ B @ M3 ) @ B6 ) ) ) ) ) ) ).
% finite_set_of_finite_maps
thf(fact_6211_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_6212_finite__Map__induct,axiom,
! [B: $tType,A: $tType,M2: A > ( option @ B ),P2: ( A > ( option @ B ) ) > $o] :
( ( finite_finite2 @ A @ ( dom @ A @ B @ M2 ) )
=> ( ( P2
@ ^ [X5: A] : ( none @ B ) )
=> ( ! [K: A,V3: B,M: A > ( option @ B )] :
( ( finite_finite2 @ A @ ( dom @ A @ B @ M ) )
=> ( ~ ( member @ A @ K @ ( dom @ A @ B @ M ) )
=> ( ( P2 @ M )
=> ( P2 @ ( fun_upd @ A @ ( option @ B ) @ M @ K @ ( some @ B @ V3 ) ) ) ) ) )
=> ( P2 @ M2 ) ) ) ) ).
% finite_Map_induct
thf(fact_6213_dom__eq__singleton__conv,axiom,
! [A: $tType,B: $tType,F3: A > ( option @ B ),X: A] :
( ( ( dom @ A @ B @ F3 )
= ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( ? [V5: B] :
( F3
= ( fun_upd @ A @ ( option @ B )
@ ^ [X5: A] : ( none @ B )
@ X
@ ( some @ B @ V5 ) ) ) ) ) ).
% dom_eq_singleton_conv
thf(fact_6214_finite__subsets__at__top__finite,axiom,
! [A: $tType,A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( finite5375528669736107172at_top @ A @ A5 )
= ( principal @ ( set @ A ) @ ( insert @ ( set @ A ) @ A5 @ ( bot_bot @ ( set @ ( set @ A ) ) ) ) ) ) ) ).
% finite_subsets_at_top_finite
thf(fact_6215_Max_Oeq__fold_H,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( lattic643756798349783984er_Max @ A )
= ( ^ [A7: set @ A] :
( the2 @ A
@ ( finite_fold @ A @ ( option @ A )
@ ^ [X5: A,Y6: option @ A] : ( some @ A @ ( case_option @ A @ A @ X5 @ ( ord_max @ A @ X5 ) @ Y6 ) )
@ ( none @ A )
@ A7 ) ) ) ) ) ).
% Max.eq_fold'
thf(fact_6216_option_Ocollapse,axiom,
! [A: $tType,Option: option @ A] :
( ( Option
!= ( none @ A ) )
=> ( ( some @ A @ ( the2 @ A @ Option ) )
= Option ) ) ).
% option.collapse
thf(fact_6217_comp__the__Some,axiom,
! [A: $tType] :
( ( comp @ ( option @ A ) @ A @ A @ ( the2 @ A ) @ ( some @ A ) )
= ( id @ A ) ) ).
% comp_the_Some
thf(fact_6218_eventually__finite__subsets__at__top__weakI,axiom,
! [A: $tType,A5: set @ A,P2: ( set @ A ) > $o] :
( ! [X17: set @ A] :
( ( finite_finite2 @ A @ X17 )
=> ( ( ord_less_eq @ ( set @ A ) @ X17 @ A5 )
=> ( P2 @ X17 ) ) )
=> ( eventually @ ( set @ A ) @ P2 @ ( finite5375528669736107172at_top @ A @ A5 ) ) ) ).
% eventually_finite_subsets_at_top_weakI
thf(fact_6219_finite__subsets__at__top__neq__bot,axiom,
! [A: $tType,A5: set @ A] :
( ( finite5375528669736107172at_top @ A @ A5 )
!= ( bot_bot @ ( filter @ ( set @ A ) ) ) ) ).
% finite_subsets_at_top_neq_bot
thf(fact_6220_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_6221_eventually__finite__subsets__at__top__finite,axiom,
! [A: $tType,A5: set @ A,P2: ( set @ A ) > $o] :
( ( finite_finite2 @ A @ A5 )
=> ( ( eventually @ ( set @ A ) @ P2 @ ( finite5375528669736107172at_top @ A @ A5 ) )
= ( P2 @ A5 ) ) ) ).
% eventually_finite_subsets_at_top_finite
thf(fact_6222_option_Osel,axiom,
! [A: $tType,X2: A] :
( ( the2 @ A @ ( some @ A @ X2 ) )
= X2 ) ).
% option.sel
thf(fact_6223_option_Oexhaust__sel,axiom,
! [A: $tType,Option: option @ A] :
( ( Option
!= ( none @ A ) )
=> ( Option
= ( some @ A @ ( the2 @ A @ Option ) ) ) ) ).
% option.exhaust_sel
thf(fact_6224_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_6225_Option_Othese__def,axiom,
! [A: $tType] :
( ( these @ A )
= ( ^ [A7: set @ ( option @ A )] :
( image2 @ ( option @ A ) @ A @ ( the2 @ A )
@ ( collect @ ( option @ A )
@ ^ [X5: option @ A] :
( ( member @ ( option @ A ) @ X5 @ A7 )
& ( X5
!= ( none @ A ) ) ) ) ) ) ) ).
% Option.these_def
thf(fact_6226_Max_Oinfinite,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A] :
( ~ ( finite_finite2 @ A @ A5 )
=> ( ( lattic643756798349783984er_Max @ A @ A5 )
= ( the2 @ A @ ( none @ A ) ) ) ) ) ).
% Max.infinite
thf(fact_6227_option_Osplit__sel__asm,axiom,
! [B: $tType,A: $tType,P2: B > $o,F1: B,F22: A > B,Option: option @ A] :
( ( P2 @ ( case_option @ B @ A @ F1 @ F22 @ Option ) )
= ( ~ ( ( ( Option
= ( none @ A ) )
& ~ ( P2 @ F1 ) )
| ( ( Option
= ( some @ A @ ( the2 @ A @ Option ) ) )
& ~ ( P2 @ ( F22 @ ( the2 @ A @ Option ) ) ) ) ) ) ) ).
% option.split_sel_asm
thf(fact_6228_option_Osplit__sel,axiom,
! [B: $tType,A: $tType,P2: B > $o,F1: B,F22: A > B,Option: option @ A] :
( ( P2 @ ( case_option @ B @ A @ F1 @ F22 @ Option ) )
= ( ( ( Option
= ( none @ A ) )
=> ( P2 @ F1 ) )
& ( ( Option
= ( some @ A @ ( the2 @ A @ Option ) ) )
=> ( P2 @ ( F22 @ ( the2 @ A @ Option ) ) ) ) ) ) ).
% option.split_sel
thf(fact_6229_eventually__finite__subsets__at__top,axiom,
! [A: $tType,P2: ( set @ A ) > $o,A5: set @ A] :
( ( eventually @ ( set @ A ) @ P2 @ ( finite5375528669736107172at_top @ A @ A5 ) )
= ( ? [X9: set @ A] :
( ( finite_finite2 @ A @ X9 )
& ( ord_less_eq @ ( set @ A ) @ X9 @ A5 )
& ! [Y10: set @ A] :
( ( ( finite_finite2 @ A @ Y10 )
& ( ord_less_eq @ ( set @ A ) @ X9 @ Y10 )
& ( ord_less_eq @ ( set @ A ) @ Y10 @ A5 ) )
=> ( P2 @ Y10 ) ) ) ) ) ).
% eventually_finite_subsets_at_top
thf(fact_6230_graph__eq__to__snd__dom,axiom,
! [B: $tType,A: $tType] :
( ( graph @ A @ B )
= ( ^ [M3: A > ( option @ B )] :
( image2 @ A @ ( product_prod @ A @ B )
@ ^ [X5: A] : ( product_Pair @ A @ B @ X5 @ ( the2 @ B @ ( M3 @ X5 ) ) )
@ ( dom @ A @ B @ M3 ) ) ) ) ).
% graph_eq_to_snd_dom
thf(fact_6231_finite__subsets__at__top__def,axiom,
! [A: $tType] :
( ( finite5375528669736107172at_top @ A )
= ( ^ [A7: set @ A] :
( complete_Inf_Inf @ ( filter @ ( set @ A ) )
@ ( image2 @ ( set @ A ) @ ( filter @ ( set @ A ) )
@ ^ [X9: set @ A] :
( principal @ ( set @ A )
@ ( collect @ ( set @ A )
@ ^ [Y10: set @ A] :
( ( finite_finite2 @ A @ Y10 )
& ( ord_less_eq @ ( set @ A ) @ X9 @ Y10 )
& ( ord_less_eq @ ( set @ A ) @ Y10 @ A7 ) ) ) )
@ ( collect @ ( set @ A )
@ ^ [X9: set @ A] :
( ( finite_finite2 @ A @ X9 )
& ( ord_less_eq @ ( set @ A ) @ X9 @ A7 ) ) ) ) ) ) ) ).
% finite_subsets_at_top_def
thf(fact_6232_filterlim__finite__subsets__at__top,axiom,
! [A: $tType,B: $tType,F3: A > ( set @ B ),A5: set @ B,F5: filter @ A] :
( ( filterlim @ A @ ( set @ B ) @ F3 @ ( finite5375528669736107172at_top @ B @ A5 ) @ F5 )
= ( ! [X9: set @ B] :
( ( ( finite_finite2 @ B @ X9 )
& ( ord_less_eq @ ( set @ B ) @ X9 @ A5 ) )
=> ( eventually @ A
@ ^ [Y6: A] :
( ( finite_finite2 @ B @ ( F3 @ Y6 ) )
& ( ord_less_eq @ ( set @ B ) @ X9 @ ( F3 @ Y6 ) )
& ( ord_less_eq @ ( set @ B ) @ ( F3 @ Y6 ) @ A5 ) )
@ F5 ) ) ) ) ).
% filterlim_finite_subsets_at_top
thf(fact_6233_Sup__fin_Oeq__fold_H,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ( ( lattic5882676163264333800up_fin @ A )
= ( ^ [A7: set @ A] :
( the2 @ A
@ ( finite_fold @ A @ ( option @ A )
@ ^ [X5: A,Y6: option @ A] : ( some @ A @ ( case_option @ A @ A @ X5 @ ( sup_sup @ A @ X5 ) @ Y6 ) )
@ ( none @ A )
@ A7 ) ) ) ) ) ).
% Sup_fin.eq_fold'
thf(fact_6234_Inf__fin_Oeq__fold_H,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ( ( lattic7752659483105999362nf_fin @ A )
= ( ^ [A7: set @ A] :
( the2 @ A
@ ( finite_fold @ A @ ( option @ A )
@ ^ [X5: A,Y6: option @ A] : ( some @ A @ ( case_option @ A @ A @ X5 @ ( inf_inf @ A @ X5 ) @ Y6 ) )
@ ( none @ A )
@ A7 ) ) ) ) ) ).
% Inf_fin.eq_fold'
thf(fact_6235_inf__Sup__absorb,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [A5: set @ A,A2: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( member @ A @ A2 @ A5 )
=> ( ( inf_inf @ A @ A2 @ ( lattic5882676163264333800up_fin @ A @ A5 ) )
= A2 ) ) ) ) ).
% inf_Sup_absorb
thf(fact_6236_Sup__fin_Osingleton,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A] :
( ( lattic5882676163264333800up_fin @ A @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= X ) ) ).
% Sup_fin.singleton
thf(fact_6237_Inf__fin_Osingleton,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X: A] :
( ( lattic7752659483105999362nf_fin @ A @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= X ) ) ).
% Inf_fin.singleton
thf(fact_6238_sup__Inf__absorb,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [A5: set @ A,A2: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( member @ A @ A2 @ A5 )
=> ( ( sup_sup @ A @ ( lattic7752659483105999362nf_fin @ A @ A5 ) @ A2 )
= A2 ) ) ) ) ).
% sup_Inf_absorb
thf(fact_6239_Inf__fin_Oinsert,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic7752659483105999362nf_fin @ A @ ( insert @ A @ X @ A5 ) )
= ( inf_inf @ A @ X @ ( lattic7752659483105999362nf_fin @ A @ A5 ) ) ) ) ) ) ).
% Inf_fin.insert
thf(fact_6240_Sup__fin_Oinsert,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic5882676163264333800up_fin @ A @ ( insert @ A @ X @ A5 ) )
= ( sup_sup @ A @ X @ ( lattic5882676163264333800up_fin @ A @ A5 ) ) ) ) ) ) ).
% Sup_fin.insert
thf(fact_6241_Inf__fin__le__Sup__fin,axiom,
! [A: $tType] :
( ( lattice @ A )
=> ! [A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ord_less_eq @ A @ ( lattic7752659483105999362nf_fin @ A @ A5 ) @ ( lattic5882676163264333800up_fin @ A @ A5 ) ) ) ) ) ).
% Inf_fin_le_Sup_fin
thf(fact_6242_Sup__fin_OcoboundedI,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A5: set @ A,A2: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( member @ A @ A2 @ A5 )
=> ( ord_less_eq @ A @ A2 @ ( lattic5882676163264333800up_fin @ A @ A5 ) ) ) ) ) ).
% Sup_fin.coboundedI
thf(fact_6243_Inf__fin_OcoboundedI,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A5: set @ A,A2: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( member @ A @ A2 @ A5 )
=> ( ord_less_eq @ A @ ( lattic7752659483105999362nf_fin @ A @ A5 ) @ A2 ) ) ) ) ).
% Inf_fin.coboundedI
thf(fact_6244_Inf__fin_Oin__idem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( member @ A @ X @ A5 )
=> ( ( inf_inf @ A @ X @ ( lattic7752659483105999362nf_fin @ A @ A5 ) )
= ( lattic7752659483105999362nf_fin @ A @ A5 ) ) ) ) ) ).
% Inf_fin.in_idem
thf(fact_6245_Sup__fin_Oin__idem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( member @ A @ X @ A5 )
=> ( ( sup_sup @ A @ X @ ( lattic5882676163264333800up_fin @ A @ A5 ) )
= ( lattic5882676163264333800up_fin @ A @ A5 ) ) ) ) ) ).
% Sup_fin.in_idem
thf(fact_6246_Sup__fin__Max,axiom,
! [A: $tType] :
( ( ( semilattice_sup @ A )
& ( linorder @ A ) )
=> ( ( lattic5882676163264333800up_fin @ A )
= ( lattic643756798349783984er_Max @ A ) ) ) ).
% Sup_fin_Max
thf(fact_6247_Inf__fin_OboundedE,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ X @ ( lattic7752659483105999362nf_fin @ A @ A5 ) )
=> ! [A11: A] :
( ( member @ A @ A11 @ A5 )
=> ( ord_less_eq @ A @ X @ A11 ) ) ) ) ) ) ).
% Inf_fin.boundedE
thf(fact_6248_Inf__fin_OboundedI,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [A4: A] :
( ( member @ A @ A4 @ A5 )
=> ( ord_less_eq @ A @ X @ A4 ) )
=> ( ord_less_eq @ A @ X @ ( lattic7752659483105999362nf_fin @ A @ A5 ) ) ) ) ) ) ).
% Inf_fin.boundedI
thf(fact_6249_Sup__fin_OboundedE,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ ( lattic5882676163264333800up_fin @ A @ A5 ) @ X )
=> ! [A11: A] :
( ( member @ A @ A11 @ A5 )
=> ( ord_less_eq @ A @ A11 @ X ) ) ) ) ) ) ).
% Sup_fin.boundedE
thf(fact_6250_Sup__fin_OboundedI,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [A4: A] :
( ( member @ A @ A4 @ A5 )
=> ( ord_less_eq @ A @ A4 @ X ) )
=> ( ord_less_eq @ A @ ( lattic5882676163264333800up_fin @ A @ A5 ) @ X ) ) ) ) ) ).
% Sup_fin.boundedI
thf(fact_6251_Inf__fin_Obounded__iff,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ X @ ( lattic7752659483105999362nf_fin @ A @ A5 ) )
= ( ! [X5: A] :
( ( member @ A @ X5 @ A5 )
=> ( ord_less_eq @ A @ X @ X5 ) ) ) ) ) ) ) ).
% Inf_fin.bounded_iff
thf(fact_6252_Sup__fin_Obounded__iff,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ ( lattic5882676163264333800up_fin @ A @ A5 ) @ X )
= ( ! [X5: A] :
( ( member @ A @ X5 @ A5 )
=> ( ord_less_eq @ A @ X5 @ X ) ) ) ) ) ) ) ).
% Sup_fin.bounded_iff
thf(fact_6253_cSup__eq__Sup__fin,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X7: set @ A] :
( ( finite_finite2 @ A @ X7 )
=> ( ( X7
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Sup_Sup @ A @ X7 )
= ( lattic5882676163264333800up_fin @ A @ X7 ) ) ) ) ) ).
% cSup_eq_Sup_fin
thf(fact_6254_Sup__fin__Sup,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic5882676163264333800up_fin @ A @ A5 )
= ( complete_Sup_Sup @ A @ A5 ) ) ) ) ) ).
% Sup_fin_Sup
thf(fact_6255_cInf__eq__Inf__fin,axiom,
! [A: $tType] :
( ( condit1219197933456340205attice @ A )
=> ! [X7: set @ A] :
( ( finite_finite2 @ A @ X7 )
=> ( ( X7
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Inf_Inf @ A @ X7 )
= ( lattic7752659483105999362nf_fin @ A @ X7 ) ) ) ) ) ).
% cInf_eq_Inf_fin
thf(fact_6256_Inf__fin__Inf,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic7752659483105999362nf_fin @ A @ A5 )
= ( complete_Inf_Inf @ A @ A5 ) ) ) ) ) ).
% Inf_fin_Inf
thf(fact_6257_Inf__fin_Oinfinite,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A5: set @ A] :
( ~ ( finite_finite2 @ A @ A5 )
=> ( ( lattic7752659483105999362nf_fin @ A @ A5 )
= ( the2 @ A @ ( none @ A ) ) ) ) ) ).
% Inf_fin.infinite
thf(fact_6258_Sup__fin_Oinfinite,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A5: set @ A] :
( ~ ( finite_finite2 @ A @ A5 )
=> ( ( lattic5882676163264333800up_fin @ A @ A5 )
= ( the2 @ A @ ( none @ A ) ) ) ) ) ).
% Sup_fin.infinite
thf(fact_6259_Inf__fin_Osubset__imp,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A5: set @ A,B6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B6 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B6 )
=> ( ord_less_eq @ A @ ( lattic7752659483105999362nf_fin @ A @ B6 ) @ ( lattic7752659483105999362nf_fin @ A @ A5 ) ) ) ) ) ) ).
% Inf_fin.subset_imp
thf(fact_6260_Sup__fin_Osubset__imp,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A5: set @ A,B6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B6 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B6 )
=> ( ord_less_eq @ A @ ( lattic5882676163264333800up_fin @ A @ A5 ) @ ( lattic5882676163264333800up_fin @ A @ B6 ) ) ) ) ) ) ).
% Sup_fin.subset_imp
thf(fact_6261_Inf__fin_Ohom__commute,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [H2: A > A,N6: set @ A] :
( ! [X4: A,Y3: A] :
( ( H2 @ ( inf_inf @ A @ X4 @ Y3 ) )
= ( inf_inf @ A @ ( H2 @ X4 ) @ ( H2 @ Y3 ) ) )
=> ( ( finite_finite2 @ A @ N6 )
=> ( ( N6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( H2 @ ( lattic7752659483105999362nf_fin @ A @ N6 ) )
= ( lattic7752659483105999362nf_fin @ A @ ( image2 @ A @ A @ H2 @ N6 ) ) ) ) ) ) ) ).
% Inf_fin.hom_commute
thf(fact_6262_Sup__fin_Ohom__commute,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [H2: A > A,N6: set @ A] :
( ! [X4: A,Y3: A] :
( ( H2 @ ( sup_sup @ A @ X4 @ Y3 ) )
= ( sup_sup @ A @ ( H2 @ X4 ) @ ( H2 @ Y3 ) ) )
=> ( ( finite_finite2 @ A @ N6 )
=> ( ( N6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( H2 @ ( lattic5882676163264333800up_fin @ A @ N6 ) )
= ( lattic5882676163264333800up_fin @ A @ ( image2 @ A @ A @ H2 @ N6 ) ) ) ) ) ) ) ).
% Sup_fin.hom_commute
thf(fact_6263_Inf__fin_Osubset,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A5: set @ A,B6: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( B6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ B6 @ A5 )
=> ( ( inf_inf @ A @ ( lattic7752659483105999362nf_fin @ A @ B6 ) @ ( lattic7752659483105999362nf_fin @ A @ A5 ) )
= ( lattic7752659483105999362nf_fin @ A @ A5 ) ) ) ) ) ) ).
% Inf_fin.subset
thf(fact_6264_Sup__fin_Osubset,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A5: set @ A,B6: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( B6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ B6 @ A5 )
=> ( ( sup_sup @ A @ ( lattic5882676163264333800up_fin @ A @ B6 ) @ ( lattic5882676163264333800up_fin @ A @ A5 ) )
= ( lattic5882676163264333800up_fin @ A @ A5 ) ) ) ) ) ) ).
% Sup_fin.subset
thf(fact_6265_Inf__fin_Oinsert__not__elem,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ~ ( member @ A @ X @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic7752659483105999362nf_fin @ A @ ( insert @ A @ X @ A5 ) )
= ( inf_inf @ A @ X @ ( lattic7752659483105999362nf_fin @ A @ A5 ) ) ) ) ) ) ) ).
% Inf_fin.insert_not_elem
thf(fact_6266_Inf__fin_Oclosed,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X4: A,Y3: A] : ( member @ A @ ( inf_inf @ A @ X4 @ Y3 ) @ ( insert @ A @ X4 @ ( insert @ A @ Y3 @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( member @ A @ ( lattic7752659483105999362nf_fin @ A @ A5 ) @ A5 ) ) ) ) ) ).
% Inf_fin.closed
thf(fact_6267_Sup__fin_Oclosed,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X4: A,Y3: A] : ( member @ A @ ( sup_sup @ A @ X4 @ Y3 ) @ ( insert @ A @ X4 @ ( insert @ A @ Y3 @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( member @ A @ ( lattic5882676163264333800up_fin @ A @ A5 ) @ A5 ) ) ) ) ) ).
% Sup_fin.closed
thf(fact_6268_Sup__fin_Oinsert__not__elem,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ~ ( member @ A @ X @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic5882676163264333800up_fin @ A @ ( insert @ A @ X @ A5 ) )
= ( sup_sup @ A @ X @ ( lattic5882676163264333800up_fin @ A @ A5 ) ) ) ) ) ) ) ).
% Sup_fin.insert_not_elem
thf(fact_6269_Inf__fin_Ounion,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A5: set @ A,B6: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B6 )
=> ( ( B6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic7752659483105999362nf_fin @ A @ ( sup_sup @ ( set @ A ) @ A5 @ B6 ) )
= ( inf_inf @ A @ ( lattic7752659483105999362nf_fin @ A @ A5 ) @ ( lattic7752659483105999362nf_fin @ A @ B6 ) ) ) ) ) ) ) ) ).
% Inf_fin.union
thf(fact_6270_Sup__fin_Ounion,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A5: set @ A,B6: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B6 )
=> ( ( B6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic5882676163264333800up_fin @ A @ ( sup_sup @ ( set @ A ) @ A5 @ B6 ) )
= ( sup_sup @ A @ ( lattic5882676163264333800up_fin @ A @ A5 ) @ ( lattic5882676163264333800up_fin @ A @ B6 ) ) ) ) ) ) ) ) ).
% Sup_fin.union
thf(fact_6271_Inf__fin_Oeq__fold,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( lattic7752659483105999362nf_fin @ A @ ( insert @ A @ X @ A5 ) )
= ( finite_fold @ A @ A @ ( inf_inf @ A ) @ X @ A5 ) ) ) ) ).
% Inf_fin.eq_fold
thf(fact_6272_Sup__fin_Oeq__fold,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( lattic5882676163264333800up_fin @ A @ ( insert @ A @ X @ A5 ) )
= ( finite_fold @ A @ A @ ( sup_sup @ A ) @ X @ A5 ) ) ) ) ).
% Sup_fin.eq_fold
thf(fact_6273_inf__Sup1__distrib,axiom,
! [A: $tType] :
( ( distrib_lattice @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( inf_inf @ A @ X @ ( lattic5882676163264333800up_fin @ A @ A5 ) )
= ( lattic5882676163264333800up_fin @ A
@ ( collect @ A
@ ^ [Uu3: A] :
? [A6: A] :
( ( Uu3
= ( inf_inf @ A @ X @ A6 ) )
& ( member @ A @ A6 @ A5 ) ) ) ) ) ) ) ) ).
% inf_Sup1_distrib
thf(fact_6274_inf__Sup2__distrib,axiom,
! [A: $tType] :
( ( distrib_lattice @ A )
=> ! [A5: set @ A,B6: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B6 )
=> ( ( B6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( inf_inf @ A @ ( lattic5882676163264333800up_fin @ A @ A5 ) @ ( lattic5882676163264333800up_fin @ A @ B6 ) )
= ( lattic5882676163264333800up_fin @ A
@ ( collect @ A
@ ^ [Uu3: A] :
? [A6: A,B5: A] :
( ( Uu3
= ( inf_inf @ A @ A6 @ B5 ) )
& ( member @ A @ A6 @ A5 )
& ( member @ A @ B5 @ B6 ) ) ) ) ) ) ) ) ) ) ).
% inf_Sup2_distrib
thf(fact_6275_sup__Inf1__distrib,axiom,
! [A: $tType] :
( ( distrib_lattice @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( sup_sup @ A @ X @ ( lattic7752659483105999362nf_fin @ A @ A5 ) )
= ( lattic7752659483105999362nf_fin @ A
@ ( collect @ A
@ ^ [Uu3: A] :
? [A6: A] :
( ( Uu3
= ( sup_sup @ A @ X @ A6 ) )
& ( member @ A @ A6 @ A5 ) ) ) ) ) ) ) ) ).
% sup_Inf1_distrib
thf(fact_6276_sup__Inf2__distrib,axiom,
! [A: $tType] :
( ( distrib_lattice @ A )
=> ! [A5: set @ A,B6: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B6 )
=> ( ( B6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( sup_sup @ A @ ( lattic7752659483105999362nf_fin @ A @ A5 ) @ ( lattic7752659483105999362nf_fin @ A @ B6 ) )
= ( lattic7752659483105999362nf_fin @ A
@ ( collect @ A
@ ^ [Uu3: A] :
? [A6: A,B5: A] :
( ( Uu3
= ( sup_sup @ A @ A6 @ B5 ) )
& ( member @ A @ A6 @ A5 )
& ( member @ A @ B5 @ B6 ) ) ) ) ) ) ) ) ) ) ).
% sup_Inf2_distrib
thf(fact_6277_Inf__fin_Oremove,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( member @ A @ X @ A5 )
=> ( ( ( ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic7752659483105999362nf_fin @ A @ A5 )
= X ) )
& ( ( ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic7752659483105999362nf_fin @ A @ A5 )
= ( inf_inf @ A @ X @ ( lattic7752659483105999362nf_fin @ A @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ) ).
% Inf_fin.remove
thf(fact_6278_Inf__fin_Oinsert__remove,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( ( ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic7752659483105999362nf_fin @ A @ ( insert @ A @ X @ A5 ) )
= X ) )
& ( ( ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic7752659483105999362nf_fin @ A @ ( insert @ A @ X @ A5 ) )
= ( inf_inf @ A @ X @ ( lattic7752659483105999362nf_fin @ A @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ).
% Inf_fin.insert_remove
thf(fact_6279_Sup__fin_Oinsert__remove,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( ( ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic5882676163264333800up_fin @ A @ ( insert @ A @ X @ A5 ) )
= X ) )
& ( ( ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic5882676163264333800up_fin @ A @ ( insert @ A @ X @ A5 ) )
= ( sup_sup @ A @ X @ ( lattic5882676163264333800up_fin @ A @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ).
% Sup_fin.insert_remove
thf(fact_6280_Sup__fin_Oremove,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( member @ A @ X @ A5 )
=> ( ( ( ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic5882676163264333800up_fin @ A @ A5 )
= X ) )
& ( ( ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic5882676163264333800up_fin @ A @ A5 )
= ( sup_sup @ A @ X @ ( lattic5882676163264333800up_fin @ A @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ) ).
% Sup_fin.remove
thf(fact_6281_pos__deriv__imp__strict__mono,axiom,
! [F3: real > real,F10: real > real] :
( ! [X4: real] : ( has_field_derivative @ real @ F3 @ ( F10 @ X4 ) @ ( topolo174197925503356063within @ real @ X4 @ ( top_top @ ( set @ real ) ) ) )
=> ( ! [X4: real] : ( ord_less @ real @ ( zero_zero @ real ) @ ( F10 @ X4 ) )
=> ( order_strict_mono @ real @ real @ F3 ) ) ) ).
% pos_deriv_imp_strict_mono
thf(fact_6282_sorted__key__list__of__set__def,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ( ( linord144544945434240204of_set @ B @ A )
= ( ^ [F4: B > A] : ( finite_folding_F @ B @ ( list @ B ) @ ( linorder_insort_key @ B @ A @ F4 ) @ ( nil @ B ) ) ) ) ) ).
% sorted_key_list_of_set_def
thf(fact_6283_strict__mono__less__eq,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( order @ B ) )
=> ! [F3: A > B,X: A,Y: A] :
( ( order_strict_mono @ A @ B @ F3 )
=> ( ( ord_less_eq @ B @ ( F3 @ X ) @ ( F3 @ Y ) )
= ( ord_less_eq @ A @ X @ Y ) ) ) ) ).
% strict_mono_less_eq
thf(fact_6284_strict__mono__leD,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [R2: A > B,M2: A,N: A] :
( ( order_strict_mono @ A @ B @ R2 )
=> ( ( ord_less_eq @ A @ M2 @ N )
=> ( ord_less_eq @ B @ ( R2 @ M2 ) @ ( R2 @ N ) ) ) ) ) ).
% strict_mono_leD
thf(fact_6285_folding__on_OF_Ocong,axiom,
! [B: $tType,A: $tType] :
( ( finite_folding_F @ A @ B )
= ( finite_folding_F @ A @ B ) ) ).
% folding_on.F.cong
thf(fact_6286_strict__mono__eq,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( order @ B ) )
=> ! [F3: A > B,X: A,Y: A] :
( ( order_strict_mono @ A @ B @ F3 )
=> ( ( ( F3 @ X )
= ( F3 @ Y ) )
= ( X = Y ) ) ) ) ).
% strict_mono_eq
thf(fact_6287_strict__mono__less,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( order @ B ) )
=> ! [F3: A > B,X: A,Y: A] :
( ( order_strict_mono @ A @ B @ F3 )
=> ( ( ord_less @ B @ ( F3 @ X ) @ ( F3 @ Y ) )
= ( ord_less @ A @ X @ Y ) ) ) ) ).
% strict_mono_less
thf(fact_6288_strict__mono__def,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ( ( order_strict_mono @ A @ B )
= ( ^ [F4: A > B] :
! [X5: A,Y6: A] :
( ( ord_less @ A @ X5 @ Y6 )
=> ( ord_less @ B @ ( F4 @ X5 ) @ ( F4 @ Y6 ) ) ) ) ) ) ).
% strict_mono_def
thf(fact_6289_strict__monoI,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F3: A > B] :
( ! [X4: A,Y3: A] :
( ( ord_less @ A @ X4 @ Y3 )
=> ( ord_less @ B @ ( F3 @ X4 ) @ ( F3 @ Y3 ) ) )
=> ( order_strict_mono @ A @ B @ F3 ) ) ) ).
% strict_monoI
thf(fact_6290_strict__monoD,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F3: A > B,X: A,Y: A] :
( ( order_strict_mono @ A @ B @ F3 )
=> ( ( ord_less @ A @ X @ Y )
=> ( ord_less @ B @ ( F3 @ X ) @ ( F3 @ Y ) ) ) ) ) ).
% strict_monoD
thf(fact_6291_strict__mono__add,axiom,
! [A: $tType] :
( ( linordered_semidom @ A )
=> ! [K2: A] :
( order_strict_mono @ A @ A
@ ^ [N2: A] : ( plus_plus @ A @ N2 @ K2 ) ) ) ).
% strict_mono_add
thf(fact_6292_strict__mono__mono,axiom,
! [B: $tType,A: $tType] :
( ( ( order @ A )
& ( order @ B ) )
=> ! [F3: A > B] :
( ( order_strict_mono @ A @ B @ F3 )
=> ( order_mono @ A @ B @ F3 ) ) ) ).
% strict_mono_mono
thf(fact_6293_folding__on_Oinsert__remove,axiom,
! [B: $tType,A: $tType,S2: set @ A,F3: A > B > B,X: A,A5: set @ A,Z: B] :
( ( finite_folding_on @ A @ B @ S2 @ F3 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X @ A5 ) @ S2 )
=> ( ( finite_finite2 @ A @ A5 )
=> ( ( finite_folding_F @ A @ B @ F3 @ Z @ ( insert @ A @ X @ A5 ) )
= ( F3 @ X @ ( finite_folding_F @ A @ B @ F3 @ Z @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ).
% folding_on.insert_remove
thf(fact_6294_folding__on_Oremove,axiom,
! [B: $tType,A: $tType,S2: set @ A,F3: A > B > B,A5: set @ A,X: A,Z: B] :
( ( finite_folding_on @ A @ B @ S2 @ F3 )
=> ( ( ord_less_eq @ ( set @ A ) @ A5 @ S2 )
=> ( ( finite_finite2 @ A @ A5 )
=> ( ( member @ A @ X @ A5 )
=> ( ( finite_folding_F @ A @ B @ F3 @ Z @ A5 )
= ( F3 @ X @ ( finite_folding_F @ A @ B @ F3 @ Z @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ).
% folding_on.remove
thf(fact_6295_strict__mono__imp__increasing,axiom,
! [F3: nat > nat,N: nat] :
( ( order_strict_mono @ nat @ nat @ F3 )
=> ( ord_less_eq @ nat @ N @ ( F3 @ N ) ) ) ).
% strict_mono_imp_increasing
thf(fact_6296_infinite__enumerate,axiom,
! [S2: set @ nat] :
( ~ ( finite_finite2 @ nat @ S2 )
=> ? [R3: nat > nat] :
( ( order_strict_mono @ nat @ nat @ R3 )
& ! [N5: nat] : ( member @ nat @ ( R3 @ N5 ) @ S2 ) ) ) ).
% infinite_enumerate
thf(fact_6297_folding__on_Ointro,axiom,
! [B: $tType,A: $tType,S2: set @ A,F3: A > B > B] :
( ! [X4: A,Y3: A] :
( ( member @ A @ X4 @ S2 )
=> ( ( member @ A @ Y3 @ S2 )
=> ( ( comp @ B @ B @ B @ ( F3 @ Y3 ) @ ( F3 @ X4 ) )
= ( comp @ B @ B @ B @ ( F3 @ X4 ) @ ( F3 @ Y3 ) ) ) ) )
=> ( finite_folding_on @ A @ B @ S2 @ F3 ) ) ).
% folding_on.intro
thf(fact_6298_folding__on_Ocomp__fun__commute__on,axiom,
! [B: $tType,A: $tType,S2: set @ A,F3: A > B > B,X: A,Y: A] :
( ( finite_folding_on @ A @ B @ S2 @ F3 )
=> ( ( member @ A @ X @ S2 )
=> ( ( member @ A @ Y @ S2 )
=> ( ( comp @ B @ B @ B @ ( F3 @ Y ) @ ( F3 @ X ) )
= ( comp @ B @ B @ B @ ( F3 @ X ) @ ( F3 @ Y ) ) ) ) ) ) ).
% folding_on.comp_fun_commute_on
thf(fact_6299_folding__on__def,axiom,
! [B: $tType,A: $tType] :
( ( finite_folding_on @ A @ B )
= ( ^ [S7: set @ A,F4: A > B > B] :
! [X5: A,Y6: A] :
( ( member @ A @ X5 @ S7 )
=> ( ( member @ A @ Y6 @ S7 )
=> ( ( comp @ B @ B @ B @ ( F4 @ Y6 ) @ ( F4 @ X5 ) )
= ( comp @ B @ B @ B @ ( F4 @ X5 ) @ ( F4 @ Y6 ) ) ) ) ) ) ) ).
% folding_on_def
thf(fact_6300_strict__mono__Suc__iff,axiom,
! [A: $tType] :
( ( order @ A )
=> ( ( order_strict_mono @ nat @ A )
= ( ^ [F4: nat > A] :
! [N2: nat] : ( ord_less @ A @ ( F4 @ N2 ) @ ( F4 @ ( suc @ N2 ) ) ) ) ) ) ).
% strict_mono_Suc_iff
thf(fact_6301_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_6302_folding__on_Oempty,axiom,
! [A: $tType,B: $tType,S2: set @ A,F3: A > B > B,Z: B] :
( ( finite_folding_on @ A @ B @ S2 @ F3 )
=> ( ( finite_folding_F @ A @ B @ F3 @ Z @ ( bot_bot @ ( set @ A ) ) )
= Z ) ) ).
% folding_on.empty
thf(fact_6303_folding__on_Oinfinite,axiom,
! [A: $tType,B: $tType,S2: set @ A,F3: A > B > B,A5: set @ A,Z: B] :
( ( finite_folding_on @ A @ B @ S2 @ F3 )
=> ( ~ ( finite_finite2 @ A @ A5 )
=> ( ( finite_folding_F @ A @ B @ F3 @ Z @ A5 )
= Z ) ) ) ).
% folding_on.infinite
thf(fact_6304_folding__on_Oeq__fold,axiom,
! [B: $tType,A: $tType,S2: set @ A,F3: A > B > B,Z: B,A5: set @ A] :
( ( finite_folding_on @ A @ B @ S2 @ F3 )
=> ( ( finite_folding_F @ A @ B @ F3 @ Z @ A5 )
= ( finite_fold @ A @ B @ F3 @ Z @ A5 ) ) ) ).
% folding_on.eq_fold
thf(fact_6305_sorted__list__of__set_Ofold__insort__key_Ofolding__on__axioms,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( finite_folding_on @ A @ ( list @ A ) @ ( top_top @ ( set @ A ) )
@ ( linorder_insort_key @ A @ A
@ ^ [X5: A] : X5 ) ) ) ).
% sorted_list_of_set.fold_insort_key.folding_on_axioms
thf(fact_6306_card__def,axiom,
! [B: $tType] :
( ( finite_card @ B )
= ( finite_folding_F @ B @ nat
@ ^ [Uu3: B] : suc
@ ( zero_zero @ nat ) ) ) ).
% card_def
thf(fact_6307_summable__mono__reindex,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [G3: nat > nat,F3: nat > A] :
( ( order_strict_mono @ nat @ nat @ G3 )
=> ( ! [N3: nat] :
( ~ ( member @ nat @ N3 @ ( image2 @ nat @ nat @ G3 @ ( top_top @ ( set @ nat ) ) ) )
=> ( ( F3 @ N3 )
= ( zero_zero @ A ) ) )
=> ( ( summable @ A
@ ^ [N2: nat] : ( F3 @ ( G3 @ N2 ) ) )
= ( summable @ A @ F3 ) ) ) ) ) ).
% summable_mono_reindex
thf(fact_6308_sums__mono__reindex,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topolo4958980785337419405_space @ A ) )
=> ! [G3: nat > nat,F3: nat > A,C3: A] :
( ( order_strict_mono @ nat @ nat @ G3 )
=> ( ! [N3: nat] :
( ~ ( member @ nat @ N3 @ ( image2 @ nat @ nat @ G3 @ ( top_top @ ( set @ nat ) ) ) )
=> ( ( F3 @ N3 )
= ( zero_zero @ A ) ) )
=> ( ( sums @ A
@ ^ [N2: nat] : ( F3 @ ( G3 @ N2 ) )
@ C3 )
= ( sums @ A @ F3 @ C3 ) ) ) ) ) ).
% sums_mono_reindex
thf(fact_6309_suminf__mono__reindex,axiom,
! [A: $tType] :
( ( ( comm_monoid_add @ A )
& ( topological_t2_space @ A ) )
=> ! [G3: nat > nat,F3: nat > A] :
( ( order_strict_mono @ nat @ nat @ G3 )
=> ( ! [N3: nat] :
( ~ ( member @ nat @ N3 @ ( image2 @ nat @ nat @ G3 @ ( top_top @ ( set @ nat ) ) ) )
=> ( ( F3 @ N3 )
= ( zero_zero @ A ) ) )
=> ( ( suminf @ A
@ ^ [N2: nat] : ( F3 @ ( G3 @ N2 ) ) )
= ( suminf @ A @ F3 ) ) ) ) ) ).
% suminf_mono_reindex
thf(fact_6310_increasing__Bseq__subseq__iff,axiom,
! [A: $tType] :
( ( real_V822414075346904944vector @ A )
=> ! [F3: nat > A,G3: nat > nat] :
( ! [X4: nat,Y3: nat] :
( ( ord_less_eq @ nat @ X4 @ Y3 )
=> ( ord_less_eq @ real @ ( real_V7770717601297561774m_norm @ A @ ( F3 @ X4 ) ) @ ( real_V7770717601297561774m_norm @ A @ ( F3 @ Y3 ) ) ) )
=> ( ( order_strict_mono @ nat @ nat @ G3 )
=> ( ( bfun @ nat @ A
@ ^ [X5: nat] : ( F3 @ ( G3 @ X5 ) )
@ ( at_top @ nat ) )
= ( bfun @ nat @ A @ F3 @ ( at_top @ nat ) ) ) ) ) ) ).
% increasing_Bseq_subseq_iff
thf(fact_6311_folding__on_Oinsert,axiom,
! [B: $tType,A: $tType,S2: set @ A,F3: A > B > B,X: A,A5: set @ A,Z: B] :
( ( finite_folding_on @ A @ B @ S2 @ F3 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X @ A5 ) @ S2 )
=> ( ( finite_finite2 @ A @ A5 )
=> ( ~ ( member @ A @ X @ A5 )
=> ( ( finite_folding_F @ A @ B @ F3 @ Z @ ( insert @ A @ X @ A5 ) )
= ( F3 @ X @ ( finite_folding_F @ A @ B @ F3 @ Z @ A5 ) ) ) ) ) ) ) ).
% folding_on.insert
thf(fact_6312_folding__idem__on_Oinsert__idem,axiom,
! [B: $tType,A: $tType,S2: set @ A,F3: A > B > B,X: A,A5: set @ A,Z: B] :
( ( finite1890593828518410140dem_on @ A @ B @ S2 @ F3 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X @ A5 ) @ S2 )
=> ( ( finite_finite2 @ A @ A5 )
=> ( ( finite_folding_F @ A @ B @ F3 @ Z @ ( insert @ A @ X @ A5 ) )
= ( F3 @ X @ ( finite_folding_F @ A @ B @ F3 @ Z @ A5 ) ) ) ) ) ) ).
% folding_idem_on.insert_idem
thf(fact_6313_comp__fun__idem__on_Ofold__insert__idem,axiom,
! [B: $tType,A: $tType,S2: set @ A,F3: A > B > B,X: A,A5: set @ A,Z: B] :
( ( finite673082921795544331dem_on @ A @ B @ S2 @ F3 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X @ A5 ) @ S2 )
=> ( ( finite_finite2 @ A @ A5 )
=> ( ( finite_fold @ A @ B @ F3 @ Z @ ( insert @ A @ X @ A5 ) )
= ( F3 @ X @ ( finite_fold @ A @ B @ F3 @ Z @ A5 ) ) ) ) ) ) ).
% comp_fun_idem_on.fold_insert_idem
thf(fact_6314_comp__fun__idem__on_Ocomp__fun__idem__on,axiom,
! [B: $tType,A: $tType,S2: set @ A,F3: A > B > B,X: A] :
( ( finite673082921795544331dem_on @ A @ B @ S2 @ F3 )
=> ( ( member @ A @ X @ S2 )
=> ( ( comp @ B @ B @ B @ ( F3 @ X ) @ ( F3 @ X ) )
= ( F3 @ X ) ) ) ) ).
% comp_fun_idem_on.comp_fun_idem_on
thf(fact_6315_folding__idem__on_Ocomp__fun__idem__on,axiom,
! [B: $tType,A: $tType,S2: set @ A,F3: A > B > B,X: A,Y: A] :
( ( finite1890593828518410140dem_on @ A @ B @ S2 @ F3 )
=> ( ( member @ A @ X @ S2 )
=> ( ( member @ A @ Y @ S2 )
=> ( ( comp @ B @ B @ B @ ( F3 @ X ) @ ( F3 @ X ) )
= ( F3 @ X ) ) ) ) ) ).
% folding_idem_on.comp_fun_idem_on
thf(fact_6316_comp__fun__idem__on_Ofun__left__idem,axiom,
! [A: $tType,B: $tType,S2: set @ A,F3: A > B > B,X: A,Z: B] :
( ( finite673082921795544331dem_on @ A @ B @ S2 @ F3 )
=> ( ( member @ A @ X @ S2 )
=> ( ( F3 @ X @ ( F3 @ X @ Z ) )
= ( F3 @ X @ Z ) ) ) ) ).
% comp_fun_idem_on.fun_left_idem
thf(fact_6317_comp__fun__idem__on_Oaxioms_I1_J,axiom,
! [B: $tType,A: $tType,S2: set @ A,F3: A > B > B] :
( ( finite673082921795544331dem_on @ A @ B @ S2 @ F3 )
=> ( finite4664212375090638736ute_on @ A @ B @ S2 @ F3 ) ) ).
% comp_fun_idem_on.axioms(1)
thf(fact_6318_folding__idem__on_Oaxioms_I1_J,axiom,
! [B: $tType,A: $tType,S2: set @ A,F3: A > B > B] :
( ( finite1890593828518410140dem_on @ A @ B @ S2 @ F3 )
=> ( finite_folding_on @ A @ B @ S2 @ F3 ) ) ).
% folding_idem_on.axioms(1)
thf(fact_6319_comp__fun__idem__on_Ocomp__comp__fun__idem__on,axiom,
! [B: $tType,A: $tType,C: $tType,S2: set @ A,F3: A > B > B,G3: C > A,R: set @ C] :
( ( finite673082921795544331dem_on @ A @ B @ S2 @ F3 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image2 @ C @ A @ G3 @ ( top_top @ ( set @ C ) ) ) @ S2 )
=> ( finite673082921795544331dem_on @ C @ B @ R @ ( comp @ A @ ( B > B ) @ C @ F3 @ G3 ) ) ) ) ).
% comp_fun_idem_on.comp_comp_fun_idem_on
thf(fact_6320_comp__fun__idem__on_Ofold__insert__idem2,axiom,
! [B: $tType,A: $tType,S2: set @ A,F3: A > B > B,X: A,A5: set @ A,Z: B] :
( ( finite673082921795544331dem_on @ A @ B @ S2 @ F3 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X @ A5 ) @ S2 )
=> ( ( finite_finite2 @ A @ A5 )
=> ( ( finite_fold @ A @ B @ F3 @ Z @ ( insert @ A @ X @ A5 ) )
= ( finite_fold @ A @ B @ F3 @ ( F3 @ X @ Z ) @ A5 ) ) ) ) ) ).
% comp_fun_idem_on.fold_insert_idem2
thf(fact_6321_folding__idem__on__def,axiom,
! [B: $tType,A: $tType] :
( ( finite1890593828518410140dem_on @ A @ B )
= ( ^ [S7: set @ A,F4: A > B > B] :
( ( finite_folding_on @ A @ B @ S7 @ F4 )
& ( finite6916993218817215295axioms @ A @ B @ S7 @ F4 ) ) ) ) ).
% folding_idem_on_def
thf(fact_6322_folding__idem__on_Ointro,axiom,
! [B: $tType,A: $tType,S2: set @ A,F3: A > B > B] :
( ( finite_folding_on @ A @ B @ S2 @ F3 )
=> ( ( finite6916993218817215295axioms @ A @ B @ S2 @ F3 )
=> ( finite1890593828518410140dem_on @ A @ B @ S2 @ F3 ) ) ) ).
% folding_idem_on.intro
thf(fact_6323_folding__idem__on__axioms__def,axiom,
! [B: $tType,A: $tType] :
( ( finite6916993218817215295axioms @ A @ B )
= ( ^ [S7: set @ A,F4: A > B > B] :
! [X5: A,Y6: A] :
( ( member @ A @ X5 @ S7 )
=> ( ( member @ A @ Y6 @ S7 )
=> ( ( comp @ B @ B @ B @ ( F4 @ X5 ) @ ( F4 @ X5 ) )
= ( F4 @ X5 ) ) ) ) ) ) ).
% folding_idem_on_axioms_def
thf(fact_6324_folding__idem__on__axioms_Ointro,axiom,
! [B: $tType,A: $tType,S2: set @ A,F3: A > B > B] :
( ! [X4: A,Y3: A] :
( ( member @ A @ X4 @ S2 )
=> ( ( member @ A @ Y3 @ S2 )
=> ( ( comp @ B @ B @ B @ ( F3 @ X4 ) @ ( F3 @ X4 ) )
= ( F3 @ X4 ) ) ) )
=> ( finite6916993218817215295axioms @ A @ B @ S2 @ F3 ) ) ).
% folding_idem_on_axioms.intro
thf(fact_6325_folding__idem__on_Oaxioms_I2_J,axiom,
! [B: $tType,A: $tType,S2: set @ A,F3: A > B > B] :
( ( finite1890593828518410140dem_on @ A @ B @ S2 @ F3 )
=> ( finite6916993218817215295axioms @ A @ B @ S2 @ F3 ) ) ).
% folding_idem_on.axioms(2)
thf(fact_6326_positive__rat,axiom,
! [A2: int,B2: int] :
( ( positive @ ( fract @ A2 @ B2 ) )
= ( ord_less @ int @ ( zero_zero @ int ) @ ( times_times @ int @ A2 @ B2 ) ) ) ).
% positive_rat
thf(fact_6327_nth__image,axiom,
! [A: $tType,L: nat,Xs: list @ A] :
( ( ord_less_eq @ nat @ L @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( image2 @ nat @ A @ ( nth @ A @ Xs ) @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ L ) )
= ( set2 @ A @ ( take @ A @ L @ Xs ) ) ) ) ).
% nth_image
thf(fact_6328_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_6329_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_6330_take0,axiom,
! [A: $tType] :
( ( take @ A @ ( zero_zero @ nat ) )
= ( ^ [Xs3: list @ A] : ( nil @ A ) ) ) ).
% take0
thf(fact_6331_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_6332_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_6333_nth__take,axiom,
! [A: $tType,I2: nat,N: nat,Xs: list @ A] :
( ( ord_less @ nat @ I2 @ N )
=> ( ( nth @ A @ ( take @ A @ N @ Xs ) @ I2 )
= ( nth @ A @ Xs @ I2 ) ) ) ).
% nth_take
thf(fact_6334_take__update__cancel,axiom,
! [A: $tType,N: nat,M2: nat,Xs: list @ A,Y: A] :
( ( ord_less_eq @ nat @ N @ M2 )
=> ( ( take @ A @ N @ ( list_update @ A @ Xs @ M2 @ Y ) )
= ( take @ A @ N @ Xs ) ) ) ).
% take_update_cancel
thf(fact_6335_nths__upt__eq__take,axiom,
! [A: $tType,L: list @ A,N: nat] :
( ( nths @ A @ L @ ( set_ord_lessThan @ nat @ N ) )
= ( take @ A @ N @ L ) ) ).
% nths_upt_eq_take
thf(fact_6336_dom__map__upds,axiom,
! [B: $tType,A: $tType,M2: A > ( option @ B ),Xs: list @ A,Ys: list @ B] :
( ( dom @ A @ B @ ( map_upds @ A @ B @ M2 @ Xs @ Ys ) )
= ( sup_sup @ ( set @ A ) @ ( set2 @ A @ ( take @ A @ ( size_size @ ( list @ B ) @ Ys ) @ Xs ) ) @ ( dom @ A @ B @ M2 ) ) ) ).
% dom_map_upds
thf(fact_6337_set__take__subset,axiom,
! [A: $tType,N: nat,Xs: list @ A] : ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ ( take @ A @ N @ Xs ) ) @ ( set2 @ A @ Xs ) ) ).
% set_take_subset
thf(fact_6338_less__rat__def,axiom,
( ( ord_less @ rat )
= ( ^ [X5: rat,Y6: rat] : ( positive @ ( minus_minus @ rat @ Y6 @ X5 ) ) ) ) ).
% less_rat_def
thf(fact_6339_take__0,axiom,
! [A: $tType,Xs: list @ A] :
( ( take @ A @ ( zero_zero @ nat ) @ Xs )
= ( nil @ A ) ) ).
% take_0
thf(fact_6340_take__Nil,axiom,
! [A: $tType,N: nat] :
( ( take @ A @ N @ ( nil @ A ) )
= ( nil @ A ) ) ).
% take_Nil
thf(fact_6341_take__update__swap,axiom,
! [A: $tType,M2: nat,Xs: list @ A,N: nat,X: A] :
( ( take @ A @ M2 @ ( list_update @ A @ Xs @ N @ X ) )
= ( list_update @ A @ ( take @ A @ M2 @ Xs ) @ N @ X ) ) ).
% take_update_swap
thf(fact_6342_take__equalityI,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ! [I3: nat] :
( ( take @ A @ I3 @ Xs )
= ( take @ A @ I3 @ Ys ) )
=> ( Xs = Ys ) ) ).
% take_equalityI
thf(fact_6343_distinct__take,axiom,
! [A: $tType,Xs: list @ A,I2: nat] :
( ( distinct @ A @ Xs )
=> ( distinct @ A @ ( take @ A @ I2 @ Xs ) ) ) ).
% distinct_take
thf(fact_6344_in__set__takeD,axiom,
! [A: $tType,X: A,N: nat,Xs: list @ A] :
( ( member @ A @ X @ ( set2 @ A @ ( take @ A @ N @ Xs ) ) )
=> ( member @ A @ X @ ( set2 @ A @ Xs ) ) ) ).
% in_set_takeD
thf(fact_6345_set__take__subset__set__take,axiom,
! [A: $tType,M2: nat,N: nat,Xs: list @ A] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ ( take @ A @ M2 @ Xs ) ) @ ( set2 @ A @ ( take @ A @ N @ Xs ) ) ) ) ).
% set_take_subset_set_take
thf(fact_6346_nth__take__lemma,axiom,
! [A: $tType,K2: nat,Xs: list @ A,Ys: list @ A] :
( ( ord_less_eq @ nat @ K2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ord_less_eq @ nat @ K2 @ ( size_size @ ( list @ A ) @ Ys ) )
=> ( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ K2 )
=> ( ( nth @ A @ Xs @ I3 )
= ( nth @ A @ Ys @ I3 ) ) )
=> ( ( take @ A @ K2 @ Xs )
= ( take @ A @ K2 @ Ys ) ) ) ) ) ).
% nth_take_lemma
thf(fact_6347_map__upd__upds__conv__if,axiom,
! [A: $tType,B: $tType,X: A,Ys: list @ B,Xs: list @ A,F3: A > ( option @ B ),Y: B] :
( ( ( member @ A @ X @ ( set2 @ A @ ( take @ A @ ( size_size @ ( list @ B ) @ Ys ) @ Xs ) ) )
=> ( ( map_upds @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ F3 @ X @ ( some @ B @ Y ) ) @ Xs @ Ys )
= ( map_upds @ A @ B @ F3 @ Xs @ Ys ) ) )
& ( ~ ( member @ A @ X @ ( set2 @ A @ ( take @ A @ ( size_size @ ( list @ B ) @ Ys ) @ Xs ) ) )
=> ( ( map_upds @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ F3 @ X @ ( some @ B @ Y ) ) @ Xs @ Ys )
= ( fun_upd @ A @ ( option @ B ) @ ( map_upds @ A @ B @ F3 @ Xs @ Ys ) @ X @ ( some @ B @ Y ) ) ) ) ) ).
% map_upd_upds_conv_if
thf(fact_6348_Rat_Opositive_Orep__eq,axiom,
( positive
= ( ^ [X5: rat] : ( ord_less @ int @ ( zero_zero @ int ) @ ( times_times @ int @ ( product_fst @ int @ int @ ( rep_Rat @ X5 ) ) @ ( product_snd @ int @ int @ ( rep_Rat @ X5 ) ) ) ) ) ) ).
% Rat.positive.rep_eq
thf(fact_6349_lex__take__index,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( lex @ A @ R2 ) )
=> ~ ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Ys ) )
=> ( ( ( take @ A @ I3 @ Xs )
= ( take @ A @ I3 @ Ys ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( nth @ A @ Xs @ I3 ) @ ( nth @ A @ Ys @ I3 ) ) @ R2 ) ) ) ) ) ).
% lex_take_index
thf(fact_6350_Nil__notin__lex,axiom,
! [A: $tType,Ys: list @ A,R2: set @ ( product_prod @ A @ A )] :
~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Ys ) @ ( lex @ A @ R2 ) ) ).
% Nil_notin_lex
thf(fact_6351_Nil2__notin__lex,axiom,
! [A: $tType,Xs: list @ A,R2: set @ ( product_prod @ A @ A )] :
~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ ( nil @ A ) ) @ ( lex @ A @ R2 ) ) ).
% Nil2_notin_lex
thf(fact_6352_Rat_Opositive__def,axiom,
( positive
= ( map_fun @ rat @ ( product_prod @ int @ int ) @ $o @ $o @ rep_Rat @ ( id @ $o )
@ ^ [X5: product_prod @ int @ int] : ( ord_less @ int @ ( zero_zero @ int ) @ ( times_times @ int @ ( product_fst @ int @ int @ X5 ) @ ( product_snd @ int @ int @ X5 ) ) ) ) ) ).
% Rat.positive_def
thf(fact_6353_listrel1__iff__update,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( listrel1 @ A @ R2 ) )
= ( ? [Y6: A,N2: nat] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( nth @ A @ Xs @ N2 ) @ Y6 ) @ R2 )
& ( ord_less @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs ) )
& ( Ys
= ( list_update @ A @ Xs @ N2 @ Y6 ) ) ) ) ) ).
% listrel1_iff_update
thf(fact_6354_not__Nil__listrel1,axiom,
! [A: $tType,Xs: list @ A,R2: set @ ( product_prod @ A @ A )] :
~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Xs ) @ ( listrel1 @ A @ R2 ) ) ).
% not_Nil_listrel1
thf(fact_6355_not__listrel1__Nil,axiom,
! [A: $tType,Xs: list @ A,R2: set @ ( product_prod @ A @ A )] :
~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ ( nil @ A ) ) @ ( listrel1 @ A @ R2 ) ) ).
% not_listrel1_Nil
thf(fact_6356_listrel1__eq__len,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( listrel1 @ A @ R2 ) )
=> ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ A ) @ Ys ) ) ) ).
% listrel1_eq_len
thf(fact_6357_rtrancl__listrel1__eq__len,axiom,
! [A: $tType,X: list @ A,Y: list @ A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Y ) @ ( transitive_rtrancl @ ( list @ A ) @ ( listrel1 @ A @ R2 ) ) )
=> ( ( size_size @ ( list @ A ) @ X )
= ( size_size @ ( list @ A ) @ Y ) ) ) ).
% rtrancl_listrel1_eq_len
thf(fact_6358_listrel1__mono,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),S: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ S )
=> ( ord_less_eq @ ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) @ ( listrel1 @ A @ R2 ) @ ( listrel1 @ A @ S ) ) ) ).
% listrel1_mono
thf(fact_6359_listrel1__rtrancl__subset__rtrancl__listrel1,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] : ( ord_less_eq @ ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) @ ( listrel1 @ A @ ( transitive_rtrancl @ A @ R2 ) ) @ ( transitive_rtrancl @ ( list @ A ) @ ( listrel1 @ A @ R2 ) ) ) ).
% listrel1_rtrancl_subset_rtrancl_listrel1
thf(fact_6360_listrel1p__def,axiom,
! [A: $tType] :
( ( listrel1p @ A )
= ( ^ [R5: A > A > $o,Xs3: list @ A,Ys3: list @ A] : ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs3 @ Ys3 ) @ ( listrel1 @ A @ ( collect @ ( product_prod @ A @ A ) @ ( product_case_prod @ A @ A @ $o @ R5 ) ) ) ) ) ) ).
% listrel1p_def
thf(fact_6361_lenlex__conv,axiom,
! [A: $tType] :
( ( lenlex @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ ( list @ A ) @ ( list @ A ) )
@ ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ $o
@ ^ [Xs3: list @ A,Ys3: list @ A] :
( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Xs3 ) @ ( size_size @ ( list @ A ) @ Ys3 ) )
| ( ( ( size_size @ ( list @ A ) @ Xs3 )
= ( size_size @ ( list @ A ) @ Ys3 ) )
& ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs3 @ Ys3 ) @ ( lex @ A @ R5 ) ) ) ) ) ) ) ) ).
% lenlex_conv
thf(fact_6362_Nil__lenlex__iff1,axiom,
! [A: $tType,Ns: list @ A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Ns ) @ ( lenlex @ A @ R2 ) )
= ( Ns
!= ( nil @ A ) ) ) ).
% Nil_lenlex_iff1
thf(fact_6363_lenlex__irreflexive,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),Xs: list @ A] :
( ! [X4: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ X4 ) @ R2 )
=> ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Xs ) @ ( lenlex @ A @ R2 ) ) ) ).
% lenlex_irreflexive
thf(fact_6364_Nil__lenlex__iff2,axiom,
! [A: $tType,Ns: list @ A,R2: set @ ( product_prod @ A @ A )] :
~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ns @ ( nil @ A ) ) @ ( lenlex @ A @ R2 ) ) ).
% Nil_lenlex_iff2
thf(fact_6365_lenlex__length,axiom,
! [A: $tType,Ms: list @ A,Ns: list @ A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ms @ Ns ) @ ( lenlex @ A @ R2 ) )
=> ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Ms ) @ ( size_size @ ( list @ A ) @ Ns ) ) ) ).
% lenlex_length
thf(fact_6366_dual__min,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( min @ A
@ ^ [X5: A,Y6: A] : ( ord_less_eq @ A @ Y6 @ X5 ) )
= ( ord_max @ A ) ) ) ).
% dual_min
thf(fact_6367_rp__inv__image__rp,axiom,
! [A: $tType,B: $tType,P2: product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ),F3: B > A] :
( ( fun_reduction_pair @ A @ P2 )
=> ( fun_reduction_pair @ B @ ( fun_rp_inv_image @ A @ B @ P2 @ F3 ) ) ) ).
% rp_inv_image_rp
thf(fact_6368_ord_Omin__def,axiom,
! [A: $tType] :
( ( min @ A )
= ( ^ [Less_eq2: A > A > $o,A6: A,B5: A] : ( if @ A @ ( Less_eq2 @ A6 @ B5 ) @ A6 @ B5 ) ) ) ).
% ord.min_def
thf(fact_6369_ord_Omin_Ocong,axiom,
! [A: $tType] :
( ( min @ A )
= ( min @ A ) ) ).
% ord.min.cong
thf(fact_6370_rp__inv__image__def,axiom,
! [B: $tType,A: $tType] :
( ( fun_rp_inv_image @ A @ B )
= ( product_case_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ ( ( B > A ) > ( product_prod @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) ) )
@ ^ [R6: set @ ( product_prod @ A @ A ),S7: set @ ( product_prod @ A @ A ),F4: B > A] : ( product_Pair @ ( set @ ( product_prod @ B @ B ) ) @ ( set @ ( product_prod @ B @ B ) ) @ ( inv_image @ A @ B @ R6 @ F4 ) @ ( inv_image @ A @ B @ S7 @ F4 ) ) ) ) ).
% rp_inv_image_def
thf(fact_6371_card__Min__le__sum,axiom,
! [A: $tType,A5: set @ A,F3: A > nat] :
( ( finite_finite2 @ A @ A5 )
=> ( ord_less_eq @ nat @ ( times_times @ nat @ ( finite_card @ A @ A5 ) @ ( lattic643756798350308766er_Min @ nat @ ( image2 @ A @ nat @ F3 @ A5 ) ) ) @ ( groups7311177749621191930dd_sum @ A @ nat @ F3 @ A5 ) ) ) ).
% card_Min_le_sum
thf(fact_6372_in__inv__image,axiom,
! [A: $tType,B: $tType,X: A,Y: A,R2: set @ ( product_prod @ B @ B ),F3: A > B] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( inv_image @ B @ A @ R2 @ F3 ) )
= ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ ( F3 @ X ) @ ( F3 @ Y ) ) @ R2 ) ) ).
% in_inv_image
thf(fact_6373_Min__singleton,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A] :
( ( lattic643756798350308766er_Min @ A @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= X ) ) ).
% Min_singleton
thf(fact_6374_Min_Obounded__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ X @ ( lattic643756798350308766er_Min @ A @ A5 ) )
= ( ! [X5: A] :
( ( member @ A @ X5 @ A5 )
=> ( ord_less_eq @ A @ X @ X5 ) ) ) ) ) ) ) ).
% Min.bounded_iff
thf(fact_6375_Min__gr__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less @ A @ X @ ( lattic643756798350308766er_Min @ A @ A5 ) )
= ( ! [X5: A] :
( ( member @ A @ X5 @ A5 )
=> ( ord_less @ A @ X @ X5 ) ) ) ) ) ) ) ).
% Min_gr_iff
thf(fact_6376_Min__const,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ B,C3: A] :
( ( finite_finite2 @ B @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( lattic643756798350308766er_Min @ A
@ ( image2 @ B @ A
@ ^ [Uu3: B] : C3
@ A5 ) )
= C3 ) ) ) ) ).
% Min_const
thf(fact_6377_minus__Min__eq__Max,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [S2: set @ A] :
( ( finite_finite2 @ A @ S2 )
=> ( ( S2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( uminus_uminus @ A @ ( lattic643756798350308766er_Min @ A @ S2 ) )
= ( lattic643756798349783984er_Max @ A @ ( image2 @ A @ A @ ( uminus_uminus @ A ) @ S2 ) ) ) ) ) ) ).
% minus_Min_eq_Max
thf(fact_6378_minus__Max__eq__Min,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [S2: set @ A] :
( ( finite_finite2 @ A @ S2 )
=> ( ( S2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( uminus_uminus @ A @ ( lattic643756798349783984er_Max @ A @ S2 ) )
= ( lattic643756798350308766er_Min @ A @ ( image2 @ A @ A @ ( uminus_uminus @ A ) @ S2 ) ) ) ) ) ) ).
% minus_Max_eq_Min
thf(fact_6379_Inf__fin__Min,axiom,
! [A: $tType] :
( ( ( semilattice_inf @ A )
& ( linorder @ A ) )
=> ( ( lattic7752659483105999362nf_fin @ A )
= ( lattic643756798350308766er_Min @ A ) ) ) ).
% Inf_fin_Min
thf(fact_6380_Min__in,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( member @ A @ ( lattic643756798350308766er_Min @ A @ A5 ) @ A5 ) ) ) ) ).
% Min_in
thf(fact_6381_Min__le,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( member @ A @ X @ A5 )
=> ( ord_less_eq @ A @ ( lattic643756798350308766er_Min @ A @ A5 ) @ X ) ) ) ) ).
% Min_le
thf(fact_6382_Min__eqI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ! [Y3: A] :
( ( member @ A @ Y3 @ A5 )
=> ( ord_less_eq @ A @ X @ Y3 ) )
=> ( ( member @ A @ X @ A5 )
=> ( ( lattic643756798350308766er_Min @ A @ A5 )
= X ) ) ) ) ) ).
% Min_eqI
thf(fact_6383_Min_OcoboundedI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,A2: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( member @ A @ A2 @ A5 )
=> ( ord_less_eq @ A @ ( lattic643756798350308766er_Min @ A @ A5 ) @ A2 ) ) ) ) ).
% Min.coboundedI
thf(fact_6384_Min__eq__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,M2: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( lattic643756798350308766er_Min @ A @ A5 )
= M2 )
= ( ( member @ A @ M2 @ A5 )
& ! [X5: A] :
( ( member @ A @ X5 @ A5 )
=> ( ord_less_eq @ A @ M2 @ X5 ) ) ) ) ) ) ) ).
% Min_eq_iff
thf(fact_6385_Min__le__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ ( lattic643756798350308766er_Min @ A @ A5 ) @ X )
= ( ? [X5: A] :
( ( member @ A @ X5 @ A5 )
& ( ord_less_eq @ A @ X5 @ X ) ) ) ) ) ) ) ).
% Min_le_iff
thf(fact_6386_eq__Min__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,M2: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( M2
= ( lattic643756798350308766er_Min @ A @ A5 ) )
= ( ( member @ A @ M2 @ A5 )
& ! [X5: A] :
( ( member @ A @ X5 @ A5 )
=> ( ord_less_eq @ A @ M2 @ X5 ) ) ) ) ) ) ) ).
% eq_Min_iff
thf(fact_6387_Min_OboundedE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ A @ X @ ( lattic643756798350308766er_Min @ A @ A5 ) )
=> ! [A11: A] :
( ( member @ A @ A11 @ A5 )
=> ( ord_less_eq @ A @ X @ A11 ) ) ) ) ) ) ).
% Min.boundedE
thf(fact_6388_Min_OboundedI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [A4: A] :
( ( member @ A @ A4 @ A5 )
=> ( ord_less_eq @ A @ X @ A4 ) )
=> ( ord_less_eq @ A @ X @ ( lattic643756798350308766er_Min @ A @ A5 ) ) ) ) ) ) ).
% Min.boundedI
thf(fact_6389_Min__less__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less @ A @ ( lattic643756798350308766er_Min @ A @ A5 ) @ X )
= ( ? [X5: A] :
( ( member @ A @ X5 @ A5 )
& ( ord_less @ A @ X5 @ X ) ) ) ) ) ) ) ).
% Min_less_iff
thf(fact_6390_Min__insert2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,A2: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ! [B4: A] :
( ( member @ A @ B4 @ A5 )
=> ( ord_less_eq @ A @ A2 @ B4 ) )
=> ( ( lattic643756798350308766er_Min @ A @ ( insert @ A @ A2 @ A5 ) )
= A2 ) ) ) ) ).
% Min_insert2
thf(fact_6391_cInf__eq__Min,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [X7: set @ A] :
( ( finite_finite2 @ A @ X7 )
=> ( ( X7
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Inf_Inf @ A @ X7 )
= ( lattic643756798350308766er_Min @ A @ X7 ) ) ) ) ) ).
% cInf_eq_Min
thf(fact_6392_Min__Inf,axiom,
! [A: $tType] :
( ( comple5582772986160207858norder @ A )
=> ! [A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798350308766er_Min @ A @ A5 )
= ( complete_Inf_Inf @ A @ A5 ) ) ) ) ) ).
% Min_Inf
thf(fact_6393_Min_Oinfinite,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A] :
( ~ ( finite_finite2 @ A @ A5 )
=> ( ( lattic643756798350308766er_Min @ A @ A5 )
= ( the2 @ A @ ( none @ A ) ) ) ) ) ).
% Min.infinite
thf(fact_6394_Min__antimono,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [M5: set @ A,N6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ M5 @ N6 )
=> ( ( M5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ N6 )
=> ( ord_less_eq @ A @ ( lattic643756798350308766er_Min @ A @ N6 ) @ ( lattic643756798350308766er_Min @ A @ M5 ) ) ) ) ) ) ).
% Min_antimono
thf(fact_6395_Min_Osubset__imp,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,B6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B6 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B6 )
=> ( ord_less_eq @ A @ ( lattic643756798350308766er_Min @ A @ B6 ) @ ( lattic643756798350308766er_Min @ A @ A5 ) ) ) ) ) ) ).
% Min.subset_imp
thf(fact_6396_inv__image__def,axiom,
! [A: $tType,B: $tType] :
( ( inv_image @ B @ A )
= ( ^ [R5: set @ ( product_prod @ B @ B ),F4: A > B] :
( collect @ ( product_prod @ A @ A )
@ ( product_case_prod @ A @ A @ $o
@ ^ [X5: A,Y6: A] : ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ ( F4 @ X5 ) @ ( F4 @ Y6 ) ) @ R5 ) ) ) ) ) ).
% inv_image_def
thf(fact_6397_mono__Min__commute,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( linorder @ B ) )
=> ! [F3: A > B,A5: set @ A] :
( ( order_mono @ A @ B @ F3 )
=> ( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( F3 @ ( lattic643756798350308766er_Min @ A @ A5 ) )
= ( lattic643756798350308766er_Min @ B @ ( image2 @ A @ B @ F3 @ A5 ) ) ) ) ) ) ) ).
% mono_Min_commute
thf(fact_6398_Min__add__commute,axiom,
! [B: $tType,A: $tType] :
( ( linord4140545234300271783up_add @ A )
=> ! [S2: set @ B,F3: B > A,K2: A] :
( ( finite_finite2 @ B @ S2 )
=> ( ( S2
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( lattic643756798350308766er_Min @ A
@ ( image2 @ B @ A
@ ^ [X5: B] : ( plus_plus @ A @ ( F3 @ X5 ) @ K2 )
@ S2 ) )
= ( plus_plus @ A @ ( lattic643756798350308766er_Min @ A @ ( image2 @ B @ A @ F3 @ S2 ) ) @ K2 ) ) ) ) ) ).
% Min_add_commute
thf(fact_6399_dual__Max,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( lattices_Max @ A
@ ^ [X5: A,Y6: A] : ( ord_less_eq @ A @ Y6 @ X5 ) )
= ( lattic643756798350308766er_Min @ A ) ) ) ).
% dual_Max
thf(fact_6400_f__arg__min__list__f,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ! [Xs: list @ A,F3: A > B] :
( ( Xs
!= ( nil @ A ) )
=> ( ( F3 @ ( arg_min_list @ A @ B @ F3 @ Xs ) )
= ( lattic643756798350308766er_Min @ B @ ( image2 @ A @ B @ F3 @ ( set2 @ A @ Xs ) ) ) ) ) ) ).
% f_arg_min_list_f
thf(fact_6401_linorder_OMax_Ocong,axiom,
! [A: $tType] :
( ( lattices_Max @ A )
= ( lattices_Max @ A ) ) ).
% linorder.Max.cong
thf(fact_6402_arg__min__list__in,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ! [Xs: list @ A,F3: A > B] :
( ( Xs
!= ( nil @ A ) )
=> ( member @ A @ ( arg_min_list @ A @ B @ F3 @ Xs ) @ ( set2 @ A @ Xs ) ) ) ) ).
% arg_min_list_in
thf(fact_6403_Min_Oeq__fold_H,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( lattic643756798350308766er_Min @ A )
= ( ^ [A7: set @ A] :
( the2 @ A
@ ( finite_fold @ A @ ( option @ A )
@ ^ [X5: A,Y6: option @ A] : ( some @ A @ ( case_option @ A @ A @ X5 @ ( ord_min @ A @ X5 ) @ Y6 ) )
@ ( none @ A )
@ A7 ) ) ) ) ) ).
% Min.eq_fold'
thf(fact_6404_min__list__Min,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( min_list @ A @ Xs )
= ( lattic643756798350308766er_Min @ A @ ( set2 @ A @ Xs ) ) ) ) ) ).
% min_list_Min
thf(fact_6405_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_6406_min__0R,axiom,
! [N: nat] :
( ( ord_min @ nat @ N @ ( zero_zero @ nat ) )
= ( zero_zero @ nat ) ) ).
% min_0R
thf(fact_6407_min__0L,axiom,
! [N: nat] :
( ( ord_min @ nat @ ( zero_zero @ nat ) @ N )
= ( zero_zero @ nat ) ) ).
% min_0L
thf(fact_6408_min_Oright__idem,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A] :
( ( ord_min @ A @ ( ord_min @ A @ A2 @ B2 ) @ B2 )
= ( ord_min @ A @ A2 @ B2 ) ) ) ).
% min.right_idem
thf(fact_6409_min_Oleft__idem,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A] :
( ( ord_min @ A @ A2 @ ( ord_min @ A @ A2 @ B2 ) )
= ( ord_min @ A @ A2 @ B2 ) ) ) ).
% min.left_idem
thf(fact_6410_min_Oidem,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A] :
( ( ord_min @ A @ A2 @ A2 )
= A2 ) ) ).
% min.idem
thf(fact_6411_take__take,axiom,
! [A: $tType,N: nat,M2: nat,Xs: list @ A] :
( ( take @ A @ N @ ( take @ A @ M2 @ Xs ) )
= ( take @ A @ ( ord_min @ nat @ N @ M2 ) @ Xs ) ) ).
% take_take
thf(fact_6412_min_Obounded__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( ord_less_eq @ A @ A2 @ ( ord_min @ A @ B2 @ C3 ) )
= ( ( ord_less_eq @ A @ A2 @ B2 )
& ( ord_less_eq @ A @ A2 @ C3 ) ) ) ) ).
% min.bounded_iff
thf(fact_6413_min_Oabsorb2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,A2: A] :
( ( ord_less_eq @ A @ B2 @ A2 )
=> ( ( ord_min @ A @ A2 @ B2 )
= B2 ) ) ) ).
% min.absorb2
thf(fact_6414_min_Oabsorb1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_min @ A @ A2 @ B2 )
= A2 ) ) ) ).
% min.absorb1
thf(fact_6415_min__less__iff__conj,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Z: A,X: A,Y: A] :
( ( ord_less @ A @ Z @ ( ord_min @ A @ X @ Y ) )
= ( ( ord_less @ A @ Z @ X )
& ( ord_less @ A @ Z @ Y ) ) ) ) ).
% min_less_iff_conj
thf(fact_6416_min_Oabsorb4,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,A2: A] :
( ( ord_less @ A @ B2 @ A2 )
=> ( ( ord_min @ A @ A2 @ B2 )
= B2 ) ) ) ).
% min.absorb4
thf(fact_6417_min_Oabsorb3,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A] :
( ( ord_less @ A @ A2 @ B2 )
=> ( ( ord_min @ A @ A2 @ B2 )
= A2 ) ) ) ).
% min.absorb3
thf(fact_6418_min__top,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [X: A] :
( ( ord_min @ A @ ( top_top @ A ) @ X )
= X ) ) ).
% min_top
thf(fact_6419_min__top2,axiom,
! [A: $tType] :
( ( order_top @ A )
=> ! [X: A] :
( ( ord_min @ A @ X @ ( top_top @ A ) )
= X ) ) ).
% min_top2
thf(fact_6420_min__bot2,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [X: A] :
( ( ord_min @ A @ X @ ( bot_bot @ A ) )
= ( bot_bot @ A ) ) ) ).
% min_bot2
thf(fact_6421_min__bot,axiom,
! [A: $tType] :
( ( order_bot @ A )
=> ! [X: A] :
( ( ord_min @ A @ ( bot_bot @ A ) @ X )
= ( bot_bot @ A ) ) ) ).
% min_bot
thf(fact_6422_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_6423_max__min__same_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( ord_max @ A @ X @ ( ord_min @ A @ X @ Y ) )
= X ) ) ).
% max_min_same(1)
thf(fact_6424_max__min__same_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( ord_max @ A @ ( ord_min @ A @ X @ Y ) @ X )
= X ) ) ).
% max_min_same(2)
thf(fact_6425_max__min__same_I3_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A] :
( ( ord_max @ A @ ( ord_min @ A @ X @ Y ) @ Y )
= Y ) ) ).
% max_min_same(3)
thf(fact_6426_max__min__same_I4_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Y: A,X: A] :
( ( ord_max @ A @ Y @ ( ord_min @ A @ X @ Y ) )
= Y ) ) ).
% max_min_same(4)
thf(fact_6427_take__replicate,axiom,
! [A: $tType,I2: nat,K2: nat,X: A] :
( ( take @ A @ I2 @ ( replicate @ A @ K2 @ X ) )
= ( replicate @ A @ ( ord_min @ nat @ I2 @ K2 ) @ X ) ) ).
% take_replicate
thf(fact_6428_min__number__of_I1_J,axiom,
! [A: $tType] :
( ( ( numeral @ A )
& ( ord @ A ) )
=> ! [U: num,V: num] :
( ( ( ord_less_eq @ A @ ( numeral_numeral @ A @ U ) @ ( numeral_numeral @ A @ V ) )
=> ( ( ord_min @ A @ ( numeral_numeral @ A @ U ) @ ( numeral_numeral @ A @ V ) )
= ( numeral_numeral @ A @ U ) ) )
& ( ~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ U ) @ ( numeral_numeral @ A @ V ) )
=> ( ( ord_min @ A @ ( numeral_numeral @ A @ U ) @ ( numeral_numeral @ A @ V ) )
= ( numeral_numeral @ A @ V ) ) ) ) ) ).
% min_number_of(1)
thf(fact_6429_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_6430_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_6431_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_6432_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_6433_Int__atMost,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A] :
( ( inf_inf @ ( set @ A ) @ ( set_ord_atMost @ A @ A2 ) @ ( set_ord_atMost @ A @ B2 ) )
= ( set_ord_atMost @ A @ ( ord_min @ A @ A2 @ B2 ) ) ) ) ).
% Int_atMost
thf(fact_6434_min__number__of_I2_J,axiom,
! [A: $tType] :
( ( ( uminus @ A )
& ( numeral @ A )
& ( ord @ A ) )
=> ! [U: num,V: num] :
( ( ( ord_less_eq @ A @ ( numeral_numeral @ A @ U ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) )
=> ( ( ord_min @ A @ ( numeral_numeral @ A @ U ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) )
= ( numeral_numeral @ A @ U ) ) )
& ( ~ ( ord_less_eq @ A @ ( numeral_numeral @ A @ U ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) )
=> ( ( ord_min @ A @ ( numeral_numeral @ A @ U ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) ) ) ) ) ).
% min_number_of(2)
thf(fact_6435_min__number__of_I3_J,axiom,
! [A: $tType] :
( ( ( uminus @ A )
& ( numeral @ A )
& ( ord @ A ) )
=> ! [U: num,V: num] :
( ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( numeral_numeral @ A @ V ) )
=> ( ( ord_min @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( numeral_numeral @ A @ V ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( numeral_numeral @ A @ V ) )
=> ( ( ord_min @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( numeral_numeral @ A @ V ) )
= ( numeral_numeral @ A @ V ) ) ) ) ) ).
% min_number_of(3)
thf(fact_6436_min__number__of_I4_J,axiom,
! [A: $tType] :
( ( ( uminus @ A )
& ( numeral @ A )
& ( ord @ A ) )
=> ! [U: num,V: num] :
( ( ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) )
=> ( ( ord_min @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) )
=> ( ( ord_min @ A @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ U ) ) @ ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) )
= ( uminus_uminus @ A @ ( numeral_numeral @ A @ V ) ) ) ) ) ) ).
% min_number_of(4)
thf(fact_6437_Int__atLeastAtMost,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A,C3: A,D3: A] :
( ( inf_inf @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) @ ( set_or1337092689740270186AtMost @ A @ C3 @ D3 ) )
= ( set_or1337092689740270186AtMost @ A @ ( ord_max @ A @ A2 @ C3 ) @ ( ord_min @ A @ B2 @ D3 ) ) ) ) ).
% Int_atLeastAtMost
thf(fact_6438_Int__atLeastAtMostR1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,C3: A,D3: A] :
( ( inf_inf @ ( set @ A ) @ ( set_ord_atMost @ A @ B2 ) @ ( set_or1337092689740270186AtMost @ A @ C3 @ D3 ) )
= ( set_or1337092689740270186AtMost @ A @ C3 @ ( ord_min @ A @ B2 @ D3 ) ) ) ) ).
% Int_atLeastAtMostR1
thf(fact_6439_Int__atLeastAtMostL1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A,D3: A] :
( ( inf_inf @ ( set @ A ) @ ( set_or1337092689740270186AtMost @ A @ A2 @ B2 ) @ ( set_ord_atMost @ A @ D3 ) )
= ( set_or1337092689740270186AtMost @ A @ A2 @ ( ord_min @ A @ B2 @ D3 ) ) ) ) ).
% Int_atLeastAtMostL1
thf(fact_6440_Int__atLeastLessThan,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A,C3: A,D3: A] :
( ( inf_inf @ ( set @ A ) @ ( set_or7035219750837199246ssThan @ A @ A2 @ B2 ) @ ( set_or7035219750837199246ssThan @ A @ C3 @ D3 ) )
= ( set_or7035219750837199246ssThan @ A @ ( ord_max @ A @ A2 @ C3 ) @ ( ord_min @ A @ B2 @ D3 ) ) ) ) ).
% Int_atLeastLessThan
thf(fact_6441_Int__greaterThanLessThan,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A,C3: A,D3: A] :
( ( inf_inf @ ( set @ A ) @ ( set_or5935395276787703475ssThan @ A @ A2 @ B2 ) @ ( set_or5935395276787703475ssThan @ A @ C3 @ D3 ) )
= ( set_or5935395276787703475ssThan @ A @ ( ord_max @ A @ A2 @ C3 ) @ ( ord_min @ A @ B2 @ D3 ) ) ) ) ).
% Int_greaterThanLessThan
thf(fact_6442_Int__greaterThanAtMost,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A,C3: A,D3: A] :
( ( inf_inf @ ( set @ A ) @ ( set_or3652927894154168847AtMost @ A @ A2 @ B2 ) @ ( set_or3652927894154168847AtMost @ A @ C3 @ D3 ) )
= ( set_or3652927894154168847AtMost @ A @ ( ord_max @ A @ A2 @ C3 ) @ ( ord_min @ A @ B2 @ D3 ) ) ) ) ).
% Int_greaterThanAtMost
thf(fact_6443_Min__insert,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798350308766er_Min @ A @ ( insert @ A @ X @ A5 ) )
= ( ord_min @ A @ X @ ( lattic643756798350308766er_Min @ A @ A5 ) ) ) ) ) ) ).
% Min_insert
thf(fact_6444_Min_Oin__idem,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( member @ A @ X @ A5 )
=> ( ( ord_min @ A @ X @ ( lattic643756798350308766er_Min @ A @ A5 ) )
= ( lattic643756798350308766er_Min @ A @ A5 ) ) ) ) ) ).
% Min.in_idem
thf(fact_6445_min__of__mono,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( linorder @ B ) )
=> ! [F3: A > B,M2: A,N: A] :
( ( order_mono @ A @ B @ F3 )
=> ( ( ord_min @ B @ ( F3 @ M2 ) @ ( F3 @ N ) )
= ( F3 @ ( ord_min @ A @ M2 @ N ) ) ) ) ) ).
% min_of_mono
thf(fact_6446_max__of__antimono,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( linorder @ B ) )
=> ! [F3: A > B,X: A,Y: A] :
( ( order_antimono @ A @ B @ F3 )
=> ( ( ord_max @ B @ ( F3 @ X ) @ ( F3 @ Y ) )
= ( F3 @ ( ord_min @ A @ X @ Y ) ) ) ) ) ).
% max_of_antimono
thf(fact_6447_min__of__antimono,axiom,
! [B: $tType,A: $tType] :
( ( ( linorder @ A )
& ( linorder @ B ) )
=> ! [F3: A > B,X: A,Y: A] :
( ( order_antimono @ A @ B @ F3 )
=> ( ( ord_min @ B @ ( F3 @ X ) @ ( F3 @ Y ) )
= ( F3 @ ( ord_max @ A @ X @ Y ) ) ) ) ) ).
% min_of_antimono
thf(fact_6448_nat__mult__min__right,axiom,
! [M2: nat,N: nat,Q2: nat] :
( ( times_times @ nat @ M2 @ ( ord_min @ nat @ N @ Q2 ) )
= ( ord_min @ nat @ ( times_times @ nat @ M2 @ N ) @ ( times_times @ nat @ M2 @ Q2 ) ) ) ).
% nat_mult_min_right
thf(fact_6449_nat__mult__min__left,axiom,
! [M2: nat,N: nat,Q2: nat] :
( ( times_times @ nat @ ( ord_min @ nat @ M2 @ N ) @ Q2 )
= ( ord_min @ nat @ ( times_times @ nat @ M2 @ Q2 ) @ ( times_times @ nat @ N @ Q2 ) ) ) ).
% nat_mult_min_left
thf(fact_6450_minus__max__eq__min,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [X: A,Y: A] :
( ( uminus_uminus @ A @ ( ord_max @ A @ X @ Y ) )
= ( ord_min @ A @ ( uminus_uminus @ A @ X ) @ ( uminus_uminus @ A @ Y ) ) ) ) ).
% minus_max_eq_min
thf(fact_6451_minus__min__eq__max,axiom,
! [A: $tType] :
( ( linord5086331880401160121up_add @ A )
=> ! [X: A,Y: A] :
( ( uminus_uminus @ A @ ( ord_min @ A @ X @ Y ) )
= ( ord_max @ A @ ( uminus_uminus @ A @ X ) @ ( uminus_uminus @ A @ Y ) ) ) ) ).
% minus_min_eq_max
thf(fact_6452_inf__min,axiom,
! [A: $tType] :
( ( ( semilattice_inf @ A )
& ( linorder @ A ) )
=> ( ( inf_inf @ A )
= ( ord_min @ A ) ) ) ).
% inf_min
thf(fact_6453_min_Ostrict__coboundedI2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,C3: A,A2: A] :
( ( ord_less @ A @ B2 @ C3 )
=> ( ord_less @ A @ ( ord_min @ A @ A2 @ B2 ) @ C3 ) ) ) ).
% min.strict_coboundedI2
thf(fact_6454_min_Ostrict__coboundedI1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,C3: A,B2: A] :
( ( ord_less @ A @ A2 @ C3 )
=> ( ord_less @ A @ ( ord_min @ A @ A2 @ B2 ) @ C3 ) ) ) ).
% min.strict_coboundedI1
thf(fact_6455_min_Ostrict__order__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less @ A )
= ( ^ [A6: A,B5: A] :
( ( A6
= ( ord_min @ A @ A6 @ B5 ) )
& ( A6 != B5 ) ) ) ) ) ).
% min.strict_order_iff
thf(fact_6456_min_Ostrict__boundedE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( ord_less @ A @ A2 @ ( ord_min @ A @ B2 @ C3 ) )
=> ~ ( ( ord_less @ A @ A2 @ B2 )
=> ~ ( ord_less @ A @ A2 @ C3 ) ) ) ) ).
% min.strict_boundedE
thf(fact_6457_min__less__iff__disj,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A,Z: A] :
( ( ord_less @ A @ ( ord_min @ A @ X @ Y ) @ Z )
= ( ( ord_less @ A @ X @ Z )
| ( ord_less @ A @ Y @ Z ) ) ) ) ).
% min_less_iff_disj
thf(fact_6458_min__add__distrib__left,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [X: A,Y: A,Z: A] :
( ( plus_plus @ A @ ( ord_min @ A @ X @ Y ) @ Z )
= ( ord_min @ A @ ( plus_plus @ A @ X @ Z ) @ ( plus_plus @ A @ Y @ Z ) ) ) ) ).
% min_add_distrib_left
thf(fact_6459_min__add__distrib__right,axiom,
! [A: $tType] :
( ( ordere2412721322843649153imp_le @ A )
=> ! [X: A,Y: A,Z: A] :
( ( plus_plus @ A @ X @ ( ord_min @ A @ Y @ Z ) )
= ( ord_min @ A @ ( plus_plus @ A @ X @ Y ) @ ( plus_plus @ A @ X @ Z ) ) ) ) ).
% min_add_distrib_right
thf(fact_6460_min_Oleft__commute,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,A2: A,C3: A] :
( ( ord_min @ A @ B2 @ ( ord_min @ A @ A2 @ C3 ) )
= ( ord_min @ A @ A2 @ ( ord_min @ A @ B2 @ C3 ) ) ) ) ).
% min.left_commute
thf(fact_6461_min_Ocommute,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_min @ A )
= ( ^ [A6: A,B5: A] : ( ord_min @ A @ B5 @ A6 ) ) ) ) ).
% min.commute
thf(fact_6462_min_Oassoc,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( ord_min @ A @ ( ord_min @ A @ A2 @ B2 ) @ C3 )
= ( ord_min @ A @ A2 @ ( ord_min @ A @ B2 @ C3 ) ) ) ) ).
% min.assoc
thf(fact_6463_of__nat__min,axiom,
! [A: $tType] :
( ( linord181362715937106298miring @ A )
=> ! [X: nat,Y: nat] :
( ( semiring_1_of_nat @ A @ ( ord_min @ nat @ X @ Y ) )
= ( ord_min @ A @ ( semiring_1_of_nat @ A @ X ) @ ( semiring_1_of_nat @ A @ Y ) ) ) ) ).
% of_nat_min
thf(fact_6464_min__max__distrib2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( ord_min @ A @ A2 @ ( ord_max @ A @ B2 @ C3 ) )
= ( ord_max @ A @ ( ord_min @ A @ A2 @ B2 ) @ ( ord_min @ A @ A2 @ C3 ) ) ) ) ).
% min_max_distrib2
thf(fact_6465_min__max__distrib1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,C3: A,A2: A] :
( ( ord_min @ A @ ( ord_max @ A @ B2 @ C3 ) @ A2 )
= ( ord_max @ A @ ( ord_min @ A @ B2 @ A2 ) @ ( ord_min @ A @ C3 @ A2 ) ) ) ) ).
% min_max_distrib1
thf(fact_6466_max__min__distrib2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( ord_max @ A @ A2 @ ( ord_min @ A @ B2 @ C3 ) )
= ( ord_min @ A @ ( ord_max @ A @ A2 @ B2 ) @ ( ord_max @ A @ A2 @ C3 ) ) ) ) ).
% max_min_distrib2
thf(fact_6467_max__min__distrib1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,C3: A,A2: A] :
( ( ord_max @ A @ ( ord_min @ A @ B2 @ C3 ) @ A2 )
= ( ord_min @ A @ ( ord_max @ A @ B2 @ A2 ) @ ( ord_max @ A @ C3 @ A2 ) ) ) ) ).
% max_min_distrib1
thf(fact_6468_of__int__min,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [X: int,Y: int] :
( ( ring_1_of_int @ A @ ( ord_min @ int @ X @ Y ) )
= ( ord_min @ A @ ( ring_1_of_int @ A @ X ) @ ( ring_1_of_int @ A @ Y ) ) ) ) ).
% of_int_min
thf(fact_6469_greaterThan__Int__greaterThan,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A] :
( ( inf_inf @ ( set @ A ) @ ( set_ord_lessThan @ A @ A2 ) @ ( set_ord_lessThan @ A @ B2 ) )
= ( set_ord_lessThan @ A @ ( ord_min @ A @ A2 @ B2 ) ) ) ) ).
% greaterThan_Int_greaterThan
thf(fact_6470_min__diff__distrib__left,axiom,
! [A: $tType] :
( ( ordered_ab_group_add @ A )
=> ! [X: A,Y: A,Z: A] :
( ( minus_minus @ A @ ( ord_min @ A @ X @ Y ) @ Z )
= ( ord_min @ A @ ( minus_minus @ A @ X @ Z ) @ ( minus_minus @ A @ Y @ Z ) ) ) ) ).
% min_diff_distrib_left
thf(fact_6471_min__diff,axiom,
! [M2: nat,I2: nat,N: nat] :
( ( ord_min @ nat @ ( minus_minus @ nat @ M2 @ I2 ) @ ( minus_minus @ nat @ N @ I2 ) )
= ( minus_minus @ nat @ ( ord_min @ nat @ M2 @ N ) @ I2 ) ) ).
% min_diff
thf(fact_6472_min__le__iff__disj,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A,Z: A] :
( ( ord_less_eq @ A @ ( ord_min @ A @ X @ Y ) @ Z )
= ( ( ord_less_eq @ A @ X @ Z )
| ( ord_less_eq @ A @ Y @ Z ) ) ) ) ).
% min_le_iff_disj
thf(fact_6473_min_OcoboundedI2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [B2: A,C3: A,A2: A] :
( ( ord_less_eq @ A @ B2 @ C3 )
=> ( ord_less_eq @ A @ ( ord_min @ A @ A2 @ B2 ) @ C3 ) ) ) ).
% min.coboundedI2
thf(fact_6474_min_OcoboundedI1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,C3: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ C3 )
=> ( ord_less_eq @ A @ ( ord_min @ A @ A2 @ B2 ) @ C3 ) ) ) ).
% min.coboundedI1
thf(fact_6475_min_Oabsorb__iff2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [B5: A,A6: A] :
( ( ord_min @ A @ A6 @ B5 )
= B5 ) ) ) ) ).
% min.absorb_iff2
thf(fact_6476_min_Oabsorb__iff1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A6: A,B5: A] :
( ( ord_min @ A @ A6 @ B5 )
= A6 ) ) ) ) ).
% min.absorb_iff1
thf(fact_6477_min_Ocobounded2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A] : ( ord_less_eq @ A @ ( ord_min @ A @ A2 @ B2 ) @ B2 ) ) ).
% min.cobounded2
thf(fact_6478_min_Ocobounded1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A] : ( ord_less_eq @ A @ ( ord_min @ A @ A2 @ B2 ) @ A2 ) ) ).
% min.cobounded1
thf(fact_6479_min_Oorder__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_less_eq @ A )
= ( ^ [A6: A,B5: A] :
( A6
= ( ord_min @ A @ A6 @ B5 ) ) ) ) ) ).
% min.order_iff
thf(fact_6480_min_OboundedI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( ( ord_less_eq @ A @ A2 @ C3 )
=> ( ord_less_eq @ A @ A2 @ ( ord_min @ A @ B2 @ C3 ) ) ) ) ) ).
% min.boundedI
thf(fact_6481_min_OboundedE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A,C3: A] :
( ( ord_less_eq @ A @ A2 @ ( ord_min @ A @ B2 @ C3 ) )
=> ~ ( ( ord_less_eq @ A @ A2 @ B2 )
=> ~ ( ord_less_eq @ A @ A2 @ C3 ) ) ) ) ).
% min.boundedE
thf(fact_6482_min_OorderI,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A] :
( ( A2
= ( ord_min @ A @ A2 @ B2 ) )
=> ( ord_less_eq @ A @ A2 @ B2 ) ) ) ).
% min.orderI
thf(fact_6483_min_OorderE,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,B2: A] :
( ( ord_less_eq @ A @ A2 @ B2 )
=> ( A2
= ( ord_min @ A @ A2 @ B2 ) ) ) ) ).
% min.orderE
thf(fact_6484_min_Omono,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,C3: A,B2: A,D3: A] :
( ( ord_less_eq @ A @ A2 @ C3 )
=> ( ( ord_less_eq @ A @ B2 @ D3 )
=> ( ord_less_eq @ A @ ( ord_min @ A @ A2 @ B2 ) @ ( ord_min @ A @ C3 @ D3 ) ) ) ) ) ).
% min.mono
thf(fact_6485_min__absorb2,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Y: A,X: A] :
( ( ord_less_eq @ A @ Y @ X )
=> ( ( ord_min @ A @ X @ Y )
= Y ) ) ) ).
% min_absorb2
thf(fact_6486_min__absorb1,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X: A,Y: A] :
( ( ord_less_eq @ A @ X @ Y )
=> ( ( ord_min @ A @ X @ Y )
= X ) ) ) ).
% min_absorb1
thf(fact_6487_min__def,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( ord_min @ A )
= ( ^ [A6: A,B5: A] : ( if @ A @ ( ord_less_eq @ A @ A6 @ B5 ) @ A6 @ B5 ) ) ) ) ).
% min_def
thf(fact_6488_min__def__raw,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( ord_min @ A )
= ( ^ [A6: A,B5: A] : ( if @ A @ ( ord_less_eq @ A @ A6 @ B5 ) @ A6 @ B5 ) ) ) ) ).
% min_def_raw
thf(fact_6489_max__mult__distrib__left,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [P: A,X: A,Y: A] :
( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ P )
=> ( ( times_times @ A @ P @ ( ord_max @ A @ X @ Y ) )
= ( ord_max @ A @ ( times_times @ A @ P @ X ) @ ( times_times @ A @ P @ Y ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ P )
=> ( ( times_times @ A @ P @ ( ord_max @ A @ X @ Y ) )
= ( ord_min @ A @ ( times_times @ A @ P @ X ) @ ( times_times @ A @ P @ Y ) ) ) ) ) ) ).
% max_mult_distrib_left
thf(fact_6490_min__mult__distrib__left,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [P: A,X: A,Y: A] :
( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ P )
=> ( ( times_times @ A @ P @ ( ord_min @ A @ X @ Y ) )
= ( ord_min @ A @ ( times_times @ A @ P @ X ) @ ( times_times @ A @ P @ Y ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ P )
=> ( ( times_times @ A @ P @ ( ord_min @ A @ X @ Y ) )
= ( ord_max @ A @ ( times_times @ A @ P @ X ) @ ( times_times @ A @ P @ Y ) ) ) ) ) ) ).
% min_mult_distrib_left
thf(fact_6491_max__mult__distrib__right,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [P: A,X: A,Y: A] :
( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ P )
=> ( ( times_times @ A @ ( ord_max @ A @ X @ Y ) @ P )
= ( ord_max @ A @ ( times_times @ A @ X @ P ) @ ( times_times @ A @ Y @ P ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ P )
=> ( ( times_times @ A @ ( ord_max @ A @ X @ Y ) @ P )
= ( ord_min @ A @ ( times_times @ A @ X @ P ) @ ( times_times @ A @ Y @ P ) ) ) ) ) ) ).
% max_mult_distrib_right
thf(fact_6492_min__mult__distrib__right,axiom,
! [A: $tType] :
( ( linordered_idom @ A )
=> ! [P: A,X: A,Y: A] :
( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ P )
=> ( ( times_times @ A @ ( ord_min @ A @ X @ Y ) @ P )
= ( ord_min @ A @ ( times_times @ A @ X @ P ) @ ( times_times @ A @ Y @ P ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ P )
=> ( ( times_times @ A @ ( ord_min @ A @ X @ Y ) @ P )
= ( ord_max @ A @ ( times_times @ A @ X @ P ) @ ( times_times @ A @ Y @ P ) ) ) ) ) ) ).
% min_mult_distrib_right
thf(fact_6493_max__divide__distrib__right,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [P: A,X: A,Y: A] :
( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ P )
=> ( ( divide_divide @ A @ ( ord_max @ A @ X @ Y ) @ P )
= ( ord_max @ A @ ( divide_divide @ A @ X @ P ) @ ( divide_divide @ A @ Y @ P ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ P )
=> ( ( divide_divide @ A @ ( ord_max @ A @ X @ Y ) @ P )
= ( ord_min @ A @ ( divide_divide @ A @ X @ P ) @ ( divide_divide @ A @ Y @ P ) ) ) ) ) ) ).
% max_divide_distrib_right
thf(fact_6494_min__divide__distrib__right,axiom,
! [A: $tType] :
( ( linordered_field @ A )
=> ! [P: A,X: A,Y: A] :
( ( ( ord_less_eq @ A @ ( zero_zero @ A ) @ P )
=> ( ( divide_divide @ A @ ( ord_min @ A @ X @ Y ) @ P )
= ( ord_min @ A @ ( divide_divide @ A @ X @ P ) @ ( divide_divide @ A @ Y @ P ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( zero_zero @ A ) @ P )
=> ( ( divide_divide @ A @ ( ord_min @ A @ X @ Y ) @ P )
= ( ord_max @ A @ ( divide_divide @ A @ X @ P ) @ ( divide_divide @ A @ Y @ P ) ) ) ) ) ) ).
% min_divide_distrib_right
thf(fact_6495_min__Suc1,axiom,
! [N: nat,M2: nat] :
( ( ord_min @ nat @ ( suc @ N ) @ M2 )
= ( case_nat @ nat @ ( zero_zero @ nat )
@ ^ [M6: nat] : ( suc @ ( ord_min @ nat @ N @ M6 ) )
@ M2 ) ) ).
% min_Suc1
thf(fact_6496_min__Suc2,axiom,
! [M2: nat,N: nat] :
( ( ord_min @ nat @ M2 @ ( suc @ N ) )
= ( case_nat @ nat @ ( zero_zero @ nat )
@ ^ [M6: nat] : ( suc @ ( ord_min @ nat @ M6 @ N ) )
@ M2 ) ) ).
% min_Suc2
thf(fact_6497_Inf__insert__finite,axiom,
! [A: $tType] :
( ( condit6923001295902523014norder @ A )
=> ! [S2: set @ A,X: A] :
( ( finite_finite2 @ A @ S2 )
=> ( ( ( S2
= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Inf_Inf @ A @ ( insert @ A @ X @ S2 ) )
= X ) )
& ( ( S2
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( complete_Inf_Inf @ A @ ( insert @ A @ X @ S2 ) )
= ( ord_min @ A @ X @ ( complete_Inf_Inf @ A @ S2 ) ) ) ) ) ) ) ).
% Inf_insert_finite
thf(fact_6498_hom__Min__commute,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [H2: A > A,N6: set @ A] :
( ! [X4: A,Y3: A] :
( ( H2 @ ( ord_min @ A @ X4 @ Y3 ) )
= ( ord_min @ A @ ( H2 @ X4 ) @ ( H2 @ Y3 ) ) )
=> ( ( finite_finite2 @ A @ N6 )
=> ( ( N6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( H2 @ ( lattic643756798350308766er_Min @ A @ N6 ) )
= ( lattic643756798350308766er_Min @ A @ ( image2 @ A @ A @ H2 @ N6 ) ) ) ) ) ) ) ).
% hom_Min_commute
thf(fact_6499_Min_Osubset,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,B6: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( B6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ A ) @ B6 @ A5 )
=> ( ( ord_min @ A @ ( lattic643756798350308766er_Min @ A @ B6 ) @ ( lattic643756798350308766er_Min @ A @ A5 ) )
= ( lattic643756798350308766er_Min @ A @ A5 ) ) ) ) ) ) ).
% Min.subset
thf(fact_6500_Min_Oclosed,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ! [X4: A,Y3: A] : ( member @ A @ ( ord_min @ A @ X4 @ Y3 ) @ ( insert @ A @ X4 @ ( insert @ A @ Y3 @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( member @ A @ ( lattic643756798350308766er_Min @ A @ A5 ) @ A5 ) ) ) ) ) ).
% Min.closed
thf(fact_6501_Min_Oinsert__not__elem,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ~ ( member @ A @ X @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798350308766er_Min @ A @ ( insert @ A @ X @ A5 ) )
= ( ord_min @ A @ X @ ( lattic643756798350308766er_Min @ A @ A5 ) ) ) ) ) ) ) ).
% Min.insert_not_elem
thf(fact_6502_Min_Ounion,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,B6: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( finite_finite2 @ A @ B6 )
=> ( ( B6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798350308766er_Min @ A @ ( sup_sup @ ( set @ A ) @ A5 @ B6 ) )
= ( ord_min @ A @ ( lattic643756798350308766er_Min @ A @ A5 ) @ ( lattic643756798350308766er_Min @ A @ B6 ) ) ) ) ) ) ) ) ).
% Min.union
thf(fact_6503_Min_Oeq__fold,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( lattic643756798350308766er_Min @ A @ ( insert @ A @ X @ A5 ) )
= ( finite_fold @ A @ A @ ( ord_min @ A ) @ X @ A5 ) ) ) ) ).
% Min.eq_fold
thf(fact_6504_Min_Oremove,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( member @ A @ X @ A5 )
=> ( ( ( ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798350308766er_Min @ A @ A5 )
= X ) )
& ( ( ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798350308766er_Min @ A @ A5 )
= ( ord_min @ A @ X @ ( lattic643756798350308766er_Min @ A @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ) ).
% Min.remove
thf(fact_6505_Min_Oinsert__remove,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,X: A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( ( ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798350308766er_Min @ A @ ( insert @ A @ X @ A5 ) )
= X ) )
& ( ( ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( lattic643756798350308766er_Min @ A @ ( insert @ A @ X @ A5 ) )
= ( ord_min @ A @ X @ ( lattic643756798350308766er_Min @ A @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ) ).
% Min.insert_remove
thf(fact_6506_lexord__take__index__conv,axiom,
! [A: $tType,X: list @ A,Y: list @ A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Y ) @ ( lexord @ A @ R2 ) )
= ( ( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ X ) @ ( size_size @ ( list @ A ) @ Y ) )
& ( ( take @ A @ ( size_size @ ( list @ A ) @ X ) @ Y )
= X ) )
| ? [I4: nat] :
( ( ord_less @ nat @ I4 @ ( ord_min @ nat @ ( size_size @ ( list @ A ) @ X ) @ ( size_size @ ( list @ A ) @ Y ) ) )
& ( ( take @ A @ I4 @ X )
= ( take @ A @ I4 @ Y ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( nth @ A @ X @ I4 ) @ ( nth @ A @ Y @ I4 ) ) @ R2 ) ) ) ) ).
% lexord_take_index_conv
thf(fact_6507_lenlex__def,axiom,
! [A: $tType] :
( ( lenlex @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A )] :
( inv_image @ ( product_prod @ nat @ ( list @ A ) ) @ ( list @ A ) @ ( lex_prod @ nat @ ( list @ A ) @ less_than @ ( lex @ A @ R5 ) )
@ ^ [Xs3: list @ A] : ( product_Pair @ nat @ ( list @ A ) @ ( size_size @ ( list @ A ) @ Xs3 ) @ Xs3 ) ) ) ) ).
% lenlex_def
thf(fact_6508_less__than__iff,axiom,
! [X: nat,Y: nat] :
( ( member @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ X @ Y ) @ less_than )
= ( ord_less @ nat @ X @ Y ) ) ).
% less_than_iff
thf(fact_6509_inf__int__def,axiom,
( ( inf_inf @ int )
= ( ord_min @ int ) ) ).
% inf_int_def
thf(fact_6510_inf__nat__def,axiom,
( ( inf_inf @ nat )
= ( ord_min @ nat ) ) ).
% inf_nat_def
thf(fact_6511_lexord__linear,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),X: list @ A,Y: list @ A] :
( ! [A4: A,B4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ B4 ) @ R2 )
| ( A4 = B4 )
| ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B4 @ A4 ) @ R2 ) )
=> ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Y ) @ ( lexord @ A @ R2 ) )
| ( X = Y )
| ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Y @ X ) @ ( lexord @ A @ R2 ) ) ) ) ).
% lexord_linear
thf(fact_6512_lexord__irreflexive,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),Xs: list @ A] :
( ! [X4: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ X4 ) @ R2 )
=> ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Xs ) @ ( lexord @ A @ R2 ) ) ) ).
% lexord_irreflexive
thf(fact_6513_lexord__Nil__right,axiom,
! [A: $tType,X: list @ A,R2: set @ ( product_prod @ A @ A )] :
~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ ( nil @ A ) ) @ ( lexord @ A @ R2 ) ) ).
% lexord_Nil_right
thf(fact_6514_pair__less__def,axiom,
( fun_pair_less
= ( lex_prod @ nat @ nat @ less_than @ less_than ) ) ).
% pair_less_def
thf(fact_6515_lexord__partial__trans,axiom,
! [A: $tType,Xs: list @ A,R2: set @ ( product_prod @ A @ A ),Ys: list @ A,Zs: list @ A] :
( ! [X4: A,Y3: A,Z3: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y3 ) @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z3 ) @ R2 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Z3 ) @ R2 ) ) ) )
=> ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( lexord @ A @ R2 ) )
=> ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ys @ Zs ) @ ( lexord @ A @ R2 ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Zs ) @ ( lexord @ A @ R2 ) ) ) ) ) ).
% lexord_partial_trans
thf(fact_6516_lexord__lex,axiom,
! [A: $tType,X: list @ A,Y: list @ A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Y ) @ ( lex @ A @ R2 ) )
= ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Y ) @ ( lexord @ A @ R2 ) )
& ( ( size_size @ ( list @ A ) @ X )
= ( size_size @ ( list @ A ) @ Y ) ) ) ) ).
% lexord_lex
thf(fact_6517_mlex__prod__def,axiom,
! [A: $tType] :
( ( mlex_prod @ A )
= ( ^ [F4: A > nat,R6: set @ ( product_prod @ A @ A )] :
( inv_image @ ( product_prod @ nat @ A ) @ A @ ( lex_prod @ nat @ A @ less_than @ R6 )
@ ^ [X5: A] : ( product_Pair @ nat @ A @ ( F4 @ X5 ) @ X5 ) ) ) ) ).
% mlex_prod_def
thf(fact_6518_List_Olexordp__def,axiom,
! [A: $tType] :
( ( lexordp @ A )
= ( ^ [R5: A > A > $o,Xs3: list @ A,Ys3: list @ A] : ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs3 @ Ys3 ) @ ( lexord @ A @ ( collect @ ( product_prod @ A @ A ) @ ( product_case_prod @ A @ A @ $o @ R5 ) ) ) ) ) ) ).
% List.lexordp_def
thf(fact_6519_set__zip,axiom,
! [B: $tType,A: $tType,Xs: list @ A,Ys: list @ B] :
( ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) )
= ( collect @ ( product_prod @ A @ B )
@ ^ [Uu3: product_prod @ A @ B] :
? [I4: nat] :
( ( Uu3
= ( product_Pair @ A @ B @ ( nth @ A @ Xs @ I4 ) @ ( nth @ B @ Ys @ I4 ) ) )
& ( ord_less @ nat @ I4 @ ( ord_min @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ B ) @ Ys ) ) ) ) ) ) ).
% set_zip
thf(fact_6520_zip__Nil,axiom,
! [B: $tType,A: $tType,Ys: list @ B] :
( ( zip @ A @ B @ ( nil @ A ) @ Ys )
= ( nil @ ( product_prod @ A @ B ) ) ) ).
% zip_Nil
thf(fact_6521_Nil__eq__zip__iff,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B] :
( ( ( nil @ ( product_prod @ A @ B ) )
= ( zip @ A @ B @ Xs @ Ys ) )
= ( ( Xs
= ( nil @ A ) )
| ( Ys
= ( nil @ B ) ) ) ) ).
% Nil_eq_zip_iff
thf(fact_6522_zip__eq__Nil__iff,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B] :
( ( ( zip @ A @ B @ Xs @ Ys )
= ( nil @ ( product_prod @ A @ B ) ) )
= ( ( Xs
= ( nil @ A ) )
| ( Ys
= ( nil @ B ) ) ) ) ).
% zip_eq_Nil_iff
thf(fact_6523_zip__replicate,axiom,
! [A: $tType,B: $tType,I2: nat,X: A,J: nat,Y: B] :
( ( zip @ A @ B @ ( replicate @ A @ I2 @ X ) @ ( replicate @ B @ J @ Y ) )
= ( replicate @ ( product_prod @ A @ B ) @ ( ord_min @ nat @ I2 @ J ) @ ( product_Pair @ A @ B @ X @ Y ) ) ) ).
% zip_replicate
thf(fact_6524_length__zip,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B] :
( ( size_size @ ( list @ ( product_prod @ A @ B ) ) @ ( zip @ A @ B @ Xs @ Ys ) )
= ( ord_min @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ B ) @ Ys ) ) ) ).
% length_zip
thf(fact_6525_nth__zip,axiom,
! [A: $tType,B: $tType,I2: nat,Xs: list @ A,Ys: list @ B] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( nth @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) @ I2 )
= ( product_Pair @ A @ B @ ( nth @ A @ Xs @ I2 ) @ ( nth @ B @ Ys @ I2 ) ) ) ) ) ).
% nth_zip
thf(fact_6526_zip__update,axiom,
! [A: $tType,B: $tType,Xs: list @ A,I2: nat,X: A,Ys: list @ B,Y: B] :
( ( zip @ A @ B @ ( list_update @ A @ Xs @ I2 @ X ) @ ( list_update @ B @ Ys @ I2 @ Y ) )
= ( list_update @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) @ I2 @ ( product_Pair @ A @ B @ X @ Y ) ) ) ).
% zip_update
thf(fact_6527_take__zip,axiom,
! [A: $tType,B: $tType,N: nat,Xs: list @ A,Ys: list @ B] :
( ( take @ ( product_prod @ A @ B ) @ N @ ( zip @ A @ B @ Xs @ Ys ) )
= ( zip @ A @ B @ ( take @ A @ N @ Xs ) @ ( take @ B @ N @ Ys ) ) ) ).
% take_zip
thf(fact_6528_zip_Osimps_I1_J,axiom,
! [B: $tType,A: $tType,Xs: list @ A] :
( ( zip @ A @ B @ Xs @ ( nil @ B ) )
= ( nil @ ( product_prod @ A @ B ) ) ) ).
% zip.simps(1)
thf(fact_6529_distinct__zipI1,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B] :
( ( distinct @ A @ Xs )
=> ( distinct @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) ) ) ).
% distinct_zipI1
thf(fact_6530_distinct__zipI2,axiom,
! [B: $tType,A: $tType,Ys: list @ A,Xs: list @ B] :
( ( distinct @ A @ Ys )
=> ( distinct @ ( product_prod @ B @ A ) @ ( zip @ B @ A @ Xs @ Ys ) ) ) ).
% distinct_zipI2
thf(fact_6531_zip__same,axiom,
! [A: $tType,A2: A,B2: A,Xs: list @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ ( set2 @ ( product_prod @ A @ A ) @ ( zip @ A @ A @ Xs @ Xs ) ) )
= ( ( member @ A @ A2 @ ( set2 @ A @ Xs ) )
& ( A2 = B2 ) ) ) ).
% zip_same
thf(fact_6532_in__set__zipE,axiom,
! [A: $tType,B: $tType,X: A,Y: B,Xs: list @ A,Ys: list @ B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) ) )
=> ~ ( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ~ ( member @ B @ Y @ ( set2 @ B @ Ys ) ) ) ) ).
% in_set_zipE
thf(fact_6533_set__zip__leftD,axiom,
! [B: $tType,A: $tType,X: A,Y: B,Xs: list @ A,Ys: list @ B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) ) )
=> ( member @ A @ X @ ( set2 @ A @ Xs ) ) ) ).
% set_zip_leftD
thf(fact_6534_set__zip__rightD,axiom,
! [A: $tType,B: $tType,X: A,Y: B,Xs: list @ A,Ys: list @ B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) ) )
=> ( member @ B @ Y @ ( set2 @ B @ Ys ) ) ) ).
% set_zip_rightD
thf(fact_6535_update__zip,axiom,
! [B: $tType,A: $tType,Xs: list @ A,Ys: list @ B,I2: nat,Xy2: product_prod @ A @ B] :
( ( list_update @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) @ I2 @ Xy2 )
= ( zip @ A @ B @ ( list_update @ A @ Xs @ I2 @ ( product_fst @ A @ B @ Xy2 ) ) @ ( list_update @ B @ Ys @ I2 @ ( product_snd @ A @ B @ Xy2 ) ) ) ) ).
% update_zip
thf(fact_6536_zip__obtain__same__length,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B,P2: ( list @ ( product_prod @ A @ B ) ) > $o] :
( ! [Zs2: list @ A,Ws: list @ B,N3: nat] :
( ( ( size_size @ ( list @ A ) @ Zs2 )
= ( size_size @ ( list @ B ) @ Ws ) )
=> ( ( N3
= ( ord_min @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ B ) @ Ys ) ) )
=> ( ( Zs2
= ( take @ A @ N3 @ Xs ) )
=> ( ( Ws
= ( take @ B @ N3 @ Ys ) )
=> ( P2 @ ( zip @ A @ B @ Zs2 @ Ws ) ) ) ) ) )
=> ( P2 @ ( zip @ A @ B @ Xs @ Ys ) ) ) ).
% zip_obtain_same_length
thf(fact_6537_list__eq__iff__zip__eq,axiom,
! [A: $tType] :
( ( ^ [Y5: list @ A,Z2: list @ A] : Y5 = Z2 )
= ( ^ [Xs3: list @ A,Ys3: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs3 )
= ( size_size @ ( list @ A ) @ Ys3 ) )
& ! [X5: product_prod @ A @ A] :
( ( member @ ( product_prod @ A @ A ) @ X5 @ ( set2 @ ( product_prod @ A @ A ) @ ( zip @ A @ A @ Xs3 @ Ys3 ) ) )
=> ( product_case_prod @ A @ A @ $o
@ ^ [Y5: A,Z2: A] : Y5 = Z2
@ X5 ) ) ) ) ) ).
% list_eq_iff_zip_eq
thf(fact_6538_in__set__impl__in__set__zip1,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B,X: A] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( 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 @ Ys ) ) ) ) ) ).
% in_set_impl_in_set_zip1
thf(fact_6539_in__set__impl__in__set__zip2,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B,Y: B] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( member @ B @ Y @ ( set2 @ B @ Ys ) )
=> ~ ! [X4: A] :
~ ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y ) @ ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) ) ) ) ) ).
% in_set_impl_in_set_zip2
thf(fact_6540_concat__injective,axiom,
! [A: $tType,Xs: list @ ( list @ A ),Ys: list @ ( list @ A )] :
( ( ( concat @ A @ Xs )
= ( concat @ A @ Ys ) )
=> ( ( ( size_size @ ( list @ ( list @ A ) ) @ Xs )
= ( size_size @ ( list @ ( list @ A ) ) @ Ys ) )
=> ( ! [X4: product_prod @ ( list @ A ) @ ( list @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ X4 @ ( set2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( zip @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) ) )
=> ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ $o
@ ^ [Y6: list @ A,Z5: list @ A] :
( ( size_size @ ( list @ A ) @ Y6 )
= ( size_size @ ( list @ A ) @ Z5 ) )
@ X4 ) )
=> ( Xs = Ys ) ) ) ) ).
% concat_injective
thf(fact_6541_concat__eq__concat__iff,axiom,
! [A: $tType,Xs: list @ ( list @ A ),Ys: list @ ( list @ A )] :
( ! [X4: product_prod @ ( list @ A ) @ ( list @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ X4 @ ( set2 @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( zip @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) ) )
=> ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ $o
@ ^ [Y6: list @ A,Z5: list @ A] :
( ( size_size @ ( list @ A ) @ Y6 )
= ( size_size @ ( list @ A ) @ Z5 ) )
@ X4 ) )
=> ( ( ( size_size @ ( list @ ( list @ A ) ) @ Xs )
= ( size_size @ ( list @ ( list @ A ) ) @ Ys ) )
=> ( ( ( concat @ A @ Xs )
= ( concat @ A @ Ys ) )
= ( Xs = Ys ) ) ) ) ).
% concat_eq_concat_iff
thf(fact_6542_in__set__zip,axiom,
! [B: $tType,A: $tType,P: product_prod @ A @ B,Xs: list @ A,Ys: list @ B] :
( ( member @ ( product_prod @ A @ B ) @ P @ ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) ) )
= ( ? [N2: nat] :
( ( ( nth @ A @ Xs @ N2 )
= ( product_fst @ A @ B @ P ) )
& ( ( nth @ B @ Ys @ N2 )
= ( product_snd @ A @ B @ P ) )
& ( ord_less @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs ) )
& ( ord_less @ nat @ N2 @ ( size_size @ ( list @ B ) @ Ys ) ) ) ) ) ).
% in_set_zip
thf(fact_6543_listrel__iff__zip,axiom,
! [B: $tType,A: $tType,Xs: list @ A,Ys: list @ B,R2: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs @ Ys ) @ ( listrel @ A @ B @ R2 ) )
= ( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys ) )
& ! [X5: product_prod @ A @ B] :
( ( member @ ( product_prod @ A @ B ) @ X5 @ ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) ) )
=> ( product_case_prod @ A @ B @ $o
@ ^ [Y6: A,Z5: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Y6 @ Z5 ) @ R2 )
@ X5 ) ) ) ) ).
% listrel_iff_zip
thf(fact_6544_map__upds__fold__map__upd,axiom,
! [B: $tType,A: $tType] :
( ( map_upds @ A @ B )
= ( ^ [M3: A > ( option @ B ),Ks: list @ A,Vs: list @ B] :
( foldl @ ( A > ( option @ B ) ) @ ( product_prod @ A @ B )
@ ^ [N2: A > ( option @ B )] :
( product_case_prod @ A @ B @ ( A > ( option @ B ) )
@ ^ [K3: A,V5: B] : ( fun_upd @ A @ ( option @ B ) @ N2 @ K3 @ ( some @ B @ V5 ) ) )
@ M3
@ ( zip @ A @ B @ Ks @ Vs ) ) ) ) ).
% map_upds_fold_map_upd
thf(fact_6545_listrel__rtrancl__refl,axiom,
! [A: $tType,Xs: list @ A,R2: set @ ( product_prod @ A @ A )] : ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Xs ) @ ( listrel @ A @ A @ ( transitive_rtrancl @ A @ R2 ) ) ) ).
% listrel_rtrancl_refl
thf(fact_6546_foldl__Nil,axiom,
! [A: $tType,B: $tType,F3: B > A > B,A2: B] :
( ( foldl @ B @ A @ F3 @ A2 @ ( nil @ A ) )
= A2 ) ).
% foldl_Nil
thf(fact_6547_foldl__cong,axiom,
! [A: $tType,B: $tType,A2: A,B2: A,L: list @ B,K2: list @ B,F3: A > B > A,G3: A > B > A] :
( ( A2 = B2 )
=> ( ( L = K2 )
=> ( ! [A4: A,X4: B] :
( ( member @ B @ X4 @ ( set2 @ B @ L ) )
=> ( ( F3 @ A4 @ X4 )
= ( G3 @ A4 @ X4 ) ) )
=> ( ( foldl @ A @ B @ F3 @ A2 @ L )
= ( foldl @ A @ B @ G3 @ B2 @ K2 ) ) ) ) ) ).
% foldl_cong
thf(fact_6548_listrel__eq__len,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B,R2: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs @ Ys ) @ ( listrel @ A @ B @ R2 ) )
=> ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys ) ) ) ).
% listrel_eq_len
thf(fact_6549_listrel_ONil,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B )] : ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( nil @ A ) @ ( nil @ B ) ) @ ( listrel @ A @ B @ R2 ) ) ).
% listrel.Nil
thf(fact_6550_listrel__Nil1,axiom,
! [A: $tType,B: $tType,Xs: list @ B,R2: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( nil @ A ) @ Xs ) @ ( listrel @ A @ B @ R2 ) )
=> ( Xs
= ( nil @ B ) ) ) ).
% listrel_Nil1
thf(fact_6551_listrel__Nil2,axiom,
! [B: $tType,A: $tType,Xs: list @ A,R2: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs @ ( nil @ B ) ) @ ( listrel @ A @ B @ R2 ) )
=> ( Xs
= ( nil @ A ) ) ) ).
% listrel_Nil2
thf(fact_6552_listrel__rtrancl__trans,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,R2: set @ ( product_prod @ A @ A ),Zs: list @ A] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( listrel @ A @ A @ ( transitive_rtrancl @ A @ R2 ) ) )
=> ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ys @ Zs ) @ ( listrel @ A @ A @ ( transitive_rtrancl @ A @ R2 ) ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Zs ) @ ( listrel @ A @ A @ ( transitive_rtrancl @ A @ R2 ) ) ) ) ) ).
% listrel_rtrancl_trans
thf(fact_6553_listrel__mono,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B ),S: set @ ( product_prod @ A @ B )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ R2 @ S )
=> ( ord_less_eq @ ( set @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) ) @ ( listrel @ A @ B @ R2 ) @ ( listrel @ A @ B @ S ) ) ) ).
% listrel_mono
thf(fact_6554_listrel__reflcl__if__listrel1,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( listrel1 @ A @ R2 ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( listrel @ A @ A @ ( transitive_rtrancl @ A @ R2 ) ) ) ) ).
% listrel_reflcl_if_listrel1
thf(fact_6555_listrel__rtrancl__eq__rtrancl__listrel1,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( listrel @ A @ A @ ( transitive_rtrancl @ A @ R2 ) )
= ( transitive_rtrancl @ ( list @ A ) @ ( listrel1 @ A @ R2 ) ) ) ).
% listrel_rtrancl_eq_rtrancl_listrel1
thf(fact_6556_rtrancl__listrel1__if__listrel,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( listrel @ A @ A @ R2 ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( transitive_rtrancl @ ( list @ A ) @ ( listrel1 @ A @ R2 ) ) ) ) ).
% rtrancl_listrel1_if_listrel
thf(fact_6557_listrel__subset__rtrancl__listrel1,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] : ( ord_less_eq @ ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) @ ( listrel @ A @ A @ R2 ) @ ( transitive_rtrancl @ ( list @ A ) @ ( listrel1 @ A @ R2 ) ) ) ).
% listrel_subset_rtrancl_listrel1
thf(fact_6558_listrel__iff__nth,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B,R2: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs @ Ys ) @ ( listrel @ A @ B @ R2 ) )
= ( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys ) )
& ! [N2: nat] :
( ( ord_less @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ ( nth @ A @ Xs @ N2 ) @ ( nth @ B @ Ys @ N2 ) ) @ R2 ) ) ) ) ).
% listrel_iff_nth
thf(fact_6559_listrel__def,axiom,
! [B: $tType,A: $tType] :
( ( listrel @ A @ B )
= ( ^ [R5: set @ ( product_prod @ A @ B )] :
( collect @ ( product_prod @ ( list @ A ) @ ( list @ B ) )
@ ( product_case_prod @ ( list @ A ) @ ( list @ B ) @ $o
@ ( listrelp @ A @ B
@ ^ [X5: A,Y6: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X5 @ Y6 ) @ R5 ) ) ) ) ) ) ).
% listrel_def
thf(fact_6560_map__of__zip__nth,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B,I2: nat] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( distinct @ A @ Xs )
=> ( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( map_of @ A @ B @ ( zip @ A @ B @ Xs @ Ys ) @ ( nth @ A @ Xs @ I2 ) )
= ( some @ B @ ( nth @ B @ Ys @ I2 ) ) ) ) ) ) ).
% map_of_zip_nth
thf(fact_6561_map__of__zip__is__None,axiom,
! [B: $tType,A: $tType,Xs: list @ A,Ys: list @ B,X: A] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( ( map_of @ A @ B @ ( zip @ A @ B @ Xs @ Ys ) @ X )
= ( none @ B ) )
= ( ~ ( member @ A @ X @ ( set2 @ A @ Xs ) ) ) ) ) ).
% map_of_zip_is_None
thf(fact_6562_dom__map__of__zip,axiom,
! [B: $tType,A: $tType,Xs: list @ A,Ys: list @ B] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( dom @ A @ B @ ( map_of @ A @ B @ ( zip @ A @ B @ Xs @ Ys ) ) )
= ( set2 @ A @ Xs ) ) ) ).
% dom_map_of_zip
thf(fact_6563_finite__dom__map__of,axiom,
! [B: $tType,A: $tType,L: list @ ( product_prod @ A @ B )] : ( finite_finite2 @ A @ ( dom @ A @ B @ ( map_of @ A @ B @ L ) ) ) ).
% finite_dom_map_of
thf(fact_6564_listrelp_ONil,axiom,
! [A: $tType,B: $tType,R2: A > B > $o] : ( listrelp @ A @ B @ R2 @ ( nil @ A ) @ ( nil @ B ) ) ).
% listrelp.Nil
thf(fact_6565_finite__graph__map__of,axiom,
! [B: $tType,A: $tType,Al: list @ ( product_prod @ A @ B )] : ( finite_finite2 @ ( product_prod @ A @ B ) @ ( graph @ A @ B @ ( map_of @ A @ B @ Al ) ) ) ).
% finite_graph_map_of
thf(fact_6566_map__of__SomeD,axiom,
! [A: $tType,B: $tType,Xs: list @ ( product_prod @ B @ A ),K2: B,Y: A] :
( ( ( map_of @ B @ A @ Xs @ K2 )
= ( some @ A @ Y ) )
=> ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ K2 @ Y ) @ ( set2 @ ( product_prod @ B @ A ) @ Xs ) ) ) ).
% map_of_SomeD
thf(fact_6567_weak__map__of__SomeI,axiom,
! [A: $tType,B: $tType,K2: A,X: B,L: list @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K2 @ X ) @ ( set2 @ ( product_prod @ A @ B ) @ L ) )
=> ? [X4: B] :
( ( map_of @ A @ B @ L @ K2 )
= ( some @ B @ X4 ) ) ) ).
% weak_map_of_SomeI
thf(fact_6568_map__of__eq__dom,axiom,
! [B: $tType,A: $tType,Xs: list @ ( product_prod @ A @ B ),Ys: list @ ( product_prod @ A @ B )] :
( ( ( map_of @ A @ B @ Xs )
= ( map_of @ A @ B @ Ys ) )
=> ( ( image2 @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ ( set2 @ ( product_prod @ A @ B ) @ Xs ) )
= ( image2 @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ ( set2 @ ( product_prod @ A @ B ) @ Ys ) ) ) ) ).
% map_of_eq_dom
thf(fact_6569_finite__range__map__of,axiom,
! [A: $tType,B: $tType,Xys: list @ ( product_prod @ B @ A )] : ( finite_finite2 @ ( option @ A ) @ ( image2 @ B @ ( option @ A ) @ ( map_of @ B @ A @ Xys ) @ ( top_top @ ( set @ B ) ) ) ) ).
% finite_range_map_of
thf(fact_6570_map__of__zip__is__Some,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B,X: A] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( member @ A @ X @ ( set2 @ A @ Xs ) )
= ( ? [Y6: B] :
( ( map_of @ A @ B @ ( zip @ A @ B @ Xs @ Ys ) @ X )
= ( some @ B @ Y6 ) ) ) ) ) ).
% map_of_zip_is_Some
thf(fact_6571_map__of__eq__None__iff,axiom,
! [A: $tType,B: $tType,Xys: list @ ( product_prod @ B @ A ),X: B] :
( ( ( map_of @ B @ A @ Xys @ X )
= ( none @ A ) )
= ( ~ ( member @ B @ X @ ( image2 @ ( product_prod @ B @ A ) @ B @ ( product_fst @ B @ A ) @ ( set2 @ ( product_prod @ B @ A ) @ Xys ) ) ) ) ) ).
% map_of_eq_None_iff
thf(fact_6572_dom__map__of__conv__image__fst,axiom,
! [B: $tType,A: $tType,Xys: list @ ( product_prod @ A @ B )] :
( ( dom @ A @ B @ ( map_of @ A @ B @ Xys ) )
= ( image2 @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ ( set2 @ ( product_prod @ A @ B ) @ Xys ) ) ) ).
% dom_map_of_conv_image_fst
thf(fact_6573_map__of__zip__upd,axiom,
! [A: $tType,B: $tType,Ys: list @ B,Xs: list @ A,Zs: list @ B,X: A,Y: B,Z: B] :
( ( ( size_size @ ( list @ B ) @ Ys )
= ( 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 @ Ys ) ) @ X @ ( some @ B @ Y ) )
= ( fun_upd @ A @ ( option @ B ) @ ( map_of @ A @ B @ ( zip @ A @ B @ Xs @ Zs ) ) @ X @ ( some @ B @ Z ) ) )
=> ( ( map_of @ A @ B @ ( zip @ A @ B @ Xs @ Ys ) )
= ( map_of @ A @ B @ ( zip @ A @ B @ Xs @ Zs ) ) ) ) ) ) ) ).
% map_of_zip_upd
thf(fact_6574_ran__map__of__zip,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( distinct @ A @ Xs )
=> ( ( ran @ A @ B @ ( map_of @ A @ B @ ( zip @ A @ B @ Xs @ Ys ) ) )
= ( set2 @ B @ Ys ) ) ) ) ).
% ran_map_of_zip
thf(fact_6575_listrelp__listrel__eq,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ B )] :
( ( listrelp @ A @ B
@ ^ [X5: A,Y6: B] : ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X5 @ Y6 ) @ R2 ) )
= ( ^ [X5: list @ A,Y6: list @ B] : ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ X5 @ Y6 ) @ ( listrel @ A @ B @ R2 ) ) ) ) ).
% listrelp_listrel_eq
thf(fact_6576_ran__distinct,axiom,
! [B: $tType,A: $tType,Al: list @ ( product_prod @ A @ B )] :
( ( distinct @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Al ) )
=> ( ( ran @ A @ B @ ( map_of @ A @ B @ Al ) )
= ( image2 @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ ( set2 @ ( product_prod @ A @ B ) @ Al ) ) ) ) ).
% ran_distinct
thf(fact_6577_map__of__is__SomeI,axiom,
! [A: $tType,B: $tType,Xys: list @ ( product_prod @ A @ B ),X: A,Y: B] :
( ( distinct @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xys ) )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ ( set2 @ ( product_prod @ A @ B ) @ Xys ) )
=> ( ( map_of @ A @ B @ Xys @ X )
= ( some @ B @ Y ) ) ) ) ).
% map_of_is_SomeI
thf(fact_6578_map__ident,axiom,
! [A: $tType] :
( ( map @ A @ A
@ ^ [X5: A] : X5 )
= ( ^ [Xs3: list @ A] : Xs3 ) ) ).
% map_ident
thf(fact_6579_map__is__Nil__conv,axiom,
! [A: $tType,B: $tType,F3: B > A,Xs: list @ B] :
( ( ( map @ B @ A @ F3 @ Xs )
= ( nil @ A ) )
= ( Xs
= ( nil @ B ) ) ) ).
% map_is_Nil_conv
thf(fact_6580_Nil__is__map__conv,axiom,
! [A: $tType,B: $tType,F3: B > A,Xs: list @ B] :
( ( ( nil @ A )
= ( map @ B @ A @ F3 @ Xs ) )
= ( Xs
= ( nil @ B ) ) ) ).
% Nil_is_map_conv
thf(fact_6581_list_Omap__disc__iff,axiom,
! [B: $tType,A: $tType,F3: A > B,A2: list @ A] :
( ( ( map @ A @ B @ F3 @ A2 )
= ( nil @ B ) )
= ( A2
= ( nil @ A ) ) ) ).
% list.map_disc_iff
thf(fact_6582_map__map,axiom,
! [B: $tType,A: $tType,C: $tType,F3: B > A,G3: C > B,Xs: list @ C] :
( ( map @ B @ A @ F3 @ ( map @ C @ B @ G3 @ Xs ) )
= ( map @ C @ A @ ( comp @ B @ A @ C @ F3 @ G3 ) @ Xs ) ) ).
% map_map
thf(fact_6583_List_Omap_Ocompositionality,axiom,
! [B: $tType,C: $tType,A: $tType,F3: B > C,G3: A > B,List: list @ A] :
( ( map @ B @ C @ F3 @ ( map @ A @ B @ G3 @ List ) )
= ( map @ A @ C @ ( comp @ B @ C @ A @ F3 @ G3 ) @ List ) ) ).
% List.map.compositionality
thf(fact_6584_list_Omap__comp,axiom,
! [B: $tType,C: $tType,A: $tType,G3: B > C,F3: A > B,V: list @ A] :
( ( map @ B @ C @ G3 @ ( map @ A @ B @ F3 @ V ) )
= ( map @ A @ C @ ( comp @ B @ C @ A @ G3 @ F3 ) @ V ) ) ).
% list.map_comp
thf(fact_6585_map__eq__conv,axiom,
! [A: $tType,B: $tType,F3: B > A,Xs: list @ B,G3: B > A] :
( ( ( map @ B @ A @ F3 @ Xs )
= ( map @ B @ A @ G3 @ Xs ) )
= ( ! [X5: B] :
( ( member @ B @ X5 @ ( set2 @ B @ Xs ) )
=> ( ( F3 @ X5 )
= ( G3 @ X5 ) ) ) ) ) ).
% map_eq_conv
thf(fact_6586_length__map,axiom,
! [A: $tType,B: $tType,F3: B > A,Xs: list @ B] :
( ( size_size @ ( list @ A ) @ ( map @ B @ A @ F3 @ Xs ) )
= ( size_size @ ( list @ B ) @ Xs ) ) ).
% length_map
thf(fact_6587_list_Omap__id0,axiom,
! [A: $tType] :
( ( map @ A @ A @ ( id @ A ) )
= ( id @ ( list @ A ) ) ) ).
% list.map_id0
thf(fact_6588_map__replicate,axiom,
! [A: $tType,B: $tType,F3: B > A,N: nat,X: B] :
( ( map @ B @ A @ F3 @ ( replicate @ B @ N @ X ) )
= ( replicate @ A @ N @ ( F3 @ X ) ) ) ).
% map_replicate
thf(fact_6589_list_Oset__map,axiom,
! [B: $tType,A: $tType,F3: A > B,V: list @ A] :
( ( set2 @ B @ ( map @ A @ B @ F3 @ V ) )
= ( image2 @ A @ B @ F3 @ ( set2 @ A @ V ) ) ) ).
% list.set_map
thf(fact_6590_map__snd__enumerate,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( map @ ( product_prod @ nat @ A ) @ A @ ( product_snd @ nat @ A ) @ ( enumerate @ A @ N @ Xs ) )
= Xs ) ).
% map_snd_enumerate
thf(fact_6591_inj__map__eq__map,axiom,
! [B: $tType,A: $tType,F3: A > B,Xs: list @ A,Ys: list @ A] :
( ( inj_on @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) )
=> ( ( ( map @ A @ B @ F3 @ Xs )
= ( map @ A @ B @ F3 @ Ys ) )
= ( Xs = Ys ) ) ) ).
% inj_map_eq_map
thf(fact_6592_map__fun__upd,axiom,
! [B: $tType,A: $tType,Y: A,Xs: list @ A,F3: A > B,V: B] :
( ~ ( member @ A @ Y @ ( set2 @ A @ Xs ) )
=> ( ( map @ A @ B @ ( fun_upd @ A @ B @ F3 @ Y @ V ) @ Xs )
= ( map @ A @ B @ F3 @ Xs ) ) ) ).
% map_fun_upd
thf(fact_6593_List_Omap_Ocomp,axiom,
! [C: $tType,B: $tType,A: $tType,F3: B > C,G3: A > B] :
( ( comp @ ( list @ B ) @ ( list @ C ) @ ( list @ A ) @ ( map @ B @ C @ F3 ) @ ( map @ A @ B @ G3 ) )
= ( map @ A @ C @ ( comp @ B @ C @ A @ F3 @ G3 ) ) ) ).
% List.map.comp
thf(fact_6594_map__comp__map,axiom,
! [B: $tType,C: $tType,A: $tType,F3: C > B,G3: A > C] :
( ( comp @ ( list @ C ) @ ( list @ B ) @ ( list @ A ) @ ( map @ C @ B @ F3 ) @ ( map @ A @ C @ G3 ) )
= ( map @ A @ B @ ( comp @ C @ B @ A @ F3 @ G3 ) ) ) ).
% map_comp_map
thf(fact_6595_size__list__map,axiom,
! [A: $tType,B: $tType,F3: A > nat,G3: B > A,Xs: list @ B] :
( ( size_list @ A @ F3 @ ( map @ B @ A @ G3 @ Xs ) )
= ( size_list @ B @ ( comp @ A @ nat @ B @ F3 @ G3 ) @ Xs ) ) ).
% size_list_map
thf(fact_6596_nth__map,axiom,
! [B: $tType,A: $tType,N: nat,Xs: list @ A,F3: A > B] :
( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ B @ ( map @ A @ B @ F3 @ Xs ) @ N )
= ( F3 @ ( nth @ A @ Xs @ N ) ) ) ) ).
% nth_map
thf(fact_6597_map__fst__zip,axiom,
! [B: $tType,A: $tType,Xs: list @ A,Ys: list @ B] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) )
= Xs ) ) ).
% map_fst_zip
thf(fact_6598_map__snd__zip,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( map @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) )
= Ys ) ) ).
% map_snd_zip
thf(fact_6599_inj__mapI,axiom,
! [B: $tType,A: $tType,F3: A > B] :
( ( inj_on @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) )
=> ( inj_on @ ( list @ A ) @ ( list @ B ) @ ( map @ A @ B @ F3 ) @ ( top_top @ ( set @ ( list @ A ) ) ) ) ) ).
% inj_mapI
thf(fact_6600_inj__map,axiom,
! [B: $tType,A: $tType,F3: A > B] :
( ( inj_on @ ( list @ A ) @ ( list @ B ) @ ( map @ A @ B @ F3 ) @ ( top_top @ ( set @ ( list @ A ) ) ) )
= ( inj_on @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) ) ) ).
% inj_map
thf(fact_6601_map__of__eq__Some__iff,axiom,
! [B: $tType,A: $tType,Xys: list @ ( product_prod @ A @ B ),X: A,Y: B] :
( ( distinct @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xys ) )
=> ( ( ( map_of @ A @ B @ Xys @ X )
= ( some @ B @ Y ) )
= ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ ( set2 @ ( product_prod @ A @ B ) @ Xys ) ) ) ) ).
% map_of_eq_Some_iff
thf(fact_6602_Some__eq__map__of__iff,axiom,
! [B: $tType,A: $tType,Xys: list @ ( product_prod @ A @ B ),Y: B,X: A] :
( ( distinct @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xys ) )
=> ( ( ( some @ B @ Y )
= ( map_of @ A @ B @ Xys @ X ) )
= ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ ( set2 @ ( product_prod @ A @ B ) @ Xys ) ) ) ) ).
% Some_eq_map_of_iff
thf(fact_6603_foldl__map,axiom,
! [A: $tType,B: $tType,C: $tType,G3: A > B > A,A2: A,F3: C > B,Xs: list @ C] :
( ( foldl @ A @ B @ G3 @ A2 @ ( map @ C @ B @ F3 @ Xs ) )
= ( foldl @ A @ C
@ ^ [A6: A,X5: C] : ( G3 @ A6 @ ( F3 @ X5 ) )
@ A2
@ Xs ) ) ).
% foldl_map
thf(fact_6604_list_Osize__gen__o__map,axiom,
! [B: $tType,A: $tType,F3: B > nat,G3: A > B] :
( ( comp @ ( list @ B ) @ nat @ ( list @ A ) @ ( size_list @ B @ F3 ) @ ( map @ A @ B @ G3 ) )
= ( size_list @ A @ ( comp @ B @ nat @ A @ F3 @ G3 ) ) ) ).
% list.size_gen_o_map
thf(fact_6605_nths__map,axiom,
! [A: $tType,B: $tType,F3: B > A,Xs: list @ B,I6: set @ nat] :
( ( nths @ A @ ( map @ B @ A @ F3 @ Xs ) @ I6 )
= ( map @ B @ A @ F3 @ ( nths @ B @ Xs @ I6 ) ) ) ).
% nths_map
thf(fact_6606_list_Osimps_I8_J,axiom,
! [A: $tType,B: $tType,F3: A > B] :
( ( map @ A @ B @ F3 @ ( nil @ A ) )
= ( nil @ B ) ) ).
% list.simps(8)
thf(fact_6607_map__concat,axiom,
! [A: $tType,B: $tType,F3: B > A,Xs: list @ ( list @ B )] :
( ( map @ B @ A @ F3 @ ( concat @ B @ Xs ) )
= ( concat @ A @ ( map @ ( list @ B ) @ ( list @ A ) @ ( map @ B @ A @ F3 ) @ Xs ) ) ) ).
% map_concat
thf(fact_6608_list_Omap__ident,axiom,
! [A: $tType,T2: list @ A] :
( ( map @ A @ A
@ ^ [X5: A] : X5
@ T2 )
= T2 ) ).
% list.map_ident
thf(fact_6609_rotate1__map,axiom,
! [A: $tType,B: $tType,F3: B > A,Xs: list @ B] :
( ( rotate1 @ A @ ( map @ B @ A @ F3 @ Xs ) )
= ( map @ B @ A @ F3 @ ( rotate1 @ B @ Xs ) ) ) ).
% rotate1_map
thf(fact_6610_map__eq__imp__length__eq,axiom,
! [A: $tType,B: $tType,C: $tType,F3: B > A,Xs: list @ B,G3: C > A,Ys: list @ C] :
( ( ( map @ B @ A @ F3 @ Xs )
= ( map @ C @ A @ G3 @ Ys ) )
=> ( ( size_size @ ( list @ B ) @ Xs )
= ( size_size @ ( list @ C ) @ Ys ) ) ) ).
% map_eq_imp_length_eq
thf(fact_6611_remdups__map__remdups,axiom,
! [A: $tType,B: $tType,F3: B > A,Xs: list @ B] :
( ( remdups @ A @ ( map @ B @ A @ F3 @ ( remdups @ B @ Xs ) ) )
= ( remdups @ A @ ( map @ B @ A @ F3 @ Xs ) ) ) ).
% remdups_map_remdups
thf(fact_6612_list_Omap__id,axiom,
! [A: $tType,T2: list @ A] :
( ( map @ A @ A @ ( id @ A ) @ T2 )
= T2 ) ).
% list.map_id
thf(fact_6613_List_Omap_Oidentity,axiom,
! [A: $tType] :
( ( map @ A @ A
@ ^ [X5: A] : X5 )
= ( id @ ( list @ A ) ) ) ).
% List.map.identity
thf(fact_6614_pair__list__eqI,axiom,
! [B: $tType,A: $tType,Xs: list @ ( product_prod @ A @ B ),Ys: list @ ( product_prod @ A @ B )] :
( ( ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xs )
= ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Ys ) )
=> ( ( ( map @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ Xs )
= ( map @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ Ys ) )
=> ( Xs = Ys ) ) ) ).
% pair_list_eqI
thf(fact_6615_enumerate__Suc__eq,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( enumerate @ A @ ( suc @ N ) @ Xs )
= ( map @ ( product_prod @ nat @ A ) @ ( product_prod @ nat @ A ) @ ( product_apfst @ nat @ nat @ A @ suc ) @ ( enumerate @ A @ N @ Xs ) ) ) ).
% enumerate_Suc_eq
thf(fact_6616_map__injective,axiom,
! [A: $tType,B: $tType,F3: B > A,Xs: list @ B,Ys: list @ B] :
( ( ( map @ B @ A @ F3 @ Xs )
= ( map @ B @ A @ F3 @ Ys ) )
=> ( ( inj_on @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) )
=> ( Xs = Ys ) ) ) ).
% map_injective
thf(fact_6617_take__map,axiom,
! [A: $tType,B: $tType,N: nat,F3: B > A,Xs: list @ B] :
( ( take @ A @ N @ ( map @ B @ A @ F3 @ Xs ) )
= ( map @ B @ A @ F3 @ ( take @ B @ N @ Xs ) ) ) ).
% take_map
thf(fact_6618_list_Omap__cong,axiom,
! [B: $tType,A: $tType,X: list @ A,Ya: list @ A,F3: A > B,G3: A > B] :
( ( X = Ya )
=> ( ! [Z3: A] :
( ( member @ A @ Z3 @ ( set2 @ A @ Ya ) )
=> ( ( F3 @ Z3 )
= ( G3 @ Z3 ) ) )
=> ( ( map @ A @ B @ F3 @ X )
= ( map @ A @ B @ G3 @ Ya ) ) ) ) ).
% list.map_cong
thf(fact_6619_list_Omap__cong0,axiom,
! [B: $tType,A: $tType,X: list @ A,F3: A > B,G3: A > B] :
( ! [Z3: A] :
( ( member @ A @ Z3 @ ( set2 @ A @ X ) )
=> ( ( F3 @ Z3 )
= ( G3 @ Z3 ) ) )
=> ( ( map @ A @ B @ F3 @ X )
= ( map @ A @ B @ G3 @ X ) ) ) ).
% list.map_cong0
thf(fact_6620_list_Oinj__map__strong,axiom,
! [B: $tType,A: $tType,X: list @ A,Xa2: list @ A,F3: A > B,Fa: A > B] :
( ! [Z3: A,Za2: A] :
( ( member @ A @ Z3 @ ( set2 @ A @ X ) )
=> ( ( member @ A @ Za2 @ ( set2 @ A @ Xa2 ) )
=> ( ( ( F3 @ Z3 )
= ( Fa @ Za2 ) )
=> ( Z3 = Za2 ) ) ) )
=> ( ( ( map @ A @ B @ F3 @ X )
= ( map @ A @ B @ Fa @ Xa2 ) )
=> ( X = Xa2 ) ) ) ).
% list.inj_map_strong
thf(fact_6621_map__ext,axiom,
! [B: $tType,A: $tType,Xs: list @ A,F3: A > B,G3: A > B] :
( ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
=> ( ( F3 @ X4 )
= ( G3 @ X4 ) ) )
=> ( ( map @ A @ B @ F3 @ Xs )
= ( map @ A @ B @ G3 @ Xs ) ) ) ).
% map_ext
thf(fact_6622_map__idI,axiom,
! [A: $tType,Xs: list @ A,F3: A > A] :
( ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
=> ( ( F3 @ X4 )
= X4 ) )
=> ( ( map @ A @ A @ F3 @ Xs )
= Xs ) ) ).
% map_idI
thf(fact_6623_map__cong,axiom,
! [B: $tType,A: $tType,Xs: list @ A,Ys: list @ A,F3: A > B,G3: A > B] :
( ( Xs = Ys )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Ys ) )
=> ( ( F3 @ X4 )
= ( G3 @ X4 ) ) )
=> ( ( map @ A @ B @ F3 @ Xs )
= ( map @ A @ B @ G3 @ Ys ) ) ) ) ).
% map_cong
thf(fact_6624_ex__map__conv,axiom,
! [A: $tType,B: $tType,Ys: list @ B,F3: A > B] :
( ( ? [Xs3: list @ A] :
( Ys
= ( map @ A @ B @ F3 @ Xs3 ) ) )
= ( ! [X5: B] :
( ( member @ B @ X5 @ ( set2 @ B @ Ys ) )
=> ? [Y6: A] :
( X5
= ( F3 @ Y6 ) ) ) ) ) ).
% ex_map_conv
thf(fact_6625_map__update,axiom,
! [A: $tType,B: $tType,F3: B > A,Xs: list @ B,K2: nat,Y: B] :
( ( map @ B @ A @ F3 @ ( list_update @ B @ Xs @ K2 @ Y ) )
= ( list_update @ A @ ( map @ B @ A @ F3 @ Xs ) @ K2 @ ( F3 @ Y ) ) ) ).
% map_update
thf(fact_6626_map__replicate__const,axiom,
! [B: $tType,A: $tType,K2: A,Lst: list @ B] :
( ( map @ B @ A
@ ^ [X5: B] : K2
@ Lst )
= ( replicate @ A @ ( size_size @ ( list @ B ) @ Lst ) @ K2 ) ) ).
% map_replicate_const
thf(fact_6627_image__set,axiom,
! [A: $tType,B: $tType,F3: B > A,Xs: list @ B] :
( ( image2 @ B @ A @ F3 @ ( set2 @ B @ Xs ) )
= ( set2 @ A @ ( map @ B @ A @ F3 @ Xs ) ) ) ).
% image_set
thf(fact_6628_map2__map__map,axiom,
! [B: $tType,A: $tType,C: $tType,D: $tType,H2: B > C > A,F3: D > B,Xs: list @ D,G3: D > C] :
( ( map @ ( product_prod @ B @ C ) @ A @ ( product_case_prod @ B @ C @ A @ H2 ) @ ( zip @ B @ C @ ( map @ D @ B @ F3 @ Xs ) @ ( map @ D @ C @ G3 @ Xs ) ) )
= ( map @ D @ A
@ ^ [X5: D] : ( H2 @ ( F3 @ X5 ) @ ( G3 @ X5 ) )
@ Xs ) ) ).
% map2_map_map
thf(fact_6629_map__zip__map2,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,F3: ( product_prod @ B @ C ) > A,Xs: list @ B,G3: D > C,Ys: list @ D] :
( ( map @ ( product_prod @ B @ C ) @ A @ F3 @ ( zip @ B @ C @ Xs @ ( map @ D @ C @ G3 @ Ys ) ) )
= ( map @ ( product_prod @ B @ D ) @ A
@ ( product_case_prod @ B @ D @ A
@ ^ [X5: B,Y6: D] : ( F3 @ ( product_Pair @ B @ C @ X5 @ ( G3 @ Y6 ) ) ) )
@ ( zip @ B @ D @ Xs @ Ys ) ) ) ).
% map_zip_map2
thf(fact_6630_zip__map__map,axiom,
! [B: $tType,A: $tType,C: $tType,D: $tType,F3: C > A,Xs: list @ C,G3: D > B,Ys: list @ D] :
( ( zip @ A @ B @ ( map @ C @ A @ F3 @ Xs ) @ ( map @ D @ B @ G3 @ Ys ) )
= ( map @ ( product_prod @ C @ D ) @ ( product_prod @ A @ B )
@ ( product_case_prod @ C @ D @ ( product_prod @ A @ B )
@ ^ [X5: C,Y6: D] : ( product_Pair @ A @ B @ ( F3 @ X5 ) @ ( G3 @ Y6 ) ) )
@ ( zip @ C @ D @ Xs @ Ys ) ) ) ).
% zip_map_map
thf(fact_6631_map__zip__map,axiom,
! [B: $tType,A: $tType,D: $tType,C: $tType,F3: ( product_prod @ B @ C ) > A,G3: D > B,Xs: list @ D,Ys: list @ C] :
( ( map @ ( product_prod @ B @ C ) @ A @ F3 @ ( zip @ B @ C @ ( map @ D @ B @ G3 @ Xs ) @ Ys ) )
= ( map @ ( product_prod @ D @ C ) @ A
@ ( product_case_prod @ D @ C @ A
@ ^ [X5: D,Y6: C] : ( F3 @ ( product_Pair @ B @ C @ ( G3 @ X5 ) @ Y6 ) ) )
@ ( zip @ D @ C @ Xs @ Ys ) ) ) ).
% map_zip_map
thf(fact_6632_zip__map2,axiom,
! [B: $tType,A: $tType,C: $tType,Xs: list @ A,F3: C > B,Ys: list @ C] :
( ( zip @ A @ B @ Xs @ ( map @ C @ B @ F3 @ Ys ) )
= ( map @ ( product_prod @ A @ C ) @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ C @ ( product_prod @ A @ B )
@ ^ [X5: A,Y6: C] : ( product_Pair @ A @ B @ X5 @ ( F3 @ Y6 ) ) )
@ ( zip @ A @ C @ Xs @ Ys ) ) ) ).
% zip_map2
thf(fact_6633_zip__map1,axiom,
! [A: $tType,C: $tType,B: $tType,F3: C > A,Xs: list @ C,Ys: list @ B] :
( ( zip @ A @ B @ ( map @ C @ A @ F3 @ Xs ) @ Ys )
= ( map @ ( product_prod @ C @ B ) @ ( product_prod @ A @ B )
@ ( product_case_prod @ C @ B @ ( product_prod @ A @ B )
@ ^ [X5: C] : ( product_Pair @ A @ B @ ( F3 @ X5 ) ) )
@ ( zip @ C @ B @ Xs @ Ys ) ) ) ).
% zip_map1
thf(fact_6634_map__inj__on,axiom,
! [A: $tType,B: $tType,F3: B > A,Xs: list @ B,Ys: list @ B] :
( ( ( map @ B @ A @ F3 @ Xs )
= ( map @ B @ A @ F3 @ Ys ) )
=> ( ( inj_on @ B @ A @ F3 @ ( sup_sup @ ( set @ B ) @ ( set2 @ B @ Xs ) @ ( set2 @ B @ Ys ) ) )
=> ( Xs = Ys ) ) ) ).
% map_inj_on
thf(fact_6635_inj__on__map__eq__map,axiom,
! [B: $tType,A: $tType,F3: A > B,Xs: list @ A,Ys: list @ A] :
( ( inj_on @ A @ B @ F3 @ ( sup_sup @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ Ys ) ) )
=> ( ( ( map @ A @ B @ F3 @ Xs )
= ( map @ A @ B @ F3 @ Ys ) )
= ( Xs = Ys ) ) ) ).
% inj_on_map_eq_map
thf(fact_6636_distinct__map,axiom,
! [A: $tType,B: $tType,F3: B > A,Xs: list @ B] :
( ( distinct @ A @ ( map @ B @ A @ F3 @ Xs ) )
= ( ( distinct @ B @ Xs )
& ( inj_on @ B @ A @ F3 @ ( set2 @ B @ Xs ) ) ) ) ).
% distinct_map
thf(fact_6637_remdups__adj__map__injective,axiom,
! [B: $tType,A: $tType,F3: A > B,Xs: list @ A] :
( ( inj_on @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) )
=> ( ( remdups_adj @ B @ ( map @ A @ B @ F3 @ Xs ) )
= ( map @ A @ B @ F3 @ ( remdups_adj @ A @ Xs ) ) ) ) ).
% remdups_adj_map_injective
thf(fact_6638_zip__map__fst__snd,axiom,
! [B: $tType,A: $tType,Zs: list @ ( product_prod @ A @ B )] :
( ( zip @ A @ B @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Zs ) @ ( map @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ Zs ) )
= Zs ) ).
% zip_map_fst_snd
thf(fact_6639_map__of__eqI,axiom,
! [B: $tType,A: $tType,Xs: list @ ( product_prod @ A @ B ),Ys: list @ ( product_prod @ A @ B )] :
( ( ( set2 @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xs ) )
= ( set2 @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Ys ) ) )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xs ) ) )
=> ( ( map_of @ A @ B @ Xs @ X4 )
= ( map_of @ A @ B @ Ys @ X4 ) ) )
=> ( ( map_of @ A @ B @ Xs )
= ( map_of @ A @ B @ Ys ) ) ) ) ).
% map_of_eqI
thf(fact_6640_map__removeAll__inj,axiom,
! [B: $tType,A: $tType,F3: A > B,X: A,Xs: list @ A] :
( ( inj_on @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) )
=> ( ( map @ A @ B @ F3 @ ( removeAll @ A @ X @ Xs ) )
= ( removeAll @ B @ ( F3 @ X ) @ ( map @ A @ B @ F3 @ Xs ) ) ) ) ).
% map_removeAll_inj
thf(fact_6641_inj__mapD,axiom,
! [B: $tType,A: $tType,F3: A > B] :
( ( inj_on @ ( list @ A ) @ ( list @ B ) @ ( map @ A @ B @ F3 ) @ ( top_top @ ( set @ ( list @ A ) ) ) )
=> ( inj_on @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) ) ) ).
% inj_mapD
thf(fact_6642_zip__left__commute,axiom,
! [B: $tType,A: $tType,C: $tType,Xs: list @ A,Ys: list @ B,Zs: list @ C] :
( ( zip @ A @ ( product_prod @ B @ C ) @ Xs @ ( zip @ B @ C @ Ys @ Zs ) )
= ( map @ ( product_prod @ B @ ( product_prod @ A @ C ) ) @ ( product_prod @ A @ ( product_prod @ B @ C ) )
@ ( product_case_prod @ B @ ( product_prod @ A @ C ) @ ( product_prod @ A @ ( product_prod @ B @ C ) )
@ ^ [Y6: B] :
( product_case_prod @ A @ C @ ( product_prod @ A @ ( product_prod @ B @ C ) )
@ ^ [X5: A,Z5: C] : ( product_Pair @ A @ ( product_prod @ B @ C ) @ X5 @ ( product_Pair @ B @ C @ Y6 @ Z5 ) ) ) )
@ ( zip @ B @ ( product_prod @ A @ C ) @ Ys @ ( zip @ A @ C @ Xs @ Zs ) ) ) ) ).
% zip_left_commute
thf(fact_6643_zip__assoc,axiom,
! [B: $tType,A: $tType,C: $tType,Xs: list @ A,Ys: list @ B,Zs: list @ C] :
( ( zip @ A @ ( product_prod @ B @ C ) @ Xs @ ( zip @ B @ C @ Ys @ Zs ) )
= ( map @ ( product_prod @ ( product_prod @ A @ B ) @ C ) @ ( product_prod @ A @ ( product_prod @ B @ C ) )
@ ( product_case_prod @ ( product_prod @ A @ B ) @ C @ ( product_prod @ A @ ( product_prod @ B @ C ) )
@ ( product_case_prod @ A @ B @ ( C > ( product_prod @ A @ ( product_prod @ B @ C ) ) )
@ ^ [X5: A,Y6: B,Z5: C] : ( product_Pair @ A @ ( product_prod @ B @ C ) @ X5 @ ( product_Pair @ B @ C @ Y6 @ Z5 ) ) ) )
@ ( zip @ ( product_prod @ A @ B ) @ C @ ( zip @ A @ B @ Xs @ Ys ) @ Zs ) ) ) ).
% zip_assoc
thf(fact_6644_zip__commute,axiom,
! [B: $tType,A: $tType] :
( ( zip @ A @ B )
= ( ^ [Xs3: list @ A,Ys3: list @ B] :
( map @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B )
@ ( product_case_prod @ B @ A @ ( product_prod @ A @ B )
@ ^ [X5: B,Y6: A] : ( product_Pair @ A @ B @ Y6 @ X5 ) )
@ ( zip @ B @ A @ Ys3 @ Xs3 ) ) ) ) ).
% zip_commute
thf(fact_6645_zip__eq__conv,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B,Zs: list @ ( product_prod @ A @ B )] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( ( zip @ A @ B @ Xs @ Ys )
= Zs )
= ( ( ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Zs )
= Xs )
& ( ( map @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ Zs )
= Ys ) ) ) ) ).
% zip_eq_conv
thf(fact_6646_distinct__insort__key,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F3: B > A,X: B,Xs: list @ B] :
( ( distinct @ A @ ( map @ B @ A @ F3 @ ( linorder_insort_key @ B @ A @ F3 @ X @ Xs ) ) )
= ( ~ ( member @ A @ ( F3 @ X ) @ ( image2 @ B @ A @ F3 @ ( set2 @ B @ Xs ) ) )
& ( distinct @ A @ ( map @ B @ A @ F3 @ Xs ) ) ) ) ) ).
% distinct_insort_key
thf(fact_6647_map__removeAll__inj__on,axiom,
! [B: $tType,A: $tType,F3: A > B,X: A,Xs: list @ A] :
( ( inj_on @ A @ B @ F3 @ ( insert @ A @ X @ ( set2 @ A @ Xs ) ) )
=> ( ( map @ A @ B @ F3 @ ( removeAll @ A @ X @ Xs ) )
= ( removeAll @ B @ ( F3 @ X ) @ ( map @ A @ B @ F3 @ Xs ) ) ) ) ).
% map_removeAll_inj_on
thf(fact_6648_eq__key__imp__eq__value,axiom,
! [A: $tType,B: $tType,Xs: list @ ( product_prod @ A @ B ),K2: A,V1: B,V22: B] :
( ( distinct @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xs ) )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K2 @ V1 ) @ ( set2 @ ( product_prod @ A @ B ) @ Xs ) )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ K2 @ V22 ) @ ( set2 @ ( product_prod @ A @ B ) @ Xs ) )
=> ( V1 = V22 ) ) ) ) ).
% eq_key_imp_eq_value
thf(fact_6649_map__of__inject__set,axiom,
! [B: $tType,A: $tType,Xs: list @ ( product_prod @ A @ B ),Ys: list @ ( product_prod @ A @ B )] :
( ( distinct @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xs ) )
=> ( ( distinct @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Ys ) )
=> ( ( ( map_of @ A @ B @ Xs )
= ( map_of @ A @ B @ Ys ) )
= ( ( set2 @ ( product_prod @ A @ B ) @ Xs )
= ( set2 @ ( product_prod @ A @ B ) @ Ys ) ) ) ) ) ).
% map_of_inject_set
thf(fact_6650_inj__on__mapI,axiom,
! [B: $tType,A: $tType,F3: A > B,A5: set @ ( list @ A )] :
( ( inj_on @ A @ B @ F3 @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ ( list @ A ) @ ( set @ A ) @ ( set2 @ A ) @ A5 ) ) )
=> ( inj_on @ ( list @ A ) @ ( list @ B ) @ ( map @ A @ B @ F3 ) @ A5 ) ) ).
% inj_on_mapI
thf(fact_6651_map__of__zip__map,axiom,
! [B: $tType,A: $tType,Xs: list @ A,F3: A > B] :
( ( map_of @ A @ B @ ( zip @ A @ B @ Xs @ ( map @ A @ B @ F3 @ Xs ) ) )
= ( ^ [X5: A] : ( if @ ( option @ B ) @ ( member @ A @ X5 @ ( set2 @ A @ Xs ) ) @ ( some @ B @ ( F3 @ X5 ) ) @ ( none @ B ) ) ) ) ).
% map_of_zip_map
thf(fact_6652_map__fst__zip__take,axiom,
! [B: $tType,A: $tType,Xs: list @ A,Ys: list @ B] :
( ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) )
= ( take @ A @ ( ord_min @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ B ) @ Ys ) ) @ Xs ) ) ).
% map_fst_zip_take
thf(fact_6653_map__snd__zip__take,axiom,
! [B: $tType,A: $tType,Xs: list @ B,Ys: list @ A] :
( ( map @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A ) @ ( zip @ B @ A @ Xs @ Ys ) )
= ( take @ A @ ( ord_min @ nat @ ( size_size @ ( list @ B ) @ Xs ) @ ( size_size @ ( list @ A ) @ Ys ) ) @ Ys ) ) ).
% map_snd_zip_take
thf(fact_6654_graph__map__of__if__distinct__dom,axiom,
! [B: $tType,A: $tType,Al: list @ ( product_prod @ A @ B )] :
( ( distinct @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Al ) )
=> ( ( graph @ A @ B @ ( map_of @ A @ B @ Al ) )
= ( set2 @ ( product_prod @ A @ B ) @ Al ) ) ) ).
% graph_map_of_if_distinct_dom
thf(fact_6655_map__of__mapk__SomeI,axiom,
! [A: $tType,B: $tType,C: $tType,F3: A > B,T2: list @ ( product_prod @ A @ C ),K2: A,X: C] :
( ( inj_on @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) )
=> ( ( ( map_of @ A @ C @ T2 @ K2 )
= ( some @ C @ X ) )
=> ( ( map_of @ B @ C
@ ( map @ ( product_prod @ A @ C ) @ ( product_prod @ B @ C )
@ ( product_case_prod @ A @ C @ ( product_prod @ B @ C )
@ ^ [K3: A] : ( product_Pair @ B @ C @ ( F3 @ K3 ) ) )
@ T2 )
@ ( F3 @ K2 ) )
= ( some @ C @ X ) ) ) ) ).
% map_of_mapk_SomeI
thf(fact_6656_set__map__of__compr,axiom,
! [B: $tType,A: $tType,Xs: list @ ( product_prod @ A @ B )] :
( ( distinct @ A @ ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Xs ) )
=> ( ( set2 @ ( product_prod @ A @ B ) @ Xs )
= ( collect @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [K3: A,V5: B] :
( ( map_of @ A @ B @ Xs @ K3 )
= ( some @ B @ V5 ) ) ) ) ) ) ).
% set_map_of_compr
thf(fact_6657_set__relcomp,axiom,
! [B: $tType,C: $tType,A: $tType,Xys: list @ ( product_prod @ A @ C ),Yzs: list @ ( product_prod @ C @ B )] :
( ( relcomp @ A @ C @ B @ ( set2 @ ( product_prod @ A @ C ) @ Xys ) @ ( set2 @ ( product_prod @ C @ B ) @ Yzs ) )
= ( set2 @ ( product_prod @ A @ B )
@ ( concat @ ( product_prod @ A @ B )
@ ( map @ ( product_prod @ A @ C ) @ ( list @ ( product_prod @ A @ B ) )
@ ^ [Xy: product_prod @ A @ C] :
( concat @ ( product_prod @ A @ B )
@ ( map @ ( product_prod @ C @ B ) @ ( list @ ( product_prod @ A @ B ) )
@ ^ [Yz: product_prod @ C @ B] :
( if @ ( list @ ( product_prod @ A @ B ) )
@ ( ( product_snd @ A @ C @ Xy )
= ( product_fst @ C @ B @ Yz ) )
@ ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ ( product_fst @ A @ C @ Xy ) @ ( product_snd @ C @ B @ Yz ) ) @ ( nil @ ( product_prod @ A @ B ) ) )
@ ( nil @ ( product_prod @ A @ B ) ) )
@ Yzs ) )
@ Xys ) ) ) ) ).
% set_relcomp
thf(fact_6658_dual__max,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( max @ A
@ ^ [X5: A,Y6: A] : ( ord_less_eq @ A @ Y6 @ X5 ) )
= ( ord_min @ A ) ) ) ).
% dual_max
thf(fact_6659_list_Oinject,axiom,
! [A: $tType,X21: A,X222: list @ A,Y21: A,Y22: list @ A] :
( ( ( cons @ A @ X21 @ X222 )
= ( cons @ A @ Y21 @ Y22 ) )
= ( ( X21 = Y21 )
& ( X222 = Y22 ) ) ) ).
% list.inject
thf(fact_6660_list_Osimps_I15_J,axiom,
! [A: $tType,X21: A,X222: list @ A] :
( ( set2 @ A @ ( cons @ A @ X21 @ X222 ) )
= ( insert @ A @ X21 @ ( set2 @ A @ X222 ) ) ) ).
% list.simps(15)
thf(fact_6661_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_6662_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_6663_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_6664_nths__singleton,axiom,
! [A: $tType,A5: set @ nat,X: A] :
( ( ( member @ nat @ ( zero_zero @ nat ) @ A5 )
=> ( ( nths @ A @ ( cons @ A @ X @ ( nil @ A ) ) @ A5 )
= ( cons @ A @ X @ ( nil @ A ) ) ) )
& ( ~ ( member @ nat @ ( zero_zero @ nat ) @ A5 )
=> ( ( nths @ A @ ( cons @ A @ X @ ( nil @ A ) ) @ A5 )
= ( nil @ A ) ) ) ) ).
% nths_singleton
thf(fact_6665_Cons__listrel1__Cons,axiom,
! [A: $tType,X: A,Xs: list @ A,Y: A,Ys: list @ A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Ys ) ) @ ( listrel1 @ A @ R2 ) )
= ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R2 )
& ( Xs = Ys ) )
| ( ( X = Y )
& ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( listrel1 @ A @ R2 ) ) ) ) ) ).
% Cons_listrel1_Cons
thf(fact_6666_horner__sum__simps_I2_J,axiom,
! [A: $tType,B: $tType] :
( ( comm_semiring_0 @ A )
=> ! [F3: B > A,A2: A,X: B,Xs: list @ B] :
( ( groups4207007520872428315er_sum @ B @ A @ F3 @ A2 @ ( cons @ B @ X @ Xs ) )
= ( plus_plus @ A @ ( F3 @ X ) @ ( times_times @ A @ A2 @ ( groups4207007520872428315er_sum @ B @ A @ F3 @ A2 @ Xs ) ) ) ) ) ).
% horner_sum_simps(2)
thf(fact_6667_lexord__cons__cons,axiom,
! [A: $tType,A2: A,X: list @ A,B2: A,Y: list @ A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ A2 @ X ) @ ( cons @ A @ B2 @ Y ) ) @ ( lexord @ A @ R2 ) )
= ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ R2 )
| ( ( A2 = B2 )
& ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Y ) @ ( lexord @ A @ R2 ) ) ) ) ) ).
% lexord_cons_cons
thf(fact_6668_lexord__Nil__left,axiom,
! [A: $tType,Y: list @ A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Y ) @ ( lexord @ A @ R2 ) )
= ( ? [A6: A,X5: list @ A] :
( Y
= ( cons @ A @ A6 @ X5 ) ) ) ) ).
% lexord_Nil_left
thf(fact_6669_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_6670_map__upds__Cons,axiom,
! [A: $tType,B: $tType,M2: A > ( option @ B ),A2: A,As: list @ A,B2: B,Bs: list @ B] :
( ( map_upds @ A @ B @ M2 @ ( cons @ A @ A2 @ As ) @ ( cons @ B @ B2 @ Bs ) )
= ( map_upds @ A @ B @ ( fun_upd @ A @ ( option @ B ) @ M2 @ A2 @ ( some @ B @ B2 ) ) @ As @ Bs ) ) ).
% map_upds_Cons
thf(fact_6671_zip__Cons__Cons,axiom,
! [A: $tType,B: $tType,X: A,Xs: list @ A,Y: B,Ys: list @ B] :
( ( zip @ A @ B @ ( cons @ A @ X @ Xs ) @ ( cons @ B @ Y @ Ys ) )
= ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ ( zip @ A @ B @ Xs @ Ys ) ) ) ).
% zip_Cons_Cons
thf(fact_6672_nth__Cons__numeral,axiom,
! [A: $tType,X: A,Xs: list @ A,V: num] :
( ( nth @ A @ ( cons @ A @ X @ Xs ) @ ( numeral_numeral @ nat @ V ) )
= ( nth @ A @ Xs @ ( minus_minus @ nat @ ( numeral_numeral @ nat @ V ) @ ( one_one @ nat ) ) ) ) ).
% nth_Cons_numeral
thf(fact_6673_take__Cons__numeral,axiom,
! [A: $tType,V: num,X: A,Xs: list @ A] :
( ( take @ A @ ( numeral_numeral @ nat @ V ) @ ( cons @ A @ X @ Xs ) )
= ( cons @ A @ X @ ( take @ A @ ( minus_minus @ nat @ ( numeral_numeral @ nat @ V ) @ ( one_one @ nat ) ) @ Xs ) ) ) ).
% take_Cons_numeral
thf(fact_6674_Cons__in__lex,axiom,
! [A: $tType,X: A,Xs: list @ A,Y: A,Ys: list @ A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Ys ) ) @ ( lex @ A @ R2 ) )
= ( ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R2 )
& ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ A ) @ Ys ) ) )
| ( ( X = Y )
& ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( lex @ A @ R2 ) ) ) ) ) ).
% Cons_in_lex
thf(fact_6675_concat__map__singleton,axiom,
! [A: $tType,B: $tType,F3: B > A,Xs: list @ B] :
( ( concat @ A
@ ( map @ B @ ( list @ A )
@ ^ [X5: B] : ( cons @ A @ ( F3 @ X5 ) @ ( nil @ A ) )
@ Xs ) )
= ( map @ B @ A @ F3 @ Xs ) ) ).
% concat_map_singleton
thf(fact_6676_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_6677_product__lists_Osimps_I2_J,axiom,
! [A: $tType,Xs: list @ A,Xss: list @ ( list @ A )] :
( ( product_lists @ A @ ( cons @ ( list @ A ) @ Xs @ Xss ) )
= ( concat @ ( list @ A )
@ ( map @ A @ ( list @ ( list @ A ) )
@ ^ [X5: A] : ( map @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X5 ) @ ( product_lists @ A @ Xss ) )
@ Xs ) ) ) ).
% product_lists.simps(2)
thf(fact_6678_list_Osimps_I9_J,axiom,
! [B: $tType,A: $tType,F3: A > B,X21: A,X222: list @ A] :
( ( map @ A @ B @ F3 @ ( cons @ A @ X21 @ X222 ) )
= ( cons @ B @ ( F3 @ X21 ) @ ( map @ A @ B @ F3 @ X222 ) ) ) ).
% list.simps(9)
thf(fact_6679_Cons__eq__map__D,axiom,
! [A: $tType,B: $tType,X: A,Xs: list @ A,F3: B > A,Ys: list @ B] :
( ( ( cons @ A @ X @ Xs )
= ( map @ B @ A @ F3 @ Ys ) )
=> ? [Z3: B,Zs2: list @ B] :
( ( Ys
= ( cons @ B @ Z3 @ Zs2 ) )
& ( X
= ( F3 @ Z3 ) )
& ( Xs
= ( map @ B @ A @ F3 @ Zs2 ) ) ) ) ).
% Cons_eq_map_D
thf(fact_6680_map__eq__Cons__D,axiom,
! [B: $tType,A: $tType,F3: B > A,Xs: list @ B,Y: A,Ys: list @ A] :
( ( ( map @ B @ A @ F3 @ Xs )
= ( cons @ A @ Y @ Ys ) )
=> ? [Z3: B,Zs2: list @ B] :
( ( Xs
= ( cons @ B @ Z3 @ Zs2 ) )
& ( ( F3 @ Z3 )
= Y )
& ( ( map @ B @ A @ F3 @ Zs2 )
= Ys ) ) ) ).
% map_eq_Cons_D
thf(fact_6681_Cons__eq__map__conv,axiom,
! [A: $tType,B: $tType,X: A,Xs: list @ A,F3: B > A,Ys: list @ B] :
( ( ( cons @ A @ X @ Xs )
= ( map @ B @ A @ F3 @ Ys ) )
= ( ? [Z5: B,Zs3: list @ B] :
( ( Ys
= ( cons @ B @ Z5 @ Zs3 ) )
& ( X
= ( F3 @ Z5 ) )
& ( Xs
= ( map @ B @ A @ F3 @ Zs3 ) ) ) ) ) ).
% Cons_eq_map_conv
thf(fact_6682_map__eq__Cons__conv,axiom,
! [A: $tType,B: $tType,F3: B > A,Xs: list @ B,Y: A,Ys: list @ A] :
( ( ( map @ B @ A @ F3 @ Xs )
= ( cons @ A @ Y @ Ys ) )
= ( ? [Z5: B,Zs3: list @ B] :
( ( Xs
= ( cons @ B @ Z5 @ Zs3 ) )
& ( ( F3 @ Z5 )
= Y )
& ( ( map @ B @ A @ F3 @ Zs3 )
= Ys ) ) ) ) ).
% map_eq_Cons_conv
thf(fact_6683_n__lists_Osimps_I2_J,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( n_lists @ A @ ( suc @ N ) @ Xs )
= ( concat @ ( list @ A )
@ ( map @ ( list @ A ) @ ( list @ ( list @ A ) )
@ ^ [Ys3: list @ A] :
( map @ A @ ( list @ A )
@ ^ [Y6: A] : ( cons @ A @ Y6 @ Ys3 )
@ Xs )
@ ( n_lists @ A @ N @ Xs ) ) ) ) ).
% n_lists.simps(2)
thf(fact_6684_List_Obind__def,axiom,
! [B: $tType,A: $tType] :
( ( bind @ A @ B )
= ( ^ [Xs3: list @ A,F4: A > ( list @ B )] : ( concat @ B @ ( map @ A @ ( list @ B ) @ F4 @ Xs3 ) ) ) ) ).
% List.bind_def
thf(fact_6685_product__concat__map,axiom,
! [B: $tType,A: $tType] :
( ( product @ A @ B )
= ( ^ [Xs3: list @ A,Ys3: list @ B] :
( concat @ ( product_prod @ A @ B )
@ ( map @ A @ ( list @ ( product_prod @ A @ B ) )
@ ^ [X5: A] : ( map @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X5 ) @ Ys3 )
@ Xs3 ) ) ) ) ).
% product_concat_map
thf(fact_6686_zip__same__conv__map,axiom,
! [A: $tType,Xs: list @ A] :
( ( zip @ A @ A @ Xs @ Xs )
= ( map @ A @ ( product_prod @ A @ A )
@ ^ [X5: A] : ( product_Pair @ A @ A @ X5 @ X5 )
@ Xs ) ) ).
% zip_same_conv_map
thf(fact_6687_zip__eq__ConsE,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B,Xy2: product_prod @ A @ B,Xys: list @ ( product_prod @ A @ B )] :
( ( ( zip @ A @ B @ Xs @ Ys )
= ( cons @ ( product_prod @ A @ B ) @ Xy2 @ Xys ) )
=> ~ ! [X4: A,Xs4: list @ A] :
( ( Xs
= ( cons @ A @ X4 @ Xs4 ) )
=> ! [Y3: B,Ys5: list @ B] :
( ( Ys
= ( cons @ B @ Y3 @ Ys5 ) )
=> ( ( Xy2
= ( product_Pair @ A @ B @ X4 @ Y3 ) )
=> ( Xys
!= ( zip @ A @ B @ Xs4 @ Ys5 ) ) ) ) ) ) ).
% zip_eq_ConsE
thf(fact_6688_inj__split__Cons,axiom,
! [A: $tType,X7: set @ ( product_prod @ ( list @ A ) @ A )] :
( inj_on @ ( product_prod @ ( list @ A ) @ A ) @ ( list @ A )
@ ( product_case_prod @ ( list @ A ) @ A @ ( list @ A )
@ ^ [Xs3: list @ A,N2: A] : ( cons @ A @ N2 @ Xs3 ) )
@ X7 ) ).
% inj_split_Cons
thf(fact_6689_set__subset__Cons,axiom,
! [A: $tType,Xs: list @ A,X: A] : ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ ( cons @ A @ X @ Xs ) ) ) ).
% set_subset_Cons
thf(fact_6690_remdups_Osimps_I2_J,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ( remdups @ A @ ( cons @ A @ X @ Xs ) )
= ( remdups @ A @ Xs ) ) )
& ( ~ ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ( remdups @ A @ ( cons @ A @ X @ Xs ) )
= ( cons @ A @ X @ ( remdups @ A @ Xs ) ) ) ) ) ).
% remdups.simps(2)
thf(fact_6691_distinct_Osimps_I2_J,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( distinct @ A @ ( cons @ A @ X @ Xs ) )
= ( ~ ( member @ A @ X @ ( set2 @ A @ Xs ) )
& ( distinct @ A @ Xs ) ) ) ).
% distinct.simps(2)
thf(fact_6692_list__update_Osimps_I2_J,axiom,
! [A: $tType,X: A,Xs: list @ A,I2: nat,V: A] :
( ( list_update @ A @ ( cons @ A @ X @ Xs ) @ I2 @ V )
= ( case_nat @ ( list @ A ) @ ( cons @ A @ V @ Xs )
@ ^ [J3: nat] : ( cons @ A @ X @ ( list_update @ A @ Xs @ J3 @ V ) )
@ I2 ) ) ).
% list_update.simps(2)
thf(fact_6693_set__ConsD,axiom,
! [A: $tType,Y: A,X: A,Xs: list @ A] :
( ( member @ A @ Y @ ( set2 @ A @ ( cons @ A @ X @ Xs ) ) )
=> ( ( Y = X )
| ( member @ A @ Y @ ( set2 @ A @ Xs ) ) ) ) ).
% set_ConsD
thf(fact_6694_list_Oset__cases,axiom,
! [A: $tType,E3: A,A2: list @ A] :
( ( member @ A @ E3 @ ( set2 @ A @ A2 ) )
=> ( ! [Z23: list @ A] :
( A2
!= ( cons @ A @ E3 @ Z23 ) )
=> ~ ! [Z12: A,Z23: list @ A] :
( ( A2
= ( cons @ A @ Z12 @ Z23 ) )
=> ~ ( member @ A @ E3 @ ( set2 @ A @ Z23 ) ) ) ) ) ).
% list.set_cases
thf(fact_6695_list_Oset__intros_I1_J,axiom,
! [A: $tType,X21: A,X222: list @ A] : ( member @ A @ X21 @ ( set2 @ A @ ( cons @ A @ X21 @ X222 ) ) ) ).
% list.set_intros(1)
thf(fact_6696_list_Oset__intros_I2_J,axiom,
! [A: $tType,Y: A,X222: list @ A,X21: A] :
( ( member @ A @ Y @ ( set2 @ A @ X222 ) )
=> ( member @ A @ Y @ ( set2 @ A @ ( cons @ A @ X21 @ X222 ) ) ) ) ).
% list.set_intros(2)
thf(fact_6697_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_6698_list__update__code_I2_J,axiom,
! [A: $tType,X: A,Xs: list @ A,Y: A] :
( ( list_update @ A @ ( cons @ A @ X @ Xs ) @ ( zero_zero @ nat ) @ Y )
= ( cons @ A @ Y @ Xs ) ) ).
% list_update_code(2)
thf(fact_6699_list__update__code_I3_J,axiom,
! [A: $tType,X: A,Xs: list @ A,I2: nat,Y: A] :
( ( list_update @ A @ ( cons @ A @ X @ Xs ) @ ( suc @ I2 ) @ Y )
= ( cons @ A @ X @ ( list_update @ A @ Xs @ I2 @ Y ) ) ) ).
% list_update_code(3)
thf(fact_6700_arg__min__list_Osimps_I1_J,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ! [F3: A > B,X: A] :
( ( arg_min_list @ A @ B @ F3 @ ( cons @ A @ X @ ( nil @ A ) ) )
= X ) ) ).
% arg_min_list.simps(1)
thf(fact_6701_listrel1I2,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,R2: set @ ( product_prod @ A @ A ),X: A] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( listrel1 @ A @ R2 ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ X @ Ys ) ) @ ( listrel1 @ A @ R2 ) ) ) ).
% listrel1I2
thf(fact_6702_map__tailrec__rev_Ocases,axiom,
! [A: $tType,B: $tType,X: product_prod @ ( A > B ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) )] :
( ! [F2: A > B,Bs2: list @ B] :
( X
!= ( product_Pair @ ( A > B ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ F2 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( nil @ A ) @ Bs2 ) ) )
=> ~ ! [F2: A > B,A4: A,As2: list @ A,Bs2: list @ B] :
( X
!= ( product_Pair @ ( A > B ) @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ F2 @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ A4 @ As2 ) @ Bs2 ) ) ) ) ).
% map_tailrec_rev.cases
thf(fact_6703_successively_Ocases,axiom,
! [A: $tType,X: product_prod @ ( A > A > $o ) @ ( list @ A )] :
( ! [P8: A > A > $o] :
( X
!= ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ P8 @ ( nil @ A ) ) )
=> ( ! [P8: A > A > $o,X4: A] :
( X
!= ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ P8 @ ( cons @ A @ X4 @ ( nil @ A ) ) ) )
=> ~ ! [P8: A > A > $o,X4: A,Y3: A,Xs2: list @ A] :
( X
!= ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ P8 @ ( cons @ A @ X4 @ ( cons @ A @ Y3 @ Xs2 ) ) ) ) ) ) ).
% successively.cases
thf(fact_6704_arg__min__list_Ocases,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ! [X: product_prod @ ( A > B ) @ ( list @ A )] :
( ! [F2: A > B,X4: A] :
( X
!= ( product_Pair @ ( A > B ) @ ( list @ A ) @ F2 @ ( cons @ A @ X4 @ ( nil @ A ) ) ) )
=> ( ! [F2: A > B,X4: A,Y3: A,Zs2: list @ A] :
( X
!= ( product_Pair @ ( A > B ) @ ( list @ A ) @ F2 @ ( cons @ A @ X4 @ ( cons @ A @ Y3 @ Zs2 ) ) ) )
=> ~ ! [A4: A > B] :
( X
!= ( product_Pair @ ( A > B ) @ ( list @ A ) @ A4 @ ( nil @ A ) ) ) ) ) ) ).
% arg_min_list.cases
thf(fact_6705_sorted__wrt_Ocases,axiom,
! [A: $tType,X: product_prod @ ( A > A > $o ) @ ( list @ A )] :
( ! [P8: A > A > $o] :
( X
!= ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ P8 @ ( nil @ A ) ) )
=> ~ ! [P8: A > A > $o,X4: A,Ys4: list @ A] :
( X
!= ( product_Pair @ ( A > A > $o ) @ ( list @ A ) @ P8 @ ( cons @ A @ X4 @ Ys4 ) ) ) ) ).
% sorted_wrt.cases
thf(fact_6706_shuffles_Ocases,axiom,
! [A: $tType,X: product_prod @ ( list @ A ) @ ( list @ A )] :
( ! [Ys4: list @ A] :
( X
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Ys4 ) )
=> ( ! [Xs2: list @ A] :
( X
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ ( nil @ A ) ) )
=> ~ ! [X4: A,Xs2: list @ A,Y3: A,Ys4: list @ A] :
( X
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X4 @ Xs2 ) @ ( cons @ A @ Y3 @ Ys4 ) ) ) ) ) ).
% shuffles.cases
thf(fact_6707_splice_Ocases,axiom,
! [A: $tType,X: product_prod @ ( list @ A ) @ ( list @ A )] :
( ! [Ys4: list @ A] :
( X
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Ys4 ) )
=> ~ ! [X4: A,Xs2: list @ A,Ys4: list @ A] :
( X
!= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X4 @ Xs2 ) @ Ys4 ) ) ) ).
% splice.cases
thf(fact_6708_remdups__adj_Oelims,axiom,
! [A: $tType,X: list @ A,Y: list @ A] :
( ( ( remdups_adj @ A @ X )
= Y )
=> ( ( ( X
= ( nil @ A ) )
=> ( Y
!= ( nil @ A ) ) )
=> ( ! [X4: A] :
( ( X
= ( cons @ A @ X4 @ ( nil @ A ) ) )
=> ( Y
!= ( cons @ A @ X4 @ ( nil @ A ) ) ) )
=> ~ ! [X4: A,Y3: A,Xs2: list @ A] :
( ( X
= ( cons @ A @ X4 @ ( cons @ A @ Y3 @ Xs2 ) ) )
=> ~ ( ( ( X4 = Y3 )
=> ( Y
= ( remdups_adj @ A @ ( cons @ A @ X4 @ Xs2 ) ) ) )
& ( ( X4 != Y3 )
=> ( Y
= ( cons @ A @ X4 @ ( remdups_adj @ A @ ( cons @ A @ Y3 @ Xs2 ) ) ) ) ) ) ) ) ) ) ).
% remdups_adj.elims
thf(fact_6709_remdups__adj_Osimps_I2_J,axiom,
! [A: $tType,X: A] :
( ( remdups_adj @ A @ ( cons @ A @ X @ ( nil @ A ) ) )
= ( cons @ A @ X @ ( nil @ A ) ) ) ).
% remdups_adj.simps(2)
thf(fact_6710_shufflesE,axiom,
! [A: $tType,Zs: list @ A,Xs: list @ A,Ys: list @ A] :
( ( member @ ( list @ A ) @ Zs @ ( shuffles @ A @ Xs @ Ys ) )
=> ( ( ( Zs = Xs )
=> ( Ys
!= ( nil @ A ) ) )
=> ( ( ( Zs = Ys )
=> ( Xs
!= ( nil @ A ) ) )
=> ( ! [X4: A,Xs4: list @ A] :
( ( Xs
= ( cons @ A @ X4 @ Xs4 ) )
=> ! [Z3: A,Zs4: list @ A] :
( ( Zs
= ( cons @ A @ Z3 @ Zs4 ) )
=> ( ( X4 = Z3 )
=> ~ ( member @ ( list @ A ) @ Zs4 @ ( shuffles @ A @ Xs4 @ Ys ) ) ) ) )
=> ~ ! [Y3: A,Ys5: list @ A] :
( ( Ys
= ( cons @ A @ Y3 @ Ys5 ) )
=> ! [Z3: A,Zs4: list @ A] :
( ( Zs
= ( cons @ A @ Z3 @ Zs4 ) )
=> ( ( Y3 = Z3 )
=> ~ ( member @ ( list @ A ) @ Zs4 @ ( shuffles @ A @ Xs @ Ys5 ) ) ) ) ) ) ) ) ) ).
% shufflesE
thf(fact_6711_insort__key_Osimps_I1_J,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F3: B > A,X: B] :
( ( linorder_insort_key @ B @ A @ F3 @ X @ ( nil @ B ) )
= ( cons @ B @ X @ ( nil @ B ) ) ) ) ).
% insort_key.simps(1)
thf(fact_6712_distinct__singleton,axiom,
! [A: $tType,X: A] : ( distinct @ A @ ( cons @ A @ X @ ( nil @ A ) ) ) ).
% distinct_singleton
thf(fact_6713_inj__on__Cons1,axiom,
! [A: $tType,X: A,A5: set @ ( list @ A )] : ( inj_on @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X ) @ A5 ) ).
% inj_on_Cons1
thf(fact_6714_length__Suc__conv,axiom,
! [A: $tType,Xs: list @ A,N: nat] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( suc @ N ) )
= ( ? [Y6: A,Ys3: list @ A] :
( ( Xs
= ( cons @ A @ Y6 @ Ys3 ) )
& ( ( size_size @ ( list @ A ) @ Ys3 )
= N ) ) ) ) ).
% length_Suc_conv
thf(fact_6715_Suc__length__conv,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( ( suc @ N )
= ( size_size @ ( list @ A ) @ Xs ) )
= ( ? [Y6: A,Ys3: list @ A] :
( ( Xs
= ( cons @ A @ Y6 @ Ys3 ) )
& ( ( size_size @ ( list @ A ) @ Ys3 )
= N ) ) ) ) ).
% Suc_length_conv
thf(fact_6716_Cons__in__shuffles__leftI,axiom,
! [A: $tType,Zs: list @ A,Xs: list @ A,Ys: list @ A,Z: A] :
( ( member @ ( list @ A ) @ Zs @ ( shuffles @ A @ Xs @ Ys ) )
=> ( member @ ( list @ A ) @ ( cons @ A @ Z @ Zs ) @ ( shuffles @ A @ ( cons @ A @ Z @ Xs ) @ Ys ) ) ) ).
% Cons_in_shuffles_leftI
thf(fact_6717_Cons__in__shuffles__rightI,axiom,
! [A: $tType,Zs: list @ A,Xs: list @ A,Ys: list @ A,Z: A] :
( ( member @ ( list @ A ) @ Zs @ ( shuffles @ A @ Xs @ Ys ) )
=> ( member @ ( list @ A ) @ ( cons @ A @ Z @ Zs ) @ ( shuffles @ A @ Xs @ ( cons @ A @ Z @ Ys ) ) ) ) ).
% Cons_in_shuffles_rightI
thf(fact_6718_distinct__length__2__or__more,axiom,
! [A: $tType,A2: A,B2: A,Xs: list @ A] :
( ( distinct @ A @ ( cons @ A @ A2 @ ( cons @ A @ B2 @ Xs ) ) )
= ( ( A2 != B2 )
& ( distinct @ A @ ( cons @ A @ A2 @ Xs ) )
& ( distinct @ A @ ( cons @ A @ B2 @ Xs ) ) ) ) ).
% distinct_length_2_or_more
thf(fact_6719_not__Cons__self2,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( cons @ A @ X @ Xs )
!= Xs ) ).
% not_Cons_self2
thf(fact_6720_ord_Omax_Ocong,axiom,
! [A: $tType] :
( ( max @ A )
= ( max @ A ) ) ).
% ord.max.cong
thf(fact_6721_ord_Omax__def,axiom,
! [A: $tType] :
( ( max @ A )
= ( ^ [Less_eq2: A > A > $o,A6: A,B5: A] : ( if @ A @ ( Less_eq2 @ A6 @ B5 ) @ B5 @ A6 ) ) ) ).
% ord.max_def
thf(fact_6722_remove1_Osimps_I2_J,axiom,
! [A: $tType,X: A,Y: A,Xs: list @ A] :
( ( ( X = Y )
=> ( ( remove1 @ A @ X @ ( cons @ A @ Y @ Xs ) )
= Xs ) )
& ( ( X != Y )
=> ( ( remove1 @ A @ X @ ( cons @ A @ Y @ Xs ) )
= ( cons @ A @ Y @ ( remove1 @ A @ X @ Xs ) ) ) ) ) ).
% remove1.simps(2)
thf(fact_6723_removeAll_Osimps_I2_J,axiom,
! [A: $tType,X: A,Y: A,Xs: list @ A] :
( ( ( X = Y )
=> ( ( removeAll @ A @ X @ ( cons @ A @ Y @ Xs ) )
= ( removeAll @ A @ X @ Xs ) ) )
& ( ( X != Y )
=> ( ( removeAll @ A @ X @ ( cons @ A @ Y @ Xs ) )
= ( cons @ A @ Y @ ( removeAll @ A @ X @ Xs ) ) ) ) ) ).
% removeAll.simps(2)
thf(fact_6724_list_Odistinct_I1_J,axiom,
! [A: $tType,X21: A,X222: list @ A] :
( ( nil @ A )
!= ( cons @ A @ X21 @ X222 ) ) ).
% list.distinct(1)
thf(fact_6725_list_OdiscI,axiom,
! [A: $tType,List: list @ A,X21: A,X222: list @ A] :
( ( List
= ( cons @ A @ X21 @ X222 ) )
=> ( List
!= ( nil @ A ) ) ) ).
% list.discI
thf(fact_6726_list_Oexhaust,axiom,
! [A: $tType,Y: list @ A] :
( ( Y
!= ( nil @ A ) )
=> ~ ! [X212: A,X223: list @ A] :
( Y
!= ( cons @ A @ X212 @ X223 ) ) ) ).
% list.exhaust
thf(fact_6727_min__list_Ocases,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X: list @ A] :
( ! [X4: A,Xs2: list @ A] :
( X
!= ( cons @ A @ X4 @ Xs2 ) )
=> ( X
= ( nil @ A ) ) ) ) ).
% min_list.cases
thf(fact_6728_transpose_Ocases,axiom,
! [A: $tType,X: list @ ( list @ A )] :
( ( X
!= ( nil @ ( list @ A ) ) )
=> ( ! [Xss2: list @ ( list @ A )] :
( X
!= ( cons @ ( list @ A ) @ ( nil @ A ) @ Xss2 ) )
=> ~ ! [X4: A,Xs2: list @ A,Xss2: list @ ( list @ A )] :
( X
!= ( cons @ ( list @ A ) @ ( cons @ A @ X4 @ Xs2 ) @ Xss2 ) ) ) ) ).
% transpose.cases
thf(fact_6729_remdups__adj_Ocases,axiom,
! [A: $tType,X: list @ A] :
( ( X
!= ( nil @ A ) )
=> ( ! [X4: A] :
( X
!= ( cons @ A @ X4 @ ( nil @ A ) ) )
=> ~ ! [X4: A,Y3: A,Xs2: list @ A] :
( X
!= ( cons @ A @ X4 @ ( cons @ A @ Y3 @ Xs2 ) ) ) ) ) ).
% remdups_adj.cases
thf(fact_6730_neq__Nil__conv,axiom,
! [A: $tType,Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
= ( ? [Y6: A,Ys3: list @ A] :
( Xs
= ( cons @ A @ Y6 @ Ys3 ) ) ) ) ).
% neq_Nil_conv
thf(fact_6731_list__induct2_H,axiom,
! [A: $tType,B: $tType,P2: ( list @ A ) > ( list @ B ) > $o,Xs: list @ A,Ys: list @ B] :
( ( P2 @ ( nil @ A ) @ ( nil @ B ) )
=> ( ! [X4: A,Xs2: list @ A] : ( P2 @ ( cons @ A @ X4 @ Xs2 ) @ ( nil @ B ) )
=> ( ! [Y3: B,Ys4: list @ B] : ( P2 @ ( nil @ A ) @ ( cons @ B @ Y3 @ Ys4 ) )
=> ( ! [X4: A,Xs2: list @ A,Y3: B,Ys4: list @ B] :
( ( P2 @ Xs2 @ Ys4 )
=> ( P2 @ ( cons @ A @ X4 @ Xs2 ) @ ( cons @ B @ Y3 @ Ys4 ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ) ).
% list_induct2'
thf(fact_6732_list__nonempty__induct,axiom,
! [A: $tType,Xs: list @ A,P2: ( list @ A ) > $o] :
( ( Xs
!= ( nil @ A ) )
=> ( ! [X4: A] : ( P2 @ ( cons @ A @ X4 @ ( nil @ A ) ) )
=> ( ! [X4: A,Xs2: list @ A] :
( ( Xs2
!= ( nil @ A ) )
=> ( ( P2 @ Xs2 )
=> ( P2 @ ( cons @ A @ X4 @ Xs2 ) ) ) )
=> ( P2 @ Xs ) ) ) ) ).
% list_nonempty_induct
thf(fact_6733_remdups__adj_Osimps_I3_J,axiom,
! [A: $tType,X: A,Y: A,Xs: list @ A] :
( ( ( X = Y )
=> ( ( remdups_adj @ A @ ( cons @ A @ X @ ( cons @ A @ Y @ Xs ) ) )
= ( remdups_adj @ A @ ( cons @ A @ X @ Xs ) ) ) )
& ( ( X != Y )
=> ( ( remdups_adj @ A @ ( cons @ A @ X @ ( cons @ A @ Y @ Xs ) ) )
= ( cons @ A @ X @ ( remdups_adj @ A @ ( cons @ A @ Y @ Xs ) ) ) ) ) ) ).
% remdups_adj.simps(3)
thf(fact_6734_list__induct2,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B,P2: ( list @ A ) > ( list @ B ) > $o] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( P2 @ ( nil @ A ) @ ( nil @ B ) )
=> ( ! [X4: A,Xs2: list @ A,Y3: B,Ys4: list @ B] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ B ) @ Ys4 ) )
=> ( ( P2 @ Xs2 @ Ys4 )
=> ( P2 @ ( cons @ A @ X4 @ Xs2 ) @ ( cons @ B @ Y3 @ Ys4 ) ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ).
% list_induct2
thf(fact_6735_list__induct3,axiom,
! [B: $tType,A: $tType,C: $tType,Xs: list @ A,Ys: list @ B,Zs: list @ C,P2: ( list @ A ) > ( list @ B ) > ( list @ C ) > $o] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( ( size_size @ ( list @ B ) @ Ys )
= ( size_size @ ( list @ C ) @ Zs ) )
=> ( ( P2 @ ( nil @ A ) @ ( nil @ B ) @ ( nil @ C ) )
=> ( ! [X4: A,Xs2: list @ A,Y3: B,Ys4: list @ B,Z3: C,Zs2: list @ C] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ B ) @ Ys4 ) )
=> ( ( ( size_size @ ( list @ B ) @ Ys4 )
= ( size_size @ ( list @ C ) @ Zs2 ) )
=> ( ( P2 @ Xs2 @ Ys4 @ Zs2 )
=> ( P2 @ ( cons @ A @ X4 @ Xs2 ) @ ( cons @ B @ Y3 @ Ys4 ) @ ( cons @ C @ Z3 @ Zs2 ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs ) ) ) ) ) ).
% list_induct3
thf(fact_6736_list__induct4,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,Xs: list @ A,Ys: list @ B,Zs: list @ C,Ws2: list @ D,P2: ( list @ A ) > ( list @ B ) > ( list @ C ) > ( list @ D ) > $o] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( ( size_size @ ( list @ B ) @ Ys )
= ( size_size @ ( list @ C ) @ Zs ) )
=> ( ( ( size_size @ ( list @ C ) @ Zs )
= ( size_size @ ( list @ D ) @ Ws2 ) )
=> ( ( P2 @ ( nil @ A ) @ ( nil @ B ) @ ( nil @ C ) @ ( nil @ D ) )
=> ( ! [X4: A,Xs2: list @ A,Y3: B,Ys4: list @ B,Z3: C,Zs2: list @ C,W: D,Ws: list @ D] :
( ( ( size_size @ ( list @ A ) @ Xs2 )
= ( size_size @ ( list @ B ) @ Ys4 ) )
=> ( ( ( size_size @ ( list @ B ) @ Ys4 )
= ( size_size @ ( list @ C ) @ Zs2 ) )
=> ( ( ( size_size @ ( list @ C ) @ Zs2 )
= ( size_size @ ( list @ D ) @ Ws ) )
=> ( ( P2 @ Xs2 @ Ys4 @ Zs2 @ Ws )
=> ( P2 @ ( cons @ A @ X4 @ Xs2 ) @ ( cons @ B @ Y3 @ Ys4 ) @ ( cons @ C @ Z3 @ Zs2 ) @ ( cons @ D @ W @ Ws ) ) ) ) ) )
=> ( P2 @ Xs @ Ys @ Zs @ Ws2 ) ) ) ) ) ) ).
% list_induct4
thf(fact_6737_shuffles_Osimps_I3_J,axiom,
! [A: $tType,X: A,Xs: list @ A,Y: A,Ys: list @ A] :
( ( shuffles @ A @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Ys ) )
= ( sup_sup @ ( set @ ( list @ A ) ) @ ( image2 @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X ) @ ( shuffles @ A @ Xs @ ( cons @ A @ Y @ Ys ) ) ) @ ( image2 @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ Y ) @ ( shuffles @ A @ ( cons @ A @ X @ Xs ) @ Ys ) ) ) ) ).
% shuffles.simps(3)
thf(fact_6738_Cons__shuffles__subset2,axiom,
! [A: $tType,Y: A,Xs: list @ A,Ys: list @ A] : ( ord_less_eq @ ( set @ ( list @ A ) ) @ ( image2 @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ Y ) @ ( shuffles @ A @ Xs @ Ys ) ) @ ( shuffles @ A @ Xs @ ( cons @ A @ Y @ Ys ) ) ) ).
% Cons_shuffles_subset2
thf(fact_6739_Cons__shuffles__subset1,axiom,
! [A: $tType,X: A,Xs: list @ A,Ys: list @ A] : ( ord_less_eq @ ( set @ ( list @ A ) ) @ ( image2 @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X ) @ ( shuffles @ A @ Xs @ Ys ) ) @ ( shuffles @ A @ ( cons @ A @ X @ Xs ) @ Ys ) ) ).
% Cons_shuffles_subset1
thf(fact_6740_insort__key_Osimps_I2_J,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F3: B > A,X: B,Y: B,Ys: list @ B] :
( ( ( ord_less_eq @ A @ ( F3 @ X ) @ ( F3 @ Y ) )
=> ( ( linorder_insort_key @ B @ A @ F3 @ X @ ( cons @ B @ Y @ Ys ) )
= ( cons @ B @ X @ ( cons @ B @ Y @ Ys ) ) ) )
& ( ~ ( ord_less_eq @ A @ ( F3 @ X ) @ ( F3 @ Y ) )
=> ( ( linorder_insort_key @ B @ A @ F3 @ X @ ( cons @ B @ Y @ Ys ) )
= ( cons @ B @ Y @ ( linorder_insort_key @ B @ A @ F3 @ X @ Ys ) ) ) ) ) ) ).
% insort_key.simps(2)
thf(fact_6741_impossible__Cons,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,X: A] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ A ) @ Ys ) )
=> ( Xs
!= ( cons @ A @ X @ Ys ) ) ) ).
% impossible_Cons
thf(fact_6742_listrelp_Osimps,axiom,
! [B: $tType,A: $tType] :
( ( listrelp @ A @ B )
= ( ^ [R5: A > B > $o,A12: list @ A,A23: list @ B] :
( ( ( A12
= ( nil @ A ) )
& ( A23
= ( nil @ B ) ) )
| ? [X5: A,Y6: B,Xs3: list @ A,Ys3: list @ B] :
( ( A12
= ( cons @ A @ X5 @ Xs3 ) )
& ( A23
= ( cons @ B @ Y6 @ Ys3 ) )
& ( R5 @ X5 @ Y6 )
& ( listrelp @ A @ B @ R5 @ Xs3 @ Ys3 ) ) ) ) ) ).
% listrelp.simps
thf(fact_6743_listrelp_Ocases,axiom,
! [A: $tType,B: $tType,R2: A > B > $o,A1: list @ A,A22: list @ B] :
( ( listrelp @ A @ B @ R2 @ A1 @ A22 )
=> ( ( ( A1
= ( nil @ A ) )
=> ( A22
!= ( nil @ B ) ) )
=> ~ ! [X4: A,Y3: B,Xs2: list @ A] :
( ( A1
= ( cons @ A @ X4 @ Xs2 ) )
=> ! [Ys4: list @ B] :
( ( A22
= ( cons @ B @ Y3 @ Ys4 ) )
=> ( ( R2 @ X4 @ Y3 )
=> ~ ( listrelp @ A @ B @ R2 @ Xs2 @ Ys4 ) ) ) ) ) ) ).
% listrelp.cases
thf(fact_6744_listrelp_OCons,axiom,
! [A: $tType,B: $tType,R2: A > B > $o,X: A,Y: B,Xs: list @ A,Ys: list @ B] :
( ( R2 @ X @ Y )
=> ( ( listrelp @ A @ B @ R2 @ Xs @ Ys )
=> ( listrelp @ A @ B @ R2 @ ( cons @ A @ X @ Xs ) @ ( cons @ B @ Y @ Ys ) ) ) ) ).
% listrelp.Cons
thf(fact_6745_foldl__Cons,axiom,
! [B: $tType,A: $tType,F3: B > A > B,A2: B,X: A,Xs: list @ A] :
( ( foldl @ B @ A @ F3 @ A2 @ ( cons @ A @ X @ Xs ) )
= ( foldl @ B @ A @ F3 @ ( F3 @ A2 @ X ) @ Xs ) ) ).
% foldl_Cons
thf(fact_6746_take__Cons,axiom,
! [A: $tType,N: nat,X: A,Xs: list @ A] :
( ( take @ A @ N @ ( cons @ A @ X @ Xs ) )
= ( case_nat @ ( list @ A ) @ ( nil @ A )
@ ^ [M3: nat] : ( cons @ A @ X @ ( take @ A @ M3 @ Xs ) )
@ N ) ) ).
% take_Cons
thf(fact_6747_Cons__in__subseqsD,axiom,
! [A: $tType,Y: A,Ys: list @ A,Xs: list @ A] :
( ( member @ ( list @ A ) @ ( cons @ A @ Y @ Ys ) @ ( set2 @ ( list @ A ) @ ( subseqs @ A @ Xs ) ) )
=> ( member @ ( list @ A ) @ Ys @ ( set2 @ ( list @ A ) @ ( subseqs @ A @ Xs ) ) ) ) ).
% Cons_in_subseqsD
thf(fact_6748_nth__Cons,axiom,
! [A: $tType,X: A,Xs: list @ A,N: nat] :
( ( nth @ A @ ( cons @ A @ X @ Xs ) @ N )
= ( case_nat @ A @ X @ ( nth @ A @ Xs ) @ N ) ) ).
% nth_Cons
thf(fact_6749_ord_Olexordp_Omono,axiom,
! [A: $tType,Less: A > A > $o] :
( order_mono @ ( ( list @ A ) > ( list @ A ) > $o ) @ ( ( list @ A ) > ( list @ A ) > $o )
@ ^ [P6: ( list @ A ) > ( list @ A ) > $o,X15: list @ A,X23: list @ A] :
( ? [Y6: A,Ys3: list @ A] :
( ( X15
= ( nil @ A ) )
& ( X23
= ( cons @ A @ Y6 @ Ys3 ) ) )
| ? [X5: A,Y6: A,Xs3: list @ A,Ys3: list @ A] :
( ( X15
= ( cons @ A @ X5 @ Xs3 ) )
& ( X23
= ( cons @ A @ Y6 @ Ys3 ) )
& ( Less @ X5 @ Y6 ) )
| ? [X5: A,Y6: A,Xs3: list @ A,Ys3: list @ A] :
( ( X15
= ( cons @ A @ X5 @ Xs3 ) )
& ( X23
= ( cons @ A @ Y6 @ Ys3 ) )
& ~ ( Less @ X5 @ Y6 )
& ~ ( Less @ Y6 @ X5 )
& ( P6 @ Xs3 @ Ys3 ) ) ) ) ).
% ord.lexordp.mono
thf(fact_6750_arg__min__list_Osimps_I2_J,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ! [F3: A > B,X: A,Y: A,Zs: list @ A] :
( ( arg_min_list @ A @ B @ F3 @ ( cons @ A @ X @ ( cons @ A @ Y @ Zs ) ) )
= ( if @ A @ ( ord_less_eq @ B @ ( F3 @ X ) @ ( F3 @ ( arg_min_list @ A @ B @ F3 @ ( cons @ A @ Y @ Zs ) ) ) ) @ X @ ( arg_min_list @ A @ B @ F3 @ ( cons @ A @ Y @ Zs ) ) ) ) ) ).
% arg_min_list.simps(2)
thf(fact_6751_Suc__le__length__iff,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( ord_less_eq @ nat @ ( suc @ N ) @ ( size_size @ ( list @ A ) @ Xs ) )
= ( ? [X5: A,Ys3: list @ A] :
( ( Xs
= ( cons @ A @ X5 @ Ys3 ) )
& ( ord_less_eq @ nat @ N @ ( size_size @ ( list @ A ) @ Ys3 ) ) ) ) ) ).
% Suc_le_length_iff
thf(fact_6752_distinct__set__subseqs,axiom,
! [A: $tType,Xs: list @ A] :
( ( distinct @ A @ Xs )
=> ( distinct @ ( set @ A ) @ ( map @ ( list @ A ) @ ( set @ A ) @ ( set2 @ A ) @ ( subseqs @ A @ Xs ) ) ) ) ).
% distinct_set_subseqs
thf(fact_6753_insort__is__Cons,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ B,F3: B > A,A2: B] :
( ! [X4: B] :
( ( member @ B @ X4 @ ( set2 @ B @ Xs ) )
=> ( ord_less_eq @ A @ ( F3 @ A2 ) @ ( F3 @ X4 ) ) )
=> ( ( linorder_insort_key @ B @ A @ F3 @ A2 @ Xs )
= ( cons @ B @ A2 @ Xs ) ) ) ) ).
% insort_is_Cons
thf(fact_6754_map__of__Cons__code_I2_J,axiom,
! [C: $tType,B: $tType,L: B,K2: B,V: C,Ps: list @ ( product_prod @ B @ C )] :
( ( ( L = K2 )
=> ( ( map_of @ B @ C @ ( cons @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ L @ V ) @ Ps ) @ K2 )
= ( some @ C @ V ) ) )
& ( ( L != K2 )
=> ( ( map_of @ B @ C @ ( cons @ ( product_prod @ B @ C ) @ ( product_Pair @ B @ C @ L @ V ) @ Ps ) @ K2 )
= ( map_of @ B @ C @ Ps @ K2 ) ) ) ) ).
% map_of_Cons_code(2)
thf(fact_6755_listrel1I1,axiom,
! [A: $tType,X: A,Y: A,R2: set @ ( product_prod @ A @ A ),Xs: list @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R2 )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Xs ) ) @ ( listrel1 @ A @ R2 ) ) ) ).
% listrel1I1
thf(fact_6756_Cons__listrel1E1,axiom,
! [A: $tType,X: A,Xs: list @ A,Ys: list @ A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Xs ) @ Ys ) @ ( listrel1 @ A @ R2 ) )
=> ( ! [Y3: A] :
( ( Ys
= ( cons @ A @ Y3 @ Xs ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y3 ) @ R2 ) )
=> ~ ! [Zs2: list @ A] :
( ( Ys
= ( cons @ A @ X @ Zs2 ) )
=> ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Zs2 ) @ ( listrel1 @ A @ R2 ) ) ) ) ) ).
% Cons_listrel1E1
thf(fact_6757_Cons__listrel1E2,axiom,
! [A: $tType,Xs: list @ A,Y: A,Ys: list @ A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ ( cons @ A @ Y @ Ys ) ) @ ( listrel1 @ A @ R2 ) )
=> ( ! [X4: A] :
( ( Xs
= ( cons @ A @ X4 @ Ys ) )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y ) @ R2 ) )
=> ~ ! [Zs2: list @ A] :
( ( Xs
= ( cons @ A @ Y @ Zs2 ) )
=> ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Zs2 @ Ys ) @ ( listrel1 @ A @ R2 ) ) ) ) ) ).
% Cons_listrel1E2
thf(fact_6758_listrel_OCons,axiom,
! [B: $tType,A: $tType,X: A,Y: B,R2: set @ ( product_prod @ A @ B ),Xs: list @ A,Ys: list @ B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y ) @ R2 )
=> ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs @ Ys ) @ ( listrel @ A @ B @ R2 ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ X @ Xs ) @ ( cons @ B @ Y @ Ys ) ) @ ( listrel @ A @ B @ R2 ) ) ) ) ).
% listrel.Cons
thf(fact_6759_listrel__Cons1,axiom,
! [B: $tType,A: $tType,Y: A,Ys: list @ A,Xs: list @ B,R2: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ ( cons @ A @ Y @ Ys ) @ Xs ) @ ( listrel @ A @ B @ R2 ) )
=> ~ ! [Y3: B,Ys4: list @ B] :
( ( Xs
= ( cons @ B @ Y3 @ Ys4 ) )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Y @ Y3 ) @ R2 )
=> ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Ys @ Ys4 ) @ ( listrel @ A @ B @ R2 ) ) ) ) ) ).
% listrel_Cons1
thf(fact_6760_listrel__Cons2,axiom,
! [B: $tType,A: $tType,Xs: list @ A,Y: B,Ys: list @ B,R2: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs @ ( cons @ B @ Y @ Ys ) ) @ ( listrel @ A @ B @ R2 ) )
=> ~ ! [X4: A,Xs2: list @ A] :
( ( Xs
= ( cons @ A @ X4 @ Xs2 ) )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y ) @ R2 )
=> ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs2 @ Ys ) @ ( listrel @ A @ B @ R2 ) ) ) ) ) ).
% listrel_Cons2
thf(fact_6761_count__list_Osimps_I2_J,axiom,
! [A: $tType,X: A,Y: A,Xs: list @ A] :
( ( ( X = Y )
=> ( ( count_list @ A @ ( cons @ A @ X @ Xs ) @ Y )
= ( plus_plus @ nat @ ( count_list @ A @ Xs @ Y ) @ ( one_one @ nat ) ) ) )
& ( ( X != Y )
=> ( ( count_list @ A @ ( cons @ A @ X @ Xs ) @ Y )
= ( count_list @ A @ Xs @ Y ) ) ) ) ).
% count_list.simps(2)
thf(fact_6762_the__elem__set,axiom,
! [A: $tType,X: A] :
( ( the_elem @ A @ ( set2 @ A @ ( cons @ A @ X @ ( nil @ A ) ) ) )
= X ) ).
% the_elem_set
thf(fact_6763_rtrancl__listrel1__ConsI1,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,R2: set @ ( product_prod @ A @ A ),X: A] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( transitive_rtrancl @ ( list @ A ) @ ( listrel1 @ A @ R2 ) ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ X @ Ys ) ) @ ( transitive_rtrancl @ ( list @ A ) @ ( listrel1 @ A @ R2 ) ) ) ) ).
% rtrancl_listrel1_ConsI1
thf(fact_6764_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,X15: list @ A,X23: list @ A] :
( ? [Y6: A,Ys3: list @ A] :
( ( X15
= ( nil @ A ) )
& ( X23
= ( cons @ A @ Y6 @ Ys3 ) ) )
| ? [X5: A,Y6: A,Xs3: list @ A,Ys3: list @ A] :
( ( X15
= ( cons @ A @ X5 @ Xs3 ) )
& ( X23
= ( cons @ A @ Y6 @ Ys3 ) )
& ( ord_less @ A @ X5 @ Y6 ) )
| ? [X5: A,Y6: A,Xs3: list @ A,Ys3: list @ A] :
( ( X15
= ( cons @ A @ X5 @ Xs3 ) )
& ( X23
= ( cons @ A @ Y6 @ Ys3 ) )
& ~ ( ord_less @ A @ X5 @ Y6 )
& ~ ( ord_less @ A @ Y6 @ X5 )
& ( P6 @ Xs3 @ Ys3 ) ) ) ) ) ).
% lexordp.mono
thf(fact_6765_zip__replicate1,axiom,
! [A: $tType,B: $tType,N: nat,X: A,Ys: list @ B] :
( ( zip @ A @ B @ ( replicate @ A @ N @ X ) @ Ys )
= ( map @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X ) @ ( take @ B @ N @ Ys ) ) ) ).
% zip_replicate1
thf(fact_6766_list_Osize_I4_J,axiom,
! [A: $tType,X21: A,X222: list @ A] :
( ( size_size @ ( list @ A ) @ ( cons @ A @ X21 @ X222 ) )
= ( plus_plus @ nat @ ( size_size @ ( list @ A ) @ X222 ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% list.size(4)
thf(fact_6767_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_6768_zip__replicate2,axiom,
! [B: $tType,A: $tType,Xs: list @ A,N: nat,Y: B] :
( ( zip @ A @ B @ Xs @ ( replicate @ B @ N @ Y ) )
= ( map @ A @ ( product_prod @ A @ B )
@ ^ [X5: A] : ( product_Pair @ A @ B @ X5 @ Y )
@ ( take @ A @ N @ Xs ) ) ) ).
% zip_replicate2
thf(fact_6769_listrel_Osimps,axiom,
! [B: $tType,A: $tType,A1: list @ A,A22: list @ B,R2: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ A1 @ A22 ) @ ( listrel @ A @ B @ R2 ) )
= ( ( ( A1
= ( nil @ A ) )
& ( A22
= ( nil @ B ) ) )
| ? [X5: A,Y6: B,Xs3: list @ A,Ys3: list @ B] :
( ( A1
= ( cons @ A @ X5 @ Xs3 ) )
& ( A22
= ( cons @ B @ Y6 @ Ys3 ) )
& ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X5 @ Y6 ) @ R2 )
& ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs3 @ Ys3 ) @ ( listrel @ A @ B @ R2 ) ) ) ) ) ).
% listrel.simps
thf(fact_6770_listrel_Ocases,axiom,
! [B: $tType,A: $tType,A1: list @ A,A22: list @ B,R2: set @ ( product_prod @ A @ B )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ A1 @ A22 ) @ ( listrel @ A @ B @ R2 ) )
=> ( ( ( A1
= ( nil @ A ) )
=> ( A22
!= ( nil @ B ) ) )
=> ~ ! [X4: A,Y3: B,Xs2: list @ A] :
( ( A1
= ( cons @ A @ X4 @ Xs2 ) )
=> ! [Ys4: list @ B] :
( ( A22
= ( cons @ B @ Y3 @ Ys4 ) )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X4 @ Y3 ) @ R2 )
=> ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ B ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ B ) @ Xs2 @ Ys4 ) @ ( listrel @ A @ B @ R2 ) ) ) ) ) ) ) ).
% listrel.cases
thf(fact_6771_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_6772_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_6773_Id__on__set,axiom,
! [A: $tType,Xs: list @ A] :
( ( id_on @ A @ ( set2 @ A @ Xs ) )
= ( set2 @ ( product_prod @ A @ A )
@ ( map @ A @ ( product_prod @ A @ A )
@ ^ [X5: A] : ( product_Pair @ A @ A @ X5 @ X5 )
@ Xs ) ) ) ).
% Id_on_set
thf(fact_6774_list_Osize__gen_I2_J,axiom,
! [A: $tType,X: A > nat,X21: A,X222: list @ A] :
( ( size_list @ A @ X @ ( cons @ A @ X21 @ X222 ) )
= ( plus_plus @ nat @ ( plus_plus @ nat @ ( X @ X21 ) @ ( size_list @ A @ X @ X222 ) ) @ ( suc @ ( zero_zero @ nat ) ) ) ) ).
% list.size_gen(2)
thf(fact_6775_shuffles_Oelims,axiom,
! [A: $tType,X: list @ A,Xa2: list @ A,Y: set @ ( list @ A )] :
( ( ( shuffles @ A @ X @ Xa2 )
= Y )
=> ( ( ( X
= ( nil @ A ) )
=> ( Y
!= ( insert @ ( list @ A ) @ Xa2 @ ( bot_bot @ ( set @ ( list @ A ) ) ) ) ) )
=> ( ( ( Xa2
= ( nil @ A ) )
=> ( Y
!= ( insert @ ( list @ A ) @ X @ ( bot_bot @ ( set @ ( list @ A ) ) ) ) ) )
=> ~ ! [X4: A,Xs2: list @ A] :
( ( X
= ( cons @ A @ X4 @ Xs2 ) )
=> ! [Y3: A,Ys4: list @ A] :
( ( Xa2
= ( cons @ A @ Y3 @ Ys4 ) )
=> ( Y
!= ( sup_sup @ ( set @ ( list @ A ) ) @ ( image2 @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X4 ) @ ( shuffles @ A @ Xs2 @ ( cons @ A @ Y3 @ Ys4 ) ) ) @ ( image2 @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ Y3 ) @ ( shuffles @ A @ ( cons @ A @ X4 @ Xs2 ) @ Ys4 ) ) ) ) ) ) ) ) ) ).
% shuffles.elims
thf(fact_6776_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_6777_nth__non__equal__first__eq,axiom,
! [A: $tType,X: A,Y: A,Xs: list @ A,N: nat] :
( ( X != Y )
=> ( ( ( nth @ A @ ( cons @ A @ X @ Xs ) @ N )
= Y )
= ( ( ( nth @ A @ Xs @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) )
= Y )
& ( ord_less @ nat @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% nth_non_equal_first_eq
thf(fact_6778_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_6779_Cons__replicate__eq,axiom,
! [A: $tType,X: A,Xs: list @ A,N: nat,Y: A] :
( ( ( cons @ A @ X @ Xs )
= ( replicate @ A @ N @ Y ) )
= ( ( X = Y )
& ( ord_less @ nat @ ( zero_zero @ nat ) @ N )
& ( Xs
= ( replicate @ A @ ( minus_minus @ nat @ N @ ( one_one @ nat ) ) @ X ) ) ) ) ).
% Cons_replicate_eq
thf(fact_6780_map__of__map__restrict,axiom,
! [B: $tType,A: $tType,F3: A > B,Ks2: list @ A] :
( ( map_of @ A @ B
@ ( map @ A @ ( product_prod @ A @ B )
@ ^ [K3: A] : ( product_Pair @ A @ B @ K3 @ ( F3 @ K3 ) )
@ Ks2 ) )
= ( restrict_map @ A @ B @ ( comp @ B @ ( option @ B ) @ A @ ( some @ B ) @ F3 ) @ ( set2 @ A @ Ks2 ) ) ) ).
% map_of_map_restrict
thf(fact_6781_map__of__map__keys,axiom,
! [B: $tType,A: $tType,Xs: list @ A,M2: A > ( option @ B )] :
( ( ( set2 @ A @ Xs )
= ( dom @ A @ B @ M2 ) )
=> ( ( map_of @ A @ B
@ ( map @ A @ ( product_prod @ A @ B )
@ ^ [K3: A] : ( product_Pair @ A @ B @ K3 @ ( the2 @ B @ ( M2 @ K3 ) ) )
@ Xs ) )
= M2 ) ) ).
% map_of_map_keys
thf(fact_6782_Cons__lenlex__iff,axiom,
! [A: $tType,M2: A,Ms: list @ A,N: A,Ns: list @ A,R2: 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 @ R2 ) )
= ( ( 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 ) @ R2 ) )
| ( ( M2 = N )
& ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ms @ Ns ) @ ( lenlex @ A @ R2 ) ) ) ) ) ).
% Cons_lenlex_iff
thf(fact_6783_map__of_Osimps_I2_J,axiom,
! [B: $tType,A: $tType,P: product_prod @ A @ B,Ps: list @ ( product_prod @ A @ B )] :
( ( map_of @ A @ B @ ( cons @ ( product_prod @ A @ B ) @ P @ Ps ) )
= ( fun_upd @ A @ ( option @ B ) @ ( map_of @ A @ B @ Ps ) @ ( product_fst @ A @ B @ P ) @ ( some @ B @ ( product_snd @ A @ B @ P ) ) ) ) ).
% map_of.simps(2)
thf(fact_6784_rtrancl__listrel1__ConsI2,axiom,
! [A: $tType,X: A,Y: A,R2: set @ ( product_prod @ A @ A ),Xs: list @ A,Ys: list @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( transitive_rtrancl @ A @ R2 ) )
=> ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( transitive_rtrancl @ ( list @ A ) @ ( listrel1 @ A @ R2 ) ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Ys ) ) @ ( transitive_rtrancl @ ( list @ A ) @ ( listrel1 @ A @ R2 ) ) ) ) ) ).
% rtrancl_listrel1_ConsI2
thf(fact_6785_Pow__set_I2_J,axiom,
! [B: $tType,X: B,Xs: list @ B] :
( ( pow2 @ B @ ( set2 @ B @ ( cons @ B @ X @ Xs ) ) )
= ( sup_sup @ ( set @ ( set @ B ) ) @ ( pow2 @ B @ ( set2 @ B @ Xs ) ) @ ( image2 @ ( set @ B ) @ ( set @ B ) @ ( insert @ B @ X ) @ ( pow2 @ B @ ( set2 @ B @ Xs ) ) ) ) ) ).
% Pow_set(2)
thf(fact_6786_sorted__list__of__set__nonempty,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( linord4507533701916653071of_set @ A @ A5 )
= ( cons @ A @ ( lattic643756798350308766er_Min @ A @ A5 ) @ ( linord4507533701916653071of_set @ A @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ ( lattic643756798350308766er_Min @ A @ A5 ) @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ) ) ).
% sorted_list_of_set_nonempty
thf(fact_6787_product__code,axiom,
! [B: $tType,A: $tType,Xs: list @ A,Ys: list @ B] :
( ( product_product @ A @ B @ ( set2 @ A @ Xs ) @ ( set2 @ B @ Ys ) )
= ( set2 @ ( product_prod @ A @ B )
@ ( concat @ ( product_prod @ A @ B )
@ ( map @ A @ ( list @ ( product_prod @ A @ B ) )
@ ^ [X5: A] : ( map @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X5 ) @ Ys )
@ Xs ) ) ) ) ).
% product_code
thf(fact_6788_set__Cons__sing__Nil,axiom,
! [A: $tType,A5: set @ A] :
( ( set_Cons @ A @ A5 @ ( insert @ ( list @ A ) @ ( nil @ A ) @ ( bot_bot @ ( set @ ( list @ A ) ) ) ) )
= ( image2 @ A @ ( list @ A )
@ ^ [X5: A] : ( cons @ A @ X5 @ ( nil @ A ) )
@ A5 ) ) ).
% set_Cons_sing_Nil
thf(fact_6789_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_6790_listset_Osimps_I2_J,axiom,
! [A: $tType,A5: set @ A,As3: list @ ( set @ A )] :
( ( listset @ A @ ( cons @ ( set @ A ) @ A5 @ As3 ) )
= ( set_Cons @ A @ A5 @ ( listset @ A @ As3 ) ) ) ).
% listset.simps(2)
thf(fact_6791_subseqs_Osimps_I1_J,axiom,
! [A: $tType] :
( ( subseqs @ A @ ( nil @ A ) )
= ( cons @ ( list @ A ) @ ( nil @ A ) @ ( nil @ ( list @ A ) ) ) ) ).
% subseqs.simps(1)
thf(fact_6792_product__lists_Osimps_I1_J,axiom,
! [A: $tType] :
( ( product_lists @ A @ ( nil @ ( list @ A ) ) )
= ( cons @ ( list @ A ) @ ( nil @ A ) @ ( nil @ ( list @ A ) ) ) ) ).
% product_lists.simps(1)
thf(fact_6793_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_6794_sorted__list__of__set__greaterThanAtMost,axiom,
! [I2: nat,J: nat] :
( ( ord_less_eq @ nat @ ( suc @ I2 ) @ J )
=> ( ( linord4507533701916653071of_set @ nat @ ( set_or3652927894154168847AtMost @ nat @ I2 @ J ) )
= ( cons @ nat @ ( suc @ I2 ) @ ( linord4507533701916653071of_set @ nat @ ( set_or3652927894154168847AtMost @ nat @ ( suc @ I2 ) @ J ) ) ) ) ) ).
% sorted_list_of_set_greaterThanAtMost
thf(fact_6795_sorted__list__of__set__greaterThanLessThan,axiom,
! [I2: nat,J: nat] :
( ( ord_less @ nat @ ( suc @ I2 ) @ J )
=> ( ( linord4507533701916653071of_set @ nat @ ( set_or5935395276787703475ssThan @ nat @ I2 @ J ) )
= ( cons @ nat @ ( suc @ I2 ) @ ( linord4507533701916653071of_set @ nat @ ( set_or5935395276787703475ssThan @ nat @ ( suc @ I2 ) @ J ) ) ) ) ) ).
% sorted_list_of_set_greaterThanLessThan
thf(fact_6796_set__Cons__def,axiom,
! [A: $tType] :
( ( set_Cons @ A )
= ( ^ [A7: set @ A,XS2: set @ ( list @ A )] :
( collect @ ( list @ A )
@ ^ [Z5: list @ A] :
? [X5: A,Xs3: list @ A] :
( ( Z5
= ( cons @ A @ X5 @ Xs3 ) )
& ( member @ A @ X5 @ A7 )
& ( member @ ( list @ A ) @ Xs3 @ XS2 ) ) ) ) ) ).
% set_Cons_def
thf(fact_6797_concat__inth,axiom,
! [A: $tType,Xs: list @ A,X: A,Ys: list @ A] :
( ( nth @ A @ ( append @ A @ Xs @ ( append @ A @ ( cons @ A @ X @ ( nil @ A ) ) @ Ys ) ) @ ( size_size @ ( list @ A ) @ Xs ) )
= X ) ).
% concat_inth
thf(fact_6798_map__upds__append1,axiom,
! [B: $tType,A: $tType,Xs: list @ A,Ys: list @ B,M2: A > ( option @ B ),X: A] :
( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( map_upds @ A @ B @ M2 @ ( append @ A @ Xs @ ( cons @ A @ X @ ( nil @ A ) ) ) @ Ys )
= ( fun_upd @ A @ ( option @ B ) @ ( map_upds @ A @ B @ M2 @ Xs @ Ys ) @ X @ ( some @ B @ ( nth @ B @ Ys @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ) ) ).
% map_upds_append1
thf(fact_6799_same__append__eq,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Zs: list @ A] :
( ( ( append @ A @ Xs @ Ys )
= ( append @ A @ Xs @ Zs ) )
= ( Ys = Zs ) ) ).
% same_append_eq
thf(fact_6800_append__same__eq,axiom,
! [A: $tType,Ys: list @ A,Xs: list @ A,Zs: list @ A] :
( ( ( append @ A @ Ys @ Xs )
= ( append @ A @ Zs @ Xs ) )
= ( Ys = Zs ) ) ).
% append_same_eq
thf(fact_6801_append__assoc,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Zs: list @ A] :
( ( append @ A @ ( append @ A @ Xs @ Ys ) @ Zs )
= ( append @ A @ Xs @ ( append @ A @ Ys @ Zs ) ) ) ).
% append_assoc
thf(fact_6802_append_Oassoc,axiom,
! [A: $tType,A2: list @ A,B2: list @ A,C3: list @ A] :
( ( append @ A @ ( append @ A @ A2 @ B2 ) @ C3 )
= ( append @ A @ A2 @ ( append @ A @ B2 @ C3 ) ) ) ).
% append.assoc
thf(fact_6803_append_Oright__neutral,axiom,
! [A: $tType,A2: list @ A] :
( ( append @ A @ A2 @ ( nil @ A ) )
= A2 ) ).
% append.right_neutral
thf(fact_6804_append__Nil2,axiom,
! [A: $tType,Xs: list @ A] :
( ( append @ A @ Xs @ ( nil @ A ) )
= Xs ) ).
% append_Nil2
thf(fact_6805_append__self__conv,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( append @ A @ Xs @ Ys )
= Xs )
= ( Ys
= ( nil @ A ) ) ) ).
% append_self_conv
thf(fact_6806_self__append__conv,axiom,
! [A: $tType,Y: list @ A,Ys: list @ A] :
( ( Y
= ( append @ A @ Y @ Ys ) )
= ( Ys
= ( nil @ A ) ) ) ).
% self_append_conv
thf(fact_6807_append__self__conv2,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( append @ A @ Xs @ Ys )
= Ys )
= ( Xs
= ( nil @ A ) ) ) ).
% append_self_conv2
thf(fact_6808_self__append__conv2,axiom,
! [A: $tType,Y: list @ A,Xs: list @ A] :
( ( Y
= ( append @ A @ Xs @ Y ) )
= ( Xs
= ( nil @ A ) ) ) ).
% self_append_conv2
thf(fact_6809_Nil__is__append__conv,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( nil @ A )
= ( append @ A @ Xs @ Ys ) )
= ( ( Xs
= ( nil @ A ) )
& ( Ys
= ( nil @ A ) ) ) ) ).
% Nil_is_append_conv
thf(fact_6810_append__is__Nil__conv,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( append @ A @ Xs @ Ys )
= ( nil @ A ) )
= ( ( Xs
= ( nil @ A ) )
& ( Ys
= ( nil @ A ) ) ) ) ).
% append_is_Nil_conv
thf(fact_6811_append__eq__append__conv,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Us: list @ A,Vs2: list @ A] :
( ( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ A ) @ Ys ) )
| ( ( size_size @ ( list @ A ) @ Us )
= ( size_size @ ( list @ A ) @ Vs2 ) ) )
=> ( ( ( append @ A @ Xs @ Us )
= ( append @ A @ Ys @ Vs2 ) )
= ( ( Xs = Ys )
& ( Us = Vs2 ) ) ) ) ).
% append_eq_append_conv
thf(fact_6812_map__append,axiom,
! [A: $tType,B: $tType,F3: B > A,Xs: list @ B,Ys: list @ B] :
( ( map @ B @ A @ F3 @ ( append @ B @ Xs @ Ys ) )
= ( append @ A @ ( map @ B @ A @ F3 @ Xs ) @ ( map @ B @ A @ F3 @ Ys ) ) ) ).
% map_append
thf(fact_6813_concat__append,axiom,
! [A: $tType,Xs: list @ ( list @ A ),Ys: list @ ( list @ A )] :
( ( concat @ A @ ( append @ ( list @ A ) @ Xs @ Ys ) )
= ( append @ A @ ( concat @ A @ Xs ) @ ( concat @ A @ Ys ) ) ) ).
% concat_append
thf(fact_6814_removeAll__append,axiom,
! [A: $tType,X: A,Xs: list @ A,Ys: list @ A] :
( ( removeAll @ A @ X @ ( append @ A @ Xs @ Ys ) )
= ( append @ A @ ( removeAll @ A @ X @ Xs ) @ ( removeAll @ A @ X @ Ys ) ) ) ).
% removeAll_append
thf(fact_6815_foldl__append,axiom,
! [A: $tType,B: $tType,F3: A > B > A,A2: A,Xs: list @ B,Ys: list @ B] :
( ( foldl @ A @ B @ F3 @ A2 @ ( append @ B @ Xs @ Ys ) )
= ( foldl @ A @ B @ F3 @ ( foldl @ A @ B @ F3 @ A2 @ Xs ) @ Ys ) ) ).
% foldl_append
thf(fact_6816_append1__eq__conv,axiom,
! [A: $tType,Xs: list @ A,X: A,Ys: list @ A,Y: A] :
( ( ( append @ A @ Xs @ ( cons @ A @ X @ ( nil @ A ) ) )
= ( append @ A @ Ys @ ( cons @ A @ Y @ ( nil @ A ) ) ) )
= ( ( Xs = Ys )
& ( X = Y ) ) ) ).
% append1_eq_conv
thf(fact_6817_length__append,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( size_size @ ( list @ A ) @ ( append @ A @ Xs @ Ys ) )
= ( plus_plus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ A ) @ Ys ) ) ) ).
% length_append
thf(fact_6818_set__append,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( set2 @ A @ ( append @ A @ Xs @ Ys ) )
= ( sup_sup @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ Ys ) ) ) ).
% set_append
thf(fact_6819_zip__append,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Us: list @ B,Ys: list @ A,Vs2: list @ B] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Us ) )
=> ( ( zip @ A @ B @ ( append @ A @ Xs @ Ys ) @ ( append @ B @ Us @ Vs2 ) )
= ( append @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Us ) @ ( zip @ A @ B @ Ys @ Vs2 ) ) ) ) ).
% zip_append
thf(fact_6820_size__list__append,axiom,
! [A: $tType,F3: A > nat,Xs: list @ A,Ys: list @ A] :
( ( size_list @ A @ F3 @ ( append @ A @ Xs @ Ys ) )
= ( plus_plus @ nat @ ( size_list @ A @ F3 @ Xs ) @ ( size_list @ A @ F3 @ Ys ) ) ) ).
% size_list_append
thf(fact_6821_bind__simps_I2_J,axiom,
! [A: $tType,B: $tType,X: B,Xs: list @ B,F3: B > ( list @ A )] :
( ( bind @ B @ A @ ( cons @ B @ X @ Xs ) @ F3 )
= ( append @ A @ ( F3 @ X ) @ ( bind @ B @ A @ Xs @ F3 ) ) ) ).
% bind_simps(2)
thf(fact_6822_nth__append__length,axiom,
! [A: $tType,Xs: list @ A,X: A,Ys: list @ A] :
( ( nth @ A @ ( append @ A @ Xs @ ( cons @ A @ X @ Ys ) ) @ ( size_size @ ( list @ A ) @ Xs ) )
= X ) ).
% nth_append_length
thf(fact_6823_nth__append__length__plus,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,N: nat] :
( ( nth @ A @ ( append @ A @ Xs @ Ys ) @ ( plus_plus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ N ) )
= ( nth @ A @ Ys @ N ) ) ).
% nth_append_length_plus
thf(fact_6824_take__append,axiom,
! [A: $tType,N: nat,Xs: list @ A,Ys: list @ A] :
( ( take @ A @ N @ ( append @ A @ Xs @ Ys ) )
= ( append @ A @ ( take @ A @ N @ Xs ) @ ( take @ A @ ( minus_minus @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) ) @ Ys ) ) ) ).
% take_append
thf(fact_6825_list__update__length,axiom,
! [A: $tType,Xs: list @ A,X: A,Ys: list @ A,Y: A] :
( ( list_update @ A @ ( append @ A @ Xs @ ( cons @ A @ X @ Ys ) ) @ ( size_size @ ( list @ A ) @ Xs ) @ Y )
= ( append @ A @ Xs @ ( cons @ A @ Y @ Ys ) ) ) ).
% list_update_length
thf(fact_6826_sorted__list__of__set__lessThan__Suc,axiom,
! [K2: nat] :
( ( linord4507533701916653071of_set @ nat @ ( set_ord_lessThan @ nat @ ( suc @ K2 ) ) )
= ( append @ nat @ ( linord4507533701916653071of_set @ nat @ ( set_ord_lessThan @ nat @ K2 ) ) @ ( cons @ nat @ K2 @ ( nil @ nat ) ) ) ) ).
% sorted_list_of_set_lessThan_Suc
thf(fact_6827_sorted__list__of__set__atMost__Suc,axiom,
! [K2: nat] :
( ( linord4507533701916653071of_set @ nat @ ( set_ord_atMost @ nat @ ( suc @ K2 ) ) )
= ( append @ nat @ ( linord4507533701916653071of_set @ nat @ ( set_ord_atMost @ nat @ K2 ) ) @ ( cons @ nat @ ( suc @ K2 ) @ ( nil @ nat ) ) ) ) ).
% sorted_list_of_set_atMost_Suc
thf(fact_6828_distinct__append,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( distinct @ A @ ( append @ A @ Xs @ Ys ) )
= ( ( distinct @ A @ Xs )
& ( distinct @ A @ Ys )
& ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ Ys ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% distinct_append
thf(fact_6829_append__eq__map__conv,axiom,
! [A: $tType,B: $tType,Ys: list @ A,Zs: list @ A,F3: B > A,Xs: list @ B] :
( ( ( append @ A @ Ys @ Zs )
= ( map @ B @ A @ F3 @ Xs ) )
= ( ? [Us2: list @ B,Vs: list @ B] :
( ( Xs
= ( append @ B @ Us2 @ Vs ) )
& ( Ys
= ( map @ B @ A @ F3 @ Us2 ) )
& ( Zs
= ( map @ B @ A @ F3 @ Vs ) ) ) ) ) ).
% append_eq_map_conv
thf(fact_6830_map__eq__append__conv,axiom,
! [A: $tType,B: $tType,F3: B > A,Xs: list @ B,Ys: list @ A,Zs: list @ A] :
( ( ( map @ B @ A @ F3 @ Xs )
= ( append @ A @ Ys @ Zs ) )
= ( ? [Us2: list @ B,Vs: list @ B] :
( ( Xs
= ( append @ B @ Us2 @ Vs ) )
& ( Ys
= ( map @ B @ A @ F3 @ Us2 ) )
& ( Zs
= ( map @ B @ A @ F3 @ Vs ) ) ) ) ) ).
% map_eq_append_conv
thf(fact_6831_replicate__app__Cons__same,axiom,
! [A: $tType,N: nat,X: A,Xs: list @ A] :
( ( append @ A @ ( replicate @ A @ N @ X ) @ ( cons @ A @ X @ Xs ) )
= ( cons @ A @ X @ ( append @ A @ ( replicate @ A @ N @ X ) @ Xs ) ) ) ).
% replicate_app_Cons_same
thf(fact_6832_split__list,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ? [Ys4: list @ A,Zs2: list @ A] :
( Xs
= ( append @ A @ Ys4 @ ( cons @ A @ X @ Zs2 ) ) ) ) ).
% split_list
thf(fact_6833_split__list__last,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ? [Ys4: list @ A,Zs2: list @ A] :
( ( Xs
= ( append @ A @ Ys4 @ ( cons @ A @ X @ Zs2 ) ) )
& ~ ( member @ A @ X @ ( set2 @ A @ Zs2 ) ) ) ) ).
% split_list_last
thf(fact_6834_split__list__prop,axiom,
! [A: $tType,Xs: list @ A,P2: A > $o] :
( ? [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
& ( P2 @ X3 ) )
=> ? [Ys4: list @ A,X4: A] :
( ? [Zs2: list @ A] :
( Xs
= ( append @ A @ Ys4 @ ( cons @ A @ X4 @ Zs2 ) ) )
& ( P2 @ X4 ) ) ) ).
% split_list_prop
thf(fact_6835_split__list__first,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ? [Ys4: list @ A,Zs2: list @ A] :
( ( Xs
= ( append @ A @ Ys4 @ ( cons @ A @ X @ Zs2 ) ) )
& ~ ( member @ A @ X @ ( set2 @ A @ Ys4 ) ) ) ) ).
% split_list_first
thf(fact_6836_split__list__propE,axiom,
! [A: $tType,Xs: list @ A,P2: A > $o] :
( ? [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
& ( P2 @ X3 ) )
=> ~ ! [Ys4: list @ A,X4: A] :
( ? [Zs2: list @ A] :
( Xs
= ( append @ A @ Ys4 @ ( cons @ A @ X4 @ Zs2 ) ) )
=> ~ ( P2 @ X4 ) ) ) ).
% split_list_propE
thf(fact_6837_append__Cons__eq__iff,axiom,
! [A: $tType,X: A,Xs: list @ A,Ys: list @ A,Xs5: list @ A,Ys6: list @ A] :
( ~ ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ~ ( member @ A @ X @ ( set2 @ A @ Ys ) )
=> ( ( ( append @ A @ Xs @ ( cons @ A @ X @ Ys ) )
= ( append @ A @ Xs5 @ ( cons @ A @ X @ Ys6 ) ) )
= ( ( Xs = Xs5 )
& ( Ys = Ys6 ) ) ) ) ) ).
% append_Cons_eq_iff
thf(fact_6838_in__set__conv__decomp,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
= ( ? [Ys3: list @ A,Zs3: list @ A] :
( Xs
= ( append @ A @ Ys3 @ ( cons @ A @ X @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp
thf(fact_6839_split__list__last__prop,axiom,
! [A: $tType,Xs: list @ A,P2: A > $o] :
( ? [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
& ( P2 @ X3 ) )
=> ? [Ys4: list @ A,X4: A,Zs2: list @ A] :
( ( Xs
= ( append @ A @ Ys4 @ ( cons @ A @ X4 @ Zs2 ) ) )
& ( P2 @ X4 )
& ! [Xa: A] :
( ( member @ A @ Xa @ ( set2 @ A @ Zs2 ) )
=> ~ ( P2 @ Xa ) ) ) ) ).
% split_list_last_prop
thf(fact_6840_split__list__first__prop,axiom,
! [A: $tType,Xs: list @ A,P2: A > $o] :
( ? [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
& ( P2 @ X3 ) )
=> ? [Ys4: list @ A,X4: A] :
( ? [Zs2: list @ A] :
( Xs
= ( append @ A @ Ys4 @ ( cons @ A @ X4 @ Zs2 ) ) )
& ( P2 @ X4 )
& ! [Xa: A] :
( ( member @ A @ Xa @ ( set2 @ A @ Ys4 ) )
=> ~ ( P2 @ Xa ) ) ) ) ).
% split_list_first_prop
thf(fact_6841_split__list__last__propE,axiom,
! [A: $tType,Xs: list @ A,P2: A > $o] :
( ? [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
& ( P2 @ X3 ) )
=> ~ ! [Ys4: list @ A,X4: A,Zs2: list @ A] :
( ( Xs
= ( append @ A @ Ys4 @ ( cons @ A @ X4 @ Zs2 ) ) )
=> ( ( P2 @ X4 )
=> ~ ! [Xa: A] :
( ( member @ A @ Xa @ ( set2 @ A @ Zs2 ) )
=> ~ ( P2 @ Xa ) ) ) ) ) ).
% split_list_last_propE
thf(fact_6842_split__list__first__propE,axiom,
! [A: $tType,Xs: list @ A,P2: A > $o] :
( ? [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
& ( P2 @ X3 ) )
=> ~ ! [Ys4: list @ A,X4: A] :
( ? [Zs2: list @ A] :
( Xs
= ( append @ A @ Ys4 @ ( cons @ A @ X4 @ Zs2 ) ) )
=> ( ( P2 @ X4 )
=> ~ ! [Xa: A] :
( ( member @ A @ Xa @ ( set2 @ A @ Ys4 ) )
=> ~ ( P2 @ Xa ) ) ) ) ) ).
% split_list_first_propE
thf(fact_6843_in__set__conv__decomp__last,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
= ( ? [Ys3: list @ A,Zs3: list @ A] :
( ( Xs
= ( append @ A @ Ys3 @ ( cons @ A @ X @ Zs3 ) ) )
& ~ ( member @ A @ X @ ( set2 @ A @ Zs3 ) ) ) ) ) ).
% in_set_conv_decomp_last
thf(fact_6844_in__set__conv__decomp__first,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
= ( ? [Ys3: list @ A,Zs3: list @ A] :
( ( Xs
= ( append @ A @ Ys3 @ ( cons @ A @ X @ Zs3 ) ) )
& ~ ( member @ A @ X @ ( set2 @ A @ Ys3 ) ) ) ) ) ).
% in_set_conv_decomp_first
thf(fact_6845_split__list__last__prop__iff,axiom,
! [A: $tType,Xs: list @ A,P2: A > $o] :
( ( ? [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Xs ) )
& ( P2 @ X5 ) ) )
= ( ? [Ys3: list @ A,X5: A,Zs3: list @ A] :
( ( Xs
= ( append @ A @ Ys3 @ ( cons @ A @ X5 @ Zs3 ) ) )
& ( P2 @ X5 )
& ! [Y6: A] :
( ( member @ A @ Y6 @ ( set2 @ A @ Zs3 ) )
=> ~ ( P2 @ Y6 ) ) ) ) ) ).
% split_list_last_prop_iff
thf(fact_6846_split__list__first__prop__iff,axiom,
! [A: $tType,Xs: list @ A,P2: A > $o] :
( ( ? [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Xs ) )
& ( P2 @ X5 ) ) )
= ( ? [Ys3: list @ A,X5: A] :
( ? [Zs3: list @ A] :
( Xs
= ( append @ A @ Ys3 @ ( cons @ A @ X5 @ Zs3 ) ) )
& ( P2 @ X5 )
& ! [Y6: A] :
( ( member @ A @ Y6 @ ( set2 @ A @ Ys3 ) )
=> ~ ( P2 @ Y6 ) ) ) ) ) ).
% split_list_first_prop_iff
thf(fact_6847_Cons__eq__appendI,axiom,
! [A: $tType,X: A,Xs1: list @ A,Ys: list @ A,Xs: list @ A,Zs: list @ A] :
( ( ( cons @ A @ X @ Xs1 )
= Ys )
=> ( ( Xs
= ( append @ A @ Xs1 @ Zs ) )
=> ( ( cons @ A @ X @ Xs )
= ( append @ A @ Ys @ Zs ) ) ) ) ).
% Cons_eq_appendI
thf(fact_6848_append__Cons,axiom,
! [A: $tType,X: A,Xs: list @ A,Ys: list @ A] :
( ( append @ A @ ( cons @ A @ X @ Xs ) @ Ys )
= ( cons @ A @ X @ ( append @ A @ Xs @ Ys ) ) ) ).
% append_Cons
thf(fact_6849_concat_Osimps_I2_J,axiom,
! [A: $tType,X: list @ A,Xs: list @ ( list @ A )] :
( ( concat @ A @ ( cons @ ( list @ A ) @ X @ Xs ) )
= ( append @ A @ X @ ( concat @ A @ Xs ) ) ) ).
% concat.simps(2)
thf(fact_6850_concat__eq__append__conv,axiom,
! [A: $tType,Xss: list @ ( list @ A ),Ys: list @ A,Zs: list @ A] :
( ( ( concat @ A @ Xss )
= ( append @ A @ Ys @ Zs ) )
= ( ( ( Xss
= ( nil @ ( list @ A ) ) )
=> ( ( Ys
= ( nil @ A ) )
& ( Zs
= ( nil @ A ) ) ) )
& ( ( Xss
!= ( nil @ ( list @ A ) ) )
=> ? [Xss1: list @ ( list @ A ),Xs3: list @ A,Xs6: list @ A,Xss22: list @ ( list @ A )] :
( ( Xss
= ( append @ ( list @ A ) @ Xss1 @ ( cons @ ( list @ A ) @ ( append @ A @ Xs3 @ Xs6 ) @ Xss22 ) ) )
& ( Ys
= ( append @ A @ ( concat @ A @ Xss1 ) @ Xs3 ) )
& ( Zs
= ( append @ A @ Xs6 @ ( concat @ A @ Xss22 ) ) ) ) ) ) ) ).
% concat_eq_append_conv
thf(fact_6851_concat__eq__appendD,axiom,
! [A: $tType,Xss: list @ ( list @ A ),Ys: list @ A,Zs: list @ A] :
( ( ( concat @ A @ Xss )
= ( append @ A @ Ys @ Zs ) )
=> ( ( Xss
!= ( nil @ ( list @ A ) ) )
=> ? [Xss12: list @ ( list @ A ),Xs2: list @ A,Xs4: list @ A,Xss23: list @ ( list @ A )] :
( ( Xss
= ( append @ ( list @ A ) @ Xss12 @ ( cons @ ( list @ A ) @ ( append @ A @ Xs2 @ Xs4 ) @ Xss23 ) ) )
& ( Ys
= ( append @ A @ ( concat @ A @ Xss12 ) @ Xs2 ) )
& ( Zs
= ( append @ A @ Xs4 @ ( concat @ A @ Xss23 ) ) ) ) ) ) ).
% concat_eq_appendD
thf(fact_6852_rev__induct,axiom,
! [A: $tType,P2: ( list @ A ) > $o,Xs: list @ A] :
( ( P2 @ ( nil @ A ) )
=> ( ! [X4: A,Xs2: list @ A] :
( ( P2 @ Xs2 )
=> ( P2 @ ( append @ A @ Xs2 @ ( cons @ A @ X4 @ ( nil @ A ) ) ) ) )
=> ( P2 @ Xs ) ) ) ).
% rev_induct
thf(fact_6853_rev__exhaust,axiom,
! [A: $tType,Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ~ ! [Ys4: list @ A,Y3: A] :
( Xs
!= ( append @ A @ Ys4 @ ( cons @ A @ Y3 @ ( nil @ A ) ) ) ) ) ).
% rev_exhaust
thf(fact_6854_Cons__eq__append__conv,axiom,
! [A: $tType,X: A,Xs: list @ A,Ys: list @ A,Zs: list @ A] :
( ( ( cons @ A @ X @ Xs )
= ( append @ A @ Ys @ Zs ) )
= ( ( ( Ys
= ( nil @ A ) )
& ( ( cons @ A @ X @ Xs )
= Zs ) )
| ? [Ys7: list @ A] :
( ( ( cons @ A @ X @ Ys7 )
= Ys )
& ( Xs
= ( append @ A @ Ys7 @ Zs ) ) ) ) ) ).
% Cons_eq_append_conv
thf(fact_6855_append__eq__Cons__conv,axiom,
! [A: $tType,Ys: list @ A,Zs: list @ A,X: A,Xs: list @ A] :
( ( ( append @ A @ Ys @ Zs )
= ( cons @ A @ X @ Xs ) )
= ( ( ( Ys
= ( nil @ A ) )
& ( Zs
= ( cons @ A @ X @ Xs ) ) )
| ? [Ys7: list @ A] :
( ( Ys
= ( cons @ A @ X @ Ys7 ) )
& ( ( append @ A @ Ys7 @ Zs )
= Xs ) ) ) ) ).
% append_eq_Cons_conv
thf(fact_6856_rev__nonempty__induct,axiom,
! [A: $tType,Xs: list @ A,P2: ( list @ A ) > $o] :
( ( Xs
!= ( nil @ A ) )
=> ( ! [X4: A] : ( P2 @ ( cons @ A @ X4 @ ( nil @ A ) ) )
=> ( ! [X4: A,Xs2: list @ A] :
( ( Xs2
!= ( nil @ A ) )
=> ( ( P2 @ Xs2 )
=> ( P2 @ ( append @ A @ Xs2 @ ( cons @ A @ X4 @ ( nil @ A ) ) ) ) ) )
=> ( P2 @ Xs ) ) ) ) ).
% rev_nonempty_induct
thf(fact_6857_list__encode_Ocases,axiom,
! [X: list @ nat] :
( ( X
!= ( nil @ nat ) )
=> ~ ! [X4: nat,Xs2: list @ nat] :
( X
!= ( cons @ nat @ X4 @ Xs2 ) ) ) ).
% list_encode.cases
thf(fact_6858_lex__append__leftI,axiom,
! [A: $tType,Ys: list @ A,Zs: list @ A,R2: set @ ( product_prod @ A @ A ),Xs: list @ A] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ys @ Zs ) @ ( lex @ A @ R2 ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs @ Ys ) @ ( append @ A @ Xs @ Zs ) ) @ ( lex @ A @ R2 ) ) ) ).
% lex_append_leftI
thf(fact_6859_lexord__append__leftI,axiom,
! [A: $tType,U: list @ A,V: list @ A,R2: set @ ( product_prod @ A @ A ),X: list @ A] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ U @ V ) @ ( lexord @ A @ R2 ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ X @ U ) @ ( append @ A @ X @ V ) ) @ ( lexord @ A @ R2 ) ) ) ).
% lexord_append_leftI
thf(fact_6860_append__listrel1I,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,R2: set @ ( product_prod @ A @ A ),Us: list @ A,Vs2: list @ A] :
( ( ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( listrel1 @ A @ R2 ) )
& ( Us = Vs2 ) )
| ( ( Xs = Ys )
& ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Us @ Vs2 ) @ ( listrel1 @ A @ R2 ) ) ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs @ Us ) @ ( append @ A @ Ys @ Vs2 ) ) @ ( listrel1 @ A @ R2 ) ) ) ).
% append_listrel1I
thf(fact_6861_append__replicate__commute,axiom,
! [A: $tType,N: nat,X: A,K2: nat] :
( ( append @ A @ ( replicate @ A @ N @ X ) @ ( replicate @ A @ K2 @ X ) )
= ( append @ A @ ( replicate @ A @ K2 @ X ) @ ( replicate @ A @ N @ X ) ) ) ).
% append_replicate_commute
thf(fact_6862_replicate__add,axiom,
! [A: $tType,N: nat,M2: nat,X: A] :
( ( replicate @ A @ ( plus_plus @ nat @ N @ M2 ) @ X )
= ( append @ A @ ( replicate @ A @ N @ X ) @ ( replicate @ A @ M2 @ X ) ) ) ).
% replicate_add
thf(fact_6863_remove1__append,axiom,
! [A: $tType,X: A,Xs: list @ A,Ys: list @ A] :
( ( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ( remove1 @ A @ X @ ( append @ A @ Xs @ Ys ) )
= ( append @ A @ ( remove1 @ A @ X @ Xs ) @ Ys ) ) )
& ( ~ ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ( remove1 @ A @ X @ ( append @ A @ Xs @ Ys ) )
= ( append @ A @ Xs @ ( remove1 @ A @ X @ Ys ) ) ) ) ) ).
% remove1_append
thf(fact_6864_remdups__append2,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( remdups @ A @ ( append @ A @ Xs @ ( remdups @ A @ Ys ) ) )
= ( remdups @ A @ ( append @ A @ Xs @ Ys ) ) ) ).
% remdups_append2
thf(fact_6865_enumerate__append__eq,axiom,
! [A: $tType,N: nat,Xs: list @ A,Ys: list @ A] :
( ( enumerate @ A @ N @ ( append @ A @ Xs @ Ys ) )
= ( append @ ( product_prod @ nat @ A ) @ ( enumerate @ A @ N @ Xs ) @ ( enumerate @ A @ ( plus_plus @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) ) @ Ys ) ) ) ).
% enumerate_append_eq
thf(fact_6866_append__eq__append__conv2,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Zs: list @ A,Ts: list @ A] :
( ( ( append @ A @ Xs @ Ys )
= ( append @ A @ Zs @ Ts ) )
= ( ? [Us2: list @ A] :
( ( ( Xs
= ( append @ A @ Zs @ Us2 ) )
& ( ( append @ A @ Us2 @ Ys )
= Ts ) )
| ( ( ( append @ A @ Xs @ Us2 )
= Zs )
& ( Ys
= ( append @ A @ Us2 @ Ts ) ) ) ) ) ) ).
% append_eq_append_conv2
thf(fact_6867_append__eq__appendI,axiom,
! [A: $tType,Xs: list @ A,Xs1: list @ A,Zs: list @ A,Ys: list @ A,Us: list @ A] :
( ( ( append @ A @ Xs @ Xs1 )
= Zs )
=> ( ( Ys
= ( append @ A @ Xs1 @ Us ) )
=> ( ( append @ A @ Xs @ Ys )
= ( append @ A @ Zs @ Us ) ) ) ) ).
% append_eq_appendI
thf(fact_6868_append__Nil,axiom,
! [A: $tType,Ys: list @ A] :
( ( append @ A @ ( nil @ A ) @ Ys )
= Ys ) ).
% append_Nil
thf(fact_6869_append_Oleft__neutral,axiom,
! [A: $tType,A2: list @ A] :
( ( append @ A @ ( nil @ A ) @ A2 )
= A2 ) ).
% append.left_neutral
thf(fact_6870_eq__Nil__appendI,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( Xs = Ys )
=> ( Xs
= ( append @ A @ ( nil @ A ) @ Ys ) ) ) ).
% eq_Nil_appendI
thf(fact_6871_comm__append__are__replicate,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( append @ A @ Xs @ Ys )
= ( append @ A @ Ys @ Xs ) )
=> ? [M: nat,N3: nat,Zs2: list @ A] :
( ( ( concat @ A @ ( replicate @ ( list @ A ) @ M @ Zs2 ) )
= Xs )
& ( ( concat @ A @ ( replicate @ ( list @ A ) @ N3 @ Zs2 ) )
= Ys ) ) ) ).
% comm_append_are_replicate
thf(fact_6872_same__length__different,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( Xs != Ys )
=> ( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ A ) @ Ys ) )
=> ? [Pre: list @ A,X4: A,Xs4: list @ A,Y3: A,Ys5: list @ A] :
( ( X4 != Y3 )
& ( Xs
= ( append @ A @ Pre @ ( append @ A @ ( cons @ A @ X4 @ ( nil @ A ) ) @ Xs4 ) ) )
& ( Ys
= ( append @ A @ Pre @ ( append @ A @ ( cons @ A @ Y3 @ ( nil @ A ) ) @ Ys5 ) ) ) ) ) ) ).
% same_length_different
thf(fact_6873_not__distinct__decomp,axiom,
! [A: $tType,Ws2: list @ A] :
( ~ ( distinct @ A @ Ws2 )
=> ? [Xs2: list @ A,Ys4: list @ A,Zs2: list @ A,Y3: A] :
( Ws2
= ( append @ A @ Xs2 @ ( append @ A @ ( cons @ A @ Y3 @ ( nil @ A ) ) @ ( append @ A @ Ys4 @ ( append @ A @ ( cons @ A @ Y3 @ ( nil @ A ) ) @ Zs2 ) ) ) ) ) ) ).
% not_distinct_decomp
thf(fact_6874_not__distinct__conv__prefix,axiom,
! [A: $tType,As: list @ A] :
( ( ~ ( distinct @ A @ As ) )
= ( ? [Xs3: list @ A,Y6: A,Ys3: list @ A] :
( ( member @ A @ Y6 @ ( set2 @ A @ Xs3 ) )
& ( distinct @ A @ Xs3 )
& ( As
= ( append @ A @ Xs3 @ ( cons @ A @ Y6 @ Ys3 ) ) ) ) ) ) ).
% not_distinct_conv_prefix
thf(fact_6875_replicate__append__same,axiom,
! [A: $tType,I2: nat,X: A] :
( ( append @ A @ ( replicate @ A @ I2 @ X ) @ ( cons @ A @ X @ ( nil @ A ) ) )
= ( cons @ A @ X @ ( replicate @ A @ I2 @ X ) ) ) ).
% replicate_append_same
thf(fact_6876_list__update__append1,axiom,
! [A: $tType,I2: nat,Xs: list @ A,Ys: list @ A,X: A] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( list_update @ A @ ( append @ A @ Xs @ Ys ) @ I2 @ X )
= ( append @ A @ ( list_update @ A @ Xs @ I2 @ X ) @ Ys ) ) ) ).
% list_update_append1
thf(fact_6877_remdups__adj__append__two,axiom,
! [A: $tType,Xs: list @ A,X: A,Y: A] :
( ( remdups_adj @ A @ ( append @ A @ Xs @ ( cons @ A @ X @ ( cons @ A @ Y @ ( nil @ A ) ) ) ) )
= ( append @ A @ ( remdups_adj @ A @ ( append @ A @ Xs @ ( cons @ A @ X @ ( nil @ A ) ) ) ) @ ( if @ ( list @ A ) @ ( X = Y ) @ ( nil @ A ) @ ( cons @ A @ Y @ ( nil @ A ) ) ) ) ) ).
% remdups_adj_append_two
thf(fact_6878_remove1__split,axiom,
! [A: $tType,A2: A,Xs: list @ A,Ys: list @ A] :
( ( member @ A @ A2 @ ( set2 @ A @ Xs ) )
=> ( ( ( remove1 @ A @ A2 @ Xs )
= Ys )
= ( ? [Ls: list @ A,Rs: list @ A] :
( ( Xs
= ( append @ A @ Ls @ ( cons @ A @ A2 @ Rs ) ) )
& ~ ( member @ A @ A2 @ ( set2 @ A @ Ls ) )
& ( Ys
= ( append @ A @ Ls @ Rs ) ) ) ) ) ) ).
% remove1_split
thf(fact_6879_lexord__append__leftD,axiom,
! [A: $tType,X: list @ A,U: list @ A,V: list @ A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ X @ U ) @ ( append @ A @ X @ V ) ) @ ( lexord @ A @ R2 ) )
=> ( ! [A4: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ A4 ) @ R2 )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ U @ V ) @ ( lexord @ A @ R2 ) ) ) ) ).
% lexord_append_leftD
thf(fact_6880_lexord__append__rightI,axiom,
! [A: $tType,Y: list @ A,X: list @ A,R2: set @ ( product_prod @ A @ A )] :
( ? [B10: A,Z4: list @ A] :
( Y
= ( cons @ A @ B10 @ Z4 ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ ( append @ A @ X @ Y ) ) @ ( lexord @ A @ R2 ) ) ) ).
% lexord_append_rightI
thf(fact_6881_lexord__sufE,axiom,
! [A: $tType,Xs: list @ A,Zs: list @ A,Ys: list @ A,Qs: list @ A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs @ Zs ) @ ( append @ A @ Ys @ Qs ) ) @ ( lexord @ A @ R2 ) )
=> ( ( Xs != Ys )
=> ( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ A ) @ Ys ) )
=> ( ( ( size_size @ ( list @ A ) @ Zs )
= ( size_size @ ( list @ A ) @ Qs ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( lexord @ A @ R2 ) ) ) ) ) ) ).
% lexord_sufE
thf(fact_6882_lex__append__left__iff,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),Xs: list @ A,Ys: list @ A,Zs: list @ A] :
( ! [X4: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ X4 ) @ R2 )
=> ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs @ Ys ) @ ( append @ A @ Xs @ Zs ) ) @ ( lex @ A @ R2 ) )
= ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ys @ Zs ) @ ( lex @ A @ R2 ) ) ) ) ).
% lex_append_left_iff
thf(fact_6883_lex__append__leftD,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),Xs: list @ A,Ys: list @ A,Zs: list @ A] :
( ! [X4: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ X4 ) @ R2 )
=> ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs @ Ys ) @ ( append @ A @ Xs @ Zs ) ) @ ( lex @ A @ R2 ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ys @ Zs ) @ ( lex @ A @ R2 ) ) ) ) ).
% lex_append_leftD
thf(fact_6884_rotate1_Osimps_I2_J,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( rotate1 @ A @ ( cons @ A @ X @ Xs ) )
= ( append @ A @ Xs @ ( cons @ A @ X @ ( nil @ A ) ) ) ) ).
% rotate1.simps(2)
thf(fact_6885_lex__append__rightI,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,R2: set @ ( product_prod @ A @ A ),Vs2: list @ A,Us: list @ A] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( lex @ A @ R2 ) )
=> ( ( ( size_size @ ( list @ A ) @ Vs2 )
= ( size_size @ ( list @ A ) @ Us ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs @ Us ) @ ( append @ A @ Ys @ Vs2 ) ) @ ( lex @ A @ R2 ) ) ) ) ).
% lex_append_rightI
thf(fact_6886_lenlex__append1,axiom,
! [A: $tType,Us: list @ A,Xs: list @ A,R: set @ ( product_prod @ A @ A ),Vs2: list @ A,Ys: list @ A] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Us @ Xs ) @ ( lenlex @ A @ R ) )
=> ( ( ( size_size @ ( list @ A ) @ Vs2 )
= ( size_size @ ( list @ A ) @ Ys ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Us @ Vs2 ) @ ( append @ A @ Xs @ Ys ) ) @ ( lenlex @ A @ R ) ) ) ) ).
% lenlex_append1
thf(fact_6887_nths__append,axiom,
! [A: $tType,L: list @ A,L3: list @ A,A5: set @ nat] :
( ( nths @ A @ ( append @ A @ L @ L3 ) @ A5 )
= ( append @ A @ ( nths @ A @ L @ A5 )
@ ( nths @ A @ L3
@ ( collect @ nat
@ ^ [J3: nat] : ( member @ nat @ ( plus_plus @ nat @ J3 @ ( size_size @ ( list @ A ) @ L ) ) @ A5 ) ) ) ) ) ).
% nths_append
thf(fact_6888_length__Suc__conv__rev,axiom,
! [A: $tType,Xs: list @ A,N: nat] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( suc @ N ) )
= ( ? [Y6: A,Ys3: list @ A] :
( ( Xs
= ( append @ A @ Ys3 @ ( cons @ A @ Y6 @ ( nil @ A ) ) ) )
& ( ( size_size @ ( list @ A ) @ Ys3 )
= N ) ) ) ) ).
% length_Suc_conv_rev
thf(fact_6889_subseqs_Osimps_I2_J,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( subseqs @ A @ ( cons @ A @ X @ Xs ) )
= ( append @ ( list @ A ) @ ( map @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X ) @ ( subseqs @ A @ Xs ) ) @ ( subseqs @ A @ Xs ) ) ) ).
% subseqs.simps(2)
thf(fact_6890_nth__append,axiom,
! [A: $tType,N: nat,Xs: list @ A,Ys: list @ A] :
( ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ A @ ( append @ A @ Xs @ Ys ) @ N )
= ( nth @ A @ Xs @ N ) ) )
& ( ~ ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ A @ ( append @ A @ Xs @ Ys ) @ N )
= ( nth @ A @ Ys @ ( minus_minus @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ) ) ).
% nth_append
thf(fact_6891_list__update__append,axiom,
! [A: $tType,N: nat,Xs: list @ A,Ys: list @ A,X: A] :
( ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( list_update @ A @ ( append @ A @ Xs @ Ys ) @ N @ X )
= ( append @ A @ ( list_update @ A @ Xs @ N @ X ) @ Ys ) ) )
& ( ~ ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( list_update @ A @ ( append @ A @ Xs @ Ys ) @ N @ X )
= ( append @ A @ Xs @ ( list_update @ A @ Ys @ ( minus_minus @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) ) @ X ) ) ) ) ) ).
% list_update_append
thf(fact_6892_listrel1E,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( listrel1 @ A @ R2 ) )
=> ~ ! [X4: A,Y3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y3 ) @ R2 )
=> ! [Us3: list @ A,Vs3: list @ A] :
( ( Xs
= ( append @ A @ Us3 @ ( cons @ A @ X4 @ Vs3 ) ) )
=> ( Ys
!= ( append @ A @ Us3 @ ( cons @ A @ Y3 @ Vs3 ) ) ) ) ) ) ).
% listrel1E
thf(fact_6893_listrel1I,axiom,
! [A: $tType,X: A,Y: A,R2: set @ ( product_prod @ A @ A ),Xs: list @ A,Us: list @ A,Vs2: list @ A,Ys: list @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R2 )
=> ( ( Xs
= ( append @ A @ Us @ ( cons @ A @ X @ Vs2 ) ) )
=> ( ( Ys
= ( append @ A @ Us @ ( cons @ A @ Y @ Vs2 ) ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( listrel1 @ A @ R2 ) ) ) ) ) ).
% listrel1I
thf(fact_6894_lexord__append__left__rightI,axiom,
! [A: $tType,A2: A,B2: A,R2: set @ ( product_prod @ A @ A ),U: list @ A,X: list @ A,Y: list @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ R2 )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ U @ ( cons @ A @ A2 @ X ) ) @ ( append @ A @ U @ ( cons @ A @ B2 @ Y ) ) ) @ ( lexord @ A @ R2 ) ) ) ).
% lexord_append_left_rightI
thf(fact_6895_lexord__same__pref__iff,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Zs: list @ A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs @ Ys ) @ ( append @ A @ Xs @ Zs ) ) @ ( lexord @ A @ R2 ) )
= ( ? [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Xs ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X5 @ X5 ) @ R2 ) )
| ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ys @ Zs ) @ ( lexord @ A @ R2 ) ) ) ) ).
% lexord_same_pref_iff
thf(fact_6896_lexord__sufI,axiom,
! [A: $tType,U: list @ A,W2: list @ A,R2: set @ ( product_prod @ A @ A ),V: list @ A,Z: list @ A] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ U @ W2 ) @ ( lexord @ A @ R2 ) )
=> ( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ W2 ) @ ( size_size @ ( list @ A ) @ U ) )
=> ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ U @ V ) @ ( append @ A @ W2 @ Z ) ) @ ( lexord @ A @ R2 ) ) ) ) ).
% lexord_sufI
thf(fact_6897_product_Osimps_I2_J,axiom,
! [A: $tType,B: $tType,X: A,Xs: list @ A,Ys: list @ B] :
( ( product @ A @ B @ ( cons @ A @ X @ Xs ) @ Ys )
= ( append @ ( product_prod @ A @ B ) @ ( map @ B @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X ) @ Ys ) @ ( product @ A @ B @ Xs @ Ys ) ) ) ).
% product.simps(2)
thf(fact_6898_snoc__listrel1__snoc__iff,axiom,
! [A: $tType,Xs: list @ A,X: A,Ys: list @ A,Y: A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs @ ( cons @ A @ X @ ( nil @ A ) ) ) @ ( append @ A @ Ys @ ( cons @ A @ Y @ ( nil @ A ) ) ) ) @ ( listrel1 @ A @ R2 ) )
= ( ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( listrel1 @ A @ R2 ) )
& ( X = Y ) )
| ( ( Xs = Ys )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R2 ) ) ) ) ).
% snoc_listrel1_snoc_iff
thf(fact_6899_horner__sum__append,axiom,
! [A: $tType,B: $tType] :
( ( comm_semiring_1 @ A )
=> ! [F3: B > A,A2: A,Xs: list @ B,Ys: list @ B] :
( ( groups4207007520872428315er_sum @ B @ A @ F3 @ A2 @ ( append @ B @ Xs @ Ys ) )
= ( plus_plus @ A @ ( groups4207007520872428315er_sum @ B @ A @ F3 @ A2 @ Xs ) @ ( times_times @ A @ ( power_power @ A @ A2 @ ( size_size @ ( list @ B ) @ Xs ) ) @ ( groups4207007520872428315er_sum @ B @ A @ F3 @ A2 @ Ys ) ) ) ) ) ).
% horner_sum_append
thf(fact_6900_nths__Cons,axiom,
! [A: $tType,X: A,L: list @ A,A5: set @ nat] :
( ( nths @ A @ ( cons @ A @ X @ L ) @ A5 )
= ( append @ A @ ( if @ ( list @ A ) @ ( member @ nat @ ( zero_zero @ nat ) @ A5 ) @ ( cons @ A @ X @ ( nil @ A ) ) @ ( nil @ A ) )
@ ( nths @ A @ L
@ ( collect @ nat
@ ^ [J3: nat] : ( member @ nat @ ( suc @ J3 ) @ A5 ) ) ) ) ) ).
% nths_Cons
thf(fact_6901_comm__append__is__replicate,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( Ys
!= ( nil @ A ) )
=> ( ( ( append @ A @ Xs @ Ys )
= ( append @ A @ Ys @ Xs ) )
=> ? [N3: nat,Zs2: list @ A] :
( ( ord_less @ nat @ ( one_one @ nat ) @ N3 )
& ( ( concat @ A @ ( replicate @ ( list @ A ) @ N3 @ Zs2 ) )
= ( append @ A @ Xs @ Ys ) ) ) ) ) ) ).
% comm_append_is_replicate
thf(fact_6902_listrel1__def,axiom,
! [A: $tType] :
( ( listrel1 @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ ( list @ A ) @ ( list @ A ) )
@ ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ $o
@ ^ [Xs3: list @ A,Ys3: list @ A] :
? [Us2: list @ A,Z5: A,Z6: A,Vs: list @ A] :
( ( Xs3
= ( append @ A @ Us2 @ ( cons @ A @ Z5 @ Vs ) ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z5 @ Z6 ) @ R5 )
& ( Ys3
= ( append @ A @ Us2 @ ( cons @ A @ Z6 @ Vs ) ) ) ) ) ) ) ) ).
% listrel1_def
thf(fact_6903_lexord__def,axiom,
! [A: $tType] :
( ( lexord @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ ( list @ A ) @ ( list @ A ) )
@ ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ $o
@ ^ [X5: list @ A,Y6: list @ A] :
? [A6: A,V5: list @ A] :
( ( Y6
= ( append @ A @ X5 @ ( cons @ A @ A6 @ V5 ) ) )
| ? [U2: list @ A,B5: A,C4: A,W3: list @ A,Z5: list @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B5 @ C4 ) @ R5 )
& ( X5
= ( append @ A @ U2 @ ( cons @ A @ B5 @ W3 ) ) )
& ( Y6
= ( append @ A @ U2 @ ( cons @ A @ C4 @ Z5 ) ) ) ) ) ) ) ) ) ).
% lexord_def
thf(fact_6904_take__Suc__conv__app__nth,axiom,
! [A: $tType,I2: nat,Xs: list @ A] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( take @ A @ ( suc @ I2 ) @ Xs )
= ( append @ A @ ( take @ A @ I2 @ Xs ) @ ( cons @ A @ ( nth @ A @ Xs @ I2 ) @ ( nil @ A ) ) ) ) ) ).
% take_Suc_conv_app_nth
thf(fact_6905_lex__conv,axiom,
! [A: $tType] :
( ( lex @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ ( list @ A ) @ ( list @ A ) )
@ ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ $o
@ ^ [Xs3: list @ A,Ys3: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs3 )
= ( size_size @ ( list @ A ) @ Ys3 ) )
& ? [Xys2: list @ A,X5: A,Y6: A,Xs6: list @ A,Ys7: list @ A] :
( ( Xs3
= ( append @ A @ Xys2 @ ( cons @ A @ X5 @ Xs6 ) ) )
& ( Ys3
= ( append @ A @ Xys2 @ ( cons @ A @ Y6 @ Ys7 ) ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X5 @ Y6 ) @ R5 ) ) ) ) ) ) ) ).
% lex_conv
thf(fact_6906_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_6907_pos__n__replace,axiom,
! [A: $tType,N: nat,Xs: list @ A,Y: 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 @ Y @ ( nil @ A ) ) @ ( drop @ A @ ( suc @ N ) @ Xs ) ) ) ) ) ) ).
% pos_n_replace
thf(fact_6908_drop0,axiom,
! [A: $tType] :
( ( drop @ A @ ( zero_zero @ nat ) )
= ( ^ [X5: list @ A] : X5 ) ) ).
% drop0
thf(fact_6909_drop__drop,axiom,
! [A: $tType,N: nat,M2: nat,Xs: list @ A] :
( ( drop @ A @ N @ ( drop @ A @ M2 @ Xs ) )
= ( drop @ A @ ( plus_plus @ nat @ N @ M2 ) @ Xs ) ) ).
% drop_drop
thf(fact_6910_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_6911_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_6912_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_6913_append__take__drop__id,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( append @ A @ ( take @ A @ N @ Xs ) @ ( drop @ A @ N @ Xs ) )
= Xs ) ).
% append_take_drop_id
thf(fact_6914_drop__replicate,axiom,
! [A: $tType,I2: nat,K2: nat,X: A] :
( ( drop @ A @ I2 @ ( replicate @ A @ K2 @ X ) )
= ( replicate @ A @ ( minus_minus @ nat @ K2 @ I2 ) @ X ) ) ).
% drop_replicate
thf(fact_6915_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_6916_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_6917_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_6918_drop__append,axiom,
! [A: $tType,N: nat,Xs: list @ A,Ys: list @ A] :
( ( drop @ A @ N @ ( append @ A @ Xs @ Ys ) )
= ( append @ A @ ( drop @ A @ N @ Xs ) @ ( drop @ A @ ( minus_minus @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) ) @ Ys ) ) ) ).
% drop_append
thf(fact_6919_drop__Cons__numeral,axiom,
! [A: $tType,V: num,X: A,Xs: list @ A] :
( ( drop @ A @ ( numeral_numeral @ nat @ V ) @ ( cons @ A @ X @ Xs ) )
= ( drop @ A @ ( minus_minus @ nat @ ( numeral_numeral @ nat @ V ) @ ( one_one @ nat ) ) @ Xs ) ) ).
% drop_Cons_numeral
thf(fact_6920_nth__drop,axiom,
! [A: $tType,N: nat,Xs: list @ A,I2: nat] :
( ( ord_less_eq @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( nth @ A @ ( drop @ A @ N @ Xs ) @ I2 )
= ( nth @ A @ Xs @ ( plus_plus @ nat @ N @ I2 ) ) ) ) ).
% nth_drop
thf(fact_6921_nth__via__drop,axiom,
! [A: $tType,N: nat,Xs: list @ A,Y: A,Ys: list @ A] :
( ( ( drop @ A @ N @ Xs )
= ( cons @ A @ Y @ Ys ) )
=> ( ( nth @ A @ Xs @ N )
= Y ) ) ).
% nth_via_drop
thf(fact_6922_drop__zip,axiom,
! [A: $tType,B: $tType,N: nat,Xs: list @ A,Ys: list @ B] :
( ( drop @ ( product_prod @ A @ B ) @ N @ ( zip @ A @ B @ Xs @ Ys ) )
= ( zip @ A @ B @ ( drop @ A @ N @ Xs ) @ ( drop @ B @ N @ Ys ) ) ) ).
% drop_zip
thf(fact_6923_drop__Nil,axiom,
! [A: $tType,N: nat] :
( ( drop @ A @ N @ ( nil @ A ) )
= ( nil @ A ) ) ).
% drop_Nil
thf(fact_6924_drop__0,axiom,
! [A: $tType,Xs: list @ A] :
( ( drop @ A @ ( zero_zero @ nat ) @ Xs )
= Xs ) ).
% drop_0
thf(fact_6925_take__drop,axiom,
! [A: $tType,N: nat,M2: nat,Xs: list @ A] :
( ( take @ A @ N @ ( drop @ A @ M2 @ Xs ) )
= ( drop @ A @ M2 @ ( take @ A @ ( plus_plus @ nat @ N @ M2 ) @ Xs ) ) ) ).
% take_drop
thf(fact_6926_in__set__dropD,axiom,
! [A: $tType,X: A,N: nat,Xs: list @ A] :
( ( member @ A @ X @ ( set2 @ A @ ( drop @ A @ N @ Xs ) ) )
=> ( member @ A @ X @ ( set2 @ A @ Xs ) ) ) ).
% in_set_dropD
thf(fact_6927_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_6928_set__drop__subset,axiom,
! [A: $tType,N: nat,Xs: list @ A] : ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ ( drop @ A @ N @ Xs ) ) @ ( set2 @ A @ Xs ) ) ).
% set_drop_subset
thf(fact_6929_distinct__drop,axiom,
! [A: $tType,Xs: list @ A,I2: nat] :
( ( distinct @ A @ Xs )
=> ( distinct @ A @ ( drop @ A @ I2 @ Xs ) ) ) ).
% distinct_drop
thf(fact_6930_drop__eq__nths,axiom,
! [A: $tType] :
( ( drop @ A )
= ( ^ [N2: nat,Xs3: list @ A] : ( nths @ A @ Xs3 @ ( collect @ nat @ ( ord_less_eq @ nat @ N2 ) ) ) ) ) ).
% drop_eq_nths
thf(fact_6931_drop__map,axiom,
! [A: $tType,B: $tType,N: nat,F3: B > A,Xs: list @ B] :
( ( drop @ A @ N @ ( map @ B @ A @ F3 @ Xs ) )
= ( map @ B @ A @ F3 @ ( drop @ B @ N @ Xs ) ) ) ).
% drop_map
thf(fact_6932_set__drop__subset__set__drop,axiom,
! [A: $tType,N: nat,M2: nat,Xs: list @ A] :
( ( ord_less_eq @ nat @ N @ M2 )
=> ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ ( drop @ A @ M2 @ Xs ) ) @ ( set2 @ A @ ( drop @ A @ N @ Xs ) ) ) ) ).
% set_drop_subset_set_drop
thf(fact_6933_append__eq__conv__conj,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Zs: list @ A] :
( ( ( append @ A @ Xs @ Ys )
= Zs )
= ( ( Xs
= ( take @ A @ ( size_size @ ( list @ A ) @ Xs ) @ Zs ) )
& ( Ys
= ( drop @ A @ ( size_size @ ( list @ A ) @ Xs ) @ Zs ) ) ) ) ).
% append_eq_conv_conj
thf(fact_6934_take__add,axiom,
! [A: $tType,I2: nat,J: nat,Xs: list @ A] :
( ( take @ A @ ( plus_plus @ nat @ I2 @ J ) @ Xs )
= ( append @ A @ ( take @ A @ I2 @ Xs ) @ ( take @ A @ J @ ( drop @ A @ I2 @ Xs ) ) ) ) ).
% take_add
thf(fact_6935_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_6936_drop__Cons,axiom,
! [A: $tType,N: nat,X: A,Xs: list @ A] :
( ( drop @ A @ N @ ( cons @ A @ X @ Xs ) )
= ( case_nat @ ( list @ A ) @ ( cons @ A @ X @ Xs )
@ ^ [M3: nat] : ( drop @ A @ M3 @ Xs )
@ N ) ) ).
% drop_Cons
thf(fact_6937_nths__drop,axiom,
! [A: $tType,N: nat,Xs: list @ A,I6: set @ nat] :
( ( nths @ A @ ( drop @ A @ N @ Xs ) @ I6 )
= ( nths @ A @ Xs @ ( image2 @ nat @ nat @ ( plus_plus @ nat @ N ) @ I6 ) ) ) ).
% nths_drop
thf(fact_6938_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_6939_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_6940_zip__append1,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ A,Zs: list @ B] :
( ( zip @ A @ B @ ( append @ A @ Xs @ Ys ) @ Zs )
= ( append @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ ( take @ B @ ( size_size @ ( list @ A ) @ Xs ) @ Zs ) ) @ ( zip @ A @ B @ Ys @ ( drop @ B @ ( size_size @ ( list @ A ) @ Xs ) @ Zs ) ) ) ) ).
% zip_append1
thf(fact_6941_zip__append2,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B,Zs: list @ B] :
( ( zip @ A @ B @ Xs @ ( append @ B @ Ys @ Zs ) )
= ( append @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ ( take @ A @ ( size_size @ ( list @ B ) @ Ys ) @ Xs ) @ Ys ) @ ( zip @ A @ B @ ( drop @ A @ ( size_size @ ( list @ B ) @ Ys ) @ Xs ) @ Zs ) ) ) ).
% zip_append2
thf(fact_6942_Cons__nth__drop__Suc,axiom,
! [A: $tType,I2: nat,Xs: list @ A] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( cons @ A @ ( nth @ A @ Xs @ I2 ) @ ( drop @ A @ ( suc @ I2 ) @ Xs ) )
= ( drop @ A @ I2 @ Xs ) ) ) ).
% Cons_nth_drop_Suc
thf(fact_6943_set__take__disj__set__drop__if__distinct,axiom,
! [A: $tType,Vs2: list @ A,I2: nat,J: nat] :
( ( distinct @ A @ Vs2 )
=> ( ( ord_less_eq @ nat @ I2 @ J )
=> ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ ( take @ A @ I2 @ Vs2 ) ) @ ( set2 @ A @ ( drop @ A @ J @ Vs2 ) ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% set_take_disj_set_drop_if_distinct
thf(fact_6944_id__take__nth__drop,axiom,
! [A: $tType,I2: nat,Xs: list @ A] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( Xs
= ( append @ A @ ( take @ A @ I2 @ Xs ) @ ( cons @ A @ ( nth @ A @ Xs @ I2 ) @ ( drop @ A @ ( suc @ I2 ) @ Xs ) ) ) ) ) ).
% id_take_nth_drop
thf(fact_6945_upd__conv__take__nth__drop,axiom,
! [A: $tType,I2: nat,Xs: list @ A,A2: A] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( list_update @ A @ Xs @ I2 @ A2 )
= ( append @ A @ ( take @ A @ I2 @ Xs ) @ ( cons @ A @ A2 @ ( drop @ A @ ( suc @ I2 ) @ Xs ) ) ) ) ) ).
% upd_conv_take_nth_drop
thf(fact_6946_lexn__conv,axiom,
! [A: $tType] :
( ( lexn @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A ),N2: nat] :
( collect @ ( product_prod @ ( list @ A ) @ ( list @ A ) )
@ ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ $o
@ ^ [Xs3: list @ A,Ys3: list @ A] :
( ( ( size_size @ ( list @ A ) @ Xs3 )
= N2 )
& ( ( size_size @ ( list @ A ) @ Ys3 )
= N2 )
& ? [Xys2: list @ A,X5: A,Y6: A,Xs6: list @ A,Ys7: list @ A] :
( ( Xs3
= ( append @ A @ Xys2 @ ( cons @ A @ X5 @ Xs6 ) ) )
& ( Ys3
= ( append @ A @ Xys2 @ ( cons @ A @ Y6 @ Ys7 ) ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X5 @ Y6 ) @ R5 ) ) ) ) ) ) ) ).
% lexn_conv
thf(fact_6947_list__encode_Oelims,axiom,
! [X: list @ nat,Y: nat] :
( ( ( nat_list_encode @ X )
= Y )
=> ( ( ( X
= ( nil @ nat ) )
=> ( Y
!= ( zero_zero @ nat ) ) )
=> ~ ! [X4: nat,Xs2: list @ nat] :
( ( X
= ( cons @ nat @ X4 @ Xs2 ) )
=> ( Y
!= ( suc @ ( nat_prod_encode @ ( product_Pair @ nat @ nat @ X4 @ ( nat_list_encode @ Xs2 ) ) ) ) ) ) ) ) ).
% list_encode.elims
thf(fact_6948_inj__list__encode,axiom,
! [A5: set @ ( list @ nat )] : ( inj_on @ ( list @ nat ) @ nat @ nat_list_encode @ A5 ) ).
% inj_list_encode
thf(fact_6949_surj__list__encode,axiom,
( ( image2 @ ( list @ nat ) @ nat @ nat_list_encode @ ( top_top @ ( set @ ( list @ nat ) ) ) )
= ( top_top @ ( set @ nat ) ) ) ).
% surj_list_encode
thf(fact_6950_list__encode__eq,axiom,
! [X: list @ nat,Y: list @ nat] :
( ( ( nat_list_encode @ X )
= ( nat_list_encode @ Y ) )
= ( X = Y ) ) ).
% list_encode_eq
thf(fact_6951_lexn_Osimps_I1_J,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( lexn @ A @ R2 @ ( zero_zero @ nat ) )
= ( bot_bot @ ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ) ) ).
% lexn.simps(1)
thf(fact_6952_list__encode_Osimps_I1_J,axiom,
( ( nat_list_encode @ ( nil @ nat ) )
= ( zero_zero @ nat ) ) ).
% list_encode.simps(1)
thf(fact_6953_lexn__length,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,R2: set @ ( product_prod @ A @ A ),N: nat] :
( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( lexn @ A @ R2 @ N ) )
=> ( ( ( size_size @ ( list @ A ) @ Xs )
= N )
& ( ( size_size @ ( list @ A ) @ Ys )
= N ) ) ) ).
% lexn_length
thf(fact_6954_lex__def,axiom,
! [A: $tType] :
( ( lex @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A )] : ( complete_Sup_Sup @ ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) @ ( image2 @ nat @ ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) @ ( lexn @ A @ R5 ) @ ( top_top @ ( set @ nat ) ) ) ) ) ) ).
% lex_def
thf(fact_6955_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_6956_upto__aux__rec,axiom,
( upto_aux
= ( ^ [I4: int,J3: int,Js: list @ int] : ( if @ ( list @ int ) @ ( ord_less @ int @ J3 @ I4 ) @ Js @ ( upto_aux @ I4 @ ( minus_minus @ int @ J3 @ ( one_one @ int ) ) @ ( cons @ int @ J3 @ Js ) ) ) ) ) ).
% upto_aux_rec
thf(fact_6957_list_Osimps_I5_J,axiom,
! [B: $tType,A: $tType,F1: B,F22: A > ( list @ A ) > B,X21: A,X222: list @ A] :
( ( case_list @ B @ A @ F1 @ F22 @ ( cons @ A @ X21 @ X222 ) )
= ( F22 @ X21 @ X222 ) ) ).
% list.simps(5)
thf(fact_6958_list_Ocase__distrib,axiom,
! [B: $tType,C: $tType,A: $tType,H2: B > C,F1: B,F22: A > ( list @ A ) > B,List: list @ A] :
( ( H2 @ ( case_list @ B @ A @ F1 @ F22 @ List ) )
= ( case_list @ C @ A @ ( H2 @ F1 )
@ ^ [X15: A,X23: list @ A] : ( H2 @ ( F22 @ X15 @ X23 ) )
@ List ) ) ).
% list.case_distrib
thf(fact_6959_list_Osimps_I4_J,axiom,
! [A: $tType,B: $tType,F1: B,F22: A > ( list @ A ) > B] :
( ( case_list @ B @ A @ F1 @ F22 @ ( nil @ A ) )
= F1 ) ).
% list.simps(4)
thf(fact_6960_remdups__adj__Cons,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( remdups_adj @ A @ ( cons @ A @ X @ Xs ) )
= ( case_list @ ( list @ A ) @ A @ ( cons @ A @ X @ ( nil @ A ) )
@ ^ [Y6: A,Xs3: list @ A] : ( if @ ( list @ A ) @ ( X = Y6 ) @ ( cons @ A @ Y6 @ Xs3 ) @ ( cons @ A @ X @ ( cons @ A @ Y6 @ Xs3 ) ) )
@ ( remdups_adj @ A @ Xs ) ) ) ).
% remdups_adj_Cons
thf(fact_6961_min__list_Osimps,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X: A,Xs: list @ A] :
( ( min_list @ A @ ( cons @ A @ X @ Xs ) )
= ( case_list @ A @ A @ X
@ ^ [A6: A,List2: list @ A] : ( ord_min @ A @ X @ ( min_list @ A @ Xs ) )
@ Xs ) ) ) ).
% min_list.simps
thf(fact_6962_transpose_Osimps_I3_J,axiom,
! [A: $tType,X: A,Xs: list @ A,Xss: list @ ( list @ A )] :
( ( transpose @ A @ ( cons @ ( list @ A ) @ ( cons @ A @ X @ Xs ) @ Xss ) )
= ( cons @ ( list @ A )
@ ( cons @ A @ X
@ ( concat @ A
@ ( map @ ( list @ A ) @ ( list @ A )
@ ( case_list @ ( list @ A ) @ A @ ( nil @ A )
@ ^ [H: A,T3: list @ A] : ( cons @ A @ H @ ( nil @ A ) ) )
@ Xss ) ) )
@ ( transpose @ A
@ ( cons @ ( list @ A ) @ Xs
@ ( concat @ ( list @ A )
@ ( map @ ( list @ A ) @ ( list @ ( list @ A ) )
@ ( case_list @ ( list @ ( list @ A ) ) @ A @ ( nil @ ( list @ A ) )
@ ^ [H: A,T3: list @ A] : ( cons @ ( list @ A ) @ T3 @ ( nil @ ( list @ A ) ) ) )
@ Xss ) ) ) ) ) ) ).
% transpose.simps(3)
thf(fact_6963_transpose_Oelims,axiom,
! [A: $tType,X: list @ ( list @ A ),Y: list @ ( list @ A )] :
( ( ( transpose @ A @ X )
= Y )
=> ( ( ( X
= ( nil @ ( list @ A ) ) )
=> ( Y
!= ( nil @ ( list @ A ) ) ) )
=> ( ! [Xss2: list @ ( list @ A )] :
( ( X
= ( cons @ ( list @ A ) @ ( nil @ A ) @ Xss2 ) )
=> ( Y
!= ( transpose @ A @ Xss2 ) ) )
=> ~ ! [X4: A,Xs2: list @ A,Xss2: list @ ( list @ A )] :
( ( X
= ( cons @ ( list @ A ) @ ( cons @ A @ X4 @ Xs2 ) @ Xss2 ) )
=> ( Y
!= ( cons @ ( list @ A )
@ ( cons @ A @ X4
@ ( concat @ A
@ ( map @ ( list @ A ) @ ( list @ A )
@ ( case_list @ ( list @ A ) @ A @ ( nil @ A )
@ ^ [H: A,T3: list @ A] : ( cons @ A @ H @ ( nil @ A ) ) )
@ Xss2 ) ) )
@ ( transpose @ A
@ ( cons @ ( list @ A ) @ Xs2
@ ( concat @ ( list @ A )
@ ( map @ ( list @ A ) @ ( list @ ( list @ A ) )
@ ( case_list @ ( list @ ( list @ A ) ) @ A @ ( nil @ ( list @ A ) )
@ ^ [H: A,T3: list @ A] : ( cons @ ( list @ A ) @ T3 @ ( nil @ ( list @ A ) ) ) )
@ Xss2 ) ) ) ) ) ) ) ) ) ) ).
% transpose.elims
thf(fact_6964_transpose_Osimps_I2_J,axiom,
! [A: $tType,Xss: list @ ( list @ A )] :
( ( transpose @ A @ ( cons @ ( list @ A ) @ ( nil @ A ) @ Xss ) )
= ( transpose @ A @ Xss ) ) ).
% transpose.simps(2)
thf(fact_6965_list_Odisc__eq__case_I2_J,axiom,
! [A: $tType,List: list @ A] :
( ( List
!= ( nil @ A ) )
= ( case_list @ $o @ A @ $false
@ ^ [Uu3: A,Uv3: list @ A] : $true
@ List ) ) ).
% list.disc_eq_case(2)
thf(fact_6966_list_Odisc__eq__case_I1_J,axiom,
! [A: $tType,List: list @ A] :
( ( List
= ( nil @ A ) )
= ( case_list @ $o @ A @ $true
@ ^ [Uu3: A,Uv3: list @ A] : $false
@ List ) ) ).
% list.disc_eq_case(1)
thf(fact_6967_transpose_Osimps_I1_J,axiom,
! [A: $tType] :
( ( transpose @ A @ ( nil @ ( list @ A ) ) )
= ( nil @ ( list @ A ) ) ) ).
% transpose.simps(1)
thf(fact_6968_transpose__map__map,axiom,
! [A: $tType,B: $tType,F3: B > A,Xs: list @ ( list @ B )] :
( ( transpose @ A @ ( map @ ( list @ B ) @ ( list @ A ) @ ( map @ B @ A @ F3 ) @ Xs ) )
= ( map @ ( list @ B ) @ ( list @ A ) @ ( map @ B @ A @ F3 ) @ ( transpose @ B @ Xs ) ) ) ).
% transpose_map_map
thf(fact_6969_transpose__empty,axiom,
! [A: $tType,Xs: list @ ( list @ A )] :
( ( ( transpose @ A @ Xs )
= ( nil @ ( list @ A ) ) )
= ( ! [X5: list @ A] :
( ( member @ ( list @ A ) @ X5 @ ( set2 @ ( list @ A ) @ Xs ) )
=> ( X5
= ( nil @ A ) ) ) ) ) ).
% transpose_empty
thf(fact_6970_zip__Cons1,axiom,
! [A: $tType,B: $tType,X: A,Xs: list @ A,Ys: list @ B] :
( ( zip @ A @ B @ ( cons @ A @ X @ Xs ) @ Ys )
= ( case_list @ ( list @ ( product_prod @ A @ B ) ) @ B @ ( nil @ ( product_prod @ A @ B ) )
@ ^ [Y6: B,Ys3: list @ B] : ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X @ Y6 ) @ ( zip @ A @ B @ Xs @ Ys3 ) )
@ Ys ) ) ).
% zip_Cons1
thf(fact_6971_zip__Cons,axiom,
! [B: $tType,A: $tType,Xs: list @ A,Y: B,Ys: list @ B] :
( ( zip @ A @ B @ Xs @ ( cons @ B @ Y @ Ys ) )
= ( case_list @ ( list @ ( product_prod @ A @ B ) ) @ A @ ( nil @ ( product_prod @ A @ B ) )
@ ^ [Z5: A,Zs3: list @ A] : ( cons @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Z5 @ Y ) @ ( zip @ A @ B @ Zs3 @ Ys ) )
@ Xs ) ) ).
% zip_Cons
thf(fact_6972_upto_Opelims,axiom,
! [X: int,Xa2: int,Y: list @ int] :
( ( ( upto @ X @ Xa2 )
= Y )
=> ( ( accp @ ( product_prod @ int @ int ) @ upto_rel @ ( product_Pair @ int @ int @ X @ Xa2 ) )
=> ~ ( ( ( ( ord_less_eq @ int @ X @ Xa2 )
=> ( Y
= ( cons @ int @ X @ ( upto @ ( plus_plus @ int @ X @ ( one_one @ int ) ) @ Xa2 ) ) ) )
& ( ~ ( ord_less_eq @ int @ X @ Xa2 )
=> ( Y
= ( nil @ int ) ) ) )
=> ~ ( accp @ ( product_prod @ int @ int ) @ upto_rel @ ( product_Pair @ int @ int @ X @ Xa2 ) ) ) ) ) ).
% upto.pelims
thf(fact_6973_upto_Opsimps,axiom,
! [I2: int,J: int] :
( ( accp @ ( product_prod @ int @ int ) @ upto_rel @ ( product_Pair @ int @ int @ I2 @ J ) )
=> ( ( ( ord_less_eq @ int @ I2 @ J )
=> ( ( upto @ I2 @ J )
= ( cons @ int @ I2 @ ( upto @ ( plus_plus @ int @ I2 @ ( one_one @ int ) ) @ J ) ) ) )
& ( ~ ( ord_less_eq @ int @ I2 @ J )
=> ( ( upto @ I2 @ J )
= ( nil @ int ) ) ) ) ) ).
% upto.psimps
thf(fact_6974_upto__Nil,axiom,
! [I2: int,J: int] :
( ( ( upto @ I2 @ J )
= ( nil @ int ) )
= ( ord_less @ int @ J @ I2 ) ) ).
% upto_Nil
thf(fact_6975_upto__Nil2,axiom,
! [I2: int,J: int] :
( ( ( nil @ int )
= ( upto @ I2 @ J ) )
= ( ord_less @ int @ J @ I2 ) ) ).
% upto_Nil2
thf(fact_6976_upto__empty,axiom,
! [J: int,I2: int] :
( ( ord_less @ int @ J @ I2 )
=> ( ( upto @ I2 @ J )
= ( nil @ int ) ) ) ).
% upto_empty
thf(fact_6977_upto__single,axiom,
! [I2: int] :
( ( upto @ I2 @ I2 )
= ( cons @ int @ I2 @ ( nil @ int ) ) ) ).
% upto_single
thf(fact_6978_nth__upto,axiom,
! [I2: int,K2: nat,J: int] :
( ( ord_less_eq @ int @ ( plus_plus @ int @ I2 @ ( semiring_1_of_nat @ int @ K2 ) ) @ J )
=> ( ( nth @ int @ ( upto @ I2 @ J ) @ K2 )
= ( plus_plus @ int @ I2 @ ( semiring_1_of_nat @ int @ K2 ) ) ) ) ).
% nth_upto
thf(fact_6979_length__upto,axiom,
! [I2: int,J: int] :
( ( size_size @ ( list @ int ) @ ( upto @ I2 @ J ) )
= ( nat2 @ ( plus_plus @ int @ ( minus_minus @ int @ J @ I2 ) @ ( one_one @ int ) ) ) ) ).
% length_upto
thf(fact_6980_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_6981_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_6982_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_6983_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_6984_atLeastAtMost__upto,axiom,
( ( set_or1337092689740270186AtMost @ int )
= ( ^ [I4: int,J3: int] : ( set2 @ int @ ( upto @ I4 @ J3 ) ) ) ) ).
% atLeastAtMost_upto
thf(fact_6985_distinct__upto,axiom,
! [I2: int,J: int] : ( distinct @ int @ ( upto @ I2 @ J ) ) ).
% distinct_upto
thf(fact_6986_upto__aux__def,axiom,
( upto_aux
= ( ^ [I4: int,J3: int] : ( append @ int @ ( upto @ I4 @ J3 ) ) ) ) ).
% upto_aux_def
thf(fact_6987_upto__code,axiom,
( upto
= ( ^ [I4: int,J3: int] : ( upto_aux @ I4 @ J3 @ ( nil @ int ) ) ) ) ).
% upto_code
thf(fact_6988_upto__split2,axiom,
! [I2: int,J: int,K2: int] :
( ( ord_less_eq @ int @ I2 @ J )
=> ( ( ord_less_eq @ int @ J @ K2 )
=> ( ( upto @ I2 @ K2 )
= ( append @ int @ ( upto @ I2 @ J ) @ ( upto @ ( plus_plus @ int @ J @ ( one_one @ int ) ) @ K2 ) ) ) ) ) ).
% upto_split2
thf(fact_6989_upto__split1,axiom,
! [I2: int,J: int,K2: int] :
( ( ord_less_eq @ int @ I2 @ J )
=> ( ( ord_less_eq @ int @ J @ K2 )
=> ( ( upto @ I2 @ K2 )
= ( append @ int @ ( upto @ I2 @ ( minus_minus @ int @ J @ ( one_one @ int ) ) ) @ ( upto @ J @ K2 ) ) ) ) ) ).
% upto_split1
thf(fact_6990_atLeastLessThan__upto,axiom,
( ( set_or7035219750837199246ssThan @ int )
= ( ^ [I4: int,J3: int] : ( set2 @ int @ ( upto @ I4 @ ( minus_minus @ int @ J3 @ ( one_one @ int ) ) ) ) ) ) ).
% atLeastLessThan_upto
thf(fact_6991_greaterThanAtMost__upto,axiom,
( ( set_or3652927894154168847AtMost @ int )
= ( ^ [I4: int,J3: int] : ( set2 @ int @ ( upto @ ( plus_plus @ int @ I4 @ ( one_one @ int ) ) @ J3 ) ) ) ) ).
% greaterThanAtMost_upto
thf(fact_6992_upto__rec1,axiom,
! [I2: int,J: int] :
( ( ord_less_eq @ int @ I2 @ J )
=> ( ( upto @ I2 @ J )
= ( cons @ int @ I2 @ ( upto @ ( plus_plus @ int @ I2 @ ( one_one @ int ) ) @ J ) ) ) ) ).
% upto_rec1
thf(fact_6993_upto_Oelims,axiom,
! [X: int,Xa2: int,Y: list @ int] :
( ( ( upto @ X @ Xa2 )
= Y )
=> ( ( ( ord_less_eq @ int @ X @ Xa2 )
=> ( Y
= ( cons @ int @ X @ ( upto @ ( plus_plus @ int @ X @ ( one_one @ int ) ) @ Xa2 ) ) ) )
& ( ~ ( ord_less_eq @ int @ X @ Xa2 )
=> ( Y
= ( nil @ int ) ) ) ) ) ).
% upto.elims
thf(fact_6994_upto_Osimps,axiom,
( upto
= ( ^ [I4: int,J3: int] : ( if @ ( list @ int ) @ ( ord_less_eq @ int @ I4 @ J3 ) @ ( cons @ int @ I4 @ ( upto @ ( plus_plus @ int @ I4 @ ( one_one @ int ) ) @ J3 ) ) @ ( nil @ int ) ) ) ) ).
% upto.simps
thf(fact_6995_upto__rec2,axiom,
! [I2: int,J: int] :
( ( ord_less_eq @ int @ I2 @ J )
=> ( ( upto @ I2 @ J )
= ( append @ int @ ( upto @ I2 @ ( minus_minus @ int @ J @ ( one_one @ int ) ) ) @ ( cons @ int @ J @ ( nil @ int ) ) ) ) ) ).
% upto_rec2
thf(fact_6996_greaterThanLessThan__upto,axiom,
( ( set_or5935395276787703475ssThan @ int )
= ( ^ [I4: int,J3: int] : ( set2 @ int @ ( upto @ ( plus_plus @ int @ I4 @ ( one_one @ int ) ) @ ( minus_minus @ int @ J3 @ ( one_one @ int ) ) ) ) ) ) ).
% greaterThanLessThan_upto
thf(fact_6997_upto__split3,axiom,
! [I2: int,J: int,K2: int] :
( ( ord_less_eq @ int @ I2 @ J )
=> ( ( ord_less_eq @ int @ J @ K2 )
=> ( ( upto @ I2 @ K2 )
= ( append @ int @ ( upto @ I2 @ ( minus_minus @ int @ J @ ( one_one @ int ) ) ) @ ( cons @ int @ J @ ( upto @ ( plus_plus @ int @ J @ ( one_one @ int ) ) @ K2 ) ) ) ) ) ) ).
% upto_split3
thf(fact_6998_transpose_Opelims,axiom,
! [A: $tType,X: list @ ( list @ A ),Y: list @ ( list @ A )] :
( ( ( transpose @ A @ X )
= Y )
=> ( ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ X )
=> ( ( ( X
= ( nil @ ( list @ A ) ) )
=> ( ( Y
= ( nil @ ( list @ A ) ) )
=> ~ ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ ( nil @ ( list @ A ) ) ) ) )
=> ( ! [Xss2: list @ ( list @ A )] :
( ( X
= ( cons @ ( list @ A ) @ ( nil @ A ) @ Xss2 ) )
=> ( ( Y
= ( transpose @ A @ Xss2 ) )
=> ~ ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ ( cons @ ( list @ A ) @ ( nil @ A ) @ Xss2 ) ) ) )
=> ~ ! [X4: A,Xs2: list @ A,Xss2: list @ ( list @ A )] :
( ( X
= ( cons @ ( list @ A ) @ ( cons @ A @ X4 @ Xs2 ) @ Xss2 ) )
=> ( ( Y
= ( cons @ ( list @ A )
@ ( cons @ A @ X4
@ ( concat @ A
@ ( map @ ( list @ A ) @ ( list @ A )
@ ( case_list @ ( list @ A ) @ A @ ( nil @ A )
@ ^ [H: A,T3: list @ A] : ( cons @ A @ H @ ( nil @ A ) ) )
@ Xss2 ) ) )
@ ( transpose @ A
@ ( cons @ ( list @ A ) @ Xs2
@ ( concat @ ( list @ A )
@ ( map @ ( list @ A ) @ ( list @ ( list @ A ) )
@ ( case_list @ ( list @ ( list @ A ) ) @ A @ ( nil @ ( list @ A ) )
@ ^ [H: A,T3: list @ A] : ( cons @ ( list @ A ) @ T3 @ ( nil @ ( list @ A ) ) ) )
@ Xss2 ) ) ) ) ) )
=> ~ ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ ( cons @ ( list @ A ) @ ( cons @ A @ X4 @ Xs2 ) @ Xss2 ) ) ) ) ) ) ) ) ).
% transpose.pelims
thf(fact_6999_transpose_Opsimps_I3_J,axiom,
! [A: $tType,X: A,Xs: list @ A,Xss: list @ ( list @ A )] :
( ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ ( cons @ ( list @ A ) @ ( cons @ A @ X @ Xs ) @ Xss ) )
=> ( ( transpose @ A @ ( cons @ ( list @ A ) @ ( cons @ A @ X @ Xs ) @ Xss ) )
= ( cons @ ( list @ A )
@ ( cons @ A @ X
@ ( concat @ A
@ ( map @ ( list @ A ) @ ( list @ A )
@ ( case_list @ ( list @ A ) @ A @ ( nil @ A )
@ ^ [H: A,T3: list @ A] : ( cons @ A @ H @ ( nil @ A ) ) )
@ Xss ) ) )
@ ( transpose @ A
@ ( cons @ ( list @ A ) @ Xs
@ ( concat @ ( list @ A )
@ ( map @ ( list @ A ) @ ( list @ ( list @ A ) )
@ ( case_list @ ( list @ ( list @ A ) ) @ A @ ( nil @ ( list @ A ) )
@ ^ [H: A,T3: list @ A] : ( cons @ ( list @ A ) @ T3 @ ( nil @ ( list @ A ) ) ) )
@ Xss ) ) ) ) ) ) ) ).
% transpose.psimps(3)
thf(fact_7000_transpose_Opsimps_I2_J,axiom,
! [A: $tType,Xss: list @ ( list @ A )] :
( ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ ( cons @ ( list @ A ) @ ( nil @ A ) @ Xss ) )
=> ( ( transpose @ A @ ( cons @ ( list @ A ) @ ( nil @ A ) @ Xss ) )
= ( transpose @ A @ Xss ) ) ) ).
% transpose.psimps(2)
thf(fact_7001_transpose_Opsimps_I1_J,axiom,
! [A: $tType] :
( ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ ( nil @ ( list @ A ) ) )
=> ( ( transpose @ A @ ( nil @ ( list @ A ) ) )
= ( nil @ ( list @ A ) ) ) ) ).
% transpose.psimps(1)
thf(fact_7002_transpose_Opinduct,axiom,
! [A: $tType,A0: list @ ( list @ A ),P2: ( list @ ( list @ A ) ) > $o] :
( ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ A0 )
=> ( ( ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ ( nil @ ( list @ A ) ) )
=> ( P2 @ ( nil @ ( list @ A ) ) ) )
=> ( ! [Xss2: list @ ( list @ A )] :
( ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ ( cons @ ( list @ A ) @ ( nil @ A ) @ Xss2 ) )
=> ( ( P2 @ Xss2 )
=> ( P2 @ ( cons @ ( list @ A ) @ ( nil @ A ) @ Xss2 ) ) ) )
=> ( ! [X4: A,Xs2: list @ A,Xss2: list @ ( list @ A )] :
( ( accp @ ( list @ ( list @ A ) ) @ ( transpose_rel @ A ) @ ( cons @ ( list @ A ) @ ( cons @ A @ X4 @ Xs2 ) @ Xss2 ) )
=> ( ( P2
@ ( cons @ ( list @ A ) @ Xs2
@ ( concat @ ( list @ A )
@ ( map @ ( list @ A ) @ ( list @ ( list @ A ) )
@ ( case_list @ ( list @ ( list @ A ) ) @ A @ ( nil @ ( list @ A ) )
@ ^ [H: A,T3: list @ A] : ( cons @ ( list @ A ) @ T3 @ ( nil @ ( list @ A ) ) ) )
@ Xss2 ) ) ) )
=> ( P2 @ ( cons @ ( list @ A ) @ ( cons @ A @ X4 @ Xs2 ) @ Xss2 ) ) ) )
=> ( P2 @ A0 ) ) ) ) ) ).
% transpose.pinduct
thf(fact_7003_remdups__adj_Opelims,axiom,
! [A: $tType,X: list @ A,Y: list @ A] :
( ( ( remdups_adj @ A @ X )
= Y )
=> ( ( accp @ ( list @ A ) @ ( remdups_adj_rel @ A ) @ X )
=> ( ( ( X
= ( nil @ A ) )
=> ( ( Y
= ( nil @ A ) )
=> ~ ( accp @ ( list @ A ) @ ( remdups_adj_rel @ A ) @ ( nil @ A ) ) ) )
=> ( ! [X4: A] :
( ( X
= ( cons @ A @ X4 @ ( nil @ A ) ) )
=> ( ( Y
= ( cons @ A @ X4 @ ( nil @ A ) ) )
=> ~ ( accp @ ( list @ A ) @ ( remdups_adj_rel @ A ) @ ( cons @ A @ X4 @ ( nil @ A ) ) ) ) )
=> ~ ! [X4: A,Y3: A,Xs2: list @ A] :
( ( X
= ( cons @ A @ X4 @ ( cons @ A @ Y3 @ Xs2 ) ) )
=> ( ( ( ( X4 = Y3 )
=> ( Y
= ( remdups_adj @ A @ ( cons @ A @ X4 @ Xs2 ) ) ) )
& ( ( X4 != Y3 )
=> ( Y
= ( cons @ A @ X4 @ ( remdups_adj @ A @ ( cons @ A @ Y3 @ Xs2 ) ) ) ) ) )
=> ~ ( accp @ ( list @ A ) @ ( remdups_adj_rel @ A ) @ ( cons @ A @ X4 @ ( cons @ A @ Y3 @ Xs2 ) ) ) ) ) ) ) ) ) ).
% remdups_adj.pelims
thf(fact_7004_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_7005_hd__remdups__adj,axiom,
! [A: $tType,Xs: list @ A] :
( ( hd @ A @ ( remdups_adj @ A @ Xs ) )
= ( hd @ A @ Xs ) ) ).
% hd_remdups_adj
thf(fact_7006_hd__append2,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( hd @ A @ ( append @ A @ Xs @ Ys ) )
= ( hd @ A @ Xs ) ) ) ).
% hd_append2
thf(fact_7007_hd__replicate,axiom,
! [A: $tType,N: nat,X: A] :
( ( N
!= ( zero_zero @ nat ) )
=> ( ( hd @ A @ ( replicate @ A @ N @ X ) )
= X ) ) ).
% hd_replicate
thf(fact_7008_hd__take,axiom,
! [A: $tType,J: nat,Xs: list @ A] :
( ( ord_less @ nat @ ( zero_zero @ nat ) @ J )
=> ( ( hd @ A @ ( take @ A @ J @ Xs ) )
= ( hd @ A @ Xs ) ) ) ).
% hd_take
thf(fact_7009_list_Oset__sel_I1_J,axiom,
! [A: $tType,A2: list @ A] :
( ( A2
!= ( nil @ A ) )
=> ( member @ A @ ( hd @ A @ A2 ) @ ( set2 @ A @ A2 ) ) ) ).
% list.set_sel(1)
thf(fact_7010_hd__in__set,axiom,
! [A: $tType,Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( member @ A @ ( hd @ A @ Xs ) @ ( set2 @ A @ Xs ) ) ) ).
% hd_in_set
thf(fact_7011_longest__common__prefix,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
? [Ps2: list @ A,Xs4: list @ A,Ys5: list @ A] :
( ( Xs
= ( append @ A @ Ps2 @ Xs4 ) )
& ( Ys
= ( append @ A @ Ps2 @ Ys5 ) )
& ( ( Xs4
= ( nil @ A ) )
| ( Ys5
= ( nil @ A ) )
| ( ( hd @ A @ Xs4 )
!= ( hd @ A @ Ys5 ) ) ) ) ).
% longest_common_prefix
thf(fact_7012_hd__append,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( Xs
= ( nil @ A ) )
=> ( ( hd @ A @ ( append @ A @ Xs @ Ys ) )
= ( hd @ A @ Ys ) ) )
& ( ( Xs
!= ( nil @ A ) )
=> ( ( hd @ A @ ( append @ A @ Xs @ Ys ) )
= ( hd @ A @ Xs ) ) ) ) ).
% hd_append
thf(fact_7013_list_Osel_I1_J,axiom,
! [A: $tType,X21: A,X222: list @ A] :
( ( hd @ A @ ( cons @ A @ X21 @ X222 ) )
= X21 ) ).
% list.sel(1)
thf(fact_7014_hd__concat,axiom,
! [A: $tType,Xs: list @ ( list @ A )] :
( ( Xs
!= ( nil @ ( list @ A ) ) )
=> ( ( ( hd @ ( list @ A ) @ Xs )
!= ( nil @ A ) )
=> ( ( hd @ A @ ( concat @ A @ Xs ) )
= ( hd @ A @ ( hd @ ( list @ A ) @ Xs ) ) ) ) ) ).
% hd_concat
thf(fact_7015_hd__zip,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B] :
( ( Xs
!= ( nil @ A ) )
=> ( ( Ys
!= ( nil @ B ) )
=> ( ( hd @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) )
= ( product_Pair @ A @ B @ ( hd @ A @ Xs ) @ ( hd @ B @ Ys ) ) ) ) ) ).
% hd_zip
thf(fact_7016_list_Omap__sel_I1_J,axiom,
! [B: $tType,A: $tType,A2: list @ A,F3: A > B] :
( ( A2
!= ( nil @ A ) )
=> ( ( hd @ B @ ( map @ A @ B @ F3 @ A2 ) )
= ( F3 @ ( hd @ A @ A2 ) ) ) ) ).
% list.map_sel(1)
thf(fact_7017_hd__map,axiom,
! [B: $tType,A: $tType,Xs: list @ A,F3: A > B] :
( ( Xs
!= ( nil @ A ) )
=> ( ( hd @ B @ ( map @ A @ B @ F3 @ Xs ) )
= ( F3 @ ( hd @ A @ Xs ) ) ) ) ).
% hd_map
thf(fact_7018_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_7019_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_7020_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_7021_list__encode_Opelims,axiom,
! [X: list @ nat,Y: nat] :
( ( ( nat_list_encode @ X )
= Y )
=> ( ( accp @ ( list @ nat ) @ nat_list_encode_rel @ X )
=> ( ( ( X
= ( nil @ nat ) )
=> ( ( Y
= ( zero_zero @ nat ) )
=> ~ ( accp @ ( list @ nat ) @ nat_list_encode_rel @ ( nil @ nat ) ) ) )
=> ~ ! [X4: nat,Xs2: list @ nat] :
( ( X
= ( cons @ nat @ X4 @ Xs2 ) )
=> ( ( Y
= ( suc @ ( nat_prod_encode @ ( product_Pair @ nat @ nat @ X4 @ ( nat_list_encode @ Xs2 ) ) ) ) )
=> ~ ( accp @ ( list @ nat ) @ nat_list_encode_rel @ ( cons @ nat @ X4 @ Xs2 ) ) ) ) ) ) ) ).
% list_encode.pelims
thf(fact_7022_horner__sum__bit__eq__take__bit,axiom,
! [A: $tType] :
( ( bit_se359711467146920520ations @ A )
=> ! [A2: A,N: nat] :
( ( groups4207007520872428315er_sum @ $o @ A @ ( zero_neq_one_of_bool @ A ) @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ ( map @ nat @ $o @ ( bit_se5641148757651400278ts_bit @ A @ A2 ) @ ( upt @ ( zero_zero @ nat ) @ N ) ) )
= ( bit_se2584673776208193580ke_bit @ A @ N @ A2 ) ) ) ).
% horner_sum_bit_eq_take_bit
thf(fact_7023_remdups__upt,axiom,
! [M2: nat,N: nat] :
( ( remdups @ nat @ ( upt @ M2 @ N ) )
= ( upt @ M2 @ N ) ) ).
% remdups_upt
thf(fact_7024_hd__upt,axiom,
! [I2: nat,J: nat] :
( ( ord_less @ nat @ I2 @ J )
=> ( ( hd @ nat @ ( upt @ I2 @ J ) )
= I2 ) ) ).
% hd_upt
thf(fact_7025_drop__upt,axiom,
! [M2: nat,I2: nat,J: nat] :
( ( drop @ nat @ M2 @ ( upt @ I2 @ J ) )
= ( upt @ ( plus_plus @ nat @ I2 @ M2 ) @ J ) ) ).
% drop_upt
thf(fact_7026_length__upt,axiom,
! [I2: nat,J: nat] :
( ( size_size @ ( list @ nat ) @ ( upt @ I2 @ J ) )
= ( minus_minus @ nat @ J @ I2 ) ) ).
% length_upt
thf(fact_7027_take__upt,axiom,
! [I2: nat,M2: nat,N: nat] :
( ( ord_less_eq @ nat @ ( plus_plus @ nat @ I2 @ M2 ) @ N )
=> ( ( take @ nat @ M2 @ ( upt @ I2 @ N ) )
= ( upt @ I2 @ ( plus_plus @ nat @ I2 @ M2 ) ) ) ) ).
% take_upt
thf(fact_7028_upt__conv__Nil,axiom,
! [J: nat,I2: nat] :
( ( ord_less_eq @ nat @ J @ I2 )
=> ( ( upt @ I2 @ J )
= ( nil @ nat ) ) ) ).
% upt_conv_Nil
thf(fact_7029_sorted__list__of__set__range,axiom,
! [M2: nat,N: nat] :
( ( linord4507533701916653071of_set @ nat @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) )
= ( upt @ M2 @ N ) ) ).
% sorted_list_of_set_range
thf(fact_7030_upt__eq__Nil__conv,axiom,
! [I2: nat,J: nat] :
( ( ( upt @ I2 @ J )
= ( nil @ nat ) )
= ( ( J
= ( zero_zero @ nat ) )
| ( ord_less_eq @ nat @ J @ I2 ) ) ) ).
% upt_eq_Nil_conv
thf(fact_7031_nth__upt,axiom,
! [I2: nat,K2: nat,J: nat] :
( ( ord_less @ nat @ ( plus_plus @ nat @ I2 @ K2 ) @ J )
=> ( ( nth @ nat @ ( upt @ I2 @ J ) @ K2 )
= ( plus_plus @ nat @ I2 @ K2 ) ) ) ).
% nth_upt
thf(fact_7032_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_7033_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_7034_upt__0,axiom,
! [I2: nat] :
( ( upt @ I2 @ ( zero_zero @ nat ) )
= ( nil @ nat ) ) ).
% upt_0
thf(fact_7035_upt__conv__Cons__Cons,axiom,
! [M2: nat,N: nat,Ns: list @ nat,Q2: nat] :
( ( ( cons @ nat @ M2 @ ( cons @ nat @ N @ Ns ) )
= ( upt @ M2 @ Q2 ) )
= ( ( cons @ nat @ N @ Ns )
= ( upt @ ( suc @ M2 ) @ Q2 ) ) ) ).
% upt_conv_Cons_Cons
thf(fact_7036_enumerate__map__upt,axiom,
! [A: $tType,N: nat,F3: nat > A,M2: nat] :
( ( enumerate @ A @ N @ ( map @ nat @ A @ F3 @ ( upt @ N @ M2 ) ) )
= ( map @ nat @ ( product_prod @ nat @ A )
@ ^ [K3: nat] : ( product_Pair @ nat @ A @ K3 @ ( F3 @ K3 ) )
@ ( upt @ N @ M2 ) ) ) ).
% enumerate_map_upt
thf(fact_7037_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_7038_map__add__upt,axiom,
! [N: nat,M2: nat] :
( ( map @ nat @ nat
@ ^ [I4: nat] : ( plus_plus @ nat @ I4 @ N )
@ ( upt @ ( zero_zero @ nat ) @ M2 ) )
= ( upt @ N @ ( plus_plus @ nat @ M2 @ N ) ) ) ).
% map_add_upt
thf(fact_7039_map__replicate__trivial,axiom,
! [A: $tType,X: A,I2: nat] :
( ( map @ nat @ A
@ ^ [I4: nat] : X
@ ( upt @ ( zero_zero @ nat ) @ I2 ) )
= ( replicate @ A @ I2 @ X ) ) ).
% map_replicate_trivial
thf(fact_7040_greaterThanLessThan__upt,axiom,
( ( set_or5935395276787703475ssThan @ nat )
= ( ^ [N2: nat,M3: nat] : ( set2 @ nat @ ( upt @ ( suc @ N2 ) @ M3 ) ) ) ) ).
% greaterThanLessThan_upt
thf(fact_7041_greaterThanAtMost__upt,axiom,
( ( set_or3652927894154168847AtMost @ nat )
= ( ^ [N2: nat,M3: nat] : ( set2 @ nat @ ( upt @ ( suc @ N2 ) @ ( suc @ M3 ) ) ) ) ) ).
% greaterThanAtMost_upt
thf(fact_7042_atLeast__upt,axiom,
( ( set_ord_lessThan @ nat )
= ( ^ [N2: nat] : ( set2 @ nat @ ( upt @ ( zero_zero @ nat ) @ N2 ) ) ) ) ).
% atLeast_upt
thf(fact_7043_atLeastLessThan__upt,axiom,
( ( set_or7035219750837199246ssThan @ nat )
= ( ^ [I4: nat,J3: nat] : ( set2 @ nat @ ( upt @ I4 @ J3 ) ) ) ) ).
% atLeastLessThan_upt
thf(fact_7044_atLeastAtMost__upt,axiom,
( ( set_or1337092689740270186AtMost @ nat )
= ( ^ [N2: nat,M3: nat] : ( set2 @ nat @ ( upt @ N2 @ ( suc @ M3 ) ) ) ) ) ).
% atLeastAtMost_upt
thf(fact_7045_distinct__upt,axiom,
! [I2: nat,J: nat] : ( distinct @ nat @ ( upt @ I2 @ J ) ) ).
% distinct_upt
thf(fact_7046_atMost__upto,axiom,
( ( set_ord_atMost @ nat )
= ( ^ [N2: nat] : ( set2 @ nat @ ( upt @ ( zero_zero @ nat ) @ ( suc @ N2 ) ) ) ) ) ).
% atMost_upto
thf(fact_7047_upt__conv__Cons,axiom,
! [I2: nat,J: nat] :
( ( ord_less @ nat @ I2 @ J )
=> ( ( upt @ I2 @ J )
= ( cons @ nat @ I2 @ ( upt @ ( suc @ I2 ) @ J ) ) ) ) ).
% upt_conv_Cons
thf(fact_7048_enumerate__eq__zip,axiom,
! [A: $tType] :
( ( enumerate @ A )
= ( ^ [N2: nat,Xs3: list @ A] : ( zip @ nat @ A @ ( upt @ N2 @ ( plus_plus @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs3 ) ) ) @ Xs3 ) ) ) ).
% enumerate_eq_zip
thf(fact_7049_map__upt__Suc,axiom,
! [A: $tType,F3: nat > A,N: nat] :
( ( map @ nat @ A @ F3 @ ( upt @ ( zero_zero @ nat ) @ ( suc @ N ) ) )
= ( cons @ A @ ( F3 @ ( zero_zero @ nat ) )
@ ( map @ nat @ A
@ ^ [I4: nat] : ( F3 @ ( suc @ I4 ) )
@ ( upt @ ( zero_zero @ nat ) @ N ) ) ) ) ).
% map_upt_Suc
thf(fact_7050_map__decr__upt,axiom,
! [M2: nat,N: nat] :
( ( map @ nat @ nat
@ ^ [N2: nat] : ( minus_minus @ nat @ N2 @ ( suc @ ( zero_zero @ nat ) ) )
@ ( upt @ ( suc @ M2 ) @ ( suc @ N ) ) )
= ( upt @ M2 @ N ) ) ).
% map_decr_upt
thf(fact_7051_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_7052_upt__add__eq__append,axiom,
! [I2: nat,J: nat,K2: nat] :
( ( ord_less_eq @ nat @ I2 @ J )
=> ( ( upt @ I2 @ ( plus_plus @ nat @ J @ K2 ) )
= ( append @ nat @ ( upt @ I2 @ J ) @ ( upt @ J @ ( plus_plus @ nat @ J @ K2 ) ) ) ) ) ).
% upt_add_eq_append
thf(fact_7053_nth__map__upt,axiom,
! [A: $tType,I2: nat,N: nat,M2: nat,F3: nat > A] :
( ( ord_less @ nat @ I2 @ ( minus_minus @ nat @ N @ M2 ) )
=> ( ( nth @ A @ ( map @ nat @ A @ F3 @ ( upt @ M2 @ N ) ) @ I2 )
= ( F3 @ ( plus_plus @ nat @ M2 @ I2 ) ) ) ) ).
% nth_map_upt
thf(fact_7054_upt__eq__Cons__conv,axiom,
! [I2: nat,J: nat,X: nat,Xs: list @ nat] :
( ( ( upt @ I2 @ J )
= ( cons @ nat @ X @ Xs ) )
= ( ( ord_less @ nat @ I2 @ J )
& ( I2 = X )
& ( ( upt @ ( plus_plus @ nat @ I2 @ ( one_one @ nat ) ) @ J )
= Xs ) ) ) ).
% upt_eq_Cons_conv
thf(fact_7055_upt__rec,axiom,
( upt
= ( ^ [I4: nat,J3: nat] : ( if @ ( list @ nat ) @ ( ord_less @ nat @ I4 @ J3 ) @ ( cons @ nat @ I4 @ ( upt @ ( suc @ I4 ) @ J3 ) ) @ ( nil @ nat ) ) ) ) ).
% upt_rec
thf(fact_7056_enumerate__replicate__eq,axiom,
! [A: $tType,N: nat,M2: nat,A2: A] :
( ( enumerate @ A @ N @ ( replicate @ A @ M2 @ A2 ) )
= ( map @ nat @ ( product_prod @ nat @ A )
@ ^ [Q5: nat] : ( product_Pair @ nat @ A @ Q5 @ A2 )
@ ( upt @ N @ ( plus_plus @ nat @ N @ M2 ) ) ) ) ).
% enumerate_replicate_eq
thf(fact_7057_map__upt__eqI,axiom,
! [A: $tType,Xs: list @ A,N: nat,M2: nat,F3: 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 )
= ( F3 @ ( plus_plus @ nat @ M2 @ I3 ) ) ) )
=> ( ( map @ nat @ A @ F3 @ ( upt @ M2 @ N ) )
= Xs ) ) ) ).
% map_upt_eqI
thf(fact_7058_upt__Suc,axiom,
! [I2: nat,J: nat] :
( ( ( ord_less_eq @ nat @ I2 @ J )
=> ( ( upt @ I2 @ ( suc @ J ) )
= ( append @ nat @ ( upt @ I2 @ J ) @ ( cons @ nat @ J @ ( nil @ nat ) ) ) ) )
& ( ~ ( ord_less_eq @ nat @ I2 @ J )
=> ( ( upt @ I2 @ ( suc @ J ) )
= ( nil @ nat ) ) ) ) ).
% upt_Suc
thf(fact_7059_upt__Suc__append,axiom,
! [I2: nat,J: nat] :
( ( ord_less_eq @ nat @ I2 @ J )
=> ( ( upt @ I2 @ ( suc @ J ) )
= ( append @ nat @ ( upt @ I2 @ J ) @ ( cons @ nat @ J @ ( nil @ nat ) ) ) ) ) ).
% upt_Suc_append
thf(fact_7060_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 )
@ ^ [I4: nat] :
( map @ nat @ A
@ ^ [J3: nat] : ( nth @ A @ ( nth @ ( list @ A ) @ Xs @ J3 ) @ I4 )
@ ( upt @ ( zero_zero @ nat ) @ ( size_size @ ( list @ ( list @ A ) ) @ Xs ) ) )
@ ( upt @ ( zero_zero @ nat ) @ N ) ) ) ) ) ).
% transpose_rectangle
thf(fact_7061_extract__Some__iff,axiom,
! [A: $tType,P2: A > $o,Xs: list @ A,Ys: list @ A,Y: A,Zs: list @ A] :
( ( ( extract @ A @ P2 @ Xs )
= ( some @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) @ ( product_Pair @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) @ Ys @ ( product_Pair @ A @ ( list @ A ) @ Y @ Zs ) ) ) )
= ( ( Xs
= ( append @ A @ Ys @ ( cons @ A @ Y @ Zs ) ) )
& ( P2 @ Y )
& ~ ? [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Ys ) )
& ( P2 @ X5 ) ) ) ) ).
% extract_Some_iff
thf(fact_7062_extract__SomeE,axiom,
! [A: $tType,P2: A > $o,Xs: list @ A,Ys: list @ A,Y: A,Zs: list @ A] :
( ( ( extract @ A @ P2 @ Xs )
= ( some @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) @ ( product_Pair @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) @ Ys @ ( product_Pair @ A @ ( list @ A ) @ Y @ Zs ) ) ) )
=> ( ( Xs
= ( append @ A @ Ys @ ( cons @ A @ Y @ Zs ) ) )
& ( P2 @ Y )
& ~ ? [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Ys ) )
& ( P2 @ X3 ) ) ) ) ).
% extract_SomeE
thf(fact_7063_extract__None__iff,axiom,
! [A: $tType,P2: A > $o,Xs: list @ A] :
( ( ( extract @ A @ P2 @ Xs )
= ( none @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) ) )
= ( ~ ? [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Xs ) )
& ( P2 @ X5 ) ) ) ) ).
% extract_None_iff
thf(fact_7064_extract__Nil__code,axiom,
! [A: $tType,P2: A > $o] :
( ( extract @ A @ P2 @ ( nil @ A ) )
= ( none @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) ) ) ).
% extract_Nil_code
thf(fact_7065_extract__Cons__code,axiom,
! [A: $tType,P2: A > $o,X: A,Xs: list @ A] :
( ( ( P2 @ X )
=> ( ( extract @ A @ P2 @ ( 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 ) ) ) ) )
& ( ~ ( P2 @ X )
=> ( ( extract @ A @ P2 @ ( 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 ) ) ) )
@ ^ [Y6: 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 ) @ Y6 @ Zs3 ) ) ) ) )
@ ( extract @ A @ P2 @ Xs ) ) ) ) ) ).
% extract_Cons_code
thf(fact_7066_lexn_Osimps_I2_J,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),N: nat] :
( ( lexn @ A @ R2 @ ( suc @ N ) )
= ( inf_inf @ ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) @ ( image2 @ ( product_prod @ ( product_prod @ A @ ( list @ A ) ) @ ( product_prod @ A @ ( list @ A ) ) ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_map_prod @ ( product_prod @ A @ ( list @ A ) ) @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) @ ( list @ A ) @ ( product_case_prod @ A @ ( list @ A ) @ ( list @ A ) @ ( cons @ A ) ) @ ( product_case_prod @ A @ ( list @ A ) @ ( list @ A ) @ ( cons @ A ) ) ) @ ( lex_prod @ A @ ( list @ A ) @ R2 @ ( lexn @ A @ R2 @ 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_7067_sum__list__map__eq__sum__count2,axiom,
! [A: $tType,Xs: list @ A,X7: set @ A,F3: A > nat] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ X7 )
=> ( ( finite_finite2 @ A @ X7 )
=> ( ( groups8242544230860333062m_list @ nat @ ( map @ A @ nat @ F3 @ Xs ) )
= ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [X5: A] : ( times_times @ nat @ ( count_list @ A @ Xs @ X5 ) @ ( F3 @ X5 ) )
@ X7 ) ) ) ) ).
% sum_list_map_eq_sum_count2
thf(fact_7068_map__prod__ident,axiom,
! [B: $tType,A: $tType] :
( ( product_map_prod @ A @ A @ B @ B
@ ^ [X5: A] : X5
@ ^ [Y6: B] : Y6 )
= ( ^ [Z5: product_prod @ A @ B] : Z5 ) ) ).
% map_prod_ident
thf(fact_7069_map__prod__simp,axiom,
! [C: $tType,A: $tType,B: $tType,D: $tType,F3: C > A,G3: D > B,A2: C,B2: D] :
( ( product_map_prod @ C @ A @ D @ B @ F3 @ G3 @ ( product_Pair @ C @ D @ A2 @ B2 ) )
= ( product_Pair @ A @ B @ ( F3 @ A2 ) @ ( G3 @ B2 ) ) ) ).
% map_prod_simp
thf(fact_7070_fst__map__prod,axiom,
! [B: $tType,A: $tType,D: $tType,C: $tType,F3: C > A,G3: D > B,X: product_prod @ C @ D] :
( ( product_fst @ A @ B @ ( product_map_prod @ C @ A @ D @ B @ F3 @ G3 @ X ) )
= ( F3 @ ( product_fst @ C @ D @ X ) ) ) ).
% fst_map_prod
thf(fact_7071_snd__map__prod,axiom,
! [B: $tType,A: $tType,D: $tType,C: $tType,F3: C > B,G3: D > A,X: product_prod @ C @ D] :
( ( product_snd @ B @ A @ ( product_map_prod @ C @ B @ D @ A @ F3 @ G3 @ X ) )
= ( G3 @ ( product_snd @ C @ D @ X ) ) ) ).
% snd_map_prod
thf(fact_7072_sum__list_ONil,axiom,
! [A: $tType] :
( ( monoid_add @ A )
=> ( ( groups8242544230860333062m_list @ A @ ( nil @ A ) )
= ( zero_zero @ A ) ) ) ).
% sum_list.Nil
thf(fact_7073_sum__list__eq__0__iff,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [Ns: list @ A] :
( ( ( groups8242544230860333062m_list @ A @ Ns )
= ( zero_zero @ A ) )
= ( ! [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Ns ) )
=> ( X5
= ( zero_zero @ A ) ) ) ) ) ) ).
% sum_list_eq_0_iff
thf(fact_7074_sum__list_OCons,axiom,
! [A: $tType] :
( ( monoid_add @ A )
=> ! [X: A,Xs: list @ A] :
( ( groups8242544230860333062m_list @ A @ ( cons @ A @ X @ Xs ) )
= ( plus_plus @ A @ X @ ( groups8242544230860333062m_list @ A @ Xs ) ) ) ) ).
% sum_list.Cons
thf(fact_7075_sum__list__append,axiom,
! [A: $tType] :
( ( monoid_add @ A )
=> ! [Xs: list @ A,Ys: list @ A] :
( ( groups8242544230860333062m_list @ A @ ( append @ A @ Xs @ Ys ) )
= ( plus_plus @ A @ ( groups8242544230860333062m_list @ A @ Xs ) @ ( groups8242544230860333062m_list @ A @ Ys ) ) ) ) ).
% sum_list_append
thf(fact_7076_map__prod__imageI,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,A2: A,B2: B,R: set @ ( product_prod @ A @ B ),F3: A > C,G3: B > D] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ R )
=> ( member @ ( product_prod @ C @ D ) @ ( product_Pair @ C @ D @ ( F3 @ A2 ) @ ( G3 @ B2 ) ) @ ( image2 @ ( product_prod @ A @ B ) @ ( product_prod @ C @ D ) @ ( product_map_prod @ A @ C @ B @ D @ F3 @ G3 ) @ R ) ) ) ).
% map_prod_imageI
thf(fact_7077_sum__list__0,axiom,
! [B: $tType,A: $tType] :
( ( monoid_add @ A )
=> ! [Xs: list @ B] :
( ( groups8242544230860333062m_list @ A
@ ( map @ B @ A
@ ^ [X5: B] : ( zero_zero @ A )
@ Xs ) )
= ( zero_zero @ A ) ) ) ).
% sum_list_0
thf(fact_7078_fst__comp__map__prod,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,F3: A > C,G3: B > D] :
( ( comp @ ( product_prod @ C @ D ) @ C @ ( product_prod @ A @ B ) @ ( product_fst @ C @ D ) @ ( product_map_prod @ A @ C @ B @ D @ F3 @ G3 ) )
= ( comp @ A @ C @ ( product_prod @ A @ B ) @ F3 @ ( product_fst @ A @ B ) ) ) ).
% fst_comp_map_prod
thf(fact_7079_snd__comp__map__prod,axiom,
! [D: $tType,C: $tType,B: $tType,A: $tType,F3: A > D,G3: B > C] :
( ( comp @ ( product_prod @ D @ C ) @ C @ ( product_prod @ A @ B ) @ ( product_snd @ D @ C ) @ ( product_map_prod @ A @ D @ B @ C @ F3 @ G3 ) )
= ( comp @ B @ C @ ( product_prod @ A @ B ) @ G3 @ ( product_snd @ A @ B ) ) ) ).
% snd_comp_map_prod
thf(fact_7080_sum__list__upt,axiom,
! [M2: nat,N: nat] :
( ( ord_less_eq @ nat @ M2 @ N )
=> ( ( groups8242544230860333062m_list @ nat @ ( upt @ M2 @ N ) )
= ( groups7311177749621191930dd_sum @ nat @ nat
@ ^ [X5: nat] : X5
@ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) ) ) ).
% sum_list_upt
thf(fact_7081_sum__list__addf,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [F3: B > A,G3: B > A,Xs: list @ B] :
( ( groups8242544230860333062m_list @ A
@ ( map @ B @ A
@ ^ [X5: B] : ( plus_plus @ A @ ( F3 @ X5 ) @ ( G3 @ X5 ) )
@ Xs ) )
= ( plus_plus @ A @ ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ F3 @ Xs ) ) @ ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ G3 @ Xs ) ) ) ) ) ).
% sum_list_addf
thf(fact_7082_member__le__sum__list,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [X: A,Xs: list @ A] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ord_less_eq @ A @ X @ ( groups8242544230860333062m_list @ A @ Xs ) ) ) ) ).
% member_le_sum_list
thf(fact_7083_map__prod__def,axiom,
! [B: $tType,D: $tType,C: $tType,A: $tType] :
( ( product_map_prod @ A @ C @ B @ D )
= ( ^ [F4: A > C,G4: B > D] :
( product_case_prod @ A @ B @ ( product_prod @ C @ D )
@ ^ [X5: A,Y6: B] : ( product_Pair @ C @ D @ ( F4 @ X5 ) @ ( G4 @ Y6 ) ) ) ) ) ).
% map_prod_def
thf(fact_7084_prod__fun__imageE,axiom,
! [B: $tType,A: $tType,D: $tType,C: $tType,C3: product_prod @ A @ B,F3: C > A,G3: D > B,R: set @ ( product_prod @ C @ D )] :
( ( member @ ( product_prod @ A @ B ) @ C3 @ ( image2 @ ( product_prod @ C @ D ) @ ( product_prod @ A @ B ) @ ( product_map_prod @ C @ A @ D @ B @ F3 @ G3 ) @ R ) )
=> ~ ! [X4: C,Y3: D] :
( ( C3
= ( product_Pair @ A @ B @ ( F3 @ X4 ) @ ( G3 @ Y3 ) ) )
=> ~ ( member @ ( product_prod @ C @ D ) @ ( product_Pair @ C @ D @ X4 @ Y3 ) @ R ) ) ) ).
% prod_fun_imageE
thf(fact_7085_map__prod_Ocomp,axiom,
! [A: $tType,C: $tType,E: $tType,F: $tType,D: $tType,B: $tType,F3: C > E,G3: D > F,H2: A > C,I2: B > D] :
( ( comp @ ( product_prod @ C @ D ) @ ( product_prod @ E @ F ) @ ( product_prod @ A @ B ) @ ( product_map_prod @ C @ E @ D @ F @ F3 @ G3 ) @ ( product_map_prod @ A @ C @ B @ D @ H2 @ I2 ) )
= ( product_map_prod @ A @ E @ B @ F @ ( comp @ C @ E @ A @ F3 @ H2 ) @ ( comp @ D @ F @ B @ G3 @ I2 ) ) ) ).
% map_prod.comp
thf(fact_7086_map__prod_Ocompositionality,axiom,
! [D: $tType,F: $tType,E: $tType,C: $tType,B: $tType,A: $tType,F3: C > E,G3: D > F,H2: A > C,I2: B > D,Prod: product_prod @ A @ B] :
( ( product_map_prod @ C @ E @ D @ F @ F3 @ G3 @ ( product_map_prod @ A @ C @ B @ D @ H2 @ I2 @ Prod ) )
= ( product_map_prod @ A @ E @ B @ F @ ( comp @ C @ E @ A @ F3 @ H2 ) @ ( comp @ D @ F @ B @ G3 @ I2 ) @ Prod ) ) ).
% map_prod.compositionality
thf(fact_7087_map__prod__compose,axiom,
! [D: $tType,C: $tType,A: $tType,E: $tType,F: $tType,B: $tType,F1: E > C,F22: A > E,G1: F > D,G22: B > F] :
( ( product_map_prod @ A @ C @ B @ D @ ( comp @ E @ C @ A @ F1 @ F22 ) @ ( comp @ F @ D @ B @ G1 @ G22 ) )
= ( comp @ ( product_prod @ E @ F ) @ ( product_prod @ C @ D ) @ ( product_prod @ A @ B ) @ ( product_map_prod @ E @ C @ F @ D @ F1 @ G1 ) @ ( product_map_prod @ A @ E @ B @ F @ F22 @ G22 ) ) ) ).
% map_prod_compose
thf(fact_7088_map__prod_Oidentity,axiom,
! [B: $tType,A: $tType] :
( ( product_map_prod @ A @ A @ B @ B
@ ^ [X5: A] : X5
@ ^ [X5: B] : X5 )
= ( id @ ( product_prod @ A @ B ) ) ) ).
% map_prod.identity
thf(fact_7089_apfst__def,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( product_apfst @ A @ C @ B )
= ( ^ [F4: A > C] : ( product_map_prod @ A @ C @ B @ B @ F4 @ ( id @ B ) ) ) ) ).
% apfst_def
thf(fact_7090_apsnd__def,axiom,
! [B: $tType,C: $tType,A: $tType] :
( ( product_apsnd @ B @ C @ A )
= ( product_map_prod @ A @ A @ B @ C @ ( id @ A ) ) ) ).
% apsnd_def
thf(fact_7091_sum__list__nonpos,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [Xs: list @ A] :
( ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
=> ( ord_less_eq @ A @ X4 @ ( zero_zero @ A ) ) )
=> ( ord_less_eq @ A @ ( groups8242544230860333062m_list @ A @ Xs ) @ ( zero_zero @ A ) ) ) ) ).
% sum_list_nonpos
thf(fact_7092_sum__list__nonneg__eq__0__iff,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [Xs: list @ A] :
( ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ X4 ) )
=> ( ( ( groups8242544230860333062m_list @ A @ Xs )
= ( zero_zero @ A ) )
= ( ! [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Xs ) )
=> ( X5
= ( zero_zero @ A ) ) ) ) ) ) ) ).
% sum_list_nonneg_eq_0_iff
thf(fact_7093_Groups__List_Osum__list__nonneg,axiom,
! [A: $tType] :
( ( ordere6911136660526730532id_add @ A )
=> ! [Xs: list @ A] :
( ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ X4 ) )
=> ( ord_less_eq @ A @ ( zero_zero @ A ) @ ( groups8242544230860333062m_list @ A @ Xs ) ) ) ) ).
% Groups_List.sum_list_nonneg
thf(fact_7094_sum__list__abs,axiom,
! [A: $tType] :
( ( ordere166539214618696060dd_abs @ A )
=> ! [Xs: list @ A] : ( ord_less_eq @ A @ ( abs_abs @ A @ ( groups8242544230860333062m_list @ A @ Xs ) ) @ ( groups8242544230860333062m_list @ A @ ( map @ A @ A @ ( abs_abs @ A ) @ Xs ) ) ) ) ).
% sum_list_abs
thf(fact_7095_map__prod__surj,axiom,
! [A: $tType,C: $tType,D: $tType,B: $tType,F3: A > B,G3: C > D] :
( ( ( image2 @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ B ) ) )
=> ( ( ( image2 @ C @ D @ G3 @ ( top_top @ ( set @ C ) ) )
= ( top_top @ ( set @ D ) ) )
=> ( ( image2 @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) @ ( product_map_prod @ A @ B @ C @ D @ F3 @ G3 ) @ ( top_top @ ( set @ ( product_prod @ A @ C ) ) ) )
= ( top_top @ ( set @ ( product_prod @ B @ D ) ) ) ) ) ) ).
% map_prod_surj
thf(fact_7096_sum__list__mono,axiom,
! [B: $tType,A: $tType] :
( ( ( monoid_add @ B )
& ( ordere6658533253407199908up_add @ B ) )
=> ! [Xs: list @ A,F3: A > B,G3: A > B] :
( ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
=> ( ord_less_eq @ B @ ( F3 @ X4 ) @ ( G3 @ X4 ) ) )
=> ( ord_less_eq @ B @ ( groups8242544230860333062m_list @ B @ ( map @ A @ B @ F3 @ Xs ) ) @ ( groups8242544230860333062m_list @ B @ ( map @ A @ B @ G3 @ Xs ) ) ) ) ) ).
% sum_list_mono
thf(fact_7097_distinct__sum__list__conv__Sum,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [Xs: list @ A] :
( ( distinct @ A @ Xs )
=> ( ( groups8242544230860333062m_list @ A @ Xs )
= ( groups7311177749621191930dd_sum @ A @ A
@ ^ [X5: A] : X5
@ ( set2 @ A @ Xs ) ) ) ) ) ).
% distinct_sum_list_conv_Sum
thf(fact_7098_sum__list__strict__mono,axiom,
! [B: $tType,A: $tType] :
( ( ( monoid_add @ B )
& ( strict9044650504122735259up_add @ B ) )
=> ! [Xs: list @ A,F3: A > B,G3: A > B] :
( ( Xs
!= ( nil @ A ) )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
=> ( ord_less @ B @ ( F3 @ X4 ) @ ( G3 @ X4 ) ) )
=> ( ord_less @ B @ ( groups8242544230860333062m_list @ B @ ( map @ A @ B @ F3 @ Xs ) ) @ ( groups8242544230860333062m_list @ B @ ( map @ A @ B @ G3 @ Xs ) ) ) ) ) ) ).
% sum_list_strict_mono
thf(fact_7099_elem__le__sum__list,axiom,
! [A: $tType] :
( ( canoni5634975068530333245id_add @ A )
=> ! [K2: nat,Ns: list @ A] :
( ( ord_less @ nat @ K2 @ ( size_size @ ( list @ A ) @ Ns ) )
=> ( ord_less_eq @ A @ ( nth @ A @ Ns @ K2 ) @ ( groups8242544230860333062m_list @ A @ Ns ) ) ) ) ).
% elem_le_sum_list
thf(fact_7100_sum_Odistinct__set__conv__list,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [Xs: list @ B,G3: B > A] :
( ( distinct @ B @ Xs )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G3 @ ( set2 @ B @ Xs ) )
= ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ G3 @ Xs ) ) ) ) ) ).
% sum.distinct_set_conv_list
thf(fact_7101_sum__list__distinct__conv__sum__set,axiom,
! [C: $tType,B: $tType] :
( ( comm_monoid_add @ C )
=> ! [Xs: list @ B,F3: B > C] :
( ( distinct @ B @ Xs )
=> ( ( groups8242544230860333062m_list @ C @ ( map @ B @ C @ F3 @ Xs ) )
= ( groups7311177749621191930dd_sum @ B @ C @ F3 @ ( set2 @ B @ Xs ) ) ) ) ) ).
% sum_list_distinct_conv_sum_set
thf(fact_7102_sum__list__map__remove1,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [X: B,Xs: list @ B,F3: B > A] :
( ( member @ B @ X @ ( set2 @ B @ Xs ) )
=> ( ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ F3 @ Xs ) )
= ( plus_plus @ A @ ( F3 @ X ) @ ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ F3 @ ( remove1 @ B @ X @ Xs ) ) ) ) ) ) ) ).
% sum_list_map_remove1
thf(fact_7103_sum__code,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [G3: B > A,Xs: list @ B] :
( ( groups7311177749621191930dd_sum @ B @ A @ G3 @ ( set2 @ B @ Xs ) )
= ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ G3 @ ( remdups @ B @ Xs ) ) ) ) ) ).
% sum_code
thf(fact_7104_sum__set__upto__conv__sum__list__int,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [F3: int > A,I2: int,J: int] :
( ( groups7311177749621191930dd_sum @ int @ A @ F3 @ ( set2 @ int @ ( upto @ I2 @ J ) ) )
= ( groups8242544230860333062m_list @ A @ ( map @ int @ A @ F3 @ ( upto @ I2 @ J ) ) ) ) ) ).
% sum_set_upto_conv_sum_list_int
thf(fact_7105_interv__sum__list__conv__sum__set__int,axiom,
! [B: $tType] :
( ( comm_monoid_add @ B )
=> ! [F3: int > B,K2: int,L: int] :
( ( groups8242544230860333062m_list @ B @ ( map @ int @ B @ F3 @ ( upto @ K2 @ L ) ) )
= ( groups7311177749621191930dd_sum @ int @ B @ F3 @ ( set2 @ int @ ( upto @ K2 @ L ) ) ) ) ) ).
% interv_sum_list_conv_sum_set_int
thf(fact_7106_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_7107_sum__list__Suc,axiom,
! [A: $tType,F3: A > nat,Xs: list @ A] :
( ( groups8242544230860333062m_list @ nat
@ ( map @ A @ nat
@ ^ [X5: A] : ( suc @ ( F3 @ X5 ) )
@ Xs ) )
= ( plus_plus @ nat @ ( groups8242544230860333062m_list @ nat @ ( map @ A @ nat @ F3 @ Xs ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ).
% sum_list_Suc
thf(fact_7108_interv__sum__list__conv__sum__set__nat,axiom,
! [B: $tType] :
( ( comm_monoid_add @ B )
=> ! [F3: nat > B,M2: nat,N: nat] :
( ( groups8242544230860333062m_list @ B @ ( map @ nat @ B @ F3 @ ( upt @ M2 @ N ) ) )
= ( groups7311177749621191930dd_sum @ nat @ B @ F3 @ ( set2 @ nat @ ( upt @ M2 @ N ) ) ) ) ) ).
% interv_sum_list_conv_sum_set_nat
thf(fact_7109_sum__set__upt__conv__sum__list__nat,axiom,
! [A: $tType] :
( ( comm_monoid_add @ A )
=> ! [F3: nat > A,M2: nat,N: nat] :
( ( groups7311177749621191930dd_sum @ nat @ A @ F3 @ ( set2 @ nat @ ( upt @ M2 @ N ) ) )
= ( groups8242544230860333062m_list @ A @ ( map @ nat @ A @ F3 @ ( upt @ M2 @ N ) ) ) ) ) ).
% sum_set_upt_conv_sum_list_nat
thf(fact_7110_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_7111_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_7112_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_7113_sum__list__map__eq__sum__count,axiom,
! [A: $tType,F3: A > nat,Xs: list @ A] :
( ( groups8242544230860333062m_list @ nat @ ( map @ A @ nat @ F3 @ Xs ) )
= ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [X5: A] : ( times_times @ nat @ ( count_list @ A @ Xs @ X5 ) @ ( F3 @ X5 ) )
@ ( set2 @ A @ Xs ) ) ) ).
% sum_list_map_eq_sum_count
thf(fact_7114_sum__list__update,axiom,
! [A: $tType] :
( ( ordere1170586879665033532d_diff @ A )
=> ! [K2: nat,Xs: list @ A,X: A] :
( ( ord_less @ nat @ K2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( groups8242544230860333062m_list @ A @ ( list_update @ A @ Xs @ K2 @ X ) )
= ( minus_minus @ A @ ( plus_plus @ A @ ( groups8242544230860333062m_list @ A @ Xs ) @ X ) @ ( nth @ A @ Xs @ K2 ) ) ) ) ) ).
% sum_list_update
thf(fact_7115_sorted__wrt__less__sum__mono__lowerbound,axiom,
! [B: $tType] :
( ( ordere6911136660526730532id_add @ B )
=> ! [F3: nat > B,Ns: list @ nat] :
( ! [X4: nat,Y3: nat] :
( ( ord_less_eq @ nat @ X4 @ Y3 )
=> ( ord_less_eq @ B @ ( F3 @ X4 ) @ ( F3 @ Y3 ) ) )
=> ( ( sorted_wrt @ nat @ ( ord_less @ nat ) @ Ns )
=> ( ord_less_eq @ B @ ( groups7311177749621191930dd_sum @ nat @ B @ F3 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( size_size @ ( list @ nat ) @ Ns ) ) ) @ ( groups8242544230860333062m_list @ B @ ( map @ nat @ B @ F3 @ Ns ) ) ) ) ) ) ).
% sorted_wrt_less_sum_mono_lowerbound
thf(fact_7116_nth__transpose,axiom,
! [A: $tType,I2: nat,Xs: list @ ( list @ A )] :
( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ ( list @ A ) ) @ ( transpose @ A @ Xs ) ) )
=> ( ( nth @ ( list @ A ) @ ( transpose @ A @ Xs ) @ I2 )
= ( map @ ( list @ A ) @ A
@ ^ [Xs3: list @ A] : ( nth @ A @ Xs3 @ I2 )
@ ( filter2 @ ( list @ A )
@ ^ [Ys3: list @ A] : ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Ys3 ) )
@ Xs ) ) ) ) ).
% nth_transpose
thf(fact_7117_filter__filter,axiom,
! [A: $tType,P2: A > $o,Q: A > $o,Xs: list @ A] :
( ( filter2 @ A @ P2 @ ( filter2 @ A @ Q @ Xs ) )
= ( filter2 @ A
@ ^ [X5: A] :
( ( Q @ X5 )
& ( P2 @ X5 ) )
@ Xs ) ) ).
% filter_filter
thf(fact_7118_filter__True,axiom,
! [A: $tType,Xs: list @ A,P2: A > $o] :
( ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
=> ( P2 @ X4 ) )
=> ( ( filter2 @ A @ P2 @ Xs )
= Xs ) ) ).
% filter_True
thf(fact_7119_filter__append,axiom,
! [A: $tType,P2: A > $o,Xs: list @ A,Ys: list @ A] :
( ( filter2 @ A @ P2 @ ( append @ A @ Xs @ Ys ) )
= ( append @ A @ ( filter2 @ A @ P2 @ Xs ) @ ( filter2 @ A @ P2 @ Ys ) ) ) ).
% filter_append
thf(fact_7120_remove1__filter__not,axiom,
! [A: $tType,P2: A > $o,X: A,Xs: list @ A] :
( ~ ( P2 @ X )
=> ( ( remove1 @ A @ X @ ( filter2 @ A @ P2 @ Xs ) )
= ( filter2 @ A @ P2 @ Xs ) ) ) ).
% remove1_filter_not
thf(fact_7121_removeAll__filter__not,axiom,
! [A: $tType,P2: A > $o,X: A,Xs: list @ A] :
( ~ ( P2 @ X )
=> ( ( removeAll @ A @ X @ ( filter2 @ A @ P2 @ Xs ) )
= ( filter2 @ A @ P2 @ Xs ) ) ) ).
% removeAll_filter_not
thf(fact_7122_set__filter,axiom,
! [A: $tType,P2: A > $o,Xs: list @ A] :
( ( set2 @ A @ ( filter2 @ A @ P2 @ Xs ) )
= ( collect @ A
@ ^ [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Xs ) )
& ( P2 @ X5 ) ) ) ) ).
% set_filter
thf(fact_7123_filter__False,axiom,
! [A: $tType,Xs: list @ A,P2: A > $o] :
( ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
=> ~ ( P2 @ X4 ) )
=> ( ( filter2 @ A @ P2 @ Xs )
= ( nil @ A ) ) ) ).
% filter_False
thf(fact_7124_length__filter__map,axiom,
! [A: $tType,B: $tType,P2: A > $o,F3: B > A,Xs: list @ B] :
( ( size_size @ ( list @ A ) @ ( filter2 @ A @ P2 @ ( map @ B @ A @ F3 @ Xs ) ) )
= ( size_size @ ( list @ B ) @ ( filter2 @ B @ ( comp @ A @ $o @ B @ P2 @ F3 ) @ Xs ) ) ) ).
% length_filter_map
thf(fact_7125_filter__insort,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F3: B > A,Xs: list @ B,P2: B > $o,X: B] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F3 @ Xs ) )
=> ( ( P2 @ X )
=> ( ( filter2 @ B @ P2 @ ( linorder_insort_key @ B @ A @ F3 @ X @ Xs ) )
= ( linorder_insort_key @ B @ A @ F3 @ X @ ( filter2 @ B @ P2 @ Xs ) ) ) ) ) ) ).
% filter_insort
thf(fact_7126_sorted__map,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F3: B > A,Xs: list @ B] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F3 @ Xs ) )
= ( sorted_wrt @ B
@ ^ [X5: B,Y6: B] : ( ord_less_eq @ A @ ( F3 @ X5 ) @ ( F3 @ Y6 ) )
@ Xs ) ) ) ).
% sorted_map
thf(fact_7127_sorted__map__same,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F3: B > A,G3: ( list @ B ) > A,Xs: list @ B] :
( sorted_wrt @ A @ ( ord_less_eq @ A )
@ ( map @ B @ A @ F3
@ ( filter2 @ B
@ ^ [X5: B] :
( ( F3 @ X5 )
= ( G3 @ Xs ) )
@ Xs ) ) ) ) ).
% sorted_map_same
thf(fact_7128_sorted__filter,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F3: B > A,Xs: list @ B,P2: B > $o] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F3 @ Xs ) )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F3 @ ( filter2 @ B @ P2 @ Xs ) ) ) ) ) ).
% sorted_filter
thf(fact_7129_filter__map,axiom,
! [A: $tType,B: $tType,P2: A > $o,F3: B > A,Xs: list @ B] :
( ( filter2 @ A @ P2 @ ( map @ B @ A @ F3 @ Xs ) )
= ( map @ B @ A @ F3 @ ( filter2 @ B @ ( comp @ A @ $o @ B @ P2 @ F3 ) @ Xs ) ) ) ).
% filter_map
thf(fact_7130_filter__concat,axiom,
! [A: $tType,P: A > $o,Xs: list @ ( list @ A )] :
( ( filter2 @ A @ P @ ( concat @ A @ Xs ) )
= ( concat @ A @ ( map @ ( list @ A ) @ ( list @ A ) @ ( filter2 @ A @ P ) @ Xs ) ) ) ).
% filter_concat
thf(fact_7131_sorted__wrt__map,axiom,
! [A: $tType,B: $tType,R: A > A > $o,F3: B > A,Xs: list @ B] :
( ( sorted_wrt @ A @ R @ ( map @ B @ A @ F3 @ Xs ) )
= ( sorted_wrt @ B
@ ^ [X5: B,Y6: B] : ( R @ ( F3 @ X5 ) @ ( F3 @ Y6 ) )
@ Xs ) ) ).
% sorted_wrt_map
thf(fact_7132_distinct__map__filter,axiom,
! [A: $tType,B: $tType,F3: B > A,Xs: list @ B,P2: B > $o] :
( ( distinct @ A @ ( map @ B @ A @ F3 @ Xs ) )
=> ( distinct @ A @ ( map @ B @ A @ F3 @ ( filter2 @ B @ P2 @ Xs ) ) ) ) ).
% distinct_map_filter
thf(fact_7133_sorted__insort__key,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F3: B > A,X: B,Xs: list @ B] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F3 @ ( linorder_insort_key @ B @ A @ F3 @ X @ Xs ) ) )
= ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F3 @ Xs ) ) ) ) ).
% sorted_insort_key
thf(fact_7134_inter__set__filter,axiom,
! [A: $tType,A5: set @ A,Xs: list @ A] :
( ( inf_inf @ ( set @ A ) @ A5 @ ( set2 @ A @ Xs ) )
= ( set2 @ A
@ ( filter2 @ A
@ ^ [X5: A] : ( member @ A @ X5 @ A5 )
@ Xs ) ) ) ).
% inter_set_filter
thf(fact_7135_sorted__map__remove1,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F3: B > A,Xs: list @ B,X: B] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F3 @ Xs ) )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F3 @ ( remove1 @ B @ X @ Xs ) ) ) ) ) ).
% sorted_map_remove1
thf(fact_7136_filter__shuffles,axiom,
! [A: $tType,P2: A > $o,Xs: list @ A,Ys: list @ A] :
( ( image2 @ ( list @ A ) @ ( list @ A ) @ ( filter2 @ A @ P2 ) @ ( shuffles @ A @ Xs @ Ys ) )
= ( shuffles @ A @ ( filter2 @ A @ P2 @ Xs ) @ ( filter2 @ A @ P2 @ Ys ) ) ) ).
% filter_shuffles
thf(fact_7137_sorted__upt,axiom,
! [M2: nat,N: nat] : ( sorted_wrt @ nat @ ( ord_less_eq @ nat ) @ ( upt @ M2 @ N ) ) ).
% sorted_upt
thf(fact_7138_sorted__wrt__upt,axiom,
! [M2: nat,N: nat] : ( sorted_wrt @ nat @ ( ord_less @ nat ) @ ( upt @ M2 @ N ) ) ).
% sorted_wrt_upt
thf(fact_7139_filter__insort__triv,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [P2: B > $o,X: B,F3: B > A,Xs: list @ B] :
( ~ ( P2 @ X )
=> ( ( filter2 @ B @ P2 @ ( linorder_insort_key @ B @ A @ F3 @ X @ Xs ) )
= ( filter2 @ B @ P2 @ Xs ) ) ) ) ).
% filter_insort_triv
thf(fact_7140_removeAll__filter__not__eq,axiom,
! [A: $tType] :
( ( removeAll @ A )
= ( ^ [X5: A] :
( filter2 @ A
@ ^ [Y6: A] : X5 != Y6 ) ) ) ).
% removeAll_filter_not_eq
thf(fact_7141_filter__remove1,axiom,
! [A: $tType,Q: A > $o,X: A,Xs: list @ A] :
( ( filter2 @ A @ Q @ ( remove1 @ A @ X @ Xs ) )
= ( remove1 @ A @ X @ ( filter2 @ A @ Q @ Xs ) ) ) ).
% filter_remove1
thf(fact_7142_sorted__wrt__filter,axiom,
! [A: $tType,F3: A > A > $o,Xs: list @ A,P2: A > $o] :
( ( sorted_wrt @ A @ F3 @ Xs )
=> ( sorted_wrt @ A @ F3 @ ( filter2 @ A @ P2 @ Xs ) ) ) ).
% sorted_wrt_filter
thf(fact_7143_sorted__wrt__true,axiom,
! [A: $tType,Xs: list @ A] :
( sorted_wrt @ A
@ ^ [Uu3: A,Uv3: A] : $true
@ Xs ) ).
% sorted_wrt_true
thf(fact_7144_partition__in__shuffles,axiom,
! [A: $tType,Xs: list @ A,P2: A > $o] :
( member @ ( list @ A ) @ Xs
@ ( shuffles @ A @ ( filter2 @ A @ P2 @ Xs )
@ ( filter2 @ A
@ ^ [X5: A] :
~ ( P2 @ X5 )
@ Xs ) ) ) ).
% partition_in_shuffles
thf(fact_7145_sorted__list__of__set_Ostrict__sorted__key__list__of__set,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A] : ( sorted_wrt @ A @ ( ord_less @ A ) @ ( linord4507533701916653071of_set @ A @ A5 ) ) ) ).
% sorted_list_of_set.strict_sorted_key_list_of_set
thf(fact_7146_sorted__replicate,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [N: nat,X: A] : ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( replicate @ A @ N @ X ) ) ) ).
% sorted_replicate
thf(fact_7147_sorted__nths,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,I6: set @ nat] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( nths @ A @ Xs @ I6 ) ) ) ) ).
% sorted_nths
thf(fact_7148_sorted__insort,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Xs: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A )
@ ( linorder_insort_key @ A @ A
@ ^ [X5: A] : X5
@ X
@ Xs ) )
= ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs ) ) ) ).
% sorted_insort
thf(fact_7149_sorted__same,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [G3: ( list @ A ) > A,Xs: list @ A] :
( sorted_wrt @ A @ ( ord_less_eq @ A )
@ ( filter2 @ A
@ ^ [X5: A] :
( X5
= ( G3 @ Xs ) )
@ Xs ) ) ) ).
% sorted_same
thf(fact_7150_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_7151_sorted__remove1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,A2: A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( remove1 @ A @ A2 @ Xs ) ) ) ) ).
% sorted_remove1
thf(fact_7152_sorted__list__of__set_Osorted__sorted__key__list__of__set,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A] : ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( linord4507533701916653071of_set @ A @ A5 ) ) ) ).
% sorted_list_of_set.sorted_sorted_key_list_of_set
thf(fact_7153_sorted__remdups__adj,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( remdups_adj @ A @ Xs ) ) ) ) ).
% sorted_remdups_adj
thf(fact_7154_sorted2,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Y: A,Zs: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( cons @ A @ X @ ( cons @ A @ Y @ Zs ) ) )
= ( ( ord_less_eq @ A @ X @ Y )
& ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( cons @ A @ Y @ Zs ) ) ) ) ) ).
% sorted2
thf(fact_7155_filter_Osimps_I2_J,axiom,
! [A: $tType,P2: A > $o,X: A,Xs: list @ A] :
( ( ( P2 @ X )
=> ( ( filter2 @ A @ P2 @ ( cons @ A @ X @ Xs ) )
= ( cons @ A @ X @ ( filter2 @ A @ P2 @ Xs ) ) ) )
& ( ~ ( P2 @ X )
=> ( ( filter2 @ A @ P2 @ ( cons @ A @ X @ Xs ) )
= ( filter2 @ A @ P2 @ Xs ) ) ) ) ).
% filter.simps(2)
thf(fact_7156_sorted1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A] : ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( cons @ A @ X @ ( nil @ A ) ) ) ) ).
% sorted1
thf(fact_7157_sorted__wrt1,axiom,
! [A: $tType,P2: A > A > $o,X: A] : ( sorted_wrt @ A @ P2 @ ( cons @ A @ X @ ( nil @ A ) ) ) ).
% sorted_wrt1
thf(fact_7158_sorted__wrt_Osimps_I1_J,axiom,
! [A: $tType,P2: A > A > $o] : ( sorted_wrt @ A @ P2 @ ( nil @ A ) ) ).
% sorted_wrt.simps(1)
thf(fact_7159_filter_Osimps_I1_J,axiom,
! [A: $tType,P2: A > $o] :
( ( filter2 @ A @ P2 @ ( nil @ A ) )
= ( nil @ A ) ) ).
% filter.simps(1)
thf(fact_7160_strict__sorted__simps_I1_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( sorted_wrt @ A @ ( ord_less @ A ) @ ( nil @ A ) ) ) ).
% strict_sorted_simps(1)
thf(fact_7161_filter__replicate,axiom,
! [A: $tType,P2: A > $o,X: A,N: nat] :
( ( ( P2 @ X )
=> ( ( filter2 @ A @ P2 @ ( replicate @ A @ N @ X ) )
= ( replicate @ A @ N @ X ) ) )
& ( ~ ( P2 @ X )
=> ( ( filter2 @ A @ P2 @ ( replicate @ A @ N @ X ) )
= ( nil @ A ) ) ) ) ).
% filter_replicate
thf(fact_7162_sorted0,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( nil @ A ) ) ) ).
% sorted0
thf(fact_7163_sorted__remdups,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( remdups @ A @ Xs ) ) ) ) ).
% sorted_remdups
thf(fact_7164_sorted__take,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,N: nat] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( take @ A @ N @ Xs ) ) ) ) ).
% sorted_take
thf(fact_7165_sorted__wrt__take,axiom,
! [A: $tType,F3: A > A > $o,Xs: list @ A,N: nat] :
( ( sorted_wrt @ A @ F3 @ Xs )
=> ( sorted_wrt @ A @ F3 @ ( take @ A @ N @ Xs ) ) ) ).
% sorted_wrt_take
thf(fact_7166_remdups__filter,axiom,
! [A: $tType,P2: A > $o,Xs: list @ A] :
( ( remdups @ A @ ( filter2 @ A @ P2 @ Xs ) )
= ( filter2 @ A @ P2 @ ( remdups @ A @ Xs ) ) ) ).
% remdups_filter
thf(fact_7167_sorted__drop,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,N: nat] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( drop @ A @ N @ Xs ) ) ) ) ).
% sorted_drop
thf(fact_7168_sorted__wrt__drop,axiom,
! [A: $tType,F3: A > A > $o,Xs: list @ A,N: nat] :
( ( sorted_wrt @ A @ F3 @ Xs )
=> ( sorted_wrt @ A @ F3 @ ( drop @ A @ N @ Xs ) ) ) ).
% sorted_wrt_drop
thf(fact_7169_sum__length__filter__compl,axiom,
! [A: $tType,P2: A > $o,Xs: list @ A] :
( ( plus_plus @ nat @ ( size_size @ ( list @ A ) @ ( filter2 @ A @ P2 @ Xs ) )
@ ( size_size @ ( list @ A )
@ ( filter2 @ A
@ ^ [X5: A] :
~ ( P2 @ X5 )
@ Xs ) ) )
= ( size_size @ ( list @ A ) @ Xs ) ) ).
% sum_length_filter_compl
thf(fact_7170_replicate__length__filter,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( replicate @ A
@ ( size_size @ ( list @ A )
@ ( filter2 @ A
@ ( ^ [Y5: A,Z2: A] : Y5 = Z2
@ X )
@ Xs ) )
@ X )
= ( filter2 @ A
@ ( ^ [Y5: A,Z2: A] : Y5 = Z2
@ X )
@ Xs ) ) ).
% replicate_length_filter
thf(fact_7171_strict__sorted__equal,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,Ys: list @ A] :
( ( sorted_wrt @ A @ ( ord_less @ A ) @ Xs )
=> ( ( sorted_wrt @ A @ ( ord_less @ A ) @ Ys )
=> ( ( ( set2 @ A @ Ys )
= ( set2 @ A @ Xs ) )
=> ( Ys = Xs ) ) ) ) ) ).
% strict_sorted_equal
thf(fact_7172_filter__cong,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,P2: A > $o,Q: A > $o] :
( ( Xs = Ys )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Ys ) )
=> ( ( P2 @ X4 )
= ( Q @ X4 ) ) )
=> ( ( filter2 @ A @ P2 @ Xs )
= ( filter2 @ A @ Q @ Ys ) ) ) ) ).
% filter_cong
thf(fact_7173_filter__id__conv,axiom,
! [A: $tType,P2: A > $o,Xs: list @ A] :
( ( ( filter2 @ A @ P2 @ Xs )
= Xs )
= ( ! [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Xs ) )
=> ( P2 @ X5 ) ) ) ) ).
% filter_id_conv
thf(fact_7174_sorted__wrt__mono__rel,axiom,
! [A: $tType,Xs: list @ A,P2: A > A > $o,Q: A > A > $o] :
( ! [X4: A,Y3: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
=> ( ( member @ A @ Y3 @ ( set2 @ A @ Xs ) )
=> ( ( P2 @ X4 @ Y3 )
=> ( Q @ X4 @ Y3 ) ) ) )
=> ( ( sorted_wrt @ A @ P2 @ Xs )
=> ( sorted_wrt @ A @ Q @ Xs ) ) ) ).
% sorted_wrt_mono_rel
thf(fact_7175_filter__set,axiom,
! [A: $tType,P2: A > $o,Xs: list @ A] :
( ( filter3 @ A @ P2 @ ( set2 @ A @ Xs ) )
= ( set2 @ A @ ( filter2 @ A @ P2 @ Xs ) ) ) ).
% filter_set
thf(fact_7176_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_7177_length__filter__less,axiom,
! [A: $tType,X: A,Xs: list @ A,P2: A > $o] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ~ ( P2 @ X )
=> ( ord_less @ nat @ ( size_size @ ( list @ A ) @ ( filter2 @ A @ P2 @ Xs ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ).
% length_filter_less
thf(fact_7178_filter__is__subset,axiom,
! [A: $tType,P2: A > $o,Xs: list @ A] : ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ ( filter2 @ A @ P2 @ Xs ) ) @ ( set2 @ A @ Xs ) ) ).
% filter_is_subset
thf(fact_7179_distinct__filter,axiom,
! [A: $tType,Xs: list @ A,P2: A > $o] :
( ( distinct @ A @ Xs )
=> ( distinct @ A @ ( filter2 @ A @ P2 @ Xs ) ) ) ).
% distinct_filter
thf(fact_7180_length__filter__le,axiom,
! [A: $tType,P2: A > $o,Xs: list @ A] : ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ ( filter2 @ A @ P2 @ Xs ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) ).
% length_filter_le
thf(fact_7181_sorted__wrt01,axiom,
! [A: $tType,Xs: list @ A,P2: A > A > $o] :
( ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) )
=> ( sorted_wrt @ A @ P2 @ Xs ) ) ).
% sorted_wrt01
thf(fact_7182_sorted__wrt__nth__less,axiom,
! [A: $tType,P2: A > A > $o,Xs: list @ A,I2: nat,J: nat] :
( ( sorted_wrt @ A @ P2 @ Xs )
=> ( ( ord_less @ nat @ I2 @ J )
=> ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( P2 @ ( nth @ A @ Xs @ I2 ) @ ( nth @ A @ Xs @ J ) ) ) ) ) ).
% sorted_wrt_nth_less
thf(fact_7183_sorted__wrt__iff__nth__less,axiom,
! [A: $tType] :
( ( sorted_wrt @ A )
= ( ^ [P4: A > A > $o,Xs3: list @ A] :
! [I4: nat,J3: nat] :
( ( ord_less @ nat @ I4 @ J3 )
=> ( ( ord_less @ nat @ J3 @ ( size_size @ ( list @ A ) @ Xs3 ) )
=> ( P4 @ ( nth @ A @ Xs3 @ I4 ) @ ( nth @ A @ Xs3 @ J3 ) ) ) ) ) ) ).
% sorted_wrt_iff_nth_less
thf(fact_7184_sorted__distinct__set__unique,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,Ys: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( ( distinct @ A @ Xs )
=> ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Ys )
=> ( ( distinct @ A @ Ys )
=> ( ( ( set2 @ A @ Xs )
= ( set2 @ A @ Ys ) )
=> ( Xs = Ys ) ) ) ) ) ) ) ).
% sorted_distinct_set_unique
thf(fact_7185_sorted__wrt__append,axiom,
! [A: $tType,P2: A > A > $o,Xs: list @ A,Ys: list @ A] :
( ( sorted_wrt @ A @ P2 @ ( append @ A @ Xs @ Ys ) )
= ( ( sorted_wrt @ A @ P2 @ Xs )
& ( sorted_wrt @ A @ P2 @ Ys )
& ! [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Xs ) )
=> ! [Y6: A] :
( ( member @ A @ Y6 @ ( set2 @ A @ Ys ) )
=> ( P2 @ X5 @ Y6 ) ) ) ) ) ).
% sorted_wrt_append
thf(fact_7186_sorted__append,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,Ys: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( append @ A @ Xs @ Ys ) )
= ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
& ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Ys )
& ! [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Xs ) )
=> ! [Y6: A] :
( ( member @ A @ Y6 @ ( set2 @ A @ Ys ) )
=> ( ord_less_eq @ A @ X5 @ Y6 ) ) ) ) ) ) ).
% sorted_append
thf(fact_7187_Cons__eq__filterD,axiom,
! [A: $tType,X: A,Xs: list @ A,P2: A > $o,Ys: list @ A] :
( ( ( cons @ A @ X @ Xs )
= ( filter2 @ A @ P2 @ Ys ) )
=> ? [Us3: list @ A,Vs3: list @ A] :
( ( Ys
= ( append @ A @ Us3 @ ( cons @ A @ X @ Vs3 ) ) )
& ! [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Us3 ) )
=> ~ ( P2 @ X3 ) )
& ( P2 @ X )
& ( Xs
= ( filter2 @ A @ P2 @ Vs3 ) ) ) ) ).
% Cons_eq_filterD
thf(fact_7188_filter__eq__ConsD,axiom,
! [A: $tType,P2: A > $o,Ys: list @ A,X: A,Xs: list @ A] :
( ( ( filter2 @ A @ P2 @ Ys )
= ( cons @ A @ X @ Xs ) )
=> ? [Us3: list @ A,Vs3: list @ A] :
( ( Ys
= ( append @ A @ Us3 @ ( cons @ A @ X @ Vs3 ) ) )
& ! [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Us3 ) )
=> ~ ( P2 @ X3 ) )
& ( P2 @ X )
& ( Xs
= ( filter2 @ A @ P2 @ Vs3 ) ) ) ) ).
% filter_eq_ConsD
thf(fact_7189_Cons__eq__filter__iff,axiom,
! [A: $tType,X: A,Xs: list @ A,P2: A > $o,Ys: list @ A] :
( ( ( cons @ A @ X @ Xs )
= ( filter2 @ A @ P2 @ Ys ) )
= ( ? [Us2: list @ A,Vs: list @ A] :
( ( Ys
= ( append @ A @ Us2 @ ( cons @ A @ X @ Vs ) ) )
& ! [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Us2 ) )
=> ~ ( P2 @ X5 ) )
& ( P2 @ X )
& ( Xs
= ( filter2 @ A @ P2 @ Vs ) ) ) ) ) ).
% Cons_eq_filter_iff
thf(fact_7190_filter__eq__Cons__iff,axiom,
! [A: $tType,P2: A > $o,Ys: list @ A,X: A,Xs: list @ A] :
( ( ( filter2 @ A @ P2 @ Ys )
= ( cons @ A @ X @ Xs ) )
= ( ? [Us2: list @ A,Vs: list @ A] :
( ( Ys
= ( append @ A @ Us2 @ ( cons @ A @ X @ Vs ) ) )
& ! [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Us2 ) )
=> ~ ( P2 @ X5 ) )
& ( P2 @ X )
& ( Xs
= ( filter2 @ A @ P2 @ Vs ) ) ) ) ) ).
% filter_eq_Cons_iff
thf(fact_7191_sorted__wrt_Osimps_I2_J,axiom,
! [A: $tType,P2: A > A > $o,X: A,Ys: list @ A] :
( ( sorted_wrt @ A @ P2 @ ( cons @ A @ X @ Ys ) )
= ( ! [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Ys ) )
=> ( P2 @ X @ X5 ) )
& ( sorted_wrt @ A @ P2 @ Ys ) ) ) ).
% sorted_wrt.simps(2)
thf(fact_7192_sorted__wrt_Oelims_I3_J,axiom,
! [A: $tType,X: A > A > $o,Xa2: list @ A] :
( ~ ( sorted_wrt @ A @ X @ Xa2 )
=> ~ ! [X4: A,Ys4: list @ A] :
( ( Xa2
= ( cons @ A @ X4 @ Ys4 ) )
=> ( ! [Xa3: A] :
( ( member @ A @ Xa3 @ ( set2 @ A @ Ys4 ) )
=> ( X @ X4 @ Xa3 ) )
& ( sorted_wrt @ A @ X @ Ys4 ) ) ) ) ).
% sorted_wrt.elims(3)
thf(fact_7193_strict__sorted__simps_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Ys: list @ A] :
( ( sorted_wrt @ A @ ( ord_less @ A ) @ ( cons @ A @ X @ Ys ) )
= ( ! [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Ys ) )
=> ( ord_less @ A @ X @ X5 ) )
& ( sorted_wrt @ A @ ( ord_less @ A ) @ Ys ) ) ) ) ).
% strict_sorted_simps(2)
thf(fact_7194_sorted__simps_I2_J,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Ys: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( cons @ A @ X @ Ys ) )
= ( ! [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Ys ) )
=> ( ord_less_eq @ A @ X @ X5 ) )
& ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Ys ) ) ) ) ).
% sorted_simps(2)
thf(fact_7195_empty__filter__conv,axiom,
! [A: $tType,P2: A > $o,Xs: list @ A] :
( ( ( nil @ A )
= ( filter2 @ A @ P2 @ Xs ) )
= ( ! [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Xs ) )
=> ~ ( P2 @ X5 ) ) ) ) ).
% empty_filter_conv
thf(fact_7196_filter__empty__conv,axiom,
! [A: $tType,P2: A > $o,Xs: list @ A] :
( ( ( filter2 @ A @ P2 @ Xs )
= ( nil @ A ) )
= ( ! [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Xs ) )
=> ~ ( P2 @ X5 ) ) ) ) ).
% filter_empty_conv
thf(fact_7197_insort__key__remove1,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [A2: B,Xs: list @ B,F3: B > A] :
( ( member @ B @ A2 @ ( set2 @ B @ Xs ) )
=> ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F3 @ Xs ) )
=> ( ( ( hd @ B
@ ( filter2 @ B
@ ^ [X5: B] :
( ( F3 @ A2 )
= ( F3 @ X5 ) )
@ Xs ) )
= A2 )
=> ( ( linorder_insort_key @ B @ A @ F3 @ A2 @ ( remove1 @ B @ A2 @ Xs ) )
= Xs ) ) ) ) ) ).
% insort_key_remove1
thf(fact_7198_inj__on__filter__key__eq,axiom,
! [B: $tType,A: $tType,F3: A > B,Y: A,Xs: list @ A] :
( ( inj_on @ A @ B @ F3 @ ( insert @ A @ Y @ ( set2 @ A @ Xs ) ) )
=> ( ( filter2 @ A
@ ^ [X5: A] :
( ( F3 @ Y )
= ( F3 @ X5 ) )
@ Xs )
= ( filter2 @ A
@ ( ^ [Y5: A,Z2: A] : Y5 = Z2
@ Y )
@ Xs ) ) ) ).
% inj_on_filter_key_eq
thf(fact_7199_sum__list__map__filter_H,axiom,
! [A: $tType,B: $tType] :
( ( monoid_add @ A )
=> ! [F3: B > A,P2: B > $o,Xs: list @ B] :
( ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ F3 @ ( filter2 @ B @ P2 @ Xs ) ) )
= ( groups8242544230860333062m_list @ A
@ ( map @ B @ A
@ ^ [X5: B] : ( if @ A @ ( P2 @ X5 ) @ ( F3 @ X5 ) @ ( zero_zero @ A ) )
@ Xs ) ) ) ) ).
% sum_list_map_filter'
thf(fact_7200_sum__list__filter__le__nat,axiom,
! [A: $tType,F3: A > nat,P2: A > $o,Xs: list @ A] : ( ord_less_eq @ nat @ ( groups8242544230860333062m_list @ nat @ ( map @ A @ nat @ F3 @ ( filter2 @ A @ P2 @ Xs ) ) ) @ ( groups8242544230860333062m_list @ nat @ ( map @ A @ nat @ F3 @ Xs ) ) ) ).
% sum_list_filter_le_nat
thf(fact_7201_filter__in__nths,axiom,
! [A: $tType,Xs: list @ A,S: set @ nat] :
( ( distinct @ A @ Xs )
=> ( ( filter2 @ A
@ ^ [X5: A] : ( member @ A @ X5 @ ( set2 @ A @ ( nths @ A @ Xs @ S ) ) )
@ Xs )
= ( nths @ A @ Xs @ S ) ) ) ).
% filter_in_nths
thf(fact_7202_sorted__iff__nth__mono__less,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
= ( ! [I4: nat,J3: nat] :
( ( ord_less @ nat @ I4 @ J3 )
=> ( ( ord_less @ nat @ J3 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ord_less_eq @ A @ ( nth @ A @ Xs @ I4 ) @ ( nth @ A @ Xs @ J3 ) ) ) ) ) ) ) ).
% sorted_iff_nth_mono_less
thf(fact_7203_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_7204_sorted__wrt_Oelims_I1_J,axiom,
! [A: $tType,X: A > A > $o,Xa2: list @ A,Y: $o] :
( ( ( sorted_wrt @ A @ X @ Xa2 )
= Y )
=> ( ( ( Xa2
= ( nil @ A ) )
=> ~ Y )
=> ~ ! [X4: A,Ys4: list @ A] :
( ( Xa2
= ( cons @ A @ X4 @ Ys4 ) )
=> ( Y
= ( ~ ( ! [Y6: A] :
( ( member @ A @ Y6 @ ( set2 @ A @ Ys4 ) )
=> ( X @ X4 @ Y6 ) )
& ( sorted_wrt @ A @ X @ Ys4 ) ) ) ) ) ) ) ).
% sorted_wrt.elims(1)
thf(fact_7205_sorted__wrt_Oelims_I2_J,axiom,
! [A: $tType,X: A > A > $o,Xa2: list @ A] :
( ( sorted_wrt @ A @ X @ Xa2 )
=> ( ( Xa2
!= ( nil @ A ) )
=> ~ ! [X4: A,Ys4: list @ A] :
( ( Xa2
= ( cons @ A @ X4 @ Ys4 ) )
=> ~ ( ! [Xa: A] :
( ( member @ A @ Xa @ ( set2 @ A @ Ys4 ) )
=> ( X @ X4 @ Xa ) )
& ( sorted_wrt @ A @ X @ Ys4 ) ) ) ) ) ).
% sorted_wrt.elims(2)
thf(fact_7206_finite__sorted__distinct__unique,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ? [X4: list @ A] :
( ( ( set2 @ A @ X4 )
= A5 )
& ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ X4 )
& ( distinct @ A @ X4 )
& ! [Y4: list @ A] :
( ( ( ( set2 @ A @ Y4 )
= A5 )
& ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Y4 )
& ( distinct @ A @ Y4 ) )
=> ( Y4 = X4 ) ) ) ) ) ).
% finite_sorted_distinct_unique
thf(fact_7207_sum__list__map__filter,axiom,
! [A: $tType,B: $tType] :
( ( monoid_add @ A )
=> ! [Xs: list @ B,P2: B > $o,F3: B > A] :
( ! [X4: B] :
( ( member @ B @ X4 @ ( set2 @ B @ Xs ) )
=> ( ~ ( P2 @ X4 )
=> ( ( F3 @ X4 )
= ( zero_zero @ A ) ) ) )
=> ( ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ F3 @ ( filter2 @ B @ P2 @ Xs ) ) )
= ( groups8242544230860333062m_list @ A @ ( map @ B @ A @ F3 @ Xs ) ) ) ) ) ).
% sum_list_map_filter
thf(fact_7208_sorted__list__of__set_Oidem__if__sorted__distinct,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( ( distinct @ A @ Xs )
=> ( ( linord4507533701916653071of_set @ A @ ( set2 @ A @ Xs ) )
= Xs ) ) ) ) ).
% sorted_list_of_set.idem_if_sorted_distinct
thf(fact_7209_set__minus__filter__out,axiom,
! [A: $tType,Xs: list @ A,Y: A] :
( ( minus_minus @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( insert @ A @ Y @ ( bot_bot @ ( set @ A ) ) ) )
= ( set2 @ A
@ ( filter2 @ A
@ ^ [X5: A] : X5 != Y
@ Xs ) ) ) ).
% set_minus_filter_out
thf(fact_7210_filter__shuffles__disjoint2_I1_J,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Zs: list @ A] :
( ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ Ys ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( member @ ( list @ A ) @ Zs @ ( shuffles @ A @ Xs @ Ys ) )
=> ( ( filter2 @ A
@ ^ [X5: A] : ( member @ A @ X5 @ ( set2 @ A @ Ys ) )
@ Zs )
= Ys ) ) ) ).
% filter_shuffles_disjoint2(1)
thf(fact_7211_filter__shuffles__disjoint2_I2_J,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Zs: list @ A] :
( ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ Ys ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( member @ ( list @ A ) @ Zs @ ( shuffles @ A @ Xs @ Ys ) )
=> ( ( filter2 @ A
@ ^ [X5: A] :
~ ( member @ A @ X5 @ ( set2 @ A @ Ys ) )
@ Zs )
= Xs ) ) ) ).
% filter_shuffles_disjoint2(2)
thf(fact_7212_filter__shuffles__disjoint1_I1_J,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Zs: list @ A] :
( ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ Ys ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( member @ ( list @ A ) @ Zs @ ( shuffles @ A @ Xs @ Ys ) )
=> ( ( filter2 @ A
@ ^ [X5: A] : ( member @ A @ X5 @ ( set2 @ A @ Xs ) )
@ Zs )
= Xs ) ) ) ).
% filter_shuffles_disjoint1(1)
thf(fact_7213_filter__shuffles__disjoint1_I2_J,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,Zs: list @ A] :
( ( ( inf_inf @ ( set @ A ) @ ( set2 @ A @ Xs ) @ ( set2 @ A @ Ys ) )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( member @ ( list @ A ) @ Zs @ ( shuffles @ A @ Xs @ Ys ) )
=> ( ( filter2 @ A
@ ^ [X5: A] :
~ ( member @ A @ X5 @ ( set2 @ A @ Xs ) )
@ Zs )
= Ys ) ) ) ).
% filter_shuffles_disjoint1(2)
thf(fact_7214_filter__eq__nths,axiom,
! [A: $tType] :
( ( filter2 @ A )
= ( ^ [P4: A > $o,Xs3: list @ A] :
( nths @ A @ Xs3
@ ( collect @ nat
@ ^ [I4: nat] :
( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ Xs3 ) )
& ( P4 @ ( nth @ A @ Xs3 @ I4 ) ) ) ) ) ) ) ).
% filter_eq_nths
thf(fact_7215_length__filter__conv__card,axiom,
! [A: $tType,P: A > $o,Xs: list @ A] :
( ( size_size @ ( list @ A ) @ ( filter2 @ A @ P @ Xs ) )
= ( finite_card @ nat
@ ( collect @ nat
@ ^ [I4: nat] :
( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ Xs ) )
& ( P @ ( nth @ A @ Xs @ I4 ) ) ) ) ) ) ).
% length_filter_conv_card
thf(fact_7216_insort__remove1,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A2: A,Xs: list @ A] :
( ( member @ A @ A2 @ ( set2 @ A @ Xs ) )
=> ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( ( linorder_insort_key @ A @ A
@ ^ [X5: A] : X5
@ A2
@ ( remove1 @ A @ A2 @ Xs ) )
= Xs ) ) ) ) ).
% insort_remove1
thf(fact_7217_sorted__iff__nth__Suc,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
= ( ! [I4: nat] :
( ( ord_less @ nat @ ( suc @ I4 ) @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ord_less_eq @ A @ ( nth @ A @ Xs @ I4 ) @ ( nth @ A @ Xs @ ( suc @ I4 ) ) ) ) ) ) ) ).
% sorted_iff_nth_Suc
thf(fact_7218_sorted__list__of__set_Ofinite__set__strict__sorted,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ~ ! [L4: list @ A] :
( ( sorted_wrt @ A @ ( ord_less @ A ) @ L4 )
=> ( ( ( set2 @ A @ L4 )
= A5 )
=> ( ( size_size @ ( list @ A ) @ L4 )
!= ( finite_card @ A @ A5 ) ) ) ) ) ) ).
% sorted_list_of_set.finite_set_strict_sorted
thf(fact_7219_sorted__iff__nth__mono,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
= ( ! [I4: nat,J3: nat] :
( ( ord_less_eq @ nat @ I4 @ J3 )
=> ( ( ord_less @ nat @ J3 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ord_less_eq @ A @ ( nth @ A @ Xs @ I4 ) @ ( nth @ A @ Xs @ J3 ) ) ) ) ) ) ) ).
% sorted_iff_nth_mono
thf(fact_7220_sorted__nth__mono,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,I2: nat,J: nat] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( ( ord_less_eq @ nat @ I2 @ J )
=> ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ord_less_eq @ A @ ( nth @ A @ Xs @ I2 ) @ ( nth @ A @ Xs @ J ) ) ) ) ) ) ).
% sorted_nth_mono
thf(fact_7221_sorted__wrt__less__idx,axiom,
! [Ns: list @ nat,I2: nat] :
( ( sorted_wrt @ nat @ ( ord_less @ nat ) @ Ns )
=> ( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ nat ) @ Ns ) )
=> ( ord_less_eq @ nat @ I2 @ ( nth @ nat @ Ns @ I2 ) ) ) ) ).
% sorted_wrt_less_idx
thf(fact_7222_distinct__length__filter,axiom,
! [A: $tType,Xs: list @ A,P2: A > $o] :
( ( distinct @ A @ Xs )
=> ( ( size_size @ ( list @ A ) @ ( filter2 @ A @ P2 @ Xs ) )
= ( finite_card @ A @ ( inf_inf @ ( set @ A ) @ ( collect @ A @ P2 ) @ ( set2 @ A @ Xs ) ) ) ) ) ).
% distinct_length_filter
thf(fact_7223_sorted__enumerate,axiom,
! [A: $tType,N: nat,Xs: list @ A] : ( sorted_wrt @ nat @ ( ord_less_eq @ nat ) @ ( map @ ( product_prod @ nat @ A ) @ nat @ ( product_fst @ nat @ A ) @ ( enumerate @ A @ N @ Xs ) ) ) ).
% sorted_enumerate
thf(fact_7224_map__sorted__distinct__set__unique,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F3: B > A,Xs: list @ B,Ys: list @ B] :
( ( inj_on @ B @ A @ F3 @ ( sup_sup @ ( set @ B ) @ ( set2 @ B @ Xs ) @ ( set2 @ B @ Ys ) ) )
=> ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F3 @ Xs ) )
=> ( ( distinct @ A @ ( map @ B @ A @ F3 @ Xs ) )
=> ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F3 @ Ys ) )
=> ( ( distinct @ A @ ( map @ B @ A @ F3 @ Ys ) )
=> ( ( ( set2 @ B @ Xs )
= ( set2 @ B @ Ys ) )
=> ( Xs = Ys ) ) ) ) ) ) ) ) ).
% map_sorted_distinct_set_unique
thf(fact_7225_sorted__list__of__set_Osorted__key__list__of__set__unique,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [A5: set @ A,L: list @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( ( sorted_wrt @ A @ ( ord_less @ A ) @ L )
& ( ( set2 @ A @ L )
= A5 )
& ( ( size_size @ ( list @ A ) @ L )
= ( finite_card @ A @ A5 ) ) )
= ( ( linord4507533701916653071of_set @ A @ A5 )
= L ) ) ) ) ).
% sorted_list_of_set.sorted_key_list_of_set_unique
thf(fact_7226_sorted__insort__is__snoc,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,A2: A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
=> ( ord_less_eq @ A @ X4 @ A2 ) )
=> ( ( linorder_insort_key @ A @ A
@ ^ [X5: A] : X5
@ A2
@ Xs )
= ( append @ A @ Xs @ ( cons @ A @ A2 @ ( nil @ A ) ) ) ) ) ) ) ).
% sorted_insort_is_snoc
thf(fact_7227_transpose__aux__filter__head,axiom,
! [A: $tType,Xss: list @ ( list @ A )] :
( ( concat @ A
@ ( map @ ( list @ A ) @ ( list @ A )
@ ( case_list @ ( list @ A ) @ A @ ( nil @ A )
@ ^ [H: A,T3: list @ A] : ( cons @ A @ H @ ( nil @ A ) ) )
@ Xss ) )
= ( map @ ( list @ A ) @ A @ ( hd @ A )
@ ( filter2 @ ( list @ A )
@ ^ [Ys3: list @ A] :
( Ys3
!= ( nil @ A ) )
@ Xss ) ) ) ).
% transpose_aux_filter_head
thf(fact_7228_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
@ ^ [X5: A] :
( ( F4 @ X5 )
!= ( none @ B ) )
@ Xs3 ) ) ) ) ).
% map_filter_def
thf(fact_7229_nth__nth__transpose__sorted,axiom,
! [A: $tType,Xs: list @ ( list @ A ),I2: nat,J: nat] :
( ( sorted_wrt @ nat @ ( ord_less_eq @ nat ) @ ( rev @ nat @ ( map @ ( list @ A ) @ nat @ ( size_size @ ( list @ A ) ) @ Xs ) ) )
=> ( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ ( list @ A ) ) @ ( transpose @ A @ Xs ) ) )
=> ( ( ord_less @ nat @ J
@ ( size_size @ ( list @ ( list @ A ) )
@ ( filter2 @ ( list @ A )
@ ^ [Ys3: list @ A] : ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Ys3 ) )
@ Xs ) ) )
=> ( ( nth @ A @ ( nth @ ( list @ A ) @ ( transpose @ A @ Xs ) @ I2 ) @ J )
= ( nth @ A @ ( nth @ ( list @ A ) @ Xs @ J ) @ I2 ) ) ) ) ) ).
% nth_nth_transpose_sorted
thf(fact_7230_rev__is__rev__conv,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( rev @ A @ Xs )
= ( rev @ A @ Ys ) )
= ( Xs = Ys ) ) ).
% rev_is_rev_conv
thf(fact_7231_rev__rev__ident,axiom,
! [A: $tType,Xs: list @ A] :
( ( rev @ A @ ( rev @ A @ Xs ) )
= Xs ) ).
% rev_rev_ident
thf(fact_7232_Nil__is__rev__conv,axiom,
! [A: $tType,Xs: list @ A] :
( ( ( nil @ A )
= ( rev @ A @ Xs ) )
= ( Xs
= ( nil @ A ) ) ) ).
% Nil_is_rev_conv
thf(fact_7233_rev__is__Nil__conv,axiom,
! [A: $tType,Xs: list @ A] :
( ( ( rev @ A @ Xs )
= ( nil @ A ) )
= ( Xs
= ( nil @ A ) ) ) ).
% rev_is_Nil_conv
thf(fact_7234_set__rev,axiom,
! [A: $tType,Xs: list @ A] :
( ( set2 @ A @ ( rev @ A @ Xs ) )
= ( set2 @ A @ Xs ) ) ).
% set_rev
thf(fact_7235_length__rev,axiom,
! [A: $tType,Xs: list @ A] :
( ( size_size @ ( list @ A ) @ ( rev @ A @ Xs ) )
= ( size_size @ ( list @ A ) @ Xs ) ) ).
% length_rev
thf(fact_7236_rev__append,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( rev @ A @ ( append @ A @ Xs @ Ys ) )
= ( append @ A @ ( rev @ A @ Ys ) @ ( rev @ A @ Xs ) ) ) ).
% rev_append
thf(fact_7237_distinct__rev,axiom,
! [A: $tType,Xs: list @ A] :
( ( distinct @ A @ ( rev @ A @ Xs ) )
= ( distinct @ A @ Xs ) ) ).
% distinct_rev
thf(fact_7238_rev__replicate,axiom,
! [A: $tType,N: nat,X: A] :
( ( rev @ A @ ( replicate @ A @ N @ X ) )
= ( replicate @ A @ N @ X ) ) ).
% rev_replicate
thf(fact_7239_remdups__adj__rev,axiom,
! [A: $tType,Xs: list @ A] :
( ( remdups_adj @ A @ ( rev @ A @ Xs ) )
= ( rev @ A @ ( remdups_adj @ A @ Xs ) ) ) ).
% remdups_adj_rev
thf(fact_7240_inj__on__rev,axiom,
! [A: $tType,A5: set @ ( list @ A )] : ( inj_on @ ( list @ A ) @ ( list @ A ) @ ( rev @ A ) @ A5 ) ).
% inj_on_rev
thf(fact_7241_rev__singleton__conv,axiom,
! [A: $tType,Xs: list @ A,X: A] :
( ( ( rev @ A @ Xs )
= ( cons @ A @ X @ ( nil @ A ) ) )
= ( Xs
= ( cons @ A @ X @ ( nil @ A ) ) ) ) ).
% rev_singleton_conv
thf(fact_7242_singleton__rev__conv,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( ( cons @ A @ X @ ( nil @ A ) )
= ( rev @ A @ Xs ) )
= ( ( cons @ A @ X @ ( nil @ A ) )
= Xs ) ) ).
% singleton_rev_conv
thf(fact_7243_rev__eq__Cons__iff,axiom,
! [A: $tType,Xs: list @ A,Y: A,Ys: list @ A] :
( ( ( rev @ A @ Xs )
= ( cons @ A @ Y @ Ys ) )
= ( Xs
= ( append @ A @ ( rev @ A @ Ys ) @ ( cons @ A @ Y @ ( nil @ A ) ) ) ) ) ).
% rev_eq_Cons_iff
thf(fact_7244_rev__concat,axiom,
! [A: $tType,Xs: list @ ( list @ A )] :
( ( rev @ A @ ( concat @ A @ Xs ) )
= ( concat @ A @ ( map @ ( list @ A ) @ ( list @ A ) @ ( rev @ A ) @ ( rev @ ( list @ A ) @ Xs ) ) ) ) ).
% rev_concat
thf(fact_7245_rev__map,axiom,
! [A: $tType,B: $tType,F3: B > A,Xs: list @ B] :
( ( rev @ A @ ( map @ B @ A @ F3 @ Xs ) )
= ( map @ B @ A @ F3 @ ( rev @ B @ Xs ) ) ) ).
% rev_map
thf(fact_7246_rev__filter,axiom,
! [A: $tType,P2: A > $o,Xs: list @ A] :
( ( rev @ A @ ( filter2 @ A @ P2 @ Xs ) )
= ( filter2 @ A @ P2 @ ( rev @ A @ Xs ) ) ) ).
% rev_filter
thf(fact_7247_sorted__wrt__rev,axiom,
! [A: $tType,P2: A > A > $o,Xs: list @ A] :
( ( sorted_wrt @ A @ P2 @ ( rev @ A @ Xs ) )
= ( sorted_wrt @ A
@ ^ [X5: A,Y6: A] : ( P2 @ Y6 @ X5 )
@ Xs ) ) ).
% sorted_wrt_rev
thf(fact_7248_sorted__upto,axiom,
! [M2: int,N: int] : ( sorted_wrt @ int @ ( ord_less_eq @ int ) @ ( upto @ M2 @ N ) ) ).
% sorted_upto
thf(fact_7249_sorted__wrt__upto,axiom,
! [I2: int,J: int] : ( sorted_wrt @ int @ ( ord_less @ int ) @ ( upto @ I2 @ J ) ) ).
% sorted_wrt_upto
thf(fact_7250_map__of__filter__in,axiom,
! [B: $tType,A: $tType,Xs: list @ ( product_prod @ B @ A ),K2: B,Z: A,P2: B > A > $o] :
( ( ( map_of @ B @ A @ Xs @ K2 )
= ( some @ A @ Z ) )
=> ( ( P2 @ K2 @ Z )
=> ( ( map_of @ B @ A @ ( filter2 @ ( product_prod @ B @ A ) @ ( product_case_prod @ B @ A @ $o @ P2 ) @ Xs ) @ K2 )
= ( some @ A @ Z ) ) ) ) ).
% map_of_filter_in
thf(fact_7251_zip__rev,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( zip @ A @ B @ ( rev @ A @ Xs ) @ ( rev @ B @ Ys ) )
= ( rev @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs @ Ys ) ) ) ) ).
% zip_rev
thf(fact_7252_map__filter__simps_I2_J,axiom,
! [B: $tType,A: $tType,F3: B > ( option @ A )] :
( ( map_filter @ B @ A @ F3 @ ( nil @ B ) )
= ( nil @ A ) ) ).
% map_filter_simps(2)
thf(fact_7253_rev_Osimps_I1_J,axiom,
! [A: $tType] :
( ( rev @ A @ ( nil @ A ) )
= ( nil @ A ) ) ).
% rev.simps(1)
thf(fact_7254_rev__swap,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( rev @ A @ Xs )
= Ys )
= ( Xs
= ( rev @ A @ Ys ) ) ) ).
% rev_swap
thf(fact_7255_rev_Osimps_I2_J,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( rev @ A @ ( cons @ A @ X @ Xs ) )
= ( append @ A @ ( rev @ A @ Xs ) @ ( cons @ A @ X @ ( nil @ A ) ) ) ) ).
% rev.simps(2)
thf(fact_7256_map__filter__simps_I1_J,axiom,
! [A: $tType,B: $tType,F3: B > ( option @ A ),X: B,Xs: list @ B] :
( ( map_filter @ B @ A @ F3 @ ( cons @ B @ X @ Xs ) )
= ( case_option @ ( list @ A ) @ A @ ( map_filter @ B @ A @ F3 @ Xs )
@ ^ [Y6: A] : ( cons @ A @ Y6 @ ( map_filter @ B @ A @ F3 @ Xs ) )
@ ( F3 @ X ) ) ) ).
% map_filter_simps(1)
thf(fact_7257_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_7258_rev__drop,axiom,
! [A: $tType,I2: nat,Xs: list @ A] :
( ( rev @ A @ ( drop @ A @ I2 @ Xs ) )
= ( take @ A @ ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ I2 ) @ ( rev @ A @ Xs ) ) ) ).
% rev_drop
thf(fact_7259_rev__take,axiom,
! [A: $tType,I2: nat,Xs: list @ A] :
( ( rev @ A @ ( take @ A @ I2 @ Xs ) )
= ( drop @ A @ ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ I2 ) @ ( rev @ A @ Xs ) ) ) ).
% rev_take
thf(fact_7260_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_7261_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_7262_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_7263_rev__update,axiom,
! [A: $tType,K2: nat,Xs: list @ A,Y: A] :
( ( ord_less @ nat @ K2 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ( rev @ A @ ( list_update @ A @ Xs @ K2 @ Y ) )
= ( list_update @ A @ ( rev @ A @ Xs ) @ ( minus_minus @ nat @ ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ K2 ) @ ( one_one @ nat ) ) @ Y ) ) ) ).
% rev_update
thf(fact_7264_nths__shift__lemma__Suc,axiom,
! [A: $tType,P2: 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] : ( P2 @ ( 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] : ( P2 @ ( product_snd @ A @ nat @ P6 ) )
@ ( zip @ A @ nat @ Xs @ ( map @ nat @ nat @ suc @ Is ) ) ) ) ) ).
% nths_shift_lemma_Suc
thf(fact_7265_sorted__rev__iff__nth__Suc,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( rev @ A @ Xs ) )
= ( ! [I4: nat] :
( ( ord_less @ nat @ ( suc @ I4 ) @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ord_less_eq @ A @ ( nth @ A @ Xs @ ( suc @ I4 ) ) @ ( nth @ A @ Xs @ I4 ) ) ) ) ) ) ).
% sorted_rev_iff_nth_Suc
thf(fact_7266_sorted__rev__nth__mono,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,I2: nat,J: nat] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( rev @ A @ Xs ) )
=> ( ( ord_less_eq @ nat @ I2 @ J )
=> ( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ord_less_eq @ A @ ( nth @ A @ Xs @ J ) @ ( nth @ A @ Xs @ I2 ) ) ) ) ) ) ).
% sorted_rev_nth_mono
thf(fact_7267_sorted__rev__iff__nth__mono,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( rev @ A @ Xs ) )
= ( ! [I4: nat,J3: nat] :
( ( ord_less_eq @ nat @ I4 @ J3 )
=> ( ( ord_less @ nat @ J3 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ord_less_eq @ A @ ( nth @ A @ Xs @ J3 ) @ ( nth @ A @ Xs @ I4 ) ) ) ) ) ) ) ).
% sorted_rev_iff_nth_mono
thf(fact_7268_nths__shift__lemma,axiom,
! [A: $tType,A5: set @ nat,Xs: list @ A,I2: 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 ) @ A5 )
@ ( zip @ A @ nat @ Xs @ ( upt @ I2 @ ( plus_plus @ nat @ I2 @ ( 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 ) @ I2 ) @ A5 )
@ ( zip @ A @ nat @ Xs @ ( upt @ ( zero_zero @ nat ) @ ( size_size @ ( list @ A ) @ Xs ) ) ) ) ) ) ).
% nths_shift_lemma
thf(fact_7269_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_7270_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_7271_transpose__column__length,axiom,
! [A: $tType,Xs: list @ ( list @ A ),I2: nat] :
( ( sorted_wrt @ nat @ ( ord_less_eq @ nat ) @ ( rev @ nat @ ( map @ ( list @ A ) @ nat @ ( size_size @ ( list @ A ) ) @ Xs ) ) )
=> ( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ ( list @ A ) ) @ Xs ) )
=> ( ( size_size @ ( list @ ( list @ A ) )
@ ( filter2 @ ( list @ A )
@ ^ [Ys3: list @ A] : ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Ys3 ) )
@ ( transpose @ A @ Xs ) ) )
= ( size_size @ ( list @ A ) @ ( nth @ ( list @ A ) @ Xs @ I2 ) ) ) ) ) ).
% transpose_column_length
thf(fact_7272_map__filter__map__filter,axiom,
! [A: $tType,B: $tType,F3: B > A,P2: B > $o,Xs: list @ B] :
( ( map @ B @ A @ F3 @ ( filter2 @ B @ P2 @ Xs ) )
= ( map_filter @ B @ A
@ ^ [X5: B] : ( if @ ( option @ A ) @ ( P2 @ X5 ) @ ( some @ A @ ( F3 @ X5 ) ) @ ( none @ A ) )
@ Xs ) ) ).
% map_filter_map_filter
thf(fact_7273_transpose__column,axiom,
! [A: $tType,Xs: list @ ( list @ A ),I2: nat] :
( ( sorted_wrt @ nat @ ( ord_less_eq @ nat ) @ ( rev @ nat @ ( map @ ( list @ A ) @ nat @ ( size_size @ ( list @ A ) ) @ Xs ) ) )
=> ( ( ord_less @ nat @ I2 @ ( size_size @ ( list @ ( list @ A ) ) @ Xs ) )
=> ( ( map @ ( list @ A ) @ A
@ ^ [Ys3: list @ A] : ( nth @ A @ Ys3 @ I2 )
@ ( filter2 @ ( list @ A )
@ ^ [Ys3: list @ A] : ( ord_less @ nat @ I2 @ ( size_size @ ( list @ A ) @ Ys3 ) )
@ ( transpose @ A @ Xs ) ) )
= ( nth @ ( list @ A ) @ Xs @ I2 ) ) ) ) ).
% transpose_column
thf(fact_7274_transpose__aux__filter__tail,axiom,
! [A: $tType,Xss: list @ ( list @ A )] :
( ( concat @ ( list @ A )
@ ( map @ ( list @ A ) @ ( list @ ( list @ A ) )
@ ( case_list @ ( list @ ( list @ A ) ) @ A @ ( nil @ ( list @ A ) )
@ ^ [H: A,T3: list @ A] : ( cons @ ( list @ A ) @ T3 @ ( nil @ ( list @ A ) ) ) )
@ Xss ) )
= ( map @ ( list @ A ) @ ( list @ A ) @ ( tl @ A )
@ ( filter2 @ ( list @ A )
@ ^ [Ys3: list @ A] :
( Ys3
!= ( nil @ A ) )
@ Xss ) ) ) ).
% transpose_aux_filter_tail
thf(fact_7275_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 )
@ ^ [X5: list @ A] :
( X5
!= ( nil @ A ) )
@ Xs ) ) ) ).
% transpose_max_length
thf(fact_7276_tl__upt,axiom,
! [M2: nat,N: nat] :
( ( tl @ nat @ ( upt @ M2 @ N ) )
= ( upt @ ( suc @ M2 ) @ N ) ) ).
% tl_upt
thf(fact_7277_foldr__append,axiom,
! [B: $tType,A: $tType,F3: B > A > A,Xs: list @ B,Ys: list @ B,A2: A] :
( ( foldr @ B @ A @ F3 @ ( append @ B @ Xs @ Ys ) @ A2 )
= ( foldr @ B @ A @ F3 @ Xs @ ( foldr @ B @ A @ F3 @ Ys @ A2 ) ) ) ).
% foldr_append
thf(fact_7278_tl__append2,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( tl @ A @ ( append @ A @ Xs @ Ys ) )
= ( append @ A @ ( tl @ A @ Xs ) @ Ys ) ) ) ).
% tl_append2
thf(fact_7279_remdups__adj__Cons__alt,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( cons @ A @ X @ ( tl @ A @ ( remdups_adj @ A @ ( cons @ A @ X @ Xs ) ) ) )
= ( remdups_adj @ A @ ( cons @ A @ X @ Xs ) ) ) ).
% remdups_adj_Cons_alt
thf(fact_7280_foldr__replicate,axiom,
! [A: $tType,B: $tType,F3: B > A > A,N: nat,X: B] :
( ( foldr @ B @ A @ F3 @ ( replicate @ B @ N @ X ) )
= ( compow @ ( A > A ) @ N @ ( F3 @ X ) ) ) ).
% foldr_replicate
thf(fact_7281_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_7282_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_7283_hd__Cons__tl,axiom,
! [A: $tType,Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( cons @ A @ ( hd @ A @ Xs ) @ ( tl @ A @ Xs ) )
= Xs ) ) ).
% hd_Cons_tl
thf(fact_7284_list_Ocollapse,axiom,
! [A: $tType,List: list @ A] :
( ( List
!= ( nil @ A ) )
=> ( ( cons @ A @ ( hd @ A @ List ) @ ( tl @ A @ List ) )
= List ) ) ).
% list.collapse
thf(fact_7285_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_7286_list_Omap__sel_I2_J,axiom,
! [B: $tType,A: $tType,A2: list @ A,F3: A > B] :
( ( A2
!= ( nil @ A ) )
=> ( ( tl @ B @ ( map @ A @ B @ F3 @ A2 ) )
= ( map @ A @ B @ F3 @ ( tl @ A @ A2 ) ) ) ) ).
% list.map_sel(2)
thf(fact_7287_map__tl,axiom,
! [A: $tType,B: $tType,F3: B > A,Xs: list @ B] :
( ( map @ B @ A @ F3 @ ( tl @ B @ Xs ) )
= ( tl @ A @ ( map @ B @ A @ F3 @ Xs ) ) ) ).
% map_tl
thf(fact_7288_sorted__tl,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( tl @ A @ Xs ) ) ) ) ).
% sorted_tl
thf(fact_7289_foldr__Cons,axiom,
! [B: $tType,A: $tType,F3: A > B > B,X: A,Xs: list @ A] :
( ( foldr @ A @ B @ F3 @ ( cons @ A @ X @ Xs ) )
= ( comp @ B @ B @ B @ ( F3 @ X ) @ ( foldr @ A @ B @ F3 @ Xs ) ) ) ).
% foldr_Cons
thf(fact_7290_foldr__map,axiom,
! [C: $tType,B: $tType,A: $tType,G3: B > A > A,F3: C > B,Xs: list @ C,A2: A] :
( ( foldr @ B @ A @ G3 @ ( map @ C @ B @ F3 @ Xs ) @ A2 )
= ( foldr @ C @ A @ ( comp @ B @ ( A > A ) @ C @ G3 @ F3 ) @ Xs @ A2 ) ) ).
% foldr_map
thf(fact_7291_tl__append,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( tl @ A @ ( append @ A @ Xs @ Ys ) )
= ( case_list @ ( list @ A ) @ A @ ( tl @ A @ Ys )
@ ^ [Z5: A,Zs3: list @ A] : ( append @ A @ Zs3 @ Ys )
@ Xs ) ) ).
% tl_append
thf(fact_7292_list_Osel_I3_J,axiom,
! [A: $tType,X21: A,X222: list @ A] :
( ( tl @ A @ ( cons @ A @ X21 @ X222 ) )
= X222 ) ).
% list.sel(3)
thf(fact_7293_tl__Nil,axiom,
! [A: $tType,Xs: list @ A] :
( ( ( tl @ A @ Xs )
= ( nil @ A ) )
= ( ( Xs
= ( nil @ A ) )
| ? [X5: A] :
( Xs
= ( cons @ A @ X5 @ ( nil @ A ) ) ) ) ) ).
% tl_Nil
thf(fact_7294_Nil__tl,axiom,
! [A: $tType,Xs: list @ A] :
( ( ( nil @ A )
= ( tl @ A @ Xs ) )
= ( ( Xs
= ( nil @ A ) )
| ? [X5: A] :
( Xs
= ( cons @ A @ X5 @ ( nil @ A ) ) ) ) ) ).
% Nil_tl
thf(fact_7295_list_Osel_I2_J,axiom,
! [A: $tType] :
( ( tl @ A @ ( nil @ A ) )
= ( nil @ A ) ) ).
% list.sel(2)
thf(fact_7296_foldr__Nil,axiom,
! [A: $tType,B: $tType,F3: A > B > B] :
( ( foldr @ A @ B @ F3 @ ( nil @ A ) )
= ( id @ B ) ) ).
% foldr_Nil
thf(fact_7297_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_7298_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_7299_tl__drop,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( tl @ A @ ( drop @ A @ N @ Xs ) )
= ( drop @ A @ N @ ( tl @ A @ Xs ) ) ) ).
% tl_drop
thf(fact_7300_foldr__cong,axiom,
! [B: $tType,A: $tType,A2: A,B2: A,L: list @ B,K2: list @ B,F3: B > A > A,G3: B > A > A] :
( ( A2 = B2 )
=> ( ( L = K2 )
=> ( ! [A4: A,X4: B] :
( ( member @ B @ X4 @ ( set2 @ B @ L ) )
=> ( ( F3 @ X4 @ A4 )
= ( G3 @ X4 @ A4 ) ) )
=> ( ( foldr @ B @ A @ F3 @ L @ A2 )
= ( foldr @ B @ A @ G3 @ K2 @ B2 ) ) ) ) ) ).
% foldr_cong
thf(fact_7301_distinct__tl,axiom,
! [A: $tType,Xs: list @ A] :
( ( distinct @ A @ Xs )
=> ( distinct @ A @ ( tl @ A @ Xs ) ) ) ).
% distinct_tl
thf(fact_7302_list_Oset__sel_I2_J,axiom,
! [A: $tType,A2: list @ A,X: A] :
( ( A2
!= ( nil @ A ) )
=> ( ( member @ A @ X @ ( set2 @ A @ ( tl @ A @ A2 ) ) )
=> ( member @ A @ X @ ( set2 @ A @ A2 ) ) ) ) ).
% list.set_sel(2)
thf(fact_7303_list_Oexpand,axiom,
! [A: $tType,List: list @ A,List3: list @ A] :
( ( ( List
= ( nil @ A ) )
= ( List3
= ( nil @ A ) ) )
=> ( ( ( List
!= ( nil @ A ) )
=> ( ( List3
!= ( nil @ A ) )
=> ( ( ( hd @ A @ List )
= ( hd @ A @ List3 ) )
& ( ( tl @ A @ List )
= ( tl @ A @ List3 ) ) ) ) )
=> ( List = List3 ) ) ) ).
% list.expand
thf(fact_7304_tl__def,axiom,
! [A: $tType] :
( ( tl @ A )
= ( case_list @ ( list @ A ) @ A @ ( nil @ A )
@ ^ [X213: A,X224: list @ A] : X224 ) ) ).
% tl_def
thf(fact_7305_foldr__filter,axiom,
! [A: $tType,B: $tType,F3: B > A > A,P2: B > $o,Xs: list @ B] :
( ( foldr @ B @ A @ F3 @ ( filter2 @ B @ P2 @ Xs ) )
= ( foldr @ B @ A
@ ^ [X5: B] : ( if @ ( A > A ) @ ( P2 @ X5 ) @ ( F3 @ X5 ) @ ( id @ A ) )
@ Xs ) ) ).
% foldr_filter
thf(fact_7306_list_Oexhaust__sel,axiom,
! [A: $tType,List: list @ A] :
( ( List
!= ( nil @ A ) )
=> ( List
= ( cons @ A @ ( hd @ A @ List ) @ ( tl @ A @ List ) ) ) ) ).
% list.exhaust_sel
thf(fact_7307_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_7308_list_Ocase__eq__if,axiom,
! [A: $tType,B: $tType] :
( ( case_list @ B @ A )
= ( ^ [F12: B,F23: A > ( list @ A ) > B,List2: list @ A] :
( if @ B
@ ( List2
= ( nil @ A ) )
@ F12
@ ( F23 @ ( hd @ A @ List2 ) @ ( tl @ A @ List2 ) ) ) ) ) ).
% list.case_eq_if
thf(fact_7309_foldl__conv__foldr,axiom,
! [B: $tType,A: $tType] :
( ( foldl @ A @ B )
= ( ^ [F4: A > B > A,A6: A,Xs3: list @ B] :
( foldr @ B @ A
@ ^ [X5: B,Y6: A] : ( F4 @ Y6 @ X5 )
@ ( rev @ B @ Xs3 )
@ A6 ) ) ) ).
% foldl_conv_foldr
thf(fact_7310_foldr__conv__foldl,axiom,
! [A: $tType,B: $tType] :
( ( foldr @ B @ A )
= ( ^ [F4: B > A > A,Xs3: list @ B,A6: A] :
( foldl @ A @ B
@ ^ [X5: A,Y6: B] : ( F4 @ Y6 @ X5 )
@ A6
@ ( rev @ B @ Xs3 ) ) ) ) ).
% foldr_conv_foldl
thf(fact_7311_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_7312_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_7313_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_7314_remdups__adj__append,axiom,
! [A: $tType,Xs_1: list @ A,X: A,Xs_2: list @ A] :
( ( remdups_adj @ A @ ( append @ A @ Xs_1 @ ( cons @ A @ X @ Xs_2 ) ) )
= ( append @ A @ ( remdups_adj @ A @ ( append @ A @ Xs_1 @ ( cons @ A @ X @ ( nil @ A ) ) ) ) @ ( tl @ A @ ( remdups_adj @ A @ ( cons @ A @ X @ Xs_2 ) ) ) ) ) ).
% remdups_adj_append
thf(fact_7315_Cons__in__shuffles__iff,axiom,
! [A: $tType,Z: A,Zs: list @ A,Xs: list @ A,Ys: list @ A] :
( ( member @ ( list @ A ) @ ( cons @ A @ Z @ Zs ) @ ( shuffles @ A @ Xs @ Ys ) )
= ( ( ( Xs
!= ( nil @ A ) )
& ( ( hd @ A @ Xs )
= Z )
& ( member @ ( list @ A ) @ Zs @ ( shuffles @ A @ ( tl @ A @ Xs ) @ Ys ) ) )
| ( ( Ys
!= ( nil @ A ) )
& ( ( hd @ A @ Ys )
= Z )
& ( member @ ( list @ A ) @ Zs @ ( shuffles @ A @ Xs @ ( tl @ A @ Ys ) ) ) ) ) ) ).
% Cons_in_shuffles_iff
thf(fact_7316_list_Osplit__sel,axiom,
! [B: $tType,A: $tType,P2: B > $o,F1: B,F22: A > ( list @ A ) > B,List: list @ A] :
( ( P2 @ ( case_list @ B @ A @ F1 @ F22 @ List ) )
= ( ( ( List
= ( nil @ A ) )
=> ( P2 @ F1 ) )
& ( ( List
= ( cons @ A @ ( hd @ A @ List ) @ ( tl @ A @ List ) ) )
=> ( P2 @ ( F22 @ ( hd @ A @ List ) @ ( tl @ A @ List ) ) ) ) ) ) ).
% list.split_sel
thf(fact_7317_list_Osplit__sel__asm,axiom,
! [B: $tType,A: $tType,P2: B > $o,F1: B,F22: A > ( list @ A ) > B,List: list @ A] :
( ( P2 @ ( case_list @ B @ A @ F1 @ F22 @ List ) )
= ( ~ ( ( ( List
= ( nil @ A ) )
& ~ ( P2 @ F1 ) )
| ( ( List
= ( cons @ A @ ( hd @ A @ List ) @ ( tl @ A @ List ) ) )
& ~ ( P2 @ ( F22 @ ( hd @ A @ List ) @ ( tl @ A @ List ) ) ) ) ) ) ) ).
% list.split_sel_asm
thf(fact_7318_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_7319_rotate1__hd__tl,axiom,
! [A: $tType,Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( rotate1 @ A @ Xs )
= ( append @ A @ ( tl @ A @ Xs ) @ ( cons @ A @ ( hd @ A @ Xs ) @ ( nil @ A ) ) ) ) ) ).
% rotate1_hd_tl
thf(fact_7320_horner__sum__foldr,axiom,
! [A: $tType,B: $tType] :
( ( comm_semiring_0 @ A )
=> ( ( groups4207007520872428315er_sum @ B @ A )
= ( ^ [F4: B > A,A6: A,Xs3: list @ B] :
( foldr @ B @ A
@ ^ [X5: B,B5: A] : ( plus_plus @ A @ ( F4 @ X5 ) @ ( times_times @ A @ A6 @ B5 ) )
@ Xs3
@ ( zero_zero @ A ) ) ) ) ) ).
% horner_sum_foldr
thf(fact_7321_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_7322_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_7323_foldr__max__sorted,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,Y: A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( rev @ A @ Xs ) )
=> ( ( ( Xs
= ( nil @ A ) )
=> ( ( foldr @ A @ A @ ( ord_max @ A ) @ Xs @ Y )
= Y ) )
& ( ( Xs
!= ( nil @ A ) )
=> ( ( foldr @ A @ A @ ( ord_max @ A ) @ Xs @ Y )
= ( ord_max @ A @ ( nth @ A @ Xs @ ( zero_zero @ nat ) ) @ Y ) ) ) ) ) ) ).
% foldr_max_sorted
thf(fact_7324_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
@ ^ [X5: list @ B] : ( ord_max @ nat @ ( minus_minus @ nat @ ( size_size @ ( list @ B ) @ X5 ) @ ( suc @ ( zero_zero @ nat ) ) ) )
@ ( filter2 @ ( list @ B )
@ ^ [Ys3: list @ B] :
( Ys3
!= ( nil @ B ) )
@ Xss )
@ ( zero_zero @ nat ) ) ) ) ) ).
% transpose_aux_max
thf(fact_7325_folding__insort__key_Ofinite__set__strict__sorted,axiom,
! [A: $tType,B: $tType,Less_eq: A > A > $o,Less: A > A > $o,S2: set @ B,F3: B > A,A5: set @ B] :
( ( folding_insort_key @ A @ B @ Less_eq @ Less @ S2 @ F3 )
=> ( ( ord_less_eq @ ( set @ B ) @ A5 @ S2 )
=> ( ( finite_finite2 @ B @ A5 )
=> ~ ! [L4: list @ B] :
( ( sorted_wrt @ A @ Less @ ( map @ B @ A @ F3 @ L4 ) )
=> ( ( ( set2 @ B @ L4 )
= A5 )
=> ( ( size_size @ ( list @ B ) @ L4 )
!= ( finite_card @ B @ A5 ) ) ) ) ) ) ) ).
% folding_insort_key.finite_set_strict_sorted
thf(fact_7326_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 )
@ ^ [X5: list @ A] :
( X5
!= ( nil @ A ) )
@ Xs ) ) ) ).
% transpose_transpose
thf(fact_7327_takeWhile__idem,axiom,
! [A: $tType,P2: A > $o,Xs: list @ A] :
( ( takeWhile @ A @ P2 @ ( takeWhile @ A @ P2 @ Xs ) )
= ( takeWhile @ A @ P2 @ Xs ) ) ).
% takeWhile_idem
thf(fact_7328_takeWhile__eq__all__conv,axiom,
! [A: $tType,P2: A > $o,Xs: list @ A] :
( ( ( takeWhile @ A @ P2 @ Xs )
= Xs )
= ( ! [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Xs ) )
=> ( P2 @ X5 ) ) ) ) ).
% takeWhile_eq_all_conv
thf(fact_7329_takeWhile__append1,axiom,
! [A: $tType,X: A,Xs: list @ A,P2: A > $o,Ys: list @ A] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ~ ( P2 @ X )
=> ( ( takeWhile @ A @ P2 @ ( append @ A @ Xs @ Ys ) )
= ( takeWhile @ A @ P2 @ Xs ) ) ) ) ).
% takeWhile_append1
thf(fact_7330_takeWhile__append2,axiom,
! [A: $tType,Xs: list @ A,P2: A > $o,Ys: list @ A] :
( ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
=> ( P2 @ X4 ) )
=> ( ( takeWhile @ A @ P2 @ ( append @ A @ Xs @ Ys ) )
= ( append @ A @ Xs @ ( takeWhile @ A @ P2 @ Ys ) ) ) ) ).
% takeWhile_append2
thf(fact_7331_takeWhile__replicate,axiom,
! [A: $tType,P2: A > $o,X: A,N: nat] :
( ( ( P2 @ X )
=> ( ( takeWhile @ A @ P2 @ ( replicate @ A @ N @ X ) )
= ( replicate @ A @ N @ X ) ) )
& ( ~ ( P2 @ X )
=> ( ( takeWhile @ A @ P2 @ ( replicate @ A @ N @ X ) )
= ( nil @ A ) ) ) ) ).
% takeWhile_replicate
thf(fact_7332_takeWhile__map,axiom,
! [A: $tType,B: $tType,P2: A > $o,F3: B > A,Xs: list @ B] :
( ( takeWhile @ A @ P2 @ ( map @ B @ A @ F3 @ Xs ) )
= ( map @ B @ A @ F3 @ ( takeWhile @ B @ ( comp @ A @ $o @ B @ P2 @ F3 ) @ Xs ) ) ) ).
% takeWhile_map
thf(fact_7333_sorted__takeWhile,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,P2: A > $o] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( takeWhile @ A @ P2 @ Xs ) ) ) ) ).
% sorted_takeWhile
thf(fact_7334_zip__takeWhile__snd,axiom,
! [A: $tType,B: $tType,Xs: list @ A,P2: B > $o,Ys: list @ B] :
( ( zip @ A @ B @ Xs @ ( takeWhile @ B @ P2 @ Ys ) )
= ( takeWhile @ ( product_prod @ A @ B ) @ ( comp @ B @ $o @ ( product_prod @ A @ B ) @ P2 @ ( product_snd @ A @ B ) ) @ ( zip @ A @ B @ Xs @ Ys ) ) ) ).
% zip_takeWhile_snd
thf(fact_7335_zip__takeWhile__fst,axiom,
! [A: $tType,B: $tType,P2: A > $o,Xs: list @ A,Ys: list @ B] :
( ( zip @ A @ B @ ( takeWhile @ A @ P2 @ Xs ) @ Ys )
= ( takeWhile @ ( product_prod @ A @ B ) @ ( comp @ A @ $o @ ( product_prod @ A @ B ) @ P2 @ ( product_fst @ A @ B ) ) @ ( zip @ A @ B @ Xs @ Ys ) ) ) ).
% zip_takeWhile_fst
thf(fact_7336_takeWhile__eq__Nil__iff,axiom,
! [A: $tType,P2: A > $o,Xs: list @ A] :
( ( ( takeWhile @ A @ P2 @ Xs )
= ( nil @ A ) )
= ( ( Xs
= ( nil @ A ) )
| ~ ( P2 @ ( hd @ A @ Xs ) ) ) ) ).
% takeWhile_eq_Nil_iff
thf(fact_7337_length__takeWhile__le,axiom,
! [A: $tType,P2: A > $o,Xs: list @ A] : ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P2 @ Xs ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) ).
% length_takeWhile_le
thf(fact_7338_distinct__takeWhile,axiom,
! [A: $tType,Xs: list @ A,P2: A > $o] :
( ( distinct @ A @ Xs )
=> ( distinct @ A @ ( takeWhile @ A @ P2 @ Xs ) ) ) ).
% distinct_takeWhile
thf(fact_7339_takeWhile__cong,axiom,
! [A: $tType,L: list @ A,K2: list @ A,P2: A > $o,Q: A > $o] :
( ( L = K2 )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ L ) )
=> ( ( P2 @ X4 )
= ( Q @ X4 ) ) )
=> ( ( takeWhile @ A @ P2 @ L )
= ( takeWhile @ A @ Q @ K2 ) ) ) ) ).
% takeWhile_cong
thf(fact_7340_set__takeWhileD,axiom,
! [A: $tType,X: A,P2: A > $o,Xs: list @ A] :
( ( member @ A @ X @ ( set2 @ A @ ( takeWhile @ A @ P2 @ Xs ) ) )
=> ( ( member @ A @ X @ ( set2 @ A @ Xs ) )
& ( P2 @ X ) ) ) ).
% set_takeWhileD
thf(fact_7341_takeWhile__eq__take,axiom,
! [A: $tType] :
( ( takeWhile @ A )
= ( ^ [P4: A > $o,Xs3: list @ A] : ( take @ A @ ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P4 @ Xs3 ) ) @ Xs3 ) ) ) ).
% takeWhile_eq_take
thf(fact_7342_takeWhile_Osimps_I1_J,axiom,
! [A: $tType,P2: A > $o] :
( ( takeWhile @ A @ P2 @ ( nil @ A ) )
= ( nil @ A ) ) ).
% takeWhile.simps(1)
thf(fact_7343_takeWhile_Osimps_I2_J,axiom,
! [A: $tType,P2: A > $o,X: A,Xs: list @ A] :
( ( ( P2 @ X )
=> ( ( takeWhile @ A @ P2 @ ( cons @ A @ X @ Xs ) )
= ( cons @ A @ X @ ( takeWhile @ A @ P2 @ Xs ) ) ) )
& ( ~ ( P2 @ X )
=> ( ( takeWhile @ A @ P2 @ ( cons @ A @ X @ Xs ) )
= ( nil @ A ) ) ) ) ).
% takeWhile.simps(2)
thf(fact_7344_folding__insort__key_Oinj__on,axiom,
! [A: $tType,B: $tType,Less_eq: A > A > $o,Less: A > A > $o,S2: set @ B,F3: B > A] :
( ( folding_insort_key @ A @ B @ Less_eq @ Less @ S2 @ F3 )
=> ( inj_on @ B @ A @ F3 @ S2 ) ) ).
% folding_insort_key.inj_on
thf(fact_7345_takeWhile__tail,axiom,
! [A: $tType,P2: A > $o,X: A,Xs: list @ A,L: list @ A] :
( ~ ( P2 @ X )
=> ( ( takeWhile @ A @ P2 @ ( append @ A @ Xs @ ( cons @ A @ X @ L ) ) )
= ( takeWhile @ A @ P2 @ Xs ) ) ) ).
% takeWhile_tail
thf(fact_7346_folding__insort__key_Odistinct__if__distinct__map,axiom,
! [A: $tType,B: $tType,Less_eq: A > A > $o,Less: A > A > $o,S2: set @ B,F3: B > A,Xs: list @ B] :
( ( folding_insort_key @ A @ B @ Less_eq @ Less @ S2 @ F3 )
=> ( ( distinct @ A @ ( map @ B @ A @ F3 @ Xs ) )
=> ( distinct @ B @ Xs ) ) ) ).
% folding_insort_key.distinct_if_distinct_map
thf(fact_7347_takeWhile__nth,axiom,
! [A: $tType,J: nat,P2: A > $o,Xs: list @ A] :
( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P2 @ Xs ) ) )
=> ( ( nth @ A @ ( takeWhile @ A @ P2 @ Xs ) @ J )
= ( nth @ A @ Xs @ J ) ) ) ).
% takeWhile_nth
thf(fact_7348_nth__length__takeWhile,axiom,
! [A: $tType,P2: A > $o,Xs: list @ A] :
( ( ord_less @ nat @ ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P2 @ Xs ) ) @ ( size_size @ ( list @ A ) @ Xs ) )
=> ~ ( P2 @ ( nth @ A @ Xs @ ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P2 @ Xs ) ) ) ) ) ).
% nth_length_takeWhile
thf(fact_7349_takeWhile__append,axiom,
! [A: $tType,Xs: list @ A,P2: A > $o,Ys: list @ A] :
( ( ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
=> ( P2 @ X4 ) )
=> ( ( takeWhile @ A @ P2 @ ( append @ A @ Xs @ Ys ) )
= ( append @ A @ Xs @ ( takeWhile @ A @ P2 @ Ys ) ) ) )
& ( ~ ! [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( P2 @ X3 ) )
=> ( ( takeWhile @ A @ P2 @ ( append @ A @ Xs @ Ys ) )
= ( takeWhile @ A @ P2 @ Xs ) ) ) ) ).
% takeWhile_append
thf(fact_7350_concat__conv__foldr,axiom,
! [A: $tType] :
( ( concat @ A )
= ( ^ [Xss3: list @ ( list @ A )] : ( foldr @ ( list @ A ) @ ( list @ A ) @ ( append @ A ) @ Xss3 @ ( nil @ A ) ) ) ) ).
% concat_conv_foldr
thf(fact_7351_length__takeWhile__less__P__nth,axiom,
! [A: $tType,J: nat,P2: A > $o,Xs: list @ A] :
( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ J )
=> ( P2 @ ( nth @ A @ Xs @ I3 ) ) )
=> ( ( ord_less_eq @ nat @ J @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( ord_less_eq @ nat @ J @ ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P2 @ Xs ) ) ) ) ) ).
% length_takeWhile_less_P_nth
thf(fact_7352_takeWhile__eq__take__P__nth,axiom,
! [A: $tType,N: nat,Xs: list @ A,P2: A > $o] :
( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ N )
=> ( ( ord_less @ nat @ I3 @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( P2 @ ( nth @ A @ Xs @ I3 ) ) ) )
=> ( ( ( ord_less @ nat @ N @ ( size_size @ ( list @ A ) @ Xs ) )
=> ~ ( P2 @ ( nth @ A @ Xs @ N ) ) )
=> ( ( takeWhile @ A @ P2 @ Xs )
= ( take @ A @ N @ Xs ) ) ) ) ).
% takeWhile_eq_take_P_nth
thf(fact_7353_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 ) )
@ ^ [X5: A] : X5 ) ) ).
% sorted_list_of_set.folding_insort_key_axioms
thf(fact_7354_filter__equals__takeWhile__sorted__rev,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F3: B > A,Xs: list @ B,T2: A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( rev @ A @ ( map @ B @ A @ F3 @ Xs ) ) )
=> ( ( filter2 @ B
@ ^ [X5: B] : ( ord_less @ A @ T2 @ ( F3 @ X5 ) )
@ Xs )
= ( takeWhile @ B
@ ^ [X5: B] : ( ord_less @ A @ T2 @ ( F3 @ X5 ) )
@ Xs ) ) ) ) ).
% filter_equals_takeWhile_sorted_rev
thf(fact_7355_folding__insort__key_Osorted__key__list__of__set__unique,axiom,
! [A: $tType,B: $tType,Less_eq: A > A > $o,Less: A > A > $o,S2: set @ B,F3: B > A,A5: set @ B,L: list @ B] :
( ( folding_insort_key @ A @ B @ Less_eq @ Less @ S2 @ F3 )
=> ( ( ord_less_eq @ ( set @ B ) @ A5 @ S2 )
=> ( ( finite_finite2 @ B @ A5 )
=> ( ( ( sorted_wrt @ A @ Less @ ( map @ B @ A @ F3 @ L ) )
& ( ( set2 @ B @ L )
= A5 )
& ( ( size_size @ ( list @ B ) @ L )
= ( finite_card @ B @ A5 ) ) )
= ( ( sorted8670434370408473282of_set @ A @ B @ Less_eq @ F3 @ A5 )
= L ) ) ) ) ) ).
% folding_insort_key.sorted_key_list_of_set_unique
thf(fact_7356_folding__insort__key_Osorted__key__list__of__set__remove,axiom,
! [A: $tType,B: $tType,Less_eq: A > A > $o,Less: A > A > $o,S2: set @ B,F3: B > A,X: B,A5: set @ B] :
( ( folding_insort_key @ A @ B @ Less_eq @ Less @ S2 @ F3 )
=> ( ( ord_less_eq @ ( set @ B ) @ ( insert @ B @ X @ A5 ) @ S2 )
=> ( ( finite_finite2 @ B @ A5 )
=> ( ( sorted8670434370408473282of_set @ A @ B @ Less_eq @ F3 @ ( minus_minus @ ( set @ B ) @ A5 @ ( insert @ B @ X @ ( bot_bot @ ( set @ B ) ) ) ) )
= ( remove1 @ B @ X @ ( sorted8670434370408473282of_set @ A @ B @ Less_eq @ F3 @ A5 ) ) ) ) ) ) ).
% folding_insort_key.sorted_key_list_of_set_remove
thf(fact_7357_linorder_Osorted__key__list__of__set_Ocong,axiom,
! [B: $tType,A: $tType] :
( ( sorted8670434370408473282of_set @ A @ B )
= ( sorted8670434370408473282of_set @ A @ B ) ) ).
% linorder.sorted_key_list_of_set.cong
thf(fact_7358_folding__insort__key_Osorted__key__list__of__set__inject,axiom,
! [A: $tType,B: $tType,Less_eq: A > A > $o,Less: A > A > $o,S2: set @ B,F3: B > A,A5: set @ B,B6: set @ B] :
( ( folding_insort_key @ A @ B @ Less_eq @ Less @ S2 @ F3 )
=> ( ( ord_less_eq @ ( set @ B ) @ A5 @ S2 )
=> ( ( ord_less_eq @ ( set @ B ) @ B6 @ S2 )
=> ( ( ( sorted8670434370408473282of_set @ A @ B @ Less_eq @ F3 @ A5 )
= ( sorted8670434370408473282of_set @ A @ B @ Less_eq @ F3 @ B6 ) )
=> ( ( finite_finite2 @ B @ A5 )
=> ( ( finite_finite2 @ B @ B6 )
=> ( A5 = B6 ) ) ) ) ) ) ) ).
% folding_insort_key.sorted_key_list_of_set_inject
thf(fact_7359_folding__insort__key_Osorted__key__list__of__set__empty,axiom,
! [A: $tType,B: $tType,Less_eq: A > A > $o,Less: A > A > $o,S2: set @ B,F3: B > A] :
( ( folding_insort_key @ A @ B @ Less_eq @ Less @ S2 @ F3 )
=> ( ( sorted8670434370408473282of_set @ A @ B @ Less_eq @ F3 @ ( bot_bot @ ( set @ B ) ) )
= ( nil @ B ) ) ) ).
% folding_insort_key.sorted_key_list_of_set_empty
thf(fact_7360_folding__insort__key_Oset__sorted__key__list__of__set,axiom,
! [A: $tType,B: $tType,Less_eq: A > A > $o,Less: A > A > $o,S2: set @ B,F3: B > A,A5: set @ B] :
( ( folding_insort_key @ A @ B @ Less_eq @ Less @ S2 @ F3 )
=> ( ( ord_less_eq @ ( set @ B ) @ A5 @ S2 )
=> ( ( finite_finite2 @ B @ A5 )
=> ( ( set2 @ B @ ( sorted8670434370408473282of_set @ A @ B @ Less_eq @ F3 @ A5 ) )
= A5 ) ) ) ) ).
% folding_insort_key.set_sorted_key_list_of_set
thf(fact_7361_folding__insort__key_Olength__sorted__key__list__of__set,axiom,
! [A: $tType,B: $tType,Less_eq: A > A > $o,Less: A > A > $o,S2: set @ B,F3: B > A,A5: set @ B] :
( ( folding_insort_key @ A @ B @ Less_eq @ Less @ S2 @ F3 )
=> ( ( ord_less_eq @ ( set @ B ) @ A5 @ S2 )
=> ( ( size_size @ ( list @ B ) @ ( sorted8670434370408473282of_set @ A @ B @ Less_eq @ F3 @ A5 ) )
= ( finite_card @ B @ A5 ) ) ) ) ).
% folding_insort_key.length_sorted_key_list_of_set
thf(fact_7362_folding__insort__key_Odistinct__sorted__key__list__of__set,axiom,
! [A: $tType,B: $tType,Less_eq: A > A > $o,Less: A > A > $o,S2: set @ B,F3: B > A,A5: set @ B] :
( ( folding_insort_key @ A @ B @ Less_eq @ Less @ S2 @ F3 )
=> ( ( ord_less_eq @ ( set @ B ) @ A5 @ S2 )
=> ( distinct @ A @ ( map @ B @ A @ F3 @ ( sorted8670434370408473282of_set @ A @ B @ Less_eq @ F3 @ A5 ) ) ) ) ) ).
% folding_insort_key.distinct_sorted_key_list_of_set
thf(fact_7363_folding__insort__key_Osorted__sorted__key__list__of__set,axiom,
! [A: $tType,B: $tType,Less_eq: A > A > $o,Less: A > A > $o,S2: set @ B,F3: B > A,A5: set @ B] :
( ( folding_insort_key @ A @ B @ Less_eq @ Less @ S2 @ F3 )
=> ( ( ord_less_eq @ ( set @ B ) @ A5 @ S2 )
=> ( sorted_wrt @ A @ Less_eq @ ( map @ B @ A @ F3 @ ( sorted8670434370408473282of_set @ A @ B @ Less_eq @ F3 @ A5 ) ) ) ) ) ).
% folding_insort_key.sorted_sorted_key_list_of_set
thf(fact_7364_folding__insort__key_Ostrict__sorted__key__list__of__set,axiom,
! [A: $tType,B: $tType,Less_eq: A > A > $o,Less: A > A > $o,S2: set @ B,F3: B > A,A5: set @ B] :
( ( folding_insort_key @ A @ B @ Less_eq @ Less @ S2 @ F3 )
=> ( ( ord_less_eq @ ( set @ B ) @ A5 @ S2 )
=> ( sorted_wrt @ A @ Less @ ( map @ B @ A @ F3 @ ( sorted8670434370408473282of_set @ A @ B @ Less_eq @ F3 @ A5 ) ) ) ) ) ).
% folding_insort_key.strict_sorted_key_list_of_set
thf(fact_7365_folding__insort__key_Osorted__key__list__of__set__eq__Nil__iff,axiom,
! [A: $tType,B: $tType,Less_eq: A > A > $o,Less: A > A > $o,S2: set @ B,F3: B > A,A5: set @ B] :
( ( folding_insort_key @ A @ B @ Less_eq @ Less @ S2 @ F3 )
=> ( ( ord_less_eq @ ( set @ B ) @ A5 @ S2 )
=> ( ( finite_finite2 @ B @ A5 )
=> ( ( ( sorted8670434370408473282of_set @ A @ B @ Less_eq @ F3 @ A5 )
= ( nil @ B ) )
= ( A5
= ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ).
% folding_insort_key.sorted_key_list_of_set_eq_Nil_iff
thf(fact_7366_folding__insort__key_Oidem__if__sorted__distinct,axiom,
! [A: $tType,B: $tType,Less_eq: A > A > $o,Less: A > A > $o,S2: set @ B,F3: B > A,Xs: list @ B] :
( ( folding_insort_key @ A @ B @ Less_eq @ Less @ S2 @ F3 )
=> ( ( ord_less_eq @ ( set @ B ) @ ( set2 @ B @ Xs ) @ S2 )
=> ( ( sorted_wrt @ A @ Less_eq @ ( map @ B @ A @ F3 @ Xs ) )
=> ( ( distinct @ B @ Xs )
=> ( ( sorted8670434370408473282of_set @ A @ B @ Less_eq @ F3 @ ( set2 @ B @ Xs ) )
= Xs ) ) ) ) ) ).
% folding_insort_key.idem_if_sorted_distinct
thf(fact_7367_folding__insort__key_Osorted__key__list__of__set__insert__remove,axiom,
! [A: $tType,B: $tType,Less_eq: A > A > $o,Less: A > A > $o,S2: set @ B,F3: B > A,X: B,A5: set @ B] :
( ( folding_insort_key @ A @ B @ Less_eq @ Less @ S2 @ F3 )
=> ( ( ord_less_eq @ ( set @ B ) @ ( insert @ B @ X @ A5 ) @ S2 )
=> ( ( finite_finite2 @ B @ A5 )
=> ( ( sorted8670434370408473282of_set @ A @ B @ Less_eq @ F3 @ ( insert @ B @ X @ A5 ) )
= ( insort_key @ A @ B @ Less_eq @ F3 @ X @ ( sorted8670434370408473282of_set @ A @ B @ Less_eq @ F3 @ ( minus_minus @ ( set @ B ) @ A5 @ ( insert @ B @ X @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ) ).
% folding_insort_key.sorted_key_list_of_set_insert_remove
thf(fact_7368_folding__insort__key_Osorted__key__list__of__set__insert,axiom,
! [A: $tType,B: $tType,Less_eq: A > A > $o,Less: A > A > $o,S2: set @ B,F3: B > A,X: B,A5: set @ B] :
( ( folding_insort_key @ A @ B @ Less_eq @ Less @ S2 @ F3 )
=> ( ( ord_less_eq @ ( set @ B ) @ ( insert @ B @ X @ A5 ) @ S2 )
=> ( ( finite_finite2 @ B @ A5 )
=> ( ~ ( member @ B @ X @ A5 )
=> ( ( sorted8670434370408473282of_set @ A @ B @ Less_eq @ F3 @ ( insert @ B @ X @ A5 ) )
= ( insort_key @ A @ B @ Less_eq @ F3 @ X @ ( sorted8670434370408473282of_set @ A @ B @ Less_eq @ F3 @ A5 ) ) ) ) ) ) ) ).
% folding_insort_key.sorted_key_list_of_set_insert
thf(fact_7369_linorder_Oinsort__key_Ocong,axiom,
! [B: $tType,A: $tType] :
( ( insort_key @ A @ B )
= ( insort_key @ A @ B ) ) ).
% linorder.insort_key.cong
thf(fact_7370_folding__insort__key_Oinsort__key__commute,axiom,
! [A: $tType,B: $tType,Less_eq: A > A > $o,Less: A > A > $o,S2: set @ B,F3: B > A,X: B,Y: B] :
( ( folding_insort_key @ A @ B @ Less_eq @ Less @ S2 @ F3 )
=> ( ( member @ B @ X @ S2 )
=> ( ( member @ B @ Y @ S2 )
=> ( ( comp @ ( list @ B ) @ ( list @ B ) @ ( list @ B ) @ ( insort_key @ A @ B @ Less_eq @ F3 @ Y ) @ ( insort_key @ A @ B @ Less_eq @ F3 @ X ) )
= ( comp @ ( list @ B ) @ ( list @ B ) @ ( list @ B ) @ ( insort_key @ A @ B @ Less_eq @ F3 @ X ) @ ( insort_key @ A @ B @ Less_eq @ F3 @ Y ) ) ) ) ) ) ).
% folding_insort_key.insort_key_commute
thf(fact_7371_extract__def,axiom,
! [A: $tType] :
( ( extract @ A )
= ( ^ [P4: A > $o,Xs3: list @ A] :
( case_list @ ( option @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) ) @ A @ ( none @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) )
@ ^ [Y6: A,Ys3: list @ A] : ( some @ ( product_prod @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) ) @ ( product_Pair @ ( list @ A ) @ ( product_prod @ A @ ( list @ A ) ) @ ( takeWhile @ A @ ( comp @ $o @ $o @ A @ (~) @ P4 ) @ Xs3 ) @ ( product_Pair @ A @ ( list @ A ) @ Y6 @ Ys3 ) ) )
@ ( dropWhile @ A @ ( comp @ $o @ $o @ A @ (~) @ P4 ) @ Xs3 ) ) ) ) ).
% extract_def
thf(fact_7372_sorted__find__Min,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,P2: A > $o] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( ? [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
& ( P2 @ X3 ) )
=> ( ( find @ A @ P2 @ Xs )
= ( some @ A
@ ( lattic643756798350308766er_Min @ A
@ ( collect @ A
@ ^ [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Xs ) )
& ( P2 @ X5 ) ) ) ) ) ) ) ) ) ).
% sorted_find_Min
thf(fact_7373_dropWhile__idem,axiom,
! [A: $tType,P2: A > $o,Xs: list @ A] :
( ( dropWhile @ A @ P2 @ ( dropWhile @ A @ P2 @ Xs ) )
= ( dropWhile @ A @ P2 @ Xs ) ) ).
% dropWhile_idem
thf(fact_7374_dropWhile__eq__Nil__conv,axiom,
! [A: $tType,P2: A > $o,Xs: list @ A] :
( ( ( dropWhile @ A @ P2 @ Xs )
= ( nil @ A ) )
= ( ! [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Xs ) )
=> ( P2 @ X5 ) ) ) ) ).
% dropWhile_eq_Nil_conv
thf(fact_7375_dropWhile__append1,axiom,
! [A: $tType,X: A,Xs: list @ A,P2: A > $o,Ys: list @ A] :
( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ~ ( P2 @ X )
=> ( ( dropWhile @ A @ P2 @ ( append @ A @ Xs @ Ys ) )
= ( append @ A @ ( dropWhile @ A @ P2 @ Xs ) @ Ys ) ) ) ) ).
% dropWhile_append1
thf(fact_7376_dropWhile__append2,axiom,
! [A: $tType,Xs: list @ A,P2: A > $o,Ys: list @ A] :
( ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
=> ( P2 @ X4 ) )
=> ( ( dropWhile @ A @ P2 @ ( append @ A @ Xs @ Ys ) )
= ( dropWhile @ A @ P2 @ Ys ) ) ) ).
% dropWhile_append2
thf(fact_7377_dropWhile__replicate,axiom,
! [A: $tType,P2: A > $o,X: A,N: nat] :
( ( ( P2 @ X )
=> ( ( dropWhile @ A @ P2 @ ( replicate @ A @ N @ X ) )
= ( nil @ A ) ) )
& ( ~ ( P2 @ X )
=> ( ( dropWhile @ A @ P2 @ ( replicate @ A @ N @ X ) )
= ( replicate @ A @ N @ X ) ) ) ) ).
% dropWhile_replicate
thf(fact_7378_takeWhile__dropWhile__id,axiom,
! [A: $tType,P2: A > $o,Xs: list @ A] :
( ( append @ A @ ( takeWhile @ A @ P2 @ Xs ) @ ( dropWhile @ A @ P2 @ Xs ) )
= Xs ) ).
% takeWhile_dropWhile_id
thf(fact_7379_dropWhile__append3,axiom,
! [A: $tType,P2: A > $o,Y: A,Xs: list @ A,Ys: list @ A] :
( ~ ( P2 @ Y )
=> ( ( dropWhile @ A @ P2 @ ( append @ A @ Xs @ ( cons @ A @ Y @ Ys ) ) )
= ( append @ A @ ( dropWhile @ A @ P2 @ Xs ) @ ( cons @ A @ Y @ Ys ) ) ) ) ).
% dropWhile_append3
thf(fact_7380_remdups__adj__Cons_H,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( remdups_adj @ A @ ( cons @ A @ X @ Xs ) )
= ( cons @ A @ X
@ ( remdups_adj @ A
@ ( dropWhile @ A
@ ^ [Y6: A] : Y6 = X
@ Xs ) ) ) ) ).
% remdups_adj_Cons'
thf(fact_7381_dropWhile_Osimps_I2_J,axiom,
! [A: $tType,P2: A > $o,X: A,Xs: list @ A] :
( ( ( P2 @ X )
=> ( ( dropWhile @ A @ P2 @ ( cons @ A @ X @ Xs ) )
= ( dropWhile @ A @ P2 @ Xs ) ) )
& ( ~ ( P2 @ X )
=> ( ( dropWhile @ A @ P2 @ ( cons @ A @ X @ Xs ) )
= ( cons @ A @ X @ Xs ) ) ) ) ).
% dropWhile.simps(2)
thf(fact_7382_dropWhile_Osimps_I1_J,axiom,
! [A: $tType,P2: A > $o] :
( ( dropWhile @ A @ P2 @ ( nil @ A ) )
= ( nil @ A ) ) ).
% dropWhile.simps(1)
thf(fact_7383_find__cong,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,P2: A > $o,Q: A > $o] :
( ( Xs = Ys )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Ys ) )
=> ( ( P2 @ X4 )
= ( Q @ X4 ) ) )
=> ( ( find @ A @ P2 @ Xs )
= ( find @ A @ Q @ Ys ) ) ) ) ).
% find_cong
thf(fact_7384_dropWhile__cong,axiom,
! [A: $tType,L: list @ A,K2: list @ A,P2: A > $o,Q: A > $o] :
( ( L = K2 )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ L ) )
=> ( ( P2 @ X4 )
= ( Q @ X4 ) ) )
=> ( ( dropWhile @ A @ P2 @ L )
= ( dropWhile @ A @ Q @ K2 ) ) ) ) ).
% dropWhile_cong
thf(fact_7385_set__dropWhileD,axiom,
! [A: $tType,X: A,P2: A > $o,Xs: list @ A] :
( ( member @ A @ X @ ( set2 @ A @ ( dropWhile @ A @ P2 @ Xs ) ) )
=> ( member @ A @ X @ ( set2 @ A @ Xs ) ) ) ).
% set_dropWhileD
thf(fact_7386_distinct__dropWhile,axiom,
! [A: $tType,Xs: list @ A,P2: A > $o] :
( ( distinct @ A @ Xs )
=> ( distinct @ A @ ( dropWhile @ A @ P2 @ Xs ) ) ) ).
% distinct_dropWhile
thf(fact_7387_length__dropWhile__le,axiom,
! [A: $tType,P2: A > $o,Xs: list @ A] : ( ord_less_eq @ nat @ ( size_size @ ( list @ A ) @ ( dropWhile @ A @ P2 @ Xs ) ) @ ( size_size @ ( list @ A ) @ Xs ) ) ).
% length_dropWhile_le
thf(fact_7388_hd__dropWhile,axiom,
! [A: $tType,P2: A > $o,Xs: list @ A] :
( ( ( dropWhile @ A @ P2 @ Xs )
!= ( nil @ A ) )
=> ~ ( P2 @ ( hd @ A @ ( dropWhile @ A @ P2 @ Xs ) ) ) ) ).
% hd_dropWhile
thf(fact_7389_dropWhile__eq__self__iff,axiom,
! [A: $tType,P2: A > $o,Xs: list @ A] :
( ( ( dropWhile @ A @ P2 @ Xs )
= Xs )
= ( ( Xs
= ( nil @ A ) )
| ~ ( P2 @ ( hd @ A @ Xs ) ) ) ) ).
% dropWhile_eq_self_iff
thf(fact_7390_sorted__dropWhile,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,P2: A > $o] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( dropWhile @ A @ P2 @ Xs ) ) ) ) ).
% sorted_dropWhile
thf(fact_7391_dropWhile__map,axiom,
! [A: $tType,B: $tType,P2: A > $o,F3: B > A,Xs: list @ B] :
( ( dropWhile @ A @ P2 @ ( map @ B @ A @ F3 @ Xs ) )
= ( map @ B @ A @ F3 @ ( dropWhile @ B @ ( comp @ A @ $o @ B @ P2 @ F3 ) @ Xs ) ) ) ).
% dropWhile_map
thf(fact_7392_find__dropWhile,axiom,
! [A: $tType] :
( ( find @ A )
= ( ^ [P4: A > $o,Xs3: list @ A] :
( case_list @ ( option @ A ) @ A @ ( none @ A )
@ ^ [X5: A,Xa4: list @ A] : ( some @ A @ X5 )
@ ( dropWhile @ A @ ( comp @ $o @ $o @ A @ (~) @ P4 ) @ Xs3 ) ) ) ) ).
% find_dropWhile
thf(fact_7393_find_Osimps_I2_J,axiom,
! [A: $tType,P2: A > $o,X: A,Xs: list @ A] :
( ( ( P2 @ X )
=> ( ( find @ A @ P2 @ ( cons @ A @ X @ Xs ) )
= ( some @ A @ X ) ) )
& ( ~ ( P2 @ X )
=> ( ( find @ A @ P2 @ ( cons @ A @ X @ Xs ) )
= ( find @ A @ P2 @ Xs ) ) ) ) ).
% find.simps(2)
thf(fact_7394_find_Osimps_I1_J,axiom,
! [A: $tType,Uu: A > $o] :
( ( find @ A @ Uu @ ( nil @ A ) )
= ( none @ A ) ) ).
% find.simps(1)
thf(fact_7395_find__None__iff,axiom,
! [A: $tType,P2: A > $o,Xs: list @ A] :
( ( ( find @ A @ P2 @ Xs )
= ( none @ A ) )
= ( ~ ? [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Xs ) )
& ( P2 @ X5 ) ) ) ) ).
% find_None_iff
thf(fact_7396_find__None__iff2,axiom,
! [A: $tType,P2: A > $o,Xs: list @ A] :
( ( ( none @ A )
= ( find @ A @ P2 @ Xs ) )
= ( ~ ? [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Xs ) )
& ( P2 @ X5 ) ) ) ) ).
% find_None_iff2
thf(fact_7397_dropWhile__eq__Cons__conv,axiom,
! [A: $tType,P2: A > $o,Xs: list @ A,Y: A,Ys: list @ A] :
( ( ( dropWhile @ A @ P2 @ Xs )
= ( cons @ A @ Y @ Ys ) )
= ( ( Xs
= ( append @ A @ ( takeWhile @ A @ P2 @ Xs ) @ ( cons @ A @ Y @ Ys ) ) )
& ~ ( P2 @ Y ) ) ) ).
% dropWhile_eq_Cons_conv
thf(fact_7398_takeWhile__eq__filter,axiom,
! [A: $tType,P2: A > $o,Xs: list @ A] :
( ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ ( dropWhile @ A @ P2 @ Xs ) ) )
=> ~ ( P2 @ X4 ) )
=> ( ( takeWhile @ A @ P2 @ Xs )
= ( filter2 @ A @ P2 @ Xs ) ) ) ).
% takeWhile_eq_filter
thf(fact_7399_dropWhile__eq__drop,axiom,
! [A: $tType] :
( ( dropWhile @ A )
= ( ^ [P4: A > $o,Xs3: list @ A] : ( drop @ A @ ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P4 @ Xs3 ) ) @ Xs3 ) ) ) ).
% dropWhile_eq_drop
thf(fact_7400_dropWhile__append,axiom,
! [A: $tType,Xs: list @ A,P2: A > $o,Ys: list @ A] :
( ( ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
=> ( P2 @ X4 ) )
=> ( ( dropWhile @ A @ P2 @ ( append @ A @ Xs @ Ys ) )
= ( dropWhile @ A @ P2 @ Ys ) ) )
& ( ~ ! [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( P2 @ X3 ) )
=> ( ( dropWhile @ A @ P2 @ ( append @ A @ Xs @ Ys ) )
= ( append @ A @ ( dropWhile @ A @ P2 @ Xs ) @ Ys ) ) ) ) ).
% dropWhile_append
thf(fact_7401_remdups__adj__append__dropWhile,axiom,
! [A: $tType,Xs: list @ A,Y: A,Ys: list @ A] :
( ( remdups_adj @ A @ ( append @ A @ Xs @ ( cons @ A @ Y @ Ys ) ) )
= ( append @ A @ ( remdups_adj @ A @ ( append @ A @ Xs @ ( cons @ A @ Y @ ( nil @ A ) ) ) )
@ ( remdups_adj @ A
@ ( dropWhile @ A
@ ^ [X5: A] : X5 = Y
@ Ys ) ) ) ) ).
% remdups_adj_append_dropWhile
thf(fact_7402_tl__remdups__adj,axiom,
! [A: $tType,Ys: list @ A] :
( ( Ys
!= ( nil @ A ) )
=> ( ( tl @ A @ ( remdups_adj @ A @ Ys ) )
= ( remdups_adj @ A
@ ( dropWhile @ A
@ ^ [X5: A] :
( X5
= ( hd @ A @ Ys ) )
@ ( tl @ A @ Ys ) ) ) ) ) ).
% tl_remdups_adj
thf(fact_7403_dropWhile__nth,axiom,
! [A: $tType,J: nat,P2: A > $o,Xs: list @ A] :
( ( ord_less @ nat @ J @ ( size_size @ ( list @ A ) @ ( dropWhile @ A @ P2 @ Xs ) ) )
=> ( ( nth @ A @ ( dropWhile @ A @ P2 @ Xs ) @ J )
= ( nth @ A @ Xs @ ( plus_plus @ nat @ J @ ( size_size @ ( list @ A ) @ ( takeWhile @ A @ P2 @ Xs ) ) ) ) ) ) ).
% dropWhile_nth
thf(fact_7404_dropWhile__neq__rev,axiom,
! [A: $tType,Xs: list @ A,X: A] :
( ( distinct @ A @ Xs )
=> ( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ( dropWhile @ A
@ ^ [Y6: A] : Y6 != X
@ ( rev @ A @ Xs ) )
= ( cons @ A @ X
@ ( rev @ A
@ ( takeWhile @ A
@ ^ [Y6: A] : Y6 != X
@ Xs ) ) ) ) ) ) ).
% dropWhile_neq_rev
thf(fact_7405_takeWhile__neq__rev,axiom,
! [A: $tType,Xs: list @ A,X: A] :
( ( distinct @ A @ Xs )
=> ( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ( takeWhile @ A
@ ^ [Y6: A] : Y6 != X
@ ( rev @ A @ Xs ) )
= ( rev @ A
@ ( tl @ A
@ ( dropWhile @ A
@ ^ [Y6: A] : Y6 != X
@ Xs ) ) ) ) ) ) ).
% takeWhile_neq_rev
thf(fact_7406_find__Some__iff2,axiom,
! [A: $tType,X: A,P2: A > $o,Xs: list @ A] :
( ( ( some @ A @ X )
= ( find @ A @ P2 @ Xs ) )
= ( ? [I4: nat] :
( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ Xs ) )
& ( P2 @ ( nth @ A @ Xs @ I4 ) )
& ( X
= ( nth @ A @ Xs @ I4 ) )
& ! [J3: nat] :
( ( ord_less @ nat @ J3 @ I4 )
=> ~ ( P2 @ ( nth @ A @ Xs @ J3 ) ) ) ) ) ) ).
% find_Some_iff2
thf(fact_7407_find__Some__iff,axiom,
! [A: $tType,P2: A > $o,Xs: list @ A,X: A] :
( ( ( find @ A @ P2 @ Xs )
= ( some @ A @ X ) )
= ( ? [I4: nat] :
( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ Xs ) )
& ( P2 @ ( nth @ A @ Xs @ I4 ) )
& ( X
= ( nth @ A @ Xs @ I4 ) )
& ! [J3: nat] :
( ( ord_less @ nat @ J3 @ I4 )
=> ~ ( P2 @ ( nth @ A @ Xs @ J3 ) ) ) ) ) ) ).
% find_Some_iff
thf(fact_7408_partition__filter__conv,axiom,
! [A: $tType] :
( ( partition @ A )
= ( ^ [F4: A > $o,Xs3: list @ A] : ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( filter2 @ A @ F4 @ Xs3 ) @ ( filter2 @ A @ ( comp @ $o @ $o @ A @ (~) @ F4 ) @ Xs3 ) ) ) ) ).
% partition_filter_conv
thf(fact_7409_lenlex__append2,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),Us: list @ A,Xs: list @ A,Ys: list @ A] :
( ( irrefl @ A @ R )
=> ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Us @ Xs ) @ ( append @ A @ Us @ Ys ) ) @ ( lenlex @ A @ R ) )
= ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( lenlex @ A @ R ) ) ) ) ).
% lenlex_append2
thf(fact_7410_lexord__same__pref__if__irrefl,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),Xs: list @ A,Ys: list @ A,Zs: list @ A] :
( ( irrefl @ A @ R2 )
=> ( ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( append @ A @ Xs @ Ys ) @ ( append @ A @ Xs @ Zs ) ) @ ( lexord @ A @ R2 ) )
= ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Ys @ Zs ) @ ( lexord @ A @ R2 ) ) ) ) ).
% lexord_same_pref_if_irrefl
thf(fact_7411_partition__filter1,axiom,
! [A: $tType,P2: A > $o,Xs: list @ A] :
( ( product_fst @ ( list @ A ) @ ( list @ A ) @ ( partition @ A @ P2 @ Xs ) )
= ( filter2 @ A @ P2 @ Xs ) ) ).
% partition_filter1
thf(fact_7412_irreflI,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ! [A4: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ A4 ) @ R )
=> ( irrefl @ A @ R ) ) ).
% irreflI
thf(fact_7413_irrefl__def,axiom,
! [A: $tType] :
( ( irrefl @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A )] :
! [A6: A] :
~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A6 @ A6 ) @ R5 ) ) ) ).
% irrefl_def
thf(fact_7414_lexord__irrefl,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( irrefl @ A @ R )
=> ( irrefl @ ( list @ A ) @ ( lexord @ A @ R ) ) ) ).
% lexord_irrefl
thf(fact_7415_irrefl__lex,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( irrefl @ A @ R2 )
=> ( irrefl @ ( list @ A ) @ ( lex @ A @ R2 ) ) ) ).
% irrefl_lex
thf(fact_7416_partition_Osimps_I1_J,axiom,
! [A: $tType,P2: A > $o] :
( ( partition @ A @ P2 @ ( nil @ A ) )
= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ ( nil @ A ) ) ) ).
% partition.simps(1)
thf(fact_7417_partition__P,axiom,
! [A: $tType,P2: A > $o,Xs: list @ A,Yes: list @ A,No4: list @ A] :
( ( ( partition @ A @ P2 @ Xs )
= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Yes @ No4 ) )
=> ( ! [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Yes ) )
=> ( P2 @ X3 ) )
& ! [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ No4 ) )
=> ~ ( P2 @ X3 ) ) ) ) ).
% partition_P
thf(fact_7418_lexl__not__refl,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),X: list @ A] :
( ( irrefl @ A @ R2 )
=> ~ ( member @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ X ) @ ( lex @ A @ R2 ) ) ) ).
% lexl_not_refl
thf(fact_7419_partition_Osimps_I2_J,axiom,
! [A: $tType,P2: A > $o,X: A,Xs: list @ A] :
( ( partition @ A @ P2 @ ( cons @ A @ X @ Xs ) )
= ( product_case_prod @ ( list @ A ) @ ( list @ A ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) )
@ ^ [Yes2: list @ A,No: list @ A] : ( if @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( P2 @ X ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Yes2 ) @ No ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Yes2 @ ( cons @ A @ X @ No ) ) )
@ ( partition @ A @ P2 @ Xs ) ) ) ).
% partition.simps(2)
thf(fact_7420_partition__filter2,axiom,
! [A: $tType,P2: A > $o,Xs: list @ A] :
( ( product_snd @ ( list @ A ) @ ( list @ A ) @ ( partition @ A @ P2 @ Xs ) )
= ( filter2 @ A @ ( comp @ $o @ $o @ A @ (~) @ P2 ) @ Xs ) ) ).
% partition_filter2
thf(fact_7421_partition__set,axiom,
! [A: $tType,P2: A > $o,Xs: list @ A,Yes: list @ A,No4: list @ A] :
( ( ( partition @ A @ P2 @ Xs )
= ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Yes @ No4 ) )
=> ( ( sup_sup @ ( set @ A ) @ ( set2 @ A @ Yes ) @ ( set2 @ A @ No4 ) )
= ( set2 @ A @ Xs ) ) ) ).
% partition_set
thf(fact_7422_min__list_Oelims,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X: list @ A,Y: A] :
( ( ( min_list @ A @ X )
= Y )
=> ( ! [X4: A,Xs2: list @ A] :
( ( X
= ( cons @ A @ X4 @ Xs2 ) )
=> ( Y
!= ( case_list @ A @ A @ X4
@ ^ [A6: A,List2: list @ A] : ( ord_min @ A @ X4 @ ( min_list @ A @ Xs2 ) )
@ Xs2 ) ) )
=> ~ ( ( X
= ( nil @ A ) )
=> ( Y
!= ( undefined @ A ) ) ) ) ) ) ).
% min_list.elims
thf(fact_7423_lists__length__Suc__eq,axiom,
! [A: $tType,A5: set @ A,N: nat] :
( ( collect @ ( list @ A )
@ ^ [Xs3: list @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs3 ) @ A5 )
& ( ( size_size @ ( list @ A ) @ Xs3 )
= ( suc @ N ) ) ) )
= ( image2 @ ( product_prod @ ( list @ A ) @ A ) @ ( list @ A )
@ ( product_case_prod @ ( list @ A ) @ A @ ( list @ A )
@ ^ [Xs3: list @ A,N2: A] : ( cons @ A @ N2 @ Xs3 ) )
@ ( product_Sigma @ ( list @ A ) @ A
@ ( collect @ ( list @ A )
@ ^ [Xs3: list @ A] :
( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs3 ) @ A5 )
& ( ( size_size @ ( list @ A ) @ Xs3 )
= N ) ) )
@ ^ [Uu3: list @ A] : A5 ) ) ) ).
% lists_length_Suc_eq
thf(fact_7424_mem__Sigma__iff,axiom,
! [B: $tType,A: $tType,A2: A,B2: B,A5: set @ A,B6: A > ( set @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ ( product_Sigma @ A @ B @ A5 @ B6 ) )
= ( ( member @ A @ A2 @ A5 )
& ( member @ B @ B2 @ ( B6 @ A2 ) ) ) ) ).
% mem_Sigma_iff
thf(fact_7425_SigmaI,axiom,
! [B: $tType,A: $tType,A2: A,A5: set @ A,B2: B,B6: A > ( set @ B )] :
( ( member @ A @ A2 @ A5 )
=> ( ( member @ B @ B2 @ ( B6 @ A2 ) )
=> ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ ( product_Sigma @ A @ B @ A5 @ B6 ) ) ) ) ).
% SigmaI
thf(fact_7426_Collect__case__prod,axiom,
! [B: $tType,A: $tType,P2: A > $o,Q: B > $o] :
( ( collect @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [A6: A,B5: B] :
( ( P2 @ A6 )
& ( Q @ B5 ) ) ) )
= ( product_Sigma @ A @ B @ ( collect @ A @ P2 )
@ ^ [Uu3: A] : ( collect @ B @ Q ) ) ) ).
% Collect_case_prod
thf(fact_7427_Sigma__empty1,axiom,
! [B: $tType,A: $tType,B6: A > ( set @ B )] :
( ( product_Sigma @ A @ B @ ( bot_bot @ ( set @ A ) ) @ B6 )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% Sigma_empty1
thf(fact_7428_Compl__Times__UNIV2,axiom,
! [B: $tType,A: $tType,A5: set @ A] :
( ( uminus_uminus @ ( set @ ( product_prod @ A @ B ) )
@ ( product_Sigma @ A @ B @ A5
@ ^ [Uu3: A] : ( top_top @ ( set @ B ) ) ) )
= ( product_Sigma @ A @ B @ ( uminus_uminus @ ( set @ A ) @ A5 )
@ ^ [Uu3: A] : ( top_top @ ( set @ B ) ) ) ) ).
% Compl_Times_UNIV2
thf(fact_7429_Compl__Times__UNIV1,axiom,
! [B: $tType,A: $tType,A5: set @ B] :
( ( uminus_uminus @ ( set @ ( product_prod @ A @ B ) )
@ ( product_Sigma @ A @ B @ ( top_top @ ( set @ A ) )
@ ^ [Uu3: A] : A5 ) )
= ( product_Sigma @ A @ B @ ( top_top @ ( set @ A ) )
@ ^ [Uu3: A] : ( uminus_uminus @ ( set @ B ) @ A5 ) ) ) ).
% Compl_Times_UNIV1
thf(fact_7430_Times__empty,axiom,
! [A: $tType,B: $tType,A5: set @ A,B6: set @ B] :
( ( ( product_Sigma @ A @ B @ A5
@ ^ [Uu3: A] : B6 )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) )
= ( ( A5
= ( bot_bot @ ( set @ A ) ) )
| ( B6
= ( bot_bot @ ( set @ B ) ) ) ) ) ).
% Times_empty
thf(fact_7431_Sigma__empty2,axiom,
! [B: $tType,A: $tType,A5: set @ A] :
( ( product_Sigma @ A @ B @ A5
@ ^ [Uu3: A] : ( bot_bot @ ( set @ B ) ) )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% Sigma_empty2
thf(fact_7432_finite__SigmaI,axiom,
! [B: $tType,A: $tType,A5: set @ A,B6: A > ( set @ B )] :
( ( finite_finite2 @ A @ A5 )
=> ( ! [A4: A] :
( ( member @ A @ A4 @ A5 )
=> ( finite_finite2 @ B @ ( B6 @ A4 ) ) )
=> ( finite_finite2 @ ( product_prod @ A @ B ) @ ( product_Sigma @ A @ B @ A5 @ B6 ) ) ) ) ).
% finite_SigmaI
thf(fact_7433_UNIV__Times__UNIV,axiom,
! [B: $tType,A: $tType] :
( ( product_Sigma @ A @ B @ ( top_top @ ( set @ A ) )
@ ^ [Uu3: A] : ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% UNIV_Times_UNIV
thf(fact_7434_fst__image__times,axiom,
! [B: $tType,A: $tType,B6: set @ B,A5: set @ A] :
( ( ( B6
= ( bot_bot @ ( set @ B ) ) )
=> ( ( image2 @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B )
@ ( product_Sigma @ A @ B @ A5
@ ^ [Uu3: A] : B6 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( B6
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( image2 @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B )
@ ( product_Sigma @ A @ B @ A5
@ ^ [Uu3: A] : B6 ) )
= A5 ) ) ) ).
% fst_image_times
thf(fact_7435_snd__image__times,axiom,
! [B: $tType,A: $tType,A5: set @ B,B6: set @ A] :
( ( ( A5
= ( bot_bot @ ( set @ B ) ) )
=> ( ( image2 @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A )
@ ( product_Sigma @ B @ A @ A5
@ ^ [Uu3: B] : B6 ) )
= ( bot_bot @ ( set @ A ) ) ) )
& ( ( A5
!= ( bot_bot @ ( set @ B ) ) )
=> ( ( image2 @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A )
@ ( product_Sigma @ B @ A @ A5
@ ^ [Uu3: B] : B6 ) )
= B6 ) ) ) ).
% snd_image_times
thf(fact_7436_set__product,axiom,
! [B: $tType,A: $tType,Xs: list @ A,Ys: list @ B] :
( ( set2 @ ( product_prod @ A @ B ) @ ( product @ A @ B @ Xs @ Ys ) )
= ( product_Sigma @ A @ B @ ( set2 @ A @ Xs )
@ ^ [Uu3: A] : ( set2 @ B @ Ys ) ) ) ).
% set_product
thf(fact_7437_insert__Times__insert,axiom,
! [B: $tType,A: $tType,A2: A,A5: set @ A,B2: B,B6: set @ B] :
( ( product_Sigma @ A @ B @ ( insert @ A @ A2 @ A5 )
@ ^ [Uu3: A] : ( insert @ B @ B2 @ B6 ) )
= ( insert @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 )
@ ( sup_sup @ ( set @ ( product_prod @ A @ B ) )
@ ( product_Sigma @ A @ B @ A5
@ ^ [Uu3: A] : ( insert @ B @ B2 @ B6 ) )
@ ( product_Sigma @ A @ B @ ( insert @ A @ A2 @ A5 )
@ ^ [Uu3: A] : B6 ) ) ) ) ).
% insert_Times_insert
thf(fact_7438_card__SigmaI,axiom,
! [B: $tType,A: $tType,A5: set @ A,B6: A > ( set @ B )] :
( ( finite_finite2 @ A @ A5 )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ A5 )
=> ( finite_finite2 @ B @ ( B6 @ X4 ) ) )
=> ( ( finite_card @ ( product_prod @ A @ B ) @ ( product_Sigma @ A @ B @ A5 @ B6 ) )
= ( groups7311177749621191930dd_sum @ A @ nat
@ ^ [A6: A] : ( finite_card @ B @ ( B6 @ A6 ) )
@ A5 ) ) ) ) ).
% card_SigmaI
thf(fact_7439_inj__on__apfst,axiom,
! [B: $tType,C: $tType,A: $tType,F3: A > C,A5: set @ A] :
( ( inj_on @ ( product_prod @ A @ B ) @ ( product_prod @ C @ B ) @ ( product_apfst @ A @ C @ B @ F3 )
@ ( product_Sigma @ A @ B @ A5
@ ^ [Uu3: A] : ( top_top @ ( set @ B ) ) ) )
= ( inj_on @ A @ C @ F3 @ A5 ) ) ).
% inj_on_apfst
thf(fact_7440_inj__on__apsnd,axiom,
! [A: $tType,C: $tType,B: $tType,F3: B > C,A5: set @ B] :
( ( inj_on @ ( product_prod @ A @ B ) @ ( product_prod @ A @ C ) @ ( product_apsnd @ B @ C @ A @ F3 )
@ ( product_Sigma @ A @ B @ ( top_top @ ( set @ A ) )
@ ^ [Uu3: A] : A5 ) )
= ( inj_on @ B @ C @ F3 @ A5 ) ) ).
% inj_on_apsnd
thf(fact_7441_Collect__case__prod__Sigma,axiom,
! [B: $tType,A: $tType,P2: A > $o,Q: A > B > $o] :
( ( collect @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [X5: A,Y6: B] :
( ( P2 @ X5 )
& ( Q @ X5 @ Y6 ) ) ) )
= ( product_Sigma @ A @ B @ ( collect @ A @ P2 )
@ ^ [X5: A] : ( collect @ B @ ( Q @ X5 ) ) ) ) ).
% Collect_case_prod_Sigma
thf(fact_7442_Times__Int__distrib1,axiom,
! [B: $tType,A: $tType,A5: set @ A,B6: set @ A,C5: set @ B] :
( ( product_Sigma @ A @ B @ ( inf_inf @ ( set @ A ) @ A5 @ B6 )
@ ^ [Uu3: A] : C5 )
= ( inf_inf @ ( set @ ( product_prod @ A @ B ) )
@ ( product_Sigma @ A @ B @ A5
@ ^ [Uu3: A] : C5 )
@ ( product_Sigma @ A @ B @ B6
@ ^ [Uu3: A] : C5 ) ) ) ).
% Times_Int_distrib1
thf(fact_7443_Sigma__Int__distrib2,axiom,
! [B: $tType,A: $tType,I6: set @ A,A5: A > ( set @ B ),B6: A > ( set @ B )] :
( ( product_Sigma @ A @ B @ I6
@ ^ [I4: A] : ( inf_inf @ ( set @ B ) @ ( A5 @ I4 ) @ ( B6 @ I4 ) ) )
= ( inf_inf @ ( set @ ( product_prod @ A @ B ) ) @ ( product_Sigma @ A @ B @ I6 @ A5 ) @ ( product_Sigma @ A @ B @ I6 @ B6 ) ) ) ).
% Sigma_Int_distrib2
thf(fact_7444_Times__Int__Times,axiom,
! [B: $tType,A: $tType,A5: set @ A,B6: set @ B,C5: set @ A,D6: set @ B] :
( ( inf_inf @ ( set @ ( product_prod @ A @ B ) )
@ ( product_Sigma @ A @ B @ A5
@ ^ [Uu3: A] : B6 )
@ ( product_Sigma @ A @ B @ C5
@ ^ [Uu3: A] : D6 ) )
= ( product_Sigma @ A @ B @ ( inf_inf @ ( set @ A ) @ A5 @ C5 )
@ ^ [Uu3: A] : ( inf_inf @ ( set @ B ) @ B6 @ D6 ) ) ) ).
% Times_Int_Times
thf(fact_7445_Sigma__Int__distrib1,axiom,
! [B: $tType,A: $tType,I6: set @ A,J4: set @ A,C5: A > ( set @ B )] :
( ( product_Sigma @ A @ B @ ( inf_inf @ ( set @ A ) @ I6 @ J4 ) @ C5 )
= ( inf_inf @ ( set @ ( product_prod @ A @ B ) ) @ ( product_Sigma @ A @ B @ I6 @ C5 ) @ ( product_Sigma @ A @ B @ J4 @ C5 ) ) ) ).
% Sigma_Int_distrib1
thf(fact_7446_Sigma__Diff__distrib1,axiom,
! [B: $tType,A: $tType,I6: set @ A,J4: set @ A,C5: A > ( set @ B )] :
( ( product_Sigma @ A @ B @ ( minus_minus @ ( set @ A ) @ I6 @ J4 ) @ C5 )
= ( minus_minus @ ( set @ ( product_prod @ A @ B ) ) @ ( product_Sigma @ A @ B @ I6 @ C5 ) @ ( product_Sigma @ A @ B @ J4 @ C5 ) ) ) ).
% Sigma_Diff_distrib1
thf(fact_7447_Times__Diff__distrib1,axiom,
! [B: $tType,A: $tType,A5: set @ A,B6: set @ A,C5: set @ B] :
( ( product_Sigma @ A @ B @ ( minus_minus @ ( set @ A ) @ A5 @ B6 )
@ ^ [Uu3: A] : C5 )
= ( minus_minus @ ( set @ ( product_prod @ A @ B ) )
@ ( product_Sigma @ A @ B @ A5
@ ^ [Uu3: A] : C5 )
@ ( product_Sigma @ A @ B @ B6
@ ^ [Uu3: A] : C5 ) ) ) ).
% Times_Diff_distrib1
thf(fact_7448_Sigma__Diff__distrib2,axiom,
! [B: $tType,A: $tType,I6: set @ A,A5: A > ( set @ B ),B6: A > ( set @ B )] :
( ( product_Sigma @ A @ B @ I6
@ ^ [I4: A] : ( minus_minus @ ( set @ B ) @ ( A5 @ I4 ) @ ( B6 @ I4 ) ) )
= ( minus_minus @ ( set @ ( product_prod @ A @ B ) ) @ ( product_Sigma @ A @ B @ I6 @ A5 ) @ ( product_Sigma @ A @ B @ I6 @ B6 ) ) ) ).
% Sigma_Diff_distrib2
thf(fact_7449_Sigma__cong,axiom,
! [B: $tType,A: $tType,A5: set @ A,B6: set @ A,C5: A > ( set @ B ),D6: A > ( set @ B )] :
( ( A5 = B6 )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ B6 )
=> ( ( C5 @ X4 )
= ( D6 @ X4 ) ) )
=> ( ( product_Sigma @ A @ B @ A5 @ C5 )
= ( product_Sigma @ A @ B @ B6 @ D6 ) ) ) ) ).
% Sigma_cong
thf(fact_7450_Times__eq__cancel2,axiom,
! [A: $tType,B: $tType,X: A,C5: set @ A,A5: set @ B,B6: set @ B] :
( ( member @ A @ X @ C5 )
=> ( ( ( product_Sigma @ B @ A @ A5
@ ^ [Uu3: B] : C5 )
= ( product_Sigma @ B @ A @ B6
@ ^ [Uu3: B] : C5 ) )
= ( A5 = B6 ) ) ) ).
% Times_eq_cancel2
thf(fact_7451_finite__cartesian__product,axiom,
! [B: $tType,A: $tType,A5: set @ A,B6: set @ B] :
( ( finite_finite2 @ A @ A5 )
=> ( ( finite_finite2 @ B @ B6 )
=> ( finite_finite2 @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A5
@ ^ [Uu3: A] : B6 ) ) ) ) ).
% finite_cartesian_product
thf(fact_7452_Times__Un__distrib1,axiom,
! [B: $tType,A: $tType,A5: set @ A,B6: set @ A,C5: set @ B] :
( ( product_Sigma @ A @ B @ ( sup_sup @ ( set @ A ) @ A5 @ B6 )
@ ^ [Uu3: A] : C5 )
= ( sup_sup @ ( set @ ( product_prod @ A @ B ) )
@ ( product_Sigma @ A @ B @ A5
@ ^ [Uu3: A] : C5 )
@ ( product_Sigma @ A @ B @ B6
@ ^ [Uu3: A] : C5 ) ) ) ).
% Times_Un_distrib1
thf(fact_7453_Sigma__Un__distrib2,axiom,
! [B: $tType,A: $tType,I6: set @ A,A5: A > ( set @ B ),B6: A > ( set @ B )] :
( ( product_Sigma @ A @ B @ I6
@ ^ [I4: A] : ( sup_sup @ ( set @ B ) @ ( A5 @ I4 ) @ ( B6 @ I4 ) ) )
= ( sup_sup @ ( set @ ( product_prod @ A @ B ) ) @ ( product_Sigma @ A @ B @ I6 @ A5 ) @ ( product_Sigma @ A @ B @ I6 @ B6 ) ) ) ).
% Sigma_Un_distrib2
thf(fact_7454_member__product,axiom,
! [B: $tType,A: $tType,X: product_prod @ A @ B,A5: set @ A,B6: set @ B] :
( ( member @ ( product_prod @ A @ B ) @ X @ ( product_product @ A @ B @ A5 @ B6 ) )
= ( member @ ( product_prod @ A @ B ) @ X
@ ( product_Sigma @ A @ B @ A5
@ ^ [Uu3: A] : B6 ) ) ) ).
% member_product
thf(fact_7455_Product__Type_Oproduct__def,axiom,
! [B: $tType,A: $tType] :
( ( product_product @ A @ B )
= ( ^ [A7: set @ A,B8: set @ B] :
( product_Sigma @ A @ B @ A7
@ ^ [Uu3: A] : B8 ) ) ) ).
% Product_Type.product_def
thf(fact_7456_Sigma__Un__distrib1,axiom,
! [B: $tType,A: $tType,I6: set @ A,J4: set @ A,C5: A > ( set @ B )] :
( ( product_Sigma @ A @ B @ ( sup_sup @ ( set @ A ) @ I6 @ J4 ) @ C5 )
= ( sup_sup @ ( set @ ( product_prod @ A @ B ) ) @ ( product_Sigma @ A @ B @ I6 @ C5 ) @ ( product_Sigma @ A @ B @ J4 @ C5 ) ) ) ).
% Sigma_Un_distrib1
thf(fact_7457_Times__subset__cancel2,axiom,
! [A: $tType,B: $tType,X: A,C5: set @ A,A5: set @ B,B6: set @ B] :
( ( member @ A @ X @ C5 )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ B @ A ) )
@ ( product_Sigma @ B @ A @ A5
@ ^ [Uu3: B] : C5 )
@ ( product_Sigma @ B @ A @ B6
@ ^ [Uu3: B] : C5 ) )
= ( ord_less_eq @ ( set @ B ) @ A5 @ B6 ) ) ) ).
% Times_subset_cancel2
thf(fact_7458_Sigma__mono,axiom,
! [B: $tType,A: $tType,A5: set @ A,C5: set @ A,B6: A > ( set @ B ),D6: A > ( set @ B )] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ C5 )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ A5 )
=> ( ord_less_eq @ ( set @ B ) @ ( B6 @ X4 ) @ ( D6 @ X4 ) ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( product_Sigma @ A @ B @ A5 @ B6 ) @ ( product_Sigma @ A @ B @ C5 @ D6 ) ) ) ) ).
% Sigma_mono
thf(fact_7459_Sigma__empty__iff,axiom,
! [B: $tType,A: $tType,I6: set @ A,X7: A > ( set @ B )] :
( ( ( product_Sigma @ A @ B @ I6 @ X7 )
= ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) )
= ( ! [X5: A] :
( ( member @ A @ X5 @ I6 )
=> ( ( X7 @ X5 )
= ( bot_bot @ ( set @ B ) ) ) ) ) ) ).
% Sigma_empty_iff
thf(fact_7460_times__eq__iff,axiom,
! [A: $tType,B: $tType,A5: set @ A,B6: set @ B,C5: set @ A,D6: set @ B] :
( ( ( product_Sigma @ A @ B @ A5
@ ^ [Uu3: A] : B6 )
= ( product_Sigma @ A @ B @ C5
@ ^ [Uu3: A] : D6 ) )
= ( ( ( A5 = C5 )
& ( B6 = D6 ) )
| ( ( ( A5
= ( bot_bot @ ( set @ A ) ) )
| ( B6
= ( bot_bot @ ( set @ B ) ) ) )
& ( ( C5
= ( bot_bot @ ( set @ A ) ) )
| ( D6
= ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ).
% times_eq_iff
thf(fact_7461_SigmaE2,axiom,
! [B: $tType,A: $tType,A2: A,B2: B,A5: set @ A,B6: A > ( set @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ ( product_Sigma @ A @ B @ A5 @ B6 ) )
=> ~ ( ( member @ A @ A2 @ A5 )
=> ~ ( member @ B @ B2 @ ( B6 @ A2 ) ) ) ) ).
% SigmaE2
thf(fact_7462_SigmaD2,axiom,
! [B: $tType,A: $tType,A2: A,B2: B,A5: set @ A,B6: A > ( set @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ ( product_Sigma @ A @ B @ A5 @ B6 ) )
=> ( member @ B @ B2 @ ( B6 @ A2 ) ) ) ).
% SigmaD2
thf(fact_7463_SigmaD1,axiom,
! [B: $tType,A: $tType,A2: A,B2: B,A5: set @ A,B6: A > ( set @ B )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ ( product_Sigma @ A @ B @ A5 @ B6 ) )
=> ( member @ A @ A2 @ A5 ) ) ).
% SigmaD1
thf(fact_7464_SigmaE,axiom,
! [A: $tType,B: $tType,C3: product_prod @ A @ B,A5: set @ A,B6: A > ( set @ B )] :
( ( member @ ( product_prod @ A @ B ) @ C3 @ ( product_Sigma @ A @ B @ A5 @ B6 ) )
=> ~ ! [X4: A] :
( ( member @ A @ X4 @ A5 )
=> ! [Y3: B] :
( ( member @ B @ Y3 @ ( B6 @ X4 ) )
=> ( C3
!= ( product_Pair @ A @ B @ X4 @ Y3 ) ) ) ) ) ).
% SigmaE
thf(fact_7465_Sigma__Union,axiom,
! [B: $tType,A: $tType,X7: set @ ( set @ A ),B6: A > ( set @ B )] :
( ( product_Sigma @ A @ B @ ( complete_Sup_Sup @ ( set @ A ) @ X7 ) @ B6 )
= ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) )
@ ( image2 @ ( set @ A ) @ ( set @ ( product_prod @ A @ B ) )
@ ^ [A7: set @ A] : ( product_Sigma @ A @ B @ A7 @ B6 )
@ X7 ) ) ) ).
% Sigma_Union
thf(fact_7466_mem__Times__iff,axiom,
! [A: $tType,B: $tType,X: product_prod @ A @ B,A5: set @ A,B6: set @ B] :
( ( member @ ( product_prod @ A @ B ) @ X
@ ( product_Sigma @ A @ B @ A5
@ ^ [Uu3: A] : B6 ) )
= ( ( member @ A @ ( product_fst @ A @ B @ X ) @ A5 )
& ( member @ B @ ( product_snd @ A @ B @ X ) @ B6 ) ) ) ).
% mem_Times_iff
thf(fact_7467_times__subset__iff,axiom,
! [A: $tType,B: $tType,A5: set @ A,C5: set @ B,B6: set @ A,D6: set @ B] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) )
@ ( product_Sigma @ A @ B @ A5
@ ^ [Uu3: A] : C5 )
@ ( product_Sigma @ A @ B @ B6
@ ^ [Uu3: A] : D6 ) )
= ( ( A5
= ( bot_bot @ ( set @ A ) ) )
| ( C5
= ( bot_bot @ ( set @ B ) ) )
| ( ( ord_less_eq @ ( set @ A ) @ A5 @ B6 )
& ( ord_less_eq @ ( set @ B ) @ C5 @ D6 ) ) ) ) ).
% times_subset_iff
thf(fact_7468_option_Othe__def,axiom,
! [A: $tType] :
( ( the2 @ A )
= ( case_option @ A @ A @ ( undefined @ A )
@ ^ [X23: A] : X23 ) ) ).
% option.the_def
thf(fact_7469_hd__def,axiom,
! [A: $tType] :
( ( hd @ A )
= ( case_list @ A @ A @ ( undefined @ A )
@ ^ [X213: A,X224: list @ A] : X213 ) ) ).
% hd_def
thf(fact_7470_trancl__subset__Sigma__aux,axiom,
! [A: $tType,A2: A,B2: A,R2: set @ ( product_prod @ A @ A ),A5: set @ A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ ( transitive_rtrancl @ A @ R2 ) )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R2
@ ( product_Sigma @ A @ A @ A5
@ ^ [Uu3: A] : A5 ) )
=> ( ( A2 = B2 )
| ( member @ A @ A2 @ A5 ) ) ) ) ).
% trancl_subset_Sigma_aux
thf(fact_7471_finite__cartesian__product__iff,axiom,
! [A: $tType,B: $tType,A5: set @ A,B6: set @ B] :
( ( finite_finite2 @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A5
@ ^ [Uu3: A] : B6 ) )
= ( ( A5
= ( bot_bot @ ( set @ A ) ) )
| ( B6
= ( bot_bot @ ( set @ B ) ) )
| ( ( finite_finite2 @ A @ A5 )
& ( finite_finite2 @ B @ B6 ) ) ) ) ).
% finite_cartesian_product_iff
thf(fact_7472_finite__cartesian__productD2,axiom,
! [A: $tType,B: $tType,A5: set @ A,B6: set @ B] :
( ( finite_finite2 @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A5
@ ^ [Uu3: A] : B6 ) )
=> ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( finite_finite2 @ B @ B6 ) ) ) ).
% finite_cartesian_productD2
thf(fact_7473_finite__cartesian__productD1,axiom,
! [B: $tType,A: $tType,A5: set @ A,B6: set @ B] :
( ( finite_finite2 @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A5
@ ^ [Uu3: A] : B6 ) )
=> ( ( B6
!= ( bot_bot @ ( set @ B ) ) )
=> ( finite_finite2 @ A @ A5 ) ) ) ).
% finite_cartesian_productD1
thf(fact_7474_finite__SigmaI2,axiom,
! [B: $tType,A: $tType,A5: set @ A,B6: A > ( set @ B )] :
( ( finite_finite2 @ A
@ ( collect @ A
@ ^ [X5: A] :
( ( member @ A @ X5 @ A5 )
& ( ( B6 @ X5 )
!= ( bot_bot @ ( set @ B ) ) ) ) ) )
=> ( ! [A4: A] :
( ( member @ A @ A4 @ A5 )
=> ( finite_finite2 @ B @ ( B6 @ A4 ) ) )
=> ( finite_finite2 @ ( product_prod @ A @ B ) @ ( product_Sigma @ A @ B @ A5 @ B6 ) ) ) ) ).
% finite_SigmaI2
thf(fact_7475_fst__image__Sigma,axiom,
! [B: $tType,A: $tType,A5: set @ A,B6: A > ( set @ B )] :
( ( image2 @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ ( product_Sigma @ A @ B @ A5 @ B6 ) )
= ( collect @ A
@ ^ [X5: A] :
( ( member @ A @ X5 @ A5 )
& ( ( B6 @ X5 )
!= ( bot_bot @ ( set @ B ) ) ) ) ) ) ).
% fst_image_Sigma
thf(fact_7476_UN__Times__distrib,axiom,
! [C: $tType,D: $tType,B: $tType,A: $tType,E5: C > ( set @ A ),F5: D > ( set @ B ),A5: set @ C,B6: set @ D] :
( ( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) )
@ ( image2 @ ( product_prod @ C @ D ) @ ( set @ ( product_prod @ A @ B ) )
@ ( product_case_prod @ C @ D @ ( set @ ( product_prod @ A @ B ) )
@ ^ [A6: C,B5: D] :
( product_Sigma @ A @ B @ ( E5 @ A6 )
@ ^ [Uu3: A] : ( F5 @ B5 ) ) )
@ ( product_Sigma @ C @ D @ A5
@ ^ [Uu3: C] : B6 ) ) )
= ( product_Sigma @ A @ B @ ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ C @ ( set @ A ) @ E5 @ A5 ) )
@ ^ [Uu3: A] : ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ D @ ( set @ B ) @ F5 @ B6 ) ) ) ) ).
% UN_Times_distrib
thf(fact_7477_swap__product,axiom,
! [B: $tType,A: $tType,A5: set @ B,B6: set @ A] :
( ( image2 @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B )
@ ( product_case_prod @ B @ A @ ( product_prod @ A @ B )
@ ^ [I4: B,J3: A] : ( product_Pair @ A @ B @ J3 @ I4 ) )
@ ( product_Sigma @ B @ A @ A5
@ ^ [Uu3: B] : B6 ) )
= ( product_Sigma @ A @ B @ B6
@ ^ [Uu3: A] : A5 ) ) ).
% swap_product
thf(fact_7478_map__prod__surj__on,axiom,
! [B: $tType,D: $tType,C: $tType,A: $tType,F3: B > A,A5: set @ B,A15: set @ A,G3: D > C,B6: set @ D,B13: set @ C] :
( ( ( image2 @ B @ A @ F3 @ A5 )
= A15 )
=> ( ( ( image2 @ D @ C @ G3 @ B6 )
= B13 )
=> ( ( image2 @ ( product_prod @ B @ D ) @ ( product_prod @ A @ C ) @ ( product_map_prod @ B @ A @ D @ C @ F3 @ G3 )
@ ( product_Sigma @ B @ D @ A5
@ ^ [Uu3: B] : B6 ) )
= ( product_Sigma @ A @ C @ A15
@ ^ [Uu3: A] : B13 ) ) ) ) ).
% map_prod_surj_on
thf(fact_7479_map__prod__inj__on,axiom,
! [D: $tType,B: $tType,C: $tType,A: $tType,F3: A > B,A5: set @ A,G3: C > D,B6: set @ C] :
( ( inj_on @ A @ B @ F3 @ A5 )
=> ( ( inj_on @ C @ D @ G3 @ B6 )
=> ( inj_on @ ( product_prod @ A @ C ) @ ( product_prod @ B @ D ) @ ( product_map_prod @ A @ B @ C @ D @ F3 @ G3 )
@ ( product_Sigma @ A @ C @ A5
@ ^ [Uu3: A] : B6 ) ) ) ) ).
% map_prod_inj_on
thf(fact_7480_snd__image__Sigma,axiom,
! [A: $tType,B: $tType,A5: set @ B,B6: B > ( set @ A )] :
( ( image2 @ ( product_prod @ B @ A ) @ A @ ( product_snd @ B @ A ) @ ( product_Sigma @ B @ A @ A5 @ B6 ) )
= ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ B @ ( set @ A ) @ B6 @ A5 ) ) ) ).
% snd_image_Sigma
thf(fact_7481_subset__fst__snd,axiom,
! [B: $tType,A: $tType,A5: set @ ( product_prod @ A @ B )] :
( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ A5
@ ( product_Sigma @ A @ B @ ( image2 @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ A5 )
@ ^ [Uu3: A] : ( image2 @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ A5 ) ) ) ).
% subset_fst_snd
thf(fact_7482_card__cartesian__product__singleton,axiom,
! [A: $tType,B: $tType,X: A,A5: set @ B] :
( ( finite_card @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) )
@ ^ [Uu3: A] : A5 ) )
= ( finite_card @ B @ A5 ) ) ).
% card_cartesian_product_singleton
thf(fact_7483_image__paired__Times,axiom,
! [C: $tType,D: $tType,B: $tType,A: $tType,F3: C > A,G3: D > B,A5: set @ C,B6: set @ D] :
( ( image2 @ ( product_prod @ C @ D ) @ ( product_prod @ A @ B )
@ ( product_case_prod @ C @ D @ ( product_prod @ A @ B )
@ ^ [X5: C,Y6: D] : ( product_Pair @ A @ B @ ( F3 @ X5 ) @ ( G3 @ Y6 ) ) )
@ ( product_Sigma @ C @ D @ A5
@ ^ [Uu3: C] : B6 ) )
= ( product_Sigma @ A @ B @ ( image2 @ C @ A @ F3 @ A5 )
@ ^ [Uu3: A] : ( image2 @ D @ B @ G3 @ B6 ) ) ) ).
% image_paired_Times
thf(fact_7484_Sigma__interval__disjoint,axiom,
! [A: $tType,B: $tType] :
( ( order @ A )
=> ! [A5: set @ B,V: B > A,W2: A] :
( ( inf_inf @ ( set @ ( product_prod @ B @ A ) )
@ ( product_Sigma @ B @ A @ A5
@ ^ [I4: B] : ( set_ord_atMost @ A @ ( V @ I4 ) ) )
@ ( product_Sigma @ B @ A @ A5
@ ^ [I4: B] : ( set_or3652927894154168847AtMost @ A @ ( V @ I4 ) @ W2 ) ) )
= ( bot_bot @ ( set @ ( product_prod @ B @ A ) ) ) ) ) ).
% Sigma_interval_disjoint
thf(fact_7485_sum_OSigma,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comm_monoid_add @ A )
=> ! [A5: set @ B,B6: B > ( set @ C ),G3: B > C > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ A5 )
=> ( finite_finite2 @ C @ ( B6 @ X4 ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X5: B] : ( groups7311177749621191930dd_sum @ C @ A @ ( G3 @ X5 ) @ ( B6 @ X5 ) )
@ A5 )
= ( groups7311177749621191930dd_sum @ ( product_prod @ B @ C ) @ A @ ( product_case_prod @ B @ C @ A @ G3 ) @ ( product_Sigma @ B @ C @ A5 @ B6 ) ) ) ) ) ) ).
% sum.Sigma
thf(fact_7486_prod_OSigma,axiom,
! [A: $tType,C: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [A5: set @ B,B6: B > ( set @ C ),G3: B > C > A] :
( ( finite_finite2 @ B @ A5 )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ A5 )
=> ( finite_finite2 @ C @ ( B6 @ X4 ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [X5: B] : ( groups7121269368397514597t_prod @ C @ A @ ( G3 @ X5 ) @ ( B6 @ X5 ) )
@ A5 )
= ( groups7121269368397514597t_prod @ ( product_prod @ B @ C ) @ A @ ( product_case_prod @ B @ C @ A @ G3 ) @ ( product_Sigma @ B @ C @ A5 @ B6 ) ) ) ) ) ) ).
% prod.Sigma
thf(fact_7487_Func__empty,axiom,
! [B: $tType,A: $tType,B6: set @ B] :
( ( bNF_Wellorder_Func @ A @ B @ ( bot_bot @ ( set @ A ) ) @ B6 )
= ( insert @ ( A > B )
@ ^ [X5: A] : ( undefined @ B )
@ ( bot_bot @ ( set @ ( A > B ) ) ) ) ) ).
% Func_empty
thf(fact_7488_Sigma__def,axiom,
! [B: $tType,A: $tType] :
( ( product_Sigma @ A @ B )
= ( ^ [A7: set @ A,B8: A > ( set @ B )] :
( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) )
@ ( image2 @ A @ ( set @ ( product_prod @ A @ B ) )
@ ^ [X5: A] :
( complete_Sup_Sup @ ( set @ ( product_prod @ A @ B ) )
@ ( image2 @ B @ ( set @ ( product_prod @ A @ B ) )
@ ^ [Y6: B] : ( insert @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X5 @ Y6 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) ) )
@ ( B8 @ X5 ) ) )
@ A7 ) ) ) ) ).
% Sigma_def
thf(fact_7489_product__fold,axiom,
! [B: $tType,A: $tType,A5: set @ A,B6: set @ B] :
( ( finite_finite2 @ A @ A5 )
=> ( ( finite_finite2 @ B @ B6 )
=> ( ( product_Sigma @ A @ B @ A5
@ ^ [Uu3: A] : B6 )
= ( finite_fold @ A @ ( set @ ( product_prod @ A @ B ) )
@ ^ [X5: A,Z5: set @ ( product_prod @ A @ B )] :
( finite_fold @ B @ ( set @ ( product_prod @ A @ B ) )
@ ^ [Y6: B] : ( insert @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X5 @ Y6 ) )
@ Z5
@ B6 )
@ ( bot_bot @ ( set @ ( product_prod @ A @ B ) ) )
@ A5 ) ) ) ) ).
% product_fold
thf(fact_7490_arg__min__list_Oelims,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ! [X: A > B,Xa2: list @ A,Y: A] :
( ( ( arg_min_list @ A @ B @ X @ Xa2 )
= Y )
=> ( ! [X4: A] :
( ( Xa2
= ( cons @ A @ X4 @ ( nil @ A ) ) )
=> ( Y != X4 ) )
=> ( ! [X4: A,Y3: A,Zs2: list @ A] :
( ( Xa2
= ( cons @ A @ X4 @ ( cons @ A @ Y3 @ Zs2 ) ) )
=> ( Y
!= ( if @ A @ ( ord_less_eq @ B @ ( X @ X4 ) @ ( X @ ( arg_min_list @ A @ B @ X @ ( cons @ A @ Y3 @ Zs2 ) ) ) ) @ X4 @ ( arg_min_list @ A @ B @ X @ ( cons @ A @ Y3 @ Zs2 ) ) ) ) )
=> ~ ( ( Xa2
= ( nil @ A ) )
=> ( Y
!= ( undefined @ A ) ) ) ) ) ) ) ).
% arg_min_list.elims
thf(fact_7491_uniformly__continuous__on__uniformity,axiom,
! [B: $tType,A: $tType] :
( ( ( topolo7287701948861334536_space @ A )
& ( topolo7287701948861334536_space @ B ) )
=> ( ( topolo6026614971017936543ous_on @ A @ B )
= ( ^ [S8: set @ A,F4: A > B] :
( filterlim @ ( product_prod @ A @ A ) @ ( product_prod @ B @ B )
@ ( product_case_prod @ A @ A @ ( product_prod @ B @ B )
@ ^ [X5: A,Y6: A] : ( product_Pair @ B @ B @ ( F4 @ X5 ) @ ( F4 @ Y6 ) ) )
@ ( topolo7806501430040627800ormity @ B )
@ ( inf_inf @ ( filter @ ( product_prod @ A @ A ) ) @ ( topolo7806501430040627800ormity @ A )
@ ( principal @ ( product_prod @ A @ A )
@ ( product_Sigma @ A @ A @ S8
@ ^ [Uu3: A] : S8 ) ) ) ) ) ) ) ).
% uniformly_continuous_on_uniformity
thf(fact_7492_min__list_Opelims,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X: list @ A,Y: A] :
( ( ( min_list @ A @ X )
= Y )
=> ( ( accp @ ( list @ A ) @ ( min_list_rel @ A ) @ X )
=> ( ! [X4: A,Xs2: list @ A] :
( ( X
= ( cons @ A @ X4 @ Xs2 ) )
=> ( ( Y
= ( case_list @ A @ A @ X4
@ ^ [A6: A,List2: list @ A] : ( ord_min @ A @ X4 @ ( min_list @ A @ Xs2 ) )
@ Xs2 ) )
=> ~ ( accp @ ( list @ A ) @ ( min_list_rel @ A ) @ ( cons @ A @ X4 @ Xs2 ) ) ) )
=> ~ ( ( X
= ( nil @ A ) )
=> ( ( Y
= ( undefined @ A ) )
=> ~ ( accp @ ( list @ A ) @ ( min_list_rel @ A ) @ ( nil @ A ) ) ) ) ) ) ) ) ).
% min_list.pelims
thf(fact_7493_infinite__cartesian__product,axiom,
! [B: $tType,A: $tType,A5: set @ A,B6: set @ B] :
( ~ ( finite_finite2 @ A @ A5 )
=> ( ~ ( finite_finite2 @ B @ B6 )
=> ~ ( finite_finite2 @ ( product_prod @ A @ B )
@ ( product_Sigma @ A @ B @ A5
@ ^ [Uu3: A] : B6 ) ) ) ) ).
% infinite_cartesian_product
thf(fact_7494_pairs__le__eq__Sigma,axiom,
! [M2: nat] :
( ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [I4: nat,J3: nat] : ( ord_less_eq @ nat @ ( plus_plus @ nat @ I4 @ J3 ) @ M2 ) ) )
= ( product_Sigma @ nat @ nat @ ( set_ord_atMost @ nat @ M2 )
@ ^ [R5: nat] : ( set_ord_atMost @ nat @ ( minus_minus @ nat @ M2 @ R5 ) ) ) ) ).
% pairs_le_eq_Sigma
thf(fact_7495_relChain__def,axiom,
! [B: $tType,A: $tType] :
( ( ord @ B )
=> ( ( bNF_Ca3754400796208372196lChain @ A @ B )
= ( ^ [R5: set @ ( product_prod @ A @ A ),As4: A > B] :
! [I4: A,J3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ I4 @ J3 ) @ R5 )
=> ( ord_less_eq @ B @ ( As4 @ I4 ) @ ( As4 @ J3 ) ) ) ) ) ) ).
% relChain_def
thf(fact_7496_natLess__def,axiom,
( bNF_Ca8459412986667044542atLess
= ( collect @ ( product_prod @ nat @ nat ) @ ( product_case_prod @ nat @ nat @ $o @ ( ord_less @ nat ) ) ) ) ).
% natLess_def
thf(fact_7497_swap__comp__swap,axiom,
! [B: $tType,A: $tType] :
( ( comp @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B ) @ ( product_prod @ A @ B ) @ ( product_swap @ B @ A ) @ ( product_swap @ A @ B ) )
= ( id @ ( product_prod @ A @ B ) ) ) ).
% swap_comp_swap
thf(fact_7498_rotate__drop__take,axiom,
! [A: $tType] :
( ( rotate @ A )
= ( ^ [N2: nat,Xs3: list @ A] : ( append @ A @ ( drop @ A @ ( modulo_modulo @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs3 ) ) @ Xs3 ) @ ( take @ A @ ( modulo_modulo @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs3 ) ) @ Xs3 ) ) ) ) ).
% rotate_drop_take
thf(fact_7499_swap__swap,axiom,
! [B: $tType,A: $tType,P: product_prod @ A @ B] :
( ( product_swap @ B @ A @ ( product_swap @ A @ B @ P ) )
= P ) ).
% swap_swap
thf(fact_7500_rotate__is__Nil__conv,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( ( rotate @ A @ N @ Xs )
= ( nil @ A ) )
= ( Xs
= ( nil @ A ) ) ) ).
% rotate_is_Nil_conv
thf(fact_7501_set__rotate,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( set2 @ A @ ( rotate @ A @ N @ Xs ) )
= ( set2 @ A @ Xs ) ) ).
% set_rotate
thf(fact_7502_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_7503_swap__simp,axiom,
! [A: $tType,B: $tType,X: B,Y: A] :
( ( product_swap @ B @ A @ ( product_Pair @ B @ A @ X @ Y ) )
= ( product_Pair @ A @ B @ Y @ X ) ) ).
% swap_simp
thf(fact_7504_distinct__rotate,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( distinct @ A @ ( rotate @ A @ N @ Xs ) )
= ( distinct @ A @ Xs ) ) ).
% distinct_rotate
thf(fact_7505_case__swap,axiom,
! [A: $tType,B: $tType,C: $tType,F3: C > B > A,P: product_prod @ C @ B] :
( ( product_case_prod @ B @ C @ A
@ ^ [Y6: B,X5: C] : ( F3 @ X5 @ Y6 )
@ ( product_swap @ C @ B @ P ) )
= ( product_case_prod @ C @ B @ A @ F3 @ P ) ) ).
% case_swap
thf(fact_7506_snd__swap,axiom,
! [B: $tType,A: $tType,X: product_prod @ A @ B] :
( ( product_snd @ B @ A @ ( product_swap @ A @ B @ X ) )
= ( product_fst @ A @ B @ X ) ) ).
% snd_swap
thf(fact_7507_fst__swap,axiom,
! [A: $tType,B: $tType,X: product_prod @ B @ A] :
( ( product_fst @ A @ B @ ( product_swap @ B @ A @ X ) )
= ( product_snd @ B @ A @ X ) ) ).
% fst_swap
thf(fact_7508_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_7509_pair__in__swap__image,axiom,
! [A: $tType,B: $tType,Y: A,X: B,A5: set @ ( product_prod @ B @ A )] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ Y @ X ) @ ( image2 @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B ) @ ( product_swap @ B @ A ) @ A5 ) )
= ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X @ Y ) @ A5 ) ) ).
% pair_in_swap_image
thf(fact_7510_rotate0,axiom,
! [A: $tType] :
( ( rotate @ A @ ( zero_zero @ nat ) )
= ( id @ ( list @ A ) ) ) ).
% rotate0
thf(fact_7511_surj__swap,axiom,
! [B: $tType,A: $tType] :
( ( image2 @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B ) @ ( product_swap @ B @ A ) @ ( top_top @ ( set @ ( product_prod @ B @ A ) ) ) )
= ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) ) ).
% surj_swap
thf(fact_7512_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_7513_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_7514_rotate__map,axiom,
! [A: $tType,B: $tType,N: nat,F3: B > A,Xs: list @ B] :
( ( rotate @ A @ N @ ( map @ B @ A @ F3 @ Xs ) )
= ( map @ B @ A @ F3 @ ( rotate @ B @ N @ Xs ) ) ) ).
% rotate_map
thf(fact_7515_rotate__append,axiom,
! [A: $tType,L: list @ A,Q2: list @ A] :
( ( rotate @ A @ ( size_size @ ( list @ A ) @ L ) @ ( append @ A @ L @ Q2 ) )
= ( append @ A @ Q2 @ L ) ) ).
% rotate_append
thf(fact_7516_rotate__conv__mod,axiom,
! [A: $tType] :
( ( rotate @ A )
= ( ^ [N2: nat,Xs3: list @ A] : ( rotate @ A @ ( modulo_modulo @ nat @ N2 @ ( size_size @ ( list @ A ) @ Xs3 ) ) @ Xs3 ) ) ) ).
% rotate_conv_mod
thf(fact_7517_inj__swap,axiom,
! [B: $tType,A: $tType,A5: set @ ( product_prod @ A @ B )] : ( inj_on @ ( product_prod @ A @ B ) @ ( product_prod @ B @ A ) @ ( product_swap @ A @ B ) @ A5 ) ).
% inj_swap
thf(fact_7518_rotate1__rotate__swap,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( rotate1 @ A @ ( rotate @ A @ N @ Xs ) )
= ( rotate @ A @ N @ ( rotate1 @ A @ Xs ) ) ) ).
% rotate1_rotate_swap
thf(fact_7519_rotate__rotate,axiom,
! [A: $tType,M2: nat,N: nat,Xs: list @ A] :
( ( rotate @ A @ M2 @ ( rotate @ A @ N @ Xs ) )
= ( rotate @ A @ ( plus_plus @ nat @ M2 @ N ) @ Xs ) ) ).
% rotate_rotate
thf(fact_7520_rotate__def,axiom,
! [A: $tType] :
( ( rotate @ A )
= ( ^ [N2: nat] : ( compow @ ( ( list @ A ) > ( list @ A ) ) @ N2 @ ( rotate1 @ A ) ) ) ) ).
% rotate_def
thf(fact_7521_rotate__add,axiom,
! [A: $tType,M2: nat,N: nat] :
( ( rotate @ A @ ( plus_plus @ nat @ M2 @ N ) )
= ( comp @ ( list @ A ) @ ( list @ A ) @ ( list @ A ) @ ( rotate @ A @ M2 ) @ ( rotate @ A @ N ) ) ) ).
% rotate_add
thf(fact_7522_product__swap,axiom,
! [B: $tType,A: $tType,A5: set @ B,B6: set @ A] :
( ( image2 @ ( product_prod @ B @ A ) @ ( product_prod @ A @ B ) @ ( product_swap @ B @ A )
@ ( product_Sigma @ B @ A @ A5
@ ^ [Uu3: B] : B6 ) )
= ( product_Sigma @ A @ B @ B6
@ ^ [Uu3: A] : A5 ) ) ).
% product_swap
thf(fact_7523_prod_Oswap__def,axiom,
! [B: $tType,A: $tType] :
( ( product_swap @ A @ B )
= ( ^ [P6: product_prod @ A @ B] : ( product_Pair @ B @ A @ ( product_snd @ A @ B @ P6 ) @ ( product_fst @ A @ B @ P6 ) ) ) ) ).
% prod.swap_def
thf(fact_7524_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_7525_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_7526_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_7527_image__split__eq__Sigma,axiom,
! [C: $tType,B: $tType,A: $tType,F3: C > A,G3: C > B,A5: set @ C] :
( ( image2 @ C @ ( product_prod @ A @ B )
@ ^ [X5: C] : ( product_Pair @ A @ B @ ( F3 @ X5 ) @ ( G3 @ X5 ) )
@ A5 )
= ( product_Sigma @ A @ B @ ( image2 @ C @ A @ F3 @ A5 )
@ ^ [X5: A] : ( image2 @ C @ B @ G3 @ ( inf_inf @ ( set @ C ) @ ( vimage @ C @ A @ F3 @ ( insert @ A @ X5 @ ( bot_bot @ ( set @ A ) ) ) ) @ A5 ) ) ) ) ).
% image_split_eq_Sigma
thf(fact_7528_listrel1__subset__listrel,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),R4: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ R4 )
=> ( ( refl_on @ A @ ( top_top @ ( set @ A ) ) @ R4 )
=> ( ord_less_eq @ ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) @ ( listrel1 @ A @ R2 ) @ ( listrel @ A @ A @ R4 ) ) ) ) ).
% listrel1_subset_listrel
thf(fact_7529_vimage__empty,axiom,
! [B: $tType,A: $tType,F3: A > B] :
( ( vimage @ A @ B @ F3 @ ( bot_bot @ ( set @ B ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% vimage_empty
thf(fact_7530_vimage__const,axiom,
! [B: $tType,A: $tType,C3: B,A5: set @ B] :
( ( ( member @ B @ C3 @ A5 )
=> ( ( vimage @ A @ B
@ ^ [X5: A] : C3
@ A5 )
= ( top_top @ ( set @ A ) ) ) )
& ( ~ ( member @ B @ C3 @ A5 )
=> ( ( vimage @ A @ B
@ ^ [X5: A] : C3
@ A5 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% vimage_const
thf(fact_7531_vimage__if,axiom,
! [B: $tType,A: $tType,C3: B,A5: set @ B,D3: B,B6: set @ A] :
( ( ( member @ B @ C3 @ A5 )
=> ( ( ( member @ B @ D3 @ A5 )
=> ( ( vimage @ A @ B
@ ^ [X5: A] : ( if @ B @ ( member @ A @ X5 @ B6 ) @ C3 @ D3 )
@ A5 )
= ( top_top @ ( set @ A ) ) ) )
& ( ~ ( member @ B @ D3 @ A5 )
=> ( ( vimage @ A @ B
@ ^ [X5: A] : ( if @ B @ ( member @ A @ X5 @ B6 ) @ C3 @ D3 )
@ A5 )
= B6 ) ) ) )
& ( ~ ( member @ B @ C3 @ A5 )
=> ( ( ( member @ B @ D3 @ A5 )
=> ( ( vimage @ A @ B
@ ^ [X5: A] : ( if @ B @ ( member @ A @ X5 @ B6 ) @ C3 @ D3 )
@ A5 )
= ( uminus_uminus @ ( set @ A ) @ B6 ) ) )
& ( ~ ( member @ B @ D3 @ A5 )
=> ( ( vimage @ A @ B
@ ^ [X5: A] : ( if @ B @ ( member @ A @ X5 @ B6 ) @ C3 @ D3 )
@ A5 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% vimage_if
thf(fact_7532_surj__vimage__empty,axiom,
! [B: $tType,A: $tType,F3: B > A,A5: set @ A] :
( ( ( image2 @ B @ A @ F3 @ ( top_top @ ( set @ B ) ) )
= ( top_top @ ( set @ A ) ) )
=> ( ( ( vimage @ B @ A @ F3 @ A5 )
= ( bot_bot @ ( set @ B ) ) )
= ( A5
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% surj_vimage_empty
thf(fact_7533_finite__vimageD,axiom,
! [A: $tType,B: $tType,H2: A > B,F5: set @ B] :
( ( finite_finite2 @ A @ ( vimage @ A @ B @ H2 @ F5 ) )
=> ( ( ( image2 @ A @ B @ H2 @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ B ) ) )
=> ( finite_finite2 @ B @ F5 ) ) ) ).
% finite_vimageD
thf(fact_7534_vimage__Suc__insert__0,axiom,
! [A5: set @ nat] :
( ( vimage @ nat @ nat @ suc @ ( insert @ nat @ ( zero_zero @ nat ) @ A5 ) )
= ( vimage @ nat @ nat @ suc @ A5 ) ) ).
% vimage_Suc_insert_0
thf(fact_7535_vimage__singleton__eq,axiom,
! [A: $tType,B: $tType,A2: A,F3: A > B,B2: B] :
( ( member @ A @ A2 @ ( vimage @ A @ B @ F3 @ ( insert @ B @ B2 @ ( bot_bot @ ( set @ B ) ) ) ) )
= ( ( F3 @ A2 )
= B2 ) ) ).
% vimage_singleton_eq
thf(fact_7536_vimage__insert,axiom,
! [A: $tType,B: $tType,F3: A > B,A2: B,B6: set @ B] :
( ( vimage @ A @ B @ F3 @ ( insert @ B @ A2 @ B6 ) )
= ( sup_sup @ ( set @ A ) @ ( vimage @ A @ B @ F3 @ ( insert @ B @ A2 @ ( bot_bot @ ( set @ B ) ) ) ) @ ( vimage @ A @ B @ F3 @ B6 ) ) ) ).
% vimage_insert
thf(fact_7537_vimage__Suc__insert__Suc,axiom,
! [N: nat,A5: set @ nat] :
( ( vimage @ nat @ nat @ suc @ ( insert @ nat @ ( suc @ N ) @ A5 ) )
= ( insert @ nat @ N @ ( vimage @ nat @ nat @ suc @ A5 ) ) ) ).
% vimage_Suc_insert_Suc
thf(fact_7538_finite__vimage__IntI,axiom,
! [A: $tType,B: $tType,F5: set @ A,H2: B > A,A5: set @ B] :
( ( finite_finite2 @ A @ F5 )
=> ( ( inj_on @ B @ A @ H2 @ A5 )
=> ( finite_finite2 @ B @ ( inf_inf @ ( set @ B ) @ ( vimage @ B @ A @ H2 @ F5 ) @ A5 ) ) ) ) ).
% finite_vimage_IntI
thf(fact_7539_finite__vimage__Suc__iff,axiom,
! [F5: set @ nat] :
( ( finite_finite2 @ nat @ ( vimage @ nat @ nat @ suc @ F5 ) )
= ( finite_finite2 @ nat @ F5 ) ) ).
% finite_vimage_Suc_iff
thf(fact_7540_finite__vimageI,axiom,
! [B: $tType,A: $tType,F5: set @ A,H2: B > A] :
( ( finite_finite2 @ A @ F5 )
=> ( ( inj_on @ B @ A @ H2 @ ( top_top @ ( set @ B ) ) )
=> ( finite_finite2 @ B @ ( vimage @ B @ A @ H2 @ F5 ) ) ) ) ).
% finite_vimageI
thf(fact_7541_refl__on__empty,axiom,
! [A: $tType] : ( refl_on @ A @ ( bot_bot @ ( set @ A ) ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ).
% refl_on_empty
thf(fact_7542_refl__onD2,axiom,
! [A: $tType,A5: set @ A,R2: set @ ( product_prod @ A @ A ),X: A,Y: A] :
( ( refl_on @ A @ A5 @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R2 )
=> ( member @ A @ Y @ A5 ) ) ) ).
% refl_onD2
thf(fact_7543_refl__onD1,axiom,
! [A: $tType,A5: set @ A,R2: set @ ( product_prod @ A @ A ),X: A,Y: A] :
( ( refl_on @ A @ A5 @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R2 )
=> ( member @ A @ X @ A5 ) ) ) ).
% refl_onD1
thf(fact_7544_refl__onD,axiom,
! [A: $tType,A5: set @ A,R2: set @ ( product_prod @ A @ A ),A2: A] :
( ( refl_on @ A @ A5 @ R2 )
=> ( ( member @ A @ A2 @ A5 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ A2 ) @ R2 ) ) ) ).
% refl_onD
thf(fact_7545_vimage__Times,axiom,
! [A: $tType,B: $tType,C: $tType,F3: A > ( product_prod @ B @ C ),A5: set @ B,B6: set @ C] :
( ( vimage @ A @ ( product_prod @ B @ C ) @ F3
@ ( product_Sigma @ B @ C @ A5
@ ^ [Uu3: B] : B6 ) )
= ( inf_inf @ ( set @ A ) @ ( vimage @ A @ B @ ( comp @ ( product_prod @ B @ C ) @ B @ A @ ( product_fst @ B @ C ) @ F3 ) @ A5 ) @ ( vimage @ A @ C @ ( comp @ ( product_prod @ B @ C ) @ C @ A @ ( product_snd @ B @ C ) @ F3 ) @ B6 ) ) ) ).
% vimage_Times
thf(fact_7546_vimage__snd,axiom,
! [B: $tType,A: $tType,A5: set @ B] :
( ( vimage @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ A5 )
= ( product_Sigma @ A @ B @ ( top_top @ ( set @ A ) )
@ ^ [Uu3: A] : A5 ) ) ).
% vimage_snd
thf(fact_7547_vimage__fst,axiom,
! [B: $tType,A: $tType,A5: set @ A] :
( ( vimage @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ A5 )
= ( product_Sigma @ A @ B @ A5
@ ^ [Uu3: A] : ( top_top @ ( set @ B ) ) ) ) ).
% vimage_fst
thf(fact_7548_Pair__vimage__Sigma,axiom,
! [B: $tType,A: $tType,X: B,A5: set @ B,F3: B > ( set @ A )] :
( ( ( member @ B @ X @ A5 )
=> ( ( vimage @ A @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X ) @ ( product_Sigma @ B @ A @ A5 @ F3 ) )
= ( F3 @ X ) ) )
& ( ~ ( member @ B @ X @ A5 )
=> ( ( vimage @ A @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X ) @ ( product_Sigma @ B @ A @ A5 @ F3 ) )
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% Pair_vimage_Sigma
thf(fact_7549_finite__vimageD_H,axiom,
! [A: $tType,B: $tType,F3: A > B,A5: set @ B] :
( ( finite_finite2 @ A @ ( vimage @ A @ B @ F3 @ A5 ) )
=> ( ( ord_less_eq @ ( set @ B ) @ A5 @ ( image2 @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) ) )
=> ( finite_finite2 @ B @ A5 ) ) ) ).
% finite_vimageD'
thf(fact_7550_inf__img__fin__dom,axiom,
! [B: $tType,A: $tType,F3: B > A,A5: set @ B] :
( ( finite_finite2 @ A @ ( image2 @ B @ A @ F3 @ A5 ) )
=> ( ~ ( finite_finite2 @ B @ A5 )
=> ? [X4: A] :
( ( member @ A @ X4 @ ( image2 @ B @ A @ F3 @ A5 ) )
& ~ ( finite_finite2 @ B @ ( vimage @ B @ A @ F3 @ ( insert @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% inf_img_fin_dom
thf(fact_7551_inf__img__fin__domE,axiom,
! [B: $tType,A: $tType,F3: B > A,A5: set @ B] :
( ( finite_finite2 @ A @ ( image2 @ B @ A @ F3 @ A5 ) )
=> ( ~ ( finite_finite2 @ B @ A5 )
=> ~ ! [Y3: A] :
( ( member @ A @ Y3 @ ( image2 @ B @ A @ F3 @ A5 ) )
=> ( finite_finite2 @ B @ ( vimage @ B @ A @ F3 @ ( insert @ A @ Y3 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% inf_img_fin_domE
thf(fact_7552_finite__finite__vimage__IntI,axiom,
! [A: $tType,B: $tType,F5: set @ A,H2: B > A,A5: set @ B] :
( ( finite_finite2 @ A @ F5 )
=> ( ! [Y3: A] :
( ( member @ A @ Y3 @ F5 )
=> ( finite_finite2 @ B @ ( inf_inf @ ( set @ B ) @ ( vimage @ B @ A @ H2 @ ( insert @ A @ Y3 @ ( bot_bot @ ( set @ A ) ) ) ) @ A5 ) ) )
=> ( finite_finite2 @ B @ ( inf_inf @ ( set @ B ) @ ( vimage @ B @ A @ H2 @ F5 ) @ A5 ) ) ) ) ).
% finite_finite_vimage_IntI
thf(fact_7553_refl__on__def,axiom,
! [A: $tType] :
( ( refl_on @ A )
= ( ^ [A7: set @ A,R5: set @ ( product_prod @ A @ A )] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R5
@ ( product_Sigma @ A @ A @ A7
@ ^ [Uu3: A] : A7 ) )
& ! [X5: A] :
( ( member @ A @ X5 @ A7 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X5 @ X5 ) @ R5 ) ) ) ) ) ).
% refl_on_def
thf(fact_7554_refl__onI,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A5: set @ A] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R2
@ ( product_Sigma @ A @ A @ A5
@ ^ [Uu3: A] : A5 ) )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ A5 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ X4 ) @ R2 ) )
=> ( refl_on @ A @ A5 @ R2 ) ) ) ).
% refl_onI
thf(fact_7555_vimage__eq__UN,axiom,
! [B: $tType,A: $tType] :
( ( vimage @ A @ B )
= ( ^ [F4: A > B,B8: set @ B] :
( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [Y6: B] : ( vimage @ A @ B @ F4 @ ( insert @ B @ Y6 @ ( bot_bot @ ( set @ B ) ) ) )
@ B8 ) ) ) ) ).
% vimage_eq_UN
thf(fact_7556_refl__on__def_H,axiom,
! [A: $tType] :
( ( refl_on @ A )
= ( ^ [A7: set @ A,R5: set @ ( product_prod @ A @ A )] :
( ! [X5: product_prod @ A @ A] :
( ( member @ ( product_prod @ A @ A ) @ X5 @ R5 )
=> ( product_case_prod @ A @ A @ $o
@ ^ [Y6: A,Z5: A] :
( ( member @ A @ Y6 @ A7 )
& ( member @ A @ Z5 @ A7 ) )
@ X5 ) )
& ! [X5: A] :
( ( member @ A @ X5 @ A7 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X5 @ X5 ) @ R5 ) ) ) ) ) ).
% refl_on_def'
thf(fact_7557_inf__img__fin__dom_H,axiom,
! [A: $tType,B: $tType,F3: B > A,A5: set @ B] :
( ( finite_finite2 @ A @ ( image2 @ B @ A @ F3 @ A5 ) )
=> ( ~ ( finite_finite2 @ B @ A5 )
=> ? [X4: A] :
( ( member @ A @ X4 @ ( image2 @ B @ A @ F3 @ A5 ) )
& ~ ( finite_finite2 @ B @ ( inf_inf @ ( set @ B ) @ ( vimage @ B @ A @ F3 @ ( insert @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) ) @ A5 ) ) ) ) ) ).
% inf_img_fin_dom'
thf(fact_7558_inf__img__fin__domE_H,axiom,
! [A: $tType,B: $tType,F3: B > A,A5: set @ B] :
( ( finite_finite2 @ A @ ( image2 @ B @ A @ F3 @ A5 ) )
=> ( ~ ( finite_finite2 @ B @ A5 )
=> ~ ! [Y3: A] :
( ( member @ A @ Y3 @ ( image2 @ B @ A @ F3 @ A5 ) )
=> ( finite_finite2 @ B @ ( inf_inf @ ( set @ B ) @ ( vimage @ B @ A @ F3 @ ( insert @ A @ Y3 @ ( bot_bot @ ( set @ A ) ) ) ) @ A5 ) ) ) ) ) ).
% inf_img_fin_domE'
thf(fact_7559_card__vimage__inj,axiom,
! [A: $tType,B: $tType,F3: A > B,A5: set @ B] :
( ( inj_on @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) )
=> ( ( ord_less_eq @ ( set @ B ) @ A5 @ ( image2 @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) ) )
=> ( ( finite_card @ A @ ( vimage @ A @ B @ F3 @ A5 ) )
= ( finite_card @ B @ A5 ) ) ) ) ).
% card_vimage_inj
thf(fact_7560_card__vimage__inj__on__le,axiom,
! [A: $tType,B: $tType,F3: A > B,D6: set @ A,A5: set @ B] :
( ( inj_on @ A @ B @ F3 @ D6 )
=> ( ( finite_finite2 @ B @ A5 )
=> ( ord_less_eq @ nat @ ( finite_card @ A @ ( inf_inf @ ( set @ A ) @ ( vimage @ A @ B @ F3 @ A5 ) @ D6 ) ) @ ( finite_card @ B @ A5 ) ) ) ) ).
% card_vimage_inj_on_le
thf(fact_7561_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_7562_set__encode__vimage__Suc,axiom,
! [A5: set @ nat] :
( ( nat_set_encode @ ( vimage @ nat @ nat @ suc @ A5 ) )
= ( divide_divide @ nat @ ( nat_set_encode @ A5 ) @ ( numeral_numeral @ nat @ ( bit0 @ one2 ) ) ) ) ).
% set_encode_vimage_Suc
thf(fact_7563_inj__vimage__singleton,axiom,
! [B: $tType,A: $tType,F3: A > B,A2: B] :
( ( inj_on @ A @ B @ F3 @ ( top_top @ ( set @ A ) ) )
=> ( ord_less_eq @ ( set @ A ) @ ( vimage @ A @ B @ F3 @ ( insert @ B @ A2 @ ( bot_bot @ ( set @ B ) ) ) )
@ ( insert @ A
@ ( the @ A
@ ^ [X5: A] :
( ( F3 @ X5 )
= A2 ) )
@ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% inj_vimage_singleton
thf(fact_7564_inj__on__vimage__singleton,axiom,
! [B: $tType,A: $tType,F3: A > B,A5: set @ A,A2: B] :
( ( inj_on @ A @ B @ F3 @ A5 )
=> ( ord_less_eq @ ( set @ A ) @ ( inf_inf @ ( set @ A ) @ ( vimage @ A @ B @ F3 @ ( insert @ B @ A2 @ ( bot_bot @ ( set @ B ) ) ) ) @ A5 )
@ ( insert @ A
@ ( the @ A
@ ^ [X5: A] :
( ( member @ A @ X5 @ A5 )
& ( ( F3 @ X5 )
= A2 ) ) )
@ ( bot_bot @ ( set @ A ) ) ) ) ) ).
% inj_on_vimage_singleton
thf(fact_7565_refl__on__singleton,axiom,
! [A: $tType,X: A] : ( refl_on @ A @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) @ ( insert @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ X ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ).
% refl_on_singleton
thf(fact_7566_inv__image__partition,axiom,
! [A: $tType,Xs: list @ A,P2: A > $o,Ys: list @ A] :
( ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
=> ( P2 @ X4 ) )
=> ( ! [Y3: A] :
( ( member @ A @ Y3 @ ( set2 @ A @ Ys ) )
=> ~ ( P2 @ Y3 ) )
=> ( ( vimage @ ( list @ A ) @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( partition @ A @ P2 ) @ ( insert @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ Ys ) @ ( bot_bot @ ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) ) ) )
= ( shuffles @ A @ Xs @ Ys ) ) ) ) ).
% inv_image_partition
thf(fact_7567_refl__on__domain,axiom,
! [A: $tType,A5: set @ A,R2: set @ ( product_prod @ A @ A ),A2: A,B2: A] :
( ( refl_on @ A @ A5 @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ R2 )
=> ( ( member @ A @ A2 @ A5 )
& ( member @ A @ B2 @ A5 ) ) ) ) ).
% refl_on_domain
thf(fact_7568_Restr__natLeq,axiom,
! [N: nat] :
( ( inf_inf @ ( set @ ( product_prod @ nat @ nat ) ) @ bNF_Ca8665028551170535155natLeq
@ ( product_Sigma @ nat @ nat
@ ( collect @ nat
@ ^ [X5: nat] : ( ord_less @ nat @ X5 @ N ) )
@ ^ [Uu3: nat] :
( collect @ nat
@ ^ [X5: nat] : ( ord_less @ nat @ X5 @ N ) ) ) )
= ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [X5: nat,Y6: nat] :
( ( ord_less @ nat @ X5 @ N )
& ( ord_less @ nat @ Y6 @ N )
& ( ord_less_eq @ nat @ X5 @ Y6 ) ) ) ) ) ).
% Restr_natLeq
thf(fact_7569_natLeq__def,axiom,
( bNF_Ca8665028551170535155natLeq
= ( collect @ ( product_prod @ nat @ nat ) @ ( product_case_prod @ nat @ nat @ $o @ ( ord_less_eq @ nat ) ) ) ) ).
% natLeq_def
thf(fact_7570_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
@ ^ [X5: nat,Y6: nat] :
( ( ord_less @ nat @ X5 @ N )
& ( ord_less @ nat @ Y6 @ N )
& ( ord_less_eq @ nat @ X5 @ Y6 ) ) ) ) ) ).
% Restr_natLeq2
thf(fact_7571_linear__order__on__singleton,axiom,
! [A: $tType,X: A] : ( order_679001287576687338der_on @ A @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) @ ( insert @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ X ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ).
% linear_order_on_singleton
thf(fact_7572_underS__def,axiom,
! [A: $tType] :
( ( order_underS @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A ),A6: A] :
( collect @ A
@ ^ [B5: A] :
( ( B5 != A6 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B5 @ A6 ) @ R5 ) ) ) ) ) ).
% underS_def
thf(fact_7573_lnear__order__on__empty,axiom,
! [A: $tType] : ( order_679001287576687338der_on @ A @ ( bot_bot @ ( set @ A ) ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ).
% lnear_order_on_empty
thf(fact_7574_natLeq__underS__less,axiom,
! [N: nat] :
( ( order_underS @ nat @ bNF_Ca8665028551170535155natLeq @ N )
= ( collect @ nat
@ ^ [X5: nat] : ( ord_less @ nat @ X5 @ N ) ) ) ).
% natLeq_underS_less
thf(fact_7575_underS__I,axiom,
! [A: $tType,I2: A,J: A,R: set @ ( product_prod @ A @ A )] :
( ( I2 != J )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ I2 @ J ) @ R )
=> ( member @ A @ I2 @ ( order_underS @ A @ R @ J ) ) ) ) ).
% underS_I
thf(fact_7576_underS__E,axiom,
! [A: $tType,I2: A,R: set @ ( product_prod @ A @ A ),J: A] :
( ( member @ A @ I2 @ ( order_underS @ A @ R @ J ) )
=> ( ( I2 != J )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ I2 @ J ) @ R ) ) ) ).
% underS_E
thf(fact_7577_total__on__singleton,axiom,
! [A: $tType,X: A] : ( total_on @ A @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) @ ( insert @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ X ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ).
% total_on_singleton
thf(fact_7578_Arg__bounded,axiom,
! [Z: complex] :
( ( ord_less @ real @ ( uminus_uminus @ real @ pi ) @ ( arg @ Z ) )
& ( ord_less_eq @ real @ ( arg @ Z ) @ pi ) ) ).
% Arg_bounded
thf(fact_7579_total__pair__less,axiom,
! [A5: set @ ( product_prod @ nat @ nat )] : ( total_on @ ( product_prod @ nat @ nat ) @ A5 @ fun_pair_less ) ).
% total_pair_less
thf(fact_7580_total__on__def,axiom,
! [A: $tType] :
( ( total_on @ A )
= ( ^ [A7: set @ A,R5: set @ ( product_prod @ A @ A )] :
! [X5: A] :
( ( member @ A @ X5 @ A7 )
=> ! [Y6: A] :
( ( member @ A @ Y6 @ A7 )
=> ( ( X5 != Y6 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X5 @ Y6 ) @ R5 )
| ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y6 @ X5 ) @ R5 ) ) ) ) ) ) ) ).
% total_on_def
thf(fact_7581_total__onI,axiom,
! [A: $tType,A5: set @ A,R2: set @ ( product_prod @ A @ A )] :
( ! [X4: A,Y3: A] :
( ( member @ A @ X4 @ A5 )
=> ( ( member @ A @ Y3 @ A5 )
=> ( ( X4 != Y3 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y3 ) @ R2 )
| ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ X4 ) @ R2 ) ) ) ) )
=> ( total_on @ A @ A5 @ R2 ) ) ).
% total_onI
thf(fact_7582_total__lenlex,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( total_on @ A @ ( top_top @ ( set @ A ) ) @ R2 )
=> ( total_on @ ( list @ A ) @ ( top_top @ ( set @ ( list @ A ) ) ) @ ( lenlex @ A @ R2 ) ) ) ).
% total_lenlex
thf(fact_7583_total__on__empty,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] : ( total_on @ A @ ( bot_bot @ ( set @ A ) ) @ R2 ) ).
% total_on_empty
thf(fact_7584_total__lexord,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( total_on @ A @ ( top_top @ ( set @ A ) ) @ R2 )
=> ( total_on @ ( list @ A ) @ ( top_top @ ( set @ ( list @ A ) ) ) @ ( lexord @ A @ R2 ) ) ) ).
% total_lexord
thf(fact_7585_Arg__correct,axiom,
! [Z: complex] :
( ( Z
!= ( zero_zero @ complex ) )
=> ( ( ( sgn_sgn @ complex @ Z )
= ( cis @ ( arg @ Z ) ) )
& ( ord_less @ real @ ( uminus_uminus @ real @ pi ) @ ( arg @ Z ) )
& ( ord_less_eq @ real @ ( arg @ Z ) @ pi ) ) ) ).
% Arg_correct
thf(fact_7586_slice__eq__mask,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [N: nat,M2: nat,A2: A] :
( ( bit_se4730199178511100633sh_bit @ A @ N @ ( bit_se2584673776208193580ke_bit @ A @ M2 @ ( bit_se4197421643247451524op_bit @ A @ N @ A2 ) ) )
= ( bit_se5824344872417868541ns_and @ A @ A2 @ ( bit_se5824344872417868541ns_and @ A @ ( bit_se2239418461657761734s_mask @ A @ ( plus_plus @ nat @ M2 @ N ) ) @ ( bit_ri4277139882892585799ns_not @ A @ ( bit_se2239418461657761734s_mask @ A @ N ) ) ) ) ) ) ).
% slice_eq_mask
thf(fact_7587_not__negative__int__iff,axiom,
! [K2: int] :
( ( ord_less @ int @ ( bit_ri4277139882892585799ns_not @ int @ K2 ) @ ( zero_zero @ int ) )
= ( ord_less_eq @ int @ ( zero_zero @ int ) @ K2 ) ) ).
% not_negative_int_iff
thf(fact_7588_not__nonnegative__int__iff,axiom,
! [K2: int] :
( ( ord_less_eq @ int @ ( zero_zero @ int ) @ ( bit_ri4277139882892585799ns_not @ int @ K2 ) )
= ( ord_less @ int @ K2 @ ( zero_zero @ int ) ) ) ).
% not_nonnegative_int_iff
thf(fact_7589_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_7590_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_7591_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_7592_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_7593_not__diff__distrib,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [A2: A,B2: A] :
( ( bit_ri4277139882892585799ns_not @ A @ ( minus_minus @ A @ A2 @ B2 ) )
= ( plus_plus @ A @ ( bit_ri4277139882892585799ns_not @ A @ A2 ) @ B2 ) ) ) ).
% not_diff_distrib
thf(fact_7594_not__add__distrib,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [A2: A,B2: A] :
( ( bit_ri4277139882892585799ns_not @ A @ ( plus_plus @ A @ A2 @ B2 ) )
= ( minus_minus @ A @ ( bit_ri4277139882892585799ns_not @ A @ A2 ) @ B2 ) ) ) ).
% not_add_distrib
thf(fact_7595_minus__eq__not__plus__1,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ( ( uminus_uminus @ A )
= ( ^ [A6: A] : ( plus_plus @ A @ ( bit_ri4277139882892585799ns_not @ A @ A6 ) @ ( one_one @ A ) ) ) ) ) ).
% minus_eq_not_plus_1
thf(fact_7596_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_7597_push__bit__mask__eq,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [M2: nat,N: nat] :
( ( bit_se4730199178511100633sh_bit @ A @ M2 @ ( bit_se2239418461657761734s_mask @ A @ N ) )
= ( bit_se5824344872417868541ns_and @ A @ ( bit_se2239418461657761734s_mask @ A @ ( plus_plus @ nat @ N @ M2 ) ) @ ( bit_ri4277139882892585799ns_not @ A @ ( bit_se2239418461657761734s_mask @ A @ M2 ) ) ) ) ) ).
% push_bit_mask_eq
thf(fact_7598_bit_Ocompl__unique,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [X: A,Y: A] :
( ( ( bit_se5824344872417868541ns_and @ A @ X @ Y )
= ( zero_zero @ A ) )
=> ( ( ( bit_se1065995026697491101ons_or @ A @ X @ Y )
= ( uminus_uminus @ A @ ( one_one @ A ) ) )
=> ( ( bit_ri4277139882892585799ns_not @ A @ X )
= Y ) ) ) ) ).
% bit.compl_unique
thf(fact_7599_bit__not__iff__eq,axiom,
! [A: $tType] :
( ( bit_ri3973907225187159222ations @ A )
=> ! [A2: A,N: nat] :
( ( bit_se5641148757651400278ts_bit @ A @ ( bit_ri4277139882892585799ns_not @ A @ A2 ) @ N )
= ( ( ( power_power @ A @ ( numeral_numeral @ A @ ( bit0 @ one2 ) ) @ N )
!= ( zero_zero @ A ) )
& ~ ( bit_se5641148757651400278ts_bit @ A @ A2 @ N ) ) ) ) ).
% bit_not_iff_eq
thf(fact_7600_cis__Arg__unique,axiom,
! [Z: complex,X: real] :
( ( ( sgn_sgn @ complex @ Z )
= ( cis @ X ) )
=> ( ( ord_less @ real @ ( uminus_uminus @ real @ pi ) @ X )
=> ( ( ord_less_eq @ real @ X @ pi )
=> ( ( arg @ Z )
= X ) ) ) ) ).
% cis_Arg_unique
thf(fact_7601_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
@ ^ [Z5: complex] :
( ( power_power @ complex @ Z5 @ N )
= ( one_one @ complex ) ) ) ) ) ).
% bij_betw_roots_unity
thf(fact_7602_bij__betw__nth__root__unity,axiom,
! [C3: complex,N: nat] :
( ( C3
!= ( 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 @ C3 ) ) ) @ ( cis @ ( divide_divide @ real @ ( arg @ C3 ) @ ( semiring_1_of_nat @ real @ N ) ) ) ) )
@ ( collect @ complex
@ ^ [Z5: complex] :
( ( power_power @ complex @ Z5 @ N )
= ( one_one @ complex ) ) )
@ ( collect @ complex
@ ^ [Z5: complex] :
( ( power_power @ complex @ Z5 @ N )
= C3 ) ) ) ) ) ).
% bij_betw_nth_root_unity
thf(fact_7603_bij__betw__add,axiom,
! [A: $tType] :
( ( cancel_semigroup_add @ A )
=> ! [A2: A,A5: set @ A,B6: set @ A] :
( ( bij_betw @ A @ A @ ( plus_plus @ A @ A2 ) @ A5 @ B6 )
= ( ( image2 @ A @ A @ ( plus_plus @ A @ A2 ) @ A5 )
= B6 ) ) ) ).
% bij_betw_add
thf(fact_7604_bij__betw__of__nat,axiom,
! [A: $tType] :
( ( semiring_char_0 @ A )
=> ! [N6: set @ nat,A5: set @ A] :
( ( bij_betw @ nat @ A @ ( semiring_1_of_nat @ A ) @ N6 @ A5 )
= ( ( image2 @ nat @ A @ ( semiring_1_of_nat @ A ) @ N6 )
= A5 ) ) ) ).
% bij_betw_of_nat
thf(fact_7605_bij__betw__funpow,axiom,
! [A: $tType,F3: A > A,S2: set @ A,N: nat] :
( ( bij_betw @ A @ A @ F3 @ S2 @ S2 )
=> ( bij_betw @ A @ A @ ( compow @ ( A > A ) @ N @ F3 ) @ S2 @ S2 ) ) ).
% bij_betw_funpow
thf(fact_7606_bij__fn,axiom,
! [A: $tType,F3: A > A,N: nat] :
( ( bij_betw @ A @ A @ F3 @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ A ) ) )
=> ( bij_betw @ A @ A @ ( compow @ ( A > A ) @ N @ F3 ) @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ A ) ) ) ) ).
% bij_fn
thf(fact_7607_bij__betw__same__card,axiom,
! [A: $tType,B: $tType,F3: A > B,A5: set @ A,B6: set @ B] :
( ( bij_betw @ A @ B @ F3 @ A5 @ B6 )
=> ( ( finite_card @ A @ A5 )
= ( finite_card @ B @ B6 ) ) ) ).
% bij_betw_same_card
thf(fact_7608_bij__betw__iff__card,axiom,
! [A: $tType,B: $tType,A5: set @ A,B6: set @ B] :
( ( finite_finite2 @ A @ A5 )
=> ( ( finite_finite2 @ B @ B6 )
=> ( ( ? [F4: A > B] : ( bij_betw @ A @ B @ F4 @ A5 @ B6 ) )
= ( ( finite_card @ A @ A5 )
= ( finite_card @ B @ B6 ) ) ) ) ) ).
% bij_betw_iff_card
thf(fact_7609_finite__same__card__bij,axiom,
! [A: $tType,B: $tType,A5: set @ A,B6: set @ B] :
( ( finite_finite2 @ A @ A5 )
=> ( ( finite_finite2 @ B @ B6 )
=> ( ( ( finite_card @ A @ A5 )
= ( finite_card @ B @ B6 ) )
=> ? [H6: A > B] : ( bij_betw @ A @ B @ H6 @ A5 @ B6 ) ) ) ) ).
% finite_same_card_bij
thf(fact_7610_bij__betw__empty1,axiom,
! [A: $tType,B: $tType,F3: A > B,A5: set @ B] :
( ( bij_betw @ A @ B @ F3 @ ( bot_bot @ ( set @ A ) ) @ A5 )
=> ( A5
= ( bot_bot @ ( set @ B ) ) ) ) ).
% bij_betw_empty1
thf(fact_7611_bij__betw__empty2,axiom,
! [B: $tType,A: $tType,F3: A > B,A5: set @ A] :
( ( bij_betw @ A @ B @ F3 @ A5 @ ( bot_bot @ ( set @ B ) ) )
=> ( A5
= ( bot_bot @ ( set @ A ) ) ) ) ).
% bij_betw_empty2
thf(fact_7612_bij__betw__finite,axiom,
! [A: $tType,B: $tType,F3: A > B,A5: set @ A,B6: set @ B] :
( ( bij_betw @ A @ B @ F3 @ A5 @ B6 )
=> ( ( finite_finite2 @ A @ A5 )
= ( finite_finite2 @ B @ B6 ) ) ) ).
% bij_betw_finite
thf(fact_7613_bij__betw__disjoint__Un,axiom,
! [A: $tType,B: $tType,F3: A > B,A5: set @ A,C5: set @ B,G3: A > B,B6: set @ A,D6: set @ B] :
( ( bij_betw @ A @ B @ F3 @ A5 @ C5 )
=> ( ( bij_betw @ A @ B @ G3 @ B6 @ D6 )
=> ( ( ( inf_inf @ ( set @ A ) @ A5 @ B6 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( inf_inf @ ( set @ B ) @ C5 @ D6 )
= ( bot_bot @ ( set @ B ) ) )
=> ( bij_betw @ A @ B
@ ^ [X5: A] : ( if @ B @ ( member @ A @ X5 @ A5 ) @ ( F3 @ X5 ) @ ( G3 @ X5 ) )
@ ( sup_sup @ ( set @ A ) @ A5 @ B6 )
@ ( sup_sup @ ( set @ B ) @ C5 @ D6 ) ) ) ) ) ) ).
% bij_betw_disjoint_Un
thf(fact_7614_notIn__Un__bij__betw,axiom,
! [A: $tType,B: $tType,B2: A,A5: set @ A,F3: A > B,A15: set @ B] :
( ~ ( member @ A @ B2 @ A5 )
=> ( ~ ( member @ B @ ( F3 @ B2 ) @ A15 )
=> ( ( bij_betw @ A @ B @ F3 @ A5 @ A15 )
=> ( bij_betw @ A @ B @ F3 @ ( sup_sup @ ( set @ A ) @ A5 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( sup_sup @ ( set @ B ) @ A15 @ ( insert @ B @ ( F3 @ B2 ) @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ).
% notIn_Un_bij_betw
thf(fact_7615_notIn__Un__bij__betw3,axiom,
! [A: $tType,B: $tType,B2: A,A5: set @ A,F3: A > B,A15: set @ B] :
( ~ ( member @ A @ B2 @ A5 )
=> ( ~ ( member @ B @ ( F3 @ B2 ) @ A15 )
=> ( ( bij_betw @ A @ B @ F3 @ A5 @ A15 )
= ( bij_betw @ A @ B @ F3 @ ( sup_sup @ ( set @ A ) @ A5 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( sup_sup @ ( set @ B ) @ A15 @ ( insert @ B @ ( F3 @ B2 ) @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ).
% notIn_Un_bij_betw3
thf(fact_7616_bij__betw__partition,axiom,
! [A: $tType,B: $tType,F3: A > B,A5: set @ A,C5: set @ A,B6: set @ B,D6: set @ B] :
( ( bij_betw @ A @ B @ F3 @ ( sup_sup @ ( set @ A ) @ A5 @ C5 ) @ ( sup_sup @ ( set @ B ) @ B6 @ D6 ) )
=> ( ( bij_betw @ A @ B @ F3 @ C5 @ D6 )
=> ( ( ( inf_inf @ ( set @ A ) @ A5 @ C5 )
= ( bot_bot @ ( set @ A ) ) )
=> ( ( ( inf_inf @ ( set @ B ) @ B6 @ D6 )
= ( bot_bot @ ( set @ B ) ) )
=> ( bij_betw @ A @ B @ F3 @ A5 @ B6 ) ) ) ) ) ).
% bij_betw_partition
thf(fact_7617_bij__betw__combine,axiom,
! [A: $tType,B: $tType,F3: A > B,A5: set @ A,B6: set @ B,C5: set @ A,D6: set @ B] :
( ( bij_betw @ A @ B @ F3 @ A5 @ B6 )
=> ( ( bij_betw @ A @ B @ F3 @ C5 @ D6 )
=> ( ( ( inf_inf @ ( set @ B ) @ B6 @ D6 )
= ( bot_bot @ ( set @ B ) ) )
=> ( bij_betw @ A @ B @ F3 @ ( sup_sup @ ( set @ A ) @ A5 @ C5 ) @ ( sup_sup @ ( set @ B ) @ B6 @ D6 ) ) ) ) ) ).
% bij_betw_combine
thf(fact_7618_finite__vimage__iff,axiom,
! [A: $tType,B: $tType,H2: A > B,F5: set @ B] :
( ( bij_betw @ A @ B @ H2 @ ( top_top @ ( set @ A ) ) @ ( top_top @ ( set @ B ) ) )
=> ( ( finite_finite2 @ A @ ( vimage @ A @ B @ H2 @ F5 ) )
= ( finite_finite2 @ B @ F5 ) ) ) ).
% finite_vimage_iff
thf(fact_7619_infinite__imp__bij__betw2,axiom,
! [A: $tType,A5: set @ A,A2: A] :
( ~ ( finite_finite2 @ A @ A5 )
=> ? [H6: A > A] : ( bij_betw @ A @ A @ H6 @ A5 @ ( sup_sup @ ( set @ A ) @ A5 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% infinite_imp_bij_betw2
thf(fact_7620_infinite__imp__bij__betw,axiom,
! [A: $tType,A5: set @ A,A2: A] :
( ~ ( finite_finite2 @ A @ A5 )
=> ? [H6: A > A] : ( bij_betw @ A @ A @ H6 @ A5 @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ).
% infinite_imp_bij_betw
thf(fact_7621_ex__bij__betw__nat__finite,axiom,
! [A: $tType,M5: set @ A] :
( ( finite_finite2 @ A @ M5 )
=> ? [H6: nat > A] : ( bij_betw @ nat @ A @ H6 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( finite_card @ A @ M5 ) ) @ M5 ) ) ).
% ex_bij_betw_nat_finite
thf(fact_7622_ex__bij__betw__nat__finite__1,axiom,
! [A: $tType,M5: set @ A] :
( ( finite_finite2 @ A @ M5 )
=> ? [H6: nat > A] : ( bij_betw @ nat @ A @ H6 @ ( set_or1337092689740270186AtMost @ nat @ ( one_one @ nat ) @ ( finite_card @ A @ M5 ) ) @ M5 ) ) ).
% ex_bij_betw_nat_finite_1
thf(fact_7623_sum_Oreindex__bij__betw__not__neutral,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comm_monoid_add @ A )
=> ! [S5: set @ B,T5: set @ C,H2: B > C,S2: set @ B,T4: set @ C,G3: C > A] :
( ( finite_finite2 @ B @ S5 )
=> ( ( finite_finite2 @ C @ T5 )
=> ( ( bij_betw @ B @ C @ H2 @ ( minus_minus @ ( set @ B ) @ S2 @ S5 ) @ ( minus_minus @ ( set @ C ) @ T4 @ T5 ) )
=> ( ! [A4: B] :
( ( member @ B @ A4 @ S5 )
=> ( ( G3 @ ( H2 @ A4 ) )
= ( zero_zero @ A ) ) )
=> ( ! [B4: C] :
( ( member @ C @ B4 @ T5 )
=> ( ( G3 @ B4 )
= ( zero_zero @ A ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A
@ ^ [X5: B] : ( G3 @ ( H2 @ X5 ) )
@ S2 )
= ( groups7311177749621191930dd_sum @ C @ A @ G3 @ T4 ) ) ) ) ) ) ) ) ).
% sum.reindex_bij_betw_not_neutral
thf(fact_7624_prod_Oreindex__bij__betw__not__neutral,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( comm_monoid_mult @ A )
=> ! [S5: set @ B,T5: set @ C,H2: B > C,S2: set @ B,T4: set @ C,G3: C > A] :
( ( finite_finite2 @ B @ S5 )
=> ( ( finite_finite2 @ C @ T5 )
=> ( ( bij_betw @ B @ C @ H2 @ ( minus_minus @ ( set @ B ) @ S2 @ S5 ) @ ( minus_minus @ ( set @ C ) @ T4 @ T5 ) )
=> ( ! [A4: B] :
( ( member @ B @ A4 @ S5 )
=> ( ( G3 @ ( H2 @ A4 ) )
= ( one_one @ A ) ) )
=> ( ! [B4: C] :
( ( member @ C @ B4 @ T5 )
=> ( ( G3 @ B4 )
= ( one_one @ A ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A
@ ^ [X5: B] : ( G3 @ ( H2 @ X5 ) )
@ S2 )
= ( groups7121269368397514597t_prod @ C @ A @ G3 @ T4 ) ) ) ) ) ) ) ) ).
% prod.reindex_bij_betw_not_neutral
thf(fact_7625_bij__betw__nth,axiom,
! [A: $tType,Xs: list @ A,A5: set @ nat,B6: set @ A] :
( ( distinct @ A @ Xs )
=> ( ( A5
= ( set_ord_lessThan @ nat @ ( size_size @ ( list @ A ) @ Xs ) ) )
=> ( ( B6
= ( set2 @ A @ Xs ) )
=> ( bij_betw @ nat @ A @ ( nth @ A @ Xs ) @ A5 @ B6 ) ) ) ) ).
% bij_betw_nth
thf(fact_7626_ex__bij__betw__strict__mono__card,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [M5: set @ A] :
( ( finite_finite2 @ A @ M5 )
=> ~ ! [H6: nat > A] :
( ( bij_betw @ nat @ A @ H6 @ ( set_ord_lessThan @ nat @ ( finite_card @ A @ M5 ) ) @ M5 )
=> ~ ( strict_mono_on @ nat @ A @ H6 @ ( set_ord_lessThan @ nat @ ( finite_card @ A @ M5 ) ) ) ) ) ) ).
% ex_bij_betw_strict_mono_card
thf(fact_7627_sum_OatLeastAtMost__reindex,axiom,
! [B: $tType,A: $tType] :
( ( ( comm_monoid_add @ A )
& ( ord @ B ) )
=> ! [H2: nat > B,M2: nat,N: nat,G3: B > A] :
( ( bij_betw @ nat @ B @ H2 @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) @ ( set_or1337092689740270186AtMost @ B @ ( H2 @ M2 ) @ ( H2 @ N ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G3 @ ( set_or1337092689740270186AtMost @ B @ ( H2 @ M2 ) @ ( H2 @ N ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ B @ A @ nat @ G3 @ H2 ) @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ) ).
% sum.atLeastAtMost_reindex
thf(fact_7628_sum_OatLeastLessThan__reindex,axiom,
! [B: $tType,A: $tType] :
( ( ( comm_monoid_add @ A )
& ( ord @ B ) )
=> ! [H2: nat > B,M2: nat,N: nat,G3: B > A] :
( ( bij_betw @ nat @ B @ H2 @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) @ ( set_or7035219750837199246ssThan @ B @ ( H2 @ M2 ) @ ( H2 @ N ) ) )
=> ( ( groups7311177749621191930dd_sum @ B @ A @ G3 @ ( set_or7035219750837199246ssThan @ B @ ( H2 @ M2 ) @ ( H2 @ N ) ) )
= ( groups7311177749621191930dd_sum @ nat @ A @ ( comp @ B @ A @ nat @ G3 @ H2 ) @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) ) ) ) ).
% sum.atLeastLessThan_reindex
thf(fact_7629_prod_OatLeastAtMost__reindex,axiom,
! [B: $tType,A: $tType] :
( ( ( comm_monoid_mult @ A )
& ( ord @ B ) )
=> ! [H2: nat > B,M2: nat,N: nat,G3: B > A] :
( ( bij_betw @ nat @ B @ H2 @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) @ ( set_or1337092689740270186AtMost @ B @ ( H2 @ M2 ) @ ( H2 @ N ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G3 @ ( set_or1337092689740270186AtMost @ B @ ( H2 @ M2 ) @ ( H2 @ N ) ) )
= ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ B @ A @ nat @ G3 @ H2 ) @ ( set_or1337092689740270186AtMost @ nat @ M2 @ N ) ) ) ) ) ).
% prod.atLeastAtMost_reindex
thf(fact_7630_prod_OatLeastLessThan__reindex,axiom,
! [B: $tType,A: $tType] :
( ( ( comm_monoid_mult @ A )
& ( ord @ B ) )
=> ! [H2: nat > B,M2: nat,N: nat,G3: B > A] :
( ( bij_betw @ nat @ B @ H2 @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) @ ( set_or7035219750837199246ssThan @ B @ ( H2 @ M2 ) @ ( H2 @ N ) ) )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G3 @ ( set_or7035219750837199246ssThan @ B @ ( H2 @ M2 ) @ ( H2 @ N ) ) )
= ( groups7121269368397514597t_prod @ nat @ A @ ( comp @ B @ A @ nat @ G3 @ H2 ) @ ( set_or7035219750837199246ssThan @ nat @ M2 @ N ) ) ) ) ) ).
% prod.atLeastLessThan_reindex
thf(fact_7631_Arg__def,axiom,
( arg
= ( ^ [Z5: complex] :
( if @ real
@ ( Z5
= ( zero_zero @ complex ) )
@ ( zero_zero @ real )
@ ( fChoice @ real
@ ^ [A6: real] :
( ( ( sgn_sgn @ complex @ Z5 )
= ( cis @ A6 ) )
& ( ord_less @ real @ ( uminus_uminus @ real @ pi ) @ A6 )
& ( ord_less_eq @ real @ A6 @ pi ) ) ) ) ) ) ).
% Arg_def
thf(fact_7632_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_7633_bij__betw__Suc,axiom,
! [M5: set @ nat,N6: set @ nat] :
( ( bij_betw @ nat @ nat @ suc @ M5 @ N6 )
= ( ( image2 @ nat @ nat @ suc @ M5 )
= N6 ) ) ).
% bij_betw_Suc
thf(fact_7634_bij__swap,axiom,
! [A: $tType,B: $tType] : ( bij_betw @ ( product_prod @ A @ B ) @ ( product_prod @ B @ A ) @ ( product_swap @ A @ B ) @ ( top_top @ ( set @ ( product_prod @ A @ B ) ) ) @ ( top_top @ ( set @ ( product_prod @ B @ A ) ) ) ) ).
% bij_swap
thf(fact_7635_some__in__eq,axiom,
! [A: $tType,A5: set @ A] :
( ( member @ A
@ ( fChoice @ A
@ ^ [X5: A] : ( member @ A @ X5 @ A5 ) )
@ A5 )
= ( A5
!= ( bot_bot @ ( set @ A ) ) ) ) ).
% some_in_eq
thf(fact_7636_bij__int__decode,axiom,
bij_betw @ nat @ int @ nat_int_decode @ ( top_top @ ( set @ nat ) ) @ ( top_top @ ( set @ int ) ) ).
% bij_int_decode
thf(fact_7637_bij__int__encode,axiom,
bij_betw @ int @ nat @ nat_int_encode @ ( top_top @ ( set @ int ) ) @ ( top_top @ ( set @ nat ) ) ).
% bij_int_encode
thf(fact_7638_ex__bij__betw__finite__nat,axiom,
! [A: $tType,M5: set @ A] :
( ( finite_finite2 @ A @ M5 )
=> ? [H6: A > nat] : ( bij_betw @ A @ nat @ H6 @ M5 @ ( set_or7035219750837199246ssThan @ nat @ ( zero_zero @ nat ) @ ( finite_card @ A @ M5 ) ) ) ) ).
% ex_bij_betw_finite_nat
thf(fact_7639_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_7640_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_7641_arg__min__SOME__Min,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ! [S2: set @ A,F3: A > B] :
( ( finite_finite2 @ A @ S2 )
=> ( ( lattic7623131987881927897min_on @ A @ B @ F3 @ S2 )
= ( fChoice @ A
@ ^ [Y6: A] :
( ( member @ A @ Y6 @ S2 )
& ( ( F3 @ Y6 )
= ( lattic643756798350308766er_Min @ B @ ( image2 @ A @ B @ F3 @ S2 ) ) ) ) ) ) ) ) ).
% arg_min_SOME_Min
thf(fact_7642_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_7643_max__ext__eq,axiom,
! [A: $tType] :
( ( max_ext @ A )
= ( ^ [R6: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ ( set @ A ) @ ( set @ A ) )
@ ( product_case_prod @ ( set @ A ) @ ( set @ A ) @ $o
@ ^ [X9: set @ A,Y10: set @ A] :
( ( finite_finite2 @ A @ X9 )
& ( finite_finite2 @ A @ Y10 )
& ( Y10
!= ( bot_bot @ ( set @ A ) ) )
& ! [X5: A] :
( ( member @ A @ X5 @ X9 )
=> ? [Y6: A] :
( ( member @ A @ Y6 @ Y10 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X5 @ Y6 ) @ R6 ) ) ) ) ) ) ) ) ).
% max_ext_eq
thf(fact_7644_bex__empty,axiom,
! [A: $tType,P2: A > $o] :
~ ? [X3: A] :
( ( member @ A @ X3 @ ( bot_bot @ ( set @ A ) ) )
& ( P2 @ X3 ) ) ).
% bex_empty
thf(fact_7645_Eps__case__prod__eq,axiom,
! [A: $tType,B: $tType,X: A,Y: B] :
( ( fChoice @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [X10: A,Y7: B] :
( ( X = X10 )
& ( Y = Y7 ) ) ) )
= ( product_Pair @ A @ B @ X @ Y ) ) ).
% Eps_case_prod_eq
thf(fact_7646_finite__Collect__bex,axiom,
! [B: $tType,A: $tType,A5: set @ A,Q: B > A > $o] :
( ( finite_finite2 @ A @ A5 )
=> ( ( finite_finite2 @ B
@ ( collect @ B
@ ^ [X5: B] :
? [Y6: A] :
( ( member @ A @ Y6 @ A5 )
& ( Q @ X5 @ Y6 ) ) ) )
= ( ! [X5: A] :
( ( member @ A @ X5 @ A5 )
=> ( finite_finite2 @ B
@ ( collect @ B
@ ^ [Y6: B] : ( Q @ Y6 @ X5 ) ) ) ) ) ) ) ).
% finite_Collect_bex
thf(fact_7647_split__paired__Eps,axiom,
! [B: $tType,A: $tType] :
( ( fChoice @ ( product_prod @ A @ B ) )
= ( ^ [P4: ( product_prod @ A @ B ) > $o] :
( fChoice @ ( product_prod @ A @ B )
@ ( product_case_prod @ A @ B @ $o
@ ^ [A6: A,B5: B] : ( P4 @ ( product_Pair @ A @ B @ A6 @ B5 ) ) ) ) ) ) ).
% split_paired_Eps
thf(fact_7648_Bex__fold,axiom,
! [A: $tType,A5: set @ A,P2: A > $o] :
( ( finite_finite2 @ A @ A5 )
=> ( ( ? [X5: A] :
( ( member @ A @ X5 @ A5 )
& ( P2 @ X5 ) ) )
= ( finite_fold @ A @ $o
@ ^ [K3: A,S8: $o] :
( S8
| ( P2 @ K3 ) )
@ $false
@ A5 ) ) ) ).
% Bex_fold
thf(fact_7649_nths__nths,axiom,
! [A: $tType,Xs: list @ A,A5: set @ nat,B6: set @ nat] :
( ( nths @ A @ ( nths @ A @ Xs @ A5 ) @ B6 )
= ( nths @ A @ Xs
@ ( collect @ nat
@ ^ [I4: nat] :
( ( member @ nat @ I4 @ A5 )
& ( member @ nat
@ ( finite_card @ nat
@ ( collect @ nat
@ ^ [I10: nat] :
( ( member @ nat @ I10 @ A5 )
& ( ord_less @ nat @ I10 @ I4 ) ) ) )
@ B6 ) ) ) ) ) ).
% nths_nths
thf(fact_7650_max__extp_Omax__extI,axiom,
! [A: $tType,X7: set @ A,Y8: set @ A,R: A > A > $o] :
( ( finite_finite2 @ A @ X7 )
=> ( ( finite_finite2 @ A @ Y8 )
=> ( ( Y8
!= ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ X7 )
=> ? [Xa: A] :
( ( member @ A @ Xa @ Y8 )
& ( R @ X4 @ Xa ) ) )
=> ( max_extp @ A @ R @ X7 @ Y8 ) ) ) ) ) ).
% max_extp.max_extI
thf(fact_7651_max__extp_Osimps,axiom,
! [A: $tType] :
( ( max_extp @ A )
= ( ^ [R6: A > A > $o,A12: set @ A,A23: set @ A] :
( ( finite_finite2 @ A @ A12 )
& ( finite_finite2 @ A @ A23 )
& ( A23
!= ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) )
& ! [X5: A] :
( ( member @ A @ X5 @ A12 )
=> ? [Y6: A] :
( ( member @ A @ Y6 @ A23 )
& ( R6 @ X5 @ Y6 ) ) ) ) ) ) ).
% max_extp.simps
thf(fact_7652_max__extp_Ocases,axiom,
! [A: $tType,R: A > A > $o,A1: set @ A,A22: set @ A] :
( ( max_extp @ A @ R @ A1 @ A22 )
=> ~ ( ( finite_finite2 @ A @ A1 )
=> ( ( finite_finite2 @ A @ A22 )
=> ( ( A22
!= ( collect @ A @ ( bot_bot @ ( A > $o ) ) ) )
=> ~ ! [X3: A] :
( ( member @ A @ X3 @ A1 )
=> ? [Xa3: A] :
( ( member @ A @ Xa3 @ A22 )
& ( R @ X3 @ Xa3 ) ) ) ) ) ) ) ).
% max_extp.cases
thf(fact_7653_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_7654_min__ext__def,axiom,
! [A: $tType] :
( ( min_ext @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A )] :
( collect @ ( product_prod @ ( set @ A ) @ ( set @ A ) )
@ ^ [Uu3: product_prod @ ( set @ A ) @ ( set @ A )] :
? [X9: set @ A,Y10: set @ A] :
( ( Uu3
= ( product_Pair @ ( set @ A ) @ ( set @ A ) @ X9 @ Y10 ) )
& ( X9
!= ( bot_bot @ ( set @ A ) ) )
& ! [X5: A] :
( ( member @ A @ X5 @ Y10 )
=> ? [Y6: A] :
( ( member @ A @ Y6 @ X9 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y6 @ X5 ) @ R5 ) ) ) ) ) ) ) ).
% min_ext_def
thf(fact_7655_map__project__def,axiom,
! [B: $tType,A: $tType] :
( ( map_project @ A @ B )
= ( ^ [F4: A > ( option @ B ),A7: set @ A] :
( collect @ B
@ ^ [B5: B] :
? [X5: A] :
( ( member @ A @ X5 @ A7 )
& ( ( F4 @ X5 )
= ( some @ B @ B5 ) ) ) ) ) ) ).
% map_project_def
thf(fact_7656_prod_Oset__conv__list,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [G3: B > A,Xs: list @ B] :
( ( groups7121269368397514597t_prod @ B @ A @ G3 @ ( set2 @ B @ Xs ) )
= ( groups5270119922927024881d_list @ A @ ( map @ B @ A @ G3 @ ( remdups @ B @ Xs ) ) ) ) ) ).
% prod.set_conv_list
thf(fact_7657_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_7658_prod_Odistinct__set__conv__list,axiom,
! [A: $tType,B: $tType] :
( ( comm_monoid_mult @ A )
=> ! [Xs: list @ B,G3: B > A] :
( ( distinct @ B @ Xs )
=> ( ( groups7121269368397514597t_prod @ B @ A @ G3 @ ( set2 @ B @ Xs ) )
= ( groups5270119922927024881d_list @ A @ ( map @ B @ A @ G3 @ Xs ) ) ) ) ) ).
% prod.distinct_set_conv_list
thf(fact_7659_lists__empty,axiom,
! [A: $tType] :
( ( lists @ A @ ( bot_bot @ ( set @ A ) ) )
= ( insert @ ( list @ A ) @ ( nil @ A ) @ ( bot_bot @ ( set @ ( list @ A ) ) ) ) ) ).
% lists_empty
thf(fact_7660_card__quotient__disjoint,axiom,
! [A: $tType,A5: set @ A,R2: set @ ( product_prod @ A @ A )] :
( ( finite_finite2 @ A @ A5 )
=> ( ( inj_on @ A @ ( set @ ( set @ A ) )
@ ^ [X5: A] : ( equiv_quotient @ A @ ( insert @ A @ X5 @ ( bot_bot @ ( set @ A ) ) ) @ R2 )
@ A5 )
=> ( ( finite_card @ ( set @ A ) @ ( equiv_quotient @ A @ A5 @ R2 ) )
= ( finite_card @ A @ A5 ) ) ) ) ).
% card_quotient_disjoint
thf(fact_7661_Cons__in__lists__iff,axiom,
! [A: $tType,X: A,Xs: list @ A,A5: set @ A] :
( ( member @ ( list @ A ) @ ( cons @ A @ X @ Xs ) @ ( lists @ A @ A5 ) )
= ( ( member @ A @ X @ A5 )
& ( member @ ( list @ A ) @ Xs @ ( lists @ A @ A5 ) ) ) ) ).
% Cons_in_lists_iff
thf(fact_7662_in__listsI,axiom,
! [A: $tType,Xs: list @ A,A5: set @ A] :
( ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
=> ( member @ A @ X4 @ A5 ) )
=> ( member @ ( list @ A ) @ Xs @ ( lists @ A @ A5 ) ) ) ).
% in_listsI
thf(fact_7663_lists__Int__eq,axiom,
! [A: $tType,A5: set @ A,B6: set @ A] :
( ( lists @ A @ ( inf_inf @ ( set @ A ) @ A5 @ B6 ) )
= ( inf_inf @ ( set @ ( list @ A ) ) @ ( lists @ A @ A5 ) @ ( lists @ A @ B6 ) ) ) ).
% lists_Int_eq
thf(fact_7664_append__in__lists__conv,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A,A5: set @ A] :
( ( member @ ( list @ A ) @ ( append @ A @ Xs @ Ys ) @ ( lists @ A @ A5 ) )
= ( ( member @ ( list @ A ) @ Xs @ ( lists @ A @ A5 ) )
& ( member @ ( list @ A ) @ Ys @ ( lists @ A @ A5 ) ) ) ) ).
% append_in_lists_conv
thf(fact_7665_lists__UNIV,axiom,
! [A: $tType] :
( ( lists @ A @ ( top_top @ ( set @ A ) ) )
= ( top_top @ ( set @ ( list @ A ) ) ) ) ).
% lists_UNIV
thf(fact_7666_listrel__refl__on,axiom,
! [A: $tType,A5: set @ A,R2: set @ ( product_prod @ A @ A )] :
( ( refl_on @ A @ A5 @ R2 )
=> ( refl_on @ ( list @ A ) @ ( lists @ A @ A5 ) @ ( listrel @ A @ A @ R2 ) ) ) ).
% listrel_refl_on
thf(fact_7667_lists__eq__set,axiom,
! [A: $tType] :
( ( lists @ A )
= ( ^ [A7: set @ A] :
( collect @ ( list @ A )
@ ^ [Xs3: list @ A] : ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs3 ) @ A7 ) ) ) ) ).
% lists_eq_set
thf(fact_7668_in__lists__conv__set,axiom,
! [A: $tType,Xs: list @ A,A5: set @ A] :
( ( member @ ( list @ A ) @ Xs @ ( lists @ A @ A5 ) )
= ( ! [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Xs ) )
=> ( member @ A @ X5 @ A5 ) ) ) ) ).
% in_lists_conv_set
thf(fact_7669_in__listsD,axiom,
! [A: $tType,Xs: list @ A,A5: set @ A] :
( ( member @ ( list @ A ) @ Xs @ ( lists @ A @ A5 ) )
=> ! [X3: A] :
( ( member @ A @ X3 @ ( set2 @ A @ Xs ) )
=> ( member @ A @ X3 @ A5 ) ) ) ).
% in_listsD
thf(fact_7670_lists_ONil,axiom,
! [A: $tType,A5: set @ A] : ( member @ ( list @ A ) @ ( nil @ A ) @ ( lists @ A @ A5 ) ) ).
% lists.Nil
thf(fact_7671_lists_Ocases,axiom,
! [A: $tType,A2: list @ A,A5: set @ A] :
( ( member @ ( list @ A ) @ A2 @ ( lists @ A @ A5 ) )
=> ( ( A2
!= ( nil @ A ) )
=> ~ ! [A4: A,L4: list @ A] :
( ( A2
= ( cons @ A @ A4 @ L4 ) )
=> ( ( member @ A @ A4 @ A5 )
=> ~ ( member @ ( list @ A ) @ L4 @ ( lists @ A @ A5 ) ) ) ) ) ) ).
% lists.cases
thf(fact_7672_lists_Osimps,axiom,
! [A: $tType,A2: list @ A,A5: set @ A] :
( ( member @ ( list @ A ) @ A2 @ ( lists @ A @ A5 ) )
= ( ( A2
= ( nil @ A ) )
| ? [A6: A,L2: list @ A] :
( ( A2
= ( cons @ A @ A6 @ L2 ) )
& ( member @ A @ A6 @ A5 )
& ( member @ ( list @ A ) @ L2 @ ( lists @ A @ A5 ) ) ) ) ) ).
% lists.simps
thf(fact_7673_listsE,axiom,
! [A: $tType,X: A,L: list @ A,A5: set @ A] :
( ( member @ ( list @ A ) @ ( cons @ A @ X @ L ) @ ( lists @ A @ A5 ) )
=> ~ ( ( member @ A @ X @ A5 )
=> ~ ( member @ ( list @ A ) @ L @ ( lists @ A @ A5 ) ) ) ) ).
% listsE
thf(fact_7674_lists_OCons,axiom,
! [A: $tType,A2: A,A5: set @ A,L: list @ A] :
( ( member @ A @ A2 @ A5 )
=> ( ( member @ ( list @ A ) @ L @ ( lists @ A @ A5 ) )
=> ( member @ ( list @ A ) @ ( cons @ A @ A2 @ L ) @ ( lists @ A @ A5 ) ) ) ) ).
% lists.Cons
thf(fact_7675_lists__IntI,axiom,
! [A: $tType,L: list @ A,A5: set @ A,B6: set @ A] :
( ( member @ ( list @ A ) @ L @ ( lists @ A @ A5 ) )
=> ( ( member @ ( list @ A ) @ L @ ( lists @ A @ B6 ) )
=> ( member @ ( list @ A ) @ L @ ( lists @ A @ ( inf_inf @ ( set @ A ) @ A5 @ B6 ) ) ) ) ) ).
% lists_IntI
thf(fact_7676_lists__mono,axiom,
! [A: $tType,A5: set @ A,B6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A5 @ B6 )
=> ( ord_less_eq @ ( set @ ( list @ A ) ) @ ( lists @ A @ A5 ) @ ( lists @ A @ B6 ) ) ) ).
% lists_mono
thf(fact_7677_lists__image,axiom,
! [A: $tType,B: $tType,F3: B > A,A5: set @ B] :
( ( lists @ A @ ( image2 @ B @ A @ F3 @ A5 ) )
= ( image2 @ ( list @ B ) @ ( list @ A ) @ ( map @ B @ A @ F3 ) @ ( lists @ B @ A5 ) ) ) ).
% lists_image
thf(fact_7678_listrel__subset,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A5: set @ A] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R2
@ ( product_Sigma @ A @ A @ A5
@ ^ [Uu3: A] : A5 ) )
=> ( ord_less_eq @ ( set @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) ) @ ( listrel @ A @ A @ R2 )
@ ( product_Sigma @ ( list @ A ) @ ( list @ A ) @ ( lists @ A @ A5 )
@ ^ [Uu3: list @ A] : ( lists @ A @ A5 ) ) ) ) ).
% listrel_subset
thf(fact_7679_quotient__empty,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( equiv_quotient @ A @ ( bot_bot @ ( set @ A ) ) @ R2 )
= ( bot_bot @ ( set @ ( set @ A ) ) ) ) ).
% quotient_empty
thf(fact_7680_quotient__is__empty,axiom,
! [A: $tType,A5: set @ A,R2: set @ ( product_prod @ A @ A )] :
( ( ( equiv_quotient @ A @ A5 @ R2 )
= ( bot_bot @ ( set @ ( set @ A ) ) ) )
= ( A5
= ( bot_bot @ ( set @ A ) ) ) ) ).
% quotient_is_empty
thf(fact_7681_quotient__is__empty2,axiom,
! [A: $tType,A5: set @ A,R2: set @ ( product_prod @ A @ A )] :
( ( ( bot_bot @ ( set @ ( set @ A ) ) )
= ( equiv_quotient @ A @ A5 @ R2 ) )
= ( A5
= ( bot_bot @ ( set @ A ) ) ) ) ).
% quotient_is_empty2
thf(fact_7682_finite__equiv__class,axiom,
! [A: $tType,A5: set @ A,R2: set @ ( product_prod @ A @ A ),X7: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R2
@ ( product_Sigma @ A @ A @ A5
@ ^ [Uu3: A] : A5 ) )
=> ( ( member @ ( set @ A ) @ X7 @ ( equiv_quotient @ A @ A5 @ R2 ) )
=> ( finite_finite2 @ A @ X7 ) ) ) ) ).
% finite_equiv_class
thf(fact_7683_finite__quotient,axiom,
! [A: $tType,A5: set @ A,R2: set @ ( product_prod @ A @ A )] :
( ( finite_finite2 @ A @ A5 )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R2
@ ( product_Sigma @ A @ A @ A5
@ ^ [Uu3: A] : A5 ) )
=> ( finite_finite2 @ ( set @ A ) @ ( equiv_quotient @ A @ A5 @ R2 ) ) ) ) ).
% finite_quotient
thf(fact_7684_quotient__diff1,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A5: set @ A,A2: A] :
( ( inj_on @ A @ ( set @ ( set @ A ) )
@ ^ [A6: A] : ( equiv_quotient @ A @ ( insert @ A @ A6 @ ( bot_bot @ ( set @ A ) ) ) @ R2 )
@ A5 )
=> ( ( member @ A @ A2 @ A5 )
=> ( ( equiv_quotient @ A @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) @ R2 )
= ( minus_minus @ ( set @ ( set @ A ) ) @ ( equiv_quotient @ A @ A5 @ R2 ) @ ( equiv_quotient @ A @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) @ R2 ) ) ) ) ) ).
% quotient_diff1
thf(fact_7685_Collect__finite__subset__eq__lists,axiom,
! [A: $tType,T4: set @ A] :
( ( collect @ ( set @ A )
@ ^ [A7: set @ A] :
( ( finite_finite2 @ A @ A7 )
& ( ord_less_eq @ ( set @ A ) @ A7 @ T4 ) ) )
= ( image2 @ ( list @ A ) @ ( set @ A ) @ ( set2 @ A ) @ ( lists @ A @ T4 ) ) ) ).
% Collect_finite_subset_eq_lists
thf(fact_7686_Collect__finite__eq__lists,axiom,
! [A: $tType] :
( ( collect @ ( set @ A ) @ ( finite_finite2 @ A ) )
= ( image2 @ ( list @ A ) @ ( set @ A ) @ ( set2 @ A ) @ ( lists @ A @ ( top_top @ ( set @ A ) ) ) ) ) ).
% Collect_finite_eq_lists
thf(fact_7687_to__nat__on__def,axiom,
! [A: $tType] :
( ( countable_to_nat_on @ A )
= ( ^ [S7: set @ A] :
( fChoice @ ( A > nat )
@ ^ [F4: A > nat] :
( ( ( finite_finite2 @ A @ S7 )
=> ( bij_betw @ A @ nat @ F4 @ S7 @ ( set_ord_lessThan @ nat @ ( finite_card @ A @ S7 ) ) ) )
& ( ~ ( finite_finite2 @ A @ S7 )
=> ( bij_betw @ A @ nat @ F4 @ S7 @ ( top_top @ ( set @ nat ) ) ) ) ) ) ) ) ).
% to_nat_on_def
thf(fact_7688_to__nat__on__finite,axiom,
! [A: $tType,S2: set @ A] :
( ( finite_finite2 @ A @ S2 )
=> ( bij_betw @ A @ nat @ ( countable_to_nat_on @ A @ S2 ) @ S2 @ ( set_ord_lessThan @ nat @ ( finite_card @ A @ S2 ) ) ) ) ).
% to_nat_on_finite
thf(fact_7689_INF__set__fold,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F3: B > A,Xs: list @ B] :
( ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ ( set2 @ B @ Xs ) ) )
= ( fold @ B @ A @ ( comp @ A @ ( A > A ) @ B @ ( inf_inf @ A ) @ F3 ) @ Xs @ ( top_top @ A ) ) ) ) ).
% INF_set_fold
thf(fact_7690_SUP__set__fold,axiom,
! [B: $tType,A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F3: B > A,Xs: list @ B] :
( ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ ( set2 @ B @ Xs ) ) )
= ( fold @ B @ A @ ( comp @ A @ ( A > A ) @ B @ ( sup_sup @ A ) @ F3 ) @ Xs @ ( bot_bot @ A ) ) ) ) ).
% SUP_set_fold
thf(fact_7691_fold__append,axiom,
! [A: $tType,B: $tType,F3: B > A > A,Xs: list @ B,Ys: list @ B] :
( ( fold @ B @ A @ F3 @ ( append @ B @ Xs @ Ys ) )
= ( comp @ A @ A @ A @ ( fold @ B @ A @ F3 @ Ys ) @ ( fold @ B @ A @ F3 @ Xs ) ) ) ).
% fold_append
thf(fact_7692_fold__replicate,axiom,
! [A: $tType,B: $tType,F3: B > A > A,N: nat,X: B] :
( ( fold @ B @ A @ F3 @ ( replicate @ B @ N @ X ) )
= ( compow @ ( A > A ) @ N @ ( F3 @ X ) ) ) ).
% fold_replicate
thf(fact_7693_fold__Cons__rev,axiom,
! [A: $tType,Xs: list @ A] :
( ( fold @ A @ ( list @ A ) @ ( cons @ A ) @ Xs )
= ( append @ A @ ( rev @ A @ Xs ) ) ) ).
% fold_Cons_rev
thf(fact_7694_rev__conv__fold,axiom,
! [A: $tType] :
( ( rev @ A )
= ( ^ [Xs3: list @ A] : ( fold @ A @ ( list @ A ) @ ( cons @ A ) @ Xs3 @ ( nil @ A ) ) ) ) ).
% rev_conv_fold
thf(fact_7695_fold__map,axiom,
! [B: $tType,A: $tType,C: $tType,G3: B > A > A,F3: C > B,Xs: list @ C] :
( ( fold @ B @ A @ G3 @ ( map @ C @ B @ F3 @ Xs ) )
= ( fold @ C @ A @ ( comp @ B @ ( A > A ) @ C @ G3 @ F3 ) @ Xs ) ) ).
% fold_map
thf(fact_7696_fold__commute__apply,axiom,
! [A: $tType,C: $tType,B: $tType,Xs: list @ A,H2: B > C,G3: A > B > B,F3: A > C > C,S: B] :
( ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
=> ( ( comp @ B @ C @ B @ H2 @ ( G3 @ X4 ) )
= ( comp @ C @ C @ B @ ( F3 @ X4 ) @ H2 ) ) )
=> ( ( H2 @ ( fold @ A @ B @ G3 @ Xs @ S ) )
= ( fold @ A @ C @ F3 @ Xs @ ( H2 @ S ) ) ) ) ).
% fold_commute_apply
thf(fact_7697_fold__commute,axiom,
! [A: $tType,C: $tType,B: $tType,Xs: list @ A,H2: B > C,G3: A > B > B,F3: A > C > C] :
( ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
=> ( ( comp @ B @ C @ B @ H2 @ ( G3 @ X4 ) )
= ( comp @ C @ C @ B @ ( F3 @ X4 ) @ H2 ) ) )
=> ( ( comp @ B @ C @ B @ H2 @ ( fold @ A @ B @ G3 @ Xs ) )
= ( comp @ C @ C @ B @ ( fold @ A @ C @ F3 @ Xs ) @ H2 ) ) ) ).
% fold_commute
thf(fact_7698_fold__Cons,axiom,
! [B: $tType,A: $tType,F3: A > B > B,X: A,Xs: list @ A] :
( ( fold @ A @ B @ F3 @ ( cons @ A @ X @ Xs ) )
= ( comp @ B @ B @ B @ ( fold @ A @ B @ F3 @ Xs ) @ ( F3 @ X ) ) ) ).
% fold_Cons
thf(fact_7699_foldr__conv__fold,axiom,
! [A: $tType,B: $tType] :
( ( foldr @ B @ A )
= ( ^ [F4: B > A > A,Xs3: list @ B] : ( fold @ B @ A @ F4 @ ( rev @ B @ Xs3 ) ) ) ) ).
% foldr_conv_fold
thf(fact_7700_fold__filter,axiom,
! [A: $tType,B: $tType,F3: B > A > A,P2: B > $o,Xs: list @ B] :
( ( fold @ B @ A @ F3 @ ( filter2 @ B @ P2 @ Xs ) )
= ( fold @ B @ A
@ ^ [X5: B] : ( if @ ( A > A ) @ ( P2 @ X5 ) @ ( F3 @ X5 ) @ ( id @ A ) )
@ Xs ) ) ).
% fold_filter
thf(fact_7701_fold__invariant,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Q: A > $o,P2: B > $o,S: B,F3: A > B > B] :
( ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
=> ( Q @ X4 ) )
=> ( ( P2 @ S )
=> ( ! [X4: A,S3: B] :
( ( Q @ X4 )
=> ( ( P2 @ S3 )
=> ( P2 @ ( F3 @ X4 @ S3 ) ) ) )
=> ( P2 @ ( fold @ A @ B @ F3 @ Xs @ S ) ) ) ) ) ).
% fold_invariant
thf(fact_7702_List_Ofold__cong,axiom,
! [B: $tType,A: $tType,A2: A,B2: A,Xs: list @ B,Ys: list @ B,F3: B > A > A,G3: B > A > A] :
( ( A2 = B2 )
=> ( ( Xs = Ys )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ ( set2 @ B @ Xs ) )
=> ( ( F3 @ X4 )
= ( G3 @ X4 ) ) )
=> ( ( fold @ B @ A @ F3 @ Xs @ A2 )
= ( fold @ B @ A @ G3 @ Ys @ B2 ) ) ) ) ) ).
% List.fold_cong
thf(fact_7703_fold__id,axiom,
! [A: $tType,B: $tType,Xs: list @ A,F3: A > B > B] :
( ! [X4: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
=> ( ( F3 @ X4 )
= ( id @ B ) ) )
=> ( ( fold @ A @ B @ F3 @ Xs )
= ( id @ B ) ) ) ).
% fold_id
thf(fact_7704_fold__simps_I1_J,axiom,
! [B: $tType,A: $tType,F3: B > A > A,S: A] :
( ( fold @ B @ A @ F3 @ ( nil @ B ) @ S )
= S ) ).
% fold_simps(1)
thf(fact_7705_fold__Nil,axiom,
! [A: $tType,B: $tType,F3: A > B > B] :
( ( fold @ A @ B @ F3 @ ( nil @ A ) )
= ( id @ B ) ) ).
% fold_Nil
thf(fact_7706_fold__simps_I2_J,axiom,
! [B: $tType,A: $tType,F3: B > A > A,X: B,Xs: list @ B,S: A] :
( ( fold @ B @ A @ F3 @ ( cons @ B @ X @ Xs ) @ S )
= ( fold @ B @ A @ F3 @ Xs @ ( F3 @ X @ S ) ) ) ).
% fold_simps(2)
thf(fact_7707_foldl__conv__fold,axiom,
! [B: $tType,A: $tType] :
( ( foldl @ A @ B )
= ( ^ [F4: A > B > A,S8: A,Xs3: list @ B] :
( fold @ B @ A
@ ^ [X5: B,T3: A] : ( F4 @ T3 @ X5 )
@ Xs3
@ S8 ) ) ) ).
% foldl_conv_fold
thf(fact_7708_union__set__fold,axiom,
! [A: $tType,Xs: list @ A,A5: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( set2 @ A @ Xs ) @ A5 )
= ( fold @ A @ ( set @ A ) @ ( insert @ A ) @ Xs @ A5 ) ) ).
% union_set_fold
thf(fact_7709_fold__rev,axiom,
! [B: $tType,A: $tType,Xs: list @ A,F3: A > B > B] :
( ! [X4: A,Y3: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
=> ( ( member @ A @ Y3 @ ( set2 @ A @ Xs ) )
=> ( ( comp @ B @ B @ B @ ( F3 @ Y3 ) @ ( F3 @ X4 ) )
= ( comp @ B @ B @ B @ ( F3 @ X4 ) @ ( F3 @ Y3 ) ) ) ) )
=> ( ( fold @ A @ B @ F3 @ ( rev @ A @ Xs ) )
= ( fold @ A @ B @ F3 @ Xs ) ) ) ).
% fold_rev
thf(fact_7710_fold__plus__sum__list__rev,axiom,
! [A: $tType] :
( ( monoid_add @ A )
=> ! [Xs: list @ A] :
( ( fold @ A @ A @ ( plus_plus @ A ) @ Xs )
= ( plus_plus @ A @ ( groups8242544230860333062m_list @ A @ ( rev @ A @ Xs ) ) ) ) ) ).
% fold_plus_sum_list_rev
thf(fact_7711_fold__remove1__split,axiom,
! [B: $tType,A: $tType,Xs: list @ A,F3: A > B > B,X: A] :
( ! [X4: A,Y3: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
=> ( ( member @ A @ Y3 @ ( set2 @ A @ Xs ) )
=> ( ( comp @ B @ B @ B @ ( F3 @ X4 ) @ ( F3 @ Y3 ) )
= ( comp @ B @ B @ B @ ( F3 @ Y3 ) @ ( F3 @ X4 ) ) ) ) )
=> ( ( member @ A @ X @ ( set2 @ A @ Xs ) )
=> ( ( fold @ A @ B @ F3 @ Xs )
= ( comp @ B @ B @ B @ ( fold @ A @ B @ F3 @ ( remove1 @ A @ X @ Xs ) ) @ ( F3 @ X ) ) ) ) ) ).
% fold_remove1_split
thf(fact_7712_foldr__fold,axiom,
! [B: $tType,A: $tType,Xs: list @ A,F3: A > B > B] :
( ! [X4: A,Y3: A] :
( ( member @ A @ X4 @ ( set2 @ A @ Xs ) )
=> ( ( member @ A @ Y3 @ ( set2 @ A @ Xs ) )
=> ( ( comp @ B @ B @ B @ ( F3 @ Y3 ) @ ( F3 @ X4 ) )
= ( comp @ B @ B @ B @ ( F3 @ X4 ) @ ( F3 @ Y3 ) ) ) ) )
=> ( ( foldr @ A @ B @ F3 @ Xs )
= ( fold @ A @ B @ F3 @ Xs ) ) ) ).
% foldr_fold
thf(fact_7713_minus__set__fold,axiom,
! [A: $tType,A5: set @ A,Xs: list @ A] :
( ( minus_minus @ ( set @ A ) @ A5 @ ( set2 @ A @ Xs ) )
= ( fold @ A @ ( set @ A ) @ ( remove @ A ) @ Xs @ A5 ) ) ).
% minus_set_fold
thf(fact_7714_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_7715_fold__append__concat__rev,axiom,
! [A: $tType,Xss: list @ ( list @ A )] :
( ( fold @ ( list @ A ) @ ( list @ A ) @ ( append @ A ) @ Xss )
= ( append @ A @ ( concat @ A @ ( rev @ ( list @ A ) @ Xss ) ) ) ) ).
% fold_append_concat_rev
thf(fact_7716_Sup__set__fold,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [Xs: list @ A] :
( ( complete_Sup_Sup @ A @ ( set2 @ A @ Xs ) )
= ( fold @ A @ A @ ( sup_sup @ A ) @ Xs @ ( bot_bot @ A ) ) ) ) ).
% Sup_set_fold
thf(fact_7717_Inf__set__fold,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [Xs: list @ A] :
( ( complete_Inf_Inf @ A @ ( set2 @ A @ Xs ) )
= ( fold @ A @ A @ ( inf_inf @ A ) @ Xs @ ( top_top @ A ) ) ) ) ).
% Inf_set_fold
thf(fact_7718_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_7719_Inf__fin_Oset__eq__fold,axiom,
! [A: $tType] :
( ( semilattice_inf @ A )
=> ! [X: A,Xs: list @ A] :
( ( lattic7752659483105999362nf_fin @ A @ ( set2 @ A @ ( cons @ A @ X @ Xs ) ) )
= ( fold @ A @ A @ ( inf_inf @ A ) @ Xs @ X ) ) ) ).
% Inf_fin.set_eq_fold
thf(fact_7720_Sup__fin_Oset__eq__fold,axiom,
! [A: $tType] :
( ( semilattice_sup @ A )
=> ! [X: A,Xs: list @ A] :
( ( lattic5882676163264333800up_fin @ A @ ( set2 @ A @ ( cons @ A @ X @ Xs ) ) )
= ( fold @ A @ A @ ( sup_sup @ A ) @ Xs @ X ) ) ) ).
% Sup_fin.set_eq_fold
thf(fact_7721_Max_Oset__eq__fold,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Xs: list @ A] :
( ( lattic643756798349783984er_Max @ A @ ( set2 @ A @ ( cons @ A @ X @ Xs ) ) )
= ( fold @ A @ A @ ( ord_max @ A ) @ Xs @ X ) ) ) ).
% Max.set_eq_fold
thf(fact_7722_Min_Oset__eq__fold,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [X: A,Xs: list @ A] :
( ( lattic643756798350308766er_Min @ A @ ( set2 @ A @ ( cons @ A @ X @ Xs ) ) )
= ( fold @ A @ A @ ( ord_min @ A ) @ Xs @ X ) ) ) ).
% Min.set_eq_fold
thf(fact_7723_comp__fun__idem__on_Ofold__set__fold,axiom,
! [A: $tType,B: $tType,S2: set @ A,F3: A > B > B,Xs: list @ A,Y: B] :
( ( finite673082921795544331dem_on @ A @ B @ S2 @ F3 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ S2 )
=> ( ( finite_fold @ A @ B @ F3 @ Y @ ( set2 @ A @ Xs ) )
= ( fold @ A @ B @ F3 @ Xs @ Y ) ) ) ) ).
% comp_fun_idem_on.fold_set_fold
thf(fact_7724_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_7725_comp__fun__commute__on_Ofold__set__fold__remdups,axiom,
! [A: $tType,B: $tType,S2: set @ A,F3: A > B > B,Xs: list @ A,Y: B] :
( ( finite4664212375090638736ute_on @ A @ B @ S2 @ F3 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( set2 @ A @ Xs ) @ S2 )
=> ( ( finite_fold @ A @ B @ F3 @ Y @ ( set2 @ A @ Xs ) )
= ( fold @ A @ B @ F3 @ ( remdups @ A @ Xs ) @ Y ) ) ) ) ).
% comp_fun_commute_on.fold_set_fold_remdups
thf(fact_7726_finite__sequence__to__countable__set,axiom,
! [A: $tType,X7: set @ A] :
( ( countable_countable @ A @ X7 )
=> ~ ! [F6: nat > ( set @ A )] :
( ! [I: nat] : ( ord_less_eq @ ( set @ A ) @ ( F6 @ I ) @ X7 )
=> ( ! [I: nat] : ( ord_less_eq @ ( set @ A ) @ ( F6 @ I ) @ ( F6 @ ( suc @ I ) ) )
=> ( ! [I: nat] : ( finite_finite2 @ A @ ( F6 @ I ) )
=> ( ( complete_Sup_Sup @ ( set @ A ) @ ( image2 @ nat @ ( set @ A ) @ F6 @ ( top_top @ ( set @ nat ) ) ) )
!= X7 ) ) ) ) ) ).
% finite_sequence_to_countable_set
thf(fact_7727_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_7728_countable__empty,axiom,
! [A: $tType] : ( countable_countable @ A @ ( bot_bot @ ( set @ A ) ) ) ).
% countable_empty
thf(fact_7729_butlast__rev,axiom,
! [A: $tType,Xs: list @ A] :
( ( butlast @ A @ ( rev @ A @ Xs ) )
= ( rev @ A @ ( tl @ A @ Xs ) ) ) ).
% butlast_rev
thf(fact_7730_countable__Diff__eq,axiom,
! [A: $tType,A5: set @ A,X: A] :
( ( countable_countable @ A @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( countable_countable @ A @ A5 ) ) ).
% countable_Diff_eq
thf(fact_7731_butlast__snoc,axiom,
! [A: $tType,Xs: list @ A,X: A] :
( ( butlast @ A @ ( append @ A @ Xs @ ( cons @ A @ X @ ( nil @ A ) ) ) )
= Xs ) ).
% butlast_snoc
thf(fact_7732_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_7733_countable__Collect__finite__subset,axiom,
! [A: $tType,T4: set @ A] :
( ( countable_countable @ A @ T4 )
=> ( countable_countable @ ( set @ A )
@ ( collect @ ( set @ A )
@ ^ [A7: set @ A] :
( ( finite_finite2 @ A @ A7 )
& ( ord_less_eq @ ( set @ A ) @ A7 @ T4 ) ) ) ) ) ).
% countable_Collect_finite_subset
thf(fact_7734_infinite__countable__subset_H,axiom,
! [A: $tType,X7: set @ A] :
( ~ ( finite_finite2 @ A @ X7 )
=> ? [C6: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ C6 @ X7 )
& ( countable_countable @ A @ C6 )
& ~ ( finite_finite2 @ A @ C6 ) ) ) ).
% infinite_countable_subset'
thf(fact_7735_countable__Collect__finite,axiom,
! [A: $tType] :
( ( countable @ A )
=> ( countable_countable @ ( set @ A ) @ ( collect @ ( set @ A ) @ ( finite_finite2 @ A ) ) ) ) ).
% countable_Collect_finite
thf(fact_7736_uncountable__infinite,axiom,
! [A: $tType,A5: set @ A] :
( ~ ( countable_countable @ A @ A5 )
=> ~ ( finite_finite2 @ A @ A5 ) ) ).
% uncountable_infinite
thf(fact_7737_countable__finite,axiom,
! [A: $tType,S2: set @ A] :
( ( finite_finite2 @ A @ S2 )
=> ( countable_countable @ A @ S2 ) ) ).
% countable_finite
thf(fact_7738_uncountable__def,axiom,
! [A: $tType,A5: set @ A] :
( ( ~ ( countable_countable @ A @ A5 ) )
= ( ( A5
!= ( bot_bot @ ( set @ A ) ) )
& ~ ? [F4: nat > A] :
( ( image2 @ nat @ A @ F4 @ ( top_top @ ( set @ nat ) ) )
= A5 ) ) ) ).
% uncountable_def
thf(fact_7739_countable__infiniteE_H,axiom,
! [A: $tType,A5: set @ A] :
( ( countable_countable @ A @ A5 )
=> ( ~ ( finite_finite2 @ A @ A5 )
=> ~ ! [G2: nat > A] :
~ ( bij_betw @ nat @ A @ G2 @ ( top_top @ ( set @ nat ) ) @ A5 ) ) ) ).
% countable_infiniteE'
thf(fact_7740_less__ccSUP__iff,axiom,
! [A: $tType,B: $tType] :
( ( ( counta3822494911875563373attice @ A )
& ( linorder @ A ) )
=> ! [A5: set @ B,A2: A,F3: B > A] :
( ( countable_countable @ B @ A5 )
=> ( ( ord_less @ A @ A2 @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A5 ) ) )
= ( ? [X5: B] :
( ( member @ B @ X5 @ A5 )
& ( ord_less @ A @ A2 @ ( F3 @ X5 ) ) ) ) ) ) ) ).
% less_ccSUP_iff
thf(fact_7741_butlast__tl,axiom,
! [A: $tType,Xs: list @ A] :
( ( butlast @ A @ ( tl @ A @ Xs ) )
= ( tl @ A @ ( butlast @ A @ Xs ) ) ) ).
% butlast_tl
thf(fact_7742_ccSup__subset__mono,axiom,
! [A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [B6: set @ A,A5: set @ A] :
( ( countable_countable @ A @ B6 )
=> ( ( ord_less_eq @ ( set @ A ) @ A5 @ B6 )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A5 ) @ ( complete_Sup_Sup @ A @ B6 ) ) ) ) ) ).
% ccSup_subset_mono
thf(fact_7743_butlast__append,axiom,
! [A: $tType,Ys: list @ A,Xs: list @ A] :
( ( ( Ys
= ( nil @ A ) )
=> ( ( butlast @ A @ ( append @ A @ Xs @ Ys ) )
= ( butlast @ A @ Xs ) ) )
& ( ( Ys
!= ( nil @ A ) )
=> ( ( butlast @ A @ ( append @ A @ Xs @ Ys ) )
= ( append @ A @ Xs @ ( butlast @ A @ Ys ) ) ) ) ) ).
% butlast_append
thf(fact_7744_ccInf__less__iff,axiom,
! [A: $tType] :
( ( ( counta3822494911875563373attice @ A )
& ( linorder @ A ) )
=> ! [S2: set @ A,A2: A] :
( ( countable_countable @ A @ S2 )
=> ( ( ord_less @ A @ ( complete_Inf_Inf @ A @ S2 ) @ A2 )
= ( ? [X5: A] :
( ( member @ A @ X5 @ S2 )
& ( ord_less @ A @ X5 @ A2 ) ) ) ) ) ) ).
% ccInf_less_iff
thf(fact_7745_ccInf__mono,axiom,
! [A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [B6: set @ A,A5: set @ A] :
( ( countable_countable @ A @ B6 )
=> ( ( countable_countable @ A @ A5 )
=> ( ! [B4: A] :
( ( member @ A @ B4 @ B6 )
=> ? [X3: A] :
( ( member @ A @ X3 @ A5 )
& ( ord_less_eq @ A @ X3 @ B4 ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A5 ) @ ( complete_Inf_Inf @ A @ B6 ) ) ) ) ) ) ).
% ccInf_mono
thf(fact_7746_ccInf__lower,axiom,
! [A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A5: set @ A,X: A] :
( ( countable_countable @ A @ A5 )
=> ( ( member @ A @ X @ A5 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A5 ) @ X ) ) ) ) ).
% ccInf_lower
thf(fact_7747_ccInf__lower2,axiom,
! [A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A5: set @ A,U: A,V: A] :
( ( countable_countable @ A @ A5 )
=> ( ( member @ A @ U @ A5 )
=> ( ( ord_less_eq @ A @ U @ V )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A5 ) @ V ) ) ) ) ) ).
% ccInf_lower2
thf(fact_7748_le__ccInf__iff,axiom,
! [A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A5: set @ A,B2: A] :
( ( countable_countable @ A @ A5 )
=> ( ( ord_less_eq @ A @ B2 @ ( complete_Inf_Inf @ A @ A5 ) )
= ( ! [X5: A] :
( ( member @ A @ X5 @ A5 )
=> ( ord_less_eq @ A @ B2 @ X5 ) ) ) ) ) ) ).
% le_ccInf_iff
thf(fact_7749_ccInf__greatest,axiom,
! [A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A5: set @ A,Z: A] :
( ( countable_countable @ A @ A5 )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ A5 )
=> ( ord_less_eq @ A @ Z @ X4 ) )
=> ( ord_less_eq @ A @ Z @ ( complete_Inf_Inf @ A @ A5 ) ) ) ) ) ).
% ccInf_greatest
thf(fact_7750_ccSup__mono,axiom,
! [A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [B6: set @ A,A5: set @ A] :
( ( countable_countable @ A @ B6 )
=> ( ( countable_countable @ A @ A5 )
=> ( ! [A4: A] :
( ( member @ A @ A4 @ A5 )
=> ? [X3: A] :
( ( member @ A @ X3 @ B6 )
& ( ord_less_eq @ A @ A4 @ X3 ) ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A5 ) @ ( complete_Sup_Sup @ A @ B6 ) ) ) ) ) ) ).
% ccSup_mono
thf(fact_7751_ccSup__least,axiom,
! [A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A5: set @ A,Z: A] :
( ( countable_countable @ A @ A5 )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ A5 )
=> ( ord_less_eq @ A @ X4 @ Z ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A5 ) @ Z ) ) ) ) ).
% ccSup_least
thf(fact_7752_ccSup__upper,axiom,
! [A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A5: set @ A,X: A] :
( ( countable_countable @ A @ A5 )
=> ( ( member @ A @ X @ A5 )
=> ( ord_less_eq @ A @ X @ ( complete_Sup_Sup @ A @ A5 ) ) ) ) ) ).
% ccSup_upper
thf(fact_7753_ccSup__le__iff,axiom,
! [A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A5: set @ A,B2: A] :
( ( countable_countable @ A @ A5 )
=> ( ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ A5 ) @ B2 )
= ( ! [X5: A] :
( ( member @ A @ X5 @ A5 )
=> ( ord_less_eq @ A @ X5 @ B2 ) ) ) ) ) ) ).
% ccSup_le_iff
thf(fact_7754_ccSup__upper2,axiom,
! [A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A5: set @ A,U: A,V: A] :
( ( countable_countable @ A @ A5 )
=> ( ( member @ A @ U @ A5 )
=> ( ( ord_less_eq @ A @ V @ U )
=> ( ord_less_eq @ A @ V @ ( complete_Sup_Sup @ A @ A5 ) ) ) ) ) ) ).
% ccSup_upper2
thf(fact_7755_less__ccSup__iff,axiom,
! [A: $tType] :
( ( ( counta3822494911875563373attice @ A )
& ( linorder @ A ) )
=> ! [S2: set @ A,A2: A] :
( ( countable_countable @ A @ S2 )
=> ( ( ord_less @ A @ A2 @ ( complete_Sup_Sup @ A @ S2 ) )
= ( ? [X5: A] :
( ( member @ A @ X5 @ S2 )
& ( ord_less @ A @ A2 @ X5 ) ) ) ) ) ) ).
% less_ccSup_iff
thf(fact_7756_butlast_Osimps_I2_J,axiom,
! [A: $tType,Xs: list @ A,X: A] :
( ( ( Xs
= ( nil @ A ) )
=> ( ( butlast @ A @ ( cons @ A @ X @ Xs ) )
= ( nil @ A ) ) )
& ( ( Xs
!= ( nil @ A ) )
=> ( ( butlast @ A @ ( cons @ A @ X @ Xs ) )
= ( cons @ A @ X @ ( butlast @ A @ Xs ) ) ) ) ) ).
% butlast.simps(2)
thf(fact_7757_butlast_Osimps_I1_J,axiom,
! [A: $tType] :
( ( butlast @ A @ ( nil @ A ) )
= ( nil @ A ) ) ).
% butlast.simps(1)
thf(fact_7758_drop__butlast,axiom,
! [A: $tType,N: nat,Xs: list @ A] :
( ( drop @ A @ N @ ( butlast @ A @ Xs ) )
= ( butlast @ A @ ( drop @ A @ N @ Xs ) ) ) ).
% drop_butlast
thf(fact_7759_in__set__butlastD,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( member @ A @ X @ ( set2 @ A @ ( butlast @ A @ Xs ) ) )
=> ( member @ A @ X @ ( set2 @ A @ Xs ) ) ) ).
% in_set_butlastD
thf(fact_7760_distinct__butlast,axiom,
! [A: $tType,Xs: list @ A] :
( ( distinct @ A @ Xs )
=> ( distinct @ A @ ( butlast @ A @ Xs ) ) ) ).
% distinct_butlast
thf(fact_7761_in__set__butlast__appendI,axiom,
! [A: $tType,X: A,Xs: list @ A,Ys: list @ A] :
( ( ( member @ A @ X @ ( set2 @ A @ ( butlast @ A @ Xs ) ) )
| ( member @ A @ X @ ( set2 @ A @ ( butlast @ A @ Ys ) ) ) )
=> ( member @ A @ X @ ( set2 @ A @ ( butlast @ A @ ( append @ A @ Xs @ Ys ) ) ) ) ) ).
% in_set_butlast_appendI
thf(fact_7762_ccInf__superset__mono,axiom,
! [A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A5: set @ A,B6: set @ A] :
( ( countable_countable @ A @ A5 )
=> ( ( ord_less_eq @ ( set @ A ) @ B6 @ A5 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ A5 ) @ ( complete_Inf_Inf @ A @ B6 ) ) ) ) ) ).
% ccInf_superset_mono
thf(fact_7763_map__butlast,axiom,
! [A: $tType,B: $tType,F3: B > A,Xs: list @ B] :
( ( map @ B @ A @ F3 @ ( butlast @ B @ Xs ) )
= ( butlast @ A @ ( map @ B @ A @ F3 @ Xs ) ) ) ).
% map_butlast
thf(fact_7764_ccSUP__mono,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A5: set @ B,B6: set @ C,F3: B > A,G3: C > A] :
( ( countable_countable @ B @ A5 )
=> ( ( countable_countable @ C @ B6 )
=> ( ! [N3: B] :
( ( member @ B @ N3 @ A5 )
=> ? [X3: C] :
( ( member @ C @ X3 @ B6 )
& ( ord_less_eq @ A @ ( F3 @ N3 ) @ ( G3 @ X3 ) ) ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A5 ) ) @ ( complete_Sup_Sup @ A @ ( image2 @ C @ A @ G3 @ B6 ) ) ) ) ) ) ) ).
% ccSUP_mono
thf(fact_7765_ccSUP__least,axiom,
! [B: $tType,A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A5: set @ B,F3: B > A,U: A] :
( ( countable_countable @ B @ A5 )
=> ( ! [I3: B] :
( ( member @ B @ I3 @ A5 )
=> ( ord_less_eq @ A @ ( F3 @ I3 ) @ U ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A5 ) ) @ U ) ) ) ) ).
% ccSUP_least
thf(fact_7766_ccSUP__upper,axiom,
! [A: $tType,B: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A5: set @ B,I2: B,F3: B > A] :
( ( countable_countable @ B @ A5 )
=> ( ( member @ B @ I2 @ A5 )
=> ( ord_less_eq @ A @ ( F3 @ I2 ) @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A5 ) ) ) ) ) ) ).
% ccSUP_upper
thf(fact_7767_ccSUP__le__iff,axiom,
! [A: $tType,B: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A5: set @ B,F3: B > A,U: A] :
( ( countable_countable @ B @ A5 )
=> ( ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A5 ) ) @ U )
= ( ! [X5: B] :
( ( member @ B @ X5 @ A5 )
=> ( ord_less_eq @ A @ ( F3 @ X5 ) @ U ) ) ) ) ) ) ).
% ccSUP_le_iff
thf(fact_7768_ccSUP__upper2,axiom,
! [A: $tType,B: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A5: set @ B,I2: B,U: A,F3: B > A] :
( ( countable_countable @ B @ A5 )
=> ( ( member @ B @ I2 @ A5 )
=> ( ( ord_less_eq @ A @ U @ ( F3 @ I2 ) )
=> ( ord_less_eq @ A @ U @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A5 ) ) ) ) ) ) ) ).
% ccSUP_upper2
thf(fact_7769_ccINF__less__iff,axiom,
! [A: $tType,B: $tType] :
( ( ( counta3822494911875563373attice @ A )
& ( linorder @ A ) )
=> ! [A5: set @ B,F3: B > A,A2: A] :
( ( countable_countable @ B @ A5 )
=> ( ( ord_less @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ A5 ) ) @ A2 )
= ( ? [X5: B] :
( ( member @ B @ X5 @ A5 )
& ( ord_less @ A @ ( F3 @ X5 ) @ A2 ) ) ) ) ) ) ).
% ccINF_less_iff
thf(fact_7770_ccINF__mono,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A5: set @ B,B6: set @ C,F3: B > A,G3: C > A] :
( ( countable_countable @ B @ A5 )
=> ( ( countable_countable @ C @ B6 )
=> ( ! [M: C] :
( ( member @ C @ M @ B6 )
=> ? [X3: B] :
( ( member @ B @ X3 @ A5 )
& ( ord_less_eq @ A @ ( F3 @ X3 ) @ ( G3 @ M ) ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ A5 ) ) @ ( complete_Inf_Inf @ A @ ( image2 @ C @ A @ G3 @ B6 ) ) ) ) ) ) ) ).
% ccINF_mono
thf(fact_7771_ccINF__lower,axiom,
! [A: $tType,B: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A5: set @ B,I2: B,F3: B > A] :
( ( countable_countable @ B @ A5 )
=> ( ( member @ B @ I2 @ A5 )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ A5 ) ) @ ( F3 @ I2 ) ) ) ) ) ).
% ccINF_lower
thf(fact_7772_ccINF__lower2,axiom,
! [B: $tType,A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A5: set @ B,I2: B,F3: B > A,U: A] :
( ( countable_countable @ B @ A5 )
=> ( ( member @ B @ I2 @ A5 )
=> ( ( ord_less_eq @ A @ ( F3 @ I2 ) @ U )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ A5 ) ) @ U ) ) ) ) ) ).
% ccINF_lower2
thf(fact_7773_le__ccINF__iff,axiom,
! [A: $tType,B: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A5: set @ B,U: A,F3: B > A] :
( ( countable_countable @ B @ A5 )
=> ( ( ord_less_eq @ A @ U @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ A5 ) ) )
= ( ! [X5: B] :
( ( member @ B @ X5 @ A5 )
=> ( ord_less_eq @ A @ U @ ( F3 @ X5 ) ) ) ) ) ) ) ).
% le_ccINF_iff
thf(fact_7774_ccINF__greatest,axiom,
! [A: $tType,B: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A5: set @ B,U: A,F3: B > A] :
( ( countable_countable @ B @ A5 )
=> ( ! [I3: B] :
( ( member @ B @ I3 @ A5 )
=> ( ord_less_eq @ A @ U @ ( F3 @ I3 ) ) )
=> ( ord_less_eq @ A @ U @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ A5 ) ) ) ) ) ) ).
% ccINF_greatest
thf(fact_7775_to__nat__on__surj,axiom,
! [A: $tType,A5: set @ A,N: nat] :
( ( countable_countable @ A @ A5 )
=> ( ~ ( finite_finite2 @ A @ A5 )
=> ? [X4: A] :
( ( member @ A @ X4 @ A5 )
& ( ( countable_to_nat_on @ A @ A5 @ X4 )
= N ) ) ) ) ).
% to_nat_on_surj
thf(fact_7776_countableE__infinite,axiom,
! [A: $tType,S2: set @ A] :
( ( countable_countable @ A @ S2 )
=> ( ~ ( finite_finite2 @ A @ S2 )
=> ~ ! [E2: A > nat] :
~ ( bij_betw @ A @ nat @ E2 @ S2 @ ( top_top @ ( set @ nat ) ) ) ) ) ).
% countableE_infinite
thf(fact_7777_sorted__butlast,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] :
( ( Xs
!= ( nil @ A ) )
=> ( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( butlast @ A @ Xs ) ) ) ) ) ).
% sorted_butlast
thf(fact_7778_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_7779_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_7780_ccSup__inter__less__eq,axiom,
! [A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A5: set @ A,B6: set @ A] :
( ( countable_countable @ A @ A5 )
=> ( ( countable_countable @ A @ B6 )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( inf_inf @ ( set @ A ) @ A5 @ B6 ) ) @ ( inf_inf @ A @ ( complete_Sup_Sup @ A @ A5 ) @ ( complete_Sup_Sup @ A @ B6 ) ) ) ) ) ) ).
% ccSup_inter_less_eq
thf(fact_7781_less__eq__ccInf__inter,axiom,
! [A: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A5: set @ A,B6: set @ A] :
( ( countable_countable @ A @ A5 )
=> ( ( countable_countable @ A @ B6 )
=> ( ord_less_eq @ A @ ( sup_sup @ A @ ( complete_Inf_Inf @ A @ A5 ) @ ( complete_Inf_Inf @ A @ B6 ) ) @ ( complete_Inf_Inf @ A @ ( inf_inf @ ( set @ A ) @ A5 @ B6 ) ) ) ) ) ) ).
% less_eq_ccInf_inter
thf(fact_7782_ccSUP__subset__mono,axiom,
! [A: $tType,B: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [B6: set @ B,A5: set @ B,F3: B > A,G3: B > A] :
( ( countable_countable @ B @ B6 )
=> ( ( ord_less_eq @ ( set @ B ) @ A5 @ B6 )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ A5 )
=> ( ord_less_eq @ A @ ( F3 @ X4 ) @ ( G3 @ X4 ) ) )
=> ( ord_less_eq @ A @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ F3 @ A5 ) ) @ ( complete_Sup_Sup @ A @ ( image2 @ B @ A @ G3 @ B6 ) ) ) ) ) ) ) ).
% ccSUP_subset_mono
thf(fact_7783_ccINF__superset__mono,axiom,
! [A: $tType,B: $tType] :
( ( counta3822494911875563373attice @ A )
=> ! [A5: set @ B,B6: set @ B,F3: B > A,G3: B > A] :
( ( countable_countable @ B @ A5 )
=> ( ( ord_less_eq @ ( set @ B ) @ B6 @ A5 )
=> ( ! [X4: B] :
( ( member @ B @ X4 @ B6 )
=> ( ord_less_eq @ A @ ( F3 @ X4 ) @ ( G3 @ X4 ) ) )
=> ( ord_less_eq @ A @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ F3 @ A5 ) ) @ ( complete_Inf_Inf @ A @ ( image2 @ B @ A @ G3 @ B6 ) ) ) ) ) ) ) ).
% ccINF_superset_mono
thf(fact_7784_mono__ccSup,axiom,
! [B: $tType,A: $tType] :
( ( ( counta4013691401010221786attice @ A )
& ( counta3822494911875563373attice @ B ) )
=> ! [F3: A > B,A5: set @ A] :
( ( order_mono @ A @ B @ F3 )
=> ( ( countable_countable @ A @ A5 )
=> ( ord_less_eq @ B @ ( complete_Sup_Sup @ B @ ( image2 @ A @ B @ F3 @ A5 ) ) @ ( F3 @ ( complete_Sup_Sup @ A @ A5 ) ) ) ) ) ) ).
% mono_ccSup
thf(fact_7785_mono__ccSUP,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( ( counta4013691401010221786attice @ A )
& ( counta3822494911875563373attice @ B ) )
=> ! [F3: A > B,I6: set @ C,A5: C > A] :
( ( order_mono @ A @ B @ F3 )
=> ( ( countable_countable @ C @ I6 )
=> ( ord_less_eq @ B
@ ( complete_Sup_Sup @ B
@ ( image2 @ C @ B
@ ^ [X5: C] : ( F3 @ ( A5 @ X5 ) )
@ I6 ) )
@ ( F3 @ ( complete_Sup_Sup @ A @ ( image2 @ C @ A @ A5 @ I6 ) ) ) ) ) ) ) ).
% mono_ccSUP
thf(fact_7786_mono__ccINF,axiom,
! [B: $tType,A: $tType,C: $tType] :
( ( ( counta4013691401010221786attice @ A )
& ( counta3822494911875563373attice @ B ) )
=> ! [F3: A > B,I6: set @ C,A5: C > A] :
( ( order_mono @ A @ B @ F3 )
=> ( ( countable_countable @ C @ I6 )
=> ( ord_less_eq @ B @ ( F3 @ ( complete_Inf_Inf @ A @ ( image2 @ C @ A @ A5 @ I6 ) ) )
@ ( complete_Inf_Inf @ B
@ ( image2 @ C @ B
@ ^ [X5: C] : ( F3 @ ( A5 @ X5 ) )
@ I6 ) ) ) ) ) ) ).
% mono_ccINF
thf(fact_7787_mono__ccInf,axiom,
! [B: $tType,A: $tType] :
( ( ( counta4013691401010221786attice @ A )
& ( counta3822494911875563373attice @ B ) )
=> ! [F3: A > B,A5: set @ A] :
( ( order_mono @ A @ B @ F3 )
=> ( ( countable_countable @ A @ A5 )
=> ( ord_less_eq @ B @ ( F3 @ ( complete_Inf_Inf @ A @ A5 ) ) @ ( complete_Inf_Inf @ B @ ( image2 @ A @ B @ F3 @ A5 ) ) ) ) ) ) ).
% mono_ccInf
thf(fact_7788_countable__as__injective__image,axiom,
! [A: $tType,A5: set @ A] :
( ( countable_countable @ A @ A5 )
=> ( ~ ( finite_finite2 @ A @ A5 )
=> ~ ! [F2: nat > A] :
( ( A5
= ( image2 @ nat @ A @ F2 @ ( top_top @ ( set @ nat ) ) ) )
=> ~ ( inj_on @ nat @ A @ F2 @ ( top_top @ ( set @ nat ) ) ) ) ) ) ).
% countable_as_injective_image
thf(fact_7789_image__to__nat__on,axiom,
! [A: $tType,A5: set @ A] :
( ( countable_countable @ A @ A5 )
=> ( ~ ( finite_finite2 @ A @ A5 )
=> ( ( image2 @ A @ nat @ ( countable_to_nat_on @ A @ A5 ) @ A5 )
= ( top_top @ ( set @ nat ) ) ) ) ) ).
% image_to_nat_on
thf(fact_7790_to__nat__on__infinite,axiom,
! [A: $tType,S2: set @ A] :
( ( countable_countable @ A @ S2 )
=> ( ~ ( finite_finite2 @ A @ S2 )
=> ( bij_betw @ A @ nat @ ( countable_to_nat_on @ A @ S2 ) @ S2 @ ( top_top @ ( set @ nat ) ) ) ) ) ).
% to_nat_on_infinite
thf(fact_7791_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_7792_butlast__list__update,axiom,
! [A: $tType,K2: nat,Xs: list @ A,X: A] :
( ( ( K2
= ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) ) )
=> ( ( butlast @ A @ ( list_update @ A @ Xs @ K2 @ X ) )
= ( butlast @ A @ Xs ) ) )
& ( ( K2
!= ( minus_minus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( one_one @ nat ) ) )
=> ( ( butlast @ A @ ( list_update @ A @ Xs @ K2 @ X ) )
= ( list_update @ A @ ( butlast @ A @ Xs ) @ K2 @ X ) ) ) ) ).
% butlast_list_update
thf(fact_7793_countable__enum__cases,axiom,
! [A: $tType,S2: set @ A] :
( ( countable_countable @ A @ S2 )
=> ( ( ( finite_finite2 @ A @ S2 )
=> ! [F2: A > nat] :
~ ( bij_betw @ A @ nat @ F2 @ S2 @ ( set_ord_lessThan @ nat @ ( finite_card @ A @ S2 ) ) ) )
=> ~ ( ~ ( finite_finite2 @ A @ S2 )
=> ! [F2: A > nat] :
~ ( bij_betw @ A @ nat @ F2 @ S2 @ ( top_top @ ( set @ nat ) ) ) ) ) ) ).
% countable_enum_cases
thf(fact_7794_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_7795_sort__key__conv__fold,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F3: B > A,Xs: list @ B] :
( ( inj_on @ B @ A @ F3 @ ( set2 @ B @ Xs ) )
=> ( ( linorder_sort_key @ B @ A @ F3 @ Xs )
= ( fold @ B @ ( list @ B ) @ ( linorder_insort_key @ B @ A @ F3 ) @ Xs @ ( nil @ B ) ) ) ) ) ).
% sort_key_conv_fold
thf(fact_7796_range__from__nat__into,axiom,
! [A: $tType,A5: set @ A] :
( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( countable_countable @ A @ A5 )
=> ( ( image2 @ nat @ A @ ( counta4804993851260445106t_into @ A @ A5 ) @ ( top_top @ ( set @ nat ) ) )
= A5 ) ) ) ).
% range_from_nat_into
thf(fact_7797_sort__upt,axiom,
! [M2: nat,N: nat] :
( ( linorder_sort_key @ nat @ nat
@ ^ [X5: nat] : X5
@ ( upt @ M2 @ N ) )
= ( upt @ M2 @ N ) ) ).
% sort_upt
thf(fact_7798_sort__upto,axiom,
! [I2: int,J: int] :
( ( linorder_sort_key @ int @ int
@ ^ [X5: int] : X5
@ ( upto @ I2 @ J ) )
= ( upto @ I2 @ J ) ) ).
% sort_upto
thf(fact_7799_sort__key__simps_I1_J,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F3: B > A] :
( ( linorder_sort_key @ B @ A @ F3 @ ( nil @ B ) )
= ( nil @ B ) ) ) ).
% sort_key_simps(1)
thf(fact_7800_set__sort,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F3: B > A,Xs: list @ B] :
( ( set2 @ B @ ( linorder_sort_key @ B @ A @ F3 @ Xs ) )
= ( set2 @ B @ Xs ) ) ) ).
% set_sort
thf(fact_7801_length__sort,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F3: B > A,Xs: list @ B] :
( ( size_size @ ( list @ B ) @ ( linorder_sort_key @ B @ A @ F3 @ Xs ) )
= ( size_size @ ( list @ B ) @ Xs ) ) ) ).
% length_sort
thf(fact_7802_distinct__sort,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F3: B > A,Xs: list @ B] :
( ( distinct @ B @ ( linorder_sort_key @ B @ A @ F3 @ Xs ) )
= ( distinct @ B @ Xs ) ) ) ).
% distinct_sort
thf(fact_7803_from__nat__into__inject,axiom,
! [A: $tType,A5: set @ A,B6: set @ A] :
( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( countable_countable @ A @ A5 )
=> ( ( B6
!= ( bot_bot @ ( set @ A ) ) )
=> ( ( countable_countable @ A @ B6 )
=> ( ( ( counta4804993851260445106t_into @ A @ A5 )
= ( counta4804993851260445106t_into @ A @ B6 ) )
= ( A5 = B6 ) ) ) ) ) ) ).
% from_nat_into_inject
thf(fact_7804_from__nat__into__inj__infinite,axiom,
! [A: $tType,A5: set @ A,M2: nat,N: nat] :
( ( countable_countable @ A @ A5 )
=> ( ~ ( finite_finite2 @ A @ A5 )
=> ( ( ( counta4804993851260445106t_into @ A @ A5 @ M2 )
= ( counta4804993851260445106t_into @ A @ A5 @ N ) )
= ( M2 = N ) ) ) ) ).
% from_nat_into_inj_infinite
thf(fact_7805_sort__key__simps_I2_J,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F3: B > A,X: B,Xs: list @ B] :
( ( linorder_sort_key @ B @ A @ F3 @ ( cons @ B @ X @ Xs ) )
= ( linorder_insort_key @ B @ A @ F3 @ X @ ( linorder_sort_key @ B @ A @ F3 @ Xs ) ) ) ) ).
% sort_key_simps(2)
thf(fact_7806_to__nat__on__from__nat__into__infinite,axiom,
! [A: $tType,A5: set @ A,N: nat] :
( ( countable_countable @ A @ A5 )
=> ( ~ ( finite_finite2 @ A @ A5 )
=> ( ( countable_to_nat_on @ A @ A5 @ ( counta4804993851260445106t_into @ A @ A5 @ N ) )
= N ) ) ) ).
% to_nat_on_from_nat_into_infinite
thf(fact_7807_filter__sort,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [P2: B > $o,F3: B > A,Xs: list @ B] :
( ( filter2 @ B @ P2 @ ( linorder_sort_key @ B @ A @ F3 @ Xs ) )
= ( linorder_sort_key @ B @ A @ F3 @ ( filter2 @ B @ P2 @ Xs ) ) ) ) ).
% filter_sort
thf(fact_7808_sort__key__stable,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ! [F3: A > B,K2: B,Xs: list @ A] :
( ( filter2 @ A
@ ^ [Y6: A] :
( ( F3 @ Y6 )
= K2 )
@ ( linorder_sort_key @ A @ B @ F3 @ Xs ) )
= ( filter2 @ A
@ ^ [Y6: A] :
( ( F3 @ Y6 )
= K2 )
@ Xs ) ) ) ).
% sort_key_stable
thf(fact_7809_sort__key__const,axiom,
! [B: $tType,A: $tType] :
( ( linorder @ B )
=> ! [C3: B,Xs: list @ A] :
( ( linorder_sort_key @ A @ B
@ ^ [X5: A] : C3
@ Xs )
= Xs ) ) ).
% sort_key_const
thf(fact_7810_from__nat__into,axiom,
! [A: $tType,A5: set @ A,N: nat] :
( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( member @ A @ ( counta4804993851260445106t_into @ A @ A5 @ N ) @ A5 ) ) ).
% from_nat_into
thf(fact_7811_inj__on__from__nat__into,axiom,
! [A: $tType] :
( inj_on @ ( set @ A ) @ ( nat > A ) @ ( counta4804993851260445106t_into @ A )
@ ( collect @ ( set @ A )
@ ^ [A7: set @ A] :
( ( A7
!= ( bot_bot @ ( set @ A ) ) )
& ( countable_countable @ A @ A7 ) ) ) ) ).
% inj_on_from_nat_into
thf(fact_7812_sorted__sort__id,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] :
( ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ Xs )
=> ( ( linorder_sort_key @ A @ A
@ ^ [X5: A] : X5
@ Xs )
= Xs ) ) ) ).
% sorted_sort_id
thf(fact_7813_sorted__sort,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] :
( sorted_wrt @ A @ ( ord_less_eq @ A )
@ ( linorder_sort_key @ A @ A
@ ^ [X5: A] : X5
@ Xs ) ) ) ).
% sorted_sort
thf(fact_7814_sorted__sort__key,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ! [F3: B > A,Xs: list @ B] : ( sorted_wrt @ A @ ( ord_less_eq @ A ) @ ( map @ B @ A @ F3 @ ( linorder_sort_key @ B @ A @ F3 @ Xs ) ) ) ) ).
% sorted_sort_key
thf(fact_7815_sorted__list__of__set__sort__remdups,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] :
( ( linord4507533701916653071of_set @ A @ ( set2 @ A @ Xs ) )
= ( linorder_sort_key @ A @ A
@ ^ [X5: A] : X5
@ ( remdups @ A @ Xs ) ) ) ) ).
% sorted_list_of_set_sort_remdups
thf(fact_7816_range__from__nat__into__subset,axiom,
! [A: $tType,A5: set @ A] :
( ( A5
!= ( bot_bot @ ( set @ A ) ) )
=> ( ord_less_eq @ ( set @ A ) @ ( image2 @ nat @ A @ ( counta4804993851260445106t_into @ A @ A5 ) @ ( top_top @ ( set @ nat ) ) ) @ A5 ) ) ).
% range_from_nat_into_subset
thf(fact_7817_sort__conv__fold,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A] :
( ( linorder_sort_key @ A @ A
@ ^ [X5: A] : X5
@ Xs )
= ( fold @ A @ ( list @ A )
@ ( linorder_insort_key @ A @ A
@ ^ [X5: A] : X5 )
@ Xs
@ ( nil @ A ) ) ) ) ).
% sort_conv_fold
thf(fact_7818_bij__betw__from__nat__into__finite,axiom,
! [A: $tType,S2: set @ A] :
( ( finite_finite2 @ A @ S2 )
=> ( bij_betw @ nat @ A @ ( counta4804993851260445106t_into @ A @ S2 ) @ ( set_ord_lessThan @ nat @ ( finite_card @ A @ S2 ) ) @ S2 ) ) ).
% bij_betw_from_nat_into_finite
thf(fact_7819_bij__betw__from__nat__into,axiom,
! [A: $tType,S2: set @ A] :
( ( countable_countable @ A @ S2 )
=> ( ~ ( finite_finite2 @ A @ S2 )
=> ( bij_betw @ nat @ A @ ( counta4804993851260445106t_into @ A @ S2 ) @ ( top_top @ ( set @ nat ) ) @ S2 ) ) ) ).
% bij_betw_from_nat_into
thf(fact_7820_sort__key__def,axiom,
! [A: $tType,B: $tType] :
( ( linorder @ A )
=> ( ( linorder_sort_key @ B @ A )
= ( ^ [F4: B > A,Xs3: list @ B] : ( foldr @ B @ ( list @ B ) @ ( linorder_insort_key @ B @ A @ F4 ) @ Xs3 @ ( nil @ B ) ) ) ) ) ).
% sort_key_def
thf(fact_7821_Bleast__code,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,P2: A > $o] :
( ( bleast @ A @ ( set2 @ A @ Xs ) @ P2 )
= ( case_list @ A @ A @ ( abort_Bleast @ A @ ( set2 @ A @ Xs ) @ P2 )
@ ^ [X5: A,Xs3: list @ A] : X5
@ ( filter2 @ A @ P2
@ ( linorder_sort_key @ A @ A
@ ^ [X5: A] : X5
@ Xs ) ) ) ) ) ).
% Bleast_code
thf(fact_7822_inter__coset__fold,axiom,
! [A: $tType,A5: set @ A,Xs: list @ A] :
( ( inf_inf @ ( set @ A ) @ A5 @ ( coset @ A @ Xs ) )
= ( fold @ A @ ( set @ A ) @ ( remove @ A ) @ Xs @ A5 ) ) ).
% inter_coset_fold
thf(fact_7823_UNIV__coset,axiom,
! [A: $tType] :
( ( top_top @ ( set @ A ) )
= ( coset @ A @ ( nil @ A ) ) ) ).
% UNIV_coset
thf(fact_7824_subset__code_I2_J,axiom,
! [B: $tType,A5: set @ B,Ys: list @ B] :
( ( ord_less_eq @ ( set @ B ) @ A5 @ ( coset @ B @ Ys ) )
= ( ! [X5: B] :
( ( member @ B @ X5 @ ( set2 @ B @ Ys ) )
=> ~ ( member @ B @ X5 @ A5 ) ) ) ) ).
% subset_code(2)
thf(fact_7825_compl__coset,axiom,
! [A: $tType,Xs: list @ A] :
( ( uminus_uminus @ ( set @ A ) @ ( coset @ A @ Xs ) )
= ( set2 @ A @ Xs ) ) ).
% compl_coset
thf(fact_7826_coset__def,axiom,
! [A: $tType] :
( ( coset @ A )
= ( ^ [Xs3: list @ A] : ( uminus_uminus @ ( set @ A ) @ ( set2 @ A @ Xs3 ) ) ) ) ).
% coset_def
thf(fact_7827_insert__code_I2_J,axiom,
! [A: $tType,X: A,Xs: list @ A] :
( ( insert @ A @ X @ ( coset @ A @ Xs ) )
= ( coset @ A @ ( removeAll @ A @ X @ Xs ) ) ) ).
% insert_code(2)
thf(fact_7828_union__coset__filter,axiom,
! [A: $tType,Xs: list @ A,A5: set @ A] :
( ( sup_sup @ ( set @ A ) @ ( coset @ A @ Xs ) @ A5 )
= ( coset @ A
@ ( filter2 @ A
@ ^ [X5: A] :
~ ( member @ A @ X5 @ A5 )
@ Xs ) ) ) ).
% union_coset_filter
thf(fact_7829_subset__code_I3_J,axiom,
! [C: $tType] :
~ ( ord_less_eq @ ( set @ C ) @ ( coset @ C @ ( nil @ C ) ) @ ( set2 @ C @ ( nil @ C ) ) ) ).
% subset_code(3)
thf(fact_7830_minus__coset__filter,axiom,
! [A: $tType,A5: set @ A,Xs: list @ A] :
( ( minus_minus @ ( set @ A ) @ A5 @ ( coset @ A @ Xs ) )
= ( set2 @ A
@ ( filter2 @ A
@ ^ [X5: A] : ( member @ A @ X5 @ A5 )
@ Xs ) ) ) ).
% minus_coset_filter
thf(fact_7831_comp__fun__idem__on__def,axiom,
! [B: $tType,A: $tType] :
( ( finite673082921795544331dem_on @ A @ B )
= ( ^ [S7: set @ A,F4: A > B > B] :
( ( finite4664212375090638736ute_on @ A @ B @ S7 @ F4 )
& ( finite4980608107308702382axioms @ A @ B @ S7 @ F4 ) ) ) ) ).
% comp_fun_idem_on_def
thf(fact_7832_comp__fun__idem__on_Ointro,axiom,
! [B: $tType,A: $tType,S2: set @ A,F3: A > B > B] :
( ( finite4664212375090638736ute_on @ A @ B @ S2 @ F3 )
=> ( ( finite4980608107308702382axioms @ A @ B @ S2 @ F3 )
=> ( finite673082921795544331dem_on @ A @ B @ S2 @ F3 ) ) ) ).
% comp_fun_idem_on.intro
thf(fact_7833_comp__fun__idem__on__axioms_Ointro,axiom,
! [B: $tType,A: $tType,S2: set @ A,F3: A > B > B] :
( ! [X4: A] :
( ( member @ A @ X4 @ S2 )
=> ( ( comp @ B @ B @ B @ ( F3 @ X4 ) @ ( F3 @ X4 ) )
= ( F3 @ X4 ) ) )
=> ( finite4980608107308702382axioms @ A @ B @ S2 @ F3 ) ) ).
% comp_fun_idem_on_axioms.intro
thf(fact_7834_comp__fun__idem__on__axioms__def,axiom,
! [B: $tType,A: $tType] :
( ( finite4980608107308702382axioms @ A @ B )
= ( ^ [S7: set @ A,F4: A > B > B] :
! [X5: A] :
( ( member @ A @ X5 @ S7 )
=> ( ( comp @ B @ B @ B @ ( F4 @ X5 ) @ ( F4 @ X5 ) )
= ( F4 @ X5 ) ) ) ) ) ).
% comp_fun_idem_on_axioms_def
thf(fact_7835_comp__fun__idem__on_Oaxioms_I2_J,axiom,
! [B: $tType,A: $tType,S2: set @ A,F3: A > B > B] :
( ( finite673082921795544331dem_on @ A @ B @ S2 @ F3 )
=> ( finite4980608107308702382axioms @ A @ B @ S2 @ F3 ) ) ).
% comp_fun_idem_on.axioms(2)
thf(fact_7836_shuffles_Opelims,axiom,
! [A: $tType,X: list @ A,Xa2: list @ A,Y: set @ ( list @ A )] :
( ( ( shuffles @ A @ X @ Xa2 )
= Y )
=> ( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( shuffles_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Xa2 ) )
=> ( ( ( X
= ( nil @ A ) )
=> ( ( Y
= ( insert @ ( list @ A ) @ Xa2 @ ( bot_bot @ ( set @ ( list @ A ) ) ) ) )
=> ~ ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( shuffles_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Xa2 ) ) ) )
=> ( ( ( Xa2
= ( nil @ A ) )
=> ( ( Y
= ( insert @ ( list @ A ) @ X @ ( bot_bot @ ( set @ ( list @ A ) ) ) ) )
=> ~ ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( shuffles_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ ( nil @ A ) ) ) ) )
=> ~ ! [X4: A,Xs2: list @ A] :
( ( X
= ( cons @ A @ X4 @ Xs2 ) )
=> ! [Y3: A,Ys4: list @ A] :
( ( Xa2
= ( cons @ A @ Y3 @ Ys4 ) )
=> ( ( Y
= ( sup_sup @ ( set @ ( list @ A ) ) @ ( image2 @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X4 ) @ ( shuffles @ A @ Xs2 @ ( cons @ A @ Y3 @ Ys4 ) ) ) @ ( image2 @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ Y3 ) @ ( shuffles @ A @ ( cons @ A @ X4 @ Xs2 ) @ Ys4 ) ) ) )
=> ~ ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( shuffles_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X4 @ Xs2 ) @ ( cons @ A @ Y3 @ Ys4 ) ) ) ) ) ) ) ) ) ) ).
% shuffles.pelims
thf(fact_7837_Field__insert,axiom,
! [A: $tType,A2: A,B2: A,R2: set @ ( product_prod @ A @ A )] :
( ( field2 @ A @ ( insert @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ R2 ) )
= ( sup_sup @ ( set @ A ) @ ( insert @ A @ A2 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( field2 @ A @ R2 ) ) ) ).
% Field_insert
thf(fact_7838_Field__empty,axiom,
! [A: $tType] :
( ( field2 @ A @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Field_empty
thf(fact_7839_underS__Field3,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A2: A] :
( ( ( field2 @ A @ R2 )
!= ( bot_bot @ ( set @ A ) ) )
=> ( ord_less @ ( set @ A ) @ ( order_underS @ A @ R2 @ A2 ) @ ( field2 @ A @ R2 ) ) ) ).
% underS_Field3
thf(fact_7840_underS__empty,axiom,
! [A: $tType,A2: A,R2: set @ ( product_prod @ A @ A )] :
( ~ ( member @ A @ A2 @ ( field2 @ A @ R2 ) )
=> ( ( order_underS @ A @ R2 @ A2 )
= ( bot_bot @ ( set @ A ) ) ) ) ).
% underS_empty
thf(fact_7841_underS__Field2,axiom,
! [A: $tType,A2: A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ A @ A2 @ ( field2 @ A @ R2 ) )
=> ( ord_less @ ( set @ A ) @ ( order_underS @ A @ R2 @ A2 ) @ ( field2 @ A @ R2 ) ) ) ).
% underS_Field2
thf(fact_7842_finite__Field,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( finite_finite2 @ ( product_prod @ A @ A ) @ R2 )
=> ( finite_finite2 @ A @ ( field2 @ A @ R2 ) ) ) ).
% finite_Field
thf(fact_7843_FieldI2,axiom,
! [A: $tType,I2: A,J: A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ I2 @ J ) @ R )
=> ( member @ A @ J @ ( field2 @ A @ R ) ) ) ).
% FieldI2
thf(fact_7844_FieldI1,axiom,
! [A: $tType,I2: A,J: A,R: set @ ( product_prod @ A @ A )] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ I2 @ J ) @ R )
=> ( member @ A @ I2 @ ( field2 @ A @ R ) ) ) ).
% FieldI1
thf(fact_7845_shuffles_Opinduct,axiom,
! [A: $tType,A0: list @ A,A1: list @ A,P2: ( list @ A ) > ( list @ A ) > $o] :
( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( shuffles_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ A0 @ A1 ) )
=> ( ! [Ys4: list @ A] :
( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( shuffles_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Ys4 ) )
=> ( P2 @ ( nil @ A ) @ Ys4 ) )
=> ( ! [Xs2: list @ A] :
( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( shuffles_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs2 @ ( nil @ A ) ) )
=> ( P2 @ Xs2 @ ( nil @ A ) ) )
=> ( ! [X4: A,Xs2: list @ A,Y3: A,Ys4: list @ A] :
( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( shuffles_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X4 @ Xs2 ) @ ( cons @ A @ Y3 @ Ys4 ) ) )
=> ( ( P2 @ Xs2 @ ( cons @ A @ Y3 @ Ys4 ) )
=> ( ( P2 @ ( cons @ A @ X4 @ Xs2 ) @ Ys4 )
=> ( P2 @ ( cons @ A @ X4 @ Xs2 ) @ ( cons @ A @ Y3 @ Ys4 ) ) ) ) )
=> ( P2 @ A0 @ A1 ) ) ) ) ) ).
% shuffles.pinduct
thf(fact_7846_Field__natLeq__on,axiom,
! [N: nat] :
( ( field2 @ nat
@ ( collect @ ( product_prod @ nat @ nat )
@ ( product_case_prod @ nat @ nat @ $o
@ ^ [X5: nat,Y6: nat] :
( ( ord_less @ nat @ X5 @ N )
& ( ord_less @ nat @ Y6 @ N )
& ( ord_less_eq @ nat @ X5 @ Y6 ) ) ) ) )
= ( collect @ nat
@ ^ [X5: nat] : ( ord_less @ nat @ X5 @ N ) ) ) ).
% Field_natLeq_on
thf(fact_7847_shuffles_Opsimps_I1_J,axiom,
! [A: $tType,Ys: list @ A] :
( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( shuffles_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Ys ) )
=> ( ( shuffles @ A @ ( nil @ A ) @ Ys )
= ( insert @ ( list @ A ) @ Ys @ ( bot_bot @ ( set @ ( list @ A ) ) ) ) ) ) ).
% shuffles.psimps(1)
thf(fact_7848_shuffles_Opsimps_I2_J,axiom,
! [A: $tType,Xs: list @ A] :
( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( shuffles_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ Xs @ ( nil @ A ) ) )
=> ( ( shuffles @ A @ Xs @ ( nil @ A ) )
= ( insert @ ( list @ A ) @ Xs @ ( bot_bot @ ( set @ ( list @ A ) ) ) ) ) ) ).
% shuffles.psimps(2)
thf(fact_7849_shuffles_Opsimps_I3_J,axiom,
! [A: $tType,X: A,Xs: list @ A,Y: A,Ys: list @ A] :
( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( shuffles_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Ys ) ) )
=> ( ( shuffles @ A @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Ys ) )
= ( sup_sup @ ( set @ ( list @ A ) ) @ ( image2 @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X ) @ ( shuffles @ A @ Xs @ ( cons @ A @ Y @ Ys ) ) ) @ ( image2 @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ Y ) @ ( shuffles @ A @ ( cons @ A @ X @ Xs ) @ Ys ) ) ) ) ) ).
% shuffles.psimps(3)
thf(fact_7850_underS__incl__iff,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A2: A,B2: A] :
( ( order_679001287576687338der_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( member @ A @ A2 @ ( field2 @ A @ R2 ) )
=> ( ( member @ A @ B2 @ ( field2 @ A @ R2 ) )
=> ( ( ord_less_eq @ ( set @ A ) @ ( order_underS @ A @ R2 @ A2 ) @ ( order_underS @ A @ R2 @ B2 ) )
= ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ R2 ) ) ) ) ) ).
% underS_incl_iff
thf(fact_7851_Under__def,axiom,
! [A: $tType] :
( ( order_Under @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A ),A7: set @ A] :
( collect @ A
@ ^ [B5: A] :
( ( member @ A @ B5 @ ( field2 @ A @ R5 ) )
& ! [X5: A] :
( ( member @ A @ X5 @ A7 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B5 @ X5 ) @ R5 ) ) ) ) ) ) ).
% Under_def
thf(fact_7852_UnderS__def,axiom,
! [A: $tType] :
( ( order_UnderS @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A ),A7: set @ A] :
( collect @ A
@ ^ [B5: A] :
( ( member @ A @ B5 @ ( field2 @ A @ R5 ) )
& ! [X5: A] :
( ( member @ A @ X5 @ A7 )
=> ( ( B5 != X5 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B5 @ X5 ) @ R5 ) ) ) ) ) ) ) ).
% UnderS_def
thf(fact_7853_Above__def,axiom,
! [A: $tType] :
( ( order_Above @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A ),A7: set @ A] :
( collect @ A
@ ^ [B5: A] :
( ( member @ A @ B5 @ ( field2 @ A @ R5 ) )
& ! [X5: A] :
( ( member @ A @ X5 @ A7 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X5 @ B5 ) @ R5 ) ) ) ) ) ) ).
% Above_def
thf(fact_7854_finite__Linear__order__induct,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),X: A,P2: A > $o] :
( ( order_679001287576687338der_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( member @ A @ X @ ( field2 @ A @ R2 ) )
=> ( ( finite_finite2 @ ( product_prod @ A @ A ) @ R2 )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ ( field2 @ A @ R2 ) )
=> ( ! [Y4: A] :
( ( member @ A @ Y4 @ ( order_aboveS @ A @ R2 @ X4 ) )
=> ( P2 @ Y4 ) )
=> ( P2 @ X4 ) ) )
=> ( P2 @ X ) ) ) ) ) ).
% finite_Linear_order_induct
thf(fact_7855_aboveS__def,axiom,
! [A: $tType] :
( ( order_aboveS @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A ),A6: A] :
( collect @ A
@ ^ [B5: A] :
( ( B5 != A6 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A6 @ B5 ) @ R5 ) ) ) ) ) ).
% aboveS_def
thf(fact_7856_splice_Opinduct,axiom,
! [A: $tType,A0: list @ A,A1: list @ A,P2: ( list @ A ) > ( list @ A ) > $o] :
( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( splice_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ A0 @ A1 ) )
=> ( ! [Ys4: list @ A] :
( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( splice_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Ys4 ) )
=> ( P2 @ ( nil @ A ) @ Ys4 ) )
=> ( ! [X4: A,Xs2: list @ A,Ys4: list @ A] :
( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( splice_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X4 @ Xs2 ) @ Ys4 ) )
=> ( ( P2 @ Ys4 @ Xs2 )
=> ( P2 @ ( cons @ A @ X4 @ Xs2 ) @ Ys4 ) ) )
=> ( P2 @ A0 @ A1 ) ) ) ) ).
% splice.pinduct
thf(fact_7857_cofinal__def,axiom,
! [A: $tType] :
( ( bNF_Ca7293521722713021262ofinal @ A )
= ( ^ [A7: set @ A,R5: set @ ( product_prod @ A @ A )] :
! [X5: A] :
( ( member @ A @ X5 @ ( field2 @ A @ R5 ) )
=> ? [Y6: A] :
( ( member @ A @ Y6 @ A7 )
& ( X5 != Y6 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X5 @ Y6 ) @ R5 ) ) ) ) ) ).
% cofinal_def
thf(fact_7858_splice_Opelims,axiom,
! [A: $tType,X: list @ A,Xa2: list @ A,Y: list @ A] :
( ( ( splice @ A @ X @ Xa2 )
= Y )
=> ( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( splice_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ X @ Xa2 ) )
=> ( ( ( X
= ( nil @ A ) )
=> ( ( Y = Xa2 )
=> ~ ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( splice_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Xa2 ) ) ) )
=> ~ ! [X4: A,Xs2: list @ A] :
( ( X
= ( cons @ A @ X4 @ Xs2 ) )
=> ( ( Y
= ( cons @ A @ X4 @ ( splice @ A @ Xa2 @ Xs2 ) ) )
=> ~ ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( splice_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X4 @ Xs2 ) @ Xa2 ) ) ) ) ) ) ) ).
% splice.pelims
thf(fact_7859_Refl__under__underS,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A2: A] :
( ( refl_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( member @ A @ A2 @ ( field2 @ A @ R2 ) )
=> ( ( order_under @ A @ R2 @ A2 )
= ( sup_sup @ ( set @ A ) @ ( order_underS @ A @ R2 @ A2 ) @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% Refl_under_underS
thf(fact_7860_split__Nil__iff,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( ( splice @ A @ Xs @ Ys )
= ( nil @ A ) )
= ( ( Xs
= ( nil @ A ) )
& ( Ys
= ( nil @ A ) ) ) ) ).
% split_Nil_iff
thf(fact_7861_splice__Nil2,axiom,
! [A: $tType,Xs: list @ A] :
( ( splice @ A @ Xs @ ( nil @ A ) )
= Xs ) ).
% splice_Nil2
thf(fact_7862_splice__in__shuffles,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] : ( member @ ( list @ A ) @ ( splice @ A @ Xs @ Ys ) @ ( shuffles @ A @ Xs @ Ys ) ) ).
% splice_in_shuffles
thf(fact_7863_length__splice,axiom,
! [A: $tType,Xs: list @ A,Ys: list @ A] :
( ( size_size @ ( list @ A ) @ ( splice @ A @ Xs @ Ys ) )
= ( plus_plus @ nat @ ( size_size @ ( list @ A ) @ Xs ) @ ( size_size @ ( list @ A ) @ Ys ) ) ) ).
% length_splice
thf(fact_7864_splice__replicate,axiom,
! [A: $tType,M2: nat,X: A,N: nat] :
( ( splice @ A @ ( replicate @ A @ M2 @ X ) @ ( replicate @ A @ N @ X ) )
= ( replicate @ A @ ( plus_plus @ nat @ M2 @ N ) @ X ) ) ).
% splice_replicate
thf(fact_7865_under__def,axiom,
! [A: $tType] :
( ( order_under @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A ),A6: A] :
( collect @ A
@ ^ [B5: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B5 @ A6 ) @ R5 ) ) ) ) ).
% under_def
thf(fact_7866_splice_Oelims,axiom,
! [A: $tType,X: list @ A,Xa2: list @ A,Y: list @ A] :
( ( ( splice @ A @ X @ Xa2 )
= Y )
=> ( ( ( X
= ( nil @ A ) )
=> ( Y != Xa2 ) )
=> ~ ! [X4: A,Xs2: list @ A] :
( ( X
= ( cons @ A @ X4 @ Xs2 ) )
=> ( Y
!= ( cons @ A @ X4 @ ( splice @ A @ Xa2 @ Xs2 ) ) ) ) ) ) ).
% splice.elims
thf(fact_7867_splice_Osimps_I2_J,axiom,
! [A: $tType,X: A,Xs: list @ A,Ys: list @ A] :
( ( splice @ A @ ( cons @ A @ X @ Xs ) @ Ys )
= ( cons @ A @ X @ ( splice @ A @ Ys @ Xs ) ) ) ).
% splice.simps(2)
thf(fact_7868_splice_Osimps_I1_J,axiom,
! [A: $tType,Ys: list @ A] :
( ( splice @ A @ ( nil @ A ) @ Ys )
= Ys ) ).
% splice.simps(1)
thf(fact_7869_splice_Opsimps_I2_J,axiom,
! [A: $tType,X: A,Xs: list @ A,Ys: list @ A] :
( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( splice_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( cons @ A @ X @ Xs ) @ Ys ) )
=> ( ( splice @ A @ ( cons @ A @ X @ Xs ) @ Ys )
= ( cons @ A @ X @ ( splice @ A @ Ys @ Xs ) ) ) ) ).
% splice.psimps(2)
thf(fact_7870_splice_Opsimps_I1_J,axiom,
! [A: $tType,Ys: list @ A] :
( ( accp @ ( product_prod @ ( list @ A ) @ ( list @ A ) ) @ ( splice_rel @ A ) @ ( product_Pair @ ( list @ A ) @ ( list @ A ) @ ( nil @ A ) @ Ys ) )
=> ( ( splice @ A @ ( nil @ A ) @ Ys )
= Ys ) ) ).
% splice.psimps(1)
thf(fact_7871_Total__subset__Id,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( total_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ ( id2 @ A ) )
=> ( ( R2
= ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
| ? [A4: A] :
( R2
= ( insert @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A4 @ A4 ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ) ) ) ).
% Total_subset_Id
thf(fact_7872_comp__fun__commute__on_Ofold__graph__insertE__aux,axiom,
! [A: $tType,B: $tType,S2: set @ A,F3: A > B > B,A5: set @ A,Z: B,Y: B,A2: A] :
( ( finite4664212375090638736ute_on @ A @ B @ S2 @ F3 )
=> ( ( ord_less_eq @ ( set @ A ) @ A5 @ S2 )
=> ( ( finite_fold_graph @ A @ B @ F3 @ Z @ A5 @ Y )
=> ( ( member @ A @ A2 @ A5 )
=> ? [Y16: B] :
( ( Y
= ( F3 @ A2 @ Y16 ) )
& ( finite_fold_graph @ A @ B @ F3 @ Z @ ( minus_minus @ ( set @ A ) @ A5 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) @ Y16 ) ) ) ) ) ) ).
% comp_fun_commute_on.fold_graph_insertE_aux
thf(fact_7873_IdI,axiom,
! [A: $tType,A2: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ A2 ) @ ( id2 @ A ) ) ).
% IdI
thf(fact_7874_pair__in__Id__conv,axiom,
! [A: $tType,A2: A,B2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ ( id2 @ A ) )
= ( A2 = B2 ) ) ).
% pair_in_Id_conv
thf(fact_7875_rtrancl__empty,axiom,
! [A: $tType] :
( ( transitive_rtrancl @ A @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) )
= ( id2 @ A ) ) ).
% rtrancl_empty
thf(fact_7876_pair__leq__def,axiom,
( fun_pair_leq
= ( sup_sup @ ( set @ ( product_prod @ ( product_prod @ nat @ nat ) @ ( product_prod @ nat @ nat ) ) ) @ fun_pair_less @ ( id2 @ ( product_prod @ nat @ nat ) ) ) ) ).
% pair_leq_def
thf(fact_7877_relpow_Osimps_I1_J,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ( compow @ ( set @ ( product_prod @ A @ A ) ) @ ( zero_zero @ nat ) @ R )
= ( id2 @ A ) ) ).
% relpow.simps(1)
thf(fact_7878_Id__def,axiom,
! [A: $tType] :
( ( id2 @ A )
= ( collect @ ( product_prod @ A @ A )
@ ^ [P6: product_prod @ A @ A] :
? [X5: A] :
( P6
= ( product_Pair @ A @ A @ X5 @ X5 ) ) ) ) ).
% Id_def
thf(fact_7879_IdD,axiom,
! [A: $tType,A2: A,B2: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ ( id2 @ A ) )
=> ( A2 = B2 ) ) ).
% IdD
thf(fact_7880_IdE,axiom,
! [A: $tType,P: product_prod @ A @ A] :
( ( member @ ( product_prod @ A @ A ) @ P @ ( id2 @ A ) )
=> ~ ! [X4: A] :
( P
!= ( product_Pair @ A @ A @ X4 @ X4 ) ) ) ).
% IdE
thf(fact_7881_fold__graph_OemptyI,axiom,
! [A: $tType,B: $tType,F3: A > B > B,Z: B] : ( finite_fold_graph @ A @ B @ F3 @ Z @ ( bot_bot @ ( set @ A ) ) @ Z ) ).
% fold_graph.emptyI
thf(fact_7882_empty__fold__graphE,axiom,
! [A: $tType,B: $tType,F3: A > B > B,Z: B,X: B] :
( ( finite_fold_graph @ A @ B @ F3 @ Z @ ( bot_bot @ ( set @ A ) ) @ X )
=> ( X = Z ) ) ).
% empty_fold_graphE
thf(fact_7883_comp__fun__commute__on_Ofold__graph__finite,axiom,
! [B: $tType,A: $tType,S2: set @ A,F3: A > B > B,Z: B,A5: set @ A,Y: B] :
( ( finite4664212375090638736ute_on @ A @ B @ S2 @ F3 )
=> ( ( finite_fold_graph @ A @ B @ F3 @ Z @ A5 @ Y )
=> ( finite_finite2 @ A @ A5 ) ) ) ).
% comp_fun_commute_on.fold_graph_finite
thf(fact_7884_comp__fun__commute__on_Ofold__graph__determ,axiom,
! [A: $tType,B: $tType,S2: set @ A,F3: A > B > B,A5: set @ A,Z: B,X: B,Y: B] :
( ( finite4664212375090638736ute_on @ A @ B @ S2 @ F3 )
=> ( ( ord_less_eq @ ( set @ A ) @ A5 @ S2 )
=> ( ( finite_fold_graph @ A @ B @ F3 @ Z @ A5 @ X )
=> ( ( finite_fold_graph @ A @ B @ F3 @ Z @ A5 @ Y )
=> ( Y = X ) ) ) ) ) ).
% comp_fun_commute_on.fold_graph_determ
thf(fact_7885_finite__imp__fold__graph,axiom,
! [A: $tType,B: $tType,A5: set @ A,F3: A > B > B,Z: B] :
( ( finite_finite2 @ A @ A5 )
=> ? [X_12: B] : ( finite_fold_graph @ A @ B @ F3 @ Z @ A5 @ X_12 ) ) ).
% finite_imp_fold_graph
thf(fact_7886_fold__graph__closed__lemma,axiom,
! [A: $tType,B: $tType,G3: A > B > B,Z: B,A5: set @ A,X: B,B6: set @ B,F3: A > B > B] :
( ( finite_fold_graph @ A @ B @ G3 @ Z @ A5 @ X )
=> ( ! [A4: A,B4: B] :
( ( member @ A @ A4 @ A5 )
=> ( ( member @ B @ B4 @ B6 )
=> ( ( F3 @ A4 @ B4 )
= ( G3 @ A4 @ B4 ) ) ) )
=> ( ! [A4: A,B4: B] :
( ( member @ A @ A4 @ A5 )
=> ( ( member @ B @ B4 @ B6 )
=> ( member @ B @ ( G3 @ A4 @ B4 ) @ B6 ) ) )
=> ( ( member @ B @ Z @ B6 )
=> ( ( finite_fold_graph @ A @ B @ F3 @ Z @ A5 @ X )
& ( member @ B @ X @ B6 ) ) ) ) ) ) ).
% fold_graph_closed_lemma
thf(fact_7887_fold__graph__closed__eq,axiom,
! [B: $tType,A: $tType,A5: set @ A,B6: set @ B,F3: A > B > B,G3: A > B > B,Z: B] :
( ! [A4: A,B4: B] :
( ( member @ A @ A4 @ A5 )
=> ( ( member @ B @ B4 @ B6 )
=> ( ( F3 @ A4 @ B4 )
= ( G3 @ A4 @ B4 ) ) ) )
=> ( ! [A4: A,B4: B] :
( ( member @ A @ A4 @ A5 )
=> ( ( member @ B @ B4 @ B6 )
=> ( member @ B @ ( G3 @ A4 @ B4 ) @ B6 ) ) )
=> ( ( member @ B @ Z @ B6 )
=> ( ( finite_fold_graph @ A @ B @ F3 @ Z @ A5 )
= ( finite_fold_graph @ A @ B @ G3 @ Z @ A5 ) ) ) ) ) ).
% fold_graph_closed_eq
thf(fact_7888_fold__graph_Ocases,axiom,
! [A: $tType,B: $tType,F3: A > B > B,Z: B,A1: set @ A,A22: B] :
( ( finite_fold_graph @ A @ B @ F3 @ Z @ A1 @ A22 )
=> ( ( ( A1
= ( bot_bot @ ( set @ A ) ) )
=> ( A22 != Z ) )
=> ~ ! [X4: A,A8: set @ A] :
( ( A1
= ( insert @ A @ X4 @ A8 ) )
=> ! [Y3: B] :
( ( A22
= ( F3 @ X4 @ Y3 ) )
=> ( ~ ( member @ A @ X4 @ A8 )
=> ~ ( finite_fold_graph @ A @ B @ F3 @ Z @ A8 @ Y3 ) ) ) ) ) ) ).
% fold_graph.cases
thf(fact_7889_fold__graph_Osimps,axiom,
! [B: $tType,A: $tType] :
( ( finite_fold_graph @ A @ B )
= ( ^ [F4: A > B > B,Z5: B,A12: set @ A,A23: B] :
( ( ( A12
= ( bot_bot @ ( set @ A ) ) )
& ( A23 = Z5 ) )
| ? [X5: A,A7: set @ A,Y6: B] :
( ( A12
= ( insert @ A @ X5 @ A7 ) )
& ( A23
= ( F4 @ X5 @ Y6 ) )
& ~ ( member @ A @ X5 @ A7 )
& ( finite_fold_graph @ A @ B @ F4 @ Z5 @ A7 @ Y6 ) ) ) ) ) ).
% fold_graph.simps
thf(fact_7890_fold__graph_OinsertI,axiom,
! [A: $tType,B: $tType,X: A,A5: set @ A,F3: A > B > B,Z: B,Y: B] :
( ~ ( member @ A @ X @ A5 )
=> ( ( finite_fold_graph @ A @ B @ F3 @ Z @ A5 @ Y )
=> ( finite_fold_graph @ A @ B @ F3 @ Z @ ( insert @ A @ X @ A5 ) @ ( F3 @ X @ Y ) ) ) ) ).
% fold_graph.insertI
thf(fact_7891_fold__graph__image,axiom,
! [C: $tType,B: $tType,A: $tType,G3: A > B,A5: set @ A,F3: B > C > C,Z: C] :
( ( inj_on @ A @ B @ G3 @ A5 )
=> ( ( finite_fold_graph @ B @ C @ F3 @ Z @ ( image2 @ A @ B @ G3 @ A5 ) )
= ( finite_fold_graph @ A @ C @ ( comp @ B @ ( C > C ) @ A @ F3 @ G3 ) @ Z @ A5 ) ) ) ).
% fold_graph_image
thf(fact_7892_comp__fun__commute__on_Ofold__graph__insertE,axiom,
! [A: $tType,B: $tType,S2: set @ A,F3: A > B > B,X: A,A5: set @ A,Z: B,V: B] :
( ( finite4664212375090638736ute_on @ A @ B @ S2 @ F3 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X @ A5 ) @ S2 )
=> ( ( finite_fold_graph @ A @ B @ F3 @ Z @ ( insert @ A @ X @ A5 ) @ V )
=> ( ~ ( member @ A @ X @ A5 )
=> ~ ! [Y3: B] :
( ( V
= ( F3 @ X @ Y3 ) )
=> ~ ( finite_fold_graph @ A @ B @ F3 @ Z @ A5 @ Y3 ) ) ) ) ) ) ).
% comp_fun_commute_on.fold_graph_insertE
thf(fact_7893_comp__fun__commute__on_Ofold__equality,axiom,
! [A: $tType,B: $tType,S2: set @ A,F3: A > B > B,A5: set @ A,Z: B,Y: B] :
( ( finite4664212375090638736ute_on @ A @ B @ S2 @ F3 )
=> ( ( ord_less_eq @ ( set @ A ) @ A5 @ S2 )
=> ( ( finite_fold_graph @ A @ B @ F3 @ Z @ A5 @ Y )
=> ( ( finite_fold @ A @ B @ F3 @ Z @ A5 )
= Y ) ) ) ) ).
% comp_fun_commute_on.fold_equality
thf(fact_7894_reflcl__set__eq,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( sup_sup @ ( A > A > $o )
@ ^ [X5: A,Y6: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X5 @ Y6 ) @ R2 )
@ ^ [Y5: A,Z2: A] : Y5 = Z2 )
= ( ^ [X5: A,Y6: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X5 @ Y6 ) @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ ( id2 @ A ) ) ) ) ) ).
% reflcl_set_eq
thf(fact_7895_Finite__Set_Ofold__def,axiom,
! [B: $tType,A: $tType] :
( ( finite_fold @ A @ B )
= ( ^ [F4: A > B > B,Z5: B,A7: set @ A] : ( if @ B @ ( finite_finite2 @ A @ A7 ) @ ( the @ B @ ( finite_fold_graph @ A @ B @ F4 @ Z5 @ A7 ) ) @ Z5 ) ) ) ).
% Finite_Set.fold_def
thf(fact_7896_Linear__order__in__diff__Id,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A2: A,B2: A] :
( ( order_679001287576687338der_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( member @ A @ A2 @ ( field2 @ A @ R2 ) )
=> ( ( member @ A @ B2 @ ( field2 @ A @ R2 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ R2 )
= ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ A2 ) @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ ( id2 @ A ) ) ) ) ) ) ) ) ).
% Linear_order_in_diff_Id
thf(fact_7897_comp__fun__commute__on_Ofold__graph__fold,axiom,
! [B: $tType,A: $tType,S2: set @ A,F3: A > B > B,A5: set @ A,Z: B] :
( ( finite4664212375090638736ute_on @ A @ B @ S2 @ F3 )
=> ( ( ord_less_eq @ ( set @ A ) @ A5 @ S2 )
=> ( ( finite_finite2 @ A @ A5 )
=> ( finite_fold_graph @ A @ B @ F3 @ Z @ A5 @ ( finite_fold @ A @ B @ F3 @ Z @ A5 ) ) ) ) ) ).
% comp_fun_commute_on.fold_graph_fold
thf(fact_7898_bsqr__def,axiom,
! [A: $tType] :
( ( bNF_Wellorder_bsqr @ A )
= ( ^ [R5: 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
@ ^ [B14: A,B23: A] :
( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ A12 @ ( insert @ A @ A23 @ ( insert @ A @ B14 @ ( insert @ A @ B23 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) @ ( field2 @ A @ R5 ) )
& ( ( ( A12 = B14 )
& ( A23 = B23 ) )
| ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( bNF_We1388413361240627857o_max2 @ A @ R5 @ A12 @ A23 ) @ ( bNF_We1388413361240627857o_max2 @ A @ R5 @ B14 @ B23 ) ) @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R5 @ ( id2 @ A ) ) )
| ( ( ( bNF_We1388413361240627857o_max2 @ A @ R5 @ A12 @ A23 )
= ( bNF_We1388413361240627857o_max2 @ A @ R5 @ B14 @ B23 ) )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A12 @ B14 ) @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R5 @ ( id2 @ A ) ) ) )
| ( ( ( bNF_We1388413361240627857o_max2 @ A @ R5 @ A12 @ A23 )
= ( bNF_We1388413361240627857o_max2 @ A @ R5 @ B14 @ B23 ) )
& ( A12 = B14 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A23 @ B23 ) @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R5 @ ( id2 @ A ) ) ) ) ) ) ) ) ) ) ) ) ).
% bsqr_def
thf(fact_7899_Linear__order__wf__diff__Id,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( order_679001287576687338der_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( wf @ A @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ ( id2 @ A ) ) )
= ( ! [A7: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A7 @ ( field2 @ A @ R2 ) )
=> ( ( A7
!= ( bot_bot @ ( set @ A ) ) )
=> ? [X5: A] :
( ( member @ A @ X5 @ A7 )
& ! [Y6: A] :
( ( member @ A @ Y6 @ A7 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X5 @ Y6 ) @ R2 ) ) ) ) ) ) ) ) ).
% Linear_order_wf_diff_Id
thf(fact_7900_wf__empty,axiom,
! [A: $tType] : ( wf @ A @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ).
% wf_empty
thf(fact_7901_wf__listrel1__iff,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( wf @ ( list @ A ) @ ( listrel1 @ A @ R2 ) )
= ( wf @ A @ R2 ) ) ).
% wf_listrel1_iff
thf(fact_7902_wf__lex,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( wf @ A @ R2 )
=> ( wf @ ( list @ A ) @ ( lex @ A @ R2 ) ) ) ).
% wf_lex
thf(fact_7903_wf__lenlex,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( wf @ A @ R2 )
=> ( wf @ ( list @ A ) @ ( lenlex @ A @ R2 ) ) ) ).
% wf_lenlex
thf(fact_7904_wf__insert,axiom,
! [A: $tType,Y: A,X: A,R2: set @ ( product_prod @ A @ A )] :
( ( wf @ A @ ( insert @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ X ) @ R2 ) )
= ( ( wf @ A @ R2 )
& ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ ( transitive_rtrancl @ A @ R2 ) ) ) ) ).
% wf_insert
thf(fact_7905_reduction__pairI,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),S2: set @ ( product_prod @ A @ A )] :
( ( wf @ A @ R )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( relcomp @ A @ A @ A @ R @ S2 ) @ R )
=> ( fun_reduction_pair @ A @ ( product_Pair @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ R @ S2 ) ) ) ) ).
% reduction_pairI
thf(fact_7906_wfI,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A5: set @ A,B6: set @ A] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R2
@ ( product_Sigma @ A @ A @ A5
@ ^ [Uu3: A] : B6 ) )
=> ( ! [X4: A,P8: A > $o] :
( ! [Xa: A] :
( ! [Y3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Xa ) @ R2 )
=> ( P8 @ Y3 ) )
=> ( P8 @ Xa ) )
=> ( ( member @ A @ X4 @ A5 )
=> ( ( member @ A @ X4 @ B6 )
=> ( P8 @ X4 ) ) ) )
=> ( wf @ A @ R2 ) ) ) ).
% wfI
thf(fact_7907_wf__if__measure,axiom,
! [A: $tType,P2: A > $o,F3: A > nat,G3: A > A] :
( ! [X4: A] :
( ( P2 @ X4 )
=> ( ord_less @ nat @ ( F3 @ ( G3 @ X4 ) ) @ ( F3 @ X4 ) ) )
=> ( wf @ A
@ ( collect @ ( product_prod @ A @ A )
@ ( product_case_prod @ A @ A @ $o
@ ^ [Y6: A,X5: A] :
( ( P2 @ X5 )
& ( Y6
= ( G3 @ X5 ) ) ) ) ) ) ) ).
% wf_if_measure
thf(fact_7908_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_7909_wf__less,axiom,
wf @ nat @ ( collect @ ( product_prod @ nat @ nat ) @ ( product_case_prod @ nat @ nat @ $o @ ( ord_less @ nat ) ) ) ).
% wf_less
thf(fact_7910_wf__lexn,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),N: nat] :
( ( wf @ A @ R2 )
=> ( wf @ ( list @ A ) @ ( lexn @ A @ R2 @ N ) ) ) ).
% wf_lexn
thf(fact_7911_wf__pair__less,axiom,
wf @ ( product_prod @ nat @ nat ) @ fun_pair_less ).
% wf_pair_less
thf(fact_7912_wf__int__ge__less__than2,axiom,
! [D3: int] : ( wf @ int @ ( int_ge_less_than2 @ D3 ) ) ).
% wf_int_ge_less_than2
thf(fact_7913_wf__int__ge__less__than,axiom,
! [D3: int] : ( wf @ int @ ( int_ge_less_than @ D3 ) ) ).
% wf_int_ge_less_than
thf(fact_7914_wf__no__loop,axiom,
! [B: $tType,R: set @ ( product_prod @ B @ B )] :
( ( ( relcomp @ B @ B @ B @ R @ R )
= ( bot_bot @ ( set @ ( product_prod @ B @ B ) ) ) )
=> ( wf @ B @ R ) ) ).
% wf_no_loop
thf(fact_7915_wfE__min_H,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),Q: set @ A] :
( ( wf @ A @ R )
=> ( ( Q
!= ( bot_bot @ ( set @ A ) ) )
=> ~ ! [Z3: A] :
( ( member @ A @ Z3 @ Q )
=> ~ ! [Y4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Z3 ) @ R )
=> ~ ( member @ A @ Y4 @ Q ) ) ) ) ) ).
% wfE_min'
thf(fact_7916_wf__iff__no__infinite__down__chain,axiom,
! [A: $tType] :
( ( wf @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A )] :
~ ? [F4: nat > A] :
! [I4: nat] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( F4 @ ( suc @ I4 ) ) @ ( F4 @ I4 ) ) @ R5 ) ) ) ).
% wf_iff_no_infinite_down_chain
thf(fact_7917_wf__no__infinite__down__chainE,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),F3: nat > A] :
( ( wf @ A @ R2 )
=> ~ ! [K: nat] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( F3 @ ( suc @ K ) ) @ ( F3 @ K ) ) @ R2 ) ) ).
% wf_no_infinite_down_chainE
thf(fact_7918_wf__induct__rule,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),P2: A > $o,A2: A] :
( ( wf @ A @ R2 )
=> ( ! [X4: A] :
( ! [Y4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ X4 ) @ R2 )
=> ( P2 @ Y4 ) )
=> ( P2 @ X4 ) )
=> ( P2 @ A2 ) ) ) ).
% wf_induct_rule
thf(fact_7919_wf__eq__minimal,axiom,
! [A: $tType] :
( ( wf @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A )] :
! [Q7: set @ A] :
( ? [X5: A] : ( member @ A @ X5 @ Q7 )
=> ? [X5: A] :
( ( member @ A @ X5 @ Q7 )
& ! [Y6: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y6 @ X5 ) @ R5 )
=> ~ ( member @ A @ Y6 @ Q7 ) ) ) ) ) ) ).
% wf_eq_minimal
thf(fact_7920_wf__not__refl,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A2: A] :
( ( wf @ A @ R2 )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ A2 ) @ R2 ) ) ).
% wf_not_refl
thf(fact_7921_wf__not__sym,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A2: A,X: A] :
( ( wf @ A @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ X ) @ R2 )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ A2 ) @ R2 ) ) ) ).
% wf_not_sym
thf(fact_7922_wf__irrefl,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A2: A] :
( ( wf @ A @ R2 )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ A2 ) @ R2 ) ) ).
% wf_irrefl
thf(fact_7923_wf__induct,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),P2: A > $o,A2: A] :
( ( wf @ A @ R2 )
=> ( ! [X4: A] :
( ! [Y4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ X4 ) @ R2 )
=> ( P2 @ Y4 ) )
=> ( P2 @ X4 ) )
=> ( P2 @ A2 ) ) ) ).
% wf_induct
thf(fact_7924_wf__asym,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A2: A,X: A] :
( ( wf @ A @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ X ) @ R2 )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ A2 ) @ R2 ) ) ) ).
% wf_asym
thf(fact_7925_wfUNIVI,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ! [P8: A > $o,X4: A] :
( ! [Xa: A] :
( ! [Y3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Xa ) @ R2 )
=> ( P8 @ Y3 ) )
=> ( P8 @ Xa ) )
=> ( P8 @ X4 ) )
=> ( wf @ A @ R2 ) ) ).
% wfUNIVI
thf(fact_7926_wfI__min,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ! [X4: A,Q8: set @ A] :
( ( member @ A @ X4 @ Q8 )
=> ? [Xa: A] :
( ( member @ A @ Xa @ Q8 )
& ! [Y3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Xa ) @ R )
=> ~ ( member @ A @ Y3 @ Q8 ) ) ) )
=> ( wf @ A @ R ) ) ).
% wfI_min
thf(fact_7927_wfE__min,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),X: A,Q: set @ A] :
( ( wf @ A @ R )
=> ( ( member @ A @ X @ Q )
=> ~ ! [Z3: A] :
( ( member @ A @ Z3 @ Q )
=> ~ ! [Y4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ Z3 ) @ R )
=> ~ ( member @ A @ Y4 @ Q ) ) ) ) ) ).
% wfE_min
thf(fact_7928_wf__def,axiom,
! [A: $tType] :
( ( wf @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A )] :
! [P4: A > $o] :
( ! [X5: A] :
( ! [Y6: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y6 @ X5 ) @ R5 )
=> ( P4 @ Y6 ) )
=> ( P4 @ X5 ) )
=> ! [X9: A] : ( P4 @ X9 ) ) ) ) ).
% wf_def
thf(fact_7929_wf__bounded__measure,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),Ub: A > nat,F3: A > nat] :
( ! [A4: A,B4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B4 @ A4 ) @ R2 )
=> ( ( ord_less_eq @ nat @ ( Ub @ B4 ) @ ( Ub @ A4 ) )
& ( ord_less_eq @ nat @ ( F3 @ B4 ) @ ( Ub @ A4 ) )
& ( ord_less @ nat @ ( F3 @ A4 ) @ ( F3 @ B4 ) ) ) )
=> ( wf @ A @ R2 ) ) ).
% wf_bounded_measure
thf(fact_7930_wf__linord__ex__has__least,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ A ),P2: B > $o,K2: B,M2: B > A] :
( ( wf @ A @ R2 )
=> ( ! [X4: A,Y3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y3 ) @ ( transitive_trancl @ A @ R2 ) )
= ( ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ X4 ) @ ( transitive_rtrancl @ A @ R2 ) ) ) )
=> ( ( P2 @ K2 )
=> ? [X4: B] :
( ( P2 @ X4 )
& ! [Y4: B] :
( ( P2 @ Y4 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ ( M2 @ X4 ) @ ( M2 @ Y4 ) ) @ ( transitive_rtrancl @ A @ R2 ) ) ) ) ) ) ) ).
% wf_linord_ex_has_least
thf(fact_7931_reduction__pair__def,axiom,
! [A: $tType] :
( ( fun_reduction_pair @ A )
= ( ^ [P4: product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) )] :
( ( wf @ A @ ( product_fst @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ P4 ) )
& ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ ( relcomp @ A @ A @ A @ ( product_fst @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ P4 ) @ ( product_snd @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ P4 ) ) @ ( product_fst @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ P4 ) ) ) ) ) ).
% reduction_pair_def
thf(fact_7932_reduction__pair__lemma,axiom,
! [A: $tType,P2: product_prod @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ),R: set @ ( product_prod @ A @ A ),S2: set @ ( product_prod @ A @ A )] :
( ( fun_reduction_pair @ A @ P2 )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R @ ( product_fst @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ P2 ) )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ S2 @ ( product_snd @ ( set @ ( product_prod @ A @ A ) ) @ ( set @ ( product_prod @ A @ A ) ) @ P2 ) )
=> ( ( wf @ A @ S2 )
=> ( wf @ A @ ( sup_sup @ ( set @ ( product_prod @ A @ A ) ) @ R @ S2 ) ) ) ) ) ) ).
% reduction_pair_lemma
thf(fact_7933_wf__eq__minimal2,axiom,
! [A: $tType] :
( ( wf @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A )] :
! [A7: set @ A] :
( ( ( ord_less_eq @ ( set @ A ) @ A7 @ ( field2 @ A @ R5 ) )
& ( A7
!= ( bot_bot @ ( set @ A ) ) ) )
=> ? [X5: A] :
( ( member @ A @ X5 @ A7 )
& ! [Y6: A] :
( ( member @ A @ Y6 @ A7 )
=> ~ ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y6 @ X5 ) @ R5 ) ) ) ) ) ) ).
% wf_eq_minimal2
thf(fact_7934_wf__bounded__set,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ A ),Ub: A > ( set @ B ),F3: A > ( set @ B )] :
( ! [A4: A,B4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B4 @ A4 ) @ R2 )
=> ( ( finite_finite2 @ B @ ( Ub @ A4 ) )
& ( ord_less_eq @ ( set @ B ) @ ( Ub @ B4 ) @ ( Ub @ A4 ) )
& ( ord_less_eq @ ( set @ B ) @ ( F3 @ B4 ) @ ( Ub @ A4 ) )
& ( ord_less @ ( set @ B ) @ ( F3 @ A4 ) @ ( F3 @ B4 ) ) ) )
=> ( wf @ A @ R2 ) ) ).
% wf_bounded_set
thf(fact_7935_finite__subset__wf,axiom,
! [A: $tType,A5: set @ A] :
( ( finite_finite2 @ A @ A5 )
=> ( wf @ ( set @ A )
@ ( collect @ ( product_prod @ ( set @ A ) @ ( set @ A ) )
@ ( product_case_prod @ ( set @ A ) @ ( set @ A ) @ $o
@ ^ [X9: set @ A,Y10: set @ A] :
( ( ord_less @ ( set @ A ) @ X9 @ Y10 )
& ( ord_less_eq @ ( set @ A ) @ Y10 @ A5 ) ) ) ) ) ) ).
% finite_subset_wf
thf(fact_7936_dependent__wf__choice,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ A ),P2: ( A > B ) > A > B > $o] :
( ( wf @ A @ R )
=> ( ! [F2: A > B,G2: A > B,X4: A,R3: B] :
( ! [Z4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Z4 @ X4 ) @ R )
=> ( ( F2 @ Z4 )
= ( G2 @ Z4 ) ) )
=> ( ( P2 @ F2 @ X4 @ R3 )
= ( P2 @ G2 @ X4 @ R3 ) ) )
=> ( ! [X4: A,F2: A > B] :
( ! [Y4: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y4 @ X4 ) @ R )
=> ( P2 @ F2 @ Y4 @ ( F2 @ Y4 ) ) )
=> ? [X_1: B] : ( P2 @ F2 @ X4 @ X_1 ) )
=> ? [F2: A > B] :
! [X3: A] : ( P2 @ F2 @ X3 @ ( F2 @ X3 ) ) ) ) ) ).
% dependent_wf_choice
thf(fact_7937_partial__order__on__well__order__on,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A5: set @ A] :
( ( finite_finite2 @ ( product_prod @ A @ A ) @ R2 )
=> ( ( order_7125193373082350890der_on @ A @ A5 @ R2 )
=> ( wf @ A @ ( minus_minus @ ( set @ ( product_prod @ A @ A ) ) @ R2 @ ( id2 @ A ) ) ) ) ) ).
% partial_order_on_well_order_on
thf(fact_7938_partial__order__on__empty,axiom,
! [A: $tType] : ( order_7125193373082350890der_on @ A @ ( bot_bot @ ( set @ A ) ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ).
% partial_order_on_empty
thf(fact_7939_dependent__wellorder__choice,axiom,
! [B: $tType,A: $tType] :
( ( wellorder @ A )
=> ! [P2: ( A > B ) > A > B > $o] :
( ! [R3: B,F2: A > B,G2: A > B,X4: A] :
( ! [Y4: A] :
( ( ord_less @ A @ Y4 @ X4 )
=> ( ( F2 @ Y4 )
= ( G2 @ Y4 ) ) )
=> ( ( P2 @ F2 @ X4 @ R3 )
= ( P2 @ G2 @ X4 @ R3 ) ) )
=> ( ! [X4: A,F2: A > B] :
( ! [Y4: A] :
( ( ord_less @ A @ Y4 @ X4 )
=> ( P2 @ F2 @ Y4 @ ( F2 @ Y4 ) ) )
=> ? [X_1: B] : ( P2 @ F2 @ X4 @ X_1 ) )
=> ? [F2: A > B] :
! [X3: A] : ( P2 @ F2 @ X3 @ ( F2 @ X3 ) ) ) ) ) ).
% dependent_wellorder_choice
thf(fact_7940_finite__Partial__order__induct,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),X: A,P2: A > $o] :
( ( order_7125193373082350890der_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( member @ A @ X @ ( field2 @ A @ R2 ) )
=> ( ( finite_finite2 @ ( product_prod @ A @ A ) @ R2 )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ ( field2 @ A @ R2 ) )
=> ( ! [Y4: A] :
( ( member @ A @ Y4 @ ( order_aboveS @ A @ R2 @ X4 ) )
=> ( P2 @ Y4 ) )
=> ( P2 @ X4 ) ) )
=> ( P2 @ X ) ) ) ) ) ).
% finite_Partial_order_induct
thf(fact_7941_chains__extend,axiom,
! [A: $tType,C3: set @ ( set @ A ),S2: set @ ( set @ A ),Z: set @ A] :
( ( member @ ( set @ ( set @ A ) ) @ C3 @ ( chains2 @ A @ S2 ) )
=> ( ( member @ ( set @ A ) @ Z @ S2 )
=> ( ! [X4: set @ A] :
( ( member @ ( set @ A ) @ X4 @ C3 )
=> ( ord_less_eq @ ( set @ A ) @ X4 @ Z ) )
=> ( member @ ( set @ ( set @ A ) ) @ ( sup_sup @ ( set @ ( set @ A ) ) @ ( insert @ ( set @ A ) @ Z @ ( bot_bot @ ( set @ ( set @ A ) ) ) ) @ C3 ) @ ( chains2 @ A @ S2 ) ) ) ) ) ).
% chains_extend
thf(fact_7942_Zorns__po__lemma,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( order_7125193373082350890der_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ! [C6: set @ A] :
( ( member @ ( set @ A ) @ C6 @ ( chains @ A @ R2 ) )
=> ? [X3: A] :
( ( member @ A @ X3 @ ( field2 @ A @ R2 ) )
& ! [Xa3: A] :
( ( member @ A @ Xa3 @ C6 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Xa3 @ X3 ) @ R2 ) ) ) )
=> ? [X4: A] :
( ( member @ A @ X4 @ ( field2 @ A @ R2 ) )
& ! [Xa: A] :
( ( member @ A @ Xa @ ( field2 @ A @ R2 ) )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Xa ) @ R2 )
=> ( Xa = X4 ) ) ) ) ) ) ).
% Zorns_po_lemma
thf(fact_7943_Chains__def,axiom,
! [A: $tType] :
( ( chains @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A )] :
( collect @ ( set @ A )
@ ^ [C8: set @ A] :
! [X5: A] :
( ( member @ A @ X5 @ C8 )
=> ! [Y6: A] :
( ( member @ A @ Y6 @ C8 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X5 @ Y6 ) @ R5 )
| ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y6 @ X5 ) @ R5 ) ) ) ) ) ) ) ).
% Chains_def
thf(fact_7944_Chains__subset_H,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( refl_on @ A @ ( top_top @ ( set @ A ) ) @ R2 )
=> ( ord_less_eq @ ( set @ ( set @ A ) )
@ ( collect @ ( set @ A )
@ ( pred_chain @ A @ ( top_top @ ( set @ A ) )
@ ^ [X5: A,Y6: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X5 @ Y6 ) @ R2 ) ) )
@ ( chains @ A @ R2 ) ) ) ).
% Chains_subset'
thf(fact_7945_list_Oin__rel,axiom,
! [B: $tType,A: $tType] :
( ( list_all2 @ A @ B )
= ( ^ [R6: A > B > $o,A6: list @ A,B5: list @ B] :
? [Z5: list @ ( product_prod @ A @ B )] :
( ( member @ ( list @ ( product_prod @ A @ B ) ) @ Z5
@ ( collect @ ( list @ ( product_prod @ A @ B ) )
@ ^ [X5: list @ ( product_prod @ A @ B )] : ( ord_less_eq @ ( set @ ( product_prod @ A @ B ) ) @ ( set2 @ ( product_prod @ A @ B ) @ X5 ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ R6 ) ) ) ) )
& ( ( map @ ( product_prod @ A @ B ) @ A @ ( product_fst @ A @ B ) @ Z5 )
= A6 )
& ( ( map @ ( product_prod @ A @ B ) @ B @ ( product_snd @ A @ B ) @ Z5 )
= B5 ) ) ) ) ).
% list.in_rel
thf(fact_7946_list__all2__Nil,axiom,
! [A: $tType,B: $tType,P2: A > B > $o,Ys: list @ B] :
( ( list_all2 @ A @ B @ P2 @ ( nil @ A ) @ Ys )
= ( Ys
= ( nil @ B ) ) ) ).
% list_all2_Nil
thf(fact_7947_list__all2__Nil2,axiom,
! [B: $tType,A: $tType,P2: A > B > $o,Xs: list @ A] :
( ( list_all2 @ A @ B @ P2 @ Xs @ ( nil @ B ) )
= ( Xs
= ( nil @ A ) ) ) ).
% list_all2_Nil2
thf(fact_7948_list__all2__rev,axiom,
! [A: $tType,B: $tType,P2: A > B > $o,Xs: list @ A,Ys: list @ B] :
( ( list_all2 @ A @ B @ P2 @ ( rev @ A @ Xs ) @ ( rev @ B @ Ys ) )
= ( list_all2 @ A @ B @ P2 @ Xs @ Ys ) ) ).
% list_all2_rev
thf(fact_7949_chains__alt__def,axiom,
! [A: $tType] :
( ( chains2 @ A )
= ( ^ [A7: set @ ( set @ A )] : ( collect @ ( set @ ( set @ A ) ) @ ( pred_chain @ ( set @ A ) @ A7 @ ( ord_less @ ( set @ A ) ) ) ) ) ) ).
% chains_alt_def
thf(fact_7950_list_Orel__map_I2_J,axiom,
! [A: $tType,C: $tType,B: $tType,Sa: A > C > $o,X: list @ A,G3: B > C,Y: list @ B] :
( ( list_all2 @ A @ C @ Sa @ X @ ( map @ B @ C @ G3 @ Y ) )
= ( list_all2 @ A @ B
@ ^ [X5: A,Y6: B] : ( Sa @ X5 @ ( G3 @ Y6 ) )
@ X
@ Y ) ) ).
% list.rel_map(2)
thf(fact_7951_list_Orel__map_I1_J,axiom,
! [A: $tType,C: $tType,B: $tType,Sb: C > B > $o,I2: A > C,X: list @ A,Y: list @ B] :
( ( list_all2 @ C @ B @ Sb @ ( map @ A @ C @ I2 @ X ) @ Y )
= ( list_all2 @ A @ B
@ ^ [X5: A] : ( Sb @ ( I2 @ X5 ) )
@ X
@ Y ) ) ).
% list.rel_map(1)
thf(fact_7952_list__all2__map1,axiom,
! [C: $tType,A: $tType,B: $tType,P2: A > B > $o,F3: C > A,As: list @ C,Bs: list @ B] :
( ( list_all2 @ A @ B @ P2 @ ( map @ C @ A @ F3 @ As ) @ Bs )
= ( list_all2 @ C @ B
@ ^ [X5: C] : ( P2 @ ( F3 @ X5 ) )
@ As
@ Bs ) ) ).
% list_all2_map1
thf(fact_7953_list__all2__map2,axiom,
! [A: $tType,B: $tType,C: $tType,P2: A > B > $o,As: list @ A,F3: C > B,Bs: list @ C] :
( ( list_all2 @ A @ B @ P2 @ As @ ( map @ C @ B @ F3 @ Bs ) )
= ( list_all2 @ A @ C
@ ^ [X5: A,Y6: C] : ( P2 @ X5 @ ( F3 @ Y6 ) )
@ As
@ Bs ) ) ).
% list_all2_map2
thf(fact_7954_list__all2__rev1,axiom,
! [A: $tType,B: $tType,P2: A > B > $o,Xs: list @ A,Ys: list @ B] :
( ( list_all2 @ A @ B @ P2 @ ( rev @ A @ Xs ) @ Ys )
= ( list_all2 @ A @ B @ P2 @ Xs @ ( rev @ B @ Ys ) ) ) ).
% list_all2_rev1
thf(fact_7955_list__all2__conv__all__nth,axiom,
! [B: $tType,A: $tType] :
( ( list_all2 @ A @ B )
= ( ^ [P4: A > B > $o,Xs3: list @ A,Ys3: list @ B] :
( ( ( size_size @ ( list @ A ) @ Xs3 )
= ( size_size @ ( list @ B ) @ Ys3 ) )
& ! [I4: nat] :
( ( ord_less @ nat @ I4 @ ( size_size @ ( list @ A ) @ Xs3 ) )
=> ( P4 @ ( nth @ A @ Xs3 @ I4 ) @ ( nth @ B @ Ys3 @ I4 ) ) ) ) ) ) ).
% list_all2_conv_all_nth
thf(fact_7956_list__all2__all__nthI,axiom,
! [A: $tType,B: $tType,A2: list @ A,B2: list @ B,P2: A > B > $o] :
( ( ( size_size @ ( list @ A ) @ A2 )
= ( size_size @ ( list @ B ) @ B2 ) )
=> ( ! [N3: nat] :
( ( ord_less @ nat @ N3 @ ( size_size @ ( list @ A ) @ A2 ) )
=> ( P2 @ ( nth @ A @ A2 @ N3 ) @ ( nth @ B @ B2 @ N3 ) ) )
=> ( list_all2 @ A @ B @ P2 @ A2 @ B2 ) ) ) ).
% list_all2_all_nthI
thf(fact_7957_list__all2__nthD2,axiom,
! [A: $tType,B: $tType,P2: A > B > $o,Xs: list @ A,Ys: list @ B,P: nat] :
( ( list_all2 @ A @ B @ P2 @ Xs @ Ys )
=> ( ( ord_less @ nat @ P @ ( size_size @ ( list @ B ) @ Ys ) )
=> ( P2 @ ( nth @ A @ Xs @ P ) @ ( nth @ B @ Ys @ P ) ) ) ) ).
% list_all2_nthD2
thf(fact_7958_list__all2__nthD,axiom,
! [A: $tType,B: $tType,P2: A > B > $o,Xs: list @ A,Ys: list @ B,P: nat] :
( ( list_all2 @ A @ B @ P2 @ Xs @ Ys )
=> ( ( ord_less @ nat @ P @ ( size_size @ ( list @ A ) @ Xs ) )
=> ( P2 @ ( nth @ A @ Xs @ P ) @ ( nth @ B @ Ys @ P ) ) ) ) ).
% list_all2_nthD
thf(fact_7959_list__all2__append2,axiom,
! [B: $tType,A: $tType,P2: A > B > $o,Xs: list @ A,Ys: list @ B,Zs: list @ B] :
( ( list_all2 @ A @ B @ P2 @ Xs @ ( append @ B @ Ys @ Zs ) )
= ( ? [Us2: list @ A,Vs: list @ A] :
( ( Xs
= ( append @ A @ Us2 @ Vs ) )
& ( ( size_size @ ( list @ A ) @ Us2 )
= ( size_size @ ( list @ B ) @ Ys ) )
& ( ( size_size @ ( list @ A ) @ Vs )
= ( size_size @ ( list @ B ) @ Zs ) )
& ( list_all2 @ A @ B @ P2 @ Us2 @ Ys )
& ( list_all2 @ A @ B @ P2 @ Vs @ Zs ) ) ) ) ).
% list_all2_append2
thf(fact_7960_list__all2__append1,axiom,
! [A: $tType,B: $tType,P2: A > B > $o,Xs: list @ A,Ys: list @ A,Zs: list @ B] :
( ( list_all2 @ A @ B @ P2 @ ( append @ A @ Xs @ Ys ) @ Zs )
= ( ? [Us2: list @ B,Vs: list @ B] :
( ( Zs
= ( append @ B @ Us2 @ Vs ) )
& ( ( size_size @ ( list @ B ) @ Us2 )
= ( size_size @ ( list @ A ) @ Xs ) )
& ( ( size_size @ ( list @ B ) @ Vs )
= ( size_size @ ( list @ A ) @ Ys ) )
& ( list_all2 @ A @ B @ P2 @ Xs @ Us2 )
& ( list_all2 @ A @ B @ P2 @ Ys @ Vs ) ) ) ) ).
% list_all2_append1
thf(fact_7961_list__all2__append,axiom,
! [A: $tType,B: $tType,Xs: list @ A,Ys: list @ B,P2: A > B > $o,Us: list @ A,Vs2: list @ B] :
( ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys ) )
=> ( ( list_all2 @ A @ B @ P2 @ ( append @ A @ Xs @ Us ) @ ( append @ B @ Ys @ Vs2 ) )
= ( ( list_all2 @ A @ B @ P2 @ Xs @ Ys )
& ( list_all2 @ A @ B @ P2 @ Us @ Vs2 ) ) ) ) ).
% list_all2_append
thf(fact_7962_list__all2__same,axiom,
! [A: $tType,P2: A > A > $o,Xs: list @ A] :
( ( list_all2 @ A @ A @ P2 @ Xs @ Xs )
= ( ! [X5: A] :
( ( member @ A @ X5 @ ( set2 @ A @ Xs ) )
=> ( P2 @ X5 @ X5 ) ) ) ) ).
% list_all2_same
thf(fact_7963_list_Orel__refl__strong,axiom,
! [A: $tType,X: list @ A,Ra2: A > A > $o] :
( ! [Z3: A] :
( ( member @ A @ Z3 @ ( set2 @ A @ X ) )
=> ( Ra2 @ Z3 @ Z3 ) )
=> ( list_all2 @ A @ A @ Ra2 @ X @ X ) ) ).
% list.rel_refl_strong
thf(fact_7964_list_Orel__mono__strong,axiom,
! [A: $tType,B: $tType,R: A > B > $o,X: list @ A,Y: list @ B,Ra2: A > B > $o] :
( ( list_all2 @ A @ B @ R @ X @ Y )
=> ( ! [Z3: A,Yb: B] :
( ( member @ A @ Z3 @ ( set2 @ A @ X ) )
=> ( ( member @ B @ Yb @ ( set2 @ B @ Y ) )
=> ( ( R @ Z3 @ Yb )
=> ( Ra2 @ Z3 @ Yb ) ) ) )
=> ( list_all2 @ A @ B @ Ra2 @ X @ Y ) ) ) ).
% list.rel_mono_strong
thf(fact_7965_list_Orel__cong,axiom,
! [A: $tType,B: $tType,X: list @ A,Ya: list @ A,Y: list @ B,Xa2: list @ B,R: A > B > $o,Ra2: A > B > $o] :
( ( X = Ya )
=> ( ( Y = Xa2 )
=> ( ! [Z3: A,Yb: B] :
( ( member @ A @ Z3 @ ( set2 @ A @ Ya ) )
=> ( ( member @ B @ Yb @ ( set2 @ B @ Xa2 ) )
=> ( ( R @ Z3 @ Yb )
= ( Ra2 @ Z3 @ Yb ) ) ) )
=> ( ( list_all2 @ A @ B @ R @ X @ Y )
= ( list_all2 @ A @ B @ Ra2 @ Ya @ Xa2 ) ) ) ) ) ).
% list.rel_cong
thf(fact_7966_list__all2__lengthD,axiom,
! [A: $tType,B: $tType,P2: A > B > $o,Xs: list @ A,Ys: list @ B] :
( ( list_all2 @ A @ B @ P2 @ Xs @ Ys )
=> ( ( size_size @ ( list @ A ) @ Xs )
= ( size_size @ ( list @ B ) @ Ys ) ) ) ).
% list_all2_lengthD
thf(fact_7967_list__all2__dropI,axiom,
! [A: $tType,B: $tType,P2: A > B > $o,Xs: list @ A,Ys: list @ B,N: nat] :
( ( list_all2 @ A @ B @ P2 @ Xs @ Ys )
=> ( list_all2 @ A @ B @ P2 @ ( drop @ A @ N @ Xs ) @ ( drop @ B @ N @ Ys ) ) ) ).
% list_all2_dropI
thf(fact_7968_list__all2__takeI,axiom,
! [A: $tType,B: $tType,P2: A > B > $o,Xs: list @ A,Ys: list @ B,N: nat] :
( ( list_all2 @ A @ B @ P2 @ Xs @ Ys )
=> ( list_all2 @ A @ B @ P2 @ ( take @ A @ N @ Xs ) @ ( take @ B @ N @ Ys ) ) ) ).
% list_all2_takeI
thf(fact_7969_list_Octr__transfer_I1_J,axiom,
! [A: $tType,B: $tType,R: A > B > $o] : ( list_all2 @ A @ B @ R @ ( nil @ A ) @ ( nil @ B ) ) ).
% list.ctr_transfer(1)
thf(fact_7970_list_Orel__distinct_I2_J,axiom,
! [A: $tType,B: $tType,R: A > B > $o,Y21: A,Y22: list @ A] :
~ ( list_all2 @ A @ B @ R @ ( cons @ A @ Y21 @ Y22 ) @ ( nil @ B ) ) ).
% list.rel_distinct(2)
thf(fact_7971_list_Orel__distinct_I1_J,axiom,
! [A: $tType,B: $tType,R: A > B > $o,Y21: B,Y22: list @ B] :
~ ( list_all2 @ A @ B @ R @ ( nil @ A ) @ ( cons @ B @ Y21 @ Y22 ) ) ).
% list.rel_distinct(1)
thf(fact_7972_list_Orel__cases,axiom,
! [A: $tType,B: $tType,R: A > B > $o,A2: list @ A,B2: list @ B] :
( ( list_all2 @ A @ B @ R @ A2 @ B2 )
=> ( ( ( A2
= ( nil @ A ) )
=> ( B2
!= ( nil @ B ) ) )
=> ~ ! [X16: A,X22: list @ A] :
( ( A2
= ( cons @ A @ X16 @ X22 ) )
=> ! [Y15: B,Y23: list @ B] :
( ( B2
= ( cons @ B @ Y15 @ Y23 ) )
=> ( ( R @ X16 @ Y15 )
=> ~ ( list_all2 @ A @ B @ R @ X22 @ Y23 ) ) ) ) ) ) ).
% list.rel_cases
thf(fact_7973_list_Orel__induct,axiom,
! [A: $tType,B: $tType,R: A > B > $o,X: list @ A,Y: list @ B,Q: ( list @ A ) > ( list @ B ) > $o] :
( ( list_all2 @ A @ B @ R @ X @ Y )
=> ( ( Q @ ( nil @ A ) @ ( nil @ B ) )
=> ( ! [A21: A,A222: list @ A,B21: B,B222: list @ B] :
( ( R @ A21 @ B21 )
=> ( ( Q @ A222 @ B222 )
=> ( Q @ ( cons @ A @ A21 @ A222 ) @ ( cons @ B @ B21 @ B222 ) ) ) )
=> ( Q @ X @ Y ) ) ) ) ).
% list.rel_induct
thf(fact_7974_list__all2__induct,axiom,
! [A: $tType,B: $tType,P2: A > B > $o,Xs: list @ A,Ys: list @ B,R: ( list @ A ) > ( list @ B ) > $o] :
( ( list_all2 @ A @ B @ P2 @ Xs @ Ys )
=> ( ( R @ ( nil @ A ) @ ( nil @ B ) )
=> ( ! [X4: A,Xs2: list @ A,Y3: B,Ys4: list @ B] :
( ( P2 @ X4 @ Y3 )
=> ( ( list_all2 @ A @ B @ P2 @ Xs2 @ Ys4 )
=> ( ( R @ Xs2 @ Ys4 )
=> ( R @ ( cons @ A @ X4 @ Xs2 ) @ ( cons @ B @ Y3 @ Ys4 ) ) ) ) )
=> ( R @ Xs @ Ys ) ) ) ) ).
% list_all2_induct
thf(fact_7975_list__all2__Cons2,axiom,
! [B: $tType,A: $tType,P2: A > B > $o,Xs: list @ A,Y: B,Ys: list @ B] :
( ( list_all2 @ A @ B @ P2 @ Xs @ ( cons @ B @ Y @ Ys ) )
= ( ? [Z5: A,Zs3: list @ A] :
( ( Xs
= ( cons @ A @ Z5 @ Zs3 ) )
& ( P2 @ Z5 @ Y )
& ( list_all2 @ A @ B @ P2 @ Zs3 @ Ys ) ) ) ) ).
% list_all2_Cons2
thf(fact_7976_list__all2__Cons1,axiom,
! [A: $tType,B: $tType,P2: A > B > $o,X: A,Xs: list @ A,Ys: list @ B] :
( ( list_all2 @ A @ B @ P2 @ ( cons @ A @ X @ Xs ) @ Ys )
= ( ? [Z5: B,Zs3: list @ B] :
( ( Ys
= ( cons @ B @ Z5 @ Zs3 ) )
& ( P2 @ X @ Z5 )
& ( list_all2 @ A @ B @ P2 @ Xs @ Zs3 ) ) ) ) ).
% list_all2_Cons1
thf(fact_7977_list__all2__Cons,axiom,
! [A: $tType,B: $tType,P2: A > B > $o,X: A,Xs: list @ A,Y: B,Ys: list @ B] :
( ( list_all2 @ A @ B @ P2 @ ( cons @ A @ X @ Xs ) @ ( cons @ B @ Y @ Ys ) )
= ( ( P2 @ X @ Y )
& ( list_all2 @ A @ B @ P2 @ Xs @ Ys ) ) ) ).
% list_all2_Cons
thf(fact_7978_list_Orel__intros_I2_J,axiom,
! [A: $tType,B: $tType,R: A > B > $o,X21: A,Y21: B,X222: list @ A,Y22: list @ B] :
( ( R @ X21 @ Y21 )
=> ( ( list_all2 @ A @ B @ R @ X222 @ Y22 )
=> ( list_all2 @ A @ B @ R @ ( cons @ A @ X21 @ X222 ) @ ( cons @ B @ Y21 @ Y22 ) ) ) ) ).
% list.rel_intros(2)
thf(fact_7979_list_Orel__inject_I2_J,axiom,
! [A: $tType,B: $tType,R: A > B > $o,X21: A,X222: list @ A,Y21: B,Y22: list @ B] :
( ( list_all2 @ A @ B @ R @ ( cons @ A @ X21 @ X222 ) @ ( cons @ B @ Y21 @ Y22 ) )
= ( ( R @ X21 @ Y21 )
& ( list_all2 @ A @ B @ R @ X222 @ Y22 ) ) ) ).
% list.rel_inject(2)
thf(fact_7980_list_Orel__mono,axiom,
! [B: $tType,A: $tType,R: A > B > $o,Ra2: A > B > $o] :
( ( ord_less_eq @ ( A > B > $o ) @ R @ Ra2 )
=> ( ord_less_eq @ ( ( list @ A ) > ( list @ B ) > $o ) @ ( list_all2 @ A @ B @ R ) @ ( list_all2 @ A @ B @ Ra2 ) ) ) ).
% list.rel_mono
thf(fact_7981_list__all2__update__cong,axiom,
! [A: $tType,B: $tType,P2: A > B > $o,Xs: list @ A,Ys: list @ B,X: A,Y: B,I2: nat] :
( ( list_all2 @ A @ B @ P2 @ Xs @ Ys )
=> ( ( P2 @ X @ Y )
=> ( list_all2 @ A @ B @ P2 @ ( list_update @ A @ Xs @ I2 @ X ) @ ( list_update @ B @ Ys @ I2 @ Y ) ) ) ) ).
% list_all2_update_cong
thf(fact_7982_list__all2__antisym,axiom,
! [A: $tType,P2: A > A > $o,Q: A > A > $o,Xs: list @ A,Ys: list @ A] :
( ! [X4: A,Y3: A] :
( ( P2 @ X4 @ Y3 )
=> ( ( Q @ Y3 @ X4 )
=> ( X4 = Y3 ) ) )
=> ( ( list_all2 @ A @ A @ P2 @ Xs @ Ys )
=> ( ( list_all2 @ A @ A @ Q @ Ys @ Xs )
=> ( Xs = Ys ) ) ) ) ).
% list_all2_antisym
thf(fact_7983_list__all2__trans,axiom,
! [B: $tType,A: $tType,C: $tType,P1: A > B > $o,P22: B > C > $o,P32: A > C > $o,As: list @ A,Bs: list @ B,Cs: list @ C] :
( ! [A4: A,B4: B,C2: C] :
( ( P1 @ A4 @ B4 )
=> ( ( P22 @ B4 @ C2 )
=> ( P32 @ A4 @ C2 ) ) )
=> ( ( list_all2 @ A @ B @ P1 @ As @ Bs )
=> ( ( list_all2 @ B @ C @ P22 @ Bs @ Cs )
=> ( list_all2 @ A @ C @ P32 @ As @ Cs ) ) ) ) ).
% list_all2_trans
thf(fact_7984_list__all2__refl,axiom,
! [A: $tType,P2: A > A > $o,Xs: list @ A] :
( ! [X4: A] : ( P2 @ X4 @ X4 )
=> ( list_all2 @ A @ A @ P2 @ Xs @ Xs ) ) ).
% list_all2_refl
thf(fact_7985_list__all2__mono,axiom,
! [A: $tType,B: $tType,P2: A > B > $o,Xs: list @ A,Ys: list @ B,Q: A > B > $o] :
( ( list_all2 @ A @ B @ P2 @ Xs @ Ys )
=> ( ! [Xs2: A,Ys4: B] :
( ( P2 @ Xs2 @ Ys4 )
=> ( Q @ Xs2 @ Ys4 ) )
=> ( list_all2 @ A @ B @ Q @ Xs @ Ys ) ) ) ).
% list_all2_mono
thf(fact_7986_list__all2__eq,axiom,
! [A: $tType] :
( ( ^ [Y5: list @ A,Z2: list @ A] : Y5 = Z2 )
= ( list_all2 @ A @ A
@ ^ [Y5: A,Z2: A] : Y5 = Z2 ) ) ).
% list_all2_eq
thf(fact_7987_list_Orel__refl,axiom,
! [B: $tType,Ra2: B > B > $o,X: list @ B] :
( ! [X4: B] : ( Ra2 @ X4 @ X4 )
=> ( list_all2 @ B @ B @ Ra2 @ X @ X ) ) ).
% list.rel_refl
thf(fact_7988_list_Orel__eq,axiom,
! [A: $tType] :
( ( list_all2 @ A @ A
@ ^ [Y5: A,Z2: A] : Y5 = Z2 )
= ( ^ [Y5: list @ A,Z2: list @ A] : Y5 = Z2 ) ) ).
% list.rel_eq
thf(fact_7989_list__all2__appendI,axiom,
! [A: $tType,B: $tType,P2: A > B > $o,A2: list @ A,B2: list @ B,C3: list @ A,D3: list @ B] :
( ( list_all2 @ A @ B @ P2 @ A2 @ B2 )
=> ( ( list_all2 @ A @ B @ P2 @ C3 @ D3 )
=> ( list_all2 @ A @ B @ P2 @ ( append @ A @ A2 @ C3 ) @ ( append @ B @ B2 @ D3 ) ) ) ) ).
% list_all2_appendI
thf(fact_7990_list_Orel__sel,axiom,
! [B: $tType,A: $tType] :
( ( list_all2 @ A @ B )
= ( ^ [R6: A > B > $o,A6: list @ A,B5: list @ B] :
( ( ( A6
= ( nil @ A ) )
= ( B5
= ( nil @ B ) ) )
& ( ( A6
!= ( nil @ A ) )
=> ( ( B5
!= ( nil @ B ) )
=> ( ( R6 @ ( hd @ A @ A6 ) @ ( hd @ B @ B5 ) )
& ( list_all2 @ A @ B @ R6 @ ( tl @ A @ A6 ) @ ( tl @ B @ B5 ) ) ) ) ) ) ) ) ).
% list.rel_sel
thf(fact_7991_subset__chain__insert,axiom,
! [A: $tType,A18: set @ ( set @ A ),B6: set @ A,B11: set @ ( set @ A )] :
( ( pred_chain @ ( set @ A ) @ A18 @ ( ord_less @ ( set @ A ) ) @ ( insert @ ( set @ A ) @ B6 @ B11 ) )
= ( ( member @ ( set @ A ) @ B6 @ A18 )
& ! [X5: set @ A] :
( ( member @ ( set @ A ) @ X5 @ B11 )
=> ( ( ord_less_eq @ ( set @ A ) @ X5 @ B6 )
| ( ord_less_eq @ ( set @ A ) @ B6 @ X5 ) ) )
& ( pred_chain @ ( set @ A ) @ A18 @ ( ord_less @ ( set @ A ) ) @ B11 ) ) ) ).
% subset_chain_insert
thf(fact_7992_subset__chain__def,axiom,
! [A: $tType,A18: set @ ( set @ A ),C10: set @ ( set @ A )] :
( ( pred_chain @ ( set @ A ) @ A18 @ ( ord_less @ ( set @ A ) ) @ C10 )
= ( ( ord_less_eq @ ( set @ ( set @ A ) ) @ C10 @ A18 )
& ! [X5: set @ A] :
( ( member @ ( set @ A ) @ X5 @ C10 )
=> ! [Y6: set @ A] :
( ( member @ ( set @ A ) @ Y6 @ C10 )
=> ( ( ord_less_eq @ ( set @ A ) @ X5 @ Y6 )
| ( ord_less_eq @ ( set @ A ) @ Y6 @ X5 ) ) ) ) ) ) ).
% subset_chain_def
thf(fact_7993_subset__Zorn,axiom,
! [A: $tType,A5: set @ ( set @ A )] :
( ! [C6: set @ ( set @ A )] :
( ( pred_chain @ ( set @ A ) @ A5 @ ( ord_less @ ( set @ A ) ) @ C6 )
=> ? [X3: set @ A] :
( ( member @ ( set @ A ) @ X3 @ A5 )
& ! [Xa3: set @ A] :
( ( member @ ( set @ A ) @ Xa3 @ C6 )
=> ( ord_less_eq @ ( set @ A ) @ Xa3 @ X3 ) ) ) )
=> ? [X4: set @ A] :
( ( member @ ( set @ A ) @ X4 @ A5 )
& ! [Xa: set @ A] :
( ( member @ ( set @ A ) @ Xa @ A5 )
=> ( ( ord_less_eq @ ( set @ A ) @ X4 @ Xa )
=> ( Xa = X4 ) ) ) ) ) ).
% subset_Zorn
thf(fact_7994_subset__Zorn_H,axiom,
! [A: $tType,A5: set @ ( set @ A )] :
( ! [C6: set @ ( set @ A )] :
( ( pred_chain @ ( set @ A ) @ A5 @ ( ord_less @ ( set @ A ) ) @ C6 )
=> ( member @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ C6 ) @ A5 ) )
=> ? [X4: set @ A] :
( ( member @ ( set @ A ) @ X4 @ A5 )
& ! [Xa: set @ A] :
( ( member @ ( set @ A ) @ Xa @ A5 )
=> ( ( ord_less_eq @ ( set @ A ) @ X4 @ Xa )
=> ( Xa = X4 ) ) ) ) ) ).
% subset_Zorn'
thf(fact_7995_subset_Ochain__empty,axiom,
! [A: $tType,A5: set @ ( set @ A )] : ( pred_chain @ ( set @ A ) @ A5 @ ( ord_less @ ( set @ A ) ) @ ( bot_bot @ ( set @ ( set @ A ) ) ) ) ).
% subset.chain_empty
thf(fact_7996_pred__on_Ochain__empty,axiom,
! [A: $tType,A5: set @ A,P2: A > A > $o] : ( pred_chain @ A @ A5 @ P2 @ ( bot_bot @ ( set @ A ) ) ) ).
% pred_on.chain_empty
thf(fact_7997_subset_OchainI,axiom,
! [A: $tType,C5: set @ ( set @ A ),A5: set @ ( set @ A )] :
( ( ord_less_eq @ ( set @ ( set @ A ) ) @ C5 @ A5 )
=> ( ! [X4: set @ A,Y3: set @ A] :
( ( member @ ( set @ A ) @ X4 @ C5 )
=> ( ( member @ ( set @ A ) @ Y3 @ C5 )
=> ( ( sup_sup @ ( ( set @ A ) > ( set @ A ) > $o ) @ ( ord_less @ ( set @ A ) )
@ ^ [Y5: set @ A,Z2: set @ A] : Y5 = Z2
@ X4
@ Y3 )
| ( sup_sup @ ( ( set @ A ) > ( set @ A ) > $o ) @ ( ord_less @ ( set @ A ) )
@ ^ [Y5: set @ A,Z2: set @ A] : Y5 = Z2
@ Y3
@ X4 ) ) ) )
=> ( pred_chain @ ( set @ A ) @ A5 @ ( ord_less @ ( set @ A ) ) @ C5 ) ) ) ).
% subset.chainI
thf(fact_7998_subset_Ochain__def,axiom,
! [A: $tType,A5: set @ ( set @ A ),C5: set @ ( set @ A )] :
( ( pred_chain @ ( set @ A ) @ A5 @ ( ord_less @ ( set @ A ) ) @ C5 )
= ( ( ord_less_eq @ ( set @ ( set @ A ) ) @ C5 @ A5 )
& ! [X5: set @ A] :
( ( member @ ( set @ A ) @ X5 @ C5 )
=> ! [Y6: set @ A] :
( ( member @ ( set @ A ) @ Y6 @ C5 )
=> ( ( sup_sup @ ( ( set @ A ) > ( set @ A ) > $o ) @ ( ord_less @ ( set @ A ) )
@ ^ [Y5: set @ A,Z2: set @ A] : Y5 = Z2
@ X5
@ Y6 )
| ( sup_sup @ ( ( set @ A ) > ( set @ A ) > $o ) @ ( ord_less @ ( set @ A ) )
@ ^ [Y5: set @ A,Z2: set @ A] : Y5 = Z2
@ Y6
@ X5 ) ) ) ) ) ) ).
% subset.chain_def
thf(fact_7999_subset_Ochain__total,axiom,
! [A: $tType,A5: set @ ( set @ A ),C5: set @ ( set @ A ),X: set @ A,Y: set @ A] :
( ( pred_chain @ ( set @ A ) @ A5 @ ( ord_less @ ( set @ A ) ) @ C5 )
=> ( ( member @ ( set @ A ) @ X @ C5 )
=> ( ( member @ ( set @ A ) @ Y @ C5 )
=> ( ( sup_sup @ ( ( set @ A ) > ( set @ A ) > $o ) @ ( ord_less @ ( set @ A ) )
@ ^ [Y5: set @ A,Z2: set @ A] : Y5 = Z2
@ X
@ Y )
| ( sup_sup @ ( ( set @ A ) > ( set @ A ) > $o ) @ ( ord_less @ ( set @ A ) )
@ ^ [Y5: set @ A,Z2: set @ A] : Y5 = Z2
@ Y
@ X ) ) ) ) ) ).
% subset.chain_total
thf(fact_8000_product__lists__set,axiom,
! [A: $tType,Xss: list @ ( list @ A )] :
( ( set2 @ ( list @ A ) @ ( product_lists @ A @ Xss ) )
= ( collect @ ( list @ A )
@ ^ [Xs3: list @ A] :
( list_all2 @ A @ ( list @ A )
@ ^ [X5: A,Ys3: list @ A] : ( member @ A @ X5 @ ( set2 @ A @ Ys3 ) )
@ Xs3
@ Xss ) ) ) ).
% product_lists_set
thf(fact_8001_list__all2I,axiom,
! [A: $tType,B: $tType,A2: list @ A,B2: list @ B,P2: A > B > $o] :
( ! [X4: product_prod @ A @ B] :
( ( member @ ( product_prod @ A @ B ) @ X4 @ ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ A2 @ B2 ) ) )
=> ( product_case_prod @ A @ B @ $o @ P2 @ X4 ) )
=> ( ( ( size_size @ ( list @ A ) @ A2 )
= ( size_size @ ( list @ B ) @ B2 ) )
=> ( list_all2 @ A @ B @ P2 @ A2 @ B2 ) ) ) ).
% list_all2I
thf(fact_8002_subset__Zorn__nonempty,axiom,
! [A: $tType,A18: set @ ( set @ A )] :
( ( A18
!= ( bot_bot @ ( set @ ( set @ A ) ) ) )
=> ( ! [C11: set @ ( set @ A )] :
( ( C11
!= ( bot_bot @ ( set @ ( set @ A ) ) ) )
=> ( ( pred_chain @ ( set @ A ) @ A18 @ ( ord_less @ ( set @ A ) ) @ C11 )
=> ( member @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ C11 ) @ A18 ) ) )
=> ? [X4: set @ A] :
( ( member @ ( set @ A ) @ X4 @ A18 )
& ! [Xa: set @ A] :
( ( member @ ( set @ A ) @ Xa @ A18 )
=> ( ( ord_less_eq @ ( set @ A ) @ X4 @ Xa )
=> ( Xa = X4 ) ) ) ) ) ) ).
% subset_Zorn_nonempty
thf(fact_8003_Union__in__chain,axiom,
! [A: $tType,B11: set @ ( set @ A ),A18: set @ ( set @ A )] :
( ( finite_finite2 @ ( set @ A ) @ B11 )
=> ( ( B11
!= ( bot_bot @ ( set @ ( set @ A ) ) ) )
=> ( ( pred_chain @ ( set @ A ) @ A18 @ ( ord_less @ ( set @ A ) ) @ B11 )
=> ( member @ ( set @ A ) @ ( complete_Sup_Sup @ ( set @ A ) @ B11 ) @ B11 ) ) ) ) ).
% Union_in_chain
thf(fact_8004_Inter__in__chain,axiom,
! [A: $tType,B11: set @ ( set @ A ),A18: set @ ( set @ A )] :
( ( finite_finite2 @ ( set @ A ) @ B11 )
=> ( ( B11
!= ( bot_bot @ ( set @ ( set @ A ) ) ) )
=> ( ( pred_chain @ ( set @ A ) @ A18 @ ( ord_less @ ( set @ A ) ) @ B11 )
=> ( member @ ( set @ A ) @ ( complete_Inf_Inf @ ( set @ A ) @ B11 ) @ B11 ) ) ) ) ).
% Inter_in_chain
thf(fact_8005_subset_Ochain__extend,axiom,
! [A: $tType,A5: set @ ( set @ A ),C5: set @ ( set @ A ),Z: set @ A] :
( ( pred_chain @ ( set @ A ) @ A5 @ ( ord_less @ ( set @ A ) ) @ C5 )
=> ( ( member @ ( set @ A ) @ Z @ A5 )
=> ( ! [X4: set @ A] :
( ( member @ ( set @ A ) @ X4 @ C5 )
=> ( sup_sup @ ( ( set @ A ) > ( set @ A ) > $o ) @ ( ord_less @ ( set @ A ) )
@ ^ [Y5: set @ A,Z2: set @ A] : Y5 = Z2
@ X4
@ Z ) )
=> ( pred_chain @ ( set @ A ) @ A5 @ ( ord_less @ ( set @ A ) ) @ ( sup_sup @ ( set @ ( set @ A ) ) @ ( insert @ ( set @ A ) @ Z @ ( bot_bot @ ( set @ ( set @ A ) ) ) ) @ C5 ) ) ) ) ) ).
% subset.chain_extend
thf(fact_8006_pred__on_Ochain__extend,axiom,
! [A: $tType,A5: set @ A,P2: A > A > $o,C5: set @ A,Z: A] :
( ( pred_chain @ A @ A5 @ P2 @ C5 )
=> ( ( member @ A @ Z @ A5 )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ C5 )
=> ( sup_sup @ ( A > A > $o ) @ P2
@ ^ [Y5: A,Z2: A] : Y5 = Z2
@ X4
@ Z ) )
=> ( pred_chain @ A @ A5 @ P2 @ ( sup_sup @ ( set @ A ) @ ( insert @ A @ Z @ ( bot_bot @ ( set @ A ) ) ) @ C5 ) ) ) ) ) ).
% pred_on.chain_extend
thf(fact_8007_finite__subset__Union__chain,axiom,
! [A: $tType,A5: set @ A,B11: set @ ( set @ A ),A18: set @ ( set @ A )] :
( ( finite_finite2 @ A @ A5 )
=> ( ( ord_less_eq @ ( set @ A ) @ A5 @ ( complete_Sup_Sup @ ( set @ A ) @ B11 ) )
=> ( ( B11
!= ( bot_bot @ ( set @ ( set @ A ) ) ) )
=> ( ( pred_chain @ ( set @ A ) @ A18 @ ( ord_less @ ( set @ A ) ) @ B11 )
=> ~ ! [B9: set @ A] :
( ( member @ ( set @ A ) @ B9 @ B11 )
=> ~ ( ord_less_eq @ ( set @ A ) @ A5 @ B9 ) ) ) ) ) ) ).
% finite_subset_Union_chain
thf(fact_8008_list__all2__iff,axiom,
! [B: $tType,A: $tType] :
( ( list_all2 @ A @ B )
= ( ^ [P4: A > B > $o,Xs3: list @ A,Ys3: list @ B] :
( ( ( size_size @ ( list @ A ) @ Xs3 )
= ( size_size @ ( list @ B ) @ Ys3 ) )
& ! [X5: product_prod @ A @ B] :
( ( member @ ( product_prod @ A @ B ) @ X5 @ ( set2 @ ( product_prod @ A @ B ) @ ( zip @ A @ B @ Xs3 @ Ys3 ) ) )
=> ( product_case_prod @ A @ B @ $o @ P4 @ X5 ) ) ) ) ) ).
% list_all2_iff
thf(fact_8009_Chains__subset,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ord_less_eq @ ( set @ ( set @ A ) ) @ ( chains @ A @ R2 )
@ ( collect @ ( set @ A )
@ ( pred_chain @ A @ ( top_top @ ( set @ A ) )
@ ^ [X5: A,Y6: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X5 @ Y6 ) @ R2 ) ) ) ) ).
% Chains_subset
thf(fact_8010_Chains__alt__def,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ( refl_on @ A @ ( top_top @ ( set @ A ) ) @ R2 )
=> ( ( chains @ A @ R2 )
= ( collect @ ( set @ A )
@ ( pred_chain @ A @ ( top_top @ ( set @ A ) )
@ ^ [X5: A,Y6: A] : ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X5 @ Y6 ) @ R2 ) ) ) ) ) ).
% Chains_alt_def
thf(fact_8011_chain__subset__alt__def,axiom,
! [A: $tType] :
( ( chain_subset @ A )
= ( pred_chain @ ( set @ A ) @ ( top_top @ ( set @ ( set @ A ) ) ) @ ( ord_less @ ( set @ A ) ) ) ) ).
% chain_subset_alt_def
thf(fact_8012_power_Opower__eq__if,axiom,
! [A: $tType] :
( ( power2 @ A )
= ( ^ [One: A,Times: A > A > A,P6: A,M3: nat] :
( if @ A
@ ( M3
= ( zero_zero @ nat ) )
@ One
@ ( Times @ P6 @ ( power2 @ A @ One @ Times @ P6 @ ( minus_minus @ nat @ M3 @ ( one_one @ nat ) ) ) ) ) ) ) ).
% power.power_eq_if
thf(fact_8013_power_Opower_Opower__0,axiom,
! [A: $tType,One2: A,Times2: A > A > A,A2: A] :
( ( power2 @ A @ One2 @ Times2 @ A2 @ ( zero_zero @ nat ) )
= One2 ) ).
% power.power.power_0
thf(fact_8014_cauchyD,axiom,
! [X7: nat > rat,R2: rat] :
( ( cauchy @ X7 )
=> ( ( ord_less @ rat @ ( zero_zero @ rat ) @ R2 )
=> ? [K: nat] :
! [M4: nat] :
( ( ord_less_eq @ nat @ K @ M4 )
=> ! [N5: nat] :
( ( ord_less_eq @ nat @ K @ N5 )
=> ( ord_less @ rat @ ( abs_abs @ rat @ ( minus_minus @ rat @ ( X7 @ M4 ) @ ( X7 @ N5 ) ) ) @ R2 ) ) ) ) ) ).
% cauchyD
thf(fact_8015_cauchyI,axiom,
! [X7: nat > rat] :
( ! [R3: rat] :
( ( ord_less @ rat @ ( zero_zero @ rat ) @ R3 )
=> ? [K4: nat] :
! [M: nat] :
( ( ord_less_eq @ nat @ K4 @ M )
=> ! [N3: nat] :
( ( ord_less_eq @ nat @ K4 @ N3 )
=> ( ord_less @ rat @ ( abs_abs @ rat @ ( minus_minus @ rat @ ( X7 @ M ) @ ( X7 @ N3 ) ) ) @ R3 ) ) ) )
=> ( cauchy @ X7 ) ) ).
% cauchyI
thf(fact_8016_cauchy__imp__bounded,axiom,
! [X7: nat > rat] :
( ( cauchy @ X7 )
=> ? [B4: rat] :
( ( ord_less @ rat @ ( zero_zero @ rat ) @ B4 )
& ! [N5: nat] : ( ord_less @ rat @ ( abs_abs @ rat @ ( X7 @ N5 ) ) @ B4 ) ) ) ).
% cauchy_imp_bounded
thf(fact_8017_cauchy__def,axiom,
( cauchy
= ( ^ [X9: nat > rat] :
! [R5: rat] :
( ( ord_less @ rat @ ( zero_zero @ rat ) @ R5 )
=> ? [K3: nat] :
! [M3: nat] :
( ( ord_less_eq @ nat @ K3 @ M3 )
=> ! [N2: nat] :
( ( ord_less_eq @ nat @ K3 @ N2 )
=> ( ord_less @ rat @ ( abs_abs @ rat @ ( minus_minus @ rat @ ( X9 @ M3 ) @ ( X9 @ N2 ) ) ) @ R5 ) ) ) ) ) ) ).
% cauchy_def
thf(fact_8018_le__Real,axiom,
! [X7: nat > rat,Y8: nat > rat] :
( ( cauchy @ X7 )
=> ( ( cauchy @ Y8 )
=> ( ( ord_less_eq @ real @ ( real2 @ X7 ) @ ( real2 @ Y8 ) )
= ( ! [R5: rat] :
( ( ord_less @ rat @ ( zero_zero @ rat ) @ R5 )
=> ? [K3: nat] :
! [N2: nat] :
( ( ord_less_eq @ nat @ K3 @ N2 )
=> ( ord_less_eq @ rat @ ( X7 @ N2 ) @ ( plus_plus @ rat @ ( Y8 @ N2 ) @ R5 ) ) ) ) ) ) ) ) ).
% le_Real
thf(fact_8019_cauchy__not__vanishes,axiom,
! [X7: nat > rat] :
( ( cauchy @ X7 )
=> ( ~ ( vanishes @ X7 )
=> ? [B4: rat] :
( ( ord_less @ rat @ ( zero_zero @ rat ) @ B4 )
& ? [K: nat] :
! [N5: nat] :
( ( ord_less_eq @ nat @ K @ N5 )
=> ( ord_less @ rat @ B4 @ ( abs_abs @ rat @ ( X7 @ N5 ) ) ) ) ) ) ) ).
% cauchy_not_vanishes
thf(fact_8020_vanishes__mult__bounded,axiom,
! [X7: nat > rat,Y8: nat > rat] :
( ? [A11: rat] :
( ( ord_less @ rat @ ( zero_zero @ rat ) @ A11 )
& ! [N3: nat] : ( ord_less @ rat @ ( abs_abs @ rat @ ( X7 @ N3 ) ) @ A11 ) )
=> ( ( vanishes @ Y8 )
=> ( vanishes
@ ^ [N2: nat] : ( times_times @ rat @ ( X7 @ N2 ) @ ( Y8 @ N2 ) ) ) ) ) ).
% vanishes_mult_bounded
thf(fact_8021_vanishes__def,axiom,
( vanishes
= ( ^ [X9: nat > rat] :
! [R5: rat] :
( ( ord_less @ rat @ ( zero_zero @ rat ) @ R5 )
=> ? [K3: nat] :
! [N2: nat] :
( ( ord_less_eq @ nat @ K3 @ N2 )
=> ( ord_less @ rat @ ( abs_abs @ rat @ ( X9 @ N2 ) ) @ R5 ) ) ) ) ) ).
% vanishes_def
thf(fact_8022_vanishesI,axiom,
! [X7: nat > rat] :
( ! [R3: rat] :
( ( ord_less @ rat @ ( zero_zero @ rat ) @ R3 )
=> ? [K4: nat] :
! [N3: nat] :
( ( ord_less_eq @ nat @ K4 @ N3 )
=> ( ord_less @ rat @ ( abs_abs @ rat @ ( X7 @ N3 ) ) @ R3 ) ) )
=> ( vanishes @ X7 ) ) ).
% vanishesI
thf(fact_8023_vanishesD,axiom,
! [X7: nat > rat,R2: rat] :
( ( vanishes @ X7 )
=> ( ( ord_less @ rat @ ( zero_zero @ rat ) @ R2 )
=> ? [K: nat] :
! [N5: nat] :
( ( ord_less_eq @ nat @ K @ N5 )
=> ( ord_less @ rat @ ( abs_abs @ rat @ ( X7 @ N5 ) ) @ R2 ) ) ) ) ).
% vanishesD
thf(fact_8024_cauchy__not__vanishes__cases,axiom,
! [X7: nat > rat] :
( ( cauchy @ X7 )
=> ( ~ ( vanishes @ X7 )
=> ? [B4: rat] :
( ( ord_less @ rat @ ( zero_zero @ rat ) @ B4 )
& ? [K: nat] :
( ! [N5: nat] :
( ( ord_less_eq @ nat @ K @ N5 )
=> ( ord_less @ rat @ B4 @ ( uminus_uminus @ rat @ ( X7 @ N5 ) ) ) )
| ! [N5: nat] :
( ( ord_less_eq @ nat @ K @ N5 )
=> ( ord_less @ rat @ B4 @ ( X7 @ N5 ) ) ) ) ) ) ) ).
% cauchy_not_vanishes_cases
thf(fact_8025_not__positive__Real,axiom,
! [X7: nat > rat] :
( ( cauchy @ X7 )
=> ( ( ~ ( positive2 @ ( real2 @ X7 ) ) )
= ( ! [R5: rat] :
( ( ord_less @ rat @ ( zero_zero @ rat ) @ R5 )
=> ? [K3: nat] :
! [N2: nat] :
( ( ord_less_eq @ nat @ K3 @ N2 )
=> ( ord_less_eq @ rat @ ( X7 @ N2 ) @ R5 ) ) ) ) ) ) ).
% not_positive_Real
thf(fact_8026_positive__Real,axiom,
! [X7: nat > rat] :
( ( cauchy @ X7 )
=> ( ( positive2 @ ( real2 @ X7 ) )
= ( ? [R5: rat] :
( ( ord_less @ rat @ ( zero_zero @ rat ) @ R5 )
& ? [K3: nat] :
! [N2: nat] :
( ( ord_less_eq @ nat @ K3 @ N2 )
=> ( ord_less @ rat @ R5 @ ( X7 @ N2 ) ) ) ) ) ) ) ).
% positive_Real
thf(fact_8027_less__real__def,axiom,
( ( ord_less @ real )
= ( ^ [X5: real,Y6: real] : ( positive2 @ ( minus_minus @ real @ Y6 @ X5 ) ) ) ) ).
% less_real_def
thf(fact_8028_Real_Opositive_Orep__eq,axiom,
( positive2
= ( ^ [X5: real] :
? [R5: rat] :
( ( ord_less @ rat @ ( zero_zero @ rat ) @ R5 )
& ? [K3: nat] :
! [N2: nat] :
( ( ord_less_eq @ nat @ K3 @ N2 )
=> ( ord_less @ rat @ R5 @ ( rep_real @ X5 @ N2 ) ) ) ) ) ) ).
% Real.positive.rep_eq
thf(fact_8029_finite__def,axiom,
! [A: $tType] :
( ( finite_finite2 @ A )
= ( complete_lattice_lfp @ ( ( set @ A ) > $o )
@ ^ [P6: ( set @ A ) > $o,X5: set @ A] :
( ( X5
= ( bot_bot @ ( set @ A ) ) )
| ? [A7: set @ A,A6: A] :
( ( X5
= ( insert @ A @ A6 @ A7 ) )
& ( P6 @ A7 ) ) ) ) ) ).
% finite_def
thf(fact_8030_lfp__funpow,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F3: A > A,N: nat] :
( ( order_mono @ A @ A @ F3 )
=> ( ( complete_lattice_lfp @ A @ ( compow @ ( A > A ) @ ( suc @ N ) @ F3 ) )
= ( complete_lattice_lfp @ A @ F3 ) ) ) ) ).
% lfp_funpow
thf(fact_8031_lfp__Kleene__iter,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F3: A > A,K2: nat] :
( ( order_mono @ A @ A @ F3 )
=> ( ( ( compow @ ( A > A ) @ ( suc @ K2 ) @ F3 @ ( bot_bot @ A ) )
= ( compow @ ( A > A ) @ K2 @ F3 @ ( bot_bot @ A ) ) )
=> ( ( complete_lattice_lfp @ A @ F3 )
= ( compow @ ( A > A ) @ K2 @ F3 @ ( bot_bot @ A ) ) ) ) ) ) ).
% lfp_Kleene_iter
thf(fact_8032_lfp__ordinal__induct,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F3: A > A,P2: A > $o] :
( ( order_mono @ A @ A @ F3 )
=> ( ! [S6: A] :
( ( P2 @ S6 )
=> ( ( ord_less_eq @ A @ S6 @ ( complete_lattice_lfp @ A @ F3 ) )
=> ( P2 @ ( F3 @ S6 ) ) ) )
=> ( ! [M8: set @ A] :
( ! [X3: A] :
( ( member @ A @ X3 @ M8 )
=> ( P2 @ X3 ) )
=> ( P2 @ ( complete_Sup_Sup @ A @ M8 ) ) )
=> ( P2 @ ( complete_lattice_lfp @ A @ F3 ) ) ) ) ) ) ).
% lfp_ordinal_induct
thf(fact_8033_def__lfp__induct,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [A5: A,F3: A > A,P2: A] :
( ( A5
= ( complete_lattice_lfp @ A @ F3 ) )
=> ( ( order_mono @ A @ A @ F3 )
=> ( ( ord_less_eq @ A @ ( F3 @ ( inf_inf @ A @ A5 @ P2 ) ) @ P2 )
=> ( ord_less_eq @ A @ A5 @ P2 ) ) ) ) ) ).
% def_lfp_induct
thf(fact_8034_le__rel__bool__arg__iff,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( ord_less_eq @ ( $o > A ) )
= ( ^ [X9: $o > A,Y10: $o > A] :
( ( ord_less_eq @ A @ ( X9 @ $false ) @ ( Y10 @ $false ) )
& ( ord_less_eq @ A @ ( X9 @ $true ) @ ( Y10 @ $true ) ) ) ) ) ) ).
% le_rel_bool_arg_iff
thf(fact_8035_lfp__induct2,axiom,
! [A: $tType,B: $tType,A2: A,B2: B,F3: ( set @ ( product_prod @ A @ B ) ) > ( set @ ( product_prod @ A @ B ) ),P2: A > B > $o] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ ( complete_lattice_lfp @ ( set @ ( product_prod @ A @ B ) ) @ F3 ) )
=> ( ( order_mono @ ( set @ ( product_prod @ A @ B ) ) @ ( set @ ( product_prod @ A @ B ) ) @ F3 )
=> ( ! [A4: A,B4: B] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A4 @ B4 ) @ ( F3 @ ( inf_inf @ ( set @ ( product_prod @ A @ B ) ) @ ( complete_lattice_lfp @ ( set @ ( product_prod @ A @ B ) ) @ F3 ) @ ( collect @ ( product_prod @ A @ B ) @ ( product_case_prod @ A @ B @ $o @ P2 ) ) ) ) )
=> ( P2 @ A4 @ B4 ) )
=> ( P2 @ A2 @ B2 ) ) ) ) ).
% lfp_induct2
thf(fact_8036_lfp__greatest,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F3: A > A,A5: A] :
( ! [U4: A] :
( ( ord_less_eq @ A @ ( F3 @ U4 ) @ U4 )
=> ( ord_less_eq @ A @ A5 @ U4 ) )
=> ( ord_less_eq @ A @ A5 @ ( complete_lattice_lfp @ A @ F3 ) ) ) ) ).
% lfp_greatest
thf(fact_8037_lfp__lowerbound,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F3: A > A,A5: A] :
( ( ord_less_eq @ A @ ( F3 @ A5 ) @ A5 )
=> ( ord_less_eq @ A @ ( complete_lattice_lfp @ A @ F3 ) @ A5 ) ) ) ).
% lfp_lowerbound
thf(fact_8038_lfp__mono,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F3: A > A,G3: A > A] :
( ! [Z9: A] : ( ord_less_eq @ A @ ( F3 @ Z9 ) @ ( G3 @ Z9 ) )
=> ( ord_less_eq @ A @ ( complete_lattice_lfp @ A @ F3 ) @ ( complete_lattice_lfp @ A @ G3 ) ) ) ) ).
% lfp_mono
thf(fact_8039_lfp__lfp,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F3: A > A > A] :
( ! [X4: A,Y3: A,W: A,Z3: A] :
( ( ord_less_eq @ A @ X4 @ Y3 )
=> ( ( ord_less_eq @ A @ W @ Z3 )
=> ( ord_less_eq @ A @ ( F3 @ X4 @ W ) @ ( F3 @ Y3 @ Z3 ) ) ) )
=> ( ( complete_lattice_lfp @ A
@ ^ [X5: A] : ( complete_lattice_lfp @ A @ ( F3 @ X5 ) ) )
= ( complete_lattice_lfp @ A
@ ^ [X5: A] : ( F3 @ X5 @ X5 ) ) ) ) ) ).
% lfp_lfp
thf(fact_8040_lfp__eqI,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F5: A > A,X: A] :
( ( order_mono @ A @ A @ F5 )
=> ( ( ( F5 @ X )
= X )
=> ( ! [Z3: A] :
( ( ( F5 @ Z3 )
= Z3 )
=> ( ord_less_eq @ A @ X @ Z3 ) )
=> ( ( complete_lattice_lfp @ A @ F5 )
= X ) ) ) ) ) ).
% lfp_eqI
thf(fact_8041_lfp__def,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ( ( complete_lattice_lfp @ A )
= ( ^ [F4: A > A] :
( complete_Inf_Inf @ A
@ ( collect @ A
@ ^ [U2: A] : ( ord_less_eq @ A @ ( F4 @ U2 ) @ U2 ) ) ) ) ) ) ).
% lfp_def
thf(fact_8042_lfp__induct,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F3: A > A,P2: A] :
( ( order_mono @ A @ A @ F3 )
=> ( ( ord_less_eq @ A @ ( F3 @ ( inf_inf @ A @ ( complete_lattice_lfp @ A @ F3 ) @ P2 ) ) @ P2 )
=> ( ord_less_eq @ A @ ( complete_lattice_lfp @ A @ F3 ) @ P2 ) ) ) ) ).
% lfp_induct
thf(fact_8043_iteratesp__def,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ( ( comple7512665784863727008ratesp @ A )
= ( ^ [F4: A > A] :
( complete_lattice_lfp @ ( A > $o )
@ ^ [P6: A > $o,X5: A] :
( ? [Y6: A] :
( ( X5
= ( F4 @ Y6 ) )
& ( P6 @ Y6 ) )
| ? [M9: set @ A] :
( ( X5
= ( complete_Sup_Sup @ A @ M9 ) )
& ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ M9 )
& ! [Y6: A] :
( ( member @ A @ Y6 @ M9 )
=> ( P6 @ Y6 ) ) ) ) ) ) ) ) ).
% iteratesp_def
thf(fact_8044_Real_Opositive__def,axiom,
( positive2
= ( map_fun @ real @ ( nat > rat ) @ $o @ $o @ rep_real @ ( id @ $o )
@ ^ [X9: nat > rat] :
? [R5: rat] :
( ( ord_less @ rat @ ( zero_zero @ rat ) @ R5 )
& ? [K3: nat] :
! [N2: nat] :
( ( ord_less_eq @ nat @ K3 @ N2 )
=> ( ord_less @ rat @ R5 @ ( X9 @ N2 ) ) ) ) ) ) ).
% Real.positive_def
thf(fact_8045_iteratesp_OSup,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ! [M5: set @ A,F3: A > A] :
( ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ M5 )
=> ( ! [X4: A] :
( ( member @ A @ X4 @ M5 )
=> ( comple7512665784863727008ratesp @ A @ F3 @ X4 ) )
=> ( comple7512665784863727008ratesp @ A @ F3 @ ( complete_Sup_Sup @ A @ M5 ) ) ) ) ) ).
% iteratesp.Sup
thf(fact_8046_iteratesp_Ocases,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ! [F3: A > A,A2: A] :
( ( comple7512665784863727008ratesp @ A @ F3 @ A2 )
=> ( ! [X4: A] :
( ( A2
= ( F3 @ X4 ) )
=> ~ ( comple7512665784863727008ratesp @ A @ F3 @ X4 ) )
=> ~ ! [M8: set @ A] :
( ( A2
= ( complete_Sup_Sup @ A @ M8 ) )
=> ( ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ M8 )
=> ~ ! [X3: A] :
( ( member @ A @ X3 @ M8 )
=> ( comple7512665784863727008ratesp @ A @ F3 @ X3 ) ) ) ) ) ) ) ).
% iteratesp.cases
thf(fact_8047_iteratesp_Osimps,axiom,
! [A: $tType] :
( ( comple9053668089753744459l_ccpo @ A )
=> ( ( comple7512665784863727008ratesp @ A )
= ( ^ [F4: A > A,A6: A] :
( ? [X5: A] :
( ( A6
= ( F4 @ X5 ) )
& ( comple7512665784863727008ratesp @ A @ F4 @ X5 ) )
| ? [M9: set @ A] :
( ( A6
= ( complete_Sup_Sup @ A @ M9 ) )
& ( comple1602240252501008431_chain @ A @ ( ord_less_eq @ A ) @ M9 )
& ! [X5: A] :
( ( member @ A @ X5 @ M9 )
=> ( comple7512665784863727008ratesp @ A @ F4 @ X5 ) ) ) ) ) ) ) ).
% iteratesp.simps
thf(fact_8048_lfp__transfer__bounded,axiom,
! [A: $tType,B: $tType] :
( ( ( comple6319245703460814977attice @ B )
& ( comple6319245703460814977attice @ A ) )
=> ! [P2: A > $o,F3: A > A,Alpha: A > B,G3: B > B] :
( ( P2 @ ( bot_bot @ A ) )
=> ( ! [X4: A] :
( ( P2 @ X4 )
=> ( P2 @ ( F3 @ X4 ) ) )
=> ( ! [M8: nat > A] :
( ! [I: nat] : ( P2 @ ( M8 @ I ) )
=> ( P2 @ ( complete_Sup_Sup @ A @ ( image2 @ nat @ A @ M8 @ ( top_top @ ( set @ nat ) ) ) ) ) )
=> ( ! [M8: nat > A] :
( ( order_mono @ nat @ A @ M8 )
=> ( ! [I: nat] : ( P2 @ ( M8 @ I ) )
=> ( ( Alpha @ ( complete_Sup_Sup @ A @ ( image2 @ nat @ A @ M8 @ ( top_top @ ( set @ nat ) ) ) ) )
= ( complete_Sup_Sup @ B
@ ( image2 @ nat @ B
@ ^ [I4: nat] : ( Alpha @ ( M8 @ I4 ) )
@ ( top_top @ ( set @ nat ) ) ) ) ) ) )
=> ( ( order_sup_continuous @ A @ A @ F3 )
=> ( ( order_sup_continuous @ B @ B @ G3 )
=> ( ! [X4: A] :
( ( P2 @ X4 )
=> ( ( ord_less_eq @ A @ X4 @ ( complete_lattice_lfp @ A @ F3 ) )
=> ( ( Alpha @ ( F3 @ X4 ) )
= ( G3 @ ( Alpha @ X4 ) ) ) ) )
=> ( ! [X4: B] : ( ord_less_eq @ B @ ( Alpha @ ( bot_bot @ A ) ) @ ( G3 @ X4 ) )
=> ( ( Alpha @ ( complete_lattice_lfp @ A @ F3 ) )
= ( complete_lattice_lfp @ B @ G3 ) ) ) ) ) ) ) ) ) ) ) ).
% lfp_transfer_bounded
thf(fact_8049_sup__continuous__lfp,axiom,
! [A: $tType] :
( ( comple6319245703460814977attice @ A )
=> ! [F5: A > A] :
( ( order_sup_continuous @ A @ A @ F5 )
=> ( ( complete_lattice_lfp @ A @ F5 )
= ( complete_Sup_Sup @ A
@ ( image2 @ nat @ A
@ ^ [I4: nat] : ( compow @ ( A > A ) @ I4 @ F5 @ ( bot_bot @ A ) )
@ ( top_top @ ( set @ nat ) ) ) ) ) ) ) ).
% sup_continuous_lfp
thf(fact_8050_lfp__transfer,axiom,
! [A: $tType,B: $tType] :
( ( ( comple6319245703460814977attice @ B )
& ( comple6319245703460814977attice @ A ) )
=> ! [Alpha: A > B,F3: A > A,G3: B > B] :
( ( order_sup_continuous @ A @ B @ Alpha )
=> ( ( order_sup_continuous @ A @ A @ F3 )
=> ( ( order_sup_continuous @ B @ B @ G3 )
=> ( ! [X4: B] : ( ord_less_eq @ B @ ( Alpha @ ( bot_bot @ A ) ) @ ( G3 @ X4 ) )
=> ( ! [X4: A] :
( ( ord_less_eq @ A @ X4 @ ( complete_lattice_lfp @ A @ F3 ) )
=> ( ( Alpha @ ( F3 @ X4 ) )
= ( G3 @ ( Alpha @ X4 ) ) ) )
=> ( ( Alpha @ ( complete_lattice_lfp @ A @ F3 ) )
= ( complete_lattice_lfp @ B @ G3 ) ) ) ) ) ) ) ) ).
% lfp_transfer
thf(fact_8051_cclfp__transfer,axiom,
! [A: $tType,B: $tType] :
( ( ( counta3822494911875563373attice @ B )
& ( counta3822494911875563373attice @ A ) )
=> ! [Alpha: A > B,F3: A > A,G3: B > B] :
( ( order_sup_continuous @ A @ B @ Alpha )
=> ( ( order_mono @ A @ A @ F3 )
=> ( ( ( Alpha @ ( bot_bot @ A ) )
= ( bot_bot @ B ) )
=> ( ! [X4: A] :
( ( Alpha @ ( F3 @ X4 ) )
= ( G3 @ ( Alpha @ X4 ) ) )
=> ( ( Alpha @ ( order_532582986084564980_cclfp @ A @ F3 ) )
= ( order_532582986084564980_cclfp @ B @ G3 ) ) ) ) ) ) ) ).
% cclfp_transfer
thf(fact_8052_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,X15: list @ A,X23: list @ A] :
( ? [Y6: A,Ys3: list @ A] :
( ( X15
= ( nil @ A ) )
& ( X23
= ( cons @ A @ Y6 @ Ys3 ) ) )
| ? [X5: A,Y6: A,Xs3: list @ A,Ys3: list @ A] :
( ( X15
= ( cons @ A @ X5 @ Xs3 ) )
& ( X23
= ( cons @ A @ Y6 @ Ys3 ) )
& ( ord_less @ A @ X5 @ Y6 ) )
| ? [X5: A,Y6: A,Xs3: list @ A,Ys3: list @ A] :
( ( X15
= ( cons @ A @ X5 @ Xs3 ) )
& ( X23
= ( cons @ A @ Y6 @ Ys3 ) )
& ~ ( ord_less @ A @ X5 @ Y6 )
& ~ ( ord_less @ A @ Y6 @ X5 )
& ( P6 @ Xs3 @ Ys3 ) ) ) ) ) ) ).
% ord_class.lexordp_def
thf(fact_8053_finite__refines__card__le,axiom,
! [A: $tType,A5: set @ A,R: set @ ( product_prod @ A @ A ),S2: set @ ( product_prod @ A @ A )] :
( ( finite_finite2 @ ( set @ A ) @ ( equiv_quotient @ A @ A5 @ R ) )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R @ S2 )
=> ( ( equiv_equiv @ A @ A5 @ R )
=> ( ( equiv_equiv @ A @ A5 @ S2 )
=> ( ord_less_eq @ nat @ ( finite_card @ ( set @ A ) @ ( equiv_quotient @ A @ A5 @ S2 ) ) @ ( finite_card @ ( set @ A ) @ ( equiv_quotient @ A @ A5 @ R ) ) ) ) ) ) ) ).
% finite_refines_card_le
thf(fact_8054_lexordp__simps_I1_J,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [Ys: list @ A] :
( ( ord_lexordp @ A @ ( nil @ A ) @ Ys )
= ( Ys
!= ( nil @ A ) ) ) ) ).
% lexordp_simps(1)
thf(fact_8055_lexordp__simps_I2_J,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [Xs: list @ A] :
~ ( ord_lexordp @ A @ Xs @ ( nil @ A ) ) ) ).
% lexordp_simps(2)
thf(fact_8056_lexordp__simps_I3_J,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X: A,Xs: list @ A,Y: A,Ys: list @ A] :
( ( ord_lexordp @ A @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Ys ) )
= ( ( ord_less @ A @ X @ Y )
| ( ~ ( ord_less @ A @ Y @ X )
& ( ord_lexordp @ A @ Xs @ Ys ) ) ) ) ) ).
% lexordp_simps(3)
thf(fact_8057_quotient__disj,axiom,
! [A: $tType,A5: set @ A,R2: set @ ( product_prod @ A @ A ),X7: set @ A,Y8: set @ A] :
( ( equiv_equiv @ A @ A5 @ R2 )
=> ( ( member @ ( set @ A ) @ X7 @ ( equiv_quotient @ A @ A5 @ R2 ) )
=> ( ( member @ ( set @ A ) @ Y8 @ ( equiv_quotient @ A @ A5 @ R2 ) )
=> ( ( X7 = Y8 )
| ( ( inf_inf @ ( set @ A ) @ X7 @ Y8 )
= ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% quotient_disj
thf(fact_8058_equiv__listrel,axiom,
! [A: $tType,A5: set @ A,R2: set @ ( product_prod @ A @ A )] :
( ( equiv_equiv @ A @ A5 @ R2 )
=> ( equiv_equiv @ ( list @ A ) @ ( lists @ A @ A5 ) @ ( listrel @ A @ A @ R2 ) ) ) ).
% equiv_listrel
thf(fact_8059_lexordp__append__rightI,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [Ys: list @ A,Xs: list @ A] :
( ( Ys
!= ( nil @ A ) )
=> ( ord_lexordp @ A @ Xs @ ( append @ A @ Xs @ Ys ) ) ) ) ).
% lexordp_append_rightI
thf(fact_8060_lexordp__append__leftD,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [Xs: list @ A,Us: list @ A,Vs2: list @ A] :
( ( ord_lexordp @ A @ ( append @ A @ Xs @ Us ) @ ( append @ A @ Xs @ Vs2 ) )
=> ( ! [A4: A] :
~ ( ord_less @ A @ A4 @ A4 )
=> ( ord_lexordp @ A @ Us @ Vs2 ) ) ) ) ).
% lexordp_append_leftD
thf(fact_8061_lexordp__append__leftI,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [Us: list @ A,Vs2: list @ A,Xs: list @ A] :
( ( ord_lexordp @ A @ Us @ Vs2 )
=> ( ord_lexordp @ A @ ( append @ A @ Xs @ Us ) @ ( append @ A @ Xs @ Vs2 ) ) ) ) ).
% lexordp_append_leftI
thf(fact_8062_lexordp__antisym,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Xs: list @ A,Ys: list @ A] :
( ( ord_lexordp @ A @ Xs @ Ys )
=> ~ ( ord_lexordp @ A @ Ys @ Xs ) ) ) ).
% lexordp_antisym
thf(fact_8063_lexordp__trans,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,Ys: list @ A,Zs: list @ A] :
( ( ord_lexordp @ A @ Xs @ Ys )
=> ( ( ord_lexordp @ A @ Ys @ Zs )
=> ( ord_lexordp @ A @ Xs @ Zs ) ) ) ) ).
% lexordp_trans
thf(fact_8064_lexordp__linear,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,Ys: list @ A] :
( ( ord_lexordp @ A @ Xs @ Ys )
| ( Xs = Ys )
| ( ord_lexordp @ A @ Ys @ Xs ) ) ) ).
% lexordp_linear
thf(fact_8065_lexordp__irreflexive_H,axiom,
! [A: $tType] :
( ( order @ A )
=> ! [Xs: list @ A] :
~ ( ord_lexordp @ A @ Xs @ Xs ) ) ).
% lexordp_irreflexive'
thf(fact_8066_lexordp__irreflexive,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [Xs: list @ A] :
( ! [X4: A] :
~ ( ord_less @ A @ X4 @ X4 )
=> ~ ( ord_lexordp @ A @ Xs @ Xs ) ) ) ).
% lexordp_irreflexive
thf(fact_8067_lexordp_OCons__eq,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X: A,Y: A,Xs: list @ A,Ys: list @ A] :
( ~ ( ord_less @ A @ X @ Y )
=> ( ~ ( ord_less @ A @ Y @ X )
=> ( ( ord_lexordp @ A @ Xs @ Ys )
=> ( ord_lexordp @ A @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Ys ) ) ) ) ) ) ).
% lexordp.Cons_eq
thf(fact_8068_lexordp_OCons,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X: A,Y: A,Xs: list @ A,Ys: list @ A] :
( ( ord_less @ A @ X @ Y )
=> ( ord_lexordp @ A @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Ys ) ) ) ) ).
% lexordp.Cons
thf(fact_8069_lexordp_ONil,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [Y: A,Ys: list @ A] : ( ord_lexordp @ A @ ( nil @ A ) @ ( cons @ A @ Y @ Ys ) ) ) ).
% lexordp.Nil
thf(fact_8070_in__quotient__imp__non__empty,axiom,
! [A: $tType,A5: set @ A,R2: set @ ( product_prod @ A @ A ),X7: set @ A] :
( ( equiv_equiv @ A @ A5 @ R2 )
=> ( ( member @ ( set @ A ) @ X7 @ ( equiv_quotient @ A @ A5 @ R2 ) )
=> ( X7
!= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% in_quotient_imp_non_empty
thf(fact_8071_in__quotient__imp__closed,axiom,
! [A: $tType,A5: set @ A,R2: set @ ( product_prod @ A @ A ),X7: set @ A,X: A,Y: A] :
( ( equiv_equiv @ A @ A5 @ R2 )
=> ( ( member @ ( set @ A ) @ X7 @ ( equiv_quotient @ A @ A5 @ R2 ) )
=> ( ( member @ A @ X @ X7 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R2 )
=> ( member @ A @ Y @ X7 ) ) ) ) ) ).
% in_quotient_imp_closed
thf(fact_8072_quotient__eq__iff,axiom,
! [A: $tType,A5: set @ A,R2: set @ ( product_prod @ A @ A ),X7: set @ A,Y8: set @ A,X: A,Y: A] :
( ( equiv_equiv @ A @ A5 @ R2 )
=> ( ( member @ ( set @ A ) @ X7 @ ( equiv_quotient @ A @ A5 @ R2 ) )
=> ( ( member @ ( set @ A ) @ Y8 @ ( equiv_quotient @ A @ A5 @ R2 ) )
=> ( ( member @ A @ X @ X7 )
=> ( ( member @ A @ Y @ Y8 )
=> ( ( X7 = Y8 )
= ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R2 ) ) ) ) ) ) ) ).
% quotient_eq_iff
thf(fact_8073_quotient__eqI,axiom,
! [A: $tType,A5: set @ A,R2: set @ ( product_prod @ A @ A ),X7: set @ A,Y8: set @ A,X: A,Y: A] :
( ( equiv_equiv @ A @ A5 @ R2 )
=> ( ( member @ ( set @ A ) @ X7 @ ( equiv_quotient @ A @ A5 @ R2 ) )
=> ( ( member @ ( set @ A ) @ Y8 @ ( equiv_quotient @ A @ A5 @ R2 ) )
=> ( ( member @ A @ X @ X7 )
=> ( ( member @ A @ Y @ Y8 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R2 )
=> ( X7 = Y8 ) ) ) ) ) ) ) ).
% quotient_eqI
thf(fact_8074_lexordp_Ocases,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [A1: list @ A,A22: list @ A] :
( ( ord_lexordp @ A @ A1 @ A22 )
=> ( ( ( A1
= ( nil @ A ) )
=> ! [Y3: A,Ys4: list @ A] :
( A22
!= ( cons @ A @ Y3 @ Ys4 ) ) )
=> ( ! [X4: A] :
( ? [Xs2: list @ A] :
( A1
= ( cons @ A @ X4 @ Xs2 ) )
=> ! [Y3: A] :
( ? [Ys4: list @ A] :
( A22
= ( cons @ A @ Y3 @ Ys4 ) )
=> ~ ( ord_less @ A @ X4 @ Y3 ) ) )
=> ~ ! [X4: A,Y3: A,Xs2: list @ A] :
( ( A1
= ( cons @ A @ X4 @ Xs2 ) )
=> ! [Ys4: list @ A] :
( ( A22
= ( cons @ A @ Y3 @ Ys4 ) )
=> ( ~ ( ord_less @ A @ X4 @ Y3 )
=> ( ~ ( ord_less @ A @ Y3 @ X4 )
=> ~ ( ord_lexordp @ A @ Xs2 @ Ys4 ) ) ) ) ) ) ) ) ) ).
% lexordp.cases
thf(fact_8075_lexordp_Osimps,axiom,
! [A: $tType] :
( ( ord @ A )
=> ( ( ord_lexordp @ A )
= ( ^ [A12: list @ A,A23: list @ A] :
( ? [Y6: A,Ys3: list @ A] :
( ( A12
= ( nil @ A ) )
& ( A23
= ( cons @ A @ Y6 @ Ys3 ) ) )
| ? [X5: A,Y6: A,Xs3: list @ A,Ys3: list @ A] :
( ( A12
= ( cons @ A @ X5 @ Xs3 ) )
& ( A23
= ( cons @ A @ Y6 @ Ys3 ) )
& ( ord_less @ A @ X5 @ Y6 ) )
| ? [X5: A,Y6: A,Xs3: list @ A,Ys3: list @ A] :
( ( A12
= ( cons @ A @ X5 @ Xs3 ) )
& ( A23
= ( cons @ A @ Y6 @ Ys3 ) )
& ~ ( ord_less @ A @ X5 @ Y6 )
& ~ ( ord_less @ A @ Y6 @ X5 )
& ( ord_lexordp @ A @ Xs3 @ Ys3 ) ) ) ) ) ) ).
% lexordp.simps
thf(fact_8076_lexordp__cases,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,Ys: list @ A] :
( ( ord_lexordp @ A @ Xs @ Ys )
=> ( ( ( Xs
= ( nil @ A ) )
=> ! [Y3: A,Ys5: list @ A] :
( Ys
!= ( cons @ A @ Y3 @ Ys5 ) ) )
=> ( ! [X4: A] :
( ? [Xs4: list @ A] :
( Xs
= ( cons @ A @ X4 @ Xs4 ) )
=> ! [Y3: A] :
( ? [Ys5: list @ A] :
( Ys
= ( cons @ A @ Y3 @ Ys5 ) )
=> ~ ( ord_less @ A @ X4 @ Y3 ) ) )
=> ~ ! [X4: A,Xs4: list @ A] :
( ( Xs
= ( cons @ A @ X4 @ Xs4 ) )
=> ! [Ys5: list @ A] :
( ( Ys
= ( cons @ A @ X4 @ Ys5 ) )
=> ~ ( ord_lexordp @ A @ Xs4 @ Ys5 ) ) ) ) ) ) ) ).
% lexordp_cases
thf(fact_8077_lexordp__induct,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [Xs: list @ A,Ys: list @ A,P2: ( list @ A ) > ( list @ A ) > $o] :
( ( ord_lexordp @ A @ Xs @ Ys )
=> ( ! [Y3: A,Ys4: list @ A] : ( P2 @ ( nil @ A ) @ ( cons @ A @ Y3 @ Ys4 ) )
=> ( ! [X4: A,Xs2: list @ A,Y3: A,Ys4: list @ A] :
( ( ord_less @ A @ X4 @ Y3 )
=> ( P2 @ ( cons @ A @ X4 @ Xs2 ) @ ( cons @ A @ Y3 @ Ys4 ) ) )
=> ( ! [X4: A,Xs2: list @ A,Ys4: list @ A] :
( ( ord_lexordp @ A @ Xs2 @ Ys4 )
=> ( ( P2 @ Xs2 @ Ys4 )
=> ( P2 @ ( cons @ A @ X4 @ Xs2 ) @ ( cons @ A @ X4 @ Ys4 ) ) ) )
=> ( P2 @ Xs @ Ys ) ) ) ) ) ) ).
% lexordp_induct
thf(fact_8078_lexordp__append__left__rightI,axiom,
! [A: $tType] :
( ( ord @ A )
=> ! [X: A,Y: A,Us: list @ A,Xs: list @ A,Ys: list @ A] :
( ( ord_less @ A @ X @ Y )
=> ( ord_lexordp @ A @ ( append @ A @ Us @ ( cons @ A @ X @ Xs ) ) @ ( append @ A @ Us @ ( cons @ A @ Y @ Ys ) ) ) ) ) ).
% lexordp_append_left_rightI
thf(fact_8079_lexordp__iff,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ( ( ord_lexordp @ A )
= ( ^ [Xs3: list @ A,Ys3: list @ A] :
( ? [X5: A,Vs: list @ A] :
( Ys3
= ( append @ A @ Xs3 @ ( cons @ A @ X5 @ Vs ) ) )
| ? [Us2: list @ A,A6: A,B5: A,Vs: list @ A,Ws3: list @ A] :
( ( ord_less @ A @ A6 @ B5 )
& ( Xs3
= ( append @ A @ Us2 @ ( cons @ A @ A6 @ Vs ) ) )
& ( Ys3
= ( append @ A @ Us2 @ ( cons @ A @ B5 @ Ws3 ) ) ) ) ) ) ) ) ).
% lexordp_iff
thf(fact_8080_finite__refines__finite,axiom,
! [A: $tType,A5: set @ A,R: set @ ( product_prod @ A @ A ),S2: set @ ( product_prod @ A @ A )] :
( ( finite_finite2 @ ( set @ A ) @ ( equiv_quotient @ A @ A5 @ R ) )
=> ( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R @ S2 )
=> ( ( equiv_equiv @ A @ A5 @ R )
=> ( ( equiv_equiv @ A @ A5 @ S2 )
=> ( finite_finite2 @ ( set @ A ) @ ( equiv_quotient @ A @ A5 @ S2 ) ) ) ) ) ) ).
% finite_refines_finite
thf(fact_8081_eq__equiv__class__iff2,axiom,
! [A: $tType,A5: set @ A,R2: set @ ( product_prod @ A @ A ),X: A,Y: A] :
( ( equiv_equiv @ A @ A5 @ R2 )
=> ( ( member @ A @ X @ A5 )
=> ( ( member @ A @ Y @ A5 )
=> ( ( ( equiv_quotient @ A @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) @ R2 )
= ( equiv_quotient @ A @ ( insert @ A @ Y @ ( bot_bot @ ( set @ A ) ) ) @ R2 ) )
= ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R2 ) ) ) ) ) ).
% eq_equiv_class_iff2
thf(fact_8082_equiv__imp__dvd__card,axiom,
! [A: $tType,A5: set @ A,R2: set @ ( product_prod @ A @ A ),K2: nat] :
( ( finite_finite2 @ A @ A5 )
=> ( ( equiv_equiv @ A @ A5 @ R2 )
=> ( ! [X17: set @ A] :
( ( member @ ( set @ A ) @ X17 @ ( equiv_quotient @ A @ A5 @ R2 ) )
=> ( dvd_dvd @ nat @ K2 @ ( finite_card @ A @ X17 ) ) )
=> ( dvd_dvd @ nat @ K2 @ ( finite_card @ A @ A5 ) ) ) ) ) ).
% equiv_imp_dvd_card
thf(fact_8083_in__quotient__imp__in__rel,axiom,
! [A: $tType,A5: set @ A,R2: set @ ( product_prod @ A @ A ),X7: set @ A,X: A,Y: A] :
( ( equiv_equiv @ A @ A5 @ R2 )
=> ( ( member @ ( set @ A ) @ X7 @ ( equiv_quotient @ A @ A5 @ R2 ) )
=> ( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X @ ( insert @ A @ Y @ ( bot_bot @ ( set @ A ) ) ) ) @ X7 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R2 ) ) ) ) ).
% in_quotient_imp_in_rel
thf(fact_8084_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_8085_UN__equiv__class__inject,axiom,
! [B: $tType,A: $tType,A5: set @ A,R2: set @ ( product_prod @ A @ A ),F3: A > ( set @ B ),X7: set @ A,Y8: set @ A] :
( ( equiv_equiv @ A @ A5 @ R2 )
=> ( ( equiv_congruent @ A @ ( set @ B ) @ R2 @ F3 )
=> ( ( ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ F3 @ X7 ) )
= ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ F3 @ Y8 ) ) )
=> ( ( member @ ( set @ A ) @ X7 @ ( equiv_quotient @ A @ A5 @ R2 ) )
=> ( ( member @ ( set @ A ) @ Y8 @ ( equiv_quotient @ A @ A5 @ R2 ) )
=> ( ! [X4: A,Y3: A] :
( ( member @ A @ X4 @ A5 )
=> ( ( member @ A @ Y3 @ A5 )
=> ( ( ( F3 @ X4 )
= ( F3 @ Y3 ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y3 ) @ R2 ) ) ) )
=> ( X7 = Y8 ) ) ) ) ) ) ) ).
% UN_equiv_class_inject
thf(fact_8086_proj__iff,axiom,
! [A: $tType,A5: set @ A,R2: set @ ( product_prod @ A @ A ),X: A,Y: A] :
( ( equiv_equiv @ A @ A5 @ R2 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( insert @ A @ X @ ( insert @ A @ Y @ ( bot_bot @ ( set @ A ) ) ) ) @ A5 )
=> ( ( ( equiv_proj @ A @ A @ R2 @ X )
= ( equiv_proj @ A @ A @ R2 @ Y ) )
= ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R2 ) ) ) ) ).
% proj_iff
thf(fact_8087_congruentI,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ A ),F3: A > B] :
( ! [Y3: A,Z3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z3 ) @ R2 )
=> ( ( F3 @ Y3 )
= ( F3 @ Z3 ) ) )
=> ( equiv_congruent @ A @ B @ R2 @ F3 ) ) ).
% congruentI
thf(fact_8088_congruentD,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ A @ A ),F3: A > B,Y: A,Z: A] :
( ( equiv_congruent @ A @ B @ R2 @ F3 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y @ Z ) @ R2 )
=> ( ( F3 @ Y )
= ( F3 @ Z ) ) ) ) ).
% congruentD
thf(fact_8089_ord_Olexordp__def,axiom,
! [A: $tType] :
( ( lexordp2 @ A )
= ( ^ [Less2: A > A > $o] :
( complete_lattice_lfp @ ( ( list @ A ) > ( list @ A ) > $o )
@ ^ [P6: ( list @ A ) > ( list @ A ) > $o,X15: list @ A,X23: list @ A] :
( ? [Y6: A,Ys3: list @ A] :
( ( X15
= ( nil @ A ) )
& ( X23
= ( cons @ A @ Y6 @ Ys3 ) ) )
| ? [X5: A,Y6: A,Xs3: list @ A,Ys3: list @ A] :
( ( X15
= ( cons @ A @ X5 @ Xs3 ) )
& ( X23
= ( cons @ A @ Y6 @ Ys3 ) )
& ( Less2 @ X5 @ Y6 ) )
| ? [X5: A,Y6: A,Xs3: list @ A,Ys3: list @ A] :
( ( X15
= ( cons @ A @ X5 @ Xs3 ) )
& ( X23
= ( cons @ A @ Y6 @ Ys3 ) )
& ~ ( Less2 @ X5 @ Y6 )
& ~ ( Less2 @ Y6 @ X5 )
& ( P6 @ Xs3 @ Ys3 ) ) ) ) ) ) ).
% ord.lexordp_def
thf(fact_8090_UN__equiv__class,axiom,
! [B: $tType,A: $tType,A5: set @ A,R2: set @ ( product_prod @ A @ A ),F3: A > ( set @ B ),A2: A] :
( ( equiv_equiv @ A @ A5 @ R2 )
=> ( ( equiv_congruent @ A @ ( set @ B ) @ R2 @ F3 )
=> ( ( member @ A @ A2 @ A5 )
=> ( ( complete_Sup_Sup @ ( set @ B ) @ ( image2 @ A @ ( set @ B ) @ F3 @ ( image @ A @ A @ R2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) )
= ( F3 @ A2 ) ) ) ) ) ).
% UN_equiv_class
thf(fact_8091_ImageI,axiom,
! [B: $tType,A: $tType,A2: A,B2: B,R2: set @ ( product_prod @ A @ B ),A5: set @ A] :
( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ R2 )
=> ( ( member @ A @ A2 @ A5 )
=> ( member @ B @ B2 @ ( image @ A @ B @ R2 @ A5 ) ) ) ) ).
% ImageI
thf(fact_8092_Image__empty2,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ B @ A )] :
( ( image @ B @ A @ R @ ( bot_bot @ ( set @ B ) ) )
= ( bot_bot @ ( set @ A ) ) ) ).
% Image_empty2
thf(fact_8093_ord_Olexordp__simps_I3_J,axiom,
! [A: $tType,Less: A > A > $o,X: A,Xs: list @ A,Y: A,Ys: list @ A] :
( ( lexordp2 @ A @ Less @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Ys ) )
= ( ( Less @ X @ Y )
| ( ~ ( Less @ Y @ X )
& ( lexordp2 @ A @ Less @ Xs @ Ys ) ) ) ) ).
% ord.lexordp_simps(3)
thf(fact_8094_ord_Olexordp__simps_I1_J,axiom,
! [A: $tType,Less: A > A > $o,Ys: list @ A] :
( ( lexordp2 @ A @ Less @ ( nil @ A ) @ Ys )
= ( Ys
!= ( nil @ A ) ) ) ).
% ord.lexordp_simps(1)
thf(fact_8095_ord_Olexordp__simps_I2_J,axiom,
! [A: $tType,Less: A > A > $o,Xs: list @ A] :
~ ( lexordp2 @ A @ Less @ Xs @ ( nil @ A ) ) ).
% ord.lexordp_simps(2)
thf(fact_8096_Image__empty1,axiom,
! [B: $tType,A: $tType,X7: set @ B] :
( ( image @ B @ A @ ( bot_bot @ ( set @ ( product_prod @ B @ A ) ) ) @ X7 )
= ( bot_bot @ ( set @ A ) ) ) ).
% Image_empty1
thf(fact_8097_Image__singleton__iff,axiom,
! [A: $tType,B: $tType,B2: A,R2: set @ ( product_prod @ B @ A ),A2: B] :
( ( member @ A @ B2 @ ( image @ B @ A @ R2 @ ( insert @ B @ A2 @ ( bot_bot @ ( set @ B ) ) ) ) )
= ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ A2 @ B2 ) @ R2 ) ) ).
% Image_singleton_iff
thf(fact_8098_listrel__Nil,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ B @ A )] :
( ( image @ ( list @ B ) @ ( list @ A ) @ ( listrel @ B @ A @ R2 ) @ ( insert @ ( list @ B ) @ ( nil @ B ) @ ( bot_bot @ ( set @ ( list @ B ) ) ) ) )
= ( insert @ ( list @ A ) @ ( nil @ A ) @ ( bot_bot @ ( set @ ( list @ A ) ) ) ) ) ).
% listrel_Nil
thf(fact_8099_equiv__class__self,axiom,
! [A: $tType,A5: set @ A,R2: set @ ( product_prod @ A @ A ),A2: A] :
( ( equiv_equiv @ A @ A5 @ R2 )
=> ( ( member @ A @ A2 @ A5 )
=> ( member @ A @ A2 @ ( image @ A @ A @ R2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% equiv_class_self
thf(fact_8100_proj__def,axiom,
! [A: $tType,B: $tType] :
( ( equiv_proj @ B @ A )
= ( ^ [R5: set @ ( product_prod @ B @ A ),X5: B] : ( image @ B @ A @ R5 @ ( insert @ B @ X5 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ).
% proj_def
thf(fact_8101_countable__Image,axiom,
! [B: $tType,A: $tType,Y8: set @ A,X7: set @ ( product_prod @ A @ B )] :
( ! [Y3: A] :
( ( member @ A @ Y3 @ Y8 )
=> ( countable_countable @ B @ ( image @ A @ B @ X7 @ ( insert @ A @ Y3 @ ( bot_bot @ ( set @ A ) ) ) ) ) )
=> ( ( countable_countable @ A @ Y8 )
=> ( countable_countable @ B @ ( image @ A @ B @ X7 @ Y8 ) ) ) ) ).
% countable_Image
thf(fact_8102_Image__singleton,axiom,
! [B: $tType,A: $tType,R2: set @ ( product_prod @ B @ A ),A2: B] :
( ( image @ B @ A @ R2 @ ( insert @ B @ A2 @ ( bot_bot @ ( set @ B ) ) ) )
= ( collect @ A
@ ^ [B5: A] : ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ A2 @ B5 ) @ R2 ) ) ) ).
% Image_singleton
thf(fact_8103_finite__rtrancl__Image,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),A5: set @ A] :
( ( finite_finite2 @ ( product_prod @ A @ A ) @ R )
=> ( ( finite_finite2 @ A @ A5 )
=> ( finite_finite2 @ A @ ( image @ A @ A @ ( transitive_rtrancl @ A @ R ) @ A5 ) ) ) ) ).
% finite_rtrancl_Image
thf(fact_8104_ord_Olexordp__append__rightI,axiom,
! [A: $tType,Ys: list @ A,Less: A > A > $o,Xs: list @ A] :
( ( Ys
!= ( nil @ A ) )
=> ( lexordp2 @ A @ Less @ Xs @ ( append @ A @ Xs @ Ys ) ) ) ).
% ord.lexordp_append_rightI
thf(fact_8105_ord_Olexordp__append__left__rightI,axiom,
! [A: $tType,Less: A > A > $o,X: A,Y: A,Us: list @ A,Xs: list @ A,Ys: list @ A] :
( ( Less @ X @ Y )
=> ( lexordp2 @ A @ Less @ ( append @ A @ Us @ ( cons @ A @ X @ Xs ) ) @ ( append @ A @ Us @ ( cons @ A @ Y @ Ys ) ) ) ) ).
% ord.lexordp_append_left_rightI
thf(fact_8106_ord_Olexordp__append__leftI,axiom,
! [A: $tType,Less: A > A > $o,Us: list @ A,Vs2: list @ A,Xs: list @ A] :
( ( lexordp2 @ A @ Less @ Us @ Vs2 )
=> ( lexordp2 @ A @ Less @ ( append @ A @ Xs @ Us ) @ ( append @ A @ Xs @ Vs2 ) ) ) ).
% ord.lexordp_append_leftI
thf(fact_8107_ord_Olexordp__append__leftD,axiom,
! [A: $tType,Less: A > A > $o,Xs: list @ A,Us: list @ A,Vs2: list @ A] :
( ( lexordp2 @ A @ Less @ ( append @ A @ Xs @ Us ) @ ( append @ A @ Xs @ Vs2 ) )
=> ( ! [A4: A] :
~ ( Less @ A4 @ A4 )
=> ( lexordp2 @ A @ Less @ Us @ Vs2 ) ) ) ).
% ord.lexordp_append_leftD
thf(fact_8108_ord_Olexordp_Ocong,axiom,
! [A: $tType] :
( ( lexordp2 @ A )
= ( lexordp2 @ A ) ) ).
% ord.lexordp.cong
thf(fact_8109_ord_Olexordp__irreflexive,axiom,
! [A: $tType,Less: A > A > $o,Xs: list @ A] :
( ! [X4: A] :
~ ( Less @ X4 @ X4 )
=> ~ ( lexordp2 @ A @ Less @ Xs @ Xs ) ) ).
% ord.lexordp_irreflexive
thf(fact_8110_finite__Image,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),A5: set @ A] :
( ( finite_finite2 @ ( product_prod @ A @ B ) @ R )
=> ( finite_finite2 @ B @ ( image @ A @ B @ R @ A5 ) ) ) ).
% finite_Image
thf(fact_8111_ord_Olexordp_OCons,axiom,
! [A: $tType,Less: A > A > $o,X: A,Y: A,Xs: list @ A,Ys: list @ A] :
( ( Less @ X @ Y )
=> ( lexordp2 @ A @ Less @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Ys ) ) ) ).
% ord.lexordp.Cons
thf(fact_8112_ord_Olexordp_OCons__eq,axiom,
! [A: $tType,Less: A > A > $o,X: A,Y: A,Xs: list @ A,Ys: list @ A] :
( ~ ( Less @ X @ Y )
=> ( ~ ( Less @ Y @ X )
=> ( ( lexordp2 @ A @ Less @ Xs @ Ys )
=> ( lexordp2 @ A @ Less @ ( cons @ A @ X @ Xs ) @ ( cons @ A @ Y @ Ys ) ) ) ) ) ).
% ord.lexordp.Cons_eq
thf(fact_8113_ord_Olexordp_ONil,axiom,
! [A: $tType,Less: A > A > $o,Y: A,Ys: list @ A] : ( lexordp2 @ A @ Less @ ( nil @ A ) @ ( cons @ A @ Y @ Ys ) ) ).
% ord.lexordp.Nil
thf(fact_8114_ord_Olexordp_Ocases,axiom,
! [A: $tType,Less: A > A > $o,A1: list @ A,A22: list @ A] :
( ( lexordp2 @ A @ Less @ A1 @ A22 )
=> ( ( ( A1
= ( nil @ A ) )
=> ! [Y3: A,Ys4: list @ A] :
( A22
!= ( cons @ A @ Y3 @ Ys4 ) ) )
=> ( ! [X4: A] :
( ? [Xs2: list @ A] :
( A1
= ( cons @ A @ X4 @ Xs2 ) )
=> ! [Y3: A] :
( ? [Ys4: list @ A] :
( A22
= ( cons @ A @ Y3 @ Ys4 ) )
=> ~ ( Less @ X4 @ Y3 ) ) )
=> ~ ! [X4: A,Y3: A,Xs2: list @ A] :
( ( A1
= ( cons @ A @ X4 @ Xs2 ) )
=> ! [Ys4: list @ A] :
( ( A22
= ( cons @ A @ Y3 @ Ys4 ) )
=> ( ~ ( Less @ X4 @ Y3 )
=> ( ~ ( Less @ Y3 @ X4 )
=> ~ ( lexordp2 @ A @ Less @ Xs2 @ Ys4 ) ) ) ) ) ) ) ) ).
% ord.lexordp.cases
thf(fact_8115_ord_Olexordp_Osimps,axiom,
! [A: $tType] :
( ( lexordp2 @ A )
= ( ^ [Less2: A > A > $o,A12: list @ A,A23: list @ A] :
( ? [Y6: A,Ys3: list @ A] :
( ( A12
= ( nil @ A ) )
& ( A23
= ( cons @ A @ Y6 @ Ys3 ) ) )
| ? [X5: A,Y6: A,Xs3: list @ A,Ys3: list @ A] :
( ( A12
= ( cons @ A @ X5 @ Xs3 ) )
& ( A23
= ( cons @ A @ Y6 @ Ys3 ) )
& ( Less2 @ X5 @ Y6 ) )
| ? [X5: A,Y6: A,Xs3: list @ A,Ys3: list @ A] :
( ( A12
= ( cons @ A @ X5 @ Xs3 ) )
& ( A23
= ( cons @ A @ Y6 @ Ys3 ) )
& ~ ( Less2 @ X5 @ Y6 )
& ~ ( Less2 @ Y6 @ X5 )
& ( lexordp2 @ A @ Less2 @ Xs3 @ Ys3 ) ) ) ) ) ).
% ord.lexordp.simps
thf(fact_8116_rev__ImageI,axiom,
! [B: $tType,A: $tType,A2: A,A5: set @ A,B2: B,R2: set @ ( product_prod @ A @ B )] :
( ( member @ A @ A2 @ A5 )
=> ( ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ A2 @ B2 ) @ R2 )
=> ( member @ B @ B2 @ ( image @ A @ B @ R2 @ A5 ) ) ) ) ).
% rev_ImageI
thf(fact_8117_Image__iff,axiom,
! [A: $tType,B: $tType,B2: A,R2: set @ ( product_prod @ B @ A ),A5: set @ B] :
( ( member @ A @ B2 @ ( image @ B @ A @ R2 @ A5 ) )
= ( ? [X5: B] :
( ( member @ B @ X5 @ A5 )
& ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X5 @ B2 ) @ R2 ) ) ) ) ).
% Image_iff
thf(fact_8118_ImageE,axiom,
! [A: $tType,B: $tType,B2: A,R2: set @ ( product_prod @ B @ A ),A5: set @ B] :
( ( member @ A @ B2 @ ( image @ B @ A @ R2 @ A5 ) )
=> ~ ! [X4: B] :
( ( member @ ( product_prod @ B @ A ) @ ( product_Pair @ B @ A @ X4 @ B2 ) @ R2 )
=> ~ ( member @ B @ X4 @ A5 ) ) ) ).
% ImageE
thf(fact_8119_Image__def,axiom,
! [B: $tType,A: $tType] :
( ( image @ A @ B )
= ( ^ [R5: set @ ( product_prod @ A @ B ),S8: set @ A] :
( collect @ B
@ ^ [Y6: B] :
? [X5: A] :
( ( member @ A @ X5 @ S8 )
& ( member @ ( product_prod @ A @ B ) @ ( product_Pair @ A @ B @ X5 @ Y6 ) @ R5 ) ) ) ) ) ).
% Image_def
thf(fact_8120_quotientI,axiom,
! [A: $tType,X: A,A5: set @ A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ A @ X @ A5 )
=> ( member @ ( set @ A ) @ ( image @ A @ A @ R2 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ ( equiv_quotient @ A @ A5 @ R2 ) ) ) ).
% quotientI
thf(fact_8121_quotientE,axiom,
! [A: $tType,X7: set @ A,A5: set @ A,R2: set @ ( product_prod @ A @ A )] :
( ( member @ ( set @ A ) @ X7 @ ( equiv_quotient @ A @ A5 @ R2 ) )
=> ~ ! [X4: A] :
( ( X7
= ( image @ A @ A @ R2 @ ( insert @ A @ X4 @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ~ ( member @ A @ X4 @ A5 ) ) ) ).
% quotientE
thf(fact_8122_wfI__pf,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A )] :
( ! [A8: set @ A] :
( ( ord_less_eq @ ( set @ A ) @ A8 @ ( image @ A @ A @ R @ A8 ) )
=> ( A8
= ( bot_bot @ ( set @ A ) ) ) )
=> ( wf @ A @ R ) ) ).
% wfI_pf
thf(fact_8123_wfE__pf,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),A5: set @ A] :
( ( wf @ A @ R )
=> ( ( ord_less_eq @ ( set @ A ) @ A5 @ ( image @ A @ A @ R @ A5 ) )
=> ( A5
= ( bot_bot @ ( set @ A ) ) ) ) ) ).
% wfE_pf
thf(fact_8124_listrel__Cons,axiom,
! [A: $tType,B: $tType,R2: set @ ( product_prod @ B @ A ),X: B,Xs: list @ B] :
( ( image @ ( list @ B ) @ ( list @ A ) @ ( listrel @ B @ A @ R2 ) @ ( insert @ ( list @ B ) @ ( cons @ B @ X @ Xs ) @ ( bot_bot @ ( set @ ( list @ B ) ) ) ) )
= ( set_Cons @ A @ ( image @ B @ A @ R2 @ ( insert @ B @ X @ ( bot_bot @ ( set @ B ) ) ) ) @ ( image @ ( list @ B ) @ ( list @ A ) @ ( listrel @ B @ A @ R2 ) @ ( insert @ ( list @ B ) @ Xs @ ( bot_bot @ ( set @ ( list @ B ) ) ) ) ) ) ) ).
% listrel_Cons
thf(fact_8125_equiv__class__eq__iff,axiom,
! [A: $tType,A5: set @ A,R2: set @ ( product_prod @ A @ A ),X: A,Y: A] :
( ( equiv_equiv @ A @ A5 @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R2 )
= ( ( ( image @ A @ A @ R2 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( image @ A @ A @ R2 @ ( insert @ A @ Y @ ( bot_bot @ ( set @ A ) ) ) ) )
& ( member @ A @ X @ A5 )
& ( member @ A @ Y @ A5 ) ) ) ) ).
% equiv_class_eq_iff
thf(fact_8126_eq__equiv__class__iff,axiom,
! [A: $tType,A5: set @ A,R2: set @ ( product_prod @ A @ A ),X: A,Y: A] :
( ( equiv_equiv @ A @ A5 @ R2 )
=> ( ( member @ A @ X @ A5 )
=> ( ( member @ A @ Y @ A5 )
=> ( ( ( image @ A @ A @ R2 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) )
= ( image @ A @ A @ R2 @ ( insert @ A @ Y @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X @ Y ) @ R2 ) ) ) ) ) ).
% eq_equiv_class_iff
thf(fact_8127_equiv__class__eq,axiom,
! [A: $tType,A5: set @ A,R2: set @ ( product_prod @ A @ A ),A2: A,B2: A] :
( ( equiv_equiv @ A @ A5 @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ R2 )
=> ( ( image @ A @ A @ R2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
= ( image @ A @ A @ R2 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% equiv_class_eq
thf(fact_8128_eq__equiv__class,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A2: A,B2: A,A5: set @ A] :
( ( ( image @ A @ A @ R2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
= ( image @ A @ A @ R2 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( ( equiv_equiv @ A @ A5 @ R2 )
=> ( ( member @ A @ B2 @ A5 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ R2 ) ) ) ) ).
% eq_equiv_class
thf(fact_8129_refines__equiv__class__eq2,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),S2: set @ ( product_prod @ A @ A ),A5: set @ A,A2: A] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R @ S2 )
=> ( ( equiv_equiv @ A @ A5 @ R )
=> ( ( equiv_equiv @ A @ A5 @ S2 )
=> ( ( image @ A @ A @ S2 @ ( image @ A @ A @ R @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( image @ A @ A @ S2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% refines_equiv_class_eq2
thf(fact_8130_refines__equiv__class__eq,axiom,
! [A: $tType,R: set @ ( product_prod @ A @ A ),S2: set @ ( product_prod @ A @ A ),A5: set @ A,A2: A] :
( ( ord_less_eq @ ( set @ ( product_prod @ A @ A ) ) @ R @ S2 )
=> ( ( equiv_equiv @ A @ A5 @ R )
=> ( ( equiv_equiv @ A @ A5 @ S2 )
=> ( ( image @ A @ A @ R @ ( image @ A @ A @ S2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( image @ A @ A @ S2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ) ).
% refines_equiv_class_eq
thf(fact_8131_Partial__order__eq__Image1__Image1__iff,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A2: A,B2: A] :
( ( order_7125193373082350890der_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( member @ A @ A2 @ ( field2 @ A @ R2 ) )
=> ( ( member @ A @ B2 @ ( field2 @ A @ R2 ) )
=> ( ( ( image @ A @ A @ R2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
= ( image @ A @ A @ R2 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( A2 = B2 ) ) ) ) ) ).
% Partial_order_eq_Image1_Image1_iff
thf(fact_8132_Image__eq__UN,axiom,
! [A: $tType,B: $tType] :
( ( image @ B @ A )
= ( ^ [R5: set @ ( product_prod @ B @ A ),B8: set @ B] :
( complete_Sup_Sup @ ( set @ A )
@ ( image2 @ B @ ( set @ A )
@ ^ [Y6: B] : ( image @ B @ A @ R5 @ ( insert @ B @ Y6 @ ( bot_bot @ ( set @ B ) ) ) )
@ B8 ) ) ) ) ).
% Image_eq_UN
thf(fact_8133_equiv__class__subset,axiom,
! [A: $tType,A5: set @ A,R2: set @ ( product_prod @ A @ A ),A2: A,B2: A] :
( ( equiv_equiv @ A @ A5 @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ R2 )
=> ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ R2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( image @ A @ A @ R2 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) ) ) ).
% equiv_class_subset
thf(fact_8134_subset__equiv__class,axiom,
! [A: $tType,A5: set @ A,R2: set @ ( product_prod @ A @ A ),B2: A,A2: A] :
( ( equiv_equiv @ A @ A5 @ R2 )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ R2 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( image @ A @ A @ R2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) )
=> ( ( member @ A @ B2 @ A5 )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ R2 ) ) ) ) ).
% subset_equiv_class
thf(fact_8135_equiv__class__nondisjoint,axiom,
! [A: $tType,A5: set @ A,R2: set @ ( product_prod @ A @ A ),X: A,A2: A,B2: A] :
( ( equiv_equiv @ A @ A5 @ R2 )
=> ( ( member @ A @ X @ ( inf_inf @ ( set @ A ) @ ( image @ A @ A @ R2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( image @ A @ A @ R2 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) ) )
=> ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ R2 ) ) ) ).
% equiv_class_nondisjoint
thf(fact_8136_singleton__quotient,axiom,
! [A: $tType,X: A,R2: set @ ( product_prod @ A @ A )] :
( ( equiv_quotient @ A @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) @ R2 )
= ( insert @ ( set @ A ) @ ( image @ A @ A @ R2 @ ( insert @ A @ X @ ( bot_bot @ ( set @ A ) ) ) ) @ ( bot_bot @ ( set @ ( set @ A ) ) ) ) ) ).
% singleton_quotient
thf(fact_8137_quotient__def,axiom,
! [A: $tType] :
( ( equiv_quotient @ A )
= ( ^ [A7: set @ A,R5: set @ ( product_prod @ A @ A )] :
( complete_Sup_Sup @ ( set @ ( set @ A ) )
@ ( image2 @ A @ ( set @ ( set @ A ) )
@ ^ [X5: A] : ( insert @ ( set @ A ) @ ( image @ A @ A @ R5 @ ( insert @ A @ X5 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( bot_bot @ ( set @ ( set @ A ) ) ) )
@ A7 ) ) ) ) ).
% quotient_def
thf(fact_8138_Image__fold,axiom,
! [B: $tType,A: $tType,R: set @ ( product_prod @ A @ B ),S2: set @ A] :
( ( finite_finite2 @ ( product_prod @ A @ B ) @ R )
=> ( ( image @ A @ B @ R @ S2 )
= ( finite_fold @ ( product_prod @ A @ B ) @ ( set @ B )
@ ( product_case_prod @ A @ B @ ( ( set @ B ) > ( set @ B ) )
@ ^ [X5: A,Y6: B,A7: set @ B] : ( if @ ( set @ B ) @ ( member @ A @ X5 @ S2 ) @ ( insert @ B @ Y6 @ A7 ) @ A7 ) )
@ ( bot_bot @ ( set @ B ) )
@ R ) ) ) ).
% Image_fold
thf(fact_8139_congruent2__implies__congruent__UN,axiom,
! [B: $tType,C: $tType,A: $tType,A16: set @ A,R1: set @ ( product_prod @ A @ A ),A26: set @ B,R22: set @ ( product_prod @ B @ B ),F3: A > B > ( set @ C ),A2: B] :
( ( equiv_equiv @ A @ A16 @ R1 )
=> ( ( equiv_equiv @ B @ A26 @ R22 )
=> ( ( equiv_congruent2 @ A @ B @ ( set @ C ) @ R1 @ R22 @ F3 )
=> ( ( member @ B @ A2 @ A26 )
=> ( equiv_congruent @ A @ ( set @ C ) @ R1
@ ^ [X15: A] : ( complete_Sup_Sup @ ( set @ C ) @ ( image2 @ B @ ( set @ C ) @ ( F3 @ X15 ) @ ( image @ B @ B @ R22 @ ( insert @ B @ A2 @ ( bot_bot @ ( set @ B ) ) ) ) ) ) ) ) ) ) ) ).
% congruent2_implies_congruent_UN
thf(fact_8140_UN__equiv__class2,axiom,
! [A: $tType,C: $tType,B: $tType,A16: set @ A,R1: set @ ( product_prod @ A @ A ),A26: set @ B,R22: set @ ( product_prod @ B @ B ),F3: A > B > ( set @ C ),A1: A,A22: B] :
( ( equiv_equiv @ A @ A16 @ R1 )
=> ( ( equiv_equiv @ B @ A26 @ R22 )
=> ( ( equiv_congruent2 @ A @ B @ ( set @ C ) @ R1 @ R22 @ F3 )
=> ( ( member @ A @ A1 @ A16 )
=> ( ( member @ B @ A22 @ A26 )
=> ( ( complete_Sup_Sup @ ( set @ C )
@ ( image2 @ A @ ( set @ C )
@ ^ [X15: A] : ( complete_Sup_Sup @ ( set @ C ) @ ( image2 @ B @ ( set @ C ) @ ( F3 @ X15 ) @ ( image @ B @ B @ R22 @ ( insert @ B @ A22 @ ( bot_bot @ ( set @ B ) ) ) ) ) )
@ ( image @ A @ A @ R1 @ ( insert @ A @ A1 @ ( bot_bot @ ( set @ A ) ) ) ) ) )
= ( F3 @ A1 @ A22 ) ) ) ) ) ) ) ).
% UN_equiv_class2
thf(fact_8141_congruent2I_H,axiom,
! [C: $tType,B: $tType,A: $tType,R1: set @ ( product_prod @ A @ A ),R22: set @ ( product_prod @ B @ B ),F3: A > B > C] :
( ! [Y15: A,Z12: A,Y23: B,Z23: B] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y15 @ Z12 ) @ R1 )
=> ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ Y23 @ Z23 ) @ R22 )
=> ( ( F3 @ Y15 @ Y23 )
= ( F3 @ Z12 @ Z23 ) ) ) )
=> ( equiv_congruent2 @ A @ B @ C @ R1 @ R22 @ F3 ) ) ).
% congruent2I'
thf(fact_8142_congruent2D,axiom,
! [A: $tType,C: $tType,B: $tType,R1: set @ ( product_prod @ A @ A ),R22: set @ ( product_prod @ B @ B ),F3: A > B > C,Y1: A,Z1: A,Y2: B,Z22: B] :
( ( equiv_congruent2 @ A @ B @ C @ R1 @ R22 @ F3 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y1 @ Z1 ) @ R1 )
=> ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ Y2 @ Z22 ) @ R22 )
=> ( ( F3 @ Y1 @ Y2 )
= ( F3 @ Z1 @ Z22 ) ) ) ) ) ).
% congruent2D
thf(fact_8143_congruent2__commuteI,axiom,
! [B: $tType,A: $tType,A5: set @ A,R2: set @ ( product_prod @ A @ A ),F3: A > A > B] :
( ( equiv_equiv @ A @ A5 @ R2 )
=> ( ! [Y3: A,Z3: A] :
( ( member @ A @ Y3 @ A5 )
=> ( ( member @ A @ Z3 @ A5 )
=> ( ( F3 @ Y3 @ Z3 )
= ( F3 @ Z3 @ Y3 ) ) ) )
=> ( ! [Y3: A,Z3: A,W: A] :
( ( member @ A @ W @ A5 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z3 ) @ R2 )
=> ( ( F3 @ W @ Y3 )
= ( F3 @ W @ Z3 ) ) ) )
=> ( equiv_congruent2 @ A @ A @ B @ R2 @ R2 @ F3 ) ) ) ) ).
% congruent2_commuteI
thf(fact_8144_congruent2I,axiom,
! [C: $tType,B: $tType,A: $tType,A16: set @ A,R1: set @ ( product_prod @ A @ A ),A26: set @ B,R22: set @ ( product_prod @ B @ B ),F3: A > B > C] :
( ( equiv_equiv @ A @ A16 @ R1 )
=> ( ( equiv_equiv @ B @ A26 @ R22 )
=> ( ! [Y3: A,Z3: A,W: B] :
( ( member @ B @ W @ A26 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ Z3 ) @ R1 )
=> ( ( F3 @ Y3 @ W )
= ( F3 @ Z3 @ W ) ) ) )
=> ( ! [Y3: B,Z3: B,W: A] :
( ( member @ A @ W @ A16 )
=> ( ( member @ ( product_prod @ B @ B ) @ ( product_Pair @ B @ B @ Y3 @ Z3 ) @ R22 )
=> ( ( F3 @ W @ Y3 )
= ( F3 @ W @ Z3 ) ) ) )
=> ( equiv_congruent2 @ A @ B @ C @ R1 @ R22 @ F3 ) ) ) ) ) ).
% congruent2I
thf(fact_8145_subset__Image1__Image1__iff,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A2: A,B2: A] :
( ( order_preorder_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( member @ A @ A2 @ ( field2 @ A @ R2 ) )
=> ( ( member @ A @ B2 @ ( field2 @ A @ R2 ) )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ R2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) ) @ ( image @ A @ A @ R2 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ A2 ) @ R2 ) ) ) ) ) ).
% subset_Image1_Image1_iff
thf(fact_8146_subset__Image__Image__iff,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A5: set @ A,B6: set @ A] :
( ( order_preorder_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( ord_less_eq @ ( set @ A ) @ A5 @ ( field2 @ A @ R2 ) )
=> ( ( ord_less_eq @ ( set @ A ) @ B6 @ ( field2 @ A @ R2 ) )
=> ( ( ord_less_eq @ ( set @ A ) @ ( image @ A @ A @ R2 @ A5 ) @ ( image @ A @ A @ R2 @ B6 ) )
= ( ! [X5: A] :
( ( member @ A @ X5 @ A5 )
=> ? [Y6: A] :
( ( member @ A @ Y6 @ B6 )
& ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y6 @ X5 ) @ R2 ) ) ) ) ) ) ) ) ).
% subset_Image_Image_iff
thf(fact_8147_preorder__on__empty,axiom,
! [A: $tType] : ( order_preorder_on @ A @ ( bot_bot @ ( set @ A ) ) @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ).
% preorder_on_empty
thf(fact_8148_Refl__antisym__eq__Image1__Image1__iff,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A2: A,B2: A] :
( ( refl_on @ A @ ( field2 @ A @ R2 ) @ R2 )
=> ( ( antisym @ A @ R2 )
=> ( ( member @ A @ A2 @ ( field2 @ A @ R2 ) )
=> ( ( member @ A @ B2 @ ( field2 @ A @ R2 ) )
=> ( ( ( image @ A @ A @ R2 @ ( insert @ A @ A2 @ ( bot_bot @ ( set @ A ) ) ) )
= ( image @ A @ A @ R2 @ ( insert @ A @ B2 @ ( bot_bot @ ( set @ A ) ) ) ) )
= ( A2 = B2 ) ) ) ) ) ) ).
% Refl_antisym_eq_Image1_Image1_iff
thf(fact_8149_finite__enumerate__initial__segment,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [S2: set @ A,N: nat,S: A] :
( ( finite_finite2 @ A @ S2 )
=> ( ( ord_less @ nat @ N @ ( finite_card @ A @ ( inf_inf @ ( set @ A ) @ S2 @ ( set_ord_lessThan @ A @ S ) ) ) )
=> ( ( infini527867602293511546merate @ A @ ( inf_inf @ ( set @ A ) @ S2 @ ( set_ord_lessThan @ A @ S ) ) @ N )
= ( infini527867602293511546merate @ A @ S2 @ N ) ) ) ) ) ).
% finite_enumerate_initial_segment
thf(fact_8150_enumerate__mono__iff,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [S2: set @ A,M2: nat,N: nat] :
( ~ ( finite_finite2 @ A @ S2 )
=> ( ( ord_less @ A @ ( infini527867602293511546merate @ A @ S2 @ M2 ) @ ( infini527867602293511546merate @ A @ S2 @ N ) )
= ( ord_less @ nat @ M2 @ N ) ) ) ) ).
% enumerate_mono_iff
thf(fact_8151_finite__enumerate__mono__iff,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [S2: set @ A,M2: nat,N: nat] :
( ( finite_finite2 @ A @ S2 )
=> ( ( ord_less @ nat @ M2 @ ( finite_card @ A @ S2 ) )
=> ( ( ord_less @ nat @ N @ ( finite_card @ A @ S2 ) )
=> ( ( ord_less @ A @ ( infini527867602293511546merate @ A @ S2 @ M2 ) @ ( infini527867602293511546merate @ A @ S2 @ N ) )
= ( ord_less @ nat @ M2 @ N ) ) ) ) ) ) ).
% finite_enumerate_mono_iff
thf(fact_8152_le__enumerate,axiom,
! [S2: set @ nat,N: nat] :
( ~ ( finite_finite2 @ nat @ S2 )
=> ( ord_less_eq @ nat @ N @ ( infini527867602293511546merate @ nat @ S2 @ N ) ) ) ).
% le_enumerate
thf(fact_8153_antisym__def,axiom,
! [A: $tType] :
( ( antisym @ A )
= ( ^ [R5: set @ ( product_prod @ A @ A )] :
! [X5: A,Y6: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X5 @ Y6 ) @ R5 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y6 @ X5 ) @ R5 )
=> ( X5 = Y6 ) ) ) ) ) ).
% antisym_def
thf(fact_8154_antisymI,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A )] :
( ! [X4: A,Y3: A] :
( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ X4 @ Y3 ) @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ Y3 @ X4 ) @ R2 )
=> ( X4 = Y3 ) ) )
=> ( antisym @ A @ R2 ) ) ).
% antisymI
thf(fact_8155_antisymD,axiom,
! [A: $tType,R2: set @ ( product_prod @ A @ A ),A2: A,B2: A] :
( ( antisym @ A @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ A2 @ B2 ) @ R2 )
=> ( ( member @ ( product_prod @ A @ A ) @ ( product_Pair @ A @ A @ B2 @ A2 ) @ R2 )
=> ( A2 = B2 ) ) ) ) ).
% antisymD
thf(fact_8156_antisym__empty,axiom,
! [A: $tType] : ( antisym @ A @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ).
% antisym_empty
thf(fact_8157_enumerate__Ex,axiom,
! [S2: set @ nat,S: nat] :
( ~ ( finite_finite2 @ nat @ S2 )
=> ( ( member @ nat @ S @ S2 )
=> ? [N3: nat] :
( ( infini527867602293511546merate @ nat @ S2 @ N3 )
= S ) ) ) ).
% enumerate_Ex
thf(fact_8158_enumerate__in__set,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [S2: set @ A,N: nat] :
( ~ ( finite_finite2 @ A @ S2 )
=> ( member @ A @ ( infini527867602293511546merate @ A @ S2 @ N ) @ S2 ) ) ) ).
% enumerate_in_set
thf(fact_8159_strict__mono__enumerate,axiom,
! [S2: set @ nat] :
( ~ ( finite_finite2 @ nat @ S2 )
=> ( order_strict_mono @ nat @ nat @ ( infini527867602293511546merate @ nat @ S2 ) ) ) ).
% strict_mono_enumerate
thf(fact_8160_antisym__singleton,axiom,
! [A: $tType,X: product_prod @ A @ A] : ( antisym @ A @ ( insert @ ( product_prod @ A @ A ) @ X @ ( bot_bot @ ( set @ ( product_prod @ A @ A ) ) ) ) ) ).
% antisym_singleton
thf(fact_8161_enumerate__step,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [S2: set @ A,N: nat] :
( ~ ( finite_finite2 @ A @ S2 )
=> ( ord_less @ A @ ( infini527867602293511546merate @ A @ S2 @ N ) @ ( infini527867602293511546merate @ A @ S2 @ ( suc @ N ) ) ) ) ) ).
% enumerate_step
thf(fact_8162_enumerate__mono,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [M2: nat,N: nat,S2: set @ A] :
( ( ord_less @ nat @ M2 @ N )
=> ( ~ ( finite_finite2 @ A @ S2 )
=> ( ord_less @ A @ ( infini527867602293511546merate @ A @ S2 @ M2 ) @ ( infini527867602293511546merate @ A @ S2 @ N ) ) ) ) ) ).
% enumerate_mono
thf(fact_8163_finite__enum__ext,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [X7: set @ A,Y8: set @ A] :
( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( finite_card @ A @ X7 ) )
=> ( ( infini527867602293511546merate @ A @ X7 @ I3 )
= ( infini527867602293511546merate @ A @ Y8 @ I3 ) ) )
=> ( ( finite_finite2 @ A @ X7 )
=> ( ( finite_finite2 @ A @ Y8 )
=> ( ( ( finite_card @ A @ X7 )
= ( finite_card @ A @ Y8 ) )
=> ( X7 = Y8 ) ) ) ) ) ) ).
% finite_enum_ext
thf(fact_8164_finite__enumerate__Ex,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [S2: set @ A,S: A] :
( ( finite_finite2 @ A @ S2 )
=> ( ( member @ A @ S @ S2 )
=> ? [N3: nat] :
( ( ord_less @ nat @ N3 @ ( finite_card @ A @ S2 ) )
& ( ( infini527867602293511546merate @ A @ S2 @ N3 )
= S ) ) ) ) ) ).
% finite_enumerate_Ex
thf(fact_8165_finite__enumerate__in__set,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [S2: set @ A,N: nat] :
( ( finite_finite2 @ A @ S2 )
=> ( ( ord_less @ nat @ N @ ( finite_card @ A @ S2 ) )
=> ( member @ A @ ( infini527867602293511546merate @ A @ S2 @ N ) @ S2 ) ) ) ) ).
% finite_enumerate_in_set
thf(fact_8166_range__enumerate,axiom,
! [S2: set @ nat] :
( ~ ( finite_finite2 @ nat @ S2 )
=> ( ( image2 @ nat @ nat @ ( infini527867602293511546merate @ nat @ S2 ) @ ( top_top @ ( set @ nat ) ) )
= S2 ) ) ).
% range_enumerate
thf(fact_8167_inj__enumerate,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [S2: set @ A] :
( ~ ( finite_finite2 @ A @ S2 )
=> ( inj_on @ nat @ A @ ( infini527867602293511546merate @ A @ S2 ) @ ( top_top @ ( set @ nat ) ) ) ) ) ).
% inj_enumerate
thf(fact_8168_finite__enumerate__mono,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [M2: nat,N: nat,S2: set @ A] :
( ( ord_less @ nat @ M2 @ N )
=> ( ( finite_finite2 @ A @ S2 )
=> ( ( ord_less @ nat @ N @ ( finite_card @ A @ S2 ) )
=> ( ord_less @ A @ ( infini527867602293511546merate @ A @ S2 @ M2 ) @ ( infini527867602293511546merate @ A @ S2 @ N ) ) ) ) ) ) ).
% finite_enumerate_mono
thf(fact_8169_finite__le__enumerate,axiom,
! [S2: set @ nat,N: nat] :
( ( finite_finite2 @ nat @ S2 )
=> ( ( ord_less @ nat @ N @ ( finite_card @ nat @ S2 ) )
=> ( ord_less_eq @ nat @ N @ ( infini527867602293511546merate @ nat @ S2 @ N ) ) ) ) ).
% finite_le_enumerate
thf(fact_8170_bij__enumerate,axiom,
! [S2: set @ nat] :
( ~ ( finite_finite2 @ nat @ S2 )
=> ( bij_betw @ nat @ nat @ ( infini527867602293511546merate @ nat @ S2 ) @ ( top_top @ ( set @ nat ) ) @ S2 ) ) ).
% bij_enumerate
thf(fact_8171_finite__bij__enumerate,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [S2: set @ A] :
( ( finite_finite2 @ A @ S2 )
=> ( bij_betw @ nat @ A @ ( infini527867602293511546merate @ A @ S2 ) @ ( set_ord_lessThan @ nat @ ( finite_card @ A @ S2 ) ) @ S2 ) ) ) ).
% finite_bij_enumerate
thf(fact_8172_finite__enumerate__step,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [S2: set @ A,N: nat] :
( ( finite_finite2 @ A @ S2 )
=> ( ( ord_less @ nat @ ( suc @ N ) @ ( finite_card @ A @ S2 ) )
=> ( ord_less @ A @ ( infini527867602293511546merate @ A @ S2 @ N ) @ ( infini527867602293511546merate @ A @ S2 @ ( suc @ N ) ) ) ) ) ) ).
% finite_enumerate_step
thf(fact_8173_enumerate__Suc_H,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [S2: set @ A,N: nat] :
( ( infini527867602293511546merate @ A @ S2 @ ( suc @ N ) )
= ( infini527867602293511546merate @ A @ ( minus_minus @ ( set @ A ) @ S2 @ ( insert @ A @ ( infini527867602293511546merate @ A @ S2 @ ( zero_zero @ nat ) ) @ ( bot_bot @ ( set @ A ) ) ) ) @ N ) ) ) ).
% enumerate_Suc'
thf(fact_8174_finite__enum__subset,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [X7: set @ A,Y8: set @ A] :
( ! [I3: nat] :
( ( ord_less @ nat @ I3 @ ( finite_card @ A @ X7 ) )
=> ( ( infini527867602293511546merate @ A @ X7 @ I3 )
= ( infini527867602293511546merate @ A @ Y8 @ I3 ) ) )
=> ( ( finite_finite2 @ A @ X7 )
=> ( ( finite_finite2 @ A @ Y8 )
=> ( ( ord_less_eq @ nat @ ( finite_card @ A @ X7 ) @ ( finite_card @ A @ Y8 ) )
=> ( ord_less_eq @ ( set @ A ) @ X7 @ Y8 ) ) ) ) ) ) ).
% finite_enum_subset
thf(fact_8175_finite__enumerate__Suc_H_H,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [S2: set @ A,N: nat] :
( ( finite_finite2 @ A @ S2 )
=> ( ( ord_less @ nat @ ( suc @ N ) @ ( finite_card @ A @ S2 ) )
=> ( ( infini527867602293511546merate @ A @ S2 @ ( suc @ N ) )
= ( ord_Least @ A
@ ^ [S8: A] :
( ( member @ A @ S8 @ S2 )
& ( ord_less @ A @ ( infini527867602293511546merate @ A @ S2 @ N ) @ S8 ) ) ) ) ) ) ) ).
% finite_enumerate_Suc''
thf(fact_8176_enumerate__Suc,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [S2: set @ A,N: nat] :
( ( infini527867602293511546merate @ A @ S2 @ ( suc @ N ) )
= ( infini527867602293511546merate @ A
@ ( minus_minus @ ( set @ A ) @ S2
@ ( insert @ A
@ ( ord_Least @ A
@ ^ [N2: A] : ( member @ A @ N2 @ S2 ) )
@ ( bot_bot @ ( set @ A ) ) ) )
@ N ) ) ) ).
% enumerate_Suc
thf(fact_8177_Least__eq__0,axiom,
! [P2: nat > $o] :
( ( P2 @ ( zero_zero @ nat ) )
=> ( ( ord_Least @ nat @ P2 )
= ( zero_zero @ nat ) ) ) ).
% Least_eq_0
thf(fact_8178_enumerate__0,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [S2: set @ A] :
( ( infini527867602293511546merate @ A @ S2 @ ( zero_zero @ nat ) )
= ( ord_Least @ A
@ ^ [N2: A] : ( member @ A @ N2 @ S2 ) ) ) ) ).
% enumerate_0
thf(fact_8179_Least__Min,axiom,
! [A: $tType] :
( ( linorder @ A )
=> ! [P2: A > $o] :
( ( finite_finite2 @ A @ ( collect @ A @ P2 ) )
=> ( ? [X_1: A] : ( P2 @ X_1 )
=> ( ( ord_Least @ A @ P2 )
= ( lattic643756798350308766er_Min @ A @ ( collect @ A @ P2 ) ) ) ) ) ) ).
% Least_Min
thf(fact_8180_Least__Suc,axiom,
! [P2: nat > $o,N: nat] :
( ( P2 @ N )
=> ( ~ ( P2 @ ( zero_zero @ nat ) )
=> ( ( ord_Least @ nat @ P2 )
= ( suc
@ ( ord_Least @ nat
@ ^ [M3: nat] : ( P2 @ ( suc @ M3 ) ) ) ) ) ) ) ).
% Least_Suc
thf(fact_8181_Least__Suc2,axiom,
! [P2: nat > $o,N: nat,Q: nat > $o,M2: nat] :
( ( P2 @ N )
=> ( ( Q @ M2 )
=> ( ~ ( P2 @ ( zero_zero @ nat ) )
=> ( ! [K: nat] :
( ( P2 @ ( suc @ K ) )
= ( Q @ K ) )
=> ( ( ord_Least @ nat @ P2 )
= ( suc @ ( ord_Least @ nat @ Q ) ) ) ) ) ) ) ).
% Least_Suc2
thf(fact_8182_LeastI2,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [P2: A > $o,A2: A,Q: A > $o] :
( ( P2 @ A2 )
=> ( ! [X4: A] :
( ( P2 @ X4 )
=> ( Q @ X4 ) )
=> ( Q @ ( ord_Least @ A @ P2 ) ) ) ) ) ).
% LeastI2
thf(fact_8183_LeastI__ex,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [P2: A > $o] :
( ? [X_1: A] : ( P2 @ X_1 )
=> ( P2 @ ( ord_Least @ A @ P2 ) ) ) ) ).
% LeastI_ex
thf(fact_8184_LeastI2__ex,axiom,
! [A: $tType] :
( ( wellorder @ A )
=> ! [P2: A > $o,Q: A > $o] :
( ? [X_1: A] : ( P2 @ X_1 )
=> ( ! [X4: A] :
( ( P2 @ X4 )
=> ( Q @ X4 ) )
=> ( Q @ ( ord_Least @ A @ P2 ) ) ) ) ) ).
% LeastI2_ex
% Type constructors (746)
thf(tcon_fun___Countable__Complete__Lattices_Ocountable__complete__distrib__lattice,axiom,
! [A9: $tType,A19: $tType] :
( ( comple592849572758109894attice @ A19 )
=> ( counta4013691401010221786attice @ ( A9 > A19 ) ) ) ).
thf(tcon_fun___Conditionally__Complete__Lattices_Oconditionally__complete__lattice,axiom,
! [A9: $tType,A19: $tType] :
( ( comple6319245703460814977attice @ A19 )
=> ( condit1219197933456340205attice @ ( A9 > A19 ) ) ) ).
thf(tcon_fun___Countable__Complete__Lattices_Ocountable__complete__lattice,axiom,
! [A9: $tType,A19: $tType] :
( ( counta3822494911875563373attice @ A19 )
=> ( counta3822494911875563373attice @ ( A9 > A19 ) ) ) ).
thf(tcon_fun___Complete__Lattices_Ocomplete__distrib__lattice,axiom,
! [A9: $tType,A19: $tType] :
( ( comple592849572758109894attice @ A19 )
=> ( comple592849572758109894attice @ ( A9 > A19 ) ) ) ).
thf(tcon_fun___Lattices_Obounded__semilattice__sup__bot,axiom,
! [A9: $tType,A19: $tType] :
( ( bounded_lattice @ A19 )
=> ( bounde4967611905675639751up_bot @ ( A9 > A19 ) ) ) ).
thf(tcon_fun___Lattices_Obounded__semilattice__inf__top,axiom,
! [A9: $tType,A19: $tType] :
( ( bounded_lattice @ A19 )
=> ( bounde4346867609351753570nf_top @ ( A9 > A19 ) ) ) ).
thf(tcon_fun___Complete__Lattices_Ocomplete__lattice,axiom,
! [A9: $tType,A19: $tType] :
( ( comple6319245703460814977attice @ A19 )
=> ( comple6319245703460814977attice @ ( A9 > A19 ) ) ) ).
thf(tcon_fun___Boolean__Algebras_Oboolean__algebra,axiom,
! [A9: $tType,A19: $tType] :
( ( boolea8198339166811842893lgebra @ A19 )
=> ( boolea8198339166811842893lgebra @ ( A9 > A19 ) ) ) ).
thf(tcon_fun___Lattices_Obounded__lattice__top,axiom,
! [A9: $tType,A19: $tType] :
( ( bounded_lattice @ A19 )
=> ( bounded_lattice_top @ ( A9 > A19 ) ) ) ).
thf(tcon_fun___Lattices_Obounded__lattice__bot,axiom,
! [A9: $tType,A19: $tType] :
( ( bounded_lattice @ A19 )
=> ( bounded_lattice_bot @ ( A9 > A19 ) ) ) ).
thf(tcon_fun___Complete__Partial__Order_Occpo,axiom,
! [A9: $tType,A19: $tType] :
( ( comple6319245703460814977attice @ A19 )
=> ( comple9053668089753744459l_ccpo @ ( A9 > A19 ) ) ) ).
thf(tcon_fun___Lattices_Osemilattice__sup,axiom,
! [A9: $tType,A19: $tType] :
( ( semilattice_sup @ A19 )
=> ( semilattice_sup @ ( A9 > A19 ) ) ) ).
thf(tcon_fun___Lattices_Osemilattice__inf,axiom,
! [A9: $tType,A19: $tType] :
( ( semilattice_inf @ A19 )
=> ( semilattice_inf @ ( A9 > A19 ) ) ) ).
thf(tcon_fun___Lattices_Odistrib__lattice,axiom,
! [A9: $tType,A19: $tType] :
( ( distrib_lattice @ A19 )
=> ( distrib_lattice @ ( A9 > A19 ) ) ) ).
thf(tcon_fun___Lattices_Obounded__lattice,axiom,
! [A9: $tType,A19: $tType] :
( ( bounded_lattice @ A19 )
=> ( bounded_lattice @ ( A9 > A19 ) ) ) ).
thf(tcon_fun___Orderings_Oorder__top,axiom,
! [A9: $tType,A19: $tType] :
( ( order_top @ A19 )
=> ( order_top @ ( A9 > A19 ) ) ) ).
thf(tcon_fun___Orderings_Oorder__bot,axiom,
! [A9: $tType,A19: $tType] :
( ( order_bot @ A19 )
=> ( order_bot @ ( A9 > A19 ) ) ) ).
thf(tcon_fun___Countable_Ocountable,axiom,
! [A9: $tType,A19: $tType] :
( ( ( finite_finite @ A9 )
& ( countable @ A19 ) )
=> ( countable @ ( A9 > A19 ) ) ) ).
thf(tcon_fun___Orderings_Opreorder,axiom,
! [A9: $tType,A19: $tType] :
( ( preorder @ A19 )
=> ( preorder @ ( A9 > A19 ) ) ) ).
thf(tcon_fun___Finite__Set_Ofinite,axiom,
! [A9: $tType,A19: $tType] :
( ( ( finite_finite @ A9 )
& ( finite_finite @ A19 ) )
=> ( finite_finite @ ( A9 > A19 ) ) ) ).
thf(tcon_fun___Lattices_Olattice,axiom,
! [A9: $tType,A19: $tType] :
( ( lattice @ A19 )
=> ( lattice @ ( A9 > A19 ) ) ) ).
thf(tcon_fun___Orderings_Oorder,axiom,
! [A9: $tType,A19: $tType] :
( ( order @ A19 )
=> ( order @ ( A9 > A19 ) ) ) ).
thf(tcon_fun___Orderings_Otop,axiom,
! [A9: $tType,A19: $tType] :
( ( top @ A19 )
=> ( top @ ( A9 > A19 ) ) ) ).
thf(tcon_fun___Orderings_Oord,axiom,
! [A9: $tType,A19: $tType] :
( ( ord @ A19 )
=> ( ord @ ( A9 > A19 ) ) ) ).
thf(tcon_fun___Orderings_Obot,axiom,
! [A9: $tType,A19: $tType] :
( ( bot @ A19 )
=> ( bot @ ( A9 > A19 ) ) ) ).
thf(tcon_fun___Groups_Ouminus,axiom,
! [A9: $tType,A19: $tType] :
( ( uminus @ A19 )
=> ( uminus @ ( A9 > A19 ) ) ) ).
thf(tcon_fun___Groups_Ominus,axiom,
! [A9: $tType,A19: $tType] :
( ( minus @ A19 )
=> ( minus @ ( A9 > A19 ) ) ) ).
thf(tcon_Int_Oint___Conditionally__Complete__Lattices_Oconditionally__complete__linorder,axiom,
condit6923001295902523014norder @ int ).
thf(tcon_Int_Oint___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_1,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___Groups_Oordered__ab__semigroup__monoid__add__imp__le,axiom,
ordere1937475149494474687imp_le @ int ).
thf(tcon_Int_Oint___Euclidean__Division_Ounique__euclidean__semiring,axiom,
euclid3128863361964157862miring @ int ).
thf(tcon_Int_Oint___Euclidean__Division_Oeuclidean__semiring__cancel,axiom,
euclid4440199948858584721cancel @ int ).
thf(tcon_Int_Oint___Divides_Ounique__euclidean__semiring__numeral,axiom,
unique1627219031080169319umeral @ int ).
thf(tcon_Int_Oint___Euclidean__Division_Oeuclidean__ring__cancel,axiom,
euclid8851590272496341667cancel @ int ).
thf(tcon_Int_Oint___Rings_Osemiring__no__zero__divisors__cancel,axiom,
semiri6575147826004484403cancel @ int ).
thf(tcon_Int_Oint___Groups_Ostrict__ordered__ab__semigroup__add,axiom,
strict9044650504122735259up_add @ int ).
thf(tcon_Int_Oint___Groups_Oordered__cancel__ab__semigroup__add,axiom,
ordere580206878836729694up_add @ int ).
thf(tcon_Int_Oint___Groups_Oordered__ab__semigroup__add__imp__le,axiom,
ordere2412721322843649153imp_le @ int ).
thf(tcon_Int_Oint___Bit__Operations_Osemiring__bit__operations,axiom,
bit_se359711467146920520ations @ int ).
thf(tcon_Int_Oint___Rings_Olinordered__comm__semiring__strict,axiom,
linord2810124833399127020strict @ int ).
thf(tcon_Int_Oint___Groups_Ostrict__ordered__comm__monoid__add,axiom,
strict7427464778891057005id_add @ int ).
thf(tcon_Int_Oint___Groups_Oordered__cancel__comm__monoid__add,axiom,
ordere8940638589300402666id_add @ int ).
thf(tcon_Int_Oint___Euclidean__Division_Oeuclidean__semiring,axiom,
euclid3725896446679973847miring @ int ).
thf(tcon_Int_Oint___Topological__Spaces_Otopological__space,axiom,
topolo4958980785337419405_space @ int ).
thf(tcon_Int_Oint___Topological__Spaces_Olinorder__topology,axiom,
topolo1944317154257567458pology @ int ).
thf(tcon_Int_Oint___Topological__Spaces_Odiscrete__topology,axiom,
topolo8865339358273720382pology @ int ).
thf(tcon_Int_Oint___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___Groups_Olinordered__ab__semigroup__add,axiom,
linord4140545234300271783up_add @ int ).
thf(tcon_Int_Oint___Bit__Operations_Oring__bit__operations,axiom,
bit_ri3973907225187159222ations @ int ).
thf(tcon_Int_Oint___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___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___Groups_Oordered__comm__monoid__add,axiom,
ordere6911136660526730532id_add @ int ).
thf(tcon_Int_Oint___Groups_Olinordered__ab__group__add,axiom,
linord5086331880401160121up_add @ int ).
thf(tcon_Int_Oint___Groups_Ocancel__ab__semigroup__add,axiom,
cancel2418104881723323429up_add @ int ).
thf(tcon_Int_Oint___Rings_Oring__1__no__zero__divisors,axiom,
ring_15535105094025558882visors @ int ).
thf(tcon_Int_Oint___Limits_Otopological__monoid__add,axiom,
topolo6943815403480290642id_add @ int ).
thf(tcon_Int_Oint___Groups_Ocancel__comm__monoid__add,axiom,
cancel1802427076303600483id_add @ int ).
thf(tcon_Int_Oint___Rings_Olinordered__ring__strict,axiom,
linord4710134922213307826strict @ int ).
thf(tcon_Int_Oint___Rings_Ocomm__semiring__1__cancel,axiom,
comm_s4317794764714335236cancel @ int ).
thf(tcon_Int_Oint___Bit__Operations_Osemiring__bits,axiom,
bit_semiring_bits @ int ).
thf(tcon_Int_Oint___Topological__Spaces_Ot2__space,axiom,
topological_t2_space @ int ).
thf(tcon_Int_Oint___Topological__Spaces_Ot1__space,axiom,
topological_t1_space @ int ).
thf(tcon_Int_Oint___Rings_Oordered__comm__semiring,axiom,
ordere2520102378445227354miring @ int ).
thf(tcon_Int_Oint___Rings_Olinordered__semiring__1,axiom,
linord6961819062388156250ring_1 @ int ).
thf(tcon_Int_Oint___Groups_Oordered__ab__group__add,axiom,
ordered_ab_group_add @ int ).
thf(tcon_Int_Oint___Groups_Ocancel__semigroup__add,axiom,
cancel_semigroup_add @ int ).
thf(tcon_Int_Oint___Rings_Olinordered__semiring,axiom,
linordered_semiring @ int ).
thf(tcon_Int_Oint___Rings_Oordered__semiring__0,axiom,
ordered_semiring_0 @ int ).
thf(tcon_Int_Oint___Rings_Olinordered__semidom,axiom,
linordered_semidom @ int ).
thf(tcon_Int_Oint___Lattices_Osemilattice__sup_2,axiom,
semilattice_sup @ int ).
thf(tcon_Int_Oint___Lattices_Osemilattice__inf_3,axiom,
semilattice_inf @ int ).
thf(tcon_Int_Oint___Lattices_Odistrib__lattice_4,axiom,
distrib_lattice @ int ).
thf(tcon_Int_Oint___Groups_Oab__semigroup__mult,axiom,
ab_semigroup_mult @ int ).
thf(tcon_Int_Oint___Rings_Osemiring__1__cancel,axiom,
semiring_1_cancel @ int ).
thf(tcon_Int_Oint___Rings_Oalgebraic__semidom,axiom,
algebraic_semidom @ int ).
thf(tcon_Int_Oint___Groups_Ocomm__monoid__mult,axiom,
comm_monoid_mult @ int ).
thf(tcon_Int_Oint___Groups_Oab__semigroup__add,axiom,
ab_semigroup_add @ int ).
thf(tcon_Int_Oint___Rings_Oordered__semiring,axiom,
ordered_semiring @ int ).
thf(tcon_Int_Oint___Rings_Oordered__ring__abs,axiom,
ordered_ring_abs @ int ).
thf(tcon_Int_Oint___Parity_Osemiring__parity,axiom,
semiring_parity @ int ).
thf(tcon_Int_Oint___Groups_Ocomm__monoid__add,axiom,
comm_monoid_add @ int ).
thf(tcon_Int_Oint___Rings_Osemiring__modulo,axiom,
semiring_modulo @ int ).
thf(tcon_Int_Oint___Rings_Olinordered__ring,axiom,
linordered_ring @ int ).
thf(tcon_Int_Oint___Rings_Olinordered__idom,axiom,
linordered_idom @ int ).
thf(tcon_Int_Oint___Rings_Ocomm__semiring__1,axiom,
comm_semiring_1 @ int ).
thf(tcon_Int_Oint___Rings_Ocomm__semiring__0,axiom,
comm_semiring_0 @ int ).
thf(tcon_Int_Oint___Groups_Osemigroup__mult,axiom,
semigroup_mult @ int ).
thf(tcon_Int_Oint___Rings_Osemidom__modulo,axiom,
semidom_modulo @ int ).
thf(tcon_Int_Oint___Rings_Osemidom__divide,axiom,
semidom_divide @ int ).
thf(tcon_Int_Oint___Num_Osemiring__numeral,axiom,
semiring_numeral @ int ).
thf(tcon_Int_Oint___Groups_Osemigroup__add,axiom,
semigroup_add @ int ).
thf(tcon_Int_Oint___Rings_Ozero__less__one,axiom,
zero_less_one @ int ).
thf(tcon_Int_Oint___Rings_Ocomm__semiring,axiom,
comm_semiring @ int ).
thf(tcon_Int_Oint___Nat_Osemiring__char__0,axiom,
semiring_char_0 @ int ).
thf(tcon_Int_Oint___Groups_Oab__group__add,axiom,
ab_group_add @ int ).
thf(tcon_Int_Oint___Countable_Ocountable_5,axiom,
countable @ int ).
thf(tcon_Int_Oint___Rings_Ozero__neq__one,axiom,
zero_neq_one @ int ).
thf(tcon_Int_Oint___Rings_Oordered__ring,axiom,
ordered_ring @ int ).
thf(tcon_Int_Oint___Rings_Oidom__abs__sgn,axiom,
idom_abs_sgn @ int ).
thf(tcon_Int_Oint___Parity_Oring__parity,axiom,
ring_parity @ int ).
thf(tcon_Int_Oint___Orderings_Opreorder_6,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_7,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_8,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_9,axiom,
ord @ int ).
thf(tcon_Int_Oint___Groups_Ouminus_10,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_11,axiom,
minus @ 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_12,axiom,
condit6923001295902523014norder @ nat ).
thf(tcon_Nat_Onat___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_13,axiom,
condit1219197933456340205attice @ nat ).
thf(tcon_Nat_Onat___Bit__Operations_Ounique__euclidean__semiring__with__bit__operations_14,axiom,
bit_un5681908812861735899ations @ nat ).
thf(tcon_Nat_Onat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_15,axiom,
semiri1453513574482234551roduct @ nat ).
thf(tcon_Nat_Onat___Euclidean__Division_Ounique__euclidean__semiring__with__nat_16,axiom,
euclid5411537665997757685th_nat @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__monoid__add__imp__le_17,axiom,
ordere1937475149494474687imp_le @ nat ).
thf(tcon_Nat_Onat___Euclidean__Division_Ounique__euclidean__semiring_18,axiom,
euclid3128863361964157862miring @ nat ).
thf(tcon_Nat_Onat___Euclidean__Division_Oeuclidean__semiring__cancel_19,axiom,
euclid4440199948858584721cancel @ nat ).
thf(tcon_Nat_Onat___Divides_Ounique__euclidean__semiring__numeral_20,axiom,
unique1627219031080169319umeral @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors__cancel_21,axiom,
semiri6575147826004484403cancel @ nat ).
thf(tcon_Nat_Onat___Groups_Ostrict__ordered__ab__semigroup__add_22,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_23,axiom,
ordere580206878836729694up_add @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add__imp__le_24,axiom,
ordere2412721322843649153imp_le @ nat ).
thf(tcon_Nat_Onat___Bit__Operations_Osemiring__bit__operations_25,axiom,
bit_se359711467146920520ations @ nat ).
thf(tcon_Nat_Onat___Rings_Olinordered__comm__semiring__strict_26,axiom,
linord2810124833399127020strict @ nat ).
thf(tcon_Nat_Onat___Groups_Ostrict__ordered__comm__monoid__add_27,axiom,
strict7427464778891057005id_add @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__cancel__comm__monoid__add_28,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_29,axiom,
euclid3725896446679973847miring @ nat ).
thf(tcon_Nat_Onat___Topological__Spaces_Otopological__space_30,axiom,
topolo4958980785337419405_space @ nat ).
thf(tcon_Nat_Onat___Topological__Spaces_Olinorder__topology_31,axiom,
topolo1944317154257567458pology @ nat ).
thf(tcon_Nat_Onat___Topological__Spaces_Odiscrete__topology_32,axiom,
topolo8865339358273720382pology @ nat ).
thf(tcon_Nat_Onat___Limits_Otopological__comm__monoid__add_33,axiom,
topolo5987344860129210374id_add @ nat ).
thf(tcon_Nat_Onat___Groups_Olinordered__ab__semigroup__add_34,axiom,
linord4140545234300271783up_add @ nat ).
thf(tcon_Nat_Onat___Topological__Spaces_Oorder__topology_35,axiom,
topolo2564578578187576103pology @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__1__no__zero__divisors_36,axiom,
semiri2026040879449505780visors @ nat ).
thf(tcon_Nat_Onat___Rings_Olinordered__nonzero__semiring_37,axiom,
linord181362715937106298miring @ nat ).
thf(tcon_Nat_Onat___Limits_Otopological__semigroup__mult_38,axiom,
topolo4211221413907600880p_mult @ nat ).
thf(tcon_Nat_Onat___Rings_Olinordered__semiring__strict_39,axiom,
linord8928482502909563296strict @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__no__zero__divisors_40,axiom,
semiri3467727345109120633visors @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__ab__semigroup__add_41,axiom,
ordere6658533253407199908up_add @ nat ).
thf(tcon_Nat_Onat___Groups_Oordered__comm__monoid__add_42,axiom,
ordere6911136660526730532id_add @ nat ).
thf(tcon_Nat_Onat___Groups_Ocancel__ab__semigroup__add_43,axiom,
cancel2418104881723323429up_add @ nat ).
thf(tcon_Nat_Onat___Limits_Otopological__monoid__add_44,axiom,
topolo6943815403480290642id_add @ nat ).
thf(tcon_Nat_Onat___Groups_Ocancel__comm__monoid__add_45,axiom,
cancel1802427076303600483id_add @ nat ).
thf(tcon_Nat_Onat___Rings_Ocomm__semiring__1__cancel_46,axiom,
comm_s4317794764714335236cancel @ nat ).
thf(tcon_Nat_Onat___Bit__Operations_Osemiring__bits_47,axiom,
bit_semiring_bits @ nat ).
thf(tcon_Nat_Onat___Topological__Spaces_Ot2__space_48,axiom,
topological_t2_space @ nat ).
thf(tcon_Nat_Onat___Topological__Spaces_Ot1__space_49,axiom,
topological_t1_space @ nat ).
thf(tcon_Nat_Onat___Rings_Oordered__comm__semiring_50,axiom,
ordere2520102378445227354miring @ nat ).
thf(tcon_Nat_Onat___Groups_Ocancel__semigroup__add_51,axiom,
cancel_semigroup_add @ nat ).
thf(tcon_Nat_Onat___Rings_Olinordered__semiring_52,axiom,
linordered_semiring @ nat ).
thf(tcon_Nat_Onat___Rings_Oordered__semiring__0_53,axiom,
ordered_semiring_0 @ nat ).
thf(tcon_Nat_Onat___Rings_Olinordered__semidom_54,axiom,
linordered_semidom @ nat ).
thf(tcon_Nat_Onat___Lattices_Osemilattice__sup_55,axiom,
semilattice_sup @ nat ).
thf(tcon_Nat_Onat___Lattices_Osemilattice__inf_56,axiom,
semilattice_inf @ nat ).
thf(tcon_Nat_Onat___Lattices_Odistrib__lattice_57,axiom,
distrib_lattice @ nat ).
thf(tcon_Nat_Onat___Groups_Oab__semigroup__mult_58,axiom,
ab_semigroup_mult @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__1__cancel_59,axiom,
semiring_1_cancel @ nat ).
thf(tcon_Nat_Onat___Rings_Oalgebraic__semidom_60,axiom,
algebraic_semidom @ nat ).
thf(tcon_Nat_Onat___Groups_Ocomm__monoid__mult_61,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_62,axiom,
ab_semigroup_add @ nat ).
thf(tcon_Nat_Onat___Rings_Oordered__semiring_63,axiom,
ordered_semiring @ nat ).
thf(tcon_Nat_Onat___Parity_Osemiring__parity_64,axiom,
semiring_parity @ nat ).
thf(tcon_Nat_Onat___Groups_Ocomm__monoid__add_65,axiom,
comm_monoid_add @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__modulo_66,axiom,
semiring_modulo @ nat ).
thf(tcon_Nat_Onat___Rings_Ocomm__semiring__1_67,axiom,
comm_semiring_1 @ nat ).
thf(tcon_Nat_Onat___Rings_Ocomm__semiring__0_68,axiom,
comm_semiring_0 @ nat ).
thf(tcon_Nat_Onat___Groups_Osemigroup__mult_69,axiom,
semigroup_mult @ nat ).
thf(tcon_Nat_Onat___Rings_Osemidom__modulo_70,axiom,
semidom_modulo @ nat ).
thf(tcon_Nat_Onat___Rings_Osemidom__divide_71,axiom,
semidom_divide @ nat ).
thf(tcon_Nat_Onat___Num_Osemiring__numeral_72,axiom,
semiring_numeral @ nat ).
thf(tcon_Nat_Onat___Groups_Osemigroup__add_73,axiom,
semigroup_add @ nat ).
thf(tcon_Nat_Onat___Rings_Ozero__less__one_74,axiom,
zero_less_one @ nat ).
thf(tcon_Nat_Onat___Rings_Ocomm__semiring_75,axiom,
comm_semiring @ nat ).
thf(tcon_Nat_Onat___Orderings_Owellorder,axiom,
wellorder @ nat ).
thf(tcon_Nat_Onat___Orderings_Oorder__bot_76,axiom,
order_bot @ nat ).
thf(tcon_Nat_Onat___Nat_Osemiring__char__0_77,axiom,
semiring_char_0 @ nat ).
thf(tcon_Nat_Onat___Countable_Ocountable_78,axiom,
countable @ nat ).
thf(tcon_Nat_Onat___Rings_Ozero__neq__one_79,axiom,
zero_neq_one @ nat ).
thf(tcon_Nat_Onat___Orderings_Opreorder_80,axiom,
preorder @ nat ).
thf(tcon_Nat_Onat___Orderings_Olinorder_81,axiom,
linorder @ nat ).
thf(tcon_Nat_Onat___Groups_Omonoid__mult_82,axiom,
monoid_mult @ nat ).
thf(tcon_Nat_Onat___Groups_Omonoid__add_83,axiom,
monoid_add @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__1_84,axiom,
semiring_1 @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring__0_85,axiom,
semiring_0 @ nat ).
thf(tcon_Nat_Onat___Orderings_Ono__top_86,axiom,
no_top @ nat ).
thf(tcon_Nat_Onat___Lattices_Olattice_87,axiom,
lattice @ nat ).
thf(tcon_Nat_Onat___GCD_Osemiring__gcd_88,axiom,
semiring_gcd @ nat ).
thf(tcon_Nat_Onat___GCD_Osemiring__Gcd_89,axiom,
semiring_Gcd @ nat ).
thf(tcon_Nat_Onat___Rings_Omult__zero_90,axiom,
mult_zero @ nat ).
thf(tcon_Nat_Onat___Orderings_Oorder_91,axiom,
order @ nat ).
thf(tcon_Nat_Onat___Rings_Osemiring_92,axiom,
semiring @ nat ).
thf(tcon_Nat_Onat___Rings_Osemidom_93,axiom,
semidom @ nat ).
thf(tcon_Nat_Onat___Orderings_Oord_94,axiom,
ord @ nat ).
thf(tcon_Nat_Onat___Orderings_Obot_95,axiom,
bot @ nat ).
thf(tcon_Nat_Onat___Groups_Ominus_96,axiom,
minus @ nat ).
thf(tcon_Nat_Onat___Power_Opower_97,axiom,
power @ nat ).
thf(tcon_Nat_Onat___Num_Onumeral_98,axiom,
numeral @ nat ).
thf(tcon_Nat_Onat___Groups_Ozero_99,axiom,
zero @ nat ).
thf(tcon_Nat_Onat___Groups_Oplus_100,axiom,
plus @ nat ).
thf(tcon_Nat_Onat___Groups_Oone_101,axiom,
one @ nat ).
thf(tcon_Nat_Onat___Rings_Odvd_102,axiom,
dvd @ nat ).
thf(tcon_Nat_Onat___Nat_Osize,axiom,
size @ nat ).
thf(tcon_Num_Onum___Orderings_Opreorder_103,axiom,
preorder @ num ).
thf(tcon_Num_Onum___Orderings_Olinorder_104,axiom,
linorder @ num ).
thf(tcon_Num_Onum___Orderings_Oorder_105,axiom,
order @ num ).
thf(tcon_Num_Onum___Orderings_Oord_106,axiom,
ord @ num ).
thf(tcon_Num_Onum___Groups_Oplus_107,axiom,
plus @ num ).
thf(tcon_Num_Onum___Nat_Osize_108,axiom,
size @ num ).
thf(tcon_Rat_Orat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_109,axiom,
semiri1453513574482234551roduct @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__monoid__add__imp__le_110,axiom,
ordere1937475149494474687imp_le @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring__no__zero__divisors__cancel_111,axiom,
semiri6575147826004484403cancel @ rat ).
thf(tcon_Rat_Orat___Groups_Ostrict__ordered__ab__semigroup__add_112,axiom,
strict9044650504122735259up_add @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__cancel__ab__semigroup__add_113,axiom,
ordere580206878836729694up_add @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__add__imp__le_114,axiom,
ordere2412721322843649153imp_le @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__comm__semiring__strict_115,axiom,
linord2810124833399127020strict @ rat ).
thf(tcon_Rat_Orat___Groups_Ostrict__ordered__comm__monoid__add_116,axiom,
strict7427464778891057005id_add @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__cancel__comm__monoid__add_117,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_118,axiom,
linord715952674999750819strict @ rat ).
thf(tcon_Rat_Orat___Orderings_Ounbounded__dense__linorder,axiom,
unboun7993243217541854897norder @ rat ).
thf(tcon_Rat_Orat___Groups_Olinordered__ab__semigroup__add_119,axiom,
linord4140545234300271783up_add @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring__1__no__zero__divisors_120,axiom,
semiri2026040879449505780visors @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__nonzero__semiring_121,axiom,
linord181362715937106298miring @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__semiring__strict_122,axiom,
linord8928482502909563296strict @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring__no__zero__divisors_123,axiom,
semiri3467727345109120633visors @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__ab__semigroup__add_124,axiom,
ordere6658533253407199908up_add @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__ab__group__add__abs_125,axiom,
ordere166539214618696060dd_abs @ rat ).
thf(tcon_Rat_Orat___Archimedean__Field_Ofloor__ceiling,axiom,
archim2362893244070406136eiling @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__comm__monoid__add_126,axiom,
ordere6911136660526730532id_add @ rat ).
thf(tcon_Rat_Orat___Groups_Olinordered__ab__group__add_127,axiom,
linord5086331880401160121up_add @ rat ).
thf(tcon_Rat_Orat___Groups_Ocancel__ab__semigroup__add_128,axiom,
cancel2418104881723323429up_add @ rat ).
thf(tcon_Rat_Orat___Rings_Oring__1__no__zero__divisors_129,axiom,
ring_15535105094025558882visors @ rat ).
thf(tcon_Rat_Orat___Groups_Ocancel__comm__monoid__add_130,axiom,
cancel1802427076303600483id_add @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__ring__strict_131,axiom,
linord4710134922213307826strict @ rat ).
thf(tcon_Rat_Orat___Rings_Ocomm__semiring__1__cancel_132,axiom,
comm_s4317794764714335236cancel @ rat ).
thf(tcon_Rat_Orat___Rings_Oordered__comm__semiring_133,axiom,
ordere2520102378445227354miring @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__semiring__1_134,axiom,
linord6961819062388156250ring_1 @ rat ).
thf(tcon_Rat_Orat___Groups_Oordered__ab__group__add_135,axiom,
ordered_ab_group_add @ rat ).
thf(tcon_Rat_Orat___Groups_Ocancel__semigroup__add_136,axiom,
cancel_semigroup_add @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__semiring_137,axiom,
linordered_semiring @ rat ).
thf(tcon_Rat_Orat___Rings_Oordered__semiring__0_138,axiom,
ordered_semiring_0 @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__semidom_139,axiom,
linordered_semidom @ rat ).
thf(tcon_Rat_Orat___Orderings_Odense__linorder,axiom,
dense_linorder @ rat ).
thf(tcon_Rat_Orat___Lattices_Osemilattice__sup_140,axiom,
semilattice_sup @ rat ).
thf(tcon_Rat_Orat___Lattices_Osemilattice__inf_141,axiom,
semilattice_inf @ rat ).
thf(tcon_Rat_Orat___Lattices_Odistrib__lattice_142,axiom,
distrib_lattice @ rat ).
thf(tcon_Rat_Orat___Groups_Oab__semigroup__mult_143,axiom,
ab_semigroup_mult @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring__1__cancel_144,axiom,
semiring_1_cancel @ rat ).
thf(tcon_Rat_Orat___Groups_Ocomm__monoid__mult_145,axiom,
comm_monoid_mult @ rat ).
thf(tcon_Rat_Orat___Groups_Oab__semigroup__add_146,axiom,
ab_semigroup_add @ rat ).
thf(tcon_Rat_Orat___Fields_Olinordered__field,axiom,
linordered_field @ rat ).
thf(tcon_Rat_Orat___Rings_Oordered__semiring_147,axiom,
ordered_semiring @ rat ).
thf(tcon_Rat_Orat___Rings_Oordered__ring__abs_148,axiom,
ordered_ring_abs @ rat ).
thf(tcon_Rat_Orat___Groups_Ocomm__monoid__add_149,axiom,
comm_monoid_add @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__ring_150,axiom,
linordered_ring @ rat ).
thf(tcon_Rat_Orat___Rings_Olinordered__idom_151,axiom,
linordered_idom @ rat ).
thf(tcon_Rat_Orat___Rings_Ocomm__semiring__1_152,axiom,
comm_semiring_1 @ rat ).
thf(tcon_Rat_Orat___Rings_Ocomm__semiring__0_153,axiom,
comm_semiring_0 @ rat ).
thf(tcon_Rat_Orat___Orderings_Odense__order,axiom,
dense_order @ rat ).
thf(tcon_Rat_Orat___Groups_Osemigroup__mult_154,axiom,
semigroup_mult @ rat ).
thf(tcon_Rat_Orat___Rings_Osemidom__divide_155,axiom,
semidom_divide @ rat ).
thf(tcon_Rat_Orat___Num_Osemiring__numeral_156,axiom,
semiring_numeral @ rat ).
thf(tcon_Rat_Orat___Groups_Osemigroup__add_157,axiom,
semigroup_add @ rat ).
thf(tcon_Rat_Orat___Fields_Ofield__abs__sgn,axiom,
field_abs_sgn @ rat ).
thf(tcon_Rat_Orat___Fields_Odivision__ring,axiom,
division_ring @ rat ).
thf(tcon_Rat_Orat___Rings_Ozero__less__one_158,axiom,
zero_less_one @ rat ).
thf(tcon_Rat_Orat___Rings_Ocomm__semiring_159,axiom,
comm_semiring @ rat ).
thf(tcon_Rat_Orat___Nat_Osemiring__char__0_160,axiom,
semiring_char_0 @ rat ).
thf(tcon_Rat_Orat___Groups_Oab__group__add_161,axiom,
ab_group_add @ rat ).
thf(tcon_Rat_Orat___Fields_Ofield__char__0,axiom,
field_char_0 @ rat ).
thf(tcon_Rat_Orat___Countable_Ocountable_162,axiom,
countable @ rat ).
thf(tcon_Rat_Orat___Rings_Ozero__neq__one_163,axiom,
zero_neq_one @ rat ).
thf(tcon_Rat_Orat___Rings_Oordered__ring_164,axiom,
ordered_ring @ rat ).
thf(tcon_Rat_Orat___Rings_Oidom__abs__sgn_165,axiom,
idom_abs_sgn @ rat ).
thf(tcon_Rat_Orat___Orderings_Opreorder_166,axiom,
preorder @ rat ).
thf(tcon_Rat_Orat___Orderings_Olinorder_167,axiom,
linorder @ rat ).
thf(tcon_Rat_Orat___Groups_Omonoid__mult_168,axiom,
monoid_mult @ rat ).
thf(tcon_Rat_Orat___Rings_Oidom__divide_169,axiom,
idom_divide @ rat ).
thf(tcon_Rat_Orat___Rings_Ocomm__ring__1_170,axiom,
comm_ring_1 @ rat ).
thf(tcon_Rat_Orat___Groups_Omonoid__add_171,axiom,
monoid_add @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring__1_172,axiom,
semiring_1 @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring__0_173,axiom,
semiring_0 @ rat ).
thf(tcon_Rat_Orat___Orderings_Ono__top_174,axiom,
no_top @ rat ).
thf(tcon_Rat_Orat___Orderings_Ono__bot_175,axiom,
no_bot @ rat ).
thf(tcon_Rat_Orat___Lattices_Olattice_176,axiom,
lattice @ rat ).
thf(tcon_Rat_Orat___Groups_Ogroup__add_177,axiom,
group_add @ rat ).
thf(tcon_Rat_Orat___Rings_Omult__zero_178,axiom,
mult_zero @ rat ).
thf(tcon_Rat_Orat___Rings_Ocomm__ring_179,axiom,
comm_ring @ rat ).
thf(tcon_Rat_Orat___Orderings_Oorder_180,axiom,
order @ rat ).
thf(tcon_Rat_Orat___Num_Oneg__numeral_181,axiom,
neg_numeral @ rat ).
thf(tcon_Rat_Orat___Nat_Oring__char__0_182,axiom,
ring_char_0 @ rat ).
thf(tcon_Rat_Orat___Rings_Osemiring_183,axiom,
semiring @ rat ).
thf(tcon_Rat_Orat___Fields_Oinverse,axiom,
inverse @ rat ).
thf(tcon_Rat_Orat___Rings_Osemidom_184,axiom,
semidom @ rat ).
thf(tcon_Rat_Orat___Orderings_Oord_185,axiom,
ord @ rat ).
thf(tcon_Rat_Orat___Groups_Ouminus_186,axiom,
uminus @ rat ).
thf(tcon_Rat_Orat___Rings_Oring__1_187,axiom,
ring_1 @ rat ).
thf(tcon_Rat_Orat___Rings_Oabs__if_188,axiom,
abs_if @ rat ).
thf(tcon_Rat_Orat___Groups_Ominus_189,axiom,
minus @ rat ).
thf(tcon_Rat_Orat___Fields_Ofield,axiom,
field @ rat ).
thf(tcon_Rat_Orat___Power_Opower_190,axiom,
power @ rat ).
thf(tcon_Rat_Orat___Num_Onumeral_191,axiom,
numeral @ rat ).
thf(tcon_Rat_Orat___Groups_Ozero_192,axiom,
zero @ rat ).
thf(tcon_Rat_Orat___Groups_Oplus_193,axiom,
plus @ rat ).
thf(tcon_Rat_Orat___Rings_Oring_194,axiom,
ring @ rat ).
thf(tcon_Rat_Orat___Rings_Oidom_195,axiom,
idom @ rat ).
thf(tcon_Rat_Orat___Groups_Oone_196,axiom,
one @ rat ).
thf(tcon_Rat_Orat___Rings_Odvd_197,axiom,
dvd @ rat ).
thf(tcon_Set_Oset___Countable__Complete__Lattices_Ocountable__complete__distrib__lattice_198,axiom,
! [A9: $tType] : ( counta4013691401010221786attice @ ( set @ A9 ) ) ).
thf(tcon_Set_Oset___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_199,axiom,
! [A9: $tType] : ( condit1219197933456340205attice @ ( set @ A9 ) ) ).
thf(tcon_Set_Oset___Countable__Complete__Lattices_Ocountable__complete__lattice_200,axiom,
! [A9: $tType] : ( counta3822494911875563373attice @ ( set @ A9 ) ) ).
thf(tcon_Set_Oset___Complete__Lattices_Ocomplete__distrib__lattice_201,axiom,
! [A9: $tType] : ( comple592849572758109894attice @ ( set @ A9 ) ) ).
thf(tcon_Set_Oset___Lattices_Obounded__semilattice__sup__bot_202,axiom,
! [A9: $tType] : ( bounde4967611905675639751up_bot @ ( set @ A9 ) ) ).
thf(tcon_Set_Oset___Lattices_Obounded__semilattice__inf__top_203,axiom,
! [A9: $tType] : ( bounde4346867609351753570nf_top @ ( set @ A9 ) ) ).
thf(tcon_Set_Oset___Complete__Lattices_Ocomplete__lattice_204,axiom,
! [A9: $tType] : ( comple6319245703460814977attice @ ( set @ A9 ) ) ).
thf(tcon_Set_Oset___Boolean__Algebras_Oboolean__algebra_205,axiom,
! [A9: $tType] : ( boolea8198339166811842893lgebra @ ( set @ A9 ) ) ).
thf(tcon_Set_Oset___Lattices_Obounded__lattice__top_206,axiom,
! [A9: $tType] : ( bounded_lattice_top @ ( set @ A9 ) ) ).
thf(tcon_Set_Oset___Lattices_Obounded__lattice__bot_207,axiom,
! [A9: $tType] : ( bounded_lattice_bot @ ( set @ A9 ) ) ).
thf(tcon_Set_Oset___Complete__Partial__Order_Occpo_208,axiom,
! [A9: $tType] : ( comple9053668089753744459l_ccpo @ ( set @ A9 ) ) ).
thf(tcon_Set_Oset___Lattices_Osemilattice__sup_209,axiom,
! [A9: $tType] : ( semilattice_sup @ ( set @ A9 ) ) ).
thf(tcon_Set_Oset___Lattices_Osemilattice__inf_210,axiom,
! [A9: $tType] : ( semilattice_inf @ ( set @ A9 ) ) ).
thf(tcon_Set_Oset___Lattices_Odistrib__lattice_211,axiom,
! [A9: $tType] : ( distrib_lattice @ ( set @ A9 ) ) ).
thf(tcon_Set_Oset___Lattices_Obounded__lattice_212,axiom,
! [A9: $tType] : ( bounded_lattice @ ( set @ A9 ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder__top_213,axiom,
! [A9: $tType] : ( order_top @ ( set @ A9 ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder__bot_214,axiom,
! [A9: $tType] : ( order_bot @ ( set @ A9 ) ) ).
thf(tcon_Set_Oset___Countable_Ocountable_215,axiom,
! [A9: $tType] :
( ( finite_finite @ A9 )
=> ( countable @ ( set @ A9 ) ) ) ).
thf(tcon_Set_Oset___Orderings_Opreorder_216,axiom,
! [A9: $tType] : ( preorder @ ( set @ A9 ) ) ).
thf(tcon_Set_Oset___Finite__Set_Ofinite_217,axiom,
! [A9: $tType] :
( ( finite_finite @ A9 )
=> ( finite_finite @ ( set @ A9 ) ) ) ).
thf(tcon_Set_Oset___Lattices_Olattice_218,axiom,
! [A9: $tType] : ( lattice @ ( set @ A9 ) ) ).
thf(tcon_Set_Oset___Orderings_Oorder_219,axiom,
! [A9: $tType] : ( order @ ( set @ A9 ) ) ).
thf(tcon_Set_Oset___Orderings_Otop_220,axiom,
! [A9: $tType] : ( top @ ( set @ A9 ) ) ).
thf(tcon_Set_Oset___Orderings_Oord_221,axiom,
! [A9: $tType] : ( ord @ ( set @ A9 ) ) ).
thf(tcon_Set_Oset___Orderings_Obot_222,axiom,
! [A9: $tType] : ( bot @ ( set @ A9 ) ) ).
thf(tcon_Set_Oset___Groups_Ouminus_223,axiom,
! [A9: $tType] : ( uminus @ ( set @ A9 ) ) ).
thf(tcon_Set_Oset___Groups_Ominus_224,axiom,
! [A9: $tType] : ( minus @ ( set @ A9 ) ) ).
thf(tcon_HOL_Obool___Countable__Complete__Lattices_Ocountable__complete__distrib__lattice_225,axiom,
counta4013691401010221786attice @ $o ).
thf(tcon_HOL_Obool___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_226,axiom,
condit1219197933456340205attice @ $o ).
thf(tcon_HOL_Obool___Countable__Complete__Lattices_Ocountable__complete__lattice_227,axiom,
counta3822494911875563373attice @ $o ).
thf(tcon_HOL_Obool___Complete__Lattices_Ocomplete__distrib__lattice_228,axiom,
comple592849572758109894attice @ $o ).
thf(tcon_HOL_Obool___Topological__Spaces_Otopological__space_229,axiom,
topolo4958980785337419405_space @ $o ).
thf(tcon_HOL_Obool___Topological__Spaces_Olinorder__topology_230,axiom,
topolo1944317154257567458pology @ $o ).
thf(tcon_HOL_Obool___Topological__Spaces_Odiscrete__topology_231,axiom,
topolo8865339358273720382pology @ $o ).
thf(tcon_HOL_Obool___Lattices_Obounded__semilattice__sup__bot_232,axiom,
bounde4967611905675639751up_bot @ $o ).
thf(tcon_HOL_Obool___Lattices_Obounded__semilattice__inf__top_233,axiom,
bounde4346867609351753570nf_top @ $o ).
thf(tcon_HOL_Obool___Complete__Lattices_Ocomplete__lattice_234,axiom,
comple6319245703460814977attice @ $o ).
thf(tcon_HOL_Obool___Topological__Spaces_Oorder__topology_235,axiom,
topolo2564578578187576103pology @ $o ).
thf(tcon_HOL_Obool___Boolean__Algebras_Oboolean__algebra_236,axiom,
boolea8198339166811842893lgebra @ $o ).
thf(tcon_HOL_Obool___Lattices_Obounded__lattice__top_237,axiom,
bounded_lattice_top @ $o ).
thf(tcon_HOL_Obool___Lattices_Obounded__lattice__bot_238,axiom,
bounded_lattice_bot @ $o ).
thf(tcon_HOL_Obool___Topological__Spaces_Ot2__space_239,axiom,
topological_t2_space @ $o ).
thf(tcon_HOL_Obool___Topological__Spaces_Ot1__space_240,axiom,
topological_t1_space @ $o ).
thf(tcon_HOL_Obool___Complete__Partial__Order_Occpo_241,axiom,
comple9053668089753744459l_ccpo @ $o ).
thf(tcon_HOL_Obool___Lattices_Osemilattice__sup_242,axiom,
semilattice_sup @ $o ).
thf(tcon_HOL_Obool___Lattices_Osemilattice__inf_243,axiom,
semilattice_inf @ $o ).
thf(tcon_HOL_Obool___Lattices_Odistrib__lattice_244,axiom,
distrib_lattice @ $o ).
thf(tcon_HOL_Obool___Lattices_Obounded__lattice_245,axiom,
bounded_lattice @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder__top_246,axiom,
order_top @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder__bot_247,axiom,
order_bot @ $o ).
thf(tcon_HOL_Obool___Countable_Ocountable_248,axiom,
countable @ $o ).
thf(tcon_HOL_Obool___Orderings_Opreorder_249,axiom,
preorder @ $o ).
thf(tcon_HOL_Obool___Orderings_Olinorder_250,axiom,
linorder @ $o ).
thf(tcon_HOL_Obool___Finite__Set_Ofinite_251,axiom,
finite_finite @ $o ).
thf(tcon_HOL_Obool___Lattices_Olattice_252,axiom,
lattice @ $o ).
thf(tcon_HOL_Obool___Orderings_Oorder_253,axiom,
order @ $o ).
thf(tcon_HOL_Obool___Orderings_Otop_254,axiom,
top @ $o ).
thf(tcon_HOL_Obool___Orderings_Oord_255,axiom,
ord @ $o ).
thf(tcon_HOL_Obool___Orderings_Obot_256,axiom,
bot @ $o ).
thf(tcon_HOL_Obool___Groups_Ouminus_257,axiom,
uminus @ $o ).
thf(tcon_HOL_Obool___Groups_Ominus_258,axiom,
minus @ $o ).
thf(tcon_List_Olist___Countable_Ocountable_259,axiom,
! [A9: $tType] :
( ( countable @ A9 )
=> ( countable @ ( list @ A9 ) ) ) ).
thf(tcon_List_Olist___Nat_Osize_260,axiom,
! [A9: $tType] : ( size @ ( list @ A9 ) ) ).
thf(tcon_Real_Oreal___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_261,axiom,
condit6923001295902523014norder @ real ).
thf(tcon_Real_Oreal___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_262,axiom,
condit1219197933456340205attice @ real ).
thf(tcon_Real_Oreal___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_263,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_264,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_265,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_266,axiom,
strict9044650504122735259up_add @ real ).
thf(tcon_Real_Oreal___Groups_Oordered__cancel__ab__semigroup__add_267,axiom,
ordere580206878836729694up_add @ real ).
thf(tcon_Real_Oreal___Groups_Oordered__ab__semigroup__add__imp__le_268,axiom,
ordere2412721322843649153imp_le @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__comm__semiring__strict_269,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_270,axiom,
strict7427464778891057005id_add @ real ).
thf(tcon_Real_Oreal___Groups_Oordered__cancel__comm__monoid__add_271,axiom,
ordere8940638589300402666id_add @ real ).
thf(tcon_Real_Oreal___Topological__Spaces_Otopological__space_272,axiom,
topolo4958980785337419405_space @ real ).
thf(tcon_Real_Oreal___Topological__Spaces_Olinorder__topology_273,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_274,axiom,
archim462609752435547400_field @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__semiring__1__strict_275,axiom,
linord715952674999750819strict @ real ).
thf(tcon_Real_Oreal___Real__Vector__Spaces_Ouniformity__dist,axiom,
real_V768167426530841204y_dist @ real ).
thf(tcon_Real_Oreal___Orderings_Ounbounded__dense__linorder_276,axiom,
unboun7993243217541854897norder @ real ).
thf(tcon_Real_Oreal___Limits_Otopological__comm__monoid__add_277,axiom,
topolo5987344860129210374id_add @ real ).
thf(tcon_Real_Oreal___Groups_Olinordered__ab__semigroup__add_278,axiom,
linord4140545234300271783up_add @ real ).
thf(tcon_Real_Oreal___Topological__Spaces_Oorder__topology_279,axiom,
topolo2564578578187576103pology @ real ).
thf(tcon_Real_Oreal___Rings_Osemiring__1__no__zero__divisors_280,axiom,
semiri2026040879449505780visors @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__nonzero__semiring_281,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_282,axiom,
topolo4211221413907600880p_mult @ real ).
thf(tcon_Real_Oreal___Topological__Spaces_Ouniform__space,axiom,
topolo7287701948861334536_space @ real ).
thf(tcon_Real_Oreal___Topological__Spaces_Operfect__space,axiom,
topolo8386298272705272623_space @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__semiring__strict_283,axiom,
linord8928482502909563296strict @ real ).
thf(tcon_Real_Oreal___Rings_Osemiring__no__zero__divisors_284,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_285,axiom,
ordere6658533253407199908up_add @ real ).
thf(tcon_Real_Oreal___Groups_Oordered__ab__group__add__abs_286,axiom,
ordere166539214618696060dd_abs @ real ).
thf(tcon_Real_Oreal___Archimedean__Field_Ofloor__ceiling_287,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_288,axiom,
ordere6911136660526730532id_add @ real ).
thf(tcon_Real_Oreal___Groups_Olinordered__ab__group__add_289,axiom,
linord5086331880401160121up_add @ real ).
thf(tcon_Real_Oreal___Groups_Ocancel__ab__semigroup__add_290,axiom,
cancel2418104881723323429up_add @ real ).
thf(tcon_Real_Oreal___Rings_Oring__1__no__zero__divisors_291,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_292,axiom,
topolo6943815403480290642id_add @ real ).
thf(tcon_Real_Oreal___Groups_Ocancel__comm__monoid__add_293,axiom,
cancel1802427076303600483id_add @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__ring__strict_294,axiom,
linord4710134922213307826strict @ real ).
thf(tcon_Real_Oreal___Rings_Ocomm__semiring__1__cancel_295,axiom,
comm_s4317794764714335236cancel @ real ).
thf(tcon_Real_Oreal___Limits_Otopological__group__add,axiom,
topolo1633459387980952147up_add @ real ).
thf(tcon_Real_Oreal___Topological__Spaces_Ot2__space_296,axiom,
topological_t2_space @ real ).
thf(tcon_Real_Oreal___Topological__Spaces_Ot1__space_297,axiom,
topological_t1_space @ real ).
thf(tcon_Real_Oreal___Rings_Oordered__comm__semiring_298,axiom,
ordere2520102378445227354miring @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__semiring__1_299,axiom,
linord6961819062388156250ring_1 @ real ).
thf(tcon_Real_Oreal___Groups_Oordered__ab__group__add_300,axiom,
ordered_ab_group_add @ real ).
thf(tcon_Real_Oreal___Groups_Ocancel__semigroup__add_301,axiom,
cancel_semigroup_add @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__semiring_302,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_303,axiom,
ordered_semiring_0 @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__semidom_304,axiom,
linordered_semidom @ real ).
thf(tcon_Real_Oreal___Orderings_Odense__linorder_305,axiom,
dense_linorder @ real ).
thf(tcon_Real_Oreal___Lattices_Osemilattice__sup_306,axiom,
semilattice_sup @ real ).
thf(tcon_Real_Oreal___Lattices_Osemilattice__inf_307,axiom,
semilattice_inf @ real ).
thf(tcon_Real_Oreal___Lattices_Odistrib__lattice_308,axiom,
distrib_lattice @ real ).
thf(tcon_Real_Oreal___Groups_Oab__semigroup__mult_309,axiom,
ab_semigroup_mult @ real ).
thf(tcon_Real_Oreal___Rings_Osemiring__1__cancel_310,axiom,
semiring_1_cancel @ real ).
thf(tcon_Real_Oreal___Groups_Ocomm__monoid__mult_311,axiom,
comm_monoid_mult @ real ).
thf(tcon_Real_Oreal___Groups_Oab__semigroup__add_312,axiom,
ab_semigroup_add @ real ).
thf(tcon_Real_Oreal___Fields_Olinordered__field_313,axiom,
linordered_field @ real ).
thf(tcon_Real_Oreal___Rings_Oordered__semiring_314,axiom,
ordered_semiring @ real ).
thf(tcon_Real_Oreal___Rings_Oordered__ring__abs_315,axiom,
ordered_ring_abs @ real ).
thf(tcon_Real_Oreal___Groups_Ocomm__monoid__add_316,axiom,
comm_monoid_add @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__ring_317,axiom,
linordered_ring @ real ).
thf(tcon_Real_Oreal___Rings_Olinordered__idom_318,axiom,
linordered_idom @ real ).
thf(tcon_Real_Oreal___Rings_Ocomm__semiring__1_319,axiom,
comm_semiring_1 @ real ).
thf(tcon_Real_Oreal___Rings_Ocomm__semiring__0_320,axiom,
comm_semiring_0 @ real ).
thf(tcon_Real_Oreal___Orderings_Odense__order_321,axiom,
dense_order @ real ).
thf(tcon_Real_Oreal___Groups_Osemigroup__mult_322,axiom,
semigroup_mult @ real ).
thf(tcon_Real_Oreal___Rings_Osemidom__divide_323,axiom,
semidom_divide @ real ).
thf(tcon_Real_Oreal___Num_Osemiring__numeral_324,axiom,
semiring_numeral @ real ).
thf(tcon_Real_Oreal___Groups_Osemigroup__add_325,axiom,
semigroup_add @ real ).
thf(tcon_Real_Oreal___Fields_Ofield__abs__sgn_326,axiom,
field_abs_sgn @ real ).
thf(tcon_Real_Oreal___Fields_Odivision__ring_327,axiom,
division_ring @ real ).
thf(tcon_Real_Oreal___Rings_Ozero__less__one_328,axiom,
zero_less_one @ real ).
thf(tcon_Real_Oreal___Rings_Ocomm__semiring_329,axiom,
comm_semiring @ real ).
thf(tcon_Real_Oreal___Nat_Osemiring__char__0_330,axiom,
semiring_char_0 @ real ).
thf(tcon_Real_Oreal___Groups_Oab__group__add_331,axiom,
ab_group_add @ real ).
thf(tcon_Real_Oreal___Fields_Ofield__char__0_332,axiom,
field_char_0 @ real ).
thf(tcon_Real_Oreal___Rings_Ozero__neq__one_333,axiom,
zero_neq_one @ real ).
thf(tcon_Real_Oreal___Rings_Oordered__ring_334,axiom,
ordered_ring @ real ).
thf(tcon_Real_Oreal___Rings_Oidom__abs__sgn_335,axiom,
idom_abs_sgn @ real ).
thf(tcon_Real_Oreal___Orderings_Opreorder_336,axiom,
preorder @ real ).
thf(tcon_Real_Oreal___Orderings_Olinorder_337,axiom,
linorder @ real ).
thf(tcon_Real_Oreal___Groups_Omonoid__mult_338,axiom,
monoid_mult @ real ).
thf(tcon_Real_Oreal___Transcendental_Oln,axiom,
ln @ real ).
thf(tcon_Real_Oreal___Rings_Oidom__divide_339,axiom,
idom_divide @ real ).
thf(tcon_Real_Oreal___Rings_Ocomm__ring__1_340,axiom,
comm_ring_1 @ real ).
thf(tcon_Real_Oreal___Groups_Omonoid__add_341,axiom,
monoid_add @ real ).
thf(tcon_Real_Oreal___Rings_Osemiring__1_342,axiom,
semiring_1 @ real ).
thf(tcon_Real_Oreal___Rings_Osemiring__0_343,axiom,
semiring_0 @ real ).
thf(tcon_Real_Oreal___Orderings_Ono__top_344,axiom,
no_top @ real ).
thf(tcon_Real_Oreal___Orderings_Ono__bot_345,axiom,
no_bot @ real ).
thf(tcon_Real_Oreal___Lattices_Olattice_346,axiom,
lattice @ real ).
thf(tcon_Real_Oreal___Groups_Ogroup__add_347,axiom,
group_add @ real ).
thf(tcon_Real_Oreal___Rings_Omult__zero_348,axiom,
mult_zero @ real ).
thf(tcon_Real_Oreal___Rings_Ocomm__ring_349,axiom,
comm_ring @ real ).
thf(tcon_Real_Oreal___Orderings_Oorder_350,axiom,
order @ real ).
thf(tcon_Real_Oreal___Num_Oneg__numeral_351,axiom,
neg_numeral @ real ).
thf(tcon_Real_Oreal___Nat_Oring__char__0_352,axiom,
ring_char_0 @ real ).
thf(tcon_Real_Oreal___Rings_Osemiring_353,axiom,
semiring @ real ).
thf(tcon_Real_Oreal___Fields_Oinverse_354,axiom,
inverse @ real ).
thf(tcon_Real_Oreal___Rings_Osemidom_355,axiom,
semidom @ real ).
thf(tcon_Real_Oreal___Orderings_Oord_356,axiom,
ord @ real ).
thf(tcon_Real_Oreal___Groups_Ouminus_357,axiom,
uminus @ real ).
thf(tcon_Real_Oreal___Rings_Oring__1_358,axiom,
ring_1 @ real ).
thf(tcon_Real_Oreal___Rings_Oabs__if_359,axiom,
abs_if @ real ).
thf(tcon_Real_Oreal___Groups_Ominus_360,axiom,
minus @ real ).
thf(tcon_Real_Oreal___Fields_Ofield_361,axiom,
field @ real ).
thf(tcon_Real_Oreal___Power_Opower_362,axiom,
power @ real ).
thf(tcon_Real_Oreal___Num_Onumeral_363,axiom,
numeral @ real ).
thf(tcon_Real_Oreal___Groups_Ozero_364,axiom,
zero @ real ).
thf(tcon_Real_Oreal___Groups_Oplus_365,axiom,
plus @ real ).
thf(tcon_Real_Oreal___Rings_Oring_366,axiom,
ring @ real ).
thf(tcon_Real_Oreal___Rings_Oidom_367,axiom,
idom @ real ).
thf(tcon_Real_Oreal___Groups_Oone_368,axiom,
one @ real ).
thf(tcon_Real_Oreal___Rings_Odvd_369,axiom,
dvd @ real ).
thf(tcon_Sum__Type_Osum___Countable_Ocountable_370,axiom,
! [A9: $tType,A19: $tType] :
( ( ( countable @ A9 )
& ( countable @ A19 ) )
=> ( countable @ ( sum_sum @ A9 @ A19 ) ) ) ).
thf(tcon_Sum__Type_Osum___Finite__Set_Ofinite_371,axiom,
! [A9: $tType,A19: $tType] :
( ( ( finite_finite @ A9 )
& ( finite_finite @ A19 ) )
=> ( finite_finite @ ( sum_sum @ A9 @ A19 ) ) ) ).
thf(tcon_Sum__Type_Osum___Nat_Osize_372,axiom,
! [A9: $tType,A19: $tType] : ( size @ ( sum_sum @ A9 @ A19 ) ) ).
thf(tcon_Filter_Ofilter___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_373,axiom,
! [A9: $tType] : ( condit1219197933456340205attice @ ( filter @ A9 ) ) ).
thf(tcon_Filter_Ofilter___Countable__Complete__Lattices_Ocountable__complete__lattice_374,axiom,
! [A9: $tType] : ( counta3822494911875563373attice @ ( filter @ A9 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Obounded__semilattice__sup__bot_375,axiom,
! [A9: $tType] : ( bounde4967611905675639751up_bot @ ( filter @ A9 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Obounded__semilattice__inf__top_376,axiom,
! [A9: $tType] : ( bounde4346867609351753570nf_top @ ( filter @ A9 ) ) ).
thf(tcon_Filter_Ofilter___Complete__Lattices_Ocomplete__lattice_377,axiom,
! [A9: $tType] : ( comple6319245703460814977attice @ ( filter @ A9 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Obounded__lattice__top_378,axiom,
! [A9: $tType] : ( bounded_lattice_top @ ( filter @ A9 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Obounded__lattice__bot_379,axiom,
! [A9: $tType] : ( bounded_lattice_bot @ ( filter @ A9 ) ) ).
thf(tcon_Filter_Ofilter___Complete__Partial__Order_Occpo_380,axiom,
! [A9: $tType] : ( comple9053668089753744459l_ccpo @ ( filter @ A9 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Osemilattice__sup_381,axiom,
! [A9: $tType] : ( semilattice_sup @ ( filter @ A9 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Osemilattice__inf_382,axiom,
! [A9: $tType] : ( semilattice_inf @ ( filter @ A9 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Odistrib__lattice_383,axiom,
! [A9: $tType] : ( distrib_lattice @ ( filter @ A9 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Obounded__lattice_384,axiom,
! [A9: $tType] : ( bounded_lattice @ ( filter @ A9 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Oorder__top_385,axiom,
! [A9: $tType] : ( order_top @ ( filter @ A9 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Oorder__bot_386,axiom,
! [A9: $tType] : ( order_bot @ ( filter @ A9 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Opreorder_387,axiom,
! [A9: $tType] : ( preorder @ ( filter @ A9 ) ) ).
thf(tcon_Filter_Ofilter___Lattices_Olattice_388,axiom,
! [A9: $tType] : ( lattice @ ( filter @ A9 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Oorder_389,axiom,
! [A9: $tType] : ( order @ ( filter @ A9 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Otop_390,axiom,
! [A9: $tType] : ( top @ ( filter @ A9 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Oord_391,axiom,
! [A9: $tType] : ( ord @ ( filter @ A9 ) ) ).
thf(tcon_Filter_Ofilter___Orderings_Obot_392,axiom,
! [A9: $tType] : ( bot @ ( filter @ A9 ) ) ).
thf(tcon_Option_Ooption___Countable_Ocountable_393,axiom,
! [A9: $tType] :
( ( countable @ A9 )
=> ( countable @ ( option @ A9 ) ) ) ).
thf(tcon_Option_Ooption___Finite__Set_Ofinite_394,axiom,
! [A9: $tType] :
( ( finite_finite @ A9 )
=> ( finite_finite @ ( option @ A9 ) ) ) ).
thf(tcon_Option_Ooption___Nat_Osize_395,axiom,
! [A9: $tType] : ( size @ ( option @ A9 ) ) ).
thf(tcon_Complex_Ocomplex___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct_396,axiom,
semiri1453513574482234551roduct @ complex ).
thf(tcon_Complex_Ocomplex___Topological__Spaces_Ofirst__countable__topology_397,axiom,
topolo3112930676232923870pology @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__div__algebra_398,axiom,
real_V8999393235501362500lgebra @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__algebra__1_399,axiom,
real_V2822296259951069270ebra_1 @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Osemiring__no__zero__divisors__cancel_400,axiom,
semiri6575147826004484403cancel @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__algebra_401,axiom,
real_V4412858255891104859lgebra @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__vector_402,axiom,
real_V822414075346904944vector @ complex ).
thf(tcon_Complex_Ocomplex___Topological__Spaces_Otopological__space_403,axiom,
topolo4958980785337419405_space @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__normed__field_404,axiom,
real_V3459762299906320749_field @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__div__algebra_405,axiom,
real_V5047593784448816457lgebra @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Ouniformity__dist_406,axiom,
real_V768167426530841204y_dist @ complex ).
thf(tcon_Complex_Ocomplex___Limits_Otopological__comm__monoid__add_407,axiom,
topolo5987344860129210374id_add @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Osemiring__1__no__zero__divisors_408,axiom,
semiri2026040879449505780visors @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__algebra__1_409,axiom,
real_V2191834092415804123ebra_1 @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Ocomplete__space_410,axiom,
real_V8037385150606011577_space @ complex ).
thf(tcon_Complex_Ocomplex___Limits_Otopological__semigroup__mult_411,axiom,
topolo4211221413907600880p_mult @ complex ).
thf(tcon_Complex_Ocomplex___Topological__Spaces_Ouniform__space_412,axiom,
topolo7287701948861334536_space @ complex ).
thf(tcon_Complex_Ocomplex___Topological__Spaces_Operfect__space_413,axiom,
topolo8386298272705272623_space @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Osemiring__no__zero__divisors_414,axiom,
semiri3467727345109120633visors @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Ometric__space_415,axiom,
real_V7819770556892013058_space @ complex ).
thf(tcon_Complex_Ocomplex___Limits_Otopological__ab__group__add_416,axiom,
topolo1287966508704411220up_add @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__vector_417,axiom,
real_V4867850818363320053vector @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Ocancel__ab__semigroup__add_418,axiom,
cancel2418104881723323429up_add @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Oring__1__no__zero__divisors_419,axiom,
ring_15535105094025558882visors @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Oreal__field_420,axiom,
real_V7773925162809079976_field @ complex ).
thf(tcon_Complex_Ocomplex___Limits_Otopological__monoid__add_421,axiom,
topolo6943815403480290642id_add @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Ocancel__comm__monoid__add_422,axiom,
cancel1802427076303600483id_add @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Ocomm__semiring__1__cancel_423,axiom,
comm_s4317794764714335236cancel @ complex ).
thf(tcon_Complex_Ocomplex___Limits_Otopological__group__add_424,axiom,
topolo1633459387980952147up_add @ complex ).
thf(tcon_Complex_Ocomplex___Topological__Spaces_Ot2__space_425,axiom,
topological_t2_space @ complex ).
thf(tcon_Complex_Ocomplex___Topological__Spaces_Ot1__space_426,axiom,
topological_t1_space @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Ocancel__semigroup__add_427,axiom,
cancel_semigroup_add @ complex ).
thf(tcon_Complex_Ocomplex___Real__Vector__Spaces_Obanach_428,axiom,
real_Vector_banach @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Oab__semigroup__mult_429,axiom,
ab_semigroup_mult @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Osemiring__1__cancel_430,axiom,
semiring_1_cancel @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Ocomm__monoid__mult_431,axiom,
comm_monoid_mult @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Oab__semigroup__add_432,axiom,
ab_semigroup_add @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Ocomm__monoid__add_433,axiom,
comm_monoid_add @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Ocomm__semiring__1_434,axiom,
comm_semiring_1 @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Ocomm__semiring__0_435,axiom,
comm_semiring_0 @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Osemigroup__mult_436,axiom,
semigroup_mult @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Osemidom__divide_437,axiom,
semidom_divide @ complex ).
thf(tcon_Complex_Ocomplex___Num_Osemiring__numeral_438,axiom,
semiring_numeral @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Osemigroup__add_439,axiom,
semigroup_add @ complex ).
thf(tcon_Complex_Ocomplex___Fields_Ofield__abs__sgn_440,axiom,
field_abs_sgn @ complex ).
thf(tcon_Complex_Ocomplex___Fields_Odivision__ring_441,axiom,
division_ring @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Ocomm__semiring_442,axiom,
comm_semiring @ complex ).
thf(tcon_Complex_Ocomplex___Nat_Osemiring__char__0_443,axiom,
semiring_char_0 @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Oab__group__add_444,axiom,
ab_group_add @ complex ).
thf(tcon_Complex_Ocomplex___Fields_Ofield__char__0_445,axiom,
field_char_0 @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Ozero__neq__one_446,axiom,
zero_neq_one @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Oidom__abs__sgn_447,axiom,
idom_abs_sgn @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Omonoid__mult_448,axiom,
monoid_mult @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Oidom__divide_449,axiom,
idom_divide @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Ocomm__ring__1_450,axiom,
comm_ring_1 @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Omonoid__add_451,axiom,
monoid_add @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Osemiring__1_452,axiom,
semiring_1 @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Osemiring__0_453,axiom,
semiring_0 @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Ogroup__add_454,axiom,
group_add @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Omult__zero_455,axiom,
mult_zero @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Ocomm__ring_456,axiom,
comm_ring @ complex ).
thf(tcon_Complex_Ocomplex___Num_Oneg__numeral_457,axiom,
neg_numeral @ complex ).
thf(tcon_Complex_Ocomplex___Nat_Oring__char__0_458,axiom,
ring_char_0 @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Osemiring_459,axiom,
semiring @ complex ).
thf(tcon_Complex_Ocomplex___Fields_Oinverse_460,axiom,
inverse @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Osemidom_461,axiom,
semidom @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Ouminus_462,axiom,
uminus @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Oring__1_463,axiom,
ring_1 @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Ominus_464,axiom,
minus @ complex ).
thf(tcon_Complex_Ocomplex___Fields_Ofield_465,axiom,
field @ complex ).
thf(tcon_Complex_Ocomplex___Power_Opower_466,axiom,
power @ complex ).
thf(tcon_Complex_Ocomplex___Num_Onumeral_467,axiom,
numeral @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Ozero_468,axiom,
zero @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Oplus_469,axiom,
plus @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Oring_470,axiom,
ring @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Oidom_471,axiom,
idom @ complex ).
thf(tcon_Complex_Ocomplex___Groups_Oone_472,axiom,
one @ complex ).
thf(tcon_Complex_Ocomplex___Rings_Odvd_473,axiom,
dvd @ complex ).
thf(tcon_Extended__Nat_Oenat___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_474,axiom,
condit6923001295902523014norder @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Countable__Complete__Lattices_Ocountable__complete__distrib__lattice_475,axiom,
counta4013691401010221786attice @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_476,axiom,
condit1219197933456340205attice @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Countable__Complete__Lattices_Ocountable__complete__lattice_477,axiom,
counta3822494911875563373attice @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Complete__Lattices_Ocomplete__distrib__lattice_478,axiom,
comple592849572758109894attice @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Ostrict__ordered__ab__semigroup__add_479,axiom,
strict9044650504122735259up_add @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Ostrict__ordered__comm__monoid__add_480,axiom,
strict7427464778891057005id_add @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Ocanonically__ordered__monoid__add_481,axiom,
canoni5634975068530333245id_add @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Lattices_Obounded__semilattice__sup__bot_482,axiom,
bounde4967611905675639751up_bot @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Lattices_Obounded__semilattice__inf__top_483,axiom,
bounde4346867609351753570nf_top @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Complete__Lattices_Ocomplete__linorder,axiom,
comple5582772986160207858norder @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Olinordered__ab__semigroup__add_484,axiom,
linord4140545234300271783up_add @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Complete__Lattices_Ocomplete__lattice_485,axiom,
comple6319245703460814977attice @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Olinordered__nonzero__semiring_486,axiom,
linord181362715937106298miring @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Osemiring__no__zero__divisors_487,axiom,
semiri3467727345109120633visors @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Oordered__ab__semigroup__add_488,axiom,
ordere6658533253407199908up_add @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Oordered__comm__monoid__add_489,axiom,
ordere6911136660526730532id_add @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Lattices_Obounded__lattice__top_490,axiom,
bounded_lattice_top @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Lattices_Obounded__lattice__bot_491,axiom,
bounded_lattice_bot @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Oordered__comm__semiring_492,axiom,
ordere2520102378445227354miring @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Complete__Partial__Order_Occpo_493,axiom,
comple9053668089753744459l_ccpo @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Lattices_Osemilattice__sup_494,axiom,
semilattice_sup @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Lattices_Osemilattice__inf_495,axiom,
semilattice_inf @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Lattices_Odistrib__lattice_496,axiom,
distrib_lattice @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Lattices_Obounded__lattice_497,axiom,
bounded_lattice @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Oab__semigroup__mult_498,axiom,
ab_semigroup_mult @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Ocomm__monoid__mult_499,axiom,
comm_monoid_mult @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Oab__semigroup__add_500,axiom,
ab_semigroup_add @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Oordered__semiring_501,axiom,
ordered_semiring @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Ocomm__monoid__add_502,axiom,
comm_monoid_add @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Ocomm__semiring__1_503,axiom,
comm_semiring_1 @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Ocomm__semiring__0_504,axiom,
comm_semiring_0 @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Osemigroup__mult_505,axiom,
semigroup_mult @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Num_Osemiring__numeral_506,axiom,
semiring_numeral @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Osemigroup__add_507,axiom,
semigroup_add @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Ozero__less__one_508,axiom,
zero_less_one @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Ocomm__semiring_509,axiom,
comm_semiring @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Orderings_Owellorder_510,axiom,
wellorder @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Orderings_Oorder__top_511,axiom,
order_top @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Orderings_Oorder__bot_512,axiom,
order_bot @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Nat_Osemiring__char__0_513,axiom,
semiring_char_0 @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Countable_Ocountable_514,axiom,
countable @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Ozero__neq__one_515,axiom,
zero_neq_one @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Orderings_Opreorder_516,axiom,
preorder @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Orderings_Olinorder_517,axiom,
linorder @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Omonoid__mult_518,axiom,
monoid_mult @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Omonoid__add_519,axiom,
monoid_add @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Osemiring__1_520,axiom,
semiring_1 @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Osemiring__0_521,axiom,
semiring_0 @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Lattices_Olattice_522,axiom,
lattice @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Omult__zero_523,axiom,
mult_zero @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Orderings_Oorder_524,axiom,
order @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Osemiring_525,axiom,
semiring @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Orderings_Otop_526,axiom,
top @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Orderings_Oord_527,axiom,
ord @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Orderings_Obot_528,axiom,
bot @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Ominus_529,axiom,
minus @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Power_Opower_530,axiom,
power @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Num_Onumeral_531,axiom,
numeral @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Ozero_532,axiom,
zero @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Oplus_533,axiom,
plus @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Groups_Oone_534,axiom,
one @ extended_enat ).
thf(tcon_Extended__Nat_Oenat___Rings_Odvd_535,axiom,
dvd @ extended_enat ).
thf(tcon_Product__Type_Oprod___Topological__Spaces_Otopological__space_536,axiom,
! [A9: $tType,A19: $tType] :
( ( ( topolo4958980785337419405_space @ A9 )
& ( topolo4958980785337419405_space @ A19 ) )
=> ( topolo4958980785337419405_space @ ( product_prod @ A9 @ A19 ) ) ) ).
thf(tcon_Product__Type_Oprod___Topological__Spaces_Ot2__space_537,axiom,
! [A9: $tType,A19: $tType] :
( ( ( topological_t2_space @ A9 )
& ( topological_t2_space @ A19 ) )
=> ( topological_t2_space @ ( product_prod @ A9 @ A19 ) ) ) ).
thf(tcon_Product__Type_Oprod___Topological__Spaces_Ot1__space_538,axiom,
! [A9: $tType,A19: $tType] :
( ( ( topological_t1_space @ A9 )
& ( topological_t1_space @ A19 ) )
=> ( topological_t1_space @ ( product_prod @ A9 @ A19 ) ) ) ).
thf(tcon_Product__Type_Oprod___Countable_Ocountable_539,axiom,
! [A9: $tType,A19: $tType] :
( ( ( countable @ A9 )
& ( countable @ A19 ) )
=> ( countable @ ( product_prod @ A9 @ A19 ) ) ) ).
thf(tcon_Product__Type_Oprod___Finite__Set_Ofinite_540,axiom,
! [A9: $tType,A19: $tType] :
( ( ( finite_finite @ A9 )
& ( finite_finite @ A19 ) )
=> ( finite_finite @ ( product_prod @ A9 @ A19 ) ) ) ).
thf(tcon_Product__Type_Oprod___Nat_Osize_541,axiom,
! [A9: $tType,A19: $tType] : ( size @ ( product_prod @ A9 @ A19 ) ) ).
thf(tcon_Product__Type_Ounit___Conditionally__Complete__Lattices_Oconditionally__complete__linorder_542,axiom,
condit6923001295902523014norder @ product_unit ).
thf(tcon_Product__Type_Ounit___Countable__Complete__Lattices_Ocountable__complete__distrib__lattice_543,axiom,
counta4013691401010221786attice @ product_unit ).
thf(tcon_Product__Type_Ounit___Conditionally__Complete__Lattices_Oconditionally__complete__lattice_544,axiom,
condit1219197933456340205attice @ product_unit ).
thf(tcon_Product__Type_Ounit___Countable__Complete__Lattices_Ocountable__complete__lattice_545,axiom,
counta3822494911875563373attice @ product_unit ).
thf(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__distrib__lattice_546,axiom,
comple592849572758109894attice @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Obounded__semilattice__sup__bot_547,axiom,
bounde4967611905675639751up_bot @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Obounded__semilattice__inf__top_548,axiom,
bounde4346867609351753570nf_top @ product_unit ).
thf(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__linorder_549,axiom,
comple5582772986160207858norder @ product_unit ).
thf(tcon_Product__Type_Ounit___Complete__Lattices_Ocomplete__lattice_550,axiom,
comple6319245703460814977attice @ product_unit ).
thf(tcon_Product__Type_Ounit___Boolean__Algebras_Oboolean__algebra_551,axiom,
boolea8198339166811842893lgebra @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Obounded__lattice__top_552,axiom,
bounded_lattice_top @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Obounded__lattice__bot_553,axiom,
bounded_lattice_bot @ product_unit ).
thf(tcon_Product__Type_Ounit___Complete__Partial__Order_Occpo_554,axiom,
comple9053668089753744459l_ccpo @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Osemilattice__sup_555,axiom,
semilattice_sup @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Osemilattice__inf_556,axiom,
semilattice_inf @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Odistrib__lattice_557,axiom,
distrib_lattice @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Obounded__lattice_558,axiom,
bounded_lattice @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Owellorder_559,axiom,
wellorder @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Oorder__top_560,axiom,
order_top @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Oorder__bot_561,axiom,
order_bot @ product_unit ).
thf(tcon_Product__Type_Ounit___Countable_Ocountable_562,axiom,
countable @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Opreorder_563,axiom,
preorder @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Olinorder_564,axiom,
linorder @ product_unit ).
thf(tcon_Product__Type_Ounit___Finite__Set_Ofinite_565,axiom,
finite_finite @ product_unit ).
thf(tcon_Product__Type_Ounit___Lattices_Olattice_566,axiom,
lattice @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Oorder_567,axiom,
order @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Otop_568,axiom,
top @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Oord_569,axiom,
ord @ product_unit ).
thf(tcon_Product__Type_Ounit___Orderings_Obot_570,axiom,
bot @ product_unit ).
thf(tcon_Product__Type_Ounit___Groups_Ouminus_571,axiom,
uminus @ product_unit ).
thf(tcon_Product__Type_Ounit___Groups_Ominus_572,axiom,
minus @ product_unit ).
thf(tcon_VEBT__Definitions_OVEBT___Nat_Osize_573,axiom,
size @ vEBT_VEBT ).
% Helper facts (4)
thf(help_If_3_1_T,axiom,
! [P2: $o] :
( ( P2 = $true )
| ( P2 = $false ) ) ).
thf(help_If_2_1_T,axiom,
! [A: $tType,X: A,Y: A] :
( ( if @ A @ $false @ X @ Y )
= Y ) ).
thf(help_If_1_1_T,axiom,
! [A: $tType,X: A,Y: A] :
( ( if @ A @ $true @ X @ Y )
= X ) ).
thf(help_fChoice_1_1_T,axiom,
! [A: $tType,P2: A > $o] :
( ( P2 @ ( fChoice @ A @ P2 ) )
= ( ? [X9: A] : ( P2 @ X9 ) ) ) ).
% Conjectures (1)
thf(conj_0,conjecture,
( ( vEBT_vebt_member @ ( vEBT_Node @ ( some @ ( product_prod @ nat @ nat ) @ ( product_Pair @ nat @ nat @ mi @ ma ) ) @ deg @ treeList @ summary ) @ ya )
| ( xa = ya ) ) ).
%------------------------------------------------------------------------------